MINISTERE DE L’ENSEIGNEMENT SUPERIEUR
UNIVERSITE DE MANOUBA
ECOLE NATIONALE DES SCIENCES DE L’INFORMATIQUE
THÈSE
Présentée...
-i-
In situations where some sellers have surplus stock while others, belonging to the same firm,
are stocked out, it may ...
-ii-
Dans des situations où certains points de vente sont en surplus de stock tandis que d'autres sont
en rupture de stock...
-iii-
Les problèmes d’optimisation numériques, en particulier ceux relatifs à la gestion de
stock, sont souvent caractéris...
-iv-
lesquelles nous nous sommes basées pour appréhender les problèmes décisionnels
inhérents à la gestion de stock à base...
-v-
objectifs. Cette région assimilée à un hyperrectangle adaptatif en fonction de l’effort
d’échantillonnage de l’algorit...
-vi-
l’existence de contraintes linéaires ou non linéaires et la stochasticité des fonctions
objectifs. Nous avons aussi m...
-vii-
sont déterminés au premier niveau, tandis que les transbordements optimaux sont
calculés dans un second temps. Une h...
-viii-
Completing my PhD degree is probably the most challenging activity of my life. The best and
worst moments of my doc...
-ix-
I would like to thank Clerot Fabrice and Seyed Taghi Akhavan Niaki, from Orange Labs
Profiling and Data mining Resear...
-x-
-xi-
ABSTRACT ...............................................................................................................
-xii-
3.1 INTRODUCTION ......................................................................................................
-xiii-
4.6.3 Validation of PHC-NSGA-II.......................................................................................
-xiv-
5.12.3 Bi-objective Cost vs. Lead time problem.........................................................................
-xv-
6.8.3 Fixing a population of overlapping hyperrectangles........................................................... 1...
-xvi-
ATM Automatic Teller Machine
CI Confidence Interval
DTLZ Deb-Thiele-Laumanns-Zitzler
EA Evolutionary Algorithm
EMO E...
-xvii-
Journal Papers
1. Nabil Belgasmi, Lamjed Ben Said, Khaled Ghédira: Evolutionary multiobjective
optimization of the ...
-1-
1.1 General context
Practical optimization problems, especially inventory management, seem to have a
multiobjective na...
1.2 Contributions
-2-
In this dissertation study, our contributions are connected to the inventory management and
optimiza...
Chapter 1 - Multiobjective benchmark problems
-3-
instantiated and studied numerically using the simulation engine we prov...
-4-
2.1 Introduction
This chapter is devoted to synthesize the existent literature related to inventory management,
modeli...
Chapter 2 - Multiobjective benchmark problems
-5-
Policies with fixed order-up-to level.
2.2.1 Structure of inventory syst...
2.2 Inventory management and optimization
-6-
costs from the source, inspection costs, etc. There are also the administrat...
Chapter 2 - Multiobjective benchmark problems
-7-
system, generates the lead-time for that particular order. It is possibl...
2.2 Inventory management and optimization
-8-
Or as an alternative to the analytical approach: For example, one might use ...
Chapter 2 - Multiobjective benchmark problems
-9-
Cycle Time (t): it is the time between consecutive inventory replenishme...
2.2 Inventory management and optimization
-10-
This model allows shortage to be backordered. This situation is illustrated...
Chapter 2 - Multiobjective benchmark problems
-11-
𝑆∗
= √
2𝑎𝐾
ℎ
√
𝑝
𝑝 + ℎ
2-6
𝑄∗
= √
2𝑎𝐾
ℎ
√
𝑝 + ℎ
𝑝
2-7
𝜏∗
=
𝑄∗
𝑎
2-8
Com...
2.2 Inventory management and optimization
-12-
𝑇𝑘 =
𝑎𝐾
𝑄 𝑘∗
+ 𝑎𝑐 𝑘 +
ℎ𝑄 𝑘∗
2
2-11
2.2.6 Stochastic inventory models
This s...
Chapter 2 - Multiobjective benchmark problems
-13-
𝑃𝑆 = 𝑃{𝑥 > 𝑧} = ∫ 𝑓(𝑥)𝑑𝑥 = 1 − 𝐹(𝑧)
∞
𝑧
2-12
𝑃𝑒 = 𝑃{𝑥 ≤ 𝑧} = ∫ 𝑓(𝑥)𝑑𝑥 =...
2.3 Newsvendor inventory models
-14-
to balance inventory holding cost and the cost of imperfect customer satisfaction. Th...
Chapter 2 - Multiobjective benchmark problems
-15-
The profit per period is
Simplifying and taking the expected value of π...
2.3 Newsvendor inventory models
-16-
the basic Newsvendor model in which the goal is to maximize the probability of achiev...
Chapter 2 - Multiobjective benchmark problems
-17-
necessary condition for the normal distribution to an equation that can...
2.4 Multi-location transshipment models
-18-
which decisions have to be made on whether to hold inventories in the form of...
Chapter 2 - Multiobjective benchmark problems
-19-
fill rates at the retailers. A group of stocking locations that share t...
2.4 Multi-location transshipment models
-20-
A significant amount of literature in transshipment assumed that complete poo...
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  1. 1. MINISTERE DE L’ENSEIGNEMENT SUPERIEUR UNIVERSITE DE MANOUBA ECOLE NATIONALE DES SCIENCES DE L’INFORMATIQUE THÈSE Présentée en vue de l’obtention du diplôme de DOCTEUR EN INFORMATIQUE EVOLUTIONARY MULTIOBJECTIVE OPTIMIZATION OF THE MULTI-LOCATION TRANSSHIPMENT PROBLEM Par Nabil BELGASMI Réalisée au sein du Laboratoire SOIE « Stratégies d’Optimisation et Informatique intelligentE » Soutenue le 16/01/2015 Devant le jury composé de : Président : Prof. Moncef TAJINA ENSI, La Manouba Rapporteur : Prof. Patrick SIARRY UPEC, Paris Rapporteur : Prof. Taïcir LOUKIL ISGI, Sfax Examinateur : Prof. Lamjed BEN SAID ISG, Tunis Directeur de thèse : Prof. Khaled GHEDIRA ISG, Tunis
  2. 2. -i- In situations where some sellers have surplus stock while others, belonging to the same firm, are stocked out, it may be desirable to share the unsold units to fulfill more unmet demands and reduce holding costs. Such practice is named “transshipment”. It represents an advanced option of inventory management. Decision-makers face some issues when trying to adopt this cooperative inventory model. They should have a complete prior knowledge about the expected performance of the transshipment-based system to decide whether to adopt or to reject it. We believe that the literature of transshipment does not provide convincing answers to such managerial problems. Most of the literature studies ignore the existence of conflicting objective functions and focus only on optimizing costs or service level separately. In this study, the contribution is twofold affecting both Inventory Management and Evolutionary Multiobjective Optimization (EMO) research fields. We focused on designing advanced transshipment models based on multiple conflicting objective functions. We studied an extended variant of the baseline multi-location Newsvendor model within a purely multiobjective framework. We proposed a challenging real-world application of this model, that is, the multiobjective ATM cash level optimization which adds to the existent real-world financial optimization benchmarks. We developed a multi-item transshipment model with linear storage constraints and multiple conflicting stochastic objectives. By solving a wide variety of bi-objective and tri-objective cases we enhanced the understanding of the problem. Detailed analysis of some problem properties gave us insight to elaborate more appropriate optimization techniques. We mainly added to the existent EMO literature in two ways. Firstly, we suggested three new real-valued Evolutionary Algorithms (EAs) for single-objective and multiobjective continuous optimization. A common characteristic of these algorithms is that they are based on reusable local search methods. Secondly, we proposed a noise handling framework based on the newly introduced concept of Multiobjective Confidence Hyperrectangles (MoCH). We applied the MoCH framework to the standard SPEA2 and performed statistical comparison with the baseline Explicit-Averaging based SPEA2. A wide variety of benchmarks are used to test the proposed approach. KEYWORDS: Evolutionary Multiobjective Optimization (EMO), Inventory management, Newsvendor inventory models, Transshipment problems, Uncertainty handling in EMO, Real-Coded Genetic Algorithms, Hybrid evolutionary algorithms, Real-world applications.
  3. 3. -ii- Dans des situations où certains points de vente sont en surplus de stock tandis que d'autres sont en rupture de stock, il pourrait être intéressant de partager les unités en excès pour satisfaire plus de demandes et réduire instantanément les coûts de possession. Cette pratique nommée « transbordement » représente une politique avancée pour la gestion coopérative des stocks. Son intégration dans le processus de gestion d’une chaine logistique fait confronter les décideurs à une multitude de problèmes décisionnels à savoir le calcul des approvisionnements optimaux et la redistribution optimale des stocks en excès. Avant de décider l’adoption d’une telle politique, les gestionnaires devraient disposer d’une simulation prédictive des mesures de performance du système implémentant le transbordement. Nous estimons que la littérature scientifique autour du transbordement ne fournit pas les outils nécessaires pour une prise de décision optimale. La plupart des études ignorent l'existence de fonctions objectifs conflictuelles et se focalisent principalement sur l'optimisation du coût ou du niveau de service séparément. Dans cette étude, nos contributions touchent à la fois le domaine de l’Optimisation Évolutionnaire Multi-objectif et celui de la Gestion des stocks et des approvisionnements. Nous nous sommes concentrés sur la conception de modèles avancés de transbordement. Nous avons étudié une extension du modèle du « marchand de journaux » dans un cadre purement multi-objectif. A l’issue de cette étude, nous avons proposé une application réelle de ce modèle concernant l’optimisation du chargement des distributeurs automatiques de billets (DABs). Nous avons développé une variante multi-dépôt et multi-produits du problème de transbordement avec des contraintes de stockages linéaires et des fonctions objectifs stochastiques conflictuelles. La résolution de plusieurs instances bi-objectif et tri-objectif du problème a révélé plusieurs propriétés intéressantes qui étaient à la base de l’élaboration de nouvelles techniques d’optimisation plus performantes. Nos contributions dans le domaine de l’optimisation évolutionnaire multi-objectif sont concrétisées par le développement de trois nouveaux algorithmes évolutionnaires à codage réel pour la résolution de problèmes continus mono et multi-objectif. Ces derniers reposent sur l’intégration intelligente d’algorithmes de recherche locale. Nous avons également introduit un nouveau concept nommé « MoCH » à base d’hyperrectangles de confiance multi-objectifs, conçu pour l’optimisation des problèmes multi-objectifs stochastiques. « MoCH » a été instancié sur l’algorithme SPEA2 et validé sur un grand nombre de benchmarks. MOTS-CLÉS: Optimisation évolutionnaire multi-objectif, Gestion de l’inventaire, Modèles du “marchand des journaux”, Problème de transbordement, Incertitude & Optimisation multi-objectif, Algorithmes génétiques à codage réel, Algorithmes hybrides, Applications réelles.
