maintenance scheduling and vehicle routing optimisation

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Maintenance Scheduling and Vehicle RoutingOptimisation with Stochastic Components

Andres Felipe Gutierrez Bonilla

To cite this version:Andres Felipe Gutierrez Bonilla. Maintenance Scheduling and Vehicle Routing Optimisation withStochastic Components. Operations Research [cs.RO]. Université de Technologie de Troyes; Universi-dad de los Andes (Bogotá), 2018. English. NNT : 2018TROY0023. tel-03212070

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THESE

pour l’obtention du grade de

DOCTEUR de l’UNIVERSITE

DE TECHNOLOGIE DE TROYES

Spécialité : OPTIMISATION ET SURETE DES SYSTEMES

présentée et soutenue par

Andres Felipe GUTIERREZ BONILLA

le 10 juillet 2018

Maintenance Scheduling and Vehicle Routing Optimisation

with Stochastic Components

JURY

M. C. BRIAND PROFESSEUR DES UNIVERSITES Président

M. C. A. AMAYA PROFESOR ASOCIADO Rapporteur

Mme L. DIEULLE MAITRE DE CONFERENCES Examinateur

M. D. FEILLET PROFESSEUR MINES SAINT-ETIENNE Rapporteur

Mme N. LABADIE MAITRE DE CONFERENCES - HDR Directrice de thèse

M. C. PRINS PROFESSEUR DES UNIVERSITES Examinateur

Mme N. VELASCO PROFESORA ASOCIADA Directrice de thèse

Acknowledgments

First of all I will like to thank You. After all you are the purpose and reason of my life. Then, I

would like to thank my advisers Nacima Labadie and Nubia Velasco. Although there are no words to

describe how grateful I am, I will give it a try. I have learned with you a lot about my thesis, solution

methods, problem modeling, and generally speaking about the scientic community. But also, I have

learned much more about life. Your support through all this years was essential for me. Even with

all my mistakes (and missed deadlines) you always had the words to motivate me. You were the best

advisers I could dream for, and my gratitude and friendship is yours forever. Also I would like to thank

Laurence Dieulle, my own third adviser. Your support and advice during the thesis was incredible.

Our discussions were always helpful for me to develop new ideas, even if sometimes I found it hard to

express the details of the methods I was thinking about. I have learned so many things with you and

I expect that you have learned at least a little bit with me.

An special word is given to thank Dominique Feillet, Ciro Amaya, Cyril Briand, and Christian

Prins for being part of the jury. It is both an honor and a motivation to present to you the results of

these years of my life. I really appreciate all the comments and discussions that we have during the

defense. They gave me lights to open my research. Besides, I owe my gratitude to the professors and

the administrative body of both LOSI laboratory at the University of Technology of Troyes, and from

Management School at Los Andes University. Special thanks to professor Carlos Davila who taught

me so much about history and critical thinking. Likewise I thank Véronique, Bernadette, Pascale,S

Isabelle, Ingrid, Juan Carlos, Fredy, and Adriana, for all the help through all these years.

Also I would love to thank my family. Dad and Mom, it was due to you that I could make it up

to here. There is still a lot more to conquer but every success in my life is to honor you. You are the

kindest people that I have ever met and I only hope to follow your steps. To my brothers, Reuben,

Juan Carlos and Miguel, their wives Sonia and Sabina, and my beautiful nephews Sebastian, Martin,

Emilio, Arian, and the coming one. You are the engine of my dreams. Camila, from all the people,

you might be the one who suered the most my frustrations, sadness and worries. You, always had

the right words to comfort me, your voice always has the power to cheer me up, and forever I will love

you. Your family will be always mine too so Ruth, Juan Jose, Majo, Camilo, Sebas, Juan Pablo, thank

you very much. I can not forget Santi and Irene, my uncle Oscar and my ants Heidy and Rosario,

even at the distance you were always at my side.

In addition, I want to acknowledge my friends that make the life at Troyes and at Bogota much

more fun. Colombia team number one, Jorge, Caro, Karen, Guille, Adri, Elyn, Syrine, Marie, and

Colombian team number two, Andrea, Felipe, David, Aleja, Samir, So, Nacef, and Mourad. It was

always great to go out with you. Back to Colombia my special gratitude to Fabian (Crazy) and

Enrique. Your advice and the challenging and stimulating conversations always keep pushing the

boundaries of my knowledge. There are a lot more people that were fundamental in this process. If

I forget somebody it was not on purpose and I hope life will let me made up to you. Additionally,

I thank the anonymous referees that reviewed my work through these years, your comments always

helped to improve our work, and though your names are nor revealed, you will never be forgotten.

Finally, this thesis was possible thanks to the nancial support of the Champagne-Ardenne Region in

France, and COLCIENCIAS in Colombia, I praise your aid to science development.

i

Contents

1 Introduction 1

2 Literature Review - Vehicle Routing Problems 7

2.1 Deterministic VRP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Capacitated Vehicle Routing Problem . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 VRP with Time Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.3 Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 VRP under uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Stochastic VRP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.2 Robust Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.3 Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Hybrid metaheuristic for the VRPSD 41

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3 Solution approach: hybrid metaheuristic . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.1 Chromosomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.2 Initial population and Restart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.3 Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3.4 Mutation and Local Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4.1 Classical testbed from Christiansen and Lysgaard . . . . . . . . . . . . . . . . . 49

3.4.2 New proposed testbed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4 Vehicle Routing Problem with stochastic travel and service times 67

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2 Problem Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3 Estimation of arrival times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3.1 Arrival and starting service times denition . . . . . . . . . . . . . . . . . . . . 71

4.3.2 Mean and Variance estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3.3 Validating the log-normality approximation . . . . . . . . . . . . . . . . . . . . 73

4.4 Multi-population Memetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.4.1 MA general structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.4.2 Chromosomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.4.3 Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.4.4 Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.4.5 Local search and mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.4.6 MPMA framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

iii

4.4.7 MA dierences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.5.1 Instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.5.2 General discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.5.3 The eects of considering multiple populations . . . . . . . . . . . . . . . . . . 83

4.5.4 MPMA + Log-normal approximation comparisons . . . . . . . . . . . . . . . . 83

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5 Wind farms maintenance 95

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2 Review on operational maintenance activities . . . . . . . . . . . . . . . . . . . . . . . 96

5.3 Operational maintenance level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.4 A multi-objective approach to the maintenance scheduling problem . . . . . . . . . . . 100

5.4.1 Problem Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.4.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.4.3 The Epsilon constraint approach . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.5 Main contributions on strategic decision level . . . . . . . . . . . . . . . . . . . . . . . 106

5.5.1 Maintenance strategy selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.5.2 Complex models for maintenance strategy selection . . . . . . . . . . . . . . . . 109

5.6 Maintenance strategies: relation with operational planning . . . . . . . . . . . . . . . . 112

5.6.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.6.2 Simulation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.6.3 Weather model - Wind speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.6.4 Schedule Modeler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.6.5 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.6.6 Model implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.6.7 Simulation model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6 Conclusions 129

Appendices 133

A Résumé en français 135

A.1 Introduction aux VRP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

A.2 Une méthode hybride pour les VRP avec demandes stochastiques . . . . . . . . . . . . 140

A.3 Un algorithme parallèle pour les VRP avec temps de trajet et temps de service stochas-

tiques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

A.4 Gestion des ressources pour la maintenance d'un parc d'éoliennes . . . . . . . . . . . . 149

A.4.1 Ordonnancement multi objective pour le problème de maintenance des éoliennes 149

A.5 Sélection des stratégies de maintenance et relation avec l'ordonnancement . . . . . . . 151

A.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

B List of contributions 159

iv

List of Figures

2.1 Solution approaches scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1 Split example for the VRPSD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Broken pairs distance example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Avg. gap of MA+GRASP against the lling coecient . . . . . . . . . . . . . . . . . . 59

3.4 Avg. gap of MA+GRASP against the number of vehicles in the BKS . . . . . . . . . . 60

3.5 Avg. gap of MA+GRASP and number of vehicles against the number nodes per instance 60

3.6 MTTT Plots for the dierent MA methods - 5% and 1% . . . . . . . . . . . . . . . . . 62

3.7 MTTT Plots for the dierent MA methods - 0.5% and 0% . . . . . . . . . . . . . . . . 63

4.1 Arrival and starting service times example. . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2 Average absolute error of 95% percentile with service and travel time noises. . . . . . . 75

4.3 Split example for the VRP with stochastic travel and service times . . . . . . . . . . . 77

4.4 Example of OX crossover. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.1 Approximate Pareto Front for Froger et al. [31] instance 10_2_1_20_B_5 . . . . . . 105

5.2 Scheme of the simulation modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.3 An example of a turbine modeled a multi-component system. . . . . . . . . . . . . . . 115

5.4 Total maintenance costs by strategy/rule for the wind farm simulation. . . . . . . . . . 118

5.5 Total number of failures by strategy/rule for the wind farm simulation. . . . . . . . . . 119

5.6 Total produced energy under dierent strategy/rules for the wind farm simulation. . . 119

5.7 Mean turbine availability for dierent strategy/rules for the wind farm simulation. . . 120

5.8 Temporal analysis for wind farm metrics under dierent strategies. . . . . . . . . . . . 120

A.1 Classication des méthodes de résolution pour les VRP. . . . . . . . . . . . . . . . . . 137

A.2 grapheique MTT pour les versions de MA - 5% and 1% . . . . . . . . . . . . . . . . . 143

A.3 grapheique MTT pour les versions de MA - 0.5% and 0% . . . . . . . . . . . . . . . . 143

A.4 Temps d'arrivée et de début des services chez un client. . . . . . . . . . . . . . . . . . 145

A.5 Solutions Pareto optimales pour l'instance 10_2_1_20_B_5 de Froger et al. [11] . . 151

A.6 Coûts par stratégie et par règle d'assignation. . . . . . . . . . . . . . . . . . . . . . . . 153

A.7 Énergie produite par stratégie et par règle d'assignation. . . . . . . . . . . . . . . . . . 153

v

List of Tables

2.1 VRPs taxonomy based on Pillac et al. [139] - Information evolution and quality . . . . 14

2.2 Summary of VRPSD literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Summary of VRPST literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1 VRPSD Christiansen and Lysgaard [7] Testbed comparison . . . . . . . . . . . . . . . 49

3.2 Christiansen and Lysgaard [7] testbed results . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 Christiansen and Lysgaard [7] testbed results: Continued . . . . . . . . . . . . . . . . 52

3.3 Summary of new testbed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 VRPSD Proposed testbed results - Costs . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5 VRPSD Proposed testbed results - Time, Gaps . . . . . . . . . . . . . . . . . . . . . . 57

3.6 Two-way test values for Friedman method . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.7 Friedman test ranks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.1 Basic experiment information per family of instances . . . . . . . . . . . . . . . . . . . 74

4.2 Arrival times average absolute gaps between simulated values and log-normal approx-

imation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3 Arrival times mean and standard deviation absolute gaps between simulated values and

three approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.4 Best solutions found by MPMA for C type instances . . . . . . . . . . . . . . . . . . . 82

4.5 Best solutions found by MPMA for R type instances . . . . . . . . . . . . . . . . . . . 82

4.6 Best solutions found by MPMA for RC type instances . . . . . . . . . . . . . . . . . . 82

4.7 Average performance of MPMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.8 Comparison of single MA1 to MPMA - 100 customer instances . . . . . . . . . . . . . 84

4.9 MPMA comparison to Miranda and Conceição [34] ILS . . . . . . . . . . . . . . . . . 84

4.10 MPMA comparison to Nguyen et al. [39] TS . . . . . . . . . . . . . . . . . . . . . . . 85

4.11 50 Customer instances best and average solutions found by MPMA per instance . . . 87

4.11 50 Customer instances best solutions and average found by MPMA per instance: Con-

tinued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.12 100 Customer instances best solutions and average found by MPMA per instance . . . 89

4.12 100 Customer instances best solutions and average found by MPMA per instance: Con-

tinued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.1 Summary of wind farms maintenance scheduling works, objectives, and types of models 98

5.2 Epsilon Constraints summary results for Froger et al. [31] instances . . . . . . . . . . 105

5.3 Summary of literature with complex models for maintenance strategy selection . . . . 110

5.4 Weibull parameters for components failures in Abdollahzadeh et al. [1] . . . . . . . . . 115

5.5 Costs, technicians, and time requirements per maintenance component based on Ab-

dollahzadeh et al. [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

A.1 Taxonomie des VRPs basée sur l'article de Pillac et al. [20] . . . . . . . . . . . . . . . 138

A.2 Synthèse des résultats sur le nouvel ensemble d'instances . . . . . . . . . . . . . . . . . 143

vii

A.3 Comparaison des diérentes MPMA Instances avec 100 clients . . . . . . . . . . . . 147

A.4 Comparaison du MPMA avec la métaheuristique ILS de Miranda et Conceição [18] . 147

A.5 Comparaison de MPMA avec la TS de Nguyen et al. [19] . . . . . . . . . . . . . . . . 148

viii

Chapter 1

Introduction

The world of science lives fairly comfortably with paradox. We know that light is a wave, and also

that light is a particle. The discoveries made in the innitely small world of particle physics

indicate randomness and chance, and I do not nd it any more dicult to live with the paradox of

a universe of randomness and chance and a universe of pattern and purpose than I do with light as

a wave and light as a particle. Living with contradiction is nothing new to the human being"

Madeleine L'Engle

The transport activities are one of the more important drivers in many of the economic activities.

In fact, according with the International Trade Administrator, logistics and transportation activities1 accounted for 8% of the Gross Domestic Product in the United States in 2015. Additionally, only

considering the logistic activities, transportation can represent up to 60% of the total costs [16],

making this topic an important area to study.

In the Operations Research context, the transportation problems and specically the Vehicle Rout-

ing Problems (VRPs) has been one of the most studied problems with more than a thousand published

papers between 1954 to 2006 [6]. Overall, the VRP consist of nding a set of minimal cost routes

performed by a set of vehicles, satisfying a set of clients, and respecting specic constraints, according

to the particular context. Furthermore, most of the works on this eld consider that all problem

parameters, travel and service times, and demands, are known in advance. Actually, Braekers et al.

[3] identify that more than 80% of 277 published articles between 2009 and 2015 in the VRPs context

are deterministic.

Many factors could aect the certitude of the information on problems parameters. For example,

the cities population concentration and their consequent trac congestion make the travel times un-

certain [11]. Furthermore, the accelerated use of Information Technologies could generate incertitude

about the requests because clients could made them more frequently and are subject to random ex-

ternal factors. Moreover, the time spent at each location visited by the vehicles can change because

of the complexity of the task to satisfy at the clients or due to the environmental conditions.

The consequence of neglecting the variability is that the solutions obtained could perform badly

in real uncertain environment [17]. Actually, based on a previously literature results, ignoring the

incertitude of the information could increase up to 10% the costs for the stochastic demands case

[8] or 4% for stochastic times [1]. Moreover, the amount of the objective (costs or utility) variation

between deterministic and stochastic solutions depends on the randomness of the problem and can be

as signicant as 20% when travel and service are uncertain [4].

In recent years, some authors have been working on stochastic VRPs, one of the most compre-

hensive study about this topic is given by Gendreau et al. [10]. The authors focus their attention

on the stochastic programming modeling which is the predominantly approach in the eld. Thus,

1Accessed the rst may 2018 - https://www.selectusa.gov/logistics-and-transportation-industry-united-states

1

only parameters which are dened by random variables or scenarios are considered2. Furthermore, to

classify the dierent problems the authors consider two main characteristics: the solution paradigm

and the stochastic parameters (customers, demands, and times). Solution paradigms are divided in

two: a priori and reoptimization paradigms. The former, consider that solutions are created before

any information is revealed (rst phase) and they are barely modied during the execution (second

phase). Meanwhile, reoptimization approach aims to modify the solution as new information becomes

available to improve. The latter approach is more related with the Dynamic problems (see Psaraftis et

al. [18] for a recent review). A priori paradigms can be further divided into Stochastic Problems with

Recourse (SPR) and Chance Constrained Problems (CCP). The SPR use recourse to react to failures,

or constraints violations during the second phase. Meanwhile, the CCP bounds the probability of

possible failures appearing during the second phase.

The a priori paradigm has been the predominant approach to solve stochastic VRPs, particularly

using SPRs. This can be explained by the fact that this type of models allow stable tactical routes

which are operationally desirable [9]. Moreover, according Bekta et al. [2], a priori paradigm is

suitable when anticipating uncertainty is crucial to nd feasible solutions and avoid penalties (economic

and reputation). Nevertheless, the a priori paradigm adds complexity of dealing with probability

calculus overhead. This makes that the size of stochastic problems that can be solved (exactly and

approximately) is rather small. For example, Gauvin et al. [8] are able to optimally solve only one

instance with 100 customers in the context of the VRP with stochastic demands. Therefore, there is

a need to develop solution methods able to tackle closer to real problems settings, in terms of size,

multiple uncertainties, and assumptions.

The literature review reveals that there is a lack of comparisons among dierent methods and

works between stochastic VRPs, which might be caused by the lack of standardized benchmarks.

Although each problem has its own characteristics (type of distribution, amount of variance, etc.)

base tests serve to prove the usefulness of new models and strategies to solve the stochastic VRPs.

Moreover, these test cases need to evolve, particularly in terms of size, to prove the capacity of the

new methods to deal with closer to life real problems. Being able to solve this type of problems will

demonstrate their capacity to fully exploit the benets of stochastic solutions over deterministic ones.

Last but not least, new stochastic models need to incorporate real life constraints, such as hard time

windows, a characteristic that has been largely studied in deterministic VRPs but not so much in the

stochastic ones.

This thesis addresses two kinds of VRPs using the stochastic programming framework and the a

priori paradigm: the rst one considers the demand as a random variable and is reported in chapter 3

where results for middle and large instances are reported, second it tackles in chapter 4 the stochasti-

city on the travel and service time. The latter under the presence of hard time windows which can

conduce to unserviced customers and using dierent types of continuous distribution for the stochastic

parameters. Both problems are presented in the context of maintenance operations for which anticipat-

ing uncertainty is imperative. The last problem studied in this thesis considers maintenance planning

on wind farms. It extends the stochastic VRPs in which technicians are to be scheduled to perform

their tasks under the appearance of uncertain new tasks (demands), random weather conditions, and

with stochastic service times. The present thesis is developed as follow:

Chapter 2 introduces an extensive review on the Vehicle Routing Problems (VRPs). Starting

with the description of deterministic VRPs, it makes its path to uncertain VRPs as their natural

evolution. Then, the attention is focused in the three main paradigms to model uncertain VRPs,

namely, Stochastic Optimization, Interval Optimization and Fuzzy Logic. Special consideration is

given to the Static Stochastic VRPs with a comprehensive review of the solution approaches and

dierent problems variants tackled in the literature. This revision shows that albeit the increase of

2Nevertheless, the parameters can be also modeled by sets (Robust Optimization) or by Fuzzy Variables. Moreinformation is given in chapter 2

2

CHAPTER 1. INTRODUCTION

research in the eld, there is still a lack of detailed results for big instances. Moreover, a lack of

research for the stochastic VRPs with hard time windows is established. Especially for the case with

stochastic travel and service times modelled by continuous random variables, so it is likely they are

in real applications.

Chapter 3 is devoted to the VRP with stochastic demands (VRPSD). In the VRPSD, customers

demands are modeled by random variables and their realization value are only known when the

vehicles arrive at the customers. A simple classical recourse action is used to model the VRPSD as

a stochastic problem with resource. To tackle the VRPSD a Greedy Randomized Adaptive Search

Procedure (GRASP) is used to restart a Memetic Algorithm (MA) and eciently solve the problem at

hand. In this chapter it is shown that large instances (up to 385 customers) can be eectively handled

by the MA+GRASP. A comparison with the state-of-the-art algorithms for the VRPSD shows that

the MA+GRASP provides better and more accurate solutions in very competitive computational

times. Moreover, the chapter establishes a new testbed of instances (based on instances already used

in deterministic context) with a higher number of customers than the traditional Christiansen and

Lysgard benchmark [5]. These results are important to open the space to discussion and further

design of other methods to VRPSD. The work presented in this chapter is under minor revision in

the Computers & Operations Research journal and an earlier version was presented at the CIE45

conference [12].

Chapter 4 focuses on a VRP with uncertain times. It presents a VRP considering stochastic travel

and service times with hard time windows. The problem is thought-out in a maintenance activities

context and the uncertainty in times are modeled through continuous probability distributions. A

model is proposed to enable the control of customers service levels but also considers the implica-

tions of missing the time windows. To overcome the problem of modeling the arrival times, it is

shown that they can be fairly approximated using a log normal distribution. To solve the problem, a

Multi-population Memetic Algorithm (MPMA) exploiting dierent characteristics in each population

(running in parallel) is proposed. Results are presented for instances with up to 100 customers derived

from the the Solomon [19] benchmark. Additionally, the MPMA is compared against state of the art

methods although these allow late services, and the proposed approach shows very good performance.

The results of this third chapter are gathered in a paper which is accepted for publication in the

Computers & Industrial Engineering journal. Preliminary results were presented at MIM2016 [14]

and CLAIO2016 [13] conferences.

Chapter 5 introduces a general review of the wind farms maintenance activities. Two related prob-

lems are further explored in this context. First, a multi-objective approach to deal with maintenance

scheduling of wind farms is addressed. In this problem the operator of the wind farm needs to decide

the order, time, and resources assignation to execute a set of maintenance tasks in a short term ho-

rizon. Moreover, the operator aims to minimize its costs while the investors want to maximize the

energy production. A linear integer model is used to model the problem and the epsilon-constraint

method is designed to approximate the optimal Pareto front. Tests are performed on the set of in-

stances proposed by Froger et al. [7] pointing out that objectives are in conict. Also it is shown

that the variation of energy production in the short term can be highly aected by the scheduling

of the activities. The second problem, extends the scheduling problem and explores the selection

of maintenance strategies for wind farms. Dierent strategies are evaluated within a event discrete

simulation approach to compare them on a long-term horizon. Furthermore, dierent ways of solving

the scheduling the maintenance tasks in the short term are compared within the simulation. Failures

appearances as well as maintenance times are considered as random variables. The proposed shows

that even simple heuristic rules are used to tackle the scheduling of technicians, they can have import-

ant eects on both the costs and the energy production. The results for the scheduling maintenance

activities problem considering multiple objectives were presented at IEOM 2017 conference held at

Bogota [15].

3

Finally the thesis ends with chapter 6 drawing the conclusions of the thesis and research clues for

future research on the uncertain vehicle routing problems, scheduling of maintenance activities, and

strategy selection in the wind farm context.

4

Bibliography

[1] N. Ando and E. Taniguchi. Travel time reliability in vehicle routing and scheduling with time

windows. Networks and Spatial Economics, 6(3):293311, 2006.

[2] T. Bektas, P. P. Repoussis, and C. D. Tarantilis. Chapter 11: Dynamic vehicle routing problems.

In Vehicle Routing: Problems, Methods, and Applications, Second Edition, pages 299347. SIAM,

2014.

[3] K. Braekers, K. Ramaekers, and I. V. Nieuwenhuyse. The vehicle routing problem: State of the

art classication and review. Computers & Industrial Engineering, 99:300 313, 2016.

[4] A. M. Campbell, M. Gendreau, and B. W. Thomas. The orienteering problem with stochastic

travel and service times. Annals of Operations Research, 186(1):6181, 2011.

[5] C. H. Christiansen and J. Lysgaard. A branch-and-price algorithm for the capacitated vehicle

routing problem with stochastic demands. Operations Research Letters, 35(6):773 781, 2007.

[6] B. Eksioglu, A. V. Vural, and A. Reisman. The vehicle routing problem: A taxonomic review.

Computers & Industrial Engineering, 57(4):1472 1483, 2009.

[7] A. Froger, M. Gendreau, J. E. Mendoza, E. Pinson, and L.-M. Rousseau. Solving a wind turbine

maintenance scheduling problem. Journal of Scheduling, 21(1):5376, 2018.

[8] C. Gauvin, G. Desaulniers, and M. Gendreau. A branch-cut-and-price algorithm for the vehicle

routing problem with stochastic demands. Computers & Operations Research, 50:141 153, 2014.

[9] M. Gendreau, O. Jabali, and W. Rei. 50th anniversary invited articlefuture research directions

in stochastic vehicle routing. Transportation Science, 50(4):11631173, 2016.

[10] M. Gendreau, O. Jabali, W. Rei, P. Toth, and D. Vigo. Stochastic vehicle routing problems.

Vehicle Routing: Problems, Methods, and Applications, 18:213, 2014.

[11] A. R. Güner, A. Murat, and R. B. Chinnam. Dynamic routing under recurrent and non-recurrent

congestion using real-time its information. Computers & Operations Research, 39(2):358 373,

2012.

[12] A. Gutierrez, L. Dieulle, N. Labadie, and N. Velasco. A memetic algorithm for the vehicle

routing problem with stochastic demands. In Proceedings of the CIE 45 International Conference

on Computers and Industrial Engineering, 2015.

[13] A. Gutierrez, L. Dieulle, N. Labadie, and N. Velasco. An approximate column generation for the

vehicle routing problem with hard time windows and stochastic travel and service times. 2016.

[14] A. Gutierrez, L. Dieulle, N. Labadie, and N. Velasco. A multi population memetic algorithm

for the vehicle routing problem with time windows and stochastic travel and service times. In

Proceedings of the 8th IFAC Conference on Manufacturing Modelling, Management & Control,

2016.

5

BIBLIOGRAPHY

[15] A. Gutierrez, L. Dieulle, N. Labadie, and N. Velasco. Wind farm maintenance scheduling model

and solution approach. In Proceedings of the International Conference on Industrial Engineering

and Operations Management, 2017.

[16] M. Hesse and J.-P. Rodrigue. The transport geography of logistics and freight distribution.

Journal of Transport Geography, 12(3):171 184, 2004.

[17] F. Louveaux. An Introduction to Stochastic Transportation Models, pages 244263. Springer

Berlin Heidelberg, Berlin, Heidelberg, 1998.

[18] P. H. N., W. Min, and K. C. A. Dynamic vehicle routing problems: Three decades and counting.

Networks, 67(1):331.

[19] M. M. Solomon. Algorithms for the vehicle routing and scheduling problems with time window

constraints. Operations Research, 35(2):254265, 1987.

6

Chapter 2

Literature Review - Vehicle Routing

Problems

Nearly 60 years have passed since the introduction of the vehicle routing problem (VRP). The rst

literature appearance of the VRP can be traced back to the seminal work of Dantzig and Ramser

[41] named The Truck Dispatching Problem. Since 1959, the number of articles and applications

have grown tremendously. Since that year, a general search for the Vehicle Routing Problem in a

searching engine such as Google Scholar gives more than 20.000 results that can be reduced to over

4600 if the words are present in the document title. Eksioglu et al. [49] reviewed nearly 1500 VRP

references from 1954 to 20061, which included journal articles, books, book chapters, technical reports,

and conference articles. The authors proposed a taxonomy to classify the vast literature and asserted

that it grew exponentially at a rate of approximately six percent per year. A more recent classication

work can be found in Braekers et al. [30] where 277 VRP journal articles from 2009 to mid-2015 were

arranged using a taxonomy similar to the one used in Eksioglu et al. [49].

The massive amount of research related to the VRP can be twofold explained. First, transportation

plays a central part in many human activities, economics, and the environment. According to Hesse

and Rodrigue [82] transportation accounted for nearly 6% of the Gross Domestic Product (GDP)

of the United States in the year 2000. Moreover, transportation transcends the purely economic

trend. In fact, the subject has been studied in the context of disaster relief and humanitarian logistics

([74, 34]), and services delivery (health care [55], technicians [147], among other). Second, the eld

has been the seed for many developments of several exact and heuristic methods for combinatorial

optimization problems [102]. These developments have an impact in other elds in the Operational

Research community [84] and therefore make the VRP research an active and important part of the

scientic development.

Besides, the nearly sexagenarian problem has seen a myriad of variants and extensions of its

basic version. Either by the addition of more characteristics, constraints, or changes on the objective

function, new problems have risen to adapt the VRP to many contexts. Within this variety, the last

few decades have seen an increment in the study of problems where the parameters information is not

certain [63]. Beyond the pure theoretical value of the works, applications deal with a reality in which

information is far from perfect and stochastic (weather, accidents, drivers skills, etc.). Moreover,

despite the usual necessary eort to solve problems with uncertain information, its value is not trivial

[6, 33, 60]. Therefore, the uncertain VRPs are an important eld to make both theoretical and

practical research.

This chapter presents a review of the VRP. It starts by introducing one of the most basic and

known version of the problem, the Capacitated Vehicle Routing Problem (CVRP). The CVRP serves

1The authors use the date 1954 as the rst VRP record in the literature considering the work of Dantzig et al. [40]on the Traveling Salesman Problem (TSP), a particular case of the VRP

7

2.1. DETERMINISTIC VRP

as proxy to introduce the mathematical formulations and a very special generalization called the VRP

with Time Windows (VRPTW). Then, the uncertain VRPs modeling and solution approaches are

explored. The chapter ends with concluding remarks on the importance of stochastic VRPs.

2.1 Deterministic VRP

2.1.1 Capacitated Vehicle Routing Problem

In its basic form, the Capacitated Vehicle Routing Problem (CVRP) can be dened by a complete

undirected graph G = (V,E) where V = 0, 1, . . . , i, . . . , n and E = [i, j]∀i, j ∈ V | i < j arethe vertex and the edge sets respectively. Moreover, let V c = V \ 0 be the customers subset. Eachcustomer has a non-negative demand qi. Vertex 0 is a depot where is located a set of homogeneous

vehicles with limited capacity Q. Furthermore, each edge [i, j] ∈ E has a non-negative cost cij . The

objective of the CVRP is to build a set of routes with minimum cost considering that each route

starts and ends at the depot, the vehicle capacity Q must be respected, and no split deliveries are

allowed. Moreover, a generic route r is dened as an ordered sequence of nodes r = r0, r1, . . . , rj ,. . . , rk, rk+1 where rj represents the jth visited node. Each vehicle starts and ends its route at the

depot, therefore, r0 = rk+1 = 0 for every route. Even more, each route r has an associated cost

Cr =∑kj=0 crj ,rj+1 .

Several other extensions and variants for the CVRP have been proposed aiming to bring the models

closer to real life applications. Among these, one can nd the Distance Constrained VRP (DVRP)

which limits the total distance traveled by each vehicle to a threshold [19, 5]; the Heterogeneous VRP

(HVRP) where the eet of vehicles is, as its name states, heterogeneous (in terms of capacity or costs)

[7, 117, 138]; the Multi Depot VRP (MDVRP) which considers multiple depots where the vehicles start

and end their routes[146, 172]; the Periodic VRP (PVRP) that requires repeated visits to customers

[57]; the open VRP (OVRP) which does not require vehicles to return to depot after serving the last

customer [116, 117]; the Orienteering Problem (OP) where customers have an associated prot (or

score) collected by a xed size eet not necessarily sucient to visit all the customers, and the objective

is to maximize the total prot [170, 98, 80]. Other variants include more additional constraints such as

the Pickup and Delivery Problems (PDP) where people or goods must be transported from dierent

origins to dierent destinations [135, 136, 17, 18]. The reader is referred to the mentioned bibliography

and to Toth and Vigo [169] for further details on variants of the CVRP.

CVRP Mathematical formulations

Three formulations are mainly used to model the CVRP [154, 101], the vehicle ow, the commod-

ity ow and the set partitioning formulations. The vehicle ow formulation uses integer variables

xij ∀i, j ∈ V to represent the number of times an edge is used in the optimal solution [107, 108].

Model M1CVRP presents a classical two-index network formulation.

M1CV RP : min∑i,j∈V

xijcij (2.1)

∑j∈V c

x0j = 2m (2.2)

∑i<p|i∈V

xip +∑

j>p|j∈V

xpj = 2 ∀p ∈ V c (2.3)

∑i∈S,j /∈Sor

i/∈S,j∈S

xij ≥ 2b(S) ∀S ⊂ V c (2.4)

8

CHAPTER 2. LITERATURE REVIEW - VEHICLE ROUTING PROBLEMS

xij ∈ 0, 1 ∀i, j ∈ V c (2.5)

x0j ∈ 0, 1, 2 ∀j ∈ V c (2.6)

In M1CVRP the objective (2.1) minimizes the total costs associated with the used edges. Con-

straint (2.2) determines the degree of the depot, using m as the number of vehicles (m can be a

variable). Constraint (2.3) ensures that any vehicle visiting a customer must leave to another node.

Constraint (2.4) serves to guarantee capacity restrictions as well as to prevent subtours formation,

that is, ensemble of connected customers without being linked to the depot. Practically, the term b(S)

can be set to⌈∑

i∈SqiQ

⌉, therefore, b(S) is a lower bound on the number of vehicles needed to satisfy

the demand of the subset of customers S. Furthermore, constraints (2.5) and (2.6) stand for variables

nature.

The second formulation, called commodity ow formulation makes use of continuous variables to

model the amount of vehicle load and empty space on the vehicle, when an edge is used. The reader is

referred to Baldacci et al. [9] for a complete formulation. The third formulation is the set partitioning

one. This one relies on the enumeration of all feasible routes which are then selected through a

set partitioning problem [11]. Associated to each route a binary variable serves to decide if it is

included within the solution or not. The reader is referred to Laporte [101] for the complete model.

Other formulations beside the three described can be used. For instance, three-index formulations

add an index to identify each vehicle separately. This types of models are useful when particular

characteristics of each route aect its feasibility or cost. One of the most common VRP extension

modeled by three-index formulation is the one with Time Windows (VRPTW), where time constraints

are imposed for the customers visits. It is now explored in more depth.

2.1.2 VRP with Time Windows

The Vehicle Routing Problem with Time Windows (VRPTW) is one of the most important and well-

studied VRPs. Usually the VRPTW uses an extended graph G =(V ,A

). The vertex set V includes

an exact copy of the depot node called n+1, therefore, V = V ∪n+ 1. Moreover A stands for the arcs

set dened as as A = (i, j)∀i, j ∈ V | i 6= j. In addition, each arc (i, j) takes a time tij ∀ (i, j) ∈ Ato be traversed. Also, each customer requires a time ti ∀ V c to be served.

The VRPTW extends the CVRP by dening a time window [ei, li]∀i ∈ V c. The vehicle must

start to service the customer during this lapse of time. Time windows constraints can be dened

as hard or soft [43]. In the hard version, the vehicles cannot start their services outside the time

windows. Nevertheless, early arrivals, i.e. arriving before ei, are possible but vehicles must wait until

the opening of the time window. Soft version allows services outside the time window at the expense

of a penalization cost. Besides, the depot often has a time window [e0, l0] representing the earliest

departure time and the latest arrival time for vehicles to the depot. The time window is the same for

the depot copy n + 1, i.e. [en+1, ln+1] = [e0, l0]. These additional constraints on service start times

add another layer of complexity when compared to the classical CVRP. When the number of vehicles

is xed, even computing a feasible solution is NP-Hard [150]. The objective of the VRPTW can dier

from that of the CVRP. According to Desaulniers et al. [43] exact approaches to the VRPTW usually

consider the same objective function as in the CVRP, i.e. the total cost of the routes. Meanwhile,

heuristic and metaheuristic methods are often designed to rst minimize the number of required

vehicles then the total cost in a hierarchical way.

Mathematical formulations

Similar to the CVRP there exist many formulations for the VRPTW. M2VRPTW presents a three-

index formulation for the version with hard time windows. In this, the binary variable xijl ∀ i, j ∈V , l ∈ L takes value one if the vehicle l uses the arc (i, j). Besides, Til stands for the time when

9


the service starts at customer i ∈ V c by vehicle l. Model M2VRPTW minimizes (2.7) the total cost

assuming a xed eet of vehicles.

M2V RPTW : min∑l∈L

∑i,j∈V

xijlcij (2.7)

∑l∈L

∑j∈V

xijl = 1 ∀i ∈ V c (2.8)

∑j∈V

x0jl = 1 ∀l ∈ L (2.9)

∑i∈V

xijl −∑i∈V

xjil = 0 ∀j ∈ V c, l ∈ L (2.10)

∑i∈V c∪0

xi,n+1,l = 1 ∀l ∈ L (2.11)

T il + ti + tij − T jl ≤M (1− xijl) ∀l ∈ L, (i, j) ∈ A (2.12)

ei ≤ T il ≤ li ∀l ∈ L, i ∈ V (2.13)∑i∈V c

qi∑j∈V

xijl ≤ Q ∀l ∈ L (2.14)

xijl ∈ 0, 1 ∀ (i, j) ∈ A, l ∈ L (2.15)

T il ∈ <+ ∀i ∈ V , l ∈ L (2.16)

Constraint (2.8) guarantees that each customer is served by only one route. Meanwhile, constraint

(2.9) species that all vehicles must leave the depot. In addition, constraint (2.10) guarantees that

a vehicle visiting a customer must leave to another node. Moreover, constraint (2.11) states that all

vehicles nish their route at node n+ 1. Note that in this formulation, the number of eectively used

vehicles can be less than |L| as far as variable x0,n+1,l can take value 1. Furthermore, constraints (2.12)

to (2.13) ensure that the times when services start, respect the nodes time windows. In constraint

(2.12) term M is a large value that let the equation holds when xijl takes value zero. Vehicles

capacity constraint is guaranteed by (2.14). Last but not least, constraints (2.15) and (2.16) stand for

the variables nature.

2.1.3 Solution Methods

Solving the VRPs is not an easy task, since the CVRP is an NP-Hard problem and most of its variants

are NP-Hard as well, including the uncertain VRPs. Solution approaches can be classied according

to the nature of the solution. In this vein, exact methods guarantee that the optimal solution will be

found but at the expense of a prohibitive running time even for medium size instances. Approximate

methods on the other hand usually provide quickly a solution but this last can be not optimal. Figure

2.1 provides a simple scheme to VRP solutions approaches. The scheme is not exhaustive but allows

to navigate through the extensive amount of methods.

10


Solution approaches for VRPs

Exact methods

Heuristics Metaheuristics

Constructive

Two-phase

Branch and Price

Branch and Price and Cut

Branch and Cut

Tabu Search

Simulated Annealing

Iterated Local Search

Matheuristics

Clarke & Wright

Nearest Neighborhood

Ant Colony Optimization

Genetic Algorithms

Cluster-first, route second

Route-first, cluster second

Approximate methods

Figure 2.1: Solution approaches scheme.

Exact methods

Exact approaches for the VRPs are highly related to the way the problem is modeled. Dierent

methods have been used for solving the CVRP and the VRPTW such as the Branch-and-price (BP),

Branch-and-cut (BC) and Branch-and-price-and-cut (BPC) [58, 43]. BC has been mainly based on

the two-index formulation (M1CVRP) or an analogous version for the VRPTW [12, 90]. Overall,

the BC works by relaxing integrality constraints and discarding the set of constraints represented by

(2.4) for the CVRP and (2.12) to (2.14) for the VRPTW. The BC solves the relaxed problem and

identies any subset of variables that violates the removed constraints. If this set is found, it generates

the violated constraints, add them to the problem and reiterates. Moreover, when no constraints are

identied, the BC branches on a fractional variable and creates problems that are solved with the

same approach.

BPC works similarly to BC methods. The dierence relies in the fact that each subproblem

(usually the Shortest Path Problem with Resource Constraints) relaxation is solved by means of a

column-generation approach. This last approach exploits the set covering formulation for both the

CVRP and the VRPTW. Dierent versions of BPC algorithms exist since dierent approaches can

be used to solve the subproblems (e.g. ng-routes, q-routes, bidirectional search), or the types of cuts

(constraints) added during the iterations. BPC has shown to be the state-of-the-art to solve the CVRP

[8, 58, 137] as well as the VRPTW [95, 42, 87]. Nevertheless, Baldacci et al. [10] has proposed the

best method based on a reduced set partitioning for the VRPTW. Further analysis and description

of the methods are available in [89, 43].

Approximate methods - Heuristic and Metaheuristics

While exact methods have seen an incredible development in the last years, the combinatorial nature

of the VRPs limit their use to relatively small instances. Nowadays, for example the CVRP can be

consistently solved for problems with up to 200 customers [137]. Meanwhile, VRPTW is consistently

solved for instances with up to 100 customers [10]. Still, since applications can easily overpass this size,

heuristics and metaheuristics are omnipresent in the literature. Heuristics are approximate algorithms

which try to nd good solutions in competitive running times.

Laporte and Semet [110] classify heuristics under constructive and two-phase methods. Labadie

et al. [99] also follow this classication. In general words, constructive heuristics work by creating an

initial solution that can be further improved [109]. Clarke and Wright savings [39] is by far the most

11


known heuristic due to its simplicity [110]. The heuristic works by iteratively merging pairs of routes

into single ones, provided that this implies a saving and guarantees feasibility. The process is repeated

until no further merges are feasible or the maximum possible saving is negative. Further information

on basic heuristics can be found in Toth and Vigo [168].

Several constructive heuristics tailored for the VRPTW are introduced by Solomon [157]. By far

the most important and successful is the insertion heuristic called I1. Iteratively I1 creates routes

starting with some seed customers. These are selected from dierent rules, such as the farthest not

visited customer or the one with the earliest initial time window. Once a seed has been selected, I1

calculates an insertion (of non visited customers) criteria based on the classical savings (distance) and

the extra time required by adding the new customer. The best customer not yet visited is added in

the best possible position. The algorithm iterates until no customers can be added to the current

route, then it starts a new one until all customers are visited. The reader is referred to Bräysy and

Gendreau [31] for other constructive approaches.

Two-phase methods can be further divided into cluster-rst, route second and route-rst, cluster

second. The rst one is based on the idea of creating groups or clusters of customers respecting

the capacity constraint. Then, customers in the cluster are ordered to completely dene a route, by

solving a Traveling Salesman Problem (TSP) for each group. Several approaches can be used to create

clusters: the sweep algorithm [70] uses angular sectors from the depot to create the necessary clusters.

Fisher and Jaikumar heuristic [56] also uses the idea of clusters around seeds aiming to minimize the

distance from customers to cluster seeds.

The route-rst, cluster second approach is mainly based on the idea of creating a giant tour (TSP

tour) without considering capacity or other constraints, and then splitting it into feasible routes. This

approach was introduced by Beasley [14] and has received more attention since the work of Prins

[140]. Indeed, Prins showed that route-rst, cluster second algorithms could be as ecient as methods

relying on classical methods at the date, such as the Tabu Search. Further examples on the route-

rst, cluster second can be found in Labadie et al. [97] with an application to the VRPTW, Prins et

al. [141] addressing the Capacitated Arc Routing Problem and the CVRP, Mendoza et al. [125] in

a multi-compartment vehicles with uncertain parameters problem, Velasco et al. [171] with a multi

objetive pick-up and delivery problem, and Mendoza et al. [126] in the context of a CVRP with

stochastic demands. A recent review on on the route-rst, cluster second approach can be found in

Prins et al. [142].

Although heuristics commonly provide a good trade-o between eciency and quality, they are

usually coupled with local search or improvement procedures. The underlying concept of local search

is the denition of neighborhoods [31, 99]. These lasts are considered as close related solutions to a

generic solution s. Neighborhoods are structured in a way such that movements can be performed

on s to achieve a new solution s′. Usually, this type of procedures contain two types of movements,

namely intra-route and inter-route ones [102]. The rst one aects only one route trying to improve

it while the second considers and changes more than one route. Moreover, neighborhoods can be

wholly explored to select the best improving movement (best acceptance) or partially explored until a

movement improves the current solution (rst acceptance) [133]. The exploration of neighborhoods is

performed in an iterative way, until reaching a stopping condition or achieving a local (global) optimal

solution. Among the most used neighborhoods one can nd the k-opt movements [119] for which 2-Opt

and 3-Opt are the most popular cases, the b-cyclic, k-transfer scheme [166], Or-opt movements [132]

and λ−interchange movements [133]. More complex movements can also be found in the literature,

e.g. GENI exchange [62] and ejection chains [144, 73]. For further details on these neighborhoods

and their characteristics, the interested reader is referred to [59, 31]. Furthermore, details on ecient

implementations of evaluation tests allowing to know either these movements are feasible or not, and

to compute the extra cost generated, can be found in [93, 173].

Despite the success of local search procedures, they are often trapped in local optima when they

12


are applied to only one initial solution obtained with a constructive heuristic. To overcome these

problems, metaheuristics are a good option. According to Bianchi et al. [24] metaheuristics are high

level procedures that combine heuristics in a more general framework. Osman and Laporte [134]

dened them as A metaheuristic is formally dened as an iterative generation process which guides

a subordinate heuristic by combining intelligently dierent concepts for exploring and exploiting the

search space, learning strategies are used to structure information in order to nd eciently near-

optimal solutions. Originally, metaheuristics were easily identied and dierentiated, however, the

increasing hybridization of such methods have blurred the lines between them [109, 29].

Still, according to Laporte et al. [109] metaheuristics can be classied into trajectory and population-

based methods. In the rst one, a solution move to another by searching in a neighborhood. Mean-

while, population-based methods use a set of solutions that interact to improve the solution quality.

Within the rst category classication one can nd methods such as the Simulated Annealing (SA)

[94], Tabu Search (TS) [72, 68], Variable Neighborhood Search (VNS) and Variable Neighborhood

Descent (VND) [129], Deterministic Annealing (DA) [47, 115], Iterated Local Search (ILS) [13, 120].

Population-based metaheuristics include Genetic Algorithms (GAs) [143, 140], Ant Colony Optimiz-

ation (ACO) [145] and Scatter Search (SS) [71].

Another important trend in solution methods for the VRP are the matheuristics [44, 123]. Math-

euristics work by hybridizing heuristics (or metaheuristics) and exact algorithms such as Integer

Programming (IP). These methods cooperate and share information to improve the solution quality.

An interesting approach is the Petal heuristics which make use of the set partitioning model. In this

one, promising routes (often called petals [102, 99] in this context) are added to a set partitioning

problem as it is done in a column generation scheme [11, 149, 146]. A similar approach is used by

Mendoza et al. [127, 126], by constructing several solutions from constructive heuristics and then

solving a set partitioning problem. Further information on solution methods for dierent variants of

VRP can be found at [28, 67, 32, 69, 99].

2.2 VRP under uncertainties

A majority of the studies on the dierent VRP variants have been carried out under the assumption

that all relevant information is known with certitude when solving the problems. That is, problems

are solved in a deterministic static way [139, 30]. Nevertheless, dealing with real life applications

implies the appearance of uncertainties. These lasts arise from many reasons, e.g. weather conditions,

accidents, customer presence, etc. Neglecting the uncertainties is not always an option since the eects

of variability can have important consequences on solutions quality. Actually, it has been shown

that deterministic solutions2 can lead to systematically bad solutions in an uncertain environment

[121, 158].

Talking about uncertainties is a discussion highly related with information. Certainly, the quality

of information and the times when it is available play a major role in both, the models and solution

approaches for this type of problems. Moreover, what can be known about the uncertain parameters

denes the framework in which an uncertain VRP can be tackled. Pillac et al. [139] dene four

categories due to sub levels of information characteristics, i.e. information evolution and information

quality. Their taxonomy is presented in table 2.1 by considering general uncertain inputs.

2Based for example on the expected value of the variable parameters

13

2.2. VRP UNDER UNCERTAINTIES

Table 2.1: VRPs taxonomy based on Pillac et al. [139] - Information evolution and qualityInformation QualityDeterministic input Uncertain inputs

Informationevolution

Input knownbeforehand

Static and deterministic Static and uncertain

Input changesover time

Dynamic anddeterministic

Dynamic and uncertain

The top-left box stands for Static and deterministic problems. This category represents the prob-

lems where all parameters are known beforehand, and their values are certain. Moreover, once the

solution is deducted it remains unchanged as far as no new information arise. The CVRP and VRPTW

lie within this category. In Dynamic and deterministic problems, part or all the inputs is revealed

dynamically during the execution of the routes [139]. More importantly, there is no exploitable in-

formation about the dynamic parameters. Therefore, solutions are construct in an online way.

Static and dynamic uncertain problems (top-right and bottom-right boxes) share the fact that at

least one input (parameter) is not known for sure. Same as the dynamic deterministic case, parameters

true value is revealed at some specic moments, e.g. when the vehicle arrives at the customer.

Nevertheless, in uncertain problems there exist exploitable information about the parameter such as

its probability distribution, the interval set for its value, moments of the distribution, etc. Therefore,

static and dynamic uncertain problems can exploit this information to devise their solutions. There

are two main dierences between static and dynamic uncertain problems. First, in static problems

all inputs (even uncertain ones) are dened before solving the problem while this is not the case in

dynamic ones. Second, both categories dier in how solutions are created and treated. In the static

uncertain problems, solutions are created before the realization of the uncertain parameters and are

barely modied. Meanwhile, dynamic problems construct and change the solution as new information

arise.

The framework to handle the uncertainties (static or dynamic) clearly depends on the type of

problem tackled and how the information evolves. Additionally, how the uncertainties are modeled

plays a major role in the approaches to solve uncertain VRPs. Three main approaches have been used

to deal with uncertainties in VRPs, namely, stochastic, robust optimization, and Fuzzy logic. They

are now explored in further detail.

