Vehicle Routing under Consideration of Driving and Working Hours: A Distributed Decision Making Perspective

Christoph Manuel Meyer Vehicle Routing under Consideration of Driving and Working Hours

GABLER RESEARCH Produktion und Logistik Herausgegeben von Professor Dr. Bernhard Fleischmann, Universität Augsburg Professor Dr. Martin Grunow, Technische Universität München Professor Dr. Hans-Otto Günther, Technische Universität Berlin Professor Dr. Stefan Helber, Universität Hannover Professor Dr. Karl Inderfurth, Universität Magdeburg Professor Dr. Herbert Kopfer, Universität Bremen Professor Dr. Herbert Meyr, Technische Universität Darmstadt Professor Dr. Thomas S. Spengler, Technische Universität Braunschweig Professor Dr. Hartmut Stadtler, Universität Hamburg Professor Dr. Horst Tempelmeier, Universität Köln Professor Dr. Gerhard Wäscher, Universität Magdeburg Kontakt: Professor Dr. Hans-Otto Günther, Technische Universität Berlin, H 95, Straße des 17. Juni 135, 10623 Berlin

Diese Reihe dient der Veröffentlichung neuer Forschungsergebnisse auf den Gebieten der Produktion und Logistik. Aufgenommen werden vor allem herausragende quantitativ orientierte Dissertationen und Habilitationsschriften. Die Publikationen vermitteln innovative Beiträge zur Lösung praktischer Anwendungsprobleme der Produktion und Logistik unter Einsatz quantitativer Methoden und moderner Informationstechnologie.

Christoph Manuel Meyer

Vehicle Routing under Consideration of Driving and Working Hours A Distributed Decision Making Perspective With a foreword by Prof. Dr.-Ing. Herbert Kopfer

RESEARCH

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de.

Dissertation University of Bremen, 2010

1st Edition 2011 All rights reserved © Gabler Verlag | Springer Fachmedien Wiesbaden GmbH 2011 Editorial Office: Stefanie Brich | Nicole Schweitzer Gabler Verlag ist eine Marke von Springer Fachmedien. Springer Fachmedien ist Teil der Fachverlagsgruppe Springer Science+Business Media. www.gabler.de No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the copyright holder. Registered and/or industrial names, trade names, trade descriptions etc. cited in this publication are part of the law for trade-mark protection and may not be used free in any form or by any means even if this is not specifically marked. Cover design: KünkelLopka Medienentwicklung, Heidelberg Printed on acid-free paper Printed in Germany ISBN 978-3-8349-2942-6

Foreword In the European Union, statutory regulations restricting truck drivers’ working and driving hours have been existing for several decades. Although these regulations have ever since been strictly prescribed by law and although their negligence in transportation planning and fulfillment was to be fined, they have only been of minor importance in the vehicle routing and scheduling literature so far. First and foremost, the techniques used for the documentation of the actual driving times and the methods for controlling the abidance by the law have changed the situation drastically. Since April 2007, the usage of digital tachographs has become mandatory in the European Union. At the same time, the new Regulation (EC) No 561/2006 on driving times of truck drivers has become effective. Due to the digital documentation, the surveillance of the driving times has become easier and can be performed efficiently by the control authorities. That is why the planning of breaks and rest periods has become a very important issue in vehicle routing and scheduling by now. The PhD thesis of Christoph Manuel Meyer is one of the very early comprehensive monographs considering driving and working hours within vehicle routing and scheduling. Its main focus is on the distributed decision making perspective in the combined problem of vehicle routing and break scheduling. In his thesis, Christoph Manuel Meyer first presents a complete mathematical optimization model for the vehicle routing and scheduling problem with time windows which complies with the European legal rules on driving and working times. This model of the so called VRPTW-EU is used for computational experiments which generate some very interesting insights concerning the impact of including break scheduling in vehicle routing. Especially the relations among different goals of vehicle routing, such as travel distance minimization, minimization of the number of used vehicles and the minimization of travel and operating times are analyzed. Based on the introduced comprehensive and full model for the compound problem of combined vehicle and break scheduling and on useful separations of the entire model, the general framework of distributed decision making is applied to the decision problem under consideration. This leads to innovative distributions of optimization tasks between dispatchers and drivers which indeed are to a

VI

Foreword

high degree relevant for operational transportation planning processes in practice. Solving and analyzing both, the decision problem of the dispatcher and that of the drivers, and combining both problems requires the definition and application of adequate anticipation functions. Several experiments with different anticipation functions are conducted and reported in this thesis, showing the advantages and disadvantages of perfect explicit, approximate explicit and implicit anticipation. Apart from a mathematical model this thesis presents a restricted dynamic programming heuristic which solves the VRPTW-EU efficiently. By means of this heuristic, valuable experimental results could be derived which are useful for the application of the decentralized planning approach to vehicle routing. This book presents and investigates the important extension of vehicle routing and scheduling by the aspects of break scheduling according to the regulations of the European Union. It proposes approaches for solving the expanded problem and analyzes the effects of including these regulations in transportation planning. Therefore, it is an essential and helpful reading for researchers and students of logistics, particularly for those with an engineering background. In addition, the contents of this book might be very interesting for executives and software engineers in the area of transportation planning.

Herbert Kopfer

Preface This work is the result of my PhD project which I carried out at the Collaborative Research Centre 637 “Autonomous Cooperating Logistic Processes” at Bremen University from November 2006 until December 2010. Many people have contributed to the success of this project to whom I would like to express my gratitude. First of all I would like to thank my supervisor Prof. Herbert Kopfer. He offered me the chance to come to Bremen and to work on this very interesting and challenging topic. With his deep knowledge in the fields of logistics and Operations Research he supported me throughout my PhD project. Apart from providing me with such a pleasant working atmosphere at the Chair of Logistics, he introduced me to the scientific community at various conferences all over the world, which I enjoyed very much. I would also like to thank Prof. Hans-Dietrich Haasis for writing such a positive review for my PhD thesis. Moreover, despite the snow chaos, Prof. Herbert Kotzab made his way to Bremen to take part in my PhD defense, which I appreciate very much. My thanks also go to my colleagues from the Chair of Logistics, especially to PD Dr. J¨orn Sch¨onberger, Xin Wang and Sebastian Sterzik for their participation in my PhD defense. Apart from that, Xin Wang took the time and effort of proof reading my PhD thesis for which I feel much obliged. In 2009 I spent three months as a guest researcher at the Department of Industrial Engineering at Pusan National University, South Korea. I would like to express my gratitude to Prof. Kap Hwan Kim for allowing me to join his team and to take part in his research seminar. And I would like to thank the German Research Foundation (DFG) for supporting this stay. During my PhD project I had a very fruitful and pleasant cooperation with researchers from the University of Twente where I was allowed to spend a couple of weeks as a visiting scholar. I would like to thank Prof. Henk Zijm and Dr. Marco Schutten for their kind invitation to Enschede. And I would like to express my heartfelt gratitude to my friend Dr. Leendert Kok for our very successful cooperation, for the nice time we had together in Enschede and Bremen, and for

VIII

Preface

showing me how to “live on the edge” from an Operations Research perspective. I would also like to express my gratitude to the Gabler Verlag team, namely Prof. Hans-Otto G¨ unther, Mrs. Stefanie Brich and Mrs. Nicole Schweitzer, for their good and reliable cooperation concerning the publication of my thesis. I would never have succeeded in finishing my PhD thesis without the support from my family and friends. Therefore, I would especially like to thank my parents Jutta and Bernd Meyer for their support during my studies and during my time as a PhD student. The certainty that I could always rely on them gave me the confidence to start this PhD project and helped me overcome many obstacles in its course. Furthermore, I would like to thank my friend Ellen Kremer for her backup during a particularly difficult time of my PhD project. Finally, my great thanks go to my friend Juliane Riese for her hint concerning the benefits of bibliography management systems, for always being there for me, and for the great time we had together when I could free myself from my PhD project.

Christoph Manuel Meyer

Contents

List of Figures

XIII

List of Tables

XV

Abbreviations

XVII

1 Introduction

1

1.1

Problem Description . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Objectives and Structure of the Thesis . . . . . . . . . . . . . . .

6

2 European Legislation on Driving and Working Hours 2.1

2.2

Regulation (EC) No 561/2006 on Driving Hours . . . . . . . . . .

2

9 9

2.1.1

Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.1.2

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.1.3

Restrictions on Driving Hours . . . . . . . . . . . . . . . .

12

2.1.4

Documentation of Driving Times . . . . . . . . . . . . . .

14

2.1.5

Comparison Between Current and Former European Legislation on Driving Hours . . . . . . . . . . . . . . . . . . .

16

Directive 2002/15/EC on Working Hours . . . . . . . . . . . . . .

20

2.2.1

Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.2.2

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.2.3

Restrictions on Working Hours . . . . . . . . . . . . . . .

21

2.2.4

Implementation into German Legislation . . . . . . . . . .

22

2.3

Structure of the European Social Legislation . . . . . . . . . . . .

22

2.4

Control Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

3 Dispatchers’ Tasks in Vehicle Routing and Break Scheduling

25

3.1

The Dispatching Process . . . . . . . . . . . . . . . . . . . . . . .

25

3.2

Impact of the Legal Rules on the Dispatching Process . . . . . . .

27

3.2.1

Effects on Vehicle Routing and Scheduling . . . . . . . . .

28

3.2.2

Examples of Effects . . . . . . . . . . . . . . . . . . . . . .

29

X

Contents 3.3

Combined Vehicle Routing and Break Scheduling in Practice . . . Methodology of the Study . . . . . . . . . . . . . . . . . .

32

3.3.2

Interviews with Freight Forwarding Companies . . . . . . .

33

3.3.3

Summary of the Findings . . . . . . . . . . . . . . . . . .

37

4 Model for the VRPTW-EU 4.1

4.2

4.3

5.2

5.3

5.4

39

Model for the VRPTW-EU with Basic Rules . . . . . . . . . . . .

40

4.1.1

Problem Definition . . . . . . . . . . . . . . . . . . . . . .

41

4.1.2

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

4.1.3

Model Formulation . . . . . . . . . . . . . . . . . . . . . .

45

Model for the VRPTW-EU with Optional Rules . . . . . . . . . .

50

4.2.1

Problem Definition . . . . . . . . . . . . . . . . . . . . . .

50

4.2.2

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

4.2.3

Model Formulation . . . . . . . . . . . . . . . . . . . . . .

52

Computional Experiments . . . . . . . . . . . . . . . . . . . . . .

58

4.3.1

Experimental Settings . . . . . . . . . . . . . . . . . . . .

59

4.3.2

Computational Results for the VRPTW-EU . . . . . . . .

60

4.3.3

Example of a Tour Plan . . . . . . . . . . . . . . . . . . .

64

5 DDM in Combined Vehicle Routing and Break Scheduling 5.1

32

3.3.1

DDM Framework . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 68

5.1.1

Classification of Distributed Decision Making Systems . . .

69

5.1.2

Coordination in Distributed Decision Systems . . . . . . .

70

Vehicle Routing as DDM Problem . . . . . . . . . . . . . . . . . .

73

5.2.1

Distribution of Tasks between Planners and Drivers . . . .

73

5.2.2

Anticipation of the Drivers’ Planning . . . . . . . . . . . .

76

Model Formulations . . . . . . . . . . . . . . . . . . . . . . . . . .

78

5.3.1

Driver’s Model . . . . . . . . . . . . . . . . . . . . . . . .

78

5.3.2

Dispatcher’s Model . . . . . . . . . . . . . . . . . . . . . .

83

5.3.3

Anticipation Functions . . . . . . . . . . . . . . . . . . . .

84

5.3.3.1

Perfect Explicit Anticipation . . . . . . . . . . .

84

5.3.3.2

Approximate Explicit Anticipation . . . . . . . .

85

5.3.3.3

Implicit Anticipation . . . . . . . . . . . . . . . .

87

Computational Experiments . . . . . . . . . . . . . . . . . . . . .

88

5.4.1

Experimental Settings and Solution Procedure . . . . . . .

88

5.4.2

Computational Results . . . . . . . . . . . . . . . . . . . .

90

5.4.2.1

Perfect Explicit Anticipation . . . . . . . . . . .

90

5.4.2.2

Approximate Explicit Anticipation . . . . . . . .

92

Contents

5.4.3

XI 5.4.2.3 Implicit Anticipation . . . . . . . . . . . . . . . . Summary of the Findings . . . . . . . . . . . . . . . . . .

6 Restricted DP Heuristic for the VRPTW-EU 6.1

DP Methods for Vehicle Routing Problems . . . . . . . . . . . . . 6.1.1 Dynamic Programming Algorithm for the Traveling Salesman Problem . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Restricted Dynamic Programming Algorithm for the Traveling Salesman Problem . . . . . . . . . . . . . . . . . . . 6.1.3 Solving Vehicle Routing Problems via Giant-Tour Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Restricted DP Heuristic for the VRPTW-EU . . . . . . . . . . . . 6.2.1 Basic Break Scheduling Method . . . . . . . . . . . . . . . 6.2.2 Extended Break Scheduling Method . . . . . . . . . . . . . 6.2.2.1 Extended Daily Driving Times . . . . . . . . . . 6.2.2.2 6.2.2.3 6.2.2.4

94 98 103 103 104 106 106 108 110 114 114

Reduced Daily Rest Periods . . . . . . . . . . . . 115 Split Breaks . . . . . . . . . . . . . . . . . . . . . 116 Split Daily Rest Periods . . . . . . . . . . . . . . 117

6.2.2.5

6.2.3 6.2.4 6.3 DDM 6.3.1

6.3.2

Overview of the Implementations of the Optional Rules . . . . . . . . . . . . . . . . . . . . . . . . Extensions to Different Time Horizons . . . . . . . . . . . Computational Experiments . . . . . . . . . . . . . . . . . using the Restricted DP Heuristic . . . . . . . . . . . . . . Computational Experiments for the Decentralized Planning Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1.1 Team Situation . . . . . . . . . . . . . . . . . . . 6.3.1.2 Non-Team Situation . . . . . . . . . . . . . . . . Summary of the Findings . . . . . . . . . . . . . . . . . .

118 119 119 125 126 126 130 132

7 Conclusions and Further Research 135 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.2 Outline of Further Research Directions . . . . . . . . . . . . . . . 137 Bibliography

141

List of Figures 2.1

Relation of the different time horizons

. . . . . . . . . . . . . . .

3.1 3.2 3.3 3.4 3.5

Customer locations and time windows . . . . . . . . . Vehicle routes without breaks . . . . . . . . . . . . . Vehicle routes including breaks . . . . . . . . . . . . Vehicle routes with breaks requiring 3 vehicles . . . . Distribution of tasks between dispatchers and drivers

. . . . .

29 30 31 32 37

4.1

Vehicle schedules for test problem 5 . . . . . . . . . . . . . . . . .

64

5.1 5.2 5.3

Classification of DDM systems . . . . . . . . . . . . . . . . . . . . Coordination in hierarchical systems . . . . . . . . . . . . . . . . DDM problem with strictly hierarchical structure . . . . . . . . .

70 71 76

5.4

DDM problem in combined vehicle routing and break scheduling .

78

6.1 6.2 6.3

Example of a dominated partial route . . . . . . . . . . . . . . . . 105 Traditional representation of a VRP solution . . . . . . . . . . . . 107 Giant-tour representation of the VRP solution . . . . . . . . . . . 107

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

12

List of Tables 2.1 2.2

Basic and optional rules of Regulation (EC) No 561/2006 . . . . . 14 Comparison of Regulation (EC) No 561/2006 with Regulation (EEC) No 3820/85 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1 4.2

Results for the VRPTW-EU with basic rules . . . . . . . . . . . . Results for criterion of minimum driving time . . . . . . . . . . .

61 63

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Dispatcher’s results for perfect explicit anticipation . . . Results for the non-team situation . . . . . . . . . . . . . Dispatcher’s results for approximate explicit anticipation Results for the non-team situation . . . . . . . . . . . . . Dispatcher’s results for implicit anticipation (55 km/h) . Results for the non-team situation . . . . . . . . . . . . . Dispatcher’s results for implicit anticipation (45 km/h) . Results for the non-team situation . . . . . . . . . . . . .

91 92 93 94 96 96 98 98

6.1

Implementation of the optional rules in the extended break scheduling method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Results for the basic method without Directive 2002/15/EC . . . 123 Results for the basic method including Directive 2002/15/EC . . 124 Results for the extended method . . . . . . . . . . . . . . . . . . 125 Results for the team situation: perfect and approximate anticipation127 Results for the team situation: implicit anticipation . . . . . . . . 129 Results for the non-team situation: perfect and approximate anticipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Results for the non-team situation: implicit anticipation . . . . . 132

6.2 6.3 6.4 6.5 6.6 6.7 6.8

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

Abbreviations AETR European Agreement concerning the Work of Crews of Vehicles Engaged in International Road Transport ArbZG Arbeitszeitgesetz DDM distributed decision making DMU decision making unit DOT Department of Transportation DP dynamic programming EC European Community EEC European Economic Community FTL full truckload LTL less-than-truckload PDP Pickup and Delivery Problem PDPTW Pickup and Delivery Problem with Time Windows PVRP Periodic Vehicle Routing Problem PVRPTW Periodic Vehicle Routing Problem with Time Windows TSP Traveling Salesman Problem TSPTW Traveling Salesman Problem with Time Windows TSPTW-EU Traveling Salesman Problem with Time Windows and European Union Social Legislation TW time window VRP Vehicle Routing Problem VRPTW Vehicle Routing Problem with Time Windows VRPTW-EU Vehicle Routing Problem with Time Windows and European Union Social Legislation

1 Introduction Driver fatigue is a major issue in road transportation. In Germany, 48,324 accidents were registered in 2008 in which trucks were involved (Federal Statistical Office, 2010). The Federal Road Research Institute (Bundesanstalt f¨ ur Straßenwesen) estimates that 19 percent of the serious truck accidents in Germany are caused by driver fatigue (Federal Road Research Institute, 2006). According to a representative study by the International Road Transport Union which was commissioned by the European Community (EC), overfatigue and falling asleep is the cause in approximately 18.6 percent of single truck accidents, i.e. accidents in which no other party is involved (International Road Transport Union, 2006). The European Commission even estimates driver fatigue to be a significant factor in approximately 20 percent of all heavy commercial vehicle crashes (European Commission, 2010). Recent studies on road safety show that after four hours of continuous driving, the accident risk is doubled, and after eight hours of continuous driving it is even ten times higher (Goudswaard et al., 2006). To counter this situation, the European Union enacted stricter laws on driving and working hours of persons engaged in road transportation. Regulation (EC) No 561/2006, which came into effect in April 2007, restricts driving hours of truck drivers. It is supposed “to improve working conditions and road safety” (Regulation, 2006, Article 1). Drivers’ working hours are restricted by Directive 2002/15/EC which was enacted in March 2002 (see Directive, 2002). Together, these laws are often referred to as European social legislation. This denotation will also be used in the following. The working time directive had to be implemented into the national laws of the member countries of the European Union to become effective. In Germany, its rules were included in §21(a) of the Law on Working Hours (Arbeitszeitgesetz (ArbZG)) which became effective in August 2006. In order to provide that drivers have enough time for recuperation, the European social legislation restricts the maximum admissible driving and working times and sets minimum requirements for breaks and rest periods. Whether the enacting of these laws really helped to improve road safety has not yet been proved. However, it is remarkable that in Germany the total number of road accidents in which trucks were involved C. M. Meyer, Vehicle Routing under Consideration of Driving and Working Hours, DOI 10.1007/978-3-8349-6732-9_1, © Gabler Verlag | Springer Fachmedien Wiesbaden GmbH 2011

2

1 Introduction

dropped by 5.4 percent from 51,075 in 2007 to 48,324 in 2008. In the years prior to that, only slight decreases could be noticed. The number of truck accidents amounted to 51,745 in 2005 and 51,334 in 2006 (Federal Statistical Office, 2010).

1.1 Problem Description The European social legislation imposes restrictions on driving and working hours of drivers engaged in road transportation, which are valid on different but interconnected time horizons, reaching from single driving periods up to periods of several weeks. These rules define a complex set of restrictions which have to be respected by all freight forwarding companies operating within the member states of the European Union, Switzerland and the countries party to the Agreement on the European Economic Area. Freight forwarding companies are forced to organize their drivers’ working times such that they can obey the legal rules (Regulation, 2006, Article 10). Through the introduction of digital tachographs, enforced by Regulation (EC) No 561/2006 in combination with Regulation No 1360/2002 (see Regulation, 2002), infringements of the legal rules can easily be detected by control authorities. Within freight forwarding companies, dispatchers or planners are responsible for the design of vehicle routes.1 They have to assign transportation requests to vehicles and their drivers such that the transportation requests are fulfilled as specified by the customers. These specifications usually comprise pickup and delivery locations, characteristics of the load to be carried, time windows during which the goods have to be loaded and delivered, and possibly other specific requirements such as temperature control. The design of vehicle routes meeting all the above requirements is in itself a complex task, which is usually referred to as the vehicle routing and scheduling problem. Apart from these requirements set by the customers, dispatchers have to create vehicle routes such that the legal rules on driving and working hours can be respected by the drivers. For drivers’ infringements of the European social legislation, also their employers can be held responsible. The integration of driving hours regulations in the design of vehicle routes presents new challenges for dispatchers at freight forwarding companies. Apart from the already highly complex problem of vehicle routing and scheduling, breaks and rest periods for each driver have to be planned. The vehicle routes have to 1

In the following the terms “dispatcher” and “planner” are used synonymously.

1.2 Related Literature

3

be constructed such that the drivers can adhere to the legal rules. Therefore, dispatchers have to incorporate the driving and working time restrictions already when creating vehicle and driver schedules. However, dispatchers are only obliged to assure that the drivers can meet the legal requirements. Usually, the final decision when to take breaks and rest periods is left to the drivers (Meyer and Kopfer, 2008a). Therefore, dispatchers need not schedule breaks and rest periods exactly within their planning process, but might use simplified planning approaches instead, which allow drivers to meet the legal rules when performing the transports. This problem of creating vehicle routes such that the legal rules on driving and working hours are respected is referred to as the problem of combined vehicle routing and break scheduling (Meyer et al., 2009). It is based in the area of operational transportation planning in road transportation.

1.2 Related Literature In order to embed this study within the body of research in the field of operational transportation planning under consideration of driving and working hours, a review of the related literature is presented in the following. First, approaches including the European social legislation are addressed. Subsequently, methods considering the US rules imposed by the Department of Transportation (DOT) are reviewed. Gietz (1994) investigates a Vehicle Routing Problem (VRP) including breaks according to the former European Economic Community (EEC) Regulation No 3820/85 which was valid until April 2007 (see Regulation, 1985a). Gietz develops construction and improvement heuristics in which he includes breaks as fictitious customers. Rochat and Semet (1994) use a similar approach for including the Swiss legislation on breaks and rest periods into a tabu search metaheuristic. Brandao and Mercer (1997) include rules on driving times according to Regulation (EEC) No 3820/85 into a tabu search heuristic for the multi-trip vehicle routing and scheduling problem. In this approach breaks are scheduled implicitly by increasing the driving time along the arcs such that slack travel time is created. Brandao and Mercer argue that explicitly incorporating breaks into their local search approach would increase its computational complexity. Stumpf (1998) also includes driving time restrictions specified by Regulation (EEC) No 3820/85 into a tabu search metaheuristic, a great deluge algorithm, and a threshold accepting

4

1 Introduction

algorithm. Goel and Gruhn (2006) introduce a large neighborhood search algorithm for a vehicle routing problem which takes into account maximum driving times according to Regulation (EEC) No 3820/85. The current European legislation on driving hours as imposed by Regulation (EC) No 561/2006 is addressed by the following studies. Kopfer et al. (2007) give a first analysis of the impact of the European social legislation on vehicle routing and scheduling and outline possibilities for integrating the legal rules into vehicle routing models. Goel and Gruhn (2007), Goel (2007) and Goel (2009) integrate parts of the current European social legislation into a large neighborhood search algorithm. Goel (2009) presents computational results based on modified problem instances of the Solomon (1987) test instances for the Vehicle Routing Problem with Time Windows (VRPTW). In his study, Goel (2009) concentrates on a set of basic rules and neglects some important optional rules of Regulation (EC) No 561/2006. Additionally, he ignores the restrictions on working times set by Directive 2002/15/EC. Z¨ apfel and B¨ogl (2008) present a mixed-integer programming model for a combined vehicle routing and crew pairing problem which considers breaks after 4.5 hours of driving. To solve the model they apply a tabu search metaheuristic and a genetic algorithm. Bartodziej et al. (2009) use a column generation approach and some local search based metaheuristics for solving a combined vehicle and crew scheduling problem which incorporates rest periods for drivers. The first solution algorithm which includes the whole set of rules imposed by the Regulation (EC) No 561/2006 and also the legislation on working hours set by Directive 2002/15/EC, is the restricted dynamic programming heuristic by Kok et al. (2010). It allows to efficiently solve vehicle routing and scheduling problems in which all rules of the current European social legislation, including both the basic and the optional rules of Regulation (EC) No 561/2006, are considered. Goel (2010) presents scheduling methods for generating truck driver schedules for a given sequence of locations to be visited within specified time windows. The methods guarantee to find a feasible break schedule if one exists. Drexl and Prescott-Gagnon (2010) describe a labeling algorithm for the elementary shortest path problem with resource constraints in which the basic and optional rules of the EU social legislation can be considered. Prescott-Gagnon et al. (2010) develop a sophisticated large neighborhood search heuristic for the vehicle routing problem with time windows and European Union driving time regulations. This heuristic shows a very good performance on the modified Solomon instances as presented by Goel (2009). The following studies address further issues of the European social legislation

1.2 Related Literature

5

apart from the development of solution algorithms. Kummer and Einbock (2007) give an estimation of the additional costs arising for freight forwarding companies through the implementation of Directive 2002/15/EC. The expected cost increase lies between 6.5 percent and 11.1 percent, differentiated according to the services provided by the company. For less-than-truckload (LTL) transports the cost increase is estimated to be 8.4 percent. Meyer and Kopfer (2008c) propose a method for assigning customer time windows to weekly planning periods as a preprocessing step for solving vehicle routing and scheduling problems including the European social legislation. Kopfer and Meyer (2009) present an integer programming model for a Traveling Salesman Problem (TSP) which considers all relevant rules of Regulation (EC) No 561/2006 for a weekly planning period. Kopfer and Meyer (2010) develop a position based mixed-integer programming model for the VRPTW including the European social legislation. Goel and Kok (2009a) present a break scheduling algorithm for team truck drivers in the European Union, which runs in quadratic time. Meyer et al. (2009) and Meyer et al. (2010) present an analysis of the problem of combined vehicle routing and break scheduling from a distributed decision making perspective. In these studies, the problem is analyzed as a decentralized decision process between dispatchers and drivers who are responsible for solving partial tasks of the overall planning problem. The US American regulations on driving and working hours, imposed by the DOT, are considered in the following studies: Savelsberg and Sol (1998) include breaks and daily rest periods into a branch and price algorithm for a pickup and delivery problem. Powell et al. (2002) consider a driver scheduling problem in which they incorporate a maximum daily driving time of 10 hours according to the US legislation. They solve this problem using an adaptive dynamic programming algorithm. Xu et al. (2003) present a column generation algorithm and some heuristics to solve a pickup and delivery problem which includes the DOT restrictions on driving times. Campbell and Savelsberg (2004) modify an insertion heuristic such that it considers maximum shift times for drivers. Erera et al. (2008) present a driver management scheme for LTL carriers. Their approach assigns drivers to trips and provides that the resulting driver schedules are conformous with the DOT regulations. Archetti and Savelsbergh (2009) analyze the trip scheduling problem considering the US hours of service regulations and present an algorithm for scheduling rest periods within a given route which runs in cubic time. Goel and Kok (2009b) also investigate truck driver scheduling satisfying the DOT rules.

6

1 Introduction

A general remark on the inclusion of breaks in vehicle routing models and algorithms is given by Cordeau et al. (2002), who suggest the use of a multi-stage network in which breaks are modeled as the transition from one stage to the next.

1.3 Objectives and Structure of the Thesis From the literature review it becomes obvious that the models and algorithms developed so far neglect some important features of the problem of combined vehicle routing and break scheduling. Apart from the algorithms by Kok et al. (2010), Drexl and Prescott-Gagnon (2010) and Prescott-Gagnon et al. (2010), none of the above algorithms considers the entire set of rules laid down in Regulation (EC) No 561/2006 and in Directive 2002/15/EC. Furthermore, most approaches assume a central planning perspective and neglect the distribution of decision rights between dispatchers and drivers, which are encountered in practice. However, it is an important feature of vehicle routing methods that they include restrictions relevant for practical freight forwarding (Ohrt, 2008). In this thesis the problem of creating vehicle routes is therefore enhanced by considering restrictions on driving and working hours which have to be respected by dispatchers in practice. Moreover, this so called problem of combined vehicle routing and break scheduling (Meyer et al., 2010) is investigated from a distributed decision making perspective to account for the different decision makers involved. Special attention is given to the dispatchers’ task of designing vehicle routes complying with the legal rules on driving and working hours. The aim is to derive strategies for dispatchers how to incorporate the legal restrictions in their planning while reducing the complexity of their overall planning task. These strategies are formulated as recommendations for dispatchers, following the understanding of management as an action-oriented science (Schneeweiss, 1992). The thesis is structured as follows: In Chapter 2, the European social legislation valid for road transportation is presented in detail. Since European directives have to be transformed into national laws to become effective, the implementation of Directive 2002/15/EC into the German Law on Working Hours is also addressed. Moreover, the control policies specified by the European Union are reviewed and the current practice of monitoring the adherence to the legal rules is analyzed. Since the European social legislation has to be considered in the dispatching process, the dispatchers’ task of creating vehicle routes including the legal rules on breaks and rest periods is outlined in Chapter 3. Subsequently, the impact of

1.3 Objectives and Structure of the Thesis

7

the legal rules on vehicle routes and schedules is discussed. Finally, the results of an empirical study are presented in which the integration of the European social legislation in practical freight forwarding is analyzed. Chapter 4 addresses the problem of combined vehicle routing and break scheduling from a central planning perspective. It presents a mathematical model for the Vehicle Routing Problem with Time Windows and European Union Social Legislation (VRPTW-EU). In contrast with the common objectives of minimizing the total travel distance or the number of vehicles (see e.g. Rieck, 2008), the minimization of the total fulfillment time is used as the objective function. Computational experiments are performed to compare the impact of this criterion with the traditional criteria. To demonstrate the structure of the resulting vehicle schedules, an example of a vehicle routing solution is presented. In Chapter 5, the problem analysis is enhanced by introducing distributed decision structures. The distribution of the planning tasks between dispatchers and drivers is modeled within the framework for distributed decision making of Schneeweiss (2003a), and different possibilities for the dispatchers to anticipate the drivers’ planning are suggested. Moreover, a situation in which the drivers act in line with the company’s objectives and a situation in which the drivers follow their own hidden objectives are modeled. The different models are evaluated using computational experiments. Chapter 6 presents a restricted dynamic programming heuristic for the VRPTWEU. This heuristic can be used to efficiently solve large scale vehicle routing and break scheduling problems. Furthermore, it was the first heuristic which could include the whole set of legal rules imposed by Regulation (EC) No 561/2006 and by Directive 2002/15/EC. The heuristic can easily be adapted for solving the problem from a distributed decision making perspective. Computational experiments are performed both for the traditional centralized planning scenario and for the more realistic situation of distributed decision structures. The main findings of this thesis are summarized in Chapter 7, and further research perspectives for combined vehicle routing and break scheduling are outlined.

2 European Legislation on Driving and Working Hours in Road Transportation In this chapter the rules of the European social legislation are presented. The restrictions to be considered by freight forwarding companies are mainly laid down in two laws: Regulation (EC) No 561/2006 restricts driving hours and Directive 2002/15/EC gives restrictions on working hours of persons engaged in road transportation. The chapter is structured as follows: First, the restrictions on driving hours and the requirements for breaks and rest periods defined by Regulation (EC) No 561/2006 are discussed. Second, the rules concerning drivers’ working hours imposed by Directive 2002/15/EC are described. European Union directives have to be implemented into national laws in order to become effective (Rang, 2007). In Germany, Directive 2002/15/EC was implemented in §21a of the ArbZG which is briefly outlined. After the introduction of these laws2 , some general remarks on the structure of and the relationship between these legal acts are given. Finally, the control policies for these laws defined by the European Union are reviewed.

