WALID KLIBI

THE DESIGN OF EFFECTIVE AND ROBUST SUPPLY CHAIN NETWORKS

Thèse présentée à la Faculté des études supérieures de l'Université Laval

dans le cadre du programme de doctorat en sciences de l'administration pour l'obtention du grade de Philosophiae Doctor (Ph.D.)

OPERATIONS ET SYSTEMES DE DECISION FACULTÉ DES SCIENCES DE L'ADMINISTRATION

UNIVERSITÉ LAVAL QUÉBEC

2009

© Walid Klibi, 2009

Résumé Pour faire face aux risques associés aux aléas des opérations normales et aux périls qui menacent les ressources d'un réseau logistique, une méthodologie générique pour le design de réseaux logistiques efficaces et robustes en univers incertain est développée dans cette thèse. Cette méthodologie a pour objectif de proposer une structure de réseau qui assure, de façon durable, la création de valeur pour l'entreprise pour faire face aux aléas et se prémunir contre les risques de ruptures catastrophiques. La méthodologie s'appuie sur le cadre de prise de décision distribué de Schneeweiss et l'approche de modélisation mathématique qui y est associée intègre des éléments de programmation stochastique, d'analyse de risque et de programmation robuste. Trois types d'événements sont définis pour caractériser l'environnement des réseaux logistiques: des événements aléatoires (ex. la demande, les coûts et les taux de changes), des événements hasardeux (ex. les grèves, les discontinuités d'approvisionnement des fournisseurs et les catastrophes naturelles) et des événements profondément incertains (ex. les actes de sabotage, les attentats et les instabilités politiques). La méthodologie considère que l'environnement futur de l'entreprise est anticipé à l'aide de scénarios, générés partiellement par une méthode Monte-Carlo. Cette méthode fait partie de l'approche de solution et permet de générer des replications d'échantillons de petites tailles et de grands échantillons. Elle aide aussi à tenir compte de l'attitude au risque du décideur. L'approche générique de solution du modèle s'appuie sur ces échantillons de scénarios pour générer des designs alternatifs et sur une approche multicritère pour l'évaluation de ces designs.

Afin de valider les concepts méthodologiques introduits dans cette thèse, le problème hiérarchique de localisation d'entrepôts et de transport est modélisé comme un programme stochastique avec recours. Premièrement, un modèle incluant une demande aléatoire est utilisé pour valider en partie la modélisation mathématique du problème et étudier, à travers plusieurs anticipations approximatives, la solvabilité du modèle de design. Une approche de solution heuristique est proposée pour ce modèle afin de résoudre des problèmes de taille réelle. Deuxièmement, un modèle incluant les aléas et les périls est utilisé pour valider l'analyse de risque, les stratégies de resilience et l'approche de solution générique. Plusieurs construits mathématiques sont ajoutés au modèle de base afin de refléter différentes stratégies de resilience et proposer un modèle de décision sous risque incluant l'attitude du décideur face aux événements extrêmes. Les nombreuses expérimentations effectuées, avec les données d'un cas réaliste, nous ont permis de tester les concepts proposés dans cette thèse et d'élaborer une méthode de réduction de complexité pour le modèle générique de design sans compromettre la qualité des solutions associées. Les résultats obtenus par ces expérimentations ont pu confirmer la supériorité des designs obtenus en appliquant la méthodologie proposée en termes d'efficacité et de robustesse par rapport à des solutions produites par des approches déterministes ou des modèles simplifiés proposés dans la littérature.

11

Abstract This thesis provides a methodology for Supply Chain Network (SCN) design under uncertainty. Due to random environmental factors such as demand, prices, exchange rates..., and to disruptive events such as natural disasters, strikes..., the SCN designed must be robust enough to cope with these events. The design problem formulation is based on the generic distributed decision-making framework proposed by Schneeweiss. It is defined as a multistage stochastic program taking into account decisions under risk, user response anticipations and resilience strategies. The methodology recognizes three event types to characterize the future SCN environment: random, hazardous and deep uncertainty events. These events shape the future environments of SCNs which are anticipated through a scenario planning approach. A Monte-Carlo method is used, outside the optimization process, to generate small sample replications and large samples of scenarios. Scenarios samples generation allow to approximate the design model to be solved with a sample average approximation program in order to produce a set of alternative designs. A multi-criteria design evaluation approach is applied to select the most effective and robust SCN among this set and the status quo design.

Several experiments are performed in order to validate the concepts introduced in the methodology proposed. The Location Transportation Problem (LTP) is defined as a hierarchical strategic decision problem to set depots location and mission and is characterized by multiple transportation options and multi-period customer demands. First, the LTP under random demand is formulated as a two-stage stochastic program with recourse and a hierarchical heuristic solution approach is proposed to solve it. This model is also used to study the impact of various types of anticipations on the quality of the solutions obtained. Second, various modeling constructs are proposed to formulate the LTP under random and hazardous events as a two-stage stochastic programs and to foster supply network resilience. Several tests on various industrial contexts, based on realistic data, enabled us to solve these design models, and to propose complexity reduction methods facilitating the solution of SCN design problems without compromising the quality of the designs obtained. Our results confirm the quality of the designs obtained when using the methodology proposed in the thesis, in terms of effectiveness and robustness, when compared to solutions produced by deterministic approaches or simplified models found in the literature.

Avant-propos Ce document est présenté sous la forme d'une thèse par articles et respecte les règles de présentation exigées par la Faculté des études supérieures. La thèse est constituée d'une introduction générale, de quatre chapitres reproduisant des articles dont je suis le premier auteur et une conclusion générale.

Le premier article, inséré au chapitre 1, s'intitule «The design of Robust Value-Creating Supply Chain Networks: A Critical Review». J'ai œuvré comme chercheur principal à la synthèse de la littérature et la rédaction de l'article. Cet article de revue a été publié dans European Journal of Operational Research (volume 203, numéro 2, pp. 283-293, 2010). De plus, une version préliminaire a été présentée à Québec (Canada) au Congrès conjoint SCRO/Journées de l'Optimisation en 2008.

Le deuxième article, inséré au chapitre 2, s'intitule «The design of Effective and Robust Supply Chain Networks». J'ai œuvré comme chercheur principal au développement des concepts méthodologiques, à leur validation et la rédaction de l'article. Des versions préliminaires de cet article ont été présentées à Saint-Etienne (France) au 12th IF AC Symposium on Information Control Problems in Manufacturing 2006 (Incom 2006) et à Madison (USA) à VInternational Conference on Information Systems, Logistics and Supply Chain 2008 (ILS08). Cet article a été récemment soumis pour publication dans Operations Research.

Le troisième article, inséré au chapitre 3, s'intitule «The Stochastic Multi-Period Location-Transportation Problem». J'ai œuvré comme chercheur principal au développement des concepts, à l'écriture du modèle et la rédaction de l'article. Cet article a été accepté pour publication dans Transportation Science en 2009. De plus, une version préliminaire a été présentée à Québec (Canada) au Congrès conjoint SCRO/Journées de l'Optimisation 2008.

Le quatrième article, inséré au chapitre 4, s'intitule «Designing Resilient Supply Networks under Disruptions». J'ai œuvré comme chercheur principal au développement des concepts, à l'écriture du modèle, à la réalisation des expérimentations et la rédaction de l'article. Cet article a été présenté à Toronto (Canada) au Congrès conjoint INFORMS/SCRO 2009. Cet article a été récemment soumis pour publication dans Transportation Science.

IV

Remerciements Cette thèse est le fruit d'un travail assidu que j 'ai mené en tant qu'étudiant au doctorat à la Faculté des sciences de l'administration de l'Université Laval. Je tiens à remercier toutes les personnes qui m'ont soutenu et aidé à concrétiser cette finalité.

D'abord, je remercie mon directeur de recherche M. Alain Martel pour tout ce que j 'ai appris à ses cotés et pour les nombreuses opportunités qu'il m'a permis de réaliser. Ses encouragements, sa disponibilité et ses qualités humaines ont été fondamentales durant ces années. Je remercie également mon co-directeur de recherche M. Adel Guitouni pour sa collaboration et ses contributions qui ont amélioré la qualité de ce travail. Je souhaite aussi remercier les professeurs M. Ossama Kettani, M. Nicolas Zuffrey, Mme Soumia Ichoua et M. Angel Ruiz pour avoir participé aux différents comités de mon cheminement doctoral. Leurs évaluations pertinentes m'ont été d'une grande aide pour accomplir ce travail. Je remercie aussi M. Samir Elhedhli, examinateur externe de l'Université de Waterloo, pour avoir accepté d'évaluer cette thèse. Je souhaite également remercier tous les professeurs du département Opérations et systèmes de décision, en particulier M. Pascal Lang, pour avoir contribué à ma formation académique. Je remercie les partenaires du projet DRESNET pour avoir financé en partie mon doctorat au sein du CIRRELT et les étudiants et professionnels de recherche de ce projet qui ont contribué à ce travail. Je remercie aussi le personnel du CIRRELT et du département pour leurs précieux services. De plus, je remercie tous mes amis pour leurs encouragements et les bons moments passés loin du bureau. En particulier, je remercie Eduardo, Sehl et Monia pour avoir partagé tant de débats, de repas et de cafés durant cette période.

Mes profonds remerciements vont surtout à tous les membres de ma famille. Ma belle famille, Fateh, Emna et Farah pour leur support important malgré la distance qui nous sépare. Mes frères Selim et Akram pour tous leurs encouragements et toutes leurs tentatives de divertissement durant ces années. Mes parents, Hédia et Hammadi, ma source d'inspiration, sans qui, je n'aurais jamais réussi ce que j 'ai entrepris. Je les remercie du fond du cœur pour la chance qu'ils m'ont offert et pour toutes les pensées et mots d'encouragements inestimables qu'ils m'ont donné durant mon doctorat. Mes immenses remerciements à ma femme Fériel pour sa précieuse présence à mes cotés, sa patience héroïque et ses encouragements continus tout au long du processus. Finalement, je remercie dieu de m'avoir préservé ainsi que toute ma famille.

À mes parents Hédia et Hammadi, et mon épouse Fériel

Table des matières Résumé i Abstract ii Avant-propos iii Remerciements iv Table des matières vi Liste des tableaux viii Liste des figures ix Introduction générale 1

1. Contexte général 1 2. Questions de recherche 4 3. Contributions de recherche 4 4. Organisation de la thèse 6

Chapitre 1: The Design of Robust Value-Creating Supply Chain Networks: A Critical Review 7 1. The Design of Robust Value-Creating Supply Chain Networks: A Critical Review.... 8

1.1 Introduction 9 1.2 Overview of the SCN Design Problem 10

1.2.1 Strategic SCN Design Decisions 10 1.2.2 Supply Chain Networks under Uncertainty 12 1.2.3 Strategic Evaluation of SCN Designs and Optimization Criteria 15

1.3 Deterministic SCN Design Models 19 1.4 SCN Design Models under Uncertainty 21

1.4.1 Randomness 23 1.4.2 Hazard 24 1.4.3 Deep Uncertainty 25

1.5 Fostering Robustness in SCN Design 26 1.5.1 Robustness 26 1.5.2 Responsiveness 28 1.5.3 Resilience 28

1.6 Conclusions 30 Chapitre 2: The Design of Effective and Robust Supply Chain Networks 33 2. The Design of Effective and Robust Supply Chain Networks 34

2.1 Introduction 35 2.2 SCN Design Methodology 36

2.2.1 Decision problem structure 36 2.2.2 Characterization of the information available 40 2.2.3 SCN Risk Analysis 44 2.2.4 SCN Design Model 55

2.3 Scenario-based SCN Design Model Solution Approach 57 2.3.1 SCN Designs Generation 59 2.3.2 Scenarios Generation 62 2.3.3 SCN Designs Evaluation 65

2.4 Conclusions 67 Chapitre 3: The Stochastic Multi-Period Location-Transportation Problem 68 3. The Stochastic Multi-Period Location-Transportation Problem 69

Vil

3.1 Introduction 70 3.2 Problem Description and Formulation 73

3.2.1 The Business Context 73 3.2.2 The Distribution Network User Problem 75 3.2.3 The Distribution Network Design Problem 77 3.2.4 Sample Average Approximation Model 79

3.3 Solution Approach 81 3.3.1 The General Scheme 81 3.3.2 The User Problem Heuristic ...81 3.3.3 Tabu Search Heuristic for the Distribution Network Design Problem 83

3.4 Computational Results 91 3.4.1 Plan of Experiments 91 3.4.2 Numerical Results 93

3.5 Conclusions 98 Chapitre 4: Designing Resilient Supply Networks under Disruptions 101 4. Designing Resilient Supply Networks under Disruptions 102

4.1 Introduction 103 4.2 The Location-Transportation Problem under Uncertainty 104

4.2.1 The Location-Transportation Problem Context 104 4.2.2 Demand and Hazard Modeling 105 4.2.3 Plausible Future Scenarios 108

4.3 Scenario-Based SN Design Approach 110 4.3.1 User Response Model I l l 4.3.2 Risk-Neutral Design Model 114 4.3.3 Design Models Based on Approximate Anticipations 117

4.4 Resilience Strategy Formulations 119 4.4.1 Optimal Back-up Formulations (RI) 121 4.4.2 Multiple Sourcing Formulations (R2) 122 4.4.3 Coverage Formulations (R3) 124

4.5 SN Design Models Solution and Evaluation Approach 125 4.6 Computational Results 127

4.6.1 Plan of experiments 128 4.6.2 SAA Models Calibration 130 4.6.3 Numerical Results 132

4.7 Conclusions 137 Conclusion générale 139 Bibliographie 143

Liste des tableaux

Table 0-1. Schéma de validation de la méthodologie proposée 5 Table 3-1. Test Problems Size 92 Table 3-2. Test Problem Cost Structures 92 Table 3-3. Ship-to-Point Demand Structure 93 Table 3-4. Heuristic Parameter Values 94 Table 3-5. Statistical Optimality Gap Values 95 Table 3-6. Comparison with CPLEX-11 Solution of SAA Model for Problem Px 96 Table 3-7. Mean Design Values for all Problem Types 97 Table 4-1. Test Problems Instances 128 Table 4-2. Test Problems Cost Structure 128 Table 4-3. Ship-to-Point Demand Structure 129 Table 4-4. Test Problems Depots Capacity Structure 129 Table 4-5. Decision-Maker Risk Attitude Parameters 131 Table 4-6. Regression Parameters by Problem Instance for M2 and M3 131 Table 4-7. Multi-Hazard Exposure Levels and Mean Inter-Arrival Times 132 Table 4-8. Model Characteristics and Average Solution Times for Pi and P^ 133 Table 4-9. Average Number of Depot Hits by Scenario 134 Table 4-10. Similarity between Design Decisions for DM Types 135 Table 4-11. Models Performance in Terms of Deviation from the Best Design 136

Liste des figures

Figure 0-1. Étude des anticipations 3 Figure 1-1. Current and Potential Supply Chain Networks 10 Figure 1-2. Supply Chain Network Vulnerability Sources 13 Figure 1-3. Example of Multi-hazard Indexes 15 Figure 1-4. Risk Matrix (Norrman and Jansson, 2004) 15 Figure 1-5. Static Design Tradeoffs for a Domestic Supply Chain Network 17 Figure 2-1. Decision Time Hierarchy for Two Planning Cycles 38 Figure 2-2. Strategic Decision Framework 40 Figure 2-3. Events Matrix ..42 Figure 2-4. Scenarios Tree for the Planning Horizon 43 Figure 2-5. Examples of Vulnerability Sources and Multihazards 46 Figure 2-6. SCN Exposure Modeling 48 Figure 2-7. Multihazard Incident Profiles Example 52 Figure 2-8. Recovery Function for a Given he H, se Sc and pe Ps 53 Figure 2-9. Distribution of the Number of Hits for a Large Sample of Scenarios 55 Figure 2-10. Generic SCN Design Methodology 59 Figure 2-11. Monte Carlo Procedure for the Generation of a Scenario CD 65 Figure 3-1. Multi-Period Location-Transportation Network 75 Figure 3-2. Ship-to-Points Stochastic Demand Process 76 Figure 3-3. Procedure MonteCarlo for the Generation of Scenario œ 80 Figure 3-4. Alternative Routes Considered in the Savings Calculations for Depot /... S3 Figure 3-5. Procedure User for Depot / in Period t under Scenario co 84 Figure 3-6. Reassignment Procedure 88 Figure 3-7. Initialization Procedure 89 Figure 3-8. Tabu Procedure 90 Figure 3-9. Exploration Strategies Design Value and Solution Time Comparisons.... 98 Figure 4-1. The LTP Structure under Uncertainty 105 Figure 4-2. Impact-Duration Functions for Depots and Ship-to-Points 107 Figure 4-3. Recovery Function Examples for Depot / and Ship-to-Point p 108 Figure 4-4. Scenario CD Generation Procedure 109 Figure 4-5. Distribution of the Number of Hits for a Large Scenario Sample 110 Figure 4-6. Demand Level at a Given Depot / under Scenario CD 112 Figure 4-7. Response Procedure for Design x under Scenario CD 115 Figure 4-8. Resilience Strategy Formulations 121 Figure 4-9. SN Design Models Solution and Evaluation Approach 126 Figure 4-10. Design Value Behavior by Hit Level for Non-Dominated Models 134 Figure 4-11. Expected Value - Mean-Semideviation Tradeoffs for Pi and P^ 135

Introduction générale

1. Contexte général La conception de réseaux logistiques implique plusieurs décisions stratégiques telles que le nombre, la localisation, la capacité et la mission d'installations d'approvisionnement, de production et de distribution, dans le but de concevoir un réseau de création de valeur capable de servir efficacement un ensemble de clients. La majeure partie de la littérature s'est concentrée sur des modèles discrets de localisation d'entrepôts, à un échelon, statiques et sous un environnement déterministe (Revelle et al., 2008). Plusieurs variantes de ces modèles classiques, concernées par la fonction objectif, les produits, la capacité des installations et/ou la nature de la demande, ont été proposées dans la littérature (Klose et Drexl, 2005). De plus, quelques modèles existants ont considéré un environnement incertain basés sur des approches probabilistes ou par scénarios dans lesquels les quantités de demande, la localisation de la demande, les temps de transit et les coûts de transport sont incertains. Lorsque l'incertitude est représentée par un ensemble de scénarios, des approches de programmation stochastique (Birge and Louveaux, 1997) et d'optimisation robuste (Kouvelis and Yu, 1997) ont été proposées. Une revue des problèmes de localisation sous incertitude a été élaborée dans Snyder (2006). Par ailleurs, le problème de localisation-allocation est difficile à résoudre dû au fait que le sous-problème d'affectation obtenu pour un sous-ensemble donné d'installations est NP-difficile (Fisher, 1986). Il est clair que les problèmes de design de réseaux logistiques sont beaucoup plus complexes que les problèmes de localisation classiques puisqu'ils ajoutent des décisions d'approvisionnement, d'acquisition de capacité, de sélection de technologie et de sélection de produit-marchés (Martel, 2005). Comme mentionné ci-haut, les modèles existants s'attaquent uniquement à un sous ensemble de ces questions et se concentrent uniquement sur un ou deux échelons de la chaîne logistique. Ceci est en grande partie dû à la complexité de modéliser ce problème et aux difficultés rencontrées pour résoudre les modèles obtenus. Dans la perspective de résoudre des problèmes réels, plusieurs stratagèmes de réduction de complexité seront employés dans cette thèse pour obtenir des modèles de design solvables sans compromettre la qualité des solutions qu'ils produisent.

De plus, les réseaux logistiques doivent être conçus pour le long terme, ce qui exige des investissements importants. L'environnement futur du réseau logistique et le rendement de ces investissements sont les préoccupations majeures au niveau stratégique. On constate aussi que la gestion du risque associé aux aléas des opérations normales et aux périls qui menacent les ressources du réseau est devenue une préoccupation sérieuse pour les entreprises. Dans ce contexte, on est amené à nous questionner sur la structure de planification qu'il faudra adopter pour le problème de design de réseaux logistiques. Les événements récents, tels que les attaques du WTC et l'ouragan Katrina, ont sérieusement

affecté la performance des réseaux logistiques actuels et ont soulevé beaucoup de questions pour les entreprises touchées (Lee, 2004; Sheffi, 2005, Hendrick and Singhal, 2005). Dans un contexte d'économie moderne, la globalisation des marchés et le déploiement international des réseaux logistiques ne font qu'augmenter leur exposition au risque d'événements catastrophiques. Pour cela, il faut clairement identifier les types d'événements qui caractérisent l'environnement des réseaux logistiques. Dans cette thèse, l'incertitude est définie comme l'incapacité, au moment de prendre les décisions de design, de déterminer le véritable état de l'environnement futur de l'usager du réseau. Cette information partielle sera modélisée à travers un ensemble de scénarios plausibles du futur et se basera sur une approche probabiliste.

Sous un environnement risqué, l'occurrence d'événements catastrophiques peut causer des ruptures dans les réseaux engendrant des coûts d'opérations supplémentaires et des recours dispendieux, ce qui met en cause la robustesse des réseaux actuels. Il est clair que le réseau doit être suffisamment réactif pour faire face à tous les facteurs aléatoires de l'environnement (demande, coûts, prix, taux de change...) qui affectent les opérations normales d'une entreprise et devrait aussi être en mesure de bien performer sous des perturbations majeures (catastrophes naturelles, ruptures de sources d'approvisionnements, incidents industriels...). Ceci nous amène à nous questionner sur les stratégies à employer pour réduire ces risques ou éviter l'exposition des réseaux logistiques à ce type d'événements. Dans ce contexte, les notions de réactivité et de resilience deviennent un élément clé pour assurer, de façon durable, la robustesse des réseaux face à leur environnement futur. La resilience est définie comme la capacité de rebondir après une rupture du réseau (Sheffi, 2005). À partir de ce concept, plusieurs stratégies ont été proposées dans la littérature (Rice and Caniato, 2003; Snyder et al., 2006; Tomlin, 2006; Tang and Tomlin, 2008). Elles se basent sur la flexibilité des ressources existantes ou bien sur la redondance d'un certain nombre de ressources clés, à utiliser uniquement en cas de ruptures. Quelques modèles basés sur la programmation stochastique (Pomper, 1976; Eppen et al., 1989; Santoso et al., 2005; Vila et al., 2007) ont aussi été proposées pour faire face aux aléas de l'environnement des réseaux logistiques. En outre, certaines approches d'optimisation robuste ont été proposées récemment pour résoudre des problèmes de localisation face à des événements imprévisibles (Gutierrez et al., 1995; Snyder et al., 2006). Cependant, les modèles proposés dans la littérature s'appliquent à des problèmes de localisation simplifiés et ne considèrent que certains types d'incertitudes, ce qui est susceptible de compromettre la robustesse de leurs solutions pour des problèmes réels. Dans cette thèse, le cadre probabiliste utilisé nous amène à l'emploi de la programmation stochastique, une approche proactive qui présente des fondements mathématiques solides pour des décisions en univers risqué et qui offre une flexibilité importante à travers les variables de recours.

D'autre part, en vue de la conception de réseaux logistiques de qualité supérieure, il est fondamental d'anticiper, au moment du design, l'environnement futur de l'usager de ces réseaux. Le concept d'anticipation est un concept clé de part son impact sur la qualité de la solution et donc sur la robustesse du réseau à déployer. Comme l'illustre la Figure 0-1, la qualité de la solution dépend de la précision du modèle construit et de la méthode utilisée pour le résoudre. Plus l'anticipation est fine, plus le modèle est précis mais sa solvabilité est problématique, et vice-versa. Étant donné la taille des modèles de conception de réseaux et leur structure combinatoire et stochastique, plusieurs options de réduction de la complexité doivent être envisagées. Pour le problème de localisation et transport, de nombreuse expériences seront effectuées sur des modèles avec différentes anticipations afin de chercher un équilibre adéquat entre toutes les dimensions concernées (ex: ensemble de routes vs nombre de scénarios) ou bien négliger certaines dimensions (ex: modèle déterministe vs modèle stochastique). Ces nuances n'ont pas été explicitement abordées dans la littérature et un des défis de cette thèse est de parvenir à trouver les meilleurs compromis entre la précision du modèle et sa solvabilité pour le problème de design de réseaux logistiques. Les anticipations approximatives s'inscrivent parmi l'ensemble de stratagèmes de réduction de complexité des modèles de design à construire.

Solvabilité du modèle

Méthode exacte (MIP) Heuristique Anticipation

exacte

Anticipation approximative

Figure 0-1. Étude des anticipations

Comme mentionné ci-haut, dans un contexte décisionnel dominé par l'incertitude, la notion de robustesse des réseaux logistiques devient fondamentale pour évaluer la qualité d'un réseau logistique face à des futurs plausibles. Dans le cadre des problèmes de design de réseaux, cette notion a souvent été négligée ou bien mal employée dans la littérature. Par conséquent, un des objectifs de cette thèse est d'abord d'introduire la notion de robustesse dans le cadre des problèmes de design de réseaux logistiques et ensuite de proposer une définition formelle qui cadre avec la méthodologie de design proposée. Il est à noter que, l'on s'intéresse à la qualité du design conçu et donc aux solutions robustes

(Wong and Rosenhead, 2000), par opposition aux notions de modèles robustes et d'algorithmes robustes traitées par ailleurs dans la littérature. Un design de réseau logistique est robuste s'il est capable de créer de la valeur de façon soutenue, le long de l'horizon de planification considéré, quelque soit le futur plausible (scénario reflétant les aléas et les périls pertinents) qui se réalisera.

Face à cette problématique, plusieurs lacunes et liens manquants dans la littérature ont été identifiés. Ceci a permis de soulever de nombreuses questions de recherche qui ont conduits aux contributions de cette thèse.

2. Questions de recherche La revue de littérature et nos recherches préliminaires sur la problématique ont permis de soulever plusieurs questions liées à la conception de réseaux logistiques efficients et robustes. Ces questions sont les suivantes:

• Quelle structure de planification utiliser pour le problème de design? • Quels types d'événements caractérisent l'environnement des réseaux logistiques? • Comment introduire la notion de robustesse des réseaux logistiques? • Quelle approche d'analyse du risque doit être utilisée? • Comment considérer des décisions sous risque dans les modèles de design? • Comment contourner la complexité reliée à la modélisation du problème sans

compromettre la qualité des solutions associées? • Quelles stratégies de réduction du risque devraient être employées pour produire

des réseaux logistiques résilients?

Comme mentionné dans la mise en contexte, ces questions ont été abordées que partiellement dans la littérature et l'objectif de cette thèse est d'apporter une réponse à chacune d'entre elles à travers plusieurs contributions de recherche.

3. Contributions de recherche Dans cette thèse, on propose de partitionner les contributions de recherche en cinq concepts clés: la structure de planification, la caractérisation de l'incertitude, les stratégies de design, les anticipations et l'approche de solution. Ces concepts sont développés et validés dans quatre articles scientifiques inter-reliés. Suite à une revue critique de la littérature dans l'article 1, la Table 0-1 illustre comment les contributions liées à ces concepts, développés dans l'article 2, sont validées graduellement dans les articles 3 et 4.

Article 1 Revue de la littérature

Article 2 Structure de

planification

Incertitude Stratégies de design Anticipations Approche de solution

générique

Article 2 Structure de

planification

Événements

aléatoires

Événement;

hasardeux

Événements

profondément

incertains

Stratégies de

déploiement

Stratégies de

resilience

Exacte Approximative Méthode

exacte

Méthode

heuristique

Article 3 1 cycle m 0 0 0 Article 4 1 cycle m a m m m 0 m S

Table 0-1. Schéma de validation de la méthodologie proposée

1. Le premier article présente une revue critique de l'état de l'art et met en évidence plusieurs lacunes à travers les questions de recherches citées ci-haut. Une analyse des sources d'incertitudes et de vulnérabilités des ressources d'une chaîne logistique est présentée. Plusieurs définitions des concepts de robustesse, de réactivité et de resilience sont revues et leur importance dans la problématique de conception de réseaux logistiques est discutée. Le titre de l'article est: «The Design of Robust Value-Creating Supply Chain Networks: A Critical Review».

2. Le deuxième article fournit un cadre conceptuel pour la conception de réseaux logistiques efficaces et robustes en univers incertain. Cinq contributions majeures y sont présentées conceptuellement : une structure de planification en univers risqué, une caractérisation de l'environnement incertain, des stratégies de déploiement et de réduction du risque, une formalisation du concept d'anticipation et une approche de solution générique. Le titre de cet article est: «The design of Effective and Robust Supply Chain Networks».

3. Le troisième article définit un problème de localisation d'entrepôts et de transport stochastique et multi-périodes. Ce problème est pertinent pour valider partiellement les concepts de la méthodologie proposée puisque c'est un problème de décision hiérarchique incluant des décisions de localisation stratégiques et une anticipation des décisions opérationnelles de transport. Dans ce problème, un cycle de planification est considéré dans lequel l'environnement futur est caractérisé par des événements aléatoires qui façonnent la demande future des clients. L'emphase est mise sur la modélisation de ce problème hiérarchique basée sur une approche de Set Partitioning et sur le développement d'une méthode de solution heuristique adaptée à la structure hiérarchique du problème. Le titre de l'article est: «The Stochastic Multi-Period Location-Transportation Problem».

4. Le quatrième article propose une modélisation du problème de localisation d'entrepôts et de transport stochastique et multi-périodes incluant des aléas et des périls. Cette recherche permet de valider l'analyse du risque des réseaux logistiques, le modèle de design pour des décisions sous incertitude et l'approche de solution générique. Premièrement, plusieurs anticipations approximatives sont étudiées et deux d'entre elles, présentant un bon compromis entre solvabilité du modèle et qualité des solutions associées, sont retenues. Deuxièmement, différentes formulations sont proposées pour modéliser plusieurs stratégies de resilience et tester leur performance dans divers contextes industriels. Cet article permet de faire des recommandations sur l'approche de modélisation à employer pour concevoir des réseaux logistiques efficaces et robustes. Le titre de l'article est: «Designing Resilient Supply Networks under Disruptions».

4. Organisation de la thèse Compte tenu de la partition des contributions de recherche de la thèse en quatre articles, ce document est organisé comme suit : le chapitre 1 présente la revue de la littérature de l'article 1. Le chapitre 2 présente la méthodologie de design proposée dans l'article 2. Les chapitres 3 et 4 couvrent la validation des concepts méthodologiques proposés dans les articles 3 et 4, respectivement. Finalement, la thèse se termine par une conclusion générale.

Chapitre 1: The Design of Robust Value-Creating Supply Chain Networks: A Critical Review

Résumé - Cet article aborde la problématique de conception de réseaux logistiques sous incertitude et présente une revue critique des modèles d'optimisation proposés dans la littérature. Plusieurs lacunes et liens manquants dans la littérature sont pointés, ce qui justifie la nécessité de développer une méthodologie pour la conception de réseaux logistiques en univers incertain. À travers une analyse des sources d'incertitude et de vulnérabilités des ressources d'une chaîne logistique, l'article révise les facteurs environnementaux clés à prendre en compte et discute la nature des événements catastrophiques qui présentent une menace pour le réseau. Par ailleurs, on identifie les critères importants pour l'évaluation stratégique de réseaux logistiques et synthétise leur usage dans les modèles existants. On discute aussi de la nécessité d'évaluer la robustesse des réseaux afin de s'assurer qu'ils créent de la valeur de façon durable. Plusieurs définitions des concepts de robustesse, de réactivité et de resilience sont revues et leur importance dans la problématique de conception de réseaux logistiques est discutée. Cet article contribue à la mise en place des fondements d'une méthodologie de design de réseaux logistiques efficaces et robustes.

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1. The Design of Robust Value-Creating Supply Chain Networks: A Critical Review

Walid Klibi1'2, Alain Martel1,2** and Adel Guitouni2'3

1 Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT)

2 Département Opérations et systèmes de décision, Faculté des sciences de l'administration, Université Laval, Québec, Canada G1V 0A6

Defense R&D Canada -Valcartier, 2459 Pie-XI Nord, Val-Bélair, Québec, Canada G3J 1X5.

Abstract. This paper discusses Supply Chain Network (SCN) design problem under uncertainty, and presents a critical review of the optimization models proposed in the literature. Some drawbacks and missing aspects in the literature are pointed out, thus motivating the development of a comprehensive SCN design methodology. Through an analysis of supply chains uncertainty sources and risk exposures, the paper reviews key random environmental factors and discusses the nature of major disruptive events threatening SCN. It also discusses relevant strategic SCN design evaluation criteria, and it reviews their use in existing models. We argue for the assessment of SCN robustness as a necessary condition to ensure sustainable value creation. Several definitions of robustness, responsiveness and resilience are reviewed, and the importance of these concepts for SCN design is discussed. This paper contributes to framing the foundations for a robust SCN design methodology.

Keywords. Supply Chain Network Design, Value Creation, Uncertainty, Network Disruptions, Robustness, Scenario Planning, Location Models, Capacity Models, Resilience Strategies.

Acknowledgement. This research was supported in part by NSERC grant no DNDPJ 335078-05.

Results and opinions in the publication attributed to named author(s) were not evaluated by CIRRELT.

Les résultats et opinions contenus dans cette publication n'engagent que leur(s) auteur(s) et n'ont pas été évalués par le CIRRELT.

* Corresponding author: alain.marte'(5)cirre't.ca Dépôt légal - Bibliothèque nationale du Québec,

Bibliothèque nationale du Canada, 2009

© Copyright Walid Klibi, Alain Martel, Adel Guitouni and CIRRELT, 2009

1.1 Introduction Supply Chain Network (SCN) design involves strategic decisions on the number,

location, capacity and mission of the production-distribution facilities of a company, or of a set of collaborating companies, in order to provide goods to a predetermined, but possibly evolving, customer base. It also involves decisions related to the selection of suppliers, subcontractors and 3PLs, and to the offers to make to product-markets. These strategic decisions must be made here-and-now but, after an implementation period, the SCN will be used on a daily basis for a long planning horizon. Day-to-day procurement, production, warehousing, storage, transportation and demand management decisions generate product flows in the network, with associated costs, revenues and service levels. The adequate design of a SCN requires the anticipation of these future activity levels. Furthermore, SCN strategic design decisions are made under uncertainty. The choice of performance metrics to assess the quality of network designs is another important challenge. Return on investment measures are often used by strategic decision makers, but the design robustness is also an important dimension to consider. Despite a rich literature on SCN design, most published models consider only a subset of these issues.

This paper presents a critical review of the SCN design problem under uncertainty, and of the available models proposed to support the design process. It points out some drawbacks and missing links in the literature, and provides motivations for the development of a comprehensive SCN design methodology. It argues that the assessment of SCN robustness is necessary to ensure sustainable value creation. The paper is organized as follows. Section 2 presents an overview of the SCN design problem. Key issues of SCN design under uncertainty are discussed, including uncertainty sources, risk exposures and available data sources. A value-based framework for SCN strategic performance evaluation is also proposed. Section 3 provides a genesis of the literature on deterministic SCN design models, starting with classical location models. Section 4 discusses uncertainty modeling and risk assessment in the context of SCN. The work published, using approaches such as stochastic programming and robust optimization, is reviewed. Section 5 discusses robustness considerations in SCN design, and explores the responsiveness and resilience strategies proposed in the literature. The paper is concluded in section 6 with a discussion on the need for a comprehensive SCN design methodology.

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1.2 Overview of the SCN Design Problem

1.2.1 Strategic SCN Design Decisions A typical SCN is shown in Figure 1-la). In short, the SCN design problem is the

reengineering of such networks to enhance value creation in the companies involved. In general, SC networks are composed of five main entity types: i) external suppliers, ii) plants manufacturing intermediate and/or finished products, iii) distribution and/or sales centers (DC), iv) demand zones, and v) transportation assets. Note that the production-distribution facilities could be subcontractors or public warehouses, and that for-hire transportation could be used. In order to reengineer an existing SCN, an alternative potential network, including all possible supply, location, capacity, marketing and transportation options, must be elaborated. This potential network can be partially represented by a directed graph as shown in Figure 1-lb). The nodes of this graph correspond to existing and potential supply sources, facilities and demand zones. The directed arcs are associated to the transportation lanes that could be used to move materials. A SCN is reengineered by selecting a feasible sub-network of the potential network that optimizes some predetermined value criterion.

a) SCN of a Pulp & Paper Company

• Production distribution Ce nter/Su be on tract or

• Distribution Center

• Demand Zone Centroïd

Uf f l

ii'

II

b) Potential SCN under Uncertainty

Plausible future environments

J_ Supply Sources

• t • • • •

To the network facilities I

□Production-Distribution \ / Distribution < Center V Center

To the demand zones

\ / Subcontractor - ' - ' /Pubtc DC

Demand Zones

Figure 1-1. Current and Potential Supply Chain Networks

The main strategic questions addressed using this generic SCN design approach are the following: Which markets should we target? What delivery time should we provide in different product-markets and at what price? How many production and distribution centers should be implemented? Where should they be located? Which activities should be externalized? Which partners should we select? What production, storage and handling

11

technologies should we adopt and how much capacity should we have? Which products should be produced/stocked in each location? Which factory/DC/demand zones should be supplied by each supplier/factory/DC? What means of transportation should be used (internal fleet, public carrier, 3PL...)? The activities of concern naturally include production and distribution, but recovery and revalorisation activities can also be considered. These strategic questions are rarely examined all together, but rather a few at a time when prompted by major events such as the launching of new products on existing or new markets, a merger, or an acquisition.