2.2.1 Stochastic VRP

Stochastic programming has been the main paradigm to deal with uncertain VRPs, driving the devel-

opment of what is known as Stochastic Vehicle Routing Problems (SVRP) [64]. The reader is referred

to the work of Birge and Louveaux [27] as a good entry point to the stochastic programming eld.

The main characteristics of SVRPs is the fact that uncertainty alters the condence on the problems

parameters, which are modeled as random variables. Thus, in static SVRPs, information is assumed

to be prior available to characterize the random parameters of the problem i.e. the probability dis-

tributions of the random parameters. As long as parameters are not certain, some constraints might

not be fully satised. Whenever a constraint is violated (due to the variability of the parameters) a

failure occurs.

SVRPs can be tracked down to the pioneer work of Tillman [167], who deals with the VRP with

stochastic demands (VRPSD) considering multiple depots. The VRPSD is an extension of the classical

CVRP where demands are modeled as random variables. Due to this fact, failures can take place

whenever a customer' demand is higher than the available remaining capacity in the vehicle. Aiming

to overcome and solve the SVRPs, two main paradigms have been mainly used [63]: the a priori

(static) and reoptimization (dynamic) paradigms.

14


The a priori optimization works by constructing a solution before the realization of the random

variables [86, 63]. Therefore, a priori paradigm can be viewed as a two-stage approach. After the

prior solution is built (rst-stage) the random parameters are revealed (second-stage). The way in

which this information is available depends on the problem. For example, many VRPSD formulations

[38, 76, 60, 126] assume that the customer demand is only known when the vehicle arrives at the

customer location. Besides, two main approaches are commonly used to optimize the a priori SVRPs,

the Stochastic Problem with Recourse (SPR) and the Chance Constraint Programming (CCP). SPR

use recourse which can be dened as policies or rules and actions that adjust the prior solution to deal

with specic situations such as failures in the second stage. Indeed, recourse can be used to respond to

failures. For example, the so called classical VRPSD recourse consists in a return to the depot to load

(unload) when a customer demand exceeds the current capacity of the vehicle and restarts the route

from this customer. It shall be noticed that recourse actions usually generate a cost which is properly

considered in the objective function. Thus, the SPR minimizes the rst-stage cost of the solution plus

the expected cost of the recourse (second-stage). CCP works by bounding the probability of failure to

a threshold. CCP can be used when recourse can be hardly dened [160] or to guarantee a service level

[118, 52, 178]. A CCP and SPR formulations for the VRPSD are introduced in the next sub-section

2.2.1.

The reoptimization paradigm on the other hand does not rely on a prior (static) solution. Con-

versely, it constructs and changes the solution as new information arises (dynamic). The increasing

amount and development of Information and Communication Technologies (ICT) has enabled to tackle

and evaluate problems in such a dynamic way [139]. Although this paradigm usually improves the

solutions quality when compared to its static counterpart, it also poses new challenges in terms of

computational eciency at operational level. Indeed, the speed in which solutions must be computed

is a limitation on such methods. Besides, a priori (static) approaches conduce to stable tactical routes,

which are operationally desirable [63]. Moreover, the a priori approach is preferable when anticip-

ating uncertainties is important for routes feasibility and to avoid costs (economic and reputational)

[15]. The main focus of the SVRPs is the static a priori approach, further information on dynamic

stochastic problems can be found at Pillac et al. [139], Bektaet al. [15], and Psaraftis et al. [130].

Given the static and stochastic context, the SVRPs literature is frequently classied with respect

to the parameters stochasticity [64]. The most common studied versions are the VRP with stochastic

demands, with stochastic customers, and with stochastic times. They are now further explored.

VRP with stochastic demands

Among the SVRP, the VRP with stochastic demands (VRPSD) is the furthermost studied problem.

The rst reported solution method for the VRPSD is proposed by Tillman [167] who used a modi-

cation of the Clarke and Wright heuristic [39]. The general VRPSD with recourse is described in

the model M3VRPSD while the CCP formulation is presented in model M5VRPSD. Both models are

proxies for the SVRPs. The notation of Gendreau et al. [64] is used. Dierences among M3VRPSD

and M1CVRP are the inclusion of the expected recourse cost (Q (x)) in objective function (2.1) and

constraint (2.4) which need to consider the demand expected value. Since demands are modeled as

random variables, they become qi∀i ∈ V c with expected value E [qi] = µi ∀i ∈ V c. The expected

recourse cost is determined by Q (x) [106].

M3V RPSD min∑i,j∈V

xijcij +Q (x, qi) (2.17)

∑i,j∈S

xij ≤ |S| −

⌈∑i∈S

µiQ

⌉∀S ⊂ V c, 3 ≤ |S| ≤ n− 1 (2.18)

(2.2)− (2.3) , (2.5)− (2.6) (2.19)

15


The objective function (2.17) minimizes the total travel costs plus the expected recourse costs.

Constraint (2.18) is equivalent to constraint (2.4), but the VRPSD version uses the expected value of

demands. M4VRPSD extends the M1CVRP but constrains the probability of a route failing at least

once to a level β. In this model, it is assumed that Vβ (S) is the minimum integer value which satises

that the subset S ⊂ V has a probability of failure not exceeding β. M4VRPSD minimizes the total

travel costs (2.20) while the routes probability of failure constraint is handled by (2.21). M3VRPSD

and M4VRPSD present the SPRs and CCPs models. The rst one considers the expected cost of

recourse (2.17), while the second one bounds failures probability (2.21).

M4V RPSD min∑i,j∈V

xijcij (2.20)

∑i∈S,j /∈Sor

i/∈S,j∈S

xij ≥ 2Vβ (S) ∀S ⊂ V c, 3 ≤ |S| ≤ n− 1 (2.21)

(2.2)− (2.3) , (2.5)− (2.6) (2.22)

SPR has been the dominant trend in the VRPSD (Bertsimas [22], Gendreau et al. [65, 66], Hjorring

and Holt [83], Christiansen and Lysgaard [38], Goodson et al. [76], Gauvin et al. [60], Mendoza et al.

[126]) when compared to the CCP formulation (Stewart and Golden [159, 160], Dror et al. [46]). The

most used recourse policy, the classical recourse is dened as follows. When the load of the vehicle is

fullled it returns to the depot to unload the charge, and resumes its assigned route from the failure

point [22]. However, other recourse policies have been studied and implemented through several

studies: preventive restocking policies are extensions of the classical recourse where return trips to the

depot are performed even if the vehicle is not empty to avoid future failures [176, 23, 25, 122, 177];

pairing strategies allowing the cooperation of multiple vehicles [3]; split deliveries between paired

routes [113] in which some customers are served by two vehicles; and backup routes [50] that receive

customers from primary routes. Although the use of more complex recourse policies can generate a

signicant saving relative to simpler ones [3], the latter have been preferred since they allow more

tractable models and stable tactical routes [63].

16


Table2.2:

Summaryof

VRPSD

literature

Author(s)

Typeofmodel

Recourse

Additionalconsiderations

Solutionapproach

Stew

artandGolden[160]

SPRandCCP

Classicalrecourse

Normaldistributeddemands,

independent

andcorrelated

ClarkeandWright[39]heuristic

DrorandTrudeau

[46]

SPRandCCP

Classicalrecourse

Direction

oftheroutes

impact

onobjectivefunction


Bertsimas

[22]

SPR

Classicalrecourse

andSkipping

custom

ers

Closed-form

expressionsfor

recourse

cost

Cyclic

heuristic

Grendeauet

al.[65]

SPR

Classicalrecourse

Stochasticcustom

ers

L-shaped

method

Hjorringet

al.[83]

SPR

Classicalrecourse

considering

exactstockout

Singlevehiclecase

L-shaped

Yanget

al.[176]

SPR

Optimalrestocking

policy

Route

duration

constraints

Route-rst-Cluster-Nextand

Cluster-First-Route-Next

heuristics

Bianchi

etal.[23]

SPR

Optimalrestocking

policy

VRPSD

andTSP

representations

Severalmetaheuristics

AkandErera

[3]

SPR

Pairedlocally

coordinated

Coordinationof

theeet

TabuSearch

ChristiansenandLysgaard

[38]

SPR

Classicalrecourse

Branch-and-price

Erera

etal.[50]

Mixed

CCPandSP

RPrimaryandback

uproutes

Hardtimewindows

HeuristicwithMonte

Carlo

simulation

Erera

etal.[51]

SPR

Classicalrecourse

andvariant

Route

duration

constraints

TabuSearch

Mendoza

etal.[125]

SPR

Classicalrecourse

Multi-com

partmentcase

Mem

eticAlgorithm

Goodson

etal.[76]

SPR

Classicalrecourse

Cyclic-Order

representation

for

solutions

Simulated

Annealing

Gauvinet

al.[60]

SPR

Classicalrecourse

TabuSearch

andbidirectional

labelalgorithm

Branch-and-cut-and-price

Mendoza

etal.[126]

SPRandCPP

Classicalrecourse

Route

duration

constraints

HybridGrasp

andheuristic

concentration

Nguyenet

al.[131]

SPR

Classicalrecourse

Hardtimewindowsand

Satiscing

Measure

Approach

TabuSearch

Luo

etal.[122]

SPR

Dynam

icrecourse

strategy

Weight-relatedcosts

AdaptiveLarge

Neighborhood

Search

17


After the VRPSD introduction, it has been widely studied. Table 2.2 shows a summary of the

literature of VRPSD. Exact methods have been used to solve the VRPSD, for instance Gendreau

et al. [65] employed an integer formulation mixed with the L-shaped method [104] to optimally

solve instances with up to 70 customers given that all customers are present. Still, their method is

restricted to discrete probability distributions [65]. Hjorring and Holt [83] tackle the VRPSD with

only one route. The classical recourse is used considering two scenarios: the rst one is the normal

stockout which represents the case when the vehicle returns to the depot and back to the customer

where the failure takes place. The second one is the exact stockout, representing the case when the

vehicle has just enough capacity to serve the current customer. In that case, the vehicle returns to

the depot and continues towards the next customer. The authors use an L-shaped approach to solve

the problem. Furthermore, the authors propose new optimality cuts that improve the performance of

the algorithm.

An important development in the eld is presented by Christiansen and Lysgaard [38]. The

authors proposed the rst branch-and-price algorithm to solve the VRPSD with classical recourse.

Moreover, they determined a testbed set of 40 instances based on Augerat sets A and P. The results

allowed to have a common benchmark. Moreover, although the method solved problems with up to

60 customers, it showed to be more powerful with a higher number of vehicles when compared to the

L-shaped method [65, 83, 106]. Gauvin et al. [60] developed an improvement of Christiansen and

Lysgaard work introducing a Branch-and-cut-and-price. The column generation is accelerated using

a TS heuristic and a bidirectional labeling algorithm. The BCP outperforms the BP of Christiansen

and Lysgaard [38]. In fact, 20 more instances are solved to optimally, so 38 of 40 instances are closed.

Furthermore, the method achieves to solve problems with up to 100 customers in less than 20 minutes.

Both works assume independent Poisson distributions. More recently, Biesinger et al. [25] propose to

use the L-shaped algorithm for the Generalized VRP with stochastic demands. More interesting is

that the recourse action corresponds to a preventive restocking policy [176] in contrast to the classical

one. Therefore, this is is the rst exact method that considers such a dierent recourse. Results are

presented for small size instances with dierent types of variations. The solution method is proven

to be eective for instances involving up to 40 nodes and with less than three expected restocking

actions.

Due to the complexity of the VRPSD, approximate methods have been mostly designed to solve

real size instances. Stewart and Golden [160] proposed a CCP and two SPR for the problem. The

SPR are dierentiated by the penalty induced by the recourse. In the rst one, a penalty is taken into

account disregarding the amount of the violation (lack of capacity). The second one induces a penalty

proportional to the amount of the violation. Using adaptations of the Clarke and Wright [39] and

the Generalized Lagrange Multipliers [159] heuristics, the authors solve some instances assuming the

demands are independent normal variables. Moreover, the authors present an approach to transform

the CCP formulation into an equivalent deterministic CVRP model. Also, solutions are provided

for the correlated-demands case. Dror and Trudeau [46] consider the models presented in [160] and

prove that in the VRPSD framework, the direction of the routes can have an important impact on

the objective function. This important fact shows the dierent structure that SVRPs take when

compared to deterministic VRPs. The Clarke and Wright [39] heuristic is adapted to incorporate the

expected recourse cost. More important, when two routes are being considered to merge, the direction

of the resulting route must be evaluated since this impacts the objective function. A comparison to

the adaptation of [160] shows that their method performs better in terms of the number of vehicles,

deterministic length and expected cost.

Bertsimas [22] considers the VRPSD as an SPR. The authors propose two strategies (recourse)

depending on the available information. Strategy a on the one hand, is the classical recourse policy

previously described. Strategy b on the other hand, assumes that demands are revealed before the

tours (routes) start, thus, customers with zero demand are omitted. Although, the last scenario seems

18


more suitable for a reoptimization paradigm, strategy b is interesting when resources do not allow

to perform a reoptimization scheme e.g when computational resources are scarce. Furthermore, this

study proposes closed-form expressions to calculate the expected costs of routes under general prob-

abilistic assumptions. Besides, under certain conditions (customer locations) the a priori strategies

perform very closely to the reoptimization paradigm. Further works of Gendreau et al. [66] led to the

development of a tabu search in the context of the VRPSD where additionally, customers might or

not be present. To calculate the expected cost, a proxy function is used to approximate the expected

solution costs.

Yang et al. [176] extended the classical recourse policy in their work: they assume that vehicles

return to depot when capacity is fullled but they can return before this event happens. When

leaving a customer, the capacity of the vehicle is compared to a threshold to determine if a visit to the

depot is valuable before attending the next customer. One threshold is associated with each customer

given a xed route. Moreover, this restocking policy is proven to be optimal given a xed route.

Two heuristics are proposed for the VRPSD with route durations limits for instances with discrete

triangular distributed demands. Bianchi et al. [23] consider the same restocking policy proposed

by Yang et al. [176] for the VRPSD. Their objective is to compare multiple metaheuristics, each of

them under two approaches for the local search: using the solution routes representation and a TSP

representation. The rst one evaluates the pertinence of local search movements given the cost changes

in the routes. The second one uses the deterministic tour length of an underlying TSP representation

of the solution to estimate if a movement is performed or not. Overall, the TSP approach can be seen

as an acceleration technique to improve the execution of the local search since recourse costs are not

calculated. Several metaheuristics are tested on instances with up to 200 customers.

Goodson et al. [76] approach the VRPSD using a cyclic-order representation of the solutions. The

classical recourse action is considered and demands are assumed to be Poisson distributed [38, 60]. An

ecient way to deal with neighborhoods under a solution cyclic-order representation is introduced.

This type of representation is based on a cyclic permutation of numbers, which is then decoded into

detailed routes. The authors use this type of representation since small changes in the permutation

can conduce to important changes in the decoded routes. Simulated Annealing is selected by the

authors to tackle the problem. The method achieves 16 out the 18 optimal solutions presented by

Christiansen and Lysgaard [38]. Later, Mendoza et al. [126] investigated the VRPSD with route

durations constraint. A hybrid metaheuristic composed of a GRASP plus Heuristic Concentration

(HC) are used. The solution method operates several simple heuristics to construct a pool of routes,

then, a set partitioning problem is solved giving a nal solution. The method solves the VRPSD with

classical recourse and achieves to nd all the best-known solutions of the Christiansen and Lysgaard

[38] testbed with average gaps of 0.02%. For the version with duration constraints two models are

considered, a CCP and an SPR.

Ak and Erera [3] proposed an alternative recourse policy called Paired locally coordinated (PLC).

The PLC extends the idea of usual recourse actions which consider vehicles separately to paired

vehicles. When the rst vehicle (type I) achieves its capacity it returns to the depot nishing its

service. Then, the second vehicle (type II) incorporates the unserved customers at the end of its

route. Besides, if the second vehicle faces a failure, it uses the classical recourse. The aim of this type

of coordination is to improve the expected total cost. A Tabu Search is used to solve instances with up

to 150 customers where customers have homogeneous discrete demands. Results show that comparing

to the classical recourse, the PLC allows signicant savings, ranging from 3% to 25% when instances

with 50 or more customers are considered. Erera et al. [50] worked on the VRPSD with hard time

windows. A limit on route duration is also considered, so drivers return to the depot respecting the

working hours. Although deliveries and analyzed data comprises a whole week, routes are created

for specic days. One of the main characteristics of the work is the fact that routes must if possible

visit the same customers. The motivation for this idea is to create long-term-relationships with the

19


customers along the horizon. Customers are classied into two groups, one that can be visited by

any driver and another group that must be visited at most by two drivers within a week. A CCP

formulation is used to constraint the probability of missing a customer' time window. The problem

is solved by means of a heuristic which uses Monte Carlo simulation to verify constraints satisfaction.

A novel recourse approach is used by creating primary and backup routes. The latter can receive

customers to guarantee feasibility or improve costs. A case study is presented and results are shown

for it.

Erera et al. [51] address the version of the VRPSD with duration constraints. Two recourses

are considered, the classical recourse and a variant where the vehicle performs a return to the depot

without servicing any demand if its current load is less than the demand met at the last customer,

then it comes back to this customer and satises its entire demand. To guarantee the route duration

constraint the authors propose to solve an Adversarial Problem. Since recourse actions generate

additional time depending on demands realization, the Adversarial Problem seeks to maximize the

additional time needed over all the possible demands values. Thus, solving this problem for a given

route allows to guarantee that in the worst case scenario, the duration constraint holds. A TS is

developed to solve the problem with independent demands which follow a discrete uniform distribution.

Results are presented for instances with up to 100 customers. Moreover, the inuence on the eet

size for the VRPSD with duration constraint is discussed. Although no general conclusions can be

drawn, the results indicate that duration constraints increase the required eet size. Mendoza et

al. [125] used a Memetic Algorithm (MA) to solve the VRPSD with multi-compartments. As far

as the calculus of failure probabilities is expensive, an approximation called Take-all-policy (TAP)

is employed. Furthermore, two heuristics are used to compare the eciency of the MA with TPA

to solve the problem. The rst one is based on an adaptation of the Clarke and Wright heuristic

proposed by Dror and Trudeau [46], the second one solves the deterministic problem with the MA

but reduces the capacity of the vehicles by a percentage. Demands are assumed to be independent

normally distributed and results are presented for instances with up to 484 customers. The MA mixed

with TAP approximation gives the best results in terms of number of vehicles and operational costs.

Other related variants of the VRPSD can also be found in the literature. A weight-related cost

version with dynamic recourse strategy is introduced in [122] while a multi-objective VRPSD is pro-

posed in [61]. Time dimension has been also considered, either in the form of time windows [114, 131]

or route duration constraints [176, 50, 51, 77, 126, 131]. Furthermore, an important version of the

VRPSD considering stochastic customers is discussed by Gendreau et al. [65, 66].

Regarding the reoptimization paradigm, the work of Dror et al. [45] introduced this approach

using a Markov Decision Process (MDP) to solve the single vehicle case. In this vein, single-vehicle

cases have been investigated in [151, 152, 153]. A multi-vehicle version was rst proposed by Goodson

[77] aiming to maximize the expected demand served.

Summing up, heuristics for the VRPSD have been mainly based on the adaptation of heuristics

initially designed for the CVRP [167, 160, 46, 125]. Similarly to the deterministic VRP, metaheuristics

have played an important role to solve the VRPSD. Tabu Search (TS) is the most used metaheuristic

in dierent variants of the VRPSD [66, 3, 51, 131]. Other methods such as the LNS [113], SA [76],

Memetic Algorithms (MA) [125], and hybridized metaheuristics e.g. Multi-Space Sampling [127, 126]

are addressed in the literature. Exact methods have been concentrated on L-shaped method [65, 83,

106, 25], branch-and-price [38] and branch-and-cut-and-price [60]. L-shaped methods have shown to

be able to solve instances with up to 100 customers but few number of vehicles [106]. Branch-and-

price and branch-and-cut-and-price on the other hand have been able to solve one instance with up to

100 customers [60]. Nevertheless, the former methods show a better performance when dealing with

multiple vehicles in the solutions [38, 60]. It shall be noticed that only the work of Biesinger et al.

[25] on the Generalized Vehicle Routing Problem with Stochastic Demands (GVRPSD) makes use of

exact methods considering a restocking policy, while the others remain on the classical recourse.

20


Moreover, it is evidenced that many of the works use their own set of instances [106, 176, 23, 3,

50, 51, 125, 126, 122] since no standard benchmark has been proposed. Meanwhile, other works have

rely on original or modied well-known data sets such as Nguyen et al. [131] and [114, 113] using

the Solomon [157] instances. Likewise, since the introduction of the Christiansen and Lysgaard [38]

benchmark other works [76, 60, 126, 122, 124] have used it as a base to compare their results (or

improve them).

VRP with stochastic customers

The VRP with stochastic customers (VRPSC) refers to the variant where the presence of the customers

is stochastic, i.e. they might be present or not. Among the classical stochastic parameters, this one is

the far less studied. The roots of the VRPSC are derived from the Traveling Salesman Problem with

Stochastic Customers (TSPSC). TSPSC was rst introduced in the thesis of Jaillet [85]. Both the

VRPSC and the TSPSC are modeled as a two-stage problem. In the rst stage, a solution to the TSP

is computed, while in the second stage, absent customers are revealed so the routes (tour) follow the

original order but skip the absent customers. Bertsimas and Howell [21] reviewed the main results of

Jaillet and proposed several heuristics based on classical neighborhoods ideas, e.g. 2-opt and 3-opt,

and others on angular sorting and space lling curves. A branch and cut approach is proposed in

[105] for the TSPSC. Bertsimas [22] also derives heuristics and bounds for the case when demand is

binary, i.e. either one or zero, and shows that the TSPSC is a special case of this problem. Further

development on the VRPSC is due to a more general problem, the VRPSC with stochastic demands

(VRPSCD). A tabu search is proposed in [66] while Gendreau et al. [65] use the L-shaped algorithm

to optimally solve the VRPSCD.

VRP with stochastic times

The Vehicle Routing Problems with Stochastic Times (VRPST) are versions of the VRP in which

the travel or/and service times are random variables. Table 2.3 presents a summary of some of the

most important works in the eld. Compared to the VRPSD, the VRPST is a more recent studied

version. The rst literature reference is presented by Laporte et al. [103] who considered the case with

stochastic travel and service times, with both CCP and SPR formulations. The authors use a two-stage

approach, designing an a priori solution before travel and service times are revealed. Moreover, in the

second stage the vehicles follow the a priori solution and a penalization for late arrivals proportional

to the length of the delay is considered. Optimal solutions are found for small instances with up to 20

customers and with at most ve travel time scenarios. An adaptation of the Clarke and Wright [39]

is used by Lambert et al. [100] for the version with stochastic travel times in a bank money collection

context. Similarly to Laporte et al. [103], a penalization is incurred on late arrivals, in this work, due

to the money lost of interest.

21


Table2.3:

Summaryof

VRPST

literature

Author(s)

StochasticParam

eters

Typeofmodel

Additionalconsiderations

Solutionapproach

Laporte

etal.[103]

TravelandServiceTimes

(Scenarios)

CCPandSP

RPenalized

late

arrivals


Lam

bertet

al.[100]

TravelTimes

(Scenarios)

SPR

Penalized

late

arrivals


WangandRegan

[174]


CCP

Analyticalapproach

underhard

timewindows

KenyonandMorton[92]

Travel(uniform

)andService

Times

(Scenarios)

Com

pletiontimes

costand

probabiity

Durationconstraints

Branch-and-cutem

beddedin

Monte

CarloSimulation.

AndoandTaniguchi

[6]

TravelTimes

(Triangular)

SPR

Timewindowspenalization

GeneticAlgorithm

Lecluyseet

al.[111]

Traveltimes

(Lognorm

al)

Routeslength

meanand

variance

Timedependent

traveltimes

TabuSearch

Liet

al.[118]


(Normal)

SPRandCCP


TabuSearch

Leiet

al.[112]

ServiceTimes

(Normal)

SPR


Generalized

Variable

Neighborhood

Search

Zhang

etal.[178]

Travel(Lognorm

al)andService

Times

(Normal)

Mixed

CCP&SP

RTimeWindows&Discrete

approximationon

ArrivalTimes

TabuSearch

Tas

etal.[163]

TravelTimes

(Gam

ma)

SPR


TabuSearch

Tas

etal.[164]

TravelTimes

(Gam

ma)

Mixed

CCP&SP

RTimewindowspenalization

Branch-and-Price

Ehm

keet

al.[48]

TravelTimes

(Normal)

CCP

Timewindows&Normal

assumptionon

initialservice

times

TabuSearch

Miranda

andConceição

[128]


(Normal)

CCP

Timewindows&Discrete

approximationon

arivaltimes

Iterated

LocalSearch

Nguyenet

al.[131]

TravelTimes

(Ambiguous

distribution)

Hierarchicalobjectives

TimeWindows&Satiscing

Measure

Approach

TabuSearch

Erricoet

al.[53]

TravelTimes

(Discrete

Triangular)

Mixed

CCP&SP

RHardTimeWindows

Branch-and-Price

Binartet

al.[26]


(DiscreteTriangular)

Mixed

CCP&SP

RHardTimeWindowson

mandatory

custom

ers

Two-phaseapproach.

22


More recently, Kenyon and Morton [92] tackled the version with stochastic service and travel

times under two dierent objectives. The rst one minimizes the maximum completion times of the

whole set of routes, while the second one maximizes the probability of completing the routes within

a pre-specied deadline. A Generalized Variable Neighbourhood Search (GVNS) for the capacitated

VRP with stochastic service times is designed in [112] using a penalization on route duration. Service

times are assumed to be normally distributed with a coecient of variation (CV) of 0.25. A closed

form to calculate the expected delay is presented under normality assumption. GVNS shows better

results at the expense of higher computational times in comparison to a VNS and a VND elaborated

by the same authors. An interesting study is conducted by Lecluyse et al. [111] for the VRPST

with stochastic time-dependent travel times. The authors consider four scenarios depending on two

relevant dimensions: speed proles and weather/road conditions. The proposed objective function

includes the expected duration of the routes but also its variability. This last is added through

the standard deviation, which is multiplied by a parameter β(β ≥ 0). Parameter β is intended to

represent the risk preferences of the decision maker. Results are obtained with a TS heuristic on 27

Augerat datasets. The authors conclude that their solution approach gives more robust and reliable

routes. In fact, their solutions compensate the increase in the expected route durations by diminishing

their variability, especially, when roads are congested for the most part of the day or when weather

conditions are bad.

When the VRPST is considered, one fundamental dimension to take into account is the time

windows. VRPST with soft time windows have been mainly studied [163, 164, 162] when compared to

hard time windows [53, 26]. However, many other works enforce the respect of the early time window

constraint but allow late services [6, 118, 178, 48, 128, 131], thus mixed soft and hard time windows are

used. One of the reasons for the use of soft time windows can be related to the convolution property.

The use of convolution property enables the deduction of closed form expressions to calculate the

recourse cost (SPRs) or the probability of constraints violations (CCPs). Hard time windows have

a truncation eect on the cumulative times, thus, convolution rules are no longer useful. Aiming to

overcome this problem some authors have used dierent techniques to model the arrival times when

hard time windows are considered: assuming that they can be modeled a well-known probability

distribution [36, 48], using discrete approximations [178, 128] to estimate their distribution or modeling

travel and service times by phase type distributions [75] and then approximate the distribution arrival

times.

Wang and Regan [174] present a theoretical development in the context of truckload transportation

industry. The authors consider the problem with stochastic travel and service times, where, travel

times incorporate the time for handling the loads. The authors show that in the context of hard

time windows, given a known distribution of the arrival time at the rst node, the same distribution

can be calculated iteratively for the next customers in the route. A CCP model is presented with

as objective maximizing the expected number of served customers, although no numerical results are

available. Ando and Taniguchi [6] studied the version with uncertainties in travel times within an

urban context in Osaka, Japan. An SPR is designed with penalization on early and late arrivals,

both costs are proportional to the time window violation. Although soft time windows are considered,

if a vehicle arrives earlier than the opening time window ei it must wait until it opens to perform

the service. Aiming to derive the distribution of the travel times, a study is conducted using probe

vehicles, recollecting time and speed information for several links. Nevertheless, since this information

is limited, they complement their inputs with a Vehicle Information Communication System (VICS)

which provides information on travel times, congestion, crashes, and car parks. A triangular distribu-

tion is proposed to model uncertainties about travel times, this is done for dierent zones of the city.

A case study is devised to compare the performance of usual routes against the ones created by a GA

which takes into account stochasticity. An improvement of around 4% is achieved for the total cost

(xed, operational and penalties costs). Moreover, the total running times were diminished by nearly

23


7%, and therefore, emissions of CO2, NOx and particle materials are reduced by around 7.5%

A similar problem is tackled in the article of Li et al. [118]. Both travel and service times are

assumed to be independent random variables following a normal distribution. Time windows are

considered in the same way as in [6], thus services are not allowed before the opening time window.

Two models are presented: a CCP and a SPR. The rst one bounds the probability of time windows

violation on both customers and the depot (route duration). The SPR, penalizes late arrivals regarding

time windows and route duration. To solve both formulations a modied TS is used. TS embeds a

Monte Carlo simulation to estimate probabilities of failure or the recourse cost. Results for own

generated instances with up to 100 customers are discussed. One of the main conclusions of the

authors is that CCP is overly constrained and harder to solve than the SPR. As a matter of fact,

when constraints on customers and route duration are simultaneously active, their method fails to

nd a feasible solution. A generalization of the models presented by Li et al. [118] is introduced by

Zhang et al. [178] proposing a combined SPR and CCP. The model allows to compare and make

trade-os between customers service and the operator costs by changing the associated constraints.

A computational study based on six Solomon [157] instances restricted to 20 customers is performed.

Travel times are assumed to be log-normal distributed while service times follow a normal distribution.

Coecients of variations range from 0.2 to 0.6 for the travel times and are xed to 0.4 for the service

times. The authors use a TS to solve the set of instances. Furthermore, a discrete approximation of

the arrival and service start time distributions, called α Discrete Approximation Method (αDAM), is

validated and used within the TS. αDAM considers a discrete set of possible values and the associated

probability mass function to represent the travel times, as well as the service times. A discrete

approximation of the arrival and service start times distributions is recursively derived. A comparison

with the normality assumption made by Chang et al. [36] shows the better performance of αDAM to

estimate the service levels, expected earliness and tardiness, and expected route costs. Nevertheless,

for the sake of fairness it must be mentioned that only one partial route involving 5 customers is

tested.

Ta³ et al. [163] deal with the pure soft time windows version of the problem under stochastic

travel times. In their work, services can start before or after the time window at the expense of a

penalization. The cost incurred by early or late services is dened as the service costs, while the

transportation costs account for vehicle travel costs and driver overtime. The two types of costs

are weighted in the objective function. Authors use the Gamma distribution to model the travel

times with various coecients of variation. To solve the problem a three-phase algorithm based on

TS is used. In the rst phase, the authors construct a solution by means of I1 heuristic proposed

by Solomon [157], without considering service costs. The initial solution is further improved by the

TS during the second phase. In the third phase, the best solution found by TS is improved by

changing the departure time of the vehicles. Since time windows are soft, the departure time can have

a signicant eect on total costs, by applying this post-optimization procedure the total costs are

reduced on average by 1.3%. Ta³ et al. [162] propose a column generation within a B&P procedure

to optimally solve the problem introduced in [163]. Using a set partitioning formulation, the authors

use a labeling algorithm to solve the associated Elementary Shortest Path Problem with Resource

Constraints (ESPPRC) [54]. Moreover, since time windows are soft, the only resource constraint is

the capacity of the vehicle. Due to this, the authors reduce the capacity of the vehicles for type 1

instances from the classical Solomon instances [157] to make them tractable. Results are provided for

20, 25, 50 some 100 customers instances. An extension of the work presented in [163] is developed

by Ta³ et al. [164] including stochastic time-dependent travel times (see also [111]). In this work,

the authors considered 5 intervals of time within a day in which travel speeds vary due to trac

conditions. Assuming that the time to travel a unit distance can be modeled by a Gamma random

variable, the mean and variance of the arrivals times are calculated in a closed form if customers

have no service times. If service times are considered, a rst order approximation is used (see also

24


[88]). Two approximate methods are proposed to solve the problem, a TS, and an Adaptive Large

Neighborhood Search (ALNS).

More recently, Ehmke et al. [48] studied the VRPST with stochastic travel times. The opening time

windows are considered hard while the time windows closure time windows are soft. No penalization is

considered for late services. The CCP formulation is solved by means of a TS. The approach used from

Ehmke et al. to estimate arrival and initial service times is quite similar to the one used by Chang et al.

[36]. Thus, arrival times are assumed to be normally distributed and initial service times are modeled

by truncated (at ei) normal random variables. A statistical study shows that even if travel times follow

shifted exponential or shifted Gamma distributions, assuming normality for the arrival times allows a

reliable estimate of metrics such as the mean, standard deviation and the 95th percentile for the start-

service times. This approach has the advantage of being easily incorporated into classical methods

for the VRPTW. Stochastic travel and service times are considered by Miranda and Conceição [128].

A CCP formulation is used so late arrivals are constrained by a desired service level. Early services

are not allowed but late services are permitted. The truncated random variables are modeled using a

discrete approximation of the arrival and service start times distributions similar to the one of Zhang

et al. [178]. An exhaustive comparison to αDAM is presented showing an improvement in various

metrics. Moreover, time consumption is greatly improved. The discrete approximation is embedded

within an ILS algorithm to solve some Solomon [157] instances.

Nguyen et al. [131] present an interesting approach for the VRPST with stochastic travel times.

A Satiscing Measure Approach (SMA) is derived to evaluate the dissatisfaction of the customers for

late arrivals. Late services are allowed and early arrivals are forbidden. Travel times are assumed to

follow an ambiguous distribution, though, for simulation purposes it is assumed that they follow a

Gamma distribution. The SMA is incorporated into the objective function as main objective (among

a hierarchy). By construction of the SMA, this implies a minimization of the expected tardiness at

customers. Their approach is embedded within a TS algorithm showing a remarkable performance;

moreover, the results show that by incrementing the number of vehicles used, as well as the distance

of the routes, the expected tardiness at customers can be signicantly reduced. Errico et al. [53]

extended their work in [52] dealing with stochastic travel times and hard time windows. In their more

recent work, the authors used a combined SPR and CCP to solve the problem. Penalties are incurred

each time a customer is not provided with a service. Furthermore, customers that are not served

are picked up based on the two proposed recourse actions. Moreover, customers may be avoided if

they induce the route to be operationally infeasible. In such a case, the current or the next customer

is selected to be omitted. Also, two constraints are added to ensure that the probability of having

no failure at a route reaches a required threshold and limit the number of recourses to at most one.

Another interesting point in this work is the way in which information is revealed. When a vehicle

arrives at a customer, it spends some xed time to determine the actual service time. Only then

the choice to avoid current or next customer is made. Service times are assumed to be discrete

random variables with triangular distribution and two cases regarding the range of possible values are

considered for experiments. The problem is modeled as a set partitioning problem and solved through

a branch-and-price.

Binart et al. [26] also studied the problem with hard time windows. In fact, their model considers

two types of customers, mandatory and optional. Only mandatory customers have an associated

time window. Uncertainty aects both the service and travel times which are modeled by discrete

triangular distributed random variables. Moreover, multiple depots are considered. A two-phase

approach is used to solve the problem. During the rst phase a skeleton of routes serving mandatory

customers is created. During the second phase, optional customers are added to the planned routes

which are modied in real time to enable time windows to be respected. Furthermore, in the second

phase, the routes are evaluated in an SPR and CCP context. Penalties are included if the time window

is missed, while the probability of success is also constrained.

25


In summary, VRPST have been solved mostly with the use of metaheuristic methods because of

the complexity of the problem. The solutions approach usually rely on a static approach. Simple

heuristic methods have been rarely used [100]. Within metaheuristics, approaches such as TS [148,

118, 48, 163, 178, 164, 131] are the foremost used, other metaheuristics such as ALNS [164], GNVS

[112], GA [6], and approximate methods with Branch-and-cut based frameworks [92, 1] have also been

used. Exact methods are limited to branch-and-cut [103], branch-and-price [162] and branch-and-cut-

and-price [53]. The use of these approaches is limited to cases where few scenarios are considered

[103], when convolution properties enable closed forms evaluations [162] or when discrete variables are

considered [53].

2.2.2 Robust Optimization

While the SVRPs have been mostly solved using the Stochastic Programming paradigm other ap-

proaches in the literature can be found. In fact, stochastic models have been criticized on three main

points: the underlying probability distributions must be known in advance, convolutions must be com-

putationally tractable and provided solutions might be infeasible for some realizations of the random

variables [156]. Actually, if the underlying probability distributions are unknown or information is

very limited this precludes the use of SVRPs. Robust Optimization (RO) oers a dierent approach to

deal with uncertainties. In fact, uncertainties become deterministic and set-based [20]. The objective

is to create solutions that are feasible for any realization of the uncertain parameters rather than being

immunized in a probabilistic sense [20]. The interested reader is referred to the book of Ben-Tal et

al. [16] and the paper of Bertsimas et al. [20] as excellent introductory points.

Although less studied than SVRPs, robust optimization has also been applied to VRP problems.

Sungur et al. [161] tackles the Robust Vehicle Routing Problem (RVRP) with demand uncertainty.

Demands are uncertain but belong to a bounded set. This bounded set is intended to represent

deviations around the expected demand values. The authors consider three sets, namely, convex hull,

box and ellipsoidal. However, most of the results are given for the convex hull set. Furthermore, it

is shown that under these bounded sets, the RVRP with uncertain demands can be formulated as a

Mixed Integer Linear Problem (MILP), using a BC method the problem is thus optimally solved. An

interesting comparison for dierent deviations in the demand uncertainty sets are presented. Moreover,

in this study CCP and SPR models are also compared to the RVRP. CCP oers similar results to the

RVRP when deviations are small, still when this parameter is increased CCP becomes infeasible. In

general terms, SPR oers solutions of equal quality to those provided by the RVRP or even better under

high deviations. The same problem is considered by Gounaris et al. [79]. Demands are assumed to be

supported on budget and factor model sets. Furthermore, the authors state the conditions to reduce

the RVRP with uncertainties in demands into a deterministic VRP. Several classical formulations for

the CVRP are examined as well as a new assignment formulation. Results present the performance

of the dierent formulations giving the use or not of robust rounded capacity inequalities. Gounaris

et al. [78] extended the previous work developing an Adaptive Memory Programming Framework

metaheuristic. This method allowed them to solve instances with up to 483 customers. A conclusion

of the work is that robust solution, which increment on average 5% the cost, are able to handle the

uncertainty even with a considerable augmentation in demands.

Solano et al. [156] tackle the robust VRP with uncertainties in the arc costs (equal to time).

The authors consider the variability in travel times using multiple scenarios. Each scenario takes into

account the cost for the whole set of arcs in the problem. An exact MILP is used to solve small size

instances. For bigger instances containing up to 100 customers and as much as 20 scenarios, several

heuristic and metaheuristics are devised. Among the solution approaches, the Multi Start ILS gives

the best results in terms of quality, averaging a gap of only 0.35%.

Robust Optimization is also used to tackle VRPs with times uncertainty. Agra et al. [2] deal

with the VRP with time windows and uncertain travel times inspired by maritime transportation.

26


Travel times are assumed to be determined by an uncertain polytope. Two formulations are proposed

based on resource and path inequalities, and are solved by means of either a branch-and-cut or a

column-row-generation algorithm. Several results are presented for instances with up to 50 nodes.

Moreover, dominance rules are presented for the path inequalities version showing that the number of

extreme points to represent the polytope can be largely reduced. Chen et al. [37] work on the version

with uncertain service times in an Arc Routing Problem context. A BCP is used to solve the problem

considering a restricted budget of uncertainty on routes. The maximum deviation of the service times

can be the double of the nominal time. Three dierent deviations are studied (0.4, 1.2, 2.0). The

results are compared to a CCP with constraints on route durations. For the considered problem, the

RO gives less costly routes which by the way, are commonly feasible in the CCP. Adulyasak and Jaillet

[1] propose to solve the RVRP with deadlines and travel times uncertainty. Moreover, the authors also

tackle the SVRP variant under the same type of uncertainties. For the RVRP a performance measure

called lateness index is dened to minimize the risk of violating deadlines, while the SVRP formulation

uses a sampling approach similar to the one in [92]. Furthermore, an extension to consider soft time

windows is described, in the case of the RVRP this is achieved by adding an earliness index. To

solve both formulations, the author use a branch-and-cut approach. Results are reported for instances

considering directed graphs with up to 80 customers and 240 arcs. A comparison between the SVRP

and the RVRP shows that SVRP heavily resides in the prior knowledge of the underlying probabilistic

distributions, while the RVRP outperforms the SVRP if this information is incorrect.

2.2.3 Fuzzy Logic

Fuzzy logic has also been applied to deal with VRP under uncertainties. In fact, given the inherent

variability of the problems parameters, fuzzy numbers and variables can be used to model these

parameters using the fuzzy logic sets. Fuzzy sets work dierently than deterministic sets. In the

former, a number has a degree of pertaining to the set with a grade of membership which ranges

between zero and one [4]. Deterministic sets are thus a particular case of the fuzzy ones where partial

membership to a set is not allowed, therefore, the grade of membership is fully dened by either one

or zero. Moreover, fuzzy numbers can be dened as convex and normalized fuzzy sets. Using fuzzy

numbers, the parameters of the problem can be represented given that the underlying probability

distribution is unknown or hard to be computed. For further details on Fuzzy variables the interested

reader is referred to [91, 155].

Teodorovi¢ and Pavkovi¢ [165] consider the VRP under demands uncertainties. Using the fuzzy

set theory, the authors model the demand at each customer as a triangular fuzzy number. Moreover,

by using fuzzy arithmetic, it is shown that the remaining capacity after visiting a customer can also be

represented by a triangular fuzzy number. Two approximate reasoning algorithms are then proposed.

The rst one, considers only the remaining capacity to decide if another customer should be serviced

by a partial route. The second one, considers both the remaining capacity and the customer demand

to decide if it should be included in a route or not. To solve the problem, the sweeping algorithm

[70] is modied to consider the fuzzy logic. The solution is further evaluated within a simulation

procedure to calculate the expected distance traveled due to failures. The classical recourse action

is considered. Results are reported for a single instance with 100 customers with an improvement

when the second reasoning algorithm is selected. A similar problem is considered by Cao and Lai

[35] addressing the Open VRP with fuzzy demands (OVRPFD). The problem is modeled as a CCP,

making use of fuzzy credibility measure to guarantee customers satisfaction. Demands are assumed to

behave as a triangular fuzzy number and the decision maker can select a preference index representing

is risk attitude towards the possible failures. Moreover, the classical recourse for the VRPSD is used

when failures arise. The OVRPFD is solved with a hybrid method combining Monte Carlo simulation

and a dierential evolution algorithm. Results are presented for instances with up to 100 customers.

Kuo et al. [96] worked on the VRP with time windows in a context where both travel times and

27

2.3. CONCLUSION

time windows are modeled through fuzzy numbers. Both parameters make use of the triangular

membership function. To solve the problem, a two phase algorithm is proposed. In the rst part, an

Ant Colony optimization procedure is used to solve the associated TSP. Then in a second phase, routes

are constructed using as input the results of the AC procedure. Routes are considered feasible if they

respect the capacity and if the time windows respect a required service level. This one is considered

as the largest membership value found in the intersection of the fuzzy arrival time and time window.

Some results are presented for a case study with 50 customers and dierent types of time windows.

He and Xu [81] deal with the VRP with uncertainties in customers demands and travel times. In this

study, the authors consider travel times as independent normal random variables, and the demands

as fuzzy variables. In fact, demands follow a normal distribution with a mean value represented by

a trapezoidal fuzzy number. The variance of the demand is assumed to be crispy. A CCP model

is deducted to guarantee a service level on the vehicles capacity and customers time windows. The

CCP model is solved by means of a GA and limited results are shown for an instance with more than

200 customers. Zheng and Liu [179] propose a VRPTW with uncertainties in the travel times. A

CCP is used to model the problem, considering a service level on customers time windows. Using

Monte Carlo simulation embedded within a GA, the authors solve a problem with 18 customers given

triangular fuzzy variables. Xu et al. [175] consider the VRPTW with soft time windows. Uncertainty

is considered for the tolerable starting time, which is modeled through a random fuzzy variable. That

is, services can start earlier than ei or later than li, but within endurable earliness and lateness

times. Therefore, uncertainty aects the time windows. The proposed model aims to minimize the

operational costs and maximize the average satisfaction level of customers. This satisfaction level

depends on the service starting time, whether it is outside the time window, during endurable times

or within the time window. A particle Swarm Optimization (PSO) is used to solve the problem and

results are presented for a case study application with 18 customers.

2.3 Conclusion

The VRP is an active research eld. The last sixty years have shown a massive amount of research

within the overwhelming dierent types of VRPs. Although most of the research has been concentrated

on deterministic problems, in the past three decades an important amount of time was devoted

to variants that consider the inherently uncertainty of information in many applications. How the

uncertainties are handled is hardly dependent on the information available, e.g. if the underlying

probability distribution of the parameters is known or not.

Stochastic optimization has been the main approach to solve uncertain VRPs. Even though SVRP

eld is still at an early stage of development [63] it has an active community. Still some issues can

be detected in the reviewed bibliography. For instance, many of the works deal with small to at most

medium instances. That is the case for the VRPSD and the Cristiansen a Lysgaard [38] benchmark

whose instances contain at most 60 customers. Incrementing the instances size of the testbeds can push

knowledge boundaries by challenging the creation or adaptation of new methods (exact and heuristic)

to solve these problems. Moreover, this implies to design strategies that exploit characteristics of the

SVRPs, while overcoming the inherent computational challenge of solving the SVRPs.

In SVRPs with stochastic times problems involving soft time windows (or mixed soft and hard

time windows) are more common while the hard time windows case is very rare. Additionally, when

hard time windows are considered, usually discrete probability distributions are used to handle the

problems complexity. Thus, SVRP with hard time windows remains a fairly unexplored area in the

eld. Besides, the largest amount of articles in the literature rely on convolution properties (also

for the VRPSD) to deal with stochastic parameters. Even if convolution can arise naturally when

dealing with random parameters, there is no proof that it is always (or recurrent) the case for SVRPs.

Therefore, solution approaches should include mechanism to be exible when convolution properties

28


do not hold. This without forgetting the tradeo between accuracy and exibility.

Given the before, this rst part of the thesis focuses on solving the SVRPs for large instances as

a way to create new comparable results. This is achieved by adapting solution methods with tailored

strategies for the SVRPs, particularly for the VRPSD in chapter 3. Then a SVRP with stochastic

travel and service times and hard time windows is considered in chapter 4.

29

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40

Chapter 3

Hybrid metaheuristic for the VRPSD

3.1 Introduction

Since its introduction by [9], the Vehicle Routing Problem (VRP) has been widely studied in the

literature, becoming a classical combinatorial problem. While the VRP comprise a broad family of

problems, it is commonly used to refer to the Capacitated Vehicle Routing Problem (CVRP) version.

The CVRP objective is to build the set of vehicle routes with minimum cost satisfying customers

demands. Stochastic Vehicle Routing Problems (SVRPs) are generalizations of the VRP where one

or more parameters of the problem are associated with random variables. Recently, more attention

has been paid to SVRPs. Their importance relies on their closeness to reality and their ability to take

into account variability of data. A recent review of the main SVRPs variants studied in the literature

can be found in [18].

To model SVRPs two stochastic approaches have been widely used in the literature: Stochastic

Programming with Recourse (SPR) and Chance Constraint Programming, using probabilistic con-

straints (CCP). Contrarily to CCP, the rst approach considers actions (called recourse) to overcome

or react to possible violations of the constraints. In fact, since parameters are random variables some-

times constraints might not hold leading to failures. SPR takes into account the cost associated with

recourse within the objective function. The CCP introduces constraints to limit the probability of

failures to a threshold aiming to guarantee a quality level of the solution. It should be noted that

SPRs and CCPs are not exclusive and can be used in mixed formulations (see for example [12]).

The Vehicle routing problem with stochastic demands (VRPSD) is an extension of the VRP in

which the demand of each customer is a random variable. VRPSD was originally proposed by [40] who

solve the problem through a modication of the well-known Clarke and Wright heuristic ([8]). Exact

methods can be found in the VRPSD literature and are divided in two approaches: the L-shaped

algorithm (see [23]) and branch-and-price based. L-shaped methods ([?], [21], [24], [22], [4], [35]) have

been the preferred approach to solve the VRPSD and are able to optimally solve instances with up

to 100 customers and few vehicles when discrete distributions are considered ([24]). In [22] normal

distributions are used to model the demands attaining optimal solutions to instances with 60 to 80

nodes and two to four vehicles.

Branch-and-price based methods have shown to solve problems with larger number of vehicles. The

rst branch-and-a-price algorithm for the VRPSD is presented in [7] to deal with instances with at

most 60 customers under a Poisson demands assumption. More recently, a branch-and-cut-and-price

algorithm is implemented by [15] and tested on instances with up to 101 customers and 15 vehicles.

