Supply chain optimization:location, production, inventory and distribution

Thèse

Maryam Darvish

Doctorat en Sciences de l’administration – opérations et systèmes dedécision

Philosophiæ doctor (Ph.D.)

Québec, Canada

© Maryam Darvish, 2017

Supply chain optimization:location, production, inventory and distribution

Thèse

Maryam Darvish

Sous la direction de:

Leandro Callegari Coelho, directeur de recherche

Résumé

L’environnement concurrentiel des affaires et la mondialisation oblige actuellement les

entreprises à accorder une importance particulière à la performance de leur chaîne

d’approvisionnement. Afin de se démarquer, les entreprises sont contraintes de prévoir et

de gérer des paramètres de performance souvent contradictoires à savoir réduire le coût de

leur chaîne d’approvisionnement et augmenter la qualité du service offert à leur clientèle. À

cet égard, la planification intégrée de la chaîne logistique c’est-à-dire que les décisions relatives

à la l’emplacement des usines, l’approvisionnement, la production, le stockage et la distribu-

tion s’avèrent majeures dans le gain d’efficience des entreprises et la réactivité d’une chaîne

d’approvisionnement.

La production et la planification de la distribution constituent deux opérations fondamentales

dans la gestion de la chaîne logistique. Couramment, elles sont traitées de manière distincte

du fait de leur complexité. Les coûts d’inventaire sont élevés dans cette approche dite aussi

linéaire ou hiérarchique, eu égard à la nécessité de respecter les délais de traitement des

commandes et d’assurer la satisfaction des clients. Cependant, il devient impératif de ne plus

négliger les liens existants entre les décisions prises pour gérer la production et la distribution,

afin de réaliser des économies de coûts dans la chaîne d’approvisionnement. Bien que l’intérêt

pour la planification intégrée de chaîne d’approvisionnement soit grandissant, les modèles

d’optimisation actuels laissent encore place à l’amélioration afin d’être plus réalistes.

Dans cette recherche, nous étudions des problèmes logistiques riches et intégrés. La perti-

nence de notre apport réside dans l’ajout de variables opérationnelles telles que les fenêtres de

temps de service ou la configuration de réseaux flexibles ainsi que de certaines caractéristiques

environnementales, dans des modèles logistiques. Notre objectif est de mettre en évidence les

valeurs d’intégration, en termes d’économies de coûts et de réduction de gaz à effet de serre.

Premièrement, nous décrivons, modélisons et résolvons le problème qui se présent chez un

partenaire industriel qui fabrique un seul produit. Dans ce système, la capacité de production

et le niveau d’inventaire sont limités, les transferts inter-usines sont permis et les fenêtres de

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délais de livraison sont flexibles. En nous appuyant sur un vaste ensemble de données réelles

collectées chez notre partenaire, nous comparons l’approche intégrée avec plusieurs scénarios

de pratique courante. Nous utilisons une méthode exacte qui permet de résoudre chacun

des scénarios et étudions les compromis entre les coûts et les niveaux de service dans une

analyse de sensibilité détaillée. Les résultats obtenus démontrent ainsi comment l’application

d’une approche synchronisée et holistique dans la prise de décision apporte de nombreuses

opportunités bénéfiques pour les systèmes logistiques en général.

Nous étendons par la suite notre étude aux systèmes de production multi-produits et multi-

échelons. Cette nouvelle configuration du problème implique que les produits sont expédiés

aux clients par l’intermédiaire d’un ensemble de centres de distribution dont le producteur

peut contrôler l’emplacement. Dans cette analyse, la conception de notre réseau de transport

est flexible puisqu’il peut varier au cours du temps. Plus le problème est riche et s’approche

de la réalité, plus le problème devient difficile et compliqué à résoudre. Les meilleures solu-

tions issues de l’approche intégrée sont obtenues au détriment d’une plus grande complexité

d’implémentation et de l’allongement du temps d’exécution. Nous décrivons, modélisons et

résolvons le problème en utilisant d’abord des approches de prise de décision intégrées puis

des approches séquentielles afin de déterminer à quel le moment l’usage d’une approche plus

complexe est avantageuse pour résoudre le problème. Les résultats confirment la pertinence

de l’approche intégrée comparativement à l’approche séquentielle.

Pour illustrer l’importance des économies réalisées grâce au caractère flexible de la conception

de réseaux et des fenêtres de temps de livraison, nous décrivons, modélisons et résolvons un

problème de localisation-tournées de vehicules intégré et flexible à deux échelons. Dans ce pro-

blème, le fournisseur livre la marchandise à ses clients grâce à un réseau d’approvisionnement

à deux échelons, avec une pénalité pour chaque demande non réalisée dans la fenêtre de li-

vraison prédéterminée. La problématique est ici traitée dans une configuration plus riche; la

livraison est planifiée en tournées de véhicules.

Le quatrième volet de cette thèse s’intéresse aux impacts environnementaux des décisions lo-

gistiques. En effet, le plus souvent, les recherches scientifiques sur l’optimisation des chaînes

d’approvisionnement se concentrent uniquement sur les aspects économiques du développe-

ment durable et tendent à ignorer les deux autres dimensions. Nous abordons donc des pro-

blématiques d’optimisation connues sous de nouveaux angles. Nous étudions deux systèmes

intégrés de production, d’inventaire, de localisation et de distribution dans lesquels une mar-

chandise produite à une usine est livrée aux détaillants dans un horizon de temps fini. Une

analyse de sensibilité élaborée nous permet d’améliorer nos connaissances sur les coûts et les

émissions dans les chaînes d’approvisionnement intégrées, en plus d’améliorer notre compré-

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hension des coûts associés à l’implantation de solutions respectueuses de l’environnement.

Dans cette thèse, nous visons non seulement une meilleure compréhension de l’approche en-

globante de logistique intégrée mais nous développons également des outils opérationnels pour

son application dans des cas complexes concrets. Nous proposons ainsi de nouveaux mo-

dèles d’affaires capables d’améliorer la performance de la chaine d’approvisionnement tout en

développant des techniques d’implémentation mathématiques efficaces et efficientes.

Mots Clés: Optimisation intégrée, Problème de dimensionnement dynamique de lots, Fe-

nêtres de temps de livraison, Problèmes de localisation, Distribution.

v

Abstract

Today’s challenging and competitive global business environment forces companies to place a

premium upon the performance of their supply chains. The key to success lies in understanding

and managing several contradicting performance metrics. Companies are compelled to keep

their supply chain costs low and to maintain the service level high. In this regard, integrated

planning of important supply chain decisions such as location, procurement, production, in-

ventory, and distribution has proved to be valuable in gaining efficiency and responsiveness.

Two fundamental operations in supply chain management are production and distribution

planning. Traditionally, mainly due to the high complexity and difficulty of these operations,

they have been treated separately. This hierarchical or sequential decision making approach

imposes high inventory holding cost, as in the traditional approach inventory plays an impor-

tant role in timely satisfying the demand. However, in the era of supply chain cost reduction,

it is becoming increasingly apparent that the interrelations between different decisions, and

especially production and distribution decisions, can no longer be neglected. Although the

research interest in the integrated supply chain planning has been recently growing, there is

still much room to further improve and make the existing models more realistic.

Throughout this research, we investigate different rich integrated problems. The richness of

the models stems from real-world features such as delivery time windows, flexible network

designs, and incorporation of environmental concerns. Our purpose is to highlight the values

of integration, in terms of cost savings and greenhouse gas emission reduction.

First, we describe, model, and solve a plant-customer, single product setting in which pro-

duction and inventory are capacitated and inter-plant transshipment is allowed. The problem

is flexible in terms of delivery due dates to customers, as we define a delivery time window.

Using a large real dataset inspired from an industrial partner, we compare the integrated ap-

proach with several current practice scenarios. We use an exact method to find the solution of

each scenario and study the trade-offs between cost and service level in a detailed sensitivity

analysis. Our results indicate how the use of a synchronized and holistic approach to decision

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making provides abundant opportunities for logistics systems in general.

We further extend our study by considering a multi-product and multi-echelon setting. In

this problem, products are shipped to customers through a set of distribution centers, and the

producer has control over their locations. In this study our network design is flexible since it

may change over time. As the problem gets richer and more realistic, it also becomes more

complex and difficult to solve. Better solutions from the integrated approach are obtained at

the expense of higher implementation complexity and execution time. We describe and model

the problem, and solve it with both integrated and sequential decision making approaches to

indicate when the use of a more complex approach is beneficial. Our work provides insights

on the value of the integrated approach compared to the sequential one.

To highlight how the two types of flexibility, from the network design and from the delivery

time windows, lead to economic savings, we describe, model, and solve an integrated flexible

two-echelon location routing problem. In this problem a supplier delivers a commodity to the

customers through a two-echelon supply network. Here, we also consider a penalty for each

demand that is not satisfied within the pre-specified time window. The problem is studied in

a richer setting, as the distribution is conducted via vehicle routing.

The fourth part of this thesis addresses the environmental impacts of logistic decisions. Tradi-

tionally, supply chain optimization has merely concentrated on costs or the economic aspects

of sustainability, neglecting its environmental and social aspects. Aiming to compare the

effect of operational decisions not only on costs but also on greenhouse gas emissions, we

reassess some well-known logistic optimization problems under new objectives. We study two

integrated systems dealing with production, inventory, and routing decisions, in which a com-

modity produced at the plant is shipped to the retailers over a finite time horizon. We provide

elaborated sensitivity analyses allowing us to gain useful managerial implications on the costs

and emissions in integrated supply chains, besides important insights on the cost of being

environmentally friendly.

In this thesis, we aim not only to better understand the integrated logistics as a whole but also

to provide useful operational tools for its exploitation. We propose new business models capa-

ble of enhancing supply chain performance while at the same time developing mathematical

and technical implementation for its effective and efficient use.

Keywords: Integrated optimization; Dynamic lot-sizing; Delivery time window; Location

analysis; Distribution

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Contents

Résumé iii

Abstract vi

Contents viii

List of Tables x

List of Figures xii

Acknowledgments xiii

Avant-propos xiv

Introduction 1

1 Literature review 51.1 Well-known integrated problems . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Integrated production-distribution problems . . . . . . . . . . . . . . . . . . 91.3 Discussions and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2 A dynamic multi-plant lot-sizing and distribution problem 232.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.4 Computational experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Sequential versus integrated optimization: production, location, inven-tory control and distribution 383.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3 Problem description and mathematical formulation . . . . . . . . . . . . . . 433.4 Sequential and lower bound procedures . . . . . . . . . . . . . . . . . . . . . 473.5 Integrated solution algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 493.6 Computational experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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4 Flexible two-echelon location routing 614.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.3 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.4 Computational experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5 Minimizing emissions in integrated distribution problems 795.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.2 Problem descriptions and formulations . . . . . . . . . . . . . . . . . . . . . 825.3 Computational experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6 Conclusion 98

Bibliography 101

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List of Tables

1.1 Supply chain integrated planning . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Single product production-distribution problems . . . . . . . . . . . . . . . . . 151.3 Multi-product production-distribution problems . . . . . . . . . . . . . . . . . 191.4 Numerical experiments and case studies on integrated production-distribution

problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1 Notations of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Total costs for different production scenarios . . . . . . . . . . . . . . . . . . . 322.3 Percentage of contribution to the total cost per component . . . . . . . . . . . 332.4 Percentage of cost changes with respect to the optimized solutions . . . . . . . 352.5 Combined effects for the transportation and setup costs . . . . . . . . . . . . . 36

3.1 Integrated production-distribution problems . . . . . . . . . . . . . . . . . . . 413.2 Notation used in the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 Input parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.4 Results from the branch-and-bound algorithm . . . . . . . . . . . . . . . . . . . 553.5 Heuristics results for r = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.6 Heuristics results for r = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.7 Heuristics results for r = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.8 Heuristics results for r = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.9 Average time for the proposed method to obtain its best solution . . . . . . . . 58

4.1 Input parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.2 DC availability in the fixed network design with r = 0 . . . . . . . . . . . . . . 724.3 Fixed vs. flexible network designs with r = 0 . . . . . . . . . . . . . . . . . . . 734.4 Value of flexibility from due dates for 3 DCs and fixed network design . . . . . 744.5 Value of flexibility from due dates for 3 DCs and flexible network design . . . . 754.6 Cost of fixed and flexible designs for 3 DCs with different due dates . . . . . . . 764.7 Comparison between the most inflexible and the most flexible scenarios . . . . 774.8 Computation time of fixed and flexible network designs . . . . . . . . . . . . . 78

5.1 Performance summary for IRP . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.2 Comparison of solutions with different objective functions for the IRP . . . . . 895.3 Performance summary for PRP . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.4 Comparison of solutions with different objective functions for PRP-Class I . . . 905.5 Comparison of solutions with different objective functions for PRP-Class II . . 915.6 Comparison of solutions with different objective functions for PRP-Class III . . 925.7 Comparison of solutions with different objective functions for PRP-Class IV . . 92

x

5.8 Average inventory KPIs for IRP . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.9 Average delivery KPIs for IRP . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.10 Average load KPIs for IRP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.11 Average inventory KPIs for PRP . . . . . . . . . . . . . . . . . . . . . . . . . . 955.12 Average delivery KPIs for PRP . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.13 Average load KPIs for PRP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

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List of Figures

2.1 An instance of the dynamic multi-plant lot-sizing and distribution problem withtwo plants and three customers . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2 Feasibility of a delivery plan with respect to the maximum lateness allowed . . 312.3 An example of production and inventory capacities in high (a) versus low (b)

demand periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1 Comparison between time (s) and gap (%) of CPLEX and the proposed matheuris-tic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

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Acknowledgments

First and foremost, I wish to thank my advisor, Professor Leandro C. Coelho. This work

would have not been possible without his support, guidance, and continuous encouragement.

The way he cared about my research and his prompt replies to my questions and emails have

been just exceptionally awesome. His passion and enthusiasm for applied research had and

will always inspire me. Thank you for showing me the light.

I will forever be thankful to my former supervisor and mentor Professor Remy Glardon. I

have been extremely lucky to have had his enormous support. In 2011 he welcomed me to

his team at École polytechnique fédérale de Lausanne, and he never stopped caring since then

and finally, he helped me to start my studies at Laval University. Thanks Remy for caring

like family.

I wish to thank my committee members: Jacques Renaud, Yan Cimon, Claudia Archetti, and

Raf Jans for their time and support.

I would like to thank administrative and technical staff members at the Faculty of Business

Administration, the Ph.D. program committee, and the CIRRELT for their support. I would

also thank the professors, my friends, and colleagues from the CIRRELT and the faculty of

business administration.

Last but not least, I would like to offer my heartfelt thanks to my friends and family who have

been with me throughout this journey. I want to express sincere gratitude to my family for all

their love, never-ending patience, and encouragement. I would like to thank my parents who

supported me in all my pursuits. Thanks for being the ultimate role models in my life.

xiii

Preface

This thesis presents my work as a Ph.D. student completed during the program of study at

the Centre Interuniversitaire de recherche sur les Réseaux d’Entreprise, la Logistique et le

Transport (CIRRELT) at the Faculty of Business Administration of Laval University. The

thesis consists of four papers; each of which is written in collaboration with other researchers,

mainly my director Leandro Callegari Coelho. One of these papers is already published and the

other three are currently under revision. In all four papers, I remain the first author and have

played the major role of setting up and conducting the research, modeling and implementation

of the the problem, analyzing the results, preparing and writing the papers.

The first paper entitled A dynamic multi-plant lot-sizing and distribution problem is written

in collaboration with Homero Larrain and Leandro C. Coelho. The paper is accepted by

the International Journal of Production Research (IJPR) on February 8, 2016 and published

online on March 2, 2016.

The second paper entitled Sequential versus integrated optimization: production, location,

inventory control and distribution is written under supervision of my director Leandro C.

Coelho.

The third paper entitled Flexible two-echelon location routing is written in collaboration with

Claudia Archetti, Leandro C. Coelho, and Maria Grazia Speranza.

The fourth paper entitled Minimizing emissions in integrated distribution problems is written

in collaboration with Claudia Archetti and Leandro C. Coelho.

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Introduction

Supply chain is an integrated network of facilities performing several logistics functions of

procurement, production, storage, and distribution of products to the market. The goal of

a supply chain is to offer the final customer, the right product, at the right time, price,

quantity, quality, and location [Christopher and Towill, 2001, Katz et al., 2003]. To attain

this goal, a set of decisions and actions are to be made and taken. Thus, a supply chain

could be considered as a network of interrelated decisions. Among all supply chain decisions,

procurement, location, production, inventory and distribution constitute the essential and

fundamental components of planning processes. Traditionally, these decisions were treated

independent from one another, mainly by considering a hierarchical approach toward decision

making. Therefore, in this approach, all the interactions among different interrelated supply

chain functions are overlooked [Cohen and Lee, 1988]. Ignoring the dependencies between

supply chain decisions results in higher costs, as any attempt in minimizing cost in one area,

mostly results in cost increase in other areas [Adulyasak et al., 2012].

With the competition among companies becoming more intense, the keys to the success of

the supply chains are known to be their efficiency and responsiveness level [Manatkar et al.,

2016]. These two have become the new strategies for supply chains; efficiency aims at deliv-

ering products at reduced costs, but competitive advantage is gained not only by reducing

supply chain costs but also by providing a faster and more flexible service to customers. As

companies have realized that dramatic improvements are to occur by exploiting integrated

production systems, the linchpin of success in today’s medium and large companies is known

to be the supply chain integration [Archetti et al., 2011]. It is not long since it was realized

how integration and coordination of supply chain decisions yield competitive and economic

advantage [Hein and Almeder, 2016]. The modern world both facilitates and necessitates a

more comprehensive and holistic view of supply chains. Coordinating and integrating decisions

of the supply chain is a complex task and mainly due to its complexity, they were investi-

gated separately [Ekşioğlu et al., 2006]. However, with the improvement of computational

ability of modern computers, we are now able to solve problems far larger and more complex

1

than previously thought in much less time. From the academic point of view, in the past

few decades, the idea of simultaneously considering different functions of the supply chain has

attracted growing attention. The new models integrate several functions of supply chains into

single models and optimize them. In most industries, transportation and distribution costs

are the main contributors to the total cost [Boudia et al., 2007]. Therefore, several supply

chain decisions are mainly integrated with distribution planning.

Building upon the insightful literature, this research contributes to the integrated optimization

literature by adding distinctive features to the problems and making them more real-world

relevant. This thesis first shows the advantages of taking an integrated approach to decision

making in complex decision settings, then it explores the idea of flexibility, in terms of delivery

to customers and the supply chain network design, and finally it investigates the trade-offs

between emission reduction and economic performance objectives in an integrated optimization

context.

A number of success stories dealing with the integration of different supply chain planning

problems are reported in the literature, e.g., Pyke and Cohen [1994], Pooley [1994], Erlebacher

and Meller [2000], Brown et al. [2001], Çetinkaya et al. [2009], Baboli et al. [2011], Degbotse

et al. [2013] and Meisel et al. [2013]. This research also supports the positive effect of the

integrated optimization on the supply chain performance [Chandra and Fisher, 1994, Coelho

et al., 2014, Acar and Atadeniz, 2015, Adulyasak et al., 2015]. Although integrating different

functions into a single model makes it more complex and difficult to solve, it mostly achieves

better results in terms of cost saving and improving the service level. Aiming to highlight

the values of the integrated approach, in this research, we compare the results obtained from

this approach with the ones from the traditional hierarchical decision making approach. We

also explore different types of flexibility: obtained from time windows and from the network

design.

The classical dynamic lot sizing problem (LSP) mainly considers a finite time horizon, a set

of customers, and deals with satisfying the demand of a particular period by delivering the

requested products to customers in the same period [Brahimi et al., 2006a, Hwang, 2010].

However, in practice, this assumption mostly does not hold true, as a delivery time window

is considered for fulfilling the demand. Lee et al. [2001] introduce the LSP with delivery time

windows and today it is recognized as one of the topics with a significant research opportunity

[Claassen et al., 2016]. In this research, we describe, model, and solve the rich integrated

problem, also considering a delivery time window.

In a typical supply chain, a number of plants produce one/several product(s); these are either

stored at the plant or shipped to one/several distribution center(s) (DCs) of the next echelon.

2

In order to fulfill the demands of customers, these products are then transported to the last

echelon (retailers, or final customers). The solution mainly needs to determine 1) where to

locate the plants and DCs, 2) the production quantities at plants and inventory allocation at

both plants and DCs, 3) assignment of DCs to plants as well as customers to DCs, and 4) the

shipment quantities from plants to DCs and from DCs to the final customers. This problem

simultaneously addresses several supply chain decisions such as lot sizing, facility location, and

distribution. Despite numerous studies on jointly optimization of production and distribution

problems, their integration with facility location decisions is not extensively studied. The

facility location problem (FLP) has long been considered as a strategic decision [Klose and

Drexl, 2005] and therefore, its integration with other tactical or operational level decisions

has received less attention [Nagy and Salhi, 2007]. However, in the era of shareconomy,

renting or leasing facilities has become a current practice. As the location of facilities can

periodically change, the FLP could be easily integrated with other operational level decisions.

A contribution of this research is to simultaneously optimize facility location, production

and/or inventory, and distribution decisions.

Integrated supply chain problems are reducible to problems such as the LSP, FLP, or vehicle

routing problem (VRP). Each of these problems are known to be very hard to solve. Most

variants of the LSP are proved to be complex and difficult to solve [Jans and Degraeve, 2007].

Florian et al. [1980] prove that the single-product capacitated problem is NP-hard, and Maes

et al. [1991] show how finding a feasible production plan for a capacitated production system

with no setup cost is an NP-complete problem. Mostly, small size instances of integrated

supply chain optimization problems could be easily solved to optimality using exact methods,

but finding a good solution in a reasonable time for large complex real-world integrated prob-

lems has been a challenge [Fahimnia et al., 2013]. Throughout this thesis, we first solve each

problem with exact methods, and if necessary, we propose approximate methods to accelerate

the solution procedure.

The remainder of this thesis is organized as follows. Chapter 1 provides an overview on

single stage integrated production and distribution optimization problems. Integration of

production and distribution decisions is an important and widely studied area in the integrated

supply chain literature. Independently, both production and transportation problems have

several well-studied variants, and so does their integration. In that chapter, the existing

research is grouped into two broad categories: single versus multi-product models and then

their characteristics and applications are elaborated.

In Chapter 2, we investigate a single-product, multi-plant production planning and distri-

bution problem for the simultaneous optimization of production, inventory control, demand

3

allocation, and distribution decisions. The objective of this rich problem is to satisfy the

dynamic demand of customers while minimizing the total cost of production, inventory and

distribution. By solving the problem we determine when the production needs to occur, how

much has to be produced in each of the plants, how much has to be stored in each of the

warehouses, and how much needs to be delivered to each customer in each period. On a large

real dataset inspired by a case obtained from an industrial partner we show that the proposed

integration is highly effective. Moreover, we study several trade-offs in a detailed sensitivity

analysis.

Chapter 3 extends the problem by considering multiple products and a leasing period for

DCs. This chapter contributes to the integrated optimization literature by simultaneously

addressing location, production, inventory, and distribution problems, and to the production

economics literature by comparing and assessing the performance of sequential and integrated

solution techniques. We develop an exact method and several heuristics, based on separately

solving each part of the problem. In this study, we show the limitations of the exact methods

in handling large size integrated optimization problems and the poor performance of sequential

approaches. Then, we introduce a hybrid adaptive large neighborhood search (ALNS) heuristic

to overcome these limitations. Our results and analysis not only compare solution costs but

also highlight the value of an integrated approach.

By a flexible two-echelon location routing problem presented in Chapter 4, we aim to assess

the value of flexibility. The chapter deals with an integrated location routing problem in

which a supplier delivers a commodity to its customers through a two-echelon supply network.

The objective is to minimize the total shipping costs from the central depot/plant to the

DCs, the traveling costs from DCs to customers, the locations costs and the penalty costs

for the unfulfilled demands. We present a mathematical formulation of the problem together

with different classes of valid inequalities. Computational results and business insights are

discussed. The results show that the combined effect of the network design and delivery

flexibility lead to considerable savings for supply chains.

Research and practice show that taking an integrative approach to supply chain planning en-

hance the business performance. Despite the ever increasing attention towards environmental

impacts of operational decisions, the research is still limited. Chapter 5 aims to identify the

trade-offs between cost and emission reduction potential within an integrated supply chain

planning system. To this end, the two well-known problems of inventory routing and produc-

tion routing are investigated. By means of several key performance indicators, we shed light

on the values of integration when it comes to emission reduction.

Finally, Chapter 6 summarizes the main contributions and highlight future research venues.

4

Chapter 1

Literature review

Supply chain is viewed as a network of facilities engaged in procurement of raw material,

production and storage of finished products, and their distribution to reach the final customers

[Hugos, 2011]. Supply chain planning processes mainly include facility location, procurement,

production, inventory, and distribution decisions [Stadtler, 2005, Christopher, 2016]. The

goal of any supply chain is to ensure that customers are provided with the right products or

services at minimum cost and time. To date, several definitions for the term supply chain

management exist but despite all their differences, generally what they have in common is

that supply chains coordinate and integrate interdependent activities and processes within

and between companies [Ellram, 1991, Cooper et al., 1997, Carter and Rogers, 2008]. Due

to the vital role supply chain planning integration plays in today’s business, it is believed to

be the new source of achieving and retaining competitive advantage [Lei et al., 2006, Hein

and Almeder, 2016]. However, despite the abundance of conceptual and empirical studies on

supply chain integration and coordination, e.g., Stevens [1989], Power [2005], Mustafa Kamal

and Irani [2014], until recently integrated models of the supply chains have been very sparse

on the operations research literature.

Simultaneous optimization of all supply chain planning decisions by integrating them into a

single model has been such a complex and difficult task that traditionally each decision was

treated separately from the others, or the problem was decomposed to smaller and easier to

solve problems. For example, location, lot sizing, and distribution have traditionally been

solved independently from one another, e.g., Erlenkotter [1978], ReVelle and Laporte [1996],

Florian et al. [1980], Barany et al. [1984], Maes et al. [1991], Desrochers et al. [1992], Gendreau

et al. [1994] and Taillard et al. [1997]. A sequential hierarchical approach was often taken in

which the solution obtained from one decision level is passed and imposed to the next. In a

hierarchical approach, the decisions at each level are taken in isolation and once decisions are

5

made in a level, they are considered to be fixed and all other decisions follow on. Since every

function tries to optimize its own decisions, this approach often leads to sub-optimal solutions

[Vogel et al., 2017]. Whereas, in integrated paradigm, various functions of the supply chain

are simultaneously taken into consideration and jointly optimized.

In this chapter we aim to provide a comprehensive summary of the well-studied integrated

production and distribution problems. The remainder of this chapter is organized as follows.

