Advanced Modeling for Transit Operations and Service Planning

ADVANCED MODELING FOR TRANSIT OPERATIONS AND SERVICE PLANNING

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ADVANCED MODELING FOR TRANSIT OPERATIONS AND SERVICE PLANNING

edited by Professor William H.K. Lam Department of Civil and Structural Engineering The Hong Kong Polytechnic University Hong Kong, China

and

Professor Michael G.H. Bell Centre for Transport Studies Imperial College London United Kingdom

2003

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CONTENTS Preface

Chapter 1 - Initial Planning for Urban Transit Systems S.C. Wirasinghe

vii

1

Chapter 2 - Public Transport Timetabling and Vehicle Scheduling Avishai Ceder

31

Chapter 3 - Designing Public Transport Network and Routes Avishai Ceder

59

Chapter 4 - Transit Path Choice and Assignment Model Approaches Agostino Nuzzolo

93

Chapter 5 - Schedule-Based Transit Assignment Models Agostino Nuzzolo

125

Chapter 6 - Frequency Based Transit Route Choice Models Michael Florian

165

Chapter 7 - Capacity Constrained Transit Assignment Models and Reliability Analysis Michael G.H. Bell

181

Chapter 8 - Dynasmart-IP: Dynamic Traffic Assignment Meso-Simulator for Intermodal Networks Hani S. Mahmassani and Khaled F. Abdelghany

201

Chapter 9 - Modeling Competitive Multi-Modal Services HongK. Lo, C W. YipandK.H. Wan

231

Chapter 10 - Modeling Urban Taxi Services: A Literature Survey and An Analytical Example Hai Yang, Min Ye, Wilson H. Tang and S C. Wong

257

Chapter 11 - The Estimation of Origin-Destination Matrices in Transit Networks S.C. Wong and C.O. Tong

287

Chapter 12 - Models for Optimizing Transit Fares Jing Zhou and William H.K. Lam

315

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PREFACE The idea for this book grew out of the organization of the Advanced Study Institute (ASI), which was sponsored by the Croucher Foundation (http://www.croucher.org.hk/) for the dissemination of knowledge and the formation of international scientific contacts on advances in modelling transit systems. While public transport (or transit) systems have arguably been in existence much longer than road traffic systems, the mathematical analysis techniques so necessary for the proper planning of transit operations have lagged far behind those for road traffic systems. For example, the body of literature available on the design of schedules for urban rail lines is miniscule in comparison to the literature on the coordination of traffic signals along an urban road. On the other hand, transit professionals appear to have disregarded most of the wealth of insights that have been available in the literature for more than a decade. The literature on transit assignment is a good example. However, public transport operators, particularly in Hong Kong and Asia, are facing ever-greater pressure in competitive markets and transit systems are congested. The need to estimate passenger demand, to monitor the performance of individual services as well as the system as a whole, to support better planning and tighter operations management, and for external reporting has increased. The optimization of transit line frequencies and transit fares has become very important for operations and service planning. Reliability and control issues are also critical in making transit systems more efficient, supported by the introduction of Intelligent Transport Systems (ITS). As tightening constraints raise serous questions about the cost-effectiveness of existing public transport services, improvements which can be implemented in the short and long term are continuously sought. Collectively, these pressures have focused attention on advanced methods and new techniques for improving transit planning and operations. In Hong Kong and other major cities in Asia, over 90% of people are using transit facilities for their daily travel. The recent rapid development and deployment of ITS makes it possible to improve the efficiency of transit operations. This book addresses the important and timely problems of how to improve transit operations and service planning by making use of new technologies and advanced modeling techniques. It will provide important references for determining the outcomes of introducing these technologies and methods, and thus assist transit professionals and scientists in resolving practical issues arising from the implementation of ITS. This book appears to be the first devoted exclusively to the topic of advanced modeling for transit operation and service planning. This book consists of 12 chapters chosen to represent the broad base of contemporary themes in modeling transit systems. Scholars from America, Europe and Asia have contributed their knowledge to produce a unique compilation of recent developments in the field. Topics both in theory and innovative applications to real world problems are included. The book covers Transit Planning and Network Design, Transit Assignment Models and Solution Algorithms, Simulation of Passenger Behaviors, Effects of ITS on Passenger Choices and Transit Service Improvements, Modeling Multi-modal Transit and Urban Taxi Services. Outline of the book contents: Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6

- Initial Planning for Urban Transit Systems - Public Transport Timetabling and Vehicle Scheduling - Designing Public Transport Network and Routes - Transit Path Choice and Assignment Model Approaches - Schedule-Based Transit Assignment Models - Frequency Based Transit Route Choice Models

Chapter 7 - Capacity Constrained Transit Assignment Models and Reliability Analysis Chapter 8 - Dynasmart-IP: Dynamic Traffic Assignment Meso-Simulator for Intermodal Networks Chapter 9 - Modeling Competitive Multi-Modal Services Chapter 10 - Modeling Urban Taxi Services: A Literature Survey and an Analytical Example Chapter 11 - The Estimation of Origin-Destination Matrices in Transit Networks Chapter 12 - Models for Optimizing Transit Fares Special appreciation is extended to Elsevier Science Ltd. who made possible the publication of all the contributions in the form of the present book in time to be available to participants attending the ASI workshop from 9th to 13th December 2002 in Hong Kong. Professor Mike Bell of Imperial College of Science, Technology & Medicine (U.K.) provided valuable oversight and guidance in enhancing the quality of the book. His support during this effort has been remarkable. Finally, I am thankful for the patience, availability, and dedication of the editorial staff at Elsevier Science Ltd., particularly Julie Neden and Chris Pringle.

William H.K. Lam Professor Department of Civil and Structural Engineering The Hong Kong Polytechnic University Yuk Choi Road Hung Horn, Kowloon HONG KONG Tel: (852) 2766-6045; Fax : (852) 2334-6389 E-mail: [email protected]

CHAPTER 1

INITIAL PLANNING FOR URBAN TRANSIT SYSTEMS S. C. Wirasinghe, Department of Civil Engineering, University of Calgary, Calgary, Alberta T2N1N4 Canada

1.

BACKGROUND

The current state of initial transit planning in many transit agencies could be described at best as an art and at worst as a collection of ad-hoc rules. There are many reasons for this situation. The complexity of the problems involved, the non-catastrophic mode of functional failure associated with transit systems, the lack of trained planners, political interference in detailed planning and the failure of people with alternative planning tools to communicate their ideas to front-line planners have contributed to the problem. In the typical transit planning problem we are concerned with providing a good transit service, which has a minimal environmental impact, at a reasonable cost to the transit agency and to the users. A good level of service is provided by a transit service which is reliable, easily accessible in time and space and provides a safe, fast and comfortable ride at a reasonable price. The precise definition of the objectives to be satisfied in providing the transit service, let alone their attainment, is a difficult task. The problem is further complicated by the conflicting nature of the objectives.

2

Advanced Modeling for Transit Operations and Service Planning

A good estimate of the future demand for public transit is necessary to plan a transit system. However, the demand, in addition to being a random quantity at any given time, is also to some extent dependent on the type of transit system and its parameters. This is another dilemma faced by planners. A transit system is a failure if the objectives with which it was planned are not met to a large degree. However, this type of functional failure, as opposed to engineering failures, is not catastrophic since the system generally continues to function and to satisfy the objectives to some degree. Further, errors in planning cannot be easily pinpointed as the major cause of failure even when this is the case.

2.

THE GENERAL PROBLEM

Consider a city or a part thereof for which a public transit system is being planned. The goal of such an exercise could be stated simply as the choice of the mix of transit modes or technologies (e.g. Bus, Light Rail, etc.) and related optimal functional designs, (routes, dispatching policies, etc.) for various areas of the city for different time periods (peak, off-peak, etc.) that maximizes the expected utility to society. However, the practical realization of the goal is not a simple matter. In theory, the above problem could be converted into four related sub-problems: (i) (ii)

Determination of the set of relevant technologies or mixes of technologies, Estimation of the present and future demand for transit given each of the possible technologies (transit systems). (iii) Optimal functional planning of each transit system for the related demand, (iv) Choice of one of the transit systems as the 'best' one. However, the sub-division does not provide us with four simple problems. Rather, each one is in itself a complex problem. 2.1

Selection of Technologies

Various available technologies can be mixed in many ways for various areas of the city, for different trip purposes and for different time periods. However, the number of possible combinations is so large that it becomes prohibitive to carry out the rest of the analysis (sub-problems ii to iv) for each technology-mix. So, a smaller perhaps more relevant set has to be chosen based on speed and capacity considerations,

Initial Planning for Urban Transit Systems

3

compatibility with technology currently in use, environmental impact, geographical constraints, etc. The interested reader is referred to the Canadian Transit Handbook (1980), Gray and Hoel (1979), Vuchic (1981), and Parajuli & Wirasinghe (2001). 2.2

Demand Estimation

An extensive literature is available on the estimation of demand. A good introductory work is that of Ortuzar and Willumsen (1994) . For further treatment of the modalsplit of demand see Domencich and McFadden (1975) and Daganzo (1979). In general, one could proceed with the functional planning aspect under the assumption that the demand for transit is given. However, in selected instances that assumption can be released in favour of the one where the demand is a random quantity with known mean and variance. 2.3

The Best System

The choice of one transit-system for implementation, out of several possible, essentially boils down to a political decision. It is the planners' duty, however, to advise the decision-maker regarding the best choice. Bayesian Decision Theory offers a rational approach by which the planner can take into account the several options available, the uncertainty regarding demand, costs, etc. and a social utility function. The reader is referred to Raiffa (1970) for an introduction to Decision Theory and to Parajuli and Wirasinghe (2001) for an example of an application to transit planning.

3.

FUNCTIONAL PLANNING

We are concerned with improving the present state of the functional planning of transit and not with introducing a completely new methodology. An attempt can be made to make the planning exercise more consistent by using analytical models based on some of the more relevant and quantifiable factors that pertain to the problem at hand. An analytical model can be optimised to obtain a theoretically sound 'initialsolution' that can then be improved and 'fine-tuned' using all available hard and soft, quantifiable and non-quantifiable constraints and other information. The 'science' of transit planning can be considered to be the analytical modelling of a real system and its optimisation, while the 'art' is the conversion of an optimal analytical solution into a practical answer to a real, complex problem.

4

Advanced Modeling for Transit Operations and Service Planning

It should be emphasised from the beginning that several of the sociological, political and geographic factors that affect a problem cannot be included easily in a model. Thus the results obtained from optimising a model should not be taken too seriously or as the "truth". A model can, however, quantify some of the more relevant aspects, serve as an initial solution and thus prevent one from proposing or implementing the ridicule. The following interrelated factors have to be considered when a transit system with a specified mix of technologies is being planned for a given area. 3.1

Network

The network is the collection of routes for each technology for each time period. In some cases, e.g. a variable route dial-a-ride service, it is sufficient to specify a zoneof-operation. 3.2

Mode of Operation

The mode of operation is defined as the type of service offered on a particular route or a network. For fixed routes, the service may be, for example, all-stop (local), fewstops (express), non-stop, zone-stop (local or express in a zone and non-stop outside the zone), or stop-on-call (e.g. dial-a-ride). For variable routes the service may be non-stop, zone-stop, stop-on-call, etc. 3.3

Dispatching Policy and Fleet-Size

Given the mode(s) of operation, the rule according to which vehicles area dispatched on the network routes is the dispatching policy. Essentially, the dispatch rate of vehicles and the travel time on each route have to be specified. The network dispatching policies, vehicle sizes and the fleet-size of each type of vehicle are interrelated and should be considered more or less together. The detailed operational aspects include the assignment of crews to vehicles and vehicles to routes. 3.4

Location of Transfer-Facilities

A transfer-facility is a location at which passengers can transfer from one mode to another. Bus stops and rail stations are the most common examples. The location of transfer-facilities is intimately related to the mode-of-operation. A terminal is a transfer-facility at an end of a route.

Initial Planning for Urban Transit Systems 3.5

5

Location of Vehicles Garages

Normally, each mode will have one or more garages at which vehicles are parked and maintenance is carried out. Occasionally, a terminal can be shared by modes with some similarities and also serve as a terminal. The location of terminals is dependent on fleet size, the network and to a large degree on the availability of land.

4.

APPROACH TO AN ACCEPTABLE APPROXIMATE METHODOLOGY

Obviously, the five factors described above are interrelated and cannot be tackled in complete isolation. However, each is a complex problem and their combination can only be described as a 'mess'. The reason for the use of ad-hoc methods and rules-of-thumb in the functional planning of transit is now fairly obvious. The fact of the matter is that the overall problem is too complex to be formulated properly as a whole for solution even by computer-based iterative methods. In any case, it would be naive to write a complex non-linear objective function with many non-linear constraints and optimise it with respect to several variables using a massive computer algorithm, since even the output from the most complex and largest possible computer program, would have to be significantly altered by planners to allow for still unmodellable factors. The basic premise of this chapter is that the 'human' planner reins supreme over computing 'machines'. The 'block-box' approach where a planner's function is simply to run local data with a 'canned' program and to accept the output as inviolable is rejected. The methodology proposed here is to break up the functional planning problem into more or less independent sub-problems and to obtain approximately optimal analytical solutions to the sub-problems. It is conceded that optimising parts of the problem does not necessarily lead to the optimal solution to the combined problem. The exercise of analytical modelling forces us to think formally about the problem at hand and to isolate the critical factors. Thus it enhances our understanding of the problem.

6

Advanced Modeling for Transit Operations and Service Planning

The analytical solutions highlight the basic interrelationships among the relevant parameters which could easily be clouded by enumerative type techniques. These solutions are also easily subjected to sensitivity analysis. Analytical solutions are ideal for planners who can use them as 'initial-solutions' to complex problems since they are easily manipulable and modifiable. 4.1

The Objective Function

To avoid issues such as the additivity of individual utilities, etc. [see Jordaan (1985)] we shall express the total disutility of a system as a total cost expressed in monetary units. The objective function in all instances unless specified otherwise will be the total cost function. The objective is to choose the relevant system parameters so as to minimize the total cost. Certain other commonly used objectives such as the minimization of the total travel time and minimization of the operating cost can be shown to be special cases of the objective chosen here. The above philosophy is illustrated in the following through the applications to a single bus route. Aspects of the dispatching policy and scheduling of travel time are discussed. They are provided for illustrative purposes and should not be considered to be sufficiently detailed for use in practice.

5.

DISPATCHING POLICIES FOR A Bus ROUTE

Most bus routes serve multiple passenger origins and destinations with a limited number of buses. The capacity of the buses, while being uniform in most cases, can vary from bus to bus. Each bus may be used for several round-trips on the route. The policy that is followed with respect to the dispatching of buses on a route is of fundamental importance since it has a direct bearing on the level of service provided to the passengers and the cost of providing service on the route. We shall explore dispatching policies for several different situations. 5.1

Newell's Policy

Newell's dispatching policy for a (public) transportation route with time-varying many to one demand [Newell (1971)], simply stated, is that the dispatch rate of vehicles is proportional to the square root of the arrival rate of passengers if the vehicles are sufficiently large, and to the arrival rate if, otherwise. The objective of

Initial Planning for Urban Transit Systems

1

the policy is to minimize total passenger waiting time subject to a fixed number of dispatches (not less than the minimum required to transport all the passengers in vehicles that are filled.) It can be written in the form

where g(t) is the dispatch rate of vehicles at time t in vehicles per unit time, p(t) is the arrival rate of passengers at time t in passengers per unit time, c is the capacity of the vehicles and K is a constant which is chosen to satisfy the constraint of a fixed number of dispatches. Newell (1973) has also pointed out that the identical policy applies if the objective is to minimize total passenger waiting time costs and dispatching costs, and how AT is a function of yw , the average value of a unit of waiting time per passenger and yD , the cost of dispatching a vehicle. The latter formulation of the policy is given by

(2)

p(f)lc The original constraint of a fixed number of dispatches is identical to fixing yw at a particular value, given y D . The dispatching policy can be in a sense unconstrained and the level of service (frequency) provided can be increased by increasing yw . If somehow the "real" value of yw were known, the related dispatch rate and hence the number of dispatches could be determined. On the other hand, one could interpret yw as the amount that the bus company is willing to pay to save one passenger one unit of waiting time. There is no evidence to date that this policy has been adopted by public transit systems, and it is not mentioned in transit manuals [Gray & Hoel (1979), Soberman & Hazard (1980), Pushkarev, et al. (1982)] and even in some reviews of dispatching policies [Furth & Wilson (1981), Chua (1984)].

8 5.2

Advanced Modeling for Transit Operations and Service Planning Uniform Dispatch Rate During Off-Peak Periods

It is common for bus companies to adopt a uniform dispatch rate during (off-peak) periods when the demand for travel is low and capacity constraints do not apply. If a uniform dispatch rate g is to be adopted during a period of duration T, in response to a time-varying demand ofpft) passengers per unit time, the sum of the waiting time cost and dispatching cost per unit time is given by

dt

(3)

if a passenger on average waits one half of a headway. This is minimized when

(4) where p is the mean demand, in passengers per unit time, during the period T. Thus the uniform dispatch rate is predicated on the square-root of the mean demand. The mean total cost per unit time during the period T under a uniform dispatch policy is obtained by substituting for g from Equation (4) in Equation (3): -

(5) The mean total cost per unit time if Newell' s Policy of a variable dispatch rate is adopted, (6)

is always lesser than the total cost given in Equation (5). However, the difference can be quite small when the demand is not highly variable with /, which is the likely situation during an off-peak period. Since there are intangible benefits in adopting a uniform dispatch rate policy for an off-period, the practice can be made consistent with Newell' s Policy if the dispatch rate in Equation (4) is used.

Initial Planning for Urban Transit Systems 5.3

9

Policy Headways

Bus companies commonly use a "policy headway" - a maximum acceptable headway, hp, as an integral part of their dispatching policy. The policy headway (usually in the range 30-60 minutes) is chosen subjectively. Newell's Policy and subjective judgement can be amalgamated by using the maximum of l/hp and, the dispatch rate given by Equation (2), as the actual dispatch rate. 5.4

Average Waiting Time

A "wait" is defined as the (absolute) deviation between the actual and desired departure times of a passenger. Consequently, the wait caused by the bus schedule could take place not only at the origin bus stop but also at the origin or destination of the trip depending on the type of trip. For example, morning peak commuters, who presumably have a specific time before which they should be at work, will likely spend the wait time at the destination (schedule-delay), subsequent to taking the last possible bus that will transport them to the destination on or before the required arrival time. Thus, we can consider the desired departure time to be independent of the schedule, and interpret p(t) to be the desired departure rate of passengers at time /. If p(t) is (approximately) uniform within a headway, then the average wait is (approximately) one half of a headway. It has been assumed implicitly in the above discussion that buses run as scheduled. 5.5

Value of Waiting Time

The Policy is based on the dual assumptions that the average wait is one half a headway and that the value of a unit of waiting time per passenger, yw, is invariant with the elapsed waiting time. The latter is an unstated assumption that is routinely made in the literature on dispatching policies and even in books on travel time [see, for example, Tarski (1987)]. Horowitz (1981), using data from a laboratory psychological scaling experiment, found that the subjective value of a unit of waiting time did not change significantly during a period of waiting for the range 5-15 minutes of waiting. However, it was higher in the range 0-5 minutes. Given that the experiments were conducted at discrete 5 minute intervals, he concluded that there is a fixed penalty attached to having to wait, which is independent of the amount of time spent waiting, and that the value of a unit of waiting time is a constant. This penalty has no effect on Newell's Policy since it is independent of the time spent waiting.

10

Advanced Modeling for Transit Operations and Service Planning

Walther (1975), using data from field studies for the range 0-6 minutes of waiting, found that the riding time equivalent to a wait is a non-linear increasing function of the wait. This is equivalent to the value of waiting time per unit time being an increasing function of the elapsed waiting time. Newell's Policy will result in nonoptimal headways if Walther's results extend to the range of higher waiting times. In the face of conflicting evidence regarding the relationship between the value of a unit of waiting time and the elapsed waiting time, it is worthwhile to investigate the relationship between headway and yw further. If we assume that the value of a unit of waiting time is a linear function Y*w+PwY (7) of the elapsed waiting time T, the total cost of passenger waiting time in a headway h(t) is

(8) where p(t) is the arrival rate of passengers and (x-y) is the elapsed waiting time up to a time * of a passenger who arrived at time y. Minimizing the sum of, the total cost of passenger waiting time/per unit time obtained by dividing the Expression (8) by h(t), and the cost of dispatching buses per unit time, yD /h(t) , we find the minimum total cost headway to be given by (l/2)p(t)y*w -yD/h2(t)-(l/3)p(t)pwh(t)=0

(9)

A unique solution for h(t) exists in h(t)>0 if fiw >0, i.e. the value of a unit of waiting time increases with elapsed waiting time, and Newell's square-root headway is an upper bound. Similar results are likely when the value of a unit of waiting time is a non- linear increasing function of the elapsed waiting time. The use of policy-headways can be interpreted as an attempt to keep headways within reasonable limits, given that the optimal headways under the assumption of a uniform value of yw can give high headways, particularly when the demand is low. However, if the values of y*w can be used instead.

an<

i Pw

are

known, the optimal headways given by Equation (9)

Initial Planning for Urban Transit Systems

11

If YW = 0, i.e. small amounts of waiting time have negligible value, the optimal headway is given by

(10) a cube-root relationship as opposed to Newell 's square-root relationship. 5.6

Stochastic Effects

Consider, for example, how the Policy can be applied to obtain a schedule for a certain period T on (say) weekdays. Data from several weekdays is obtained from the arrival rate of passengers, as it varies with time during the period T, and the mean arrival rate at time t over several days is used as the numerical value of the function p(t) at t. Subsequently, the dispatch rate at time t, g(t), is obtained from Equation (2). The actual dispatch times are those at which the integral ofgft) with respect to time is equal to integer values. The above procedure can be improved upon, if one considers that p(t) varies from weekday to weekday at each / and consequently can be treated as a random variable at each t. Then, g(t), given by Equation (2), is a function of a random variable, and consequently has a mean and a variance that can be estimated approximately by applying a theorem given by Lindley (1965). The mean dispatch rate at time t,

(p(t)/c and the related variances of the dispatch rate at time t,

I a2P(t) /c2 where the mean, variance and coefficient of variation of (t) are given by ~p(t), a2P(t) and C2P(t) respectively.

12

Advanced Modeling for Transit Operations and Service Planning

A more appropriate schedule is obtained if the procedure described in the first paragraph of this section is used with the mean dispatch rate g(t) given by Equation (11) instead ofg(t). The probability that passengers could be left behind by full buses (due to underestimation of the demand) can be reduced by choosing a dispatch rate, g(t) + Acrg(t) , with the constant A>0, when the capacity constraint governs the dispatch rate. The value of C2p(t) can be high during certain off-peak periods when the demand at a time / can vary significantly from day to day through p(t) is low. For example, if p(t) is distributed Negative Exponentially at time (t), and hence C2p(l)= 1, the dispatch rate is reduced by 12.5% when Equation (11) is used in place of Equation (2). However, the values of C2p(t) are likely to be small during peak-periods and even during most off-peak periods. Thus there is no significant difference in the schedule if Equation (11) is used in schedule development instead of Equation (2).

6.

MANY TO MANY DEMAND ROUTES

If all boardings precede all alightings (e.g. a commuter bus that picks people at several CBD stops, runs non-stop to a residential area and makes several stops where passengers may alight), each seat or passenger-space is used only by one passenger. Thus the analysis in Newell (1971) will apply. It is also applicable to routes with one to one, many to one and one to many type demand. In the following the word "seat" will be used to describe a "passenger-space". We consider a route where passengers can board and alight at any stop and counts of boardings and alightings from each bus in the current schedule are available. If the demand to board does not exceed the capacity of a vehicle, when the dispatching policy is determined by the square root rule, there are sufficient seats to allocate a particular seat to each passenger and to no one else. Consequently, no new dispatching rules are necessary. However, if the demand to board does exceed the capacity of a vehicle, it does not necessarily follow that a lower headway based on capacity considerations should be used. Since passengers can alight at any stop, the maximum of the actual number of people in the vehicle at any time (load), could be less than its capacity c.

Initial Planning for Urban Transit Systems

13

Let the load in a bus, which varies with the elapsed bus travel time z, be denoted by L(z) as shown in Figure 1. If c>L(z)for all (z) during the trip, the dispatch can be made at time, say, t0 according to the square-root policy. However, if c < L(z) for any z, when the dispatch is made at t0, the dispatch time should be moved forward, by the smallest possible increment, to IA, until c>L(z) for all z, as illustrated in Figure 2. Thus the bus is dispatched so as to be filled to capacity at the maximum load point and all passengers will be able to board the bus without being left behind. This dispatch time IA cannot be earlier than the dispatch time, tc, given by Newell's capacity constraint. However, dispatching the bus at tc guarantees that the demand for boarding can be accommodated, even if data on the alighting pattern is unavailable. Newell's Policy thus provides a safe lower bound headway when the capacity constraint applies. Newell's Policy can be extended to allow for boarding and alighting at any point if the capacity constraint is applied to the demand for travel given in terms of seats per unit time, since a seat can be used in series by more than one passenger. Thus the policy given by Equation (2) can be modified to

s(t)/c

(12)

where s(t) is the demand at time t in terms of seats per unit time. The rate s(t) can be estimated for an existing route as follows:

Figure 1. Variation of Load with Dispatch Time and Elapsed Travel Time

14

Advanced Modeling for Transit Operations and Service Planning

Figure 2. Variation of Passengers Boarding A Bus with Dispatch Time

Let the cumulative number of people who board and alight a bus / dispatched at time /, be By and Ay respectively up to stop j. Then the maximum load on the bus and hence the demand for seats in the bus is

The cumulative demand, in terms of seats, for all buses up to time tt is

We can plot the cumulative demand M(tj) versus ?, as a step function, smooth it out to obtain M(f), and take the slope at any time as s(t) [Figure 3]. The demand at time t in terms of seats per unit time is

Initial Planning for Urban Transit Systems

15

For the special case of an existing route which has a fixed maximum load point, s(t) is the rate at which passengers pass that point at time t + zm where zm is the bus travel time between the route origin and the maximum load point. Consequently, s(t) can be measured easily be placing one observer at that point. The rate s(t) cannot be estimated for a planned route without detailed time-dependent data on origins and destinations of passengers. As a practical matter it is best to use Newell's Policy and/or policy headways for scheduling a planned route until the route becomes operational and the data for estimating s(t) becomes available. The application of the modified policy given by Equation (12) is illustrated in Figure 4. The dispatch time ts for vehicle 2 based on the square root rule is modified to t'2(c, because the capacity constraint is not violated. Under the original policy of Newell, vehicle 3 would have been dispatched at t] (< t}) to satisfy the capacity constraint.

Figure 3. Demand and Cumulative Demand in Terms of Seats

The dispatching policy for many to many time-varying demand can be extended to allow for policy headways and the day-to-day variations in traffic. Then the modified dispatching policy is given as follows:

16

Advanced Modeling for Transit Operations and Service Planning

The dispatch rate

and the related variances

where s(t)and <72s(t), are the mean and variance of s(t). The actual schedule is obtained using the procedure described briefly in the section entitled Stochastic Effects. Newell's Policy is applicable to a two-way route if it is considered as a round trip route, with a single dispatching point. The fleet-size is then the maximum number of dispatches during a round trip period.

Figure 4. Modified Capacity Constraint for Many to Many Time Varying Demand

Initial Planning for Urban Transit Systems

7.

17

PRACTICAL HEADWAYS AND SCHEDULES

Several difficulties are associated with applying Newell's Policy in the original or modified form in practice. The commonest one encountered by front-line planners is the lack of knowledge regarding yw, the average value of a unit of passenger waiting time. Methods available for estimating yw, (e.g. [Bruzelius (1978)] ) are expensive and data intensive. It may also be more appropriate to interpret yw, as the amount that the bus company is willing to pay to save one passenger one unit of waiting time. In any case the policy is not too sensitive to the value of yw and approximate figures could be used. The allocation of buses to routes could be made equitable by using the same value of yw on all routes, particularly when the available fleet is limited. In other words, the value of YW can be varied until the fleet size required is equal to that available. Another perceived problem is that of providing headways so that (i) buses depart at the same time of the hour every hour to facilitate remembering the schedule and the operation of the system and (ii) so that headways are certain "acceptable" numbers such as 5, 10 and 15 minutes. For example, 7 minute headways are not usually provided. One cannot see why peak period headways should follow the above dictums since a regular commuter would presumably take the same bus each day and it would make no difference if the bus was say the 7:11 bus or the 7:10 bus. During the off-peak when there may be many passengers who are not "regulars" and when it is convenient to the transit system to provide service according to a simple schedule, one can use the variation of Newell's Policy given in the section on Uniform Dispatch Rate During Off-Peak Periods and an acceptable headway that is close to the "optimal" one. A computer program that facilitates the implementation of the modified policy has been developed [Wirasinghe and Roose (1987)]. For an existing one or two-way route for which the demand to board and alight each vehicle at each stop, the actual dispatch time of each vehicle, the round trip time, the mean value yw of a unit of passenger waiting time, the seated and standing capacity of the vehicles and the cost per dispatch is known, the program is designed to provide a schedule, the necessary fleet size and the in-service and out of service time for each bus for the three different level-of-service scenarios (a), (b) and (c) given below in decreasing order of level-ofservice:

18

Advanced Modeling for Transit Operations and Service Planning

(a)

Yw

=

(b)

yw - yw

bus capacity = seated capacity + half of

(c)

yw = yw

bus capacity = seated and standing capacity

Yw + 7w'

bus capacity = seated capacity standing capacity

Options are provided so that the schedule could be based on actual or "acceptable" headways. The standard deviation of yw is substituted for yw'/2 if available. Under level of service (c) headways will be relatively high and passengers are likely to be left behind if random increases in demand occur, while under scenario (a), such fluctuations could most likely be absorbed by the standing spaces and the headways will be relatively low. An intermediate level of service is provided by (b).

8.

CONCLUDING COMMENTS ON DISPATCHING POLICY

The square-root dispatching policy with a capacity constraint proposed by Newell (1971) for time-varying many to one passenger demand is applicable, with some modifications, under most conditions. The square-root formulation is applicable even if the average waiting time is not one half the headway but is some other fraction of headway. A fixed penalty associated with having to wait for a bus has no effect on the policy. However, if the value of a unit of waiting time, yw, is an increasing function of the elapsed waiting time, the square-root formula is only an upperbound, and a unique optimum headway that satisfies the objective exists. Further study regarding the possible variation of yw with elapsed waiting time is required, particularly for high headway situations. Newell's Policy is easily adapted to the case of stochastic demand (variations inp(t) over several days) and can be applied directly during peak periods and most off-peak periods when the coefficient of variation ofp(t) is small. Newell's Policy can be modified for use on many to many demand routes with variable passenger trip lengths and variable maximum load points by using the demand measured in terms of seats per unit time at t, s(t), in the capacity constraint. This quantity can be easily estimated for existing routes. The capacity constraint

Initial Planning for Urban Transit Systems

19

applied to the demand in terms of people per unit time,p(t), provides a lower-bound headway which could be used in new routes, until s(t) can be determined. The dispatching policy of Newell is robust and applicable under a variety of circumstances with few modifications. It could be used easily by transit planners for obtaining reasonable and practical schedules.

9.

SCHEDULING TRAVEL TIME ON A Bus ROUTE

9.1

Background

Consider a bus route with many stops on which buses are operated according to a schedule. Such schedules cannot be maintained exactly because of the delays that occur at the terminal from which the buses are dispatched, the random natures of the travel time between stops and the time lost at stops while processing (loading, unloading) passengers. However, extensive passenger surveys have repeatedly shown that reliability is ranked first among level-of-service attributes [Gray (1992)]. Transit planners and operators have used the following procedure for maintaining a high degree of adherence to the schedule: (i) Buses are dispatched from the terminal on time, as much as possible; (ii) Several stops enroute are chosen to be "time-points" (check-points), i.e. the departure times from these stops are scheduled. The scheduled departure time from a time-point are chosen by taking the mean travel (including passenger processing) time between two adjacent time-points, combining with some slack time and adding the sum to the scheduled departure time from the upstream time-point. Buses that are early (i.e. they are ready to depart a time-point prior to the scheduled time) are held until the scheduled time, while those that are late are released immediately upon completion of passenger processing. If a bus falls significantly behind schedule, it may be taken out of service, if it is possible to insert a spare bus into service. Essentially, the departure time of each bus from the terminal and each time-point, and the arrival time at the final destination is the "schedule". When the schedule is published, passengers know that the buses will not leave the time-points prior to the scheduled time and consequently can arrange to arrive at time-points accordingly. At intermediate (non-time) stops, passengers can estimate the departure time by interpolation. With advances in technology it is now possible in some cities to obtain periodic updates of the estimated arrival time of buses at every bus-stop, by

20

Advanced Modeling for Transit Operations and Service Planning

telephone. Consequently, the uncertainty regarding the bus arrival time is reduced and passengers can wait "at home" instead of at the bus stop if a bus is delayed relative to the schedule and even manage to catch buses that are running slightly ahead of schedule. These schemes have not negated the need for a schedule. If a bus on a scheduled route is delayed for any reason, the tendency is for it to process extra passengers and fall further behind, in some instances causing the following bus to catch-up to it to form a pair. This phenomenon was first analysed by Newell and Potts (1964). The proper scheduling of a route does not prevent the bunching of buses but makes it a low probability event. The planning of a bus-route involves the selection of the number of time-points, their locations at selected stops and the slack-times to be added to the mean travel time between adjacent time-points. In this section, we shall not be concerned with determining the locations of timepoints. Rather, we wish to examine the amount of slack time that should be added to the mean travel time (including passenger processing time) for scheduling travel to time-points on a route. Presently, there is no agreement on the answer to this question. Various simulation studies have suggested a range of zero to two standard deviations. Others have suggested that slack time be chosen based on an arbitrarily chosen reliability (probability of being on time) such as 0.75 to 0.83 [Lesley (1975)] and 0.95 [Abkowitz and Engelstein (1984)]. Our intent is to examine the costs affected by the choice of slack time and therefore to obtain additional insight regarding its choice. 9.2

Travel Time Parameters

Consider the special case of a single link (non-stop) route where passengers board at a terminal and vehicles run non-stop to the destination. The scheduled travel time on the link is defined as the mean travel time plus slack time. Given the schedule, passengers and the transport company must allocate or budget at least that amount of time for the trip. Delays from the schedule cause additional travel time and inconvenience to the passengers and extra operating costs to the operator. In some cases the operator and even the passengers may also be subject to additional penalties for being late. The scheduled travel time should be chosen to optimise an objective function that reflects the above disutilities.

Initial Planning for Urban Transit Systems

21

Consider a vehicle following a vehicle that has been run exactly as scheduled. Let the travel time (including passenger processing time) of the following vehicle, if it were unscheduled, be a random variable, T, with a unimodal probability density function f(t) in t3tm , where tm is the minimum travel time. If a scheduled travel time, S, is imposed on the route, that amount of travel time is budgeted by the operator and the passengers for the trip. Early arrival does not cause the vehicle to be utilized more. It merely causes the vehicle to be held and extra layover time for the crew. Passengers who have budgeted the scheduled amount of time for the trip are unlikely in most cases to be able to make use of the early arrival of the vehicle. Though there are some exceptions, we shall take the travel time to be the budgeted travel time S, even if the vehicle is nominally early. The expectation of the actual travel time, E[A] , is greater than S. Formally,

and, the rate at which it varies with S,

where F(-) is the cumulative probability distribution function of T. The expected delay:

is decreasing function of S, as illustrated in Figure 5. Minimization of the budgeted travel time leads to setting the scheduled travel time, S, to the minimum value of tm, while minimization of the expected delay leads to very large values of S (say mean plus two standard deviations). Therein lies the essence of a trade-off for the choice of S. 9.3

Cost of Travel Time

Let the cost of a unit of budgeted travel time per vehicle, which includes the cost of travel time of all passengers as well as vehicle owning and operating costs be known. If a vehicle is

22

Advanced Modeling for Transit Operations and Service Planning

Figure 5. Expected Actual Travel Time and Delay delayed beyond the scheduled travel time, the additional time spent by passengers is assumed to cost more per unit time because of the inconvenience, stress, etc. This effect has been seen in empirical studies undertaken by Horowitz (1981) and Walther (1975) for passenger waiting time. Certainly, the delay will not cost any less per unit time than budgeted time. The crew cost can also increase in some cases due to "overtime" rates. The cost of a unit of travel time per vehicle is formulated as: y' for budgeted travel time

and y for delayed travel time with the understanding that y >y' when passengers have similar trip purposes. Thus, if the vehicle is on time or early (t > S), the cost per trip is y'S and if the vehicle is delayed (t > S), it is y'S + y(t - S). Then the expected cost of travel time is

(19) 9.4

Delay Penalty

The cost related to a vehicle being behind schedule can, in many cases, have another component that is unrelated to the actual amount of delay. For example, if a transfer is missed, the related cost to passengers is independent of the amount of time by which it was missed. This component, the delay penalty, which includes the

Initial Planning for Urban Transit Systems

23

inconvenience of missed appointments, etc. may be modelled as a cost equal to zero for on time vehicles and to y for delayed vehicles. Therefore, the expected penalty

9.5

Expected Total Cost

The expected total cost for the trip, E[Z/(S)J, is the sum of the expected travel time cost (Equation 19) and the expected penalty cost (Equation 20):

10.

OPTIMAL SCHEDULED TRAVEL TIME

10.1

Optimisation

We define the optimal scheduled travel time as that which minimizes the expected total cost given by Equation 21. The respective first and second order conditions for the optimum are:

F(s}-(yp/y}f(s}+y'/y

=i

(22a)

and

f(s)-(yp/y)f'(s)>0

(22b)

respectively. The solutions to Equation 22a depend on the exact form of the probability density function (f(t) and the various cost parameters [Figure 6]. The slope of £'[z(5')] isy'/y'1 (<0) when S = tm and y1 /y (<0) when S-»oo. Consequently, £[z(s)] must have a minimum in S> tm. Any unimodal density, with the mode loaded at t\f, will satisfy Equation 22b for all S>tM. Thus any root of Equation 22a that is greater than IM will be a relative minimum of ^[z^)]. However, under certain conditions -£"[z(s)] can also have a relative minimum in S < IM- The function, /?[z(,S)] has only one minimum and no internal maxima in all cases with one or two roots. In the case

24

Advanced Modeling for Transit Operations and Service Planning

of three roots, there are two relative minima and the minimum has to be selected by enumeration.

Figure 6. Solutions to Equation (22a) The optimal scheduled travel time, S*, is dependent only on the probability density function of T and on the two unit cost ratios: y /y, the unit cost of budgeted travel time as a fraction of the unit cost of delayed travel time. We now consider some special situations to investigate the properties of S*. 10.2

No Delay Penalty (yp = o)/ Recreational Trips

Consider a recreational trip in which there is no specific penalty in arriving late (y = 0) and no advantage in arriving early. However, a delay means that time that was planned to be spent in recreational activities is pent travelling in a vehicle. Setting y = 0 in Equation 22a gives:

(23) For S* to be the median value of T, y' /y- 0.5, i.e. a unit of delayed travel time is worth two units of budgeted travel time. It is common, however, for S* to be set by adding k (> 0) standard deviations (a) of "slack time" to the mean. If T is assumed to be normally distributed, setting k = Vi, for example, is equivalent to the unlikely situation where a unit of delayed travel time is valued at 3.25 units of budgeted travel time. The equivalent probability of being on time (reliability) is about 0.7. A rather

Initial Planning for Urban Transit Systems

25

unconventional insight is that S* can be set at less than the median value of T for trips with no penalty if y' I y > 1A. For example, if y' / y - 2/3, S* = F "7(0.33). 10.3

Delay Penalty Only (y' = y)

Consider a feeder trip where any delay; regardless of the amount, results in a penalty equivalent to a missed transfer. If all disutilities related to the delay are absorbed in yp and y' = y, Equation 24 gives

(24) For example, S* = E[T] + 0.5 a when T is normally distributed and the penalty is equivalent to the cost of 1.96
Thus S* can be estimated by substituting the headway for yp /y in

Equation 24 if the probability of two or more successive transfers being missed is neglected, and there are no other penalties. For intercity trips, yp

can include the costs to the operator for meals and

accommodation for stranded passengers. Also, local bus operators that refund the fare if a bus is not on schedule can consider the fare to be a part of yp per passenger. 10.4

No Disutility of Delay (y = y' and yp = 0)

If the cost of travel time is uniform regardless of any delay (y = y') and there is no penalty to being delayed ( y p =0), vehicles must be scheduled to run in the minimum possible travel time (S* = t^) and the probability of delay is one. The above scenario will not occur in a public transportation route. However, it can occur in a route operated by an unregulated monopoly if the operator ignores the disutilities of passengers and pays no overtime.

26 10.5

Advanced Modeling for Transit Operations and Service Planning Case of y' < y and yp > )AVork Trips

For work trips, y is likely to be > y' due to the stress to workers of being late, and yp can be >0 due to work related penalties of being late. For example, some workers may loose a half hour of pay for being late by less than or equal to one half hour. Consequently, both y'/y and yp /y will be factors in the choice of S*. It is unlikely that y' /y < Y2. Hence from two sub-sections above, it is apparent that S* will be greater than the median value of T, mostly when yp also exists in addition to y>y'. For example, if

y'/y

= 2/3,

yp/y^

1.75
distributed, S* = E[T] + 0.675 a and the reliability is 0.75. 10.6

Case of /' > y and yp > 0/Split Trip Purposes

It is possible for y' > y , if passengers in a vehicle have different trip purposes. For example, consider a bus in which half the passengers are transfers and the other half are workers at the destination. The value of y' will be calculated based on all the passengers, while y will be based only on the group of passengers who work at the destination, causing y' > y in some cases. However, the penalty cost now includes the sum of the cost of delayed travel time spent in the bus by transferring passengers and the cost of additional time spent out of the bus while waiting to transfer. Consequently, a minimum total cost solution for S is still available [Figure 6] and is given by S*(2) which is the higher one of two roots.

11.

CONCLUDING COMMENTS ON SCHEDULING TRAVEL TIME

An idealized situation (a single link) was analysed with respect to the scheduled travel time estimation problem. However, the cost based approach is insufficiently developed for application to most real routes with many time-points. It was shown that an optimal scheduled travel time that minimizes the sum of the expected costs of budgeted and delayed travel time and expected penalty costs exists for a link. The scheduled travel time (and the related reliability) are dependent to a large extent on the trip type. For example some recreational trips can have negative

Initial Planning for Urban Transit Systems

27

optimal slack times and reliabilities < 0.5 while work trips have positive optimal slack times and reliabilities around 0.75. Further investigation is required regarding the assertion that delayed travel time "costs" more per unit time than budgeted travel time, because of the inconvenience and stress associated with delays. The Equation (22a) for S* cannot be easily used in practice until y/y', the factor by which the cost of (budgeted) travel time per passenger per unit time is increased for delayed travel time, is estimated for various trip purposes using techniques such as disaggregate mode choice analysis. The model proposed here can be extended to allow for the case where early arrival of a vehicle allows passengers to use the extra time (S-t). However, it is then necessary to estimate the cost of "early time' per passenger per unit time which is presumably less than the equivalent cost of budgeted in-vehicle travel time. When required, the proposed method can be used solely with either operator costs or passenger costs. The estimation of the probability distribution of the travel time of unscheduled vehicles, f(t), is also problematic. Since most existing routes operate under some control strategy, the above travel times cannot be easily observed. In a scheduled single link route, all travel times will be appropriate observations if the drivers have not intentionally "killed" time enroute. In the case of multiple time-points, observations must be made at each time-point and any holding time enroute deducted from the total, again with the stipulation that time not be killed between time-points. Thus cooperation of the vehicle operators is required in running without killing time enroute, in not leaving time-points early, and in leaving time points as soon as passenger processing is complete, if running late. Regardless of the above, and except in pre-booked or large headway routes, the observations are strictly valid only if the vehicle is following a vehicle that was run exactly on schedule. Application of the above methodology to a normal many to many demand bus route with several time points is given in Wirasinghe and Liu (1995) and extended in Liu and Wirasinghe (2001).

ACKNOWLEDGEMENTS This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant number A4711. The sections on Dispatching Policies

28

Advanced Modeling for Transit Operations and Service Planning

and on Scheduling Travel Time were essentially taken from Wirasinghe (1990) and Wirasinghe (1993) respectively.

REFERENCES Abkowitz, M. and I. Engelstein (1984). Methods for maintaining transit service regularity. Transportation Research Record, 961, 1-8. Bruzelius, N. (1978). The Value of Travel Time — Theory and Measurement. Nationalekonomiska Institution, Stockholm University, Sweden, Skrift No. 3. Canadian Transit Handbook (1980). (R. M. Soberman and H. A. Hazard, eds.), University of Toronto-York University Joint Programme in Transportation, Toronto, 767. Chua, T. A. (1984). The planning of urban bus routes and frequencies: A survey. Transportation., 12, 147-172. Daganzo, C. (1979). Multinomial Probit. Academic Press, New York, 222 . Domencich, T. A. and D. McFadden (1975). Urban Travel Demand. North Holland, Amsterdam, 215 . Furth, P. G. and N. H. M. Wilson (1981). Setting frequencies on bus routes: Theory and practise. Transportation Research Record, 818, 1-7. Gray, G. E. and L. A. Hoel (1979). Public Transportation: Planning, Operations and Management. Prentice-Hall, New Jersey, 749. Gray, G. E. (1992). Perceptions of public transportation. In: Public Transportation 2nd Edition (G.E. Gray and L.A. Hoel, eds.), pp. 617-635. Horowitz, A. J. (1981). Subjective value of time in bus transit travel. Transportation, 10, 149-164. Jordaan, I. J. (1985). Probability and Decision for Civil Engineers. Ellis-Horwood, U.K. Lesley, L. J. S. (1975). The role of the timetable in maintaining bus service reliability. Proceedings of the Symposium on Operating Public Transport, University of Newcastle Upon Tyne, U.K., 36-53. Lindley, D. B. (1965). Introduction to Probability and Statistics. Cambridge University Press, Cambridge. Liu, G. and S. C. Wirasinghe (2001). A simulation model of reliable schedule design for a fixed transit route. Journal of Advanced Transportation, 35(2), 145-174. Newell, G. F. and R. B. Potts (1964). Maintaining a bus schedule. Proceedings of the Second Conference, Australian Road Research Board, 2(1), 388-393.

Initial Planning for Urban Transit Systems

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Newell, G. F. (1971). Dispatching policies for a transportation route. Transportation Science, 5, 91-105. Newell, G. F. (1973). Scheduling, location, transportation and continuous mechanics: Some simple approximations to optimization problems. SIAM Journal of Applied Mathematics, 25(3), 346-360. Ortuzar, J. de D. and L. G. Willumsen (1994). Modelling Transportation. Wiley, 439. Parajuli, P. and S. C. Wirasinghe (2001). A line haul transit technology selection model. Transportation Planning and Technology, 24, 271-308. Pushkarev, B. S., et al. (1982). Urban Rail in America - An Exploration of Criteria for Fixed Guideway Transit. Indiana University Press, Bloomington. Raiffa, H. (1970). Decision Analysis. Adison Wesley, Mass., 309. Soberman, R. M. and H. A. Hazard (eds.) (1980). Canadian Transit Handbook, (University of Toronto - York University Joint Program in Transportation, Toronto. Tarski, P. (1987). The Time Factor in Transportation Processes. Elsevier, Amsterdam. Vuchic, V. R. (1981). Urban Public Transportation: Systems and Technology. Prentice Hall, 673. Walther, K. (1975). Die fahrzeitaquivalente reisezeit im offentlichen personennahverkehr. Verkehr und Technik, 7, 271-274. Wirasinghe, S. C. and R. Roose (1987). Modified DISPOL Program and Users Manual. Department of Civil Engineering, the University of Calgary, Calgary. Wirasinghe, S. C. (1990). Re-examination of NewelFs dispatching policy and extension to a public transportation route with many to many time varying demand. In: Transportation and Traffic Theory, (M. Koshi, ed.), pp. 363-378. Elsevier. Wirasinghe, S. C. (1993). Cost based approach to scheduling travel time on a public transportation route. In: Transportation and Traffic Theory, (C.F. Daganzo, ed.), pp. 205-216. Elsevier. Wirasinghe, S. C. and G. Liu (1995). Determination of number and locations of time, points in transit schedule design - Case of a single run. Annals of Operations Research, 60, 161-191.

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CHAPTER 2

PUBLIC TRANSPORT TIMETABLING AND VEHICLE SCHEDULING Avishai Ceder, Cn il Engineering Department, Transportation Research Institute, Technion-Israel Institute of Technology, Haifa, Israel 32000.

1.

INTRODUCTION

The bus, railway and passenger ferry operational planning process includes four basic components performed usually in sequence: (1) network route design, (2) setting timetables, (3) scheduling vehicles to trips, and (4) assignment of drivers (crew). The framework of this process appears in Figure 1. It is desirable for all the four components to be planned simultaneously to exploit the system's capability to the greatest extent and maximize the system's productivity and efficiency. However this planning process is extremely cumbersome and complex, and therefore seems to require separate treatment of each component, with the outcome of one fed as an input to the next component. In the last twenty years, a considerable amount of effort has been invested in the computerization of the four components mentioned above, in order to provide more efficient controllable and responsive schedules. The best summary as well as the accumulative knowledge of this effort was presented in the second through the eight International Conferences on Public Transport Scheduling, and appear in the books edited by Wren (1981), Rousseau (1985), Daduna and Wren

32

Advanced Modeling for Transit Operations and Service Planning

Figure 1. Framework of public transport operational planning process with emphasis on this chapter's components

Public Transport Timetabling and Vehicle Scheduling

33

(1988), Desrochers and Rousseau (1992), Daduna et al. (1995), Wilson (1999) and Voss and Daduna (2001). This chapter focuses on two scheduling components: timetabling, and vehicle scheduling while assuming that the public transport network is unchanged. The first component in Figure 1 deals with the establishment of public transport interchanges (meeting point of various public transport modes), terminals (trip start 'and end points), and routes. This first component is covered in the next chapter. The fourth scheduling component is usually divided into creating crew duties and crew rosters (rotation of duties among the drivers) and will not be dealt with in this book. The timetable component in Figure 1 is aimed to meet the general public transportation demand. The demand varies during the hours of the day, the day of the week, from one season to another, and even from one year to another. This demand reflects the business, industrial, cultural, educational, social and recreational transportation needs of the community. It is the purpose of this component to set appropriate timetables for each transit route to meet the variation in the public demand. Determination of timetables is performed on the basis of passenger counts, and must comply with service frequency constraints. The vehicle scheduling component appearing in Figure 1 is to schedule vehicles to trips according to given timetables. A public transport trip can be either planned to transport passengers along its route or to make a dead-heading trip in order to connect efficiently two service trips. The scheduler's task is to list all daily chains of trips (some dead-heading) for a vehicle, ensuring the fulfillment of the timetable requirements and the operator requirements (refueling, maintenance, etc.). The major objective of his task is to minimize the number of vehicles required. All the components in Figure 1 are very sensitive to internal and external factors, sensitivity which could easily lead toward an inefficient solution. The complexity involved in the public transport operational planning process challenges researches to develop automated computerized procedures which led to number of software packages available in the market. It is worth mentioning that the evaluation module of such a software package should be based on an external input related to cost coefficients, and performance criteria. The cost coefficients include: vehicle cost (fixed and variable), crew cost (fixed and variable), service benefit, and other costs. The performance criteria include: measures of passenger service, measures for vehicle and crew schedules, and measures for duty rosters, and other criteria. This chapter includes two main parts. The first covers the determination of vehicle frequencies and the construction of alternative public timetables for both cases of

34

Advanced Modeling for Transit Operations and Service Planning

even headway and even average load. The second part provides an overview on exact solutions to the vehicle scheduling problem and describes a graphical heuristic procedure for the determination of minimum fleet size and its lower bound. The chapter ends with concluding remarks and list of references.

2.

CONSTRUCTION OF ALTERNATIVE PUBLIC TIMETABLES

2.1

Background

Mathematical programming methods for determining frequencies and timetables have been proposed by Furth and Wilson (1981), Koutsopoulos et al. (1985) and Ceder and Stern (1984). The objective in Furth and Wilson (1981) is to maximize the net social benefit, consisting of ridership benefit and wait time saving, subject to constraints on total subsidy, fleet size and passenger loading levels. Koutsopoulos et al. (1985) extended this formulation by incorporating crowding discomfort costs in the objective function and treating the time dependent character of transit demand and performance. Their initial problem comprises a non-linear optimization program relaxed by linear approximations. Ceder and Stern (1984) addressed the problem with an integer programming formulation and heuristic person-computer interactive procedure. The latter approach focuses on reconstructing timetables when the available vehicle fleet is restricted. Public transport timetable is commonly constructed for given sets of derived frequencies. The basic criteria for the determination of frequencies are: (a) to provide adequate vehicle's space to meet passenger demand, and (b) to assure a minimum frequency (maximum-policy headway) of service. Ceder (1984) described four different methods for calculating the frequencies. Two are based on point-check (counting the passengers on-board the public transport vehicle at certain point(s)), and two - on ride-check (counting the passengers along the entire public transport route). In the point-check methods the frequency is the division between passenger load at the maximum (max) load point (either the one across the day or in each hour) and the desired occupancy or load factor. In the ride-check methods the frequency is the division between the average or restricted-average passenger load and the desired occupancy. The average load is determined by the area under the load profile (in passenger-km) divided by the route length (km), and the restricted average is a higher value than the average one, in order to assure that in certain percentage of the route length the load does not exceed the desired occupancy. This desired occupancy (or load factor) is the desired level of passenger load on each vehicle, in each time period (e.g. number of seats). Commonly, across almost all the public transport agencies, the

Public Transport Timetabling and Vehicle Scheduling

35

frequency is determined by the maximum load procedure. This max load procedure is established to ensure adequate space to accommodate the maximum number of onboard passengers, along the entire route, for a given time period (e.g. one hour). That is,

where lj is the average (over days) maximum number of passenger (max load) observed on-board in period j, dj is the desired occupancy (load factor) in period j, and Fmj is the minimum required frequency (number of vehicles) in period j. In a follow-up study Ceder (1987) analyzed optional ways for generating public timetables. This analysis allows for establishing a spectrum of alternative timetables, based on three categories of options: (a) selection of type of headway, (b) selection of frequency determination method for each period, and (c) selection of special requests. In category (a) the headway (time interval between adjacent departures) can be equal or balanced. Equal headway refers to the case of evenly spaced headways and balanced headway - to the case of unevenly spaced headways while treating each vehicle separately at the hourly maximum load point. These cases are being extended in this chapter. In category (b) it is possible to select for each time period one of the four frequency determination methods (two point-check, and two ride-check) mentioned above, or a given frequency by the scheduler. In category (c) it is possible to request clock headways (departure times that repeat themselves in each hour, easyto-memorize) and/or certain number of departures (usually for cases with limited resources). The outcome of these analyses is a set of optional timetables in terms of vehicle's departure times at all specified timepoints, using passenger load data. Each timetable is accompanied by two comparison measure which are used as an evaluation indicator in conjunction with resource saving. The first measure is the total required vehicle runs (departures) and the second is an estimate for the minimum required fleet size at the route level only. Public transport timetable is perhaps the main reference for defining unreliable public transport service. The assumption that passengers will adjust themselves to given timetables (with headways of, say, longer that 10 minutes) instead of adjusting the timetables to the passenger demand is one of the largest sources of unreliable service. When passenger demand is not met, the vehicles are slowing down (increased dwell time), behind the schedule and entering the inevitable process of further slow down. This will eventually lead to the known bunching phenomenon with the vehicles behind. Opposite to that is the situation of overestimating the demand which may result in vehicles running ahead of time. Both situations are not observed when the

36

Advanced Modeling for Transit Operations and Service Planning

service is highly frequent and characterized by low variance of the headway distribution. 2.2

Three Procedures

In this section of the chapter three different procedures are proposed and analyzed for better matching the passenger demand with a given timetable while attempting to minimize the number of departures (leads to reduce the number of vehicles which is one of the main resources). This will result in a more reliable and comfortable service. Procedure 1 produces departure times with evenly spaced headways while considering a smooth transition between adjacent hours. This procedure is based on the given standards dj and Fmj for each hour j and on the j-th hourly max load, lj . Procedure 2 determines departure times such that, in average sense, vehicles will carry on even dj loads at the hourly max load point. This procedure 2 is based on d j , Fmj and on individual vehicle loads at the hourly max load point where lj is observed. Procedure 3 derives the departure times such that, in average sense, the on-board passenger load will not exceed d j , and will be equal to dj at each individual vehicle max load point (as opposed to the lj points in procedure 2). Example Problem and Initial Analysis The example problem is used as an explanatory device for three procedures. Table 1 contains the necessary information and data for a 2-hour example j = 1, 2 of a bus route from A to C with one stop at B. There are 5 departures observed. For each departure Table 1 contains the average observed on-board passenger at both boarding points A and B. The minimum required frequency is Fmj = 2, j - 1, 2. The desired occupancy is di = 50 and d2 = 55 passenger per bus, and the average travel time from A to B is 18 minutes. The estimated hourly demand is also included in Table 1. It is based on two basic assumptions: (a) the average load observed is a representative value of the actual demand and it is independent of the exact setting of departure times; (b) the passengers observed on-board are accumulated at a uniform rate. The first assumption can be realized when using the vast amount of data anticipated from equipment like APC (Automated Passenger Counters), or when the schedulers have reliable sources of information provided by road inspectors and supervisors. The second assumption usually holds when the observed headways are relatively small. For headways greater than 30 minutes, part of the passengers may time their arrival and, if data is available, this second assumption may not be needed.

37

Public Transport Timetabling and Vehicle Scheduling Table 1. Input and observed data of the example Route:

G>

Average Travel Time:

18 min

Average Observed Number of Passengers on-board the Bus Dept Time at A at Time A 7-8 a.m. 8-9 a.m.

7:15 7:45

30

8:10 8:30 8:50

25 94

80

Derived Values Headway Frequency (bus/hr) (min) H H I I A M A M M M L L

at B

Hourly Average Max. Hourly Demand Ind. Load at at Max. (HAML) A Point B

65 35

125

148

193

50

2.96

3.86

20

16

80 72 67

192

177

214

55

3.49

3.89

17

15

Desired Occ. (pass/ bus)

Minimum Frequency: 2 buses per hour Referring to the example in Table 1, the hourly demand for the first hour, between 7 8 a.m., j = 1, is based not only on the average loads observed, but also on part of the load observed on the first bus in the second hour, j = 2. That is, the average load on the first bus in j = 2 is divided proportionally in order to reflect the demand at the end to the period j = 1. Therefore, at point A and B the loads of 25 and 80 associated with the 8:10 departure, are divided into 3/5 and 2/5 where the 3/5 portion is related to the j = 1 demand. This proportion is stem from the 25 minutes difference between the last departure of the period] = 1 (7:45) and 8:10 where 15 minutes of this time difference belongs to j = 1, and 10 minutes to j = 2. The hourly demands at A and B are 125 and 148 passengers, respectively, for j = 1, and 192 and 171, respectively, for j = 2. This means that for j = 1 the hourly max load point is B with 1, = 148, and - point A for j = 2 with 12 = 192. hi addition, the third column under hourly demand, in Table 1, includes a newly element called individual max hourly demand. This demand reflects the sum of the max on-board loads observed on each bus, in each hour, while considering also the proportion of max demand associated with the first bus of the next hour. It results are 193 and 214 passenger demand for j = 1, 2, respectively, where, for the example of j=l, one

38

Advanced Modeling for Transit Operations and Service Planning

obtains 193 = 65+80+(3/5)x80. The interpretation of this element is clarified under the description of procedure 3. Finally in the last four columns in Table 1 there are the derived frequencies and headways based on equation (1). The headway Hj for hour j, is simply the inverse of the frequency, and in minutes:

Therefore, for j = 1, 2 the frequencies based on the hourly max load points are 2.96 and 3.49, respectively, and are 3.86 and 3.89 buses per hour for the individual max load hourly demand, respectively. Procedure 1 One characteristic of existing transit timetables is the repetition of the same headway in each time period. The scheduler, using Hj, is facing, however, a problem on how to set the departure times in the transition segments between adjacent time periods. In addition, the scheduler (or existing software) usually rounds-up the frequencies F, to the next integer, prior the use of equation (2). It is shown here that in order to save resources there is no need to round-up Fj and moreover the transition between hours (or any other time periods) can be carried out in a simple and accurate manner. The Underlining Principle of Procedure 1 The simple way, used by many bus agencies, to smooth the headways during the transition time is to consider an average headway between two adjacent hours. This average rule may result in either undesirable overcrowding or underutilization. For example, using equations (1) and (2) one obtains HI - 25 and H2 = 9 minutes with average of 17 minutes. Thus, a timetable can be set to 7:00, 7:25, 7:50, 8:07, 8:16, .... By assuming uniform arrival rate with di = 50 and d2 = 60, j - 1 contributes for the 8:07 departure (10/25) x 50 = 20 passengers, for the remaining 10 minutes between 7:50 and 8:00, and j - 2 contributes (7/7) x 60 = 60 passengers. The total is 20 + 60 = 80 average passengers, on the 8:07 departure, representing overcrowding. In order to overcome this undesirable situation the following principle is employed. Principle 1: establish a curve representing the accumulative frequency versus the time (adding the non-integer value of the frequency determined with respect to time). Moving horizontally, for each departure, until intersecting the accumulative curve, and then vertically, results with the required departure times.

Public Transport Timetabling and Vehicle Scheduling

39

Proposition 1: Principle 1 provides the required evenly spaced headways with a transition load approaching the average of du and d u +i , where du and d u +i are the desired occupancies for two consecutive time periods. Proof; Figure 2 illustrates Principle 1 using the information in Table 1. Since the slopes of the lines are 2.96 and 3.49 for j = 1 and j = 2, respectively, the resultant headways are those required. The transition load is the one determined for the 8:01 departure, and is comprised of 20 minutes arrivals for j = 1, and 1 minute arrival for j = 2. Therefore (20/20) x 50 +(1/17) x 55 = 53 approximately. This transition load is not the exact average between d] = 50 and d2 = 55 since departures are made in integer minutes. That is, the exact determined departure after 8:00 is (3-2.96) x 60/3.49 = 0.688 minutes, and inserting this value instead of 1 minutes in the above calculation yields a closer value to the exact average. Basically, the proportions considered satisfy the proof-by-construction of Proposition 1. Figure 2 exhibits the resultant six departures for procedure 1 where the determined frequencies are kept non-integer. Principle 1, therefore, allows for saving some unnecessary bus runs and also stabilizes the average load during the transition segment between time periods. Procedure 2 While arriving with procedure 1 to a satisfactory timetable, with even headways, it is still unclear if the loads on individual buses will not exceed d j , for all j. It is wellknown that passenger demand varies even within a single time period, reflecting the business, industrial, educational, cultural, social and recreational transit needs of the community. This dynamic behavior provides a basis for the scheduler to adjust the departure times. These adjustments are not done frequently unless there is a clear cut information (e.g. from the road supervisions) to support it. Nonetheless with the anticipated vast amount of passenger load data (e.g. from APCs) it is possible to construct procedures to better match the timetables (departure times) with the variable demand. This and the next section provide such procedures. The Undergoing Principle of Procedure 2 The results of procedure 1 starts with the 7:20 and 7:40 departures for j = 1. The frequency required, based on the hourly max load point, is 2.96 for j = 1 (as is shown in Table 1). This frequency aimed at 50 passengers per bus while considering the entire hourly max demand. However, the assumption of uniform passenger arrival rate, between the observed departures, results in 65/15 = 4.3 passengers/minute between 7:00 and 7:15 and 35/30 = 1.2 pass/min between 7:15 and 7:45 at the hourly max load point B. Therefore, the 7:20 departure (by procedure 1) may result in 65 +

40

Advanced Modeling for Transit Operations and Service Planning

1.2 x 5 = 71 passengers; significantly above the desired 50 passengers. In order to avoid this imbalanced situation the following principle is exploited.