2.1 Regulation (EC) No 561/2006 on Driving Hours Regulation (EC) No 561/2006 lays down restrictions of driving hours and defines minimum requirements for breaks and rest periods. It postulates that transport undertakings have to organize the work of their drivers in such a way that they can adhere to the restrictions set by this regulation. For infringements committed by the drivers, the transport undertaking can be held responsible. Moreover, the regulation demands that every party involved in the transportation process, i.e. the transport undertakings, consignors, forwarders, tour operators, principal contractors, subcontractors and even driver employment agencies ensure that the driver schedules comply with the legal requirements (Regulation, 2006, Article

2 In the following, also European Union directives are referred to as laws. The terms law and legal act are used synonymously.

C. M. Meyer, Vehicle Routing under Consideration of Driving and Working Hours, DOI 10.1007/978-3-8349-6732-9_2, © Gabler Verlag | Springer Fachmedien Wiesbaden GmbH 2011

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2 European Legislation on Driving and Working Hours

10). Therefore, the dispatchers’ task of planning vehicle routes is affected by the driving time regulation.

2.1.1 Scope The rules of Regulation (EC) No 561/2006 encompass drivers of vehicles with a total mass exceeding 3.5 tons or of vehicles constructed to carry more than nine persons including the driver. It applies to transports undertaken exclusively within the EC, between the EC and Switzerland or the countries party to the Agreement on the European Economic Area. Besides all 27 member countries of the European Union these are Iceland, Liechtenstein and Norway. The regulation’s restrictions are valid irrespective of the country of the vehicle’s registration. For international road transports undertaken partly outside the member countries of European Economic Area and Switzerland, the European Agreement concerning the Work of Crews of Vehicles Engaged in International Road Transport (AETR) applies for the whole journey if the vehicle is registered in the EC or in a member country of the AETR (see AETR, 1997). For vehicles registered in a third country, the AETR applies only for the part of the journey undertaken in the EC and the countries party to the AETR. Regulation (EC) No 561/2006 states that the provisions of the AETR, which are already similar to the European social legislation, shall be fully adapted to the Regulation (EC) No 561/2006 such that its restrictions on driving times are valid for all transports undertaken in the EC. Nevertheless, its adaptation has been blocked by certain member countries (Rang, 2008). The regulation does not apply for urban transports, in which the routes do not exceed 50 kilometers, for emergency transports and for transportation tasks performed by vehicles designed for special purposes. These are vehicles with a maximum speed of no more than 40 km/h, vehicles of the armed forces, vehicles used in the non-commercial transport of humanitarian aid and specialized vehicles for medical purposes. Moreover, drivers of specialized breakdown vehicles do not need to adhere to the legislation as long as they operate within a radius of no more than 100 kilometers of their depot. Furthermore, vehicles or combinations of vehicles with a maximum permissible mass of less than 7.5 tons, used for the non-commercial carriage of goods, and vehicles with a historic status used for the non-commercial carriage of passengers or goods do not fall under the scope of Regulation (EC) No 561/2006. Finally, transportation tasks carried out in total on private premises are not affected by this law.

2.1 Regulation (EC) No 561/2006 on Driving Hours

11

2.1.2 Definitions To better understand the restrictions on driving hours, certain terms are defined in Regulation (EC) No 561/2006. These definitions have to be clarified in order to include the legislation into mathematical models and algorithms for vehicle routing and scheduling. The first set of important definitions concerns breaks and rest periods. A break is defined as a period in which a driver does not have to work and can freely dispose of his time. Rest period means an uninterrupted period which can be used for recuperation. These rests are distinguished into daily and weekly rest periods. A daily rest period is defined as a “daily period during which a driver may freely dispose of his time” (Regulation, 2006, Article 4). However, the term “daily” is not defined in Regulation (EC) No 561/2006. It is not even necessarily related to a day comprising the time period from 00:00 to 24:00. On the contrary, a daily rest period can stretch over two days and there can be more than one daily rest period on a single day. Daily rest periods can be distinguished into regular daily rest periods and reduced daily rest periods according to their minimum duration. A regular daily rest period denotes a rest period of at least 11 hours whereas the minimum duration of a reduced daily rest period equals 9 hours. A weekly rest period is a “weekly period during which a driver may freely dispose of his time” (Regulation, 2006, Article 4). The regulation also distinguishes regular and reduced weekly rest periods. A regular weekly rest period is a recreational period of at least 45 hours while the duration of a reduced weekly rest period lies between 24 hours and 45 hours. In the context of Regulation (EC) No 561/2006, a week is defined as the period between Monday 00:00 and Sunday 24:00. However, even though a week is defined in this act, in fact the same is true for weekly rest periods as for daily rest periods. A week can comprise more than one weekly rest period and a weekly rest period can stretch over two weeks, for example if a driver begins his regular weekly rest period with a duration of 45 hours on Sunday at 00:00 and finishes it on Tuesday at 01:00. The second important set of definitions laid down in Regulation (EC) No 561/2006 concerns driving times. The driving time is defined as the duration of the driving activity which has to be recorded either digitally or manually.

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2 European Legislation on Driving and Working Hours

Figure 2.1: Relation of the different time horizons (Kopfer et al., 2007) A driving period refers to the accumulated driving time between a break or rest period and the next break or rest period taken by a driver. The daily driving time is defined as the accumulated driving time between the end of a daily or weekly rest period and the beginning of the following daily or weekly rest period. Again, the term “daily” is not related to the common usage of the term “day” since the daily driving time is constituted by the rest periods surrounding it. It is clear that this definition allows daily driving times to comprise two days. The introduction of another definition not included in the regulation is required for modeling purposes. The weekly phase means the accumulated driving time between two weekly rest periods.

2.1.3 Restrictions on Driving Hours Regulation (EC) No 561/2006 covers different but interconnected time horizons. The relationship between the restrictions on these different time horizons is depicted in Figure 2.1. Moreover, the regulation provides a set of basic rules, which can be regarded as the standard restrictions. It is sufficient to create driver schedules satisfying these standard rules to comply with the legislation on driving hours. However, Regulation (EC) No 561/2006 also offers a set of optional rules which relax some of the basic rules. Therefore, the exploitation of these optional rules provides dispatchers and drivers with more flexibility. The regulation restricts the total driving time of a single driving period to a maximum duration of 4.5 hours. Drivers are obliged to take a break of at least 45 minutes to enter a new driving period. These breaks can optionally be divided into two parts, of which the first part must last for at least 15 minutes and the

2.1 Regulation (EC) No 561/2006 on Driving Hours

13

second part for at least 30 minutes. A driving period ends, when a break of sufficient length is taken, in the case of the division when the 30-minute part of the break is scheduled. Therefore, a driving period consists of the complete time interval between two breaks and the total driving time of that period comprehends all particular driving times between these two breaks. If a driver takes a break of 45 minutes before driving 4.5 hours, he enters a new driving period. The daily driving time is restricted to 9 hours. Twice a week drivers may extend their daily driving time to 10 hours. A daily driving time ends when a regular or a reduced daily rest period is taken or when a weekly rest period starts. Within 24 hours after the end of a daily or weekly rest period the next daily rest period must have been taken. The regulation allows drivers to take up to three reduced daily rest periods between two weekly rest periods. Furthermore, it provides drivers with the option of splitting daily rest periods into two parts. The first part must have a minimum duration of 3 hours and the second part of at least 9 hours. Regulation (EC) No 561/2006 restricts the weekly driving time to a maximum of 56 hours. Moreover, it forces freight forwarding companies to provide that the weekly driving time does not exceed an average of 45 hours over a four month period. Additionally, it states that the maximum driving time of any two consecutive weeks must not exceed 90 hours such that the average driving time of 45 hours per week is maintained. This means that if a driver wants to extend his weekly driving time to 56 hours, he may not only have a maximum driving time of 34 hours in the following week, but also in the week previous to the week considered. After reaching his maximum weekly driving time, a driver has to take a regular or a reduced weekly rest period. A driver is allowed to take one reduced weekly rest period in any two consecutive weeks. Reductions have to be compensated by equal extensions of other rest periods of at least 9 hours, i.e. of daily or weekly rest periods. A new weekly rest period has to be taken after at most 144 hours, that is after six days, after the end of the previous weekly rest period. Regulation (EC) No 561/2006 provides some relaxations of the above restrictions in case of multi-manning, i.e. the presence of two drivers on a truck. Its most important advantages are that one driver may take a break while the other is driving, and that the time until which the next daily rest period of at least 9 hours must have been taken by the drivers is extended to 30 hours instead of 24 hours. An overview of the basic and optional rules of Regulation (EC) No 561/2006 is

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2 European Legislation on Driving and Working Hours

shown in Table 2.1. Scope

Basic rule

Optional rule

Driving period

Maximum duration of

No exception

Break

4.5 hours Minimum duration of 45 minutes

Two parts of 15 minutes and 30 minutes respectively

Daily driving time

Maximum duration of 9 hours

Extension to 10 hours twice a week

Daily rest period

Minimum duration of 11 hours

Weekly driving time

Maximum duration of 56 hours

Reduction to 9 hours three times between two weekly rest periods; split into two parts of at least 3 hours and 9 hours duration No exception

Weekly rest period

Minimum duration of 45 hours

Reduction to 24 hours once in any two consecutive weeks; reduction has to be compensated by an equal extension before the end of the third week following the week considered

Table 2.1: Basic and optional rules of Regulation (EC) No 561/2006

2.1.4 Documentation of Driving Times The restrictions on driving hours are enforced by the national authorities of the EC member countries. Driving hours are documented by on-board instruments

2.1 Regulation (EC) No 561/2006 on Driving Hours

15

installed in the vehicles. The former rules of Regulation (EEC) No 3821/85 postulated the use of an analogous tachograph (see Regulation, 1985b). Regulation (EC) No 561/2006 in combination with Regulation No 1360/2002, Annex Ib, enforces the usage of a digital tachograph for new vehicles registered after 1. May 2006 (see Regulation, 2002). For vehicles registered before this date but after 1. January 1996 the equipment with a digital tachograph is only required if their total mass exceeds 12 tons, or in case of passenger transports of more than eight persons and a vehicle mass of more than 10 tons. These vehicles need to be equipped with a digital tachograph if their analogous tachograph has to be replaced due to a defect (Rang, 2007).

The digital tachograph can be used with different cards allowing selective access to the data recorded: driver card, company card and garage card. The data on driving hours is recorded both in the memory of the tachograph and on the driver card.

Drivers of vehicles equipped with a digital tachograph have to carry their driver cards with them, and also all manual documentations and printouts produced during the past 28 days. Drivers are obliged to present their cards and all other relevant documents to inspecting officers immediately. The companies have to download the recorded data regularly and have to store it for at least one year. Official authorities have to be given access to this data directly or remotely.

In case of a defect of the driver card, the driver has to make a printout of his vehicle’s details from the tachograph and enter his name and the number of the defective card. Thereafter, he has to manually document his driving hours, times for other work and periods of availability. When finishing his tour, another printout from the tachograph has to be made and again the required driver data has to be entered.

Infringements of the regulation by drivers and companies are fined by national authorities. Penalties shall be “effective, proportionate, dissuasive and non-discriminatory” (Regulation, 2006, Preamble). They can comprise financial fines, the immobilization of the vehicle, the suspension of the driver’s driving license or even of the company’s license.

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2 European Legislation on Driving and Working Hours

2.1.5 Comparison Between Current and Former European Legislation on Driving Hours The enacting of Regulation (EC) No 561/2006 caused enormous discontent in the European freight forwarding industry.3 The complaints where mainly related to the inflexible and strict provisions of the new regulation. Therefore, it is interesting to recognize that in fact little has changed between the former restrictions set by Regulation (EEC) No 3820/85 and those set by Regulation (EC) No 561/2006. The most remarkable changes can be seen in the compulsory equipment of new vehicles with digital tachographs and the enforcement of stricter control policies of driving times. With respect to the actual restrictions on driving times, only very few fortifications and even one relaxation can be realized. A comparison of the restrictions of both regulations is given in Table 2.2. Restriction on:

Regulation (EEC) No 3820/ 85

Regulation (EC) No 561/ 2006

Driving periods

Maximum duration of 4.5 hours before a break has to be taken

Unchanged

Daily driving time

Maximum duration of 9 hours; twice a week extension

Unchanged;

Weekly driving time

Weekly phase

3

to 10 hours allowed No restriction, weekly

unchanged Maximum duration of

driving times up to 74 hours possible; maximum of 90 hours in any two consecutive weeks

56 hours;

Maximum of 56 hours

No restriction, maximum of 67.5 hours possible

This section is based on Meyer and Kopfer (2008b).

unchanged

2.1 Regulation (EC) No 561/2006 on Driving Hours Break

Minimum duration of 45 minutes after no

Unchanged;

more than 4.5 hours of driving;

Regular daily rest period

Reduced daily rest period

Regular weekly rest period

division in three parts of at least 15 minutes each allowed Minimum duration of 11 hours; division into two or three parts allowed if one part has a duration of not less than 8 hours and the total duration is at least 12 hours Minimum duration of 9 hours; can be taken three times a week, if the reduction is compensated until the end of the following week Minimum duration of 45 hours; has to be taken the latest after the minimum of six daily driving times and of six days; for passenger transports a weekly rest period can be transfered to the following week

division in two parts of 15 and 30 minutes allowed Unchanged; division into two parts allowed if the first part is at least 3 and the second part at least 9 hours

Unchanged; can be taken three times in a weekly phase; no compensation required Unchanged; has to be taken the latest after six 24 hour periods after the end of the previous weekly rest period; no exceptional rule for passenger transports

17

18

2 European Legislation on Driving and Working Hours Reduced weekly rest period

Differences in case of multi-manning

Minimum duration of 36 hours at the loca-

One reduced weekly rest period is allowed

tion of the vehicle’s registration, 24 hours

in any two consecutive weeks regardless

at other locations;

of the location where it is taken; unchanged

the reduction has to be compensated by the end of the following three weeks After no more than 30 hours after the end of a daily or weekly rest period a new daily rest period of at least 8 hours has to be taken

After no more than 30 hours after the end of a daily or weekly rest period a new daily rest period of at least 9 hours has to be taken

Table 2.2: Comparison of Regulation (EC) No 561/2006 with Regulation (EEC) No 3820/85

Concerning the restrictions on driving times, on the first sight the new regulation brings a fundamental change. This change lies in the restriction of the weekly driving time to 56 hours whereas in the former legislation no explicit border was mentioned. Therefore, driving times of up to 74 hours were theoretically possible (Rang, 2007). However, the working time restrictions enforced a maximum working time of no more than 60 hours per week. Since driving times are regarded as working times, driving times were thus implicitly restricted to a maximum of 60 hours per week by the legislation on working times. The maximum durations of driving periods and daily driving times remain unchanged. In both laws the maximum duration of a single driving period is restricted to 4.5 hours and the maximum duration of a daily driving time is restricted to 9 hours which can be extended to 10 hours twice a week. In the former Regulation (EEC) 3820/85 the maximum driving time of a weekly phase was restricted to 56 hours. The new regulation does not include any explicit restriction of the driving time in a weekly phase. However, as the maximum duration of a weekly phase is restricted to 144 hours and a weekly phase can be

2.1 Regulation (EC) No 561/2006 on Driving Hours

19

spread over two weeks, a maximum driving time of 67.5 hours can be calculated as follows. A driver uses seven daily driving times of nine hours each including the required minimum breaks of 45 minutes for each daily driving period and a further single driving period of 4.5 hours, followed by the new weekly rest period required after 144 hours. After each daily driving time a daily rest period has to be taken. Therefore, three reduced and four regular daily rest periods have to be scheduled by the driver. The total time required for this schedule amounts to 143.75 hours, followed by the required weekly rest period. The total driving time of this weekly phase therefore equals 67.5 hours. Regarding breaks and rest periods only minor changes can be detected. The minimum duration of a break remains restricted to 45 minutes. The difference that it can no longer be split into three parts of at least 15 minutes each, but only into two parts of 15 and 30 minutes, is negligible. The minimum duration of regular daily rest periods still remains 11 hours. According to Regulation (EC) No 561/2006 they can be split into two parts instead of three parts which was possible according to Regulation (EEC) No 3820/85. In both regulations the reduced daily rest periods have to last for at least 9 hours. The former legislation permitted three reduced daily periods per week and required compensation for the reductions. The new legislation allows them to be taken three times in a weekly phase without compensation. Concerning weekly rest periods the new legislation allows for more flexibility than the old law. Now the reduction to 24 hours can be taken irrespective of the vehicle’s location. This was not possible according to Regulation (EEC) No 3820/85. The time required for a regular weekly rest period still amounts to 45 hours. A new daily rest period now has to be scheduled after at most 144 hours after the end of the previous weekly rest period. Before, it had to be taken after no more than six daily driving periods and no more than 144 hours after the end of the previous rest period. The current legislation allows only one reduced weekly rest period in any two consecutive weeks. Both acts require the compensation of the time of the reduction. Only for passenger transport there has been a fundamental change. According to Regulation (EEC) No 3820/85 it was allowed to transfer a weekly rest period to the following week resulting in the possibility to carry out driving activities on 12 days in a row without taking a weekly rest period in between. Regulation (EC) 561/2006 does no longer include this optional rule. In Germany, the abolishment of the so called “12-Tage-Regel” has attracted many protests from bus tour operators (e.g. Bundesverband Deutscher Omnibusunternehmer e.V., 2007).

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2 European Legislation on Driving and Working Hours

With respect to multi-manning the only change is that according to the new legislation the minimum requirement for a daily rest period was extended to 9 hours whereas according to the former legislation it had to amount to 8 hours.

2.2 Directive 2002/15/EC on Working Hours Directive 2002/15/EC supplements the rules on driving times laid down in Regulation (EC) No 561/2006 by restricting working times of persons engaged in road transportation. As driving times are part of the total working time, these laws are interdependent and both have to be considered by dispatchers. 2.2.1 Scope Directive 2002/15/EC is valid for persons performing mobile road transport activities falling under the scope of Regulation (EEC) No 3820/85 or the AETR, and who are employed by undertakings based in a member state of the EC. It encompasses transports performed on roads open to the public by vehicles with a total mass of more than 3.5 tons, used for the carriage of passengers or goods. Transports carried out in total on a company’s premises do not fall under this legislation. Moreover, transportation tasks performed by vehicles constructed to carry less than nine persons, buses in urban transport, vehicles with a maximum speed of less than 30 km/h and certain emergency services are excluded from the scope of this directive. Directive 2002/15/EC states that its restrictions on working hours shall also become valid for self-employed drivers from 23 March 2009, subject to ratification by the European Parliament and the Council. Although the integration of self-employed drivers seemed to be a matter of fact (Rang, 2008), some obstacles had to be overcome. A study commissioned by the European Commission recommends “not to include the self-employed within the scope of the Directive” (Goudswaard et al., 2006). In their report to the Council and the European Parliament of 23 May 2007, the European Commission does not give a final advice on whether to apply the regulations imposed by the Directiv2002/15/EC to self-employed drivers (Report, 2007). Subsequently, the Commission followed the study’s recommendations and suggested not to include self-employed drivers within the scope of Directive 2002/15/EC. This proposal was rejected by the European Parliament, which decided in 2009 that the rules of the working time directive shall also apply to self-employed drivers. Subsequently, the European

2.2 Directive 2002/15/EC on Working Hours

21

Commission again suggested to exclude self-employed drivers from the scope of Directive 2002/15/EC. However, this proposal was finally rejected by the European Parliament in June 2010. Therefore, the member countries of the EC now have to extend the scope of their working time legislation to self-employed drivers.

2.2.2 Definitions The rules on working hours of Directive 2002/15/EC supplement the provisions set by Regulation (EC) No 561/2006. In the directive, working time is defined as “the time devoted to all road transport activities” (Directive, 2002, Article 3). Besides driving times, which are also restricted by Regulation (EC) No 561/2006, working time comprises time for loading and unloading, time for assisting passengers while boarding and disembarking from the vehicle, and time for cleaning and technical maintenance. Additionally, all times which a driver cannot freely dispose of and during which his presence at the working place is required are regarded as working time. This includes the time drivers have to wait in line at a loading ramp to unload their vehicles. According to interviews carried out with dispatchers of freight forwarding companies, these periods can be very long resulting in extensive nonproductive working times. In line with Regulation (EC) No 561/2006, a week is defined as the period between Monday 00:00 and Sunday 24:00.

2.2.3 Restrictions on Working Hours Directive 2002/15/EC provides restrictions on breaks and weekly working times. It postulates that persons performing mobile road transport activities must not work for more than 6 hours without taking a break. This break has to amount to at least 30 minutes if the total working time lies between 6 hours and 9 hours. If the working time exceeds 9 hours, the total break time has to be extended to at least 45 minutes. Drivers have the option of dividing the total break time into parts of no less than 15 minutes each. The weekly working time is restricted to a maximum of 60 hours. Furthermore, the directive postulates that the average weekly working time shall not exceed 48 hours over a period of four months.

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2 European Legislation on Driving and Working Hours

2.2.4 Implementation into German Legislation The restrictions set by Directive 2002/15/EC had to be implemented into the national laws of all member countries of the EC. In Germany this was achieved by the German Bundestag by enacting §21(a) of the Arbeitszeitgesetz which became effective on August 14, 2006. It concerns the working hours of drivers in road transportation. The law states that the weekly working time may not exceed 48 hours. It can be extended up to 60 hours if over a period of 16 weeks or four months the average working time does not exceed 48 hours. The German law allows for deviations of the maximum weekly working hours for technical reasons and for reasons concerning the organization of working hours (see Arbeitszeitgesetz, 2009). In this case the average working time may not exceed 48 hours over a period of six months. The Arbeitszeitgesetz also states that rest periods shall comply with the restrictions of Regulation (EC) No. 561/2006. It forces employers to record their employees’ working hours and to store those records for a period of at least two years.

2.3 Remarks on the General Structure of the European Social Legislation The European social legislation imposes a complex set of restrictions on driving and working hours of drivers engaged in road transportation. This complexity arises both from the different but interconnected time horizons of the legal rules, and from the different structures of their boundaries. Single driving time intervals and daily driving times are confined by breaks and daily rest periods. This means that their boundaries also result from the process of establishing driver schedules. A daily driving time does not align with a day, i.e. the time between 00:00 and 24:00. Daily driving times are constituted by the single driving time intervals in between the respective daily rest periods. The end of a daily driving time, marked by the beginning of a daily rest period, also terminates a single driving time interval. This definition of driving intervals and daily driving times allows drivers to take up to 32 single driving intervals a day, each of a duration of 1 minute, and up to 3 daily driving times, separated by two daily rest periods. Taking into account the rule that drivers must have taken a new daily rest period within 24 hours after the end of the previous rest period, they can still take up to 20 driving periods before scheduling a reduced daily rest period exactly 15 hours after the end of the previous rest.

2.4 Control Policies

23

On the other hand, the boundaries of the weekly driving time are defined by exact points in time, namely as the time from Monday 00:00 until Sunday 24:00. The boundaries of this interval do not result from a planning process, but are input data for the planning of driving and working times. The end of a weekly driving time does not enforce the end of a daily driving time or even of a single driving period. For example, if a driver finishes a daily rest period on Sunday at 23:00, he may start driving until Monday 03:30 until a break is required and the driving period ends. Since the new week begins on Monday at 00:00, the 1 hour of driving performed before is counted towards his weekly driving time of the first week, whereas the 3.5 hours are counted towards his driving time of the second week. Within the maximum time of 144 hours between two weekly rest periods, a driver can take up to 14 daily driving times, each of a very short duration of 1 minute, separated by three reduced and 10 regular daily rest periods.

Since driving times are also regarded as working times, dispatchers have to make sure that the driver schedules comply both with the legislation on driving hours and the legislation on working hours. This means that a schedule which adheres to the rules of Regulation (EC) No 561/2006 also has to comply with the restrictions on working hours if the additional working times such as times for unloading and unforeseen waiting times are considered. Both laws force drivers to take breaks and rest periods. These breaks do not have to be taken separately due to the fact that breaks satisfying the restrictions of both laws can also be considered for both. This means that e.g. a 30-minute break taken by a driver can be considered as the second part of a break according to Regulation (EC) No 561/2006, and as a break in the sense of the Directive 2002/15/EC. Consequently, a sequence of a 15-minute and a 30-minute break satisfying the driving time regulation also meets the requirements of the working time directive. This similar structure of the break times can be exploited by dispatchers when planning vehicle routes and driver schedules.

2.4 Control Policies On 15 March 2006, the European Parliament and the Council enacted Directive 2006/22/EC on stricter control policies for drivers’ working hours (see Directive, 2006). The directive postulates that all member states of the EC shall organize a

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system of regular checks on driving and working hours both at the roadside and at the premises of the enterprises. This directive had to be implemented into the national laws of the member countries, and the member states have to provide a “coherent national enforcement strategy” (Directive, 2006, Preamble) for their territories. The directive sets minimum requirements for the amount of controls in road transportation. It postulates that member states shall organize the checks in such a way that, from May 2006 on, one percent of the working days of the drivers falling within the scope of Regulation (EEC) No 3820/85 and of Regulation (EEC) No 3821/85 had to be investigated. From January 2008 on, the minimum percentage of working days checked had to be increased to 2 percent and from 2010 on to 3 percent. Under certain conditions, the percentage can again be increased to 4 percent starting from January 2012. Moreover, the directive postulates that from 2006 on, at least 15 percent of the total number of the working days reviewed had to be checked at the roadside and 30 percent at the premises of undertakings. From January 2008 on, no less than 30 percent of the total number of the working day checks shall be performed at the roadside and 50 percent at the premises. Roadside checks must be organized in various places and at varying times to ensure that they cannot be circumvented by the drivers. These checks comprise the daily and weekly driving times, breaks and daily and weekly rest periods. Furthermore, the adherence to the vehicles’ speed limits and the correct functioning of the recording equipment have to be investigated. Checks at the premises shall be planned according to past experience with the transportation companies. If serious infringements of the working time restrictions committed by a company’s drivers are detected at the road side, the premises of the respective company shall also be checked by the control authorities. Controls at the premises shall additionally comprise the observance of all biweekly driving time limits of 90 hours and the checks of record sheets, vehicle units and the driver card data and printouts. Violations of the maximum daily, weekly or fortnightly driving times, the minimum daily or weekly rest periods or the minimum break times are regarded as infringements of the European social legislation, and can be fined by the control authorities.

3 Dispatchers’ Tasks in Combined Vehicle Routing and Break Scheduling The rules on driving and working times set by the European social legislation have to be respected by dispatchers when creating vehicle routes, which is a partial task of the order dispatching process. In this chapter the impact of the legal rules on order dispatching is analyzed. Furthermore, the integration of the legal rules on driving and working hours in practical freight forwarding is investigated. The chapter is structured as follows: In Section 3.1 the dispatching process is outlined and the dispatchers’ tasks are derived. In particular, the planning task of vehicle routing and scheduling is embedded in the dispatching process. Section 3.2 analyzes the impact of the legal rules on the design of vehicle routes and gives examples of their integration in vehicle routing and scheduling. Finally, Section 3.3 presents the results of a recent study in which dispatchers of freight forwarding companies were interviewed on how they account for the European social legislation when creating vehicle schedules.

3.1 The Dispatching Process A freight forwarding company receives transportation requests from its clients. These transportation requests specify origin and destination, time windows for pickup and delivery, the type of good to be transported and the required capacity (Kopfer and Pankratz, 1999). The sellers of the freight forwarding company are responsible for selling the company’s transportation services to its clients. They have to decide whether the freight forwarding company is willing to fulfill the transportation requests, and they have to negotiate their prices. If the sellers accept these requests for fulfillment, they have to be processed by dispatchers (Jurczyk et al., 2006). The dispatchers’ task is then to assign these transportation requests to the available transportation resources such that the requests are fulfilled as specified by the customers, and the costs of using these resources are minimized (adapted from Gudehus, 2007a,b). The available resources hereby comprise both internal C. M. Meyer, Vehicle Routing under Consideration of Driving and Working Hours, DOI 10.1007/978-3-8349-6732-9_3, © Gabler Verlag | Springer Fachmedien Wiesbaden GmbH 2011

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and external means of transport (see e.g. Kopfer and Wang, 2009a; Krajewska and Kopfer, 2009). Internal means of transport refer to the company’s own fleet of vehicles. The external means of transport are constituted by external carriers. Moreover, if the freight forwarding company belongs to a collaboration of independent freight forwarders, the collaborating partners are also regarded as external resources (Kopfer and Wang, 2009b). Order dispatching encompasses several decisions. First, the mode of fulfillment has to be chosen for each request. According to Krajewska (2007), these fulfillment modes comprise the deployment of the own vehicle fleet, the utilization of vehicles from partner carriers that are employed on a long-term basis, and the vending of the requests on the spot market. For the requests assigned to the own vehicle fleet and also for the requests subcontracted to partner carriers, the dispatchers have to create vehicle routes. For the requests sold on the spot market, the dispatchers have to solve a freight flow consolidation problem (e.g. Kopfer, 1984, 1990, 1992) since the carriers are paid on the basis of the flow of loading (Krajewska, 2007). These planning problems, i.e. the mode selection, the routing and the flow consolidation, can be solved either simultaneously or sequentially. If they are addressed simultaneously, the planning problem is referred to as the integrated operational transportation planning problem (e.g. Kopfer and Krajewska, 2007; Krajewska and Kopfer, 2009). In practice the problem is often solved sequentially following a hierarchical process (Jurczyk et al., 2006). First, requests offering a high contribution margin are assigned to the own vehicle fleet. The own vehicles are first utilized since their fixed costs have to be covered. Subsequently, the remaining requests are assigned to external carriers. For the own fleet and for the dependent carriers, the dispatchers have to build vehicle routes. This problem of creating vehicle routes can be described as the problem of clustering the customer requests and finding a sequence in which they have to be serviced, such that the requests of each cluster can be served by a single vehicle, located at a central depot (see e.g. Rieck, 2008; Domschke, 1997). If the customers specify certain time intervals during which the pickup and delivery operations have to be performed, dispatchers have to schedule the operations such that the vehicles can meet these time windows. Possible objectives of the dispatchers can be the minimization of the total distance or the driving time, the minimization of the number of routes or the balanced capacity utilization of the vehicles (Vahrenkamp, 2007). Consequently, vehicle routing and scheduling comprises the tasks of clustering the customer requests and routing the vehicles

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such that all constraints are fulfilled. After creating the vehicle routes, the dispatchers instruct the drivers with their tours and routes, and the drivers have to fulfill the transport requests as specified by the customers, hereby respecting the legal rules on driving and working hours.

3.2 The Impact of the European Social Legislation on the Dispatching Process As described in Chapter 2, dispatchers have to make sure that the vehicle routes and schedules generated in the dispatching process also satisfy the legal rules on driving and working hours for the drivers. Therefore, dispatchers have to take these legal restrictions into account when creating vehicle routes. This results in an additional task for the dispatchers, namely the scheduling of breaks and rest periods which have to be taken by the drivers along their routes. This extended problem is referred to as the problem of combined vehicle routing and break scheduling (Meyer et al., 2009). It can be defined as the problem of finding vehicle routes to service a set of transportation requests such that a cost criterion is minimized and legal rules on driving and working hours are observed. The problem of combined vehicle routing and break scheduling thus comprises the following three partial planning problems which have to be addressed by the dispatchers: • Clustering of transportation requests • Routing of vehicles • Scheduling of breaks and rest periods for the drivers The sets of customer locations (or the sets of pickup and delivery locations of the transportation requests), resulting from the clustering process, are called tours, and the sequence in which a vehicle visits its assigned customer locations is called the vehicle’s route (Z¨apfel and B¨ogl, 2007). The sequence of breaks and rest periods taken along a vehicle route and their properties, such as their durations and start and end times, are referred to as the driver’s break schedule. In practice not all three partial problems are solved by dispatchers. Especially the scheduling of breaks and rest periods is usually left to the drivers (see Section 3.3).