  4. 4. -iii- Les problèmes d’optimisation numériques, en particulier ceux relatifs à la gestion de stock, sont souvent caractérisés par la nature conflictuelle de leurs fonctions objectifs. En effet, l’optimisation des inventaires consiste à trouver des solutions qui corrigent les défauts de prévision dues à la stochasticité de la demande des clients, et ce en réduisant les coûts de gestion tout en préservant un niveau de service compétitif. Au cours des dernières années, l'émergence des nouvelles technologies et des systèmes d'information avancés a rendu plus facile l’accès rapide et fiable aux données de l’entreprise. Ce qui a fortement influencé les approches adoptées par les décideurs des chaines logistiques pour définir leurs stratégies de gestion des stocks. Dans ce contexte, les transbordements latéraux, définis comme étant le mouvement contrôlé de stock entre les sites d’un même échelon, représentent une option avancée dans la gestion collaborative des stocks. Sur le plan pratique, le transbordement impliquent les sites disposant d’un excès de stock à partager leurs stocks physiques aux les sites déficitaires. Cette redistribution de stock réduit les coûts de sur-stockage et de rupture de stock à la fois moyennant un coût de transport additionnel à prendre en considération lors du calcul du transbordement optimal. Les modèles de transbordement basiques ont été largement étudiés et appliqués dans divers domaines tels que la médecine, l’industrie du textile, le commerce des pièces de rechange, etc. Étant donné que la mise en place d’un système à base de transbordement génère des coûts non négligeables relatifs au processus de redistribution des stocks (acquisition de véhicules, maintenance, frais de déplacement, etc.), les décideurs auraient nécessairement besoin d’une estimation à priori du rendement du futur système. À notre connaissance, la littérature de transbordement ne fournit pas les éléments nécessaires pour évaluer efficacement la pratique de transbordement. La plupart des études suggèrent des modèles mono-objectifs basiques qui se limitent à la minimisation des coûts ou à la maximisation du niveau de service. Sous certaines hypothèses simplificatrices, ces modèles proposent des solutions optimales via des méthodes de résolution exactes. Le domaine de l’optimisation multi-objectif est devenu de plus en plus actif. Les algorithmes évolutionnaires multi-objectifs (AEMOs) ont montré un succès remarquable dans la résolution de problèmes complexes sans avoir besoin de connaitre au préalable leurs propriétés mathématiques (problèmes continus, discrets, différentiables, convexes, multimodaux, etc.). Ce sont les principales motivations sur
  5. 5. -iv- lesquelles nous nous sommes basées pour appréhender les problèmes décisionnels inhérents à la gestion de stock à base de transbordement dans un contexte multi- objectif. Dans cette thèse, nos contributions concernent principalement deux axes : (1) la gestion des stocks et (2) l’optimisation évolutionnaire multi-objectif. En premier lieu, nous avons élaboré plusieurs modèles multi-objectifs du transbordement. Nous avons étudié une variante du problème de référence du Marchand de journaux à plusieurs objectifs. Nous avons également proposé une application réelle qui implémente le modèle étudié : l’optimisation de l’approvisionnement des guichets automatiques de billets. Nous avons justifié le rattachement de l’application au problème générique du marchand des journaux. Nous avons ensuite repris le problème de transbordement dans sa version basique. Nous avons traité à tour de rôle une version mono-objectif et une version multi-objectif du problème en prenant en considération les contraintes sur la capacité de stockage. Finalement, nous avons proposé une forme plus générale du problème de transbordement, à savoir le cas multi-produits. Le modèle mathématique suggéré est stochastique à deux-niveaux introduisant des contraintes linéaires pour traduire la capacité de stockage locale par produit et par dépôt. Les réapprovisionnements optimaux sont calculés au premier niveau du problème, tandis que les transbordements optimaux sont déterminés au second niveau. Sur le plan optimisation, nos contributions s’étendent aux problèmes stochastiques mono et multi-objectifs. Nous avons mis au point un algorithme génétique à codage réel hybride (H-RCGA) dans le sens où l’opération de croisement repose sur une descente de gradient. Nous avons souligné l’intérêt de généraliser cette hybridation au cas multi-objectif en proposant un nouvel algorithme évolutionnaire hybridé avec une recherche locale guidée par le quasi-gradient multi-objectif (H-SPEA2). Nous avons aussi proposé un nouvel algorithme évolutionnaire hybride nommé PHC-NSGA-II conçu pour l’optimisation des problèmes continus. Cet algorithme repose sur l’utilisation de la recherche locale Hill Climbing adaptée au cas multi-objectif. Pour mieux gérer l’incertitude dans les problèmes de transbordement multi-produits, nous avons introduit le nouveau concept des hyperrectangles de confiance multi-objectifs (MoCH). En effet, lorsque les fonctions objectifs du problème sont stochastiques, leur représentation dans l’espace des objectifs ne se réduit plus à un point. Les simulations stochastiques des fonctions objectifs du problème permettent de construire des intervalles de confiance dans le but de définir la région de l’espace des objectifs qui a la plus forte probabilité de contenir l’espérance mathématique de toutes les fonctions
  6. 6. -v- objectifs. Cette région assimilée à un hyperrectangle adaptatif en fonction de l’effort d’échantillonnage de l’algorithme hôte. Nous avons établi plusieurs propriétés, théorèmes et corolaires pour dégager les caractéristiques les plus intéressantes introduites par ce concept. Sur la base de cette idée, nous avons pris comme algorithme hôte SPEA2. Une étude expérimentale très détaillée montre la capacité de la technique proposée à réduire d’une façon dynamique et efficace l’effet de la stochasticité du problème sur plusieurs indicateurs de performance multi-objectif tels que l’hyper- volume, l’espacement, la distance de convergence, etc. Tous les algorithmes et les problèmes de transbordement proposés dans cette thèse ont été regroupés dans une librairie d’optimisation entièrement développée en Java et représentant une extension avancée du projet open-source jMetal. Organisation du rapport Le présent rapport comporte sept chapitres, y compris l’introduction générale (Chapitre 1) et la conclusion (Chapitre 7). Il est complété par deux annexes. Dans le chapitre 2, nous passons en revue l’état de l’art relatif aux modèles de transbordement et de gestion d’inventaires. Le chapitre débute par des généralités sur la structure des systèmes d’inventaires et sur leur modélisation, déterministe ou stochastique. Le problème classique du marchand de journaux est décrit en examinant les principales extensions. Nous nous sommes focalisés ensuite sur les modèles de systèmes d’inventaires multi-sites avec transbordement, qui incorporent la possibilité de transfert de marchandises invendues dans un autre site, où l’offre est insuffisante pour répondre à la demande. Nous avons décrit les politiques de transbordement existantes en catégorisant les problèmes en fonction de paramètres tels que le nombre d’articles différents, le nombre de sites et la nature des contraintes. Enfin, nous avons exposé les méthodes de résolution analytiques, ou à base de simulations des problèmes de transbordement. En conclusion, nous avons dégagé les limitations des approches existantes, ce qui nous amené à préconiser les algorithmes évolutionnaires dans les cas les plus complexes, en particulier dans les situations multi-objectif, fortement contraintes et entachées d’incertitudes. Le chapitre 3 est consacré à l’optimisation multi-objectif en particulier aux algorithmes évolutionnaires. Après une présentation des concepts de base, une brève revue de la littérature sur les techniques d’optimisation évolutionnaire est présentée, en commençant par les approches les plus classiques reposant sur les mécanismes conventionnels de reproduction et de sélection. Le chapitre aborde aussi des approches plus avancés où coexistent plusieurs sources de complexité telles que : la dimensionnalité du problème, la continuité de l’espace de recherche, la multi-modalité,
  7. 7. -vi- l’existence de contraintes linéaires ou non linéaires et la stochasticité des fonctions objectifs. Nous avons aussi mis l’accent sur une classe spéciale de l’optimisation évolutionnaire multi-objectif qui prend en considération la présence du bruit dans les fonctions objectifs du problème. Nous avons également passé en revue des métriques de performance usuelles qui permettent de mesurer l’efficacité des algorithmes multi- objectifs en termes de convergence et de diversité. Le chapitre 4 représente notre première contribution. Il s’agit de proposer une modélisation multi-objectif du problème du marchand de journaux, étudié généralement dans un contexte monocritère. Le modèle ainsi proposé est résolu par deux algorithmes classiques SPEA2 et NSGA-II. Ensuite, une amélioration de ces algorithmes par l’hybridation avec une méthode de recherche locale (Pareto Hill Climbing) est proposée en mettant l’accent sur les avantages de l’algorithme hybride résultant (PHC-NSGA-II) par rapport à d’autres approches de résolutions (NSGA- SQP, SPEA-SQP, etc.). Une fois validée sur plusieurs fonctions tests, l’algorithme hybride proposé a été appliqué avec succès dans la résolution de plusieurs instances du problème multi-objectif du marchand de journaux. Le chapitre se termine par une application réelle du secteur bancaire : la gestion de guichets automatiques de retrait/dépôt de billets. Nous avons montré le rattachement de cette application au problème générique du marchand de journaux. Le chapitre 5 représente notre deuxième contribution relative au problème principal de transbordement sous sa forme dite basique. Le chapitre commence par la modélisation du problème mono-objectif avec des contraintes sur la capacité de stockage exprimées sous forme d’un programme linéaire sous des hypothèses très strictes. Pour résoudre ce problème, un algorithme génétique hybride à codage réel a été mis au point. Il repose sur l’utilisation d’un opérateur de croisement à base de descente de gradient qui accentue dynamiquement la recherche vers les régions les plus prometteuses de l’espace de recherche. Ensuite, le modèle multi-objectif est étudié en discutant tous les paramètres et les fonctions objectifs qui s’y rattachent. L’étude expérimentale présentée porte sur l’application des algorithmes SPEA2 et NSGA-II. En exploitant certaines propriétés du problème étudié, une amélioration de SPEA2 a été proposée. Il s’agit d’une hybridation avec un opérateur de mutation polynomiale guidé par le quasi- gradient multi-objectif (H-SPEA2). Le chapitre se termine par une étude comparative entre les différents algorithmes proposés. Le chapitre 6 représente un modèle avancé du problème de transbordement. En effet, il s’agit d’une généralisation au cas multi-produits et multi-objectif à la fois en mettant l’accent sur l’aspect bi-niveaux du problème, où les réapprovisionnements optimaux
  8. 8. -vii- sont déterminés au premier niveau, tandis que les transbordements optimaux sont calculés dans un second temps. Une heuristique polynomiale a été élaborée pour traiter les instances de grandes tailles, sans contraintes. Les premiers résultats expérimentaux sont fournis par SPEA2 et NSGA-II. Il a été démontré empiriquement que pour les instances de grandes tailles, la prise en compte de l’incertitude devient une obligation pour maintenir de bonnes performances. Dans ce contexte, nous avons introduit le concept des hyperrectangles de confiance multi-objectif (MoCH) qui nous a permis de contourner l’effet du bruit sur la résolution du problème peu importe son intensité. Le chapitre se termine par une étude expérimentale exhaustive du concept introduit. Les résultats montrent une nette supériorité du framework proposé. Des conclusions et des perspectives sont tirées à la fin dans le chapitre 7.