7:00

Figure 2. Determination of the example departure times (at A) for evenly spaced headways with a smoothing process between time periods 71 passengers; significantly above the desired 50 passengers. In order to avoid this imbalanced situation the following principle is exploited. Principle 2: construct a curve representing the accumulative loads observed on individual buses at the hourly max load points. Moving horizontally per each dj for all j, until intersecting the accumulative curve, and then vertically, results with the required departure times.

Public Transport Timetabling and Vehicle Scheduling

41

Proposition 2: Principle 2 results in departure times such that the average max load on individual buses, at the hourly j max load point, approaches the desired occupancy Proof; Figure 3 illustrates Principle 2 for the example problem appearing in Table 1. The derived departure times are unevenly spaced to obtain even loads at points B for j = 1 , and point A for j = 2. These even loads are constructed on the accumulative curve to approach d\ and da for j = 1, j = 2, respectively. Assuming uniform passenger arrival rate between each two observed departures shows that the load (at B) of the 7:45 departure (at A), for example, is comprised of the arrival rate between 7:12 and 7:15 (65/15 = 4.3) and the rate between 7:15 and 7:45 (35/30 = 1.2). Thus, 4.3 x 3 + 1.2 x 30 = 49 which is approaching di = 50. Moreover, in the transition between j = 1 and j = 2, the value of da = 55 is considered since the resultant departure is after 8:00. The load of the bus departing A on 8:16 at its hourly max load point A, is comprised of (25/25) x 25 + (94/20) x 6= 53.2 which is approaching d2 = 55. The exact value of da can be obtained only for departures with non-integer minutes. This completes the proof-by-construction of Proposition 2. Figure 3 includes the results of procedure 2 with six departures. The last departure at 8:52 is determined using a slight extrapolation of the uniform passenger arrival rate between 8:30 and 8:50. Procedure 3 While procedure 2 ensures even average loads of dj at the j-th max load point, it does not guarantee that in other bus stops the average load will not exceed dj and, therefore, may result in overcrowding. The purpose of the procedure presented below is to derive the bus timetable provided that in an average sense all buses will have even loads (equal to the desired occupancy) at the max load stop of each bus. That is, for a given time period each bus may have a different max load point across the entire bus route with a different observed average load. The objective set forth is to change the departure times such that all observed average max loads will be same and equal to dj during all j. Certainly the adjustments in the timetable are not intended for highly frequent urban services where the headway is less than say, 10 minutes, or an hourly frequency of about 6 vehicles or more. Behind this procedure is the notion that passenger overcrowding situations (loads greater than dj) should be avoided. The Underling Principle of Procedure 3 The results of procedure 2 are exhibited in Figure 3. Considering in that Figure, for example, the resultant departure at 7:45 with 50 average passengers on-board at point B. From Table 1 it is clear that point B is the j=l max load point. However, one does not know what is the average load in the other stops, and in the example problem, it is referred to point A. Since the first departure is at 7:12, the accumulative load at A

42

Advanced Modeling for Transit Operations and Service Planning

between 7:12 and 7:45 is of interest. For that purpose the data in Table 1 is used while constructing an accumulative curve of the observed loads at A. The average load at A for the 7:11 departure results in 22 passengers (30/15 = 2 pass/min arrival rate). The average load at 7:45 is combined from the remaining passengers between the observed 30 at 7:15 and 22, and those observed on the 7:45 departure. That is (30 22) +80 = 88 passengers. No doubt that the 7:45 departure faces, in an average sense, overcrowding at A while complying with di = 50 at B. In order to overcome this undesirable possible overcrowding the following principle is employed.

Figure 3. Determination of the example departure times (at A) with even loads at the hourly maximum load point

Public Transport Timetabling and Vehicle Scheduling

43

Principle 3: construct an accumulating passenger load curve at each stop (except the arrival point). Moving horizontally per each dj , for all j, on each curve, until intersecting each of the accumulative curves, and then vertically to establish a departure time for each curve. The required departure time is the minimum one across all curves. Using the last determined departure time, set the loads across all the curves and add the considered or next dj. Repeat until the end of the time span. Proposition 3: Principle 3 results in departure times such that the average max load observed on individual buses approaches the desired occupancy dj. Proof: Figure 4 illustrates Principle 3 for the first three departure of the example problem in Table 1. Figure 4 shows the accumulative load curves of the three buses where the curve at B is shifted by 18 minutes to allow for an equal time basis (at the

122 r-72

Figure 4. Determination of the first three departure times (at A) considering even loads at the individual bus maximum load point

44

Advanced Modeling for Transit Operations and Service Planning

route's departure point) in the analysis. At the initialization the value of 50 is coordinated with the two accumulative curves to obtain: 7:11 at B and 7:22.5 at A. According to Principle 3 one selects the minimum time between the two to be the first departure at 7:11 (emphasized in Figure 4). It means that the first bus is shifted backward by 4 minutes to have at B, in an average sense, 50 instead of 65 passengers. Then one adds di = 50 to 50 at stop B curve, and to 22 at stop A curve. This results in 7:31 and 7:45 departures. Hence, 7:31 is the next departure, and the procedure continues and results in 7:56 as the last departure at the period [7:00 - 8:10]. Adding dj =50 to 122 (at A) or to 134 (at B) results in departures beyond 8:10. The bus of 7:11 has its d]=50 passengers at B and the bus of 7:31 -at A. This completes the proof-by-construction of Proposition 3. Figure 4 includes at its bottom the complete set of departure times of the example problem. If extrapolating the accumulative curve, another departure can be set at approximately 9:00. Comparison The comparison between the observed data of the example problem and the results of the three procedures is summarized in Table 2. In this table the associated individual average max load and its corresponding stop appear in brackets under each departure. It can be seen, as expected, that only procedure 3 complies with balanced loads at the critical individual max load points. Table 2. Departure times and loads of the observed data and for the three procedures 5th 6th Departure 1st 2nd 3rd 4th 7th Characteristic Observed 7:15 7:45 8:30 8:50 Observed 8:10 (65,B)* (80,A) (803) (94,A) (88,A) Procedure 1 7:20 7:40 8:35 8:52 Even Headways 8:01 8:18 (81,A) (75,A) (583) (723) (54,A) (573) Procedure 2 7:11 7:45 8:28 8:40 8:53 Even Load at 8:16 (88,A) (993) (79,A) (52,B) (61,A) _ Hourly Max (503) Load Point 8:24 Procedure 3 7:11 7:31 7:56 8:12 8:36 8:48 Even Load at (50,A) (50,B) (55,B) (55,A) (55,A) (5 5, A) Individual Max (503) Load Point "(i, j) in bracket means: i = average individual max load associated with the cell's departure time, j = the stop where i is observed or determined

Public Transport Timetabling and Vehicle Scheduling

3.

OPTIMAL DETERMINATION OF VEHICLE SCHEDULES

3.1

Background

45

Figure 1 presents the public transport operations planning framework as a multistep process. Due to the complexity of this process each step is normally conducted separately, and sequentially fed into the other. In order for this process to be costeffective and efficient, it should embody a compromise between passenger comfort and cost of service. For example, a good match between vehicle supply and passenger demand occurs when vehicle schedules are constructed so that the observed passenger demand is accommodated while the number of vehicles in use is minimized. Following the construction of an adequate public timetable in section 2 above, the next step is to determine vehicle schedules or chains of trips carried out by individual vehicles so as to reach the minimum number of vehicles required to cover the entire timetables. It is assumed that each vehicle has the same number of seats and same capacity (seats plus standees). This section provides an overview on exact solutions to the vehicle scheduling problem and describes a graphical heuristic procedure for the determination of minimum fleet size and its lower bound. 3.2

Exact solutions

The problem of scheduling vehicles in a multi-depot scenario is known as the MultiDepot Vehicle Scheduling Problem (MDVSP). This problem is complex (NP-hard) and considerable effort is devoted to solve it in an exact way. Review and description of some exact solutions can be found in Desrosiers et al. (1995), Daduna and Paixao (1995), Lobel (1999), and Mesquita and Paixao (1999). An example formulation of the MDVSP is as follows: [„ + !„ + ,

objective function: wm^VVc..^.. ^^ ^^ J J

(3)

where / is the event of-ending of a trip at time ai , j is the event of-start of a trip at time b • , and y;j -

ending is connecting to start! 0, otherwise

For i = n + 1 then yn+lj -I if a depot supplies a vehicle for the j'-th trip. For / = n + 1 then yi>n+l - 1 if after the i-th trip end, the vehicle returns to a depot, and yn+i,n+\ ~ No. of vehicles remain unused at a depot.

46

Advanced Modeling for Transit Operations and Service Planning

The cost function cfj takes the form c, =

(4)

where: K = the saving incurred by reducing the fleet size by one vehicle, Ly- direct dead-heading cost from event i toy , and Ey = cost of idle time of a driver between / andy. This formulation which appears in a similar form in Gavish et al. (1978) covers the chaining of vehicles in a sequential order from the depot to the transit routes alternating with idle time and dead-heading trips, and back to the depot. This is a zero-one integer programming problem that can be converted to a large scale assignment problem. In addition, the assignment of vehicles from the depots to the vehicle schedule generated in the above chaining process can be formulated as a "transportation problem" known in every operations research literature. Lobel (1999) is using a branch-and-cut method for MDVSP with the generation of upper bounds and the use of Lagrangean relaxations and pricing. Mesquita and Paxiao (1999) are comparing in this problem the linear relaxation based on multicommodity network flow approach. This review covers exact solutions to the vehicle scheduling component while realizing the complexity of the problem. This leads to look for some heuristics, hi the next section a heuristic solution is described. 3.3

Deficit Function (DF) approach

Following is a description of a step function approach described by Ceder and Stern (1981), for assigning the minimum number of vehicles to allocate for a given timetable. The step function is called Deficit Function (DF) as it represents the deficit number of vehicles required at a particular terminal in question in a multiterminal transit system. That is, DF is a step function that increases by one at the time of each trip departure and decreases by one at the time of each trip arrival. To construct a set of deficit functions, the only information needed is a timetable of required trips. The main advantage of the DF is its visual nature. Let d(k,t,S) denote the DF for the terminal k at the time t for the schedule S. The value of d(k,t,S) represents the total number of departures minus the total number of trip arrivals at terminal k, up to

Public Transport Timetabling and Vehicle Scheduling

47

and including time t . The maximal value of d(k,t,S) over the schedule horizon [T} , T2 ] is designated D(k, S) . Let t's and t'e denote the start and end times of trip i, i e S . It is possible to partition the schedule horizon of d(k,t,S) into sequence of alternating hollow and maximal intervals. The maximal intervals [sf ,ef ]/' = l,...,n(k) define the interval of time over which d(k,i} takes on its maximum value. Note that the S will be deleted when it is clear which underlying schedule is being considered. Index / represents the z'th maximal intervals from the left and n(k) represents the total number of maximal intervals in d(k,t}. A hollow interval Hf , l=Q,\,2,...,n(k) is defined as the interval between two maximal intervals. Hollows may consist of only one point, and if this case is not on the schedule horizon boundaries (7^ or 7^) , the graphical representation of d(k,t} is emphasized by clear dot. If the set of all terminals is denoted as T, the sum of D(k) for all k e T is equal to the minimum number of vehicles required to service the set T. This is known as the fleet size formula. Mathematically, for a given fixed schedule 5: max < / M

(5)

Where D(S) is the minimum number of buses to service the set T. When Deadheading (DH) trips are allowed, the fleet size may be reduced below the level described in Equation 5. Ceder and Stern (1981) described a procedure based on the construction of a Unit Reduction DH Chain (URDHC), which, when inserted into the schedule, allows a unit reduction in the fleet size. The procedure continues inserting URDHCs until no more can be included or a lower boundary on the minimum fleet is reached. The lower boundary G(S) is determined from the overall deficit function defined as g(t,S) = ^d(k,t,S) keT

where G(S) = max g(t , S) . This >el \' 2\

function represents the number of trips simultaneously in operation. Initially, the lower bound was determined to be the maximum number of trips in a given timetable that are in simultaneous operation over the schedule horizon. Stem and Ceder (1983) improved this lower bound, to G(S') > G (S) based on the construction of a temporary

48

Advanced Modeling for Transit Operations and Service Planning

timetable, S', in which each trips is extended to include potential linkages reflected by DH time consideration in S. This lower bound is further improved in this work. The algorithms of the deficit function theory are described in detail by Ceder and Stern (1981). However, it is worth mentioning the next terminal (NT) selection rule and the URDHC routines. The selection of the NT in attempting to reduce its maximal deficit function may rely on the basis of garage capacity violation, or on a terminal whose first hollow is the longest, or on a terminal whose overall maximal region (from the start of the first maximal interval to the end of the last one) is the shortest. The rationale here is to try to open up the greatest opportunity for the insertion of the DH trip. In the URDHC routines there are four rules: R=0 for inserting the DH trip manually in a conversational mode, R=l for inserting the candidate DH trip that has the minimum travel time, R=2 for inserting a candidate DH trip whose hollow starts farthest to the right, and R.-3 for inserting a candidate DH trip whose hollow ends farthest to the right. In the automatic mode (R-1,2,3), if a DH trip cannot be inserted and the completion of a URDHC is blocked, the algorithm backs up to a DH candidate list and selects the next DH candidate on that list. Figure 5 presents an example with 9 trips and four terminals (a, b, c, and d). In its upper part the 9 trips are shown with respect to time with departure and arrival terminals. Note that trip 4 starts and ends in the same terminal. Four DFs are constructed along with the overall DF. The maximal areas of the DFs are emphasized with a heavy line. Based on the NT procedure terminal a (whose maximal region is the shortest) is selected for possible reduction in D(a). Given that all DH times are 3 units of time, and using R=2, a DH trip is inserted from terminal c to a, DH]. This will increase d(c,i) at t=3 from -1 to 0, reduce d(a,f) at t=6 from 3 to 2, but will also increase d(c,t) at t=10 from 0 to 1. In order to eliminate the increase ofD(c) from 0 to 1 another DH trip is inserted, DH2 from a to c. The result is that D(a) is reduced from 3 to 2, and the DFs of a and c are updated with d(a,f)=2 between t=6 and t=7, and d(c,i)=Q between t=3 and t=10. One can see that no more DH trips (with trip time of 3 units) can be further inserted to reduce D(k), k=a,b,c,d. Hence D(S)=5. The sum of all the DFs, g(t), is illustrated at the bottom of Figure 5 and has G=3 (maximal number of vehicles in simultaneous operation). It will be used in a following section for the lower bound improvement. Finally, all of the trips, including the DH trips, are chained together for constructing the vehicle schedule (blocks). Two rules can be applied for creating the chains: first in-first out (FIFO), and a chain-extraction procedure described by Gertsbach and Gurevich (1977). The FIFO rule simply links the arrival time of a trip to the nearest

Public Transport Timetabling and Vehicle Scheduling

49

departure time of another trip (at the same location), and continues to create a schedule until no connection can be made. The trips considered are deleted and the process continues. The chain-extraction procedure allows an arrival-departure connection for any pair within a given hollow (on each deficit function). The pairs considered are deleted and the procedure continues. Figure 6 illustrates for clarity one hollow (between two peaks of the deficit function) with arrivals of trips 1 , 2 , 3 and departures of trips 4, 5, 6. Below the figure there is the FIFO chain (within this hollow) as well as other alternatives, where in all the minimum the fleet size is maintained. 1 2 3 4 I I I I

Figure 5. Nine-trip example with deadheading trip insertion for reducing the fleet size

50

Advanced Modeling for Transit Operations and Service Planning

An example of creating chains of trips within a hollow using FIFO rule and all other possibilities 3.4

Lower Bound

The initial lower bound on the fleet size with DH trip insertions was proved by Ceder and Stern (1981) to be G.(S). An improved lower bound of this problem was established and proved later by Stem and Ceder (1983), and Ceder (2002) using the following procedure: 1. extend each trip's arrival time to the time of the first feasible departure time of a trip with which it may be linked to T2 (the ending time of the finite time horizon). 2. given that the extended schedule is S', construct the overall DF, g'(t,S'), and determine its maximum value as G'(S'). While creating 5" it is possible that several trips' arrival points will be extended forward to the same departure point being their first feasible connection. Nonetheless

Public Transport Timetabling and Vehicle Scheduling

51

in the final solution of the minimum fleet size problem only one of these extensions will be linked to the single departure point. This observation opens an opportunity to look into further artificial extensions of certain trips' arrival points without violating the generalization of all possible combinations needed to prove that the resultant boundary on the fleet size is its lower bound. A stronger lower bound than G'(S') is found and proved in Ceder (2002). However it is beyond the scope of this chapter and hence not presented here. The stronger the lower bound is, the closer it is to the minimum fleet size required. Also, the stronger the lower bound is the better it serves the public transport decision makers on how far the fleet size can be reduced via DH trip insertions. Figure 7 presents the schedule of Figure 5 with 5" in its upper part and two overall deficit functions: g(t) and g '(t). All trips in S' are extended either to their first feasible connection (with all DH times are 3 units of time) or to the time horizon, t=18. The improved lower bound is therefore G'(Sr) = 4.

3

•o 0>

I

Figure 7. The example of Figure 5 with artificial extensions of each trip to its first feasible connection which results with the improved lower bound, G'=4

52 3.5

Advanced Modeling for Transit Operations and Service Planning Shifting Departure Times with Given Tolerances

Another factor considered in a manually produced public transport schedule is related to the shifting of trip departure times. A general description of a technique to reduce the fleet size for a variable departure time scheduling problem can be found in Gerstbach and Stern (1978). This technique for job schedule utilizes the deficit function representation as a guide for local minimization in maximal intervals, M"VueT . However when considering variable departure times along with a possible insertion of DH trips, the problem becomes more complex. The scheduler who performed shifting in trip departure times is not always aware of the consequences which could arise from these shifts. Ceder (2002) describes a formal algorithm to handle the complexities of shifting departure times. The algorithm is intended for both automatic and man-computer conversational modes. Figure 8 illustrates an example of two terminals and seven trips using the DH representation in part (i). Part (ii) shows how to reduce the fleet size using shifting tolerances of % time unit (forward or backward) where the shifts are shown with small arrows and the update DF is marked by a dashed line. Part (iii) shows how to apply only the URDHC procedure with DH times of 2 time units, and part(iv) presents a modified URDHC (mixed with the shifting) procedure. As can be seen in Figure 8(i) the fixed schedule without DH considerations requires 5 vehicles. Using shifting allows for reducing the number of vehicles to 3. The use of URDHC allows for reducing it to 4, and the use of a combined approach requires 3 vehicles. The viewpoint of the public transport operator will lead to use first the shifting procedure while wishing to minimize the operational cost (reducing DH mileage). However there is also the issue of passenger comfort while trying to accommodate the observed demand. Changes in departure times may result in imbalance passenger loads and reduction in the service reliability. Past experience in applying the DF approach at several bus properties shows that best is to first identify small shifts in departure times, enabling the reduction of the fleet size, without noticeable changes in the timetable. Second is to apply the combined approach of URDHC and shifting departure times.

Public Transport Timetabling and Vehicle Scheduling

Figure 8. An example with seven trips and two terminals using three procedures

53

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Advanced Modeling for Transit Operations and Service Planning

4.

CONCLUDING REMARKS

4.1

Timetabling

Different public transport agencies use different scheduling strategies based primarily on their own schedulers' experience, and secondarily on their scheduling software (if any). As the result, it is unlikely that two independent bus agencies will use exactly the same scheduling procedures, at the detailed level. In addition, even at the same public transport agency, the schedulers may use different scheduling procedures for different groups of routes. Consequently, there is a need when developing computerized procedures to supply the schedulers with alternative schedule options along with interpretation and explanation of each alternative. Three such alternatives are presented in this chapter. Also, undoubtedly, it is desirable that one of the alternatives will coincide with the scheduler manual procedure. In this way, the scheduler will be in a position not only to expedite manual tasks but also to compare the different procedures regarding the trade-off between passenger's comfort and operating cost. The first part of this chapter presents the creation of public transport timetables with even headways and even average passenger loads on individual vehicles. Average even loads on individual vehicles can be approached by relaxing the evenly spaced headways pattern (rearrangement of departure times). It is known that passenger demand varies even within one hour, reflecting the business, industrial, educational, cultural, social and recreational public transport needs of the community. This dynamic behavior can be detected through passenger load counts, and information provided by road supervisors. The adjustments of departure times, made in this chapter by three procedures, form the basis to improve the correspondence of vehicle departure times with the fluctuated passenger demand. These adjustments, resulting in a balanced load timetables, are based on a given vehicle desired occupancy at the maximum load point of each vehicle. The keyword here is to be able to control the loading instead of being exposed repeatedly to an unreliable service resulted from imbalance loading situations. With the growing problems of cultural, social and recreational public transport needs of the community. This dynamic behavior can be detected through passenger load counts. With advance in the technology of passenger information system the importance of even and clock headways is reduced. This allows for introducing optional timetables with the consideration of even average loads on individual vehicles. The construction of such timetables takes into account, in essence, the

Public Transport Timetabling and Vehicle Scheduling

55

passenger perspective. The controlled procedures for adjusting the timetable, will eventually reduce one of the major sources of unreliable service, resulting also in the reduction of wait and travel times. 4.2

Vehicle Scheduling

This chapter, in its second part, describes a highly informative graphical technique for the problem of finding the least number of vehicles required to service a given timetable of trips. The technique used is a step function, called a deficit function, which was introduced in the last 20 years as an optimization tool for minimizing the number of vehicles in a fixed trip schedule. The step function is called Deficit Function (DF) as it represents the deficit number of vehicles required at a particular terminal in question in a multiterminal public transport system. That is, DF is a step function that increases by one at the time of each trip departure and decreases by one at the time of each trip arrival. The second part of this chapter presents both fixed and variable trip schedules, where in the latter possible shifts in departure times are allowed. Also this part describes an improved lower bound to the fixed schedule fleet size problem. The use both the fixed and variable schedules allows for the combination of deadheading trip insertions and shifts in departure times in the fleet size minimization problem. It is achieved by the exploitation of the fixed-and variable scheduling characteristics. That is, a possible deadheading trip can be inserted and reduce the fleet size only if a certain shift in departure time is made, and a time shift can reduce the fleet size only if a deadheading trip is inserted. It is worth mentioning that there are several example applications of the deficit function use in designing better public transport services. These example applications are: network route design, short-turn design of individual and group of routes, the design of operational public transport parking spaces, vehicle scheduling with different vehicle types, and the task of crew scheduling. Overall the DF approach provides immediate feedback on the value of shifting departure times, within given tolerances, as well as combining these shifts with the insertion of deadheading trips for reducing the fleet size. The value of embarking on such a technique is to achieve the greatest vehicle saving while complying with passenger demand. This saving is attained through a procedure incorporating a man/computer interface which would allow the inclusion of practical considerations that experienced public transport schedulers may wish to introduce in the schedule.

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REFERENCES Ceder, A. and H. I. Stern (1981). Deficit function bus scheduling with deadheading trip insertion for fleet size reduction. Transportation Science, 15(4), 338-363. Ceder, A. and H. I. Stern (1984). Optimal transit timetables for a fixed vehicle fleet. Proceedings of the ICf International Symposium on Transportation and Traffic Theory. UNU Science Press, Holland, 331-355. Ceder, A. (1984). Bus frequency determination using passenger count data. Transportation Research, ISA (5/6), 439-453. Ceder, A. (1987). Methods for creating bus timetables. Transportation Research, 21A (1), 59-83. Ceder, A. (2002). A step function for improving transit operations planning using fixed and variable scheduling. Transportation & Traffic Theory, (15th ISTTT), (M. A. P.Taylor, ed.) Elsevier Science & Pergamon Pub, 1-21. Daduna, J. R. and A. Wren (Eds.) (1988). Computer-Aided Transit Scheduling. Lecture Notes in Economics and Mathematical Systems, 308, Springer-Verlag, Berlin. Daduna, J. R., I. Branco and J. M. P. Paixao (Eds.) (1995). Computer-Aided Transit Scheduling. Lecture Notes in Economics and Mathematical Systems, 410, Springer-Verlag, Berlin. Daduna, J. R and J. M. P. Paixao (1995). Vehicle scheduling for public mass transitand overview. (J. R. Daduna, I. Branco and J. M. P. Paixao, Eds.). ComputerAided Transit Scheduling, Springer-Verlag, Berlin. Desrosiers, J., Y. Dumas, M. M. Solomon and F. Soumis (1995). Time constrained routing and scheduling. (M. O. Ball, T. L. Magnanati, C. L. Monma, G. L. Nemhauser, Eds.). Network Routing Volume 8 of Handbooks in Operations Research and Management Science. Elsevier Science B.V., 35-39. Desrochers, M. and J. M. Rousseau (Eds.) (1992). Computer-Aided Transit Scheduling. Lecture Notes in Economics and Mathematical Systems, 386, Springer-Verlag, Berlin. Furth, P. G. and N. H. M. Wilson (1981). Setting frequencies on bus routes: Theory and practice. Transportation Research Board, 818, 1-7. Gavish, B., P. Schweitzer and E. Shlifer (1978). Assigning buses to schedules in a metropolitan area. Computers and Operations Research, 5, 129-138. Gertsbach, I. and H. I. Stem (1978). Minimal resources for fixed and variable job schedules. Operations Research, 26, 68-85. Gertsbach, J. and Y. Gurevich (1977). Construction an optimal fleet for a transportation schedule. Transportation Science, 11, 20-36.

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Koutsopoulos, H. N., A. Odoni and N. H. M. Wilson (1985). Determination of headways as function of time varying characteristics on a transit network. (J. M.Rousseau, Ed.). Computer Scheduling of Public Transport 2. North-Holland, Amsterdam, 391-414. Lobel, A. (1999). Solving large scale multiple-depot vehicle scheduling problems. (N. H. M. Wilson, ed.). Computer-Aided Scheduling of Public Transport. Lecture Notes in Economics and Mathematical Systems, 471, Springer-Verlag, Berlin, 192-220. Mesquita, M. and J. M. P. Paixao (1999). Exact algorithms for the multi-depot vehicle scheduling problem based on multicommodity network flow type formulations. (N. H. M. Wilson, ed.). Computer-Aided Scheduling of Public Transport. Springer-Verlag, Berlin, 221-243. Rousseau, J. M. (Ed.) (1985). Computer Scheduling of Public Transport 2. NorthHolland, Amsterdam. Stern, H. I. and A. Ceder (1983). An improved lower bound to the minimum fleet size problem. Transportation Science, 17(4), 471-477. Voss, S. and J. R. Daduna (Eds.) (2001). Computer Scheduling of Public Transport. Lectures Notes in Economics and Mathematical Systems, 505, Springer-Verlag, Berlin. Wilson, N. H. M. (Ed.) (1999). Computer-Aided Scheduling of Public Transport. Lecture Notes in Economics and Mathematical Systems, 471, Springer-Verlag, Berlin. Wren, A. (Ed.) (1981). Computer Scheduling of Public Transport: Urban Passenger Vehicle and Crew Scheduling. North Holland Publishing Co.

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CHAPTER 3

DESIGNING PUBLIC TRANSPORT NETWORK AND ROUTES Avishai Ceder, Civil Engineering Department, Transportation Research Institute, Technion-Israel Institute of Technology, Haifa, Israel 32000.

1.

INTRODUCTION

In the previous chapter it is shown that the public transport operational planning process includes four basic components performed usually in sequence: (1) network route design, (2) setting timetables, (3) scheduling vehicles to trips, and (4) assignment of drivers (crew). The framework of this process appears in Figure 1 with an emphasize on the first component which is the interest of this chapter. The problem addressed here is a general one of how to design a new public transport network or redesign an existing network, given no a-priori specifications as to the desired network structure. From a practical perspective it is desirable that the route design procedures include interaction and feedback loops between the selection of effective routes and the operational scheduling components: setting frequencies and timetables and schedule vehicles to the established trips with a special attention to the fleet size required. This is the reason for firstly describing the scheduling components

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Advanced Modeling for Transit Operations and Service Planning

Figure 1. Framework of public transport operational planning process with emphasis on this chapter's component

Designing Public Transport Network and Routes

61

in the previous chapter, and secondly - the network route design component in this chapter. This chapter focuses on two public transport route design parts: (a) design of routes at the network level, and (b) creating efficient operations of each route using the shortturn strategy. In part (a) the presented method generates all feasible routes and transfers which connect every place (node) in the network to all others. From this vast pool of possible routes and transfers it then generates smaller subsets which maintain network connectivity. For each subset thus generated the analysis meets transportation demands by calculating the appropriate frequency for each route. Next, it calculates pre-specified optimization parameters for each subset. Based on the specific optimization parameter desired by the user, it is then possible to select the most suitable subset. The method has been designed as a tool for the planning of future public transport networks as well as the maintenance of existing ones. The presented method ensures flexibility by allowing the user to either input his own data or to run the analysis automatically. Part (b) of this chapter presents a set of procedures to design efficiently individual public transport routes with trips that are initiated beyond the route departure point and/or terminated before the route arrival point. These trips are called short-turn trips, hi practice, transit frequency is determined at the heaviest load route segment, whereas at other segments, the operation may be inefficient due to situations characterized by empty seats. The public transport schedulers attempt to overcome this problem by constructing manually short-turn trips with the objective to reduce the number of vehicles required to carry on the timetable. The purpose of this second part is to show methods on how to improve and automate this task. These methods identify feasible short-turn points; derive the minimum fleet size required to carry on a given schedule; adjust the number of departures in each short-turn point to that required by the load data, minimize the number of short-turn trips while ensuring that the minimum fleet size is preserved; and create vehicle schedules.

2.

PUBLIC TRANSPORT NETWORK DESIGN

2.1

Background

Generally speaking only a few researchers have studied the interrelationship between the scheduling components and the network design element. The interrelationship

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exists in two directions: (i) each set of routes yields, based on the demand, a different set of frequencies and timetables, and ultimately, the required fleet size, and (ii) the operational cost derived from the scheduling components and the passenger level of service affect the search for the optimal route design while relying on a compromise between the operator and the user. Practical network design focuses almost entirely on individual routes which have been identified as candidates for change. However, it may be possible that the overall public transport network could be improved through restructuring of the entire network. For many public transport properties which have not been reappraised in this respect since the 1950s, it is high time to consider precisely such an undertaking. Such considerations motivate to seek an efficient network route design method, based on certain objective functions and a set of constraints. The main purpose of the method presented in this part of the chapter is to transport a given origin-destination demand through the public transport network in the most cost-effective way. The special characteristics of route design problems are: (a) passenger demand is spread throughout the entire network where it is generated and terminated at many points along the network's links, and can be grouped in terms of an origin-destination matrix; (b) the demand is to be transported simultaneously; (c) over a given planning horizon, it is impossible to reconstruct the routes, i.e., once the route network is designed, it will remain as it is over an entire planning period. Prior approaches to the public transport network design problem can be grouped into those which simulate passenger flows, those which deal with ideal networks and those based on mathematical programming. Simulation models are presented in Dial and Bunyan (1968), Heathington et al. (1968) and Vandebona and Richardson (1985). These models require a considerable amount of data, and their proximity to optimality is uncertain. Ideal network methods are based on a broad range of design parameters and a choice of objectives reflecting user and operator interest. Such methods appear in Kocur and Henrickson (1982), Tsao and Schonfeld (1984) and Kuah and Perl (1988). These methods are adequate for screening or policy analyses in which approximate design parameters are to be determined rather than a complete design. Thus these methods cannot represent real situations. Mathematical programming models are divided to generalized network design models and PT specific networks models. Known generalized network models are well summarized and review in Kim andBarnhart (1999) and as an example one can look at the heuristics developed by Farovlden and Powell (1994). The public transport network design models are inevitably heuristic due to the extremely high computational effort required. These

Designing Public Transport Network and Routes

63

partial optimization approaches appear in Hasselstrom (1981), Dubois et al. (1979), Lampkin and Saalmans (1967), Silman et al. (1974), Rea (1971), Mandl (1979), Marwah et al. (1984), Sharp (1974) and Keudel (1988). Apart from Hasselstrom's model, which is included in the Volvo public transport planning package, all other models have not been actually applied. However, the Hasselstrom model is quite complex, non-user oriented and expensive, both in terms of the data required and the direct cost and staff time needed for the process. The disadvantages of the existing mathematical programming models can be summarized in five points: (i) cannot handle large size public transport networks; (ii) do not consider optimal objectives functions and constraints; (iii) vehicle frequency determination is based on economic parameters rather than on passenger counts (as in done in practice, and described by Ceder (1987) and in the previuos chapter); (iv) cannot incorporate simultaneously three out of the four planning components: network design, setting timetables and vehicle scheduling. In particular, the models cannot evaluate the network without defining the vehicle requirements for each route and thus lacks precision in evaluating the cost effectiveness of design; (v) cannot incorporate non-quantitative constraints such as imposing certain links to be included in public transport routes, and considering operational strategies. Another stream of articles related to the public transport network design problem is the area of public transport assignment problem. A good review of the literature of this assignment problem appears in Spiess and Florian (1989) and Wu et al. (1994). The public transport assignment problem solutions consider only the user perspective while assuming that passenger behavior reflects the minimization of walk, wait and in-vehicle times, or a weighted sum of these times. The problem input is a given public transport network of routes, and the question is how to distribute a given public transport demand on this given network of routes. Spiess and Florian (1989) developed a two-part algorithm to assign the passenger demand. Their algorithm's first part computes the total travel times between origin and destination while considering possible transfers. In the second part, the passenger demand is assigned based on a defined optimal strategy. Wu et al. (1994) propose a model for the public transport equilibrium assignment problem. Their approach considers the effect of traffic congestion on the passenger route choice decision with the modeling of flow dependent waiting times. These assignment procedures could be especially useful to

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describe passenger behavior for a given network of routes while evaluating different traffic control and road management strategies. However the examination of an extensive amount of networks of routes, by these procedures, is time consuming. 2.2

Methodology

The overall methodology or system comprises six elements shown in Figure 2. Its entire formulation and interpretation appear in Ceder and Israeli (1992), Israeli and Ceder (1995, 1996) and Ceder (2001). In the first element, the system generates every feasible route and transfer (throughout the entire network), from all terminals including shortest path computation. Initially the network contains average travel times covering a time window, which is usually the peak period. These measured average travel times are then input into the calculation of shortest path between each origin-destination (O-D). Each determined candidate route meets the route length factor constraint. In other words, one procedure in this element screens out routes according to given boundaries on the route length, hi addition, there is a limit on the public transport route average travel time between each origin-destination (O-D) pair. That is, a given public transport demand, usually during peak hours, cannot be assigned to a candidate route if its average travel time exceeds the shortest path travel time by more than a given percentage. The feasible transfers are based on the following method: establishing additional direct routes between O-D pairs characterized by high O-D demands (predetermined O-D). These direct routes are actually initiated and/or terminated at non-terminal nodes and consequently, deadheading trips are responsible for their connection to the terminals. Also, a low OD demand, without a direct route, is not considered for obtaining service. The transfers are created using a mapping algorithm (branching routing possibilities along with constraints check). This applies to the disconnected O-D pairs, as well as to all the O-D pairs. hi the second element the system creates minimal set(s) of routes and their related transfers, such that connectivity between nodes is maintained and their total deviation from the shortest path is minimized. This problem is defined as a Set Covering Problem (SCP), which is hard to solve (Minieka, 1978). The SCP can determine the minimal set of routes from the matrix of the feasible routes, hi this matrix each row represents either a feasible route or a feasible transfer, hi the third element the entire O-D demand is assigned to the chosen set of routes. The assignment algorithm which

Designing Public Transport Network and Routes Input: Network nodes and arcs ; Average travel time on each arc ; public transport Origin -Destination demand ; set of terminals

Maximal deviation from shortest path ; Maximum route length

Additional routes and transfers given by user

^w

Element 1: Creation of public transport routes and determination of transfers

Element 2: Creation of minimal set of routes and transfers (Set Covering Problem)

Policy headway (inverse of min. frequency); Vehicle desired occupancy ; Set of routes and transfers given by user (e.g. existing); weights of passenger wait, transfer and in-vehicle times Monetary weights for the optimization criteria elements

Constraints on the effectiveness of each set of solutions

Element 3: Demand assignment procedure for frequency calculation

Element 4: Calculation of the optimization criteria, Z j and Z 2 (in terms of passenger hours, passenger hours difference from shortest path, waiting time, empty passenger hours, fleet size)

Element 5: Generation of more sub-optimal solutions (Z i, Z2)

Element 6: Multi-objective user decision Figure 2. A methodology for designing public transport routes

65

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has been developed (Israeli and Ceder, 1996) includes steps which are related to a route-choice decision investigation, i.e. the algorithm includes a probabilistic function for passengers who are able to select the public transport vehicle that arrives first, or alternatively, wait for a faster vehicle. The passengers' strategy is to minimize the total weights of wait, transfer and in-vehicle times. The methodology used is similar to that developed by Marguier and Ceder (1984) but with a different probabilistic function. The approaches used in Spiess and Florian (1989) and Wu et al. (1994) can be inserted in this element as well though they require extensive computation time and do not intend to be executed on a vast amount of alternative public transport networks. The fourth element represents the optimization criteria from the passengers, operator, and community perspectives. It is detailed below in the next section where Z, is the criteria in passenger-hours units and Z2 comes in terms of the fleet size required. The fifth element is responsible for constructing alternative sets of routes in order to search for additional (Z\, 7.2) values in the vicinity of their optimal setting. The procedure for this search is based on incremental changes in the set of routes, much like the reduced gradient methods. Given the set of routes associated with the minimum Z\ value, the single route which is the worst contributor to Z\ is deleted and then the SCP is solved in the second component, followed by the execution of the third and fourth elements. This process could continue, but there is no guarantee that a previous alternative will not be repeated. In order to overcome this problem, a new matrix is constructed with the idea of finding the minimal and worst set of candidate routes for possible deletion in each iteration, i.e., a new SCP matrix is constructed in which the candidate routes are the columns and each row represents a previous set of routes which was already identified in the vicinity of the optimal (Z\, Z2) setting. The solution to this new SCP matrix is a set of rejected routes so as not to repeat a previous alternative solution. During this process, a number of unique collections of routes are termed "prohibited columns" as they are the only ones which can transport a certain demand. These prohibited columns are assigned an artificially high cost value, so as not to be included in the solution. This process also involves some bounds in order to converge on a desired number of iterations, or number of (Z\, Z2) solutions. The sixth and final element of the system involves multi-objective programming of the two objective functions Zj and Z2- Given the alternative sets of routes derived in the fifth element, the purpose is to investigate the various alternatives regarding the most efficient (Z\, Z2) solution. The method selected in this element is called the

Designing Public Transport Network and Routes

67

compromise set method (Zeleny, 1973, 1974). It fits linear objective functions (Duckstein and Opricovic, 1980) for discrete variables. This multi-objective method can be also found in a recent book by Coello et al. (2002). The outcome of this method is the theoretical point in which (Z\, 7.2) attains its relatively minimal value. The results can be presented in a table or a two-dimensional graph which shows the trade-off between Z\ and Z2- These results also indicate the optimal zone or the socalled pareto front. The decision-maker can then decide whether or not to accept the proposed solution. In the latter case, for example, the decision-maker can see how much Zj is increased by decreasing Z2 to a certain value and vice versa. 2.3

Formulation

The two objective functions Z\ and Z2 mentioned above can be formulated as follows:

where, APH(i,j) = Passenger Hours difference, for the demand between nodes i and j, i, j e N, between the public transport route and the shortest (automobile) path (defined as passengers' riding time in a public transport vehicle on an hourly basis minus the total passengers' time if riding the shortest route. It measures how much excessive time is spent by passengers on public transport vehicles between the two nodes); WH(ij) = Waiting Time between nodes i and j, i, j e N, (defined as the amount of wait time at public transport stops and transfer points, on an hourly basis. It measures how much wait time is spent by passengers between the two nodes); EHr = Empty Space-Hours on route r (defined as the unused seats in a public transport vehicle on an hourly basis. Empty Space-Hours measures to what capacity public transport vehicle are used); FS — Fleet Size (number of public transport vehicles required to provide all trips along the chosen set of routes); <*i=i, 2,3 = Possible Monetary weights (to minimize the cost involved). The objective functions set forth takes into account three perspectives: the passengers, the operator and the community. A good public transport route is defined as an

68

Advanced Modeling for Transit Operations and Service Planning

attractive one from all the three perspectives. That is, APH(i,j) represents the passengers and the community views, WH(ij) represents the passengers view, and EHr and FS represent the operator view where FS can also represts the community view (e.g. the objective to minimize the pollution coming out from the public transport vehicles, like in Hong Kong). The complete formulation and explanation, including the constraints of the network design problem can be found in Ceder (2001). The nature of the overall formulation is non-linear (non-linear and mixed integer programming). Its analog problem is the generalized network design problem described by Magnanti and Wong (1984) with an NP - hard computational complexity. Thus, conventional approaches are incapable of providing a solution even with a relatively high degree of simplification. 2.4

Example

A simple 8-node example is used as an expeditory device to demonstrate the procedures used. The basic network with two terminals (from which trips can be initiated) -is shown in Figure 3 with the input demand presented in Table 1. It is related to a bus service to be established on this network.

Route diversion (from shortest path) factor (a) = Maximum degree of transfers = Bus capacity =

0.40 2 50

Figure 3. Example of an 8-node network with its basic input

Designing Public Transport Network and Routes Table 1.

1 2 3 4 5 6 7 8

69

The demand between nodes for the example problem (assumed to be symmetrical)

1 0 80 70 160 50 200 120 60

2 80 0 120 90 100 70 250 70

3 70 120 0 180 150 120 30 250

4 160 90 180 0 80 210 170 230

5 50 100 150 80 0 250 40 130

6 200 70 120 210 250 0 130 120

7 120 250 30 170 40 130 0 120

8 60 70 250 230 130 120 70 0

The outcome of the first element, in Figure 2, is presented in Table 2, while using the maximum deviation from the shortest path as a = 0.4 (no route length or its portion can exceed its associated shortest travel time by more than 40%). The first element, in Figure 2, is based on an algorithm that mainly produces feasible transfers throughout the entire network. The first step of the algorithm is to establish additional direct routes between O-D pairs characterized by high O-D demands (a predetermined O-D demand value). These direct routes are actually initiated and/or terminated at non-terminal nodes and consequently, deadheading trips are responsible for their connection to the terminals. Also, a low O-D demand without a direct route is not considered for obtaining service. The transfers are created using a mapping algorithm and are shown in Table 3 for the example problem, where the numbers in parentheses are the route numbers (see Table 2) which comprise the transfers. In the transfer path description, the numbers outside the parentheses represent nodes while those inside the parentheses represent all routes. hi the second element the system creates minimal set(s) of routes and their related transfers, and is defined as a Set Covering Problem (SCP). The SCP can determine the minimal set of routes from the matrix of the feasible routes. In this matrix each row represents either a feasible route or a transfer. The "1" in the matrix is inserted whenever an O-D demand can be transported by the route or transfer, and "0" otherwise. The word to "covering" refers here to at least one column with "1" in each row. The transfers are combined columns in the SCP matrix and therefore increase the complexity of the problem. The algorithm used here follows Israeli and Ceder (1996). It has been developed and tested with a random network. Its solution was compared with: (i) integer programming optimization, without considering transfers;

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(ii) with non-linear programming using relaxation methods on the integer variables for bus networks with transfers; and (iii) with a complete enumeration of all possible covering scenarios. The heuristic search algorithm used includes three covering variations: (a) covering only the shortest paths between all O-D pairs, (b) covering all the O-D paths while minimizing the number of paths that are not the shortest, and (c) covering all the O-D paths while minimizing the combined paths' travel time. Nine sets of routes are generated in the example problem, based also on the third, fourth and the fifth element of the introduced methodology. These sets are presented in Table 4 and refer to the routes and transfers indicated in Tables 2 and 3.

Table 2. All routes of the example problem generated by the first element

Route # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Description 1 -» 2 1 -» 2 -» 1 -» 2 •* 1 -» 2 -» 1 -> 2 -* 1 -> 2 -» 1 -» 2 -> 1 •» 3 1 -> 3 -» 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

4 5 5 5 5 5

-> 6 -» 7 ^7 -> 7

-> 8 ^8

-> 3 ^3

^6

6

2 2 -» 1 2 -» 1 2 -» 3 2 ^ 3

-> 6

5 ^ 2 ^ -> 2 -> 5

-> 6

-> -» -» -» ->

-> 5 -> 5 ^ 3 -> 5 -» 6

^ -> -> -> ^ -> -> •> ->

5 5 5 7 7 7 7 7 7

7

^ ^ 7 ^ 7 -> -> ^ -> ->

5 5 5 8 8

-> 8 ^8

^ 3 -> 6 -> 6

^6

^6

71

Designing Public Transport Network and Routes Table 3. All routes of the example problem generated by the first component

Transfer #

Description

29(5,18,27) 30(5,18,28) 31(6, 18) 32(6, 25) 33(7, 18) 34(7, 25) 35(18,20,27) 36(18,20,27) 37(18,21) 38(18,22) 39(18,24,27) 40(18,24,28) 41(18,26,27) 42(18,26,28) 43(21,25) 44(22, 25) 45(25, 27) 46(25, 28)

3(18) 3(18) 3(18) 3(25) 3(18) 3(25) 3(18) 3(18) 3(18) 3(18) 3(18) 3(18) 3(18) 3(18) 3(25) 3(25) 3(25) 3(25)

-> -> -> -» -* -> -» -* -* -> -» -> -> -» -» -» -* -*

5(5) 5(5) 5(6) 5(25,6) 5(7) 5(25,7) 5(20) 5(20) 5(21) 5(22) 5(24) 5(24) 5(26) 5(26) 5(25,21) 5(25,22) 5(25) 5(25)

» 7(27) » 7(28) * » * -» »

7(6) 7(6) 7(7) 7(7) 7(27)

•> » ^ » » » > -» -» » »

7(28) 7(21) 7(22) 7(27) 7(28) 7(27) 7(28) 7(21) 7(22) 7(27) 7(28)

8 ^ 8 ^ 8 ^ -> 8 -> 8 -> 8 -> 8 8 ^ 8 ^ -> 8 -> 8 8 ^ -> 8 -> 8 -> 8 8 ^ -> 8 ^ 8

Table 4. All subsets generated in the example problem Set

Description {4.6,9, 11.25,28,32,46} {7,9, 11,19,25,27,34,45} {7,9, 11,25,28,34,46} {6,9. 11,16,25,28,32,46} {6, 12,19,25,28,32,46} {7,12,25,27,34,45} {7, 12, 25, 28, 34, 46} {4, 6, 12, 25, 28, 32, 46} {6, 12, 16,25,28,32,46}

In the third element the entire O-D demand is assigned to the chosen set of routes. The assignment algorithm which has been developed in Israeli and Ceder (1996) includes steps which are related to a route-choice decision investigation. The function used considers the route length and the length of a transfer path between each O-D

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pair, where both the routes and transfer paths are divided into "slow" and "fast" categories. The bus frequency is used in the function as a variable, and is derived from the passenger load profile of each route. The load on each route is determined by the demand and assignment method. The heart of the whole element is a set of equations (3rd order degree), which are solved by an iterative method developed especially for this problem. The outcome of the element is: bus frequencies of the set of routes, passenger load profiles and demand assignment across the set of routes. The fourth element calculates the optimization parameters APH, WH, EH for computing Zj,and determines the minimum fleet size required to meet the demand, Z2. The method used for evaluating the fleet size is based on the Deficit Function theory proposed by Ceder and Stern (1981) and extended in Ceder (2002). This theory is explained and interpreted in the previous chapter. The fifth element is responsible for constructing alternative sets of routes in order to search for additional (Zj, Z2) values in the vicinity of their optimal setting. In the example problem ai= 0.2 = 0.3 = 1 in the objective function, Equation (1), and based on Stern and Ceder (1983) the lower bound on the fleet size is used to estimate FS in Equation (2). Nine "good" sets were produced by this element, as shown in Table 4, along with their (Z\. Z2) values which appear in Table 5. Table 5. Alternative sets of routes in the example problem SETS# 1 2 3 4 5 6 7 8 9

Zi 788 900 1105 866 937 997 1213 869 961

Z2=FS 106 109 117 102 103 101 113 103 105

The sixth and final element of the system involves multi-objective programming of the two objective functions Z\ and Z2- Given the alternative sets of routes derived in the fifth element, the purpose is to investigate the various alternatives regarding the most efficient (Z\, Z2) solution. The trade-off situation regarding the example problem is depicted in Figure 4. The lower left corner of the envelope contour of the nine

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solutions (Pareto front) represents the best sets. The user is then able to choose his desired solution. In the example case, the choice is between [Zi=788, Z2==106] and [Zi=866, Z2=102].

1200

1300

Figure 4. Trade-off between Z\ and the minimal fleet size, Zi in the example problem

3.

SHORT-TURN STRATEGY FOR INDIVIDUAL ROUTES

3.1

Background

Once the public transport routes are design or exist in operation it is useful to look at each route characteristics for possible improvement. The public transport planners or schedulers certainly understand the need to accommodate the observed passenger demand as well as possible. However, at the same time, their effort is also directed to the minimization of vehicle and driver costs. The trade-off between increasing

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passenger comfort and reducing the cost of service makes the schedulers' task extremely cumbersome and complex. Therefore, there is a need to supply the schedulers' computerized procedures with alternative schedule options along with interpretation and explanation of each alternative. The operational planning process described in Figure 1 shows that based on the demand the next step after establishing the route is the timetable construction. The alternative timetables described in the previous chapter (Ceder, 1984, 1987) are created for each route separately and moreover for the entire length of the route, hi this second part of the chapter it is shown that not all the derived trips in the timetable should cover the entire route in order to make the public transport service more efficient and productive. These trips are called short-turn trips. A short-turn trip is initiated beyond the route departure terminal and/or terminated before the route arrival terminal. The possibility to generate short lines opens the opportunity to further save vehicles while ensuring that the passenger load in each route segment will not exceed the desired occupancy (load factor). hi fact, the schedulers at most public transport properties usually include short-turn operating strategy in their efforts to reduce the cost of service. The procedures commonly used by them are based only on visual observation of the load profile (the distribution of the loads along the entire route). That is, a potential turn point is determined at the nearest adequate time point (major stop) to a stop in which a sharp decrease or increase in the passenger load is observed. While this procedure is intuitively correct, the schedulers do not know if all the short-turn trips are actually needed to reduce the fleet size. On the other hand, each short-turn trips limits the service and hence, tends to reduce the passenger's level-of-service. Furth et al. (1984) reported on an overview of operating strategies on major downtown-oriented bus routes. Among the strategies discussed are the short-turn trips where the service trip is initiated further down along the route, but the arrival point of all the trips is the same. This part of this chapter is based on Ceder (1990, 1991) and describes a methodology to designs all the possible categories of short-turn trips for any type of public transport lines (cross town routes, downtown-oriented routes, feeder routes, etc.). It is to note that only a few articles were published concerning the short-turn strategy. The major objectives set forth for the methodology are: (i) to identify feasible short-turn points based on passenger load profile data; (ii) to derive the minimum fleet

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size required to carry on a given timetable (including the consideration of deadheading - non revenue trips); (iii) to adjust the number of departures in each short-turn point to that required by the load data, provided that the maximum headway to be obtained is minimized (this objective results in the maximum possible short-turn trips and the minimum required fleet size); (iv) to minimize the number of short-turn trips provided that the minimum fleet size is maintained (for a given timetable, this objective results in increasing the level-of-service seen by the passengers); (v) to create vehicle schedules (blocks) for the final derived timetable (a block is a sequence of revenue and non revenue activities for an individual vehicle). In order to satisfy the objectives, several methods were developed. These methods are based on procedures and algorithms which utilize data commonly inventoried or collected by most transit properties. Furth (1983) uses origin-destination (O-D) data to assess short-turn strategies for route 16 in Los Angeles (SCRTD route) between West Hollywood and downtown. While the O-D data can improve the scheduling of shortturn trips, it is commonly unavailable at the public transport agencies. The methodology below is not based on O-D data, but its modules can be extended to include such data whenever it is available. 3.2

Framework

The initial information required for constructing the short-turn trips is based on: (i) a complete timetable at all the route timepoints; (ii) passenger loads for each time period across all the timepoints; (iii) minimum frequency or policy headway; and (iv) set of candidate short-turn points. This information is given for both directions of the route, each with its own data. The complete timetable can be provided by the scheduler or be derived from the passenger load information (Ceder, 1984, 1987). The candidate short-turn points are usually all the route major stops (timepoints) in which the public timetable is posted. In some cases, the scheduler may limit these points to only a set of timepoints in which the vehicles can actually turn back. This set represents the candidate short-turn points. The overall program to accomplish the objectives of the presented methodology is described in a flow chart form in Figure 5. It starts with a procedure to determine the set of feasible short-turn points Rj among the candidate points. Then the deficit function theory, explained in the previous chapter, is used to derive the minimum number of vehicles required to carry on all the trips in the complete two-direction timetable, N m j n . The required number of departures is determined at each of the feasible short-turn point, and then the so called minimax H algorithm is applied. It is

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based on eliminating some departures from the complete timetable, in order to obtain the number of departures required. In that procedure the algorithm minimizes the maximum difference between two following departure times (headway). At this stage, as shown in Figure 5, the deficit function method derives the minimum fleet size required with short-turns, Nmin. If this minimum is less than the size required without short-turns, then another procedure is applied. The latter inserts back the maximum possible departures among those previously eliminated, provided that the minimum fleet size, N m j n , is maintained. The final step of the overall program is to create vehicle schedules to cover all the trips which appear in the last version of the twodirection timetable. 3.3

Feasible Short-Turn Points

The short-turn points are usually selected among the route timepoints provided that at these points the vehicle can turn back without interfering with the traffic flow. It is anticipated, therefore, that for each route, the initial set of candidate short-turn points is given. Let the set of candidate snort-turn points be designated as set RI for one direction and R2 for the opposite route direction. Note that RI does not necessarily coincide with R2. More specifically,

where r^ the jth candidate short-turn point in the jth direction (j =1, 2) are n and q such points for directions 1 and 2, respectively. For a given time period, the fluctuation of a passenger load along the entire route (load profile) may reveal that some short-turn points are actually redundant. Theoretically, each segment between two adjacent short-turn points can be treated independently with respect to its required frequency. This frequency is determined by the maximum observed load in the segment. However, in the short-turn strategy, all the trips must serve the heaviest load segment of the route and as the result some short-turn points may be redundant (more trips will cross them than actually required by their passenger load). The exclusion of the redundant points at each time period j results in a set of feasible short-turn points, Rj and this analysis is important from a computational time viewpoint. For subsequent analyses, the union of all Rj for all time periods j, is denoted as the set R.

Designing Public Transport Network and Routes

Figure 5. A flow-chart to describe the design of efficient short turn trips

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Example The deficit function theory described in the previous chapter is used to determine the minimum number of vehicles required to cover the complete timetable without short-turn trips. This minimum size is designated Nmjn as shown in Figure 5. A simple example is used as an expository device to illustrate the deficit function approach and the procedures developed. This example appears in Figure 6. It is based on a given timetable that covers two-hour schedule. These hours refer to the departure times at the maximum load points. The route (set R) is comprised from three timepoints: A, B, C and the average travel times for service and deadheading trips are also given in Figure 6. Based on the deficit function approach (see previous chapter), d(k,t) denotes the deficit for point k at time t. Point k can be either a terminal or a timepoint provided that some trips are initiated and/or terminated at this point. The value of d(k,t) represents the total number of departures less the total number of trip arrivals up to and including time t. The maximal value of d(k,t) over the schedule horizon is designated D(k), and the fleet size formula is:

where N is the minimum number of vehicles to service the set E. In the example in Figure 6 it is possible to construct d(A,t) and d(C,t). The minimum number of vehicles required without deadheading trips is D(A) + D(C) = 6. No DH (Deadheading) trip can be inserted. Hence, Nmjn = 6. 3.5

A Procedure to Create Short-Turn Trips (Minimax-H Algorithm)

The basic information required to consider short-turns is the load profile along the entire route. This data is available at most public transport properties world-wide and called ride check information (loads and running times along the entire route). Based on this load profile information, each route segment between two adjacent short-turn points can be treated separately. That is, the required number of trips between the (k l)th and kth short- turn points for a given direction and time periods is:

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where PI< is the maximum load observed between the two adjacent short-turn points, d is the desired occupancy (load standard) and Fmin is the minimum required frequency (the inverse of what is known as the policy headway). The complete timetable in current practice is based on the maximum load, Pm, observed along the entire route in a given time period. Hence, the formula for frequencies higher than the minimum level, F min , is:

The manual procedure done by the scheduler to create short-turn trips is simply to exclude departure times in order to set the frequency at each short-turn point k to Fk instead of Fm. The exclusion of departure times is usually performed without any systematic instructions, with the belief that by doing so, it is possible to reduce the number of vehicles required to carry on the timetable. The result of excluding certain departure times is that some passengers will have to extend their wait at the short-turn points. In order to minimize this adverse effect, it is possible to set the following (minimax H) criterion: Delete Fm -Fk departure times at k with the objective to minimize the maximum headway to be obtained. It is known that in a deterministic passenger arrival pattern, the wait time is half the headway. Therefore, the above criterion attempts to achieve the minimization of maximum wait. This criterion is called minimax H, and it may represent an adequate passenger level-of-service whenever the scheduler's strategy allows for elimination of some departure times. In order to solve the optimization problem with the minimax H criterion, a theory was developed in Ceder (1991). It is based on: (i) representation of the problem on a directed network with a special pattern; (ii) applying a modified shortest-path algorithm on the network to determine the minimax headway; and (iii) applying an algorithm to ensure that the exact number of required departures will be included in the optimal solution. What follows is the outline of the minimax H algorithm and its application to the example problem presented in Figure 6.

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Figure 6. Example of 2-hour, 2-way schedule with its deficit functions

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Let Gm = {Nm, Am} be the special network consisting of a finite node set Nm and a finite set Am of directed arc s. hi general, n departures are given from the complete timetable, and it is required that only m< n will remain while satisfying the minimax H criterion. The construction of Gm is based on m-2 equally spaced departure times between the first and last given departures: ti, and tn respectively. These equally spaced departure times are denoted by t 2 , t3 ,...., tm.i and have the equal headway of te=(tn-ti)/(m-l). The Gm network has the following six characteristics: (1) Gm consists of m rows where the first and last rows are only nodes ti and tn, respectively, and there is a row for each tj, j = 2,3,..., m-1. (2) Each node in Nm represents a departure time in the given set of departures; however, it is not necessary that all the given departures be included in Nm. (3) The nodes in each' row are organized in increasing time order from left to right with respect to their associated t'j. That is, all the given nodes tj such that tj < tj < tk+i are positioned twice: once to the right of tk and once to the left of tk+i, where tk, tk+i are two adjacent equally spaced departure times. An exception is that in the second and the (m-l)th rows, only one node is positioned left to t2 and right to tm-i , respectively. These single nodes, ts and t n _2 are selected such that ts is the closest node to t2 provided that t2 > ts, and (4)

(5) (6)

tn-2 is the closest node to tm-i provided that t m _i < tn.2 The directed arcs in Am connect only nodes from the kth row to the (k+1 )th row, k =1,2,..., m-1. A directed arc from tj to tj is included in Am if tj > tj and an arc from tj to t; if and only if without this arc Gm is disconnected. The length of an arc from tj to tj is exactly tj - t j .

After constructing Gm a modified shortest-path algorithm is applied. This is a modified version of an efficient algorithm like the one initially proposed by Dijkstra (1959). The Dijkstra method is based on assigning temporary labels to nodes, the label on the node being an upper bound on the path length from the origin node to that node. These labels are then updated (reduced) by an iterative procedure. At each iteration exactly one of the temporary labels becomes permanent, implying that it is no longer the upper bound but the exact length of the shortest path from the origin to the considered node. The modification of the Dijkstra method is in the computation step in which the labels are updated. It is modified from:

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n (to = min [n (tj), n*( t k ) + (t; - t k )]

(8)

n (tj) = min [D (tj), max {D*( t k ) , ft - t k )}]

(9)

to

where lift), n*( tk ) are temporary and permanent labels of nodes tj and tk respectively. This algorithm is applied to Gm where the origin node is ti and the algorithm terminates when the temporary label on the node tn becomes permanent. For example given 9 departures at 7:00, 7:10, 7:20, 7:45, 8:00, 8:20, 8:40, 8:50, and 9:00, and only 5 departures are required (four short-turn trips). The network is then constructed around the 5 equally spaced departure times which are 7:00, 7:30, 8:00, 8:30, and 9:00 (in the second row, after node 7:00 in the first row, there will be 3 nodes: 7:20, 7:30, and 7:45). The times 7:10 and 8:50 are not part of the network. Applying the modified Dijkstra with Equation (9) results in minimax H=40 minutes with 2 solutions: (7:00, 7:20, 7:45, 8:20 and 9:00), and (7:00, 7:20, 8:00, 8:40 and 9:00). Since in the second solution the minimax H is reached twice there is some advantage to the first solution. The third part of the minimax H algorithm ensures that the optimal result will include exactly the required number of departures. While the modified shortest-path algorithm on Gm determines the value of the minimax headway, it does not ensure that the result will include all the required number of departures. The detailed description of this third part appears in Ceder (1991) along with a procedure to treat also multiple departures (same and more than one departure time in the given timetable). The minimax H algorithm is now applied to the example problem presented in Figure 6 though in that simple example the results are obtained straightforward by inspection. The required number of departures for each direction of travel can be easily calculated from the data in Figure 6. On the direction A-C at A only one departure is required (50/50=1) and it will be at 6:15. On the direction C-A at B two departures are required (90/50=1.8) to be at 6:45 and 7:15 (from B to A). All the 3 departures must leave B to C and C to B (B,C are max load points, in the directions A-C and C-A, respectively). Figure 7 presents the timetable and deficit functions for the case of maximum (three) short-turns. The deficit functions include also the function at B since trips are departing and arriving to B. The minimum fleet size then is 5 (no deadheading trip can be inserted), and therefore it is worth implementing short-turns (saving of one vehicle). The next step is to check possible extensions of these three short-turn trips without increasing the minimum of 5 vehicles.