Section 1.1 provides a brief overview on the well-known integrated problems of the supply

chain. The integrated production-distribution problem is presented in Section 1.2. Section 1.3

presents directions for further research and our conclusions are given in Section 1.4.

1.1 Well-known integrated problems

Generally, integrated models combine decisions of at most two functions from the supply chains

and due to their nature, operational level decisions have been the main targets for integration.

To date, a number of review papers have attempted to provide a comprehensive overview of

each integrated problem. In what follows, we present the integrated problems and the existing

reviews on them. In Table 1.1 we present the well-studied integrated supply chain planning

models, adopting from the work of Adulyasak et al. [2012].

Table 1.1: Supply chain integrated planning

Problem Location Production Inventory RoutingLocation-Routing Problem (LRP) X XLocation-Inventory Problem (LIP) X X

Lot Sizing Problem (LSP) X XInventory-Routing Problem (IRP) X X

Production-Routing Problem (PRP) X X X

Location-routing problem (LRP): includes both facility location planning (FLP) and vehicle

routing problem (VRP). Being closely related to the supply chain network design, has long

been regarded as a critical strategic issue in the literature [Aikens, 1985] dealing with decisions

such as establishing a new facility, relocating the existing ones, or any capacity expansion plans

[Melo et al., 2006]. In FLPs, a set of alternative locations are available to serve geographically

dispersed customers and the goal is to select locations such that the cost and/or time of reach-

ing the customer is minimized [Melo et al., 2009]. Literature reviews published on the FLP

classify the models and algorithms and provide a comprehensive picture for a large variety of

studies on FLP (e.g., Francis et al. [1983], Brandeau and Chiu [1989], Owen and Daskin [1998],

Klose and Drexl [2005], Snyder [2006], Şahin and Süral [2007], Melo et al. [2009], Farahani et al.

[2012, 2014]). Due to its strategic nature a considerable capital investment as sunk costs has

6

been associated to location decisions; therefore, integrating this problem with other tactical

or operational decisions was less prevalent in the integrated optimization literature [Nagy and

Salhi, 2007]. However, joint efforts, such as the one between Proctor & Gamble and Walmart,

to manage the distribution flows efficiently and collaboratively have recently become more

common [Amiri, 2006], to the extent that the dynamic selection of partners in supply chains

is identified as one of the new trends and research opportunities in the supply chain man-

agement [Speranza, 2018]. Adapting to the reality of the new business environment, location

decisions could now enjoy more flexibility and should be revised periodically, and hence, they

could easily be combined with other tactical/operational level decisions. Even if considered

as a long term decision, the facility location is interrelated with other decisions, in particular

with transportation decisions [Drexl and Schneider, 2015]. Boventer [1961] recognizes the re-

lationship between transportation costs and location rent and introduces the location-routing

problems (LRPs) as the combination of facility location and routing decisions. A few papers

such as Nagy and Salhi [2007], Prodhon and Prins [2014], and Drexl and Schneider [2015]

discuss different variants and extensions of the LRP. In LRP the questions of which facilities

should be selected, which customers should be served, and in which order are to be answered

[Drexl and Schneider, 2015]. The objective is to minimize the total costs of locating facilities

and distributing the products.

Location inventory problem (LIP): the LIP combines the strategic location decisions with

the operational inventory management. The literature deals with the inventory and location

problems separately. First the location and number of DCs or warehouses are decided solving

facility location problems and then assuming these decisions to be fixed, the optimal inventory

replenishment policies are determined [Daskin et al., 2002]. By decomposing the LIP into two

separate problems, all the interrelated costs are ignored; this approach becomes questionable

specially in dealing with uncertain demands [Shen et al., 2003]. Farahani et al. [2015] provide a

review on the models that jointly consider facility location decisions and inventory management

problems. In the LIP, the questions of which facilities to be selected and how much inventory

to be kept at each facility are to be answered. The objective is to minimize the total costs

including locating facilities and inventory holding.

Lot sizing problem (LSP): as an important and challenging problem in production planning,

the LSP integrates the production and inventory decisions and deals with the trade-off between

production and storage costs [Karimi et al., 2003]. The objective is to minimize the total costs

of production, setup, and inventory [Jans and Degraeve, 2007]. Since the seminal paper of

Wagner and Whitin [1958], several studies have investigated the LSP. Florian et al. [1980]

prove that the single-product capacitated problem is NP-hard, and Maes et al. [1991] show

that finding a feasible production plan for a capacitated production system with no setup

7

cost is an NP-complete problem; even multi-plant, uncapacitated lot sizing problem is NP-

complete [Sambasivan and Schmidt, 2002]. Numerous reviews of the literature (e.g., Karimi

et al. [2003], Brahimi et al. [2006b], Jans and Degraeve [2007, 2008], Robinson et al. [2009],

Buschkühl et al. [2010], Glock et al. [2014], Axsäter [2015]) have provided an overview of the

variants, models and formulations, and algorithms of LSPs.

Inventory-routing problem (IRP): as the name indicates, the IRP is the integration of inven-

tory management and routing decisions. An indispensable part of many integrated supply

chain models entails the delivery/transportation decisions. On one hand, in most industries,

transportation cost is the major component of the total logistics cost [Boudia et al., 2007],

and on the other hand, the ever-rising transportation costs, and increasing customer sensitiv-

ity to lead time have become the salient reasons behind the supply chain literature emphasis

on transportation cost reduction and performance increase. The road-based transportation

methods mentioned in different existing integrated optimization problems could be classified

into two broad categories of direct shipment and routing decisions. A very well-known and

well-researched area in distribution and transportation planning is the VRP [Toth and Vigo,

2014]. A number of papers review the variants, models and algorithms of this problem (e.g.,

Laporte [1992], Eksioglu et al. [2009], Laporte [2009], Pillac et al. [2013], Lahyani et al. [2015c],

Koç et al. [2016a], Psaraftis et al. [2016], Ritzinger2016). The problem concerns the design of

vehicle routes to make deliveries to customers in each period. The IRP first appeared on the

literature as a variant of the VRP [Coelho et al., 2014]. The reviews on the IRP of Campbell

et al. [1998], Moin and Salhi [2007], Andersson et al. [2010], and Coelho et al. [2014] provide an

exhaustive overview on its variants, applications, models and formulation, and solution meth-

ods. In the IRP, the questions of how much of the inventory needs to be kept at each center,

which customers should be served and in which order are to be answered. The objective is to

minimize the total of the inventory and transportation costs.

Production-routing problem (PRP): the PRP is an integration of the IRP with production

decisions [Coelho et al., 2014] or equivalently, the LSP with the VRP [Adulyasak et al., 2015].

The most recent reviews on the PRP [Adulyasak et al., 2015, Díaz-Madroñero et al., 2015]

thoroughly investigate the various solution techniques, formulations, applications and classi-

fications of the problem. The PRP answers the questions of how much to be produced in

each period, how much of the inventory needs to be kept at each facility, and which customers

should be served and in which order. The PRP jointly minimizes the production (setup

and variable), inventory, and transportation costs. It belongs to the vast class of integrated

production-distribution problems.

The integration of production and distribution decisions is an important and widely studied

8

area of supply chain literature, as according to Chen [2004], production and distribution are

the most important operational decisions of supply chains. In order to better understand,

classify the broad integrated production-distribution literature, and provide directions for

future research, in what follows we review several existing research on the topic and discuss

the assumptions, main problems, methods, and results. In this work, we also review studies in

which the strategic decisions, such as facility location, are simultaneously optimized with short-

term and operational level decisions, such as transportation and distribution are addressed.

1.2 Integrated production-distribution problems

Independently, both production and transportation problems have several well-studied vari-

ants, and so does their integration. Here, we focus only on integrated approaches. Depending

on the variant of the integrated problem, it is known by different names in the literature. In

some cases, despite the difference in what they are called, they tackle the same problem. Chan-

dra and Fisher [1994] deal with production scheduling and distribution problem (PSD). The

integration of production and routing is called the production-inventory-distribution-routing

problem (PIDRP) in Bard and Nananukul [2009b], Lei et al. [2006] and Bard and Nananukul

[2010], it is also considered as integration of an inventory-routing problem with a produc-

tion planning problem and is called integrated production-distribution problem (IPDP) by

Armentano et al. [2011] and Boudia and Prins [2009] or PRP by Absi et al. [2014], Adulyasak

et al. [2012], and Adulyasak et al. [2015]. Karaoğlan and Kesen [2017] study the integrated

production and transportation scheduling problem (PTSP). The lot-sizing problem with pro-

duction and transportation (LSPT) is studied in Hwang and Kang [2016], the same problem

is called operational integrated production and distribution problem (OIPDP) in Belo-Filho

et al. [2015]. Finally, when location decisions are incorporated into the production-distribu-

tion model, the problem becomes a production-distribution system design problem (PDSDP)

[Elhedhli and Goffin, 2005].

As of now, a number of reviews on the coordination and integration of production and trans-

portation decisions exist. Cohen and Lee [1988] and Sarmiento and Nagi [1999] survey inte-

grated production and distribution systems, Mula et al. [2010] review mathematical program-

ming models for supply chain production and transport planning, Chen [2010] and Meinecke

and Scholz-Reiter [2014] review integrated production and outbound distribution scheduling

(IPODS), integrated production-distribution planning is reviewed in Fahimnia et al. [2013].

In the what follows, we review the literature that present mathematical models to solve single

stage integrated optimization problems. Therefore, integrated production-distribution models

that schedule the production are out of the scope of this research. With regards to the integra-

9

tion of production and distribution, it is present in two distinct streams of research. One deals

with integrated facility location and production planning decisions and the other integrates

production with distribution decisions. Looking at the cost structure of the integrated facility

location and LSP in single echelon, single product/period models, the distinction between

these two streams of literature becomes less evident. By putting the common elements of

both objective functions aside, i.e., production variable, transportation, and inventory han-

dling costs, in both streams a binary decision is present to decide whether or not a product

is produced in a plant. This decision is called facility location allocation in FLP, and setup

decision in LSP. Therefore, we consider facility location as a characteristic of the model only if

a distinct binary decision is defined to indicate whether a facility is operational in a period or

to select a/several facility(ies) among a set of available ones. Our criteria for selection of the

paper is that both production (either variable or fixed setup) and transportation costs must

be present in the objective function. Generally, the common assumptions of most papers deal-

ing with integration of production and distribution are that the plants are multi-functional,

therefore, all products can be produced in any of the plants; when routing is the delivery

method, each customer can be visited at most once per day; a fleet of homogeneous vehicles

with limited capacity are considered; the demand cannot be split; no shortage, stockouts or

backlogging is allowed; and finally, transfer between sites is not allowed. The cost functions

are often linear, however fixed-charge and general concave cost structures are also considered

in some studies. In what follows, when economies of scale are present, we call the cost function

concave. Unless otherwise specified, these are the assumptions of the problems studied here.

We organize our revision around two broad categories of single versus multi-product models,

and then categorize them based on the following characteristics:

• the number of echelons (single or multiple);

• the number of plants (single or multiple);

• the number of periods (single or multiple);

• whether production/inventory capacities exist;

• whether the demand is deterministic or stochastic;

• whether the production setup cost is considered in the objective function;

• whether location decision is addressed in the model.

Variants of production-distribution models with single product are presented in Section 1.2.1

and models that consider multiple products are reviewed in 1.2.2

10

1.2.1 Single-product production-distribution models

In this section we review integrated production-transportation models for single product prob-

lems. To better analyze the variants and characteristics of each problem, we divide them into

two categories of models with direct shipment and those with routing. We notice that in

production integrated with direct shipment more variants of the production problem are con-

sidered, but the joint production and routing literature has been mainly focused on finding

faster and more efficient solution algorithms. Therefore, majority of the PRP studies compare

their results and performance of the algorithms with the existing methods on the literature.

van Hoesel et al. [2005] present a model to integrate production, inventory and transportation

decisions in a serial multi-echelon supply chain. All cost functions are concave and a produc-

tion capacity is present. They model the problem as a capacitated minimum-cost network

flow problem and study different transportation and inventory holding cost structures in the

integrated problem. In general, their dynamic programming algorithms runs in polynomial

time in the number of periods and echelons. Ekşioğlu et al. [2006] also formulate the produc-

tion and transportation planning problem as a network flow and propose a primal-dual based

heuristic to solve it. In their model, the plants are multi-functional, the production and setup

cost vary from one plant to another as well as from one period to the next, and transportation

costs are concave. They claim that their problem is a special case of the facility location

problem and an extension of the classical lot sizing problem as the facility selection decision is

also present in the model. Ahuja et al. [2007] study a multi-period single-sourcing problem in

a dynamic environment in which they consider production, inventory and throughput, as well

as the perishable products constraints. The single-sourcing suggests that the customer de-

mand during the whole planning horizon is fulfilled from the same facility and cannot be split

among different facilities. They formulate the problem as a nonlinear assignment problem to

link retailers to facilities, taking timing, location and production quantities into consideration.

They first propose a greedy heuristic and then a very-large-scale-neighborhood (VLSN) search

method to improve the greedy solutions. Hwang [2010] investigates integrated economic LSP

with production and transportation. Using stepwise cost for transportation, the model con-

siders economies of scale in shipment and consequently production. The number of vehicles

is assumed to be unlimited and the production cost is concave. While backlogging is allowed,

the results are provided for both cases of with and without backlogging assumptions. Romeijn

et al. [2010] study the integration of facility location and production planning decisions. They

introduce the idea of generalizing LSP by integrating it with facility location decision [Liang

et al., 2015]. The objective function is to minimize the location, production, inventory, and

transportation costs. They study a new approximation method for cases with special produc-

tion and inventory cost structures and seasonal demand patterns. Akbalik and Penz [2011]

11

combine distribution decisions and LSP with delivery time windows. They aim to compare

the just-in-time and time window policies. Under the just-in-time policy, the customer re-

ceives a fixed quantity on the due date of the demand but with the time window policy the

deliveries are constrained by the time windows and advanced shipment is possible. In their

model, costs change over time and they assume a fixed transportation cost per vehicle. A

dynamic programming (DP) algorithm is used to solve the problem. They show that the time

window policy has a lower cost than the just-in-time one and they also compare the Mixed

Integer Linear Programming (MILP) and DP methods, and conclude that even for their large

size instances, the DP outperforms the MILP. Sharkey et al. [2011] apply a branch-and-price

algorithm for an integration of the location and production planning problem similar to the

one studied in Romeijn et al. [2010]. In their model, single sourcing is considered. Their find-

ings show the potential benefits of integrating facility location decisions with the production

planning. The proposed branch-and-price algorithm works better when the ratio of the num-

ber of customers to the number of plants is low. Hwang et al. [2016] reduce the complexity

of the algorithm proposed in van Hoesel et al. [2005] by utilizing only the information on the

aggregated production quantities and considering only those periods in which transportation

occurs. A concave transportation cost consisting of a fixed and variable cost of shipment, and

a linear holding cost function is assumed. Later Hwang and Kang [2016] propose a stepwise

transportation function and consider a production-distribution problem in which backlogging

is allowed. They further improve the O(T 3) algorithm of Hwang et al. [2016] and reduce its

complexity to O(T 2 log T) where T is the number of periods.

Lei et al. [2006] consider integration of production, inventory, and distribution routing prob-

lems and associate a heterogeneous vehicle to each plant. They propose a two-phase heuristic

approach to solve the PRP. In the first phase, the routing decisions are relaxed and the prob-

lem is solved considering direct shipments, in the second phase, they propose a heuristic for

the routing part of the problem. Single product PRP with capacity constraints is studied in

Boudia et al. [2007, 2008] and Boudia and Prins [2009]. In their application, the customers

are served at most once a day based on a first-in-first-out (FIFO) policy by a limited fleet

of capacitated vehicles. The customers cannot receive a late service, however, if the capacity

permits, their demand can be fulfilled in advance. Although the inventory held at the plant

and customer levels is capacitated, the holding cost at the customer is negligible compared to

the one at plant. Boudia et al. [2007] suggest a greedy randomized adaptive search procedure

(GRASP) to solve the PRP but in order to change production and delivery days for some

of the demands the local search is used and to reinforce the combination either a reactive

mechanism or a path relinking is added. Boudia et al. [2008] propose two greedy heuristics

followed by two local search procedures to solve the problem. The same problem is solved using

12

a memetic algorithm with population management (MA|PM) and with dynamic population

management in Boudia and Prins [2009]. Their memetic algorithm yields better results than

the GRASP, and shows 23% saving in cost compared to the classical hierarchical approach

[Boudia and Prins, 2009]. The same problem is addressed in Bard and Nananukul [2009a,b,

2010], in which a single plant serves a set of customers over a multi-period time horizon. The

demand is satisfied either from the inventory held at the customer or daily distribution of

the product. Two cases are considered for the distribution: to fulfill the demand of the day

by vehicle routing or routing is replaced by allocation and aggregated vehicle capacity con-

straints replace the routing constraints [Bard and Nananukul, 2010]. In Bard and Nananukul

[2009a], they solve the problem with a reactive tabu search which is followed by a path re-

linking procedure to improve the solution. Compared to Boudia et al. [2007], the results from

Bard and Nananukul [2009a] are slightly better. Bard and Nananukul [2009b] use branch-

and-price algorithm and compare several heuristics for the IRP in the context of PIDRP. To

take advantage of the efficiency of a heuristic and accuracy of the branch-and-price, in Bard

and Nananukul [2010], they improve their former method by proposing a hybrid algorithm

combining exact and heuristic methods within the branch-and-price framework. Within 30

minutes of run time, the algorithm is able to find optimal solutions only for instances with up

to 10 customers, two periods, and five vehicles. Moreover, the lower bounds they obtain are

not strong [Adulyasak et al., 2014]. The problem discussed in Ruokokoski et al. [2010] con-

siders a single uncapacitated vehicle. As in Bard and Nananukul [2010], the maximum level

(ML) inventory policy is used for the quantities delivered to each retailer [Absi et al., 2014].

They introduce several strong reformulations for the problem, inequalities to strengthen them

and a branch-and-cut algorithm to solve them. Their results support the cost saving benefits

of the coordinated approach compared to the uncoordinated one. The proposed algorithm

can solve instances with up to 80 customers and eight periods, but finds larger instances still

challenging. Archetti et al. [2011] compare the ML and order-up to level (OU) inventory

policies in the PRP context. The demand is delivered to the customers using an unlimited

fleet of capacitated vehicles. Using a branch-and-cut algorithm, they conclude that the ML

policy outperforms OU in short time horizons, but with increase in the number of periods, the

difference between the costs obtained from these two policies also reduces. However, within

two hours of execution, the proposed branch-and-cut approach does not provide optimal solu-

tion for all instances. Adulyasak et al. [2012] compare the performance of the adaptive large

neighborhood search (ALNS) heuristic against GRASP [Boudia et al., 2007], MA|PM [Boudia

and Prins, 2009], reactive tabu search [Bard and Nananukul, 2009a], tabu search with path re-

linking [Armentano et al., 2011], and the branch-and-cut approach proposed in Archetti et al.

[2011]. Their proposed heuristic outperforms the former heuristics. Absi et al. [2014] propose

a two-phase iterative heuristic approach for PRP with a limited fleet of capacitated vehicles

13

and an ML inventory policy. Their decomposition method addresses the lot sizing decisions in

the first phase and determines the routing in the second one. The comparisons of this method

against the ALNS proposed by Adulyasak et al. [2012], and the other five heuristics compared

in Adulyasak et al. [2012] reveal that while the second phase outperforms all existing methods

using less running time, developing fast heuristics that can yield good results for the lot sizing

phase still remains a challenge. Adulyasak et al. [2014] consider both PRP with ML or OU

policies. A main difference between their model and that of Archetti et al. [2011] is that prod-

ucts could be delivered to the customers on the same period as the demand happens, and there

is no need to wait until the products are replenished in the facility. Using a single core and

within two hours of execution, instances with up to three periods and three vehicles, and up

to 25 customers are solved to optimality. The PRP with demand uncertainty is addressed for

the first time in Adulyasak et al. [2015]. They propose and compare branch-and-cut algorithm

with Benders decomposition method to solve the two-stage and multi-stage stochastic PRP.

Solyalı and Süral [2017] use a multi-phase heuristic to solve a single echelon PRP with pro-

duction and inventory capacities. They evaluate the performance of their proposed heuristics

on the benchmark instances of Boudia et al. [2007] and Archetti et al. [2011]. Although the

multi-phase heuristic finds new best solutions for 65% of instances, for larger instances, the

better solution is found at the cost of higher computation time.

Table 1.2 presents the papers that consider only one product in their models. As indicated

in Table 1.2, only Adulyasak et al. [2015] consider the demand to be stochastic. Except for

van Hoesel et al. [2005], models address a single plant-retailer echelon. In a few number of

papers, the model does not consider the fixed or setup cost of production but they assume the

production cost to be a function of the quantities produced. Exact methods (mostly DP) and

metaheuristics are exploited when the direct shipment is the delivery method. Prior to 2010,

the common solution approach to tackle the PRP was metaheuristics, however, matheuristics

are more common since they proved to be better in terms of efficiency and performance.

1.2.2 Multi-product production-distribution models

In this section we present the integrated production-distribution models with multiple prod-

ucts. The number of publications in this category is less than the ones with single product

models. Obviously adding multiple products to the model makes it more difficult to solve,

therefore, in this section fewer papers consider routing as the shipment method. Here, we first

present those studies that consider only one echelon in the model, mostly considering a set of

plants and customers. Then, multi-echelon problems are surveyed.

14

Tab

le1.

2:Sing

leprod

uctprod

uction

-distributionprob

lems

Referen

ceNumber

ofCap

acity

Inventory

Dem

and

Setup

Location

Shipment

Solution

method

Echelon

sPeriods

Plants

Produ

ction

Inventory

vanHoesele

tal.[20

05]

MM

SX

P,DC,C

DDS

Dyn

amic

prog

ramming

Ekşioğluet

al.[20

06]

SM

MP

DX

DS

Primal-dua

lbased

heuristic

Ahu

jaet

al.[20

07]

SM

MX

XP

SDS

Greed

yad

aptive

search

proced

ure/very-la

rge-scale-ne

ighb

orho

od-searchmetho

dHwan

g[201

0]S

MS

PD

DS

Dyn

amic

prog

ramming

Rom

eijn

etal.[20

10]

SM

MP

SX

DS

Greed

yalgo

rithm

andcost-scalin

gAkb

alik

andPen

z[201

1]S

MM

XP,C

DX

DS

Dyn

amic

prog

ramming

Sharkeyet

al.[20

11]

SM

MP

DX

DS

Branch-an

d-price

Hwan

get

al.[20

16]

SM

SX

P,C

DDS

Dyn

amic

prog

ramming

Hwan

gan

dKan

g[201

6]S

MM

PD

DS

Geometrictechniqu

ewithresidu

alzoning

Leie

tal.[20

06]

SM

MX

XP,DC

DR

Decom

position

metho

dBou

diaet

al.[20

07]

SM

SX

XP,C

DX

RGreed

yad

aptive

search

proced

ure

Bou

diaet

al.[20

08]

SM

SX

XP,C

DX

RGreedyhe

uristics

andlocals

earch

Bou

diaan

dPrins

[200

9]S

MS

XX

P,C

DX

RMA|PM

Bardan

dNan

anuk

ul[2009a

]S

MS

XX

P,C

DX

RReactivetabu

search

Bardan

dNan

anuk

ul[2009b

]S

MS

XX

P,C

DX

RBranchan

dprice

Ruo

koko

skie

tal.[20

10]

SM

SP,C

DX

RBranchan

dcu

tBardan

dNan

anuk

ul[2010]

SM

SX

XP,

CD

XR

Branchan

dprice

Arche

ttie

tal.[20

11]

SM

SX

P,C

DX

RBranchan

dcu

tAdu

lyasak

etal.[20

12]

SM

SX

XP,C

DX

RALN

San

dMCNF

Absie

tal.[20

14]

SM

SX

XP,C

DX

RTwo-ph

aseiterativemetho

dAdu

lyasak

etal.[20

14]

SM

SX

XP,C

DX

RALN

San

dbran

ch-and

-cut

Adu

lyasak

etal.[20

15]

SM

SX

XP,C

SX

RBranch-an

d-cu

tan

dBen

ders

decompo

sition

Solyalıa

ndSü

ral[20

17]

SM

SX

XP,C

DX

RA

multi-pha

sehe

uristic

Num

berof

echelons,p

eriods

andplan

ts:S:

Sing

le-M:M

ultiple

Dem

and:

D:D

eterministic;S

:Stochastic

Inventoryat:P:P

lant

-DC:D

istributioncenter

-C:C

ustomer

Shipment:

DS:

Directshipment-R:R

outing

15

Chandra and Fisher [1994] are among the firsts to compare the hierarchical decision making

approach with the integrated one. Their PRP model considers an unlimited number of vehicles

split deliveries to the customers, a storage capacity but no inventory holding cost at the

plant [Boudia and Prins, 2009]. The integrated problem is solved using a local improvement

heuristic. They investigate the value of integrating production and routing decisions and show

that compared to the sequential approach, PRP yields to 3–20% cost savings. The model

presented in Fumero and Vercellis [1999] differs from other PRP studies as partial delivery to

customers is allowed, the transportation cost is obtained considering both distances and loads,

and a fixed cost per each identical capacitated vehicle and finally, the demand of the customer

may be fulfilled in advance. They propose a Lagrangian relaxation approach to solve the

problem. Comparing results from the synchronized and the decoupled approach, they show

the value of the integrated approach. Armentano et al. [2011] consider a multi-product PRP

with a fleet of identical vehicles, and evaluate the performance of the proposed tabu search

with path relinking approach exploiting their own generated instances for the multi-product

case and the single-product instances generated by Boudia et al. [2007]. On single product

instances, their approach outperforms the MA|PM [Boudia and Prins, 2009] and reactive tabu

search [Bard and Nananukul, 2009a]. They also show that for multiple product instances,

tabu search with path relinking always yields better solutions than the tabu search, but on

large instances it requires a very high running time. Brahimi and Aouam [2016] are the firsts

to consider backordering in the context of PRP. They combine a decomposition relax-and-fix

method with the local search heuristic. They compare their approach with the performance

of a commercial solver, concluding that for most cases, the hybrid relax-and-fix heuristic

outperforms the commercial solver.