  9. 9. -viii- Completing my PhD degree is probably the most challenging activity of my life. The best and worst moments of my doctoral journey have been shared with many people. It has been a great privilege to spend several years in the Modeling and Simulation Research Group (MoS) of the SOIE Laboratory (Stratégies d’Optimisation et Informatique IntelligentE), and its members will always remain dear to me. My first debt of gratitude must go to my advisor, Pr. Khaled Ghédira. He patiently provided the vision, encouragement and advice necessary for me to complete my dissertation. I express my gratitude to my advisor Dr. Lamjed Ben Said for his encouragement, constructive criticism and especially his moral support; whose expertise added considerably to my graduate experience. I would like to thank my PhD committee members for accepting taking part of it and furnishing efforts for examining this work. Special thanks to the thesis reviewers Pr. Patrick Siarry and Pr. Taïcir Loukil for agreeing to read and judge my thesis and for their valuable feedback. Members of SOIE Laboratory also deserve my sincerest thanks, their friendship and assistance has meant more to me than I could ever express. I should express my gratitude to: Dr. Slim Bechikh for being very able colleague and providing me useful documents and manuscripts; he is a true “gold mine” of scientific materials, ideas and imagination; Mr. Mohamed Hmiden, for all the rewarding discussions we made together about some common scientific topics. My colleagues in the Bank of Tunisia (B.T) deserve to be thanked infinitely. They never stopped encouraging me to achieve my researches. Special thanks to the Director of the Computer Science Department, Mr. Lotfi Kefi for his patience and comprehensiveness. He was always interested in my research field, and believes that it would certainly be innovative if it can be applied to solve some of the challenging real-world banking problems. I am also grateful to the Deputy Director of the Computer Science Department and Information System, Mr. Zouheir Abdessamed for his continuous support of my scientific activities. He never refused my request for leave to participate in regular scientific events organized by our laboratory, despite the workload and the criticality of certain periods of the year. Special thanks to my colleagues: Khabeb Ferjani, Hassene Ayari, Khaled Jabeur, Assad Fayache, Mohamed Ali Cherif, Mohamed Ben Miled, Abdejlil Abaidia, Aboulbaba Bennassr, Ghaleb Sabri, just to cite few.
  10. 10. -ix- I would like to thank Clerot Fabrice and Seyed Taghi Akhavan Niaki, from Orange Labs Profiling and Data mining Research Group (France) and the Sharif University of Technology (Tehran, Iran) respectively, for their help and very interesting discussions via the ResearchGate scientific network. Very special thanks are due to my wife “Mouna Belgasmi née Balti” for her support and encouragement through my PhD, which allowed me to finish this journey. Thank you for the congratulatory gifts you have offered me when my little twin daughters Nour and Nada were born in June 2014. I will like to thank my stepparents especially Mr. “Moncef Balti” for their excellent guidance, constructive discussions and constant encouragement. Finally, I would like to thank my parents “Dalila” and “Ali”; their love and sincere support provided my inspiration and were my driving force. I owe them everything and I wish that this work makes them proud.
  11. 11. -x-
  12. 12. -xi- ABSTRACT ............................................................................................................................................I RÉSUMÉ ..............................................................................................................................................II RÉSUMÉ ÉTDENDU .............................................................................................................................III ACKNOWLEDGMENTS ......................................................................................................................VIII CONTENTS..........................................................................................................................................XI ACRONYMS ......................................................................................................................................XVI PUBLICATIONS.................................................................................................................................XVII CHAPTER 1. INTRODUCTION ............................................................................................................1 1.1 GENERAL CONTEXT............................................................................................................................... 1 1.2 CONTRIBUTIONS .................................................................................................................................. 1 1.3 OUTLINE OF THE THESIS......................................................................................................................... 3 CHAPTER 2. LITERATURE REVIEW OF NEWSVENDOR AND TRANSSHIPMENT MODELS.....................4 2.1 INTRODUCTION.................................................................................................................................... 4 2.2 INVENTORY MANAGEMENT AND OPTIMIZATION.......................................................................................... 4 2.2.1 Structure of inventory systems................................................................................................ 5 2.2.2 Methods of determining policies for inventory systems ......................................................... 6 2.2.3 Analytical models review ........................................................................................................ 7 2.2.4 Various uses of simulation ...................................................................................................... 7 2.2.5 Deterministic inventory models............................................................................................... 8 2.2.6 Stochastic inventory models.................................................................................................. 12 2.3 NEWSVENDOR INVENTORY MODELS ....................................................................................................... 13 2.3.1 The basic newsvendor model ................................................................................................ 14 2.3.2 Extensions of the basic newsvendor model........................................................................... 15 2.3.3 Handling different objective functions .................................................................................. 15 2.3.4 Extensions to random yields.................................................................................................. 17 2.3.5 Extensions to constrained multi-products............................................................................. 17 2.3.6 Extensions to multi-echelon systems..................................................................................... 17 2.4 MULTI-LOCATION TRANSSHIPMENT MODELS............................................................................................ 18 2.4.1 Assumptions.......................................................................................................................... 19 2.4.2 Transshipment policies.......................................................................................................... 20 2.4.3 Inventory replenishment policies .......................................................................................... 21 2.4.4 Classification of transshipment problems ............................................................................. 22 2.4.5 Methodology......................................................................................................................... 24 2.5 CONCLUSIONS AND DISCUSSIONS........................................................................................................... 25 CHAPTER 3. EVOLUTIONARY MULTIOBJECTIVE OPTIMIZATION OVERVIEW ................................... 27
  13. 13. -xii- 3.1 INTRODUCTION ..................................................................................................................................27 3.2 BASIC CONCEPTS RELATED TO PROBLEMS OPTIMIZATION.............................................................................27 3.2.1 Dimensionality.......................................................................................................................28 3.2.2 Multimodality ........................................................................................................................28 3.2.3 Discontinuity..........................................................................................................................28 3.2.4 Epistasis .................................................................................................................................28 3.2.5 Constraints.............................................................................................................................28 3.3 MULTIOBJECTIVE OPTIMIZATION............................................................................................................29 3.4 EVOLUTIONARY MULTIOBJECTIVE OPTIMIZATION.......................................................................................30 3.5 CONSTRAINT-HANDLING METHODS ........................................................................................................31 3.5.1 Penalty based approaches for boundary constraints ............................................................32 3.5.2 Repair methods of boundary constraints ..............................................................................32 3.6 HYBRID AND MEMETIC EMOAS ............................................................................................................33 3.7 NOISE HANDLING IN EVOLUTIONARY MULTIOBJECTIVE OPTIMIZATION............................................................34 3.7.1 Approaches based on fixed sample size.................................................................................34 3.7.2 Approaches based on adaptive sampling..............................................................................35 3.7.3 Modified Pareto ranking scheme...........................................................................................35 3.7.4 Domination-dependent lifetime ............................................................................................37 3.7.5 Fitness inheritance.................................................................................................................37 3.8 PERFORMANCE ASSESSMENT OF EMOAS................................................................................................37 3.8.1 Error Ratio (ER) ......................................................................................................................38 3.8.2 Generational Distance (GD)...................................................................................................38 3.8.3 Hypervolume and Ratio (HV) .................................................................................................38 3.8.4 Spacing (S) .............................................................................................................................39 3.8.5 Epsilon-Indicator (Iϵ) ..............................................................................................................39 3.9 CONCLUSIONS AND DISCUSSIONS ...........................................................................................................40 CHAPTER 4. SOLVING THE MULTIOBJECTIVE NEWSVENDOR PROBLEM ......................................... 42 4.1 INTRODUCTION ..................................................................................................................................42 4.2 MOTIVATIONS ...................................................................................................................................42 4.3 PROBLEM DESCRIPTION........................................................................................................................44 4.3.1 Parameters ............................................................................................................................44 4.3.2 Decision variables ..................................................................................................................44 4.3.3 Assumptions ..........................................................................................................................44 4.4 OBJECTIVE FUNCTIONS FORMULATION ....................................................................................................44 4.4.1 Newsvendor holding cost.......................................................................................................45 4.4.2 Newsvendor shortage cost ....................................................................................................45 4.4.3 Newsvendor aggregate cost..................................................................................................45 4.4.4 Newsvendor cost variance.....................................................................................................46 4.4.5 Newsvendor fill rate...............................................................................................................47 4.5 OPTIMIZATION RESULTS USING SPEA2 AND NSGA-II................................................................................48 4.6 PHC-NSGA-II: ENHANCING NSGA-II WITH A PARETO ILL CLIMBING LOCAL SEARCH.......................................49 4.6.1 Description.............................................................................................................................49 4.6.2 The algorithm .......................................................................................................................51