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83

Optimal extension of the determined short-turn trips

No doubts that extensions of the short-turn trips improve the passenger's level-ofservice. These extensions are made toward the original trip end points. A few observations can be made with respect to the arrival/departures to an intermediate point (Im), like point B in the example. The following observations hold: (a) any trip departure at Im that increases D(Im,t) by one unit (see the deficit function theory in the previous chapter) can be extended to its original departure point. That is, if a given trip departure increases D(Im,t) by one unit then by extending it to its original (A) departure point, D(Im,t) will be reduced by one with the possibility to increase D(A,t) by one (net increase of zero vehicles in the fleet size in the worst case). (b) any trip arrival at Im that will not increase D(Im,t) by one unit, if extended to its original departure point, can be extended . That is, if a given trip arrival time is after the maximal region of D(Im,t) and if by extending it D(Im,t) will remain same then this extension (to A) can be made since D(A,t) will be never increased by a trip arrival, hi other words the fleet size will remain same. (c) any combination of DH trip with either an extension of a trip arrival or a trip departure from d(Im,t), such that the fleet size is preserved, can be performed. The formal algorithm of this extension theory appears in Ceder (1990). hi the example problem in Figure 7 it can be easily observed that the departure at 6:30 from B can be extended to A with a net zero change in the fleet size. That is, D(A,t) becomes 2 instead of 1, and D(B,t) becomes zero instead of 1. The dashed line, in Figure 7, represents the situation after the extension and shows that neither the 7:00 arrival nor the 7:15 departure from B could be also extended. Consequently the 6:00 departure is back and only 2 short-turn trips are required to preserve the minimum of 5 vehicles. In order to demonstrate the extension possibilities of the determined short-turn trips Figure 8 exhibits another example of a deficit function at d(k,t) which is given as an intermediate point (the example in Figure 8 is not the example in Figures 6,7). Three extensions can be performed in Figure 8 without increasing the minimum fleet size. The first two represent extensions of departure times and the third-an extension of an arrival time. The minimum number of vehicles required at point k, D(k,t)=D(k) changes from D(k)=2 to D(k)=0. Figure 8 shows how the deficit function changes following each extension and what opportunities are opened and closed for further extensions.

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Direction Timepoint A Departure (6:00) and Arrival 6:15 Times Departure point at B.

A-C

E 6:30* 6:45 7:15*

C-A

C

03

7:15

6:00

7:30 6:30 8:00 6:45 Arrival point at B.

B 6:45

A 7:00 _

7:00** 7:15 7:30 ( ) Extented departure time eficit function — — Deficit function after extension

Figure 7. Maximum (and final) short-turn following one possible departure time extension with their assiciated deficit function

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Figure 8. Example of updated deficit functions at an intermediate short-turn point after each of the three indicated extensions

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At the end of the extension procedure each of the N^ vehicles is assigned to a group of trips in the final schedule. A single group of trips, called vehicle schedule or block, exhibits a sequence of service and deadheading trips for an individual vehicle. The task of scheduling vehicles to chains of trips can be carried out by the first-in-first-out (FIFO) rule or by a chain extraction procedure described in the previous chapter.

4.

CONCLUDING REMARKS

This chapter focuses on two public transport route design parts: (a) design of routes at the network level, and (b) creating efficient operations of each route using the shortturn strategy. 4.1

Network Design

The optimization parameters (subject for minimization) used in the first part of this chapter are: (i) total passengers' riding time in public transport vehicles, on an hourly basis, minus the total passengers' time if riding the shortest route. It measures how much excessive time is spent by passengers on public transport vehicles between each two nodes on the network; (ii) the total amount of wait time at public transport stops and transfer points, on an hourly basis. It measures how much wait time is spent by passengers between each two nodes on the network; (iii) the total unused seats in public transport vehicles, on an hourly basis. It measures the unutilized capacity of the public transport vehicles between each two nodes on the network; and (iv) the total number of vehicles required to provide all trips along the chosen set of routes. The parameters of (i), (ii) and (iii) are combined into the Z\ objective function, and parameter (iv) is the Z2 objective function. The solution approach of the network design problem can be comprised of two stages: In the first stage, the problem dimension is reduced through the construction of skeleton feasible route network which meets a maximum travel time constraint. The skeleton network is the basis for an optimization routine to determine the shortest direct and indirect (via transfers) paths between each pair of nodes. The second stage relies on a procedure which incorporates optimization and enumeration processes in order to derive the minimal Z\ objective function. This procedure, while searching for min Z\, creates various Z2 solutions - each associated with a different Z2 solution.

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Finally, the most desirable set of Z\, Z2 is derived through known techniques in multi-objective programming (Coello et al, 2002). The framework described in the first part of this chapter comprises six main elements, each of which uses the data created by its predecessor (or data given by the user) to carry on the next step in the analysis (see Figure 2). This framework is believed to be a useful toolset for the following applications: (a) optimal design for a new public transport network; (b) optimal design for expansion or curtailment of an existing public transport network; (c) assessment of the performance of an existing public transport network from the aspects of: operator efficiency; passenger level of service; and sensitivity analysis of public transport network performance for a variety of system parameters. 4.2

Short-Turn Strategy

The second part of this chapter presents a set of procedures to efficiently design individual public transport routes and timetables with trips that are initiated beyond the route departure point and/or terminated before the route arrival point (short-turn trips). In practice, public transport frequency is determined at the heaviest load route segment, whereas at other segments, the operation may be inefficient due to situations characterized by empty seats. The public transport schedulers attempt to overcome this problem by constructing manually short-turn trips with the objective to reduce the number of vehicles required to carry on the public transport timetable. The purpose of this second part of this chapter is to improve and automate this task through the following objectives: to identify feasible short-turn points; to derive the minimum fleet size required to carry on a given schedule; to adjust the number of departures in each short-turn point to that required by the load data, provided that the maximum headway (associated with passenger wait time) to be obtained is minimized; to minimize the number of short-turn trips while ensuring that the minimum fleet size is preserved; and to create vehicle schedules (blocks). Future work in this subject can be concentrated along the following lines: (a) extension of the methods to handle also Origin-Destination (O-D) data whenever it is available. (b) the inclusion of more than two end route points. That is, a public transport route may consist of branches, and the procedures developed can easily be extended to consider such cases.

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Advanced Modeling for Transit Operations and Service Planning (c)

modification of the procedures to handle a network of interlining routes (in which a vehicle can traverse from one route to another in its block). It is noticed in the previous chapter that when interlining routes are allowed, the minimum fleet size can be further reduced in comparison to the operation of independent routes.

A remark is pertinent about the future task (a). In some practical circumstances, a vehicle which performs a complete trip followed by a short-turn trip may experience 'higher load than the load on short-turn trip. The consideration of O-D data can eliminate, to some extent, this passenger load imbalance situation. However, the O-D data seldom exist in public transport properties. Meanwhile, the methodology presented here can be practically implemented in such load imbalance situations through the assistance of the public transport property mobile supervisors and road inspectors, as well as the drivers. They can be advised to examine whether or not a significant load imbalance is observed on certain trips in a systematic manner, hi such cases, practically speaking, the short-turn trip can be regulated at certain stops and/or a skip-stop strategy can be utilized for the complete trip. Finally, it is noted that methodology presented enables identification of the minimum, and hence, the crucial allowed short-turn trips which are required to save operational cost via the reduction of the fleet size. The current manual procedure in many public transport properties to create short turns (mainly based on visual inspection of the route load profile), may result in certain route segments, in an inefficient operation characterized by empty seats and in an unnecessary increase of passenger wait time. It is believed that the automated procedures presented are capable of improving the scheduler's manual procedure.

REFERENCES Ceder, A. and Y. Israeli (1992). Scheduling consideration in designing transit routes at the network level, hi: Computer-Aided Transit Scheduling, (M. Desrochers and J. M. Rousseau, eds.), pp. 113-136. Springer-Verlag Pub. Co., BerlinHeidelberg. Ceder, A. and H. I. Stern (1981). Deficit function bus scheduling with deadheading trip insertion for fleet size reduction. Transportation Science, 15 (4), 338-363. Ceder, A. (1984). Bus frequency determination using passenger count data. Transportation Research, ISA (5/6), 439-453.

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Ceder, A. (1987). Methods for creating bus timetables. Transportation Research, 21A (1), 59-83. Ceder, A. (1990). Optimal design of transit short-turn trips. Transportation Research Record, 1221, 8-22. Ceder, A. (1991). A procedure to adjust transit trip departure times through minimizing the maximum headway. Computers and Operations Research Journal, 18 (5), 417-431. Ceder, A. (2001). Operational objective functions in designing public transport routes. Journal of Advanced Transportation, 35(2), 125-144. Ceder, A. (2002). A step function for improving transit operations planning using fixed and variable scheduling. Transportation & Traffic Theory, (15th ISTTT), M. A. P. Taylor (ed.) Elsevier Science & Pergamon Pub. 1-21. Coello, C. A. C., D. A. Van Veldhuizen and G. B. Lamont (2002). Evolutionary Algorithms for Solving Multi-Objective Problems. Kluwer Academic/Plenum Pub.New-York. Dial, R. B. and R. E. Bunyan (1968). Public transit planning system. Socio-Economic Planning Science, 1, 345-362. Dijkstra, E. W. (1959). A note on two problems in connection with graphs. Numerische Mathematik, 1, 269-271. Dubois, D., G. Bel and M. Libre (1979). A set of methods in transportation network synthesis and analysis. Operations Research, 30(9), 797-808. Duckstein, L. and S. Opricovic (1980). Multiobjective optimization in river basin development. Water Resources Research, 16(1), 14-20. Farvolden, J. M. and W. B. Powell (1994/ Subgradient methods for the service network design problem. Transportation Science, 28(3), 256-272. Furth, P. G. (1983). Bus Route & Service Design: Application of Methods and Procedures. Multisystems, Report prepared for U.S. Department of Transportation. Furth, P. G., D. Brian and J. Attanucci (1984). Bus Route and Service Design: An Overview of Strategies for Major Radial Bus Routes. Multisystems, Final Report prepared for U.S. Department of Transportation. Hasselstrom, D. (1981). Public Transportation Planning. Ph.D. Thesis, Department of Business Administration, University of Gothenburg, Sweden. Heathington, K. W., J. Miller, R. R. Knox, G. C. Hoff and J. Bruggman (1968). Computer simulation of a demand scheduled bus system offering door to door service. Highway Research Record, 91(251), 26-40.

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Israeli, Y. and A. Ceder (1995). Transit route design using scheduling and multiobjective programming techniques. In: Computer-Aided Transit Scheduling, (J.R. Daduna, I. Branco and J. M. P. Paixao, eds.), pp. 56-75. Springer-Verlag Pub. Israeli, Y. and A. Ceder (1996). Public transportation assignment with passenger strategies for overlapping route choice. Transportation and Traffic Theory (13th ISTTT). J. B. Lesort (ed.) Elsevier Science & Pergamon Pub. 561-588. Keudel, W. (1988). Computer-aided line network design (DIANA) and minimization of transfer times in networks (FABIAN). In: Computer-Aided Transit Scheduling. Lecture Notes in Economics and Mathematical Systems, (J. R. Daduna and A. Wren, eds.), 308, Springer-Verlag, Berlin. Kim, D. and C. Bamhart (1999). Transportation service network design: Models and algorithms. Proceeding of the 7th International Workshop on Computer-Aided Scheduling of public transport. Lectures notes in Economics and Mathematical Systems 471, Springer-Verlag, 259-283. Kocur, G. and C. Hendrickson (1982). Design of local bus service with demand equilibration. Transportation Science, 16(2), 149-170. Kuah, G. K. and J. Perl (1988). Optimization of feeder bus routes and bus-stop spacing. Journal of Transportation Engineering, 114(3), 341-354. Lampkin, W. and P. D. Saalmans (1967). The design of routes, service frequencies and schedules for a municipal bus undertaking: A case study. Operations Research, 18(4), 375-397. Magnanti, T. L. and R. L. Wong (1984). Network design and transportation planning: Models and algorithms. Transportation Science, 18(1), 1-55. Mandl, C. E. (1979). Evaluation and optimization of urban public transportation networks. European Journal of Operation Research, 5, 396-404. Marguier, P. H. J. and A. Ceder (1984). Passenger waiting strategies for overlapping bus routes. Transportation Science, 18(3), 207-230. Marwah, B. R., F. S. Umrigar and S. B. Patnaik (1984). Optimal design of bus routes and frequencies for Ahmedabad. Transportation Research Record, 994, 41-47. Minieka, M. (1978). Optimization Algorithms for Networks and Graphs. Marcel DekkerInc.,N.Y. Rea, T. C. (1971). Designing Urban Transit Systems: An Approach to the Route Technology Selection Problem. Seattle, Washington, PB 204 881, University of Washington. Sharp, G. P. (1974). Public transit system network models: consideration of guideway construction, passenger travel, delay time and vehicle scheduling. Georgia Institute of Technology, Ph.D. Thesis, Atlanta, Georgia, USA.

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Silman, L. A., Z. Barzily and U. Passy (1974). Planning the route system for urban buses. Computers and Operations. Research, 1, 201-211. Spiess H. and M. Florian (1989). Optimal strategies: A new assignment model for transit networks. Transportation Research, 23B(2), 83-102. Stern, H. and A. Ceder (1983). An improved lower bound to the minimum fleet size problem. Transportation Science, 17(4), 471-483. Tsao, S. and P. Schonfeld (1984). Branched transit services: An analysis. Journal of Transportation Engineering, 110(1), 112-128. Vandebona, U. and A. J. Richardson (1985). Simulation of transit route operations. Transportation Research Record, 1036, 36-40. Wu, J. H., M. Florian and P. Marcotte (1994). Transit equilibrium assignment: A model and solution algorithms. Transportation Science, 28(3), 193-203. Zeleny, M. (1973). Compromise programming. In: Multiple Criteria Decision Making, (Cochrane J. L. and M. Zeleny, eds.) Columbia: University of South Carolina Press. Zeleny, M. (1974). A concept of compromise solutions and the method of the displaced ideal. Computers and Operation Research, 1(4), 479-496.

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CHAPTER 4

TRANSIT PATH CHOICE AND ASSIGNMENT MODEL APPROACHES(O)

Agostino Nuzzolo, "Tor Vergata " University of Rome, Italy, EU

1.

INTRODUCTION

Simulation models for public transport networks, as those for road networks, include supply models, path choice models and assignment models (see Figure 1). However, such models can be quite different from those for road ones, and in this chapter a general overview is reported, showing the main approaches that can be used. Section 2 describes the two possible representations of transit supply (line-based and run-based), section 3 reports the different path choice models in relation to the different path choice behaviour that users have for both high-frequency (section 3.1) and low-frequency (section 3.2) services, while section 4 describes transit assignment models using the two possible approaches (frequency-based and schedule-based) and the application fields of such models. Finally, section 5 reports a reference note about supply, path choice and assignment models presented in literature for public transport networks. Francesco Russo is co-author of sections 2.2, 3.1.2, 3.2.2; Umberto Crisalli is co-author of sections 3.1.1.. 4. 5

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Figure 1. Transit assignment models: general scheme

2.

TRANSIT SUPPLY MODELS

Scheduled transportation services, such as those provided by aeroplanes, trains, buses are modelled as discrete in both time and space as they can only be accessed at certain times and certain locations, such as airports, rail stations, and bus stops. A timetable is made of stop locations and arrival/departure times of scheduled services, and with respect to a given timetable, individual runs and service lines can be defined (see Table 1). A run r represents an individual connection with a given scheduled time (e.g. a given train connection), while a line I is a set of runs with the same characteristics (e.g. stops, travel times, quality of services, and so on). Table 1. Timetable, lines and runs

Run

Line

1 2 3 4

AA BB AA CC

Service type Intercity Regional Intercity Intercity

Departure Time 9.30 9.50 10.30 11.30

Initial Station A A A A

Intermediate Stops B/C D

Terminal Station D D D E

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In general, supply models consist of a network model (graph plus cost functions) and of a set of relationships connecting link costs to path costs and link flows to path flows. In order to study public transport services, two different modelling approaches can be used: the first one refers to services represented in terms of lines (line-based supply model), while in the second one considers services represented as single runs (runbased supply model), as described in the following sections. 2.1

Line-Based Supply Models

In the line-based approach, the whole supply model derives from the join of two submodels: the first is relative to the service representation; the second is relative to the access/egress network to/from the public transport system. Transit services are represented by the line subgraph in which nodes represent stops and events that happen at stops. The access/egress subgraph is made of a set of nodes and links which allow the connection between centroids (origin and/or destination of trips) and stops. Figure 2 reports a possible stop representation for high-frequency services. In this case, the stop node represents the stop in space and the user decision to use transit services in time; this node is usually connected with the stop pedestrian node which represents the stop in the access/egress subgraph. Line nodes represent line vehicles available at stops. Other links represent the user waiting/boarding at stops (waiting/boarding links), the alighting from line vehicles (alighting links) and the travel from one stop to another stop of the same line (line links). The whole network model (see Figure 3) is obtained connecting the two subgraphs through proper links introduced at nodes representing stops and considering a generalised cost associated with each link. 2.2

Run-Based Supply Models

Given the service timetable (see Figure 4), in the run-based approach each run is explicitly taken into account in both space and time.

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In the following, three different types of supply models will be described: the diachronic network, the dual network and a mixed line-based/database supply model. The diachronic graph O consists of three different sub-graphs in which each node has an explicit time coordinate: - a service sub-graph ng in which each run of each line is defined both in space, through its stops, and in time, according to its arrival/departure times at stops (see Figure 5); - a demand sub-graph Qi, in which each node represents a temporal centroid, in order to simulate the space-time characteristics of the trip (see Figure 5); - an access/egress sub-graph Oae which allows the connection of the demand subgraph with access/egress stops and stops between them.

Figure 2. Line-based stop representation for high-frequency services line 1 +*V-

c

)„.„„„„,„„<;?

J- - - - :

ine 1

r—'

!! <<

line 2

line 3 line 4

'

^C = = =

=

F4

F..

G

»*

£

?C

line 5

,<J

>• r line 6

D

£

c

)= = = = =

Hne5

J -fr.~< >-===-"=*"""^ line 5 jj line 5

ine 7

•s

O

line 7

£ stop pedestrian nodes • stop nodes • line nodes

0

lineS

ine 8

-A- centroids O pedestrian nodes

Jk

•= * () *

^

.

line 7 line 8

pedestrian links boarding/alighting links ''ne linlcs

Figure 3. Line-based network representation for high-frequency transit services

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97

TIMETABLE run

Terminal A

Terminal B

arr.

dep.

08.30

10.10

10.15

11.00

12.35

12.37

arr.

dep.

IC634

08.55

09.00

IC640

08.25

IC741

10.58

Terminal C

arr.

dep.

11.15

11.18

12.00

12.05

14.00

14.02

Figure 4. The run-based representation of transit services The service sub-graph Qg consists, in turn, of different sub-graphs ngr (one for each run of the transit services). Referring to the generic run r, the relative sub-graph Qgif consists of nodes representing the arrival and departure times at stops, and links representing travel from one stop to another (run section) or the dwelling of the vehicle at a given stop. Other nodes represent the time in which users board or alight from each run at the stop. These nodes are connected to the nodes representing run arrival and departure through boarding and alighting links. Finally, the whole subgraph Qg is built by connecting all the sub-graphs ngr through links representing the users' transfer from one run to the next one at the same stop (stop axis).

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The demand sub-graph Qd representing temporal demand segmentation is made up, in turn, by the same number of sub-graphs Qj.c as centroids. For each spatial centroid c, the sub-graph Od.c consists of temporal centroids, that are nodes located spatially in the position of the spatial centroid c and temporally according to the user origin departure times TotThe access/egress sub-graph Qae is made up of: - links connecting origin temporal centroids to nodes on the first boarding stop axis, to represent the access to the services; - links connecting alighting nodes of the stop axis to the destination spatial centroids, to represent the egress from the transit system; - links connecting stops, to represent possible interchanges between different stops, too. Finally, the global diachronic graph Q is obtained connecting the three sub-graphs by using proper links. The diachronic graph is very useful as it allows efficient use of standard network algorithms (as least-cost paths) and allows more straightforward treatment of congestion, when it is considered. Note that, in the line-based approach, all runs belonging to the same line are represented by a single sequence of links (see Figure 4) to which average values of run attributes belonging to the line are associated with. This representation is simpler than the run-based one but allows us to obtain only line average values (e.g. average on-board loads) as assignment results. The dual graph was initially developed to model turn prohibitions/restrictions in road networks and then was extended to represent public transport networks. The extension to public transport networks is done by replacing each transit segment (which represents a line or a single run according to the supply representation approach) with a node. Nodes are connected by links considering time congruence relative to arrival/departure times of transit vehicles (runs) at stops. An example of the dual graph representation of a stop with multiple transit routes and runs is reported in Figure 6. Note that, for simplicity, Figure 6 only shows the dual link for one particular link/route, but all other combinations can be created in the same way according to the arrival/departure times of scheduled runs. This implies that for large transit networks we have to consider an high number of links to represent all possible connections in

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99

terms of arrival/departure runs at stops, in addition to the fact that the transformation from the original network to the dual network and vice versa have to be considered at least at the starting and ending phases of the simulation procedure.

temp oral centroids

stop axis s

Figure 5. The diachronic run-based representation of transit services

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Figure 6. The dual graph representation of transit services A different approach is the mixed line-based/database one, which uses a database to consider the timetable together with a traditional line-based network model. It uses a line-based approach (see section 2.1) to describe spatially the service network topology in terms of routes, lines and stops, while data associated with nodes and links define the clock-dependent movements of transit vehicles (runs) on the network. This approach is more complex with respect to the above described ones, even if considering some simplifications made by expanding on-line the strictly necessary part of the network, in order to define the time-dependent congruence of services on the line-based representation. The disadvantage of this approach is that the direct use of traditional network algorithms to carry out path search and network loading is not allowed, as clockdependent variables have to be considered through the database (e.g. using modified Dijkstra type algorithms).

3.

USER PATH CHOICE MODELS FOR TRANSIT SERVICES

User decisions in path choice can be classified according to two types of choice behaviour:

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-

101

pre-trip choice behaviour, which underlies user choices before departure. It includes the comparison of possible alternative strategies and the choice of one of them on the basis of expected characteristics, or attributes. en-route choice behaviour, which underlies user choices during the trip. This behaviour describes how users respond to unknown or unpredictable events. The type of en-route choice behaviour can be defined as: + indifferent: if users board the first arriving vehicle belonging to the set of alternatives; + intelligent: if users, when a vehicle belonging to the attractive set arrives, compare the disutility of the arriving vehicle with the disutility of the next arriving vehicles belonging to the set of alternatives.

Moreover, path choice model have to be specified according to user behavioural hypotheses that depend on specific user and service characteristics. The main service factors affecting user behaviour in path choice are frequency, regularity and information available to users. Service frequency can be related directly to the frequency of line / in the reference period, i.e. the number of runs belonging to line / in such a period or, for overlapping lines, to the cumulative frequency, i.e. the sum of the frequencies of all "attractive" lines connecting the O/D pair od. Usually service frequency is high if the average headway of vehicles is less than 12-15 minutes, while frequency is defined low if the average headway exceeds 15 minutes. Service regularity is a measure of to what extent the schedule is observed. If regularity is used to make assumptions on user behaviour in line-based systems, such as buses and trains, deviations from the schedule should be related to the average headway of runs belonging to the same line. Usually regular services are associated with low service frequencies, typical of extraurban systems (e.g. railways). On the other hand, irregular services generally refer to high frequencies as in urban or metropolitan areas, e.g. bus or underground lines. Pre-trip and/or en-route real-time information on services can be available to the user in different places (for example at stops or at home) and concerns at least waiting times of arriving vehicles at the chosen access stop, while more advanced information systems could give information on travel times and on-board occupancy, too. Static information on run schedule is traditionally available with timetable. Intelligent Transportation Systems (ITS) have significantly expanded the range of information available to the traveller through Advanced Traveller Information Systems (ATIS).

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ITS also improves the performance of transit services, in terms of regularity, through the use of Advanced Public Transportation Systems (APTS). As regards user characteristics, the main difference to consider concerns whether they are frequent or occasional users. Frequent users travel frequently and know routes and scheduled timetable as well as the real system functioning based on previous experience. Occasional users use sometimes transit services, so they only know some line routes (the most important) and their scheduled timetable. Of course different types of users that behave differently from those described above can be specified, but in the following we refer to these two user types that can be considered at the extremes of the possible range of user type to be taken into account. In addition, user behaviour is often related to trip purpose. Two main approaches can be used in transit path choice modelling: the frequencybased approach and the schedule-based one. The frequency-based approach considers services in terms of sets of runs (lines). In this case run scheduled times are not considered explicitly, but we refer to the line headways, or to their inverse (the service frequencies), from which the name of the approach derives. Therefore we are not able to calculate explicitly attributes that users consider in relation to single runs, but we can refer only to average values relative to lines. The schedule-based approach refers to services in terms of runs using the real vehicle arrival/departure time, and hence all the values of level of service attributes, evaluated at time in which users make their choices, can be explicitly taken into account. This approach allows us to take into account the evolution in time of both supply and demand, as well as run loads and level of service attributes. Given the modelling approach (frequency-based or schedule-based), as the main factor that greatly affects user behaviour is service frequency, the following sections will describe path choice models for both high and low frequency services, considering either the frequency-based and the schedule-based approach. User behaviour hypotheses and path choice models will be specified in the framework of (random) utility theory, in which users are assumed rational and behave with the aim of maximising their perceived utility.

Transit Path Choice and Assignment Model Approaches 3.1

103

Path Choice Models for High-Frequency Services

In relation to the high frequency of services we can assume that the origin departure time coincides with the desired origin departure time, so user arrival at the stop is not related to run departure scheduled times. Furthermore, it is assumed that users do not have full information before starting their trip and they follow a mixed pre-trip/enroute choice behaviour. The choice of boarding stops is considered to be made before starting the trip, since it is not influenced by unknown events, while en-route choices occur at stops and are relative to the decision to board a particular run or to wait for another run of the attractive set. 3.1.1 Frequency-Based Path Choice Models for High-Frequency Services In the sphere of the frequency-based approach, path choice for high-frequency services implies the choice of the boarding stop and that of the line (or sequence of lines) leading users to their destination. In the case of frequency-based models, the traditional approach in path choice modelling is that based on the concept of optimal strategy, in which users do not choose a precise path, but they rather identify with a pre-trip choice behaviour a set of attractive paths, within which they choose during the trip on the basis of an indifferent en-route choice behaviour at stops, minimising the perceived average trip cost, hi the following of this section we will refer to the line-based representation of the transit network reported in Figure 7, which is a convenient way to represent the line-based supply model reported in Figure 3.

Figure 7. Example of line-based transit networks

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Advanced Modeling for Transit Operations and Service Planning

Under the assumptions of mixed pre-trip/en-route choice behaviour, a set of attractive paths can be represented on the transit network through a subgraph known as hyperpath. An hyperpath example for the transit network of Figure 7 is reported in Figure 8.

Figure 8. Example of hyperpath Given the en-route choice behaviour and the service functioning conditions, an average generalised cost C, can be associated with each hyperpath j. This cost is equal to the average generalised costs C* of paths belonging to hyperpath j, weighted by the probabilities q^j of choosing path k in the effective trip:

0) kej

kej

lek

where c\ represents the generalised cost of the generic link / belonging to the generic path k. To each diversion node i of the hyperpath j will correspond a diversion set L/j of attractive lines belonging to the considered hyperpath. To the boarding links l=(i,m) connecting the diversion node z with nodes m of the lines belonging to Lfj, it is possible to assign a diversion probability, rjij, that is the probability of using the line corresponding to link / connecting node z with node m of hyperpath j. Such a probability depends on the assumed user en-route choice behaviour and on service characteristics. In case of indifferent en-route choice, according to the hypothesis that users board the first arriving vehicle of a line belonging to the attractive set, and with the assumption that user and vehicle arrivals at stops can be modelled as Poisson random process, the

Transit Path Choice and Assignment Model Approaches

1 05

probability of boarding line / belonging to the set of attractive lines at stop z, L,j, can be expressed as:

where
User chooses the hyperpath according to random utility theory, for which the user is a rational decision-maker who associates with each hyperpath j, belonging to the set /„«/ of hyperpaths that connects the O/D pair od, a perceived utility of the hyperpath Uj sum of a systematic utility Vj, equal to the average cost of the hyperpathy, Q, and of a random term £/

choosing the alternative of the maximum perceived utility, or of the minimum perceived average cost. The choice among the available alternatives (hyperpaths) can be expressed in a general form as the probability/?^ that the hyperpath y is that of maximum perceived utility:

Different models can be specified according to the hypotheses on the random term distribution. We can have then deterministic utility models (£/=0), which assign the whole demand to the hyperpath of minimum generalized cost, and random utility models (^0).

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Advanced Modeling for Transit Operations and Service Planning

In case of transit networks, the use of random utility models of logit type is not much acceptable in case of hyperpaths that may include a high number of lines in common (as usually happens for traditional transit networks of medium-sized towns), as the hypothesis of the Independence of the Irrelevant Alternative (HA) that is at the basis of logit models leads to relevant errors in path choice simulation. Then the use of probit models, which allow taking into account different covariances between overlapping paths, can be considered. Finally once calculated the hyperpath choice probabilities it is possible to obtain the choice probability of the generic path k as:

The hypotheses of the hyperpath model are fully acceptable for services with high frequency, very low punctuality and low user information, but can generate considerable approximations when used in different contexts, in particular for lowfrequency services and when ITS (Intelligent Transportation Systems) are present. In fact in the case of high-frequency and not random services, and in particular with real time information at stops about vehicle arrival times, user behaviour can be assumed intelligent en-route (i.e. the choice of the run at the stop, in the attractive choice set, is made by comparing the attributes of alternatives, particularly the waiting time), so the real arrival and departure times of transit vehicles have to be taken into account. Hence, the use of the frequency-based approach is not the best one, since it does not let us take into account the degree of regular functioning of services and users' information, producing errors in the simulation of path choice. 3.1.2 Schedule-Based Path Choice Models for High-Frequency Services In the sphere of the schedule-based modelling approach for high-frequency services, generally several boarding stops can be reached for the same origin and departure time and, even if considering the same user arrival time at stop, many runs can be available. Thus path choice implies choice of access stop and choice of run (or sequence of runs) leading users to their destination. Figure 9 reports an example of path choice set: considering the origin departure time TDI, three different access stops are available, from which different runs of different lines can be used to reach the destination (e,g., from stop 2 run 3 of line/or the combination of run 2 of line g and run 7 of line h can be considered).

Transit Path Choice and Assignment Model Approaches

107

In the schedule-based approach a path between origin o and destination d, departing from o at a given time TDi, is defined by the space-time sequence of links which includes: origin o with origin departure time Tut, access links to access stop s with relative arrival time TDIS, run (or sequence of runs) with run departure time from access stop and run arrival time to egress stop s', egress links from stop 5' to destination d with relative arrival time at destination.

Figure 9. Example of path choice set Schedule-based path choice models give the probability poci[r,s\ TDI] of choosing a path including run r at boarding stop s, given the O/D pair od and the origin departure time

If Tas is the average access time at stop s, the user with origin departure time TDI will reach this stop s at time TDiS=TDi+Tas. The path choice probability p0d[r,s\Toi} can be expressed as Pod[r,s\TDi] = pod[r\s,

TDis]pod[s\TDJ

(7)

which is the product of the probability of choosing run r at stop s, given the arrival time TDJS, by the probability of choosing stop s, given the origin departure time TDI- In the following the index od, when not reported, is understood. The probability p[r\s, TDIS] of choosing the arriving run r, belonging to the run choice set K5, at stop s can be expressed as:

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Advanced Modeling for Transit Operations and Service Planning

where the perceived utility Ur of the generic run r can be expressed as

in which ft are the weights of attributes Xjr, which make up the systematic utility Vr, and sr is a random residual. The service attributes Xjr that can be adopted are: waiting time, on-board time, transfer time, number of transfers, "route" on-board comfort (function of on-board crowding on the following links), "stop" boarding comfort (function of on-board crowding at the stop), monetary cost. The run choice model can be deterministic or stochastic if random residuals er are assumed null or otherwise, and different random utility models (logit, probit, etc.) can be specified according to the distribution of random residuals er. The random residuals er usually take into account aggregation errors (e.g., zoning, network model), missing attributes (e.g., scenic quality, habit), dispersion of user behaviour (e.g., value of time) and user perception errors (e.g., travel time). Moreover, some models include in the random terms some aspects they do not consider explicitly, like service irregularity. Users assess in different ways attributes of Equation (9) if they are frequent users or occasional ones, and if an user information system, especially on waiting time of arriving runs, is available at stops. Attributes are typically forecast by users at the time in which they decide on the basis of available information and past experience. Anyway, if an user information system is working, some attributes can be assessed directly through real-time information at stops (e.g., the waiting time of next arriving runs provided to users). Moreover, in the case of high-frequency services, sometimes there is a substantial difference between real arrival/departure times of vehicles at stops and scheduled times (irregular services), which implies different values of attributes with respect to the ones evaluated on the basis of the timetable. This aspect, jointly with the type of user (frequent or occasional), leads to considerable different run choice mechanisms. For example, in the case of irregular services with user information at stops about waiting times, and considering frequent users, the run choice at stop is typically enroute and can be simulated through a sequential mechanism that considers an intelligent en-route choice behaviour. When a run r of the path choice set fC arrives at

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stop s (see Figure 10), the user chooses to board r if the perceived utility Ur is greater than the utility Ur' of all other runs r'&K* yet to arrive.

Figure 10. Example of path choice set In this case, Ur- includes the waiting time of the next run r\ given by the difference between the arrival time of run r' and the arrival time of run r (supplied by the user information system), while in Ur the waiting time of run r is replaced by the time already spent at the stop (equal to the difference between arrival time of run r and the user arrival time TDIS at stop s) simulating a possible "impatience effect". Other attributes of Equation (9) are considered as previously described. Of course, if the user does not choose run r, the choice is reconsidered when the next run arrives and so on (sequential run choice mechanism with intelligent en-route behaviour). Occasional users behave in the same way, but they consider a reduced run choice set (i.e., they only consider the most important runs connecting the O/D pair od). Finally, one of the most important element of classification in path choice modelling is the dynamic feature of the model. In the sphere of schedule-based approach, path choice models consider at least the within-day dynamic, which allows the system evolution (on-board loads and level of service attributes) within the reference period to be taken into account. Schedule-based path choice models can also consider the day-to-day dynamic (i.e., the evolution of system characteristics from one day to another) if they include a learning process on attributes of Equation (9). Path choice models that consider both within-day and day-to-day dynamics are usually referred in literature as doubly dynamic path choice models. For what concerns the stop choice, it is usually assumed to be fully pre-trip, as no real-time information is typically available at origin. The probability p[s\ rDi] of choosing the boarding stop s, within a choice set of boarding stops, S0d, can be expressed as

110

Advanced Modeling for Transit Operations and Service Planning P[S\TDI] =prob(Us > Us-) = prob(Vs + es> Vs- + es-)

s'*s, s'eSod

(10)

where Us is the perceived utility, sum of the systematic utility Vs and of a random residual es. The systematic stop utility is a function of the stop-specific attributes (e.g. access or egress times, presence of shops, etc.) and "inclusive utility" expressing the average utility associated with all runs available at stop s. As described for the run choice, deterministic or stochastic stop choice models can be specified according to the assumptions on random residuals ss. 3.2

Path Choice Models for Low-Frequency Services

From a practical point of view, low-frequency services, as regional bus and intercity railways, are usually characterised by regular service functioning and we hypothesise users have full information before starting their trip. In this case there is no difference in user behaviour for frequent and occasional users, and the presence of user information is unnecessary to support user choices, assuming that they at least know routes and timetable. Furthermore, as low-frequency services are typical of extraurban areas, it is common that only one access terminal as well as a single egress terminal are available, so stop choice can be easily simulated. The run choice is assumed fully pre-trip and, in addition to the other service attributes, we need to consider the disutilities that occur because of the difference (that can be considerable) between desired user departure time and vehicle scheduled departure time or between desired user arrival time at destination and run scheduled arrival. In the literature, this difference is called early schedule penalty or late schedule penalty. Figure 11 reports an example of early/late schedule penalty for a desired departure time (DDT) from origin at 1.30pm. Considering the connected arrival time at the stop at 1.50pm, the closest runs that allow us to reach the destination are train IR310, departing at 1.30pm, and train IR312, departing at 2.30pm. Hence users who will board train IRS 10 have to bring forward their origin departure time to 1.10pm, which implies an early departure penalty of 20 min to be taken into account in the utility function as an early schedule penalty (ESP) attribute. On the other hand, if the user boards train IR312 departing at 2.30pm, a late schedule penalty of 40min as a late schedule penalty (LSP) attribute has to be considered. The weights of early and late schedule penalties and hence the probabilities of choosing trains mainly depend on trip purpose, as will be shown in chapter 5.

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Figure 11. Example of early/late schedule penalty 3.2.1 Frequency-Based Path Choice Models for Low-Frequency Services In the case of low-frequency services, the main problem of the frequency-based approach is that it does not allow to consider explicitly the above described early/late schedule penalties connected to departure/arrival in early/delay in relation to desired departure/arrival time. Moreover, the frequency-based approach in the case of low-frequency services leads to great approximations when attributes for path alternatives that include interchanges have to be assessed. In fact the use of a line-based representation of services does not allow a precise evaluation of level of service attributes at single run level, for which average values have to be considered. It implies relevant errors in path choice probability prediction. Furthermore, in the case of low-frequency services, the frequency-based approach is modelled by using the concept of optimal strategy or hyperpath (see section 3.1.1) that, as it is based on quite different assumptions about user and vehicle arrivals at stops with respect to those assumed in the low-frequency case, is not suitable to be used.

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3.2.2 Schedule-Based Path Choice Models for Low-Frequency Services In order to take into account explicitly the early/late schedule penalties, the schedulebased approach is required. The probability p[r\ToJ of choosing run r, given the O/D pair od and desired departure time TDI (or desired arrival time r^/) can be written as

Attributes that can be considered in the systematic utility Vr are: access and egress times and costs; on-board times; transfer times; number of transfers; monetary cost; comfort; early/late schedule penalty. Most of schedule-based path choice models for low-frequency services take into account user target times (desired departure times at origins or destination arrival times at destinations), considering as path choice set alternatives the two "nearest" paths in terms of minimum early and late times with respect to the user target time (see Figure 12).

Figure 12. Example of path choice alternatives

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Some models consider maximum earliness and lateness values to define a time slice around user target time, within which the path choice set is defined, while other models consider only the late alternative with respect to the origin target time or the early alternative in the case of desired arrival time at destination. If the path choice model is deterministic the minimum disutility alternative is considered, while if it is stochastic a certain probability to all alternatives of the choice set is associated with. Deterministic and stochastic path choice models can be specified according to the hypothesis of null er or otherwise. Assuming er different from zero, different random utility models can be specified. In particular, in the case of multi-class multi-service transit systems (e.g. railways), the existing correlation between alternatives have to be taken into account, and nested-logit or probit models should be used. Moreover, considering a learning process on attributes composing the systematic utility, day-to-day dynamic schedule-based path choice models for low-frequency services can be specified to take into account the day-to-day evolution of on-board loads and level of service attributes.

4.

ASSIGNMENT MODELS FOR TRANSIT SERVICES

Assignment models allow the on-board load of transit vehicles to be obtained. The level of output results depends on the path choice modelling approach (frequencybased or schedule-based). In particular, using a schedule-based approach, in which all runs of the transit services are explicitly considered, it is possible to obtain very detailed results in terms of on-board loads and level of service attributes for each vehicle. As this approach allows us to take into account the evolution in time of supply, demand and on-board loads, these models are intrinsically within-day dynamic. More aggregate results can be obtained by using a frequency-based approach that, considering services in terms of sets of runs (lines), allows to refer only to average values relative to lines and to compute only average line flows. The use of average line on-board flows, do not allow us to take into account peaks of loads on vehicles that can occur inside the reference period. Finally, the use of a constant average rate of user arrivals at stops, when this rate changes significantly during the period of analysis, could lead to relevant errors in vehicle load calculations. Whatever the used modelling approach is, the classification of assignment models usually refers to the classical one adopted for road networks, in which different

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assignment models can be specified according to the type of behavioural path choice model (deterministic or stochastic), the type of link performance functions (flowdependent or otherwise, which lead to congested or uncongested networks), the assignment approach (network loading, user equilibrium, dynamic process, and so on), and the dynamic evolution (within-day and/or day-to-day) they take into account. Assignment models for transit networks were traditionally developed using the frequency-based approach on the basis of the optimal strategy or hyperpath concept to simulate path choice, while recently the innovative schedule-based approach, based on single space-time paths, have been considered. Hence, in the following description, when we talk about paths, we refer to an hyperpath in the frequency-based approach and to a single space-time path (which includes a run or a sequence of runs) in the schedule-based approach. Table 2 reports the classification parameters of assignment models for transit networks. Table 2. Classification of transit assignment models Path Choice Model Stochastic Transit network Assignment Deterministic approach Network uncongested SNL AON Loading [c=cost] congested

Equilibrium

DUE

SUE

[c=c(f)l

Dynamic

DDP or SDP

DDP or SDP

AON=A11 or Nothing; SNL=Stochastic Network Loading; DUE=Deterministic User Equilibrium; SUE=Stochastic User Equilibrium; DDP=Deterministic dynamic process; SDP=Stochastic dynamic process.

4.1

Frequency-Based Approach

In the framework of the frequency-based modelling approach, the AON (All or Nothing) assignment model allows us to load only the lines belonging to the least cost hyperpath, while using a probabilistic path choice model, the SNL (Stochastic Network Loading) assignment model allows to calculate the choice probability for all hyperpaths belonging to the hyperpath choice set, according to which lines belonging to hyperpaths are loaded.

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For congested networks both equilibrium and dynamic models can be used. UE (User Equilibrium) assignment approach has been traditionally used to simulate congested transit networks using, at first, the variational inequality approach, and then the fixedpoint one, for which it is possible to demonstrate existence and uniqueness of equilibrium flows and costs. Recently, SUE (Stochastic User Equilibrium) assignment models have been specified on the basis of fixed point models. Stochastic models are more coherent with the problem, as the travel time associated with most links is better modelled as a random variable. For example, in addition to the user perception errors, link travel time dispersion depends on the traffic congestion on the transit mode and on the other interacting modes, as well as waiting time at bus stops depends on service irregularity. Day-to day dynamic process assignment models, both deterministic and stochastic, can be specified by considering a learning process on attributes and an updating mechanism of user choices. 4.2

Schedule-Based Approach

The type of assignment models presented in the framework of the frequency-based approach can be specified in the schedule-based approach, too. In particular, for uncongested transit networks, both AON (All or Nothing) and SNL (Stochastic Network Loading) assignment models can be considered to calculate on-board loads, as well as DUE (Deterministic User Equilibrium) and SUE assignment models can be specified for congested transit networks. For what concern the dynamic evolution, as in the schedule-based approach the timetable is explicitly considered, the within-day dynamic is a native characteristic of the schedule-based approach, in the sense that all schedule-based assignment models are at least within-day dynamic. Moreover, considering a learning process on attributes, the day-to-day dynamic can also be considered, leading to doubly dynamic (within-day and day-to-day dynamic) assignment models. The dynamic evolution can be simulated by dynamic process models, both stochastic and deterministic. A detailed description of the schedule-based assignment models, as well as a concise state of the art will be deepened in chapter 5.

116 4.3

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In the last ten years the improvements in computer science and the increasing computation capacity have allowed the schedule-based approach to be developed and applied, sometimes replacing the commonly used frequency-based approach. The schedule-based approach was initially developed for low-frequency systems and has recently been proposed for urban transit systems in order to consider more coherent user behavioural hypotheses in relation to the characteristics of the analysed public transport system. Frequency-based models are suitable to be used when an high degree of detail is not necessary and results have to be obtained with few input data. For example in the case of the strategic planning of a new transit network, we need to specify only the line paths and stops in addition to service frequencies, and the average line loads, obtained as output, are sufficient for the aims of the project. In this case, the definition of the timetable and the output in terms of load for each run is not consistent with the level of detail of the other components of the simulation models. By contrast, in the sphere of operative planning (e.g. with the aim of supporting the definition of a new timetable or to assess the introduction of ITS systems) we need more precise and detailed results that can be obtained only through a schedule-based approach. Moreover, an example of problems that arise using the frequency-based approach, when bus arrivals are not uniform for both constant and time-varying user arrivals at stops, is reported in Figure 13. Let us consider a stop served by two lines (A and B) with the same service frequency (10 bus/h), but line B arrives just one minute after line A (see Figure 13). Even if assuming a constant user arrival rate at stop (e.g., 3 users/min), according the frequency-based approach, we have that users will board lines A and B according line frequency (that is, 50% on each line), while it is clear that all users will board line A, if congestion phenomenon does not arise, except for those arrived immediately after line A departure and before the departure of line B, leading to choice probability quite different from the 50%. This difference depends on user and vehicle arrivals at stops, requiring a modelling approach that takes into account both service timetable and time-dependent user arrivals.

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Figure 13. Example of frequency-based approach disadvantages For what concern the operative application of such models, although all possible combinations of frequency, regularity and information to users are possible, three type of public transport systems (reference systems), which define three situations of greater operative interest, can be considered: - low-frequency regular services (extraurban systems); - high-frequency services without user information at stop (urban traditional systems); - high-frequency services with user information at stop (urban innovative systems). A summary of user behavioural hypotheses for the reference transit systems are reported in Table 3. Each of these three cases may require different supply network models and path choice models. They should be simulated using a scheduled-based approach, except for the case of high frequency services without user information that, under the hypothesis of constant user arrival rate and random vehicle arrivals as a Poisson process, can be simulated without distinction through a schedule-based or frequency-based approach, as it is possible to demonstrate that the hypothesis of indifferent en-route choice is fully acceptable. Table 3. Service characteristics and user behaviour transit system low- frequency high-frequency & information high-frequency & no information

stop choice pre-trip pre-trip pre-trip

line/run choice user arrival timetable-related pre-trip not en-route (intelligent) timetable related not en-route (intelligent or indifferent) timetable related

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REFERENCE NOTES

This section reports some references for public transport networks in the sphere of supply and path choice models for both the frequency-based approach and the schedule-based one. References to assignment models will be given only for the frequency-based approach, as chapter 5 will report in details that for the schedulebased approach. An effort has been made to give credit to the proper authors quoted from the literature. Nevertheless, for well-known results credits may be unintentionally omitted or misplaced. Apologies are submitted in advance for any error of this type. 5.1

Supply Models

In the sphere of the supply modelling, the diachronic network was proposed by Nuzzolo and Russo (1993) and was used in the schedule-based path choice and assignment models for low-frequency services presented by Cascetta et al. (1996), Crisalli (1999), Nuzzolo and Russo (1994, 1998b) and Nuzzolo et al. (2000). The diachronic network, with some changes in the demand subgraph and access/egress subgraph to take into account different demand and access/egress characteristics, was also used to represent high-frequency services (Nuzzolo and Russo, 1998a; Nuzzolo etal, 1999; 2001a; 2001b). The dual graph representation is described in Anez et al. (1996) and in de La Barra (1989). It was used in the integrated land use-transport model TRANUS (Modelistica, 1993) and in the initial phase of the set up of the transit network model of the largescale model system for the Copenhagen-Ringsted railway project (Nielsen and Jovicic, 1999). The mixed line-based/database supply representation have been widely used in literature (Carraresi et al., 1996; Florian, 1998; Hickman and Wilson, 1995; Hickman and Bemstain, 1997; Nguyen et al., 1997, 2001; Nielsen et al, 2000; Tong and Richardson, 1984; Wong and Tong, 1999), as it can be obtained by slight modifications of the traditional line-based approach to represent transit services, even if for large networks some computation problems could occur. This approach was initially used by Tong and Richardson (1984) and is described in Wong and Tong (1999). They use a line-based graph associating with line segments a data set which considers the number of runs of the transit line and their relative departure times at both upstream and downstream nodes. A similar approach is used in EMME/2 release

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9, described in Florian (1998), in which to reduce memory problems a mixed linebased/database supply model is used, but the clock-dependent congruence of run vehicles on the network is assured by expanding in real-time the strictly necessary part of the network to define the time-dependent congruence of single runs on the line-based representation. The mixed line-based/database approach is also used in Nielsen et al. (2000) to reduce computation times with respect to the use of the dual graph used at a first stage of their research (Nielsen and Jovicic, 1999). 5.2

Path Choice Models

The modelling approach (frequency-based or schedule-based) is the main criterion for the analysis and classification of existing path choice models. For this reason, section 5.2.1 reports path choice models developed by using the frequency-based approach, while section 5.2.2 presents path choice models specified through the schedule-based approach. Then, as user behaviour and choice dimensions depend on user and service characteristics (mainly the service frequency), classification considers models specified for high and low frequency transit systems 5.2.1 Frequency-Based Path Choice Models In the frequency-based approach, the simulation of path choice for high-frequency transit systems has been commonly performed on the basis of the concept of optimal strategies presented by Spiess and Florian (1989). Nguyen and Pallottino (1988) characterised and studied in detail the notion of the optimal strategies by using a graph theoretical approach, and developed the concept of hyperpath. Starting from these deterministic path choice models, Cantarella and Vitetta (200la) recently proposed a stochastic path choice model, which considers perception errors of path attributes. In the framework of low-frequency services, frequency-based path choice models are not present in literature and the same models specified for high-frequency services (optimal strategies) are often used by practitioners, even if they imply relevant approximations in path choice simulation due to different user behavioural hypotheses. 5.2.2 Schedule-Based Path Choice Models In the sphere of high-frequency transit services, Table 4 reports some of the existing path choice models, classified in relation to the possibility to take into account service

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regularity, user information, and different user type, as well as the used approach in utility and dynamic specifications. Table 4. Schedule-based path choice models for high-frequency transit systems user choice service user dynamic utility type approach approach mechanism regularity info Y D Hickman and Wilson (1995) FU SIM R-I WD Y SEQ S FU R-I WD Hickman and Bernstein (1997) FU S Y-N SIM R-I WD Nuzzolo and Russo (1998a) Y FU S SEQ R-I WD-DD Nuzzolo era/. (1999) S FU N SIM R WD Wong and Tong (1999,2001) FU-OU Y S SEQ R-I WD-DD Nuzzolo ef al. (200 la) FU S SEQ R-I Y-N WD-DD Nuzzolo et al. (200 Ib) R= regular services; 1= irregular services; N=without user information; Y=with user information; D=deterministic utility path choice model; S=random utility path choice model; SIM=simultaneous choice mechanism; SEQ=sequential choice mechanism; WD=within-day dynamic; DD=day-to-day dynamic; FU=frequent users; OU=occasional users. models

Although one of the first path choice models was the one introduced by Tong and Richardson in 1984 (described in Wong and Tong, 1999), the schedule-based approach in path choice and assignment was initially developed for low-frequency transit systems and only recently it has been proposed for urban transit systems. Hickman and Wilson (1995) specified deterministic path choice models, while stochastic ones were presented by Hickman and Bernstein (1997), Nuzzolo and Russo (1998a), Wong and Tong (1999, 2001), Nuzzolo et al. (2001a). Stochastic path choice models differ among them for the different random utility models (logit, nested-logit, probit) they use, and if the path choice set is considered explicitly or implicitly. Considering the dynamic feature of existing models, Hickman and Wilson (1995), Hickman and Bernstein (1997), Nuzzolo and Russo (1998a), Wong and Tong (1999, 2001) presented within-day dynamic path choice models that allow time variation in supplied services to be considered. The Nuzzolo and Russo (1998a) path choice models were extended in a doubly dynamic stochastic path choice model (Nuzzolo et al., 1999; 200la; 200Ib) that explicitly considers the within-day and day-to-day variations of services and user learning on attributes. For the simulation of low-frequency services, many schedule-based path choice models have been presented in literature (Nuzzolo and Russo, 1994; Carraresi et al., 1996; Cascetta et al., 1996; Nguyen et al., 1997; Florian, 1998; Nielsen and Jovicic, 1999; Nielsen et al., 2000; Nuzzolo et al., 2000); they differs among them for the way in which the path choice set and the path choice probabilities are defined.

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Most schedule-based path choice models for low-frequency services take into account DDT (desired departure times at origins) and DAT (desired arrival times at destinations) user target times, considering as path choice set alternatives the path of minimum disutility departing/arriving before and after user desired departure/arrival time (Nuzzolo and Russo, 1994; Cascetta et a/., 1996; Nielsen and Jovicic, 1999; Nuzzolo et al., 2000). Florian (1998) considers maximum earliness and lateness values to define a time slice around user target time, within which the path choice set is defined, while Carraresi et al. (1996) and Nguyen et al. (1997, 2001) consider only the late alternatives of minimum disutility with respect to the origin desired departure time or the early alternatives of minimum disutility in the case of desired arrival time at destination. Given the choice set, Florian (1998), Carraresi et al. (1996), and Nguyen et al. (1997, 2001) select the least generalised cost path in a deterministic way. Nuzzolo and Russo (1994), Cascetta et al. (1996), Crisalli (1999), Nguyen et al. (1997), Nielsen and Jovicic (1999), Nielsen (2000), Nielsen et al. (2000), Nuzzolo et al. (2000) use stochastic path choice models, both logit and nested-logit or probit, according to the hypothesis on the distribution of perceived utility random term. In particular, in the case of multi-class multi-service transit systems (e.g., railways), the existing correlations between alternatives have to be taken into account, and nestedlogit or probit models should be used (Nuzzolo and Russo, 1994; Nuzzolo et al, 2000; Nielsen, 2000). Table 5 reports a classification of some existing models for low-frequency transit services in relation to the random utility approach in path choice (deterministic or stochastic), and to the service type (monoservice or multiservice). Table 5. Schedule-based path choice models for low-frequency transit systems service utility type approach MONO Nuzzolo and Russo (1994) S MONO Carraresi et al. (1996) D MULTI Cascetta et al. (1996) S Nguyen et al. (1997,2001) MONO D MONO Florian (1998) D Nielsen and Jovicic (1999) MONO S Nuzzolo et al. (2000) MULTI S MONO Nielsen et al. (2000) S D=deterministic utility path choice model; S=random utility path choice model; MONO=monoservice; MULTI=multiservice. models

122 5.3

Advanced Modeling for Transit Operations and Service Planning Assignment Models

The above described path choice models are the core of the assignment models presented in literature for public transport networks. As schedule-based assignment models will be deepened in the next chapter, the following reference notes will concern the frequency-based approach only. In the framework of the frequency-based approach for high-frequency services, deterministic network loading models have been presented on the basis of the optimal strategy or hyperpath concept (Spiess and Florian, 1989; Nguyen and Pallottino, 1988), while deterministic and stochastic equilibrium models have been presented in order to take into account congestion and perception errors of path attributes (Nguyen and Pallottino, 1988; De Cea and Fernandez, 1993; Wu and Florian, 1993; Wu et al., 1994; Lam et al., 1999; Cantarella and Vitetta, 2001 a). Dynamic process assignment models have been formalised by Cantarella and Vitetta (200 Ib) in order to take into account the day-to-day variation of on-board flows. Further references on the frequency-based approach can be deepened in BouzaieneAyarietal. (1998).

REFERENCES Anez, J., T. de la Barra, and B. Perez (1996). Dual graph representation of transport networks. Transportation Research, 30B, 209-216. Bouzaiene-Ayari, B., M. Gendreau, and S. Nguyen (1998). Passenger assignment in congested transit networks: a historical perspective. In: Equilibrium and advanced transportation modelling (P. M. Marcotte and S. Nguyen, eds.), pp. 47-71. Kluwer Academic Publishers. Cantarella, G. E. and A. Vitetta (2001 a). Stochastic assignment to high frequency transit networks: models, algorithms, and applications with different perceived cost distributions. In: Mathematical methods on optimization in transportation systems (M. Pursula and J. Niittymaki, eds.), pp. 109-129. Kluwer Academic Publishers. Cantarella, G. E. and A. Vitetta (2001b). Frequency-based transit assignment models: stochastic equilibrium and day-to-day dynamic process, Teaching material of the Advanced course on transit networks, Dept Civil Engineering, "Tor Vergata" University of Rome, Italy.

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Carraresi, P., F. Maluccelli, and S. Pallottino (1996). Regional Mass Transit Assignment with resource constraints. Transportation Research, 30B, 81-98. Cascetta, E., L. Biggiero, A. Nuzzolo. and F. Russo (1996). A system of within-day dynamic demand and assignment models for scheduled intercity services. Proceedings of 24th European Transportation Forum Seminar D-E part 2, London. Crisalli, U. (1999). User's behaviour simulation of intercity rail service- choices. Simulation Practice & Theory, 17, 233-249. De Cea, J. and E. Fernandez (1993). Transit assignment for congested public transport system: An equilibrium model. Transportation Science, 27(2), 133-147. de La Barra, T. (1989). Integrated land use and transport modelling. Cambridge University Press. Florian, M. (1998). Deterministic time table transit assignment. Preprints of PTRC seminar on National models, Stockholm. Hickman, M. D. and D. H. Bernstein (1997). Transit service and path choice models in stochastic and time-dependent networks. Transportation Science, 31, 129146. Hickman, M. D. and N. H. M. Wilson (1995). Passenger travel time and path choice implications of real-time transit information. Transportation Research, 3C, 211-226. Lam, W. H. K., Z. Y. Gao, K. S. Chan, & H. Yang (1999). A stochastic user equilibrium model for congested transit networks. Transportation Research, 33B, 351-368. Modelistica (1993). TRANUS Technical description. Caracas, Venezuela. Nguyen, S. and S. Pallottino (1988). Equilibration traffic assignment for large scale transit networks. European Journal of Operational Research, 37, 176-186. Nguyen, S., S. Pallottino and M. Gendreau (1997). Implicit enumeration of hyperpaths in logit models for transit networks. Transportation Science, forthcoming. Nguyen, S., S. Pallottino and F. Malucelli (2001). A modeling framework for the passenger assignment on a transport network with time-tables. Transportation Science, 35, 238-249. Nielsen, O. A. (2000). A stochastic transit assignment model considering differences in passengers utility functions, Transportation Research, 34B, 377-402. Nielsen, O. A. and G. Jovicic (1999). A large scale stochastic timetable-based transit assignment model for route and sub-mode choices. Proceedings of 27th European Transportation Forum, Seminar F, 169-184, Cambridge, England.

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Nielsen, O. A., C. O. Hansen and A. Daly (2000). A large-scale model system for the Copenhagen-Ringsted railway project. Proceedings of 9th IATBR Conference, Gold Coast, Australia. Nuzzolo, A. and F. Russo (1993). Un modello di rete diacronica per 1'assegnazione dinamica al trasporto collettivo extraurbano, Ricerca Operativa, 67, 37-56. Nuzzolo, A. and F. Russo (1994). An equilibrium assignment model for intercity transit networks, Proceedings of TRISTAN II Conference, Capri, Italy, 1994. Nuzzolo, A. and F. Russo (1998a). A dynamic network loading model for transit services, Proceedings of TRISTAN III Conference, San Juan, Puerto Rico. Nuzzolo, A. and F. Russo (1998b). Departure time and path choice models for intercity transit assignment. In: Travel Behaviour Research: updating the state of play (J. Ortuzar, D. Hensher and S. Jara-Diaz, eds.), pp. 385-399. Pergamon Elsevier Science Ltd, Amsterdam, The Netherlands. Nuzzolo, A., U. Crisalli, and F. Gangemi, (2000). A behavioural choice model for the evaluation of railway supply and pricing policies. Transportation Research, 35A, 211-226. Nuzzolo A., F. Russo and U. Crisalli (1999). A doubly dynamic assignment model for congested urban transit networks. Proceedings of 27th European Transportation Forum, Seminar F, 185-196, Cambridge, England. Nuzzolo, A., F. Russo and U. Crisalli (2001a). Doubly dynamic path choice models for urban transit systems. In: Travel behaviour research. The leading edge (D. Hensher, ed.), pp. 797-812. Pergamon. Nuzzolo, A., F. Russo and U. Crisalli (2001b). A doubly dynamic schedule-based assignment model for transit networks. Transportation Science, 35, 268-285. Spiess, H. and M. Florian (1989). Optimal strategies: a new assignment model for transit networks. Transportation Research, 23B, 83-102. Tong, C. O. and A. J. Richardson (1984). Estimation of time-dependent origindestination matrices for transit networks. Journal of Advanced Transportation, 18, 145-161. Wong, S. C. and C. O. Tong (1999). A stochastic transit assignment model using a dynamic schedule-based network. Transportation Research, 33B, 107-121. Wong, S. C., and C. O. Tong (2001). Planning an urban rail transit system using a schedule-based network model. Proceedings of 9th WCTR Conference, Seoul, Korea. Wu, J. H. and M. Florian (1993). A simplicial decomposition method for the transit equilibrium assignment problem. Annals of Operations Research, 44, 245-260. Wu, J. H., M. Florian and P. Marcotte (1994). Transit equilibrium assignment: a model and solution algorithms. Transportation Science, 28, 193-203.

CHAPTER 5

SCHEDULE-BASED TRANSIT ASSIGNMENT MODELS0 Agostino Nuzzolo, "Tor Vergata" University of Rome, Italy, EU

1.

INTRODUCTION

In the last ten years improvements in computer science and increasing computation capacity have allowed the schedule-based approach to be developed and applied. The schedule-based approach was initially developed for low-frequency systems and has recently been proposed for urban transit systems. It considers the coherent user behavioural hypotheses in relation to user and service characteristics with respect to the well-known frequency-based approach which is based on the concept of optimal strategy or hyperpath (see chapter 4). As described in chapter 4, the schedule-based approach requires an explicit treatment for - the time segmentation of the origin/destination matrix, as user departure or arrival time distribution, both within-day and day-to-day, has to be considered; - supply modelling, as space-time characteristics of services during the considered periods have to be taken into account; Umber to Crisalli is co-author of sections 2, 3, 4, 5.1, 5.2, 5.3, 5.5; Pierluigi Coppola is coauthor of sections 5.4, 5.6

126

Advanced Modeling for Transit Operations and Service Planning - path choice model, with explicit within-day and day-to-day timedependencies.

This chapter describes some characteristics and the state of the art of the schedulebased approach in transit assignment modelling, and some application examples to realistically-sized networks.

2.

USER TARGET SEGMENTATION

TIME

AND

DEMAND

TEMPORAL

As previously mentioned, the times in which users desire to start or end their trips plays a key role in the schedule-based approach. These times are usually called user target times (TT) and can be classified into desired departure times (DDT), which represent the times in which users would depart from origin, and their desired arrival times (DAT), which represent the times which users would arrive at the destination. User target times TT can also be related to trip purpose: for exan p'e, a DAT can be considered for home-departure trips for commuter, business and st> idy trip purposes, while a DDT is generally used for home-destined trips. On account of the high frequency of urban transit services, only DDT, which are assumed to coincide with the user departure time TDI, are considered. For low frequency services, which are typical of extra-urban areas, either DDT or DAT, which are usually quite different from departure/arrival times of services, are used. This temporal characterisation of trips implies a temporal segmentation of the demand, in addition to the space features in terms of origin-destination zones. The reference period t is usually discretised in n elementary time intervals (e.g. of one minute each for high-frequency systems or ten minutes each for low-frequency ones), and given the generic time interval i. It is assumed that all user target times TT (DDT and/or DAT) in this time interval are aggregated into a single point TTTI (?Di and/or TAJ), e.g. in the middle point of the time interval /. An example of demand segmentation for both urban and extra-urban trips is reported in Figure 1. In the framework of the schedule-based approach, the O/D matrix should consider user target times in addition to the traditional space characterisation in origin and destination zones. For this reason, considering the reference period divided in time

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slices, an O/D matrix for each user target time is defined. For example, Figure 2 reports time-dependent O/D matrices in relation to origin departure times for each time slice of the reference period. In order to obtain time-dependent O/D matrices when they are not available, some procedures to estimate or update time-dependent transit O/D matrices from timevarying on-board counts can be used. The dynamic estimation problem can also be specified, thus extending the relationships between flows, counts and demand, which are formalised in the static approach in order to explicitly consider the time dependencies. In particular, it is necessary to describe the relationship between time-varying link counts and demand referring to the time interval /', in which demand relative to the O/D pair (od) leaves the origin and the time interval j, in which link counts^ are measured.