Heuristics and metaheuristics methods are more often used to solve the VRPSD. In [39] a CCP and

two SPR models are provided for the problem at hand, which is then solved by means of the Clarke

and Wright heuristic [8] and a Lagrangian Relaxation based heuristic for instances with normally

independent and correlated demands. In this study, a transformation of the VRPSD into CVRP

41

3.1. INTRODUCTION

under certain conditions is also discussed. [?] propose a Tabu Search called tabustoch to tackle the

extension of VRPSD where additionally, customers are present or not with a given probability.

[41] consider the VRPSD under restocking possibilities, thus adopting strategies for preventive

restocking. That is, return trips to the depot to restock even if the vehicle is not empty to avoid future

failures. The authors develop and embed an optimal restocking policy in the route design. In [3],

hybridization of ve metaheuristics with dierent objective function approximations and preventing

restocking is devised. More recently, [20] propose a Simulated Annealing procedure to solve the

VRPSD that uses a cyclic-order encoding to represent solutions. The authors designed a two-phase

technique embedded in the solution method. In the rst phase they consider only deterministic costs,

while the recourse cost is explicitly integrated into the second phase.

[28] use a Memetic Algorithm to solve the multi compartment VRPSD which is a generalization

of the classic VRPSD. In the multi compartment VRPSD, the customers are associated to several

products demands that cannot be mixed. This constraint imposes to load each product in a dierent

compartment. In a more recent study, [29] designed a Greedy Randomized Adaptive Search Procedure

(GRASP) enhanced with Heuristic Concentration (HC) to solve the VRPSD with maximum route

duration constraints. To the best of our knowledge, the method of [29] reports the best overall results

for [7] testbed.

Other variants of the VRPSD have also been considered in the literature. [27] addressed the

VRPSD with weight-related costs and solved it by means of an Adaptive Large Neighborhood heuristic

using several approximate methods. [37] dealt with the VRPSD with dynamic requests meaning that

previously unknown customers can be received and scheduled over time. A Variable Neighborhood

Search (VNS) based approach was proposed by the authors to solve both stochastic and dynamic

cases.

[32] study the VRPSD with time windows using a Satiscing Measure Approach (SMA). The SMA

is embedded in a tabu search showing very competitive results in small computational time. A multi-

objective version of the VRPSD considering total traveling distance, total driver remuneration, number

of vehicles and drivers remuneration balance is proposed in [16] and solved using a multi-objective

evolutionary algorithm.

Concerning the stochastic models, SPR formulations have been dominant in the VRPSD literature

compared with CCP formulations ([38, 39, 10]). The most used recourse policy, called from now

onward in this paper the classical recourse, is dened as follows. When the charge of the vehicle is

emptied (fullled) it returns to the depot to replenish (unload) the charge, and resumes its assigned

route from the failure point ([2]). However, other recourse policies have been studied and implemented

through several studies: preventive restocking policies are extensions of the classical recourse where

return trips to the depot are performed even if the vehicle is not empty to avoid future failures

([41, 3, 4, 27, 42, 35, 36]); pairing strategies allow the cooperation of multiple vehicles ([1]); split

deliveries between paired routes ([26]) in which some customers are served by two vehicles; and

backup routes ([11]) that receive customers from primary routes.

Although the use of more complex recourse policies can represent a signicant saving relative to

simpler ones ([1]), the latter have been preferred since they allow more tractable models and stable

tactical routes ([17]). For this reason only the works of [4] and [35] deal with exact methods for the

VRPSD using recourse actions dierent than the classical one. [4] consider a restocking policy for the

generalized VRPSD using a single vehicle. In [35] optimal restocking policies allow vehicles to decide

between a visit to the depot to replenish or proceeding to the next customer. Decisions are made

using an optimal remaining capacity threshold for each customer within the route. The authors show

that under arbitrary discrete probability distributions, instances with up to 60 customers and four

vehicles can be solved. Moreover, it is shown that restocking policies can reduce the recourse cost to

half of those achieved by the classical recourse.

This chapter is dedicated to the vehicle routing problem with stochastic demands using the clas-

42

CHAPTER 3. HYBRID METAHEURISTIC FOR THE VRPSD

sical recourse. We propose a Memetic Algorithm hybridized with a GRASP (MA+GRASP) to solve

eciently the VRPSD. The GRASP framework is embedded within a MA and is used as a way of

restarting the algorithm. In this chapter, it is shown that the MA+GRASP is a valid and ecient

method to solve the VRPSD. Moreover, a new testbed built from instances originally designed for the

CVRP with up to 385 customers is proposed. The obtained results on new large instances serve as

benchmark for testing future methods in real life scale problems. The remainder of this chapter is

organized as follows. The problem formulation is introduced in section 3.2. Section 3.3 presents the

developed solution approach for the problem. Numerical results are given and discussed in section

3.4. Finally, section 3.5 concludes the chapter.

3.2 Problem formulation

The traditional CVRP can be described as follows. Let G = (V,E) be a complete undirected graph

where V = 0, 1, . . . , i, . . . , n and E = [i, j]∀i, j ∈ V | i < j are the vertex and the edge sets

respectively. Moreover, V c = V \ 0 is the customers subset, each customer has a non-negative

demand qi ∀i ∈ V c. Vertex 0 stands for a central depot where a homogeneous set of vehicles with

a limited capacity Q each are initially located. Furthermore, each edge [i, j] ∈ E has a non-negative

cost cij . The objective is to build a set of routes with minimum cost considering that each route must

start and end at the depot, the maximal capacity Q must be respected and, no split deliveries are

allowed. This last constraint means that each customer is serviced once by a vehicle which deliver (or

pickup) its whole demand.

The problem tackled in here presumes that the demand qi of each customer i follows a probability

distribution ψ, with expected value and variance noted E [qi] > 0 and V ar [qi] > 0 respectively. We

assume that probability function ψ is known and demands are mutually independent. Furthermore, we

consider as other authors ([7, 15, 20, 29]), that ψ distribution has a cumulative property, i.e. the sum

of demands probability functions is also ψ distributed. Furthermore, the vehicles eet is assumed to be

unlimited and no xed cost per vehicle is involved. This problem is formulated in this work as an SPR

in which the rst recourse policy of Bertsimas [2] is employed. This recourse (classical) assumes that

whenever a route achieves its maximum capacity Q, the vehicle returns to the depot to load/unload,

then it returns to the customer where the capacity was fullled to complete the unserviced demand

and then continues its route from the failure point. Let r be a route dened as a sequence of nodes

r = r0 = 0, r1, . . . , ri, . . . , rk, rk+1 = 0, the cumulative demand up to a client ri in a route r can be

dened as Dri =∑ij=1 qrj with E [Dri ] =

∑ij=1E [qrl ], V ar [Dri ] =

∑ij=1 V ar [qrl ]. The expected

recourse for a given customer ri ∈ V c in a route r is estimated by equation 4.1, as done by [15].

ERCri = 2 · c0ri ·

[ ∞∑u=1

P(Dri−1

≤ uQ)− P (Dri ≤ uQ)

](3.1)

Given a route r, the term P(Dri−1

≤ uQ)stands for the probability of cumulative demand up to

customer ri being less than or equal to a multiple of the capacity of the vehicle. Therefore, the

probability part of the expression represents the sum of having the uth failure of the route at client ri.

It shall be noticed that this expression does not consider the case when the remaining capacity equals

the demand of a customer (exact stock-out). In such case the vehicle can return to the depot and

then, continue towards the next customer in the route. Indeed, Hjorring and Holt [21] have already

addressed the exact stock-out recourse. Nevertheless, we keep expression (4.1) since probability of such

events might be rather low and its consideration can make the calculation computationally inecient.

Hence, the expected cost of a given route r can be calculated using equation 4.2.

E [Cr] =

k∑j=0

crjrj+1 +

k∑j=1

ERCrj (3.2)

43

3.3. SOLUTION APPROACH: HYBRID METAHEURISTIC

In order to avoid multiple failures in a route r, the expected demand is limited to be at most equal to

the maximum capacity Q, this assumption has been already used by many authors such as Laporte

et al. [24], Christiansen and Lysgaard [7], and Gauvin et al. [15]. Indeed, if constraint (4.3) is not

included, the optimal solution tends to be composed by only one route which visit all the customers.

k∑i=1

E [qri ] ≤ Q ∀r (3.3)

The objective is thus to create a set of routes with minimum expected cost calculated by means of

equation 4.2 and respecting constraint 4.3.

3.3 Solution approach: hybrid metaheuristic

A hybrid metaheuristic, combining a Memetic Algorithm (MA) and a GRASP, is proposed to solve

large VRPSD instances. The proposed method is described in Algorithm 1. The MA basis is borrowed

from the ideas of Prins [33] and it works with a xed population size. MA starts by creating an ordered

initial population called Pop (line 1 Algorithm 1). New individuals are created from the crossover

of two chromosomes selected from Pop (line 5 Algorithm 1). Moreover, a mutation procedure can

be performed on the new individual with probability pmp (line 6 Algorithm 1). A local search is

then executed with an associated probability of pls (line 8 Algorithm 1) on the resulting solution. A

procedure called Split is used to evaluate an individual tness (lines 7 and 10 Algorithm 1) and is

presented in section 3.3.1. It allows also to convert chromosomes to VRPSD solutions by computing

the detailed routes. Furthermore, when the local search is carried out, the routes are concatenated

before using Split (line 10 Algorithm 1). To ensure a diversity on the population, clones are not

allowed (line 12 Algorithm 1). Indeed, a new chromosome is kept for the next iteration only when

its distance to the current population is not null. The distance measure used is the broken pairs [6]

that counts the number of times a pair of consecutive customers in a rst individual is broken in a

second one. When the new individual is accepted to enter the population, it is added to this last in a

position that keeps the population ordered (line 13 Algorithm 1).

Since the MA works with a xed size population, when a new chromosome is entering the popula-

tion, another one already in Pop is removed. This last is randomly selected among those with tness

superior to the median (line 13 Algorithm 1). Moreover, the MA uses a restart procedure. Indeed,

after each φ iterations without improving the best solution the MA discards all the individuals ex-

cept the best one (line 17 Algorithm 1). The population is completed using a GRASP procedure as

explained in section 3.3.2. The algorithm stops when a time limit τ is achieved or if ρ iterations have

been performed without improving the best solution found so far (line 4 Algorithm 1).

44


Algorithm 1 MA + GRASP1: Pop← Initialize population2: φ← Constant > 03: i← 14: while not (stop) do5: c← crossover (Pop)6: Mutation, pmp (c)7: Split (c)8: Local Search, pls (c)9: if Executed Local Search then10: Concatenate and Split (c)11: end if12: if Is Not Clone (c) then13: Update Population (c,Pop)14: end if15: Update(i)16: if i ≥ φ then17: Restart with GRASP (Pop)18: i← 119: end if20: end while

3.3.1 Chromosomes

The MA + GRASP uses a twofold representation for each individual. The rst one follows the

idea of Prins [33] and consists in representing a solution by a permutation of the V c customers.

This representation have been already used by authors such as Mendoza et al. [28] for the multi-

compartment VRP with stochastic demands, Mendoza et al. [30] and Goodson et al. [20] for the

VRPSD, and Mendoza et al. [29] for the VRPSD with time time duration constraints. The second

one gives the detailed routes composing the solution. In order to decode the permutation of customers

into a set of routes, the Split procedure presented in [33] is employed.

Split works by constructing a directed graph H = (W,Y ) composed by its vertex set W =

W 0 = 0,W 1, · · · ,W i, · · · ,Wn, where W 0 serves as a dummy auxiliary vertex while the rest of

vertex W 1, · · · ,W i, · · · ,Wn ∀i ∈ V c are the permutation of the V c customers. The arcs set Y is

built in way that every arc (W i,W j) | j > i represents a feasible route starting at the depot, visiting

customers W i+1, · · · ,W j and returning to the depot. Since each arc is associated with a feasible

route r, the arcs have an associated weight equal to E [Cr]. Therefore, it is during the construction

of routes (arcs) composing graph H that stochasticity is considered, by properly calculating the costs

using equation (4.2). After constructing the set of all feasible routes (respecting constraint 4.3) the

goal is to nd the shortest path from vertex W 0 to Wn and thus the arcs composing the shortest path

are the optimal partition for the permutation of customers in the ordered sequence to routes. The set

of routes is, in fact, the second representation of the individual, and is used when the local search is

performed. For more details about Split method the reader is referred to [33].

Figure 3.1 presents an example of the Split procedure on a VRPSD instance composed by ve

customers (V c = 1, 2, 3, 4, 5) and where vehicles have a limited capacity Q = 50. Moreover, it is

assumed that all clients have a stochastic demand which follows a Poisson distribution with mean

20 units, that is qi ∼ Poisson (20)∀i ∈ V c and costs (cij) are equal to the distances. Part (a) of

gure 3.1 represents a permutation of clients that is going to be decoded. Part (b) gives the necessary

distances information to decode the permutation. Part (c) stands for the Split auxiliary graph, note

that each arc has an associated cost (calculated by means of equation (4.2)), furthermore, the dashed

arcs are the ones used in the solution of the shortest path. Take for example arc (2, 1), it represents

the route r = 0, 4, 1, 0 and has an expected cost of E [Cr] = 88.68, the cost can be further divided

into deterministic cost 85 units, and expected recourse cost 3.68 = 88.68 − 85. Figure 3.1 part (d)

shows the optimal decoding of the permutation into routes, with the cost of the whole set of routes

which is the tness or cost of the individual.

45


0

2

4

1

5

3

2 4 1 5 30 1 52 4 3

15

10 10

5

20

40

35

20

10

Clients permutation (a)

40 80 70 40 20

79.21 67.11

88.68 36.05

Split auxiliary graph (c)

Decoded routes from permutation (d)Individual cost: 40.00 + 88.68 + 36.05 = 164.74

Distance information (b)

0

2

4

1

5

3

Figure 3.1: Split example for the VRPSD.

0

1

2 34

5

0

1

23

4

5

Solution 1 Solution 2

Figure 3.2: Broken pairs distance example.

Concatenation procedure allows to pass from a set of routes representation to a permutation one.

It is achieved by removing the depot node from the start and end of the routes. Then, the customers

sequences of the routes are added one after another. See for example the part (d) of gure 3.1, where

routes 0− 2− 0, 0− 4− 1− 0, and 0− 5− 3− 0 derive to the customers permutation 2− 4− 1− 5− 3.

Besides, the calculus of the broken pair distance [6] is performed using the routes representation.

Figure 3.2 shows two solutions for which the number of broken pairs is to be calculated. Solution 1

serves as base for the comparison while solution 2 is a candidate to enter into the population. The

distance measure is two since the arcs 2− 3 and 4− 5 present in solution 2 are not present in solution

1.

3.3.2 Initial population and Restart

Population Pop is initially lled with three individuals created with the well-known heuristics: Clarke

and Wright [8], Gillet and Miller [19] and best insertion. Each of the aforementioned heuristics is

executed twice with slightly dierences in the capacity Q. In the rst call to the heuristics, the full

capacity of the vehicles (Q) is considered, while in the second one the capacity of vehicles is reduced

to Q′ = 0.9 ·Q. That is, constraint (4.3) right side is changed to 0.9 ·Q. By doing so, it is expected

that constructed routes will have a lower probability of failures (see [28]), introducing important

information to the population. These heuristics are modied to consider the associated recourse costs.

The routes obtained by each heuristic are then concatenated to obtain a chromosome which is then

evaluated by the splitting procedure described in 3.3.1. Only the best three individuals among the

46


six generated ones are kept in the initial population, the other three are discarded. To complete the

population Pop, the remaining chromosomes are created from completely random permutations of

customers which are next evaluated with the Split procedure. The population size is denoted Popsize

afterward in the paper.

Restart

Restart procedure is executed after performing φ iterations without improving the best solution found

so far. Algorithm 2 shows the pseudo-code of the procedure. It starts by deleting all the individuals

of Pop except the best one. In order to generate Popsize − 1 missing individuals the next strategy

based on a GRASP procedure is used. The GRASP generates⌈(Popsize−1)

2

⌉individuals. Meanwhile,

the remaining ones required to achieve Popsize are created from completely random permutations

of customers, which are next evaluated with the Split procedure. This approach is used to ensure

diversication within the restart.

Greedy Randomized Adaptive Procedure

Greedy Randomized Adaptive Search Procedure (GRASP) is a metaheuristic introduced by Feo and

Rasende [13]. GRASP consists in using a randomized constructive procedure to build solutions that

are improved after by a local search approach. This process is repeated through a number of iteration.

The behavior of the constructive procedure is controlled thanks to a parameter called greediness which

permit to balance between greediness and randomness.

The proposed GRASP works as follows. Iteratively, the Greedy procedure of the GRASP, which is

based on the Nearest Neighbor (RNN) heuristic is called. The RNN works by creating a permutation

of the customers. At each iteration, the RNN picks randomly a client among the k nearest neighbors

to the last visited customer (initially it starts from the depot), and add it to the permutation. After

several preliminary tests we picked k with the expression Max(2,⌈|V |60

⌉) and iterations to a value of

10. The RNN works without considering recourse costs as in [3, 20]. After generating a permutation

by using the RNN, it is decoded using the Split procedure. The resulting solution undergoes just

after the local search procedure (see section 3.3.4) considering only deterministic costs to avoid the

overhead of considering recourse costs. The best solution found by this steps is selected and kept.

Then, the local search procedure considering the recourse cost is called to improve the individual.

This last is added to the population and the process is repeated until⌈(Popsize−1)

2

⌉individuals are

added.

47


Algorithm 2 GRASP RestartRequire: Population Pop, iterations1: BestIndividual← Pop[0]

2: NumRuns←⌈(Popsize−1)

2

⌉3: Clear Population(Pop)4: i← 15: while i ≤ NumRuns do6: j← 17: BestCost←∞8: BestSol← null9: while j ≤ iterations do10: c← Generate Individual with RNN11: Split (c)12: Local Search (c) . Only deterministic costs13: if Cost (c) < BestCost then14: BestCost← Cost (c)15: BestSol← c16: end if17: j ← j + 118: end while19: Local Search (BestSol)20: Add to population (BestSol,Pop)21: i← i+ 122: end while23: Add to population (BestIndividual,Pop)24: Complete Population with Random Individuals25: Sort(Pop)

3.3.3 Crossover

Crossover procedure is performed in order to create new individuals, this is done by means of the OX

crossover. Two individuals p1 and p2 from the population are selected from binary tournaments as

well as two random positions i, j | i 6= j, j > i. Using the permutation representation of solutions,

the information from position i to position j (included) are copied from p1 to the new individual in

the same positions. Additionally, p2 is circularly traversed from position j + 1 to j completing the

ospring from j + 1 to i, with clients in p2 not included yet. By changing the roles of p1 and p2,

another individual can be created using the same procedure. Among the two new chromosomes one

is randomly retained.

3.3.4 Mutation and Local Search

After being created, a new chromosome can be modied by the mutation procedure (line 6 Algorithm

1). It consists in moving nm random selected customers from their current positions to new ones

randomly selected. In our implementation, after preliminary tests the value of nm is set to two. This

operation is performed on the chromosome. The proposed local search (LS) for the MA is organized

as a Variable Neighborhood Descent (VND), a variant of the Variable Neighborhood Search proposed

in [31]. The LS is performed on the routes representation. The neighborhoods used for the VND

(line 8 Algorithm 1) are the Or-opt and 2-opt movements in their intra and inter route versions, and

the inter-CROSS movement. The order in which neighborhoods are explored is randomly selected

each time the LS procedure is called. For a further related review on neighborhoods structures and

types, the reader is referred to [5]. The LS procedure starts by exploring the neighborhoods trying

to improve a solution. Whenever an improving movement is found in a current neighborhood it is

executed, and the procedure passes to the next one when no improving movement is possible. Each

time a solution has been improved, the procedure restarts from the rst neighborhood, running so

on until any of them is able to enhance the solution. Inter route Or-opt movements are limited to

sequences of at most three customers, and inter-cross movements exchange to at most two costumers

48


Table 3.1: VRPSD Christiansen and Lysgaard [7] Testbed comparisonMethod

MetricMA +GRASP

GRASP+ HC

SA

Avg. Gap < 0.01% 0.02% 0.35%Max. Gap 0.14% 0.19% 1.89%Avg. Best Gap 0.00% 0.00% 0.04%NBKS 40/40 40/40 33/40Max. CPU (s) 10.13 102.43 603.80Min. CPU (s) 0.69 1.69 9.00Avg. CPU (s) 7.39 36.09 268.66

per route.

3.4 Numerical results

The tests are carried on two groups of instances: the rst group is already used in the VRPSD literature

and is due to Christiansen and Lysgaard [7]. The second one, with larger graphs, is elaborated in this

work to asses the performance of the developed approach on real size benchmarks which can be used

for future works. The detailed results are provided in the following sections.

3.4.1 Classical testbed from Christiansen and Lysgaard

In order to assess the performance of the hybrid MA+GRASP, the tests are carried on 40 existing

instances proposed in [7]. In this benchmark, the number of customers varies from 16 to 60 and the

minimum number of vehicles needed to satisfy the customers demands is comprised between two and

fteen. Christiansen and Lysgaard [7] testbed is based on Augerat test sets A and P and Christodes

and Elion test set E. Demands are assumed to be Poisson distributed as in [7, 20, 30, 15, 29], with

expected values equal to the deterministic demand values. Moreover, travel costs are calculated as the

Euclidean distance between two nodes rounded to the nearest integer as done in [15]. For Christiansen

and Lysgaard [7] benchmark, the best Known Solutions (BKS) are either taken from [15] in which 38

solutions are proven to be optimal, or from [29] and [20] which use heuristic approaches. All tests were

conducted on a Dell Latitude E6420 personal computer with Intel Core i7-2760QM 2.4 GHz, running

under Windows 7 Professional 64 bits. The algorithms were coded on Java and compiled with JavaSE-

1.845 with maximum allocated memory of 1 Gb. Random variables and computation probabilities were

generated by the library of Stochastic Simulation in Java ([25]). Preliminary tests were performed to

select the parameters used by the heuristic, local search rate and mutation probability. These two last

parameters are set to 0.15 and 0.2 respectively. The hybrid MA+GRASP stops after 5000 iterations

without improving the best solution or after running for 10 seconds.

Table 4.1 summarizes the results of the proposed MA+GRASP compared to those of Mendoza et

al. [29] (GRASP-HC), and Goodson et al. [20] obtained with a Simulated Annealing (SA) procedure.

For each method are reported: the average gap on 10 runs (Avg. Gap), the maximum average gap

across the 40 instances (Max Gap), the average gap of the best solution found over the 10 runs (Avg.

Best Gap), the number of best known solutions found considering the 10 runs (NBKS), the average

time over 10 runs for the 40 instances (Avg. CPU), and the maximal and minimal average time over

the 40 instances (Max. CPU Min. CPU).

The MA+GRASP and the GRASP-HC achieve to nd all the BKS (40 out of 40) whereas SA

nd 33 BKS. Moreover, our method reaches an average gap below 0.01% (0.004%) which is nearly

four times lower than the GRASP-HC. The low average gap also shows the stability of the method

when dealing with dierent types of instances. The Max. Gap of 0.14% conrms the ability of the

MA+GRASP method to regularly nd near optimal solutions, this metric shows also that our results

49

3.4. NUMERICAL RESULTS

are better in comparison to all the published methods in the literature. Although times are not scaled

due to dierences in programming languages, operating systems, compiler versions and characteristics

of the computers, MA+GRASP seems to oer the best performance in terms of execution time. The

time metric is specially important in its Max CPU version since the tested instances were small (at

most 60 nodes). The MA+GRASP among the three methods oers the best results quality with

reasonable running times, suggesting it as an ecient method to deal with real life size problems.

Detailed results for MA+GRASP and GRASP+HC are given in table 3.2.

50


Table3.2:

ChristiansenandLysgaard[7]testbed

results

MA+GRASP

GRASP

+HC

Instance

BKS

Avg.Cost

BestCost

Avg.Time(s)

Avg.Gap

Avg.Cost

BestCost

Avg.Time(s)

Avg.Gap

A-n32-k5

853.60*

853.60

853.60

5.32

0.00%

853.60

853.60

14.79

0.00%

A-n33-k5

704.20*

704.20

704.20

4.93

0.00%

704.20

704.20

13.15

0.00%

A-n33-k6

793.90*

793.90

793.90

4.77

0.00%

793.90

793.90

13.34

0.00%

A-n34-k5

826.87*

826.87

826.87

5.37

0.00%

826.27

826.87

14.48

0.00%

A-n36-k5

858.71*

858.71

858.71

7.43

0.00%

858.71

858.71

20.26

0.00%

A-n37-k5

708.34*

708.34

708.34

7.47

0.0%

708.34

708.34

23.23

0.00%

A-n37-k6

1030.73*

1030.73

1030.73

6.81

0.00%

1030.86

1030.73

20.14

0.01%

A-n38-k5

775.13*

775.13

775.13

8.08

0.00%

775.13

775.13

20.11

0.00%

A-n39-k5

869.18*

869.18

869.18

8.26

0.00%

869.18

869.18

27.89

0.00%

A-n39-k6

876.60*

876.60

876.60

8.15

0.00%

876.60

876.60

25.33

0.00%

A-n44-k6

1025.48*

1025.48

1025.48

10.10

0.00%

1025.92

1025.48

33.93

0.04%

A-n45-k6

1026.73*

1026.73

1026.73

10.02

0.00%

1026.81

1026.73

31.93

0.01%

A-n45-k7

1264.83*

1264.83

1264.83

10.06

0.00%

1267.05

1264.83

38.47

0.18%

A-n46-k7

1002.22*

1002.22

1002.22

10.10

0.00%

1002.22

1002.22

46.23

0.00%

A-n48-k7

1187.14*

1187.14

1187.14

10.14

0.00%

1187.32

1187.14

55.05

0.02%

A-n53-k7

1124.27*

1124.27

1124.27

10.04

0.00%

1124.27

1124.27

80.22

0.00%

A-n54-k7

1287.07*

1287.07

1287.07

10.09

0.00%

1287.41

1287.07

86.17

0.03%

A-n55-k9

1179.11*

1179.11

1179.11

10.10

0.00%

1179.11

1179.11

66.16

0.00%

A-n60-k9

1529.82

1529.88

1529.82

10.09

0.00%

1529.82

1529.82

102.43

0.02%

E-n22-k4

411.57*

411.57

411.57

1.67

0.00%

411.57

411.57

1.24

0.00%

E-n33-k4

850.27*

850.27

850.27

5.67

0.00%

851.87

850.27

24.66

0.19%

E-n51-k5

552.26

552.26

552.26

10.07

0.00%

552.26

552.26

56.75

0.00%

P-n16-k8

512.82*

512.82

512.82

0.69

0.00%

512.82

512.82

1.69

0.00%

P-n19-k2

224.06*

224.06

224.06

1.72

0.00%

224.06

224.06

3.51

0.00%

51


Table3.2:

ChristiansenandLysgaard[7]testbed

results:Continued

MA+GRASP

GRASP

+HC

Instance

BKS

Avg.Cost

BestCost

Avg.Time(s)

AvgGap

Avg.Cost

BestCost

Avg.Time(s)

Avg.Gap

P-n20-k2

233.05*

233.05

631.58

1.86

0.00%

233.05

233.05

4.76

0.00%

P-n21-k2

218.96*

218.96

218.96

2.53

0.00%

218.96

218.96

6.04

0.00%

P-n22-k2

231.26*

231.26

231.26

2.83

0.00%

231.26

231.26

7.10

0.00%

P-n22-k8

681.06*

681.06

681.06

1.18

0.00%

681.06

681.06

4.72

0.00%

P-n23-k8

619.52*

619.53

619.52

1.21

0.00%

619.53

619.52

5.45

0.00%

P-n40-k5

472.50*

472.50

472.50

8.72

0.00%

472.50

472.50

26.49

0.00%

P-n45-k5

533.52*

533.52

533.52

10.03

0.00%

533.83

533.52

36.25

0.06%

P-n50-k10

758.76*

759.04

758.76

10.09

0.03%

758.76

758.76

40.40

0.00%

P-n50-k7

582.37*

582.37

582.37

10.09

0.00%

582.37

582.37

44.17

0.00%

P-n50-k8

669.23*

669.33

669.23

10.05

0.01%

669.33

669.23

39.7

0.02%

P-n51-k10

809.70*

809.70

809.70

10.08

0.00%

809.70

809.70

52.72

0.00%

P-n55-k10

742.41*

742.41

742.41

10.08

0.00%

742.41

742.41

56.26

0.00%

P-n55-k15

1068.05*

1068.05

1068.05

9.38

0.00%

1068.05

1068.05

72.10

0.00%

P-n55-k7

588.56*

588.56

588.56

10.11

0.00%

588.76

588.56

64.54

0.03%

P-n60-k10

803.60*

803.73

803.60

10.06

0.01%

803.60

803.60

73.36

0.00%

P-n60-k15

1085.49*

1087.02

1085.49

10.11

0.14%

1085.49

1085.49

85.52

0.00%

BKScolumn:

Markedwith*whenoptimalproven

52


3.4.2 New proposed testbed

To conrm the ability of MA+GRASP to eciently solve the VRPSD, we propose a new testbed of

39 instances composed by the instances of Augerat test sets A and P, which are not considered in [7]

(a total of 22 instances), instances from the Cristodes, Mingozi and Toth CMT test set (4 instances)

and the Rochat and Taillard instances (13 instances). This new benchmark contains from 22 to 385

customers and from 3 to 46 vehicles necessary to satisfy customers demands. The average number

of customers per instance is almost 97. It shall be noticed that some of these have been already

addressed by Gauvin et al. [15]. The proposed testbed has on average nearly 2.4 more customers

than the benchmark of [7] which oers the opportunity for future comparisons of new methods dealing

with the VRPSD. Moreover, travel costs are calculated as the Euclidean distance between two nodes

rounded to the nearest integer as done in [15]

Some adjustments were made for the stopping condition of our hybrid method, for instances with

|V | < 100 the time limit is set to ten seconds, it is increased to 80 seconds for instances with |V | <= 200

and 160 seconds for instances with |V | > 200. Demands are assumed to be Poisson distributed, with

expected values equal to the deterministic demand values. Moreover, travel costs are calculated as

the Euclidean distance between two nodes rounded to the nearest integer.

Table 3.3 presents a summary of the new testbed instances. For each instance is reported: the

number of nodes (|V |), the minimum number of vehicles (Min veh) needed to satisfy the customers

demands, the vehicles capacity (Q), the lling coecient (FC) which stands for∑i∈V E[qi]

Min veh·Q , the best

known solution (BKS) provided either by [15] or among the several preliminary runs and the reported

results, the expected cost of the optimal deterministic solution (BDS)1 in the presence of uncertain

demands2, and the value of the stochastic solution (VSS) which stands for the percentage of improve-

ment among the BDS and the BKS. Furthermore, for the BKS is presented: the number of vehicles

in the BKS (Veh), the total expected cost (Total), the deterministic cost (Det), and the recourse cost

(Rec). The BDS also presents its total expected cost (Total) and its recourse cost (Rec).

As shown in table 3.3 most of the instances (38 out 39) show a positive VSS. Instance E-n23-k3

present a null VSS since the optimal deterministic solution is also optimal in the stochastic scenario.

Overall the VSS rounds the 5.37%, showing the importance of considering the stochasticity. VSS

seems to grow with the number of nodes within each instance, for instances with less than 100 nodes

its average value is 4.67%. Meanwhile, this value increases to 6.39% for instances with more than 100

nodes. As well, instances with more than 150 nodes achieve a VSS of 7.93%. Even if these results are

not conclusive in a direct relation between the VSS and the number of nodes, it seems that the larger

are the instances, the higher is the VSS. Additionally, the BKS cost (Total) is mainly composed by the

deterministic cost averaging 97.38%, while the recourse only achieves 2.66%. Indeed, the positive VSS

can be explained as follows. The BDS is optimal in the deterministic component of the cost. However,

the recourse value of BDS solutions becomes far more important than it is in the BKS. Indeed, the

recourse cost averages 10.88% of the Total BDS in contrast of the 2.55% of the Total BKS. Therefore,

the trade-o between the deterministic and recourse cost is better in the BKS, generating a positive

VSS.

Tables 3.4 and 3.5 illustrate the results on the new set of instances using the MA+GRASP and two

alternative versions, namely MA+RANDOM and NR-MA. MA+RANDOM only changes the way new

individuals are created during the restart procedure, using completely random customers permutations

while NR-MA is a simpler version of the algorithm without considering the restart procedure. For

each instance on this testbed are provided in table 3.4: the best known solution (BKS) with the same

considerations as in table 3.3, the best cost found out on the 10 runs (Best Cost) and the average cost

1Optimal deterministic solutions retrieved from http://vrp.atd-lab.inf.puc-rio.br or https://www.coin-or.org/SYMPHONY/branchandcut/VRP/data/index.htm for all instances except tai385 for which the deterministicBKS from the rst link is used.

2We use the same approach of Gauvin et al. [15], by evaluating the routes in the optimal deterministic solution tothe right and the reverse, retaining the one with the minimum cost

53


Table 3.3: Summary of new testbedBKS BDS

Instance |V | Min veh Q FC Veh Total Det Rec Total Rec VSSA-n61-k9 61 9 100 0.98 10 1144.23 1084 60.23 1215.37 181.37 5.85%A-n62-k8 62 8 100 0.92 9 1430.81 1375 55.81 1533.07 245.07 6.67%A-n63-k10 63 10 100 0.93 11 1459.49 1412 47.49 1581.17 267.17 7.70%A-n63-k9 63 9 100 0.97 10 1847.69 1734 113.69 1991.75 375.75 7.23%A-n64-k9 64 9 100 0.94 10 1569.85 1534 35.85 1699.68 298.68 7.64%A-n65-k9 65 9 100 0.97 10 1313.30 1266 47.30 1421.52 247.52 7.61%A-n69-k9 69 9 100 0.94 10 1259.35 1214 45.35 1339.55 180.55 5.99%A-n80-k10 80 10 100 0.94 11 1987.17 1918 69.17 2109.56 346.56 5.80%E-n23-k3 23 3 4500 0.75 3 569.72 569 0.72 569.72 0.72 0.00%E-n30-k3 30 3 4500 0.94 4 504.55 503 1.55 569.92 32.92 11.47%E-n76-k10 76 10 140 0.97 11 885.11 861 24.11 911.70 81.70 2.92%E-n76-k14 76 14 100 0.97 16 1118.90 1086 32.90 1187.18 166.18 5.75%E-n76-k7 76 7 220 0.89 7 698.95 692 6.95 723.96 41.96 3.46%E-n76-k8 76 8 180 0.95 8 771.23 744 27.23 791.88 56.88 2.61%E-n101-k14 101 14 112 0.93 15 1164.149 1116 48.15 1233.84 166.84 5.65%E-n101-k8 101 8 200 0.91 8 839.47 824 15.47 878.36 63.36 4.43%P-n55-k8 55 7 160 0.93 7 607.71 581 26.71 631.82 43.82 3.82%P-n65-k10 65 10 130 0.94 10 854.06 802 52.06 861.52 69.52 0.87%P-n70-k10 70 10 135 0.97 11 882.01 851 31.01 929.82 102.82 5.14%P-n76-k4 76 4 350 0.97 4 609.54 593 16.54 609.62 16.62 0.01%P-n76-k5 76 5 280 0.97 5 648.11 628 20.11 649.10 22.10 0.15%P-n101-k4 101 4 400 0.91 4 686.81 684 2.81 694.47 13.47 1.10%CMT12 101 10 200 0.91 10 982.80 827 155.81 984.31 164.31 0.15%CMT11 121 7 200 0.98 8 1201.15 1187 14.15 1226.62 189.62 2.08%CMT4 151 12 200 0.93 12 1072.19 1036 36.19 1130.31 114.31 5.02%CMT5 200 16 200 1.00 18 1378.85 1355 23.85 1486.96 209.96 7.27%Tai75a 76 10 1445 0.95 11 1653.85 1649 4.85 1718.39 102.39 3.76%Tai75b 76 9 1679 0.99 10 1353.09 1343 10.09 1438.62 99.62 5.95%Tai75c 76 9 1122 0.94 9 1349.87 1332 17.87 1396.98 110.98 3.37%Tai75d 76 9 1699 0.93 9 1392.86 1391 1.86 1444.39 87.39 3.57%Tai100a 101 11 1409 0.98 12 2107.71 2089 18.71 2388.31 350.31 11.75%Tai100b 101 11 1842 0.96 12 1985.33 1986 29.33 2118.93 181.93 6.31%Tai100c 101 11 2043 0.93 11 1421.66 1413 8.66 1526.73 126.73 6.88%Tai100d 101 11 1297 0.95 12 1602.53 1599 3.53 1748.94 174.94 8.37%Tai150a 151 15 1544 0.94 15 3211.55 3188 23.55 3574.95 530.95 10.10%Tai150b 151 14 1918 0.95 15 2792.39 2789 3.39 3113.52 397.52 10.31%Tai150c 151 14 2021 0.99 15 2406.13 2380 26.13 2546.29 201.29 5.50%Tai150d 151 14 1874 0.97 15 2718.39 2692 26.39 3044.91 406.91 10.64%Tai385 386 46 65 1.00 55 29364.03 28007 1357.03 31360.45 7008.45 6.22%

(Avg. Cost). Additionally, in table 3.5 are reported: the computational time (Time) in seconds, the

gap between the average cost and the BKS (Avg. Gap) and the gap between the best solution found

and the BKS (Best Gap). Moreover, ten out of the 39 instances have a proven optimal solution given

by Gauvin et al. [15]3, these values are marked with an asterisk in table 3.4.

Overall, the MA+GRASP shows a good performance presenting 33 out of 39 instances with an

average gap bellow one percent. Furthermore, the average gap is as low as 0.380%. MA-RANDOM

ranks second in this metric with 0.418% and the NR-MA achieves 0.434%. MA+GRASP presents an

average gap of 0.64% for instances with 100 customers or more, while instances with fewer customers

show an average gap of 0.20%. The dierence can be explained by the inherent combinatorial nature of

the problems at hand. MA+GRASP accomplishes to nd nine out of ten proven optimal solutions, still

instance P-n65-k10 is the only one which cannot be found by this method. Furthermore, MA+GRASP

as well as NR-MA reach 22 of the BKS while MA+RANDOM nds 21 BKS. In addition, the Best

gap metric averages only 0.16% with a maximum of 1.21% in instance CMT5 for the MA+GRASP.

MA+RANDOM and NR-MA perform similarly with an average best gap of 0.19%. Computational

times present a similar behavior, the MA+GRASP uses 40.92 seconds on average to solve each instance,

MA+RANDOM uses slightly less time with an average of 40.43 seconds, while the NR-MA reaches

the lowest value with 39.81 seconds. Although computational times cannot be directly compared to

those of table 4.1, obtained results suggest that our method has competitive times even for larger

instances.

3Instance CMT12 appears in Gauvin et al. [15] as M-n101-k10

54


Table3.4:

VRPSD

Proposed

testbed

results-Costs

BKS

MA+GRASP

MA+RANDOM

NR-M

A

Instance

BestCost

Avg.Cost

BestCost

AvgCost

BestCost

AvgCost

A-n61-k9

1144.23

1144.23

1144.72

1144.23

1146.00

1144.23

1146.37

A-n62-k8

1430.81*

1430.81

1431.36

1430.81

1431.56

1430.81

1431.05

A-n63-k10

1459.49*

1459.49

1459.49

1459.49

1459.57

1459.49

1459.78

A-n63-k9

1847.69

1848.69

1851.74

1852.12

1852.21

1851.23

1852.28

A-n64-k9

1569.85

1569.85

1570.07

1569.85

1570.07

1569.85

1570.07

A-n65-k9

1313.30*

1313.30

1313.30

1313.30

1313.30

1313.30

1313.30

A-n69-k9

1259.35*

1259.35

1259.81

1259.35

1259.95

1259.35

1259.37

A-n80-k10

1987.17

1987.17

1993.07

1987.17

1995.33

1988.25

1994.28

E-n23-k3

569.72*

569.72

569.72

569.72

569.72

569.72

569.72

E-n30-k3

504.55*

504.55

504.55

504.55

504.55

504.55

504.55

E-n76-k10

885.11*

885.11

886.77

885.11

889.03

885.11

887.71

E-n76-k14

1118.90

1118.90

1120.34

1118.90

1121.03

1118.90

1119.52

E-n76-k7

698.95

698.95

699.30

698.95

698.95

698.95

699.06

E-n76-k8

771.23

771.94

774.48

771.94

773.64

771.94

774.52

E-n101-k8

839.47

839.47

840.37

839.47

839.89

839.47

840.12

E-n101-k14

1164.15

1164.15

1167.61

1164.15

1170.72

1164.15

1170.38

P-n55-k8

607.71

607.71

607.71

607.71

607.71

607.71

607.71

P-n65-k10

854.06*

858.30

859.57

859.17

859.71

858.30

859.43

P-n70-k10

882.01*

882.01

883.20

882.01

883.83

882.01

884.37

P-n76-k4

609.54

611.59

616.56

614.04

616.44

614.04

616.48

P-n76-k5

648.11

652.16

652.71

652.16

653.80

652.16

653.02

P-n101-k4

686.81

686.81

687.78

687.67

688.57

686.81

687.91

CMT12

982.80*

982.80

982.80

982.80

982.80

982.80

982.80

CMT11

1201.15

1202.26

1208.66

1202.75

1207.03

1202.98

1208.14

55


Table3.4:

VRPSD

Proposed

testbed

results-Costs:Continued

BKS

MA+GRASP

MA+RANDOM

NR-M

A

Instance

BestCost

Avg.Cost

BestCost

AvgCost

BestCost

AvgCost

CMT4

1072.19

1076.84

1084.79

1082.70

1085.47

1082.55

1088.21

CMT5

1378.85

1395.52

1404.44

1391.27

1400.47

1388.90

1399.96

Tai75a

1653.85

1653.85

1654.30

1653.85

1654.30

1653.85

1653.86

Tai75b

1353.09

1353.09

1355.74

1353.09

1355.93

1353.09

1357.38

Tai75c

1349.87

1354.42

1354.55

1354.42

1354.55

1354.42

1357.47

Tai75d

1392.86

1392.86

1393.15

1392.86

1393.19

1393.20

1393.22

Tai100a

2107.71

2109.39

2112.72

2113.50

2121.39

2111.09

2120.53

Tai100b

1985.33

1988.67

1991.05

1988.67

1990.84

1988.67

1990.14

Tai100c

1421.66

1422.54

1422.84

1422.75

1423.38

1421.66

1423.05

Tai100d

1602.53

1602.53

1603.19

1602.53

1602.62

1602.53

1602.53

Tai150a

3211.55

3243.06

3257.09

3225.34

3257.29

3239.82

3255.23

Tai150b

2792.39

2794.43

2810.26

2792.51

2807.16

2792.40

2803.75

Tai150c

2406.13

2410.29

2416.04

2407.62

2423.69

2407.58

2423.64

Tai150d

2718.39

2721.86

2755.41

2731.58

2758.45

2738.25

2777.99

Tai385

29364.03

29597.42

29798.48

29632.52

29828.55

29582.63

29827.15

Average

2074.41

2081.2

2089.7

2082.

12

2091.

12081.2

2091.4

Max

29364.0

2959

7.0

29798.5

29632.

529828.

629582.6

29827.2

Min

504.6

504.6

504.6

504.

6504.

6504.6

504.6

BKScolumn:Marked

with*when

optimalproven.BestCostcolumns:bold

valueifBKSfound

56


Table3.5:

VRPSD

Proposed

testbed

results-Time,Gaps

MA+GRASP

MA+RANDOM

NR-M

A

Instance

Time(s)

Avg.Gap

BestGap

Time(s)

Avg.Gap

BestGap

Time(s)

Avg.Gap

BestGap

A-n61-k9

10.12

0.04%

0.00%

10.03

0.15%

0.00%

10.02

0.19%

0.00%

A-n62-k8

10.05

0.04%

0.00%

10.02

0.05%

0.00%

10.02

0.02%

0.00%

A-n63-k10

10.17

0.00%

0.00%

10.02

0.01%

0.00%

10.01

0.02%

0.00%

A-n63-k9

10.02

0.22%

0.05%

10.01

0.24%

0.24%

10.01

0.25%

0.19%

A-n64-k9

10.36

0.01%

0.00%

10.02

0.01%

0.00%

10.02

0.01%

0.00%

A-n65-k9

10.35

0.00%

0.00%

10.02

0.00%

0.00%

10.01

0.00%

0.00%

A-n69-k9

10.42

0.04%

0.00%

10.03

0.05%

0.00%

10.02

0.00%

0.00%

A-n80-k10

10.17

0.30%

0.00%

10.04

0.41%

0.00%

10.05

0.36%

0.05%

E-n23-k3

2.84

0.00%

0.00%

2.21

0.00%

0.00%

2.23

0.00%

0.00%

E-n30-k3

5.90

0.00%

0.00%

4.49

0.00%

0.00%

4.18

0.00%

0.00%

E-n76-k10

10.15

0.19%

0.00%

10.03

0.44%

0.00%

10.02

0.29%

0.00%

E-n76-k14

10.12

0.13%

0.00%

10.02

0.19%

0.00%

10.01

0.06%

0.00%

E-n76-k7

10.51

0.05%

0.00%

10.06

0.00%

0.00%

10.02

0.02%

0.00%

E-n76-k8

10.16

0.42%

0.09%

10.03

0.31%

0.09%

10.02

0.43%

0.09%

E-n101-k8

80.18

0.11%

0.00%

80.18

0.05%

0.00%

80.07

0.08%

0.00%

E-n101-k14

80.22

0.30%

0.00%

76.74

0.56%

0.00%

64.30

0.54%

0.00%

P-n55-k8

10.14

0.00%

0.00%

10.01

0.00%

0.00%

10.01

0.00%

0.00%

P-n65-k10

10.20

0.64%

0.50%

10.01

0.66%

0.60%

10.01

0.63%

0.50%

P-n70-k10

10.29

0.13%

0.00%

10.02

0.21%

0.00%

10.01

0.27%

0.00%

P-n76-k4

10.22

1.15%

0.34%

10.04

1.13%

0.74%

10.05

1.14%

0.74%

P-n76-k5

10.20

0.71%

0.62%

10.07

0.88%

0.62%

10.03

0.76%

0.62%

P-n101-k4

80.59

0.14%

0.00%

80.16

0.26%

0.12%

80.19

0.16%

0.00%

CMT12

80.69

0.00%

0.00%

79.38

0.00%

0.00%

71.74

0.00%

0.00%

CMT11

80.16

0.63%

0.09%

80.22

0.49%

0.13%

80.25

0.58%

0.15%

57


Table3.5:

VRPSD

Proposed

testbed

resultsTime,Gaps:Continued

MA+GRASP

MA+RANDOM

NR-M

A

Instance

Time(s)

Avg.Gap

BestGap

Time(s)

Avg.Gap

BestGap

Time(s)

Avg.Gap

BestGap

CMT4

80.65

1.18%

0.43%

80.39

1.24%

0.98%

80.24

1.49%

0.97%

CMT5

80.53

1.86%

1.21%

80.54

1.57%

0.90%

80.46

1.53%

0.73%

Tai75a

10.24

0.03%

0.00%

10.02

0.03%

0.00%

10.03

0.00%

0.00%

Tai75b

10.04

0.20%

0.00%

10.02

0.21%

0.00%

10.25

0.32%

0.00%

Tai75c

10.18

0.35%

0.34%

10.03

0.35%

0.34%

10.02

0.56%

0.34%

Tai75d

10.19

0.02%

0.00%

10.03

0.02%

0.00%

10.04

0.03%

0.02%

Tai100a

80.24

0.24%

0.08%

79.48

0.65%

0.27%

78.00

0.61%

0.16%

Tai100b

80.92

0.29%

0.17%

77.57

0.28%

0.17%

75.76

0.24%

0.17%

Tai100c

80.41

0.08%

0.06%

78.71

0.12%

0.08%

79.99

0.10%

0.00%

Tai100d

80.62

0.04%

0.00%

80.08

0.01%

0.00%

80.06

0.00%

0.00%

Tai150a

80.19

1.42%

0.98%

80.27

1.42%

0.43%

80.19

1.36%

0.88%

Tai150b

80.93

0.64%

0.07%

80.16

0.53%

0.00%

80.19

0.41%

0.00%

Tai150c

81.15

0.41%

0.17%

80.16

0.73%

0.06%

80.25

0.73%

0.06%

Tai150d

81.09

1.36%

0.13%

80.31

1.47%

0.48%

80.20

2.19%

0.73%

Tai385

164.64

1.48%

0.79%

165.52

1.58%

0.91%

163.91

1.58%

0.74%

Average

40.93

0.38

0%0.

16%

40.44

0.418%

0.18%

39.8

20.4

34%

0.18%

Max

164.64

1.86

%1.

21%

16552

1.57%

0.98%

163.9

12.1

9%

0.97%

Min

2.84

0.00

%0.