  14. 14. -xiii- 4.6.3 Validation of PHC-NSGA-II..................................................................................................... 54 4.6.4 Using PHC-NSGA-II to solve the multiobjective newsvendor problem .................................. 59 4.7 REAL-WORLD APPLICATION: MULTIOBJECTIVE ATM CASH LEVEL OPTIMIZATION ............................................. 60 4.7.1 Analogy with the multi-location multiobjective newsvendor model..................................... 60 4.7.2 Main optimization results ..................................................................................................... 62 4.8 CONCLUSIONS AND DISCUSSIONS........................................................................................................... 66 CHAPTER 5. OPTIMIZATION OF THE BASIC CAPACITATED TRANSSHIPMENT PROBLEM.................. 67 5.1 INTRODUCTION.................................................................................................................................. 67 5.2 SINGLE OBJECTIVE MODEL DESCRIPTION.................................................................................................. 67 5.2.1 Parameters............................................................................................................................ 68 5.2.2 Decision variables.................................................................................................................. 69 5.2.3 Modeling assumptions .......................................................................................................... 69 5.3 DETERMINING THE OPTIMAL TRANSSHIPMENT QUANTITIES ......................................................................... 69 5.3.1 Linear programming ............................................................................................................. 69 5.3.2 A proposed tractable analytical solution of problem K ......................................................... 70 5.4 SINGLE OBJECTIVE MODEL FORMULATION................................................................................................ 72 5.4.1 Aggregate cost function........................................................................................................ 72 5.4.2 Properties.............................................................................................................................. 72 5.5 APPROXIMATION BY RESAMPLING TECHNIQUE ......................................................................................... 73 5.6 A NEW HYBRID R.C.G ALGORITHM FOR THE TRANSSHIPMENT PROBLEM........................................................ 73 5.6.1 Chromosome structure.......................................................................................................... 74 5.6.2 Fitness evaluation and error rate.......................................................................................... 74 5.6.3 Initialization .......................................................................................................................... 74 5.6.4 Selection................................................................................................................................ 74 5.6.5 Mutation ............................................................................................................................... 75 5.6.6 The proposed gradient-descent crossover ............................................................................ 75 5.6.7 Constraints handling ............................................................................................................. 75 5.7 SINGLE OBJECTIVE OPTIMIZATION RESULTS .............................................................................................. 76 5.7.1 Case study ............................................................................................................................. 76 5.7.2 Experimental design.............................................................................................................. 76 5.7.3 Validation with a benchmark ................................................................................................ 77 5.8 DISCUSSION OF THE PROPOSED SINGLE-OBJECTIVE MODEL.......................................................................... 79 5.9 MULTI-OBJECTIVE TRANSSHIPMENT MODELING........................................................................................ 79 5.9.1 Description ............................................................................................................................ 79 5.9.2 Additional parameters and assumptions .............................................................................. 80 5.10 ADDITIONAL OBJECTIVE FUNCTIONS FORMULATION................................................................................. 80 5.10.1 Fill rate function .................................................................................................................. 80 5.10.2 Lead time function .............................................................................................................. 81 5.10.3 Shared inventory quantity function..................................................................................... 82 5.11 SYSTEM SIMULATION PROCESSES......................................................................................................... 82 5.12 MULTI-OBJECTIVE OPTIMIZATION RESULTS (NSGA-II & SPEA2)............................................................... 82 5.12.1 Detailed example ................................................................................................................ 83 5.12.2 Bi-objective Cost vs. Fill rate problem ................................................................................. 85
  15. 15. -xiv- 5.12.3 Bi-objective Cost vs. Lead time problem..............................................................................86 5.12.4 Bi-objective Fill rate vs. Lead time problem.........................................................................86 5.12.5 Three-objective Cost vs. Fill rate vs. Lead time (C/F/L) problem..........................................87 5.12.6 Varying costs and demand structures .................................................................................87 5.13 H-SPEA2: HYBRIDIZATION OF SPEA2 WITH A QUASI-GRADIENT GUIDED LOCAL SEARCH.................................88 5.13.1 The algorithm ......................................................................................................................89 5.13.2 The Guided Multiobjective Mutation Operator ...................................................................91 5.14 SOLVING THE TRANSSHIPMENT PROBLEM WITH H-SPEA2 ........................................................................91 5.14.1 Optimization results.............................................................................................................91 5.14.2 Cost vs. Fill rate problem .....................................................................................................92 5.14.3 Cost vs. Lead time................................................................................................................93 5.15 CONCLUSIONS AND DISCUSSIONS .........................................................................................................95 CHAPTER 6. THE CAPACITATED MULTIOBJECTIVE MULTI-ITEM TRANSSHIPMENT PROBLEM ......... 96 6.1 INTRODUCTION ..................................................................................................................................96 6.2 PROBLEM DESCRIPTION........................................................................................................................97 6.2.1 Parameters ............................................................................................................................97 6.2.2 Decision variables ..................................................................................................................98 6.2.3 Assumptions ..........................................................................................................................98 6.3 DETERMINING THE OPTIMAL MULTI-ITEM TRANSSHIPMENT QUANTITIES.........................................................99 6.3.1 Linear programming..............................................................................................................99 6.3.2 Scalability.............................................................................................................................100 6.3.3 Level of stochasticity of problem K ......................................................................................100 6.4 OBJECTIVE FUNCTIONS FORMULATION ..................................................................................................101 6.4.1 Cost minimization ................................................................................................................102 6.4.2 Fill rate function...................................................................................................................102 6.4.3 Lead time function...............................................................................................................102 6.4.4 Global initial stock level.......................................................................................................103 6.4.5 Shared inventory quantity ...................................................................................................103 6.4.6 Problem constraints.............................................................................................................103 6.5 AN EFFICIENT POLYNOMIAL HEURISTIC TO SOLVE HIGHER DIMENSION UNCONSTRAINED INSTANCES ...................104 6.5.1 Algorithm.............................................................................................................................104 6.5.2 Complexity analysis .............................................................................................................104 6.5.3 Empirical validation .............................................................................................................105 6.6 OPTIMIZATION RESULTS USING SPEA2 AND NSGA-II..............................................................................106 6.7 CONFIDENCE HYPERRECTANGLE BASED FRAMEWORK TO HANDLE UNCERTAINTY IN EVOLUTIONARY MULTIOBJECTIVE PROBLEMS ............................................................................................................................................109 6.7.1 Related Background.............................................................................................................109 6.7.2 SMOP mathematical formulation........................................................................................110 6.7.3 Basic confidence intervals definitions..................................................................................111 6.7.4 Multiobjective confidence hyperrectangles definitions .......................................................114 6.8 MULTIOBJECTIVE ADAPTIVE SAMPLING OF HYPERRECTANGLES....................................................................121 6.8.1 Additional parameters and assumptions.............................................................................122 6.8.2 Proposed strategies to fix two overlapping hyperrectangles ..............................................122
  16. 16. -xv- 6.8.3 Fixing a population of overlapping hyperrectangles........................................................... 124 6.9 MULTIOBJECTIVE CONFIDENCE HYPERRECTANGLE BASED EMOA.............................................................. 126 6.9.1 Basic algorithm ................................................................................................................... 126 6.9.2 MoCH-SPEA2: Multiobjective Confidence Hyperrectangle SPEA2....................................... 126 6.10 EXPERIMENTAL CETUP..................................................................................................................... 127 6.10.1 Proposed inter-sets distance indicator.............................................................................. 127 6.10.2 Injecting noise in multiobjective test problems................................................................. 131 6.11 COMPUTATIONAL RESULTS............................................................................................................... 133 6.11.1 Empirical analysis of noise effects on resampling based SPEA2 ....................................... 133 6.11.2 MoCH-SPEA2 performance assessment ............................................................................ 140 6.11.3 MoCH parameters sensitivity analysis .............................................................................. 158 6.12 APPLICATION TO THE MULTIOBJECTIVE MULTI-ITEM TRANSSHIPMENT INSTANCES ........................................ 160 6.13 SUMMARY AND CONCLUSIONS.......................................................................................................... 161 CHAPTER 7. CONCLUSIONS AND FUTURE WORKS........................................................................ 164 7.1 CONCLUSIONS ................................................................................................................................. 164 7.2 DIRECTIONS FOR FUTURE RESEARCH..................................................................................................... 166 APPENDIX A. MULTIOBJECTIVE BENCHMARK PROBLEMS.......................................................... 168 1. ZITZLER-DEB-THIELE (ZDT) BENCHMARK SUITE........................................................................................ 168 2. DEB-THIELE-LAUMANS-ZITZLER (DTLZ) BENCHMARK SUITE ....................................................................... 169 APPENDIX B. DETAILED OPTIMIZATION OUTPUT ...................................................................... 172 1. NON-NORMALITY OF THE OBJECTIVE FUNCTIONS DISTRIBUTIONS.................................................................. 172 2. IMPACT OF THE MAXIMUM NUMBER OF SAMPLES 𝑺𝑴𝒎𝒂𝒙 ...................................................................... 175 3. IMPACT OF DIFFERENT NOISE LEVELS ...................................................................................................... 177 4. MOCH-SPEA2 TO SOLVE THE MULTI-ITEM TRANSSHIPMENT PROBLEM....................................................... 178 BIBLIOGRAPHY ................................................................................................................................ 179
  17. 17. -xvi- ATM Automatic Teller Machine CI Confidence Interval DTLZ Deb-Thiele-Laumanns-Zitzler EA Evolutionary Algorithm EMO Evolutionary Multiobjective Optimization EMOA Evolutionary Multiobjective Optimization Algorithm GD Generational Distance H-RCGA Hybrid Real-Coded Genetic Algorithm HV Hypervolume IGD Inverted Generational Distance ISD Inter-Sets Distance LS Local Search MO Multiobjective Optimization MoCH Multiobjective Confidence Hyperrectangle MoCH-SPEA2 MoCH based SPEA2 MOP Multiobjective Optimization Problem NSGA-II Elitist Non-dominating Sorting Genetic Algorithm NV Newsvendor SBX Simulated Binary Crossover SMOP Stochastic Multiobjective Optimization Problem SOP Single-objective Optimization Problem SPEA2 Strength Pareto Evolutionary Algorithm SSOP Stochastic Single-Objective Problem TR Transshipment ZDT Zitzler-Deb-Thiele
  18. 18. -xvii- Journal Papers 1. Nabil Belgasmi, Lamjed Ben Said, Khaled Ghédira: Evolutionary multiobjective optimization of the multi-location transshipment problem. Operational Research 8(2): pp. 167-183 (2008) 2. Nabil Belgasmi, Lamjed Ben Said, Khaled Ghédira: Multiobjective Analysis of the Multi- Location Newsvendor and Transshipment Models. International Journal of Information System and Supply Chain Management (IJISSCM) 6(4): pp. 42-60 (2013) Papers in Conference Proceedings 1. Nabil Belgasmi, Lamjed Ben Said, Khaled Ghédira: Genetic Optimization of the Multi- Location Transshipment Problem with Limited Storage Capacity. European Conference on Artificial Intelligence (ECAI) 2008: pp. 563-567. 2. Slim Bechikh, Nabil Belgasmi, Lamjed Ben Said, Khaled Ghédira: PHC-NSGA-II: A Novel Multi-objective Memetic Algorithm for Continuous Optimization. International Conference on Tools with Artificial Intelligence (ICTAI) 2008: pp. 180-189. 3. Nabil Belgasmi, Lamjed Ben Said, Khaled Ghédira: Evolutionary optimization of the multiobjective transshipment problem with limited storage capacity. Winter Simulation Conference (WSC) 2009: pp. 2375-2383. 4. Nabil Belgasmi, Lamjed Ben Said, Khaled Ghédira: Greedy Local Improvement of SPEA2 Algorithm to Solve the Multiobjective Capacitated Transshipment Problem. Learning and Intelligence OptimizatioN Conference (LION-5) 2011: pp. 364-378.