The fact that the capacity is extendable, distinguishes the model presented in Jolayemi and

Olorunniwo [2004] from others. In their profit maximization model, any shortfall in demand

can be overcome by extension in the capacity or subcontracting. If the resources are not suf-

ficient to satisfy the demand, the model identifies where and how much extension in capacity

is required. They introduce a procedure to reduce the size of the zero-one MILP, and using

a numerical example, they show that the reduced and full-size models generate exactly the

same results. Another paper with a profit maximization objective function is that of Park

[2005]. Plants and retailers are controlled by an OU inventory policy and the model allows

stockouts; identical capacitated vehicles are used for direct shipments. A two-phase heuristic

is developed: in the first phase the production and distribution plans are identified and in

the second phase, the plans are improved by trying to consolidate the deliveries into the full

truckloads. This heuristic generates good results only for the small instances. The paper inves-

tigates the benefits of the integrated approach compared to the decoupled planning procedure

16

concluding that with the integrated approach the profit increases on average by 4.1% and

the demand fill rate by 2.1%. The sensitivity analysis reveals that the integrated approach

is more advantageous when production capacity, fixed cost per vehicle, and unit stockout

costs are high and the vehicle capacity is small. Ekşioğlu et al. [2007] extend the problem

studied in Ekşioğlu et al. [2006] by considering multiple rather than single-product integrated

production-transportation problem. They apply the Lagrangean decomposition heuristic to

solve the problem. The problem investigated by Melo and Wolsey [2012] is similar to that of

Park [2005]. Melo and Wolsey [2012] develop formulations and heuristics that yield solutions

with 10% gap for instances with limited transportation capacity but up to 40% for instances

with joint production/storage capacity restrictions. Nezhad et al. [2013] tackle an integration

of location, production with setup costs, and distribution decisions. This is one of the few

papers in which only one period is considered. In their problem all plants are uncapacitated

and single-source. The only fixed cost incurred in the model is the setup cost of producing a

certain product in a plant.

The integrated production-distribution problem addressed in De Matta et al. [2015] assumes

that each plant uses either direct shipment or a consolidated delivery mode using a third party

logistics firm. They use Benders decomposition to solve integrating consolidated deliveries in

the production and distribution problem. Liang et al. [2015] extend the model presented in

Romeijn et al. [2010] by allowing backlogging and propose a hybrid column generation and

relax-and-fix method. Th exact approach provides the lower bounds and the decomposition

yields the upper bounds of the problem. Comparing their results to the ones from CPLEX,

it is superior in obtaining lower and upper bounds and it is not as sensitive as CPLEX to the

number of facilities in the problem size. Over a long planning horizon, Pirkul and Jayaraman

[1996] present a Lagrangean relaxation based heuristic to tackle a production-distribution,

and facility location-allocation problem. They present a single source model, therefore each

customer is served from a single warehouse selected among a limited number of available

ones. The number of open plants is also limited. They test the performance of a Lagrangean

relaxation based heuristic over randomly generated set of instances. The solution time is

between 46 and 76 seconds and the gap ranges between 0.8% to 2.7%. A similar approach

is also applied in Barbarosoğlu and Özgür [1999]. They present an integrated model for

production and distribution decisions, but propose a decomposition technique to divide the

problem into two subproblems. The focus is to find good solutions for each of the decomposed

sub problems. As an extension of Pirkul and Jayaraman [1996], Jayaraman and Pirkul [2001]

incorporate procurement of the raw material and supply side of the problem into the model.

Generating several instances, first they compare the bounds from the Lagrangean approach

with the optimal solution obtained from a commercial solver. Then, they apply the method

17

to the data obtained from a real case. The gap between the proposed method and the feasible

solution ranges between 1.36% and 2.65% and is obtained between 45 and 88 seconds. A

production-distribution system design is addressed in Elhedhli and Goffin [2005]. The stable

demand of customers is fulfilled from a single DC (single-sourcing), but the system design

is reconfigured by opening and closing the DCs. The problem is considered over a very

large planning horizon. They integrate Lagrangean relaxation, interior-point methods, and

branch-and-bound to solve the complex problem. They obtain better results compared to the

classical Lagrangean approach. Also, in the context of supply chain network design, Correia

et al. [2013] consider integration of production/storage facility costs, fixed maintenance cost,

unit operation costs per product family, and unit shipment costs. They use a branch-and-cut

algorithm to solve their randomly generated instances.

Hein and Almeder [2016] highlight the benefits of the integrated approach toward decision

making using numerical experiments, but their study differs from the former papers in which

they also include the supply side routing to their model, and therefore, combine lot sizing

and supply scheduling. They minimize the setup, inventory holding, and transportation costs

while ignoring the variable cost of production. The transportation costs are load and distance-

based. They consider both just-in-time and keeping inventories at the plants scenarios. Their

results show that the potential savings are higher when inventories are not involved.

A summary of the integrated production-distribution models with multiple products is pre-

sented in Table 1.3.

1.3 Discussions and analysis

Findings from several studies confirm the many benefits of the integrated approach in sup-

ply chain decision making and we have also identified a rapidly growing interest in these

approaches. A substantial number of papers, especially on PRP, are published recently. How-

ever, the literature still poses several challenges need to be further investigated. We summarize

them as follows.

Demand uncertainty: only very few studies have considered the stochasticity of the

dynamic environments. The majority of studies assume the demand to be known in

advance, and plan the production and distributions based on this assumption. A future

research opportunity would be to investigate stochastic production-distribution integra-

tion.

Effective exact algorithms: integrated lot sizing with distribution problems are

mostly solved exploiting decomposition or relaxation techniques. The set of binary

18

Tab

le1.

3:Multi-produ

ctprod

uction

-distributionprob

lems

Referen

ceNumber

ofCap

acity

Inventory

Dem

and

Setup

Location

Shipment

Solution

method

Echelon

sPeriods

Plants

Produ

ction

Inventory

Cha

ndra

andFisher[1994]

SM

SP,C

DX

RLo

calimprovem

entheuristic

Fumeroan

dVercellis[1999]

SM

SX

P,C

DX

RLa

gran

gean

relaxa

tion

Arm

entano

etal.[2011]

SM

SX

XP,C

DX

RTab

usearch

withpa

threlin

king

Brahimia

ndAou

am[2016]

SM

SX

XP,C

DX

RRelax

-and

-fixheuristic

Jolayemia

ndOlorunn

iwo[2004]

SM

MX

XC

DX

DS

MILP

Park[2005]

SM

MX

XP,C

DX

DS

Localimprovem

entheuristic

Ekşioğluet

al.[2007]

SM

MX

PD

XDS

Lagran

gean

decompo

sition

heuristic

Meloan

dWolsey[2012]

SM

SX

XC

DX

DS

Hyb

ridheuristic

Nezha

det

al.[2013]

SS

M–

DX

DS

Lagran

gian

relaxa

tion

DeMatta

etal.[2015]

S/M

MM

DC

DX

DS

Benders

decompo

sition

andDP

Lian

get

al.[2015]

SM

MX

CD

XX

DS

Colum

ngeneration

Pirku

land

Jaya

raman

[1996]

MS

MX

XP,

DC

DX

DS

Lagran

gian

relaxa

tion

Barba

rosoğluan

dÖzgür

[1999]

MM

SX

PD

XDS

Lagran

gian

relaxa

tion

Elhedhlia

ndGoffi

n[2005]

MM

MX

–D

XDS

Lagran

gian

relaxa

tion

andbran

ch-and

-price

Jaya

raman

andPirku

l[2001]

MS

MX

–D

XX

DS

Lagran

gian

relaxa

tion

Correia

etal.[2013]

MM

MX

XP,

DC

DX

XDS

Branch-an

d-cut

Heinan

dAlm

eder

[2016]

MM

SX

PD

XR

Branch-

and-bo

und

Num

berof

echelons,p

eriods,a

ndplan

ts:S:

Sing

le-M:M

ultiple

Dem

and:

D:D

eterministic-S:

Stocha

stic

Inventoryat:P:P

lant

-DC:D

istributioncenter

-C:C

ustomer

Shipment:

DS:

Directshipment-R:R

outing

19

decisions are considered first and then, one solves a network flow problem. Future re-

search could develop faster and more efficient methods to solve large instances of LSPs

[Absi et al., 2014].

Multiple products, multi-echelon settings: as the number of products, echelons,

facilities, and periods increase, the problem becomes more complex and therefore, more

difficult to solve. Hence, most papers consider a single echelon and single product

problems. A future research avenue would be to develop fast and efficient methods to

solve multi-echelon, multi-product, large instances of the problem.

Integrating decisions from different supply chain functions: location and pro-

curement decisions of the supply chains could also be integrated into the production-dis-

tribution models.

LSP with time windows: integration of the production and inventory decisions,

known as the LSP, has been vastly studied. A number of variants and solution algorithms

are investigated and proposed in the literature [Jans and Degraeve, 2008]. Among them,

variants such as the LSP with time windows, and the LSP for perishable products are

considered as the recent hot topics [Clark et al., 2011]. Future research may consider

integrating these variants of LSP with distribution planning.

Heterogeneous vehicle routing: the PRP has the routing decisions as a component,

however, different variants of the VRP such as heterogeneous vehicles, time-dependent

travel time, delivery time windows are still to be explored.

Non-linear cost structures: in most studies costs are linear and no economies of

scale arise. Economies of scale in production, transportation and inventory could also

be investigated in future research.

Environmental issues: incorporating environmental issues in optimization research is

receiving growing attention. However, the current integrated optimization literature has

been mainly focused on increasing the service level and reducing the total costs.

Decision making under flexible network design: Speranza [2018] identifies the

flexible network design as a new trend and research opportunity in logistics. Most of the

research integrating production and distribution considers fixed settings. For example,

the upstream and downstream supply partners are always known and fixed. The location

of facilities rarely changes during the planning horizon.

20

Considering shortages and backordering in the models: Brahimi and Aouam

[2016] are among the few to consider backordering in a PRP model. A prospective

research opportunity is to incorporate stockout, shortage, and backordering.

Real-world applications: Table 1.4 lists the applications of the studies cited in the

previous section. As indicated in this table, a great number of papers published on

production-distribution integration use numerical experiments on randomly generated

instances to test the performance of the solution algorithms. Real-world applications of

the models and algorithms are still lacking in the literature.

1.4 Conclusions

This chapter provides a comprehensive, but not exhaustive overview of the broad literature

on integrated supply chain decisions, with main focus on integration of production and distri-

bution decisions. We have classified the papers presented on the literature based on different

characteristics, including the number of products, echelons, periods and plants considered, the

presence of capacity constraints, setup costs, location decisions in their mathematical model,

the shipment methods, and finally type of demand. We have also reviewed their solution

algorithms and suggested some interesting venues for the future research.

21

Tab

le1.

4:Num

erical

expe

riments

andcase

stud

ieson

integrated

prod

uction

-distributionprob

lems

Autho

r(year)

App

lication

Cha

ndra

andFisher[199

4]13

2rand

omly

gene

ratedinstan

ceswithup

to10

prod

ucts,5

0custom

ersan

d10

period

sPirku

land

Jayaraman

[199

6]Ran

domly

generatedinstan

ceswithup

to10

0custom

ers,

20warehou

ses,

10plan

tsan

d10

prod

ucts

Fumeroan

dVercellis[199

9]Instan

ceswithup

to12

custom

ers,

10prod

ucts

and8pe

riod

sBarba

rosoğluan

dÖzgür

[199

9]12

0instan

ceswithup

to10

custom

ers,

5prod

ucts,1

0depo

ts,a

nd12

period

sJo

layemia

ndOlorunn

iwo[200

4]A

real

case

with2raw

materials,2

vend

ors,

10prod

ucts,5

plan

ts,7

5custom

ers,

and30

availablewareh

ouses

Park[200

5]63

instan

ceswithup

to5plan

ts,7

0custom

ers,

5prod

ucts

and12

period

sElhedhlia

ndGoffi

n[200

5]19

8instan

ceswithup

to10

prod

ucts,1

5plan

ts,1

0DCs,

and50

custom

ers

Leie

tal.[20

06]

Areal

supp

lynetw

orkof

achem

ical

compa

ny,w

ith12

period

s,2plan

tsan

d13

DCs,

3heterogeneou

svehicles

Ekşioğluet

al.[20

06]

300instan

ceswithup

to40

facilitiesan

d38

4pe

riod

sEkşioğluet

al.[20

07]

110sm

alla

ndlargeinstan

ceswithup

to10

plan

ts,5

0prod

ucts,3

5pe

riod

san

d60

custom

ers

Ahu

jaet

al.[20

07]

25instan

ceswithup

to5facilitiesan

d30

0custom

ers,

and6pe

riod

sBou

diaet

al.[20

07]

90rand

omly

gene

ratedinstan

ceswith50

,100

and20

0cu

stom

ersan

d20

period

sBardan

dNan

anuk

ul[200

9b]

150instan

ceswithup

to8pe

riod

san

d50

custom

ers

Ruo

kokoskie

tal.[20

10]

Instan

ceswithup

to40

custom

ersan

d15

period

sor

with80

custom

ersan

d8pe

riod

sBardan

dNan

anuk

ul[201

0]15

0instan

ceswithup

to8pe

riod

san

d50

custom

ers

Akb

alik

andPenz[201

1]87

0instan

ceswith50

period

s,an

dtimewindo

wsbe

tween0an

d4

Arm

entano

etal.[20

11]

180instan

ceswithup

to10

0custom

ers,

10prod

ucts

and24

period

sArchettie

tal.[20

11]

480instan

ceswith6pe

riod

san

d14

,50an

d10

0retailers

Sharkeyet

al.[20

11]

240instan

ceswithup

to40

custom

ers,

20plan

tsan

d10

period

sMeloan

dWolsey[201

2]Differentclassesof

rand

ominstan

ceswithup

to10

prod

ucts,5

plan

ts,4

0custom

ers,

and24

period

sCorreia

etal.[20

13]

72instan

ceswithup

to25

prod

ucts,4

period

s,an

d50

custom

ers

Nezha

det

al.[20

13]

47instan

ceswithup

to31

8custom

ers,

100plan

ts,a

nd40

prod

ucts

Adu

lyasak

etal.[20

14]

336instan

ceswithup

to35

custom

ers,

3pe

riod

s,an

d3vehicles

DeMatta

etal.[20

15]

160instan

ceswitheither

8pe

riod

sor

12pe

riod

san

ddiffe

rent

costsan

dcapa

cities

Lian

get

al.[20

15]

96instan

ceswithup

to12

0prod

ucts,2

4pe

riod

san

d8facilities

Brahimia

ndAou

am[201

6]Instan

ceswithup

to25

prod

ucts,1

2pe

riod

s,an

d50

custom

ers

Heinan

dAlm

eder

[201

6]3,88

8instan

ceswithup

to6supp

liers,1

2inpu

tmaterials,4

prod

ucts,a

nd10

period

sSo

lyalıa

ndSü

ral[20

17]

Instan

cesfrom

Archettie

tal.[20

11]a

ndBou

diaet

al.[20

07]

22

Chapter 2

A dynamic multi-plant lot-sizing and

distribution problem

Chapter information A paper based on this chapter is published in International Journal of

Production Research: Darvish, M., Larrain, H., & Coelho, L. C. (2016). A dynamic multi-plant

lot-sizing and distribution problem. International Journal of Production Research, 54(22),

6707-6717.

In this chapter we investigate a multi-plant, production planning and distribution problem

for the simultaneous optimization of production, inventory control, demand allocation, and

distribution decisions. The objective of this rich problem is to satisfy the dynamic demand

of customers while minimizing the total cost of production, inventory and distribution. By

solving the problem we determine when the production needs to occur, how much has to be

produced in each of the plants, how much has to be stored in each of the warehouses, and how

much needs to be delivered to each customer in each period. On a large real dataset inspired

by a case obtained from an industrial partner we show that the proposed integration is highly

effective. Moreover, we study several trade-offs in a detailed sensitivity analysis. Our analyses

indicate that the proposed scenarios give the company competitive advantage in terms of

reduced total logistics costs, and also highlight more possibilities that become available taking

advantage of an integrated approach toward logistics planning. These abundant opportunities

are to be synergized and exploited in an interconnected open global logistics system.

2.1 Introduction

The world is changing rapidly and so do the supply chain management paradigms. Envi-

ronmental concerns, economical crisis and rapid pace of technological changes have called for

23

more efficient and effective supply chains. In response to the need for better supply chain

management, the Physical Internet (PI) idea is introduced as an interconnected open global

logistics system, which is increasingly gaining attention from research and practice. Inspired

by the digital internet, researchers are developing the PI to move, store, realize, supply and

use the goods similarly to how data is treated in the digital internet [Montreuil, 2011, Mon-

treuil et al., 2013]. Going global and exploiting a system as gigantic and holistic as the PI also

requires a comprehensive view of different activities within a supply chain. In order to be able

to exploit the opportunities the PI has to offer and to be an active member in this global open

interconnected web, it is required that each company uses its own resources more efficiently.

Short and medium-term decisions such as production planning and distribution might be the

ones to benefit the most from or serve the PI better. By avoiding over or under production

and improving transportation and distribution efficiencies, coordination and integration of

different supply chain members are becoming the new source for competitive advantage. A

real-world example of such effort in integrating supply chain decisions is the Kellogg Company

which could save $35 to $40 million per year as a result of the joint optimization of production,

inventory and distribution [Brown et al., 2001]. Another example comes from IBM that has

developed an integrated plan for production and shipment of semiconductors and could gain

15% increase for on-time delivery, 2-4% increase in asset utilization, and 25-30% decrease in

inventory [Degbotse et al., 2013]. Despite the fact that any cost minimization effort which is

focused on only one area of supply chain and ignores other areas will often lead to increase

in cost and not necessarily a reduction of global cost due to the lack of synergy [Adulyasak

et al., 2012], there is abundant literature available on any of the individual problems and all

their well-known generalization. Integrated problems often present advantages from a prac-

tical point of view and are becoming the new standard [Coelho and Laporte, 2013b, Coelho

et al., 2014, Adulyasak et al., 2015].

Having seen the advantages gained by this holistic approach in supply chain management,

researchers have shifted their interests towards integrating supply chain decisions that have

typically been treated individually. For instance, the vehicle routing problem (VRP) and

the lot-sizing problem (LSP) are two of the most well-known classical problems in supply

chain management that have been studied separately for decades [Adulyasak et al., 2012].

In the VRP several capacitated vehicles deliver products to a number of customers from a

warehouse with the objective of minimizing the delivery cost, while the LSP is focused on

the quantities and timing of production with the objective of production and inventory cost

minimization [Karimi et al., 2003]. Obviously, in practice, these two problems should not be

treated separately, as production planning and distribution are interdependent and can benefit

from a joint decision making.

24

2.1.1 Literature review

Since the seminal paper of Wagner and Whitin [1958], several studies have investigated the

dynamic LSP, some of which report close to optimal solutions for single plant, multi-product

capacitated LSPs, but they have not been successful in solving large size instances [Sambasivan

and Yahya, 2005]. In what follows we review some papers that have solved the LSP with the

aim of finding a balance between the setup cost and the inventory holding cost [Clark et al.,

2011].

Florian et al. [1980] proved that the single-product capacitated problem is NP-hard, and

Maes et al. [1991] showed that finding a feasible production plan for a capacitated production

system with no setup cost is an NP-complete problem. The LSP encompasses a wide variety

of problems from continuous time scale, constant demand and infinite time horizon to discrete

time scale, dynamic demand and finite time horizon lot-sizing models [Jans and Degraeve,

2007]. Several classifications have been presented in the literature for the LSP, for example,

single or multi-level problems [Bahl et al., 1987], with the most simple form of the dynamic

LSP being the single-product uncapacitated problem [Jans and Degraeve, 2008]. LSPs are also

categorized based on the number of products and number of plants considered, ranging from

single to multi-product [Bahl et al., 1987, Sambasivan and Yahya, 2005, Jans and Degraeve,

2008, Clark et al., 2011, Absi et al., 2011], and single or multi-plant [Sambasivan and Yahya,

2005], or based on how capacity constraints are treated [Sahling et al., 2009, Karimi et al.,

2003], i.e., uncapacitated [Absi et al., 2011] versus capacitated LSP [Bruno et al., 2014].

Li et al. [2012] identify two lines of research in LSP: in the first, the effort is to develop exact

methods with mathematical programming, valid inequalities or tighter formulations; in the

second, matheuristics are developed for large scale problems. Due to its complexity, various

methods and techniques have been exploited to solve LSP; a Lagrangean-based heuristic for

scheduling of lot sizes in a multi-plant, multi-product, multi-period capacitated environment

with inter-plant transfers [Sambasivan and Yahya, 2005], matheuristic through a series of

mixed-integer linear programs solved iteratively in a fix-and-optimize approach to solve a

multi-level capacitated LSP with multi-period setup carry-over [Sahling et al., 2009], and

dynamic programming for the single-product LSP [Chen et al., 1994, Shaw and Wagelmans,

1998]. A hybrid of an ant-based algorithm and mixed-integer linear programming method was

used by Almeder [2010] to solve a multi-level capacitated LSP; a tabu search and a variable

neighborhood search heuristic was developed by Almada-Lobo and James [2010] to solve the

multi-product capacitated lot-sizing and scheduling problem with sequence-dependent setup

times and costs; finally, a hybrid of genetic algorithms and a fix-and-optimize heuristic was

used to solve the capacitated LSP with setup carry-over [Goren et al., 2012].

25

The problem studied in this paper is on one hand closely related to the production-routing

problem (PRP), which integrates lot-sizing, inventory and distribution decisions simultane-

ously and on the other hand, it deals with the LSP with time windows. Both delivery and

production time windows are considered in the literature [Absi et al., 2011, Akbalik and Penz,

2011], and the PRP has been thoroughly studied recently [Absi et al., 2014, Adulyasak et al.,

2015]. The PRP is solved mostly by heuristics: greedy randomized adaptive search procedure

[Boudia et al., 2007], memetic algorithm [Boudia and Prins, 2009], reactive tabu search [Bard

and Nananukul, 2010] are to name a few. Archetti et al. [2011] propose a hybrid algorithm

to solve a production and distribution of a single-product single-plant system to minimize the

total cost of production, inventory replenishment of a set of retailers and transportation costs.

A common recent approach in dealing with integrated planning problems is to decompose the

problems into subproblems and to exploit heuristics to solve them. For instance, Adulyasak

et al. [2014] use an adaptive large neighborhood search combined with a network flow algo-

rithm. Chen [2015] proposes a heuristic approach to solve dynamic multi-level capacitated

lot-sizing problems with and without setup carryovers. Steinrücke [2015] uses relax-and-fix

decomposition methods to solve a multi-stage production, shipping and distribution schedul-

ing problem in the aluminum industry. Camacho-Vallejo et al. [2015] consider a two-level

problem for planning the production and distribution in a supply chain in which the upper

level minimizes the transportation cost while the lower level deals with the operation costs

occurring at plants for which a scatter search-based heuristic is proposed. The reduce and

optimize approach is used by Cárdenas-Barrón et al. [2015] to solve the multi-product multi-

period inventory lot-sizing with supplier selection problem. Akbalik and Penz [2011] propose a

mixed integer linear program and a pseudo-polynomial dynamic program to solve time window

LSP and ultimately to compare time window with just-in-time policy. Finally, by means of

dynamic programming, Absi et al. [2011] solve single item uncapacitated LSP with production

time windows.

2.1.2 Contributions and organization of the paper

In this paper we solve a problem inspired by a real case with an integrated cost saving ap-

proach, in which we optimize a multi-plant multi-period problem to determine production

scheduling and lot sizes, inventory quantities as well as aggregated plans for demand delivery

within a promised time window. This problem arises in a company that produces and sells

furniture, who has shared data and insights with us. Inventory and production capacities are

limited and once a changeover takes place at a production site, a fixed setup cost is incurred.

Thus, we consider a rich generalization of the dynamic LSP. This paper contributes to the

literature by considering a multi-plant capacitated LSP in which one has to fulfill the demand

26

of customers within a promised time window, determining production dates and quantities, in-

ventory levels, demand satisfaction and distribution. This problem generalizes many variants

of the transportation problem and network flow problems.

Making the right lot-sizing decisions becomes even more significant in interconnected logistic

networks. By pooling capacities and enabling horizontal collaboration, interconnected logistics

networks seek higher performance levels [Sarraj et al., 2014]. By optimizing lot-sizing decisions,

we show the benefits of an integrated approach to supply chain decision making. Besides, we

demonstrate the resource sharing potentials a case company could offer to the interconnected

logistic networks. Finally we also highlight the importance of the cost of transportation which

justifies the need for utilizing the synergy in an open interconnected network, such as that

advocated by the PI initiative.

The remainder of this paper is organized as follows. In Section 2.2 we describe the problem

and introduce the real case study. Section 2.3 presents the mathematical formulation of

the problem. We present the results of extensive computational experiments and sensitivity

analysis in Section 2.4, followed by our conclusions in Section 2.5.

2.2 Problem description

The motivation for this problem stems from a company’s needs that produces and sells furni-

ture. The plants are located in Canada and the US and its customers are spread all over North

America. The demand is categorized in two groups: first, from large retail stores where the

products are sold directly to customers, and second, online customers who visit the company’s

website and place orders over the internet. In this paper we focus only on the second group,

in which all demand is satisfied from one of the warehouses associated with each of the plants.

Whenever a plant starts production, a fixed setup cost is incurred. Therefore, the company

must determine how to plan its daily production and inventory levels across different plants.

By considering a dependable demand forecast, and production and warehouse capacities, the

company develops a production plan to fulfill the demand. Since this demand forecast is highly

precise, we consider the demand to be deterministic by relying on these accurate forecasts.

Moreover, given the very long planning horizon of our problem, we can assume that the time it

takes to produce an item as well as the lead-time to be negligible. A solution to this problem

identifies the periods in which production takes place as well as the production quantities, the

amount that has to be stored and which customers to serve in each of the periods from each

of the warehouses.

We define a plant set Np and a customer set Nc, such that N = Np ∪ Nc. The plant set Np

27

contains p plants, and a fixed setup cost fi (i ∈ Np) is incurred whenever a production batch

starts, and a variable cost ui is paid for each unit produced. A warehouse located close to

each plant stores all the products. A holding cost hi is incurred per unit held in warehouse

i ∈ Np per period. Production at the plants are limited to bi units per period and inventory

at the warehouses are limited to si units. The production is sent to customers i ∈ Nc from

the warehouses. Customers represent the final demand, meaning that they do not hold any

inventory and there is no transshipment between them. The planning horizon is e periods long,

typically days over one year, and at the beginning of each time period t ∈ T = {1, . . . , e},the demand of customer i is dti, which needs to be fulfilled within r periods, i.e., r is the time

window or the maximum lateness allowed. Inventories are not allowed to be negative and all

demand must be satisfied, i.e., backlogging is not permitted. We also assume, without loss

of generality, that the initial inventory is zero for all warehouses. Once the delivery dates for

each customer is determined, the products are posted to the customer using a third party

logistics provider. As a consequence no consolidation is allowed and the company pays cij as

the postal fee from warehouse i to customer j for each unit of product. Products can be sent

directly to the final customers from any of the warehouses, and transfers between warehouses

are also allowed, costing cij , with i and j ∈ Np.

A schematic view of the model for an instance with two plants and three customers is depicted

in Figure 2.1. This figure shows how inventories evolve in each plant, and how dispatches are

allowed between plants or from plants to customers.

Figure 2.1: An instance of the dynamic multi-plant lot-sizing and distribution problem with twoplants and three customers

The objective of this problem is to minimize the total of production, inventory holding and

delivery costs. All demand must be satisfied by the end of the planning horizon, all capacities

and the maximum lateness must be respected.