Figure 1. Example of origin departure times

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Figure 2. Example of time-varying origin/destination matrix The flow on link / during the time slice j (fij) can be written as 0)

;=/ k

and considering all link counts in a matrix form, we have

f (d—T D1 ,.....,d—T

—

T

T

,d—T Dn) = Z—l—l.J Q T

/

Pd

=

(2) ^ '

i=J

where hj is the path flow vector, whose generic element hitk represents flow on path k relative to the O/D pair od and departing in the time interval /; Pj is the path choice matrix relative to time interval /', whose generic element p[k/od, ToJ represents the fraction of d°

>TD

' using path k; Qij is the crossing fraction matrix, whose generic

element q^1. represents the fraction of d° 'TD' using path k and contributing to flow on link / in time interval j; Mtj is the assignment matrix, whose generic element m° ' TD/ represents the fraction of demand d° 'TD' contributing to flow on link / during time interval/ The link flow vector^ can be calculated through a dynamic assignment model and is usually different from the vector of link counts relative to time interval j, f . , whose generic element // • represents the on-board count of vehicles represented by link / during time interval j. The estimation of the O/D trip demand vector d_ = (d_T

,d_T

, ...... ,d_T

) can be

carried out by "efficiently" combining traffic counts with all other available information. Estimators can be classified under classic estimators (like the Maximum

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Likelihood or the Generalised Least Squares), if they combine experimental information (sample surveys) with traffic counts, and Bayesian estimators, if they combine non-experimental information ("a priori" information) with traffic counts. As formalised in the traditional static approach, the estimation of time-varying O/D matrices from time-varying on-board counts can be written as

x,

xn>0

(3)

~+z2[(f],...,fn);(f],...,fn)J} Equation (3) represents an optimisation problem made of the minimisation of the "distance" between the unknown time-varying demand x = (xj,...,xn) and the a priori time-varying demand dL = (d_T

,..-,d_r

), represented by the function z/f.J,

and between time-varying link flows f = (f,(x),...,f

(x)) , obtained using a

dynamic assignment model, and the observed link flows (time-varying on-board counts) f = (f.,..., f

), represented by the function z^/7- Functions zi[.J and z^/7

can be differently specified according to the chosen estimator. In the sphere of the dynamic O/D estimation, a further classification in simultaneous and sequential estimators can be done. Simultaneous estimators allow the joint estimation of O/D matrices, one for each time slice, to be obtained as reported in Equation (3). Sequential estimators provide a sequence of estimated O/D matrices, in which the estimation of the O/D matrix for the time slice j depends on the estimation of the O/D matrices relative to the previous time slices j-l,j-2,... ,j-n. Hence, the estimation of the demand vector — d.TDj , given the estimates V(d^ ,dr D2 , to _ T D I —T

,d ) —rTpj-1 '

relative to previous time intervals, can be written as

Xj;d

J +z2[f(Xj/drDl,...d

);f

1}

(4)

*^

Whatever the formulation and used estimators are, the core of the problem is the computation of the link flow vector fa of Equation (2) through the estimation of the assignment matrix MJJ, that, for transit networks, can be carried out using a schedulebased approach.

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3. CLASSIFICATION OF SCHEDULE-BASED TRANSIT ASSIGNMENT MODELS The classification of schedule-based transit assignment models follows that introduced in chapter 4, in which different assignment models can be specified according to the type of behavioural path choice model (deterministic or stochastic), the type of link performance functions (flow-dependent or otherwise, which lead to congested or uncongested networks), the assignment approach (network loading, user equilibrium, dynamic process), and the dynamic evolution (within-day and/or day-today) they take into account. Note that, as in the schedule-based approach, the timetable is explicitly considered. The within-day dynamic is a native characteristic of the schedule-based approach in the sense that all schedule-based assignment models is at least under the within-day dynamic. Moreover, considering a learning process on attributes, the day-to-day dynamic can also be taken into account, leading to doubly dynamic (within-day and day-to-day dynamic) assignment models. In the case of transit, a further important element of classification concerns the system function, for which regular or irregular services can be taken into account (as a different run arrival sequence, with respect to the scheduled one, can produce quite different on-board loads). Service irregularity can be taken into account implicitly or explicitly. Implicit simulation of service irregularity is made by including the path choice model values of attributes calculated on the scheduled configuration of services. In addition, a further component to the random term is added in the perceived utility function of path alternatives in order to take into account service irregularity. Explicit simulation of service irregularity can be carried out by calculating values of attributes in path choice models on the basis of service configurations, generated starting from the scheduled timetable, and taking into account changes induced by service irregularity, in terms of arrival/departure times of run vehicles at stops. Table 1 reports the classification features of assignment models based on the abovedescribed criteria. The reader should consider the adjectives "deterministic" and stochastic reported in Table 1 as a peculiarity of the dynamic process features that, apart from the path choice model, can be either deterministic or stochastic.

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Table 1. Classification of schedule-based transit assignment models Transit system functioning regular services

Transit Network uncongested [c=cost] Equilibrium approach congested Dynamic [c=c(f)] Process approach

irregular services implicit explicit simulation simulation

Dynamic Network Loading

Stochastic Supply Dynamic Network Loading User Equilibrium

Deterministic or Stochastic Dynamic Process (Deterministic Supply)

Deterministic or Stochastic Dynamic Process (Stochastic Supply)

In the case of regular services, or irregular services with an implicit treatment of irregularity, a DNL (Dynamic Network Loading) approach can be specified for uncongested transit networks; it can use a deterministic or a stochastic path choice model. For congested networks, if the day-to-day evolution is neglected, an UE (User Equilibrium) approach can be considered. DUE (Deterministic User Equilibrium) or SUE (Stochastic User Equilibrium) models can be specified according to the use of deterministic or stochastic path choice models. If a day-to-day dynamic evolution of the system has to be taken into account, a DP (Dynamic Process) assignment model, that can be Deterministic or Stochastic in the Demand component, is required. Different assignment models can be specified in the case of transit networks in which service irregularity is explicitly dealt with. For uncongested networks, a SSDNL (Supply Stochastic Dynamic Network Loading) can be used by considering average run loads calculated through successive averages of run loads carried out by different service configurations. In the case of congested networks, service irregularity can be explicitly treated using a SSDP (Stochastic Supply Dynamic Process) model.

4.

REFERENCE NOTES

In the sphere of schedule-based assignment models for high-frequency services for uncongested networks, Hickman and Wilson (1995) used a Deterministic Network Loading model to assess results of their path choice models, while Stochastic Network Loading models were used by Crisalli (1998), Nuzzolo and Russo (1998a), Nuzzolo et al. (1999) and by Hickman and Bernstein (1997). Congested networks have been studied using both the equilibrium and the dynamic process approach. Stochastic user equilibrium assignment models have been presented by Crisalli

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(1999a), Nielsen (2000) as well as Wong and long (1999, 2001): they used an MSA (Method of Successive Averages) algorithm based on logit, hybrid logit and estimation by numerical simulation path choice models. Dynamic Process assignment models, Supply Deterministic and Stochastic, have been presented by Nuzzolo et al. (1999, 200la). In the case of low-frequency schedule-based assignment models, deterministic network loading models have been presented by Nuzzolo and Russo (1993), Carraresi et al. (1996), Florian (1998). Cascetta et al. (1996), Crisalli (1999b), Nguyen et al. (1997), Nuzzolo and Russo (1994, 1996, 1998b), Nuzzolo et al, (2000) proposed stochastic network loading models. Congested networks have been studied by Nuzzolo and Russo (1993) through a DUE assignment model, while Nielsen et al. (2000) extended to the stochastic user equilibrium assignment model proposed by Nielsen (2000) the extra-urban case. Moreover, considering a learning process on attributes, day-to-day dynamic schedulebased path choice models for low-frequency services could be specified to take into account the day-to-day evolution of level of service attributes, even if, to the author's knowledge, models of this type have not been presented in literature. The theoretical framework for the dynamic estimation of O/D matrices from traffic counts was defined in Cascetta et al. (1993) for road networks. In the framework of the schedule-based approach, only a few have been presented for transit networks: Wong and Tong (1998) presented an estimation method for urban transit networks on the basis of a maximum entropy estimator, while Nuzzolo and Crisalli (2001) used a generalized least squares estimator to solve the estimation problem for uncongested low-frequency transit services.

5.

APPLICATION EXAMPLES

As reported previously, the schedule-based dynamic approach requires a more detailed description of services and demand; it is more time-consuming with respect to the frequency-based approach, but obtains more accurate results in terms of vehicle loads and level of service attributes. Applications to real cases show the advantages of this class of path choice and assignment models and highlight the importance of software tools, like the Decision Support Systems, for the operations planning of transit networks.

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133

In the following some application examples of schedule-based assignment models will be presented. In particular, section 5.1 reports an application to regional transit services; section 5.2 describes the application to a national railway network; section 5.3 shows some applications to urban transit systems; section 5.4 reports an application in the field of short-term forecasting of on-board loads and run arrival times at stops for urban transit services; sections 5.5 and 5.6 describe a possible use of schedule-based assignment models to estimate transit O/D matrices and to support the timetable setting. 5.1

Regional transit services

In the following, some application examples of a system of schedule-based assignment models (at the regional district level) will be described. These models were implemented in DY-RT, a software tool that simulates regional transit services by using a schedule-based dynamic approach on the basis of the system of models proposed by Nuzzolo et al. (2000). The DY-RT mode-run choice model considers car, park & ride (car + train), bus and train mode alternatives. For transit services, it takes into account two run alternatives: the ones of minimum generalised cost (including early and late schedule penalties) departing before and after the user target time. The run choice is fully assumed pretrip, and the structure of the mode-run choice model of DY-RT is reported in Figure 3.

Early Train (bus+train)

Early Park-Ride

Late Bus

Figure 3. The structure of the mode-run choice model In the framework of the random utility theory, the probability pff/TrrJ of choosing mode-run alternative j, given the od pair and target time TTT, can be written as = prob(Vj/~TrrJ+£j>

Vj

for each alternative y '^j belonging to the mode-run choice set.

(5)

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Attributes composing the systematic utility F/ for each alternative are those reported in Table 2. Table 2. Path choice model - alternatives and attributes alternatives

attributes

description

CAR

EB

ET

EPR

LB

LT

LPR

car

early bus

early train

early park&ride

late bus

late train

late park&ride

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X X

X

X

X

X

X

TB

transit travel time

TC

car travel time

X

C

travel cost

X

EP

early schedule penalty

LP

late schedule penalty

A/E

access/egress time

HI

high income user

PR

park&ride dummy

CAR

car dummy

EBUS

early bus dummy

X

X

X

X X

X

X X

Figure 4 describes the functional architecture of DY-RT. On-board loads for each run of transit services, and relative revenues, are computed as the result of a sequence of phases consisting of: -

definition of operating characteristics and timetable of transit services; construction of the supply model, both road and transit; calculation of path choice sets and level of service attributes; estimation of time-dependent O/D matrices for each mode (car, bus, train); calculation of path choice probabilities for each od pair and target time TTT', assignment of O/D matrices for each target time TTT to the diachronic

-

network /2 (within-day dynamic network loading procedure) according to path choice probabilities computed in the previous step; determination of on-board run loads and revenues.

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Figure 4. DY-RT functional architecture Given a design scenario, which includes a supply configuration (routes and timetable of transit services) and a space-time O/D demand, DY-RT can compute the on-board loads for each run of each transit mode (bus and train) as the result of two types of within-day dynamic assignment procedures: - run choice simulation, in which the demand of each mode (a priori estimated) is assigned to the network to calculate on-board loads and revenues for each run of bus and rail services; - mode-run choice simulation, in which the total demand of the scenario is given. At first, the mode choice is simulated to estimate the O/D matrices for each mode in relation to level of service attributes of the design scenario. Subsequently, on-board loads and revenues for each run are calculated in relation to the estimated demand. DY-RT has been widely used to support operations planning of many regional transit networks in Italy (e.g., Lazio and Veneto districts). In particular, this software allows us to consider effects of rail-bus integration for regional transit trips and to assess the effects of network rationalisations, in order to reduce costs and improve service quality. In the following, two application cases will be briefly described. The definition of minimum regional bus services in Lazio district Lazio study area was divided into 395 traffic zones at municipality level, except for the metropolitan area of Rome, which was divided into 9 different zones. The regional bus services were made of 8892 runs per day, 2769 of which were relative to the morning peak reference period (6.00-10.00am). The railway services were made of 1764 train runs (405 of which were scheduled from 6.00am to 10.00am).

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Considering the relevant number of bus and railway runs, diachronic supply networks have great dimensions: for example the bus diachronic graph is made of about 230000 nodes and 560000 links. The demand relative to the reference period (6.00-10.00am) was divided into 8 time slices of 30 minutes to take into account different user target times. Given a possible configuration of the regional transit network (see Figure 5), the within-day dynamic assignment allows us to obtain on-board loads for each run of each scheduled service over each section of the network. An example is reported in Figure 6 for the six runs of line Rieti-Rome, passing over section Poggio NativoNerola.

Figure 5. Example of regional transit services (Lazio district)

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137

Figure 6. Example of run on-board loads Figure 6 also shows the advantages of the schedule-based approach, through which the variation in the reference period of on-board loads for different runs of the same line over the same section can be taken into account. On-board run loads are quite different from the average on-board line value, which can be obtained using the frequency-based approach. Moreover, the calculation of level of service attributes at the single run level allows us to support the route design. This allows the definition of the minimum set of main and feeder lines to cover the Lazio district, as well as to assess the timetable set up that minimises the interchange times between runs of main and feeder lines as well as between bus lines and trains. The integration of regional rail-way-bus services in Veneto district Aiming to design regional bus and rail services in a strictly integrated network, Veneto district was divided into 195 zones, served by 237 trains and 1776 bus. The diachronic networks, which represent these services, were made of about 25000 nodes and 55800 links for regional bus services, while railway services were represented by 3300 nodes and 10600 links, approximately. With respect to the reference scenario, the design was built after considering the railbus integration through 112 new railway runs, 10 new terminals and 28 park & ride areas. In particular, bus and rail timetables were changed to minimize the impacts of interchanges through the synchronization of bus and train arrivals and departures. The

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train timetable was designed using the "clock-headway" concept, which is characterized by a timetable of easy-to-memorize repetitions of departure time, usually in multiples of an hour or less (e.g. at 5, 10, 20, 30, 40, 45, 50 and 60 minutes). In addition to on-board loads for each scheduled run of transit services described in the previous example, DY-RT allows us to calculate the level of service attributes and the performance indexes at the single run level, as well as to assess integration policies implemented in the design scenario (e.g., see Table 3 for railway services). This result can be obtained using the schedule-based approach only. Table 3. Example of comparison between level of service attributes railway comparison (reference period 6. 30am- 10.30 am)

attributes access time (min) egress time (min) on-board time (min) transfer time (min) number of runs for O/D

5.2

design

reference

variation

10.0 8.2 27.2 10.1 8.0

11.8 10.6 30.2 13.5 4.5

-15.3% -22.6% -9.9% -25.2% 77.8%

Long-distance railway services

This section describes a system of within-day dynamic railway service choice and assignment models, which are used as the modelling basis for a large Decision Support System (DSS) for the operational planning of rail services, named SASM, that was recently developed by the Italian Railways Company. The decisional variables considered by the DSS concerns the operating characteristics (paths, stops, timetables and train composition) and price of the service. The support system was designed to help by both analysing the current situation and defining marketing strategies. SASM computes the performance of railway and competitive mode services, passenger loads and related returns of each single train for a design scenario through a sequence of phases consisting of (Figure 7): - formulation of the design scenario, with the definition of operating characteristics and prices of railway services; - construction of supply models of scenario rail/services; - calculation of alternative runs and level of service attributes for demand models;

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Schedule-Based Transit Assignment Models

calculation of probabilities of each alternative mode-service run configuration for each demand segment and for each O/D pair od, by means of the system of demand models; application of such probabilities to "current" rail O/D matrices for each demand segment and computation of new O/D matrices for each railway service (high speed, slow speed); assignment of O/D matrices to networks of the relative railway services (within-day dynamic network loading procedure); determination of the rail service passenger flows; determination of railway revenues.

Rail service passenger flows

Figure 7. Functional architecture of the Decision Support System SASM The core of the system is the path choice model, which was calibrated on the most important railway corridor in the northern part of Italy that runs from Turin to Venice through Milan. The calibration phase of the path choice model is briefly described. In order to apply the service, run and class choice model, current railway demand was subdivided into market segments given by the combination of: - 9 trip purposes: commuter home-work and vice-versa, home-business and v.v., home-school/university and v.v., home-other purposes and v.v., non home-based trips;

140

Advanced Modeling for Transit Operations and Service Planning - 15 desired arrival times at destination (DAT) for home-departure trips and/or non home-based trips and 15 origin desired departure times (DDT) for homebound trips; - 3 user categories: with reimbursed travel cost, with medium-high family gross income (> 20000 €), with a low family gross income (< 20000 €).

To obtain the distribution of users for the various categories starting from current demand data, on the basis of analyses conducted on the user sample of the on-board train survey, the following assumptions were made: - distribution by purpose depends on the service and class currently used and on the type of O/D relation; six relation types were considered combining three zone types (Metropolitan Area, Provincial Town and Other) with two O/D distance band (less than or greater than 100 km); - distribution by target time depends on the purpose and O/D distance band (<100km, 100-200 km, >200 km); - distribution by user category depends on service, class and trip purpose. An on-board train survey was conducted in order to estimate the rail O/D matrix, to calibrate the path choice model and to obtain the market segment distribution. The survey consisted of: - counts of passengers (approx. 50000) boarding high speed (intercity) or slow speed (regional) trains on a workday, with interviews on origin and destination; - survey of a sample of about 10000 users, with questions about used terminals, access and egress mode, trip purpose, target time and socioeconomic characteristics. In order to compute the level of service attributes, a supply model was implemented. Italy was divided into 484 traffic zones (with about 100000 inhabitants per zone). The road infrastructure network model, which also includes the zone centroids and passenger service terminals, has 4969 nodes and 16414 links. The railway infrastructure network model consists of 448 nodes, 441 of which are railway terminals, and 1086 links. The best specification obtained for the service/run/class choice model is that reported in Figure 8, and the probability p[cl/ser,r,m] of choosing the elementary path alternative identified by railway service type ser, run (or sequence of runs) r, class cl, given the O/D pair od and the market segment m (which defines the user target time TTT) can be written as:

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141

where:

in which symbols assume the meaning reported in Table 4. The same table reports parameter estimations and test statistics for commuter work purpose. All estimated parameters have correct signs and most of them are statistically significant. The scale parameters Sser and 8r assumes values quite different from one, proving the hypothesised correlation among alternatives.

Figure 8. SASM choice model structure

VOTs (Values Of Time) have the expected values in relation to trip purpose; they also respect the difference between the two income classes. Indeed, VOTs relative to the medium-high income class are greater than those for the low-income class. The significant difference between the two classes could be explained by the fact that the average income of low class railway users, especially for commuter work purpose, is less than the average income of other mode users. As we can see in Table 4, the early (late) schedule delay parameters have considerable weight. The expected symmetry between early and late penalties for outgoing and returning trips is clear: we can observe a greater disutility, for outgoing trips, associated with a late arrival schedule delay at destination rather than with an early one. It is related to trip purpose (commuter work) and these results are also confirmed in other studies. The opposite phenomenon can be observed for returning trips, where

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a late schedule delay, with respect to the desired departure time, is perceived less than an early one. The scheduled delays are considered in relation to trip distance, in order to take into account their greater influence for shorter trips than for longer ones. Table 4. Coefficients and VOTs Attribute

symbol

units

P

t-statistics

TV

hours

-3.752

-3.92

VOT (€)

h. 7.20 Total travel time

1. 2.82 Total travel cost for mediumhigh (h) income class

CH

€

-0.521

-8.49

Total travel cost for low (1) income class

CL

€

-2.552

-24.2

-5.816

-7.78

Early arrival schedule delay with respect to DAT, divided by the distance (outgoing trips)

ESPG

Late arrival schedule delay with respect to DAT, divided by the distance (outgoing trips)

LSPG

Early departure schedule delay with respect to DDT, divided by the distance (returning trips)

ESPR

Late departure schedule delay with respect to DDT, divided by the distance (returning trips)

LSPR

hours/ 100km

hours/ 100km

hours/ 100km hours/ 100km

h. 11.17 1. 4.38 h. 16.32

-8.5

-8.15 1. 6.39 h. 20.20

-10.52

-11.53 1.7.91

h. 13.75 -7.161

-10.52 1.5.39

Access (egress) logsum for outgoing (returning) trips

AC

1.481

3.11

Egress (access) logsum for outgoing (returning) trips

EG

0.22

0.93

High-speed train dummy

1C

0/1

-0.225

-0.74

Second class of high-speed train dummy

IC2

0/1

0.341

1.83

Run scale parameter

5r

0.79

2.65

Service scale parameter

5ser

0.81

2.19

2

Rho squared

P

0.575

Rho squared correct

p corr

0.569

Number of observations

N. obs.

1275

2

The dummy variables are relative to the efficient combination (in terms of model response) in order to avoid correlations in the calibration phase. A negative value for high-speed train dummy (1C) and a positive value for second class of high-speed train

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143

dummy (/C2) showed user preference for the cheaper service (due to the daily trip feature for commuter work purpose). Moreover, a limited significance of the egress variable coefficient can be observed. This is probably due to the calibration process which was based only on railway choices, where in many cases, there are no real alternatives for the egress terminal, thus reducing the explanatory power of this variable. Obviously, logsum values of access/egress mode choice could play a different role in a model that also considers other modes (car, bus and aeroplane). The model was validated using the goodness of fit p statistic showed in Table 4 and reproducing the sample distribution among services, runs and classes, through the computation of RMSE% values. These results show a good capacity to describe the sample choices, as reported in Table 5. Table 5. RMSE% values Choice

RMSE%

service

4.8%

service/class

8.7%

service/run/class

16.6%

A more sophisticated model structure which includes other choice dimensions, implicitly considered in the structure of Figure 8, could be taken into account. For example, the access/egress mode choice could be explicitly considered, but the introduction of a further level in the nested-logit structure might induce a very large number of theoretical possibilities. An example of access-egress choice models for railway services can be found in Crisalli and Gangemi (1997). Some test applications of the DSS were carried out in order to verify its capability to reproduce the actual state of the railway services. The current distribution of train loads was reproduced (an example is reported in Figure 9 that shows the load of train IC640 from Turin to Venice), comparing the total number of users boarding each train by the model and the number of users counted in the corridor surveys. The comparison between counted and simulated passenger is shown in Figure 10 and appears quite satisfactory for operational uses.

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Figure 9. Passenger flow on train 1C 640 from Venice to Turin 800

700600

500 --

1400 S

300200 100

0

100

200

300

400

500

600

700

800

counts

Figure 10. Comparison between train counts and model results 5.3

Urban transit services

This section presents assignment results carried out considering the system evolution both during a single day period (within-day variation) and in following periods (day-to-day variation) on the basis of the doubly dynamic schedule-based assignment models described in Nuzzolo et al (200Ib). These models were implemented in a software tool (DY-BUS) that can be used to simulate urban transit services Given a design scenario, which includes a supply configuration (paths and timetable of transit services) and a transit space-time demand, DY-BUS can compute the on-board loads for each run of the transit services as the result of different types of assignment procedure:

Schedule-Based Transit Assignment Models •

•

•

1 45

Within-day dynamic assignment, which simulates the evolution of on-board loads within a single-day period on the basis of the supply configuration of a given day t; the assignment model is a Stochastic Network Loading (SNL) model, since it uses a stochastic path choice model. Stochastic user equilibrium assignment, which allows the calculation of onboard loads for each run in the case of congested networks with the regular service functioning. Dynamic process assignment that simulates the evolution of on-board loads on different days in which user choice is influenced by the learning process on congestion-related attributes and by the different supply at their disposal due to service irregularity, leading to different level of service attributes from one day to another. Service irregularity is explicitly considered through a sequence of supply configurations (one per day), obtained by using Monte Carlo simulation techniques, starting from the scheduled timetable, as described in Nuzzolo et al. (200 Ib).

In order to simulate the day-to-day dynamic evolution, a learning process on attributes experimented and forecast on previous days is considered. It is specified using an exponential filter in which the attribute value X{° forecast on day t is expressed by a convex combination of the attribute forecast on day t-1, X{°} , and of the value realised on the same day t-1, X**j , that is:

(8) where ye [0,1] is the weight given by users to attributes realised on day t-1. In order to verify the treatment of realistically-sized urban transit networks to support operations planning, results of applications of the above described tool are here reported. The test network was developed on the topological basis of the transit network of the town of Salerno, situated in southern Italy. The service supply was changed to simulate different functional conditions. The study area was divided into 62 traffic zones characterised by a transit demand of about 32000 users, which was assumed constant during the simulation period (morning rush hour: 7:30-8:30), which was, in turn, subdivided into 60 time slices of one minute each. On the basis of the 58 lines of the transit system a service timetable with 523 runs was defined. Such a supply allowed us to hypothesise a high frequency service for all O/D pairs and random

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arrival of users at stops. The diachronic network used to represent the service supply consists of 44846 nodes and 81654 links. At the moment DY-BUS uses a logit specification, even if some problems related to the IIA assumption could occur. Given an od pair and an origin departure time TDI, path choice probabilities were calculated considering a sequential choice mechanism described in chapter 4, in which the specification of the systematic utilities of the arriving run r and of the next ones r' are: (9)

(10)

where TWr- is the waiting time (equal to the difference between the arrival time of run r' and the arrival time of run r, supplied by the information system), TBr and TBr- are on-board times, TCr and TCr- are transfer times, NTr and NTr' are the number of transfers; CFBr and CFBr' are the "route" on-board comfort (function of on-board crowding degree in the following links, up to egress stop), CFWr and CFWr' are the on-board comfort at stops when users have not yet boarded (function of run crowding degree at stop), TPr is the time already spent at stop (equal to the difference between arrival time of run r and the user arrival time Tots at stop). The choice of stop s is calculated considering a logit model in which the systematic utility is given by: V,=PTATAS+PHHS

(11)

where TAS is the access time of stop s from the considered origin o, Hs is the inclusive utility of runs at stop s. A value 7=0.05 for the day-to-day learning process of Equation (8) is considered, while attribute parameters of Equations (9, 10, 11) are those reported in Nuzzolo et al. (200 Ib). In the following the results of different assignment models will be presented. Within-day dynamic assignment The use of a schedule-based stochastic path choice model in a within-day dynamic network loading procedure allows us to perform an SNL (Stochastic Network Loading) assignment, which computes on-board loads for each run in the reference period.

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147

Figure 11. Example of within-day dynamic assignment results As an example, Figure 11 reports on-board loads for runs of line 20 over section 115127, where several bus lines are in competition for the same od pairs. Single run loads differ from one another and from the average line value for many reasons, such as different bus headways, presence of competitive runs of other lines and so on. Stochastic user equilibrium assignment In order to take into account congestion, it is possible to explore assignment results using both equilibrium and dynamic process approaches. In the framework of regular service functioning, the day-to-day utility variations are exclusively due to comfort attribute dependence on on-board loads, and an equilibrium approach can be used. Using a SUE (stochastic user equilibrium) assignment model, Figure 12 reports onboard loads for all runs of line 20 passing over section 169-173. The results are based on the application of an MSA (Method of Successive Averages) algorithm on costs. The capacity, defined the same for all vehicles, is equal to 100 users.

runs Figure 12. Example of equilibrium assignment results

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Figure 12 shows the capability of the model to smooth (when it is possible) congestion effects. The values of on-board loads exceeding capacity, obtained with the MSA algorithm, are due to severe high saturation conditions. It is because the model cannot balance both for the lack of alternative runs in relation to user departure time, and for the impossibility to use capacity-explicit bounds in the equilibrium assignment approach without damaging the existence and uniqueness of the problem solution. Such values, even if higher than capacity are, in any case, still lower than the correspondent values obtained through the SNL model. Dynamic process assignment For the day-to-day dynamic, the results of applications to transit services with regular and irregular functioning, respectively, are reported in the following. In the case of regular service functioning, as an example of results of a deterministic dynamic process assignment model, Figure 13 reports the on-board load of line 20 run 22 and of line 1 run 18 over the same section 115-127 obtained for a 90-day simulation. Such values show system convergence towards points that seem to be fixed point attractors. In fact, the load value of line 20 run 22, which was initially near capacity (100 users/vehicle), decreases because of the learning process on the comfort attribute, that tend to disadvantage this run use when compared with alternative runs. This can be observed in the increase of line 1 run 18 over the same section in Figure 13.

Figure 13. Example of day-to-day assignment for regular services In the case of irregular service functioning, the on-board load changes from day to day mainly due to service irregularity (i.e. if a run is delayed, alternative runs are much more heavily loaded compared to days when they arrived earlier) and congestion (i.e. high on-board load are considered through learning and updating mechanisms that lead to changes in user choices on different days, even if runs are on

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149

time). An example is reported in Figure 14 where, for the same run described in Figure 13, the on-board load in the case of service irregular functioning is pictured.

Figure 14. Example of day-to-day assignment for irregular services 5.4

Short-term forecasting of on-board loads and arrival times at stops for urban transit systems

An interesting application of schedule-based assignment models is to support the short-term forecasting of on-board loads and run arrival times at stops by the use of Advanced Traveller Information System (ATIS) to provide to users en-route descriptive information on the network state in the attempt to facilitate their travel choices. Conceived essentially for private transport networks, in recent years ATIS has increasingly been applied to transit networks, where users may be interested in: (1) real-time waiting times at stops or equivalently the arrival time of runs at the stop; (2) degree of occupancy of upcoming runs at the stops. The latter information can affect to great extent en-route traveler's choices, especially in congested transit systems, where travellers may choose to skip overloaded runs and wait for less crowded ones by trading off between longer waiting-time and higher on-board comfort. In order to provide information of the type 1 (i.e. runs arrival time), it is necessary to develop an algorithm able to predict real-time link travel time on the network that is based on current vehicle location and historical data needs. On the other hand, to provide information of type 2 (e.g. loads or runs occupancy) a modelling procedure of the overall system needs to be set up. In fact, the real-time prediction of loads on runs of transit network requires the simulation of travellers' path choice and the way in which they propagate on the network. The prediction of the O/D travel demand

150

Advanced Modeling for Transit Operations and Service Planning

pattern is also required. To this aim, a modelling procedure which explicitly simulates within-day dynamics in transit networks from both the demand and supply side, has been built up. This is based on a schedule-based approach and is described in the following of this chapter. The conceptual scheme of the overall system is depicted in Figure 15, where the main components of the 775 (Transit Information System) are clearly outlined.

Figure 15. Schematic representation of the considered Transit Information System The surveillance system, by means of monitoring and communication devices (e.g. DGPS, radio modem, etc.), detect and transmit to the Operations Control Centre (OCC) the current location of the vehicles and the number of passenger that boarded or alighted at stops. The OCC elaborates the real-time data gathered to predict arrival times and occupancy of runs at stops, and transmits them to user interfaces through communication devices (e.g. long-range radio communication systems, and so on). Arrival times and runs occupancy predictions are based on the modelling framework schematically depicted in Figure 16. In principle, arrival time prediction is simpler than run occupancy prediction since it does not require the simulation of traveller behaviour with respect to the current network condition and to the information provided. Examples of algorithms to predict link travel times, and hence arrival times at stops, are widely reported in the literature. On the other hand, to predict run occupancy, a comprehensive modelling framework based on the schedule-based approach has been built up. It consists of (see Figure 16): - a time-varying O/D matrices estimation procedure based on real-time observation of the number of passengers boarding and alighting from vehicles at stops; - a supply model to represent a time-dependent transit network, whose temporal coordinates are updated real-time in relation to the information on vehicle location; - a sequential path choice model based on random utility theory, simulating public transport traveller behaviour;

Schedule-Based Transit Assignment Models -

151

a within-day dynamic assignment procedure following a schedule-based approach, estimating the loads on each run of the transit system at any time of the reference period.

The system of schedule-based supply, path choice and assignment models are those presented by (Nuzzolo et al., 200la). The algorithm adopted to implement the modelling framework follows an event-based simulation approach. An event is defined by the arrival of a signal from the surveillance system to the OCC. This happens when new information on vehicle location and/or passengers who boarded/alighted on/from a given run at a given stop are available. When an event occurs (say at time T), the input variables of the schedule-based transit assignment models previously described are updated, that is: - the service configuration (diachronic supply model) is updated, based on the current vehicles location and on the link travel times estimates; -

the O/D matrix is estimated for each time slice TDJ, based on the passenger boarded/alighted counts and current network performances. Then, traveller path choice is simulated and the loads on the runs for the remaining time of the whole reference period (i.e. the period of time from T to the end of the reference period) are estimated.

Figure 16. Modelling components of the Operations Control Centre (OCC)

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Advanced Modeling for Transit Operations and Service Planning

In the following, a preliminary application example is described that is used to test the approach and its potential use to support operations planning. The considered test network is schematically depicted in Figure 17. Four centroids have been considered; these are connected to nodes 1,2,8 and 9; the transit services consist of 3 lines and 9 runs; their diachronic representation consists of about 65 nodes and 200 links. Given a uniform demand pattern (i.e. constant arrival rates at stops) within a reference period of 1 hour (from 7:00 to 8:00), the experiment was aimed at investigating travelers' path choice variations for two cases of information provision: case 1: travellers are provided only with waiting times of upcoming runs at the stops; case 2: travellers are provided with waiting times and loads (i.e. run occupancy) of upcoming runs at stops.

Figure 17. Test network Let us consider travel demand from centroid 2 to centroid 9: path choice alternatives are represented by line 1 (on-board travel time equal to 30 min) and line 2 (on-board travel time equal to 20 min). In fact, the alternative of taking line 1 and then transferring to line 3, in this case, is dominated by the alternative of taking line 1 straightforward to destination. Moreover, let us suppose that, according to the transit schedule, runs of line 1 are expected to arrive at stop 2 at 7:10, 7:30 and 7:40, while runs of line 2 at 7:14 and 7:44.

Schedule-Based Transit Assignment Models

153

Case 1 Let us consider travellers arriving at stop 2 at 7:15 (i.e. right after the departure of first run of line 2). These travellers are provided, through the Transit Information System (e.g. by means of a VMS or other User-Interface device), with the following information: they have to wait 15 minutes for line 1 and 29 minutes for line 2. The effects of the information can be seen using the graph depicted in Figure 18, which was derived from the path choice model considered here (adapted to the present case study from Nuzzolo et al., 200Ib). From the graph, it can be seen that assuming the arrival time of the runs remain on schedule (i.e. estimated waiting times at stop linearly decrease with absolute time), 85% of travellers would choose line 1 and 15% would choose line 2. This represents the trade-off between more waiting time with less on-board time. At time 7:20, waiting times have been perturbed in the attempt to simulate irregularity of service. In particular, the waiting time of line 1 has been increased until it is greater than the waiting time of line 2. Accordingly, path choice probabilities vary: at time 7:44, when the run of line 2 arrives at stop 2 (i.e. when the travellers waiting at the stop actually make their choice), then 95% of the travellers would choose to board line 2, as shown in Figure 18. Case 2 Let us suppose now that travellers at stop 2 also have information on passenger loads in the upcoming runs, these being 25 passengers for line 1 and 75 passengers for line 2. The path choice probabilities are slightly different with respect to case 1. In fact, according to the scheduled services (e.g. at time 7:15), choice probability of line 1 would be equal to 91% (vs. 85% of case 1) due to the greater attractiveness of boarding a less loaded and, hence, more comfortable vehicle (see Figure 18). Similarly, when the travellers actually make their choice (i.e. at time 7:44) choice probability of line 1 is equal to 8% (vs. 5% of case 1). Note that due to service irregularity, the estimates of loads on line 1 vary with time. In fact, as a result of the delay in line 1, there is an overall increase of loads on line 1 (from 25 to 43 passengers). This results from the fact that traveller arrival at stops have been assumed to be uniformly distributed over time: This means that the longer a run is delayed, the number of travellers arriving at the stop (and boarding the vehicle) will also increase.

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Advanced Modeling for Transit Operations and Service Planning

P"

U-

10 5

5

7

c

70 60

a40 10

5

7

Figure 18.

5.5

Information provided by TIS (i.e. waiting time and loads) and path choice probability for travellers departing at time 7:15 from centroid 2 to centroid 9

Estimation of time-varying transit O/D matrices from time-varying traffic counts

This section reports a method and an application of a Generalised Least-Squares (GLS) estimator, which can be used to carry out transit O/D matrices segmented for user target time, on the basis of the methodology described in section 2. In the sphere of uncongested transit networks and considering user origin departure time fixed, the use of a simultaneous estimator is fully acceptable. Using a GLS estimator (Cascetta, 1984), Equation (3) can be written as

(12)

+ If IM, / •*/ -/JJ0TYIM,- / •*,- -/J j; j=i 1=1

-

i=i

~

where W_ is the variance-covariance matrix relative to vector e_ = (e_1>

,£,•,-•••,£„)

of assignment model and counting errors, characterised by E(s)=0_ and var[£j=W;

Schedule-Based Transit Assignment Models

155

d_j is an available estimation of the unknown demand vector x, obtained through a random sample such as dj=xj+ilj

(13)

in which JJ/ is a random vector that considers demand sample errors, characterised by =0_ and var[rjj]=Zj. In order to solve problem (12), it is necessary to calculate the assignment matrix MJJ defined as Mij-Qij-Pi

(14)

The use of the schedule-based dynamic transit modelling approach allows a more precise and direct way of calculating the crossing fraction matrix Qtj and the path choice matrix Pj. In fact, as each link of the diachronic graph represents a precise location in both space and time, it is possible to calculate, in a simpler way, the crossing fraction matrix QJJ, which is made up of elements qf'1. , because each path k is individuated on the graph by a sequence of links that allowed the fraction of path load /?,-,k and the crossing link a in time interval j to be defined directly. Elements of path choice matrix Pj, which represents the probability of choosing path k (individuated by run r) for users travelling on the O/D pair od and departing in time interval /., can be calculated using the schedule-based path choice models described in chapter 4. This approach allows problem (12) to be easily solved by using a traditional project gradient algorithm. This method is applied to estimate the Italian railway O/D matrices at a national scale, which is made of 13420 O/D pairs for each service (high speed and low speed trains) and travel class (first and second). The estimation is carried out on the basis of about 868 time- varying counts on 257 of the 334 trains of the railway services over several sections. The reference period (one-day) was divided into 15 time intervals of one hour each in the rush hours, while wider intervals for other periods were defined. An origin departure time for each time interval was considered. In order to calculate path choice probabilities, the path choice model structure, attributes and parameters are those reported in section 5.2.

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Advanced Modeling for Transit Operations and Service Planning

This methodology is actually used in the Decision Support System of the Italian Railways to estimate the railway O/D matrices at national scale. The starting O/D matrices are the railway O/D matrices of sold tickets, which are available from automatic station reports. These O/D matrices differ from the real ones as they do not consider tickets sold on-board, railway passes and free tickets. In addition, it also does not fully account for tickets users who purchase tickets that could be used over the next 60 days. For this reason, in order to simulate in a more precise way railway services, the updating of railway O/D matrices of sold tickets from traffic counts is crucial. The accuracy of a solution is evaluated by the difference between the generic O/D vector d and the real one d_. Different statistics like the root mean square error (rmse), the relative mean error (rme) and the mean square error (mse) are used. Performance of the used estimator have been considered by computing average values of statistics on the basis of different starting matrices d_ obtained through random perturbations of the real O/D vector d_ as df = df

+ (p(u-0.5) df

(15)

where u is a value extracted by an uniform (0,1) random variable and ^ is a parameter that is 0.7, 1.4, 2.0 to which corresponds to variation coefficients of 0.35, 0.70 and 1, respectively. Estimated O/D matrices have been carried out using a project gradient algorithm, in which the optimal solution is reached when the difference between two successive iterations does not exceed 1 % (£=0.01). Table 6 reports statistics for the three different estimations ((f>= 0.7, 1.4, 2.0). It can be seen that the method has succeeded in improving the starting vector, which showed absolute reductions are larger when the perturbation is higher. Different applications with different number of time intervals and perturbations have been considered, showing that the percentage of the reduction is approximately constant and improvements are slightly larger for higher number of intervals (i.e. smaller interval widths).

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157

Table 6. Comparison between starting and estimated demand matrices

* 0.7 1.4 2.0

5.6

rme 0.03 0.05 0.09

mse 794.0 1216.7 2300.4

Rmse

0.26 0.32 0.44

Timetable setting

Assessing an optimal timetable can improve, to a great extent, effectiveness and efficiency of the transit network, both on the demand-side and on the operator perspective. In fact, an optimal timetable is the result, on the one hand, of a good match between bus supply and passenger demand and, on the other hand, of the minimization of the number of the operating bus fleet. In the light of these objectives it is worthwhile to distinguish between high-frequency and low-frequency systems. In high-frequency systems, the high frequency of transit lines and the random arrival of travellers at stops means that the marginal benefit of improvements (in terms of waiting and transfer time) is minimal. Thus, typical timetable schedules are derived from the optimisation of crew and vehicle schedules which may induce a higher marginal benefit in terms of operating cost. For low-frequency systems, the problem of constructing timetables can be addressed as a problem of defining the scheduled departure-times of the runs of the transit lines. Once the frequency of each line is determined, i.e. the total number of runs departing in the reference period (e.g. the morning peak-hour), the number of runs of each line can be determined at prior planning levels. The method of determination can be grouped into those that simulate passenger flows, those that deal with ideal networks and those based on mathematical programming. A review of these methods can be found in chapter 2. Given the frequency of each line, the problem of optimal timetable setting is that of determining the departure-time of each run r of the line /, (Ort /. This can be achieved through different criteria such as maximizing synchronization of rides at transfer nodes, tuning ride departure-times versus user demand bases (from traffic counts) at stops, or simulating user choice. Among the latter methods, a method for optimal transit setting (Coppola, 2002) which is based on the schedule-based dynamic transit assignment modelling (Nuzzolo et al, 200la) is described. Typically the following assumption are made, i.e.:

158 -

Advanced Modeling for Transit Operations and Service Planning routes and stops of all transit lines are fixed, as well as the roundtrip travel time of each line;

-

capacity of the bus operating on each line is a priori determined;

-

the origin-destination demand profile in the reference period is known and fixed with respect to the transit timetable.

The timetable setting problem can then be defined as the minimization of an objective function R(co) specified according to different perspectives: the users' and the operator perspective. From the operator perspective, the goal is to minimize operative costs which in turn means minimizing the operating fleet size on one hand and maximizing the load factors on the other. Assuming that all the buses are operating within the system, the operative cost does not depend on run departure times and the objective function results in a function only of the perceived users' costs. The perceived users' cost, which is assumed to be the total travel time the users spent on the network, can be written as:

(16) where: -

d° 'TD' is the demand flow on the O/D pair od leaving the origin during time interval /';

~ Pk/odTDi (^) *s me probability of choosing path k connecting origin-destination -

pair od departing in time slice /. WTk/odTD.(G}) is the waiting time on path k connecting O/D pair od departing in time slice /';

~

TTk/odToi(Q) *s me transfer time on path k connecting O/D pair od departing in

-

time slice /; BTj(/ociTD.(a>) is the on-board time on path k connecting O/D pair od departing

-

in time slice /. the sum is extended to all paths of all O/D pairs and time slices of the reference period.

In order to assure the consistency between link and path flows, the following constraining expression of the flow on each link of the diachronic network is the sum of all the flows of paths including that link, and can be considered as:

Schedule-Based Transit Assignment Models

159

where - fj the flow on the generic link 7' of the diachronic network; - Fk/0(jTD.(&) is the flow on the generic path k of the diachronic network, -

connecting the O/D pair od of users departing in time slice /; 8jtk the element of the link-path incidence matrix, equal to 1 if linky belongs to path k, 0 otherwise.

Furthermore, technical constraints can be added to the optimization problem. To guarantee a minimal space distance between the buses of two subsequent runs, a minimum value of the headway is introduced (MinHdw): MinHdw
Vr,/

(18)

In addition, to have a balanced distribution of the load of the lines among the respective scheduled runs, a constraint on the maximum load on each bus is introduced: f}((^
Vy

(19)

being Capveh\de the capacity of the bus. Concerning solution methods, the case of a single run can be solved numerically since the objective function results are piecewise linear. The minimum can thus be found by computing R(o)) in the point of discontinuity of R'(co). On the other hand, the algorithms for the solution of the problem presented above are based on a two-step iterative procedure. In the first step, the probabilities Pk/odTot (^ are calculated through a schedule-based dynamic transit assignment, given an initial vector of run departure-times (e.g. with evenly spaced headway). In the second step, given the path choice probabilities, it is possible to express all the components of travel time (i.e. WT, TT, BT) as function of a>. Therefore, it is possible to determine the optimal vector a>* of run departure times by means of an algorithm based on greedy heuristics and local search, which optimises departure time of one run while maintaining fixed departure time of all the others. The modelling framework above presented has been applied to the test network depicted in Figure 19.

160

Advanced Modeling for Transit Operations and Service Planning

Figure 19. Test network The reference period has been assumed to be equal to 3 hours (i.e. from 7 am to 10 am). The test network consists of four lines: one (i.e. line 1) consisting of five departing runs while the other lines consisting of three departing runs in the considered reference period. The O/D demand flow distribution has been assumed to be asymmetric within the reference period, as depicted in Figure 20.

Figure 20. Departure time demand distribution From the initial vector of evenly-spaced run departure times (i.e. clock-headways), the algorithm gives the optimal timetable schedule reported in Table 7. A significant reduction of the disutility-pro-capita can be observed with respect to the initial timetable configuration (see Figure 21).

161

Schedule-Based Transit Assignment Models

Table 7. Optimal timetable for the given demand pattern

Line 1

Line 2

Line 3

Line 4

Stops A B I

A D H I

Run 1

Run 2

7:24 7:41

7:54 8:11

8:04

8:34

Run 3 8:24

9:14

8:41 9:04

9:31 9:54

7:34

8:24

9:26

7:46

8:36 9:01 9:12

9:38

10:03 10:14

8:36 8:49

9:38 9:51

9:03

10:05

8:11 8:22

D E F

7:46

B E H

7:41

8:41

9:53

7:49 8:03

8:49 9:03

10:01 10:15

7:59 8:13

Run 4

Run 5 9:36 9:53

10:16

disutility pro-capita

clock-

step 1

step 2

step 3

step 4

step 5

step 6

Figure 21. Disutility pro-capita at each step of the optimisation procedure

REFERENCES Carraresi, P., F. Maluccelli and S. Pallottino (1996). Regional mass transit assignment with resource constraints. Transportation Research, 30B, 81-98. Cascetta, E. (1984). Estimation of trip matrices from traffic counts and survey data: A generalised least squares estimator. Transportation Research, 18B, 289-299. Cascetta, E., D. Inaudi and G. Marquis (1993). Dynamic estimators of origindestination matrices using traffic counts. Transportation Science, 27, 363-373.

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Cascetta, E., L. Biggiero, A. Nuzzolo and F. Russo (1996). A system of within-day dynamic demand and assignment models for scheduled intercity services. Proceedings of 24th European Transportation Forum Seminar D-E part 2, London. Crisalli, U. (1998). A stochastic network loading model for ITS urban transit networks. In: Urban Transport and the Environment for the 21st century IV, (C. Borrego and L. Sucharov, ed.), pp. 481-490. Computational Mechanics Publications. Crisalli, U. (1999a). Dynamic transit assignment algorithms for urban congested networks. In: Urban Transport and the Environment for the 21st century V, (L. J. Sucharov, ed.), pp. 373-382. Computational Mechanics Publications. Crisalli, U. (1999b). User's behaviour simulation of intercity rail service choices. Simulation Practice & Theory, 17, 233-249. Crisalli, U. and F. Gangemi (1997). The access/egress mode choice to railway terminals. In: Urban Transport and Environment for the 21s' Century III, (L. J. Sucharov and G. Bidini, ed.), pp. 43-52. Computational Mechanics Publications. Florian, M. (1998). Deterministic time table transit assignment. Preprints of PTRC Seminar on National Models, Stockholm. Hickman, M.D. and D.H. Bernstein (1997). Transit service and path choice models in stochastic and time-dependent networks. Transportation Science, 31, 129-146. Hickman, M.D. and N.H.M. Wilson (1995). Passenger travel time and path choice implications of real-time transit information. Transportation Research, 3C, 211-226. Nguyen, S., S. Pallottino and M. Gendreau (1997). Implicit enumeration of hyperpaths in logit models for transit networks. Transportation Science, forthcoming. Nielsen, O.A. (2000). A stochastic transit assignment model considering differences in passengers utility functions. Transportation Research, 34B, 377-402. Nielsen, O.A., C.O. Hansen and A. Daly (2000). A large-scale model system for the Copenhagen-Ringsted railway project. Proceedings of 9th IATBR Conference, Gold Coast, Australia. Nuzzolo, A. and F. Russo (1993). Un modello di rete diacronica per 1'assegnazione dinamica al trasporto collettivo extraurbano. Ricerca Operativa, 67, 37-56. Nuzzolo, A. and F. Russo (1994). An equilibrium assignment model for intercity transit networks. Proceedings of TRISTAN II Conference, Capri, Italy, 1994.

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Nuzzolo, A. and F. Russo (1996). Stochastic assignment models for transit low frequency services: Some theoretical and operative aspects. In: Advanced Methods in Transportation Analysis, (L. Bianco and P. Tom, ed.), pp. 321339. Springer Verlag, Berlin. Nuzzolo, A. and F. Russo (1998a). A dynamic network loading model for transit services. Proceedings of TRISTAN III Conference, San Juan, Puerto Rico. Nuzzolo, A. and F. Russo (1998b). Departure time and path choice models for intercity transit assignment. In: Travel Behaviour Research: updating the state of play, (J. Ortuzar, D. Hensher and S. Jara-Diaz, ed.), pp. 385-399. Elsevier Science Ltd., Amsterdam, The Netherlands. Nuzzolo, A. and U. Crisalli (2001). Estimation of transit origin/destination matrices from traffic counts using a schedule-based approach. Proceedings of European Transportation Forum 2001, Cambridge, UK. Nuzzolo, A., U. Crisalli and F. Gangemi (2000). A behavioural choice model for the evaluation of railway supply and pricing policies. Transportation Research, 35A, 211-226. Nuzzolo, A., F. Russo and U. Crisalli (1999). A doubly dynamic assignment model for congested urban transit networks. Proceedings of 27th European Transportation Forum, Seminar F, 185-196, Cambridge, England. Nuzzolo, A., F. Russo and U. Crisalli (2001a). A doubly dynamic schedule-based assignment model for transit networks. Transportation Science, 35, 268-285. Nuzzolo, A., F. Russo and U. Crisalli (2001b). Doubly dynamic path choice models for urban transit systems. In: Travel Behaviour Research. The Leading Edge, (D. Hensher, ed.), pp. 797-812. Pergamon. Wong, S. C. and C. O. Tong (1998). Estimation of time-dependent origin-destination matrices for transit networks. Transportation Research, 32B, 35-48. Wong, S.C. and C. O. Tong (1999). A stochastic transit assignment model using a dynamic schedule-based network. Transportation Research, 33B, 107-121. Wong, S.C. and C. O. Tong (2001). Planning an urban rail transit system using a schedule-based network model. Proceedings of 9th WCTR Conference, Seoul, Korea.

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CHAPTER 6

FREQUENCY BASED TRANSIT ROUTE CHOICE MODELS Michael Florian, Centre for Research on Transportation, Universite de Montreal, Montreal, Canada

1.

INTRODUCTION

Transit route choice models, or transit assignment models, aim to represent the decisions made by transit travelers which lead to the traffic flows which occur on a network of transit lines which operate at known frequencies. The modeling of a transit trip requires a relatively detailed network representation which involves the walking time to a stop, the waiting time for a transit vehicle, the transfers between lines if more than one line is taken and the in-vehicle time. The simplest model of transit route choice considers the travel times and the frequencies to be constant. This is the linear case model and is described in the next section. If the travel times model congestion aboard the transit vehicles, a convex cost optimization model results; it is described in the third section of this chapter. The most complex transit route choice model considers both congestion aboard the vehicles and the limited capacity of transit vehicles. If the arriving vehicle has limited free capacity, then not all passengers waiting at a stop can board the first arriving vehicle. As a consequence, the frequency perceived by the passengers is not the nominal frequency, since they have to wait for one of the following vehicles which

166

Advanced Modeling for Transit Operations and Service Planning

would have spare capacity in order to board. This model is the subject of the fourth section of this chapter. A conclusion section outlines some future research and application challenges.

2.

THE LINEAR COST MODEL

The following exposition is based on the work of Spiess and Florian (1989). Consider a transit network which consists of a set of nodes, a set of transit lines, each defined by an ordered list of nodes at which boarding and alighting are permitted, and a set of walk links, each defined by two nodes. The time associated with each walk link and each transit line segment is constant. At each node that is on the itinerary of a transit line, the distribution of the interarrival times of the vehicles is known for each line which serves the node. As a consequence one can compute the combined expected time for the arrival of the first vehicle, for any subset of lines incident at a node, as well as the probability that each line arrives first. In order to state the mathematical model which corresponds to the transit route choice selection, it is noted that each walk link may be replaced (conceptually) by a transit line of one link with a zero waiting time (infinite frequency). Also, it is assumed that the underlying network is strongly connected. The objective is taken to be the minimization of expected waiting and travel time, or the expected total generalized cost if waiting times and travel times may have different weights (e.g. waiting is more onerous than in-vehicle time). The network is composed of four types of arcs: wait arcs (no travel time), in vehicle (no waiting), alighting (no travel time, no waiting) and walk arcs (travel time, no waiting). Thus, the segment of a transit line is an arc that is served by a vehicle at given intervals and the transit traveler waits for the link to be served by a vehicle. The arcs that will be included in a solution of the model are denoted by A e A, where A is the set of arcs and N is the set of nodes. Thus the solution for a single destination q is a subgraph ^=(^,,4). The demand for travel from nodes i,ieN to the destination q is denoted g(.. Among the links included in a solution A, at each node /, i e N, a traveler boards the first vehicle that serves any of the line in the A*(A = u / y4, + ). The set A\ corresponds to the lines that will be chosen by the traveler to yield one or more routes from i to q in a solution of the model. At each stop i, it is convenient to refer to the set A * as the set of attractive lines.

Frequency Based Transit Route Choice Models

167

Let W(A*} be the expected waiting time for the arrival of the first vehicle serving any of the links a & A ] , which is denoted as the combined waiting time of links a e A *. Let Pa (A *) be the probability that link a is the first line to be served among the links A ] . If an exponential distribution of interarrival time is assumed then

and

where fa is the frequency of link (line) a . Since A is not known a priori the single destination model is formulated by using binary variables xa

The optimization model may be stated now as follows. For simplicity of exposition only one destination q is considered and the index q is not used:

subject to ^

-^

where sfl is the travel cost on link a and Vi is the total volume at node / . At first sight, the problem (4)-(7) is a mixed integer nonlinear optimization problem.

168

Advanced Modeling for Transit Operations and Service Planning

Fortunately, the problem may be reduced to a much simpler linear programming problem by making the following observation. (6) may be replaced by the nonnegativity constraints of the link volumes va > 0, a e A since ^ va = Vt, i e TV. aeA +

Then, by introducing new variables w(., which denote the total waiting time of all trips at node /, w =

V. ' , i € TV, one obtains the equivalent problem Z-/X at A*

subject to

The objective function (8) is now linear and the 0-1 variables are only used in constraints (9), which are the only nonlinear constraints. These constraints may be relaxed by replacing (9) with

which yields the linear programming problem (8), (12), (10), (11). It may be shown by using the extreme point properties of the solutions of a linear programming model, that this problem is equivalent to (8)-(12). The dual problem of this last linear program is g.H,.

subject to

where ut, Uj are the dual variables corresponding to (10) and jua are the dual variables corresponding to (9).

Frequency Based Transit Route Choice Models

169

Let (v*,W) and (u*,ju') denote the optimal solutions of the primal and dual problems. The weak complementary slackness conditions are and

In both the primal and dual formulations, the transit route choice model has a close resemblance to the shortest path route choice model. The latter corresponds to the shortest path problem when none of the links of the network involves waiting; thus fa —> oo and vt>. -> 0. The solution algorithm which solves the transit route choice model bears a strong resemblance to the label setting algorithm for computing shortest paths. The solution algorithm is composed of two parts. In a first pass, from the destination nodes to all origins, the arcs which carry flow, A', and the expected travel times u* from each node i, ieN, to the destination nodes are computed. In a second pass, from all origins to the destination, the demand is assigned to the arcs a, a e A *. The algorithm is stated below (the convention 0 oo = 1 is used):

Transit Route Choice Algorithm

Part l:Find A* Step 1. 1 (Initialization) : ut <- oo, / e N - [q] ; uq <- 0

Step 1.2 (Choose next link) : If S - $ STOP; otherwise find a = (/,/) e S such that

Step 1.3 (Update node labels) : If ui > ui+sa

Go to Step 1.2;

1 70

Advanced Modeling for Transit Operations and Service Planning

Part 2: Assign demand to A* Step 2.1 (Initialization) : V^T., ieN Step 2.2 (Loading) :

For every link a e A in decreasing order of (u} + sa ) do : If aeA then v <- — F,

otherwise va <— 0. The auxiliary variables ff, ieN contain the combined frequencies of all selected links at node i . By using the primal and the dual formulations of the transit route choice model, one can prove that the algorithm indeed finds the solution of (8), (12), (10), (11). The algorithm is applied for each destination in turn. It is worth mentioning the work of Wu and Florian (1993) and Wu et al. (1994) which present algorithmic variations for solving this problem.

3.

THE NONLINEAR COST MODEL

If the link travel times ca are not constant, but are continuous monotone increasing functions ca(v), a e A, if the vector of flows v = {va },a e A , the transit assignment models may consider the discomfort "cost" of passengers in crowded vehicles, or the travel time of vehicles under different load conditions. When ca(y) = ca(va), aeA there exists an equivalent convex cost optimization problem which corresponds to an equilibrium transit assignment. The total volume on link a, va , is the sum of the link volumes vqa to all destination nodes q, q e Q qeQ

The problem is no longer separable by destination, as is the case for the linear cost model. The demand from node /, i e I to destination q,q eQ, is denoted g? and h\ is the part of the demand gf which is allocated to the network using strategy k, k e K . The conservation of flow equations is

Frequency Based Transit Route Choice Models

171

It is hypothesized that travellers choose strategies which minimize the expected travel times of their trips and, in the case of congestion, do not choose strategies of larger cost. This leads to a statement of equilibrium conditions akin to Wardrop's (1952) user optimal principle. If s] denotes the expected travel time of strategy k^Kq from model i to destination q , the equilibrium conditions maybe stated as follows:

By using a similar development to that used by Smith (1979), it may be restated in the form of a variational inequality. For general costs ca(v) this variational inequality may not have a unique solution since the corresponding cost mapping is not necessarily monotone. When ca(v) = ca(va), a&A it is straightforward to show that there exists an equivalent convex cost optimization problem which is

?

(22)

subject to (19) and

This problem may be solved by any of the standard algorithms for convex cost optimization problems. However, the application of the linear approximation algorithm (Frank and Wolfe, 1956) is appropriate here since the auxiliary subproblems that need to be solved are linear cost transit assignment problems. The linear approximation of the objective function (22) results in a problem which is separable by destination q. The resulting algorithm is as follows:

172

Advanced Modeling for Transit Operations and Service Planning Transit Route Choice Algorithm with Convex Costs

Step 1. (Initialization): Find feasible (v°, w°) where vv° = ^T ^T w? is the total waiting time and v° is a vector of link flows. Step 2. (Subproblem): I = 1 + 1 Compute (v,w) by applying the linear cost route choice model with link costs ca = ca(v'~l) for each destination q. Step 3. (Line Search): Find A,1 which minimize the objective function (22) on the line segment (1 - A)(v M , WM ) + /l(v, w),0 < /I < 1. Step 4. (Update):

( v ', vv') = (1 - A1 )(v'"', w1'1) + A' (v, w)

Step 5. (Stopping Criterion): If a stopping criterion is satisfied, STOP. Otherwise, return to Step 2. The stopping criterion that may be used are a maximum number of iterations, a relative gap which is available as a by product of Step 2 of the algorithm or a measure of differences in flows and waiting times such as (v', w! j - (v1'1, w'~l One should note that in this model all passengers experience the same level of discomfort, even though in reality, those who board the bus first have a better chance to find a seat and may be better off. Also, the waiting time is not affected by the transit volumes. All passengers may board the first vehicle to arrive and queuing delays at stops are not considered. These phenomena are the subject of the next section.

4.

THE VARIABLE FREQUENCY (OR CAPACITATED) MODEL

If the transit network is congested it often happens that passengers can not board the first vehicle to arrive at a stop, which was the assumption made for the development of the models presented in the two previous sections. The notion of effective frequency is central to the development of an appropriate model. It is the actual frequency of the service perceived by passengers which wait at a stop. The following exposition is based on the work of Cominetti and Correa (2001) and Cepeda (2002).

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173

The effective frequency of a line a is a strictly decreasing differentiable function of the total flow v f l , that is fa :[0,v fl )-+(0,oo), such that / ' ( - ) < 0 and / a (v f l )->0 when va —» va ,' where v"a is the saturation flow of line a. In order to formulate the corresponding equilibrium model some additional notation is required. Qy is the space of feasible link flows, Qv =[0, oo)1 and Q^ is the space of destination feasible link flows, Q;f=[0,oo) f

respectively.