On top of these strategic questions and of the number of potential internal and external entities involved, many factors contribute to the complexity of SCN decision models. The first one is industry structure and decoupling points. For example, problems involving complex manufacturing processes in assemble-to-order or make-to-order industries are much more difficult than problems involving single-stage production and/or distribution in a make-to-stock context. A second dimension is the multinational or global coverage of a SCN. When several countries are involved, additional factors such as exchange rates, transfer prices, tariffs, tax regulations and trade barriers must be taken into account (Martel et al., 2005, 2006). A third important aspect is the long-term impact of the design decisions. It may be reasonable to use a static one-year model when the decisions are limited to the selection of public warehouses, as most of the literature suggests. However, when supply agreements and manufacturing facilities last several decades, as in the forest product industry, static one-year models are far from suitable. This leads to a fourth complexity factor: uncertainty. Most models proposed in the literature are not only static, but deterministic. When long planning horizons are involved, the problem becomes dynamic and non-deterministic (e.g., stochastic). In addition, it is not sufficient to consider business-as-usual random variables such as demands, prices and exchange rates, but one should include extreme events such as natural disasters or terrorist attacks that may seriously affect the capabilities and the operations of the supply network.

Important investments are often required to implement strategic SCN decisions. Usually, executives and board members require an assessment of return on investments before making these decisions. The return comes from the net revenues generated by using a SCN during the planning horizon considered: sales revenues less SCN operating expenditures associated to day-to-day procurement, production, warehousing, inventory, transportation and demand fulfilment decisions. These operating revenues and expenditures must be anticipated in the SCN design model. This is usually done using aggregate production, inventory and flow variables, which provides only a raw estimation

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of real operating revenues and costs. With this in mind, the following sections focus on uncertainty, performance evaluation and related issues in the context of SCN design.

1.2.2 Supply Chain Networks under Uncertainty The future business environment under which a SCN will operate is generally

unknown (see Figure 1-lb). At best, several plausible future environments may be considered. Under stochastic assumptions, these future environments are shaped by the random variables associated to business-as-usual factors such as raw material prices, energy costs, product-market demands, labour costs, finished product prices, exchange rates, etc. Recent history has shown that a large spectrum of catastrophic events can be the source of major SCN deficiencies. Catastrophic events have been ignored by most businesses in the past, but a growing interest has been observed recently (Martha and Vratimos, 2002; Semchi-Levi et al., 2002; Helferich and Cook, 2002; Christopher and Lee, 2004; Chopra and Sodhi, 2004 and Sheffi, 2005). Several categories of SCN risk sources were identified (Christopher and Peck, 2004; Kleindorfer and Saad, 2005; Wagner and Bode, 2006 and Tang, 2006b), and Chopra and Sodhi (2004) proposed an extended list of SC risk drivers. In what follows, we examine the sources of uncertainty shaping future business environments from the point of view of a firm or SCN, and not from the point of view of the entire economy. Totally destructive events causing irreversible damages to the entire business are excluded from the analysis.

The suppliers, facilities and ship-to-points of SCN are typically dispersed across large geographical regions, possibly involving several countries, and adverse events may be associated directly to SCN assets/partners, or to the territory over which they are deployed. Three broad categories of SCN vulnerability sources are distinguished in Figure 1-2: endogenous assets, SC partners and exogenous geographical factors. Endogenous assets include the equipments, vehicles, human resources and inventories of production, distribution, recovery, revalorisation and service centers. SC partners include customers, raw material and energy suppliers, subcontractors, and third-party logistics providers (3PLs). In addition to the random business-as-usual factors discussed previously, SCN assets and partners may fail: industrial accidents or fires may destroy or break equipments, vehicles and inventoried products; labour disputes may stop work during a period of time; partner bankruptcy, strikes or accidents may limit raw-material supply or decrease customer demand; etc. A review of potential impacts of these uncertainty sources on SC operations is found in Helferich and Cook (2002).

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Assets and partners are located in specific geographical locations and regions. These regions and their associated public infrastructures (travel ways, terminals, ports, telecommunication networks, utilities...) are themselves exposed to natural disasters (hurricanes, earthquakes, blizzards, floods, forest fires...), major accidents (epidemics, chemical/nuclear spills...) and wilful attacks (terrorist attacks, political coup...). All these possible extreme events are important sources of SC uncertainty. Little or no information is usually available to determine what could go wrong, and the likelihood of asset, partner or infrastructure failures. Based on recorded past events and/or professional expert opinions, for a given SCN design project, a portfolio of plausible extreme event types could be built, hazard zones differentiating exposure levels could be elaborated, and an event type arrival process per zone could be modeled (Banks, 2006, Gogu et al., 2005). Moreover, in network design projects, only vulnerability sources having a serious impact on the strategic performance of the SCN should be considered. Sheffi (2005) proposed to build an enterprise vulnerability map to categorize and prioritize different possible disruptions, and Haimes (2004) suggested an a priori filtering, based on a qualitative assessment, to eliminate low consequence event types.

Production equipment

Storage equipment

Ressources / Handling equipment Production centers r X T .

H u m a n resources

vehicle» t

Endogenous Assets I \ inventories . Distribution centers ^ ■ o

Recovery/Revalonzation centers V Service centers

L * Suppliers

External providers / Subcontractors Supp ly Chain Partners / ^ ~ v

V 3PL.S ^ o

Demand zones Nature

o Travel ways

Exogenous Geograph ica l Factors

'-

! Terminals/ports Public infrastructures V °

V Telecommunication networks

\_ Socio-econormc-political factors

Figure 1-2. Supply Chain Network Vulnerability Sources

Another important aspect is the consequence of high impact disruptions on a SCN. Recently, Craighead et al., (2007) argued that the severity of a supply chain disruption is related to SC density, SC complexity and SC nodes criticality. Several authors reported the impact of such catastrophic events on companies in terms of monetary losses based on direct costs of repair and market share loss (Rice and Caniato, 2003; Lee, 2004; Sheffi,

14

2005, Hendricks and Singhal, 2005). However, facilities are generally insured. Thus, rebuilding and repairing costs are not necessarily relevant for SCN design. On the other hand, the indirect losses related to business interruptions and to temporary relocation and/or rerouting of materiel are crucial. In fact, the cost of any recourse used by the SC to continue operating during the crisis must be taken into account. Unfortunately, to our knowledge, no work to date has proposed a disruption severity modeling approach adequate for SCN design. The work done on SCN vulnerabilities (Helferich and Cook 2002; Kleindorfer and Saad, 2005; Sheffi, 2005) suggest that damages caused to assets/partners should be estimated in term of design parameters such as capacity loss, supply loss or demand surge. Banks (2006) also suggested mapping severity with duration-impact curves, which seems adequate to model assets/partners availability in SCN design.

Natural, accidental and wilful hazards data are generally available, but not always adequate for SCN design purposes. This data can be used relatively easily to compute exposure level indexes by geographical zones, for specific multi-hazard classes. Figure l-3a) for example provides a natural catastrophes exposure index based on data provided by the Centre for Research on the Epidemiology of Disasters1. The Failed States Index presented in Figure l-3b) is a similar multi-hazard index designed to reflect the political stability of a country2. Other relevant multi-hazard indexes such as global competitiveness3, industrial accident4 and public infrastructure quality5 scores may be relevant. An example of an empirical SC disruptions study involving multiple data sources is found in Craighead et a i , (2007).

The risk matrix proposed by Norrman and Jansson (2004) summarizes key elements of the previous discussion (see Figure 1-4). The impact on a SCN of business-as-usual random variables is relatively minor, and it can be modeled using standard probabilistic approaches. However, network threats are difficult to predict and may have serious or catastrophic consequences, which makes them much harder to model in the SCN design process. Intuitively, many natural and man-made phenomena follow the Pareto law: a small fraction of the events cause most of the damage (Sheffi, 2005). This is why the risk

1 See www.cred.be. Other organizations such as the Federal Emergency Management Agency (www.fema.gov) and the U.S. Geological Survey (www.usgs.gov) provide similar information.

2 The Failed States Index is compiled by Foreign Policy (www.foreionpolicy.com) and the Fund for Peace (yvww.fundforpeace.org) based on 12 economical, political, social and ethnic indicators. The Opacity Index published by the Milken Institute (www.milkeninstitute.org) is another political stability measure.

3 See the World Competitiveness Scores of the International Institute for Management Development (www.imd.ch) or the Global Competitiveness Index of the World Economic Forum (www.weforum.org).

4 These indexes are based on the claims made to insurance companies (www.munichre.com). 5 Calculated from databases such as the CIA World Factbook (www.cia.gov/cia/puhlications/facthook).

15

exposure to such events is typically measured by its probability of occurrence multiplied by its business impact (or severity). Extreme events occurrences are predictable when they occur repeatedly, but they can also be sudden, unique and unpredictable. Little a priori information is typically available on non-repetitive extreme events such as sabotage, sudden currency devaluations or political coups (Banks, 2006). The occurrence of such events remains very difficult to predict (Sheffi, 2001; Kaplan, 2002; Lambert et a i , 2005).

a) Natural Disasters by Country (1974-2003)

i MX' \ I H I I I .

Source: www.emdat.be

b) The Failed States Index 2008

Total Number of D t M e M n 0 X

STATES INDEX 2008

■ ■ .i É — ^ — i j ^ l Source: www.foreignpolicv.com

Figure 1-3. Example of Multi-hazard Indexes

»

i s I

I I I

Very high

High

Medium

Minor

No No Medium Catastrophic

Serious Minor

- - Business Impact - *•

Figure 1-4. Risk Matrix (Norrman and Jansson, 2004)

1.2.3 Strategic Evaluation of SCN Designs and Optimization Criteria

It can be argued that the paramount goal of a business should be the sustainable creation of shareholder value, and that this goal implicitly provides a mechanism to reach a proper balance between the conflicting objectives of the various stakeholders of a firm (Yucesan, 2007). Value is defined as the sum of all the future residual cashflows (RCF) generated by a firm, discounted at the firm's weighted average cost of capital, where

RCF = (Revenues - Operating expenses)(l - Tax rate) - Capital expenditures

16

In order to obtain value-creating supply chains, one should therefore select a SCN design maximizing the present value of all future RCF generated by the SCN, and discounted at the firm's cost of capital, which is easier said than done. Often, value-driven businesses use static strategic performance indicators such as the economic profit (EP), also referred to as the economic value added (EVA), and the return on capital employed (ROCE)6. They also break these strategic metrics into financial and operational performance indicators that are more appropriate for mid-level and operations managers (Yucesan, 2007). A comprehensive review of performance measures and metrics in SC management is found in Gunasekaran and Kobu (2007).

The definition of residual cash flows above implies that three broad categories of value drivers must be taken into account in SCN design, namely: revenue drivers, cost drivers and capital expenditures. Tax rates are an important consideration mainly for multinational SCN. Cost drivers can be associated with SCN procurement, production, warehousing, storage, transportation and sale activities using Activity-Based Costing (ABC) concepts (Terrance, 2005; Shapiro, 2008). Revenue drivers are related to the notion of order winners introduced by Hill (1989). Order winners are value criteria enabling a firm to win orders in its product-markets, and thus to increase its market share and its revenues. These order winning criteria include product range, product prices, product quality and reliability, delivery speed and reliability, volume and design flexibility, agility (often defined as the combination of speed and flexibility), market coverage, ecological footprint, etc. (Lefrançois et al., 1995; Vidal and Goetschalckx, 2000; Gunasekaran et al., 2004). Several of these criteria are directly related to the firm SC capabilities. Capital expenditures capture the investments required to develop the SCN as well as the market value of current assets. They may also be influenced by the financing mechanism used by the firm. They are associated to the various location/capacity options considered in the SCN design process. The net present value (NPV) of these revenues and costs over the life of the SCN must be calculated to evaluate the value of a SCN design.

The value drivers discussed above are not necessarily all relevant for SCN design. They are relevant only if they are affected by the various design options considered. Much of the SCN design literature considers simplified static and deterministic models for which the demand for a typical future period (usually a year) is assumed known. Under this assumption, the revenues are a constant and the objective reduces to the minimization of total network costs (relevant operating expenses and capital charges). The capital charges

6 Since ROCE = (Revenues - Operating expenses)/Capital employed , its use as an objective in a design model would however lead to a fractional program (Barros, 1995).

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must then be expressed as a fixed yearly rent associated to binary facilities/technology selection variables. Some authors have proposed bi-criterion models aiming to minimize total network costs and an order winning criterion such as response time (Ballou, 1992) and volume flexibility (Sabri and Beamon, 2000). This is typically done by incorporating a constraint in the model imposing qualifying requirements on the order winner considered, and by parametrizing this requirement to construct an efficient frontier. This is illustrated in Figure l-5a), where each point in the graph gives the total network cost and the maximum response time provided by a design such as the one in Figure 1-la). One of the designs on,the efficient frontier can then be selected by management (Rosenfield et a l , 1985). If an explicit relationship can be established between demand, product prices and some order winners depending on the network structure (Ho and Perl, 1996; Vila et a i , 2007), or if sales in demand zones are considered as decisions variables bounded by penetration targets and potential market shares (Cohen et a l , 1989; Martel, 2005), then revenues depend on design variables and, as illustrated in Figure l-5b), the objective must be to maximize residual cash flows.

When a finite planning horizon is considered, as opposed to a single planning period, the timing of structural SCN adaptations (opening/closing of facilities or of systems within facilities) and the consideration of real options (Trigeorgis, 1996) become important issues. The SCN design objective then becomes the maximization of the present value of the cash inflows and outflows generated by the SCN during the planning horizon, and of the residual value of the SCN assets at the end of the horizon, i.e. of the RCF generated by the SCN assets after the planning horizon. Clearly, for any realistic planning horizon, these cash flows and residual values are not known with certainty at the time when SCN design decisions are made.

a) Efficient Frontier

Total Network

Cost

Dominated Design

7°^ Qualifying y requirement

b) Residual Cash Flows Maximization

Costs (Revenues)

Response Time Response Time

Figure 1-5. Static Design Tradeoffs for a Domestic Supply Chain Network

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Static financial or operational performance indicators such as EVA, ROCE, assets turnover, resource utilization rates, market shares, service levels, etc. are easy to compute from historical data when looking at the past, but they are not of much use when looking at the future. Since future RCF values are uncertain, the measures employed to evaluate future SCN performances depend on the approach used to model uncertainty. They normally involve a measure of central tendency, such as the expected value, and measures of dispersion, such as the variance or the maximum regret. The way in which these measures are combined to arrive at a global strategic valuation measure (or return measure) depends on the way uncertainty is modeled and also on the attitude toward risk of the decision-maker. A risk neutral decision-maker would base his decisions purely on a central tendency measure, but when considering strategic issues such as SCN design, most decision-makers are risk averse. Two types of aversion to risk must also be distinguished in SCN design, namely aversion to RCF variability and aversion to high-impact catastrophic events. Some authors have also advocated the elaboration of an efficient value-risk frontier, by incorporating maximum risk constraints in their model (Hodder and Jucker, 1985; Eppen et a i , 1989). Note that instead of trying to elaborate an adequate combined return measure, a multi-criteria decision approach may be used.

Several authors have proposed SCN performance measures or attributes to value sustainable returns in a perturbed business environment. These include downside risk (Eppen et a i , 1989), which is commonly used in finance to assess the risk of potential investments, operational flexibility (Dornier et a i , 1998), agility (Lee, 2004), reliability (Vidal and Goetschalckx, 2000; Snyder and Daskin, 2005; Berman et al., 2007), robustness (Snyder and Daskin, 2006; Kouvelis and Yu, 1997; Dong, 2006), responsiveness (Bertrand, 2003; Graves and Willems, 2003) and resilience (Sheffi, 2005). There is a considerable overlap in these concepts, and the notions of robustness, responsiveness and resilience are sufficient to consider all the nuances they bring. Intuitively, robustness is the quality of a SCN to remain effective for all plausible futures, responsiveness is the capability of a SCN to respond positively to variations in business conditions, and resilience is the capability of a SCN to avoid disruptions or quickly recover from failures. These three concepts are discussed in detail in what follows.

It is clear that the performance of a firm depends on its SCN design strategy: an adequate capacity deployment (network structure) provides valuable order winners and lowers costs; appropriate responsiveness and resilience strategies maintain value creation under uncertainty. So, the challenge is to design SCN that are capable of providing sustainable shareholder value for any plausible future business environment, i.e. to design

19

robust value-creating SCN. Therefore, the approach used to analyse SC vulnerabilities, and to model SCN structures, future business uncertainties and SCN responsiveness/resilience strategies is crucial. To the best of our knowledge, no comprehensive SCN design approach considering all these issues has been proposed in the literature. This paper charts directions for the development of a comprehensive SCN design approach through a representative review of the relevant literature. Although much of our discussion is cast in a business context, it is also directly relevant for non-business SCN such as military (Girard et al., 2008) or emergency relief (Tovia, 2007) logistics networks.

1.3 Deterministic SCN Design Models Facility location models (Daskin, 1995; Drezner, 1995; Sule, 2001; Drezner and

Harnacher, 2002; Daskin et a l , 2003; Revelle and Eiselt, 2005), and in particular discrete facility location models (Mirchandani and Francis, 1990), can be considered as the foundation of SCN design models. They deal with the location of facilities in some given geographical area. Basic facility location problems (FLP) consider a single product and a single production/distribution echelon with uncapacitated (UFLP) or capacitated (CFLP) facilities. Their original formulation goes back to Balinski (1961) and are still being studied (Revelle et a l , 2008). In the CFLP, demand can be supplied from more than one source. When it is required that each demand zone is supplied from a single source (CFLPSS), the problem is much more difficult to solve. In fact, the generalized assignment sub-problem obtained for a given set of facilities is NP-hard (Fisher, 1986). Kaufman et al. (1977) studied an extended version of the UFLP incorporating a production and a distribution echelons. Several authors also studied multi-product extensions of the one or two echelon CFLP and CFLPSS. Geoffrion and Graves (1974) proposed a Benders decomposition approach to solve a path-based formulation of a multicommodity CFLPSS, with fixed production facilities and location-allocation decisions for the distribution echelon. Hindi and Basta (1994) solved an arc-based formulation of a similar problem with a Branch and Bound algorithm. Hindi et a l (1998), Klose (2000) and Pirkul and Jayaraman (1996, 1998) proposed Lagrangian relaxation procedures to solve two-echelon CFLPSS's and CFLP's. Several heuristics were also proposed to solve these problems, based on interchange procedures (Kuehn and Hamburger, 1963; Zhang et a l , 2005), tabu search (Al-Sultan and Al-Fawzan,1999; Michel and Van Hentenryck, 2004), genetic methods (Kratica et a l , 2001), randomized rounding (Barahona and Chudak, 2005) and very large-scale neighborhood (VLSN) search (Ahuja et a l , 2004). Owen and Daskin (1998) and Klose and Drexl (2005) present detailed reviews of the large literature available on these problems.

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The static FLP models reviewed in the previous paragraph are based on the following assumptions: i) facilities capacity is predetermined, ii) at most one production stage is considered, iii) nodes and arcs of the network are within the same country, iv) fundamental tradeoffs are between facilities fixed capital/operating charges and variable linear production, warehousing and transportation expenditures, the later being crudely approximated via aggregate flow decisions. Several extensions were proposed to relax these assumptions. They can be classified in two categories: extensions to model SCN design decisions more closely, and extensions to anticipate operating decisions more precisely.

The importance of capacity as a decision variable in location problems was recognized early by Elson (1972). Nonetheless, explicitly integrating capacity decisions as SCN design variables is more recent. Some models consider capacity expansion as a continuous variable (Verter and Dincer, 1995) but, others more realistically consider discrete facility capacity options (Paquet et a l , 2004; Amiri, 2006) or alternative facility configurations (Amrani et a l , 2008). The extended formulations proposed to model multi-stage production-distribution networks are based on the use of aggregate bill-of-material structures (Cohen and Moon, 1990; Arntzen et a l , 1995; Paquet et a l , 2004; Martel, 2005), or on the use of generic activity graphs with recipes (Brown et a l , 1987; Dogan and Goetschalckx, 1999; Lakhal et a l , 2001; Philpott and Everett, 2001; Vila et a l , 2006). Extensions covering product development and recycling (Fandel and Stammen, 2004), and alternative transportation modes were also considered (Cordeau et a l , 2006). Some authors have proposed extensions to take into account economies of scale in production/handling (Soland, 1974; Kelly and Khumawala, 1982; Cohen and Moon, 1990), inventory (Martel and Vankatadri, 1999; Martel, 2005; Ballou, 2005) and transportation (Fleischmann, 1993) costs. Finally, several authors have proposed extensions to maximize residual cash flows in an international context (Cohen et a l , 1989; Arntzen et a l , 1995; Vidal and Goetschalckx, 2001; Goetschalckx et a l , 2002; Bhutta, 2004; Kouvelis et a l , 2004; Martel, 2005; Meixell et a l , 2005). The static-deterministic SCN models proposed by Arntzen et al. (1995), Fandel and Stammen (2004), Martel (2005), Cordeau et al. (2006) and Vila et al. (2006) are among the most comprehensive presented to date.

In the last few years major efforts have been devoted to the development of location models with a much more detailed anticipation of network users' transportation and inventory management decisions. Shen (2007) has reviewed integrated location-routing, location-inventory and location-routing-inventory models. The first classification of location-routing problems is found in Laporte (1988). Several papers have studied different

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aspects of this problem (Nagy and Salhi, 1996; Prins et a l , 2007; Berger et a l , 2007). A comprehensive review of location-routing models and of their applications can be found in Nagy and Salhi (2007). Other contributions have considered the risk pooling effects of network cycle and safety stocks in location-inventory models (Ho and Perl, 1996; Daskin et a l , 2002; Shen et a l , 2003; Ambrosino and Scutella, 2005; Shen, 2007). Recently, Romeijn et al. (2007) has integrated inventory and transportations decisions into a two echelon SCN design model. Sabri and Beamon (2000) also proposed an integrated approach to take strategic and operational planning decisions into account.

Several deterministic multi-period SCN design models were also proposed in the literature. Some of these models are static, in that they involve design decisions only at the beginning of the planning horizon, but they use several planning periods to anticipate more closely operational decisions (Cohen et a l , 1989; Arntzen et a l , 1995; Dogan and Goetschalckx, 1999; Martel, 2005; Vila et a l , 2006). Some dynamic models allowing the revision of design decisions (number, location, technology and capacity of facilities; sourcing and marketing policies) at the beginning of each planning period were also proposed. Dynamic location problems were studied by Erlenkotter (1981), Shulman (1991) and Daskin et al. (1992). Capacity expansion problems are by definition multi-period (Julka etal., 2007). Dynamic SCN design models were proposed by Bhutta et al. (2003), Melo et al. (2005) and Paquet et al. (2008).

Several particular exact and heuristic methods were proposed to solve basic location-allocation problems. Decomposition methods have been proposed to solve more elaborated SCN design models (Geoffrion and Graves, 1974; Dogan and Goetschalckx, 1999; Paquet et a l , 2004, Cordeau et a l , 2006). Others have proposed to include valid inequalities in their model (Dogan and Goetschalckx, 1999; Paquet et a l , 2004). In our opinion, most static deterministic SCN design models can now be solved efficiently with the recent versions of commercial solvers such as CPLEX and Xpress-MP. Melo et al. (2009) provide a recent review of the literature on the various extensions of location models discussed in this section.

1.4 SCN Design Models under Uncertainty The deterministic models discussed in the previous section provide a solid foundation

for SCN design. Nonetheless, any design obtained based on these models has no guarantee of performance for any plausible futures. These models do not handle uncertainties and information imperfections about expected plausible future business environments. So, uncertainty modeling becomes an important challenge for more realistic SCN design.

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Uncertainty has different meaning and implications in a number of different fields. We therefore start this section with a relatively general discussion on various approaches used to model uncertainty. Then, we address SCN design models under different types of uncertainty.

The distinction between uncertainty, risk and certainty is an old issue of crucial importance (Knight, 1921). Rosenhead et al. (1972) proposed to distinguish between decision-making under certainty, risk and uncertainty. This characterization was subsequently adopted by several authors (Kouvelis and Yu, 1997; Snyder, 2006). According to these authors, certainty corresponds to the case where no element of chance intervenes between decisions and outcomes. Risky situations are those where the link between decisions and outcomes is governed by probability distributions. Uncertainty describes situations where it is impossible to attribute probabilities to the possible outcomes of a decision. This distinction between risk and uncertainty is however not universally accepted. In classical risk management, risk refers to the product of the probability and the severity of extreme events (Haimes, 2004; Grossi and Kunreuther, 2005), and probabilities are not the only way to model the likelihood of possible future events. Fuzzy sets (Zadeh, 1965), possibilities (Zadeh, 1978), belief functions (Shafer, 1990), rough sets (Pawlak, 1991) are example of other uncertainty modelling paradigms. Therefore, we suggest characterizing decision-making situations based on the quality of the information available: decisions are made under certainty when perfect information is available and under uncertainty when one has only partial (or imperfect) information (French, 1995; Zimmermann, 2000; Roy, 2005; Stewart, 2005). The term uncertain under this paradigm is value neutral, i.e. it includes the chance of gain and, conversely, the chance of damage or loss. As explained by Stewart (2005), uncertainty leads to risk and this term refers to the possibility that undesirable outcomes could occur. The risk increases as the likelihood and the negative impact of possible outcomes increases, as illustrated by Normann's risk matrix in Figure 1-4.

Under uncertainty, different quality of information may be available. The worst case is total uncertainty or complete ignorance. Three types of uncertainties may be distinguished when partial information is available: randomness, hazard, and deep uncertainty. Randomness is characterized by random variables related to business-as-usual operations, hazard by low-probability high-impact unusual events, and deep uncertainty (Lempert et a l , 2006) by the lack of any information to assess the likelihood of plausible future extreme events. For hazards, as indicated previously, it may be very difficult to obtain sufficient data to assess objective probabilities and subjective probabilities must often be

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used. Note that although these definitions of randomness and hazard are based on probabilistic notions, other formalisms such as fuzzy sets, possibilities or rough sets could be used to model outcome likelihood. However, since most of the literature on non-deterministic SCN models and on risk assessment is based on a probabilistic approach, we will pursue our discussion using a probabilistic language.

1.4.1 Randomness Under randomness, some of the SCN design model parameters (demands, prices,

exchange rates, raw material/energy costs...) are considered as random variables with known probability distributions. The joint-events associated to the possible values of the random variables can be considered as plausible future scenarios, and each of these scenarios has a probability of occurrence. One approach often used to deal with these problems is to elaborate an "average scenario", and then solve the resulting deterministic model. It is known though that the solution thus obtained is not necessarily optimal. Moreover, such solutions may be very bad or even unfeasible under specific scenarios (Sen and Higle, 1999). An alternative is to solve the resulting deterministic model for a subset of representative scenarios, and to evaluate the designs obtained using Monte Carlo sensitivity analysis (Saltelli et a l , 2004; Ridlehoover, 2004). The difficulty with this approach is to determine which among the solutions found is the best. A method to select a solution is presented in Lowe et al. (2002): they propose a screening procedure using a number of filtering criteria such as Pareto optimality, mean-variance efficiency and stochastic dominance. Good examples of how this approach works are found in Kôrksalan and Sural (1999), Mohamed (1999) and Vidal and Goetschalckx (2000). This is a reactive solution approach because random variables are only considered during the a posteriori evaluation step. To consider the random variables explicitly in the SCN design model, a proactive stochastic programming (Birge and Louveaux, 1997; Ruszczynski and Shapiro, 2003, Shapiro, 2007) approach must be used.

Most of the static deterministic models reviewed previously can be transformed into two-stage stochastic programs with recourse relatively easily (Santoso et a l , 2005). The models thus obtained typically consider that the design variables must be implemented before (first stage variables) the outcome of the random variables is observed, but that the network usage variables (second stage variables) provide the recourses necessary to make sure that the design obtained is feasible. The objective is to optimize the expected value of the design and recourse decisions. These models can also be extended to consider risk aversion through the use of risk measures such as mean-variance functions and conditional

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value at risk functions (Mulvey et a l , 1995; Shapiro, 2007). Dynamic problems can also be modeled using multi-stage stochastic programs. A major difficulty of the stochastic programming approach is to deal with the possibly infinite number of possible scenarios. A random sample of scenarios selected with Monte Carlo methods may be used to overcome this difficulty (Shapiro, 2003). Scenario generation techniques were also proposed for multi-stage programs (Ducapova et a l , 2000; Hoyland and Wallace, 2001).

Stochastic location models were proposed by Birge and Louveaux (1997) and Snyder and Daskin (2006). A comprehensive review of simple location models under uncertainty is found in Snyder (2006). Fine and Freund (1990) developed a stochastic program for capacity planning. A review of recent relevant developments in the capacity management literature is found in Van Mieghem (2003). Two-stage stochastic SCN design models were proposed by Tsiakis et al. (2001), Santoso et al. (2005), Vila et al. (2007, 2008) and Azaron et al. (2008). Some models incorporating mean-variance objective functions to measure design robustness were also elaborated (Hodder and Jucker, 1985). Following the pioneering work of Pomper (1976), some authors have also proposed multi-stage SCN design models (Eppen et a l , 1989; Huchzermeier and Cohen, 1996; Ahmed and Sahinidis, 2003).

1.4.2 Hazard

High-impact extreme events should not be treated the same way as low-impact business-as-usual events. Moreover, identifying potential threats and assessing their risk are very challenging undertakings. Catastrophe models have been used to estimate the location, severity and frequency of potential future natural disasters (Grossi and Kunreuther, 2005). They are usually based on a catastrophe arrival process, and they provide tradeoffs between economic loss (a severity evaluation measure) and the probability that a certain level of loss will be exceeded on an annual basis (Haimes, 2004; Grossi and Kunreuther, 2005; Banks, 2006). This type of assessment is practical for the insurance industry, but it is not adequate for SCN design; considering each type of hazard separately is too cumbersome, and economic loss is not an adequate severity measure because it is not directly related to design variables. The first difficulty can be avoided by using multi-hazards, i.e. aggregate extreme events incorporating all types of recurrent natural, accidental and wilful hazards (Gogu et al. 2005; Scawthorn et a l , 2006). However, adequate severity measures for SCN design would have to be related to key design variables/parameters such as facility/supplier capacity and customer demand. Qualitative

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SC disruptions risk identification and assessment approaches are proposed by Kleindorfer and Saad (2005) and Manuj and Mentzer (2008).

The relative importance of extreme events versus business-as-usual events is related to the issue of the aversion of decision-makers to extreme events. Models using expected value objective functions completely miss this important problem dimension, because they give the same weight to these two types of events. A multi-objective partitioning approach was proposed by Haimes (2004) to avoid this pitfall. It uses a set of conditional expected value assessment functions taking the impact of various types of events into account. Despite the fact that the importance of extreme events in SCN design is now well documented (Helferich and Cook, 2002; Christopher and Lee, 2004; Sheffi, 2005; Craighead et a l , 2007), to the best of our knowledge, no formal SCN design models currently take hazards into account.

1.4.3 Deep Uncertainty It is possible to elaborate plausible future scenarios under deep uncertainty. However,

the information available is not sufficient to estimate an objective or subjective probability for these scenarios. There is a large literature on the elaboration of narrative scenarios to support strategic decision-making (Godet, 2001; Van der Heijden, 2005). Lempert et al. (2006) suggests the use of narrative scenarios in deep uncertainty situations and shows how to use these scenarios to enhance solution robustness. Scenarios can be elaborated through structured brainstorming sessions and/or expert interviews related to SCN opportunities and threats. Qualitative forecasting approaches, such as the Delphi method, can be used to support the process (Boasson, 2005). Some companies, such as Shell, push this approach very far: they produce and regularly revise scenarios of what the world might look like over the next twenty years (Shell, 2005). This approach can be used to produce likely scenarios, but also to imagine "worst case" scenarios.

Narrative scenarios can be streamlined to obtain quantitative scenarios about the business future. When this is done, robust optimization methods (Mulvey et a l , 1995; Kouvelis and Yu, 1997) can be used to find adequate SCN designs. The robust optimization approach proposed by Mulvey et al. (1995) can be seen as an extension of stochastic programming, but it can be used with a min-max regret criterion, which would be done in the case of deep uncertainty. With the approach proposed by Kouvelis and Yu (1997), the most common robustness criteria used are the minimization of the maximum cost and the minimization of the maximum regret across all possible scenarios. Robust optimization has been applied to different versions of the facility location problem under

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uncertainty (Gutierrez et a l , 1996; Kouvelis and Yu, 1997; Yu and Li, 2000; Snyder and Daskin, 2006), as well as to capacity expansion problems (Bok et a l , 1998).

To conclude this discussion of non-deterministic models, note that fuzzy sets were used by some authors to model site selection problems (Sule, 2001; Kahraman et a l , 2003) and SCN design problems (Chen and Lee, 2006). A few papers based on the possibility approach were also published on SC problems (Wang and Shu, 2007; Torabi and Hassini, 2008). A relevant review of uncertainty models is found in Matos (2007). It should also be noted that all the location and SCN design papers reviewed in this section assume that the SC modelled is either in a randomness context or a deep uncertainty context. In real life, elements of plausible future business environments can fall under any of the three types of uncertainties discussed, namely: randomness, hazard and deep uncertainty. To the best of our knowledge, no comprehensive SCN design approach, dealing with all uncertainty types, has been proposed to date.

1.5 Fostering Robustness in SCN Design

1.5.1 Robustness The concept of robustness has raised a lot of discussion in the literature on decision­

making under uncertainty. Roy (2002) suggested that the term robust can have different meanings depending on the decision-making context considered. A first distinction needs to be made between model robustness (Mulvey et a l , 1995; Vincke, 1999), algorithm robustness (Sorensen, 2004) and solution (or decision) robustness (Rosenhead et a l , 1972; Mulvey et a l , 1995; Kouvelis and Yu, 1997; Wong and Rosenhead, 2000; Roy, 2002; Hites et a l , 2006). In our case, we are clearly concerned with solution robustness, or more specifically SCN design robustness. Rosenhead et al. (1972) and Wong and Rosenhead, (2000) state that robustness is a measure of the useful flexibility maintained by a decision so as to leave many options for the choices to be made in the future, which is representative of the generic definitions found in the literature. It is interesting to note that robustness is associated with the notion of solution flexibility, which is congruent with the recent emphasis on flexibility and agility in the SC literature (Bertrand, 2003; Lee, 2004). Several authors have discussed robustness in a supply chain context (Rosenblatt and Lee, 1987; Gutierrez et a l , 1996; Mo and Harrison, 2005; Sheffi, 2005; Dong, 2006; Snyder and Daskin, 2006). They define robustness as the extent to which the SCN is able to carry its functions for a variety of plausible future scenarios.

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Linking these definitions to our previous discussion on the evaluation of supply chain performances, it can be stated that a SCN design is robust, for the planning horizon considered, if it is capable of providing sustainable value creation under all plausible future scenarios (normal business conditions as well as major disruptions). To evaluate the sustainability of a design, one must work with the discounted sum of the residual cash flows generated over a multi-period planning horizon, and take the three types of uncertainties identified into account. When considering a set of plausible future scenarios, resulting partly from the random, hazard and deeply uncertain environmental elements considered, the revenues and costs of all the operational and contingency actions required to satisfy customers demands with a given network design must be evaluated. One necessarily selects a robust design, under randomness and hazards, by maximizing the expected value of these discounted cash flows. This is the approach taken by stochastic programming through the modelling of recourses. To take aversion to value variability into account, one must use risk measures such as mean-variance or conditional value at risk functions (Mulvey et a l , 1995; Shapiro, 2007) instead of expected value. If scenario probabilities are not available (deep uncertainty) a robust optimization model can be used (Kouvelis and Yu, 1997). If probabilistic and non-probabilistic scenarios are considered, which is desirable in most practical situations, then the scenario set must be partitioned accordingly and, as suggested by Haimes (2004), a multi-criteria approach based on conditional expectations and min-max regrets could be used. Hites et al. (2006) introduced a multicriteria evaluation of robustness. Some authors have also suggested incorporating a regret constraint (p-robustness) in their model (Snyder and Daskin, 2006). This partitioning approach can also be used to take aversion to extreme events into account. Currently, no model available in the literature considers all these robustness criteria.

Our previous discussion provides means to evaluate the robustness of a SCN design. But, what kind of SCN structure is likely to be robust? More specifically, what kind of risk mitigation constructs should be incorporated in our optimization models to obtain robust SCN designs? To answer these questions we look more closely at the notions of SCN responsiveness and resilience. At the operational level, short-term mitigation actions are required to deal with the variability of low-impact, as well as high-impact, business events: these are the domain of responsiveness policies. However, to deal with network threat situations, mitigation postures related to the SCN structure, but going beyond the standard design decisions discussed previously, are required: these are the domain of resilience strategies. Currently, most supply networks are incapable of coping with emergencies (Lee, 2004). According to Chopra and Sodhi (2004) most companies develop plans to protect

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against recurrent low-impact events, but they neglect high-impact low-likelihood disruptions.

