Document [original]

Dissertation

Topics in Integrated

Vehicle and Crew Scheduling

in Public Transport

Dipl. Wirt.-Inform. Ingmar Steinzen

Schriftliche Arbeit zur Erlangung des akademischen Grades

doctor rerum politicarum (dr. rer. pol.)

im Fach Wirtschaftsinformatik

eingereicht an der

Fakult¨at f¨ur Wirtschaftswissenschaften der

Universit¨at Paderborn

Paderborn, im Juli 2007

Datum der m¨undlichen Pr¨ufung: 23.11.2007

Gutachter:

1. Prof. Dr. Leena Suhl

2. Prof. Dr. Knut Haase

It is not the mountain

we conquer – but ourselves.

Sir Edmund Hillary

Acknowledgements

The thesis in front of you is a result of the research that I conducted as member of

the International Graduate School of Dynamic Intelligent Systems and the Deci-

sion Support & Operations Research Lab (DSOR) at the University of Paderborn.

This dissertation would not have been possible to complete in the last three years

without the precious support of so many people.

First and foremost, I am very grateful to my supervisor Prof. Dr. Leena

Suhl for giving me the opportunity to work in her working group and to write

this thesis. It was her valuable guidance, support, and ever-friendly nature that

added a great deal to the completion of this work.

Furthermore, I thoroughly enjoyed the friendly atmosphere and innumerable

interesting discussions that I had with my colleagues at the working group. In

particular, I would like to thank Vitali Gintner for providing such a great company

and the general support during the first two years. Moreover, several students

and student assistants made valuable contributions in the implementation of the

optimization system. Among others I wish to thank Bastian, Boris, and Viktor.

I am indebted to the International Graduate School of Dynamic Intelligent

Systems, its director PD Dr. Eckhard Steffen, and its support staff who accom-

panied and financially supported my work. I hope many future students will have

the opportunity to participate in their excellent PhD program.

Finally, I wish to sincerely thank my father and his wife for their constant

support and encouragement in all my private and professional endeavors. Most

importantly, I thank Andrea. I am very fortunate to have the opportunity to

share everything with you that worries me, bothers me, delights me, or just

makes me laugh.

Ingmar Steinzen

Paderborn, July 2007

Contents

1. Introduction 1

1.1. Planning Process of Public Transport Companies . . . . . . . . . 2

1.1.1. Vehicle Scheduling . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2. Crew Scheduling . . . . . . . . . . . . . . . . . . . . . . . 6

1.2. Integrated Vehicle and Crew Scheduling . . . . . . . . . . . . . . 8

1.3. Irregular Timetables . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4. Selected Combinatorial Optimization Problems . . . . . . . . . . 12

1.4.1. Network Flow Problems . . . . . . . . . . . . . . . . . . . 13

1.4.2. Set Partitioning/Covering Problem . . . . . . . . . . . . . 16

1.5. Selected Combinatorial Optimization Techniques . . . . . . . . . . 17

1.5.1. Lagrangian Relaxation . . . . . . . . . . . . . . . . . . . . 17

1.5.2. Dantzig-Wolfe Decomposition and Column Generation . . 20

1.5.3. Lagrangian Relaxation based Column Generation . . . . . 24

1.5.4. Branch-and-Bound . . . . . . . . . . . . . . . . . . . . . . 26

1.5.5. Metaheuristics . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.6. Scope and Purpose of the Thesis . . . . . . . . . . . . . . . . . . 28

2. Integrated Vehicle and Crew Scheduling: State-of-the-Art 31

2.1. Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2. LiteratureReview........................... 33

2.2.1. Sequential Vehicle and Crew Scheduling . . . . . . . . . . 33

2.2.2. Partial Integration . . . . . . . . . . . . . . . . . . . . . . 38

2.2.3. Complete Integration . . . . . . . . . . . . . . . . . . . . . 40

2.3. Modelingapproach .......................... 47

2.4. Solution Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.4.1. The Master Problem . . . . . . . . . . . . . . . . . . . . . 55

2.4.2. The Column Generation Pricing Problem . . . . . . . . . . 57

2.4.3. Integer Solutions . . . . . . . . . . . . . . . . . . . . . . . 59

3. New Approaches to Integrated Vehicle and Crew Scheduling 61

3.1. Modeling the Column Generation Pricing Problem . . . . . . . . . 62

3.1.1. Modeling Approaches . . . . . . . . . . . . . . . . . . . . . 64

vii

3.1.2. Network Models for a Decomposed Pricing Problem . . . . 66

3.2. Solving the Column Generation Pricing Problem . . . . . . . . . . 74

3.2.1. Dynamic Programming Algorithms . . . . . . . . . . . . . 76

3.2.2. Preprocessing ......................... 78

3.2.3. Acceleration Techniques . . . . . . . . . . . . . . . . . . . 85

3.3. IntegerSolutions ........................... 93

3.3.1. Sequential Approach . . . . . . . . . . . . . . . . . . . . . 94

3.3.2. Branch-and-Bound with MIP-Solver . . . . . . . . . . . . . 96

3.3.3. Heuristic Branch-and-Price . . . . . . . . . . . . . . . . . . 102

3.4. Integrated Planning with Unrestricted Changeovers . . . . . . . . 110

3.5. Computational Results . . . . . . . . . . . . . . . . . . . . . . . . 114

3.5.1. Real-world Data Instances . . . . . . . . . . . . . . . . . . 116

3.5.2. Randomly Generated Data Instances . . . . . . . . . . . . 117

3.6. Summary ............................... 122

4. A Hybrid Evolutionary Algorithm 125

4.1. Problem Decomposition . . . . . . . . . . . . . . . . . . . . . . . 126

4.2. Components of Evolutionary Algorithm . . . . . . . . . . . . . . . 127

4.2.1. Initialization.......................... 128

4.2.2. Fitness Calculation . . . . . . . . . . . . . . . . . . . . . . 128

4.2.3. Genetic Operators . . . . . . . . . . . . . . . . . . . . . . 131

4.2.4. Termination.......................... 132

4.3. Computational Results . . . . . . . . . . . . . . . . . . . . . . . . 132

4.4. Summary ............................... 135

5. Practical Extensions 137

5.1. Rules and Regulations in Germany . . . . . . . . . . . . . . . . . 137

5.2. Extensions of Modeling and Solution Approach . . . . . . . . . . 139

5.2.1. Driving Time Constraints . . . . . . . . . . . . . . . . . . 140

5.2.2. Block and Ratio Break Rules . . . . . . . . . . . . . . . . 141

5.2.3. Break Positions . . . . . . . . . . . . . . . . . . . . . . . . 142

5.2.4. DutyMix ........................... 143

5.3. SystemOverview ........................... 145

5.4. Computational Results . . . . . . . . . . . . . . . . . . . . . . . . 147

5.5. Summary ............................... 149

6. Ex-Urban Vehicle and Crew Scheduling with Irregular Timetables 151

6.1. ProblemDefinition .......................... 152

6.2. Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

viii

6.3. Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . 157

6.4. Solution Approaches . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.4.1. Local Branching and Branching Rules . . . . . . . . . . . . 158

6.4.2. Bi-Objective Metaheuristics . . . . . . . . . . . . . . . . . 163

6.5. Computational Results . . . . . . . . . . . . . . . . . . . . . . . . 166

6.6. Summary ............................... 172

7. Summary and Concluding Remarks 177

A. Definitions and Abbreviations 181

List of Figures 184

List of Tables 186

List of Algorithms 187

Bibliography 189

1. Introduction

Vehicle and crew scheduling are two major planning problems arising in public

bus transport companies. Briefly stated, these problems aim at assigning vehicle

itineraries to scheduled trips and crews itineraries to tasks resulting from the

vehicle schedule.

For several years now, Operations Research (OR) has been successful in solving

vehicle and crew scheduling optimization problems in public transport. Several

commercial software systems have been developed and used by public transport

companies to support planning and to run their operation. Many companies in

Germany and other countries have adapted these tools, mainly for three reasons:

•These companies face an increasing cost pressure and competition due to

market deregulation. Furthermore, the level of subsidy is being gradually

reduced. As a consequence, efficient utilization of available resources is

more and more important.

•Scheduling vehicles and crews has become increasingly complex due to

larger problem sizes and an increased complexity of labor rules.

•The power of computers and algorithms has advanced remarkably.

The main objective of a vehicle and crew schedule is to offer a given service

that allows passengers to travel easily at a low fare while minimizing asset and

operating costs. Traditionally, both planning steps have been approached se-

quentially where vehicle schedules are determined before crew schedules. In this

thesis, however, we focus on the integrated consideration of vehicle and crew

scheduling. The integration of both planning steps discloses additional flexibility

that can lead to gains in efficiency compared to sequential planning.

However, we do not only focus on how to conduct operations at minimum

cost but also on another aspect which is related to the quality of crew schedules.

In particular, we consider the case where schedules consist of trips serviced ev-

ery day as well as trips that do not repeat daily. The traditional approach to

crew scheduling usually produces irregular crew schedules which are undesired in

practice. Regularity is an important aspect for crew schedules in public transport

1. Introduction

since regular solutions are less error-prone and are easier to implement and to

manage.

In the remainder of this chapter, we describe the planning process of a public

transport company (Section 1.1), the integrated treatment of vehicle and crew

scheduling in Section 1.2, and the effect of irregular timetables on the regularity

of crew schedules (Section 1.3). In Sections 1.4 and 1.5 we discuss some well-

known combinatorial optimization problems and techniques, respectively. The

chapter is concluded in Section 1.6 with a discussion of the scope and purpose of

the thesis.

1.1. Planning Process of Public Transport

Companies

Most public transport companies aim to offer a service of good quality that allows

passengers to travel easily at a low fare while utilizing the resources at their dis-

posal as efficient as possible. The complete planning process in public transport

is very complex and is computationally not tractable as a whole. Consequently,

it is traditionally divided into a strategic,tactical, and operational phase. Each

phase is further split into several subproblems that are successively solved. Fig-

ure 1.1 depicts the entire planning process. In the following, we briefly describe

each subproblem.

In strategic planning, we are concerned with network design and line planning.

The planning horizon is typically several years. It is generally assumed that an

origin-destination (O-D) matrix is available for strategic planning. Each entry in

the matrix gives the number of passengers that want to travel between any two

points in the network by time of the day. Based on this demand data, the network

design problem is to determine the links of the network such that construction

costs are minimized. Of course, these links must provide sufficient capacity to

transport the estimated number of passengers. The line planning problem consists

of choosing a set of line routes (see Figure 1.2) and their frequencies for a given

transportation network such that passenger demand can be satisfied. There are

two conflicting objectives: maximize passenger comfort and minimize operating

costs of the lines. Passenger comfort can be measured by the total transit time

or the number of direct connections.

The tactical planning phase aims at timetable construction. The timetabling

problem is solved on a seasonal basis for given line routes, their frequencies, trav-

eling times along the lines, and any potential layover times at stations. The task

is to convert the desired frequency of a line into a detailed timetable. A timetable

1.1. Planning Process of Public Transport Companies

relief points

Strategic

timetabling timetable

vehicle scheduling vehicle blocks

crew scheduling crew duties

crew rostering crew rosters

TacticalOperational

O-D matrix

work regulations

network design transportation

network

service trips

tasks

line planning line network /

frequencies

Figure 1.1.: Planning Process of a Public Transport Company

corresponds to a set of (service) trips with start and end locations and times.

A common objective is maximum synchronization of trips such that transfers

within the network are well-timed and passengers can transfer between lines with

minimum waiting time. Similar to [Desaulniers and Hickman, 2006] we include

the timetabling step in the tactical level (and not in the operational phase), since

it is primarily focused on service quality (and not on cost minimization).

At the operational level, planning is concerned with constructing vehicle and

crew itineraries that minimize total costs while considering all operational con-

straints and work regulations. In other words, this phase solves the problem of

how to conduct operations to offer the proposed service at minimum cost.

The vehicle scheduling problem is to assign vehicles to trips resulting in vehicle

1. Introduction

PaderSprinter-Liniennetzplan

* Inbetriebnahme nach baulicher Ferigstellung

Elsen

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1 5 11

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Dahl

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Wewer

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Gartenstadt

Sande

Gesseln

Thuner Siedlung

Ingolstädter Weg

Marienloh

Neuenbeken

Friedhof

Auf dem Dören

Kaukenberg

Schöne

Aussicht

Heinz Nixdorf

Wendeschleife

Auf der Lieth

Lemgoer Str.

Mönkeloh

Pastorskamp

Zollhaus

Infanterieweg

Gartenstadt

Hagebuttenweg

Sunderkampstr.

Elsen Heide

Obernheideweg

Lerchenweg

Ingolstädter Weg

Von-Dript-Weg

Friedhof

Auf dem Dören

BussardwegBussardwegBussardweg

Mistelweg

Vinsebecker Weg

Bahneinschnitt

Heinz Nixdorf

Wendeschleife

Lemgoer Str.

Schulzentrum

Niesenteich

Vereinshaus

Westerntor

Karl-Korthaus-Str.

Kilianplatz

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burger

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str.

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friedhof

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zur Heide

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weg

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Habichtsweg

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str.

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lohstr.

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TÜV

An der Talle

Württemberger Weg

Schleswiger Weg

Haustenbecker Str.

Pirolweg

Ausbesserungswerk

Bonifatiusweg

Zum Kampe

Im Vogtland

Schlesierweg

Tegelweg

Füllers Heide

Gerold

Anhalter Weg

Bonifatiuskirche

G.-Schultze-Str.

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weg

Senneweg

Marienloh Mitte

Cheruskerstr.

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derstr.

SB-Zentral-

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Bekscher Berg

Roncalliplatz

Klostergarten

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haus

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Im Spirings-

felde

Ostfriedhof

Auf der Lieth

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Post

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Dahler Heide

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weg

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platz

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station

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str.

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teich

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Schützenplatz

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Wilhelmshöhe

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Almerfeld

Goerdelerstr.

Mentropstr.

Mittelweg

Gesamtschule

Mühlenteichstr.

Karl-Arnold-Str.

Sande, Friedhof

Ostenländer

Str.

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Sande, Schule Waldweg

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Stadtheide

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Schützenplatz

Nord Nordbahnhof

Wilseder Weg

Rotheweg

Langer Weg

Piepen-

turm-

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Bhf. Kasseler Tor

Corveyer Weg

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Schulstr.

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gymnasium

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Aue

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haus

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weg

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denstr.

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Warburger Str.

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teich

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str. Abzw. Fischteiche

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kamp

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landweg

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Dahler Weg

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Fixberg

Weikenweg

Mastbruch Gaststätte

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haus

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Hauptlinien

Hauptbahnhof – Sennelager

Wewer – Marienloh

Mönkeloh – Lemgoer Straße

(Ahorn Sportpark) – Heinz Nixdorf

Wendeschleife – Dahl

Nebenlinien

Thuner Siedlung – Hauptbahnhof

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Hauptbahnhof – Kaukenberg

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Sande – Auf der Lieth

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Hauptlinie

Nebenlinie

Fahrtrichtung

Haltestellen

Umsteigehaltestellen

Tarifgebiet Paderborn Innenbereich

Tarifgebiet Paderborn Außenbereich

AnrufLinienFahrt

Legende

Figure 1.2.: Excerpt from line network of PaderSprinter, Paderborn (Germany)

blocks. A vehicle block is the daily schedule for one vehicle. Each vehicle block

is a sequence of tasks where each task needs to be covered by a crew duty in

the crew scheduling problem. A crew duty is the workload of an anonymous

driver for one day that must satisfy a number of work regulations. The crew

rostering problem consists of constructing long-term (monthly) work schedules

(called crew rosters) from short-term (daily) crew duties considering several work

regulations. Unlike crew duties, crew rosters are assigned to individual drivers.

The planning horizon for both vehicle and crew scheduling is usually one day,

while crew rostering problems are solved on a monthly basis. In the following

subsections, we describe vehicle and crew scheduling in more detail since this

thesis is mainly focused on these problems.

1.1.1. Vehicle Scheduling

A bus depot is basically a maintenance and storage facility where buses may be

parked and serviced when not in use. A depot may have a maximum storage

capacity. Vehicles start and terminate their daily schedule in a depot. A fleet

1.1. Planning Process of Public Transport Companies

may consist of different vehicle types that differ in capacity, speed or, equipment.

In practice, there are often trips that must be operated by a specific vehicle type

or a subset of vehicle types. Furthermore, the number of vehicles in a facility

of a particular type must be equal at the beginning and end of a day. As a

consequence, we will treat a combination of facility and vehicle type as depot.

A timetable defines a set of trips that are used to carry passengers. Generally,

it is assumed that start and end location for all trips are fixed as well as their start

and end times. Given a set of timetabled trips, the vehicle scheduling problem

(VSP) can be stated as follows: find an assignment of trips to vehicles such that

•each trip is assigned exactly once,

•each vehicle performs a feasible sequence of trips,

•each sequence starts and ends at the same depot, and

•asset and operational costs are minimized.

Two trips are said to be compatible if they can be covered by the same vehicle.

Trips operated in sequence by the same vehicle are linked by deadheads. Dead-

heads are vehicle movements or idle times (or both) without carrying passengers.

A vehicle is idle if it stands (idle) at a location other than the depot. A vehicle

block is a sequence of compatible trips that starts with a pull-out trip and ends

with a pull-in trip. A pull-out trip connects the depot with the start location

of the first trip while a pull-in trip moves a vehicle from the end location of the

last trip to the depot. A daily schedule for one vehicle can thus include several

vehicle blocks. Figure 1.3 depicts an example of a daily schedule for one vehicle

with two blocks.

depot

Figure 1.3.: Schedule of one vehicle consisting of two blocks

1. Introduction

While asset costs usually correspond to investment and maintenance costs,

public transport companies define operational cost in different ways such as dis-

tance driven without passengers or waiting time. In most practical situations,

companies try to minimize their asset costs first and leave operational cost min-

imization as a secondary objective.

In the case that there is only one depot, a homogeneous fleet, and no route

constraints, we have the standard single depot vehicle scheduling problem (SD-

VSP). If there are multiple depots, we have the multiple-depot vehicle scheduling

problem (MDVSP). In the latter case, each vehicle is assigned to a given depot

and, possibly, some trips have to be serviced by vehicles from a certain subset of

depots.

It is well known that SDVSP corresponds to a minimum cost flow problem that

can be solved in polynomial time while its multiple-depot counterpart MDVSP

is NP-hard (see [Bertossi et al., 1987]). Additionally, L¨obel [L¨obel, 1997] shows

that an -approximation of MDVSP is also NP-hard. The complexity of a specific

problem instance mainly depends on the number of depots, the number of trips,

and the number of potential deadheads. Of course, additional constraints, e.g.

depot capacities or route time constraints, can be imposed that make instances

more challenging.

1.1.2. Crew Scheduling

Crew scheduling plays an important role in the operational planning process since

crew costs generally dominate vehicle costs (see [Bodin et al., 1983], [Leuthardt,

1998]). Instead of assigning trips to vehicles as in the preceding phase, we now as-

sign tasks to crews. A basic assumption is that all crews are equal since individual

crew members are not considered.

The crew scheduling problem (CSP) is defined as follows: find a set of duties

for a given set of tasks such that

•each task is covered by a duty that can be performed by a single driver,

•each duty satisfies a wide variety of federal laws, safety regulations, and

(collective) in-house agreements, and

•labor costs are minimized.

A task is a sequence of activities (such as performing trips or deadheading) be-

tween two consecutive relief points and represents an elementary portion of work

that can be assigned to a driver. A relief point defines a location and time where

a driver may change his vehicle. In traditional crew scheduling, i.e., a vehicle first

1.1. Planning Process of Public Transport Companies

- crew second approach, relief points subdivide vehicle blocks that were obtained

in the preceding phase.

Apiece of work is a sequence of tasks without a (long) break for which a driver

stays with the same vehicle. Consequently, duties are composed of pieces of work

separated by breaks. Duties start with a sign-on and end with a sign-off activity.

Typically, there are several duty types in practical applications, each with a dif-

ferent rule set. Examples of working rules are minimum/maximum driving time,

minimum break length, allowed start and end time, or maximum spread (length)

of a duty. Moreover, companies often limit the (minimum/maximum) number or

percentage of duties of a particular type. For instance, the percentage of split

duties that have two pieces of work - one in the early morning and another in

the late afternoon with a long break in the middle - is often restricted. Figure

1.4 shows the schedule of one crew that consists of two pieces of work. Note that

the first two tasks remain unassigned.

vehicle block I vehicle block II

ABB BC BAA C BAAB

vehicle duty

depot

time

trip from A to C

C A

relief point

deadhead from B to C

piece of work I piece of work II

task VItask I task II task III task IV task V

crew duty

piece of work III

Figure 1.4.: Schedule of one vehicle and one crew where a piece of work remains

unassigned

The objective is often to first minimize the number of duties and second the

total working time. Therefore, high fixed crew costs and an hourly rate for

working time are taken into account. Crew scheduling problems, however, are

often subject to non-linear costs, e.g. overtime bonuses.

[Fischetti et al., 1987] and [Fischetti et al., 1989] show that the CSP with either

working time or spread time constraints is NP-hard. Although duty constraints

1. Introduction

differ from application to application, we assume that the CSP has at least one

of these constraints and is therefore NP-hard.

1.2. Integrated Vehicle and Crew Scheduling

Vehicle and crew scheduling are traditionally approached in a sequential manner

which means that vehicle schedules are determined before crew schedules. In this

section, we discuss potential benefits of integrating vehicle and crew scheduling.

Although scheduling vehicles independently of crews was seriously criticized

in the early eighties by [Bodin et al., 1983], most commercial software packages

still use the sequential approach or offer integration at user level. However, an

integrated approach as sketched in Figure 1.5 discloses additional flexibility in

crew scheduling leading to savings as we will show in the following with a small

example.

timetabling timetable

integrated

vehicle and crew

scheduling

vehicle blocks

crew duties

crew rostering

work regulations

crew roster

tactical

operational

Figure 1.5.: Planning process for integrated vehicle and crew scheduling

Example 1 Let us assume a timetable with five trips f1, f2, f3, f4, and f5is given

as follows: f1from A to B (08:15-09:40), f2from B to A (09:50-10:15), f3from

A to C (10:15-10:55), f4from B to A (11:15-12:15), and f5from C to C (10:45-

11:30). Furthermore, the travel times in minutes between all stations and both

depots (deadhead matrix) are given in Table 1.1. Note that the travel time between

two locations is shorter than a service trip between the same locations since the

bus does not need to stop for passengers. A duty is feasible if the duration of

each piece of work is less than or equal to 4 hours. A duty has either one or

two pieces of work with a break of at least 45 minutes in between. A duty may

only contain tasks of a single depot and must start and end in that depot. If a

1.2. Integrated Vehicle and Crew Scheduling

depot 1 depot 2 A B C

depot 1 – 15 15 41 23

depot 2 15 – 22 35 10

A 15 22 – 40 17

B 41 35 40 – 20

C 23 10 17 20 –

Table 1.1.: Deadhead matrix

duty does not start/end in its depot, additional walking time (as defined in the

deadhead matrix) is added to the time of the duty. Both depots and station B are

relief points. Finally, we define the cost structure as follows: a fixed cost of 1000

per vehicle, 1000 per duty, and variable vehicle costs of 60 per hour that a vehicle

is without passengers outside its depot.

The optimal vehicle schedule (see Figure 1.6) consists of two vehicles where

one vehicle operates trips f1, f2, f3and f4and the other trip f5. Obviously, the

first vehicle block cannot be covered by a single crew leading to a crew schedule

with three duties.

depot 2

depot 1

Figure 1.6.: Optimal vehicle and crew schedule for sequential approach consist of

two blocks and three duties.

However, scheduling vehicles and crews together leads to an overall optimal

solution of two blocks and two duties (see Figure 1.7). Instead of one long and

one short vehicle block as illustrated in Figure 1.6, we obtain two vehicle blocks

of almost same length that allow a better assignment to duties. Notice that the

overall optimal solution incurs higher operational costs for the vehicle schedule

but saves one duty.

1. Introduction

depot 2

depot 1

Figure 1.7.: Optimal vehicle and crew schedule for integrated approach consist of

two blocks and two duties.

Of course, problems are much more complex in reality than in this example.

Nevertheless, this example shows that scheduling vehicles independently of crews

can lead to inefficient solutions and that the overall solution can be improved by

using an integrated approach.

Moreover, applying an integrated approach is essential when relief opportuni-

ties are rare, i.e., in extra- or sub-urban public transport systems, since efficient

vehicle schedules may lead to poor or even infeasible crew scheduling solutions. If

we construct an optimal vehicle schedule in an extra-urban scenario, it is likely to

contain pieces of work that are too long to meet break regulations. In an urban

setting with many relief points drivers can often move to another relief point by

foot or other means of transport. Thus, many pieces of work can be combined to

form a duty. In other settings, distances are such that drivers are virtually tied

to their vehicle in order to reach relief opportunities. As a consequence, integrat-

ing vehicle and crew scheduling is essential in sub- or extra-urban settings since

vehicle scheduling strongly affects crew scheduling.

The integrated vehicle and crew scheduling problem (VCSP) for a given set of

timetabled trips, depots, and relief points can be stated as follows: find mini-

mum cost sets of vehicle blocks and crew duties such that both vehicle and crew

schedule are feasible and mutually compatible. Vehicle and crew schedule are

compatible if each trip is covered and each deadhead used in the vehicle schedule

is also covered by exactly one duty while all deadheads not contained in the ve-

hicle schedule are not part of any duty. Feasible vehicle and crew schedules are

defined in Section 1.1.1 and 1.1.2, respectively. VCSP is NP-hard since (at least)

the crew scheduling part is NP-hard.

1.3. Irregular Timetables

In the preceding section, we focused on how to conduct operations of a given

timetable at minimum cost. In this section, however, we will address another

aspect which is related to the quality of crew schedules.

In practice, timetables consist of many trips serviced every day and some excep-

tions that do not repeat daily. In particular, service trips to schools, production

facilities, or public swimming baths are often subject to change, e.g., trips may

be operated on every day except Sunday or on Monday only. Unless specifically

imposed, traditional vehicle and crew scheduling usually produces irregular crew

schedules which are undesired in practice. A crew schedule is called irregular if

it cannot be repeated many times. Similar to airline crew scheduling, regularity

is an important aspect for crew schedules in public transport since regular solu-

tions can improve operational reliability and reduce training costs. Furthermore,

regular solutions are less error-prone and crews often prefer to repeat itineraries.

In current practice, companies often try to increase regularity of crew scheduling

solutions by applying one of the following heuristic procedures:

•All first - irregular second: First, the planner solves a crew scheduling

problem for a particular period with both regular and irregular trips. In

a second step, he fixes the subset of crew duties that can be operated the

whole period and reoptimizes all unfixed trips. Notice that the second

problem can also contain some regular trips.

•Regular first - irregular second: The set of service trips is divided into

regular and irregular trips. First, a crew scheduling problem for the set

of regular trips is solved while the irregular trips are left for subsequent

optimization.

In both cases, the second problem has a sparse schedule and, thus, likely requires

extensive deadheading and even its optimal solution yields a high costs. On the

other hand, if the second problem contains many trips, the corresponding solution

has low cost but low regularity as well.

In the following, we will evaluate the impact of irregular timetables on the

regularity of vehicle and crew scheduling solutions. We consider the timetables

for Wednesday and Thursday of a small-town in Germany where 5% of the trips

are different on Thursday. A reference crew schedule is given and we seek a crew

schedule for Thursday that is similar to Wednesday’s schedule. First, we perform

traditional vehicle and crew scheduling (vehicle first – crew second). Second, we

solve an independent crew schedule problem for Thursday where the set of tasks

1. Introduction

corresponds to the set of service trips (see Section 6.1 for a detailed description).

In both cases, we compare the crew schedule for Thursday with Wednesday’s

crew schedule.

In Figure 1.8 we depict the impact of an irregular timetable on the regularity

of the crew scheduling solution if independent crew scheduling is performed. In

timetable

Wednesday timetable

Thursday

5% of trips different

crew schedule

peform independent

crew scheduling

crew schedule

peform independent

crew scheduling

93% of duties different

Figure 1.8.: Impact of an irregular timetable on the regularity of the crew schedul-

ing solution if independent crew scheduling is performed.

our setting, 5% of trips are either in the Wednesday or the Thursday timetable,

but not in both. As we can see in Figure 1.8, only 7% of the duties in the

Wednesday crew scheduling solution are also part of the Thursday solution. The

impact is even stronger for traditional vehicle and crew scheduling: none of the

duties in the Wednesday solution could be preserved in the Thursday solution.

Therefore, we conclude that, unless specifically imposed, small modifications of

the timetable can destroy the structure of the crew scheduling solution.

Finally, notice that public transport companies face a similar situation when-

ever they change their timetable, e.g., scheduled timetable changes in summer

or winter. Typically, the changes involve only a small portion of the complete

timetable. Again, a traditional approach to vehicle and crew scheduling is likely

to produce solutions for the new timetable that do not have much in common

with the former solution.

1.4. Selected Combinatorial Optimization Problems

For more than 50 years now, many researches have studied the field of combinato-

rial optimization. It involves the problem of minimizing or maximizing a function

of discrete decision variables subject to equality or inequality constraints. Vehicle

and crew scheduling problems have been formulated as combinatorial optimiza-

tion problems for more than thirty years now. Therefore, we review selected

1.4. Selected Combinatorial Optimization Problems

well-known combinatorial optimization problems that we will use in this thesis.

We do not describe all mathematical theory, but rather provide the necessary

background for the models and algorithms used in the remainder of this the-

sis. For extensive surveys of integer and combinatorial optimization we refer

to [Nemhauser and Wolsey, 1988] and [Wolsey, 1998].

1.4.1. Network Flow Problems

In network flow problems ”we wish to move some entity (electricity, a consumer

product, a person or a vehicle, a message) from one point to another in an

underlying network, and to do so as efficiently as possible, both to provide good

service to the users of the network and to use the underlying (and typically

expensive) transmission facilities effectively” ( [Ahuja et al., 1993]).

In this subsection we describe three related network flow problems that will

occur as subproblems in the remainder of the thesis, namely the minimum cost

flow, the multicommodity cost flow, and the resource constrained shortest path

problem.

Minimum Cost Flow Problem

The minimum cost flow problem (MCFP) is a very fundamental network flow

problem. The problem aims at finding a minimum cost shipment of a commodity

through a network that satisfies the demand at certain nodes from available

supplies at other nodes (see [Ahuja et al., 1993]).

Let G= (N, A) be a directed graph with Nas the set of nodes and Aas the

set of directed arcs. We associate a cost cij with each directed arc (i, j)∈Athat

corresponds to the cost per unit flow on that arc and varies linearly with the

amount of flow. Furthermore, we define an upper (lower) bound uij (lij) on the

amount of flow for each arc (i, j)∈A. Let bidenote the supply/demand of node

i∈N. If bi>0 (bi<0), node iis a supply (demand) node. We call each node

iwith bi= 0 transshipment node. Finally, we associate a decision variable xij

with the amount of flow on arc (i, j)∈Aand formulate MCFP as follows:

(i,j)∈A

cijxij →min (1.1)

s.t. X

{j:(i,j)∈A}

xij −X

{j:(j,i)∈A}

xji =bi∀i∈N, (1.2)

lij ≤xij ≤uij ∀(i, j)∈A. (1.3)

The objective function (1.1) aims at minimizing total costs such that total outflow

minus total inflow of each node is equal to the supply/demand of that node (1.2).

1. Introduction

The flow on each arc must satisfy the lower and upper bound of that arc (1.3).

We refer to constraint set (1.2) as flow conservation constraints and to (1.3) as

flow bound constraints.

Notice that it is not necessary to impose integrality on the flow variables if all

data (supplies, demands, and bounds) are integral since the constraint matrix

is totally unimodular (see e.g. [Nemhauser and Wolsey, 1988]) and, thus, each

solution to the linear program above is integral. However, algorithms that exploit

the structure of the underlying network, such as capacity scaling or network

simplex algorithms, are often considerably faster than general purpose linear

programming algorithms (e.g. primal or dual simplex). Many of these algorithms

run in polynomial time.

The shortest path problem (SP) can be stated as a minimum cost flow problem

and aims at finding a path of minimum cost (length) from a source sto a sink t

in network G. If we set bs= 1, bt=−1 and bi= 0 for all other nodes, we will

send unit flow from node sto t. Additionally, we set uij = 1 for all (i, j)∈Aif

Gcontains directed cycles of negative length. The all-pairs shortest path problem

is a generalization of the shortest path problem where we would like to find the

shortest paths between all pairs of nodes.

Multicommodity Flow Problem

The multicommodity flow problem (MFP) is a generalization of the minimum cost

flow problem. MFP is composed of several commodities each with its origin and

destination that use the same underlying network whereas MCFP considers only

a single commodity. Each commodity has separate flow conservation constraints

while all commodities share the same flow bound constraints.

The formulation of the multicommodity flow problem is related to the minimum

cost flow problem in the previous subsection. However, we introduce Kas the

set of commodities and separate flow variables, supply/demand, and costs by

commodity k∈K. Furthermore, we have a set of flow bound constraints that are

separated by commodity and another set for all commodities with lower bounds

1.4. Selected Combinatorial Optimization Problems

Lij and upper bounds Uij, respectively.

k∈KX

(i,j)∈A

ijxk

ij →min (1.4)

s.t. X

{j:(i,j)∈A}

ij −X

{j:(j,i)∈A}

ji =bk

i∀i∈N, ∀k∈K(1.5)

ij ≤xk

ij ≤uk

ij ∀(i, j)∈A, ∀k∈K(1.6)

Lij ≤X

k∈K

ij ≤Uij ∀(i, j)∈A(1.7)

ij ∈N∀(i, j)∈A(1.8)

As opposed to minimum cost flow problems, solutions to multicommodity flow

problems are not necessarily integral. We must impose integrality on the flow

variables in order to obtain integral solutions - even if all data is integral. The

integral multicommodity flow problem has been proven to be NP-complete (see

[Garey and Johnson, 1979]) if there are at least two commodities. As stated

earlier, the single commodity flow problem can be solved in polynomial time.

Resource Constrained Shortest Path Problem

The resource constrained shortest path problem (RCSP) is an extension of the

shortest path problem. It consists of finding the minimum cost path between a

source sand a sink twhile respecting constraints on resource consumption.

Again, we define a directed network G= (N, A) as in the previous subsections.

However, we do not only associate a traversal cost cij with each arc (i, j)∈A, we

also define a resource consumption dr

ij ≥0 for each resource r∈R. Consequently,

each path Paccumulates P(i,j)∈Pdr

ij of resource r∈R. We say that a path is

resource feasible if and only if the resource consumption along the path is greater

or equal to lower bound lrand less or equal to upper bound ur. RCSP can be

formulated as follows:

(i,j)∈A

cijxij →min (1.9)

s.t. X

{j:(i,j)∈A}

xij −X

{j:(j,i)∈A}

xji =









1 for i=s

0 for i∈N\ {s, t}

−1 for i=t

,(1.10)

lr≤X

(i,j)∈A

ijxij ≤ur∀r∈R, (1.11)

xij ∈ {0,1} ∀(i, j)∈A. (1.12)

1. Introduction

Note that a resource constrained shortest path problem with cost and resource

consumption on nodes can be easily transformed to a problem with consumption

on arcs. The resource constrained shortest path problem is NP-hard if there

is least one resource (see [Garey and Johnson, 1979]), even though it can be

solved in pseudo-polynomial time with a simple dynamic programming formula-

tion. [Hassin, 1992] shows that a fully polynomial -approximation scheme exists.

1.4.2. Set Partitioning/Covering Problem

The set partitioning problem (SPP) can be defined as follows: given a finite set

M, constraints defining a set Nof feasible subsets of M, and a cost associated

with each member in N, find a minimum cost subset of Nthat is a partition of

M(see [Balas and Padberg, 1976]).

Let xjbe a binary decision variable that equals 1 if subset j∈Nis part

of the solution and 0 otherwise. We associate a cost cjwith each variable xj.

Furthermore, we set aij = 1 if subset j∈Ncontains element i∈Mand aij = 0

otherwise. Now, the set partitioning problem can be expressed as follows:

j∈N

cjxj→min (1.13)

s.t. X

j∈N

aijxj= 1 ∀i∈M, (1.14)

xj∈ {0,1} ∀j∈N. (1.15)

The set covering problem (SCP) is a relaxation of SPP that does not require

to partition set Mbut to cover it. In other words, a feasible solution may consist

of several (but at least one) subsets j∈Nthat contain a particular element

i∈M. Clearly, each solution that is feasible for SPP is also feasible for SCP.

The optimal solution of the set covering problem is a lower bound of the set

partitioning problem. SCP can be obtained from SPP by replacing the equality

A great variety of scheduling problems from practice can be formulated as

set partitioning or covering problems. Well-established applications that use

this formulation are (bus/airline) crew scheduling and vehicle routing problems.

Typically, these problems have a huge number of variables.

It is well known that both the set partitioning and set covering problem are

NP-hard (see [Garey and Johnson, 1979]).

1.5. Selected Combinatorial Optimization Techniques

1.5. Selected Combinatorial Optimization

Techniques

The purpose of this section is to discuss algorithms and techniques to solve com-

binatorial optimization problem that will be used in the remainder of this thesis.

1.5.1. Lagrangian Relaxation

The general idea behind Lagrangian relaxation is to remove complicating con-

straints from a (combinatorial) optimization problem by penalizing their violation

in the objective function. In the following we will briefly discuss this relaxation

method and refer to [Geoffrion, 1974] and [Fisher, 1981] for an extensive discus-

sion of the theory of Lagrangian relaxation. We consider the following problem

Z(P) = min X

j∈N

cjxj(1.16)

s.t. X

j∈N

aijxj=di∀i∈M1,(1.17)

j∈N

bijxj=ei∀i∈M2,(1.18)

xj∈Z+∀j∈N. (1.19)

Suppose that constraints (1.17) are hard constraints in the sense that the opti-

mization problem without these constraints is easy to solve. Dualizing the hard

constraints leads to the Lagrangian subproblem

Φ(π) = min X

j∈N

cjxj+X

i∈M1

πi(di−X

j∈N

aijxj) (1.20)

s.t. X

j∈N

bijxj=ei∀i∈M2,(1.21)

xj∈Z+∀j∈N, (1.22)

where π= (πi)i∈M1represents the Lagrangian multipliers associated with the

dualized hard constraints. When we dualize inequality constraints of the form

Pj∈Naijxj≤di(Pj∈Naij xj≥di) for all i∈M1, the corresponding Lagrangian

multipliers are restricted in sign πi≤0 (πi≥0), i∈M1. Note that Φ(π) can be

1. Introduction

rewritten as

Φ(π) = min X

j∈N

¯cjxj+X

i∈M1

πidi|

j∈N

bijxj=ei,∀i∈M2;xj∈Z+,∀j∈N(1.23)

where we call ¯cj=cj−Pi∈M1aijπiLagrangian cost of column j∈N.

Φ(π) defines a lower bound on the original problem Pfor any fixed vector

πsince each feasible solution for the original problem is also feasible for the

Lagrangian subproblem (but not vice versa) and Φ(π) equals the objective value

of such feasible solution in the original problem.

We obtain the best possible lower bound by solving the Lagrangian dual problem

(LDP)

Z(LDP) = max

πΦ(π).(1.24)

It can be shown (see [Geoffrion, 1974]) that the optimal solution of the Lagrangian

dual problem always provides a lower bound on the original problem that is at

least as good as the objective value of the linear relaxation LP:

Z(LP)≤Z(LDP)≤Z(P).(1.25)

Furthermore, the objective values of Lagrangian and linear relaxation are equal

when the integrality constraints (1.22) of the Lagrangian subproblem can be re-

placed with xj≥0, j ∈Nand the solution of the subproblem remains unchanged

for all possible multipliers π. Then, the Lagrangian subproblem is said to have

the integrality property.

The Lagrangian dual problem maximizes a piecewise linear concave, but non-

differentiable function Φ(π) which implies that LDP is also nondifferentiable. In

the context of combinatorial optimization LDP is typically solved by a subgradi-

ent algorithm that was introduced by [Held and Karp, 1971] and will be described

in the following.

Subgradient Algorithm

The subgradient algorithm is an iterative search procedure to optimize nondiffer-

entiable functions. It is well-known that a differentiable function fcan be opti-

mized by an iterative gradient method like the steepest ascent method: starting

with an initial solution u0the sequence

ut+1 =ut+wt∇f(ut) (1.26)

1.5. Selected Combinatorial Optimization Techniques

eventually converges to an optimal solution with ∇f(ut) as the gradient of fat

utand wtas suitable step length. However, for nondifferentiable functions we

cannot use a gradient method since some points do not have a gradient. Instead

we use a subgradient method which is a generalization of the gradient method to

the nondifferentiable case where gradients are replaced by subgradients.

Asubgradient at π0of a concave function Φ : R|M1|→R1is a vector s∈R|M1|

such that Φ(ϕ)≤Φ(π0) + s(ϕ−π0) for ϕ∈R|M1|. In other words, a subgradient

in π0is the slope of a straight line through point (π0,Φ(π0)) that runs above

function Φ for |M1|= 1. Let ∂Φ(π0) denote the non-empty set of all subgradients

(subdifferential) of Φ at π0that reduces to the gradient if Φ is differentiable at

π0. At points where the function is nondifferentiable the subgradient method

chooses an arbitrary subgradient from the subdifferential. Figure 1.9 shows a

subgradient skof Φ(π) at the nondifferentiable point π0(arbitrarily) chosen from

the subdifferential which is represented by the grey area. It is easy to see that

sk(

Figure 1.9.: Subdifferential and subgradient skof a concave, nondifferentiable

function Φ(π) at π0for |M1|= 1.

a subgradient s∈R|M1|for the Lagrangian subproblem is given by si=di−

Pj∈Naijxj, i ∈M1with xas optimal solution to this problem.

A basic version of the subgradient method to solve the Lagrangian dual problem

is depicted in Algorithm 1. Notice that the formula used to compute the step size

in step 4 does not assure that LDP converges to the global maximum. However,

an empirical justification for the rule used in step 4 is given by [Held et al., 1974].

Furthermore, this rule has a low computational burden as opposed to other step

1. Introduction

length rules with proven convergence (e.g. [Polyak, 1967]). As proposed by [Held

and Karp, 1971] we initially set the step size parameter λ= 2 and halve it

whenever Φ(πt) fails to improve in a certain number of iterations. Parameter 

basically steers the accuracy of the solution and, thus, the number of iterations.

Algorithm 1: Subgradient Algorithm

(Step 1) Initialization

Initialize multipliers π0, parameter and set t= 0.

Compute upper bound UB.

(Step 2) Solve Lagrangian subproblem

Solve Φ(πt) and store optimal solution xt

(Step 3) Compute search direction δt

Compute subgradients st

i=di−Pj∈Naijxt

jfor all i∈M1.

Set δt=st.

(Step 4) Compute step size wt

wt=λUB−Φ(πt)

||δt||2

(Step 5) Update Lagrangian multipliers

λt+1

i=λt

i+wtδt

ifor all i∈M1

(Step 6) Check termination criteria

Terminate if one of the following criteria is satisfied:

st= 0,

Pi∈M1(δt

i)2≤

λ≤

t≥tmax

UB −Φ(πt)≤

otherwise set t=t+ 1 and return to step 2

1.5.2. Dantzig-Wolfe Decomposition and Column Generation

Column generation is a method to solve linear programs that involve a large

number of columns. Formulations of problems with a large number of variables

arise in many real-life situations, e.g crew scheduling or vehicle routing, where one

likes to select a minimum cost (maximum profit) subset from a very wide choice of

subsets. Typically, the chosen subset must satisfy a number of constraints. Below

we show how such formulations (master problems) of a combinatorial optimization

problem may arise by reformulation and how linear relaxations of master problems

can be solved by (delayed) column generation.

1.5. Selected Combinatorial Optimization Techniques

Consider the following (compact) integer program P

Z(P) = min{c(x) : Ax=d,x∈X}(1.27)

where X={x∈Z|N|

+:Bx=e}. We assume that Xis non-empty and contains

a large (but finite) number of elements. It is well known that replacing Xby

the convex hull conv(X) does not change the optimal objective value Z(P). The

Minkowski-Weyl theorem (see [Nemhauser and Wolsey, 1988]) states that each

x∈Xcan be represented as a convex combination of extreme points {xp}p∈P plus

a non-negative combination of extreme rays {xr}r∈R of conv(X) (see Figure 1.10).

Pdenotes the set of extreme points and Rthe set of extreme rays. As we assumed

conv(x)

Figure 1.10.: The convex hull conv(X) of the unbounded polyhedron Xwith two

extreme points and two extreme rays.

polyhedron Xto be bounded, the convex hull conv(X) can be represented by

a convex combination of a finite number of extreme points {xp}p∈P , i.e. x=

Pp∈P xpϑpwith Pp∈P ϑp= 1 and ϑ∈ {0,1}|P|. Next, we replace xin the

original problem with the internal representation of X, define cp=c(xp) as well

as ap=Axpwith p∈ P, and obtain the following extensive formulation P0with

1. Introduction

a large number of columns

Z(P0) = min X

p∈P

cpϑp(1.28)

s.t. X

p∈P

apϑp=d(1.29)

p∈P

ϑp= 1 (1.30)

ϑ∈ {0,1}|P| (1.31)

which we refer to as master problem. Solving the extensive formulation is equiv-

alent to solving the original compact formulation. Any fractional solution to

the LP relaxation of the compact formulation is a feasible solution to the LP

relaxation of the extensive formulation if and only if it can be expressed by a

convex combination of extreme points of conv(X), but not vice versa. In par-

ticular, [Geoffrion, 1974] shows that the extensive formulation provides a tighter

LP relaxation if conv(X) does not only have integral extreme points.

The reformulation process is also known as Dantzig-Wolfe decomposition (see

[Dantzig and Wolfe, 1960]) and is one way of obtaining a formulation with a large

number of variables. Problems may also have a ”natural” formulation with a huge

number of variables. However, [Villeneuve et al., 2005] show that a compact

formulation to such a natural (extensive) formulation exists under very mild

assumptions.

Below we will describe a column generation algorithm to solve the linear pro-

gramming relaxation of the following master problem MP

Z(MP) = min X

j∈N

cjxj(1.32)

s.t. X

j∈N

aijxj=di∀i∈M, (1.33)

xj∈ {0,1} ∀j∈N. (1.34)

where |N|is huge. Notice that in the linear programming (LP) relaxation of MP

the integrality constraints (1.34) are replaced by xj≥0,∀j∈N. As in most

practical situations we assume that it is impossible to explicitly keep all columns

in main memory and, as a result, to solve the master linear program from scratch.

Instead, we solve a sequence of restricted master linear programs (RMP) where

each problem contains only a small subset of all columns.

We initialize the algorithm (see Algorithm 2) with an initial column set N0⊆N

that contains a feasible solution. This initial solution can either be generated

1.5. Selected Combinatorial Optimization Techniques

Algorithm 2: Column Generation Algorithm

(Step 1) Initialization

Choose initial column set N0.

Set t= 0.

(Step 2) Solve (restricted) master problem

Solve RMP(Nt) and store duals πt.

(Step 3) Solve pricing problem

Solve pricing problem P(πt) and obtain columns N0with

negative reduced costs.

If |N0|= 0 terminate.

(Step 4) Add column(s) to restricted master problem

Set Nt+1 = (N0∪Nt) and t=t+ 1.

Goto Step 2.

by a heuristic or by adding appropriate artificial variables to the RMP. After the

restricted problem is solved, the dual information of the solution is used to price

out new columns with negative reduced costs, i.e., columns that can improve the

objective value of the RMP. Given an optimal dual solution πof the current RMP

the pricing problem (subproblem) can be stated as follows

¯c∗= min {cj−X

i∈M1

aijπi:j∈N}.(1.35)

If the pricing problems returns ¯c∗<0, the column jwith least negative reduced

costs ¯cj=cj−Pi∈M1aijπiis added to the RMP. The process iterates until

¯c∗≥0. Then, the current solution xto the RMP solves the linear relaxation of

MP without having all columns explicitly available. The complete set of columns

Nis only implicitly available since new columns are only generated (and kept in

memory) when needed. The pricing problem (1.35) is an optimization problem

itself that in our application corresponds to a resource constrained shortest path

problem (see Section 3.1).

The solution to the restricted master problem satisfies all constraints of our

master problem except for the integrality constraints. We obtain integer solutions

by integrating column generation and branch-and-bound (see Section 1.5.4). A

column generation algorithm is used to solve the linear relaxation of the master

problem in each node of the branch-and-bound tree. This solution approach is

often referred to as branch-and-price. For a general discussion on integer pro-

gramming (IP) column generation see [Barnhart et al., 1998].

Excellent surveys on column generation are given by [Desaulniers et al., 2005]

1. Introduction

and [L¨ubbecke and Desrosiers, 2005]. Acceleration techniques often applied in

column generation algorithms are described in [Desaulniers et al., 2002].

1.5.3. Lagrangian Relaxation based Column Generation

Typically, column generation is used to solve the LP-relaxation of the master

problem, but it can also be combined with Lagrangian relaxation as we will

discuss in this section. In this thesis Lagrangian relaxation in combination with

column generation will be used to solve integrated vehicle and crew scheduling

problems (see Section 2.4).

There is a strong relationship between Dantzig-Wolfe decomposition and La-

grangian relaxation. Let us consider the Lagrangian relaxation of the compact

formulation (1.27) with constraints Ax=drelaxed. The corresponding La-

grangian subproblem reads

Φ(π) = {min X

j∈N

(cj−X

i∈M1

aijπi)xj+X

i∈M1

πidi|x∈X}.(1.36)

It is well known (see [Nemhauser and Wolsey, 1988]) that the associated La-

grangian dual is the dual of the LP relaxation of the extensive formulation

(1.28)-(1.31). In other words, we can solve the Lagrangian dual either by ap-

plying the subgradient method to it or by solving the linear relaxation of the

extensive formulation (RMP) with a column generation approach. Consequently,

the optimal lower bound of the restricted linear master program (RMP) and the

best Lagrangian dual are the same. Moreover, the Lagrangian subproblem is

of the form of the column generation pricing problem and, consequently, solu-

tions of the Lagrangian subproblem can be added as new columns to the RMP.