  19. 19. -1- 1.1 General context Practical optimization problems, especially inventory management, seem to have a multiobjective nature much more frequently than a single objective one. The major reason for managing inventory is to cope with the demand randomness and to achieve lower costs while ensuring higher service levels. In recent years, the emergence of advanced information systems in most companies has encouraged the real-time sharing of data and information. This greatly impacted the way decision makers design their inventory optimization strategies. In this context, lateral transshipments, which are recognized as the monitored movement of material among locations at the same echelon, represent an advanced option of inventory management. They are designed to ensure inventory pooling between the system locations. That is, locations having unsold inventory are encouraged to help locations in shortage to fulfill their unmet customers’ demands (Figure 1-1). Transshipment based models have gained increasingly attention in medicine, apparel, and fashion goods, or critical repairable spare parts. Some issues emerge when using transshipments. A manager may decide whether it is interesting to allow lateral transshipments or no. He must be sure in advance that his investment is beneficial and not risky. We believe that the literature of transshipment does not provide convincing answers to such managerial problems. Most of the existent works ignore the existence of conflicting objective functions and only focus on optimizing costs or fill rate separately. Moreover, such models become intractable rather quickly if an arbitrary number of locations or cost setting is used. On the other hand, the field of Multiobjective Optimization (MO) became a very rich and active area. Evolutionary Multiobjective Optimization Algorithms (EMOAs) have proven an outstanding success in solving complex multiobjective real-world problems without a prior knowledge of the problem. These are the main reasons we believe that the transshipment is worth being investigating in a multiobjective context. Figure 1-1 Illustrative example of inventory pooling based system 1.2 Contributions
  20. 20. 1.2 Contributions -2- In this dissertation study, our contributions are connected to the inventory management and optimization area as well as the EMO research field (Figure 1-2). First, advanced inventory models based on multiple conflicting objective functions are proposed. We studied an extended variant of the reference multi-location Newsvendor model within a purely multiobjective framework, and suggested a challenging real-world application of this model, that is, the multiobjective ATM cash level optimization problem which adds to the existent real-world financial applications (Belgasmi, Ben Said, & Ghédira, 2013). Both single and multiobjective modeling of the basic transshipment model enriched with new storage capacity constraints is achieved (Belgasmi, Ben Said, & Ghédira, 2008). Detailed analysis of some objective functions properties are discussed which gave us insight to elaborate more efficient optimization techniques. Simulation-based optimization is used to solve the proposed multiobjective problems due to the two-stage nature of most of the objective functions and the absence of tractable analytical form of the optimal inventory pooling sub-problem (Belgasmi, Ben Said, & Ghédira, 2009). By solving a wide variety of bi- objective and tri-objective we enhanced the understanding of the problem (Belgasmi, Ben Said, & Ghédira, 2008). Finally, we developed a multi-item transshipment model with linear storage constraints and multiple conflicting stochastic objectives. Special interest was devoted to the scalability and the stochasticity. Interested issues behind the proposed model were captured and discussed. From an EMO viewpoint, we added to the existent literature in several ways. Besides the use of well-known EMOAs such as NSGA-II and SPEA2 to solve the problems, new EAs were proposed and extensively tested on a wide range of benchmarks. We introduced a hybrid real- coded genetic algorithm named H-RCGA to solve continuous single objective optimization problems. It is based on a quasi-gradient mutation operator designed to guide the search to promising regions of interest. New hybrid EMOA named PHC-NSGA-II designed to solve continuous multiobjective problems is suggested (Bechikh, Belgasmi, Ben Said, & Ghédira, 2008). It is based on a Pareto-guided hill climbing technique that can escape from local optima traps. New hybrid EMOA, named H-SPEA2 (Belgasmi, Ben Said, & Ghédira, 2011), designed to solve multiobjective continuous problems that have smooth landscapes is proposed. It incorporates a new multiobjective quasi-gradient mutation operator into SPEA2 that generates approximate multiobjective descent directions and use them as suggested line searches. Insight given from an empirical study of the multi-item transshipment problem encouraged us to introduce a noise handling framework based on the newly introduced concept of Multiobjective Confidence Hyperrectangles (MoCH). It is designed to improve the ability of an arbitrary selected EMOA to deal with stochastic multiobjective problems without prior restriction on the noise properties. It is based on the efficient adaptive sampling of noisier solutions (or hyperrectangles) in order to reduce the number of overlaps. We applied the MoCH framework to the standard SPEA2 and performed statistical comparison with the baseline Explicit-Averaging based SPEA2. A Java-based inventory optimization toolbox is developed, and used in all the experiments. All the proposed transshipment models can be
  21. 21. Chapter 1 - Multiobjective benchmark problems -3- instantiated and studied numerically using the simulation engine we provided in the toolbox. An advanced extension of the well-known open-source jMetal EMO framework was achieved. Figure 1-2 Methodology and main contributions of the thesis 1.3 Outline of the thesis Including this introduction, the thesis consists of seven chapters. Chapter 2 is concerned with a literature review of classical inventory management and optimization. Both Newsvendor and Transshipment models are discussed. Chapter 3 gives a basic understanding of evolutionary multiobjective optimization concepts. A brief review of memetic algorithms is presented. The common techniques to handle noisy multiobjective optimization problems are also discussed. Chapter 4 focuses on investigating the Newsvendor problem in multiobjective context. A novel hybrid EMOA (PHC-NSGA-II) is proposed. The real-world financial ATM optimization problem is mathematically formalized and solved using common EMOAs. Chapter 5 suggests a single-objective and multi-objective modeling of the basic transshipment problem. A Hybrid Real-Coded Genetic Algorithm (H-RCGA) is suggested to solve the single-objective variant. Whereas a hybrid EMOA designed for multiobjective continuous optimization is proposed to solve more efficiently the transshipment problem. Chapter 6 presents an advanced multiobjective multi-item transshipment model. Both scalability and stochasticity issues are addressed. It is also proposed a new noise handling framework based on the concept of Confidence Hyperrectangle, which is validated through the MoCH-SPEA2 algorithm. Finally, the conclusions of the thesis, as well as some possible paths for future research, are presented in chapter 7.