28

2.3 Mathematical formulation

In order to model the problem as an integer linear program, let yti be a binary variable equal

to one if and only if plant i ∈ Np is active in period t. Let pti be the quantity produced at plant

i in period t, qtij be the quantity delivered from warehouse i ∈ Np to customer or warehouse

j ∈ N in period t, and Iti be the inventory level of warehouse i ∈ Np measured at the end of

period t.

Table 2.1 summarizes the notation used in our model:

Table 2.1: Notations of the model

Parametersbi Production capacity of plant i (unit of products)cij Shipping cost from node i to node j ($/unit)dti Demand of customer i in period t (unit of products)fi Fixed setup cost in plant i ($)hi Unit holding cost in warehouse i ($/period)r Time window or the maximum lateness allowed (periods)si Inventory capacity of warehouse i (unit of products)ui Unit variable cost of production in plant i ($)SetsNc Set of customersNp Set of plants

N = Np ∪Nc Set of all nodesT Set of periods

Variablespti Quantity produced at plant i in period tqtij Quantity delivered from warehouse i ∈ Np to customer or warehouse j ∈ N in period tIti Inventory level at warehouse i ∈ Np in period tyti Binary variable equal to one if and only if plant i ∈ Np is active in period t

Indicesk, t Period indexi, j Node index

The problem can be formulated as follows:

minimize∑t∈T

∑i∈Np

uipti + fiyti + hiI

ti +

∑j∈N ,i 6=j

cijqtij

(2.1)

subject to

pti ≤ biyti i ∈ Np t ∈ T (2.2)

29

Iti ≤ si i ∈ Np t ∈ T (2.3)

Iti = It−1i + pti −∑

j∈N ,i 6=j

qtij +∑

j∈Np,i 6=j

qtji i ∈ Np t ∈ T (2.4)

I0i = 0 i ∈ Np (2.5)t∑

k=1

∑j∈Np

qkji ≥t−r∑k=1

dki i ∈ Nc t ∈ T (2.6)

t∑k=1

∑j∈Np

qkji ≤t∑

k=1

dki i ∈ Nc t ∈ T (2.7)

∑k∈T

∑j∈Np

qkji =∑k∈T

dki i ∈ Nc (2.8)

yti ∈ {0, 1} (2.9)

pti, Iti , q

tij ∈ Z∗. (2.10)

The objective function (2.1) minimizes the total cost of production setup and variable costs,

inventory holding cost and transportation cost. Constraints (2.2) ensure that the production

capacity is respected, while constraints (2.3) guarantee that the inventory level does not ex-

ceed the capacity of the warehouse. Constraints (2.4) and (2.5) are respectively inventory

conservation and initial conditions, and constraints (2.6) and (2.7) enforce that the customer

demand is satisfied within the r periods time window. Specifically, constraints (2.6) impose

that the total demand up to period t − r must be delivered by time t, thus allowing the de-

mand satisfaction to a maximum lateness of r periods, and constraints (2.7) guarantee that

no demand is satisfied in advance. Finally, constraints (2.8) ensure that all the demand is

satisfied by the end of the planning horizon. Constraints (2.9) and (2.10) define the domain

and nature of the variables. Figure 2.2 shows how a feasible delivery plan looks like, satisfying

constraints (2.6)–(2.8). Graphically, the accumulated curve of a feasible delivery plan for a

given customer has to move between the accumulated demand and the accumulated expiring

demand curves; the latter corresponds to the accumulated demand, displaced by r time units.

The accumulated delivery curve has to match the total demand by the end of the horizon.

30

Figure 2.2: Feasibility of a delivery plan with respect to the maximum lateness allowed

2.4 Computational experiments

The mathematical model was implemented into a branch-and-bound algorithm of CPLEX

coded in C++, using IBM CPLEX 12.6 as the MIP solver. All computations were executed

on a grid of Intel Xeon™ processors running at 2.66 GHz with up to 48 GB of RAM installed

per node, with the Scientific Linux 6.1 operating system. A time limit of one hour was imposed

on the execution of each run.

An industrial partner has provided us with a large amount of data regarding production costs,

production and inventory capacities, shipping costs, and daily demand for a year-long planning

horizon. We have used this information to evaluate our method under different production

scenarios. In our dataset, periods represent days and we have a year long planning horizon of

available data. Customers are spread all over North America; there are in total 66 customers

aggregated over the US states and Canadian provinces. The company operates two plants,

significantly far away from each other. Company practices dictate that the time window r

varies between three and four days, depending on the area. We test different values of r for

the whole customer set. In our testbed, there exists 364 daily periods. All data has been

obfuscated for confidentiality reasons, and our resulting instances can be shared upon request.

31

We start our computational analysis by comparing some feasible production plans inspired

by those a manager could employ to control the production. These do not represent current

company practices, but are inspired from them. We report in Table 2.2 different scenarios,

that range from producing every period, every other period, every three periods, and the

global optimized plan. Other plans, such as every four periods, have been tested as suggested

by our industrial partner, however, no feasible solution could be obtained for any other time

window scenarios, which could be due to the fact that a bigger time window for production

makes the problem significantly bigger and consequently the problem becomes very difficult

to be solved to optimality. Table 2.2 reveals that the obtained optimal scenario gives the

company competitive advantage in terms of serving customers faster since for any promised

time windows r the optimized scenario yields lower total cost compared to the other three

possible scenarios. The optimized scenario provides solutions which are on average 17%,

4% and 4% less costly in comparison to the other three production scenarios. We observe

that when producing every day, the optimal solution is to immediately ship the products

to customers so as not to incur any inventory costs, thus the solution cost does not change

with respect to r. When the production is intermittent, the best cases are observed for

slightly bigger time windows. We observe that due to production and inventory capacities,

it is impossible to produce every three days and guarantee deliveries within a tight time

window. Finally, as expected, the optimized scenario yields the best production plan such as

to minimize total costs. Moreover, we observe from Table 2.2 that the optimized plan can

offer next-day delivery guarantee with a slight increase in the total cost with respect to the

other time windows, only 5% more costly than the minimum cost obtained with r = 10. This

constitutes the first important insight obtained from our research.

Table 2.2: Total costs for different production scenarios

Production scenario Time window r1 2 3 5 7 10

Every day 6,366,591 6,366,591 6,366,591 6,366,591 6,366,591 6,366,591Every other day 5,713,911 5,703,504 5,696,257 5,688,965 5,687,841 5,687,841Every three days — — — 5,662,420 5,571,270 5,540,990Optimized 5,625,782 5,527,674 5,453,789 5,380,042 5,349,342 5,331,363Gap (%) 2.95 2.52 1.74 0.81 0.50 0.16

The results reported in Table 2.2 were obtained after solving the model described in Section 2.3

using CPLEX for one hour. Although, given the size of the problem, CPLEX was not always

able to yield an optimal solution, the optimality gap is very low, on average only 1.45%, with

a maximum of 2.95%.

In order to better understand the impact of each part of the objective function to the costs

obtained in Table 2.2, we provide a decomposition of each individual cost as a percentage of

32

the total cost as presented in Table 2.3.

Table 2.3: Percentage of contribution to the total cost per component

Production scenario Cost component Time window r1 2 3 5 7 10

Every day

Inventory holding 0.00 0.00 0.00 0.00 0.00 0.00Setup 21.44 21.44 21.44 21.44 21.44 21.44

Variable production 5.36 5.36 5.36 5.36 5.36 5.36Transportation 73.20 73.20 73.20 73.20 73.20 73.20

Every other day

Inventory holding 0.10 0.02 0.02 0.01 0.00 0.00Setup 12.01 12.03 12.05 12.06 12.07 12.07

Variable production 6.07 6.06 6.04 6.01 6.00 6.00Transportation 81.82 81.89 81.89 81.92 81.93 81.93

Every three days

Inventory holding — — — 1.57 0.23 0.11Setup — — — 8.08 8.21 8.26

Variable production — — — 6.39 6.47 6.46Transportation — — — 83.96 85.09 85.18

Optimized

Inventory holding 1.74 1.15 0.50 0.16 0.04 0.01Setup 9.23 8.28 7.74 6.87 6.45 6.12

Variable production 6.02 6.12 6.24 6.32 6.38 6.42Transportation 83.00 84.45 85.52 86.66 87.13 87.44

In all scenarios the transportation cost is the main contributor to total cost, given high quanti-

ties and distances. Although the same number of units needs to be produced and transported,

neither transportation nor variable production costs are constant due to the choice of plant,

their different parameters, and distances to customers. As expected, when plants produce

every day the setup costs are significant, but no inventory is kept at the warehouses. Once

plants operate only every other day, the setup costs decrease significantly, giving more relative

weight to the transportation cost. We observe that when production is set to happen at every

three periods, then production capacities come into play. Thus, these cases with r = 1, 2 and

3 yield infeasible solutions. Finally, the gap between the variable cost of production and the

setup cost is the minimum in the optimized solution while the inventory holding cost has the

biggest percentage of the total cost in the optimized scenario. It is in this scenario that the

transportation cost plays a bigger role, representing more than 85% of the total cost of the

system.

Further analysis of the optimized solution for the time window currently used by the company,

i.e., r = 3, reveals interesting and useful information justifying the potential gains the company

could obtain offering its idle capacities on an open logistics network. We have selected a plant

randomly and have identified two distinct periods of high versus low usage. This has allowed

us to measure how much of the production and inventory capacities were used by the company

and how much could be offered to an open logistics networks as that proposed by the PI. As

depicted in Figure 2.3 during the high demand period (a), the production is up to the capacity

33

while the inventory capacity is available; on the other hand, in low production period (b), the

idle inventory and production capacities are ready to be offered to other companies on the

interconnected logistics network.

Figure 2.3: An example of production and inventory capacities in high (a) versus low (b) demandperiods

In order to analyze how changes in cost or capacity parameters of the model affect the total

cost, a number of alternative parameter scenarios are now assessed. Table 2.4 summarizes the

percentage change in the objective function when either decreasing the costs and production

capacity by half, or increasing them by a factor of two. We observe that decreasing costs

by half will have a beneficial impact on the total cost, but on different degrees. Since the

transportation cost is the main contributor to the total cost, any reduction in transportation

costs will have the most noticeable impact on the bottom line. We note that the slight increase

in cost when the inventory holding cost is halved in the case of a time window of r = 10 periods

is due to the noise in the computations because not all optimal solutions could be obtained, as

explained earlier. By decreasing the production capacity by half, the plants need to be active

more often in order to satisfy the demand and this incurs an increase in the total cost of about

5%. Likewise, when costs are doubled, the total costs always increase, but in different degrees.

Finally, if the capacities are doubled, the opposite effect is observed: due to relatively low

inventory costs, plants can operate less often, reducing the total cost.

34

Table 2.4: Percentage of cost changes with respect to the optimized solutions

Parameters Cost component Time window r1 2 3 5 7 10

0.5

Inventory holding −1.31 −1.08 −0.61 −0.15 −0.12 0.03Setup −5.17 −4.59 −4.10 −3.40 −3.20 −3.09

Variable production −3.16 −3.13 −3.07 −3.13 −3.06 −3.17Transportation −41.32 −42.80 −42.87 −43.27 −43.46 −43.74

Production capacity 3.08 3.51 4.42 5.12 5.57 5.88

2.0

Inventory holding 0.97 0.28 0.26 0.21 0.12 0.00Setup 8.69 7.67 7.16 6.58 6.18 6.07

Variable production 6.34 5.73 5.98 6.37 6.46 6.38Transportation 82.85 83.96 85.43 86.54 87.11 87.44

Production capacity −0.84 −1.09 −1.13 −1.64 −2.09 −2.41

The sensitivity analysis shown in Table 2.4 only considers the individual effects of the cost

components. In order to detect possible synergies arising when these factors are combined, we

have performed experiments on all possible pairwise combinations of cost components. In Ta-

ble 2.5 we show that of the transportation and setup costs, as these are the two most significant

cost components identified in Table 2.4. We have solved the instances where transportation

and setup costs are altered simultaneously. In Table 2.5 we compare the combined effects with

the sum of individual effects. A quick look at this table shows that the combined effects for

smaller time windows are almost linear, i.e., the combined effects are very close to the sum

of the individual effects. However, when the time windows get larger, the cost increase by

combined effects can be even 10% smaller than the sum of the individual costs increase in

our instances. Hence, once more we witness the positive effect of integrated decision making

approach.

35

Table 2.5: Combined effects for the transportation and setup costs

Time window r Trasnportation cost Setup cost Additive effect (%) Combined effect (%) Difference (%)

10.5 0.5 −46.49 −46.35 −0.14

2 −32.63 −32.45 −0.18

2 0.5 79.37 77.71 1.662 91.53 91.31 0.22

20.5 0.5 −47.39 −47.81 0.41

2 −35.13 −35.96 0.83

2 0.5 79.37 76.53 2.842 91.64 88.76 2.87

30.5 0.5 −46.97 −48.40 1.43

2 −35.72 −37.42 1.70

2 0.5 81.33 75.94 5.392 92.59 86.98 5.60

50.5 0.5 −46.67 −49.07 2.40

2 −36.69 −39.54 2.85

2 0.5 83.14 75.15 8.002 93.12 84.86 8.26

70.5 0.5 −46.66 −49.41 2.75

2 −37.66 −40.41 2.75

2 0.5 84.35 74.87 9.482 93.51 83.95 9.57

100.5 0.5 −46.83 −49.57 2.75

2 −37.66 −40.92 3.26

2 0.5 84.35 74.73 9.632 93.51 83.50 10.01

2.5 Conclusions

We have solved a real multi-plant, production planning and distribution problem in which

production, inventory, demand and distribution decisions are optimized simultaneously. A

mathematical formulation for a rich dynamic LSP problem with delivery time windows is

proposed. The problem is defined using a large real dataset inspired by a case obtained from an

industrial partner. We have shown that our matheuristics method is highly effective; elaborate

and extensive sensitivity analysis are presented to compare various production scenarios as

well as changes in the parameters of the model.

The contributions of this work to PI literature are twofold. First, we demonstrate the po-

tentials of resource sharing in a real case company. Total of idle capacities from different

companies gathered together in an interconnected logistics network provides enormous op-

portunities for businesses to benefit economically, environmentally and socially. Second, the

results indicate that the most important cost component in this problem is the transportation

cost. Exploiting PI-enabled open distribution centers to store products closer to the final

customers and PI-enabled mobility web is expected to dramatically reduce the storage and

transportation costs.

Further extensions of our work could include multi-products cases and transition costs, which

would appear as an additional setup cost when switching products, as well of integrating

36

different lead times per product and per facility. Another interesting extension is related

to the consolidation of the transportation costs when delivering the goods to account for

economies of scale.

37

Chapter 3

Sequential versus integrated

optimization: production, location,

inventory control and distribution

Chapter information A paper based on this chapter is currently submitted for publication:

M. Darvish and L.C. Coelho. Sequential versus integrated optimization: production, location,

inventory control and distribution. Document de travail du CIRRELT-2017-39, Tech. Rep.,

2017.

In this chapter we study an integrated lot sizing problem with distribution and delivery time

windows in which the producer has the choice to locate distribution centers from where prod-

ucts are stored and shipped to customers. Motivated by a real case, we describe, model, and

solve this rich integrated production-distribution problem. In this integrated optimization

application, we consider a multi-plant, multi-product, and multi-period setting. The goal is to

minimize fixed and variable production costs, inventory, and distribution costs while satisfying

demands within a promised delivery time window. Our work contributes to the integrated

optimization literature by simultaneously addressing location, production, inventory, and dis-

tribution problems, and to the production economics literature by comparing and assessing the

performance of sequential and integrated solution techniques. We develop an exact method

and several heuristics, based on separately solving each part of the problem, as well as a gen-

eral integrated matheuristic. Our results and analysis not only compare solution costs but

also highlight the value of an integrated approach.

38

3.1 Introduction

The ultimate goal of any production system is to fulfill the demand of its customers quickly

and efficiently. This goal is achieved through effective and efficient supply chain planning.

Historically, supply chain planning has been conducted in a sequential or hierarchical fashion.

This approach treats each supply chain decision separately from the others. Therefore, in such

a disintegrated planning system, even despite the high cost associated with holding stocks,

inventory plays an important role in satisfying the demand in a timely manner and linking

different functions of the supply chain.

In recent years, the increasing competition among supply chains has forced companies to seek

solutions that result in saving cost and improving the efficiency on the one hand, and offering

even faster and more flexible service to the customers on the other. Inventory optimization

has become the main target for cost reduction initiatives. The emphasis on inventory cost

reduction coupled with the growing transportation cost and competitive delivery dates accen-

tuate the importance of coordination and integration of supply chain functions and decisions

[Fumero and Vercellis, 1999]. Under an integrated approach, various functions and decisions

within a supply chain are simultaneously treated and jointly optimized. In the sequential

approach, typically known as management in silos, the solution obtained from one level is

imposed to the next one in the hierarchy of decisions [Vogel et al., 2017]. Ignoring the links

between decisions, this approach results in sub-optimal solutions. On the contrary, most re-

search and case studies on supply chain integration confirm the positive effect of integration

on business performance [Adulyasak et al., 2015, Coelho et al., 2014]. Hence, supply chain

integration is recognized as the linchpin of success for today’s companies [Archetti et al., 2011].

In this paper, we describe, model, and solve a multi-product, multi-plant, multi-period, multi-

echelon integrated production, inventory, and distribution problem. This integrated problem

has three distinct features of direct shipment, delivery time windows, and dynamic location

decisions for distribution centers (DCs).

The transportation decision in integrated production and distribution literature is mainly

considered as either direct shipment or vehicle routes. With the large number of firms out-

sourcing the transportation function to third party logistics service providers [Amorim et al.,

2012], direct shipment is considered in this paper.

Owing to its significant research and practical potential, much attention is devoted to time

windows. We consider a delivery time window, meaning that the demand must be satisfied

within a specific time frame.

Facility location planning has always been a critical strategic decision. Once the locations

39

are determined, all other decisions such as production quantities, inventories, and transporta-

tion can be made. In modern days, customers always impose tighter delivery time windows,

therefore, keeping a high service level and managing inventory require simultaneous produc-

tion and dynamic facility location planning. The integrated production-location problem has

become so prevalent that flexible network integration is identified as one of the important

recent trends in logistics [Speranza, 2018]. Hence, following this trend, in this paper, we study

a flexible supply network by considering geographically dispersed DCs available to be rented

for a specific period of time.

The objective is to operate a production-distribution system that minimizes production, lo-

cation, inventory, and distribution costs while satisfying demands within a predetermined

delivery time window. To the best of our knowledge, this rich problem has not yet been stud-

ied in the literature. The problem is inspired by a real-world case. Our industrial partner is

facing a steady but gradual increase in demand, which requires expanding the operations. To

date, the company has invested abundant capital on its production and storage facilities and

therefore, production capacities exceed the demand of the company for the moment. However,

with the increasing demand growth rate, capacity constraints seem to be fated. Currently, the

production manager makes decisions on the production scheduling and quantities, which are

later used by the transportation manager to plan the distribution. At this point, the company

is interested in how to conduct production planning to save on costs, but at the same time to

maintain a high service level.

To solve the problem in a sequential manner, we exploit three commonly used procedures.

These procedures decompose the main problem into easier subproblems and then solve each

of them separately. Two of these procedures mimic the current situation in companies while

one is a lower bound procedure used as a benchmark. Taking an integrative approach, we

solve the problem by both an exact method and a matheuristic. Our matheuristic combines

an adaptive large neighborhood search (ALNS) heuristic with an exact method.

In summary, the main contributions of this paper are as follows. First, we describe and model

a real-life problem in which production, inventory, distribution, and facility location decisions

are simultaneously taken into consideration. Second, sequential and integrated optimization

approaches are applied. We exploit an exact and a heuristic method to solve the integrated

problem. Finally, we demonstrate the value of the integrated approach by comparing its costs

with those obtained from the sequential approach. Moreover, we evaluate the quality and

performance of all these methods by comparing them with the solutions obtained from the

exact methods.

The remainder of this paper is organized as follows. Section 3.2 provides an overview of the

40

relevant integrated production-distribution literature. In Section 3.3, we formally describe and

model the problem at hand. This is followed by a description of the procedures used to solve

the problem sequentially in Section 3.4. Our proposed integrated matheuristic is explained

in Section 3.5. We present the results of extensive computational experiments in Section 3.6,

followed by the conclusions in Section 3.7.

3.2 Literature review

Despite the abundance of conceptual and empirical studies on supply chain integration and

coordination, e.g., Power [2005], Mustafa Kamal and Irani [2014], until recently, integrated

models of supply chains have been sparse in the operations research literature. Simultaneous

optimization of critical supply chain decisions, by integrating them into a single problem, has

been such a complex and difficult task that the common approach to solving any integrated

problem was to treat each decision separately. Mainly due to their nature, operational level

decisions are the targets for integration, among which production and distribution decisions

are the most important ones. Independently, both production and distribution problems have

several well-studied variants, and so does their integration. As of now, few reviews on various

integrated production-distribution models exist, e.g., Sarmiento and Nagi [1999], Mula et al.

[2010], Chen [2010], Fahimnia et al. [2013], and Adulyasak et al. [2015]. Focusing on the

studies that integrate production with direct shipment, in what follows, we briefly review the

relevant literature. A list of these papers with their features is presented in Table 3.1.

Table 3.1: Integrated production-distribution problems

Reference Number of Inventory Setup LocationProducts Echelons Periods PlantsEkşioğlu et al. [2006] S S M M P XAkbalik and Penz [2011] S S M M P,C XSharkey et al. [2011] S S M M P XDarvish et al. [2016] S S M M P X XJolayemi and Olorunniwo [2004] M S M M C XPark [2005] M S M M P,C XEkşioğlu et al. [2007] M S M M P XMelo and Wolsey [2012] M S M M C XNezhad et al. [2013] M S S M – X XDe Matta et al. [2015] M S/M M M DC XLiang et al. [2015] M S M M C X XBarbarosoğlu and Özgür [1999] M M M S P XJayaraman and Pirkul [2001] M M S M – X XThis paper M M M M P,DC X X

Number of products, echelons, periods and plants: S: Single - M: MultipleInventory at: P: Plant - DC: Distribution center - C: Customer

Ekşioğlu et al. [2006] formulate the production and transportation planning problem as a

network flow and propose a primal-dual based heuristic to solve it. In their model, plants are

multi-functional, production and setup costs vary from one plant to another as well as from

41

one period to the next, and transportation costs are concave. Aiming to compare just-in-time

and time window policies, Akbalik and Penz [2011] consider delivery time windows. With the

just-in-time policy, customers receive a fixed amount whereas, with the time window policy

the deliveries are constrained by the time windows. In their model, costs change over time and

a fixed transportation cost per vehicle is assumed. A dynamic programming (DP) algorithm is

used to solve the problem. The results show that the time window policy has lower cost than

the just-in-time one, furthermore, by comparing the mixed integer linear programming (MILP)

and DP methods, the authors show that even for large size instances the DP outperforms the

MILP. Sharkey et al. [2011] apply a branch-and-price method for an integration of location

and production planning in a single sourcing model. The findings show the potential benefits

of integrating facility location decisions with the production planning. The proposed branch-

and-price algorithm works better when the ratio of the number of customers to the number of

plants is low. Darvish et al. [2016] investigate a rich integrated capacitated lot sizing problem

(LSP) with a single-product, multi-plant, and multi-period setting. They incorporate direct

shipment, delivery time windows, and facility location decisions. They use a branch-and-bound

approach to solve the problem. Assessing the trade-offs between costs and fast deliveries, they

show the competitive advantage of the integrated approach, both in terms of total costs and

service level.

In the profit maximization model presented by Jolayemi and Olorunniwo [2004], any shortfall

in demand can be overcome by either increasing capacity or subcontracting. They introduce a

procedure to reduce the size of the zero-one MILP and, using a numerical example, they show

that the reduced and full-size models generate exactly the same results. Another paper with a

profit maximization objective function is that of Park [2005]. The model allows stockout and

uses homogeneous vehicles for direct shipments. They develop a two-phase heuristic; in the

first phase, the production and distribution plans are identified while in the second, the plans

are improved by consolidating the deliveries into full truckloads. Only for the small instances,

the heuristic generates good results. The paper also investigates the benefits of the integrated

approach compared to the decoupled planning procedure, concluding that with the integrated

approach both the profit and the demand fill rate increase. Ekşioğlu et al. [2007] extend the

problem studied in Ekşioğlu et al. [2006] by considering multiple products. They apply a

Lagrangian decomposition heuristic to solve the problem. The problem investigated by Melo

and Wolsey [2012] is similar to that of Park [2005]. They develop formulations and heuristics

that yield solutions with 10% gap for instances with limited transportation capacity but up

to 40% for instances with joint production/storage capacity restrictions. Nezhad et al. [2013]

tackle an integration of location, production with setup costs, and distribution decisions. In

their problem plants are single-source and not capacitated. They propose Lagrangian-based

42

heuristics to solve the problem. The integrated production-distribution problem addressed in

De Matta et al. [2015] assumes that each plant uses either direct shipment or a consolidated

delivery mode provided by a third party logistics firm. They use Benders decomposition to

select the delivery mode and to simultaneously schedule the production. Liang et al. [2015]

allow backlogging in the model and propose a hybrid column generation and relax-and-fix

method, the exact approach provides the lower bounds while the decomposition yields the

upper bounds.

In Barbarosoğlu and Özgür [1999], a Lagrangian-based heuristic is applied to solve an inte-

grated production-distribution problem. They propose a decomposition technique to divide

the problem into two subproblems and to optimize each of them separately. Jayaraman and

Pirkul [2001] incorporate procurement of the raw material and supply side decisions into the

model. Generating several instances, first, they compare the bounds from the Lagrangian

approach with the optimal solution obtained by a commercial solver. Then, they apply the

proposed method to the data obtained from a real case.

3.3 Problem description and mathematical formulation

We now formally describe the integrated production, facility location, inventory management,

and distribution with delivery time windows problem. We consider a set of plants, available

over a finite time horizon, producing multiple products. Starting a new lot incur a setup cost at

each plant where a variable cost of production is also considered. Each plant owns a warehouse

where the products are stored. An inventory holding cost is due for the products kept at these

warehouses. The products are then sent to DCs, to be shipped to the final customers. There

is a set of potential DCs from which some are selected to be rented. A fixed cost is due and

the DC remains rented for a given duration of time. DCs charge an inventory holding cost per

unit per period. The products are finally shipped to the geographically scattered customers to

satisfy their demand. There is a maximum allowed lateness for the delivery of these products

to customers, meaning that the demand must be met within the predetermined delivery time

window. A service provider is in charge of all shipments, from plants to DCs and from DCs

to final customers. The transportation cost is proportional to the distance, the load, and the

type of product being shipped.