1.5.2 Responsiveness Usually, responsiveness policies aim at providing an adequate response to short-term

variations in supply, capacity and demand. They provide a hedge against randomness and hazards to increase the SCN expected value. For a given network structure, these policies shape the means that can be used to satisfy demands from internal resources and with preselected external providers. Responsiveness policies are typically associated to resource flexibility mechanism, such as capacity buffers (Sabri and Beamon, 2000 and Chopra and Sodhi, 2004), production shifting (Graves and Tomlin, 2003), overtime and subcontracting (Bertrand, 2003); safety stock pooling and placement strategies (Graves and Willems, 2003); flexible sourcing contracts (Kouvelis, 1998; Semchi-Levi et a l , 2002; Lee, 2004; Sheffi, 2005; Tomlin, 2006); and shortage response actions, such as product substitution, lateral transfers, drawing products from insurance inventories, buying products from competitors, rerouting shipments or delaying shipments (Shen et a l , 2003; Gunasekaran et a l , 2004; Tomlin, 2006; Tang and Tomlin, 2008). SCN design models usually assume that responsiveness policies are elaborated beforehand. When using stochastic programming, these policies are reflected in the recourse anticipation structure of the model. For example, if lateral transfers are permitted, then second stage flow variables between production-distribution centers would be defined; if overtime is permitted within certain bounds, then recourse variables and constraints would be added to reflect this policy; if dual sourcing is permitted then flow variables from suppliers would be defined accordingly.

1.5.3 Resilience Resilience is directly related to the SCN structure and resources, and hence to first-

stage design variables. It can be seen as a strategic posture of deployed resources (facilities, systems capacity and inventories), suppliers and product-markets, as a physical insurance against SC risk exposure, providing the means to avoid disruptions as much as possible, as well as the means to bounce back quickly when hit. More general discussions of enterprise resilience are found in Van Opstal (2007) and on the Web site of the Center for Resilience who defines resilience as "the capacity of a system to survive, adapt, and grow in the face of unforeseen changes, even catastrophic incidents". Rice and Caniato (2003), Christopher and Peck (2004) and Sheffi (2005) conclude from empirical studies

7 www.resilience.osu.edu

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that business is in need of resilience strategies to deal effectively with unexpected disruptions. The main challenge is to elaborate resilience strategies providing an adequate protection from disruptions without reducing the SCN effectiveness in business-as-usual situations.

Resilience strategies aim at obtaining a SCN structure reducing risks and providing capabilities for the efficient implementation of the responsiveness policies previously discussed. This can be done by avoiding or transferring risks (Manuj and Mentzer, 2008), and/or by investing in flexible and redundant network structures (Rice and Caniato, 2003; Sheffi, 2007). Avoidance strategies are used when the risk associated to potential product-markets, suppliers or facility locations is considered unacceptable, due for example to the instability of the associated geographical area. This may involve closing some network facilities, delaying an implementation, or simply not selecting an opportunity. Another way to avoid risks may be through vertical integration, i.e. the internalisation of activities. This may reduce risk through an improved control, but it converts variable costs into fixed costs. This is an incitation to produce internally for low risk product-markets and to outsource production for higher risk product-markets, thus transferring risks to suppliers. These are important tradeoffs that must be captured in SCN design models.

Responsiveness capabilities development may be flexibility or redundancy based. Flexibility based capabilities are developed by investing in SCN structures and resources before they are needed. Examples of design decisions providing such capabilities include selecting production/warehousing systems that can support several product types and real­time changes, choosing suppliers that are partially interchangeable, and locating distribution centers to ensure that all customers can be supplied by a back-up center with a reasonable service level if its primary supplier fails. Redundancy based capabilities involve a duplication of network resources in order to continue serving customers while rebuilding after a disruption. An important distinction between flexibility and redundancy based capabilities is that the latter may not be used (Rice and Caniato, 2003). Examples of redundancy based capabilities include insurance capacity, that is maintaining production systems in excess of business-as-usual requirements, and insurance inventory dedicated to serve as buffers in critical situations (Sheffi, 2005). The consideration of such responsiveness capabilities complicates SCN design models considerably. Although a few reliability models for location decisions have investigated these concepts (Snyder et al., 2006; Murray and Grubesic, 2007), much remains to be done to address the problem adequately.

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1.6 Conclusions The body of literature on SCN design problems is extensive. However, in our opinion,

several aspects of the problem are overlooked. Most design models make significant assumptions and simplifications falling short of current business needs. Several shortcomings and opportunities for research were identified in the previous pages, and we argued that a comprehensive methodology dealing with all relevant problem facets is in need. The main research directions proposed to develop a comprehensive methodology for SCN design under uncertainty are the following:

• SCN risk analysis. Numerous environmental uncertainties and SC vulnerabilities, ranging from business-as-usual randomness to major asset/partner failures, were discussed in the literature. However, in a specific context, the consequences of these event types can vary from catastrophic to low. For SCN design purposes, the random variables and vulnerability sources explicitly considered must be reduced to a manageable number. This requires the development of a multi-criteria filtering process, based on a subjective evaluation of the likelihood and severity of possible event types, to select the sources of uncertainty to incorporate in the SCN design model.

• SCN hazards modeling. A large literature exists on the modeling of various types of catastrophes however it is not adequate for SCN design. Considering each type of hazard separately is too cumbersome and one must rather work with multi-hazards having generic impacts on SCN resources/markets. The definition of multi-hazard arrival processes over multi-hazard zones is in itself a challenging problem. Adequate disruption severity metrics and recovery functions, related to key design variables/parameters such as facility/supplier capacity and customer demand, must also be elaborated. Very little work has been done in this area.

• Scenario development and sampling. Multi-period plausible future scenarios must be developed to support the optimization and the evaluation of SCN designs. In most practical situations, an infinite number, of scenarios are possible but, given the complexity of the problem, only a few of them can be considered in the optimization process. Monte-Carlo methods can be used to generate some scenarios from the random variable distributions and the multi-hazard models, but this is not sufficient. An "importance" based sampling approach must be developed to ensure that all important plausible future facets (random, hazard and worst case events, evolutionary paths...) are covered in the small sample of scenarios selected.

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• Value based SCN design models. A large proportion of the SCN design models proposed in the literature minimize costs. This is not sufficient to help a business create and sustain a competitive advantage. To this end, the objective should be sustainable value creation. This has several implications: i) the relationship between order winners and SC capabilities has to be understood and used to formulate demand and revenue functions, ii) revenues and expenditures should be anticipated over a multi-period planning horizon, iii) capital expenditures should be modelled closely and the firm financing constraints need to be taken into account. The ecological footprint of the SC is also increasingly linked to value creation. Design models considering these issues need to be developed. Current SCN design models anticipate revenues and expenditures through crude production, inventory and flow aggregations. The adequacy of this approach is to be challenged and alternative anticipation schemes should be considered.

• Modeling for robustness. SCN design models should be based on representative samples of plausible future scenarios, using stochastic programming and/or robust optimization approaches. Moreover, the objective should not be purely the maximization of expected value, but rather a strategic valuation measure incorporating aversion to RCF variability and to high-impact catastrophic events. This measure should weight scenarios based on random, hazard and worst case events adequately, and in an integrated way. To our knowledge, no SCN design model of this type is currently available.

• Modeling resilience and responsiveness. Deterministic SCN design models do not take responsiveness and resilience into consideration, and most stochastic models take them into account only partially. The explicit incorporation of risk mitigation constructs, such as back-up suppliers or insurance capacity, in our optimization models would lead to more robust SCN designs. Although some of these concepts were investigated with simple location models, much remains to be done to address this opportunity adequately.

• Solution methods. Although static deterministic SCN design models can often be solved with modern commercial solvers, this is far from being true for realistic multi-period stochastic models. Very few efficient heuristic methods have been developed to solve these models and this is another promising research direction.

All these elements make the elaboration of SCN design models capturing the essence of real problems quite complex. We recognise though that the models formulated should strike a balance between realism and tractability, or solvability, using data available in typical practical contexts. Achieving this objective remains a considerable challenge. In

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our opinion, however, the research directions proposed in this paper provide a path towards a SCN design methodology fostering sustainable value creation.

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Chapitre 2: The Design of Effective and Robust Supply Chain Networks

Résumé - Cet article fournit une méthodologie pour le design de réseaux logistiques en univers incertain. Le problème est initialement définit comme un processus décisionnel à deux niveaux : les décisions de design doivent être prisent maintenant, mais le réseau ne sera utilisé qu'après une période d'implantation. Le long de l'horizon de planification considéré, un ensemble de décisions d'usage du réseau et d'adaptation structurelles doivent être anticipées par le modèle de design. La méthodologie reconnaît trois types d'événements pour caractériser l'environnement futur des réseaux logistiques : aléatoires, hasardeux et profondément incertain. Au moment du design, l'environnement futur est anticipé à travers une approche par scénarios. La génération d'échantillons de scénarios permet de résoudre le problème de design à l'aide d'un modèle approximé afin de produire un ensemble de designs alternatifs. Une approche d'évaluation est appliquée pour choisir le design le plus efficace et robuste parmi cet ensemble et le statuquo.

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2. The Design of Effective and Robust Supply Chain Networks

Walid Klibi12 and Alain Martel1'2

1 Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT)

2 Département Opérations et systèmes de décision, Faculté des sciences de l'administration, Université Laval, Québec, Canada G1V 0A6

Abstract. This paper provides a methodology for Supply Chain Network (SCN) design under uncertainty. The problem is initially defined as a two level organizational decision process: the design decisions must be made here and now, but the SCN can be used only after an implementation period. During a multistage planning horizon a set of user response decisions and a set of planned structural adaptation decisions must be anticipated. The methodology recognizes three event types to characterize the future SCN environment: random, hazardous and deep uncertainty events. At the design time, future environments are anticipated through a scenario planning approach. Scenario samples generation allows approximating the design model to be solved with a sample average approximation program in order to produce a set of alternative designs. A design evaluation approach is then applied to select the most effective and robust SCN among this set and the status quo design.

Keywords. Supply Chain Network Design, Uncertainty, Robustness, Anticipation, Scenario Planning, Network Disruptions, Multihazard, Stochastic Programming

Acknowledgement. This research was supported in part by NSERC grant no DNDPJ 335078-05, by Defence R&D Canada and by Modellium Inc. Results and opinions in the publication attributed to named author(s) were not evaluated by CIRRELT.

Les résultats et opinions contenus dans cette publication n'engagent que leur(s) auteur(s) et n'ont pas été évalués par le CIRRELT.

* Corresponding author: alain.martel(3>cirrelt.ca Dépôt légal - Bibliothèque nationale du Québec,

Bibliothèque nationale du Canada, 2009

© Copyright Walid Klibi, Alain Martel and CIRRELT, 2009

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2.1 Introduction Supply Chain Network (SCN) design involves strategic decisions on the number,

location, capacity and mission of the supply, production and distribution facilities of a supply chain in order to provide goods to a predetermined, but possibly evolving, customer base. Location models have been studied extensively in the literature under deterministic, dynamic and stochastic environments. Detailed reviews of location models are found in Owen and Daskin (1998) and Klose and Drexl (2005). Location models based on stochastic optimization (Birge and Louveaux, 1997; Snyder and Daskin, 2006) and robust optimization approaches (Kouvelis and Yu, 1997) were also proposed to take uncertainty into account. A review of location models under uncertainty is found in Snyder (2006). When classical location models are extended to design SCNs, other strategic decisions on sourcing, capacity acquisition, technology selection and market policies must be considered. The problem then is much more complex: the number of echelon in the network increases, objectives become heterogeneous, new complex constraints, to deal with international issues for example, are needed and the environment uncertainty increases. Management's ultimate goal is to maximize the effectiveness and competitiveness of the SCN. The predominant approach to solve these problems has been to use deterministic mathematical programming models with appropriate sensitivity analysis and scenario analysis. An integrated deterministic modeling framework incorporating most of the aspects of the problem studied to date is presented in Martel (2005).

Since SCNs must be designed to last for several years, it is clear that they should be robust enough to cope with all the random environmental factors (demand, prices, exchange rates...) affecting the normal operations of a company. In addition, SCNs should perform well under major disruptions. In view of recent events, such as the 9/11 terrorist attacks on WTC and hurricane Katrina, companies are aware that they should prepare for the next disaster, but in reality only a few do (Lee, 2004; Sheffi, 2005). At a time when management efforts strive to make supply chains as lean as possible such events may have serious impacts on company performances (Hendricks and Singhal, 2005). Clearly, this type of event should be taken into account in SCN design models. Moreover, in most real life projects, one has to compare the design proposed with the status quo network. This is often done by calculating the economic value of the two solutions with the objective function of the mathematical model used to obtain the proposed design, which is inadequate. It is then legitimate for management to question the precision and validity of

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such results. As far as we know, this question has not been addressed explicitly in the SCN literature.

A few authors have proposed stochastic linear programming (SLP) models (Pomper, 1976; Eppen et a l , 1989; Santoso et a l , 2005; Vila et a l , 2007) and robust optimization (RO) models (Gutierrez et al , 1995; Snyder et a l , 2006) to deal with environment uncertainty in SCN design. The models proposed however apply either to simplified location problems, or they consider only certain types of uncertainties, which compromise their solution robustness for real life problems. No comprehensive approach dealing with all the issues raised above currently exist. A critical review of major drawbacks and missing links in the current SCN design literature is found in Klibi et al. (2009a).

This paper proposes a SCN design methodology taking into account the various types of uncertainties that may affect a supply chain. A scenario based solution approach to design and evaluate SCNs under uncertainty is also proposed. The paper is organized as follows. Section 2 presents the SCN design methodology. It also proposes an approach to take high-impact disruptions into account in SCN design models. Section 3 proposes a generic solution approach to obtain robust value-creating SCN designs. Finally, conclusions and future research directions are provided.

2.2 SCN Design Methodology

2.2.1 Decision problem structure SCN design problems deal with strategic decisions such as facility location,

technology selection, capacity acquisition and deployment issues that are the responsibility of top management. At that level, a major preoccupation is the long term financing of the investments required, the expected return on these investments, risk management and, more generally, the impact of the SCN design decisions on the value of the firm, in a business context, or on the effectiveness with which the organization can accomplish its mission, in other contexts (government, military, NGOs...). However, design decisions impose resource availability and utilization constraints on the users of the SCN which, through their daily supply, production and distribution actions, in response to customer demands, determine the return that will be obtained from the investment. Note that although much of the following discussion is cast in a business context, the methodology proposed applies as well to non-business contexts.

37

Clearly, design decisions cannot be made without anticipating how the users will use the SCN to respond to daily events. The timing of the decisions made at the design and user response levels must also be taken into account. At the beginning of the planning horizon, SCN design decisions are made and after an implementation period the network designed or reengineered becomes available for use during several usage periods. During these usage periods, users serve customers, and react to disruptions, on an ongoing basis with the SCN designed. Although events occur continuously, we assume that the users make daily or weekly decisions, and thus that it is sufficient to observe the environment at the beginning of discrete working periods r e Tu . Furthermore, additional design decisions will be taken in time to adapt the SCN to its environment, which leads to replications of the design and response planning cycle along the planning horizon considered. This gives rise to the multi-stage decision process illustrated in Figure 2-1 for two planning cycles. However, in a rolling horizon framework, the only decisions implemented when the problem is solved at the beginning of the horizon are the first design decisions. Subsequent design decisions can be considered as future opportunities to adapt the network to its environment. During the planning horizon some disruptions may also affect the SCN. Unfortunately, at the beginning of the horizon, the future is not known. The best that can be done is to anticipate, with the information currently available, what the users and the designer will subsequently do to respond to the business environment that will prevail and to adapt the structure of the SCN. In order to avoid any ambiguity, in what follows, we use the expressions design decision only for the decisions to be implemented at the beginning of the horizon. Subsequent design decisions are referred to as structural adaptation decisions.

This paper proposes a SCN design methodology based on the explicit modeling of the design and user response levels over a multi-stage planning horizon. Each level is described by a decision model depending on a, possibly multi-criterion, preference structure C , on a decision space (X for the design level and Y for the user response level), and on the information available / at the time a decision is made. It is assumed however that the two levels consider themselves as part of a team (i.e. they do not have an antagonistic behaviour) and that the information asymmetry is due mainly to the fact that the decisions are not made at the same time. The anticipation pertains, first, to the response of the user to short term events within the network provided by the design level for each planning cycle (stage). In addition, the anticipation covers the SCN structural adaptation decisions for future planning cycles. This leads to the formulation of an anticipated adaptation-response model. A perfect anticipation is not possible, however. The response

38

and design models could be used in the anticipation but, because of the decision time lag, the information cannot be the same. In most cases, the anticipated adaptation-response model is based on aggregate information and on simplified response and design models. It must be realized that the anticipation used has a major impact on the quality of the SCN designed. The role of anticipations in SCN design is studied in Klibi et al. (2009b). They investigate the impact of various response anticipation sub-models on SCN design quality, and they show that there is an (anticipation accuracy, model solvability) trade-off to consider in order to obtain good SCN design models.

Design decision

Structural adaptation decision

i

x2eX2 | /"(£) 1

x , e X ,

Design level

/n(3)

1 I Q Deployment

lead lime

Deployment § + A lead lime

A

ï \*y -S A-2 : .

1- r=7jur 2

Usage period T{

y, e Y;<

Usage period T2

y2 e Y*>

Planning periods

User response level Tu=T luuT2

u

Working periods y T e \ ^ \ l " ( r )

Figure 2-1. Decision Time Hierarchy for Two Planning Cycles

As illustrated in Figure 2-1, we assume in this text that the planning horizon considered covers a set N of planning cycles also referred to as decision stages. At the user response level, for stage ne N , decisions are made each working period r e 7n" (days or weeks). At the design level, these working periods are usually aggregated into quarterly or

m - \

yearly planning periods teT n . Each planning cycle ne N starts with a design (adaptation) decision at date 8n . A known implementation lead time of A planning periods is then incurred before the new design is available. The planning cycle includes the set of planning periods t e Tn defined to cover the working periods t e 7n". The complete planning horizon considered is thus defined by f - f { uf 2 . . .u7L at the design level or by T" =T" KJT"...uTG at the user response level. In what follows, n(t) is used to denote the planning cycle n containing period t.

39

At the beginning of each working period r e T", when the user has to make his decisions, the information available /" (r) is almost perfect, but at time <5j design decisions are made under uncertainty. The information available at time Sn is denoted by In(S„) and we assume that it relates to a set <7l of plausible future scenarios. The fundamental structure of this strategic decision problem is presented in Figure 2-2. The formulation used is based on the generic distributed decision-making framework proposed by Schneeweiss (2003). It is assumed that design decisions are made on a rolling horizon basis.

In the figure, the hat ' A ' is used to indicate terms in the anticipation, and 72-{...|/} is a generalized future return measure depending on the nature of the information available /. The superscripts d and u are used to denote the design and user response levels, respectively. At the design level, x, is the vector of the location, technology, capacity, mission and resilience strategy decisions made at time S], X, is the feasible design space for this decision vector, and Xj is the design selected for implementation. Cd and Cd" are respectively the private criterion and the top-down criterion of the design model. The former captures mainly investment costs. The later is the part of the design-criterion taking future response decisions and structural adaptation decisions into account; it captures the revenues and expenses generated by using the SCN and the additional investment costs necessary to adapt the SCN during the planning horizon. The anticipated design criterion Cd is used to evaluate the structural adaptation decision vector xn (CD) under scenario eve Q, and X*"-' (CD) is the feasible structural adaptation decisions space for stage n. Note that the later depends on the state of the system at the beginning of cycle n, i.e. on the design decisions of the previous period.

At the user response level, yT is the vector of tactical, operational and/or recourse procurement, production, warehousing and transportation decisions made for each working period r e 7n". The user model return measure TZ{C (yT) /" (r)} depends on the nature of the information available I " ( T ) . At the design level, all these tactical and response decisions can usually not be considered explicitly: they are replaced by aggregate surrogate decision vectors y, (CD) , with value depending on the scenario eve Q, for each planning periods t e T . The anticipated response criterion C and decision space Y,"("(â>) are constructed conceptually from the user response level model and/or statistically from past behaviour observations. The adaptation-response model typically involves aggregations over products, customers, means of transportation and working periods. However, these anticipation decisions cannot be implemented and are only used to anticipate the revenues

40

and expenses of the adaptation-response model. For a given design x, E X, and a given scenario coe Çl, the anticipated adaptation-response model optimizes the value of surrogate response decisions and structural adaptation decisions over the planning horizon. In addition, non-anticipativity constraints must be added to ensure that the decisions xn (CD) are identical for all the scenarios CO incorporating the same events for the previous periods.

Design Model

max72.{c(x„.)|/n (£,)}, C(XI,Û)) = CJ (xx,a) + Cd" (x^a)),^^ ii x,eX,

Cd t t(x„co),coeÇi

C-(Xl,«)= opt I C " ( y » ) + £ C'(x>)) + £c"(y,H) Anticipated h(-),J^-) Adaptation- s t ^ ^ e ^ ( ^ V n > 1, y, (®)e Y,x"<" (co)\/t Response Model

& non-anticipativity of x„ (cv) Q)_Z Q

< a ( î )

User Response opt 7^{c" (y r ) | / " (t)} Decisions „ cv"*<r> ^ J ^ I L

r(r) t e T u

Figure 2-2. Strategic Decision Framework

In the design model, the future value of design x, is assessed using a return measure 72.{c(x,,.) / n ( J , ) J taking all the scenarios coe Q. into account and which, as we shall see, may be defined to reflect both expected value and aversion to risk. This return measure would also normally incorporate a discount factor to take the timing of C into account.

2.2.2 Characterization of the information available A supply network must be designed to cope with its future environment, but at the

point in time when it is engineered (or reengineered) the future is not known with certainty. Uncertainty is defined here as the inability to determine the true state of the future business environment which may be partially known or completely unknown. When some

• information is available, three types of uncertainties can be distinguished: randomness, hazard and deep uncertainty. Randomness is characterized by random variables related to

41

business as usual operations, hazard by low probability unusual situations with a high impact and deep uncertainty by the lack of any information to assess the probability of plausible future events. For hazards, it may be very difficult to obtain sufficient data to assess objective probabilities and subjective probabilities must often be used. A detailed discussion of the relevance of these three types of uncertainties for supply chain design is found in Klibi et al. (2009a).

During a planning horizon, the SCN evolves under varying environments. An environment is defined as the internal and external conditions under which the SCN operates during a given period of time. The future is considered at the design level by specifying possible sequences of environments over the planning periods t e f . Each possible sequence of environments defines a scenario. An event is a measurable (i.e. having observable consequences) factor or incident influencing the business environment during a given time period. An event is defined over an adjacent subset of planning periods in T. The environment of planning period t is a compound event, i.e. the result of all the events occurring during period t. From our characterization of uncertainty, it is seen that three types of events shape SCN environments: random, hazardous and deeply uncertain events.

Random events are assumed to be defined over a single period t e T, and they describe factors with a probability of occurrence which can be estimated. Historic information on supply, demand, costs, lead times, exchange rates, etc., can be used to estimate the probability distribution of the random variables related to the business as usual operations of the SCN. These events include the degenerate case of certain events that occur when perfect information exists.

Hazardous events describe factors or incidents affecting a number of adjacent planning periods in T and creating SCN disruptions. Hazards are rare but repetitive events which may be characterized by formal location, severity and occurrence processes. Hazardous events involve natural, accidental or wilful incidents affecting SCN resources. They include accidental disruptions in operations such as major equipment breakdowns, strikes and discontinuities in supply due to supplier bankruptcy, for example. They also include disruptions arising from natural hazards affecting a geographical region, such as earthquakes, floods, windstorms, volcanic eruptions, droughts, forest fires, heat waves, freezes and cold waves. For such events, catastrophe models have been used to provide likelihood of occurrence and/or likelihood of associated monetary losses, based on historical data and/or professional expert opinions (Grossi and Kunreuther, 2005).

42

Deeply uncertain events are incidents affecting a number of adjacent planning periods in f for which no directly relevant information exists. These events include isolated, non repetitive, extreme events for which a likelihood of occurrence cannot be evaluated (Banks, 2006). Events related to terrorism (sabotage, bombing...) and political instability (sudden currency devaluation, coup...), with unpredictable time of occurrence, severity and location, are usually considered as deeply uncertain. In the recent past, some of these disruptions, like the 9/11 WTC attack and the SARS epidemic, have lead to major business failures. Lempert et al. (2006) suggest the use of narrative scenarios in deep uncertainty situations and show how to use these scenarios to enhance solution robustness.

The events matrix presented in Figure 2-3 is a crossover between our information-based classification of events and their expected severity. Light zones correspond to random events having normal impacts on SCNs. As seen in the introduction, several deterministic models and a few stochastic programming models were proposed in the literature to deal with SCN design problems under these types of events. Dark zones correspond to hazardous and deeply uncertain events. As indicated previously, some robust optimization approaches were proposed to deal with simple location problems under these types of events. Our aim here is to propose an integrated SCN design methodology to take all these types of events into account. The methodology proposed is based on recent work in stochastic programming (Shapiro, 2007), catastrophe modeling (Grossi and Kunreuther, 2005), scenarios planning (Van der Heijden, 2005) and risk analysis (Haimes, 2004). It builds on the fact that in all these modeling approaches, the information available on the future can be presented in the form of a set of scenarios about how the future may unfold.

c o X! g u O

Certainty

Randomness

Hazard Deep Uncertainty

Deterministic Models

Stochastic Programming Models

Catastrophe Models Robust Optimization Catastrophe Models Robust Optimization

None Moderate Serious

Catastrophic Normal

Impact Figure 2-3. Events Matrix

From our previous definitions, it is clear that a scenario is a compound event. Each scenario is the result of the juxtaposition of one or more event types that shape the

43

environment of SCNs. All scenarios include random events associated to business-as-usual conditions, but they do not necessarily include the hazardous or deeply uncertain events associated to the SCN threats discussed previously. Hereafter, totally destructive events causing irreversible damages to an entire business are excluded from the analysis. Also, in what follows, in order to analyse the various sources of risk properly, it is necessary to partition the set of scenarios Q into two mutually exclusive and collectively exhaustive subsets: Qp including all probabilistic scenarios without deeply uncertain events (P-

scenarios), and Slu including all other scenarios ((/-scenarios). In principle, it should be possible to evaluate the probability p(û)) of scenarios coe Of. However, the probability of (/-scenarios cannot be evaluated. A conceptual representation of the scenarios tree thus obtained, and of its relationship with SCN decisions, is provided in Figure 2-4. Each path in this tree correspond to a scenario coe Q..

x2(co) N M > l a a__ a a a a a a a ■ a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a : a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a .a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a m a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a

Probabilistic scenarios cvs ÇïF

Deeply uncertain scenarios eve £l u

± ± 4-n >

? e r Planning horizon

Figure 2-4. Scenarios Tree for the Planning Horizon

Businesses and organizations operate in a complex world and, when looking far away, it cannot be assumed that the future will unfold in the tracks of the past. When developing their strategies, companies like Shell study significant events, they analyse political, social and economic actors and their motivations, they explore what the world might look like over the next twenty years, and the impact of alternative views of the future on their business environment (Shell Global Scenarios to 2025, 2005). In other words, they define possible evolutionary paths. The scenarios in Figure 2-4 must consider such evolutionary paths. The scenarios in Q are possible realizations of a set of underlying stochastic processes with known (for P-scenarios) or unknown (for (/-scenarios) parameters. In what

44

follows, it is assumed that a set K of evolutionary paths with probability p k , ke K, can be defined and that the parameters of the scenario generating stochastic processes depend on evolutionary paths. It is thus seen that the set of scenarios Q is the union of the scenario sets Qp*, D.Uk associated to the evolutionary paths ke K.

2.2.3 SCN Risk Analysis Supply chain networks are usually geographically dispersed across regions and

countries which increase their risk exposure and, in order to design robust SCNs, the impact of random, hazardous and deeply uncertain events must be taken into account. Using historical data, classical forecasting and statistical analysis methods can be used to estimate the probability distributions associated to random events. However, the case of hazards and deep uncertainty deserves further analysis. The disruptions which may affect a supply chain can take several forms and it is important to find a practical way of taking them into account without getting lost into a maze of possible incident types. This can be done by classifying hazards into a small number of meta-events with generic impacts on SCN resources (multihazards) and, by considering deep uncertainty through the use of imaginative scenarios. To embed this in our SCN design methodology, we must provide an answer to the three fundamental questions associated to risk analysis: 1) What can go wrong? 2) What are the consequences? 3) What is the likelihood of that happening? For deep uncertainty events, only the two first questions can be partially answered. For hazards, this leads to a three phase approach to model SCN exposures. It combines concepts from catastrophe analysis (Haimes, 2004; Grossi and Kunreuther, 2005; Banks, 2006) and SCN vulnerability analysis (Helferich and Cook., 2002; Kleindorfer and Saad, 2005; Sheffi, 2005; Craighead et a l , 2007, Wagner and Bode, 2008).

The next paragraphs describe the three phases of the SCN hazard modeling approach proposed. The role of each of these phases is the following:

1) Characterization of multihazards and vulnerability sources. The SCN vulnerability sources to take into account in the study are identified and related to relevant multihazards to specify threat domains. The territory over which the network is deployed is partitioned into hazard zones, which are related to exposure levels or regions. When the phase is completed, each network location is associated to a vulnerability source, a hazard zone and an exposure level.

2) Modeling of multihazard processes. A compound stochastic process is defined to describe how multihazards occur in space and in time, and to specify incident's

45

intensity and duration. This phase is independent of the SCN considered. We assume that each incident occurs in a hazard zone at the beginning of a working period. The impact intensity and duration variables are however associated to exposure levels.

3) Modeling the impact of hits on the SCN. The occurrence of an incident in a hazard zone does not necessarily result in a hit of all the SCN locations in that zone. Attenuation probabilities are defined to reflect hits likelihood. When a location is hit, the impact on the network capacity and demand is modelled using recovery functions based on intensity and time to recovery variables.

In what follows the approach is described in generic terms and examples are given to illustrate particular cases.

Multihazards and vulnerability sources

To perform its activities the SCN exploits internal resources, it does business with SC partners, and it uses public infrastructures. Examples of typical resources, partners and infrastructures are given in Figure 2-5. These resources/partners are associated to specific geographical locations. Moreover, when modeling a SCN, some of these locations may be aggregated into geographical zones with a computable centroid. For example, in a business context, ship-to points are usually aggregated into demand zones and, in a military context, demand is naturally associated to regions where conflicts of various types may develop. Let L be the set of all the SCN locations considered. When an extreme event occurs, all locations are not affected in the same way. For example, a fire in a plant may decrease production capacity but an earthquake in a demand zone may increase demand for first-aid products drastically, but decrease demand for luxury products. For this reason, depending on their nature, locations / E L are classified in vulnerability source subsets with similar impacts and time to recovery. Let S be the set of all relevant vulnerability sources. The notation s(l) is used to denote the vulnerability source se S characterizing location l e L. In a SCN, transportation means are also used to move materials between locations. The potential locations and moves considered when designing a supply chain define a network similar to the one illustrated on the vulnerability source layer of Figure 2-6.

46

Production centers „

Endogenous

Assets

Storagt tquÉpnvtnl Handhng tquipment

Ressources r — Dstnbutoori centers r ~ — V . hfaman rwourcts •<

ReccweryiR&vatorrzation centers

Service centers

Vu lnerab i l i t y suppliers

Sources External providers f SuOcontractors

J Supply Chain 3PLS

Partners First-aid product-markets

\ Customers j Sustamment proouct-martets

\ Luxury proauct-marKets

Airports

X Natural disasters

Geopolit ical failures

Market fai lures

Industrial accidents

Public Infrastructures

Terminals Seaports

Rail stations

Travefcvays

T s e S h e H

Figure 2-5. Examples of Vulnerability Sources and Multihazards

When considering potential risks arising from natural, accidental and wilful hazards on the SCN, a large set of vulnerability sources can be identified (Helferich and Cook, 2002). However, the impact of hazards on these vulnerability sources can vary from catastrophic to low. At the strategic decision-making level, the number of vulnerability sources considered should be reduced to a manageable level. A filtering process based on a subjective evaluation of the vulnerability identified leads to the selection of the sources with potential strategic consequences to be included in the set S. The vulnerability sources retained usually include the main internal production, distribution and service resources influencing capacity (plants, warehouses, stores...), the main product-markets or service-offers influencing demand, and the main vendors influencing supply (raw-material suppliers, energy suppliers...). It is assumed that all strategic vulnerabilities come from the SCN locations l e L and not from its arcs. The overriding criterion for the definition of a vulnerability source se S is that all the locations l e Ls it covers must have a similar behaviour in terms of impact intensity, time to recovery and recovery pattern when hit by a multihazard, so that they can all be described in terms of the same metrics. They must also be defined so that the sets Ls c L, s e S, are mutually exclusive and collectively exhaustive. This may lead to the definition of more than one location / for a same geographical region. For example, if the sales of two product categories in the same region (say first-aid products and luxury products) are not affected in the same way by a multihazard (one may increase and the other decrease), then they must be distinguished by

47

associating them to different locations. Similarly, in a military context, potential humanitarian relief missions and peace-keeping missions in a same geographical area must be distinguished because they do not require the same material.

Natural, accidental and wilful hazards cover large classes of incidents which do not necessarily affect SCN vulnerability sources in the same way. Also, depending on the scope of the study, some hazard types may not be relevant. For example, when designing an American network, natural disasters are relevant, but the risk of armed conflicts resulting from a political failure is negligible. However, when designing an international SCN, potential state failures must be taken into account. Finally, even if a hazard type is relevant, for some parts of the world the data required to characterize it may not be available. For all these reasons, for a given SCN design study, a set H of multihazards to consider must be specified. Such a multihazard set is illustrated in Figure 2-5. Multihazards can be elaborated from the data provided by several public sources such as the Centre for Research on the Epidemiology of Disasters (www.cred.be), the Heidelberg Institute for International Conflict (www.hiik.de), the Federal Emergency Management Agency (www.fema.gov) and the U.S. Geological Survey (www.usgs.gov), and private sources such as Swiss Re (www.swissre.com) and Munich Re Group (www.munichre.com). Vulnerability source threat domains must also be defined by specifying the subset Hs _zH of multihazards which have an impact on each vulnerability source se S . -

In what follows, we assume that extreme event threats are not directly related to the resources/partners involved in the SCN but rather to the vulnerability source they are associated to and to their geographical location. In order to map threats, the geographical territory in which the SCN performs must be partitioned into a set of hazard zones Z. Using geographical coordinates, the hazard zone z( l)eZ of a location l e L can be identified, as illustrated in Figure 2-6. Hazard zones delineate areas with similar geological, meteorological, political, economical and critical infrastructure characteristics. These zones may correspond to counties, to states/provinces, to countries, to 3-digit zip-codes, or to a combination of those, depending on the level of precision desired and the data available. They must be constructed, however, to make sure that the SCN location aggregates defined fit uniquely in a hazard zone, and they must be large enough to consider the occurrence of extreme events in different zones as independent. They must also be defined so that the sets L.cz L of locations in the zones z e Z are mutually exclusive and collectively exhaustive. The zonation process is a key issue since the zone granularity

48

determines the realism of the multihazard incidents considered in the SCN optimization model.

Vulnerability sources layer

Network le Vulnerability

mmmÎ] SOUTCCS SCt

s(l),te L

Exposure levels

g„ (zUeZ

Figure 2-6. SCN Exposure Modeling

Unfortunately, with the data available, it is often difficult to estimate hazard arrival and impact processes directly at the hazard zone level. For each multihazard he H , this leads to the introduction of a set Gh of zone aggregates called exposure levels. The notation gh(z) is used to denote the exposure level g e G h including hazard zone ze Z , and ZgczZ the set of zones in exposure level g e G h . Exposure levels can be defined top-down or buttom-up, depending on the context. Exposure levels are sometimes associated to geographical regions, such as continents. The states in the continent then provide the relationship gh(z) between zones and levels. Alternatively, levels can be constructed by evaluating an exposure index for each zone, and then associating levels to adjacent index value intervals. Zones are then assigned to levels based on their index value. For a multihazard he H , this defines an exposure map such as the one illustrated on the multihazard exposure layer in Figure 2-6. The exposure index used to do this can be based on failed state (www.foreignpolicy.com) and/or opacity (www.opacityindex.com) indexes designed to reflect the political stability of a region, natural catastrophes exposure indexes calculated from the data provided by CRED, FEMA or USGS, economic performance indexes such as the World Competitiveness Scores of IMD (www.imd.ch) or the Global

49

Competitiveness Index of WEF (www.weforum.org), industrial accident indexes related to the claims made to insurance companies, public infrastructure quality indexes calculated from databases such as the CIA World Factbook (www.cia.gov/cia/publications/factbook), or on a combination of those. The exposure level gh(l) - gh(z(l)) of a location l e L can be uniquely determined for each multihazard he H . This initial analysis phase thus leads to the specification of multihazard classes (s,g)e SxGh, he H , with associated mutually exclusive and collectively exhaustive location subsets Êsg = {/■$(/) = s,gh (l) = g) .