Both solution methods for the Lagrangian dual have advantages and disadvan-

tages, hence [Barahona and Jensen, 1998] use a hybrid method that combines

the advantages of both approaches. Note that the Lagrangian multiplier vec-

tor πcorresponds to the dual variables associated with the linking constraints

Pp∈P(Axp)ϑp=dof the LP relaxation of the extensive formulation.

Lagrangian relaxation can also be applied to the extensive formulation in order

to obtain approximate dual solutions and lower bounds to the restricted master

problem. Instead of solving restricted problems with the simplex method to opti-

mality, a subgradient method is used to solve the Lagrangian dual approximately.

At the end of the subgradient phase, the Lagrangian multipliers are an approx-

imation of the optimal dual variables for the current restricted master problem.

The multipliers can be used to price out new columns. There are different reasons

to choose this approach:

1.5. Selected Combinatorial Optimization Techniques

•The subgradient method is fast, easy to implement, and does not require a

commercial LP solver.

•When solving the RMP with a simplex method, we obtain a basic dual

solution that corresponds to a vertex of the optimal face of the dual poly-

hedron. Basically, a new column of the RMP may cut that vertex while a

dual solution interior (in the center) of the dual face allows stronger dual

cuts (i.e. better primal columns). [Bixby et al., 1992] and [Barnhart et al.,

1998] note that this may improve the convergence of a column generation

algorithm and reduce degeneracy. The subgradient method naturally pro-

vides non-basic solutions with many non-zero elements. [Jans and Degraeve,

2004] provide computational results indicating that Lagrangian multipliers

are beneficial.

•During the subgradient phase possibly feasible solutions are generated.

However, since the Lagrangian multipliers are not exact, columns in the restricted

master problem may have negative reduced costs. Thus, they should be modified

before generating new columns in order to prevent that columns are generated

twice. [Carraresi et al., 1995] and [Freling, 1997] describe a greedy heuristic that

modifies Lagrangian multipliers in such a way that all columns in the RMP

have non-negative reduced costs and that the lower bound does not decrease.

In Algorithm 3 we describe their heuristic with πas approximate dual variables

to the linear relaxation of master problem (1.32)-(1.34) and Ntas the current

column set.

Algorithm 3: Lagrangian multiplier adjustment heuristic

(Step 1) Find negative reduced cost columns

Define negative reduced cost columns j∈R⊆Ntwith

cj−Pi∈M1aijπi<0

(Step 2) Update multipliers and reduced costs

foreach r∈Rdo

Define δ=cr−Pi∈M1airπi

Pi∈M1air

foreach i∈M1with air = 1 do

Update multiplier πi=πi+δ

foreach j∈Rwith j > r do

Update reduced costs ¯cj=cj−Pi∈M1aijπi

1. Introduction

A thorough discussion on how to combine Lagrangian relaxation and column

generation with some examples can be found in [Huisman et al., 2005b].

1.5.4. Branch-and-Bound

Branch-and-bound is an exact solution approach for combinatorial optimization

problems that is based on the divide-and-conquer principle (see [Wolsey, 1998]).

Basically, the problem is decomposed into a series of smaller problems that are

easier to solve than the original. These smaller problems are solved and their

solution information is later put together to solve the original problem. Branching

divides the original solution space into two or more parts. Each of the smaller

solution spaces is split again into two or more parts and so on. As a result

we obtain a search tree where each solution space is represented by a node.

Without bounding this procedure corresponds to a complete enumeration which

is impossible for most problems of practical size. Bounding is used to prune

nodes of the search that cannot contain a solution better than the best solution

(incumbent) found so far. For minimization problems we need to find lower

bounds on each solution space (node). Typically, lower bounds are calculated by

solving the LP relaxation of each node. A combinatorial optimization problem

can be solved with branch-and-bound based on the LP relaxation by successively

separating fractional solutions from the feasible solution space. Algorithm 4

depicts a branch-and-bound method to solve minimization problem P.

Basically, branch-and-bound algorithms leave two choices: how to branch (Step

6) and which (sub)problem to select next (Step 3). In LP-based branch-and-

bound algorithms branching can only be performed by adding linear inequalities

to the problem or by modifying bounds on variables. Inequalities or bound mod-

ifications correspond to a valid branching rule if they split the problem, cut the

current fractional feasible solution (but no integer solution), and result in in-

teger solutions in each leaf node after a finite number of separations. Ideally,

a good branching rule is not only valid but also takes the performance of the

algorithm into account. Recall that nodes are pruned either by optimality, by

bound, or by infeasibility. Thus, a branch-and-bound algorithm works in two

directions: construct an integer solution and provide a tight lower bound that

possibly proves optimality of the incumbent. Of course, we do not seek infeasible

subproblems. Consequently, we like to select that branching rule among several

alternatives that maximizes the minimum lower bound and has a good chance to

generate many integer solutions. The same reasoning holds for good node selec-

tion strategies. For a recent comparison of general branching rules see [Achterberg

et al., 2005]. For a comprehensive survey on general branch-and-bound strategies

1.5. Selected Combinatorial Optimization Techniques

Algorithm 4: Branch-and-bound

(Step 1) Initialization

Set upper bound Z∗=∞.

Add original problem to set of unprocessed nodes N.

(Step 2) Check termination criteria

If N=∅terminate and output incumbent x∗with

objective value Z∗.

(Step 3) Select next node

Choose node p∈Nand delete it from N.

(Step 4) Calculate bound of current node p

Solve LP relaxation of current problem pwith dual bound

Zand store solution b

If b

xis empty prune by infeasibility and goto Step 2.

(Step 5) Bounding

If b

Z≥Z∗prune by bound and goto Step 2.

If b

xis integer and b

Z < Z∗set Z∗=b

Zand store new

incumbent x∗=b

x. Then, prune by optimality and goto

Step 2.

(Step 6) Branching

Create two subproblems p1and p2that separate the

current LP solution and add both to list of unprocessed

nodes N.

Goto Step 3.

1. Introduction

we refer to [Linderoth and Savelsbergh, 1999]. We will discuss problem-specific

branching rules for the crew scheduling problem in Section 6.4.1.

In a branch-and-cut algorithm we tighten lower bounds by adding cutting

planes to a subproblem. A branch-and-price algorithm uses a column genera-

tion algorithm to solve linear relaxation with an enormous number of columns

in each node (see Section 1.5.2). A branch-and-cut-and-price method combines

both extensions.

1.5.5. Metaheuristics

Aheuristic applied to a combinatorial optimization problem seeks to find a good

approximate solution in reasonable time while an exact method guarantees to find

the global optimum in a potentially long time. [Osman and Laporte, 1996] define

ametaheuristic formally as ”an iterative generation process which guides a sub-

ordinate heuristic by combining intelligently different concepts for exploring and

exploiting the search space, learning strategies are used to structure information

in order to find efficiently near-optimal solutions.” Well-known metaheuristics

include ant colony optimization (ACO), tabu search (TS), simulated annealing

(SA), and evolutionary algorithms (EA). For an overview of metaheuristics the

reader is referred to [Reeves, 1993]. In this thesis, we will propose a novel hybrid

evolutionary algorithm for the integrated vehicle and crew scheduling problem

(see Chapter 4). Moreover, we will use different metaheuristics to solve combina-

torial optimization problems with two competing objectives (see Section 6.4.2).

1.6. Scope and Purpose of the Thesis

In the previous sections we have introduced both the practical and the theoretical

foundation for the remainder of this thesis. We will now define the scope and

purpose of the thesis.

Until recently it was not possible to solve real-world integrated vehicle and crew

scheduling problems with several depots within reasonable time and guaranteed

solution quality. Still, large instances with complex duty feasibility rules cannot

be tackled in an integrated manner. In addition to cost reduction the quality of

crew schedules is an important aspect. Therefore, we will consider the regular-

ity of crew schedules as one aspect of quality. All together, our main research

objectives are threefold:

•To develop models and techniques for the integration of vehicle and crew

scheduling in public transit that allow to tackle large problem instances.

1.6. Scope and Purpose of the Thesis

Moreover, to efficiently model complex crew duty feasibility rules aris-

ing from German federal laws, safety regulations, and (collective) in-house

agreements.

•To develop models and techniques to increase the regularity of crew sched-

ules for the integration of vehicle and crew scheduling with irregular timeta-

bles.

•To test the applicability of the proposed techniques in practice.

The thesis comprises seven chapters and is set up as follows. The first chap-

ter corresponds to this introduction which outlines both the practical and the

theoretical foundation of the thesis.

In Chapter 2 we define the integrated multiple-depot vehicle and crew schedul-

ing problem in public transport. We review models and solution techniques that

are used in literature for sequential, partially integrated, and fully integrated ve-

hicle and crew scheduling. Furthermore, we thoroughly describe the modeling

approach, mathematical formulation, and solution approach that provides the

starting point for the following chapters of the thesis. The solution approach is

based on column generation in combination with Lagrangian relaxation.

Chapter 3 presents new approaches for the integrated vehicle and crew schedul-

ing problem. More specifically, we propose a novel approach for the column

generation pricing problem that includes both modeling approach and solution

method. Furthermore, we discuss different solution approaches to construct in-

teger solutions. Finally, we propose a new model where drivers are allowed to

change their vehicle whenever there is a relief point. The chapter is concluded

with a computational study using real-world and randomly generated benchmark

instances in order to evaluate the effectiveness of our approaches.

In Chapter 4 we deal with a novel hybrid evolutionary algorithm to tackle

integrated vehicle and crew scheduling problems. Our method combines math-

ematical programming techniques with an evolutionary algorithm. We compare

different versions of the evolutionary algorithms with each other, with the tra-

ditional sequential approach, and with an integrated treatment of both planning

steps.

In Chapter 5 we consider practical rules and regulations arising in public trans-

port companies in Germany. We suggest enhancements and modifications of our

modeling and solution approach from Chapter 3 to cover these practical exten-

sions. Furthermore, we give an overview of how our implementation is being

integrated in the commercial software tool interplanr. Finally, we test the appli-

cability of the proposed techniques using randomly generated and real-life data

1. Introduction

instances.

We address the ex-urban vehicle and crew scheduling problem with irregu-

lar timetables in Chapter 6. We discuss the impact of irregular timetables on

the regularity of crew scheduling solutions. Regularity is an aspect which is re-

lated to the quality of crew schedules. We suggest a novel combination of local

and follow-on branching to construct cost-effective and regular crew schedules.

Furthermore, we show how bi-objective metaheuristics can be used to quickly

estimate the quality of the solution generated with the latter approach. The

chapter is concluded with a computational study that involves real-world and

randomly generated ex-urban scenarios with a single depot.

Finally, in Chapter 7 we conclude the thesis with a summary and some final

remarks.

2. Integrated Vehicle and Crew

Scheduling: State-of-the-Art

In this chapter, we define the integrated vehicle and crew scheduling problem

with multiple depots. Furthermore, we review state-of-the-art models and so-

lution methods for traditional (sequential), partially integrated and integrated

vehicle and crew scheduling. In particular, we thoroughly describe the modeling

approach, mathematical formulation and solution approach that we will use in

the remainder of this thesis.

This chapter is organized as follows. In Section 2.1, we give a problem def-

inition for the integrated vehicle and crew scheduling problem and discuss the

state-of-the-art for both sequential and simultaneous treatment of vehicle and

crew scheduling in Section 2.2. In the next section, we describe a mathematical

formulation from literature for the integrated vehicle and crew scheduling that

we will use in the following chapters. Finally, we describe the column generation

algorithm in Section 2.4 that we use to solve integrated models.

2.1. Problem Definition

The integrated vehicle and crew scheduling problem (VCSP) for a given set of

timetabled trips within a fixed planning horizon, given traveling times between

all pairs of locations, and a fleet of vehicles assigned to several depots can be

stated as follows: find minimum cost sets of vehicle blocks and crew duties such

that both vehicle and crew schedule are feasible and mutually compatible. Vehicle

and crew schedule are compatible if each vehicle activity in the vehicle schedule is

also covered by exactly one duty while all deadheads not contained in the vehicle

schedule are not part of any duty.

A vehicle schedule is feasible if each trip is assigned to exactly one vehicle and

each vehicle performs a feasible sequence of trips. A sequence of trips (vehicle

block) is feasible if each pair of consecutive trips can be executed in sequence and

each block starts and ends at the same depot. The vehicle costs comprise fixed

(asset) costs for every vehicle and variable costs for travel and idle time outside

2. Integrated Vehicle and Crew Scheduling: State-of-the-Art

the depot.

A crew schedule is feasible if each task (either deadhead or trip) of the vehicle

schedule is covered by a duty that can be performed by a single driver, and if

each duty complies with a wide variety of federal laws, safety regulations, and

collective in-house agreements. A task is a sequence of activities on a vehicle

(such as performing trips and/or deadheading) between two consecutive relief

points (see Section 1.1.2) and represents an elementary portion of work that can

be assigned to a driver. A relief point defines a location and time where a driver

may change his vehicle. A piece of work is a sequence of tasks without a (long)

break for which a driver stays with the same vehicle. Consequently, duties are

composed of pieces of work separated by breaks. The duty cost usually consists

of a fixed component for wages and variable costs for working time or overtime

bonuses.

If there are multiple depots some trips possibly have to be assigned to vehicles

and drivers from a certain (sub)set of depots. It is easy to see that a problem

with multiple depots reduces to several single depot problems if every trip can

only be serviced from a single depot.

We make the same assumptions as [Huisman, 2004] in order to obtain com-

parable results in Chapter 3. However, we will relax and change some of the

assumptions in Chapter 5 in order to apply our approach on practical scenarios

arising in Germany.

•Each vehicle is assigned to the depot where its daily schedule starts and

ends. Each depot is unlimited in capacity. That is, it can store an unlimited

number of vehicles.

•Each crew is assigned to a depot and may only conduct tasks on vehicles

from this particular depot. However, a duty does not necessarily start and

end in this depot.

•A piece of work is only restricted by its duration. It may have a minimum

and maximum duration.

•A vehicle returns to its depot if the idle time between two consecutive trips

is long enough to perform a round-trip to the depot.

•Each trip has exactly two relief points: one at the beginning and the other

at the end of the trip.

•A driver is required to be present if a bus is outside of a depot (continuous

attendance) while there is no driver needed inside a depot.

2.2. Literature Review

•A driver may only change his or her vehicle during a break, i.e., between

two pieces of work. The change of a vehicle of a driver is called changeover.

Notice that the last two assumptions imply that a second driver must be present

during the break of a driver, if the original driver has no changeover and the

vehicle is outside the depot. Otherwise, nobody would attend the vehicle during

the break.

2.2. Literature Review

The purpose of this section is to review state-of-the-art models and approaches

for solving sequential and (partially) integrated vehicle and crew scheduling prob-

lems.

2.2.1. Sequential Vehicle and Crew Scheduling

In the sequential or traditional planning procedure, the crew scheduling problem

is solved after the vehicle scheduling problem. That is, we first assign trips to

vehicles and schedule crews based on the vehicle blocks obtained before.

A thorough understanding of traditional vehicle and crew scheduling provides

a useful introduction to the integrated problem since the integrated problem

includes traditional vehicle and crew scheduling problems as subproblems. Fur-

thermore, we will use the traditional approach to evaluate the efficiency gain by

an integrated treatment. We will first discuss single- and multiple-depot vehi-

cle scheduling problems and then review models and approaches for the crew

scheduling problem.

Single-Depot Vehicle Scheduling

The single-depot vehicle scheduling problem (SDVSP) arises for small to medium-

sized public transport companies that have a single depot and a homogeneous fleet

of vehicles. Additionally, it may appear as a subproblem for the multiple-depot

case. It is well known that the SDVSP corresponds to a minimum cost flow prob-

lem (see [Bodin et al., 1983]) that can be solved in polynomial time. The SDVSP

has also been formulated as linear assignment problem [Orloff, 1976] and trans-

portation problem [Gavish and Shlifer, 1978]. The formulation as transportation

problem is also known as quasi-assignment formulation. All formulations can be

solved in polynomial time.

[L¨obel, 1996] describes an efficient implementation of the network simplex

method for a minimum cost flow formulation. [Paix˜ao and Branco, 1987] propose

2. Integrated Vehicle and Crew Scheduling: State-of-the-Art

an algorithm based on the Hungarian method for a quasi-assignment and as-

signment formulation. Their algorithm performs better on the quasi-assignment

formulation and outperforms all other algorithms at that time. [Freling, 1997]

and, subsequently, [Freling et al., 2001b] solve a quasi-assignment formulation

efficiently with a combined forward and reverse auction algorithm (see [Bertsekas

and Casta˜non, 1992]). Additionally, they propose a two phase approach that is

valid when a special cost structure can be used. They test their algorithms on

both real-world and artificial data with up to 1,500 trips and show that their algo-

rithms outperform other approaches proposed before. [Silva et al., 1999] present

an arc generation approach for a quasi-assignment formulation that is initialized

with short deadhead arcs. Further arcs are added to the master problem by a

column generation approach until optimality is proven.

For surveys on the SDVSP and its practical extensions we refer to [Daduna

and Paix˜ao, 1995], [Desrosiers et al., 1995], and [Bunte and Kliewer, 2006].

Multiple-Depot Vehicle Scheduling

The multiple-depot vehicle scheduling problem (MDVSP) often arises in medium-

sized public transport companies and is inevitable in larger ones. The company

operates its (homogeneous) fleet out of several depots where each vehicle is as-

signed to a single depot. The MDVSP can be extended by multiple vehicle types

and by the constraint that some trips have to be serviced by vehicles from a

certain subset of depots. The MDVSP has been extensively studied for more

than 25 years now. Since this is a NP-hard problem (see [Bertossi et al., 1987]),

early works mainly focused on heuristic algorithms. For an overview on heuristic

methods for the MDVSP we refer to [Dell’Amico et al., 1993] and [L¨obel, 1997].

A recent comparison and computational tests of different heuristic approaches to

the MDVSP can be found in [Pepin et al., 2006].

[Fischetti et al., 2001] categorize exact solution approaches to the MDVSP by

the mathematical formulation used:

1. Single-commodity formulations,

2. Multicommodity formulations,

3. Set partitioning formulations.

The first exact solution approach is proposed by [Carpaneto et al., 1989] and

belongs to the first category. They add subtour breaking constraints and derive

lower bounds by an additive lower bounding method (see [Fischetti and Toth,

1989]). Subsequently, they use a branch-and-bound method with user-defined

2.2. Literature Review

branching rules to obtain integer feasible solutions. The weak lower bound of the

formulation can be improved by adding path elimination cuts in a branch-and-

cut framework (see [Fischetti et al., 1999]). This approach is further extended

by [Fischetti et al., 2001]. A single-commodity flow formulation with special

assignment variables is discussed in [Mesquita and Paix˜ao, 1992].

The multicommodity flow formulation is a generalization of the network flow

approach for single-depot problems to the multiple-depot case where a network is

set up for each depot. The multicommodity formulation combines these networks

to form a multigraph that contains an arc for each depot between two nodes. Sev-

eral authors including [Bodin et al., 1983], [Bertossi et al., 1987], [Forbes et al.,

1994], [Ribeiro and Soumis, 1994], [L¨obel, 1997], [L¨obel, 1998], and [Kliewer et al.,

2006b] use this type of formulation. Some authors use two types of variables

(one for the assignment of trips to depots and another to obtain a feasible flow,

e.g., [Bertossi et al., 1987]) while other propose a more compact model with only

one type of variables (e.g. [Ribeiro and Soumis, 1994]). [Mesquita and Paix˜ao,

1999] show that both variants lead to the same LP-relaxation. Moreover, they

prove that multicommodity flow formulations have a tighter LP-relaxation than

single-commodity formulations. Multicommodity flow formulations can be fur-

ther classified by the underlying network structure. Among others, [L¨obel, 1997]

and [L¨obel, 1998] use a connection-based network where each feasible connection

between two trips corresponds to an explicit arc in the network. Notice that the

size of the network grows quadratically with the number of trips. Since the num-

ber of connection arcs can be vast, they propose an arc generation approach with

a special Lagrangian pricing technique to solve large instances to proven optimal-

ity. Recently, [Kliewer et al., 2006b] propose a multicommodity flow model based

on a time-space network that does not explicitly consider all possible connections

between trips. The network structure exploits the transitivity property of partial

ordered sets and aggregates connections between groups of compatible trips. In

fact, this approach reduces the number of connection arcs dramatically (by 97 to

99%) if the number of start and end locations is small compared to the number

of trips. They report solving large scale real-world instances with up to 7,068

trips and five depots to optimality with an off-the-shelve mixed-integer program-

ming (MIP) optimization software. In [Gintner et al., 2005] the authors propose

a two-phase heuristic that first fixes some connections based on the solutions of

easier subproblems and optimizes the reduced model with a standard MIP solver.

Their results indicate that close to optimal solutions can be found for very large

scale instances in reasonable time, e.g., for an instance with 11,062 trips and 55

depot/vehicle type combinations in about five hours.

In contrast to the latter formulations columns in set partitioning models for the

2. Integrated Vehicle and Crew Scheduling: State-of-the-Art

MDVSP correspond to feasible vehicle routes. The basic idea is to enumerate all

feasible vehicle routes and choose a subset of routes that partitions the set of trips.

Set partitioning formulations can be derived from multicommodity flow formu-

lations by applying the Dantzig-Wolfe decomposition principle (see [Ribeiro and

Soumis, 1994]). Recently, [Hadjar et al., 2006] discuss a branch-and-bound algo-

rithm for a set partitioning formulation that includes column generation, variable

fixing, and cutting planes. As originally proposed by [Ribeiro and Soumis, 1994]

they use a column generation algorithm to solve the LP-relaxation in each node

of the branch-and-bound tree since the number of feasible columns is enormous.

Furthermore, they apply a variable fixing strategy similar to [Bianco et al., 1994].

Their experiments on randomly generated instances involve up to 850 trips and

6 depots. However, their results are difficult to compare with results obtained

from real-world instances since the average number of trips per vehicle route is

small (approximately 4). Apparently, such instances are not very realistic.

For a recent survey on models for multiple-depot vehicle scheduling problem

the reader is referred to [Bunte and Kliewer, 2006].

Crew Scheduling

Crew scheduling is similar to vehicle scheduling but is more complex due to duty

feasibility constraints. It has received considerable attention in the operations

research literature since the late eighties. An overview of earlier works on crew

scheduling can be found in [Carraresi and Gallo, 1984]. In the following we

will not only review literature from public bus transport, we will rather include

sources from airline crew scheduling since similar models and solution approaches

are used there. Recall from section 1.1.2 that the crew scheduling problem (CSP)

is NP-hard even if only working time or spread time constraints are imposed

(see [Fischetti et al., 1987] and [Fischetti et al., 1989]).

The CSP is usually formulated as a set partitioning problem and solved with

a column generation approach since the number of columns is huge in real-world

problems. In such a model columns represent feasible crew duties while the con-

straints ensure that each vehicle activity (task) outside the depot is covered by

exactly one driver. Consequently, duty feasibility constraints have to be con-

sidered in the pricing problem only. Several authors formulate the CSP as set

covering problem that allow tasks to be over-covered. In practice, this over-

covering is often not acceptable, but solutions of this model often contain little

or no over-covers (since it is cheaper to assign only one driver to a task). The

main advantage of a set covering over a set partitioning formulations is that

continuous and integer solutions can be easier computed.

2.2. Literature Review

In [Desrochers and Soumis, 1989] column generation was applied to the CSP in

public transport for the first time. They propose to solve the LP relaxation of a

set covering formulation with column generation and model the pricing problem

as a resource constrained shortest path problem. A dynamic programming algo-

rithm similar to [Desrochers and Soumis, 1988] is used to generate new negative

reduced cost columns. Moreover, in order to obtain integer solutions, the column

generation algorithm is used to solve the linear relaxation of the master problem

in each node of the branch-and-bound tree. A similar approach for airline crew

scheduling (pairing) is proposed by [Lavoie et al., 1988] where the pricing problem

corresponds to a pure shortest path problem on a specific state-expanded net-

work structure. Other successful applications of a column generation approach

to solve the LP relaxation of a set partitioning/covering formulation are [Falkner

and Ryan, 1992] and [Desrochers et al., 1992]. In the following fifteen years,

branch-and-price approaches have been further refined by acceleration strate-

gies (see [Desaulniers et al., 2002]) , stabilization (see [Du Merle et al., 1999]

and [Ben Amor et al., 2006]), and heuristics (see [Barnhart et al., 1998], [Vance

et al., 1997a], and [Danna and Le Pape, 2005]) in order to tackle huge real-world

crew scheduling problems.

Instead of solving the linear relaxation of the master problem by column gen-

eration, [Carraresi et al., 1995] and [Freling, 1997] approximately solve the La-

grangian relaxation by column generation. A branch-and-price heuristic with

speed-up techniques is proposed by [Gr¨otschel et al., 2003]. Instead of solving

the Lagrangian dual with a subgradient method, they solve the dual of the mas-

ter problem by a coordinate ascent method (e.g. [Wedelin, 1995]) in combination

with a boxstep stabilization technique (see [Marsten et al., 1975]). Both subgra-

dient and coordinate ascent methods are based on Lagrangian relaxation. The

pricing problem is a resource constrained shortest path problem that is solved

by a two-phase algorithm. First, Lagrangian distance labels are generated. In

a second step, these labels serve as backtracking criterion in an enumerative

algorithm. Finally, they propose a heuristic variable fixing strategy within a

branch-and-generate framework that does not allow to backtrack.

However, duty feasibility constraints often arise in practice that cannot be cov-

ered with a resource constrained shortest path formulation. A simple workaround

in order to deal with this problem is to ignore these rules in the RCSP pricing

problem and skip all duties that violate them in a second phase. Another way of

overcoming this difficulty has been suggested by [de Silva, 2001], [Fahle, 2002],

and [Yunes et al., 2005] who apply constraint programming techniques to solve the

pricing problem. Constraint programming models allow to model a wide variety

of complex work rules that cannot be covered by a resource constrained shortest

2. Integrated Vehicle and Crew Scheduling: State-of-the-Art

path formulation. In particular, the results of [Yunes et al., 2005] indicate that in

a column generation context a combination of mathematical programming and

constraint programming performs better than isolated approaches.

Many heuristic approaches to the crew scheduling problem have been sug-

gested: see [Wren and Rousseau, 1995] for a survey and, more recently, [Fores

et al., 2002]. Most of the approaches heuristically generate a subset of feasi-

ble duties and solve a set partitioning/covering model with these duties after-

wards. Furthermore, several metaheuristics have been proposed for solving the

crew scheduling problem. Genetic algorithms are used by, among others, [Wren

and Wren, 1995], [Kwan et al., 1999], [Kwan et al., 2001], [Marchiori and Steen-

beek, 2003], and [Li and Kwan, 2005]. [Cavique et al., 1999] and [Shen and Kwan,

2001] suggest tabu search algorithms. [Louren¸co et al., 2001] propose a genetic

and tabu search algorithm that involves multiple objectives.

[Barnhart et al., 2003] and [Gopalakrishnan and Johnson, 2005] provide ex-

tensive surveys on the state-of-the-art of airline crew scheduling.

2.2.2. Partial Integration

Scheduling vehicles independently of crews was seriously criticized in the early

eighties by [Bodin et al., 1983] since crew costs mostly dominate vehicle costs.

Although integrated vehicle and crew scheduling problems have been proposed

in literature at that time (see [Ball et al., 1983] and [Patrikalakis and Xerocostas,

1992]), it was not until recently that problems of considerable size and multiple

depots could be solved in an integrated manner (see [Huisman et al., 2005a]).

Consequently, most approaches until the late nineties were based on a heuristic

integration of both problems since a fully integrated consideration was compu-

tationally intractable. Such a heuristic integration is called a partial integration

of vehicle and crew scheduling. Similar to [Freling, 1997] we distinguish between

two types of partial integration:

•perform crew scheduling but include vehicle scheduling considerations and

construct a feasible vehicle schedule afterwards (crew first - vehicle second),

•perform vehicle scheduling but include crew scheduling considerations and

subsequently generate a feasible crew schedule (vehicle first - crew second).

Most approaches of the first category are inspired by the landmark contribution

[Ball et al., 1983]. The authors define a multigraph that shares the same set

of nodes but contains two types of arcs: one type for combined vehicle-crew

activities and another for crew-only activities (such as waiting or walking). The

2.2. Literature Review

set of nodes corresponds to a source, a sink, and the set of tasks (called d-

trips), i.e., elementary portions of work that must be operated by one driver and

one vehicle. The single depot is represented by the source and the sink. Each

vehicle schedule corresponds to a set of node disjoint paths from the source to the

sink that partitions the node set and that uses only combined vehicle-crew arcs.

Similarly, a feasible crew schedule corresponds to a set of node disjoint paths

from the source to the sink that partitions the node set and that may contain

both types of arcs. Furthermore, both sets of paths must be compatible to make

an overall feasible solution. Vehicle and crew schedule are compatible if each

vehicle-crew arc in the vehicle schedule is also covered in the crew schedule while

all vehicle-crew arcs not contained in the vehicle schedule are not part of the

crew schedule. However, the integrated model is not practical due to prohibitive

network dimensions. Thus, the solution process is decomposed into three parts

that emphasize the crew scheduling phase: heuristically construct a set of pieces

of work, improve the set of pieces by recombination, and generate a feasible crew

schedule. The pieces of work are constructed in such a way that a feasible vehicle

schedule can always be derived. To sum up, the model is integrated while the

solution approach is sequential. Other heuristic approaches of the first category

have been suggested by [Tosini and Vercellis, 1988], [Falkner and Ryan, 1992],

and [Patrikalakis and Xerocostas, 1992]. All these approaches use a network

structure similar to [Ball et al., 1983].

Heuristic approaches of the second category have been suggested by [Scott,

1985] and [Darby-Dowman et al., 1988]. In the latter, an interactive decision

support system is described that allows to include crew scheduling considerations

while performing vehicle scheduling. However, no details on the models and

algorithms are provided. The system is part of a planning system developed for

the Rome transport agency.

Recently, another approach of the second category is proposed by [Bornd¨orfer

et al., 2002]. The authors modify the costs in the vehicle scheduling problem

in such a way that pull-in/out trips are encouraged while connections between

long service trips are discouraged. Furthermore, the authors impose constraints

on the length of vehicle blocks. These modifications aim at generating vehicle

blocks with many relief opportunities for drivers. As a consequence, there is more

flexibility when crews are scheduled.

Another interesting approach that belongs to neither category was proposed

by [Gintner et al., 2004, Gintner et al., 2006a] and [Gintner, 2007]. The basic

idea is to change a given optimal vehicle schedule without loss of optimality in

the crew scheduling phase. They set up a time-space network that allows to

recombine parts of vehicle blocks in order to disclose additional flexibility in crew

2. Integrated Vehicle and Crew Scheduling: State-of-the-Art

scheduling while maintaining the optimality of vehicle schedule. In other words,

the crew scheduling approach does not only consider a single optimal vehicle

schedule, but a set of optimal vehicle schedules with minimum fleet size and

minimum operational costs. The authors report savings of 8-24% on real-world

instances as compared with a sequential approach.

2.2.3. Complete Integration

As reviewed in the previous subsection, only few partially integrated approaches

have been suggested for the integrated vehicle and crew scheduling problem in the

eighties and the early nineties. However, the problem has lately attracted several

researchers who developed models and solution techniques mainly based on de-

composition approaches for mathematical programming. In the following, we will

first discuss integrated models for the single depot and then for the multiple-depot

case.

Single Depot Case

The first mathematical formulation for the integrated vehicle and crew schedul-

ing problem with a single depot is given in [Patrikalakis and Xerocostas, 1992].

However, the model is computationally intractable and, thus, the authors resort

to a partially integrated solution method.

[Freling, 1997] propose the first integrated treatment of vehicle and crew

scheduling in terms of model and solution approach. The model consists of three

components: a quasi-assignment formulation for vehicle scheduling, a set parti-

tioning formulation for crew scheduling, and additional linking constraints that

ensure the compatibility of vehicle and crew schedule. He suggests a solution

approach that is based on column generation in combination with Lagrangian re-

laxation. The set partitioning and linking constraints are relaxed such that two

independent Lagrangian subproblems remain: a single depot vehicle scheduling

problem and an easy selection problem. The Lagrangian dual problem is solved

by a subgradient algorithm while a novel two-phase pricing method is proposed

to generate new columns (duties). Finally, he applies several heuristics to obtain

feasible integer solutions. The approach is tested on real-world and randomly

generated instances with up to 148 trips on a Pentium 90 PC with 32MB of main

memory. The largest instance is solved in approximately one hour where 96%

of the time is spent on the column generation pricing problem. The solution

methodology of [Freling, 1997] has inspired a series of publications (e.g. [Freling

et al., 2001a], [Freling et al., 2003], and [Huisman, 2004]) and also forms the basis

2.2. Literature Review

of our solution approach (see Section 2.4).

[Friberg and Haase, 1999] propose an integrated mathematical formulation

that combines two set partitioning models: the vehicle scheduling model of

[Ribeiro and Soumis, 1994] with the crew scheduling model of [Desrochers and

Soumis, 1989]. They propose an exact branch-and-cut-and-price algorithm where

they solve the LP-relaxation in each node of the search tree by column gener-

ation. Furthermore, clique cuts (see [Hoffman and Padberg, 1993]) are derived

to tighten the LP-relaxation. However, only small sized instances with up to 20

trips are solved within one hour of computational time on a SUN Sparc-10/40.

[Haase et al., 2001] introduce an interesting crew scheduling formulation with

side constraints that involves duty flow variables and a bus counter variable. The

set partitioning formulation with flow conservation and bus count constraints

(similar to the plane count constraints of [Klabjan et al., 2002]) guarantees that

an optimal vehicle schedule can always be derived afterwards. They propose an

elaborate branch-and-price algorithm that relies on several acceleration strategies,

e.g., dynamic generation of bus count constraints and appropriate substitution

of partitioning constraints in order to reduce column density. Computational

results with randomly generated data show that instances with at most 150 trips

(300 tasks) can be solved in 82 minutes on a SUN Ultra-10/400 and an average

optimality gap of 0.3%. In order to tackle larger instances they suggest a heuristic

version where multiple branching decisions are made at every node of the search

tree. With this approach they solve instances with up to 350 trips (700 tasks) in

approximately two hours on average and with an average integrality gap of 0.3%.

[Bornd¨orfer et al., 2002] suggest a formulation for the integrated problem with

a single depot that combines the vehicle scheduling model of [L¨obel, 1998] with

the crew scheduling approach of [Bornd¨orfer et al., 2001,Gr¨otschel et al., 2003].

They propose a column generation algorithm based on Lagrangian relaxation to

solve a linked multicommodity network flow and set partitioning formulation.

The Lagrangian dual problem is solved with a subgradient method. They apply

two primal heuristics to compute feasible solutions. Their computational tests

involve three scenarios with up to 1,457 trips and one depot and is performed on

a dual Intel Xeon PC 1.7GHz with 1GB of main memory. They solve the largest

instance in approximately 6.5 hours where 50% of the cpu time is spent on the

column generation pricing problem. However, the authors do not compute a valid

lower bound, and, thus, cannot assess the quality of their solutions.

[Valouxis and Housos, 2002] describe a combined vehicle and crew scheduling

problem that is actually a crew scheduling problem since drivers are tied to their

vehicle. They propose a fast heuristic which is based on column generation and

solve instances with up to 350 trips within a given 30 minute timeframe. Another

2. Integrated Vehicle and Crew Scheduling: State-of-the-Art

elaborate heuristic is proposed by [Rodrigues et al., 2006] where vehicle and crews

are scheduled in an integrated way. Unlike other approaches mentioned so far,

the set of trips is not given in advance. Instead, the timetable is heuristically

constructed to meet an estimated passenger demand. They test their algorithm

on a real-world scenario from Sao Paulo in Brazil with up to 395 trips. How-

ever, they do not compute lower bounds and, consequently, the quality of their

solutions cannot be assessed.

Multiple-Depot Case

The main difference between single and multiple-depot integrated vehicle and

crew scheduling is that the vehicle scheduling subproblem with multiple depots

is NP-hard unlike the single depot case (see Section 2.2.1).

The integrated vehicle and crew scheduling problem with multiple depots has

been introduced by [Gaffi and Nonato, 1999]. Their approach is developed for

ex-urban public transport systems where crews are virtually tied to their vehicle

or crew-deadheading (by foot) is highly constrained. In particular, crews may

only be relieved in depots and, thus, vehicle blocks correspond to pieces of work.

These assumptions make the problem computationally much more attractive than

the general case that we consider in this thesis. Their formulation consists of two

parts: a quasi-assignment model for scheduling vehicles and a set of linking con-

straints to ensure compatibility of the crew schedule. Furthermore, the number

of vehicle blocks per depot and the number of duties of a specific type can be

restricted. The authors develop a heuristic column generation algorithm based

on a Lagrangian relaxation with all linking constraints relaxed. They test their

approach on a Power PC 604/180MHz with real-world ex-urban and sub-urban

instances. Their results with instances of up to 257 trips and 28 depots in the ex-

urban setting show that their algorithm finds feasible solutions to all instances

while the planning system currently used could not. The cpu time is greater

than 24 hours for the largest instance and between 2 and 6 hours on average.

Their approach seems to be less suitable for the sub-urban setting with more

relief opportunities and smaller distances between the depots. However, they do

not compute lower bounds which makes it difficult to assess the quality of their

solutions.

[Huisman, 2004,Huisman et al., 2005a] investigate two formulations that gen-

eralize the single depot models of [Freling, 1997,Freling et al., 2003] and [Haase

et al., 2001] to the multiple depot case. Moreover, they propose two similar

adaptations of the solution approach that was suggested for the single depot case

by [Freling, 1997]. In the following we will review the formulation of [Huisman,

2.2. Literature Review

2004,Huisman et al., 2005a] that is based on the single depot formulation of [Frel-

ing, 1997]. We will compare this formulation with the formulation of [Gintner,

2007] in Section 2.3.

Let T={1,2, . . . , n}be the set of ntimetabled trips where trip istarts at

time stiand ends at time eti. We assume that the set of trips is ordered by

increasing start times with sti≤sti+1. Furthermore, we denote by τij the travel

time between the end location of trip iand the start location of trip j. Two

trips iand jare said to be compatible if they can be covered consecutively by

the same vehicle, that is eti+τij ≤stjholds. Now, let us define H={(i, j)|i <

j, i and jcompatible, i ∈ T , j ∈ T } as the set of deadheads (including waiting

activities outside the depot). Let D={1,2, . . . , m}be the set of depots. For

each depot d∈ D we define an acyclic vehicle scheduling network Gd= (Nd, Ad)

with nodes Nd=Td∪{rd, td}and arcs Ad=Hd∪(rd×T d)∪(Td×td) where both

rdand tdrepresent depot d. The set of trips and deadheads that can be serviced

from depot d∈ D is denoted by Tdand Hd, respectively. Pull-out (pull-in) trips

are denoted by rd× T d(Td×td). We associate vehicle costs cd

ij with each arc

(i, j)∈Adthat are typically a function of travel and idle time. Moreover, we add

a fixed (asset) cost for using a vehicle to the cost of each pull-out arc. As stated

earlier, we assume that a vehicle returns to its depot if the idle time between two

trips is long enough to perform a round-trip to the depot. Deadhead arcs between

trips that allow a round-trip to the depot are called long arcs Ald ⊂Ad. All other

deadhead arcs between trips are short arcs Asd ⊂Ad. Finally, we introduce two

types of decision variables: flow variables and duty variables. Flow variable yd

indicates whether arc (i, j)∈Adis used and assigned to depot d∈ D or not.

Likewise, duty variable xd

k∈Kdwith associated cost fd

kindicates whether duty k

is selected for depot d∈ D or not. Furthermore, Kd(i) denotes the set of duties

that cover trip i∈ T dwhile Kd(i, j) denotes the set of duties covering deadhead

task (i, j)∈Asd. Note that this implicitly assumes that a trip corresponds to

exactly one task. [Huisman, 2004] proposes to state the integrated vehicle and

crew scheduling problem with multiple depots (MDVCSP-H) as follows.

d∈D X

(i,j)∈Ad

ijyd

ij +X

d∈D X

k∈Kd

kxd

k→min (2.1)

s.t. X

d∈D X

j:(i,j)∈Ad

ij = 1 ∀i∈ T (2.2)

d∈D X

i:(i,j)∈Ad

ij = 1 ∀j∈ T (2.3)

i:(i,j)∈Ad

ij −X

i:(j,i)∈Ad

ji = 0 ∀d∈ D,∀j∈Nd(2.4)

2. Integrated Vehicle and Crew Scheduling: State-of-the-Art

j:(i,j)∈Ad

ij −X

k∈Kd(i)

k= 0 ∀d∈ D,∀i∈Nd(2.5)

ij −X

k∈Kd(i,j)

k= 0 ∀d∈ D,∀(i, j)∈Asd (2.6)

itd+X

j:(i,j)∈Ald

ij −X

k∈Kd(i,td)

k= 0 ∀d∈ D,∀i∈Nd(2.7)

rdj+X

i:(i,j)∈Ald

ij −X

k∈Kd(rd,j)

k= 0 ∀d∈ D,∀j∈Nd(2.8)

ij ∈ {0,1} ∀d∈ D,∀(i, j)∈Ad(2.9)

k∈ {0,1} ∀d∈ D,∀k∈Kd(2.10)

The objective function (2.1) minimizes the sum of vehicle and duty costs. Con-

straint sets (2.2)-(2.4) correspond to a multicommodity flow formulation for the

vehicle scheduling problem. Constraint set (2.5) imposes that each trip will be

covered by a duty from a depot if and only if the trip is covered by a vehicle

from the same depot. Similarly, constraints (2.6)-(2.8) establish the link between

vehicle and crew deadheads where deadheads corresponding to short and long

arcs are considered separately.

The solution approach to solve MDVCSP-H consists of two phases: the first

phase computes a lower bound on the optimal solution value while a feasible so-

lution is constructed in the second phase. To obtain a lower bound he solves the

linear relaxation of model MDVCSP-H using a column generation method in com-

bination with Lagrangian relaxation. The author relaxes constraints (2.4)-(2.8)

and, hence, obtains a large single depot vehicle scheduling problem and a triv-

ial selection problem as Lagrangian subproblems. The Lagrangian dual problem

is solved with a subgradient method while the vehicle scheduling subproblem is

solved with the combined forward and reverse auction algorithm of [Freling, 1997].

As originally proposed in [Freling, 1997] he uses a two phase (column generation)

pricing problem to generate new negative reduced cost columns where pieces of

work are generated in the first and feasible duties in the second phase. Finally,

a heuristic solution to model MDVCSP-H is constructed where constraints (2.5)-

(2.8) are relaxed in a Lagrangian way. Again, the Lagrangian dual problem is

solved with a subgradient method where the Lagrangian subproblem corresponds

to a multiple-depot vehicle scheduling problem. Hence, each feasible solution to

the subproblem constitutes a feasible vehicle schedule. Feasible crew schedules

are constructed by solving a crew scheduling problem based on the current so-

lution of the Lagrangian subproblem. However, only columns generated in the

lower bounding phase are considered, that is no new columns are constructed in

the second phase.

2.2. Literature Review

A series of tests on real-world and randomly generated data with up to 653

trips indicates that both integrated approaches lead to efficiency gains compared

to sequential planning (vehicle first - crew second). The average number of depots

which a trip can be assigned to lies between 1.27 and 2.47 for real-world and is

either 2 or 4 for artificial instances. The cpu times to compute lower bounds

for the real-world instances varies a lot and takes at most six hours on an Intel

Pentium III PC 450MHz with 128 MB main memory. The results on artificial

data show that instances with up to 200 trips and 2 depots can be solved in

approximately 2 hours on average while it takes about 2.5 hours on average for

160 trips and 4 depots. The cpu time is limited to 3 hours per randomly generated

instance. About 85% percent of the cpu time is spent in the column generation

pricing problem. The gap between best lower and best upper bound lies between

5.31% and 8.11% for the approach based on model MDVCSP-H. Additionally,

neither of the integrated approaches can outperform the other. However, the

approach based on model MDVCSP-H regularly provides tighter lower bounds.

In order to tackle large instances, [de Groot and Huisman, 2004] suggest several

heuristics that split large instances into several smaller ones which can be solved

by an integrated or sequential vehicle and crew scheduling method. They use

the same formulation and solution approach for integrated problems as stated

above. They test their heuristics on real-world instances with up to 1,372 trips

and 6 depots on a Intel Pentium IV PC 1.8GHz/512MB main memory. In their

setting each trip can be assigned to 1.27-3.64 depots on average. Their results

show that large (previously unsolved) instances can be solved now. Furthermore,

they show that their heuristics disclose efficiency gains compared to a simple

sequential approach. Interestingly, the best heuristic outperformed an integrated

approach with a given time limit in terms of solution quality and time. The

largest instance is solved in about 1 hour with a heuristic.

Another approach that relies on model MDVCSP-H is proposed by [Bornd¨orfer

et al., 2004]. They use a solution approach similar to the one sketched above since

they aim at computing a lower bound first and subsequently generate an integer

feasible solution. However, the authors solve the Lagrangian dual problem with

an inexact adaptation of a proximal bundle method (see [Kiwiel, 1995]) that

produces dual and additional primal information as opposed to a subgradient

algorithm. The inexact bundle method is embedded in a backtracking proce-

dure to produce integer solution in the second phase. The procedure utilizes the

primal information produced by the bundle method to iteratively fix deadhead

(flow) variables until the complete vehicle schedule is fixed. A compatible crew

schedule is generated as described in [Gr¨otschel et al., 2003]. The authors re-

port computational results with both real-world and artificial data. All tests are

2. Integrated Vehicle and Crew Scheduling: State-of-the-Art

performed on a dual Intel Xeon PC 3GHz/4GB main memory. The largest real-

world instance contains 1414 trips with 1 depot and 3 vehicle types and is solved

in about 125 hours. Furthermore, they compare their approach with [Huisman,

2004] on the same set of artificial instances. Using the same assumptions their

approach clearly outperforms Huisman’s method and solves instances with up to

400 trips and 2 (4) depots in 3.3 (12) hours.

Very recently, [Mesquita and Paias, 2006] propose two mathematical formula-

tions similar to MDVCSP-H but with fewer constraints. Both models involve

a multicommodity network flow model for vehicle scheduling while the crew

scheduling part either relies on a pure set partitioning or on a combined set

partitioning/covering formulation. They develop a column generation algorithm

where the LP relaxation of their formulation is solved with a commercial LP

solver. The column generation subproblem corresponds to a resource constrained

shortest path problem that is either solved exactly or approximately by a dy-

namic programming algorithm similar to [Desrochers and Soumis, 1988]. If the

optimal solution of the LP relaxation is not integer, they use a commercial IP

(branch-and-bound) solver to find an integer solution over a subset of feasible

crew duties. They show that integer solutions can be obtained by branching on

one type of decision variables, i.e., either flow or duty variables. They report

computational results for the randomly generated instances that have also been

used by [Bornd¨orfer et al., 2004] and [Huisman, 2004, Huisman et al., 2005a].

However, it should be mentioned here that they make different assumptions and,

therefore, their results cannot be directly compared for the following reasons.

•They do not assume that a vehicle returns to its depot if the idle time

between two consecutive trips is long enough to perform a round-trip to

the depot. Consequently, it is easy to construct a piece of work that is

feasible in their definition but not in Huisman’s.

•The authors only provide computational results for the case where they

do not assume that a crew may only conduct tasks on vehicles from a

single depot, i.e., changeovers are allowed during a piece of work (and not

only during a break). This makes the problem computationally much more

attractive. Furthermore, they provide computational results where drivers

may walk on deadhead connections that are not part of the vehicle schedule.

•A different set of duty types is used that expands the solution space.

Nevertheless, all computational tests are performed on an Intel Pentium IV

3.2GHz. The authors are able to solve instances with up to 400 trips and 4

depots in less than 4 hours. It still remains an open question whether their

2.3. Modeling approach

method is also effective under the assumptions as stated by [Huisman, 2004].

In [Mesquita et al., 2006] the authors compare different branching strategies for

the model and solution approach sketched above. They are able to solve the same

randomly generated instances as above with up to 100 trips in about 3.5 hours

using an exact branch-and-price algorithm. The results of the heuristic branching

schemes correspond to the results presented in [Mesquita and Paias, 2006].

[Hollis et al., 2006] present a new set covering formulation with side constraints

for an integrated vehicle and crew scheduling problem with multiple depots faced

by Australia Post mail distribution. The main difference to all approaches dis-

cussed before is that the set of trips is not given in advance. Instead the set of

trips (routes) is heuristically constructed by solving a vehicle routing problem

prior to the actual integrated problem. Furthermore, as opposed to other formu-

lations they include crew and vehicle capacity constraints separated by depot.

They use a heuristic column generation procedure to solve instances with up to

1,181 shipments. We do not provide more details on their results since they are

not comparable with those given above.

Finally, [Gintner, 2007] proposes a model that is based on a time-space network

and leads to a mathematical formulation with fewer constraints and variables

compared to approaches previously exposed in literature. Since we will use this

formulation in the remainder of this thesis, we devote the next section to describe

this model in detail. Moreover, we describe his solution approach in Section 2.4.

2.3. Modeling approach

In this section we discuss a modeling approach and mathematical formulation for

the integrated vehicle and crew scheduling problem with multiple depots under

the assumptions stated in Section 2.1. The formulation is introduced by [Gint-

ner, 2007] and combines a multicommodity network flow formulation for vehicle

scheduling with a set partitioning formulation for crew scheduling. The main

advantage of this formulation is the structure of the underlying vehicle schedul-

ing network that leads to models with fewer constraints and variables compared

to approaches previously exposed in literature. Recall from Section 2.2.1 that

multicommodity network flow formulations for multiple-depot vehicle scheduling

problems can be classified by the underlying network structure. In a connection-

based network (CBN), each feasible connection between two trips corresponds to

an explicit arc in the network while in a time-space network (TSN) only con-

nections between groups of compatible trips are considered. In fact, a time-

space network approach reduces the number of connection arcs dramatically if

2. Integrated Vehicle and Crew Scheduling: State-of-the-Art

the number of start and end locations is small compared to the number of trips. A

time-space network structure for the multiple depot vehicle scheduling problem in

public transport has been introduced by [Kliewer, 2005,Kliewer et al., 2006b] and

adapted for integrated vehicle and crew scheduling problems by [Gintner, 2007].