  22. 22. -4- 2.1 Introduction This chapter is devoted to synthesize the existent literature related to inventory management, modeling and optimization. The key components of an inventory system’ structure is presented. The common methods of determining policies of inventory management are introduced. The most common analytical models were reviewed. We also highlighted the various use of simulation to address more complex inventory models. We review both deterministic and stochastic inventory models while focusing on the impact of demand uncertainty on the form of optimal solutions. Next, the basic Newsvendor inventory model is described as well as its different extensions. Transshipment based models are emphasized referring to the most relevant related studies. The last section of the chapter is dedicated to discuss the main advantages and limitations of the existent models, and to provide insights on the extensions propounded in this thesis. 2.2 Inventory management and optimization Recently, the new fields of operations research and management science highlighted the use of quantitative techniques, and have produced a substantial literature on inventory problems. Mathematicians, economists, and statisticians have become interested in theoretical aspects of inventory problems which initially were the concern only of business managers and engineers. A number of books dealing with inventory problems have appeared in recent years, with varying levels of mathematical sophistication and abstraction. A great number of articles on inventory problems have been published. Not all of them were directly connected to practical applications. They were developed instead to exhibit certain interesting mathematical properties of the system. Some other proposed models were formulated explicitly for purposes of practical application. According to (Viale, 1996), the three most important questions to be answered by an inventory policy are: When to review stocks? A distinction is made between: Periodic review policies where stocks are reviewed at fixed time intervals, the review periods. Continuous review policies where stocks are reviewed after each transaction. When to order? A distinction is made between: Periodic review policies where orders can only be placed at the periodic review instants. Continuous review policies which use reorder points in inventory positions. What to order? A distinction is made between: Policies with a fixed order quantity;
  23. 23. Chapter 2 - Multiobjective benchmark problems -5- Policies with fixed order-up-to level. 2.2.1 Structure of inventory systems In describing the structure of any inventory system, it is helpful to include the following components (Viale, 1996): The spatial configuration: It refers to the physical structure of the inventory. For example, demands may occur at a single location or at a number of locations. Outlets may be confined to a single city or scattered over several continents. In addition, an inventory system may consist of a single-echelon or multi-echelon levels. In a single-echelon system all stocking points are the same in the sense that no point serves as a warehouse for any other point. However in multi-echelon system one or more stocking points do serve as warehouses. Figure 2-1 depicts a typical two-echelon system, a source-warehouse-retailer structure (the source is not usually considered an echelon). Figure 2-1 A two-echelon inventory system The type of control: An analysis of the way the system is controlled should include the following points: whether control is centralized or decentralized, when decisions are made, what the rules are for making decisions, what is known about the system at any point in time, and how the flow of materials, information, and funds can occur. For multi-location and/or multi-echelon systems, the extent to which decision-making is centralized or decentralized has an important impact on the system analysis. Decisions are also influenced by whether or not pooling is allowed between various stocking points at a given echelon, and if so, by what means of transportation. For multi-echelon structures, it is necessary to know whether any warehouse can help ship to any stocking point at next lowest level or whether it ships only to certain ones. The kind of items carried: Some inventory’ systems handle only a single item while others carry as many as thousands (see Chapter 6for an advanced multi-item model). These items may substitutable; they may be reparable or non-reparable; they may be perishable or non-perishable (Li, Lan, & Mawhinney, 2010); they may become obsolete very rapidly or they may be indispensable items. All these characteristics have an important influence on the systems’ behavior. A final item characteristic is the nature of the costs incurred by the inventory system carrying it. First is the item’s variable cost, which may or may not depend in an important way on the quantity procured. Then there are the transportation
  24. 24. 2.2 Inventory management and optimization -6- costs from the source, inspection costs, etc. There are also the administrative costs involved in placing orders and perhaps setup costs at the factory. Various expenses are also associated with holding the item in inventory. These include insurance, cost of floor space, rate of return requirements, etc. Finally there are the costs incurred if demands occur when the system is out of stock. These costs are very difficult if not impossible to measure in many cases; however, they must be taken into account implicitly, if not explicitly, when one analyzes an inventory system (refer to 5.9.1 Description for more details about “shortage costs”). The processes generating the stochastic variables: The time pattern of demands impinging upon an inventory system cannot be predicted with certainty (Hadley & Whitin, 1964). However, if there is sufficient regularity in the demand, it can profitably be treated as deterministic. In most cases, though, the demand must be described in probabilistic terms. Important variations in the system operation arise depending on whether the mean rate of demand remains constant over time, whether the demands are correlated in time, and on the absolute level of the mean rate of demand. In the real world, the system will almost never know, a priori, what sort of process generates demands. Information about the nature of this process is usually gained only from historical data. Arising from the nature of the demands, we should distinguish between two situations. First, demands arising when the system is out of stock may be lost forever (i.e., the customer goes elsewhere). This is the lost-sales case. Second, demands occurring when the system is out of stock may be backordered and met when procurements arrive. This is the backorder case. In some situations, it is also possible to have some sales lost and some backordered when the system is out of stock, depending perhaps on the backorders existing when a demand occurs. 2.2.2 Methods of determining policies for inventory systems Three different procedures are available as aids in developing a set of operating rules for any inventory systems (Beamon, 1998). These may be referred to as: Analytical approaches: they consist of constructing mathematical models of the system to be studied, which will be an abstraction of the real world and will require the introduction of a number of simplifying assumptions in order to handle it analytically. Simulation-based approaches: they are encouraged by the development of the high- speed digital computers which are able to obtain numerical solutions to much more complex analytical models than would otherwise be possible. To perform a simulation, one begins with the model of the system and a complete set of rules and/or assumptions for operating the system. Then the computer generates the inputs (stochastic or deterministic). These include times between demands, number of units needed per demand, and perhaps the stochastic or deterministic leadtimes. The computer uses the inputs to simulate the behavior of the system through time, and determines the inventory levels, backorders, and all other quantities of interest. It determines when to place an order; places the order at the appropriate time; and if the procurement lead-time is not determined by other parts of the
  25. 25. Chapter 2 - Multiobjective benchmark problems -7- system, generates the lead-time for that particular order. It is possible in this way to simulate many months or years of systems operation. Simulation-based approaches have been useful in studying very complex inventory systems that are difficult or impossible to examine analytically. Simulation is used especially when the analytical work becomes difficult to carry out. 2.2.3 Analytical models review There are a limited number of inventory systems that have been studied analytically. When attempts are made to optimize the system, almost all effort has been restricted to studying systems consisting of a single stage and a single outlet. A classification of the existent analytical models can be done according to the following criteria (Huang, Leng, & Parlar, 2013): Stochastic Processes Demand Deterministic, stochastic, fuzzy Constant or varying mean rate Backorders or lost sales with order size or no Markovian or non-Markovian Leadtimes Constant or stochastic Independent of or dependent upon each other Control Periodic review Continuous review Costs Ordering cost Fixed or not fixed Unit cost of item Linear or Nonlinear Backorder Cost Linear or nonlinear Shortage costs allowed or not Items Storage Constrained or unconstrained Interaction Allowed or not allowed Table 2-1 Criteria used in the classification of analytical inventory models 2.2.4 Various uses of simulation Simulation has already been mentioned as useful tool for studying inventory systems. There are varieties of ways in which simulation has been or can be used in studying inventory systems (refer to: (Evers & Wan, 2012), (Becerril-Arreola, Leng, & Parlar, 2013)). These may be conveniently classified according to how simulation is used: As a complement to analytical analysis: often simulation can provide a graphic description of the way in which some operating policy obtained from analytical models will behave when installed in the real world system. Another way in which simulation can be used along with analytical methods is to study parameter variations or to make sensitivity analyses which are difficult to do analytically.
  26. 26. 2.2 Inventory management and optimization -8- Or as an alternative to the analytical approach: For example, one might use simulation to find the optimal values of one or more parameters in an operating doctrine, in such a way as to minimize some cost expression (refer to: (Lange, Herrmann, & Claus, 2013), (Muñoz, Muñoz, & Ramírez‐López, 2013), (Akcay, 2012), ). 2.2.5 Deterministic inventory models In a deterministic model, it is often assumed that items are withdrawn from the inventory at even rate a. Replenishments are of a fixed size 𝑄, and lead time is zero or constant. The resulting behavior of the system is shown in Figure 2-2. For several operating assumptions, an analytical optimal solution of the problem exists (Pentico, 2011). Figure 2-2 Variation of inventory level in a deterministic model The common notation used in deterministic inventory models is the following (Hestenes, 2013): Ordering Cost – c(z): it is the cost of placing an order. The amount ordered is z, and the function c(z) if often nonlinear. Setup Cost – (K): making an order is usually associated to a fixed cost that is independent of the amount ordered. This fixed cost is called the “setup cost”. Product Cost (c): it is the unit cost of purchasing the product as part of an order. In some models, if the cost is independent of the amount ordered, the total cost is “c.z”. Alternatively, the product cost may be a decreasing function of the amount ordered. Holding Cost (h): it is the cost of holding an item in inventory. It usually includes the lost investment income. Other elementary costs are associated with the holding cost: cost of storage, insurance, and other factors that are proportional to the amount stored. It is an important component of the cost of inventory. Shortage Cost (p): it is an estimation of the cost paid when the on-hand inventory cannot fulfill a customer demand. This parameter is not considered in most of the deterministic models. Demand Rate (a): it is the constant rate at which the product is withdrawn from inventory (units/time). Lot Size (Q): it is the fixed quantity received at each replenishment action. Order Level (S): it is the maximum level reached by the inventory. When backorders are not allowed, it is less than Q (units).
