Transportation Planning: State of the Art

Transportation Planning

Applied Optimization Volume 64

Series Editors: Panos M. Pardalos University of Florida, U.S.A. Donald Hearn University of Florida, U.S.A.

The titles published in this series are listed at the end of this volume.

Transportation Planning State of the Art

Edited by

Michael Patriksson Department of Mathematics, Chambers University of Technology, Gothenburg, Sweden and

Martine Labbé ISRO and SMG, Université de Bruxelles, CP 210-01, Bruxelles, Belgium

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

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Contents

Preface

ix

1 Origin-Based Network Assignment Hillel Bar-Gera, David Boyce Introduction 1.1 Problem statement 1.2 1.3 Review of solution methods for TAP An origin-based method for TAP 1.4 Experimental results 1.5 1.6 Discussion Conclusions 1.7 2 On Traffic Equilibrium Models with a Nonlinear Time/Money Relation Torbjörn Larsson, Per Olov Lindberg, Michael Patriksson, Clas Rydergren 2.1 Introduction 2.2 The time-based traffic equilibrium problem 2.3 Solution approaches 2.4 A route generation algorithm 2.5 Numerical tests 3 Stochastic Network Equilibrium Under Stochastic Demand David Watling Introduction 3.1 Notation 3.2 3.3 Critique of SUE in the context of day-to-day variability 3.4 Equilibrium conditions: fixed demand Equilibrium conditions: stochastic demand 3.5 3.6 Solution algorithm 3.7 Numerical tests 3.8 Conclusion 4 Stochastic Assignment with Gammit Path Choice Models Giulio Erberto Cantarella, Mario Giuseppe Binetti Introduction 4.1 4.2 Review of stochastic assignment

v

1 1 2 4 6 7 13 14

19 20 21 23 26 27 33 33 35 37 40 44 45 46 49 53 53 55

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Probabilistic path choice models Numerical examples Conclusions

5 Estimation of Travel Time Reliability Chris Cassir, Michael G.H. Bell Introduction 5.1 Logit SUE model 5.2 Logit SUE sensitivity analysis 5.3 5.4 Approximation of travel times variances Example 5.5 Conclusion 5.6 6 A Joint Model of Mode/Parking Choice with Elastic Parking Demand Pierluigi Coppola 6.1 Background and objectives The parking choice sub-model 6.2 The mode choice sub-model 6.3 Simulation of realistic parking policies 6.4 7 A New General Equilibrium Model Yanling Xiang, Michael J. Smith, Miles Logie Introduction 7.1 DREAM—The general equilibrium model 7.2 An outline of the DREAM model 7.3 Features of the general equilibrium model 7.4 Test Results 7.5 8 Macroscopic Flow Models J.P. Lebacque, M.M. Khoshyaran 8.1 Introduction 8.2 The basic model LWR model for a link Partial flow models for links 8.3 8.4 Intersection modeling 8.5 Intersection models as solutions of optimization problems 8.6 An experimental validation Conclusion 8.7 9 AIMSUN 2 Simulation of a Congested Auckland Freeway John T Hughes Introduction and objectives 9.1 9.2 Simulation model 9.3 AIMSUN 2 simulation process 9.4 Study area and scope Model development 9.5 Geometric information 9.6 9.7 Traffic flow information

59 63 66 69 69 72 72 77 77 83 85 85 88 95 97 105 106 108 109 113 116 119 119 120 122 126 133 136 137 141 141 142 144 144 146 146 147

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Contents 9.8 9.9 9.10 9.11 9.12 9.13 9.14 9.15 9.16 9.17 9.18 9.19

Trip matrices Driver and vehicle information Maximum vehicle acceleration Motorway model Model outputs Lane utilisation Motorway speeds Greenlane Northbound on-ramp Calibration parameters Run times Conclusion Postscript

10 Fuzzy Traffic Signal Control Jarkko P. Niittymäki 10.1 Introduction 10.2 Fuzzy traffic signal control 10.3 Fuzzification interface 10.4 Defuzzification of outputs 10.5 Conclusions 11 An Urban Bus Network Design Procedure S. Carrese, S. Gori 11.1 Introduction 11.2 The main transit network (MTN) 11.3 The main transit lines (MTL) 11.4 Feeder lines 11.5 Model application and results 11.6 Conclusions 12 The Cone Projection Method Michael J. Smith, A. Battye, A. Clune, Y. Xiang 12.1 Introduction 12.2 Achieving the complementarity formulation 12.3 A cone field method of calculating equilibria 12.4 The cone projection method 12.5 A simple method 12.6 Conclusion 13 A Park & Ride Integrated System Chafik Allal, Benoit Colson, Bernard Fortz 13.1 Introduction 13.2 A Park & Ride Integrated system 13.3 Routing model 13.4 Travel time prediction 13.5 Computational results 13.6 Conclusion

147 147 149 151 153 153 154 155 156 159 159 160 163 163 163 166 170 174 177 177 178 181 185 186 193 197 198 198 203 205 207 209 213 214 215 217 221 225 227

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TRANSPORTATION PLANNING 14 229 Longitudinal Analysis of Car Ownership in Different Countries Akli Berri 14.1 Introduction 229 230 14.2 An age-cohort-period model 233 14.3 A multinational comparison 14.4 A comparative analysis for homogeneous zones 239 241 14.5 Long term forecasting using the demographic approach 242 14.6 Summary and conclusions Index

247

Preface

This book collects selected presentations of the Meeting of the EURO Working Group on Transportation, which took place at the Department of Mathematics at Chalmers University of Technology, Göteborg (or, Gothenburg), Sweden, September 9–11, 1998. [The EURO Working Group on Transportation was founded at the end of the 7th EURO Summer Institute on Urban Traffic Management, which took place in Cetraro, Italy, June 21–July, 1991. There were around 30 founding members of the Working Group, a number which now has grown to around 150. Meetings since then include Paris (1993), Barcelona (1994), and Newcastle (1996).] About 100 participants were present, enjoying healthy rain and a memorable conference dinner in the Feskekôrka. The total number of presentations at the conference was about 60, coming from quite diverse areas within the field of operations research in transportation, and covering all modes of transport: Deterministic traffic equilibrium models (6 papers) Stochastic traffic equilibrium models (5 papers) Combined traffic models (3 papers) Dynamic traffic models (7 papers) Simulation models (4 papers) Origin–destination matrix estimation (2 papers) Urban public transport models (8 papers) Aircraft scheduling (1 paper) Ship routing (2 papers) Railway planning and scheduling (6 papers) Vehicle routing (3 papers)

Traffic management (3 papers) Signal control models (3 papers) Transportation systems analysis (5 papers) ix

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Among these papers, 14 were eventually selected to be included in this volume. A further selection of papers from the Meeting are being edited for a special issue of Transportation Research, B, devoted to public transport problems. The 14 papers are described shortly as follows. Chapter 1, written by BarGera and Boyce, describes a very promising approach to the traffic assignment problem which is founded on a fundamental property of user equilibrium flows, that for each origin the flows describe an ayclic subnetwork. Chapter 2, written by Larsson et al., illustrates that when combining time delays and money outlays into a generalized cost, user equilibrium flows resulting from the conversion time money will be different from the result of using the conversion money time. A convex programming formulation is presented for the latter case, together with an algorithm for its solution. Chapter 3, written by Watling, proposes a variation of the stochastic user equilibrium (SUE) model to represent day–to–day stochastic variations in travel demands, and shows that the effect of such variations is a relatively stable mean link flow, but an increased link flow and total travel cost variance. Chapter 4, written by Cantarella and Binetti, analyzes the additive Gammit SUE model which is based on a Gamma distribution of route cost perception. This model satisfies stipulated conditions on probabilistic route-choice models, and is shown to yield results close to probit-based SUE models. Chapter 5, written by Cassir and Bell, presents a methodology for evaluating the reliability of transportation networks, which could be used to support the design of networks that are robust to everyday disturbances. In the case of the logit SUE route-choice model, reliability measures related to travel times are shown to be available efficiently, and are illustrated by means of numerical examples. Chapter 6, written by Coppola, notes that in previous models of parking management policies, parking search time is a neglected attribute which in reality should be a function of parking demand. A joint, nested logit model of mode and parking choice is then developed, and evaluated with data taken from an EC project. Chapter 7, written by Xiang et al., describes a general multi-modal traffic equilibrium model which embraces the classical four-step procedure into one combined model, including controls and stochastic travel costs. The main motivation for developing such a comprehensive model is to support the assessment of demand management strategies. Chapter 8, written by Lebacque and Khoshyaran, adapts the continuous Lighthill-Whitham-Richards model to dynamic traffic assignment. Included in the adaptation are specifications of partial and inhomogenous flows on links as well as several alternatives for intersection modeling. Chapter 9, written by Hughes, describes the application of the AIMSUN2 microscopic traffic simulation package to a section of a congested urban freeway in Auckland, New Zealand. Preliminary results show a good reproduction of speed and flow relationships, but less so for transient effects. Chapter 10, written by

PREFACE

xi

Niittymäki, develops a fuzzy traffic signal control model for a signal control scenario with several conflicting optimization criteria. Several approaches to the fuzzification and defuzzification phases of fuzzy traffic signal control are discussed. Chapter 11, written by Carrese and Gori, describes a coordinated process for the design of a bus transit network, including both lines and frequences. Heuristic methods construct the final plan through the sequential identification of a skeleton network, the main transit lines, and feeder lines. The model is applied to the Rome transit network system. Chapter 12, written by Smith et al., describes a descent algorithm for bilevel optimization in multi-modal equilibrium transportation models for the optimization of traffic control parameters. Convergence towards equilibrium points is ensured, and an intuitive motivation for the convergence of the control parameters towards points satisfying necessary optimality conditions are provided. Chapter 13, written by Allal et al., describes a demand-responsive park & ride transport system. Its real-time aspects are analyzed, and simulations results are reported. Chapter 14, written by Berri, analyzes household car ownership in seven countries, characterized by different economic and cultural contexts, by means of demographic modeling. Differences in long term forecasting results between countries and zones are explained by two main factors: the history of car ownership development and population density. The Swedish Communications Research Board (KFB) supported the conference financially, as did Chalmers, for which we are greatful. We would like to thank the participants of the Meeting for making it enjoyable, and the referees for their duly work. MICHAEL PATRIKSSON, GÖTEBORG, DECEMBER 2000

Chapter 1 ORIGIN-BASED NETWORK ASSIGNMENT Hillel Bar-Gera [email protected]

David Boyce [email protected] Department of Civil and Materials Engineering University of Illinois at Chicago 842W Taylor St. (mc 246), Chicago, IL 60607, USA

Abstract

Most solution methods for the traffic assignment problem can be categorized as either link-based or route-based. Only a few attempts have followed the intermediate, origin-based approach. This paper describes the main concepts of a new, origin-based method for the static user equilibrium traffic assignment problem. Computational efficiency in time and memory makes this method suitable for large-scale networks of practical interest. Experimental results show that the new method is especially efficient in finding highly accurate solutions.

Keywords:

Origin-based traffic assignment, user equilibrium, network optimization

1.

Introduction

Given the demand for travel (vehicles/hour) between pairs of origins and destinations, the traffic assignment problem is to allocate those flows to specific routes of a given network of nodes and links according to a given behavioral hypothesis. A common hypothesis in transportation research is that users seek to minimize the cost associated with their chosen routes. These flow-dependent costs are assumed to be known perfectly in advance. Under these assumptions, known as Wardrop’s user equilibrium principle, for every pair, origin and destination a positive flow on route implies the cost of route is not greater than the cost of any other route from to The term cost is used here in the most general way, and can be interpreted as travel time, monetary cost, some 1

M. Patriksson and M. Labbé (eds.), Transportation Planning, 1–17. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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combination of those, or any other measure of disutility of using the specific route. The solution of the Traffic Assignment Problem (TAP) is relatively challenging. The main difficulty is that link costs are typically increasing non-linear functions of link flows. Solving TAP for large-scale networks of practical interest requires computationally efficient solution methods. Many computational methods are available to approximate the equilibrium solution. All of these methods are iterative; i.e. they start by considering some initial assignment, calculate the costs using the flows of the considered assignment, then modify the assignment and update the costs. One way to categorize solution methods is by the level of aggregation in which they store previous solutions. The most aggregated approach is the linkbased approach of storing total link flows, aggregated over all origin-destination pairs. The main advantage of this approach is its relatively modest memory requirements. The most disaggregated approach is the route-based approach of storing all used routes and the flow on each. Route-based methods have been shown to achieve better solutions; the main disadvantage is their large memory requirements. We propose an intermediate approach of storing link flows by origin. This approach maintains the main advantages of the route-based approach while reducing its memory requirements substantially. The next section provides a formal statement of the traffic assignment problem, and introduces the notation used in this paper. Section 3 reviews previous work. A schematic description of an origin-based method is presented in section 4. More details about this method can be found in Bar-Gera (1999). Experimental results are presented in section 5. Section 6 discusses the main characteristics of the origin-based approach. Finally, conclusions and plans for future research are presented.

2.

Problem statement

Let the graph G = {N, A} represent a transportation network, where N is the set of nodes and A is the set of directed links (arcs) on that network. Let be the set of origins, and be the set of destinations. Let be the travel demand matrix, where denotes the travel demand (assumed fixed) from origin to destination in vehicles/hour. The total demand is defined as the sum of flows over all O-D pairs, that is: Traffic on the network is described by an originbased link flow array where is the flow on link of travelers originating at The total flow on link is the aggregation of origin-based link flows over all origins, and is the vector of total link flows. Let be the vector of link costs, where is the cost function of link In this paper we assume that link costs are separable, that is

Origin-based network assignment

3

strictly positive, strictly increasing and continuously differentiable (these assumptions may be relaxed). Let E be the link-node incidence matrix, where if i is the initiation node of link if i is the termination node of link and otherwise. Let be the expanded demand matrix, where if and if A feasible traffic assignment is an origin-based link flow array f, that maintains non-negativity, conservation of flow, and satisfies the demand. In other words, the set of feasible assignments is: The user equilibrium Traffic Assignment Problem (TAP) is to assign the travel demand onto specific routes in the network, assuming that users seek to minimize their own costs. Under this assumption at equilibrium the cost of any used route is not greater than the cost of any alternative route for the same O-D pair. When

is well defined, i.e. path-independent, (our assumptions of monotonic and separable cost functions are sufficient), TAP is equivalent to the following minimization problem:

The evaluation of solution methods requires consideration of convergence measures. There are several possible measures of convergence; the ones used in this paper are easier to define using route-based notation. Let be the set of simple routes from to and the set of all simple routes. Let be the cost of route and the flow on route Denote the minimum O-D cost by: The basic measure of the violation of Wardrop’s equilibrium conditions is the route excess cost defined as the difference between the route cost and the minimum O-D cost, that is: Clearly, at equilibrium the excess cost of all used routes must be zero. The main global (aggregate) measure of convergence used here is the average excess cost, weighted by route flow over all used routes of all O-D pairs, Average excess cost is equivalent to the difference between the objective function and the lower bound obtained from the solution to the linearized subproblem, divided by the total demand. This is a common measure that can be calculated using link-based, origin-based or route-based solutions. It is

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also common to consider the relative gap, which is the difference between the current solution objective function and the best lower bound obtained so far, divided by the absolute value of the best lower bound. Another possible measure of convergence is the maximum excess cost over all used routes of all O-D pairs, Maximum excess cost is a sensitive and effective measure for solution accuracy; however, it requires a detailed solution, origin-based or route-based, and cannot be calculated from a link-based solution.

3.

Review of solution methods for TAP

The most common solution method for TAP is the nonlinear optimization method of Frank and Wolfe (FW). In each iteration, a subproblem of minimizing the linearized objective function is solved by assigning all traffic to minimum cost routes defined on the current link flow solution of the main problem. The new solution is obtained by finding a convex combination of the current solution and the subproblem solution that minimizes the original objective function. The objective function can be evaluated using total link flows only. An aggregated link-based representation of the current solution is therefore sufficient for this method. As a result memory requirement of this method is relatively small, which is its main advantage. The main drawback of FW is its slow convergence rate. See Patriksson (1994) for a detailed discussion. Related link-based methods were proposed by Florian and Spiess (1983), Fukushima (1984), LeBlanc et al. (1985), and Lupi (1986). In all cases some combination of previous solutions and the subproblem solution is used as a search direction. The Restricted Simplicial Decomposition (RSD) method of Hearn et al. (1987) suggests performing a multi-dimensional search over the convex hull of all previous subproblem solutions. That is if are subproblem solutions from previous iterations, the main problem solution at iteration is obtained by solving the following multidimensional nonlinear problem:

The nonlinear simplicial decomposition of Larsson et al. (1998) is a similar method in the sense that solutions to the main problem are obtained by solving a similar multidimensional nonlinear problem, where are still subproblem solutions, only that the subproblems are nonlinear, rather than linear, approximations of the main problem. All of the above methods are link-based methods; that is, only total link flows aggregated over all O-D pairs must be stored between iterations. More recently, increased attention was devoted to route-based methods. These methods assume that all used routes, and the flow on each route, are known for the

Origin-based network assignment

5

current solution. Using that information, flows can be shifted from high cost routes to low cost routes in order to achieve equilibrium. The first method proposed to solve TAP, in fact, was a route-based method. In this method, for each O-D pair considered in a cyclic order, flows are shifted from the maximum cost used route to the minimum cost route until both routes have the same cost. This idea was suggested by Dafermos (1968, 1969) and implemented by Gibert (1968). Bothner and Lutter (1982) implemented a similar route-based method that is used in practice in Germany. When link cost derivatives are known, they can be used to approximate flow shifts from all routes to the minimum cost route of every O-D pair. The aggregation of flow shifts over all O-D pairs is used as a search direction, and the next solution is chosen as the minimum point of the objective function along that direction. Larsson and Patriksson (1992) refer to this approach as Disaggregated Simplicial Decomposition (DSD); they also provide encouraging experimental results. Jayakrishnan et al. (1994) proposed another route-based method, where shifts are based on Gradient Projection (GP). In general, route-based methods seem to achieve high accuracy levels. The detailed information provided by a route-based solution has some additional merits. It allows re-optimization with respect to changes in problem conditions like demand, cost parameters, network topology, road pricing, etc. Detailed route flow information is also important for certain analyses, like emission estimation. In general, route flow solutions for TAP are not unique; however, Rossi et al. (1989), Janson (1993), and others proposed maximum entropy as a criterion for the most likely route flow solution, subject to user equilibrium. The third category of solution methods is the origin-based approach. An origin-based formulation of the traffic assignment problem was first proposed by Beckmann et al. (1956). To the best of our knowledge, there have been few attempts to pursue this approach in developing computational methods. Bruynooghe, Gibert and Sakarovitch (1968) made an attempt to develop such a method; however, Gibert (1968) subsequently concluded that the presence of cycles makes origin-based methods quite complicated. In the late 70’s and early 80’s Gallager and Bertsekas developed destinationbased algorithms for routing in communication networks, a problem that is mathematically equivalent to the traffic assignment problem. (See Gallager, 1977; Bertsekas, 1979; Bertsekas et al., 1979; Bertsekas et al., 1984; Bertsekas, 1998, pp. 390–391.) The reader may find certain similarities between the concepts of the algorithm proposed here, which was developed independently by the authors, and the algorithms proposed by Gallager and Bertsekas.

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An origin-based method for TAP

For every origin we consider a specific restricting subnetwork In this paper all restricting subnetworks are a-cyclic. For a given set of (acyclic) subnetworks we wish to consider only solutions that are restricted by that set. In other words we consider the following restricted feasible set:

The restricted traffic assignment problem is formulated as:

The general scheme of the algorithm is as follows:

1 Find an initial solution nothing assignment).

(trees of minimum cost routes, all or

2 Initialize an outer loop index:

inner loop index:

3 Set/Update restrictions 4 Find a feasible descent direction

5 Find the step size 6 Set 7 Repeat steps 4-6 with

update link flows and link costs. to restricted convergence.

8 Eliminate residual flows. 9 Repeat steps 3-8 with

to global convergence.

In order to choose the restricting subnetwork for a certain origin we first calculate the maximum cost over used routes from to each node using current link costs. When there are no used routes from to an extended maximum cost, as defined in the appendix, is used. The link is included in the new restricting subnetwork if and only if The resulting subnetwork is a-cyclic, since implies a topological order (precedence order) for that subnetwork, that is a function such that if link belongs to the a-cyclic network, then This method of restriction update also guarantees the feasibility of the previous solution. Once a set of restricting subnetworks is chosen, the restricted problem is solved iteratively. In each iteration, for every origin and every node, link costs

Origin-based network assignment

7

and cost derivatives are used in a quasi-Newton fashion to find flow adjustments that seek to equalize the cost of travel from that origin to that node through all approaches. The aggregation of those adjustments over all nodes and all origins yields a search direction. Using a line search, an optimal step size that minimizes the objective function along the search direction is found. Finally the solution is updated accordingly. A critical step in this algorithm is the elimination of residual flows (step 8). This step is similar to a regular iteration, except that only flow adjustments that cause flow on some link to be zero are considered. Those adjustments are chosen so that objective function value will decrease, thus avoiding disruption to overall convergence. The same adjustments may have been considered in previous regular iterations; however, the step size in regular iterations is typically less than 1.0. As a result, a positive (probably small) residual flow is likely to remain on some links during all iterations, even though the restricted equilibrium flow on these links is zero. These residual flows have a negligible impact on link costs, and on the value of the objective function. However, when updating restrictions, any flow from to (originating at ) prohibits the introduction of the link from to (for origin ), as it causes a cycle. Therefore, eliminating residual flows before the next restriction update is crucial for global convergence.

5.

Experimental results

Experimental comparisons between Frank-Wolfe and the proposed originbased method have been conducted using two networks, Sioux Falls (LeBlanc et al. 1975 1), and a sketch (aggregate) network for the Chicago region for the year 1990. The Frank-Wolfe method used the L-deque minimum cost routes algorithm of Pape (1974), considered by Pallottino and Scutella (1998) to be one of the best choices for transportation networks at the current state-of-theart. All experiments were conducted with double precision arithmetic on a Pentium II, 265MHz Dell server. Sioux Falls is a small network of 24 nodes, each representing an origin and a destination, 528 O-D pairs, and 76 links. The difficulty in finding equilibrium solutions for Sioux Falls is probably due to the high level of congestion on the network. At equilibrium, the flows on 59 out of 76 links exceed their capacities. (Capacity values are derived assuming a BPR interpretation of the link cost functions.) Figure 1.1 shows relative gap results for the Sioux Falls network; Figure 1.2 shows excess cost results for the Sioux Falls network; average route costs are also given for comparison. After 3 seconds of CPU time, 71 restriction updates, and 1003 line searches, the origin-based assignment method reached an objective function value of 42.313352871074300, relative gap of 1.54E-15, average excess cost of 3.09E-

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Origin-based network assignment

9

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16 hours, and maximum excess cost of 2.41E-14 hours. These results are likely affected by the machine precision of approximately 2.6E-16. In fact additional iterations exhibit instability, probably due to truncation errors. The purpose of solving the problem to such high accuracy is to examine the behavior of the method, which we found to be quite pleasing. In the same CPU time, the Frank-Wolfe method performed 964 iterations, and reached a relative gap of 2.23E-04, and average excess cost of 2.80E-05 hours. Even after 10,000 iterations, and 33.4 seconds of CPU time, the relative gap for the Frank-Wolfe method was 2.50E-05, and the average excess cost was as high as 4.00E-06 hours. The results indicate the clear advantage of the origin-based assignment method over the Frank-Wolfe method. The equilibrium origin-based solution used 770 routes for the 528 O-D pairs with positive flow, averaging 1.46 routes per O-D pair. Examples from the resulting solution are shown in Figure 1.3. The flows from origin 1 (Figure 1.3a) form a relatively simple network, with only two additional links; that is, a tree may be obtained from this network by eliminating two links. Figure 1.3b shows the flows from origin 12 for that solution, which form a relatively complicated network with seven additional links. One may observe eight different routes from origin 12 to destination 16, which is the maximum number of routes for one O-D. The Chicago sketch network is a medium size network of 317 zones, each representing an origin and a destination, 76,267 O-D pairs, 1,088 nodes, and 3,008 links. The Chicago network is less congested; at equilibrium only 502 out of 3,008 links have flows that exceed their capacities. Relative gap results are shown in Figures 1.4 and 1.5. Excess costs for that network are shown in Figures 1.6 and 1.7, and average route costs are also given for comparison. After 30 minutes of CPU time, 63 restriction updates, and 8,715 line searches, the originbased assignment method reached a relative gap of 1.08E-13, average excess cost of 1.31E-13, and maximum excess cost of 5.01E-10. In the same CPU time, the Frank-Wolfe method performed 2,300 iterations, and reached a relative gap of 1.70E-05, and average excess cost of 2.80E-05. After 5,000 iterations (66.5 minutes of CPU time) the relative gap for the Frank-Wolfe method was 8.00E-06, and the average excess cost was 1.10E-05. On this network, the performance of the two methods is similar for the first 20-30 seconds (25-40 Frank-Wolfe iterations). However additional CPU time allows the origin-based assignment method to improve solution accuracy at a reasonable rate, while convergence of the Frank-Wolfe method is quite slow. The equilibrium origin-based solution used 121, 189 routes for the 76,267 O-D pairs with positive flow, averaging 1.59 routes per O-D pair. The memory requirement for storing the origin-based solution (excluding input data) with the first set of restrictions is about 4.9MB, but as the algorithm converges, the memory requirement decreases to about 1.9MB at equilibrium. (In the current

Origin-based network assignment

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Origin-based network assignment

13

implementation, the minimum memory requirement for a network of that size is about 1.4MB, while the upper bound is about 26.0MB.) In comparison, the input data requires about 0.8MB which is used mainly to store the trip table in double precision accuracy. The Frank-Wolfe method may be implemented with an almost negligible additional amount of memory.

6.

Discussion

There are two main reasons to prefer origin-based algorithms over the stateof-practice Frank-Wolfe method for practical applications: detailed solution, and substantially lower CPU time when higher accuracy is required. For many applications the detail provided by an origin-based solution is practically equivalent to the detail of a route-based solution. See Bar-Gera and Boyce (1999) for an elaborated discussion. Such detail is not provided by link-based methods. As noted in section 3, such detail is needed for certain analyses, like emission estimation, which have important practical implications. Detailed solutions also allow re-optimization with respect to changes in the demand, cost parameters, network topology, road pricing etc. For example, an origin-based solution for a given demand, can be easily adapted to provide a feasible solution for a different demand. This feature makes origin-based algorithms highly suitable in cases where the traffic assignment problem is one component of a larger transportation modeling problem. Route-based methods also provide detailed solutions; however, origin-based methods are more suitable for practical large-scale applications because of their reasonable memory requirements. The theoretical upper bound on the memory requirements of an origin-based solution is on the order of the number of origins times the number of links. The minimum requirement, if no alternative routes are used, is one integer per origin per node. For practical networks memory requirements are likely to be closer to the minimum requirements than to the upper bound. According to the results, the origin-based method is faster than the FrankWolfe method when higher accuracy is desired. Accuracy requirements in practice depend on computer technology. The number of FW iterations that can be computed in a practical sense for large-scale networks has increased from 5-10 in 1980 to 20-40 presently. As computer speed and memory increase further, additional iterations will likely be performed, as more highly converged solutions are desired. At that time we suggest that origin-based algorithms will become the preferred method. From a theoretical point of view there are several reasons for the computational efficiency of origin-based algorithms. First, the restriction to a-cyclic origin-based networks allows the definition of topological order. Using the topological order, the computation time of the minimum, maximum, and av-

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erage cost from the origin to all destinations is linear in the number of links in the subnetwork. The computation time of the search direction per origin is also on the order of the number of links in the subnetwork of that origin. The topological order of an a-cyclic network can be found in a time which is a linear function of the total number of links. Another contribution to the efficiency is achieved by reducing optimization complexity, which may be measured by the number of independent variables (decision variables) in the solution. In a route-based algorithm, the number of independent variables is the number of alternative routes in the system, that is the difference between the total number of used routes and the number of O-D pairs. In an origin-based algorithm, the number of independent variables is the number of additional links in the subnetworks, defined by: In general the number of independent variables in an origin-based representation is expected to be substantially lower than the number of independent variables in an equivalent route-based representation. For example, our equilibrium solution for the Chicago sketch network, suggests that there are 44,922 alternative routes, while the same solution requires only 8,428 additional links in an origin-based representation.

7.

Conclusions

The origin-based approach provides a highly accurate, memory conserving, and computationally efficient solution method for the traffic assignment problem. The current implementation, even though not computationally optimal, produces encouraging results. In addition to coding improvements, the implementation includes several parameters that may be optimized to improve computational efficiency. The authors plan to conduct careful experimental comparisons of the origin-based method with other methods in the near future. Finally, the authors hope to apply this method to larger problems, especially the combined model for travel demand and traffic assignment, and examine the method’s efficiency in that context.

Appendix: extended maximum costs Restriction updates are based on the maximum cost from the origin to each node The maximum cost over used routes is defined only if there exists at least one used route from to Following Hagstrom (1997), we suggest an extended definition of maximum cost for all nodes that may be described as follows. For a given origin link is a used link if the origin-based link flow from is strictly positive, A used node is the termination node of a used link. An extended used route is a route that starts as a used route, and continues

REFERENCES

15

through unused links and unused nodes only. An extended maximum cost used route is an extended used route, for which the used part has the maximum cost among all alternative used routes. The extended maximum cost from the origin to used node i is defined naturally as the maximum cost over all used routes from to For unused nodes the extended maximum cost is defined as the minimum cost over all extended maximum cost used routes.

Acknowledgments We are grateful for the financial support of the National Science Foundation through the National Institute of Statistical Sciences, Research Triangle Park, NC. Comments and suggestions of Professors Jane Hagstrom and Robert Abrams are appreciated. Chicago network data were provided by the Chicago Area Transportation Study, Chicago, IL.

Notes 1. The trip matrix was divided by 10, to reproduce results in previous literature.

References [1] H. Bar-Gera. Origin-based Algorithms for Transportation Network Modeling. PhD thesis, Civil Engineering, University of Illinois at Chicago, 1999. [2] H. Bar-Gera, and D. Boyce. Route flow entropy maximization in originbased traffic assignment. Transportation and Traffic Theory, Proceedings of the 14th international symposium on transportation and traffic theory. Jerusalem, Israel, 20-23 July, 1999, A. Ceder, ed. Elsevier Science, Oxford, UK, 397–415. [3] M. Beckmann, C.B. McGuire, and C.B. Winston. Studies in the Economics of Transportation. Yale University Press, New Haven, CT, 1956. [4] D. P. Bertsekas. Algorithms for nonlinear multicommodity network flow problems. in Proceedings of the International Symposium on Systems Optimization and Analysis. A. Bensoussan and J. L. Lions, eds., SpringerVerlag, New-York, 210–224, 1979. [5] D. P. Bertsekas, E. M. Gafni, and K. S. Vastola. Validation of algorithms for optimal routing of flow in networks. in Proceedings of the 1979 IEEE conference on Decision and Control, San Diego, CA, January 10-12, 1979, 220–227, 1979. [6] D. P. Bertsekas, E. M. Gafni, and R. G. Gallager. Second derivative algorithms for minimum delay distributed routing in networks. IEEE Transactions on Communications, COM-32:911–919, 1984.

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[7] D. P. Bertsekas. Network Optimization - continuous and discrete models, Athena Scientific, Belmont, Massachusetts, USA, 1998. [8] P. Bothner and W. Lutter. Ein direktes verfahren zur verkehrsumlegung nach dem 1. prinzip von wardrop. Forschungsbereich: Verkehrssysteme Arbeitsbericht 1, Universitaet Bremen, 1982. [9] M. Bruynooghe, A. Gibert, and M. Sakarovitch. Une méthode d’affectation du trafic. In Proceedings of the 4th International Symposium on the Theory of Road Traffic Flow, Karlsruhe, 1968, W. Leutzbach and P. Baron, editors, Beiträge zur Theorie des Verkehrsflusses Strassenbau und Strassenverkehrstechnik, Heft 86, Herausgegeben von Bundesminister für Verkehr, Abteilung Strassenbau, Bonn, pp. 198–204, 1969.

[10] S.C. Dafermos. Traffic Assignment and Resource Allocation in Transportation Networks. PhD thesis, Johns Hopkins University, Baltimore, MD, 1968. [11] S.C. Dafermos and F.T. Sparrow. The traffic assignment problem for a general network. Journal of Research of the National Bureau of Standards, 73B:91–118, 1969. [12] M. Florian and H. Spiess. Transport networks in practice. In Proceedings of the Conference of the Operations Research Society of Italy, Napoli, pp. 29–52, 1983. [13] M. Fukushima. A modified Frank-Wolfe algorithm for solving the traffic assignment problem. Transportation Research, 18B:169–177, 1984. [14] R. G. Gallager. A minimum delay routing algorithm using distributed computation. IEEE Transactions on Communications, COM-25:73–85, 1977. [15] A. Gibert. A method for the traffic assignment problem. Report LBSTNT-95, Transportation Network Theory Unit, London Business School, London, 1968. [16] J.N. Hagstrom. Computing tolls and checking equilibrium for traffic flows. University of Illinois at Chicago, preprint, 1997. [17] D.W. Hearn, S. Lawphongpanich, and J.A. Venture. Restricted simplicial decomposition: computation and extensions. Math. Program. Study, 31:99–118, 1987. [18] B.N. Janson. Most likely origin-destination link uses from equilibrium assignment. Transportation Research, 27B:333–350, 1993. [19] R. Jayakrishnan, W.K. Tsai, J.N. Prashker, and S. Rajadhyaksha. A faster path-based algorithm for traffic assignment. Transportation Research Record, 1443:75–83, 1994.

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[20] T. Larsson and M. Patriksson. Simplicial decomposition with disaggregated representation for the traffic assignment problem. Transportation Science, 26:4–17, 1992. [21] T. Larsson, M. Patriksson, and C. Rydergren. Application of simplicial decomposition with nonlinear column generation to nonlinear network flows. In licentiate thesis by Clas Rydergren, Thesis No. 702, Linköping Institute of Technology, Linköping, Sweden, 1998. [22] L.J. LeBlanc, R.V. Helgason, and D.E. Boyce. Improved efficiency of the Frank-Wolfe algorithm for convex network programs. Transportation Science, 19:445–462, 1985. [23] L.J. LeBlanc, E.K. Morlok, and W.P. Pierskalla. An efficient approach to solving the road network equilibrium traffic assignment problem. Transportation Research, 9:309–318, 1975. [24] M. Lupi. Convergence of the Frank-Wolfe algorithm for solving the traffic assignment problem. Civil Engineering Systems, 3:7–15, 1986. [25] S. Pallottino and M.G. Scutella. Shortest path algorithms in transportation models: classical and innovative aspects. In P. Marcotte and S. Nguyen, editors, Equilibrium and Advanced Transportation Modelling. Kluwer Academic Publishers, Boston, 1998. [26] U. Pape. Implementation and efficiency of moore-algorithms for the shortest route problem. Mathematical Programming, 7:212–222, 1974. [27] M. Patriksson. The Traffic Assignment Problem, Models and Methods. VSP, Utrecht, Netherlands, 1994. [28] T. F. Rossi, S. McNeil, and C. Hendrickson. Entropy model for consistent impact fee assessment. Journal of Urban Planning and Development/ASCE, 115:51–63, 1989.

Chapter 2 ON TRAFFIC EQUILIBRIUM MODELS WITH A NONLINEAR TIME/MONEY RELATION Torbjörn Larsson [email protected]

Per Olov Lindberg [email protected] Department of Mathematics, Linköping University SE-581 83 Linköping, Sweden

Michael Patriksson [email protected] Department of Mathematics, Chalmers University of Technology SE-412 96 Gothenburg, Sweden

Clas Rydergren [email protected] Department of Mathematics, Linköping University SE-581 83 Linköping, Sweden

Abstract

We consider a traffic equilibrium problem in which each route has two attributes, time delay and monetary outlay, which are combined into a generalized time through a nonlinear relation. It is shown that this problem can be stated as a convex optimization model. Two simplicial decomposition type methods are proposed for its solution. The subproblem of these methods, which is a twoattribute shortest route problem, can be efficiently solved by the multi-labelling technique which has previously been applied to resource-constrained shortest path problems. Our numerical experiments show that both methods are feasible approaches to the equilibrium problem.

Keywords:

Traffic equilibrium, nonlinear value-of-time 19

M. Patriksson and M. Labbé (eds.), Transportation Planning, 19–31. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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1.

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Introduction

In models for traffic equilibria one usually assumes that travellers choose the routes from their origins to their destinations that are preferred according to a cost criterion (the Wardrop principle, e.g., [10, Section 2.1], or [11]). The routes typically have several attributes, such as time, monetary outlays, distance etc. Usually these attributes are combined linearly to a generalized cost. However, empirical studies, e.g., [7] and [6], indicate that travelers value travel time changes nonlinearly rather than linearly, in that short changes have lower value of time than longer ones. At first sight, this might seem puzzling, but it is in fact easy to envisage situations where this is the case. Consider, for example, a young commuter heading for his workplace. Arriving a quarter of an hour late to work is probably then more than three times worse than coming five minutes late. Conversely, having to wake up a quarter of an hour earlier, similarly, is experienced to be more than three times worse than waking up five minutes earlier. In the present paper, we will study a situation with two route attributes: time and money. In a subsequent paper, we will hopefully return to the case of more than two attributes. The two-attribute case has already been studied by Bernstein and Gabriel (B&G for short) in [4] and [2]. They assume that route choice is based on a (nonlinear) generalized cost. In particular they assume that the (generalized) cost for route r is

with and being monetary outlay and travel time, respectively, and the nonlinear value of time for route To compute equilibria of their model, B&G utilize a specialized code based on a Gauss-Newton type method for nonlinear complementarity problems. It is not a priori obvious why route choice should be based on (generalized) cost rather than on time. Thus, in the present paper we consider to instead base route choice on time rather than on cost, At first sight, the time-based cost (2) seems to be just a reformulation of the money-based one (1), with It turns out, however, that they are not equivalent. For the case with flow independent monetary outlays, the time based equilibrium problem can be stated as an equivalent optimization problem. This will give the benefit of usually faster computational methods as well as better control of convergence. The development of this equivalent optimization model and the techniques to solve it are the theme of this paper. The paper is structured as follows. In Section 2 we state the time-based traffic equilibrium problem. Section 3 is devoted to the simplicial decomposition

On Traffic Equilibrium Models with a Nonlinear Time/Money Relation

21

approaches to solve this problem; on the one hand a disaggregate version, where all route flows are stored, on the other hand an aggregate version, where only link flows are stored. In Section 4 we describe how to generate new routes to be used in the simplicial decomposition schemes, using a multi-label shortest path method [3]. Section 5, finally, is devoted to some computational testing on the example of B&G and on the classical Sioux Falls network.

2.

The time-based traffic equilibrium problem

We will first introduce some further notation. The set of commodities (i.e., the origin-destination pairs) is denoted by and denotes the traffic demand for commodity (More general, could denote a segment of travellers in a certain travel relation.) Let be the set of simple routes for commodity and let be the set of all routes. Similarly, let denote the routes that pass link with being the set of links in the network. Let denote the flow along route and let be the travel time at link flow on link Thus, the travel time for route will be Similarly, let be the monetary outlay for link which is assumed to be independent of the traffic flow. If the monetary outlay for route is assumed to be additive, then (We can allow for more complex structures for the monetary outlay for a route. The important thing is that they are monotone and can be computed recursively.) Travellers are assumed to base their route choice on some generalized times on the routes, and based on these times they always choose the minimal time route from origin to destination. The situation can be expressed by the well known equilibrium condition, the Wardrop principle (e.g., [10]),

where is the generalized time on route If travel time is the only measure of generalized time, we have and it is well known that the flows and that fulfill (3) can be obtained by solving the following convex optimization problem (see e.g., [10], Section 2.2).