Formally, the problem is defined on a graph G = (N ,A) where N = {1, . . . , n} is the node setand A = {(i, j) : i, j ∈ N , i 6= j} is the arc set. The node set N is partitioned into a plant

set Np, a DC set Nd, and a customer set Nc, such that N = Np ∪Nd ∪Nc. Let P be the set

of P products, and T be the set of discrete periods of the planning horizon of length T . The

inventory holding cost of product p at node i ∈ Np ∪ Nd is denoted as hpi, the unit shipping

43

cost of product p from the plant i to the DC j is cpij , and the unit shipping cost of product

p from the DC j to the customer k is c′pjk. Let also fi be the fixed rental fee for DC i; once

selected, the DC will remain rented for the next g periods. Let spi be the fixed setup cost

per period for product p in plant i, vpi be the variable production cost of product p at plant

i, and dtpi be the demand of customer i for product p in period t. The demand occurring in

period t must be fulfilled until period t+ r, as r represents the delivery time window. For ease

of representation let D be the total demand for all products from all customers in all periods,

i.e., D =∑t∈T

∑p∈P

∑i∈Nc

dtpi.

To solve this rich integrated problem, in each period of the planning horizon, one needs to

determine: the product(s) and quantities to be produced in each plant, the DCs to be selected,

the inventory levels in plants and DCs, the quantity of products sent from plants to DCs, if

the demand of customer is satisfied or delayed, and the quantity of products sent from DCs

to customers.

We formulate the problem with the following binary variables. Let θtpi be equal to one if

product p is produced at plant i in period t, and zero otherwise; λti be equal to one if and

only if DC i is chosen to be rented in period t, to be used for g consecutive periods, and ωti

be equal to one to indicate whether DC i in period t is in its leasing period. Integer variables

to represent quantities produced and shipped are defined as follows. Let αtt′pij be the quantity

of product p delivered from DC i to customer j in period t to satisfy the demand of period

t′, with t ≥ t′, ρtpi represent the quantity of product p produced at plant i in period t, βtpijrepresent the quantity of product p delivered from plant i to DC j in period t, κtpi as the

amount of product p held in inventory at DC i at the end of period t, and µtpi, the amount of

product p held in inventory at plant i at the end of period t.

Table 3.2 summarizes the notation used in our model.

44

Table 3.2: Notation used in the model

Parametershpi inventory holding cost of product p at node i ∈ Np ∪Nd

cpij unit shipping cost of product p from plant i to DC jc′pjk unit shipping cost of product p from DC j to customer kfi fixed renting cost for DC ispi fixed setup cost per period for product p in plant ivpi variable production cost of product p in plant idtpi demand of customer i for product p in period tSetsNc Set of customersNp Set of plantsNd Set of DCsT Set of periodsP Set of productsVariablesθtpi equals to one if product p is produced at plant i in period tλti equals to one if DC i is chosen in period t to be used for g consecutive periodsωti equals to one to indicate whether DC i in period t is in its leasing periodαtt′pij quantity of product p delivered from DC i to customer j in period t, to satisfy the demand of period t′

ρtpi quantity of product p produced at plant i in period tβtpij quantity of product p delivered from plant i to DC j in period tκtpi amount of product p held in inventory at DC i at the end of period tµtpi amount of product p held in inventory at plant i at the end of period tIndicesp, Product indext′, t Period indexi, j Node index

The problem is then formulated as follows:

min∑p∈P

∑i∈Np

∑t∈T

vpiρtpi +

∑p∈P

∑i∈Np

∑t∈T

spiθtpi +

∑p∈P

∑i∈Nd

∑t∈T

hpiκtpi +

∑p∈P

∑i∈Np

∑t∈T

hpiµtpi+∑

i∈Nd

∑t∈T

fiλti +

∑p∈P

∑i∈Np

∑j∈Nd

∑t∈T

cpijβtpij +

∑p∈P

∑i∈Nd

∑j∈Nc

∑t∈T

∑t′∈T

c′pijαtt′

pij

(3.1)

subject to:

ρtpi ≤ θtpiD i ∈ Np, t ∈ T , p ∈ P (3.2)

µtpi = µt−1

pi + ρtpi −∑j∈Nd

βtpij p ∈ P, i ∈ Np, t ∈ T \ {1} (3.3)

µ1pi = ρ1pi −

∑j∈Nd

β1pij p ∈ P, i ∈ Np (3.4)

κtpi = κt−1pi +∑j∈Np

βtpji −

∑j∈Nc

t∑t′=t−r

αtt′

pij p ∈ P, i ∈ Nd, t ∈ T \ {1}(3.5)

κ1pi =∑j∈Np

β1pji −

∑j∈Nc

α11pij p ∈ P i ∈ Nd (3.6)

45

∑p∈P

κtpi ≤ ωtiD i ∈ Nd t ∈ T (3.7)

∑p∈P

κtpi ≤ ωt+1i D i ∈ Nd t ∈ T \ {T} (3.8)

t∑t′=t−g+1

t′≥1

λt′

i ≥ ωti i ∈ Nd t ∈ T (3.9)

t′=t+g−1, t′≤T∑t′=t

ωt′

i ≥ λtimin(g, T − t) i ∈ Nd t ∈ T (3.10)

t′=t+g−1, t′≤T∑t′=t

λt′

i ≤ 1 i ∈ Nd t ∈ T (3.11)

∑i∈Nd

s∑t=1

t∑t′=1

αtt′

pij ≤s∑

t=1

dtpj p ∈ P j ∈ Nc s ∈ T (3.12)

αtt′

pij = 0 p ∈ P i ∈ Nd j ∈ Nc t ∈ T t′ ∈ {0, ..., t− r − 1} ∪ {t+ 1, ..., T} (3.13)∑i∈Nd

min(T,s+r)∑t=1

s∑t′=1

αtt′

pij ≥s∑

t=1

dtpj p ∈ P j ∈ Nc s ∈ T (3.14)

∑p∈P

∑j∈Nc

t∑t′=1

αtt′

pij ≤ Dωti i ∈ Nd t ∈ T (3.15)

∑i∈Nd

∑t∈T

αtt′

pij = dt′

pj p ∈ P j ∈ Nc t′ ∈ T (3.16)

ωti , θ

tpi, λ

ti ∈ {0, 1} (3.17)

ρtt′

pij , κtpi, α

tt′

ijp, βtijp ∈ Z∗. (3.18)

The objective function (3.1) minimizes the total cost consisting of the production setup and

variable costs, inventory holding costs, rental fees, and transportation costs, from plants to

DCs and also from DCs to final customers. Constraints (3.2) guarantee that only products set

up for production are produced. Constraints (3.3) and (3.4) ensure the inventory conservation

at each plant. Similarly, constraints (3.5) and (3.6) are applied to DCs. Constraints (3.7)

and (3.8) guarantee that the remaining inventory at the DC is transferred to the next period

only if the DC is rented in the next period. Constraints (3.9)–(3.11) ensure that once a DC

is selected, it will remain rented for the next g consecutive periods. Constraints (3.11) make

sure that the rental fee for each g period is paid only once. Constraints (3.12) and (3.13)

guarantee that no demand is satisfied in advance, while constraints (3.14) impose r periods as

the maximum allowed lateness for fulfilling the demand. Thus, the total demand up to period

t must be delivered by period t+ r. No delivery to customers can take place from a DC if it

is not rented, as ensured by constraints (3.15). Redundant constraints (3.16) make sure that

every single demand is delivered to the customers. By adding new cuts, redundant constraints

reduce the number of nodes to be explored and improve the performance of the MIP solver

[Gendron and Crainic, 1994, Jena et al., 2015b, Coelho and Laporte, 2014, Jena et al., 2015a,

46

Lahyani et al., 2015a]. Finally, constraints (3.17) and (3.18) define the domain and nature of

the variables.

3.4 Sequential and lower bound procedures

In this section we propose three sequential procedures to solve the problem. Their motivation

is twofold. First, we want to mimic production systems managed in silos, as inspired and

currently conducted by our industrial partner. Second, we want to assess how a sequential

algorithm performs compared to the integrated one proposed in this paper. These compar-

isons are presented in Section 3.3. In what follows, in Section 3.4.1 we present a Top-down

procedure, for the cases in which production is the most important part of the process and

has priority in determining how the system works. This decision is then followed by inven-

tory allocation to DCs and finally by distribution decisions. In Section 3.4.2 we describe a

Bottom-up procedure, simulating the alternative scenario in which distribution has priority.

The distribution decisions are followed by DC allocation, and lastly by production decisions.

Finally, in Section 3.4.3 we describe an Equal power procedure, in which all three departments

would have similar positions in the hierarchy of power; we explain how this procedure yields

a lower bound on the optimal cost.

3.4.1 Top-down procedure

In the Top-down procedure, production managers have the most power and therefore, they

can determine how the rest of the system works. This method, which observed as the current

practice of our industrial partner, works as follows.

First, minimize only production costs∑p∈P

∑i∈Np

∑t∈T

vpiρtpi +

∑p∈P

∑i∈Np

∑t∈T

spiθtpi subject to (3.2)–

(3.18). An optimal solution to this problem determines the best production plan without

any interaction with downstream decisions. Let these optimal decision values be θtpi and ρtpi.

Note that because these production decisions are made considering the whole feasible region,

determined by (3.2)–(3.18), feasibility is ensured.

The second phase works by considering a minimization objective function consisting of only

DC-related terms, namely∑

i∈Nd

∑t∈T

fiλti, subject to (3.2)–(3.18), and to θtpi = θtpi and ρ

tpi = ρtpi.

In this problem, inventory allocation decisions are made subject to the feasible region of the

overall problem and the production decisions that had priority over the inventory ones. Let

the value of these decision variables be λti and ωti .

The final phase consists of determining the best way to distribute the inventory to the cus-

tomers, given fixed production and allocation plans. This is accomplished by minimizing

47

∑p∈P

∑i∈Nd

∑j∈Nc

∑t∈T

∑t′∈T

c′pijαtt′pij , and subject to the feasible region of the original problem (3.2)–

(3.18), and to θtpi = θtpi, ρtpi = ρtpi, λ

ti = λti and ω

ti = ωt

i . By putting together all three levels

of decisions, one can obtain the overall solution and easily compute the cost of the solution

yielded by the Top-down procedure. A pseudocode of this procedure is presented in Algorithm

1.

Algorithm 1 Top-down procedure1: Consider all constraints of the problem formulation from Section 3.3, (3.2)–(3.18).2: Build an objective function with production variables θtpi and ρtpi:

∑p∈P

∑i∈Np

∑t∈T

vpiρtpi +∑

p∈P

∑i∈Np

∑t∈T

spiθtpi.

3: Optimize the problem, obtain optimal values θtpi and ρtpi.4: Fix θtpi and ρtpi to their obtained values.5: Add DC-related variables λti to the objective function:

∑i∈Nd

∑t∈T

fiλti.

6: Optimize the problem, obtain optimal values λti and ωti .

7: Fix variables λti and ωti to their obtained values.

8: Add all variables to the objective function, as it is defined in (3.1).9: Optimize the problem, obtain optimal values for all variables.

10: Return the objective function value.

3.4.2 Bottom-up procedure

In the Bottom-up procedure, we suppose that the distribution managers have the most power

and can, therefore, determine how the rest of the system works. This is done by taking

all constraints (3.2)–(3.18) into account, but optimizing the objective function only for the

distribution variables. Once distribution decisions are made and fixed, inventory allocation

decisions, namely when and which DCs to rent, are optimized. As mentioned earlier, feasibility

is guaranteed. We now solve the same problem with a new set of fixed decisions (related to

distribution), and optimize only DC-related costs. When this part is determined, all the

decisions are fixed and no longer change. Finally, once DCs have been selected, and all

distribution and DC variables are known, we can optimize the remaining variables of the

problem. By putting together all three levels of decisions, one can obtain the overall solution

and easily compute the cost of the solution yielded by the Bottom-up procedure. A pseudocode

of this procedure is presented in Algorithm 2.

3.4.3 Equal power procedure

In the Equal power procedure, we assume that all three decision levels have equal power.

Therefore, information is shared with all departments at the same time but decisions are

made in parallel and each department optimizes its own decisions. This procedure will likely

yield an infeasible solution since each part of the problem is optimized individually. However,

48

Algorithm 2 Bottom-up procedure1: Consider all constraints of the problem formulation from Section 3.3, (3.2)–(3.18).2: Build an objective function with distribution variables αtt′

pij :∑p∈P

∑i∈Nd

∑j∈Nc

∑t∈T

∑t′∈T

c′pijαtt′

pij .

3: Optimize the problem, obtain optimal values αtt′

pij .4: Fix αtt′

pij to their obtained values.5: Add DC-related variables λti to the objective function:

∑i∈Nd

∑t∈T

fiλti.

6: Optimize the problem, obtain optimal values λti and ωti .

7: Fix variables λti and ωti to their obtained values.

8: Add all variables to the objective function, as it is defined in (3.1).9: Optimize the problem, obtain optimal values for all variables.

10: Return the objective function value.

the sum of the costs of all three levels indicates the optimal decision for each level, when

the costs of the other levels are not considered. Having all three decision levels put together,

if these yield a feasible solution, it is obviously optimal, otherwise, their sum constitutes a

valid lower bound on the costs of the problem. Algorithm 3 describes the pseudocode for this

procedure.

Algorithm 3 Equal power procedure1: Consider all constraints of the problem formulation from Section 3.3, (3.2)–(3.18).2: Build an objective function with distribution variables αtt′

pij .3: Optimize the problem, obtain optimal values αtt′

pij , and optimal distribution solution zc.4: Build an objective function with DC-related variables λti, βt

pij , and κtpi.5: Optimize the problem, obtain optimal values for λti, ωt

i , βtpij , and κtpi, and optimal DC solution

zd.6: Build an objective function with plant-related variables θtpi, ρtpi, and µt

pi.7: Optimize the problem, obtain optimal values for θtpi, ρtpi, and µt

pi, and optimal production solutionzp.

8: if the combination of all three decisions is feasible then9: Return optimal solution and its cost z∗ = zp + zd + zc.

10: else11: Return lower bound value z = zp + zd + zc.12: end if

3.5 Integrated solution algorithm

The problem at hand is reducible to the multi-plant uncapacitated LSP and also the joint-

replenishment problem, an extension of the uncapacitated fixed charge network flow. The

joint-replenishment problem is known to be NP-hard [Cunha and Melo, 2016], as are most

variants of the LSP. Although the uncapcitated LSP is easier to solve, the multi-plant version

is still NP-complete [Sambasivan and Schmidt, 2002]. As is the case of many other NP-hard

problems, exact methods can solve small-size instances to optimality in a reasonable time but

to obtain good solutions for larger instances, one must develop ad hoc heuristic algorithms. To

49

solve the problem at hand, we propose a matheuristic based on a hybrid of ALNS and exact

methods. The ALNS introduced by Ropke and Pisinger [2006] has shown outstanding results

in solving various supply chain problems. ALNS, as a very efficient and flexible algorithm,

explores large complex neighborhoods and avoids local optima. Hence, because of its generality

and flexibility, it is highly suitable for the problem at hand. Our contribution, however, lies

in customizing and applying this method to our problem.

We propose a three-level matheuristic approach in which the problem is divided into two

subproblems that are then solved in an iterative manner. In the first level, we apply the

ALNS heuristic in order to decide which plants and DCs should be selected, and to determine

which products have to be produced in any of the selected plants. In the second level, all the

other remaining decisions on deliveries from selected plants to rented DCs, and from rented

DCs to the customers, as well as the inventory level held at plants and rented DCs are obtained

exactly by solving an integer linear programming sub-problem. Finally, if needed and to avoid

local optimum solutions, we improve the obtained solution and move it toward the global

optimal by solving the model presented in Section 3.3 with exact methods for a very short

period of time. The detailed algorithmic framework is as follows.

• Initial solution: we start with generating a feasible initial solution by making all plants

and DCs selected in all periods. This feasible initial solution is quickly improved by

deselecting as many facilities as possible while maintaining feasibility. At this step,

costs are not yet of concern and in order to improve the solution, we take all the con-

straints of (3.2)–(3.18) and solve the problem with the following objective function:

min∑p∈P

∑i∈Np

∑t∈T

θtpi +∑

i∈Nd

∑t∈T

λti. We obtain the initial solution s and its corresponding

cost z(s) to be improved.

• Large neighborhood: at each iteration, one operator from the list described in Section

3.5.1 is selected. Operators work for any type of facility; therefore, plants and DCs have

the same chance of being selected. To diversify the search, following Pereira et al. [2015],

each operator is repeated n times, n being drawn from a semi-triangular distribution

and bounded between [1,a]. We compute n as in (3.19) where b is a random number in

the [0,1] interval and a is an integer number.

n =⌊a−

√(1− b)(a− 1)2 + 0.5

⌋(3.19)

• Adaptive search engine: the operators are selected according to a roulette-wheel mech-

anism. A weight is associated to previous performances of each of the operators, modu-

lating their chances of being selected.

50

• Acceptance criteria: To diversify the solutions, a simulated annealing-based acceptance

rule is applied. The current solution s is accepted over the incumbent solution s′ with the

probability of e

z(s′)− z(s)H

, where H is the current temperature. The temperature

is decreased at every iteration by α, where 0 < α < 1. Once the temperature reaches

the final temperature, Hfinal, it is reset to the initial temperature, Hstart.

• Adaptive weight adjustment: A score and a weight are assigned to each of the operators.

The weight matrix, which has an initial value of one, is updated at every ϕ iterations. It

is updated using the scores each operator has accumulated. The score matrix is initially

set to zero, and the better the operator performs, the higher score it accumulates. We

define σ1 > σ2 > σ3 > 0. If an operator finds a solution better than the best solution

obtained so far, a score of σ1 will be assigned to it. If the obtained solution by the

operator is not the best but it is better than the incumbent solution, the score will be

updated by σ2. Finally, if the solution is no better than the incumbent solution but it

still satisfies the acceptable criteria, the operator will be given a σ3 score.

• Periodic post-optimization: if no improvement is achieved for more than 2ϕ iterations,

we use the best solution as an input to the model of Section 3.3 and solve it for 20

seconds with the exact method. If this post-optimization attempt yields an optimal

solution, the algorithm stops, as the global optimum has been found; otherwise, if it

improves the solution, the improved solution is passed to the ALNS framework and the

procedure continues.

• Stopping criteria: the algorithm will stop, if either the maximum number of iterations

itermax or the maximum allotted time is reached, we limit the running time to one hour.

It will also stop, if the solution does not change in more than itermax2 iterations. Moreover,

it will stop when the optimal solution is obtained in the periodic post-optimization step.

3.5.1 List of operators

The operators we have designed to explore the search space with the ALNS framework are as

follows.

1. Random: this operator selects a plant, a product and a period (or a DC and a period),

and flips its current status; if the facility is not in use it becomes in use, and vice-versa.

2. Based on shipping costs: first, for each product we compare the shipping costs from

plants that are not producing any product to all currently rented DCs, and then we

51

identify the combination of product, plant, and period with the lowest cost. The corre-

sponding product is then assigned to be produced at that plant in that period. Similarly,

the highest shipping cost induces a product to have its production stopped at the given

plant and period.

3. Based on unit costs: among all plants, we identify the plant and the product with the

highest unit production cost; we stop production of the identified product in the selected

plant; for DCs, we stop renting the one with the highest unit inventory cost.

4. Based on demand: first, we identify the product and period with the highest demand,

then we make all the plants produce that product in that period. Similarly, we identify

the product and period with the minimum demand, and stop its production in the

identified period.

5. Based on delivery quantity to DCs: we identify the plant delivering the least (most) and

the DC receiving the least (most) per period. Facilities with the least usage will not be

in use; for those with the largest usage, a random DC is rented in the same period, and

production for the same plant is set up for all products in the following period.

6. Based on inventory level: we identify the plant and period with the maximum inventory,

and ensure it stays in use in that period. If the plant is already in use, we keep it in

use also in the next period. For DCs, we stop renting the one with the lowest inventory

level during its g leasing periods.

7. Based on production quantity: we identify the product/plant/period combination in

which the maximum (minimum) production occurs; we stop production of that product

in the plant with the smallest production in the identified period but assign the plant

and the product with the maximum production to its next period. We also identify the

period with the highest production, and rent an extra DC.

8. Based on delivery quantity to customers: we identify all DC/period combinations with

deliveries lower than a percentage of the total demand and among them, we select a DC

and end its lease for that period (and consequently the next g periods). Similarly, for

plants, we select a random one and stop production of all products in the previously

identified period.

3.5.2 Parameter settings and the pseudocode

We have tested different combinations of parameters and tuned them mainly by trial and error.

The initial temperature Hstart is set to (r + 1) × 100,000. This initial temperature is cooled

down until it reaches the final temperature Hfinal = 0.01. The cooling rate, α, is tuned to

52

0.999. In our implementation, iteration count is one of the stopping criteria, and it is satisfied

once 3,000,000 iterations are performed. We set ϕ to 1,000 iterations and update the scores

with σ1 = 10, σ2 = 4, and σ3 = 3.

The pseudocode for the proposed matheuristic is provided in Algorithm 4.

Algorithm 4 Proposed matheuristic1: Initialize weights to 1, scores to 0, H ← Hstart.2: s← sbest ← initial solution.3: while stopping criteria are not met do4: s′ ← s5: Select an operator and apply it to s′6: Solve the remaining problem, obtain solution z(s′)7: if z(s′) < z(s) then8: if z(s′) < z(sbest) then9: sbest ← s′

10: update the score for the operator used with σ111: else12: s← s′

13: update the score for the operator used with σ214: end if15: else16: if s′ is accepted by the simulated annealing criterion then17: update the scores for the operator used with σ318: s← s′

19: end if20: end if21: H ← α×H22: if iterations is a multiple of ϕ then23: update weights and reset scores of all operators24: if no improvement found in last 2ϕ iterations then25: if H < Hfinal then26: H ← Hstart

27: if no improvement found for z(s′) then28: Input sbest into the MIP in Section 3.3 and solve it for 20 seconds29: end if30: end if31: else32: s← sbest33: end if34: end if35: end while36: Return sbest

3.6 Computational experiments

We now describe the details related to the computational experiments used to evaluate our

algorithms. All computations are conducted on Intel Core i7 processor running at 3.4 GHz

with 64 GB of RAM installed, with the Ubuntu Linux operating system. A single thread was

53

used for up to one hour, i.e., a time limit of 3600 seconds was imposed on all algorithms. The

algorithms are coded in C++ and we use IBM Concert Technology and CPLEX 12.6.3 as the

MIP solver. Section 3.6.1 describes how the instances are generated, detailed computational

results are provided in Section 3.6.2, and sensitivity analysis and the managerial insights are

provided in Section 3.6.3.

3.6.1 Generation of the instances

By consultation with our industrial partner, we have generated a large data set by varying

the number of products, periods, plants, DCs, and customers. Our test bed is generated as

shown in Table 3.3. The number of plants and DCs are determined by the number of periods:

if T = 5, then Nd = 8 and Np = 5, if T = 10, then Nd = 15 and Np = 10, and finally if

T = 50, then Nd = 25 and Np = 15. For each of ten combinations, we generate five random

instances. For each instance we consider a delivery time window r = 0, 1, 2, or 5 periods.

Thus, we solve 200 instances in total.

Table 3.3: Input parameter values

Name Parameter ValuesProducts P 1, 5, 10Periods T 5, 10, 50Plants Np 5, 10, 15DCs Nd 8, 15, 25Customers Nc 20, 50, 100Delivery time window r 0, 1, 2, 5DC active period g T

5Demand dtpk [0, 2]Plant setup cost spi [1,000, 1,500]Plant variable cost vpi [10, 50]Fixed DC renting cost fj [10,000, 15,000]Inventory holding cost hpj [1, 4]Shipping cost (plants-DC) cpij [10, 100]Shipping cost (DC-customers) c′pjk [10, 1,000]

3.6.2 Results of the computational experiments

We now present the results of extensive computational experiments carried out to evaluate the

performance of all algorithms, and to draw meaningful conclusions for the problem at hand.

We first describe the results of the experiments with the mathematical model proposed in

Section 3.3. This is followed by the comparison of the performance of the sequential procedures

proposed in Section 3.4, and our integrated hybrid matheuristic from Section 3.5 with that of

the exact algorithms.

54

Average computational results using the CPLEX branch-and-bound algorithm are presented

in Table 3.4. For each instance, we report the average of the gaps (G) with respect to the

lower bound obtained by CPLEX, calculated as 100 × Upper Bound− Lower BoundLower Bound

, the

number of cases solved to optimality (O), and the average running time (T ) in seconds for

each predetermined time window r. As presented in Table 3.4, only the small instances, mostly

those with fewer than five products or periods, could be solved to optimality. The parameter

controlling the number of periods seems to have a strong effect on the performance of the

exact method. Indeed, it has a huge effect on the size of the problem as measured by the

number of variables and constraints. Moreover, the length of the delivery time window affects

the number of instances solved to optimality, the average gap, and the running time.

Table 3.4: Results from the branch-and-bound algorithm

Instance r = 0 r = 1 r = 2 r = 5

P -T -Nc-Nd-Np G(%)(O) T (s) G(%)(O) T (s) G(%)(O) T (s) G(%)(O) T (s)

1-5-20-8-5 0.00(5) 2 0.00(5) 2 0.00(5) 1 0.00(5) 01-10-100-15-10 2.43(1) 3,113 20.15(0) 3,606 25.97(0) 3,603 27.63(0) 3,6031-10-50-15-10 0.00(5) 610 10.94(0) 3,616 18.13(0) 3,606 16.50(0) 3,6011-50-100-25-15 30.69(0) 3,640 94.85(0) 3,626 131.83(0) 3,624 247.84(0) 3,6345-5-20-8-5 0.00(5) 342 0.13(4) 1,321 0.81(3) 1,703 0.00(5) 2215-10-100-15-10 14.48(0) 3,612 28.41(0) 3,601 32.95(0) 3,601 39.21(0) 3,6015-10-50-15-10 11.56(0) 3,610 22.49(0) 3,602 23.19(0) 3,602 25.65(0) 3,60210-5-20-8-5 2.08(0) 3,602 3.66(0) 3,604 4.27(0) 3,604 0.05(5) 1,27510-10-100-15-10 14.60(0) 3,602 26.93(0) 3,602 33.88(0) 3,610 34.43(0) 3,60110-10-50-15-10 7.19(0) 3,605 21.38(0) 3,603 29.74(0) 3,602 20.83(0) 3,611Average 8.30(0.32) 2,574 22.89(0.18) 3,018 30.08(0.16) 3,056 41.21(0.30) 2,675

To evaluate the performance of the sequential procedures versus our proposed matheuristics

and to gain insight into managerial decisions related to the problem at hand, we present their

results in Tables 3.5–3.8, one table per value of the delivery time window r. The improvements

with respect to the solution obtained from the exact algorithm by the Top-down, Bottom-up,

and matheuristic algorithms are presented along with their running times. For each method,

this improvement is obtained as I(%) = 100× Upper BoundCPLEX − Costmethod

Upper BoundCPLEX.