  27. 27. Chapter 2 - Multiobjective benchmark problems -9- Cycle Time (t): it is the time between consecutive inventory replenishments. For the models of this section, it is equal to “Q/a” (time). Cost per time (T): it is the total of all costs related to the inventory system that are affected by the decision under consideration. Optimal Quantities (Q*, S*, T*): the quantities defined above that maximize profit or minimize costs for a given model are the optimal solution. 2.2.5.1 Lot Size Model with no Shortages This model is based on several assumptions: (a) inventory level ranges between 0 and the amount Q, (b) shortages are not allowed, (c) periodically an order is placed for replenishment, (d) and replenishment leadtimes are negligible. That is, the inventory level is periodically shifted from 0 to the amount Q. Between orders the inventory decreases at a constant rate a. The time between orders is called the cycle time, and is the time required to use up the amount of the order quantity, or Q/a. The total cost expressed per unit time is: Cost/unit time = Setup cost + Product cost + Holding cost 𝑇 = 𝑎𝐾 𝑄 + 𝑎𝑐 + ℎ𝑄 2 2-1 Setting to zero the derivative of T with respect to Q we obtain: 𝑑𝑇 𝑑𝑄 = − 𝑎𝐾 𝑄2 + ℎ 2 = 0 2-2 Solving for the optimal policy, 𝑄∗ = √ 2𝑎𝐾 ℎ 𝑎𝑛𝑑 𝜏∗ = 𝑄∗ 𝑎 2-3 Substituting the optimal lot size into the total cost expression, we obtain: 𝑇∗ = 𝑎𝑐 + √2𝑎ℎ𝐾 2-4 At the optimum, the holding cost is equal to the setup cost. We see that optimal inventory cost is a concave function of product flow. The optimal policy does not depend on the unit product cost. The optimal lot size increases with increasing setup cost and flow rate and decreases with increasing holding cost. 2.2.5.2 Shortages Backordered
  28. 28. 2.2 Inventory management and optimization -10- This model allows shortage to be backordered. This situation is illustrated in Figure 2-3. The inventory level decreases below the 0 level. A portion of the demand can be backlogged. The maximum inventory level is S, and occurs after replenishments. The maximum backorder level is “Q – S”. Figure 2-3 Lot size model with shortages allowed The total cost per unit time is: Cost/time = Setup cost + Product cost + Holding cost + Backorder cost 𝑇 = 𝑎𝐾 𝑄 + 𝑎𝑐 + ℎ𝑆2 2𝑄 + 𝑝(𝑄 − 𝑆)2 2𝑄 2-5 The factor multiplying h in this expression is the average on-hand inventory level. This is the positive part of the inventory curve shown in Figure 2-4. Figure 2-4 The first cycle of the lot size with backorders model Setting to zero the partial derivatives of T with respect to Q and S yields:
  29. 29. Chapter 2 - Multiobjective benchmark problems -11- 𝑆∗ = √ 2𝑎𝐾 ℎ √ 𝑝 𝑝 + ℎ 2-6 𝑄∗ = √ 2𝑎𝐾 ℎ √ 𝑝 + ℎ 𝑝 2-7 𝜏∗ = 𝑄∗ 𝑎 2-8 Comparing these results to the no shortage case, we see that the optimal lot size and the cycle times are increased by √(𝑝 + ℎ) ℎ⁄ . The ratio between the order level and the lot size depends only on the relative values of holding and backorder cost. 2.2.5.3 Quantity Discounts This deterministic model incorporates quantity discount prices that depend on the amount ordered. For this model, no shortages are allowed. We assume there are 𝑁 different prices: c1, c2… cN, with the prices decreasing with the index. The quantity level at which the k-th price becomes effective is qk, with q1 equal to zero. To determine the optimal policy for this model we observe that the optimal order quantity for the no backorder case is not affected by the product price, c. Therefore we compute the optimal lot size Q* using the parameters of the problem. 𝑄∗ = √ 2𝑎𝐾 ℎ 2-9 Find for each k the value of Qk*, such that - If Q* > qk+1 then Qk * = qk+1 - If Q* < qk then Qk * = qk, - If qk ≤ Q* < qk+1 then Qk * = Q* Let n* be the price level for which Q* lies within the quantity range: 𝑇𝑛∗ = 𝑎𝐾 𝑄∗ + 𝑎𝑐 𝑛∗ + ℎ𝑄 2 ∗ 2-10 For each level k > n*, the total cost Tk for the lot size Qk is given by:
  30. 30. 2.2 Inventory management and optimization -12- 𝑇𝑘 = 𝑎𝐾 𝑄 𝑘∗ + 𝑎𝑐 𝑘 + ℎ𝑄 𝑘∗ 2 2-11 2.2.6 Stochastic inventory models This section is concerned with inventory systems with stochastic demands. Leadtimes are assumed to be constant, and equal to L units of time. It is assumed also that each customer demands one unit and that unfulfilled demands are backlogged. The expected demand per unit of time will be denoted by E{D} and the standard deviation of the demand per unit of time by 𝜎{D}. The stochastic nature of the demand is explicitly recognized. Several models are presented later in this chapter, and are only abstractions of the real world. The interest of discussing them is about the guidance and insights they provide to understand the advanced inventory models proposed in this study. 2.2.6.1 Demand probability distribution The one feature of uncertainty considered in this section is the demand. Leadtimes are assumed to be deterministic (Schmitt, Snyder, & Shen, 2010). Without loss of generality, the demand is considered unknown, but that its probability distribution is known. The following notion is commonly used: Random Variable for Demand (x) Discrete Demand Probability Distribution Function (P(x)) Discrete Cumulative Distribution Function (F(b)) Continuous Demand Probability Density Function (f(x)) Continuous Cumulative Distribution Function (F(b)) Standard Normal Distribution Function In the following we abbreviate probability distribution function or probability density function as pdf. We abbreviate the cumulative distribution function as CDF. Later in this report, we are often concerned about the relation of demand during some time period. If the demand is less than the initial inventory level, it is a condition of excess. If the Stochastic demand is greater than the initial inventory level, we have the condition of shortage. At some point, assume the inventory level is a positive value 𝑧. During some interval of time, the demand is a random variable 𝑥 with pdf, 𝑓(𝑥) , and CDF, 𝐹(𝑥) . With the given distribution, we compute the probability of a shortage, 𝑃𝑠, and the probability of excess, 𝑃𝑒. For a continuous distribution, we have the following formulas:
  31. 31. Chapter 2 - Multiobjective benchmark problems -13- 𝑃𝑆 = 𝑃{𝑥 > 𝑧} = ∫ 𝑓(𝑥)𝑑𝑥 = 1 − 𝐹(𝑧) ∞ 𝑧 2-12 𝑃𝑒 = 𝑃{𝑥 ≤ 𝑧} = ∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑧) 𝑧 0 2-13 2.2.6.2 Continuous review models The most commonly used continuous review policies are (Bijvank & Vis, 2011): Single period model with no setup cost: (s; Q) policy: whenever the inventory position drops at or below the reorder point ‘s’ a fixed quantity Q is ordered; (s; S) policy: whenever the inventory position drops at or below the reorder point s an order is placed of a size that brings the inventory position to the order-up-to level S. Policies with an order-up-to level are also called base-stock policies. Remark that in case of unit demand per customer, the (s; Q) and (s; S) policies are equivalent when Q = S - s. 2.3 Newsvendor inventory models The Newsvendor problem (or newsboy or single-period (William, 2009)) is one of the fundamental problems in Inventory Management. It is a mathematical model in operations management and applied economics used to determine optimal inventory levels in the presence of uncertain demand and simple cost structure. It is often considered as the basic model on which were based most of the proposed Transshipment models (Qin, Wang, Vakharia, Chen, & Seref, 2011). The newsboy or Newsvendor problem is to find the order quantity which maximizes the expected profit (or minimize the expected cost) in a single period probabilistic demand framework. A decision maker orders inventory before the starting of a selling season characterized by a stochastic demand. If too much is ordered, stock is left over at the end of the period and the retailer incurs additional holding costs. The manager must dispose of the remaining stock at a loss. If the order quantity is lower than the realized demand, the manager forgoes some profit. Customers' disappointment is implicitly considered as a paid penalty cost. Therefore, in choosing an order quantity, the manager must balance the costs of “ordering too much” against the costs of “ordering too little”. The Newsvendor problem applies in a large array of cases. In the fast-moving computer business, International Business Machine produced $700 million of excess inventory of their Value-Point line, but in another year, they under produced their Aptiva® PC line, and lost potential revenues of more than $100 million (Ziegler, 1994). Both oversized inventories and low fill rates are costly management strategies. Business may be lost through canceled orders, and the company's reputation may be severely damaged. It is therefore in a company's interest
  32. 32. 2.3 Newsvendor inventory models -14- to balance inventory holding cost and the cost of imperfect customer satisfaction. The trade-off inventory vs. customer satisfaction is one of the classic issues of logistics and supply chain management. The objective of the basic Newsvendor problem is to find a product's order quantity that maximizes the expected profit (or minimizes the expected cost) under probabilistic demand. The model assumes that if any inventory remains at the end of the period, a discount is used to sell it or it is disposed of (Nahmias S. , 1996). If the order quantity is smaller than the realized demand, the newsvendor forgoes some profit. The model is reflective of many real life situations and is often used to aid decision making in the fashion and sporting industries, both at the manufacturing and retail levels (Gallego & Moon, 1993). The model can also be used in managing capacity and evaluating advanced booking of orders in service industries such as airlines and hotels (Weatherford & Pfeifer, 1994). Interest in the Newsvendor has increased in the last decade with hundreds of papers published since 1988. The related literature is very large and complete coverage is beyond the scope of a single section (refer to the following references for more details: (Petruzzi & Dada, 2010), (Wu, Zhu, & Teunter, 2013), (Käki, Liesiö, Salo, & Talluri, 2013)). Recent studies were interested in additional aspects of the problem such as mean-variance modeling. Special interest has been assigned to profit maximization or expected cost minimization in a single or multiple independent newsvendors. (Chen, Sim, Simchi-Levi, & Sun, 2003) proved that without stockout cost, the variance function of the stochastic profit is a monotone increasing function of order quantity, so the mean-variance trade-offs can be carried out efficiently. However, if stockout cost is considered, the variance function will lose this monotonicity property and the mean-variance trade-offs becomes much more complicated. (Wu J. , Li, Wang, & Cheng, 2009) applied a mean-variance approach to analyze the risk-averse problem with stockout cost. They derived an explicit form of the cost variance and analyzed the mean variance trade-offs under stockout situation. 2.3.1 The basic newsvendor model Researchers have followed two approaches to solving the problem (Qin, Wang, Vakharia, Chen, & Seref, 2011). In the first approach, the expected costs of overestimating and underestimating demand are minimized. In the second approach, the expected profit is maximized. Both approaches yield the same results. Define the following notation:
  33. 33. Chapter 2 - Multiobjective benchmark problems -15- The profit per period is Simplifying and taking the expected value of π gives the following expected profit: Let the superscript * denote optimality. Using Leibniz's rule to obtain the first and second derivatives shows that E(π) is concave. The sufficient optimality condition is the well-known fractile formula: Identical results can be obtained by minimizing the expected underage and overage costs. Many authors describe the overage cost (Co) as a cost of holding inventory which is charged to the ending inventory. 2.3.2 Extensions of the basic newsvendor model The basic model has wide applicability especially in service industries (Choi T. M., 2012). Many extensions were investigated: Extensions to different objectives and utility functions. Extensions to different supplier pricing policies. Extensions to different Newsvendor pricing policies and discounting structures. Extensions to random yields. Extensions to different states of information about demand. Extensions to constrained multi-product. Extensions to multi-product with substitution. Extensions to multi-echelon systems. Extensions to multi-location models. Extensions to models with more than one period to prepare for the selling season. 2.3.3 Handling different objective functions Researchers observed that maximizing E(π) may not reflect reality. Actually, maximizing the probability of achieving a target profit was empirically found to be more consistent with the actions of many managers (Lanzilotti, 1958). Subsequently, researchers proposed extensions to