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subject to

If we want to include a monetary cost component for each route, and to express a situation with a nonlinear relation between travel time and monetary cost, we can express the generalized time on route for given link flows according to

Since the time equivalent of money, probably is dependent on socioeconomic factors, it is natural to assume that it depends not on but on the commodity corresponding to The function is naturally assumed to be nonnegative and increasing. The Wardrop conditions for the time-based traffic equilibrium problem are expressed by (3) using the definition of according to (4). An equivalent optimization model for this problem can be formulated as

subject to

The only difference in comparison with problem (P) is the extra linear term in the objective. Thus, is a convex problem, since (P) is. Further, since has a compact feasible set it attains its minimum, and since the constraints are linear an Abadie constraint qualification is fulfilled, whence there are Lagrange

On Traffic Equilibrium Models with a Nonlinear Time/Money Relation

23

multipliers such that the Karush–Kuhn–Tucker (KKT) conditions are satisfied (see [1], Theorem 5.1.3 and Lemma 5.1.4). Letting be the multiplier for the demand constraint and substituting the expression for into the objective, we get for the KKT condition

with equality if

Thus

Hence, the KKT conditions state, for that if that is fulfills the Wardrop conditions. Conversely, assuming that fulfills the Wardrop condition and letting it follows that fulfills the KKT conditions, and hence is an optimal solution to If the link travel time functions are strictly increasing, then the solution to the time-based traffic equilibrium problem has unique link flows. This follows from the fact that the objective of is in this case strictly convex in the link flows. In summary we have shown the following. Proposition 1 (Optimization formulation) The time-based traffic equilibrium problem admits an equivalent optimization formulation.

3.

Solution approaches

In practice, the route set is too large to be handled explicitly. Instead, one has to generate profitable routes systematically and iteratively. We propose to solve the problem using either the disaggregate (DSD) or aggregate form (ASD) of the simplicial decomposition method (cf. [10, 8]). The DSD method, which alternates between a route generation phase and a master problem phase where the problem is solved using the routes generated so far, is analogous to the original DSD method for the standard traffic equilibrium problem proposed in [8]. Suppose we have generated a subset of the routes, where is the subset of the routes for commodity We then solve a restriction of replacing with This problem (the master problem) is denoted , and it gives the solution giving an upper bound to the optimal value of Next, we want to find out whether we need to include more routes. This is accomplished through the solution of a subproblem, a linearized version of The solution of this subproblem (the route generation problem) amounts

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to checking whether some non-generated route than zero, i.e., to check whether

where

has a reduced cost less

is the multiplier for demand constraint

Thus, we

have to solve

which is to find the route with minimal generalized time, and then we have to check whether this generalized time is lower than A solution procedure for finding the new routes is given in next section. Since there are only finitely many routes for each O/D–pair we have the following result. Proposition 2 (Convergence) The DSD method will terminate with an optimal solution after a finite number route generations. Similarly to classic traffic assignment problems, the shortest path subproblems in the route generation arises from a linearization of the objective of e.g., ([10], Section 4.1.2). If we linearize the objective, then the cost coefficients will be flow independent, and all traffic can be sent along the shortest paths. Due to the convexity of the objective function of the solution of the linearized problem yields a lower bound of the optimal value, as stated below. Proposition 3 (Lower bound) paths in the route generation for

be the set of generated shortest Then,

is a lower bound on the optimal value of The lower bound together with the upper bound from solving the master problem gives the possibility to terminate the algorithm when a prescribed error is achieved rather than when all routes with nonzero flows are generated. In large networks, the DSD strategy may be too memory consuming. Hence, there is need for more aggregate versions. In classical (aggregate) simplicial decomposition (ASD) methods for traffic assignment, one takes convex combinations of link flow patterns arrived at by assigning all demand in each commodity to the shortest path obtained from the route generation subproblems (e.g., [10]). Exactly the same can be done in the current situation.

On Traffic Equilibrium Models with a Nonlinear Time/Money Relation

25

To be specific, assume that the shortest route for commodity in the iteration is and that this route has monetary outlay Assigning all demand in each commodity to the shortest route, we get the link flow pattern vector where for each link we have

Further, let be the total time equivalent of the monetary outlays for this flow pattern. The aggregate master problem for the time based traffic assignment problem is then to minimize the objective of the original problem over convex combinations of generated link flow patterns up to the current iteration I, that is

subject to

This problem gives new link flows and hence travel times Given these travel times, we solve a new route generation subproblem. Assigning all flow to the generated routes, we get a new link flow pattern etc. The theoretical results for the ASD method are totally in parallel with those of the DSD method. The solution to whose feasible set corresponds to a restriction of gives an upper bound to the optimal value of Proposition 4 The ASD method will terminate after a finite number of route generations. Proof Follows from the fact that there are only a finite number of link flow patterns that can be generated, since they are composed of flows along a finite number of routes. The observation that the generated shortest paths solve a linearization of the true objective, is valid also for the aggregate master the only difference

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being that we linearize at link flows that are convex combinations of generated link flow patterns. In practice, the number of generated link flow patterns will become very large. Hence, if we have a limit on how many patterns we can store, we will have to drop patterns (possibly after aggregation), i.e., we will use restricted simplicial decomposition [5]. This will destroy the finite convergence, giving convergence in the limit.

4.

A route generation algorithm

The minimization problem to solve for finding new routes is a generalized shortest path problem, where the links have multiple attributes, in our case travel time and monetary outlay. Due to the nonlinear transformation of money to time, it cannot be solved using standard procedures for shortest paths. In particular it does not adhere to the optimality principle of dynamic programming. If the “shortest” path from an origin to a destination passes node that does not imply that the sub-path from the origin to node is the shortest path between the origin and node Our shortest path problem can however be solved using a multi-label shortest path method, known from the solution of resource constrained shortest path problems [3]. Each node gets labels (usually several) of the form where and are the accumulated travel time and monetary outlay to get to the node from the origin via predecessor node (or or more generally, label) For a given origin, one starts by giving the origin node the label (0, 0, –). For each labeled node, one labels all nodes that can be reached from it. If for instance node can be reached via a link with time delay and monetary outlay from node with label then node gets the label Thus, the labels will indicate the total time and total monetary outlay to get to the nodes along paths indicated by the predecessor indices. The embryonic version of multi-label shortest path methods just described, will in fact generate all paths in a network. Possibly, a node can get one label for each predecessor. Since the number of labels in practice can be very large, we need to use domination tests to keep down the number of labels. If in a given node there are two labels and with and then the label can be deleted, since any path continuing from that label will be no better than the corresponding path from Since both travel time delays and monetary outlays for the links are nonnegative, domination implies that no path corresponding to the labels will contain cycles. Hence, the labeling process is finite. When the process stops, each destination node has received a set of labels It is then easy to find the best label, i.e., the one minimizing and the best O/D path can

On Traffic Equilibrium Models with a Nonlinear Time/Money Relation

27

be traced using the predecessor indices. The multi-label shortest path method is summarized in the rudimentary pseudo code in Table 2.1.

5.

Numerical tests

We have tested the disaggregate as well as the aggregate version of the proposed simplicial decomposition method on two small test networks which are summarized in Table 2.2. Network 1 is taken from [2]; we have used the inverse of their time-to-money transformation,

to describe the nonlinear relation between monetary outlay and travel time, i.e. Moreover, since G&B in contrast to us use elastic demand, we have used their equilibrium demand as our demand. Network 2 is a modification of the classical Sioux Falls network, where we have used the tolled links of [9] and the same function G as for Network 1. Our experiments were not aiming at obtaining computational efficiency, but rather at verifying the validity of the algorithms. Hence, for both networks, and for both methods, we have solved the master problems to a quite small relative error before generating new routes. This accuracy bound is computed in a similar way as for the global accuracy. For the minimization in the master problem, we have used gradient projection in the DSD method and Newton’s method in the ASD method. We initialize the algorithm by assigning all travel demand to the time optimal routes.

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The disaggregate version terminates, for both test networks, in a few iterations, as shown in Tables 2.3 and 2.4. (Note that the relative gap, or error, in a given iteration is in fact not determined until the route generation of the next iteration.) The solution procedure generates 131 and 1175 routes for Network 1 and 2, respectively, and the number of routes with positive flow in the optimal solutions is 84 and 663 for Network 1 and 2, respectively. In the tests of the aggregate version we have used non-restricted simplicial decomposition, i.e., no traffic patterns were discarded. As expected, convergence is much slower for the aggregate version. Figures 2.1 and 2.2 display the iteration histories, giving iteration-wise upper and lower bounds of the objective. We achieve a gap of 0.1% at iterations 23 and 45 for the network 1 and 2, respectively. Gabriel and Bernstein give no computing times and do not mention how many routes they have generated. However, in order to compare their results to ours, we chose our money-to-time transform G as the inverse of the time-

On Traffic Equilibrium Models with a Nonlinear Time/Money Relation

29

to-money transform in [2]. To our initial surprise, the solutions did not coincide. A closer look shows that this is in fact quite natural. Consider the example in Figure 2.3. Using we get a money-based cost of and a generalized time of It is then easy to verify that the money

30

based equilibrium is

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and that the time based equilibrium is

Observation 1 (Non-equivalence of time- and money-based equilibrium problems) Evaluating routes in a money based way using generalized cost does not necessarily give the same equilibrium as evaluating routes in a time based way, using generalized time with In a later paper we will come back to how one can utilize the equivalent optimization formulation of the time based approach to solve money based problems and more general multi attribute traffic assignment problems.

References [1] M.S. Bazaraa, H.D. Sherali and C.M. Shetty. Nonlinear Programming: Theory and Algorithms. John Wiley & Sons, New York, NY, 1993. [2] D. Bernstein and S.A. Gabriel. Solving the nonadditive traffic equilibrium. In P.M. Pardalos, D.W. Hearn, and W. Hager, editors, Network Optimization, volume 450 of Lecture Notes in Economics and Mathematical Systems, pages 72–102. Springer–Verlag, 1997. [3] J. Desrosiers, Y. Dumas, M.M. Solomon, and R. Soumis. Time constrained routing and scheduling. In M.O. Ball, T.L. Magnanti, C.L. Monma, and G.L. Nemhauser, editors, Handbook in Operations Research and Management Science, Network Models. North–Holland, 1995. [4] S.A. Gabriel and D. Bernstein. The traffic equilibrium problem with nonadditive path costs. Transportation Science, 31:337–348, 1997. [5] D.W. Hearn, S. Lawphongpanich, and J.A. Ventura. Restricted simplicial decomposition: Computation and extensions. Mathematical Programming Study, 31:99–118, 1987.

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[6] D.A. Hensher and T.P. Truong. Valuation of travel savings. Journal of Transport Economics and Policy, pages 237–260, 1985. [7] L. Hultkranz and R. Mortazavi. The value of travel time changes in a random nonlinear utility model. CTS working paper 1997:16., Submitted. [8] T. Larsson and M. Patriksson. Simplicial decomposition with disaggregated representation for the traffic assignment problem. Transportation Science, 26:4–17, 1992. [9] T. Larsson, M. Patriksson, and A-B. Strömberg. Ergodic, primal convergence in dual subgradient schemes for convex programming. Mathematical Programming, 86:283–312, 1999. [10] M. Patriksson. The Traffic Assignment Problem – Models and Methods. VSP, Utrecht, 1994. [11] Y. Sheffi. Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods. Prentice–Hall, Englewood Cliffs, NJ, 1985.

Chapter 3 STOCHASTIC NETWORK EQUILIBRIUM UNDER STOCHASTIC DEMAND David Watling [email protected] Institute for Transport Studies, University of Leeds, UK

Abstract

A generalisation of the conventional stochastic user equilibrium (SUE) model is developed in order to represent day-to-day variability in traffic flows due to stochastic variation in both a) the inter-zonal trip demand matrix, and b) the route choice proportions conditional on the demands. The equilibrated variables in this new problem are the link flow means and covariance matrix. A heuristic solution algorithm is proposed, based on the solution of a sequence of SUE subproblems. Numerical results are reported from the application of this technique to a realistic network, under the assumption of probit-based choice probabilities. In these tests, as the level of demand variability is increased (but the mean demand held fixed), the link flow variances predicted by the proposed model are seen to increase, but the effect on mean flows is relatively small. The increased variation in flows is, however, seen to have an inflationary effect on one of the prime indicators of network congestion, mean total travel cost.

Keywords:

Networks, route choice, equilibrium, uncertainty, stochastic demand

1.

Introduction

The established family of network equilibrium models for representing driver route choice over congested traffic networks consists of a variety of techniques, the most well-known being the deterministic user equilibrium (DUE) and stochastic user equilibrium (SUE) models, which have been formulated for both the ‘steady state’ [13] and ‘within-day dynamic’ [11] cases. A characteristic feature of all models in this family is their use of a deterministic representation for both: the key variables of predictive interest, namely link flows and travel times; and 33

M. Patriksson and M. Labbé (eds.), Transportation Planning, 33–51. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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the primary input variables, namely the inter-zonal travel demand matrix and network characteristics (e.g. free-run speeds/times, capacitites). The essential difference between the ‘steady state’ and ‘dynamic’ models is that in the latter these input/output quantities are disaggregated into shorter time periods within the day. The more important distinction for the present paper is that between DUE and SUE (whether steady state or dynamic). In the SUE model, drivers are assumed to have perceptual differences in their evaluation of a given travel cost, these differences being most conveniently represented by a pre-specified perceptual probability distribution, distributed across the population of drivers. DUE, on the other hand, assumes a single, mean perception of travel cost. That is to say, in terms of the input/output quantities mentioned above, SUE is no more stochastic than DUE (a point also made in [6]). This latter comment on the inherent determinism of SUE is not merely a theoretical nicety. There is, of course, extensive evidence that the factors above may vary significantly from day-to-day (e.g., [9]; [5]; [14]; [17]; [8]), but the deterministic assumption has prevailed on the basis of their being no clear, tractable way of including it, and based on the belief that variability need not be considered in order to approximate mean conditions. There are, however, a number of drawbacks to such a pragmatic approach. Firstly, there are theoretical arguments to suggest that – due to the non-linearity of the system under consideration – neglecting variability will lead to a systematic bias in the estimate of mean travel times/costs. Moreover, it is not clear that the impact of such biases will be consistent across the network or with respect to different policy measures. Secondly, the models provide only mean output measures (traffic flow, network performance, etc.), whereas information on the variations in traffic flows and travel times would also be of value to the planner. Thirdly, these existing models are poorly suited to testing policies that are designed to respond to variability. The study of driver information systems has provided a primary example of such a policy; the major complications that have arisen in matching model predictions with empirical evidence of response could be said to be a difficulty in separating ‘subjective variation’ (individuals’ preferences and constraints) from the ‘uncertainty’ (the degree to which an individual fails to satisfy their own preferences/constraints due to predictive errors). With these comments in mind, the objectives of the present paper are: 1 To formulate a modified version of SUE that is able to represent stochastic variation in travel demand, and the effect of such variation on route choice. 2 To propose a solution algorithm for this modified model, and to present preliminary numerical results from its application to a realistic network.

Stochastic Network Equilibrium Under Stochastic Demand

35

In particular, it is noted that the objective is not to build an entirely new modelling paradigm, but to seek a minimal extension to SUE in order to include such variations. The approach is based on extending the network equilibrium model presented in [15] from the case of deterministic to stochastic demand. In fact, the approach may be further extended to include the effect of stochastic variations in network attributes such as capacities, provided their distribution is independent of traffic conditions. However, in the present paper, the focus will be on stochastic demand (rather than supply) variations, as it is the demand variations that are the most complex to represent within a consistent equilibrium framework. The paper begins by introducing some basic notation and definitions (section 2), and then goes on to make a critique of the existing SUE model in the context of daily variations (section 3). In section 4, a summary is given of the salient details of a recently proposed extension to SUE, that is able to model variable traffic flows but under deterministic demand. In section 5, this method is subsequently extended to the case of stochastic demand. In section 6, a heuristic solution algorithm is presented, and in section 7 preliminary simulation results are reported.

2.

Notation

We suppose the network consists of A links indexed and W inter-zonal (origin-destination) movements indexed The N possible routes that pass through a link at most once, across all such inter-zonal movements, is indexed by the set {1,2,..., N}, in such a way that the subset of routes relating to inter-zonal movement are indexed by the set

The demand rates (vehicles per hour) for each of the W inter-zonal movements are held in the column vector of dimension W, with elements 1,2,... , W). Define to be an N × W path-movement incidence matrix with elements (for and

Then is an N × N diagonal matrix, with diagonal entries from the vector such that each row relates to a route, and the diagonal entry for that row is the demand appropriate to that route. The column vector of dimension N denotes an assignment of flow to each of the possible routes, with the convex set of demand-feasible non-negative

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route flow rates denoted by

where denotes N-dimensional non-negative real space, and where here and throughout the paper, denotes the transpose of the matrix The corresponding convex set of demand feasible link flow rates is

where is an A × N link-path incidence matrix with elements 1 , 2 , . . . , A and

The cost of travelling along link at a given link flow rate vector is denoted These functions may also themselves be arranged in a column vector, These link performance functions imply corresponding route cost-flow performance functions

by

Suppose further that for each movement is a route choice model describing the probability of a randomly-selected driver on interzonal movement choosing each of the alternative routes when the perceived route costs (averaged across the driver population) are and that denotes these functions across all movements, arranged in a column vector of dimension N. For example, for each movement it may be assumed that is a random utility model:

where

follow some given joint probability distribution.

We may then state the following well-known definition [13]. Definition 1 The route flow rate vector (SUE) if and only if

is a stochastic user equilibrium

Alternatively, the link flow rate vector

is termed a SUE if and only if

Stochastic Network Equilibrium Under Stochastic Demand

37

Corresponding to the usual flow rate variables and as defined above) it will also prove useful to define respective absolute flow variables and with elements in the discrete (integer) units of “vehicles" or “drivers". These absolute flows relate to a particular period of the day, of duration hours. For example, Throughout the paper, the capitalised versions of and namely F, V, , , T and C will be used to denote vector random variables of the relevant flow and cost quantities. In terms of the absolute flows, the discrete demand-feasible route flows are given by

where denotes the N-dimensional space of non-negative integers. Similarly, the demand-feasible link flows are given by:

3.

Critique of SUE in the context of day-to-day variability

As noted in the introduction above, observations of traffic volumes and travel times indicate considerable day-to-day variability, and so it is not difficult to make a case that these quantities are most appropriately represented as stochastic variables. From a purely deductive philosophy, the fact that SUE neglects significant sources of variability (as DUE does) is perhaps sufficient criticism to warrant the investigation of more sophisticated modelling tools. This is true even if our only interest is in mean network performance, due to the non-linear nature of the interactions between traffic flow, travel times and travel choice, refuting any naive claim that SUE/DUE necessarily represent mean performance to which variations may subsequently be added. Taking a more pragmatic viewpoint, however, it is known from the long experience with DUE/SUE models that they have a good degree of explanatory power; that is to say, they “explain" a good deal of the performance of traffic networks. It is therefore relevant to ask in what way can they be built upon for the purpose of this paper, and in what circumstances can they said to characterise approximate mean performance. Answering this question is a deceptively difficult problem. An appealing pragmatic approach is to assume the SUE link flows represent the mean of independent (between links) Poisson variables, and one could then compute mean costs/times corresponding to these variable flows. One problem with this approach is that is does not represent the covariances between links that are likely to occur due to the network structure. Two obvious sources of covariance are: in the case of a fixed demand matrix, for two links that are part of alternative routes for a particular inter-zonal movement, the contributions

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to these link flows from that movement will be negatively correlated (since the route flows are negatively correlated), and for links that are part of the same route there will be a positive correlation; in the case of stochastic demand, the correlations above must be balanced against positive correlations between the components of link flow for all links used by a particular inter-zonal movement (when demand is high, all route flows are likely to be high). A second problem with this approach is that it pre-supposes SUE correctly predicts mean flows under such variable conditions. An argument often cited for this is that one can suppose that the long-run variable route flows (for each inter-zonal movement) are multinomially distributed, assuming drivers make choices independently, with choice probabilities based on some form of random utility model evaluated at SUE costs. It is then argued that as all inter-zonal demands become large, the distribution of the flow proportions will become increasingly narrow, and focused on the SUE flows; in the limit, then, as the flow probability distribution becomes focused on a single point, SUE can be viewed as an equilibrium condition on the flow probability distribution. There are a number of difficulties with this line of reasoning: 1 If drivers build up their predictions of travel times/costs from a finite number of past experiences (these experiences are themselves subject to random variation), then while the distribution of route flows conditional upon the past may be multinomial, the unconditional equilibrium distribution will in general be an over-dispersed multinomial, since the choice probabilities are then stochastic [7]. 2 Even if it is possible to assume that drivers’ experiences are sufficiently long that they are able to predict long-run expected costs, so that the choice probabilities are deterministic, then for non-linear link cost-flow performance relationships, SUE - in effectively substituting costs at expected flows for expected costs - contains an inherent systematic bias [2]. For convex cost-flow relationships, expected costs are systematically underestimated. 3 As the absolute number of travellers on all inter-zonal movements approaches infinity, then we expect SUE increasingly to approximate SUE mean flow rates, in spite of the misgivings above [4]. In typical urban, peak period, traffic assignment applications, the zoning system is likely to be sufficiently fine that many inter-zonal movements will have quite a “small" demand, and so the usefulness of the asymptotic result is not clear. In addition, it is important not to confuse the absolute demand with the typical demand input to a traffic assignment model, which is

Stochastic Network Equilibrium Under Stochastic Demand

39

the mean flow rate per hour. The absolute demand depends on both the demand rate and the length of the time period over which this rate is assumed to be valid. A large demand rate on its own is not sufficient, particularly when one considers the current trend in traffic assignment models, towards dynamic models that assign a number of multiple, short time periods. In response to some of the issues raised here, a radically new approach to traffic assignment modelling was proposed by Cascetta [2], and extended further by Davis [4] and in [1]. This approach models the dynamic, day-to-day evolution of travel choices as a discrete time stochastic process, explicitly representing variability in flows and travel times/costs and their effect on (future) travel choices. Equilibrium in this setting refers to a fixed point condition on the joint probability distribution of network flows. The flexibility of this approach makes it extremely appealing, yet it leaves a practical dilemma. Should we discard the many years of research on understanding and applying traditional network equilibrium approaches (offering an albeit limited but well-controlled modelling environment), in favour of a new approach (the outputs of which are significantly more complex)? Can no use be made of the understanding of traditional equilibrium solution methods? This is a difficult decision, particularly since we would probably be most comfortable selecting conventional equilibrium for some policy tests, and the new approach for others, though this leaves a problem of an inconsistent evaluation framework. With these issues in mind, the present author recently proposed an intermediate modelling framework, termed a Generalised Stochastic User Equilibrium of order and denoted [15]. This is effectively based on equilibrating the moments of a natural joint probability distribution of network flows, the model having as active equilibrated variables the moments of order and below. A GSUE(1) model therefore equilibrates means only, and turns out to be an SUE model (regardless of the demand levels, i.e. this is not only a large sample result), whilst a GSUE(2) model equilibrates the flow mean and covariance matrix. This approach, which will form the basis of the remainder of the paper, will shortly be introduced in detail. However, it is useful first to describe the philosophy adopted in the context of random utility theory. In the SUE model (presuming its aim is to estimate mean traffic flows), the cost of each alternative (route) is essentially set to:

The cost at mean traffic flows is a deterministic quantity given by the costflow performance relationships, and the perceptual differences are randomly

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distributed (and, in realistic models such as probit, correlated between alternatives). It turns out that the structure of the choices made in the GSUE(2) model could be described as being equivalent to assuming:

where the uncertainty is in general a random quantity (between days and between drivers) and is due to the actual variance in traffic conditions. At this level, the objective of the paper can be viewed as a technique for fitting alternative error structures for a random utility model, with mean flows subsequently predicted by SUE based on this modified error structure. This description of the approach needs, however, to be clarified in two ways: 1 If the true variance in costs were and drivers’ predictions are assumed to be formed from a large number say) of experiences, then neglecting any correlations between experiences/alternatives, the variance in the “Uncertainty" is and so as this variance tends to zero. This does not, however, imply that (5) approaches (4), since unlike “Perceptual difference", the “Uncertainty" does not have a zero mean; the variability in traffic flows affects not only the variance in actual costs, but also mean actual costs. 2 From the argument in 1., determining the appropriate correction factor is non-trivial, since in order to determine mean costs at variable flows, the whole flow probability distribution is required. But this leads to a circular argument, since in order to determine the random utility error structure, the (equilibrium) flow distribution is required, but the whole point of specifying the error structure is to determine the equilibrium flows. The approach will therefore be to deduce conditions that must simultaneously be satisfied by the error structure and the equilibrium flow allocation. This is first considered, in section 4, in the case of a fixed (deterministic) demand, and is then generalised in section 5 to the case of stochastic demand.

4.

Equilibrium conditions: fixed demand

The first task is to define equilibrium in a more general setting in which there is random variation in the route and link flows, which in turn induces random variability in the actual travel costs. This is quite a complex issue, and is dealt with in two stages. Firstly, equilibrium conditions are presented on the joint probability distribution of network flows. Secondly, an approximation to these conditions, based on first and second order moments only, is deduced. This

Stochastic Network Equilibrium Under Stochastic Demand

41

latter approximation will be the defining conditions for our generalised model. In this section we assume that demand is fixed (i.e. inelastic and deterministic). For a further elaboration of the proofs and analysis of all the results in this section, the reader is referred to [15]. Let denote the column vector which has elements given by the (unknown) probabilities thus has dimension equal to the cardinality of and is simply a representation of the joint probability distribution of the absolute link flow vector variable . This distribution is related to the route flow probability distribution (a column vector of probabilities o f dimension by where is a matrix with elements

Suppose that as in the SUE definition (2)/(3), a function is given, which relates the probabilities of a randomly selected driver choosing each of the alternative routes at given route costs Then define partitions of this function and the route flow vector according to the different inter-zonal movements, such that:

Now suppose that the route costs where is a vector of link costs, so that conditionally on , for each inter-zonal movement independently, each ofthe drivers independently chooses between the available routes with probabilities Then the distribution of movement route flows conditional on random link costs Y, is given by (independently for

The following consistency (equilibrium) condition on the distribution then be established.

may

Theorem 1 (Asymptotic equilibrium condition). Suppose that drivers form estimates of actual costs from a random sample of the link costs from their previous travel experiences, where is given,

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and is a sample of independent, identically distributed, demand-feasible link flow vectors. The given cost-flow performance functions are assumed to be bounded for Suppose further that, at given route costs the population of drivers chooses independently between the available routes with probabilities Then asymptotically, as the link flow probability distribution satisfies the equilibrium condition:

where

is a vector of dimension

with elements the probabilities

where

denotes that has a given probability distribution where denotes the expectation operator with respect to the distribution of and where the conditional distribution of is given by (8) based on the partition (7). Theorem 1 essentially hinges on the fact that as the number of experiences tends to infinity, the variance in the mean of these i.i.d. experiences will tend to zero, and in the limit implies that the distribution of the mean of the experiences is focused on the true long run expectation. Hence the corresponding choice probabilities will be deterministic, and the unconditional distribution of can be approximated by the conditional distribution (10), which by hypothesis is formed from a combination of multinomials (8). The derivation of the condition above is very much along the lines of conventional analyses of traffic networks, in the sense that we seek a consistency condition that should reasonably be satisfied, without any specific reference as to how a network may arrive at such a state. For example, in the SUE model (1) defined earlier, the hypothesis is that is in equilibrium if the route proportions given by the behavioural choice model at costs are consistent with flows of In addition, by allowing the number of experiences to become very large in Theorem 1, we are effectively assuming a “well-informed" driver population, again something that is consistent with conventional analyses. Having presented the underlying equilibrium conditions, a more tractable approximation may then be deduced. Theorem 2 (Approximation to Equilibrium Conditions). Consider a network with twice-dijferentiable link cost-flow functions Then an approximation to the mean and covariance matrix of an equilibrium probability distribution (9) is given by an A-vector and A × A matrix satisfying the fixed point

Stochastic Network Equilibrium Under Stochastic Demand

43

conditions:

where

is an A-vector with elements

where is the A × A Hessian matrix of 1 , 2 , . . . , A), where the scalar product of any two is denoted by

evaluated at matrices X and Y

and where is a function whose result is an N × N block diagonal matrix, with blocks the matrices of dimension

where is partitioned as in (7). A pair satisfying (11) is termed a Generalised Stochastic User Equilibrium of order 2, and is written GSUE(2). The essential steps in the proof are firstly to deduce a second order Taylor series approximation to the link cost-flow functions, whereby (12) may be deduced as an approximation to the expected costs in (10). The effect is that on the right hand side of (9), only appears through its mean and covariance matrix, and so these are the only ‘active’ elements of to equilibrate. Hence, on the left hand side, we also deduce the link flow rate mean and covariance matrix, which is related to the absolute link and absolute route flow mean and covariance matrix. This latter mean is diag (note while the covariance matrix is block diagonal, with blocks this latter being an expression for the covariance matrix of a multinomial variable. The particular advantage of conditions (11) over condition (9) is in the development of solution algorithms for large realistic networks. In such cases, the enormous dimension of makes it unappealing to deal with directly, whereas (11) may be solved by an efficient heuristic method. A discussion of this heuristic method will be postponed to section 6, where it is extended to the case of stochastic demand.

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Equilibrium conditions: stochastic demand

Having set up a model in which driver choices and flows may be stochastic, it is a natural extension to allow the inter-zonal demands to be stochastic: Theorem 3 (Stochastic Demand Equilibrium Conditions). Suppose that the hypotheses of Theorem 2 hold, except that now demand is assumed to be stochastic. Let the given W-vector denote the potential demand (number of potential travellers) on each of the inter-zonal movements, and suppose that on any given day, each such potential traveller decides independently to travel with given constant movement-specific probabilities in the W-vector which has elements Conditional on the choice to travel, a route is subsequently chosen according to probabilities Then, relative to (11), the modified equilibrium conditions on are:

where where blocks:

and are as previously defined, and is a function whose result is a block diagonal matrix, with

and where the mean demand rate

is related to

by

A pair satisfying (14) is termed a Stochastic Demand Generalised Stochastic User Equilibrium of order 2, and is written SDGSUE(2). Proof The conditions may be derived from Theorem 2, by introducing a dummy “no travel" route joining each inter-zonal movement. Under the assumptions of Theorem 3, the potential travellers on each movement are multinomially distributed between the non-dummy routes with probabilities Hence, by the application of Theorem 2, we obtain equilibrium conditions of:

where substitution (16).

Expression (14) is then obtained from (17) by the

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Stochastic Network Equilibrium Under Stochastic Demand

One point that is worth noting about (14) is what happens as In that case, with all other variables held fixed, tends to the zero matrix. Then, from (12), and the first condition in (14) reduces to an SUE condition on Now as for a given mean trip demand rate of and travel probability then by (16) the absolute number of potential travellers That is to say, asymptotically (as all absolute trip demands approach infinity), a SDGSUE(2) mean flow vector is an SUE. This is consistent with the findings of Davis [4] in the context of stochastic process models. Of course, in real networks many interzonal demand levels will be rather small, and so the practical applicability of this result is limited.

6.

Solution algorithm

One of the main advantages of the SDGSUE(2) formulation (14) is that it allows the direct computation of moments of the equilibrium probability distribution, without having to refer to the underlying distribution. Examining (14), it is notable that the first condition has the appearance of an SUE condition (3), and this observation is the motivation for the proposed heuristic solution algorithm. In particular, for given the first condition in (14) is indeed an SUE condition on based on modified link cost functions This leads to the obvious strategy of alternately solving an SUE sub-problem in for given and then updating according to the second SDGSUE(2) condition for the equilibrium route proportions output by the SUE sub-problem. The SUE sub-problem is solved by the method of successive averages (MSA), as described in many standard texts (e.g. [13]). Formally, the algorithm is as follows: Initialisation Set matrix.

to the A-vector of zeroes, and

to the A × A zero

Then, for Auxiliary solution Solve an SUE sub-problem in

conditional on

denoting the solution by and the SUE route proportions by Obtain the corresponding estimate of from:

The pair

is the iteration

auxiliary solution.

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46

Update Estimates Update the SDGSUE(2) estimates according to:

The algorithm is therefore based on two (an ‘inner’ and ‘outer’) MSA updating schemes. The inner iterations are used to solve an SUE sub-problem, conditional on the current estimates of the link flow covariance matrix. The outer iterations use the auxiliary solution from the SUE sub-problem to form an updated estimate of a satisfying the SDGSUE(2) conditions; at any given outer iteration, this estimate is the average of all auxiliary solutions computed to date. By initialising the covariance matrix to zero, the first outer iteration computes a conventional SUE solution (i.e. based on link cost functions This seems a sensible starting point given the asymptotic correspondence, noted in section 5, between SDGSUE(2) mean flows and SUE. In the ‘separable’ cost function case, if each is twice continuously differentiable, is strictly increasing and has a non-decreasing second derivative, then conditional on the modified link cost functions are continuous and monotonically increasing in This, together with some technical conditions on the joint probability distribution of perceptual errors, guarantees the existence of a unique solution to each SUE sub-problem, and the convergence of the MSA algorithm to this solution ([3]; [13]). However, the convergence of the outer iterations is not guaranteed, but if the outer iterations do converge, the resulting estimate will, by construction, be a SDGSUE(2) solution.

7. Numerical tests 7.1. Test networks The algorithm presented in section 6 was implemented in the C language on a personal computer, in which a user-specified number of inner and outer iterations are performed. Previous experience with the algorithm in the deterministic demand case had found it to be an efficient and reliable procedure, provided that a relatively large number of inner iterations were permitted. These are the iterations used to solve each SUE sub-problem. In tests on the network considered below, thirty outer and one hundred inner iterations was found to be a reasonable compromise between computation speed and reproducibility of the results for different random number seeds ([15]). A probit-based choice probability model was used in the tests reported here, implicitly defined by assuming that the link cost perceptual errors were independent between links, the error for each link following a Normal distribution with a mean of zero and a standard deviation where is

Stochastic Network Equilibrium Under Stochastic Demand

47

a link-independent dispersion parameter and is the free-flow travel cost on link A value of was assumed throughout. Actual travel time was assumed to be synonymous with actual travel cost. Separable, BPR-type link cost-flow performance functions were used in all cases, of the form:

where and are user-specified constants, and vehicles/hour is the capacity of link All cases were run for a short time period duration of hours, in order to emphasise the differences (relative to, say, convergence error). A choice of a larger value for would generally cause the differences between SUE and SDGSUE(2) flow rates reported below to decrease. The test network considered represents the Weetwood area of the city of Leeds, a commuting corridor consisting of some 70 zones, 440 links and 174 nodes. The inter-zonal demand matrix represents morning peak period trips, consisting of a total of some 39692 passenger car equivalents per hour. The link-specific BPR powers range from 1.6 to 12.2.

7.2.

Test results

Running the algorithm for 100 inner and 30 outer iterations on the test network required around 45 minutes of run-time on a 120MHz PC. Convergence, as monitored by various flow similarity measures between successive iterations, was achieved with little difficulty in all the tests performed. By construction, any such converged point is a SDGSUE(2) solution, establishing empirically that such solutions do indeed exist. The main results of the tests are summarised in Table 3.1. The definitions of the column headings are as follows: Probability of a randomly selected driver choosing to travel on any given day. Mean total travel time: Mean total travel time across the network

vehicle-hours/hour).

Adjusted SUE: An SUE model in which the final travel times are adjusted to approximate the effect of variability (see below). AAD(flows): Average absolute difference, on a link-by-link basis, between SDGSUE(2) mean flows and SUE flows. AAD(times): As AAD(flows), except for mean travel times. S: A measure of flow variation, simply the unweighted sum across all links of the SDGSUE(2) link flow standard deviations.

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The comparisons in Table 3.1 are made for different values of the travel probability (a common value across all inter-zonal movements), but with the matrix adjusted according to equation (16) so as to maintain a constant mean demand matrix across all tests. In this way, the smaller the value of the higher is the demand variability. The adjusted SUE model is a modified form of SUE in which the flow means and covariance matrix are estimated. The flow means are precisely SUE flows. The covariance matrix is computed by assuming that the SUE route proportions are multinomial probabilities:

where

is given by (15). The adjusted mean travel times are then given by based on (12). In fact, this is equivalent to performing a single iteration of the SDGSUE(2) solution algorithm, i.e. with no iterative feedback of the new expected costs. It is presented for comparative purposes, and is not intended to be a suggested model in its own right. For comparison, the total travel time under a conventional SUE model was estimated as 648.0. The SDGSUE(2) mean total travel time is therefore between 5% and 7% higher than the SUE prediction, the discrepancy increasing with increasing variability in demand. This inflationary effect on mean total travel time of allowing for flow variability is what one would expect under convex link performance relationships, as discussed in section 3. However, it should be clarified that the anticipated inflationary effect typically described in the literature (e.g. [2]) is with respect to a fixed flow probability distribution, whereas in the SDGSUE(2) model the flow probability distribution must effectively equilibrate. In contrast, a comparison of the results for the SUE and Adjusted SUE models is effectively a comparison with respect to a fixed flow probability distribution,

ACKNOWLEDGMENTS

49

and it is notable that the inflationary effect is considerably greater than that predicted by the SDGSUE(2) model. That is to say, in the SDGSUE(2) model drivers are able to compensate for some of the inflationary effect on mean travel times by changing to less “risky" routes. This seems reasonable, since it would be precisely these mean travel times that drivers would be expected to perceive. The Adjusted SUE model, on the other hand, will tend to over-estimate the effect of variability on mean conditions. This is considered to be an empirical justification for the SDGSUE(2) approach, relative to the Adjusted SUE model and, indeed, other approaches based on post hoc modifications of a conventional equilibrium model (see [16] for a review of these latter approaches). Turning attention, then, to a comparison of the SUE and SDGSUE(2) link flow and travel time predictions, table 3.1 indicates hardly any change in the AAD between the mean link flows of the two models as the demand variability is increased. In absolute terms, on a link-by-link basis, the differences in mean flows range from around -95 to +95 vehicles/hour, this range hardly changing with The main effect of is on the variance in link flows, as can be seen from the increase in with increasing variability. This in turn has an effect on the SDGSUE(2) mean travel times, with the discrepancy with SUE travel times (as measured by the AAD) increasing with an increase in demand variability.

8.

Conclusion

It has been demonstrated that stochastic, day-to-day variation in route choice and trip demand can be formulated within the context of an extended network equilibrium framework. The resulting model, which equilibrates link flow means and covariance matrix, is in this way a natural extension of conventional modelling techniques. The heuristic solution algorithm proposed has been shown to be computationally feasible for large realistic networks. Further research in this area can focus on a number of directions. Firstly, more elaborate models may be developed, taking account of factors such as finite driver learning processes, in which the multinomial route flow assumption breaks down, as well as within-day variations in traffic flows and travel choices. Secondly, more numerical simulations should be made; a particularly interesting area is the comparison with stochastic process models. Thirdly, attempts should be made to establish theoretical properties of these new models, such as conditions to guarantee existence and uniqueness of equilibrium and the convergence of solution algorithms. Fourthly, the practical use of higher order moments, such as variances and covariances, may be studied in the context of scheme evaluation; the present paper has focused on mean outputs, but how would knowledge of variances affect decision-making? Fifthly, applications of these techniques should be developed, which seem to have particular relevance to policies that respond to variability, such as driver information systems.

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Acknowledgments This research was carried out under the support of an Advanced Fellowship from the UK Engineering and Physical Sciences Research Council.

References [1] Cantarella G.E. and Cascetta E. (1995). Dynamic Processes and Equilibrium in Transportation Networks: Towards a Unifying Theory. Transpn Sci 29(4), 305-329. [2] Cascetta E. (1989). A stochastic process approach to the analysis of temporal dynamics in transportation networks. Transpn Res B 23B(1), 1-17. [3] Daganzo C.F. (1982). Unconstrained Extremal Formulation of Some Transportation Equilibrium Problems. Transpn Sci 16(3), 332-360. [4] Davis G.A. and Nihan N.L. (1993). Large population approximations of a general stochastic traffic assignment model. Operations Research 41(1), 169-178. [5] Hanson S. and Huff J. (1988). Repetition and day-to-day variability in individual travel patterns. In: Behavioural Modelling in Geography and Planning, ed. by R.C.Golledge and H.Timmermans, Croom Helm, Kent, U.K. [6] Hazelton M. (1998). Some remarks on stochastic user equilibrium. Transpn Res 32B(2), 101-108. [7] Hazelton M. and Watling D.P. (1999). Approximation methods for overdispersion and learning processes in markov models of route choice. In preparation. [8] Mohammadi R. (1997). Journey time variability in the London area. Traffic Engineering and Control 38(5), 250-257. [9] Montgomery F.O. and May A.D. (1987). Factors affecting travel times on urban radial routes. Traffic Engineering and Control, September 1987, 452-458. [10] Ortuzar J. and Willumsen L.G. (1994). Modelling Transport. Second edition. John Wiley and Sons, Chichester, U.K. [11] Ran B. and Boyce D.E. (1993). Dynamic Urban Transportation Network Models. Springer-Verlag, Berlin. [12] Rathi A.K. (1992). The use of common random numbers to reduce the variance in network simulation of traffic. Transpn Res 26B, 357-363. [13] Sheffi Y. (1985). Urban transportation networks. Prentice-Hall, New Jersey.