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3.2.1 Effects on Vehicle Routing and Scheduling Regulation (EC) No 561/2006 states that each transport undertaking has to organize the work of “drivers it employs or who are put at its disposal” (Regulation, 2006, Article 10) such that they can comply with the European social legislation. The drivers’ working times are mainly determined through the tasks assigned to them by the dispatchers. Consequently, for the drivers of the own fleet and for the dependent carriers, dispatchers have to organize their working schedules such that they comply with the legal rules on driving and working hours. For transportation requests which are sold on the spot market, the subcontracting company is responsible for the fulfillment of these requests, and therefore for the compliance with the legal rules. Since dispatchers have to create vehicle routes for the own fleet, the legislation forces them to provide that these routes and the resulting driver schedules comply with the European social legislation. Therefore, apart from the restrictions set by the customer specifications for their transport requests, dispatchers have to face additional legal restrictions for the design of the vehicle routes. However, these restrictions only have an impact on the dispatchers’ planning if the pickup and delivery operations are time constrained, which is usually the case in practice. Otherwise, the ex post integration of breaks and rest periods into the already constructed vehicle routes does always result in feasible vehicle schedules, and even maintains the optimality of the solutions considering traditional criteria such as the minimum number of vehicles or the minimum driving distance. The legal restrictions force planners to add additional time required for breaks and rests to the vehicle routes if the expected driving time of a route exceeds certain time limits. Compared with the routes created by neglecting breaks and rest periods, the arrival times are delayed by the time required for these breaks, resulting in an increased route duration. For vehicle routes crossing the maximum daily driving times, this impact is enormous since daily rest periods of 9 hours respectively 11 hours have to be scheduled. The ex post planning of breaks and rest periods and the postponement of the arrival times at customer locations can lead to a violation of the specified time windows. Consequently, the vehicle routes may become infeasible. Apart from that, dispatchers have to make sure that the drivers’ weekly driving and working times do not exceed their time limits. This results in a restriction of the maximum route durations which has to be considered by the planners.

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Figure 3.1: Customer locations and time windows 3.2.2 Examples of Effects The above analysis shows that the ex post planning of breaks and rest periods in the presence of time windows causes the risk of creating infeasible vehicle routes. The optimal vehicle routes including driving and working time restrictions may therefore differ from the routes derived without consideration of the legal rules. The following examples demonstrate possible effects of the legal rules on the resulting vehicle routes.4 In the following, a VRPTW is considered in which all freight to be transported is located at a central depot, based in Bremen. From this central depot four customers, located in Cuxhaven, Hamburg, Hannover and Nienburg, have to be supplied with goods. Each transportation request comprises half a truckload. The distances and the delivery time window (TW) of each customer are specified in Figure 3.1. The dispatcher’s task is to find vehicle routes such that each customer is visited only once and supplied with his requested goods, and the delivery operations start within the time windows specified by the customers. Each unloading operation has a duration of 30 minutes. The dispatcher’s primary objective is to minimize the number of vehicles required, and his secondary objective is to minimize the total travel distance. Solving this VRPTW without consideration of the European social legislation 4

This section is based on Kopfer et al. (2007).

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Figure 3.2: Vehicle routes without breaks results in the vehicle routes depicted in Figure 3.2. The minimum number of required vehicles is 2, which can also be computed as a lower bound by dividing the total capacity of the requests by the vehicle capacity. The minimum travel distance equals 665 km. Since the driving time from Bremen to Cuxhaven via Hamburg amounts to 5 hours, according to Regulation (EC) No 561/2006 a break of 45 minutes has to be scheduled before the arrival at Cuxhaven. This break postpones the driver’s arrival time at Cuxhaven to 14:15. However, as the customer requests the delivery between 9 and 13:30, this vehicle route does no longer meet the specifications of the transport request, and is therefore infeasible. The optimal routes including the legal rules on driving and working hours are depicted in Figure 3.3. Although the dispatcher’s primary criterion remains unchanged, i.e. only 2 vehicles are required, his second criterion, the total travel distance, increases to 717 km. This means a deterioration of 7.8 %. If the problem settings depicted in Figure 3.1 are changed such that the delivery time window of the customer located in Hannover is tighter, namely from 09:00 until 13:00, still the vehicle routes depicted in Figure 3.2 are feasible and, of course, optimal for the problem without consideration of the European social legislation. However, the routes depicted in Figure 3.3 are no longer feasible since the arrival time at Hannover violates the customer time window. In this case 3 vehicles are required to service all customers on time and comply with the legal

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Figure 3.3: Vehicle routes including breaks rules. The optimal vehicle routes are depicted in Figure 3.4. The total distance now amounts to 674 km which means a deterioration of both criteria compared with the optimal routes without consideration of driving time restrictions. Of course, this vehicle plan is also feasible for the first example with the larger time window. The deployment of an additional vehicle does of course deteriorate the dispatcher’s main objective. However, it is remarkable that the total travel distance is reduced through the utilization of an additional vehicle. This shows that the positive correlation between these criteria, which is usually encountered in vehicle routing problems, does not hold for time window constrained problems. The above example clearly demonstrates two effects of the European social legislation on vehicle routing and scheduling. First, the total travel distance may increase through the integration of breaks and rest periods, since the additional travel times may lead to a violation of time windows. Consequently, the optimal solution without driving time restrictions is rendered infeasible and a rerouting of the vehicles has to be performed. Second, additional vehicles might be required in the case that a rerouting of the already used vehicles does not allow for a feasible solution. In this case again the decision on the mode of fulfillment has to be taken by the dispatchers. As shown in the example above, the restrictions of the European social legislation have an enormous impact on the planning of vehicle routes. Consequently, planners should integrate these rules into the process of vehicle routing and scheduling, since their ex post planning may lead to infeasibilities.

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Figure 3.4: Vehicle routes with breaks requiring 3 vehicles

3.3 Combined Vehicle Routing and Break Scheduling in Practice The question of how dispatchers account for the European social legislation in practice was investigated in an empirical study by Onken (2009). For the study, expert interviews with dispatchers of five medium-sized freight forwarding companies in northern Germany were conducted. The main results are reported in the following. 3.3.1 Methodology of the Study In the course of the study, dispatchers were visited at their place of work, and were observed while carrying out their tasks. Afterwards, semi-structured interviews were conducted with the dispatchers concerning the fulfillment of their tasks. Special attention was given to the following issues: • Which logistics services are offered by the company? • How is the dispatching department organized and which are the dispatchers’ tasks? • What is the distribution of tasks between dispatchers and drivers? • Is the task of vehicle routing and scheduling carried out manually or ITbased?

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• How many vehicles are already equipped with a digital tachograph? • To which extent are the legal rules of the European social legislation accounted for when planning vehicle routes? • How do dispatchers provide that the drivers can adhere to the legal rules on driving and working hours? • Did the enacting of Regulation (EC) No 561/2006 result in cost increases for the company? • Did the enacting of Regulation (EC) No 561/2006 lead to an increased use of multi-manning and relief persons?

3.3.2 Interviews with Freight Forwarding Companies In the following the main findings from the interviews are summarized for each freight forwarding company. For reasons of confidentiality, the company names are anonymized.

Freight Forwarding Company 1

Freight forwarding company 1 has about 450 employees and has several affiliates in different regions in Germany. It offers line hauls as well as heavy duty and distribution services. At the time of this study, about half of its 250 trucks were equipped with digital tachographs. Customers can submit their transport requests via an online portal. The dispatching department is responsible for the administration of these customer requests and for creating vehicle tours and routes. However, as there are usually at most three customers belonging to a tour, the routing is trivial. The software “Automotive Daimler” is utilized for supporting the administration and the clustering of the customer requests. For the monitoring of the vehicles’ progress on their routes and also of the driving times, the planners use telematic systems. The drivers are responsible for planning breaks along their routes. The planning of daily rest periods and the extension of daily driving times is carried out jointly by planners and drivers. For long routes, the company uses semitrailers which can easily be switched between trucks. Only very rarely, the company applies multimanning. Tours requiring multi-manning are always forwarded to subcontractors.

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If due to unforeseen delays along the route, a driver cannot deliver his goods to a customer on time without exceeding the allowed daily driving time, the dispatchers sometimes advise the drivers to perform the delivery, accepting the violation of the legislation. Freight forwarding company 1 cannot estimate the cost impact resulting from the enacting of the new European social legislation. Freight Forwarding Company 2 The second freight forwarding company has 15 branch offices in Germany and mainly offers services for the following business areas: automotive, cargo, food/ non food and volume. It has about 1,200 employees and owns 280 trucks and more than 1,000 semitrailers. At the time of the study, about one third of the trucks was equipped with digital tachographs. The dispatchers are responsible for building tours and also for the routing of the vehicles. When performing these tasks, they already take times for breaks and rest periods into account. To support their planning, they use software which can include some basic rules of the legislation. The optional legal rules (see Chapter 2) are not considered. The drivers are informed about the proposed scheduling of breaks. However, they can deviate from these recommendations and take breaks according to their preferences. For tours going to regions where traffic jams are likely (e.g. the Ruhr Area), time buffers are included to allow for possible additional breaks. The company advises its drivers to adhere to the legislation even if this causes delayed deliveries. For some special line hauls relief persons are used, and one line haul is performed using multi-manning. The enacting of Regulation (EC) No 561/2006 did not result in cost increases for freight forwarding company 2. Nevertheless, the restriction of the average weekly working time to 48 hours by Directive 2002/15/EC increased the costs for personnel. Freight Forwarding Company 3 Freight forwarding company 3 employs about 1,200 people and owns about 170 trucks of which about one quarter is equipped with digital tachographs. Its headquarters are located in Northern Germany and it has affiliates in the Netherlands,

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Australia, China, Indonesia and in Central and South America. Besides road transportation, its services comprise sea, inland water and railway transportation. Its customers mainly belong to the automotive, steel and chemical industry. All its vehicles are equipped with telematic systems and use the software “Daimler Fleetboard”. Those vehicles that are equipped with a digital tachograph can document driving times and rest periods in real time. In cases of traffic jams, the company always documents the reason for exceeding the allowed non-stop driving times and has never been fined by the control authorities. Due to its main areas of business, this freight forwarding company fulfills many full truckload (FTL) transports for which the tasks of clustering and routing are trivial. For these transport requests the whole planning of the tour is left to the drivers. For the LTL transports, the clustering is carried out by the dispatchers. The tasks of routing and break scheduling are performed by the drivers. The company did not face any direct cost effects through the enacting of Regulation (EC) No 561/2006. However, some line hauls had to be redesigned. The restriction of the weekly working time to an average of 48 hours through Directive 2002/15/EC had an enormous impact on the costs of freight forwarding company 3. Especially the waiting times at loading ramps, which are considered as working times, lead to a bottleneck in the drivers’ availability.

Freight Forwarding Company 4

Freight forwarding company 4 is a family-owned enterprise with about 8,000 employees and 24 affiliates distributed over Germany. It offers logistics services in various modes of transport such as road, sea, inland water and air cargo. A central dispatching department for national and international FTL and LTL transports is based in its headquarters in Northern Germany. The headquarters own 55 vehicles of which 48 are equipped with digital tachographs. It handles between 500 and 700 FTL and LTL transports per day. Dispatchers are supported by the dispatching software “Dispotool” which facilitates the administration of transport requests. Moreover, a telematic software is employed to monitor the routes. The dispatchers are responsible for the clustering of customer requests and for the decision whether a request should be carried out by the own vehicle fleet or by means of subcontracting. For tours carried out by the own fleet, also the routing is performed by the dispatchers. Apart from that, they monitor the driving times

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of the drivers. The drivers are responsible for scheduling breaks and rest periods. The management does not allow drivers to exceed their maximum driving times since the company tries to avoid fines caused by these violations. Due to the enacting of Regulation (EC) No 561/2006, some line hauls had to be redesigned. Moreover, buffer times had to be included in order to allow for the adherence to the European social legislation while still meeting the due dates. Freight forwarding company 4 does not apply multi-manning. Due to the new legislation, some charter tours had to be redesigned. Therefore, freight forwarding company 4 had to face decreased revenues. However, the company cannot give a good estimation of their magnitude.

Freight Forwarding Company 5

Freight forwarding company 5 is based in Northern Germany and employs about 720 people. It does not have any affiliates, but belongs to a European logistics cooperation which offers standardized LTL services all over Europe. The company owns about 210 vehicles and operates 56 line hauls per day as well as FTL and LTL services. Two thirds of its trucks are equipped with digital tachographs. The company uses its own dispatching software which supports dispatchers in creating customer clusters according to predefined delivery areas. Based upon these clusters, the dispatchers carry out the routing. To account for the drivers’ breaks, the dispatchers include buffer times in the routes. The final scheduling of the breaks is done autonomously by the drivers. The company gives high priority to the legal rules on driving and working times. Drivers are never advised to neglect the legislation on breaks and rest periods in order to keep due dates. Instead multi-manning is used if required. To guarantee the adherence to the legislation, the company also checks the documentation of the driving times recorded by the digital tachographs. Telematic software has been bought to facilitate a real-time monitoring of driving times, but it had not been fully implemented at the time of the study. The company has not experienced costs increases through the implementation of Regulation (EC) No 561/2006. Nevertheless, the restriction of the weekly working times to 48 hours by Directive 2002/15/EC has led to an estimated loss in the drivers’ productivity of about 25 %.

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Figure 3.5: Distribution of tasks between dispatchers and drivers 3.3.3 Summary of the Findings The interviews show that dispatchers of all five freight forwarding companies have to face the problem of including driving and working time restrictions in the process of vehicle routing and scheduling. Although different approaches are used for solving this problem, it is obvious that in all cases the problem is not solely solved by the dispatchers. Instead, the tasks are distributed between dispatchers and drivers. Apart from the studies by Meyer et al. (2009, 2010), this distribution of tasks has not been accounted for in the scientific literature on combined vehicle routing and break scheduling so far. The distribution of tasks between dispatchers and drivers is similar in all five companies. In all cases the clustering of customer requests is performed by the dispatchers. This is reasonable since the decisions to be taken here affect several drivers. Therefore, it is not reasonable to have the drivers performing this task since they cannot oversee the whole impact of their actions. Also the routing of the vehicles is mainly carried out by the dispatchers. The final decision on when to schedule breaks is always taken by the drivers. Therefore, the expert interviews show that the distribution of tasks between dispatchers and drivers and the relationship between their planning activities is as depicted in Figure 3.5. Another interesting finding is that three out of five freight forwarding companies (company 2, 4 and 5) use time buffers to provide drivers with slack time within their routes during which they can schedule breaks and rest periods. It can be concluded that the utilization of buffer times is a common principle of addressing the problem of combined vehicle routing and break scheduling in practice. The integration of time buffers is an implicit way for the dispatchers to account for the drivers’ break scheduling (Meyer et al., 2009), since they do not need to consider breaks and rest periods explicitly. In the following, this implicit way of anticipating the drivers’ planning will be evaluated. Furthermore, it will be

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compared with planning approaches in which the dispatchers explicitly schedule breaks and rest periods to be taken by the drivers. Four out of five freight forwarding companies experienced cost increases through the enacting of the new European social legislation. This cost impact can be directly assigned to the restriction of the average weekly working times to 48 hours, imposed by Directive 2002/15/EC. The new rules imposed by Regulation (EC) No 561/2006 did not influence the costs of the freight forwarding companies significantly, compared with the rules specified in Regulation (EEC) No 3820/85. This finding is in line with the discovery that the changes resulting from the new driving time regulation are actually negligible (see Section 2.1.5).

4 Model for the Vehicle Routing Problem with Time Windows and European Social Legislation In this chapter the problem of combined vehicle routing and break scheduling is investigated from a central planning perspective.5 The problem of combined vehicle routing and break scheduling is defined as the problem of finding vehicle routes to service a set of transportation requests such that a cost criterion is minimized and legal rules on driving and working hours are observed (Meyer et al., 2010). This definition allows for different underlying vehicle routing problems such as the VRP, the VRPTW, the Pickup and Delivery Problem (PDP) or the Pickup and Delivery Problem with Time Windows (PDPTW). Moreover, different sets of legal rules can be considered such as the US regulations set by the DOT or the European social legislation. In the following, mathematical models for the VRPTW-EU are presented. According to Dinkelbach (1973), a mathematical model can be defined as an isomorphous mapping of a preprocessed part of a real problem or system. Hence, mathematical models are constructed by abstracting away from certain aspects of the real system, hereby generating a simplified part of the real problem (see also Gr¨ unert and Irnich, 2005; Shiflet and Shiflet, 2006). This process is referred to as abstraction (Schneeweiss, 1992). The aim of mathematical modeling is to give a formal representation of a real-life problem which can be investigated and solved by means of analytical or numerical solution techniques. In vehicle routing it is important include problem features relevant for practice (Ohrt, 2008), since abstracting from them might lead to the rejection of the model’s solution by practitioners (Hatzopoulou and Miller, 2009). Therefore, it seems to be worthwhile to include the legal rules on driving and working hours into vehicle routing models. The vehicle routing problem was first introduced by Dantzig and Ramser (1959). Since then it has become one of the most important and most studied combinatorial optimization problems (Toth and Vigo, 2002b). For simple VRPs without 5

This chapter is based on Kopfer and Meyer (2010).

C. M. Meyer, Vehicle Routing under Consideration of Driving and Working Hours, DOI 10.1007/978-3-8349-6732-9_4, © Gabler Verlag | Springer Fachmedien Wiesbaden GmbH 2011

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time window restrictions the legal rules on driving and working hours do not affect the optimality of the solutions with respect to the minimum driving time or the minimum number of vehicles. Consequently, the problem can be solved without consideration of the legal rules, and beaks and rest periods can be scheduled ex post (Gietz, 1994). This solution property does not necessarily hold for VRPTWs due to the fact that the ex post integration of breaks can lead to infeasibilities (see Section 3.2). Therefore, in the following a VRPTW-EU is investigated. First, a model for the VRPTW-EU containing only the basic rules of the European social legislation is derived. This model represents the state-of-the-art since most previous studies have only focused on the integration of a basic set of the legal rules into solution approaches for vehicle routing problems (e.g. Z¨apfel and B¨ogl, 2008; Goel, 2009; Bartodziej et al., 2009). Goel (2009) even claims that neglecting the optional rules of Regulation (EC) No 561/2006 results in more robust vehicle schedules. However, according to Hatzopoulou and Miller (2009) it is a misunderstanding that simple models are more robust. In contrast, Hatzopoulou and Miller (2009) argue that added complexity is an opportunity for making models more accurate. In fact, the inclusion of the optional legal rules allows for more flexibility since these rules are less restrictive than the basic rules. Hence, their integration leads to an enlarged solution space and may result in improved vehicle schedules. Consequently, a full model including all optional rules of the European social legislation is presented. Finally, computational experiments are performed to investigate the effects of considering driving and working hours in vehicle routing and scheduling.

4.1 Model for the VRPTW-EU with Basic Rules In contrast with the common understanding of a vehicle routing problem, which only covers a single day (Francis et al., 2008), the VRPTW-EU covers a planning horizon of several days up to a whole week. Therefore, it resembles the Periodic Vehicle Routing Problem (PVRP) (see e.g. Beltrami and Bodin, 1974; Chao et al., 1995; Cordeau et al., 1997; Francis et al., 2008). The PVRP can be described as the problem of servicing customers on several days with a fleet of vehicles, such that the vehicles start and end their daily tours at a central depot and the customers are serviced according to their desired visiting schedules. Furthermore, in the Periodic Vehicle Routing Problem with Time Windows (PVRPTW), visits can only be carried out during predefined time windows for each day (Cordeau

4.1 Model for the VRPTW-EU with Basic Rules

41

et al., 2001). The VRPTW-EU also covers a planning period of several days. In contrast with the PVRPTW, in the VRPTW-EU each vehicle is only allowed to carry out one route which might span over several days. In the VRPTW-EU, days are defined by the legal restrictions, but not through the modeling features of the underlying vehicle routing problem.

4.1.1 Problem Definition The model presented in the following assumes a central planning perspective which is common for traditional vehicle routing problems. This means that the planner creates vehicle routes for a given set of vehicles and simultaneously schedules breaks and rest periods for the drivers. To include the restrictions on driving and working hours for the drivers into the vehicle routes, a fixed assignment of drivers to vehicles is assumed. Each vehicle is single manned and the driver stays with his vehicle for the whole planning period. Consequently, the vehicle’s driving time equals the driver’s driving hours and the constraints imposed on drivers can be embedded into the restrictions for the vehicles as suggested by Toth and Vigo (2002a). Therefore, the problem of assigning drivers to vehicles, the so called crew scheduling problem (see e.g. Park and Ryu, 2006), needs not be addressed here. To derive a mathematical model formulation for the VRPTW-EU, the following modeling assumptions for vehicle routing problems are made, which are commonly used for most problems in the vehicle routing literature (e.g. Laporte, 1992; Toth and Vigo, 2002a; Cordeau et al., 2002; Lecluyse et al., 2009). The VRPTW-EU is defined on a complete graph G = (V, A), where V represents the set of vertices and A represents the set of arcs. The set V contains n + 2 vertices where vertices 0 and n + 1 represent a central depot and vertices 1 to n represent the customer locations. The distance between any two vertices, denoted by dij , is nonnegative, with dii = 0 for all vertices i ∈ V . Each customer node is associated with a nonnegative demand li which is known in advance. In the case of a VRPTW-EU, possible service times at the customers are time restricted. This means that the service at each customer node i has to start within a given time window [ai , ei ], and the vehicle must remain at the customer node during service. The parameter ai denotes the opening time of the time window and ei denotes its closing time. If a vehicle arrives at a customer

42

4 Model for the VRPTW-EU

node i before the opening of the customer’s time window, it has to wait. If the arrival time at node i is denoted by ti , the minimum waiting time equals ai − ti . A homogeneous fleet of vehicles k, each of capacity C with li ≤ C, is located at the central depot and available to service the customer nodes. Each vehicle may at most make one route. The VRPTW-EU consists of finding the minimum cost vehicle routes such that each customer is visited exactly once and serviced within his time window, the demand of each customer is satisfied, the vehicle capacity is not exceeded and the legal rules on driving and working hours are respected. Real vehicle routing problems are usually characterized by a large number of additional constraints (Rieck et al., 2007). Apart from the above assumptions, which are required for modeling a standard VRPTW, the VRPTW-EU with basic rules comprises the following restrictions on driving and working hours as laid down by Regulation (EC) No 561/2006 and Directive 2002/15/EC. The VRPTW-EU covers a weekly phase, i.e. the time between two weekly rest periods. In order to include restrictions on driving and working times, the distances dij have to be converted into travel times. Travel times are assumed to be deterministic, time-independent and known in advance. For the model it is required that the driving time between any two distinct nodes i and j is short enough such that not both a break and a daily rest period or two daily rest periods are needed on this arc. This assumption can be justified for VRPs since vehicle routing problems are usually considered as short- to medium-distance distribution problems which incur shorter travel distances than for example PDPs (Pankratz, 2002). Since a deterministic setting is considered, all waiting times are known in advance. Therefore, according to Directive 2002/15/EC, they need not be considered as working hours. In the model, drivers’ working times only comprise driving and service times. It is assumed that all drivers start their work directly after finishing a weekly rest period. On the lowest time horizon (see Figure 2.1, Chapter 2), single driving and working periods are considered. According to Regulation (EC) No 561/2006 single driving periods are restricted to a maximum duration of 4.5 hours. Drivers have to take a break of at least 45 minutes after driving at most 4.5 hours unless they take a daily or weekly rest period. After a working time of no more than 6 hours drivers have to take a break of at least 30 minutes according to Directive 2002/15/EC. If the working time since

4.1 Model for the VRPTW-EU with Basic Rules

43

the last daily rest period exceeds 9 hours, the total break time has to be extended to at least 45 minutes. It is important to note that breaks satisfying the requirements of the driving time regulation also satisfy the working time directive (see Section 3.2). For example, if a driver drives for 4.5 hours and takes a break of 45 minutes, which is enforced by Regulation (EC) No 561/2006, this break also counts as a break for his working time if it is taken before a working time of 6 hours. In this case, he has already satisfied the break time requirements for his whole working day imposed by Directive 2002/15/EC. Therefore, the requirement for breaks imposed by Directive 2002/15/EC is modeled such that drivers have to take a break of at least 45 minutes after a working time of no more than 6 hours unless a daily rest period is taken. In this case, the rule of the working time directive is modified, such that it is aligned with the break time restrictions imposed by Regulation (EC) No 561/2006. On the intermediate time horizon, daily driving times are restricted to 9 hours. When drivers reach their maximum daily driving times, they have to take a daily rest period of at least 11 hours. On the highest time horizon, the weekly driving time is restricted to 56 hours and the weekly working time to 60 hours. The model also allows for adjustments towards lower weekly driving and working times by prespecified parameters. In traditional vehicle routing problems, common cost criteria are the minimization of the total travel distance or travel time, or the minimization of the number of vehicles (e.g. Vahrenkamp, 2007). However, in the case of combined vehicle routing and break scheduling, not only travel distances are considered, but also breaks and rest periods have to be determined. The times required for breaks and rest periods should also be included in the objective function in order to avoid an extensive and time consuming use of breaks and rest periods. Moreover, waiting times are an undesirable feature of vehicle routes, since they incur additional costs for the carriers (Erera et al., 2008). Hence, waiting times should also be included in the objective function. Consequently, in the following model formulation, the objective of minimizing the total fulfillment time is chosen.

4.1.2 Notation A position-based model formulation (see e.g. Dethloff, 1994) is chosen for the VRPTW-EU since it allows to calculate driving times of partial routes. The

44

4 Model for the VRPTW-EU

variables contain an index for the position q, at which a node j is visited along a route. For the model formulation, the following sets are required: V : set of locations i ∈ {0, . . . , n + 1} such that: i = 0 and i = n + 1: depot, and i = 1, . . . , n: customer nodes P : set of customer nodes, i.e. P = {1, . . . , n} Q: set of positions q, at which node j ∈ V can occur (the position of node 0 is set to 1) Q(i, j): set of positions q, at which node j ∈ V can occur, if j is reached from node i ∈ V , with: Q(0, j) = {2} and 2 ∈ / Q(i, 0) K: set of vehicles available at the depot, K = {1, ..., k max }

The following variables are used: ⎧ ⎪ ⎪ 1 if vehicle k drives from node i ∈ V to node j ∈ V such that ⎪ ⎨ k wijq : = node j is reached at position q of its route ⎪ ⎪ ⎪ ⎩0 else b45kijq : number of 45-minute breaks taken by vehicle k on its way from node i ∈ V to node j ∈ V if j is reached at position q of its route ⎧ ⎪ ⎪ 1 if a daily rest period is taken by vehicle k on its way from node i ∈ V ⎪ ⎨ k drijq : = to node j ∈ V such that j is reached at position q of its route ⎪ ⎪ ⎪ ⎩0 else ttkij : travel time of vehicle k from node i ∈ V to node j ∈ V including breaks and daily rest periods k : duration of vehicle k’s daily rest period between node i ∈ V and node T Rijq j ∈ V if j is reached at position q of its route

DTqk q : driving time of vehicle k from the node at position q  to the node at position q  of its route W Tqk q : working time of vehicle k’s driver between the node at position q  ∈ Q to the node at position q  ∈ Q of his route f : objective value ti : arrival time at node i ∈ P tk0 : departure time of vehicle k from the depot

4.1 Model for the VRPTW-EU with Basic Rules

45

tkreturn : return time of vehicle k to the depot

The following parameters have to be defined: dij : driving time between node i ∈ V and node j ∈ V , i = j, and (i, j) = (0, n+1) si : service time required to service customer i ∈ P ai : beginning of node i’s time window, i ∈ V ei : end of node i’s time window, i ∈ V li : load to be transported to customer i ∈ P C : vehicle capacity n : number of customer nodes to be visited k max : number of vehicles available at the depot   M : big number, e.g. M = i∈V j∈V dij k W DTw−1 : weekly driving time of vehicle k in the previous week w − 1

Δw : desired deviation of the weekly working time from its limit of 60 hours

4.1.3 Model Formulation Using the above notation, a model for the VRPTW-EU with basic rules can now be formulated as follows. The objective of the VRPTW-EU consists in minimizing the total fulfillment time of the vehicle routes. Therefore, the difference between the vehicles’ start times at the depot and their return times to the depot is minimized. This results in the Objective Function (4.1).

M in f =



(tkreturn − tk0 )

(4.1)

k∈K

This objective function leads to an automatic delay of a vehicle’s departure time from the depot if the vehicle would otherwise arrive early at the first customer node. The model for the VRPTW-EU comprises the following restrictions. First, the restrictions for a standard VRP in the position-based formulation are presented. Second, the time window restrictions are introduced. Third, the restrictions on

46

4 Model for the VRPTW-EU

driving and working hours are presented according to their planning time horizon.

  

k wijq = 1 ∀j ∈ P

(4.2)

i∈V q∈Q(i,j) k∈K



k w0j2 =

j∈P



k win+1q ∀k ∈ K

(4.3)

i∈P q∈Q(i,n+1)



k wisq =

i∈V \{n+1}





k wsj,q+1 ∀s ∈ P, q ∈ Q, k ∈ K

(4.4)

j∈V \{0}



 

k lj wijq ≤ C ∀k ∈ K

(4.5)

i∈V \{n+1} j∈P q∈Q(i,j)

Restrictions (4.2) to (4.5) are the standard restrictions for VRPs in the positionbased model formulation. Restriction (4.2) requires that each customer node is visited exactly once. The fact that each vehicle leaving the depot also has to return to the depot is guaranteed by (4.3). Restriction (4.4) states that if a node s is reached at position q of the route, it must be left such that its successor is reached at position q + 1. Finally, the vehicles’ capacity restrictions are modeled in (4.5). ttkij = dij +



k (0.75 ∗ b45kijq + T Rijq )

(4.6)

q∈Q(i,j)

∀i ∈ V, j ∈ V, k ∈ K 

tj ≥ ti + si + ttkij − M (1 −

k wijq ) ∀i ∈ P, j ∈ P, k ∈ K

(4.7)

q∈Q(i,j)

tj ≥ tk0 + si + ttk0j − M (1 −



k w0jq ) ∀j ∈ P, k ∈ K

(4.8)

q∈Q(0,j)

tkreturn ≥ ti + si + ttkin+1 − M (1 −



k win+1q ) ∀i ∈ P, k ∈ K

(4.9)

q∈Q(i,n+1)

ti ≥ ai ∀i ∈ P

(4.10)

4.1 Model for the VRPTW-EU with Basic Rules

47

ti ≤ ei ∀i ∈ P

(4.11)

tk0 ≥ a0 ∀k ∈ K

(4.12)

tkreturn ≤ e0 ∀k ∈ K

(4.13)

In Restrictions (4.6) to (4.13), the time window constraints are modeled. Since in the VRPTW-EU not only driving times but also times for breaks and rest periods along the arcs need to be considered, Equation (4.6) defines the travel times between any two nodes i ∈ V and j ∈ V . The travel times ttij consist of the driving times dij , the times required for 45-minute breaks b45kijq and the k durations of daily rest periods T Rijq taken on these arcs. If ti > 0 denotes the arrival time at node i ∈ P , and tk0 and tkreturn the departure and return times of vehicle k respectively, Restrictions (4.7), (4.8) and (4.9) compute the arrival times at all nodes. Restrictions (4.10) and (4.11) postulate that the service at each customer i has to be started within its time window [ai , ei ]. Restrictions (4.12) and (4.13) define the planning time horizon by restricting the times at which the vehicles may start their routes and their return times to the depot. 

q  

k dij wijq ∀q  , q  ∈ Q, q  ≤ q  , k ∈ K

(4.14)

q   k (dij + si )wijq ∀q  , q  ∈ Q, q ≤ q  , k ∈ K

(4.15)

DTqk q =

q=q 

i∈V j∈V



W Tqk q =

q=q i∈V j∈V

A vehicle’s total driving time required to travel from the node at position q  to the subsequent node at position q  of its route can be derived according to Equation (4.14). The total working time between the node at position q  can be computed accordingly (see Equation (4.15)). On the lowest time horizon considered by the European social legislation, i.e. single driving and working periods, the following restrictions are valid. 