Modeling of multihazard processes

The SCN designed must cope with the future and thus the modeling of future extreme events must take possible evolutionary paths into account. We assume that multihazards occur independently in hazard zones, and that the time between the occurrences of successive multihazards in a zone is characterized by a non-stationary stochastic arrival process depending on the evolutionary path considered. More specifically, under evolutionary path ke K, if an incident occurs in working period TeT", then the time before the arrival of the next multihazard he H m zone z e Z is a random variable Xh

Ax

with cumulative distribution function F*k_(.). In practice, catastrophe models often use Poisson processes to determine the number of extreme events that can occur in a given period (Banks, 2006). Accordingly, we consider that in most cases it is sufficient to assume that F£T(.) is an exponential distribution Exptji^) with an expected time between multihazards p!_\x. Let $(//**, r) be a function elaborated by experts to superimpose a time pattern for evolutionary path k on fihzS , the historical mean time between multihazards he H in hazard zone z e Z estimated at the beginning of the planning horizon (i.e. at time Sl ). Then, the required probability distributions are obtained simply by calculating ju^t = </>h

k(MhZ8 )̂ for all h, z, k and r.

When designing a domestic SCN in America, the data required to estimate arrival processes directly at the hazard zone level can be obtained relatively easily. However, when designing a global SCN, the data provided by organizations such as CRED and HIIK is not sufficiently detailed to support such an approach. A hierarchical modeling approach based on exposure level arrival processes and conditional hazard zone hit probabilities must then be used. Let X.x be a random variable, with cumulative distribution function F*( . ) , representing the time before the arrival of the next multihazard he H in exposure level (region) g e G h under evolutionary path ke K when an incident occurs in working period t e T " . Also, proceeding as in the previous paragraph, let /ih

gk. = $(jlgs.>T) De m e

mean time between multihazards he H in exposure region g e Gh under evolutionary

50

path k when in working period r. This process models the arrival of incidents in the exposure regions, but it does not specify in which hazard zone within the region the multihazard occurs. In order to specify this zone, subjective conditional hit probabilities can be estimated from public or constructed indexes l h , z e Z , h e H . For example, for geopolitical failures the Failed State Index published yearly by Foreign Policy (www.foreignpolicy.com) can be used, and for natural disasters an incident occurrence frequency calculated from CRED data can be used. Using such indexes, for a given multihazard he H and exposure region g e G h , the following conditional probability mass function can be calculated: ph

:lg = I./_y_ Ihz, z e Z g .

Intuitively, it appears that the impact intensity and duration of hazards are usually highly correlated. We thus assume that when a multihazard he H occurs in a zone z e Z , its duration (in working periods) and its intensity (in a generic measure such as the loss level, or the casualty level, per period) are characterized by two correlated random variables related to the zone exposure level g(z)e Gh, namely: the impact intensity /3g, with cumulative distribution function Ff (.) and the duration 0H. The duration is related to the intensity through incident impact-duration functions 0g - fh (j3g) + £h, he H , estimated by regression, and with a random error term £h~ Normal(0,cr*). These distribution functions and incident impact-duration functions can be estimated from the data provided by organizations such as CRED, HIIK, FEMA and USGS.

Modeling the impact of hits on the SCN

The occurrence of an extreme event in hazard zone z does not necessarily imply that all the SCN locations / e Lz will be hit. When the hazard zones are large (countries or states), it is likely that only a part of the zone locations will be hit. Also, when considering the impact on product-markets, the SCN does not necessarily respond to all incidents. When designing a pre-positioning supply network for a humanitarian or military organization, for example, the organization's response to a natural disaster may depend on its policies, on UN solicitations and on other commitments (Girard et al., 2008). In such cases, a demand surge for first-aid products in a hazard zone does not necessarily generate demands in the corresponding SCN demand zone. This leads to the estimation of attenuation probabilities af which are conditional probabilities that location / is hit when a multihazard he H occurs in zone z(l). It is clear that these probabilities are related to the hazard zones granularity. Large zones lead to small attenuation probabilities, and vice versa. Attenuation probabilities can be estimated by experts for each SCN location, based on experience and data available.

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When the SCN is hit, this has impacts on the network capacity and demand. In order to model these impacts, we need to refine our representation of the SCN. A hit on vulnerability sources such as plants, distribution centers (DCs) and suppliers result mainly in capacity loss, but a hit on product-markets affects demand processes. To reflect this, we partition the vulnerability source set S in two subsets: capacity-based sources 5C and demand-based sources Sd. Also, in SCN design projects, the products manufactured and sold are usually aggregated into a set of product families p e P , and the subset of product families Ps c P associated to each vulnerability source se S needs to be identified. Finally, to model impacts, we need to define a parameter c, denoting the capacity of location l e Ls, s e Sc, for product p e Ps, and a random variable dlpt, with cumulative distribution function F d ( . ) , specifying the normal operations demand of location l e Ls, se Sà, for product p e P. in period z e T " , under evolutionary path ke K.

When a location l e L in zone z(l) is hit by a multihazard he H , the severity of the incident is characterized on two correlated dimensions: the impact intensity and the time to recovery (Sheffi, 2005). Clearly, these dimensions are related to the generic multihazard intensity and duration variables /?* and 0h defined previously. However, the SCN impact severity must be expressed in units related to the capacity and demand of the vulnerability sources. It is assumed that the metrics used to characterize these two severity dimensions are the same for all the locations associated to a given vulnerability source, i.e. for all l e L s . Hence, for each vulnerability source s e S , incident profiles such as the ones illustrated in Figure 2-7 must be specified for all locations leL s , products p e Ps and multihazards he H s . Damage on suppliers is typically assessed using an unfilled rate (% of material ordered during the incident not delivered) and the time required to restore supplies, whereas damage on production-distribution resources is usually assessed using a capacity loss rate and the time before production/distribution can resume. For vulnerability sources affecting demand, damage is usually assessed using an inflation or deflation rate expressing a demand surge or drop for a given period of time. Note that the evaluation of incidents severity may also be influenced by the state of the resources/partners associated to a vulnerability source. In some cases, an engineering analysis may be required to establish the fragility of vulnerability source resources depending on the building type, age, etc.

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Capacity- based VuherabirySources 5r ={1 ,2 ,3 } Demand-based VukierabilirySources S

d = {4,5.6} 1)

Suppliers

2)

Plants

3)

DCs

4) 5) 6) First-aid Sustainment L u x u r y

Product-markets Product-markets Product-markets

Impact intensity

r-■o

-. ■x s - I

£ 3

II

a) Natural disasters

Unfilled supply rate

Capacity loss rate

Capacity loss rate

Demand inflation H H Demand deflation rate rate

Impact intensity

r-■o

-. ■x s - I

£ 3

II

b) Market failures

Unfilled supply rate

H Demand deflation Demand deflation rate rate

Impact intensity

r-■o

-. ■x s - I

£ 3

II c) Industrial 1 accidents |

Capacity loss rate

Time to recovery

■s.

■s

J 3 II

o) Natural disasters

Time to restoring supplies

Time to restarting production

Time to restarting distribution

Surge duration 1 1 Drop duration

Time to recovery

■s.

■s

J 3 II

b) Market failures

Time to restoring supplies

1 Drop duration Drop duration Time to recovery

■s.

■s

J 3 II

c) Industrial 1 accidents |

Time to restarting production

1 Drop duration Drop duration

H

Figure 2-7. Multihazard Incident Profiles Example

Let çf* be a discrete random variable giving the time to recovery, in working periods, of location le L when hit by a multihazard h e Hs(l). We assume that this time to recovery can be related to the multihazard duration 0gU) using an adequate translation function çf* = #*(„(#*(,)) specified for each vulnerability source se S and multihazard he H s . This function may be based on a proportion estimated from past instances or provided by experts. Consider a multihazard he H hitting location le L at the beginning of working period r ' e T " . Then, the impact of the hit lasts during working periods T = T', . . . ,T'+tf-l .

When a multihazard he H hits a location /, its impact is not necessarily felt uniformly during the time to recovery çf* (Sheffi, 2005). Several phases are usually observed, depending on the nature of the multihazard and of the vulnerability source. For example, when a manufacturing plant is hit by a natural disaster, production capacity drops quickly during a first phase, then there may be a stagnation period while recovery measures are organized, and during a third phase the capacity is gradually restored. On the other end, when a disaster relief organisation initiates an assistance mission, it typically involves the three following phases: deployment, sustainment and redeployment. Such phase-dependent impacts can be characterized by defining discrete recovery functions p = rs*(/?,çf,p), he H , s e S,pe Ps, where p = [pT,,...,pT,+^_]]is a vector of capacity/demand amplification percentages for the çf working periods affected by the multihazard. The p.,,...,pm,+ç_x values used as an argument in the function reflect amplification percentages before the hit and the function returns percentages after the hit, as illustrated in Figure 2-8. If the working periods affected by the multihazard are not still recuperating from a previous incident, then the a priori percentages are pT=l00%, r = r ' , . . . , r '+çf-l . The amplitude of the amplification percentages depends on /?, the multihazard generic impact

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intensity measure. Multihazard recovery functions are defined by experts for each vulnerability source and product family, based on experience and data available.

Using these recovery functions, the capacity available or the demand can be calculated for specific working periods and locations. More specifically, the behaviour of the capacity c', or the demand d', resulting from a multihazard he H is described by the following relations:

c\P.=Pipr%, T = TG.. ,T'+tf-ï , P,p = rsh

p(j3hg{l),^,plp); s e S c , p e P s , l e L s ( 1 )

d' lpT=P,pApT, T = z \ . . . , f + % - \ ; p l p = r ^ hg ( l ) , ^ , p l p ) ; s e S d , p e P s , l e L s ( 2 )

This SCN impact modeling approach is based on a simplified representation of SCN resources, but it should be relatively easy to adapt to the specificities of real life cases. In particular, expressions (1) and (2) reflect multiplicative impacts, which is typically appropriate in business contexts. However, for humanitarian relief or military organizations, the demand is usually more adequately described using additive impact relationships because dlpt can be zero when there is no incident. Also, we assumed that multihazard recovery functions are not affected by evolutionary paths, which is not always the case.

Amplification percentage

100%

Amp itude based on p

A priori percentages

_i ^J

.IT

i i i i i i i i i

Pr

I I I

Recovery function

-I—I-1 t ' + ç - l Working periods

Time to recovery

Figure 2-8. Recovery Function for a Given he H , se Sc and p e Ps

Plausible future scenarios

The SCN hazard modeling framework proposed in the previous paragraphs is based on a number of key concepts: the identification of evolutionary paths K, the classification of SCN locations L into vulnerability sources S and of hazards into multihazards H, the

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zonation of the territory into hazard zones Z and their classification into exposure levels G, the definition of incident profiles in terms of impact intensity and time to recovery with associated recovery functions, and the characterization of multihazards likelihood through the use of incident arrival stochastic processes, impact intensity probability distribution functions, incident impact-duration functions and attenuation probabilities. The superposition, during the planning horizon, of a specific instance of this hazard occurrence process over specific instances of the business-as-usual random variables used to model the SCN yield a probabilistic scenario coe Q.p . Some of these plausible future scenarios may involve only a few multihazard over the planning horizon but others may be much more chaotic. An intuitive measure to assess the risk associated to a scenario coe Q p is the number of hits y(co) it undergoes during the planning horizon. Figure 2-9 illustrates the distribution of the number of hits for a large sample of scenarios with exponential multihazard inter-arrival times. In order to distinguish between the scenarios a decision maker would consider as acceptable, in term of the risks involved, and those that would raise a serious concern, we define a hazard tolerance level K. This level is the maximum number of hits the decision maker can tolerate without serious concern. This tolerance level is used to partition the set of probabilistic scenario Q.p in two subsets, namely QA

the set of acceptable-risk scenarios and Q.s the set of serious-risk scenarios.

For a given SCN design project, the sets, measures and functions required to characterize hazards are necessarily defined based on the information and experience available and, consequently, they may completely overlook some potential extreme events for which no information and experience exist. It is to cope with these potential threats that imaginative deeply uncertain scenarios must be elaborated. Some uncertain extreme events associated with these scenarios can be identified through structured brainstorming sessions and/or expert interviews related to SCN threats and vulnerabilities (Van der Heijden, 2005). However, for our purposes, the resulting scenarios must be expressed quantitatively in terms of the parameters used in the design model. This can be achieved by following the structured process described in this section but by replacing probability distributions and impact functions by human inputs for multihazards which cannot be described probabilistically. Also, these scenarios necessarily include random events and they may also include probabilistic hazards so they are most easily created by perturbating probabilistic scenarios. In what follows, our interest in deep uncertainty scenarios will be mainly related to our need to generate worst case scenarios. These would typically be probabilistic scenarios in the tail of the distribution of the number of hits, as illustrated in Figure 2-9, or serious-risk scenarios perturbated by deep-uncertainty events imagined by

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experts. Our challenge now is to elaborate a SCN design modeling framework taking all this into account.

Acceptable-risk scenarios Serious-risk scenarios A

Worst-case scenarios

\

2 3 L 4 5 6

Hazard tolerance level («•=3)

10 11 12 Number of hits

Figure 2-9. Distribution of the Number of Hits for a Large Sample of Scenarios

2.2.4 SCN Design Model The generic design model proposed in Figure 2-2 does not take the nuances

introduced in the previous section into account explicitly. More specifically, in Figure 2-2, the generalized future return measure H{.\IÇI (6X)\ used is defined over the set of all scenarios Çl and it does not take the partitioning into acceptable-risk, serious-risk, and deeply uncertain scenarios into account. A fundamental argument of risk analysis is that this should not be done because it gives the same weight to normal impact and serious impact events. To avoid this pitfall, in risk analysis, traditional expected value assessment functions are replaced by a set of conditional expected value assessment functions taking the impact of various types of events into account (Haimes, 2004). Along this line of thinking, in our SCN design methodology, to take into account the quality of information available and the impact intensity of events (as described in Figure 2-3), we propose to replace our original future return measure by three conditional return functions defined over the scenario subsets QA, £ls and Qu respectively.

This transforms our original model into the following multiobjective program:

(3)

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^{Ch.0}-«a^{C(«,,0}+f4l^{C(« | f.)},f i l6[O,l] (4) ^|5{C(x.-)}=EnS|5{c(x1,.)) + çJsPn5|5{c(x1,.)},^,e[0,l] (5)

Kapp {C(x»-)} = Jgj *></ {C(x„»)} (6) where TZ^.^C), TZ.nss(C) and T^clvu(C) are conditional return functions for scenarios in QA, Q.s and Q y , respectively, defined in terms of the conditional expected value EaJiA(C) and E n S . (C) of random variable C, of the conditional measures of dispersion (variability) ^aAA(C) and T?nSj(C) of random variable C and of the weights <pA and <pH. When T>aAA(C) and T?QS (C) aie coherent risk measures,7tnAA(C) and TZ. s (C) are also coherent risk measures (Rockafellar, 2007). Coherent risk measures must satisfy a number of convexity, monotonicity, translation equivalence and positive homogeneity conditions. The most convenient coherent dispersion measures to use in our context are

T>{C} = E{\E(C)-C]+} or T>{C) = minE{[C-r]+ +V[r-C]+}, v > 0

where [c]+ = max (c, 0) and v is a constant. The former is a mean-semideviation risk function and the later the so-called conditional value at risk function (Shapiro, 2007).

Since the probability of occurrence of scenario coe Of is not known, expected value and dispersion measures cannot be defined for [/-scenarios. For this reason, for U-scenarios, we use conditional return functions TZ. y (C) based on the robust optimization criteria proposed by Kouvelis and Yu (1997). In our context, it is most convenient to use the absolute robustness or robust deviation criteria defined respectively by:

T>l]{c(xx,co)}=-C(xi,co) or Vu{c(x„co)} = \c(x^,co)-C(x„co)\

where xf is the optimal single scenario design obtained when scenario CD is realized, i.e. where xJu=arg{maxxeX C(x{,co)}. From a computational point of view, the absolute robustness criterion is more attractive because it does not require the optimal decision for each scenario but, according to Kouvelis and Yu (1997), it leads to very conservative decisions. The robust deviation criterion is more adequate in our context but it requires more computations.

Several methods are proposed in the literature to solve multiobjective programming problems. Given the complexity and size of model (3), an adequate approach here is to simply convert it into the multiparametric program:

ng«-a-r)[wA^{c(«i,o}+w5^{c(xlfo}]+r^{c(xI,o} (7)

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with subjective weights 0 < wA < 1, ws = 1 - wA ,0 < y/ < 1. Since R is a convex combination of coherent risk measures, it is also a coherent risk measure (Rockafellar, 2007). Such an approach leads to satisfactory designs only if program (7) is solved parametrically for different weight values. Note that when the probabilities p(eo) for all probabilistic scenarios can be evaluated, and when the decision maker is neutral to risk, we have <pA = <p5 = 0 in (4) and (5), and 1//-Q. Under these conditions, by defining the weight wA-LœaAp(CD) as the probability of acceptable-hazard scenarios, (7) reduces to the maximization of the unconditional expected value of C(x,,.). On the other end, a decision maker averse to extreme events would define weights wA <EMQ/,/?(#>) and y/>0, and a decision-maker averse to dispersion would set <pA > 0 and <ps > 0 . Note however that, in most practical cases, the number of possible scenarios \Çï\ is extremely large, if not infinite, and thus it is impossible to obtain the set Qp and the probabilities p(co), eve Slp

explicitly.

Our previous discussion of extreme events has another impact on the formulation of SCN design models. When facing such threats, one would like to design the SCN to avoid risky locations as much as possible and to be able to bounce back quickly when hit, i.e. to favour network structures and response policies helping the user to react efficiently when hit. This is the domain of resilience strategies (Sheffi, 2005). Clearly, resilient designs improve the SCN robustness. In order to obtain resilient designs, additional decision variables and constraints may have to be included in the formulation of the solution sets X,, X*" ' (co),ne N\{ 1}. For example, one may want to provide instructions on the backup depot to use to supply customers when the primary depot is hit, or to impose primary and backup distance covers to ensure an adequate response to all customers (Klibi and Martel, 2009). Unfortunately, this further complicates the design model. Note finally that the anticipated adaptation-response model in Figure 2-2 must incorporate an evaluation of the recourses necessary to obtain a feasible solution under any scenario coe Q. All this certainly lead to extremely complex optimization models under uncertainty. In what follows, we propose a generic SCN design approach based on reasonable approximations of model (7).

2.3 Scenario-based SCN Design Model Solution Approach SCN design model (3) is a multi-stage stochastic program with an infinite set of

scenarios, a multiobjective reward function, and an anticipation of adaptation-response decisions. Unfortunately, this model is intractable in its current form, and the objective of this section is to propose a complexity reduction approach to obtain solvable SCN design

58

models capturing the essence of the problem. Our purpose is not so much to obtain the optimal design as it is to identify practical ways of hedging risks that are largely overlooked in classical SCN models. The approach is based on several accuracy-solvability tradeoffs likely to yield effective and robust SCN designs.

A first complexity reduction avenue is to use approximate anticipations of adaptation-response decisions to simplify the combinatorial structure of the design model. A second simplification is to reduce the multiobjective function to a multiparametric function based on (7), and capturing only the primordial expected value and risk aversion criteria associated to probabilistic scenarios. A third opportunity comes from the fact that SCN design problems are usually solved on a rolling horizon basis so that the only decisions implemented when the model is solved are the first design decisions x*. This suggests that the model can be reduced to a multi-cycle two-stage stochastic program with recourse without loosing its hedging capabilities, which simplifies both the generation of scenarios and the resolution of the model. Finally, the resulting stochastic program can be solved using several samples of scenarios generated using Monte Carlo methods. The approach then reduces to solving a set of large MIPs, as done when solving stochastic programs with the Sample Average Approximation (SAA) method (Shapiro, 2003). Additional designs can be obtained by varying the anticipation granularity and the objective function weights. The designs obtained are then compared using performance measures (4)-(7), evaluated with a more precise adaptation-response model than the one incorporated in the design model and a larger scenario sample.

The SCN design approach thus obtained is summarized in Figure 2-10. It includes three phases: scenario generation, design generation and design evaluation. The first phase involves the generation of several plausible future scenario samples for the design generation and evaluation phases. It produces / independent small Monte Carlo samples of mA acceptable-risk scenarios and ms serious-risk scenarios, ft™ =0?* \jClf1, i = l,...,/, as well as estimates ifA and n s of the probabilities nA and n s , for the design generation phase. It also provides larger samples QM*, £lMs of probabilistic scenarios, and a sample ClM" of worst-case scenarios to the design evaluation phase. The second phase of the approach involves the resolution of the MIPs SAA( Q|" ), i = 1,..., / , resulting from the approximations made. For a given resilience and response anticipation formulation, this yields a set of distinct designs x{,j = l,...J ( J < / ) . The third phase of the approach compares the performance of these designs, and of the status quo design x°, by solving an approximate adaptation-response model for each of the scenarios in

59

ÇlM =QM* u Q " s KJÇIMU KJ{CO°}, where of is an historical scenario. The set of scenario specific design values c(x{,cv), eve £lM , thus obtained are used to evaluate performance measures 7en„4|A{C(x/,.)}, 7en«J|5{C(x/,.)}, n a^{C(x{,.)} andfltx/) based on (4)-(7). Finally, classical multicriteria filtering and selection techniques can be used to select the design x* to implement. In what follows each phase of this generic design approach is discussed and explained in more details.

Multi-criteria evaluation

Effective and Robust Design

Design Generation (G SAA(n?),i = l,...,I

SCN design models - Resilience formulation -Anticipation - Solution method

Status quo Xj

L

Risk-attitude weights based on XA and x s

(W„i=l I),P=A,S

Small samples replications W,i=\,..J),P=AS

x{,j=\,...J 0 Plausible Future

Scenario Generation

Design Evaluation & Selection (G • Adaptation-response optimization

C* ,(x{,mo),û.eiïM

• Performance measures c(xl,co) = CJ(x{,a>) + C*'(x{,mO),co<-Ll."

H a . r ] p{c(x(, . j \ ,P = A,S,U;R(.xi)

• Filtering and selection

Monte-Carlo evaluation samples

Çf',P=A,S

Hazard tolerance level (K)&

Risk-aversion overshoot (A)

Worst-case scenarios Çf

Historical scenario (ai)

Figure 2-10. Generic SCN Design Methodology

2.3.1 SCN Designs Generation As indicated previously, four complexity reduction stratagems can be used to

formulate solvable SCN design models. The first one involves the incorporation of approximate anticipations of adaptation-response decisions in the design model. In classical SCN design models user decisions are anticipated using continuous throughput and flow variables. The anticipation variables are typically aggregates over several products, customers, transportation means and working periods. These decision variables are used to anticipate revenues and expenses, but they cannot be implemented. For example, in practice, flow decisions take the form of daily shipments in response to specific customer orders, and not the form of an annual quantity of products to ship between two locations. The later is used as a crude approximation of the former. In fact,

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such approximations are often so crude, that they raise issues about the validity of the model. Some authors have proposed models with more accurate anticipations: a model incorporating detailed market response anticipations is proposed by Vila et a l , (2007), and more elaborated transportation and inventory costs anticipations are proposed in the location-routing and location-inventory models reviewed in Shen (2007). Klibi et a l , (2009b) studied various anticipations for the stochastic location-transportation problem, based on different demand representations (multiple scenarios vs average demand) and on different aggregations of transportation decisions (route vs flows), customers (ship-to-point vs demand zones) and time (working periods vs planning periods). The results obtained show that although significant gains can be made by using more precise anticipations, given the computational power currently available, some tradeoffs are necessary. The best approach seems to be to seek an adequate equilibrium between all the dimensions involved instead of neglecting some dimensions (ex: using a deterministic model to be able to anticipate transportation costs with route-variables). The use of adequate approximate anticipations reduces the size of the multi-stage stochastic program to solve.

A second complexity reduction avenue is the replacement of the multiobjective function in model (3) by a simplified version of the composite return function (7), which is a convex combination of conditional return functions (4)-(6). The return functions (4) and (5) are defined in term of an expected value and a dispersion measure. Since the recourse variables included in the anticipation sub-model tend to be very expensive, the stochastic program tries to eliminate any extreme behaviour, which naturally reduces variability even if the dispersion terms are not included in these return functions. For this reason, a convenient complexity reduction mean is to set <pA = cps = 0 in functions (4) and (5), respectively. Also, the aim of return function (7) is to reflect the consequences of extremely hazardous deeply uncertainty scenarios. However, since the probabilistic scenarios were separated into acceptable and serious risk scenarios, one can give more importance to extreme events if desired by increasing weight ws. For this reason, another reasonable complexity reduction opportunity is simply to set y/ = Q. This reduces the original design model to the simpler multi-stage stochastic program:

^ [ w A \ y { C ( x ^ ) } + ws\sls{C(x,,.)]j (8) Recall that, to take the risk attitude of decision makers into account, the weights wA and ws should be based on the probabilities 7tA and 7ts. These probabilities are not available but the weights can be based on the estimates 7tA and ifs provided by the scenario generation procedure.

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The dynamics of the multi-stage decision structure described in Figure 2-1 and Figure 2-2 is an important source of complexity. When several planning cycles are considered, first-stage design decisions x, are made here and now, and the subsequent structural adaptation decisions x2,...,xn are implementable policies (Shapiro, 2007) elaborated to respect non-anticipativity conditions. When this is taken into account explicitly, the size of the problem tends to blow up. However, since design decisions are made on a rolling horizon basis, the structural adaptation decisions x2,...,xn will in fact be revised before they are implemented. In our context, they are essentially anticipation variables. Under these conditions, a reasonable complexity reduction assumption is to consider that the decisions x2,...,xn must be made at the beginning of the planning horizon. This eliminates non-anticipativity constraints and transforms the model into a multi-cycle two-stage stochastic program. In most practical cases, the number of possible scenarios |£2| is extremely large and, in order to solve (8), one needs to limit the number of scenarios considered and to avoid the explicit use of the probabilities p(CD), me Q?. Another complexity reduction method is to replace the population sets £lp in design model (8) by representative Monte Carlo samples Slm" and Qm' of mA equiprobable acceptable-risk scenarios and ms equiprobable serious-risk scenarios, respectively. Clearly, the quality of the design obtained with the resulting SAA program depends on the number m = mA+ms of scenarios considered. To get better designs, the model can be solved with / scenario sample replications Çl™ = Q™* u Cl™s, i = 1,...,/. Statistical gaps can be calculated to evaluate the quality of the solutions obtained with these scenario sets (Shapiro, 2003), and they can be used to calibrate the size of the scenario samples to generate. The SAA model to solve for a given scenario sample Œ™ is the following:

SAA'a?) max £ i £ \cd(x^+^C"(x„,û>)+YC"(y,(û>)) } (9) P=A,S Ptu_mm^ [ n>\ { € f J

s.t. x ,eX, ; xneXxn"G ne N\{1) (10)

y t(to)eY^'>(co),teT,(oeÇl™ (11)

Despite all the simplifications proposed, for real SCN design problems, program S A A ( Q " ) may still be extremely large and difficult to solve. Santoso e ta l . (2005) proposed the use of Benders decomposition to solve this type of SCN design models. Recent commercial solvers, such as CPLEX-11, incorporate generic heuristics (ex: the feasibility pump) to find good initial solutions and they are able to solve surprisingly large SCN design problems. Also, since our objective here is to generate good potential SCN

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designs, and since commercial solvers tend to take a lot of time to prove optimality after they found the optimal solution, larger optimality gap parameter values can be used to reduce computation times. Note finally that several heuristic methods were proposed in the literature to solve deterministic location-allocation problems (see Klibi et al. (2009a) for a review) and some of them can be extended relatively easily to solve stochastic versions of the problem.

By solving SAA(£l™ ),i -1, . . . , I , a set of potentially effective and robust designs are obtained. However, the SAA program solved is based on a specific resilience and adaptation-response anticipation formulation, and on given risk-attitude weights. Other potential designs can be generated by modifying the model formulation or the risk-attitude weights. Modifying the model formulation may be cumbersome, but reformulations such as a change in the granularity of some anticipation variables are possible without excessive efforts. Changing risk-attitude weights is easy and, since these weights are not hard data, varying them is an adequate approach to generate alternative designs x ( , j - \,...,J ( / < / ) . One way to take into account the fuzzy nature of these weights, and to make sure that the designs obtained by solving the / SAA programs are all distinct, is to consider them as random variables with ws-Uniform[ifs,ifs + A], wA = l - w s and A<7TA. Since the overshoot parameter A determines the maximum value of ws, its value is selected to reflects the risk-aversion of the decision-maker. The weights w'A and w's used for model SAA( Q™ ) are generated randomly from this distribution.

2.3.2 Scenarios Generation Plausible future scenario samples are required by the two other phases of the design

methodology in Figure 2-11. As indicated before, scenarios are juxtapositions of random, hazardous and deeply uncertain events over the planning horizon f, and they are shaped by possible evolutionary paths ke K. When the decision process is approximated by a two-stage stochastic program, plausible futures can be represented as a fan of individual scenarios, as illustrated in Figure 2-4, and it is sufficient to generate particular event type realizations and to concatenate them to obtain a scenario. Importance sampling techniques (Ducapova et a l , 2000) can also be used to obtain scenario samples adequately covering all scenario types and evolutionary paths. In this section, we provide a procedure to generate individual scenarios and we discuss the generation of the various scenario samples required to obtain effective and robust SCN design.

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As explained previously, random and hazardous events can be characterized by random variables with distribution functions depending on working periods r e T" and on evolutionary paths ke K. Also, some of the problem data may be considered as known but affected by hazards. To illustrate this, in the section on the modelling of hazards, we introduced a known constant capacity parameter c, and a time-dependent random demand variable dlpT, both being subjected to the effects of hazards. To simplify the presentation we also assume in this section that capacity and demand are the only two variables affected by hazards. Other random variables related to prices, costs, exchange rates... may be influenced by evolutionary paths, but not by hazards. Let E be the set of all these random variables, denoted by Çe

T, e e E, and let Fkr(.), ee E, be their cumulative distributions for working period t e T " under evolutionary path ke K. For a given scenario CO, the value taken by these variables is denoted by clpT(co), dlpT(co) and ÇT(co). The Monte Carlo procedure required to generate these values is given in Figure 2-11. In the procedure, u denotes a pseudorandom number, and <P~x(u) the inverse of the standardized Normal variate.

The procedure includes five main steps. First, an evolutionary path is randomly selected. Then, a chronological list Tz of all the multihazards arrival periods is constructed for every hazard zone z e Z . Third, the intensity and duration of the incidents are generated and used to calculate amplification factors using the recovery functions. Forth, the amplification factors are used to calculate the working period's capacity and demand. The value of the hazard-independent random variables is also computed. We assume here that the random variables Ç*,eeE, aie independent. If they are not, the generation process is more complicated but straightforward. The last step aggregates the working period values obtained into planning period values. This is required because the design generation phase needs scenarios expressed in terms of planning periods t e T . The design evaluation phase however usually uses scenarios expressed in terms of working periods r e T " . Note that this aggregation process does not always involve a simple sum over all the working periods t e T " , t e T . For the capacity, for example, in order to take congestion into account properly, this may involve period sampling or the application of a correcting factor.

The procedure in Figure 2-11 can be used to generate all the scenarios, probabilities and risk-attitude weights required by the design generation phase. To do this, a large sample of Md scenarios £lM is generated and partitioned into acceptable and serious hazard subsets QM* and QMs , using the hazard tolerance level «"(see Figure 2-9). From

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these sub-samples, the probability estimates KA = M d /M d and 7ts = \ - x A are calculated. The small scenario samples Q.™\ QTS, i = l,...,/,are then randomly selected in QMA and Çl s , respectively. Through this hierarchical sampling procedure, one makes sure that all the scenarios in £1™* and ÎΙ5 are equiprobable, with probability \/mA and \ /m s

respectively. Based on irs and on the risk-aversion overshoot factor A, the weights w'A, w's, i - l,...,I, can also be generated from the Uniform f^,^ + A] distribution. Note that the samples obtained include scenarios coming from all the evolutionary paths ke K. If mA and ms aie relatively large, then each evolutionary path is well represented in the samples. However, if the sample size is small, one may want to force a good representation of each evolutionary path by hierarchically sampling mM scenarios for path k to get the samples Q|"u, Q , y , k e K , i — 1,...,/. The objective function (9) must then be replaced by:

max 2 ^ 1 ^ I \cd(xvco) + ZCd(xn,co) + Z&(Uo>))} (12) P=A,S kmK m k P oXmim^ [ n>\ t e f J

The procedure in Figure 2-11 can also be used to generate all the scenarios required by the design evaluation phase. To this end, another large sample of scenarios ÇlM is independently generated and partitioned into acceptable-hazard scenarios QM* and serious-hazard scenarios Q. s , based again on the hazard tolerance level K. From these samples, two moderate size subsets of scenarios are randomly selected to perform the design evaluation: a subset Q.M* c Q.M* of MA acceptable-hazard scenarios, and a subset ÇlMs c Q.Ms of Ms serious-hazard scenarios. In order to obtain worst-case scenarios, a subset QMw c QMs of tail scenarios is also selected in the distribution of the number of hits (see Figure 2-9). These scenarios are then taken as is, or modified manually by adding imaginative elements, to get the required set of worst-case scenarios £lMv.

65

1) Select an evolutionary path k randomly using p k , k e K 2) For all h e H and g e Gh, do:

/7 = 0 While Tj < IT" 1 do:

,* - i Compute the next multihazard arrival moment TJ = TJ + F' (u)

Select a hazard zone z randomly in Zg using p à g , z e Zg

Insert the pair (| n\ ,h) chronologically in the list T_ End While

End For 3) For all z e Z , do:

Set plpT = 1,1 e L , p e P , t e T "

For all (T ' ,h)eT z ,do:

Compute ft = F j £ (u) and t f = f h ( $ ) + C£h<P^ ( u )

For all / e L_ \u < Otf, do:

Compute Çf=q hm (6 h

m )

Compute plpT = r hm p(f i h

z ( l^,P l p r) , t = t ' , . . . ,T'+ g - 1 , p e P{1]

End For End For

End For 4) For all s e S c , l e L s , p e Ps and t e T " : Compute the capacity clpT (CD) = plpxclp

For all s e S d , I e Ls, p e Ps and t e T " : Generate the demand dlpT(CD) = p,pTFlpkT ' («)

For all e e E and T e T " : Compute Çt(co) = F k ~ \ u )

5) Aggregate these values over periods t e T", t e T , to obtain clpl (to) ,dlpt (CD) and CH Figure 2-11. Monte Carlo Procedure for the Generation of a Scenario co

2.3.3 SCN Designs Evaluation The aim of the design evaluation phase is to select the best SCN design among those

generated (x ( , j = l,...,J ) and to compare them to the status quo x°. If we were applying the standard SAA approach, this would be done by solving the second stage program, obtained by fixing x( and xJ

n,ne N\{ 1}, in (9)-(l 1), with the scenarios £1MA UQ.M S , and then by comparing the designs expected value. However, since the SAA model is based on several approximations, there is no reason to restrict ourselves to such a gross assessment. The evaluation of the designs should be based on a response optimization model as close as possible to the real user model. Moreover, to obtain the SAA( Çlm ) model, we assumed

66

that the design adaptation decisions needed to be made at the beginning of the planning horizon. However, these decisions can be reoptimized to improve the assessment process. Finally, to obtain SAA( Q™ ), we simplified the objective function, but when comparing the designs, there is no reason not to use the performance evaluation measures (4)-(7).

Consequently, for a given design x{ and a given scenario coe ÇlM (recall that QM _ Q « , U QMS U ^M„ u {^ J ) m e mathematical program to solve to obtain the net revenues provided by the design under this scenario is the following:

C * ( X » = m a x J jCd(xn,co) + £ Cu(yr„co) (13) ï2 i 3 " n>\ reT" J 1 . J 2 - "

s.t. x ,6X; - ' , »6 iV\{ l} (14)

y r .eY'G'(CD), r ' e T " (15)

In this model, the response variables y f , , sets Y*:'T'(a)) and functions C"(yT,,co) aie accentuated with a ' - ' instead of a ' A ' to reflect the fact that the user response can be anticipated more precisely than in the design model, even if an exact anticipation is usually not possible. Also, the index t ' instead of t is used to indicate that the time unit used can be a compromise between the working period r and the planning period t. For example, one could use months or seasons as in tactical planning models. Since this model is solved for a single scenario at the time, it is much easier to solve than SAA( Q™ ). Note finally that Cdu(x(,CD) does not include the investment costs associated to design x{. The design values for the sample of evaluation scenarios are thus provided by:

c(x{,CD) = Cd(x{,CD) + Cdu(x{,co), coeÇlM (16)

These design values can be used to evaluate performance measures based on (4)-(7). Given the evaluation scenario samples generated, this yield the following measures:

We are left with a classical multicriteria decision making problem to determine the most effective and robust design x*. Formal multicriteria decision making techniques (Triantaphyllou, 2000) can be used to reach a decision, but simpler filtering and pegging methods can also help to examine the designs from different points of view. Filtering

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techniques can be used to eliminate dominated designs. Pegging can be performed to compare specific solutions with the status quo x° for specific scenarios. In practice, managers particularly like to make such comparisons for the historical scenario of and for worst case scenarios coe <7lMu. Sensitivity analysis can be performed for the various risk-attitude weights <pA, q>s, wA, ws and y/. In other words, several multicriteria/multi-scenario views can be elaborated to help select the best design.