00%

2.21

0.00%

0.00%

2.2

30.0

0%

0.00%

58


0,00%

0,20%

0,40%

0,60%

0,80%

1,00%

1,20%

1,40%

1,60%

1,80%

2,00%

0,88 0,90 0,92 0,94 0,96 0,98 1,00

Avg

. Gap

-M

A+

GR

ASP

Filling capacity

0,00%

0,20%

0,40%

0,60%

0,80%

1,00%

1,20%

1,40%

1,60%

1,80%

2,00%

0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00

Avg

. Gap

-M

A+G

RA

SP

Filling capacity* - BKS number of vehicles

Figure 3.3: Avg. gap of MA+GRASP against the lling coecient

General performance discussion

Authors like Laporte et al. [24] or Jabali et al. [22] had reported the diculty of their exact ap-

proaches to solve the VRPSD when the number of vehicles increases or when the lling coecient

(FC) approaches one. Although in our approach the eet is considered unlimited instead of xed,

we test if such parameters had an impact on our solution method. We use as proxy the average gap

retrieved by the MA+GRASP to compare it against both the FC and the number of vehicles. Figure

3.3 presents the relation between the FC and the average gap. In the left gure the FC reported in

table 3.3 is used while the right part represents a FC calculated using the actual number of vehicles

used in the BKS. Instance E-n23-k3 was discarded for the graphs as far as this point is a very extreme

point4. The graph with the original FC shows a slightly positive trend between the variables while

the modied FC presents no relation at all.

The same approach to FC is used to compare the avg gap of MA+GRASP with the number

of vehicles used in the BKS. Figure 3.4 presents the graph of both variables. The graph uses the

information of all instances but Tai385 which also is marked as an extreme point when compared to

the rest of the testbed5. A clearer pattern of a positive relation between the variables is devised for

this relation. Nevertheless, a further inspection on gure 3.5 shows that the number of vehicles is

highly correlated with the number of nodes per instance. Moreover, this same gure shows a very

clear pattern in the relation of the average gap against the number of nodes in the instance. The

results are not surprising as far as this parameter has an important inuence on the speed of the local

searches of the MA+GRASP and is of importance in the framework of the MA. Consequently, new

methods relying on local search procedures should aim to minimize the complexity of the task.

Addressing the eect of GRASP restart

In order to test the impact of the GRASP and the restart procedure included in the MA, the variance

analysis method of Friedman is used to compare the MA+GRASP against the MA+RANDOM. The

results presented in table 3.4 are intended to perform two comparisons in terms of the best solution

found, and the average cost. The Friedman tests works as follows. For each methods comparison,

results are ranked within each instance giving one to the best value to two to the worst one, average

values are used in case of ties (table 3.7). Let R (Xij) be the rank of the heuristic j in instance i,

b the number of instances, k the number of tested heuristics and Rj =∑bi=1R (Xij)∀j). Moreover,

A =∑bi=1

∑kj=1 (R (Xij))

2, C = bk(k+1)2

4 , T 1 =(k−1)

∑kj=1(Rj−

b(k+1)2 )2

A−C , T 2 = (b−1)T 1

b(k+1)−T 1. In addition

t1−(α2 ) stands for the 1 − α2 quantile of the Student t distribution with (b− 1) (k − 1) degrees of

freedom and F1−(α2 ) the 1 − α2 quantile of the F distribution with (k − 1) numerator degrees of

freedom and (b − 1)(k − 1) denominator degrees of freedom. The test to set if the heuristics present

4E-n23-k3 instance has a FC which is nearly 5 standard deviations from the mean FC5Tai385 has a number of vehicles in the BKS which is over 5 standard deviations from the mean

59


0,00%

0,20%

0,40%

0,60%

0,80%

1,00%

1,20%

1,40%

1,60%

1,80%

2,00%

0 5 10 15 20

Avg

. Gap

-M

A+G

RA

SP

Number of vehicles in BKS

Figure 3.4: Avg. gap of MA+GRASP against the number of vehicles in the BKS

0,00%

0,20%

0,40%

0,60%

0,80%

1,00%

1,20%

1,40%

1,60%

1,80%

2,00%

0 50 100 150 200

Avg

. Gap

-M

A+G

RA

SP

Number of nodes

0

2

4

6

8

10

12

14

16

18

20

0 50 100 150 200

Nu

mb

er o

f ve

hic

les

in B

KS

Number of nodes

Figure 3.5: Avg. gap of MA+GRASP and number of vehicles against the number nodes per instance

a dierence within a metric is dened as:

T 2 > F 1−(α2 ) (3.4)

By setting α = 1% the right side of the equation 4.4 becomes 7.35 for every comparison, the left side

is calculated with the values presented in table 3.7 giving 8.05, and 4.34 for the average cost, and the

best cost respectively. That means, that there exist statistically dierences between the methods for

the average cost. If α = 5% then the right side of the equation 4.4 becomes 4.09 and thus at 5% level

of signicance there exist statistically dierences in the best cost metric. Provided that the methods

have dierent performances, the test to set if two heuristics are statistically dierent within a metric

can be expressed as follows:

|Rl −Rm| > t1−(α2 )

√√√√2(bA−

∑kj=1Rj

2)

(b− 1) (k − 1)(3.5)

Let R1, R2 be the Rj metrics for the MA+GRASP, and MA+RANDOM respectively. Table 4.5

summarizes the necessary values for performing the tow two-pair comparisons. Moreover, the right

side of equation 4.5 has values of 14.33 and 7.77 provided that α = 1% for the average cost and

α = 5% for the best cost metric. As far as equation 4.5 holds for every |Rl −Rm| we can conclude the

statistical dierence the heuristics within each metric and therefore, MA+GRASP can be determined

as the best alternative.

Indeed, results show that the MA+GRASP presents the best performance and it is statistically

signicant on the average cost at a condence level of α = 1%, and for the best solution found at

60


Table 3.6: Two-way test values for Friedman methodMetric

AverageCost

Best Cost

|R1 −R2| 15.0 8.0

Table 3.7: Friedman test ranksMA+GRASP MA+RANDOM

InstanceAvg. Cost

rank

Best Cost

rank

Avg. Cost

rank

Best Cost

rank

A-n61-k9 1 1 2 2A-n62-k8 1 1.5 2 1.5A-n63-k10 1 1.5 2 1.5A-n63-k9 1 1 2 2A-n64-k9 1.5 1.5 1.5 1.5A-n65-k9 1 1.5 2 1.5A-n69-k9 1 1.5 2 1.5A-n80-k10 1 1.5 2 1.5CMT12 1.5 1.5 1.5 1.5CMT11 2 1 1 2CMT4 1 1 2 2CMT5 2 2 1 1E-n101-k14 1 1.5 2 1.5E-n101-k8 2 1.5 1 1.5E-n23-k3 1.5 1.5 1.5 1.5E-n30-k3 1.5 1.5 1.5 1.5E-n76-k10 1 1.5 2 1.5E-n76-k14 1 1.5 2 1.5E-n76-k7 2 1.5 1 1.5E-n76-k8 2 1.5 1 1.5P-n101-k4 1 1 2 2P-n55-k8 1.5 1.5 1.5 1.5P-n65-k10 1 1 2 2P-n70-k10 1 1.5 2 1.5P-n76-k4 2 1 1 2P-n76-k5 1 1.5 2 1.5Tai100a 1 1 2 2Tai100b 2 1.5 1 1.5Tai100c 1 1 2 2Tai100d 2 1.5 1 1.5Tai150a 1 2 2 1Tai150b 2 2 1 1Tai150c 1 2 2 1Tai150d 1 1 2 2Tai75a 1 1 2 2Tai75b 1 1.5 2 1.5Tai75c 1.5 1.5 1.5 1.5Tai75d 1 1.5 2 1.5Tai385 1 1 2 2Rj 51.0 54.5 66.0 62.5

61


0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 10 20 30 40 50 60 70 80

Cu

mu

lati

ve p

rob

abili

ty

Time (seconds)

MTTT Plot - Target Value 5% gap to BKS

MA+GRASP NR - MA MA+RANDOM

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Cu

mu

lati

ve p

rob

abili

ty

Time (seconds)



Figure 3.6: MTTT Plots for the dierent MA methods - 5% and 1%

α = 5%. Overall MA+GRASP presents the best quality results, and the Friedman tests conrms

the enhancement of the method with the restart procedure driven by a GRASP. Besides, a multiple

instances and targets time-to-target plot (mttt-plot) was used to compare the three solution versions.

The mttt-plot method is an extension of the time-to-target plots (ttt-plots) introduced in [14]. Ac-

cording to Reyes and Ribeiro [34] the plots can be used to compare the running times of stochastic

algorithms or dierent strategies for solving a given problem. The approach is based on the construc-

tion by simulation presented in [34]. The resulting graphs represents the probability of nding the

proposed target values (one per instance) for the set of instances in a specic amount of time.

To extract the necessary information, we run again the three methods for each of the new testbed

instances. The stopping criteria is set to a maximum time of half an hour or when a BKS is achieved.

During the execution time, for each instance run and method, the best solution is always surveyed, so

when a new best solution is found the necessary time to nd it is recorded. Ten runs are performed

per instance and per method. Using this information we estimate the probability distribution of the

time required to nd dierent target values. Indeed, we vary the target value for each instance as a

function of a gap to the BKS reported in table 3.36. This allows to see how the methods perform over

dierent targets giving a more comprehensive story of the performance of the algorithms.

As far as we imposed a limit time, it may happen that a method cannot meet the required target.

In such case we let the maximum time as record. The rationale behind this is that even if we do

not know the actual time required to nd the target value, it stills contains valuable information.

Indeed, more stable methods should be able to nd the BKS (or very near BKS) targets more often.

Therefore, letting the maximum time as record when the target value is not found accomplishes the

aim of characterizing the solution method times. Moreover, this allows to limit the computational

times required to perform a mttt-plot analysis, as far as some methods can require huge amount of

time to nd certain targets for some instances.

Figure 3.6 shows the mttt-plots for target values not farther than 5% and 1% of the BKS. The

5% gure concludes that the NR-MA is better than MA+RANDOM and MA+GRASP, and with less

time it has higher probabilities of achieving a 5% gap to BKS for the whole benchmark of instances.

Nevertheless, the 1% gure displays a completely dierent behavior. In this last, the MA+RANDOM

and MA+GRASP present a very similar behavior which is far better than NR-MA. For example, the

probability of nding a target value within 1% of the BKS (for the whole set of instances) within 2500

seconds is one for MA+GRASP and MA+RANDOM, for NR-MA is around 70%.

Going further to determine the best method, gure 3.7 presents the mttt-plots for target val-

ues of 0.5% and 0% of the BKS. In both targets it is shown that MA+GRASP outperforms the

MA+RANDOM and the NR-MA showing a higher probability of nding the matches for a xed

amount of time. It is interesting that as the target approaches the BKS the performance of MA+RANDOM

6Except for instances CMT4, Tai 150a, 150c, 150d, and 385 for which the BKS reported was found during this longrun, a slightly bigger BKS was used during this tests

62


0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 2000 4000 6000 8000 10000 12000 14000 16000

Cu

mu

lati

ve p

rob

abili

ty

Time (seconds)

MTTT Plot - Target Value 0.5% gap to BKS


0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 5000 10000 15000 20000 25000 30000 35000 40000

Cu

mu

lati

ve p

rob

abili

ty

Time (seconds)



Figure 3.7: MTTT Plots for the dierent MA methods - 0.5% and 0%

and NR-MA become very similar. This shows that the random restart might have a very little impact

on the solution approach in long runs. However, the restart based on the GRASP does show an

important impact, giving the edge to MA+GRASP as the best option among the tested ones to solve

the VRPSD.

3.5 Conclusions

In this chapter the Vehicle Routing Problem with Stochastic Demands (VRPSD) was studied. In order

to solve this problem a MA+GRASP method is proposed. The obtained results on a classical testbed

from Christiansen and Lysgaard [7] show that our method outperforms state-of-the-art algorithms in

terms of quality and eciency. Moreover a new testbed with nearly 2.4 more customers on average per

instance is derived. The results on this new benchmark conrm the pertinence of the MA+GRASP

which can eciently solve instances with as much as 385 customers. The new instances, which are

closer to real life size problems, and the presented results can be used for future comparison. Research

currently underway includes the evaluation of new recourse actions, the extension of the problem to

consider other stochastic parameters in order to solve problems closer to reality and the introduction of

correlation among the demands. Finally, undergoing work also search the means to combine heuristic

methods with exact algorithms to tackle large instances of dierent SVRPs.

Contributions

Preliminary results of this chapter were presented at CIE45 conference:

Gutierrez, A., Dieulle, L., Labadie, N., Velasco, N. (2015)

A Memetic Algorithm for the Vehicle Routing Problem with Stochastic Demands

In Proceedings of the 45th International Conference on Computers & Industrial Engineering CIE45

Metz, France, 2830 October, 2015.

An article version of this chapter has been published in the Computers & Operations Research

journal. Please cite it as follows:

Gutierrez, A., Dieulle, L., Labadie, N., Velasco, N. (2018). A Hybrid metaheuristic algorithm for

the vehicle routing problem with stochastic demands. Computers & Operations Research, 99, 135-147.

https://doi.org/10.1016/j.cor.2018.06.012.

63

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66

Chapter 4

Vehicle Routing Problem with

stochastic travel and service times

4.1 Introduction

The Vehicle Routing Problem (VRP) is one of the most studied combinatorial problems; its importance

is inherent in the number of real applications where it arises, as well as its signicant theoretical

contributions to combinatorial optimization eld. First proposed by Dantzig and Ramser [10], many

variants of the problem has been considered, e.g. Vehicle Routing Problem with Time Windows

(VRPTW), Distance Constrained Vehicle Routing Problem (DVRP), Orienteering Problem, among

others. Each of the proposed variants aims to address the problem in a more realistic way to nd more

reliable solutions in real world applications. Nevertheless, one of the most important drawbacks in

classical models is that available information can be limited because many of the input parameters are

uncertain, such as travel times, service times, customers demands, or customer presence. Neglecting

the lack of complete information can conduce to, for example, suboptimal solutions, so implying

higher operational costs or customer dissatisfaction. Stochastic Vehicle Routing Problems (SVRP)

arise when parameters are modeled as random variables, making models closer to reality but harder

to solve than their deterministic counterpart.

SVRP are usually solved by means of stochastic programming. Two approaches are often used

to model and solve a stochastic optimization problem: Chance Constrained Programming (CCP)

and Stochastic Programming with Recourse (SPR). CCP aims to solve the problem by bounding the

probability of constraints violations to a threshold. SPR uses recourse which are actions to recover

the feasibility of the solution when failures occur. The expected costs related to these actions are

taken into account in the objective function. Both approaches (CCP and SPR) rely on a two-stage

approach: in the rst stage a priori solution is created and then at the second stage the parameters

are revealed. However, since CCP does not consider the cost associated with failures, the quality of

the solutions might be inferior to those provided by SPR models.

This chapter deals with maintenance scheduling and routing problems where groups of technicians

must be assigned to visit a set of customers to execute repairing tasks within given time windows.

The goal is to build a minimum cost set of routes subject to the following constraints: every route

starts and ends at the depot; each customer is visited and serviced once by only one route within its

time window, and the sum of customer demands on each route does not exceed the vehicle capacity.

With regard to time window constraints, there are two cases that are encountered in the literature:

hard time windows exist when the service cannot start outside the time interval; early arrivals are

permitted but the vehicle must wait until the opening of the time window. In contrast, soft time

windows services are allowed outside the time windows usually with a penalty cost. In this study, the

67

4.1. INTRODUCTION

vehicle routing problem with hard time windows and stochastic travel and service times (SVRPTW)

is considered. In fact, travel times modeled as random variables consider the uncertainty in time due

to trac, weather, accidents, driving skills, etc. Furthermore, the stochastic service time accounts for

the complexity of maintenance tasks, which may only be known when technicians arrive at customer

locations.

A recent survey on SVRPs is presented in Gendreau et al. [16] reviewing dierent variants,

e.g. stochastic demands, customers, travel and service times. The version with stochastic demands

(VRPSD) was rst proposed by Tillman [45] with later works on exact methods in [28, 8, 15] and

heuristic methods in [42, 19, 33]. A VRPSD where the presence of customers is also a random variable

is presented by Gendreau et al. [17] and solved with a Tabu Search.

Earlier related works concerning stochastic travel and/or times problems include those of Laporte

et al. [27] for the Vehicle Routing Problem with Stochastic Travel and Service times, in which the

authors propose three models, including a CCP and an SPR, solving instances with up to 20 nodes

where travel times are restricted to ve discrete states. Lambert et al. [26] propose an adaptation of

the Clarke and Wright [9] saving heuristic to solve the VRP with stochastic travel times for money

collection. Later, two models for the VRP with stochastic travel and service times are presented in

Kenyon and Morton [24]. The rst minimizes the expected completion time, while the second model

maximizes the probability of completion within a pre-specied deadline. To solve small instances, the

authors used a branch-and-cut algorithm; for larger ones, the branch-and-cut scheme was embedded

in a Monte Carlo sampling approach.

A Genetic Algorithm reporting signicant improvements in terms of costs and delay penalties is

referred to in [2] for the SVRP with time windows and stochastic travel times. More recently, the

SVRP with stochastic travel and service times is modeled by Li et al. [31] as a CCP and an SPR.

The problem is solved using a Tabu Search, and Monte Carlo simulation is used to check stochastic

constraint feasibility and estimate the expected recourse value. The authors test this approach on

their own instances with up to 100 clients and soft time windows, concluding that CCP models are

harder to solve than SPR. Zhang et al. [49] also address the SVRP with stochastic travel and service

times with soft time windows. The authors propose a CCP model to guarantee a service level to

customers services, arrival at the depot and on the total duration of the routes. To estimate the

vehicles arrival times, a discrete approximation method is embedded into a Tabu search heuristic to

solve some Solomon [41] instances with up to 20 customers. A Tabu Search is also proposed in Ta³ et

al. [43] for the SVRP with stochastic travel times and soft time windows. The objective is to minimize

a weighted cost composed of two parts: the rst relates to the service (early and late arrivals) and

the second to transportation (distance, xed vehicle costs and overtime). Ta³ et al. [44] analyze the

same problem but deal with stochastic time-dependent travel times. The authors deduced a way of

calculating exactly the mean and variance of arrival times provided that service times for customers

equal zero. Conversely, when service times are non-zero, an approximation is used. Both, exact and

approximation schemes are derived if travel times are Gamma distributed.

A Satiscing Measure Approach (SMA) to mitigate the dissatisfaction experienced by customers

is proposed in Nguyen et al. [39]. The SMA is realized for both the SVRP with stochastic demands

and time windows and the SVRP with stochastic travel times and time windows. The latter model

considers the early time window (ready time) as a strict requirement and the late time window (due

time) as a soft requirement. The results for Solomon [41] instances show small computational times.

Errico et al. [13] eectively solve the SVRP with hard time windows and stochastic service times

with a branch-cut-and-price algorithm for instances with up to 50 customers. The model proposed

by the authors is an SPR, nevertheless two conditions (constraints) are imposed: a service level on

being operationally feasible for each route is required and a maximum of one recourse per route is

allowed. A variant of the problem that considers multiple depots and client priorities is addressed by

Binart et al. [4]. The problem incorporates two types of customers: optional and mandatory. The

68

CHAPTER 4. VEHICLE ROUTING PROBLEM WITH STOCHASTIC TRAVEL AND SERVICETIMES

former customers have an associated hard time window that must be respected. The problem is solved

with a two-stage method and is proved eective in instances containing up to 50 customers and three

vehicles.

Jula et al. [22] develop approximations for the mean and variance of the arrival times for the

Traveling Salesman Problem with time windows (TSPTW) which is solved via dynamic program-

ming. The arrival times mean and variance are estimated using a rst-order Taylors series expansion.

The authors guarantee a service level requirement on clients time windows using the Chebyshev and

Cherno bounds. In the same vein, Ehmke et al. [11] estimate both the arrival and service times for

the VRP with stochastic travel times and time windows using a normal approximation. The authors

embed their approximation within a Tabu Search method to solve a CCP formulation on Solomon

[41] instances where late services are allowed. The work of Gomez et al. [18] tackles the estimation of

arrival times using phase-type (PH) distribution for the Distance-Constrained Capacitated VRP with

Stochastic Travel and Service Times. Although no time windows are considered, the authors conclude

that PH distributions can be used to handle them. [34] focus on the VRP with time windows and

stochastic travel and service times modeled as a CCP and solved using an Iterative Local Search pro-

cedure mixed with a discrete approximation on arrival times. Travel and service times are considered

as normally distributed. Results were presented for seven Solomon [41] instances with 100 customers.

The purpose of this study is: (1) to tackle the SVRP with hard time windows, instead of soft time

windows as is frequently executed in the literature; (2) to propose a recursive approach to estimate

mean and variance of arrival times, including the eect of late arrivals at previous customers; (3) given

the fact that arrival times probability distribution are in general unknown, and in most cases expensive

to calculate, a log-normal approximation is selected among other probability distributions; and (4) to

propose a Multi-Population Memetic Algorithm (MPMA) exploiting dierent characteristics in each

population to enhance its overall performance. Moreover, the contributions found in this chapter

are: it is shown that despite the eects of hard time windows and the sum of random variables with

dierent probability distributions, the proposed approximation of the arrival times is a valid and fast

approach to guide a solution algorithm. Moreover, the experiments made with the MPMA using

dierent populations reveal that tackling the problem with dierent assumptions on the parameters,

while the populations share their solutions, greatly improves the best and average solutions found.

Furthermore, the use of a combined model (CCP + SPR) allows it to easily adapt the problem to

comprise dierent objectives, such as costs and customers satisfaction.

This chapter is organized as follows. The SVRP with hard time windows and stochastic service

and travel times is introduced in section 4.2. The approach to estimate arrival times is assessed in

section 4.3. In section 4.4, the Multi-Population Memetic Algorithm developed to solve the problem

is described. Numerical results are reported in section 4.5, and lastly a conclusion is put forward in

section 4.6.

4.2 Problem Denition

The Vehicle Routing Problem with hard Time Windows and Stochastic Travel and Service Times

(SVRPTW) is a generalization of the VRPTW where travel and service times are modeled by random

variables. The VRPTW is dened by a vertex set V = 0, 1, . . . , i, . . . , n and an edge set E =

[i, j]∀i, j ∈ V | i < j that composes of a complete graph G = (V, E). Each vertex i ∈ V is

characterized by a coordinate (xi, yi), and a time window [ei, li] in which the service must start; eibeing the opening time and li the closure time. Furthermore, Vc = V\ 0 is the customers subset,each of which has a non-negative demand qi and a specied service time si. Vertex 0 represents a

depot where a eet of homogenous vehicles with a limited capacity Q is located. In addition, to each

edge [i, j] ∈ E there is an associated non-negative cost cij and a travel time tij .

Let M be the xed cost associated to each used vehicle, and the vertex n + 1 a dummy copy

69

4.2. PROBLEM DEFINITION

of the depot 0 with the same location and time window, moreover allow V ′ = V ∪ n+ 1 and

E ′ = E ∪ [i, n+ 1] ∀i ∈ Vc with Vc = V\ 0. Service times and travel times are random variables

denoted by si(∀i ∈ Vc) and tij (∀ [i, j] ∈ E ′) respectively.To model the SVRPTW, a combined CCP and SPR formulation is used to deal with dierent

settings. On the one hand, the CCP part lets managerial decisions to be considered when solving the

problem (by controlling the condence levels). Conversely, the SPR adds the recourse cost component,

hence ensuring a more reliable measure of the quality of the solution. Our model integrates stochastic

constraints (CCP) to guarantee the condence levels α, β and γ for hard time windows constraints

at clients, depot time window, and success for the whole set of routes respectively. For simplicity, it

is assumed that every client in Vc has the same required condence level α. Let r be a route dened

as an ordered sequence of clients r = r0 = 0, r1, . . . , rj , . . . , rk, rk+1 = n+ 1 where rj represents thejth visited customer. Furthermore, AT rj stands for the arrival time to client rj ∈ Vc on the route r

and P (A) the probability of an event A. AT rj is a random variable because it depends on travel and

service times which are dened as random variables. Aiming to guarantee the condence levels to the

above-mentioned, the following constraints are imposed on every route r:

P(AT rj ≤ lrj

)≥ α ∀rj ∈ Vc (4.1)

P(AT rk+1

≤ l0)≥ β (4.2)

Constraint (4.1) enforces a service level for every customer, stating that the customer service must

start within its time window a with probability of at least α. Also by (4.2) it is guaranteed that

vehicles will return within the depot time window with a probability of at least β. Even when a route

r meets equations (4.1) and (4.2), sometimes it can miss the customer or depot time window closure;

these events are called failures. Let Ur be the probability of having no failures in a route r. A solution

for the SVRPTW will normally be composed of more than one route, therefore let s be a solution for

the SVRPTW composed by a set K of routes; the following constraint is imposed for every solution:∏r∈s

Ur ≥ γ (4.3)

Equation (4.3) guarantees a service level on the whole route plan rather than only on customers

[12], i.e. none of the routes in the solution will miss time a window constraint with customers or at the

depot, with a probability γ. This formulation is valid if, and only if, the set of routes in the solution

are independent. This assumption holds if travel times are independent and also if every client is

visited by only one vehicle, i.e. the probability of no failures is not related from one route to others.

The recourse action is considered as follows: if a vehicle arrives at a customer i ∈ Vc later than the

closure of its time window, the vehicle will continue its route (without performing the service) towards

the next client. The recourse is founded in the idea of rescheduling the visit of customer i later. The

associated recourse cost can be seen as a penalization for missing the customer time window, this

penalty is represented by the cost of a vehicle visiting exclusively customer i. The recourse albeit

simple, ts the maintenance scheduling problem. As far as service and travel times can only be known

until they are completed, technicians cannot anticipate failures. Therefore, a failure is only known at

the customer location where it takes place. Furthermore, this simple recourse lets us introduce the

cost of missing a service while keeping the computations of probabilities tractable.

The proposed recourse has already been addressed in the literature: Nguyen et al. [39] described

this recourse action, nevertheless, it is not used in their approach; Wang and Regan [48] also proposed

this recourse but the authors did not associate a cost to it. In this work we dene the expected cost

of a route r as:

cr = M +

k∑j=0

crjrj+1 +

k∑j=1

P(AT rj > lrj

)·(2 · c0,rj +M

)(4.4)

70


The rst part of (4.4) stands for the xed vehicle costs M ; the second part is the cost of edges

traversed by the route and the third represents the expected recourse cost. It is important to note

that the recourse cost does not depend on the length of the delay for the late arrival as it only uses

the probability of missing the time window.

In here, the eet size is considered unlimited and homogenous. The probability density functions

ψi (∀i ∈ Vc) associated with service times are known, and service and travel times are assumed to be

mutually independent. The probability density function φij∀ [i, j] ∈ E of every travel time is known.

We set the capacity of the vehicles to innity (Q =∞) considering that it does not limit the technicians

capacity to provide their services. We assume that service times are identically Gamma distributed

and travel times are identically distributed with a Log-normal distribution. Kaparias et al. [23] and

Lecluyse et al. [29] have already used log-normal distributions, recognizing the importance of skewed

distributions to model travel times. Furthermore, Gamma distribution is selected for the service times

as long as it respects the principle of increasing repair rate.

4.3 Estimation of arrival times

4.3.1 Arrival and starting service times denition

Since the vehicles arrival times to customers and depot depends on travel and service times which are

by denition random variables, thus arrival times become random variables too. The same condition

applies to the starting time when the service is performed at a customer, as long as it depends on the

arrival times and time windows. Let ST rj denote the random starting time of the service at client

rj ∈ Vc in a route r. Because of the problems dened, the service is performed only if AT rj ≤ lrj .

Otherwise, if a failure takes place (AT rj > lrj ) the vehicle continues towards the next node in its route,

i.e. the service is not performed. Also, because time windows are hard, the service for a customer can

only start at or after the opening of its time window. Let 1Z be an indicator function which takes

value one if a condition Z holds or zero otherwise, the time when a service starts at a customer can

thus be set as:

ST rj = erj · 1AT rj<erj + AT rj · 1erj≤AT rj≤lrj (4.5)

Aiming to dene arrival times and considering the used recourse, it is mandatory to check if a failure

took place at the last customer visited or not. Therefore, arrival times can be dened by equation (4.6).

It should be noted that recursively using equation (4.6) implies that failures aect the distribution of

the arrival and initial service times.

AT rj =

ST rj−1 + srj−1 + trj−1rj , AT rj−1 ≤ lrj−1

AT rj−1 + trj−1rj , AT rj−1 > lrj−1

(4.6)

Figure 4.1 presents a graphical example of both arrival times and initial service times. The example

presents a technician arriving at customer i (a), performing its service if he arrives before li (b), or

continuing its route, and then going to node j (e). Furthermore, gure 4.1 part (a) presents vertical

lines representing ei and li at time 500 and 540. The parts (c) and (d) of the gure show the density

function of service time for customer i and the travel time from i to j, respectively. The arrival

time at node j presented in part (e) also shows the eects of time windows. Since time windows

are hard, the use of convolution properties are precluded, making it harder to properly model the

arrival times. Thus, AT rj is usually approximated by a random variable AT rj∀j ∈ V with a known

distribution allowing tractable computations. The route cost equation can then be rewritten in terms

of the approximated arrival times.

cr = M +

k∑j=0

crjrj+1 +

k∑j=1

P(AT rj > lrj

)·(2 · c0,rj +M

)(4.7)

71

4.3. ESTIMATION OF ARRIVAL TIMES

420 440 460 480 500 520 540 560

Arrival Time at i (a)

Time

Den

sity

500 510 520 530 540

Start Service Time at i (b)

Time

Den

sity

20 40 60 80 100

Maintenance time (c)

Time

Den

sity

0 20 40 60 80

Travel Time i to j (d)

Time

Den

sity

520 540 560 580 600 620 640 660

Arrival Time at j (e)

Time

Den

sity

Figure 4.1: Arrival and starting service times example.

4.3.2 Mean and Variance estimation

To estimate the mean and variance of the arrival times an approach similar to the one used by Ehmke

et al. [11] is proposed, nevertheless, we do take into account the impact of possible service failures on

AT rj as well as the fact that service times are random.

Let µX = E[X]be the expected value of a variable X and σ2

X= V ar

[X]its variance. By deni-

tion the standard deviation is set to σX =√σ2X. For simplicity let P 1 = P

(erj ≤ AT rj ≤ lrj | AT rj ≤ lrj

)and P 2 = 1 − P1 = P

(AT rj < erj | AT rj ≤ lrj

). Applying the laws of total expectation and total

variance to the equation (4.5), the mean and variance for the initial service times at customer j are

presented in equations (4.8) and (4.9).

µST rj |AT rj≤lrj

= µAT rj |erj≤AT rj≤lrj

· P 1 + erj · P 2 (4.8)

σ2ST rj |AT rj≤lrj

= σ2AT rj |erj≤AT rj≤lrj

· P 1 + e2rj · P 1 · P 2

+µ2AT rj |erj≤AT rj≤lrj

· P 1 · P 2 − 2 · erj · µAT rj |erj≤AT rj≤lrj · P 1 · P 2 (4.9)

Using the results from equations (4.8) and (4.9), we now apply the laws of total expectation and total

variance to equation (4.6). Again for simplicity let P 3 = P(AT rj−1

≤ lrj−1

)and P 4 = 1 − P 3 =

P(AT rj−1 > lrj−1

). The mean and variance for the arrival times at a node j are dened by equations

(4.10) and (4.11).

µAT rj

=(µST rj−1

|AT rj−1≤lrj−1

+ µsrj−1+ µtrj−1rj

)· P 3

+(µAT rj−1

|AT rj−1>lrj−1

+ µtrj−1rj

)· P 4 (4.10)

72


σ2AT rj

=

(σ2ST rj−1


+ σ2srj−1

+ σ2trj−1rj

)· P 3

+

(σ2AT rj−1

|AT rj−1>lrj−1

+ σ2trj−1rj

)· P 4

+ (P 3 · P 4) ·(µST rj−1


+ µsrj−1− µ

AT rj−1|AT rj−1

>lrj−1

)2(4.11)

Although the equations (4.8) to (4.11) are computed for AT rj , they are valid for AT rj too, since no

assumption is made on the distribution. Moreover, by using AT rj instead of AT rj two problems are

solved. The rst one is that the calculus of probabilities can be easily made. The second is that the

calculation of the mean and variance of the truncated variables, e.g. the mean of the upper trun-

cated arrival time (µAT rj |AT rj≤lrj

), can be evaluated, for example using the closed forms expressions

gathered in [21] for several distributions.

Summing up, by iteratively applying equations (4.8) to (4.11) using AT rj , one can calculate the

parameters of the AT rj distribution for a given route. Assuming a technician starts its route from

the depot at time zero with no variance, and using equations (4.8) and (4.9), the mean and variance

of the starting service time at the rst customer are derived. This can be easily achieved since AT r1depends only on the travel time from the depot to r1. Next, the parameters of the arrival time at the

second customer can be calculated using equations (4.10) and (4.11), all the more, after performing

this step, equations (4.8) and (4.9) are used to compute the parameters of the starting service time

at the second customer. The process is repeated until the whole route is evaluated. Additionally, by

replacing AT rj in constraints (4.1) and (4.2), the feasibility of the route can be checked.

Multiple authors have used dierent distributions and assumptions to estimate the arrival times

(readers can refer to [7, 11, 18, 34]). It is assumed that AT rj follows a log-normal probability dis-

tribution. The log-normal assumption is twofold motivated. First, an experiment using Monte Carlo

simulation shows that statistically, the log-normal distribution tted the arrival times better than

other distributions (e.g. normal, Gamma). Second, although normality assumption has been proven

to be eective in previous works [7, 11]; skewness is an important factor to be considered in stochastic

vehicle routing algorithms to lead to reliable routing decisions [18]. The presence of left truncation

due to early arrivals induces more asymmetry in the time when service starts which is transferred to

arrival times, and because normal distribution has zero-skewness it does not appear to be the best

distribution to approximate arrival times. Based on these reasons and the tractable computational

times, the log-normal assumption is retained.

4.3.3 Validating the log-normality approximation

An experiment to validate the log-normality assumption is conducted as follows. A solution is rst

created for each instance (see section 4.5.1 for further detail on the instances) using the Clarke and

Wright [9] heuristic combined with the simulation procedure to verify constraints (4.1, and 4.2). For

each tested instance, we considered the two routes in the solution with the greatest number of visited

clients. The reason for this choice is that larger routes will be harder to t and to approximate as

far as more truncation eects are summed up. Furthermore, for the whole set of selected routes a

test is performed to compare the mean, standard deviation, and percentiles of the arrival times at

each node. This is estimated through simulation (10000 trials) against the values obtained while con-

sidering the variables AT i log-normally distributed (estimated). The reported gaps are calculated as|Xsimulation−Xestimated|

Xsimulationwhere Xsimulation and Xestimated are replaced by the mean, standard deviation

and each percentile of the arrival times. Table 4.1 shows basic information of the experiments, includ-

ing the number of evaluated routes, the total number of evaluated nodes, the average number of nodes

per route and the percentage of arrival times that tted log-normal, gamma and normal distributions.

Although more classic probability distributions were tested e.g. exponential, Weibull, chi-squared,

Poisson, Pareto, triangular, uniform, Cauchy, logistic, Laplace, and Erlang, we reported the results

73

4.3. ESTIMATION OF ARRIVAL TIMES

Table 4.1: Basic experiment information per family of instances

InstanceFamily

Num.routes

Totalnodes

Avg.customersper route

% Fittedlog-normal

% FittedGamma

% FittedNormal

C1 18 195 9.38 35.03% 36.16% 15.82%C2 16 460 27.75 23.42% 21.17% 10.81%R1 24 219 8.13 46.67% 40.00% 18.97%R2 22 363 15.5 40.47% 34.31% 23.17%RC1 16 143 7.94 28.35% 18.11% 9.45%RC2 16 241 14.06 30.67% 25.33% 16.44%

Table 4.2: Arrival times average absolute gaps between simulated values and log-normal approximationNotable percentiles 1st to 99th percentiles

InstanceFamily

Mean SD 90% 95% 99% Mean Min. Max.

C1 0.10% 1.41% 0.21% 0.31% 0.70% 0.26% 0.15% 0.97%C2 0.06% 2.92% 0.19% 0.36% 0.78% 0.21% 0.10% 0.78%R1 0.07% 0.76% 0.13% 0.28% 0.73% 0.20% 0.11% 0.73%R2 0.03% 0.73% 0.06% 0.12% 0.29% 0.08% 0.04% 0.29%RC1 0.08% 1.57% 0.25% 0.58% 1.45% 0.44% 0.17% 1.74%RC2 0.03% 0.77% 0.08% 0.17% 0.41% 0.11% 0.04% 0.42%

only for the best three distributions. A total of 112 routes were used in the analysis accounting for a

total of 1621 evaluated nodes. The number of routes diers from family to family since the number

of instances in each family is dierent (except for C2, RC1, and RC2 with 8 instances). Instances of

type 2 have a larger number of clients per route as expected because they have larger time windows.

Table 4.1 presents the percentage of nodes that tted log-normal, Gamma and normal distributions

respectively. The standard Kolmogorov-Smirnov test was used for testing the distributions. The rst

customer of each route was not considered because the arrival time distribution will be the same dis-

tribution of travel times between the depot and the customer. The test concludes that nearly one in

three of the arrival times are log-normal distributed while this number decrease to around 15% when

tting a normal distribution. Gamma distribution achieves to t around 28% of arrival times. Based

on these results it appears that the log-normal approximation can be used as a compromise solution

to model the arrival times while allowing tractable computation of the cost function and probabilities.

Thus, we selected it to model AT rj and continued testing its pertinence. As shown in Table 4.2, the

proposed approximation displays an average gap of the arrival times mean of at most one tenth of

one percent, showing good performance. The lowest average gap is reached by families R2 and RC2

where the mean has an average gap of only 0.03% while the highest gap of 0.1% is achieved in family

C1. On this, standard deviations (SD) have bigger gaps, yet, half of the studied families present an

average standard deviation gap below one percent, and only one family of instances exceeds a gap of

two percent.

Given that equations (4.1) and (4.2) make use of probabilities, it is mandatory to test how accurate

the estimations of dierent percentiles are. Therefore, table 4.2 also presents the average absolute gap

for some notable percentiles and the mean, minimum, and maximum, average absolute gap when the

1st to 99th percentiles are evaluated with increments of one percent.

Results in table 4.2 show the pertinence of the log-normality assumption. The 90th percentile has

an average gap under a quarter of one percent, while the 95th percentile estimation is on average

quite bellow one percent error rate for the whole set of instances. Moreover, the 99th percentile

gives an average gap less than one percent for all families except for RC1 for which the maximum is

reached at 1.45%. It should be noted that the errors grow with the percentile value. Table 4.2 also

reects the accuracy of the estimations, with a mean gap across the families and from the 1st to 99th

percentile inferior to half a percent. Again, RC1 shows the highest gaps in terms of mean, minimum

and maximum average gap when compared with the rest of the families. Type 2 instances (C2, R2,

RC2) look to have lower gaps than their counterparts (C1, R1, RC1) which might be explained by

74


0,00%

1,00%

2,00%

3,00%

4,00%

5,00%

6,00%

0,00% 5,00% 10,00% 15,00% 20,00% 25,00% 30,00%

Mea

n a

bso

lute

err

or

per

cen

tile

95

% of noise Travel Times

30,00%

25,00%

20,00%

15,00%

10,00%

5,00%

0,00%

Figure 4.2: Average absolute error of 95% percentile with service and travel time noises.

the fact that larger time windows prevent the truncation of the arrival times.

Table 4.3: Arrival times mean and standard deviation absolute gaps between simulated values andthree approximations

Log-Normal Normal GammaInstanceFamily

Mean SD Mean SD Mean SD

C1 0.10% 1.41% 0.22% 2.96% 0.22% 3.09%C2 0.06% 2.92% 0.15% 4.79% 0.15% 4.51%R1 0.07% 0.76% 0.11% 1.99% 0.11% 1.80%R2 0.03% 0.73% 0.03% 0.90% 0.03% 0.84%RC1 0.08% 1.57% 0.12% 3.11% 0.11% 2.40%RC2 0.03% 0.77% 0.03% 1.21% 0.03% 1.06%

The proposed approach is very exible and can easily modify the assumption that AT rj is log-

normally distributed to other probability distributions, e.g. Normal, Gamma, Exponential. In fact,

the mean and standard deviation of AT rj were also calculated assuming AT rj is normally and Gamma

distributed. However, log-normal approach presents the best results, improving by more than 80%

of the absolute gap error of arrival times' mean when compared to the normality assumption (see

table 4.3). Indeed, these results with those reported in table 4.2 show that the log-normal distribution

gives better results in terms of parameters and percentiles estimations. Therefore, we retained this

distribution as the best option. To check the robustness of the log-normal approximation we conducted

an additional experiment. During the simulation of the routes, we added white noise to the realization

of the travel and service times. The aim was to inspect if the results held up under these scenarios,

thus representing the uncertainties in the probability distributions of both the travel and service times.

Noise was added using a normal random variables X with parameters µX = 0 and standard deviations

as a percentage of the expected value in travel and service times. This percentage varied from 0 to

0.3 with steps of 0.05. Figure 4.2 shows the average absolute gap error of percentile 95% for dierent

levels of noise in travel and service times. Percentile 95% is selected since it is commonly used as a

standard service level. The results show that the maximum absolute error is just above 5% when noise

of travel and service times is 30% of their means. However, if noise levels are at 15% or less, the error

is below 2%. Indeed, even at noise values of 20% the error stays under the 3% mark. Therefore, even

with the presence of noise in times, the log-normal approximation reaches small gaps at percentiles

estimation.

4.4 Multi-population Memetic Algorithm

To solve the SVRPTWwe propose to use a Multi-Population Memetic Algorithm (MPMA) framework.

The choice of a MA is due to its exibility and performance in a variety of VRPs (for example see

[40, 47]). For the sake of clarity this section is structured as follows: rst the MA general structure

75

4.4. MULTI-POPULATION MEMETIC ALGORITHM

is presented, this is followed by details of the main components of the proposed MA. Finally we will

present the Multi-Population MA general framework.

4.4.1 MA general structure

A Memetic Algorithm (MA) can be described as the hybridization of a Genetic Algorithm with local

search procedures [36]. Our MA structure follows the ideas of Prins [40] and is summarized in algorithm

3. The MA starts by creating a population (line 1 of Algorithm 3), which is a set of encoded solutions

to the problem at hand. The MA iterates until a stopping criteria is met (line 12 Algorithm 3). At

each iteration a new chromosome (C) is processed. The chromosome C can suer a mutation process

(line 4 Algorithm 3) or it can be decoded to perform a local search (lines 5 and 6 of Algorithm 3).

Then at line 11 of Algorithm 3 the population is updated considering C. Each of the components of

the MA and its procedures is now explained in detail.

Algorithm 3 MA1: Pop← Generate initial population2: repeat

3: C ← Crossover(Pop)4: C ← Mutate(C) with probability pm.5: D ← Decode(C).

6: D′← LocalSearch(D) with probability pls.

7: if Costs(D′) < Costs(D) then

8: C ← Encode(D′)

9: D ← Decode(C).10: end if

11: Process(C,D,Pop)12: until Stopping criteria

4.4.2 Chromosomes

Encoding

A chromosome is an encoded solution to the tackled problem. The MA chromosomes that we use are

coded as permutations of the n clients. The decoded version of a chromosome is composed of a set

of routes that satises the problem constraints. Henceforth we use the term chromosome to make

reference to the permutation representation while the term individual is used to dene its associated

decoded solution.

Decoding

To decode a chromosome, we use the Split method of Prins [40]. Split works by considering an auxiliary

directed graph H = (W,Y ) with vertex set W = W 0 = 0,W 1, · · · ,W i, · · · ,Wn. W 0 represents a

dummy vertex and verticesW 1 . . .Wn ∈ Vc characterizes an ordered sequence of customers dened bya chromosome. An arc (W i,W i+d) represents a feasible route for visiting the customers fromW i+1 to

W i+d, with its associated cost. The Split procedure nds the shortest path from vertex W 0 to vertex

Wn and subsequently the optimal set of routes associated to the chromosome. Normally this can be

done using a shortest path algorithm. Nevertheless, constraint (4.3) precludes this implementation

for the SVRPTW. As far as constraint (4.3), it is dependent on the set of routes which comprises a

solution, this constraint must be guaranteed when decoding a chromosome.

∑r∈K

ζr ≤ θ (4.12)

To overcome this, we rewrite constraint (4.3) as done by Errico et al. [12]. Note that setting

ζr = − ln (Ur) and θ = − ln (γ), constraint (4.3) is equivalent to equation (4.12). By considering

76


0

2

4

1

5

3

2 4 1 5 3

0 1 52 4 3

𝜇 ሚ𝑡02: 25

Customers permutation (Figure 2a)

Ƹ𝑐𝑟=75.54𝑈𝑟 = 0.97

0

2

4

1

5

3

Split auxiliary graph (Figure 2c)

Decoded routes from permutation (Figure 2d)Times information (Figure 2b)

[25,40][𝑒𝑖 , 𝑙𝑖] [40,90] [60,95] [30,60] [35,85]

𝜇 ሚ𝑡04: 30

𝜇 ሚ𝑡01: 20

𝜇 ሚ𝑡05: 15

𝜇 ሚ𝑡03: 25

𝜇 ሚ𝑡24: 20

𝜇 ሚ𝑡41: 20𝜇 ሚ𝑡15: 20

𝜇 ሚ𝑡53: 20

𝜇 ǁ𝑠2: 20

𝜇 ǁ𝑠2: 20

𝜇 ǁ𝑠2: 20

𝜇 ǁ𝑠2: 20

𝜇 ǁ𝑠2: 20

Ƹ𝑐𝑟=71.78𝑈𝑟 = 0.93

Ƹ𝑐𝑟=60.75𝑈𝑟 = 0.98

Ƹ𝑐𝑟=50.34𝑈𝑟 = 0.99

Ƹ𝑐𝑟= 60𝑈𝑟 = 1

Ƹ𝑐𝑟= 40𝑈𝑟 = 1

Ƹ𝑐𝑟= 30𝑈𝑟 = 1

Ƹ𝑐𝑟= 50𝑈𝑟 = 1

Chromosome cost: 75.54+40+60.75 = 176.29Probability no failures: 0.97x1x0.98 = 95.06%

Figure 4.3: Split example for the VRP with stochastic travel and service times

equation (4.12), the shortest path problem involved in the Split comes back to solve a Constrained

Shortest Path Problem (CSPP). In this CSPP the scarce resource is θ and each edge consumes ζr units

of the resource, where r is the route associated with the edge. The CSPP is solved using a Labeling

Algorithm as the one employed in [14]. Thus, Split allows to decode the optimal routes partition for

a chromosome, giving a solution which is feasible, i.e. respect constraints (4.1) to (4.3).

An example of the split method using the log-normal approximation is reported in gure 4.3. For

this example it is assumed that α, β, γ are set to 95%. Customer time windows are reported in part

(a) of the gure along with the chromosome. Moreover, a coecient of variation of 0.2 is used for the

travel and service times. Their mean values are reported in part (b). Distances are assumed to be

equal to the mean of travel times. For simplicity, the xed cost of each vehicle is assumed to be zero.

Besides, the time windows for the depot are dened by e0 = 0 and l0 = 140. Part (c) of gure 4.3

shows the auxiliary graph of the Split, and the bold arcs represent the optimal solution to the CSPP.

In this auxiliary graph the arc (2, 1) for example, represents a route starting at the depot, visiting

customers 4 and 1, and then returning to the depot. This route has an expected cost (see equation

4.7) of 71.8 and a probability of 93% of having no failures. The reader shall notice that only the arcs

representing feasible routes (respecting the service levels α and β) are considered in gure 4.3 part

(c). Lastly, part (d) displays the individual, or the decoded routes with the cost and the probability

of having no failures in the solution.

4.4.3 Population

The population (Pop) is dened as an ensemble of chromosomes, thus, an ensemble of coded solutions.

Pop is composed of PopSize chromosomes which are ordered in a decreasing way with respect to the

cost of their detailed corresponding solutions. PopSize is constant, so there are always the same

number of chromosomes in Pop. The diversity of the population is controlled by mean of Campos

et al. [6] distance measure. The latter is computed at the detailed solution level. To enhance

diversication, clones (distance zero) are discarded, and chromosomes with a positive distance are

allowed to enter. It shall be noted that the although this makes the MA a version with population

control, the distance measure is only used to avoid clones. Further improvements can be performed

by dynamically adjusting the minimum distance to accept the chromosomes willing to enter the

population1. Furthermore, as populations are of a xed size, a random chromosome among the worst

half of the population is deleted before the new one is inserted.

The initial population is created as follows: Four individuals are computed by using heuristics.

1I thank professor Christian Prins for pointing this out during the dissertation questions and comments.

77

4.4. MULTI-POPULATION MEMETIC ALGORITHM

1 6 4 2 3 7 5 8Parent 1 (𝜋1)

Parent 2 (𝜋2)New individual

2 1 5 6 4 8 3 7

5 6 4 2 3 8 7 1

i j

Figure 4.4: Example of OX crossover.