In the following, we will first describe the time-space network used and, subse-

quently, present the mathematical formulation that we will use in the remainder

of this thesis. The exposition in this section follows [Gintner, 2007].

In a time-space network each node represents a specific location at a particular

time while each arc corresponds to a transition in time and, possibly, space. In

order to ease the exposition we first assume that there is only one depot and

one vehicle type. The vehicle scheduling solution must satisfy flow conservation

constraints that force the vehicles to circulate through a network of service trips

where each vehicle must return to the depot where its daily schedule has started.

In a time-space network, flow conservation is enforced by modeling the activity at

each station (including the depot) with a timeline. Each timeline contains nodes

that either represent arrivals or departures from the station. Each departure

(arrival) splits an edge of the timeline and adds a node to the timeline at the

departure (arrival + minimum layover) time (see Figure 2.1). Notice that each

Figure 2.1.: Timeline of a station with four arrivals and two departures.

arrival node has only one outgoing waiting arc in a time-space network while in

a connection-based network each arrival node entails an arc to each (feasible)

subsequent departure node. A trip arc is used to connect the corresponding

departure and arrival nodes at the start and end location of a trip. Furthermore,

a pull-out (pull-in) trip arc is added from (to) the depot for each trip including the

corresponding departure (arrival) nodes in the depot timeline. The timeline of

the depot is made a cycle to force a circulation through the network that arises

from the flow conservation constraints and the lack of source and sink nodes.

Since the network has a timespan of one day, the circulation flow defined by a

solution defines a daily vehicle schedule. A flow along a timeline (on waiting

arcs) represents a vehicle waiting at the station while a flow on pull-in/out and

2.3. Modeling approach

trip arcs corresponds to vehicle movements. Each unit of flow on the circulation

arc from the last to the first node of the depot timeline corresponds to a vehicle

used. Consequently, each path from the first to the last depot node represents a

daily schedule for one vehicle.

Vehicle movements without passengers (deadheads) are virtually unrestricted

in public bus transport. Thus, a deadhead arc is added between two compatible

trips that require a deadhead from the end location of the first trip to the start

location of the second one. However, it is not necessary to connect all pairs of

compatible trips explicitly as in a connection-based network. Instead each arrival

node is connected with the next compatible departure node of the start location of

the second trip. All subsequent connections at the station are implicitly included

by traversing the timeline.

As an example, Figure 2.2 shows the deadhead arcs of a connection-based and

time-space network for 11 trips that arrive at station Aor depart from station

B. In time-space networks, there is at most one deadhead arc to connect a

trip with all subsequent trips at a different station as opposed to a connection-

based network where there is an explicit arc for each pair of compatible trips.

Moreover, in time-space networks, not all arrival nodes at station Ahave an

outgoing deadhead arc to station B, e.g., arrival node of trip t2. There is no

benefit of adding a deadhead arc between the arrival node of t2and the departure

node of t9since both trips can be connected by following the timeline at station

A(and using the deadhead between t3and t9). The same reasoning holds for

connecting trip t4and t10. In other words, if a group of arrivals at station A

has the same first compatible departure at station B, only the latest arrival

of the group must be connected with the first compatible departure. In our

example, we have 14 deadhead arcs in the connection-based network, but only 3

in the time-space network. In time-space networks, the small number of deadhead

arcs generally outweighs the overhead generated by introducing waiting arcs in

timelines. We will discuss the network complexity of both network structures in

the next paragraph.

Let nbe the number of service trips and mthe number different start and

end locations. Typically, the number of different start and end locations is small

compared to the number of service trips in real-world settings. As stated earlier

there is at most one deadhead arc to connect a trip with all subsequent trips at a

different station in time-space networks as opposed to a connection-based network

where there is an explicit arc for each pair of compatible trips. While the number

of deadhead arcs in a connection-based network is O(n2), time-space networks

only contain O(nm) deadhead arcs with nm. Notice that the number of

waiting arcs grows linearly with the number of tasks. As discussed in [Gintner,

2. Integrated Vehicle and Crew Scheduling: State-of-the-Art

connection-based

network

t11

t10

station A

station B

t8t9

time-space

network

t2t6

t11

t10

station A

station B

t8t9

Figure 2.2.: Deadhead arcs in a connection-based and time-space network be-

tween two stations

2007], Table 2.1 provides the number of deadhead arcs for instances with 100 to

2,000 trips for both network formulations. It is easy to see that the number

of deadhead arcs can be dramatically reduced (up to 99%) with a time-space

network formulation. It is important to mention that both networks are directed

acyclic graphs and contain all compatible connections between trips.

Finally, the costs and capacities associated with each arc are defined as follows.

Generally, vehicle costs consist of both fixed and variable costs where variable

costs reflect operating time outside the depot and distance covered. Thus, the

sum of operating time and distance costs is assigned to pull-in/out, trip, and

deadhead arcs while only operating time costs are considered for waiting arcs

outside the depot. A vehicle waiting inside the depot does not incur any costs.

In order to reflect asset costs of vehicles, the circulation arc takes a fixed cost for

each unit of flow. The maximum capacity of pull-in/out and trip arcs is set to 1

2.3. Modeling approach

network type # trips

100 200 400 800 2,000

connection-based network 4,043 16,396 65,788 269,462 1,879,262

time-space network 362 946 2,106 4,589 17,086

Table 2.1.: Number of deadhead arcs of a connection-based and time-space net-

work structure as presented in [Gintner, 2007]

while all other arcs have a maximum capacity equal to the number of available

vehicles.

Figure 2.3 depicts an example of a time-space network with three stations, six

trips, and one depot. Notice that there is no waiting arc at station Bbetween

Figure 2.3.: Time-space network with six trips

trips t3and t5since we assumed in Section 2.1 that a vehicle returns to its depot if

the idle time between two consecutive trips is long enough to perform a round-trip

to the depot.

So far, a directed acyclic time-space network structure is defined for scheduling

vehicles where each path from the first to the last depot node corresponds to a

daily schedule for one vehicle. Now, we will turn our attention to scheduling

crews. As assumed in Section 2.1 each trip starts and ends with a relief point.

As a result, each node in the vehicle scheduling network corresponds to a relief

point for drivers and each arc (except waiting arcs in the depot) represents a task.

Moreover, each path between two nodes is associated with a piece of work if it

satisfies piece feasibility constraints. Additionally, such a path must not contain

2. Integrated Vehicle and Crew Scheduling: State-of-the-Art

waiting arcs in the depot since each stop in the depot terminates the current

piece of work. A duty consists of at least one such piece of work (path). A duty

is feasible if all duty feasibility constraints are satisfied and, clearly, incurs costs.

To sum up, the integrated vehicle and crew scheduling problem aims at finding

a (vehicle) network flow solution and a set of duties such that each each flow

unit on the arcs (vehicle activity) of the network flow solution is also covered by

exactly one duty while all deadheads not contained in the vehicle schedule are

not part of any duty. Furthermore, the sum of vehicle and crew costs is to be

minimized.

If there are multiple depots,a network for each depot is set up where each trip

is represented by several trip arcs. Each trip arc corresponds to a depot-trip

combination. Of course, only one depot-trip combination can be selected for a

trip. Next, we discuss the mathematical formulation introduced by [Gintner,

2007] that is based on the time-space network described above.

Let T={1,2, . . . , n}be the set of ntimetabled trips and D={1,2, . . . , m}

be the set of depots. The set of trips that can be serviced from depot d∈ D is

denoted by Td. For each depot d∈ D,Gd= (Nd, Ad) defines an acyclic vehicle

scheduling network as described earlier in this section with Ndas the set of nodes

and Adas the set of arcs. e

Ad⊂Addenotes the set of arcs that requires both

vehicle and crew activities, i.e., all arcs except waiting arcs of depot timelines.

Let Ad(t) : T → Adbe a function that returns the set of trip arcs (i, j)∈Ad

for trip t∈ T and depot d∈ D. Note that Ad(t) = ∅if tcannot be operated

from depot d. A vehicle cost cd

ij is associated with each arc (i, j)∈Adwhich

is typically a function of travel and idle time. Moreover, a fixed (asset) cost

for using a vehicle from depot dis put on each circulation arc. The maximum

capacity ud

ij of pull-in/out and trip arcs (i, j)∈Ad,∀d∈ D is set to 1 while all

other arcs have a maximum capacity udequal to the number of vehicles available

in depot d∈ D. Finally, two types of decision variables are introduced: flow

variables and duty variables. Flow variable yd

ij indicates whether arc (i, j)∈Ad

is used and assigned to depot d∈ D or not. Likewise, the binary duty variable

k, k ∈Kdwith associated cost fd

kindicates whether duty kis selected for depot

d∈ D or not. Furthermore, Kddenotes the set of duties that can be operated

from depot d∈ D while Kd(i, j)⊂Kddefines the set of duties covering arc

(i, j)∈e

Ad. The integrated vehicle and crew scheduling problem with multiple

depots (MDVCSP) can be stated as follows:

d∈D X

(i,j)∈Ad

ijcd

ij +X

d∈D X

k∈Kd

kfd

k→min (2.11)

2.4. Solution Approach

s.t. X

d∈D X

(i,j)∈Ad(t)

ij = 1 ∀t∈ T (2.12)

{j:(j,i)∈Ad}

ji −X

{j:(i,j)∈Ad}

ij = 0 ∀d∈ D,∀i∈Nd(2.13)

k∈Kd(i,j)

k−yd

ij = 0 ∀d∈ D,∀(i, j)∈e

Ad(2.14)

0≤yd

ij ≤ud

ij, yd

ij ∈N∀d∈ D,∀(i, j)∈Ad(2.15)

k∈ {0,1} ∀d∈ D,∀k∈Kd(2.16)

The objective (2.11) minimizes the sum of vehicle and crew costs. Constraints

(2.12)-(2.13) correspond to a multicommodity flow formulation for the vehicle

scheduling problem where the set of trip tasks must be partitioned among the

depots (2.12) and flow conservation is ensured for each depot (2.13). Constraint

set (2.14) establishes the link between vehicle and crew schedule: each arc covered

by vehicle(s) must also be covered by the same number of duties assigned to the

depot from which the vehicle(s) originate(s). Constraints (2.15) guarantee that

the maximum capacity of the flow variables is satisfied.

A feasible solution to MDVCSP consists of a network flow solution and a

compatible set of duties. It is important to mention that any feasible solution

to the multicommodity flow formulation (2.12)-(2.13) represents several feasible

vehicle schedules since only connections between groups of trips are considered

in our network. However, a feasible vehicle schedule, that is also compatible to

the crew schedule, can always be constructed using a decomposition algorithm

(see [Gintner, 2007] for a description of the algorithm).

When the number of variables and constraints in models MDVCSP-H and

MDVCSP is compared, we see that in both formulations a flow variable and a

constraint are defined for each arc in the underlying vehicle scheduling network.

As shown in Table 2.1 the number of arcs is considerably smaller with a time-space

network structure. As a result, model MDVCSP is much more promising from

a computational point of view since it contains fewer variables and constraints

than model MDVCSP-H. Of course, models with smaller dimensions are not

necessarily more attractive. However, [Gintner, 2007] shows that MDVCSP is

indeed beneficial compared to MDVCSP-H.

2.4. Solution Approach

In this section, we discuss the solution approach to solve model MDVCSP as de-

scribed in [Gintner, 2007]. Our exposition in this section follows [Gintner, 2007].

2. Integrated Vehicle and Crew Scheduling: State-of-the-Art

The solution method is a combination of column generation and Lagrangian re-

laxation and has been inspired by [Freling, 1997,Huisman, 2004]. The benefit of

combining these methods is described in Section 1.5.3. Basically, the method will

be used in the remainder of this thesis. However, in Chapter 3 we will propose

new approaches for the column generation pricing problem as well as new meth-

ods for finding integer solutions. Furthermore, in Chapter 5 the general approach

will be extended to include further requirements from practice. An outline of the

approach is shown in Algorithm 5.

Algorithm 5: Solution method for model MDVCSP

(Step 1) Initialization

Solve MDVSP and, subsequently, CSP for each depot.

Take CSP solution as initial column set K0.

Set t= 0.

(Step 2) Solve restricted master problem

Solve a Lagrangian dual problem with the current set of

columns K0.

Store lower bound for the current set of columns and dual

information.

(Step 3) Solve pricing problem

Generate new columns K00 with negative reduced costs.

If |K00|= 0 terminate.

(Step 4) Perform column management

Add new columns to restricted master problem:

K0:= K0∪K00.

Delete columns with high positive reduced costs from K0

if |K0|is large.

(Step 5) Check termination criteria

Terminate if one of the following criteria is satisfied:

t≥tmax

No significant improvement of lower bound.

otherwise set t=t+ 1 and return to step 2.

(Step 6) Construct feasible solution

Use Lagrangian heuristic to construct feasible vehicle and

crew schedules.

First, a feasible solution is generated by using the sequential approach where

a multiple-depot vehicle scheduling problem (MDVSP) is solved using the so-

lution approach of [Kliewer et al., 2006b] and standard optimization software

such as MOPSr(see [Suhl, 2000]) or ILOG CPLEXr(see [ILOG, 2006]). Af-

2.4. Solution Approach

terwards, a crew scheduling problem (CSP) is solved for each depot based on

the vehicle schedule for that depot. The method we used to solve the CSP is

described in [Gintner, 2007]. The set of columns obtained by solving the series of

crew scheduling problems serves as initial column set for the column generation

algorithm.

The main part of the algorithm (Step 2-5) computes a lower bound using a

column generation algorithm in combination with Lagrangian relaxation. The

Lagrangian relaxation and subgradient method used to obtain dual information

and a lower bound on the current set of columns (Step 2) will be described in

Section 2.4.1. In order to improve the lower bound the algorithm tries to find

new duties with negative reduced costs in the pricing problem (Step 3). The

pricing problem is the topic of Section 2.4.2. When the pricing algorithm finds

new columns, these columns are added to the restricted master problem and,

possibly, columns with high positive reduced costs are deleted in Step 4. The

algorithm deletes columns only if the number of columns in the restricted master

problem exceeds a given threshold value. The method iterates from Step 2 to 5

as long as new negative reduced cost columns are found, the number of iterations

does not exceed tmax, and the lower bound improved significantly over the last n

iterations. Finally, a feasible solution is computed using a Lagrangian heuristic

(Step 6) which we will discuss in Section 2.4.3.

2.4.1. The Master Problem

As described in Section 1.5.3 Lagrangian relaxation can be applied to the exten-

sive formulation (in a column generation context) in order to obtain approximate

dual solutions and lower bounds to the restricted master problem. Instead of

solving restricted problems with the simplex method to optimality, a subgradient

method is used to solve the Lagrangian dual approximately. At the end of the

subgradient phase, the Lagrangian multipliers are an approximation of the op-

timal dual variables for the current restricted master problem. In the following,

we discuss the Lagrangian relaxation of model MDVCSP that we will use in the

remainder of this thesis.

The linking constraints (2.14) in model MDVCSP require a simultaneous treat-

ment of vehicle and crew scheduling. If these constraints are relaxed in a La-

grangian way, the model decomposes into a multiple-depot vehicle and crew

scheduling problem that can be solved separately. Both problems are linked

by penalizing incompatible vehicle and crew scheduling solutions in the objective

function. While the crew scheduling part of the decomposed problem can be

solved efficiently (by pricing out xvariables), the remaining vehicle scheduling

2. Integrated Vehicle and Crew Scheduling: State-of-the-Art

part still constitutes an NP-hard problem. Consequently, constraints (2.12) are

additionally dualized which results in several small single depot vehicle schedul-

ing problems as vehicle scheduling subproblem. Recall that single depot vehicle

scheduling problems can be solved in polynomial time. Additionally, we use the

relaxation of model MDVCSP where greater or equal signs replace the equality

about partitioning versus covering formulations.

Next, Lagrangian multipliers µd

ij and πtare associated with constraints (2.14)

and (2.12), respectively. The objective function (2.11) now reads:

min X

d∈D X

(i,j)∈Ad

ijcd

ij +X

d∈D X

k∈Kd

kfd

d∈D X

(i,j)∈e

µd

ij yd

ij −X

k∈Kd(i,j)

k!(2.17)

t∈T

πt 1−X

d∈D X

(i,j)∈Ad(t)

ij!.

Furthermore, the Lagrangian subproblem results in:

Φ(µ, π)=Φy(µ, π)+Φx(µ) + X

t∈T

πt(2.18)

with

Φy(µ, π) = min X

d∈D X

(i,j)∈Ad

ij ¯cd

ij |(2.19)

{j:(j,i)∈Ad}

ji =X

{j:(i,j)∈Ad}

ij,∀d∈ D,∀i∈Nd,

0≤yd

ij ≤ud

ij,∀d∈ D,∀(i, j)∈Ad

as vehicle scheduling subproblem and

Φx(µ) = min X

d∈D X

k∈Kd

k¯

k|(2.20)

k∈ {0,1},∀d∈ D,∀k∈Kd

as crew scheduling subproblem. The reduced cost ¯cd

ij on arc (i, j)∈Adof the

2.4. Solution Approach

vehicle scheduling network of depot d∈ D is defined as

¯cd

ij =









ij +µd

ij −πtfor (i, j)∈e

Adand ∃t∈ T : (i, j)∈Ad(t)

ij +µd

ij for (i, j)∈e

Adand @t∈ T : (i, j)∈Ad(t)

ij for (i, j)/∈e

(2.21)

while

k=fd

k−X

(i,j)∈e

Ad(k)

µd

ij (2.22)

denotes the reduced cost of duty k∈Kdwhere e

Ad(k)⊆e

Adcorresponds to the

set of arcs that is covered by duty k∈Kd.

Given multipliers µand πthe vehicle scheduling subproblem Φy(µ, π) results

in |D| minimum cost flow problems (see Section 1.4.1). Note that integrality is

not imposed on the flow variables since all bounds are integral and, thus, each

solution to the linear program above is integral (see Section 1.4.1). As stated

earlier each minimum cost flow problem can be solved in polynomial time. For

given multipliers µ, the crew scheduling subproblem Φx(µ) can easily be solved

by setting xd

k= 1 if and only if ¯

k≤0. Notice that both subproblems have the

integrality property (see Section 1.5.1).

We obtain a lower bound by approximately solving the Lagrangian dual prob-

lem with a subgradient algorithm. However, there are some modifications of

the standard subgradient algorithm. In particular, small norm subgradients are

constructed as described in [Caprara et al., 1999] and the search direction is cal-

culated similar to [Camerini et al., 1975]. Furthermore, columns in the restricted

master problem may have negative reduced costs since the Lagrangian multipliers

are not exact. Thus, they should be modified before generating new columns in

order to prevent that columns are generated twice. We refer to Section 1.5.3 for

a description of the method that is used to adjust the multipliers.

2.4.2. The Column Generation Pricing Problem

After the restricted master problem is solved, the dual information of the solution

is used to price out new columns with negative reduced costs, i.e., columns that

can improve the objective value of the master. However, the number of feasible

pieces of work (and, thus, the number of feasible duties) is vast since vehicle

blocks are not known in advance. Therefore, [Freling, 1997,Huisman, 2004] have

proposed a two phase pricing procedure for the column generation pricing prob-

lem: in the first phase, a piece generation network is set up to generate a set of

2. Integrated Vehicle and Crew Scheduling: State-of-the-Art

pieces of work. These pieces serve as input for the second phase where duties are

generated. Finally, notice that all work regulations concerning duty feasibility

must be considered in the pricing problem.

Generation of Pieces of Work

Recall that a piece of work is defined as a sequence of tasks without a (long) break

for which a driver stays with the same vehicle, and that this sequence is only

restricted by its duration. The piece generation network ¯

Gd= ( ¯

Nd,e

Ad) is similar

to the vehicle scheduling network Gd= (Nd, Ad) from the previous section. More

precisely, ¯

Gdis an acyclic directed time-space network where e

Ad⊂Adspecifies

the set of trip arcs, deadhead arcs, and waiting arcs outside the depot. Note that

we assumed in Section 2.1 that each trip has exactly two relief points: one at the

beginning and the other at the end of the trip. Thus, each node in ¯

Nd⊂Nd

corresponds to a relief point. Figure 2.4 shows the piece generation network that

is associated with the vehicle scheduling network depicted in Figure 2.3.

Figure 2.4.: Piece generation network

Let gd

ij be the crew cost associated with arc (i, j)∈e

Ad. The reduced cost of arc

(i, j)∈e

Adis then defined as ¯gd

ij =gd

ij −µd

ij where µd

ij are the multipliers associated

with linking constraints (2.14) that represent trip, deadhead, or waiting arcs

outside the depot. Hence, the reduced cost of a path is equal to the reduced

cost of the associated piece of work. Each path between two nodes n1and n2in

network ¯

Gdis a feasible piece of work if and only if the minimum and maximum

duration is satisfied. However, it is not necessary to enumerate all feasible pieces

of work since it suffices to prove that no duties with negative reduced costs are left

(in order to reach column generation optimality). The sufficient subset of pieces

is generated by considering the minimum reduced cost path between each pair

of nodes that meet the duration constraints. These paths are generated using an

2.4. Solution Approach

all-pairs-shortest-path algorithm (e.g. Floyd/Warshall method). Furthermore,

three additional pieces of work are considered for each path: a pull-in trip is

added at the beginning, a pull-out trip at the end, and both. As shown by

[Freling, 1997] it suffices to generate only this subset of pieces to assure column

generation optimality. However, there are further constraints concerning piece of

work feasibility in practice. In Chapter 5 we will show how further requirements

can be considered in the piece generation phase.

Generation of Duties

In the second phase of the pricing algorithm, feasible duties are constructed using

the pieces of work generated in the previous phase. A duty is feasible if it satisfies

a number of constraints such as minimum/maximum working time or spread time.

Furthermore, the number of pieces of work is limited. [Huisman, 2004, Gintner,

2007] consider the case where a duty consists of at most two pieces of work. As a

result they simply enumerate all possible combinations of pieces and check duty

feasibility of such a combination. Under the assumption of continuous attendance

(see Section 2.1) the reduced cost of a duty can be computed by adding up the

reduced costs of the pieces. The authors stop enumerating when a specified

number of negative reduced cost duties is found or all combinations are checked.

However, this approach becomes impractical if more than 2 pieces of work per

duty are allowed. In Chapter 5 we will show how duties with more than 2 pieces

of work can be efficiently computed using a resource constrained shortest path

formulation.

2.4.3. Integer Solutions

The final step of the solution method aims at finding a pair of feasible and com-

patible vehicle and crew schedules with a Lagrangian heuristic. Only constraints

(2.14) are relaxed in a Lagrangian way. Thus, the solution of the vehicle schedul-

ing subproblem gives a feasible vehicle schedule. However, the subproblem corre-

sponds to a MDVSP which is an NP-hard problem. As in the column generation

phase, a subgradient algorithm is used to solve the associated Lagrangian dual

problem. Since the subgradient method is initialized with good multipliers (from

the last iteration of the column generation phase) only few iterations are needed

to obtain good multipliers. Finally, for the last kfeasible vehicle schedules, the

associated CSP is solved for each depot in order to obtain a feasible and com-

patible crew schedule. Notice that this method always yields solutions with the

minimum number of vehicles.

2. Integrated Vehicle and Crew Scheduling: State-of-the-Art

In Section 3.3 we will propose three different ways of obtaining a feasible so-

lution to MDVCSP, namely a modified version of the method just described,

different branching approaches in a branch-and-bound framework, and a heuris-

tic branch-and-price approach.

3. New Approaches to Integrated

Vehicle and Crew Scheduling

Integrated vehicle and crew scheduling with multiple depots has received increas-

ing attention over the past years. However, large problem instances still require

an enormous computational effort to determine adequate solutions. Therefore, we

will modify and extend the solution approach for the integrated vehicle and crew

scheduling problem with multiple depots that we have described in the previous

chapter.

The solution approach is based on Lagrangian relaxation in combination with

column generation (see Section 2.4). More precisely, column generation is used

to compute a lower bound where Lagrangian relaxation is applied to solve the

master problem. In this chapter, we will propose a novel approach for the column

generation pricing problem. We will discuss three network models for the column

generation pricing problem that are based on the decomposed pricing scheme as

described in Section 2.4.2. In particular, we compare a connection-based duty

generation network with two novel aggregated time-space networks for duty gen-

eration. To the best of our knowledge, subproblem decomposition has not been

applied in combination with a time-space network for duty generation before.

Then, we will describe methods to solve the resource constrained shortest path

problems that appear in the duty generation phase of the decomposed pricing

problem. In the third part of the chapter, we propose different methods for find-

ing integer solutions: a modified version of the algorithm described in Section

2.4.3, novel adaptations of branching schemes in a branch-and-bound framework,

and a novel heuristic branch-and-price method. Furthermore, we will propose a

novel modification of model MDVCSP (see Section 2.3) where we allow drivers

to use vehicles from all depots and to change their vehicle whenever possible.

Finally, we test our solution approaches on real-world instances and on randomly

generated instances from literature.

This chapter is organized as follows. In Section 3.1 we discuss different net-

work formulations for the column generation pricing problem and propose two

new formulations for decomposed pricing. We describe our solution method for

solving the associated resource constrained shortest path problem with dynamic

3. New Approaches to Integrated Vehicle and Crew Scheduling

programming in Section 3.2. Section 3.3 deals with methods to find feasible solu-

tions. We relax some of our assumptions concerning changeovers in Section 3.4.

Finally, we provide extensive computational results on real-world and randomly

generated instances in Section 3.5.

3.1. Modeling the Column Generation Pricing

Problem

In the previous chapter, we described the basic column generation algorithm

to solve the model MDVCSP. For the following discussion we will assume that

crews are identical. However, our approach can easily be extended to the case

with non-identical crews where we repeatedly solve pricing problems for each

crew type/duty type combination. To simplify the exposition we will first recall

some definitions and notations.

For integrated vehicle and crew scheduling problems it is impossible to ex-

plicitly keep all columns in main memory and, as a result, to solve the master

linear program from scratch. Instead, we solve a sequence of restricted master

programs (RMP) where each problem contains only a small subset of all columns.

After the restricted master problem has been solved, the dual information of the

solution is used to price out new columns that can improve the objective value

of the RMP. In other words, the purpose of the pricing problem (subproblem) is

to find variables with negative reduced costs. In our case, variables correspond

to feasible crew duties.

Consider the piece generation network ¯

Gd= ( ¯

Nd,e

Ad) for depot d∈ D and

model MDVCSP (2.11)-(2.16) as defined in Section 2.3. Furthermore, recall from

Section 2.4 that the reduced cost of duty kfrom depot d∈ D was defined by

k=fd

k−X

(i,j)∈e

Ad(k)

µd

ij (3.1)

where fd

kcorresponds to the cost of the duty, µd

ij ∈Rare the Lagrangian mul-

tipliers associated with linking constraints (2.14), and e

Ad(k) defines the set of

vehicle activities that are covered by duty kof depot d. Consequently, for given

multipliers µthe pricing problem (subproblem) for each depot d∈ D can be

stated as follows:

f∗=min ¯

kxd

k|xd

k∈e

Kd(3.2)

where e

Kdcorresponds to the set of feasible duties that can be operated from depot

d∈ D. Notice that the set e

Kd⊇Kddefines the set of all feasible duties for depot

3.1. Modeling the Column Generation Pricing Problem

dwhile Kdcorresponds to the set of duties for depot dthat are currently in the

RMP. If the pricing problem returns ¯

f∗<0, the column xd

kwith least negative

reduced cost ¯

k≤0 is added to the RMP. The process stops when ¯

f∗≥0.

Of course, the structure of the pricing problem is independent of the way

the dual information is obtained. In our case, an approximated dual solution

is computed by solving a Lagrangian dual problem with a subgradient method.

However, a dual solution can also be computed by using a simplex method on

the linear relaxation of model MDVCSP where the integrality of variables xd

kis

relaxed.

In the following, we will describe a mathematical formulation for the pricing

problem (3.2) in more detail. As stated earlier, a separate pricing problem must

be solved for each depot. For notational convenience, however, we describe the

pricing problem for a single depot.

Basically, all work regulations concerning duty feasibility must be considered

in the pricing problem. Recall that the set of all duties e

Kdcan be huge. Thus,

the subproblem cannot be solved by sorting set e

Kdwith increasing reduced costs

and selecting the least cost column. Instead, the subproblem (3.2) is usually

formulated as a resource constrained shortest path problem (RCSP, see Section

1.4.1) where each feasible path from the source to the sink represents a feasible

duty (see e.g. [Desrochers and Soumis, 1989]). The cost of a path is defined in

such a way that it is equal to the reduced cost of the corresponding duty. We

will refer to an acyclic network H= (N, A) with source sand sink tthat is used

to generate negative reduced cost paths as duty generation network. We obtain

f∗= min X

(i,j)∈A

fijzij (3.3)

s.t. X

{j:(i,j)∈A}

zij −X

{j:(j,i)∈A}

zji =









1 for i=s

0 for i∈N\ {s, t}

−1 for i=t

(3.4)

lr≤X

(i,j)∈A

ijzij ≤ur∀r∈R(3.5)

zij ∈ {0,1} ∀(i, j)∈A(3.6)

where the binary flow variable zij indicates whether there is a flow on arc (i, j)∈

A,¯

fij =fij −µij is the reduced cost of arc (i, j)∈A, and µij is the dual variable

(Lagrangian multiplier) of arc (i, j)∈Athat corresponds to the associated linking

constraint (2.14). However, we do not only associate a traversal cost ¯

fij with each

arc (i, j)∈A, we also define a resource consumption dr

ij ≥0 for each resource

r∈R. Consequently, each path Paccumulates P(i,j)∈Pdr

ij of resource r∈R.

3. New Approaches to Integrated Vehicle and Crew Scheduling

We say that a path is resource feasible if and only if the resource consumption

along the path is greater or equal to lower bound lrand less or equal to upper

bound ur. As described in Section 1.4.1 the RCSP is NP-hard when there is at

least one resource even though pseudo-polynomial algorithms have been proposed

in literature.

Typically, it is assumed that the resource extension function (REF) is non-

decreasing, and reduced cost and resource consumption are separable functions

of pieces of work or tasks, respectively. Basically, a REF is associated with an arc

and defines how the resources are updated along that arc. Then, reduced cost and

resource consumption can be accumulated during path construction. We refer to

[Irnich and Desaulniers, 2005,Irnich, 2006] for a thorough description of resource

extension functions. Notice that not all constraints arising in practice fit into the

structure of the objective function (3.3) and constraints (3.5). Some feasibility

constraints can be dealt with when constructing the network (e.g. minimum rest

time between two tasks or pieces of work) while others cannot (e.g. maximum

working time in any 4-hour period of a duty). Furthermore, public transport

companies often apply several duty types that differ in the feasibility constraints

imposed. Thus, it might be necessary to solve a separate pricing problem for each

duty type, or, if the problem involves multiple depots, for each depot-duty type

combination. However, in chapter 5 we will discuss how constraints arising from

German regulations can be incorporated into a resource constrained shortest path

model.

In the next section, we discuss two modeling approaches for duty generation

network Hwhere either tasks or pieces of work serve as network elements. In

Section 3.1.2, we describe three different piece-of-work-based network models for

a decomposed pricing problem. Furthermore, we give an example how resources

can be used to model duty constraints from practice such as minimum/maximum

working time or maximum spread time.

3.1.1. Modeling Approaches

A duty is a sequence of pieces of work separated by breaks where a piece of work

is a sequence of tasks without a (longer) break on the same vehicle. Further-

more, there are constraints concerning each piece of work and the entire duty.

Consequently, we can either use tasks or pieces of work as network elements for

the duty generation network H. The choice between tasks and pieces of work

involves a trade-off between the size of the network and the resource constraints

on the paths. In particular, there are much more feasible pieces of work than

tasks. Therefore, piece-based networks are larger, but piece feasibility is checked

3.1. Modeling the Column Generation Pricing Problem

in advance (during piece construction). On the other hand, task-based networks

are small, but piece feasibility must be checked during duty construction which

usually requires additional resources. Additional resources make the RCSP more

difficult (and time-consuming) since more constraints must be checked. To sum

up, there is a trade-off between memory and time consumption.

Table 3.1 provides several applications of task- and piece-of-work-based network

models for crew scheduling both in airline and public transport settings. All

authors use the corresponding network model in a column generation context

that involves a resource constrained shortest path problem as pricing problem.

airline public transport

tasks [Vance et al., 1997a] [Friberg and Haase, 1999]

[Desaulniers et al., 1999] [Haase et al., 2001]

[Bornd¨orfer et al., 2006] [Gr¨otschel et al., 2003]

[Sandhu and Klabjan, 2006] [Bornd¨orfer et al., 2004]

[Mesquita and Paias, 2006]

pieces of work [Desaulniers et al., 1997] [Desrochers and Soumis, 1989]

[Vance et al., 1997a] [Desrochers et al., 1992]

[Vance et al., 1997b] [Carraresi et al., 1995]

[Galia and Hjorring, 2004] [Freling, 1997]

[Sandhu and Klabjan, 2006]

Table 3.1.: Network modeling approaches for crew scheduling in literature

[Vance et al., 1997a] test both approaches in an airline setting. However, a

direct comparison is not possible since different rule sets are used and the tests

are executed on different machines. Even though a direct comparison is not valid

the authors observe that the task-based version consumes more time but does not

involve prohibitive network dimensions for large problem data as piece-of-work-

based networks do. In contrast, [Sandhu and Klabjan, 2006] also describe both

modeling approaches, but finally use a piece-of-work-based network in their com-

putational experiments. The authors claim that this network inherently captures

more feasibility rules and, thus, requires less resources and less computational

time. In conclusion, it remains an open question whether there is a beneficial

modeling approach in an airline setting.

In contrast to airline planning there has been no direct comparison between

3. New Approaches to Integrated Vehicle and Crew Scheduling

both modeling approaches in public transport settings. As presented in Table 3.1

some authors use a task-based approach while others prefer a piece-of-work-based

network.

However, most approaches of the second category simply enumerate all pieces

of work which may lead to a huge number of pieces for integrated vehicle and

crew scheduling problems. Hence, there is much room for improvement if not all

pieces are enumerated. As shown by [Freling, 1997] the number of pieces can be

dramatically reduced when the subproblem is decomposed into a piece and duty

generation phase (see Section 2.4.2). In the first phase, a piece generation network

is used to generate a subset of all pieces of work. These pieces serve as input

for the second phase where duties are generated. Recall that this decomposition

scheme can only be applied if the resource consumption is equal for all paths

between two relief points. Despite this decomposition, up to 96% of the total time

in column generation is spent on the pricing problem for large integrated single-

depot vehicle and crew scheduling problems. The time for the piece generation

phase can be neglected. Recall that [Huisman, 2004,Huisman et al., 2005a] use

a procedure similar to [Freling, 1997] for duties with up to two pieces of work,

but enumerate all feasible piece combinations during pricing for a multiple-depot

integrated vehicle and crew scheduling problem.

When we compare a task-based network with a piece-of-work-based formulation

in combination with the decomposed pricing scheme of [Freling, 1997], we see that

the total number of feasible paths is higher in the task-based version. In a task-

based network for duty generation, there are multiple paths between each pair

of relief points while there is exactly one path in a piece-based representation

with decomposed pricing (the piece that was generated in the piece generation

phase). Consequently, the solution space of the pricing problems is smaller with

a piece-based formulation and, thus, appears to be beneficial.

In the following section, we will describe the duty generation network as pro-

posed by [Freling, 1997] for a decomposed pricing problem. Furthermore, we will

suggest two novel formulations that have a lower network complexity than the

model of [Freling, 1997].

3.1.2. Network Models for a Decomposed Pricing Problem

Let us consider piece generation network ¯

G= ( ¯

N, ¯

A) as defined in Section 2.4.2,

where nodes correspond to relief points. Arcs in ¯

Arepresent either deadhead,

trip, or non-depot waiting tasks. Recall that ¯

Gis acyclic. The cost associated

with each arc (i, j)∈¯

Ais defined in such a way that the cost of each path

equals the reduced costs of the corresponding piece. Since the feasibility of a

3.1. Modeling the Column Generation Pricing Problem

piece is only restricted by its duration, we generate the set of pieces by solving a

shortest path problem between each pair of nodes in ¯

Nthat satisfy the duration

constraint. Let νbe the number of relief points, then the number of pieces of

work is in O(ν2).

In the following, we will describe the connection-based model of [Freling, 1997],

a time-space, and an aggregated time-space duty generation network. To com-

plete the description of the models, we define the resource consumption and

resource constraints that are necessary to cover the following constraints: max-

imum working time, maximum spread time (duty length), minimum start time,

maximum end time, minimum break length, and minimum/maximum number of

pieces. A piece of work is only restricted by its duration. Notice that the piece of

work related constraint has already been checked in the piece generation phase.

The same set of duty regulations is used in [Huisman, 2004]. Moreover, we will

give the network complexity of each model. Finally, we will compare all network

representations.

Connection-based Model

We describe the connection-based duty generation network Ht

c= (Nt

c, At

c) for each

duty type tas proposed by [Freling, 1997]. Nodes Nt

ccorrespond to the feasible

pieces from the preceding phase. Source and sink represent the depot. Arcs

in At

ceither represent breaks between two pieces or sign on/sign off activities.

Furthermore, breaks can be combined with walking, i.e., the driver first takes a

break and then walks (or takes a bus) from the arrival station of the first piece to

the departure station of the second piece. A break arc represents the connection

between two pieces whose connection time is greater or equal the minimum break

length plus (if necessary) the additional walking time. A duty starts (ends) with

a sign-on (sign-off) arc. Consequently, sign on arcs originate from the source

while sign off arcs terminate at the sink. We only add sign on (sign off ) arcs to

the network if they are within the minimum start (maximum end) time. Figure

3.1 depicts a connection-based duty generation network with five pieces of work.

Notice that each piece of work is shown as a node where the arrival station of the

piece is given in the middle of the node. The node of a piece is located on the

timeline of the start station at the start time while the end time is not directly

shown.

In order to cover the remaining constraints within the duty construction pro-

cess, the connection-based network requires three resources: number of pieces of

work, working time, and spread time. Table 3.2 shows the resource consumption

for a connection-based network by arc type with Ftas fixed cost of duty type t,

3. New Approaches to Integrated Vehicle and Crew Scheduling

piece of work with

arrival station

break

source

sink

station A

station B

station C

depot

departure

station

B A

start time

of piece of work

break combined

with walking

sign-on

sign-off

A B

Figure 3.1.: Connection-based duty generation network

vtas variable costs of duty type tper minute, dias length (working time) of piece

iin minutes, and lij as length of arc (i, j)∈At

cin minutes. Notice that cost and

resource consumption defined on nodes can always be transferred to arcs.

type of arc cost working spread number of

(i, j)∈At

ctime time pieces

sign-on Ft+djvtdjlij +dj1

sign-off 0 0 lij 0

break djvtdjlij +dj1

break with walking djvtdjlij +dj1

Table 3.2.: Resource consumption for a connection-based network

The connection-based network contains O(ν2) nodes, O(ν4)break arcs, and

O(ν2)sign on/sign off arcs. As we have seen, we need three resources to cover

the rules stated above.

Time-Space Model

Next, we define a time-space duty generation network Ht

s= (Nt

s, At

s) for each

duty type twhere a pair of piece start and piece end nodes is associated with

each piece of work. Source and sink represent the depot as before. We have four

types of arcs in At

s:sign on-, sign off -, piece- and break-arcs. Piece-arcs connect

piece start with piece end nodes while break-arcs have the opposite direction.

Again, breaks can be combined with walking, but must satisfy minimum break

time. Sign on arcs originate from source and terminate at a piece start node

3.1. Modeling the Column Generation Pricing Problem

while sign off arcs emanate from a piece end node and end at sink. We only

add sign on (sign off ) arcs to the network if they are within the minimum start

(maximum end) time. Note that there is no direct connection between two piece-

or break-arcs, respectively. The major difference between a connection-based and

time-space network is that in a connection-based network pieces are represented

by nodes while in a time-space network pieces correspond to arcs. Figure 3.2

shows the corresponding time-space network to the connection-based network in

Figure 3.1.

piece of work start

piece of work

source

sink

station A

station B

station C

depot

departure

station time

break

sign-on

sign-off

piece of work end

break combined

with walking

Figure 3.2.: Time-space duty generation network

Similar to the connection-based network, three resources are necessary to val-

idate duties: working time, spread time, and number of pieces. Table 3.3 shows

the resource consumption for a time-space network by arc type.

type of arc cost working spread number of

(i, j)∈At

stime time pieces

sign-on Ft0lij 0

sign-off 0 0 lij 0

piece of work lijvtlij lij 1

break 0 0 lij 0

break with walking 0 0 lij 0

Table 3.3.: Resource consumption for a time-space network

The time-space representation has O(ν2) nodes, O(ν2)piece-, O(ν4)break-,

and O(ν2)sign on/sign off -arcs. However, the time-space network can be ag-

gregated by exploiting the structure of the underlying piece generation network.

3. New Approaches to Integrated Vehicle and Crew Scheduling

Observe that there are O(ν) pieces of work that start at the same relief point.

Since a break-arc always follows a piece-arc, we can aggregate all piece start

nodes with the same origin and the same time without changing the set of feasi-

ble paths. The same reasoning holds for piece end nodes. Due to aggregation the

number of nodes reduces to O(ν) and, thus, the number of arcs to O(ν2). The

overall network size O(ν2) is therefore considerably smaller than the connection-

based formulation of [Freling, 1997]. Nevertheless, the time-space network still

has O(ν2) break arcs that can be further aggregated to O(ν) as we will show in

the following.

Aggregated Time-Space Model

We define an aggregated time-space duty generation network Ht

a= (Nt

a, At

a) that

modifies the time-space network in the following way. We introduce a departure

timeline for each station that connects two subsequent piece start nodes of a

station by a waiting arc. Furthermore, we use a break arc to connect a piece end

with a piece start node. However, there is at most one break arc to connect

apiece end with all subsequent piece start nodes of a station as opposed to a

time-space network where we have an explicit arc for each compatible pair of

piece end and piece start nodes. In an aggregated model, all connections not

explicitly present at a station are implicitly included by traversing the timeline.

Figure 3.3 depicts an aggregated time-space network with timelines at stations

A and B. In order to illustrate timelines, Figure 3.3 does not contain the same

set of pieces as the preceding figures.

piece of work start

piece of work

source

sink

station A

station B

station C

depot

departure

station time break

sign-on

sign-off

piece of work end

break combined

with walking

waiting

Figure 3.3.: Aggregated time-space duty generation network

Similar to the previous time-space model three resources are necessary to check

3.1. Modeling the Column Generation Pricing Problem

duty feasibility: working time, spread time, and number of pieces. Table 3.4

shows the resource consumption for an aggregated time-space network by arc

type. However, we need an additional resource if there is a maximum break

duration. Furthermore, in many settings duties must not start with a waiting

time which cannot be inherently modeled in the network structure. Consequently,

we must perform an additional check during duty construction.

type of arc cost working spread number of

(i, j)∈Aa

stime time pieces

sign-on Ft0lij 0

sign-off 0 0 lij 0

piece of work lijvtlij lij 1

break 0 0 lij 0

break with walking 0 0 lij 0

waiting 0 0 lij 0

Table 3.4.: Resource consumption for an aggregated time-space network

Basically, the aggregated time-space representation has the same network di-

mensions as the time-space network. However, the number of break arcs reduces

to O(ν) since there is at most one break arc from each piece end node to a station.

The number of waiting arcs grows linearly with the number of relief points.

Comparison

Table 3.5 summarizes the network complexities of the three network representa-

tions described earlier. The number of break arcs includes both types of breaks:

”pure” breaks and breaks combined with walking. As a result, connection-based

networks have the largest number of network elements followed by time-space

and aggregated time-space networks. Moreover, all network representations need

three resources to validate the feasibility constraints according to [Huisman,

2004].

We will now investigate the actual network dimensions of the formulations for

integrated vehicle and crew scheduling problems. We use randomly generated

instances available at [Huisman, 2003]. A detailed description of the instances

and how they were generated is given by [Huisman, 2004,Huisman et al., 2005a].

The instances have been classified into two classes according to the travel speed

where the speed is lower for problems in class B. As a consequence, trips in class

B are longer. The instances in class A are considered more demanding than those

3. New Approaches to Integrated Vehicle and Crew Scheduling

network type

element type connection time-space aggr. time-space

arcs sign-on O(ν2)O(ν2)O(ν2)

sign-off O(ν2)O(ν2)O(ν2)

piece of work – O(ν2)O(ν2)

break O(ν4)O(ν2)O(ν)

waiting – – O(ν)

nodes piece of work O(ν2) – –

piece start/end – O(ν)O(ν)

resources – 3 3 3

Table 3.5.: Network dimensions of different duty generation networks

in class B. We will only report computational results for class A that all involve 4

depots and groups with ntrips where n= 80,100,160,200,320 and 400. For each

group 10 instances are available. Table 3.6 gives average network dimensions for

duty types with 2 pieces of work as described in [Huisman, 2004,Huisman et al.,

2005a] including the assumptions stated in Section 2.1. Of course, all network

formulations have the same solution space.

As can be seen from Table 3.6 we were not able to set up the network for the

connection-based formulation with more than 200 trips due to excessive memory

consumption. Furthermore, the total number of arcs and nodes reduces dramat-

ically when a time-space network is used (by 99.66% to 99.94%) instead of the

connection-based representation of [Freling, 1997]. The network size can be fur-

ther reduced by approximately 20% when an aggregated time-space network is

used. Notice that the average number of arcs in the aggregated time-space net-

works can be smaller than the number of pieces since minimum/maximum start

and end time constraints are imposed. Due to prohibitive memory consumption

we will not consider the connection-based network in the remainder of this thesis.

In Table 3.7 we compare the time-space (tsn) with the aggregated time-space

(atsn) network representation for duty generation. All tests were executed on

a Pentium IV 3.40GHz personal computer (2GB RAM) using the algorithm de-

scribed in Section 2.4 with the following settings. In accordance with [Huisman,

2004,Huisman et al., 2005a] we consider five different types of duties: one tripper

type with one piece of work between 30 minutes and 5 hours, and four types con-

sisting of two pieces of work. We use the instances available at [Huisman, 2003]

that have also been used and described above. However, we randomly chose three

instances out of each group (10 instances). We assign a fixed cost of 1,000 for

3.1. Modeling the Column Generation Pricing Problem

trips elements network type

connection time-space aggr. time-space

080 nodes 8,666 405 405

arcs 3,254,370 10,822 8,652

100 nodes 13,678 507 507

arcs 7,277,958 16,684 13,493

160 nodes 39,658 812 812

arcs 67,950,380 47,546 38,158

200 nodes 57,687 1,017 1,017

arcs 123,414,953 73,383 59,597

320 nodes – 1,548 1,548

arcs – 192,717 157,381

400 nodes – 1,934 1,934

arcs – 290,899 240,993

Table 3.6.: Average network dimensions of different duty generation networks for

integrated vehicle and crew scheduling problems

each vehicle and duty and a small variable cost of 1 for each minute a vehicle

is outside the depot and 0.1 for each minute a crew is working. In other words,

we minimize the total number of vehicles and drivers first and leave operational

cost minimization as secondary objective. For all settings, we terminate if the

improvement of the lower bound is less than 0.5% in the last 10 iterations or if

the computational time is more than 5,400 seconds for the lower bound phase.

All data given corresponds to the average over three instances. In order to get a

realistic picture of the performance, we applied all preprocessing and acceleration

techniques that we will describe in the following sections. Finally, in Table 3.7

we report the number of iterations (#iter), the cpu time in seconds spent on the

master (cpu ma) and pricing problem (cpu pr), the lower bound (lb), and the

total number of vehicles and drivers (v+d) of the best feasible solution. Recall

that the approach to compute integer solutions (see Section 2.4.3) always yields

vehicle schedules with the minimum number of vehicles.

The aggregated version seems to require considerably more column generation

iterations (see groups with 160 and 200 trips) and more time to price out new

columns. For larger instances the algorithm terminated due to the time limit and

returned worse lower bounds with the aggregated version. Therefore, we conclude

that (except for the smallest instances) the time-space network outperforms the

aggregated time-space formulation in terms of pricing time and solution quality.

3. New Approaches to Integrated Vehicle and Crew Scheduling

network #trips

080 100 160 200 320 400

tsn #iter 21.3 21.7 28.0 28.0 33.0 28.7

cpu ma 172 246 729 941 3,989 4,268

cpu pr 12 70 350 361 1,451 1,243

lb 29,649 37,849 48,331 63,477 77,282 108,366

v+d 27.3 35.7 46.3 61.3 76.0 105.7

atsn #iter 18.3 22.3 33.5 36.3 31.0 26.7

cpu ma 126 227 956 1,177 4,283 4,623

cpu pr 16 42 635 1,381 1,322 1,848

lb 29,813 37,712 48,208 63,340 78,206 110,865

v+d 28.0 36.3 46.7 61.3 77.3 107.0

Table 3.7.: Comparison of the time-space and aggregated time-space network rep-

resentation for duty generation

Notice that the average path length from the source to the sink is longer in the

aggregated version. In connection with the dynamic programming algorithm that

we use to solve the associated RCSP (see Section 3.2), a longer path length could

cause more labels to be generated and evaluated. The more labels the algorithm

has to evaluate, the more time is consumed. Due to the worse performance

and the reasoning stated above, we will not consider the aggregated time-space

network for duty generation in the remainder of this thesis.

In the next section, we will describe methods to solve the resource constrained

shortest path problems that appear in the duty generation phase of the decom-

posed pricing problem.