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[14] To D.K.B. (1990). What happens when it rains. M.Sc. thesis, Institute for Transport Studies, University of Leeds, U.K. [15] Watling D.P. (1999a). A Second Order Stochastic Network Equilibrium Model. First revision, submitted to Transpn Sci. [16] Watling D.P. (1999b). Traffic Assignment with Stochastic Flows and the Estimation of Travel Time Reliability. Paper presented at Second Workshop on Network Reliability, July 27th 1999, Newcastle, U.K. [17] Willumsen L. and Hounsell N.B. (1994). Simple models of highway reliability: supply effects. Paper presented at Seventh Int Conf on Travel Behaviour Research, Santiago, Chile, June 1994.

Chapter 4 STOCHASTIC ASSIGNMENT WITH GAMMIT PATH CHOICE MODELS Giulio Erberto Cantarella [email protected] Dept of Comp. Sci., Math., Electr., Transport.- Univ. of Reggio Calabria - Italy tel: +39-0965-875227, fax: +39-0965-875297

Mario Giuseppe Binetti [email protected] Dept. of Highways and Transportation - Politecnico di Bari - Italy tel: +39-080-5460485, fax: +39-080-5460329

Abstract

Traffic assignment models simulate transportation systems, where flows resulting from user choice behaviour are affected by transportation costs, and costs may be affected by flows due to congestion. Several path choice behaviour models can be specified through random utility theory. Probabilistic path choice models, where perceived path costs are modelled as random variables, lead to stochastic assignment. In this paper, reasonable modelling requirements are proposed to assure a realistic simulation ofpath choice behaviour through probabilistic choice models. Then, additive Gammit path choice models based on Gamma distribution are introduced and deeply analysed. These models satisfy all the proposed modelling requirements, and can be effectively embedded within existing models and algorithms for stochastic assignment.

Keywords:

Stochastic assignment, gammit choice model, gath choice, stochastic user equilibrium, stochastic network loading.

1.

Introduction

Traffic assignment simulate transportation systems, where flows resulting from user choice behaviour are affected by transportation costs, and costs may be affected by flows due to congestion. Three sub-models make up an assignment model: 53

M. Patriksson and M. Labbé (eds.), Transportation Planning, 53–67. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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supply model which simulates (whether and) how user behaviour affects network performances, such as travel times, etc; demand model which simulates how network performances affects user behaviour; supply-demand interaction model which simulates their interaction. Transportation supply is usually simulated through a congested network model. In this case, demand-supply interaction is mostly simulated through equilibrium models. These models search for mutually consistent flows and costs, defining a state in which no user can reduce the (perceived) cost of his/her choice by unilaterally changing it (reviews and references in [13]; [12]; [6]). Recently, dynamic process models have been proposed, which generalise the equilibrium approach (see [5] for a general framework and a review). If the user behaviour do not affect network performances, transportation supply is simulated through a simpler non-congested network model, leading to network loading models, which also play a relevant role in the formulation and solution of assignment models for congested networks. Generally,the user choice behaviour, simulated by the demand model, refers to several choice dimensions, such as destination, mode, path, etc. In this paper, for simplicity sake, only path choice behaviour will be explicitly dealt with, assuming that other choice dimensions, like mode and destination, are not affected by (congested) network performances. Several path choice behaviour models can be specified through random utility theory. Probabilistic path choice models, where perceived path costs are modelled as random variables lead to stochastic user equilibrium (SUE) effectively formulated through fixed-point models ([8]; [4]). This paper explicitly deal with (multi-user) mono-modal equilibrium assignment with rigid demand (briefly reviewed in section 2 for unfamiliar readers). Extensions to multimodal assignment, and/or elastic demand, as well as pre-trip/en-route path choice behaviour (relevant for urban transit systems) are quite straightforward as discussed in [4]. Cantarella and Cascetta [5] discuss extensions to dynamic process assignment. This paper, after presenting notations and the necessary background in section 2, first provides a general framework to path choice behaviour modelling (subsections 3.1-3), embedding it within stochastic assignment theory. In particular, reasonable modelling requirements are proposed to assure a realistic simulation of path choice behaviour through probabilistic choice models, as well as link-based and additive path choice models. Then, Gammit path choice models, based on Gamma distribution, are analysed in subsection 3.4; the use of Gamma distribution has been suggested by other authors ([13]; [2]; [11]), but resulting choice models have never been analysed. Whilst additive Gammit

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model fulfils all the modelling requirements, this is never the case for other existing models. Gammit choice models can be embedded within existing models and algorithms for stochastic assignment and applied to real-size networks, as shown in section 4.

2.

Review of stochastic assignment

A transportation system is generally analysed by concentrating origins and destinations of journeys into centroids, and grouping users into user classes. Each user class, denoted by a single index such as is a set of users travelling between an origin-destination pair with a common set of relevant paths and common behavioural parameters. Throughout this paper, it is assumed that at least one path connects each O-D pair, that is each set is non-empty. Moreover, only elementary (say loop-free) paths are considered (for further comments on this issue see section 3.1), thus only a finite number of paths exists for each user class, that is each set is finite. It seems worth noting that a non-elementary path surely contain an elementary path, whilst an elementary path may not contain another elementary path.

2.1.

Basic notations and definitions

Transportation supply is usually simulated through a congested network model, which expresses how user behaviour affects network performances. Let be the link-path incidence matrix for user class with entries link a belongs to path and otherwise; be the path flow vector for user class

if

with entries

be the link flow vector, with entries be the link cost vector, with non negative entries by travel time); be the path cost vector for user class with entries negative,

(for example given

assumed non

The path flow propagation model defines the relation between path and link flows:

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Moreover, link flows depend on link costs through link cost functions, which can be specified through traffic engineering models:

The path cost model defines the relation between link and path costs:

The transportation demand model simulates how network performances affects user behaviour. As already stated in section 1, for simplicity’s sake, only path choice behaviour is explicitly dealt with. Let be the demand flow for users belonging to class b e the path choice probability vector class with entries

and

for user

The path choice model simulates user path choice behaviour through a relation between path flows and costs:

The demand flow conservation equation assures that the sum of path flows is equal to the demand flow for each user class:

2.2.

Path choice models from random utility theory

Most path choice models are specified through random utility theory (introduced by Domencich and McFadden [9]; for a comprehensive review see [1]). It is assumed that each user within class • examines all paths in the (non empty and finite) set •

associates to each path within set a perceived utility modelled through a random variable, due to several causes, such as aggregation errors, fluctuations of attributes, missing attributes, dispersion of user behaviour, user perception errors, etc;

• chooses the maximum utility path. According to the above assumptions the probability, that a user of class chooses path within set is given by the probability that path is the maximum perceived utility path. The perceived utility (p.u.), of path for a user of class can be expressed by the sum of its expected value

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Stochastic assignment with gammit path choice models

the systematic utility (s.u.), plus a random residual (r.r.) Thus, each random residual has zero mean, and variance equal to the p.u. variance The above assumption yield:

Hence, the path choice probabilities depend on the expected values of perceived utilities, and of the distribution parameters as well. By using vectorial notations, let be the path perceived utility vector for user class

with entries

be the path systematic utility vector for user class

with entries

be the path random residual vector for user class

with entries

be the (symmetric positive semi-definite) path covariance matrix for user class relative to p.u. or r.r. vectors. Assuming that costs are measured in the same units of utility, systematic utility is usually assumed equal to the opposite of cost (plus other attributes not explicitly introduced for simplifying notations):

The relationship between the path choice probability vector and the vector of systematic utility is called the path choice map:

The path choice map, combined with the utilityfunction between systematic utility and cost allows specifying the path choice model (2.4). In particular, probabilistic path choice models are obtained when path covariance matrix is non singular, In this case any path may be used, and the resulting relation between path costs and choice probabilities is a function (otherwise it may be a point-to-set map), which also depends on parameters of p.u. or r.r. distribution. Probabilistic path choice models will be discussed in details in the next section 3.

2.3.

Models and algorithms for stochastic assignment

The application of probabilistic choice models leads to stochastic assignment. In particular, if link costs do not depend on link flows (non congested networks) the resulting stochastic assignment model can be described by

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the stochastic network loading (SNL) function obtained combining equations (2.1),(2.3),(2.4)and(2.5):

where is the feasible link flow set, non empty (if at least one path is available to each user), compact (since bounded and closed), and convex. If link costs depend on link flows (congested networks) stochastic assignment is usually based on the equilibrium approach, searching for consistency between flows and costs. Stochastic user equilibrium (SUE) can be effectively analysed through fixed-point models obtained combining the SNL function (2.8) with the link cost flow functions (2.2), such as the model proposed by Daganzo [8]:

Other fixed-point models are described in [4]. Existence of stochastic user equilibrium flow or cost pattern can be stated through the Brouwer theorem, mainly requiring that SNL function and link cost functions are continuous, since set is non-empty, compact and convex, and condition is always assured. Uniqueness of SUE flow or cost pattern mainly requires that the SNL function is non-increasing (or quasi strictly decreasing) monotone and link cost functions are strictly increasing (non-decreasing) monotone. If existence and uniqueness conditions hold, stochastic user equilibrium link flow pattern can be found through two algorithms which are simple, since their application only requires the computation of SNL and cost functions, and feasible, since they produce a sequence of feasible link flow vectors. The Flowaveraging algorithm (MSA-FA), whose convergence is assured if the Jacobian of link cost functions is symmetric, is described by the recursive equations below ([8]; and [4]):

The Cost-averaging algorithm (MSA-CA), whose convergence is assured if the Jacobian of SNL functions is symmetric, is described by the recursive equations

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below, cf. [4]:

The MSA-CA algorithm can be proved converging also if Jacobian of link cost functions is not symmetric, but its convergence is generally slower than the MSA-FA algorithm. Thus, two-stage algorithms should be preferred, where the starting solution, is obtained through MSA-FA scheme and final steps are performed through MSA-CA scheme. Multi-stages algorithms are also useful to prevent step becoming too small.

3.

Probabilistic path choice models

Different probabilistic path choice models can be specified according to different assumptions on the joint probability density function of perceived utilities or random residuals, as discussed in the rest of this section. In particular, requirements introduced in the following subsection are useful to analyse different specifications of the path choice models (some of them are also in [2]; [11]).

3.1.

Requirements for probabilistic path choice models

Mathematical requirements, presented below with reference to path choice behaviour, are useful to effectively model any choice behaviour. In particular, the path choice model (2.4), is specified by a function if perceived utility values and random residuals, as said before, are assumed distributed as continuous random variables with non singular covariance matrix, Continuity of the path choice function, assures that small changes of path costs induce small changes of choice probabilities. If it is also continuously differentiable it has a continuous Jacobian, This feature, assured by commonly used joint probability density functions, guarantees continuity of the resulting SNL function, thus it is useful to state existence of stochastic user equilibrium. Monotonicity of the path choice function, assures that an increase of the cost of a path induces a decrease of the corresponding choice probability. More generally, the path choice function, should be non-increasing monotone with respect to path costs. This feature guarantees monotonicity of the resulting SNL function, thus it is useful to state uniqueness of solutions of stochastic user equilibrium. Independence from linear transformations of utility assures that any change of the scale of the utility does not affect the model (as guaranteed by Normal or Gamma distribution).

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Beside the above presented mathematical requirements, modelling requirements, presented below, are useful to effectively simulate path choice behaviour. Similarity of perception of partially overlapping paths rules out counterintuitive results. Indeed two partially overlapping paths are likely not perceived as two totally separated paths. Introducing a positive co-variance between any two overlapping paths can simulate similarity. Independence from link segmentation (within the network model) assures that if a link is further divided into sub-links redefining link costs so that path costs are not affected, path p.u. or r.r. distribution is not affected too, and thus choice probabilities. This requirement makes reference to features of the distribution of the sum of random variables. Further considerations are reported in subsection 3.3. Negativity of perceived utility assures that no user perceives a positive utility to travel along any path. This feature can be assured by assuming lower bounded random distributions (for instance Log-Normal, or Gamma). According to this feature a non-elementary path is always a worse choice than the elementary path within it, thus supporting the assumption of considering elementary paths only (section 2). On the other hand, if this feature is not presented, a nonelementary path may be a better choice than the elementary path within it. Hence, non-elementary paths should be included within the path choice set (which is non longer finite), possibly leading to unrealistic situations (a part from some algorithmic drawbacks).

3.2.

Path versus link formulations of probabilistic path choice models

In path or direct formulations of probabilistic path choice models the distribution of path perceived utility or random residuals, introduced in subsection 2.2, is explicitly specified. The analysis of path choice behaviour should be carried out at the path level, but path p.u. or r.r. distribution can be specified by introducing link perceived utilities and specifying their distribution, using link or indirect formulations. Let be the link perceived utility vector, with entries equal to the opposite of link cost vector,

and expected value

be the link r.r. vector, with entries with null expected value, and variance equal to the variance of link perceived utility, be the (symmetric positive semi-definite) link (p.u. or r.r.) covariance matrix. Path perceived utilities and random residuals, introduced in section 2., are assumed given by the sum of the corresponding link perceived utilities and

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random residuals:

Hence, link p.u. or r.r. distribution specifies path perceived utility or random residuals distribution with path covariance matrix given by:

According to equation (3.3), even if link p.u.’s and r.r.’s are independently distributed, that is covariance matrix is diagonal, the perceived utilities, and random residuals, of any two overlapping paths have a positive covariance, given by the sum of variance of the perceived utilities, or random residuals, of common links. Thus, through link formulated choice models similarity of perception of paths partially overlapping can be easily simulated. The independence from link segmentation mainly requires the use of reproductive random variables. For instance, the sum of several independently distributed Normal random variables is still a Normal random variable, with mean given by the sum of the means and variance by the sum of the variances. This feature is also shown by independently distributed Gamma random variables with the same variance to mean ratio. In both case if links in a path are further segmented, provided that the mean path cost and the variance are not affected, the resulting path perceived utility distribution is not affected, and then the choice probabilities. Generally, the SNL function with any link-formulated choice model can be easily computed, if link perceived utilities are assumed independently distributed, through MonteCarlo techniques (introduced by Burrell [3]; see also [13]). In this case the result is an unbiased estimate of obtained by averaging several shortest path loading corresponding to different pseudo-realisations of with If all elementary paths are considered, explicit path enumeration can be avoided by using anyone of the many shortest path algorithms. (It is worth noting that effective shortest path algorithms require that link costs are non-negative, paths containing negative cost loops.)

3.3.

Additive probabilistic path choice models

A particular class of choice models are additive models, which are (continuous and continuous differentiable) probabilistic choice models where the p.u. or r.r. distribution does not depend on the path costs (neither set ). In other worlds the parameters of p.u. or r.r. distribution, such covariance matrix, do not vary with the path costs These models can be easily specified assuming, for instance, that parameters of p.u. or r.r. distribution do not depend on costs at all or depend on zero-flow costs. Choice probabilities resulting from additive

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choice models only depend on differences between path costs, rather than their absolute values. Some empirical results seem indicating that these assumptions on user choice behaviour are quite realistic (see [2]). Additive probabilistic path choice models, are specified by non increasing monotone functions with a symmetric (negative semi-definite) Jacobian (as shown by Cantarella [4], starting from results in [7]). Strictly positive additive models are specified by quasi strictly monotone functions ( [4]). Thus, these models lead to continuous and monotone SNL functions (with symmetric Jacobian) features useful to guarantee existence and uniqueness of stochastic user equilibrium (and convergence of MSA algorithm). Additive path-formulated choice models always assures independence from link segmentation, whilst additive link-formulated choice models have this feature only for reproductive random variables (for instance Normal or Gamma variables). From all results presented above, additive choice models seem appealing both from theoretical and practical points-of-view. However, additivity is only a sufficient condition to assure relevant features, whilst characteristics of non-additive choice models are still an open issue.

3.4.

The Gammit choice model

In this subsection, choice models based on Gamma distribution are introduced and deeply investigated. Nielsen [11], starting from an earlier suggestion by Sheffi [13], also considered Gamma distribution without analysing it in details (see also [2]). The Gammit model is obtained assuming that perceived disutilities are jointly distributed as a non negative "shifted" MultiVariate Gamma, with mean equal to the path costs and path covariance matrix Gammit path choice probabilities cannot be expressed in a closed form, and their computation requires MonteCarlo techniques. The Gammit model allows taking into account similar perception of paths partially overlapping through covariance between paths. Path-formulations of Gammit model may be undetermined, since the Covariance matrix alone does not completely define the joint probability density function of a MultiVariate Gamma random variable. On the other hand, the link formulation described below allows overcoming this drawback, and at the same time assures independence of link segmentation and rule out positive perceived utilities for any paths. Let be the (strictly positive) reference cost on link a, assumed not greater than the link cost, say The perceived disutility of link a, is assumed distributed (independently of the perceived disutility of any other link) as a non negative shifted Gamma variable with mean given by the link cost, variance by proportional to (strictly positive) reference

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63

link costs and shifting factor given by the difference between link cost and reference cost,

In other words, the link perceived disutility is the sum of a non-negative deterministic term possibly depending on link flows, and a non-negative stochastic term independent from link flows. The assumption on link reference costs yields that the corresponding reference cost on path is strictly positive and not greater than the path cost, Thus, the perceived disutility on path k is marginally distributed as a non negative shifted Gamma variable:

Path perceived disutility vector is distributed as a non negative shifted Multi Variate Gamma, with covariance matrix since the link covariance matrix is given by In this case independence from link segmentation is assured. In fact, let be the reference path cost vector, with (strictly positive) entries the variance for path is given by and the covariance between paths and is given by where is the costs of links shared by paths and The described specification leads to an additive choice model if the variance parameter and reference link costs do not vary with link costs (for instance are zero-flow costs), and the resulting additive Gammit path choice model fulfils all the requirements proposed in subsection 3.1. Aggregate calibration parameter is briefly addressed in section 4, but a detailed analysis of this issue is out of the scope of this paper.

4.

Numerical examples

The main aim of this section is showing that stochastic assignment with Gammit path choice model can be applied to a real size network, thus results of the application as such will only briefly discussed. In particular, the O-D matrix, obtained through calibration of demand models and/or estimation from counts, may be worth of further analysis, as well as the parameters within link cost functions. The graph simulates road facilities of the city of Salerno (Italy), with roughly 200,000 inhabitants, it contains 685 nodes, including 89 centroids, and 1147 links. Traffic counts are available for 65 links. The number of relevant O-D pairs is, in the morning peak hour, 3844 out of 7921 (89x89); whilst the total demand flow is 23,825 vehicles/h. Path choice behaviour has been modelled through a link-formulated additive Gammit model with zeroflow costs as reference costs. Separable link cost functions have been used.

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65

The SNL function has been computed through MonteCarlo techniques, with 50 pseudo-random realisations of link random residuals, generated through analytical approximations, using the same draws for all the O-D pairs. Stochastic user equilibrium has been solved through the MSA-FA algorithm, with convergence error 0.10. The total number of shortest-path loading steps can be reduced through two-stage algorithms, if during the first stage the SNL is computed with only one pseudo-realisation of costs. Figure 4.1 shows the (relative absolute) maximum and the (relative absolute) average difference between computed and observed link flows against variance parameter Whilst the average difference seems only slightly affected by the value of this parameter, the maximum difference gets its minimum for This quite strange result may depend on O-D matrix estimate (and possibly the link cost functions calibration). The joint calibration of the variance parameter together with the update of O-D matrix will be addressed in future work.

Figure 4.2 shows a detailed comparison between observed and computed link flows, relative to the best fit variance parameter. Generally, observed link flows are well reproduced. Figure 4.3 shows a comparison between Gammit and Probit-based assignment. It can be easily recognised that the two models provide close values.

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different from the well-established Probit model. Anyhow, it is worth noting that the use of Probit model requires some care to avoid negative perceived costs, which may causes troubles to standard shortest path algorithms. Some issues seem worth of further research work. First of all the efficiency of stochastic network loading algorithms based on the proposed choice models strongly relies on draws from a Gamma random variable. Thus, more efficient algorithms not based on Monte Carlo methods could be looked for (following an approach similar to the numerical approximation proposed by Maher and Hughes [10] for Probit-based assignment). Furthermore, the calibration of the variance parameter, possibly jointly with the update of the O-D matrix, needs to be addressed. More generally, features of non-additive path choice models, such as Gammit (or Probit) models with flow dependent reference costs, are still an open issue.

Acknowledgments Authors wish to thank an anonymous referee and the editor who rose several relevant issues helpful to improve the paper.

References

REFERENCES

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[1] Ben Akiva, M. and Lerman, S. R. (1987). Discrete Choice Analysis. MIT Press, Cambridge MA. [2] Bovy, P. H. L. and Stern, E. (1990). Route Choice: Wayfinding In Transport Networks. Kluwer Acad. Pub, Dordrecht, The Netherlands. [3]

Burrell, J. E. (1968). Multiple route assignment and its application to capacity restraint. In Leutzbach, W. and Baron, P., editors, Proceedings of 4th International Symposium on the Theory of Road Traffic Flow. Karlsruhe, Germany. [4] Cantarella, G. E. (1997). A General Fixed-Point Approach to MultiModde Multi-User Equilibrium Assignment with Elastic Demand. Transportation Science, 31:107–128. [5]

Cantarella, G. E. and Cascetta, E. (1995). Dynamic Processes and Equilibrium in Transportation Networks: Towards a Unifying Theory. Transportation Science, 9:305–329.

[6]

Cascetta, E. (1998). Teoria trasporto. UTET, Torino, Italy.

[7]

Daganzo, C. F. (1979). Multinomial Probit: The Theory and Its Application to Demand Forecasting. Academic Press, New York, NY.

[8]

Daganzo, C. F. (1983). Stochastic Network Equilibrium with Multiple Vehicle Types and Asymmetric, Indefinite Link Cost Jacobians. Transportation Science, 17:282–300.

metodi dell’ingegneria dei sistemi di

[9]

Domencich, T. A. and McFadden, D. (1975). Urban Travel Demand: Behavioral Analysis. North Holland, Amsterdam. [10] Maher, M. J. and Hughes, P. C. (1997). A Probit-Based Stochastic User Equilibrium Assignment Model. Transportation Research -B, 31B:341– 355. [11] Nielsen, O. A. (1997). On the distributions of the stochastic components in SUE traffic assignment models. In Proceedings of 25th European Transport Forum Annual Meeting,Seminar F, pages 77–93. [12] Patriksson, M. (1994). The Traffic Assignment Problem: Model and Methods. VSP, Utrecht, The Netherlands. [13] Sheffi, Y. (1985). Urban Transportation Networks. Prentice Hall, Englewood Cliffs, NJ.

Chapter 5 ESTIMATION OF TRAVEL TIME RELIABILITY USING STOCHASTIC USER EQUILIBRIUM ASSIGNMENT SENSITIVITY Chris Cassir [email protected]

Michael G.H. Bell Transport Operation Research Group University of Newcastle-upon-Tyne Newcastle, UK [email protected]

Abstract

1.

This paper presents a methodology for evaluating the reliability of transportation networks, which could be used to support the design of networks that are robust to everyday disturbances, in the sense that an acceptable level of network performance will normally be maintained. While tools already exist to determine the expected benefits of travel demand management or new infrastructure, tools have yet to be developed which take into account unlikely disbenefits arising from disturbances (like gridlock, to take a dramatic example). This paper focuses on the performance reliability of transportation networks in the face of normal variations .It is proposed to use a logit Stochastic User Equilibrium assignment model for obtaining reliability measures related to travel times. It is shown that logit SUE sensitivity expressions can be computed and applied in order to estimate travel time distributions. Computational results are also discussed.

Introduction

The efficient functioning of any society depends critically on networks of various kinds, such as water, electricity, gas, sewage, communication and of course transportation. The importance of the transportation network is perhaps best appreciated when it is severely disrupted, for example by an earthquake. 69

M. Patriksson and M. Labbé (eds.), Transportation Planning, 69–84. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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The potential sources of disruption to transportation networks are numerous, ranging at one extreme from natural or man-made disasters (like earthquakes, floods, landslides, terrorist attacks, mining subsidence, bridge or tunnel collapses, and major accidents), which tend to occur rather infrequently, to at the other extreme events (like congestion, road maintenance, badly parked vehicles, and minor collisions), which occur on a daily basis. The scale, impact, frequency and predictability of such events will of course vary enormously. While little can be done about their scale, frequency or predictability, particularly where natural disasters are concerned, it should be possible to design transportation networks so as to minimise the disruption such events can cause. While many tools exist for studying the impact of new transport infrastructure or travel demand management measures on traffic flows (for example, the widely-used CONTRAM and SATURN programs), there are no tools for assessing the impact of such measures on the reliability of transportation networks. As an example, the reallocation of road space from private to public transport through the creation of a bus lane will on average provide benefits for public transport which may be assessed using existing tools. However, the bus lane may also increase the risk of queues of cars blocking upstream junctions (referred to as blocking back), which may exceptionally cause severe disruption to public transport services. To make a good judgement, the decision-maker should set unlikely disbenefits against likely benefits in a way that duly reflects priorities and risk adversity. Network reliability has two dimensions. The first relates to the connectivity reliability of a network. When links fail in unfavourable configurations it may no longer be possible to reach a given destination from a given origin, in which case the network becomes disconnected. However, even a connected network may fail to provide an adequate level of service. For example, random events may for a given network cause unacceptable variation in origin-to-destination travel times, making it difficult for travellers to arrive at their destinations on schedule. The second dimension of reliability is therefore the performance reliability of a network. Previous work on network reliability (Du and Nicholson, 1993; Iida and Wakayabashi, 1989) has focused principally on connectivity in degradable transportation networks, and the field of performance reliability in normal conditions appears to be under-researched. Assuming OD flows are normally distributed with known parameters, Asakura and Kashiwadani (1991) solved a static User Equilibrium (UE) assignment problem several times, with a demand sampled from these normal distributions, in order to estimate OD travel time distributions, from which the probability of reaching one’s destination within some acceptable time can be estimated, thereby providing some measures of network performance reliability.

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71

The study in this paper attempts to achieve the same latter objective, namely estimating some travel time distributions in normal conditions, but it proposes to use sensitivity analysis of a logit Stochastic User Equilibrium (SUE) assignment model, instead of simulation. This approach would have the potential advantage of delivering reproducible results in reasonable computational time. The principles of the method are based on approximating linearly the relationship between equilibrium OD or route travel times and exogenous factors such as OD trips or link capacities, which in normal conditions are subject to fluctuations. The factors in this linear approximation can be obtained by deriving sensitivity expressions for the equilibrium assignment model. Tobin and Friesz (1988) obtained expressions for the sensitivity of deterministic user equilibrium (DUE) link flows to perturbations in origin-destination demands and link travel times, however because DUE route flows are not unique, they had to determine one route flow solution among the many in order to carry out the sensitivity analysis. SUE assignment presents the advantage of providing one unique route flow solution which is essential for sensitivity analysis, and the logit version of SUE offers the advantage of providing a tractable analytical solution. Taking Fisk (1980) route based formulation as the non-linear program corresponding to a logit SUE problem, Bell and Iida (1997) presented sensitivity expressions for the equilibrium route flows, in response to perturbations in origin-destination flows and link travel times. The route flow sensitivities can then be used to evaluate link flow, link travel time and route travel time sensitivities. However their method requires inverting matrices whose dimensions are determined by the number of routes defined in the network. Since this number can be quite large in large realistic networks, this may make computation cumbersome. A more economical approach would be to reduce the dimension of the system for sensitivity analysis by deriving the logit SUE link flow sensitivity expressions directly, and subsequently using those to calculate link travel time and route travel time sensitivities. It will be shown below that such an approach is possible with SUE, provided however that the route flows are explicitly defined in the base solution, around which sensitivity analysis is being carried out. That is a reason which makes route-based formulations for SUE particularly attractive in the context of sensitivity analysis. This paper will first present the general mathematical formulation of a route based logit SUE model. It will then be shown how sensitivity expressions for SUE link flows are derived , along with link ,route and minimum expected OD travel times sensitivities, and how those sensitivity expressions can be used to estimate variances of travel time distributions. Finally, results for a small network will be presented and discussed.

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2.

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Logit SUE model

Given a vector of OD flows a SUE solution yields route flows and route travel times consistent with each other by virtue of the equilibrium principle, while the equilibrium assignment of flows to routes is governed by the logit route choice model as follows, for any route flow between an OD pair

with

the route choice proportion factor given by

where denotes the set of routes between OD pair is the logit dispersion parameter and a vector of link saturation flows.The relationship between route costs and route flows occurs because route travel times are the summation of link travel times and that the travel time on a particular link depends on the flow going through that link and on its saturation flow, via some monotonically increasing function. Using the linear relationship between link flows and route flows where A is the link-route incidence matrix (and denotes the transpose of A), we have: with The SUE solution can be interpreted as the mean value of a network equilibrium where users minimise their perceived travel times. Due to random errors in travel time perception, the model is stochastic and therefore the estimated flows and travel times are also random variables. However the mean values provided by the SUE are meant to provide a reasonable estimation of average network traffic conditions on a particular period of the day (like peak hour) as it allows for some non homogeneous driver behaviour in route choice (as opposed to DUE) while taking congestion effects into account. We shall hereafter refer to the ‘base solution’ as being the logit SUE solution obtained for a given vector of OD flows considered to be the average trip table, and the vector of average link saturation flows. From this base we can then look at the effects of random variations in both demand and saturation flows on route travel times and expected minimum OD travel times necessary for calculating reliability measures.

3.

Logit SUE sensitivity analysis

We start this section by denoting the number of OD pairs by by the number of links in the network, and the number of routes defined in the

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network.The logit SUE solution for the link flows satisfy the following system of non-linear equations:

where represents the probability choice matrix, whose elements are the logit model continuous functions of route cost and are defined by: if route j belongs to OD pair otherwise. This system of non-linear equations can be rewritten in the following form:

Each elements of represent the differences between some given link flows and the link flows that would result from a logit assignment with costs depending on the given flows. At equilibrium these quantities should be equal to zero.

3.1.

Sensitivity of SUE link flows

Suppose, without loss of generality, that we are interested in variations of the trip table only. We can then drop as a variable and look at the following system of non-linear equations:

For a given fixed it is known that this fixed point problem has a unique solution if the Jacobian of link travel times is a positive-definite matrix. This is a sufficient condition for a unique SUE solution (see Sheffi ,1985) and is satisfied if the link cost functions are strictly increasing with link flows. Suppose the trip table is perturbed in an infinitesimal way , with an vector of unity, and an infinitesimal small positive number.We then want to find the new corresponding SUE solution, that will solve the perturbed system:

By differentiation, we obtain:

where

is the

matrix with elements

matrix with elements

and

is the

Taking one perturbation at a time, we have, for

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with yields:

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the

th column of

If we take the limit

exists, that is provided

is non-singular.

Lemma 1 is non-singular for all link travel times is a positive-definite matrix. Proof For all links

equation (3.1)

if the Jacobian of

we have:

W represents the set of OD pairs and represents some element the link-route incidence matrix for OD pair So for the derivatives of are given by:

of and

using

since for a route belonging to OD pair equation (3.2) can be written as:

In matrix notation

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Estimation of travel time reliability

where is the Jacobian of the vector of route-choice proportion for OD pair with respect to route travel times. Now to prove that is non-singular and therefore invertible, we first note that being positive definite is also non-singular.Also being the choice probability Jacobian is also the Hessian of the expected minimum travel time for OD pair

Since this function is concave with respect to travel times it follows that is negative semi-definite, or equivalently – is positive semi-definite. We then notice that is also positive semi-definite, since it is a symmetric matrix with semi positive elements (each element results from a quadratic form with a positive semi-definite matrix). Therefore, as a sum of positive semi-definite matrices factored by a positive quantity is also positive semi-definite. If we then multiply from the right by we get:

The first term on the right is positive definite, and the second term is semipositive, since it is a symmetric matrix with semi positive elements, each resulting from a quadratic form with a positive semi-definite matrix. Adding a positive definite matrix with a positive semi-definite one gives a positive definite matrix, therefore is positive definite, which implies that it is also non-singular. being non-singular means that has to be non-singular. If we regroup all the vectors together in a matrix, we obtain the following matrix of sensitivity expressions of SUE link flows with respect to perturbations of the trip table:

Likewise, the sensitivity expressions of SUE link flows with respect to perturbations of link capacities are given by:

where

is the Jacobian of

with respect to

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3.2.

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Sensitivity of route and expected minimum OD travel times

Sensitivity of route travel times, between each OD pair can then be easily computed via the Jacobian of link travel times and the route-link incidence matrix using the chain rule of differentiation:

Overall we can see that the main difficulty in computing these expressions, consists in calculating the inverse of the (number of links squared) matrix necessary for the SUE link flow sensitivities. This is much more manageable in general than having to do the same type of inversion operation for a route-based matrix. However, the route structure still needs to be retained for calculating as can be seen from the proof of the lemma. If we assume that there is no re-routing induced by the fluctuations of demand and saturation flow, then it is no longer appropriate to consider variations of flow due to re-routing. Therefore the SUE sensitivities and are no longer required. Instead we can look at distributions of route travel times around the equilibrium caused by the demand- and supply-side perturbations only (the route choice proportions remain fixed). The following gradients then reflect the route travel time sensitivities:

Another useful measure is the sensitivity of expected minimum OD travel time vector defined as follows according to the logit model, for Since the logit route-choice proportion, the sensitivities of are given by:

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4.

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Approximation of travel times variances

By making the following first order approximation of the relationship between equilibrium route/OD travel times and OD flows/capacity, around a base solution

and by assuming some independent normal random distributions for each elements of and around and we can obtain the following approximate variance and covariance expressions for route travel times and OD expected minimum travel times.

5.

Example

We were able to use the expressions derived in the previous sections for a fairly large network (8000 OD pairs, 2000 links) of York and obtained the estimates of route travel time variances in about 8 hours, which is a considerable improvement on the method of finite differences reported in Bell et al (1999). However, to illustrate the type of results obtained, the methodology was applied to a much smaller network, thereby allowing a better overview of the results to be presented in this paper.

5.1.

The network

The network considered here is a small part of an urban network in Leicester, England. It consists of 103 links (including micro-links at junctions), 9 origins and 9 destinations. A representation of the network is shown in Fig. 5.1. The topological data (including signal timings), along with a trip table for a peak period were made available for a Phd project on OD estimation based on SCOOT traffic counts. We did not use the link detector data here.

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5.2.

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Computation of time variances and reliability measures with respect to OD flow fluctuations

We used our route-based SUE model, the Path Flow Estimator (PFE), on the Leicester network with the trip table available for the afternoon peak-hour to obtain a base solution, and applied the sensitivity expressions defined in section 2, with and without route response, in order to compute first-order estimates of variances for route travel times and expected minimum OD travel times, with respect to OD flow fluctuations only. For lack of information on OD flow fluctuations, we assumed normal distributions of OD flows centred around the trip table values and chose arbitrary standard deviations equal to However, since some of the OD flows were larger than 100 veh/hr, we put a limit of 30 veh/hr on the standard deviation. This was to avoid very large deviations, which given the first order approximation used in the method, could have lead to unrealistically large variances of travel times. Note that a deviation of 30 veh/hr is not strictly speaking small enough to warrant a valid use of the first order approximation for the relation between the equilibrium solution and the OD flows. However it was conjectured that for this moderately congested network, the second-order derivatives of the equilibrium flows would be sufficiently small so as to make the second order terms relatively negligible. This conjecture is based on the network structure (it is a relatively small network) and on the Kimber and Hollis (1979) link-cost function that is used in our model: the first and second-derivatives of this cost function yield very small values in the base case, and since these terms feature prominently in the second-order approximation terms for route travel times, it appears that, with a maximum deviation of 30 veh/hr for OD flows, an upper bound for the second order terms might be found to be about one minute, for all route and OD travel times, which would be an acceptable error for a crude approximation. A rigorous study about the acceptability of the first-order approximation, which depends on the network, the base demand and the link cost functions employed in the model, would however be necessary in the future. 5.2.1 OD expected minimum time variances. Table 5.1 shows the standard deviation, with and without SUE re-routing, of the expected minimum OD travel times, for the 20 most unreliable OD pairs. We can see that, for all OD pairs, the standard deviation is greater, and thus the reliability lower, when equilibrium route response is not taken into account. This suggests that the effects of drivers re-routing is to lessen to impacts of variations. We can see that all OD pairs connecting DES_B have a substantially higher variability in travel time compared to all other pairs. A quick look at the network shows why: to get to DES_B, it is impossible to avoid either link 215L or link 215K. These two

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links happen to be the only over-saturated links in the whole network in the base solution. (The same observation could be said for DES_A on the evidence of Fig. 5.1; however in reality link 215K is divided into 2 sub-links, and it is only the right-turn sub-link leading to DES_B that is actually over-saturated.). This evidence points to a correlation between congestion in the average equilibrium situation and high variations in travel times.

5.2.2 Route travel time variances and reliability. The normal distributions for the input OD flows mean that the output travel times are also normally distributed, because of the linear approximation of the relationship between OD flows and route travel times. We could then obtain a measure of reliability for routes by calculating the probability that a route travel time is less than some performance threshold. The travel time threshold for reliability, was arbitrarily taken at 110% of the mean route travel time.

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Thus for each route the travel time reliability defined as the probability that the travel time will be less than 110% of the mean travel time was calculated as follows:

with 1.1 and where is the cumulative probability of the unit normal distribution . Table 5.2 shows the travel time deviations (with SUE re-routing) and reliability results obtained for the 20 most ‘unreliable’ routes, of which 10 are shown in Fig. 5.2. Again, we see that the over-saturated links 215L and 215K (highlighted) feature quite strongly in the most ‘unreliable’ routes. We also noted that links 212S and 222A, which appear in some of the routes, are operating close to capacity, with high delays, in the base case. All those links appear to account for the significant variations in travel time. Overall we can see that, given our definition of reliability, the most unreliable routes are those with short travel time and relatively high deviation due to the presence of congested links.

While the correlation between high variability and congestion appears to make sense, it still prompts a question about some limitations of the method; the most unreliable routes could indeed have been identified before the calculations

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of the variances just by finding out the routes containing congested links in the base solution.This limitation comes from the first order approximation, which is only valid in the vicinity of the base solution. Therefore it would be useful to account more accurately for the effects of larger variations, which tend to happen in reality anyway. However to model the impacts of these larger variations the non-linear effects in the relationship between input and output need to be included. The potential problems in this case would concern first the evaluation of the second-order approximation, and secondly the subsequent estimation of the output distributions.

6.

Conclusion

We have presented a method for deriving logit SUE sensitivity expressions which can be used in order to estimate travel time reliability measures. Experimentation on a small network showed an expected link between congestion and travel time variability, under normal variation of the demand. Future work shall be carried out in the following directions: a) calculation of error bounds to check the validity of the first order approximation in the relationship between exogenous input and the model output b) inclusion of second-order terms where appropriate c) combination of several variable exogenous factors (demand and supply) with appropriate random distribution in order to improve reliability assessment.