Table 3.5 presents the results obtained with no delivery time window, i.e., r = 0. On average,

the proposed method gets slightly better solutions than CPLEX. When only one product

is involved, no matter of the number of customers or periods, our proposed method always

outperforms CPLEX. Both the Top-down and Bottom-up procedures are very fast, but the

costs obtained by these methods are much higher than the ones from CPLEX. As indicated in

Table 3.5, the Bottom-up procedure outperforms the Top-down on almost all large instances

with multiple products, more than five periods and 50 customers. Although on average the

Bottom-up procedure takes less running time, the results obtained by this procedure are about

1.5 times worse than the ones from the Top-down.

55

Table 3.5: Heuristics results for r = 0

Instance Top-down Bottom-up proposed methodP -T -Nc-Nd-Np1 I (%) T(s) I (%) T(s) I (%) T(s)1-5-20-8-5 −10.03 0 −315.27 0 0.00 11-10-100-15-10 −46.41 3 −157.86 2 0.10 3,6061-10-50-15-10 −33.05 2 −238.78 2 0.00 2,0481-50-100-25-15 −123.86 558 −123.01 245 8.14 3,6025-5-20-8-5 −18.42 1 −107.79 0 0.00 1,1505-10-100-15-10 −119.74 34 −42.70 13 −1.25 3,6025-10-50-15-10 −70.89 12 −74.98 5 −0.74 3,60410-5-20-8-5 −32.37 1 −50.13 1 −0.16 3,46210-10-100-15-10 −179.37 53 −23.22 38 −3.69 3,60510-10-50-15-10 −116.55 51 −38.67 21 −0.66 3,610Average −75.09 72 −117.24 33 0.17 2,829

When r = 1, as indicated in Table 3.6, our approach always outperforms CPLEX, with an

average improvement of 4.17%. For a large instance with one product, 50 periods, and 100

customers, this difference is up to 27.03%. Although the solutions obtained by both sequential

methods have slightly worsened, the extra delivery period has dramatically increased the

running time for the Top-down procedure, with almost no significant effect on the Bottom-up.

Table 3.6: Heuristics results for r = 1

Instance Top-down Bottom-up proposed methodP -T -Nc-Nd-Np I (%) T(s) I (%) T(s) I (%) T(s)1-5-20-8-5 −19.34 0 −423.38 0 0.00 31-10-100-15-10 −54.70 17 −201.61 1 2.21 3,6081-10-50-15-10 −43.01 8 −301.56 3 0.71 3,4741-50-100-25-15 −84.26 1,391 −162.38 313 27.03 3,6015-5-20-8-5 −39.68 2 −149.74 0 0.00 2,6275-10-100-15-10 −142.26 1,012 −68.09 16 2.75 3,4195-10-50-15-10 −92.443 289 −109.26 6 4.10 3,60410-5-20-8-5 −54.21 3 −89.97 2 0.08 3,60810-10-100-15-10 −195.44 1,292 −46.50 38 1.58 3,60110-10-50-15-10 −129.44 1,054 −60.89 25 3.27 3,604Average −85.45 507 −161.34 40 4.17 3,115

Table 3.7 shows the results obtained by considering two-day delivery time window, i.e., r = 2.

On average our algorithm improves the solution by 5.29%. As before, the biggest improvement

is observed for the large instance with one product, 50 periods and 100 customers, but small

instances are either solved to optimality as CPLEX or has been slightly improved. As the

time window grows, the performance of both Top-down and Bottom-up procedures declines

but compared to the r = 1 case, the running time slightly increases.

56

Table 3.7: Heuristics results for r = 2

Instance Top-down Bottom-up proposed methodP -T -Nc-Nd-Np I (%) T(s) I (%) T(s) I (%) T(s)1-5-20-8-5 −20.41 0 −459.31 0 0.00 41-10-100-15-10 −65.78 20 −200.37 3 2.87 3,6081-10-50-15-10 −51.46 9 −282.99 2 0.57 3,3711-50-100-25-15 −80.81 1,549 −206.21 317 31.44 3,6015-5-20-8-5 −45.66 2 −178.08 1 0.04 3,1495-10-100-15-10 −162.31 1,244 −84.87 24 1.80 3,6025-10-50-15-10 −102.42 431 −125.29 10 4.87 3,60310-5-20-8-5 −76.32 3 −109.47 6 0.75 3,60610-10-100-15-10 −201.60 1,291 −61.37 66 4.74 3,60110-10-50-15-10 −170.87 1,034 −105.63 24 5.86 3,601Average −97.76 558 −181.36 45 5.29 3,174

The best results of our proposed method are obtained for r = 5. As presented in Table 3.8,

our method improves the results by 7.62%. The difference in performance of the two methods

becomes even more evident for the big instances with 50 periods and 100 customers, in which

our proposed method improves the solution obtained by CPLEX up to 49.66%. As before, the

two sequential procedures can quickly provide feasible solutions, but of very poor quality.

Table 3.8: Heuristics results for r = 5

Instance Top-down Bottom-up proposed methodP -T -Nc-Nd-Np I (%) T(s) I (%) T(s) I (%) T(s)1-5-20-8-5 −36.13 0 −514.97 0 0.00 21-10-100-15-10 −90.91 19 −219.95 4 4.65 3,6071-10-50-15-10 −84.37 6 −269.28 3 1.13 3,6081-50-100-25-15 −39.37 1,769 −273.87 412 49.66 3,6015-5-20-8-5 −76.86 0 −199.40 1 0.00 3,2055-10-100-15-10 −198.27 470 −109.05 30 3.47 3,6015-10-50-15-10 −138.67 104 −163.85 12 9.31 3,60110-5-20-8-5 −113.01 1 −126.65 1 0.00 3,60510-10-100-15-10 −257.30 1,151 −62.30 98 0.89 3,60110-10-50-15-10 −170.87 459 −105.63 43 7.14 3,601Average −120.58 398 −204.50 60 7.62 2,685

3.6.3 Sensitivity analysis and managerial insights

We now perform sensitivity analysis to derive important managerial insights. From Table

3.4, we observe that the more flexible the delivery time windows gets, the harder to solve the

problem becomes. Also, as the number of products, periods, and customers increases, the

problem becomes harder to be solved to optimality. Small instances with P = 1, T = 5, and

Nc = 20 are easily solved to optimality, however, instances with only one product but T > 5

cannot be solved to optimality under the presence of any delivery time window.

This difficulty in solving the problem when delivery time windows exist shows two interesting

57

aspects of the business problem. The first one is related to the potential cost saving if one is

to properly exploit the added flexibility of time windows. This is evident since all solutions

without time windows are still valid to the cases in which they are considered. However, to

take advantage of such flexibility, using a tailored method seems necessary. As shown already,

modeling the problem into a commercial solver or using a sequential method does not yield

any good solutions. In fact, the quality of solutions degrades as the size of the problem

and the added flexibility increase. Figure 3.1 provides an overview on the comparison of our

matheuristic and CPLEX for different delivery time windows. We compare the performance

of both methods over the lower bound obtained by CPLEX. As observed in this figure, on

average over all instances, our proposed algorithm works better when the delivery time window

enlarges. The results reveal that for large instances our matheuristic outperforms the exact

algorithm. The highest average improvement is obtained for r = 5. For all instances that

could be solved to optimality by CPLEX, our algorithm also obtains the optimal solution.

r0 r1 r2 r52,000

2,500

3,000

3,500

2,574

3,018 3,056

2,675

2,829

3,1153,174

2,685

Time (s)

CPLEX

Matheuristic

r0 r1 r2 r5

10

20

30

40

8.3

22.89

30.08

41.21

7.94

15.12

19.06 19.82

Gap (%)

CPLEX

Matheuristic

Figure 3.1: Comparison between time (s) and gap (%) of CPLEX and the proposed matheuristic

Considering the processing time, CPLEX performs slightly better, mainly because the iterative

heuristic reaches the time limit to search the solution area, aiming to improve the solution

obtained. However, as presented in Table 3.9, our algorithm takes on average less than 20

minutes to find its best solution, which is often better than the ones from the exact algorithm.

Table 3.9: Average time for the proposed method to obtain its best solution

Time window r = 0 r = 1 r = 2 r = 5 AverageAverage time (s) 1,160 1,159 1,022 909 1,063

Regarding our proposed matheuristic, Tables 3.5–3.8 also reveal that taking an integrative

58

approach towards production, location, inventory, and distribution decisions can lead to enor-

mous cost reductions. For all time windows, the average results obtained from the proposed

method are always better than the sequential ones. It is interesting to note that on average

the solutions obtained by Top-down procedure are lower than the ones from the Bottom-up

approach; however, the Bottom-up procedure is much faster. As presented in Tables 3.5–3.8,

the Bottom-up procedure generates better results, in less time, than the Top-down when P > 1

and Nc > 20.

As expected, applying the Equal power procedure, where each department of the company is

focused only on its own decisions, results in not even one instance with a feasible solution.

Comparing the solutions obtained by this procedure to the lower bounds of the exact algorithm,

on average this infeasible solution from the Equal power procedure is 48.11% worse than the

lower bound, which forgoes any hopes that this approach would yield any good solution. For

this reason we do not provide detailed results from this method.

3.7 Conclusions

This paper investigates a challenging and practical problem of integrated production, location,

inventory, and distribution, in which multiple products are produced over a discrete time hori-

zon, stored at the DCs before being shipped to final customers. The paper contributes to the

integrated optimization literature as it combines distinct features of delivery time windows,

distribution with direct shipment, and dynamic location decisions. A state of the art commer-

cial solver is able to find optimum solutions for very small instances of our problem, however,

it does not prove optimality in a reasonable time for larger instances. To achieve better solu-

tions in an acceptable computation time, we have proposed a matheuristic algorithm. Several

instances are generated and the solutions are compared to the optimal ones (if any) obtained

by the exact method. On average the solutions obtained with our algorithm improve the ones

from the exact method by up to 49.66%, generally in only a third of the running time.

In this paper, we have also evaluated how a typical management in silos would perform,

by deriving and implementing sequential solution methods. Our results confirm the cost

benefits of the integrated approach towards decision making. Both Top-down and Bottom-up

procedures perform worse than the exact methods as well as our proposed method. However,

between these two procedures, the Bottom-up works better for instances with larger planning

horizons and more products and customers, while Top-down is preferred when there is only

one product and fewer than 20 customers.

Using our randomly generated instances validated by an industrial partner, we have shown

59

the benefits of an integrated management, as opposed to the sequential one. Moreover, we

have shown that for complex and rich integrated problems inspired by real-world cases, such

as the one studied here, neither a hierarchical solution approach nor modeling and solving the

problem by a commercial solver yield good solutions in a reasonable time. We have proposed

a flexible and very powerful method which is capable of effectively handling all aspects of the

problem in an efficient manner.

60

Chapter 4

Flexible two-echelon location routing

Chapter information A paper based on this chapter is currently submitted for publication:

M. Darvish, Archetti, C., Coelho, L.C., Speranza, M.G. Flexible two-echelon location routing.

Document de travail du CIRRELT-2017-64, Tech. Rep., 2017.

This chapter deals with an integrated routing problem in which a supplier delivers a commodity

to its customers through a two-echelon supply network. The commodity is first sent from a

single depot to a set of distribution centers (DCs). Then, from the DCs, it is delivered to

customers on the basis of their requests. A limited planning horizon is considered and the

objective is to minimize the total cost consisting of the sum of the shipping costs from the depot

to the DCs, the traveling costs from DCs to customers, the location costs, and the penalty

costs for any unmet demand. On top of this basic setting, we study two sources of flexibility:

flexibility in due dates and flexibility in the network design. The former establishes an interval

within which the customer requests can be satisfied while the latter is related to the possibility

of deciding which DCs are convenient to be rented at each period of the planning horizon.

We present a mathematical formulation of the problem together with different classes of valid

inequalities. Extensive computational tests are made on randomly generated instances to show

the value of the two kinds of flexibility. Computational and business insights are discussed.

The results show that the combined effect of the two kinds of flexibility leads to total average

savings of up to almost 35%.

4.1 Introduction

The recent literature on routing problems is evolving to the study of more and more complex

problems. This complexity stems from different sources, among which integration and flexibil-

ity are the most investigated. By integration, we mean to include broader parts of the decision

61

systems, and not only the one focused on the pure stand-alone routing. One of the most clas-

sical examples of integrated routing problem is the Inventory Routing Problem (IRP) where

routing is integrated with inventory management (see Bertazzi and Speranza [2012, 2013],

Coelho et al. [2014] for surveys and tutorials and Archetti and Speranza [2016] for a study on

the value of integration in IRPs). Other important examples of integrated routing problems

are the ones in which network design issues are integrated in routing decisions, an example

being location routing problems (see Prodhon and Prins [2014] for a recent survey) and two-

echelon vehicle routing problems (see Cuda et al. [2015] for a recent survey and Guastaroba

et al. [2016] for a more general survey on transportation problems with intermediate facilities).

Flexibility is related to the possibility of relaxing some constraints in order to save costs. For

example, in a distribution problem where customers requests have to be satisfied within a

planning horizon, one may achieve cost savings if flexibility in the due dates is allowed, as

shown in Archetti et al. [2015]. Another study related to the advantage of flexibility in routing

problems is provided in Archetti et al. [2017] where the authors study the flexible periodic

vehicle routing problem, that is a generalization of the periodic vehicle routing problem in

which no visiting schedule is considered.

In this paper we study a routing problem where both integration and flexibility are considered.

In particular, we study a problem coming from a real application where a supplier has to build

a distribution plan to serve the customers through a two-layer distribution network. A single

commodity is produced at a production plant, or stocked at the depot, and is distributed from

there to a set of distribution centers (DCs). Then, the commodity is delivered to customers

from the DCs. A planning horizon is considered which is discretized in periods, typically days.

The supplier has the possibility to choose among the available DCs on a daily basis. In fact,

we consider the DCs as the rented space in physical facilities shared with other companies and

managed by a third party. Daily customers requests are known and dynamic. Moreover, each

order has a due date, which represents the latest delivery date. Each order has to be entirely

fulfilled in one delivery. A penalty is related to an unmet demand, i.e., to orders which are

not satisfied by the delivery due date. Products are shipped from the depot to the selected

DCs via direct shipment, and from DCs to customers via milk runs. The supplier has to take

four simultaneous decisions: which DCs to use in each period, when to satisfy the orders of

customers, from which of the selected DCs to ship to the customers, and how to create vehicle

routes from the selected DCs to the customers.

We call this problem the Flexible Two-Echelon location routing Problem (F-2E-LRP), in which

the objective is to minimize the total costs consisting of the sum of the shipping costs from

the depot to the DCs, the delivery cost from the DCs to the customers, the renting cost of

62

DCs, and the penalty cost for the unmet demand. The F-2E-LRP merges integration issues

related to the decision of which DCs to rent, and flexibility issues coming from two sources:

• the possibility of selecting amongst the available DCs on a daily basis;

• the possibility of selecting the day when customer orders are satisfied, provided that

either the due date is respected or a penalty is paid.

The F-2E-LRP is motivated by recent interest in collaborative business. As an important

variant of the sharing economy, in collaborative business, companies gain by sharing their

assets, capacities or in general their infrastructure with others [Savelsbergh and Van Woensel,

2016]. While Savelsbergh and Van Woensel [2016] name some advantages for sharing assets,

in this paper we highlight the potential advantages of optimizing a supply chain over different

kinds of flexibility. From the academic point of view, as mentioned earlier, the F-2E-LRP

is also related to several well known problems, including location routing, inventory routing,

and multi-depot vehicle routing problems. Location routing is the problem of determining the

location of facilities that are then used to distribute goods to customers (see Prodhon and

Prins [2014] and Drexl and Schneider [2015] for recent surveys). The F-2E-LRP extends the

location routing problem as it considers a two-echelon network and a planning horizon where

the location decision is taken on a daily basis. The link with inventory routing is due to the

fact that DCs may be used to store goods from one day to another before being delivered

to customers. Finally, the F-2E-LRP is related to the multi-depot vehicle routing problem

[Renaud et al., 1996, Cordeau et al., 1997, Lahyani et al., 2015b] as routes serving customers

depart from different DCs.

In addition to the above mentioned problems, the F-2E-LRP is an extension of the works

presented in Archetti et al. [2015] and Darvish et al. [2016]. Archetti et al. [2015] study the

multi-period vehicle routing problem with due dates. They propose several formulations, solve

them through branch-and-cut and compare their performance. This work extends the work of

Archetti et al. [2015] by first adding intermediate facilities (DCs) where goods are stored and

second, by considering the possibility of choosing among several DCs on a daily basis. Darvish

et al. [2016] study a multi-echelon integrated lot sizing-distribution problem considering both

delivery time windows and facility location decisions. A key difference between the F-2E-LRP

and that problem is the use of vehicle routes to manage the distribution to customers instead

of direct shipments, which significantly enriches the problem setting investigated by Darvish

et al. [2016].

The contributions of this paper are summarized as follows. We introduce the F-2E-LRP and

propose a mathematical formulation along with different classes of valid inequalities. We run

63

a large set of experiments on randomly generated instances to show the value of flexibility,

both in terms of due dates, in terms of network design, and on their combined effect. The

results highlight the cost saving advantages of both types of flexibility. In particular, we show

that the combined effect of the two kinds of flexibility leads to a saving in total cost of up to

almost 35%. We also provide computational and business insights based on this analysis.

The remainder of the paper is organized as follows. In Section 4.2 we formally describe the

problem while in Section 4.3 we present a mathematical formulation together with different

classes of valid inequalities. We present the results of the computational experiments in Section

4.4, followed by our conclusions in Section 4.5.

4.2 Problem description

In the F-2E-LRP a supplier delivers a single commodity to its customers through a two echelon

supply chain which consists of the supplier plant, referred to as the depot, and a set of DCs.

The supplier decides on a daily basis which subset of DCs to use in order to distribute the

goods to its final customers. DCs are replenished by direct shipment from the depot. Goods

are consolidated at the DCs and distributed to final customers via milk runs. Without loss

of generality, each day all DCs are available to be rented for a fee. The paid fee covers the

fixed cost to use a vehicle, for which only routing costs are due. Dynamic customer orders

are known in each period, and a due date is associated with them, i.e., each order must be

satisfied within its due date, otherwise it is subject to a penalty per period of delay per unit.

Let T indicate the discretized planning horizon, typically days, of length T . Let C represent

the set of customers and D the set of potential DCs, each with a single vehicle available for

the distribution (if the DC is selected). Each customer c ∈ C has a known demand dtc for each

period t ∈ T . Once the customer places an order, the demand could be fulfilled from any of

the selected DCs within a due date r, which is fixed for all customers in all periods. Late

orders are not lost but any demand fulfilled after the due date is subject to a penalty cost p

per period. Although the demand is known a priori, no demand can be satisfied in advance.

Let fi be the daily fee for DC i. Each selected DC is rented for one day. If the same DC is

selected for two or more consecutive days, it can hold inventory from one day to another up

to a capacity Ci, i ∈ D. When a DC is not rented in a given day, any remaining previous

inventory is lost. We assume that the fee fi covers all the handling costs of products kept in

the DCs, hence, no inventory holding cost is due.

All products are stored in DCs before being sent to the customers. Each DC possesses a

vehicle with capacity Q. The vehicle may visit several customers per day in a single trip,

64

starting and ending at the same DC. No partial shipment of an order is possible. Different

orders from the same customer in different periods may be either bundled together or shipped

separately from the same or different DCs, and/or in different periods, but one order from a

customer in a period cannot be split in different periods.

Transportation costs are accounted as follows. Each shipment from the depot to DC i costs

si and has a transportation capacity W . Vehicle routes from each DC to any of the customers

incur a cost which is based on the distance traveled. A distance matrix cij is known, i, j ∈ C∪D,where cij is the cost of traveling from location i to location j. No transshipment between DCs

is allowed, i.e., goods stored at a DC are distributed to customers only.

The objective of the F-2E-LRP is to minimize the total cost of distribution, including the DC

rental fees, the transportation costs from the depot to the DCs and from DCs to the final

customers as well as the late delivery penalty.

4.3 Mathematical formulation

In this section we propose the mathematical programming formulation for the F-2E-LRP. It

extends a commodity-flow formulation initially proposed by Garvin et al. [1957] and exten-

sively used in Koç et al. [2016b], Lahyani et al. [2015b], and Salhi et al. [2014].

The commodity flow formulation for the F-2E-LRP makes use of the following variables:

• binary variables xdtij indicate whether a vehicle from DC d traverses arc (i, j) in period

t;

• binary variables ydti take value one if and only if a vehicle from DC d visits node i in

period t;

• continuous variables ztij represent the remaining load on the vehicle when traversing arc

(i, j) in period t, i.e., after visiting node i and before visiting node j;

• continuous variables qtid indicate the quantity delivered to customer i from DC d in

period t;

• continuous variables Sti represent the amount of unserved orders for customer i in period

t;

• binary variable wtd take value one if DC d is rented in period t;

• continuous variables Itd represent the amount of inventory in DC d in period t;

65

• continuous variable gtd represent the quantity shipped to DC d in period t;

• binary variables αtpi indicate whether the demand of customer i in period t is satisfied

in period p. These will be used to ensure that the delivery of a demand will not be split

over several periods.

The F-2E-LRP is formulated as follows:

minimize∑t∈T

∑i∈D

fiwti +∑i∈D

sigti +

∑d∈D

∑i∈D∪C

∑j∈D∪C

cijxdtij

+T+1∑t=1

∑i∈C

pSti (4.1)

subject to

∑d∈D

ydti ≤ 1 i ∈ C, t ∈ T (4.2)

ydti ≤ ydtd i ∈ C, d ∈ D, t ∈ T (4.3)∑j∈D∪C

xdtij +∑

j∈D∪Cxdtji = 2ydti i ∈ C, d ∈ D, t ∈ T (4.4)

∑j∈D∪C

xdtij =∑

j∈D∪Cxdtji i ∈ C, i 6= j, d ∈ D, t ∈ T (4.5)

xdtij = 0 i ∈ C, j ∈ D, d ∈ D, j 6= d, t ∈ T (4.6)

xdtij = 0 i ∈ D, j ∈ C, d ∈ D, i 6= d, t ∈ T (4.7)∑i∈C∪D

ztij −∑

i∈C∪Dztji =

∑d∈D

qtjd j ∈ C, t ∈ T (4.8)

∑i∈D

∑j∈C

ztij =∑j∈C

∑d∈D

qtjd t ∈ T (4.9)

ztij ≤∑d∈D

Qxdtij i, j ∈ C ∪ D, t ∈ T (4.10)

qtid ≤ Qytid i ∈ C, d ∈ D, t ∈ T (4.11)∑d∈D

∑t′≤t

qt′id ≤

∑t′≤t

dt′i i ∈ C, t ∈ T (4.12)

St+1i ≥

∑t′≤t

dt′i −

∑d∈D

∑t′≤t+r

qt′id i ∈ C, t ∈ T (4.13)

∑i∈C

S0i = 0 (4.14)

66

T+1∑p≥t

αtpi = 1 i ∈ C, t ∈ T (4.15)

∑d∈D

qpid =∑

t∈T ,t≤pαtpi d

ti i ∈ C, p ∈ T (4.16)

ydti ≤ wtd i ∈ C ∪ D, d ∈ D, t ∈ T (4.17)

Itd ≤ Cdwtd d ∈ D, t ∈ T (4.18)

Itd = It−1d + gtd −∑i∈C

qtid d ∈ D, t ∈ T \ {0} (4.19)

I1d = I0d + g0d −∑i∈C

q0id d ∈ D (4.20)

gtd ≤Wwtd d ∈ D, t ∈ T (4.21)

ztij = 0 i, j ∈ D, t ∈ T (4.22)

xdtij = 0 i, j, d ∈ D, t ∈ T (4.23)

wtd, y

dti , α

tpi , x

dtij ∈ {0, 1} (4.24)

Sti , z

tij , I

td, g

td, q

tid ∈ Z∗. (4.25)

The objective function (4.1) minimizes the total cost composed of the fixed renting costs of

the DCs, transportation costs to the DCs, distribution costs to the customers, and the late

delivery penalties. Constraints (4.2) impose that a customer is visited at most once per period,

and constraints (4.3) ensure that customers are visited only from the rented DCs. Constraints

(4.4) and (4.5) are degree constraints. Constraints (4.6) and (4.7) forbid a vehicle to start a

route from a DC and finish at another. Constraints (4.8) ensure the connectivity of a route,

while constraints (4.9) ensure that the quantity loaded on vehicles from all DCs is delivered

to customers in the same period. Constraints (4.10) impose a bound on the z variables and

ensure that vehicle capacities are respected. Constraints (4.11) link the delivery quantities

with the DC used for delivery to that customer. Constraints (4.12) impose that no demand

can be satisfied in advance. Constraints (4.13) and (4.14) determine the amount of stockout.

Constraints (4.15) and (4.16) ensure that each demand of each customer is delivered exactly

once. Constraints (4.17) allow routes to start only from rented DCs, while constraints (4.18)

impose capacity constraints on the selected DCs. Constraints (4.19) set the inventory level

at each DC and (4.20) indicate that the initial inventory is equal to zero. Constraints (4.21)

guarantee that only rented DCs receive deliveries from the depot and the delivery respects the

transportation capacity. Constraints (4.22) and (4.23) forbid vehicles to travel between DCs.

Constraints (4.24)–(4.25) define the nature and bounds of the variables.

We also propose the following valid inequalities to strengthen formulation (4.1)–(4.25):

67

ztij = 0 i ∈ C, j ∈ D, t ∈ T (4.26)

ztii = 0 i ∈ C ∪ D, t ∈ T (4.27)

xdtii = 0 i ∈ C ∪ D, d ∈ D, t ∈ T (4.28)

xdtid ≤ ydti i ∈ C, d ∈ D, t ∈ T (4.29)

xdtdi ≤ ydti i ∈ C, d ∈ D, t ∈ T (4.30)

xdtij + ydti +∑

h∈D,h6=d

yhtj ≤ 2 i, j ∈ C, i 6= j, d ∈ D, t ∈ T (4.31)

xdtij + xdtji ≤ 1 i, j ∈ C, d ∈ D, t ∈ T (4.32)∑i∈C

∑t′∈T ,t′≤t

∑d∈D

qt′id ≤

∑d∈D

∑t′∈T ,t′≤t

Qwt′d t ∈ T (4.33)

∑i∈C

T+1∑t′=t

St′i ≥

∑i∈C

T∑t′=t

dt′i −

∑d∈D

T∑t′=t

Qwt′d , t ∈ T (4.34)

∑i∈C

t2∑t′=t1

qt′id ≤

∑d∈D

t2∑t′=t1

Qwt′d , t1, t2 ∈ T , t1 ≥ t2 (4.35)

2ydtd ≤∑

j∈D∪Cxdtdj +

∑j∈D∪C

xdtjd d ∈ D, t ∈ T . (4.36)

Constraints (4.26) impose that the vehicles return empty to the DCs, breaking symmetries

in the solutions that differ only in the quantity loaded. Constraints (4.27) and (4.28) forbid

links between a node and itself. Constraints (4.29) and (4.30) strengthen the link between

routing and visiting variables. Inequalities (4.31) exclude infeasible vehicle routes that visit

customers assigned to two different DCs, and (4.32) are two-cycle elimination constraints.