3.2. Solving the Column Generation Pricing

Problem

The methods developed for solving resource constrained shortest path problems

can be classified into Lagrangian relaxation, constraint programming, and dy-

namic programming approaches. However, not all RCSP algorithms are suitable

in a column generation context. In column generation pricing, an algorithm does

not necessarily need to find the most negative reduced cost column. In order

to guarantee convergence of column generation, it suffices that an algorithm re-

turns any negative reduced cost column and returns no column if and only if

3.2. Solving the Column Generation Pricing Problem

there is no negative reduced cost path. Furthermore, algorithms should return

multiple paths with negative lengths (multiple pricing) since this usually accel-

erates the convergence of column generation. In the following, we will briefly

review methods for the RCSP that satisfy these conditions. For an extensive

survey on shortest path problems with resource constraints we refer to [Irnich

and Desaulniers, 2005].

Lagrangian relaxation (see Section 1.5.1) methods for RCSP assume that re-

source consumption is additive along the path and that resource consumption is

only constrained as a whole (see [Irnich and Desaulniers, 2005]). In other words,

resource constraints cannot vary from node to node and the resource extension

function (REF) cannot take the structure of a partial path into account. Recall

that a REF is associated with an arc and defines how the resources are updated

along that arc. [Beasley and Christofides, 1989] and [Gr¨otschel et al., 2003] pro-

pose to compute lower bounds for the RCSP with Lagrangian relaxation. In a

second step, they exploit these bounds in a tree search procedure. Additional

constraints that could not directly be covered in the RCSP can always be consid-

ered in the search phase. However, these constraints cannot be covered directly

as in a dynamic programming approach.

Constraint programming (see [Marriot and Stuckey, 1998]) approaches allow to

tackle a wide range of complex constraints where some cannot be modeled using

resources or simple structural constraints: for instance, a crew may not drive

more than 5 hours in any 8-hour period. [de Silva, 2001] and [Fahle et al., 2002]

use constraint programming to tackle the RCSP as pricing problem. However,

it remains an open question if constraint programming is also beneficial when

all constraints can be modeled using resources or inherently covered by the duty

generation network.

Dynamic programming (see [Ahuja et al., 1993]) is widely used and the most

successful approach to solve RCSP in a column generation context. Successful

applications include, among others, [Desrochers and Soumis, 1989], [Haase et al.,

2001], [Xu et al., 2003], and [Dell’Amico et al., 2006]. As described in [Irnich

and Desaulniers, 2005,Irnich, 2006] dynamic programming allows to cover many

constraints from practice using specialized (non-decreasing) resource extension

functions. This approach appears to be more flexible than Lagrangian methods

since the number of applicable REFs is comparatively high.

Since most constraints in public transport that we are aware of can be modeled

with resources or inherently covered by the duty generation network described in

Section 3.1, we also use a dynamic programming approach (see Section 3.2.1). As

stated earlier, in column generation it is only necessary to solve RCSPs to proven

optimality to show that no negative reduced cost paths exist. Therefore, it suffices

3. New Approaches to Integrated Vehicle and Crew Scheduling

to approximately (heuristically) solve the RCSP in all but the final iteration and

obtain arbitrary negative reduced cost path(s). We will describe preprocessing

and further acceleration techniques in Section 3.2.2 and 3.2.3, respectively, that

can be heuristically adapted.

3.2.1. Dynamic Programming Algorithms

The basic idea of dynamic programming for the RCSP is to iteratively construct

paths starting from source suntil sink tis reached. [Joksch, 1966] gives a recursion

for the RCSP (3.3)-(3.6) with a single resource r∈R, i.e., |R|= 1. Let Fj(tr) be

the cost of the shortest path from node sto jin network H= (N, A) where the

resource consumption of the path is less than or equal to tr. We assume that the

arcs (i, j)∈Aare numbered in such way that i<jholds. Then, the recursion

reads

Fj(tr) = minFj(tr−1),min

(i,j)∈A|dr

ij ≤tr{Fi(tr−dr

ij) + cij}(3.7)

where we set F1(tr) = 0 for 0 ≤tr≤urand Fj(0) = ∞for j= 2,...,|N|.

We compute the minimum cost path from sto tby solving Ft(ur) where uris

an upper bound on the consumption of resource r. The running time of the

algorithm is in O(|A|ur) while the space consumption is in O(|N|ur). Hence,

Fj(tr) is a pseudo-polynomial algorithm for RCSP. Furthermore, the dynamic

programming recursion can be used to obtain a fully polynomial -approximation

for RCSP by rounding and scaling (see [Hassin, 1992]).

Labeling approaches improve pure dynamic programming methods in such a

way that they identify and discard inefficient paths that cannot be part of the

optimal solution. Labeling methods use labels (states) to represent feasible (par-

tial) paths where a label Lk

npat node np∈Ncorresponding to path Pk

np=

(s, . . . , np−1, np) is linked with its predecessor label Lj

np−1at node np−1. Linking

allows to reconstruct the path of a label without storing the complete path in each

label. Furthermore, each label Lk

np= (Ll

np−1, c(Pk

np), d1(Pk

np), . . . , d|R|(Pk

np)) con-

tains the cost c(Pk

np) and resource consumption dr(Pk

np) for each resource r∈R.

We denote the set of states at node npby Lnp. Basically, a pulling dynamic pro-

gramming algorithm pulls labels from all possible predecessor nodes to a node

while updating cost and resource consumptions. A new label lat node jpulled

from label kat node iis given by

j= (Lk

i, c(Pk

i) + cij, d1(Pk

i) + d1

ij, . . . , d|R|(Pk

i) + d|R|

ij ).(3.8)

If dr(Pk

i)+dr

ij < lrwe set dr(Pl

j) = lr. A new label Ll

jis accepted if it corresponds

to a feasible partial path Pl

j, i.e., dr(Pl

j)≤urfor each r∈R. Additionally, we

3.2. Solving the Column Generation Pricing Problem

only store labels that are not dominated by any other label at that node. A label

idominates label Ln

iif c(Pm

i)≤c(Pn

i) and dr(Pm

i)≤dr(Pn

i) for each r∈R.

Notice that two labels dominate each other if cost and resource consumptions

are equal. We call a non-dominated label efficient and denote the set of efficient

labels at node iby L∗

i. Algorithm 6 describes a (pulling) label setting algorithm

where N−(j) = {i: (i, j)∈A}defines the set of predecessors of node j∈N.

Algorithm 6: Basic label setting algorithm for the RCSP

(Step 1) Initialization

Set Ls= (nil,0,...,0).

Set Li=∅for each i∈N\{s}.

(Step 2) Path extensions and dominance checks

foreach j∈N\{s}do

// loop all predecessor nodes

foreach i∈N−(j)do

// loop all efficient labels

foreach l∈L∗

ido

if ∃r∈R:dr(l) + dr

ij > urthen

next l

// create new label

j= (l, c(l) + cij, d1(l) + d1

ij, . . . , d|R|(l) + d|R|

ij )

Lj=Lj∪Lm

// remove dominated labels

L∗

j=Efficient Labels(Lj)

Procedure Efficient Labels removes dominated states from set Lj. The minimum

cost path in L∗

tcorresponds to the optimal path. In addition to this generic

algorithm, there are label setting and label correcting versions. In the latter

case, a label can be corrected several times in the course of the algorithm while

in a label setting method all labels are permanent and cannot be changed. Notice

that our network is acyclic and that the nodes are treated in topological order

and, thus, labels in L∗

iare permanent. However, neither label correcting nor label

setting variants improve the pseudo-polynomial worst case complexity of function

(3.7).

The running time of the algorithm depends on the implementation of the dom-

inance tests in procedure Efficient Labels. In a naive implementation where

each pair of labels is compared, we have a complexity of O(|R|(Lmax)2) with

Lmax ≤Qr∈R(ur−lr) as the maximum number of labels at any node (see [Frel-

ing, 1997]). The computational effort for dominance tests can be reduced to

3. New Approaches to Integrated Vehicle and Crew Scheduling

O(Lmax(log Lmax)|R|−2) with a divide-and-conquer algorithm (see [Kung et al.,

1975]).

[Joksch, 1966] already noted that Fj(tr) is a step function and that it suffices

to locate its steps. Furthermore, the author observed that the list of efficient

labels are the non-differentiable points of the step function and, thus, only these

have to be considered to obtain the optimal solution.

In the following, we will describe the label setting approach of [Desrochers,

1986] which is also described in [Desrosiers et al., 1995]. The method differs

from Algorithm 6 in the order how paths are extended. The method has been

used by several authors in a column generation context, e.g., [Haase et al., 2001],

[Mesquita and Paias, 2006]. The algorithm is a multi-dimensional generalization

of a pulling dynamic programming algorithm for the shortest path problem with

time-windows (see e.g. [Desrochers and Soumis, 1988]).

The algorithm assumes that each arc (i, j)∈Ahas positive cost or at least

one positive resource consumption: ∃r∈R:dr

ij >0. In our setting, we always

have at least one positive resource consumption for each arc (see Section 3.1).

Without loss of generality, we order the resources in such a way that d1

ij ≥$ > 0

holds for each (i, j)∈Awhere $is a lower bound on the resource consumption

of the first resource. For each node i∈N,Lidenotes the set of labels and

Li⊆Lithe subset of permanent labels. Furthermore, set ¯

Liis characterized by a

variable bound ηion the consumption of the first resource where ¯

Lidefines labels

such that l1≤maxl∈¯

Lid1(l)≤ηi≤u1. Algorithm 7 shows the label setting

algorithm of [Desrochers, 1986]. The algorithm chooses unprocessed nodes with

a ”small” resource consumption first. It guarantees in combination with strictly

positive resource consumption d1

ij >0 that all path extensions have a higher

resource consumption than the previously created labels. Thus, labels in ¯

Liare

permanent.

In our computational experiments we found that the dynamic programming

method often generated only few new columns, esp. in the final iterations. There-

fore, we do not apply dominance tests when pulling labels at the sink node t. Fur-

thermore, the performance of the standard version can be considerably improved

by using preprocessing and further acceleration techniques as we will describe in

the following sections.

3.2.2. Preprocessing

The purpose of this section is to describe network reductions for duty generation

network H= (N, A) in order to improve the performance of the pricing algorithm.

In the following, we will discuss generic node and arc reductions introduced by

3.2. Solving the Column Generation Pricing Problem

Algorithm 7: Label setting algorithm of [Desrochers,

1986]

(Step 1) Initialization

Set Ls=¯

Ls= (nil,0,...,0) and ηs=u1.

Set Li=¯

Li=∅for each i∈N\{s}and ηi=l1.

(Step 2) Path extensions and dominance checks

while true do

// select node

if ∀i∈N\{s}:ηi=u1then

exit

else

Select j∈arg mini∈N\{s}{ηi|ηi< u1}.

// pull labels at node j

foreach i∈N−(j)do

// loop all labels

foreach l∈¯

Lido

if ∃r∈R:dr(l) + dr

ij > urthen

next l

if not ηj≤d1(l) + d1

ij ≤min{u1, ηj+$}then

next l

// create new label

j= (l, c(l) + cij, d1(l) + d1

ij, . . . , d|R|(l) + d|R|

ij )

Lj=Lj∪Lm

// remove dominated labels

Lj=Efficient Labels(Lj∪¯

Lj)

ηj= min{u1, ηj+$}

3. New Approaches to Integrated Vehicle and Crew Scheduling

[Aneja et al., 1983]. Furthermore, we propose a novel problem-specific filtering

technique that discards arcs based on their reduced costs and does not require to

set up network Hbeforehand.

Generic preprocessing

In [Aneja et al., 1983] the number of arcs and nodes is reduced by computing min-

imum resource paths from the source to each node in the network and from each

node in the network to the sink. The minimum resource paths are computed for

each resource and used to identify nodes and arcs that violate the resource limits.

Each node and arc that violates resource limits can be discarded. [Freling, 1997]

describes essentially the same method but computes minimum and maximum

resource paths since there are lower and upper limits on resource consumption.

More formally, let Pr

ij (¯

ij) be the path with minimum (maximum) consump-

tion of resource r∈Rfrom node ito jwith i, j ∈N, and denote Pc

ij as the

least cost path from ito jwhen resource constraints are not considered. Notice

that finding the shortest paths from sto each node i∈Nrequires the same

computational effort as finding the shortest path from sto tusing a simple O(m)

dynamic programming algorithm. Thus, the overall complexity for computing all

shortest paths is O(|R|m).

Now, we can delete all nodes i∈Nfrom network Hthat either violate resource

limits (3.9) or cannot be part of a negative reduced cost path (3.10):

∃r∈R:dr(Pr

si) + dr(Pr

it)> ur∨dr(¯

si) + dr(¯

it)< lr(3.9)

c(Pc

si) + c(Pc

it)>0.(3.10)

Likewise, we can remove all arcs (i, j)∈Afrom network Hwhere at least one of

the following conditions is satisfied:

∃r∈R:dr(Pr

si) + dr

ij +dr(Pr

jt)> ur∨dr(¯

si) + dr

ij +dr(¯

jt)< lr(3.11)

c(Pc

si) + cij +c(Pc

jt)>0.(3.12)

[Beasley and Christofides, 1989] and, more recently, [Mehlhorn and Ziegel-

mann, 2000] extend the method of [Aneja et al., 1983] by considering lower and

upper bounds which they compute by a Lagrangian relaxation approach. We do

not consider these Lagrangian methods here since they require to solve an addi-

tional Lagrangian dual. Notice that we would have to solve a separate Lagrangian

dual problem for each duty generation network in each column generation itera-

tion. We expect that this method is too time-consuming.

Recently, [Dumitrescu and Boland, 2003] propose a simplification of the method

used in [Beasley and Christofides, 1989] and [Mehlhorn and Ziegelmann, 2000].

3.2. Solving the Column Generation Pricing Problem

In particular, the authors use the initial cost instead of performing Lagrangian

relaxation and using the resulting reduced costs. Their computational results

show that their method is at least as good as the approach of [Beasley and

Christofides, 1989] in terms of the degree of reduction.

Problem-specific preprocessing

Table 3.6 showed that between 79-83% of all arcs in a time-space network are piece

of work arcs. Therefore, we propose a novel problem-specific reduction technique

to discard piece of work arcs that does not require to set up the duty generation

network. The arcs are dynamically discarded based on dual information.

Recall that a duty consists of several pieces of work separated by breaks. Fur-

thermore, breaks have a minimum and/or maximum duration. In order to sim-

plify the exposition, we assume that a duty consists of at most two pieces of work.

However, our approach can be easily extended to the case with more than two

pieces of work.

Let us consider an arbitrary piece of work p∗. Given that a duty may consist

of at most two pieces of work, piece p∗can be either at the first or the second

position of a duty. Furthermore, it can be connected with other pieces of work

that end (start) within the maximum break duration blmax but not within the

minimum break length blmin (see Figure 3.4).

piece of work start

piece of work

arrivals

station A

departures

station A

break

piece of work end

break combined

with walking

*****

*multiple arcs

arrivals

station B

time

departures

station B

* * * *

blmin

blmax

blmin

blmax

Figure 3.4.: Compatible pieces of work in a time-space duty generation network

Let stibe the start time of piece iand etiits end time. Furthermore, we denote

by ωij the walking time between the end location of piece iand the start location

of piece j. Two pieces iand jare said to be compatible if they can be covered

consecutively by the same crew, that is blmin ≤stj−eti−ωij ≤blmax holds. We

3. New Approaches to Integrated Vehicle and Crew Scheduling

denote by Pp∗the set of pieces that are compatible to piece p∗. Now, we can

discard all pieces of work p∗that satisfy the following condition

ft+ min

p∈Pp∗

fp+¯

fp∗≥0 (3.13)

where ftdenotes the fixed cost of the corresponding duty type and ¯

fpthe reduced

cost of piece p. However, the set of compatible pieces is in O(ν2) for each piece

with νas number of relief points. As a result, the computational effort for

checking condition (3.13) can be prohibitively high.

In order to reduce the computational time to check the condition, we define

time slots s∈Sof length σfor each station (e.g. 15 minutes). Furthermore,

for each time slot s∈Swe store the arriving piece with minimum reduced cost

sand the departing piece with minimum reduced cost ¯

s(see Figure 3.5). We

piece of work start

piece of work

arrivals

station A

departures

station A

break

piece of work end

break combined

with walking

*multiple arcs

arrivals

station B

time

departures

station B

blmin

blmax

blmin

blmax

...

time-slot

Figure 3.5.: Compatible time slots in a time-space duty generation network

define the set of forward and backward compatible time slots s∈Sof piece p∗

by Sf

p∗and Sb

p∗, respectively.

p∗={s∈S|blmin ≤stp∗−sts−ωsp∗∨0≤stp∗−ets−ωsp∗≤blmax}(3.14)

p∗={s∈S|blmin ≤ets−etp∗−ωp∗s∨0≤sts−etp∗−ωp∗s≤blmax}(3.15)

Similar to condition (3.13) we can discard each piece p∗where

ft+ min

s∈Sf

p∗

s+¯

fp∗≥0∧ft+ min

s∈Sb

p∗

s+¯

fp∗≥0 (3.16)

holds. The length of the time slots σ > 0 can be directly used to control the

number forward/backward compatible time slots:

1 + blmax −blmin

σ.(3.17)

3.2. Solving the Column Generation Pricing Problem

It can easily be seen that we discard the same piece arcs with condition (3.13) and

(3.16) if we set σ= 1. Notice that condition (3.16) requires less computational

effort than (3.13) and can be straightforwardly adjusted. Furthermore, we see

that problem-specific preprocessing with σ= 1 discards at least as many piece

arcs (i, j)∈Arepresenting piece p∗as generic preprocessing (3.12) since ft+

minp∈Pp∗¯

fp+¯

fp∗≥c(Pc

si) + cij +c(Pc

jt) is satisfied.

Computational tests

In the following, we will evaluate whether the time spent on problem reduction

actually pays off. We use the test set as described in the Section 3.1.2 which

comprises 18 instances.

In Table 3.8 we show the impact of problem-specific (ts), and problem-specific

plus generic preprocessing (ts+gen) on the overall performance. We report the

number of iterations (#iter), the cpu time in seconds spent on the master (cpu ma)

and pricing problem including the time for network reduction (cpu pr), total

master and pricing time (cpu ma+pr), the lower bound (lb), and the average

arc reduction in percent (arc red%) for duties with two pieces of work. All data

given corresponds to the average over three instances. Notice that the given lower

bound does not necessarily correspond to a valid lower bound. In order to get a

realistic picture of the performance, we applied all acceleration techniques that

we will describe in the following section except restricted networks. As we can

see from Table 3.8, both preprocessing techniques considerably reduce the cpu

time for the pricing problem. However, the time and network reduction are more

favorable for instances with up to 200 trips. A possible explanation why our

reduction performs better on small instances is that cost-based reductions are

particularly useful in the final stage of column generation. The final stage is not

reached for instances with 320 and 400 trips since column generation terminates

due to the time limit.

Figure 3.6 shows the number of arcs (no of arcs) in the column generation

process for the first depot of instance 320A09 (see [Huisman, 2003]) and three

different duty types. Tripper duties consist of exactly one piece of work while

split and normal duties have exactly two pieces of work (see [Huisman, 2004] for a

complete description of the duty types). We performed 60 iterations and did not

use restricted networks as described in the following section. As we can easily see

the number of arcs can be almost halved in the course of the column generation

process. Furthermore, the results support our claim that the problem-specific

(cost-based) reduction works better in the final phase of column generation.

3. New Approaches to Integrated Vehicle and Crew Scheduling

50000

100000

150000

200000

0 10 20 30 40 50 60

no of arcs

cg iteration

tripper

split

normal

Figure 3.6.: Network reduction in the column generation process for first depot

of instance 320A09 and three different duty types

3.2. Solving the Column Generation Pricing Problem

reduction #trips

080 100 160 200 320 400

none #iter 21.3 26.7 28.3 28.3 29.3 25.0

cpu ma 160 282 723 936 3,721 3,708

cpu pr 78 342 1,957 3,186 1,647 1,603

cpu ma+pr 238 624 2,680 4,122 5,368 5,311

lb 29,672 37,872 48,314 63,460 78,192 110,849

arc red% 0 0 0 0 0 0

ts #iter 21.3 22.7 29.3 28.3 29.3 26.0

cpu ma 173 244 763 973 3,977 4,084

cpu pr 35 124 1,364 1,232 1,490 1,319

cpu ma+pr 208 368 2,127 2,205 5,467 5,403

lb 29,647 37,811 48,321 63,456 78,278 110,705

arc red% 24 22 20 21 14 14

ts+gen #iter 20.7 21.3 27.0 28.7 29.7 25.0

cpu ma 157 229 707 947 4,022 4,174

cpu pr 22 69 531 522 1,772 1,119

cpu ma+pr 179 298 1,238 1,496 5,794 5,293

lb 29,651 37,840 48,320 63,458 78,386 110,670

arc red% 42 40 34 36 22 23

Table 3.8.: Impact of different network reduction techniques on the overall

performance

3.2.3. Acceleration Techniques

There are numerous acceleration techniques for dynamic programming algorithms

applied to column generation pricing in literature. For an excellent survey on mas-

ter and pricing techniques to accelerate the overall performance we refer to [De-

saulniers et al., 2002]. While the results in literature indicate the importance

of acceleration methods, specific comparisons demonstrating actual savings are

rare. The only attempts we are aware of are [Gamache et al., 1999] for the airline

crew rostering problem, [Gr¨onkvist, 2005] for the airline fleet assignment prob-

lem, and [Westerlund et al., 2006] for the traveling salesman subtour problem.

However, only [Gamache et al., 1999] evaluate the impact of acceleration meth-

ods in column generation pricing. The purpose of this section is twofold: first,

we describe pricing techniques that were particularly useful in accelerating the

overall performance and second, we evaluate the impact of these techniques. Ac-

celeration methods used to improve the performance of the master problem are

3. New Approaches to Integrated Vehicle and Crew Scheduling

discussed in [Gintner, 2007].

Recall that the column generation method does not require the selection of

the most negative reduced cost variable (duty). Furthermore, there are many

feasible variables with negative reduced cost in the first iterations of the column

generation process and only few in the final phase. To reduce the computational

effort per iteration, we apply partial pricing where we heuristically reduce the

search space and gradually increase it if we cannot find any (or enough) columns

in the reduced search space. Finally, we perform full pricing to prove column

generation optimality or to produce a column with negative reduced cost. We

define a sequence of pricing heuristics H1, . . . , Hpwhere the search space of Hi

is smaller than that of Hi+1. We denote the exact pricing algorithm by Hp+1.

Algorithm 8 describes a generic partial pricing algorithm where procedure Pric-

ing(Hi) returns negative reduced cost columns using heuristic Hi. Notice that

this algorithm may increase the number of iterations and the overall effect may

also be unfavorable.

Algorithm 8: Generic partial pricing algorithm

(Step 1) Initialization

Set pricing level i= 1.

Define column set C=∅.

Define column threshold t.

(Step 2) Pricing

while |C| ≤ t∧i≤p+ 1 do

C=Pricing(Hi)

i=i+ 1

return C

Essentially the same idea is used in the pricing step of the simplex algorithm

where the reduced costs of the nonbasic variables are computed and one of the

negative reduced cost columns (if any) is selected to enter the basis. [Dantzig,

1963] originally proposed the so-called Dantzig rule where all nonbasic variables

are checked (full pricing) and the least negative reduced cost column is selected

(in case of a minimization problem). [Orchard-Hays, 1968] proposed to check only

a part of the nonbasic variables (partial pricing) and select the best variable from

this part. The search space is changed when no entering variable with negative

reduced cost can be found.

In the following, we will describe the acceleration strategies that we used in

our pricing algorithm. We will present a combination of known and novel tech-

niques to further improve the performance of the algorithm. In particular, we

3.2. Solving the Column Generation Pricing Problem

will propose a novel way of generating balanced restricted networks, suggest cus-

tomized dominance rules, and show how we can strengthen label-pruning bounds

previously exposed in literature. Furthermore, we will evaluate the impact on

the overall performance using the same assumptions as described in Section 2.1.

Multiple pricing

A well-known strategy to accelerate column generation is to create multiple neg-

ative reduced cost columns in each pricing step. With a dynamic programming

algorithm we can easily compute multiple paths per iteration since all states at

the sink node correspond to feasible paths. Basically, multiple pricing increases

the computational effort to solve the master problem, but may decrease the num-

ber of iterations.

Similar to the pivot strategy in the simplex algorithm there is probably not

a best pricing strategy in column generation. A disadvantage of the common

strategy to select columns with minimum reduced costs is the following. If there

is a (small) subset of rows with high dual values, it is likely that the pricing

algorithm returns many columns covering these rows while only one of these

columns can be in the final integer solution. Consequently, we use an adapted

version of the disjoint column method of [Gamache and Soumis, 1998] to avoid

generating similar columns in an iteration. Finally, we refer to [Vanderbeck, 1994]

for a thorough discussion on how to select ”good” columns. [Freling, 1997] also

reviews different strategies for column selection.

In our computational experiments, we found that adding only the least reduced

cost column leads to a very poor performance of the overall algorithm. Based on

limited computational experience [Vanderbeck, 1994] states that multiple pric-

ing works better if the subproblem solution is computationally expensive (as in

our case). Our results seem to support his understanding and, thus, we will not

consider adding the least reduced cost column again. Furthermore, our compu-

tational results indicate that adding at most 5,000 columns per depot/duty type

combination works best for a wide range of problem sizes.

Restricted networks

As stated earlier there are many feasible variables with negative reduced cost in

the first iterations of the column generation process and only few in the final

phase. To speed up the pricing step, we heuristically reduce the network size

in the first iterations by ignoring some arcs and nodes. In particular, we ignore

arcs according to the dual information or randomly. In our implementation, we

3. New Approaches to Integrated Vehicle and Crew Scheduling

initially set the maximum number of piece of work arcs to 200|T | where Tcorre-

sponds to the set of trips. If the improvement of the lower bound is less than 10%

in the last five iterations, we subsequently reintroduce the discarded arcs until

we have the complete network in the final iteration(s). We do not reintroduce

arcs that cannot be part of a negative reduced cost path after problem-specific

preprocessing (see Section 3.2.2). Notice the similarity to traditional scaling al-

gorithms (see [Ahuja et al., 1993]). Examples of restricted networks in column

generation can be found in [Barnhart et al., 1995] and [Gamache et al., 1999].

Next, we describe how we actually ignore arcs. Basically, the strategy to simply

ignore all piece of work arcs with poor reduced costs can lead to unbalanced duty

generation networks. The rationale behind this observation can be understood in

the following way. If there is a (small) subset of rows with high dual values, it is

very likely that pieces covering these rows will be included in the duty generation

network. Thus, the network contains comparatively many arcs covering these

rows, and the pricing algorithm is likely to return an unbalanced set of duties.

As discussed earlier this may lead to poor convergence of the column generation

algorithm. Therefore, we propose to discard pieces in a balanced way using time

slots. To the best of our knowledge, this has not been tried before. In this

context, we define time slots s∈Sof length σfor each station similar to those

in Section 3.2.2. Furthermore, we compute the initial acceptance rate by

ρ= min1.0,200|T |

|P| (3.18)

where Pcorresponds to the set of pieces of work. Then, we discard (1 −ρ)∗100

percent of all pieces of a time slot either randomly or according to the dual

information. As a result, we obtain a balanced network where the reduction is

evenly spread among the complete network. In our computational experiments,

we found that discarding pieces randomly outperforms discarding pieces according

to the dual information. This was particularly true for large instances. Our

interpretation is that randomly restricted networks provide more balanced duty

sets.

State space reduction

The basic idea of state space reduction in dynamic programming is to increase

dominance between labels (states) and, hence, reduce the solution time. We have

tested different ways to increase dominance between labels. However, we will only

describe those techniques that proved to work best on a wide range of instances

and feasibility constraints.

3.2. Solving the Column Generation Pricing Problem

[Gamache et al., 1999] increase dominance by reducing the range of the units

of the labels. In other words, the authors use less precise units of measurement

to describe some of the resources. For instance, resources related to time can

be measured in hours instead of minutes or can be rounded to the nearest ten

minutes. As a consequence, labels that originally had different consumption for

some resources are now represented by the same value. This increases the chance

that one label dominates another and that fewer labels have to be evaluated.

Since the restricted state space does not guarantee column generation optimality,

we gradually adapt the unit measurement until the initial resource vectors are

used. However, the feasibility of a label is checked at each node always using the

initial unit of measurement.

Additionally, dominance can be increased by using stronger dominance rules.

In particular, we use four different dominance rules where our dynamic pro-

gramming algorithm starts with level 1 and ends with level 4 to prove column

generation optimality. We increase the dominance level if we have not found

enough columns in the current level. The computational effort increases with the

level of dominance. Consider the following feasibility constraints for duties as

defined in Section 3.1.2: maximum working time, maximum spread time (duty

length), minimum start time, maximum end time, minimum break length, and

minimum/maximum number of pieces. Furthermore, recall that we need three

resources and costs to cover these constraints: working time, spread time, and

number of pieces of work. We propose the following dominance rules that are

customized for the vehicle and crew scheduling problem.

•Level 1: We ignore the spread time (duty length) information in all domi-

nance checks. Thus, we apply the dominance process over three-dimensional

vectors instead of four-dimensional ones. The underlying assumptions are

that most of the spread time is also working time and that maximum work-

ing time is more restrictive than maximum spread time. Therefore, we

substitute the spread time information of a label by working time. More-

over, we apply dominance tests for labels that do not satisfy the lower limit

for the number of pieces of work.

•Level 2: We do not allow dominance tests between labels with a different

number of pieces of work. At level 1, we allow to compare two labels that

represent two partial paths with a different number of pieces. Furthermore,

it is likely that the label with the higher number of pieces also consumes

more resources than the label with a smaller number of pieces. Conse-

quently, there is a good chance to discard labels with a higher number of

pieces. In other words, we do not prefer labels with a small number of

3. New Approaches to Integrated Vehicle and Crew Scheduling

pieces at level 2. However, we have to check more labels compared to level

1. Again, we ignore the spread time information of the labels.

•Level 3: We apply the dominance process over all resources, i.e., a four-

dimensional vector, but still ignore lower limits for resource consumption.

•Level 4: At the final level, we check all resources considering upper and

lower limits for resource consumption. Given a lower limit is a hard con-

straint, we can consider this limit in two different ways. We either do not

apply dominance tests to labels that do not satisfy the lower limit or we

introduce a new resource to strictly enforce it. The new resource is the

negative of the original resource to enforce the upper limit (see [Gamache

et al., 1999]).

Table 3.9 shows the performance of our dynamic programming algorithm in the

first column generation iteration. We report the cpu time in seconds to compute

the 100 least reduced cost columns (cpu pr) with all other acceleration strategies

disabled except network reduction. Furthermore, we give the lower bound (lb 1)

obtained with a subgradient approach (see Section 2.4.1) after the columns have

been added. We use the same test set as described in Section 3.2.2. As we can

see from Table 3.9 the cpu time can be dramatically reduced by using inexact

dominance tests while there is only a small increase of the lower bound (less than

3%) at level 1 and 2. In other words, it suffices to solve the pricing problem in

the initial phase of column generation with inexact dominance rules.

dominance #trips

080 100 160 200 320 400

level 1 cpu pr 0.1 0.2 0.7 0.9 6.2 7.6

lb 1 38,931 45,209 68,669 76,191 105,125 144,323

level 2 cpu pr 0.1 0.2 0.9 1.2 7.9 10.6

lb 1 38,881 45,206 68,281 75,764 105,440 142,525

level 3 cpu pr 0.5 0.8 6.6 9.7 183.8. 239.0

lb 1 38,798 45,197 67,302 75,522 102,452 141,883

level 4 cpu pr 1.9 4.6 67.3 136.7 3024.1 4399.7

lb 1 38,798 45,198 67,333 75,522 102,117 142,405

Table 3.9.: Results of dynamic programming algorithm with different dominance

tests in the first column generation iteration

3.2. Solving the Column Generation Pricing Problem

Label pruning

Similar to state space reduction, label pruning also relates to the question how

the number of evaluated states can be reduced. In this section, however, we

consider pruning techniques to reduce the number of labels that do not refer to

increasing dominance.

In order to control the number of unprocessed labels, we can set a maximum

size κmax of the label list for each node. We only keep the best κmax labels

at each node with respect to reduced cost. A disadvantage of this approach

may be that the pricing algorithm returns many similar columns (given that we

have a small subset of rows with high dual values). Therefore, we use a similar

method to [Mesquita and Paias, 2006] where labels are randomly discarded. The

authors argue that only a small percentage of the columns generated in earlier

iterations will be part of the basis of the current linear restricted master problem.

Therefore, they prefer to generate a small set of columns with a greater diversity.

The authors show that this method actually reduces cpu time while maintaining

the quality of the linear relaxation. Our method differs from the one suggested

by [Mesquita and Paias, 2006] in the following way. Let κjbe the number of

labels at node j∈N. When pulling labels at node jwe can easily estimate the

final number of labels ¯κjat that node by

¯κj=ρX

i∈N−(j)

|¯

Li|(3.19)

where ρcorresponds to the average acceptance rate of a label after the dominance

process has been applied. If the estimated number of labels at that node exceeds

a given threshold κmax we pull a label from node iwith probability κmax/¯κj. With

our method, we heuristically reduce the search space only if the number of labels

becomes large. For instance, if the number of discarded arcs during preprocessing

is large and, hence, the pricing problem is comparatively easy, we still use the

exact pricing algorithm.

Furthermore, we use lower bounds to reduce the state space similar to [L¨ubbecke,

2005]. We first describe the method of [L¨ubbecke, 2005] and then discuss our

adaptations in order to strengthen the lower bounds. Consider label Lk

(Ll

h, c(Pk

i), d1(Pk

i), . . . , d|R|(Pk

i)) at node i∈Nthat represents path Pk

i. Further-

more, let A+(Pk

i) = {(m, n)∈A|m≥i}be the set of arcs that are compatible

with path Pk

i. Recall that the nodes in the duty generation network are sorted

by increasing time. According to [L¨ubbecke, 2005] a lower bound lbk

ifor label Lk

can be computed by

lbk

i=c(Pk

i) + X

(m,n)∈A+(Pk

fmn.(3.20)

3. New Approaches to Integrated Vehicle and Crew Scheduling

If lbk

i≥0 holds, label Lk

ican be discarded. Of course, the lower bound is not

very tight and discards only few labels since it does not take the structure of

the network and path Pk

iinto account. Therefore, we propose a novel way to

strengthen bound lbk

iin the following.

Recall from Section 3.2.2 that we defined time slots s∈Sof a specific length

for each station and stored the least reduced cost ¯

sof all pieces departing within

that time slot. Again, we consider the case where a duty consists of at most two

pieces of work. However, the approach can be generalized to the n-piece case in

a straightforward way. Basically, label elimination is most effective when applied

early in the construction of a particular path. Therefore, we consider the cases

where (1) the first piece arc and (2) the first break is to be added to path Pk

of label Lk

i. In the first case, we cannot derive a tighter bound when problem-

specific preprocessing has been applied before the network was set-up. In the

latter case, however, we will not extend label Lk

iby arc (i, j)∈Aif

c(Pk

i) + cij + min

s∈Sb

(i,j)

s≥0 (3.21)

where Sb

(i,j)denotes the set of backward compatible time slots of break arc (i, j).

If we use a time-space duty generation network (see Section 3.1.2), |Sb

(i,j)|= 1

holds since only the time slot of the piece departure node has to be considered.

Furthermore, we strengthen the pruning conditions in the early iterations in

such a way that we only accept labels if the estimated lower bound is significantly

below zero.

Computational tests

We use the test set and settings as described in the Section 3.1.2 which comprises

18 instances. In Table 3.10 we show the performance of our dynamic programming

algorithm with and without acceleration techniques. We report the number of

iterations (#iter), the cpu time in seconds spent on the master (cpu ma) and

pricing problem (cpu pr), the lower bound (lb), and the total number of vehicles

and drivers (v+d) of the best feasible solution. Notice that the given lower bound

does not necessarily correspond to a lower bound for the overall problem (see

Section 3.5). All data given corresponds to the average over three instances. With

acceleration techniques the time spent on the pricing problem can be dramatically

reduced. For instances with more than 160 trips, column generation terminates

due to excessive computation time in the pricing problem when no acceleration

is used. As we can see for instances with at most 100 trips, acceleration increases

the number of iterations, but the cpu time for master and pricing can be reduced

by 30%-52%. Time savings for instances with 160 trips are even higher where 80%

3.3. Integer Solutions

of the cpu time to obtain a comparable lower bound can be saved. For larger

problems, we can considerably improve the lower bound by using acceleration

techniques since more column generation iterations can be performed within the

given timeframe.

type #trips

080 100 160 200 320 400

normal #iter 15.3 15.7 14.3 10.0 6.3 5.0

cpu ma 149 207 319 283 14 9

cpu pr 111 446 5,103 5,154 5,386 5,391

lb 29,647 37,832 48,814 66,723 86,682 120,649

v+d 27.7 35.7 46.3 62.3 79.7 110.0

acceleration #iter 21.3 21.7 28.0 28.0 33.0 28.7

cpu ma 172 246 729 941 3,989 4,268

cpu pr 12 70 350 361 1,451 1,243

lb 29,649 37,849 48,331 63,477 77,282 108,366

v+d 27.3 35.7 46.3 61.3 76.0 105.7

Table 3.10.: Results of dynamic programming algorithm with and without accel-

eration techniques

3.3. Integer Solutions

In Section 2.4 we described the solution process for model MDVCSP (see Section

2.3). Basically, the solution approach consists of two stages: a lower bound phase

and a final phase where a feasible solution is constructed. In the lower bound

phase, we apply column generation in combination with Lagrangian relaxation. In

the preceding sections we have described how we modeled and solved the column

generation pricing problem while this section is devoted to the final phase where

we compute feasible solutions to model MDVCSP.

This section is organized as follows. In Section 3.3.1, we use the approach

of [Huisman, 2004] (see Section 2.4.3), but compare different ways of constructing

feasible solutions. In Section 3.3.2 we describe different branching rules when

feasible solutions are generated with a commercial MIP solver such as ILOG

CPLEX [ILOG, 2006] or MOPS [Suhl, 2000]. Finally, in Section 3.3.3, we propose

a novel integer procedure which enhances the approach of [Huisman, 2004]. We

enhance the approach in the sense that we regenerate columns in the integer

3. New Approaches to Integrated Vehicle and Crew Scheduling

phase and apply depth-first (heuristic) branching in combination with different

fixing strategies.

3.3.1. Sequential Approach

In order to ease the exposition, we will describe the Lagrangian heuristic of

Huisman (see [Huisman, 2004]) to construct integer solutions in detail. The final

step (integer phase) of the solution method (see Algorithm 5, Section 2.4) aims

at finding a pair of feasible and compatible vehicle and crew schedules with a

Lagrangian heuristic. Only the linking constraints (2.14) of model MDVCSP

(2.11)-(2.16) are relaxed in a Lagrangian way. The objective function now reads

min X

d∈D X

(i,j)∈Ad

ijcd

ij +X

d∈D X

k∈Kd

kfd

d∈D X

(i,j)∈e

µd

ij yd

ij −X

k∈Kd(i,j)

k!(3.22)

where µd

ij correspond to the Lagrangian multipliers associated with the linking

constraints. Furthermore, the Lagrangian subproblem results in

Φ0(µ)=Φ0

y(µ)+Φx(µ) (3.23)

with

Φ0

y(µ) = min X

d∈D X

(i,j)∈Ad

ij ¯cd

ij |(3.24)

d∈D X

(i,j)∈Ad(t)

ij = 1 ∀t∈ T

{j:(j,i)∈Ad}

ji =X

{j:(i,j)∈Ad}

ij,∀d∈ D,∀i∈Nd,

0≤yd

ij ≤ud

ij, yd

ij ∈N,∀d∈ D,∀(i, j)∈Ad

as vehicle scheduling subproblem and

Φx(µ) = min X

d∈D X

k∈Kd

k¯

k|(3.25)

k∈ {0,1},∀d∈ D,∀k∈Kd

3.3. Integer Solutions

as crew scheduling subproblem. The reduced cost ¯cd

ij on arc (i, j)∈Adof the

vehicle scheduling network of depot d∈ D is defined as

¯cd

ij =(cd

ij +µd

ij for (i, j)∈e

ij for (i, j)/∈e

Ad(3.26)

while

k=fd

k−X

(i,j)∈e

Ad(k)

µd

ij (3.27)

denotes the reduced cost of duty k∈Kdwhere e

Ad(k)⊆e

Adcorresponds to the set

of arcs that is covered by duty k∈Kd. In contrast to the lower bound phase, the

vehicle scheduling subproblem corresponds to a multiple-depot vehicle scheduling

problem that neither has the integrality property nor can be solved in polynomial

time. However, the solution of the vehicle scheduling subproblem gives a feasible

vehicle schedule. Each feasible vehicle schedule can be used to construct a feasible

crew schedule using traditional (sequential) crew scheduling. As in the lower

bound phase, we use a subgradient algorithm to solve the associated Lagrangian

dual problem.

Unlike [Huisman, 2004] we do not only perform 10 subgradient iterations in the

final phase since our computational experiments indicated that better solutions

can be found if more iterations are performed. Therefore, we solve the associated

CSP for each depot every i-th subgradient iteration after kiterations have been

performed in order to obtain a feasible and compatible crew schedule. Similar to

the integrated setting, we use column generation in combination with Lagrangian

relaxation to solve the associated crew scheduling problems. Furthermore, we

apply dynamic programming to solve the pricing problem with the acceleration

techniques described earlier. Integer solutions are computed with the commercial

MIP solver ILOG CPLEX 10.0 using the columns generated before. Basically,

the same approach is used by [Gintner, 2007], but the author uses a different way

of pricing new negative reduced cost columns.

In addition to sequential crew scheduling as used by [Huisman, 2004] we can

compute feasible crew schedules using a partially or fully integrated approach

(see Sections 2.2.2 and 2.2.3, respectively). In particular, we apply the partially

integrated method of [Gintner et al., 2006a, Gintner, 2007] to generate feasible

solutions. The basic idea is to change a given optimal vehicle schedule without

loss of optimality in the crew scheduling phase. The authors set up a time-space

network that allows to recombine parts of vehicle blocks in order to disclose addi-

tional flexibility in crew scheduling while preserving vehicle schedule optimality.

3. New Approaches to Integrated Vehicle and Crew Scheduling

In other words, the crew scheduling approach does not only consider a single

optimal vehicle schedule, but a set of optimal vehicle schedules with minimum

fleet size and minimum operational costs. For the remainder of the thesis, we call

this crew scheduling approach adaptive crew scheduling.

Furthermore, we tested a fully integrated approach where the trip-depot as-

signment instead of the complete vehicle schedule serves as input. Notice that

this results in a (single depot) integrated vehicle and crew scheduling problem

for each depot that discloses even more flexibility in the crew scheduling phase

than adaptive crew scheduling. However, in our computational experiments we

found that this method has a poor performance in terms of solution quality and

computational time. Therefore, we will not consider this approach again.

In Table 3.11 we compare the performance of sequential and adaptive crew

scheduling to construct integer solutions. We use the same test set (18 problem

instances) and settings as described in the preceding sections except that we

extend the cpu time for the lower bound phase to 36,000 seconds (10 hours).

Furthermore, we try to find an upper bound in a column generation iteration

if more than 200 subgradient iterations have been performed. In that case, we

solve a multiple-depot vehicle scheduling problem using the best multipliers of

that iteration as costs on the arcs. In Table 3.11 we give the number of iterations

(#iter), the cpu time in seconds spent on the master problem (cpu ma), on

pricing problem (cpu pr), on the integer phase (cpu ip), and the total cpu time

(cpu tot). The cpu time of the master problem includes the time to find new

upper bounds in the lower bound phase. Furthermore, we provide the lower

bound (lb) and the total number of vehicles and drivers (v+d) of the best feasible

solution. Notice that the best lower bound obtained is not necessarily a valid

lower bound.

The sequential approach to find integer solutions outperforms the adaptive

one in terms of solution time, but generates worse solutions. Notice that no

integer phase was executed with the adaptive method for the group with 80 trips

since the best upper bound obtained in the lower bound phase already had the

minimum number of duties. From our point of view, the gain in solution quality

outweighs the additional computational effort. Therefore, we will not consider

the sequential approach to generate feasible solutions in the remainder of this

chapter.

3.3.2. Branch-and-Bound with MIP-Solver

In this section, we apply a branch-and-bound approach (see Section 1.5.4) in the

integer phase instead of the method described in the preceding section. First, we

3.3. Integer Solutions

type #trips

080 100 160 200 320 400

sequential #iter 19.0 19.7 26.0 25.7 34.7 33.0

cpu ma 130 188 643 832 5,190 5,501

cpu pr 9 35 245 195 5,987 7,958

cpu ip 222 473 1,671 2,473 6,140 7,688

cpu tot 362 698 2,565 3,506 17,422 21,288

lb 29,643 37,814 48,304 63,437 77,137 108,001

v+d 27.3 35.7 46.3 61.7 76.3 106.0

adaptive #iter 18.0 19.0 23.3 24.7 35.3 32.3

cpu ma 153 222 1,170 1,712 7,560 8,318

cpu pr 11 32 302 202 5,477 8,120

cpu ip 0 578 1,676 2,621 7,635 9,808

cpu tot 165 834 3,155 4,541 20,794 26,384

lb 29,664 37,834 48,382 63,470 77,191 107,922

v+d 26.7 35.0 45.3 60.7 76.0 105.0

Table 3.11.: Results of sequential and adaptive crew scheduling to find integer

solutions

generate a set of promising columns in the lower bound phase (steps 1 to 5 of

Algorithm 5). Then, we apply an LP-based branch-and-bound method on model

MDVCSP with the restricted set of columns to compute a feasible solution.

Recall that our model MDVCSP has two types of decision variables: flow and

duty variables. In the following, we propose three different branching schemes for

our model that prioritize either flow or duty variables. Furthermore, we suggest

to apply the well-known Ryan-Foster branching rule on our model. To the best

of our knowledge, these branching schemes have not been used in combination

with model MDVCSP. Furthermore, follow-on branching has not been applied on

integrated vehicle and crew scheduling problems. We will conclude this section

with computational results comparing the branch-and-bound methods with our

approach from the preceding section.

Branching on variables

Several authors have proposed branching rules for integrated vehicle and crew

scheduling problems with multiple depots. In the following, we will briefly review

these approaches.

The approach of [Bornd¨orfer et al., 2004] relies on model MDVCSP-H (see

3. New Approaches to Integrated Vehicle and Crew Scheduling

Section 2.2.3). They use a solution approach similar to Algorithm 5 since they

aim at computing a lower bound first and subsequently generate an integer fea-

sible solution. However, the authors solve the Lagrangian dual problem with an

inexact adaptation of a proximal bundle method. The inexact bundle method is

embedded in a backtracking procedure to produce integer solution in the second

phase. The procedure utilizes the primal information produced by the bundle

method to iteratively fix deadhead (flow) variables until the complete vehicle

schedule is fixed. In their model fixing a deadhead determines the successor of a

service trip, but also implicitly assigns that sequence to a depot.

The approach of [Mesquita et al., 2006] is based upon a model similar to

MDVCSP-H that contains fewer constraints. They propose to solve the linear re-

laxation of a combined multi-commodity flow and mixed set partitioning/covering

model with column generation. If the linear relaxation of the root node is not inte-

ger, the authors suggest a branch-and-bound and two branch-and-price schemes.

The branch-and-bound method branches over the set of feasible duties generated

while solving the linear relaxation of the root node. The authors have compared

two branching schemes. In one strategy they branch on duty variables first while

in the other strategy they first branch on flow variables. Although the authors

do not provide computational results for both schemes, they state that the first

scheme performs better on their model.

To sum up, [Bornd¨orfer et al., 2004] branch on flow variables while [Mesquita

et al., 2006] prefer to branch on duty variables first. Consequently, we propose

three branching schemes that prioritize either flow or duty variables of model

MDVCSP for branching. Recall that our model is based on a time-space network

while those of the authors stated above rely on a connection-based network.

We define the following priority function that returns the branching priority of

an arbitrary flow or duty variable

Ψ : {yd

ij|(i, j)∈Ad, d ∈ D} ∪ {xd

k|k∈Kd, d ∈ D} → R+

0(3.28)

with the following properties.

1. Ψ(z) = 0: Variable zis not selected for branching.

2. Ψ(z)>0: Variable zcan be selected for branching if zis not integer in the

current solution of the linear relaxation.

3. Ψ(z1)>Ψ(z2): If z1is not integer, z1will be selected for branching before

z2is chosen. If z1is integer, z2will be chosen no matter what priority z1

has.

3.3. Integer Solutions

Furthermore, we denote the set of trip arcs for depot d∈ D by Ad

T=St∈T Ad(t).

We define the following branching schemes that first branch on flow variables:

ps : Ψ(yd

ij)>Ψ(yd

rs)>Ψ(xd

k)>0,

∀(i, j)∈Ad

T,∀(r, s)∈Ad\Ad

T,∀xd

k∈Kdand ∀d∈ D,(3.29)

pv : Ψ(yd

rs)>Ψ(yd

ij)>Ψ(xd

k)>0,

∀(i, j)∈Ad

T,∀(r, s)∈Ad\Ad

T,∀xd

k∈Kdand ∀d∈ D.(3.30)

Branching scheme ps first branches on flow variables that correspond to service

trips. In other words, we assign trips to depots before we decide about the

sequence of trips or about the crew scheduling part. If we assign a trip tito

depot dj, we can discard all other trips arcs of tithat belong to another depot.

However, it does not suffice to assign trips to depots to completely define a

vehicle schedule. Thus, deadhead connections must be fixed in a second step.

In branching rule pv we give a higher priority to flow variables that correspond

to deadheads. We first decide which trips are operated in sequence before we

assign these sequences to a depot. In a time-space network, however, we cannot

directly fix a connection between two particular trips since deadhead and waiting

activities are aggregated (see Section 2.3). Therefore, we must also branch on

(trip) flow variables in order to obtain a complete vehicle schedule.

Finally, we propose branching scheme pd that first branches on duty variables.