References Asakura, Y and Kashiwadani, M (1991). Road network reliability caused by daily fluctuation of traffic flow. In: Proceedings of the PTRC Summer Annual Meeting in Brighton, Seminar G, pp. 73–84. Asakura, Y and Kashiwadani, M (1992). Road network reliability measures based on statistic estimation of day-to-day fluctuation of link traffic. In: Proceedings of the World Conference on Transport Research, Lyon, France, June. Asakura, Y and Kashiwadani, M (1995). Traffic assignment in a road network with degraded links by natural disasters. Journal of the Eastern Asia for Transport Studies, Vol. 1, No. 3, pp. 1135–1152. Asakura, Y (1996). Reliability measures of an origin and destination pair in a deteriorated road network with variable flow. In: Proceedings of the Meeting of the EURO Working Group, Newcastle-upon-Tyne, UK, September.

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Bell, M G H, Cassir, C, Iida, Y and Lam, Y (1999). A sensitivity based approach to network reliability assesment. To appear in Proceedings of the International Symposium on Transportation and Traffic Theory. Jerusalem, Israel, July. Bell, M G H, Lam, W and Iida, Y (1996). A time-dependent multiclass route flow estimator. In: Proceedings of the International Symposium on Transportation and Traffic Theory, Lyon, France, July. Bell, M G H and Iida, Y (1997), Transportation Network Analysis, Wiley, England. Du, Z P and Nicholson, A J (1993). Degradable transportation systems performance, sensitivity and reliability analysis. Research Report, No.93-8, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand. Du, Z P and Nicholson, A J (1997). Degradable transportation systems: Sensitivity and reliability analysis. Transportation Research B, 31, No 3, 225–237. Fisk, C (1980). Some developments in equilibrium traffic assignment. Transportation Research, 14B, 243-255. Iida, Y and Wakayabashi, H (1989). An approximation method of terminal reliability of road network using partial minimal route and cut set. In: Proceedings of the World Conference, Vol. IV, Yokohama, Japan, pp. 367–380. Kimber, R M and Hollis, E M (1979). Traffic queues and delays at road junctions. TRRL Laboratory Report 909, Transport and Road Research laboratory, Crowthorne, England. Sheffi, Y (1985) Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods, Prentice-Hall, Englewoods Cliffs, NewJersey. Tobin, S J and Friesz T L (1988). Sensitivity analysis for equilibrium network flow. Transportation Science, 22, 242-250.

Chapter 6 A JOINT MODEL OF MODE/PARKING CHOICE WITH ELASTIC PARKING DEMAND Pierluigi Coppola [email protected] Dipartimento d’Ingegneria dei Trasporti Università degli Studi di Napoli “Federico II” Via Claudio 21, 80125 NAPOLI (Italy)

1. Background and objectives In the last years it has been widely recognized by analysts that Parking Management policies have the great potential to divert Travel Demand from Private modes (car, Motorbikes,...) towards Public Transportation System (bus, metro,...) and, thus, to contribute to reduce congestion (and other traffic-related undesired impacts) in urban areas, especially in their Central Business District (CBD). In fact, the problem of increasing parking demand in CBD is approached in a different way: the possibility of finding a parking place at the end of the trip is no more considered a service debt to drivers, but a useful tool to influence their travel choices. In order to define optimal parking management policies a model simulating the performance of the Parking System and the behavioral responses of the travelers, is necessary. But which are the requirements for parking modeling ? Parking Management deals substantially with the global dimension of parking supply and with the location of parking places within the study area. The parking places, furthermore, have to be differentiated according to different parking typologies (free, charged, time-limited, illegal,...). Impacts of changes in parking supply pattern, on the other hand, can affect both Parking Demand System itself (e.g. occupation ratio of parking infrastructures, revenues from parking fares,...) and the global 85 M. Patriksson and M. Labbé (eds.), Transportation Planning, 85–104. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Travel Demand System (e.g. modal split, trip distribution and trip frequency within a certain area, tour timing within a day and land-use). Therefore, a model aiming at simulating parking choice has to be able to simulate some of these interactions. Moreover, there are two other modeling requirements that a parking model should have. Firstly, the model should be multi-user, since behavioral responses to changes in parking system could strongly vary with travelers’ characteristics (e.g. the desired parking duration, trip purpose,...): it is rather intuitive to understand that an increase of parking fares will affect much more the choice of travelers with long parking stays than those with short parking stays. Secondly, since parking demand flows are not transient such as the link-flows on a road network, but persistent in the parking infrastructure for the whole parking duration, to model the performance of the parking system we must consider the effects of accumulation and dissipation of occupancy over the day. Thus, the temporal dependency between the successive time periods over the day is an additional model requirement for parking modeling (Polak and Vythoulkas, 1993). The parking models found in literature mainly follow two approaches: a network and a non-network approach. In a network approach (Eldin et al., 1981, Gur and Beimborn, 1986, Bifulco, 1991), a standard privatemode-network is expanded by means of parking links (simulating the cost of parking infrastructure) connected to the centroids by pedestrian links (simulating the walking distance to the final destination) and to the original network by access-egress links (see Fig. 1). Following a network approach, parking choices are simulated within the assignment procedure.

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In a non-network approach, on the other hand, parking choices are modeled through Random Utility Theory: at each parking alternative (i.e. a certain parking infrastructure) a value of Utility, sum of a deterministic term (i.e. the Systematic Utility) and a random residual, is associated. Different specifications are possible according to the specification of the systematic utility and to the hypotheses on the residual. Most of the specifications found in literature are Logit or Nested Logit (Van Der Goot, 1982, Hunt and Teply, 1993, Bifulco, 1996). Previous researches have been carried on in order to simulate explicitly the impacts of parking policies on mode choice. Bradley et al. (1993) and Miller (1993) propose to model jointly mode and parking choices following Random Utility theory, through a Nested Logit model specification. In these models, however, the (attribute) searching time for an empty parking space which should depend on parking occupation ratios (so in turn on parking demand) is calculated off-line on an average basis. In other words, there is a circular dependence between parking demand and parking generalized costs, which in the above models is not taken into account. The lack of feedback between parking supply performances and parking demand leads to a model system in which both parking demand and performances are derived from exogenous variable (fixed by the analyst) which are not the results of users’ choice process. In this paper a joint model of mode and parking choice with elastic parking demand is presented. The overall framework consists of the

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Nested Logit specification depicted in Fig. 2: users who have chosen to travel by car or motorbike, will next have to choose parking typology and location. Consistency between two choice dimensions (i.e. mode and parking) is achieved through the satisfaction variable (i.e. the Logsum variable) which is explicitly included as an attribute of the mode choice sub-model. In Section 2 a non-network parking choice sub-model, including the functions adopted to compute the generalized parking cost, is presented, while in Section 3 we report the results of a partial information estimation of the mode choice sub-model. In Section 4, finally, the results of the application of the model to Salerno (a mediumsize city of South of Italy) in order to simulate realistic parking policies, are shown.

Although the model focuses on the effects of Parking Management on modal split, the adopted approach can be easily extended to deal with other travel choices like destination and trip frequency.

2.

The parking choice sub-model

The parking choice sub-model aims at simulating interactions between Parking Supply and Parking Demand Systems. In the adopted parking choice sub-model, the existing parking places are grouped according to parking typology and according to their location. The elementary parking alternative is defined as the total parking supply of a given type in a given zone. Considered parking typologies are:

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free parking; illegal parking; time limited parking (i.e. free parking allowed for a certain period then parking not allowed); charged parking (where a fare structure is applied); limited + charged parking (i.e. charged parking allowed only for a certain period).

Each parking elementary alternative is characterized by several attributes. The following parking attribute classification is proposed:

where indicates the users’ class and F the parking occupancy. According to the above classification, the attributes included in the model are the following.

Attributes of type C1 is time spent walking from the parking infrastructure to the final destination. Assuming a walking speed of 1 meter/second, it is given by the distance (in meters) from the parking places to the centroid of the destination zone. All the parking places of a given typology in a given zone have the same pedestrian distances. In doing so, the effective distribution of parking places of the same typology in a given zone is lost. This is, however, consistent with the discretization of the trips, whose “real” origin and destination are assumed to be the centroids of origin and destination zones.

Attributes of type is the monetary cost for users’ class i related to parking fare .

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where is desired parking duration (in hours) of users’ class is the fare for each parking hour after the first (in EURO per hour) and fix is the fare for the first parking hour (in EURO). Since parking monetary cost depends on parking duration, a segmentation of the demand according to the desired parking duration is required. Therefore, parking demand is segmented in four user-classes according to four time intervals of desired parking duration. An average parking duration is associated to each of them, as shown in Tab. 1.

is the perceived monetary cost for users’ class i of illegal staying or overstaying:

where Tmax (in hours) is the maximum permitted parking stay (for illegal parking Tmax = 0); fine is the amount of the fine for illegal staying or overstaying and risk is the probability of being fined for each illegal (staying or overstaying) hour.

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According to the level of enforcement the proposed values for risk are those reported in Tab 2.

Attributes of type C3(F) is the time spent looking for an available parking space. It is, in general , a function of the capacity of the parking infrastructure, of the occupancy at the beginning of the simulation period and of the number of arrivals and departures during the simulation period. The adopted mathematical expression of the searching time function (Bifulco, 1991) gives rise to cost functions like the ones depicted in Fig. 3, where the capacity of the considered parking infrastructure is assumed to be 100 vehicles.

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The curves are parameterized with respect to the parking occupancy at the beginning of the simulation period. The independent variable represents the parking occupancy due to the parking loading mechanism within the simulation period and the dependent variable is the searching time.

Attributes of type none, since the specification of attributes contemporaneously depending on users’ class and parking occupancy could give rise to nonuniqueness and instability to the solution of parking choice equilibrium problems (see below). As shown above, the number of users that choose a certain parking infrastructure during the simulation period, depends on the searching time for an empty parking place. In our model, searching time is a function of parking infrastructure occupancy, which in turns depends on the number of users choosing that infrastructure. In other words there is a “circular dependence” between parking demand and searching time (i.e. parking cost), as shown in Fig. 4. This gives rise to a “fixed point”

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problem similar to the fixed-point problem related to assignment to congested road network (see Cantarella, 1997) which here is solved using an MSA (Method of Successive Averages) algorithm (Sheffi, 1985). Moreover, it is possible to prove the existence and the uniqueness of the stated fixed-point problem because of the absence of the costs simultaneously depending on both users’ class and parking congestion.

The user’s choice set (i.e. the set of all the alternatives the user considers to be attractive for his own choice) consists of all the elementary parking alternatives of his/her destination zone and of those owing to “neighboring” zones. The parking demand directed to a certain zone can, thus, be satisfied by the parking supply of the destination zone and by the parking supply of “neighboring” ones. Two zones are assumed to be “neighboring” if, for instance, the (average) walking distance lying between them is not greater than 500 meters. The systematic utility functions of each parking alternative are assumed to be linear in of the above attributes. However, since the parking choice sub-model has not been calibrated, the coefficients used in the applications have been taken from the (calibrated) mode choice model. In practice we have proceed as follows. Starting from value of betas taken from literature (Bradley et al., 1993), we have used such values to compute the inclusive variables of parking choice (i.e. Logpark variable, see section 3), which are attributes

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of the mode choice sub-model. With these values of logsum we have calibrated the mode choice model. It has been assumed, then, that the values of searching time, walking time and monetary cost in parking choice are perceived by the user as equal respectively to in-vehicle time, walking time and monetary cost in mode choice. As consequence, according to Nested-Logit theory (see Mc Fadden, 1978), the values of for such attributes have been obtained by dividing the values of corresponding parameter in the mode choice model by the value of In doing so, the values obtained are not exactly the same of those according to which the inclusive variables have been calculated. Therefore, starting from these values new inclusive variables are calculated and accordingly a new calibration of the mode choice is carried out. This iteration procedure stops when consistency between inclusive variable and beta’s of parking attributes is achieved.

The trip purposes, here considered, are: Workplace Other constrained: purposes constrained in destination (i.e. those characterized by destination that cannot be daily chosen), such as Business, Study, and so on Other unconstrained: purposes that are not constrained in destination (i.e. those characterized by destination that can be easily changed), such as shopping, leisure, and so on. For different trip purposes the reported in Tab 3.

estimated as described above, are those

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3. The mode choice sub-model The modal alternatives to be included in the model specification come out of the analysis of the data available for the calibration. In fact, as the estimation of the model is based on an RP (Revealed-preference) survey, any mode that either doesn’t exist or that is coarsely utilized within the study area (e.g. Metro and bicycle, in Salerno) must be left out of the model specification. As consequence, the modal alternatives considered are: Moto Car

Walk Bus

The systematic utilities of each mode are the following:

where: is the travel time by motorbike on the minimum path between origin zone and destination zone is the travel time by car on the minimum path between and

is the travel time to walk from

to

is the waiting time at the bus stop on the minimum hyperpath (see Guyen e Pallottino, 1988) connecting

to

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is the access-egress time to/from the bus stop on the minimum hyperpath connecting

to are the monetary costs to go from

to

respectively by car, motorbike and bus;

Age is a dummy variable which is equal to 1 if the user is not older than 30 years, 0 otherwise; is the inclusive variable of the parking choice alternatives (i.e. Logsum variable), computed as:

being

and

respectively the Systematic Utility of the generic

parking alternative and the choice set of users directed to zone (i.e. all the parking alternatives within the zone plus those within the “neighboring” ones, as already explained in section 2) Note that Alternative Specific Constant (ASC) have been introduced for “Auto”, “Moto” and “Walk” modes. The following rules are introduced in order to define which subset of the four potential alternatives is feasible for each user: Moto is available only if the pedestrian time on the same O/D pair does not exceed 5 minutes. Auto is available if the pedestrian time on the same O/D pair does not exceed 5 minutes and the user owns a car and a license driving. Walk mode is available only for trips not exceeding 30 minutes. Bus is available if the Access+Egress pedestrian time does not exceed 30 minutes, the waiting time does not exceed 30 minutes and the pedestrian time on the same O/D pair does not exceed 5 minutes.

The mode choice model has been calibrated through the “maximum likelihood” method. In Tab. 4 the and the corresponding values of T-ratio are reported for different trip purposes.

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All the coefficient estimates have the expected sign and are significantly different from zero at the 5% level of confidence. As expected, the “Other constrained” value of time (VOT) is higher than other VOT’s since this purpose includes also “Business” trip (remark that “workplace” here is synonymous of “commuters”).

4. Simulation of realistic parking policies The model so far presented is part of a comprehensive modeling system developed within the EC project AIUTO. It has been used to evaluate the impacts of Parking Management on the modal split in the city of Salerno, one of the Italian AIUTO test-sites. In this section some

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realistic parking policies will be simulated. It is worth noting that Parking Management could also affect destination and activity-time choices, however these choice levels here are not covered. Moreover, only the morning peak period is simulated. Salerno is a medium-size city of the South of Italy that extends for about narrow between the mountains and the homonymous gulf. With a population of more than 150.000, Salerno suffers from heavy congestion problem due to high and “concentrated” travel demand. The travel demand is here segmented per parking average duration (i.e. 0.5, 2, 5, 9 hours), which directly affects parking choice, and per age (i.e. > or <= 30 years old), which directly affects mode choice. Considered travel purposes are: workplace, other purposes constrained in destination (e.g. Business) and other purposes not constrained in destination (e.g. Shopping). The (percentage) trip distribution across trip purposes over the morning peak period and over the whole day is reported in Tab. 6.

The study area is defined as the city center of Salerno plus the traffic zones adjacent to the center. Here most of the congestion is located. The traffic zones within such area have been grouped into two macro-zones: the so-called “red” zone, consisting of the zones of the city center, the “yellow” zone, consisting of the zones adjacent to the city center (see Fig. 5).

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Different parking measures have been introduced for these two groups of zones with respect to a base-scenario derived from a survey of 1993. It is worth to remark that the perceived cost of illegal parking is here calculated as the product of fine and probability of being fined. Therefore, increasing fines could correspond to increase either the fine itself or the probability of being fined (e.g. by means of enforcement of parking restrictions). In the “red zone” the applied parking measures consist of: the free parking spaces are reduced to 100 spaces for each traffic zone transforming the rest in charged parking with the costs indicated in Tab. 7, corresponding to a fare of 0.75 EURO per hour in the base scenario and 1.5 EURO per hour after applying parking policies; the illegal parking is controlled with the fines indicated in Tab. 7, corresponding to a “high” level of enforcement (see Tab. 2) and a fine of 25 EURO in the base scenario, and to a “very high” level of enforcement and a fine of 50 EURO after applying of parking policies; the time limited parking becomes charged parking with the fares indicated in Tab. 7.

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In the “yellow zone”, the parking measures applied, consist of: the free parking spaces are reduced to 100 spaces for each traffic zone transforming the rest in time limited parking with the fines (in case of overstaying) indicated in Tab. 8; the illegal parking is controlled with the fines indicated in Tab. 8, corresponding to a “medium” level of enforcement (see Tab. 2) and a fine of 25 EURO in the base scenario, and to a “high” level of enforcement and fine of 50 EURO after applying of parking policies; to the already existing charged parking the fares applied are those indicated in Tab. 8, corresponding to a fare of 0.75 EURO per hour in the base scenario and 1.0 EURO per hour after applying parking policies; to the already existing time-limited parking the fines, in case of overstaying, are those indicated in Tab. 8.

The main impact of such parking measures is that of diverting car travel

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demand to alternative modes, as it is shown in Tab. 9. where the (percentage) trip changes for each mode is reported.

Percentage trip changes, for the generic mode

are here calculated as:

being and the demand on mode m respectively after and before the introduction of parking measures. It can be noted that the reduction of the number of trips by car seems to be comparable for red and yellow zones, even tough fares and parking restriction in the red zones are stronger. This can be explained by the fact that the parking model takes into account that users can reach a given destination (e.g. in the red zone) also by parking in neighboring zones (e.g. in the yellow zone). This yields a first modeling consideration: a parking model which allows drivers directed to a certain zone to park in the neighboring zones, leads to a sort of homogenization of the impacts of parking policies across all the zones of the understudied area. In Tab. 10 the percentage reduction of trip-by-car for different trip purposes is reported: as expected the parking policy impact (both in red and yellow zone) is strongly different according to travel purpose. While trip purposes characterized by long average parking duration (i.e. workplace) are strongly discouraged by parking measures, medium stay purposes (i.e. other constrained) are much less affected and short term purposes (i.e. other unconstrained) are almost not discouraged at all by using car.

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This is explainable as follows. When parking measures are applied, in fact, users travelling for purposes implying long parking duration (e.g. workplace) have to pay much more than in the base-scenario, therefore, most of them find more convenient diverting towards other modes. On the other hand, the relatively low monetary cost increase for users travelling for purposes characterized by a short (or medium) parking duration is compensated (at least partially) by the reduction in searching time (reduction which is due to the fact that long duration stay trips have diverted). As consequence, those with low desired parking duration might find parking even more attractive (e.g. other unconstrained in the red zone, see table above) when, for instance, the reduction in searching time is perceived as higher than the increase in parking monetary cost. The increase of monetary parking costs can be compensated by decrease of searching time (depending on parking congestion) and vice versa, since parking model and the mode choice model are framed into an “elastic” modeling procedure. The parking demand (i.e. the number of users choosing the car mode) and the parking Level Of Service (LOS) are (at equilibrium) mutually consistent.

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Acknowledgements

The author would like to thank the Salerno research laboratory ELASIS for having given access to the database through which the calibration of the model has been possible.

6. References BIFULCO G.N., (1996) - Stochastic models for the simulation of parking choices: a non-network approach - Proceedings of the EUF-PTRC BIFULCO G.N., (1991) - A stochastic user equilibrium assignment model for the evaluation of parking policies - Proceedings of the PTRC Summer Annual Meeting BRADLEY M. -KROES E. and HINLOOPEN E., (1993) - A joint model of mode/parking type choice with supply constrained application Proceedings of the PTRC Summer Annual Meeting CANTARELLA G.E.,(1997) - A General Fixed-Point Approach to MultiMode Multi-User Equilibrium Assignment with Elastic Demand Transportation Science, vol.31 (2). ELDIN N.-EL REEDY T.J. and ISMAIL H.K., (1981) - A combined parking and traffic assignment model -Traff. Eng. and Contr., 22(10) pp. 524-530 GUYEN S. and PALLOTTINO S., (1988) – Equilibration Traffic Assignment for large scale transit Network – European Journal of Operational Research - pp. 176-186 GUR Y.J. and BEIMBORN E.A., (1986) - Analysis of parking in urban centres : equilibrium assignment approach - Transportation Research Record 957 - pp. 55-62 HUNT J.D. and TEPLY S., (1993) - A Nested Logit model of parking location choice - Transp. Res.-B, Vol. 27B, No.4 - pp. 253-265 McFADDEN D., (1978) – Modeling the choice of residential location – in Spatial Interaction Theory and Planning models by Karlqvist A. et al.,

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North Holland Publishing Company MILLER E.J., (1993) - Central area mode choice and parking demand Transportation Research Record 1413 - pp. 60-69 POLAK J. and VYTHOULKAS P. (1993) – An assessment of the state-ofthe-art in the modeling of parking behavior – Report n.752 of Transport Study Unit SHEFFI Y. (1985) – Urban transportation networks – Prentice-Hall ed. VAN DER GOOT D., (1982) - A model to describe the choice of parking place - Transp. Res., Vol. 16A, No.2 - pp. 109-115

Chapter 7 A NEW GENERAL EQUILIBRIUM MODEL Yanling Xiang [email protected]

Michael J. Smith [email protected] York Network Control Group Mathematics Department, York University York YO10 5DD, UK

Miles Logie [email protected] MVA Limited MVA House, Victoria Way, Woking, Surrey GU21 1DD, UK

Abstract

In this paper we present a new general equilibrium model appropriate for multimodal networks. A solution of the model gives the equilibrium distribution of travellers and vehicles over a transportation network. The model is expressed in terms of inverse cost-flow functions; and delays are explicitly modelled. The paper outlines an equilibration algorithm and convergence results for a small network are provided. The equilibrium model has been designed in such a way that optimisation procedures may naturally be added to the equilibration algorithm.

Keywords:

Equilibrium, assignment, multicopy network, bottleneck delay, monotone cost functions

105 M. Patriksson and M. Labbé (eds.), Transportation Planning, 105–118.

© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Introduction The General Equilibrium Model described in this paper

The General Equilibrium Model specified in this paper has been designed and a solution method has been implemented within software as part of a UK Government LINK project with partners MVA Limited, University College London and the University of York. The LINK project was supported by the Department of the Environment, Transport and the Regions (DETR), the Engineering and Physical Sciences Research Council (EPSRC), the Engineering and Sciences Research Council (ESRC) in the United Kingdom and MVA Limited. The model is general for a number of reasons. Firstly, it provides an integrated framework for consideration of both transport demand and supply. It does this in a form of a standard economic model, suggesting the potential for making the transport model an element of a wider economic model. Secondly, its form embraces the four stages of trip generation, distribution, modal split, and assignment of classical transport modelling. Thirdly, the model allows for complex network modelling features such as capacity constraints and interactions where flows on one link affect costs and flows on another. These interactions may extend across modes and need not be symmetrical. Finally, the model may consider costs calculated from deterministic and stochastic functions, and offers a very natural potential for extension to the dynamic case. The model and the context motivating its development have been described by Logie et al. [5]. In this paper we describe (i) some technical details of the model, and (ii) some initial convergence results generated by the model. It should be noted that “the equilibrium objective function" in (3.6) which is driven to zero at equilibrium is completely different to the usual “equilibrium objective function". This departure; which appears to be necessary so as to allow for interactions; means that the model is novel within transportation software although the theoretical ideas have been around for a while. The different equilibrium objective function also allows bilevel optimisation procedures to be readily implemented within the model and one way of doing this is outlined in Smith et al. [12]. Thus the “optimising" version of this model will itself make suggestions to the transportation planner and so assist with the achievement of targets.

1.2.

Issues for transport modelling

The generality of the model described here is of significance to many of the issues which current trends in transport policy, planning, and modelling are emphasising. These trends give rise to changing targets and increasingly so-

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phisticated controls which provide the transport planner with a correspondingly difficult task. Policies in the UK, for instance, increasingly emphasise demand management, and local authorities face traffic reduction targets. This has been accompanied by research into the impact of highway capacity reductions ([2]) and into elasticities associated with demand suppression and induction (MVA [6], and previously SACTRA [8]). We may note that these considerations match the original work by Beckmann et al. [1], who allowed for demand elasticity and strict capacity limits. Thus a pure Beckmann equilibrium model is suitable at least to assess these issues. We may observe also that the demand for transport is derived from the need to conduct other activities; it is not an end in itself. Other research is aimed at modelling needs for activities. These needs (to go to work, shopping, collecting children from school, etc.) may also form an input to the General Equilibrium Model which, depending on the implied costs, can determine whether travel or alternatives, such as telecommunications, are required to meet the needs. Currently, computer models of transportation are used to assess strategies, leaving the planner to devise strategies for testing against new and changing targets. This is a difficult task and there is a corresponding need for an effective and proven mathematical optimisation methodology; implemented with helpful and easy-to-use software. The ability to allow bilevel optimisation, where the upper level is used to optimise a strategy, is therefore significant. This is most clearly seen in respect of traffic signal controls where signal timings for a network (particularly in the UK but also elsewhere in Europe) are often chosen to minimise delay for the observed traffic pattern, and neither consider nor seek to improve long-run effects. Yet it is easy to see that settings which are best for the present may cause problems in future as route and mode choices change to take advantage of these settings. It is therefore valuable to be able to optimise controls such as signal settings when assessing schemes and policies. Prices are also important controls of economic and transportation systems, and their varied application commands increasing attention as part of transport policies. Methods proposed for setting prices are often based on marginal cost prices which pay little regard to practical constraints. It is usually more appropriate to solve the problem, known as the second best problem, namely; what should prices be set at when some prices cannot be marginal cost prices? This arises because it is either practically or politically difficult to alter some prices (say, bus fares and road tolls) and not others (say, parking charges). Similarly, theoretical System-Optimal flow patterns only give the right prices if all links are charged; as soon as some links are not charged we have a second best problem. Therefore a more general optimisation procedure is required

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which may incorporate constraints reflecting the fact that certain links must not be subject to charge. These pricing issues further the value of integrating transportation assessment and optimisation models as the General Equilibrium Model framework allows.

2.

DREAM—The general equilibrium model

The model described here incorporates controls; including prices and greentimes; within a model with inelastic and elastic demand, and deterministic and stochastic route and mode-choice. Like Beckmann et al. [1] the model embodies explicit capacity constraints and like Payne and Thompson [7] explicit queueing delays. Here we describe the model without stochastic choice and with fixed controls; give an equilibration procedure, and some initial convergence results.

2.1.

Monotonicity

The model will be monotone in the flow, delay and cost variables for each fixed control vector This will allow a fairly wide range of interactions between flows on one link and costs on other links but will also mean that there is at least one algorithm which is bound to converge to equilibrium for any fixed signal control settings and prices.

2.2.

DREAM

The model is called DREAM (Demand Responsive equilibrium assignment model) and has the potential for optimisation facilities, including second-best pricing facilities, essentially built in. The need for optimisation has been a central feature of the design of the equilibrium procedures. We believe it is the first equilibrium model to have been constructed beginning with this point of view.

2.3.

The basic structure of DREAM

Smith [9] introduced an equilibrium as a point

such that

This formulation of equilibrium as a complementarity problem is fundamental to DREAM. DREAM also has an explicit representation of delays at link exits following Payne and Thompson [7]. The model, specified in a route space, has been developed from that described in Smith et al. [12], in Smith et al. [11], and in Xiang and Smith [14] which is in a link flow framework. Three descent algorithms for equilibration are

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implemented: the Smith Algorithm (D) direction ([10]), the steepest descent direction, and the conjugate gradient direction; and various line searches are implemented including Armijo type and quadratic type. Many non-linear flow-delay functions may be used in DREAM. For example, the flow-delay function corresponding to the transition curve due to Kimber and Hollis [4] has been implemented. Figures 7.2 and 7.3 illustrate the performance of DREAM on a small network by using this curve. The convergence of DREAM to equilibrium is guaranteed for monotone non-decreasing flow-delay functions.

3. 3.1.

An outline of the DREAM model Notation

The notation adopted in DREAM is given below; there is a base network (a model of the actual road network) and a multicopy network ([3]). The idea of the multicopy network is explained below before the notation is shown. In a multicopy network, within each copy the commodity flowing on that copy must have a single specific destination node. If there are K commodities the multicopy network has K copies of the basic network. The simplest commodity comprises all travellers with the same destination. Then if there are K destinations there are K commodities. We may also consider different types of vehicle as different commodities. Here, for example, we may consider all heavy goods vehicles with a specific destination. In this case the copy of the basic network may have no links which correspond to link in the basic network – the heavy vehicles may be prohibited from certain road links. Thus the copies may in fact essentially be copies of subsets of the base network. In what follows all the copies are considered together as a single multicopy network. At an initial reading think of the as 1. Variables to be found. = flow along route

in the multicopy network;

= least cost of reaching the destination from node in the multicopy network and is to be zero if is that destination; and = bottleneck delay at exit of link in the base network. Fixed control variables. = proportion of time stage

is green;

= price to be paid to traverse route

in the multicopy network; and

= price to be paid to traverse link in the base network.

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Fixed given data. = free-flow cost of traversing route = saturation flow at exit of link (may be infinite); = passenger car unit equivalent of a unit of flow in that commodity whose network contains route and = passenger car unit equivalent of a unit of flow in that commodity whose network contains node Multicopy network structure data.

if node

is the upstream node of route

before

and 0

otherwise; if link belongs to route in the multicopy network and 0 otherwise; if link is in stage if links in stage 0 otherwise.

and 0 otherwise; and

have downstream node

in the base network and

Immediately derived variables. passenger car unit, or normalised, flow along link in the base network so that (it is this normalised flow which causes congestion); and the nominal capacity of link Demand and flow-delay functions. flow generated at node if the cost to destination vector is (with inelastic demand constant and for all if is a destination); and maximal possible average flow on link if the average bottleneck delay is and the nominal link capacity is

3.2.

Supply, Demand and User Equilibrium

Supply. The links in DREAM have performances which may be written: and

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This says that the realistic capacity of a link is never exceeded and that if the outflow is less than the realistic capacity then the queueing delay is zero there is no holding back. Demand constraints. An appropriate (elastic or inelastic) demand constraint may be written similarly:

This implies that for any non-destination node the net inflow to node (allowing for the properly generated by the cost vector ) is never positive; or that (allowing for the there is no net outflow from the network at any non-destination node. If this condition holds then it is automatically true that the outflow at a destination equals the sum of the inflows at all the non-destination nodes for that commodity and that this sum is at least the sum of the relevant (The user-equilibrium condition below then implies that the inflow to the network at each equals User equilibrium. We shall use the following version of Wardrop’s user equilibrium condition ([13]): for each route the (least) cost to the destination from the upstream node (before is no more than the least cost to the destination via route and if it is less then no flow will enter route or

3.3.

The Equilibrium Problem

The problem here is the problem of finding an equilibrium within the demand and capacity constraints. The equilibrium conditions (3.1), (3.2) and (3.3) may be written as a non-linear complementarity problem (multiplying (3.2) and (3.3) through by the positive and as follows:

is normal at

to

for each

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is normal at

to

for each

is normal at to for each Let the vector with coordinate vector with coordinate be may be written:

be and let the Then the above conditions

is normal at where is the number of routes (a variable in DREAM) in the multicopy network, is the number of OD pairs in the multicopy network, and is the number of links in the base network.

3.4.

Optimisation Formulations of the Equilibrium Conditions

Rewrite (3.4) in the form:

where

and

Generally will be the co-ordinate of and will be the co-ordinate of We must augment the condition (3.5) with feasibility constraints on and These we will here assume to be:

and

(all and

where A is a matrix.

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Consider the energy

where and If and for all then satisfies (3.5) and is an equilibrium provided is feasible. Thus, finding an equilibrium is now the same as finding a feasible minimum of at which certain upper bounds are not binding. (We are now solving this model in a way which avoids the need for the upper bounds. See Smith et al. (2000) and the objective function therein.)

4. 4.1.

Features of the general equilibrium model Gap function for optimisation

In this paper, the assignment problem is considered by letting the control variables be fixed. To be specific we suppose to be linear here. Then let (for fixed controls)

where are concerned with flow-delay functions used in DREAM as explained below, is the Jacobian matrix of equation (4.0), and is a constant vector.

4.2.

Flow-delay functions

Flow-delay functions currently available in DREAM are of the form:

Formula (4.2) is a function of the bottleneck delay at the exit of link and the (fixed) nominal capacity of link and are two parameters specified by users. Typically, and In these cases, flow-delay function (4.2) is a non-decreasing function of if is fixed. For fixed functions (4.2) may be written as

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where is a diagonal matrix with elements and is a vector consisting of elements is the total number of links in the network). As described earlier in the paper, congestion costs normally represented by a cost-flow function will in fact occur in DREAM as bottleneck delays which arise from equilibrating via thefunctions The may be thought of as the inverse of given cost-flow functions.

4.3.

The outer loop and inner loops

DREAM alternates between two phases: an outer loop and inner loops. In the outer loop, an active route set is generated or augmented by Dijkstra’s shortest path algorithm. The active route set is generated automatically and route enumeration is not needed. In the inner loops, a capacity-constrained user equilibrium assignment problem is solved as an optimisation problem within the route set generated so far.

4.4.

Assignment problem in the inner loops

From equations (4.0) and (4.3) it is clear that (the Jacobian matrix of is positive semi-definite when flow-delay function (4.2) is non-decreasing, and then is a monotone function. According to Smith [10], gap function (3.6) has for fixed always a descent direction while away from zero where

The gradient vector

of the gap function

in (3.6) is

Apart from the algorithm (D) direction (4.4), the steepest descent direction and the conjugate gradient direction can be easily obtained from the above gradient vector (4.5). The three directions are implemented along with three distinct line searches: the constant step length, the quadratic line search, and the Armijo-type line search.

4.5.

Active route set generation in the outer loop

DREAM uses Dijkstra’s algorithm to calculate shortest paths between one origin and all destinations, for the active route initialisation and also later when new cheapest routes are added to the active route set. When a new shortest path is found according to current link costs for an origin-destination pair the number of routes for the OD pair is increased by

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one, and the total number of routes in the active set is then updated. The system will be re-equilibrated in the new (expanded) route set. When the active route set is updated, the gap function (3.6) will jump upwards to a (new) peak point.

4.6.

Upper bounds

The gap function used in DREAM is given in (3.6), where are lower bounds (always zero) on variables and are upper bounds on variables Route flow variables are naturally bounded above by demands. For link delays and least origin-destination pair costs, upper bounds must be specified initially by users as input parameters. When a variable is found to reach its upper bound during iterations, DREAM will automatically update the corresponding upper bound by increasing a certain amount. When this happens, the gap function jumps upwards; as happens when new routes are added into the route set.

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Test Results

Figure 7.1 illustrates a single-modal example network with twenty-eight links and twelve nodes. Nodes 31 – 38 are signalised junctions (31 – 34 are also destinations), and nodes 39 – 42 are origins. There are four origindestination pairs with demand 1000 veh/hr each: 39 to 33, 40 to 34, 41 to 31, and 42 to 32. Signalised junctions 31 – 38 have a fixed cycle time of 120 seconds with two stages and for each of the junctions, the green-times of the two stages are all fixed at 60 and 60 seconds respectively. Link labels, saturation flows, free flow times, and the stage numbers for the Dring network are given in Table 7.1. Figure 7.2 illustrates the convergence of the (energy) gap function (3.6) to zero. The three peaks indicate where the active route set has been updated, that is, new cheapest routes have been added into the route set. Figure 7.3 illustrates the convergence of the total journey times for two flow-delay functions: and (the Kimber and Hollis formula).

Acknowledgments We thank UCL for their collaboration. Our results presented here are obtained within a UK LINK project, supported by DETR, EPSRC, and ESRC in the UK, and MVA Limited. We are very grateful for their financial support.

References [1] Beckmann, M., C. B. McGuire, and C. B. Winsten: 1956, Studies in the Economics of Transportation. Yale University Press. [2] Cairns, S., C. Hass-Klau, and P. Goodwin: 1998, Traffic Impact of Highway Capacity Reductions: Assessment of the Evidence. Landor Publishing.

[3] Charnes, A. and W. W. Cooper: 1961, ‘Multicopy Traffic Network Models’. In: Herman R. (ed.): Proceedings of the Symposium on the Theory of

REFERENCES

117

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Traffic Flow,. held at the General Motors Research Laboratories, Elsevier, Amsterdam. [4] Kimber, R. M. and E. M. Hollis: 1979, ‘Traffic Queues and Delays at Road Junctions’. Technical Report LR909, TRRL Laboratory, UK. [5] Logie, M., B. G. Heydecker, and M. J. Smith: 1998, ‘Generalised Equilibrium Modelling: An Integrated Framework for Transport Planning’. In: European Transport Forum. [6] MVA Limited: 1998, Traffic Impact of Highway Capacity Reductions: Report on Modelling. Landor Publishing. [7] Payne, H. J. and W. A. Thompson: 1975, ‘Traffic Assignment on Transportation Networks with Capacity Constraints and Queueing’. Paper presented at the 47th National ORSA/TIMS North American Meeting. [8] SACTRA: 1994, ‘Trunk Roads and the Generation of Traffic’. Technical report, Department of Transport, UK. [9] Smith, M. J.: 1979, ‘The Marginal Cost Taxation of a Transportation Network’. Transportation Research, B13, 237-242. [10] Smith, M. J.: 1984, ‘A Descent Method for Solving Monotone Variational Inequalities and Monotone Complementarity Problems’. Journal of Optimisation Theory and Applications 44, 485–496. [11] Smith, M. J., Y. Xiang, R. Yarrow, and M. O. Ghali: 1998, ‘Bilevel and other Modelling Approaches to Urban Traffic Management and Control’. In: Marcotte P. and Nguyen. S (eds.): Equilibrium and Advanced Transportation Modelling. pp. 283–325. [12] Smith, M. J., Battye, A., A. Clune, and Y. Xiang: 2000, ‘Cone Fields and the Cone Projection Method of Designing Signal Settings and Prices for Transportation Networks’. In: Proceedings of the 6th Meeting of the EURO Working Group on Transportation, Gothenburg.

[13] Wardrop, J. G.: 1952, ‘Some Theoretical Aspects of Road Traffic Research’. In: Proceedings, Institute of Civil Engineers II, 1, pp. 235–278. [14] Xiang, Y. and M. J. Smith: 1998, ‘A General Equilibrium Model, Solution Algorithms and Convergence Results’. Paper presented at the First International Transportation Symposium, Newcastle.

Chapter 8 FIRST ORDER MACROSCOPIC TRAFFIC FLOW MODELS FOR NETWORKS IN THE CONTEXT OF DYNAMIC ASSIGNMENT J.P. Lebacque [email protected] CERMICS-ENPC, France

M.M. Khoshyaran TASC Consultant, USA

Abstract

1.

The purpose of the paper is to adapt the classical LWR (Lighthill-WhithamRichards) model, in its continuous version, to networks, in the context of dynamic assignment. This implies several specific adaptations of the basic model: introduction of partial flows, possibly inhomogeneous flows on links, and intersection modeling. The latter proves particularly difficult, and we discuss three different modeling approaches: extended versus pointwise intersection models, and flow maximization. We show that all three types of models are actually closely related, and compatible with the link flow models. The concepts of local traffic supply and demand prove to be essential, both for link and for intersection modeling. A brief comparison with experimental merge data gives some support to the phenomenological models introduced in the paper.

Introduction

Modeling traffic flow for dynamic assignment is a difficult task, because a compromise has to be struck between conflicting imperatives. The model should be realistic, accomodate partial flows, be suitable for large networks and at the same time be very simple in order to be compatible with intensive numerical computation. The general trend in traffic models for dynamic assignment has been one of increasing sophistication, starting with the simple exit function models [23], moving on to travel time function models [7], [25], point queue models [11] and finally macroscopic models ([20], [12], [6], [27], [2], [1]). 119

M. Patriksson and M. Labbé (eds.), Transportation Planning, 119–140. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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These macroscopic models rely on the discretization of their basic equations, and on heuristics for the intersection description. It is therefore very difficult to obtain for instance optimality conditions for assignment problems with such macroscopic models. This has only been attempted in the simplest of cases [9]. The object of the proposed paper is, in the case of the simpler LWR (LighthillWhitham-Richards [21], [26]) model, to provide a general methodology for a rigorous description of networks in the context of dynamical assignment. Our choice of the LWR model is justified by: its simplicity, the availability of nontrivial analytical solutions, the absence of inconsistent features ([4]). The outline of the paper is the following. After recalling the basic LWR model and introducing the local traffic supply and demand concept, we describe boundary conditions and link models for partial flows, i.e. flows disagregated according to some attribute. We then proceed to intersection modeling, beginning with the so-called zone model in which the physical extension of the intersection is taken into account. By neglecting the dimension of the intersection, we can deduce pointwise intersection models. These can be shown to result also from some flow-maximization principle. We conclude with a brief comparison between model prediction and experimental data in the case of a simple merge.