Inequalities (4.33) state that total deliveries to all customers from all DCs up to period t′

should not exceed the total capacities of all vehicles used during the t′ periods. Constraints

(4.34) establish that the unserved order has to be at least equal to the exceeding demand with

respect to the capacity of the vehicles used, while (4.35) state that the quantity delivered from

a DC is bounded by the vehicle capacity multiplied by the number of days in which the DC

is used. Finally, (4.36) impose that at most one route can start and end at a DC in each day,

in case the DC is rented.

68

4.4 Computational experiments

The formulation presented in Section 4.3, together with its valid inequalities, has been solved

through CPLEX 12.7.0 and IBM Concert Technology. No separation of constraints or valid

inequalities is needed as they are all in polynomial number. All computations are conducted

on Intel Core i7 processor running at 3.4 GHz with 64 GB of RAM installed, with the Ubuntu

Linux operating system. The maximum execution time is 10,800 seconds.

The goal of our experiments is to assess the value of two types of flexibility: the one arising

from the possibility of modifying the design of the supply chain network, and the one gained by

relaxing due dates on customers requests. In order to highlight the values of these two types

of flexibility we have conducted experiments on randomly generated instances. In Section

4.4.1 we explain how the instances are generated. In Section 4.4.2 we assess the value of the

network design flexibility, and in Section 4.4.3 we show the value of the flexibility obtained

by relaxing the due date. Finally, in Section 4.4.4 we combine and analyze the effects of both

type of flexibility.

4.4.1 Instance generation

We randomly generated instances for the F-2E-LRP according to the parameter values speci-

fied in Table 4.1. The instance generation is done as follows. For each combination of number

of customers and number of days (10 combinations), we first generate instances with only one

DC. Then, instances with two and three DCs are created by using instances with one DC and

adding DC locations at random. This way, the advantage of having multiple DCs, if any, is

completely imputable to their availability and not to different customer and DC locations. In

addition, we considere three different values for the capacity of the vehicle performing deliv-

eries from DCs to customers: tight, normal or loose. The capacity value of the truck shipping

the goods from the depot to the DCs (W ) is set to a sufficiently large value so that flexibility

in network design and/or due dates is fully exploited. Penalty cost p is also set to a high

value in order to force deliveries within due dates if possible. Finally, concerning due dates,

we considere three cases: no due date (r = 0, i.e., the customer request has to be satisfied

when it is released), next day delivery (r = 1) or delivery within two days (r = 2). For

each combination of the above mentioned parameters we generate five instances by randomly

choosing the values of Ci, dti, fi, si, Xi and Yi, as specified in Table 4.1, for a total of 1,350

instances.

69

Table 4.1: Input parameter values

Name Parameter ValuesPeriods T {3, 6}DCs D {1,2,3}

Customers C {5,10,15,20,25}Vehicles K One per facilityDue dates r {0,1,2}Demands dti [0, 5]

DC rental fees fi [100,150]Shipping costs (plant-DC) si [1,5]

X coordinates Xi [0,100]Y coordinates Yi [0,100]

Shipping costs (DC-customers) cij

⌊√(Xi −Xj)2 + (Yi − Yj)2 + 0.5

⌋Penalty cost p 1000

Inventory capacities Ci [2,3]×Dmax

Full truckload capacity W Dmax

Tight vehicle capacity QT Dmin

Normal vehicle capacity QN

⌈Dmin +Dmax

2

⌉Loose vehicle capacity QL Dmax

where Dmax equals the total demand of the peak day (maxt∑i∈C

dti) and

Dmin equals the total demand of the day with the lowest demand (mint∑i∈C

dti )

4.4.2 Flexibility from changing the network design

In order to assess the value gained from the flexibility in supply chain network design, we solve

each instance under two different scenarios, allowing the model to choose among the available

DCs. In the first scenario, called fixed network design, we impose the DCs selected in the first

period to remain unchanged throughout the planning horizon, whereas in the second scenario,

called flexible network design, the model re-evaluates the decision on which of the DCs should

be rented in each period. Thus, in the first scenario we have wtd = w1

d for each DC, i.e., if we

decide to rent (not rent) a DC, it remains rented (not rented) for the entire planning horizon,

while in the second scenario wtd remains flexible per period as defined in Section 4.3. As the

goal in this section is to compare the two network designs, we set r = 0 in order to exclude

any effect of due dates.

Table 4.2 presents, for the first scenario, the average costs and optimality gaps over all five

instances with the capacity specified in the first column and the number of customers and

days specified in the second and third columns, respectively. The table compares the results

for different number of DCs. For each number of DCs we report the cost of the best solution

found and the optimality gap. In addition, for the case where the number of DCs is equal to

2 and 3, we report, in column ‘guaranteed savings’, the gap between the cost of the solution

70

with the corresponding number of DCs and the lower bound of the solution with one DC.

This way, we provide an upper bound on the savings that are achieved by introducing new

DCs. Note that, when the instance with one DC is solved to optimality, this upper bound

corresponds to the exact value of the savings achieved. The guaranteed savings are calculated

as 100 × Cost− LBCost

, where Cost is the value reported in column ‘Cost’ and LB is the lower

bound of the solution of the same instance with one DC.

Table 4.2 shows that the cost savings become more relevant when the vehicle capacity tightens.

While the global average cost for all three vehicle capacity scenarios has a decreasing trend as

the number of available DCs increases, the biggest influence of adding extra DCs is observed

under the tight capacity scenario. Moreover, the savings are more substantial when moving

from 1 to 2 DCs than moving from 2 to 3 DCs. In fact, the average savings achieved with 2

DCs are 60.81% while in the case of 3 DCs they are 62.46%. Concerning solution time, we

see that, depending on the size of the instance and the capacity scenario, the average CPU

time over all five instances varies, but in general when more DCs are available, the problem

becomes more difficult to solve. Small size instances such as the ones with three periods, five

customers and three DCs are solved in less than a second for all capacity scenarios. However,

even after three hours of computation, instances with six periods, 25 customers, and two or

three DCs under tight capacity could only be solved with 1.44% optimality gap. Moreover,

when the vehicle capacity decreases, the problem takes on average 60% more time to be solved.

Table 4.3 compares the best solutions obtained by the fixed network design with the ones

obtained by the flexible network. The guaranteed savings are calculated as in Table 4.2 with

LB being equal to the lower bound of the solution related to the fixed network design. The last

column of the table reports the average reduction in the number of rented DCs. In particular,

for each solution, the number of rented DCs is calculated as the sum of the number of DCs

rented in each day of the planning horizon. Then, the reduction in the number of rented DCs

is calculated as 100 × Number of rented DCFixed −Number of rented DCFlexible

Number of rented DCFixed. Note that,

when calculating the reduction in the number of DCs, we considered the best solution found

in both the fixed and the flexible network design case, which corresponds to the optimal

solution when the optimality gap is equal to 0. As the table indicates, the flexible network

design always yields lower costs. However, this difference is more evident for the cases with

normal and tight vehicle capacities. In these cases, the model not only reduces the cost but

also decreases the number of DCs rented throughout the planning horizon. On average the

flexible network design reduces costs by almost 6% and, at the same time, it uses 15% fewer

DCs. The difference between the two designs is more significant under the normal vehicle

capacity scenario, for which we observe 11% savings in total cost and 28% reduction in the

number of rented DCs.

71

Table 4.2: DC availability in the fixed network design with r = 0

Instances # of DC = 1 # of DC = 2 # of DC = 3Periods Customers Cost Gap Cost Gap Guaranteed savings Cost Gap Guaranteed savings

(%) (%) (%) (%) (%)

Loose

3 5 1,005.60 0.00 1,001.20 0.00 0.44 986.40 0.00 1.913 10 1,469.60 0.00 1,436.20 0.00 2.27 1,390.60 0.00 5.383 15 1,591.40 0.00 1,500.40 0.00 5.72 1,469.20 0.00 7.683 20 1,613.60 0.00 1,561.00 0.00 3.26 1,535.60 0.00 4.833 25 2,131.40 0.00 1,762.40 0.18 17.31 1,757.80 0.09 17.536 5 2,221.40 0.00 2,108.00 0.00 5.10 2,085.40 0.00 6.126 10 3,023.00 0.00 2,953.00 0.00 2.32 2,813.40 0.00 6.936 15 3,232.20 0.00 3,150.60 0.00 2.52 3,079.60 0.00 4.726 20 3,468.40 0.00 3,385.60 0.47 2.39 3,372.60 0.00 2.766 25 4,200.60 0.00 3,978.00 0.16 5.30 3,788.60 0.23 9.81

Average 2,395.72 0.00 2,283.64 0.02 4.66 2,227.92 0.03 6.77

Normal

3 5 4,965.00 0.00 1,341.00 0.00 72.99 1,334.60 0.00 73.123 10 5,621.40 0.00 1,819.20 0.00 67.64 1,744.40 0.00 68.973 15 12,690.96 0.01 1,896.00 0.00 85.06 1,863.80 0.00 85.313 20 13,697.00 0.01 1,950.40 0.00 85.76 1,903.20 0.00 86.103 25 18,598.20 0.01 2,205.40 0.18 88.14 2,174.80 0.82 88.316 5 23,216.60 0.01 2,938.80 0.00 87.34 2,807.20 0.00 87.916 10 49,408.40 0.01 3,799.80 0.00 92.31 3,603.60 0.00 92.716 15 27,217.60 0.01 3,917.20 0.00 85.16 3,838.00 0.00 85.906 20 46,887.40 0.01 4,220.80 0.47 91.00 4,172.20 0.55 91.106 25 46,174.00 0.01 4,804.80 0.52 89.59 4,565.00 0.82 90.11

Average 24,847.73 0.01 2,889.34 0.12 84.54 2,800.68 0.22 84.95

Tight

3 5 17,669.00 0.00 2,757.40 0.00 84.39 1,456.40 0.00 91.763 10 22,485.20 0.00 2,242.40 0.00 90.03 1,841.00 0.00 91.813 15 35,300.60 0.01 1,938.20 0.00 94.51 1,894.40 0.00 94.633 20 43,081.60 0.01 1,981.00 0.39 95.40 1,941.20 0.29 95.493 25 71,617.60 0.01 2,269.80 0.21 96.83 2,215.80 1.04 96.916 5 95,632.60 0.01 14,302.40 0.00 85.04 4,953.60 0.00 94.826 10 184,902.80 0.01 11,712.00 0.03 93.67 3,889.80 0.00 97.906 15 191,139.40 0.01 5,531.20 0.55 97.11 4,159.40 1.08 97.826 20 205,257.60 0.01 4,333.00 1.15 97.89 4,269.20 1.32 97.926 25 197,439.60 0.02 4,922.80 1.44 97.51 4,645.20 1.41 97.65

Average 106,452.60 0.01 5,199.02 0.38 93.24 3,126.60 0.51 95.67Global average 44,565.35 0.01 3,457.33 0.17 60.81 2,718.40 0.25 62.46

72

Table 4.3: Fixed vs. flexible network designs with r = 0

Instances Fixed Flexible

Periods Customers Cost Gap Cost Gap Guaranteed saving Reduction in(%) (%) (%) # DCs

Loose

3 5 986.40 0.00 984.20 0.00 0.22 0.003 10 1,390.60 0.00 1,389.00 0.00 0.12 0.003 15 1,469.20 0.00 1,469.20 0.00 0.00 0.003 20 1,535.60 0.00 1,533.40 0.00 0.14 0.003 25 1,757.80 0.09 1,756.60 0.12 0.00 0.006 5 2,085.40 0.00 2,071.80 0.00 0.65 0.006 10 2,813.40 0.00 2,809.20 0.00 0.15 0.006 15 3,079.60 0.00 3,079.20 0.00 0.01 0.006 20 3,372.60 0.00 3,372.60 0.01 0.00 0.006 25 3,788.60 0.23 3,785.00 0.28 0.00 0.00

Average 2,227.92 0.03 2,225.02 0.04 0.13 0.00

Normal

3 5 1,334.60 0.00 1,142.00 0.00 14.43 30.003 10 1,744.40 0.00 1,544.00 0.00 11.49 30.003 15 1,863.80 0.00 1,664.20 0.00 10.71 26.673 20 1,903.20 0.00 1,725.60 0.00 9.33 26.673 25 2,174.80 0.82 1,946.60 0.93 9.75 30.006 5 2,807.20 0.00 2,405.60 0.00 14.31 30.006 10 3,603.60 0.00 3,277.20 0.00 9.06 23.336 15 3,838.00 0.00 3,354.00 0.00 12.61 35.006 20 4,172.20 0.55 3,828.20 0.53 7.74 23.336 25 4,565.00 0.82 4,154.60 0.29 8.23 26.67

Average 2,800.68 0.22 2,504.20 0.18 10.77 28.17

Tight

3 5 1,456.40 0.00 1,306.60 0.00 10.29 21.213 10 1,841.00 0.00 1,643.60 0.00 10.72 27.273 15 1,894.40 0.00 1,767.20 0.00 6.71 16.673 20 1,941.20 0.29 1,802.60 0.27 6.87 20.003 25 2,215.80 1.04 2,084.60 0.87 4.93 16.676 5 4,953.60 0.00 4,592.60 0.00 7.29 19.446 10 3,889.80 0.00 3,648.80 0.09 6.20 15.156 15 4,159.40 1.08 3,956.20 1.38 3.85 13.646 20 4,269.20 1.32 4,068.60 1.52 3.43 13.336 25 4,645.20 1.41 4,527.00 1.84 1.16 8.33

Average 3,126.60 0.51 2,939.78 0.60 6.14 17.17Global average 2,718.40 0.25 2,556.33 0.27 5.68 15.11

4.4.3 Flexibility from due dates

In this section we evaluate the value of the flexibility obtained by relaxing due dates. We

compare the costs and difficulty in solving the problems when no due dates are considered,

i.e., r = 0, and when due dates are r = 1 and r = 2. We separate our analysis for fixed

and flexible network designs in order to interpret the benefits coming from due dates only.

Results are presented in Table 4.4 for the fixed network design and in Table 4.5 for the flexible

network design. In both cases we consider three DCs. In both tables, the guaranteed savings

are calculated by comparing the best solution value for the cases with r = 1 and r = 2 with

the lower bound obtained with r = 0.

In general, delivery with larger due dates reduces the costs but makes the problem more

difficult to solve. For both network designs, larger savings are achieved when changing from

the case with no due dates (r = 0) to next day delivery (r = 1), rather than from the next day

delivery to a two-day delivery (r = 2). However, comparing Tables 4.4 and 4.5, this difference

is more significant in the flexible network design. In other words, when the location of DCs

is fixed, offering larger due dates does not have as significant cost saving effect as it has in

flexible networks. Overall, while serving the demand the next day rather than on the same

73

day reduces the cost by 16% on fixed design, on a flexible network design the savings go up

to 22%. Changing from next day delivery to delivery within two days leads to 4% additional

savings in fixed networks and 10% in flexible ones.

Table 4.4: Value of flexibility from due dates for 3 DCs and fixed network design

Instances r = 0 r = 1 r = 2

Periods Customers Cost Gap Cost Gap % guaranteed savings Cost Gap % guaranteed savings(%) (%) over r = 0 (%) over r = 0

Loose

3 5 986.40 0.00 904.20 0.00 8.33 897.20 0.00 9.043 10 1,390.60 0.00 1,277.60 0.00 8.13 1,248.00 0.00 10.253 15 1,469.20 0.00 1,344.40 0.00 8.49 1,320.20 0.00 10.143 20 1,535.60 0.00 1,378.20 0.34 10.25 1,337.40 0.82 12.913 25 1,757.80 0.09 1,541.00 3.04 12.26 1,523.00 2.87 13.286 5 2,085.40 0.00 1,714.40 0.00 17.79 1,609.40 0.00 22.836 10 2,813.40 0.00 2,244.40 0.00 20.22 2,065.20 0.00 26.596 15 3,079.60 0.00 2,408.40 0.59 21.80 2,200.60 0.52 28.546 20 3,372.60 0.00 2,820.20 2.91 16.38 2,678.80 2.67 20.576 25 3,788.60 0.23 3,203.40 6.25 15.25 3,169.40 12.21 16.15

Average 2,227.92 0.03 1,883.62 1.31 13.89 1,804.92 1.91 17.03

Normal

3 5 1,334.60 0.00 1,096.60 0.00 17.83 1,071.60 0.00 19.713 10 1,744.40 0.00 1,428.40 0.00 18.12 1,364.80 0.00 21.763 15 1,863.80 0.00 1,553.00 0.00 16.68 1,439.60 0.00 22.763 20 1,903.20 0.00 1,539.20 1.64 19.13 1,452.80 2.13 23.673 25 2,174.80 0.82 1,726.60 2.43 19.95 1,681.40 4.02 22.056 5 2,807.20 0.00 2,423.80 0.00 13.66 2,273.60 0.00 19.016 10 3,603.60 0.00 2,818.40 0.00 21.79 2,649.40 0.24 26.486 15 3,838.00 0.00 2,854.40 1.26 25.63 2,777.20 2.65 27.646 20 4,172.20 0.55 3,453.40 4.96 16.77 3,255.40 6.74 21.546 25 4,565.00 0.82 3,917.40 11.17 13.47 3,556.00 9.79 21.46

Average 2,800.68 0.22 2,281.12 2.15 18.30 2,152.18 2.56 22.61

Tight

3 5 1,456.40 0.00 1,319.20 0.00 9.42 1,267.80 0.00 12.953 10 1,841.00 0.00 1,564.80 0.00 15.00 1,499.80 0.00 18.533 15 1,894.40 0.00 1,630.60 0.41 13.93 1,507.80 0.35 20.413 20 1,941.20 0.29 1,672.80 2.59 13.58 1,543.80 3.14 20.243 25 2,215.80 1.04 1,958.40 7.58 10.69 1,891.00 10.12 13.766 5 4,953.60 0.00 2,795.60 0.00 43.56 2,685.60 0.00 45.786 10 3,889.80 0.00 3,227.80 0.16 17.02 3,046.80 0.56 21.676 15 4,159.40 1.08 3,495.80 4.50 15.04 3,282.40 4.02 20.226 20 4,269.20 1.32 3,645.80 7.53 13.46 3,382.40 6.73 19.716 25 4,645.20 1.41 3,991.00 10.18 12.86 3,659.60 8.55 20.09

Average 3,126.60 0.51 2,530.18 3.29 16.46 2,376.70 3.35 21.34Global average 2,718.40 0.25 2,231.64 2.25 16.22 2,111.27 2.60 20.33

74

Table 4.5: Value of flexibility from due dates for 3 DCs and flexible network design

Instances r = 0 r = 1 r = 2

Periods Customers Cost Gap Cost Gap % guaranteed savings Cost Gap % guaranteed savings(%) (%) over r = 0 (%) over r = 0

Loose

3 5 984.20 0.00 835.40 0.00 15.12 733.80 0.00 25.443 10 1,389.00 0.00 1,162.20 0.00 16.33 991.80 0.00 28.603 15 1,469.20 0.00 1,227.40 0.00 16.46 1,032.60 0.00 29.723 20 1,533.40 0.00 1,254.60 0.68 18.18 1,040.60 0.85 32.143 25 1,756.60 0.12 1,454.40 1.65 17.10 1,280.80 4.81 27.006 5 2,071.80 0.00 1,554.40 0.00 24.97 1,336.40 0.00 35.506 10 2,809.20 0.00 2,090.00 0.00 25.60 1,807.60 0.73 35.656 15 3,079.20 0.00 2,255.00 1.67 26.77 1,925.20 2.71 37.486 20 3,372.60 0.01 2,620.00 2.71 22.31 2,359.40 5.33 30.046 25 3,785.00 0.28 2,872.40 6.54 23.90 2,601.40 9.85 31.08

Average 2,225.02 0.04 1,732.58 1.33 20.67 1,510.96 2.43 31.26

Normal

3 5 1,142.00 0.00 945.60 0.00 17.20 874.40 0.00 23.433 10 1,544.00 0.00 1,234.40 0.00 20.05 1,056.00 0.00 31.613 15 1,664.20 0.00 1,312.00 0.00 21.16 1,140.00 0.00 31.503 20 1,725.60 0.00 1,327.80 1.02 23.05 1,117.20 1.77 35.263 25 1,946.60 0.93 1,537.60 2.82 20.27 1,370.60 7.28 28.936 5 2,405.60 0.00 1,821.40 0.00 24.29 1,654.20 0.00 31.246 10 3,277.20 0.00 2,329.00 1.48 28.93 2,050.60 1.71 37.436 15 3,354.00 0.00 2,474.80 3.80 26.21 2,216.60 7.63 33.916 20 3,828.20 0.53 2,897.80 7.69 23.90 2,663.20 10.97 30.066 25 4,154.60 0.29 3,145.60 9.31 24.06 3,144.20 20.27 24.10

Average 2,504.20 0.18 1,902.60 2.61 22.91 1,728.70 4.96 30.75

Tight

3 5 1,306.60 0.00 1,066.60 0.00 18.37 1,006.20 0.00 22.993 10 1,643.60 0.00 1,337.80 0.00 18.61 1,210.60 0.00 26.343 15 1,767.20 0.00 1,394.40 0.95 21.10 1,233.40 1.13 30.213 20 1,802.60 0.27 1,434.60 3.08 20.20 1,249.80 3.44 30.483 25 2,084.60 0.87 1,782.60 11.03 13.74 1,558.20 9.82 24.606 5 4,592.60 0.00 2,475.80 0.00 46.09 2,276.80 0.00 50.426 10 3,648.80 0.09 2,835.40 3.98 22.22 2,565.20 3.11 29.636 15 3,956.20 1.38 3,026.80 9.14 22.42 2,756.00 8.68 29.366 20 4,068.60 1.52 3,204.60 10.44 20.02 2,879.40 10.01 28.146 25 4,527.00 1.84 3,503.00 12.99 21.17 3,414.40 19.86 23.17

Average 2,939.78 0.60 2,206.16 5.16 22.39 2,015.00 5.60 29.53Global average 2,556.33 0.27 1,947.11 3.03 21.99 1,751.55 4.33 30.51

4.4.4 Fixed versus flexible network design with due dates

In this section we analyze the combined effect of both types of flexibility. Table 4.6 provides

a general overview on the flexibility gained from the network design and the due dates. We

compare the solutions obtained from both fixed and flexible network designs by assuming

different due dates. As shown in the table, the cost for both designs decreases as the due date

increases. The cost of the flexible design is always lower than the one of the fixed design and

the difference between the costs increases when the value of r increases. This is consistent

with the results shown in the previous section. Moreover, the gap between the costs of fixed

vs. flexible network are more relevant for the cases of normal and loose vehicle capacity.

Concerning optimality gaps, we see that they increase with the value of r and with a tight

vehicle capacity. In all cases, the flexible case is always more difficult to solve and presents

larger optimality gaps.

In order to have a better estimation of the savings achieved by combining the two types of

75

flexibility, in Table 4.7 we compare the most inflexible case, i.e., with r = 0 and a fixed network

design, against the most flexible one which has two-day delivery due date and a flexible network

design. Although the most flexible problem is harder to solve to optimality, we could observe

cost savings of 35% with 27% reduction in the number of used DCs, on average. While in

cases with tight or normal capacity this reduction in the total cost is related to a remarkable

fewer number of DCs rented (32% and 43% less, respectively), for the cases with loose vehicle

capacity, an average of 31% less cost is gained with a reduction in the number of rented DCs

of only 5%.