  34. 34. 2.3 Newsvendor inventory models -16- the basic Newsvendor model in which the goal is to maximize the probability of achieving a target profit. Other authors used different effectiveness criterion, risk tolerance and utility functions (refer to: (Anvari, 1987), (Choi S. &., 2011)). Kabak and Schi (Kabak & Schi, 1978) solved the basic Newsvendor problem under the objective of maximizing the probability of achieving a target profit of B, denoted PB. Kabak and Schi derived the necessary condition for Q* and provided a closed-form solution for exponentially distributed demand. Work on a variation of the Newsvendor model was also being carried out under the Cost-Volume-Profit (C-V-P) analysis. The C-V-P states: Shih (Shih W. , 1979) observed that a deficiency in the stochastic C-V-P is that even though demand is treated as a random variable, the effects of any unsold units on profit were not taken into account. Shih considered the effects of over production and derived a general probability distribution of p, its expected value, and its variance as a function of Q. For normally distributed demand, Shih derived the probability distribution, the mean and variance of p. Also, Shih derived an expression for Q which maximizes PB. Finley and Liao (Finley & Liao, 1981) claimed that Shih's analysis may be erroneous and proposed an analysis which fixes the error. Lau and Lau provided simpler formulas for computing the mean and variance of p and the order quantity which maximizes PB. Lau and Lau (Lau & Lau, 1988) revisited the Newsvendor problem under the objective of maximizing PB for the two-product case. Lau and Lau considered S=0 and S>0 separately because the latter is more complex. Lau and Lau identified three approaches to estimating PB and Qi*, i=1,2: (1) simulate the problem, (2) develop an expression for PB and find Qi*, i=1, 2 using a “hill-climbing” procedure and (3) analytically solve the first order conditions. Approach 2 was found to be the only practical one. Lau and Lau provided numerical solution to a two-product Newsvendor problem with uniform and normal demands and provided some insights into management behavior. Lau and Lau then derived general expressions for PB and closed-form expressions for it under uniform demand distributions. Lau and Lau derived Qi*, i=1,2 for two identical products. Lau and Lau found some counter-intuitive results. For example, if a firm has two single-product divisions and each division will receive a bonus for achieving a certain profit, it is beneficial for the divisions to cooperate if the targets are lax and profit margins are high, but not if the targets are high and margins are low. Thakkar et al. (Thakkar, Finley, & Liao, 1983) argued that the maximizing E(π) or PB criteria do not consider the investment that must be made to attain that profit and used Return On Investment (ROI) as a criteria. Thakkar et al. solved the Newsvendor problem under two objectives: (a) maximizing E(ROI) and (b) maximizing the probability of achieving a target ROI, PROI. Using incremental analysis to maximize E(ROI), Thakkar et al. showed there is a unique Q* and derived the necessary optimality condition. Thakkar et al. simplified the
  35. 35. Chapter 2 - Multiobjective benchmark problems -17- necessary condition for the normal distribution to an equation that can be manually solved. For maximizing PROI, Thakkar et al. showed that a simple iterative procedure can be used to solve the discrete demand case and a search procedure can be used for continuous demand and obtained a closed-form solution for normally distributed demand. 2.3.4 Extensions to random yields Scholars have suggested an extension of the basic Newsvendor model in which 𝑄 contains defective units (Ehrhardt & Taube, 1987) or the available production capacity is a random variable. (Shih W. , 1980) assumed defective units are unsalable and are returned to the manufacturer at his/her expense. He also assumed that the percentage of defectives (ρ) is a random variable with known probability distribution. Shih derived the expected cost function and provided proof of its convexity and derived the necessary optimality condition for Q for any distribution of x and ρ. For uniformly distributed demand, Ehrhardt and Taube provided a closed-form expression for Q*. Ehrhardt and Taube provided a heuristic when (ρ) has a Beta distribution and demand follows a negative binomial distribution. 2.3.5 Extensions to constrained multi-products Hadley and Whitin (Hadley & Whitin, 1963) solved the multi-product Newsvendor under a storage (or budget) constraint. Hadley and Whitin developed two algorithms. The first is based on a search for the Lagrangian multiplier that satisfies the necessary conditions. Results of this algorithm will have to be rounded to integers and thus are suitable when the Qi* are large. For the case when the Qi* are small and rounding may have a significant impact on E(π*), Hadley and Whitin developed a marginal analysis approach to find an integer solution. Nahmias and Schmidt (Nahmias & Schmidt, 1984) provided four heuristics for solving the single-constraint Newsvendor under normally distributed demand. One of the heuristics required fewer computations and provided good solutions relative to the Lagrange method. The procedure is useful for continuous Q and is thus appropriate for moderate-to high demand items. For a 5000-item problem, Nahmias and Schmidt's heuristic required a computing time of 5 seconds versus 2 hours for the Lagrange method on a DEC 2060 system. Lau and Lau (Lau & Hing-Ling Lau, 1995) solved a multi-product multi-constraint Newsvendor problem. Since evaluating 𝐸(𝜋) involves many integrals which is time consuming, a direct search procedure to numerically evaluate 𝐸(𝜋) is inappropriate. Also, since a typical newsstand will have a large number of products and much smaller number of constraints, the N-variable “primal problem” should be converted to M-variable “dual problem”. Lau and Lau developed a procedure based on the “active set methods”, tested the procedure against state-of-the-art nonlinear programming software and found that their procedure was much faster and provided better quality solutions. 2.3.6 Extensions to multi-echelon systems Gerchak and Zhang (Gerchak & Zhang, 1992) developed a two-echelon Newsvendor model in
  36. 36. 2.4 Multi-location transshipment models -18- which decisions have to be made on whether to hold inventories in the form of raw materials or finished products. Holding raw material is less costly but if demand turns out to be high then a fraction of demand is lost since some customers might not be willing to wait for the conversion of raw materials into finished goods. Gerchak and Zhang assumed that there is an initial inventory at both stages and derived 𝑬(𝜋) as a function of the inventory levels and proved its concavity. Gerchak and Zhang then derived the optimal inventory policy. Their conclusion can be summarized as: if both initial stocks are high nothing will be ordered, and, depending on the proportion of finished products in the total stock, some raw material might be processed. If the stock of finished products is high but the combined stock is low, only raw materials should be ordered. If both stocks are low, both items should be ordered up to constant level derived by Gerchak and Zhang. 2.4 Multi-location transshipment models Unlike the Newsvendor model, a system based on lateral transshipments allows the unsold inventories to be moved from locations with surplus inventory to fulfill more unmet demands at stocked out locations (Figure 2-5). Both models were thoroughly studied and researches were usually confined to cost minimization or profit maximization. Physical pooling of inventories has been widely used in practice to reduce cost and improve customer service (Herer, Tzur, & Yücesan, 2006). Transshipments are recognized as the monitored movement of material among locations at the same echelon. It affords a valuable mechanism for correcting the discrepancies between the locations’ observed demand and their on-hand inventory. Subsequently, Transshipments may reduce costs and improve service without increasing the system-wide inventories. Figure 2-5 Relation between the basic Newsvendor and Transshipment models The main precondition of successful implementation of transshipment is well-established information systems. At present many large modern companies connected by information systems can control the relationships of many branches, and thus they may be ready to gain cost reduction and service improvement associated with lateral transshipment (Paterson, Kiesmüller, Teunter, & Glazebrook, 2011). The overall performance of the distribution network, whether evaluated in economic terms or in terms of customer service, can be substantially improved if the retailers collaborate in the occurrence of unexpectedly high demand, which may result in shortages in one or more retailing outlets. Collaboration usually takes the form of lateral inventory transshipment from a stock outlet with a surplus of on-hand inventory to another outlet that faces a stockout. Since the cost of transshipment in practice is generally lower than both the shortage cost and the cost of an emergency delivery from the designated warehouse and the transshipment time is shorter than the regular replenishment lead time, lateral transshipment simultaneously reduces the total system cost and increases the
  37. 37. Chapter 2 - Multiobjective benchmark problems -19- fill rates at the retailers. A group of stocking locations that share their inventory in this manner is to form a pooling group, since they effectively share their stock to reduce the risk of shortages and provide better service at lower cost. In this chapter we mainly focus on presenting a comprehensive description, classification, methodologies and solution procedures of transshipment models in supply chain system. 2.4.1 Assumptions There are several basic assumptions that are commonly seen in the literature of transshipment (Paterson, Kiesmüller, Teunter, & Glazebrook, 2011) such as the behaviors of demand occurrence, transshipment time, repair time, and transshipping priority rule, etc… They are stated as follows. 2.4.1.1 Demands The behaviors of demand occurrence are usually characterized by the time between demands and the distribution of demand size. The time between demands is commonly assumed to follow an Exponential or Gamma distribution. However, the distributions of demand size per each demand occurrence depend on the characteristics of the investigated industry. For example, it was taken as Weilbull distribution for spare parts which have slow-moving, expensive and lumpy demand pattern (Kukreja & Schmidt, 2005). (Needham & Evers, 1998) assume the normal distribution truncated at zero for military spare parts. A drawback of using the normal distribution is that it is less appropriate for low volume items; however, it does not place any restriction on the values of the mean and variance (compared to, say, the Poisson distribution which requires that they must be equal). In addition, its properties are well known and it is typically the basis for examining continuous demand. In a large amount transshipment literature the behaviors of demand are alternatively characterized by assuming what distribution the average demand per time period. 2.4.1.2 Transshipment and replenishments leadtimes The majority literature assumes transshipment leadtime to be negligible especially if it does not exceed one day. At present only some papers account for the non-negligible transshipment time. The transshipment time are assumed to be shorter than emergency supply. In other words, lateral transshipments are faster and cheaper than emergency supplies because all firms in the pooling group should be at close distance to each other. Otherwise it makes no sense to pool the item inventories. (Gong & Yucesan, 2006) formulated a multi-location transshipment problem with positive replenish lead time. They used simulation optimization by combining an LP/Network flow in corporate with infinitesimal perturbation analysis (IPA) to analyze the problem, and obtains the optimal base stock quantities through sample path optimization. 2.4.1.3 Complete/Partial pooling
  38. 38. 2.4 Multi-location transshipment models -20- A significant amount of literature in transshipment assumed that complete pooling policy is to be applied. This is part of the agreement between the cooperating companies. When the demand at a location cannot be met from on-hand inventory, it is met via transshipment(s) from other outlet(s) in a way that minimizes the transshipping cost. A unit demand is backordered if it cannot be satisfied via transshipment, in other words when there are no units in the system. 2.4.2 Transshipment policies As the literature and practice suggested, there are two classes of transshipment. (Lee, Jung, & Jeon, 2007) proposed that lateral transshipment can be divided into two categories: emergency lateral transshipment (ELT) and preventive lateral transshipment (PLT). ELT directs emergency redistribution from a retailer with ample stock to a retailer that has reached stockout. However, PLT reduces risk by redistributing stock between retailers that anticipate stockout before the realization of customer demands. In short, ELT responds to stockout while PLT reduces the risk of future stockout. Figure 2-6 Lateral transshipments (Paterson, Kiesmüller, Teunter, & Glazebrook, 2011) 2.4.2.1 Lateral transshipments based on availability (TBA) policy This implies that either all current transshipment needs have been met, or the total available transshipment quantity among all the excess locations has been exhausted. TBA, or called ELT, allows the transshipment decisions to be made more than once during a review cycle, based on the transshipment order point signal (Chiou, 2008). 2.4.2.2 Lateral transshipments for inventory equalization (TIE) policy Under this policy, the transshipment decisions are based on the concept of inventory balancing or equalization through stock redistribution. One of the first such models is due to (Gross, 1963) who characterized an optimal policy for a two-location system in which replenishment and transshipment decisions are taken together at the beginning of each period. (Das, 1975) analyzed a variant of this model in which the transshipment decision is taken at a fixed point

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