2.

The basic model LWR model for a link The basic LWR model can be writen as:

or simpler as:

with

the flow, K the density, V the speed,

resp. speed-density relationships and denote position and time. The aspect of

and

the equilibrium flow-

As usually, is the following:

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Macroscopic flow models

In the sequel we only consider the continuous version of the LWR model. The classical calculation of the analytical entropy solution of this equation is wellknown and need not be recalled here, the discretization of this equation relies on the Godunov scheme [8] (ch 3), [3], [15]. The following questions must be addressed in order to extend this basic model to networks and to assignment problems: define proper boundary conditions for links, modelize intersections with emphasis on their relationship with link boundary conditions, introduce partial densities and flows into the link and intersection models. Following [15], let us introduce now the equilibrium supply and demand functions, which are functions of the density and the position

(the symbols + and - representing right- and left-hand limits). The following illustration describes these functions.

Then we define the local supply and demand

and

which represent respectively the greatest possible inflow and outflow at point It can be shown that the entropy solution of the LWR model satisfies the natural condition:

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(the flow as the minimum between upstream demand and downstream supply). The formula (2.5) would be trivial, but for the circumstance that it is valid at all discontinuity points, whether related to density (shock-waves), or to the equilibrium flow-density function (boundary conditions, incidents, intersections). The importance of the local traffic supply and demand concept stems from this fact. Supply and demand are the basic tools required to describe discontinuities. Boundary conditions express themselves trivially in terms of local supply and demand. Let us consider a link such as the following:

the boundary data is clearly the downstream supply at extremity and the upstream demand at extremity These quantities, combined with the link traffic supply and demand, yield the link inflow and link outflow according to:

Thus: An intersection model should therefore provide adequate supply values to the upstream links of the intersection, and demand values to links downstream of the intersection.

3.

Partial flow models for links

For assignment models, the flow has to be disaggregated according to some assignment attribute d (destination, path, vehicle type, information availability). Let us denote the flow, density and speed of vehicles d (the partial flows, densities and speeds). Trivial equations for partial flows are:

which must be supplemented with a behavioral equation for

3.1.

A FIFO partial flow model

The simplest possible model of partial speeds is the FIFO model:

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Macroscopic flow models

which will be privileged in dynamic assignment models. composition coefficients:

Introducing the

it follows that and the following advection equation results for the composition coefficients:

This equation means that the traffic composition remains constant along a trajectory. Notably, if is the travel time experienced by the user entering the link at time the following equations holds for compositions: which shows that, under the FIFO hypothesis, the present composition of the link exit flow is determined by the composition of the link inflow at some past instant. This is a very important fact because it implies strong constraints on intersection models. The reader is referred to [1] and [16] for the numerical computation of the variables T and which all satisfy advection equations.

3.2.

Local assignment models

The problem we consider here is that vehicles may have restricted access to lanes according to some assignment attribute d. Let I be the set of lanes, the set of lanes accessible to vehicles the maximum density of lanes the density of vehicles in lanes Then are the unknowns of the lane-assignment problem and are subject to the following constraints:

Implicitly The

if The constraints express the split of constraints express that the total density

into the in lane

cannot exceed the maximum density of this lane. The constitute the dynamic data (analogue to OD data in classical deterministic assignment on networks problems) and the and constitute the geometric data of the lane assignment problem. The unknowns can be determined by solving either:

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(Wardrop user optimum), or:

(Wardrop system optimum), subject to constraints (3.4) in both instances. The lane assignment resulting of (3.4), (3.5) is such that the speed of vehicles is maximal in lanes such that and identical in all those lanes. This fact can be easily checked by writing the Karush-Kuhn-Tucker (KKK) necessary optimality conditions for (3.4), (3.5), which are also sufficient in this case, and using the property of of being a decreasing function. The formulation (3.5) is actually an analogue of the Beckman transformation, which is hardly surprising: the problem is one of assignment between lanes. Specifically, (3.5) realizes locally (at each location and instant) a user (Wardrop) optimum in which the inverse of speed replaces path travel times or costs, and lanes replace paths. The model [5] appears as a special case of the model (3.4), (3.5), with two lanes two vehicle types and The lane assignment resulting of (3.4), (3.6) is such that the wave speed of vehicles is maximal in lanes such that and identical in all those lanes. By definition the global flow is maximized locally, which would correspond to an entropy-like solution in which transient dynamics in the traffic flow are neglected. To use once more the analogy between static deterministic assignment on networks and assignment between lanes, (3.6) could be viewed as yielding locally system optimal solutions to the lane assignment problem. Only field experiments would allow to choose between models (3.5) or (3.6), the difference between the two being essentially a difference in user behavior. Both models might well be correct, under different conditions. At this point, to the authors knowledge, no proper boundary conditions have been stated for the lane-assignment model.

3.3.

Supply-Demand models

These models are based on ideas sketched in [14] and [1], and developed in [15]. They generalize formulas (2.4), (2.5) and (2.6). The principle is to calculate partial supplies and demands for all superscripts and to determine the corresponding partial flows by comparing partial demands to partial flows. Partial demands are defined as:

which is the usual FIFO-like model (partial demands proportional to the compositions). The partial supply model really defines the user behavioral model.

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Macroscopic flow models

Let us first define coefficients which determine the maximum density (i.e. of the lanes available to vehicles d. If we refer to the notations of the preceding subsection,

We propose the following two models for the partial supplies: Model 1: Model 2:

(linear model), (homogeneous section model).

Recall that is the total supply, and is the equilibrium supply function at location Model 1 is extremely simple but allows to exceed Model 2 does not have this drawback but still does not take partial flow overlapping into account as precisely as the lane-assignment models of the preceding section, since the data formed by the sets and the coefficients has been simplified and only coefficients are left. It would also be natural to define the partial flows as:

Nevertheless, since

is usually > 1 (because of the partial flow overlap-

ping), it is possible that partial flows calculated according to (3.8) satisfy:

thus implying a model inconsistency. In keeping with the idea of constructing entropy solutions, we propose to calculate the partial flows as solutions of the following program:

with

some concave increasing functions. For instance, with:

the KKT optimality conditions for (3.9) result in the following expression for the partial flows:

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and the total flow is then given by:

It should be noted that although (3.10) yields a relatively simple closed form for the partial flows, it does not constitute an entirely satisfactory model since the functions thus defined are not intrinsic. They depend on the local partial supplies and demands instead of the local geometric attributes such as the or the maximum flow. As such, model (3.10) is a purely ad hoc phenomenological model. The idea behind the choice (3.10) and the resulting expressions (3.11) is that: the partial flows

should be equal to their natural values, the quantities whenever possible, i.e. provided that the total flow does not exceed the total supply,

the partial flows natural values

should be as large as possible and proportional to their otherwise.

The supply-demand models of the present subsection provide natural boundary conditions: partial supplies and demands, which are fully compatible with network modeling. Another remark: all models of this section yield systems of conservation laws, the solutions of which have the usual properties (families of characteristics, Shockwaves ... ).

3.4.

Other models

A few other models exist in the literature. We may cite [22], which relies on a linear relaxation-type interaction between lanes. Another example is [10], although this last model constitutes a discretization of the LWR model only in a very loose sense.

4. 4.1.

Intersection modeling Exchange zones

Macroscopic modeling of intersections is difficult, since macroscopic models are based on the continuum hypothesis, i.e. the assumption that the traffic flow can be described by the macroscopic functions K, V. This assumption is only valid at a sufficiently large space and time scale, say 50 to 100 meters as far as the space scale is concerned, which exceeds the size of most intersections. The basic difficulty in modeling an intersection is that various movements may occupy the same physical location. Two answers are possible: consider overlapping cells (the SSMT solution, [13] and [6]) or consider exchange

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Macroscopic flow models

zones [1], in which movements overlap, and the details of this interaction are accounted for by global zone behavioral functions. Basically, an exchange zone is similar to a cell or segment (as in the Godunov scheme), but has multiple entry points and exit points and the following dynamic variables: point

the number of vehicles having entered the zone through entry exiting it through exit point the number of vehicles having entered the zone

through entry point the number of vehicles about to exit the zone through exit point the total number of vehicles of the zone, all of which, if need be, can be disaggregated according to some assignment attribute The mixing of movement flows in the zone is described by global zone supply and demand equilibrium functions, defined as link equilibrium functions, with the difference that their argument is N instead of and for a zone These functions yield the global zone supply and demand:

The zone dynamical variables are calculated by simple conservation relationships, for which the following ingredients are necessary: zone inflows zone outflows assignment coefficients and inflow composition. The inflows are obtained by comparing the zone partial supplies and demands to the upstream link (or zone) demands and downstream supplies in the usual way. The only difficulty is to deduce the zone partial supplies and demands from the available zone variables, which constitute the behavioral component of the model. For the partial demands, we

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recommend the following FIFO-like model:

Other models are possible (see [17]). For partial supplies, several models are possible depending on driver behavior. Hereafter we give three examples, relying on so-called supply-split coefficients equal for example to the relative number of lanes available in the zone for traffic entering through entry point model 1:

(linear model),

model 2: model 3: ual storage capacity).

(homogeneous zone model), (partial supply proportional to resid-

For complex intersections, several zones may be required, for simpler ones, such as merges and diverges, a simple zone is sufficient. All these partial supply and demand models rely on global zone supply and demand, which recapture the global behavior of the zone. The dynamic equations of traffic in a zone are given by:

with the assignment coefficients of the zone. is the proportion of users entering the zone through entry point at time and exiting it through exit point The above formula also relies on the usual asumption on the composition of the exit flow, supposedly the same as that of the zone itself. The outflows are given by

and the inflows are given by :

a formula similar to (3.11), and based on the same idea. Specifically, the inflows should be as great as possible, considering the total inflow, should be less than the zone supply. The inflows should also be proportional to their “natural values”

Macroscopic flow models

4.2.

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Pointwise models

Of course, to solve complex optimization problems related to dynamic assignment, these zone models are far too complicated and expensive in terms of computational requirements. A drastic simplification can be obtained by considering pointwise intersections. For instance in [2], pointwise merges and diverges were considered. In this section we adopt the following approach: we consider the intersection as a single zone, and study the limiting case in which the dimension of the zone is neglected. This way, part of the physical properties of the intersection, such as they are described by the zone equilibrium supply and demand functions and the supply and demand split coefficients, are carried over in a consistent way to the pointwise model. In order to calculate the zone supplies and demands say and in this case, as a function of upstream demands and downstream supplies the best approach is to calculate a stationary state of the zone, by considering the upstream supplies and downstream demands as constant. This means we consider the time-scale of the variation speed of upstream demand and downstream supply as infinitely large in regard to the variation speed of variables internal to the zone. Such an approach was applied successfully for the study of moving singularities in [19]. 4.2.1 Stationary solution of zone dynamics: the case of a merge. Let us consider the following merge:

and the simple linear supply split model giving the partial supplies: We shall consider only a stationary, i.e. a time-independent solution of the zone dynamics, for stationary data (downstream supply) and (upstream demands). We shall denote the maximum flow through the intersection, as resulting from the zone equilibrium functions. The main internal unknowns are the total number N and partial numbers of vehicles in the zone, but the main unknowns from the point of view of the intersection workings are the inflows and the total outflow The total zone supply and demand are given by:

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and the zone inflows

are given by:

and the total zone outflow is given by:

The equilibrium of the zone can be expressed by the following: and the partial numbers actually result from the partial inflows (at equilibrium): Therefore the only effective unknown is N. Now and are respectively concave decreasing and increasing functions of N, with Further, if N = 0, and and if (storage capacity of the zone). The existence of a solution results immediately. Supply Regime. This regime is characterized by is given and that concave and piecewise linear function of

The greatest possible value of regime is:

The equation yielding

being

and It follows that can be represented as an increasing which has the following aspect.

the validity domain of the supply

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can be solved by a Newton-like algorithm in O(I), with I the number of zone entry points which is described in detail in [18]. The algorithm is straightforward by virtue of the piecewise linearity of Demand Regime. This regime is characterized by and resulting from (4.3) with is given and that concave and trivially piecewise linear function of

It follows that is an increasing

The unknown is now which is determined by the equation: which admits a solution iff:

This last condition constitutes the validity range of the demand regime. Let us summarize these results:. and

results from (4.6), if the supply regime (4.5) applies.

and results from the equation demand regime (4.7) applies.

if the

global supply and demand, and the various flows are given by (4.2), (4.3) and (4.4). 4.2.2 Stationary solution of zone dynamics: the case of a diverge. Let us consider the following zone, modelling a diverge

The data are the downstream supplies and the upstream demand the total capacity of the diverge is the assignment coefficients are and the basic unknowns are the total number of vehicles N and the partial numbers All dynamic quantities, including of course the data, are assumed here to be stationary. As usual, the partial zone demands are given by the FIFO-like model: where we introduce the zone composition coefficients

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These coefficients satisfy the constraints:

The inflow is given by and the outflows are given by Supposing now that the flow in the upstream link satisfies the FIFO rule, it follows that the zone inflow has composition thus yielding the following stationarity conditions:

The unknowns are N and the Supply Regime. It is characterized by:

and the data the

and

and the

Hence:

The outflows are proportional to the assignment coefficients which are generically independent from the downstream supplies This observation is the key to the resolution. Let us define:

Generically, or as shown in [18]. The case is obviously a limit case of the demand regime, and will not be considered anymore. Let us consider that which determines the outflows which yields the validity range of the supply regime:

The solution is given by:

Demand Regime. It is characterized by and the zone equilibrium conditions are and All these quantities can be deduced from the data. The equilibrium condition at each zone exit point is:

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with unknowns

The above equilibrium condition implies that:

thus yielding the validity range of the demand regime. An argument similar to the one applied for the supply regime shows that:

This completes the analysis of the demand regime.

5.

Intersection models as solutions of optimization problems

Other models can be built according to a slightly different concept, which is the following. One way to express the nature of the entropy solution is to maximize flow, which then appears as the minimum of the upstream supply and downstream demand. This is the concept we extended to intersections by introducing partial supply and demand models in the intersection. Another way to recapture the concept of an entropy solution at intersections would be to maximize some concave increasing flow function, subject to supply and demand constraints. As usual, this means that transitory states, in the present case both inside and around the intersection are neglected. For instance in the case of a merge, with upstream demands and downstream supply as depicted hereafter,

we could find the flows

by considering the following problem:

with concave increasing functions describing the behavior of users at the intersection. The constraint could include the maximum throughflow of the merge, by replacing by if need be.

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Let be the KKT coefficient of the

with otherwise

constraint. It follows:

a decreasing function. If

then

results from the following equation:

The right-hand-side of that equation being a monotone increasing function in the numerical resolution is trivial. Now, let us consider the specific function

with split of the supply merge. If

The coefficients describe as usual the according to the linear model, i.e. the geometry of the then and

Otherwise, the equation yielding

becomes:

It follows:

is smaller than and thus equal to summarized by the single formula:

Both solutions can be

As already mentionned, an obvious drawback of these functions is that they are not intrinsic but depend on the data and of the problem. The above solution (5.1) is of course similar to (3.11), but also to (4.3) which were obtained by a very different approach. Let us note nevertheless that in formula (4.3) the downstream supply is replaced by the equilibrium zone supply which is the solution of (4.5), i.e.:

and therefore greater than for large supply values, and equal to for small supply values. The above model (5.1) is slightly different in this respect from the stationary zone model (4.3).

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A similar optimization problem can be considered for the diverge problem

Let us denote as usual the downstream supplies, the upstream demand, the diverge outflows, and the assignment coefficients. We shall also denote the total diverge inflow. We would try to optimize some function:

subject to supply and demand constraints, with the concave increasing functions. The supply and demand constraints are trivial:

In order to complete the model we have to relate the total diverge inflow to the diverge outflows which is to say we have to complete the behavioral part of the model. The simplest possibility is to assume FIFO behavior of upstream vehicles. This would be the case if the upstream link has only one lane, or no preselection lanes. It follows:

and the constraints can be reduced to:

The resulting optimization problem admits the following solution, considering that the functions are increasing:

This model was already obtained with the stationary zone model of 4.2.2. It is also similar to the discretized diverge model of [3]. A non FIFO model can be obtained by assuming that the outflow constraints should be:

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expressing that the coefficients are demand composition coefficients. Such a behavioral model requires a non FIFO flow model for the upstream link, such as the lane assignment or the partial supply/demand models. The resulting constraints are and the (trivial) solution of the corresponding optimization problem is, (since the functions are increasing):

For the diverge models we have seen, the main behavioral elements are the demand split coefficients and models, the assignment coefficients and the type of upstream flow, FIFO or non FIFO. It should also be noted that the solutions (5.2) and (5.3) do not depend on any specific choice of the functions. The different macroscopic approaches to intersection modelling we have developed in the present paper appear to be consistent. The pointwise models of subsection 4.2 result from the zone models at the vanishingly small timespace scale limit, and yield formulas similar to the flow maximization models of the present subsection. For practical purposes, the simpler flow optimization approach should therefore be preferred in most cases. Another interesting point: the models developed in this section are completely consistent with the supply-demand link flow models.

6.

An experimental validation

In this section we shall interpret some experimental obtained and analyzed by Oltra and Jardin [24]. The several experimental sites these authors considered were all of following the merge type:

and the traffic conditions were oversaturated. The authors made experimental observations of the flows and and tried to calculate linear relationships between these flows through linear interpolation of the data. In order to validate the merge model developed in section 5, we assume a linear supply split model with coefficients chosen as the ratio between the number of lanes of the upstream link and the downstream link. Assuming one of the links to be oversaturated, the following inequalities result (see [18], an expanded version

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of the present paper, for details):

and, if both links

are oversaturated, the following identity results:

The deterministic scatter predicted by the model is the expression of a hidden variable: the demand upstream of the link that is not oversaturated. This prediction fits the data much better than the linear relationships considered by the authors. The following figure illustrates the data points as compared to the model predictions:

7.

Conclusion

We have shown that the LWR model can be extended to networks. There is no unique extension, even by sticking to entropy solutions: indeed, user behavior plays a major role both in link modeling (FIFO behavior should be assumed or not depending on infrastructure and traffic management) and in intersections, in which it determines how upstream demand and downstream supply interact. The discretization of the models described in this paper is trivial and in some cases available ([1]), but the investigation of analytical solutions is yet to be done even in the simplest cases. Further, little field evidence is available at this point as far as intersection modeling goes. There is some support for the split-demand and split-supply intersection models [24], as we have seen, nevertheless experimental validation should be an essential objective for future investigations.

References

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[1]

C. Buisson, J.P. Lebacque, J.B. Lesort, H. Mongeot “The STRADA model for dynamic assignment”. Proc. of the 1996 ITS Conference. Orlando, USA. 1996.

[2]

C.F. Daganzo, “The cell transmission model 2: network traffic simulation”. Transportation Research 29B. 2: 79-93. 1995.

[3]

C.F. Daganzo, “A finite difference aproximation of the kinematic wave model”. Transportation Research 29B. 261-276. 1995.

[4]

C.F. Daganzo, “Requiem for second order fluid approximations of traffic flow”. Transportation Research 29B. 277-286. 1995.

[5]

C.F. Daganzo, “A continuum theory of traffic dynamics for freeways with special lanes”. Transportation Research 31 B. 83-102. 1997.

[6]

E. Elloumi, H. Hadj Salem, M. Papageorgiou, “METACOR, a macroscopic modelling tool for urban corridors”. Proc. of the TRISTAN II Int. Conf.. V1, 135-149. Capri. 1994.

[7] T.L. Friesz, D. Bernstein, T.E. Smith, R.L. Tobin, B.W. Wie, “A variational inequality formulation of the dynamic network user equilibrium problem”. Operations Research 41, 179-191. 1993. [8]

E. Godlewski, P.A. Raviart, “Hyperbolic systems of conservation laws”. SMAI. Ellipses (Paris). 1991.

[9]

B.G. Heydecker, J.D. Addison, “Analysis of traffic models for dynamic equilibrium traffic assignment, in “Transportation networks: recent methodological advances” (M.G.H. Bell ed.). 35-50. Pergamon. 1997.

[10] M. Hilliges, “Ein phänomenologisches Modell des dynamischen Verkehrsflusses in schnellstraßennetzen”. PHD dissertation. Institut für theoretische Physik der Universität Stuttgart. Shaker Verlag. 1995. [11] M. Kuwahara, T. Akamatsu, “Decomposition of the reactive dynamic assignments with queues for a many to many origin-destination pattern.” Transportation Research B. B 31, 1-10. 1997 [12] R. Jayakrishnan, H.S. Mahmassani, T.Y. Hu, An evaluation tool for advanced traffic information and management in urban networks”. Transportation Research C, C2: 129-147. 1994. [13] J.P. Lebacque, “Semimacroscopic simulation of urban traffic”. Proc. of the Int. 84 Minneapolis Summer Conference. AMSE. V4, 273-291. 1984. [14] J.P. Lebacque, “L’échelle des modèles de trafic, du microscopique au macroscopique”. Annales des Ponts, 74. 48-68. 1995. [15] J.P. Lebacque, “The Godunov scheme and what it means for first order traffic flow models”. Proc. of the 1996 ISTTT(J.B. Lesort ed.). 647-677. 1996.

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[16] 1. J.P. Lebacque, “Instantaneous travel times for macroscopic traffic flow models”. CERMICS Report 59-96. 1996. 2. C. Buisson, J.P. Lebacque, J.B. Lesort, “Travel times computation for dynamic assignment modelling”, in “Transportation networks: recent methodological advances” (M.G.H. Bell ed.). 309-319. Pergamon. 1997. [17] J.P. Lebacque, “Rapport du Contrat LICIT-CERMICS 1997”. 1997. [18] J.P. Lebacque, M.M. Khoshyaran. “First order macroscopic traffic flow models for networks in the context of dynamic assignment.” CERMICS Report. To be Published. [19] J.P. Lebacque, J.B. Lesort, F. Giorgi, “Introducing buses into first order macroscopic traffic flow models”. Transportation Research Record 1644: 70-79. National Academy Press. 1998 [20] D.R. Leonard, P. Gower, N.B. Taylor, “CONTRAM, structure of the model”. TRRL Report RR 178. Crowthorne (U.K.). 1989. [21] M.H. Lighthill, G.B. Whitham, “On kinematic waves II: A theory of traffic flow on long crowded roads”. Proc. Royal Soc. (Lond.) A 229: 317-345. 1955. [22] P.G. Michalopoulos, D.E. Beskos, Y. Yamauchi. “Multilane traffic flow dynamics: some macroscopic considerations”. Transportation research 18B, 377-395. 1984. [23] D.K. Merchant, G.L. Nemhauser, 1. “A model and an algorithm for the dynamic traffic assignment problem”. 2. “Optimality conditions for a dynamic assignment model”. Transportation Science 12: 183-199 & 200-207. 1978 . [24] C. Oltra, P. Jardin. “Etude des écoulements de trafic sur les voies rapides pendant la congestion”. RTS 54. 3-14. 1997. C.Oltra, “Etude des écoulements de trafic sur les voies rapides pendant la congestion: approche macroscopique”. Rapport de DEA. ENPC - Université Paris XII. 1994. [25] B. Ran, D.E. Boyce, “Modeling dynamic transportation networks”. Second revised edition. Springer Verlag. 1996. [26] P.I. Richards, “Shock-waves on the highway”. Op. Res. 4: 42-51. 1956. [27] M. Van Aerde, “INTEGRATION: A model for simulating integrated traffic networks”. Transportation Systems Research Group. Queen’s University, Kingston, Ontario. 1994.

Chapter 9 AIMSUN2 SIMULATION OF A CONGESTED AUCKLAND FREEWAY

John T Hughes

[email protected] Regional Transportation Engineer Transit New Zealand, P. O. Box 1459 Auckland, New Zealand

Abstract

The Spanish AIMSUN2 microscopic traffic simulation package is being used to model a 9.7 km section of a congested urban freeway in Auckland, New Zealand. An outline of the model building process is presented along with selected findings. Flow profiles based on field data from a four-hour morning commuter traffic peak have been applied to the modelled network. Preliminary results show that modelled speed and flow relationships realistically reproduce those found in field data from the freeway. Exact reproduction of transient effects, such as timevarying lane utilisation ratios and queue lengths have not been fully achieved in the model. Further investigation into appropriate values for sensitive lane-change parameters are expected to improve this aspect of the model’s performance.

1. Introduction and objectives The objective of this paper is to present an outline of the model building process and selected findings from a traffic simulation model of a congested 9.7 km section of an urban freeway in Auckland, New Zealand. Field data were collected to provide a comprehensive set of New Zealand traffic characteristics for the freeway micro-simulation model. The data set constitutes a detailed "snapshot" of traffic conditions over one week (23 to 29 September 1997), with additional detail on particular days. The data, which include traffic speeds, volumes, headways, accelerations and lane change counts, are being used to 141

M. Patriksson and M. Labbé (eds.), Transportation Planning, 141–161. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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build an AIMSUN2 traffic simulation model. This is a new tool that can provide input into transport investment decisions in Auckland.

2.

Simulation model

AIMSUN2 (Advanced Interactive Microscopic Simulator for Urban and Non-Urban Networks) is a microscopic, stochastic model for simulating traffic on road networks. It is part of the GETRAM (Generic Environment for Traffic Analysis and Modelling) software suite developed at the Universitat Politècnica de Catalunya in Barcelona, Spain. The model described in this paper was produced using version 3.0 of GETRAM. The software has since been updated, the current version 3.3 was released in May 1999 and contains several new features which were not used in the work reported here. GETRAM (Barceló and Ferrer 1997, Montero et al 1998) consists of a userfriendly graphical interface, a traffic network graphical editor (TEDI, Traffic Editor) supporting any kind of road type or network geometry, a network database and a module for storing and presenting results. It includes an animated simulation display that shows vehicles moving through the network.

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3. AIMSUN 2 simulation process The logic of the AIMSUN 2 simulation is shown in Figure 1. It can be regarded as a hybrid simulation combining scheduled events with activity scanning. At each simulation time step unconditional events such as traffic signal changes, are updated as per a scheduling list. After this a set of nested loops update the states of the entities (road sections and intersections) and vehicles in the model. Once all entities have been updated the remaining operations are performed i.e. input of new vehicles entering the model and collection of output data. If a route based model is being used a shortest route component periodically calculates the new shortest routes according to travel times provided by the simulator. The route selection model assigns the vehicles to these routes during the current time interval. Vehicles continue to follow these routes from the origin to their destination unless they have been identified as guided vehicles that can then dynamically change their route midway along them. The model can simulate a range of traffic management features including incident detection and surveillance systems, variable message signs and wide area traffic control strategies. Simulating predictive control and guidance strategies are also feasible. AIMSUN2 has a wide variety of possible applications in traffic management of Auckland’s congested arterial road network. In particular it may be an important enabling technology to serve as a testbed for an Advanced Traffic Management System (ATMS) being developed in Auckland. Transit New Zealand (Transit), the national State Highway authority, is implementing the ATMS, the first portions of which are scheduled to be operational by the year 2000. The traffic data collected on Auckland’s Southern Motorway are being used to calibrate the model and determine the accuracy with which it can represent real traffic flows. If it is shown that the model is accurate over a section of motorway by comparison with extensive measured data then confidence can be had in simulations of other motorway sections and features where less comprehensive field data is available.

4.

Study area and scope

Auckland is located towards the north of New Zealand’s North Island. With a population of some 1.1 million people and is growing at 2.5% per year it is the country’s largest urban centre. The study area (Figure 2) is located south of the central business district (CBD) and passes through the region’s core industrial areas of Penrose and Mt Wellington. The

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study section is a 9.7 km length of the Southern Motorway extending from Panama Road (just south of Mt Wellington Highway) in the south to Khyber Pass in the north. Traffic in New Zealand drives on the left side of the road.

The motorway section crosses relatively flat terrain with an isolated maximum grade of 4.0% and the balance at 3.0% or less. In 1998 it carried bi-directional Average Annual Daily Traffic (AADT) volumes ranging from 101,000 vehicles per day (vpd) in the south to 194,000 vpd at Khyber Pass. To obtain information that was comprehensive, yet at a sufficiently high level of detail, different types of data were collected simultaneously at varying intervals along the motorway (Hughes 1998). Automatic count data were collected over a full 7-day week. Resource-intensive data collection methods, including video taping, aerial photography and laser-gun speed profiles, were measured on a single day, mostly within that week. Most of the data are being used to calibrate the AIMSUN2 model. Some data not used in the calibration will be used as benchmarks to which simulation model output is compared in order to validate the model.

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5. Model development The work reported in this paper is a preliminary investigation into the ability of the model to reproduce traffic flows in the northbound direction of this motorway corridor. The general approach taken was to define the motorway links, apply field-measured traffic flows at the network boundaries and then calibrate the model by tuning parameters to seek agreement with measured data at intermediate points within the corridor. In common with many simulation models (Wang and Cassidy 1995, Hua Heng 1989, Quadstone 1996) AIMSUN2 requires input information defining the road network geometry, traffic stream conditions and driver and vehicle characteristics.

6.

Geometric information

For this project the motorway geometric layout was obtained as a CAD (“dxf”) file showing kerb lines and edges of the road pavement. Additional information on the widths and number of lanes and the lane configurations at ramps was obtained from a variety of historic construction drawings and aerial photography. The dxf map file was imported into TEDI and roadway links were created using the section drawing tools over the map background. Alternatively it would also have been possible to import an EMME/2 network model, with its centriod connector structure, directly into GETRAM using an optional module available for this purpose (Montero et al 1998). Table 1 shows the basic road section parameters adopted in this model from field studies of the southern motorway. The capacities are from maximums observed in the field (Hughes 1998).

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Traffic flow information

Traffic flow information includes trip demands across the network, fleet composition (cars, trucks, buses etc.) and traffic control mechanisms such as traffic signals and intersection priority signage. Between 23 and 29 September 1997 traffic data were collected on the study section of motorway. Detector sites included permanent loop classifiers and temporary video classification on the motorway through lanes and unclassified counts collected on all 17 ramps entering and leaving the motorway study section (Hughes 1998). In the past micro-simulation models have tended to require defined traffic flows on each entry link to the network and specified turning percentages (by vehicle type) at each intersection or off-ramp. With this approach the vehicles entering the network have no "knowledge" of their intended route or destination. The more recent trend in microscopic simulation is for traffic inputs to be defined as time-sliced origin-destination matrices. This allows greater flexibility in modelling traffic scenarios and problems involving route assignment can be investigated at the microscopic level. AIMSUN2 allows both methods of traffic data input.

8.

Trip matrices

Trip matrices were obtained from a 1992 origin-destination postcard survey for three different vehicle types during the 7.00am to 9.00am morning commuter peak. These were manually adjusted by factoring in a spreadsheet to approximately match measured traffic flows entering and leaving the motorway section on 26 September 1997. The result was a total of 48 matrices (ie. 3 vehicle types per quarter hour from 6am to 10am) which were applied to the study network.

9. Driver and vehicle information A range of characteristics affect the way vehicles and drivers travel through a road network. These include mechanical attributes of the vehicle (eg: size, performance) and aspects of driver behaviour (eg: desired speed, acceleration and gap preferences). In GETRAM this type of information is input as parameters pertaining to vehicle types, where any number of types can be defined by the user.

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The data that may be entered for each vehicle type are shown in Figure 4. These include desired speed, acceleration, normal and emergency deceleration, maximum yield time and minimum vehicle spacing when stopped in a queue. The queuing up and queue leaving speeds control whether or not a vehicle will enter an intersection that contains vehicles which are "queued", as defined by these parameters. In a traffic stream these data may vary stochastically between vehicles. For each data item (desired speed, acceleration etc.) the values attributed to individual vehicles are considered normally distributed and the user may define the distribution parameters (mean, standard deviation, minimum and maximum values). The three vehicle types currently being used for this study are shown in Table 2.

AIMSUN2 does not use vehicle weight as a model parameter. However, pending more specific data, the weight classes have been assumed to correspond to the length classes shown in Table 2. This enabled trip matrices from the earlier postcard Origin – Destination survey to be used in the study. The following data were obtained from individual vehicles recorded at three of the motorway detector sites on 26 September 1997.

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Free speed data were derived from a sample of vehicles travelling at a time gap of 10 seconds or more behind the vehicle ahead. A small number of data records were discarded as outliers. These were 27 doubtful length records (ie. < 1.9 or > 23.0m) and 17 suspect free-speed records (ie. < 60 km/h).

10. Maximum vehicle acceleration As each simulated vehicle enters the modelled road network it is assigned three speed-change parameters. These are its maximum acceleration rate and its normal and maximum deceleration (or emergency braking) rates. Although it is difficult to measure speed change parameters of a whole fleet and driver population, some information was gleaned from three local sources. These were a study of traffic decelerating on a motorway off-ramp (Bennett 1993), instrumented vehicle trials for development of a fleet emissions control strategy (Ministry of Transport 1997), and laser gun speed change measurements by the author. Bennett recorded average deceleration rates from 2,000 vehicles on Auckland’s Grafton off ramp of between and for various approach and final speeds. He also cites a study on urban streets in Palmerston North, New Zealand (ATS 1990) which reported the following maximum rates for vehicles travelling at less than 70km/hr.

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The New Zealand Ministry of Transport (MOT) conducted an extensive series of trials using 23 instrumented vehicles which followed qualitatively defined drive cycles on Auckland area roads (Ministry of Transport 1997). The vehicles were all cars of various ages and conditions with engine capacities ranging from 1.3 to 4.1 litres. The results included the following:

The author measured accelerations at Dilworth footbridge on Auckland’s Southern Motorway and at a signalised urban arterial intersection on Quay Street. Speed changes were measured over successive pairs of observations of each vehicle. Scatter plots of the data are shown in Figure 3.

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Each of the three sources of field data gives an indication of the range of speed-change values appropriate for the model. However none of them closely corresponds to the desired statistics, namely the probability distributions for the maximum accelerations and decelerations experienced by the population of vehicles and drivers. Figure 4 shows the vehicle parameters adopted for the model runs reported in this paper.

11.

Motorway model

A model has been constructed in GETRAM of the northbound lanes of the study section of the Southern Motorway. It consists of the three motorway through lanes and a short length of each interchange on- and off-ramp. Traffic has been applied to the network as three vehicle types (Cars, LCVs and HCVs). A trip matrix was produced for each vehicle type for each 15 minute time period during an extended morning commuter peak, from 6.00am to 10.00am. The traffic flows applied to the model are an approximation to the actual flows that existed on the motorway on Friday 26 September 1997. The raw traffic data measured in the field were manually adjusted to make up for several deficiencies. These included the lack of length classification on the ramps, under - counting due to equipment faults and the fact that the video classifier

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sites operated only for the middle two hours of the four hour extended peak. These gaps in the field data were filled by comparison with flow data from other days and missing length classification percentages on the ramps were assumed from the 1992 postcard survey. An accident just south of the study area blocked lane 1 (the leftmost lane) for about 5 minutes at 7.30am on the Friday morning. This resulted in flow reductions entering the study area and a corresponding increase in speeds. The effect was removed from the traffic entering the model by averaging the flow rates before and after the blockage period. After entering the road sections and trip matrices the model was run and some parameters adjusted by trial and error to try to replicate traffic conditions observed in the field.

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12. Model Outputs In Figure 5 average speeds and flows over 5-minute intervals from virtual traffic detectors in the model are compared with field data measured at the same location on the motorway in the morning and afternoon peak periods on Friday 26 September. There is reasonable agreement between the model and field data at three of the four sites. The model under-predicted speeds by some 20 km/hr in Lane 1 at Panama Road. The model output for Lane 2 at Panama Road realistically demonstrates both free-flow and congested traffic regimes. However the transition between the two conditions is more abrupt than indicated by the field data.

13. Lane Utilisation Figure 6 shows modelled and field flow rates in each motorway through lane and on-ramp at the Ellerslie Interchange. Although the modelled ramp and lane flows generally follow the field data the model is over-predicting traffic counts in Lane 1 and under-predicting them in Lanes 2 and 3 for the

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first 1.5 hours of the simulation period. It is noted that subsequent runs using enhanced lane change parameters available in version 3.3 of GETRAM have shown closer agreement between field and modelled data than indicated in Figure 6.

14. Motorway Speeds Comparisons between field measured speeds and model outputs on the motorway through lanes are shown in Figure 7. Significant congestion develops at Greenlane in the model from 7.10 am when the average speed drops from 80 to 40 km/hr. The severity and duration of congestion exceeds that observed in the field. The queues propagate upstream until they are observed in the model as a sudden breakdown of flow at Mt Wellington lasting from 9.05 am to 9.50 am. This model does not currently reproduce the exact timing of queues observed in the field on that particular day. However it exhibits a qualitative realism in reproducing these transient dynamic phenomena. This suggests that a better fit should be able to be achieved with further tuning of the calibration parameters.

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Greenlane Northbound On-Ramp

Figure 8 shows an aerial photo of the motorway north of the Greenlane on-ramp and the graphical model displays at the same moment in the simulation. The bunching of traffic as on-ramp vehicles merge into the through lane flows can be clearly seen in both the photo and the simulated outputs. There is good agreement between the field and modelled vehicle lane densities.

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16. Calibration Parameters In addition to the vehicle characteristics noted previously there are several other parameters to be considered in building an AIMSUN2 model. These parameters and their values currently adopted in the Southern Motorway model (GETRAM version 3.0) are shown in Table 7.

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The GETRAM default values for Driver Reaction Time and Cruising Tolerance have been used in this model. By default the desired speed applied to turning vehicles is calculated automatically by reference to the deflection angle of the turn. The user may, if desired over-ride this by specifying a fixed value on any section. The Visibility Distance defines the distance before the end of each road section at which the gap acceptance model is applied (if required). For example, suitable gaps in conflicting traffic streams will be sought for each vehicle approaching within this distance of an intersection. Although, in reality, sight distance is generally good on the Southern Motorway the value of 200m used in this model may be excessive and shorter distances should be evaluated. Distance Zones 1 and 2 are lengths ahead of the end of a section within which vehicles will respectively seek, or force, a lane change to accomplish a required turn or merge. Vehicle behaviour in the model, especially lane occupancy, is clearly sensitive to this parameter and further investigation of a range of values is required. Queuing Up and Queue Leaving speeds control whether vehicles will enter an intersection which is partially blocked by other queued vehicles. Although not applicable in a motorway-only model, values much lower than shown in Table 7 have been found to lead to gridlock in a dense urban street network model.

17.

Run times

The 4 hour simulation of this 9.7 km section of northbound traffic on the Southern Motorway took 9 minutes and 6 seconds to run in batch mode on a Pentium 166MMX personal computer.

18.

Conclusion

At this initial stage of the model building and validation process, AIMSUN2 appears to provide a high degree of realism in simulating congested traffic flows on Auckland’s Southern Motorway. Basic speedflow relationships are reproduced well but detailed simulation of transient lane utilisation and queuing effects has not yet been achieved. Further investigation is required into appropriate values for sensitive lane-change

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parameters, especially section “Visibility Distances” and “Distance Zones 1 and 2”, in order to improve these aspects of the model’s performance.

19. Postscript Since the model runs reported above several software changes have been made in version 3.3 of GETRAM (released in May 1999). The “Distance Zone” parameters, which were global variables (where the same values were applied on all sections in the model) are now local variables and can have different values on each road section. The definition and behaviour of the vehicle “Speed Acceptance” parameter (see Figure 4), which controls the degree to which vehicles comply with posted speed limits, has been changed and an enhanced lane-change algorithm has also been added. Recent runs of this Southern Motorway model using version 3.3 of GETRAM have shown closer agreement between field and simulated traffic speed and flow data than indicated in Figures 5 to 12.