Mid

f-\fa\,a&A

f(v) = {fa(v)}aeA

. t(a\ h(a)

are the tail and head nodes of an arc,

\

is the vector of nominal line frequencies and

is the vector of effective frequencies. c(v) = {ca(v)}aeA is the vector

of arc travel costs, which are assumed to be continuous. In the following development it is convenient to refer to a node strategy which is a subset of the arcs A*. Each node /, i&N is an origin when a flow traverses it towards destination q. A strategy can be decomposed into a set of node strategies one for each node in TV*, the nodes of the network which carry flow for strategy k. ( ( N k , A k j is the subgraph associated with strategy k, or the hyperpath k, such named by Nguyen and Pallotino (1982)). Qs is the space associated with node strategies, Q^ -R^ms where ms = ^\Kf\, where Kt is the set of non-empty subsets of ,4*. & e A\ is a node ieN

strategy for node / and hqk is the flow of the node strategy k towards destination q. Sqk(v) is the expected time for travel from i to q by using node strategy k and r J(v) is the expected cost of the shortest hyperpath from i to q with frequencies fa(y)

and

time ca(v). Last but not least, A^v) (respectively A'iq{v), A^g(v), A*q(v)) is the set of arcs a e / 4 ^ such that f 0 (v) + r* ( f l ) ). The set of feasible flows Q* is defined by the conservation of flow and nonnegativity conditions

Z < - Z V ' = ^ / ' / e A r ' ^Q

c€/4, +

aeA~

(26>

vqa>0,aeA, q&Q

(27)

/ a (v)>0, aeA

(28)

and

while the set Q s xQ^ consists of all the vectors (/z,v)eQ s xQ^ such that v e Q ^ and in addition

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If the network is not over capacity and can carry all the demand, that is X* v« < ^'

ae

^' ^ e 2'

men an

equilibrium solution exists (refer to the proof

provided by Cominetti and Correa (2001) for the case of common transit lines for one origin-destination pair). For all v e Q^ such that /(v) > 0 the variables rf are computed recursively for each destination by:

(see the algorithm for the linear cost model). It was shown by Cominetti and Correa (2001) that T is an implicit continuous function of / and c. But /(v) and c(v) are explicit functions and hence T is an implicit function of v. hi the following all values of T will be denoted rf(v). For each v e Q ^ such that /(v)>0 the cost of a node strategy k&A* may be computed recursively as

The equilibrium model is formulated by stating Wardrop's (1952) user optimal principle as:

Finally, let V(K) = {v e Q^ : (h, v) e Q5 x Q^}. There are several properties which allow the characterization of minimal cost node strategies. They demonstrate that in a general network, the common lines problem

Frequency Based Transit Route Choice Models

175

studied by Cominetti and Correa (2001) is solved implicitly for each node destination pair. These are stated without proofs, which may be referenced in Cepeda (2002). If k,k} and k2 are subsets of A] such that k = kluk2,qeQ and v e Q ^ then the strategy costs are relatively

If k is a node strategy of minimal cost for the pair (19) and ve Q^, then k n A^(v) . In addition, k n A^(v) - . Finally, the node strategy k = A
Initialization : while

" A: := |a e ^ ^: afl > OJ in non empty" do

8=

This algorithm decomposes aa=vqal fa(v)

into the quantities $ = hl/^lbekfb(v)

for

strategy k which includes arc a, that is

which yields v e V(h) and (h, v) e Qs x Q^. The computation of the cost of the assignment h for the node-destination pair (i,d} is done by the expression

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Advanced Modeling for Transit Operations and Service Planning q

This may be shown by using the definition of Sqk which is

which after some algebraic manipulations results in

V

V

Let be A+ be such that —-— = max fle + — 2 —. Arc b belongs to all strategies k ^' generated by the algorithm and hence

and (34) is so demonstrated. A consequence of this development is that the term

v" maxae^ —2— represents the total waiting time at node i for all travellers towards destination q. The central theoretical result, which eventually leads to a computational method for solving the variable frequency model, is the following. A flow v e Q ^ is an equilibrium flow if and only if there exists a vector a = (a ?)f€e^ such that for every destination q e Q and each node /

To show the necessity of (36) let v be an equilibrium flow with strategy flow h and arc flows v e V(h). Then

. But if a e Alq(v) then a belongs to each strategy k e A] such that hqk> 0 and hence v9, / fa(v) - aj. Similarly if a e A^(v) belongs to no strategy k e A] with h\> 0 and hence the third term of (36) equals zero.

Frequency Based Transit Route Choice Models

1 77

On the other hand of v e Qf and a satisfies (36) then one can find h e Qs by the application of the above algorithm. In order to show that h is an equilibrium flow one notes that the strategies with hqk>0 generated by the algorithm are such that A^(v) e k e A~iq(v). By (36) each arc a e A^(v) has aa - vqj fa(v) = a"i, hence it is present in all k strategies. All arcs a e A ] ( v ) satisfy aa = 0 and are not considered by the algorithm. The inclusion A^(v)
The proof is omitted here and may be referenced in Cepeda (2002). This characterization of the equilibrium flow leads to the interpretation of (38) as a gap function which has a value of zero for an equilibrium flow. Hence if one can solve the problem

then one has obtained an equilibrium flow for the variable frequency model. Unfortunately, this problem is not easy to solve by any of the classical non differentiable optimization algorithms. However, it may be solved by the method of successive averages to £ optimality. The algorithm will be stated after a short discussion of (39) as a generalization of the linear and nonlinear cost models. When the costs are linear and the frequencies are constant, that is c" = c(v) and / - /(v)

men

(39) is the statement of the difference between the primal and dual

objective functions.

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Advanced Modeling for Transit Operations and Service Planning

J y'

/

—\

where vqa < faw? implies wi < max — and r M c , f J = u ] . Ja

When the costs are nonlinear and the frequencies are constant then (39) may be interpreted as the gap function of convex cost optimization problem which is (22-25). In order to state the solution algorithm it is useful to restate (39) in the equivalent form

subject to

If v is interpreted as today's flows then /(v) and c(v) represent the information that serves to select tomorrow's route choice. The flow obtained by solving the internal problem, which is a linear cost problem, corresponds to a shortest strategy choice by all travellers. This is the intuitive interpretation of the following algorithm. Algorithm for Variable Frequency Model Step 1. (Initialization): / = 0; Find a feasible flow v by solving a linear cost model with linear costs and fixed frequencies c, / to obtain v° Step 2. (Update costs): 1 = 1 + 1. Compute c(v'), /(v'). Step 3. (Line Search): Find a new flow v' by solving a linear cost model with

Step 4. (Update) :

New solution : v' =

v'"1 +-(vl).

Step 5. (Stopping Criterion): If the gap function (40) is sufficiently close to zero STOP. Otherwise, return to Step 2. The algorithm has been applied successfully on several large scale projects. The empirical convergence obtained is very good in spite of the fact that some of the sufficient conditions for the convergence of the MSA algorithm could not be demonstrated for this problem.

Frequency Based Transit Route Choice Models

5.

179

CONCLUSION

The linear cost model transit assignment model has been implemented in several transportation planning software packages including EMME/2 (Babin et al., 1982) and is widely used. Since the nonlinear cost model and the variable frequency model use the solution of the linear cost as a subproblem in the respective algorithms, it is relatively easy to implement algorithms for these more complex problems. Certainly the variable frequency model is the most appropriate for the analysis and planning of congested transit systems, such as in many Asian, European and South American cities.

ACKNOWLEDGEMENTS The variable frequency model description is based on the recently completed doctoral thesis of Cepeda (2002) which was co-directed by Roberto Cominetti, University of Chile and the authors. Technical articles based on Cepeda's thesis are forthcoming in the near future.

REFERENCES Babin, A., M. Florian, L. James-Lefebvre and H. Spiess (1982). EMME/2: Interactive graphic method for road and transit planning. Transportation Research Record, 86B, 1-9. Cepeda, M. (2002). Modele d'equilibre dans les reseaux de transport en commun : Le cas des capacites explicites des services. PhD thesis, Departement d'informatique et de recherche operationnelle, Universite de Montreal, Montreal, Quebec, Canada. Cominetti, R. and J. Correa (2001). Common-lines and passenger assignment in congested transit networks. Transportation Science, 35, 250-267. Frank, M. and P. Wolfe (1956). An algorithm for quadratic programming. Naval Research Logistics Quarterly, 3, 95-110. Smith, M. J. (1979). The existence, uniqueness and stability of traffic equilibria. Transportation Research, 13B, 295-304. Spiess, H. and M. Florian (1989). Optimal strategies: a new assignment model for transit networks. Transportation Research, 23B, 83-102. Wardrop, J. G. (1952). Some theoretical aspects of road traffic research. Proceedings of the Institution of Civil Engineers, Part II, 325-378.

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Wu, J. H., M. Florian and P. Marcotte (1994). Transit equilibrium assignment: a model and solution algorithms. Transportation Science, 28, 193-203. Wu, J. H., and M. Florian (1993). A simplicial decomposition method for the transit equilibrium assignment problem. Annals of Operations Research, 44, 245-260.

CHAPTER 7 CAPACITY ASSIGNMENT ANALYSIS

CONSTRAINED MODELS AND

TRANSIT RELIABILITY

Michael G. H. Bell, Department of Civil and Environmental Engineering, Imperial College London, UK

1.

INTRODUCTION

The problem of full transit vehicles is experienced daily by passengers in cities like Hong Kong, London and Osaka (to mention a few recently visited by the author), as well as in many parts of the developing world. Even where transit over-crowding is not endemic, events like football matches can lead to temporary overloads. Where there is no alternative route, a passenger may simply have to wait until a vehicle arrives that he can board. Where there are route alternatives, a passenger may choose a less direct or slower route to avoid a long wait. The passenger route choice strategy may therefore be influenced by the fear of having to wait. Much has been written on passenger line selection strategies under the heading of the common lines problem. This problem concerns the definition of a set of attractive lines with the assumption that among these lines the passenger chooses the first vehicle to arrive at his stop. Where there is a choice, the probability of choosing an attractive line is therefore proportional to its frequency. The set of all paths that a passenger may use to get from his first stop, which will be in the vicinity of his origin, to his last stop, which will be in the vicinity of his destination, is referred to as a hyperpath. The problem of full transit vehicles has received comparatively little attention in the literature. Where it has been considered, the usual approach is to define an effective frequency for an

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Advanced Modeling for Transit Operations and Service Planning

attractive line, which falls with increasing probability that the passenger encounters a full vehicle when trying to board. However, the set of attractive lines may also increase with the probability of encountering a full vehicle. Moreover, the probability of encountering a full vehicle at a given stop depends on the amount of boarding and alighting activity both at that stop and upstream of that stop. An alternative approach to the problem based on network reliability concepts is proposed. This chapter begins by looking at the alternative network definitions found in the literature. It then reviews the work on the common lines problem, and points out that the problem only arises where passengers make decisions at the stop based on the first vehicle to arrive. Where different lines use different platforms, difficulty in moving between platforms may force passengers to choose a line before arriving at the stop. Alternatively, pre-trip information may allow passengers to choose departure time and line before arriving at the stop. The network loading problem, which has received almost no attention in the literature, is considered. A Markov chain approach which incorporates line capacity constraints through failure-to-board probabilities is formulated and illustrated by numerical example. While schedule-based assignment is briefly reviewed, the focus of attention is frequency-based transit assignment.

2.

EARLY WORK

Early approaches to transit assignment consisted largely of attempts to adapt vehicle assignment methods to transit (Dial, 1967; Fearnside and Draper, 1971). However, the problem is intrinsically more complex. In addition to in-vehicle travel time, the passenger has to take walking time, waiting time or service frequency, transfer penalties and fares into account. Rather than simply choosing a route, as in the case of the car driver, the passenger has to choose an origin stop, a destination stop, and then a line or sequence of lines by which to reach the destination stop. The fare structure may be distance related, time related, zonal or flat (or some combination of these). In his Pathfinder program, Dial (1967) inputs the transit system as link and line files. The former resembles the network description used by existing traffic assignment tools, while the latter contains line frequency information. To economize on storage, the transit network is stored as "trunk line links" consisting of a pair of nodes, a travel time, and a set of line numbers. Figure 1 shows such a network. The stops are represented by rectangles, the line numbers (two in this case) are represented by Roman numeral(s) beside each link, and the travel time is given by an Arabic numeral, also beside each link. At each node, transfer between any pair of lines is possible, unless precluded by the user, subject to a transfer penalty based on the frequency of the recipient line. The transfer penalty is equal to the expected waiting time if the arrival time is random, recipient services are regular, and the user

Capacity Constrained Transit Assignment Models and Reliability Analysis

183

takes the first line to arrive. All lines traversing the link are assumed to offer the same level of service, so only one link travel time is specified; if some services offer a different level of service, additional links must be added to the network.

Dial (1967)'s Pathfinder: "trunk line links"

I, II

15 15 II

Figure 1. Transit network using trunk line links An important characteristic of transit networks, in contrast to car networks, is that they violate the Markov property of route choice. This point is illustrated in Figure 1. The best route from Home to C is not independent of the destination, because of the transfer penalties. Traffic from Home to A is divided between the two lines in proportion to their frequencies. Traffic from Home to C would prefer line I, as would traffic from Home to D. However, traffic from Home to E would use only line II, since a transfer from line I to line II would add 15 minutes to the travel time (assuming a 30 minute headway). Dial modifies Moore's minimum path algorithm (Moore, 1957) to include the transfer penalties. An alternative approach, proposed by Fearnside and Draper (1971), involves the introduction of line-specific links and nodes. A stop is then represented by a set of nodes connected by transfer links. This leads to the multiplicity of links and nodes that Dial (1967) was attempting to avoid. The representation of the transit network in Figure 1 in terms of line-specific links and nodes is shown in Figure 2. This representation, however, tends to disguise the common line problem. Where a number of lines share a stop, the passenger may be faced with a difficult choice between taking the next line to arrive at the stop or waiting for a faster line. Where lines offer the same level of service (in this case travel time to the destination), the probability of taking each line would be proportional to its frequency.

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Advanced Modeling for Transit Operations and Service Planning

Figure 2. Transit network with node-specific nodes and links An early model for transit assignment was TRANSEPT (Last and Leak, 1976), which had some interesting innovations. Links are defined which would typically include a number of stops. Each link starts at a stop and ends at a stop, with interchange allowed only at link ends. A dual network is then constructed by connecting the mid points of links by new links, referred to as "direct links". The representation of the transit network in Figure 1 in terms of direct links is shown in Figure 3; for clarity, only direct links not involving a line transfer are shown. Least cost paths are constructed from the direct links, whose generalised costs include waiting times and interchange penalties. The waiting times are calibrated functions of the average headways, which in turn are calculated from the sums of the frequencies of the lines using a direct link. Paths run from mid link to mid link in the primal network. Trips run from zone centroid to zone centroid. For each zone centroid, those links with at least one stop within a preset maximum walking distance of the zone centroid are found. Access and egress links are created by connecting the mid points of these links to the zone centroids, and generalised costs are calculated for the access and egress links taking walking time, invehicle time and fare into account. For a given pair of zone centroids, the paths with the least generalised cost are found for each pair of access and egress links. The best n of these are retained for assignment. Traffic is assigned to the best n paths according to a logit path choice model.

Capacity Constrained Transit Assignment Models and Reliability Analysis

185

Figure 3. Transit network with direct links TRANSEPT takes capacity constraints into account. An iterative loading process is used to calculate the probabilities of overloading. The sequence in which the links are loaded is such that, when a link is loaded, all the upstream links have already been loaded, ensuring the priority of those on board over those seeking to board. For each primary link and route, a probability histogram for the number of seats available is constructed. This is fed into the waiting time sub-model together with an initial probability histogram of the number of seats demanded. By convoluting probability distributions for seat availability, passengers arriving and passengers alighting, the sub-model calculates the probability that an individual arriving at a stop for a particular route will not be able to get on the first vehicle that arrives and on subsequent vehicles. From this, an estimate of average waiting time is obtained, which leads to an updated probability histogram for the number of seats demanded. Waiting times and route demands are successively updated in this way until they stabilise. When a stable loading for the route is achieved, the seat availability histograms for the route are updated. This process must be repeated for each route many times until the overall stability is achieved.

3.

COMMON LINES, HYPERPATHS AND STRATEGIES

A passenger at a stop frequently has a choice between a number of lines, referred to as common lines, which will get him directly or indirectly to his destination. The lines may differ in their attractiveness, perhaps due to the travel time to the destination, the number of changes, the probability of seat availability, etc. A dilemma frequently faced is whether to

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Advanced Modeling for Transit Operations and Service Planning

take the next vehicle on a relevant line or to wait for a more attractive line. Often the choice is determined by the first vehicle to arrive. This family of issues is referred to generically as the common lines problem. Lampkin and Saalmans (1967) assumed that the passenger at the stop ignores lines that are obviously bad (for example, lines where the travel time to the destination is more than by the best route plus half the headway for the best route) and chooses the first vehicle to arrive from among the other routes. This introduces the notion of a strategy, which consists of a choice set of attractive lines and a selection rule. Chriqui and Robillard (1975) presented a probabilistic framework for studying the common lines problem. The passenger at a stop selects the sub-set of lines which minimises his expected travel time on the assumption that he selects the next vehicle serving a line within that sub-set. To solve this problem where the number of lines precludes the complete enumeration of the sub-sets, Chriqui and Robillard propose the following heuristic. The lines are ranked according to the increasing expected travel time and the choice set is empty. The line with the minimum expected travel time is added to the choice set. The next ranked line is then added to the choice set if by so doing the expected travel time is reduced. This process continues until the expected travel time is not reduced any further or until all lines have been included. They conjecture that this greedy type algorithm produces the optimal choice set. Spiess and Florian (1989) combined the common lines problem and the equilibrium assignment problem in a linear programming framework. Passengers choose a set of routes to minimize their expected travel times. They always board the first vehicle to arrive provided it serves one of their chosen routes. To find the solution, a non-linear mixed integer program with a total travel time objective plus flow conservation and non-negativity constraints was first formulated for each destination. Non-trivial variable transformations produced a compact linear program for each destination. The dual of this linear program may then be solved efficiently by a generalisation of Dijkstra's algorithm. Capacity constraints were not included, although it was noted that the effect of not being able to board the first vehicle on an attractive line would be to reduce the "effective frequency" of the line. The approach of Spiess and Florian was given a graph theoretic framework by Nguyen and Pallottino (1988) who introduced the concept ofhyperpaths. A hyperpath connecting an origin to a destination includes all the elemental paths that could be used by a passenger, and thus encapsulates his strategy. Costs consist of link travel costs and node delay costs. The share of traffic on each link leaving any node in a hyperpath is proportional to the respective service frequencies on those links, so the distribution of traffic across the elemental paths can be calculated sequentially. Nguyen and Pallattino (1988) noted that similar sequential procedures have been obtained by Dial (1971) for the probabilistic assignment of the road traffic. However, the distribution of hyperpath flow does not suffer from the "independence of irrelevant alternatives" disadvantage that besets logit assignment.

Capacity Constrained Transit Assignment Models and Reliability Analysis

4.

187

CAPACITY CONSTRAINTS

Having presented their fixed link cost model, Spiess and Florian (1989) then developed a model where the link costs depend on the passenger flows, to reflect the greater inconvenience of crowded vehicles and the longer average waiting times due to the greater variance in the vehicle headways caused by the "bus bunching" phenomenon. They do not specify the cost function, but assume it to be continuous and increasing with flows. They acknowledge, however, that this extension does not allow for full vehicles. To accommodate the possibility of not being able to board the next vehicle due to insufficient capacity, they suggest reducing the "effective service frequency" on the relevant link. The first transit network model to incorporate a limited line capacity appears to be due to De Cea and Fernandez (1993). Passengers follow a sequence of transfer nodes from origin to destination. Transfer nodes are connected by links referred to as "route sections". The representation of the transit network in Figure 1 in terms of route sections is shown in Figure 4. A common lines problem is solved between each pair of successive transfer nodes. Waiting times at transfer nodes, as well as the passenger distribution between transfer nodes, depend not only on the flow boarding but also on the flow already on the line. Rather than using flowdependent link costs to capture the "discomfort" effect of congestion, De Cea and Fernandez concentrate congestion at the transfer nodes. A BPR-type function is used to estimate the wait at each transfer node in terms of the "through flow"-to-capacity ratio. However, this is not based on queuing theory and still allows capacity to be exceeded (of course, at a cost). Cominetti and Correa (2001) present a framework for congested transit assignment that can incorporate congestion functions obtained from queuing models. Passengers are assumed to travel by fastest (or more generally least cost) hyperpaths. Link waiting time is defined to be equal to the inverse of the "effective frequency", which is itself expressed as a function of the vector of link flows. The distribution of traffic across the links of a hyperpath that leave any node is proportional to the effective service frequencies for those links. The literature reviewed thus far has considered service frequencies but not timetables. Tong and Wong (1999) highlight the difference between frequency-based and schedule-based models, which consider timetables explicity. The Tong and Wong model is a form of simulation, as passengers or packets of passengers are moved through the network in time steps. The passengers are either on a walking link, queuing on a platform or on a transit link. The shortest paths are found with a time-dependent branch and bound method, as first described in Tong and Richardson (1984). The path costs include the weighted waiting time, walking time, in-vehicle time and a line change penalty.

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Advanced Modeling for Transit Operations and Service Planning

Figure 4. Transit network with route sections In an extension, Poon et al. (2001) look at the case where the transit network is congested. The time-step approach allows the prediction of the number of passengers on board and how many more passengers want to board. If there is not enough spare capacity on the vehicle for all the passengers waiting on the platform, passengers have to wait for the next vehicle. Using the Method of Successive Averages, some passengers are allocated to other routes, which are now quicker because of the increased waiting time on their preferred routes. If not all passengers can board the vehicle, it is assumed that those passengers who arrive first board first (First-in-First-out, FIFO).

5.

LEVELS OF INFORMATION

The review reveals the amount of attention given in the literature to the common lines problem. While this problem is clearly encountered in most urban bus networks, particularly in developing countries (De Cea and Fernandez, 1993), it does not arise where passengers have prior knowledge of the schedule and services run to schedule. In this case, passengers could plan in advance which service to use and time their departures accordingly. There would be no frequency related wait. Many commuter rail services will be of this type. The spread of information services may make on-line planning possible too. Another situation where the common lines problem does not apply is where stops are line-specific. Much (but not all) of the London Underground is like this. Taking another line implies a walk (sometimes lengthy) between platforms.

Capacity Constrained Transit Assignment Models and Reliability Analysis

1 89

NOTATION Let / /*

A A* L L *(i) IN(i) OUT(i) IN*(i) OUT*(i) ca fa fi wa va

Sia

6.

Set of nodes in the transit network Set of nodes on hyperpaths to destination s Set of links in the transit network Set of links on hyperpaths to destination s Set of lines Set of attractive lines (lines on hyperpaths) passing through node i*s Set of links that lead into node i^s Set of links that lead out of node i^s Set of links on hyperpaths to s that lead into node i^s ( IN*(i)^fN(i) ) Set of links on hyperpaths to 5 that lead out of node i^s ( OUT*(i)cOUT(i) ) Travel time on link a Frequency of service on link a (fa > 0) Frequency of service on line / (// > 0) Waiting time for link a Flow on link a Trips destined for destination s originating at node i*s Probability that a passenger fails to board at node i^s Generalized cost of travel from node i^s to node j^s 1 if line / operates on link a, and 0 otherwise Capacity of line / in terms of number of passengers per unit time passed a given point

NETWORK REPRESENTATION AND CONSTRAINTS

We adopt the network representation proposed by De Cea and Fernandez (1993), elaborated to require a link for each line. A trip follows a path consisting of a series of at least two nodes representing the stops, the first being the boarding stop and the last being the alighting stop. At stops where a number of lines are attractive with regard to the given destination s, referred to as transfer nodes, a common lines problem is solved. By definition the destination is not a transfer node. Where travel is possible between nodes without changing line, links, referred to as route sections, are added to the network. There is one route section for each line. The division of traffic at transfer nodes is in proportion to the service frequencies for the attractive lines, so

190

Advanced Modeling for Transit Operations and Service Planning

vfl = (Zfl-eoi/rYfl va)fa I lLa-eovT(i)fa; a=(ij) eA *

(1)

The assumption made here is that arrivals at node / are random, and therefore uncoordinated with departures. The cost of a trip consists of the wait before boarding, the in-vehicle time and the wait at any transfer node (transfer nodes along the route where there is no transfer are by-passed by route sections). For either boarding or transferring the average wait is Wa = ail^aeOUT*(i)fa

, d=(ij) eA*

(2)

measured in units of hours per passenger. The parameter a, has a value between 0.5 and 1, depending on whether the service headways are uniformly or exponentially distributed respectively. Services are assumed to uncoordinated, so the time of arrival at node / is treated as random. An assumption made frequently (see, for example, Spiess and Florian, 1989) is that there is only one value of a, for the whole network, so by choosing an appropriate interval of time over which to calculate the frequencies, this value can be normalized to 1. Note that if line / is an attractive line into a transfer node /' it cannot logically be an attractive line out, for a given destination 5.

7.

HYPERPATHS

Note that

fa = IjeL^a However, not all lines operating on link a will be attractive (i.e. lie on hyperpaths to destination 5). The set of attractive lines may be determined by solving the following nonlinear mixed integer programming problem: PI : Minimise Zfl&4 (ca + w^vfl with respect to va, a eA, and xa, a eA, subject to ^LaeOUT(i) Va ~ T,aeIN(i) Va = Oj , i^S

(3)

Wa = ai/T.a-eOUT(i)Xa'fa-,Cl=(i,j)eA

(4)

Va =

(5)

(Za'eOUT(i) V«-) Xafa

\Va I a,

xa = (0,1), a a4

(6)

va > 0, aeA

(7)

Capacity Constrained Transit Assignment Models and Reliability Analysis

191

Constraint (3) represents flow conservation at each node while (4) constrains the wait at each stop to be inversely proportional to the sum of the relevant service frequencies. Constraint (5) divides the flow leaving a node between relevant links according to the relative frequencies of service on those links. Finally, (6) and (7) are binary variable and non-negativity constraints respectively. Following Spiess and Florian (1989), PI may be transformed into the following compact linear program: ?2: Minimise Za&4 cava + Z/e/ W( with respect to va, a eA, and Wj, iel, subject to

where

is the total wait (as opposed to the wait per passenger) at node i measured in units of passenger-hours per hour. The logic of (8) is that for passengers leaving node i the probability of choosing link a=(i,j) eA is either proportional to the frequency of service on that link (if the constraint is active) or 0 (if the constraint is inactive). In the former case, the link is on the hyperpath, while in the latter case, the link is not on the hyperpath. Cases where the constraint is not active but va > 0 do not arise (see Spiess and Florian, 1989). At the solution to ?2 (denoted by *), we can determine the sets IN*(i) and OUT*(i) as follows: a=(i,j)eIN*(j) and a=(ij) eOUT*(i) if and only if v*a > 0. This is so because every link in the hyperpath will be used, since/, > 0. Having determined IN*(i) and OUT*(i), (1) and (2) allow the determination of transition probabilities and nodal delays. In order to solve ?2, it helps to formulate the dual problem: PS: Maximize Z/e/ 0,-w,- with respect to U{, iel, and wa, a eA, subject to

PS may be solved by a Dijkstra-like procedure (see Spiess and Florian, 1989).

192

8.

Advanced Modeling for Transit Operations and Service Planning

LINE LOADING BY MARKOV CHAIN

A Markov chain is characterized by a transition matrix defining the probability of an entity (in this case a passenger) moving from one state to another state. Conservation requires the rows of this matrix to sum to one. In this context, the states represent the transfer nodes of the transit network, the origins, a destination and a notional "bin" where trips that "fail" collect. In this context, a trip "fails" when there is insufficient capacity on the line on which the passenger seeks to board. The transition probabilities are both line- and destination-specific. The destination and the bin constitute absorbing states, in the sense that this state once entered is not left. In the event of multiple destinations, these have to be considered one at a time. In order to establish the available capacity at each stop it is necessary to load the lines in the chronologically correct sequence. Destination-specific transition probabilities may be used to preserve the boarding sequence. Let: tfji =

Probability of transition from stop i to stop j by line / on hyperpaths to destination 5

then

Further define

and

From the definition

or in matrix notation

Capacity Constrained Transit Assignment Models and Reliability Analysis

193

where y = [yj, o = [oj, T = [tij+], Tt = [tyj and / is an identity matrix of appropriate dimension. This series is convergent provided T" —> 0 as n —> oo. If the row sums of T are bounded by 1 and at least one row sum is strictly less than 1, and provided each destination is reachable from any node (in which case it is an ergodic system), it follows from the PerronFrobenius Theorem (see Cox and Miller, 1965) that all the eigen values of T are strictly less than 1 and so T" —> 0 as n —> oo.

9.

ENFORCING CAPACITY CONSTRAINTS

So far, a capacity constraint has not been enforced. Current thinking (see Spiess and Florian, 1989; De Cea and Fernandez, 1993; Cominetti and Correa, 2001) seems to be that, in order to enforce this constraint, the "effective frequency" of the line should be reduced at the point where the overloading first arises. The justification for this is that some passengers will not be able to board the first vehicle that arrives at that stop due to the overloading. De Cea and Fernandez (1993), the first offer a solution, suggest using BPR-type waiting time functions at the stop, where the capacity available at a stop on each line depends on the line capacity and the passenger "through traffic" on that line at that point. The effective frequency is proportional to the inverse of the waiting time. Cominetti and Correa (2001) then look at replacing the BPR-type functions by queuing models. As a result of the capacity constraint, we require that the boarders at each transfer node / should not exceed the available capacity across all lines serving the stop that are in the attractive set.

To prevent capacity being infringed, some passengers will not be able to board. Hence

Note that (12) and (13) are simultaneous equations for a given set of hyperpaths. However, one would expect the probability of not boarding to influence the set of hyperpaths in some way.

1 0.

CAPACITY CONSTRAINED TRANSIT ASSIGNMENT

The risk of failure to board can be included in the determination of hyperpaths by defining the following generalised cost of travel:

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where ft is a weight representing the risk averseness of the traveller. The following problem finds a risk-averse transit assignment for given failure-to-board probabilities: P4: Minimise Za=(«,/;>&4 gijVa with respect to va, a<=A, and Wh iel, subject to

If all ca and Wi were zero, the minimum cost route would be the most reliable route, hence the functional form chosen for the risk component of cost (sometimes referred to as the safety margin).

11.

EXAMPLE

This example is designed to illustrate the effect of the capacity constraint on routing for various levels of risk averseness with respect to the failure-to-board probabilities. For the sake of simplicity, the hyperpaths are assumed to be singular. Figure 5 shows an example network with 4 stops and 4 lines. Figure 6 shows the movements that are allowed in the network. Note that U-tums are not permitted and that travel is either from left to right or from right to left. Where interchange is allowed, passengers can interchange between lines at the stops at a cost of 1 unit. The flow between each station is assumed to be 100 passengers per unit time. To ensure congestion, the capacity of each line is assumed to be 100 passengers per unit time. The cost function given in (14) is used. Tables 1 to 4 show the results for different /? values. In all cases, the capacity constraints are fully kept. Note that passengers from A to D would prefer to transfer at C (namely alight at 9 and board at 7 in Figure 5), if they were not averse to the risk of failing to board at 7. As /? increases from 0 to 100, we see the number alighting at 9 declines and the occupancy at 9 increases correspondingly. In the case of 5, the effect of greater risk averseness is for the passengers from A who are routed via B due to limited capacity on the direct line to C to squeeze out passengers wishing to board at B to C or D.

Capacity Constrained Transit Assignment Models and Reliability Analysis

195

Figure 5. Capacity constrained network example

Figure 6. Network represented by transfer links

12.

CONCLUSIONS AND DISCUSSION

The chapter presents a capacity constrained transit network loading method which respects the sequence in which passengers board transit vehicles. When combined with a route choice model that takes into account the probability of not being able to board, this leads to a capacity constrained transit assignment method. The network loading method does not require any simulation and is computationally efficient. Failure probabilities for each node are defined so that the vehicle is as full as possible at every platform. It can be proven that these failure probabilities are unique for non-circular lines.

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Advanced Modeling for Transit Operations and Service Planning

The model as presented can be used to assess capacity problems of an existing transit network. For network planning purposes the effect of line capacity changes (like timetable or vehicle size/train length changes) or changes in the infrastructure can be analysed. The equilibrium failure-to-board probability is a good measure of capacity problems. The example shows that when passengers are averse to waiting, they will use costlier routes, if these are available. As the example also shows, this may be at the expense of passengers trying to board down stream. The model allows an analysis of the inefficient use of capacity when short distance travellers block access to lines by the long distance travellers.

ACKNOWLEDGEMENTS The authors gratefully acknowledge the helpful comments of Alan Nicholson and Markus Friedrich on an early draft of this chapter and the financial support of EPSRC that enabled much of the background work to be carried out. The chapter also benefited from the constructive criticism of participants at a workshop in Hong Kong, in particular (in alphabetical order) David Boyce, Ben Heydecker, William Lam, Hong Lo, S. C. Wong and Hai Yang.

REFERENCES Chriqui, C. and P. Robillard (1975). Common bus lines. Transportation Science, 9, 115-121. Cominetti, R. and J. Correa (2001). Common lines and passenger assignment in congested transit networks. Transportation Science, 35, 250-267. Cox, D. R. and H. D. Miller (1965). The Theory of Stochastic Processes. Chapman and Hall: London. De Cea, J. and E. Ferbandez (1993). Transit assignment for congested public transport system: An equilibrium model. Transportation Science, 27(2), 133-147. Dial, R. B. (1967). Transit pathfinder algorithm. Highway Research Record, 205, 67-85. Dial, R. B. (1971). A probabilistic multipath traffic assignment model which obviates path enumeration. Transportation Research, 5, 83-111. Fearnside, K. and D. P. Draper (1971). Public transport assignment - a new approach. Traffic Engineering and Control, 12, 298-299. Lampkin, W. and P. D. Saalmans (1967). The design of routes; service frequencies and schedules of a municipal bus undertaking: A case study. Operational Research Quarterly, 18(4), 375-397. Last, A. and S. E. Leak (1976). TRANSEPT: A bus model. Traffic Engineering and Control, 17,14-17,20.

Capacity Constrained Transit Assignment Models and Reliability Analysis

197

Moore, E. F. (1957). The shortest path through a maze. Proceedings of International Symposium on the Theory of Switching, Harvard University, April. Nguyen, S. and S. Pallottino (1988). Equilibrium traffic assignment for large scale transit networks. European Journal of Operational Research, 37, 176-186. Poon, M. H., S. C. Wong and C. O. Tong (2001). A dynamic schedule-based model for congested transit networks. Submitted to Transportation Research. Spiess, H. and M. Florian (1989). Optimal Strategies: A new assignment model for transit networks. Transportation Research, 23B, 83-102. Tong, C. O. and A. J. Richardson (1984). A computer model for finding the time-dependent minimum path in a transit system with fixed schedules. Journal of Advanced Transportation, 18, 145-161. Tong, C. O. and S. C. Wong (1999). A schedule-based time-dependent trip assignment model for transit networks. Journal of Advanced Transportation, 33(3), 371-388.

198

^ico CM

T-

8

o o 0 o

R 8

o

c

n

CN

o

Advanced Modeling for Transit Operations and Service Planning

o o

co co in

0

I6 8

8

o in

o

8 ~

o

~

o

0

8

o o o

o

o o o o o T T

o

co co o co

O)

oo

0

o

o o o o

o 0 o o o

^ o o o o o CD

in

8 o o o

pCO

0

m

88

0) o o CO m Q. Q. CO D)

o

o

^ o co CM

0)

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o

Table 4. Results for p= 100 Occupancy

0

0

0

0

100

100

100

0

34.9

68

63.8

65.4

0

0

0

0

65.1

32

36.2

34.6

100

100

100

100

69.8

100

62.4

32

36.2

34.6

0

100

0

100

69.8

100

65.1

32

36.2

34.6

97.3

0

100

0

1

1

0.32 0.962 0.347

1

0.68

Space

100

Boarding

100

0

100

0

Alighting

0

100

0

100

1 0.349

Reliability

0.698

1 0.638

I

100

30.2

69.81

1 0.654

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CHAPTER 8

DYNASMART-IP: DYNAMIC TRAFFIC ASSIGNMENT MESO-SIMULATOR FOR INTERMODAL NETWORKS Hani S. Mahmassani, University of Maryland, College Park, USA. and Khaled F. Abdelghany, United Airlines, Elk Groove, USA.

1.

INTRODUCTION

Private car use continues to increase in most urban areas around the world, exacerbating various associated problems such as traffic congestion, environmental degradation and high fuel consumption. With the dispersion of land-use activity patterns in many urban areas, serious challenges face the design of public transportation systems to provide an effective substitute for the private car. A more plausible approach would be to design and promote an intermodal transportation system, which integrates the private car with existing or planned transit modes. Considerable research over the past decade has been

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Advanced Modeling for Transit Operations and Service Planning

directed towards modeling route choice dynamics in urban transportation networks. Most existing and proposed Dynamic Traffic Assignment (DTA) models have focused on passenger cars as the principal, if not the sole component of urban traffic, and the main source of traffic congestion. These models do not provide the capability of assigning transit/intermodal trips and capturing the interaction between mode choice and traffic assignment. Ignoring such interaction limits the applicability of these models in congested urban networks that include a considerable number of transit and intermodal trips. For instance, the DYNASMART simulation-assignment model (Jayakrishnan et al, 1994; Mahmassani et al., 1998) includes buses as part of the vehicular mix; these follow pre-specified routes and timetables. However, the assignment of individual trips to the transit lines and the choice of mode of travel are exogenous to the model. As such, the model's capabilities are not sufficient to evaluate a range of ITS user services targeting Advanced Public Transportation Systems (APTS) as well as certain Advanced Traveler Information Systems (ATIS) intended to serve transit and intermodal system users. Such capabilities require explicit representation of (1) the supply characteristics of the system, in the form of a multidimensional network model of the available modes and their interaction along links as well as at nodes; and (2) the user decision processes that govern the choice of mode or intermodal combination along with the associated path. This Chapter presents a dynamic traffic assignment-simulation model for intermodal urban transportation networks, DYNASMART-IP, which provides the above capabilities. The model framework is a generalization of the approach underlying the DYNASMART simulation-assignment model. The implementation is a major re-engineered extension of the previous capabilities of the software. However, the intermodal framework is not limited to this particular simulation-assignment tool, and could be implemented using alternative platforms. The new model approach considers intermodal transportation networks consisting of different travel modes such as private cars, buses, metro/subway and High Occupancy Vehicles (HOV). The model captures the interaction between mode choice and traffic assignment under different traveler information provision strategies. It implements a multi-objective assignment procedure in which travelers choose their modes and routes based on a range of choice criteria. The model assumes a stochastically diverse set of travelers in terms of underlying preferences (i.e. relevant choice criteria and associated trade-off rules), as well as in terms of access and response to the supplied information. The modeling tool is intended primarily for operational planning applications, and it could also be used in conjunction with real-time traffic management systems. As a within-day dynamic assignment procedure, it overcomes many of the known limitations of static tools used in current practice. These limitations relate to the

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203

types of alternative measures that may be analyzed and to the policy questions that planning agencies are increasingly asked to address. The Chapter is organized as follows. A review of mode-route choice modeling is presented in the next section. The conceptual framework and structure of the model are then presented, followed by a description of the vehicle simulation components, especially those pertaining to the newly implemented transit and intermodal elements. Two sections present the assignment components of the model, which include the multiobjective shortest path algorithm and the mode-route choice procedures. A set of simulation experiments designed to illustrate the capabilities of the model are presented, followed by discussion of the results. Concluding comments are outlined in the final section.

2.

MODE-ROUTE CHOICE DYNAMICS IN INTERMODAL NETWORKS

Several approaches have been proposed over the past two decades to model and analyze transportation networks with several interacting modes. These approaches have addressed primarily static assignment problems, jointly with mode choice. Early contributions considered the "multimodal" version of the problem, in which the entire trip takes place via a single mode of travel from origin to destination (Florian and Nguyen, 1978; Florian and Los 1978; Florian, 1977; Abdulaal and LeBlanc, 1979; LeBlanc and Fahrangian, 1981; Florian and Spiess, 1983; Sheffi, 1985). Trips are hence separated by mode, and separately assigned to each mode's sub-network, though with possible interactions captured through the link performance functions when transit vehicles and passenger cars share the same right of way. In these approaches, intermodal trips would be modeled as a sequence of independent trips, generated respectively at the initial origin node and at intervening transfer nodes where travelers switch between the different modes. The transfer phenomenon inherent in intermodal trips is not represented in these models. In another similar approach, intermodal trips are considered in the formulations by defining every feasible combination of modes, or a subset of the more meaningful combinations of modes as a new mode that is added to the mode choice set (Fernandez et al, 1994). Different assignment procedures have been proposed in the literature for transit trips, typically for the static case. Baaj (1990) gives a review of these procedures. An early approach, similar to the auto assignment, assigns passengers to the minimum expected

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Advanced Modeling for Transit Operations and Service Planning

travel time path in the transit network (Le Clercq, 1972; Dial, 1986). A more reasonable behavioral representation of the choice process defines a set of feasible paths from which passengers could choose to minimize their expected travel times (Nguyen and Pallottino, 1988; De Cea et al, 1988; Spiess and Florian, 1989; Hickman and Beradtien, 1997). For example, Spiess and Florian (1989) used the term "optimal strategy" to describe a feasible path in the traveler's choice set. The number of strategies that a traveler may choose from depends mainly on the information that is available during the trip. The method proposed by Andersson (1977) moves toward more realistic behavior representation of the passengers' choice than simply minimizing the travel time. It enumerates a set of paths among which one is selected by a rather elaborate set of heuristic rules that seem to work quite well in practice. An example of these heuristic rules is also presented by Han and Wilson (1982), and implemented by Baaj and Mahmassani (1990) and Shih and Mahmassani (1994) in their transit network design model. The brief overview of existing approaches reveals several types of limitations. Foremost among these is the incomplete representation of intermodal trips, by not allowing explicitly for transfers between modes, associated waiting times and other unique attributes of intermodal travel. Second, limitations exist in modeling the supply interactions among the various available modes, especially when the infrastructure is shared. The third set of limitations arises from the assumption of static versus timevarying network conditions, including time-varying demand and associated congestion patterns. The limitations of static assignment models for congested network modeling, for both autos and transit modes, are well recognized in the literature (Srinivas, 1994), particularly with regard to ATIS/ATPS applications. The last but not least set of limitations arises from the behavioral assumptions governing user's choice of mode or intermodal combinations, along with the selection of specific path in the associated intermodal network. For example, most existing formulations considers travel time as the only criterion for mode-route choice, thereby ignoring the trade-offs among the conflicting criteria that travelers usually consider in their decision process, such as access and waiting times, parking cost, and number of transfers. Dial (1997) proposed an algorithm for the bicriterion traffic assignment problem. The two criteria considered are assumed to be flow-dependent and integrated through a generalized cost function. However, the problem is formulated only for the single-mode static case. Introducing these considerations in the interest of greater behavioral realism in mode choice and assignment procedures generally introduces mathematical complications that preclude direct application of mathematical programming models to well-behaved formulations.

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205

Hence, a simulation-based approach to the dynamic assignment problem in intermodal networks is adopted, in an attempt to combine behavioral realism with tractability, efficiency and relevance to practice.

3.

CONCEPTUAL MODEL

The model represents several modal networks through a single integrated multidimensional network. Associated with each link are two state vectors for each time interval, representing respectively the number of vehicles of each class on the link, and the associated cost incurred by each class in traversing that link (when the link is entered during the specified time interval). There is no restriction on the number and types of vehicle classes that may be considered in the model. Typical classes of relevance to the study of intermodal networks include autos, trucks and various types of transit modes. They may also include HOV vehicles. The associated cost vector provides the principal mechanism for designating certain links for particular classes. For example, a very high cost for a single occupant auto on a certain link, coupled with the actual travel time for a HOV, could indicate a special HOV facility. Similarly, a transit network may be represented to allow both exclusive (e.g. underground rail) and shared right of way (e.g. buses). Transfer penalties at major transfer nodes in the network are explicitly modeled. For each traveler, the waiting time till the arrival of the next vehicle that serves the chosen transit line, and the parking cost at the park-and-ride facility are considered while evaluating the different travel options. The model captures explicitly the dynamic interactions between mode choice and traffic assignment in addition to the resulting evolution of the network conditions. It determines the time-dependent assignment of individual trips to the different mode-routes in the network, including the corresponding arc flows and transit vehicles loading. Figure 1 illustrates the modeling framework and the different components that are designed to address the above problem requirements. As noted, the model can accept as demand input a file listing the population of travelers, their attributes and travel plans (including origins, destinations, time of departure), and mode choice if known. However, a more likely way of applying the model is to generate travelers on the basis of prespecified time-dependent Origin-Destination (OD) zonal demands. Each generated traveler is assigned a set of attributes, which include his/her trip starting time, generation

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Advanced Modeling for Transit Operations and Service Planning

link, final destination and a distinct identification number. A binary indicator variable is also assigned to each traveler to denote car ownership status. In parallel, transit vehicles are generated according to a pre-determined timetable and follow pre-determined routes. Prevailing travel times on each link are estimated using the vehicle simulation component, which moves vehicles capturing the interaction between autos and transit vehicles, as described later. The model also estimates other measures that may be used by travelers as criteria to evaluate the different mode-route options, including travel distances, parking cost, highway tolls, transit fares, out of vehicle time, and number of transfers along the route. A mode-route decision module is activated at fixed intervals to provide travelers with a superior set of mode-route options. The activation interval (usually in the range of 3 to 5 minutes) is set such that the variation in network conditions is captured, while retaining desirable computational performance for the procedure. The route-mode decision module consists of a multi-objective shortest path algorithm designed for large-scale intermodal transportation networks, which is described separately. This multi-objective shortest path algorithm generates a set of superior paths in terms of the set (or a suitable subset) of attributes listed above. Considering diverse set of travelers' behavioral rules as well as different levels of information availability and response, travelers evaluate the different mode-route options and choose a preferred one. These behavior rules and response mechanisms are implemented through a behavior component within the model as described in a subsequent section. Each option represents an initial plan that a traveler follows (unless he/she receives enroute real-time information of a better plan) to reach his/her final destination. This plan describes the used mode(s) and the route to be followed including any transfer node(s) along this route. Based on the available options, a traveler may choose a "pure" mode or a combination of modes to reach his/her final destination. If a traveler chooses private car for the entire trip or for part of it, a car is generated and moved onto the network with a starting time equals to its driver's starting time. Each newly generated vehicle is assigned an ID number that is unique to this vehicle. Vehicles are then moved in the network subject to the prevailing traffic conditions until they reach their final destinations or the next transfer node along the pre-specified route (in the case of an intermodal trip).

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Figure 1. Overall Modeling Framework

If a traveler chooses a transit mode, he/she is assigned to a transit line such that the destination of this passenger is a node along the route followed by the bus line. If no single line is found or if the passenger is not satisfied with the available single line, the passenger is assigned to a path composed of two lines with one transfer node, such that

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Advanced Modeling for Transit Operations and Service Planning

the destination of the passenger is a node along the route followed by the second bus. If no such two lines are found, the search is continued for three lines with two transfers. It is assumed that no passenger would be willing to incur more than two transfers in his/her trip. Thus, if no path with a maximum of two transfers is available, the trip is indicated as infeasible. Given the passenger's origin node, the nearest transit stop along the first line in the passenger's path is determined, and he/she waits until the arrival of the next vehicle that serves that transit line. When a transit vehicle arrives at a certain stop, all passengers waiting for a vehicle serving this specific line board this vehicle (subject to a capacity constraint) and head towards either their final destination or the next transfer node along their route. Upon the arrival of a vehicle (private car or transit vehicle) to a certain destination node, this destination is compared to the final destinations of the travelers on board. If it matches the final destination of a traveler, the current time is recorded for this traveler as his/her arrival time. If they are different, the traveler transfers to the next transit line in his/her plan. The nearest stop is again determined and the traveler waits for his/her next transit vehicle. The time difference between arrival at the transfer node and boarding of the next line is calculated as the waiting time at the current transfer node for this traveler. This process is continued until all vehicles reach their final respective destinations. If a traveler misses the initially assigned transit vehicle because of late arrival or because the vehicle does not have enough space, the model allows the traveler to re-plan his/her trip. The available options are regenerated for this traveler and he/she makes a selection according to the decision process described in a subsequent section.

4.

VEHICLE MOVEMENT SIMULATION

The vehicular traffic flow simulation logic in DYNASMART has been adapted to better represent interactions among transit vehicles and autos. The essential features of the traffic flow simulation logic have been described elsewhere, and will not be repeated here (Jayakrishnan et al, 1994; Mahmassani et al., 1998). For completeness, those elements pertaining to transit vehicle simulation in the context of the overall traffic flow simulation are briefly described. This simulation component is a time-based simulation which moves individual vehicles along links according to local speeds determined consistently with macroscopic traffic

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stream models (i.e. a speed-density relation, of modified Greenshield's form, is used in this implementation). Every time step, the number of vehicles on each link is calculated using conservation principles; numbers in each class of vehicles in the traffic mix are kept separately. Consistently with the macroscopic logic for modeling vehicle interactions, average passenger car equivalent factors are used to convert each vehicle type to the equivalent passenger car units. The resulting equivalent-car concentration is then calculated for each link, and used to estimate the corresponding speed through the speed-density relation. These speeds, updated continually to reflect prevailing conditions, determine vehicular movement on that link. Queuing and turning maneuvers at junctions are explicitly modeled, thereby ensuring adherence to first-in-first-out principles as well as traffic control devices at junctions. Vehicles that reach the end of the link and are unable to move to a downstream link because of capacity limitations join the back of a queue of vehicles at the downstream end of the link. The physical size of the queue is explicitly represented in the simulation resulting in the division of the link into a moving part and a queuing part. Vehicles that reach the back of the queue must wait until vehicles ahead of them are discharged. All inflow and outflow constrains that limit the number of vehicles entering and leaving each link, under the prevailing traffic control, are enforced. The right of way among competing movements is allocated according to the existing control element at every intersection. The outflow constraints limit the maximum number of vehicles allowed to leave any given approach of an intersection, reflecting the available vehicles in queue and outflow capacities of the approach under the prevailing control. The inflow constraints bound the total number of vehicles that are allowed to enter a link. These constraints bound the total number of vehicles from all approaches that can be accepted by the receiving link, which reflects both physical storage consideration and inflow throughput capacity. If a bus stop is located along a particular link, and a bus is stopped at this location, the storage capacity of the link is reduced accordingly to represent the bus stopping effect. In addition, the inflow and outflow rates of this link are adjusted based on the location of the bus stop within the link. A near-end stop (i.e. located at the upstream end of the link) reduces the link inflow rate while the far-end stop (i.e. located at the downstream end of the link) reduces the link outflow rate. The factors by which the storage capacity and the flow rates are reduced could vary from one complete lane blockage to zero lane blockage in the event of a special-purpose bus bay.

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Advanced Modeling for Transit Operations and Service Planning

DYNASMART allows representation of complete transit networks, with both exclusive and shared infrastructure. This flexibility is allowed by its integrated multidimensional network representation as described earlier. A set of bus lines is defined in terms of the constituent routes, for which the average headway, stop locations, and vehicle capacities are specified. Different bus capacities may be specified for the different routes. Given a timetable, buses are generated from their origin terminals and moved in the network along their pre-specified routes following prevailing traffic conditions. The model tracks all buses along their routes and records their respective arrival times at each stop. Upon arrival at a bus stop, buses are held to allow passengers to board and alight. The number of passengers on board (bus-occupancy) is updated, representing the new bus occupancy, which is also tracked along the vehicles' routes. If a vehicle is full, no passengers are allowed to board and all waiting passengers are reassigned to the next bus or to another trip plan. The model is capable of simulating special bus services such as express service with limited stops and bus services with different deadheading strategies in which some stops could be skipped under certain conditions. The metro/subway service model is quite similar to the bus service model in most of its features. However, metro vehicles are assumed to have their separate right of way and hence move with predetermined speeds.

5.

MULTI-OBJECTIVE SHORTEST PATH ALGORITHM

As mentioned earlier, the model applies a multi-objective assignment procedure. This is implemented through a multi-objective shortest path algorithm which is designed especially for large-scale intermodal networks. This section defines briefly the multiobjective shortest path problem and describes the algorithm implemented within the current version of the model. More detail can be found in Abdelghany and Mahmassani (1999) and Abdelghany (2001). Compared to the traditional single-objective shortest path (SOSP) problem, the multiobjective shortest path (MOSP) problem has two main characteristics: First, in the normal sense of optimal solution, the multi-objective shortest path problem generally has no "optimal" solution, in that there no guarantee that a single path that simultaneously optimizes all objectives exist. For instance, in a transportation network, the least expensive paths usually have the longest travel time, and the fastest ones are usually the most expensive. Nevertheless, the set of Pareto Optimal or non-dominated paths can be determined. This set has the following characteristics: for any non-dominated path, it is

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not possible to determine a path which improves some of its objectives without at least one of the remaining ones getting worse. Second, the MOSP is known to be a NP-hard problem (Hansen, 1980), which means that there exists a family of graphs for which the number of optimal paths grows exponentially with the number of nodes in the network. Listing these paths would require an exponential number of operations and thus, no polynomial behavior can be expected. As shown in Figure 2, the implemented MOSP algorithm defines an intermodal transportation network as a set of layers such that each layer represents the subnetwork Gf(Nj, At) of a certain mode, where Nt is the set of nodes and Af is the set of arcs in the subnetwork of mode i . It determines the non-dominated paths from all nodes in the network to all destinations. As described below, the logic of the implemented algorithm is based on the "divide and conquer" technique in which the non-dominated sub-paths between every two transfer nodes in the network are first determined. The search in this step includes all modes' sub-networks (layers) connecting these two transfer nodes. These non-dominated sub-paths are then composed together to generate the non-dominated (single mode and intermodal) paths between the origin and the final destination. Multi-Objective Shortest Path Algorithm in Intermodal Networks Step 0: Initialize Identify the transfer nodes in the intermodal network. Step 1: Divide Do for every two-transfer nodes Do for every mode's sub-network Generate the non-dominated set of sub-paths between these two transfer nodes. End do End do Step 2: Conquer Compose these sub-paths to generate the non-dominated paths from the origin to the final destination. End Different algorithms have been suggested in the literature to determine the nondominated paths between two nodes (Hansen, 1980; Sancho, 1988; Dial, 1979; Climaco and Martin 1982). Abdelghany and Mahmassani (1999) implemented and compared two of these algorithms for the purpose of application to intermodal dynamic traffic assignment: the multi-labeling shortest path algorithm and the k-shortest path algorithm. The idea of the multi-labeling algorithm is that the function equation is extended from a

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scalar function to a vector-valued function such that all objectives under consideration are included. In addition, the standard minimization performed at each node is replaced by the selection of the non-dominated paths. The idea of using the k-shortest path algorithm is that paths are being determined by non-decreasing order of one of the objectives until all or at least a considerable number of the non-dominated paths are determined. The number of non-dominated paths in the final set depends mainly on the value of the parameter k.

G3(N3,A3)

Figure 2. Multi-layer Representation of Intermodal Networks The multi-labeling algorithm was found to outperform the k-shortest path algorithm in terms of the number of determined non-dominated paths in the experiments conducted. However, the k-shortest path algorithm succeeded in finding a considerable number of these non-dominated paths (at low value of k) in much less execution time. For this reason, the k-shortest path algorithm is implemented in this model. The k-shortest path implementation guarantees polynomial time execution on the model subnetworks. It also allows the generation of k paths from each origin to all destinations when performing a conventional single-objective assignment.

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213

MODE-ROUTE CHOICE MODELING

The model assumes a stochastically diverse set of travelers with different relevant choice criteria and response mechanisms to externally supplied information. The model framework allows implementation of different mode-route choice models that might adequately represent the travelers' behavior. The implemented model could be deterministic or stochastic and could be based on compensatory or non-compensatory choice rules. Deterministic models assume availability of perfect information for travelers and that travelers choose the best alternative based on the available information. Stochastic models (e.g. logit form or probit form), on the other hand, take into consideration that information might not be perfect and that travelers may have different perception to the supplied information. The mode choice model formulated and estimated for the experiments presented in this Chapter is based on a data set from a household activity survey conducted by the Central Planning Staff for the Boston metropolitan region in April 1991. An unordered, nonnested multinomial logit model was estimated to determine transit and single-occupant and high-occupancy vehicle (SOV and HOV) mode splits. The only exogenous variables included in the original model were total travel time and travel cost. These two variables were then combined to form a generalized cost term, using the value of travel time determined from the model. To facilitate the interface of the mode choice model with the shortest path calculation, the generalized cost is expressed in the units of time (minutes). The resulting systematic utility equations are:

where SR = shared ride (HOV), DA = drive alone (SOV), and TR = transit. The drive alone mode was treated as the base case. As expected, the negative constants in Equations (2) and (3) indicate that when the generalized cost is the same for all modes, the drive alone mode is preferred to the HOV and transit options. These negative constants incorporate all of the characteristics of the traveler and the travel mode not explained by travel time and cost.

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The choice probabilities for each modal alternative are then obtained from the usual multinomial logit model form:

where Prn(i) is the probability of individual n choosing alternative /, Vin is the utility for alternative i, VJn is the utility for alternativey, and J is the set of all possible alternatives (reduced to two in this application). Two main attributes are considered in the following experiments: total travel time (invehicle travel time and out-of-vehicle time) and total travel cost (travel cost, parking cost, transit fare, highway toll) for the trip. A fixed value of time across all travelers, taken as $6.0/hour, is used to calculate the generalized cost measure. All travelers are assumed to own a car, and to consider transit and intermodal travel options that involve at most one transfer along the trip. Thus, four modal options are assumed to be available for each individual as follows: private car, one bus line, two bus lines with one connecting transfer, and park-and-ride with one intermodal transfer.

7.

EXPERIMENTAL DESIGN

Different sets of simulation experiments are designed to illustrate the functionality and the capabilities of the model and to show the significance of including the mode choice dimension in the dynamic traffic assignment framework. Figure 3 depicts the test network used in these experiments, which represents the south central corridor in the Fort Worth area. The network consists of a freeway (1-3 5W) surrounded by a street network with a total of 178 nodes and 441 arcs, distributed over 13 zones. All arcs are twodirectional arcs and are actually modeled as two-way arcs. An arc could be a freeway link, HOV/HOT lane, on-ramp link, off-ramp link, arterial street or local street. The number of lanes along each arc corresponds to those of the real network. The freeway links have a mean free speed of 65 mph and all other links have a 35 mph mean free speed. With regard to intersection control, all signalized intersections (61 intersections) are assumed to operate under vehicle-actuated control. The maximum green value was set as 25 seconds for the 4-phase intersections and 55 seconds for the 2-phase intersections. The network does not contain three-phase signalized intersections. The minimum green is

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set as 10 seconds for all cases. The vehicle-actuated signal control supports different bus preemption strategies such as phase extension, red truncation and phase advance. The unsignalized intersections are set as follows: no control (62 intersections), yield sign control (24 intersections) and stop sign control (31 intersections). A set of bus lines is assumed to connect the different zones in the network through the main corridors as shown (as bold lines) in Figure 3. While the highway network is the actual system presently in operation in the area, the transit network is introduced here for illustrative purposes only, and does not correspond to available service (which is very limited). All bus lines are characterized in terms of route layout, location of stops, dispatch headways from the starting terminal, and the fare structure. All buses are assumed to share the same roadway infrastructure as the vehicular traffic. In other words, buses are assumed to move on the links with the prevailing speeds, which determine the buses' arrival times at the different stops along their routes. The experimental factors considered in this study can be separated into the following categories: Demand Levels: The intermodal network performance in terms of modal split and average travel and waiting times are compared under different congestion levels. Three different congestion levels are considered in the experiments, namely, congested, mild congested and light congested. Table 1 shows the number of travelers generated for each of these three cases for a loading period of 20 minutes in all cases (no travelers are generated afterward). With regard to the spatial distribution, of the O-D trip desires, a morning peak travel pattern is assumed. In this pattern, travelers head to the CBD area (the upper part of the network) from all other zones.

Table 1. The number of generated travelers for the different Congestion Levels Level Of Congestion Light Congestion Mild Congestion High Congestion

Number of Generated Travelers 6659 8874 11044

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Information Provision Strategies: The effect of different information supply strategies is examined in this set of experiments. Four different information schemes are considered, which are partly confounded with the awareness level of different modal and intermodal routing alternatives. Providing travelers with information on transit availability, opportunities for easy transfer and parking availability might induce many private car users to leave their cars and use a transit mode. The first scheme represents the full information scheme where travelers are assumed to have perfect information on transit lines, opportunities for transfer between these lines and park-and-ride facilities in the network (full info). The second scheme considers the case of not having information on the park-and-ride facilities in the network (no park-and-ride info). Thus, travelers cannot switch from their private cars to buses. The third information scheme assumes that travelers have no information on the available transfer options between the different transit lines. Passengers use buses only if there is a single line connecting their origin to the desired final destination (no bus transfer info). The last scheme assumes that travelers have no information on possibility of transfer either between different transit lines or from their private car to a transit line. All travelers in this case are assumed to use one mode for travel (no transfer info). The mild congestion level shown in Table 1 is used in this set of experiments. Vehicle operating cost is set to be $0.50/link and parking cost is designed as flat fare of $0.50 regardless of the parking period. All bus lines are assumed to operate with a frequency level of 12 buses/hour, and under a flat fare taken as $0.50. Vehicle Operating/Toll Cost: This set of experiments studies the effect of private car operating/toll cost on trip modal split and resulting overall network performance. The case in which trip travel time is considered as the only choice criterion is compared with other cases in which both trip time and trip cost are considered in evaluating the different travel options. Fixed cost per link of $0.00, $0.20 and $0.50 are assumed. This cost represents out-of-pocket cost due to tolls and/or hidden vehicle operating cost that travelers might consider. In this set of experiments, all bus lines are assumed to operate with a frequency of 12 buses/hour, and under a flat fare of $0.50. Free parking is assumed for all private car users. In addition, all travelers are assumed to have full information on the modal and intermodal routing alternatives in the network (full information scheme). Parking Cost: The effect of parking availability and pricing on travelers' mode-route choice and on overall network performance are also studied. Flat fees (i.e. that do not depend on the parking duration) of $0.00, $1.00 and $5.00 are assumed at all parking facilities in the network. For the $5.00 fee, two cases are compared: First, fees are imposed on all private car drivers who use their cars either for the whole trip or only part

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of it (park-and-ride trip); second, fees are imposed only on drivers who use their cars for the whole trip. Drivers willing to leave their car and complete the trip using a transit mode are guaranteed free parking. In these experiments, all bus lines are operated with a frequency rate of 12 buses/hour and under a $0.50 flat fare. Vehicle operating/toll costs are ignored in this set of experiments. The mild congestion level is considered. A full information scheme in which all travelers are assumed to have complete information on all possible travel mode options is used. Bus Frequency: This set of experiments examines the effect of transit availability on modal split and resulting overall network performance. The network performance is compared under three bus frequency levels: 3, 6, and 12 buses/hour. These frequencies are assumed to be the same for all the bus lines. The results are obtained the mild demand congestion case. The vehicle operating cost/toll cost of $0.50 per link is assumed across all private car users. A flat bus fare, taken as $0.50, is assumed for the twelve bus lines. In addition, no parking fees are imposed at any of the parking facilities in the network, and all travelers are also assumed to have full information on all possible travel options. Adding/Removing Transit Lines: The effect of adding or removing new transit service is considered in this set of experiments. The twelve bus lines are reduced to only eight lines. The four longitudinal bus lines are removed as shown in Figure 3. In both cases, buses are assumed to operate with a frequency of 12 buses/lines under a $0.50 flat fare. The demand level is considered to be mildly congested. In addition, the vehicle operating cost/toll cost of $0.50 per link is assumed across all private car users. HOV/HOTLanes: This set of experiments shows the effect of including HOV/HOT lanes as a new travel option in addition to transit. HOV lanes are special lanes on the freeway (or the arterial) that allow only high occupancy vehicles to pass through. They are designed to provide higher level of service than the other regular lanes of the facility. Such lanes provide travelers with an incentive to carpool, and could increase the personcarrying throughput of the facility. Recognizing that HOV lanes may have residual vehicular capacity, overall throughput and system performance can be improved by shifting a certain number of vehicles from the general-purpose (non-HOV) lanes to the HOV lane with no degradation in its level of service. One possible way to do so is to sell the residual capacity of the HOV lane to Lower Occupancy Vehicles (LOV) at a fair market price, resulting in so-called HOV lanes with hybrid operation or High Occupancy Toll (HOT) lanes. This is the rationale behind the HOT lanes, which allow LOV to purchase the privilege to travel through the HOV lanes at a higher service level than the

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regular lanes. At the same time, travelers who remain in the un-tolled lanes could see a reduction in congestion due to some vehicles shifting to the HOV lanes, thereby achieving better overall utilization of the physical capacity. The price mechanism is a plausible tool to allocate the use of the residual capacity, generating revenues for the operating entity as a side benefit. In addition, more people might be induced to carpool to save the toll and at the same time enjoy the advantages of the HOT lanes. The fees (tolls) charged for the use of the HOT lanes could be fixed or demand-dependent (high during peak periods and low during the off-peak). In both cases, the charged fees should be designed such that a certain level of service on the HOT lanes is maintained to keep the advantage for high occupancy vehicles. Two different sets of paths are generated for the LOV class. In generating the first set of paths, the algorithm ignores all the HOT lanes in the network. Hence, the generated paths do not include any HOT lane and the LOV are assigned only to the general-purpose network. For the second set of paths, the algorithm scans the entire network including the HOT lanes, after considering the charged fees if any. hi this case, the LOV driver can choose between using the general-purpose network without paying any fees, or using the HOT lane at additional out of pocket cost but with higher level of service. Because the HOV vehicles are not charged for using the HOV lanes, the algorithm generates only one set of paths for this class. The paths in this set may or may not encompass HOV lanes. Two HOV lanes are added to links 41-37 and 35-39, respectively. Four different operation scenarios are considered for these two HOV lanes. In the first scenario, no charges are applied to either HOV or LOV travelers, hi the second and third scenarios, the HOV travelers use the lanes at no cost. However, the LOV are charged to use these lanes. Fixed tolls of $2.50 and $5.00 are applied, respectively. In the last scenario, no LOV travelers are allowed to use the HOV lanes. Bus Preemption: In this set of experiments, bus preemption at signalized intersections is evaluated in the context of its potential network-level effects, including modal shifts by travelers to take advantage of improved transit service. Three main bus preemption strategies are widely applied: phase extension, red truncation and phase advance. The phase advance strategy is used as an example to these preemption strategies. In the phase advance case, if a bus is detected on any of the intersection approaches, one of the following two cases could be encountered. First, the phase that serves this bus is already active. In this case, the phase is directly extended until the detected bus is discharged. Second, the phase that serves the detected bus is not the active phase. In this case, the phase that serves the detected bus is advanced to minimize bus delay at the intersection.