The rst is created with the Clarke and Wright[9] heuristic, the second and third from Solomon [41]

insertion heuristic (using two sets of parameters), and the fourth from Algorithm D proposed by

Nagata and Bräysy [37] using as a starting solution the best individual among the Clarke and Wright

and Solomon heuristics. All heuristics are run for the deterministic problem using the mean travel

and service times as the true values. The four heuristic solutions are converted to chromosomes by

concatenating their routes and then added to Pop. Meanwhile, to achieve the value PopSize, the

population is lled with chromosomes created with random permutations of the customers. If clones

are built during this procedure they are discarded.

4.4.4 Crossover

Crossover procedure is used to create new chromosomes from those already present in Pop. The

crossover used in this work is the well-known Ordered Crossover (OX). OX works as follows: two

chromosomes π1, π2 are selected from the population and two random positions i, j | i < j ≤ n are

chosen. The information between positions i, j is copied from π1 to the new ospring. To complete

the latter, π2 is circularly traversed from position j + 1 to position j. The customers which are not

already with the new child are then copied circularly from position j + 1 to i. The roles of π1, π2 are

exchanged to produce another ospring. Before performing the crossover, the MA selects π1, π2 with

binary tournaments, and then one of the two children generated by OX is randomly picked.

4.4.5 Local search and mutation

The local search (LS) is designed to improve the objective function of a solution by performing dierent

modications on the solution itself. The proposed local search is based on the Variable Neighborhood

Search (VNS) [35]. In this context Neighborhoods are structured in such a way that movements can

be performed on a solution s to achieve a new solution s′. Further explanation on neighborhoods

denitions for VRPs can be found in Labadie et al. [25] and Bräysy et al. [5]. The neighborhoods

used for the local search are: Or-opt, 2-opt movements, both intra and inter routes, and the CROSS

exchanges in their inter-routes version.

LS starts by searching for a movement in the rst neighborhood which improves the solution. If

such movement does not exist, it passes to the next neighborhood, however, if a new best solution

is found in the current neighborhood, the LS starts again from the rst neighborhood. The process

is repeated until no neighborhood can improve the solution. The order in which neighborhoods are

explored is randomized and it changes every time the LS is performed. A movement is executed

immediately if it improves the solution, therefore a rst accept criterion is used.

Since LS can be expensive in terms of running time, the neighborhoods exploration is constrained,

i.e. Or-opt movements are limited to movements involving, at most, three customers, and CROSS

exchanges use sequences to at most two clients. Also, LS is performed to an individual with probability

pls. Mutations are performed by selecting a random position among the chromosome, then the selected

customer and the following are relocated to another random position. Mutations are executed with a

probability of pm.

78


4.4.6 MPMA framework

Besides the exibility and performance of MAs they also allow a multi-population approach, e.g.

using distributed evolutionary algorithms. In the latter, several populations can be used to enhance

the diversication [1]. Our MPMA aims to enhance the diversication but also to speed-up and

improve the solutions through the introduction of dierences between the problems tackled in each

MA.

Algorithm 4 presents the framework of the proposed MPMA, it uses a set of MAs with the same

presented structure, but with dierences in the problem handled by each one. In the rst line of

Algorithm 4 the MAs are created. Then, each MA starts to solve its problem (in parallel) in lines

three to six of Algorithm 4. While the MAs are working they cooperate by letting chromosomes be

copied from one MA to another (line 8 of Algorithm 4). The aim of communication is to transfer

valuable information from one MA to another and to enhance diversication.

Once all the MAs nish, the MPMA continue its procedure. Since the MAs solutions are created

through the approximation of arrival times (see section 4.4.7), they may not be feasible in the un-

certain environment. Therefore, to guarantee a feasible solution we proceed as follows. The best ve

chromosomes of each MA are extracted (lines 10 to 12 of Algorithm 4) and decoded into individuals

using Split (lines 14 to 17 of Algorithm 4). Monte Carlo simulation is used to check the feasibility of

the routes, the probability of having no failures, and to estimate their costs.

Algorithm 4 MPMARequire: α, β, γ, τ, numSimulations, runT ime, F1: Create(MAf ) ∀f = 1...F2: initialT ime← CurrentTime3: In parallel : // The MAs start and run in parallel4: for all f = 1...F do

5: run(MAf (α, β, γ, τ, runT ime)) // See algorithm 36: end for

7: while CurrentTime - initialT ime ≤ runT ime do8: Communicate MAf ∀f = 1...F9: end while

10: for all f = 1...F do

11: Best← Best ∪Get best Chromosomes(MAf )12: end for

13: Pool← 14: for all C ∈ Best do15: Split(C,α, β, γ, numSimulations) // Using Monte Carlo Simulation16: Pool← Pool ∪ Simulated routes during Split17: end for

18: return SolveSetPartitioning(Pool, γ)

Moreover, since Split using simulation is a highly time consuming task, we take advantage of every

route simulated during the Split of the best MA chromosomes. This is done by saving each route into

a pool, which acts as a list of feasible routes that have been validated by Monte Carlo Simulation.

After all selected chromosomes have been evaluated, the routes in the pool are used to solve a set

partitioning problem (see for example [33]) which uses equation (4.12) to guarantee the feasibility of

the solution. By solving the set partitioning using the list of Monte Carlo Simulation validated routes,

we guarantee that the routes in the solution respect the problem constraints. The solution retrieved

by the set partitioning problem is returned as the best solution found by the MPMA.

4.4.7 MA dierences

To allow the communication of the MAs some changes are performed to the base MA. Every τ seconds,

each MA sends two copies of its own chromosomes. One is randomly selected from the best half of

Pop, and the other from the worst. The destination MA of each child is randomly picked. Moreover,

79


destination MAs use the received chromosome in the next iteration of algorithm 3 instead of creating

a new one by a crossover procedure.

Our MPMA uses three MAs (K = 3), namely MA1, MA2, MA3 with the same general structure

presented in section 4.4.1 but with its own particularities. MA1 tackles the SVRPTW by embed-

ding the log-normal modeling presented in section 4.3 while MA2 and MA3 work on a deterministic

VRPTW. The idea behind MA2 and MA3 is that even if deterministic solutions might not perform

well on stochastic environments, they may serve as good seeds to potential good stochastic solutions

([3]).

MA2 and MA3 dier since MA2 travel and service times are set to their respective mean values,

whereas MA3 service and travel times are set to the 75th percentile that is si = ψi−1 (75%) ∀i ∈ Vc and

tij = φij−1 (75%) ∀ [i, j] ∈ E where ψi−1∀i ∈ Vc and φij−1∀ [i, j] ∈ E represent the inverse probability

functions of the service and travel times respectively. The 75th percentile is selected after performing

some preliminary tests. Increasing the service and travel times provides solutions with low probability

of failures. A similar idea has already been used in the context of the VRP with stochastic demands

([32]). Therefore, MA3 is used to provide robust solutions which can be fast improved upon to perform

better in a stochastic environment. Moreover, MA2 and MA3 use labels to enhance the LS procedure

(see [47]) and allow unfeasible solutions with the returns in time (time wraps as explained in [38]).

Additionally, MA2 and MA3 use the Bellman algorithm during the Split procedure (section 4.4.3) as

done by Prins [40].

MA1 presents its own important considerations. Log-normal modeling allows MA1 to check feas-

ibility, estimate the costs, and the underlying probabilities when using Split (section 4.4.3) and while

performing the LS. This increases the complexity of the LS since feasibility checks require the evalu-

ation of the whole route. Therefore, MA1 uses an additional strategy. Indeed, MA1 alternates its plseach % seconds by changing the probability of LS from zero to a given pls

′and from pls

′to zero value.

When MA1 pls is greater than zero, it uses deterministic labels to eciently evaluate the pertinence

of a movement (see [47]). The idea is that if a movement is unfeasible in a deterministic environment

(times equal to their mean values), it is very likely that it will violate constraint (4.1) or (4.2). Sub-

sequently it will be a waste of time to reevaluate the route using the log-normal approximation. If the

movement is feasible for the deterministic model, then the log-normal approach is used to evaluate its

feasibility and pertinence.

4.5 Numerical Results

4.5.1 Instances

To test the MPMA, results are gathered for modied Solomon [41] instances. Customer service times

are set as Gamma distributed variables with mean equal to the deterministic service time given in

the original instances, and a standard deviation inferred for a coecient of variation of 0.2. Edges

travel times are set as log-normally distributed variables with mean equal to the length of the edge

and with standard deviation derived from a coecient of variation of 0.2. This value is used when

considering authors like Turner et al. [46] (as cited in [11]) previously stated that it varies from 0.15

to 0.25 when dealing with freeways and from 0.20 to 0.25 for principal and secondary roads. As the

original instances do not state the kind of roads represented by the edges, the value 0.2 works overall

for any type of road. Solomon [41] proposed 56 les divided among six families, C1, R1, RC1, C2, R2,

RC2 for the VRPTW. Families C1, R1, and RC1 have tight time windows, so routes in these instances

tend to have less number of customers. On the other hand C2, R2, and RC2 have larger time windows

therefore more clients are visited by each vehicle. Another classication can be done based on the

position of customers in the space. Families C1 and C2 have clustered sets of customers, while they

are randomly positioned in R1 and R2. RC1 and RC2 have a mix of clustered and randomly located

80


clients. All 56 instances are comprised of a depot and 100 customers.

While considering that Solomon [41] instances are designed for the VRPTW, we performed some

modications to adapt them to deal with the stochastic nature of the SVRPTW. The service levels

α, β, γ are set to 95%, 95%, and 90% respectively. To guarantee a feasible solution, the elementary

routes (routes visiting only one customer) dened as r = r0 = 0, r1 = i, r2 = n+ 1 ∀i ∈ V c mustat least respect constraints (4.1) and (4.2). To guarantee the feasibility of these routes, each one was

tested through one million trials by simulation. If a given route r is unfeasible because of constraint

(4.1), the customer time window closure li is set to li = li + 5. If infeasibility arises because of

constraint (4.2) then, the depot time window closure is set to ln+1 = ln+1 + 5. This process is

repeated until feasibility is reached for every route visiting a single customer.

4.5.2 General discussion

The MPMA presented in section 4.4 was tested on the modied instances of Solomong [41] introduced

in section 4.5.1. We used the instances including the rst 50 customers and the complete instances.

MPMA was implemented on a Dell Latitude E6420 personal computer with Intel Core i7-2760QM

@2.4 GHz, running Windows 7 Professional 64 bits. The algorithms were coded in Java and compiled

with JavaSE-1.8_45, with maximum allocated memory of 1 Gb. Random variables and computation

probabilities were generated by the library of Stochastic Simulation in Java ([30]). Parameter values

were selected after several preliminary tests. The vehicle cost M is xed to 1000, this value allows the

MPMA to minimize the number of vehicles as the rst objective followed by the total distance plus

recourse cost. MA populations size PopSize is set to 25 for the three MAs. When simulation is used

during the MPMA it performs 1000 trials to evaluate the feasibility, estimate the cost and required

probabilities for each route. The MPMA performs 10 runs for the whole set of instances (50 and 100

customers). % is set to n/10 seconds, while τ value is 0.05 seconds. The pls and pm are set to 0.2

for all MAs although MA1 starts without performing LS. The MAs stopping condition is set to 20

seconds for the 50 customer instances while this value is increased to 100 seconds for the instances

with 100 customers. Set partitioning problem is solved by means of Gurobi 6.0 ([20]).

Tables 4.4 to 4.6 presents the best solutions found out of the 10 runs for C, R, and RC instances

respectively. Meanwhile, average results can be found at tables 4.11 and 4.12. For each instance the

expected route cost (ERC), the number of vehicles (# Veh.), the distance (DC), and the recourse cost

(RC) are reported. Type 1 instances show higher recourse costs than type 2 instances. This is quietly

pronounced in family R (100 customers) since the average recourse cost is increased from R2 to R1

by 605%; similarly families C and RC display the same behavior. This is completely coherent since

type 1 instances have tighter time windows which imply that failures are more likely to occur, and

consequently recourse costs are higher.

Furthermore, the family C instance shows an increment of nearly 106% in the cumulative number

of vehicles (CNV) required for 100 customer instances when compared to 50 customer instances. The

rise of the total costs is near to 105%, while distance and recourse increases by 96.9% and 83.1%

respectively. Family R on the other hand, shows the minimum increments (among C, R, and RC

families), with an augmentation of nearly 72.8% in the CNV, 70.3% in terms of cost and 50.8% and

71.1% for distance and recourse metrics. Moreover, RC family raises the CNV by 79.5% and the total

cost by 78.1%. The distance is increased by nearly 69.2%, and recourse presents the lowest increment

of only 26.1%. Also, it should be noted that the best CNV for 50 customer instances is 302 while

the same metric for 100 customer instances is 554. This dierence of 252 vehicles is translated into

an increase of 83.4%. A similar behavior was observed for the deterministic best solutions found by

MA2, which incremented the CNV by a factor of three quarters when passing from 50 customer to

100 customer instances. Thus, the increments in the CNV seem to be driven mostly by the inherent

combinatorial nature of the problems whether the stochastic or deterministic versions are considered.

Table 4.7 summarizes the results attained by MPMA. For each family of instances they are reported

81


Table 4.4: Best solutions found by MPMA for C type instancesBest Solution - 50 Customers Best Solution - 100 Customers

Instance ERC # Veh. DC RC ERC # Veh. DC RC

C101 9741.22 9.00 642.90 98.32 21519.83 20.00 1497.54 22.29C102 8635.27 8.00 590.81 44.46 18391.69 17.00 1330.67 61.02C103 7534.46 7.00 514.58 19.88 15227.97 14.00 1193.63 34.33C104 5382.54 5.00 369.45 13.10 11986.00 11.00 961.25 24.75C105 7555.61 7.00 522.88 32.73 16292.06 15.00 1199.72 92.35C106 8609.68 8.00 564.34 45.34 16259.28 15.00 1212.13 47.15C107 6506.86 6.00 484.18 22.68 14171.11 13.00 1095.82 75.29C108 6467.17 6.00 440.31 26.86 13069.71 12.00 956.64 113.07C109 5369.22 5.00 364.11 5.11 10846.15 10.00 836.91 9.24C201 3432.76 3.00 417.42 15.34 5821.75 5.00 717.08 104.67C202 2462.64 2.00 433.43 29.20 5755.96 5.00 718.78 37.18C203 2514.33 2.00 501.01 13.32 4715.41 4.00 679.56 35.85C204 2384.27 2.00 372.21 12.06 4667.72 4.00 654.73 12.99C205 2464.49 2.00 460.92 3.57 4688.49 4.00 671.74 16.75C206 2460.33 2.00 456.51 3.82 4666.89 4.00 656.51 10.38C207 2449.09 2.00 448.27 0.82 4655.02 4.00 649.24 5.78C208 2366.15 2.00 364.61 1.54 4632.91 4.00 625.22 7.69

Total 86336.10 78.00 7947.94 388.16 177367.9 161.00 15657.17 710.78

Table 4.5: Best solutions found by MPMA for R type instancesBest Solution - 50 Customers Best Solution - 100 Customers


R101 16232.47 15.00 1191.51 40.96 28960.67 27.00 1934.28 26.38R102 14082.91 13.00 1049.13 33.78 25744.05 24.00 1725.49 18.56R103 10890.01 10.00 867.52 22.49 19409.84 18.00 1371.98 37.86R104 7734.53 7.00 722.66 11.88 14154.45 13.00 1128.20 26.25R105 12007.66 11.00 986.34 21.32 19602.65 18.00 1527.25 75.40R106 9898.11 9.00 867.16 30.95 17420.11 16.00 1369.40 50.71R107 8795.05 8.00 775.87 19.18 14258.45 13.00 1194.59 63.86R108 6659.90 6.00 631.58 28.32 12068.34 11.00 1040.52 27.82R109 9863.25 9.00 849.35 13.90 15308.71 14.00 1270.43 38.28R110 8779.34 8.00 762.32 17.02 14239.67 13.00 1193.42 46.25R111 8751.49 8.00 745.91 5.58 14184.60 13.00 1151.82 32.78R112 7678.27 7.00 672.68 5.58 13035.89 12.00 1029.60 6.29R201 3898.66 3.00 892.30 6.36 6333.61 5.00 1306.96 26.65R202 3745.80 3.00 739.81 5.99 6090.63 5.00 1079.22 11.41R203 2726.72 2.00 703.38 23.34 4927.94 4.00 916.71 11.23R204 2512.02 2.00 511.28 0.74 3792.57 3.00 792.53 0.03R205 2774.60 2.00 768.96 5.64 6011.41 5.00 1011.05 0.36R206 3642.05 3.00 642.02 0.04 3981.86 3.00 981.50 0.35R207 2587.16 2.00 587.16 0.00 3835.66 3.00 835.66 0.00R208 2493.69 2.00 493.69 0.00 3715.32 3.00 710.28 5.04R209 2694.99 2.00 693.19 1.80 4926.19 4.00 918.85 7.34R210 2697.69 2.00 692.05 5.63 4993.53 4.00 992.32 1.21R211 2561.32 2.00 561.28 0.03 4778.41 4.00 778.20 0.20

Total 153707.6 136.00 17407.14 300.54 261774.5 235.00 26260.27 514.25

Table 4.6: Best solutions found by MPMA for RC type instancesBest Solution - 50 Customers Best Solution - 100 Customers


RC101 12133.54 11.00 1091.58 41.96 20920.01 19.00 1819.27 100.73RC102 9947.87 9.00 901.19 46.68 18714.96 17.00 1660.41 54.55RC103 8883.27 8.00 848.98 34.29 15475.45 14.00 1427.97 47.48RC104 6797.88 6.00 728.56 69.32 13300.61 12.00 1260.09 40.51RC105 10989.11 10.00 969.13 19.98 18783.43 17.00 1704.47 78.96RC106 8935.65 8.00 904.74 30.91 16627.30 15.00 1597.60 29.70RC107 7827.67 7.00 803.86 23.80 15433.93 14.00 1387.96 45.97RC108 6738.07 6.00 710.82 27.25 14321.21 13.00 1309.87 11.34RC201 4825.24 4.00 820.78 4.46 7402.04 6.00 1387.02 15.02RC202 3836.37 3.00 830.57 5.80 6184.45 5.00 1181.73 2.72RC203 3640.20 3.00 633.32 6.88 5098.22 4.00 1096.44 1.78RC204 2498.75 2.00 490.95 7.80 4801.08 4.00 800.55 0.53RC205 4684.67 4.00 676.94 7.73 7245.54 6.00 1229.47 16.07RC206 2914.94 2.00 856.71 58.23 5229.15 4.00 1195.54 33.60RC207 3617.36 3.00 614.62 2.74 5071.24 4.00 1067.12 4.11RC208 2531.56 2.00 525.87 5.68 4879.98 4.00 866.94 13.04

Total 100802.1 88.00 12408.62 393.52 179488.5 158.00 20992.47 496.11

Table 4.7: Average performance of MPMA50 Customer instances 100 Customer instances

InstanceFamily

Avg. #Veh.

% TimesBSF

Avg.Time(s)

Avg. #Veh.

% TimesBSF

Avg.Time(s)

C1 7.04 52.3% 37.61 14.33 57.7% 132.14C2 2.38 57.5% 61.00 4.28 70.0% 188.82R1 9.36 65.0% 35.14 16.48 29.2% 128.05R2 2.63 26.4% 57.94 4.51 14.54% 186.06RC1 8.31 61.25% 36.18 15.54 38.8% 127.65RC2 3.24 32.50% 51.49 5.29 11.25% 170.59

82


as follows: the average number of vehicles (Avg. # Veh.), the percentage of times the best solution is

found (% Times BSF), and the average from 10 runs. The average time stands for the time expended

by MPMA in one run of one instance. It comprises from the start time of the algorithm until the

best solution is retrieved, i.e. it does not take into account the total computational time generated

by parallel threads. To establish if two solutions are equal, we simulate the best solution found at

each run of the MPMA, which allows us to construct a condence interval of 95% on the total costs2.

If two condence interval solutions overlap, it is assumed that they are, on average, equal. Detailed

results of the best an average solutions are presented in tables 4.11 to 4.12.

Type 2 instances with random located customers are harder to solve than type 1 instances when

considering the percentage of times that the best solution is found. This is evident for family RC2

with 100 customers, where the best solution is only found 11.25% of the time among the 10 runs.

Still, 4 out of 6 families found the best solution almost more than 30% of the times in 100 customer

instances. In contrast, C2 family (with 100 customers) achieves 70% of the time for the best solution.

Overall the MPMA shows a good performance, retrieving on average 49.1% and 35.5% of the time the

best solution for instances with 50 and 100 clients respectively. In terms of computational time, the

MPMA needs at most three minutes on average to solve any instance (100 customers). The algorithm

achieves these low times thanks to the log-normal approach presented in section 4.3 and because

simulation is used limitedly. Indeed, the Split with simulation plus the set partitioning model take on

average around a third of the total time for 100 customer instances.

Moreover, during the whole set of experiments it was evidenced the importance of considering the

stochasticity of the parameters for the proposed problem. When the best chromosomes retrieved by

MA2 were decoded in the uncertain environment (using Split with Monte Carlo simulation) they used

more vehicles than the best solutions found by MPMA. Indeed, the CNV of the best MPMA solutions

is just over half of the CNV of the best MA2 chromosomes. Thus, neglecting the stochasticity and

using deterministic solutions can conduce to a huge increment of the number of vehicles. This is

specially important for type 1 instances in which due to tight time windows, deterministic solutions

perform very poorly.

4.5.3 The eects of considering multiple populations

To assess the eects of using multiple populations in the MPMA we compare the performance on 100

customer instances of the single MA1 and the MA1 combined with MA2 and MA3, to the full MPMA.

Table 4.8 presents the results for the dierent congurations. It is shown that using the MPMA with

the three populations improves the solutions in terms of the average number of vehicles by around 6%

when compared to the MA1 used alone. Moreover, the best solutions of the full MPMA reduces by

more than 4% the same metric of MA1, MA1+MA2 and MA1+MA3. In general, congurations with

more than one population achieves to nd solutions with less vehicles than the single MA1. Besides

this, the best costs in terms of distance are achieved by the MPMA with three populations, however

it presents the higher recourse costs. Despite the increase of nearly 15% on the average total time of

the full MPMA in relation to MA1, the former conguration has the best overall results.

4.5.4 MPMA + Log-normal approximation comparisons

To further test the proposed MPMA + log-normal approximation, we compared it to the works of

Miranda and Conceição [34] and Nguyen et al. [39]. These are two of the closest problems related

to the problem at hand. [34] consider the SVRPTW with stochastic travel and service times and

time windows. Also, late services are allowed but the service must start at or after the opening time

window (ei∀i ∈ Vc), furthermore the authors set the customers service level at 80%. Their Iterated

2The number of trials was increased for this test to 30,000 to have a better accuracy on the condence intervals, andthus conclude if two solutions are dierent on average

83


Table 4.8: Comparison of single MA1 to MPMA - 100 customer instances

Metric MA1 MA1 + MA2 MA1 + MA3MA1 + MA2

+ MA3

# Vehicles Best Solutions 579 575 576 554# Vehicles Average 616.7 612.7 608.4 577.3Average Distance Cost 1156.56 1158.88 1138.24 1123.87Average Recourse Cost 3.86 3.90 3.88 31.16Avg. Time on pure MA (s) 100 100 100 100Avg. Time on simulation+setpartitioning (s)

35.28 65.06 61.85 54.70

Avg. Total Time (s) 135.28 165.06 161.85 154.80

Table 4.9: MPMA comparison to Miranda and Conceição [34] ILSInstance Method

Avg. #Veh.

Avg.DC

Avg.Min SL

Avg. SLAvg.SLE

Max.SLE

Time(s)

R105 ILS 17.67 1615.75 81.66% 99.31% 0.18% 1.81% 18.40MPMA 17.30 1470.89 80.55% 97.30% 0.39% 2.81% 31.82

R109 ILS 15.00 1488.81 79.53% 95.45% 0.17% 0.89% 6.36MPMA 15.00 1270.27 80.96% 98.00% 0.28% 3.44% 32.21

C101 ILS 17.00 2284.77 83.82% 95.52% 0.22% 0.87% 29.49MPMA 17.00 1372.27 80.42% 95.42% 0.64% 3.56% 31.77

C106 ILS 14.67 1722.32 81.33% 98.65% 0.18% 0.70% 5.56MPMA 14.00 1141.17 80.57% 96.01% 0.72% 4.78% 32.25

RC101 ILS 19.67 2012.76 78.32% 100.22% 0.16% 1.42% 15.18MPMA 19.60 1841.32 80.98% 97.53% 0.46% 4.98% 31.77

RC106 ILS 14.67 1584.63 82.26% 95.57% 0.16% 0.82% 24.43MPMA 15.30 1534.95 81.07% 97.87% 0.40% 4.64% 32.08

RC107 ILS 14.00 1569.10 83.03% 96.23% 0.16% 0.86% 34.99MPMA 13.70 1368.53 81.46% 98.44% 0.23% 3.01% 32.40

Avg. ILS 16.10 1754.02 81.42% 97.28% 0.18% 1.05% 19.20MPMA 15.99 1428.49 80.86% 97.20% 0.44% 3.89% 32.04

Local Search (ILS) is used to solve some of the Solomon [41] instances with 100 customers. Service

times are assumed to be normally distributed with mean equals to the deterministic service time and

standard deviation derived from a coecient of variation generated by a uniform law U [0.1; 0.6]. Travel

times are also assumed to be normally distributed with mean equals to the euclidean distance between

the nodes and standard deviation derived from a coecient of variation generated by a uniform law

U [0.1; 0.6]. Some modications were made to our MPMA to compare it with the ILS proposed by

Miranda and Conceição [34]. Service levels α, β, γ were set to 80%, 0%, and 0% respectively. No

recourse action is considered for the problem so the cost of the route (equation 4.7) only takes into

account both distance and xed costs of the vehicle. Travel and service times were modeled as normal

variables same with the same parameters used in [34]. The MAs stopping condition is set to 25

seconds running time and late services are allowed as in [34]. Table 4.9 presents the results of the

two methods, which are gathered over 10 runs. For each instance the method, the average number

of vehicles (Avg. # Veh.), the average distance (Avg. DC), the average minimum service level (Avg.

Min SL), i.e. minP (AT rj ) ≤ li | rj = i, i ∈ Vc, the average service level for all customers (Avg. SL),and the average and maximum service level error (Avg. SLE - Max. SLE) are reported. Service level

errors are dened as the absolute dierence between service level estimated by simulation minus the

service level calculated with the approximation.

Results show that MPMA uses near to one percent fewer vehicles than ILS. This decrease does

not negatively impact the operational costs (distance) which are improved upon by almost 23%.

Concerning the service levels, the two methods appear to give similar results to the extent that the

average service level of MPMA is inferior to ILS by only 0.08%. Moreover, it should be noted that

MPMA ends up using Monte Carlo simulation enabling the method to validate the chance constraints.

In fact, MPMA guarantees a service level of α in all tested instances rather than the ILS method which

guarantees this constraint in only ve out of seven instances. In terms of service level errors, the metric

tends to be higher in MPMA although solutions with a higher number of vehicles and distance cost

have a signicant reduction in service level errors. It cannot be concluded which approximation

performs better using the service level errors as far as this metric is gathered from the best solution

84


Table 4.10: MPMA comparison to Nguyen et al. [39] TSInstance Method Avg. # Veh. Avg. DC Avg. ET Time (s)

C1 TS (AD1) 10.33 944.55 2.82 9.20TS (AD2) 10.33 929.85 3.43 7.94TS (AD3) 10.22 918.61 4.19 8.62MPMA 9.89 908.73 5.55 51.41


R1 TS (AD1) 15.67 1521.12 30.77 16.10TS (AD2) 15.83 1495.02 24.02 14.96TS (AD3) 15.58 1447.70 26.02 18.95MPMA 15.51 1409.47 19.25 51.76


RC1 TS (AD1) 15.25 1736.57 49.79 11.24TS (AD2) 15.38 1713.98 34.81 10.78TS (AD3) 15.13 1700.99 41.07 9.46MPMA 15.86 1630.91 22.55 51.71


Avg. TS (AD1) 8.91 1241.83 15.91 23.48TS (AD2) 8.95 1198.80 11.35 19.46TS (AD3) 8.82 1181.67 12.92 19.98MPMA 9.04 1131.34 9.19 52.16

found by each algorithm at each run, and they are also structurally dierent (see operational costs).

Computational times were not scaled up since operating systems, language programming and hardware

characteristics were dierent, however, it can safely be said that ILS is faster than MPMA using nearly

40% less time.

Nguyen et al. [39] work on the SVRPTW with stochastic travel times and time windows, using

a Satiscing Measure Approach (SMA) to guide its solution process. The authors use a tabu search

(TS) with their SMA and tested it on Solomon [41] instances. Travel times are assumed to follow

an ambiguous distribution, however, for simulation purposes, Gamma distribution is used with a

mean equivalent to the deterministic distance between two nodes and standard deviation derived

from a coecient of variation of 0.5. Hierarchical objectives are used by the authors, in descending

order of importance: maximize the total number of customers served, maximize the overall satiscing

measure, minimize the total number of vehicles used, and minimize total distance traveled. The

following changes were made to our MPMA by aiming to compare it with the results in Nguyen et

al. [39]. Service levels α, β, γ were set to 50%, 0%, and 0% respectively since these constraints are

not considered in [39], therefore we let MPMA be driven as an SPR. Travel times are assumed to

be Gamma distributed with the same parameters used in [39]. The α value of 50% serves to discard

routes with high probability of failures and thus to accelerate the MPMA local search. Furthermore,

as a result of the associated travel times variability is high, the MA3 is set to the 60th percentile that

is tij = φij−1 (60%) ∀ [i, j] ∈ E where φij−1∀ [i, j] ∈ E represents the inverse probability functions of

the travel times. The MAs stopping condition is set to 40 seconds running time and late services

were allowed as in [39]. Table 4.10 presents the summary of results gathered for MPMA (out of 10

runs), and TS (out of 1 run). TS runs only once since it does not have any randomized components.

For each family of instances the method, the average number of vehicles (Avg. # Veh.), the average

distance or expected travel time if travel times are set to their mean (Avg. DC), the average expected

tardiness (Avg. ET), and the average running time (Time) are reported.

Results show that MPMA is very competitive. If compared to TS (AD1) the MPMA increments

the number of vehicles by only 1.5%, additionally MPMA can reduce distance costs by almost 10%

and more importantly, reduces the expected tardiness by more than 40%. The MPMA presents

the best results among the compared methods in both distance (total travel time) and expected

85

4.6. CONCLUSIONS

tardiness. Although MPMA needs more time than TS, it nds better results on average, with low

expected tardiness and barely incrementing the number of vehicles used. Times are not scaled up

since dierences in the operating system, language programming, and hardware, yet, TS procedures

work on average 2.5 times faster than MPMA.

4.6 Conclusions

This chapter presented a Multi-population Memetic Algorithm (MPMA) to solve vehicle routing

problems with time windows (SVRPTW) and stochastic travel and service times. The approach

incorporates the fact that failures can take place at the same time that a service level must be satised.

The estimation of the arrival times is based on the assumption that they can be approximated by a

log-normal distribution.

The MPMA exploits the characteristics of dierent populations guaranteeing high service levels

for customers service. The MPMA was tested by comparing it to two recent variants of the SVRPTW

where late services are permitted. The MPMA found on average better solutions for the two compar-

isons albeit MPMA has higher computational times.

The results conrms that our formulation and resolution method are exible enough to deal with

some variants of the SVRPTW and that the MPMA + log-normal approximation is a valid and

eective method to solve SVRPs. Undergoing work is concentrating on the use of the proposed

approximation within exact methods and a multi-objective version of the algorithm where service

levels are considered as objectives rather than constraints.

Contributions

Preliminary results of this chapter were presented at MIM2016 conference:


A multi population memetic algorithm for the vehicle routing problem with time windows and

stochastic travel and service times

In 8th IFAC Conference on Manufacturing Modelling, Management and Control MIM 2016 Troyes,

France, 2830 June 2016

Extensions of the problem at hand were also presented at CLAIO2016 conference:


An approximate column generation for the vehicle routing problem with hard time windows and

stochastic travel and service times. In XVIII CLAIO, the Latin-Iberoamerican Conference on Opera-

tions Research Santiago de Chile, Chile, 26 October 2016.

An article version of this chapter has been published in the Computers & Industrial Engineering

journal. Please cite it as follows:

Gutierrez, A., Dieulle, L., Labadie, N., Velasco, N. (2018). A multi-population algorithm to solve

the VRP with stochastic service and travel times. Computers & Industrial Engineering, 125, 144-156.

https://doi.org/10.1016/j.cie.2018.07.042.

MPMA Tables with average results for Solomon instances

86


Table4.11:50

Customer

instancesbestandaveragesolutionsfoundby

MPMAper

instance

BestSolution

Average

Instance

TotalCost

#Vehicles

Distance

Recourse

TotalCost

#Vehicles

Distance

Recourse

Avg.Time(s)

C101

9741.22

9.00

642.90

98.32

10568.66

9.90

643.79

24.87

33.87

C102

8635.27

8.00

590.81

44.46

9250.17

8.63

592.84

32.33

35.02

C103

7534.46

7.00

514.58

19.88

7643.30

7.10

518.78

24.52

37.20

C104

5382.54

5.00

369.45

13.10

5391.23

5.00

376.50

14.73

42.13

C105

7555.61

7.00

522.88

32.73

7557.89

7.00

518.46

39.43

37.31

C106

8609.68

8.00

564.34

45.34

9280.85

8.67

583.46

30.73

34.76

C107

6506.86

6.00

484.18

22.68

6721.72

6.20

473.09

48.63

38.69

C108

6467.17

6.00

440.31

26.86

6471.70

6.00

441.35

30.35

39.13

C109

5369.22

5.00

364.11

5.11

5369.68

5.00

364.40

5.28

40.41

C201

3432.76

3.00

417.42

15.34

3437.15

3.00

421.77

15.38

55.02

C202

2462.64

2.00

433.43

29.20

2941.41

2.50

418.47

22.94

59.62

C203

2514.33

2.00

501.01

13.32

3227.11

2.80

410.43

16.68

59.12

C204

2384.27

2.00

372.21

12.06

2387.06

2.00

372.09

14.97

67.22

C205

2464.49

2.00

460.92

3.57

2745.95

2.30

435.88

10.06

59.35

C206

2460.33

2.00

456.51

3.82

2461.65

2.00

456.44

5.22

64.03

C207

2449.09

2.00

448.27

0.82

2916.79

2.50

412.22

4.57

58.92

C208

2366.15

2.00

364.61

1.54

2366.67

2.00

365.46

1.20

64.69

R101

16232.47

15.00

1191.51

40.96

16251.20

15.00

1193.00

58.20

30.37

R102

14082.91

13.00

1049.13

33.78

14387.81

13.30

1054.86

32.96

32.37

R103

10890.01

10.00

867.52

22.49

10894.64

10.00

873.06

21.58

34.18

R104

7734.53

7.00

722.66

11.88

8341.19

7.60

725.32

15.87

38.77

R105

12007.66

11.00

986.34

21.32

12012.63

11.00

991.21

21.43

32.00

R106

9898.11

9.00

867.16

30.95

9912.10

9.00

874.72

37.38

34.24

R107

8795.05

8.00

775.87

19.18

8806.54

8.00

776.63

29.91

36.06

R108

6659.90

6.00

631.58

28.32

7069.71

6.40

644.84

24.87

40.86

R109

9863.25

9.00

849.35

13.90

9873.92

9.00

850.63

23.29

33.76

R110

8779.34

8.00

762.32

17.02

8795.09

8.00

772.41

22.69

35.21

R111

8751.49

8.00

745.91

5.58

8752.46

8.00

745.84

6.62

35.71

R112

7678.27

7.00

672.68

5.58

7683.13

7.00

672.59

10.54

38.10

87

4.6. CONCLUSIONS

Table4.11:50

Customer

instancesbestsolutionsandaveragefoundby

MPMAper

instance:Continued

BestSolution

Average

Instance

TotalCost

#Vehicles

Distance

Recourse

TotalCost

#Vehicles

Distance

Recourse

Avg.Time(s)

R201

3898.66

3.00

892.30

6.36

4390.74

3.50

880.36

10.38

46.90

R202

3745.80

3.00

739.81

5.99

3751.01

3.00

746.78

4.23

51.04

R203

2726.72

2.00

703.38

23.34

3552.54

2.90

648.06

4.48

53.28

R204

2512.02

2.00

511.28

0.74

2514.82

2.00

513.20

1.62

71.17

R205

2774.60

2.00

768.96

5.64

3748.57

3.00

746.25

2.32

50.60

R206

3642.05

3.00

642.02

0.04

3642.61

3.00

642.54

0.07

52.79

R207

2587.16

2.00

587.16

0.00

2785.60

2.20

585.40

0.20

63.16

R208

2493.69

2.00

493.69

0.00

2495.53

2.00

495.50

0.04

75.37

R209

2694.99

2.00

693.19

1.80

3541.97

2.90

641.44

0.54

53.32

R210

2697.69

2.00

692.05

5.63

3096.76

2.40

691.66

5.11

56.89

R211

2561.32

2.00

561.28

0.03

2565.12

2.00

564.91

0.21

62.85

RC101

12133.54

11.00

1091.58

41.96

12136.53

11.00

1092.18

44.35

32.45

RC102

9947.87

9.00

901.19

46.68

9951.28

9.00

901.64

49.64

34.65

RC103

8883.27

8.00

848.98

34.29

8886.01

8.00

847.77

38.24

36.93

RC104

6797.88

6.00

728.56

69.32

7427.68

6.67

712.08

48.94

40.61

RC105

10989.11

10.00

969.13

19.98

10992.42

10.00

966.46

25.96

33.39

RC106

8935.65

8.00

904.74

30.91

9144.53

8.20

909.40

35.13

35.29

RC107

7827.67

7.00

803.86

23.80

8191.04

7.33

831.58

26.12

36.86

RC108

6738.07

6.00

710.82

27.25

7125.15

6.38

715.42

34.73

39.23

RC201

4825.24

4.00

820.78

4.46

5431.86

4.70

730.02

1.84

40.38

RC202

3836.37

3.00

830.57

5.80

4527.63

3.80

725.19

2.44

45.76

RC203

3640.20

3.00

633.32

6.88

3846.56

3.20

643.27

3.30

49.40

RC204

2498.75

2.00

490.95

7.80

2501.56

2.00

491.83

9.73

70.03

RC205

4684.67

4.00

676.94

7.73

4722.71

4.00

712.35

10.35

42.31

RC206

2914.94

2.00

856.71

58.23

3594.25

2.90

688.42

5.82

51.52

RC207

3617.36

3.00

614.62

2.74

3623.11

3.00

620.07

3.04

51.93

RC208

2531.56

2.00

525.87

5.68

2836.99

2.30

530.66

6.33

60.63

Total

340845.92

302.00

37763.70

1082.22

354543.96

316.27

37254.92

1022.37

2586.86

88


Table4.12:100Customer


MPMAper

instance

BestSolution

Average

Instance

TotalCost

#Vehicles

Distance

Recourse

TotalCost

#Vehicles

Distance

Recourse

Avg.Time(s)

C101

21519.83

20.00

1497.54

22.29

21521.67

20.00

1492.13

29.54

122.91

C102

18301.69

17.00

1330.67

61.02

18408.91

17.00

1355.34

53.58

126.97

C103

15227.97

14.00

1193.63

34.33

15810.35

14.56

1200.19

54.61

135.82

C104

11986.00

11.00

961.25

24.75

12746.69

11.71

1005.74

26.66

139.63

C105

16292.06

15.00

1199.72

92.35

16635.03

15.33

1210.43

91.27

132.15

C106

16259.28

15.00

1212.13

47.15

16661.56

15.38

1239.61

46.95

128.98

C107

14171.11

13.00

1095.82

75.29

14299.95

13.11

1100.08

88.76

132.54

C108

13069.71

12.00

956.64

113.07

13084.16

12.00

986.37

97.80

134.11

C109

10846.15

10.00

836.91

9.24

10847.90

10.00

837.01

10.89

136.14

C201

5821.75

5.00

717.08

104.67

6034.01

5.20

757.33

76.68

182.17

C202

5755.96

5.00

718.78

37.18

5766.79

5.00

715.54

51.25

182.42

C203

4715.41

4.00

679.56

35.85

4720.62

4.00

683.64

36.98

188.48

C204

4667.72

4.00

654.73

12.99

4673.33

4.00

656.36

16.97

195.69

C205

4688.49

4.00

671.74

16.75

4692.55

4.00

667.81

24.74

188.10

C206

4666.89

4.00

656.51

10.38

4668.69

4.00

658.98

9.71

193.42

C207

4655.02

4.00

649.24

5.78

4656.70

4.00

644.04

12.66

187.30

C208

4632.91

4.00

625.22

7.69

4646.78

4.00

641.82

4.96

192.96

R101

28960.67

27.00

1934.28

26.38

29874.72

27.90

1936.55

38.17

117.27

R102

25744.05

24.00

1725.49

18.56

25771.42

24.00

1746.29

25.13

120.44

R103

19409.84

18.00

1371.98

37.86

19524.64

18.10

1379.08

45.55

124.89

R104

14154.45

13.00

1128.20

26.25

14581.18

13.40

1147.77

33.41

134.78

R105

19602.65

18.00

1527.25

75.40

19950.73

18.33

1530.22

87.18

121.79

R106

17420.11

16.00

1369.40

50.71

17640.71

16.20

1378.77

61.94

126.33

R107

14258.45

13.00

1194.59

63.86

15033.80

13.78

1212.86

43.16

130.68

R108

12068.34

11.00

1040.52

27.82

12954.66

11.90

1036.49

18.16

139.89

R109

15308.71

14.00

1270.43

38.28

16318.27

15.00

1283.37

34.90

126.05

R110

14239.67

13.00

1193.42

46.25

15020.88

13.80

1194.13

26.75

128.51

R111

14184.60

13.00

1151.82

32.78

14642.54

13.44

1165.17

32.92

130.73

R112

13035.89

12.00

1029.60

6.29

13042.87

12.00

1034.17

8.70

135.19

89

4.6. CONCLUSIONS

Table4.12:100Customer


MPMAper

instance:Continued

BestSolution

Average

Instance

TotalCost

#Vehicles

Distance

Recourse

TotalCost

#Vehicles

Distance

Recourse

Avg.Time(s)

R201

6333.61

5.00

1306.96

26.65

7657.52

6.40

1239.12

18.40

150.98

R202

6090.63

5.00

1079.22

11.41

6628.18

5.50

1112.76

15.42

162.23

R203

4927.94

4.00

916.71

11.23

5589.65

4.67

914.40

8.58

184.70

R204

3792.57

3.00

792.53

0.03

4674.21

3.90

771.75

2.46

212.88

R205

6011.41

5.00

1011.05

0.36

6238.60

5.20

1034.17

4.43

166.24

R206

3981.86

3.00

981.50

0.35

4834.63

3.90

931.97

2.66

190.35

R207

3835.66

3.00

835.66

0.00

4392.29

3.56

834.28

2.45

206.23

R208

3715.32

3.00

710.28

5.04

3722.45

3.00

718.06

4.39

236.11

R209

4926.19

4.00

918.85

7.34

5713.55

4.80

910.65

2.90

171.43

R210

4993.53

4.00

992.32

1.21

5750.90

4.78

967.41

5.72

173.25

R211

4778.41

4.00

778.20

0.20

4793.44

4.00

790.98

2.46

192.24

RC101

20920.01

19.00

1819.27

100.73

21592.35

19.67

1825.71

99.97

121.58

RC102

18714.96

17.00

1660.41

54.55

18949.66

17.22

1665.70

61.74

124.16

RC103

15475.45

14.00

1427.97

47.48

15490.04

14.00

1441.47

48.57

128.98

RC104

13300.61

12.00

1260.09

40.51

13309.50

12.00

1264.60

44.91

137.39

RC105

18783.43

17.00

1704.47

78.96

19486.15

17.70

1718.26

67.89

122.96

RC106

16627.30

15.00

1597.60

29.79

17481.11

15.89

1559.34

32.88

125.24

RC107

15433.93

14.00

1387.96

45.97

15641.16

14.20

1411.37

29.79

128.74

RC108

14321.21

13.00

1309.87

11.34

14591.50

13.25

1313.79

27.72

132.16

RC201

7402.04

6.00

1387.02

15.02

8256.29

6.90

1344.75

11.55

150.34

RC202

6184.45

5.00

1181.73

2.72

6735.37

5.50

1226.10

9.26

163.14

RC203

5098.22

4.00

1096.44

1.78

6122.41

5.11

1005.18

6.12

172.14

RC204

4801.08

4.00

800.55

0.53

4810.54

4.00

808.65

1.89

204.69

RC205

7245.54

6.00

1229.47

16.07

7894.78

6.67

1214.48

13.63

152.72

RC206

5229.15

4.00

1195.54

33.60

6146.14

5.00

1133.96

12.19

166.81

RC207

5071.58

4.00

1067.12

4.11

5825.83

4.78

1036.10

11.95

171.22

RC208

4879.98

4.00

866.94

13.04

5405.15

4.56

844.39

5.20

183.62

Total

618334.60

554.00

62829.36

1505.24

641965.52

577.28

62936.72

1745.07

8668.88

90

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94

Chapter 5

Wind farms maintenance

Let us be careful! If in our climates, the industry can avoid to use the direct solar heat,

necessarily, a day will come when, for the lack of fuel, it will be forced to return to the work of

other natural agents. That the deposits of coal and petroleum will still supply for a long time their

caloric power, we do not doubt it. But these deposits will undoubtedly exhaust themselves...(about

coal energy) One can not refrain from concluding that it is prudent and wise not to fall asleep with

respect to this in a misleading security

Augustine Mouchot

5.1 Introduction

Wind energy is one of the most important sources of renewable energies. According to the Global

Wind Energy Council (GWEC) [17], the year 2016 nished with an installed capacity of nearly 487

GW, representing a 12.6% growth when compared to 2015. In 2016 wind power also achieved to

account for almost a quarter of the worldwide renewable energies capacity (including hydropower)

[65]. Moreover, the GWEC projections show that wind energy installed capacity will increase by

almost 70% in the following ve years reaching nearly 830 gigawatts (GW). Despite the rapid growth

of oshore installations [17] onshore still represents nearly the 97.2% of the total wind worldwide

capacity [16] in 2015.

In this context, Operation and Maintenance (O&M) activities represent an important cost [77, 71,

23, 64] in wind farm projects representing as much as 25% to 30% of the total energy production costs

(Ding et al. [23]). In the oshore context Shaee [72] and Raknes et al. [64], state that O&M costs

range from 25-33% of the total life cycle cost. According to Scheu et al. [71], El-Thalji and Liyanage

[27], and Racknes et al. [64], O&M costs include transportation costs, technician salaries and cost

of repair actions and spare parts, as well as loss of revenue caused by production stops. Therefore,

optimizing the O&M activities is a mandatory task to improve the competitiveness of prices and

attractiveness of the wind energy projects.

This chapter is divided in two parts. The rst one starts by making a review about maintenance

activities in wind farms with a focus on operational decision level of analysis (section 5.2). Then,

section 5.4 introduces a multi-objective maintenance scheduling problem in the context of onshore

wind farms. The multicriteria approach is motivated by the presence of dierent stakeholders with

conicting objectives in this kind of applications. The second part explores the strategic level of

maintenance activities in wind farms in section 5.5. Then, in section 5.6 the eects of the operational

scheduling problem within a long horizon and its relation with the maintenance strategy are studied.

The chapter ends in section 5.7 with concluding remarks on the found results and gives some research

clues.

95

5.2. REVIEW ON OPERATIONAL MAINTENANCE ACTIVITIES

5.2 Review on operational maintenance activities

Wind turbines are the systems that allow to transform the kinetic energy in the air into electric energy

[50, 10]. Although wind turbines can be used as a stand alone system to provide the energy (or part of

it) for particular houses, farms etc, most of the capacity is installed in wind farms to produce energy in

an industrial scale [50]. Wind farms are the collection of several turbines working on grids to produce

energy.

Maintenance activities are performed to keep systems working, and to x and prevent possible

failures [73]. The fact that wind farm projects are devised for long periods of time [60] (20 to 40 years)

makes the maintenance and repair activities a central issue to keep the turbines working. Maintenance

actions can include inspection, changes of consumables (greasing, lubrication, oil lters), oil sampling,

re-tightening of the bolts, repair, overhaul, and replacement of parts [5]. According to Manwell [50]

the scheduled and unscheduled maintenance reduce the availability1 of wind turbines. Furthermore,

maintenance activities have an inuence in the failure rates of wind turbines, since poorly maintained

turbines can have higher failure rates [15]. Moreover, both maintenance activities and failures imply

losses in production due to downtime [71, 27] and generate important costs (spare parts, technicians,

etc.).

The interaction of multiple actors involved in wind farms has eects on how maintenance activities

are carried. Markard and Petersen [51] recognize ve stakeholders on wind farms projects, namely:

turbine manufacturer, project developer, investor, operator, and load management and power distrib-

utor. The exact conguration of the stakeholders roles depends on each project. Still, dierent actors

can play multiple roles in the project. For instance, an investor can also manage the operation of the

wind farm. Situations in which roles are held by dierent entities might derive in cases where, one

actor pursuing its objectives harms other actors targets.