20. About the author The author leads Transit New Zealand’s Transportation Planning Section in Auckland. In this role he manages the transportation planning phases of new State highway and freeway projects in the Auckland urban area. This motorway simulation-modelling project is the subject of the author’s Master of Engineering thesis at the University of Auckland.

21. Disclaimer The opinions expressed in this paper are the author’s and do not necessarily represent the views of Transit New Zealand.

22.

Acknowledgements

Transit New Zealand funded the work described in this paper. Development of the motorway model is part of a collaborative research project with Transportation Simulation Systems Ltd (TSS) of Barcelona. Post-graduate student M. Kamruzzaman assisted with the model building and collection and reduction of the field data.

References

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ATS (1990). Acceleration/Deceleration Profiles at Urban Intersections. Report to Transit New Zealand, Australasian Traffic Surveys, Victoria, Australia. Barcelo, J. and Ferrer, J. L. (1997). An Overview of AIMSUN2 Microsimulator, Department of Statistics and Operations Research, Universitat Politèchnica de Catalunya. 16 pp. Benekohal, R. F., (1997) Procedure for Validation of Microscopic Traffic Flow Simulation Models, Transportation Research Record No 1320, Department of Civil Engineering, University of Illinois, TRR 1320, pp 190 – 202. Bennett, C. R. (1993) Revision of Project Evaluation Manual Speed Change Cycle Costs, N. D. Lea International NZ Ltd., Doc. No 8341, 53 pp. Hua Heng, T. (1989) Simulation of Traffic Flow on Dense Urban Street Networks: A Study of the Calibration Requirements of the FHWA-Netsim Model for New Zealand Conditions, ME Thesis, Department of Civil Engineering, University of Auckland Hughes, J. Intensive Traffic Data Collection for Simulation of a Congested Auckland Motorway (1998). Proceedings 19th ARRB Transport Research Conference, Sydney, Australia, 15pp. Ministry of Transport (1997) Vehicle Fleet Emissions Control Strategy for Local Air Quality Management. Stage 1, New Zealand Ministry of Transport, 201pp. Montero, L, Codina E., Barcelo, J., and Barcelo, P (1998). Combining Macroscopic and Microscopic Approaches for Transportation Planning and Design of Road Networks. Department of Statistics and Operational Research, Technical University of Catalonia (Barcelona, Spain) (Prepublication draft). Quadstone Ltd. (1996) Paramics, Wide-Area Microscopic Traffic Simulation, UK Motorway Validation Report. Quadstone Ltd. 25 pp. Transit New Zealand (1996). Highway Information Sheets Region 2 Auckland. Transit New Zealand. 28 pp. Transportation Research Board (1994). Highway Capacity Manual.

Chapter 10 FUZZY TRAFFIC SIGNAL CONTROL Jarkko P. Niittymäki Helsinki University of Technology [email protected]

1.

Introduction

Traffic management is an integral part of urban management, and transport planners have traditionally concentrated on the movement of vehicles as the major aim of this process. The traffic signal design can be viewed as measures of performance of intersection operation criteria or, in other words, desirable outcomes: hence a decrease in each of delay, number of stops, fuel consumption, pollutant emissions, noise, vehicle operating costs, queue length and personal time as well as an increase in consideration for pedestrian, bicycle and transit traffic and safety are all desirable. The functioning of traffic signal control has a significant effect on the environment. The general main goal is that the number of stops has to be minimized at the level of transportation system, while at the level of one intersection the delays have to be minimized. The aim of this paper is to present a systematic of fuzzy traffic signal control, and to discuss and compare different fuzzification and defuzzification methods for our FUSICO-project at the Helsinki University of Technology.

2. 2.1.

Fuzzy traffic signal control Fuzzy set theory in traffic signal control

Fuzzy logic allows linguistic and inexact data to be manipulated as a useful tool in designing signal timings. It also provides a mean of converting a linguistic control strategy, which is expressed by if-then-else-statements, into a control algorithm. Fuzzy logic has the ability to comprehend linguistic instructions and to generate control strategies based on verbal communication. The motivation for designing a fuzzy controller is that there is a fairly direct relationship between the loose linguistic expressions of a traffic control strategy and its manual implementation. It is important that the fuzzy algorithms have 163

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the distinct advantage of not relying on a mathematical transfer function for formulating a control strategy. Instead, the design of a fuzzy signal controller requires the expert knowledge and experience of traffic control in formulating the linguistic protocol, which generates the control input to be applied to the traffic signal control system ([1]). The fuzzy statement protocol is a fruitful technique for modeling the knowledge and experience of a human operator. Thus, traffic signal control is a suitable task for fuzzy control. Indeed, one of the oldest examples of the potentials of fuzzy control is a simulation of traffic signal control in an intersection of two one-way streets. Even in this very simple case the fuzzy control was at least as good as the traditional adaptive control ([2]). In general, fuzzy control is found to be superior in complex problems with multi-objective decisions ([3]). In traffic signal control several traffic flows compete for the same time and space, and different priorities are often set to different traffic flows or vehicle groups. Normally, the optimization of traffic signals includes several simultaneous conflict criteria, like the average delays of vehicles or pedestrians, maximum queue lengths and percentage of stopped vehicles. So it is very likely that fuzzy control is very competitive in complicated real intersections where the use of traditional optimization methods is problematic. In reality, we cannot handle the approaching traffic exactly. The control possibilities are complicated, and, especially, to handle them is too complicated. Maximizing safety, minimizing environmental aspects and minimizing delays are the objectives of control, but it is not possible to handle them together in the traditional traffic signal control. The causal-relationship is also not possible to explain in traffic signal control. In adaptive traffic signal control the increase in flexibility increases the number of overlapping green phases in the cycle, thus making the mathematical optimization very difficult. Particularly for that reason, the adaptive signal control in most cases is not based on precise optimization but on the green extension principle.

2.2.

Structure of fuzzy signal control process

The fuzzy signal control process consists of seven parts; the current traffic situation with signal status, the detection or measuring part (crisp input), traffic situation modeling, the fuzzification interface, fuzzy inference (fuzzy decision making), defuzzification, and signal control actions (for example, extension or termination of signal group) (Figure 10.1). Fuzzy control has been developed in the context of fuzzy inference. Fuzzy inference is the inference process based on the multi-value logic of inference; in other words, the truth values of input and rules of the inference process are not singular (yes or no), rather, they are multi-valued. As a result, the truth of

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the conclusion is given a value between 0 and 1, which more closely resembles the inference performed by a human - the ambivalence of the human decision process. The essence of this inference is the use of fuzzy set for representation of the input and the rules (relation). A number of reference materials related to fuzzy sets, inference and control are available ([4]). The fuzzy traffic signal inference is based on the generalized modus ponens (GMP), which states: “If A is true, and A implies B, then B is true”. In this statement, A implies B is the inference rule, where A is the antecedent and B is the consequence

The values of fuzzy sets (A,B) can be partially true. In a fuzzy inference system, the rule base consists of several implications Input : Rule : Consequence :

x is A' and y is B' if x is A and y is B then z is C z is C'

and in terms of the fuzzy relations

Our application examples in this paper are an isolated signalized pedestrian crossing and two-phase control. In the signalized pedestrian crossing case, the main goal of fuzzy control is to give pedestrians an opportunity to cross the street safely, and with minimum waiting time, but also that the risk of rear-end collisions is minimized. The input parameters WT (pedestrian waiting time) and A (approaching vehicles) were chosen, because they represented the main goals of rule base. The S (discharging queue indicator) was chosen, because it is not common to terminate vehicle green while the queue is discharging. The total number of rules is 18, and the final decision is the extension (E) or the termination of the green of the vehicle signal group. The second example is two-phase control. The rule-base structure has been divided into extension intervals. Each interval has only four rules and two input parameters. The main goals of these rules are to adjust the cycle time and, to divide the cycle into the green times of phases.

The input parameters A (approaching vehicles, green phase), Q (queuing vehicles, red phase) and the rule structure try to emulate the decision algorithm of a human (an experienced policeman controlling traffic).

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Our FUSICO-rule base also contains fuzzy inferences for multi-phase control, public transport priority, pedestrian signal group control and isolated control at arterial roads. The first fuzzy traffic signals are working in Oulunkylä, Helsinki, using the fuzzy inference of the two-phase control. Otherwise, the testing and development is done using the microscopic simulation. The aim of this paper is to present different methods for fuzzification and defuzzification (dark parts). This study gives some recommendations to our FUzzy SIgnal Control-project (FUSICO).

3. 3.1.

Fuzzification interface Basic principles of fuzzification

In everyday situations, linguistic terms whose definitions are not so clear, are used for easy and efficient communication. Equivalent expressions with exactly defined terms are very difficult to achieve. The linguistic terms are normally easy to select and to use in our daily lives. However they include some kind of uncertainty, because we understand them in common terms (not exact) and in a context dependent way. The fuzzification process involves the scale mapping of the measured input variables into the corresponding universes of discourse. This process includes an evaluation of the membership ratio of the crisp input with respect to each fuzzy set x of the input universe

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where fuzzifier represents a fuzzification operator. The operator converts the crisp values into suitable linguistic labels of fuzzy sets. The most important aspects relating to the fuzzification are the universe dividing into a certain number of segments (fuzzy variables) and the membership functions of the fuzzy sets. The fuzzy sets describe terms of linguistic variables. The meaning of a fuzzy linguistic term is defined by the membership function, because it indicates a grade of membership of each element in a fuzzy linguistic set of interest. This means that the physical meaning of a linguistic term is characterized by the membership function, which is assigned by a person intending to use this term. The shape of a membership function is quite free, but the typical feature is some kind of smoothness (Z-shaped, Bell-shaped, S-shaped). The triangleshaped (trapezoidal) functions, as in our traffic signal control application, are also commonly used. When the shape of the fuzzy sets are determined, several other parameters have to be adjusted. The choice of fuzzy variables has a substantial influence on the sensitivity of control, but there is no unique solution for that. The optimal partition can be achieved by a heuristic method, but the basic principle could be to use our real life linguistic terms. For example, the pedestrian waiting time at the signalized intersection can be short, long or very (too) long. The membership functions representing linguistic values of a linguistic variable should describe the nature and properties of the linguistic variable. The methods of constructing membership functions can be divided into direct and indirect methods. The direct method means that experts try to find answers to the questions what is the membership degree of x in S and which elements x have the degree of membership By answering these questions, a set of pairs can be defined, and the membership functions can be constructed using some curve-fitting method. On the other hand, sometimes it is easier to compare the degrees to which elements belong to S than to give the actual degree of membership for each element as in the direct methods. An expert makes pairwise comparisons between elements of the universal set U with respect to how much they belong in S.

3.2.

Fuzzification Examples at FUSICO-project

In our FUSICO-project, the membership functions have been defined by using the direct method. The triangle-shaped membership functions are used in our project. This means that three parameters [p1,p2,p3] define the corners of the membership functions.

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The membership function example of our application of the isolated pedestrian crossing is shown in Figure 10.2. Niittymäki and Kikuchi [5] prepared membership functions of the pedestrian waiting time (WT; short, long, too long) for three cases: (1) normal, (2) pedestrian friendly, and (3) vehicle friendly. As seen, the difference between the three cases is the location of the membership function along the time(X)-axis. “Friendliness” to either party can be controlled by moving the membership functions along the time axis. The results are shown in Figure 10.3. According to the results, when the pedestrian friendly membership function is assumed, the pedestrian delay is clearly lower. Naturally, the vehicle delay varies depending on the pedestrian volume. Obviously a higher pedestrian volume gives a similar pedestrian delay, but the vehicle delay is shorter when the vehicle friendly membership functions are used for WT. Correspondingly, the percentage of stopped vehicles increases with pedestrian volume and with the use of pedestrian friendly membership functions. This finding increases the adaptivity of the fuzzy controller. For example you can change your control policy for different traffic situations: 1 When pedestrian handling is important, then use the pedestrian friendly membership functions.

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2 When the vehicle volume is large, then use the vehicle friendly membership functions. Our second example is traditional two-phase control. Bingham (1998) studied neuro-fuzzy methods for traffic signal control in the FUSICO-project. The objective was to construct a neural learning algorithm, which modifies the parameters of fuzzy membership functions. The advantages of the calibration of the membership function using the neuro-fuzzy-technique are the better efficiency and the systematic handling of the membership functions ([6]). The first results show that the practical application of the neuro-fuzzy-technique can be the adjustment of the membership functions for different detector locations. In our FUSICO-project, the ideal detector location is 100 m before the stop line, but in reality, the detectors can be located even 40–60 m before stop line. This finding helps us to develop a general fuzzy rule base for traffic signal control, because the calibrated membership functions adapt the current traffic infrastructure.

4. 4.1.

Defuzzification of outputs Defuzzification principles

Because the outputs of decision rules are fuzzy, we need some kind of defuzzification method to achieve a crisp output for the final control action. The defuzzification process is an important step. In general, there is no systematic procedure to choose a defuzzification strategy. The defuzzification process involves a mapping from a space of fuzzy control actions into a space of nonfuzzy control actions. This procedure is inverse to that of the fuzzifier. The initial data value, y, consists of the membership values of the current output with respect to all the output fuzzy subsets of the output space,

where is the nonfuzzy control output and defuzzifier is the defuzzification operator. Several techniques have been developed to produce an output ([6]). The three most used are 1 Mazimizer, by which the maximum output is selected; the max criterion method (MC). 2 Average (or measured average), which averages measured possible outputs; the mean of maximum method (MOM). 3 Centroid (and its variations), which finds the output’s center of mass; the center of gravity (COG).

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In the MC method, the crisp output is the point where the membership function reaches maximum value. The control action may be expressed as

In the MOM, the produced action corresponds to the mean value of all local control actions, whose membership functions reach their maximum. In the case of a discrete universe, it can be expressed as

where and m is the number of points in the universe of discourse with maximum membership function value. The center of the area of the membership function of the output control section fuzzy set is produced in the COG method. For the discrete case, it may be written as

where m is the number of points in the universe of discourse with maximum membership function value. The example of Figure 10.4 shows that it is not possible to find an unequivocal solution, and we have to have some practical experiences to decide the defuzzification methods for our FUSICO-project.

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Defuzzification examples of FUSICO-project

The output of the defuzzification process should be a signal group action. The method of defuzzification is dependent on the desired goal of the traffic signal control. Our examples are also in this case the signalized pedestrian crossing and two-phase control. The output (decision) variables of pedestrian crossing control are to extend the same phase (E) or to terminate the current phase (T). The first tested defuzzification method was the max criterion method (fuzmax). The weak point of this method was that only one rule (total 18) defines the decision. The strong point was that the main goal of our fuzzy rule base was to find the best moment for the termination of the vehicle signal group. Using the MC-method, the green termination was done immediately when one T-rule had a stronger membership value than any E-rule had. The second alternative could be the average method, where we calculate the average membership values of the E- and T-rule sets (9 rules in each set). The third proposed defuzzification method will be called “fuzzy” similarity, which is based on Lukasiewicz logic. The aim is to calculate the similarity value for each rule

Then the final decision is This method (fuzsim) is a combination of average and weighted methods, and the biggest advantage is that you can give different weights (w) for different inputs based on their importance in signal control. The method testing is going on. The first simulated results have shown that the total delay of fuzsim was slightly smaller than the total delay of fuzmax. The difference was not significant. The sensitivity of the different weights will be tested in near future. According to preliminary results, if traffic volume is high then the use of fuzsim saves 2-4 minutes/hour in total delay. Figure 10.5 shows the differences in total delay (fuzmax-fuzsim). The second example is two-phase control. In Figure 10.4, three different control methods are compared. The final defuzzification method is dependent on the goals of the rule structure. Our suggested method is the center of gravity (COG), because in that way the final extension interval is dependent on all four rules. Pappis and Mamdani [2] had their own rule structure with five interventions. Each intervention had 5 rules. At each intervention, the five rules are fired in ten compared extension (1–10 s). Actually 5*10 = 50 values are compared using the max-min-method. In their method only the strongest

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rule defines the next extension and the selected extension corresponds to the maximum degree of confidence. In other words, fuzzy predictive decision making implies that the selected action minimizes fuzziness.

4.3.

Defuzzification criteria

However, one important aspect of defuzzification is also discussed in the paper of Pappis and Mamdani. The extension criteria of membership functions. The criteria are very important in all cases, the main result of which is the degree of membership functions. The recommended criterion of Pappis-Mamdani was 0.5. The best known criteria in scientific journals are 0.0, 0.3, 0.5 and 0.7, but it is very difficult to give any general recommendations. The experiences of the FUSICO-project have shown that if we estimate the length of the next extension interval (0–9 s), then we do not need any criteria. This method was used in our multi-objective control application, in which we had three different (quite simple) rule-bases for delay minimizing (D), traffic safety (S) and energy minimizing (E). The final defuzzified extension was defined using the max-min-max-min-method over extension intervals (0–9 s). In the first phase, the membership values of each rule-base were calculated using the max-min-method (like in Pappis-Mamdani). In the second phase, the matrix with 10 extensions (0–9 s) and 3 rule-bases was calculated. Then the minimum value of each extension was selected, and the maximum membership value defined the length of the next extension. The results of this application were promising. In some cases, the efficiency of this kind of control was

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better than the efficiency of the fuzzy two-phase control. Working with multiobjective decision making will continue, and also in this approach, the fuzzy similarity can be an available method. On the other hand, we are just deciding step by step (n times per second) to extend or terminate, and we have some kind of uncertainty modeled in our fuzzy inference, for example terminate certainly (0.0), terminate probably (0.25),..., extend certainly (1.0). In this case, it is very clear that the extension criterium is 0,5 (both are equal), and we extend if and we terminate if If then we do not know the best result, but our recommendation is that the state of signals will be steady. The reason is that we try to limit the number of changes in our algorithm (typical civil engineer decision).

5.

Conclusions

A new method, fuzzy logic, has been presented as a possibility for the traffic signal control of the future. All important steps of a fuzzy problem solving can be seen in this FUSICO-project: 1 Define the fuzzy problem in detail. 2 Identify all important variables and their ranges. 3 Determine membership profiles for each variable range. 4 Determine rules (statements), including action needed. 5 Select the defuzzification methodology. The sixth step is the system testing, which is done by testing different alternatives to collect enough experience for a general rule base at the isolated intersections. This step still continues. One important part of fuzzy operating procedure is fuzzification part. Basically, it means that the physical meaning of a fuzzy linguistic term is characterized by a membership function, which is intuitively assigned by a person intending to use this term. Neural networks can be a helpful tool for the membership function defining. The shape of a membership function is quite free, but the typical feature is some kind of smoothness (Z-shaped, Bell-shaped, S-shaped). The triangle-shaped (trapezoidal) functions, as in our traffic signal control application, are also commonly used. The fifth step, called defuzzification, is very important in the operating procedure of fuzzy controllers. Its purpose is to convert the fuzzy set representing the overall conclusion obtained before into a real number that, in some sense, best represents the fuzzy set. Although, there are various methods, each justified in some way, the most common method is to determine the value for which the area under the fuzzy set of the membership functions is equal divided

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(COG-method). Our recommendation is that COG-method is the most suitable, but if the theoretical background or the objectives have different systematical structure, then other methods should be known and tested. The results of the first applications and installations have indicated that the application area of fuzzy control is very wide, and many benefits can be achieved using fuzzy logic based signal control. The discussed methods of this paper play an important part in connecting fuzzy set theory and practical application together. Without a deep knowledge of the different opportunities and without the alternative method comparisons our expertise of fuzzy traffic signal control would be limited.

References [1] Kim S. (1994). Applications of PETRI Networks and Fuzzy Logic to Advanced Traffic Management Systems. Ph.D. thesis, Polytechnic University, USA. [2] Pappis C. and Mamdani E. (1977). A Fuzzy Logic Controller for a Traffic Junction. IEEE Transactions on Systems, Man and Cybernetics. Vol. SMC7, No. 10, pp. 707–717. [3] Niittymäki J. (1997). Isolated Traffic Signals–Vehicle Dynamics and Fuzzy Control. Helsinki University of Technology, Transportation Engineering. Publication 94. [4] Yager R. and Filev D. (1994). Essentials of Fuzzy Modeling and Control. John Wiley & Sons, Inc, New York, USA. [5] Niittymäki J. and Kikuchi S. (1998). Application of Fuzzy Logic to the Control of a Pedestrian Crossing Signal. Transportation Research Record 1651, Transportation Research Board, Washington D.C. pp. 30–38. [6] Bingham E. (1998). Neurofuzzy Traffic Signal Control. Master thesis, Helsinki University of Technology, Department of Engineering Physics and Mathematics.

Chapter 11 AN URBAN BUS NETWORK DESIGN PROCEDURE S. CARRESE AND S. GORI [email protected] Department of Civil Engineering, Università di Roma Tre via C. Segre 60, 00146 Rome – Italy.

Abstract

This paper describes a coordinated process to configure a bus transit network with its set of lines and frequencies using heuristic approaches. In this context, the planning procedure consists of the following phases: in the first phase, the main skeleton of the network of the public transport system, not articulated in lines, is defined through heuristic procedures based on the demand matrix and the road network, integrated by “fixed supply” such as underground, urban railways, and those transit lines identified by Authority and located on the road network as fixed links with characteristics constant with flows; in the second and third phases the lines of main and feeder networks are designed respectively. In regard to this classification, the 2 phases are applied to define express lines initially, and subsequently main, in which express lines become “fixed supply” for following main lines design. This model has been applied to the Public Transit Network Design of the city of Rome, Italy.

1. Introduction This paper describes a coordinated process to design a bus transit network with a set of routes and frequencies. The nature of the problem precludes a solution by exact optimization models (Miller & Goodnight 1973, Morlok 1978, Newell 1979); therefore heuristic approaches to find “reasonable” result are utilized, but it is not possible to guarantee an optimal solution (Rea 1972, Rhome 1972, Mandl 1979, Marvah et al. 1984, Axhausen & Smith 1984, Van Nes et al. 1988, Ceder & Wilson 1986, List 1990). For a complete review of transit network design models we refer to Baaj & Mahmassani (1995), which also presents a new bus network design model. 177

M. Patriksson and M. Labbé (eds.), Transportation Planning, 177–195. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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The hierarchical procedures of modern transit network design have involved bus lines too, thus leading to a classification into express, main and feeder lines. In this paper only the urban transit network has been taken into account: for low demand areas other systems can be considered, for example Timed Transfer Systems (Carrese et al. 1998a) or Dial a Bus (Carrese et al. 1997). In particular, express lines are characterized by high frequency, a distance between two consecutive stops greater than 800 m, and the opportunity of overtaking main lines. Main line qualifications are high frequency, bus stop distances between bus stops equal to or longer than 400 m. Feeder lines, whose function is to lead passengers to Main Transit Lines (MTL), offer low-medium frequency, and distances between bus stops never greater than 400 m. The urban public transit network design procedure has been organized into three steps: 1. definition of the skeleton of a Main Transit Network (MTN); 2. identification of the network of Main Transit Lines (MTL); 3. definition of the feeder lines to improve MTL accessibility.

2. The main transit network (MTN) The model to define the MTN is based on the attempt to exploit the economies of scale of the public transit system (PT) (Rea 1972, Morlok 1978, Newell 1979), where supply function is characterized by a Level of Service (LoS) which improves as demand increases; this is due to the decrease in headways necessary to meet growing demand and to the possibility of using higher performing PT systems (figure 1). This statement is not true, if the model is used in a verification problem. Figure 1 states, if the project phase is taken into account, where the number of people in a square metre can be imposed as a constraint. In this case, it could be useful to develop transportation system performances in terms of sub-modes. In particular, the flow range up to 7,000 pass./h can be divided into the three service classes already introduced. In this paper only the bus system is considered. For the selection of an optimal system, the reader is invited to refer to Carrese et al., (1998c).

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The objective of the MTN algorithm is the concentration of the highest number of passengers on the same link so as to let them use a higher performance system. The mechanism of the model consists of an iterative assignment procedure of the OD matrix on the possible network (which is defined as the set of road links where the passage of a bus is possible), similar to the fixed capacity models used for car flows. Usually, this procedure is applied to avoid traffic congestion in private car assignment. Instead, in the public transport system the link function's performance increases as the flow grows. Furthermore, the optimal assignment technique would be All-or-Nothing. A consequence of this observation is that for a given demand on a possible network, where every link is associated to a speed flow relation like the one in fig. 1, the proposed algorithm for the MTN calculation is the following: 1) The calculation of minimum paths among every OD pair, assuming maximum link speeds corresponding to maximum flows; 2) All or Nothing (AoN) assignment of the total OD matrix over minimum paths; 3) Recalculation of link speeds to flows calculated in step 2, through speed flow design function; 4) Recalculation of a minimum paths set, according to link speeds of step 3; 5) Assignment of the total OD matrix to the minimum paths of step 4; 6) The iterative process of steps 3, 4, 5 ends when no more variation is revealed in link speeds otherwise go to step 3. The main differences between the MTN algorithm and a fixed capacity one for private traffic are the assignment of the total OD matrix at each iteration (with the aim of low concentration), the choice of the maximum speed in step 1, and the stopping criterion. This part of the procedure aims at concentrating user flows on a small number of links, in order to focus on the main flow routes. For this purpose, special procedures have been used which produce this concentration of flows in an efficient way. Methods of Successive Averages (MSA) for iterative computation of link speed variation and Stochastic User Equilibrium (SUE) assignment tend to create smooth processes. Then

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MSA and SUE can be used in those cases, where simple iterative concentration procedures produce flow level on links, which cannot be served by the best transit system proposed. The flow chart of the procedure to define the skeleton is shown in figure 2.

The algorithm can be interpreted by considering a scenario where the links compete against each other to reach the highest LoS (often expressed in terms of speed). As a result of this competition, some links will prevail and obtain a higher LoS, while other links will lose and be limited to offering a lower LoS, up to the "walking" LoS. The elimination of the links, or, in more positive terms, the definition of the MTN configuration, is obtained by fixing a condition of flows equilibrium on the network. Since this is a heuristic procedure, in order to assess the model's capability to aggregate and converge in a logical way, it is necessary to resort to a series of indicators aimed at indicating the trends of the model as: flow concentration; the Level of Service reached in the various iterations; creation of main traffic paths; stability evaluation of the solution obtained with the various iterations. The results of the heuristic procedure to define a skeleton of the main transit network for the test case of the city of Rome can be found in figure 3. The complete set of indicators is: I. Sum of link flows in successive iterations; II. Total time spent by the users on the network; III. Maximum link flow increase in each iteration;

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IV. V. VI.

Maximum link flow decrease in each iteration; Absolute total volumes exchanged between the links in successive iterations; Number of links with flow higher than 3000/4000 passengers /hour for each iteration; VII. Sum of link volumes with a value higher than 3000/4000 passengers /hour for each iteration; VIII. Average link speed for each iteration. The first and fifth indicators evaluate the stability of the algorithm. The second and eighth indicate the trend of the LoS offered in the different iterations. The third and fourth characterize the trend of the concentration of flows. The sixth and seventh suggest the possible direction of the routes.

3.

The main transit lines (MTL)

The algorithm for the definition of the MTL generates a route network on the basis of an objective function (OF) depending on users and company costs. The OF is then defined as:

where

and are weights and the main components of the OF are:

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1. the difference between the effective time spent on buses by all the users to perform their trips and the time necessary to perform the same trips on minimum paths; 2. all user waiting time defined as the sum of the waiting times for all the routes 3. operative service cost, measured from the number of vehicles/hour required to satisfy the demand for all journeys, which represents the network capacity equal to the sum of all routes travel times multiplied by their frequency. The network capacity is the sum of two components: used capacity (pass/h) and unused capacity seats/h on the route The problem can be submitted to some constraints, such as the complete satisfaction of demand, the guarantee of a minimum LoS, the maximum route length and winding path of the route, the maximum number of transfers and door to door travel time, the maximum link flow and the fleet size. The OF weights have been introduced to balance different components in the service utility according to user evaluation. Usually a higher value is assigned to waiting time compared to on-board time. If a different weight were assigned to transfer compared to initial waiting time, it would be necessary to distinguish in the second term of the OF between these two time categories. Regarding the three OF components the following observations can be done: 1. which is on-board travel time exceeding minimum travel time, including both user and company costs; 2. which is waiting time, including only user costs; 3. which is unused capacity and an exclusive component of the company’s cost. The first term of the OF is assumed to be constant, because of the particular structure of the first part of the algorithm, which aims at achieving a base network construction. Only in the successive improvement procedures will be considered as variable. It is worth noting that a route has an impact on the OF which is as high as the flow of passengers served by the route itself. In fact, in order to ensure the capability, more loaded lines correspond to higher frequencies and, as a consequence, to shorter waiting time. On the other hand, the lines with a higher capacity represent the bearing structure of the network.

3.1.

Route Construction

On the basis of these remarks, the algorithm has been conceived to form at first the main lines by performing a sequential choice of the links (figure 4).

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The algorithm consists of the following steps: 1) From the distribution of link flows obtained in the first phase (MTN), a classification of the links as a function of the load itself is formed. 2) Selection of the link with the highest flow value and the choice of this link as a primary component of the route. 3) Computation of the route frequency necessary to satisfy the demand as a function of the vehicle capacity. 4) Expansion of the route to include other links following the criterion of total cost reduction. For this purpose: 4a) Computation of the number of passengers toward the m links incident on each end node of the link is considered. 4b) It is proposed that the link with the highest flow be incorporated into the route.

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5) Check if the constraint of the route obtained is respected: if the constraint is not respected, the link is not added, the procedure is interrupted and goes back to step 2; if the constraint is respected, it goes forward to the next step. 6) Comparison between the transfer costs of passengers present on the vehicles and those for the unused capacity the route formation is interrupted, the link is not incorporated and one goes back to step 2, otherwise the link is incorporated into the route and it goes forward to the next step. 7) Computation of the number of passengers towards the links incident at the new end node of the route; comparison of these values and those already calculated for the node at the other end of the route and one goes back to step 4. During the procedure, the origin-destination flows that are served are deleted. Taking into account transfers from the developing line to the others, the route is constructed, node by node, in the direction that reduces the number of transfers. It is easy to observe how the route interruption is due to the reduction in on-board passengers. The problem can also be seen as the determination of a balance between operating and user costs; the solution obtained depends heavily on the regulating criterion adopted to choose the links and the weights assigned to the costs of the operator and of the users.

3.2.

Improvement procedures

The heuristic algorithm MTL has to be completed with two procedures aimed at improving the solution found. The first procedure consists of systematic changes to each route in order to reduce the number of transfers, the total travel time and, in particular the transfer time. The second procedure tries to reduce the operating costs of each line. Before moving to the improvement procedure, it is necessary to complete the OF, which now also includes walking and transfer time, neither of which is considered in the first phases of the algorithm proposed. Then, the lines are modified by the procedure; the aim of the changes is to improve the OF working on the balance between user and operating costs. The basic principle of the procedure is that a small saving for many people is equal to a larger saving for a few. Distortions caused by this principle could be appropriately reduced.

3.3.

First procedure: transfer reduction

This is a sequential procedure for the analysis and reduction of transfers between lines, which works at network level. The demand of a line (which needs more transfers to be served), and the worst node on it, (which is identified through ad hoc methods considering the number of transfers and users involved), are selected. The transfer is eliminated adding a new route which is composed of parts of the two lines - and incident on The optimal frequencies of the three lines and are recalculated and an assessment of the changes made, with the standard transit assignment based on the concept of optimal strategies (Spiess and Florian, 89), is carried out at the

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level of the whole network. These changes are accepted only if the OF improves. With this method, waiting and travel time will be reduced and a reduction of the number of vehicles currently necessary to satisfy the demand could be obtained. This is also consistent with the assumption that transfers are not well accepted by users. The procedure must be repeated for all worst nodes

3.4.

Second procedure: route expansion

The second procedure works at line level in order to reduce operating costs, through an increase in the line length between two consecutive nodes, if a “sufficient” demand increase occurs. It consists of the following steps: 1. A value is fixed for the maximum percentage increment of the travel time D. 2. The route nodes are classified in decreasing order, in relation to the number of onboard users exiting from the same node. 3. Starting from the node with the least number of on-board passengers, the link with as the next node is eliminated. 4. The new minimum path between and is determined. 5. The increment of on-board passenger travel time is verified, in order to check if it is smaller than D; otherwise the route is unchanged and then goes back to step 3, at the first node of the list, at point 2. 6. Computation the new value of the OF. 7. If OF improves, then the new path is included in the route, otherwise the route is unchanged, then go back to step 3, at first node of the list at point 2. The procedure stops either when all route nodes have been considered or when the travel time constraint is no more respected. The procedure obviously stops when all the lines are processed. The initial classification for increasing use is aimed at increasing travel times on the less used links.

3.5.

Express and main lines

The design of the MTL and the improvement procedures are performed in two subsequent phases. In the first phase, the express line network is built up using a long distance trip O/D matrix (in the test case, over 6 Km). In the second phase, the express line network resulting from the first phase is treated as existing lines (such as underground lines) and is considered as “fixed supply”. In this context, the main line network is built up assigning the full O/D matrix.

4.

Feeder lines

The main role of feeder lines is to provide a high level of accessibility to the main transit lines (MTL); however, in the real world it is not possible always to carry out a clear demarcation between these two kinds of networks; thus, sets of lines appear with intermediate characteristics.

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Existing literature contains many different methodologies to realize a feeder network, according to whether journey origins and destinations are concentrated or widespread (many-to-many or many-to-one) and according to flow levels. Many-to-many feeder line design is directly derived as an MTL sub product, that is a set of lines, which does not reach the standards of the main lines. Of the remaining ones, once the MTL main attractive poles (AP) are identified, the procedure is built up as a search for the balance between user and operating costs. The model deals with the calculation of route costs between origin node (bus stop) and the nearest AP on the minimum path, and viceversa. The cost consists of three terms, respectively: bus operating, users in vehicle and waiting costs. Each line is built up selecting the bus stop with highest direct cost in the feeder network. Afterwards, a line is extended by the inclusion of bus stop not yet assigned, on the basis of the maximum savings criterion. The assignment of the bus stop to the line involves a saving which is expressed as:

The conditions a) and b) set the alternative choice of sequence to reach bus stops and k. In this way, the line is characterized by coincidence of the two paths: to go and to come back. Instead, the conditions c) and d) are related to circular lines. The heuristic nature of the proposed algorithm makes it necessary to add an adjustment procedure that bypasses the limits of the model itself: 1. The algorithm builds up routes in a sequential order, so that the order of nodes for each line cannot be the optimal one. 2. A node, which has been assigned in a step of the procedure, cannot be considered in a successive step for the formation of a new route. The first adjustment procedure aims at reducing the length of each line. The length reduction in fact brings benefits both to the users and to the company. The second adjustment procedure improves the solution with node shifts, one at a time, from one route to the other. Both procedures use the beta optimality criterion: “a route is defined beta optimum if it is impossible to obtain a lower cost by shifting each one of the beta arcs (or nodes)”.

5.

Model application and results

The proposed procedure has been applied to a general plan arising from the participation of the authors in the Public Transit Design for Rome. The model was first used in 1991 for the Design of the Main Transit Network. Its application to the executive design for a large zone of Rome (Montesacro, 250.000 inhabitants) has been completed in 1999. In 1998 the model has also been used for the

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Design of the shuttle service (as integration of the tourist buses plan) in the historic Centre of Rome for the Christian Jubilee in the year 2000. The city of Rome is characterized by high densities in an area of with more than 3 million inhabitants and 2 million cars. Its area is divided into 560 zones. The graph is composed of around 1,400 nodes and 7,700 links and the peak hour matrix adds up to more than 550,000 trips. It is useful to observe the indicators (fig. 5-12) utilized in the MTN procedure to verify the results obtained. The figures confirm the effectiveness of the proposed heuristic procedure: in fact, the indicator trends are substantially coherent except for some slight fluctuations. In particular, the sum of link flows (figure 5) presents a trend that rapidly increases in the first five iterations, increasing in the successive three and almost constant in the last ones.

Referring to the total time spent on the network, figure 6 shows a sudden decrease in the first iteration, a slow decrease in the next four and an almost constant trend in the remaining iterations.

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The maximum link flow increase and decrease (figures 7 and 8) present the same trend: a highly unstable variation in the early iterations, which becomes smaller towards the end (values close to zero).

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In the same way, the absolute total volumes exchanged between the links (figure 9) rapidly decreases in the first five iterations with a slight reduction in the last ones.

Figure 10 shows the number of links with a flow higher than 3000/4000 passengers /hour for each iteration. In both cases it presents a constant increase in the first five iterations and remains almost stable in the last ones (with some slight fluctuation).

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Figure 11 shows that the sum of flows on the evaluated links presents an analogous trend for both indicators.

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From the average link speed indicator in fig. 12 it is possible to deduce the obtained LoS resulting from the model: a sudden increase in the first two iterations with almost constant values in the last ones.

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The effectiveness of the procedure can be appreciated by comparing the proposed network with the one that already exists. A comparison between the two networks shows that: a) The total length of the transit network has been considerably reduced in the design plan (1,450 km compared to 3,000 km). b) The number of vehicle-Km of the design network decreases by 33% compared to the current network (14,000 versus 21,000). c) The design network presents a higher use of metro and train lines (49%) compared to the current network (36%); the same is true for the use of tram lines (12% versus 6%). d) The number of vehicles necessary for the current network during the AM peak hour is equal to 2,000; while in the design network only 1,200 vehicles are necessary. e) The number of design network lines is nearly half that for the existing network. f) The high frequency lines (up to 8 bus/h) of the design network are 55% of the current one. g) The existing network presents a high concentration of lines and terminals in the historical Centre of Rome. It has to be noted that in the design network it will be necessary to introduce a set of “social” lines to service the “other aims trips” (i.e. different from those to school and work). These lines have to be considered according to political criteria, rather than in terms of increased efficiency. A first estimate shows an increase in the number of both vehicle-km and vehicles of about 20%. The high number of vehicles and vehicle-km of the existing network depends on the use of a network spread of low frequency lines. The design network, on the other hand, offers high frequency and straight routes. The modal split model is a multinomial logit for two user classes; home to work and home to school. In the design scenario the modal split presents an increase of the transit demand of 4%; this shift depends mainly on the new transit supply, and on the

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introduction of parking pricing in the historic Centre of Rome. The number of parking places is relatively high (12,000) but represents a small share (2%) compared to the total O/D demand of the AM peak hour. The obtained shift of the modal split involves an increase in transit users; in fact, the design network shows a higher number of boardings, in particular on tram lines (from 42,000 boardings under the current network to 76,000 boardings under the design network), and a decrease of the utilisation of the low frequency lines. The “lost demand” of the design network is equal to 3,300 users, who do not have any “utility” to use the transit service anymore, but the majority of them (2,700) is related to the very small O/D flows (less than 10 units for each O/D). This part of demand was partially attracted by social lines. Another important indicator is the average total time spent on the transit network. This shows a decrease of 22% in the design plan with respect to the current 63 minutes. This is mainly due to the reduction in time spent on-board the vehicle and reduced waiting time. Also, the average number of transfers presents a small reduction when comparing the two networks, although the decrease of number and length of design lines. The last observation concerns the uniform flow distribution on each transit line, as shown by link flow values resulting from the assignment model. The indication of balanced flows indicates a utilisation of vehicles along most routes with a high loading factor. In table 1 the main characteristics of the actual and design networks are listed.

6.

Conclusions

In the operative models found in the technical literature, the design of the public transport system is developed by choosing single lines from a wide set, built up on the a priori basis of a predefined criterion (for example linking the most distant ODs or the ones with the highest flow),often with a constraint on their maximum number. In the proposed model systems, the main characteristics are as follows: 1. In order to find the main transit route in the design phase, a speed flow curve has been used which increases the LoS at routes with higher flows. 2. Lines are built one by one, based on a general utility criterion (minimization of transfer number) without constraints on terminals or number of lines, because the

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evaluation process balances users costs (in particular those connected with transfer) with the operating costs of the transit company. 3. The sequential order of models can define line typology hierarchically: express, main and feeder. The heuristic nature of the model requires the use of improvement procedures for results. In particular the aim is to further reduce both transfer number and operating costs. The modelling system has been applied to several test cases, at different scales in both urban and regional contexts. Applications have been a useful test phase for the models. Results, supported by the comparison between the performance of design and existing lines, have been successfully evaluated. Furthermore, a survey has been recently conducted between inhabitants of a district in Rome where a PT network designed with the models has been implemented in the last few months. The sample of around 1000 people, 60% of whom are every day users, demonstrates a generally positive reaction to the new service. Finally, results analysis confirmed the suitability of the proposed model for the automatic identification of design choices at network level, their reliability to different application contexts and the simplicity to take care of design constraints. The development of the proposed procedure follows the path of explicitly introducing interrelations between PT network design, parking measures (Park+Ride, Toll Parking) and Land-use control (Carrese S., Gori S., 1998b). In particular, although Land-use interactions are only studied on a theoretical level, the implementations of Park+Ride and Toll Parking models are at an advanced stage and they should be validated in real world applications.