The rationale behind that rule can be understood in the following way. The

crew scheduling problem is usually more constrained than the vehicle scheduling

problem since many work regulations must be considered. Therefore, it may be

beneficial to first decide about the crew schedule and later construct a compatible

vehicle schedule.

pd : Ψ(xd

k)>Ψ(yd

ij) = Ψ(yd

rs)>0,

∀(i, j)∈Ad

T,∀(r, s)∈Ad\Ad

T,∀xd

k∈Kdand ∀d∈ D (3.31)

A disadvantage of rule pd may be that there are many similar duty variables.

Thus, forbidding the use of a variable may only have a minor impact since a

similar column can be selected at the child node. As a consequence, many nodes

must be evaluated until a good solution is found. In the following subsection, we

will describe a branching scheme based on the Ryan-Foster rule that overcomes

this shortcoming.

Finally, the branching schemes stated above only determine the type of variable

that is to be preferred. The schemes do not define which particular variable is

chosen. In our computational experiments, however, we found that it performs

best if we leave this decision to the MIP solver such as CPLEX.

3. New Approaches to Integrated Vehicle and Crew Scheduling

Branching on follow-ons

Branching on follow-ons relies on a general branching strategy for set partitioning

problems that was introduced by [Ryan and Foster, 1981]. The branching scheme

is based on the following property. Given a fractional solution to a set partitioning

problem, we can identify two rows iand jsuch that the subset C(i, j) of columns

that contain iand jhas the property

0<X

c∈C(i,j)

xc<1.(3.32)

The remaining fraction of cover for each constraint must be provided by columns

that do cover both rows at the same time. Thus, an effective constraint branching

scheme is to require to cover two rows iand jby the same column on one branch

and by different columns on the other. [Vance et al., 1997a] slightly modify the

scheme to maintain tractability. They only consider trips (rows) iand jthat

correspond to trips operated consecutively in a duty (column). Furthermore, the

authors show that this modification still constitutes a correct branching scheme.

We refer to this strategy as branching on follow-ons since we impose which trips

can follow trip iin the solution. Moreover, we refer to the trip pair (i, j) as

follow-on.

In the following we will describe how we adapt branching on follow-ons for the

integrated vehicle and crew scheduling problem with multiple depots. Consider

two trip arcs yd

ij ∈Ad

Tand yd

rs ∈Ad

Tfrom depot d∈ D and the set of du-

ties Kd(yd

ij, yd

rs) where both trips are covered consecutively. Now, we define our

branching scheme for two compatible service trips. Two trips must be operated

consecutively from depot don one branch and not consecutively from depot don

the other.

k∈Kd(yd

ij ,yd

rs)

k≥1∧yd

ij = 1 ∧yd

rs = 1 1-branch (3.33)

k∈Kd(yd

ij ,yd

rs)

k≤0 0-branch (3.34)

Notice that arcs yd

ij and yd

rs are not necessarily consecutive in the final vehicle

and crew schedule solution since deadhead arcs may be in between.

Given a fractional LP solution there are usually many candidate follow-ons

that can be used. Hence, we define the support of a follow-on (yd

ij, yd

rs) similar

to [Vance et al., 1997a].

f(yd

ij, yd

rs) = X

k∈Kd(yd

ij ,yd

rs)

k(3.35)

100

3.3. Integer Solutions

Clearly, 0 ≤f(yd

ij, yd

rs)≤1 is satisfied. The support of a follow-on can be inter-

preted as the probability of including that follow-on in the solution. Furthermore,

we define two branching schemes where fo-flf selects the least fractional while

fo-fmf chooses the most fractional follow-on among all candidate follow-ons.

fo-flf : (yd

ij, yd

rs) = arg max f(yd

ij, yd

rs) (3.36)

fo-fmf : (yd

ij, yd

rs) = arg min |0.5−f(yd

ij, yd

rs)|(3.37)

Branching rule fo-flf seems to be particularly useful in combination with a depth-

first tree search (see [Vance et al., 1997a]). Notice that selecting a follow-

on with f(yd

ij, yd

rs) = 1 will not eliminate the current fractional LP solution.

Therefore, [Vance et al., 1997a] propose to fix all perfect follow-on pairs where

f(yd

ij, yd

rs) = 1 is satisfied. Of course, additional fixings at a branch-and-bound

node are heuristic. In the following section we evaluate both follow-on branching

schemes with and without additional fixings.

In contrast to the original branching scheme on set partitioning models, we

cannot guarantee that we always find a follow-on for a fractional solution for

model MDVCSP. In particular, we cannot find a follow-on if all trips have been

assigned to a depot. In such a case, we perform the default branching decision of

the MIP solver. However, such a situation never occurred in our computational

experiments.

Computational results

In Table 3.12 we compare the performance of our branching rules with the ap-

proaches from the preceding sections. We make the same assumptions as before.

However, we only report results for five instances with 100 trips from the Huis-

man test set (see [Huisman, 2003]). For larger instances, we hardly found integer

solutions in the set of columns generated in the lower bound phase. In our com-

putational experiments, we tried to generate different numbers of columns, but

all settings lead to a very poor performance of the MIP solver. We tested our

implementation on a Pentium IV 2.3GHz/2GB RAM personal computer using

ILOG CPLEX 10.0. We terminated the branch-and-bound search if the cpu time

exceeded 3,600 seconds.

We give results for the sequential (seq) and adaptive (adap) approach described

in Section 3.3.1, using the default setting of CPLEX (cpxdef ), and the branching

schemes from this section. In Table 3.12 we report the total number of vehicles

and drivers (v+d) of the best feasible, the cpu time in seconds spent on the integer

phase (cpu ip) and the number of processed nodes (nodes). Additionally, we

present the number of problems where the MIP solver found the optimal solution

101

3. New Approaches to Integrated Vehicle and Crew Scheduling

over the current column set (#opt), could improve the best known upper bound

(#impr), and could not improve the upper bound (#nimpr).

type v+d cpu ip nodes #opt #impr #nimpr

seq 36.0 1,342 – – – –

adap 35.6 1,273 – – – –

cpxdef 36.6 3,096 747 2 2 1

fo-flf fix 35.4 2,097 454 4 1 0

fo-fmf fix 35.8 2,414 298 2 3 0

fo-flf 36.0 2,479 575 3 2 0

fo-fmf 36.6 2,698 297 2 3 0

ps 37.6 2,801 394 2 2 1

pd 37.6 3,600 2,873 0 4 1

pv 38.8 2,607 266 2 0 3

Table 3.12.: Results of user-defined branching rules and sequential approaches on

model MDVCSP over five instances with 100 trips

The test results indicate that follow-on branching performs much better than

using branching priorities. Furthermore, selecting the least fractional follow-on

gives better results than choosing the most fractional follow-on for branching.

The solution quality can be improved by fixing perfect follow-ons no matter what

follow-on scheme is applied. Using CPLEX with its default settings performs

better than setting branching priorities, but returns worse results compared to

follow-on branching. The adaptive approach gives slightly worse results than the

best follow-on variant, but consumes only 60% of its cpu time. Therefore, we

conclude that the sequential and adaptive approach are basically more suited to

generate integer solutions for model MDVCSP.

3.3.3. Heuristic Branch-and-Price

In this section we propose a novel extension of the sequential approach for the

integer phase described in Section 3.3.1. In the sequential approach, we have

solved the Lagrangian dual problem where the Lagrangian subproblems corre-

spond to a multiple-depot vehicle scheduling problem and a trivial problem for

crew scheduling, respectively. However, in the lower bound phase, we have re-

placed the equality signs of the linking constraints (2.14) by greater or equal

signs. The corresponding Lagrangian multipliers are restricted in sign in the

102

3.3. Integer Solutions

lower bound phase while they are not in the integer phase. The underlying as-

sumption was that the Lagrangian multipliers that we have found in the lower

bound phase are a good approximation of the multipliers required in the inte-

ger phase. However, it is an open question whether the columns generated in

the lower bound phase are also suitable to obtain good multipliers (and vehicle

schedules) in the integer phase. Therefore, we propose to perform multiple it-

erations in the integer phase where we generate new columns in each iteration.

Furthermore, our computational experiments revealed that it is beneficial to fix

parts of the vehicle scheduling problem in each iteration of the integer phase.

Our method can be understood as a heuristic branch-and-price procedure. In

an exact branch-and-price approach, the linear relaxation of the root node of the

branch-and-bound tree is solved with column generation to optimality. If the

solution of the continuous relaxation is not integer, two subproblems are created

(branching), one of unprocessed subproblems (nodes) is selected, and the linear

relaxation of that node is solved with column generation to optimality. The pro-

cess iterates until the optimal solution is found or the complete search tree has

been investigated. In our method, however, we do not solve the associated lin-

ear relaxation of each node to optimality. Instead, we perform only one column

generation iteration after a subproblem was selected. Notice that the pricing

problem can be solved as described in Sections 2.4.2 and 3.2, respectively. Fur-

thermore, we perform multiple fixings decisions in each node of the search tree

in order to (heuristically) reduce the search space. However, if the increase in

the lower bound is too large, we reverse fixing decisions, i.e., we perform simple

backtracking. The search tree is traversed in a depth-first manner. This type

of branch-and-price approach is often referred to as fix-and-price procedure. In

Algorithm 9 we give an overview of our approach as stated above. If backtracking

is performed in step 3, we must guarantee that different fixings are performed in

the subsequent branching step to prevent cycling.

In step 4 of our method we fix parts of the vehicle schedule, i.e., a subset of the

flow variables of model MDVCSP. Recall that we apply a subgradient algorithm

to approximate the values of the dual variables for the current set of columns.

Therefore, we do not have primal information in order to separate the current

fractional solution. Of course, there are other methods to solve the Lagrangian

dual that also provide primal information, e.g., bundle methods (see [Kiwiel,

1995]) or the volume algorithm (see [Barahona and Anbil, 2000]). [Bornd¨orfer

et al., 2004] propose to use an inexact adaptation of a proximal bundle method

in a fix-and-price (branch-and-generate) framework. We will compare their ap-

proach with our method on a set of randomly generated instances in Section 3.5.

Furthermore, we tested the volume algorithm. However, early computational ex-

103

3. New Approaches to Integrated Vehicle and Crew Scheduling

Algorithm 9: Heuristic branch-and-price approach for

model MDVCSP

(Step 1) Initialization

Select initial column set from lower bound phase N0.

Define maximum allowed increase of lower bound .

Set t= 0.

(Step 2) Perform column generation

Solve Lagrangian dual lt= maxµtΦ0(µt) with the current

set of columns Nt.

Solve pricing problem ¯

f∗(µt) and obtain columns N0\Nt

with negative reduced costs.

Set Nt+1 = (N0∪Nt) and t=t+ 1.

(Step 3) Backtracking

If lt−lt−1≥reverse fixing decisions of last iteration and

return to step 2.

(Step 4) Branching

Terminate if all flow variables are fixed.

Fix subset of flow variables.

Set t=t+ 1 and return to step 2.

periments revealed that the primal information of the volume algorithm lead to

worse fixings than the method we describe in the following.

Basically, branching decisions must not destroy the structure of the column

generation subproblem. To illustrate this consider a branching scheme based on

dichotomy of variables xd

k. In one branch we fix a duty into the solution (xd

k= 1)

while we ban it from the solution (xd

k= 0) in the other branch. Observe that

fixing a duty variable into the solution of one depot implies to ban all duties from

other depots covering service trips of the fixed duty. However, banning a specific

duty from the solution is difficult since we must forbid the specific path from being

generated in the pricing problem of the corresponding depot. To prevent ”zero”

columns from being re-generated significantly complicates the pricing problem

(see [L¨ubbecke and Desrosiers, 2005] and references therein). Branching schemes

(fixing decisions) are said to be compatible if they do not considerably complicate

the pricing problem. In terms of column generation, branching schemes based

on the original variables of the compact formulation are compatible (see Section

1.5.2 and [L¨ubbecke and Desrosiers, 2005]). In our case, the original variables

correspond to flow variables yd

ij.

[Holmberg and Yuan, 2000] proposed to perform fixings based on the solution

104

3.3. Integer Solutions

of the Lagrangian subproblems for a capacitated network design problem. The

authors also use a subgradient method to solve the corresponding Lagrangian dual

and apply two different fixing schemes denoted by α- and β-fixing, respectively.

In the remainder of this section, we propose to use a novel adaptation of α- and

β-fixing for model MDVCSP. In particular, we fix flow variables to a depot that

either correspond to service trips or connections between service trips (follow-

ons). We will conclude this section with computational results concerning our

fixing schemes.

Fixing service trips to depots

In this subsection, we will describe how service trips can be fixed to depots. To

assign service trips to depots appears to be reasonable since the complexity of

the Lagrangian subproblem for vehicle scheduling Φ0

y(µ) can be reduced. When

all trips are assigned to a depot, the vehicle scheduling subproblem can be solved

in polynomial time while its multiple-depot counterpart is NP-hard (see Sections

1.1.1 and 1.4.1). In other words, the more trips are fixed to a depot the less CPU

time should be consumed for the corresponding Lagrangian subproblem.

Fixing a service trip t∈ T to a depot di∈ D involves the following modifica-

tions in the Lagrangian subproblems and the pricing problem, respectively. In

the vehicle scheduling subproblem we set ydi

ij = 1,(i, j)∈Adi(t) or, alternatively,

ydi

ij = 0,(i, j)∈Ad(t),∀d∈ D\{di}. Furthermore, we disable all duties kfrom the

current set of duties Kd(i, j) where (i, j)∈Ad(t),∀d∈ D\{di}. In the pricing

problem, we must prevent duties that cover edge (i, j)∈Ad(t),∀d∈ D\{di}

from being regenerated. This can easily be done by setting µd

ij =−Mfor

(i, j)∈Ad(t),∀d∈ D\{di}where Mcorresponds to a sufficiently high value.

Then, all duties covering that edge will not have negative reduced cost according

to (3.27). In the following, we describe how we actually decide which trips to fix.

The basic idea of α-fixing is to fix those trips that often appear in the solution

of the vehicle scheduling subproblem Φ0

y(µ). If a flow variable yd

ij is constantly set

to one in vehicle scheduling subproblem, it indicates that arc (i, j) is likely to be

included in the optimal solution. Likewise, arc (i, j) is probably not included if the

Lagrangian subproblem suggests that the value of yd

ij is zero. The straightforward

approach would be to fix arcs to one that are part of all subproblem solutions and

fix arcs to zero that are not used in any solution. However, the approach is more

flexible by introducing parameter α∈[0,0.5] to allow deviations. In particular,

αcorresponds to the deviation rate from the straightforward approach sketched

105

3. New Approaches to Integrated Vehicle and Crew Scheduling

above. Our α-fixing scheme now reads

ij =(1 if PL

l=1 yd,(l)

ij ≥(1 −α)·L

0 if PL

l=1 yd,(l)

ij ≤α·L(3.38)

where yd,(l)

ij corresponds to the value of yd

ij in the l-th (subgradient) iteration and

Lis equal to the number of subgradient iterations performed in that column

generation iteration. Obviously, if α= 0 we only fix arcs that have the same

value in all subgradient iterations.

β-fixing is based on the reduced cost of the flow variables instead of the so-

lutions of the Lagrangian subproblems. Notice that there is a relation between

the reduced cost of the flow variables and the solution of the subproblems. Flow

variables with high negative reduced costs ¯cd

ij are more likely to be selected for

the subproblem solution than variables with positive reduced costs. Parameter

β∈[0,1] defines the ratio of service trips to be fixed to one in one column gener-

ation iteration. The number of arcs to be fixed is equal to dβ|T ∗|e where T∗⊆ T

denotes the set of unfixed service trips. Notice that we cannot predict the num-

ber of fixed arc when α-fixing is applied. Furthermore, we cumulate the reduced

costs of all arcs (i, j)∈Ad(t),∀t∈ T ,∀d∈ D in the following way:

ij =(¯cd

ij in the first iteration,

γ·Rd

ij + ¯cd

ij if the lower bound improved. (3.39)

Similar to [Holmberg and Yuan, 2000] we consider reduced costs associated with

an improved lower bound to be more reliable and, thus, should have a higher

impact (i.e. γ∈[0,1]). In our implementation, we set γ= 0.5. Finally, we fix

dβ|T ∗|e arcs that have the smallest (cumulated) reduced costs. However, we only

fix a service trip to a particular depot if there is a significant difference to any

other depot.

Fixing follow-ons to depots

In the preceding section, we proposed to perform fixing based on information

from the vehicle scheduling subproblem while we will concentrate on the crew

scheduling subproblem in this section. In particular, we fix variables based on

follow-ons (see Section 3.3.2) that appear in the crew scheduling solution of the

Lagrangian subproblem. As we will see in Section 3.5 our solutions almost al-

ways have the minimum number of vehicles while there is room for improvement

concerning the number of duties. Furthermore, it is easier to construct a feasi-

ble vehicle schedule when a subset of trips is fixed than to obtain a feasible crew

106

3.3. Integer Solutions

schedule. Therefore, we hope that variable fixing based on information from crew

scheduling solutions will overall improve the solution quality.

Recall that we refer to the pair of trips arcs (yd

ij, yd

rs) with yd

ij, yd

rs ∈Ad

Tas

follow-on if the corresponding service trips are operated consecutively. Moreover,

Kd(yd

ij, yd

rs) denotes the set of duties where both trips are covered consecutively.

In the following, we only consider follow-ons that occur on the same vehicle and

are serviced by the same crew. In other words, follow-ons must not have a crew

break in between since a changeover may occur between two pieces of work. This

assumption is necessary to maintain tractability.

Fixing a follow-on (ydi

ij , ydi

rs) to a depot di∈ D involves the following modi-

fications in the Lagrangian subproblems and the pricing problem, respectively.

Clearly, fixing a follow-on to a depot implies to assign each trip of the pair to the

depot. Thus, all modifications of the preceding subsection also apply here.

Additionally, we modify the vehicle scheduling subproblem Φ0

y(µ) to guarantee

that both arcs are connected (operated by the same vehicle). Recall that in a

time-space network we only have connections between groups of trips (see Sec-

tion 2.3). As a consequence, we cannot directly fix a connection between two

trips. However, we can enforce a flow between arcs ydi

ij and ydi

rs by modifying the

minimum capacity ldi

ij of the shortest path between the arcs. In each subgradient

iteration, we compute the shortest path between the end node of ydi

ij and the start

node of ydi

rs. Then, we increase the minimum capacity of all arcs on the shortest

path by one flow unit. As a result, we can always decompose the flow solution of

Φ0

y(µ) into a set of paths where at least one path covers both trips consecutively.

The reasoning still holds if different follow-ons require the same connection arc.

In such a case, we simply set the minimum capacity of the connection arc to the

number of follow-ons that require the arc. Furthermore, we disable all duties

k∈Kdifrom the current set of duties with

(Kdi(i, j)∪Kdi(r, s))\Kdi(ydi

ij , ydi

rs).(3.40)

In the column generation pricing problem, we must guarantee that only duties

can be constructed for depot dithat either contain follow-on (ydi

ij , ydi

rs) or none

of the follow-on arcs. Since we assumed that only follow-ons within a piece of

work are considered, it suffices to adapt the piece generation phase (see Section

2.4.2). Pieces of work for depot dimust operate the trips either consecutively

or not at all. Figure 3.7 shows a sample piece generation network to illustrate

how trips can be connected. In our sample, we have four service trips denoted

by t1, . . . , t4that start at station A, B, and C. Clearly, the shortest path between

node iand jis to use trip arcs t2and t4with cost 30. However, if t2is part of

another follow-on it must not be connected with trip t4and the shortest path

107

3. New Approaches to Integrated Vehicle and Crew Scheduling

station A

station B

station C

depot

t2t4

waiting

with arc cost

trip

time

t110 20 20

10 20

node i

node j

deadhead

with arc cost

Figure 3.7.: Sample piece generation network for follow-on fixing

then includes t3and t4with cost 60. As a consequence, it does not suffice to

use a simple label correcting algorithm to compute the shortest path between all

compatible pairs of nodes where only cost are considered. Similar to the dynamic

programming algorithms described in Section 3.2.1, a label must also include the

last service trip traversed. Furthermore, a label may only be corrected if the new

label has lower cost and the last service trip of the old label equals that of the

new one. In our example, we have two labels at the start node of t4: (30, t2) and

(60, t3). At that node, we check whether t2is a valid predecessor of t4and finally

chose label (60, t3) for propagation. Likewise we have two labels at the end node

of t4for further propagation: (60, t4) and (70, t1).

Similar to the preceding subsection, we apply α-fixing to decide which follow-

ons to fix. We fix follow-on (yd

ij, yd

rs) to 1 if

l=1

f(l)(yd

ij, yd

rs)≥(1 −α)·L(3.41)

where f(l)corresponds to the support of the follow-on in the l-th subgradient

iteration. We do not fix follow-ons to 0 since this lead to poor results in our

experiments. Out of the same reason we also do not perform β-fixing.

Computational results

In Table 3.13 we compare the performance of different fixing schemes in our

heuristic branch-and-price framework. FO refers to follow-on fixing while ST

denotes service trip fixing schemes with either α- or β-fixing. Finally, we report

results when no fixings are made (IPCG5). For all settings, we perform five

column generation iterations in the integer phase. We use the same test set and

108

3.3. Integer Solutions

settings as described in Section 3.3.1 and use the adaptive method as reference.

Furthermore, we try to find an upper bound in the lower bound phase if more than

200 subgradient iterations have been performed in a column generation iteration.

In that case, we solve a multiple-depot vehicle scheduling problem using the best

multipliers of that iteration as costs on the arcs. We disable backtracking in order

to assess the quality of the fixings.

In Table 3.13 we give the cpu time in seconds spent on the integer phase

(cpu ip) for each approach. Furthermore, we provide the last lower bound (lb)

and the total number of vehicles and drivers (v+d) of the best feasible solution.

Notice that the last lower bound obtained is not necessarily a valid lower bound.

type #trips

080 100 160 200 320 400

reference cpu ip 0 578 1,676 2,621 7,635 9,808

lb 29,664 37,834 48,382 63,470 77,191 107,922

v+d 26.7 35.0 45.3 60.7 76.0 105.0

IPCG5 cpu ip 374 442 2,840 5,212 16,802 25,909

lb 29,605 37,839 48,273 63,485 76,964 107,877

v+d 26.7 34.3 45.7 60.0 74.3 104.0

FOα=0.01 cpu ip 215 359 2,009 3,658 10,257 18,256

lb 30,254 39,269 50,892 67,547 82,894 112,359

v+d 27.0 35.1 45.7 60.8 74.9 105.8

STα=0.15 cpu ip 286 388 2,073 3,174 16,043 22,413

lb 29,634 37,962 48,274 63,429 77,071 107,955

v+d 26.7 34.3 45.7 60.0 74.3 104.0

STα=0.1cpu ip 328 432 2,337 4,075 15,298 26,582

lb 29,578 37,887 48,266 63,573 77,133 107,856

v+d 27.0 34.7 45.7 60.0 74.3 104.7

STβ=0.12 cpu ip 206 299 2,119 3,489 11,580 16,374

lb 30,448 39,353 51,430 68,046 81,659 113,902

v+d 27.0 35.0 45.7 60.7 75.0 106.0

STβ=0.1cpu ip 216 335 2,389 3,892 12,050 17,332

lb 30,198 39,174 50,767 66,440 81,966 113,844

v+d 27.0 35.0 45.7 60.7 75.0 106.0

Table 3.13.: Results of heuristic branch-and-price algorithms

As expected, the reference (sequential method) requires less computational

time than the other approaches since there is no column generation in the in-

teger phase. Notice that no integer phase was performed for the group with 80

109

3. New Approaches to Integrated Vehicle and Crew Scheduling

trips in the reference method since the best upper bound obtained in the lower

bound phase already had the minimum number of duties. Both α-follow-on fix-

ings and β-service trip fixings reduce the computational burden compared to the

method without fixings. Unfortunately, there is a remarkable increase in the lower

bound as well. Hence, we conclude that those fixing methods are inappropriate

for our problem since the total number of vehicles and drivers also increased.

Furthermore, the results show that α-service trip fixing does neither worsen the

computational time nor the solution quality or the lower bound. Instead, solution

times are better than without fixings when α= 0.15 is set. To sum up, we infer

that column generation in the integer phase improves (with the exception of the

160 trips group) the solution quality but increases the computational burden.

Finally, (bold) α-fixings of service trips allow to fix parts of the problem without

increasing the lower bound and without worsening the solution quality.

3.4. Integrated Planning with Unrestricted

Changeovers

In Section 2.1 we assumed that each crew is assigned to a depot and may only

conduct tasks on vehicles from this particular depot. Furthermore, we assumed

that a driver may only change his vehicle during a break, i.e., between two pieces

of work. In other words, a changeover is only allowed between vehicles from

the same depot and the driver must take a break after leaving his vehicle. In

this section, however, a driver may change between two vehicles of different de-

pots whenever there is a relief point (no matter if he takes a break or not).

In the following, we say that changeovers are unrestricted while in the original

setting changeovers were restricted. Similar assumptions were (implicitly) used

by [Mesquita and Paias, 2006]. However, in their setting, drivers may addition-

ally walk on deadhead connections that are not part of the vehicle schedule. Of

course, this is not very realistic since this assumes that a driver can deadhead by

foot within the same time as a driver does by bus. Therefore, in our setting, we

will not allow drivers to use deadheads that are not part of the vehicle schedule

(except at the beginning and end of a duty - see Section 2.1).

In the following, we propose a novel modification of model MDVCSP that

allows unrestricted changeovers as described above. In model MDVCSP, the

duty variables and linking constraints are separated by depot. However, if we

allow unrestricted changeovers drivers may use vehicles from all depots. As a

consequence, duty variables and linking constraints do not need to be separated

by depot. In addition to the notation used in Section 2.3 we define Kas the

110

3.4. Integrated Planning with Unrestricted Changeovers

set of duties, xk, k ∈Kas binary duty variables, and e

A=Sd∈D e

Adas the set

of arcs that require both vehicle and crew activities. Without loss of generality,

we assume that the nodes of the |D| vehicle scheduling networks are numbered

in such a way that nodes with the same time and location have the same index

in all networks. We consider two arcs to be equal if they have the same start

and end index as well as the same type (see Figure 2.3). As a consequence,

A| ≤ Pd∈D |e

Ad|holds. The model that allows unrestricted changeovers can be

stated as follows (MDVCSP-C).

d∈D X

(i,j)∈Ad

ijcd

ij +X

k∈K

xkfk→min (3.42)

s.t. X

d∈D X

(i,j)∈Ad(t)

ij = 1 ∀t∈ T (3.43)

{j:(j,i)∈Ad}

ji −X

{j:(i,j)∈Ad}

ij = 0 ∀d∈ D,∀i∈Nd(3.44)

k∈K(i,j)

xk−X

d∈D

ij = 0 ∀(i, j)∈e

A(3.45)

0≤yd

ij ≤ud

ij, yd

ij ∈N∀d∈ D,∀(i, j)∈Ad(3.46)

xk∈ {0,1} ∀k∈K(3.47)

The objective (3.42) minimizes the sum of vehicle and crew costs. Constraints

(3.43)-(3.44) correspond to a multicommodity flow formulation for the vehicle

scheduling problem where the set of trip tasks must be partitioned among the

depots (3.43) and flow conservation is ensured for each depot (3.44). Constraint

set (3.45) establishes the link between vehicle and crew schedule: each arc cov-

ered by a vehicle/vehicles must also be covered by the same number of duties.

Constraints (3.46) guarantee that the maximum capacity of the flow variables is

satisfied. If a driver may walk on deadhead connections that are not part of the

vehicle schedule, we replace the equality signs of the linking constraints (3.45)

by greater or equal signs. Similar to our solution approach for model MDVCSP

we apply column generation in combination with Lagrangian relaxation to solve

model MDVCSP-C.

The Master Problem

Basically, we relax the same constraints in a Lagrangian way as stated in Section

2.4.1. We associate Lagrangian multipliers µij and πtwith constraints 3.45 and

3.43, respectively. The Lagrangian subproblem

Φ(µ, π)=Φy(µ, π)+Φx(µ) + X

t∈T

πt(3.48)

111

3. New Approaches to Integrated Vehicle and Crew Scheduling

remains unchanged except the definition of the reduced cost. Notice that the

vehicle scheduling subproblem Φy(µ, π) still constitutes a single depot vehicle

scheduling problem for each depot. The reduced cost ¯cd

ij for arc (i, j)∈Adof the

vehicle scheduling network of depot d∈ D is defined as

¯cd

ij =









ij +µij −πtfor (i, j)∈e

Aand ∃t∈ T : (i, j)∈Ad(t)

ij +µij for (i, j)∈e

Aand @t∈ T : (i, j)∈Ad(t)

ij for (i, j)/∈e

(3.49)

while

fk=fk−X

(i,j)∈e

A(k)

µij (3.50)

denotes the reduced cost of duty k∈Kwhere e

A(k)⊆e

Acorresponds to the set

of arcs that is covered by duty k.

The Pricing Problem

After the restricted master problem is solved, the dual information of the solution

is used to price out new columns with negative reduced costs. As described in

Section 2.4.2 we use a two phase pricing procedure. In the first phase, we set up

a piece generation network to generate a set of pieces of work. These pieces serve

as input for the second phase where duties are generated (see 3.1.2). However,

the column generation pricing problem differs from the original version in the

way the pieces of work are generated.

Clearly, we no longer have a separate pricing problem for each depot since in

model MDVCSP-C the duty variables are not separated by depot. Consequently,

we set up a single piece generation network ¯

Gc= ( ¯

Nc,e

A). In particular, ¯

Gcis

an acyclic directed time-space network where the set of arcs e

Acorresponds to all

activities that require both vehicle and crew. Note that we assumed that each

trip has exactly two relief points: one at the beginning and the other at the end

of the trip. Thus, each node in ¯

Nc⊂Sd∈D Ndcorresponds to a relief point.

Let gij be the crew cost associated with arc (i, j)∈e

A. The reduced cost of arc

(i, j)∈e

Ais then defined as ¯gij =gij −µij where µij are the multipliers associated

with linking constraints (3.45) that represent trip, deadhead, or waiting arcs

outside the depot. Recall that each path corresponds to a piece of work. Hence,

the reduced cost of a path is equal to the reduced cost of the associated piece of

work. We compute the shortest path between each pair of nodes that meet the

(piece) duration constraint. Furthermore, we consider additional pieces for each

112

3.4. Integrated Planning with Unrestricted Changeovers

path and for each depot: we add a pull-in trip at the beginning, a pull-out trip

at the end, and for each depot combination both. Notice, however, that a driver

does not necessarily remain on the same vehicle for the duration of the piece of

work. Clearly, this requires to adapt the initial definition of a piece of work (see

Section 2.1) where the driver stays with the vehicle for the entire piece.

Computational Results

In Table 3.14 we compare our result on model MDVCSP-C with unrestricted

changeovers (unrestricted) with those of model MDVCSP (reference). We use

the same test set and settings as described in Section 3.3.1 and use the adaptive

method as reference. In Table 3.14 we give the number of iterations (#iter), the

cpu time in seconds spent on the master problem (cpu ma), on pricing problem

(cpu pr), on the integer phase (cpu ip), and the total cpu time (cpu tot). The

cpu time of the master problem includes the time to find new upper bounds in the

lower bound phase. Furthermore, we provide the best lower bound obtained (lb)

and the total number of vehicles and drivers (v+d) of the best feasible solution.

Notice that the best lower bound obtained is not necessarily a valid lower bound.

type #trips

080 100 160 200 320 400

reference #iter 18.0 19.0 23.3 24.7 35.3 32.3

cpu ma 153 222 1,170 1,712 7,560 8,318

cpu pr 11 32 302 202 5,477 8,120

cpu ip 0 578 1,676 2,621 7,635 9,808

cpu tot 165 834 3,155 4,541 20,794 26,384

lb 29,664 37,834 48,382 63,470 77,191 107,922

v+d 26.7 35.0 45.3 60.7 76.0 105.0

unrestricted #iter 25.7 29.3 38.0 37.7 50.7 46.7

cpu ma 138 215 596 754 2,039 2,688

cpu pr 9 20 161 172 1,370 2,037

cpu ip 60 136 482 579 6,659 8,386

cpu tot 208 372 1,243 1,513 10,338 13,455

lb 29,573 37,884 48,195 63,177 76,433 106,889

v+d 27.0 35.0 45.0 59.7 73.7 103.3

Table 3.14.: Results of integrated planning with restricted and unrestricted

changeovers

113

3. New Approaches to Integrated Vehicle and Crew Scheduling

As we can see in Table 3.14 there is a considerable speed-up and a better

solution quality if changeovers are not restricted. As expected, most of the time

is saved in the lower bound phase. In fact, for large instances approximately 70%

of the time is saved in the lower bound phase. Recall that model MDVCSP-C has

fewer constraints and, thus, fewer Lagrangian multipliers. Furthermore, there is

only one pricing problem per iteration in the unrestricted case while there is a

separate pricing problem for each depot for model MDVCSP. Additionally, we

obtained better solutions for instances with more than 80 trips if changeovers are

not restricted. In Section 3.5.2, we will give additional results and compare our

approach with other methods previously exposed in literature.

3.5. Computational Results

The purpose of this section is to summarize our computational results on real-

world problem data from Connexxion and randomly generated instances. All

experiments in this section were conducted on a Dell OptiPlex GX620 personal

computer running Windows XP with an Intel Pentium IV 3.4 GHz processor and

2 GB of main memory. Our integrated method ICOPT is implemented in C#

and has been compiled using the .NET framework version 2.0.50727.

We test our integrated approach on real-world instances from Connexxion

which is the largest bus operator in the Netherlands. The instances have been

kindly provided by the author of [Huisman, 2004]. The total set involves 1,104

trips and four depots and has been split into 8 smaller instances by [Huisman,

2004, Huisman et al., 2005a]. In the original setting, not all trips were allowed

to be driven by a vehicle from every depot. The average number of depots a

trip can be operated from was 1.71. However, we will consider a different setting

where every trip may be serviced from every depot. Obviously, this makes the

problems more difficult since the solution space is expanded.

Moreover, we use the randomly generated instances and settings that have also

been used and described in the preceding sections. In order to ease the exposition

we recall the basic properties. The instances available at [Huisman, 2003] have

been classified into two classes according to the travel speed where the speed

is lower for problems in class B. As a consequence, trips in class B are longer.

However, we will only report computational results for class A that all involve 4

depots and groups with ntrips where n= 80,100,160,200,320 and 400. For each

group 10 instances are available. Furthermore, we generated 10 instances with

640 trips and four depots according to [Huisman, 2004, Huisman et al., 2005a].

The instances are available at the web page [Steinzen, 2007]. To the best of

114

3.5. Computational Results

our knowledge, randomly generated instances of that size have not been tackled

before.

In accordance with [Huisman, 2004, Huisman et al., 2005a] we consider five

different types of duties: one tripper type with one piece of work between 30

minutes and 5 hours, and four types consisting of two pieces of work. Each duty

starts with a sign-on and ends with a sign-off. If the first (last) duty activity

starts (ends) at the depot we impose a sign-on (sign-off) time of 10 and 5 min-

utes, respectively. If the duty starts (ends) at another relief point both sign-on

and sign-off time increase to 15 minutes plus the deadhead time between the start

(end) location and the depot. The time a driver spends on the vehicle is work-

ing time. The duty length corresponds to the total duty duration including all

activities such as sign-on/off, pieces of work, and breaks. Table 3.15 summarizes

the settings we use in this section.

early day late split

Type min max min max min max min max

Start time 8:00 13:15

End time 16:30 18:14 19:30

Piece length 0:30 5:00 0:30 5:00 0:30 5:00 0:30 5:00

Break length 0:45 0:45 0:45 1:30

Duty length 9:45 9:45 9:45 12:00

Working time 9:00 9:00 9:00 9:00

Table 3.15.: Properties of different duty types

The objective is to minimize the total sum of vehicles and drivers. We assign a

fixed cost of 1,000 for each vehicle and duty and a small variable cost of 1 for each

minute a vehicle is outside the depot and 0.1 for each minute a crew is working. In

other words, we minimize the total number of vehicles and drivers first and leave

operational cost minimization as secondary objective. Furthermore, we used the

following parameter settings.

1. We solved the pricing problem independently for each depot and duty type

combination where we generated at most 5,000 duties for each combination.

2. The column generation algorithm is terminated if the improvement of the

lower bound is less than 0.2% in the last 10 iterations. Notice that we do

not have a valid lower bound unless

−Pd∈D Pk∈Kd

max Φf(µ, π)≤0.002 (3.51)

115

3. New Approaches to Integrated Vehicle and Crew Scheduling

is satisfied where Kd

fis the set of duties added in the final column generation

iteration and where max Φf(µ, π) is the best lower bound. Furthermore, all

duties with negative reduced must have been added in the final iteration in

order to obtain a valid lower bound for the overall problem. However, our

computational experiments indicated that terminating according to (3.51)

led to higher computational times without improving the final integer so-

lution.

3. The maximum number of iterations in the subgradient algorithm is 1,000

for every column generation iteration.

4. We limit the computational time for the lower bound phase to 21,600 sec-

onds (6 hours).

This section is organized as follows. In Section 3.5.1 we give computational

results for the integrated approach on real-world data instances. In Section 3.5.2,

we also report results of our method on randomly generated benchmark instances.

We present results for the case with restricted and unrestricted changeovers.

3.5.1. Real-world Data Instances

In Table 3.16 we give computational results of our integrated approach on Con-

nexxion data instances. For each of the 8 instances we report the number of

trips and the number of depots where each trip can be serviced from every de-

pot. Notice that this setting is different to the results published in [Huisman,

2004, Huisman et al., 2005a] since the solution space is expanded. As a conse-

quence, the results cannot be directly compared. However, the deadhead matrix

remains unchanged and is such that some connections to and from depots are

not allowed. Furthermore, we give the number of iterations (#iter), the CPU

time in seconds spent on the master (cpu ma), the pricing problem (cpu pr), the

integer phase (cpu ip), and the total time including all initializations (cpu tot).

Additionally, we report the final upper bound (ub), the number of vehicles (vehi-

cles), drivers (drivers), total number of vehicles and drivers for integrated (v+d)

and sequential vehicle and crew scheduling (v+d seq.). Finally, we give the to-

tal number of duties and drivers obtained by [Huisman et al., 2005a] (v+d ref.).

Recall that their results cannot be directly compared with ours since, in their

setting, not all trips can be serviced from each depot.

The results show that real-world instances with up to 653 trips and 4 depots

can be solved. Furthermore, there is an efficiency gain compared to sequential

planning. Observe that the computational time does not always increase with the

116

3.5. Computational Results

instance

1 2 3 4 5 6 7 8

#trips 194 210 220 237 304 386 451 653

#depots 4 4 4 4 4 4 4 4

#iter 38 25 37 27 30 52 48 62

cpu ma 965 572 1,008 741 1,204 2,932 3,468 7,251

cpu pr 1,358 118 729 157 483 10,936 1,763 17,058

cpu ip 964 869 634 859 2,062 17,282 8,097 31,648

cpu tot 3,329 1,566 2,379 1,762 3,763 31,195 13,388 56,097

ub 48,905 87,978 64,803 93,650 111,034 89,459 118,107 189,185

vehicles 19 33 28 34 40 32 47 67

drivers 26 48 32 52 62 51 62 108

v+d 45 81 60 86 102 83 109 175

v+d seq. 52 89 77 95 114 92 138 191

v+d ref. 49 83 68 89 105 90 124 184

Table 3.16.: Results on Connexxion data instances

problem size. The difference can be understood in the following way. The struc-

ture of the problems are different since the average trip lengths are comparatively

long for instances 2,4, and 5. For these instances, the average number of trips

per duty is less than 5 while for all other instances there are at least 6 trips per

duty. Therefore, we conclude that our algorithm performs better if the density of

the columns is small. Similar results were obtained by [Oukil et al., 2007]. The

authors report a strong impact of column density on the computational burden of

a column generation algorithm for a multiple-depot vehicle scheduling problem.

Finally, recall that for our approach the number of vehicles is always minimal,

i.e., equals the number of vehicles when sequential planning is performed.

3.5.2. Randomly Generated Data Instances

In this section, we give computational results of our solution approach for ran-

domly generated data instances. We will first concentrate on the setting where

changeovers are restricted. In the second part we focus on the case with unre-

stricted changeovers.

Restricted Changeovers

In Table 3.17 we report results of our integrated approach on the Huisman data

instances type A with 80 to 640 trips and 4 depots. Changeovers are restricted

117

3. New Approaches to Integrated Vehicle and Crew Scheduling

as stated in Section 2.1. Each trip can be operated from every depot. We use the

same parameter settings for each group of 10 instances. The structure of Table

3.17 is similar to Table 3.16.

#trips

080 100 160 200 320 400 640

#iter 20.5 22.0 25.8 28.2 34.7 33.0 46.3

cpu ma 153 234 528 754 1,825 2,330 5,207

cpu pr 11 24 228 651 1,593 2,074 9,007

cpu ip 70 110 819 1,318 10,877 15,772 42,588

cpu tot 235 369 1,579 2,730 14,325 20,320 57,235

ub 31,338 37,347 52,165 62,568 89,084 109,750 190,488

vehicles 9.2 11.0 14.8 18.4 26.7 32.9 59.9

drivers 19.1 22.7 31.8 38.8 55.8 67.9 120.4

v+d 28.2 33.7 46.6 57.2 82.5 100.8 177.3

v+d seq. 35.3 41.3 53.6 63.9 89.8 108.8 211.3

Table 3.17.: Detailed results on Huisman data instances type A with four depots

and restricted changeovers

Similar to our results on real-world problem instances, the total number of

vehicles and drivers can be remarkably reduced if integrated planning is per-

formed. For instances with more than 200 trips, the CPU time spent in the

integer phase jumps up since we use our heuristic branch-and-price method (see

Section 3.3.3). However, we only perform three column generation iterations in

the integer phase. We also tested our method with more than three iterations,

but the strong increase in solution time did not justify the improvement of the

final solution.

In Table 3.18 we compare our results with results from literature. In particular,

we summarize the results of our implementation from Table 3.17 (ICOPT) and

give results of [Gintner et al., 2006b] (GSS05), [Bornd¨orfer et al., 2004] (BLW04),

and [Huisman et al., 2005a] (HFW05). For each group of instances, we report

the total number of vehicles and drivers (v+d) and the total computational time

(cpu tot) in seconds. Notice that we do not give the computational time for

[Huisman et al., 2005a] since it cannot be compared with the other approaches.

Table 3.18 shows that our approach clearly outperforms all other approaches from

literature in terms of solution quality and solution time. Furthermore, we have

so far tackled the largest instances with 4 or more depots.

[Gintner et al., 2006b] also use model MDVCSP while the other approaches

118

3.5. Computational Results

rely on model MDVCSP-H (see Section 2.2.3). As can be seen from Table 3.18

approaches based on model MDVCSP are beneficial compared to the classic

connection-based model MDVCSP-H. As stated earlier our implementation also

relies on model MDVCSP, but uses a different pricing scheme and a modified in-

teger phase compared to [Gintner et al., 2006b]. The modifications we proposed

in this chapter improve the results of [Gintner et al., 2006b]: the solution quality

is better for all groups and the computational time can be reduced.

source #trips

080 100 160 200 320 400 640

ICOPT cpu tot1235 369 1,579 2,700 14,325 20,320 57,235

v+d 28.2 33.7 46.6 57.2 82.5 100.8 177.3

GSS05 cpu tot2420 660 1,620 3,300 – – –

v+d 29.3 35.0 47.4 58.3 – – –

BLW04 cpu tot3780 1,260 2,640 6,360 19,680 43,200 –

v+d 29.6 35.7 47.7 59.0 82.8 102.0 –

HFW05 cpu tot4– – – – – – –

v+d 29.6 36.2 48.9 60.0 – – –

1,2on Dell OptiPlex GX620 with Intel Pentium IV 3.4 GHz/2GB

3on Dell Precision 650 PC with Intel Dual Xeon 3.0 GHz/4GB

4CPU times not comparable

Table 3.18.: Comparison on Huisman data instances type A with four depots and

restricted changeovers

Figure 3.8 illustrates that our approach requires between 29 and 73 percent

of the computational time of [Bornd¨orfer et al., 2004]. Furthermore, the results

indicate that ICOPT is the overall fastest known method for integrated vehicle

and crew scheduling problems under the assumptions stated in [Huisman, 2004].

Unrestricted Changeovers

In Table 3.19 we report results of our integrated approach on the Huisman data

instances type A with 80 to 640 trips and 4 depots. Changeovers are unrestricted

as described in Section 3.4. Each trip can be operated from every depot. We

use the same parameter settings for each group of 10 instances. The structure of

Table 3.19 is similar to Table 3.17.

119

3. New Approaches to Integrated Vehicle and Crew Scheduling

100

120

080

100

160

200

320

400

computational time (%)

number of trips

BLW04

GSS05

ICOPT

Figure 3.8.: Comparison of computational times in percent on Huisman data in-

stances type A with four depots

120

3.5. Computational Results

#trips

080 100 160 200 320 400 640

#iter 23.2 25.5 36.1 38.9 44.4 47.3 49.3

cpu ma 126 182 529 789 1,735 2,717 4,116

cpu pr 9 17 139 195 1,085 2,155 2,805

cpu ip 62 82 747 996 5,427 6,789 15,724

cpu tot 197 282 1,422 1,990 8,481 11,947 23,125

ub 31,636 37,347 50,725 61,477 88,394 108,201 190,182

vehicles 9.2 11.0 14.8 18.4 26.7 32.9 56.9

drivers 19.2 22.7 31.3 37.8 55.2 67.6 120.3

v+d 28.4 33.7 46.1 56.2 81.9 100.5 177.2

v+d seq. 34.0 38.9 51.1 61.5 88.0 105.5 184.3

Table 3.19.: Detailed results on Huisman data instances type A with four depots

and unrestricted changeovers

Basically, the results show that there is an efficiency gain if vehicle and crew

scheduling are treated in an integrated way. Similar to the restricted case, most

of the time is spent in the integer phase. Except the 80 trips group the number of

vehicles and drivers is smaller compared to our results with restricted changeovers.

Thus, we conclude that model MDVCSP-C is computationally more attractive

than model MDVCSP. Furthermore, we believe it is worthwhile for planners in

practice to allow unrestricted changeovers since the additional flexibility results

in efficiency gains.

In Table 3.20 we compare the results of our implementation (ICOPT C ) where

unrestricted changeovers are allowed with the results of [Mesquita and Paias,

2006] (MP06). Recall from Section 2.2.3 that the authors allowed crews to walk

on deadhead connections that were not part of the vehicle schedule. Due to

the reasons stated in Section 3.4 we did not allow deadheading by foot which

obviously reduces the solution space compared to [Mesquita and Paias, 2006].

The structure of Table 3.20 is similar to Table 3.18.

The results show that our approach outperforms the method of [Mesquita and

Paias, 2006] in terms of computational time and/or solution quality. Only for the

80 trips group ICOPT C consumes more time. Furthermore, we solved instances

with 640 trips and 4 depots that, to the best of our knowledge, have not been

tackled before. Finally, we would like to mention that we can basically compute

a valid lower bound with our method while this is not possible for the method

of [Mesquita and Paias, 2006] (since the set of tasks is heuristically defined).

121

3. New Approaches to Integrated Vehicle and Crew Scheduling

source #trips

080 100 160 200 320 400 640

ICOPT C cpu tot1197 282 1,422 1,990 8,481 11,947 23,125

v+d 28.4 33.7 46.1 56.2 81.9 100.5 177.2

MP06 cpu tot272 428 2,436 3,064 11,023 13,453 –

v+d 28.7 35.5 46.1 56.9 82.5 101.8 –

1on Dell OptiPlex GX620 with Intel Pentium IV 3.4 GHz/2GB

2on PC with Intel Pentium IV 3.2 GHz

Table 3.20.: Comparison on Huisman data instances type A with four depots and

unrestricted changeovers

3.6. Summary

In this chapter, we discussed solution approaches for the integrated multiple-

depot vehicle and crew scheduling problem. More precisely, we addressed the

column generation pricing problem and methods to construct integer solutions.

Moreover, we proposed a novel model variation where drivers can use vehicles

from all depots and can change their vehicles whenever possible.

The column generation pricing problem corresponds to a resource constrained

shortest path problem. We proposed two novel network formulations for a de-

composed pricing problem. Moreover, we showed that the network complexity

of our models is superior to models previously exposed in literature. We applied

dynamic programming to solve the resource constrained shortest path problems.

We suggested novel, problem-specific reduction techniques that considerably sped

up the solution process. Furthermore, we presented a combination of known and

novel techniques to further improve the performance of the algorithm.

We discussed three methods to compute integer solutions: a Lagrangian heuris-

tic (sequential approach), a branch-and-bound approach with novel branching

schemes, and a novel heuristic branch-and-price algorithm (fix-and-optimize).

Our computational tests revealed that the latter approach generates high-quality

solutions while the sequential method finds good solutions in a short timeframe.

Furthermore, we found the branch-and-bound method inappropriate to find good

quality solutions for medium-sized instances in a reasonable timeframe.

We proposed a novel model variation where drivers are not tied to vehicles of a

single depot and where vehicle changes may be performed during a piece of work.

We concluded this chapter with an extensive computational study on real-

world and randomly generated benchmark instances. Our results indicate that

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3.6. Summary

medium-sized with about 640 trips and 4 depots could be solved efficiently. In

fact, our method outperformed other approaches previously exposed in literature

in terms of solution quality and computational time. For well-known benchmark

instances, we presented previously unknown solutions and were able to tackle the

largest instances so far.

123

3. New Approaches to Integrated Vehicle and Crew Scheduling

124

4. A Hybrid Evolutionary Algorithm

In this chapter, we present a novel hybrid evolutionary algorithm for the multiple-

depot integrated vehicle and crew scheduling problem that combines mathemat-

ical programming techniques with an evolutionary algorithm. The algorithm is

novel since an evolutionary algorithm has not been applied to integrated vehicle

and crew scheduling problems before. This section is partly based on [Steinzen

et al., 2007a].