The challenge raised by improving the maintenance of wind farms has attracted a large number

of academic communities with dierent perspectives on how to achieve this goal. Some studies have

focused on the materials used for the components (particularly the blades to avoid failures) of the wind

turbines [9, 43], wind farm topology [44], fault diagnostics techniques [26, 29, 37], prognostics [81, 45,

13, 85, 82, 12, 91], scheduling of activities and resources [8, 40, 79, 19, 35, 31, 30], or maintenance

strategy selection [81, 14, 11, 62, 68, 25]. These studies aim to improve the maintenance activities

according to criteria in dierent moments of the wind farms life-cycle. Therefore, a powerful way to

classify the dierent scientic contributions in this eld is to consider which decision-making level is

involved.

As in production management, maintenance problems can be categorized in dierent decision levels

according to the time horizon involved. Shaee [72] uses a three-echelon classication of decision-

making with strategic, tactical and operational echelons as its levels. Strategic decisions aect in the

long-term the maintenance activities. In this echelon, one can nd aspects such as the location [67, 89],

layout [2, 32, 88], or the type of maintenance strategy [42, 23, 11, 76]. Tactical level is related to mid-

term decisions which can vary from year to year, e.g. spare parts inventory management [36, 82].

Operational echelon deals with short-term maintenance decisions. Within this, one can include the

scheduling of resources to perform the required maintenance tasks [8, 63, 79, 19, 35, 31, 64].

The attention given by the Operations Research community to the Operational level (short-term

maintenance decisions) focused mainly on the maintenance scheduling problem. As pointed out by

Kovàcs et al. [40], maintenance scheduling plays a major role in the turbines availability and costs.

The rst study presented in this chapter is dedicated to such types of operational problems and starts

with a review of studies dealing with this decision level.

1Refers to the percentage of time that a wind turbine is available to produce power

96

CHAPTER 5. WIND FARMS MAINTENANCE

5.3 Operational maintenance level

Operational decision level makes reference to day-to-day decisions. Within this level, maintenance

planning refers to the problems in which maintenance tasks are scheduled to be performed on a xed

short length horizon. The detailed scheduling may include constrained resources, such as technicians,

vessels, vehicles, etc.

In this context, it is usually assumed that the maintenance actions are known prior to their

execution [8, 40, 31]. Additional information on the tasks such as their release dates and due dates

[72] are also known. Since the horizon is limited to very few days, it is a common practice to assume

that stochastic parameters such as wind speed or the waves height can be perfectly predicted. In cases

where parameters can change, a roll over approach is used, re-optimizing each current day scheduling

with the new predictions of stochastic parameters and the realizations over previous periods.

Table 5.1 shows a summary of the principal works dealing with the maintenance scheduling prob-

lem. It focuses on the types of objectives used in the dierent literature works. The column costs

makes reference to objectives associated with costs, while the energy one refers to objectives associated

with the energy production. The following conventions are used for the table: Onshore (ONS), o-

shore (OFF), penalties (PEN), transportation (TRAN), technicians wage (TWAG), technicians extra

wage (EWAG), production loss (PL), energy produced (EP), and revenue (REV).

From table 5.1 one can see that few onshore works have relied on energy objectives. In this context,

Kovàcs et al. [40] assume that crews formed by two technicians are used to perform maintenance tasks.

A Mixed Integer Linear Problem (MILP) on a rolling horizon is designed to minimize the total loss of

production due to downtime of the turbines, or when they are still working but in a degraded state.

Two types of degradations are considered, general and peak, the rst one diminishes the power output

by a percentage in any operation condition, while the second one reduces the production during high

speed winds but not on lower speed ones. Albeit no detailed computational results are given, the

authors claim to solve instances with up to 50 maintenance tasks, 4 teams, and 7 wind farms. More

recently Froger et al. [31] have also considered the onshore wind farm maintenance scheduling. The

authors dealt with multiple technician skills, dierent types of execution modes for the tasks, as well

as dierent farm locations. Moreover, the objective in the proposed model is to maximize electricity

production over a short-time horizon. It must be noticed that maximizing the revenue and the energy

produced are only equivalent when the energy prices are constant. Since this is the case in [31, 30] both

objectives are selected. To solve the problem, two formulations based on Integer Linear Programming

(ILP) are proposed. A constraint programming large neighborhood approach is devised to solve the

problem. The same authors worked a second paper dealing with this previous problem. They tackled

it in this last study [30] with a branch-and-check approach. This method can consistently produce

optimal or near optimal solution for instances with up to 80 tasks, several modes, skills, and farms

locations.

97

5.3. OPERATIONAL MAINTENANCE LEVEL

Table5.1:

Summaryof

windfarm

smaintenance

schedulingworks,objectives,andtypes

ofmodels

Con-

text

Costs

Energy

Reference

ONS

OFF

PEN

TRAN

TWAG

EWAG

PL

EP

REV

Typeof

model/SolutionApproach

Results

Com

ments

Kovacset

al.[40]

xx

MILP

Optimal

Twotypes

ofwindturbines

degradations

Froger

etal.[31]

xx

xMILP/ConstraintProgram

ming

Optimal

Tasks

modes

denetheirduration

Froger

etal.[30]

xx

xMILP/Brand-and-check

Optimal

Instanceswithup

to80

tasks,severalmodes,skillsandlocationssolved.

Besnard

etal.[8]

xx

xx

MILP/Simulation

Approximated

Transportation

cost

reducedthanks

toopportunisticmaintenance

Kennedy

etal.[39]

xx

xx

MILP/G

eneticAlgorithm

-Simulation

Approximated

Costscanbereduceddelaying

thescheduleof

preventive

tasks

Stålhane

etal.[79]

xx

xx

MILP

Optimal

Noweather

modeling.

Daiet

al.[19]

xx

xx

MILP

Optimal

Limited

toinstanceswithless

than

8tasks

Rakneset

al.[64]

xx

xx

MILP-Simulation

Optimal/A

pproximated

Simulationisused

toaddressadynamicversionof

theproblem

Iraw

anet

al.[35]

xx

xx

xMILP

Optimal

Use

ofmultipleO&M

bases

98


In the oshore context minimizing the transportation costs is omnipresent. This can be explained

by the high costs associated to vessels and helicopters. Besnard et al. [8] propose a stochastic

optimization model for wind farms maintenance planning. Stochasticity is incorporated through

scenarios in which the wind, waves and production take dierent values. The model is based on

a rolling horizon which considers opportunistic and corrective maintenance actions. The horizon

planning is constituted of seven days and is discretized in steps of one day. Scenarios are used to

characterize the expected hourly production power, the wind speed and wave height; and the objective

is to minimize production losses, transportation costs, and extra hours penalties. A similar model is

proposed by Kennedy et al. [39]. The authors consider only one vessel and one maintenance team.

Using a Genetic Algorithm, the authors optimize the order and time at which tasks must be carried

out. The objective is to minimize the production losses, transportation and crew costs. Also, it is

shown that a signicant saving (from 13% to 21%) can be achieved when the maintenance schedule is

optimized instead of repairing the turbines as fast as weather conditions allow access to the turbines.

Problems involving more than one vessel are the most common type in oshore context. Two

models based on arc-ow and path-ow formulations are presented in Stålhane et al. [79] for the

routing and scheduling of vessels that perform maintenance tasks at oshore wind farms. The presen-

ted models consider as objective the minimization of transportation, downtime, and penalties costs.

Instances considering a workday and at most eight tasks and ve vessels are solved to optimality.

A similar problem is tackled in Dai et al. [19]. The authors use a four index MILP where vessels

daily availability depends on the type of the vessel and the weather conditions. Numerical results are

provided for instances with eight turbines and an horizon of three days, aiming to minimize the costs

and production loss. A combined vessel routing and maintenance scheduling problem is tackled by

Raknes et al. [64]. The model consider multiple wind farms which are managed by the same O&M

enterprise. Dierent types of vessels are included, some of which can stay oshore for longer periods,

and other need to return to the depot at the end of each shift. The authors propose a large Mixed

Integer model to minimize the sum of production loss, transportation costs, and penalty costs related

to tasks that are not carried out. Furthermore, a dynamic version of the problem is considered dealing

with new task arrivals and updated weather forecasts. The results show the importance of evaluating

the strategies by simulation rather than considering static models in dynamic contexts.

The maintenance routing and scheduling at oshore wind farms is tackled by Irawan et al. [35]

thanks to a model that considers multiple vessels, periods (days), bases, and wind farms. Weather

conditions are considered by dening maximum working hours for a vessel. A Dantzig-Wolfe decom-

position is used, where for each vessel and each period, all the feasible scheduled routes (customers

visits) are generated. The time horizon varies from three to seven days, with three types of technicians.

The objective function contains the cost of technicians, transports, and penalty costs when turbines

are visited after a given deadline. Results are presented for literature instances with eight turbines

and three periods. The authors also show that O&M with multiple bases and attending multiple wind

farms can produce savings of around 12% when compared to solutions with single O&M base and wind

farm pairs. Yet, it is interesting that Irawan et al. [35] nd that costs are mostly composed by crew

costs. Certainly, that behavior could be expected to appear in the onshore case where technicians are

the most costly resource (apart from spare parts).

From the explored literature almost all the works have used exact methods to solve the problems.

This presents the drawback of ignoring the stochastic parameters to have tractable models. Con-

sequently, the works assume that weather conditions, or the time to perform a task are known in

advance. Moreover, no work has addressed the multi-objective case in which both minimization of the

costs, and the maximization of energy related objectives are tackled at the same time.

99

5.4. A MULTI-OBJECTIVE APPROACH TO THE MAINTENANCE SCHEDULING PROBLEM

5.4 A multi-objective approach to the maintenance scheduling

problem

Based on the considerations presented in sections 5.2, in this section a model and a resolution approach

are presented to deal with the multi-objective maintenance scheduling problem for onshore wind

farms. It is assumed that O&M operator and investors are the two parts involved in the maintenance

scheduling. O&M operator is paid to perform the maintenance activities, and it is assumed that he

seeks to minimize its costs. Meanwhile, the investors expect to maximize the energy production. The

problem is forthwith formally dened.

5.4.1 Problem Denition

A set of J turbines indexed by j are considered. Turbines might be distributed among dierent wind

farms but it is assumed that they are close enough to be reached in despicable times. Furthermore,

turbines might require more than one type of maintenance. Each task i ∈ I represents a maintenanceactivity. It is assumed that the set of tasks is specied prior to solving the problem. Also, each task

is associated with a turbine j ∈ J , and Ij dene the subset of tasks to be performed in turbine j.

To execute each task i a number of χis technicians with the skill s are necessary. Actually, each

task requires one or dierent skills held by technicians to perform the maintenance, e.g. mechanical,

electrical, electro-mechanical, etc. S denes the whole set of skills. Besides, an execution time βicharacterizes every task i. Time βi corresponds to the time elapsed from stopping the turbine until

it is restarted and veried again. It is assumes that all maintenance activities in I requires that the

associated turbine is stopped during the task execution. Additionally, every started task is performed

until nished, and all tasks must be performed during the planning horizon.

Furthermore, every task is associated to a time window [ei, li], where the maintenance should take

place. The opening time window is a hard constraint, so the task must start at ei or latter. The

enforcement of this condition accounts for the necessity of spare parts or any special equipment (e.g.

cranes) to perform the task. Other spare parts or consumables are assumed to be available at ei. The

time window closure is a soft constraint; therefore, the task should preferably be nished at most at

li. In the case it nishes later, a penalization cost α is considered and is proportional to the delay.

Usually, periodic tasks can be performed all along the planning horizon, while corrective maintenance

can be constrained for special equipment or special considerations based on their impacts.

Time is discretized in equally length periods t ([19, 31, 35]). The set of all time periods is dened

by T . Every time period stands for the same amount of time, e.g. one minute, two hours, etc. During

each time period t, a turbine j generates an amount of energy or utility per production denoted as

Θjt. Moreover, the additional representation of time as done by Froger et al. [31] is also used. In

this one, a set D denes the set of days in the planning horizon. Every day d ∈ D has a number

of workable t periods called τd. Furthermore, the time intervals within a day are divided as normal

working and extra working periods. The subset of time periods representing extra working periods is

dened as Te. At the end of each working day in the horizon plan, turbines continue to produce an

amount of energy or utility per production Υjd ∀ j ∈ J until the next day. Besides, for safety reasons,

maintenance task can be only carried out during certain periods of time when weather allows it, e.g.

wind speeds are below a given value. To address this fact, the parameter ρit takes value one if the

safety conditions are met and zero otherwise. If a task i is being performed during time t − 1 and

ρit = 0, the technicians must stop until the safety conditions are met again.

A limited set of technicians is dened by P . Technicians travel to the turbines to perform their

assigned maintenance tasks. All technicians perceive a xed salary wp, which is linearly dependent on

the amount of skills they possess. Furthermore, a technician receives an extra wage ewp for each extra

working hour. It is assumed that a worker with a higher salary will enjoy a higher extra payment, i.e.

100


wp > wp′ then ewp > ewp′∀p, p′ ∈ P . Besides, the parameter πps takes value one if the technician

p has the skill s and zero otherwise. Let us also dene ls as the minimum wage among the workers

p ∈ P | πps = 1 ∀s ∈ S. That is, the less costly wage for a technician with the skill s. When a

technician is assigned to perform a task, he must be present at the turbine until the task is complete.

Moreover, technicians can only work at one task in each time period.

It is considered that O&M operator costs comprise the spare parts, the special equipment to

perform tasks and the technician's wages. Moreover, three main components are examined. The rst

one is the dierence of wages perceived by technicians assigned to tasks that could be performed by

less costly ones. The second one, is the extra hour wages paid to technicians, and the third one, is a

penalization due to maintenance tasks outside the time window. Spare parts and special equipment

costs are disregarded since they are assumed to be unavoidable by the O&M operator. Hence, the

objective is to minimize the costs of the O&M operator while maximizing the amount/prot of energy

production.

5.4.2 Mathematical model

The problem described in section 5.4.1 is now formalized as an Integer Linear Problem (ILP). The

following variables are thus dened:

xit :

1 if task i is scheduled to begin at time t

0 otherwise

zit :

1 if task i is scheduled at time t

0 otherwise

eit :

1 if task i is nished at time t

0 otherwise

ypi :

1 if resource p is assigned to task i

0 otherwise

vpti :

1 if resource p is scheduled at time t to task i

0 otherwise

uid :

1 if task i is scheduled in day d

0 otherwise

γjt :

1 if turbine j is able to produce energy at period t

0 otherwise

ηjd :

1 if turbine j is able to produce at the end of day d

0 otherwise

ζi : The number of days task i is delayed with respect to bi

Using the abovementioned variables and the characteristics described in section 5.4.1, the ILP is

dened as follows:

maxZ1 =∑j∈J

∑t∈T

Θjtγjt +∑j∈J

∑d∈D

Υjdηjd (5.1)

minZ2 =∑i∈I

∑p∈P

∑s∈S

κiypi(wp − ls) +∑p∈P

∑t∈Te

∑i∈I

ewpvpti +∑i∈I

αζi

∑t∈T

xit = 1 ∀ i ∈ I (5.2)

101


∑t∈T

eit = 1 ∀ i ∈ I (5.3)

zit = xit + xit−1 − eit−1 ∀ i ∈ I, t ∈ T (5.4)

xi0 + ei0 + zi0 = 0 ∀i ∈ I (5.5)∑t∈T

πpsypi ≥ χis ∀i ∈ I, s ∈ S (5.6)

∑t∈T

zitρit=βi ∀i ∈ I (5.7)

dτd∑t=1+(d−1)τd

zit ≤ τduid ∀ i ∈ I, d ∈ D (5.8)

uid = 0 ∀ i ∈ I, d ∈ D | d < ei (5.9)

ζi ≥ duid − li ∀ i ∈ I, d ∈ D (5.10)

zit + ypi ≤ 1 + vpti ∀ p ∈ P. t ∈ T, i ∈ I (5.11)

vpti ≤ zit ∀ p ∈ P. t ∈ T, i ∈ I (5.12)∑t∈T

vpit ≤Mypi ∀ p ∈ P, i ∈ I (5.13)

γjt ≤ (1− zit) ∀j ∈ J, t ∈ T, i ∈ Ij (5.14)

ηjd ≤ 2− (uid + uid+1) ∀ j ∈ J, d ∈ D, i ∈ Ij (5.15)

xit, zit, eit ∈ 0, 1 ∀i ∈ I, t ∈ T (5.16)

ypi ∈ 0, 1 ∀ p ∈ P, i ∈ I (5.17)

vpti ∈ 0.1 ∀ p ∈ P, t ∈ T, i ∈ I (5.18)

uid ∈ 0, 1 ∀ i ∈ I. d ∈ D (5.19)

γjt ∈ 0, 1 ∀ j ∈ J. t ∈ T (5.20)

ηjd ∈ 0, 1 ∀ j ∈ J. d ∈ D (5.21)

ζi ∈ Z+ ∪ 0 ∀ i ∈ I (5.22)

The objectives in equation (5.1) state that the amount/utility of energy produced is to be maxim-

ized while cost must be minimized2. This last is composed by three parts, the total dierence between

the salaries of technicians assigned to a task and the minimum wage of a technician oering the same

skill. The second part accounts for the extra time periods wages, and the third one is the penalization

for not nishing the task within the time window. Constraints (5.2) and (5.3) ensure that all tasks are

started and nished within the time horizon. The consistance between the start, the execution, and

the end of a task is guaranteed by constraints (5.4), which ensure that tasks are executed until nished

when they have been started. Constraint (5.5) state the initial conditions by considering that no task

starts, nishes, or is assigned during time period zero. The number of technicians with the required

skills to perform each task is imposed in constraints (5.6). These equations have a sense of greater

or equal for some special cases, e.g. consider a task i which requires two skills s1, s2 with a number

of technicians of two and one respectively. If the equation is set to equality two technicians p1, p2both with skills s1, s2 cannot be simultaneously assigned to the task since it will violate the equation

for skill s2. Still, this scenario where p1 and p2 are assigned to perform task i is valid and therefore

the constraint sense is kept. Constraints (5.7) guarantee that the amount of time periods spent on

2Note that minZ2 is equivalent to max−Z2

102


a task will be sucient to nish it, the parameter ρit ensures that the weather allows to execute

the task during these periods. The coupling between tasks executed during a day and the scheduled

time periods is considered in constraint (5.8). Time windows are dened in constraints (5.9) and the

(5.10), the rst ones safeguard that tasks must start after their associated opening time window, and

the second ones, keep track of the number of days a task is delayed with respect to its time window

closure. Equations (5.11) to (5.13) couple the technicians assignment to a task with its execution

times. The parameter M in constraint (5.13) is used as a big M value. Moreover, constraints (5.14)

and (5.15) determine if a turbine can produce energy at normal time periods, and at the end of the

days respectively. Finally, equations (5.16) to (5.22) x the decision variables.

5.4.3 The Epsilon constraint approach

Multi-Objective Optimization Problems (MOOP) are dened as problems where at least two, often

conicting objectives are to be optimized at the same time. Multi-objective optimization methods

dier from single objective ones in which only one solution is found to minimize/maximize the criterion

at hand. In fact, in MOOP more than one solution can be found, especially when the method used

deals with Pareto-optimization. Thus, solving MOOP is a process of nding the ensemble of solutions

called Pareto ecient or non-dominated. These solutions are those for which no objective can be

improved without worsening at least another objective. The whole set of non-dominated solutions is

called Pareto Front [48].

Several approaches are discussed in the literature to solve MOOP, e.g. weighted global criterion

[52], goal programming [80], epsilon constraints [55], etc. To address the model the last method is

selected. This approach is employed to construct an ensemble of Pareto Ecient Solutions (PES).

In general terms, the epsilon constraint works by iteratively solving single objective problems. Con-

sider the following multi-objective problem maxZ (x) = (z1 (x) , z2 (x) , . . . , zn (x)) , x ∈ Ω. Each

zi(x) ∀ i = 1 . . . n represents an objective, and the condition x ∈ Ω stands for the feasible set of

solutions. Epsilon constraint method solves a group of problems max zj (x) | zi (x) ≥ ∆i ∀ i =

1..n ∧ i 6= j, x ∈ Ω by changing the values of ∆i. Each solution found is ecient and kept devising

(partially) the Pareto Front. Although simpler approaches can be used such as the Weighting Method

in which the dierent objectives are reduced to a simple objective using weights for each compon-

ent, the epsilon constraint is preferred for the following reasons. First, epsilon constraint can nd

non-supported solutions, i.e. solutions not in the convex envelope of the Pareto Front. Second, there

is no need to scale the objectives. Third, epsilon constraint iterations can be coded so new ecient

points are found at each iteration, this can avoid iterations which need considerable amounts of time

for solving the ILPs.

Algorithm 5 Epsilon Constraints adapted method for maximizing Z = (Z1,−Z2)

Require: numSteps1: Solutions← 2: z1∗, z2. ← maxZ1

3: z1∗, z2∗ ← max−Z2| (Z1 = z1∗)

4: Solutions← Solutions ∪ z1∗, z2∗5: z1., z2∗∗ ← max−Z2

6: stepSize← (z2∗−z2∗∗)

numSteps

7: i← 18: while (z2

∗ − i · stepSize) > z2∗∗ do

9: z1∗, z2. ← maxZ1| (Z2 ≤ z2∗ − i · stepSize)10: z1∗, z2∗∗∗ ← max−Z2| (Z1 = z1

∗)11: Solutions← Solutions ∪ z1∗, z2∗∗∗12: i← i+ 113: end while

14: Sort and check(Solutions)

Algorithm 5 shows the general steps of the used implementation of the epsilon constraint method.

103


The procedures maxZ1 and max−Z2 solve the ILP described in section 5.4.2 maximizing the produced

energy and minimizing the costs respectively. Moreover, the procedures retrieve both the values of

Z1 and Z2 given the solution of the ILP. Line 1 starts by initializing an empty array in which the

solutions will be kept. In line 2, the quantity/utility of the energy production is maximized, while line

3 minimizes the costs with the additional constraint that energy production must match the one found

in line 2. The solution is then saved in the proper array of ecient solutions. It shall be noticed that

it is necessary to solve both problems (energy and costs), to retrieve an ecient point. Preliminary

tests show that solving only the problem maxZ1 gives suboptimal solutions in terms of costs. The

algorithm continues solving the problem max− Z2 in line 5, this is done to dene the interval within

the objective Z2 is comprised, i.e. [z2∗, z2

∗∗]. In line 6, a step size is calculated to use it for the epsilon

constraint method. This value depends on the minimum and maximum value attained by Z2 as well

as a parameter called numSteps. This last allows to control the number of iterations performed during

the loop. Higher values for the number of steps permit to better determine the Pareto Front at the

expense of higher computational times. Nevertheless, the value for this parameter does not guarantee

that a dierent ecient point will be found for each possible step value, it works as an upper bound.

Between lines 8 and 13 the main loop iteratively solves the ILPs for energy and costs using the proper

constraints, while saving the ecient solutions. Finally, in line 14 the ensemble of solutions is sorted

and solutions are checked for dominance. If solutions are optimal, they are ecient, nevertheless, if

optimality is not guarantee, solutions are only potentially ecient.

5.4.4 Results

To test the model, the instances proposed by Froger et al. [31] are used. This testbed is composed

by 160 instances grouped in families (5 instances per family) described as a_b_c_d_e where a, b,

c, d, and e refer to the number of time periods in the planning horizon, the number of periods per

day, the number of skills considered, the number of tasks and technician-to-work ratio, respectively.

Accordingly each core characteristics can have dierent values such as: time horizon lengths (10, 20

or 40), time periods per day (2 or 4), number of tasks (20, 40 or 80), number of skills (1 or 3), and

the technician-to-work-ratio (A and B). Among these, only the subset of 40 instances with at most 20

tasks and 20-time periods are used. Moreover, for instances with two-time periods per day additionally

period to stand for extra hours is considered. This extra time period has an implicit duration of four

hours. The same procedure is performed for instances with four-time periods, however for these last,

two extra time periods are added per day (each one representing 2 hours). To assign an energy utility

to these time periods, the following procedure is employed. The original Υjd for each day and turbine

is taken and divide by 16 (number of hours between workable days). Then, extra periods use this

coecient multiplied by the number of hours they stand for, to determine the amount/utility of energy

production. It is assumed that weather conditions are safe to perform maintenance tasks during extra

time periods, i.e. ρit = 1. Additionally, since original instances consider multiple modes or ways to

perform each task, a single mode is randomly selected. This picked mode includes the information on

the number of technician per skill required to perform the task, and its duration (number of periods).

Besides, it is assumed that transport times between every pair of turbines are negligible. Tasks time

windows are not considered in Froger et al. (2017) instances, thus ei = 0, li = |D| ∀i ∈ I.To constraint the computational eort required to solve the problems the amount of time that

epsilon constraint method expends on the ILPs is limited. For problems in lines 2, 3 and, 5 in

Algorithm 1, the time limit is set to 2000 seconds. ILPs in lines 9 and 10 are limited to 500 seconds.

The rst problems are led to run for more time since they are used to create the interval in which

epsilon constraints will be dened. Moreover, the parameter associated to the number of steps is set

to 50. This value guarantees a good trade-o between the approximation of the Pareto Front and the

running times. Both time limits and the number of steps were selected after several preliminary tests.

Since times are constrained, optimal solutions are not guaranteed, therefore, the GAP metric for each

104


89000

89500

90000

90500

91000

91500

92000

92500

93000

100 150 200 250 300 350

Energy

Costs

Figure 5.1: Approximate Pareto Front for Froger et al. [31] instance 10_2_1_20_B_5

Table 5.2: Epsilon Constraints summary results for Froger et al. [31] instancesFamily Avg. Time (h) Avg. Solutions Avg. E. Solutions Avg. Gap C Avg. Gap E Avg. CNOP Avg. ENOP

10_2_1_20_A 1.85 12.2 4.4 4.7% 0.7% 2 7.210_2_1_20_B 1.22 10.2 4 0% 0.3% 0 610_2_3_20_A 1.96 10.8 5 5.7% 0.6% 2 4.610_2_3_20_B 2.39 13.4 6.2 9.5% 0.3% 1.2 620_4_1_20_A 2.81 12.8 4.2 16.2% 0.5% 4.6 7.620_4_1_20_B 3.41 15.2 3 12.1% 0.1% 5.2 10.820_4_3_20_A 2.65 8.4 1 28.4% 1.6% 5.0 5.120_4_3_20_B 2.70 7.8 1 34.6% 1.4% 5.4 4.6

Total 2.37 11.35 3.6 13.9% 0.6% 3.2 6.5

single objective problem is saved.

All runs are conducted on a Dell Latitude E6420 personal computer with Intelr CoreTM i7-

2760QM @2.4 GHz, running Windows 7 Professional 64 bits. The algorithms were coded on Java and

compiled with JavaSE-1.8_45, with maximum allocated memory of 1 Gb. To solve linear problems,

the Java interface with Gurobi 7 (2017) optimizer is used.

Figure 5.1 presents an example of the solutions found. The instance originally contains 10 period

times but this value was increased to consider extra time periods. Three periods are dened for

each day and only one skill is considered. Moreover, the instance contains 20 tasks with a regular

technician-to-work-ratio, that is, technicians can perform all the tasks during the planning horizon.

16 non-dominated points (solutions) are found, although only two are proven optimal, thus ecient.

Moreover, 12 points present a small gap in the energy component. Still, this gap is on average of

0.03%. The other two points present an average gap of 0.04% in the cost component. The graphic

shows an overall concave behavior, displaying a smaller energy utility/production when costs are small

and bigger production at the expense of higher costs. However, it shall be noticed that not all points

rely on the convex hull, e.g. point nine, counting from the left-below part of the graph, showing hence

the conicting nature of the objectives. All the tested instances present a similar behavior in terms of

shape. Furthermore, it shall be noticed that for gure 5.1, an increase of 115% in the minimum cost

can have an impact of almost 4% in the energy utility/production.

Table 5.2 summarizes the results found for testbed instances. For each family of instances is

reported, the average time (Avg. Time) per solved instance, the average number of non-dominated

solutions found (Avg. Solutions), the average number of solutions proven to be ecient (Avg. E.

Solutions), the average gap for non-optimal solutions in terms of costs (Avg. Gap C), the average gap

for non-optimal solutions for the energy objective (Avg. Gap E), the average number of solutions in

which costs are not optimal (Avg. CNOP), and the number of solutions in which energy is not optimal

(Avg. ENOP).

Table 5.2 shows that on average, instances take around 2.37 hours to be solved. A signicant

105

5.5. MAIN CONTRIBUTIONS ON STRATEGIC DECISION LEVEL

increase in computational time is seen when the number of periods is incremented. The reason for

this behavior is the increase in the size of the ILP models. In terms of the number of solutions, a

very limited number is found for the whole set of instances, averaging 11.35 points. A reasonable

explication to this limited number of potentially non-dominated solutions is that solution space of

the problems is highly constrained. Therefore, the number of solutions, and more important, ecient

solutions are limited.

Among the solutions found by the epsilon-constraint method, nearly one third are proven to be

ecient. As little as this number might be seen, this fact can be explained by the complexity in solving

the single objective ILP. One can see that the cost objective is by much the one with the higher gaps,

averaging 13.9%. Furthermore, the gaps on cost component show a considerable rise when more skills

are considered in the problems. This is especially important in instances with higher number of time

periods, for instance family 20_4_3_20_B reaches a maximum of 34.6%.

Energy objective contrary to costs objective, show an excellent performance. Through the 40

instances, the average gap is only 0.6%. Type B instances (regular technician-to-work-ratio) consist-

ently outperform type A (tight technician-to-work-ratio) instances in this metric. Despite the better

performance in terms of energy gaps, results show that the number of solutions not proven ecient is

mostly due to energy objective function. The average number of solutions for which energy presents

a positive gap doubles the same metric for costs objective. Therefore, it is safe to say that single ob-

jective Z2 is solved to optimality more consistently than Z1. However, when dealing with sub-optimal

solutions, the average costs (Z2) gap is over 20 times bigger than the energy (Z1) average gap.

The results reect the importance of operational decisions with the wind farm performance indic-

ators. The fact that schedules can change in as much as 4% the amount of energy produced in the

current context is a matter of most importance. Actually, in closer to real life decisions, the time

required to perform the activities, or the activities themselves might be stochastic. In such situations

the changes in the production can be more pronounced. The same eect may be produced when

bigger instances are solved.

Toward maintenance strategy selection

Evaluating which maintenance strategy is better adjusted is one of the most important decisions

concerning O&M for a project [42, 23, 72, 76]. Having no strategy on how to perform maintenance

tasks usually derives in a very bad performance. Actually, Van Bussel and Schöntag [84] showed

that a maintenance strategy is required to enhance the availability of wind farms. Additionally,

the maintenance strategy policy denes scheduling parameters, in consequence the selection of this

strategy will aect the solution of the maintenance scheduling at operational level. To evaluate the

strategies, this part analyzes the eect of dierent maintenance strategies on the operational cost and

energy production. Before presenting the developed model, a quick revision on strategic maintenance

problems is presented in the following section.

5.5 Main contributions on strategic decision level

Strategies for maintenance activities rely essentially on two main decisions: when to take maintenance

actions and what kind of maintenance (preventive,corrective,opportunistic, etc.) to perform. In the

particular context of wind farms, Shaee [72] uses a general classication based on two major classes:

failure-based (reactive response), and proactive maintenance.

Failure-based strategy uses the idea of performing maintenance tasks only when failures take place,

deriving in corrective actions. This actions can be performed immediately or delayed to a future time

[5, 86]. Nevertheless, there is no standardized way for dening a failure in the wind energy. For

example, Caroll et al. [15] dene a failure as a visit to a turbine, outside the scheduled operation, in

106


which material is consumed. Meanwhile, Wilkinson et al. [87] assume failures as events that require

manual intervention to restart the machine generating a downtime of at least one hour. Consequently,

it is important to dene what failures are in advance so maintenance strategies can be eciently

selected.

Alternatively, Shaee [72] denes proactive maintenance as any maintenance that is performed

before a failure occurs, aiming to avoid possible unexpected failures which usually derive in more

costly actions [41, 53]. Within this classication, two categories can be found: preventive and pre-

dictive maintenance. Preventive maintenance is used to refer to time-based maintenance schemes,

whereas predictive maintenance is equivalent to condition-based maintenance schemes (decision is

made according to the observed health state of the system).

Predictive maintenance usually requires condition-monitoring data. The collected information

used to prevent failures before they become more costly to x. Condition-maintenance-systems are

analyzed in Nilsson and Bertling [59] showing that an increase on the availability of around 0.43% can

compensate the costs of such systems. The eects of false alarm of condition monitoring systems are

studied by May and McMillan [56] showing that even with a decrease in condition monitoring system

reliability, the availability of a wind farm is hardly changed. A recent review on condition-based

maintenance optimization for stochastically deteriorating systems in a general context can be found

in [3].

Prognostics can also be used in maintenance decision making. Examples of maintenance strategies

based on the estimation of the Remaining Useful Life (RUL) of the asset can be found in [12, 46].

Zhao et al. [91] analyze Supervisory control and Data Acquisition Systems (SCADA) information

from turbines to detect normal and anomaly data of turbines generators during their runtime. This

allows to make good estimations of the RUL, and provides sucient time before the appearance of

failures so that a schedule (or a plan) can be implemented before they take place. A recent review on

prognostics techniques for wind turbines can be found at Leite et al. [21].

When dealing with multi-unit systems, another type of maintenance is the one called opportunistic.

This type of maintenance aims to take advantage of breakdowns or stops of the system, due to a failure

or a proactive maintenance, to perform additional proactive maintenance tasks on other components

[81, 22, 12, 1, 47]. In the oshore context, Tian et al. [81] comment the possibility to perform

additional maintenance tasks on nearby turbines when one was selected for a maintenance task. The

main ideas is that grouping maintenance activities can lead to a cost reduction by sharing maintenance

setup costs and downtime duration.

5.5.1 Maintenance strategy selection

Qualitative and quantitative approaches can be used to select the more appropriate strategy according

to Andrawus et al. [6]. The former relies on subjective opinions [7, 70] and may not be adequate

to select the best possible strategy. Quantitative approaches aim to pick the best strategy based on

mathematical models and are by far more studied than the qualitative ones.

In quantitative models, strategies are usually compared together. To evaluate which strategy is

the best, two criteria are often considered: reliability (and availability, see [49]) maximization and cost

minimization [23]. Other metrics such as environmental ones [53] has been used to compare dierent

policies.

Strategies such as failure-based has been studied by Ding [24] given that corrective maintenance

are performed after more than one failure occur. The author shows that this can conduce to an

improvement in terms of cost compared to executing maintenance actions immediately after the rst

failure. Nevertheless, corrective actions have shown to have considerable negative eects [42, 41] on

wind farms costs and availability. Alsyouf and El-Thalji [4] concludes that proactive maintenance is

more suitable for wind farm maintenance problems. Condition-based and opportunistic maintenance

are proposed in Van Bussel et al. [83] adapted to oshore context. Thus, it is safely to conclude that

107


relying only on failure-based strategies is far from optimal.

Nowadays, most of the proposed strategies are hybrids in which corrective maintenance is combined

with some proactive maintenance, either preventive and predictive approaches [78]. For example, Lu

et al. [47] studied a condition-based maintenance strategy coupled with opportunistic maintenance

for oshore wind farms. In this work, an articial neural network (ANN) approach is used to predict

the component life percentage based on monitoring information. Also, by using the information from

the ANN, a conditional failure probability is determined. The authors use a two failure probability

thresholds to decide when to perform a preventive or an opportunistic maintenance task. The authors

conclude that the proposed strategy reduces the cost by nearly 30% when compared to a time-based

(preventive) strategy in onshore and oshore contexts. A very similar method can be found in [81], the

authors compare a condition-based maintenance against a periodic maintenance policy. The condition-

based maintenance use two threshold values to decide when the turbines needs to be maintained. The

authors conclude that a condition-based strategy can reduce by more than 40% the maintenance cost

for a ve turbines wind farm example.

Shaee et al. [75] propose an opportunistic condition-based maintenance for wind turbine blades.

The authors optimize two decisions: the length of a crack at which a major repair must be performed,

and the operational age of the blades that triggers a preventive maintenance. In major repairs and

preventive actions, the other blades also receive a maintenance. The proposed method achieves to

reduce by more than 20% the O&M costs when compared to a reactive strategy. A higher saving

(30%) on O&M costs is generated by the proposed strategy against an individual strategy (per blade)

without using opportunistic maintenance. A similar work is presented in Shaee et al. [74] where

the intermediate speed shaft and the high speed shaft are considered. The same parameters (crack

threshold and operational age) are optimized, although, each component has its own threshold. In

Hameed and Vatn [33] the opportunistic strategy is extended to consider components of a turbine and

among turbines. Still, opportunistic maintenance is preferred to be performed on the same turbine to

avoid traveling costs from turbine to another. By using this strategy setup costs are shared among

dierent maintenance activities, thus reducing the total costs. The approach to group the tasks works

as follows. First, an optimal interval for individual components for individual activities is found.

Then, using a heuristics the groups are created. An example of three wind turbines each with eight

components for a horizon length of four years is analyzed. The results show that approximately the

group of activities have a periodicity of around three months. However, further experimentation is

needed to evaluate the pertinence of this type of grouping. Especially, since the appearance of failures

or bad weather times can completely change the planning.

Other studies on strategies selection involving opportunistic maintenance can be found in [22, 69,

92, 90, 28]. In [90] two level thresholds are used to decide on the basis of the reliability functions of

component if minimal maintenance is performed or if a replacement is preferable. Moreover opportun-

istic maintenance is also considered as soon as a minimal maintenance or a replacement takes place.

In Erguido et al. [28] a dynamic opportunistic policy is evaluated. Dynamism is added through reli-

ability thresholds to perform activities, changing in function of the weather conditions. The authors

conclude that using dynamic thresholds surpass the use of static thresholds.

Although the previous works address the comparison of dierent strategies, they usually rely on

many simplications. For example, considerations of the weather, the number of resources (techni-

cians, vessels, etc), the time to perform the maintenance tasks, the necessary spare parts, among

others are not fully addressed issues, or not considered at all. Therefore, other works have focused

on incorporating such considerations so the complexity of wind farms operation and maintenance is

better modeled. In this context, simulation has been the preferred approach to enable more tractable

comparisons.

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5.5.2 Complex models for maintenance strategy selection

Dierent authors have employed advanced simulation techniques to evaluate maintenance strategies

under the interaction of wind farms elements. Normally, wind farms are simulated as an ensemble

of turbines dened as single or multi-components systems. Moreover, weather models are used to

recreate the wind speeds and wave heights (oshore) faced by the turbines. Weather data has two

purposes, rst it enables to deduct the amount of the energy produced (wind speeds), and second

it allows to check if maintenance safety conditions are met to conduct the tasks. The amount of

energy produced depends on the rating of the turbine, speeds, air density, and many components,

and is usually included as a function or Power Model. Furthermore, the inherent characteristics of

each context creates dierences in the considered elements of the models. For example, maintenance

operations are not constrained by transportation resources when the authors consider onshore wind

farms. Conversely, in oshore this is an important characteristic since dierent types of vessels,

corresponding to scarce resources are needed to perform the maintenance.

Table 5.3 presents a summary of the works using simulations approaches to evaluate maintenance

strategies for wind farms maintenance. These references are chosen dependently in if they consider

limited resources, incorporate weather models, or other operational or tactical levels. The selection

criteria is used as this complex models allow to further research the interaction between strategy

selection and lower levels decision problems. For each work the following nomenclature is used to

classify the main aspects of the works: Onshore (ONS), oshore (OFF), number of turbines (NT),

turbines power generation model (PG), weather model (WM), number of components per turbine

(NC), failures modeling (FM), corrective maintenance (CM), preventive maintenance (PM), predictive

maintenance (PAM), limited technicians (LT), limited resources for transportation (LRT), the presence

of energy metrics to evaluate the strategy (EO), the evaluation of strategies using cost metrics (CO),

and the consideration of inventories (IC). The abbreviation NS stands for non specied.

How the turbines are modeled is an important aspect of the models dealing with how the failures

appear (and are solved). Either a single or a multi-component system are used as seen in column

NC, table 5.3. The former considers only one of the components of the turbine (or aggregate them all

in a single one). Meanwhile, multi-component systems permit to consider the individual degradation

of each selected subsystem, allowing to consider strategies of opportunistic maintenance. Single-

component systems have been addressed by Carlos et al. [14] who propose to model the turbine using

a reliability model in a onshore context. A multi-objective problem is designed by the authors using

maintenance costs and the amount of energy produced as the two selected criteria. The failures process

is considered to be Weibull distributed. Moreover, the authors consider that maintenance activities

are imperfect and use a Proportional Age Set-back as in [54] while Wind velocity is modeled using

a Weibull distribution. Furthermore, Carlos et al. [14] consider that the maintenance time are also

random variables following an uniform distribution. Although the authors do not justify this choice, it

can be explained by the deterioration of the system, or the technicians eciency. Using a simulation

of one turbine, the authors derive an optimum time interval to perform maintenance activities.

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Table5.3:

Summaryof

literaturewithcomplex

modelsformaintenance

strategy

selection

Reference

ONS

OFF

NT

PG

WM

NC

FM

CM

PM

PAM

LTLRT

EO

CO

ICAndrawus

etal.[6]

x26

4Weibulldistribution

xx

xx

xMcM

illan

andAult[58]

x1

xx

4Markovmodel

xx

xCarloset

al.[14]

x1

xx


xx

xByonet

al.[11]

x100

xx

1Markovmodel

xx

xx

xx

Perez

etal.[62]

x100

xx

4HiddenMarkovmodel

xx

xx

xx

xHofmannandSp

erstad[34]

x50

xx

NS

NS

xx

xx

xx

xSahnounet

al.[66]

x80

xx

1Scalefunction

xx

xx

xx

xDalgicet

al.[20]

x150

xx

NS

Rateper

year

xx

xx

xx

Santos

etal.[68]

x1

xx


xx

xx

Abdollahzadeh

etal.[1]

NS

29x

x4

Weibulldistribution

xx

xx

xx

Dahaneet

al.[18]

x80

xx

2Scalefunction

xx

xx

xx

xx

110


Similarly, Sahnoun et al. [66] design turbines as a single component for which degradation is

modeled with ten status scale in an oshore context. Turbines status changes according to a function

that includes, an exponential variable representing the components time to failure, and the eect of

weather on turbines degradation. This is very important since among the studied works, this is the

only one that considers the degradation as a function of the weather. Three strategies are compared

using a multi-agent systems simulation: periodic, condition-based maintenance, and a hybrid version

combining both. Also, maintenance task are scheduled based on rules such as the turbine with the

maximum degradation, or if a time window is available to perform the task. Simulating a wind farm

with 80 turbines, the authors show the hybrid strategy is the most eective considering the costs

and the produced energy. Although Byon et al. [11] also model turbines as one component, this last

stands for a subsystem of the turbine. This contrast with the approach of [14, 66] where the single

component aimed to model the whole turbine. Byon et al. picked the gearbox since the turbines most

critical failures are related to this component. The gearbox degradation is modeled through a Hidden

Markov Chain with a transition matrix of one week periodicity. This type of model allow the authors

to integrate the incomplete information received through sensors output. That is, the real state of the

components are hidden and the sensor outputs can be generated by multiple of these states. Using

a Discrete Events System Specication (based on the work of Perez et al. [61]) they simulate 100

turbines letting to conclude that condition-based maintenance enables more wind power generation

as it reduces the number of failures.

Multi-component systems have been preferred to model the turbines. Half of the reviewed works

use four components to represent the turbines (see column NC, table 5.3). These usually include

the gearbox and the generator, and other components such as the blades, shafts, or electrical ones.

These components are usually selected since they represent most of the downtime experienced by

a turbine [15], or have the larger impacts in terms of costs. Andrawus et al. [6] include turbines

subsystems such as the main shaft, main bearing, gearbox and generator. The last two include also

some sub-components such as the bearings. The lifetime of subsystems and components are modeled

using Weibull distributions. The parameters are derived from maximum likelihood estimators from

reported data failures. A simulation is carried out for 26 wind turbines composing a wind farm.

Maintenance crews are limited and inventories policies are also addressed. Maintenance strategies are

selected for each subsystem based on the parameters of the Weibull distribution, but no comparison

among dierent strategies is given. Meanwhile, the benet of installing condition monitoring systems

for the onshore context is tackled by McMillan and Ault [58, 57]. Similarly to [6], the authors consider

turbines as a four-component model which includes: the generator, gearbox, blades, and electronic

related parts. This study uses the items presenting the most important failures, but also focuses

on monitored components. It is assumed in [58] that monitoring reveals the true component state.

Moreover, the turbine is modeled with a Markov Chain taking account the state of each component.

The authors conclude that for most onshore wind turbine components, condition-monitoring is cost-

eective. Furthermore, according to the authors the same conclusions should hold in the oshore

case.

An extension of the work in [11] is presented by Perez et al. [62] to consider multiple components

in the onshore context. As in other related works [6, 58, 81], the number of components is limited

to four, namely the gearbox, power generator, blades, and control system. Moreover, the authors

include the inherent constraints for schedules due to limited maintenance teams and to lead times in

spare parts. Condition-based-maintenance presents better performance when compared to periodic

maintenance. Additionally, a condition-based-maintenance strategy which also includes opportunistic

maintenance, presents the best results in terms of costs and number of failures. Santos et al. [68]

consider a one turbine model. The turbine is simulated by a four-component that includes the gearbox,

generator, pitch system, and rotor. The authors optimize the maintenance strategies through the use

of generalized stochastic Petri nets coupled with Monte Carlo simulation. The authors optimize two

111

5.6. MAINTENANCE STRATEGIES: RELATION WITH OPERATIONAL PLANNING

parameters: the preventive repair threshold and the age reduction ratio. The rst is a proportion of

the mean time to failure of the component to perform a preventive task, while the second stands for

how much of the age of the component is reduced with the maintenance task.

A multi-objective opportunistic maintenance optimization considering limited resources is tackled

by Abdollahzadeh et al. [1]. Turbines are dened by multiple components (rotor, main bearing,

gearbox, generator) with lifetime modeled by a Weibull distribution. Using the reliability of the

components, the authors dene several thresholds to execute opportunistic maintenance. By mean of

a Particle Swarm Optimization, the threshold values are optimized to maximize the energy produced

while minimizing the maintenance costs. Several conclusions can be retained from this work. First,

it is shown that opportunistic maintenance coupled with preventive actions outperforms the classical

corrective strategies. Second, the number of maintenance teams has an important eect in the Pareto

ecient solutions and the addition of maintenance teams can greatly increase the energy produced.

Third, maintenance strategies must be compared with basis on more than one objective, as far as

these can be in conict and lead to dierent conclusions.

A variant of the work of Sahnoun et al. [66] is introduced by Dahane et al. [18] considering the

impact of spare parts re-manufacturing. Turbines are modeled by two components, a single turbine

representation (as in [66]) and the gearbox. Both elements use similar degradation functions that

include the eects of weather or the accelerated degradation of re-manufactured components. The

comparison is performed on a simulated 80 turbines wind farm. A concluding remark of the authors

is that whichever the strategy, the average production of energy does not vary. The NOWIcob tool

is introduced by Hofmann and Sperstad [34]. The tool uses a multi-component approach, although

the authors do not mention the number of modeled components. The example presented shows

the importance of transportation vehicles in the oshore context. The authors show that the use

of a mothership can increment by more than 3% the availability when compared to the use of a

platform. Dalgic et al. [20] also model turbines as a series of subsystems. Each subsystem is modeled

by its reliability function derived from historical failures rates and expert judgment. Nevertheless,

no information is given about the number of components. The authors also model other uncertain

parameters such as the weather (wind and waves). Although the maintenance strategy is not the focus

of the work, the authors conclude that remote-monitoring (and thus condition-based maintenance)

can lead to important improvements on the oshore wind farms performance metrics.

Overall, the results of [25, 64, 62, 1] show that under the presence of limited resources, maintenance

strategy outputs can signicantly vary. This follows also the conclusion of Van Horenbeek [85] who

claims that assuming maintenance operations duration as negligible can conduce to bad decisions.

Therefore, one can conclude that although models are simplication of the real wind farms, several

aspects must be considered to compare dierent maintenance strategies. Moreover, most of the studied

works consider both the costs and energy related objectives, but only Abdollahzadeh et al. [1] explicitly

construct a set of solutions for which costs and energy production change. It is also interesting to

see how the scheduling of limited resources can be tackled. Nearly all of the cited works schedule the

resources as fast as they can be assigned, ignoring the possible gains (and eects) of optimizing the

resources. Only the work of Sahnoun et al. [66] consider a rule to prioritize some of the tasks. These

two issues, the multiple objectives and the rules to schedule operational resources, are considered in

section 5.6 to evaluate the impact of operational decisions on dierent metrics in a long term horizon.

5.6 Maintenance strategies: relation with operational planning

In this section, the maintenance scheduling problem presented in section 5.4, is embedded in a long-

horizon simulation model to evaluate how costs and energy production objectives perform through the

life cycle of a wind farm project. Nevertheless, since the model and solution approach of section 5.4

requires much time to be practically often solved in a long-horizon evaluation, two actions are taken.

112


First, the model is simplied to consider only one type of technicians and no extra-hours. Second, it

incorporates straightforward rules to schedule the limited resources. Therefore, these rules are used

to heuristically solve the scheduling problem instead of using an exact solution method.