2.4 Conclusions This paper proposes a new methodology to design effective and robust SCNs. It

underlines the temporal hierarchy between design time and utilization time, and it proposes to evaluate robustness through a high-quality anticipation of user decisions for a sample of adequately selected plausible future scenarios. In order to design superior SCNs, it is not sufficient to maximize overall effectiveness under normal operations, as is usually done in the literature: robustness under unpredictable disruptions must also be considered. An approach is proposed to take such disruptions into account in the design process. In addition to considering expected values, the approach considers the risk attitude of the decision maker. However, incorporating all these elements in the SCN design model yields an intractable multi-stage stochastic program. Given that, an approximate design methodology is proposed to capture the essence of the problem while preserving solvability. Complementary work performed to test the approach (Klibi et a l , 2009b; Klibi and Martel, 2009) indicates that it offers a judicious accuracy-solvability trade-off. We also believe that our framework provide ample opportunities for additional research.

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Chapitre 3: The Stochastic Multi-Period Location-Transportation Problem

Résumé - Cet article étudie un problème de localisation d'entrepôts et de transport stochastique caractérisé par plusieurs options de transport, plusieurs périodes de demande et une demande aléatoire. Le problème porte sur des décisions stratégiques sur le nombre, la localisation et la mission des entrepôts requises pour satisfaire la demande d'un ensemble de clients. Il est formulé comme un programme stochastique à deux étapes et une approche de solution heuristique est proposée pour le résoudre. Cette approche hiérarchique incorpore une procédure de recherche tabou, une formule d'approximation des longueurs de routes et une procédure Clark and Wright modifiée. Trois stratégies d'exploration du voisinage sont proposées et comparées sur plusieurs instances de problèmes basées sur des données réalistes.

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3. The Stochastic Multi-Period Location-Transportation Problem

Walid Klibi1'2, Francis Lasalle1'2, Alain Martel1'2'* and Soumia Ichoua1'3

1 Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT)

2 Département Opérations et systèmes de décision, Faculté des sciences de l'administration, Université Laval, Québec, Canada G1V 0A6

Department of Computer Charlotte, NC 28216, USA.

3 Department of Computer Science & Engineering, Johnson C. Smith University,

Abstract. This paper studies a Stochastic Multi-Period Location-Transportation Problem (SMLTP) characterized by multiple transportation options, multiple demand periods and a stochastic demand. We consider the determination of the number and location of the depots required to satisfy customer demand, and the mission of these depots in terms of the subset of customers they must supply. The problem is formulated as a stochastic program with recourse, and a hierarchical heuristic solution approach is proposed. It incorporates a Tabu search procedure, an approximate route length formula, and a modified Clark and Wright procedure. Three neighbourhood exploration strategies are proposed and compared with extensive experiments based on realistic problems.

Keywords. Location Problem, Transportation Problem, Stochastic Customer Order Process, Stochastic Programming, Monte Carlo Scenarios, Tabu Search.

Acknowledgement. This research was supported in part by NSERC grant no DNDPJ 335078-05, by Defence R&D Canada and by Modellium Inc.

Results and opinions in the publication attributed to named author(s) were not evaluated by CIRRELT.

Les résultats et opinions contenus dans cette publication n'engagent que leur(s) auteur(s) et n'ont pas été évalués par le CIRRELT.

* Corresponding author: alain.martel(5>cirrelt.ca Dépôt légal - Bibliothèque nationale du Québec,

Bibliothèque nationale du Canada, 2009

i Copyright Walid Klibi, Francis Lasalle, Alain Martel, Soumia Ichoua and CIRRELT, 2009

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3.1 Introduction Depot location decisions arise at the strategic planning level of distribution networks.

Fundamentally, in a business context, location-allocation problems involve the determination of the number and location of the depots required to satisfy customer demand, and the mission of these depots in terms of the subset of customers they must supply. Deterministic, dynamic and stochastic models have been proposed in the last decade to solve variants of the depot location problem. Comprehensive reviews of the literature on these models are found in Owen and Daskin (1998) and Klose and Drexl (2005). Three formulations of the problem under uncertainty were studied based respectively on stochastic programming (Birge and Louveaux, 1997; Snyder and Daskin, 2005), robust optimization (Kouvelis and Yu, 1997) and queuing theory (Berman et a l , 1995). In most of these formulations, the demand is the main random variable considered.

When the distribution network designed is in operation, on a daily basis, the depots must ship the products ordered by their customers to specified ship-to-points. Moreover, nowadays, an increasing number of companies rely on external transportation resources to ship their products to customers and they do not have their own vehicle fleet. In this context, depending on the size of the orders received, they may be shipped in single-customer partial truckloads (STL) or full truckloads (FTL), on multi-drop truckload routes (MTL), or via less-than-truckload (LTL) transportation, and the profits of the distribution network depends heavily on the efficiency of the transportation decisions made. These transportation options are defined more precisely in the following pages. The vehicle routing problems (VRP) encountered for the MTL case were studied extensively in the literature (see Laporte and Osman (1995) for a review). Although exact solution methods were developed for deterministic routing problems (Toth and Vigo, 1998) their complexity led most researchers in this field to propose heuristic solution methods (Laporte et al , 2000). A few stochastic versions of the VRP problem were also studied (Laporte and Louveaux, 1998). Despite the fact that location and transportation problems are clearly interrelated, the large classical literature on these problems assumes that they are independent. In the last few years, however, major efforts have been devoted to the development of integrated models, such as location-routing and inventory-routing models. A recent review of these integrated models is found in Shen (2007).

The first classification of location-routing problems (LRPs) is found in Laporte (1988) who proposes various deterministic formulations of the problem. More recent papers

71

(Chien, 1993; Tuzun and Burke., 1999; Barreto et a l , 2007; Prins et a l , 2007) present heuristic methods to solve deterministic LRPs. A multi-echelon version with inventory is addressed in Ambrosino and Scutella (2005). An uncapacitated LRP with distance constraints is studied in Berger et al. (2007); they propose a set partitioning formulation and a branch and price solution approach. The hierarchical structure of the problem is stressed in Nagy and Salhi (1996a, b) who develop a heuristic in which a routing phase is embedded into the location method. A dynamic LRP defined over a planning horizon is examined in Laporte and Dejax (1989) and in Salhi and Nagy (1999). A stochastic LRP is studied in Laporte et a l , (1989) and in Albareda-Sambola et al. (2007). In the stochastic program considered, depot locations and a priori routes must be specified in the first stage, and second stage recourse decisions deal with first-stage route failures. Recently Shen (2007) proposed a stochastic LRP model based on routing cost estimations. An approach to solve continuous LRP's is presented in Salhi and Nagy (2007). Comprehensive reviews of location-routing models and of their applications are provided in Min et al. (1998) and Nagy and Salhi (2007).

The LRP models found in the literature have two main shortcomings when compared to the strategic needs of distribution businesses using external transportation resources:

• First, most LRP models assume that the distributor has its own vehicle fleet and that the transportation problems encountered are pure VRP problems. When common or contract-carriers are used, more options are available, namely: single-customer full (FTL) or partial (STL) truckload, multi-drop truckload (MTL) and/or less-than-truckload (LTL) transportation. The FTL, STL and MTL options are provided by TL carriers. The distinction comes from the way the truck is used by the shipper: FTL refers to the case where a full trailer is delivered to a single destination, STL to the case where the trailer shipped is not loaded to full capacity, and MTL to the case where the truck delivery route involves more than one destination. In the latter case, the route is elaborated by the shipper as if it was its own truck. This leads to Location-Transportation Problems (LTP) instead of location-routing problems. These problems must not be confused with the class of problems known as transportation-location problems which are in fact transshipment-location problems, as pointed out by Nagy and Salhi (2007).

• Second, most LRP models implicitly assume that the location decisions and the routing decisions can be made simultaneously for the planning horizon considered (a year, for example), i.e. that the routes do not change on a daily basis and that customer

72

demand is static. In the business context considered here, the location and mission of depots must be fixed for the planning horizon, but transportation decisions are made on a daily basis in reaction to the customer orders received. This gives rise to what we call Stochastic Multi-period location-Transportation Problems (SMLTP). Most LRP models do not consider the customer order arrival process explicitly. Salhi and Nagy (1999) introduce a deterministic multi-period location-routing problem to analyse the consistency of the solutions provided by static location-routing methods. Laporte and Dejax (1989) consider a related dynamic location-routing problem but they do not require that the location and mission of the depots is fixed for the planning horizon. In our context, a random number of customers order a random amount of products on a daily basis, and these orders must be delivered on the next day. The dynamic customer demand pattern is therefore not known when the problem has to be solved. We assume however that the demand process is stationary.

The aim of this paper is to provide a more precise definition of the SMLTP, to formulate it as a stochastic program with recourse and to propose a heuristic method to solve it.

Laporte (1988) has shown that the LRP is NP-hard. For each time period, the transportation sub-problem considered in the SMLTP is an open VRP problem having the same NP property as a basic VRP (Sariklis and Powell, 2000). Thus the SMLTP is a NP-hard stochastic combinatorial optimization problem. Consequently, although exact solution approaches may be used to solve small size instances of the SMLTP optimally, they are not capable of handling realistic instances. This justifies the development of heuristics that better achieve a good trade-off between computation time and solution quality.

The paper is organized as follows. The next section provides a detailed description of the SMLTP and formulates the problem as a stochastic program with recourse. The following section shows how this stochastic program can be solved with a sample average approximation (SAA) MIP based on a Monte Carlo scenario sampling scheme. In the following section, a solution approach is proposed to solve the SMLTP: it combines efficiently, in a nested schema, a Monte Carlo scenario generation procedure, a transportation heuristic, and a location-allocation Tabu search procedure. The section also proposes various neighbourhood exploration strategies. Finally, in the last section, experiments are designed to evaluate the various versions of the Tabu heuristic proposed, for problems with different realistic characteristics, and computational results are presented and discussed.

73

3.2 Problem Description and Formulation 3.2.1 The Business Context

To start with, let us examine the business context of the SMLTP more closely. A company purchases (or manufactures) a family of similar products, considered as a single product, from a number of supply sources. This product is sold to customers located in a large geographical area and hence it must be shipped to a large number of ship-to-points. In order to provide a good service, the company cannot satisfy customer orders directly from the supply sources and it must operate a number of uncapacitated distribution centers (also referred to as depots). The customers order a varying quantity of product on a daily basis and the company wants to provide next day delivery using common or contract carriers. For a given day t, at each depot, when all the orders are in, the company plans its transportation for the next day, and it requests from its carriers the trucks required to deliver products to ship-to-points through FTL, STL, MTL or LTL transportation. When the orders are delivered, the trucks need not return to the depot. Let L be the set of depots considered, P the set of ship-to-points where orders can be delivered, LpczL the subset of depots which are able to provide next day delivery service to ship-to-point p e P , and, conversely, P, czP the subset of ship-to-points which could be served by depot /. Also, for day t, let Kjf be the set of feasible TL (STL or MTL) delivery routes from depot l e L, considering ship-to-points demands, service requirements and vehicle characteristics; and let P k e .P be the ordered set of ship-to-points in route k e Kjf. The TL-tariff wk charged by the carriers to assign a specific truck to route k is based on the following formula:

wk = max( rk mk ; TL, ) + a, (| Pk | -1)

where

mk = total mileage of route k

rk = transportation cost rate per mile for the vehicle associated to route k

TL, = minimum transportation charge for any TL route starting at depot /

a, = drop charge for any additional stop on a route starting at depot /

Note that the charge wk must be paid to the carrier independently of the load shipped in the vehicle. When a customer orders more than a truckload on a given day, we assume that the depot ships as much as possible in full truckloads. Note also that several routes ke Kjf could have identical ship-to-point sets Pk if vehicle types with different capacities and

74

rates can be used. However, if this occurs, there is no reason not to use the route with the lowest charge wk. We therefore assume, in what follows, that the sets Kjf contain only non-dominated TL routes.

If the vehicle on route k has a load close to its capacity, then multi-drop TL transportation is usually cheaper than LTL transportation. For the orders received on a given day, if it is not possible to construct routes with a good load/capacity ratio for all orders, then it may be cheaper to send some orders by LTL transportation. In particular, if a TL route is more expensive than LTL shipments, then it should not be used. More precisely, a route ke Kjf would not be used if wk > Hl^p LTL(l,p;dpl), where dpl is the load to be dropped at ship-to-point p e Pk on day t, and LTL(l,p;d) is the LTL charge for the shipment of a load d from depot / to ship-to-point p . If its alternative LTL routes dominate such a TL route, it can be removed from the set of possible routes a priori. We assume, from now on, that all the routes in the set Kjf are also non-dominated by their alternative LTL routes. Conversely, on a day t, a LTL shipment on lane (/, p) would be considered only if it is cheaper than the least-cost STL shipment k(l . on this lane, i.e. only if LTL(l,p;dpl)< max(rk mk ;TL,). This implies that, in the daily transportation planning process at depot /, in addition to the non-dominated feasible TL-routes k e Kjt

L, all the economic depot to ship-to-point LTL routes k e K^TL must be considered, i.e. the set of routes to consider is Klt = Kjf u K^TL. The LTL-tariff paid for a single destination route k e K^TL is given by wk = LTL(l, p k ; d p l ) , where pk is the ship-to-point of route k.

Given the distribution network user context described previously, the strategic decisions to make here involve the selection of a subset of the depots L ' c L to operate during the planning horizon T considered and the assignment of ship-to-points Pf c ij), l e L*, to these depots, in order to maximize total expected profits. Note that, in

what follows, the notation L is used to represent a subset of open depots in L, and L = LXL its complement. Similarly, P, c P, is used to represent the subset of ship-to-points assigned to depot l e L . An important aspect of the problem is that the mission of the selected depots, defined by their customer sets, /J*, / e L* must remain the same for each day t e T of the planning horizon. When a depot / e L is used, a fixed operating cost A, is incurred, and the unit value of products shipped from that depot is v,. The value v, takes into account the product production/procurement costs, inbound shipment costs, warehousing costs and inventory holding costs. The unit price of products sold to ship-to-point p is up. The structure of the multi-period location-transportation network examined is illustrated in Figure 3-1.

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3.2.2 The Distribution Network User Problem As previously explained, for a given distribution network with depots set L c L and

mission /J, l e L, on a daily basis, the depots l e t receive orders from their customers p e P, and they make shipping decisions for the next day. It is assumed that the demands

of the ship-to-points p e P follow a compound stationary stochastic process with a random order inter-arrival time qp and a random order size op . The cumulative distribution functions of inter-arrival times and order sizes are denoted respectively by FH.) and Fp(.). A possible realization of these compound stochastic processes over planning horizon T is illustrated in Figure 3-2, for exponential inter-arrival times and Normal order sizes. Such realizations constitute demand scenarios and the set of all demand scenarios associated to the compound demand processes considered is denoted by Q.. The probability that demand scenario coe Q, will eventually be observed is denoted by K(CO) . For a given scenario coe Q, the set of ship-to-points ordering products on day t is denoted by Pt (CD) , and the shipments to make on day t e T at depot l e L are defined by the loads dpl(co), p e P,, (co), where P,t (co) = P, nP.(co) is the set of depot / ship-to-

points which order products on day t .

Design levet Location decisions Allocation decisions

User level Daily transportation decisions

Potential DC locations

leL éG

y (SourcesJ) > o _ <

^ -->.

Potential DC locations

leL X, . . .

Potential DC locations

leL

vlp I Compound Demand Process

Days 3W t z T

-*■ MTL Route STL Route FTL Shipment ♦ LTL Shipment

Figure 3-1. Multi-Period Location-Transportation Network

76

Time between consumer orders Ordered quantities

Ship-to point/

Ship-to point 2

Ship-to point

IPI

U i LL

1 Multi-period demand scenario

A.

I I I — l - H — h - 1 — \ — I — I — I — 1 - H — I — I — I — Y — \ — \ — I — I — \ — I — I " T Day t

Figure 3-2. Ship-to-Points Stochastic Demand Process

Given the loads dpl (co), p e 1_\ (co), I e L, to deliver on day t, shipping decisions are made by the network depots users in two steps. First, for the loads that are larger than a truckload, a decision is made to ship as much as possible in full truckloads. Let KfJt

L (co) be the set of vehicle types (routes) selected to make full truckload shipments to point p. Then the residual loads to be inserted in the STL, MTL or LTL shipments of depot / on day rare:

*„(®HU«)--I « w O l pe&(») (1) kmK^(CO)

where yfJtL (co) is the number of truckloads shipped to point p from depot / on route k, and

bk is the capacity of route k vehicles. We assume that yhJtL (co) is selected to ensure that

dpl (co) is non-negative.

Next, the best delivery routes must be constructed. Let,

Ku(co): Set of non-dominated feasible delivery routes (i.e. such that Pk c /J, (co) and £ » d p t ( c o ) < b k , ke Ku (CD)) from depot /, on day t, under scenario CD; 8^ : Binary coefficient taking the value 1 if ship-to point p is covered by route k

(i.e. if p e Pk ), and 0 otherwise;

ykll (to) : Binary decision variable equal to 1 if route k is used from depot / on day t under scenario to, and to 0 otherwise.

For demand scenario to, the best routes are obtained at depot / on day t by solving the following transportation sub-problem:

77

C;(tv) = Min 2 "WfeM (2) y keKu{(0)

subject to

S Skpym(co) = l pe£.(*,) 0)

vw»e{0,l} keKu( to) (4)

where y denotes the vector of all the routing decisions, and C"t(to) is the cost of the optimal shipments made by depot / on day t under scenario CD. This model is similar to the classical set partitioning formulation of the deterministic VRP (Toth and Vigo, 1998).

Furthermore, the shipments made on a daily basis generate sales revenues. Taking these into account, as well as depot production/procurement, warehousing, inventory holding and customer shipment costs, the net revenues R" (to) generated at the distribution network user level for demand scenario to aie given by:

*» = II leL leT

I [ K - V , K » ] - X I HH£>)-C:(*) /*&(«») PSË,{mm)keK%-(mO)

(5)

These net revenues are an important element to take into account in the distribution network design model.

3.2.3 The Distribution Network Design Problem The SMLTP is a hierarchical decision problem due to the temporal hierarchy between

the location decisions and the transportation decisions. At the strategic level, the only decisions made here and now aie the selection of the subset of facilities L cz L to use during the planning horizon T considered, and the mission Ff, l e L* , of these facilities. After a deployment lead time, on a daily basis, the transportation decisions discussed previously are made by the network users. However, the network design decisions must be considered when taking daily transportation decisions and, conversely, adequate network design decisions cannot be made without anticipating the net revenues (5) generated by daily sales and transportation decisions, for a given distribution network, during the network usage horizon T. The best possible anticipation involves the explicit inclusion, in the design model, of the transportation model (2-4) and of the net revenue expression (5), but with the information available at the time the network design decisions are made. Since the ship-to-points demands for the horizon T are not known when the design decisions are made, this information takes the form of the set of potential demand scenarios i l

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previously defined. This leads to the formulation of the SMLTP as a two-stage stochastic program with recourse (Ruszczynski and Shapiro, 2003), where the first stage deals with depot location and mission decisions, and the second stage with daily transportation decisions. The following first stage decision variables are required to formulate the model:

JC, : Binary variable equal to 1 if depot / is opened, and to 0 otherwise; x. : Binary variable equal to 1 if ship-to point p is assigned to depot /, and to 0 otherwise;

and the notation x is used to denote the vector of all these decision variables, and X. the vector of depot / ship-to-point assignment variables. Note that a given binary vector x specifies unique design sets L and /} , namely that: L = y \x, = l j , L=|/Lc, =0J and 5 = { p \ x „ , = - } . l e L . The stochastic programming model to solve is the following:

R*=Max R(x)=Yj7U(C0)Rdu(x,co)-Y_.Alxl OJmm- l .L

subject to

fc£„

x l p ^ x l

x,,x,pe{0,l}

p e P

le L, p e P,

l e L, p e P,

(6)

(7)

(8)

(9)

where, based on (2-5), the optimal value R " (x,co) of the second stage program for design x and scenario co is given by:

**(**)-ZZ with

leL teT peP,(a>) k-v,k,H- Z rfH^-CM

***£»

Cdu(x„û))=Min Y. w4y„,(©) y k<_Ku{a>)

subject to

Z skPyk,i{û>)=x,P kmK„(mO)

y„,H6{o, i}

peP.(to)

keK„(to)

(10)

(11)

(12)

(13)

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In the first term of objective function (6), expected net revenues are calculated and, in the second term, depot fixed costs are subtracted to get expected profits. Constraints (7) in the first stage program enforce single depot assignments for ship-to-points, and constraints (8) limit ship-to-point assignments to opened depots. Constraints (12) in the second stage program are coupling relations ensuring that daily route selections respect depot mission decisions.

Notwithstanding the inherent combinatorial complexity of this model, it would be virtually impossible to solve because the set of demand scenarios Q is usually extremely large. In fact, when the inter-arrival times and order size distribution functions FU.) and Fp(.) are continuous, there is an infinite number of possible demand scenarios. This is the case, for example, when inter-arrival times are exponential and order sizes are log-Normal, a frequent case in practice. This difficulty can be alleviated, however, through the use of Monte Carlo scenario sampling methods.

3.2.4 Sample Average Approximation Model The approach proposed to reduce the stochastic complexity of our problem is based on

the Monte Carlo sampling methods presented in Shapiro (2003), and applied to the VRP in Verweij et al. (2003) and Rei et al. (2007), and to supply chain network design problems in Santoso et al. (2005) and Vila et al. (2007). A random sample of scenarios is generated outside the optimization procedure and then a sample average approximation (SAA) program is constructed and solved. The idea is first to generate an independent sample of n equiprobable scenarios ^cà,...,af\ = Q.n czÇl from the initial probability distributions of order inter-arrival times and order sizes, which also removes the necessity of explicitly computing the scenario probabilities n(to). Then, based on (6-13), the SAA program obtained is the following:

I p . P,{w)

rk -v/k \ ~ X wkykll(co)\ k.Km(Cm) lmL

(14)

subject to p e P (15)

x l p ^ x l l e L, p e P, (16)

Z < ^ H = x l p coeQ", l e L , t e T , p e Pt(to) (17)

*€/C„(<u)

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xi>xiP,yu,{o>)z{W} C û e Ç l n , l e L , p e P , t e T , k e K u ( c o ) (18)

The scenarios in sample Q." czQ. used in the model are generated directly from the cumulative distribution functions of inter-arrival times and order sizes Fq(.) and F°(.), pe P . Assuming that the customer orders are independent of each other, to sample a

scenario coe Q., we generate independent pseudorandom numbers « and u0 uniformly distributed on the interval [0;l], and we compute the inverse, F* (« ) and F°A(u0), of the distributions of inter-arrival times and order sizes. The MonteCarlo procedure used to generate the daily demands d (to), p e P, t e T , of the ship-to-points for scenario co is presented in Figure 3-3. In this procedure, the continuous variable r is used to denote order-arrival times. Order arrivals are generated in the interval l0, | r | l and mapped onto the corresponding planning periods t e T . More than one order can arrive in a given planning period. Repeating this Monte Carlo sampling procedure n times yields the required sample of scenarios Q". Note that all the Procedures presented in the paper use the following syntax:

Procedure(input_variableI,...; procedure_parameterl,...; output__variablel,...)

MonteCarlo((F;(.),F;(.),p eP) ,T ;d p l (û ) ) ,peP , teT) For all p e P , do: T = 0; dpt(tv) = 0, t e T

While T < \T\, do: Generate the Uniform [0,1 ] random numbers u and u o

Compute the next order arrival time r = T+F ' " 1 (M ) and r-M Compute the planning period / demand dpl (to) - dpl (to) *-F;X)

End While End Do

Figure 3-3. Procedure MonteCarlo for the Generation of Scenario to

Clearly, the quality of the solution obtained with this approach improves as the size n of the sample of scenarios used increases. The SAA model above has a structure similar to the deterministic location-routing problem (Berger et a l , 2007), but it separates explicitly assignment and routing decisions and it is much larger. It uses an exponential number of binary decision variables for route selection. A pre-established set of non-dominated routes can be generated as input to the model, but only small problem instances can be solved to optimality this way with commercial solvers. A better approach is to use column generation which avoids the explicit consideration of all the possible routes. When

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\T\* Q" is large, however, this optimal approach can be used only for relatively small problems. Our aim in the next section is to propose a heuristic method to solve realistic problem instances.

3.3 Solution Approach

3.3.1 The General Scheme The SMLTP is a hierarchical decision problem for which a hierarchical heuristic

solution approach is a natural fit. Nagy and Salhi (2007) present a review of sequential, clustering, iterative and hierarchical heuristic approaches to solve the LRP. The difference between these heuristics relates mainly to how the solution method treats the relationship between the location and the routing sub-problems. The heuristic solution approach proposed in this section is a nested method that integrates location-allocation and transportation decisions in a hierarchical manner. In sync with the bi-level problem definition provided in Figure 3-1, the solution approach proposed builds on a user level transportation heuristic and a design level location-allocation heuristic, combined into an efficient nested procedure. For the design problem, the Tabu search heuristic proposed locates depots and assigns ship-to-points to the opened depots. It is a local search procedure; it explores neighbouring depot configurations and perturbs the ship-to-point assignments using a set of restricted moves. For the user transportation problem, a modified and extended Clarke and Wright savings heuristic is proposed. This user heuristic is also used to evaluate potential moves in the design heuristic.

Clearly, several hundred moves may be considered during the solution process, and an exact evaluation of each potential solution, using the user heuristic, would be too time consuming. This is particularly true given the stochastic nature of our problem. To solve the SAA model (14-18), the evaluation of the solutions considered must be based on an estimation of their expected value for the scenario sample £1" generated. To reduce the calculation effort, we propose a fast approximate move evaluation procedure based on a route cost estimation formula. In the taxonomy proposed by Talbi (2002), our solution approach could be classified as a low-level hybrid heuristic. The following sections present our user level heuristic and our design level heuristic.

3.3.2 The User Problem Heuristic Consider the user problem for a given distribution network with depots set LczL and

mission P,, l e L . At the user level, under scenario to, for all the ship-to-point orders

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P„ (to) received by depot / on day /, the objective of depot / is to select the FTL, STL, MTL and LTL shipments minimizing its transportation costs. As explained previously, this is achieved with a two-step procedure. The first step determines the number of full truckload shipments to make and residual ship-to-point loads. The number of full trucks of each type ke K ^ ( t u ) to ship on day t from depot / is found by solving the following simple integer programs by inspection:

( \ [ ^ ( ^ ) l ^ m M = a r g max £ «.% Z b

kyk ^dpl (to), yk =0,1,... i keK^ (o j )

\ ' teJfrJH

,peP„( to) (19)

Then the residual loads dpl (to), p e P„ (to), to be shipped are computed with (1).

The second step solves program (2-4), i.e. it finds the best TL (STL or MTL) or LTL shipments to deliver the residual loads. To solve this transportation problem, we propose a modification of the simple and efficient VRP heuristic, based on perturbed Clarke and Wright (CW) savings and 2-opt improvements, developed by Girard et a l , (2006). The main differences between our problem and a classical VRP are that i) two different direct delivery transportation modes can be used (STL or LTL), and ii) for all modes considered, the vehicle used does not return to the depot after its last drop. Clearly, the best direct delivery mode for a given ship-to-point can be determined a priori by comparing their respective costs, so that the cost of the best direct shipment from depot / to ship-to-point p is:

w0p) = min\_LTL(l,p;dpt(to))\max(r(lp)m(Ip)-,TL,)~],pe P„(to) (20)

Then, as illustrated in Figure 3-4, for two ship-to-points p and p ' , the savings associated to using a multi-drop TL route (l,p,p'), instead of the best direct shipments (l,p) and (l,p'), can be calculated with the following expression:

V = w(/.P)+M;(/.^-H '('.^v P * £»(<*>), P ' Z P > , ( G > ) \ { P } (21)

where vv( ;ppl = max(r(/p p^m(/ pp l;TL,) + a,. The perturbed CW heuristic works as the original CW algorithm but uses the following modified savings formula:

V = Mfcrt + w0>„ - V W « . / M O . ^ ^ H - P ' ^ » ^ } (22) where the weight A , is randomly selected between two predetermined limits A' and A+

for every pair (p,p'). The 2-opt heuristic is then applied to improve each route of the solution obtained. The procedure is repeated / times and the best solution found is

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retained. The total transportation cost of the best solution found is denoted by Cdu (P„ (to)) . The net revenue generated by depot l e L on day t e T of scenario toe Cl" is given by:

*?(Pt,(co))= Z pefi,(<»)

(«,-v,K»- Z ^ i 7 » -cf (£(»)) (23)

Figure 3-4. Alternative Routes Considered in the Savings Calculations for Depot /

The user problem heuristic proposed is summarised in Figure 3-5, in the procedure User. This procedure can be used to evaluate any given network design x under any given scenario coe Q.. The total net revenues generated by design x , for the scenario CD considered, is obtained simply by summing net revenues over all depots and days, i.e. by

calculating ^ " ( x ^ ) = Z t e z .Z, e r^ ' " (^" (®))- W h e n a s a m P l e Q " o f n M o n t e C a r l ° scenarios is used, an estimate Rn (x) of the expected value of the design considered is thus given by:

^w4z„Zte,Z,6^"(^(^))-Z/eiA (24)

3.3.3 Tabu Search Heuristic for the Distribution Network Design Problem

The heuristic proposed to solve the SAA program (14-18) is based on Tabu search, a local iterative approach that is able to escape from local optima by allowing a degradation of the objective function, as opposed to pure descent methods. Since non-improving moves are allowed, solutions previously encountered during the search may be re-visited. To avoid this cycling phenomenon, a short term memory, called Tabu list, keeps track of the most recent moves. The interested reader will find more details about this approach in Glover and Laguna (1997).

At each iteration of the Tabu heuristic, the current design xc available is improved. Potential, non-Tabu, solutions in the neighbourhood of xf are evaluated with a fast, but

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approximate, route cost estimation formula. The best potential solution found is then evaluated more precisely with User to determine if it is better than the best solution x* found to date. The following paragraphs describe the main features of the Tabu search heuristic proposed, as well as three alternative neighbourhood exploration strategies.

Vser((bk,keK™(a>)),d,<a),peP l.<œ)i y,AGA';Rd;{Pu(co)))

S1 : a) Solve ( 19) by inspection to obtain the FTL shipments y™(co) ,keKZL ( to) ,peP, , (co)

b) Compute the residual loads dpl (to), p e Pu(co), with (1) S2: a) Select the best direct delivery transportation modes (LTL vs STL), and compute

their costs wu >, p e Pu (to), with (20) b) Solve the resulting open routing problem y times with the modified Clark and

Wright algorithm, using the savings epp.,pe Plt(co), p ' e P„ (to)\{p} computed with (22), and retain the best transportation solution.

S3: Compute the net revenue Rdu (P ;,(Û>)) with (23)

Figure 3-5. Procedure User for Depot / in Period t under Scenario CD

Neighbourhood Structure

At each iteration, a new design xm is generated from the current design xc using one of the following three moves: (i) Drop: close an opened distribution center and re-assign its ship-to-points; (ii) Add: open a closed distribution center and assign some ship-to-points to it; (iii) Shift: close an opened distribution center and open a closed one while modifying some ship-to-point assignments.

Let Mc be the set of all possible moves from the current design xc at an iteration of the algorithm. Then, using the operator © to denote a move, the General Neighbourhood of design xc is defined by the feasible solution set GAf(xc) = {xm xm =xc ®m,me M c ] . Unfortunately, the size of GN(xc) increases rapidly with the number of ship-to points and potential locations, and it would be too time consuming to assess all the potential solutions in G N ( X C ) . We therefore restrict the search for a better design to moves associated to dropping, adding or shifting depots, combined with a greedy reassignment of ship-to-points based on a unitary net revenue maximisation rule. Also, using the concept of area of influence introduced by Nagy and Salhi (1996a), the moves considered are restricted to adjacent depots. Two depots are neighbours, if there exists at least one ship-to-point for which they are respectively the nearest and the second nearest depots. Let N (l) be the set

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of neighbours of depot /. Then, for each open depot l e V specified by the current design x f , we define a region IZ(l) including depot /, the depots in M ( l ) , and the ship-to points Pc

r, l ' e{ l}^ j ( j \ f ( l )nL) , assigned to these depots. In our algorithm, only drop, add or shift moves within regions are considered. These features limit the search for a better design to a Restricted Neighbourhood RN(X C } czGN(xc ) defined by limited move subsets M(l) c M c , l e Lc. For a given region 1Z(l), the feasible moves considered are:

i. Drop move (wdrop(/)): If N(l)<~^L_c * 0 , close depot / and assign each ship-to point p e TZ(l)nP to depot lp =argmax;,ejV(/)n4 (up -v , ) -w ( l . p ) .

ii. Add moves (M^^l ) ) : If A/"(/)nLc*0, for all l'e M(l ) r \ Z?, open depot /' and assign each ship-to point p e l Z ( l ) n P to depot /=a rgmax , ,, e, , „ (u_ -v, .)-w„. ,.

iii. Shift moves ( Jv\m(I)): If J\f(l)nlc * 0 , for all l 'e _\f(l)nLc , open depot /' and close depot / simultaneously, and assign each ship-to point p e l Z ( l ) n P to depot /„ = are max - ,, ,, ,„ \u_ -v,.)—w„. ..

r P b re( jV( l )nL c . r j { l ' ] V P ' > (' -P)

In what follows, the following sets of possible moves are used:

-Hop ={wd f o p( /)}^,Aid d=(J t e ic>U.(/) . M m = \ J * e W m ( f )

M = Mdrop U M d d U H m . Mil) = {/ndrop (/)} u Madd (/) u Mhl„ (/)

Note that only the moves leading to a feasible solution are considered. For example, if a drop move creates an isolated ship-to-point, i.e. a point too far from other depots to permit next day delivery, then it is not considered. Note also that during the neighbourhood exploration the moves in M are evaluated only if they are not Tabu.

Neighbourhood Evaluation

The evaluation of a design xm e RN (xc) involves the estimation of its expected value for the sample Q" of Monte Carlo scenarios generated. This could be done by applying the User heuristic to all (l,to,t)e LmxQ." xT , and then by computing É„(xm) with (24). However, evaluating all possible moves using the User heuristic would be too time consuming. Possible moves must thus be evaluated using a fast approximation. The evaluation function proposed is based on a linear regression route length estimator (introduced by Daganzo (1984) and extended by Nagy and Salhi (1996b)) modified to account for LTL transportation. It estimates the total transportation cost Cd"(P™(co)) of depot / on day t under scenario to, as a function of the set of ship-to-points P_™,(co) visited

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from depot / on that day under design xm . The function is obtained with a multiple regression model based on three explanatory variables: i) a linehaul variable (£;(F;7(<y))/NS, J, where Ç,(P) is the sum of the distances from depot / to all p e P , and NS, the average number of stops on the routes from depot /; ii) a detour variable \£,(P"(co))lJ\P_";.(co)l\; and iii) a LTL variable (p"TLtp,(P™(CD))), where tp,(P) is the sum of the load-distances from depot / to all p e P , and p\ J L the proportion of load-distances from depot / on LTL routes. This leads to the following cost approximation formula:

6(£,(«>)P = 0. NS, + Â 6 (#(«))

J\p: M + A (P,"1*, (*:(•))) (25)

where p,, /32 and yff, are regression coefficients associated to the linehaul, detour and LTL variables, respectively. These regression coefficients are estimated using a sample of historical daily delivery routes, or a sample of daily routes obtained with User for different network designs. The parameters NS, and pfTL are initially estimated with the same route sample, but they are updated in the algorithm every time the User heuristic is used to evaluate a new design.

An adequate approximation of Rn(xm)is then obtained by replacing Cf;(P™(cu)j in (23) by Cdu (P7,(CD)) , and by substituting in (24), to get:

k-v,k,H- Z rfW *? ; (P: (<»))= z

4«-èZ r fZ^2Lr« ,(iSW)-Z^.A

-Çf (£(»)) (26)

(27)

Restricted Neighbourhood Exploration

Given the restricted neighbourhood structure previously defined, three different strategies are considered to explore the neighbourhood RN(xc) of the current solution xc . These strategies specify the order in which the drop, add and shift moves are made during each iteration of the algorithm. These three strategies have the following characteristics.

Strategy 1: This strategy is inspired from the Tabu search approach proposed by Nagy and Salhi (1996a) to solve a deterministic location-routing problem. It is a straightforward

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version of the Tabu search method. At each iteration, all the drop, add and shift moves in Ai are considered to generate a new solution.

Strategy 2: This strategy is an extension of the three-phase hill-climbing method proposed by Kuehn and Hamburger (1963) to solve a deterministic location-allocation problem. Assuming that a minimum number of depots are initially opened, the first phase explores add moves M.iA only, the second phase considers drop moves A4drop exclusively, and the third phase concentrates on shift moves -M,hift. In our implementation, only non-Tabu shift moves are considered.