Evolutionary algorithms (EA) are adaptive heuristic search methods that are

based on the idea of evolutionary processes in nature. In evolutionary processes,

populations evolve in accordance with the principle of natural selection or, in

other words, the ”survival of the fittest”. Individuals that are successful in adapt-

ing to their environment have a better chance of surviving and reproducing than

individuals with a worse fitness. As a result, genes from highly successful in-

dividuals will spread across the population from generation to generation. The

combination of good genes from different individuals can yield even more fit off-

spring.

An EA simulates this processes by creating an initial population of individuals

and applying genetic operators in each generation/reproduction. Each individual

is represented by a string or chromosome and corresponds to a possible solu-

tion to the (combinatorial) optimization problem. The fitness of an individual

represents the value of the objective function. Furthermore, individuals with a

high fitness get the opportunity to reproduce among each other by exchanging

genetic information. Algorithm 10 shows the basic steps of a simple evolutionary

algorithm. For an extensive survey on evolutionary algorithms the reader is

referred to [B¨ack, 1996].

This chapter is organized as follows. In Section 4.1 we discuss a decomposition

approach of model MDVCSP (see Section 2.3). The decomposition approach pro-

vides the basis for the hybrid evolutionary algorithm that we describe in Section

4.2. Finally, we compare the computational results of our evolutionary algorithm

with other integrated approaches in Section 4.3.

125

4. A Hybrid Evolutionary Algorithm

Algorithm 10: Basic Evolutionary Algorithm

(Step 1) Initialization

Generate initial population and evaluate fitness of each

individual.

(Step 2) Evolutionary process

repeat

Select parents from population.

Recombine genes of parents to produce children.

Evaluate fitness of children.

Replace (parts of the) population by children.

until sufficiently good solution is found

4.1. Problem Decomposition

Our solution approach decomposes the MDVCSP (see Section 2.3) into different

subproblems as shown in Figure 4.1. First, we assign each trip t∈ T to a

depot d∈ D. Thus, we obtain a trip-depot vector θ∈ {1,...,|D|}|T | where

each trip is assigned to exactly one depot. In a second phase, we compute the

optimal solution of each single depot vehicle scheduling problem and, afterwards,

we solve a crew scheduling problem for each depot given the vehicle schedule

for that depot. The main advantage of this decomposition is that the vehicle

Assign trips to depots

Construct vehicle schedule

Construct crew schedule

Figure 4.1.: Problem decomposition for evolutionary algorithm

scheduling problem with multiple depots is NP-hard unlike the single depot case

that appears in our second phase.

In the second phase, we schedule vehicles independently of crews which cor-

responds to a traditional (sequential) approach to vehicle and crew scheduling.

However, we also consider a partial integration similar to [Gintner et al., 2006a]

where we allow to recombine parts of vehicle blocks in order to disclose addi-

126

4.2. Components of Evolutionary Algorithm

tional flexibility in crew scheduling while preserving vehicle schedule optimality.

The additional flexibility often results in vehicle schedules that allow better crew

schedules (with less duties) compared to the sequential approach.

Furthermore, vehicle and crew scheduling can be considered in an integrated

way for a given trip-depot vector. That is, we solve |D| integrated vehicle and

crew scheduling problems with a single depot. Typically, the additional free-

dom in scheduling vehicles dependent on crews (and vice versa) leads to better

solutions compared to the sequential or partially integrated method.

In summary, there are three different methods to construct a feasible vehicle

and crew schedule for a given trip-depot vector: sequential, partially integrated,

and fully integrated. Of course, there is a strong relationship between the level

of integration and the computational time needed to solve the corresponding

problems. The computational time increases with the level of integration. The

evolutionary algorithm we propose in the next section is based on this decompo-

sition approach where we first make a trip-depot assignment.

4.2. Components of Evolutionary Algorithm

Our evolutionary algorithm (EA) is based on a non-binary representation that is

equal to the trip-depot vector θfrom the previous section. A chromosome is a

string of length equal to the number of trips where the i-th entry contains (the

index of) the depot the i-th trip is assigned to. We use an evolutionary algorithm

to find a good trip-depot assignment where the fitness of a chromosome (individ-

ual) is evaluated using mathematical programming techniques. In particular, we

use the all-purpose MIP solver CPLEX [ILOG, 2006] and column generation in

combination with Lagrangian relaxation.

The size of the EA search space with the non-binary representation is |T||D|.

Notice that a feasible vehicle schedule can always be constructed from a given

trip-depot vector since each trip can always be covered by a short vehicle block:

pull-out trip - service trip - pull-in trip. Furthermore, duty constraints in practice

are such that almost all vehicle blocks can be covered by a feasible crew duty. As a

consequence, virtually all chromosomes represent feasible trip-depot assignments

and, thus, we do not use a repair mechanism to transform infeasible to feasible

solutions. However, if a trip-depot assignment θdoes not yield both feasible

vehicle and crew schedules, the individual will be discarded after the evaluation

of the fitness function. Moreover, the search space can be reduced if the number

of trips assigned to a depot must be greater or equal a lower limit ρ < |T|.

Assigning a small number of trips to a depot often leads to inefficient vehicle and

127

4. A Hybrid Evolutionary Algorithm

crew schedules since many deadheads and/or vehicles are needed to cover trips

that are long way/time away from each other.

In the following, we will describe the essential components of our evolutionary

algorithm. After discussing how the initial population is set up, we describe three

different methods to calculate the fitness of an individual. Finally, we define the

genetic operators and the termination criteria we use in our algorithm.

4.2.1. Initialization

In the first step of the algorithm an initial population is generated to serve as

seed for the evolutionary process. We create our solutions (1) in areas where good

solutions are likely to be found and (2) randomly in order to cover a wide range

of the solution space. We apply four heuristics of the first category that analyze

the geographical structure of the problem, i.e., the start and end location of the

service trips. The first three heuristics have been proposed by [de Groot and

Huisman, 2004] in combination with a heuristic to split large problem instances

such that each split problem can be solved with an integrated approach.

•Assign a service trip to the depot closest to its start location,

•Assign a service trip to the depot closest to the combination of start and

end location,

•Solve the multiple-depot vehicle scheduling problem and assign a service

trip to the depot where it is assigned to in the optimal solution,

•Assign a service trip to the depot either closest to its start or end location.

The rationale behind these heuristics can be understood in the following way. If

the trips assigned to the same depot are operated in the same geographical area,

it is likely that many of these trips can be covered without extensive deadhead-

ing. Furthermore, few deadheads result in a low unproductive overhead since a

vehicle and driver outside of the depot spend most of the time on transporting

passengers. As stated earlier, we require that at least ρ= 10 trips are assigned to

a depot. However, if one of the heuristics above leads to an assignment violating

the minimum assignment, we randomly shift these trips to other depots.

4.2.2. Fitness Calculation

As described in Section 4.1 there are three different methods of constructing

vehicle and crew schedules for a given trip-depot assignment: sequential, par-

128

4.2. Components of Evolutionary Algorithm

tially integrated, and fully integrated. We apply mathematical programming

techniques in order to assess the quality of a particular trip-depot assignment.

In the sequential and partially integrated method, we solve a single depot vehi-

cle scheduling problem for each depot with the network simplex implementation

of CPLEX. The solution for each depot d∈ D consists of a set of arcs ¯

Ad⊂Adand

the corresponding flow values. Thus, the total fitness fv

θof the vehicle schedule

of an individual θreads:

θ:= X

d∈D X

(i,j)∈¯

ijcd

ij.(4.1)

Now, we construct a crew schedule based on the vehicle schedule either sequen-

tially or partially integrated. Although we omit the mathematical details, in

both cases model MDVCSP reduces to a separate, generalized set partitioning

problem (SPP) for each depot. If the vehicle schedule is given (the yvariables

are fixed) only constraints (2.14) and (2.16) remain where the right-hand side of

(2.14) is a constant pd

ij equal to the flow value of yd

ij. Our computational experi-

ments indicate that it suffices to compute a lower bound on the SPP instead of

constructing a feasible crew schedule for each individual.

Since the number of duty variables can be vast even for small-sized problems,

we apply a column generation algorithm to compute a lower bound of the SPP.

Traditionally, column generation (see Section 1.5.2) is a method to solve linear

programs that involve a large number of columns. Instead of solving a large

problem with all feasible columns (duties), a sequence of restricted master lin-

ear programs (RMP) is solved where each problem contains only a small subset

of all columns. As described in Section 2.4 for the integrated case we found it

very promising to solve the master problem with Lagrangian relaxation (with

constraints (2.14) relaxed) instead of the linear relaxation. That is, we solve the

Lagrangian dual with a subgradient method in order to obtain an approximate

dual solution and a lower bound for the current RMP. The dual information

πd

ij ≥0, d ∈ D,(i, j)∈¯

Adis used to price out new columns that are added to

the RMP. To solve the pricing problem we set up a duty generation network and

solve an associated resource constrained shortest path problem with a dynamic

programming algorithm (see Sections 3.1 and 3.2). The column generation pro-

cess iterates until no new columns can be found. Finally, we end up with a |D|

separate sets of columns and an approximate solution to the Lagrangian dual (i.e.

129

4. A Hybrid Evolutionary Algorithm

the lower bound) for each depot d∈ D:

zd:= max

π{min X

k∈Kd

k(fd

k−X

(i,j)∈¯

Ad(k)

πd

ij)

(i,j)∈¯

πijpd

ij|xd

k∈ {0,1}}.(4.2)

Furthermore, the overall fitness fθfor an assignment θis defined as the sum of

vehicle fitness and crew fitness for each depot:

fθ:= fv

θ+X

d∈D

zd.(4.3)

Notice that the set of columns that we obtain while solving the Lagrangian

dual for each depot can be used to construct a feasible integer solution (feasible

crew schedule). We use the branch-and-bound implementation of CPLEX to

generate integer solutions. However, it can be quite time consuming to construct

an integer solution for each individual. Therefore, we only compute an integer

solution in the final phase of our EA for individuals with a good overall fitness.

The integrality gap between the lower bound and the final integer solution was

always less than 2%.

Finally, we would like to mention that both sequential and partially integrated

fitness calculation do not require an elaborate solution method for integrated

problems (such as [Bornd¨orfer et al., 2004] or [Huisman, 2004]). Instead, we

only need a sequential vehicle and crew scheduling algorithm that is used in

most commercial software packages for public transport companies. The partially

integrated evaluation only differs in the way duties are generated in the column

generation pricing problem (but is essentially a set partitioning model as in the

sequential crew scheduling problem). Of course, a sequential approach is much

easier to implement than a fully integrated one.

So far, we have defined the fitness for a sequential and partially integrated

evaluation of a trip-depot assignment. Now, we will specify a fully integrated

evaluation. For a given trip-depot assignment model MDVCSP reduces to a

minimum cost flow problem in combination with a set partitioning problem. Al-

though this is an NP-hard problem, the vehicle scheduling subproblem can be

solved in polynomial time. Again, we first compute a lower bound in order to

assess the quality of trip-depot assignment.

We relax constraints (2.12) and (2.14) in a Lagrangian way and use a column

generation algorithm similar to the one described earlier. The Lagrangian sub-

problem constitutes a single depot vehicle scheduling problem (with flow conser-

vation constraints (2.13)) that has to be solved in each iteration of the subgradient

130

4.2. Components of Evolutionary Algorithm

method. Notice that the Lagrangian subproblem in the previous subsection was

a trivial selection problem since no constraints remained after relaxing (2.14).

Again, we end up with |D|separate sets of columns and an approximate solution

to the Lagrangian dual (i.e. the lower bound) for each depot d∈ D. As described

earlier the sets of columns can be used to compute an integer feasible solution.

However, it turned out that the computational time to generate an integer

solution with a fully integrated model can be prohibitive. On the other hand, the

lower bound derived from a fully integrated evaluation gives a better indication

on the quality of an assignment than the methods described earlier. This is due

to the fact that in this approach vehicles are scheduled dependent on crews (and

vice versa).

Therefore, we use the fully integrated fitness calculation only in the first phase

of our algorithm. That is, after creating the initial population we use the fully

integrated method for fitness assignment in the first iterations of the EA. In our

computational experiments we found that using the full integrated evaluation

in the first d|T |/3eiterations provides robust results. Later we recalculate the

fitness of each individual and iterate using one of the other methods. Our tests

indicate that this improves the overall quality of the population.

4.2.3. Genetic Operators

The parents are selected based on the tournament selection discussed in [Beasley

and Chu, 1996]. In tournament selection two pools of randomly selected indi-

viduals from the population are constructed. In a second step, we choose the

individual with the best fitness from each pool. Our computational tests indicate

that forming two pools with 2 individuals each performs best (binary tournament

selection).

The recombination performed is based on the fusion operator proposed by

[Beasley and Chu, 1996]. The fusion operator produces a single child and takes

both the structure of the parents and their fitness into account. The basic idea

is to copy an assignment (gene) to the child if both parents assign the trip to the

same depot. If the gene values are different in both parents, it is more likely to

inherit the gene from the parent with the better fitness. We compare two parents

θ1and θ2and apply the following rules to obtain child θ0:

θ1[i] = θ2[i] then θ0[i] := θ1[i] = θ2[i] (4.4)

θ1[i]6=θ2[i] then θ0[i] := (θ1[i] with prob. p:= fθ2

fθ1+fθ2

θ2[i] with prob. 1 −p. (4.5)

Our computational test showed that the quality of the final solution is very

131

4. A Hybrid Evolutionary Algorithm

sensitive to the mutation rate. In particular, a high mutation rate almost always

led to a bad solution quality. Therefore, we use mutation only to eliminate

duplicate individuals.

The evaluation of the fitness of an individual is the most time consuming step

in our algorithm. Thus, we propose a tabu list in order to store individuals that

have been evaluated before. If we generate a child in the recombination phase

that has been constructed before, we randomly shift trips between non-empty

depots until a new individual is generated. Tests showed that in most cases 2

trip shift suffice to create a new individual.

4.2.4. Termination

We have two different termination criteria. The most obvious is to terminate

when a given time limit is exceeded. Furthermore, we terminate whenever there

has been no significant improvement of the fitness of the currently best individual

within the last 3|T | iterations.

4.3. Computational Results

In this section, we report computational results for the hybrid evolutionary algo-

rithm described in this chapter. All experiments in this section were conducted

on a Dell OptiPlex GX620 personal computer running Windows XP with an In-

tel Pentium IV 3.4 GHz processor and 2 GB of main memory. Our algorithm is

implemented in C# and has been compiled using the .NET framework version

2.0.50727.

We use the randomly generated instances and settings that have also been

used and described in the preceding chapter. In order to ease the exposition

we recall the basic properties. The instances available at [Huisman, 2003] have

been classified into two classes according to the travel speed where the speed

is lower for problems in class B. As a consequence, trips in class B are longer.

However, we will only report computational results for class A that all involve 4

depots and groups with ntrips where n= 80,100,160 and 200. For each group

10 instances are available. In accordance with [Huisman, 2004, Huisman et al.,

2005a] we consider five different types of duties: one tripper type with one piece

of work between 30 minutes and 5 hours, and four types consisting of two pieces

of work. A detailed description of the duty types is given in Section 3.5.

Furthermore, we used the following parameter settings. In all tests we set the

population size to 20 and the number of children per iteration to 2. We also

tested other settings, but the configuration above appears to be very insensitive

132

4.3. Computational Results

to the problem size. The CPU time limit is set to 30 minutes for instances with

80 to 100 trips, to 60 minutes for 160 trips, and 120 minutes for 200 trips. In

the following, all results given for our evolutionary algorithms correspond to the

average of five runs.

First, we compare our EA methods with the traditional sequential approach to

vehicle and crew scheduling. However, we do not use the fully integrated evalua-

tion in the first d|T |/3eiterations of the EA. Consequently, we do not require an

integrated solution method (such as [Huisman et al., 2005a]). We rather apply a

method similar to the traditional sequential approach to compute the fitness of

the individuals. Table 4.1 reports the average solution values for each problem

class for the sequential approach (vehicle first - crew second), the EA with the

sequential fitness calculation (EA-S), and the EA with partially integrated fitness

calculation (EA-PI ). For each solution approach the number of vehicles, drivers,

and the sum of vehicles and drivers (v+d) is given. Furthermore, we present the

relative deviation of the EA solutions from the sequential approach.

#trips

Method 80 100 160 200

Seq.

vehicles 9.2 11.0 14.8 18.4

drivers 25.8 29.9 38.8 47.1

v+d 35.0 40.9 53.6 65.5

EA-S

vehicles 9.3 +1.0% 11.3 +2.7% 15.2 +2.7% 18.6 +1.1%

drivers 23.2 -10.0% 27.6 -7.6% 35.5 -8.5% 42.5 -9.8%

v+d 32.5 -7.1% 38.9 -4.8% 50.7 -5.4% 61.1 -6.7%

EA-PI

vehicles 9.5 +3.2% 11.5 +4.5% 15.3 +3.4% 18.6 +1.1%

drivers 23.2 -10.0% 27.4 -8.4% 36.8 -5.1% 42.5 -9.8%

v+d 32.7 -6.6% 38.9 -4.9% 52.1 -2.8% 61.1 -6.7%

Table 4.1.: Comparison of sequential vehicle and crew scheduling and evolution-

ary algorithms on Huisman data instances type A

It is easy to see that both evolutionary approaches significantly reduce the

number of duties between 5.1% and 10.0%. Although the number of vehicles is

slightly higher in EA-S and EA-PI, the total number of vehicles and drivers can

be decreased between 2.8% and 7.1%. Furthermore, crew costs generally domi-

nate vehicle costs in practice (see [Bodin et al., 1983]) and, thus, the total savings

will be even higher than the numbers indicate. Finally, EA-S appears to perform

133

4. A Hybrid Evolutionary Algorithm

better than EA-PI. One reason may be that the number of evaluated individuals

is much higher in EA-S. To sum up, the EA approaches use an essentially se-

quential approach for fitness evaluation, but outperform a stand-alone sequential

approach.

Next, we compare EA-S with fully integrated approaches from literature. In

Table 4.2 we report the average results of the evolutionary algorithm EA-S* where

we use the fully integrated evaluation in the first d|T |/3eiterations, [Huisman,

2004,Huisman et al., 2005a] (HFW05), and [Bornd¨orfer et al., 2004] (BLW04).

We do not consider the results of [Mesquita and Paias, 2006] here since they use

different assumptions (see Section 2.2.3). We give the total number of vehicles

and drivers (v+d), the average computation time (cpu tot), and, for the EA, the

average standard deviation (avgsdev) of the sum of vehicles and drivers. Notice

that CPU times cannot be directly compared.

#trips

80 100 160 200

EA-S* v+d 32.1 37.4 49.2 60.8

avgsdev 0.38 0.36 0.35 0.36

cpu tot11,800 1,800 3,600 3,600

ICOPT v+d 28.3 33.7 46.8 57.2

cpu tot2235 369 1,740 2,700

BLW04 v+d 29.6 35.7 47.7 59.0

cpu tot3780 1,260 2,640 6,360

HFW05 v+d 29.6 36.2 49.5 60.4

cpu tot4– – – –

1,2on Dell OptiPlex GX620 with Intel Pentium IV 3.4 GHz/2GB

3on Dell Precision 650 PC with Intel Dual Xeon 3.0 GHz/4GB

4CPU times not comparable

Table 4.2.: Comparison of evolutionary algorithm EA-S* with other approaches

from literature on Huisman data instances type A

It can be seen that our algorithm performs worse than the best known fully

integrated algorithm ICOPT, but the solution quality of EA-S* increases with the

problem size. Furthermore, we can conclude that EA-S* is competitive with the

integrated approach of HFW05 for instances with 160 and 200 trips, respectively.

In particular, EA-S* was able to find better solutions than HFW05 for the 160

trips class.

134

4.4. Summary

We have suggested a hybrid evolutionary algorithm to tackle multiple-depot in-

tegrated vehicle and crew scheduling problems in public transport. The evo-

lutionary algorithm uses Lagrangian heuristics based on column generation to

compute the fitness of the individuals. The algorithm is novel since an evolu-

tionary algorithm has not been applied to integrated vehicle and crew scheduling

problems before. The algorithm is based on a problem decomposition that first

assigns trips to depots and, thus, reduces the multiple-depot integrated problem

to several integrated problems with a single depot. Unlike the multiple-depot

case the single depot case has a vehicle scheduling subproblem that can be solved

in polynomial time.

The results reported in the previous section indicate that medium-sized prob-

lem instances with multiple depots can be solved by using the proposed evolution-

ary algorithm. Furthermore, our approach discloses significant savings compared

to the traditional sequential approach without requiring a fully integrated solution

method. Although our algorithm performs worse than the best known integrated

algorithm, it proved to be competitive with other integrated approaches from

literature especially for medium-sized instances.

In addition to partitioning trips among the depots, trips must be assigned to

vehicle blocks and crew duties. A current limitation of our approach is that we

do not take this assignment into account. Further research will focus on how to

partition the trips assigned to a depot among vehicles and drivers with a local

search heuristic.

135

4. A Hybrid Evolutionary Algorithm

136

5. Practical Extensions

In the preceding chapters we considered rules and regulations for duty genera-

tion that were relatively simple. We had at most two pieces of work in a duty

with a break in between. Furthermore, the break had to be taken outside of

a vehicle and, thus, there was a changeover after each break. A piece of work

was only restricted by its duration. Finally, the following constraints were im-

posed: maximum working time, maximum spread time (duty length), minimum

start time, maximum end time, minimum break length, and minimum/maximum

number of pieces. In this chapter, however, we will extend and change some of

the assumptions in order to apply our approach on practical scenarios arising in

Germany. However, regulations differ from company to company, and it is almost

impossible to list all rules that may occur. Therefore, we will consider only those

regulations that are either based on federal regulations or are widely used. In

particular, we will consider two different types of break rules: block rules and

ratio rules. Moreover, a piece of work is not only restricted by its duration, but

also by its driving time. We use an external black-box verifier that discards all

duties which do not comply with the complete set of requirements for a company.

Our approach of Chapter 3 and the modifications that we propose in this chapter

are being integrated in the commercial software tool interplanrof the PTV AG

(see [PTV AG, 2007]).

This chapter is organized as follows. In Section 5.1 we give a description of rules

and regulations arising in Germany. We discuss the extensions and modifications

of our modeling and solution approach to cover these rules in Section 5.2. In

Section 5.3 we give an overview of how our implementation is being integrated

in the commercial software tool interplanr. We conclude this chapter (Section

5.4) with computational results and a case study on the local public transport

company in Paderborn (see [PaderSprinter, 2007]).

5.1. Rules and Regulations in Germany

In this section, we will describe common rules and regulations arising in Germany.

The basic regulations are defined in the federal regulations for drivers of trucks

137

5. Practical Extensions

and buses (in German: Fahrpersonalverordnung (FPersV ), see [Bundesminis-

terium f¨ur Verkehr, Bau- und Wohnungswesen, 2005]). In the following, we will

introduce new definitions and redefine those that are different to the definitions

in Chapter 3.

Layover time Layover time is the time a driver is waiting with a vehicle between

two service trips. The driver does not steer the vehicle.

Driving time The driving time includes at least all activities of a driver where

the driver is scheduled to steer a vehicle. These activities include but are

not limited to service trips, deadheads, and turnarounds. However, the

term driving time is not clearly defined in FPersV.

Continuous driving time A sequence of activities is counted as continuous driv-

ing time if there is no layover time of a specified minimum length therein.

A layover that is longer than the specified minimum length is called long

layover.

Working time The working time involves at least all activities of a driver that

the driver spends on the vehicle either driving or waiting. However, only

waiting tasks up to a given length are counted.

Each duty must satisfy at least one break rule if the driving time of that duty

exceeds 270 minutes. There are two groups of break rules: block and ratio break

rules. It is often subject to in-house agreements which rules may actually be

applied.

Block break rules A duty satisfies a block break rule if a duty either contains 1

break of at least 30 minutes, 2 breaks of at least 20 minutes, or 3 breaks

of at least 15 minutes. In the following, we denote these rules by block30,

block20, and block15, respectively.

Ratio break rules A duty satisfies a ratio break rule if the total layover time of

a duty amounts to at least 1

6(1

5) of the total driving time where layovers

of less than 10 (8) minutes do not count. In the following, we denote these

rules by ratio6 and ratio5, respectively.

A duty is said to be connected if it complies with a block or ratio break rule and

the longest break/layover does not exceed a given threshold. If the threshold is

exceeded, we have a split duty that consist of two parts: one in the early morning

and another in the late afternoon with a long break in the middle. If the total

driving time of such a duty part exceeds 270 minutes, this part must satisfy a

138

5.2. Extensions of Modeling and Solution Approach

break rule. If each part has more than 270 minutes of driving time, the parts

may satisfy two different break rules. In addition to the break rules we consider

the following constraints for the construction of a feasible duty where the actual

parameter values may differ by duty type.

•minimum/maximum working time

•minimum/maximum spread time (duty length)

•maximum continuous driving time

•maximum total driving time

•earliest/latest start time of duty

•earliest/latest end time of duty

•minimum/maximum number of pieces of work

•minimum working time before first break

•minimum working time after last break

•sign-on/off time inside/outside of depot

In practice, not only the construction of a single duty is constrained but also

specific requirements for groups of duties must be met. In particular, companies

often limit the (minimum/maximum) number or ratio of duties of a particular

type in the final crew scheduling solution. For instance, the ratio of split duties is

often restricted or the average working time of duties is limited. Rules concerning

groups of duties or the crew schedule as a whole are called global constraints while

regulations for a single duty are said to be local constraints.

5.2. Extensions of Modeling and Solution Approach

In this section, we will discuss the extensions and modifications of our modeling

and solution approach to cover the rules and regulations described in the preced-

ing section. We will first describe modifications due to local constraints and then

discuss our extensions for global constraints. Recall that the following constraints

have already been considered in the preceding chapter: minimum/maximum

working time, minimum/maximum spread time (duty length), earliest/latest

start time of duty, earliest/latest end time of duty, minimum/maximum num-

ber of pieces of work, and sign-on/off time inside/outside of depot.

139

5. Practical Extensions

5.2.1. Driving Time Constraints

There are two types of driving time constraints: maximum total and maximum

continuous driving time. Recall that we apply a two-phase pricing scheme (see

Section 2.4.2) where we set up a piece generation network in the first phase to

generate a set of pieces of work. These pieces serve as input for the second phase

where duties are generated. In Section 3.1.2, we discussed three network formu-

lations for the duty generation phase where the maximum total driving can easily

be covered by introducing another resource (total driving time). Furthermore,

we compute the total driving time of each piece generated in the first phase. All

arcs associated with a piece of work consume the driving time of the correspond-

ing piece. However, notice that the introduction of driving time destroys our

assumption that a piece of work is only restricted by its duration. Clearly, not all

paths between two relief points necessarily consume the same amount of driving

time since a path may contain waiting activities. In fact, it no longer suffices

to compute the least reduced cost piece of work for each pair of relief points in

order to prove column generation optimality. Instead, we must compute the set

of non-dominated (see Section 3.2.1) pieces of work where both driving time and

reduced cost are considered. The set of non-dominated pieces can be generated

using a simple dynamic programming algorithm.

In so far as the maximum continuous driving time is concerned, the piece

generation phase needs further modifications. In the preceding chapter, we used

the piece generation network ¯

Gd= ( ¯

Nd,¯

Ad) for each depot d∈ D as defined

in Section 2.4.2 where nodes ¯

Ndcorrespond to relief points. For a given node

ni∈¯

Nd, we computed the shortest path from node nito each node nj∈¯

Ndthat

satisfied the duration constraint. In particular, δmin ≤tnj−tni≤δmax holds with

δmin (δmax) as the minimum (maximum) duration of a piece of work and tnias

the time of node ni. Thus, the set of (destination) nodes ¯

Nf(ni)⊂¯

Nthat we

consider for node niread

f(ni) = {nj∈¯

Nd|δmin ≤tnj−tni≤δmax}.(5.1)

As described in the preceding section, a piece of work is not restricted by its

duration, it is rather limited by the maximum continuous driving time. As a

consequence, we have to modify the definition of set ¯

f(ni). Before we start with

column generation, we compute the shortest path from node nito all nodes nj

with tnj> tni. The costs on the arcs are defined in such a way that the costs equal

the driving times of the corresponding activity. Finally, the set ¯

f(ni) consists

of all nodes njwhere the length of shortest path is less or equal to the maximum

continuous driving time. For column generation, we redefine the costs on the arcs

140

5.2. Extensions of Modeling and Solution Approach

such that they correspond to the reduced cost. Furthermore, it does not suffice

to compute the shortest paths between node ni∈¯

Ndand nodes ¯

f(ni) in a

column generation iteration since the maximum continuous driving time may be

violated. Thus, the construction of pieces relies on a k-shortest-path enumeration

where we only accept non-dominated paths that satisfy the maximum continuous

driving time. The total number of pieces can be prohibitive for duty generation

since there can be multiple paths between each pair of relief points (instead of

a single one). However, the number of pieces can be reduced by applying state

space reduction (see Section 3.2.3). A simple heuristic is to construct only a

single piece for each pair of relief points: the first feasible piece returned by the

k-shortest-path algorithm.

Clearly, not all activities in the piece generation network correspond to activi-

ties where the crew actually drives the vehicle. As a consequence, the set ¯

f(ni)

may contain more destination nodes than necessary since the shortest paths can

contain long layovers that reset the maximum continuous driving time. How-

ever, there may be other paths without long layovers between these nodes that

constitute the shortest path in a subsequent column generation iteration.

5.2.2. Block and Ratio Break Rules

The purpose of this section is to describe the modifications necessary to cover

block and ratio break rules. We will first discuss changes due to block break rules

and concentrate on ratio break rules afterwards. Clearly, the modifications only

involve the duty generation network. Thus, we will discuss the changes of the

three network formulations for decomposed pricing described in Section 3.1.2,

namely the connection-based, time-space, and aggregated time-space model.

Recall that a duty satisfies a block break rule if a duty either contains 1 break

of at least 30 minutes (block30), 2 breaks of at least 20 minutes (block20), or

3 breaks of at least 15 minutes (block15). Furthermore, drivers must take their

breaks at given break locations. We assume that a transfer matrix is given where

the walking time from all start/end locations of service trips to all break locations

is defined. For all three network models, we modify the definition of break arcs.

In particular, we allow only those break arcs where the minimum break length

(30, 20, or 15 minutes) combined with walking to/from the break location is

satisfied. Furthermore, we introduce a new type of arcs that connects two pieces

of work: layover arcs. In a connection-based network, layover arcs originate and

terminate at a piece node while they connect a piece end with a piece start node

in a time-space network. As opposed to break arcs, layover arcs must correspond

to a long layover (see Section 5.1), but must not be longer than the minimum

141

5. Practical Extensions

break length. Additionally, traversing a layover arc does not increase the number

of pieces of work. In this context, the resource that counts the number of pieces

is used to count the number of pieces that succeed a valid break. For all network

models, the minimum number of pieces is 2 for block30, 3 for block20, and 4 for

block15.

For ratio break rules, we must track both driving and layover time. Recall that

the total layover time of a duty must amount to at least 1

6(1

5) of the total driving

time where layovers of less than 10 (8) minutes do not count. Thus, we introduce

two additional resources: total layover and total driving time. Notice that we

have already defined a resource to count the total driving time if the maximum

total driving time is restricted. The pricing algorithm must check the ratio of

layover to driving time at the sink node.

5.2.3. Break Positions

In this section, we consider the situation where a minimum working time $b

before the first and a minimum working time $aafter the last break is imposed.

In the following, we will describe the modifications when a decomposed pricing

scheme is used. As stated in the preceding section, the modifications only involve

the duty generation network. Thus, we will again discuss the changes of the three

network formulations.

Recall that a duty starts (ends) with a sign-on (sign-off) activity. In the

connection-based model, we have sign-on (sign-off) arcs from the source to piece

nodes (from piece nodes to the sink). Let Pddenote the set of pieces of work of

depot d∈ D. Furthermore, recall that the set of pieces corresponds to the set

of nodes in the duty generation network. We compute the working time ˇwd

iof a

piece of work i∈Pdbefore the driver takes his first break (block break rules) or

has a long layover (ratio break rules). Similarly, we calculate the working time

ˆwd

iof a piece i∈Pdafter the driver took his last break or had the last long

layover, respectively. A piece of work node i∈Pd, d ∈ D may only be connected

with the sink if ˆwd

i≥$aholds while it may be connected with the source only if

ˇwd

i≥$bis satisfied. Clearly, the network complexity is still O(ν4).

For the (aggregated) time-space model, the network formulation is modified

in a different way. Recall that multiple piece of work arcs originate from a

piece start node and terminate at a piece end node. As a consequence, we

cannot simply modify the definition of sign on/sign off arcs as we have done

for the connection-based model. Instead, we modify the (aggregated) time-space

model in the following way. We introduce a duty start (duty end) node for each

piece start (piece end) node. Furthermore, sign on arcs originate from source

142

5.2. Extensions of Modeling and Solution Approach

and terminate at a duty start node while sign off arcs emanate from a duty end

node and end at sink. In addition to the piece arcs that connect piece start with

piece end nodes, we introduce further piece arcs. If ˇwd

i≥$bis satisfied for piece

i∈Pd, there is piece arc from the corresponding duty start to the piece end

node. Likewise, we introduce another piece arc for piece i∈Pdfrom the associ-

ated piece start node to the duty end node if ˆwd

i≥$aholds. Figure 5.1 depicts

a time-space duty generation network with 5 pieces of work where a minimum

working time before the first and after the last break is imposed. In our example,

the first piece can only be at the beginning of a duty while the next two can

be both at the beginning and end of a duty. The last piece must not be at the

beginning of a duty. Despite the additional piece arcs, the network complexity

of both models remains O(ν2).

piece of work start

piece of work

source

sink

station A

station B

station C

depot

departure

station time break

sign-on

sign-off

piece of work end

break combined

with walking

piece of work

duty start

duty end

*multiple arcs

****

Figure 5.1.: Modified time-space duty generation network to consider specific

break positions

5.2.4. Duty Mix

As stated earlier global constraints deal with groups of duties at once while local

constraints define the feasibility of a single duty. In this section, we discuss duty

type constraints that belong to global constraints. In particular, we limit the

(minimum/maximum) number or ratio of duties of a particular type in the final

crew scheduling solution (duty mix).

We extend model MDVCSP (see Section 2.3) to impose a maximum number

143

5. Practical Extensions

max of duties of a specific type t∈Tfor each depot d∈ D as follows.

k∈Kd

kxd

k≤td

max ∀d∈ D (5.2)

kis equal to 1 if duty kis of type tand 0 otherwise. Notice that the same

constraints can be used to restrict the ratio νtof duties of type t∈ T in the final

solution. We set td

k= 1 −νtif duty k∈Kdhas type tand td

k=−νtotherwise.

Furthermore, we define td

max = 0,∀d∈ D.

As described in Section 2.4.1 we use Lagrangian relaxation in combination with

column generation to compute a lower bound. Therefore, we associate Lagrangian

multipliers δd≤0,∀d∈ D with constraints (5.2) and dualize the constraints. The

Lagrangian subproblem now reads

Φm(µ, π, δ)=Φy(µ, π)+Φm

x(µ, δ) + X

t∈T

πt+X

d∈D

δdtd

max.(5.3)

The vehicle scheduling subproblem remains unchanged (see Equation (2.20))

while the crew scheduling subproblem now involves the multipliers associated

with the duty mix constraints:

Φm

x(µ, δ) = min X

d∈D X

k∈Kd

k¯

k|(5.4)

k∈ {0,1},∀d∈ D,∀k∈Kd.

The reduced cost of duty k∈Kdis denoted by

k=fd

k−δdtd

max −X

(i,j)∈e

Ad(k)

µd

ij (5.5)

where e

Ad(k)⊆e

Adcorresponds to the set of arcs that is covered by duty k∈Kd.

When the sequential approach is used (see Sections 2.4.3 and 3.3.1) to compute

feasible solutions we only relax constraints (2.14) in a Lagrangian way. Thus, the

solution of the vehicle scheduling subproblem gives a feasible vehicle schedule.

Each feasible vehicle schedule can be used to construct a feasible crew sched-

ule using sequential crew scheduling for each depot. However, the duty mix

constraints must be considered in the crew scheduling problem. In our imple-

mentation, we use Lagrangian relaxation in combination with column generation

to generate a promising set of columns. Finally, we use the branch-and-bound

implementation of ILOG CPLEX to find integer solutions.

Clearly, equation (5.2) can be adjusted if the duty mix is restricted for all

depots at once. Notice that this requires to perform sequential crew scheduling

in the integer phase for all depots at once.

144

5.3. System Overview

Even though we can handle most duty construction rules directly, some regula-

tions are too complex to be covered efficiently. As a consequence, we ignore the

rule in our pricing algorithm and check duties with an external black-box verifier.

The verifier either accepts or rejects a duty, but does not expose details of the

evaluation process to the rest of the system. In case that the duty is accepted

the verifier also computes accurate planned (operational) cost. Accurate cost

calculation can be a very challenging task since it is often subject to in-house

regulations and may strongly differ from company to company. The duty costs

in our pricing algorithm are defined in such a way that we never overestimate the

(operational) cost of a duty. If we would overestimate the cost of a duty in our

pricing algorithm, the duty might be discarded by our pricing scheme although

the duty is valuable (has negative reduced cost). However, the planned cost re-

turned by the verifier can turn a negative reduced cost duty into a non-negative

reduced cost duty. In such a case, we discard the duty. Similar approaches are

used by [Bornd¨orfer et al., 2006] and [Galia and Hjorring, 2004] for airline crew

scheduling. However, the authors use a rule verification oracle where only pairing

(duty) feasibility is checked.

Furthermore, we allow the verifier to propose duties for addition to the re-

stricted master problem (RMP). More precisely, the verifier is regularly called to

propose duties that are added to a column pool. For example, we can add feasible

columns that are part of the current solution from practice or similar to duties

of the current solution. Basically, a column pool contains a set of feasible duties

that are explicitly kept in computer memory, but that are not part of the current

restricted master problem. Recall that in the original version of our pricing algo-

rithm, columns are only implicitly available unless they are added to the RMP.

Columns in the column pool can easily be priced in subsequent iterations and are

added to the RMP if they have negative reduced cost. Clearly, columns added to

the RMP are deleted from the pool. Furthermore, we add columns to the pool

that were deleted from the RMP (due to high positive reduced cost). To sum up,

we use a combination of implicit and explicit column generation.

The system described in this section is being integrated in the commercial

software tool interplanrof the PTV AG (see [PTV AG, 2007]). Figure 5.2 depicts

an overview of the system with our optimization system ICOPT. In addition

to the interaction described above, our optimization system transfers feasible

solution(s) to interplan. Interplan displays the solution(s) on a graphical user

interface. In this context, the planner can decide whether he would like to accept

a solution as final solution or whether further optimization is required. In other

145

5. Practical Extensions

ICOPT

lower bound phase

Computes lower bound

using column generation in

combination with

Lagrangian relaxation

Combines implicit and

explicit column generation

column pool

black-box verifier

Checks feasibility of

duties

Computes accurate

(operational) cost of

duties

generator

Proposes user-defined

duties

integer phase

Computes feasible

solution(s)

Transfers solution(s) to ptv

interplan

graphical user interface

Displays solution(s) found

by ICOPT

Figure 5.2.: Integration of ICOPT with PTV interplanr

146

5.4. Computational Results

words, the purpose of ICOPT is not to provide a single final solution, it should

rather generate several solutions. As a consequence, the planner can select that

solution among the set of generated solutions that meets his requirements in

terms of costs and other soft (social) factors.

5.4. Computational Results

The purpose of this section is to summarize our computational results on real-

world problem data from Connexxion and randomly generated instances. All

experiments in this section were conducted on a Dell OptiPlex GX620 personal

computer running Windows XP with an Intel Pentium IV 3.4 GHz processor

and 2 GB of main memory. Our integrated method ICOPT is implemented in

C# and has been compiled using the .NET framework version 2.0.50727. For a

description of the problem instances we refer to Section 3.5.

We use the same parameter settings and cost function as described in Section

3.5. Recall that we assigned a fixed cost of 1,000 for each vehicle and duty and

a small variable cost of 1 for each minute a vehicle is outside the depot and 0.1

for each minute a crew is working. However, we consider a different set of duty

types and work regulations. More precisely, we consider three block break rules

with at least two pieces of work and one tripper type with a single piece. The

tripper duty type does not require a break, but the total driving time is limited to

04:30 hours. Table 5.1 summarizes the duty types that require at least one break.

All duty types arise from federal regulations in Germany. Notice the difference

between total and continuous driving time: a sequence of vehicle activities is

counted as continuous driving time if there is no long layover therein while the

total driving time is additive over the whole duty. In other words, the continuous

driving time is reset to zero if the driver has a long layover.

In Table 5.2 we give computational results of our integrated approach on Con-

nexxion data instances. For each of the 8 instances we report the number of

trips and the number of depots where each trip can be serviced from every depot.

We give the number of iterations (#iter), the CPU time in seconds spent on

the master (cpu ma), the pricing problem (cpu pr), the integer phase (cpu ip),

and the total time including all initializations (cpu tot). Moreover, we report

the final upper bound (ub), the number of vehicles (vehicles), drivers (drivers),

total number of vehicles and drivers for integrated (v+d) and sequential vehicle

and crew scheduling (v+d seq.). Finally, we give the total number of duties and

drivers that we obtained with the comparatively easy duty types in Section 3.5.1

(v+d ref.).

147

5. Practical Extensions

block30 block20 block15 split

Type min max min max min max min max

Break length 00:30 01:00 00:20 01:00 00:15 01:00 01:00 –

No. of breaks 1 – 2 – 3 – 1 –

Driving time – 09:30 – 09:30 – 09:30 – 09:30

Driving time (continuous) – 04:30 – 04:30 – 04:30 – 04:30

Long layover time 00:10 – 00:10 – 00:10 – 00:10 –

Duty length 07:00 12:00 07:00 12:00 07:00 12:00 07:00 14:00

Working time – 09:00 – 09:00 – 09:00 – 09:00

Sign-on 00:08 00:08 00:08 00:08 00:08 00:08 00:08 00:08

Sign-off 00:06 00:06 00:06 00:06 00:06 00:06 00:06 00:06

Sign-on after break – – – – – – 00:05 00:05

Sign-off before break – – – – – – 00:05 00:05

Table 5.1.: Properties of different duty types

148

5.5. Summary

instance

1 2 3 4 5 6 7 8

#trips 194 210 220 237 304 386 451 653

#depots 4 4 4 4 4 4 4 4

#iter 26 23 31 24 25 35 28 31

cpu ma 794 641 1,003 729 1,138 2,233 2,605 4,676

cpu pr 2,260 584 1,750 567 1,261 10,383 8,661 16,521

cpu ip 10,097 1,300 1,537 4,656 8,377 5,292 5,982 20,107

cpu tot 13,163 2,535 4,306 5,959 10,792 18,346 17,719 41,954

ub 48,834 82,824 65,610 87,640 105,976 86,397 120,898 183,017

vehicles 19 33 28 34 40 32 47 67

drivers 26 43 33 46 57 48 65 103

v+d 45 76 61 80 97 80 112 170

v+d seq. 49 85 70 91 106 85 120 175

v+d ref. 45 81 60 86 102 85 109 174

Table 5.2.: Results on Connexxion data instances with practical extensions

As can be seen from Table 5.2 we could solve problem instances with up to 653

trips and four depots using rather complex duty types. Furthermore, the results

show that the total number of vehicles and drivers can be significantly reduced

when an integrated approach is used. As expected the time spent on the pricing

problem increases compared to the relatively easy duty types in Chapter 3 (see

Table 3.16). Except for problem 7 the number of duties is equal or smaller than

the reference where the easy duty types were used. Thus, we conclude that our

method in combination with the modifications described in this chapter can also

efficiently cover complex duty types. However, a thorough investigation would

require to compute valid lower bounds for both duty sets and compare these

bounds with the best corresponding upper bounds.

5.5. Summary

In this chapter we considered practical rules and regulations arising in public

transport companies in Germany. We suggested extensions and modifications

of our modeling and solution approach from Chapter 3 to cover these practical

extensions. The enhancements included driving time constraints, complex break

rules where many pieces of work are allowed, break positions, and duty mix con-

straints. Furthermore, we gave an overview of how our implementation is being

integrated in the commercial software tool interplanr. We tested the applicabil-

149

5. Practical Extensions

ity of the proposed techniques using real-life data instances with up to 653 trips

and four depots. The results indicate that our approach can efficiently cover duty

types with many pieces of work and complex feasibility rules.

150

6. Ex-Urban Vehicle and Crew

Scheduling with Irregular

Timetables

In the preceding chapters, we focused on how to conduct operations of a given

timetable at minimum cost. In this chapter, however, we will consider another

aspect which is related to the quality of crew schedules.

We will discuss the case where timetables consist of many trips serviced every

day together with some exceptions that do not repeat daily. In particular, service

trips to schools, production facilities, or public swimming baths are often subject

to change, e.g. trips may be operated on every day except Sunday or on Mon-

day only. Unless specifically imposed, traditional methods for vehicle and crew

scheduling usually produce schedules that contain irregularities which are not de-

sired in practice. A crew schedule is called irregular if it cannot be repeated many

times. Similar to airline crew scheduling (see [Klabjan et al., 2001]), regularity is

an important aspect for crew schedules in public transport since regular solutions

can improve operational reliability and can reduce training costs (see [Dallaire

et al., 2004]). Furthermore, regular solutions are less error-prone and crews often

prefer to repeat itineraries. In current practice, companies often try to increase

regularity of crew scheduling solutions by one of the following heuristic two-phase

procedures:

•All first - irregular second: First, the planner solves a crew scheduling

problem for a particular period with both regular and irregular trips. In a

second step, he or she fixes the subset of crew duties that can be operated

the whole period and reoptimizes all unfixed trips. Notice that the second

problem can also contain some regular trips.

•Regular first - irregular second: The set of service trips is divided into

regular and irregular trips. First, a crew scheduling problem for the set

of regular trips is solved while the irregular trips are left for subsequent

optimization.

151

6. Ex-Urban Vehicle and Crew Scheduling with Irregular Timetables

In both cases, the second problem has a sparse schedule and, thus, likely requires

extensive deadheading and even its optimal solution yields a high costs. On the

other hand, if the second problem contains many trips, the corresponding solution

has low cost but low regularity as well.

As stated earlier, we are concerned with the regularity of crew schedules and not

with the regularity of vehicle schedules. In fact, vehicles are rather insensitive

to the quality of their schedules as opposed to drivers. In order to test our

approaches, we will concentrate on scenarios where crew scheduling plays the

major role. This holds particularly for ex-urban scenarios as we will see in the

following section.

To the best of our knowledge, solution approaches to improve the regularity

of crew schedules in public bus transport have not been described in literature

before. In this chapter, we propose two approaches that capture both costs and

regularity of crew scheduling solutions. In particular, we propose a novel com-

bination of local branching and follow-on branching that improves the regularity

of crew schedules while cost optimality is maintained. Furthermore, we compare

four bi-objective metaheuristics that include both cost and regularity as objective

functions. The latter approach can be used to get a quick estimate of the solution

quality obtained with the first approach.

This chapter is organized as follows. In Section 6.1, we give a formal problem

definition for the ex-urban vehicle and crew scheduling problem with irregular

timetables. We discuss other approaches related to public (bus) transport from

literature in Section 6.2. In the next section, we describe how local branching

and user-defined branching rules can be used to steer the solution method to reg-

ular crew scheduling solutions. In the same section, we describe four bi-criteria

metaheuristics from literature that try to approximate the set of Pareto optimal

solutions. Finally, we provide computational results on real-world and randomly

generated instances in Section 6.5. The chapter is concluded with a short sum-

mary (Section 6.6).

6.1. Problem Definition

In this section, we will first describe the ex-urban vehicle and crew scheduling

problem and discuss regularity of crew schedules afterwards.

Public transport scenarios can be categorized according to the structure of the

underlying transportation network. Urban service provides connections within

the city while ex-urban (regional) service connects the city with the suburbs

and minor towns in the region of the city. Of course, many companies offer

152

6.1. Problem Definition

a mixture of both categories. Many regional scenarios have in common that

the line network is star-shaped around the depots with only few relief points.

Furthermore, distances between relief points are such that drivers are virtually

tied to their vehicle in order to reach the relief points. In other words, pieces

of work often correspond to vehicle blocks. When traditional vehicle and crew

scheduling (vehicles first - crew second) is applied in an ex-urban setting, vehicle

blocks are likely to be too long to meet break requirements, or drivers cannot

return to their home depot. Conclusively, crews must be scheduled at the same

time as vehicles or before vehicles in order to guarantee the feasibility of the crew

schedule. In the remainder of this chapter, we will assume that drivers may only

change their vehicles in depots (ex-urban scenario).

Crews can easily be scheduled before vehicles if there is a single depot and

vehicle changes outside the depot are not allowed (or drivers can walk from all

relief opportunities to the depot). In such a case, we first solve an independent

crew scheduling problem (ICSP) that we define as follows. Given the traveling

times between all pairs of locations and a set of tasks which corresponds to the set

of service trips1, find a minimum cost set of duties such that all tasks are covered

by feasible duties (see also [Huisman, 2004]). Since each duty starts and ends at

the depot, the vehicle rotations that result from the crew scheduling solution can

be put together to form a feasible vehicle schedule (using a vehicle scheduling

method). The approach to schedule crews before vehicles is also referred to as

partial integration (see [Bornd¨orfer et al., 2004]). However, the number of vehicles

is not necessarily minimal in contrast to a fully integrated approach (see Section

2.3). Notice that a feasible vehicle schedule can also be constructed when there

are multiple depots and duties that start and end at the same depot. If continuous

attendance is required (see Section 2.1), and a driver must not stay on his (idle)

vehicle during a break, each piece of work must start and end at the same depot.

As a result, drivers spend their breaks in a depot and take the same or a different

vehicle for the consecutive piece of work.