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If the red truncation logic is implemented, the minimum green value is given to the remaining phases in the cycle in order to advance the bus phase. If the phase advance is implemented, the green of the active phase is immediately cut off and the bus phase is activated. In both cases, after the bus phase is terminated due to either max out or gap out, the normal phase sequence is resumed until the cycle ends. In this set of experiments, a scenario in which bus preemption is allowed at all signalized intersections is compared with the do-nothing scenario. In both cases, buses are assumed to operate with a frequency of 12 buses/lines at a $0.50 flat fare. The demand level is considered to be mildly congested. In addition, the vehicle operating hidden cost of $0.50 per link is assumed across all private car users.

Figure 3. The Test Network Showing the Simulated Transit Lines

8.

SIMULATION RESULTS AND ANALYSIS

8.1

Demand Levels

Table 2 shows the effect of network congestion on modal split and overall network performance. With the increase in network congestion, more travelers shift to use transit

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either for the entire trip or in conjunction with park-and-ride. For example, the percentage of private car usage decreases from 70.8% in the light congestion scenario to 68.10% in the congested scenario. Similarly, the one-bus travel alternatives increased from 3.50% to 5.00%. The increase in the congestion level makes it equally attractive for travelers to use either their private cars or to use a transit mode. At the lower congestion level, the waiting and the delays associated with the transit service lead to the dominance of private car usage. A near linear increase pattern in average vehicle travel time and average traveler travel time is observed with the increase in the congestion level. Comparing the lightly congested scenario with the congested one, the average vehicle travel time, and the average traveler travel time increase by about 80% and 50%, respectively. Network congestion also leads to an increase in the passenger travel times due to bus delays. The average passenger travel time increases by about 11 minutes due to congestion. This increase in passenger travel time is also associated with an increase in the average passenger waiting time, as shown in Table 2.

Table 2. Effect of Network Congestion Level on Modal Split and Network Performance Congestion Level

Light

Mild

Congested

Private car % One bus line % Park-and-ride % Two bus lines % Average Vehicle Travel Time (min) Average Traveler Travel Time (min) Average Passenger Travel Time (min) Average Passenger Waiting Time (min)

70.80 3.50 24.00 2.70 12.32 15.86 27.87 6.56

70.40 4.10 24.10 2.40 16.83 19.28 29.80 7.23

68.10 5.00 24.40 2.50 22.06 23.60 38.37 8.30

8.2

Information Provision Strategies

Table 3 presents the network performance under the four information schemes mentioned above. Four different information strategies are examined namely: (1) Full Information, (2) No Park-and-ride Information, (3) No bus transfer Information, and (4) No Transfer Information. As shown in the table, the best network performance in terms of the average vehicle travel time and the average passenger travel time is obtained under the full information scheme. When no information about the parking opportunities is available, which provides the opportunity for easy transfer between auto and the different transit modes, no travelers considered the park-and-ride alternative. This also results in

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more private car usage and more network congestion; the private car share increased from 69% to 96% and the average vehicle travel time increased by about 31%. Similarly, when no transit-transfer information is available, the transit trip share decreases, resulting in more private car usage and more network congestion. An increase of 13% is observed in the average traveler travel time in the absence of this information. Finally, the absence of any transfer information increases the private car usage and accordingly increases the average vehicle travel time and the average passenger travel time by about 33%, and 20%, respectively.

Table 3. Effect of Different Information Supply Strategies on Modal Split and Network Performance Information Strategy

Full Information

Private car % One bus line % Park-and-ride % Two bus lines % Average Vehicle Travel Time (min) Average Traveler Travel Time (min) Average Passenger Travel Time (min) Average Passenger Waiting Time (min)

8.3

69.10 3.90 25.30 1.70 15.80 20.46 31.89 6.01

No Parkand-Ride Info. 96.10 2.10 -1.80 20.81 20.96 45.63 15.52

No BusTransfer Info. 70.10 4.50 25.40 -17.72 22.77 28.39 6.35

No Transfer Info. 94.70 5.30 20.86 21.61 36.39 6.48

Vehicle Operating/Toll Cost

Table 4 presents the effect of private car operating/toll cost on modal split and network performance. When trip cost is ignored, and travel time is considered as the only choice criterion, the model predicts less than 2% of the travelers using transit, and only 13% share for park and ride. Including trip cost as a relevant choice criterion in addition to the travel time increases the predicted transit share. This share increases linearly with the increase in the trip cost. At $0.20 per link, the model predicts about 6% of the travelers to use transit and about 24% park and ride. This percentage increases to 9% when $0.50 per link is assumed, and the park and ride option jumps to 36%. A significant improvement in network performance is observed with the increase in the transit share. For example, the average traveler travel" time decreases by about 5% when $0.20 per link is assumed. Corresponding savings of about 6% and 8% are predicted in the average vehicle travel

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time and the average passenger travel time, respectively. The model appears to adequately capture sensitivity of mode choice to pricing, which is an important property to evaluate measures such as congestion pricing and high occupancy toll lanes. While the model would require calibration to the specific conditions of the study area, the results examined here suggest some likely degree of success of pricing mechanisms to induce some modal shifts.

Table 4. Effect of Vehicle Hidden Operating and Toll Cost on Modal Split and Network Performance Vehicle Hidden Operating and Toll Cost

$0.00

$0.50/link

Sl.OO/link

Private car % One bus line % Park-and-ride % Two bus lines % Average Vehicle Travel Time (min) Average Traveler Travel Time (min) Average Passenger Travel Time (min) Average Passenger Waiting Time (min)

85.20 0.30 13.40 1.10 18.70 20.39 26.19 7.35

70.40 4.10 24.10 1.40 16.83 19.28 29.80 7.23

55.00 5.70 36.60 2.70 16.28 18.46 '2.27 8.10

8.4

Parking Availability and Cost

The effect of imposing parking fees on private car users is presented in Table 5. It presents the case when the parking fees are imposed at the final destination of the trip and also at all park-and-ride facilities in the network. Considering such fees in the travelers' mode-route choice process reduces the private car share and increases the transit share. The results in Table 5 show that free parking at the park-and-ride facilities induces many private car users to leave their cars in order to avoid the high parking fees at their final destination. For example, when a $1.00 parking fee is used, about 6% of the private car drivers shift to a park-and-ride travel mode. This percentage jumps to 39% when $5.00 is used. Although, the number of private car drivers (i.e. pure private cars and the park-andride travelers) is not reduced, total private car usage is significantly reduced. This results in significant reduction in the average vehicle travel time. For example, at $5.00 parking fee, the average vehicle travel time is improved by about 10%. It is noted that this reduction in the average vehicle travel time is not accompanied by significant reduction in the average passenger travel time. The extra waiting time at the park-and-ride facilities increases the average passenger travel time compared to the same case in Table 5.

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Table 5. Effect of Parking Cost on Modal Split and Network Performance

Parking cost Private car % One bus line % Park-and-ride % Two bus lines % Average Vehicle Travel Time (min) Average Traveler Travel Time (min) Average Passenger Travel Time (min) Average Passenger Waiting Time (min)

8.5

$0.00 Final Destination

$1.00 Final Destination

$5.00 Final Destination

85.20 1.00 13.40 0.40 18.70

78.90 1.40 19.20 0.50 18.07

43.30 3.50 52.30 0.90 17.73

$5.00 Park-andride & Final Destination 80.60 4.20 13.30 1.90 18.49

20.39

19.55

20.85

19.27

26.19

26.81

31.61

28.89

7.35

7.04

7.39

7.44

Bus Frequency

Table 6 shows the effect of increasing the bus service frequency on the bus trip share and overall network performance. Based on the obtained results, a slight increase in the transit share is observed with the increase in the bus frequency. For example, doubling the service frequency from 3 buses/hour to 6 buses/hour appears to have only a slight effect on the transit share (an increase of only 2.50%, which does not include the parkand-ride share). However, improving the transit service encourages many private car users to park their cars and use transit for a part of the trip. The park-and-ride share is nearly doubled with the increase in the bus frequency from 3 buses/hour to 6 buses/hour. The increase in bus frequency to 12 buses/hour does not lead to a significant change in the transit share. However, some savings in the average passenger waiting and travel times are observed. The average passenger waiting time is decreased by about two minutes, while the average passenger travel time is decreased by one minute. 8.6

Adding/Removing Transit Lines

Table 7 shows the results for the case when the number of transit lines is reduced from 12 lines to 8 lines as shown in Figure 3. Surprisingly, the transit share does not change

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significantly with this change in the transit network structure. The transit share decreases by about 4%, which is substituted by a 2% increase in the park-and-ride share. Eliminating the connectivity between the transit lines also reduces the possibility of using more than one transit line to complete one trip, as shown in the table. Table 6. Effect of Bus Frequency on Modal Split and Network Performance Bus Frequency (Buses/Hour) Private car % One bus line % Park-and-ride % Two bus lines % Average Vehicle Travel Time (min) Average Traveler Travel Time (min) Average Passenger Travel Time (min) Average Passenger Waiting Time (min)

Table 7.

No Bus Service 100

21.11

3 85.90 1.70 12.10 0.30 18.55

6 71.40 3.40 24.20 1.00 17.15

12 70.40 4.10 24.10 1.40 16.83

21.11

20.33

20.04

19.28

32.56

30.70

29.80

12.66

9.16

7.23

Effect of Adding/Removing Transit Service on Modal Split and Network Performance

Number of Transit Lines

8 Bus Lines

12 Bus Lines

Private car % One bus line % Park-and-ride % Two bus lines % Average Vehicle Travel Time (min) Average Traveler Travel Time (min) Average Passenger Travel Time (min) Average Passenger Waiting Time (min)

71.90 1.40 26.70 0.00 16.04 18.80 18.81 5.92

70.20 4.10 24.00 1.70 16.83 19.28 29.80 7.23

8.7

HOV/HOT Lanes

As explained earlier, the model is capable of modeling HOV/HOT lanes and representing the usage pattern of these lanes. This set of experiments is designed to show this capability. As shown in Table 8, in addition to the transit shares, the HOV lane usage

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percentage is recorded for different pricing schemes. Two extreme scenarios are set as a benchmark to compare the results. In the first scenario, The HOV lane is used as a regular lane with no pricing scheme. In the last scenario, no LOV's are allowed to use the HOV lane. Scenarios two and three allow LOV users to use the HOV lane after paying some toll. Therefore, the HOV lanes are used as HOT lanes. As seen in Table 8, a $2.50 fee reduces the percentage of LOV vehiqles in the HOV lane from 25% to 18%. This percentage decreases further to 10% when the toll is increased to $5.00. Efficient usage of the HOV lanes by allowing use by some of the LOV vehicles leads to improved network performance. For example, under scenarios two and three, the average vehicle travel time and the average traveler travel time are slightly better than under the two extreme scenarios. The slight improvement in overall network performance is due to the limited number of HOV lanes added to the system, which means that only a fraction of the total population of travelers is impacted.

Table 8. Effect of HOV Operation on Network Performance HOV Pricing Scenario Private car % One bus line % Park-and-ride % Two bus lines % LOV users in HOV Lane*/ (LOV \ HOV) '/„ HOI'' users in HOV Lanes/iLOV- HOV) % Average Vehicle Travel Time (min) Average Traveler Travel Time (min) Average Passenger Travel Time (min) Average Passenger Waiting Time (min)

8.8

No Toll 90.80 1.20 7.00 1.00 25.00 75.00 18.23 19.39 28.55 6.45

$2.50 90.80 1.20 7.10 0.90 IS. 00 82.00 18.05 19.12 27.86 6.55

$5.00 90.50 1.30 7.30 0.90 10.00 90.00 18.10 19.17 28.95 6.53

NO LOV 90.50 1.30 7.30 0.90 -100 18.16 19.18 28.49 6.24

Bus Preemption

The last set of experiments address the effect of bus preemption at signalized intersections on the modal share and corresponding network performance. Providing priority to buses at signalized intersections reduces their delays and leads to more reliable transit service. Fast and reliable transit service could induce some private car users to leave their cars and start depending on transit. Table 9 shows the results of this set of experiments. Introducing the phase advance as bus preemption strategy at all signalized intersections in the network reduces the private car share by about 4%. This 4% decrease

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in the private care share translates into increases in the share of other transit modes. The decrease in private car usage improves overall network performance. For example, improvements of about one minute are observed in the average vehicle travel time and the average traveler travel time. Table 9. Effect of Bus Preemption on Modal Split and Network Performance Bus Preemption Availability Private car % One bus line % Park-and-ride % Two bus lines % Average Vehicle Travel Time (min) Average Traveler Travel Time (min) Average Passenger Travel Time (min) Average Passenger Waiting Time (min)

9.

Without Bus Preemption 69.10 3.90 25.30 1.60 17.60 20.46 31.89 6.01

With Bus Preemption 65.30 5.20 26.90 2.60 16.62 19.43 31.12 6.07

SUMMARY

This chapter has presented a dynamic trip assignment model for urban intermodal transportation networks. The model captures the dynamic interaction between mode choice and traffic assignment and also estimates the effect of this interaction on the overall network performance. The model implements a multi-objective dynamic trip assignment procedure in which travelers choose their mode-route based on a range of choice criteria. A set of simulation experiments is designed to illustrate the different capabilities of the model. These experiments illustrate the significance of including the mode choice dimension in the DTA framework and also show the importance of the multi-objective assignment procedure incorporated in this model. One set of experiments studies the effect of different traveler information provision strategies. Providing information on transit service availability, opportunity of easy transfer between the different transit lines, and parking availability is found to induce many private car users to leave their cars and use a transit mode. Other measures such as imposing roadway tolls and parking fees on private car users and improving transit service are also examined. The effect of introducing bus preemption at signalized intersections shows a significant improvement in the bus share and reduces the network congestion. It should be noted that

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the results in this chapter are intended to illustrate the application of a dynamic simulation-assignment methodology to the analysis of intermodal transportation networks. As such, the specific conclusions should be viewed with proper caution, as they reflect several aspects that may be specific to the test bed network, transit service and demand loading and characteristics that are considered in this chapter.

REFERENCES Abdelghany, K. and H. S. Mahmassani (1999). Multi-objective shortest path algorithm for large scale intermodal networks. Presented at the INFORMS Fall Annual Meeting, Philadelphia, USA. Abdelghany, K. (2001). Stochastic Dynamic Traffic Assignment for Intermodal Transportation Networks with Consistent Information Supply Strategies. Ph.D. Dissertation, The University of Texas at Austin, USA. Abdulaal, M. and L. LeBlanc (1979). Methods for combining modal split and equilibrium assignment models. Transportation Science, 13, 292-314. Andersson, J. (1977). A method for the analysis of transit networks. In: Advances in Operation Research (M. Roubens, ed.), pp. 1-8. Stockholm, Sweden. Baaj, M. H. (1990). The Transit Network Design Problem: An AI-B as ed Approach. Ph.D. Dissertation, The University of Texas at Austin, USA. Baaj, M. H. and H. S. Mahmassani (1990). TRUST: A lisp program for the analysis of transit route configuration. Transportation Research Record, 1283, 125-135. Climaco, J. N. and E. Q. Martin (1982). A bicriterion shortest path algorithm. European Journal of Operation Research, 11, 399-404. De Cea, J., J. P. Bunster, L. Zubieta and M. Florian (1988). Optimal strategies and optimal routes in public transit assignment models: An empirical comparison. Traffic Engineering and Control, 29, 520-526. Dial, R. B. (1979). A model and algorithm for multicriteria route-mode choice. Highway Research, 138,311-316. Dial, R. B. (1986). Transit pathfinder algorithm. Highway Research Record, 205, 67-85. Dial, R. B. (1997). Bicriterion traffic assignment: Efficient algorithms plus examples. Transportation Research, 31B, 357-379. Fernandez, E., J. De Cea, M. Florian and E. Cabrera (1994). Network equilibrium models with combined modes. Transportation Science, 28, 182-192.

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Florian, M. (1977). Traffic equilibrium model of travel by car and public transit modes. Transportation Science, 11, 166-179. Florian, M. and S. Nguyen (1978). A combined distribution modal split and trip assignment model. Transportation Research, 12B, 241-246. Florian, M. and M. Los (1978). Determining intermediate origin-destination matrices for the analysis of composite mode trips. Transportation Research, 13B, 91-103. Florian, M. and H. Spiess (1983). On binary mode choice/assignment models. Transportation Science, 17, 32-47. Han, A. F. and N. H. Wilson (1982). The allocation of buses in heavily utilized networks with overlapping routes. Transportation Research, 16B, 221-232. Hansen, P. (1980). Bicriterion path problems. In: Multiple Criteria Decision Making: Theory and Applications (G. Fandel and T. Gal, eds.), pp. 109-127, Springer, Heidelberg. Hickman, M. and D. Berndtien (1997). Transit service and path choice models in stochastic and time-dependent networks. Transportation Science, 31, 129-146. Jayakrishnan, R., H. S. Mahmassani and T. Y. Hu (1994). An evaluation tool for advanced traffic information and management systems in urban network. Transportation Research, 2C, 129-147. LeBlanc, L. and K. Fahrangian (1981). Efficient algorithms for solving elastic demand traffic assignment problems and modal split assignment problems. Transportation Science, 15, 306-317. Le Clercq, F. (1972). A public transport assignment method. Traffic Engineering and Control, 14, 91-96. Mahmassani, H. S. et al. (1998). DYNASMART-X: Real-time dynamic traffic assignment system - system implementation and software design. Technical Report ST067-85-Volume III, Center for Transportation Research, The University of Texas at Austin. Nguyen, S. and S. Pallottino (1988). Equilibrium traffic assignment for large scale transit networks. European Journal of Operation Research, 37,176-186. Sancho, N. F. (1988). A new type of multi-objective routing problem. Engineering Optimization, 14, 115-119. Sheffi, Y. (1985). Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Models. Prentice-Hall me, USA. Shih, M. C. and H. S. Mahmassani (1994). A Design Methodology for Bus Transit Networks with Coordinated Operations. Center for Transportation Research, The University of Texas at Austin.

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Spiess, H. and M. Florian (1989). Optimal strategies: A new assignment model for transit networks. Transportation Research, 23B, 83-102. Srinivas, P. (1994). System Optimal Dynamic Traffic Assignment in Congested Networks with Advanced Information Systems. Ph.D. Dissertation, The University of Texas at Austin, 1994.

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CHAPTER 9

MODELING COMPETITIVE SERVICES

MULTI-MODAL

Hong K. Lo1, C. W. Yip and K. H. Wan, Department of Civil Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong

1.

INTRODUCTION

Public transit system plays a pivotal role in serving the transportation needs of many major cities. In Hong Kong, for example, over 90% of the 11 million daily trips are provided by public transit services. The planning and design of these services, therefore, are an important welfare consideration. Over-supply of these services is not only wasteful in terms of resource allocation but adds to road congestion. Undersupply, on the other hand, causes long waiting times, over-crowed services, and, eventually, shifts to auto trips, which in turn creates a demand for more road-space. The situation is particularly intriguing if these services are to be provided by the private sector whose primary objective is not to ensure welfare gains or an efficient utilization of the road-space but to maximize profit. Analyses of this problem would have important implications on government regulations.

1

[email protected]

232

Advanced Modeling for Transit Operations and Service Planning

In metropolitan areas where multi-modal trips are common, modeling modal transfers is an important issue. In Hong Kong, for example, travelers often need to transfer up to three or four times to complete a trip. Yet not all (theoretically possible) transfers make sense to travelers; for example, rarely would travelers transfer from the subway to a bus route and then switch back to the same subway again despite that this is feasible and perhaps in some cases even a competitive choice in cost terms. In another example, a combined-mode trip of auto-subway-auto is impossible since the auto has already been parked or left at the end of the first segment and is not available for the third segment. These examples illustrate the need of attending to the transfers that would be adopted by travelers. Many of the existing transit assignment studies modeled travelers' route-line (or mode) choices as a common lines problem - exemplified by the example of passengers choosing not to board the first arrival bus but to wait for the one with express service so as to minimize their total journey time. This involves determining the route-line strategies or hyperpaths with the minimum expected cost (Nguyen and Pallottino 1988; Spiess and Florian, 1989). De Cea and Fernandez (1993) considered this approach for congested transit systems. Wu et al. (1994) extended Spiess and Florian's approach to asymmetric cost functions while considering both the waiting and in-vehicle costs as a function of the transit flows. They represented a transit stop as a combination of walk, wait, and transfer arcs. Lam et al. (1999) introduced line capacity constraints and extended the approach with a stochastic user-equilibrium condition. Traffic assignment in a multi-modal network is more complicated than the assignment of pure vehicular or bus trips. It involves combined-mode trips in which travelers choose not only the routes but also the transport modes and the kinds and locations of transfers. Fernandez et al. (1994) developed a framework for integrating trips with combined modes, proposing three levels of integrating mode and route choices. Nevertheless, in all of these studies, no explicit consideration is given to the number and kinds of transfers taking place. The solutions of these studies can include unrealistic transfers and mode choices as discussed earlier. Another difficulty associated with transit assignment is the common practice of nonlinear fare structures. Fares are often district-based, sometimes origin-destinationbased, but in most cases not directly proportional to the travel distance or travel time. They result in non-additive route costs, which cannot be determined by simply adding up the corresponding link costs, making the assignment procedure non-trivial.

Modeling Competitive Multi-modal Services

233

In this study, we propose a formulation to overcome these two difficulties, namely the number and kinds of transfers are explicitly considered and nonlinear fare structures accommodated. Through a state augmentation technique, we transform a multi-modal network to one we call State-Augmented Multi-modal (SAM) network. The probable transfer rules as well as nonlinear route fares or utilities are automatically captured in the SAM network. One can simply treat it as a simple network for trip assignment purposes or combine it with a deterministic or stochastic user-equilibrium approach to suit the applications at hand. The SAM network provides a structure to model travelers' combined modal-route choices in a network of multi-modal transit services, given the fare and frequency characteristics of the services. On the other hand, one can treat the fare or frequency of the services as decision variables to study the outcomes of various scenarios of competition and regulation, on network congestion, profitability of the services, the eventual fares, etc. Such information will provide insights on impacts of certain policies. As an example, in this study we incorporate a game theoretic approach to analyze the effect of deregulating the transit fares, allowing all transit services to engage freely in a fare competition. A case study of the ground transportation system connecting the Hong Kong International Airport to the downtown area in Kowloon West is provided. The results show that more, freer competition may not always be in the best interest of travellers, at least as far as this case study is concerned. The outline of this chapter is the following. Section 2 discusses the formation of the SAM network. Section 3 provides the fare competition formulation. To illustrate this approach, a case study of transit services between the Hong Kong International Airport (HKIA) and downtown area Kowloon West is provided in Section 4. Finally, concluding remarks are given in Section 5.

2.

STATE-AUGMENTED MULTI-MODAL (SAM) NETWORK

Consider a multi-modal transportation network M = (U,V), where U,V respectively are the sets of physical nodes and links, be partitioned into w modal sub-networks, Mb = (U 6 , V6), b e B, U A c U, \b c V, where b represents an individual mode, B is the set of transport modes such that w = B , U6 and Vb, respectively, are the sets of nodes and links associated with the sub-network Mb. Through the state augmentation approach (Bertsekas, 1995), we transform these modal sub-networks

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Advanced Modeling for Transit Operations and Service Planning

into one combined network, G = (N,A) where N is a set of nodes and A a set of links. These resultant sets of nodes and links are augmented with state variables to encode modal transfer rules and the maximum number of allowable transfers, and to model nonlinear fare structures and utility functions. We refer to this general network G as the state-augmented multi-modal (SAM) network. Specifically, in the SAM network, four state variables are included to describe each node, including: 1. Location ( i ) : specifies the origin or destination of travelers and the possibility of transfers between specific modes; transfers can only occur at collocated terminals or stops; 2. Transfer state ( s ) : is related to the adopted transfer rules, as explained in the section of Probable Transfers; 3. Number of prior transfers ( n ) : indicates the number of prior modal transfers, and is used for specifying a hard constraint on the maximum number of transfers possible; 4. Alight or aboard indicator (/): specifies the status of a trip segment and is used for modeling modal transfer. It is equal to 1 (0) at the start (end) of an invehicle link; or, correspondingly, at the end (start) of a transfer link. Each node in N is represented as ( i , s , n , f y , containing the four state variables defined above. The notations of origin and destination nodes, where passengers enter and leave the network respectively, require special notations. We denote an origin node as (i,0,0,0)and a destination as (/', 0,0,1) where / ' s in both cases designate their physical locations. The middle two O's in these two cases do not carry any physical meaning other than specifying them as an origin or a destination. The last state variable denotes the status of starting or ending a transfer link. 2.1

Probable Transfers

The set of links connecting the nodes in the SAM network, A , is divided into two subsets, i.e., A = A, u \d where A, denotes the set of transfer links and Ad the set of in-vehicle links. In a multi-modal network, travelers may transfer a number of times before reaching their final destinations, hi constructing the multi-modal network, one must be careful in establishing the connectivity. That is to say, not all

Modeling Competitive Multi-modal Services

235

transfers make sense to travelers or are probable. This study develops a network structure to automatically limiting the modal transfers to a set of probable ones by embedding the possible transfer state transitions to the SAM network. What kinds of transfers are probable obviously is a function of the locality and its practices or habits. This study imposes ho restrictions on how these rules are defined. For exposition purposes, we define a set of probable transfer states as shown in Figure 1. Each node in Figure 1 represents a transfer state with its associated transport mode, whereas each arrow indicates a probable transfer. Transfer state 0 is reserved for the origin or destination of a trip. In addition to carrying information about the current transport mode being considered, states are defined to implicitly capture the prior modal usage; such information is important in deciding the subsequent transfers. For example, even though both transfer states 2 and 5 use the subway mode, they belong to different states due to different prior modal usages, which in turn affect the subsequent transfer rules. Travelers in transfer state 2 may consider bus (transfer state 4) as a probable transfer option; whereas travelers in transfer state 5, given that they already finished a bus segment, will terminate their journeys with the subway mode.

Figure 1. Probable transfer states diagram

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Advanced Modeling for Transit Operations and Service Planning

Formally, let s be an element of the set of transfer states, S, of a multi-modal network. Each transfer state s is associated with a transport mode, denoted as 77 (5) = b e B. We define the set of probable transfer states from s to be £" (s) e S. In Figure 1, 4"(0) = {1,2,3} and £"(l) = {0,2,3}. Furthermore, we notate an element in £" (s) to be g such that g e £" (s). Implicitly defined in the probable transfer states is the maximum number of probable transfers permitted. In Figure 1, for example, a traveler cannot transfer more than 2 times or travel on more than 3 modes. Care must be exercised in defining the maximum number of transfers and constructing the probable transfer states to strike a balance between capturing realistic transfer behavior and the proliferation of nodes in the SAM network. 2.2

Transfer Links between Modes

Consider a location i where transfers between two or more modes can occur. Two columns of nodes are constructed, as shown in Figure 2. The left hand nodes represent those alighting at i as indicated by the alighting/boarding variable / = 0 and the right hand ones those boarding at / with / = 1. Let a, e A, be a transfer link in the SAM network. Graphically, transfer links are shown as arrows between the left and right hand nodes in Figure 2. Associated with each transfer link is its head node and tail node. Mathematically, from each alighting node (z,s,n,0), a set of transfer links is constructed to connect to a set of boarding nodes (/,g,n + l,l) where (a) ge£(s} and (b) n +1 < R. Only those boarding nodes with probable transfer states from 5 are connected (see Figure 1). Moreover, the number of transfers is increased from n at the alighting node to n +1 at the boarding nodes, which must be less than or equal to the maximum number of allowable transfers (/?). In the case if the constraint of maximum number of transfers is not imposed, one can simply ignore criterion (b) or put R to be a very large number. Two special nodes, origin and destination, as discussed earlier, are represented as (z, 0,0,0) and (z, 0,0,1) in Figure 2. Their associated departing or arriving links are shown dotted; whereas en route transfers are shown as solid arrows.

Modeling Competitive Multi-modal Services

237

Figure 2. Augmented nodes and transfer links in the SAM network at location i Congestion effect during transfers or entering the multi-modal network can be modeled by assigning an appropriate link performance function for each transfer link. Generally each transit mode operates with a specific frequency and a limited pervehicle capacity. Waiting for the arriving vehicle is unavoidable. In the case of a congested system, passenger may have to wait for a few vehicles before they can board one. In any case, waiting time can be expressed as a function of the queue length, capacity and frequency of the transit vehicles, and the volume of passengers already in the transit vehicles. More details on the treatment of congestion in transfer links can be found in Lo et al. (2002a). The utility function of a transfer link is represented as: 0) S

where Q is a penalty term for each transfer from state s to state g at location / , ^, is the coefficient for transfer waiting time and ca is the waiting time of the transfer link. The more transfers a route has, the more penalty terms it will incur.

238 2.3

Advanced Modeling for Transit Operations and Service Planning Direct In-vehicle Links within Modal Sub-network

Most transit services or highway tolls are not directly proportional to the travel distance or time. It is common for transit fares to be roughly based on an inter-zonal structure, which cannot be expressed by a simple additive function. That is, one cannot simply add up the individual link fares to obtain the origin-destination fare. In addition, travelers may have nonlinear valuation of travel time - small amounts of time have lower value whereas large amounts of time are very valuable. Gabriel and Bernstein (1997) and Lo and Chen (2000) referred to these situations as non-additive route costs. To allow for flexibility in capturing these nonlinear or non-additive route cost structures, we construct a direct in-vehicle link between each pair of connected nodes in each modal sub-network. Mathematically, for each boarding node (i,s,n,i) on the sub-network of transport mode b, where b = Tj(s}, a direct in-vehicle link a^sn e Ad is constructed to connect each of the reachable alighting nodes with the same transition state s : that is, (j,s,n,0) where i,j e U A , i* j. The second and third state variables s and n remain unchanged as they involve no change of state or transfer. The fourth state variable changes from / = 1 at the start of the direct in-vehicle link to 1 = 0 at the end of it. A direct link on the SAM network G is defined by the location of its head and tail ( / , j), its state (s), and the number of prior transfer (n). It does not specify the physical route(s) from its boarding location to its alighting location. We define two types of direct in-vehicle links according to the number of physical route(s) available for each direct link. Type-1 refers to cases wherein only a unique route is available on the corresponding modal sub-network linking i j , such as a fixed bus or rail line. For illustration purposes, Figure 3 shows the formation of Type-1 direct links. Figure 3 (a) shows the route of a transport mode b along four of its stations or stops. The schematic diagram in Figure 3(b) shows the augmented nodes at the same four locations3. Nodes 1, 3, 5, 7 are alighting nodes; whereas nodes 2, 4, 6, 8 are boarding ones. The boarding node at z, is connected to each of the alighting nodes at /2, /3, z'4 via direct links #1, #4, #6, respectively. The same holds for the boarding node at z' 2 ,i 3 , 2

The terms direct in-vehicle link and direct link are used in the text interchangeably for the same meaning. 3 For simplicity of illustration, the state variables are not shown here. Moreover, only one alighting and one boarding node per location are shown in Figure 3(b).

Modeling Competitive Multi-modal Services

239

and / 4 . Note that each direct link on the SAM network is associated with one or more physical links: e.g., Link #4 covers vl and v2 ; Link #6 covers vl, v2 , and v3 .

Figure 3.

Formation of Type-1 direct in-vehicle links

(a) Physical modal network

(b) SAM network

vl

#4 sub-network

#5 sub-network

#6 sub-network

(c) Type-2 direct links

Figure 4.

Formation of Type-2 direct in-vehicle links

Type-2 direct links, on the other hand, refer to situations wherein multiple physical routes are available between the boarding and alighting nodes. Each Type-2 direct link is associated with a physical sub-network in the corresponding mode. Auto and

240

Advanced Modeling for Transit Operations and Service Planning

taxi direct links are typically of such nature. An illustration of Type-2 direct links for private car or taxi is shown in Figure 4. Figure 4(a) shows the physical private car network. Figure 4(b) shows the resultant SAM network. Each boarding node is connected to each reachable alighting node, via direct links #1 through #6. Figure 4(c) shows three of the Type-2 direct links in the SAM network, i.e. #4, #5 and #6, and their associated physical sub-networks. Both #4 and #5 consist of two routes in the physical network, whereas #6 has 3 routes. These mappings from the direct links in the SAM network to the physical modal subnetworks are important. Whereas traffic assignment, as discussed later, is accomplished at the level of the SAM network, congestion is accrued at the level of physical links. Based on the travel costs accrued on the physical links, the mappings help determine the node-to-node cost or utility function for each direct link aijdsn. In this manner, as the travel costs are represented directly from node to node, any cost or utility functions can be accommodated. The tradeoff is an increased number of links in the sub-network. For a fully connected sub-network with k nodes, this requires k(k-\) direct links. To quantify the utility of mode-specific direct links, transport modes are divided into three sub-classes based on their effect and interactions on network congestion (Lo et al, 2002a). Class-1 transport mode has exclusive right-of-ways without congestion interactions with other transport modes, such as subway. Classes-2 and -3 transport modes share the same roadway segments and hence the congestion together. Specifically, we distinguish the Class-2 modes as transit services with fixed routes and frequencies, such as buses, whereas the Class-3 modes as those without, such as taxi or auto. Classes-1 and -2 transport modes lead to Type-1 direct links; Class-3 transport modes generally result in Type-2 direct links. For Classes-1 and -2 transport modes, their perceived travel times are modified by discomfort functions to reflect crowding effects (Nielsen, 2000). For Class-3 transport modes, both their route choices and hence the link traffic volumes are related to the network congestion and the traffic assignment procedure adopted. In terms of notations, the three classes of transport mode are specifically distinguished4 as blf b2, irrespectively. The utility function of transport modes bm, m = 1,2, on a direct in-vehicle link from / to j is thus represented as: 4

For notational simplicity, at places where the distinction between the three classes is unimportant, we will use b rather than bm to represent a transport mode.

Modeling Competitive Multi-modal Services V-L = f t + 7 V c » + 7 V / ^ » ViJeU,, to*. 8.1.17(5) = *

241 (2)

where yb is the mode specific constant for the in-vehicle link; 7, and y2, respectively, are the coefficients for modified travel time cf and fare pi. Note that the fare is defined based on the starting and ending locations of the trip segment in the transport mode; any nonlinear fare structure can be accommodated in this direct link representation. Unlike Classes-1 and -2 modes, each direct link a^sn between i —»y, V / , y e U 3 o n a Class-3 mode represents a set of route(s) K^ linking, i- j. In the special case if the Class-3 direct link a%sn belongs to Type-1, then K^ contains only a single element; its treatment is similar to the direct links of Classes-1 and -2 transport modes. On the other hand, if the Class-3 direct link belongs to Type-2, the utility function of a^sn depends on the flow proportions among the routes in K| and is expressed as a composite utility term. In this study, we adopt a logit-based stochastic user equilibrium (SUE) assignment approach and the composite utility is taken as the expected maximum utility of the route set K^ (Bell and lida, 1997): ^ =^[max^] = iln £ exp(^f), L

e

*J

J

to*.

8.1.17(5) = 6,

(3)

kzK'', *>

where 9 is the coefficient of perceptional variation. n\ is the utility associated with route k in the associated physical sub-network, expressed as: < = n , + 7 V ' f + ?Vpf,

ViJsU^

(4)

where y^ is a mode-specific constant; t\ is the travel time on route k; and pf is the monetary cost associated with route k, which can be specific to the particular mode in Class-3 as taxi charge or gasoline cost etc. 2.4

Multi-modal Utility Functions

A route in this SAM network generally consists of a sequence of node-to-node direct links ad on a transport mode plus transfer links a, between modes. In-vehicle travel times, transit fares, and modal preferences constitute the utility of the node-to-node direct links ad; whereas the transfer time and transfer penalties are considered in the transfer links at. To simplify the analysis, we further consider that link utilities i//a,

242

Advanced Modeling for Transit Operations and Service Planning

a e A = A r u Arf , in the SAM network are additive. Let A" be the utility of route p between origin-destination pair rs in the SAM network: )-¥a,

(5)

where S(p,a] = 1 if link a in the SAM network is on route p ; zero otherwise. Each link in the SAM network is either a transfer or direct link, whose utility is determined from (1), (2), or (3). Within this SAM network, we can further divide travelers into different classes with different utility functions to capture their specific travel choice characteristics. For simplicity, in the example shown in Section 4, we only consider the case of different coefficients $,, $ , yb, 7, and y2 in (1), (2) and (4) for different traveler classes, as will be discussed more in Section 4.

Figure 5. The original 3-modal network The above discussions lead to the formation of the SAM network. For illustration purposes, consider the multi-modal network shown in Figure 5. Passengers departing from node 1 to node 4 have three transport modes and transfer possibilities at nodes 2 and 3. Adopting the transfer rules in Figure 1, the associated SAM network is formulated as shown in Figure 6. 2.5

Multi-modal Network Assignment

After a multi-modal network is transformed to a SAM network, it can be considered as a simple network without the need to attend to transfer feasibility, nonlinear fares, and related issues. A route in the SAM network automatically combines the modetransfer choices, which can be decoded for the specific modes used and transfer locations selected. In the following example, we formulate the multi-modal assignment problem as a SUE route choice problem.

Modeling Competitive Multi-modal Services

Figure 6.

243

The resultant SAM network of the original 3-modal network

The perceived utility (A") of route p between origin-destination (OD) pair rs in the SAM network is modeled as the sum of a systematic term (A") and an error term (e") that is independently and identically distributed: A" = A1 + srps

VrsellS,p€ Prs

(6)

where RS is the set of OD pairs in the SAM network; Pn is the set of routes linking OD pair rs. Assuming that the random term is Gumbel distributed, the resultant SUE pattern follows this logit relationship: expf/l-A") „" _

* \

"I

Wr, ,- D« V

RS

(7)

teP"

where w" is the proportion of passengers between rs using route p; /I is a parameter on travel utility perception variation. Moreover, the flow on route p (h") satisfies the demand constraint: h^-w^-q"=Q, V~rse'RS,VpeP7s

(8)

where q" is the total demand on OD pair rs. By multiplying h™ to (8), we obtain the following complementary form:

244

Advanced Modeling for Transit Operations and Service Planning RS,VpePs

(9)

If h" > 0 , then (8) must be satisfied or h™ is apportioned according to (7). Theoretically, the nature of the SUE assignment would assign a positive flow to each route, rendering the case of h™ = 0 nonexistent. Thus, this SUE problem can be written as a nonlinear complementarity problem (NCP) in the form of:

h>0 where h = (/*;, \/7s e~RS,\/p&P7s)

and F(h) = (/£ -wj ^

This NCP can be solved with a mathematical programming technique (Lo and Chen, 2000; Lo and Szeto, 2002) or an approach based on the Method of Successive Averages (MSA). For brevity, we do not describe the solution procedure here.

3.

FARE COMPETITION AMONG TRANSIT SERVICES

The SAM network provides a structure to model travelers' combined route-modal choices in a network of multi-modal transit services. Using the SAM network as a base, it enables various kinds of sensitivity analysis of transit service modifications and competitions. One example is studying the impact of fare competition on the services' revenues, the resultant fares, and network congestion. This analysis has important implications on government regulations. Our previous study (Lo and Yip, 2002) examined this problem in a simplified fashion by adjusting the fare of each service until the fare elasticity of each service becomes negative one. This simplified approach cannot be extended for an intertwined multi-modal network with many transfers between services. In this study, we propose a formulation via the Variational Inequality Problem (VIP) that seeks a Nash equilibrium among the transit services, in which no operator can increase its revenue further by unilaterally changing its fare. This equilibrium fare structure represents the result of an unregulated fare competition among the transit services. Let p = (/?,,••-,p ( 0 ,-~PK) be the fares of K transit services engage in a fare competition, and y/ = (^],...,^(B,...,^) be the services' revenues. According to (2), changing the fare of a service will affect the utilities of its associated direct links, and

Modeling Competitive Multi-modal Services

245

hence its patronage and revenue, therefore one can express y/ = \y (p, h (p)) = y (p) . +

/*

*

K

* \

A Nash equilibrium is achieved when a fare vector p = \p\ ,...,/9 (U ,.../? Ar Je9I + is found such that:

where p*^ = (p\,...,p*ia_^p'(a^,...,p'K \ . In other words, for each individual service CD , charging any fare that is different from p*u will result in a lower revenue. When this occurs, none of the services will have an incentive to deviate from this equilibrium fare structure. To find p" , we assume that the revenue for each service co , y/^ , is a continuously differentiate function and concave with respect to pa ("n" shape). This assumption is reasonable as it implies that the optimal fare for service a> is neither zero (which will result in zero revenue) nor infinitely high due to the presence of competition, but somewhere in between. Mathematically, for each service co, this concave property with respect to p^ can be written as:

.-/O^.(p)

02)

where p ', p e 9?f are any two arbitrary fare vectors. Setting p ' = p* , we have:

M*>n^(A.V.)*r.(p).

(13)

As long as one can find p* such that: or -

then (13) becomes: ^ (/?*)> y/m (p) , which is the same as (1 1) - a fare that service co does not want to deviate from. Repeating this argument for each service and summing its corresponding condition (14), we have the variational inequality problem VTP(F,9lf): finding a vector //e9^ such that: -V¥(p}T(p-p)>V,

Vpe^f,

(15)

246

Advanced Modeling for Transit Operations and Service Planning

, v where -Vy/{p ]=

OI//AP i oy/^p i o\i/K\p i -—'-,..., -—'-,..., -—'- . More discussions on

this type of oligopolistic market equilibrium can be found in Nagumey (1993). We reiterate the relationship that y/(p) = y/"(p,h(/?)) or any change in the fare structure will affect the route flows in the multi-modal assignment, which in turn affect the revenue distribution. To solve this VIP(F,9tf), a number of projection methods can be employed. In this study, we adopt the projection method developed in Han and Lo (2002). hi the interest of space, we omit the details but only summarize the main algorithmic steps as below:

(16) M P

=pk-tke(pk,pk)

(17)

where P^H is the projection onto SRf; pk refers to the fare vector at the kth iteration; flk, tkare parameters to be set, which can be set to vary from iteration to iteration; and e(pk,j3k\ is a measure of the error of convergence at the kth iteration. Through applying (16)-(17) repeatedly, the algorithm gradually reduces the error measure e(pk,j3k]

to zero, ensuring that the solution obtained at the last iteration,

pk, fulfills the VIP (15).

4.

CASE STUDY

To illustrate the applicability of the SAM network, we study the impact of fare competition among services connecting the Hong Kong International Airport (HKIA) to the downtown area in Kowloon West. After establishing and calibrating the SAM network of the study area with field data, we determine the market equilibrium under fare competition using the VIP formulation discussed in Section 3. 4.1

HKIA - Kowloon West SAM Network

A sample of seven hundred travelers at the arrival hall of the HKIA was surveyed on 18 and 19 January 2000. The survey revealed the variety of transit modes and combinations selected by travelers. In addition to auto, travelers chose among the rail

Modeling Competitive Multi-modal Services

247

services, bus services (A-Bus, E-Bus & S-Bus), shuttle services (H-Bus), and taxis. The survey also recorded passengers' transfer locations. The model considers three pairs of OD demands from the airport, to: Tsim-Sha-Tsui (TST), Mongkok (MK), and Kowloon West (KW). Travelers are stratified into two classes: "arriving passengers": who have just finished a trip overseas and arrived at the airport and "domestic passengers": who went to the airport to pick up friends or work at the airport. The hourly demands of both traveler classes are shown in Table 1. The multi-modal network is shown in Figure 7(a). Each roadway link is labeled (which will be referred to in result discussions). Links 1 to 5 are highways that have spare capacities; whereas links 6, 7,9, 10 are congested urban roadways. Shown along each node (or location) are the alighting or boarding modes available. We define the probable transfer states as those that were actually used by travelers according to the survey, as shown in Figure 7(b). Note that: (i) the Airport Express Line (AEL) cannot reach nodes 4, 6, and 7 directly, so transfer is necessary for trips heading for those destinations; (ii) some AEL passengers chose to walk from the AEL station at node 5 to the destination node 7, the walk link 8 between nodes 5 and 7 is thus introduced; (iii) States 8 and 9 both correspond to the Mass Transit Railway (MTR) line. They are designated different states due to different prior mode usages. State 8 refers to trips transferred from AEL; State 9 refers to those transferred from SBus. For trips transferred from the AEL service, the MTR trip is free. This demonstrates how states in the SAM network can capture collaborative fare discounts between transit operators. The disutility functions include these attributes: fare, in-vehicle travel time, and waiting time. Their associated coefficients are calibrated from the survey data, as shown in Table 2. To simplify the calibration procedure, we assume the same attribute coefficients but different mode specific constants for the two traveler classes with different destinations. Also, as auto and taxi costs vary as a function of travel distance and time, we convert their associated monetary costs into equivalent in-vehicle times, combine with the actual in-vehicle time, and calibrate the in-vehicle time attribute accordingly. This is why the coefficients of in-vehicle time of these two modes are different from the others. A generic transfer penalty term is added to the utility for each transit transfer. Using the maximum likelihood estimation procedure developed in Lo and Yip (2002), we calibrate the disutility functions. By combining the calibrated disutility functions with the SAM network analysis, we obtained the hourly passenger flows by class on each network link, as shown in

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Advanced Modeling for Transit Operations and Service Planning

Figure 8. The loads on each roadway link, each bus line, and each rail service are depicted. Table 1.

Average hourly demands

O-D pair Origin

HKIA (node 1)

Destination

Arriving passengers

Domestic passengers

TST (node 7)

1167

652

MK (node 6)

288

621

KW (node 4)

121

152

Table 2.

Utility

Demand

Attribute coefficients of the disutility functions

Class f Class 1 Class 1 Class 2** Class 2 Class 2 (for (for (for (for (for (for MK) KW) TST) TST) MK) KW)

AEL

-2.3

-3.6

-9.2

-1.5

-1.3

-21.5

MTR

-5.4

1.5

-4.5

-2.0

-4.6

A-Bus

-7.3

-0.8

11.5

-1.9

-3.4

Fare

In-Veh Waiting time time 0.0470

0.1787

1.8

0.0387 0.0387

0.0470

0.1787

5.0

0.0387

0.0470

0.1787

0.0470

0.1787

E-Bus

36.2

0.6

-11.4

36.7

17.3

-18.2

0.0387

Auto

-7.5

-1.0

-9.9

-2.0

-4.5

11.5

0

0.1507

0.1787

H-Bus

-6.7

NA

NA

-3.2

NA

NA

0.0387

0.0470

0.1787

Taxi

-6.6

-0.3

14.2

-2.3

-5.0

-0.4

0

0.1905

0

S-Bus

23.0

-1.5

13.5

28.8

22.3

-21.7

0.0387

0.0470

0.1787

Walking

-2.6

NA

NA

50.57

NA

NA

0

0.0470

0

Transfer Penalty: Note:

2.62

Class 1 refers to arriving passengers; Class 2 refers to domestic passengers. NA is denoted if no passengers of the specific class took this mode.

Modeling Competitive Multi-modal Services

249

o.

(a) The P h y s i c a l M ulti-m o d a l N e t w o r k of the case s t u d y s c e n a r i o

Figure 7.

(b) State T r a n s i t i o n D i a g r a m of the s a m e n e t w o r k w i t h Identified Probable Transfers Physical network and probable transfer states for the case study

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Advanced Modeling for Transit Operations and Service Planning

Figure 8. 4.2

Passenger volumes on the exiting multi-modal, network

Transit Fare Competition

We study the scenario wherein these privately operated services engage in a fare competition, including AEL, MTR, A-Bus, E-Bus and S-Bus. Other modes are excluded from this competition because they are either private services (e.g. auto and H-Bus) or tightly regulated (e.g. taxi). The results in Tables 3 - 4 and Figure 9 show the impact of this competition on fare, revenue, and travel pattern. Table 3 depicts the fare and aggregate revenue of each service as a result of the competition. All modes raise their fares, ranging from 10% for AEL, 27% for S-Bus, and doubling or more for A-Bus, E-Bus and MTR. Note that the existing AEL fare is higher than the other modes5. The competition, if unregulated, would allow A-Bus, E-

5

The rail services must recover the right-of-way and infrastructure costs, whereas the bus services use the roadway infrastructure that is provided and maintained for free by the government. Therefore, justifiably the Hong Kong government allows the rail services with a greater flexibility to define their fares.

Modeling Competitive Multi-modal Services

251

Bus and AEL to all raise their fares so that they become about the same. This of course defeats the purpose of intending the bus services as affordable modes. This result indicates the possible occurrence of collusion if allowed, wherein all transit operators simultaneously raise their fares to exploit travelers. In the end, most of the operators are able to increase their revenues from the current levels, especially for the rail service MTR and to a great extent for the other rail services AEL and E-Bus. On the other hand, S-Bus will lose out in this competition despite a substantial fare increase. Table 3. Impacts on Fare and Revenue initiated by Transit Fare Competition S-Bus Mode AEL MTR A-Bus E-Bus Fare Change (%) +26.9 +10.3 +147.3 +91.3 +222.0 -25.3 Revenue Change (%) +59.3 +4.1 +14.3 +19.9 Figure 9 shows the passenger volumes as a result of the fare competition. In crossreference with Figure 8, it shows a major shift of travelers (especially domestic passengers) from E-Bus to AEL, A-Bus, and MTR, after E-Bus more than doubles its fare. E-Bus, originally designed as a service with low fare but a long detour route, has a relatively high mode-specific disutility (Table 2). This large fare increase leads to a significant drop in E-Bus's patronage. This of course defeats its purpose as an affordable mode. Due to substantial fare increases of all transit modes, a considerate amount of passengers now prefer to take taxis or autos, resulting in an increase in network congestion. Table 4 shows the travel time of each roadway link of the network. It shows an increase in the travel time for all the links. The percentage changes in travel times are particularly substantial for links 7 and 8, which are already congested main roads leading to the downtown area. This shift to taxi and auto trips further exacerbates the congestion problem. All in all, the fare competition leads to these outcomes: • • •

Travelers generally pay higher fares for all of the transit services The network gets more congested, especially on the already congested roadways Most of the transit service operators enjoy high revenues.

This case study indicates that if the system were allowed to engage in a totally unregulated fare competition, both travelers and the network would suffer. It challenges the notion that more, freer competition would always benefit consumers.

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Advanced Modeling for Transit Operations and Service Planning

Table 4. Travel Time of each Roadway Link of the Network PART 1 - Auto/Taxi Travel Time (min) 1

1

Road

Current auto travel time

10

7.0

15.0

15.0

10.1

12.2

13.1

12.9

Travel time after competition

7.0

15.1

15.0

10.4

12.7

13.1

Percentage Increase (%)

0.4

0.4

0.0

2.4

4.4

0.2

5.2

9.0

15.4

8.2

9.1

9.5

19.9

56.0

1.8

7.1

PART 2 - Bus (A-Bus, E-Bus & S-Bus) Travel Time (min) Road

1

1

10 2

Current bus travel time

15.0

20.0

20.0

15.2 N.A.

N.A.

12.9 N.A.

11.2 N.A.

Travel time after competition

15.1 20.1

20.0

15.5 N.A. N.A.

15.4 N.A.

11.4 N.A.

Percentage Increase (%)

0.4

0.0

2.4

N.A. N.A.

19.9 N.A.

1.8 N.A.

Notes: 2

0.4

See Figure 7(a) for reference. No bus routes on links 5, 6, 8 or 10.

Figure 9.

Passenger volumes as a result of fare competition

Modeling Competitive Multi-modal Services

5.

253

CONCLUDING REMARKS

The paper presented a framework to model multi-modal networks. Two key considerations addressed by the SAM network approach are the incorporation of probable transfer states and the introduction of nonlinear fare and utility structures. We have coupled this SAM network with a logit assignment approach. For combinedmode trips with transfers, mode-segment overlaps between transfers are likely to happen, which may violate the assumption of independent and irrelevant alternatives (IIA) of the standard logit approach. To avoid this problem, one can extend the consideration to a nested logit approach. On example of such an extension is shown in Lo et al. (2002b). In general, how to couple the SAM network structure with the nested logit approach so as to better describe travelers' route-modal choice behavior, while maintaining the model to be tractable and conveniently calibrated is a future research direction. As demonstrated in this study, combining this SAM network framework with game theoretic approaches forms a very rich platform for analyzing the competitions between operators as well as for studying the case of regulations. Such analysis would shed light on creating and maintaining the thin balance between corporate profitability, traveler disutility, and overall network congestion. In broader terms, how should the government create an environment such that the private sector can be relied upon to provide for the services while maintaining the public welfare? Is there a role for government regulations on fare and service quality so as to create "win-win" situations, in which private investors are enticed into quality services provision and travelers continue to enjoy affordable services? Analysis leading to answering this question are of significant interest and potential application to many transit-oriented cities.

ACKNOWLEDGEMENT This study is sponsored by the Competitive Earmarked Research Grants, HKUST6083/OOE and HKUST6161/02E, of the Hong Kong Research Grant Council.

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Advanced Modeling for Transit Operations and Service Planning

REFERENCES Bell, M. and Y. lida (1997). Transportation network analysis. John Wiley & Sons, Inc., Bertsekas, D. (1995). Dynamic Programming and Optimal Control. Belmont, Mass. Athena Scientific. De Cea, J. and E. Fernandez (1993). Transit assignment for congested public transport systems: An equilibrium model. Transportation Science, 27, 133-147. Fernandez, E., J. De Cea, M. Florian, and E. Cabrera (1994). Network equilibrium models with combined modes. Transportation Science, 28, 182-192. Gabriel, S. and D. Bernstein (1997). The traffic equilibrium problem with nonadditive path costs. Transportation Science, 31, 337-348. Han, D. and H. Lo (2002). Solving nonadditive traffic assigment problems: A decent method for co-coercive variational inequalities. European Journal of Operational Research. Submitted. Lam, W. H. K., Z. Y. Gao, K. S. Chan, and H. Yang (1999). A stochastic user equilibrium assignment model for congested transit networks. Transportation Research, 33E, 351-368. Lo, H. and A. Chen (2000). Traffic equilibrium problem with route-specific costs: Formulation and algorithms. Transportation Research, 34B, 493-513. Lo, H. and W. Y. Szeto (2002). A methodology for sustainable traveler information services. Transportation Research, 36B, 113-130. Lo, H. and C. Yip (2002). Deregulating transit services: Winners and losers in a competitive market. Journal of Advanced Transportation, 35, 215-235. Lo, H., C. Yip, and K. Wan. (2002a). Modeling transfers and nonlinear fare structure in multi-modal transit network. Transportation Research B. In press. Lo, H., C. Yip, and K. Wan. (2002b). Modeling competitive multi-modal transit services: A nested logit approach. Transportation Research C. In press. Nagumey, A. (1993). Network Economics: A Variational Inequality Approach. Kluwer Academic Publishers. Nielsen, O. A. (2000). A stochastic transit assignment model considering differences in passengers utility functions. Transportation Research, 34B, 377-402. Nguyen, S. and S. Pallottino (1988). Equilibrium traffic assignment for large scale transit networks. European Journal of Operational Research, 37, 176-186.

Modeling Competitive Multi-modal Services

255

Spiess, H. and M. Florian (1989). Optimal strategies: A new assignment model for transit network. Transportation Research, 23B, 83-102. Wu, J., M. Florian and P. Marcotte (1994). Transit equilibrium assignment: A model and solution algorithms. Transportation Science, 28, 193-203.

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CHAPTER 10

MODELING URBAN TAXI SERVICES: A LITERATURE SURVEY AND AN ANALYTICAL EXAMPLE Hai Yang, Min Ye, Wilson H. Tang, Department of Civil Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, P.R. China, and S. C. Wong, Department of Civil Engineering, The University of Hong Kong, Hong Kong, P.R. China

1.

INTRODUCTION

In most large cities, taxis are an important transportation mode that offers a speedy, comfortable and direct transportation service. Unlike regular transit such as trains, trams, and buses, taxis are usually operated by a large number of private firms, frequently owned and operated by individuals. With such a market setting, individual taxi drivers or firms can freely choose their working schedule in response to market profitability and operating cost as well as opportunity cost of being in service in different time of the day. A remarkable characteristic of taxi service in most

258

Advanced Modeling for Transit Operations and Service Planning

metropolitans is the hourly variation of customer demand and taxi service intensity throughout the whole day. hi addition, cruising rather than dispatch taxi service is the norm; this is particularly true in metropolitans in developing countries. The taxi market is highly regulated. Nearly all over the world, local governments limit the number of taxi licenses issued, set the fares to be charged, and impose and enforce standards of quality and rules of conduct. In the urban area of Hong Kong, taxis currently form about 25% of the traffic stream, hi some critical locations, taxis form as much as 50% to 60% of the traffic stream (Transport Department, 1986-2000). Taxis make considerable demands on limited road space and contribute significantly to traffic congestion even when empty (cruising for customers). In general the taxi industry is subject to various types of regulation such as entry restriction and price control, hi Hong Kong, taxi operations are subject to service area demarcation as well. The urban taxis operate throughout the territory, while the others are fundamentally confined to the rural areas of the New Territories and Lantau Island. Currently, annual taxi service surveys (surveys at sampled taxi stands and roadside observation points) have been conducted since 1986 to gather the information on customer/taxi waiting time, taxi utilization and taxi availability for the city of Hong Kong (Transport Department, 1986-2000). These types of information have been used for the evaluation of taxi services and government decision-making with respect to the increase in the number of taxis and/or adjustment of taxi fares. Furthermore, a simple modal split model of taxis has been developed in Hong Kong for the prediction of taxi person and vehicle origin-destination matrices, which has been incorporated as an important component in comprehensive transport studies (Transport Department, 1993). Traditionally, many economists have examined the models and economics of urban taxi services under various types of regulation, such as entry restriction and price control in an aggregate way. It was only recently that urban taxi services was modelled in a network context. A realistic method has been proposed to describe vacant and occupied taxi movements in a road network as well as taxi drivers' search behavior for customers. A few extensions have been made to deal with demand elasticity, multiclass taxi services with service area regulation, and congestion effects together with development of efficient solution algorithms. The models have been extended to the multi-period dynamic demand-supply equilibrium of taxi services. Calibration and

Modeling Urban Taxi Services: A Literature Survey and An Analytical Example 259 validation of the network taxi service models have been conducted towards their practical applications. This chapter presents a comprehensive survey of the static and dynamic modeling methods for urban taxi services. Both the conventional aggregative demand-supply equilibrium model and the recent network equilibrium models are covered, with emphasis placed on the interplay among taxi supply, customer demand and externality of service consumption on customer waiting time and taxi utilization. The models and analysis presented here are useful to understand the manner in which the demand and supply are equilibrated in the presence of regulations in terms of entry restriction and price control, thereby providing information for government decision-making in terms of such regulations.

2.

AGGREGATE MODELING

2.1

Overview

The taxi market is heavily regulated mainly by local government in most large cities. Regulation makes the taxi market interesting, because price does not play the usual role of clearing the market, and taxi supply and customer demand are brought in balance through two intermediate variables: taxi availability and taxi utilization. The interest of economists for the analytical aspects of the taxi market can be traced back to the early sixties. In an appendix of his first provisional edition of "Price Theory" (Friedman, 1962), Friedman included the issue of "licensing taxicabs": a problem of delightful subtlety. This problem soon attracted interest by professional economists (Lipsey and Steiner, 1966; Orr, 1969). A subsequent stream of papers followed the topic continually kept up to date (Douglas, 1972; Beesley, 1973; De vany, 1975; Shrieber, 1975, 1977; Abe and Brush, 1976; Manski and Wright, 1976; Foerster and Gilbert, 1979; Beesley and Glaister, 1983; Schroeter, 1983; Frankena and Pautler, 1986; Gallick and Sisk, 1987; Rometsch and Wolfsteter, 1993; Hackner and Nyberg, 1995). Early works centered on the general recognition of pervasive market failures in this sector and the analysis of the effect of regulation of fares and also investigated entry under alternative assumptions regarding the market structure and the organization of the service. Recent notable studies in the topic have made significant improvements in our understanding of the market mechanism (Arnott, 1996, Cairns and Liston-Heyes, 1996, Bergantino and Longobardi, 2001; Fernandez et a/., 2001). In addition, there have been many contributions towards the study of empirical

260

Advanced Modeling for Transit Operations and Service Planning

aspects of taxi regulation/deregulation around the world (Teal and Berglund, 1987; Garling et al, 1995; Dempsey, 1996; Gaunt, 1996; Gaunt and Black, 1996; Morrison, 1997; Radbone, 1998; Schaller, 1999; Flath, 2002). The economics of taxi service has been overwhelmingly examined in an aggregative manner and the general analytical framework can be described below:

,

dF

-

,

dW

(1)

W=W(V\ — <0, dV

(2)

C = c(U + V)

(3)

where Q is the demand for taxi rides, U, occupied or utilized taxi-hours (equal to the product of average ride time and demand Q), V, vacant taxi-hours, F, the expected fare or money price of a taxi ride, W expected waiting time, C, total taxi costs, and c cost per hour of taxi service time. Equation (1) states that the demand for taxi rides is a decreasing function of the expected fare and the expected customer waiting time (a proxy of service quality); (2) states that expected waiting time decreases when the total vacant taxi-hours increases; and (3) states that the cost of operating a taxi is a constant per hour. This highly aggregate model was originally proposed by Douglas (1972) and has been adopted by subsequent studies on the economics of taxis, without consideration of the spatial structure of the market. It is commonly realized that there are two principal characteristics that distinguish the taxi market from the idealized market of conventional economic analyses: the role of customer waiting time and the complex intervening relationship between users (customers) and suppliers (firms) of the taxi service. In the taxi market, the equilibrium quantity (total taxi-hours) of service supplied will be greater than the equilibrium quantity (occupied taxi-hours) demanded by a certain amount of slack (vacant taxi-hours). It is this amount of slack that governs the average customer waiting time. The expected customer waiting time is generally considered as an important value or quality of the services received by customers. This variable affects customers' decision as to whether or not to take a taxi, and thus plays a crucial role in the determination of the price level and the resulting equilibrium of the market (De vany, 1975; Abe and Brush, 1976; Manski and Wright, 1976; Foerster and

Modeling Urban Taxi Services: A Literature Survey and An Analytical Example 261 Gilbert, 1979). A reduction in expected waiting time increases the demand for taxi service. However, from the point of view of each individual taxi firm, expected customer waiting time is different from the quality of the typical product. In most markets where quality is a variable, each firm can decide what quality to produce. In most taxi markets, expected waiting time is not amenable to differentiation, but depends on the total number of vacant taxi-hours. An individual firm cannot offer customers an expected waiting time different from that offered by other firms, although a large firm may be able to affect expected waiting time (Frankena and Pautler, 1986). Furthermore, the demand and supply of taxi services are interrelated through two intervening variables: taxi availability (as measured by expected customer waiting time) and taxi utilization (as measured by expected fraction of time a taxi is occupied). On the demand side, potential customers will consider taxi availability as well as fare in making their mode choice decisions. From the supply perspective, taxi firms will operate in response to taxi utilization rate as well as trip revenues and costs. Moreover, taxi availability, through its influence on the level of taxi use, indirectly affects the taxi utilization rate; the utilization rate, through its influence on the level of supply, in turn affects taxi availability. Recently, Yang et al. (1998) developed an aggregate simultaneous equations model based on the Hong Kong taxi survey data. Number of taxis, taxi fare and disposable income are used as exogenous variables; while customer and taxi waiting times, taxi utilization in terms of the percentage of occupied taxis on the roads, taxi availability in terms of vacant taxi headway and customer demand are used as endogenous variables. The functional form of the structural model is generally difficult to specify and has to be built in a heuristic manner. The nonlinear simultaneous equation model developed by Yang et al. (1998) is found to be able to predict general outcomes of introducing new taxi policies (issue of new taxi licences and change of taxi fare), but the accuracy of prediction for certain variables needs to be enhanced. With the same set of data, Xu et al. (1999) applied a neural network approach for the analysis of the complex nonlinear relationships among the above endogenous and exogenous variables. They compared the performance of the neural network model and the simultaneous equations model, and found that both models can be used to predict the demand-supply equilibrium for given values of exogenous variables and thus can be used for policy sensitivity analysis. The neural network model, however, performs better in terms of the accuracy of replicating the observed outcomes.