Strategy 3: This strategy starts with the initial solution obtained by assigning each ship-to-point to the feasible depot yielding the maximum marginal net revenue. Clearly, in this solution, all the potentially interesting depots are opened. Then, at each iteration, a drop move in MdI0p is performed, followed by shift moves in -M.hift • This process continues until the specified maximum number of iterations is reached. It is worth noting that the shift moves made at each iteration can be viewed as an intensification phase in the region of the search space that contains solutions having the same number of opened depots as the current solution. In addition, following each shift move, a reassignment procedure for borderline points is applied. This intensification phase is a Tabu search because non-improving moves are allowed.

Note that the moves m(l)e M considered are based on changes to the status of depots, followed by a greedy adjustment to ship-to-point assignments to ensure that the resulting design is feasible. There is no guarantee, however, that the assignments adjustment made is optimal. It may therefore be profitable, when all moves are performed, to refine the adjustments made to get the new solution xc . To do this, we elaborated a reassignment procedure for borderline points, inspired from a heuristic proposed by Zainuddin and Salhi (2007) to solve the capacitated Weber problem. A point p assigned to depot l e IT (i.e. in P,m), is considered borderline if its distance from depot / exceeds a predefined target distance mmax, i.e. if m,,p) > mmax. Let B be the set of borderline ship-to-points. A ship-to point p is reassigned from depot / to depot lp e L_mp if its distance ratio p, =m{, p)/m(, p) is the lowest among the eligible depots, provided that it does not exceed a predefined value pmax. When this is done for all the borderline points p e B associated to xm , a new solution vector x' results. Several alternative solution vectors x' can be generated by considering a set Y ^ of pm3X values. The solutions thus generated can then be evaluated with (27) to determine which one is the best. This gives rise to the reassignment procedure presented in Figure 3-6 and applied in the three strategies.

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Tabu Lists

During the search procedure, two Tabu lists of varying length are kept: when a depot / is added or dropped in the new current solution, / is inserted in a drop/add list T t . On the other hand, when the new current solution is obtained by shifting two depots / and / ' , the pair (/,/') is inserted in a shift list T2. In both cases, the oldest element of the list is removed. Note that when a depot / is inserted in TJ, all the shifts involving this depot are also inserted in T2. At each iteration, the length of the relevant list is randomly generated in the interval [a, I LI/2,1 LI/2] for Tx and [a 2 \Ll ,2 lLl] for T2 where a x ,a 2e]0, l [ are predefined parameters. Note that these interval bounds have been adequately fixed after several preliminary tests and are close to those considered in Nagy and Salhi (1996a).

Reassign (xc ; mmax, Tmax ; x , Rn (x)) Set x' = xc and compute Rn (x') with (27) Construct the borderline point set B For all p ^ e T ^ do

x m =x ' For all p e B do

lp =argmin /H„ [p,. =ma.p)/m0iP)\p,- < ̂ J , X£ =0, x£p =1

Compute R„(xm) with (27) If ( Rn (xm ) > Rn ( x ) ), then x = xm and Rn ( x ) = R_ (xm )

End do Figure 3-6. Reassignment Procedure

Solution Algorithm Initialization Before the Tabu search is started, the solution process must be initialized. This first

involves fixing the various parameters required by the procedures used. In addition to the parameters already defined, the following two parameters are required to control the Tabu search: MI = maximum number of iterations, MNI = the maximum number of iterations without improvement.

Next, a sample of n demand scenarios d(co)=[dpl(co)] PieT, coe ÇI", must be generated using procedure MonteCarlo n times, and the neighbour sets M ( l ) , l e L, must be created. Initial solutions must then be constructed for the neighbourhood exploration strategy considered. For strategy 2), an initial solution x° is obtained by sequentially opening the closed depot / e L maximizing the marginal net revenues of the ship-to-points P not yet allocated. For strategy 1 ) and 3), L° is initially set to L. Then for all cases the

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missions are obtained by assigning each ship-to point p e P to the depot l e L maximizing marginal net revenues. The resulting design x0 is then evaluated with the User heuristic and the expected value function (24). Figure 3-7 presents the initialization procedure thus obtained.

Initialize (Strategy; n,y,A,A\a,,a2,mma,TVKixMIMNI,(d(co),coe £l"),(M(l),leL),x°jln(x0))

Set the heuristic parameters (n,y,A~,A+,al,a2,mnax,TmaxMI,MNI) For all toe ft", do MonteCarlo ((Fq

p(.),F° (.),p e P),T ; d pl(to), p e P , te T) Obtain the neighbour sets N ( l ) , l&L If Strategy = 2, then

Set L ° = 0 , P = P While P * 0 , do

/ ' = argmax - V _ (u -v . ) -w, , , t" lm-? -i--mPf_PnP i\ P U (l,P)

Set L0=L°u{/ '} , P,, = P n P r and P = P\P,. End While

Else Set L°=L Construct an initial design x° by assigning each ship-to-point p e P to the depot lp=aigmax i_L 0(up-v,)-w ( ,p )

For all (I,to,t)eL°xQ.nxT, do VSr . r [ (b k ,keK F ; ; (co) ) ,d p . (œ) ,pe^

Compute the expected value Rn (x° ) with (24)

Figure 3-7. Initialization Procedure Tabu Search Procedure

The Tabu search procedure proposed appears in Figure 3-8. The iterations of the algorithm are controlled by two parameters: iter, the current number of iterations, and iterjfii, the current number of iterations without improvement. Step S2 examines different moves in the neighbourhood of the current solution x c . Step S3 applies the reassignment procedure to the current solution x c . Step S4 evaluates the best move and updates the parameters of the transportation costs estimation function. Step S5 manages the Tabu lists. Step S6 performs a shift-move intensification when the parameter Improve=l. Step S7 checks if the last iteration improved the best solution to date x*. Finally, steps S8 and S9 control the iterations of the algorithm. If either of the iteration limits is not reached, the exploration continues from S2 with the current solution xc . Else, if MNI iterations were made without improvement, the exploration continue from SI with the best solution to date

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x*. Otherwise, the algorithm stops. Note that, to simplify the exposition, we used the parameters MI and MNI when calling Tabu in S6 as in the main procedure. In the implementation, however, the iteration limits parameters used in S6 are not the same as in the main procedure.

Using the Initialize and the Tabu procedures described previously, the neighbourhood exploration strategies considered are implemented as follows:

Tabu (x°, Rn (x° ), M; MI, MNI, Improve; x ,Rn (x )) SO: Set x* = x° and Rn(x') = Rn(x°) and initialize the Tabu lists ( Tx and T2 ) SI: Set xc = x* and Rn (xc ) = Rn (x* ) S2: / ? „ ( x > 0

For all / e Lc, do Construct the region 1Z(l) For all m(l)e M(l) and m(l) not Tabu, do

Set xm = xc ®m(l) and compute Rn (xm) with (27) If (Rn (xm) > Rn (x')), then x' = xm and Rn (x) = R„(*m)

End do End do

c '

x =x S3: Reassign (xc; wmax,Tmax; x',Rn (x)) and set xc = x' x' = argmaxzeNçx-){fapprox(z)}S4: For all (l,co,t)e LxÇl"xT, do

Vser((bk,ke K™(co)),dpl(co),pe Pc„(co); y,AGA+; Rdu (pc„(co))) Compute the expected value Rn (xc ) with (24) Update the means NS, and p\TL, I e Lc, using the augmented set of generated routes

S5: Update the Tabu lists ( % and T2 ) S6: If ( Improve= 1 ), then

T - m b ^ x ^ R . & ^ M s ^ M h M N I & x ^ R ^ x ) ) x<=xandR_(xc) = Rn(x)

S7: If ( Rn (xc) > Rn (x )), then x* =xc, Rn (x* ) =Rn (xc) and iter_ni = 0 Else iter_ni = iter__ni + 1

S8: iter = iter + 1 S9: If (iter < MI) and (iter_ni < MNI), then go to S2

Else If (iter < MT), then set iter_ni = 0 and go to SI Else stop

Figure 3-8. Tabu Procedure

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Strategy 1 (S_):

Initialize^; n,y,A,A\ax,a2,mmM,Tm^,MI,MNI,(d(co),CDe Çl"),(M(l),le L),x°Jîn(x0))

TSmbu(xo,Rn(xo),M;MI,MNI,0;x',Rn(x*))

Strategy 2 (S2):

Initialize(2; n,y,AGA+,ava2,mmax,TmiX,MI,MNI,(d(to),toe Q"),(M(l),le L),x°,R_(x°))

Tiibu(x%RB(xo),MM-,MI,MNI,0-,xadd,Rn(xadd))

TBbu(x^,Rn(x^),Md^-,MI,MNI,0-,xdro ' ',Rn(xdrop))

Tabu(xdmp,R_(xdr°p),M%m; MI, MNI, 0; x ,Rn (x* ))

Strategy 3 (53):

Initialize(3; n,y,A,A\ava2,mm!iX,TmaMl,MNl,(d(co),coe Çl"),(N(l),le L),x°,Rn(x°))

Tabu(x",Rn(x0),M±o_;MI,MNI,l;x,Rn(x))

The next section evaluates and compares these strategies.

3.4 Computational Results

3.4.1 Plan of Experiments In order to test the heuristic approach proposed to solve the SMLTP, several problem

instances were generated based on the following four dimensions: the problem size, the cost structure, the demand process and the network characteristics. The problem instances were generated randomly, but they were based on realistic parameter value ranges obtained partially from the Usemore case documented in Ballou (1992), and from the data of a real case. The problems were defined over various US regions and all distances were calculated with PC*MILER (www.alk.com), for the current US road network. For all cases, it was assumed that the order inter-arrival times are exponentially distributed with a mean inter-arrival time A, and that order sizes are log-Normal with a mean p and a standard-deviation a . A one year planning horizon, assumed to include \T\ = 200 working days, was used.

Problems of three different sizes were tested, small (Pi), medium (F2) and large (P_.), as defined in Table 3-1. In each case, the problem size varies in terms of the number of

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depots and ship-to-points considered, and in terms of the geographical region in the US covered. Based on the realistic industrial problems examined, the number of ship-to-points in the problems is much larger than the number of potential depots, and the latter was fixed at about 3%|P|. In order to capture different cost structures, two levels of fixed and variable costs were considered, as specified in Table 3-2. The fixed operating costs, A,, and the unit value of products, v,, were selected randomly in the interval specified in the table. The products price on the market, up, was fixed equal to the value in the table for all ship-to-points.

Problem instance

Geographical Area

Number of depots

Number of ship-to-points

Pi Central North Eastern US States 7 206

P i North Eastern & Midwest US States 15 706

P_ North Eastern & Midwest US States 28 1206

Table 3-1. Test Problems Size

High product value/price Low product value/price

High fixed costs (a):[230jST,2501jC];[l9,2l];23 (fe): [230 A", 250 AT] ; [9,11] ; 13

Low fixed costs (c): [130K, \50K]; [19,21]; 23 (ûO:[l30AT,150jft:];[9,ll];13

(Cost structure): [Fixed cost (-4,) range]; [Product value (v, ) range]; Product price (I* ) Table 3-2. Test Problem Cost Structures

Next, demand processes are associated to the geographical coordinates of the ship-to-points in a problem. These demand processes are calibrated to represent Large, Medium or Small customers. Two types of network are generated: 1) networks composed mainly of large and medium size customers (LAO, and 2) networks including mainly small and medium size customers (SN). Table 3-3 provides the proportion of each type of customers in LN and SN networks, as well as the (A,p,a) parameter values range used to generate specific instances. Finally, two random replications (DS], DS2) aie generated for each network structure considered.

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Ship-to-point size and characteristics Large Medium Small

Larger Ship-to-points Network (LN) 15% 65% 20%

Smaller Ship-to-points Network (SN) 10% 30% 60%

//(cwt) [480,580] [300,400] [120,220]

a(%p) 7% 10% 16%

A (days) [2.5,4.5] [5.5,15.5] [20.5,35.5] Table 3-3. Ship-to-Point Demand Structure

The combination of these four dimensions yields 48 problem instances. Each instance is denoted as follows:

(i,j,k,l) ie{P l ,P2 ,P3},je{a,b,c,d], ke{LN,SN}, le{DS l ,DS2}

The three neighbourhood exploration strategies proposed in the previous section were tested for all these problem instances, and the numerical results obtained are presented in the next paragraphs.

3.4.2 Numerical Results The heuristics proposed were implemented in VB.Net 2005, and the experiments

reported in this section were performed on a 2 GHz Dual Core workstation with 3 GB of RAM. This section starts with a discussion of the calibration of the several procedures used in the heuristics: preliminary tests on various problem instances were performed to fix the algorithm parameters, and to study the stochastic behaviour of the solutions obtained. We then provide a comprehensive analysis of performances of the three neighbourhood exploration strategies considered, for the 48 problem instances. In addition, for the small problem (Pi) instances, the heuristic is compared to the optimal solution of the SAA model (14-18) obtained with CPLEX-11, when using all possible non-dominated routes. These problems were solved on a 64-bit server with a 2.5 GHz Intel XEON processor and 16 GB of RAM.

Procedures Calibration

The solution approach is based on several procedures including a number of parameters which were calibrated with a set of preliminary experiments. Using several Pi and P2 instances, the three strategies were executed alternatively in order to fix the heuristic parameters specified in the initialization procedure. Table 3-4 presents, for each

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parameter, the range of values tested and provides the value selected. Note that these values were fixed for the three strategies and present the best trade-off in terms of algorithm search speed and solution accuracy.

Procedure User Reassign Tabu

Parameters y r A+ "»*» T

max « i a2 MI MNI

Values range

[1.50] [.5,15] [1.2] [50,400] [0,1.5] [o.i] [0,1] [20,200] [0,20]

Selected value

10 1 1.2 250 {1,-75,-5} .25 .25 100 10

Table 3-4. Heuristic Parameter Values

Another important parameter to calibrate is the number n of scenarios to generate (with the MonteCarlo procedure) to obtain good solutions. It is important to note that since our demand process is stationary, since the user problem must be solved by each depot on a daily basis, and since the planning horizon includes 200 days, when n scenarios are used, 200n user model instances sampled from the same probability distributions are solved by each depot. For this reason, a relatively small number of scenarios is required to obtain good results. To determine the best value of n, different sample size were tested and the quality of the solutions obtained was evaluated using a statistical optimality gap, as is usually done when the SAA method is used to solve stochastic programs. This calibration was done using problem Pi with larger ship-to-points (LN), and it was based on the optimal solution of the SAA model (14-18) obtained with CPLEX-11. Values of n = 1, 2, 4, 6, 8 were tested. We were not able to solve larger problems to optimality without truncating the set of non-dominated routes.

To estimate the statistical optimality gap for a sample size n, several SAA models based on independent samples of size n must be solved. Let Rj

n and x'n, j = l,...,m, be the optimal value and an optimal solution of the SAA model for the m samples used. Well known results in stochastic programming are that E[Rn ]>/?*, where E[ ] denotes the expected value, Rn is the optimal value of ( 14) and R* is the optimal value of (6), and that Rnm =Zm=i^" ^ m *s a n unbiased estimator of E[Rn] (Shapiro, 2003). A statistical upper bound on the optimal value is thus provided by R_m>R*. Also, the true objective function value R(xJ

n) < R' of the feasible solutions x'n, j = l,...,m, can be estimated with

(28) cn_i." l_L

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where £1" c Q is a sample of n' scenarios generated independently of the sample used to obtain x'_, and with n' » n . A statistical lower bound on the optimal value is thus provided by Rn.(x

Jn)< R'. Furthermore, if the User heuristic is used to solve the second

stage program, its value R"(x]_,co) in (28) must be replaced by the value Rdu (xJ

n,co) < Rdu (xi,co) calculated with User, to get Rn.(xl)< Rn.(K) ■ A n estimate of the optimality gap of the solution x'n can thus be calculated as follows: BaPnJn,n(xD = Pr.sn~Pn(x-D • Since m SAA models with a sample of size n are solved, an estimate of the average gap for a sample of size n is given by: gapn.n=Z™=,8aP«."..«^«)/m-

In order to evaluate alternative sample sizes, we calculated this average gap with m = 4 sample replications, and we evaluated (28) with the User heuristic, using scenario samples of size n' = 100. The average gap values obtained for different sample size are provided in Table 3-5. These values are expressed as a percentage of the objective function value of the best design found. It can be seen that samples of 6 or 8 scenarios provide very good results. When inspecting the design found with these sample sizes, we noted that they were all very similar: they opened the same depots and only one or two ship-to-point assignments were different. However, the solution time increases considerably with the sample size. For these reasons, a sample size of n = 6 scenarios has been selected as the best trade-off to use in the experiments. This means that 1200 user model instances, sampled from the same probability distributions, are solved by each depot every time a design is evaluated. The gaps in Table 3-5 suggest that the approach proposed is extremely accurate. This is due partly to the fact that, for the problem instance considered, the profits generated are relatively high and the objective function value has low variability, which could mean that the value of stochastic solution (Birge and Louveaux, 1997) is relatively low.

Sample size (n) 1 2 4 6 8 gap„,ioo(in%) 2.32% 3.42% 2.21% 0.67% 0.07%

Table 3-5. Statistical Optimality Gap Values

Analysis of Results

This section discusses the quality of the supply network designs obtained with the three neighbourhood exploration strategies proposed, as well as their respective solution times. First, in order to determine how close to the optimum the solutions obtained are, the SAA model (14-18) was solved with CPLEX-11 for problem Pi instances, using a MIP

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Relative Tolerance of 0.001 and a sample size n = 6. These are the largest SAA problems we were able to solve to optimality. The results obtained with our three exploration strategies and with CPLEX are provided in Table 3-6. The table values are the estimation of the true objective function value R(x) provided by Rn (x), with a sample size n' = 100. The last row provides the %-difference between the value of the CPLEX solution and the best design obtained with our heuristic search strategies. It can be seen that the differences are very small. For problem (Pi, b, LN), the design provided by SI is slightly better than the solution provided by CPLEX. This is possible because the SAA is solved to optimality with a sample size n = 6, which is an approximation. For a given sample size, our heuristic can therefore provide a better solution of the original stochastic programming model (6-13) than the SAA model. The SAA models solved include 2 501 456 binary variables for the larger ship-to-point networks (LAO, and they were solved on average in 939 seconds by CPLEX-11 on our 64-bit server. It took on average 10 seconds to obtain the best solution found with our heuristics on a 32-bit workstation.

Problem (a,I_N) (b.LN) (c,LN) (d,LN) (a, SAO (b,SN) (c.SiV) (d,SN)

SI 2 689 489 2 493 083 4 204 796 4 145 389 1430 919 1 306 351 2 396 894 2 360 981 S2 2 689 162 2 492 898 4 204 796 4 145 389 1430 919 1 306 142 2 396 894 2 360 981 S3 2 632 805 2 439 453 4 203 591 4 144 280 1430 919 1 306 142 2 396 894 2 360 981

CPLEX-11 2 689 481 2 493 069 4 204 851 4 145 394 1 431 210 1 306 349 2 397 181 2 361 277 %-difference 0.000% -0.001% 0.001% 0.000% 0.020% 0.000% 0.012% 0.013%

Table 3-6. Comparison with CPLEX-11 Solution of SAA Model for Problem Pi

Next, our three exploration strategies were compared for all problem instances. The true objective function value of a design x obtained was estimated with R„(x), using a sample of n' = 100 scenarios. For a given problem instance, to ensure that solution strategies are compared on the same basis, the sample of n = 6 scenarios used in the heuristic, and the sample of n' = 100 scenarios used to estimate the design value, were the same for the three exploration strategies. The mean design value obtained for different subsets of problem instances are presented in Table 3-7. The dot in the problem subset labels denotes all instances corresponding to a problem attribute. (Pi, a, ., .), for example, is the subset of 4 problems of size Pi with cost structure a. The value of the best search strategy for each problem subset is highlighted. More detailed comparisons of the three strategies' design value and solution times are provided in Figure 3-9. In the three first plots of this figure, for Pi, P2 and Py respectively, the design value %-deviation from the best-known solution is given for all the instances solved. The fourth plot provides a

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comparison of average time (in seconds) required to find the best solution, by problem size. Pl (Pi, a , . , . ) ( P u b , . , . ) (P,,c,.,.) ( P u d , . , . ) (Pu.,LN,.) (Pu.,SN,.) (P.,.,., •) SI S2 S3

2 119 638 1 957 886 2 426 406 2 884 254 2 426 062 2 883 577 2 412 939 2 883 854

3 255 870 1 908 016 2 581943 SI S2 S3

2 119 556 2 048 156

1 957 788 1 927 984

2 426 406 2 884 254 2 426 062 2 883 577 2 412 939 2 883 854

3 255 103 1 907 990 3 204 709 1 907 974

2 581546 2 556 341

Pi (Pi , a , . , . ) (Pi, b,.,.) (P2, c,.,.) (P2, d,.,.) (P2,.,LN,.) (P2,.,SN,.) (Pi, ) SI S2 S3

9 578 586 8 845 539 8 599 984 9 631837 10 712 395 5 780 462 8 246 428 SI S2 S3

9 521073 9 560 879

8 836 155 8 809 231

8 595 897 9 633 208 8 570 404 9 616 201

10 677 916 5 782 614 10 691 160 5 758 618

8 230 265 8 224 889

P3 (P., a , . , . ) ( P . , b , . , . ) (P3, c , . , . ) (P_, d,.,.) (Pm,.,LN,.) (Pm,.,SN,.) (Ps, -, -, ■) SI S2 S3

20 752 448 15 848 841 16 485 836 17 554 079 23 447 483 13 701596 18 574 540 SI S2 S3

20 645 080 20 640 702

15 971 828 16 540 209 17 547 339 23 437 516 13 726 087 18 581 802 SI S2 S3

20 645 080 20 640 702 15 663 930 16 389 856 17 530 314 23 325 136 13 638 200 18 481668

Table 3-7. Mean Design Values for all Problem Types

When looking at the detailed results, the first observation that comes out is that, even if none of the proposed strategies is completely dominated, search strategies SI and S2 provide better results. Strategy 1 provides the best solution for 65% of the problem instances solved, and it is within 0.5% of the best solution found for 96% of the problems solved. This is mainly because this strategy converges quickly to a solution with a good number of depots, and then performs several moves to improve this solution. It usually finds the best solution during the initial search iterations. Strategy 1 is also very fast: the average solution times for Pi, P2 and P3 are 13, 43 and 292 seconds, respectively. Strategy 2 provides very good results for P2 and P_, but it is relatively dependent on the quality of its initial solution. It provides the best solution for 52% of the problem instances solved, and it is within 0.5% of the best solution found for 90% of the problems solved. It gives excellent results for large problems (P3) and for instances with smaller ship-to-points (SN). It is also the fastest strategy. Strategy 3 finds the best design only for 27% of the problem instances. It gives better results for small networks with smaller ship-to points (SN). Due to the intensification phase added in Strategy 3, the number of shift moves grows rapidly and, for large problems, the solution time is significantly larger than for the two other strategies.

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o.oo% -1,00%

-2.00*

-3.00%

-4.00%

-5,00%

-6,00%

-7.00%

-8.00%

-9.00%

0 00-

oso

1 2 3 A 5 6 7 8 9 10 11 12 13 14 IS 16

S2 S3 P, Instincts

> -

I 1 5 0 '

2 oa-

'■■J f: V v y "X

1 \ 1

1

■i

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

-s i S2 S3 P. Instincts

0,00%

-1,00%

-2,00%

-3,00%

-4,00%

-5,00%

-6,00%

—v*—•—\:/—V—

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

-SI S2 S3 P. Instincts

Figure 3-9. Exploration Strategies Design Value and Solution Time Comparisons

When the network density is low, the number of borderline ship-to-points is high and the Reassign procedure improves the design obtained significantly. When fixed depot costs are low, the design obtained includes more depots in order to save on transportation costs. Conversely, when ship-to-points are smaller, the design obtained includes fewer depots. The cost structure does not have a marked impact on the performance of the search strategies. For large problems (P3), however, SI performs better for instances with large fixed and variable costs (a) and S2 for problems with extreme fixed to variable costs ratios (b, c).

Our results also show that the route-length estimation formula (27) is very accurate. Its use to evaluate trial moves provides very good designs. It gives values between [99%, 104%] of the exact transportation costs, computed with the user model, in the case of larger ship-to-point networks (LAO, and between [98%, 108%] in the case of smaller ship-to-point networks (SAO.

3.5 Conclusions This paper defines and formulates an important strategic planning problem for

distribution businesses using external transportation resources: the Stochastic Multi-period

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Location-Transportation Problem (SMLTP). The problem is characterized as a hierarchical decision problem involving a design level taking network location and allocation decisions, and a user level taking transportation decisions, and it is formulated as a two-stage stochastic program with recourse. We show how a sample of multi-period demand scenarios can be generated from the stochastic demand processes of customers, and used to solve the problem. Since, the resulting sample average approximation MIP is extremely large, it cannot be solved to optimality with commercial solvers for realistic problems, and a hierarchical heuristic solution approach is proposed to solve it. It is based on a user level transportation heuristic and a design level location-allocation heuristic. Three different strategies, based on drop, add and shift moves, were proposed to explore the neighbourhood of a solution. A revised route length approximation formula is also proposed to speed up calculations.

In order to test the quality of the heuristic developed, several industrial cases were examined and used to construct 48 realistic test-problem instances. The experiments made showed that the solution approach proposed provides good results in terms of solution quality and solution time. We found that neighbourhood search strategies SI and S2 give the best results. The excellent performance of the nested solution approach proposed is principally due to the good quality of the anticipation of transportation costs in the design heuristic, and to the efficient combination of several procedures and heuristics. We believe that the approach as it stands is sufficiently evolved to solve most practical cases efficiently and effectively. Some room is left however for additional research. Following are some future research avenues that could yield significant payoffs:

• We considered a context where a ship-to-point must be allocated to the same depot for the entire planning horizon. In other contexts, a flexible allocation of ship-to-points to depots may be possible. The User problem would then become a multi-depot transportation problem and it would be more difficult to solve.

• We assumed that the demand process is stationary. If it was non-stationary, the size n of the scenario samples to use to obtain good statistical optimality gaps would be much larger, and the SAA model would be more difficult to solve.

• We were able to solve the SAA model to optimality with CPLEX-11 only for small problems (Pi), and we did not develop a specialized decomposition method to try to solve larger problems to optimality. This may be very difficult because, as indicated earlier, the number of binary variables for route selection grows exponentially, but it deserves further investigation.

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The three search strategies examined could be refined and fine-tuned, and other search methods, such as variable neighbourhood search, could be used to elaborate alternative heuristics.

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Chapitre 4: Designing Resilient Supply Networks under Disruptions

Résumé - Cet article étudie plusieurs approches de modélisation afin de concevoir des réseaux d'approvisionnement résilients pour le problème de localisation d'entrepôts et de transport sous incertitude. L'environnement futur du réseau est caractérisé par des scénarios de demande aléatoire et de périls perturbants la capacité des entrepôts et la demande. L'approche de design propose plusieurs formulations pour ce problème basées sur la programmation stochastique et en utilisant des anticipations approximatives et des modélisations alternatives de stratégies de resilience. Une méthode de solution générique est proposée pour la génération d'échantillons de scénarios plausibles par une approche Monte-Carlo, et pour l'élaboration et l'évaluation d'un ensemble de designs alternatifs. Plusieurs expériences sont effectuées afin de comparer les modèles proposés et fournir des recommandations sur l'approche à utiliser afin de concevoir des réseaux logistiques efficaces et robustes.

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4. Designing Resilient Supply Networks under Disruptions

Walid Klibi1'2 and Alain Martel1,2**

1 Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT)

2 Département Opérations et systèmes de décision, Faculté des sciences de l'administration, Université Laval, Québec, Canada G1V 0A6

Abstract. This paper studies various modeling approaches to design resilient Supply Networks (SN) for the location-transportation problem under uncertainty. The future environment of the SN is assumed to be shaped by random demands, and by hazardous events perturbating depots capacity and ship-to-point demand processes. The paper proposes several anticipation-based stochastic programming models incorporating alternative resilience seeking formulations. A generic approach to generate plausible future scenarios, and to elaborate and evaluate SN designs is also proposed. Experiments are made to compare the SN design models proposed, and recommendations are drawn on the approach to use to design effective and robust supply networks.

Keywords. Uncertainty, Resilience, Scenario Planning, Network Disruptions, Multihazard, Stochastic Programming

Acknowledgement. This research was supported in part by NSERC grant no DNDPJ 335078-05, by Defence R&D Canada and by Modellium Inc.

Results and opinions in the publication attributed to named author(s) were not evaluated by CIRRELT.

Les résultats et opinions contenus dans cette publication n'engagent que leur(s) auteur(s) et n'ont pas été évalués par le CIRRELT.

* Corresponding author: alain.martel@cirrelt.ca Dépôt légal - Bibliothèque nationale du Québec,

Bibliothèque nationale du Canada, 2009

© Copyright Walid Klibi, Alain Martel and CIRRELT, 2009

4.1 Introduction A Supply Network (SN) is a configuration of endogenous resources and supply chain

partners geographically deployed in order to serve a customer base. Strategic SN design decisions involve the determination of the number, location and mission of a set of facilities. At that level, the main objective of the firm is the design of a SN maximising shareholder value and maintaining this value at its best along a planning horizon. However, at design time the future environment under which the SN will evolve is unknown which complicates the strategic evaluation of potential designs. Traditional SN design approaches assume that the environment is deterministic, which give rise to classical location models (Klose and Drexl, 2005). Typical extensions of these models take into account random factors using stochastic programming (Birge and Louveaux, 1997) or depot failures using robust optimisation (Kouvelis and Yu, 1997). However, no comprehensive SN design methodology considering plausible future scenarios has been proposed. A recent review on supply chain networks design problems under uncertainty is found in Klibi et a l , (2009a).

A major preoccupation of contemporary businesses is the consideration of risk management issues when designing SNs. In addition to the random variables associated to business-as-usual factors, several catastrophic events have recently disrupted supply chain networks. Rice and Caniato (2003) and Christopher and Peck (2004) investigate network vulnerability to extreme unforeseen events such as natural disasters and strikes, and Sheffi (2005) examines the case of several companies who suffered from fires, earthquakes, floods, intentional attacks, etc. SNs are geographically dispersed across large regions which increase their exposure to extreme events and, in order to design robust SNs, the impact of such events must be considered. To this end, this paper considers the random, hazardous and deeply uncertain events that shape future SN environments. The challenge here is to elaborate a SN design methodology taking all these event types into account while remaining sufficiently synthetic to be practical. The approach proposed is based on the modeling of multihazard exposures and it involves the definition of network vulnerability sources and exposure levels, the estimation of multihazard arrival processes, and the assessment of multihazard consequences. The risk modeling concepts applied are based on Haimes (2004), Grossi and Kunreuther (2005), Kleindorfer and Saad (2005) and Banks (2006).

In this context, responsiveness and resilience become key elements to enhance the robustness of SN designs. Responsiveness policies are operational rules to deal with random supply and demand factors as well as disruptions. Sheffi (2005) defines resilience

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as the ability to bounce back from disruptions. Resilience strategies are predispositions of network resources favouring risk avoidance and mitigation. Currently most SNs have difficulty coping with emergencies (Lee, 2004), and they do not develop plans to protect against high-impact low likelihood events (Chopra and Sodhi, 2004). Recently, SN risk management strategies incorporating redundancy and flexibility have been proposed in the literature (Chopra and Sodhi, 2004; Lee, 2004; Sheffi, 2005; Tang, 2006; Snyder et a l , 2006). However, these strategies were never explicitly incorporated in SN design models. Using the location-transportation problem under uncertainty as a typical SN design problem, this paper introduces responsiveness and resiliency constructs into SN design models, and it investigates the performance of the resulting SN designs.

The rest of the paper is organized as follows. Section 2 describes the location-transportation problem under uncertainty and characterizes underlying demand and hazard processes. Section 3 presents a scenario-based SN design approach and proposes two anticipation-based design models using stochastic programming. Section 4 discusses risk avoidance and SN resilience, and it proposes three design models to improve SN resilience. Section 5 proposes a generic approach to generate plausible future scenarios, and to elaborate and evaluate SN designs. Computational results are presented and analysed in section 6. Finally, section 7 concludes the paper.

4.2 The Location-Transportation Problem under Uncertainty

4.2.1 The Location-Transportation Problem Context The company considered purchases a product family from a number of supply sources.

This product is sold to customers located in a large geographical area and hence it must be shipped to a large number of ship-to-points. In order to serve its customers, the company must implement a number of capacitated depots with similar processes and technology. For a .given day, the capacity of a depot reflects the maximum throughput sustained by its resources. In addition to its regular capacity level, we assume that under normal business conditions, the depot can provide an additional capacity per day using local recourses (ex. overtime).

Customers order a varying quantity of product and the company wants to provide next day delivery from a single source using common or contract carriers. To this end, several transportation options are available, namely: single customer full truckloads (FTL), single customer partial truckloads (STL), multi-drop truckloads (MTL) or less than truckload (LTL) transportation. For a given day, when all the orders are in, the company plans its

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transportation for the next day and it requests from its carriers the trucks required to deliver products to ship-to-points. However, the network's depots are under the threat of disruptions and, consequently, their capacity to respond adequately to ship-to-point's orders can be perturbed. Therefore, in order to complete the orders received for a given day, the company relies first on its regular capacity, and second on a local recourse such as overtime. If this is not sufficient, external resources can be used to satisfy all its customers. Figure 4-1 illustrates the Location Transportation Problem (LTP) under uncertainty. This problem is an extension of the LTP under stochastic demand studied by Klibi et al. (2009).

Design level • Location decisions • Allocation decisions

Potential depot locations

s /

(SourcesJ)

^ ^ ^ . . leL x t . . .

Compound stochastic

hazard process

Compound Demand Process

User level Daily transportation decisions

MTL Route STL Route FTL Shipment ♦ LTL Shipment

Figure 4-1. The LTP Structure under Uncertainty

Let L be the set of potential depots considered to perform distribution operations, P the set of all ship-to-points and P, the subset of ship-to-points that could be served by depot l e L. Also, let a, be the capacity of depot /, i.e. the quantity of products it could ship during a day. At design time, strategic decisions are made on the subset of depots I c Z - to use during the planning horizon, and on their mission P,<zP, , leL. These decisions are denoted by the vector x. However, at design time, these decisions are taken under uncertainty and they must consider plausible future scenarios over a discrete planning horizon r e T "

4.2.2 Demand and Hazard Modeling Based on the information available at design time, two types of events shaping the

business environment can be distinguished, namely random and hazardous events. Random events are related to business-as-usual operations, and hazardous events to low-probability high-impact disruptions. In this paper, random customer's demand and network's multihazards are modeled as compound stochastic processes. We assume that the demand

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of the SN ship-to-points p e P follows a compound process with a random order inter-arrival time qp and a random order size op . The cumulative distribution functions of inter-arrival times and order sizes are denoted respectively by FH.) and F°(.). The approach used to model hazards is based on Klibi and Martel (2009). Hazards are first amalgamated into meta-events with generic impacts called multihazards, and SN vulnerability sources are identified. Second, a compound stochastic process is defined to describe how multihazards occur in space and in time, and to specify incident's intensity and duration. Third the impact of hits on the SN is modelled using recovery functions.

In our context, the depots L and ship-to-points P constitute a set of network locations N = P K J L . We assume that depots and ship-to points have different incident profiles in terms of impact and time to recovery and thus constitute two distinct vulnerability sources, denoted by sL and sp. The notation s(n) is used to identify the vulnerability source of location ne N . To map potential threats, the geographical territory in which the SN operates is partitioned into a set of hazard zones Z delineating areas with similar exposure characteristics. Using an exposure measure, each hazard zone z e Z is assigned to a discrete exposure level g ( z ) , geG . Based on its geographical position each network location n e N is positioned in a hazard zone z(n)eZ and it has an exposure level g(n) = g(z(n)). We assume that multihazards occur independently in zones z e Z , and that the time between their successive occurrences is a random variable A, characterized by a stochastic arrival process with cumulative distribution function F/(.) .

When zone z e Z is hit by a multihazard, the severity of the incident is characterized by two correlated random variables, expressed in terms of metrics depending on the vulnerability source se{sL , s p ) , namely the impact intensity J3z_, with cumulative distribution function Ff(z)s(.), and the time to recovery &__. The time to recovery is related to the impact intensity through an impact-duration function 9,.= f s(Px) + £s, where e. is a random error term with probability distribution function F/ (.). Figure 4-2 illustrates this function for depots, in %-capacity loss, and for ship-to-points, in %-demand variation. Note that, following a hit on ship-to-points, first necessity products would see their demand raising but luxury products would see their demand dropping. As a single product family is considered here, the impact intensity provides a net effect for the entire product family. We can have a demand surge for some ship-to-points and a drop for others, however.