References Axhausen. K.W., Smith R.L.,(1984). Evaluation of heuristic transit network optimization algorithms. Transportation Research Record 976, pp.7-20. Baaj M.H. and Mahmassani H.S. (1995). Hybrid route generation heuristic algorithm for the design of transit networks. Transportation Research C 3C, 31-50. Carrese S., Cipriani E., Gori S. (1997). Urban Transit Service for Elderly Persons. International Symposium on Urban Areas and an Aging Population. INRETS Arles en Provence, France. October 8-10. Carrese S., Fusco G., Gori S. (1998a). Timed Transfer Systems Strategies in Low Demand Areas. Proceedings of the 8th World Conference on Transport Research. Antwerp Belgium July 1217. Carrese S., Gori S. (1998b). Bus Network Design and Parking Policies Interactions. Proceedings of the TRISTAN III Conference Puerto Rico June 17-23, Volume II. Carrese S., Gori S., Grossi Gondi F. (1998c). A System Selection Model of an Urban Transit Route. Proceedings of the 8th World Conference on Transport Research. Antwerp Belgium July 12-17. Ceder A. and Israeli Y. (1997).Creation of Objective functions for transit network design. Proceedings of IFAC Transportation Systems Chania Greece pp.684-689 Ceder A. and Wilson N.H.M. (1986). Bus Network Design. Transportation Research B 20B, 331-344.

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List G.F. (1990). Toward optimal sketch-level transit service plans. Transportation Research B 24B, 325-344. Mandl C.E. (1979). Evaluation and optimization of urban public transportation networks. Paper presented at the 3rd European Congress on Operations Research Amsterdam, Netherlands. Marwah B.R., Umrigar F.S., Patnaik S.B., (1984). Optimal Design of Bus Lines and Frequencies for Ahmedabad, Transportation Research Record No.994. Miller N.C., Goodknight J.C. (1973). Policies and procedures for planning transit systems in small urban areas, Highway Research Record, No.449. Morlok E.K. (1978). Introduction to Transportation Engineering and Planning. McGraw-Hill, New York. Newell G. (1974). Optimal network geometry. Proceedings of the Sixth International Symposium on Transportation and Traffic Theory, D. J. Buckley, Editor. New South Wales, Australia. Newell G. (1979). Some issue relating to the optimal design of bus lines. Transportation Science Vol.13, 20-35. Rea J.C. (1972). Designing urban transit systems: an approach to the route-technology selection problem. Highway Research Record 417, pp.48-59. Rhome R.C. (1972). A Strategy for Urban Mass Transportation Route-Technology Selection NTIS, UMTA. Spiess, H. & M. Florian (1989). Optimal Strategies: A New Assignment Model for Transit Networks. Transportation Research B, Vol. 23B, pp. 83-102. Van Nes R., Hamerslag R., Immers B.H. (1988). Design of Public Transport Networks, Transportation Research Record, No. 1202.

Chapter 12 CONE FIELDS AND THE CONE PROJECTION METHOD OF DESIGNING SIGNAL SETTINGS AND PRICES FOR TRANSPORTATION NETWORKS Michael J. Smith [email protected]

A. Battye A. Clune

Y. Xiang York Network Control Group, Department of Mathematics University of York Heslington, York, YO1 5DD United Kingdom

Abstract

This paper builds on ideas in Smale [13] and Smith et. al. [11, 12]. The paper utilises Smale’s cone fields rather than vector fields to impel disequilibrium steady state traffic-price-green-time distributions; and applies these ideas to the design of steady state signal controls and prices on transportation networks. The work is applied within a multi-modal equilibrium transportation model which contains elastic demands and deterministic choices. The model may readily be extended to include some stochastic route-choice or mode choice. Capacity constraints and queueing delays are permitted; and signal green-times and prices are explicitly included. The paper shows that, under natural linearity and monotonicity conditions, for fixed control parameters the set of equilibria is the intersection of convex sets. Using this result the paper outlines a cone field method of calculating equilibria and also an associated method of designing appropriate values for the control parameters; taking account of travellers’ choices by supposing that the network is in equilibrium. The method is shown to apply to certain non-linear monotone problems by linearising about a current point. 197

M. Patriksson and M. Labbé (eds.), Transportation Planning, 197–211. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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A rigorous proof of convergence to the set of equilibria is provided, for linear and some non-linear monotone problems. But only an outline of a potential proof of convergence to a (flow, control) pair which satisfies a Karush-Kuhn-Tucker necessary condition for local optimality is provided.

1. 1.1.

Introduction Inspiration and motivation

This paper follows up ideas introduced in Smith [8, 9]. These were partially inspired by cone fields introduced in Smale [13]. Smale introduced dynamical systems whose solution trajectories are not uniquely defined; they merely move in “roughly” the right direction, rather than exactly the right direction; and argued that such dynamical systems may be more appropriate for the study of the evolution of economic systems. The “solution trajectories” in these systems have their direction of motion at each point confined to be within a cone, instead of being confined to be in a precise direction. The ideas here were also motivated by Von Neumann and Morgenstern [14], Nash [5], Brown and Von Neumann [2], Nikaido [6] and Arrow and Hurwicz [1]. Arrow and Hurwicz has proved to be a constantly inspiring reference. Smith [8] formulated an elastic traffic equilibrium as a point x which satisfies

The vector field in [8] was specified by a rather general network excess demand function and the convex set was the set of flows all of whose coordinates are non-negative.

2.

Achieving the complementarity formulation

The notation adopted is shown below (in Table 12.1); there is a base network and a multi-copy (Charnes and Cooper [3]) version of this. Within each copy the travellers or vehicles all have a single destination node. This network structure is very similar to that in Smith [10].

2.1.

Equilibrium, demand and capacity constraints

We use Wardrop’s [15] condition: more costly routes carry no flow. But we choose to write this in the following form: for each route the (least) cost to the destination from the node upstream of route is no more than the least cost to the destination via route and if it is less then no flow will enter

The Cone Projection Method

route

199

or

Here is just a way of writing the cost at the node upstream of route This sum comprises just the single cost at that node at the entrance of route The (elastic or inelastic) demand constraints may be written as: the total route-flow out of node equals the demand Rewriting

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this in a slightly weaker and more artificial way we obtain:

This is in fact, under natural conditions (which include the Wardrop condition above), equivalent to the stronger condition above. For the capacity constraint condition we suppose here that for any average bottleneck delay and nominal link capacity there is a maximum possible flow consistent with the delay Then the capacity constraint may be written:

As specified here this condition will ensure that congestion costs normally represented by a cost-flow function will in fact occur as “bottleneck” delays which arise from equilibrating via the functions The may be thought of as the inverse of given cost-flow functions.

2.2.

The complementarity formulation

First we define network response functions

and for fixed Y, P; writing above as:

for

etc; we rewrite the conditions

and and and

Then we put to obtain:

as follows:

and

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The Cone Projection Method

(The and the

2.3.

are “new” suffices and are unrelated to the previous and

)

The equilibrium set (with fixed ) as the intersection of convex sets

Let be a feasible control vector (in fixed for the moment. The feasibility and equilibrium conditions (2.1) or (2.2) are plainly equivalent to:

The first two of these constraints will be called feasibility constraints. Suppose that for each fixed the function is monotone and linear in Then for some square positive semi-definite matrix and some which both depend on Since is positive semidefinite (that is for each is a convex function of (the positive semi-definite is the Hessian of this function);and so is a convex function of for each fixed By linearity each is convex too. Thus each inequality in (2.3) specifies a convex set and the set of equilibria is the intersection of the convex sets in (2.3). (Monotonicity alone ensures convexity of the intersection itself).

2.4.

Including control constraints

Since we wish to vary in we need constraints on the set of possible values. Let these constraints be (Capacity constraints, which will sometime involve and together, are included in the conditions in (2.3).) Now we let (for stand for and the “old” and stand for and Then the control-augmented set of constraints is:

2.5.

Summary of assumptions required

1 All are differentiable, their gradients in and non-zero and the are all linear, 2

is monotone in

for each fixed

3

is linear (or, more properly, affine).

space are continuous

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2.6.

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Min-max conditions for equilibrium

Given a feasible

Then a feasible

put

satisfies all the constraints in (2.4) if and only if

So to find an equilibrium we need only: (i) find a feasible i.e. one which satisfies for (ii) maintain this feasibility throughout; (iii) reduce to a minimum; and (iv) check that this For each fixed let is feasible}. Theorem 1 Let the assumptions (1,2,3) above hold, and let Then there are directions at which reduce and maintain feasibility. Proof Since is linear and monotone all the constraints in (2.4) are convex and so is convex in for each fixed Since there is a feasible such that Since is convex and the feasibility constraints are convex the line joining and is an M-reducing feasible direction. Thus there are M-descent directions at which preserve feasibility. Now suppose is monotone but non-linear. Linearise denote for the new linearised at

at

and let and

If assumption 1 holds, a direction is a feasible direction which reduces if and only if it is a direction which reduces at because all functions are differentiable and their gradients are continuous. This yields the following theorem. Theorem 2 Let assumptions 1 and 2 hold and let Then there are directions at which reduce sibility.

and maintain fea-

Proof By theorem 1 there is a direction which reduces at This direction will then be a feasible descent direction for the original at because all gradients are continuous, using the remark preceding the statement of the theorem. Corollary Let assumptions 1 and 2 hold and suppose that each linearised problem has a solution. Then away from equilibrium there are feasible directions which reduce

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The Cone Projection Method

Proof The existence of solutions to the linearised problems implies that away from a solution of (2.4) and theorem 2 then says that there are directions which reduce and maintain feasibility.

3.

A cone field method of calculating equilibria

In the light of theorem 1 (and theorem 2) it becomes natural to define the cone field F to be the function which assigns to any feasible the cone of directions in which does not increase at Suppose is defined for all and has a right derivative at all Suppose further that

In this case we shall call the trajectory an assignment process following Smith [8] who followed Smale [13]. Thus an assignment process is a path along which This notion of an assignment process is a generalisation of the solution of a differential equation. It is very useful here because (i) it perhaps more precisely reflects travellers likely choices on the basis of current information by being less precise and (ii) it permits a wide class of “solution trajectories”, including polygonal trajectories to be defined below, even if the underlying cone field is not very smooth. We are able here to get away with just continuity of the constraint gradients rather than standard Lipshitz continuity ordinarily imposed on vector fields to ensure the existence of solutions of differential equations. Our cone field is just piecewise continuous. Given for all satisfying:

for if the relative interior of is nonempty; and if the interior of is empty. Given also a starting point then define a polygonal assignment process which begins at goes to where construction:

unless the process stops. Usually we obtain an infinite sequence

3.1.

Gain

The gain

at

is defined by

we may and then etc. By

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3.2.

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Complete assignment processes

Smale defined a complete process as one such that if was approached by it then there would automatically be no non-trivial process starting from In our context an assignment process will be complete if it is such that any point approached by it is automatically an equilibrium.

3.3.

A class of complete assignment processes

Theorem 3 Let D(.,.) satisfy the above condition and also be such that the gain is a continuous function of Then any bounded assignment process generated by D(.,.) will be complete. (These conditions may be made weaker. In particular it is sufficient to suppose that is bounded below by a continuous function which is positive at non-equilibria and this makes the specification of a suitable D much easier and more natural). Proof Let D be such that is continuous. It is automatically true that will be positive at non-equilibria. Let an assignment process arising from D be bounded. Then if it has only finitely many positive steps it must terminate at (say) where and so and we are at an equilibrium. On the other hand let the bounded polygonal path contain an infinite number of corners These corners are all distinct as M decreases. Hence the set is infinite, and being bounded has a limit point We shall show that such a limit point is an equilibrium. Suppose that is a limit point and that this is not an equilibrium. Then the interior of is non-empty, D is non-zero and so Since ing to

is a limit point there is a subsequence of convergCall this subsequence Now tends to as tends to infinity since is continuous. Hence there is an such that for all Thus M is reduced by at least between and for (for any choice of Over steps M is thus reduced by at least which may be made more than (the initial value of M) by choosing sufficiently large. However M decreases at each step and so if is so chosen then

However M is intrinsically positive and so we have a contradiction.

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The Cone Projection Method

This contradiction arises from the assumption that has a limit point which is a feasible non-equilibrium. Thus all limit points of the sequence must be equilibria and the assignment process defined by D is complete. 3.3.1

Notes.

1 For practical implementation 2 Only continuity at the limit point

must be specified. is required.

3 Even then continuity is only required with reference to a subsequence of points of the original this will be important later.

4.

The cone projection method

Now we specify a which not only ensures convergence to an equilibrium (that is gives rise to a complete assignment process) but also seeks to do the best for any given smooth objective function. Suppose given a smooth (continuously differentiable) objective function where is the vector of flows, delays and costs, and is the vector of signal green-time proportions and prices (including any feasible road prices). The general form of “the cone projectionmethod” is to begin at any feasible and continually follow a polygonal path which at each step follows a direction D which reduces while “approximately doing the best for” the given Z. As motivated here such a trajectory may (under natural conditions) be expected to converge to equilibrium and a weak variety of local-optimality simultaneously. Let be feasible but not an equilibrium. If we will say that constraint is very violated; otherwise it is not very violated. Let be the cone of directions in which do not cause any constraint (in (2.4)) to become violated and let be the cone of those directions at along which no very violated (in (2.4)) becomes more violated. Then is the cone of directions (in space) along which if Directions in the relative interior of reduce the violation of all very violated constraints in (2.4), at simultaneously and so reduce M. Let the vector be the “centre-line” of the cone (the zero vector if and only if the cone is empty), and be projected onto the cone

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The cone projection method follows the assignment process generated by D where D has the following form:

At each step step. Now (and hence and hence

and (both positive) are to be chosen so that M decreases each is non-zero unless it is forced to be zero by having and also is normal at to and so a

trajectory following D may be continued (it has a direction to go in) unless both the above conditions hold. The solution method is thus in outline to follow a direction at each This is intended to be a refinement of the bi-level method proposed in Smith et. al. [10, 11, 12]; replacing half-spaces with cones to narrow the search region and reduce numerical/computational problems. So far, however we have not specified the centre-line and we have not defined The “centre-line” of the cone is obtained by solving the problem shown below (for simplicity of notation is omitted from the statement of the problem): Problem

Minimise

1

if

subject to and if there is a

such that

then

and, 2

if

and if there is a

such that

then Now at each feasible non-equilibrium where and solve is non-zero at non-equilibria. Consider also problem below. Problem

By virtue of theorem 1 above

Minimise

1

and

if

2

and

if

Now

let

subject to and

is defined by

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where and solve Direction may now be defined as a weighted sum of these two vectors as in (4.1) This generalises the corresponding direction in Smith et. al. [10, 11, 12] which involved projections onto half-spaces. Smith et. al. [11] give an initial result of the method using half-spaces.

5.

A simple method

Suppose a feasible starting point is given and is given as in (4.1) above. Then D and are discontinuous where changes. But D and are continuous if the active constraints do not change. Let etc. as before. Thus we follow a standard “M-reducing” polygonal trajectory. We obtain a (usually infinite) sequence

5.1.

An outline proof that the simple method converges to the set of equilibria

The proof relies on and is similar to theorems 1 and 2. Let the polygonal path be bounded. Then if it has only finitely many steps it must terminate at (say) where is zero. At such a point and is an equilibrium. On the other hand let the bounded polygonal path have an infinite number of corners where Since the path is bounded the set being infinite (no duplicates as M declines), has a limit point We shall show that every such limit point is an equilibrium. Suppose that is any such limit point and that this is not an equilibrium. For each let be the set of those suffices and such that and occur in (These suffices correspond to active constraints.) Then there is at most a finite number of possible and some must be repeated infinitely often. Consider a subsequence of which converges to and for which (say) is always the same set of active suffices. Call this subsequence Then in this subsequence all involve the same constraints and the constraint functions are continuous; and so converges (in an obvious sense) to (say) with again the same active constraints. Now, since the constraint functions are continuous,

so being a non-equilibrium implies is nonempty which implies is non-empty (this is a “bare hands” proof of the necessary one-sided continuity of and

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Hence M must decline in direction determined as is determined but using only those and constraints with suffices in A. Let be the greatest reduction in M possible in direction beginning at Then and it follows that, since tends to as tends to infinity, there is an such that for all Thus M is reduced by at least between and for (for any choice of Over steps M is thus reduced by at least which may be made more than (the initial value of M) by choosing sufficiently large. However M decreases at each step and so if is so chosen then

M is intrinsically positive and so we have a contradiction. This contradiction arises from the assumption that has a limit point which is a non-equilibrium. Thus all limit points of the sequence must be equilibria.

5.2.

Problems with the simple method

The step lengths are chosen solely to reduce M to zero and so there is no reason to think that the polygonal path will converge to a minimum of Z within the set of equilibria. This problem may be resolved in several ways but here we propose to interrupt the previous M reducing scheme periodically and to follow a modified direction with a constant small step length aiming to reduce Z. This interruption may instead use a standard constrained minimisation procedure: minimising Z subject to a relaxed equilibrium condition, allowing M to increase somewhat as Z is minimised. Thus we come by the implementation proposed below.

5.3.

Implementation and outline justification

We have defined

but now we put

is a measure of the degree to which departs from satisfying a Karush-Kuhn-Tucker (KKT) condition. (We say that satisfies a KKT condition iff We follow the two-stage method in Smith et. al. [10] and Clegg and Smith et. al. [4]. Beginning at let and In the first stage we follow the polygonal path above each step of which begins in

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direction stage we follow

given by (4.1); until given by (4.1), with

Then in the second small, so that eventually

This will be a polygonal path and both and must be chosen so that both conditions hold at the termination of stage 2. We are confident that this may be done by choosing sufficiently small but we have no rigorous proof as yet. Repeating these two stages yields a sequence Let be any limit point of this sequence. Then 1

and so

and

belongs to the

equilibrium set; and 2

and it follows by continuity that

is an asymp-

totic KKT point.

5.4.

Conjecture and alternative second stage

We conjecture that for appropriate the second stage of the algorithm does halve N (at each iteration) without losing control of M. An alternative here would be to use a constrained minimisation algorithm instead of the direction above.

6.

Conclusion

The paper has outlined a method for calculating signal timings and prices in an urban transportation network which uses directions confined to cones. The method applies to linear monotone multi-modal deterministic elastic and inelastic problems; and may be extended to include stochastic elements. The linearity may be relaxed for some problems by linearising the given problem to establish the non-emptyness of certain cones. Monotonicity is however essential. We have proved convergence to equilibria and we have given an outline justification of the method; suggesting that the method yields an equilibrium which satisfies an asymptotic Karush-Kuhn-Tucker condition or a KarushKuhn-Tucker condition. This is a weak necessary condition: a truly optimal must satisfy it. Further work is needed (i) to assess the practical efficiency of the cone projection method, comparing with other design methods, (ii) to convert the “outline justification” given here to a rigorous proof of convergence to a local optimum or at least a critical point, and (iii) to relax certain other conditions specified in this paper.

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Acknowledgments We are grateful for financial support from: DETR, EPSRC, ESRC, DGVII (E2), and for the support from our LINK partners at UCL and MVA.

References [1] Arrow, K. J. and Hurwicz, L. (Eds) (1977), Studies in Resource Allocation Processes. Cambridge University Press. [2] Brown G. W. and Von Neumann J. (1950), Solution of games by a differential equation, Ann. Math. Stud. 24, 73 – 79. [3] Charnes A. and Cooper W. W. (1961), Multi-copy traffic network models. Proceedings of the Symposium on the Theory of Traffic Flow, held at the General Motors Research Laboratories, 1958 (Editor: R Herman), Elsevier, Amsterdam. [4] Clegg J. and Smith M. J. (1998), Bilevel optimisation in transportation networks. Paper presented at the IMA Conference on Mathematics in Transport Planning, Cardiff. [5] Nash J. (1950), Non-cooperative Games, Annals of Mathematics, 54, 286 – 295. [6] Nikaido H. (1959), Stability of equilibrium by the Brown-Von Neumann Differential Equation, Econometrica, 26, 522 – 552. [7] Payne H. J. and Thompson W. A. (1975). Traffic assignment on transportation networks with capacity constraints and queueing. Paper presented at the 47th National ORSA/TIMS North American Meeting. [8] Smith M. J. (1979a), The marginal cost taxation of a transportation network, Transportation Research, 237–242. [9] Smith M. J. (1979b), The existence, uniqueness and stability of traffic equilibria, Transportation Research 13B,295 –304. [10] Smith M. J., Xiang Y., Yarrow R. and Ghali M. O. (1996), Bilevel and other modelling approaches to urban traffic management and control, Paper presented at a Symposium at the Centre for Transport Research, University of Montreal, and in Equilibrium and Advanced Transportation Modelling (ed. P. Marcotte and S. Nguyen) (1998), Kluwer Academic Publishers, Mass., 283–325. [11] Smith M. J., Xiang Y., and Yarrow R. (1997a), Bilevel optimisation of signal timings and road prices on urban road networks. Preprints of the IFAC/IFIP/IFORS Symposium, Crete, 628 – 633. [12] Smith M. J., Xiang Y., and Yarrow R. (1997b), Descent Methods of Calculating Locally Optimal Signal Controls and Prices in Multi-modal and

REFERENCES

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Dynamic Transportation Networks, Presented at the Fifth Euro Meeting on Transportation, University of Newcastle-on-Tyne, and to appear in the Proceedings. [13] Smale S. (1976), Exchange Processes with Price Adjustment, Journal of Mathematical Economics, 3, 211 – 226. [14] Von Neumann J. and Morgenstern O. (1944), Theory of Games and Economic Behaviour, Princeton University Press. [15] Wardrop J. G. (1952) Some Theoretical Aspects of Road Traffic Research, Proceedings, Institution of Civil Engineers II, 1, 235 – 278.

Chapter 13 A PARK & RIDE INTEGRATED SYSTEM Chafik Allal [email protected] Service de Mathématiques de la Gestion Université Libre de Bruxelles Brussels, Belgium

Benoit Colson [email protected] Groupe de Recherche sur les Transports Facultés Universitaires Notre-Dame de la Paix Namur, Belgium

Bernard Fortz [email protected] Service de Mathématiques de la Gestion Institut de Statistique et de Recherche Opérationnelle Université Libre de Bruxelles Brussels, Belgium

Abstract

This paper describes the Park & Ride Integrated System, a new concept of demand-responsive transport system, and the underlying problems to be solved to manage this system. We first present its functionalities and its global architecture. The first feature we examine is the real-time aspects having to be dealt with. We then propose a routing algorithm to compute itineraries in order to satisfy the demands. The input of this algorithm consists among others of travel times whose prediction is in itself another problem of interest. Finally, we present simulations results performed with the software developed.

Keywords:

Park & Ride, demand-responsive, dynamic, routing, prediction, travel time.

213 M. Patriksson and M. Labbé (eds.), Transportation Planning, 213–228. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Introduction

Among urban transportation systems, public services like buses, trams and underground are efficient because they do not need a large amount of space per passenger. However, they lack many of the conveniences of the private car like comfort and most of all flexibility and availability. A good combination of both these systems could be an advantageous alternative. For example, such an idea can lead to a multimodal system during peak hours for passengers moving from a low-density zone to a high-density zone. Several novel concepts were based on this approach. These concepts have been widely studied and developed in new transportation systems (see for instance [2, 5]). To be attractive, a system based on the Park & Ride concept should run small vehicles (for instance minibuses) and provide a demand-responsive service making use of dynamic information. The design of such a concept must emphasize the complementarity with existing urban transport systems. It must also include the development of an easy-to-use interface providing information about multimodality. Moreover, in order to give an incentive to customers, every passenger should be served within a minimal delay by a minibus in which a seat has been reserved. These ideas have been used to develop several Park & Ride systems such as the ones whose functioning is evaluated in [7]. However all these systems are quasi-static and do not take into account changes in traffic conditions. This paper describes a Park & Ride Integrated System (PRIS) using algorithms based on dynamic aspects. It is based on a multi-modal approach with the following components: a number of Park & Ride stops (PR-stops), some of them being located at peri-urban parking lots and the others being similar to bus-stops equipped with terminals in the town-centre, a minibus service allowing the users to reach any PR-stop located within the considered area, and especially the ones of the town-centre. To ensure the technical feasibility of this system as well as the efficiency of the algorithms we use, we have been developing a simulation software called PRsim. It provides the kernel of the management system in view of a real-world implementation of the PRIS. This software will be the basis to develop the PR dispatch centre described in Figure 13.1.

The software performs event-driven simulations. The events are among others the introduction of a demand via a terminal, the arrival of passengers at a PR-stop, the arrival of a minibus at a PR-stop, and so on. The

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simulation tool recognizes the different types of events and performs the adequate tasks to treat them. The Routing Unit is in charge of computing the vehicle routes in a dynamic way. It creates routes satisfying a number of demands and updates these routes in order to integrate new demands. The routing algorithm is described in Section 3. The Travel Time Prediction Unit computes estimations of the travel times for any period of the day. It thus controls the access to historical information and also updates it whenever a new travel time is collected. It also takes some modifications of the network state into account to adapt historical data when refreshing travel times. Details about this are given in Section 4. In the next section, we describe the operational aspects of the system.

2.

A Park & Ride Integrated system

Customers have the possibility to book a place in a peri-urban parking at a given time of the day, to be taken by a minibus a short time after arrival at the parking, to reach the desired PR-stop and to make the return-trip to the parking at the desired time. These services are summarized in Figure 13.2. Let us now describe the sequence of operations taking place when a customer wants to use the system.

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Booking. This operation sets off the mechanisms allowing the system to take a demand into consideration. It must record every feature that identifies the demand: number of persons, desired destination, time components (desired arrival time at destination and expected arrival time at parking after acceptance), payment information (credit card number, customer number, ... ). Acceptance. In response to his request, the customer is given a keycode and the place (parking and transfert area) where he is expected to enter the system. Modal transfert. Once the customer arrives at the parking, he introduces his keycode at a terminal. This has the effect to confirm the demand to the dispatch centre. In response to this confirmation, the customer receives information about the minibus which will bring him/her to his/her destination. Four elements are necessary to implement the PRIS in a real-world environment. Dispatch Centre. This is the key element of the PRIS. It collects all the information about minibuses, customers and traffic conditions. This centre consists of a central computer linked to all PR-stops (through telecommunication lines) and to all minibuses (through radio links). The central computer is in charge of computing the best minibus itineraries to satisfy the demands, displaying in real-time the minibus position and occupancy and the waiting time of customers at PR-stops, the billing and fare management, and finally the management of the parkings.

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Parking lot management. The parking lots are appropriately distributed throughout the periphery and each of them is equipped with a terminal providing the following information: minibus arrival times, information about other transportation modes (e.g. one can ask for a taxi), city information, trip information (fares, traffic conditions, … ), and booking facilities. Since the number of places in a parking lot is limited, a customer may be asked to change his/her plans when booking and to go to another lot than the one he/she was thinking of. PR-stops. Terminals are also installed at PR-stops, e.g. allowing to book for a return-trip. These terminals are similar to the ones that equip parking lots. Transportation demands. Each transportation demand corresponds to a trip. Trips are of two categories: in-trips: these are for customers entering the system. They only give their destination. A place in a parking is booked for them and they are assigned to this parking, the latter being considered as the origin of their trip; out-trips: these are for customers leaving the system. In this case, they choose the origin, but a fixed destination is attached to their demand. This destination is the parking where they left their car.

3.

Routing model

We are given a graph where is the set of nodes corresponding to PR-stops (the number of such nodes is denoted by N) and is the set of directed arcs. On the graph we are given a set of D transportation demands. Each demand corresponds to the transportation of persons from the origin PR-stop to the destination PR-stop For each demand the customer specifies the desired time for arrival at destination. The actual arrival time is denoted by To serve these customers, a fleet of M vehicles is available. A vehicle is supposed to be depot-based, starting its route from depot and ending it at Each vehicle has a capacity The routing cost for each

hour of service is The itineraries of the vehicles define a set of routes, being the route of vehicle described as an ordered sequence of nodes of the graph, i.e. being the number of PR-stops visited by vehicle The construction of the routes must satisfy the operational constraints of the system. These constraints are bounds on:

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the vehicle capacities; the waiting time for a vehicle at a PR-stop; the travel time of a passenger; the ratio between the travel time and the shortest travel time from the origin to the destination; time windows for the arrival at destination: desired service times are very restrictive constraints, which implies the use of a large number of vehicles to satisfy the demands; in order to avoid such situations, we use the classical approach of relaxing the arrival at desired time by a time window around therefore the constraint is

Two categories of routing problems arise: we construct static routes for some vehicles (stream routes), using forecast demands given by a historical database while the routes of other vehicles are dynamic (sweep routes). By doing this, we reduce the number of dynamic routes to be constructed. We explain hereafter the stream route construction. The sweep route construction is currently being investigated and is left for a forthcoming paper. To solve the problem of stream routes, we use a classical two-step approach: route construction and route improvement. The way these two steps are handled is described below.

3.1.

Routes construction

We adapt the insertion heuristic developed by Toth and Vigo [8]. The method consists in initializing a small set of routes satisfying a fixed part of the demand, every route containing a pivot trip, and then in assigning the remaining trips to existing routes when possible; otherwise, a new route is started with the unrouted trip. Initialization: to compute the stream vehicle routes, we first give an efficiency score to every solution. This score is a linear combination of: a penalty for violating the vehicle capacity constraint; a quadratic penalty term for a late arrival at destination; a penalty for the waiting time at origin; a penalty for an early arrival at destination; the total travel time.

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These different terms are given in decreasing order of importance. Hence the efficiency score can be written:

where is the number of customers in vehicle is the waiting time at origin for customer is the number of vehicles in use, and denotes the travel time on any arc The associated weights are positive and satisfy The number of initial routes is determined by an estimate of the minimum number of routes needed to serve a given fraction of the number of trips. This is done by solving the problem:

The choice of a pivot trip for each route consists of assigning the most difficult trips (in a sense that will be defined later) to the initialized routes. This is done as follows. We first define a difficulty degree for each trip d depending on the average time needed to connect to trip for all and the excess of maximum travel time with respect to direct travel time. The first component is the average of the travel times between origins and destinations of and in all possible combinations, as suggested by Figure 13.3. It is thus equal to

The second component is the difference between the actual arrival time and the desired arrival time, that is

where is the maximum travel time of trip d. The difficulty degree can then be written as

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where and are weights. For a trip we define its attractiveness as the number of trips having their origin or destination within a defined time from the origin or destination of

The pivot trip for vehicle is – among all unrouted trips – the one maximizing the score defined as a linear combination of the difficulty degree and the attractiveness of trip Therefore:

where

and

are again weights.

Trip insertion: after having initialized a set of routes with pivot trips, we still have unrouted trips. At this stage, we iteratively insert the unrouted trips in the existing routes whenever it is possible as follows. Let be the number of such trips. These trips are assigned to the current routes by solving a min-cost assignment problem, where the insertion matrix is defined by rows corresponding to routes and columns corresponding to unrouted trips. Each entry of the matrix corresponds to the extra cost of the best feasible insertion of a trip in a route; this cost takes routing and inconvenience penalties into account. At each iteration of the insertion phase, a chosen number of best assignments is performed. Consequently, some terms (corresponding to the routes) of the assignment matrix need to be recomputed. If the cost of inserting of a trip is greater than some value (corresponding to a prescribed delay), we open a new route (if still possible) with some unused vehicle.

3.2.

Route improvement

The route improvement procedure is a Minimum cost Insertion with General Unstringing and Stringing (MINGUS), which uses ideas of the postoptimiza-

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tion procedures used by Gendreau, Hertz and Laporte for the traveling salesman problem [4]. We first explain the insertion method used for one node then we describe the complete improvement algorithm. The insertion of a node in the route takes place between two consecutive nodes. Generalized unstringing and stringing consists of iteratively: considering one of the constructed routes; removing a trip from this route by removing both its origin and its destination; inserting it back in one of the existing routes using classical node insertion. Insertion concerns both origin and destination in the same route. If is the cost of the constructed solution and is the corresponding set of routes then the different steps of the algorithm can now be described: MINGUS Algorithm

4.

Travel time prediction

Route computation depends largely on the travel time of each route. Thus, in order to compute good routes, the routing unit must have reliable information about a future state of the network. This future can be immediate or delayed as the system must be able to compute routes for demands known a long time in advance as well as for demands arriving a few minutes before the desired time of loading.

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However it is difficult to implement a travel time prediction unit on the basis of “link requests”, i.e. one procedure call for each link each time this link is taken into account for an envisaged route. Such a solution would have the enormous disadvantage of requiring thousands of almost simultaneous calls for the computation of each piece of route (shortest paths computation), which would of course significantly slow down the progress of the software computations. We made the decision to work with timeslices at the end of which a dedicated unit predicts travel times for the next few timeslices. In this way, route computations and travel time predictions become independent processes, which allows to implement them in parallel. As many models (see for instance Aron [1]), our travel time prediction method uses two types of information: historical data, that is the summary of information collected since the system was brought into service – these are grouped in so-called historical matrices (see later); data collected in real time, that is travel times transmitted by the minibuses as and when they progress along the network. Intuitively, it seems natural to work with the links of the network: roughly speaking, a link corresponds to a street and one computes travel times for each street (in both directions). Travel time for getting from PR-stop to PR-stop is then obtained as the result of summing the travel times of all links of the path chosen to reach from This is what we call a link-based model. The drawback to such a model comes from the simplification it carries out when assuming that a route is “only” a succession of links. Indeed it ignores the necessary time taken to join the links which can often be significant as it may correspond to the crossing of crossroads and/or the waiting at traffic lights. A possible way to reduce the inaccuracy of the link-based model is to work with routes (or paths) instead of working with links. We thus compute priori a certain number (say of short paths between each origin-destination pair. These origins and destinations correspond to PR-stops as every minibus journey is the juxtaposition of routes joining these places. The route computation consists then in choosing the shortest path (among the retained) for each one of the O-D pairs constituting an itinerary. The main advantage of this path-based model is that it allows to somewhat redistribute the inaccuracy of the link-based model on the whole path between two places. Moreover, when working with paths, one handles higher values than in a link-based framework and an inaccuracy of a few seconds has a lower impact on the solution. We now describe a way to classify historical information. It seems natural to think that, on a same link (and a fortiori on a same path), similar travel times are oberved for similar traffic conditions. These conditions can be defined by a set

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of observable factors such as the day of the week, the time of the day, weather conditions, type of day (schoolday, holiday, ...) and so on. In order to take these factors into account, the historical database is constituted of a number of historical matrices equal to the number of typical days. For example, one can consider four types of days: schooldays, working (non school-)days, Saturdays and Sundays plus bank holidays. Each such matrix has rows and columns, where is the number of abovementioned O-D pairs, is the number of shortestpaths retained for each O-D pair, and is the number of timeslices per day. In what follows, we use the following notation: for a fixed type of day, denotes the “historical” value of the travel time on path for the timeslice At the beginning of each day, the system loads the historical matrix corresponding to the appropriate type of day. The entries of the matrix are used to predict travel times (see below). However, some entries are also modified as and when new travel time values are transmitted to the system. This is the updating operation, for which we use the following formula:

where denotes the new (collected) travel time value (for path and timeslice and where This formulation is quite frequent in the framework of dynamic traffic models (see Ben Akiva et al. [3]). Generally, the value of the parameter is 0.5, but it can vary due to the following circumstances: during the initialization phase of the system (i.e. during its early days of functioning), is greater than 0.5 in order to give a larger weight to new values, which should allow the calculation of historical values more rapidly; the second case corresponds to the so-called “incidents” and is described in detail later. We now describe the prediction of travel times, for which we use the following notation: denotes the (computed) value of the predicted travel time on path for timeslice We speak about short-term prediction when it has to be made for a horizon of one hour at most. For instance, if every timeslice has a duration of 15 minutes, and if the last timeslice is the short-term predictions are related to timeslices and Beyond this time window, we speak about long-term prediction. In the case of a short-term prediction, one may use the most recent information. This means that the predicted value (with if one considers timeslices of 15 minutes) will be a combination of the historical

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value (with suppose that the differences

and the information received recently. We

are kept in memory for the previous timeslices, that is for prediction is given by

The

where we give a larger weight to the information of the most recent timeslices. For a long-term prediction (more than one hour), it is more difficult to make use of recent information (except in the case of an incident as described below). We have to rely on historical information exclusively, which means that in the case of a long-term prediction we perform a simple assignment:

We now explain some ideas for the management of the situations caused by an incident. By “incident” we mean the case in which a new (collected) travel time value is much greater than the predicted one. For the sake of clarity, we ignore the case for which a new travel time is much smaller than the predicted one but this other type of incident is managed in the same way. So let us suppose that during a certain timeslice and for a given path one has

This kind of event can happen when there has been a traffic accident on the considered path or it can be the result of congestion, road works, the presence of a lorry loading or off-loading goods, and so on. The problem with these incidents is that one must take them into account in order to react appropriately and at the same time one must avoid overestimating their impact. For instance, one must not give too much weight to a new (large) travel time value if this increase is due to an exceptional event which has a low probability of happening again at the same time in the next days because this would deteriorate the quality of historical information. On the other hand attention must be paid to the fact that these large values may be the result of longer term changes in the traffic conditions (e.g. restricted number of available lanes for a road under repair) which have to be passed on historical data, allowing a possible rerouting. The only way to make short work of this difficulty is to have further information at one’s disposal allowing to classify the incident in one of the following two categories: exceptional incidents,

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incidents which are likely to cause traffic changes in a longer term (for the next 24 hours at least).

To do this, we suppose that the minibuses can send additional information – the driver may for instance use a cellular phone to describe the situation – to help the system adapting the value of parameter in (1): in the first case, its value decreases in order to minimize the consequences of the exceptional incident while in the second case it increases so as to pass the travel time increase on the historical database. One also must be conscious that the management of incidents is largely site-dependent and we thus need more elaborate experimentations to possibly modify this strategy. Moreover, we could imagine to have another matrix at our disposal in order to store perturbations (or events which are likely to affect travel times) known in advance. We close these considerations with two remarks. The first one is that, given the possible variations of parameter in the case of an incident on a path and on that path only, there must be one such parameter per path. Second, when using a path-based model such as ours, another problem may arise if there is an incident on each one of the paths retained for one O-D pair. This can happen if is small and/or if the paths have many common links. This is why we must keep track of the links of the network in order to be able to recompute alternative paths for a given O-D pair after an incident has been detected.

5.

Computational results

In this section, we show some computational results obtained with the described algorithms. We have solved a set of four instances of the problem. These instances correspond to a chosen number of randomly generated demands for which all information are known in advance when used in the routing algorithm. The PR-stops are spread over 30 nodes covering the municipality of Charleroi, Belgium (Figure 13.4). Among these PR-stops, there are 5 parking lots. When we choose a number of demands, we mean actually twice this number or requested trips (we consider that every customer books an in-trip and an out-trip). The number of persons of a demand is between 1 and 5. The desired service times are such that there are peak-hour clusters and off-peak-hour clusters (similar distributions of demands are observed for all instances tested). We have used the following values for the penalty parameters of the routing algorithm: and Maximum travel time for each trip is proportional to the direct travel time from the origin to the destination of the trip. The proportionality factor is randomly chosen between 1.5 and 2. This is used to define the time windows.