DTqk q

≤ 4.5 + 4.5

q   q=q

i∈V j∈V



b45kijq

+M

q   q=q

i∈V j∈V

k drijq

(4.16)

48

4 Model for the VRPTW-EU ∀q  , q  ∈ Q, q ≤ q  , k ∈ K 

W Tqk q ≤ 6 + 6

q   q=q



b45kijq + M

q   q=q

i∈V j∈V

k drijq

(4.17)

i∈V j∈V

∀q , q  ∈ Q, q ≤ q  , k ∈ K k ∀i, j ∈ V, q ∈ Q(i, j), k ∈ K b45kijq ≤ M ∗ wijq

(4.18)

Restriction (4.16) considers the single driving times restricted by Regulation (EC) No 561/2006. It guarantees that a break of 45 minutes is taken for each partial route from the node at position q  to the node at position q  for which the driving time exceeds 4.5 hours. However, if a daily rest period of 11 hours k is taken on the considered partial route, i.e. if drijq = 1 for q ≤ q ≤ q  , the restriction is relaxed such that no break has to be scheduled. The requirements for breaks according to the working time directive are modeled in Restriction (4.17). In this model a break of at least 45 minutes after no more than 6 hours of working time is considered. The restriction states that a working period must not exceed 6 hours unless a break is taken. If a driver takes a daily rest period, again the restriction is relaxed. Breaks satisfying the driving time regulation are also considered as breaks for the working time directive. This restriction already meets the break time requirements of Directive 2002/15/EC for a whole working day. Finally, Restriction (4.18) makes sure that breaks can only be taken on arcs which are actually used. The rules on daily driving times are modeled in Restrictions (4.19) to (4.23). 

DTq q

q   k ≤ 9 + 9( drijq )

(4.19)

q=q  i∈V j∈V

∀q  , q  ∈ Q, q < q  , k ∈ K k k ≥ 11 − M (1 − drijq ) ∀i, j ∈ V, q ∈ Q(i, j), k ∈ K T Rijq



tj  − tk0 ≤ 24 + 24

q  i∈V j∈V q=2

(4.20)



k drijq + M (1 −

q  i∈V q=2

k wij q )

(4.21)

4.1 Model for the VRPTW-EU with Basic Rules

49

∀j  ∈ P, q  ∈ Q, k ∈ K

tkreturn − tk0 ≤ 24 + 24

 

k drijq ∀k ∈ K

(4.22)

i∈V j∈V q∈Q(i,j)

k k k + T Rijq ≤ M ∗ wijq ∀i, j ∈ V, q ∈ Q(i, j), k ∈ K drijq

(4.23)

Restriction (4.19) models the minimum requirement for daily rest periods. After a driving time of no more than 9 hours a daily rest period has to be taken. Restriction (4.20) provides that the duration of a daily rest period equals at least 11 hours. Restrictions (4.21) and (4.22) model the rule that after no more than 24 hours after the end of the previous daily rest period a new daily rest period has to be taken. Equation (4.23) states that daily rest periods may only be taken on used arcs. Finally, the legal rules on weekly driving and working hours are modeled in Restrictions (4.24) to (4.26).  

k dij wijq ≤ 56 ∀k ∈ K

(4.24)

k k dij wijq ≤ 90 − W DTw−1 ∀k ∈ K

(4.25)

k (dij + si )wijq ≤ 60 − Δw ∀k ∈ K

(4.26)

i∈V j∈V q∈Q(i,j)

  i∈V j∈V q∈Q(i,j)

  i∈V j∈V q∈Q(i,j)

Restriction (4.24) restricts the weekly driving time to 56 hours. Furthermore, drivers are not allowed to drive for more than 90 hours in any two consecutive k weeks which is modeled in Restriction (4.25). The term W DTw−1 is a parameter

representing the driving time of the previous week w − 1 and restricts the allowed driving time of the current week w. This parameter allows to include initial conditions of rules exceeding the planning time horizon of a weekly phase into the model as input data. Restriction (4.26) sets the maximum admissible weekly working time to 60 hours. Again, the term Δw allows for adjustments towards the long term average of 48 hours and gives the possibility for including initial conditions for each driver.

50

4 Model for the VRPTW-EU

The Objective Function (4.1) and Restrictions (4.2) to (4.26) give a complete model for the VRPTW-EU including the basic rules of Regulation (EC) No 561/2006 and of Directive 2002/15/EC.

4.2 Model for the VRPTW-EU with Optional Rules In the following an extended model formulation for the VRPTW-EU will be presented which covers the whole European social legislation including also the optional rules. The extended model provides planners with more flexibility and an increased solution space. To derive the model formulation for the VRPTWEU with optional rules, the problem definition has to be extended and additional notation has to be introduced.

4.2.1 Problem Definition Since the VRPTW-EU with optional rules is an extension of the model presented in Section 4.1 in which the considered legal rules are enhanced, the basic assumptions for the VRPTW remain unchanged. Again the VRPTW-EU with optional rules covers a weekly planning period and drivers start their work after finishing a weekly rest period. Therefore, they have not yet taken any reduced daily rest periods or driving time extensions during the current planning period. On the lowest time horizon, drivers can split breaks into two parts of at least 15 minutes and 30 minutes, respectively. The 15-minute part has to be taken before the 30-minute part. If both partial breaks are taken on the same arc (i, j), they can be either combined to a 45-minute break, or be taken separately. After a working time of no more than 6 hours, drivers have to take a break of at least 30 minutes if the daily working time lies between 6 hours and 9 hours. If the working time since the last daily rest period exceeds 9 hours, an additional break of 15 minutes has to be scheduled. On the intermediate time horizon, drivers may extend their daily driving times up to 10 hours twice during the weekly planning period. In the extended model, drivers can choose between taking a regular daily rest period of at least 11 hours or reducing their daily rest period to 9 hours. This is possible in two cases. First, up to three times during the planning period, a

4.2 Model for the VRPTW-EU with Optional Rules

51

driver can reduce his daily rest periods without compensation (reduced daily rest period). Second, a driver can split daily rest periods into two parts. The first part must last for at least 3 hours and the second part for at least 9 hours (split daily rest period). This option is modeled such that drivers are allowed to take an unrestricted number of daily rest periods of 9 hours. Each driver may take up to three 9-hour rest periods without compensation. Each additional 9-hour daily rest period has to be preceded by an additional 3-hour rest during the current daily driving period. On the highest planning time horizon, weekly driving and working hours are restricted in the same way as in the model with basic rules.

4.2.2 Notation Apart from the sets, variables and parameters introduced for the VRPTW-EU with basic rules, the following variables are required for the VRPTW-EU including the optional rules of the legislation: b15kijq : number of 15-minute breaks taken by vehicle k on its way from node i ∈ V to node j ∈ V if j is reached at position q ∈ Q of its route b30kijq : number of 30-minute breaks taken by vehicle k on its way from node i ∈ V to node j ∈ V if j is reached at position q ∈ Q of its route ⎧ ⎪ ⎪ 1 if a daily rest period of at least 9 hours is taken by vehicle k ⎪ ⎨ k dr9ijq : = on its way from node i ∈ V to node j ∈ V ⎪ ⎪ ⎪ ⎩0 else

dr11kijq

⎧ ⎪ ⎪ 1 if a daily rest period of at least 11 hours is taken by vehicle k ⎪ ⎨ : = on its way from node i ∈ V to node j ∈ V ⎪ ⎪ ⎪ ⎩0 else

taken by vehicle k on its way from node i ∈ V to node j ∈ V ⎪ ⎪ ⎪ ⎩0 else ⎧ ⎪ ⎪ 1 if vehicle k extends its daily driving time on its way from ⎪ ⎨ : = node i ∈ V to node j ∈ V ⎪ ⎪ ⎪ ⎩0 else

dr3kijq : =

bdrivekijq

⎧ ⎪ ⎪ 1 if a the first part of a daily rest period of at least 3 hours is ⎪ ⎨

52

4 Model for the VRPTW-EU

k ST Rijq : duration of vehicle k’s first part of a split daily rest period between node i ∈ V and node j ∈ V if j is reached at position q ∈ Q of its route

4.2.3 Model Formulation Using the above notation, a model for the VRPTW-EU with all optional rules can be formulated as follows. The objective of the VRPTW-EU again consists in minimizing the total fulfillment time of the vehicle routes (4.27).

M in f =



(tkreturn − tk0 )

(4.27)

k∈K

The model for the VRPTW-EU comprises the following restrictions.

  

k wijq = 1 ∀j ∈ P

(4.28)

i∈V q∈Q(i,j) k∈K



k w0j2 =

j∈P



k wisq =

i∈V \{n+1}





k win+1q ∀k ∈ K

(4.29)

i∈P q∈Q(i,n+1)



k wsj,q+1 ∀s ∈ P, q ∈ Q, k ∈ K

(4.30)

j∈V \{0}



 

k lj wijq ≤ C ∀k ∈ K

(4.31)

i∈V \{n+1} j∈P q∈Q(i,j)

Restrictions (4.28) to (4.31) are again the standard restrictions for VRPs in the position-based model formulation.

ttkij = dij +



k k (0.25 ∗ b15kijq + 0.5 ∗ b30kijq + T Rijq + ST Rijq )

(4.32)

q∈Q(i,j)

∀i ∈ V, j ∈ V, k ∈ K

tj ≥ ti + si + ttkij − M (1 −

 q∈Q(i,j)

k wijq ) ∀i ∈ P, j ∈ P, k ∈ K

(4.33)

4.2 Model for the VRPTW-EU with Optional Rules tj ≥ tk0 + si + ttk0j − M (1 −



53

k w0jq ) ∀j ∈ P, k ∈ K

(4.34)

k win+1q ) ∀i ∈ P, k ∈ K

(4.35)

q∈Q(0,j)

tkreturn ≥ ti + si + ttkin+1 − M (1 −

 q∈Q(i,n+1)

ti ≥ ai ∀i ∈ P

(4.36)

ti ≤ ei ∀i ∈ P

(4.37)

tk0 ≥ a0 ∀k ∈ K

(4.38)

tkreturn ≤ e0 ∀k ∈ K

(4.39)

Restrictions (4.32) to (4.39) model the travel times and time windows. Equation (4.32) defines travel times ttij such that in this model they consist of the driving times dij , the times required for 15-minute breaks b15kijq , for 30-minute breaks k b30kijk , the durations of daily rest periods T Rijq and the durations of the first k taken on these connections. Restrictions parts of split daily rest periods ST Rijq (4.33) to (4.39) are the same as in the model with basic rules.



q  

k dij wijq ∀q  , q  ∈ Q, q  ≤ q  , k ∈ K

(4.40)

q   k (dij + si )wijq ∀q  , q  ∈ Q, q ≤ q  , k ∈ K

(4.41)

DTqk q

=

q=q 

i∈V j∈V



W Tqk q =

q=q i∈V j∈V

Equations (4.40) and (4.41) again define the total driving and working times between the nodes at positions q  and q  . On the lowest time horizon considered by the European social legislation the following restrictions include the option of splitting breaks. These restrictions are now substantially different from and more complex than those in the model with basic rules.

54

4 Model for the VRPTW-EU 

DTqk q ≤ 4.5 + 4.5

q   q=q



b30kijq + M (

i∈V j∈V

q   q=q

dr9kijq + dr11kijq + dr3kijq )

i∈V j∈V

(4.42) ∀q , q  ∈ Q, q ≤ q  , k ∈ K



W Tqk q ≤ 6 + 6

q   q=q



b30kijq + M

q   q=q

i∈V j∈V

(dr9kijq + dr11kijq + dr3kijq )

i∈V j∈V

(4.43) ∀q , q  ∈ Q, q ≤ q  , k ∈ K



W Tqk q ≤ 9 + 9

q   q=q i∈V j∈V



b15kijq + M

q  

(dr9kijq + dr11kijq + dr3kijq )

q=q i∈V j∈V

(4.44) ∀q , q  ∈ Q, q ≤ q  , k ∈ K Restriction (4.42) considers the single driving times restricted by Regulation (EC) No 561/2006. It guarantees that for each partial route from the node at position q  to the node at position q  of vehicle k’s route for which the driving time exceeds 4.5 hours, a break of 30 minutes is taken since this second part of a split break ends the current driving period. However, if a daily rest period of at least 9 hours or a first part of a split rest period of at least 3 hours duration is taken on the considered partial route, i.e. if dr9kijq = 1, dr3kijq = 1 or dr11kijq = 1 for q  ≤ q ≤ q  , the restriction is relaxed such that no break has to be taken. The requirements for breaks according to the working time directive are modeled in Restrictions (4.43) and (4.44). In this model a break of at least 30 minutes is scheduled after no more than 6 hours of working time. Restriction (4.43) states that a working period must not exceed 6 hours unless a break is taken. However, if a driver takes a daily rest period or the first part of a split rest period, the restriction is relaxed. After 9 hours of working time an additional break of at least 15 minutes has to be taken according to Restriction (4.44). Breaks satisfying the

4.2 Model for the VRPTW-EU with Optional Rules

55

driving time regulation are also considered as breaks according to the working time directive.



q  



b30kijq −

q=2 i∈V j∈V

q  

b30kijq

(4.45)

q=2 i∈V j∈V



q  

b15kijq ≤ 0 ∀q  ∈ Q, k ∈ K

+ (1 −

q=2 i∈V j∈V





dr9kijq )

≥

q  

b15kijq

(4.46)

q=2 i∈V j∈V

i∈V j∈V

∀q ∈ Q, k ∈ K 

q  

b30kijq + (1 −





dr11kijq  ) ≥

q  

b15kijq

(4.47)

b15kijq

(4.48)

k ∀i, j ∈ V, q ∈ Q(i, j), k ∈ K b15kijq + b30kijq ≤ M ∗ wijq

(4.49)

q=2 i∈V j∈V

q=2 i∈V j∈V

i∈V j∈V

∀q ∈ Q, k ∈ K 

q   q=2 i∈V j∈V

b30kijq

+ (1 −





dr3kijq )

≥

i∈V j∈V

q   q=2 i∈V j∈V

∀q ∈ Q, k ∈ K

Restrictions (4.45) to (4.48) arrange the combination of 15- and 30-minute breaks to regular breaks satisfying the requirements of the driving time regulation. Restriction (4.45) makes sure that for each partial route starting at the depot and ending at the node at position q  , at least as many 15-minute breaks are taken as there are 30-minute breaks. Restrictions (4.46), (4.47) and (4.48) require that on each of these sub-routes only one 15-minute break more may be taken than the number of 30-minute breaks in order to prevent the model from using additional 15-minute breaks later on. Additionally, (4.46) to (4.48) make sure that the number of 15- and 30-minute breaks must be equal if the sub route ends with a daily rest period or a split daily rest period at position q  . This is required in order to prevent the model from planning the 15-minute part of a split break before a (partial) rest period and to count it as first part for a split

56

4 Model for the VRPTW-EU

break taken after the daily rest period. Restriction (4.46) accounts for reduced and split daily rest periods, (4.47) accounts for regular daily rest periods and (4.48) for first parts of split daily rest periods. Restriction (4.49) ensures that breaks can only be taken on arcs which are part of the vehicle’s route. The rules on daily driving times are modeled in Restrictions (4.50) to (4.60). Again, the restrictions required to capture the whole set of legal rules are much more complex than in the model with basic rules.



DT

q  q 



q q     ≤9+9 (dr9kijq + dr11kijq ) + 1 bdrivekijq q=q i∈V j∈V

(4.50)

q=q  i∈V j∈V

∀q  , q  ∈ Q, q < q  , k ∈ K  

bdrivekijq ≤ 2 ∀k ∈ K

(4.51)

dr9kijq ≤ 1 ∀i, j ∈ V, q  , q  ∈ Q, q  < q  , k ∈ K

(4.52)

dr11kijq ≤ 1 ∀i, j ∈ V, q  , q  ∈ Q, q  < q  , k ∈ K

(4.53)

i∈V j∈V q∈Q(i,j)



q 



bdrivekijq −

q=q 

q=q



q 

q 



bdrivekijq −

q=q 

q  q=q

k T Rijq ≥ 9 − M (1 − dr9kijq ) ∀i, j ∈ V, q ∈ Q(i, j), k ∈ K

k ≥ 11 − M (1 − dr11kijq ) ∀i, j ∈ V, q ∈ Q(i, j), k ∈ K T Rijq

k ≥ 3 − M (1 − dr3kijq ) ∀i, j ∈ V, q ∈ Q(i, j), k ∈ K ST Rijq



q  i∈V j∈V

q=q

(4.54)

(4.55)

(4.56)



dr9kijq ≤

q  i∈V j∈V

q=q 

dr3kijq + 3 ∀q  , q  ∈ Q, q  < q  , k ∈ K

(4.57)

4.2 Model for the VRPTW-EU with Optional Rules 

tj  − tk0 ≤ 24 + 24

q 

57 

(dr9kijq + dr11kijq ) + M (1 −

i∈V j∈V q=2

q 

k wij q )

(4.58)

i∈V q=2

∀j  ∈ P, q  ∈ Q, k ∈ K

tkreturn − tk0 ≤ 24 + 24

 

(dr9kijq + dr11kijq ) ∀k ∈ K

(4.59)

i∈V j∈V q∈Q(i,j)

k k k bdrivekijq + dr3kijq + dr9kijq + dr11kijq + T Rijq + ST Rijq ≤ M ∗ wijq

(4.60)

∀i, j ∈ V, q ∈ Q(i, j), k ∈ K Restrictions (4.50) to (4.60) formulate the conditions for the positioning and length of the daily rest periods. They include the possibilities of extending a daily driving time to ten hours and of using reduced and split daily rest periods. Reduced and split daily rests are modeled similarly since in both cases the 9-hour part marks the end of a daily driving time. The difference between the two is that in case of a split daily rest the 9-hour rest period has to be preceded by an additional partial rest of at least 3 hours. Restriction (4.50) requires that a driver may only drive for 9 hours unless a daily rest period is taken. For each reduced, split or regular daily rest period taken on a partial route, his driving time on this sub route may be extended by another 9 hours. The variable bdrivekijq allows the extension of daily driving times by one hour. This extension can only be taken twice in a weekly period which is guaranteed by (4.51). Restrictions (4.52) and (4.53) ensure that only one variable bdrivekijq may take the value 1 between two daily rest periods guaranteeing that the daily driving time cannot exceed 10 hours. As described in Chapter 2, regular daily rest periods have to last for at least 11 hours. Reduced rest periods have a minimum duration of 9 hours and split daily rest periods consist of two parts of which the first part has to last for at least 3 hours and the second part for at least 9 hours. Hence, both in the case of a reduced and a split daily rest period, the duration of the part ending the daily driving time is 9 hours. Therefore, both of them can be modeled using a variable dr9kijq indicating that a daily rest period of a duration of at least 9 hours has

58

4 Model for the VRPTW-EU

been taken on the arc (i, j) (see Equation (4.54)). In the case of a split daily rest period, the model has to provide that an additional 3-hour rest is taken before. Restriction (4.55) sets the minimum duration of regular daily rest periods to 11 hours. If drivers split their daily rest periods into two parts, Equation (4.56) sets the duration of the first part of a split daily rest period to at least 3 hours. Since drivers are allowed to reduce their daily rest periods up to three times a week without compensation, Equation (4.57) allows three daily rest periods of 9 hours to be taken without the corresponding 3 hours part of split daily rest periods. For every additional rest period of less than 11 hours, drivers have to take a first part of a split rest period of at least 3 hours. Restrictions (4.58) and (4.59) ensure that after no more than 24 hours after the end of the previous daily rest period a new daily rest period has to be taken. Finally, (4.60) allows driving time extensions and daily rest periods only on used connections. The legal rules on weekly driving and working hours are modeled in Restrictions (4.61) to (4.63). They are the same as in the basic model.  

k dij wijq ≤ 56 ∀k ∈ K

(4.61)

k k dij wijq ≤ 90 − W DTw−1 ∀k ∈ K

(4.62)

k (dij + si )wijq ≤ 60 − Δw ∀k ∈ K

(4.63)

i∈V j∈V q∈Q(i,j)

  i∈V j∈V q∈Q(i,j)

  i∈V j∈V q∈Q(i,j)

The Objective Function (4.27) and Restrictions (4.28) to (4.63) give a complete model for the VRPTW-EU including all basic and optional rules of Regulation (EC) No 561/2006 and of Directive 2002/15/EC.

4.3 Computional Experiments In this section the models for the VRPTW-EU are analyzed using computational experiments. In particular, the structure of the resulting weekly driver schedules is investigated.

4.3 Computional Experiments

59

4.3.1 Experimental Settings For testing the models and for deriving results relevant for real life decision making, test problems were created which allow for a realistic representation of the real problem. The test data for the VRPTW-EU presented by Goel (2009), which is based on the problem sets introduced by Solomon (1987), is not appropriate here since first problems with 100 customers cannot be solved to optimality, and second, their setting is too artificial. The customer nodes are located in a (100 x 100)-matrix and travel times are based on the assumption that each vehicle travels 5 distance units per hour. In order to have a more realistic setting, 22 test problems were created as follows: In a first step, potential customer locations were chosen within real cities in Germany. To obtain a wide geographical diversification, all capitals of the German Federal States were selected, i.e. Berlin, Bremen, Dresden, D¨ usseldorf, Erfurt, Hamburg, Hannover, Kiel, Magdeburg, Mainz, Munich, Potsdam, Saarbr¨ ucken, Schwerin, Stuttgart and Wiesbaden. Additionally, some large cities in North Rhine-Westphalia were included to allow for a clustering of the customer locations. These are Duisburg, Essen and Cologne. The depot was chosen to be in Frankfurt am Main since it is located in the center of Germany and it is also an important logistics center due to its international airport. This procedure resulted in 20 nodes of which 19 are possible customer locations. The distances between these nodes was calculated from their geographical coordinates. A vehicle is assumed to travel at an average speed of 65 km/h and the travel distances were converted into travel times accordingly. The demand of every node was generated as a random number between 1 and 10 and the vehicle capacity was restricted to 50. The service time at each customer node was set to 30 minutes and is regarded as working time for the drivers. Using these settings, two different problem sets were generated. The first set comprises 10 problem instances, test problems 1 to 10, for which the planning period was set to a whole working week. The planning period comprises the time from Monday 00:00 to Friday 24:00, i.e. 120 hours, and drivers have just finished a weekly rest period at the beginning of the planning period. To restrict the planning period to 5 days, the depot’s time window was set to a0 = 0 and e0 = 120. For this problem set, the customer time windows were generated following the approach of Solomon (1987): The center of each time window was chosen as a random number within the planning period. Subsequently, the half width of the time window was chosen as normally distributed random number.

60

4 Model for the VRPTW-EU

The second problem set consists of 12 test problems which comprise a shorter time horizon. For this problem set, problem numbers 11 to 22, only half a working week was considered. Therefore, the depot’s time window was set to a0 = 0 and e0 = 60. Again, the center of each customer’s time window was generated as a random number between 0 and 60 and half the width of the time window was generated as a normally distributed random number. For each test problem, the depot was located in Frankfurt and 10 customer nodes were chosen randomly. The number of vehicles available at the depot k max was set to 3. For each driver the allowed weekly driving and working time were initialized with their maximum values, i.e. 56 hours and 60 hours, respectively.

4.3.2 Computational Results for the VRPTW-EU The model for the VRPTW-EU with basic rules was implemented in OPL 6.3 and was solved using CPLEX 12.1 on a Core 2 Duo, 1.33 GHz CPU and 1.0 GB of RAM. To obtain a lower bound for the objective function, Restriction (4.64) was added to the model formulation. It states that the total fulfillment time has to be at least equal to the total time required for driving and service times, breaks and rest periods.



(tkreturn − tk0 ) ≥

 

k k (dij wijq + b45kijq ∗ 0.75 + T Rijq )

(4.64)

k∈K i∈V j∈V q∈Q(i,j)

k∈K

The results of the 22 test problems for the VRPTW-EU are reported in Table 4.1. For each test problem the table shows the total fulfillment time, the number of vehicles, the total driving time, the times required for breaks and for rest periods, the total waiting and service times. The CPU-times ranged between a minimum of 4:52 minutes (problem 21) and a maximum of 29:16 hours (problem 6).

Test

Total

No. of

Dr.

Breaks

Rests Wait.

Serv.

problem

dur.

vehi-

time

(h)

(h)

time

time

(h)

cles

(h)

(h)

(h)

1

82.67

3

40.27

2.25

33

2.1

5

2

102.21

3

50.15

2.25

44

0.80

5

3

101.36

3

44.96

4.5

33

13.90

5

4

117.80

3

51.64

3

55

3.15

5

4.3 Computional Experiments

61

5

121.25

3

57.54

3

55

0.69

6

90.89

3

45.55

2.25

33

5.09

5 5

7

112.93

3

50.68

2.25

55

0

5

8

96.69

3

41.02

3

33

14.67

5

9

125.17

3

60.67

4.5

55

0

5

10

80.61

3

40.36

2.25

33

0

5

11

102.56

3

47.87

4.5

22

23.18

5

12

92.26

3

43.00

3.75

33

7.51

5

13

80.34

3

38.95

3

33

0.38

5

14

106.43

3

47.21

3

44

7.22

5

15

104.33

3

50.98

3

44

1.35

5

16

66.48

3

28.38

2.25

22

8.85

5

17

105.65

3

50.91

3

44

2.73

5

18

105.79

3

52.44

4.5

33

10.85

5

19

105.32

3

51.84

3.75

33

11.73

5

20

107.69

3

49.05

3.75

44

5.88

5

21

60.47

2

31.22

2.25

22

0

5

22

59.45

2

30.20

2.25

22

0

5

Table 4.1: Results for the VRPTW-EU with basic rules

A first remarkable finding is that for the first problem set (problem 1 to 10), in the optimal solution always all 3 vehicles are used although each problem can also be solved using only 2 vehicles. Therefore, in the first problem set it is obvious that by using more vehicles, the total fulfillment times can be reduced. For the second problem set, there are 2 out of 12 problems for which the optimal solution requires 2 instead of 3 vehicles (problems 21 and 22). However, the differences in the objective values when deploying 2 or 3 vehicles are very small (0.3% and 0.1%). In total, only 4 of these problems can be solved with only 2 vehicles (problems 13, 16, 21 and 22). Therefore, when solving the VRPTW-EU using the criterion of minimizing the total fulfillment time, there seems to be a tendency that the utilization of more vehicles leads to a reduction of the required fulfillment times. This result contrasts the results for standard VRPs, in which usually a positive correlation between the number of vehicles and the travel distance can be detected. However, when considering the total fulfillment time, driving times

62

4 Model for the VRPTW-EU

only account for 47.3% of the total fulfillment time on average, whereas breaks amount to 3.2% and rest periods to 38.4%. When driving a longer route, drivers may be forced to take additional breaks and rest periods. Utilizing an additional truck might result in a higher travel distance but probably allows to reduce the required number of breaks and rest periods. Therefore, in the VRPTW-EU the increased driving time can be compensated by a reduction of the times required for breaks and rest periods. The remaining fulfillment time is constituted by service times (5.4% on average) and waiting times (5.7% on average). Considering the productive times of a driver, i.e. the driving times and times for loading and unloading the vehicles, they only account for 52.7% of the total fulfillment time. This means that in the scenarios chosen here, drivers’ activities only account for about half of the total time the vehicle is on route. In this model formulation, breaks and rest periods are chosen such that the total fulfillment time is minimized. These experiments indicate that, since driving times only amount to about half of the actual fulfillment time, planners should not neglect breaks and rest periods which make up a substantial share of the fulfillment time. To further analyze this finding, the same 22 test problems were solved using the objective of minimizing the total driving time. The results are depicted in Table 4.2. Test

Total

No. of

Dr.

Breaks

Rests Wait.

Serv.

problem

dur.

vehi-

time

(h)

(h)

time

time

(h)

cles

(h)

(h)

(h)

1

178.82

2

34.44

0.75

66

72.63

5

2

258.69

3

45.93

2.25

88

117.50

5

3

264

3

36.75

2.25

88

131.99

5

4

237.08

2

43.74

2.25

88

98.08

5

5

240

2

48.78

2.25

88

95.96

5

6

187.5

2

34.06

1.5

66

80.93

5

7

252

3

45.92

2.25

88

110.82

5

8

211.46

3

39.10

2.25

66

99.11

5

9

252

3

49.92

6.75

88

102.32

5

10

141.79

2

38.04

12

55

31.74

5

11

169.88

3

45.61

2.25

55

62.01

5

12

195.80

3

42.31

3.75

77

67.74

5

13

164.34

3

36.28

1.5

44

77.55

5

14

180.47

3

47.21

2.25

55

71.01

5

4.3 Computional Experiments

63

15

198.07

3

50.29

7.5

66

69.27

5

16

144

2

27.39

0

44

67.60

5

17

180.70

3

45.44

1.5

55

73.76

5

18

240

3

50.06

9

77

98.93

5

19

169.48

3

43.66

3

55

62.82

5

20

142.68

3

47.17

10.5

44

36.01

5

21

94.47

2

30.49

1.5

44

13.48

5

22

125.98

2

29.53

1.5

44

45.94

5

Table 4.2: Results for criterion of minimum driving time

A first remarkable effect when using the new criterion lies in the reduction of the CPU-times. CPU-times were now between a minimum of 7 seconds (problem 1) and a maximum of 7 minutes (problem 2). Therefore, the problem seems to be much easier to solve using this rather simple criterion. In contrast with the previous results, Table 4.2 shows that half of the test instances of the first problem set now require only 2 vehicles. In the second problem set, one additional problem (problem 16) is now solved using only 2 vehicles. Therefore, applying the criterion of minimizing the driving time, the relationship between the objective criterion and the number of vehicles seems to be ambiguous. In this set of experiments, the driving times only amount to 22.4% and the service times only require about 2.8% of the total fulfillment times on average. The time required for breaks takes a share of about 2.0% and the rest periods take about 34.2% of the total fulfillment times. The rest of the time is spent as waiting time, which now amount to 38.6% of the total fulfillment times on average. This is more than the share of the drivers’ productive times, i.e. driving and service times, which equal only 25.2% of the fulfillment times. Compared with the results obtained by minimizing the total fulfillment time, here the fulfillment times increase enormously. The average fulfillment time almost doubles from 96.8 hours to 192.2 hours. However, the average driving time only decreases by about 9.2% from 45.6 hours to 41.4 hours. Waiting times increase enormously since they are no longer considered in the objective function. The average waiting time in the first scenario amounts to 5.5 hours on average, whereas the average waiting in this scenario is about 76.7 hours which is 13 times

64

4 Model for the VRPTW-EU

Figure 4.1: Vehicle schedules for test problem 5 more. The times required for breaks increase moderately by 15% and the times required for rest periods by 74% on average. The computational experiments clearly demonstrate that when creating vehicle schedules considering breaks and rest periods, the criterion of minimizing the travel distance, which is often encountered in the vehicle routing literature (see e.g. Vahrenkamp, 2007; Laporte, 2009), is limited. The resulting vehicle schedules are not practicable since unproductive times take a major share of the total fulfillment times. 4.3.3 Example of a Tour Plan To better illustrate the resulting vehicle and driver schedules, in the following the vehicle routes for test problem 5 are presented in detail. Table 4.1 shows that the total fulfillment time amounts to 121.3 hours. The total route duration of the first vehicle is 43.9 hours, of the second 47.9 hours and of the third 29.5 hours. It is assumed that the planning period starts on Monday 00:00. The schedules for the three vehicles are depicted in Figure 4.1. In this histogram, for each vehicle the sequence of the customers on its route and the corresponding start times of the drivers’ activities are depicted. For example, the first vehicle starts its route 3.6 hours after the beginning of the planning period, i.e. on Monday at 03:36, from the depot in Frankfurt (F). After a driving time of 5.1 hours and a break of 45 minutes along this route, it reaches the customer in Hamburg (HH) at 09:30, directly before the closing time of the time window. After half an hour of service time the vehicle continues its route to Duisburg (DU). Since the driving time from Hamburg to Duisburg equals 4.3 hours, the driver exceeds his maximum daily driving time of 9 hours. Therefore,

4.3 Computional Experiments

65

a rest period of 11 hours is scheduled along the route. Note that the driving time does not exceed 10 hours and that the customer could be reached if the optional rule of extending the daily driving time was included in the model. The vehicle arrives at the customer node at time 25.3, i.e. on Tuesday at 01:18. Since the opening of the time window is at 02:00, the driver has to wait for 42 minutes before starting its unloading operations. At 02:30 the vehicle continues its route towards Schwerin (SN). Since the time window at this customer node opens at time 42, i.e. on Tuesday at 18:00, the driver has to take another daily rest period due to the 24-hour rule. His last daily rest period has ended before Tuesday at 01:18, and after no more than 13 hours after the end of the previous rest period a new daily rest period has to be started. Therefore, a new daily rest has to be started the latest at time 38.3, i.e. on Tuesday at 14:18, which is before the opening of the time window. After servicing the customer on Tuesday from 19:36 until 20:06, the vehicle returns to the depot where it finishes its route on Wednesday at 03:30. The second vehicle’s route comprises the customer nodes Hannover (H), Bremen (HB), Kiel (KI) and Berlin (B). The third vehicle visits customers in Mainz (MZ), Saarbr¨ ucken (SB) and Dresden (DD). Their schedules can be read similar to the schedule of vehicle 1. This example demonstrates the vehicle routes and break schedules obtained by the dispatcher if he addresses all three partial planning tasks simultaneously. In the following, this central planning perspective is abandoned in favor of a more realistic distributed decision making perspective.