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f.(fl) 120

0 10 20 30 40 50 60 70 80 90 100

Physical Impact P (in %) Depots Ship-to-points

Figure 4-2. Impact-Duration Functions for Depots and Ship-to-Points

The occurrence of an incident in a hazard zone z e Z does not necessarily result in a hit of all the SN locations in that zone. Conditional attenuation probabilities ctn, ne Nz, are defined to reflect location hit likelihood. When a location n e Nz associated to vulnerability source s is hit, the impact intensity and the time to recovery are provided respectively by fin = /?z(nWn) and 9n = 9zinHn) ■ Consider a multihazard hitting location ne Nz at the beginning of period T'e T", and, to simplify the presentation, let the indexes x = q,o,a be associated respectively to the customer order size (q) and order arrival (o) processes, and to the depot capacity process (a). The impact of a hit is not necessarily felt uniformly during the time to recovery: several phases can be observed, depending on the nature of the multihazard and of the vulnerability source (Sheffi, 2005). Such phase-

dependent impacts can be characterized by defining discrete recovery functions px

m = rx([m-n,0_,pxnt),'T = T',...,T'+0n-l, where px

T is a capacity/demand amplification percentage for process x at location n in period x. The px

m value used as an argument in the function reflects amplification percentages before the hit and the function returns percentages after the hit. As illustrated in Figure 4-3, if the periods affected by the multihazard are not still recuperating from a previous incident, then the a priori percentages are px

T=l00%, \fx,n, T = T ' , . . . , T ' + 0 _ - \ . The amplitude of the amplification depends on the multihazard impact intensity fin. Using these recovery functions, the capacity and the demand can be calculated for specific periods and locations. For the order inter-arrival times and sizes, this gives rise to the perturbed random variables qpT = pq

pTqp

and opT = p"pzop, T = T',..., t '+ 9 - 1 , and to their associated distributions functions FpT (.) and F° t(.). For the depots, this yield perturbed capacity levels au=a lp

a,x, T = T ' , . . . ,T '+9 , - \ .

Amplification percentage

100%

Amplitude based on fi

I I I

Amplification percentage

r? >-_ | - " \ / \ A priori percentages

» — f Capacity loss \ . f ^ T recovery function 100%

Smgnalio pha A ,

I I I I ,l I ! I

Demand surge recovery function ery tunc

s r '+ 0, - 1 Periods

deplovmenl i Sustainment [ Linen

phase ! recovi

i i i i i i

108

Amplitude based on fL

T ' + 0 - I Periods

Time to recovery Time to recovery

Figure 4-3. Recovery Function Examples for Depot / and Ship-to-Point p

4.2.3 Plausible Future Scenarios The instantiation of the demand and multihazard processes described previously over

all the possible values of the random variables involved yields a set Q of plausible future scenarios with associated probabilities JT(CD), toeQ.. The Monte Carlo procedure in Figure 4-4 can be used to generate a scenario instance toe Q.. The procedure uses independent pseudorandom numbers u, uniformly distributed on the interval [0,1], first to generate multihazard arrivals, second to generate recovery functions, and third to generate daily ship-to-point demands d(co) = [d (to)]

l>eP, TeT" and depot capacities

a(to) = [a,,(to)] . The input parameters a and a are capacity and attenuation probability vectors, respectively, and F denotes the set of all the previously defined probability distributions.

Some of the plausible future scenarios in Q may involve only a few multihazard over the planning horizon but others may be much more chaotic. An intuitive measure to assess the risk associated to a scenario toe £1 is the number of hits y(co) it undergoes during the planning horizon. Figure 4-5 illustrates, for two SNs of different size, the distribution of the number of hits for a large sample of scenarios generated using procedure MonteCarlo with exponential multihazard inter-arrival times. In order to distinguish between the scenarios a decision maker would consider as acceptable, in term of the risks involved, and those that would raise a serious concern, we define a hazard tolerance level K. This level is the maximum number of hits the decision maker can tolerate without serious concern. This tolerance level is used to partition £l in two subsets: the set of low-hazard scenarios £lL = jto\y(to)<K\ with associated low-hazard probability nL ='Lc^nLx(co) and conditional scenario probabilities nL(to),toe Q.L, and the set of high-hazard scenarios Cl" =£l\ÇlL with associated high-hazard probability nH =L H7r(to) and conditional scenario probabilities nH (to),toe Q.H. The risk associated to a SN depends on the size of the territory it covers and on its density in terms of number of depots/points per unit area

109

(Craighead et a l , 2007). For this reason, as illustrated in Figure 4-5, the distribution of hits is increasingly skewed towards the right as the size of the network increases. Consequently, the value K selected also tends to increase with the size of the network considered. Also, the worst-case scenarios in the tail of the distribution will be of particular interest to us. They will be useful to assess the robustness of the SN designs considered.

MonteCarlo((Nz,ze Z),T",a,a,F,(rx^c = q,o,a);d(CD),n(to)) 1) For all z e Z , do:

Using the cumulative distribution of A., generate multihazard arrival moments T __\T" Set px

t = 1 for all ne Nz, r e T", x = q,o,a For all r ' e T , d o :

Compute A, = F ^ ( « ) and 6U =\fs ( £ > F / ' («)] , 56 {sL,sp}

For all ne Nz\u<an, do: PXnr^rx(PzWsW,9z(n)sM,px

nz), T = T ' , . . . , T ' + 9 M ) - l , x = q,o,a End For

End For 2) For all pe P, do:

?j = 0;dpT(to) = 0, r e r ; r = l ; F ; r = F ; While 77<|ru|,do:

Compute the next order arrival time TJ = tj + FpT (u) and t = \ î ] \ For x = q,o: Derive Fpr from Fp and p x

Compute the daily demand dpr (to) = dp. (co) + Fp°r' (u) End While

End For For all l e L and Te T" : Compute the daily capacity a,T (to) = a,p".

Figure 4-4. Scenario co Generation Procedure

Finally, note that for a given scenario CO, on day T, the depots capacity a,T(co), l e L, is known and the set of depots L can be partitioned into operational depots L,(to)={/| alT (cv) = a,} and partially operational depots IfT (cv) = {/1 a,z (cv) < a,}. Similarly, the ship-to-points demand dpT(co),pe P, is known and the set of ordering ship-to-points P. (to) = j p l d ^ (to) > O] can be specified. The next section proposes a design approach for the LTP under uncertainty based on the risk modeling and scenario generation procedures presented here.

110

0,3

0,25

Low-hazards Scenarios High-hazards Scenarios

2 / 3 Hazard tolerance level

( r = 2)

Low-hazards Scenarios -j^m.

6 7 8 9 Number of hits fora medium size problem

High-hazards Scenarios rKm.

2 3 / 4 S 6 Hazard tolerance level

7 8 9 10 11 12 Number of hits for a large size problem

Figure 4-5. Distribution of the Number of Hits for a Large Scenario Sample

4.3 Scenario-Based SN Design Approach The LTP under uncertainty is a hierarchical decision problem due to the temporal

hierarchy between the location decisions and the transportation decisions. At design time depots location and mission decisions x must be made. However, these strategic decisions impose resource constraints on the network user's transportation decisions and they may restrict recourse actions in response to customer demands and hazards. On a daily basis, depot / users make shipping decisions, denoted by the vectors y , r , t e T u , in response to the orders received from customers. These decisions must be anticipated at design time since it is through them that net sales revenues are generated. In addition, due to hazards, operations can be perturbed by demand and/or depots disruptions. The recourse actions then employed to provide efficient service must also be anticipated to design a resilient network structure. This can be done by formulating the SN design problem as a two stage stochastic program with recourse. When based on exact anticipations, these SN design models are extremely complex and their solvability is an issue. To produce good designs, adequate precision-solvability tradeoffs must thus be made (Klibi et al., 2009b). To this end, in what follows, we examine two approximate anticipations of user responses involving scenario-period sampling and transportation decisions aggregation.

I l l

4.3.1 User Response Model For a given distribution network x, a depots set LczL and depots business-as-usual

missions P, <zP,, l e L, aie set for all the days r e T" of the planning horizon. In a single sourcing delivery context, each ship-to-point p is served by a unique depot denoted l ( p ) . Due to hazards, short-term variations in capacity and demand occur and recourse actions are necessary on a daily basis to provide an adequate response to customers. To respond adequately, the company must decide on a daily basis if the primary mission of depots is maintained or adapted. In the latter case, the company makes internal order reassignment decisions and, ultimately, recourse to external supply sources. Tomlin (2006) proposed operational tactics for managing disruptions based on rerouting ships. Once the order reassignment is done, the company makes shipping decisions, at each depot, with the objective to maximise sales revenues. To implement this, based on the response policy of the company, order assignment and transportation procedures must be elaborated.

For a given day T under scenario to, the set of ordering ship-to-points PT(co), their demand dpT(co),pe PT(co), and the depots daily capacity a,T(co),le L, are known. Thus, based on depots primary missions, revised daily assignment decisions P,T(to), l e L, must be made to make sure that the depots capacity constraints _5_] . .d^ (cv)<a,T (cv),le L, aie respected. Additionally, internal recourses (ex: overtime) or external resources can be used to satisfy unfulfilled orders. When a depot / is operational on day r under scenario to, additional capacity Ç,a,, is available for use, where Ç, is a fixed proportion of regular daily capacity. As illustrated in Figure 4-6, the stochastic demand level and depot capacity on each day dictate the kind of response decisions to take. When depot / is operational it serves its primary orders and it can process reassigned orders from other depots. When depot / is partially operational, however, it serves only a subset of its primary ship-to-point orders and the remaining ones are transferred.

Let P be a priority list ranking ship-to-points p e P in decreasing order of their importance. When all the SN orders have been received on a given day, we assume that list V is used to assign the most important customers to their primary depot l (p) and to transfer the remaining orders to an alternative depot l 'e L \{l(p)} or to an external supply source (identified using index / = 0 ). We assume that the reassignment policy used is based on a proximity rule (default policy), or on explicit or implicit instructions provided by the design model decisions. More specifically, we assume that the default reassignment policy is to supply ship-to-point p orders not shipped by l (p) from the nearest depot in Lp\{l(p)}, that is to set /'(/?) = argminreL .̂. ^m,-p, where mlp is the distance in miles between depot / and ship-to point p . If depot l ' (p) cannot ship the order, the company

112

asks the external supply source / = 0 to make a direct shipment to ship-to-point p. On day T, the ship-to-points supplied from the external depot are thus P0r (co) = PT (tu)\u,€ LQT(to).

Depot capacity

Depot partially operational Depot totally operational

}zlr(cv) n Local

recourse capacity

Demand level

I djcv)

Hazard duration

Figure 4-6. Demand Level at a Given Depot / under Scenario CD

Once reassignments have been made, the set P,.(to) and the loads d^(co) , p<e P,T(CD)

to deliver on day t are known for each depot / e L u {0}. Then, the company can plan its transportation for the next day, and requests the trucks required for each depot from its carriers. These shipping decisions are made by the SN depot users in two steps. First, for loads that are larger than a truckload, a decision is made to ship as much as possible in FTL. We assume that a single type of vehicle is available to make full truckload shipments to point p. Let,

yfpJL i®) '• The number of truckloads shipped to point p on day T

b F : The capacity of vehicles used for FTL shipments wlp : The cost of a FTL shipment to point p from depot /

To determine the FTL shipments to make to point peP,T(co) , problem (1) below is solved by inspection. Then the residual loads to be inserted in the STL, MTL or LTL shipments are given by (2).

y^ ( to ) = aig(inaxyy\b fy<dp T( to)) ( 1 )

d M = dptH-bFy7TL(co) (2)

Next, the best delivery routes must be constructed. Let,

Pk : Ordered set of ship-to-points in route k

113

K,t(to): Set of non-dominated feasible STL, MTL or LTL routes (i.e. such that Pk <zP,r(co) and WZ^pdpT(co)<bk, ke'K, t(to), where bk is the capacity of route k vehicles) from depot /, on day r , under scenario CO

wk : Transportation cost of route k e K,T (co) 8kp : Binary coefficient taking the value 1 if ship-to-point p is covered by route k,

and 0 otherwise yUt(co) : Binary decision variable equal to 1 if route k is used from depot / on day t

under scenario to, and 0 otherwise

For scenario CD, the best routes are obtained at depot / on day T by solving the following transportation sub-problem:

C;r(û?) = min X W*>WM ™

subject to

Z 3p3WH = l PePlrit») (4) *€*„(»)

yklT(to)e{0,l} keK,T(co) (5)

where C"T(to) is the cost of the optimal shipments made by depot / on day t under scenario to. In this paper, the heuristic proposed by Klibi et al. (2009) to solve this problem is used. It is based on perturbed Clarke and Wright savings and 2-opt improvements. The cost of the transportation solution obtained with this heuristic is denoted by C"r(co).

Furthermore, the shipments made on a daily basis generate sales revenues, and additional costs are incurred for recourse actions. Let,

v; : Unit value of products shipped from depot / (taking into account the product production/procurement costs, inbound shipment costs, warehousing costs and inventory holding costs under normal operations)

up : Unit price of products sold to ship-to-point p Z,T(CD): Recourse capacity needed at depot / on day t (z,T(to) = V dp.(co)—a,

if le L.(cv) and 0 otherwise) cz, : Additional unit cost incurred when recourse capacity is needed to ship

products from depot /, with the consequence that the unit value of the products shipped becomes vf = v, + cf

ve: Unit cost of products supplied from the external emergency source using STL, MTL or LTL shipments

114

vH : Unit cost of products supplied from the external source using FTL shipments

Since the emergency supply source would be used only in extreme cases, the costs ve

and v* are assumed to be the same for all ship-to-points. These costs are external recourse penalties with values higher than product values and prices, i.e. such that v, < vf <v h <v e

for all / and up < vh < ve for all p . From this, unit external supply loss can be defined as: ce

p = ve —u and chp = vh —u , for all p . Under scenario CD, for day r , the net revenues

R1T(CD) generated by depot l e L and the network loss for external recourses C0T(CD) can be calculated as follows:

*/»= I ([(«,-Vi)^W]-^W-^?W)-c;(») (6)

C 0 »= X ( c ^ W + c y y ^ W ) (7)

The UserResponse procedure described above is summarized in Figure 4-7, and it can be used to calculate the net revenues R" (x, co) generated over the planning horizon for a given SN design x , under a scenarios CD.

4.3.2 Risk-Neutral Design Model The strategic decisions to make here involve the selection of a subset of depots

L* c L to operate during the planning horizon T", and the assignment of ship-to-points jÇ* <= P„ l e Ê to these depots, to maximize total expected profits. An important aspect of

the problem is that the mission of the selected depots, defined by their customer sets, P,*, l e L* must remain the same for each day T e T " of the planning horizon. To take adequate design decision we must anticipate how the company response policy, materialized in the response procedure implemented, will perform with these decisions. This implies that the transportation sub-problem (3)-(5) and the revenues and loss functions (6) and (7) must be incorporated in the design model. Let,

x, : Binary variable equal to 1 if depot / is opened, and 0 otherwise x,p : Binary variable equal to 1 if ship-to point p is assigned to depot /, and 0

otherwise elpr (co) : Binary variable equal to 1 if under scenario CO the ship-to point p order for

day T is shipped from the emergency source instead of its primary depot / ,

and 0 otherwise hlpT (to) : Integer variable giving the number of FTL shipments made to ship-to point p

from the emergency source, instead of depot /, for day T under scenario CD

Also, the additional unit costs incurred when products are shipped from the emergency

115

supply source instead of from supply depot / are denoted by cf = ve - v, for STL, MTL or LTL shipments and by cf =vh -v , for FTL shipments.

UserResponse(x,(/'(p),/76P),a(û>),d(«),(4(û>),Pr(û;),rer'),P;^"(x,û;)) For all TeT", do:

Set P,T(to) = 0 , l e L u { 0 } ; alT(to) = (l + Ç,)a,, leLz(tv)nL Assign orders to depots

For all p e P-(to) in order of the priority in V, do 1 { a , ( p ) r H ^ d

P r H t h e n

S e t ^ { p ) T n = p l { P ) r ( ( 0 ) ^ { p } a n d a,(P)A0>)=yP)Aû>)-dpA<») Else If a,.{p)i(to)>dpr(to) then

S e t P , { p ) r n = Pt(p)A<°)v{p} and a,,{p)r(to) = ar(p)T(to)-dpT(co) Else: P0T(to) = P0T(to)Kj{p}

End do Compute depot revenues and network loss

For all / e L, do Solve the transportation problem (3)-(5) with Klibi et al. heuristic (2009) Compute depot / net revenues R,r (to) with (6)

End do, Compute the network loss for external recourses C0r (to) with (7)

End do

Compute the SN net revenues R" (x, to) = ]£ (X / e z .^ {œ) ~ Qr (ft;))

Figure 4-7. Response Procedure for Design x under Scenario co

For a risk-neutral decision-maker, a SN design maximizing expected net revenues would be obtained by solving the following two-stage stochastic program with recourse:

fl = max ]£ xsy£7rs(û))Rdu(x,to)-]y jA,x, (8) " S=L,H ffiû! l<_L

subject to 2X=1 P*p W leL„ x l p<X l l e L , p e P , (10)

x, ,x l pe{0,l] l e L , p e P , (11)

Based on (3)-(6), the optimal value Rdu (x,to) of the second stage program for design x and scenario to is given by:

116

**■(*,«)= Z Z Z ([(«,-v,)^»]^-^[^W\-V(«)])-C(^) lmL PmP.^my^P,

(12)

with

Cf(x,û») = minZ lmL

subject to

Z wtyHr(û>)+cl%(û>)+ Z [<^Wv(«) + < yVW]

Z ^ * ( » K W H *€£„(«)

/e L,p€Pr(û;)n/}

/eL,/?eP ri»n^

*e/r, r((») peP.(m-)c.P, p*Pt(<i-)r.Pl

<a;r(û>) + z,rl» + Z è \ H P*l_[t-)rMl

leL

z,Aco)< ,̂a„ leL.(co); z,r(»<0, le^co)

y H. N» V M e i0»1}» z,.{<°)>h,pT i°>) integer keK,T(to), l e L ,

P eP T ( to )nP ,

(13)

(14)

(15)

(16)

(17)

(18)

In the first term of objective function (8) the expected net revenues are calculated and in the second term the depot fixed costs are subtracted to get expected profits. Constraints (9) in the first stage program enforce single depot assignments for ship-to-points and constraints (10) limit ship-to-point assignments to opened depots. For design x, under scenario CD, expression (12) estimates the anticipated net revenues based on allocation decisions, depots FTL shipment costs and other transportation and recourse costs obtained by solving the second stage program (13)-(18). The objective function (13) computes STL, MTL or LTL shipment costs, depot overcapacity costs and emergency supply costs. Constraints (14) are route coverage and also coupling relations ensuring that daily routes selection respects depots mission decisions for the second stage. Constraints (15) insure that FTL recourses are employed for day T only if ship-to-point p is assigned to depot /. Constraints (16) ensure that each depot / capacity is respected given the demand on day T for the assigned ship-to-points. Constraints (17) limit local recourse proportionally to the depot capacity.

117

4.33 Design Models Based on Approximate Anticipations The stochastic program (8)-(18) is intractable due to the infinite number of plausible

future scenarios and the extremely large number of possible transportation routes. Thus, approximate anticipations must be used to obtain solvable SN design models. To this end, based on the work of Klibi et al. (2009b), two approximate anticipations providing good quality-solvability trade-offs are considered in this paper. The first one is obtained through scenario and period sampling, and the second one through transportation decision aggregations transforming the stochastic LTP into a stochastic location-allocation problem.

To reduce the stochastic complexity of the problem, it can be reformulated using a random sample of scenarios generated with the MonteCarlo procedure presented in Figure 4-4. This type of Sample Average Approximation (SAA) method (Shapiro, 2003) has been successfully applied to solve several SN design problems (Santoso et a l , 2005; Vila et a l , 2007; Klibi et a l , 2009). Given our partition of the plausible future scenarios in two subsets £lL and Q." , the idea in our context is first to generate a large independent sample of M equiprobable scenarios ClM c Q using procedure MonteCarlo, and to partition it into the subset QML of ML low-hazard scenarios and the subset Q.M" of MH

high-hazard scenarios. An estimate of the probabilities nL and 7tH is then given by JiL = ML/M and WH = l - i r L . Second, a small sample QT' of mL scenarios is randomly selected in Q.ML and a small sample ÇÏ"" of mH scenarios is randomly selected in £lM" to get Qm = QG1 u Clm" . These sets of equiprobable scenarios can then be used to formulate a SAA model. Unfortunately, the multi-period SAA model obtained is still extremely difficult to solve with current solvers. Since the demand processes of the stochastic LTP are stationary, Klibi et a l (2009b) proposed to further simplify the problem by period sampling, i.e. by considering only a subset TczT" of daily periods (for example, one randomly selected day per week) with associated ship-to-point demands dp l(to),pe P.(to), toe Q.m,te f , and depot capacities a„(co), l e L , toe CT, te f . The net revenues must then be multiplied by the period shrinking factor \T"lIlTl to obtain an adequate approximation of the total expected profits.

By using the probabilities nL and 7tH in the objective function (8), the total expected profit is obtained, which is adequate for a risk-neutral decision-maker. However, if the decision-maker is risk-averse, these probabilities need to be replaced by weights nH > KH

and nL = 1 - ftH to give more importance to high-hazard scenarios (Klibi and Martel, 2009). For the SAA model, the value of these weights are based on the estimated probability WH . Also, since generating all possible routes yield extremely large models, the SAA model proposed is based on adequately generated (Klibi et a l , 2009b) subsets of

118

routes K,m(co)(z K,T(co), l e L , r e T u , toe £lm. Given these elements, program (8)-( 18) is transformed into the following approximate anticipation location-transportation model (Ml):

*M1 = m « ^ l £ I I I JL J[(v' lKW]v^[^Wv\.H])

-IA*.

p-P,(a>)nP,

(19) - I ^y.i,(^)-Gz„((v)- I [Gdp,(oy)eip,(cv) + ch,bFhlpl(co)]

k*k_{a) PmP,(m))nP,

subject to

Z a Z M,(*>)hM)+ Z *')£»** *€*„(<») psPriattnP, psPM^P,

<a„(a>) + zM+ Z ^ » t l f S ' ( 22 )

Z/r (<») ^ £>,» ' e A (»)»' Zf, (<») ^ 0, / e Lf (©) 0 D e Ç r , t e î (23)

, coeOTjef and to the location-allocation constraints (9)-(l 1).

Model Ml is much simpler than the original model but it is still difficult to solve when the set of plausible future scenarios £lm is large. Additional simplifications are possible when the transportation sub-problems are replaced by flow variables between depots and ship-to-points. When this is done, routing costs are replaced by unit flow costs w, between depots l e L and ship-to-points p e Pt. These costs are estimated by regression using daily historical data (Klibi et a l , 2009b). This introduces second-stage binary variable x,pl (to) which takes value 1 if ship-to-point p is served by depot / in period t under scenario CD, and 0 otherwise. Also, under scenario CD, the model needs a binary recourse variable e (to) to specify if products are supplied to ship-to-point p by the emergency source in period t. An average unit net revenue (u - v e ) , with ve =(ve +vh) 2, is associated to these flows. The SAA formulation of the stochastic location-allocation model thus obtained has the following form (M2):

119

I* I S=L,H m S fflen"' t - T

subject to

lm\ . ( 0 >) + ep<(U)) = l

leL.

(

vpe/>,(<»)n/V Z | Z \S} t p- v i - \ ) d 0^ a ' \ \ \A(o)-cfz ,X(o)

-ZA*.

p e P t ( c o ) , c o e i T , t e f

+ /eZ.

(25)

(26)

X ,P ,{°>)^X ,P

Z M®)** M- û * H + **(«»)

l e L , p e P l ( t o ) n P , , t o e Q . m , t e f (27)

l e L , c o e i T , t e f (28)

zu(<o)<Ç,a„ leL,(co); z„(to)<0, l e Ldt (co) coeQT, te f (29)

^ ( © ^ ( ^ ( û ^ O / e L , / ; e ^ ( ( y ) n ^ û > 6 a m , / € f (3°)

x lpAû))e{0,ï} l e L , p e P . ( i o ) n P , , t o e i T , t e T OD

and to the location-allocation constraints (9)-(l 1).

Since M2 is solved relatively easily, larger scenario samples can be used. Note that a gross anticipation could also be obtained by neglecting uncertainty, and by using the familiar capacitated location-allocation model found in the literature. This requires the

j r \

calculation of annual average demands Dp from the demand process parameters, and the estimation of average unit transportation costs w,p by regression (Klibi et a l , 2009b). The depots capacity can be set to IT"la,, l e L , which is clearly an overestimation. The following formulation results (M3):

^M3 = max Z Z {{UP ~v'~ K ) VPK " Z A*/ X leLpeP, leL

subject to Z DPXIP ^ \T"

pzp, l e L

(32)

(33)

and to location-allocation constraints (9)-(l 1).

4.4 Resilience Strategy Formulations As mentioned, due to demand randomness and hazards, SN user operations can be

perturbed and response actions must be tailored to occurring events to maintain business continuity. However, response policies are rarely anticipated in SN design models. The stochastic programming models proposed in the previous section anticipate response

120

policies through the use of the second stage variables yWr(û>), z;r(<w), ^ p r (^) and h,pT(cD) . Due to the costs associated to these variables, the models position the depots and specify their mission to avoid risks as much as possible. Moreover, by considering the risk attitude of the decision-maker, the models can be more or less drastic in their effort to avoid hazards. For these reasons, models Ml and M2 are appropriate mainly when pursuing a Risk Avoidance (RA) strategy. A discussion of risk avoidance in supply chain management is found in Manuj and Mentzer (2008).

Despite efforts to avoid risks as much as possible, it is clear that the SN designed will be hit by occasional disruptions. Given this, the question then becomes: what kind of risk mitigation constructs could be incorporated in the design models to obtain more robust SN designs? In other words, how can we design the network to make sure that it will bounce back quickly when hit? This can be done by investing in flexible and redundant network structures before they are needed (Sheffi, 2005), and by providing better resilience policy instructions. This is the domain of resilience strategies. A main concern is to design robust SN not only to hedge against major disruptions but also to improve business-as-usual operations. Despite the growing number of papers on the importance of the resilience concept, little has been done to incorporate it in SN design models. However, several modeling constructs found in the literature on coverage (Church and Revelle, 1974), vector assignment (Weaver and Church, 1985) and reliability (Snyder et a l , 2006; Tomlin, 2006; Murray and Grubesic, 2007) models can help to foster resilience.

Our aim in this section is to propose three LTP network design models to improve resilience when a single sourcing policy is applied. These models distinguish themselves from Ml and M2 by the fact that they strive to provide either explicit or implicit instructions on the back-up depot l ' (p) to use when applying the response procedure described in Figure 4-7. The first model finds the optimal primary and back-up depots to use for each ship-to-point. The second model allows multiple sourcing for each ship-to-point in order to reduce the risk of disruption when the primary depot is hit. The idea behind the third model is to offer a better network coverage by ensuring that at least two depots are geographically located within desired distances of each ship-to-point. Figure 4-8 illustrates the three resilience strategy formulations proposed. The next subsections extent models Ml and M2 for each of these formulations.

a) Optimal back-up b) Multiple sourcing c) Coverage

Primary ( □ ;

Figure 4-8. Resilience Strategy Formulations

121

4.4.1 Optimal Back-up Formulations (RI) The aim of these formulations is to specify the backup depot to use when the primary

depot assigned to a ship-to-point cannot supply its orders (Figure 4-8a). The formulations proposed are inspired by the work of Weaver and Church (1985) and Snyder et a l , (2006) on the vector assignment problem and on the reliable fixed charge location problem, respectively. Binary variables are defined to specify primary allocation decisions x\ and backup allocation decisions xf . The variables xf, r-1,2, take the value 1 if ship-to-point p is allocated to depot / as level r (primary or back-up) supply facility, and 0 otherwise. In the models, these mission specification variables are first stage variables and thus they remain the same for all the periods of the planning horizon. However, for each period t of scenario toe Q.m, a second stage binary variable xlpt(cv) needs to be introduced: it takes the value 1 if ship-to-point p is supplied by depot / in period t under scenario to, and 0 otherwise. As before, an external emergency source takes over when both the primary and backup depots are unable to supply a given ship-to-point. This leads to the transformation of model Ml into the following SAA model (Ml-Rl):

A i n v- n< /?„,„,= max - p - Z m -\T\ rff jç III

œ n " - i-f I-I, PmP,(m))nP,

I J [K- V /K(Û»)]^M-^[CM^N-VN])

IA*/

subject to

S4=i leL.

Z*i-*/ r=l,2

- I wtyu[tu)-cfzk{co)- I [c'dt,(tv)e¥(tv)-cfbFf\,(0)] kek^to) p-P,(<D)nP,

p e P , r = l,2

p e P,,le L

(34)

(35)

(36)

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r=l,2

K,(G>)<y™(co)x,pl(c>)

l e L ^ e P ^ o ^ c ^ P ^ t o e Ç r ^ e î

le L,pe P l(to)nP,,toeÇlm,tef

Z « Z V » ^ » ) + Z bFyPJ'(<»K(o>) kmKl,(m.) PmPXrO)^ Pm^OJ^P,

Z ôkPykiAû>)+eiPAû>) = xiPA0)) *»(«•)

Z-vN=1

and to constraints (23)-(24).

/€ L,toeiY <a,,(co) + zu((o)+ Z * % » ' € Î

p.P,(ù))nPi

l e L , p e P l(co)r^P,,CDeiT,tef kmKAm)

peP.(to), œ e i T , t e f

l e L , p e P,(û))nP,,toeQ.m , tef r = l,2

(37)

(38)

(39)

(40)

(41)

(42)

Constraints (35) and (36) ensure that each ship-to point has distinct primary and backup depots. Constraints (37) guarantee that shipments are made only from primary or backup depots. Constraints (41) guarantee that the single sourcing rule is respected for every period.

By proceeding similarly, M2 can be transformed into the following backup optimization model (M2-R1):

*M2-R1 = m a X -rf? L, \T\ st?Hms

ZZ tU£imm' l e f I zLype Pu(a>)

+ PmP, (lO)

-ZA*, (43)

I^I

subject to constraints (23), (26)-(28), (30), (35)-(37) and (42).

Note that these models include a large number of binary variables, which complicates their resolution. The instruction provided by the models to the response procedure is the following: l (p) = l\x]p = l and l'(p) = l\xfp = 1 , p e P .

A A.l Multiple Sourcing Formulations (R2)

The idea behind these formulations is to allow multiple sourcing for each ship-to-point (Figure 4-8b) to identify natural supply depots backups, as opposed to forced backups as

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in the previous section. For formulation Ml, this is done essentially by removing constraints (9) from the first-stage program, and by introducing second-stage assignment variables x,pl (to) with value 1 when depot / supplies ship-to-point p in period t under scenario CO. The following model results (M1-R2):

4..Rj = max ^ 1 ^ I H I ' I S=L,H

mS M = Q - re f leL

I ([K-vl)<'»]x,(ffl)-^[C(«)v(»)-VMl p*rH»)m\

- I "W» (•)-«?«•(•)- I [c.d„(eo)e¥((o)+cfb%((k>)] k.K_{a) peP,{t-)nP,

(44)

IA*J

subject to constraints (10)-(11), (23)-(24), (27), (31) and (38)-(41).

When this model is solved, several depots may be used to serve a given ship-to-point during the planning horizon. The primary depot for a ship-to-point is then defined as the one shipping the largest quantity of products to this point during the planning horizon. The depot with the second largest shipment quantity is specified as the backup depot. More specifically, we define:

l (p) = m_igmax-_^]T^d (to)x (to) m d 1 ' ( P ) = ar8, FH^,-ZZM®)** H l e L m „ tmL.\\i(P)\ m „

Note that the backup depot is specified only for ship-to-points supplied by more than one depot during the planning horizon.

For formulation M2, the second-stage program is defined in terms of continuous flow variables instead of binary assignment variables, namely: x, (cv) the quantity of product supplied by depot / to ship-to point p in period t under scenario to, and epl(cv) the quantity supplied to p by the emergency supply source. This leads to the following model (M2-R2):

Z Z [«,-Vi-tfJjfc(»)-<fo(») / W =max -Zf- Z — j Z Z l i l S=l.,H ^ fflgfl"' Imt \peP,[m)nP, (45)

subject to

mZxipAû>) + eP , M = d

PAû>) l e L

■vM^M*/,

+{

uP~

v e) Z

eM )

p-P,(a>) -ZA*,

lmL

peP,( to) , te f ,coeQ.m (46)

l e L , p e P . ( t o ) n P , , t e f , (47)

œe QT

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z ^w^w^w pef*.*)

V (û>)>0

l e L , t e f , t o e Ç l m (48)

/ eL ,pe /? (u ; )n /> , fe7 \ f t>eQ m (49)

and to constraints (10)-(11) and (29)-(30).

The instruction provided by the model in this case is:

l(p) = argmax-ZZ^M ^ ''(p) = arg ^ r X Z ^ »

4.4.3 Coverage Formulations (i?J> As in classical covering problems (Church and Revelle, 1974), the idea behind the

third formulation is to offer better network coverage by using proximity criteria specifying maximum primary/backup depot to ship-to-point distances (Figure 4-8c). Let Ûp cz L be the set of depots located within the backup distance specified for ship-to-point p, and Èp c L2

p be the set of depots located within the primary distance specified. This formulation imposes that for each ship-to-point p e P , at least one depot is in Lp and at least 2 depots are in l i . This tends to increase the number of opened depots and to spread them more evenly on the territory. This leads to the transformation of model Ml into the following SAA model (M1-R3):

£M,.R3=™x ^pl-f-i I I I l i I S=L,H r n S aeST* l e t leL pePt(a>)r.P,

IA*I

(50)

subject to

Z ^ 1 fel'

Z ^ 2

kmk_(a>) PmP,(Cm)l^P,

p e P

p e P

(51)

(52) fell

and to constraints (9)-(l 1) and (20)-(24).

It also leads to the transformation of model M2 into the following model (M2-R3):

f, \TU\ ^ # , R m RJ = m a x — ^ /__, zz Z Z [(up-vi-K)dAa)_\xiA<°)-cizu.(0)

lmL\PmP t ( tO)r \P l

+(«P-^) Z dMeA<°) p_P,(c.)

—ZA*. (53)

i - t .

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subject to constraints (9)-(l 1), (26)-(31) and (51)-(52).

These models are easier to solve than the previous ones. The instruction they provide to the response level is l (p) = l x,p =1 . l '(p) is not specified and the default backup policy is used.

4.5 SN Design Models Solution and Evaluation Approach The approach used to solve the SAA models formulated previously, and to evaluate

their performance in terms of value creation and robustness, is based on the SN design methodology proposed by Klibi and Martel (2009). As illustrated in Figure 4-9, the approach involves three phases: scenarios generation, design generation and design evaluation. It can be seen as an extension of the SAA method used to solve stochastic programs (Shapiro, 2003). The scenario generation is done using procedure MonteCarlo presented in Figure 4-4, and it provides scenarios to the design generation and evaluation phases. A large sample of scenarios £lM is generated and partitioned into low-hazard scenarios ÇlM' and high-hazard scenarios Q.M", based on the hazard tolerance level K. From these samples, two subsets of scenarios are randomly selected to perform the designs evaluation: a subset ÇlMt c QM' of M [ low-hazard scenarios, and a subset £lM" c Q.M" of Me

H high-hazard scenarios. A subset QMw c Q.M" of Mew worst-case scenarios is also

selected in the tail of the distribution of the number of hits (Figure 4-5).

The design generation phase involves the solution of the SAA models. As normally done with the SAA method, each model is used to generate several designs using N replications of small scenario samples. To do this, another large sample of scenarios SlM

is independently generated and partitioned into low and high hazard subsets QM L and £lM " . From these samples, the probabilities JtL and ifH aie estimated, and N replications of small scenario samples Q.fGQ.f", i = l,...,N,aie randomly selected to construct the SAA models. Based on 7tL and 7iH the risk aversion weights ftL and 7tH are also specified. The MIPs obtained are then solved for each sample replication, using a commercial solver such as CPLEX-11 or a specialized solution algorithm. Given the different models proposed and the N replications made, the design generation phase produces a set of alternative SN designs xJ ,7 = 1,..., J . Each SN design obtained specifies the set of depots L to open and their primary mission l ( p ) , p e P . For some models, an instruction l ' ( p ) , p e P , on the backup depot to use is also provided. When no backup depot is specified, the default reassignment policy applies.

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Risk-aversion weights

Design Generation (usingCPLEX-11)

Locat ion-Transportation SAA Models

Location-Allocation SAA Models

X -<-Multi-criteria

evaluation

Efficient & Resilient Design

Small sample replications

ÏÏsGiX\i = \,...,N),S = L,H

X , J — 1,..., J ' . ' • Depot locations • Primary assignments • Backup instructions

Plausible Future Scenario Generation

Design Evaluation

• User response procedure

• Performance measures

Evaluation samples

Hazard tolerance level (K)

QM'L QM'„ Ç.M.