We will now formally define the vehicle and crew scheduling problem with

irregular timetables. Let Fbe a timetable with tasks f1, . . . , fnwhere task fi

starts earlier than fi+1. Furthermore, a reference crew schedule R={R1, . . . , Ru}

with duties Ri={fi1, . . . , fip}that is compatible to timetable Fis given. The

integrated vehicle and crew scheduling problem with irregular timetables (VCSP-

IT) for timetable F06=Fand given depots, relief points, and a reference crew

schedule Rcan be stated as follows: find minimum cost sets of vehicle blocks

and crew duties such that both vehicle and crew schedule are feasible and mu-

1In order to stay consistent with the terminology of the preceding chapters, we modify the

definition of a task: a task no longer needs to start and end with a relief point.

153

6. Ex-Urban Vehicle and Crew Scheduling with Irregular Timetables

tually compatible. Furthermore, crew schedule D={D1, . . . , Dv}should have a

small distance to reference schedule R. A crew schedule with a small distance to

reference Ris called similar or regular. However, minimizing costs remains the

primary objective.

The perception of distance between two crew schedules can differ from company

to company. A very simple distance measure is to count the number of duties

in the new crew schedule that could not be preserved from the reference crew

schedule. In the following, we will describe a more elaborate distance measure

that basically counts the number task sequences not preserved from the reference.

Let Q=F∩F0be the set of regular tasks that are part of both timetables. A

regular pair S⊆Qis an ordered pair of regular tasks (fi, fi+k) that are operated

consecutively in both reference Rand new crew schedule D. We denote by S1

the first task of regular pair Swhile S2corresponds to the second task. Notice

that an irregular trip may be operated between fiand fi+k, but no regular trip.

Clearly, a regular trip to cannot be at the first (second) position of more than

one regular pair. However, it may be at the first position in one pair and at

the second in another pair. Furthermore, a regular chain T= (S1, . . . , Sj) =

((S1

1, S2

1),...,(S1

j, S2

j)) with j≥1 and S2

i=S1

i+1,1≤i < j −1 is an ordered

sequence of interconnected regular pairs. e

Tdenotes the number of regular tasks

of regular chain T. Furthermore, let ¯

Sand ¯

Tdenote the set of all regular pairs

and chains, respectively. We define distance measure σp(σc) that corresponds to

the number of regular tasks that are not part of a regular pair (chain).

σp=|Q| − 2|¯

S|(6.1)

σc=|Q| − X

T∈¯

T(6.2)

Of course, there are numerous other distance measures possible. However, we be-

lieve that our measures give an intuitive approach to regularity of crew schedules.

Therefore, we will focus on σpand σcin the remainder of this chapter. However,

our approaches also work with other distance measures.

6.2. Literature Review

In this section, we review state-of-the-art models and solution methods for crew

scheduling with irregular timetables from both public transport (bus and railway)

and airline perspectives. Since we are concerned about the regularity of crew

schedules, we do not consider vehicle scheduling in our literature review. As we

will see, the literature on irregular timetables in public bus transport is virtually

non-existent. Therefore, we include railway and airline settings in our review.

154

6.2. Literature Review

Solution approaches can mainly be categorized into regularity and rescheduling

approaches. Regularity approaches build a solution from scratch for a given (long)

period where the solution should inherently contain as many regular patterns

as possible. In rescheduling methods, a reference schedule is given and a new

solution for a (short) period is constructed where the new solution should be as

similar as possible to the reference. In the following, we will review models and

solution methods based on both approaches.

Regularity Approaches

[Tajima and Misono, 1997] describe an airline crew scheduling problem with

many irregular flights. The authors seek to find a set of pairings (duties) that

cover all flights in the planning period (one month) where essentially the total

number of man-days is minimized. The number of man-days of a pairing is

equal to the number of days it lasts. The secondary objective is to minimize

costs. Furthermore, a large portion (between 9% and 54%) of all flights is not

flown on every day of the planning period. The authors propose a heuristic that

systematically merges irregular flights into pairings that only consist of regular

flights. Their computational tests involve two real-world data instances with

8,876 and 9,504 flights where the ratio of irregular flights was 54% and 9%,

respectively. Their experiments revealed that the instances could be solved in 41

and 92 minutes on an IBM RS/6000 model 900. Moreover, their method could

find better solutions than manual planning by experienced engineers. Although

the primary objective was to minimize the number of man-days, the approach

manages to produce regular crew schedules. For the first instance, 81% of the

pairings were regular while 92% of the pairings were flown every day for the

second one. However, the authors do not report the impact on operational costs

since regular pairings may contain a lot of (paid) waiting time.

[Klabjan et al., 2001] introduce the weekly airline crew scheduling model

with regularity. The model captures the trade-off between regularity and costs

in a weekly schedule. The set of flights is partitioned into groups in such a

way that regularity is easily obtainable in each group. A g-regular group for

g= 4,...,7 contains flights that can be repeated gconsecutive days of the week.

By definition, regular flights ifrom a g-regular group have gi≥g. Each g-

regular group is subsequently partitioned by g-regular pairings. All flights not

assigned to a g-regular group, g=4,. . . ,7, are called irregular flights and must be

assigned to irregular pairings. In their model, the authors assign penalty costs

to irregular flights. Penalty costs decrease with increasing regularity. However,

the complete regularity model is intractable and, thus, the authors resort to

155

6. Ex-Urban Vehicle and Crew Scheduling with Irregular Timetables

an approximate model and solution methodology. In particular, pairings are

produced in decreasing order of regularity. 7-regular pairings are produced first

and an appropriate subset is computed to form 7-regular pairings in the final

weekly solution. The flight schedule is reduced by all flights already covered by

7-regular pairings. In the next stage, the remaining flights can only be covered by

6-regular pairings. The process iterates until irregular pairings are generated and

the complete flight schedule is partitioned. Computational results with three real-

world data instances show that problems with at most 492 flights can be solved

in 47 hours computational time. The tests were performed on two clusters: one

consisting of 16 machines each with Quad Pentium Pro 200MHz/256 MB main

memory and the other comprised of 48 machines each with Dual Pentium II

300MHz/512 MB main memory. The solutions reported improve on existing

solutions used by the airline both in terms of regularity and costs.

Rescheduling Approaches

We distinguish between unplanned and planned rescheduling. Unplanned resche-

duling of crews is necessary when the planned crew schedule cannot be executed

due to irregular operations or disruptions. Planners usually aim to determine

new crew assignments that make as few changes to the original schedule as pos-

sible. In other words, planners like to find a new solution with a small distance

to the original (reference) solution. Unplanned crew rescheduling is also referred

to as crew recovery. Typically, the underlying flight schedule may be changed in

crew recovery problems, i.e., flights may be delayed or even canceled, if no fea-

sible recovery scheme is found in a given timeframe. Notice that the underlying

timetable must not be altered in the problem stated in the preceding section.

Furthermore, typical scenarios for crew recovery include local disruptions while

irregular trips are often spread over the complete timetable. In conclusion, solu-

tion approaches for crew recovery do not seem to be well suited for our problem

stated in Section 6.1. However, recent approaches to airline crew rescheduling

(recovery) include, among others, [Lettovsky et al., 2000], [Guo et al., 2005], [Nis-

sen and Haase, 2006], and [Medard and Sawhney, 2007]. A recent survey can be

found in [Clausen et al., 2005].

In planned crew rescheduling the changes in the underlying timetable are typi-

cally known in advance. [Huisman, 2007] describes the planned crew rescheduling

problem in a railway setting at NS which is the largest passenger railway opera-

tor in the Netherlands. At NS crew scheduling is performed in two stages. First,

solutions for an annual plan are constructed, i.e., for a general Monday, Tuesday,

and so on. In a second phase, the general days are adapted to individual days

156

6.3. Mathematical Formulation

where specific changes in the timetable for those days are considered. The author

states that the changes in the timetable are mainly due to track maintenance or

extra service trips that are both usually known in advance. He suggests a set

covering formulation where original duties are replaced by new (similar) duties

such that all tasks of the modified timetable are covered and total costs of the

new duties are minimized. He used a heuristic based on column generation in

combination with Lagrangian relaxation and an elaborate set covering heuristic

to compute integer solutions. The computational experiments involved two real-

world scenarios and were performed on personal computer with a Pentium IV 3.0

GHz processor/512 MB main memory. The instances with 5,683 and 7,740 tasks

had 355 (6.2%) and 827 (10.6%) expired tasks, respectively. For the first instance,

only 12.6% of the original duties needed modifications while the ratio increased

to 29.5% for the second instance. The author could solve the first instance in

approximately 9 hours and the second one in less than 16 hours.

The only approach for public bus transport we are aware of is described in

[Dallaire et al., 2004]. However, the authors do not provide any details on their

approach which is part of the commercial software package HASTUS/CrewOpt

(see [GIRO Inc., 2007]). They rather emphasize the practical importance of

generating efficient solutions that are similar to a reference crew schedule (when

the underlying timetable is changed).

6.3. Mathematical Formulation

In this section, we will give the mathematical formulation that we will use in

the remainder of this chapter. Recall that we assumed that drivers may only

change their vehicles in depots (ex-urban scenario). Therefore, we propose to

solve the independent crew scheduling problem (ICSP - see Section 6.1) first and,

then, put the vehicle rotations from the crew scheduling solution together such

that the vehicle schedule is feasible (see Section 2.1). In Section 6.4 we will seek

to improve the regularity of crew schedules for the independent crew scheduling

problem.

Let Tbe the set of tasks. Furthermore, we define Kas the set of all feasible

duties and K(t), t ∈ T as the set of duties that cover task t. The cost of duty

k∈Kis denoted by ck. Finally, decision variables xkindicate whether duty kis

selected in the solution or not. The ICSP can be formulated as set partitioning

157

6. Ex-Urban Vehicle and Crew Scheduling with Irregular Timetables

problem:

k∈K

ckxk→min (6.3)

s.t. X

k∈K(t)

xk= 1 ∀t∈ T ,(6.4)

xk∈ {0,1}.(6.5)

The objective (6.3) is to minimize the total costs of the selected duties, and

constraints (6.4) assure that each task will be covered by exactly one duty. When

the equality sign in constraints (6.4) is replaced by a greater or equal sign ”≥”, we

obtain a set covering formulation. Then, tasks may be assigned to more than one

driver where the additional drivers are passengers. The set covering formulation

is computationally more attractive than the set partitioning formulation (see

[Vanderbeck, 1994]). In the remainder of this chapter, we will consider a set

covering formulation.

6.4. Solution Approaches

The purpose of this section is to present two solution approaches that improve the

regularity of crew schedules compared to traditional crew scheduling. For both

approaches we use model (6.3)-(6.5) and apply a column generation algorithm in

combination with Lagrangian relaxation. We solve the corresponding Lagrangian

dual with a subgradient algorithm (see Section 1.5.1) to obtain approximate dual

values. The column generation pricing problem corresponds to a resource con-

strained shortest path problem (see Section 3.1) and is solved with a dynamic

programming algorithm (see Section 3.2).

The columns generated in the column generation phase serve as input to the

second phase where an appropriate integer solution is sought. In the following,

we suggest two methods for the second phase that take the trade-off between

costs and regularity into account. In particular, we propose a novel combination

of local branching and follow-on branching in Section 6.4.1 while we discuss four

bi-objective metaheuristics in Section 6.4.2.

6.4.1. Local Branching and Branching Rules

Our solution approach is based on the observation that (independent) crew

scheduling problems have thousands of optimal solutions. This is mainly due

to degeneracy.

158

6.4. Solution Approaches

In Table 6.1 we give the average number of optimal solutions for independent

crew scheduling problems with 80,100, and 160 trips (tasks). We used the same in-

stances and duty type definitions as in chapters 3 and 4. However, we enumerated

at most 2,500 different optimal solutions per instance with the branch-and-bound

implementation of ILOG CPLEX 9.1.3. The root node of the branch-and-bound

tree was solved with a column generation algorithm, i.e., we did not regenerate

columns during tree search. As we can see in Table 6.1, the average number of

different optimal solutions can be very high in independent crew scheduling prob-

lems. Furthermore, the number of optimal solutions increases if a mere 0.01%

deviation to the optimal solution value is allowed.

#trips #instances opt. tolerance

solved 0.00% 0.01%

80 10 1,052 1,115

100 9 723 945

160 9 1,807 2,046

Table 6.1.: Average number of optimal solutions for independent crew scheduling

on Huisman data instances type A

The basic idea of our solution method is to systematically search an optimal

solution among all optimal solutions that is as similar as possible to a given

reference solution. In particular, we use local branching cuts to select suitable

solution subspaces and explore these subspaces with an adapted version of follow-

on branching. The exposition in this section is partly based on [Steinzen et al.,

2007b].

Local Branching to Find Regular Crew Schedules

Local branching (see [Fischetti and Lodi, 2003]) is an exact solution method for

general mixed integer programs. The basic idea of local branching is to define

suitable solution subspaces that are efficiently explored with a generic MIP solver.

In other words, local branching cuts are added to strategically define subspaces

that are tactically explored with a black-box solver. The procedure can be viewed

as a two-level branching scheme that aims at finding good incumbent solutions

at early stages of the computation. The underlying assumption is that small

instances of a problem can be efficiently solved with a generic solver while large

instances cannot.

159

6. Ex-Urban Vehicle and Crew Scheduling with Irregular Timetables

Given a feasible start solution ¯x∈ {0,1}|K|of ICSP we define the Hamming

distance

∆(x, ¯x) = X

k∈L0

(1 −xk) + X

k∈K\L0

xk(6.6)

where L0={k∈K: ¯xk= 1}denotes the support of ¯x. The distance ∆(x, ¯x)

counts the number of variables in xthat flip their values with respect to ¯x(either

from 1 to 0 or from 0 to 1). For a given neighborhood parameter κ∈N+, the

solution space can be partitioned with local branching cuts:

∆(x, ¯x)≤κ(left branch),(6.7)

∆(x, ¯x)≥κ+ 1 (right branch).(6.8)

For an appropriate value κ, subspace ∆(x, ¯x)≤κcan be efficiently explored

with a generic MIP solver. If the subspace contains a new incumbent ¯x2, the

scheme is reapplied to the right branch where two new subspaces are constructed:

∆(x, ¯x2)≤κand ∆(x, ¯x2)≥κ+ 1. On the other hand, if subspace ∆(x, ¯x)≤κ

does not contain a new incumbent, the remaining (large) subspace ∆(x, ¯x)≥

κ+ 1 has to be explored with a MIP solver. Notice that the concept of local

branching is quite different to standard branching: the solution method first

explores promising solution subspaces instead of cutting fractional solutions. For

further details on local branching we refer to [Fischetti and Lodi, 2003].

For independent crew scheduling, we use a local branching scheme to first

explore regions of the solution space that contain solutions similar to a given

reference crew schedule R. Similar to equation (6.1) let σp

kbe the number of

tasks of duty kthat are not part of a regular pair. Then, we solve ICSP (possibly

to optimality) with a modified objective function to obtain a start solution ¯xas

basis for local branching. The start solution should be similar to the reference

crew schedule and should have sufficiently low costs. Therefore, we replace the

original cost ckof column kby ˆck=ck+ασp

kand define αin such a way that σp

dominates the modified cost. Finally, we restore the objective function and use

¯xto define the initial neighborhood for local branching.

We would like to mention that the choice of parameter αis crucial for the

performance of the solution procedure. If parameter αis too small, we get a

start solution with low costs and low similarity. As a consequence, it is difficult

to improve the similarity with local branching. On the other hand, if parameter

αis too large, the computational burden to find a minimum cost solution can be

very high. In our computational experiments we found that α∈[150,400] is a

robust parameter setting.

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6.4. Solution Approaches

Follow-On Branching to Find Regular Crew Schedules

In order to simplify the exposition, we will briefly recall the basic idea of follow-

on branching. Branching on follow-ons relies on a general branching strategy for

set partitioning problems that was introduced by [Ryan and Foster, 1981]. The

branching scheme is based on the following property. Given a fractional solution

to a set partitioning problem, we can identify two rows (tasks) ti∈ T and tj∈ T

such that the subset K(ti, tj) of columns that contain tiand tjhas the property

0<X

k∈K(ti,tj)

xk<1.(6.9)

The remaining fraction of cover for each constraint must be provided by columns

that do cover both rows at the same time. Thus, an effective constraint branch-

ing scheme is to require to cover two rows tiand tjby the same column on one

branch and by different columns on the other. [Vance et al., 1997a] slightly mod-

ify the scheme to maintain tractability. They only consider trips (tasks/rows)

tiand tjthat correspond to trips operated consecutively in a duty (column).

Furthermore, the authors show that this modification still constitutes a correct

branching scheme. We refer to this strategy as branching on follow-ons since

we impose which task can follow task tiin the solution. Moreover, we refer to

the task pair (ti, tj) as follow-on. Notice that each regular pair Si∈¯

Sis also a

follow-on. In the following, we will describe how follow-on branching is used to

construct regular crew schedules.

A regular crew schedule contains as many regular pairs and chains as possible.

We modify the follow-on branching scheme in such a way that an (cost) optimal

solution has a high regularity as well. In the following, we will propose three novel

adaptations of follow-on branching: branching on regular pairs (fo-r1), regular

chains (fo-r2), and pieces of work (fo-r3).

Similar to our definition in Section 3.3.2 we define the support of a regular pair

(ti, tj)∈¯

g(ti, tj) = X

k∈K(ti,tj)

xk.(6.10)

Since we aim at generating regular crew schedules we branch on a candidate

regular pair (ti, tj)∈¯

Swhere 0 < g(ti, tj)<1 is satisfied. Branching scheme

fo-r1 selects the regular pair with the best support among all regular pairs.

fo-r1 : (ti, tj) = arg max

(ti,tj)∈¯

Sg(ti, tj) (6.11)

However, if ¯

S=∅we choose the follow-on with ti, tj∈ T and max g(ti, tj).

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6. Ex-Urban Vehicle and Crew Scheduling with Irregular Timetables

Branching scheme fo-r2 does not rely on the support of single regular pairs, but

tries to fix regular chains of maximum length. Recall that ¯

Tis associated with

the set of regular chains. Furthermore, we associate K(Ti) with the set of duties

that cover regular chain Ti. The set of candidate regular chains ¯

Tccontains all

regular chains Ti∈¯

Twhere 0 < g(Ti)<1 with g(Ti) = Pk∈K(Ti)xkis satisfied.

Algorithm 11 depicts branching scheme fo-r2 where we try to branch on a regular

chain of maximum length if there are candidate chains.

Algorithm 11: Branching on regular chains (fo-r2)

Find candidates

Compute set of candidate regular chains ¯

Tc={Ti: 0 < g(Ti)<1}.

Branching

if ¯

Tc6=∅then

Branch on follow-on ti, tj∈ T with max g(ti, tj)

else

Initialize ¯

Tmax

c={Ti∈¯

Tc:|Ti|= maxTj∈¯

Tc|Tj|}

Branch on regular chain Ti∈¯

Tmax

cwith max g(Ti)

Notice that scheme fo-r2 corresponds to the latter scheme fo-r1 if the set of

candidate regular chains ¯

Tconly consists of chains of length two.

Finally, we propose branching scheme fo-r3 where we branch on a piece of work

whenever that piece of work forms a regular chain. If several pieces correspond to

candidate regular chains, we select the piece with the maximum number of tasks.

Algorithm 12 presents how branching on regular pieces of work is performed.

Local and Follow-On Branching to Find Regular Crew Schedules

Local branching and follow-on branching can be combined. In particular, we

embed follow-on schemes fo-r1 to fo-r3 into local branching to explore neighbor-

hoods ∆(x, ¯x)≤κ. We hope to explore neighborhoods ∆(x, ¯x)≤κin such a way

that (1) an new incumbent is found fast and (2) the new incumbent has a smaller

distance than other solutions in the neighborhood. If the reference solution is of

high quality, a valuable follow-on might be selected first and might reduce the

computational time to explore the neighborhood. To sum up, we strategically

define subspaces with local branching and tactically explore them with follow-on

branching.

162

6.4. Solution Approaches

Algorithm 12: Branching on regular pieces of work (fo-r3)

Find candidates

Compute set of candidate regular chains ¯

Tc={Ti: 0 < g(Ti)<1}.

Branching

if ¯

Tc6=∅then

Branch on follow-on ti, tj∈ T with max g(ti, tj)

else

if ∃Ti∈¯

Tc:Tiis piece of work then

Initialize ¯

Tcp ={Ti∈¯

Tc:Tiis piece of work}

Branch on regular chain Ti∈¯

Tcp with |Ti|= maxTj∈¯

Tcp |Tj|and

max g(Ti)

else

Initialize ¯

Tmax

c={Ti∈¯

Tc:|Ti|= maxTj∈¯

Tc|Tj|}

Branch on regular chain Ti∈¯

Tmax

cwith max g(Ti)

6.4.2. Bi-Objective Metaheuristics

In this section, we explicitly consider regularity as second objective function in

the integer phase instead of implicitly seeking regular crew schedules as in the

preceding section. In particular, we extend model ICSP by a second objective

function. The exposition in this section is partly based on [Suhl et al., 2007].

The consideration of both cost and regularity as objective functions leads to

the following bi-objective set partitioning problem (2ICSP):

k∈K

ckxk→min (6.12)

k∈K

σkxk→min (6.13)

s.t. X

k∈K(t)

xk= 1 ∀t∈ T ,(6.14)

xk∈ {0,1},(6.15)

where σkdenotes the distance of duty kto reference schedule R. Model 2ICSP

corresponds to model ICSP except the additional objective function (6.13) where

we minimize the distance to the reference solution.

In multicriteria (multiobjective) optimization (see [Ehrgott, 2005]), we look for

Pareto optimal solutions instead of seeking optimal solutions as in the single ob-

jective case. We call a solution exPareto optimal, if there is no other solution that

is at least as good as exwith respect to both objective functions and strictly better

163

6. Ex-Urban Vehicle and Crew Scheduling with Irregular Timetables

with respect to one objective. If exis Pareto optimal, ez= (Pk∈Kckexk,Pk∈Kσkexk)

is said to be efficient. In other words, we want to identify a set of efficient crew

schedules where for each crew schedule a reduction in one objective would nec-

essarily lead to an increase in the other objective. Optimization problems with

multiple objectives usually have many efficient solutions while our approach from

the preceding section returns a single (at least cost-effective) crew schedule.

We now have to choose an appropriate solution approach to solve our bi-

objective problem. Multiobjective metaheuristics have been successfully applied

to multiobjective optimization problems in general (see [Gandibleux et al., 2004])

and multiobjective crew scheduling problems in particular (see [Louren¸co et al.,

2001]). In the following, we will briefly describe four well-known metaheuristics

from literature and adapt the methods to the bi-objective set covering prob-

lem stated above. All methods approximate the set of efficient solutions. We

discuss an evolutionary algorithm, a tabu search method, a simulated anneal-

ing approach, and an ant colony optimization algorithm. First and foremost,

the purpose of our study is to compare the performance of different multiobjec-

tive metaheuristics on our particular problem. For a recent survey on heuristic

methods for multiobjective optimization the reader is referred to [Ehrgott and

Gandibleux, 2004].

Strength Pareto Evolutionary Algorithm (SPEA2) The improved Strength Pa-

reto Evolutionary Algorithm (SPEA2) proposed by [Zitzler et al., 2002] is

an evolutionary algorithm (see Chapter 4) to approximate the set of Pareto

optimal solutions. Their method is an enhancement of [Zitzler and Thiele,

1999]. The basic idea of their approach is to use the dominance criterion for

fitness calculation and selection of solutions. Furthermore, non-dominated

solutions are stored in an external archive, i.e., independent from the cur-

rent population. The authors present promising results of their algorithm

as compared with other evolutionary approaches. In our implementation,

we use a binary representation of solutions and apply the genetic operators

proposed by [Beasley and Chu, 1996].

Multiobjective Tabu Search Tabu search (see [Glover and Laguna, 1993]) is a

local search method where a selective history of the search states is stored.

In its simplest form a tabu list is used to prevent the search method to get

stuck in local optima. Our implementation is based on [Louren¸co et al.,

2001] who suggested a multiobjective tabu search method for the crew

scheduling problem. Basically, they use a weighted scalarizing function

to represent multiple objectives. Furthermore, the authors propose an op-

timized intensification strategy and insert as well as remove tabu lists. In

164

6.4. Solution Approaches

addition to tabu lists, we apply the greedy heuristic of [Caprara et al., 1999]

to construct new feasible solutions. The construction heuristic requires to

initially solve a Lagrangian dual problem with a subgradient method (see

Section 1.5.1).

Multiobjective Simulated Annealing Simulated Annealing (see [Dowsland, 1993])

is a local search method that explores the neighborhood of an incumbent

solution and allows a worse solution to be accepted as starting point for

further exploration. The acceptance rate is based on the quality of the so-

lution and the computational time spent. Accepting worse solutions allows

a simulated annealing method to backtrack from local optima. The first

multiobjective version of simulated annealing was introduced by [Serafini,

1992] where the author considered several multiobjective acceptance rules.

In our implementation, we considered acceptance rule M which is a mixture

of the Chebyshev and product rule. Furthermore, we apply the heuristic

of [Caprara et al., 1999] to construct neighboring solutions.

Multiobjective Ant Colony Optimization Ant Colony Optimization (see [Dorigo

and St¨utzle, 2004]) is a metaheuristic that is inspired by the behavior of

real ants. In nature, ants indirectly communicate by means of pheromone

trails in order to find the shortest path between their ant hill and the food

source. The shorter the path between nest and food source is, the more ants

can use it in a given timeframe, and, as a consequence, the stronger the

pheromone trail will be. Our implementation is based on the population-

based ant colony optimization approach of [Guntsch and Middendorf, 2003].

However, our implementation differs from theirs in the following way. We

alternately construct new solutions using cover costs (see [Marchiori and

Steenbeek, 2003]) and the greedy heuristic of [Caprara et al., 1999].

In Section 6.5 we provide computational results concerning the quality of the

approximated Pareto fronts of the heuristics stated above. In the following, we

illustrate how the Pareto front can be used to support planners while assessing

the trade-off between cost and regularity.

A Basic Decision Support System

In Figure 6.1 we present a basic interface of a decision support system to evaluate

the trade-off between costs and regularity. The major purpose of the system

is to illustrate the trade-off in a straightforward way. In the upper right part

of the system, the solutions of the current Pareto front are shown where the

165

6. Ex-Urban Vehicle and Crew Scheduling with Irregular Timetables

distance measure is shown on the vertical axis and costs are displayed on the

horizontal axis. Notice that a high regularity corresponds to a small distance,

i.e., in our example low cost solutions have a low regularity as well. Furthermore,

the approximated Pareto front allows the planner to estimate the actual trade-off

between costs and regularity while the branching approach described in Section

6.4.1 basically provides a single (at least cost-effective) solution.

In addition to the front shown in the upper right part of the interface, the pa-

rameters of the metaheuristics can be altered in the upper left part. The current

status is displayed in the lower part when the system is running. During runtime

the current front is regularly updated. Moreover, if the heuristic uses a scalarizing

weighted objective function, the planner may interactively direct the search of

the metaheuristics by changing the relation of cost and regularity/distance (see

Figure 6.2).

6.5. Computational Results

We test our approaches on real-world and randomly generated data instances.

We consider two real-world and eight randomly generated data instances. The

artificial instances were generated as described in [Huisman, 2004]. However,

all instances have a single depot and drivers may only change their vehicle in

that depot. We make these assumptions in order to reflect a typical ex-urban

scenario (see Section 6.1). Furthermore, we assume that a reference crew schedule

is known for each data instance.

In Table 6.2 we give details on the data instances that result from solving

the linear relaxation of the ICSP with a column generation algorithm. The last

two instances correspond to real world problems while the others were randomly

generated. We report the ratio of irregular trips in percent (%irr), the number of

rows (#rows), columns (#cols), and non-zeros (#nnz). For each data instance

the ratio of irregular trips refers to number of new trips, i.e. trips that are not

in the reference schedule, compared to the total number of trips. In the second

part of the table we give details on the column generation phase: the number of

iterations (#iter), and the computational time spend on master (cpu ma) and

pricing problem (cpu pr). To maintain comparability between both approaches,

we used operating costs as single objective in the column generation phase.

Notice that a direct comparison between both approaches is not possible. Our

branching scheme provides a single solution while the bi-objective metaheuristics

return a set of Pareto optimal solutions. Therefore, we first give results on our

branching scheme and then on the bi-objective metaheuristics.

166

6.5. Computational Results

Figure 6.1.: Basic decision support system to estimate the trade-off between costs and regularity

167

6. Ex-Urban Vehicle and Crew Scheduling with Irregular Timetables

Figure 6.2.: The search direction of the metaheuristic can be interactively

changed if a scalarizing weighted objective function is used

Local Branching and Branching Rules

In addition to the assumptions stated above we apply the following parameter

settings for our branching approach:

•The computational time to find an integer solution is limited to 2 hours

(7,200 seconds).

•In our local branching implementation, at most 20% of the variables of the

incumbent may flip their values. Furthermore, the computational time to

explore subspaces ∆(x, ¯xi)≤κ(left branches) is limited to 15 minutes (900

seconds). If the time limit is reached and no new incumbent is found, we

reduce the size of the subspace by 50% to speed-up its exploration. For

further details we refer to [Fischetti and Lodi, 2003].

All computational experiments with the branching schemes were performed on

a personal computer running Windows XP with an Intel Pentium IV 2.2 GHz

processor and 2 GB of main memory.

In Table 6.3 we show results on the regularity of crew schedules when we

apply local branching (locbr) and follow-on branching (fo-r1, fo-r2, fo-r3) as de-

scribed in Section 6.4.1. Furthermore, we compare our method with the default

branch-and-bound implementation of ILOG CPLEX 9.1.3 (cpx-def ) and local

branching in combination with default branching of CPLEX (locbr cpx-def ). For

each method we give the average over the ten instances described in Table 6.2.

168

6.5. Computational Results

instance %irr #rows #cols #nnz #iter cpu ma cpu pr

art320 1 5.0 320 100,944 857,215 31 245 140

art320 2 5.0 320 60,128 384,478 21 143 85

art400 1 5.0 400 72,673 459,906 22 125 122

art400 2 5.0 400 57,769 352,592 21 130 77

art640 1 5.0 640 156,044 1,227,320 41 1,006 1,673

art640 2 5.0 640 104,595 643,113 28 572 695

art800 1 5.0 800 135,572 852,337 37 1,060 2,054

art800 2 5.0 800 162,209 1,158,539 39 1,773 2,887

real430 4.4 430 98,710 1,204,084 31 391 297

real433 4.8 433 103,516 1,236,954 31 411 257

Table 6.2.: Description of data instances

In Table 6.3 we report the computational time in seconds spent in the second (in-

teger) phase (cpu ip), the optimality gap in percent (%gap) and three regularity

measures. The regularity measures are defined as follows. The percentage of pre-

served duties (%prd) refers to the percentage of duties in the new crew schedule

that could be (exactly) kept from the reference crew schedule. Likewise we de-

fine the percentage of preserved regular pairs (%prp). The average regular chain

length of a crew schedule corresponds to the average number of regular tasks

in a duty. In this context, the percentage of the average chain length (%avgcl)

refers to the average regular chain length of the new crew schedule compared with

average regular chain length of the reference crew schedule. For example, if the

reference schedule has on the average 8 regular tasks per duty, and the average

regular chain length in the new crew schedule is 4 tasks, then avgcl =4

8= 50%.

As can be seen from Table 6.3 branching scheme fo-r1 provides the best results

in terms of solution time and solution quality. Recall that objective function

and, thus, solution quality refer to operational costs. On the other hand, local

branching considerably improves the regularity of the new crew schedules, e.g.,

the number duties that can be kept from the reference. Basically, we generally

observe an increase of solution time and decrease of solution quality if local

branching is used. However, local branching in combination with scheme fo-

r1 gives a better solution quality than the default version of CPLEX. To sum

up, we conclude that local branching effectively improves the regularity while

follow-on branching scheme fo-r1 is well suited to improve solution quality and

time. The combination of both methods leads to improved solutions in terms of

169

6. Ex-Urban Vehicle and Crew Scheduling with Irregular Timetables

regularity measures

method cpu ip %gap %prd %prp %avgcl

cpx-def 2,437 1.93 6.3 53.5 31.0

fo-r1 2,095 0.42 7.7 54.4 31.2

fo-r2 3,649 2.20 8.2 56.8 33.7

fo-r3 4,247 2.81 6.6 55.0 32.5

locbr cpx-def 6,420 2.60 27.4 79.0 50.1

locbr fo-r1 5,492 1.55 28.0 80.2 51.2

locbr fo-r2 5,806 3.81 32.3 81.1 54.5

locbr fo-r3 6,270 3.70 25.6 80.0 51.2

Table 6.3.: Results on regularity of crew schedules for branching approaches

both cost and regularity compared to a traditional approach with CPLEX.

Bi-Objective Metaheuristics

Typically, performance of optimization algorithms is assessed on both computa-

tional time consumed and solution quality. For the single objective case, it is

common practice to monitor the computational time and to define quality by

means of the value of the objective function. As to the computational effort,

multiobjective optimization algorithms can be evaluated in the same way as sin-

gle objective methods. However, if there are multiple objectives, an algorithm

returns a set of non-dominated solutions. Obviously, we cannot compare two sets

of solutions in the same straightforward way as two single solutions: solutions

in either set may be dominated by solutions in the other set and other solutions

may be incomparable.

The hypervolume measure (see [Zitzler and Thiele, 1998] and [Fleischer, 2003]

for a multidimensional generalization) is one of the most commonly applied mea-

sures to compare the results of multiobjective optimization algorithms. If there

are two minimizing objective functions and upper bounds for both objectives

given, the hypervolume measures the area covered by the non-dominated so-

lutions. In Figure 6.3 we illustrate the hypervolume for five non-dominated

solutions and two objective functions with upper bounds u1and u2. The hy-

pervolume indicator allows to infer that an approximate set is not worse than

another, but it does not provide an indication how much better the approxi-

mation actually is (see [Zitzler et al., 2003]). In the following, we will compare

the bi-objective metaheuristics described in Section 6.4.2 using the hypervolume

170

6.5. Computational Results

hypervolume

non-dominated solution

Figure 6.3.: Example of hypervolume measure for five non-dominated solutions

and two objectives

measure. All computational experiments were performed on a personal computer

running Windows XP with an Intel Pentium IV 3.0 GHz processor and 2 GB of

main memory. The computational time for each run was limited to 10 minutes

since our primary goal is to provide a quick estimate for the trade-off.

In order to restrict the impact of random effects, we repeated our experiments

ten times for each test problem and algorithm. The hypervolume of a particular

algorithm corresponds to the average over all ten runs. We build a list for each

instance ranking the algorithms where we put the algorithm with the largest

volume on the first position, the second largest volume on the second position,

and so on. In Table 6.4 we report how often a metaheuristic held a particular

position in the hypervolume ranking. In particular, we give results for simulated

annealing (SA), tabu search (TS), evolutionary algorithm (EA), and ant colony

optimization (ACO).

While there is a clear performance gap between the EA and ACO as well as

ACO and TS/SA, the fronts achieved by SA and TS are rather close together.

In Figures 6.4 to 6.7 we present an approximate Pareto front generated by the

metaheuristics for each of four instances. Furthermore, we give the solutions ob-

tained with CPLEX (cpx-def ) and local branching in combination with follow-on

branching version 1 (locbr fo-r1) as described in the preceding section. Finally,

171

6. Ex-Urban Vehicle and Crew Scheduling with Irregular Timetables

algorithm #hv positions

1st 2nd 3rd 4th

SA – – 5 5

TS – – 5 5

EA 10 – – –

ACO – 10 – –

Table 6.4.: Number of positions held in hypervolume ranking by bi-objective

metaheuristics on test set (10 instances)

we give grey lines for the minimum distance and operational cost if the corre-

sponding second objective is omitted. Notice that all figures are differently scaled

and that GA corresponds to the results of the evolutionary algorithm SPEA2.

Approximate Pareto fronts computed with the EA contain many high-quality

solutions that are well-spread. The fronts of the ACO approach comprises many

solutions that are usually of average quality but better spread than those of the

EA. The simulated annealing algorithm produces many low cost solutions. As

a consequence, the approximated Pareto front is primarily located in low cost

areas. The tabu search method generates approximated fronts with few low-cost

solutions that are very close to each other. We conclude that the evolutionary

algorithm SPEA2 is well-suited to provide an estimate for the trade-off between

costs and regularity in a short timeframe. Finally, the figures show that the

default version of CPLEX and local branching in combination with follow-on

branching provide (almost) optimal solutions concerning costs. However, espe-

cially SPEA2 could always find solutions with lower distance (but higher costs).

As a consequence, we believe that the bi-objective metaheuristics can provide

reasonable, additional information for the planner to assess the quality of a so-

lution concerning regularity. This additional information can be generated in a

short timeframe.

6.6. Summary

In this chapter, we discussed the ex-urban vehicle and crew scheduling problem

with a single depot and irregular timetables. Unless specifically imposed, tra-

ditional vehicle and crew scheduling usually produces irregular crew schedules

which are undesired in practice. We presented two solution approaches that im-

prove the regularity of crew schedules compared to traditional crew scheduling.

172

6.6. Summary

100

120

50000 60000 70000 80000 90000 100000

distance

operational cost

art320_1

ACO

locbr_fo-r1

cpx-def

Figure 6.4.: Approximate Pareto fronts for instance art320 1 generated by meta-

heuristics compared to branching schemes

100

120

140

90000 100000 110000 120000 130000 140000 150000

distance

operational cost

art400_1

ACO

locbr_fo-r1

cpx-def

Figure 6.5.: Approximate Pareto fronts for instance art400 1 generated by meta-

heuristics compared to branching schemes

173

6. Ex-Urban Vehicle and Crew Scheduling with Irregular Timetables

100

150

200

250

300

350

120000 140000 160000 180000 200000 220000

distance

operational cost

art640_1

ACO

locbr_fo-r1

cpx-def

Figure 6.6.: Approximate Pareto fronts for instance art640 1 generated by meta-

heuristics compared to branching schemes

100

150

200

250

300

350

400

180000 200000 220000 240000 260000 280000 300000 320000

distance

operational cost

art800_1

ACO

locbr_fo-r1

cpx-def

Figure 6.7.: Approximate Pareto fronts for instance art800 1 generated by meta-

heuristics compared to branching schemes

174

6.6. Summary

In particular, we proposed a novel combination of local branching and follow-on

branching. Furthermore, we showed how bi-objective metaheuristics can be used

to quickly estimate the quality of the solution generated with the latter approach.

Finally, a computational study that involved randomly generated and real-life

data showed the applicability of the proposed techniques. In fact, our branching

scheme lead to improved solutions in terms of both cost and regularity compared

to a traditional approach with CPLEX.

175

6. Ex-Urban Vehicle and Crew Scheduling with Irregular Timetables

176

7. Summary and Concluding

Remarks

In this thesis we addressed the integrated multiple-depot vehicle and crew schedul-

ing problem in public bus transport. Vehicle and crew scheduling are two major

planning problems that basically aim at assigning scheduled trips of a given

timetable to vehicle and crew itineraries. For several years now, Operations Re-

search has been successful for solving both planning problems. Traditionally,

both planning steps have been approached sequentially where vehicle schedules

are determined before crew schedules. However, the integrated consideration

of vehicle and crew scheduling has received considerable attention over the past

years. Several authors have shown that the integrated treatment of both planning

steps discloses additional flexibility that can lead to gains in efficiency compared

to sequential planning.

In Chapter 1 we introduced the vehicle and crew scheduling problem and pro-

vided the necessary background for combinatorial optimization problems and so-

lution techniques. In the following Chapter 2 we defined the integrated multiple-

depot vehicle and crew scheduling problem in public transport. We reviewed

models and solutions techniques that are used in literature for sequential, par-

tially integrated, and fully integrated vehicle and crew scheduling. Furthermore,

we thoroughly described the modeling approach, mathematical formulation, and

solution approach that provided the starting point for the following chapters of

the thesis. The two-phase solution approach is based on column generation in

combination with Lagrangian relaxation. In the first phase a lower bound is

computed while feasible solutions are constructed in the second phase.

The main contributions of this thesis for the integrated vehicle and crew

scheduling problem with multiple depots were described in Chapter 3. More

specifically, we proposed an approach for the column generation pricing problem

that involved two novel network formulations for a decomposed pricing prob-

lem. We showed that the network complexity of our approach is beneficial com-

pared to other approaches previously exposed in literature. We applied a dy-

namic programming method to solve the pricing problem. In this context, we

discussed known as well as novel adaptations of preprocessing and acceleration

177

7. Summary and Concluding Remarks

techniques that were essential to solve large problem instances. Furthermore,

we discussed three solution methods to construct integer solutions, namely a

Lagrangian heuristic, a branch-and-bound method, and a novel heuristic branch-

and-price method. Basically, the Lagrangian heuristic generated good quality

solutions in a short timeframe while the branch-and-price heuristic provided high

quality solutions at high computational costs. The branch-and-bound method

appeared to be inappropriate for solving large instances. Finally, we presented

a new model variation where drivers are not tied to vehicles from a single de-

pot and can change their vehicle whenever there is a relief point (unrestricted

changeovers). Our computational study involved real-world and randomly gener-

ated benchmark instances with up to 653 trips and four depots. The experiments

showed the effectiveness of our approach. In this context, we presented previously

unknown solutions for the widely used benchmark instances of [Huisman, 2003].

In fact, the results indicated that our method outperformed other approaches

from literature in terms of computational time and solution quality. Further-

more, we solved benchmark instances with 640 trips and four depots. To the

best of our knowledge, randomly generated instances of that type and size have

not been tackled before. We obtained similar results for the model variation with

unrestricted changeovers.

In Chapter 4 we dealt with a novel hybrid evolutionary algorithm to tackle inte-

grated vehicle and crew scheduling problems. Our method combined mathemati-

cal programming techniques with an evolutionary algorithm. We applied an evo-

lutionary algorithm to find a good trip-depot assignment where the fitness of an

individual is evaluated using column generation in combination with Lagrangian

relaxation. The computational experiments were performed with the randomly

generated benchmark instances that have been used in the preceding chapter. We

compared different versions of the evolutionary algorithms with each other, with

the traditional sequential approach, and with an integrated treatment of both

planning steps. The results indicated that medium-sized problem instances with

multiple depots can be solved by using the evolutionary algorithm. Furthermore,

our approach disclosed significant savings compared to the traditional sequential

approach without requiring a fully integrated solution method. Although our

algorithm performed worse than the best known integrated algorithm, it proved

to be competitive with other integrated approaches from literature especially for

medium-sized instances.

In Chapter 5 we considered practical rules and regulations arising in public

transport companies in Germany. We suggested extensions and modifications

of our modeling and solution approach from Chapter 3 to cover these practical

extensions. The enhancements included driving time constraints, complex break

178

rules where many pieces of work are allowed, break positions, and duty mix con-

straints. Furthermore, we gave an overview of how our implementation is being

integrated in the commercial software tool interplanr. We tested the applica-

bility of the proposed techniques using real-life data instances. The results on

instances with up to 653 trips and four depots indicate that our approach can

efficiently cover duty types with many pieces of work and complex feasibility

rules.

In Chapter 6 we did not only focus on how to conduct operations at minimum

cost but also on another aspect which is related to the quality of crew schedules.

In practice, timetables consist of many trips serviced every day and some excep-

tions that do not repeat daily. In other words, timetables in practice are irregular

and, unless specifically imposed, traditional vehicle and crew scheduling usually

produces irregular crew schedules which are undesired in practice. Therefore,

we addressed the ex-urban vehicle and crew scheduling problem with irregular

timetables. We proposed two approaches that capture both costs and regularity

of crew scheduling solutions. More specifically, we suggested a novel combination

of local branching and follow-on branching that improves the regularity of crew

schedules while cost optimality is maintained. Furthermore, we compare four bi-

objective metaheuristics that take both cost and regularity as objective functions.

The latter approach can be used to get a quick estimate of the solution quality

obtained with the first approach. Our computational study with real-world and

artificial instances showed that the branching approach led to improved solutions

in terms of both cost and regularity compared to a traditional approach.

At the beginning of this thesis (see Section 1.6) we stated three research ob-

jectives. In short, our objectives were (1) to develop models and techniques for

the integration of vehicle and crew scheduling that allow to tackle large problem

instances, (2) to develop models and techniques to increase the regularity of crew

schedules when timetables are irregular, and (3) to test the applicability of the

proposed techniques in practice. From our perspective these objectives have been

achieved. In Chapters 3 and 5 we approached the first objective. We obtained

promising results concerning the effectiveness of our methods for large problem

instances. Furthermore, we modeled complex duty feasibility rules in Chapter

5. With respect to the second objective, we suggested models and techniques to

increase the regularity of crew scheduling solutions in Chapter 6. Our computa-

tional results for the partially integrated (ex-urban) vehicle and crew scheduling

indicate that the regularity can be improved while maintaining cost optimality.

However, we left a fully integrated consideration for future research. We devoted

Chapter 5 and in part Chapter 3 to achieve the last objective. We tested our

approaches on real-world instances and showed their effectiveness. Furthermore,

179

7. Summary and Concluding Remarks

our methods are being integrated in the commercial software package interplanr

for public transport companies.

Finally, we would like to make some suggestions for future research in the field

of vehicle and crew scheduling. Although some progress has been made over the

past years, we are not aware of an approach that could deal with 1,000s or even

10,000s of trips. However, problem instances of such size with many depots are

common in big cities such as the German towns of Munich, Hamburg, or Berlin.

Therefore, we suggest to pursue further research on faster solution procedures

for integrated problems. Moreover, the partial integration of vehicle scheduling

and timetabling results in gains in efficiency (see [Kliewer et al., 2006a]). Hence,

we deem it worthwhile to include timetable considerations into the integrated

treatment of multiple-depot vehicle and crew scheduling. Finally, we suggest to

continue research on aspects related to the quality of vehicle and crew schedules

such as robustness or quality of work conditions.

180

A. Definitions and Abbreviations

In this appendix we summarize definitions and abbreviations that we have intro-

duced throughout this thesis.