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The minibus capacity is chosen equal to 15. For every simulation, we consider that the municipality owns a fixed number of minibuses. The aim of the optimization is to minimize the number of minibuses in use. In order to show the influence of travel times on the deviation w.r.t. desired service times, we performed two series of tests: the first one consists in applying the routing algorithm to show the efficiency in the case of deterministic travel times; the second is a case with a perturbation of the travel times. This perturbation is obtained in the following way: if is the a priori (historical) travel time on the arc then the actual value of the travel time we use to move the minibuses is:

where rand

is a random value uniformly distributed such that

The characteristics of the instances we used are described in Table 13.1. Table 13.2 shows the results obtained with deterministic travel times (the last two lines show the improvement of the total cost we obtain by using the local search) while Table 13.3 shows the results obtained in the case of perturbation. Note that the number of used minibuses, the average distance and the cost remain unchanged when using perturbed travel times. The results of these simulations show two things: the first one is that routing algorithms prove to be efficient, since the average delay for all simulation

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tests of Table 13.2 is negative (i.e. customers arrive in average early at their destination) and the maximal delay is not very large; second, it shows the lack of robustness of the algorithm when perturbations of the travel time are used. Table 13.3 shows a large maximal delay. It is therefore necessary to use a travel time prediction algorithm in a dynamic way when computing the itineraries.

6.

Conclusion

The Park & Ride Integrated System (PRIS) is intended to be a multi-modal answer to the inefficiency and lack of flexibility of public transportation systems at hours of low demand. We addressed here the key issues in the success of such

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a system, as well as a presentation of a possible architecture for a real-world implementation and comparisons with a classical bus transportation system. To convince urban transportation operators and urban decision makers to use such a system, we have to prove its feasibility, both technically and economically, and its performance. A useful tool to achieve this goal is our simulation software PRSim. In this paper, we describe some important tasks to be performed by this software, and in particular, we emphasize on the routing unit and on the prediction of travel times. The next two major challenges will be to test our algorithms in a real-world environment and to compare the efficiency of the PRIS with the one of existing transport systems. This will be the subject of a forthcoming paper.

Acknowledgments The authors wish to thank Martine Labbé, Thierry Martens and Philippe L. Toint for their helpful comments.

References [1] A RON M., Le modèle Mithra – Extrapolation à court terme des débits observés sur un réseau autoroutier, à partir de la connaissance de la structure des déplacements ainsi que des temps de parcours, INRETS Report Number 205, December 1995. [2] A UGELLO A., B ENEJAM E., N ERRIÈRE J.-P., P ARENT M., Complementarity between public transport and a car sharing service, World Congress on Application of Transport Telematics and Intelligent Vehicle Highway Systems, 1994. [3] BEN AKIVA M., DE P ALMA A., K AYSI I., Dynamic network models and driver information systems, Transportation Research, 25 A(5): 251-266, 1991. [4] G ENDREAU M., H ERTZ A., L APORTE G., New insertion and postoptimization procedures for the traveling salesman problem, Operations Research, Vol. 40-6, pp 1086-1094, 1992. [5] P ARENT M., TEXIER P.-Y., A public transport system based on light electric cars, International conference on Automated People Movers, IrvingTexas, 1993. [6] SAVELSBERGH M.W.P., SOL M., The general pickup and delivery problem, Transportation Science, Vol. 29-1, pp 17-29, 1995. [7] TAS Partnership limited, Bus based Park & Ride in Great Britain – A survey and report, 64 pages, January 1997. [8] TOTH P. , V IGO D., Heuristic algorithms for the handicapped persons transportation problem, Transportation Science, Vol. 31-1, pp 60-71, 1997.

Chapter 14 LONGITUDINAL ANALYSIS OF CAR OWNERSHIP IN DIFFERENT COUNTRIES Akli Berri [email protected] Institut National de Recherche sur les Transports et leur Sécurité (INRETS) Département Economic et Sociologie des Transports (DEST) 2, Avenue du Général Malleret-Joinville 94114 Arcueil cedex, France

Abstract

1.

Household car ownership behavior in seven countries, characterized by different economic and cultural contexts, is analyzed by means of demographic modeling. This approach provides a powerful tool for long term forecasting of the car fleet. Using data from series of household surveys, pseudo-panels are constructed according to the generation of household head; this is also done for homogeneous zones with respect to population density in France. Estimates are made for age and generation effects, and for the effects of the general economic context faced by households through income and price variables. The differences between countries and zones can be explained by two main factors: the history of car ownership development and population density. Changes are likely to be occurring in the car ownership behavior of young American generations: they seem to be less motorized than the older ones.

Introduction

Nowadays, the individual motorization rate (i.e. the average number of cars per adult) continues to rise almost linearly in most industrialized countries.1 Such a tendency questions the relevance of a priori saturation thresholds. Saturation thresholds fixed a priori in car ownership forecasting models have successively been exceeded (particularly in USA), due to the rise of multiequipment (urban cars, vans, etc.) and to complex individual attitudes characterized by some inertia. Indeed, even during a period of collapse of the new car market, the lengthening lifetime of owned cars reduces, and perhaps offsets, the recession impact on car ownership.2 Not only do the threshold estimates 229 M. Patriksson and M. Labbé (eds.), Transportation Planning, 229–245. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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rely heavily on the set of observations considered, but they may also differ significantly according to the error structure assumed [2]. Hence, in forecasting car ownership, it is necessary to use a flexible modeling which endogenizes the diffusion process in order to avoid an a priori fixed saturation threshold, to evaluate the effective role of economic factors (particularly income), and to take into account the dynamic heterogeneity of the individual behaviors. Indeed, under the assumption of a dynamic homogeneity of behavior, each agent plans his life-cycle consumption and maybe modifies his choices according to his expectations, but the trajectories are considered as stable in the long term. Yet, translations of these profiles can occur in the case of modifying preferences over time. Bonus [1] showed that the diffusion of durables is accompanied by temporal shifts of the quasi-Engel curves, relating equipment rates to income. This study covers seven countries characterized by different economic and cultural contexts. A longitudinal analysisLongitudinal analysis, tracking subsequent generations over time, is adopted. Using data from household surveys, a pseudo-panel is constructed for each country by grouping households from a series of independent cross-sections of the same survey, according to the birth band of the head. On the basis of the “parallelism” of the curves associated to the cohorts, an analysis scheme founded on an additive mechanism of the age and generation effects is used. Economic indicators and period variables are introduced, in order to identify the effect of the current economic context. This approach has great advantages. Indeed, it avoids the use of a predefined functional form, hence a priori fixed saturation levels; it conciliates the effects of economic and demographic factors; and it uses reliable variables (i.e. demographic projections) for long term forecasts. Let us first present the methodology (section 2), and then the results obtained for the countries studied (section 3). In section 4, we examine the effects of population density as shown by the comparison between homogeneous zones in France. Section 5 briefly presents the mechanism of long term forecasting of the car fleet using the demographic approach, and contrasts it with the models used by some countries in their respective national transportation planning software. Section 6 concludes.

2. 2.1.

An age-cohort-period model Why a demographic approach to car ownership?

The profound structural changes which have accompanied the rapid growth of individual mobility in developed countries underline the necessity of studying the transportation demand not in a context of equilibrium, but in a context of historical evolution [8]. Only a longitudinal analysis of behavior, centered on

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the temporal follow-up of individuals or cohorts, permits us to identify the factors determining this evolution. The justification for this approach is two-fold: Firstly, a critique of the traditional econometric models used for long term projection of motorization. If, in a cross-section analysis, income remains one of the main variables explaining the rate of motorization of the households, it cannot be considered, at the current stage of diffusion of the private car, as the only factor explaining the rise in the rate of car ownership. Secondly, the necessity of locating the analysis of motorization in a precise temporal setting, by explicitly taking into account the history of the diffusion of the automobile, and thus incorporating the phenomenon of saturation which starts to appear, without having to establish an a priori level of saturation. We know that the history of the car diffusion is different in Europe and in North America, as well as in capital regions compared to provincial regions. For example, in the early fifties, the level of motorization in Quebec was triple that of France [4].

2.2.

Decomposition of time into three dimensions

The longitudinal approach highlights the complex impact of age which, in a dated temporal context, consists of the combination of three linked dimensions: the stage in the life cycle, which indicates the importance of age in car ownership decisions; the generation (or cohort), which identifies the behavior of individuals born during the same period, and therefore sharing a common life experience; and the period, which indicates the impact of the global socio-economic context. The evaluation of the effect of the stage in life cycle gives us a characteristic curve indicating the evolution of the motorization rates related to age, which corresponds to a definite pattern. The introduction of the generation effects constitutes a first amendment to the vision of equilibrium, and permits us to place this profile in a historical perspective. In the case of the acquisition of durable goods, this approach is quite relevant, since it shows us the importance of diffusion effects linked, for example, to the evolution of life styles, institutional constraints, consumers needs, or characteristics of supply. Finally, taking into account the period effects permits us to measure short term or medium term factors of disequilibrium which simultaneously affect all the individuals or households.

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The model

Let us denote a measure of car ownership behavior for households whose heads are years old at date and belong to the generation defined by their birth year. Obviously, there is an exact relationship between these parameters:

Let:

where and are respectively dummy variables for age and generation and where denotes period effects which we will try to explain by economic variables. Thus, we adjust by variance analysis the following model:

where is real total expenditure for all households (explaining general economic growth) and is a relative price index influencing car ownership (total cost or, preferably, purchase costs including new and second-hand cars). Even adjusting without intercept, dummy variables cannot cover all age groups and all generations, because of identification problems due to the relation 2.1. Thus, the age variables cover the life cycle, whereas the generation variables characterize all cohorts except one: the “ reference generation ” (here 1936-40), for which the coefficient is set to 0. Hence, each generation coefficient can be interpreted as a gap between the cohort and the “ reference generation ”. Moreover, the set of age coefficients can be interpreted as the motorization curve along the life cycle for the “ reference generation”. For the same reasons, the variables representing period effects have to be set to 0 at the same date (here generally in 1994). It seems difficult to introduce individual income and dummy variables for age in the same model [7]. This is probably due to the heterogeneity of needs over the life cycle. Estimates of the changes in income and car ownership for different generations between 1977-79 and 1992-94 illustrate this in the case of France.3 For young households, the number of cars per adult grows much faster than income. It increases also rapidly when the head of household is around 50 (grown up children acquiring their first car while they still live with their parents). For old-age households, the number of cars per adult decreased rapidly, during a period where income per consumption unit grew because of real increases in retirement pensions. These heterogeneity problems at different

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stages of life cycle have led us to introduce global income (or expenditure) to explain period effects in an age-cohort-period model.

3. 3.1.

A multinational comparison The data

Most of our data comes from Household Expenditure surveys. These surveys convey crucial information about household socio-economic characteristics, motorization and transport expenditures. Their use allows handling both economic and demographic factors in analyzing transport behavior. Pseudopanels were constructed using time-series of cross-sections, and the cohorts were formed according to the birth year of household head (or household reference person). For France, we used data from annually repeated cross-sections of the INSEE Household Conjuncture Survey, covering the period 1977-94 (about 10,000 households interviewed each year). Households are interviewed twice, with a one-year interval, in October. The German data is from the 1983, 88 and 93 Budget surveys. The final sample for the 1988 survey contained 45,074 households. Households with very high net income are excluded from the survey. Households formed by foreign people were included for the first time in 1993. From the 1993 survey only data for the former FRG is used in this study. The Japanese data is from the 1979, 84, 89 and 94 issues of the Statistics Bureaupublications on the National Survey of Family Income and Expenditure. The 1994 survey was designed to sample about 59,800 households including 4,700 one-person households. The Dutch data covers 1985, 90 and 95. The average number of cars per household is the division of the number of registered cars and the number of households from the Household Statistics (EBB survey). The proportions of households with one, two or more cars are estimated from the National Travel Survey. For Poland, the data is from the Household Budget Survey carried out since 1982, with a rotation method and on a quarterly basis. Each year, around 30,000 households are surveyed. The data used cover the periods 1987-90, 1992, and 1994-95. The UK data is drawn from the Family Expenditure Survey, carried out annually with a sample of around 10,000 households ; about 70% of them respond. For comparison with France, the econometric results are based on data from 1977 to 1994. The US data is from the Consumer Expenditure Survey, a rotating panel (renewed by one fifth every quarter) conducted since 1980. The households are interviewed during 5 consecutive quarters. Each quarter of data is processed

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independently. Around 6,000 households are surveyed each quarter. This study covers the period 1980-89, except 1982, 1983 and 1986 for which the data sets are incomplete. As we are concerned with car ownership at a given date, we chose to use data from the first quarter interview in each year.

3.2.

Car ownership patterns

Car fleets continue to grow almost linearly in all the countries studied.4 Making long term forecasts of these fleets raises the question about the suitable kind of data to use. The following graphs display the evolution of the number of cars per household in three countries, as shown by the survey data, for the defined cohorts. Figure 14.1 represents the evolution of the number of cars per household in USA for the different cohorts. The first, second and third points of each curve refer to the surveys in 1980, 1984 and 1989, respectively. For the generation 1966–70, only two points are represented. On one hand, we see the differences in motorization levels according to age: linking all points pertaining to the same survey year, we obtain what could be interpreted as a motorization profile over the life cycle (such a profile is shown for 1984 by the bold dotted line). The number of cars owned by a household increases until the head is around 50 years old and then it decreases. On the other hand, the comparison of the average numbers of cars per household of two or three successive cohorts interviewed at a given date shows the differences in motorization at the same age, thus reflecting the generation effect. Hence, the use of aggregate time-series is not relevant, since they do not take into account the heterogeneity across individuals. Nor is the use of only one crosssection of individual data, since the heterogeneity of behaviors over time is not accounted for [10]. This reinforces the necessity of the longitudinal analysis adopted in this study. Figures 14.1, 14.2 and 14.3 show the different situations in terms of motorization levels, life cycle profiles and discrepancies between generations, reflecting the diversity of economic and socio-cultural contexts and of the historical developments of car ownership across countries.

3.3.

A comparative analysis of the results

In this section, we present the results for the average number of cars per household5 and the proportion of motorized households. The oldest generation includes all households whose heads were born before 1910. The last age group was defined as 75 years and over. Households whose heads are under 20 have been omitted before estimating the model, because they do not form a total cohort. Indeed, children leave their parents’ household at different ages, depending on social groups and on the period.

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Figure 14.1.

235

Number of cars per household in USA: 1980, 84 and 89

The figures plotted in the following graphs are the values of the two car ownership indicators as simulated by the model. The life cycle profiles for the reference generation are obtained by plotting the age coefficients against age bands of the household head. Adding to each generation coefficient the estimated coefficient for a given age interval (here 35-39), we obtain a representation of the gaps between the generations at the same age. 3.3.1 Number of cars per household. Let us first consider, on the example of the generation 1936-40, how the number of cars per household varies along the life cycle (Figure 14.4). At all stages of the life cycle, this proportion is highest in the US and lowest in Poland. Excepting Poland, the lowest motorization levels are observed in Japan for young households and in the Netherlands for old ones ; France and the UK show quite the same pattern.

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In all cases, the average number of cars increases at the beginning of the life cycle until it reaches its maximum and declines thereafter. The maximum is reached when the head of household is around 50 years old in France and in the UK, about 55 in the Netherlands and Poland, and about 60 in Japan. In USA, this occurs earlier (at the age of 45 to 49). This maximum is around 1.3 cars per household in the UK and in Japan, 1.2 in France, 1.1 in the Netherlands, and only 0.38 in Poland. At old age (for instance, 70-74 years), the average number of cars per household is about the same in France, Japan and the UK (0.8). The gaps between generations can be shown from their motorization simulated by the model for the same age (35–39), although not at the same date (Figure 14.5). As noted in the examples of Paris and Montreal [3], there seems to be a generation with maximum motorization rates in almost all the countries under review: at the same age, those born after 1950 have significantly fewer cars in the UK and in France, while in Japan, there is almost no difference between generations born after 1955. Compared to this most motorized generation, the cohort born during the second half of the 1960’s has 0.15 less cars in France and 0.1 less in the UK. As one could expect, the most motorized generation in USA was born earlier (in the 1930’s). The gap between the most motorized generation and the generations born at the beginning of the century is more important than the gaps between new cohorts (notably less in the US where car ownership developed 20 years earlier than in Western Europe). The case of Poland is different since car ownership developed much later (in the 1990’s) than in the other countries. Thus, the number of cars per household is still higher for the new generations.

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Concerning period effects (Table 14.16), the total expenditure elasticity is significantly lower for USA (0.15) and Japan (0.29) than for France (0.50) or the UK (0.58). Price elasticities are almost the same for France and UK. 3.3.2 Proportion of motorized households. On the example of generation 1936-40 (Figure 14.6), we note again that, at all stages of the life cycle, the highest and lowest percentages of motorized households are observed in USA and Poland respectively. We notice that the proportion increases rapidly at the beginning of a household’s life cycle, but its highest level is reached

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at different ages from one country to another. For instance, in France and in the US, it reaches its maximum (around 85%) when the head of household is between 30 and 40 years old. In the UK and West Germany, this maximum is observed slightly later (40-44 years), and at a lower proportion (about 82% and 84%, respectively). In Japan, there are much less motorized young households, and their proportion increases until the 55-59 age range where it reaches about 80%. At old age (for instance, 70-75 years), and apart from Poland, the least motorized are the Japanese (45% of households without car), followed by the British and the French (about 40%) and, finally, the Americans (only 20%).

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The gaps between generations represented by the proportion of motorized households when the head of household is 35-39 years old (Figure 14.7), show a cohort with a maximum proportion in all the countries under review, except Japan and Poland: these generations were born between 1910 and 1950 in the US, between 1930 and 1955 in the UK and in France. Compared to this generation, the proportion of motorized households is 14 points lower in the US, 11 points in France and 7 points in the UK, for the generation born during the second half of the 1960’s. In West Germany and the Netherlands, there is almost no difference between the youngest generations. Period effects are higher (in absolute value) in France than in the UK, but the confidence intervals overlap. The total expenditure elasticity is even higher in Japan (0.45), which is consistent with the continually increasing gaps between the generations, thus reflecting the fact that Japan is further from saturation than are France and the UK. From the examination of the generation gaps for the two motorization indicators considered, it seems that in some countries young generations are less motorized compared to the older ones (especially, the most equipped ones). Such a fact is quite clear in the case of USA. One might explain it by an income effect, but scrutiny of the average real income per Consumption Unit (Oxford scale) calculated from the survey data leads to a different interpretation. Indeed, it is not clear cut that young cohorts are becoming poorer, at the same age, than the preceding ones. One could think of the effect of the replacement of cars with vans or light trucks, but when we consider all of the vehicles owned (not only cars) we obtain the same features (though the generation gaps are somewhat smaller). Thus, changes are likely to be occurring in the behavior of young generations regarding car ownership. However, other factors such as the age at which young households are formed and couples start having children have to be investigated. Further analyses, covering a longer survey period, are necessary to confirm this conclusion.

4.

A comparative analysis for homogeneous zones

Factors other than those already mentioned can be analyzed by adjusting different models for different categories. High population density is likely to lower car ownership. Indeed, alternative transport modes are more widely available in high population density areas compared to less dense ones. Besides, congestion problems may lead to a restrained use of cars, either by users themselves or due to specific political measures. Such an influence can be observed, for the two car ownership indicators, in the cases of Japan and of the Netherlands for which population density levels are the highest among the seven countries considered.

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In what follows, we perform the Age-Cohort-Period model for three homogeneous zones in France, with respect to population density. We consider concentric zones: city-centers, inner suburbs, and outer suburbs and rural areas.

4.1.

Age and generation effects

The curves representing the average number of cars per adult have the same shape for each zone: it increases rapidly until 35-39, then stays almost constant at a slightly lower level until 70 years old and decreases thereafter (Figure 14.8). Until the age of 65 to 69, car ownership is higher in outer suburbs and rural areas than in inner suburbs and city-centers.

In all zones, there is almost no significant difference between generations born after 1945 (Figure 14.9). For all generations, the number of cars per adult is higher in outer suburbs or rural areas than in inner suburbs and city-centers.

4.2.

Income and price effects

The main contrast between zones is for period effects (Table 14.2). Income elasticity is significantly lower in city-centers and inner suburbs than in outer suburbs or rural areas. Let us mention that this result is obtained when explaining economic growth by changes in the average disposable income per Consumption Unit observed in each zone, which has increased more rapidly in less densely populated areas. As for purchase prices, the elasticity is lower in absolute value in outer suburbs and rural areas than in the two other zones. These results reflect the differences, pointed out above, between high population density areas and less dense ones regarding the availability of alternative modes and congestion problems, and their consequences on car ownership and use.

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5.

241

Long term forecasting using the demographic approach

The demographic approach provides a powerful tool for making long term forecasts of the car fleet, in that it avoids fixing saturation thresholds, it conciliates the effects of economic and demographic factors, and it relies on stable population projections. The long term projection model thus comprises two parts: first, a projection of the age structure of the population of driving age, which allows us to take into account purely demographic phenomena, in relation, for example, to aging which is foreseeable in most industrialized societies; and second, and fundamental to the model, the estimation of a standard profile of the life cycle and its evolution through time, as described above. From a comparison between the population structures in 1995 and those projected for 2010,19 a general feature in all the countries considered is the decrease in the proportion of persons under 40 years old and the increase in that of those aged 60 and over. Such a tendency would be of great importance as to the evolution of the car fleet. The models used to deal with car ownership in national transportation planning software differ, each in some aspect, from that outlined above. For instance, in the Polish prognosis of transport development,20 the population age structure was not accounted for in forecasting the number of cars; instead, the size of total population was used, along with other factors such as the projected changes in the structure of expenditures. The car ownership model used in forecasting the national road traffic for Great Britain21 incorporated saturation levels (which differ from one household type to another) and was estimated on cross-section data. In the Dutch National Model System,22 the car ownership model relies on the results of the driving license model. Both are individual choice models and are estimated on the basis of cross-section data from the National Travel Survey. Projections of the total number of private

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cars in the Netherlands based on the Van den Broecke/Social Research car ownership model are taken as control totals. Constants in the disaggregated car ownership model are adjusted equally for each zone so as to make the results as close as possible to the national totals.

6.

Summary and conclusions

A multinational comparison of household car ownership has been carried out by means of a longitudinal analysis. Using a flexible modeling, the approach endogenizes the car diffusion process and accounts for both demographic and economic factors. In addition, it provides a powerful tool for long term forecasting of the car fleet. Two major insights can be drawn from the results: Firstly, the differences between countries and zones can be explained by two main factors: the history of car ownership development: the US, where the diffusion of the automobile started before World War II, is closer to saturation than the West European countries or Japan, and especially Poland where motorization rates are still growing rapidly. Indeed, there are smaller differences between generations in USA, and new cohorts seem less motorized than those born in the 1930’s. Moreover, elasticities with respect to total expenditure are lower than in Europe and Japan; and population density: saturation thresholds are likely to be lower in more densely populated areas (city-centers or metropolitan areas) or countries. During the life cycle, maximum levels of motorization seem to be observed later in more densely populated countries (e.g., Japan and the Netherlands). Secondly, the shape of the curves representing the generation gaps in USA for two motorization indicators might reflect changes in the behavior of the young generations regarding car ownership: they seem to be less motorized compared to older ones. However, this has to be confirmed by further analyses.

Acknowledgments This study was done within the SCENARIOS and SCENES projects, funded by the European Commission under the Transport RTD Program of the Framework Program. It also forms part of the project “Dynamic Analysis of Car Ownership and Transport Expenditures” being carried out in collaboration with the ESRC Transport Studies Unit in the UK. Material from the Family Expenditure Survey is Crown copyright; has been provided by the Central Statistical Office through the ESRC Archive Data; and has been used by permission. Neither the CSO nor the ESRC Data Archive bears any responsibility for the analysis or interpretation reported here.

ACKNOWLEDGMENTS

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The US data used in this study were made available by the Inter-university Consortium for Political and Social Research (ICPSR). The Consumer Expenditure Survey was conducted by the U.S. Department of Labor, Bureau of Labor Statistics (BLS). The data was reorganized for household-level analysis by Julie A. Nelson, University of California, Davis. Neither the original source or collectors of the data, nor the organizer or distributors of the data bear any responsibility for the analyses or interpretations presented here. I am grateful to J. A. Dargay (UCL) for providing me with the UK pseudopanel she constructed using FES data. I am also indebted to T. Yamamoto (Kyoto University), R. Konen (CBS), C. Ortmann (DLR) and C. Starzec (INSEE) for the Japanese, Dutch, German and Polish data, respectively. I thank J. A. Nelson (Brandeis University) for valuable suggestions about the CEX data, and M. Gray and W. Marshall (BLS) for documentation on the survey. I also thank M. Bak (University of Gdansk) for documentation on the Polish prognosis of transport development. I express my gratitude to Drs. J.-L. Madre and C. Gallez (INRETS-DEST), Prs. J. A. Dargay (UCL), F. Gardes and L. Lévy-Garboua (Université de Paris 1) and an anonymous referee for their helpful comments and suggestions. Responsibility for any remaining errors or omissions is solely mine.

Notes 1. See Demand Descriptors of Passenger Transport, Deliverable No. D5 of the SCENARIOS project, October 1997. 2. In France, the mean duration of car keeping decreased from 3.82 years in 1985 to 3.72 in 1990 and then grew to 4.08 years in 1994. See [5]. 3. Source: INSEE Household Conjuncture Survey, 1977-94. 4. See the different issues of World Road Statistics, IRF, Geneva. 5. Not available for Germany 6. Germany, the Netherlands and Poland are not represented in this table. Neither income nor price variables were included in the models, since there are too few points in time to estimate them. 7. The adjusted is that of the regression including an intercept. 8. Economic growth is taken into account through real total expenditure (per capita in Japan and France, per household in the UK). 9. Purchase relative prices include both new and second-hand cars: it is a price index for the UK and the average value per car for France. For Japan, there are not enough data (only 4 points in time) to take into account the price factor. 10. The elasticities are estimated, as implied by the model specification, by the ratio of the coefficient of the explanatory variable over the total population average of the dependent variable in a certain period (1980 in the case of USA, and 1994 for the remaining countries). Confidence intervals (with a probability of 95%) for elasticities are shown between brackets. Only one bound of an interval is mentioned if the other bound has the wrong sign (negative for income elasticities and positive for price elasticities). 11. For USA, the estimates of period effects (real disposable income per capita and relative price index for car purchase, at the national level and in the previous year, since car ownership is observed during the first quarter of each year) are either not precise or with the wrong signs (as indicated by the symbol (**)). 12. The adjusted is that of the regression including an intercept.

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13. Income is accounted for by including real total expenditure for the whole of France, and disposable income per Consumption Unit for the different zones. 14. Purchase prices are average values of new and second-hand car prices. 15. The elasticities are estimated, as implied by the model specification, by the ratio of the coefficient of the explanatory variable over the total population average of the dependent variable in 1994. Confidence intervals (with a probability of 95%) are shown between brackets. 16. City-centers. 17. Inner suburbs. 18. Outer suburbs and rural areas. 19. Sources: Eurostat for European Union countries; the International Data Base of the US Bureau of the Census for USA, Japan and Poland. 20. Elaborated in 1997 under the direction of Pr. Burnewicz (University of Gdansk) for the Ministry of Transport. 21. National Road Traffic Forecasts (Great Britain) 1997, Department of the Environment, Transport and the Regions. 22. See The National Model System for Traffic and Transport: Outline, Ministry of Transport and Public Works, Rijkswaterstaat, Transportation and Traffic Research Division, Rotterdam, January 1992.

References [1]

Bonus, H.: 1973, ‘Quasi-Engel curves, diffusion, and the ownership of major consumer durables’, Journal of Political Economy, Vol. 83, pp. 655– 677.

[2]

Brooks, R. J., Dawid, A. P., Galbraith, J. I., Galbraith R. F., Stone, M. and Smith, A. F. M.: 1978, ‘A note on forecasting car ownership’, Journal of the Royal Statistical Society, Vol. 141A, Part 1, pp. 64–68.

[3]

Bussière, Y., Madre, J.-L., Armoogum, J., Gallez, C. and Girard, C.: 1994, ‘Longitudinal approach to motorization: long term dynamics in three urban regions’, paper presented at the International Association of Travel Behavior Research (IATBR) Conference, Valle Nevado, Chile.

[4] Bussière, Y.: 1989, ‘L’automobile et l’expansion des banlieues: le cas de Montréal, 1901-2001’, Urban History Review/Revue d’Histoire Urbaine, No. 2 (Oct.), pp. 159–165. [5]

CCFA (Comité des Constructeurs Français d’Automobiles): 1994, L’Industrie Automobile en France - 1994, Paris.

[6]

Deaton, A.: 1985, ‘Panel data from time series of cross-sections’, Journal of Econometrics, 30, pp. 109–126.

[7]

Gallez, C.: 1994, Modèles de Projection à Long Terme de la Structure du Parc et du Marché de l’Automobile, Doctoral Dissertation, Université de Paris 1.

[8]

Goodwin, P. B., Dix, M. C., and Layzell, A.D.: 1987, ‘The case for heterodoxy in longitudinal analysis’, Transportation Research, Vol. 21A, No. 4/5, pp. 363-376.

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[9] Nelson, J. A.: 1994, Consumer Expenditure surveys, 1980-1989: Interview surveys, for household-level analysis, (computer file), Washington, DC: US Department of Labor, BLS (producer), 1992. Ann Arbor, MI: ICPSR (distributor), 1994. [10] Pendyala, R. M., Kostyniuk, L. P. and Goulias, K. G.: 1995, ‘A repeated cross-sectional evaluation of car ownership’, Transportation, 22, pp. 165184.

Index

A-cyclic network, 6 Active route set, 24, 114 Age-Cohort-Period model, 230 Average excess cost, 3 Bottleneck, 200 Capacity constraints, 108, 200 Car diffusion, 231 Complementarity problem, 108, 200 Complete assignment process, 204 Cone field, 203 Cone projection method, 205 Continuous cost function, 59 Continuum hypothesis, 126 Control constraints, 201 Cost-averaging algorithm (MSA-CA), 58 Day-to-day variations, 34 Defuzzification, 164 Demand responsive model, 108, 213 Disaggregate simplicial decomposition (DSD), 5, 23 Diverge problem, 135 Dynamic heterogeneity, 230 Dynamic traffic assignment, 120 Economic and demographic factors, 230 Equilibrium conditions, 21, 40, 44, 198 Equilibrium travel time variance, 80 Extended maximum cost, 6 FIFO, 122 Flow-averaging algorithm (MSA-FA), 58 Flow-delay functions, 113 Fuzzification, 164 Fuzzy similarity, 172 Fuzzy traffic signal control, 163 Gammit model, 62 General equilibrium model, 113 Generalized travel cost, 20 Historical database, 218, 223 Household car ownership, 229 Intersection modeling, 126 Lighthill-Whitham-Richards model, 120 Long term forecasting, 229

Monotone cost function, 59, 108, 201 Multi-label shortest path, 26 Multicopy network, 109 Nonlinear money-time relation, 20 Optimization formulation, 23, 112 Park & Ride, 213 Partial flows, 120 Path choice map, 57 Path choice model, 56 Path cost model, 56 Path flow propagation model, 55 Perceived cost, 39 Pointwise intersection, 129 Population density, 229 Pseudo-panels, 229 Reliability connectivity, 70 performance, 70 Route construction, 217 Route excess cost, 3 Route travel time variance, 81 Routing, 217 dynamic, 215 Saturation, 229 Sensitivity analysis, 72 Simplicial decomposition, 5 Stationary solution, 129 Stochastic demands, 34 Stochastic network loading (SNL), 58 Stochastic user equilibrium (SUE), 33, 58, 72 sensitivity analysis, 72 Traffic assignment, 53 link-based, 4, 25 origin-based, 2 route-based, 5, 23 Transport system demand-responsive, 213 Travel time prediction, 215, 221 Unreliable routes, 81 Variational inequality, 198 Vector field, 198 Wardrop conditions, 21, 111

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V. Kreinovich, A. Lakeyev, J. Rohn and P. Kahl: Computational Complexity and Feasibility of Data Processing and Interval Computations. 1998 ISBN 0-7923-4865-6

11.

J. Gil-Aluja: The Interactive Management of Human Resources in Uncertainty. 1998 ISBN 0-7923-4886-9

12.

C. Zopounidis and A.I. Dimitras: Multicriteria Decision Aid Methods for the Prediction of Business Failure. 1998 ISBN 0-7923-4900-8

13.

F. Giannessi, S. Komlósi and T. Rapcsák (eds.): New Trends in Mathematical ProISBN 0-7923-5036-7 gramming. Homage to Steven Vajda. 1998

14.

Ya-xiang Yuan (ed.): Advances in Nonlinear Programming. Proceedings of the ’96 International Conference on Nonlinear Programming. 1998 ISBN 0-7923-5053-7

15.

W.W. Hager and P.M. Pardalos: Optimal Control. Theory, Algorithms, and Applications. 1998 ISBN 0-7923-5067-7 Gang Yu (ed.): Industrial Applications of Combinatorial Optimization. 1998 ISBN 0-7923-5073-1

16. 17.

D. Braha and O. Maimon (eds.): A Mathematical Theory of Design: Foundations, Algorithms and Applications. 1998 ISBN 0-7923-5079-0

Applied Optimization 18.

O. Maimon, E. Khmelnitsky and K. Kogan: Optimal Flow Control in Manufacturing. Production Planning and Scheduling. 1998 ISBN 0-7923-5106-1

19.

C. Zopounidis and P.M. Pardalos (eds.): Managing in Uncertainty: Theory and Practice. 1998 ISBN 0-7923-5110-X

20.

A.S. Belenky: Operations Research in Transportation Systems: Ideas and Schemes of Optimization Methods for Strategic Planning and Operations Management. 1998 ISBN 0-7923-5157-6

21.

J. Gil-Aluja: Investment in Uncertainty. 1999

22.

M. Fukushima and L. Qi (eds.): Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smooting Methods. 1999 ISBN 0-7923-5320-X

23.

M. Patriksson: Nonlinear Programming and Variational Inequality Problems. A Unified Approach. 1999 ISBN 0-7923-5455-9

24.

R. De Leone, A. Murli, P.M. Pardalos and G. Toraldo (eds.): High Performance Algorithms and Software in Nonlinear Optimization. 1999 ISBN 0-7923-5483-4

25.

A. Schöbel: Locating Lines and Hyperplanes. Theory and Algorithms. 1999 ISBN 0-7923-5559-8

26.

R.B. Statnikov: Multicriteria Design. Optimization and Identification. 1999 ISBN 0-7923-5560-1

27.

V. Tsurkov and A. Mironov: Minimax under Transportation Constrains. 1999 ISBN 0-7923-5609-8 ISBN 0-7923-5610-1 V.I. Ivanov: Model Development and Optimization. 1999

28.

ISBN 0-7923-5296-3

29.

F.A. Lootsma: Multi-Criteria Decision Analysis via Ratio and Difference Judgement. ISBN 0-7923-5669-1 1999

30.

A. Eberhard, R. Hill, D. Ralph and B.M. Glover (eds.): Progress in Optimization. Contributions from Australasia. 1999 ISBN 0-7923-5733-7 T. Hürlimann: Mathematical Modeling and Optimization. An Essay for the Design of Computer-Based Modeling Tools. 1999 ISBN 0-7923-5927-5 J. Gil-Aluja: Elements for a Theory of Decision in Uncertainty. 1999 ISBN 0-7923-5987-9 H. Frenk, K. Roos, T. Terlaky and S. Zhang (eds.): High Performance Optimization. 1999 ISBN 0-7923-6013-3

31. 32. 33. 34.

N. Hritonenko and Y. Yatsenko: Mathematical Modeling in Economics, Ecology and the Environment. 1999 ISBN 0-7923-6015-X

35.

J. Virant: Design Considerations of Time in Fuzzy Systems. 2000 ISBN 0-7923-6100-8

Applied Optimization 36.

G. Di Pillo and F. Giannessi (eds.): Nonlinear Optimization and Related Topics. 2000 ISBN 0-7923-6109-1

37.

V. Tsurkov: Hierarchical Optimization and Mathematical Physics. 2000 ISBN 0-7923-6175-X

38.

C. Zopounidis and M. Doumpos: Intelligent Decision Aiding Systems Based on ISBN 0-7923-6273-X Multiple Criteria for Financial Engineering. 2000

39.

X. Yang, A.I. Mees, M. Fisher and L.Jennings (eds.): Progress in Optimization. Contributions from Australasia. 2000 ISBN 0-7923-6286-1 D. Butnariu and A.N. Iusem: Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization. 2000 ISBN 0-7923-6287-X J. Mockus: A Set of Examples of Global and Discrete Optimization. Applications of Bayesian Heuristic Approach. 2000 ISBN 0-7923-6359-0 H. Neunzert and A.H. Siddiqi: Topics in Industrial Mathematics. Case Studies and Related Mathematical Methods. 2000 ISBN 0-7923-6417-1 K. Kogan and E. Khmelnitsky. Scheduling: Control-Based Theory and PolynomialTime Algorithms. 2000 ISBN 0-7923-6486-4 E. Triantaphyllou: Multi-Criteria Decision Making Methods. A Comparative Study. 2000 ISBN 0-7923-6607-7 S.H. Zanakis, G. Doukidis and C. Zopounidis (eds.): Decision Making: Recent DevelISBN 0-7923-6621 -2 opments and Worldwide Applications. 2000

40. 41. 42. 43. 44. 45. 46. 47.

G.E. Stavroulakis: Inverse and Crack Identification Problems in Engineering Mechanics. 2000 ISBN 0-7923-6690-5 A. Rubinov and B. Glover (eds.): Optimization and Related Topics. 2001 ISBN 0-7923-6732-4

48.

M. Pursula and J. Niittymäki (eds.): Mathematical Methods on Optimization in Transportation Systems. 2000 ISBN 0-7923-6774-X

49.

E. Cascetta: Transportation Systems Engineering: Theory and Methods. 2001 ISBN 0-7923-6792-8

50.

M.C. Ferris, O.L. Mangasarian and J.-S. Pang (eds.): Complementarity: Applications, Algorithms and Extensions. 2001 ISBN 0-7923-6816-9 V. Tsurkov: Large-scale Optimization – Problems and Methods. 2001 ISBN 0-7923-6817-7

51. 52.

X. Yang, K.L. Teo and L. Caccetta (eds.): Optimization Methods and Applications. ISBN 0-7923-6866-5 2001

53.

S.M. Stefanov: Separable Programming Theory and Methods. 2001 ISBN 0-7923-6882-7

Applied Optimization 54.

S.P. Uryasev and P.M. Pardalos (eds.): Stochastic Optimization: Algorithms and Applications. 2001 ISBN 0-7923-6951-3

55.

J. Gil-Aluja (ed.): Handbook of Management under Uncertainty. 2001 ISBN 0-7923-7025-2

56.

B.-N. Vo, A. Cantoni and K.L. Teo: Filter Design with Time Domain Mask ConISBN 0-7923-7138-0 straints: Theory and Applications. 2001 ISBN 0-7923-7139-9 S. Zlobec: Stable Parametric Programming. 2001

57. 58.

M.G. Nicholls, S. Clarke and B. Lehaney (eds.): Mixed-Mode Modelling: Mixing Methodologies for Organisational Intervention. 2001 ISBN 0-7923-7151-8

59.

F. Giannessi, P.M. Pardalos and T. Rapcsák (eds.): Optimization Theory. Recent ISBN 1-4020-0009-X Developments from Mátraháza. 2001

60.

K.M. Hangos, R. Lakner and M. Gerzson: Intelligent Control Systems. An IntroducISBN 1-4020-0134-7 tion with Examples. 2001

61. 62.

D. Gstach: Estimating Output-Specific Efficiencies. 2002

ISBN 1-4020-0483-4

J. Geunes, P.M. Pardalos and H.E. Romeijn (eds.): Supply Chain Management: ISBN 1-4020-0487-7 Models, Applications, and Research Directions. 2002

63.

M. Gendreau and P. Marcotte (eds.): Transportation and Network Analysis: Current ISBN 1-4020-0488-5 Trends. Miscellanea in Honor of Michael Florian. 2002

64.

M. Patriksson and M. Labbé (eds.): Transportation Planning. State of the Art. 2002 ISBN 1-4020-0546-6

65.

E. de Klerk: Aspects of Semidefinite Programming. Interior Point Algorithms and Selected Applications. 2002 ISBN 1-4020-0547-4

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