Figure 4-9. SN Design Models Solution and Evaluation Approach

In the evaluation phase, the alternative designs x J , j = l,...,J are compared using the plausible future scenario sample Q.M =Q.M L uf l M " u£lMw . Since the models proposed incorporate approximate anticipations of the user response, it is not adequate to use their objective function to determine the best design. Moreover, these objective functions do not cover all the value creation and robustness dimensions that decision-makers may want to explore. For these reasons, the evaluation of the SN designs is based on a set of measures related to the net revenues R" (xJ,to), toe Q.M , j = I,..., J , provided by the UserResponse procedure. More specifically, for a given design x J , these performance measures are based on the value added during the planning horizon under the scenarios considered, that is:

R(xj,co) =RU (x j , to) - Z A (•*/ X 0)e a M ' ( 5 4 )

1-1.

An adequate SN design evaluation must be based on expected value and robustness measures, and it must take the decision-makers risk attitude into account. The expected return R(xJ) of a design xJ is provided by:

tf(x7)=Zs=i>^A(x7); R s ^ ) = - ^ 7 l L ^ s k ^ \ û > ) , S = L,H (55)

where RL(xJ) and RH(x') are conditional expected returns for low and high hazard scenarios, respectively. Robustness is related to the variability of the returns obtained under different scenarios. Since downside deviations from mean returns are undesirable, an

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adequate variability measure to assess a design x ; is the mean-semideviation MSD(xj) given by:

MSD(x i)=^sLHïïsMSDs(x i)

M S D s ( x J ) = ^ ^ ^ s m a x [ ( R s ( x n - k x \ t o ) ) - , 0 l s = L f H (56)

Ms

where MSDL(xJ) and MSDH(xJ) are conditional mean-semideviations for low and high hazard scenarios, respectively. The mean-semideviation is a coherent risk measure (Shapiro, 2007). Decision-makers are also interested by the behaviour designs under extreme conditions. Using worst-case scenarios, this is often evaluated with the absolute robustness criteria proposed by Kouvelis and Yu (1997). For design xJ this measures the minimum return Rw(xJ) under all worst case scenarios, calculated as follows:

Rw(x j)= min {â(x;,co)} ( 5 7 )

Measures (55)-(57) provide the basis for a multi-criteria evaluation of the designs considered. Note that these measures can be used to compare the designs provided by the N replications of a given SAA model, as well as the best designs obtained from the different SAA models.

The previous performance measures can also be used to construct a compound return measure reflecting the decision-makers aversion to variability and to extreme events. Such a measure is provided by the following expression:

ft = 0-V) Z *,(%(x')+?,Mffl),(x0) + r5r<*') (58) S=L,H

where <ps e [0,1], S = L,H, aie variability aversion weights for low and high hazard scenarios, and where y/e [0,1] is an extreme event aversion weight. Note that if we set 9L ~ PH = ¥ ~ 0, the return function obtained corresponds to the objective function of the SAA models used in the design generation phase.

4.6 Computational Results This section presents the experiments made to compare the models proposed in the

previous sections and it analyse the results obtained. In addition to the eight SAA formulations elaborated, two additional models where included to provide conventional SN design approaches benchmarks. One of them is the deterministic model M3 which completely neglects uncertainty. The second one is a variant of Ml neglecting hazards: it's

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a SAA model considering only the randomness in demand and denoted by Ml-D. Also, for the eight SAA formulations considering hazards, two types of decision-makers (DM) are considered: a risk neural DM and a risk averse DM. Consequently, in what follows, 18 alternative SN design models are compared.

4.6.1 Plan of experiments In order to test the SN design models proposed, several problem instances were

generated based on the following four dimensions: the SN breadth, the cost structure, the demand characteristics and the depots size. First, two problem instances are considered with different number of potential depots and ship-to points scattered over different geographical areas, as specified in Table 4-1. The distances between the network nodes are based on existing road networks and they are calculated with PC*MILER (www.alk.com). A one year planning horizon including IT" =240 working days is used. A next day delivery service requirement is implemented through a 400 miles limit on the distance between depots and ship-to points. Exceptionally, when the number of incident lanes of a given ship-to-point is less than three, depots with distances larger than 400 miles are also considered.

Problem instance

Geographical Area Potential depots

Number of ship-to-points

Pi North-Eastern US States 7 206 P i North-Eastern & Midwest US States 15 706

Table 4-1. Test Problems Instances

Based on the cost structure of a real case, high level fixed costs (a) and low level fixed costs (b) are defined. The fixed cost for each depot A, is randomly generated in the intervals given in Table 4-2. Also, the unit product value, v,, aie selected randomly in [19,21] and the products price, up, aie fixed to 23 for all ship-to-points. The values vh and ve of the products coming from the external supplier are fixed to 25 and 24, respectively, and the local capacity unit recourse cost cf is fixed to 1 for all the depots.

M'W-Up High product value/price

High fixed costs (a):[180tf,200/n;[19,21];23

Low fixed costs (b):[60tf,80/n;[19,21];23

Table 4-2. Test Problems Cost Structure

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To adequately capture the ship-to points demand variability, daily orders are modelled as a compound stationary stochastic process. We assume that the distribution Fq(.) of the inter-arrival times is exponential with an expected time between orders 77 . Also a log-normal distribution with a mean p p and standard deviation ap is used for the order quantity distribution Fp(.). Three ship-to-points size (Large, Medium and Small) are defined to generate two types of network: larger ship-to-points networks (LAO dominated by large and medium size customers and smaller ship-to-points networks (SN) dominated by small customers. The proportion of ship-to-points of different size in each network type is given in Table 4-3. The table also provides the probability distribution parameters used to generate orders for each ship-to-point size. For model M3 the annual average demand of ship-to-points is given by Dp = \T"\jup/7]p .

Ship-to-point size: Large Medium Small Larger ship-to-points Network (LN) 15% 65% 20% Smaller ship-to-points Network (SN) 10% 30% 60%

p: (cwt) [480,580] [300,400] [120,220] <x(%//) 7% 10% 16% TJ (days) [2.5,4.5] [5.5,15.5] [20.5,35.5]

Table 4-3. Ship-to-Point Demand Structure

Finally, problems with large capacity depots (LD) and tight capacity depots (TD) are tested. Each depot / has a capacity level given by a, = vl.p.ppp I T}p, where v is a factor randomly generated in the intervals, depending on the network structure, given in Table 4-4. The additional capacity, available for local recourse, is fixed to 25 % of the regular capacity level. Note also that all the vehicles capacity (bF and bk ,ke K ) are fixed to 400 cwt.

Capacity factor v range LN SN

Tight depots structure (TD) [0.75; 1] [1/1.25]

Large depots structure (LD) [1.25; 1.5]

Table 4-4. Test Problems Depots Capacity Structure

The combination of these four dimensions yields 16 problem instances. Each instance is denoted as follows: ( i , j ,k, l) ; ie{PvP2},je{a,b], ke{LN,SN], le{TD,LD).

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4.6.2 SAA Models Calibration The solution and evaluation approach presented in Figure 4-9 was implemented in

VB.Net 2005, and the experiments reported in this section were performed on a 64-bit server with a 2.5 GHz Intel XEON processor and 16 GB of RAM. All the models were generated with OPL Studio 6.1 and solved with CPLEX-11. This section discusses the calibration of SN design model's parameters, taking into account the solvability of the MIPs obtained. All the design models were solved to optimality with the following CPLEX parameters: MIP Emphasis = Optimality, Aggressive Cuts strategy, MIP Relative Tolerance = 0.005 and Time Limit = 10 hours.

Recall that the SAA models are formulated using a subset of m = mL+mH scenarios, a subset of planning periods T, as well as route subsets K,T (to), l e L , TeT", coe £lm, for the location-transportation models based on Ml. The size of these subsets is therefore an important issue and, clearly, the models become much more difficult to solve as the size increases. Solvability is consequently a major concern, and the subsets were selected as large as possible to remain solvable with CPLEX-11. For all the instances solved, one day per week was sampled in the planning horizon yielding IT I = 48. For the models based on Ml, using the set of all routes and a sample of 10 scenarios (with mL = mH = 5) produced SAA design models that could be solved to optimality for Pi. For P^, the largest SAA models we were able to solve to optimality were obtained with 6 scenarios samples (with mL=mH =3) using a reduced subset of non-dominated routes. Based on preliminary results, the total number of routes was limited to 1 500 000, including all routes with at most one drop and a subset of interesting routes with at least two drops. Note that since the demand and hazard processes are stationary, and since the planning horizon includes 48 days, for each scenario the user response sub-model is replicated 48 times for demands and hazards generated from the same processes. For this reason, what appears here to be a very small number of scenarios gives reasonable results. Also, for each SAA model, N = 4 replications were solved with different scenario samples. These subsets size and replication parameters were used for models Ml, Ml-Rl, M1-R2, M1-R3 and Ml-D. The absence of binary route selection variables in the location-allocation SAA models based on M2 makes them more tractable, and the number of scenarios can be significantly increased to improve the statistical optimality gap. After several preliminary tests on various sample size, m was fixed to 50 for Pi and to 30 for P2 (with m L - m H = m/2) . This sample size was applied to models M2, M2-R1, M2-R2 and M2-R3, with if I = 48 and N = 4.

Samples of M= 1000 scenarios were generated for each problem instance to obtain the smaller design generation and evaluation scenario samples, and to estimate the probability

131

7iL and ïïH of low and high hazard scenarios. Histograms, for one of these samples for each problem instance, are represented in Figure 4-5. These samples and probabilities were used to calibrate the hazard tolerance level K and the weight of high-hazards scenarios ÈH for risk averse decision-makers. The values obtained/retained for Pi and P2 are given in Table 4-5. The table also provides the weights applied to evaluate the designs with (58), for a risk neutral and a risk averse decision-maker.

Risk Neutral DM Risk Averse DM K *a <PL a n d <PH ¥ K _%

<PL a n d <PH ¥ Pl 2 0.28

0 0 2 0.35

0.2 0.2 Pl 3 0.38 0 0 3 0.45 0.2 0.2

Table 4-5. Decision-Maker Risk Attitude Parameters

The models based on M2 and M3 require transportation cost approximation functions estimated by regression. Using a representative sample of daily routes generated with procedure UserResponse, a linear regression function w,p =Ç0+ çxm,p was estimated for each model type. The regression parameter values obtained for Pi and P2 are provided in Table 4-6. In addition, for models M1-R3 and M2-R3, we need to fix a priori the cover-radius to use to define the sets Êp and Ê . Several values were tested and the inner and outer radiuses retained for Pi were 200 and 400 miles respectively. For P2, the values used were 300 and 400 miles.

Pi P i M2 M3 M2 M3

£0 0,1091 0,1009 0,2714 0,1009

£1 0,0057 0,0044 0,0058 0.005

Table 4-6. Regression Parameters by Problem Instance for M2 and M3

The hazard model parameters also needed to be estimated. The US states in the regions covered by problems Pi and P2 were used as hazard zones. The exposure levels and the multihazard arrival process for each zone (state) was estimated from historical data on major disasters provided by FEMA (www.fema.gov). We assume that the inter-arrival time distribution F.x(.) is exponential, and the mean inter-arrival times A estimated for each state z, from the FEMA data, are provided in Table 4-7. The table also gives the state exposure level g(z) estimated on a scale from 1 to 4. We assume that the impact intensity distribution Ffz)s(.) is Uniform for all z and s. For depots it is expressed in terms of capacity loss with parameters [0,0.25), [0.25,0.5), [0.5,0.75), [0.75,1] for exposure levels

132

g= 1,2,3,4, respectively. For ship-to-points it is expressed in terms of demand surge or reductions with amplitude parameters [0,0.1), [0.1,0.2), [0.2,0.3), [0.3,0.4] for exposure levels g =1,2,3,4, respectively. The sign of the amplitude was randomly selected to get 50% surge proportion. For all nodes, the attenuation probability an is randomly generated in an interval [0.1,0.2), [0.2,0.3), [0.3,0.4) or [0.4,0.5] depending on the relative area of zone z(n). The impact duration functions (Figure 4-2) used are 9, = 0.001 fif +0.4709$ +eL for depots (sL) and 9p =0.8419^ +e p for ship-to points (s p ) . The recovery functions used are similar to the ones illustrated in Figure 4-3, with a stagnation phase of [0.250,.] periods for depots, and instantaneous deployment and recovery phases and a sustainment phase of 9 periods for ship-to-points.

State (z) VT DE DC MA NY NJ WV KY OH IN Az (in days) 537 430 567 578 293 609 391 358 344 466

8(z) 4 4 4 4 4 4 4 4 4 3 State (z) ME NH RI CT PA VA IL MD Wl MI Az (in days) 405 577 757 703 355 340 371 607 412 611

M 3 3 3 3 3 3 3 2 2 1

Table 4-7. Multi-Hazard Exposure Levels and Mean Inter-Arrival Times

4.6.3 Numerical Results Given the 16 problem instances defined previously, this section discusses the

solvability of the SN design models proposed, and it studies the quality of the design they provide using performance measures (55)-(58). For each formulation, Table 4-8 shows the design models characteristics for problem instances (-,a,LN,TD), the mean solution time (MST in seconds) and the solution time standard deviation (STSD in seconds) for P\ and P2. Models based on Ml have a very tight LP relaxation which helps reduce solution times significantly even with the large number of binary variables (> 1 000 000) involved. The solution times are smaller for P2 than Pi for Ml-Rl and M1-R2 because a smaller number of scenarios and routes are used. Even if the models based on M2 include a mix of binary and continuous variables, their solution times are sometimes longer than for the corresponding Ml-model because the scenarios sample used is five times larger. The deterministic model M3 is trivial to solve. Note that there is a lot of variability in solution times from one instance to another. This is explained partly by the fact that CPLEX-11 incorporates a number of heuristics to reduce solution times, and that these heuristics do not work as well on all problem structures and instances.

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Ml Ml-Rl M1-R2 M1R3 M2 M2-R1 M2-R2 M2-R3 MID M3

Pl

Variables 1060375 1429062 1269822 1304038 416031 397725 393549 416031 1325159 1453

Pl Constraints 128811 207914 199431 128006 433030 413690 459742 433442 125708 1658 Pl MST is. 786 46% 4448 241 5093 1852 5620 6288 437 <1

Pl

STSD(s) 1224 8870 4722 127 8259 2099 3201 8057 631 0

P2

Variables 1801570 1952566 1947790 1801570 883766 888542 883766 883766 1796446 10609

P2 Constraints 276312 444865 443453 277724 906057 906763 905351 907469 271238 11314 P2 MST (s) 2108 2509 3102 612 1297 939 5200 343 1463 <1

P2

STSD(s) 3943 2366 3357 643 1108 1075 3561 209 1283 0

Table 4-8. Model Characteristics and Average Solution Times for Pi and Pi

An important issue examined in this paper is the impact of the precision of the user response anticipation incorporated in the design model on the quality of the designs obtained. The location-transportation formulations (Ml-models) proposed include a relatively precise anticipation, but the location-allocation formulations (M2-models) included a more approximate anticipation. Our results show that the two formulations never produce the same design, although in most cases, the location decisions are similar. The difference comes mainly from the assignment of ship-to-points to depots. Ml-models provide different designs for each scenario sample replications which is to be expected because the sample size is relatively small. M2-models often produce at least two similar designs among the four replications solved which, again, is normal since larger scenario samples are used. For the smaller problem Pi, M2 performs extremely well since it gives the best design half of the time. This is explained by the fact that we were able to use large samples of scenarios in this case (50 scenarios, comparatively to 10 for Ml-models). However, for P2, the scenario sample size had to be reduced and Ml-models almost always perform better than M2-models. For some instances, the difference in expected design value reaches 7%. This shows that significant gains may be made by using more precise anticipations and larger scenario samples. A more detailed study of the impact of anticipations in SN design is found in Klibi et al. (2009b).

We stressed earlier that our models try to avoid risky depot locations as much as possible. Let y(xJ,to) be the total number of hits on depots under scenario to when design xJ is implemented. For a given problem instance, the average number of hits on design x ;

for the scenarios evaluated is given by y(x') = Z „"' Yfr',CD) l£lM I. Table 4-9 reports the mean of these values by problem type ( yp (x) and yp̂ (x) ) for the optimal design provided by each model. The lowest average number of hits is obtained with Ml, which is congruent with the fact that Ml is a risk avoidance model. The classical deterministic and stochastic models M3 and Ml-D also perform very well from this point of view. This indicates that for the regions of the USA covered by Pi and P2, the optimal solutions of M3 and Ml-D provide a natural cover against hazards. This would certainly not be the case in general. Note that the designs provided by the resilience-seeking models are hit more

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often. This is happening, because there is a demand surge for some ship-to-points when they are hit. Resilient networks can then make additional profits by providing a good support to the victims of extreme events. Finally, it can be seen that M2-based models yield a higher average number of hits than Ml-based models.

Ml Ml-Rl M1-R2 M1-R3 M2 M2-R1 M2-R2 M2-R3 Ml-D M3 7PM) 0.89 1.19 0.90 1.33 1.12 1.31 1.24 1.35 0.90 0.94

n-2(x) 1.62 2.19 1.72 2.14 2.18 2.44 2.41 2.44 1.65 1.80

Table 4-9. Average Number of Depot Hits by Scenario

The behaviour of the designs when hit can be analysed further by examining the average value of a design (calculated from R(xJ,to), toeÇlM ) for scenarios including y =0,1,2,... hits. Figure 4-10 presents a value-hit graph of non-dominated risk-mitigation formulations (RA, RI, R2 and R3) for (-,a,LN,TD) problem instances. In terms of value creation under a given number of hits, RA, RI, M3 and Ml-D models are dominated either by R2 or R3 formulations. For low-hazard scenarios, R2-models tend to give better designs than R3-models. However, for high-hazard scenarios, R3-models create more value. A similar behaviour is observed for other problem instances. Note however that RA-

models are often almost as good as R2-models.

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Robust SN designs should also exhibit low value added mean-semideviations (MSD). Figure 4-11 provides value-MSD tradeoffs graphs for Pi and P2. The points shown on the graphs for a given risk-mitigation formulation are average values over all the models solved. The graphs show that from a value variability point-of-view, RI-model s are very conservative: they provide lower value but with lower variability. At the other extreme, RA, R2, M3 and Ml-D models are more aggressive: they provide more value but with more variability. R3-models provide a compromise between these two extremes. Note

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however that the MSD is relatively low, in comparison with expected values, for all models. In other words, for the cases considered, variability does not stand out as discriminating factor.

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All our models were solved with risk-neutral DM weights and risk-averse DM weights (see Table 4-5). Table 4-10 presents the percentage of similar design decisions obtained with each risk attitude weights. The table shows that sensitivity to risk attitude depends very much on the model. Rl-models are much more sensitive to risk attitude than the other models. In general, however, it is clear that the weights selected have an impact on the SN design obtained.

Ml Ml-Rl M1-R2 M1-R3 M2 M2-R1 M2-R2 M2-R3 P. 69% 0% 56% 44% 63% 0% 41% 59% P2 19% 0% 19% 22% 88% 3% 47% 81%

Table 4-10. Similarity between Design Decisions for DM Types

Table 4-11 provides more detailed results on the performance of the models for the sixteen problem instances generated. The results are expressed in terms of %-deviation from the return of the best design obtained. When the models are solved with risk-neutral DM weights, the return is evaluated with performance measure (55). When the models are solved with risk-averse DM weights, the results are compared using compound return measure (58), with the weights in Table 4-5. Note that evaluations using higher weights were also made, but it did not have a significant impact on the models ranking. For the SAA models, the table provides the %-deviation of the best design obtained with the 4 scenario sample replications generated. The best design value for each problem instance is highlighted.

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P l a,LN,TD a,LN.LD b.LN.TD b.LN.LD a,SN,TD a.SN.LD b.SN.TD h.SN.LD All Expected Return ( / ? ) for Risk Neutral Models

M l -2,25% -0,94% -1,06% -1,87% -0,30% -0,68% -1,10% -0,17% -0,38% Ml-Rl -12,66% -21,74% -8,14% -8,30% -20,16% -24,84% -11,31% -12,04% -12,86% M1-R2 0,00% -0,94% -1,42% -1,90% -0,14% -0,72% 0,00% 0,00% 0,00% M1-R3 -4,51% -4,47% -3,21% -0,56% -8,72% -9,35% -4,73% -2,74% -3,25% M2 -2,69% -3,05% -1,86% -2,07% 0,00% -4,39% -3,88% -1,43% M2-R1 -15,61% -14,29%l -8,68% -8,67% -19,95% -30,23% -19,48% -19,57% -14,27% M2-R2 -1,39% -1,07%| 0 ,00* 0,00% -0,48% -2,29% -2,89% -0,05% M2-R3 -4,43% -3,26% -2,92% -1,86% -8,57% -8,37% -4,63% -3,88% -3,27% Ml-D -4,25% -1,05% -0,92% -1,89% -2,44% -0,82% -0,96% -0,53% -0,85% M3 -4,31% -1,91% -0,96% -1,87% -2,74% -0,73% -0,30% -0,47% -0,93%

Compound Return Measure CR.) for Risk Averse Models M l -2,18% -4,25% -0,18% -5,47% -3,08% -4,86% -1,31% -0,23% -2,04% Ml-Rl -11,69% -24,77% -8,41% -7,71% -15,65% -24,39% -11,33% -14,23% -13,02% M1-R2 0,00% -4,25% -5,51% -0,13% -4,89% U,UU?b -1,33% M1-R3 -6,17% -6,26% -4,24% -3,52% -11,10% -12,54% -4,97% -3,47% -5,14% M2 -3,62% -3,05% -2,07% -2,42% -3,12% -1,83% -4,41% -4,93% -2,36% M2-R1 -11,24% -11,20% -8,30% -9,67% -21,01% -30,96% -15,85% -15,10% -12,80% M2-R2 -1,26% 0,00% -0,17% 0,00% 0,00% -2,09% -3,12% 0,00% M2-R3 -6,04% -3,26% -2,15% -3,57% -10,73% -9,25% -6,23% -4,93% -4,29% Ml-D -10,85% -4,36% -0,19% -5,50% -9,35% -5,04% -0,97% -0,85% -3,82% M3 -10,96% -4,26% -0,24% -5,47% -9,79% -4,92% -0,33% -0,76% -3,78%

P i a.LN.TD a.LN.LD b.LN.TD b.LN.LD a.SN.TD a,SN,LD b.SN.TD b.SN.LD All

Expected Return ( R ) for Risk Neutral Models M l -0,12% -0,09% -0,28% 0,00% -0,70% -0,11% -0,36% -0,04% -0,10% Ml-Rl -19,35% -21,40% -14,55% -14,91% -25,78% -23,14% -19,12% -17,44% -18,73% M1-R2 0,00% 0,00% -0,37% -0,14% 0,00% 0,00% 0,00% 0,00% 0,00% M1-R3 -2,22% -2,03% -0,60% -6,33% -4,76% -1,76% -0,70% -1,85% M2 -2,39% -1,90% -0,70% -1,20% -2,05% -0,14% -1,74% -0,51% -1,27% M2-R1 -25,19% -25,58% -19,92% -18,31% -28,56% -25,52% -26,97% -23,22% -23,44% M2-R2 -1,93% -1,83% -0,70% -1,19% -7,09% -4,27% -1,65% -1,11% -2,05% M2-R3 -3,61% -3,04% -0,67% -1,19% -7,69% -4,73% -2,01% -0,94% -2,58% Ml-D -0,20% -0,08% -0,23% -0,03% -0,68% -0,07% -0,87% -0,03% -0,15% M3 -0,82% -0,08% -0,27% -0,62% -1,95% -0,08% -1,41% -0,33% -0,54%

Compound Return Measure (TZ) for Risk Averse Models M l -0,16% -0,43% -0,69% -0,15% 0,00% -0,18% -0,47% -0,18% -0,18% Ml-Rl -17,42% -20,76% -14,74% -13,13% -25,70% -26,00% -16,79% -18,47% -18,18% M1-R2 0,00% 0,00% -0,41% 0,00% -0,68% 0,00% 0,00% 0,00% 0,00% M1-R3 -1,87% -2,20% -0,74% -5,23% -4,71% -1,50% -0,84% -1,69% M2 -1,48% -2,10% -0,85% -1,21% -0,87% -0,09% -2,18% -0,62% -1,10% M2-R1 -25,03% -25,70% -19,37% -17,89% -26,59% -24,09% -24,05% -22,86% -22,62% M2-R2 -0,63% -2,00% -0,90% -1,21% -6,12% -3,55% -2,06% -0,62% -1,73% M2-R3 -2,29% -3,34% -0,64% -1,21% -6,56% -4,52% -2,31% -1,06% -2,31% Ml-D -0,31% -0,40% -0,60% -0,19% -0,69% -0,12% -1,72% -0,17% -0,36% M3 -0,72% -0,38% -0,60% -1,01% -2,57% -0,10% -2,24% -0,54% -0,80%

Table 4-11. Models Performance in Terms of Deviation from the Best Design

Several observations can be drawn from this table. Note first that the different models never give the same SN design. Model M1-R2 performs extremely well for larger problems (P2): it usually provides the best design and, when it does not, it is very close to the best. For smaller problems (Pi), when it is not the best, it is usually dominated by M2-R2. This confirms that risk-mitigation formulation R2 outperforms other formulations. Our results also show clearly that RI does not perform very well. Although this risk-mitigation approach is conceptually appealing (Snyder et a l , 2006), it is too conservative and it

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provides poor returns. Since R3 tends to open more depots, it does not provide the best expected returns but, as shown in Figure 4-10, it may remain attractive to decision-makers wishing to obtain reasonable returns under high disruptions, mainly when depots fixed costs are low.

For P2, the risk-avoidance model Ml gives the best result for two problem instances, and it is generally not too far from the best model. Models Ml-D and M3 are dominated, but they also provide surprisingly good results for models not considering hazards explicitly. This can be explained as follows. First, in our evaluation process, we use a default depot reassignment policy based on a proximity rule. It turns out that this response policy is excellent and that it copes well with hazards even for designs not optimized for resilience. Second, our results show that the designs obtained are very sensitive to the problem size and topology (Pi vs P2). For some problem topology, the designs provided by models not seeking resilience explicitly (Ml, M2, Ml-D and M3), are naturally resilient. Note finally that, except for R3-formulations which are sensitive to cost structure, the other models are relatively insensitive to cost structure, customer size and depot size.

4.7 Conclusions This paper studies a SN design approach under uncertainty. In the context of the multi-

period location-transportation problem, it proposes design models incorporating resilience-seeking formulations. A generic solution approach is also proposed to produce effective and resilient SN designs. The models formulated are based on approximate anticipations of the response procedure implemented by the SN users. Our results show that the quality of the user response anticipation incorporated in a design model is a critical issue: significant gains can be made by using more precise representations of delivery decisions (routes vs flows) and larger scenario samples. Given the computational power currently available, tradeoffs are however necessary. The best approach seems to be to seek an adequate equilibrium between all the dimensions involved (ex: route set cardinality vs scenario sample size) instead of neglecting some dimensions (ex: using a deterministic model to be able to incorporate more routes).

Our results also show that more robust designs are obtained by modeling hazards explicitly. This is particularly important when additional revenues can be generated by providing the demand surge goods required by customers under extreme events. The models proposed to cope with hazards try to avoid risk and to provide resilient network structures. However, the incorporation of resilience-seeking constructs in the design models may induces biases that are not necessarily congruent with decision-makers

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objectives, mainly when the user response procedures are only roughly anticipated. The Rl-models proposed to select backup depots are good examples of this phenomenon. This also stresses the importance of evaluating potential SN designs with response procedures that are as close as possible to those used by the company considered. By doing this, we found that a good approach to design effective and robust SN is given by R2-models which assume that customers can be served from multiple depots. These models provide resilient networks, even when the user operates under a single-sourcing policy. For decision-makers averse to disruptions, the R3-models provide an interesting alternative, particularly when the depots fixed costs are low. In some context, models simply trying to avoid risks (Ml and M2) are also a good alternative.

This paper sheds some light on some important SN design issues, but it also raises several questions to address in future research. Our analysis was based on a single product two-echelon location-transportation problem centered on location decisions at the design level and transportation decisions at the user level. More complex multi-product multi-echelon problems incorporating sourcing, capacity and market selection decisions at the design level, as well as inventory and production decisions at the user level, should be studied. Also, our experiments were based on two realistic SN with similar topologies in the north-east of the USA. Experiments should be made with more varied network topologies and multihazard processes in different parts of the world. Also, other modeling approaches to get resilient networks can certainly be investigated. Finally, as the design problems considered become more complex, the models formulated become more difficult to solve. This raises the need for the elaboration of heuristic methods to solve the problems, and the issue of the tradeoffs between the models precision and solvability.

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Conclusion générale Dans cette thèse, une méthodologie de design de réseaux logistiques efficaces et robustes a été élaborée. Les contributions de recherche proposées ont été présentées à travers les quatre articles inclus dans le document.

Le premier article de la thèse a exposé une revue critique de la littérature et a permis de souligner les lacunes de celle-ci pour le problème de design de réseaux logistiques en univers incertain. Ces lacunes se résument à négliger l'environnement incertain ou certains types d'événements qui ont un impact sérieux sur la performance du réseau, et aussi à utiliser des modèles de design simplifiés incluant une anticipation rudimentaire de leur usage futur. Les contributions principales de cet article de revue sont les suivantes :

Une revue de la littérature récente pour le problème de conception de réseaux logistiques, notamment des articles qui traitent de l'incertitude et du risque. L'identification des critères importants pour l'évaluation stratégique de réseaux logistiques et la synthèse de leur usage dans les modèles existants. L'introduction et la définition des notions de réactivité, de resilience et de robustesse dans le cadre de la problématique de design de réseaux logistiques.

L'article 2 de la thèse a proposé une méthodologie de design de réseaux en univers incertain. D'abord, une structure de planification en univers incertain a été proposée pour le problème de design de réseaux. Ensuite, une caractérisation de l'environnement des réseaux logistiques a été effectuée en définissant trois types d'événements : les événements aléatoires, hasardeux et profondément incertains. Ces événements ont été modélisés dans l'environnement futur des réseaux logistiques en s'appuyant sur une approche par scénarios. Afin de se prémunir contre tout scénario plausible, des stratégies de resilience ont été considérées dans la formulation du modèle de design. De plus, une étude sur la qualité des anticipations a été effectuée et a permis de tirer des conclusions sur les meilleurs compromis entre la solvabilité du modèle de design et la qualité de la solution associée. Finalement, une approche de solution générique a été proposée afin de produire des designs efficaces et robustes. Elle inclut une phase de génération de scénarios, une phase de génération de designs et une phase d'évaluation et de sélection de designs. Cette méthodologie a la qualité d'être riche en termes de concepts et flexible en terme de modèles et méthodes qu'elle peut faire intervenir. A noter que pour valider ces concepts méthodologiques, de nombreuses expérimentations ont été effectuées dans les articles 3 et 4 de la thèse. Les contributions principales de cet article sont les suivantes :

La proposition d'une structure de planification en univers incertain. - La caractérisation de l'environnement incertain des réseaux logistiques.

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La proposition de stratagèmes de réduction de complexité du modèle conceptuel de design et l'introduction du concept d'anticipation. La proposition d'une approche de solution générique composée de trois phases: génération de scénarios, génération de designs et évaluation de designs.

L'article 3 étudie un problème hiérarchique de localisation d'entrepôts et de transport avec une demande aléatoire qui n'avait jamais été étudié. Ce problème a été modélisé par un programme stochastique à deux étapes et la demande a été considérée à travers un ensemble de scénarios. Pour des tailles de problème réalistes, une approche de solution heuristique a été proposée pour résoudre le modèle obtenu. Cette approche utilise une procédure de recherche tabou pour la phase de localisation et les procédures de Clark and Wright et d'approximation des longueurs des routes pour la phase de transport. Trois stratégies d'exploration du voisinage ont été proposées et comparées afin de calibrer une approche rapide, efficace et qui produit des solutions de design de bonne qualité. Des expériences basées sur plusieurs contextes industriels réalistes ont été réalisées et les résultats ont confirmé les bonnes performances de l'heuristique proposée. Les contributions principales de cet article sont les suivantes :

La formulation du problème stochastique et multi-périodes de localisation d'entrepôts et de transport. La prise en compte de plusieurs options de transport fournies par les transporteurs publics à partir des entrepôts. La compression du nombre potentiellement infini de scénarios à l'aide de méthode d'échantillonnage Monte-Carlo. L'inclusion d'une formule d'approximation des longueurs des routes afin d'accélérer les calculs d'évaluation des solutions partielles. Le développement de stratégies de recherche efficaces dans le voisinage pour améliorer la qualité des solutions et la rapidité de l'approche.

L'article 4 a utilisé le problème de localisation d'entrepôts et de transport pour proposer une approche d'analyse du risque qui tient compte des aléas et des périls. Une modélisation des catastrophes naturelles dans le temps et dans l'espace a été proposée et leur impact sur la capacité des entrepôts et la demande des clients a été capturé. Une approche par scénarios a été élaborée pour représenter les aléas et les périls sur l'horizon de planification du réseau et une approche Monte-Carlo a été utilisée pour générer des échantillons de scénarios. Pour le programme stochastique avec recours obtenu, des construits mathématiques ont été proposés pour inclure des stratégies de resilience dans le modèle de design. Plusieurs expériences numériques ont été menées pour tester la solvabilité des modèles obtenus. À partir de ces modèles, plusieurs solutions de designs ont été générées et évaluées sur différents contextes industriels réalistes. Les résultats obtenus nous ont permis de valider l'approche de solution et d'affirmer la supériorité des designs obtenus

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lorsque la resilience est prise en compte. Les contributions principales de cet article sont les suivantes :

La modélisation des périls et de leur impact sur les ressources du réseau. - La proposition de plusieurs construits mathématiques pour prendre en compte la

resilience dans le modèle de design de réseaux. La validation empirique et la comparaison de plusieurs stratégies de resilience en étudiant leur impact sur la robustesse des solutions de design.

En plus des travaux menés dans le cadre de cette thèse, la méthodologie proposée permet d'entrevoir plusieurs avenues de recherche futures intéressantes:

• La modélisation du problème de design. Par rapport au problème de localisation d'entrepôts et de transport utilisé pour la validation des concepts dans cette thèse, d'autres décisions stratégiques telles que le choix de sources d'approvisionnement, l'acquisition de capacité, la sélection de technologie et la sélection de produit-marchés devraient être incorporées dans le modèle de design. Ceci donnera lieu à un problème multi-échelons et multi-produits très complexe qu'il faudra modéliser adéquatement.

• L'étude des anticipations. Pour mieux évaluer la performance des décisions stratégiques, des anticipations approximatives des décisions de l'usager du réseau sur le transport, la production et les inventaires devraient être incluses. Il est évident que la complexité engendrée par l'inclusion de ces décisions conduira à des modèles intégrés difficiles à résoudre et qu'il faudra essayer de trouver les meilleurs compromis entre la précision du modèle de design et sa solvabilité à travers plusieurs expérimentations.

• La structure de planification. Plusieurs cycles de planification devraient être considérés afin d'enrichir l'horizon de planification. Ceci pourrait permettre de prendre en considération des scénarios de l'environnement futur des réseaux logistiques issus de plusieurs tendances évolutives. Ces scénarios pourraient alors exprimer des évolutions climatiques, écologiques, socio-économiques, politiques, etc.

• La caractérisation de l'incertitude. Des données plus riches pourraient être collectées et utilisées pour raffiner l'analyse de risque des réseaux logistiques en termes d'indices d'exposition par région, de processus d'arrivés des catastrophes, d'estimation de l'impact des périls et de la fonction de récupération des sources de vulnérabilité. Également, une méthode de génération d'événements imaginatifs

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basée sur les jugements d'experts pourrait être développée pour enrichir leurs contenus.

La génération de scénarios plausibles. Dans la plupart des situations, un nombre réduit de scénarios peuvent être considérés dans le processus d'optimisation. En plus de la méthode Monte-Carlo, d'autres techniques d'échantillonnage pourraient être développées pour s'assurer que tous les futurs plausibles soient couverts par les échantillons de scénarios générés.

Les stratégies de resilience. Nos résultats ont prouvés que l'inclusion explicite de construits mathématiques visant la resilience conduit à des réseaux logistiques plus robustes. En fonction du contexte étudié, d'autres stratégies de resilience basées sur la flexibilité et/ou la redondance pourraient être imbriqués dans les modèles d'optimisation.

Les mesures de robustesse. À partir de la notion de robustesse définie dans le cadre du problème de design de réseaux logistiques, plusieurs pistes sont possibles sur comment mesurer la robustesse d'un design et comment évaluer le design le plus robuste selon une ou plusieurs mesures de robustesse. Les mesures utilisées d'efficacité et de resilience et d'autres mesures complémentaires de dispersion, de flexibilité et de stabilité peuvent contribuer à mesurer adéquatement la robustesse d'une solution de design le long d'un horizon de planification.

L'approche de solution. Compte tenu de la complexité du problème de design, l'élaboration de méthodes de solutions sophistiquées, exactes ou heuristiques est nécessaire pour résoudre des problèmes de taille réaliste. Plusieurs des méthodes existantes, développées pour des modèles déterministes, pourraient être adaptées et testées pour résoudre efficacement les modèles stochastiques de design de réseaux logistiques.

Tous ces éléments représentent des pistes de recherche intéressantes et leur réalisation reste un défi considérable.

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•

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