5.6.1 Problem Description

Let T to be the planning period for a wind farm. During this time the O&M performs its activities

while a set J of turbines produce energy. Each turbine j ∈ J produces PEjt units of energy that

depends on the wind speed during a period t ∈ T . Two dierent strategies to perform the maintenance

are considered: "'On failure mode"' and "Preventive mode" strategies. On failure mode is a reactive

strategy that considers a maintenance tasks if a failure takes place. Besides, the "'Preventive mode"'

considers maintenance tasks based on xed time periods. Furthermore, in this mode if a failure takes

place then a corrective maintenance task is performed. Besides, whichever the strategy, twice a year

the turbines are visited to execute some maintenance on the components not explicitly considered and

to change consumables. The dierent types of maintenance are identied by the set U indexed by u.

Additionally, each turbine j is composed by a set of components K indexed by k. Moreover, turbines

(and their components) are subject to deterioration process which derive in failures. The degradation

process of one component is independent from the others. Turbines components are maintained by

a limited set of technicians (P ) who perform the maintenance. Technicians travel to the turbines to

perform their assigned maintenance tasks within their shifts.

To perform a maintenance task type u on component k, a number of χku technicians are required,

with an associated cost cku. Proactive maintenance is considered as an imperfect repair, that is,

a preventive maintenance action does not restore the component to an as good as new condition.

Conversely, a replacement brings the component back to a state as good as new. Furthermore, an

execution time βku characterizes every task. Time βku corresponds to the time elapsed from stopping

the turbine until it is restarted and veried again. It is assumed in this section that βku is a random

variable with known probability distribution. When a task is started by a set of technicians it is not

stopped until nished, therefore, the assigned technicians will not work on other tasks until they nish

the started activity. Still, at the end of each shift, the technicians stop their work and continue on

the next workable shift. Moreover, if the conditions are not safe (high wind speeds) the technicians

wait until they become safe again.

Maintenance Strategies are evaluated through several metrics gathered from the whole planning

horizon: the average availability (A) of the wind farm, the amount of total energy produced (E),the total number of failures (NF), and the total costs (T C). The objective is to compare the dif-

ferent strategies, taking as decision variables how and when to perform the schedule of the resources

(operational problem). Again, this schedule could be executed with the model presented in section

5.4. Nevertheless, this is computationally intractable due to the length of the planning horizon (and

thus the number of operational problems to solve). For this reason, other characteristics such as the

technicians skills are not considered in this part. Also, to overcome the computational problem simple

priority rules to dene the schedules are devised.

5.6.2 Simulation model

The simulation model is formed by four big modules: the wind turbines, the weather model, the

scheduler model and the resources module. A scheme of the modules is presented in gure 5.2.

Wind Turbines

This module represents the turbines, their degradation process, the time between failures, and the

power generation. It also allows to keep information as availability and energy production. In this

module a set of |J | identical turbines are considered, each of them composed of four components

113


Wind Turbines

Weather model

Wind speeds

Energy producedAvailability

Components

Schedule modeler

Resources

Wind speeds

Available

Power Generation

Status

Schedule

Repair

Wind speed –safety conditions

Degradation

CostsNumber of maintenance tasksNumber of failures

Figure 5.2: Scheme of the simulation modules.

(k = 4) namely C1, C2, C3, and C4 (see gure 5.3). Each of the components can be also composed

by sub-components, for instance, the rst component (C1) may represent the blades, which are indi-

vidually represented: SC 1.1 (blade 1), SC 1.2 (blade 2), and SC 1.3 (blade 3). If any sub-component

presents a failure then the component itself will fail, generating a shut-down of the turbine. Thus,

a turbine is designed as a serial connected components. This type of representation allows to model

the turbines as a single component or multi-component system. Furthermore, it enables to explore

dependencies between components such as the eects of one component deterioration over other com-

ponents. Notwithstanding, as explained in section 5.6.1 the model in this part assumes independent

components.

Modeled components

A turbine is modeled by four key sub-components3: the rotor, the main bearing, gearbox and gener-

ator. Such components have proven to be one of the more important cost inductors for maintenance

operations. It is also assumed as reported by [81, 22, 6, 68, 1] that the component times between

failures can be modeled using a two-parameters Weibull distribution. The parameters reported by

Abdollahzadeh et al. [1] for three dierent types of turbines as shown in table 5.4 are used. Since

imperfect preventive maintenance is considered the virtual age vajk of each component k in turbine j

(∀k ∈ K, j ∈ J) [28]. As time pass the virtual age of each component continues to increase. It is only

modied in two cases: if a failure takes place, in which case the component is replaced and the age is

restarted (vajk = 0), and in the case of preventive maintenance. In the second case the new vajk′is

set to vajk′

= vajk (1− q). That is, the virtual age is reduced as percentage of the old age. The term

q (age reduction ratio) can be component dependent. For simplicity in this work it is assumed that q

is the same for every component in every turbine. It is further presumed that q follows a continuous

uniform distribution [14] q ∼ U(0.8, 0.95).

Power generation

Equation (5.23) shows the power generated by a turbine j based on the formulas of Karki and Patel

[38] given a wind speed vt at time t in the turbine location l. Furthermore, v0 represents the cut-in

3Each of the sub-components is not further divided into sub-components

114


Turbine

C4C3C2C1

SC1.1

SC1.2

SC1.3

SC2.1 SC2.2

SC 2.2.1 SC 2.2.2

SC 4.1

SC 4.1.1

SC 4.1.1.1

Figure 5.3: An example of a turbine modeled a multi-component system.

Table 5.4: Weibull parameters for components failures in Abdollahzadeh et al. [1]Component Scale parameter α (days) Shape Parameter β

Type 1 Type 2 Type 3Rotor (k=1) 3000 2400 1750 3

Main Bearing (k=2) 3750 3100 2400 2Gearbox (k=3) 2400 1200 1200 3Generator (k=4) 3300 2200 1800 2

115


wind speed, v1 the rated wind speed, and v2 the cut-out speed.

PEjt :

0, 0 ≤ vt ≤ v0P r(A+Bvt + Cvt

2), v0 ≤ vt < v1

P r, v1 ≤ vt < v2

0, vt ≥ v2

(5.23)

A =1

(v0 − v1)2

[v0 (v0 + v1)− 4v0v1

(v0 + v1

2v1

)3]

B =1

(v0 − v1)2

[4 (v0 + v1)

(v0 + v1

2v1

)3

− (3v0 + v1)

]

C =1

(v0 − v1)2

[2− 4

(v0 + v1

2v1

)3]

Thus, when the weather module provides a speed to each turbine, this one can estimate by means

of equation (5.23) the amount of energy produced.

5.6.3 Weather model - Wind speeds

Performing simulation of wind farms require to address a fundamental aspect, that is, a synthetic

creation of wind data speeds. Several ways of dealing with this are reported in the literature, for

example Carlos et al. [14] use a Weibull distribution to model wind velocity. Byon et al. [11] and

Perez et al. [62] use a spatio-temporal model to generate wind speed sequences at dierent locations

and heights. Meanwhile, McMillan and Ault [58] use an auto-regressive model (AR(1)) to model wind

speeds using the equation (5.24).

wst − µ = Λ (wst−1 − µ) + εt (5.24)

Although the approach presented in [11] seems to oer a more robust way to generate correlated

winds for many locations, the auto-regressive model of McMillan and Ault [57] is kept for its simplicity.

Moreover, it allows to consider the season patterns depending on the month, by varying the value of

Λ, and the value of µ.

5.6.4 Schedule Modeler

Whichever the type of strategy used, at the operational decision level the resources utilization needs

to be optimized. To the best of our knowledge, no other study has before considered to optimize the

maintenance scheduling plan during a life cycle simulation of a wind farm. This is mainly due to

the computationally demanding time to carry on such a task. Certainly, to embed the whole model

and solution approach presented in section 5.4 in a long-term simulation model is unrealizable. This

is why an approximate solution approach based on simple heuristics rules to assign the maintenance

tasks is used.

In this part, failure-based maintenance, and periodic maintenance are compared. Strategy (CM)

performs only maintenance tasks when a failure occurs. Meanwhile, (PM) policy also use corrective

tasks when failures occur, but also, it performs periodic maintenance to avoid possible future failures.

To optimize the schedules a rolling basis to make operational decisions is devised. That is, each

day the assignment of resources to maintenance tasks is redened. Nevertheless, tasks which have

already begun are not rescheduled. Each day, at 7 a.m. it is assumed that the wind speeds for the

116


Table 5.5: Costs, technicians, and time requirements per maintenance component based on Abdol-lahzadeh et al. [1]

Ck1 Ck2 χk1 χk2 µβk1 µβk2Rotor 28 112 2 3 8 16Main Bearing 15 60 2 3 9.6 16Gearbox 38 152 2 3 16 24Generator 25 95 2 3 12 32

day are known to the O&M operator in a hourly basis. Moreover, next ve days previsions are also

known with certitude. At this time the operator also knows which tasks are needed to be performed

(except for possible imminent failures).

The following rules are tested to design the tasks scheduling:

O1 : schedule the maintenance tasks as fast as possible when there are available resources

O2 : schedule the maintenance tasks to start at times where the wind speeds are low and always

in the morning

5.6.5 Resources

The resources considered are technicians working in a shift starting at 8:00h in the morning and

nishing at 17:00h from Monday to Friday. Technicians perform the maintenance tasks if wind speeds

are below 10 m/s. At higher speeds the technicians need to wait. All other resources as vehicles, spare

parts, cranes or any other requirement are always available to perform the maintenance tasks.

5.6.6 Model implementation

The model described in section 5.6.1 was implemented using a Discrete Events Simulation program

designed for this purpose. All runs are conducted on a Dell Latitude E6420 personal computer with

Intelr CoreTM i7-2760QM @2.4 GHz, running Windows 7 Professional 64 bits. The horizon length is

set to a 15 years horizon length assuming the wind farms working 365 days of the year and 24 hours

per day. A total of 1000 replications are run for each combination of strategy and rule of decision

from the scheduler.

The model is tested on a virtual instance with the following characteristics. A total of 100 turbines

with rated power of 2MW are considered. Each turbine is modeled following the assumptions presented

in in section 5.6.2. For the sake of simplicity, turbines are located in a square location of 9 square

kilometers in a grid with intervals of 300 meters.

Additionally, consider Ck1 and Ck2 to be the preventive and corrective costs for a component

k ∈ K. Table 5.5 presents this costs and the number of technicians (persons) and the expected

time required to carry out the activities. It is assumed as in [1] that βk1 ∼ N(µβk1 , σ2βk1

) and

βk2 ∼ N(µβk2 , σ2βk2

). The values of σ2βk1

and σ2βk2

are inferred from a coecient of variation of 0.2

and 0.3 respectively [1]. Table 5.5 values are gathered from [6, 81, 1, 28] and are reported for Type 1

turbine from table 5.4. In addition, the number of required technicians for the recurrent visits (u = 3)

is two, with a duration uniformly distributed between two and four hours4. It should be noticed that

even if most of the data is gathered from the literature, it is not claimed that it to be representative

for any particular wind farm.

5.6.7 Simulation model results

To gather the results 1000 simulations are run for each of the strategies and scheduling rules: CM-O1,

CM-O2, PM-O1, PM-O2. The "`CM"' strategies are those based on the reactive strategy, meanwhile

"`PM"' use a preventive approach. The "`O1"' and "`O2"' parts stand for the rules presented in

4Based on Carlos et al. [?] and meetings with experts on the eld.

117


Figure 5.4: Total maintenance costs by strategy/rule for the wind farm simulation.

section 5.6.4 to schedule the activities as fast as possible or to improve energy production respectively.

Figure 5.4 presents the box-plot for the total costs measured in monetary units (MU) calculated as

the sum of the cost of all maintenance tasks performed during the planning period. It is noticeable

that PM type strategies present lower costs than CM strategies. These results are expected since the

costs of failures are much larger than those due to preventive activities (nearly four times larger, see

table 5.5). Moreover, O2 shows higher costs than O1 specially for the PM strategy. Due to that O2

tries to schedule activities more widespread in time, to exploit low wind periods. This eect might

be also amplied by selecting mornings to schedule the activities and thus, avoid between days loss of

production. Therefore, preventive activities can be executed on larger horizon times (several weeks),

augmenting the probability of failures.

Figure 5.5, conrms that the costs are increased due to the total number of failures. One should also

note that O2 strategies present more variability. This appears to be the result of the vulnerability

of O2 subject to longer scheduling periods. It also oers a clue to derive new rules in which the

uncertainties (appearance of failures) are considered so the plan of preventive maintenance follow an

intelligent order. This analysis lead us to another characteristic to include in the model, the condition

base maintenance strategies. Due to the imperfect maintenance tasks, each component of each turbine

presents dierent degradation levels and therefore, dierent probabilities of stopping working. Such

information can be integrated so new improved rules exploit it to make better decisions.

Figure 5.6 presents the total energy production for all maintenance strategies at the end of the

simulation. The O2 policy seems to oer the higher production. The dierence in the total energy

production between PM-O1 and PM-O2 gives an improvement of nearly 1.7% for the latter. This

is further conrmed with the fact that PM-O2 shows the highest availability (see gure 5.7). The

explanation of this dierences is again, the way in which O2 plans the activities. The avoidance of

scheduling tasks, for example on a Friday afternoon, imply that the turbine will be operative during

the weekend, and the tasks will begin preferably on the Monday.

To further explore the dierences between the tested strategies and scheduling rules, a life cycle

analysis is conducted. The same four metrics (costs, energy production, availability, and number

of failures) are gathered at the end of each year during the simulation of each strategy. Figure 5.8

presents the values averaged from the 1000 runs across 15 years, and for each one the cumulative value

is reported.

118


Figure 5.5: Total number of failures by strategy/rule for the wind farm simulation.

Figure 5.6: Total produced energy under dierent strategy/rules for the wind farm simulation.

119


Figure 5.7: Mean turbine availability for dierent strategy/rules for the wind farm simulation.

0

10000

20000

30000

40000

50000

60000

70000

80000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

TOTA

L C

OST

S (M

U)

YEAR

CM-01 CM-02 PM-01 PM-02

0

100

200

300

400

500

600

700

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

NU

MB

ER O

F FA

ILU

RES

YEAR

CM-01 CM-02 PM-01 PM-02

0

1000000

2000000

3000000

4000000

5000000

6000000

7000000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

TOTA

L EN

ERG

Y M

WH

YEAR

CM-01 CM-02 PM-01 PM-02

94,00%

95,00%

96,00%

97,00%

98,00%

99,00%

100,00%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

AV

AIL

AB

ILIT

Y

YEAR

CM-01 CM-02 PM-01 PM-02

Figure 5.8: Temporal analysis for wind farm metrics under dierent strategies.

120


Around the sixth and seventh year of operation, the number of failures start to increase in strategies

that do not use preventive maintenance, this puts in evidence the eect of preventive strategies in

the long term. This fact coincides roughly with mean time to failure of the components (see table

5.4), time at which failures probability becomes more important. The reduction in the number of

failures is reected one or two years on the costs and the variation between strategies becomes more

prominent at the tenth and eleventh year. The lag between failures and costs can be explained by the

preventive costs matching the corrective costs of CM strategies. Nevertheless, after one to two years

PM strategies take the edge and become less costly. Other important aspect to remark is that after

the tenth year the total costs seems to be more stable for the PM strategy while the CM strategy

exhibit a steady increase.

Figure 5.8 shows the behavior of the wind farm availability. This metric is mainly aected by the

scheduling rule, particularly for the preventive strategy PM-O1. Such strategy greatly diminishes the

availability metric around the sixth year of its life cycle. This is due to the preventive behavior of

that strategy since turbines are stopped to execute the maintenance. Nevertheless, after a minimum

value of 95% in the eight year, the availability increases for PM-O1 since very few failures occur.

Meanwhile, strategies based on O2 rule show a more stable behavior, decreasing only by 1% during

the whole horizon-length. Even though O2 allows a bigger availability this also comes with increasing

costs for the PM strategy.

The presented results conrm the idea that maintenance strategies are importantly dependent on

how operational activities are planed and executed. Although the results were gathered for a virtual

example in which resources were highly constrained5 the conclusions might just hold for more realistic

environments. Actually, the reduced number of technicians compensates with the lack of lead times

result of unavailable spare parts or need for special cranes.

5.7 Conclusions

In this chapter Maintenance Planning problems in the wind farm context are presented. The rst part

tackled the operational level problem, in which the resources demanded to perform maintenance tasks

are assigned and scheduled. In the second part the integration between the operational and strategic

level is shown and the eects of dierent maintenance strategies on the operational cost and energy

production is studied.

To solve the maintenance scheduling problem, a bi-criteria mixed integer program is proposed and

solved within an epsilon constraint method for the scheduling problem. The results show that O&M

costs and the energy produced objectives are in conict an have signicant consequences on each

other. Moreover, these results imply that the resources will be assigned depending on the objective

function. Thus, the eects on the long-term of using schedules based on one type of objective need to

be addressed.

Using a Monte Carlo simulation within an Discrete Events framework we compared two classical

strategies widely used in the literature and in real systems: corrective and preventive strategies are

compared. Moreover, two scheduling rules at the operational level are tested within each strategy.

The results shown that the preventive strategy impacts in a positive way the costs and number of

failures comparing with the corrective strategy: the cost are reduced on nearly a quarter and failures

by half. Meanwhile, O2 scheduling rule benets the energy production and availability over rule O1

showing increments of nearly 2% on both metrics at the expense of a 9% increment in costs.

Undergoing work is concentrated on integrating the complete model presented in section 5.4 within

the simulation approach. Moreover, it is intended to incorporate the uncertainties in the scheduling

problem, for example due to weather predictions errors. Besides, new rules to schedule resources and

new maintenance strategies including opportunistic and condition based need to be tested. Finally,

5Only four technicians were considered.

121

5.7. CONCLUSIONS

the studied models will be extended to the oshore case to bring more understanding on the how to

select the best strategy to investigate the maintenance tasks in dierent environments.

Contributions

The results of the multi-objective problem in this chapter were presented at IEOM conference:


Wind farm maintenance scheduling model and solution approach

In Proceedings of the International Conference on Industrial Engineering and Operations Management

Bogota, Colombia, 2526 October, 2017.

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Chapter 6

Conclusions

This thesis is dedicated to the Vehicle Routing Problems (VRP) under uncertainties and Maintenance

Planning on wind farms. Although most of the works on these elds are devoted to deterministic

problems, recently published literature has shown an increasing interest on both research subjects.

This growing attention is due to the fact that in real world applications, data are seldom known

with precision or with certitude when the decisions have to be taken. Furthermore, the eects of

neglecting that data are imperfect when designing maintenance schedule or vehicles routes can have

important impacts in terms of costs, systems unavailability, or customers' dissatisfaction. Within this

context, this thesis makes the following contributions: rst it presents ecient solution approaches

based on memetic algorithms with strategies specially tailored for two Stochastic VRPs variants:

namely the VRP with stochastic demands and the VRP with random travel and service times in

which maintenance tasks on distributed assets must begin within hard time windows. New results are

exhibited to these two stochastic VRPs including new best solutions. Moreover, a natural extension

of such problem to maintenance planning for wind farms is devised for a deterministic case, and then

explored within a simulation approach.

First, a hybridized Memetic Algorithm with a restarting procedure based on a Greedy Randomized

Adaptive Procedure Search (GRASP) was proposed to tackle the VRP with stochastic demands. The

method is proven to be ecient in a classical available benchmark showing better results than current

state-of-the-art methods. During this thesis, the literature review showed a lack of detailed results for

bigger instances involving larger number of customers, as can be encountered in real life applications.

Therefore, we proposed a new data set including 39 instances based on a benchmark originally designed

for the deterministic VRP. The average number of customers in this new set doubles the previous

classical set, and comprises some instances with more than 200 customers. New results were reported

for 29 instances while the remaining ten have been already optimally solved by other authors. The new

benchmark and best solutions achieved by our algorithm enable in the future comparison on middle to

big size instances for which results are available. In addition, new recourse actions and strategies such

as re-optimization can be assessed against the classical recourse policy used in our methods. Currently,

our results can serve as the baseline for evaluating the tradeo between algorithms eciency and the

use of more complex policies to face uncertainties.

Second, a VRP with stochastic travel and service times and hard time windows was introduced since

the majority of related works considered soft time windows, or coupled soft and hard time windows.

This problem also considers continuous random variables with dierent probability distributions which

is fairly uncommon in the related literature. A parallel Memetic Algorithm framework is used to tackle

the problem. Although the base of each thread of the algorithm is based on the same principles, each

one undertakes the uncertainties in a dierent way. Some of the populations are created to solve

deterministic problems for which the uncertain parameters are replaced by the mean values or a

percentile of the considered probability distributions. To the best of our knowledge, this is the rst

129

time a SVRP is solved using such a solution approach. The results conrm a signicant improvement

on the eciency due to the multi-populations parallel algorithm. Moreover, the proposed algorithm

is compared to other works published recently, and the results proved that it is very competitive.

Besides this, it can easily be extended to other SVRPs which make the designed parallel framework a

powerful and exible tool.

The frameworks of the methods designed to solve the previous problems allowed us to state that

good solutions in a deterministic related problem can be used as departure points for good stochastic

solutions. Nevertheless, they require to be further improved by integrating adequately the uncertain-

ties. In our approach, this strategy led to improvements on the algorithms performances. In particular,

it increased the probability of nding near to best known solutions in shorter running times.

In the wind farm turbine maintenance context, a multi-objective optimization problem dealing

with scheduling of maintenance resources is studied. The two objectives taken into account in our

problem are the costs and energy production. A bi-criteria mixed integer program is proposed and

solved within an epsilon constraint method. To the best of our knowledge, this is the rst time

that this kind of approaches is conducted in the wind farm operational decision level context. The

computational results suggest that with our formulation small size instances can be solved by providing

the exact Pareto-front. Nevertheless, they also show that augmentation of the costs, mainly driven by

overtime, can signicantly aect the amount of produced energy. Therefore, we extended this work to

evaluate how these objectives (costs and energy production) behave through the life cycle of a wind

farm project under dierent maintenance strategies. To do so, as is common in the literature, we

devised a Monte Carlo simulation approach based on discrete events simulations. The appearance

of failures and the time required to carry out maintenance activities are considered as stochastic.

Unlike what is usually used in other works, we derived a policy to schedule maintenance tasks in short

term horizon, considering the wind speed for the subsequent days as well as the energy production.

This allows to emulate good solutions to the operational scheduling problem. The results show an

important eect in the life cycle maintenance costs and produced energy. Although other works have

employed simulation to compare maintenance strategies, most of them rely on assigning the resources

to execute the maintenance tasks as fast as they are available. Further research must be devoted to

the introduction of more information to schedule the resources, such as the system status, components

history of failures, etc.

Further work on both SVRPs and wind farm maintenance planning should focus on the independ-

ence assumption of the random variables. Concerning stochastic demands, the interdependent scenario

can arise when demands of close geographical assets are aected by localized events. Moreover, one

can expect that variables representing travel times are highly correlated due to factors such as trac

jams. Similarly, in the wind farms context, the degradation processes of the turbines components are

usually assumed to be independent. This assumption is rather questionable.

Other interesting topics of research are related to the evaluation of the distributions used to model

uncertain parameters. For example, the widespread devices carrying Geographical Positions Systems

(GPS) can be used to conduct several analysis of travel times in dierent areas. This will allow to

evaluate the accuracy and pertinence of using some distributions. In this sense, more experimental

studies are needed to conrm the importance of SVRP solution in real applications. In theoretical

works the Value of the Stochastic Solution (VSS) has proven to be considerable. Our studies showed

that the VSS increases with the size of the problems and also when they are more constrained.

We expect to see an increase in the number of metaheuristic approaches to deal with SVRPs.

The experience of deterministic combinatorial problems must be capitalized and benet to stochastic

variants. This must be accompanied with new strategies to incorporate the parameters variability.

Currently, one of the main problems in such methods remains the time overload generated by prob-

ability calculation to asses either a recourse action or a probabilistic constraint. This is particularly

important in local search procedures which are recognized as time-consuming components. Besides,

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CHAPTER 6. CONCLUSIONS

branch-and-price methods can be used to increment the amount of exact methods on SVRPs. Ap-

proximations of the recourse costs based on sampling procedures distributions (simulation) can be

used to solve the pricing problems. Additionally, more recourse actions are expected to appear but

they will likely be more complex.

To enable comparisons among dierent recourses, standardized benchmarks are needed to be set

so that fair competition should contribute to improve the results and highlight new properties on

these problems. Finally, although it was not the focus of this dissertation we expect to see an increase

in the number of articles dealing with robust optimization as an important methodology to handle

uncertainties when the parameters probability distributions are unknown. The trends of the literature

on SVRPs and Maintenance Planning make us believe that it will continue to increase in the following

years. We hope that the contributions of this thesis and the research clues identied through it will

help future developments.

131

Appendices

133

Annexe A

Résumé en français

Les activités de transport jouent un rôle très important dans l'économique. En eet, la logistique et

les activités de transport ont généré 8% du Produit Intérieur Brut aux États-Unis pendant l'année

2015. Le transport rien qu'à lui même peut représenter jusqu'à 60% des coûts logistiques, ce qui le

rend un sujet d'étude important. Sur le plan académique, la communauté de la Recherche Opération-

nelle a largement contribué à la résolution de problèmes soulevés dans le domaine du transport, et en

particulier les problèmes de tournées de véhicules (VRP en anglais pour Vehicle Routing Problem).

Ceci a donné lieu à un grand nombre de publications depuis l'introduction du VRP en 1954. Toute-

fois, la plupart des travaux continuent à considérer que les informations, et donc les paramètres des

problèmes sont connus à l'avance. Cette supposition est rarement vraie dans la réalité, en eet il existe

plusieurs facteurs qui peuvent remettre en cause cette supposition sur la certitude des paramètres.

Dans le contexte du transport urbain par exemple, les temps de trajets peuvent être aectés par les

embouteillages. Les temps nécessaires pour servir les clients peuvent aussi dépendre de la complexité

des services à eectuer, etc.

Les problèmes de type VRP avec incertitudes ont sucité un intérêt grandisant ces dernières années.

Gendreau et al. [13] ont proposé récemment une des revues les plus complètes sur le sujet, démontrant

une activité de recherche croissante. Dans cet article, les auteurs se sont concentré sur les diérents

modèles de programmation stochastique dédiés aux problèmes de tournées avec incertitudes. Pour

analyser et classier les travaux, deux caractéristiques principales sont en général considérées : le type

de paradigme de résolution considéré et les paramètres entâchés par les incertitudes (demandes, pré-

sence de clients, temps...etc.). Les paradigmes de résolution se divisent en deux catégories : l'approche

d'optimisation à priori et la réoptimisation. L'optimisation à priori est liée au cas statique dans le-

quel des décisions doivent être prises ici et maintenant bien avant la révélation des réalisations des

paramètres stochastiques. C'est le paradigme préféré quand les incertitudes peuvent avoir des consé-

quences importantes sur la solution, et donc il est préférable de les anticiper. Parmi ces paradigmes

à priori, nous citons les problèmes avec recours (SPR en anglais pour Stochastic Programming with

Recourse) et les problèmes avec contraintes probabilistes (CCP en anglais pour Chance Constraint

Programming).

Les modèles type SPR utilisent des actions appelées recours qui permettent de réagir face aux

situations de violation de contraintes (échecs) suite à la révélation des paramètres incertains. Les

modèles CCP quant à eux visent à limiter la probabilité de violation des contraintes. Les deux modèles

présentent cependant l'inconvénient d'êtres très lourd à résoudre en termes du temps nécessaire pour

le calcul des coûts relatifs aux actions de recours, et des probabilités de satisfaction des contraintes.

Ainsi, il est crucial de développer des méthodes de résolution qui permettent de gérer les incertitudes

d'une façon ecace et ceci aussi bien pour les modèles SPR que CCP.

Cette thèse utilise la programmation stochastique et le paradigme à priori pour étudier deux

VRP stochastiques et un problème de planication de la maintenance de parcs d'éoliennes. Dans

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A.1. INTRODUCTION AUX VRP

le chapitre 2 une introduction générale aux problèmes VRPs et une revue des VRPs stochastiques

sont présentées. Le chapitre 3 est consacré au VRP avec demandes stochastiques (VRPSD). Pour

ce dernier une métaheuristique hybride composée d'un algorithme mémétique et d'une procédure de

redémarrage, par une méthode de type Greedy Randomized Adaptive Search Procedure (GRASP), est

proposée. Les résultats de cette approche hybride montrent son ecacité en comparaison avec d'autres

méthodes publiées dans la littérature. De plus un nouvel ensemble d'instances de grande taille, inspiré

d'un benchmark dédié au cas déterministe, est proposé pour servir de base à des comparaisons futures.

Le chapitre 4 se concentre sur le VRP avec temps de trajets et de services stochastiques et fenêtres

de temps dures. Le modèle proposé intégre l'impact de la violation des fenêtres de temps sous forme

de recours, mais impose de garantir des niveaux de services. En outre, pour estimer les temps d'arrivée

chez les clients, une approximation par une loi Log-normale est proposée et démontrée ecace par des

tests statistiques. Pour résoudre le problème, un algorithme méméthique parrallèle et à populations

multiples a été développé. Cette méthode a permis d'obtenir de très bons résultats en comparaison

avec ceux disponibles dans la littérature.

Le chapitre 5 présente une revue de la littérature consacrée à la planication des activités de main-

tenance pour un parc d'éoliennes abordé du point de vue décision opérationnelle. Dans ce problème

nous considérons deux critères de décision : l'opérateur du parc veut minimiser les coûts de mainte-

nance tandis que l'investisseur du parc veut produire la plus grande quantité d'énergie possible. Un

modèle mathématique bi-objectif est utilisé pour modéliser le problème sous forme d'un programme

linéaire à variables mixtes. Ce dernier est ensuite résolu par une méthode de type epsilon-contraintes.

Dans la seconde partie de ce chapitre, le problème précédent est étendu sur un horizon de planication

long tout en considérant les stratégies de maintenance. Ce dernier problème est étudiée et abordée

par une méthode basée sur la simulation. Les resultats obtenus montrent l'impact du choix des règles

de priorité utilisées pour l'ordonnancement des tâches de maintenance sur les coûts et la quantité

d'énergie produite. La thèse se cloture avec le chapitre 6 en orant des pistes pour des recherches

futures sur l'ensemble des problèmes étudiés.

A.1 Introduction aux VRP

Dans le chapitre 2 une introduction générale aux problèmes de tournées des véhicules (VRP en anglais)

est présenté ainsi qu'une étude plus détaillée des problèmes de type VRP avec incertitudes. Les VRPs

ont été largement étudiés depuis leur introduction par Dantzig et Ramser [7]. Ceci peut être expliqué

par deux raisons : l'importance du transport dans les activités humaines (distribution des produits et

services), et le fait que le domaine a été l'origine du développement de diérentes méthodes, exactes

et approchées, pour la résolution de problèmes combinatoires.

Dans sa version de base, le VRP avec contraintes de capacité (CVRP) a comme objectif de

construire un ensemble des tournées de coût minimal qui respectent les contraintes de capacité des

véhicules. Le problème est déni sur un graphe complet non orienté G = (V,E). L'ensemble de n÷uds

est noté V = 0, 1, . . . , i, . . . , n et l'ensemble d'arêtes est E = [i, j]∀i, j ∈ V | i < j. Le n÷ud 0

est associé à un sommet particulier appelé dépôt, et le reste des n÷uds V c = V \ 0 représentent lesclients. De plus, un ensemble de véhicules ayant la même capacité Q sont disponibles au dépôt. Par

ailleurs, chaque arête dans E est associée à un coût non négatif cij , et chaque client dans V c est asso-

cié à une demande qi. La solution du problème est un ensemble de tournées visitant une et une seule

fois chaque client. Chaque tournée est une séquence ordonnée de n÷uds r = r0 = 0, r1 = 0, . . . , rj ,

. . . , rk, rk+1 = 0 qui démarre et nit au dépôt. Ainsi, le coût d'une tournée particulière r est calculé

par Cr =∑kj=0 crj ,rj+1 .

Beaucoup de travaux sont consacrés au CVRP, cependant l'existence de plusieurs cas particuliers

dans la réalité a donné naissance à de très nombreuses variantes. Le lecteur peut se référer au livre

de Toth et Vigo [25] pour une revue de la littérature sur les variantes du CVRP. Une de ces variante

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ANNEXE A. RÉSUMÉ EN FRANÇAIS

est le VRP avec fenêtres de temps (VRPTW en anglais). Le VRPTW généralise le CVRP par l'ajout

des durées de trajet et de service, ainsi que par la présence de fenêtres de temps [ai, bi] sur le début

de service pour tout n÷ud ∀i ∈ V . Comme le CVRP, le VRPTW peut être dénit sur le graphe G. Le

VRPTW rajoute un temps de trajet tij ∀ i, j ∈ V pour chaque arête et un temps de service ti ∀ V cpour chaque client. Les fenêtres de temps sont classées en deux types : dures et souples [8]. La version

avec fenêtres de temps dures considère le cas où les services chez les clients doivent impérativement

commencer à l'intérieur de la fenêtre de temps. Par conséquent, si un véhicule arrive chez le client

avant l'ouverture de la fenêtre de temps, il doit attendre jusqu'à ce moment-là. Dans le cas où le

véhicule arrive après la fermeture de la fenêtre de temps, aucun service ne peut être eectué. Dans

le VRPTW avec fenêtres souples, les services en dehors des fenêtres sont autorisés mais une pénalité

proportionnelle à l'écart entre la date de début de service et la borne de la fenêtre de temps est souvent

considérée pour ces évènements. La fenêtre de temps pour le dépôt pose une contrainte sur la date de

départ et de retour à ce dernier.

Solution approaches for VRPs

Exact methods

Heuristics Metaheuristics

Constructive

Two-phase

Branch and Price

Branch and Price and Cut

Branch and Cut

Tabu Search

Simulated Annealing

Iterated Local Search

Matheuristics

Clarke & Wright

Nearest Neighborhood

Ant Colony Optimization

Genetic Algorithms

Cluster-first, route second

Route-first, cluster second

Approximate methods

Figure A.1 : Classication des méthodes de résolution pour les VRP.

Pour résoudre le VRPTW (et en général les VRP) beaucoup de méthodes ont été proposées. En

eet, résoudre les VRPs n'est pas une tâche facile car cette catégorie de problèmes fait partie des

problèmes NP-Diciles. Par conséquent, il n'existe pas d'algorithmes de complexité polynomiale qui

peut les résoudre pour toute taille de problème. Les diérentes approches utilisées pour obtenir des

solutions peuvent être classiés en méthodes exactes et méthodes approchées comme le montre la

gure A.1. Les méthodes exactes permettent de trouver la solution optimale, cependant leurs temps

d'exécution sont très élevés à partir d'une taille donnée. Ceci est d'ailleurs le principal inconvénient de

ce type de méthodes. Actuellement, le CVRP est résolu pour des instances allant jusqu'à 200 n÷uds,

alors que ce chire diminue à 100 n÷uds pour le VRPTW. Les méthodes approchées essaient de

trouver un compromis entre la qualité de la solution et le temps d'exécution. Néanmoins, la plupart

de ces méthodes n'orent aucune garantie d'optimalité de la solution obtenue. Parmi les méthodes

approchées, les métaheuristiques sont très utilisées car elles donnent des solutions souvent assez proches

de l'optimum.

La littérature dédiée aux VRPs continue de s'accroitre mais une grande partie des travaux suppose

que les paramètres des problèmes sont connus à l'avance ou d'une façon déterministe [20]. Cependant,

dans la réalité certains paramètres ne peuvent pas être connus avec certitude à cause des conditions

météorologiques, les accidents, de la présence ou non de la clientèle, etc. Les travaux publiés dans

la littérature montrent qu'ignorer les incertitudes conduit à des solutions infaisables et couteuses. Le

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A.1. INTRODUCTION AUX VRP

tableau A.1 présente une classication des VRPs, telle que suggérée par Pillac et al. [20] selon la

qualité et l'évolution de l'information.

Table A.1 : Taxonomie des VRPs basée sur l'article de Pillac et al. [20]Qualité de l'informationDonnées déterministes Données incertaines

Évolution del'information

Données connuesà l'avance

Statiques et déterministes Statiques et incertaines

Donnéesdynamiques

Dynamiques etdéterministes

Dynamiques et incertaines

Dans le cas statique et déterministe, les paramètres sont considérés comme connus dès la plani-

cation de la solution. Dans le cas déterministe et dynamique les paramètres (ou une partie d'entre

eux) sont complètement inconnus et sont seulement révélés à des moments spéciques. Le cas avec

incertitudes partage une caractéristique avec le cas dynamique, étant donné que les vraies valeurs

des paramètres ne deviennent connues qu'à des moments précis. Dans le cas statique et incertain, on

dispose d' informations exploitables sur l'incertitude des paramètres (leurs lois de probabilité, les inter-

valles dont lesquels ils prennent leurs valeurs, etc.). Ces informations sont donc utilisées pour résoudre

le problème. Finalement dans le cas dynamique et incertain, les paramètres (ou une partie d'entre

eux) sont inconnus, mais comme dans le cas statique et incertain, il existe des informations relatives

à ces derniers. Une autre diérence fondamentale entre les problèmes statiques et dynamiques est la

façon dont laquelle les solutions sont calculées. Dans le cas statique, une solution reste non modiable

quelques soient les vraies valeurs des paramètres. Dans le cas dynamique en revanche, la solution peut

être constamment modiée pour s'adapter aux informations qui arrivent au fur et à mesure.

Trois approches ont été principalement utilisées pour modéliser les incertitudes des paramètres

des VRPs, à savoir : la programmation stochastique, l'optimisation par intervalles, et la logique oue.

L'optimisation par intervalles modélise les paramètres incertains par des intervalles de valeurs pos-

sibles. L'objectif poursuivi par l'optimisation par intervalles est de trouver des solutions qui sont

faisables pour toutes les réalisations possibles des paramètres [2]. Des VRPs avec incertitudes sur les

demandes et les temps apparaissent dans la littérature. Adulyasak et Jaillet [1] traitent la version avec

temps de trajets incertains. Les auteurs proposent une comparaison entre l'optimisation stochastique

et l'optimisation par intervalle. Les auteurs montrent que les solutions robustes surpassent largement

les solutions issues de l'optimisation stochastique dans le cas où les paramètres ne sont pas modélisés

par la bonne loi de probabilité. Enn, la logique oue permet aussi de représenter les incertitudes en

utilisant des variables oues. Certains travaux combinent deux approches en utilisant par example des

lois de probabilité pour modéliser les paramètres du problème et des nombres ous pour modéliser

leur espérance ou leur variance [15].

Parmi les trois approches, la programmation stochastique est la plus répandue pour résoudre

les VRPs avec incertitudes. Dans cette approche, les paramètres sont modélisés par des variables

aléatoires de lois connues. De ce fait, les contraintes des problèmes peuvent ne plus être respectées

par les solutions. Par exemple, dans le VRP avec demandes stochastiques (VRPSD), la réalisation

de la demande d'un client peut dépasser la capacité restante du véhicule. Si une contrainte n'est pas

respectée par la réalisation des parametres du problème un "`échec"' est dire de se produire. Pour les

considérer, deux types des modèles sont utilisés [12] : la réoptimisation et les approches à priori avec

recours.

Les modèles qui utilisent la réoptimisation ne reposent pas sur une solution xe. En eet la solution

est construite et modiée au fur et à mesure que les informations apparaissent. Dès révélation de ces

dernières, les tournées peuvent-être réoptimisées à nouveau. Toutefois, ce type d'approche rend la

coordination des véhicules assez ardue et la vitesse à laquelle les solutions nécessitent d'être fournies

reste un problème.

138


Les méthodes du type à priori sont basées sur des solutions statiques et peuvent être divisées en

deux catégories : les problèmes avec recours (SPR) et les problèmes avec contraintes probabilistes

(CCP). Les premiers utilisent des actions appelées "`recours"' qui permettent de rétablir la faisabilité

de la solution quand des échecs se produisent. Un exemple pour le VRPSD est de revenir au dépôt

quand la demande d'un client dépasse la capacité disponible du véhicule. Après le passage au dépôt,

le véhicule reprend la tournée depuis le client où l'échec s'est produit. En outre, le véhicule complète

la demande restante avant de servir les clients suivants planiés dans la tournée. Les coûts associés

à ce type d'actions sont rajoutés dans la fonction-objectif du problème. D'autre part, les problèmes

avec des contraintes probabilistes cherchent à limiter la probabilité des échecs à un seuil. Les CCP

sont recommandés quand la dénition du recours est trop dicile ou quand un niveau de service doit

être garanti.

Une façon commune de classier les VRP stochastiques (SVRP) consiste à considérer les paramètres

entâchés par les incertitudes. Selon Gendreau et al. [13] trois catégories peuvent être considérées :

les VRPs avec demandes stochastiques (VRPSD pour VRP with Stochastic Demands), VRPs avec

incertitudes sur la présence des clients (VRPSC), et les VRPs avec temps stochastiques (VRPST).

Le problème avec demandes stochastiques a été le plus étudié et les modèles avec recours ont été

privilégiés par apport aux modèles avec contraintes probabilistes. Le recours classique considère que

lorsque la capacité d'un véhicule est épuisée, le véhicule fait un retour au dépôt pour s'approvisionner

puis revient chez le client où l'échec s'est produit. D'autres recours existent pour le VRP avec la

même politique de réapprovisionnement mais des visites au dépôt pouvant être eectuées avant que la

capacité du véhicule ne soit atteinte. De cette façon, des économies en temps et en distance parcourue

peuvent être réalisées. Ils existent d'autres recours plus complexes toutefois les recours simples ont été

favorisés. Pour résoudre le VRPSD des méthodes exactes ont été proposés pour les cas où des recours

simples sont utilisés. Cependant c'est les mpethodes apporchées les plus utilisés. Ces méthodes sont

souvent testés sur un ensemble standard d'instances, toutefois la taille de ces dernières demeurent

petites.

Les problèmes avec clientes stochastiques sont ceux dans lesquels la présence des clients est incer-

taine. Quand le véhicule arrive chez le client, ce dernier peut-être présent ou non avec une certaine

probabilité. Les VRPCS sont les moins étudiés parmi les VRP stochastiques. En eet, les VRP avec

temps (trajet ou service) stochastiques (VRPST) ont reçu plus d'attention que les VRPCS mais de-

meurent moins étudiés que les VRPSD. La plupart des travaux consacrés à cette catégorie de problèmes

considèrent des fenêtres de temps sur le service. Ces fenêtres de temps peuvent être souples ou

dures , les premières étant largement favorisées. Ceci peut être expliqué par l'eet des fenêtres de

temps dures sur les temps d'arrivée chez les clients. Généralement, les temps de trajets sont représentés

par des lois qui ont des propriétés de convolution, mais les fenêtres dures empêchent l'utilisation de

ces propriétés. Des problèmes qui prennent en compte les deux types des fenêtres sont les plus étudiés.

Ces travaux considèrent que la date au plus tôt de service (début de fenêtre de temps) doit absolu-

ment être respectée mais autorise le service après la date de fermeture des fenêtres. La complexité des

VRPST explique le faible nombre de publications utilisant des méthodes exactes. L'utilisation de ces

dernières est limitée aux cas dans lesquels les variables aléatoires sont discrètes, ou quand l'espace de

scénarios réduit, où au cas de variables aléatoires additives (avec des convolutions possibles à calculer).

De même que pour les autres VRP stochastiques, les méthodes approchées restent les plus privilégiées

pour résoudre le VRP avec temps stochastiques.

La complexité des VRP avec incertitudes a limité le développement des recherches sur ce sujet ainsi

que la taille des instances résolues. Toutefois, comme pour le cas déterministe, il existe un réel besoin

de méthodes puissantes pour résoudre les problèmes dans des conditions réelles, autant en termes

de taille, de nature de paramètres aléatoires et de contraintes complexes. Donc, dans cette thèse on

propose d'étudier les VRP de nature stochastique et de développer des approches adaptées capables

de résoudres des instances de grande taille. Ce travail est l'un des rares à considérer des fenêtres de

139

A.2. UNE MÉTHODE HYBRIDE POUR LES VRP AVEC DEMANDES STOCHASTIQUES

temps dures sur le service tout ayant des temps de trajet et service stochastiques.

A.2 Une méthode hybride pour les VRP avec demandes sto-

chastiques

Le chapitre 3 présente une méthode de résolution approchée pour le VRP avec demandes stochastiques

(VRPSD). Un nouvel ensemble d'instances est proposé pour permettre des comparaisons futures. La

méthode de résolution développée pour ce problème est un algorithme mémétique (MA) hybridé

avec une méthode gloutonne du type Greedy Randomized Adaptive Search Procedure (GRASP).

L'algorithme proposé montre une meilleure performance que les méthodes trouvées dans la littérature.

En eet, pour les instances "`classiques"' de Christiansen et Lysgaard [4] notre méthode, trouve toutes

les meilleures solutions connues dans un temps de calcul réduit.

Le VRPSD est une généralisation du CVRP dans lequel les demandes des clients sont représentées

par des variables aléatoires. Le problème étudié dans ce chapitre considère une version du VRPSD

avec les caractéristiques suivantes. La demande qi de chaque client i est modélisée par une variable

aléatoire qui suit une loi de probabilité ψ avec espérance E [qi] > 0 et variance V ar [qi] > 0. Il est

supposé que la loi ψ est connue et que les demandes sont indépendantes entre elles. De plus, il est

considéré que la loi de probabilité de la somme des variables ψ est aussi une variable aléatoire de

même nature ψ.

E [Cr] =

k∑j=0

crjrj+1 +

k∑j=1

ERCrj (A.1)

Pour modéliser le VRPSD nous avons utilisé la programmation stochastique avec recours. Le

recours utilisé est le même que celui proposé par Bertsimas [3]. Celui-ci prévoit que lorsqu'un véhicule

atteint sa capacité Q, il retourne au dépôt pour déchargement. Ensuite, le véhicule reprend sa tournée

depuis le client où la rupture de charge est survenue pour compléter la demande qui n'a pas pu être

satisfaite, puis le véhicule continue avec sa tournée. L'objectif du VRPSD avec recours est de minimiser

le côut total des tournées y compris ceux relatifs aux recours. L'évaluation du coût d'une tournée est

donnée par l'équation A.1. Elle inclut les coûts déterministes et la valeur moyenne du recours (ERC).

Pour cette dernière, la demande cumulée jusqu'au client ri est dénie par Dri =∑ij=1 qrj . Du fait de

l'hypothèse d'independence des variables aléatoires relatives aux demandes, l'espérance et la variance

de la demande cumulée se calcule par E [Dri ] =∑ij=1E [qrl ], V ar [Dri ] =

∑ij=1 V ar [qrl ]. La valeur

moyenne du recours est donc calculée par l'équation A.2. Il faut remarquer que la otte est supposée

illimitée, et que le nombre de véhicules n'a pas d'impact sur la fonction objectif. Ainsi, pour éviter des

solutions avec des tournées avec des échecs fréquents, la contrainte E [Drk ] < Q ∀r est aussi rajoutée.

ERCri = 2 · c0ri ·

[ ∞∑u=1

(P(Dri−1

≤ uQ)− P (Dri ≤ uQ)

)](A.2)

La méthode de résolution proposée pour résoudre le VRPSD est une hybridation entre un algo-

rithme mémétique (MA) [21] et une méthode de type Greedy Randomized Adaptive Search Procedure

(GRASP) [10]. L'objectif de la méthode baptisé MA+GRASP est de trouver des solutions de qualité

en peu de temps même pour des grandes instances. Pour échapper aux optima locaux, le MA utilise

une méthode de re-démarrage basé sur la méthode GRASP. Le but de cette procédure est d'accroître

l'exploration de l'espace de recherche tout en controlant son impact négatif sur les temps d'exécution.

L'algorithme 6 présente un aperçu global du fonctionnement de la méthode. Elle utilise une po-

pulation Pop à taille xe dans laquelle chaque individu représente une solution du problème. L'algo-

rithme exécute les lignes 4 et 19 jusqu'à ce que un critère d'arrêt soit satisfait. À chaque itération,

MA+GRASP crée une nouvelle solution en croisant deux solutions présentes dans la population. Ainsi,

140


Algorithm 6 MA + GRASP1: Pop← Initialize population2: φ← Constant > 03: i← 14: while not (stop) do5: c← crossover (Pop)6: Mutation, pmp (c)7: Split (c)8: Local Search, pls (c)9: if Executed Local Search then10: Concatenate and Split (c)11: end if12: if Is Not Clone (c) then13: Update Population (c,Pop)14: end if15: Update(i)16: if i ≥ φ then17: Restart with GRASP (Pop)18: i← 119: end if20: end while

cette nouvelle solution peut être modiée par des procédures de mutation et de recherche locale (lignes

6 et 8). Comme MA+GRASP est basé sur un algorithme génétique, des méthodes permettant de co-

der et de décoder une solution sont utilisés. Le décodage utilise la méthode Split de Prins [21], alors

que le codage concatène toutes les tournées de la solution sans considérer le dépôt. Dans la ligne 17,

et si le nombre d'itérations sans amélioration de la meilleure solution trouvée dépasse une valeur φ,

l'algorithme redémarre à nouveau le MA. Cette action implique la réinitialisation de la population de

MA+GRASP.