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Advanced Modeling for Transit Operations and Service Planning

3.

A SPECIFIC AGGREGATE MODEL

3.1

Model Assumptions

A specific aggregative model is presented to help understand the general nature of taxi demand and supply equilibrium. Let Q = G(P) = Qexp{-a(F + -clW + t2T)}, (<x>0)

(4)

be the number of customers per unit time (one hour) demanding taxi rides over the whole service area, where P = F + T,r + i2W is the total trip cost or full trip price, F is the expected fare or money price of a taxi ride, W and T are the expected customer waiting time for engaging a taxi and average taxi ride time, respectively, T, and T 2 are the values of waiting time and in-vehicle time of customers, respectively, Q is the potential customer demand per hour, and a (a > 0) is a demand sensitivity parameter. Let TV be the number of taxis in service. Note that here a taxi in service can be either vacant or occupied by a customer. Given a reasonably large number of taxis and customers, and no differentiation of individual taxis, the average rides per hour of each taxi in service in a given one-hour period is

»-£ (5) N By neglecting the congestion effects, we assume the average taxi ride time T (a fraction of one hour) is constant. Then, the average number of vacant taxis, Nv, available (in service but not occupied) at any given instant is NV = N-TQ

(6)

As ATV > 0, we must have Q < N/T. Average customer waiting time is assumed to be decreasing when the total vacant taxi hours (equal to the average number of vacant taxis as one hour unit period is considered) increases (Yang et al., 2002a): W=—-—, ( P > 0 ) N-TQ^ '

(7)

where P (P > 0) is a positive parameter whose value depends on the size of the service area and the distribution of taxi stands over the service area. Substitute Equation (7) into the demand function (4), we have

Modeling Urban Taxi Services: A Literature Survey and An Analytical Example 263 0

(8)

Note that O (Q) is a continuous function of Q e (0, N/T) . Furthermore, .0 > 0

(N-TQY and _ f f TP ^1 / \ N Lim <&(()) - — Qe\p< —a LF + —— + i2T \> < 0 and Z-/w O(2) = — > 0

We thus conclude that for a given fleet size N and fare F per taxi ride, there exists one and only one equilibrium value of Q from the above equation within the feasible domain (0, N/T).

We can thus write the equilibrium demand function as

Q = Q(F,N,
do J ( i i.pro -^---Q — + — 8F

*(a

V1

-, <0 (N-TQ)2)

(9)

\ ~^

dN

~{ ~

T.ccp

J

(10)

The equilibrium customer demand, as expected, always decreases with fare and increases with taxi fleet size. 3.2

Market equilibrium

With the aforementioned aggregate model, we now consider how to find the equilibrium output, capacity, and utilization of capacity in the taxi market under the following scenarios: free entry in a monopoly market with fixed price; free entry in a competitive market with fixed price; first-best social optimum and second-best social optimum. 3.2. J

Monopoly solution

Assume the city government grants a single firm monopoly rights through a charter to pick up customers within a market area, and the government also sets the fare to be

264

Advanced Modeling for Transit Operations and Service Planning

F*. Under this monopoly system, the single expected profit-maximizing firm would select a fleet size Nm to maximize: max n(F\Nm,
(11)

where F* is the fixed charge rate per unit service time set by the government, and Nm represents the monopoly solution of taxi fleet. The monopolist will operate at a solution where marginal revenue of taxis at the regulated price equals marginal cost of taxis. If fare is unconstrained, the monopolist will choose a fare-fleet combination that maximizes profits: max n(Fm,Nm,u>) = FmQ(Fm,Nm,y)-cNm

Fm,Nm

3.2.2

^

'

^

(12)

'

Competitive solution

Assume that the market is comprised of owner-operated taxis, one taxi per owner or firm. The fare to be charged is set by the regulator, but entry is unrestricted, hi this competitive free-entry market, the resultant supply will satisfy the market equilibrium where the marginal revenue obtained by the last unit of taxi service just covers its cost (profits are nil). It is at this point that the individual incentive to join the taxi industry disappears. Hence equilibrium occurs at c F'o(F*,N ,(p] V -^-c = 0 c N

(13) V '

where F* is again regulated fare and Nc represents the competitive solution of taxi fleet. Evidently, the number of taxis at the competitive free-entry market is a function of the price set by the regulator. If the fare is unconstrained as well, then the competitive equilibrium solution ( F C , N C } in this unregulated case is determined by the following break-even equation: FCQ(F{ C,NC,
Modeling Urban Taxi Services: A Literature Survey and An Analytical Example 265 3.2.3 First-best social optimum In the taxi market, social surplus per hour is defined to be the sum of consumers' and producers' surpluses. Assuming equal value of time for all customers, the first-best social optimum is derived by maximizing social surplus S(F,N,
max s(Ff,Nf,
J

P(a)du)-PQ(Ff ,Nf ,
,
where P = G~' (Q) from Equation (9) and (pj , N f } represents the first-best solution of taxi fare and fleet size. Note that at the social optimum, price or marginal willingness-to-pay for one-hour taxi service is set equal to the marginal cost per taxihour, and fleet size should be such that the marginal benefit from adding one additional taxi (which stems from reduced customer waiting time and thus induced customer demand) equals marginal cost (cost per taxi-hour). Therefore, the social optimum would generally generate an efficient but unfeasible (deficit) equilibrium in the sense that taxi revenues may just cover the cost of occupied taxi-hours so that, in the aggregate, taxi operation may make a loss equal to the cost of vacant taxi-hours. 3. 2. 4 Second-best social optimum Running a private industry requires restricting price and entry so that profits are nonnegative. The subsequent second-best social optimum problem is that if given the constraint that revenues cover costs (profits are nil), what would determine an optimal price and number of taxis supplied in this market? This second-best social optimum can be obtained by the following constrained welfare maximization problem: max

S

,NS,($>]=

J

P(a)d(d

0

Fs,N5,y-cN5

(16)

subject to FsQ(Fs,Ns,q) = cN5

(17)

where the index V means the second-best solution. The problem becomes one where that social surplus (in this case social surplus equals consumer surplus since total taxi costs are held equal to the total taxi revenues), is maximized subject to a zero-profit

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constraint. Note that the zero-profit constraint (17) means that the objective function is actually simplified into Q^Fs,Ns,^/a. Since a is a constant, the second-best problem is equivalent to the problem that maximizes the total customer demand subject to the zero-profit constraint, and so the efficient price maximizes total realized customer demand (note that this is true in the case of homogeneous customers with identical value of time). 3.3

Numerical Example

To illustrate the complicated relationship between the demand for and the supply of the taxis, we consider the operation characteristics of the taxi market under a taxi fleet-fare two-dimensional space. The numerical example is based on the following values of parameters in the demand function: 0 = 100,000 (ride/h), 7 = 0.3(h), T, = 60.0 (HKD/h), T 2 = 35.0 (HKD/h), <x = 0.03(l/h), p = 400.0 (veh-h/km2). The cost per taxi hour of service time is assumed to be c = 50 (HKD/h). 3.3.1 Results and discussions There are four combinations of price and entry limitations in each (competitive and monopoly) market. These scenarios are shown in Table 1. Table 1. Scenarios of taxi regulations Competitive Market Monopoly Market

w X

Fare Unconstrained

Free Limited

Fixed

I-C

III-M

II-C II-M

I-M III-C

IV-C IV-M

The principal operational characteristic of the taxi market is portrayed graphically in a fleet (TV) - fare ( F ) space shown in Figures 1 and 2. The various combinations of fleet and fare would result in various social surpluses, taxi profits and customer demand, which are represented by iso-social surplus, iso-profit and iso-demand contours, respectively. A number of representative solutions with and without fare

Modeling Urban Taxi Services: A Literature Survey and An Analytical Example 267 and/or entry controls in monopoly and competitive contexts can be identified and discussed in subsequent subsections (also refer to Yang et al., 2002a).

20000

Figure 1. The iso-profit and iso-social surplus contours under a fleet (7V)-fare (F) twodimensional space

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15000

2000C

Figure 2. The iso-demand contours under a fleet (7V)-fare (F) two-dimensional space Case I: free entry, unconstrained fare In this case, the competitive equilibrium solution is determined by the nonlinear breakeven Equation (13) and indicated by the bold zero-profit counter curve in the figures. There is a feasible interval of fare within which the equation has solution. This feasible interval is given by the lowest and highest fare along the zero-profit contour.

Modeling Urban Taxi Services: A Literature Survey and An Analytical Example 269 In the fully unregulated competitive market, if the initial supply of taxis is in the left side of the zero-profit curve, there will be positive profits in the industry. Supply will be expanded toward zero-profit contours until the attractiveness of this action disappears - that is until profits become zero. No service will be offered on the right side of the zero-profit contours. However, there is no compelling evidence that any combination of fare and fleet solution values in the zero-profit curve will occur. Nevertheless it is conceivable that the most probable stable equilibrium occurs at point E where the competitive fleet size is maximized. This can be explained below. If an equilibrium occurs somewhere below point E at a fare lower than that at E (Figure 3), a single taxi driver would be able to increase his own profit by slightly raising his fare, thus entering the positive profit region. This can be represented by a short vertical vector pointing upwards from the zero-profit curve below point E. This profitable action will be perceived by all drivers and more taxi firms will be attracted to join the market. The influx of additional taxis can be represented by a short horizontal vector pointing right to reach the zero profit line. Eventually, the progression of fare and fleet increases will drive the equilibrium point to move upwards. If an equilibrium occurs somewhere above point E at a fare higher than that at E, a single taxi driver would be able to increase his own profit by slightly reducing his fare, again entering the positive profit region. This can be represented by a short vertical vector pointing downwards from the zero-profit curve above point E. The reason for making more profit by reducing fare is that the drivers could do more business (see the iso-demand contour in Figure 2), and hence increase the profit even with a lower fare. Again, existence of such a profitable business would attract more taxi firms to enter the market until the marginal profit of the last driver to enter becomes zero. This can be represented by a short horizontal vector pointing right to reach the zero profit curve. As a result, progression of fare decreases and fleet increases will drive the equilibrium point to move downwards. The only point, at which the incentive for all individual firms to change fare and/or for the number of taxis in service to change disappears, is point £", where the derivative of total market revenue with respect to fare vanishes, because total revenue = FQ = cN reaches a maximum value at TV" = A r max . Therefore, point E is considered a stable competitive equilibrium point in a long run. This observation indicates that a competitive market will eventually operate at point E if the market is fully deregulated.

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We must emphasize that the existence of the aforementioned downward and upward pressures on price is an empirical matter and probably indeterminate. Effective information communication between taxi drivers and customers about price levels is necessary to make possible upward and downward price competition. We now consider the monopoly solution in the absolutely unregulated market where there exists a single firm. The profit-maximizing solution is given by solving the monopoly problem (11) with respect to both selection of taxi fleet and setting of taxi fare. The solution point to the single monopoly firm occurs at ( N m , F m j = (7,000veh, 50HKD/ride), giving rise to a maximum profit of HK$366,509 HKD per hour corresponding to point D in Figures 1 and 2. Clearly the monopoly fare is artificially high and the monopoly fleet size is low. If there were more than one firm, the firms might try to lobby the government to impose entry and fare regulations that would move the industry away from the zero-profit contour toward this profit-maximizing point. Case II: fixed fare, free entry Now we consider a partially regulated market where the regulator sets fare, hi a competitive market, the solution of fleet size is unique and given by the point located at the zero-profit curve corresponding to the regulated fare. Thus the point at which the industry would operate depends on the fare set by the regulator. Two representative points are worth mentioning here. The first is the social surplusmaximizing point B located at ( N S , F S ) = (14,000veh, 20HKD/ride). This welfaremaximizing point is referred to as the second-best solution defined by (15) and (16), which gives a maximum social surplus (1,254,720 HKD per hour), this point also leads to a maximum realized demand (35,085 customer rides per hour), which means that social surplus and customer demand are maximized simultaneously under zeroprofit constraint in this case. Efficient fare regulation would require that fare be set at point B so that a feasible second-best solution is realized. As the fare is lowered below point B, the equilibrium utilization ratio is high, implying that the number of vacant taxis is low and the waiting time long, thus taxi demand declines because the increased waiting time more than nullifies the advantage of the lower fare for every customer. Similarly, as the fare is raised above point B, customer demand also declines, because the reduction in waiting time is insufficient to offset the higher fare for every customer. The extent of this tradeoff depends on the value of time of customers. Note that if there are two or more groups with different values of time, a

Modeling Urban Taxi Services: A Literature Survey and An Analytical Example 271 conflict situation will emerge in determining the second-best solution. Those with higher value of waiting time would prefer a higher price than those whose waiting time has less value. There is no easy answer for defining a unique combination of price and service level that is optimal for all groups of customers in this case. In practice, a comprise solution in price setting that maximize the welfare of most group of users can be envisioned. The second point worth mentioning is the fleet sizemaximizing point E. This point is inefficient from the society point of view because the maximum offered quantity of taxi-hours of service is not utilized efficiently. Additionally, as we can see from Figure 2, the maximization of service quality (minimum average customer waiting time) occurs at point H on the zero-profit curve with a somewhat higher fare than at point E. On the other hand, in a monopoly market the monopolist operates where the marginal revenue of taxis at the regulated price equals marginal cost of taxis, and the solution depends on the regulated fare, which is displayed by the curve of "monopoly solution under constrained fare". There would exist a regulated fare where customer demand (and thus total consumption of service) and the social surplus are maximized, respectively. These fares need not coincide with each other, but both are located in the neighborhood of point C where the quantity of taxi-hours of service provided by the monopolist is maximized. Within the lower fare range below point C, the monopolist will increase its taxi supply as fare increases. This action also leads to the concomitant increase in social surplus, profit and realized customer demand. At higher regulated fare above point C, the monopolist operates where its fleet size declines with further increase in fare. This would lead to concomitant decrease in customer demand, but increase in its profit as long as the fare does not exceed point D where maximum profit is achieved. The social surplus will increase initially with increase in fare from C to point G (where the social surplus is a maximum along the curve of "monopoly solution under constrained fare) and then decrease with further increase in fare above point G. Wherever the fare is fixed, the fleet size selected by the monopolist is always less than the fleet size that maximizes social surplus at the second-best solution point B. We now compare the demand and social surplus associated with competitive and monopoly industry at the-same regulated fare. This can be checked by drawing a horizontal line at the regulated fare, and finding its two intersections with the curve of monopoly solution under constrained fare and the zero-profit curve, respectively. Then the values of iso-social surplus (Figure 1) and iso-demand contours (Figure 2) going through the intersections can be identified. The following observations are

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made from the comparison. The competitive fleet size is always greater than the monopoly fleet size, thus the competitive industry demand curve (demand versus regulated fare) lies above the monopoly demand curve because of the higher service quality or lower waiting time offered by provision of service of larger taxi-hours. Therefore, the competitive outcome is superior to the monopoly outcome at every proper fare up to and including the second-best efficient level. This is mainly due to the use of different decision rules by the competitive firms and monopolist under price-fixing. Namely, the competitive firms will not select taxi-hours of operation as a function of customer demand; rather, it will behave as if it can supply an unlimited number of taxi hours at a taxi utilization rate where the marginal cost of additional taxi equals price (stay in the zero-profit curve). However, the monopoly firm will internalize the interdependencies that make the utilization rate of a given taxi dependent on the number of taxis operating to maximize its total profit at a given, constrained fare. If, on the other hand, prices are set too high (well above the secondbest efficient level), the reduction in taxi supply experienced under monopoly decision-making becomes relatively efficient in terms of total market conditions (social surplus). Namely, marginal cost of taxi-hour under competition will be in excess of marginal value. This is caused by the regulator's setting the price above the efficient level, which results in wasteful competition among firms. Case III: limited entry and unconstrained fare The case of limited entry and unconstrained fare seldom exists in reality. Figures 1 and 2 indicate that entry controls, when set at binding, will normally have an effect on both fare and demand levels in an either competitive or monopoly market. In a monopoly market, if the entry limit is set below the level at point D (it is conceivable that no regulatory authority would like to set such a low entry limit), the monopolist will be willing to supply as many taxi-hours as possible as profit increases with fleet size in this range. Then the fare movement will occur along the curve of "monopoly solution under entry limit". If the taxi-hour limit is greater than the level at D, the monopolist will choose point D as the optimal level of supply conditional on unconstrained fare, and thus the entry limit has no impact. Nevertheless, there might be one possibility that holding a monopoly right granted by the regulatory authority would require the monopoly firm to operate a minimum number of taxis in order to sustain a minimum service standard. In this case the monopolist will select a fare to maximize monopoly profit, and the fare movement occurs along the curve of "monopoly solution under minimum fleet requirement" between points D and E. The

Modeling Urban Taxi Services: A Literature Survey and An Analytical Example 273 monopoly solution curve could be further extended beyond point E in a short run where the monopoly firm operates at a loss, but in a long run, it may make profit. On the other hand, in a competitive market with entry limit, the market becomes protected, and will operate in the positive profit region at given entry limit. The fare could probably take any value in between the lowest and the highest zero-profit fares at the given entry limit. But it tends to self-adjust upwards or downwards and eventually stays at average profit-maximizing point at the entry limit. If the entry limit is set below profit-maximizing fleet size at point D, the competitive solution coincides with the monopoly solution under entry limit. This explains that the regulation of entry limit would create a monopoly market where individual taxi drivers coordinate and cooperate to fix their fare level (say through a taxi driver union or association) because they all share the common interest: profit maximization. If the entry limit is set in between points D and E, competitive solution will stay on the curve of "monopoly solution under minimum fleet requirement". Although in this case competitive and monopoly solutions are identical, the forces that sustain their equilibrium are different: the force for the former is due to active limit on the entry of new competitive firms (one taxi per firm), while that for the latter is due to minimum fleet size that a monopoly firm has to provide in order to hold a monopoly right from the government. Furthermore, under effective fare competition, the resultant fare is much higher and the realized demand is much lower in comparison with the secondbest efficient solution, it seems fair to conclude that entry controls should only be used in conjunction with fare controls. Case IV: limited entry and fixed fare This is the usual situation with most large city taxi regulations. Regulatory authorities allocate a fixed number of operating permits, and prices are fixed on a meter basis. A single firm in a monopoly market will operate at a level of taxi-hours as close as possible to the profit-maximizing point D. The competitive market will produce a fleet size defined by the price and/or entry limit. The entry limit, if set in the right side of the zero-profit curve, will have no effect, and, if set in the left side of the zero-profit curve, will become a determining factor. In the latter case, the competitive firms will earn supernormal profits that could be reflected by a high taxi transaction price (medallion or license price). On the other hand, the fare constraint, if set above dashed line D-E, will generally have no impact, and, if set below D-E, will determine the competitive outcome in conjunction with entry limit. Therefore, when fare and entry are regulated, either regulation may be the determining factor.

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Whenever restriction in fare and entry is binding, relaxation of such a restriction could result in either an increase or a decrease of social surplus. Now suppose that the authority initially imposes fare and entry regulations so that the taxi market is operating in the left-side of zero-profit curve, there would be efficiency gain if the authority reduced the fare to the second-best efficient level and eliminated entry restrictions, since the competitive market would then move to the second-best point B, which is on a higher iso-social surplus contour. If the fare was reduced but the entry restriction was not changed, the market would move down to stay in a higher isosocial surplus contour (a smaller positive efficiency gain) if the entry restriction is set relatively high (say higher than or equal to the second-best efficient level), and the market would move down to stay in an either higher or lower iso-social surplus contour (either positive or negative efficiency gain) if the entry restriction is set well below point B. In a similar manner, it can be checked that elimination of the entry restriction without a reduction in fare might either reduce or increase social surplus, depending on the initial location of market operation. Therefore, regulation of a taxi market should be implemented cautiously to improve social surplus. The second-best social optimum can be realized by restriction of fare only in a competitive market. Nevertheless, in a monopoly market, realization of the second-best social optimum requires both fare restriction and minimum fleet requirement. Leader-follower or Stackelberg game solution Some cases of the taxi service regulation and market equilibrium problem examined so far can be described as a leader-follower, or a Stackelberg game (Varian, 1992) where the regulator is the leader, and the taxi firm(s) is the follower. In a competitive or monopoly market, suppose the regulator has control of either fare or entry (fleet size), but not both, in light of any control decision, taxi firm(s) decides whether to enter the market (selection of fleet size in the case of monopoly) or chooses a fare to maximize profit. Thus, the Stackelberg game solution can be characterized by that the regulator chooses an optimal control decision to maximize social surplus, while taking into account the reaction of the taxi firm(s). We first consider the case where the regulator has control of fare only. In a competitive market, individual taxi firms (drivers) decide whether or not to enter the market for a given fare, and the aggregate reaction curve of taxi firms is the curve of "competitive solution". Point B (the second-best point) is the Stackelberg game solution that leads to a maximum social surplus along this reaction curve, hi a

Modeling Urban Taxi Services: A Literature Survey and An Analytical Example 275 monopoly market, the reaction curve of the monopolist is the curve of "monopoly solution under constrained fare", which describes how the monopolist chooses a taxi fleet size to maximize profit for a given regulated fare. The Stackelberg game solution corresponds to point G at which the reaction curve is tangent to one of the social surplus contours. Clearly, point G is near point C (the maximum monopoly fleet size solution). We next consider the case where the regulator has control of entry or fleet size only. Then the reaction curve of taxi firm(s) will be the broken line D-E in both competitive and monopoly market. As mentioned before, this reaction curve is due to the regulator's entry limit control in a competitive market and minimum fleet requirement in a monopoly market. The Stackelberg game solution in this case depends where the reaction curve is tangent to the social surplus contours. If tangent at point E or any point in its left side (the nonnegative profit domain), the tangent point will be the Stackelberg solution. Otherwise, point E will be the solution because a tangent point associated with a negative profit is infeasible. In the current example, it was found that the Stackelberg game solution is located at about the middle point between point D and E. Social optimum Finally, we briefly discuss the first-best social optimum. From the point of view of the economy as a whole, the efficient allocation would be to choose a fleet-fare combination to maximize social surplus defined by (14). The first-best social optimum is obtained at point ( N f , F f }

= (17,000veh, 15HKD/ride), which is located

at the right side of the zero-profit contour. The first-best taxi pricing entails taxi operation at a loss; more specifically at an aggregate loss of 219,661 HKD per hour, but resulting in a social surplus of 1,281,145 HKD per hour in the market. Therefore, attainment of the full first-best would require introduction of a mechanism that subsidizes taxi travel to cover the deficit that is related to the cost of vacant taxi-hours at the optimum. The improvement in social surplus gained by driving the market from the second-best to the first-best is equal to 26,425 HKD per hour (from 1,254,720 HKD/h of the second-best to 1,281,145 HKD/h of the first-best). Clearly, this improvement is very marginal, giving rise to the question: Is it appropriate to try to attain a social optimum at the expense of introducing a complex subsidy mechanism?

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4.

MULTI-PERIOD DYNAMIC AGGREGATE MODELING

4.1

Hourly Variation of Customer Demand and Taxi Service Intensity

Existing modeling of taxi services involves a single static period in which all licensed taxis are assumed in service (either vacant or occupied), and demand and supply are distributed uniformly. A remarkable characteristic of taxi service in most metropolitans is the hourly variation of customer demand and taxi service intensity throughout the day. In Hong Kong, a taxi driver generally owns his own taxi or hires one from a taxi owner, and majority of taxis are cruising taxis. Taxi drivers provide customer services in response to the profitability of the industry under given government regulations such as fare control. Taxi drivers vary the extent of service by changing the number of hours per day they work and their work schedule. In general, taxi drivers work on a shift basis with 8 hours per shift and most taxis operate on two or three shifts each day, and shift schedule varies across individual taxis. Although the three 8-hour shifts span the whole 24 hours demand period, not all taxis operate for three shifts, and a taxi driver may not work for 8 hours during each shift. Taxi drivers have to spend time for other activities such as a meal break. Taxi service hours or service intensity is strongly governed by the market profitability. When the business is unprofitable, owners may simply park taxis. Profitability of the market depends on the actual utilization of taxis in service, while taxi utilization depends on the customer demand and taxi supply, all variables exhibit substantial time of variations. In general the total number of daily working hours and the actual time of day working schedule of each taxi are endogenous variables, depending on the time distribution of customer demands. 4.2

Necessity to Model Period-Dependent Taxi Services

As aforementioned, customer demand and the actual number of taxis operating for a given number of licensed taxis changes dramatically during time of day. Thus precise understanding of the essential industry equilibrium requires modeling the service intensity of individual taxis in response to the customer demand in each period, thereby generating more useful and suitable information for regulating the industry, such as temporal fare differentiation, as introduced in many large cities (e.g., Tokyo) where taxi fares in the middle night are usually higher. The development of a multiperiod taxi service model has important application in practice as well. In most large cities, taxi industries are almost always regulated, and the regulation is likely to remain a reality. A common rationale for regulating the industry has been to make

Modeling Urban Taxi Services: A Literature Survey and An Analytical Example 211 transport available at times when demand is low and in areas where population is dispersed. Hence it is important to predict the impact of both price and entry regulations relevant to a given industry. What can be observed is the number of taxis operating, the ratio of occupied to empty (unoccupied but in service) taxis, trips made and waiting times. These four quantities have been prominent in discussions of the trade. Indeed, in Hong Kong, roadside observation surveys and taxi stand surveys covers the whole day in order to capture the characteristics of both spatial and temporal variations of taxi supply and customer demand. All these characteristics point to the necessity and importance to develop a taxi model capable of analyzing the multi-period characteristics, thereby acquiring more useful measures of service quality and market profitability. 4.3

A Clock-Network Equilibrium Modeling Approach

Here we briefly describe an aggregate multi-period dynamic taxi service model with endogenous taxi service and utilization intensity (Yang et al., 2002b). One possible way to model time of day taxi services is to divide the whole service period (one whole day) into a number of sub-periods. During each sub-period, taxi supply and demand characteristics are assumed to be uniform. Customer demand is period specific and described as a function of waiting time for a given fare structure. Taxi operating cost consists of two components: one component being a function of total service time and the other component being period dependent. Each taxi driver can work for one or more shifts each day and freely choose starting and ending time of each shift. Equilibrium of taxi services obtains when taxi drivers cannot increase their individual profits by changing their individual work schedule (extending total service time or shifting service periods). As shown in Figure 3, the dynamic multi-period taxi service demand and supply equilibrium problems can be characterized by a clock network equilibrium problem. Assume that a typical day of taxi services is divided into 24 periods of equal length (one hour) over which customer demand varies. Path flow on the clock network represents taxi workflow throughout the whole day. There are three groups of links connecting a single origin-destination pair. Along the clock, the number of each node corresponds to a clock time; each time period is represented by a link with start node (time) and end node (time), respectively. This group of links along the clock is termed as clock links and the corresponding nodes as clock nodes. The second group of links is entry links connecting the origin node 25 and each of the clock nodes. The third group of link is exit links connecting each of the clock node and the destination node

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26. Clearly, a path flow in the network from the origin 25 to destination 26 represents a shift of taxi services. For example, a path flow, 25 —» 8 —> 15 —> 26, represents a shift starting at 8:00am and ending at 3:00pm. In addition, flow on a clock link represents the total number of taxis in service during the relevant time period.

Figure 3. The proposed clock network for equilibrium characterization of multiperiod aggregate taxi services with endogenous service intensity With the above construct, the multi-period taxi service equilibrium is formulated as network equilibrium model with path-specific cost and can be solved using conventional nonlinear network optimization methods. The proposed model can ascertain at equilibrium the intensity of use of taxis, utilization rate for taxi and level of service quality in each period. This information is useful for prediction of the effects of alternative regulations such as time of day differentiation of service charge on the taxi industry.

Modeling Urban Taxi Services: A Literature Survey and An Analytical Example 279

5.

NETWORK EQUILIBRIUM MODELING

5.1

Overview

To precisely understand the equilibrium nature of urban taxi services and assess traffic congestion due to (both vacant and occupied) taxi movements together with normal traffic, it is necessary and important to model taxi services in a network context. In this respect, Yang and Wong and their collaborators have developed a substantial stream of researches in recent years and moved the literature a long way forward in terms of scope for application of the taxi models. In this section we present a brief review of their recent works on network equilibrium modeling of urban taxi services. Yang and Wong (1997, 1998) made an initial attempt to characterize taxi movements in a road network for a given and fixed customer origin-destination demand pattern. To model taxi traffic, it is assumed that a customer, who has taken a taxi once before (similar to the normal traffic), will try to minimize their individual travel cost from his origin to destination; and a (vacant) taxi, who has dropped a customer once before, will try to minimize their individual expected search cost required to meet the next customer. The probability that a vacant taxi meeting a customer at a particular zone is specified by a logit model by assuming that the expected search time in each zone is an identically distributed random variable due to variations in perceptions of taxi drivers and the random arrival of customers. A simultaneous equations system is proposed to describe the movements of both empty and occupied taxis and solved by a fixed-point algorithm. Wong and Yang (1998c) reformulated the taxi service problem in networks as an optimization problem that leads to a more efficient and convergent iterative balancing algorithm for the case without traffic congestion. The simple network model of urban taxi services was further enhanced and extended by Wong and Yang (1998a) and Wong et al. (2001) and Yang et al. (2002a). The extensions include incorporation of congestion effects, customer demand elasticity, reformulation of the problem as two simultaneous optimization problems and development of a new solution algorithm, hi their models, normal traffic is included and assumed to follow conventional equilibrium routing behavior. Instead of the characterization of pure taxi movements in a network by a system of nonlinear equations, a simultaneous optimization of two equilibrium problems is proposed for taxi movements in congested road networks (Wong et al., 2001), and an improved sensitivity-based algorithm was presented in Wong et al. (2002a). One problem is a combined network equilibrium model that describes simultaneous movements of

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vacant and occupied taxis as well as normal traffic in a user-optimal manner for given total customer generation from each origin and total customer attraction to each destination. The other problem is a set of linear and nonlinear equations ensuring that the relation between taxi and customer waiting times and the relation between customer demand and taxi supply are satisfied. The model can be used for the assessment of road traffic congestion due to both taxi and normal traffic movements and evaluation of the level of taxi services in a network context. The potential applications of the model have been demonstrated by several case studies with the urban area of Hong Kong. Wong and Yang (1998b), Wong et al. (1999) and Yang et al. (2001) reported the results of their case study for the calibration and validation of the simple network model for urban area in Hong Kong using the Hong Kong annual taxi survey data and annual traffic census data (Transport Department, 1986-2002, 1993, 1998). Yang et al. (2002a) investigated the nature of demand-supply equilibrium under competition and regulation with their calibrated taxi service model, hi particular, they found that the current regulated fare and entry are reasonably close to the second best solution, which reflects the fact that the policy set by the government has acted quite effectively in response to the market outcome. They also found that the most probable stable equilibrium in a competitive market is, in fact, very close to the second best solution, which means that the government has successfully created a socially effective but also highly competitive market environment for taxi services through a good mix of company and individual taxi owners and the competitive bidding for taxi licenses (Hong Kong Transport Advisory Committee, 1992). 5.2

Multi-Class Taxi Model with Service Area Regulation

Existing economic analyses of the regulation or deregulation of taxi services have overwhelmingly focused on price and entry control. There remains one important issue to be examined: service area regulation, which has to be examined in a context of multi-class taxi services. For instance, in Hong Kong New Territories taxis can only take customers to as far as Tsuen Wan (the western part of Kowloon) or Shatin (the northern part of Kowloon), but cannot pick up customers in the urban area, whereas urban taxis can pick up customers in both urban areas and New Territories. There are two primary objectives for this service area regulation: 1) to guarantee the taxi service quality in terms of taxi availability in rural areas, and 2) to differentiate license fee and unit charge since urban and rural taxis contribute to traffic congestion to different extents at different locations. Evidently, in this case spatial structure of the

Modeling Urban Taxi Services: A Literature Survey and An Analytical Example 281 taxi market is essential, and the existing single class network equilibrium models for taxi services have to be extended to the multi-class taxi service situation to describe how various kinds of taxis (vacant and occupied taxis in both urban and New Territories service areas) cruise in a road network to search for various kinds of customers and provide transportation services. In two companion papers, Wong et al. (2002b and 2002c) developed and solved the taxi service models with service area regulation in congested networks with elastic customer demands. The extended model explicitly takes into account multiple user classes, multiple taxi classes and hierarchical modal choice of customers for taxi services together with normal traffic. It is assumed that there are several classes of customers with different values of time and money and several classes of taxi services with distinct combinations of service area restrictions and fare levels. For service area restrictions, the taxi services are divided into several classes geographically by restricting certain classes of taxis to pick up or set down passengers in certain areas (i.e. they may not operate within some zones). For taxi fares, it allows different classes of taxis to charge at different levels (for instance, the operation of luxury taxis), but is independent of the types of customers taking the service. A simultaneous optimization formulation of two equilibrium sub-problems is proposed for the model with multiple user and taxi classes. One sub-problem is a combined network equilibrium model (CNEM) that describes the hierarchical logit mode choice model between occupied taxis and normal traffic together with the vacant taxi distributions in the network searching for customers. The other subproblem is a set of linear and nonlinear equations (SLNE) ensuring that the relation between taxi- and customer-waiting times, the relation between customer demand and taxi supply for each taxi class and the taxi service time constraints are satisfied. The CNEM is formulated as a mathematical program that is solvable by a partial linearization algorithm. On the other hand, the SLNE is solved by a Newtonian algorithm with line search (Press et al., 1992). The model can be used for the assessment of the inter-relationships between customer demand and taxi utilizations for different classes of taxis providing services in the network.

6.

CONCLUSIONS

This chapter has shown that modeling urban taxi service under competition and regulation is an intriguing issue worth of analytical and computational studies by

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economists and transportation researchers. Indeed, the taxi market is not amenable to the usual demand-supply analysis and has been one of the prototypical examples for analytical study of the economic consequences of regulation and regulatory choices used by economists. The interest of economists for the analytical aspects of the taxi market has lasted for more than forty years and still continues today. Recent developments of network and dynamic modeling of taxi services have made significant progress toward more practical applications and broader economic analyses of taxi service regulation in both time and space dimension. As a natural development of the studies described in this chapter, it is delightful to combine the multi-period aggregate equilibrium model on the artificial clock network and the static demandsupply equilibrium model on the physical road network to investigate both temporal and spatial aspects of urban taxi services, and then a graphical representation described in this chapter will offer all the possibilities of the demand-supply equilibria under alternative regulatory choices. It is particularly meaningful to investigate how regulations in terms of price control and/or entry restriction should be implemented for achieving the second-best optimum in a competitive market when congestion externality is present and customers are heterogeneous in terms of their values of waiting time and in-vehicle ride times. We also intend to model the bilateral search behaviors of customers and taxi drivers and spell out their resulting search friction and equilibrium movement decisions. All these studies are being conducted by the authors and results will be reported in subsequent papers.

ACKNOWLEDGMENTS This study was substantially supported by research grants from the Research Grants Council of the Hong Kong Special Administrative Region (Project No. HKUST6033/02E and HKU7019/99E) and an Outstanding Young Research Award 2000 from the University of Hong Kong.

REFERENCES Abe, M. A. and B. C. Brush (1976). On the regulation of price and service quality: The taxicab problem. Quarterly Review of Economics and Business, 16, 105111. Arnott, R. (1996). Taxi travel should be subsidized. Journal of Urban Economics, 40, 316-333.

Modeling Urban Taxi Services: A Literature Survey and An Analytical Example 283 Beesley, M. E. (1973). Regulation of taxis. Economic Journal, 83, 150-169. Beesley, M. E. and S. Glaister (1983). Information for regulation: The case of taxis. The Economic Journal, 93, 594-615. Bergantino, A. S. and E. Longobardi (2001). A critical appraisal of deregulation in the taxi market. Paper presented at the 9th World Conference on Transportation Research, 22-27 July 2001, Seoul, Korea. Cairns, R. D. and C. Liston-Heyes (1996). Competition and regulation in the taxi industry. Journal of Public Economics, 59, 1-15. Dempsey, P. S. (1996). Taxi industry regulation, deregulation and reregulation: The paradox of market failure. Transportation Law Journal, 24, 73-120. De vany, A. S. (1975). Capacity utilization under alternative regulatory constraints: An analysis of taxi markets. Journal of Political Economy, 83, 83-94. Douglas, G. W. (1972). Price regulation and optimal service standards: The taxicab industry. Journal of Transport Economics and Policy, 20, 116-127. Fernandez, J. E., J. De Cea, and J. Briones (2001). Evaluation of policies for the operation of the market of urban taxi services: the case of Santiago de Chile. Paper presented at the 9* World Conference on Transportation Research, 22-21. July 2001, Seoul, Korea Flath, D. (2002). Taxicab regulation in Japan. Kyoto Institute of Economic Research, Kyoto (working paper). Foerster, J. F. and G. Gilbert (1979). Taxicab deregulation: Economic consequences and regulatory choices. Transportation, 8, 371-387. Frankena, M. W. and P. A. Pautler (1986). Taxicab regulation: An economic analysis. Research in Law and Economics, 9, 129-165. Friedman, M. (1962). Price theory: A provisional text. Aldine, Chicago: Aldine Publishing Co. Gallick, E. C. and D. E. Sisk (1987). A reconsideration of taxi regulation. Journal of Law, Economics and Organization, 3(1), 117-128. Gaunt, C. (1996). Information for regulators: the case of taxicab license prices. International Journal of Transport Economics, XXIII(3), 331-345. Gaunt, C. and Black, T. (1996). The economic cost of taxicab regulation: The case of Brisbane. Economic Analysis and Policy, 26(1), 45-58 Garling, T., T. Laitila, A. Marell and K. Westin (1995). A note on the short-term effects of deregulation of the Swedish taxicab industry. Journal of Transport Economics and Policy, 29(2), 209-214. Hackner, J. and S. Nyberg (1995). Deregulating taxi services: A word of caution. Journal of Transport Economics and Policy, 29, 195-207.

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Hong Kong Transport Advisory Committee (1992). Consultative Paper on Taxi Policy Review. Hong Kong Transport Advisory Committee, Hong Kong. Lipsey, R. G. and P. Steiner (1966). Economics, Harper-Row, New York. Manski, C. F. and J. D. Wright (1976). Nature of equilibrium in the market for taxi services. Transportation Research Record, 619, 296-306. Morrison, P. S. (1997). Restructuring effects of deregulation: The case of the New Zealand taxi industry. Environment and Planning , 29A, 913-928. Orr, D. (1969). The taxicab problem: A proposed solution. Journal of Political Economy, 77, 141-147. Press, W. H., S. A. Teukolsky, W. T. Vetterling and B. P. Flannery (1992). Numerical Recipes in Fortran 77: The Art of Scientific Computing. Second Edition, Cambridge University Press, Cambridge, New York. Radbone, I. (1998). Looking at Adelaide's taxi industry. Road and Transport Research, 7(2), 52-59. Rometsch, S. and E. Wolfstetter (1993). The taxicab market: An elementary model. Journal of Institutional and Theoretical Economics, 149(3), 531-546. Schaller, B. (1999). Elasticities for taxicab fares and service availability. Transportation, 26, 283-297. Schroeter, J. R. (1983). A model of taxi service under fare structure and fleet size regulation. Bell Journal of Economics, 14, 81-96. Shrieber, C. (1975). The economic reasons for price and entry regulation of taxicabs. Journal of Transport Economics and Policy, 9, 268-293. Shrieber, C. (1977). A rejoinder. Journal of Transport Economics and Policy, 11, 298-304. Teal, R. F. and B. Berglund (1987). The impacts of taxicab deregulation in the USA. Journal of Transport Economics and Policy, 21, 37-56. Transport Department (1986-2000). The Level of Taxi Services. TTSD Publication Series, Hong Kong Government, Hong Kong. Transport Department (1993). Updating of Second Comprehensive Transport Study Final Report. Hong Kong Government, Hong Kong. Transport Department (1998). The Annual Traffic Census - 1998. Hong Kong SAR, Hong Kong. Transport Department with MVA Asia (1993). Travel Characteristics survey: Final Report. Hong Kong Government, Hong Kong. Varian, H. R. (1992). Microeconomic Analysis, W.W. Norton & Company, Inc., New York.

Modeling Urban Taxi Services: A Literature Survey and An Analytical Example 285 Yang, H., Y. W. Lau, S. C. Wong and H. K. Lo (1998). A macroscopic taxi model for passenger demand, taxi utilization and level of services. Transportation, 27, 317-340. Yang, H. and S. C. Wong (1997). Modeling urban taxi services: A network equilibrium approach. Proceedings of the Second Conference of Hong Kong Society for Transportation Studies, Hong Kong, 6 December 1997, 15-20. Yang, H. and S. C. Wong (1998). A network model of urban taxi services. Transportation Research, 32B, 235-246. Yang, H., S. C. Wong and K. I. Wong (2001). Modeling urban taxi services in road networks: Progress, problem and prospect. Journal of Advanced Transportation, 35, 237-258. Yang, H., S. C. Wong and K. I. Wong (2002a). Demand-supply equilibrium of taxi services in a network under competition and regulation. Transportation Research, 36B, 799-819. Yang, H., M. Ye, W. H. Tang and S. C. Wong (2002b). An aggregate multi-period dynamic taxi service model with endogenous service intensity (working paper). Wong, K. I., S. C. Wong, H. Yang and C. O. long (2002a). A sensitivity-based solution algorithm for the network model of urban taxi services. Proceedings of the 15th International Symposium on Transportation and Traffic Theory (ISTTT1S) (edited by M. A. P. Taylor), Pergamon, 16-18 July 2002, Adelaide, Australia. 23-42. Wong, K. I., S. C. Wong, H. Yang and J. H. Wu (2002b). Modeling urban taxi services with multiple user and vehicle classes, I: model formulation. Transportation Science (submitted). Wong, K. I., S. C. Wong, H. Yang and J. H. Wu (2002c). Modeling urban taxi services with multiple user and vehicle classes, II: solution algorithm and numerical examples. Transportation Science (submitted). Wong, K. I., S. C. Wong and H. Yang (2001). Modeling urban taxi services in congested road networks with elastic demand. Transportation Research, 35B, 819-842. Wong, K. I., S. C. Wong and H. Yang (1999). Calibration and validation of a network equilibrium taxi model for Hong Kong. Proceeding of the Fourth Conference of Hong Kong Society for Transportation Studies, Hong Kong, 4 December 1999, 249-258. Wong, S. C. and H. Yang (1998a). Modeling the level of taxi services in congested road networks. Proceedings of Tristan III, Volume II, Puerto Rico, 17-23 June 1998.

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Wong, S. C. and H. Yang (1998b). Calibration of a network equilibrium model for urban taxi services. Proceedings of the First Asia Pacific Conference on Transportation and the Environment, Singapore, 13-15 May 1998, 122-130. Wong, S. C. and H. Yang (1998c). Network model of urban taxi services: Improved algorithm. Transportation Research Record, 1623, 27-30. Xu, J. M., S. C. Wong, H. Yang and C. O. long (1999). Modeling the level of urban taxi services using a neural network. Journal of Transportation Engineering, ASCE, 125(3), 216-223.

CHAPTER 11

THE ESTIMATION OF ORIGIN-DESTINATION MATRICES IN TRANSIT NETWORKS S. C. Wong and C. O. Tong, Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong SAR, P.R. China

1.

INTRODUCTION

The steady-state transit assignment problem is usually modeled by a headway-based approach, in which the arrivals of transit vehicles on a transit line are described by means of its average headway. Le Clerq (1972) was among the first to point out the problem of common lines on transit networks. A large body of literature has since been devoted to solving this problem (Chriqui and Robillard, 1975; Last and Leak, 1976; Nguyen and Pallottino, 1988; De Cea et al, 1988; De Cea and Fernandez, 1989, 1993; Spiess, 1987; Wu et al., 1994). However, little effort has been spent on dynamic trip assignment for transit networks. The headway-based approach assumes that each transit line operates at a constant headway, and that the speed of a vehicle is determined by a volume/delay function. The passenger waiting time for boarding a line is a probabilistic function of the vehicle headway. This is a static model that assumes a steady state situation and predicts the average performance of the transit system within a specified period, e.g. at peak hour or off peak hour.

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Recently, a promising schedule-based approach for transit assignment, in which vehicle headway and speed are determined from line schedules, has received much attention (Tong and Richardson, 1984; Tong, 1986; Carraresi et al., 1996; Tong and Wong, 1999; Nguyen et al, 2001; Nuzzolo et al, 2001). The passenger waiting time for boarding a line is a deterministic function of line schedules and the arrival time of the passenger at the station. This schedule-based approach describes the clockdependent movement of vehicles within the network, and models vehicle profiles and headways exactly as specified in line schedules. When the transit vehicles are known to operate quite close to pre-announced schedules and the vehicle speeds are not greatly affected by traffic conditions, there are advantages in adopting a schedulebased approach. If the schedules are stored in a computer database, then a network description can be generated automatically with little manual coding. Given a specified period of analysis, which could be one hour, several hours, or the whole day, the model analyzes the schedules and compiles appropriate network descriptions. Hence, it is particularly suitable for time-dependent analysis. As the schedule-based approach can be used as an itinerary builder, it is often used as part of passenger information systems. It can also be employed to develop a dynamic trip assignment model. As the optimal paths that are computed are clock-dependent, the assignment procedure can keep track of the vehicle schedules and assign passengers who are waiting at stations (or platforms) to the appropriate vehicles that are departing the stations. The output from the model shows the number of passengers who are assigned to each vehicle at different times during the period of analysis. It also shows the number of passengers who are waiting at platforms or walking between platforms in walk links. This type of information can be used to evaluate the performance of an existing transit system that is operating to schedules, such as a railway network. As in many dynamic assignment models, time-dependent origin-destination (O-D) matrices, which are usually difficult and expensive to obtain, form one of the key inputs to the model. Hence, in this chapter we present a low cost methodology to estimate these matrices based on the observed passenger counts at some locations in the network. The estimation of stationary O-D matrix from traffic counts in highway networks has been thoroughly studied in the literature (Wilson, 1970; Van Zuylen and Willumsen, 1980; Bell, 1983, 1991; Cascetta, 1984; Spiess, 1987; Lam and Lo, 1990; Yang et al, 1992, 1994). For dynamic O-D estimation for highway networks, Cascetta et al. (1993) provided a theoretical framework for the problem, in which the problems of dynamic O-D estimation were broadly divided into two categories: simultaneous estimators and sequential estimators. Simultaneous estimators produce

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joint estimates of the whole set of O-D matrices, one for each time slice, whereas sequential estimators produce a sequence of O-D estimates for successive time slices. While simultaneous estimators are a direct extension of the state-of-the-art static O-D matrix estimation, Ashok and Ben-Akiva (1993) proposed a sequential estimator, in which the problem of dynamic O-D matrix estimation was formulated as a Kalman Filter where the state vector consists of deviations of O-D flows from prior estimates based on historical data. In this chapter we propose a simultaneous estimator to estimate the time-dependent OD matrices for transit networks. The entropy-based approach is employed here, given the fact that there is no strong evidence of which type of model is superior, but the entropy-based model has certain computational advantages that will be discussed in a later section. In the next section, the schedule-based dynamic transit network model is described. We then introduce the time-dependent optimal path algorithm and the network loading procedure. The entropy-based formulation of the estimation of timedependent matrices in transit networks is given, and the solution of the resultant problem by means of a very efficient sparse algorithm are discussed (Wong and Tong, 1998). The Mass Transit Railway (MTR) system in Hong Kong is used as a case study to illustrate the effectiveness of the methodology, in which the estimated O-D matrices are compared to the true matrices that were obtained from a sophisticated electronic fare collection system on the MTR. Cases with and without prior information are considered for this network, and a good agreement of results is found. The methodology that is presented in this chapter provides a low cost solution to the estimation of O-D matrices, and hence allows the sensitivity analysis of the performance of a transit system with respect to the changes in transit schedules and the addition/removal of transit lines.

2.

THE DYNAMIC SCHEDULE-BASED MODEL

A transit system is represented by a graph G = (TV, A, L), where N is the set of nodes, A is the set of links, and L is the set of transit lines. A node is used to represent a source or sink of passengers, or to represent a transit stop, or a platform in a transit station. While a small transit stop is represented by a single node, a large transit station can be represented by many nodes, with one node to represent the ticketing hall and other nodes to represent the platforms.

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Let T be the period of analysis. The entire period of analysis T is divided into Nt equal intervals each of length T (say 1 minute). All of the time related variables in the model are then specified as a multiple of T. During each time interval /, the passenger demand for each O-D pair is assumed to be uniformly distributed. The time dependent O-D trip matrix is given by (1)

where drst is the number of passengers who are departing from origin node r to destination node s, who leave their origin node during the time interval /, and Nr and Ns are the numbers of origin and destination nodes in the transit system. This timedependent O-D matrix forms one of the key inputs into the dynamic model, which can be obtained from a field survey, or alternately estimated by a time-dependent O-D matrix estimator for the transit networks that are presented in the sequel. A transit line is a fixed path through which transit vehicles run periodically at fixed schedules and a fixed run profile. A run is a trip made by a transit vehicle from the beginning to the end of a transit line. A run profile is defined by the time space graph of a vehicle when making a complete run. It is assumed that vehicles always operate on schedule. For each transit line / e L , a timetable is defined as follows. Let It and Jf be the number of transit stops and the number of runs of vehicles for the transit line /. Denote n^ and h^ as the node number of the rth stop and the departure time from the /th stop for they'th run of vehicle, for transit line /. Note that the runs of a transit line are numbered in ascending order of the departure times from the first node, i.e. (2)

Each pair of adjacent stops along a transit line defines a unidirectional transit link in the network. When there are several common lines directly joining two nodes, each of them forms a distinct transit link. For a transit link a, denote a~ and a+ as the upstream and downstream nodes, and a ° as the transit line to which the link belongs. As each transit line / is operated according to a specified schedule, the number of runs of vehicles traversing a transit link is also equal to J{ . We further assume that no overtaking is allowed for the transit vehicles. Therefore, they'th run of vehicles in the timetable is also they'th run of vehicle on a transit link, and the (j+l)lh. run of vehicle must always depart from the stop later than theyth run of vehicle. Let M • and v be

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the departure time of they'th run of vehicle from the upstream node of the transit link a and the time this vehicle departs at the downstream node of the link. Hence, we have

and v

aj=hlj-HJ>

when a" - «/, and a+ = «/ J+1 . The travel time of they'th run of vehicle on a transit link a can then be determined as

where the superscript "z" denotes that the variable is associated with a transit line. The loading condition of transit link a is represented by a vector \za = (xzaj , V/) , where xzaj is the number of passengers whom are present in they'th run of a vehicle while traversing transit link a. Apart from the transit links, there is a set of walk links in the transit system. A walk link is a unidirectional link for passengers to board, alight, or transfer within the transit network. Let t™ be the walking time along a walk link a, which is calculated from the length of the link and an assumed constant walking speed. The superscript "w" denotes that the variable is associated with a walk line. The loading condition for a walk link is represented by a vector x^ = (x™{ , V?) , where x% is the number of passengers walking on link a during time interval t.

3.

TIME-DEPENDENT OPTIMAL PATH ALGORITHM

While others used hyper-network schemes to expand the network for time-dependent analysis (Nguyen et al, 2001; Nuzzolo et al, 2001), Tong and Wong (2000) adopted a simple network representation that made use of a specially developed branch and bound algorithm to generate the time-dependent minimum path with non-additive generalized costs in the assignment procedure. Passengers are assumed to travel on a path with minimum generalized cost that consists of four components: in-vehicle time, waiting time, walking time, and a time penalty for each line change. The

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assignment procedure is conducted over a period in which both passenger demand and train headway vary. A time-dependent optimal path algorithm that is associated with the above network was developed in Tong and Richardson (1984). A path is considered to be optimal if it consumes the least weighted time, taking into account waiting time, walking time, invehicle time, and a line change penalty. In this chapter, the weights for the various trip components are defined relative to the in-vehicle time of a journey. Let P j and P 2 be the weighting factors for waiting time and walking time respectively, and P 3 be the line change penalty in weighted time units. It is assumed that P ] , P2 ^ 1 and P3 > 0 . The algorithm consists of three stages. • Stage 1 : Forward pass of quickest path • Stage 2: Backward pass of quickest path • Stage 3 : A branch and bound method The first two stages basically follow a Dijkstra type algorithm for the path (Dijkstra, 1959), and are employed to define the time windows and sub-networks for use in the branch and bound method in Stage 3. 3.1

Forward pass of quickest path

For a particular time t, the quickest paths from an origin node r to all destination nodes are determined by a Dijkstra type algorithm. The term "quickest" here means that the paths which are obtained are the shortest in clock time. For a passenger who is leaving origin node r at time t , let n trk be the time at which they arrive at node k along the quickest path from node r, and K.(rk be the back link of node k (i.e. K.*rk=k and K^ is the back node of k along the quickest path). The Dijkstra algorithm branches progressively outwards from r until all other nodes in the network have been fully explored. Suppose that node A is reached at time co. All links including walk and transit links from node A are then explored. Consider link a extending from A with a downstream node B. Let £,a be the travel time that a passenger would spend on link a. If the link is a walk link, then we have

(6)

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and the waiting time vanishes, i.e. w = 0. However, if the link is a transit link, the travel time that this passenger would spend on the link is determined by locating the wth run of the vehicle, such that

The travel time becomes

Note that the waiting time for the wth run of the vehicle, w, can be determined by

and there will be a line change if

During the forward pass, the arrival time and back link at B are updated by

and

The above procedure is repeated until the quickest paths from origin r to all other nodes have been identified. 3.2

Backward pass of quickest path

From the forward pass in the previous section, the quickest path from an origin r to any destination node s is determined. However, this path is not necessarily the optimal path from r to s when weighted time is considered. Nevertheless, it forms an upper bound for the optimal path to be searched. To obtain the weighted time along the quickest path, the path is examined and the components of waiting time \\i j , walking time \\i 2, number of line change vj/ 3 , and in-vehicle time vy 4 are determined from

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Equations (6-10) along the path. The clock and weighted arrival times at node 5 can be determined by

and

for a passenger from node r who embarked at time t. To determine the optimal path between O-D pair r-s, a reversed network is defined as follows. All links including walk and transit links are reversed. The timetables of the run profiles of all transit lines are also reversed. The departure time of a run of vehicle in the reversed network is equal to M minus the original departure time in the normal network, where M is an arbitrary large number. The nodes along a transit line and the runs of the vehicles are numbered in the reversed order. With this reversed network, moving forward from 5 to r is analogous to backtracking from 5 to r in the normal network. While a normal network is useful for finding the quickest paths from the origin to all destinations when the starting time at the origin is given, a reversed network is useful for finding the quickest paths from all origins to one destination when the arrival time at the destination is specified. Using this reversed network, the quickest paths from node s starting at a clock time, M - § trs , to all other nodes are determined in the same manner as is the forward pass in the previous section. Let x, tsk ^e me latest departure time from node k such that a passenger can arrive at the destination node s by clock time §trs along the quickest path from the backward pass, where x, tsk

can

be determined as M minus the quickest

travel time from node s to node k in the reversed network. Proposition: The optimal path from origin node r the destination node s when a passenger embarks onto the network at time t must not pass through any node k with

Proof: Let Y = § trs - 1 be the total weighted time from node r to node s along the quickest path in the forward pass, y^ be the weighted time along a path from node r to node k, and y2 be the weighted time along a path from node k to node s. As P i , 02 - 1 >

an

d Ps ^ 0 , we have y} >ntrk-t for any path from r to k, and

y2 > §trs ~ %tsk

=

Y+ 1 ~ Titsk f°r anv Pam fr°m £ to s. Therefore, the weighted time

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along any path from r to s via k becomes Y = y\ +y2 + PI& , where 8 = 1 when there is a line change at node k; and 8 = 0 otherwise. As P 3 8 > 0 , we have Y > y\ +y2 ^ (ntrk -t} + (Y+t-itsk) > Y, from the condition that n(rk >%tsk- However, as the quickest path in the forward pass forms the upper bound for the optimal path, any path with weighted time exceeding the weighted time associated with the quickest path must not be optimal. It therefore follows that any path from r to 5 via k must not be the optimal path from r to s. This completes the proof. 3.3

The branch and bound method

From the proposition, it follows that all the nodes with n(rk > %tsk , and all links connected to these nodes, can be deleted from the network to form a sub-network, while searching for an optimal path between O-D pair r-s that is embarked on at time /. This greatly reduces the number of possibilities in the branch and bound searching, and hence substantially decreases the computational effort that is required. Denote Y trkp ' ^trkpi

an

^ Ktrkp

as me

clock arrival time, weighted arrival time, and back link

at node k along path p from node r that is embarked on at time t. During the path extension process, the clock time at node y tr^p is used to determine which vehicle run will be taken to traverse a transit link, the weighted time r\trkp keeps track of the accumulated weighted time up to a node, and the back link variable K.trkp records the trajectory of a path. In the branch and bound method, the interval (t,$trs) forms the time window for optimal path searching, where §lrs is used as the bound of any feasible paths. To avoid looping, a path is extended by a new branch only if the branch is considered to be efficient. The definition of path efficiency is similar to that of Dial (1971), but some modifications have been made to improve the performance of the algorithm in this application. Consider a branch of a path at node A extending to node B via link a in the network. A path is considered as efficient if it moves progressively further away from the origin, or if it travels along the same transit line (i.e. K^. A/ , = a°). Starting from the origin node r, the paths branch progressively outward from the node r within the sub-network, guided by the path efficiency requirement. A path is terminated whenever this path extends to a node with weighted arrival time greater than the bound (i.e. r\trkp >§trs) or the destination node s is reached. The branch and bound procedure is stopped when all branching paths are terminated. The optimal

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path p

is then identified as the complete path from r to s with the minimum

weighted arrival time, min T),

. The optimal path can also be backtracked by

means of the set of back link variables, (K

4.

« , V& e p ).

NETWORK LOADING

For given values of p t , (3 2 , and (33, the trip demand is loaded onto the transit network by means of an all-or-nothing assignment process. Initially, we set all link loading values \-(\2a,\^,V#) = 0. These link loading values are then updated during the loading process. Consider a particular O-D pair r-s and time interval /. The number of passengers who embark during the time interval and travel from the origin node r to the destination node s is given by the time-dependent O-D matrix as drst. This packet of passengers is then loaded onto the network along the minimum path p , which is determined from the time-dependent optimal path algorithm in the previous section. Denote p -(a],a2,...,an) as the sequence of links that constitutes the minimum path, where aj~ =r, a* = s, and a,+ =aJ+\, i-1,2,... ,n-\. This sequence of links is obtained from the set of back links along the optimal path, (K^ *, V& e p ).

If link af along the optimal path is a walk link, then the departure time of the passengers from the starting node is exactly the time that those passengers arrive at the node from the upstream link. Let a be the arrival time of these passengers at the starting node of a,. The passengers will then walk on link at during the period t™.. The link loading value is then updated as x

w*-xw+d'«'

o = a,a + l,...,o + ^ - l ,

(15)

and the arrival time at the starting node of the downstream link ai+\ along the path is updated as cr + 1 ™ .

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However, if link a( along the optimal path is a transit link, then the passengers who have arrived at the starting node of the link have to wait until the next vehicle run arrives. Again, let a be the arrival time of these passengers at the starting node of a, . The next vehicle run on this transit link that passengers can take to continue the journey is determined by locating m, such that

The passengers will then board the mth run of vehicle, and the waiting time for the vehicle becomes ua.m - a . The link loading value is updated as

and the arrival time at the starting node of the downstream link a (+1 along the path is updated as va.m .

5.

ESTIMATION OF ORIGIN-DESTINATION MATRICES

The transit model that is described above requires for input multi-interval trip matrices to represent the time-dependent station to station passenger demand. If an electronic fare collection system is installed at the stations, then the data that is collected by the system can be used to compile the required trip matrices. However, when such a system is not installed, or when the model deals with a hypothetical demand scenario, other methods will have to be used. This section gives an entropybased model to estimate these time-dependent O-D matrices. Let Cat, a = 1,2,..., Na, t = 1,2,..., Nt, be the observed passenger counts arriving at the end node of link a during time interval t, where Na is the number of observed links. From the transit model that was described in the previous section, the time-dependent optimal paths based on the weighted time for all O-D pairs in the network can be determined. Using these results, for each O-D pair in the transit network, the graphs of the clock arrival time of a passenger at the end node of an observed link (if they pass while on the way from their origin to their destination) against the clock departure time at the origin node can be plotted. A typical graph of the arrival time at

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the end node of link a against the departure time at origin node r for a particular 0-D pair r-s is shown in Figure 1.