5 Distributed Decision Making in Combined Vehicle Routing and Break Scheduling After the analysis of the problem of combined vehicle routing and break scheduling from a centralized planning perspective, the problem is now enhanced by introducing decentralized decision structures. The overall planning problem comprises three interconnected partial problems: clustering of transportation requests, routing of vehicles and break scheduling. In practice, these tasks are usually not solved by a single decision maker, but are distributed over dispatchers and drivers (see Section 3.3). Dispatchers are responsible for the clustering of the requests, and in many cases also perform the routing. The task of break scheduling is always delegated to the drivers. Consequently, the problem is solved sequentially by two different decision makers and the planning situation is characterized by hierarchies in distributed decision making (DDM). When performing their planning tasks, the European social legislation forces dispatchers to provide that the drivers can comply with the legal rules (see Regulation, 2006, Article 10). Since planners and drivers are linked in their planning restrictions and the planners’ degree of economic success also depends on the drivers’ performance, the need for coordination of their decisions arises (Laux and Liermann, 2005). Therefore, dispatchers have to anticipate the drivers’ subsequent decisions in this sequential planning process. To reduce the complexity of this task, dispatchers might use approximations of the drivers’ planning models. To account for the above organizational structure, the problem of combined vehicle routing and break scheduling is now analyzed from a DDM perspective. The DDM framework developed by Schneeweiss (2003a,b) is applied to model the planning problem. Using distributed decisions structures and delegating decisions to lower planning levels is a common method for addressing complex problems (Laux and Liermann, 2005; Frese, 2005). Providing lower levels with more autonomy does not only help to reduce the overall complexity of a problem, but may also lead to a better accomplishment of the task (Windt and H¨ ulsmann, 2007). In combined vehicle routing and break scheduling, dispatchers always delegate the task of break scheduling to the drivers since this task can be better accomplished C. M. Meyer, Vehicle Routing under Consideration of Driving and Working Hours, DOI 10.1007/978-3-8349-6732-9_5, © Gabler Verlag | Springer Fachmedien Wiesbaden GmbH 2011

68

5 DDM in Combined Vehicle Routing and Break Scheduling

on the drivers’ planning level. For example, when planning the vehicle routes the dispatchers cannot foresee the availability of motorway stations at the time their plans are executed by the drivers. The complexity of the dispatchers’ planning task can be further reduced by providing the drivers with more decision competencies. Consequently, in the following conditions are investigated under which the dispatchers can delegate the task of routing to the drivers. The clustering should then again be performed by the dispatchers since this task requires decisions which affect several drivers. If this task was also delegated to the drivers, they would have to take decisions without being able to oversee their impact on other decision makers. The delegation of decision competencies provides drivers with more autonomy. Since delegation of decisions leads to a loss of control for the central planning unit (Arndt and M¨ uller-Christ, 2007), this situation may give rise to a misapplication of the autonomy (Frese, 2005). To account for the possibility of antagonistic behavior, apart from a situation in which dispatchers and drivers work as a team, a non-team situation is investigated in which the drivers deviate from the dispatchers’ objectives. The aim is to derive strategies for the planners to cope with this principal-agent situation. The chapter is structured as follows: Based on an introduction of the DDM framework, the problem of combined vehicle routing and break scheduling is modeled from a DDM perspective. Different degrees of anticipation are proposed, resulting in different planning models for the dispatchers. Furthermore, a principal-agent setting is investigated. Finally, the effects of different degrees of anticipation are analyzed using computational experiments.

5.1 The Distributed Decision Making Framework This section provides an introduction to the framework for DDM suggested by Schneeweiss (2003a,b) which is applied for further analysis. This DDM concept was first developed as a framework for hierarchical planning by Schneeweiss (1992, 1994, 1995). This framework has already been successfully applied to such diverse areas as repair process planning (Schneeweiss and Schr¨oder, 1992), production planning (Carravilla and de Sousa, 1995; Gfrerer and Z¨ apfel, 1995), supply chain management (Zimmer, 2001; Schneeweiss and Zimmer, 2004), strategic network planning (H¨onscheidt and Sucky, 2007), capacity coordination (Fransoo et al., 1995), job-shop scheduling (Paulli, 1995), lot sizing (S¨ ohner and Schneeweiss,

5.1 DDM Framework

69

1995), resource planning (Pesenti, 1995), managerial accounting (Eichin and Schneeweiss, 2001), financial planning (Goedhart and Spronk, 1995), budget allocation (Corbett et al., 1995), organizational design (Joernsten and Leisten, 1995), and contract design (Schenk-Mathes, 1995). It was enhanced to a framework for DDM by Schneeweiss (2003a,b). In the following, this framework is used to model the problem of combined vehicle routing and break scheduling with distributed decision structures. Distributed decision making is defined as “the design and coordination of connected decisions” (Schneeweiss, 2003a). As in practice the decisions in combined vehicle routing and break scheduling are usually distributed over several decision makers, this framework can be applied to analyze the division of tasks between dispatchers and drivers.

5.1.1 Classification of Distributed Decision Making Systems There are several types of DDM systems which can be classified according to the number of decision makers involved, the existence of a team situation and the conflict potential between the decision makers (Schneeweiss, 2003b). This classification is depicted in Figure 5.1. In DDM situations decisions can be taken either by one decision making unit (DMU) or by several DMUs. In case there is only one DMU, the decisions are usually taken at different points in time or on different levels of aggregation in order to simplify the planning task. Consequently, artificial hierarchies are introduced into the planning problem in order to reduce the computational effort for solving it (Schneeweiss, 1995). It is clear that situations with only one DMU are conflict-free since there are no conflicting objectives. For several DMUs team and non-team situations have to be distinguished. A team is a group of persons, working together in an organization in order to achieve a common goal (Laux and Liermann, 2005). Team situations are characterized by complementary or at least not conflicting objectives of the DMUs. Therefore, these situations are also conflict-free. In contrast, in non-team situations the DMUs follow their own conflicting goals. In enforced team situations one DMU has the power to make the others follow its instructions. If no DMU can force the others to adhere to its instructions, incentives can be offered to align the other parties with one party’s goals. This is typically the case for principal agent settings where the leading DMU, the so called principal, takes a single coordinating decision first. Knowing the principal’s decision, the other DMUs, the so called agents, take

70

5 DDM in Combined Vehicle Routing and Break Scheduling

Figure 5.1: Classification of DDM systems (Schneeweiss, 2003a) their decisions following their own private objectives. However, as the principal’s objective function is dependent on the agents’ actions, the principal can try to influence the agents’ behavior by offering them incentives to act in a way that is favorable for the principal. In the case that several coordinating decisions are taken over time, the conflict can be resolved by negotiations between the DMUs. These cases may result either in proper conflict resolutions where at least one DMU adjusts its objectives, or in a compromise, which allows all DMUs to maintain their preferences. Finally, a number of other game theoretic settings are possible.

5.1.2 Coordination in Distributed Decision Systems In the following only DDM systems with two decision levels are considered. Since the relationship between dispatchers and drivers investigated in this thesis is characterized by a hierarchical structure due to the DMUs’ different decision competencies, the presentation is confined to hierarchical settings. In these settings the levels are referred to as top-level and base-level. In this context both levels are integrated through planning activities (Schneeweiss, 2003a). Both levels are characterized by their decision model M which can be described by its decision field A representing possible actions to be taken, by its preference criterion C and the respective information status I (Schneeweiss, 2003b). Therefore, the top-level’s decision model can be written as M T = M T (C T , AT , ItT0 ).

5.1 DDM Framework

71

Figure 5.2: Coordination in hierarchical systems (Schneeweiss, 2003b) The base-level’s decision model has the form M B = M B (C B , AB , ItB1 ) where t0 represents the time at which the top-level takes its decision and t1 represents the time of the base-level’s decision, with t0 < t1 . The two levels are coupled by a bottom-up influence called anticipation and by a top-down influence called instruction. The structure of a hierarchical DDM system is depicted in Figure 5.2 in which the different states of information are neglected for the sake of simplicity. In this hierarchical planning setting the top-level takes its decision first and communicates its instruction to the base-level. When making its decision it can anticipate the base-level’s expected reaction. This is done by the so called anticipation function AF (IN ). To model the base-level’s behavior, the top-level applies a planning model Exp(M B ) which it expects the base-level to use. Using this model the toplevel calculates the expected optimal decision a ˆB∗ of the base-level as a reaction to the instruction IN = IN (aT ) the top-level deduces from its decision aT . In this way the top-level derives its optimal top-decision aT ∗ ∈ AT and gives the resulting optimal instruction IN ∗ = IN (aT ∗ ) to the base-level. Having obtained the instruction IN ∗ , the base-level takes the decision aB∗ ∈ AB which best fulfills its criterion. This hierarchical coordination scheme can be modeled as follows: For the toplevel the optimal decision can be calculated by Equation (5.1).  

aT ∗ = arg optaT ∈AT E C T C T T (aT ), C T B (AF (IN (aT ))) |ItT0

(5.1)

The expression arg opt stands for the argument which optimizes the term and   E . . . |ItT0 stands for the mathematical expectation with respect to the infor-

72

5 DDM in Combined Vehicle Routing and Break Scheduling

mation status ItT0 . The top-level’s criterion C T can be decomposed into the two criteria C T T and C T B where C T T represents the top-level’s private criterion and C T B the top-down criterion which takes the base-level’s actions into account. Therefore, C T B depends on the anticipation function, C T B = C T B (AF ), whereas C T T is independent of the base level’s behavior. It is important to note that the top-down criterion C T B does not necessarily equal the base criterion C B (Schneeweiss, 1994). The top-level’s optimal decision aT ∗ results in the instruction IN ∗ = IN (aT ∗ ) which sets the frame for the base-level’s decision by affecting the base-level’s criterion, decision field and information status. Hence, the base equation can be written according to Equation (5.2).   B B B aB∗ = arg optaB ∈ABIN ∗ E CIN ∗ (a )|IIN ∗ ,t 1

(5.2)

In this hierarchical planning context no differentiation between the base-level’s private criterion and a bottom-up criterion is required since no further decision by the top-level is taken which has to be anticipated by the base-level. The Equations (5.1) and (5.2) describe the general hierarchical planning system. The top-down influence is taken into account by the top-level’s instruction IN ∗ . The bottom-up influence is given by the anticipation function AF (IN ) which enters the top-level’s criterion. This anticipation function can be stated as in Equation (5.3).

B B (ˆ aB )|IˆIN AF (IN ) = aˆB∗ (IN ) = arg optaˆB ∈AˆB E CˆIN IN

(5.3)

B is the expected information status the top-level The information status IˆIN expects in t = t0 that the base-level will have in t = t1 . The Equations (5.1) to (5.3) are called coupling equations (Schneeweiss, 2003b).

The Anticipation Function (5.3) is an approximation of the expected base-level’s planning model used by the top-level. Schneeweiss (2003b) distinguishes between four different degrees of anticipation: perfect explicit anticipation, approximate explicit anticipation, implicit anticipation and non-reactive anticipation. The first three are also referred to as reactive anticipation since they consider the base-level’s behavior via some sort of anticipation function whereas non-reactive anticipation does not utilize any anticipation function. Perfect explicit anticipation: The planning model of the base-level is known

5.2 Vehicle Routing as DDM Problem

73

exactly to the top-level and is used without any approximations. Approximate explicit anticipation: The base-level’s planning model is taken into account only approximately, e.g. by making simplifying assumptions. Implicit anticipation: Only some features of the base-level are taken into account. The anticipation function does no longer explicitly describe the base-levels decision model. Non-reactive anticipation: No anticipation function is utilized but some general features of the base-level may be taken into account in the top level’s objective function. In the following section the problem of combined vehicle routing and break scheduling is embedded into this framework for DDM.

5.2 The Problem of Combined Vehicle Routing and Break Scheduling as a Problem of Distributed Decision Making After introducing the DDM framework of Schneeweiss (2003a,b), this framework is now applied to analyze the problem of combined vehicle routing and break scheduling under consideration of distributed decision structures.6 In the following, the planning problem is described from a DDM perspective and the modeling assumptions are outlined.

5.2.1 Distribution of Tasks between Planners and Drivers In combined vehicle routing and break scheduling, two groups of decision makers are considered, namely planners and drivers. For simplicity, in the following a strictly hierarchical system (Schneeweiss, 1991) is assumed in which each driver only receives instructions from one dispatcher and one single dispatcher is responsible for coordinating the fulfillment of all customer requests. The dispatcher takes his decisions first, and the drivers have to perform their planning within the frame set by the dispatcher’s decisions. Therefore, within the above DDM framework, the dispatcher represents the top-level and the drivers can be regarded as the base-level. 6

This section is based on Meyer et al. (2010).

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5 DDM in Combined Vehicle Routing and Break Scheduling

In line with Chapter 4, a distribution problem is considered in which all freight to be delivered to the customers is located at a central depot. Nevertheless, the models can easily be adapted to different transportation problems. An organizational structure is investigated in which the dispatcher is responsible for the clustering of customer requests and can also suggest routes to the drivers. The final decision on the route to be taken and also the scheduling of breaks and rest periods are delegated to the drivers. In this stage of the planning process, all customer requests have to be fulfilled using the company’s own fleet. The dispatcher’s task is to cluster the customer requests such that each request is assigned to exactly one customer cluster CLk . This task can be regarded as an assignment problem in which customer requests have to be assigned to vehicles such that the vehicles’ capacity is not exceeded. Therefore, the set of all customer clusters which comply with the vehicles’ capacity restriction constitutes the dispatcher’s decision field AT . In this study, the dispatcher strives to achieve the freight forwarding company’s objectives. A company’s objectives in vehicle routing are usually related to the overall business goal of minimizing costs (Vahrenkamp, 2007). To support this objective, the dispatcher tries to minimize the total fulfillment costs for all customer orders. These fulfillment costs consist of fixed and variable costs. Fixed costs arise for example from the deployment of trucks whereas variable costs may arise from the costs for fuel and from depreciations for wear and tear. Hence, in the follow  ing the dispatcher’s criterion C T is decomposed accordingly, C T = C T T , C T B , such that both fixed and variable costs are accounted for. His primary objective, his private criterion C T T , is chosen to be the minimization of the number of vehicles required to service the customers. This objective supports the company’s objectives since by minimizing the number of vehicles, the tied-up capital for the fleet is minimized. Consequently, this objective contributes to the minimization of the fixed costs. As a secondary objective, the dispatcher tries to minimize the total travel distance which can be seen as a main driver of transportation costs (Daganzo, 2005). This objective is his top-down criterion C T B , which depends on the drivers’ planning, since the routes are chosen by the drivers. It clearly fosters the reduction of the variable costs of serving the customer requests by reducing vehicle depreciations and fuel consumption. Furthermore, distance based road tolls, which have been introduced e.g. in Germany, give a high priority to the objective of minimizing the travel distance (Rieck et al., 2007).

5.2 Vehicle Routing as DDM Problem

75

These two objectives followed by the dispatcher are widely used in the literature on vehicle routing (see e.g. Toth and Vigo, 2002a; Laporte, 2009), and they are common in hierarchical solution approaches for vehicle routing problems (see e.g. Marchetti Spaccamela et al., 1984). The dispatcher uses a lexicographical ordering of his criteria. This means that he first optimizes his primary objective, i.e. the number of vehicles. If this procedure does not yield a unique solution, his secondary objective, the travel distance, is minimized under the restriction that the first objective takes its optimal value (Laux, 2005). However, for the computation of the travel distance, the planner needs to anticipate the drivers’ routings since the solution of his assignment problem only comprises sets of customer requests and does not encompass the calculation of travel distances. Having derived the optimal clusters CL∗k for all vehicles k ∈ K with respect to the dispatcher’s objectives, each driver is instructed with the customers of his tour. Therefore, the dispatcher’s optimal instruction is IN ∗ = {CL∗k |k ∈ K}. Drivers whose duties are not required are instructed with an empty customer cluster. Based on this instruction, each driver performs the routing and break scheduling such that each customer is visited exactly once, the customer time windows are respected and the route complies with the European legislation on driving and working hours. These restrictions define the driver’s decision field AB . Each driver k has to solve a Traveling Salesman Problem with Time Windows and European Union Social Legislation (TSPTW-EU) for his customer cluster CL∗k . When deriving their routes and break schedules, the drivers always exploit the optional rules of the European social legislation in their planning process. The drivers may follow different objectives when performing their planning tasks. If they act according to the company’s objectives, which they are instructed to follow, their objective C B is to minimize the travel distance in order to reduce variable transportation costs. In the following, this situation is referred to as the team situation between planner and drivers. However, since the drivers can decide upon their routes autonomously, they might also follow their own private objectives, even if they are not in line with the company’s objectives. For example, instead of minimizing the travel distance, each driver might try to minimize his return time to the depot in order to maximize his leisure. This objective C B does not support the overall goal of reducing the transportation costs and can even be conflicting with the planner’s objectives. Since the purpose of delegating decisions to the drivers is to unburden

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5 DDM in Combined Vehicle Routing and Break Scheduling

Figure 5.3: DDM problem with strictly hierarchical structure the dispatcher, it is assumed that the drivers’ performance is not permanently monitored. Therefore, at least comparably small increases in the travel distances are not observed by the dispatcher. This antagonistic setting is referred to as the non-team situation. The coordination structure between the dispatcher and the drivers described so far is depicted in Figure 5.3.

5.2.2 Anticipation of the Drivers’ Planning In order to avoid creating customer clusters which are infeasible with respect to the drivers’ planning, the dispatcher can account for the drivers’ planning process via an anticipation function AF (IN ). Infeasibilities may result from the presence of time windows which have not yet been considered by dispatcher. When generating the customer clusters, the dispatcher can account for the drivers’ planning by anticipating the routing and break scheduling that will be performed subsequently. The anticipation of the drivers’ routing also allows the dispatcher to estimate his top-down criterion C T B , the travel distance. In all cases the dispatcher assumes a team situation, i.e. he uses the minimization of the travel distance within each customer cluster as the anticipated drivers’ criterion Cˆ B . The dispatcher also suggests the routes derived by him to the drivers. However, the drivers can deviate from the dispatcher’s advice. Since the overall planning problem is very complex, dispatchers might not anticipate the drivers’ exact planning model but rather use some simplified approach by relaxing certain model features. In the following, three different degrees of anticipation are proposed and analyzed according to their impact on the resulting vehicle schedules (see Meyer et al., 2009). Only reactive anticipation functions are suggested. Non-reactive anticipation does not seem reasonable in this case, since the planner’s top-down criterion can only be computed through the utilization of

5.2 Vehicle Routing as DDM Problem

77

an anticipation function which estimates the travel distance. Perfect explicit anticipation is modeled such that the dispatcher considers the full planning model used by the drivers without any relaxations. In this case, the dispatcher expects each driver to solve a full TSPTW-EU including the optional legal rules. By anticipating this driver model, the dispatcher is able to instruct customer clusters that allow the drivers to find feasible routes and break schedules. Moreover, if he solves the problem to optimality, in the case of a team situation the resulting vehicle routes are optimal with respect to the drivers’ criterion of minimizing the travel distance. Therefore, the drivers cannot improve the routes considering this criterion. This situation is consequently used as the benchmark scenario. In the case of a non-team situation however, it might be possible for the drivers to find better routes with respect to their hidden private criterion, the minimization of their return time. In case of approximate explicit anticipation, the planner simplifies the drivers’ planning model. A reasonable relaxation is to neglect the optional rules of the European social legislation. By including only the basic legal rules, the dispatcher can still guarantee that the clusters he derives allow for feasible routes and break schedules. Nevertheless, his planning task is facilitated by neglecting the complex set of optional rules. In this case the dispatcher’s anticipated base model is reduced to a TSPTW-EU with basic rules. Note that the drivers still exploit the optional legal rules in their planning process. Since the inclusion of the optional rules increases the solution space, the drivers might find routes with shorter travel distances than the dispatcher and improve the dispatcher’s top-down criterion in the case of a team situation. In the non-team situation, the drivers might also improve the routes suggested by the planner regarding their private criterion. Implicit anticipation means that the planner only considers some features of the base model. Therefore, this planning approach is modeled such that the dispatcher does not explicitly consider the task of break scheduling and only models the drivers’ routing task which he needs to calculate his top-down criterion. Since the dispatcher wants to account for breaks and rest periods implicitly, he includes slack travel time which can be used by the drivers to schedule breaks and rest periods. A reasonable method for including slack travel time which is proportional to the travel distance is to use a lower velocity than the average travel speed (see e.g. Bartodziej et al., 2009). Consequently, the anticipated base model is a basic Traveling Salesman Problem with Time Windows (TSPTW) in which a speed reduction is utilized. Applying this relaxation of the drivers’ model does no longer guarantee the feasibility of the resulting customer clusters.

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5 DDM in Combined Vehicle Routing and Break Scheduling

Figure 5.4: DDM problem in combined vehicle routing and break scheduling The planning process in this hierarchical DDM system described so far is depicted in Figure 5.4.

5.3 Model Formulations In the following a mathematical model formulation for the DDM system depicted in Figure 5.4 is derived. Since the drivers’ model is crucial for the understanding of the dispatcher’s anticipation functions, it is presented first. Second, the dispatcher’s problem of clustering the customer requests is modeled. Finally, the anticipation functions of the drivers’ planning models are presented.

5.3.1 Driver’s Model As described above, each driver is instructed with the set of customers he has to visit. Within his customer cluster CLk , the planning problem of vehicle k  ’s driver can be modeled as a TSPTW-EU including all optional rules of the European social legislation. It is assumed that the drivers always exploit the optional legal rules in their planning process.7 For modeling the driver’s planning problem, the following additional notation is required: CL0k := CLk ∪ {0} CLn+1 k := CLk ∪ {n + 1} 7

The model for the TSPTW-EU is based on Kopfer and Meyer (2009).

5.3 Model Formulations

79

CL0,n+1 := CLk ∪ {0, n + 1} k The driver’s model M B (C B , AB ) in the case of a team situation can be stated as follows: C B : M in







dij wijq

(5.4)

q∈Q(i,j) i∈CL0k j∈CLn+1 k

In the team situation the driver’s criterion C B corresponds with the planner’s top-down criterion. The driver chooses his route and break schedule such that his travel distance is minimized (Equation (5.4)). The driver’s decision space AB is determined by the following Restrictions (5.5) to (5.34). Apart from the negligence of index k, the decision variables are the same as in Chapter 4. 

w0j2 = 1

(5.5)

j∈CLk







i∈CL0k

q∈Q(i,j)

wisq =

i∈CL0k

wijq = 1 ∀j ∈ CLn+1 k



wsj,q+1 ∀s ∈ CLk , q ∈ Q

(5.6)

(5.7)

j∈CLn+1 k

Equation (5.5) forces the vehicle to leave the depot, (5.6) states that each customer in the cluster CLk has to be visited exactly once, and (5.7) defines the positions at which a customer is visited along the route.

ttij = dij +



(0.25 ∗ b15ijq + 0.5 ∗ b30ijq + T Rijq + ST Rijq )

(5.8)

q∈Q(i,j)

∀i, j ∈ CL0,n+1 k

tj ≥ ti + si + ttij − M (1 −



wijq ) ∀i ∈ CL0k , j ∈ CLn+1 k

(5.9)

q∈Q(i,j)

ti ≥ ai ∀i ∈ CL0,n+1 k

(5.10)

80

5 DDM in Combined Vehicle Routing and Break Scheduling ti ≤ ei ∀i ∈ CL0,n+1 k

(5.11)

Restrictions (5.8) to (5.11) again model the time window restrictions. The variables t0 and tn+1 denote the depot’s time windows and define the planning time horizon for the TSPTW-EU. On the lowest time horizon, the driver has to schedule breaks according to the driving and working time regulations. This is modeled in Restrictions (5.12) to (5.21). 

DTq q =

q   q=q

q   q=q 



(5.12)

(dij + si )wijq ∀q  , q  ∈ Q, q < q 

(5.13)

i∈CL0k j∈CLn+1 k



DTq q ≤ 4.5 + 4.5

dij wijq ∀q  , q  ∈ Q, q  < q 

i∈CL0k j∈CLn+1 k



W Tq q =



q  





b30ijq + M

q=q  i∈CL0  j∈CLn+1 k k

q  



(dr9ijq (5.14)

q=q i∈CL0  j∈CLn+1 k k

+dr11ijq + dr3ijq ) ∀q  , q  ∈ Q, q ≤ q 



W Tqk q ≤ 6 + 6

q  





b30kijq + M

q=q i∈CL0  j∈CLn+1 k k

q  



(dr9ijq

(5.15)

(dr9ijq

(5.16)

q=q i∈CL0  j∈CLn+1 k k

+dr11ijq + dr3ijq ) ∀q  , q  ∈ Q, q ≤ q 



W Tqk q ≤ 9 + 9

q  





b15ijq + M

q=q i∈CL0  j∈CLn+1 k k

q  



q=q i∈CL0  j∈CLn+1 k k

+dr11ijq + dr3ijq ) ∀q  , q  ∈ Q, q ≤ q  

q  



q=2 i∈CL0  j∈CLn+1 k k



b30ijq −

q  



q=2 i∈CL0  j∈CLn+1 k k

b15ijq ≤ 0 ∀q  ∈ Q

(5.17)

5.3 Model Formulations 

q  



81

b30ijq + (1 −

q=2 i∈CL0  j∈CLn+1 k k







dr9ijq ) ≥

i∈CL0k j∈CLn+1 k

q  



b15ijq

q=2 i∈CL0  j∈CLn+1 k k

(5.18) ∀q ∈ Q



q  





b30ijq + (1 −

q=2 i∈CL0  j∈CLn+1 k k





dr11ijq ) ≥

q  

i∈CL0k j∈CLn+1 k



b15ijq

q=2 i∈CL0  j∈CLn+1 k k

(5.19) ∀q ∈ Q



q  



b30ijq + (1 −

q=2 i∈CL0  j∈CLn+1 k k







dr3ijq ) ≥

i∈CL0k j∈CLn+1 k

q  



b15ijq

q=2 i∈CL0  j∈CLn+1 k k

(5.20) ∀q ∈ Q

, q ∈ Q(i, j) b15ijq + b30ijq ≤ M ∗ wijq ∀i, j ∈ CL0,n+1 k

(5.21)

On the intermediate time horizon, Restrictions (5.22) to (5.31) model the planning of daily driving times and rest periods.



DTq q ≤ 9 + 9

q  





(dr9ijq + dr11ijq ) +

q=q  i∈CL0  j∈CLn+1 k k

q  



1 ∗ bdriveijq

q=q  i∈CL0  j∈CLn+1 k k

(5.22) ∀q , q  ∈ Q, q < q  





q∈Q(i,j) i∈CL0k j∈CLn+1 k

bdriveijq ≤ 2

(5.23)

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5 DDM in Combined Vehicle Routing and Break Scheduling 

q 



bdriveijq −

q=q 

dr9ijq ≤ 1 ∀i, j ∈ CL0,n+1 , q  , q  ∈ Q, q  < q  k

(5.24)

q=q



q 

q 



bdriveijq −

q=q 

 i∈CL0k

q 

dr11ijq ≤ 1 ∀i, j ∈ CL0,n+1 , q  , q  ∈ Q, q < q  , k ∈ K (5.25) k

q=q

T Rijq ≥ 9 − M (1 − dr9ijq ) ∀i, j ∈ CL0,n+1 , q ∈ Q(i, j) k

(5.26)

T Rijq ≥ 11 − M (1 − dr11ijq ) ∀i, j ∈ CL0,n+1 , q ∈ Q(i, j) k

(5.27)

, q ∈ Q(i, j) ST Rijq ≥ 3 − M (1 − dr3ijq ) ∀i, j ∈ CL0,n+1 k

(5.28)



q   j∈CLn+1 k

q=q

dr9ijq ≤





q  

i∈CL0k

tj − t0 ≤ 24 + 24

j∈CLn+1 k



dr3ijq +3 ∀q  , q  ∈ Q, q < q  (5.29)

q=q



q  

(dr9ijq + dr11ijq )

(5.30)

q=2 i∈CL0k j∈CLn+1 k  ∀j ∈ CLn+1 k , q ∈ Q

bdriveijq + dr3ijq + dr9ijq + dr11ijq + T Rijq + ST Rijq ≤ M ∗ wijq

(5.31)

∀i, j ∈ CL0,n+1 , q ∈ Q(i, j) k On the highest planning time horizon, weekly driving times have to be planned by the driver according to Restrictions (5.32) to (5.34). 





q∈Q(i,j) i∈CL0k j∈CLn+1 k

dij wijq ≤ 56

(5.32)

5.3 Model Formulations 



83 

dij wijq ≤ 90 − W DTw−1

(5.33)

(dij + si )wijq ≤ 60 − Δw

(5.34)

q∈Q(i,j) i∈CL0k j∈CLn+1 k







q∈Q(i,j) i∈CL0k j∈CLn+1 k

The Objective Function (5.4) and Restrictions (5.6) to (5.34) give a full description of the driver’s planning model in a team situation. In the case of antagonistic behavior, the driver uses his private criterion C B , which is to minimize his return time. Since the driver’s decision space remains unchanged, the driver’s planning model M B (C B , AB ) in a non-team situation can be formulated using Criterion (5.35) and again Restrictions (5.6) to (5.34). C B : M in tn+1

(5.35)

5.3.2 Dispatcher’s Model The dispatcher’s problem of clustering the customer requests can be modeled as an assignment problem with capacity restrictions. The following new variables are introduced: ⎧ ⎨1 if customer i is served by vehicle k xik : = ⎩0 else ⎧ ⎨1 if vehicle k is used zk : = ⎩0 else The dispatcher’s top-model M T (C T T , AT ) can now be formulated by Equations (5.36) to (5.38).

C T T : M in



zk

(5.36)

k∈K

The planner’s private criterion C T T is the minimization of the number of vehicles required to service all customer requests (Equation (5.36)). The planner’s decision space AT is constituted by Restriction (5.37), which states that each customer has to be assigned to one vehicle, and by Restriction

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5 DDM in Combined Vehicle Routing and Break Scheduling

(5.38), which makes sure that if a customer is assigned to a vehicle, the vehicle is also utilized, and its capacity restriction is respected. 

xik = 1 ∀i ∈ P

(5.37)

k∈K

Czk ≥



li xik ∀k ∈ K

(5.38)

i∈P

After deriving the customer clusters CLk ⊆ {1, ..., n} for each vehicle k, the planner passes them on to the corresponding drivers who constitute the base-level. Therefore, the instruction equals the customer clusters, i.e. IN ∗ = {CL∗k |k ∈ K}, with CL∗k = {i ∈ P |x∗ik = 1} and x∗ik the optimal assignments in (5.36) - (5.38). Note that the top-model does not account for drivers’ routing and break scheduling so far. In the presence of time windows, M T may result in infeasible clusters for the drivers. This danger can only be circumvented if the dispatcher anticipates the drivers’ subsequent planning. However, if the unloading operations are not time constrained, this simple top-model always allows the drivers to derive feasible routes and break schedules. 5.3.3 Anticipation Functions In this section, the anticipation functions AF (IN ) of the driver’s planning model are formulated. The term “function” is appropriate here since the driver’s routes and therefore also the realization of the dispatcher’s top-down criterion C T B are dependent on his instructions IN = {CLk |k ∈ K}. The dispatcher always assumes a team situation and expects the drivers to follow the company’s objectives. This means that the anticipated base model ˆ B (Cˆ B , AˆB ), such that Cˆ B = C B . The used by dispatcher is Exp(M B ) = M anticipation functions therefore only differ in the decision space AˆB anticipated by the planner. 5.3.3.1 Perfect Explicit Anticipation In the case of perfect explicit anticipation, the dispatcher anticipates the driver’s planning model exactly and uses it without relaxations. Hence, he considers each driver to solve a TSPTW-EU including the optional legal rules. His anticipated ˆ B therefore equals the driver’s model M B presented in Section 5.3.1. base-model M

5.3 Model Formulations

85

5.3.3.2 Approximate Explicit Anticipation In the case of approximate explicit anticipation, the dispatcher utilizes a TSPTWEU considering only the basic rules of the European social legislation as the anticipated base model Exp(M B ). Again, the anticipated base criterion Cˆ B is the minimization of the travel distance (Equation (5.39)). 