Definitions

depot maintenance and storage facility where buses may

be parked and serviced when not in use

(service) trip vehicle activity with passengers and defined by

start and end locations and times

deadhead (trip) vehicle activity without carrying passengers such

as movements or idle times outside the depot (or

both)

pull-in trip moves a vehicle from the depot to the start location

of the first trip of a vehicle block

pull-out trip moves a vehicle from the end location of the last

trip of a vehicle block to the depot

compatible trips two trips that can be covered consecutively by the

same vehicle

vehicle block sequence of compatible trips that can be executed

by a single vehicle, starts with a pull-in and ends

with a pull-out trip

relief point defines a location and time where a driver may

change his vehicle

task elementary portion of work between two relief

points that can be assigned to a driver

piece (of work) sequence of tasks without a (long) break for which

a driver stays with the same vehicle

duty sequence of pieces of work that can be assigned to

an anonymous driver and satisfies a wide variety

of regulations

changeover change of a vehicle of a driver

181

A. Definitions and Abbreviations

continuous attendance a driver is required to be present if a bus is outside

of a depot

Abbreviations

ACO ant colony optimization

CSP crew scheduling problem

EA evolutionary algorithm

IP integer program

LDP Lagrangian dual problem

LP linear program

MCFP minimum cost flow problem

MDVSP multiple-depot vehicle scheduling problem

MDVCSP integrated multiple-depot vehicle and crew

scheduling problem (formulation (2.11)-(2.16))

MDVCSP-H integrated multiple-depot vehicle and crew

scheduling problem (formulation (2.1)-(2.10))

MDVCSP-C integrated multiple-depot vehicle and crew

scheduling problem with unrestricted changeovers

(formulation (3.42)-(3.47))

MFP multicommodity flow problem

MP master problem

RCSP resource constrained shortest path problem

REF resource extension function

RMP restricted master problem

SA simulated annealing

SCP set covering problem

SDVSP single-depot vehicle scheduling problem

SP shortest path problem

SPP set partitioning problem

TS tabu search

TSN time-space network

VCSP integrated vehicle and crew scheduling problem

182

List of Figures

1.1. Planning Process of a Public Transport Company . . . . . . . . . 3

1.2. Excerpt from line network of PaderSprinter, Paderborn (Germany) 4

1.3. Schedule of one vehicle consisting of two blocks . . . . . . . . . . 5

1.4. Schedule of one vehicle and one crew where a piece of work remains

unassigned............................... 7

1.5. Planning process for integrated vehicle and crew scheduling . . . . 8

1.6. Optimal vehicle and crew schedule for sequential approach consist

of two blocks and three duties. . . . . . . . . . . . . . . . . . . . . 9

1.7. Optimal vehicle and crew schedule for integrated approach consist

of two blocks and two duties. . . . . . . . . . . . . . . . . . . . . 10

1.8. Impact of an irregular timetable on the regularity of the crew

scheduling solution if independent crew scheduling is performed. . 12

1.9. Subdifferential and subgradient skof a concave, nondifferentiable

function Φ(π) at π0for |M1|=1. .................. 19

1.10. The convex hull conv(X) of the unbounded polyhedron Xwith

two extreme points and two extreme rays. . . . . . . . . . . . . . 21

2.1. Timeline of a station with four arrivals and two departures. . . . . 48

2.2. Deadhead arcs in a connection-based and time-space network be-

tweentwostations .......................... 50

2.3. Time-space network with six trips . . . . . . . . . . . . . . . . . . 51

2.4. Piece generation network . . . . . . . . . . . . . . . . . . . . . . . 58

3.1. Connection-based duty generation network . . . . . . . . . . . . . 68

3.2. Time-space duty generation network . . . . . . . . . . . . . . . . 69

3.3. Aggregated time-space duty generation network . . . . . . . . . . 70

3.4. Compatible pieces of work in a time-space duty generation network 81

3.5. Compatible time slots in a time-space duty generation network . . 82

3.6. Network reduction in the column generation process for first depot

of instance 320A09 and three different duty types . . . . . . . . . 84

3.7. Sample piece generation network for follow-on fixing . . . . . . . . 108

183

List of Figures

3.8. Comparison of computational times in percent on Huisman data

instances type A with four depots . . . . . . . . . . . . . . . . . . 120

4.1. Problem decomposition for evolutionary algorithm . . . . . . . . . 126

5.1. Modified time-space duty generation network to consider specific

breakpositions ............................ 143

5.2. Integration of ICOPT with PTV interplanr............ 146

6.1. Basic decision support system to estimate the trade-off between

costsandregularity.......................... 167

6.2. The search direction of the metaheuristic can be interactively changed

if a scalarizing weighted objective function is used . . . . . . . . . 168

6.3. Example of hypervolume measure for five non-dominated solutions

andtwoobjectives .......................... 171

6.4. Approximate Pareto fronts for instance art320 1 generated by meta-

heuristics compared to branching schemes . . . . . . . . . . . . . 173

6.5. Approximate Pareto fronts for instance art400 1 generated by meta-

heuristics compared to branching schemes . . . . . . . . . . . . . 173

6.6. Approximate Pareto fronts for instance art640 1 generated by meta-

heuristics compared to branching schemes . . . . . . . . . . . . . 174

6.7. Approximate Pareto fronts for instance art800 1 generated by meta-

heuristics compared to branching schemes . . . . . . . . . . . . . 174

184

List of Tables

1.1. Deadheadmatrix ........................... 9

2.1. Number of deadhead arcs of a connection-based and time-space

network structure as presented in [Gintner, 2007] . . . . . . . . . 51

3.1. Network modeling approaches for crew scheduling in literature . . 65

3.2. Resource consumption for a connection-based network . . . . . . . 68

3.3. Resource consumption for a time-space network . . . . . . . . . . 69

3.4. Resource consumption for an aggregated time-space network . . . 71

3.5. Network dimensions of different duty generation networks . . . . . 72

3.6. Average network dimensions of different duty generation networks

for integrated vehicle and crew scheduling problems . . . . . . . . 73

3.7. Comparison of the time-space and aggregated time-space network

representation for duty generation . . . . . . . . . . . . . . . . . . 74

3.8. Impact of different network reduction techniques on the overall

performance.............................. 85

3.9. Results of dynamic programming algorithm with different domi-

nance tests in the first column generation iteration . . . . . . . . 90

3.10. Results of dynamic programming algorithm with and without ac-

celeration techniques . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.11. Results of sequential and adaptive crew scheduling to find integer

solutions................................ 97

3.12. Results of user-defined branching rules and sequential approaches

on model MDVCSP over five instances with 100 trips . . . . . . . 102

3.13. Results of heuristic branch-and-price algorithms . . . . . . . . . . 109

3.14. Results of integrated planning with restricted and unrestricted

changeovers .............................. 113

3.15. Properties of different duty types . . . . . . . . . . . . . . . . . . 115

3.16. Results on Connexxion data instances . . . . . . . . . . . . . . . . 117

3.17. Detailed results on Huisman data instances type A with four de-

pots and restricted changeovers . . . . . . . . . . . . . . . . . . . 118

185

List of Tables

3.18. Comparison on Huisman data instances type A with four depots

and restricted changeovers . . . . . . . . . . . . . . . . . . . . . . 119

3.19. Detailed results on Huisman data instances type A with four de-

pots and unrestricted changeovers . . . . . . . . . . . . . . . . . . 121

3.20. Comparison on Huisman data instances type A with four depots

and unrestricted changeovers . . . . . . . . . . . . . . . . . . . . . 122

4.1. Comparison of sequential vehicle and crew scheduling and evolu-

tionary algorithms on Huisman data instances type A . . . . . . . 133

4.2. Comparison of evolutionary algorithm EA-S* with other approaches

from literature on Huisman data instances type A . . . . . . . . . 134

5.1. Properties of different duty types . . . . . . . . . . . . . . . . . . 148

5.2. Results on Connexxion data instances with practical extensions . 149

6.1. Average number of optimal solutions for independent crew schedul-

ing on Huisman data instances type A . . . . . . . . . . . . . . . 159

6.2. Description of data instances . . . . . . . . . . . . . . . . . . . . . 169

6.3. Results on regularity of crew schedules for branching approaches . 170

6.4. Number of positions held in hypervolume ranking by bi-objective

metaheuristics on test set (10 instances) . . . . . . . . . . . . . . 172

186

List of Algorithms

1. Subgradient Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 20

2. Column Generation Algorithm . . . . . . . . . . . . . . . . . . . . 23

3. Lagrangian multiplier adjustment heuristic . . . . . . . . . . . . . . 25

4. Branch-and-bound........................... 27

5. Solution method for model MDVCSP . . . . . . . . . . . . . . . . 54

6. Basic label setting algorithm for the RCSP . . . . . . . . . . . . . 77

7. Label setting algorithm of [Desrochers, 1986] . . . . . . . . . . . . 79

8. Generic partial pricing algorithm . . . . . . . . . . . . . . . . . . . 86

9. Heuristic branch-and-price approach for model MDVCSP . . . . . 104

10. Basic Evolutionary Algorithm . . . . . . . . . . . . . . . . . . . . . 126

11. Branching on regular chains (fo-r2) ................. 162

12. Branching on regular pieces of work (fo-r3) ............. 163

187

LIST OF ALGORITHMS

188

Bibliography

[Achterberg et al., 2005] Achterberg, T., Koch, T., and Martin, A. (2005).

Branching rules revisited. Operations Research Letters, 33:42–54.

[Ahuja et al., 1993] Ahuja, R. K., Magnanti, T. L., and Orlin, J. B. (1993). Net-

work Flows: Theory, Algorithms, and Applications. Prentice Hall, New Jersey.

[Aneja et al., 1983] Aneja, Y., Aggarwal, V., and Nair, K. (1983). Shortest chain

subject to side constraints. Networks, 13(2):295–302.

[Balas and Padberg, 1976] Balas, E. and Padberg, M. (1976). Set partitioning:

A survey. SIAM Review, 18(4):710–760.

[Ball et al., 1983] Ball, M., Bodin, L., and Dial, R. (1983). A matching based

heuristic for scheduling mass transit crews and vehicle. Transportation Science,

17(1):4–31.

[Barahona and Anbil, 2000] Barahona, F. and Anbil, R. (2000). The volume al-

gorithm: producing primal solutions with a subgradient method. Mathematical

Programming, 87(3):385–399.

[Barahona and Jensen, 1998] Barahona, F. and Jensen, D. (1998). Plant location

with minimum inventory. Mathematical Programming, 83:101–111.

[Barnhart et al., 2003] Barnhart, C., Cohn, A. M., Johnson, E. L., Klabjan, D.,

Nemhauser, G. L., and Vance, P. H. (2003). Airline crew scheduling. In

Hall, R. W., editor, Handbook of Transportation Science, volume 56 of Inter-

national Series in Operations Research & Management Science, pages 517–560.

Springer, New York.

[Barnhart et al., 1995] Barnhart, C., Hatay, L., and Johnson, E. L. (1995). Dead-

head selection for the long-haul crew pairing problem. Operations Research,

43(3):491–499.

[Barnhart et al., 1998] Barnhart, C., Johnson, E. L., Nemhauser, G. L., Savels-

bergh, M. W., and Vance, P. H. (1998). Branch-and-price: Column generation

for solving huge integer programs. Operations Research, 46(3):316–329.

189

Bibliography

[B¨ack, 1996] B¨ack, T. (1996). Evolutionary algorithms in theory and practice:

evolution strategies, evolutionary programming, genetic algorithms. Oxford

University Press, Oxford, UK.

[Beasley and Christofides, 1989] Beasley, J. and Christofides, N. (1989). An al-

gorithm for the resource constrained shortest path problem. Networks, 19:379–

394.

[Beasley and Chu, 1996] Beasley, J. and Chu, P. (1996). A genetic algorithm

for the set covering problem. European Journal of Operational Research,

94(2):392–404.

[Ben Amor et al., 2006] Ben Amor, H., Desrosiers, J., and de Carvalho, J. V.

(2006). Dual-optimal inequalities for stabilized column generation. Operations

Research, 54(3):454–463.

[Bertossi et al., 1987] Bertossi, A. A., Carraresi, P., and Gallo, G. (1987). On

some matching problems arising in vehicle scheduling models. Networks,

17:271–281.

[Bertsekas and Casta˜non, 1992] Bertsekas, D. P. and Casta˜non, D. A. (1992). A

forward/reverse auction algorithm for asymmetric assignment problems. Com-

putational Optimization and Applications, 1(3):277–297.

[Bianco et al., 1994] Bianco, L., Mingozzi, A., and Ricciardelli, S. (1994). A

set partitioning approach to the multiple depot vehicle scheduling problem.

Optimization Methods and Software, 3:163–194.

[Bixby et al., 1992] Bixby, R. E., Gregory, J. W., Lustig, I. J., Marsten, R. E.,

and Shanno, D. (1992). Very large-scale linear programming: A case study

in combining interior point and simplex methods. Operations Research,

40(5):885–897.

[Bodin et al., 1983] Bodin, L., Golden, B., Assad, A., and Ball, M. (1983). Rout-

ing and scheduling of vehicles and crews: the state of the art. Computers &

Operations Research, 10(2):63–211.

[Bornd¨orfer et al., 2001] Bornd¨orfer, R., Gr¨otschel, M., and L¨obel, A. (2001).

Scheduling duties by adaptive column generation. Technical Report ZIR 01-

02, Konrad Zuse Zentrum f¨ur Informationstechnik, Berlin.

[Bornd¨orfer et al., 2002] Bornd¨orfer, R., L¨obel, A., and Weider, S. (2002). In-

tegrierte Umlauf- und Dienstplanung im Nahverkehr. Technical Report ZIB

Report 02-10, Konrad-Zuse Zentrum, Berlin. in German.

190

Bibliography

[Bornd¨orfer et al., 2004] Bornd¨orfer, R., L¨obel, A., and Weider, S. (2004). A bun-

dle method for integrated multi-depot vehicle and duty scheduling in public

transit. Technical Report ZR 04-14, Konrad-Zuse Zentrum f¨ur Information-

stechnik, Berlin, Germany.

[Bornd¨orfer et al., 2006] Bornd¨orfer, R., Schelten, U., Schlechte, T., and Wei-

der, S. (2006). A column generation approach to airline crew scheduling. In

Haasis, H.-D., Kopfer, H., and Sch¨onberger, J., editors, Operations Research

Proceedings 2005, pages 343–348, Berlin. Springer.

[Bundesministerium f¨ur Verkehr, Bau- und Wohnungswesen, 2005]

Bundesministerium f¨ur Verkehr, Bau- und Wohnungswesen (2005). Verord-

nung zur Durchf¨uhrung des Fahrpersonalgesetzes (Fahrpersonalverordnung -

FPersV). Bundesgesetzblatt Teil I, 40:1882–1933. (in German).

[Bunte and Kliewer, 2006] Bunte, S. and Kliewer, N. (2006). An overview on

vehicle scheduling models. Technical Report 11/2006, University of Paderborn,

DS&OR Lab. Presented at 10th International Conference on Computer-Aided

Scheduling of Public Transport (CASPT2006), Leeds, UK, June 21-23, 2006.

[Camerini et al., 1975] Camerini, P., Fratta, L., and Maffioli, F. (1975). On im-

proving relaxation methods by modified gradient techniques. Mathematical

Programming Study, 3:26–34.

[Caprara et al., 1999] Caprara, A., Fischetti, M., and Toth, P. (1999). A heuristic

method for the set covering problem. Operations Research, 47(5):730–743.

[Carpaneto et al., 1989] Carpaneto, G., Dell’Amico, M., Fischetti, M., and Toth,

P. (1989). A branch and bound algorithm for the multiple depot vehicle

scheduling problem. Networks, 19:531–548.

[Carraresi and Gallo, 1984] Carraresi, P. and Gallo, G. (1984). Network models

for vehicle and crew scheduling. European Journal of Operational Research,

16(2):139–151.

[Carraresi et al., 1995] Carraresi, P., Nonato, M., and Girardi, L. (1995). Net-

work models, lagrangean relaxation and subgradients bundle approach in crew

scheduling problems. In Computer-Aided Transit Scheduling, Proceedings of

the Sixth International Workshop, volume 430 of Lecture Notes in Economics

and Mathematical Systems, pages 188–212, Berlin. Springer.

191

Bibliography

[Cavique et al., 1999] Cavique, L., Rego, C., and Themido, I. (1999). Subgraph

ejection chains and tabu search for the crew scheduling problem. Journal of

the Operational Research Society, 50(6):608–616.

[Clausen et al., 2005] Clausen, J., Larsen, A., and Larsen, J. (2005). Disruption

management in the airline industry - concepts, models and methods. Technical

Report 2005-01, Informatics and Mathematical Modelling, Technical Univer-

sity of Denmark, Lyngby, Denmark.

[Daduna and Paix˜ao, 1995] Daduna, J. R. and Paix˜ao, J. M. P. (1995). Vehicle

scheduling for public mass transit - an overview. In Daduna, J., Branco, I.,

and Paix˜ao, J., editors, Proceedings of the Sixth International Workshop on

Computer-Aided Scheduling of Public Transport, volume 430 of Lecture Notes

in Economics and Mathematical Systems, pages 76–90. Springer, Heidelberg.

[Dallaire et al., 2004] Dallaire, A., Fleurent, C., and Rousseau, J.-M. (2004). Dy-

namic constraint generation in crewopt, a column generation approach for tran-

sit crew scheduling. Technical report, GIRO Inc., Montr´eal, Canada. (submit-

ted to 9th International Conference on Computer-Aided Scheduling of Public

Transport (CASPT), San Diego).

[Danna and Le Pape, 2005] Danna, E. and Le Pape, C. (2005). Branch-and-price

heuristics: A case study on the vehicle routing problem with time windows. In

Desaulniers, G., Desrosiers, J., and Solomon, M., editors, Column Generation,

chapter 4, pages 99–129. Springer, New York.

[Dantzig, 1963] Dantzig, G. (1963). Linear Programming and Extensions. Prince-

ton University Press, Princeton.

[Dantzig and Wolfe, 1960] Dantzig, G. B. and Wolfe, P. (1960). Decomposition

principle for linear programs. Operations Research, 8(1):101–111.

[Darby-Dowman et al., 1988] Darby-Dowman, K., Jachnik, J. K., Lewis, R. L.,

and Mitra, G. (1988). Integrated decision support systems for urban trans-

port scheduling: Discussion of implementation and experience. In Daduna,

J. R. and Wren, A., editors, Proceedings of the Fourth International Work-

shop on Computer-Aided Transit Scheduling, Lecture Notes in Economics and

Mathematical Systems, pages 226–239, Berlin. Springer.

[de Groot and Huisman, 2004] de Groot, S. W. and Huisman, D. (2004). Vehicle

and crew scheduling: Solving large real-world instances with an integrated

approach. Technical Report EI2004-13, Erasmus University Rotterdam, San

Diego.

192

Bibliography

[de Silva, 2001] de Silva, A. (2001). Combining constraint programming and lin-

ear programming on an example of bus driver scheduling. Annals of Operations

Research, 108:277–291.

[Dell’Amico et al., 1993] Dell’Amico, M., Fischetti, M., and Toth, P. (1993).

Heuristic algorithms for the multiple depot vehicle scheduling problem. Man-

agement Science, 39(1):115–125.

[Dell’Amico et al., 2006] Dell’Amico, M., Righini, G., and Salani, M. (2006). A

branch-and-price approach to the vehicle routing problem with simultaneous

distribution and collection. Transportation Science, 40(2):235–247.

[Desaulniers et al., 1997] Desaulniers, G., Desrosiers, J., Dumas, Y., Marc, S.,

Rioux, B., Solomon, M. M., and Soumis, F. (1997). Crew pairing at air france.

European Journal of Operational Research, 97:245–259.

[Desaulniers et al., 1999] Desaulniers, G., Desrosiers, J., Lasry, A., and Solomon,

M. M. (1999). Crew pairing for a regional carrier. In Wilson, N. H., editor,

Computer-Aided Transit Scheduling, volume 471 of Lecture Notes in Economics

and Mathematical Systems, pages 19–41, Berlin. Springer.

[Desaulniers et al., 2002] Desaulniers, G., Desrosiers, J., and Solomon, M. M.

(2002). Accelerating strategies in column generation methods for vehicle rout-

ing and crew scheduling problems. In Ribeiro, C. and Hansen, P., editors,

Essays and Surveys in Metaheuristics, pages 309–324. Kluwer, Boston.

[Desaulniers et al., 2005] Desaulniers, G., Desrosiers, J., and Solomon, M. M.,

editors (2005). Column Generation. Springer, New York.

[Desaulniers and Hickman, 2006] Desaulniers, G. and Hickman, M. D. (2006).

Public transit. In Barnhart, C. and Laporte, G., editors, Transportation, Hand-

books in Operations Research and Management Science, pages 69–127. North

Holland, The Netherlands.

[Desrochers, 1986] Desrochers, M. (1986). La fabrication d’horaires de travail

pour les conducteurs d’autobus par une m´ethode de g´en´eration de colonnes.

PhD thesis, Universit´e de Montr´eal, Montr´eal, Canada. (in French).

[Desrochers et al., 1992] Desrochers, M., Gilbert, J., Sauv´e, M., and Soumis, F.

(1992). Crew-opt: Subproblem modeling in a column generation approach to

urban crew scheduling. In Desrochers, M. and Rousseau, J., editors, Computer-

Aided Scheduling, volume 386 of Lecture Notes in Economics and Mathematical

Systems, pages 395–406, Berlin. Springer.

193

Bibliography

[Desrochers and Soumis, 1988] Desrochers, M. and Soumis, F. (1988). A gener-

alized permanent labelling algorithm for the shortest path problem with time

windows. INFOR, 26(3):191–212.

[Desrochers and Soumis, 1989] Desrochers, M. and Soumis, F. (1989). A column

generation approach to the urban transit crew scheduling problem. Transporta-

tion Science, 23(1):1–13.

[Desrosiers et al., 1995] Desrosiers, J., Dumas, Y., Solomon, M. M., and Soumis,

F. (1995). Time Constrained Routing and Scheduling, volume 8 of Handbooks

in Operations Research and Management Science, chapter 2, pages 35–139.

Elsevier Science, Amsterdam.

[Dorigo and St¨utzle, 2004] Dorigo, M. and St¨utzle, T. (2004). Ant Colony Opti-

mization. MIT Press (Bradford Books).

[Dowsland, 1993] Dowsland, K. A. (1993). Simulated Annealing, chapter 2, pages

20–69. Blackwell, London.

[Du Merle et al., 1999] Du Merle, O., Villeneuve, D., Desrosiers, J., and Hansen,

P. (1999). Stabilized column generation. Discrete Mathematics, 194:229–237.

[Dumitrescu and Boland, 2003] Dumitrescu, I. and Boland, N. (2003). Improved

preprocessing, labeling and scaling algorithms for the weight-constrained short-

est path problem. Networks, 42(3):135–153.

[Ehrgott, 2005] Ehrgott, M. (2005). Multicriteria Optimization, volume 491 of

Lecture Notes in Economics and Mathematical Systems. Springer, Berlin, 2nd

edition.

[Ehrgott and Gandibleux, 2004] Ehrgott, M. and Gandibleux, X. (2004). Ap-

proximative solution methods for multiobjective combinatorial optimization.

TOP, 12(1):1–63.

[Fahle, 2002] Fahle, T. (2002). Integrating Concepts from Constraint Program-

ming and Operations Research Algorithms. PhD thesis, University of Pader-

born.

[Fahle et al., 2002] Fahle, T., Junker, U., Karisch, S. E., Kohl, N., Sellmann, M.,

and Vaaben, B. (2002). Constraint programming based column generation for

crew assignment. Journal of Heuristics, 8:59–81.

194

Bibliography

[Falkner and Ryan, 1992] Falkner, J. and Ryan, D. M. (1992). EXPRESS: Set

partitioning for bus crew scheduling in Christchurch. In Desrochers, M. and

Rousseau, J., editors, Computer-Aided Scheduling, volume 386 of Lecture Notes

in Economics and Mathematical Systems, pages 359–378, Berlin. Springer.

[Fischetti and Lodi, 2003] Fischetti, M. and Lodi, A. (2003). Local branching.

Mathematical Programming, 84(1):23–47.

[Fischetti et al., 1987] Fischetti, M., Lodi, A., Martello, S., and Toth, P. (1987).

The fixed job schedule problem with spread-time constraints. Operations Re-

search, 35(6):849–858.

[Fischetti et al., 1989] Fischetti, M., Lodi, A., Martello, S., and Toth, P. (1989).

The fixed job schedule problem with working-time constraints. Operations

Research, 37(3):395–403.

[Fischetti et al., 2001] Fischetti, M., Lodi, A., Martello, S., and Toth, P. (2001).

A polyhedral approach to simplified crew scheduling and vehicle scheduling

problems. Management Science, 47(6):833–850.

[Fischetti et al., 1999] Fischetti, M., Lodi, A., and Toth, P. (1999). A branch-

and-cut algorithm for the multi depot vehicle scheduling problem. Technical

report.

[Fischetti and Toth, 1989] Fischetti, M. and Toth, P. (1989). An additive bound-

ing procedure for combinatorial optimization problems. Operations Research,

37(2):319–328.

[Fisher, 1981] Fisher, M. L. (1981). The lagrangian relaxation method for solving

integer programming problems. Management Science, 27(1):1–19.

[Fleischer, 2003] Fleischer, M. (2003). The measure of pareto optima. In Fonseca,

C. M., Fleming, P. J., Zitzler, E., Deb, K., and Thiele, L., editors, Evolutionary

Multi-Criterion Optimization: Proceedings of the Second International Confer-

ence on Evolutionary Multi-Criterion Optimization (EMO2003), volume 2632

of Lecture Notes in Computer Science, pages 519–533, Berlin. Springer.

[Forbes et al., 1994] Forbes, M., Holt, J. N., and Watts, A. M. (1994). An exact

algorithm for multiple depot bus scheduling. European Journal of Operational

Research, 72:115–124.

[Fores et al., 2002] Fores, S., Proll, L. G., and Wren, A. (2002). Tracs ii: a

hybrid ip/heuristic driver scheduling system for public transport. Journal of

the Operational Research Society, 53(10):1093–1100.

195

Bibliography

[Freling, 1997] Freling, R. (1997). Models and Techniques for Integrating Vehicle

and Crew Scheduling. PhD thesis, Erasmus University of Rotterdam.

[Freling et al., 2001a] Freling, R., Huisman, D., and Wagelmans, A. P. (2001a).

Applying an integrated approach to vehicle and crew scheduling in practice. In

Voß, S. and Daduna, J., editors, Computer-Aided Scheduling of Public Trans-

port, volume 505 of Lecture Notes in Economics and Mathematical Systems,

pages 73–90, Berlin. Springer.

[Freling et al., 2003] Freling, R., Huisman, D., and Wagelmans, A. P. (2003).

Models and algorithms for integration of vehicle and crew scheduling. Journal

of Scheduling, 6:63–85.

[Freling et al., 2001b] Freling, R., Wagelmans, A. P., and Paix˜ao, J. M. P.

(2001b). Models and algorithms for single-depot vehicle scheduling. Trans-

portation Science, 35(2):165–180.

[Friberg and Haase, 1999] Friberg, C. and Haase, K. (1999). An exact branch and

cut algorithm for the vehicle and crew scheduling problem. In Wilson, N. H.,

editor, Computer-Aided Transit Scheduling, volume 471 of Lecture Notes in

Economics and Mathematical Systems, pages 63–80, Berlin. Springer.

[Gaffi and Nonato, 1999] Gaffi, A. and Nonato, M. (1999). An integrated ap-

proach to ex-urban crew and vehicle scheduling. In Wilson, N. H., editor,

Computer-Aided Transit Scheduling, volume 471 of Lecture Notes in Economics

and Mathematical Systems, pages 103–128, Berlin. Springer.

[Galia and Hjorring, 2004] Galia, R. and Hjorring, C. (2004). Modelling of com-

plex costs and rules in a crew pairing column generator. In Ahr, D., Fahrion,

R., Oswald, M., and Reinelt, G., editors, Operations Research Proceedings 2003

- Selected Papers of the International Conference on Operations Research (OR

2003), pages 133–140, Berlin. Springer.

[Gamache and Soumis, 1998] Gamache, M. and Soumis, F. (1998). A method for

optimally solving the rostering problem. In Yu, G., editor, Operations Research

in the Airline Industry, volume 9 of International Series in Operations Research

& Management Science, pages 124–157. Kluwer Academic Publishers, Boston.

[Gamache et al., 1999] Gamache, M., Soumis, F., Marquis, G., and Desrosiers,

J. (1999). A column generation approach for large-scale aircrew rostering prob-

lems. Operations Research, 47(2):247–263.

196

Bibliography

[Gandibleux et al., 2004] Gandibleux, X., Sevaux, M., S¨orensen, K., and T’kindt,

V., editors (2004). Metaheuristics for Multiobjective Optimisation, volume 535

of Lecture Notes in Economics and Mathematical Systems. Springer, Heidel-

berg, Germany.

[Garey and Johnson, 1979] Garey, M. and Johnson, D. (1979). Computer and

Intractability: A Guide to NP-Completeness. Freeman, San Francisco.

[Gavish and Shlifer, 1978] Gavish, B. and Shlifer, E. (1978). An approach for

solving a class of transportation scheduling problems. European Journal of

Operational Research, 3:122–134.

[Geoffrion, 1974] Geoffrion, A. (1974). Lagrangean relaxation for integer pro-

gramming. Mathematical Programming Study, 2:82–114.

[Gintner, 2007] Gintner, V. (2007). Integrierte Umlauf- und Dienstplanung im

OPNV. PhD thesis, University of Paderborn, Germany (to be submitted).

[Gintner et al., 2004] Gintner, V., Kliewer, N., and Suhl, L. (2004). A crew

scheduling approach for public transit enhanced with aspects from vehicle

scheduling. Technical Report WP0407, University of Paderborn, Decision Sup-

port & Operations Research Lab, Paderborn.

[Gintner et al., 2005] Gintner, V., Kliewer, N., and Suhl, L. (2005). Solving large

multiple-depot multiple-vehicle-type bus scheduling problems in practice. OR

Spectrum, 27(4):507–523.

[Gintner et al., 2006a] Gintner, V., Kramkowski, S., Steinzen, I., and Suhl, L.

(2006a). Adaptive Dienst- und Umlaufplanung im ¨

OPNV. In Haasis, H.-

D., Kopfer, H., and Sch¨onberger, J., editors, Operations Research Proceedings

2005, pages 55–60, Berlin. Springer.

[Gintner et al., 2006b] Gintner, V., Steinzen, I., and Suhl, L. (2006b). A time-

space network based approach for integrated vehicle and crew scheduling in

public transport. In Binetti, M., Civitella, F., Liddo, E. D., Dell’Orco, M., and

Ottomanelli, M., editors, Proceedings of the EWGT2006 Joint Conferences,

pages 371–377, Bari, Italy.

[GIRO Inc., 2007] GIRO Inc. (2007). Hastus – transit scheduling and operations.

available at http://www.giro.ca/en/products/hastus/index.htm (last ac-

cess on July 9th, 2007).

197

Bibliography

[Glover and Laguna, 1993] Glover, F. and Laguna, M. (1993). Tabu Search, chap-

ter 3, pages 70–150. Blackwell, London.

[Gopalakrishnan and Johnson, 2005] Gopalakrishnan, B. and Johnson, E. L.

(2005). Airline crew scheduling: State-of-the-art. Annals of Operations Re-

search, 140(1):305–337.

[Gr¨onkvist, 2005] Gr¨onkvist, M. (2005). The Tail Assignment Problem. PhD

thesis, Chalmers University of Technology and G¨oteborg University.

[Gr¨otschel et al., 2003] Gr¨otschel, M., Bornd¨orfer, R., and L¨obel, A. (2003).

Duty scheduling in public transit. In J¨ager, W., editor, Mathematics - key

technologies for the future, pages 653–674. Springer, Berlin.

[Guntsch and Middendorf, 2003] Guntsch, M. and Middendorf, M. (2003). Solv-

ing multi-criteria optimization problems with population-based ACO. In Goos,

G., Hartmanis, J., and van Leeuwen, J., editors, Proceedings of Second Interna-

tional Conference on Evolutionary Multi-Criterion Optimization (EMO2003),

volume 2632 of Lecture Notes in Computer Science, pages 464–478, Berlin.

Springer.

[Guo et al., 2005] Guo, Y., Suhl, L., and Thiel, M. P. (2005). Solving the air-

line crew recovery problem by a genetic algorithm with local improvement.

Operational Research – An International Journal, 5(2).

[Haase et al., 2001] Haase, K., Desaulniers, G., and Desrosiers, J. (2001). Simul-

taneous vehicle and crew scheduling in urban mass transit systems. Trans-

portation Science, 35(3):286–303.

[Hadjar et al., 2006] Hadjar, A., Marcotte, O., and Soumis, F. (2006). A branch-

and-cut algorithm for the multiple depot vehicle scheduling problem. Opera-

tions Research, 54(1):130–149.

[Hassin, 1992] Hassin, R. (1992). Approximation schemes for the restricted short-

est path problem. Mathematics of Operations Research, 17(1):36–42.

[Held and Karp, 1971] Held, M. and Karp, R. M. (1971). The travelling salesman

problem and minimum spanning trees: Part ii. Mathematical Programming,

1:6–25.

[Held et al., 1974] Held, M., Wolfe, P., and Crowder, H. (1974). Validation of

subgradient optimization. Mathematical Programming, 6:62–88.

198

Bibliography

[Hoffman and Padberg, 1993] Hoffman, K. L. and Padberg, M. (1993). Solving

airline crew scheduling problems by branch-and-cut. Management Science,

39(6):657–682.

[Hollis et al., 2006] Hollis, B., Forbes, M., and Douglas, B. (2006). Vehicle rout-

ing and crew scheduling for metropolitan mail distribution at australia post.

European Journal of Operational Research, 173:133–150.

[Holmberg and Yuan, 2000] Holmberg, K. and Yuan, D. (2000). A lagrangean

heuristic based branch-and-bound approach for the capacitated network design

problem. Operations Research, 48(3):461–481.

[Huisman, 2003] Huisman, D. (2003). Random data instances

for multiple-depot vehicle and crew scheduling. available at

http://people.few.eur.nl/huisman/instances.htm (last access on

April 11th, 2007).

[Huisman, 2004] Huisman, D. (2004). Integrated and Dynamic Vehicle and Crew

Scheduling. PhD thesis, Erasmus University of Rotterdam.

[Huisman, 2007] Huisman, D. (2007). A column generation approach to solve

the crew re-scheduling problem. European Journal of Operational Research,

180(1):163–173.

[Huisman et al., 2005a] Huisman, D., Freling, R., and Wagelmans, A. P. (2005a).

Multiple-depot integrated vehicle and crew scheduling. Transportation Science,

39(4):491–502.

[Huisman et al., 2005b] Huisman, D., Jans, R., Peeters, M., and Wagelmans,

A. P. (2005b). Combining column generation and lagrangian relaxation. In

Desaulniers, G., Desrosiers, J., and Solomon, M., editors, Column Generation,

chapter 9, pages 247–270. Springer, New York.

[ILOG, 2006] ILOG (2006). CPLEX 10.0.1 User’s Manual. ILOG, Gentilly

Cedex, France.

[Irnich, 2006] Irnich, S. (2006). Resource extension functions: Properties, in-

version, and generalization to segments. Technical Report 2006-01, Deutsche

Post Endowed Chair of Optimization of Distribution Networks, RWTH Aachen

University, Aachen, Germany.

[Irnich and Desaulniers, 2005] Irnich, S. and Desaulniers, G. (2005). Shortest

path problems with resource constraints. In Desaulniers, G., Desrosiers, J., and

199

Bibliography

Solomon, M., editors, Column Generation, chapter 2, pages 33–65. Springer,

New York.

[Jans and Degraeve, 2004] Jans, R. and Degraeve, Z. (2004). An industrial ex-

tension of the discrete lot-sizing and scheduling problem. IIE Transactions,

36(1):47–58.

[Joksch, 1966] Joksch, H. (1966). The shortest route problem with constraints.

Journal of Mathematical Analysis and Applications, 14:191–197.

[Kiwiel, 1995] Kiwiel, K. (1995). Approximations in proximal bundle methods

and decomposition of convex programs. Journal of Optimization Theory and

Applications, 84(3):529–548.

[Klabjan et al., 2001] Klabjan, D., Johnson, E. L., Nemhauser, G. L., Gelman,

E., and Ramaswamy, S. (2001). Airline crew scheduling with regularity. Trans-

portation Science, 35(4):359–374.

[Klabjan et al., 2002] Klabjan, D., Johnson, E. L., Nemhauser, G. L., Gelman,

E., and Ramaswamy, S. (2002). Airline crew scheduling with time windows

and plane-count constraints. Transportation Science, 36(3):337–448.

[Kliewer, 2005] Kliewer, N. (2005). Optimierung des Fahrzeugeinsatzes im

¨offentlichen Personennahverkehr. PhD thesis, University of Paderborn, Ger-

many.

[Kliewer et al., 2006a] Kliewer, N., Bunte, S., and Suhl, L. (2006a). Time win-

dows for scheduled trips in multiple depot vehicle scheduling. In Binetti, M.,

Civitella, F., Liddo, E. D., Dell’Orco, M., and Ottomanelli, M., editors, Pro-

ceedings of the EURO Working Group on Transportation (EWGT) Joint Con-

ferences 2006, pages 340–346, Bari, Italy.

[Kliewer et al., 2006b] Kliewer, N., Mellouli, T., and Suhl, L. (2006b). A time-

space network based exact optimization model for multi-depot bus scheduling.

European Journal of Operational Research, 175(3):1616–1627.

[Kung et al., 1975] Kung, H., Luccio, F., and Preparata, F. (1975). On finding

the maxima of a set of vectors. Journal of the Association for Computing

Machinery, 22(4):469–476.

[Kwan et al., 1999] Kwan, A., Kwan, R. S., and Wren, A. (1999). Driver schedul-

ing using genetic algorithms with embedded combinatorial traits. In Wilson,

200

Bibliography

N. H., editor, Computer-Aided Transit Scheduling, volume 471 of Lecture Notes

in Economics and Mathematical Systems, pages 81–102, Berlin. Springer.

[Kwan et al., 2001] Kwan, R. S., Kwan, A., and Wren, A. (2001). Evolutionary

driver scheduling with relief chains. Evolutionary Computation, 9:445–460.

[Lavoie et al., 1988] Lavoie, S., Minoux, M., and Odier, E. (1988). A new ap-

proach for crew pairing problems by column generation with an application to

air transportation. European Journal of Operational Research, 35(1):45–58.

[L¨ubbecke, 2005] L¨ubbecke, M. (2005). Dual variable based fathoming in dy-

namic programs for column generation. European Journal of Operational Re-

search, 162:122–125.

[L¨ubbecke and Desrosiers, 2005] L¨ubbecke, M. and Desrosiers, J. (2005). Se-

lected topics in column generation. Operations Research, 53(6):1007–1023.

[L¨obel, 1996] L¨obel, A. (1996). Solving large-scale real-world minimum-cost flow

problems by a network simplex method. Technical Report SC96-7, Konrad-

Zuse Zentrum f¨ur Informationstechnik (ZIB), Berlin.

[L¨obel, 1997] L¨obel, A. (1997). Optimal vehicle scheduling in public transit. PhD

thesis, Technical University Berlin, Germany.

[L¨obel, 1998] L¨obel, A. (1998). Vehicle scheduling in public transit and la-

grangean pricing. Management Science, 44(12):1637–1650.

[Lettovsky et al., 2000] Lettovsky, L., Johnson, E. L., and Nemhauser, G. L.

(2000). Airline crew recovery. Transportation Science, 34(4):337–348.

[Leuthardt, 1998] Leuthardt, H. (1998). Kostenstrukturen von Stadt-, ¨

Uberland-

und Reisebussen. Der Nahverkehr, 6:19–23. (in German).

[Li and Kwan, 2005] Li, J. and Kwan, R. S. (2005). A self-adjusting algorithm

for driver scheduling. Journal of Heuristics, 11(4):351–367.

[Linderoth and Savelsbergh, 1999] Linderoth, J. T. and Savelsbergh, M. W.

(1999). A computational study of search strategies for mixed integer pro-

gramming. INFORMS Journal on Computing, 11(2):173–187.

[Louren¸co et al., 2001] Louren¸co, H. R., Paix˜ao, J. M. P., and Portugal, R.

(2001). Multiobjective metaheuristics for the bus driver scheduling problem.

Transportation Science, 35(3):331–343.

201

Bibliography

[Marchiori and Steenbeek, 2003] Marchiori, E. and Steenbeek, A. (2003). An

evolutionary algorithm for large scale set covering problems with application

to airline crew scheduling. In et al., S. C., editor, Real-World Applications of

Evolutionary Computing: EvoWorkshops 2000: EvoIASP, EvoSCONDI, Evo-

TEL, EvoSTIM, EvoRob, and EvoFlight, Edinburgh, Scotland, UK, volume

1803 of Lecture Notes in Computer Science, pages 367–381, Berlin. Springer.

[Marriot and Stuckey, 1998] Marriot, K. and Stuckey, P. (1998). Programming

with Constraints. MIT Press, Cambridge, Massachusetts.

[Marsten et al., 1975] Marsten, R. E., Hogan, W., and Blankenship, J. (1975).

The boxstep method for large-scale optimization. Operations Research,

23(3):389–405.

[Medard and Sawhney, 2007] Medard, C. P. and Sawhney, N. (2007). Airline

crew scheduling: From planning to operations. European Journal of Opera-

tional Research, 183(3):1013–1027.

[Mehlhorn and Ziegelmann, 2000] Mehlhorn, K. and Ziegelmann, M. (2000). Re-

source constrained shortest paths. In Paterson, M., editor, Proc. 8th Annual

European Symposium on Algorithms (ESA2000), volume 1972 of Lecture Notes

in Computer Science, pages 326–337, Berlin. Springer.

[Mesquita and Paias, 2006] Mesquita, M. and Paias, A. (2006). Set

partitioning/covering-based approaches for the integrated vehicle and crew

scheduling problem. Computers & Operations Research, in press.

[Mesquita et al., 2006] Mesquita, M., Paias, A., and Resp´ıcio, A. (2006). Branch-

ing approaches for the integrated vehicle and crew scheduling. Technical Report

9/2006, Operations Research Center at the University of Lisbon (CIO), Lisbon,

Portugal.

[Mesquita and Paix˜ao, 1992] Mesquita, M. and Paix˜ao, J. M. P. (1992). Multiple

depot vehicle scheduling problem: A new heuristic based on quasi-assignment

algorithms. In Desrochers, M. and Rousseau, J.-M., editors, Proceedings of the

Fifth International Workshop on Computer-Aided Scheduling of Public Trans-

port (CASPT), volume 386 of Lecture Notes in Economics and Mathematical

Systems, pages 167–180, Berlin. Springer.

[Mesquita and Paix˜ao, 1999] Mesquita, M. and Paix˜ao, J. M. P. (1999). Exact

algorithms for the multi-depot vehicle scheduling problem based on multicom-

modity network flow type formulations. In Wilson, N. H., editor, Computer-

202

Bibliography

Aided Transit Scheduling, volume 471 of Lecture Notes in Economics and Math-

ematical Systems, pages 221–243, Berlin. Springer.

[Nemhauser and Wolsey, 1988] Nemhauser, G. and Wolsey, L., editors (1988).

Integer and Combinatorial Optimization. Wiley, New York.

[Nissen and Haase, 2006] Nissen, R. and Haase, K. (2006). Duty-period-based

network model for crew rescheduling in european airlines. Journal of Schedul-

ing, 9(3):255–278.

[Orchard-Hays, 1968] Orchard-Hays, W. (1968). Advanced Linear-Programming

Computing Techniques. McGraw-Hill, New York.

[Orloff, 1976] Orloff, C. S. (1976). Route constrained fleet scheduling. Trans-

portation Science, 10(2):149–168.

[Osman and Laporte, 1996] Osman, I. H. and Laporte, G. (1996). Metaheuris-

tics: A bibliography. Annals of Operations Research, 63(5):513–623.

[Oukil et al., 2007] Oukil, A., Amor, H. B., Desrosiers, J., and Gueddari, H. E.

(2007). Stabilized column generation for highly degenerate multiple-depot ve-

hicle scheduling problems. Computers & Operations Research, 34:817–834.

[PaderSprinter, 2007] PaderSprinter (2007). PaderSprinter - public transport

company in Paderborn. available at http://www.padersprinter.de/ (last

access on June 13th, 2007).

[Paix˜ao and Branco, 1987] Paix˜ao, J. M. P. and Branco, I. (1987). A quasi-

assignment algorithm for bus scheduling. Networks, 17:249–269.

[Patrikalakis and Xerocostas, 1992] Patrikalakis, I. and Xerocostas, D. (1992). A

new decomposition scheme of the urban public transport scheduling problem.

In Desrochers, M. and Rousseau, J.-M., editors, Proceedings of the Fifth Inter-

national Workshop on Computer-aided Scheduling of Public Transport, volume

386 of Lecture Notes in Economics and Mathematical Systems, pages 407–425,

Berlin. Springer.

[Pepin et al., 2006] Pepin, A.-S., Desaulniers, G., Hertz, A., and Huisman, D.

(2006). Comparison of heuristic approaches for the multiple depot vehicle

scheduling problem. Technical Report EI2006-34, Econometric Institute, Eras-

mus University Rotterdam.

[Polyak, 1967] Polyak, B. (1967). A general method of solving extremum prob-

lems. Soviet Mathematics Doklady, 8:593–597.

203

Bibliography

[PTV AG, 2007] PTV AG (2007). ptv interplan - advanced op-

erational planning for public transport companies. available at

http://www.english.ptv.de/cgi-bin/traffic/traf ip.pl (last access on

June 13th, 2007).

[Reeves, 1993] Reeves, C. R., editor (1993). Modern heuristic techniques for

combinatorial problems. Blackwell, London.

[Ribeiro and Soumis, 1994] Ribeiro, C. and Soumis, F. (1994). A column gener-

ation approach to the multiple-depot vehicle scheduling problem. Operations

Research, 42(1):41–52.

[Rodrigues et al., 2006] Rodrigues, M. M., de Souza, C., and Moura, A. (2006).

Vehicle and crew scheduling for urban bus lines. European Journal of Opera-

tional Research, 170:844–862.

[Ryan and Foster, 1981] Ryan, D. M. and Foster, B. (1981). An integer pro-

gramming approach to scheduling. In Wren, A., editor, Computer Scheduling

of Public Transport: Urban Passenger Vehicle and Crew Scheduling, pages

269–280, Amsterdam. North-Holland.

[Sandhu and Klabjan, 2006] Sandhu, R. and Klabjan, D. (2006). Integrated air-

line fleeting and crew pairing decisions. Operations Research, to appear.

[Scott, 1985] Scott, D. (1985). A large scale linear programming approach to the

public transport scheduling and costing problem. In Rousseau, J.-M., editor,

Computer Scheduling of Public Transport 2, pages 473–491. Elsevier Science

Publishers, Amterdam.

[Serafini, 1992] Serafini, P. (1992). Simulated annealing for multiple objective

optimization problems. In Proceedings of the 10th International Conference on

Multiple Criteria Decision Making, pages 87–96, Taipeh, Taiwan.

[Shen and Kwan, 2001] Shen, Y. and Kwan, R. S. (2001). Tabu search for driver

scheduling. In Voß, S. and Daduna, J., editors, Computer-Aided Scheduling of

Public Transport, volume 505 of Lecture Notes in Economics and Mathematical

Systems, pages 121–135, Berlin. Springer.

[Silva et al., 1999] Silva, G. P., Wren, A., Kwan, R. S., and Gualda, N. D. F.

(1999). Bus scheduling based on an arc generation - network flow approach.

Technical report, University of Leeds - School of Computer Studies.

204

Bibliography

[Steinzen, 2007] Steinzen, I. (2007). Instances for multiple-depot vehicle and

crew scheduling. available at http://dsor.upb.de/visteinzen/ (last access

on April 11th, 2007).

[Steinzen et al., 2007a] Steinzen, I., Becker, M., and Suhl, L. (2007a). A hy-

brid evolutionary algorithm for the vehicle and crew scheduling problem in

public transit. In Procs. of the IEEE Congress on Evolutionary Computation

(CEC2007), Singapore. IEEE Press. (accepted for publication).

[Steinzen et al., 2007b] Steinzen, I., Gintner, V., and Suhl, L. (2007b). Local

Branching und Branching-Strategien f¨ur Umlauf- und Dienstplanung im Re-

gionalverkehr mit unregelm¨aßigen Fahrpl¨anen. In G¨unter, H.-O., Mattfeld,

D., and Suhl, L., editors, Management logistischer Netzwerke: Entscheidung-

sunterst¨utzung, Informationssysteme und OR-Tools, pages 407–424. Physica-

Verlag, Heidelberg.

[Suhl et al., 2007] Suhl, L., Kliewer, N., and Steinzen, I. (2007). Opti-

mierungssysteme f¨ur die Dienstplanung im ¨

OPNV. In Oberweis, A., Wein-

hardt, C., Gimpel, H., Koschmieder, A., Pankratius, V., and Schnitzler, B., ed-

itors, eOrganisation: Service-, Prozess, Marketengineering - 8. Internationale

Tagung Wirtschaftsinformatik 2007, pages 447–464, Karlsruhe, Germany. Uni-

versit¨atsverlag Karlsruhe.

[Suhl, 2000] Suhl, U. (2000). MOPS - Mathematical Optimization System. OR

News, 8:11–16.

[Tajima and Misono, 1997] Tajima, A. and Misono, S. (1997). Airline crew-

scheduling with many irregular flights. In Leong, H. W., Imai, H., and Jain,

S., editors, Proceedings of the 8th International Symposium on Algorithms and

Computation - ISAAC97, volume 1350 of Lecture Notes in Computer Science,

pages 2–11, Heidelberg. Springer.

[Tosini and Vercellis, 1988] Tosini, E. and Vercellis, C. (1988). An interactive

system for extra-urban vehicle and crew scheduling problems. In Daduna,

J. R. and Wren, A., editors, Proceedings of the Fourth International Work-

shop on Computer-Aided Transit Scheduling, Lecture Notes in Economics and

Mathematical Systems, pages 41–53, Berlin. Springer.

[Valouxis and Housos, 2002] Valouxis, C. and Housos, E. (2002). Combined bus

and driver scheduling. Computers & Operations Research, 29(3):243–259.

205

Bibliography

[Vance et al., 1997a] Vance, P. H., Atamt¨urk, A., Barnhart, C., Gelman, E.,

Johnson, E. L., Krishna, A., Mahidhara, D., and Rebello, R. (1997a). A heuris-

tic branch-and-price approach for the airline crew pairing problem. Technical

Report LEC-97-06, Georgia Institute of Technology, Atlanta, USA.

[Vance et al., 1997b] Vance, P. H., Barnhart, C., Johnson, E. L., and Nemhauser,

G. L. (1997b). Airline crew scheduling: A new formulation and decomposition

algorithm. Operations Research, 45(2):188–200.

[Vanderbeck, 1994] Vanderbeck, F. (1994). Decomposition and Column Genera-

tion for Integer Programs. PhD thesis, Universit´e Catholique de Louvain.

[Villeneuve et al., 2005] Villeneuve, D., Desrosiers, J., L¨ubbecke, M., and

Soumis, F. (2005). On compact formulations for integer programs solved by

column generation. Annals of Operations Research, 139(1):375–388.

[Wedelin, 1995] Wedelin, D. (1995). An algorithm for large scale 0-1 integer

programming with application to airline crew scheduling. Annals of Operations

Research, 57:283–301.

[Westerlund et al., 2006] Westerlund, A., G¨othe-Lundgren, M., and Larsson, T.

(2006). A stabilized column generation scheme for the traveling salesman sub-

tour problem. Discrete Applied Mathematics, 154(15):2212–2238.

[Wolsey, 1998] Wolsey, L. A. (1998). Integer Programming. Wiley Interscience,

New York.

[Wren and Rousseau, 1995] Wren, A. and Rousseau, J.-M. (1995). Bus driver

scheduling - an overview. In Daduna, J., Brance, I., and Paixao, J., editors,

Computer-Aided Transit Scheduling, volume 430 of Lecture Notes in Economics

and Mathematical Systems, pages 173–187, Berlin. Springer.

[Wren and Wren, 1995] Wren, A. and Wren, D. O. (1995). A genetic algorithm

for public transport driver scheduling. Computers & Operations Research,

22(1):101–110.

[Xu et al., 2003] Xu, H., Chen, Z.-L., Rajagopal, S., and Arunapuram, S. (2003).

Solving a practical pickup and delivery problem. Transportation Science,

37(3):374–364.

[Yunes et al., 2005] Yunes, T., Moura, A., and de Souza, C. (2005). Hybrid

column generation approaches for urban transit crew management problems.

Transportation Science, 39(2):273–288.

206

Bibliography

[Zitzler et al., 2002] Zitzler, E., Laumanns, M., and Thiele, L. (2002). Spea2:

Improving the strength pareto evolutionary algorithm for multiobjective opti-

mization. In Giannakoglou, K., Tsahalis, D., P´eriaux, J., Papailiou, K., and

Fogarty, T., editors, Evolutionary Methods for Design, Optimisation and Con-

trol, pages 95–100, Barcelona, Spain. CIMNE.

[Zitzler and Thiele, 1998] Zitzler, E. and Thiele, L. (1998). Multiobjective opti-

mization using evolutionary algorithms - a comparative case study. In Eiben,

A., B¨ack, T., Sch¨onauer, M., and Schwefel, H., editors, Parallel Problem Solv-

ing from Nature — PPSN V, volume 1498 of Lecture Notes in Computer Sci-

ence, pages 292–301, Berlin, Germany. Springer.

[Zitzler and Thiele, 1999] Zitzler, E. and Thiele, L. (1999). Multiobjective evo-

lutionary algorithms: a comparative case study and the strength pareto ap-

proach. IEEE Transactions on Evolutionary Computation, 3(4):257–271.

[Zitzler et al., 2003] Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C., and

da Fonseca, V. G. (2003). Performance assessment of multiobjective optimiz-

ers: An analysis and review. IEEE Transactions on Evolutionary Computation,

7(2):257–271.

207