Figure 1. A typical graph of the clock arrival time at the end node of an observed link against the clock departure time at an origin node. Reprinted from Transportation Research Part B, 32B, S. C. Wong and C. 0. Tong, Estimation of time-dependent origin-destination matrices and transit networks, 35-48, Copyright 1998, with permission from Elsevier Science

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From the graph, for each time interval t at the end node of link a we can determine the period T^U within the time interval u at the origin node, during which a passenger who leaves origin node r will arrive at the end node of link a within the interval t. As we assume that the passenger demand at each origin node during each time interval is uniformly distributed, the ratio of the period T^D to the length of an interval T defines the proportion of passenger demand leaving origin node r within interval u that will arrive at the end node of link a within the interval t, which is denoted by

(18) If an observed link is not on the optimal path between an O-D pair at any time, the values of p™^ obviously vanish. From these proportions, the number of passengers who arrive at the end node of each observed link during each time interval can be determined, and should be equal to the observed counts as follows,

This forms a set of equality constraints in the entropy-based formulation of O-D estimation that is shown below. 5.1

Maximum entropy formulation

The standard maximum entropy formulation is employed to estimate the timedependent O-D matrices for this transit system as follows (Wilson, 1970; Van Zuylen and Willumsen, 1980):

subject to

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where drst are the prior time-dependent O-D matrices, which can be obtained from the out-dated matrices in the system, a sample survey of the O-D matrices, or any direct/indirect means of observation. The constraints in Equation (20b) ensure that the time-dependent O-D matrices reproduce the observed passenger counts in the network. Equation (20c) provides the non-negativity constraints of O-D demand, and the equality constraints model the logical condition that no traveler will have the same origin and destination in making a trip in the transit system. When no prior information on the time-dependent O-D matrices is available, the values of drst are substituted as unity in Equation (20a) (see Van Zuylen and Willumsen, 1980). The formulation so far assumes that the counts are error free. The relaxation of this limitation might lead to a potential extension of the proposed methodology (Cascetta etal., 1993). The problem is now formulated as a convex program with linear constraints, and can therefore be solved by any standard routine (Luenberger, 1973). However, the major difficulty that is encountered in solving the problem is in the large number of unknowns that are involved. This renders many standard algorithms not so efficient when applied to the problem. For example, a transit system with 200 origin/destination nodes and 8 time intervals will have 200 x 200 x 8 = 320,000 variables. To tackle this difficulty, the next section proposes an efficient sparse algorithm to solve the entropy maximization problem in Equation (20).

6.

A SPARSE ALGORITHM

Ignoring the non-negativity constraints in Equation (20c) at this stage, the convex program can be rewritten in the following form.

subject to

where nc (= NaxNt) is the number of equality constraints, d} is the /th element of the unknown vector d of size nd (=NrxNsxNt), which is a collection of all the OD passenger demand in all intervals, g^ is the Ath row and /th column element of the coefficient matrix for the set of linear equality constraints in Equation (20), and fk is

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the Ath element of the constant vector for the linear constraints. Using this representation, a Lagrangian can be formed for the objective function and the equality constraints as follows,

(22)

where Kk is the Lagrangian multiplier of the Ath constraint in Equation (21b). For the entropy function in Equation (2la) to be maximized, we have

(23)

The variables dt can then be expressed in terms of the Lagrangian multipliers as

Substituting Equation (24) into Equation (22), the Lagrangian IT(A,) is solely dependent on the set of Lagrangian multipliers, A, = Col(A,/t,VA:). Now, the nonnegativity constraints in Equation (20c) come in. From Equation (24), no matter what values of Kk that we obtain as a solution to the problem or any intermediate values during the iteration process that is discussed later, the non-negativity constraints are automatically satisfied. In other words, the exponential transformation in Equation (24) relaxes the inequality constraints that are imposed on the problem. This enables a more efficient Newtonian algorithm to be used to determine the solution (Bell, 1983). The problem becomes one of finding vector 'k, such that 11

=0, VJt.

(25)

Expanding by Taylor's series, the Lagrangian can be approximated by r\

T>

I

T

1

(26)

302 where Fl

Advanced Modeling for Transit Operations and Service Planning is the Lagrangian as evaluated at a current point of A , AA represents a

small change in vector A from the current point, R is the residual vector, and H is the Hessian matrix of the Lagrangian. The z'th element of the residual vector can be obtained by

and the z'th row andy'th column element of the Hessian matrix H is calculated by

where d{ in Equations (27 and 28) are evaluated from Equation (24). The problem can be solved by the following recursive Newtonian iterative equation,

where the superscript / denotes the 7th iteration. The iterative equation is solved repeatedly until a certain convergence criterion is satisfied, for say |R|| < s, an acceptable error. This procedure was also employed by Bell (1983) in solving the entropy-based O-D matrix estimation problem for highway networks. The most time consuming part in the above solution algorithm is on the evaluation of the residual vector and Hessian matrix in Equations (27 and 28). Moreover, the storage of the full matrix g = [g/y,Vfc,/], which is a ncxnd matrix, requires a large amount of memory space. In the following, a sparse algorithm is proposed for the evaluation of R and H. To save space, the matrix g is stored in its sparse form. Let &k be the number of non-zero elements of the Mi row of matrix g, and mki and qk! represent the column number and value of the /th non-zero element along the Mi row. With this representation, only non-zero elements of g are stored. The vector d is then calculated by the following algorithm.

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Algorithm Al

Set d = 0 For k = 1 to nc : For / = 1 to Qk d

mkl <-dmk,+^k
Endfor Endfor

For / = 1 to n^ : dj <— df exp(-dt ) Endfor The evaluation of R can be conducted by the following sparse algorithm. Algorithm A2

For i = 1 to n :

For / = 1 to R

i

Endfor Endfor Let fi of size nd be a temporary array. The Hessian matrix H is evaluated by the following algorithm. Algorithm A3

Set \i = 0 For i - 1 to nc : For j = 1 to i: For / = 1 to & :

Endfor Hy

^

304

Advanced Modeling for Transit Operations and Service Planning For / - 1 to Sy : H

U

^-Hij+VjlVrnj,

Endfor For / = 1 to » ( :

»mu ^ 0 Endfor Endfor Endfor Note that only the lower (or upper) triangular part of matrix H is needed for the evaluation, as the matrix itself is symmetric. Not only is the memory space that is needed to store the matrix reduced almost by half, the solution part of a particular iteration in Equation (29) can also take advantage of this symmetric property in matrix algebra. Notwithstanding the fact that algorithms A1-A3 are given here to solve the problem of estimating time-dependent O-D matrices in a transit network system, they can also be applied to other related O-D estimation problems for highway networks with similar entropy-based formulations (see Van Zuylen and Willumsen, 1980). These algorithms provide a more efficient than usual solution procedure in terms of memory space and computing time requirements.

7.

CASE STUDY

In this section, a transit network from the Mass Transit Railway (MTR) system in Hong Kong is employed as an example to demonstrate the effectiveness of the proposed methodology in the estimation of time-dependent O-D matrices. The MTR system, consisting of 38 stations, is shown in Figure 2, and is one of the busiest railway systems in the world, carrying up to 2.5 million passengers a day. With its sophisticated electronic fare collection system, full information on the time-dependent O-D distribution and the in- and out-flows at all stations can be obtained. To show the effectiveness of the proposed method, the time-dependent O-D matrices are estimated from the in- and out-flows at all stations for the cases with and without prior information on O-D distribution. These predicted matrices are then compared to the true matrices that are observed.

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Figure 2. The Hong Kong Mass Transit Railway. Reprinted from Transportation Research Part B, 32B, S.C. Wong and C.O. Tong, Estimation of timedependent origin-destination matrices and transit networks, 35-48, Copyright 1998, with permission from Elsevier Science

The analysis period is from 6:00 a.m., when the system starts operating, until 10:00 a.m., well after the morning peak demand. The period is divided into eight intervals, each of which lasts for half an hour. With the in- and out-flows shown in Figures 3 and 4 as constraints and the assumption that nobody will have the same origin and destination, there are a total of 38 x 38 x 8 - 11,552 O-D pairs in eight intervals and 38 x 8 x 3 = 912 equality constraints.

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Figure 3. The plot of in-flows at stations. Reprinted from Transportation Research Part B, 32B, S.C. Wong and C.O. Tong, Estimation of time-dependent origin-destination matrices and transit networks, 35-48, Copyright 1998, with permission from Elsevier Science

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6:30am - 7:00am

9000- 10500 >10500

Figure 4. The plot of out-flows at stations. Reprinted from Transportation Research Part B, 32B, S.C. Wong and C.O. Tong, Estimation of time-dependent origin-destination matrices and transit networks, 35-48, Copyright 1998, with permission from Elsevier Science

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With the sparse algorithms that were described in the previous section, the problem is transformed into solving the Lagrangian multipliers of 912 unknown variables. The problem size is now reduced by over ten times. It is also found that the sparsity of matrix g is less than 0.5%, and therefore the sparse algorithms are extremely efficient. The errors between true and predicted time-dependent O-D matrices for the case without prior information is shown in Figure 5. For the case with prior information on O-D distribution patterns, outdated prior time-dependent matrices that were obtained two years before the true matrices are input into the model, and the errors between true and predicted matrices are plotted in Figure 6. It can be seen that with prior information on O-D distribution pattern, the proposed method produces much better results. The discrepancy between the two sets of matrices are also measured by the "mean absolute error" as follows:

and

where e is the total mean absolute error between the two sets of O-D matrices, and et is the mean absolute error for a particular time interval t. The errors for the cases with and without prior information are listed in Table 1. Without prior information on O-D distribution, a total error of 33% between the true and predicted matrices is obtained. However, when prior information is provided, the error is significantly reduced to only 6%.

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6

7;00am - 7:30am

Figure 5. The error plot between true and predicted matrices without prior information. Reprinted from Transportation Research Part B, 32B, S.C. Wong and C.O. Tong, Estimation of time-dependent origin-destination matrices and transit networks, 35-48, Copyright 1998, with permission from Elsevier Science

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Figure 6. The error plot between true and predicted matrices with prior information. Reprinted from Transportation Research Part B, 32B, S.C. Wong and C.O. Tong, Estimation of time-dependent origin-destination matrices and transit networks, 35-48, Copyright 1998, with permission from Elsevier Science

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Table 1. The mean absolute errors between true and predicted matrices Interval

6:00 - 6:30 a.m. 6:30 - 7:00 a.m. 7:00 - 7:30 a.m. 7:30- 8:00 a.m. 8:00 - 8:30 a.m. 8:30 - 9:00 a.m. 9:00 - 9:30 a.m. 9:30- 10:00 a.m. 6:00- 10:00 a.m.

8.

Total number of trips 14,000 27,000 59,000 95,000 126,000 156,000 103,000 45,000 625,000

Mean absolute error (%) With prior Without prior information information

43.0 43.0 38.1 34.8 29.6 27.8 27.8 55.7 33.2

0.2 0.4 0.8 3.1 8.9 7.8 9.0 4.9 6.2

CONCLUSIONS

A methodology for the estimation of time-dependent O-D matrices for transit networks based on observed passenger counts has been given. Using the dynamic assignment framework that was proposed by Tong (1986), an entropy-based approach has been developed to estimate the matrices. To reduce the memory storage and computational efforts in the solution stage of the problem, an efficient sparse algorithm has been proposed. This algorithm can also be effectively applied to other related entropy-based O-D matrix estimation problems. A transit network from the Mass Transit Railway system in Hong Kong has been employed to test the methodology, and good agreement between the predicted and true matrices for this network has been obtained.

ACKNOWLEDGEMENTS The study was supported by research grants HKU536/96E and HKU7018/99E from the Hong Kong Research Grants Council, and an Outstanding Young Researcher Award 2000 from the University of Hong Kong. The Hong Kong Mass Transit Railway Corporation is gratefully acknowledged for providing trip matrices for preparing the case study. This chapter is essentially taken from Wong and Tong (1998) and Tong and Wong (1999).

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REFERENCES Ashok, K. and M. Ben-Akiva (1993). Dynamic O-D matrix estimation and prediction for real-time traffic management systems. In: Transportation and Traffic Theory (C. F. Daganzo, ed.), pp. 465-484. Elsevier Science Publishing Company Inc. Bell, M. G. H. (1983). The estimation of an origin-destination matrix from traffic counts. Transportation Science, 17, 198-217. Bell, M. G. H. (1991). The estimation of origin-destination matrices by constrained generalised least squares. Transportation Research, 25B, 13-22. Carraresi, P., F. Malucelli and S. Pallottino (1996). Regional mass transit assignment with resource constraints. Transportation Research, 30B, 81-98. Cascetta, E. (1984). Estimation of trip matrices from traffic counts and survey data: A generalized least squares estimator. Transportation Research, 18B, 289-299. Cascetta, E., D. Inaudi and G. Marquis (1993). Dynamic estimators of origindestination matrices using traffic counts. Transportation Science, 27, 363-373. Chriqui, C. and P. Robillard (1975). Common bus lines. Transportation Science, 9, 115-121. De Cea, J. and E. Fernandez (1993). Transit assignment for congested public transport systems: An equilibrium model. Transportation Science, 27, 133-147. De Cea, J. and E. Fernandez (1989). Transit assignment to minimal routes: An efficient new algorithm. Traffic Engineering and Control, 30, 491-494. De Cea, J., J. P. Bunster, L. Zubieta and M. Florian (1988). Optimal strategies and optimal routes in public transit assignment models: An empirical comparison. Traffic Engineering and Control, 29, 520-526. Dial, R. B. (1971). A probabilistic multipath assignment model which obviates path enumeration. Transportation Research, 5, 83-111. Dijkstra, E. W. (1959). Note on two problems in connection with graphs (spanning tree, shortest path). Numerical Mathematics, 1, 269-271. Lam, W. H. K. and H. P. Lo (1990). Accuracy of O-D estimates from traffic counts. Traffic Engineering and Control, 31, 358-367. Last, A. and S. E. Leak (1976). Transept: A bus model. Traffic Engineering and Control, 18, 14-20. Le Clerq, F. (1972). A public transport assignment method. Traffic Engineering and Control, 14, 91-96. Luenberger, D. G. (1973). Introduction to Linear and Nonlinear Programming. Addison-Wesley, Reading, Mass. Nguyen, S. and S. Pallottino (1988). Equilibrium traffic assignment for large scale transit networks. European Journal of Operational Research, 37, 176-186.

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Nguyen, S., S. Pallottino, and F. Malucelli (2001). A modeling framework for passenger assignment on a transport network with timetables. Transportation Science, 35, 238-249. Nuzzolo, A., F. Russo, and U. Crisalli (2001). A doubly dynamic schedule-based assignment model for transit networks. Transportation Science, 35, 268-285. Spiess, H. (1987). A maximum likelihood model for estimating origin-destination matrices. Transportation Research, 21B, 395-412. Tong, C. O. (1986). A schedule-based transit network model. Ph.D. Thesis, Monash University, Australia. Tong, C. O. and A. J. Richardson (1984). A computer model for finding the timedependent minimum path in a transit system with fixed schedules. Journal of Advanced Transportation, 18, 145-161. Tong, C. O. and S. C. Wong (1999). A stochastic transit assignment model using a dynamic schedule-based network. Transportation Research, 33B, 107-121. Tong, C. O. and S. C. Wong (2000). A schedule-based dynamic assignment model for transit networks. Journal of Advanced Transportation, 33, 371-388. Van Zuylen, H. J. and L. G. Willumsen (1980). The most likely trip matrix estimated from traffic counts. Transportation Research, 14B, 281-293. Wilson, A. G. (1970). Entropy in Urban and Regional Modelling. Pion, London. Wong, S. C. and C. O. Tong (1998). Estimation of time-dependent origin-destination matrices and transit networks. Transportation Research, 32B, 35-48. Wu, J. H., M. Florian, and P. Marcotte (1994). Transit equilibrium assignment: A model and solution algorithm. Transportation Science, 28(3), 193-203. Yang, H., Y. lida and T. Sasaki (1994). The equilibrium-based origin-destination matrix estimation problem. Transportation Research, 28B, 23-33. Yang, H., T. Sasaki, Y. lida and Y. Asakura (1992). Estimation of origin-destination matrices from traffic counts on congested networks. Transportation Research, 266,417-433.

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CHAPTER 12

MODELS FOR OPTIMIZING TRANSIT FARES Jing Zhou, Graduate School of Management Science & Engineering, Nanjing University, Nanjing, 210093, P. R. China and William H. K. Lam, Department of Civil & Structural Engineering, The Hong Kong Polytechnic University, Hong Kong, P. R. China

1.

INTRODUCTION

Over the past three decades, a substantial amount of attention was given to the problem of transit network design (see Chapter 3). A number of scholars (e.g. Nihan and Morlok, 1976; Turnquist, 1985) had conducted research on the different aspects of the transit system design problem. This includes attempts to employ transit service planning to optimize the system performance of scheduled passenger carriers, generally with the goal of fulfilling various objectives that included system operation under specific constraints as well as under different institutional environments. Focusing on urban transit networks, Ceder and Wilson (1986) described the transit system model as a multi-stage process that can be used to solve a number of transit system design problems, which ranges from network design to crew scheduling problems. List (1990) presented a transit model for sketch-level planning that allowed major changes in the service patterns of a transit network to be assessed quickly. The trans-shipment equations adopted in the model were then used to determine optimal transit vehicle flows in response to passenger flows, service frequency requirements, line segment capacities, system fleet size, and storage node (or station) capacities. In

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fact, extensive work on the scheduling of urban transit lines can be found in literature (Ceder, 1986; Chakroborty etal., 1995). However, it is noted that the aforementioned models do not consider the passengers' response, in the form of changes to their route choice, resulting from changes brought upon by the operators' decisions. Moreover, a common assumption used in transit models is that transit fares remain fixed regardless of their effects on passenger volume. Nevertheless, transit operators are aware that service frequency and fare structure of a transit line are important decision variables when planning for their services. It is also known that improvements in transit services, particularly fare reductions, can lead to an increase in passenger demand and market share. This issue is particularly relevant to developing economies such as China and many Asian countries, and to some modernized cities with high population densities such as Hong Kong and Singapore, where a large proportion of the population use transit services as the main mode of transportation. As an example, Hong Kong, a city of more than 6.8 million people and a land area of only 1060 square kilometres, has a well-established and extensive transit network with a variety of transit services. Of these, over 90% of the 11 million daily person trips are served by privately operated public transit modes. Relying on private sector operators to provide transit services raises some interesting issues because the principal objective of private transit operators is neither welfare gain nor efficient utilization of road space, but is to maximize individual profit. The trend of providing the community with extensive transit services is not so pronounced in Western cities. This can be attributed to the relatively low population densities within large Western cities, which makes it difficult to justify the provision of extensive transit services, hi fact, the transit industry is often subsidised by the government because ridership is insufficient for the industry to operate profitably. As a result, priority is placed on the minimization of transit system operating costs through the optimization of transit line frequencies rather than any form of profit maximization. However, this is not the case in some Eastern Asian cities, particularly those with high-density population development such as Hong Kong, where the transit industry can operate profitably without government subsidies. In addition, because the total number of transit vehicles is generally predetermined to cater for passenger demand during peak periods, transit operators have fewer options when trying to optimize transit line frequencies. Thus, transit operators are more concerned with the optimization of the transit fare structure, where they exercise more control over when trying to maximize profits. Moreover, passenger demand in Hong Kong is relatively sensitive to transit line fare. On the basis of survey results (Lai et al., 2000; Allain, 1993), a percentage point reduction in transit fare generally results in a 1.33-1.45 percent increase in transit usage in Hong Kong as compared to a 0.3 percent increase

Models for Optimizing Transit Fares

31 7

in usage for a similar fare reduction in Canada. The demand elasticity effects of the transit fares on the revenues of the transit operators can be illustrated as follows. Suppose passenger demand v can be described by a monotonic decreasing function with respect to line fare/? in the following form: v = v0eap where v0 and a are initial passenger demand and function parameters respectively. The elasticity of ridership with respect to fare is then dv p

8 = — — = ap dp v

Thus we can identify the parameter a as a = s/p. Note that the elasticity of ridership will typically be a negative value as the ridership is related inversely to the line fare. Let R be the revenue of the transit line, where: R = vp = pv0eaf The derivative of revenue R with respect to the line fare/7 is x.

Thus, when passenger demand elasticity e is less than -1 (and hence a<-l/p), an increase in fare will result in a decrease in revenue, and vice versa. Operators can thus increase revenue by decreasing fares as long as the demand elasticity with respect to the line fare remains less than -1.0 (i.e. «<-!//?). This explains why fare competition between transit service providers in Hong Kong is so keen and why the fare structure of a transit line has been advocated as an efficient means of coordinating the transit passenger flows and alleviating congestion in Hong Kong. It also presents an important and interesting aspect for a deregulated transit market within which all transit fares can be adjustable. This provides the reason to study the competition between transit service operators who attempt to optimize profits by changing their fare structures rather than their line frequencies. This chapter is focused on the transit fare optimization problem in transit networks with one operator as well as when multiple operators are present. A bi-level programming approach is adopted here to determine the optimal fare structure of the transit network, where the passengers' route choice behaviour is taken into account using a stochastic user equilibrium (SUE) transit assignment model. In the upper-level problem, we will first investigate the optimal fare problem for the case of a transit system with one operator and subsequently extend it to the deregulated transit market with multiple operators. In the lower-level problem, the capacity restraint SUE transit assignment models with elastic demand and elastic line frequency can be incorporated respectively.

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The rest of the chapter is organized as follows. The notations are given and some useful concepts for transit networks are introduced briefly in Section 2. A bi-level programming formulation for the transit fare optimization problem is presented in Section 3. In Section 4, the stochastic user equilibrium (SUE) transit assignment model with elastic demand and elastic line frequency are proposed respectively. The optimal fare problem with one operator is discussed together with a heuristic solution algorithm in Section 5. Finally, a network equilibrium model with bi-level framework for a deregulated transit system is presented in section 6 to describe the fare competition between transit operators whilst incorporating the interaction between transit operators and passengers.

2.

SOME USEFUL CONCEPTS AND NOTATIONS

2.1

Basic concepts and general assumptions

A transit network constitutes a set of transit lines and a set of stations (nodes) where passengers can board, alight or change vehicles. A transit line can be described by the frequency of the vehicles and the vehicle types (e.g. bus or underground train). A line segment is a portion of a transit line between two consecutive stations. In this chapter, we adopt the transit route concept suggested by De Cea and Fernandez (1993). Instead of defining the path in terms of consecutive segments of transit lines, the path between an Origin-Destination (OD) pair is defined as a sequence of transfer stops that is denoted as the "transit route". The transit route is defined by a sequence of nodes including the origin node, the destination node and all intermediate nodes representing the transfer points. The section between two consecutive transfer nodes is referred to as a route section or link, which is associated with a set of attractive lines or common lines and determined as described in De Cea and Fernandez's model (1993). The attractive set of lines is the set of transit lines chosen by passengers to minimize their expected total travel time. It is assumed here that all paths chosen by passengers are line sections in the attractive set of lines. A line section is the section of a transit line passing through a route section (see Lam and Zhou, 2000). We will describe a transit network by a set of nodes (transfer stations) N and a set of links (route section) S. The notations used throughout this chapter are given as follows.

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319

L: set of transit lines. E: set of all line segments. W: set of Origin-Destination (OD) pairs. Rw: set of passenger's transit routes (or path) between OD pair w. N,: the number of transit line / . Tl: total round trip (cycle travel) time on transit line / . gw: demand between OD pair \v. h™: flow on transit route r between OD pair w. f , : frequency on transit line /. //max maximal frequency on transit line /. f : vector of line frequencies. k, :unit vehicle capacity of transit line /. pi: fare of transit line /. p: vector of line fares. As: set of feasible transit line sections passing through route section or link s. ts: in-vehicle travel time on route section or link 5. us: waiting time for passengers boarding on route section or link s. ds: equilibrium passenger overload delay, i.e. the time penalty that passengers will wait for the next coming vehicle when they can't board the first coming vehicle because of insufficient capacity of in-vehicle on route section or link 5. ke: capacity on line segment e. k: vector of capacities on line segments. vs: passenger flow on route section or link s. v: vector of link flows. x's: proportion of passengers choosing line section / on route section or link s. \s: vector with elements x[. X: matrix with elements \s. v's: passenger flow choosing line section / on route section or link s . ve: passenger flow on line segment e. i+ ( s ) : origin node on route section or link s. i~ (s}: destination node on route section or link s. In practice, various line fares at different levels are charged to transit passengers. A wide range of transit fare structures can exist, and virtually all transit systems have

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some differential fares. However, in order to facilitate the presentation of essential ideas, only single-class passengers and single-service transit lines are considered in this chapter. For simplicity, a flat fare structure as well as a steady state assumption is also adopted. Nevertheless, the steady state assumption will not limit the application of the built model for the purpose of strategic transit planning. Of course, a temporary saturated steady-state may exist for a short duration during the peak period, for example, when passenger flows are constrained by limited actual capacities on a set of links and a temporary steady-state passenger queuing holds. It is also assumed that passengers do not have perfect knowledge of the transit line timetable and would select the path that minimizes their perceived total travel cost (including both passenger travel time and transit fare).

3.

A BI-LEVEL PROGRAMMING FORMULATION

An efficient transit fare structure for a transit operator should optimize the objective function of interest. At the same time, passengers' route choice behaviour in response to changing fare charges should also be taken into account. The transit fare structure problem can be represented as a leader-follower or Stackelberg game problem where the transit operator is the leader, and the passengers are the followers. It was assumed that the operator can influence, but cannot control, the passengers' mode choice (in terms of elastic demand) and route choice behaviour by adopting alternative fare policies, hi any transit fare structure, passengers choose their travel path in the SUE manner. This interaction game can be described using the following bi-level programming problem (Yang and Lam, 1996; Yang and Bell, 1997). Mm F(x,y(x)) Subject to

G(x,y(x))<0

Where y(x) is implicitly defined by Min f(x,y) y

Subject to

g(x,y)<0

where F is the objective function of the upper-level decision-maker (transit operator); x is the decision vector (transit fare) of the upper-level decision-maker; G is the constraint set of upper-level decision vector; f is the objective function of the lowerlevel decision-maker (passengers); y is the decision vector (passenger flow) of the lower-level decision-maker; and g is the constraint set of the lower-level decision vector.

Models for Optimizing Transit Fares

4.

THE SUE ASSIGNMENT PROBLEMS CONSTRAINTS

4.1

Flow conservation

WITH

321

CAPACITY

Let B be the OD-path incidence matrix with elements bw being equal to 1 if path r connects OD pair w and 0 otherwise. The relationship between the OD flows and passenger path flows is expressed as follow:

On the other hand, the relationship between link flows and path flows can be written as:

where asr is the element of the link-path incidence matrix A, which equals 1 if link s lies on path r and is 0 otherwise. In practice, different route sections may share the same line segment of a transit line. Moreover, the absolute capacity on each line should not be exceeded. It is obvious that a route section is not overloaded if and only if individual transit lines passing through the route section are not overloaded. We should therefore impose line segment capacity constraints so that the resultant line flow will not exceed its capacity. It is assumed that each line segment on a line must have the same capacity as the line. Let F = (yes ) denote the line segment-route section incidence matrix, which equals 1 if line segment e lies on route section s and is 0 otherwise. According to the assignment proportion of route section flows, the passenger flows on line segment e of line / can be obtained by the following equation:

For simple expression, let yes =yesx's denote the proportion of passengers choosing line segment e on line / passing through route section s. Equation (3) can thus be rewritten as below:

Substituting Equation (2) into (4), the relationship between line segments and paths can be obtained: (5)

The line capacity constraint can thus be described by the following condition:

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Advanced Modeling for Transit Operations and Service Planning e

^E

(6)

Let F be the matrix with elements yes , the feasible conditions for path flows can be expressed in the form of vectors as below:

(7d) For convenience, let H and V be the feasible set of path flows and link flows satisfying the constraints (7a)~(7d) respectively. 4.2

The SUE assignment model with capacity constraints

Given the transit network G= (N, S), we now consider passenger behaviour on transit route choice. In a congested transit network, there are variations in individual perception as path travel costs are perceived differently by each passenger. Thus, perceived total travel cost on each transit route should be treated as a random variable. As a result, the SUE assignment in a transit network is more realistic in practice. Let c? be the passenger actual total travel cost (including travel time and transit fares) on transit route reRw. According to Sheffi (1985), the passenger perceived travel cost can be expressed as:

(g) where £^is a random error term associated with the route under consideration. Furthermore, it is assumed that E[%™ ] = 0 , or equivalentlyJE'fC^] = cwr . It means that the average perceived path travel cost is equal to the actual travel cost on the route. Suppose that the errors E% in Equation (8) are identically and independently distributed (iid) Gumbel random variables. The choice probabilities can then be specified as the logit route choice probabilities. We note that according to this logittype choice model, the path flows are determined uniquely from the path costs, and correspondingly, from the link costs. Definition A SUE with a logit choice model is achieved in a congested transit network when the allocation of passengers between alternative transit routes conforms to the following logit model (9)

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where r and r' are the alternative transit routes connecting the same OD pair \v, and 6»0 is a given parameter that is used to measure the different degree of passengers' perception on the path travel time. In general, the corresponding 0 value for a bus network should be smaller than that for an underground transit system as the latter has higher reliability on travel time. As 9 -» oo, the results of SUE approximates that of the deterministic user equilibrium (DUE). A simplified expression of the passenger total travel cost on route section or link s is now presented. Let: where A is a conversion factor (converting transit fare to equivalent travel time). It can be proved that the SUE transit assignment problem with capacity restraint is equivalent to the following form of Fisk's (1980) optimization problem:

Subject to

Proposition 1 The problem (NP-1) yields a SUE transit assignment with capacity constraints if and only if the Lagrangian multipliers me associated with line capacity constraints (lid) are equal to -de, where de is the equilibrium passenger overload delay (see Lam et al. (2002) for proof). The solution algorithm proposed by Lam and Zhou (1999) can be applied to solve the above nonlinear programming problem (11). Let Me =exp(me) and£w =exp(/ w ). The solution algorithm for the above SUE transit assignment problem based on the concept of transit routes is proposed as follows: Algorithm 1: Step 1 (Initialization) Set Me(n) =1 for all links e e E, and L(? =1 for all OD pairs w. Set «=1. Step 2

(Iteration)

324

Advanced Modeling for Transit Operations and Service Planning If the values of Me(n) obtained by consecutive steps are close enough to each other, then stop. Otherwise, for each e e E , calculate

For each w e W , calculate

Step 3 Step 4

Set n=n+l, back to Step 2. (Output passenger link flows and overload delays) For each r e R w , calculate

For each route section s e S, wzW

rtRw

For each line segment e e E, calculate

It is known that the set of possible choice of optimal multipliers can be unbounded under Wardrop's equilibrium condition (see Larsson and Patriksson, 1998, 1999). However, Bell (1995) has proved that under logit-type SUE assignment, delays (and hence the optimal multipliers) are unique whenever binding capacity constraints are linearly independent. Bell's conclusion can be extended readily to our case because the logit-type SUE assignment model is also used here. 4.3

The SUE assignment model with elastic demand

The above SUE assignment model (11) can easily be extended to the case with elastic demand. The demand between each OD pair is taken to be a continuous and strictly monotonically decreasing function of the expected minimum perceived travel cost between that particular OD pair. Thus: gw = Gw[Sw(cJ\

(12)

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325

where Sw(cw ) is the expected minimum perceived travel cost between OD pair w e W. With a logit-type SUE assignment, the expected minimum travel cost between an OD pair can be expressed as (see, for example, Sheffi, 1985): S(cw) = ^minlc;}] = --In V exp(-6brw) I'6*J 9 ,t£

(13)

Demand functions that are commonly found in the literature have the following exponential form:

]Sw)

(14)

where g° is the maximum demand between OD pair w and /? is the sensitivity to the expected travel cost. Lam and Zhou (1999) have proved that the SUE transit assignment problem with capacity constraints and elastic demand is equivalent to the following minimization problem and the Lagrange multipliers associated with the line capacity constraints (lie) are equivalent equilibrium passenger overload delays on the transit lines:

(NP-2)

hetf

t/

wsW

reK^

U w&w g

(15) For simplicity, this nonlinear programming problem (NP-2) is expressed as: (NP-2) MmZ(h,p) he//

4.4

(16)

The SUE assignment model with elastic line frequency

The elastic line frequency. It is well known that the cycle journey time for a transit vehicle on a transit line is dependent on the network traffic condition and dwelling delays at stations. Wirasinghe and Szplett (1984) and Lam et al, (1998) have found that the vehicle dwelling time at a transit station was governed by passenger boarding and alighting, i.e. total interchanging passenger volume. This is clearly related to the equilibrium passenger flows on transit lines. As the number of vehicles on each transit line is assumed to be constant, the line frequencies are thus dependent on the cycle journey times for transit vehicles and are also related to equilibrium passenger flows. Given link flows or route section flows v, the number of alighting and boarding passengers at node n from transit line / can be determined as follows:

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where

It should be noted that if node n is the origin of line /, then Al'n - 0; if node n is the destination of line /, then Bo'n = 0. According to Wirasinghe and Szplett (1984) and Lam et al. (1998), the dwelling time for a transit vehicle on line / at node n can be a function with respect to boarding and alighting volumes as below: (22)

Substituting Equations (17) and (18) to Equation (22) directly, we can obtain the relationships between the dwelling time of transit vehicle and passenger flow v, i.e. (23)

Let t'm represent the travel time on line segment m on transit line /, then the cycle journey time for a transit vehicle on transit line / is:

where w e / and n e / imply that line segment m and transfer station (node) n lie on transit line / respectively, and t'0 is the terminal time on line /. When there is only one terminal on the circular line, p = 1; when there are two terminals on the same line: p = 2. The frequency on transit line / can then be obtained: (")

On the other hand, the relationship between link flows and path flows is written as:

(26) where asr is the element of the link-path incidence matrix A, which equals 1 if link s lies on path r, otherwise 0. Equation (26) can also be expressed in the form of vector: (27) Thus, Equation (25) can be rewritten as follow:

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<28) As N, is assumed to be fixed, we have (29)

The fixed point problem. According to De Cea and Fernandez (1993), the waiting time for passengers boarding route section s can be expressed as:

where the parameter a may be chosen to approximate the distributions assumed for vehicle headway. The value a -1 corresponds to an exponential distribution assumed for vehicle headway and the value a = 0.5 approximates a uniform distribution (see Lam and Morrall, 1982). The route section flows are assigned to line sections in direct proportion to their frequencies on route sections, i.e. (31)

According to the definition of yes and Equation (31), the left term of the capacity constraint (6) also depends on line frequency. The capacity of line segment e on line / is the same as the capacity on line /, i.e. where k, is the capacity of each transit vehicle on line /. On the other hand, it can be seen that line frequency f is dependent on the passenger flows from Equation (25). The line frequency f thus becomes an endogenous variable that is to be determined from the SUE transit assignment model with capacity constraints. Now we rewrite the problem (NP-1) as follows: V

Subject to

/

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For simplicity, substituting Equation (33c) into the objective function (3 3 a) and capacity constraint (33d), the above problem can be expressed as a general parametric nonlinear programming problem: h

Subject to IS}

where Z : R

w

h € H(i ) w

(34b)

x F -> R , F = {f | 0 < /, < /;

niax

, / e L} , and H : F

]L}

M

-> ¥(R ) is

a point-to-set mapping or a constraint mapping defined by the constraints (33b)~(33e), M is the dimension of the path flow vector h.
W(RM}

can be referred to as the power set of RM . The problem (34a)~(34b) is therefore a minimization problem with respect to h for any given value of f. Let us define the optimal function Z : F|L| -> R, and the optimal point-to-set mapping H : Fw -* ¥(RM ) , i.e.,

Continuity Theorem: If //(f ) is non-empty, the solution of path flow h to problem (34a)~(34b) is a continuous function for any line frequency f e Fw . Proof: Since both ke(i) and Jes(f)

are continuous functions with respect to line

frequency f, the constraint mapping H defined by a set of inequalities and equalities is continuous at any f e F]i| if the set H(f ) is non-empty. Obviously, the objective function Z is continuous with respect to (f ,h € //(f )) . As H is continuous, the optimal set mapping //(f) is thus continuous with respect to h e H(f)

in view of the results derived from stability and sensitivity analysis of

nonlinear programming problems (see Hogan, 1973; Yang, 1998). It can be seen in the objective function (3 3 a) that the second term is strictly convex and the first term is linear with respect to path flow h. The objective function Z is therefore strictly convex at anyh e H(f ) . As a result, the solution h* (f * ) to problem

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329

(34a)~(34b) is unique for a given value f *. The optimal set mapping //(f) is thus single-valued and becomes a continuous function with respect to f (Hogan, 1973). It can be concluded that the equilibrium path flow h for problem (34a)~(34b) depends continuously on the line frequency f. This completes the proof. From Equation (29), it can be seen that the line frequency f is dependent on the path flow h. Therefore we have the following equilibrium condition: f = 0(f) (37) where

Existence Theorem: If point-to-set map H is non-empty, then there is at least one solution for the equilibrium condition (38) (see Lam et al. (2002) for proof). However, it should be noted that the set H(i) could be empty if and only if the total capacity of the transit network is not sufficient to cater for passenger demand. The aforementioned assumptions can guarantee the condition of the above theorems. 4.5

Solution algorithm

For any given value of line frequency f, the equilibrium path flow h can be solved by the above Algorithm 1. The iterative approach can be applied to obtain the equilibrium point for Equation (38). Therefore, the solution algorithm for the SUE transit assignment problem with elastic line frequency is proposed as follows: Algorithm 2: Step 1

Determine an initial value of line frequency 0 < //0) < /,max for any / e L,

and set n=0. Step 2

Perform Algorithm 1 to obtain h (n) (f ((I) ).

Step 3

According to Equations (17)~(24), calculate the 7T/") (h ( B ) ). For all / e Z,, calculate -^v,* (n) = m(n\ , /*',(n\, ^ *, Update //"*'> = //-> +--(y™ - //">) H +l

Step 5

if | //(n+1) - f,(n) \< T for all / e L , where T is a predetermined tolerance, then stop, otherwise let n=n+l and go to Step 2.

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It is worth mentioning that the above model with elastic line frequency (NP-3) can easily be extended to the case with elastic demand (Lam and Zhou, 1999; Lam et al, 2002).

5.

THE OPTIMAL FARE STRUCTURE WITH A SINGLE TRANSIT OPERATOR

5.1

The upper-level problem

Generally, the transit network design problem can be formulated with different forms of decision variables and objective functions that are dependent on the characteristics of the particular problem of interest. For instance, if the transit operator (public section) is expected to coordinate the transit passenger flows and to alleviate congestion in the transit network with fixed OD demand, the system objective is to minimize the total network travel time while the transit fares are adjustable. The upper-level problem can therefore be formulated as:

Subject to where /?/

is the lower bound of fare charges and might be set to ,be zero or some

other predetermined value to cover part or the whole operating cost of the transit line concerned; p™* is the upper bound of fare charges which would be accepted by passengers. Since the transit fare influences the passenger's route choice behaviour, the link flows (or route section flows) and passenger overload delays are therefore dependent on the fare of each transit line. The objective function of the upper-level problem, which is to minimize the total network travel time, is implicitly related to the transit fare. When the transit network with elastic OD demand is considered, the optimization objectives should include passengers' surplus and total revenues raised from fares. Here it is assumed that the operator would prefer to maximize revenue. The objective function can then be described as: ^2(P,v(p)) = ZIXv/PM

(40)

seS IeAs

As passenger demand is elastic to transit fare, the upper bound of fares is applied implicitly. The transit fare optimization problem can therefore be formulated as: (P2)

Max F2(p,v(p)) p

(41a)

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331

Subject to where p™n is the lower bound of fare charges and might be set to zero or another predetermined value to cover part or the full operating cost of the transit line concerned. 5.2

The bi-level programming model and sensitivity analysis based algorithm

By combining problem (PI) with (NP1) or (NP-3) and combining problem (P2) with (NP-2), we can obtain the fare design problems in the bi-level programming form. In the bi-level programming problem, the leader (operator) determines a fare structure and the followers (passengers) will make mode choice (in terms of elastic demand) and route choice decisions in response to the leader's strategy. The leader can predict the response of followers (passengers) before he makes decision, but cannot control the followers' decision. Sensitivity analysis has become an effective approach to solve the bi-level programming problem over the past two decades. Based on the earlier method of Tobin and Friesz (1988), Yang (1995a) conducted sensitivity analysis of the network assignment problem with both queue and congestion. One of the important applications of these sensitivity analysis methods is the development of a solution algorithm for some bi-level transportation optimization problems in which the traffic equilibrium problem is taken as the lower-level problem. Such applications include the network design (Tobin and Friesz, 1988), OD demand matrix estimation (Yang, 1995b) as well as the congestion pricing problem (Yang and Lam, 1996; Yang and Bell, 1997) and so on. A gradient projection descent algorithm (Bertsekas, 1982) is suggested to solve the upper-level problem (PI or P2). The derivative of the objective function with respect to the fare structure p is calculated using sensitivity analysis information:

Therefore, a heuristic solution algorithm based on sensitivity analysis can be used to solve the bi-level programming problem (Zhou and Lam, 2000; Lam and Zhou, 2000).

332

6.

Advanced Modeling for Transit Operations and Service Planning

THE BI-LEVEL FRAMEWORK WITH COMPETITION BETWEEN OPERATORS

In recent years, the number of private operators to enter the market for transit services has been increasing in Hong Kong as well as some cities in China. Optimization of transit fares is an interesting and critical issue because, as mentioned previously, the principal interest of these private operators is neither welfare gain nor an efficient utilization of the road-space, but profit maximization. Consequently, a Nash game would occur between these different transit operators (Allsop, 2001). In addition, a Stackelberg game also occurs between each transit operator (leader) and the passengers (followers) as described in the previous section. In this section, we present a bi-level framework in which the upper-level problem represents the Nash game between transit operators and the lower-level problem is concerned with the passengers' response to the fare structure of the transit operators. Within the upperlevel problem, each operator anticipates the effect of variations in his fare on passengers' choices, so that the relationship between the two levels retains the Stackelberg form (i.e. leader-follower problem). 6.1

The competition mechanism between operators

The operators' strategies have significant effects on passenger path choice and flow patterns on transit lines. These interactions also result in competition between operators as each operator seeks to maximize his own revenue. However, this process is not as straightforward as it may originally seem since revenue raised by passenger fare is directly related to the volume of boarding passengers which is, in turn, dependent on the fares charged. In addition, there can be several line sections belonging to different operators passing through a route section (or a link). It means that there can be overlapping transit lines between some consecutive transfer nodes. This leads to the competition between operators serving the same route section. Most of the existing transit models assume that passengers would board the first arriving vehicle as long as it is among the feasible options irregardless of in-vehicle travel time and fare charge (De Cea and Fernandez, 1993; Wu et al, 1994). The route section flows or link flows are thus assigned to line sections according to the ratio of line frequency divided by the combined frequency of the feasible lines as shown in Equation (31). However, this assumption is not always realistic as passengers may choose not to board the first arriving vehicle at a station but wait for the express line or cheaper service that provides a lower total journey cost. Passengers will then choose their desired (or preferred) transit lines according to their past travel

Models for Optimizing Transit Fares

333

experience. Because of these effects, the following logit-type choice model (43) with the utility theory interpretation is more appropriate to describe passenger behaviour in choosing these alternative transit lines. For the given passenger flow vs on route section s, the probability that a passenger chooses line section / e As for travelling on route section s is calculated according to the following logit-type choice model:

(43)

6.2

The operator utility function

Consider a transit network where each operator provides services on several transit lines. The in-vehicle travel time of each transit line is assumed to be constant and predetermined. The operator utility or pay-off function is the revenue raised by the fares paid. With equation (43), we can estimate the passenger flow on line / as:

The revenue from route section s for operator k can then be obtained:

(«) where / e As n Lk means that line / served by operator k passes through the route section s. When Equation (44) is substituted into Equation (45), it is difficult to determine whether the revenue function is concave with respect to line fare. Note that passenger waiting time u\ is related to the headway (the inverse of frequency) of line

Therefore Equation (44) is established with taking account the relationships between the three variables that are known; namely, flow v's, fare p\ and frequency // of line /. Once any two of three variables are known, the remaining variable will then be determined by Equation (44). As the line frequency is assumed to be fixed, the fare p's can then be determined by the passenger flow v'. Let:

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Advanced Modeling for Transit Operations and Service Planning

where fare is treated as a constant for other operators. Let:

(47) Then, Equation (44) can be rewritten as: (48)

Let v* and p* be the passenger flow vector and the fare vector for operator k on route section s respectively. If we sum up the line flows of the Ath operator on route section s, i.e. the sum of elements of vector v* , and denote the sum by v/ , then:

Solving Equations (48) and (49), and eliminating^' > we nave:

Substituting Equation (50) into Equation (51), we have:

where u' =!/// is fixed, and A''* is also related to the decision variables p~* of other operators. For route section s, given other operators' strategy vector p^*, this equation can be expressed as:

Noted that p* can be obtained by Equation (50) for a given v*. For brevity, let v~* be the passenger flow vector excluding the kth operator. If frequency and operating cost of the transit line are assumed to be fixed, the latter can then be omitted for simplicity. The utility maximization model for operator k can then be expressed as:

Models for Optimizing Transit Fares

335

wherev* = {v*},v~* ={v~*},p* = {p*},p~* = {p~*}, and v~k is the sum of other operators' passenger flows on route section s, i.e. the sum of elements of vector v~ . In fact, the constraints (53b) and (53c) can be combined as: 0 < v\
lzAsr^Lk,s&S

(54)

Let Q* be the feasible set of strategies that are subject to constraint (54) for the Ath operator. It should be noted that the constraint (54) is related to passenger flows on other operators' lines. The set Q* is therefore implicitly associated with v~*. The feasible set for the utility maximization problem (JVP*) of the Ath operator that is determined by constraint (54) can be specified as Q*(v"*). For convenience, let y*

= ( v * , p * ) , y-* = (V*,p-*), y = (y*,y~*), U = (Uk,U~k). The problem (NPk)

can then be written as: (NPk)

MaxkUk(yk,y-k)

/enV)

(55)

It can be proved that the utility function Uk of (53a) is strictly concave with respect to y* (Zhou et al, 2002). 6.3

The formulation of generalized Nash equilibrium

In a typical formulation of a non-cooperative game, it is assumed that the feasible set of the game comprises the full Cartesian product of the individual strategy sets. Thus it is assumed that players can only affect the utilities of the other players but not their feasible sets. In the past two decades, attention has been given to the kind of games in which the feasible set is a proper subset of the full Cartesian product of the individual players' strategy sets (Ichiishi, 1983; Marker, 1990 and 1991). In transit service market competition, which is investigated here, the strategy set of each operator is affected by the others' strategies. This leads to a generalized Nash game presented by Harker (1990), which can be defined formally as follows: Definition: Consider a set K of K players (operators) in a non-cooperative game where each player k e K is represented by a strategy vector y* e Q* c R"k (nk is the dimension of y*), a point-to-set mappingOA : Q — > Q k , and a utility function Uk :y -» R where O = I1O*. A generalized Nash equilibrium y* e Q of the game k

GNE(<$,y,U) is defined as a point at which no player can increase his utility by taking unilateral action:

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Advanced Modeling for Transit Operations and Service Planning

Thus, the multi-function O represents the constraints that are imposed across the players of the game. Marker (1991) has established the following results: Proposition 2 Let Q* be a non-empty, convex and compact subset of Rn*, let O* :Q-»Q*

be non-empty and closed mapped, and let Uk : y — » / ? be once

continuously differentiate and pseudo-concave with respect to y* for all k e K . Then y* e O(y*) is an optimal solution to the generalized Nash equilibrium problem GNE(<&,y,U) if and only if y* e ®(y*) is an optimal solution to the following quasivariational inequality problem QVI(<&,G):

where G* is the negative gradient of the utility Uk, i.e.

Clearly, the expression GNE(,y,U) that we consider here satisfies the conditions of this proposition because the utility function is concave and the constraints in Q* are composed of linear equalities and inequalities. Defining and

Then the above quasi-variational inequality (?F/(<X>,G) can be expressed as:

Chan and Pang (1982) and Marker (1991) have discussed the existence of solutions to the finite-dimensional QVI problem. According to their theorems, the solution of the QVI(<&,G) problem exists and is equivalent to a solution of an equivalent VI problem under certain conditions. Marker (1991) has also shown that the generalized Nash equilibrium problem with the linear mapping and polyhedral strategy set can be solved as an equivalent variational inequality. Here, let 7 be the intersection of the operators' strategy sets. Clearly, it consists of the group of linear constraints (53), and can be expressed in the general form: The mapping O* from Q to Q* is given below:

Models for Optimizing Transit Fares

337

where BJ is the column of the matrix B associated with y y , and b is equal to the vector of route section flows v in Equation (53). With Equation (60), the set y and the mapping O satisfies the conditions of Barker's (1991) theorems. On the basis of his corollary, we therefore have the following proposition and implication. Proposition 3 Suppose that the set Y and the mapping O* in the QVI formulation of the generalized Nash equilibrium (61) are defined by Equations (62) and (63) respectively. Then, for each solution to the following VI problem: find y* e Y such that: is a generalized Nash equilibrium. The key consequence of this proposition is that it leads to a new class of solution algorithms for the QVI problem. In this special case of the QVI problem, any VI algorithm such as the projection method or the diagonalisation method (Harker, 1984) can be employed to solve this problem (64). 6.4

The bi-level transit fare equilibrium problem

As stated above, given operator strategy y, passengers seek to minimize their total travel cost so that the route section passenger flows v = {vs} can be obtained by solving the SUE transit assignment problem (NP-2). On the other hand, for a given v = {vs}, the generalized Nash equilibrium between operators described by Equation (61) can be achieved, which is equivalent to a solution of the VI problem (64). The Stackelberg game is basically a kind of non-cooperative decision problem wherein one player has the ability to enforce his strategy on other players. For decision problems of this kind, a hierarchical equilibrium solution approach has been introduced. In recent years, the bi-level model (Stackelberg form) is considered to be more suitable to describe network design problems where the decision-maker is the leader (or transit operator) and the passengers or travellers are the followers. Therefore, the bi-level transit fare equilibrium problem (BTFEP) can be identified as an extension of a non-cooperative game between two systems of players. The transit operators seek individually to maximize their own revenues, which are affected by the behaviour of transit passengers. Passengers are in turn influenced by the decisions of the transit operators and will change their travel choices so as to minimize their total

338

Advanced Modeling for Transit Operations and Service Planning

travel costs accordingly. As a result, the solution to BTFEP is the vector y* of the operators' strategies satisfying a generalized Nash equilibrium between operators together with the vector v of the equilibrium passenger flow. This equilibrium is achieved when each operator within the transit network cannot increase his/her profit (or revenue) by changing his line fares unilaterally after the effects of his change on the other operators and the responses of the passengers using the transit network have been accounted for. With the use of the equation v = Ah, the transit fare equilibrium can therefore be described in the following Stackelberg form: (i) find y* e Y such that

(ii) where, h(y) e H is such that

In this formulation, the upper-level problem (65a) is a VI problem which gives an equivalent solution to the generalized Nash game between transit operators, while the lower-level problem (65b) is the SUE transit assignment model which is adopted to represent the passengers' response to the changes in the travel costs (including fare changes). An equilibrium state for both passengers and operators consists of a vector of strategies y* satisfying condition (65a) and a vector of path flows h* obtained by problem (65b). For transit operators under the equilibrium condition, no operator can increase revenue by adopting different fares if the other operators keep their fares constant. For passengers, the equilibrium state means that no strategy other than the optimal one can reduce their perceived travel cost for each OD pair. 6.5

Solution algorithm

In the Stackelberg formulation described above, transit operators dominate the decision process and there is an asymmetry between the roles of transit operators and passengers. This leads to a hierarchical decision process in which the transit operators have to anticipate the passengers' responses in order to determine their own strategies. As passengers' route choice behaviour is modelled by the logit-type SUE assignment problem (65b), the passengers' responses to each equilibrium strategy y of the transit operators is unique and can be described completely by a response curve h(y).

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339

Consequently, a Stackelberg equilibrium solution can be obtained by solving the VI problem (65a) with the lower-level problem (65b), which can be solved by the solution algorithm developed by Lam and Zhou (1999). As discussed above, for a given link flow or route section flow, the existence of solutions to the finite-dimensional QVI problem (64) can be guaranteed by the concavity of the utility function. According to the fixed point theorem, the existence of at least one equilibrium solution to the transit fare equilibrium problem is guaranteed because of the compactness of the set of feasible solutions and the continuity of the function involved. The VI problem (65a), which follows an asymmetric Jacobian, can be solved by the diagonalisation method (Marker, 1984). As the passengers' responses are considered, according to (55), we have:

Because Uk is a strictly concave function with respect to y*, the variational inequality (65 a) can be diagonalised to give the following nonlinear problem for each k:

with solution y* o+1) , which represents the strategy vector of operator k in iteration (/+!), and where y'k(j) denotes the strategy vector excluding operator k's in the iteration j. A gradient descent algorithm can be used to solve the problem (66). The derivative of the objective function Uk with respect to decision variable y* can be calculated using the chain rule after incorporating information obtained from sensitivity analysis, which is derived from the solution to the lower-level problem: l

'

where the derivative of the path flows h with respect to decision variables y* can be calculated according to the logit-type SUE transit assignment model (NP-2) adopted here for the lower-level problem. The inclusion of this information in the second term on the right-hand side of (67) represents the operator's anticipation of travellers' response to variations in his decision variables, and typifies a Stackelberg formulation of the transit fare design problem.

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Advanced Modeling for Transit Operations and Service Planning

Based on the methods used by Tobin and Friesz (1988), Yang (1995a) conducted the sensitivity analysis of the network assignment problem with both queues and congestion. One of the important applications of the sensitivity analysis methods mentioned here is the development of a solution algorithm for some bi-level transportation optimization problems in which the network equilibrium problem is taken as the lower-level problem. Noted that the sensitivity analysis formula is not applicable when the number of OD pairs is larger than the number of links in the network because certain matrixes involved become non-inevitable (Bell and lida, 1997). It is therefore important to recognise the limitations of the sensitivity analysis approach in practice. However, we can take some measures to reduce the possibility of obtaining a non-invertible matrix. For example, the OD demand can be aggregated further to reduce the number of OD pairs. In so doing, the maximum number of path variables in the SUE transit assignment problem using the concept of transit routes as presented in this Chapter can also be pre-determined and reduced greatly. Finally, an iterative heuristic solution algorithm that employs the Method of Successive Averages (MSA) is proposed for the hierarchical formulation as follow: AlgorithmS:

Step 1 Set an initial transit fare structure y (0) , z=0. Step 2 Solve the following variational inequality problem fory (l) = (y* (l) ,y~* (l) ) e Y : find y(/+1) = (y*(/+1),y~*(l'+1)) e F such that

That is equivalent to solve the following mathematical programme for each k to obtain y*(/+I) such that

Step 2.1 SeU=l; (For each operator k, solve problem (55) by the Frank-Wolfe method) Step 2.2 Let x*(0) = y*(0 ,y=0; Step2.3 For the known fare vector (x* O) ,y~* (l) ), solve the problem (65b) with sensitivity analysis and obtain the following gradient of Uk according to (67), k '

Models for Optimizing Transit Fares

341

Step 2.4 Formulate local linear approximation of the objective function of problem (56) using the derivative information and solve the resulting linear programme to obtain an auxiliary solution z; Step2.5 Computex*°'+I) = x*(y) +a(\k(i) -z),

where a e (0,1) and can be

obtained by linear search, or simply set to be

; 7+1

Step 2.6 If| x*(y+1) - x*(y) | > T, a preset tolerance, theny—y+1 return to Step 2.3; Otherwise, yk = x* (y+l) , and if k
7.

CONCLUSION

This chapter presents the transit fare optimization problem in transit networks with one operator only as well as when multiple operators are present. A bi-level approach is adopted here to determine the optimal fare structure of the transit network, where passenger route choice behaviour is represented by the stochastic user equilibrium (SUE) transit assignment model. In the upper-level problem, the optimal fare design problem was presented for the case of the transit system with one operator and subsequently extended to the deregulated transit market with multiple operators, hi the lower-level problem, the capacity restraint SUE transit assignment model was

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used and accounted for the situations where demand and/or frequency were elastic respectively. The concept of transit routes was also discussed, where a transit route is defined as a sequence of transfer stops from origins to destinations rather than as consecutive segments of transit lines. The SUE assignment model with elastic demand is proposed for transit networks with bottlenecks and a solution algorithm was presented. The stochastic effects of passenger behaviour and overcrowded vehicle arrivals are incorporated in the proposed model, together with the elastic transit demand. A mathematical programming problem is formulated and equivalent to the SUE assignment problem with elastic demand in congested transit networks. When the invehicle link capacity constraint is reached, it can be proven that the Lagrange multipliers of the mathematical problem are equivalent to the equilibrium passenger overload delays in the transit network (Lam and Zhou, 1999). A new formulation was also presented for the capacity restraint transit assignment problem with congestion and elastic line frequency, in which the frequency of each transit line is considered to be dependent on vehicle dwelling time at stations. Consequently the line frequency is related to passenger flows on the transit line. Based on the concept of transit routes, a SUE transit assignment model taking into account congestion and elastic line frequency was proposed and the equivalent mathematical programming problem was formulated. Since passenger waiting time and line capacity are dependent on line frequency, a fixed point problem with respect to line frequency was devised. At the same time, on the basis of work by Lam et al. (2002), a solution algorithm for the proposed model was suggested. The model formulation and solution algorithm have also been presented for the design problem of transit fare structure with one operator and multiple operators. The transit operator aims to determine an optimal transit fare structure so as to maximize total revenues obtained from fares. The transit fare structure problem was represented here as a leader-follower or Stackelberg game problem, where passengers are allowed to choose their travel path according to the SUE formulation. A bi-level programming method was developed to determine the optimal transit fare structure. In particular, the competition mechanism between transit operators is formulated in the upper-level problem of the proposed bi-level model. It was found that there exists a generalized Nash game between transit operators (Zhou et al., 2002). In order to assess the revenues and/or profits of operators in an oligopolistic transit market, the utility function of each operator must be defined explicitly to describe the interaction between operators. It was established that the utility function is strictly concave to

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guarantee the existence of the generalized Nash equilibrium. According to game theory, a generalized Nash game can be formulated as a quasi-variational inequality (QVI) problem. It is also proved that this QVI problem is equivalent to a variational inequality (VI) problem that can be solved by the diagonalisation method that was presented here. As a result, the bi-level transit fare equilibrium problem between transit operators and passengers is described in the Stackelberg form, in which transit operators (i.e. the leaders) dominate the decision-making process while passengers (i.e. the followers) would react to the operators' decisions by changing their travel choices. A heuristic solution algorithm using MSA technique was proposed for solving this bi-level problem.

ACKNOWLEDGEMENTS The work described here was substantially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region (Project No. PolyU/5040/02E) and a grant from the National Natural Science Foundation of the P. R. China (Grant No. 70071049).

REFERENCES Allain, J. R. (1993). The Canadian Transit Handbook (third edition). Canadian Urban Transit Association, Toronto, Canada. Allsop, R. E. (2001). Modelling the provision of transport with planned or marketdetermined public transport services. Journal of Advanced Transportation, 35, 97104. Bell, M. G. H. (1995). Stochastic user equilibrium assignment in networks with queues. Transportation Research, 29B, 125-137. Bell, M. G. H. and Y. lida (1997). Transportation Network Analysis, Wiley, UK. Bertsekas, D. P. (1982). Constrained Optimization and Lagrange Multiple Method. Academic Press, New York. Ceder, A. (1986). Methods for creating bus timetable. Transportation Research, 20A, 59-83. Ceder, A. and N. H. M. Wilson (1986). Bus network design. Transportation Research, 20B, 331-344. Chan, D. and J. S. Pang (1982). The generalized quasi-variational inequality problem. Mathematics of Operations Research, 7, 211-222.

344

Advanced Modeling for Transit Operations and Service Planning

Chakroborty, P., K. Deb and P. S. Subrahmanyam (1995). Optimal scheduling urban transit systems using genetic algorithms. Journal of Transportation Engineering, 121,544-553. De Cea, J. and E. Fernandez (1993). Transit assignment for congested public transport system: An equilibrium model. Transportation Science, 27, 133-147. Fisk, C. (1980). Some developments in equilibrium traffic assignment. Transportation Research, 14, 243-255. Marker, P. T. (1984). A variational inequality approach for the determination of oligopolistic market equilibrium. Mathematical Programming, 30, 105-111. Marker, P. T. (1990). Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Mathematical Programming Research, 48, 161-220. Marker, P. T. (1991). Generalized Nash games and quasi-variational inequalities. European Journal of Operational Research, 54, 81-94. Hogan, W. W. (1973). Point-to-set maps in mathematical programming. SIAMReview. 15, 591-603. Ichiishi, T. (1983). Game Theory for Economic Analysis. Academic Press, New York. Lai, M. F., K. H. Wan and H. K. Lo (2000). Multi-objective analysis of competition transit services: case study of the Hong Kong international airport. Proceedings of the 5th meeting of Hong Kong Society for Transportation Studies, 461-469. Lam, W. H. K. and J. Morrall (1982). Bus passenger walking distances and waiting times: a summer-winter comparison. Transportation Quarterly, 36, 407-421. Lam, W. H. K., C. Y. Cheung and Y. F. Poon (1998). A study of train dwelling time at the Hong Kong Mass Transit Railway system. Journal of Advanced Transportation, 32, 285-296. Lam, W. H. K. and J. Zhou (1999). Stochastic transit assignment with elastic demand. Journal of Eastern Asia Society for Transportation Studies, 3(2), 75-87. Lam, W. H. K. and J. Zhou (2000). Optimal fare structure for transit networks with elastic demand. Transportation Research Record, 1733, 8-14. Lam, W. H. K., J. Zhou and Z. H. Sheng (2002). A capacity restraint transit assignment with elastic line frequency. Transportation Research B (in press). Larsson, T. and M. Patriksson (1998). Side constrained traffic equilibrium modelstraffic management through link tolls. Equilibrium and Advanced Transportation Modelling (P. Marcotte and S. Nguyen, eds.), Proceedings of the Equilibrium and Advanced Transportation Modelling Colloquium, Centre de recherche sur les transport, Universite de Montreal, Montreal, Canada, 10-11 October, 1996, Kluwer Academic, Boston, MA, 125-151. Larsson, T. and M. Patriksson (1999). Side constrained traffic equilibrium modelsanalysis, computation and applications. Transportation Research, 33, 233-264.

Models for Optimizing Transit Fares

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List, G. F. (1990). Toward optimal sketch-level transit service plans. Transportation Research B, 24, 325-344. Nihan, N. L. and E. K. Morlok (1976). A planning model for multiple-mode transportation system operations. Transportation Planning and Technology, 3, 5973. Sheffi, Y. (1985). Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Models. Prentice Hall, Englewood Cliffs, NJ. Turnquist, M. (1985). Research opportunities in transportation system characteristic and operations. Transportation Research, 19A, 357-366. Tobin, R. L. and T. L. Friesz (1988). Sensitivity analysis for equilibrium network flow. Transportation Science, 22, 243-250. Wirasinghe, S. C. and D. Szplet (1984). An investigation of passenger interchange and train standing time at LRT stations: (ii) Estimation of standing time. Journal of Advanced Transportation, 18, 13-24. Wu, J. H., M. Florian and P. Marcotte (1994). Transit equilibrium assignment: A model and solution algorithms. Transportation Science, 28, 193-203. Yang, H. (1995a). Sensitivity analysis for queuing equilibrium network flow and its application to traffic control. Math. Comp. Modeling, 22, 247-258. Yang, H. (1995b). Heuristic algorithms for the bi-level origin-destination matrix estimation problem. Transportation Research, 29B, 231-242. Yang, H. and W. H. K. Lam (1996). Optimal road tolls under conditions of queuing and congestion. Transportation Research, 30A, 463-486. Yang, H and M. G. H. Bell (1997). Traffic restraint, road pricing and network equilibrium. Transportation Research, 31B, 303-314. Yang H. (1998). Multiple equilibrium behaviors and advanced traveler information systems with endogenous market penetration. Transportation Research, 32B, 205218. Zhou, J. and W. H. K. Lam (2000). A bi-level programming approach - optimal transit fare under line capacity constraints. Journal of Advanced Transportation, 35, 105124. Zhou, J., W. H. K. Lam and B. G. Heydecker (2002). The generalized Nash equilibrium model for oligopolistic transit market with elastic demand. Resubmitted to Transportation Research B.