Cˆ B : M in





dij wijq

(5.39)

q∈Q(i,j) i∈CL0k j∈CLn+1 k

The anticipated decision space AˆB is constituted by Restrictions (5.40) to (5.58), which form the restrictions for a TSPTW-EU with basic rules. 

w0j2 = 1

(5.40)

j∈CLk







i∈CL0k

q∈Q(i,j)



wisq =

i∈CL0k

wijq = 1 ∀j ∈ CLn+1 k

(5.41)

wsj,q+1 ∀s ∈ CLk , q ∈ Q

(5.42)

(0.75 ∗ b45ijq + T Rijq )

(5.43)

j∈CLn+1 k

ttij = dij +

 q∈Q(i,j)

∀i, j ∈ CL0,n+1 k

tj ≥ ti + si + ttij − M (1 −



wijq ) ∀i ∈ CL0k , j ∈ CLn+1 k

(5.44)

q∈Q(i,j)



DTq q =

ti ≥ ai ∀i ∈ CL0,n+1 k

(5.45)

ti ≤ ei ∀i ∈ CL0,n+1 k

(5.46)

q  



q=q i∈CL0  j∈CLn+1 k k

dij wijq ∀q  , q  ∈ Q, q  ≤ q 

(5.47)

86

5 DDM in Combined Vehicle Routing and Break Scheduling 

W Tq q =

q  



(dij + si )wijq ∀q , q  ∈ Q, q  ≤ q

(5.48)

q=q  i∈CL0  j∈CLn+1 k k



DTq q ≤ 4.5 + 4.5

q  





b45ijq + M

q=q i∈CL0  j∈CLn+1 k k

q  



drijq

(5.49)

q=q i∈CL0  j∈CLn+1 k k

∀q  , q  ∈ Q, q  ≤ q 



W Tq q ≤ 6 + 6

q  





b45ijq + M

q=q i∈CL0  j∈CLn+1 k k

q  



drijq

(5.50)

q=q  i∈CL0  j∈CLn+1 k k

∀q  , q  ∈ Q, q  ≤ q 

b45ijq ≤ M ∗ wijq ∀i, j ∈ CL0,n+1 , q ∈ Q(i, j) k 

q   DTq q ≤ 9 + 9(



drijq )

(5.51)

(5.52)

q=q  i∈CL0  j∈CLn+1 k k

∀q , q  ∈ Q, q  < q 

T Rijq ≥ 11 − M (1 − drijq ) ∀i, j ∈ CL0,n+1 , q ∈ Q(i, j) k

tj − t0 ≤ 24 + 24



(5.53)



q  

drijq

(5.54)

q=2 i∈CL0k j∈CLn+1 k  ∀j ∈ CLn+1 k , q ∈ Q

drijq + T Rijq ≤ M ∗ wijq ∀i, j ∈ CL0,n+1 , q ∈ Q(i, j) k 





q∈Q(i,j) i∈CL0k j∈CLn+1 k

dij wijq ≤ 56

(5.55)

(5.56)

5.3 Model Formulations 

87





dij wijq ≤ 90 − W DTw−1

(5.57)

(dij + si )wijq ≤ 60 − Δw

(5.58)

q∈Q(i,j) i∈CL0k j∈CLn+1 k







q∈Q(i,j) i∈CL0k j∈CLn+1 k

5.3.3.3 Implicit Anticipation In the case of implicit anticipation, the dispatcher implicitly anticipates the driver’s task of break scheduling. Consequently, he uses a simple TSPTW and neglects the exact modeling of restrictions on driving and working hours. However, he applies reduced driving times dij , with dij < dij , in order to include time buffers. The anticipated base criterion Cˆ B is the minimization of the travel distance (Equation (5.59)). 





i∈CL0k

j∈CLn+1 k

q∈Q(i,j)

Cˆ B : M in

dij wijq

(5.59)

The anticipated decision space AˆB is constituted by Restrictions (5.60) to (5.65), which represent the restrictions for a simple TSPTW in the position based model formulation. 

w0j2 = 1

(5.60)

j∈CLk





wijq = 1 ∀j ∈ CLn+1 k

(5.61)

i∈CL0k q∈Q(i,j)

 i∈CL0k

wisq =



wsj,q+1 ∀s ∈ CLk , q ∈ Q

(5.62)

j∈CLn+1 k

tj ≥ ti + si + dij − M (1 −



wijq ) ∀i ∈ CL0k , j ∈ CLn+1 k

(5.63)

q∈Q(i,j)

ti ≥ ai ∀i ∈ CL0,n+1 k

(5.64)

88

5 DDM in Combined Vehicle Routing and Break Scheduling ti ≤ ei ∀i ∈ CL0,n+1 k

(5.65)

5.4 Computational Experiments for the Decentralized Approaches The different approaches of anticipating the drivers’ planning behavior are now tested using computational experiments. For all three degrees of anticipation, both a team and a non-team situation are analyzed. The effects on the dispatcher’s and the drivers’ objectives are investigated using the 22 test problems presented in Chapter 4. Remember that problems 1 to 10 comprise a planning time horizon of a whole week (120 hours), whereas problems 11 to 22 comprise a time horizon of only 60 hours.

5.4.1 Experimental Settings and Solution Procedure In the DDM situation, the dispatcher takes his decision first. His problem can be modeled as an assignment problem with capacity restrictions, which is enhanced with further restrictions through the anticipation of each driver’s routing and break scheduling. In the case of perfect and approximate explicit anticipation, the dispatcher’s planning problem can be addressed by solving a VRPTW-EU in which he considers all drivers simultaneously. The solution of the VRPTWEU comprises both the assignment of customers to vehicles and the design of vehicle routes including break schedules. In the case of implicit anticipation, the planner can solve a VRPTW with a reduced travel speed. Thus, the dispatcher generates the customer clusters CL∗k , k ∈ K, and instructs each driver k with his cluster CL∗k . Furthermore, the planner communicates the routes he derives to the drivers. However, the drivers can deviate from the dispatcher’s suggestions if they find better routes with respect to their criterion. To create the customer clusters, the dispatcher has to solve a multi objective optimization problem in which his objectives are ordered lexicographically. The dispatcher’s primary objective is to minimize the number of vehicles required, his private criterion C T T . He uses his top-down criterion C T B of minimizing the travel distance to break ties if there are several solutions optimizing his private criterion. Since there is a clear priority for the objective of minimizing the number of vehicles, the problem can be solved iteratively by optimization in priority order.

5.4 Computational Experiments

89

This means that the main objective is optimized first, and in the next iteration the second objective is optimized, while the optimum value for the higher priority objective function is added as a constraint (Rosenthal, 1985). However, in this case the so called “ − Constraint M ethod”(Miettinen, 2008) in which one objective function is optimized and the others are converted into additional constraints for the optimization problem seems to be more efficient. It is already known from Chapter 4 that all test problems can be solved using at most 3 vehicles. Therefore, to consider the different objectives, it is tried to solve problems 1 to 22 using only 2 vehicles. Only if no feasible solution can be found, the number of vehicles is reset to 3 and the calculations are performed again. This solution procedure guarantees that the solutions are pareto optimal, which is a desirable feature for solutions of multi objective optimization problems (Branke et al., 2008). The primary objective is thus transformed into a constraint, and the minimization of the travel distance is chosen as objective function. This procedure of setting target values for all but one objective and including these target values as constraints when optimizing the remaining objective is a commonly used principle of multi objective optimization (Rosenthal, 1985). Having thus derived the optimal clusters CL∗k , the drivers’ planning model is solved within each customer cluster. For the case of perfect explicit anticipation, the drivers cannot improve the routes already generated by the planner in terms of travel distance. This is due to the fact that each route generated by the dispatcher using the above procedure is already optimal with respect to the travel distance. Therefore, in the team situation the drivers accept the routes generated by the dispatcher and do not need to perform a rerouting. For the team situation the case of perfect anticipation is considered as the benchmark for the other scenarios. In contrast with the case of perfect explicit anticipation, in the cases of approximate explicit anticipation and implicit anticipation, an improvement of the travel distance by the drivers might be possible. Consequently, a rerouting by the drivers within their customer clusters is performed using the drivers’ planning model. The results of these cases are compared with the case of perfect anticipation which constitutes the overall optimum of the dispatcher’s problem. In the non-team situation the dispatcher does not consider the drivers’ criterion correctly. Hence, a rerouting by the drivers might lead to improvements with respect to their private criterion for all different anticipation strategies. At the same time, the deviation from the routes suggested by the dispatcher might deteriorate the latter’s top-down criterion.

90

5 DDM in Combined Vehicle Routing and Break Scheduling

5.4.2 Computational Results 5.4.2.1 Perfect Explicit Anticipation The results for the 22 test problems in the case of perfect explicit anticipation are depicted in Table 5.1. The second column reports the dispatcher’s private criterion C T T , i.e. the number of vehicles required. The first problem set (problem 1 to 10) with a large planning time horizon requires only two vehicles for each test problem. Only one test problem with shorter planning horizon and denser time windows requires three vehicles (test problem 20). The total number of vehicles over all test problems equals 45. The third column reports the dispatcher’s topdown criterion C T B , the anticipated total travel distance for all vehicles. Since the drivers’ planning model is perfectly anticipated, this value equals the actual realization of the base-level’s objective value in the case of a team situation. The accumulated travel distance over all 22 test problems equals 55299.34 km. Columns 4 to 6 report the dispatcher’s anticipation of the drivers’ return times. These return times are calculated as the sum of the arrival time at the last customer on the driver’s route, the service time at this node and the travel time to the depot including breaks and rest periods. Test

CT T

problem

CT B

Return

Return

(km)

dr. 1

dr. 2

dr. 3

Return

1

2

2238.76

114.82

107.99

-

2

2

2655.64

116.99

118.87

-

3

2

2533.27

107.99

119.77

-

4

2

2836.12

119.77

107.99

-

5

2

2675.00

119.77

50.41

-

6

2

2214.01

119.37

109.49

-

7

2

2805.01

107.99

116.82

-

8

2

2550.43

109.49

119.77

-

9

2

2851.56

102.79

99.40

-

10

2

2350.69

107.99

77.19

-

11

2

2668.39

61.45

62.29

-

12

2

2433.25

52.16

46.17

-

13

2

2482.55

55.34

58.70

-

14

2

2617.10

63.81

32.90

-

15

2

2683.14

35.11

75.46

-

16

2

1780.70

52.16

71.00

-

17

2

2636.76

61.76

58.70

-

5.4 Computational Experiments

91

18

2

2842.56

46.00

63.05

-

19

2

2530.12

65.81

58.70

-

20

3

3066.51

57.00

44.63

66.49

21

2

1982.22

45.34

54.81

-

22

2

1919.68

73.46

30.98

-

Table 5.1: Dispatcher’s results for perfect explicit anticipation

In case of a team situation, the dispatcher perfectly anticipates the drivers’ planning model. Since the routes generated by the planner are optimal with respect to the travel distance, the drivers cannot improve the planner’s routes. Therefore, both the planner’s private criterion and his top-down criterion remain unchanged, and the drivers’ criterion, i.e. the sum of the travel distances, equals the planner’s top-down criterion. In the non-team situation, the drivers can perform a rerouting which possibly improves their return times. Therefore, for each cluster derived by the dispatcher, the drivers’ planning model has to be solved with the criterion of minimizing the return time. For each of the test problems 5, 7, 11, 13, 17, 20, 21 and 22, one driver can improve his private objective by choosing a different route than the one suggested by the dispatcher. For example, for test problem 5, the planner anticipates the second driver to return to the depot at time 50.4 and to use the route Frankfurt - Duisburg - Saarbr¨ ucken - Mainz - Frankfurt. This results in a minimum travel distance of 601.2 km. By changing the sequence of the customer locations into Frankfurt - Duisburg - Mainz - Saarbr¨ ucken - Frankfurt, the driver can improve his private criterion C B and finish his duty already at time 46.9. However, the new route results in an increased travel distance of 671.3 km. The changes in the total travel distances and in the return times for each test problem in which a rerouting is performed are reported in Table 5.2. Test

Change

Change

problem

in

in

dist.

ret.

(%)

time (%)

5

2.5

-6.9

7

17.6

-8.7

11

0.3

-15.1

13

1.0

-21.7

92

5 DDM in Combined Vehicle Routing and Break Scheduling 17

0.5

-15.5

20

17.7

-52.8

21

1.5

-4.8

22

2.0

-13.3

Table 5.2: Results for the non-team situation

The average deterioration of the dispatcher’s top-down criterion, the total travel distance, equals 5.4% for the eight test problems in which a rerouting is performed by the drivers. The overall deterioration of the dispatcher’s top-down criterion over all 22 test problems amounts to 1.9%. At the same time, the drivers can achieve an average improvement of 18.3% in their return times for the eight routes considered here. The average improvement over all 45 routes equals 3%.

5.4.2.2 Approximate Explicit Anticipation The results for the 22 test problems in the case of approximate explicit anticipation are depicted in Table 5.3. For seven out of 22 test problems (problems 1, 6, 13, 16, 20, 21 and 22), the dispatcher finds the same clusters as in the benchmark scenario of perfect explicit anticipation. These problems are emphasized with a * in the first column. Test

CT T

problem

CT B

Return

Return

(km)

dr. 1

dr. 2

dr. 3

Return

1∗

2

2238.76

114.82

107.99

-

2

2

2999.63

110.07

107.99

-

3

2

2572.44

119.77

107.99

-

4

2

2843.63

119.77

107.99

-

5

2

3171.15

110.07

119.77

-

6∗

2

2214.01

119.37

109.49

-

7

2

3224.31

108.87

107.99

-

8

2

2567.46

120

109.49

-

9

2

3363.41

114.32

101.19

-

10

2

2473.16

109.49

103.09

-

11

3

2965.1387 25.88

60.7

72

12

3

2750.22

66.41

43.92

31.88

5.4 Computational Experiments

93

13∗

2

2482.55

55.34

58.7

-

14

3

3068.71

60.47

26.76

54.81

15

3

3269.37

31.91

54.07

70.77

16∗

2

1780.70

52.16

71

-

17

3

2953.73

31.88

60.7

52.76

18

3

3254.38

77.92

57

68

3

2837.99

59.53

25.48

60.7

∗

3

3066.51

44.63

57

66.49

21∗

2

1982.22

54.81

45.34

-

22∗

2

1919.68

30.98

73.46

-

19 20

Table 5.3: Dispatcher’s results for approximate explicit anticipation

In this planning scenario, for eight out of 22 test problems, the planner requires three vehicles. The other problems can be solved using only two vehicles. It is remarkable that all test problems for which three vehicles have to be utilized belong to the second problem set, i.e. the problem set with short planning horizon and dense time windows. Again, test problem 20 requires three vehicles to service all customers on time. The total number of vehicles required for all test problems amounts to 52. The total travel distance equals 60218.76 km. In case of a team situation, the drivers might improve the travel distance by exploiting the optional rules of the European social legislation which are neglected in the dispatcher’s planning. Therefore, the drivers’ planning problem has to be solved within each customer cluster CL∗k . For test problem 2, the first driver can improve the distance traveled from 1697.15 km to 1691.27 km. The total travel distance of this test problem can hence be reduced by 0.2% from 2999.63 km to 2993.75 km. The route anticipated by the planner goes from Frankfurt via Hamburg, D¨ usseldorf, Duisburg, Hannover, Kiel and Bremen back to the depot, whereas the driver decides to use the route Frankfurt, Hamburg, Duisburg, D¨ usseldorf, Hannover, Kiel, Bremen, Frankfurt. In the non-team situation, the drivers’ planning problem has to be solved within each CL∗k , using the criterion C B of minimizing the drivers’ return time. For 10 test problems (problems 3, 5, 7, 9, 11, 13, 17, 20, 21 and 22) the drivers can improve their private criterion by deviating from the routes suggested by the dispatcher. In each of these test problems, only one driver performs a rerouting. The changes in the travel distances and in the return times compared with the dispatcher’s anticipations are depicted in Table 5.4.

94

5 DDM in Combined Vehicle Routing and Break Scheduling Test

Change

Change

problem

in

in

(%)

dist.

ret.

time (%)

3

11.5

-0.2

5

0.1

-0.2

7

0.1

-9.6

9

8.1

-11.0

11

0.3

-27.5

13

1.0

-21.7

17

0.4

-1.1

20

17.6

-52.8

21

1.5

-4.8

22

1.9

-13.3

Table 5.4: Results for the non-team situation

The average deterioration of the dispatcher’s top-down criterion over these 10 test problems equals 4.2%. The overall deterioration of the dispatcher’s top-down criterion over all 22 test problems amounts to 1.9%, the same as in the case of perfect explicit anticipation. The average improvement of the return times for the 10 drivers amounts to 14.2% and the average improvement over all 52 routes equals 2.7%.

5.4.2.3 Implicit Anticipation In the scenario of implicit anticipation, the planner creates time buffers along the routes, which the drivers can use to schedule breaks and rest periods. To include slack times which are proportional to the driving distance, a lower average speed is applied in the dispatcher’s planning process. In the following, two different speed reductions are tested. The drivers travel at an average speed of 65 km/h which is known to the planner. For his planning purposes, travel speeds of 55 km/h and 45 km/h are investigated in the following. Higher speed reductions are not possible since the time windows would not allow the drivers to reach all customers on time. For the 22 test problems, the dispatcher’s results for implicit anticipation with a speed of 55 km/h are depicted in Table 5.5. Using a reduced speed of 55 km/h,

5.4 Computational Experiments

95

the dispatcher anticipates that all test problems can be solved using only two vehicles. Therefore, his anticipated private criterion over all test problems takes the value 44 which is even better than in the benchmark scenario. This fact already indicates that some of the clusters generated by the dispatcher must be infeasible. For 13 test problems (1 to 10, 16, 19 and 21), the planner finds the same clusters as in the case of perfect anticipation. For test problem 2 the anticipated travel distance equals 2625.12 km whereas in the benchmark scenario, it amounts to 2655.64 km, which is also realized by the drivers. Clearly, the route anticipated by the dispatcher is infeasible for the drivers since it does not allow them to find a feasible break schedule. Nevertheless, within the clusters there is enough slack time to derive a feasible route. The same is true for problems 13 and 22 where the dispatcher estimates the travel distance to be 2391.41 km and 1830.58 km, respectively. The minimum travel distances that can be found by the drivers amount to 2516.26 km and 1948.74 km, respectively. The overall travel distance of 53896.30 km clearly underestimates the minimum travel distance since it contains infeasible routes. Test

CT T

problem

CT B

Return

Return

Return

(km)

dr. 1

dr. 2

dr. 3

∗

1

2

2238.76

109.58

99.21

-

2∗

2

2625.12

109.58

99.99

-

3∗

2

2533.27

122.71

109.58

-

4∗

2

2836.12

122.71

109.58

-

5∗

2

2675

47.85

122.71

-

6∗

2

2214.02

107.19

109.58

-

7∗

2

2805.01

99.21

109.58

-

8∗

2

2550.43

109.58

122.71

-

9∗

2

2851.56

93.86

105.46

-

10∗

2

2350.69

109.58

75.04

-

11

2

2408.95

50.81

67.46

-

12

2

2426.47

56.53

44.09

-

13

2

2391.41

55.32

58.11

-

14

2

2550.29

23.44

55.32

-

15

2

2308.34

26.78

65.02

-

16∗

2

1780.70

50.81

52.72

-

17

2

2615.11

50.81

53.34

-

18

2

2571.05

67.46

26.78

-

19∗

2

2530.12

50.81

55.32

-

96

5 DDM in Combined Vehicle Routing and Break Scheduling 20

2

2875.42

65.02

25.73

-

21∗

2

1982.22

55.32

36.77

-

22

2

1830.59

26.79

52.72

-

Table 5.5: Dispatcher’s results for implicit anticipation (55 km/h)

In the team situation, the drivers cannot improve the travel distances estimated by the dispatcher. In the three cases mentioned above, the drivers’ reroutings even deteriorate the planner’s criterion. This is due to the fact that the routes suggested by the dispatcher are infeasible. Moreover, for seven test problems (problems 11, 12, 14, 15, 17, 18, 20), one driver in each problem cannot find a feasible route within his customer cluster. In the non-team situation, in 6 cases one driver can improve his return time by deviating from the route suggested by the dispatcher. The changes in the travel distances and in the return times compared with the dispatcher’s anticipations are listed in Table 5.6. Test

Change

Change

problem

in

in

dist.

ret.

(%)

time (%)

5

4.0

-2.0

11

1.7

-22.6

17

1.7

-2.2

18

1.1

-5.5

21

1.5

-5.7

22

6.6

-5.5

Table 5.6: Results for the non-team situation

The average deterioration of the dispatcher’s top-down criterion is more difficult to calculate in the case of implicit anticipation since half of the routes improved by the drivers belong to problems where another driver cannot find a feasible route within his cluster. For these test problems, the increase in the travel distance was only calculated for the feasible clusters. Over all six test problems, the

5.4 Computational Experiments

97

deterioration of the travel distance equals 2.7%. Over all feasible routes of the 22 test problems, it is only 0.7%. The average improvement of the return times for the routes above amounts to 7.2%. The average improvement over all 37 feasible routes equals 1.1%. The planner’s results for implicit anticipation with a reduced speed of 45 km/h are presented in Table 5.7. For all test problems except of number 13, the dispatcher requires only two vehicles. Over all test problems, 45 vehicles are required. In 16 out of 22 test problems (problems 1 to 10, 12, 16, 17, 18, 19 and 21) the same clusters are generated as in the case of perfect anticipation. Furthermore, for all of these test problems, the anticipated travel distance also equals the distance of the benchmark case. The total anticipated travel distance equals 55145.51 km. Only for test problem 22, the planner underestimates the travel distance, since he expects the drivers’ travel distance to be 1904.62 km whereas the minimum travel distance realized by the drivers is 1971.94 km. Only three of the 45 clusters generated by the dispatcher are infeasible, one cluster in test problems 14, 15 and 20 each. CT T

CT B (km)

dr. 1

dr. 2

dr. 3

1∗

2

2238.76

109.71

102.08

-

2∗

2

2655.64

102.95

109.71

-

3∗

2

2533.27

124.43

109.71

-

4∗

2

2836.12

124.43

109.71

-

∗

2

2675

59.71

124.43

-

6∗

2

2214.02

111.64

109.71

-

7∗

2

2805.01

109.71

102.08

-

8∗

2

2550.43

124.43

109.71

-

9∗

2

2851.56

106.22

95.83

-

10∗

2

2350.69

76.27

109.71

-

11

2

2785.72

46.06

65.82

-

12

2

2433.25

57.43

45.43

-

13

3

2450.19

56.06

59.71

29.04

14

2

2550.29

24.67

56.06

-

15

2

2636.33

47.06

65.82

-

16∗

2

1780.70

59.71

57.43

-

17∗

2

2636.76

54.17

52.41

-

18∗

2

2842.56

68.22

37.16

-

19∗

2

2530.12

56.06

52.41

-

Test problem

5

Return

Return

Return

98

5 DDM in Combined Vehicle Routing and Break Scheduling 20

2

2956.41

59.71

65.82

-

21∗

2

1982.22

54.17

43.89

-

22

2

1904.62

26.92

65.82

-

Table 5.7: Dispatcher’s results for implicit anticipation (45 km/h)

Again, in the case of a team situation, the drivers cannot improve the routes generated by the drivers. For test problem 22, the planner’s anticipated travel distance is even deteriorated by the rerouting. This is due to the fact that the route suggested by the dispatcher does not allow for a feasible break schedule. In the non-team situation, in two cases the drivers can improve the routes with respect to their private criterion, the return time. The changes in the travel distances and return times are depicted in Table 5.8. Test

Change

Change

problem

in

in

dist.

ret.

(%)

time (%)

5

4.0

-21.4

17

0.5

-3.7

Table 5.8: Results for the non-team situation

The average deterioration of the travel distance over these two test problems equals 2.2%. Over all 22 test problems, it is only 0.2%. The average improvement of the return times for the routes above amounts to 12.5%. The average improvement over all 42 feasible routes equals 0.6%.

5.4.3 Summary of the Findings A comparison of the different degrees of anticipation has to regard both the complexity of the dispatcher’s task and the feasibility and quality of the resulting solutions. In the following the different approaches are evaluated according to their performance on the planner’s objectives which represent the company’s goals.

5.4 Computational Experiments

99

As argued already, the more complex scenarios in which the dispatcher explicitly anticipates the drivers’ task of break scheduling always result in feasible customer clusters. This is not necessarily true for the implicit approaches. For both reduced velocities, several customer clusters turned out to be infeasible on the base-level. The most complex scenario of perfect explicit anticipation yields the best results for the dispatcher in terms of his private criterion, the number of vehicles. Only in the case of implicit anticipation, the reduced speed of 55 km/h gives a lower anticipated number of vehicles. However, within some of the clusters the drivers cannot find a feasible route and break schedule. To resolve this problem, the planner would have to deploy an additional vehicle to service these customers on time (see Section 3.2). Therefore, the ex post (i.e. after the drivers perform their planning) realization of his private criterion deviates from its anticipated value. In the explicit planning approaches, they are always equal. In the case of approximative anticipation, the dispatcher’s private criterion C T T takes the value of 52. Compared with the 45 vehicles required in the case of perfect explicit anticipation, this means an increase of 15.5%. Moreover, the anticipated total travel distance over all 22 test problems increases from 55353.5 km to 59999.2 km. This equals a deterioration of the dispatcher’s top-down criterion by 8.3%. It is remarkable that all overestimations of the number of vehicles are encountered in the second problem set. This means that in the case of dense time windows and a short planning time horizon, the additional flexibility provided by the optional legal rules allows for substantial improvements both in terms of the dispatcher’s private and in terms of his top-down criterion. In the first problem set, the planner’s private criterion is equal for both explicit anticipation functions. However, the planner’s top-down criterion, the travel distance, is still 7.6% higher when applying the approximate anticipation function. In the non-team situation only eight routes are changed by the drivers if the dispatcher uses a base model including the optional legal rules. In the case of approximate anticipation the drivers perform a rerouting for 10 customer clusters. Although for these routes the average deterioration of the travel distance is higher in the case of perfect anticipation (5.4% instead of 4.2%), the deterioration over all routes is equal in both cases (1.9%). Furthermore, in the case of a non-team situation the ex post accumulated travel distance over all 22 test problems is 8% less in the scenario with perfect anticipation. If the dispatcher applies a reduced velocity during the planning process in order

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5 DDM in Combined Vehicle Routing and Break Scheduling

to generate slack time along the routes, he can no longer guarantee the feasibility of the resulting customer clusters. Especially in the case of a comparably high average speed of 55 km/h, which equals a speed reduction of about 15%, the share of infeasible clusters of almost 16% is not acceptable. In the case of a lower speed of 45 km/h, which means a speed reduction of about 30%, the share of infeasible clusters only amounts to 6.6%. These infeasibilities have to be resolved by deploying further vehicles. Moreover, only for test problem 13, the dispatcher overestimates the number of vehicles required. For 16 test problems the dispatcher derives the optimal clusters. Using a reduced speed of 55 km/h, the planner only derives the optimal clusters for 13 test problems. In both cases, for the first problem set the optimal clusters are derived. This means that for all test problems with large time windows and a long planning time horizon, the dispatcher derives feasible clusters, which are even optimal with respect to both his private and his top-down criterion. Regarding only the first problem set, both planning approaches show an equal performance. Taking into account the second problem set, the lower average speed of 45 km/h clearly outperforms the higher speed of 55 km/h. It even turns out that implicit anticipation outperforms approximate anticipation with respect to the first problem set. Implicit anticipation results in the pareto optimal solutions also derived with perfect anticipation for all 10 test problems, whereas approximate anticipation only results in pareto optimal solutions twice. For both degrees of anticipation the dispatcher’s private criterion takes the value of 20 over these test problems. Nevertheless, the total travel distance is 7.6% higher using approximate anticipation. The above analysis yields the following recommendations for dispatchers: 1. In a planning context with a short time horizon and dense customer time windows, it is beneficial for the dispatcher to apply perfect explicit anticipation in order to exploit the additional flexibility provided by the optional rules of the European social legislation. This is true for the team and for the non-team situation. 2. In the case of a large planning time horizon and wide customer time windows, it is possible for the dispatcher to reduce the complexity of his planning task by applying a simplified anticipation function. In this case it turns out that implicit anticipation using an average speed reduction of about 30% even outperforms approximate explicit anticipation. However, the feasibility of the resulting customer clusters cannot be guaranteed.

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Especially the second result is remarkable since it suggests that for this planning situation the dispatcher does not face a trade-off between complexity and solution quality.

6 Restricted Dynamic Programming Heuristic for the Problem of Combined Vehicle Routing and Break Scheduling Solving large-scale vehicle routing problems with complex additional constraints such as the problem of combined vehicle routing and break scheduling is very difficult. Even large-scale vehicle routing problems with time windows without additional constraints can only be solved to optimality for problems with tight time windows and a structure which allows only few possible solutions (Pisinger and Ropke, 2007). Therefore, heuristics and metaheuristics are mainly applied to solve these problems (Funke et al., 2005). In the following the restricted dynamic programming (DP) heuristic developed by Kok et al. (2010) is presented and utilized for solving the problem of combined vehicle routing and break scheduling both from a centralized and from a distributed decision making perspective. The chapter is structured as follows: In Section 6.1 the basic principles of restricted dynamic programming are introduced. Section 6.2 presents a break scheduling heuristic, which is integrated into the restricted dynamic programming heuristic such that the algorithm can be used for solving the problem of combined vehicle routing and break scheduling from a centralized planning perspective. In Section 6.3 the heuristic is modified such that it can be utilized to solve the distributed decision making problem.

6.1 Dynamic Programming Methods for Vehicle Routing Problems DP is based on the partitioning of a problem into different stages (Nemhauser, 1966). Therefore it can be applied to optimization problems which have a natural structure that comprises different stages (Gr¨ unert and Irnich, 2005). The traveling salesman problem is an example of such a problem. Here, the partial routes of different lengths represent the “natural” stages of the problem and the addition of a customer node to a partial route can be regarded as the transition C. M. Meyer, Vehicle Routing under Consideration of Driving and Working Hours, DOI 10.1007/978-3-8349-6732-9_6, © Gabler Verlag | Springer Fachmedien Wiesbaden GmbH 2011

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from one stage to the next. This section gives a brief introduction into the concept of DP and its application to traveling salesman problems. Using a giant-tour representation of vehicle routes, DP can also be used for solving vehicle routing problems. Additional timing constraints such as time windows can be included into the solution procedure via the control of the solution properties. 6.1.1 Dynamic Programming Algorithm for the Traveling Salesman Problem DP is an optimization principle which can be used for solving large-scale integer optimization problems. When solving a problem with DP, the problem has to be divided into partial problems for which optimal solutions have to be found. These optimal solutions are then extended until the overall optimal solution is derived (Ohse, 1998). DP can be applied to problems which are characterized by the following features (Hans, 2001): The problem can be divided into stages such that at each stage a decision has to be taken. At each stage different states (S,j) are possible. Each state holds all the information to take a decision at this stage. This decision determines the characteristics of the subsequent stage including its objective value. Given the current state, the optimal decision for each of the subsequent stages does not depend on the decisions taken previously. To obtain the final solution, a recursion function is required which relates the costs of the current stage to the costs of its previous stage. These characteristics hold for the TSP where each stage represents the addition of a new customer node at the end of a partial route. Therefore, DP can be used to solve TSPs as proposed by Bellmann (1962) and Held and Karp (1962). The solution procedure is based on the observation that partial routes which are dominated by other partial routes cannot be part of the overall optimal route. In the case of a TSP, a partial route r1 is dominated by another partial route r2 if both sub-routes have the same start and end node and contain the same set of customer nodes, but the cost of r2 is less than the cost of r1. The difference in the costs is due to the different sequences of the visited nodes. An example of a dominated partial route is depicted in Figure 6.1. Both partial routes start at node 0 and end at node 3 visiting nodes 1 and 2 in between. The first route r1 has a length of 9 whereas the length of the second route r2 is only 7. Therefore, r2 dominates r1.