Algorithm 7 GRASP RestartRequire: Population Pop, iterations1: BestIndividual← Pop[0]

2: NumRuns←⌈(Popsize−1)

2

⌉3: Clear Population(Pop)4: i← 15: while i ≤ NumRuns do6: j← 17: BestCost←∞8: BestSol← null9: while j ≤ iterations do10: c← Generate Individual with RNN11: Split (c)12: Local Search (c) . Only deterministic costs13: if Cost (c) < BestCost then14: BestCost← Cost (c)15: BestSol← c16: end if17: j ← j + 118: end while19: Local Search (BestSol)20: Add to population (BestSol,Pop)21: i← i+ 122: end while23: Add to population (BestIndividual,Pop)24: Complete Population with Random Individuals25: Sort(Pop)

Pour le redémarrage, MA+GRASP utilise une méthode GRASP qui se charge de remplacer l'an-

cienne population. Cette méthode exécute la plupart de temps une recherche locale qui considère juste

le coût déterministe. De cette manière, les temps d'exécution sont contrôlés. L'idée derrière cette façon

141

A.2. UNE MÉTHODE HYBRIDE POUR LES VRP AVEC DEMANDES STOCHASTIQUES

de procéder est que, même si les solutions déterministes n'ont pas la meilleure performance dans le

contexte stochastique, elles peuvent servir de base pour être améliorées rapidement. Alors, une fois

que des bonnes solutions déterministes ont été trouvées, elles sont améliorées avec la recherche locale

qui prend en compte les coûts du recours. Il faut remarquer qu'une partie de la population est com-

plétée par des solutions aléatoires pendant le redémarrage. Cette partie a comme but d'augmenter la

diversité et donc l'exploration de l'espace de recherche.

Notre approche MA+GRASP permet d'obtenir de très bons résultats. Une comparaison avec les

travaux de Mendoza et al. [17] basé sur un GRASP, et ceux de Goodson et al. [14] ( présentant un récuit

simulé) conrme la supériorité de notre méthode. MA+GRASP arrive à trouver toutes les meilleures

solution connues (BKS en anglais). De plus, elle présente un écart moyen de juste 0.004%. Cela veut

dire que l'algorithme retrouve des solutions très proches du BKS à chaque exécution et pour chaque

instance (10 exécutions par instance). De plus, MA+GRASP utilise moins de dix secondes pour trouver

ces valeurs, ce qui permet de montrer que MA+GRASP est très ecace pour résoudre le VRPSD.

Pour tester la méthode sur des instances plus diciles, on a proposé un nouvel ensemble d'instances

basé sur les ensembles A et P d'Augerat qui n'ont pas été considérés par Christiansen and Lysgaard

[4] (22 instances), ainsi que des chiers test de Cristodes, Mingozi et Toth CMT (4 instances), et de

Rochat et Taillard (13 instances) 1. Le nouvel ensemble présente en moyenne 2.4 plus de clients que

l'ensemble original de Christiansen and Lysgaard. Les caractéristiques et les BKS du nouvel ensemble

sont reportées dans le tableau A.2.

On constate comme illustré dans le tableau A.2 que le coût des solutions BKS est principalement

composé du coût déterministe (Det) sur les instances testées. Cela explique pourquoi les solutions qui

considèrent juste le coût derministent sont de bonnes solutions aussi pour le problème stochastique.

Cependant, si seuls les meilleurs solutions du problèmes déterministe (BDS) sont évaluées, une forte

augmentation intervient dans les coûts du recours lorsque ce dernier est considéré. On peut également

clairement noter que pour des instances de plus grande, les économies générées par les solutions

stochastiques sont plus importantes.

Finalement, pour analyser l'impact de la méthode GRASP sur les perfomances de la méthode

hybride MA+GRASP, un ensemble de tests numériques a été eectué. Deux versions modiées de

MA+GRASP ont été utilisées pour résoudre le VRPSD pour le nouvel ensemble d'instances. La pre-

mière considère le redémarrage mais, ne fait pas appel au GRASP (MA+RANDOM). Dans ce dernier

algorithme, tous les individus crées pour remplacer l'ancienne population sont générés aléatoirement.

La deuxième variante est une version sans redémarrage (NR-MA). Une série de calculs eectuée sur les

trois algorithmes a été réalisée, le but étant d'analyser le temps que chacun prend pour atteindre les

valeurs cibles pour chaque instance. Cet ensemble de données a servis pour construire des graphiques

de probabilités cumulés comme proposé par Reyes and Ribeiro [22] (graphiques MTT). Les gures

A.2 et A.3 montrent les graphiques pour diérentes valeurs cibles en fonction de l'écart à la meilleure

solution connue. Il est évident que la méthode MA+GRASP a l'avantage quand il s'agit de trouver

des solutions proches du BKS. En eet, MA+GRASP a plus de chance de trouver tous les BKS avec

une même quantité de temps en comparaison avec le MA+RANDOM ou NR-MA. Donc, ceci conrme

que le redémarrage avec la méthode GRASP, qui utilise de "`bonnes"' solutions déterministes comme

base pour le contexte incertain, génère un eet positif sur les résultats trouvés par la méthode.

On peut conclure par les résultats obtenus que MA+GRASP a une performance supérieure à celle

des autres méthodes de la littérature. En comparaison avec les versions sans redémarrage ou avec

redémarrage complètement aléatoire, le MA+GRASP retrouve des solutions de meilleure qualité en

moins de temps. Dans la continuité de ce travail, on peut s'intéresser à l'utilisation de nouveaux

recours, à la considération d'autres paramètres stochastiques (comme le temps), et l'extensions du

problème pour considérer des demandes corrélées.

1Toutes les instances sont disponibles dans la page : http ://vrp.atd-lab.inf.puc-rio.br

142


Table A.2 : Synthèse des résultats sur le nouvel ensemble d'instancesBKS BDS

Instance |V | Min veh Q FC Veh Total Det Rec Total Rec VSSA-n61-k9 61 9 100 0.98 10 1144.23 1084 60.23 1215.37 181.37 5.85%A-n62-k8 62 8 100 0.92 9 1430.81 1375 55.81 1533.07 245.07 6.67%A-n63-k10 63 10 100 0.93 11 1459.49 1412 47.49 1581.17 267.17 7.70%A-n63-k9 63 9 100 0.97 10 1847.69 1734 113.69 1991.75 375.75 7.23%A-n64-k9 64 9 100 0.94 10 1569.85 1534 35.85 1699.68 298.68 7.64%A-n65-k9 65 9 100 0.97 10 1313.30 1266 47.30 1421.52 247.52 7.61%A-n69-k9 69 9 100 0.94 10 1259.35 1214 45.35 1339.55 180.55 5.99%A-n80-k10 80 10 100 0.94 11 1987.17 1918 69.17 2109.56 346.56 5.80%E-n23-k3 23 3 4500 0.75 3 569.72 569 0.72 569.72 0.72 0.00%E-n30-k3 30 3 4500 0.94 4 504.55 503 1.55 569.92 32.92 11.47%E-n76-k10 76 10 140 0.97 11 885.11 861 24.11 911.70 81.70 2.92%E-n76-k14 76 14 100 0.97 16 1118.90 1086 32.90 1187.18 166.18 5.75%E-n76-k7 76 7 220 0.89 7 698.95 692 6.95 723.96 41.96 3.46%E-n76-k8 76 8 180 0.95 8 771.23 744 27.23 791.88 56.88 2.61%E-n101-k14 101 14 112 0.93 15 1164.149 1116 48.15 1233.84 166.84 5.65%E-n101-k8 101 8 200 0.91 8 839.47 824 15.47 878.36 63.36 4.43%P-n55-k8 55 7 160 0.93 7 607.71 581 26.71 631.82 43.82 3.82%P-n65-k10 65 10 130 0.94 10 854.06 802 52.06 861.52 69.52 0.87%P-n70-k10 70 10 135 0.97 11 882.01 851 31.01 929.82 102.82 5.14%P-n76-k4 76 4 350 0.97 4 609.54 593 16.54 609.62 16.62 0.01%P-n76-k5 76 5 280 0.97 5 648.11 628 20.11 649.10 22.10 0.15%P-n101-k4 101 4 400 0.91 4 686.81 684 2.81 694.47 13.47 1.10%CMT12 101 10 200 0.91 10 982.80 827 155.81 984.31 164.31 0.15%CMT11 121 7 200 0.98 8 1201.15 1187 14.15 1226.62 189.62 2.08%CMT4 151 12 200 0.93 12 1072.19 1036 36.19 1130.31 114.31 5.02%CMT5 200 16 200 1.00 18 1378.85 1355 23.85 1486.96 209.96 7.27%Tai75a 76 10 1445 0.95 11 1653.85 1649 4.85 1718.39 102.39 3.76%Tai75b 76 9 1679 0.99 10 1353.09 1343 10.09 1438.62 99.62 5.95%Tai75c 76 9 1122 0.94 9 1349.87 1332 17.87 1396.98 110.98 3.37%Tai75d 76 9 1699 0.93 9 1392.86 1391 1.86 1444.39 87.39 3.57%Tai100a 101 11 1409 0.98 12 2107.71 2089 18.71 2388.31 350.31 11.75%Tai100b 101 11 1842 0.96 12 1985.33 1986 29.33 2118.93 181.93 6.31%Tai100c 101 11 2043 0.93 11 1421.66 1413 8.66 1526.73 126.73 6.88%Tai100d 101 11 1297 0.95 12 1602.53 1599 3.53 1748.94 174.94 8.37%Tai150a 151 15 1544 0.94 15 3211.55 3188 23.55 3574.95 530.95 10.10%Tai150b 151 14 1918 0.95 15 2792.39 2789 3.39 3113.52 397.52 10.31%Tai150c 151 14 2021 0.99 15 2406.13 2380 26.13 2546.29 201.29 5.50%Tai150d 151 14 1874 0.97 15 2718.39 2692 26.39 3044.91 406.91 10.64%Tai385 386 46 65 1.00 55 29364.03 28007 1357.03 31360.45 7008.45 6.22%

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 10 20 30 40 50 60 70 80

Cu

mu

lati

ve p

rob

abili

ty

Time (seconds)



0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Cu

mu

lati

ve p

rob

abili

ty

Time (seconds)



Figure A.2 : grapheique MTT pour les versions de MA - 5% and 1%

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 2000 4000 6000 8000 10000 12000 14000 16000

Cu

mu

lati

ve p

rob

abili

ty

Time (seconds)

MTTT Plot - Target Value 0.5% gap to BKS


0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 5000 10000 15000 20000 25000 30000 35000 40000

Cu

mu

lati

ve p

rob

abili

ty

Time (seconds)



Figure A.3 : grapheique MTT pour les versions de MA - 0.5% and 0%

143

A.3. UN ALGORITHME PARALLÈLE POUR LES VRP AVEC TEMPS DE TRAJET ETTEMPS DE SERVICE STOCHASTIQUES

Contributions

Des résultats préliminaires dans ce chapitre ont été présentés à la conférence CIE45 :


A Memetic Algorithm for the Vehicle Routing Problem with Stochastic Demands

In Proceedings of the 45th International Conference on Computers & Industrial Engineering CIE45

Metz, France, 2830 October, 2015.

Un article complet dédié au VRPSD fait l'objet d'une révision mineure dans le journal Computers

& Operations Research.

A.3 Un algorithme parallèle pour les VRP avec temps de trajet

et temps de service stochastiques

Dans le chapitre 4, on a étudié la version du VRP avec temps de trajet et temps de service stochastiques

dans un contexte des tournées de maintenance. Les fenêtres de temps considérées dans ce travail sont

dures. Le caractère aléatoire des temps de trajets reète les eets des conditions climatiques, l'état

des rues, accidents, etc. Les temps de service stochastiques représentent la variabilité de la durée de

réparation d'une panne. Pour résoudre le problème un algorithme mémétique qui utilise plusieurs

populations, chacune étant utilisée pour résoudre le problème avec des considérations spéciques sur

pour l'intégration des incertitudes. Des informations sont partagées entre les diérentes populations,

dans un paradigme d'algorithme parralèle, par transfert d'individus d'une population vers une autre.

Le VRP avec temps de trajet et de service stochastique et fenêtres de temps (SVRPTW en anglais)

utilise le VRPTW comme base. La diérence est que les temps de service et de trajet sont des variables

aléatoires désignées comme si (∀i ∈ Vc) et tij (∀ (i, j) ∈ E ′). En plus l'utilisation d'un véhicule a un

coût xe M . Un modèle combinant des contraintes probabilistes et recours (CCP + SPR) est utilisé

pour dénir des pénalités quand le service n'est pas eectué dans les fenêtres des clients, tout en

garantissant que ces événements soient rares. La partie CCP considère trois niveaux de service dans

notre modèle. Le premier (α) garantit une probabilité de service pour chaque client, β est utilisé pour

la probabilité de retour des véhicules au dépôt avant la date limite. Le niveau de service γ assure

que tous les clients dans la solution soient servis avec une probabilité γ. En dénissant AT i comme

le temps d'arrivée chez le client i, les conditions précédentes exprimées pour les clients et dépôt sont

décrites par les contraintes (A.3) et (A.4) pour chaque tournée. Ainsi, le niveau de service pour une

solution s est assuré par (A.5) dans laquelle Ur est la probabilité de servir tous les clients de la tournée

r.

P(AT rj ≤ brj

)≥ α ∀rj ∈ Vc (A.3)

P(AT rk+1

≤ b0)≥ β (A.4)∏

r∈sUr ≥ γ (A.5)

Le recours proposé pour le problème a été déjà décrit dans la littérature, cependant il n'existe pas

des résultats pour ce type de recours. Ce dernier considère que si un véhicule arrive chez un client

i ∈ V c plus tard que la date de fermeture de sa fenêtre de temps bi, le véhicule continuera sa route

sans eectuer le service. Un nouveau véhicule viendra eectuer le service plus tard chez ce client. Ce

recours conduit à une pénalité équivalente à l'utilisation d'un véhicule dédiée pour servir le client i

tout seul. Le recours est simple mais, il faut retenir que tous les temps sont stochastiques, donc les

échecs ne peuvent pas être connus à l'avance. La vraie valeur des temps de trajets n'est connue qu'une

fois la traversée du lien est terminée, de même pour le temps de service.

Le coût d'une tournée peut être déni par l'équation (A.6). Trois parties composent le coût de la

144


420 440 460 480 500 520 540 560

Arrival Time at i (a)

TimeD

ensi

ty

500 510 520 530 540

Start Service Time at i (b)

Time

Den

sity

20 40 60 80 100

Maintenance time (c)

Time

Den

sity

0 20 40 60 80

Travel Time i to j (d)

Time

Den

sity

520 540 560 580 600 620 640 660

Arrival Time at j (e)

Time

Den

sity

Figure A.4 : Temps d'arrivée et de début des services chez un client.

tournée, d'abord la valeur xe liée à l'utilisation d'un véhicule, le coût des arêtes (cij) parcourues dans

la tournée, et le coût moyen du recours. Dans ce chapitre le coût (M) est choisie comme étant très

élevé. À cet eet, l'optimisation du problème est du type hiérarchique, d'abord minimisant le nombre

de tournées, et ensuite les coûts tournée et des recours.

cr = M +

k∑j=0

crjrj+1 +

k∑j=1

P(AT rj > brj

)·(2 · c0,rj +M

)(A.6)

Il faut noter que les temps d'arrivées chez les client sont aussi des variables aléatoires. En eet,

AT j dépend des temps de trajets et de services précédant l'arrivée chez le client j. La gure A.4

expose une telle situation. La partie (4.a) représente la répartition des dates d'arrivée chez un client

i, les lignes verticales représentent la fenêtre de temps. La répartition des dates de début de service

(compte tenu de la contrainte de fenêtre de temps) est tracée dans la gure (A.4.b). Les gures A.4.c

et A.4.d représentent la répartition des temps de service (maintenance) et des temps de trajet entre

i et j. Finalement, la partie A.4.e est le temps d'arrivée chez le client j. La gure illustre la diculté

de modéliser les temps d'arrivée dans les problèmes avec temps stochastiques puisque les contraintes

de fenêtres de temps rendent impossible l'utilisation de la convolution et de ses propriétés.

La loi de probabilité de AT i est nécessaire pour la vérication des contraintes probabilistes et le

calcul de la fonction-objectif. Dans ce travail AT i est aprochée par la variable AT ri de distribution

Log-normale à l'issue d'une étude comparative de plusieurs lois. En utilisant les instances de Solomon

[24], une série de tournées ont été construites avec l'heuristique de Clarke et Wright. Les temps de

trajet ont été simulés en utilisant la loi Log-normale et les temps de service avec la loi Gamma.

Les espérances de ces variables sont égales aux valeurs déterministes des instances originalles, et les

variances ont été calculées en utilisant des coecients de variation de 0.2. Les résultats de cette analyse

montrent que la loi Log-normale présente la meilleure performance parmi plusieurs lois testées. En

eet, les écarts entre les variables AT i et AT ri restent petits même si le temps d'arrivée ne sont pas

statistiquement représentés par des lois log-normales.

L'algorithme proposé pour résoudre le SVRPTW est un algorithme mémétique (MA) avec plusieurs

populations (MPMA Multi Population Memetic Algorithm en anglais). Chaque population fonctionne

comme un algorithme mémétique individuel mais communique des informations aux autres popula-

145


tions à des moments précis. La structure de base des algorithmes mémétiques est la même que celle

décrite dans l'algorithme 6. Cependant, la gestion de la diversité est controlée par la communication

entre les populations. De plus les MA individuels n'utilisent pas une procédure de redémarrage. La

communication entre les MA se fait grace au transfert des solutions d'un MA vers un autre MA.

Quand une solution est échangée entre deux MA, celui qui la reçoit l'utilise comme s'il s'agissait d'un

nouvel individu créé par la procédure de croissement. Comme les MA fonctionnent en parallèle, la

communication s'exécute d'une manière asynchrone controlée par une mesure de temps. Quand les

diérents MAs se terminent, le MPMA maître prend les meilleures solutions de chaque MA et les

décode avec la simulation Monte Carlo. Cette étape est très importante car la simulation permet

d'estimer AT ri et donc le "`vrai"' coût des tournées ainsi que leur faisabilité.

Pour ce chapitre on dénit trois (K = 3) MA, appelés MA1, MA2 et MA3. En plus, chaque MA

est en charge de résoudre son propre problème. MA2 résout le VRPTW avec les temps de trajet et de

service égaux à leur valeurs moyennes. MA3 est en charge d'un VRPTW mais les valeurs des temps

de trajets et de service sont plus grandes que les valeurs moyennes (choisis comme le percentile 75%

de chaque loi). Le but de MA2 et MA3 est de trouver de bonnes solutions de base pour MA1. Ce

dernier est en charge de résoudre le SVRPTW en utilisant la loi Log-normale comme approximation

de AT ri .

Algorithm 8 MPMARequire: α, β, γ, τ, numSimulations, runT ime, F1: Create(MAf ) ∀f = 1...F2: initialT ime← CurrentTime3: In parallel : // The MAs start and run in parallel4: for all f = 1...F do

5: run(MAf (α, β, γ, τ, runT ime)) // See algorithm 36: end for

7: while CurrentTime - initialT ime ≤ runT ime do8: Communicate MAf ∀f = 1...F9: end while

10: for all f = 1...F do

11: Best← Best ∪Get best Chromosomes(MAf )12: end for

13: Pool← 14: for all C ∈ Best do15: Split(C,α, β, γ, numSimulations) // Using Monte Carlo Simulation16: Pool← Pool ∪ Simulated routes during Split17: end for

18: return SolveSetPartitioning(Pool, γ)

L'algorithme MPMA est testé sur des instances modiées de Solomon [24]. Pour valider l'impor-

tance de la présence des trois populations, diérentes versions de MPMA ont été testées. Ces résultats

sont présentés dans le tableau A.3. Selon les données, on voit que l'utilisation des diérentes MA

réduit le nombre de véhicules cumulés pour les instances de Solomon. De plus, en moyenne le nombre

de véhicules et le coût des tournées est réduit par rapport aux autres versions. Cependant le MPMA

complet (avec 3 populations) présente des coûts de recours plus élevés. On peut en conclure que le

fait d'utiliser plusieurs populations génère un impact positif dans les résultats. Ceci montre que les

méthodes de résolution pour les problèmes stochastiques doivent être adaptés pour améliorer leur

performance.

Bien que les résultats du MPMA sont intéressants, il est nécessaire de le comparer avec d'autres

méthodes publiées dans la littérature. Parmi les problèmes récemment étudiés, les travaux les plus

proches du problème proposé sont ceux de Miranda et Conceição [18], et Nguyen et al. [19]. Les

deux travaux considèrent le SVRPTW avec temps de trajets stochastiques. Miranda et Conceição [18]

prennent également en compte des temps de service stochastiques. Les services après la fermeture des

fenêtres de temps sont autorisés dans les deux travaux. Cela veut dire que les services sont toujours

exécutés par les véhicules même en cas d'arrivées tardives. Les résultats sont donnés sur les instances

146


Table A.3 : Comparaison des diérentes MPMA Instances avec 100 clients

Metric MA1 MA1 + MA2 MA1 + MA3MA1 + MA2

+ MA3

# Véhicules Meilleuressolutions

579 575 576 554

# Véhicules moyenne 616.7 612.7 608.4 577.3Coût déterministe moyen 1156.56 1158.88 1138.24 1123.87Coût du recours moyen 3.86 3.90 3.88 31.16Temps moyen pendant MA (s) 100 100 100 100Temps moyen simulation (s) 35.28 65.06 61.85 54.70Temps total moyen (s) 135.28 165.06 161.85 154.80

Table A.4 : Comparaison du MPMA avec la métaheuristique ILS de Miranda et Conceição [18]

InstanceMé-thode

# Veh.Moyen

CDMoyen

NS MinMoyen

NSMoyen

ENSMoyen

Max.ENS

Temps

(s)

R105 ILS 17.67 1615.75 81.66% 99.31% 0.18% 1.81% 18.40MPMA 17.30 1470.89 80.55% 97.30% 0.39% 2.81% 31.82

R109 ILS 15.00 1488.81 79.53% 95.45% 0.17% 0.89% 6.36MPMA 15.00 1270.27 80.96% 98.00% 0.28% 3.44% 32.21

C101 ILS 17.00 2284.77 83.82% 95.52% 0.22% 0.87% 29.49MPMA 17.00 1372.27 80.42% 95.42% 0.64% 3.56% 31.77

C106 ILS 14.67 1722.32 81.33% 98.65% 0.18% 0.70% 5.56MPMA 14.00 1141.17 80.57% 96.01% 0.72% 4.78% 32.25

RC101 ILS 19.67 2012.76 78.32% 100.22% 0.16% 1.42% 15.18MPMA 19.60 1841.32 80.98% 97.53% 0.46% 4.98% 31.77

RC106 ILS 14.67 1584.63 82.26% 95.57% 0.16% 0.82% 24.43MPMA 15.30 1534.95 81.07% 97.87% 0.40% 4.64% 32.08

RC107 ILS 14.00 1569.10 83.03% 96.23% 0.16% 0.86% 34.99MPMA 13.70 1368.53 81.46% 98.44% 0.23% 3.01% 32.40

Avg. ILS 16.10 1754.02 81.42% 97.28% 0.18% 1.05% 19.20MPMA 15.99 1428.49 80.86% 97.20% 0.44% 3.89% 32.04

classiques de Solomon [24] dédiées au VRPTW. Les deux articles considèrent des lois continues pour

les temps de trajet et de service.

Toutefois chacune des deux études a ses propres particularités. Miranda et Conceição [18] modé-

lisent le SVRPTW comme un CCP avec un taux de service α égal à 80%. Ainsi, les temps de service

et de trajets suivent des lois normales avec des espérance égales à leurs valeurs déterministes et la

variance est déduite à partir des coecients de variations générées par une loi uniforme U [0.1; 0.6].

Le tableau A.4 montre les métriques suivantes : nombre de véhicules, la distance moyenne parcourue,

les niveaux de services (NS) et les écarts entre le niveau de service (ENS) calculé par simulation et

celui prédit par l'approximation de AT i. On peut remarquer que MPMA utilise moins de véhicules

en moyenne qu'ILS avec une distance parcourue beaucoup plus réduite. En plus MPMA est le seul

capable de garantir le niveau de service requis pour toutes les instances (α = 80%).

D'une autre part, le travail de Nguyen et al. [19] utilise des temps de trajets avec une distribution

de probabilité Gamma. L'espérance des variables est égale à la valeur déterministe et l'écart type

est de 50% l'espérance. Le tableau A.5 résume les résultats pour le nombre moyen de véhicules,

le coût de la solution (distance), et le retard moyen (RA) chez chaque client. Dans un objectif de

comparaison, le MPMA est modié pour n'utiliser que le coût du recours sans tenir compte des

contraintes probabilistes. Le retard minimal moyen est atteint par MPMA de même que le coût

moyen minimale. Ceci est réalisé au détriment d'une petite augmentation sur le nombre de véhicules

en comparaison avec les diérentes versions de la Recherche Tabu (TS) de Nguyen et al. [19]. Ces

auteurs considèrent le retard moyen comme objectif principal. On peut conclure que MPMA donne

les meilleurs résultats par rapport à ces deux travaux.

Le modèle proposé dans le chapitre 4 ainsi que la méthode de solution MPMA ont montré leur

exibilité. Les comparaisons avec d'autres méthodes de la littérature donnent l'avantage à MPMA,

même si la méthode utilise des temps d'exécution légèrement supérieurs. Il est également important de

noter que l'approximation de AT i avec des loi Log-normales permet d'avoir des résultats compétitifs

pour les problèmes même si les temps de trajet (et de service) sont distribués par des loi normales où

147


Table A.5 : Comparaison de MPMA avec la TS de Nguyen et al. [19]

InstanceMéthode# Veh. MoyenDC MoyenRA MoyenTemps (s)C1 TS (AD1) 10.33 944.55 2.82 9.20

TS (AD2) 10.33 929.85 3.43 7.94TS (AD3) 10.22 918.61 4.19 8.62MPMA 9.89 908.73 5.55 51.41






Avg. TS (AD1) 8.91 1241.83 15.91 23.48TS (AD2) 8.95 1198.80 11.35 19.46TS (AD3) 8.82 1181.67 12.92 19.98MPMA 9.04 1131.34 9.19 52.16

Gamma2. Il reste dans la continuité de ce travail de tester d'autres types de recours, et l'extensions

du problème pour considérer les temps de trajets corrélés. Aussi, l'approximation de AT i peut être

exploitée dans des méthodes exactes de type Branch-and-Price. Finalement, on prévoit d'étudier des

problèmes multi objectif dont le niveau de service est un objectif et pas une contrainte.

Des résultats présentés dans ce chapitre ont été présentés à la conférence MIM2016 :


A multi population memetic algorithm for the vehicle routing problem with time windows and sto-

chastic travel and service times

In 8th IFAC Conference on Manufacturing Modelling, Management and Control MIM 2016 Troyes,

France, 2830 June 2016

Une extension de ce travail pour résoudre le problème avec une méthodes exacte a été présentée à

la conférence CLAIO2016 :


An approximate column generation for the vehicle routing problem with hard time windows and sto-

chastic travel and service times.

In XVIII CLAIO, the Latin-Iberoamerican Conference on Operations Research Santiago de Chile,

Chile, 26 October 2016.

Un article regroupant les résultats de ce chapitre a été accepté pour publication dans le journal

Computers & Industrial Engineering.

2Avec des coecients de variation de plus de 35%

148


A.4 Gestion des ressources pour la maintenance d'un parc d'éo-

liennes

Le chapitre 5 introduit un problème d'aectation des ressources pour des activités de maintenance

pour un parc d'éoliennes. Le problème étudié sert pour montrer la présence d'objectifs conictuels

associés à diérentes parties prenantes aux parcs éoliens.

L'énergie éolienne est l'une des sources les plus importantes d'énergies renouvelables. Selon le

Conseil Mondial de l'Énergie Éolienne (GWEC en anglais) [6], l'année 2016 s'est achevée avec une

puissance installée de près de 487 GW, soit une croissance de 12,6% par rapport à 2015. Les projections

du GWEC montrent que la capacité d'énergie éolienne augmentera de près de 70% au cours des cinq

prochaines années. Ainsi, on prévoit que d'ici 2021, la capacité installée d'énergie éolienne atteindra

près de 830 GW. Malgré la croissance rapide des installations oshore [6], l'onshore représentait en

2015 près de 97,2% de la capacité éolienne mondiale totale [5]. Les activités d'exploitation et de

maintenance (O&M en anglais) pour les parcs éoliens sont très importantes. En fait, selon [9] l'O&M

représente entre 25% et 30% du coût total de la production d'énergie, et de 25% à 33% du coût total

du cycle de vie d'un parc éolienne. Il est donc indispensable d'optimiser ces coûts pour améliorer

la compétitivité des prix d'énergie et l'attractivité de ces projets. Les activités de maintenance sont

eectuées pour garantir le fonctionnement du système. Leur objectif est non seulement de corriger

mais aussi de prévenir les pannes du système an de retarder le plus possible leur apparition.

Toutefois, pendant ces activités les éoliennes doivent être arrêtés, ce qui conduit à une perte de

production d'énergie. Ainsi, l'exécution de ces activités peut aussi être couteuse si elles sont exécutées

très régulièrement. Par conséquent, il faut trouver un équilibre entre minimiser le risque de défaillance

et le coût total de maintenance (y compris la perte de production liée à l'indisponibilité). De plus

l'interaction de multiples acteurs impliqués dans les parcs éoliens peut impacter la façon dont les

activités de maintenance sont menées. Par exemple, l'agent en charge des O&M veut minimiser ses

coûts tandis que l'investisseur a intérêt à produire la plus grande quantité d'énergie [11]. De manière

générale on peut distinguer trois niveaux de décision, stratégique, tactique et opérationnel dans ce

domaine.

Le niveau opérationnel concerne des décisions quotidiennes. Par exemple, l'ordonnancement des

tâches de maintenance concerne ce niveau de décision. Dans ce dernier, la disponibilité des ressources

joue un rôle central. Kovàcs et al. [16] décrivent l'ordonnancement des tâches de maintenance comme

un composant majeur qui aecte la disponibilité et les coûts des turbines. Dans cet échelon il est

supposé que les actions de maintenance sont connues avant leur exécution car elles auraient été choisis

par une stratégie de maintenance. Aussi, puisque l'horizon de plan est limité à très peu de jours, il

est courant de supposer que les paramètres stochastiques tels que la vitesse du vent ou la hauteur

des vagues sont parfaitement prédits. Des travaux pour l'onshore et l'oshore ont été publiés dans la

littérature. Les objectifs sont normalement associés à la minimisation des coûts. Certaines références

traitent la maximisation de la production énergétique (ou la minimisation des pertes de production)

surtout dans le contexte onshore. La plupart des travaux utilisent des méthodes exactes (surtout

l'optimisation linéaire en nombres entiers), en revanche les paramètres tels que la durées des activités

ou les conditions météorologiques sont déterministes. Aussi, il n'existe pas de travaux qui considèrent

plusieurs objectifs au même temps. Pour combler cet écart de la littérature, le problème suivant est

presenté.

A.4.1 Ordonnancement multi objective pour le problème de maintenance

des éoliennes

Le problème d'ordonnancement des ressources dans un contexte onshore est traité dans ce chapitre.

Deux acteurs sont considérés dans ce travail : l'opérateur O&M et l'investisseur du parc. Chacun

149

A.4. GESTION DES RESSOURCES POUR LA MAINTENANCE D'UN PARC D'ÉOLIENNES

cherche à satisfaire son propre objectif, à savoir, minimiser les coûts d'une part et maximiser l'énergie

produite de l'autre part. Le problème considère un ensemble d'éoliennes pour lesquelles des tâches

de maintenance doivent être exécutées. Un nombre limité de techniciens ayant chacun leurs propres

compétences est dédié à la réalisation de ces tâches. Pendant l'exécution des tâches de maintenance,

les éoliennes sont arrêtées et ne produisent donc pas d'énergie. De plus, toute tâche commencée doit

être achevée. Ainsi, à la n de chaque journée, si la tâche n'est pas encore nie, les techniciens la

poursuivront le lendemain.

Chaque tâche est associée à une fenêtre de temps dont l'ouverture peut correspondre par exemple à

la date de disponibilité des matériaux nécessaires à l'éxécution de la tâche de maintenace. La fermeture

de la fênetre est considérée comme une date à laquelle la tâche devrait nir. Si la fenêtre de temps n'est

pas respectée, une pénalité est appliquée. Chaque tâche nécessite un certain nombre de compétences

et de techniciens par compétence. Le temps est discrétisé, et pendant chaque créneau de temps les

éoliennes qui ne sont pas en maintenance produisent de l'énergie. Pour chaque créneau, la vitesse du

vent est supposée connue ce qui permet de savoir si les conditions permettent ou non d'eectuer la

tâche prévue. Dans le cas où la vitesse est trop élevée, les techniciens doivent arrêter leur travail.

En plus le salaire des techniciens dépend de leurs compétences. Trois coûts sont considérés dans ce

chapitre, le coût lié à l'aectation de techniciens surqualiés, le coût lié aux heures supplémentaires,

et la pénalité en cas de retard d'exécution de la tâche de maintenance.

Algorithm 9 Epsilon Constraints adapté pour la maximisation de Z = (Z1,−Z2)

Require: numSteps1: Solutions← 2: z1∗, z2. ← maxZ1

3: z1∗, z2∗ ← max−Z2| (Z1 = z1∗)

4: Solutions← Solutions ∪ z1∗, z2∗5: z1., z2∗∗ ← max−Z2

6: stepSize← (z2∗−z2∗∗)

numSteps

7: i← 18: while (z2

∗ − i · stepSize) > z2∗∗ do

9: z1∗, z2. ← maxZ1| (Z2 ≤ z2∗ − i · stepSize)10: z1∗, z2∗∗∗ ← max−Z2| (Z1 = z1

∗)11: Solutions← Solutions ∪ z1∗, z2∗∗∗12: i← i+ 113: end while

14: Sort and check(Solutions)

Un modèle bi-objective est présenté pour résoudre le problème. Les deux objectifs sont dénis par

Z1 et Z2 et cherchent à maximiser la quantité d'énergie produite tout en minimisant l'ensemble des

coûts respectivement. Il existe diérentes façons de résoudre les problèmes avec plusierus objectifs.

Dans cette étude une adaptation de la méthode epsilon constraints expliquée dans l'algorithme 9

est utilisée. La méthode itère pendant un nombre prédéni de pas. À chaque itération (lignes 8 à

13) l'algorithme cherche à optimiser chaque objectif (séparément) en utilisant des contraintes pour les

valeurs de l'autre objectif. Ceci permet de trouver des solutions Pareto optimales ou des solutions pour

lesquelles aucun objectif ne peut être amélioré sans endommager un autre. Les lignes 2 à 6 permettent

de limiter la recherche entre les bornes dénies par la quantité maximale d'énergie produite et le coût

minimal.

L'approche Epsilon Contraintes est testée sur les instances proposés par Froger et al. [11]. Certaines

modications ont été introduites pour tenir compte des heures supplémentaires. La gure A.5 présente

un exemple de solutions trouvées pour trois périodes (créneaux) par jour et une seule compétence.

De plus, l'instance contient 20 tâches et considère une période de cinq jours. Un total de 16 solutions

non dominées sont trouvés mais seulement 2 sont prouvées optimales. Cela signie que l'exécution

des problèmes linéaires pour optimiser z1 et z2 s'est arrêtée à cause de la limite imposée sur le temps

d'exécution avant de garantir l'optimalité. Toutefois, les écarts moyen sont très faibles et c'est possible

150


89000

89500

90000

90500

91000

91500

92000

92500

93000

100 150 200 250 300 350

Energy

Costs

Figure A.5 : Solutions Pareto optimales pour l'instance 10_2_1_20_B_5 de Froger et al. [11]

que toutes les 16 solutions soient ecaces. La plupart des instances présentent des résultats similaires.

Il convient de noter que pour l'exemple, une augmentation de 115% des coûts peut avoir un impact de

près de 4% sur la production d'énergie. Ces résultats montrent le conit entre les objectifs des diérents

acteurs du projet. Ils permettent également de mettre en évidence l'importance de la planication et

l'ordonnancement des tâches de maintenance pour produire la quantité maximale d'énergie. Sur la

base de ces résultats il est proposé d'étendre ce problème sur un horizon de planication long qui

inclus les stratégies de maintenance.

Les résultats préliminaires de ce chapitre ont été présentés à la conférence IEOM :


Wind farm maintenance scheduling model and solution approach

In Proceedings of the International Conference on Industrial Engineering and Operations Management

Bogota, Colombia, 2526 October, 2017.

A.5 Sélection des stratégies de maintenance et relation avec

l'ordonnancement

Le problème d'ordonnancement introduit dans le chapitre 5 est étendu pour étudier les interactions

entre les stratégies de maintenance et les problèmes opérationnels. En eet, la sélection de la politique

de maintenance peut avoir des eets considérables sur le coût d'O&M et la production énergétique.

En même temps la composante opérationnelle (ordonnancement des tâches) peut altérer le plan de

maintenance. Une procédure de simulation est conçue pour évaluer ces interactions. Toutefois, car le

problème du chapitre 5 est trop complexe, il est simplié et résolu par de simples règles heuristiques

pour aecter les ressources aux tâches.

Le niveau stratégique concerne des décisions prises pour plusieurs années ou même pour le cycle

de vie du projet. Ce niveau inclus l'emplacement du parc, la disposition des éoliennes, ou le type de

politique de maintenance. Une politique de maintenance détermine à quelles dates intervenir et quel

type de maintenance doit être fait sur les éoliennes. Deux types principaux de stratégies existent selon

Shaee [23] : la maintenance préventive et corrective. La dernière consiste à intervenir lorsqu'une

défaillance est constatée. Les activités de maintenance préventives sont réalisées avant la survenue

d'une panne dans le but de retarder son apparition. La maintenance préventive peut être divisée en

maintenance systématique, exécutée périodiquement et maintenance conditionnelle qui est réalisée en

151

A.5. SÉLECTION DES STRATÉGIES DE MAINTENANCE ET RELATION AVECL'ORDONNANCEMENT

fonction de l'état du système. En plus, des actions opportunistes peuvent être aussi considérées. Il

s'agit des activités préventives qui cherchent à proter des arrêts du système à cause d'une panne ou

à d'une autre activité préventive. Cela permet de diminuer les coûts de transport, ou de la mise en

place d'équipement pour eectuer les entretiens. Les activités opportunistes peuvent se faire dans la

même éolienne ou sur d'autres à proximié. Pour sélectionner quel type de stratégie est meilleure, la

plupart des études utilisent de la simulation pour estimer les coûts et la production énergétique.

Le problème est déni avec un ensemble de turbines qui nécessitent des tâches de maintenance. De

plus, chaque éolienne est composée de plusieurs sous-systèmes. Chaque sous-système est soumis à un

processus de détérioration conduisant à une défaillance. Les sous-systèmes sont considérés indépen-

dants entre eux. Deux types de maintenance sont considérés : corrective et systématique. Quand des

tâches correctives sont mise en place, les composants en panne sont remis à neuf. En revanche, quand

les activités sont de type préventif, la réparation est imparfaite.

Un ensemble limité de techniciens ayant tous les mêmes compétences est considère. Les techniciens

se déplacent vers les éoliennes pour eectuer les entretiens. Les considérations de sécurité et exécution

des tâches de maintenance présentés pour le problème d'ordonnancement sont aussi prises en compte

dans cette partie. Il faut rajouter que le temps d'exécution de chaque tâche est une variable aléatoire.

Un horizon de 15 ans est utilisé et deux stratégies de maintenance sont examinées. La première est

une stratégie purement corrective qui planie des actions de maintenance lorsqu'une panne se produit

dans une éolienne. La deuxième stratégie combine des action correctives et des actions préventives

périodiques. Quelque soit le type de stratégie, il est supposé que deux fois par an des activités d'en-

tretien son faites pour chaque éolienne. Ces activités considèrent les changements de consommables

comme l'huile, etc.

Une méthode de simulation à événements discrets est utilisée pour simuler et comparer les deux

stratégies. En plus, deux règles de décisions sont utilisées pour aecter les techniciens aux tâches de

maintenance. La première aecte les ressources dès qu'elles sont disponibles pour eectuer les tâches

sans aucune autre considération (O1). La deuxième utilise les prévisions de vitesses de vent pour

sélectionner le moment où il est préférable d'exécuter les tâches pour réduire les pertes de production

d'énergie. En plus les tâches doivent toujours démarrer en début de période.

Des modules pour simuler la vitesse de temps, les techniciens, et les composants (et leurs pannes)

des éoliennes ont été programmés. Ainsi un module est responsable de la gestion des ressources selon

les règles d'aectation. En combinant les deux stratégies de maintenance avec les deux règles de

décisions quatre scénarios sont considères : "`CM-O1"', "`CM-O2"', "`PM-O1"', et "`PM-O2"'. Chaque

combinaison est simulée pour un parc éolien constitué de 100 éoliennes. Des simulations du parc pour

un horizon de quinze ans sont considérées, de plus, une centaine de répliques sont faites pour collecter

les résultats.

La comparaison des stratégies se fait par rapport aux coûts de maintenance, énergie produite,

le nombre de pannes et le pourcentage du temps que les turbines sont disponibles pour produire

de l'énergie. Il est supposé que les coûts associés aux tâches correctives sont plus élevés que ceux

correspondant aux préventives. Les résultats montrent que les stratégies préventives conduisent à une

réduction des coûts par rapport aux stratégies correctives. En eet, en comparant chaque stratégie,

les coûts sont plus élevés pour la règle O2, donc, celle-ci conduit à une augmentation des pannes des

composantes. Cette situation peut être expliquée par le fait que les techniciens sont souvent aectés

d'une façon plus étendue sur le temps avec O2 et non pas dès qu'une tâche de maintenance arrive.

La règle O2 génére une production d'énergie plus grande que la règle O1 ce qui n'est pas sur-

prenant en regardant le principe de chacune. En eet, ces résultats peuvent être expliqué par le

fait que O2 empêche de planier des tâches tardives. Par exemple, la réglé O2 n'aectera jamais les

techniciens pour démarrer une tâche le vendredi après-midi, ce qui conduira à une perte d'énergie

produite pendant le week-end. En revanche, O1 pourrait parfaitement exécuter cette planication.

Ces résultats montrent un point très important : la quantité d'énergie produite et les coûts subissent

152


Figure A.6 : Coûts par stratégie et par règle d'assignation.

Figure A.7 : Énergie produite par stratégie et par règle d'assignation.

153

A.6. CONCLUSIONS

des changements considérables en fonction de la règle choisie.

De plus les décisions opérationnelles ainsi que les stratégiques ont un impact important sur ces

métriques. Donc les régles de décision utilisées pour faire l'aectation des ressources aux tâches de

maintenance semblent être une piste importante à explorer. D'autres règles de décision sont requises

d'être testées pour améliorer les tâches de maintenance.

A.6 Conclusions

Cette thèse est dédiée aux problèmes de tournées de véhicules avec incertitudes et les problèmes

de planication de la maintenance pour les éoliennes. La plupart des travaux traitant ces problèmes

considèrent le cas déterministe en supposant que les informations sont connues à l'avance. Récemment,

une plus grande attention a été donnée aux versions stochastiques du fait de leur importance aussi

bien sur le plan académique qu'applicatif. En eet, il a été démontré que les conséquences d'ignorer

la nature stochastique des problèmes sont importantes et peuvent impacter les coûts, l'indisponibilité

des systèmes, ou l'insatisfaction des clients. Dans ce contexte, les contributions de cette thèse sont

les suivantes : D'abord, des méthodes de résolutions ecaces, basées sur des algorithmes mémétiques,

sont présentées pour deux VRP avec incertitudes. Pour gérer la nature stochastiques du problème,

des procédures adéquates ont été développées pour ces derniers. Il s'agit du VRP avec demandes

stochastiques, et du VRP avec temps de trajet et de services stochastiques et de fenêtres de temps

dures. De plus, de nouveaux meilleurs résultats sont présentés pour les deux problèmes. Le problèmes

de planication de tâches de maintenance pour un parc d'éoliennes est également étudié dans cette

thèse dans un contexte multicritère. Après avoir proposé une formulation mathématique du problème,

une méthode de type epsilon contraintes a été conçue pour le résoudre. Une variante intégrant la

sélection des stratégies de maintenance à l'aectation des ressources (techniciens) aux tâches est

également analysée en utilisant une méthode basée sur la simulation à évènements discrets.

Pour le VRP avec demandes stochastiques, un algorithme méméthique combiné à une procédure de

redémarrage basée sur une méthode de type GRASP a été proposée. Des résultats sont donnés aussi

bien sur les instances couremment utilisés dans la littérature que sur un nouvel ensemble d'instances

comportant un plus grand nombre de clients. Ces derniers résultats pourront servir de points de com-

paraison pour des travaux futurs, il pourront également être utilisés pour évaluer d'autres approches

telles que la réoptimisation ou des méthodes faisant appel à de nouvelles formes de recours.

Le VRP avec temps de trajet et de service stochastiques et fenêtres de temps dures a été également

examiné dans cette thèse. Plusieurs populations ont été utilisées dans un algoritme mémétique parallèle

conçu pour résoudre ce problème. Ces populations considèrent chacune des choix d'approximation des

paramètres spéciques, et partagent des informations en échangeant des individus à des temps précis

dans l'algorithme. Des comparaisons à d'autres méthodes publiées sur le sujet démontrent que notre

approche est ecace et éxible pour résoudre le problème considéré ainsi que d'autres variantes pour

lesquelles des résultats sont disponibles dans la littérature.

Le problème de planication de la maintenance pour les éoliennes a été d'abord étudié dans un

contexte déterministe. En utilisant un modèle linéaire bi-critère à variables mixtes résolu grace à

la méthode epsilon contraintes, il a été démontré que les coûts de maintenance et la production

énergétique peuvent être en conit même dans le cas opérationnel. Une intégration de ce problème

au niveau de décision stratégique permet de conclure que la façon de gérer la planication de la

maintenance et la sélection des stratégies de maintenance ont un impact important sur les coûts et la

production énergétique. Dans ce dernier travail, l'apparition des pannes et le temps nécessaire pour

la réparation sont considérés comme stochastiques.

Les travaux futurs autour de VRP stochastiques et la planication de la maintenance des éoliennes

doivent se concentrer sur l'interdépendance des variables (demandes, dégradation, temps de trajet). De

plus, plus d'approches statistiques pour valider les lois de probabilités utilisées pour les paramètres des

154


problèmes seront bienvenues. Des études de cas seront nécessaires pour montrer la valeur des solutions

stochastiques dans des applications réelles. Pour ces dernières, nous nous attendons à une croissance

du nombre de méthodes approchées pour les résoudre. La conception d'instances standardisées pour

les VRP stochastiques est également essentielle pour permettre une comparaison rigoureuse entre

diérents modèles et méthodes de résolution.

Enn, nous pensons que l'optimisation robuste connaitra un développement important pour com-

plémenter et enrichir ce domaine de recherche quand les lois de probabilités sont inconnues. Nous

espérons que les pistes de recherche proposées et que les travaux présentés dans cette thèse serviront

pour de noubreux développements futurs.

155

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158

Appendix B

List of contributions

International peer-reviewed journal papers

Gutierrez, A., Dieulle, L., Labadie, N., Velasco, N. (2018). A multi-population algorithm to solve the

VRP with stochastic service and travel times. Computers & Industrial Engineering, 125, 144-156.

https://doi.org/10.1016/j.cie.2018.07.042. http://www.sciencedirect.com/science/article/

pii/S0360835218303668

Gutierrez, A., Dieulle, L., Labadie, N., Velasco, N. (2018). A Hybrid metaheuristic algorithm

for the vehicle routing problem with stochastic demands. Computers & Operations Research,

99, 135-147. https://doi.org/10.1016/j.cor.2018.06.012. http://www.sciencedirect.com/science/

article/pii/S0305054818301667

Peer-reviewed conference proceedings

Gutierrez, A., Dieulle, L., Labadie, N., Velasco, N. Wind farm maintenance scheduling model and

solution approach, Proceedings of the IEOM Conference Bogota, Colombia, 2526 October 2017

Gutierrez, A., Dieulle, L., Labadie, N., Velasco, N. A multi population memetic algorithm for

the vehicle routing problem with time windows and stochastic travel and service times, 8th IFAC

Conference on Manufacturing Modelling, Management and Control MIM 2016 Troyes, France,

2830 June 2016

Gutierrez, A., Dieulle, L., Labadie, N., Velasco, N. A Memetic Algorithm for the Vehicle Routing

Problem with Stochastic Demands, Proceedings of the 45th International Conference on Computers

& Industrial Engineering (CIE45) Metz, France, 2830 October 2015

Conference presentations

Gutierrez, A., Dieulle, L., Labadie, N., Velasco, N. An approximate column generation for the vehicle

routing problem with hard time windows and stochastic travel and service times. XVIII CLAIO, the

Latin-Iberoamerican Conference on Operations Research Santiago de Chile, Chile. 26 October

2016.

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maintenance scheduling and vehicle routing optimisation

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