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Figure 6.1: Example of a dominated partial route All solutions containing dominated partial routes may be excluded from further consideration. In this way an exact algorithm for solving TSPs can be derived. To describe a DP algorithm for the TSP, a network is considered which consists of n + 2 nodes {0, ..., n + 1} where 0 and n + 1 again represent the depot and nodes 1 to n the customer nodes to be visited. Let dij be the travel distance from node i to node j. Moreover, let S be a set of customers S ⊆ {1, ..., n}, j ∈ S, and D(S, j) be the minimum distance required to reach customer j after visiting all other nodes in S. Then for |S| = 1, i.e. for the first customer to be visited, Equation (6.1) holds. D({j} , j) = d0j ∀j = 1, ..., n

(6.1)

In each successive stage, the cost of each state (S, j), with j the last node visited after visiting node i ∈ S \ {j}, can be computed using the Recurrence Relation (6.2). D(S, j) = mini∈S\{j} {D(S \ {j} , i) + dij }

(6.2)

The travel distance of the optimal TSP route can then be calculated according to Equation (6.3). D∗ = mini∈{1,...,n} {D({1, ..., n} , i) + din+1 }

(6.3)

To obtain the sequence in which the customers are visited on the optimal route, for each state (S, j) the node i ∈ S \ {j} that minimizes D(S \ {j} , i) + dij has to be stored. Knowing the predecessor of each node, the optimal route can be derived recursively. This exact solution algorithm is only suitable for solving small problem instances. For larger problem sizes, the above DP algorithm can be transformed into a heuristic by restricting the number of states to be expanded at each stage.

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6.1.2 Restricted Dynamic Programming Algorithm for the Traveling Salesman Problem Malandraki and Dial (1996) apply the principle of DP in the design of a restricted dynamic programming algorithm for the TSP with time dependent travel times dij (ti ). In this way the exact algorithm presented above is transformed into a heuristic algorithm to avoid the exponential increase of computation time. This is achieved by considering at each stage where |S| = i only the H most promising partial solutions, i.e. those with the smallest durations T (S, k). The parameter H has to be defined by the dispatcher in advance. Only the H best solutions are expanded and transformed into the next stage where |S| = i + 1. The reason for this approach is that these solutions are more likely to result in the route with the lowest total duration. For H = 1 the algorithm turns into the simple nearest neighbor heuristic.

6.1.3 Solving Vehicle Routing Problems via Giant-Tour Representations To apply the idea of DP for the TSP to VRPs where more than only one route has to be found, the giant-tour representation of the VRP introduced by Funke et al. (2005) can be utilized (see Gromicho et al., 2008). This representation allows problems with complex timing constraints such as time windows and routeduration constraints to be modeled within one conceptual framework (Funke et al., 2005). For a formal deduction of this method, again a directed graph G = (V, A) is considered. The set V can be divided into the disjoint sets of origin nodes O, destination nodes D and request nodes R, which represent the customer locations. There are kmax vehicles available to service the customer requests. To apply the DP approach to the VRP, the vehicles are ordered. Consequently, the vehicle routes rk , k = 1, ..., k max , are also ordered in a routing solution. Then the gianttour representation of this solution is a cycle in G in which the end node of a vehicle route rk is equal to the start node of the subsequent vehicle’s route rk+1 , i.e. the destination node dk of vehicle route k equals the origin node ok+1 of vehicle route k + 1. In a simple VRP all origin and destination nodes are represented by the depot. The cycle is closed by connecting the destination node dkmax of the last route with the origin node o1 of the first route. An example for the a VRP solution comprising two vehicle routes is presented

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Figure 6.2: Traditional representation of a VRP solution

Figure 6.3: Giant-tour representation of the VRP solution

in Figure 6.2 (adapted from Kok, 2010). To derive a giant-tour representation, origin and destination nodes are introduced for each vehicle route. The giant-tour representation for the example is depicted in Figure 6.3. The VRP in its traditional form is a clustering and sequencing problem. Through the giant-tour representation the VRP is transformed into a pure sequencing problem. However, not all possible expansions for this sequencing problem result in feasible solutions for the VRP. To obtain feasible giant-tour representations for a VRP it has to be secured that for each state (S, dv ) only the expansion ov+1 is allowed. Moreover, node ov+1 must only be reached from node dv . Therefore, a feasibility check for every expansion needs to be performed.

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6.2 Restricted Dynamic Programming Heuristic for the Vehicle Routing Problem with Time Windows and European Social Legislation In this section a restricted DP heuristic for the VRPTW-EU is described.8 This heuristic can be used to solve large-scale vehicle routing and break scheduling problems efficiently. Two different break scheduling methods are suggested. The basic version only considers the basic rules of Regulation (EC) No. 561/2006 and of Directive 2002/15/EC as presented in Chapter 2. The extended method also allows for the inclusion of all optional legal rules. Both algorithms can solve VRPTWs including the European social legislation for a weekly period. Moreover, the solution methods can be extended to longer time horizons and can easily be adapted to a rolling horizon framework. The underlying vehicle routing problem is solved using the restricted DP framework proposed by Gromicho et al. (2008). Within this framework, customers are sequentially added to the end of partial vehicle routes. Feasibility of such additions, for example checking whether the added customer is visited within his time window, is controlled by extra state dimensions. The compliance with the European social legislation is also checked through the introduction of extra state dimensions. For this purpose, break scheduling methods are proposed which schedule breaks and rest periods at the customer or on the route to the customer to be added. The restricted DP heuristic for the VRP, developed by Gromicho et al. (2008), is based on the exact DP algorithm for the TSP of Held and Karp (1962) and Bellmann (1962). The DP algorithm for the TSP is applied to the VRP through the giant-tour representation of vehicle routing solutions introduced by Funke et al. (2005). Using this representation, the DP algorithm is applied to the extended node set including the vehicle origin and destination nodes, where a feasibility check has to be performed for each node addition. These feasibility checks ensure that an origin node of a vehicle ok can be added to a partial route represented by a state if and only if the last visited node is dk−1 . Furthermore, these checks only allow dk to be added if ok is already in the visited node set S. Further state dimensions can be added to account for other restrictions, such as the vehicle capacity or time windows. For example, in the case of capacity restric8

This section is based on Kok et al. (2010).

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tions a state dimension c is added which keeps track of the capacity utilization of the active vehicle k. Active vehicle refers to the last vehicle for which ok has been added to the set of visited nodes. Each time a vehicle origin node ok is added to a state, c is reset to zero. A customer addition is only allowed if the current capacity utilization c together with the new customer’s demand do not exceed the capacity of the active vehicle. Many other restrictions such as time windows, sequencing restrictions (pickup and delivery), multiple depots, and heterogeneous vehicle fleets can be incorporated by adding state dimensions or control via the input. Therefore, the restricted DP heuristic of Gromicho et al. (2008) forms a general framework for solving vehicle routing and scheduling problems. Since the (unrestricted) DP algorithm does not run in practically acceptable computation times for problem instances of realistic sizes, Gromicho et al. (2008) restrict the state space by parameters H and E. The value of H specifies the maximum number of states to be taken to the next iteration, where the smallest cost states are maintained, as proposed by Malandraki and Dial (1996). The value of E restricts the number of state expansions of a single state, such that only the E nearest, unvisited neighbors are considered which allow feasible state expansions. Since in good VRP solutions successive nodes are in general near neighbors of each other, this restriction cuts off less promising parts of the state space. These restrictions on the number of state expansions result in the following running time complexity of the restricted DP heuristic. In each stage, at most H states are expanded to at most nH states. Since only the H best states are expanded to the next stage, each stage requires O(nHlog(H)) time. The total number of stages equals the number of nodes in the network, which is O(n). Therefore, the running time complexity of the restricted DP heuristic is O(n2 Hlog(H)) (Gromicho et al., 2008). The European social legislation can be incorporated in this restricted DP heuristic by adding further state dimensions. For this purpose, two break scheduling heuristics are proposed. They decide locally, i.e. at the next customer node or on the route to the customer to be added, when and where breaks have to be scheduled. There are two main reasons to use a local view for scheduling breaks and rest periods. First, a local criterion allows to schedule breaks in constant time. Therefore, the running time complexity of the restricted DP heuristic does not increase. This even holds when the complex optional legal rules, which are generally ignored in

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the literature, are incorporated. Second, using this local view results in intuitive rules for scheduling breaks which are both easy to implement, and easily acceptable by dispatchers and operations managers in practice. If a global scheduling algorithm is used, then breaks and rest periods may be scheduled and extended prematurely, such that the benefits are less clear. For example, it may turn out that a state expansion results in an early arrival at a customer j, such that the active nonrest period reaches its maximum of 13 hours before the opening of customer j’s time window, requiring a rest period before serving customer j. If due to this rest period the time window at customer j cannot be met, then this state expansion is infeasible. However, if there is waiting time at some predecessing customer i, then it might be possible to schedule an early rest period partially during this waiting time without violating any of the time windows between customers i and j. This may postpone the start of the active nonrest period when arriving at customer j, allowing to serve customer j before having to schedule a rest period. This global view in which also breaks and rest periods at predecessing customers are considered requires at least linear time, which increases the running time complexity of the restricted DP heuristic. Note that the utilization of such local break scheduling criteria does not guarantee to find a feasible break schedule, even if one exists. In the following, two break scheduling methods are proposed: a basic method and an extended method. The basic method is an extension of the “naive label setting method” proposed by Goel (2009), which is improved by allowing for more local flexibility of customer additions. This is done by first minimizing the start of service time of the added customer. For this minimum start time the accumulated time since the last rest, and the accumulated driving and working time since the last break are minimized by trying to schedule daily rest periods or breaks during waiting time at customer nodes. The extended method enhances the basic method by incorporating the optional rules of the legislation. The same methodology of increasing the local flexibility at the last visited customer is applied. Both break scheduling methods are now described in detail.

6.2.1 Basic Break Scheduling Method For the basic approach, the requirements on breaks set by Directive 2002/15/EC are modified such that a break of 45 minutes (instead of 30 minutes) is scheduled after no more than 6 hours of working time. This modification is similar to the approach chosen for modeling the VRPTW-EU with basic rules, presented in

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Section 4.1. It ensures that the second requirement of Directive 2002/15/EC on the break length between working periods is also satisfied, which states that the total break time on a day should be at least 45 minutes if daily working time exceeds 9 hours. Furthermore, it fulfills the requirements of Regulation (EC) No 561/2006 on the break length between two driving periods, such that also a new driving period is initiated. To include the legislation on driving and working hours into the restricted DP heuristic, it has to be ensured that the partial route represented by each state is feasible with respect to the legal restrictions. For this purpose, six new state dimensions are introduced: nonbreak working time, nonbreak driving time, nonrest time, daily driving time, weekly working time, and weekly driving time.

tnbw : accumulated nonbreak working time. This variable denotes the total amount of working time since the last break of at least 45 minutes. tnbd : accumulated nonbreak driving time. This variable denotes the total amount of driving time since the last break of at least 45 minutes. tnr : accumulated nonrest time. This variable denotes the total amount of time passed by since the last rest period of at least 11 hours. tdd : accumulated daily driving time. This variable denotes the total amount of driving time since the last rest period of at least 11 hours. tww : accumulated weekly working time. This variable denotes the total amount of working time since the last rest period of at least 45 hours. twd : accumulated weekly driving time. This variable denotes the total amount of driving time since the last rest period of at least 45 hours. First, a planning period of one week is considered, i.e. the time between Monday, 00:00, and Sunday, 24:00. Furthermore, it is assumed that the planning starts right after a weekly rest period has been taken by all drivers. This results in all state dimensions tnbw , tnbd , tnr , tdd , tww , and twd being zero for all vehicles at the start of the planning period. Section 6.2.3 discusses extensions to longer time horizons and state dimensions not being zero at the start of the planning period, allowing for a rolling horizon framework. When a new vehicle is deployed, the break scheduling method has to check whether the first customer to be visited can be reached from the depot on time. This might not be the case if a vehicle starts from the depot at time zero and

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requires a break or rest period before starting service, since this might violate the customer time window. If the customer cannot be served by a vehicle leaving the depot at time zero, the vehicle’s departure time is delayed such that the vehicle arrives at the customer node exactly at the opening of the time window. Within the basic approach the optional rules of the legislation are not considered. Whenever a state (S, i) is expanded with a customer j, the arrival time arj at this customer is determined first, considering possible breaks and rest periods that have to be scheduled along the route from i to j. For this purpose, arj is first set to the departure time (service completion time) from customer i, and a variable δij is introduced, denoting the remaining driving time to customer j. This variable is initially set to the total driving time dij from customer i to customer j. A variable Δ is defined as Δ = min(δij , 6 − tnbw , 4.5 − tnbd , 13 − tnr , 9 − tdd ), which represents the minimum driving time until a break or rest period needs to be scheduled, or the next customer is reached. Subsequently, the algorithm recursively checks whether δij = Δ holds. If δij =  Δ, then either a break or a rest period has to be scheduled along the route. It has to be checked whether Δ equals 13 − tnr or 9 − tdd . If so, an 11-hour rest period is scheduled and the values of tnbw , tnbd , tnr , and tdd are set to zero. Otherwise, either 6 − tnbw or 4.5 − tnbd equals Δ. Thus, a 45-minute break has to be scheduled and the values of tnbw and tnbd are reset to zero. However, if a break has to be scheduled, the algorithm checks whether the break fits within the remaining available nonrest time. Otherwise a daily rest period is scheduled instead of a break since in this case a rest period would have to be scheduled directly after the break. After scheduling a daily rest period or a break, the remaining driving time δij is updated, and in case of a break also the values of tnr and tdd , as follows:

δij := δij − Δ

(6.4)

tdd := tdd + Δ

(6.5)

tnr := tnr + Δ + 0.75

(6.6)

The remaining driving time to the next customer is reduced by Δ (see Equation (6.4)). In the case that a break is scheduled, the accumulated daily driving time is increased with Δ (6.5), and the accumulated nonrest time is increased with Δ and with the break time (6.6).

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After determining the arrival time arj , the algorithm checks whether the accumulated nonbreak working time and the accumulated nonrest time allow to service the customer without scheduling another break or rest period at customer j. To check this, the service time sj of customer j and the time window {aj , ej } in which service must start at customer j are required. If tnbw + sj > 6, then a break is scheduled and arj , tnr , tnbw and tnbd are updated. Next, if tnr + max {0, aj − arj } + sj > 13, then a daily rest period is scheduled. However, if both inequalities hold, then the algorithm extends the 45-minute break forced by the nonbreak working time to an 11-hour rest period instead of scheduling a 45-minute break and an 11-hour rest period directly after each other. Finally, if arj ≤ ej , the customer j can be reached on time and is added to the partial route. To decide whether the addition of customer j is feasible with respect to all rules of the European social legislation, it remains to check whether the vehicle can return to the depot without violating the restrictions on weekly driving and working times. The expansion is forbidden if after visiting the customer a return to the depot would be infeasible in order to avoid including infeasible states. Consequently, an expansion is only allowed if Restrictions (6.7) and (6.8) are satisfied.

dij + dj0 ≤ 56 − twd

(6.7)

dij + sj + dj0 ≤ 60 − tww

(6.8)

To improve this scheduling procedure by increasing the local flexibility at customer j, a number of scheduling features is introduced which reduce the values of tnbw , tnbd , tnr , and tdd without delaying the start of service at customer j. The highest priority is attributed to reducing the accumulated nonrest time, since in VRPTWs, large waiting times often cause this to be the tightest restriction. Large waiting times are also considered a major issue in practical freight forwarding (see Section 3.3). Therefore, in a first attempt, the heuristic tries to schedule a daily rest period during waiting times if this does not delay the start of service at a customer node. This means that a rest period is scheduled before serving a customer node whenever the waiting time until the opening of the customer’s time window is more than 11 hours. In this case all values tnr , tdd , tnbw , and tnbd can be reset to zero. If after taking the rest period there is still waiting time left, the rest period is extended until the opening of the customer time window, such

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that tnr is not increased before starting service. If it is not possible to schedule a rest period during waiting time, but there is a rest scheduled along the route to customer j, then this rest is extended by the waiting time at customer j (if any). This reduces the value of tnr at the start of service at customer j without affecting the other variables. This feature might even reduce the start of service time, if otherwise the additional waiting time would force another rest period to be scheduled before starting service, caused by the value of tnr reaching its maximum value. This additional rest period might postpone the start of service after aj , or even after ej , rendering the expansion infeasible. If the first two cases do not apply, but there is waiting time at the customer, then it is checked whether a 45-minute break can be scheduled in order to reset tnbw and tnbd to zero. This increases the flexibility of adding customers afterwards.

6.2.2 Extended Break Scheduling Method To adapt the above presented algorithm to more realistic planning purposes and to allow for an enlargement of the solution space, in the following the optional rules of Regulation (EC) No 561/2006 are incorporated in the DP heuristic. Furthermore, the splitting of breaks according to Directive 2002/15/EC is also taken into consideration. Again, the extended set of rules is only exploited if they allow for a local improvement of the current partial solution. In the following, the implementation of the optional legal rules is described.

6.2.2.1 Extended Daily Driving Times Regulation (EC) No 561/2006 allows drivers to extend their daily driving time up to 10 hours twice a week while the basic rule restricts the daily driving time to 9 hours. Driving 9 hours can be accomplished with only one break if the driver takes this break exactly after 4.5 hours and afterwards continues driving for another 4.5 hours. However, the extension to more than 9 hours forces the driver to take at least two breaks as the daily driving time exceeds the maximum length of two driving periods. Therefore, this extension might cause a delayed arrival at a customer due to the additional break. On the other hand, a driving time extension might allow drivers to arrive earlier at a customer, because they do not have to schedule a rest period already after 9 hours of driving.

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In the extended break scheduling algorithm, a driving time extension is utilized if it reduces the start time of the service at the customer to be added. Besides, if the driving time extension increases the waiting time at the customer such that a rest period can be scheduled during this waiting time, the extension is also used. To compute the arrival time at the customer in case of extending the driving time, a similar procedure is used as described in the basic method. The only adaptation lies in the fact that the maximum daily driving time is set to 10 hours. This arrival time is compared with the arrival time calculated in the traditional way and then it is decided whether a driving time extension is profitable. Since drivers can extend their driving times only twice a week, the number of driving time extensions which have already been used has to be accounted for. For this purpose, a new state dimension ndte is introduced: ndte : number of driving time extensions taken by the active vehicle. The state dimension ndte is initialized with zero and each time a driving time extension is scheduled it is increased by one. Moreover, it is restricted to two and when the current node is the depot, ndte is updated to zero since a new vehicle is used.

6.2.2.2 Reduced Daily Rest Periods Reducing daily rest periods can be beneficial in two ways. First, it might allow an earlier start of the next nonrest period. Second, it might extend the current nonrest period by at most 2 hours. The latter case appears, since this daily rest period must have been taken within 24 hours after the end of the previous rest period, while this rest is reduced by at most 2 hours. When a rest period must be taken on a route, then the algorithm checks whether it is beneficial to schedule a reduced rest period. This is done by calculating the arrival time at the customer to be added if the rest period is reduced. Should this arrival time permit an earlier start of service or increase the waiting time allowing for another (reduced) rest period, then a reduced daily rest period is scheduled. This procedure is similar to the procedure applied for checking the profitability of driving time extensions. Since extended driving times can be combined with reduced rest periods, there are four different scenarios to consider when a rest has to be scheduled during a travel. Therefore, first the arrival times for each of these scenarios have to be calculated. Subsequently, it has to be detected whether some of the arrival times

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allow for another (reduced) rest period during waiting time. If this is the case, the break schedule with the least number of optional rules is selected. In case of having to choose between extending driving times and reducing rest periods, the procedure gives priority to using driving time extensions. This is due to the fact that there is a limited number of times each optional rule can be applied, and rest reductions increase the available time for all working activities. Therefore, it seems to be beneficial to retain as many rest reductions as possible. If none of the scenarios allows to schedule a daily rest period during waiting time, the procedure selects the scenario which minimizes the start time of service. Again, if different scenarios result in this minimum start of service time, then the one is chosen which contains the least number of optional rules. Since a driver is only allowed to reduce daily rest periods three times between two weekly rest periods, the number of remaining reduced rest periods has to be accounted for. For this purpose, a new state dimension nrr is introduced which indicates the number of rest reductions taken by the active vehicle. nrr : number of rest reductions taken by the active vehicle. Whenever a rest reduction is scheduled, nrr is increased by one and if the current node visited is the depot then nrr is reset to zero. Upon arrival at a customer node, the procedure also checks whether it is beneficial to reduce the next rest period. This is the case if a nonrest time of 13 hours does not allow to service the customer before taking a rest period, while a nonrest time of 15 hours would make immediate service possible. Consequently, the next rest period is reduced, thereby permitting the delivery without having to schedule a rest before service and expediting the start of service at this customer.

6.2.2.3 Split Breaks Both Regulation (EC) No 561/2006 and Directive 2002/15/EC allow drivers to split their breaks. The regulation on driving times allows to divide breaks of at least 45 minutes into two parts. The first part has to last for at least 15 minutes and the second part for at least 30 minutes. Besides, the directive on working hours allows to split the total time required for breaks into parts of at least 15 minutes each. In the extended break scheduling method, the optional rule of splitting breaks is applied whenever waiting times of at least 15 minutes but less than 45 minutes

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occur at a customer node. This waiting time is not sufficient to schedule a regular break as required by Regulation (EC) No 561/2006. Therefore, a 15-minute break is scheduled and extended until the ready time of the customer. If the break lasts for at least 30 minutes, it counts as a full break for the nonbreak working time and tnbw is set to zero. If it is less than 30 minutes, then it counts as a 15-minute break and another break of 15 minutes is required when tnbw reaches its maximum value of 6 hours. If such a partial break is taken, either during waiting time or forced by the accumulated nonbreak working time, it can be counted as the 15-minute part of a break required by the nonbreak driving time. Therefore, when in this case the nonbreak driving time tnbd reaches its maximum of 4.5 hours, an additional break of only 30 minutes has to be scheduled. Note that the break scheduling method does not schedule a 45-minute break anymore when the accumulated nonbreak working time reaches its maximum value. This is due to the fact that a 30-minute break now also counts as a 15minute break for the nonbreak driving time. Therefore, if subsequently a break is forced by the nonbreak driving time, then it benefits from this 30-minute break, as opposed to the case where the optional rules are neglected. Directive 2002/15/EC also requires that if the daily working time exceeds 9 hours, the total break time on that day should be at least 45 minutes, instead of 30 minutes if the working time is between 6 and 9 hours. To account for this rule, the state variable tdw is introduced, which indicates the daily working time: tdw : daily working time of the active vehicle. Whenever this state dimension reaches its maximum of 9 hours, another break of at least 15 minutes is scheduled if the total break time during this daily driving period does not add up to at least 45 minutes already. In the latter case namely, the total duration of breaks satisfies the working time directive. Since only breaks of at least 15 minutes are scheduled also the required structure of the breaks is satisfied. 6.2.2.4 Split Daily Rest Periods Regulation (EC) No 561/2006 allows drivers to split daily rest periods into two parts of which the first must last for at least 3 hours and the second for at least 9 hours. It has to be noticed that the total time required for split rest periods equals 12 hours instead of 11 hours as required for a regular rest period. Therefore, in

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the extended break scheduling method this option is only used if the 3 hours part of a reduced rest period can be scheduled during waiting times in order to avoid an increased time required for rests. This is reasonable if the waiting time lies between 3 and 9 hours such that no reduced daily rest period can be taken during waiting time, or, in case that the state variable nrr has reached its maximum value, if the waiting time lies between 3 and 11 hours such that no regular daily rest period can be scheduled. In this case scheduling a 3-hour rest period is beneficial, since it allows an extension of the nonrest period to 15 hours. The 3-hour part of a split rest period is only scheduled if no such part has been scheduled already, and it is extended until the opening of the customer’s time window. As the rest time of 3 hours lies above 45 minutes, the state dimensions tnbw and tnbd can be reset to zero when the service starts. When the next daily rest period is required by tnr or tdd , then only the second part of the split daily rest period of 9 hours is scheduled. Furthermore, the maximum nonrest period is extended to 15 hours until this next rest is scheduled. After taking the second part of the split rest all state dimensions tnr , tdd , tnbw , tnbd , and tdw are set to zero. There is one further scenario in which a split rest period may be beneficial. This is, when the waiting time at a customer is less than 3 hours, and the accumulated nonrest period would exceed 13 hours if no rest is taken before serving this customer, while it would not exceed 15 hours. If in this situation the maximum number of reduced rest periods has already been taken, while a split rest of 3 hours together with the customer service time still fits within the 15 hour nonrest period, then a split rest of 3 hours is scheduled.

6.2.2.5 Overview of the Implementations of the Optional Rules Table 6.1 summarizes all implementations of the optional rules in the extended break scheduling method. Optional rule

Implementation

Extended driving time

Applied if it reduces the start of service time; applied if it increases the waiting time, allowing for a rest period before service

Reduced rest period

Applied if it reduces the start of service time; applied if it increases the waiting time, allowing for a rest period before service

6.2 Restricted DP Heuristic for the VRPTW-EU Split breaks

Scheduling of a 15-minute break if there is enough waiting time

Split rest periods

Scheduling of a 3-hour rest period if there is

119

enough waiting time; scheduling of a 3-hour rest period if this allows service without taking a daily rest period before, and no rest reductions are left Table 6.1: Implementation of the optional rules in the extended break scheduling method

6.2.3 Extensions to Different Time Horizons So far, a weekly planning horizon and a situation in which all drivers have just completed a weekly rest period were considered. However, the break scheduling method can easily be adapted to longer time horizons and to a rolling horizon framework. If a longer planning time horizon has to be considered, a state dimension tnwr can be added to account for the maximum working period of 144 hours between two weekly rest periods. As soon as one of the state dimensions tww , twd , or tnwr reaches its maximum value, a weekly rest period is scheduled. Afterwards, all state dimensions are reset to zero. To use the optional rules on the weekly rest period, the weekly driving time, and the weekly working time, a similar methodology can be used as described for the extended break scheduling method. In planning scenarios in which drivers have not just completed a weekly rest period at the start of the planning period - which is typically the case in a rolling horizon framework - the state dimensions for each vehicle can be initialized with the initial conditions of the corresponding drivers. Consequently, the algorithm can also be used in a rolling horizon framework.

6.2.4 Computational Experiments The restricted DP heuristic including the break scheduling methods is tested on the modified Solomon instances proposed by Goel (2009). The Solomon (1987) instances consist of 6 problem sets: in the c1 and c2 instances customer locations

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are clustered, in the r1 and r2 instances customer locations are chosen randomly, and in the rc1 and rc2 instances they are semi-clustered; the 2-instances have a relatively longer time horizon and larger vehicle capacities than the 1-instances allowing for larger vehicle routes in terms of number of customers. Goel suggests the following modifications of the Solomon instances for the VRPTW-EU. He proposes to consider the depot opening times as a period of 144 hours and to scale the customer time windows accordingly. Goel uses a driving speed of 5 distance units per hour, and he sets all service times to 1 hour. Due to the required breaks and rest periods it may be impossible to reach certain customers before their due dates, or the vehicle may not be able to return to the depot on time after serving a customer immediately after the opening of his time window. Therefore, Goel suggests to adjust these time windows such that the ready time equals the earliest time the vehicle can reach the customer, and the due date is such that starting service at this due date and directly returning to the depot results in a return time at the depot’s due date, respectively. The restricted DP heuristic including the break scheduling methods was implemented in Delphi 7 and the experiments were performed on a Pentium M, 2.00 GHz CPU and 1.0 GB of RAM. First, the results for the basic break scheduling method are reported, in which the optional rules are neglected. These results are compared with the best results found by Goel. Since the method proposed by Goel does not consider Directive 2002/15/EC on drivers’ working hours, the basic break scheduling method is relaxed by setting the maximum nonbreak working period to 13 hours, i.e. the maximum period between two daily rest periods in the basic method. Second, computational results on the impact of Directive 2002/15/EC are reported. Finally, the computational results for the extended break scheduling method including all optional legal rules are presented. As described in Gromicho et al. (2008), the value of H, which restricts the stage width after each iteration of the restricted DP heuristic, has a large impact on computation time and solution quality. For the experiments the value of H is set to 10, 000 since this gives an average computation time of 65 seconds (with a maximum of 107 seconds) per problem instance, which is practically acceptable. Furthermore, the number of state expansions of a single state is not restricted (the maximum number of state expansions E of a single state is set to n, the number of customers). As in Goel (2009), the primary objective is to minimize the number of vehicles, and the secondary objective is the minimization of the total travel distance. In order to implement this hierarchy of objectives, a large cost M is added to a state each time a vehicle returns to the depot and a new

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vehicle has to be used. Table 6.2 presents the results of the basic break scheduling method without consideration of the working time restrictions imposed by Directive 2002/15/EC, and the best results found by Goel. Note that Goel allows significantly larger computation times: Goel’s results are the best out of five runs of half an hour each per problem instance. Table 6.2 shows that the restricted DP heuristic clearly outperforms the large neighborhood search algorithm proposed by Goel. Only for one problem instance (r103) it requires one more vehicle, while for 47 other problem instances the number of vehicles can be reduced. On average over all problem instances, the above method finds solutions requiring 18.26% less vehicles. Also with respect to the second objective, the total travel distances, the restricted DP heuristic shows significant improvements. Only for the r1 problem instances no improvement is found, on average. In total, the travel distances of 37 problem instances could be reduced with an average reduction of 5.41% over all 56 problem instances.

Problem c101 c102 c103 c104 c105 c106 c107 c108 c109

DP heuristic Vehicles Distance CPU(s) 11 923.66 43 11 1,097.97 53 10 1,080.04 72 10 1,053.27 89 10 839.99 42 11 900.10 42 10 874.03 47 10 892.71 51 10 1027.19 57

Best in Goel (2009) Vehicles Distance 13 1,143.32 13 1,198.82 11 971.11 10 1,101.42 11 908.29 11 1,079.24 10 1,023.77 10 975.20 11 1,088.87

Change Vehicles Distance -15.38% -19.21% -15.38% -8.41% -9.09% 11.22% 0.00% -4.37% -9.09% -7.52% 0.00% -16.60% 0.00% -14.63% 0.00% -8.46% -9.09% -5.66%

c1

10.33

965.44

55.04

11.11

1,054.45

-7.00%

-8.44%

c201 c202 c203 c204 c205 c206 c207 c208

6 5 5 4 5 5 5 5

941.60 866.09 810.74 768.19 711.96 677.79 709.36 677.62

55 71 80 107 62 66 64 68

9 9 9 8 8 7 9 8

1,064.57 990.03 982.49 873.22 973.53 838.91 966.19 948.21

-33.33% -44.44% -44.44% -50.00% -37.50% -28.57% -44.44% -37.50%

-11.55% -12.52% -17.48% -12.03% -26.87% -19.21% -26.58% -28.54%

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Problem c2

DP heuristic Vehicles Distance CPU(s) 5.00 770.42 71.60

Best in Goel (2009) Vehicles Distance 8.38 954.64

Change Vehicles Distance -40.30% -19.30%

r101 r102 r103 r104 r105 r106 r107 r108 r109 r110 r111 r112

13 13 11 8 11 9 9 8 9 8 9 8

1,483.95 1,398.59 1,256.53 1,023.47 1,207.87 1,162.18 1,068.90 1,011.90 1,094.14 1,061.92 1,085.39 973.86

39 44 54 74 47 53 65 77 60 77 64 96

15 13 10 10 12 11 10 9 11 10 10 10

1,413.43 1,296.16 1,251.81 1,024.13 1,276.23 1,150.95 1,098.62 1,047.53 1,058.01 1,062.43 1,008.31 1,043.10

-13.33% 0.00% 10.00% -20.00% -8.33% -18.18% -10.00% -11.11% -18.18% -20.00% -10.00% -20.00%

4.99% 7.90% 0.38% -0.06% -5.36% 0.98% -2.71% -3.40% 3.42% -0.05% 7.64% -6.64%

r1

9.67

1,152.39

62.51

10.92

1,144.23

-11.45%

0.71%

r201 r202 r203 r204 r205 r206 r207 r208 r209 r210 r211

10 10 9 6 8 7 7 6 7 7 6

1,337.07 1,258.97 1,130.86 913.46 1,136.25 1,084.71 1,024.53 918.88 1,104.62 1,185.38 1014.32

41 46 62 91 53 62 76 92 64 59 77

13 12 10 9 12 9 11 8 10 10 9

1,335.17 1,215.88 1,122.58 1,013.70 1,183.14 1,068.91 1,064.22 1,088.12 1,067.09 1,076.23 943.45

-23.08% -16.67% -10.00% -33.33% -33.33% -22.22% -36.36% -25.00% -30.00% -30.00% -33.33%

0.14% 3.54% 0.74% -9.89% -3.96% 1.48% -3.73% -15.55% 3.52% 10.14% 7.51%

r2

7.55

1,100.83

65.69

10.27

1,107.14

-26.55%

-0.57%

rc101 rc102 rc103 rc104 rc105 rc106 rc107 rc108

12 11 10 9 12 10 9 9

1,454.01 1,403.06 1,278.33 1,188.22 1,426.29 1,253.11 1,189.06 1,212.69

49 57 73 96 59 67 81 89

13 11 11 9 12 12 11 10

1,599.01 1,434.52 1,268.81 1,263.25 1,405.72 1,297.67 1,243.08 1,269.90

-7.69% 0.00% -9.09% 0.00% 0.00% -16.67% -18.18% -10.00%

-9.07% -2.19% 0.75% -5.94% 1.46% -3.43% -4.35% -4.50%

rc1

10.25

1,300.60

71.27

11.13

1,347.75

-7.87%

-3.50%

rc201 rc202

10 9

1,554.93 1,356.14

46 60

11 10

1,510.67 1,415.67

-9.09% -10.00%

2.93% -4.21%

6.2 Restricted DP Heuristic for the VRPTW-EU

Problem rc203 rc204 rc205 rc206 rc207 rc208 rc2

DP heuristic Vehicles Distance CPU(s) 8 1,295.72 72 6 975.56 104 9 1,437.07 56 8 1,220.06 59 8 1,234.27 66 7 1.059.39 87 8.13

1,266.64

68.68

Best in Goel (2009) Vehicles Distance 10 1,274.45 9 1,264.73 11 1,521.10 11 1,418.40 10 1,171.94 8 1,201.13 10.00

1,347.26

123

Change Vehicles Distance -20.00% 1.67% -33.33% -22.86% -18.18% -5.52% -27.27% -13.98% -20.00% 5.32% -12.50% -11.80% -18.75%

-5.98%

Table 6.2: Results for the basic method without Directive 2002/15/EC The main reason for this remarkably large improvement with respect to the solutions found by the large neighborhood search algorithm proposed by Goel (2009) is presumably the following. Determining the feasibility of neighborhood solutions with respect to the rules of Regulation (EC) No 561/2006 requires significantly larger computation times than when this regulation is ignored. Therefore, the number of neighborhood solutions which can be evaluated in a fixed amount of time is reduced significantly when respecting this regulation. In contrast, the running time complexity of the restricted DP heuristic does not increase when including the rules of Regulation (EC) No 561/2006. Therefore, the number of states that can be investigated during a fixed amount of computation time does not significantly decrease when this regulation is respected. This advantage of the restricted DP heuristic over solution approaches based on local search does not hold for the classical VRPTW, since no difficult (but realistic) timing restrictions such as driving time regulations are included in this problem type. This is demonstrated by applying the restricted DP heuristic to the classical Solomon (1987) instances, which results in 1.2 more vehicles and 8.8% more travel distance compared with the best known solutions for these problem instances, on average. If a practical application allows for more computation time, then this would be beneficial for the above break scheduling method. For example, if H is set to 100, 000, then the average computation time increases to 11 minutes (which is still much smaller than the computation times allowed by Goel, 2009), but with an average additional reduction of the number of vehicles and travel distance of 1.46% and 1.90%, respectively. Restrictions on drivers’ working hours imposed by Directive 2002/15/EC are generally ignored in the literature. However, they do reduce the solution space

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and, therefore, may have a significant impact on the solution quality. This impact was tested by solving the benchmark problems of Goel (2009) with the basic break scheduling method. Table 6.3 shows the average results of the basic break scheduling method including Directive 2002/15/EC for the six problem sets. Columns five and six present the changes of the two criteria caused by the inclusion of Directive 2002/15/EC. It is obvious that these changes are significant (3.89% on average for the number of vehicle routes and 0.96% on average for the distance traveled). Therefore, Directive 2002/15/EC has a significant impact on the resulting vehicle schedules.

Problem c1 c2 r1 r2 rc1 rc2 a

Incl. Vehicles 10.33 5.75 9.67 7.91 10.25 8.50

working hours Distance CPU(s) 949.31 52.80 834.47 72.18 1155.89 59.22 1097.26 62.82 1300.14 67.57 1264.52 67.56

Change a Vehicles Distance 0.00% -1.67% 15.00% 8.31% 0.00% 0.30% 4.82% -0.32% 0.00% -0.04% 4.62% -0.17%

Change with respect to the results without Directive 2002/15/EC

Table 6.3: Results for the basic method including Directive 2002/15/EC Finally, the impact of the optional legal rules on the quality of vehicle routing solutions is investigated. These optional rules have been ignored in the literature, since they are hard to incorporate in existing solution methods for the VRPTW. However, in practice they are usually exploited (Onken, 2009), since they provide dispatchers and drivers with an enlarged decision space. Table 6.4 reports the average objective values for the six problem sets using the restricted DP heuristic with the extended break scheduling method. In columns five and six the results are compared with the results of the basic break scheduling method (see Table 6.3). These columns indicate the profitability of using the optional rules. The average results for all problem sets are improved. There is a significant reduction in the number of vehicles used (4.28% on average) and in the total distance traveled (1.54% on average). Therefore, again the benefits of using the optional rules are significant and these rules should be accounted for when constructing vehicle routes. This finding supports the results of Chapter 5. The computation times are larger than with the basic break scheduling method (111 seconds on average versus 65 seconds, and 241 seconds maximum versus 107

6.3 DDM using the Restricted DP Heuristic

Problem c1 c2 r1 r2 rc1 rc2 a

Incl. Vehicles 10.11 5.25 9.33 7.36 10.00 8.13

optional Distance 937.08 773.80 1142.62 1084.70 1322.41 1247.37

rules CPU(s) 85.90 138.25 102.15 105.20 122.69 120.51

125 Change a Vehicles Distance -2.15% -1.29% -8.70% -7.27% -3.45% -1.15% -6.90% -1.15% -2.44% 1.71% -4.41% -1.36%

Change with respect to the results of the basic break scheduling method

Table 6.4: Results for the extended method seconds). This can be explained by the additional checks in the extended break scheduling method which have to be preformed to exploit the optional rules. Moreover, the algorithm has to check whether a return to the depot is possible after adding a customer by invoking the extended break scheduling method. The computational results show that the restricted DP heuristic is well suited to address combined vehicle routing and break scheduling problems, since it clearly outperforms state-of-the-art heuristics for the VRPTW-EU.

6.3 Distributed Decision Making Using the Restricted Dynamic Programming Heuristic In this section the restricted DP heuristic is used to investigate the planning scenario with distributed decision structures between dispatchers and drivers introduced in Chapter 5.9 Since the algorithm allows problem instances of a more realistic size to be solved heuristically, its adaptation to distributed decision structures might offer new insights into the effects of different anticipation functions and of the team character of the planning situation. The DDM setting does not only require the VRPTW-EU to be solved. In the case of implicit anticipation, the dispatcher has to solve a VRPTW. Moreover, each driver has to solve a TSPTW-EU applying either the criterion of minimizing the travel distance or of minimizing his return time. The restricted DP heuristic can be modified such that it can be utilized to solve all the above problems. The algorithm is designed for the VRPTW-EU for which it has proven to be an efficient solution method. Furthermore, it can easily be adapted to the VRPTW by relaxing all break scheduling constraints, e.g. by setting the allowed accumulated 9

This section is based on Meyer et al. (2009).

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6 Restricted DP Heuristic for the VRPTW-EU

driving and working times to high values (e.g. to the planning time horizon of the problem instance). Finally, since the VRPTW-EU is a generalization of the TSPTW-EU, the restricted DP algorithm can also solve the TSPTW-EU. Again the benchmark instances proposed by Goel (2009) are used for testing the different anticipation functions. As required by the hierarchical nature of the planning process, the dispatcher’s problem is solved first for each problem instance and anticipation function. Subsequently, the clusters derived by the planner are stored and within each cluster a TSPTW-EU including the optional rules is solved using the restricted DP heuristic. Again, an H value of 10, 000 is used for all experiments.

6.3.1 Computational Experiments for the Decentralized Planning Approach To quantify the impact of the different anticipation functions, various computational experiments were conducted. This quantification also allows a comparison of different strategies of scheduling breaks and rest periods within vehicle routing. In the following the results for the team and for the non-team situation are presented.

6.3.1.1 Team Situation First the results of the team situation are presented, in which the drivers’ objective is to minimize their travel distance. For perfect and approximate anticipation, the dispatcher’s solution results in feasible vehicle routes for the drivers. Again it is assumed that a driver only changes the route suggested by the dispatcher, if he is able to derive a better route and break schedule in terms of his criterion. Table 6.5 presents the results for perfect and approximate anticipation. The first column indicates the different problem sets and, in brackets, the number of problem instances. Remember that in the C-instances, customer locations are clustered, in the R-instances they are random, and in the RC-instances they are semi-clustered; the 2-instances have a relatively longer time horizon and larger vehicle capacities than the 1-instances, allowing for longer vehicle routes in terms of number of customers. The results for the situation with perfect anticipation and the situation with approximate anticipation contain three columns each: the average (over all problem instances in each problem set) number of clusters found by the planner, the average travel distance of the routes found by the planner,

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127

and the average distance of the final routes found by the drivers. Note that, even in the case of perfect anticipation, the drivers may find better routes than the dispatcher, since the state space of each driver’s problem is smaller than the state space of the dispatcher’s problem, while the state space restrictions for both problems are the same (H = 10, 000). Problem set c1 (9) c2 (8) r1 (12) r2 (11) rc1 (8) rc2 (8)

Clusters 10.00 5.50 9.42 7.27 10.25 7.88

Perfect ant. Disp. dist. 947.39 787.00 1157.75 1092.68 1333.28 1219.54

Dr. dist. 946.38 785.17 1154.37 1091.36 1331.68 1218.99

Clusters 10.33 5.63 9.75 7.73 10.13 8.50

Approx. ant. Disp. dist. Dr. dist. 951.84 949.50 817.24 811.39 1158.67 1152.60 1106.28 1102.40 1297.58 1290.00 1269.66 1261.23

Table 6.5: Results for the team situation: perfect and approximate anticipation The results demonstrate that perfect anticipation clearly outperforms approximate anticipation. For all but one problem set, the average number of clusters is smaller in case of perfect anticipation. On average over all problem instances, perfect anticipation results in 3.5% less clusters than approximate anticipation. Also the travel distances are smaller in case of perfect anticipation than in case of approximate anticipation. On average, the difference is 0.90% for the routes found by the dispatcher, and 0.56% for the final routes found by the drivers. The improvements in terms of reduced travel distance found by the drivers are relatively small. However, for a significant number of customer clusters, the drivers find better routes than the dispatcher: 7.7% in the case of perfect anticipation and even 19% in the case of approximate anticipation. The larger portion in case of approximate anticipation is due to the larger solution space that the drivers consider by including the optional rules, which are ignored by the dispatcher in this case. The average reductions of the travel distances for these customer clusters are 1.4% and 2.4% in case of perfect and approximate anticipation, respectively. Table 6.6 presents the results for implicit anticipation. Different speed reductions are applied to create time buffers, which can be utilized by the drivers for scheduling breaks and rest periods. The speeds of 2, 3, 4, and 5 distance units per hour are used for the experiments, where 5 is the reference speed for each problem instance. The case in which the speed is set to the reference speed is the extreme situation in which the dispatcher neglects all breaks and rest periods which have to be scheduled by the drivers. However, since the dispatcher still

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6 Restricted DP Heuristic for the VRPTW-EU

does account for the routing including time windows, this is also a case of implicit anticipation. Since with implicit anticipation certain customer clusters may not allow for feasible routes and break schedules, such infeasibilities have to be considered. Therefore, for each problem set the average number of clusters found by the dispatcher is reported (column three), the average number of routes found by the dispatcher that allow feasible break schedules (column four), and the average number of clusters for which the drivers can find feasible routes and break schedules (column six). To make a fair comparison between the travel distances found by the dispatcher and by the drivers, the average total travel distance of each problem set is presented for those customer clusters, for which the routes found by the dispatcher permit feasible break schedules. Column five states these travel distances for the routes found by the dispatcher, and column seven reports them for the routes found by the drivers. The travel distances of the routes found by the dispatcher and the drivers are not reported for those customer clusters for which the routes found by the planner do not allow feasible break schedules, because it is not reasonable to compare travel distances of infeasible vehicle routes with other (in)feasible vehicle routes. The results indicate that the smaller the speed reduction, the smaller the number of clusters, but also the larger the number of infeasible clusters. The smallest and largest speeds do not seem to be suitable, since a speed of 2 results in too many clusters (on average 80% more than with perfect reactive anticipation), while a speed of 5 results in many infeasible clusters (for 30% of the customer clusters the drivers cannot find a feasible route and break schedule). However, the speeds of 3 and 4 show an interesting trade-off between solution quality (12% less clusters with speed 4 than with speed 3) against feasibility (13% of infeasible clusters with speed 4 against 4% with speed 3). In contrast with the findings of Chapter 5, there is no clear distinction between the clusters with shorter planning time horizon (1-problems) and with larger planning time horizon (2-problems). This might be due to the fact that for the 2-problems, less vehicles are required and each vehicle services more customers on its route. Therefore, the vehicle schedules are also tight in these scenarios, and additional time for breaks and rest periods can result in infeasibilities. For all speeds, there are customer clusters for which the routes found by the dispatcher do not allow feasible break schedules, while the drivers do find feasible routes and break schedules. On average, the drivers’ routing results in 2.4%, 15%,

6.3 DDM using the Restricted DP Heuristic

Speed

2

3

4

5

Problem set c1 c2 r1 r2 rc1 rc2 c1 c2 r1 r2 rc1 rc2 c1 c2 r1 r2 rc1 rc2 c1 c2 r1 r2 rc1 rc2

Clusters 13.89 14.38 14.25 13.82 18.75 19.38 10.00 8.00 10.08 9.00 11.00 10.50 10.00 6.00 9.08 7.64 10.13 8.38 10.00 5.25 8.58 6.55 9.50 7.63

Dispatcher Feas.a Dist.b 13.89 1219.67 13.00 1285.17 13.50 1229.34 13.18 1212.38 18.63 1840.88 19.13 1894.00 9.89 891.23 6.88 764.10 7.92 897.43 6.64 805.11 9.13 1074.80 8.63 1030.28 9.22 875.84 4.25 575.20 5.25 621.87 3.64 492.16 6.75 844.85 3.50 438.56 8.22 801.89 3.75 510.07 3.67 453.58 1.82 270.02 4.63 629.32 2.25 272.97

129

Drivers Feas.c Dist.d 13.89 1219.67 13.50 1284.76 14.00 1228.93 13.82 1209.57 18.75 1839.77 19.38 1889.82 10.00 891.05 7.25 764.10 9.42 895.66 8.91 803.54 10.75 1073.47 9.88 1027.50 9.78 875.24 5.00 574.72 7.58 621.00 6.09 490.63 9.63 843.14 6.50 437.20 9.22 799.46 4.13 510.04 5.92 452.32 3.18 269.58 7.88 629.14 4.13 272.62

a

Average number of routes found by the dispatcher that allow feasible break schedules Average total travel distance per problem instance for the feasible routes found by the dispatcher c Average number of customer clusters for which the drivers could find feasible routes and break schedules d Average total distance per problem instance for the routes found by the drivers for those customer clusters for which the routes found by the dispatcher allow feasible break schedules b

Table 6.6: Results for the team situation: implicit anticipation

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37%, and 43% more feasible vehicle routes than the routes found by the planner with speeds 2, 3, 4, and 5, respectively. The reductions in travel distances are similar to the case with perfect anticipation.

6.3.1.2 Non-Team Situation After investigating the situation in which the drivers follow the company’s objective of minimizing the travel distance, now the non-team situation is addressed in which the drivers’ (hidden) objective is to minimize their return times. This case can be easily implemented within the restricted DP algorithm by adjusting the objective function for the TSPTW-EU. The objective of minimizing the return time can be included by changing the cost of each state from the total distance traveled to the completion time of the last visited customer. The dominance criterion has to be changed accordingly. Table 6.7 presents the results for the non-team situation with perfect and approximate anticipation. The number of clusters is the same as in the team situation, since the dispatcher’s problem does not change, and each customer cluster allows for a feasible route and break schedule. Table 6.7 presents the average total travel distance and the average return time for the routes found by the dispatcher, and for the routes found by the drivers. Anticipation function

Perfect

Approximate

Problem set c1 c2 r1 r2 rc1 rc2 c1 c2 r1 r2 rc1 rc2

Dispatcher Dist. Ret. 947.39 979.08 787.00 684.13 1157.75 959.97 1092.68 821.21 1333.28 1041.40 1219.54 921.74 951.84 1008.52 817.24 702.24 1158.67 1014.32 1106.28 878.76 1297.58 1064.81 1269.66 1015.06

Drivers Dist. Ret. 1045.13 956.58 791.67 681.16 1256.69 938.34 1151.91 805.82 1473.23 1012.01 1279.46 914.37 1055.97 965.57 850.74 689.08 1350.24 960.17 1200.71 847.73 1472.51 1021.15 1389.64 985.26

Table 6.7: Results for the non-team situation: perfect and approximate anticipation The results indicate that the drivers can improve the routes found by the dispatcher with respect to their (hidden) objective. The average return time reductions over all problem instances are 1.9% and 3.9% in case of perfect and approx-

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imate anticipation, respectively. However, by improving the routes according to their private criterion, the drivers deteriorate the dispatcher’s secondary objective, the total travel distance, by 7.1% and 11%, respectively. The percentage of routes that could be improved by the drivers in terms of their criterion is 30% in case of perfect anticipation and 57% in case of approximate reactive anticipation. Since the dispatcher’s top-down criterion conflicts with the base level’s criterion, there is much more room for improvement by the drivers than in the team situation, in which these criteria are in line. Another interesting observation for the non-team situation is that the difference between perfect and approximate anticipation is bigger with respect to the secondary objective than in the team situation. The difference between perfect and approximate anticipation in terms of travel distance for the routes found by the drivers is 4.71%. This is much larger than the 0.56% in the case of the team situation. This difference is due to the fact that perfect anticipation results in tighter routes found by the dispatcher than approximate anticipation. Therefore, if drivers take different routes, which better fulfill their private objective, it is unlikely that the total travel distance is much bigger. Less tight routes, derived through approximate anticipation, may result in larger increases of the travel distance. The difference in return time between perfect and approximate anticipation is also significant: 5.2% for the routes found by the dispatcher and 3.0% for the routes found by the drivers. These results indicate that a more precise representation of the base level’s planning model within a non-team situation has a positive impact on the quality of the planning, both in terms of the dispatcher’s top-down criterion and in terms of the base-level’s criterion. In a non-team situation, this impact is even stronger than in a team situation. Table 6.8 presents the results for the implicit anticipation functions in the nonteam situation. The results for the number of feasible clusters are similar as in the team situation. However, there are some small differences in the number of feasible routes found by the drivers, due to the different objective function (this is caused by the fact that all problems are solved using a heuristic and not an exact solution method). The reductions of the return times found by the drivers with respect to the feasible vehicle routes found by the dispatcher are 1.7%, 1.8%, 1.2%, and 2.0% for speeds of 2, 3, 4, and 5, respectively. These improvements of the drivers’ objective result in a deterioration of the dispatcher’s top-down criterion, the total travel distance, of 7.7%, 8.9%, 6.4%, and 6.5%, respectively. Therefore, they are similar to the case of perfect anticipation. The percentage of routes that could be improved by the drivers are 29%, 25%, 13%, and 13%, respectively. The

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fact that larger speeds allow for less improvements by the drivers is probably due to tighter routes with respect to the time windows using these higher speeds. Apparently, introducing less slack travel time not only reduces the number of routes found by the planner for which a feasible break schedule exists, but also allows less improvement possibilities for the drivers on such routes. Speed

2

3

4

5

Problem set c1 c2 r1 r2 rc1 rc2 c1 c2 r1 r2 rc1 rc2 c1 c2 r1 r2 rc1 rc2 c1 c2 r1 r2 rc1 rc2

Clusters 13.89 14.38 14.25 13.82 18.75 19.38 10.00 8.00 10.08 9.00 11.00 10.50 10.00 6.00 9.08 7.64 10.13 8.38 10.00 5.25 8.58 6.55 9.50 7.63

Feas. 13.89 13.00 13.50 13.18 18.63 19.13 9.89 6.88 7.92 6.64 9.13 8.63 9.22 4.25 5.25 3.64 6.75 3.50 8.22 3.75 3.67 1.82 4.63 2.25

Dispatcher Dist. Ret. 1219.67 1188.06 1285.17 1333.05 1229.34 1230.14 1212.38 1242.96 1840.88 1588.59 1894.00 1784.48 891.23 954.75 764.10 778.55 897.43 791.89 805.11 685.09 1074.80 899.56 1030.28 899.53 875.84 866.41 575.20 494.01 621.87 527.80 492.16 378.42 844.85 686.45 438.56 369.49 801.89 765.99 510.07 455.07 453.58 357.64 270.02 198.71 629.32 477.33 272.97 196.32

Feas. 13.89 13.63 13.58 13.82 18.50 18.88 10.00 7.25 9.42 9.00 10.75 10.00 9.78 4.50 7.58 5.82 9.50 5.88 9.22 2.88 6.17 2.82 7.75 3.75

Drivers Dist. 1268.54 1336.36 1343.04 1377.49 1965.17 2026.05 968.36 774.90 1012.43 869.38 1165.78 1145.04 929.03 580.71 672.39 506.27 944.07 464.81 852.89 518.24 483.27 284.53 711.27 279.27

Ret. 1171.79 1320.98 1201.93 1216.44 1555.87 1758.44 935.59 775.11 773.58 671.63 878.53 885.49 850.78 493.61 523.75 375.64 670.53 366.82 747.99 454.26 349.53 194.92 461.92 192.98

Table 6.8: Results for the non-team situation: implicit anticipation

6.3.2 Summary of the Findings In both the team and the non-team situation the explicit anticipation functions are superior to the implicit anticipation functions both in terms of the dispatcher’s and the drivers’ objectives. Even when only a small speed reduction is applied (speed 4, for which the percentage of infeasible clusters is still rather large: 12%), the average number of customer clusters is still larger than in the case of perfect anticipation. However, since explicit anticipation is also more complex, dispatchers face a clear trade-off between the solution quality and the complexity of their

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planning task. Furthermore, if dispatchers resort to implicit anticipation of the drivers’ planning behavior, they face a trade-off between the solution quality in terms of the number of vehicles used, and the feasibility of the resulting tours. If the drivers do not follow the company’s objectives, but optimize their own hidden criterion instead, the dispatcher’s primary objective - the number of vehicles used - is not affected. However, the dispatcher’s secondary objective - the total travel distance - is deteriorated significantly by the drivers’ hidden actions. Here it turns out that a more precise anticipation of the drivers’ planning model results in a less severe deterioration of the dispatcher’s top-down criterion. This is due to the fact that drivers do not have so many possibilities for deviating from the routes found by the dispatcher. These results suggest that in practice instead of solely creating time buffers to schedule breaks and rest periods, dispatchers should try to use a more precise representation of the drivers’ planning model. A more precise anticipation will help to better fulfill the company’s objectives, both in case of a team and in case of a non-team situation. However, speeds of 3 or 4 distance units per hour, i.e. speed reductions between 20% and 40%, seem to offer reasonably good results. Especially the speed of 3 performs quite well with respect to feasibility (only 4 % of the clusters were infeasible) and also solution quality. Therefore, if the dispatcher resorts to using time buffers, such a medium speed reduction seems to be an acceptable option. Since in practical freight forwarding the integration of time buffers is a common principle of dealing with the problem of combined vehicle routing and break scheduling (see Section 3.3), this is a valuable guideline for dispatchers at freight forwarding companies.

7 Conclusions and Further Research Towards the end of this thesis it is time to reflect on the main results obtained through this study. This reflection process also reveals aspects which were left out of the scope of this work. They leave room for further research in the field of combined vehicle routing and break scheduling.

7.1 Conclusions for Vehicle Routing Under Consideration of Driving and Working Hours In this thesis the problem of vehicle routing under consideration of driving and working hours was analyzed both from a central and from a distributed decision making perspective. In practical freight forwarding, the partial tasks involved are usually divided among dispatchers and drivers. A study showed that usually the clustering of customer requests and also the routing is performed by the dispatchers, whereas the scheduling of breaks and rest periods is left to the drivers. Especially since the introduction of the new European social legislation, planners are forced to build vehicle routes and schedules which respect the legal restrictions on driving and working hours. Therefore, the problem of creating vehicle routes such that the drivers can subsequently schedule breaks and rest periods satisfying the legal rules presents a major challenge for dispatchers of freight forwarding companies operating in the European Union. To obtain a formal problem representation, first a mathematical model for the VRPTW-EU was developed which includes all basic and optional rules of the European social legislation. This model assumes a central planning perspective in which a single decision maker is responsible for all partial planning tasks. Computational experiments showed that the times required for breaks and rest periods take an enormous share of the total fulfillment time of the resulting vehicle schedules. Therefore, in a practical context, times required for breaks and rest periods should not be neglected in the process of vehicle routing and scheduling. The central planning model represented a first formal representation of the problem of combined vehicle routing and break scheduling. Since this central C. M. Meyer, Vehicle Routing under Consideration of Driving and Working Hours, DOI 10.1007/978-3-8349-6732-9_7, © Gabler Verlag | Springer Fachmedien Wiesbaden GmbH 2011

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planning perspective does not align with the reality in practical freight forwarding, the problem was modeled from a DDM perspective. This perspective accounts for the different decision makers involved in solving the problem in practice, namely dispatchers and drivers. Based on an analysis of the dispatching process, ways for planners to integrate the legal restrictions on driving and working hours into their planning were suggested. In practice, a common way of accounting for breaks and rest periods is to incorporate time buffers into the vehicle routes which can be used by the drivers for their subsequent planning of breaks and rest periods (see Section 3.3). Within the DDM model, the inclusion of time buffers was modeled as an implicit anticipation function by applying a reduced speed within the planner’s routing. Compared with the explicit anticipation functions, this approach is rather simple since it does not require the explicit consideration of breaks and rest periods by the dispatcher. The dispatcher’s tasks can therefore be solved solely by performing the clustering of customer requests and the routing of the vehicles, as in traditional vehicle routing approaches. This convenience is also one major reason for its wide application in practice. However, the reduced complexity comes at the cost of not being able to guarantee the feasibility of the resulting vehicle schedules. Infeasible schedules may result, which do not satisfy the legal rules or the time windows defined by the customers. This is a major drawback compared with the explicit approaches which do guarantee the feasibility of the resulting routes. Despite its simplicity and the danger of creating infeasible vehicle schedules, in the computational experiments the usage of time buffers did not perform as poor as one might have expected. Different speed reductions were tested and an average speed reduction of between 30 % and 40 % actually showed a quite reasonable performance (45 km/h instead of 65 km/h and speeds of 3 or 4 distance units per hour instead of the reference speed of 5). This also corresponds with the findings of the central planning model, where the share of time required for breaks and rest periods amounted to approx. 36 % of the total fulfillment times on average (with the objective function of minimizing the total driving time). Especially in scenarios with a large planning time horizon and customers with relatively wide time windows, the application of buffer times seems to be a reasonably well suited approach. When solving the DDM model in Chapter 5, for all such test problems even the same vehicle schedules could be derived as in the benchmark case of perfect explicit anticipation. Therefore, a first conclusion is that the practical approach of using time buffers seems to be well suited for

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such a planning scenario. Nevertheless, for planning scenarios with a short planning time horizon and relatively dense time windows, the application of buffer times resulted in unacceptably many infeasible vehicle routes. In such a planning context, dispatchers should better revert to explicit anticipation of the drivers’ break scheduling since these explicit approaches guarantee the feasibility of the resulting vehicle schedules. Only considering the basic legal rules allows planners to create feasible schedules without facing the full complexity of the legal rules. However, this approach resulted in highly deteriorated vehicle schedules compared with the utilization of the entire set of legal rules. The inclusion of the optional rules enhances the dispatcher’s decision space and allows him to find vehicle schedules which better fulfill his criteria. However, this comes at the cost of facing a higher planning complexity. In Chapter 6, a restricted DP heuristic was introduced which can be used for solving combined vehicle routing and break scheduling problems. It allows dispatchers to consider both the basic and the optional rules of the European social legislation. Even the integration of the optional rules does not result in an increased running time complexity of the break scheduling heuristic. Since this algorithm thus allows dispatchers to design vehicle routes including the optional legal rules efficiently, the trade-off between the increased complexity and the solution quality is negligible. Therefore, a second conclusion is that in the above planning scenario, dispatchers should always consider breaks and rest periods explicitly and incorporate the optional legal rules when creating vehicle schedules.

7.2 Outline of Further Research Directions As already pointed out in the previous section, this thesis could provide some valuable insights into the wide research field of vehicle routing under consideration of driving and working hours. However, lots of work remains to be done, out of which some possibilities for extending the results presented in this thesis are now outlined. First, possible extensions of the work presented here can focus on the problem setting. In this thesis the focus was on a setting in which the dispatchers are responsible for clustering the customer requests and they can also give advice on the routes and breaks to be taken. However, the final decision on the routing and break scheduling is left to the drivers. Of course, other distributions of tasks

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are possible. For example, it is common, although less interesting from a mathematical perspective, that the planners perform both the clustering and routing, and the drivers have to use the routes they are instructed by the planners. On the other hand, it might also be possible to have autonomous cooperating drivers who perform all three partial planning tasks in combined vehicle routing and break scheduling by themselves. In this setting the dispatcher’s role is reduced to administrative tasks. The clustering of the transportation requests among the drivers would then have to be organized by some coordination scheme such as auctions or negotiations, possibly using multi-agent systems. Throughout this thesis, a fixed assignment of drivers to vehicles was assumed. Problems of crew scheduling could therefore be neglected. However, in practice one truck can be used by different drivers, and also multi-manning can be encountered. Therefore, the problem could be enhanced by allowing for a flexible assignment of drivers to vehicles and taking into account the scheduling of the crews. Moreover, a means of making the problem even more realistic would be to include real-time information on travel times into the planning process at the driver level. In practice, drivers are often faced with unforeseen road conditions such as traffic jams, which render the plans derived by the planners infeasible. Including such real-time information and finding strategies for the planners to cope with changing problem characteristics in a non-deterministic setting would also be an interesting and challenging field for further study. Apart from those extensions of the problem outset, also different methods for addressing the problems are worth further effort. In Chapter 4, a position-based model formulation for the VRPTW-EU was developed. The positions at which a customer is visited along the route are used to compute the driving times up to any customer node on the route. However, this approach does not allow to schedule breaks and rest periods at exact points of time, but rather to extend the travel times by the time required for breaks and rest periods. The major drawback of this modeling approach is that it can only be used for problems with rather short time horizons and travel distances. The problem formulation can easily be extended to a pickup and delivery problem, which usually comprises longer haulages. Nevertheless, the resulting model formulation is not suitable for addressing problems with arcs requiring long driving times. A problem in which a truck has to pick up freight e.g. in Spain and deliver it directly to a customer in Germany can therefore not be solved by this position-

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based model. Consequently, different modeling approaches have to be developed in order to adapt the models to long distance haulage. The restricted DP heuristic presented in Chapter 6 yielded very promising results for the VRPTW-EU which were documented by computational experiments. Furthermore, in contrast with local search techniques the break scheduling heuristic presented in this thesis runs even in constant time. To date, the restricted DP heuristic is the only heuristic which can account for all the optional rules of the European social legislation and therefore allows to solve both the dispatcher’s and the drivers’ planning problems in the DDM setting. However, it is desirable to have some benchmark solutions which are solved to optimality. Therefore, a major challenge is to develop exact solution algorithms for the VRPTW-EU such as e.g. branch and cut algorithms which have already performed well on other vehicle routing problems with complex timing constraints. It is obvious that lots of work remains to be done in the field of vehicle routing under consideration of driving and working hours. The results presented in this thesis can be regarded as a contribution to the investigation of the general problem and as a first step towards accounting for the different decision makers involved in solving this problem in practice.

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