Table 4.6: Cost of fixed and flexible designs for 3 DCs with different due dates

Instances r = 0 r = 1 r = 2

Periods Customers Fixed Flexible Fixed Flexible Fixed FlexibleCost Gap (%) Cost Gap (%) Cost Gap (%) Cost Gap (%) Cost Gap(%) Cost Gap (%)

Loose

3 5 986.40 0.00 984.20 0.00 904.20 0.00 835.40 0.00 897.20 0.00 733.80 0.003 10 1,390.60 0.00 1,389.00 0.00 1,277.60 0.00 1,162.20 0.00 1,248.00 0.00 991.80 0.003 15 1,469.20 0.00 1,469.20 0.00 1,344.40 0.00 1,227.40 0.00 1,320.20 0.00 1,032.60 0.003 20 1,535.60 0.00 1,533.40 0.00 1,378.20 0.34 1,254.60 0.68 1,337.40 0.82 1,040.60 0.853 25 1,757.80 0.09 1,756.60 0.12 1,541.00 3.04 1,454.40 1.65 1,523.00 2.87 1,280.80 4.816 5 2,085.40 0.00 2,071.80 0.00 1,714.40 0.00 1,554.40 0.00 1,609.40 0.00 1,336.40 0.006 10 2,813.40 0.00 2,809.20 0.00 2,244.40 0.00 2,090.00 0.00 2,065.20 0.00 1,807.60 0.736 15 3,079.60 0.00 3,079.20 0.00 2,408.40 0.59 2,255.00 1.67 2,200.60 0.52 1,925.20 2.716 20 3,372.60 0.00 3,372.60 0.01 2,820.20 2.91 2,620.00 2.71 2,678.80 2.67 2,359.40 5.336 25 3,788.60 0.23 3,785.00 0.28 3,203.40 6.25 2,872.40 6.54 3,169.40 12.21 2,601.40 9.85

Average 2,227.92 0.03 2,225.02 0.04 1,883.62 1.31 1,732.58 1.33 1,804.92 1.91 1,510.96 2.43

Normal

3 5 1,334.60 0.00 1,142.00 0.00 1,096.60 0.00 945.60 0.00 1,071.60 0.00 874.40 0.003 10 1,744.40 0.00 1,544.00 0.00 1,428.40 0.00 1234.40 0.00 1,364.80 0.00 1,056.00 0.003 15 1,863.80 0.00 1,664.20 0.00 1,553.00 0.00 1,312.00 0.00 1,439.60 0.00 1,140.00 0.003 20 1,903.20 0.00 1,725.60 0.00 1,539.20 1.64 1,327.80 1.02 1,452.80 2.13 1,117.20 1.773 25 2,174.80 0.82 1,946.60 0.93 1,726.60 2.43 1,537.60 2.82 1,681.40 4.02 1,370.60 7.286 5 2,807.20 0.00 2,405.60 0.00 2,423.80 0.00 1,821.40 0.00 2,273.60 0.00 1,654.20 0.006 10 3,603.60 0.00 3,277.20 0.00 2,818.40 0.00 2,329.00 1.48 2,649.40 0.24 2,050.60 1.716 15 3,838.00 0.00 3,354.00 0.00 2,854.40 1.26 2,474.80 3.80 2,777.20 2.65 2,216.60 7.636 20 4,172.20 0.55 3,828.20 0.53 3,453.40 4.96 2,897.80 7.69 3,255.40 6.74 2,663.20 10.976 25 4,565.00 0.82 4,154.60 0.29 3,917.40 11.17 3,145.60 9.31 3,556.00 9.79 3,144.20 20.27

Average 2,800.68 0.22 2,504.20 0.18 2,281.12 2.15 1,902.60 2.61 2,152.18 2.56 1,728.70 4.96

Tight

3 5 1,456.40 0.00 1,306.60 0.00 1,319.20 0.00 1,066.60 0.00 1,267.80 0.00 1,006.20 0.003 10 1,841.00 0.00 1,643.60 0.00 1,564.80 0.00 1,337.80 0.00 1,499.80 0.00 1,210.60 0.003 15 1,894.40 0.00 1,767.20 0.00 1,630.60 0.41 1,394.40 0.95 1,507.80 0.35 1,233.40 1.133 20 1,941.20 0.29 1,802.60 0.27 1,672.80 2.59 1,434.60 3.08 1,543.80 3.14 1,249.80 3.443 25 2,215.80 1.04 2,084.60 0.87 1,958.40 7.58 1,782.60 11.03 1,891.00 10.12 1,558.20 9.826 5 4,953.60 0.00 4,592.60 0.00 2,795.60 0.00 2,475.80 0.00 2,685.60 0.00 2,276.80 0.006 10 3,889.80 0.00 3,648.80 0.09 3,227.80 0.16 2,835.40 3.98 3,046.80 0.56 2,565.20 3.116 15 4,159.40 1.08 3,956.20 1.38 3,495.80 4.50 3,026.80 9.14 3,282.40 4.02 2,756.00 8.686 20 4,269.20 1.32 4,068.60 1.52 3,645.80 7.53 3,204.60 10.44 3,382.40 6.73 2,879.40 10.016 25 4,645.20 1.41 4,527.00 1.84 3,991.00 10.18 3,503.00 12.99 3,659.60 8.55 3,414.40 19.86

Average 3,126.60 0.51 2,939.78 0.60 2,530.18 3.29 2,206.16 5.16 2,376.70 3.35 2,015.00 5.60Global average 2,718.40 0.25 2,553.77 0.27 2,231.64 2.25 1,947.11 3.03 2,111.27 2.60 1,751.55 4.33

76

Table 4.7: Comparison between the most inflexible and the most flexible scenarios

Instances Most inflexible Most flexible

Periods Customers Cost Gap Cost Gap Guaranteed savings % reduction(%) (%) (%) in # DCs

Loose

3 5 986.40 0.00 733.80 0.00 25.61 0.003 10 1,390.60 0.00 991.80 0.00 28.68 0.003 15 1,469.20 0.00 1,032.60 0.00 29.72 0.003 20 1,535.60 0.00 1,040.60 0.85 32.23 0.003 25 1,757.80 0.09 1,280.80 4.81 27.07 0.006 5 2,085.40 0.00 1,336.40 0.00 35.92 20.006 10 2,813.40 0.00 1,807.60 0.73 35.75 10.006 15 3,079.60 0.00 1,925.20 2.71 37.49 16.676 20 3,372.60 0.00 2,359.40 5.33 30.04 3.336 25 3,788.60 0.23 2,601.40 9.85 31.18 3.33

Average 2,227.92 0.03 1,510.96 2.43 31.37 5.33

Normal

3 5 1334.60 0.00 874.40 0.00 34.48 40.003 10 1,744.40 0.00 1,056.00 0.00 39.46 43.333 15 1,863.80 0.00 1,140.00 0.00 38.83 40.003 20 1,903.20 0.00 1,117.20 1.77 41.30 43.333 25 2,174.80 0.82 1,370.60 7.28 36.46 46.676 5 2,807.20 0.00 1,654.20 0.00 41.07 45.006 10 3,603.60 0.00 2,050.60 1.71 43.10 45.006 15 3,838.00 0.00 2,216.60 7.63 42.25 48.336 20 4,172.20 0.55 2,663.20 10.97 35.81 41.676 25 4,565.00 0.82 3,144.20 20.27 30.55 38.33

Average 2,800.68 0.22 1,728.70 4.96 38.33 43.17

Tight

3 5 1,456.4 0.00 1,006.2 0.00 30.91 33.333 10 1,841.0 0. 1,210.6 0.00 34.24 36.363 15 1,894.4 0.00 1,233.4 1.13 34.89 33.333 20 1,941.2 0.29 1,249.8 3.44 35.43 33.333 25 2,215.8 1.04 1,558.2 9.82 28.94 30.006 5 4,953.6 0.00 2,276.8 0.00 54.04 30.566 10 3,889.8 0.00 2,565.2 3.11 34.05 27.276 15 4,159.4 1.08 2,756.0 8.68 33.02 34.856 20 4,269.2 1.32 2,879.4 10.01 31.65 33.336 25 4,645.2 1.41 3,414.4 19.86 25.45 30.00

Average 3,126.60 0.51 2,015.00 5.60 34.26 32.24Global average 2,718.40 0.25 1,751.55 4.33 34.65 26.91

Finally, Table 4.8 provides the computation times. We can see that they increase for both

network designs as the due date increases and, in general, the flexible network design takes

slightly longer.

77

Table 4.8: Computation time of fixed and flexible network designs

Instances Fixed FlexiblePeriods Customers r = 0 r = 1 r = 2 r = 0 r = 1 r = 2

Loose

3 5 0 1 1 0 1 13 10 14 82 37 6 41 663 15 8 1,726 1,372 7 285 4013 20 53 4,176 6,998 26 3,753 5,3323 25 2,436 10,566 1,0051 2,717 9,122 1,08016 5 3 11 9 2 25 296 10 19 331 1,342 16 2,992 5,0756 15 153 7,177 7,911 54 10,800 9,5646 20 647 10,800 10,800 2,552 8,751 10,8006 25 6,066 10,800 10,801 10,800 10,800 10,800

Average 940 4,567 4,932 1,618 4,657 5,287

Normal

3 5 0 1 1 1 4 33 10 9 116 166 10 128 1113 15 34 2,417 3,682 27 902 5,2153 20 192 6,599 7,085 233 6,901 7,6133 25 5,679 9,036 8,767 4,938 10,485 1,08016 5 3 26 15 5 48 1066 10 361 949 3,384 74 7,530 8,5946 15 571 10,541 10,800 937 10,800 10,8006 20 6,706 10,801 10,801 7,072 10,800 10,8006 25 10,377 10,800 10,800 7,746 10,800 10,800

Average 2,393 5,129 5550 2,104 5,840 6,484

Tight

3 5 0 2 1 0 4 33 10 13 104 168 10 217 5003 15 102 5,091 5,250 159 5,222 8,1383 20 3,541 10,380 10,800 4,058 10,786 9,0343 25 8,949 10,801 10,801 7,221 10,801 10,8016 5 9 130 201 8 1,040 4,0096 10 9,559 5,637 6,659 2,445 10,800 10,8006 15 1,413 10,800 10,800 10,801 10,800 10,8006 20 10,378 10,801 10,800 10,458 10,800 10,8006 25 10,801 10,801 10,800 10,801 10,800 10,801

Average 4,476 6,455 6,628 4,596 7,127 7,569Global average 2,603 5,383 5,703 2,773 5,875 6,447

4.5 Conclusions

In this paper we have introduced the F-2E-LRP and proposed a mathematical formulation

along with different classes of valid inequalities for it. Inspired by recent works in the sharing

economy and extending several classes of the routing problems, the F-2E-LRP combines inte-

gration issues related to the decision of which facilities to rent, and flexibility issues coming

from two sources: network design and delivery due dates. The results obtained from the ex-

periments on randomly generated instances show the value of flexibility, both in terms of due

date and network design. The results highlight the cost saving advantages of both types of

flexibility.

This study opens different avenues for future research. In particular, being the F-2E-LRP a

highly complex problem which can find applications in real distribution settings, it would be

worthwhile to propose heuristic algorithms that can handle large size instances.

78

Chapter 5

Minimizing emissions in integrated

distribution problems

Chapter information A paper based on this chapter is currently submitted for publication:

M. Darvish, Archetti, C., Coelho, L.C. Minimizing emissions in integrated distribution prob-

lems. Document de travail du CIRRELT-2017-41, Tech. Rep., 2017.

The integration of operational decisions of different supply chain functions is an important

success factor in minimizing their total costs. Traditionally, supply chain optimization has

merely concentrated on costs or the economical aspects of sustainability, neglecting its envi-

ronmental and social aspects. However, with the growing concern towards green operations,

the impact of short term decisions on the reduction of carbon emissions can no longer be

overlooked. In this paper, we aim to compare the effect of operational decisions not only on

costs but also on emissions, and we reassess some well-known logistic optimization problems

under new objectives. We study two integrated systems dealing with production, inventory,

and routing decisions, in which a commodity produced at the plant is shipped to the retailers

over a finite time horizon. These two problems are known as the production-routing and the

inventory-routing problems. We define and measure several metrics under different scenarios,

namely by minimizing total costs, routing costs only, or minimizing emissions. Each solution is

evaluated under all three objective functions, and their costs and business performance indica-

tors are then compared. We provide elaborated sensitivity analyses allowing us to gain useful

managerial insights on the costs and emissions in integrated supply chains, besides important

insights on the cost of being environmentally friendly.

79

5.1 Introduction

Increasing environmental concerns have changed the definition of competitive advantage. To

be competitive, companies need to make a proper balance between economical, environmen-

tal, and social dimensions of their business. Today the market urges companies not only

to be efficient in terms of cost but also to consider the environmental and social impacts of

their operations. To be efficient, they have to make decisions that boost their margins, make

them fulfill their customers demands in a fast and flexible manner and to be sustainable; they

have to evaluate the trade-offs between efficient solutions versus sustainable ones. Recently,

the negative impacts of operational decisions on the environment have become more impor-

tant, especially in terms of greenhouse gases (GHGs) emissions. All production, inventory, and

distribution activities contribute to environmental problems, however, transportation and par-

ticularly road-based transportation is considered as one of the principal sources of GHG [Jabali

et al., 2012], mainly measured by carbon dioxide (CO2) emissions from burning fossil fuels.

Nonetheless, to reduce emissions, one cannot simply avoid using trucks, as the distribution

of many products and services relies on their operation [Coelho et al., 2016]. To date, much

research is devoted to evaluate how truck usage affects the environment, such as estimating the

amount of GHG emitted [Demir et al., 2011], or to develop new approaches to tackle the tradi-

tional vehicle routing problem (VRP), such as the green VRP [Bektaş and Laporte, 2011]. A

number of variants of these problems has recently appeared [Dabia et al., 2017, Franceschetti

et al., 2017]. They reinforce the need of better understanding how business operations can

affect the environment, and also what can be done to be environmentally friendly while at the

same time be efficient from a business perspective. Despite the significant recent literature

on the green VRP, the incorporation of the evaluation of GHG emissions into multi-echelon

supply chain models is not extensively studied [Absi et al., 2013]. This paper revisits classi-

cal inventory-routing and production routing problems (IRP and PRP) taking both economic

and environmental objectives into consideration. These problems already combine multiple

contradicting objectives, such as minimizing transportation, inventory and production costs.

In the following, we briefly describe these two classical problems.

The IRP is a multi-period problem in which a supplier must distribute a commodity from its

depot to a set of customers. Both transportation and inventory costs are considered, and all

demand must be satisfied without backlogging. The problem was initially proposed by Bell

et al. [1983] and many algorithms have been proposed for its resolution, including heuristics

methods (e.g., Bertazzi et al. [2002], Archetti et al. [2012], Coelho et al. [2012], Coelho and

Laporte [2013a]) and exact algorithms (e.g., Archetti et al. [2007], Desaulniers et al. [2015]).

A set of benchmark instances exists and has been vastly used to assess the performance of the

solution algorithms. A review of the practical aspects of the IRP is available in Andersson

80

et al. [2010] while methodological aspects are surveyed in Coelho et al. [2014].

Adding production optimization into the IRP and simultaneously determining production, in-

ventory, and distribution decisions creates the PRP [Absi et al., 2014]. Introduced by Chandra

[1993], in this problem the objective is to minimize total costs, including fixed and variable

production costs, inventory holding, and transportation costs. Several heuristics (e.g., Bard

and Nananukul [2009a], Armentano et al. [2011], Adulyasak et al. [2012], Absi et al. [2014])

and exact algorithms (e.g., Bard and Nananukul [2010], Archetti et al. [2011], Adulyasak et al.

[2014]) exist for its resolution, as well as a set of benchmark instances. A recent survey of the

PRP can be found in Adulyasak et al. [2015].

In the integrated logistics literature, Benjaafar et al. [2013] highlight the potential impact

of operational supply chain decisions on carbon emissions and the need to incorporate them

in integrated optimization models. Taking emissions into consideration can be performed in

two different ways. The first one is to add carbon capacity constraints but without changing

the goal of the optimization (e.g., Benjaafar et al. [2013], Absi et al. [2013], Helmrich et al.

[2015], Absi et al. [2016], Qiu et al. [2017]). This method is not preferable as one simply

aims to respect the carbon capacity but ignores the hidden potential reductions. The other

approach, indeed, aims at minimizing the emissions (e.g., Soysal et al. [2015], Kumar et al.

[2016], Soysal et al. [2016]). In this category, the authors consider distance, load or speed

factors when minimizing emissions or fuel consumption. Among these factors, distance has

been the traditional factor directly influencing emissions, even though the literature provides

comprehensive emission models that optimize distribution decisions using very rich and de-

tailed fuel consumption estimation models. These advanced models take into consideration

many vehicle related factors (aerodynamics, engine specifications, etc.) and environment re-

lated factors (type and quality of the pavement, gradient of the roadway, etc.). See Demir

et al. [2014] for a comprehensive list of factors affecting the fuel consumption.

A very important factor affecting the emissions is the payload [Demir et al., 2014] as each load

carried by the vehicle increases the fuel consumption and ultimately the emissions (Bektaş

and Laporte [2011], Suzuki [2011], Kopfer et al. [2014]). A straightforward and commonly

used approach to estimate vehicle emissions is presented by Kara et al. [2007]. They compare

distance minimization models with energy minimization ones. Here, energy consumption is

a function of the load of the vehicle and the distance being traveled. The rationale is that

the heavier a truck, the more fuel it consumes. Xiao et al. [2012] consider the same approach

to estimate fuel consumption and develop an optimization model for the VRP. Comparing

different factors affecting fuel consumption, they argue that all other factors being constant,

emissions mainly depend on the distance traveled and the load.

81

Our goal in this paper is to understand and measure the existing trade-offs between solutions

that are economically optimized (i.e., the traditional objective) versus fully environmental

friendly solutions, as well as other alternatives in between. We compare how much pollution a

minimum cost solution would emit, whereas how much an emission optimized solution would

cost financially. We study the trade-offs between three different objectives: reducing the total

cost, the total distance, and the emissions. We contribute to the literature by modeling and

solving emission minimization integrated distribution problems, conducting extensive com-

putational experiments, and sensitivity analysis in order to better understand the trade-offs

between the components of each model. Based on sensitivity analysis of several performance

indicators, we provide managerial insights for production and distribution systems.

The remainder of this paper is structured as follows. Section 5.2 provides a general description

of the problems being solved as well as their formal formulations. Extensive computational

experiments are presented in Section 5.3 which is followed by conclusions and managerial

insights in Section 5.4.

5.2 Problem descriptions and formulations

We now formally describe the mathematical formulation of the IRP and the PRP. The defi-

nition of the IRP is based on Coelho and Laporte [2013b] and that of the PRP on Archetti

et al. [2011]. Both problems are defined on a complete undirected graph G = (V, E), where

V = {0, ..., n} is the vertex set and E = {(i, j) : i, j ∈ V, i < j} is the edge set. Vertex

0 represents the supplier and the remaining vertices of V ′ = V \{0} represent n retailers.

The set T = {1, ...,H} denotes the discrete planning horizon. A single commodity is pro-

duced/stored at the supplier and shipped to the retailers through a single vehicle. A vehicle

based at the supplier has a capacity Q and can perform one route per period. Let hi (i ∈ V)represent the unitary inventory holding cost per period at vertex i, and assume that each

retailer has a maximum inventory holding capacity Ci. Initial inventories at the supplier and

the retailers (I0i , i ∈ V) and the demand of each retailer i (dti) are known. Backlogging is

not allowed. A routing cost cij satisfying the triangle inequality is associated with each edge

(i, j) ∈ E . Finally, for the PRP, let s and u be the fixed set-up and the variable production

costs, respectively.

The following decision variables are used to formulate the problems. Routing variable xtij are

equal to the number of times edge (i, j) is used in period t. If vertex i is visited in period t,

variables yti is set to one. Variables Iti represent the inventory level at vertex i at period t.

The quantity delivered to retailer i in period t is given by qti . Let rt represent the quantity

produced at the supplier. For the IRP it is a parameter of the problem, while for the PRP, it

82

is a decision variable. For the PRP, variable zt is equal to one if production occurs in period

t.

In Section 5.2.1, we provide the mathematical models under the traditional cost minimization

objectives. Our different proposed objective functions are elaborated in Sections 5.2.2 and

5.2.3, where we show how the total cost minimization formulations can be adapted.

5.2.1 Total cost minimization models

We now present the mathematical models for the IRP in Section 5.2.1.1 and for the PRP in

Section 5.2.1.2. These models use the traditional total cost minimization objective function.

5.2.1.1 IRP

The total cost minimization formulation for IRP is shown below [Coelho and Laporte, 2013b]:

minimize∑i∈V

∑t∈T

hiIti +

∑(i,j)∈E

∑t∈T

cijxtij , (5.1)

subject to

It0 = It−10 + rt−1 −∑i∈V ′

qt−1i t ∈ T \ {1} (5.2)

It0 ≥∑i∈V ′

qti t ∈ T (5.3)

Iti = It−1i + qt−1i − dt−1i i ∈ V ′ t ∈ T \ {1} (5.4)

qti ≤ Ci − Iti i ∈ V ′ t ∈ T (5.5)

qti ≤ Ciyti i ∈ V ′ t ∈ T (5.6)∑

i∈V ′

qti ≤ Qyt0 t ∈ T (5.7)

∑j∈V,i<j

xtij +∑

j∈V,j<i

xtji = 2yti i ∈ V t ∈ T (5.8)

∑i∈S

∑j∈S,i<j

xtij ≤∑i∈S

yti − ytm S ⊆ V ′ t ∈ T m ∈ S (5.9)

yti ≤ 1 i ∈ V ′ t ∈ T (5.10)

Iti , qtj ≥ 0 i ∈ V j ∈ V ′ t ∈ T (5.11)

xt0j ∈ {0,1,2} j ∈ V ′ t ∈ T (5.12)

xtij ∈ {0,1} (i, j) ∈ E t ∈ T (5.13)

yti ∈ {0,1} i ∈ V t ∈ T . (5.14)

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The objective function (5.1) minimizes the total cost including inventory and transportation

costs. Constraints (5.2) and (5.3) are the inventory conservation constraints at the supplier

and constraints (5.4) have the same function for the retailers. Constraints (5.5) limit the

delivery to each retailer assuring that the inventory capacity is respected. Constraints (5.6)

guarantee that only visited retailers receive deliveries. The total quantity loaded in each

vehicle cannot exceed its capacity as imposed by constraints (5.7). Constraints (5.8) are

degree constraints and constraints (5.9) eliminate subtours. Constraints (5.10) prevent split

deliveries. Constraints (5.11)−(5.14) enforce integrality and non-negativity conditions on the

variables.

5.2.1.2 PRP

The total cost minimization formulation for PRP is shown below [Archetti et al., 2011]:

minimize∑i∈V

∑t∈T

hiIti +

∑(i,j)∈E

∑t∈T

cijxtij +

∑t∈T

(szt + urt) , (5.15)

subject to

It0 = It−10 + rt −∑i∈V ′

qti t ∈ T \ {1} (5.16)

Iti = It−1i + qti − dti i ∈ V ′ t ∈ T \ {1} (5.17)

Iti ≤ Ci i ∈ V t ∈ T (5.18)∑i∈V ′

qti ≤ Qyt0 t ∈ T (5.19)

qti ≤ (Ci + dti)yti i ∈ V ′ t ∈ T (5.20)

pt ≤ zt∑i∈V ′

∑t∈T

dti t ∈ T (5.21)

∑j∈V,i<j

xtij +∑

j∈V,j<i

xtji = 2yti i ∈ V t ∈ T (5.22)

∑i∈S

∑j∈S,i<j

xtij ≤∑i∈S

yti − ytm S ⊆ V ′ t ∈ T m ∈ S (5.23)

yti ≤ 1 i ∈ V ′ t ∈ T (5.24)

Iti , qtj , rt ≥ 0 i ∈ V j ∈ V ′ t ∈ T (5.25)

xt0j ∈ {0,1,2} j ∈ V ′ t ∈ T (5.26)

xtij ∈ {0,1} (i, j) ∈ E t ∈ T (5.27)

zt, yti ∈ {0,1} i ∈ V t ∈ T . (5.28)

84

The objective function (5.15) minimizes the total cost including production, inventory, and

transportation costs. Constraints (5.16) and (5.17) define the inventory conservation con-

straints at the supplier and the retailers, respectively. Constraints (5.18) impose the maximal

inventory level at the retailers, and constraints (5.19) and (5.20) limit the delivery to each cus-

tomer assuring that the vehicle capacity is respected. Constraints (5.21) force the production

setup. Constraints (5.22) are degree constraints and constraints (5.23) eliminate subtours.

Constraints (5.24) prevent split deliveries. Constraints (5.25)−(5.28) enforce non-negativity

and integrality conditions on the variables.

5.2.2 Distance minimization

We now formulate the problems by considering that the more mileage a truck covers, the more

emissions it generates. Hence, from transportation and environmental perspectives, it means

to fulfill all demand by driving as few miles as possible. In terms of mathematical formulation,

this can be easily done as follows. For each problem (IRP or PRP), one should consider all

the constraints presented in Sections 5.2.1.1 and 5.2.1.2 with the following objective function:

minimize∑

(i,j)∈E

∑t∈T

cijxtij . (5.29)

This objective function ignores all costs except the transportation one and aims to satisfy all

demand by driving the least distance.

5.2.3 Emission minimization

As mentioned in the introduction, a measure which is widely used to evaluate the emissions

is the one proposed in Kara et al. [2007] and Xiao et al. [2012], i.e., the distance traveled

multiplied by the vehicle load. In order to use this objective function, one needs to know

the flow of goods traversing each edge and the corresponding direction. Thus, we propose

a directed load-based formulation for the IRP and the PRP. For this purpose, we define a

directed graph G′ = (N ,A), where N = {0, ..., n} is the node set and A = {(i, j) : i, j ∈ N}is the arc set. In addition, we define binary variables ωt

ij representing the directed arcs. We

also define σtij as the renaming load on the vehicle when traveling from node i to j, i.e., after

visiting node i and before visiting node j. Therefore, the objective functions of both problems

become:

minimize∑

(i,j)∈A

∑t∈T

cijσtij , (5.30)

and we add the following constraints:

85

σtij ≤ Qωtji i, j ∈ N t ∈ T (5.31)∑

j∈N ′

σt0j ≤∑j∈N ′

qtj t ∈ T (5.32)

∑i∈N

σtij −∑i∈N

σtji = qtj j ∈ N ′ t ∈ T (5.33)

σti0, σtii = 0 i ∈ N t ∈ T . (5.34)

Moreover, we replace constraints (5.8) and (5.9) of the IRP and constraints (5.22) and (5.23)

of the PRP with the following:

∑j∈N

ωtij = yti i ∈ N t ∈ T (5.35)

∑j∈N

ωtij =

∑j∈N

ωtji i ∈ N t ∈ T (5.36)

ωtij + ωt

ji = xtij i, j ∈ N , i < j t ∈ T (5.37)

ωtij ∈ {0,1} (i, j) ∈ A t ∈ T . (5.38)

5.3 Computational experiments

A branch-and-cut algorithm is used to solve the problems. First, excluding the subtour elimi-

nation constraints, the model is solved by a general purpose mixed-integer programming solver.

Then violated subtour inequalities are identified and added to the formulation and the model

is reoptimized. For details on the branch-and-cut algorithm and on how subtour elimination

constraints are separated, the reader is referred to Archetti et al. [2011] and Coelho and La-

porte [2013b]. The formulations presented in Section 5.2 have been solved through CPLEX

12.7.0 and IBM Concert Technology. All computations are conducted on Intel Core i7 pro-

cessor running at 3.4 GHz with 64 GB of RAM installed, with the Ubuntu Linux operating

system. The maximum execution time is set to 3,600 seconds.

5.3.1 Instances

We used the benchmark instance sets created by Archetti et al. [2007] for IRP and Archetti

et al. [2011] for PRP.

The attributes of the IRP instances are as follows: the time horizon is either three or six

periods; in the instances with three periods, up to 50 retailers exist and in the ones with six

periods, up to 30 retailers are considered. Inventory costs are low or high, and, in this paper,

86

we identify the instances by the number of periods and their inventory holding cost category,

therefore, 3-High, represents 3 periods and high holding cost. There is a total of 160 instances

for the IRP.

The PRP instances are characterized by six time periods, 14 retailers, two levels for inventory

cost, two intervals for the distance coordinates, and three values for the capacity of the vehicle.

Four classes of instances are identified: (1) base case, (2) high variable production costs, (3)

high transportation cost, and (4) no inventory cost at the retailers. There is a total of 480

instances for the PRP.

5.3.2 Computational results

The computational results for IRP and PRP are presented in Sections 5.3.2.1 and 5.3.2.2,

respectively.

5.3.2.1 Computational results for the IRP

Table 5.1 presents the performance summary for IRP. We report the average optimality gap

at the end of computation and the average solution time. Each row reports the average results

over the 5 instances with the same number of customers, horizon and inventory cost.

While all instances with total cost and distance objective functions are solved to optimality

within a few seconds, after an hour of execution the case with emission objective function is

not solved to optimality.

In order to study the trade-offs between different objectives for IRP, for each instance, we

take the solution obtained from one objective function and evaluated it in terms of the other

two objectives. Table 5.2 summarizes the corresponding results. Columns are grouped in

three clusters of two columns each. Each cluster refers to one objective function, i.e., total

cost, distance, emissions. For each cluster, we evaluate the percentage increase value of the

corresponding objective function when considering the solution obtained by minimizing the

other two objective functions. For example, considering the first cluster and the first row, in

the first column we show that a solution obtained by minimizing the distance incurs 0.35%

increase in the total cost, when compared to the solution that minimized the total cost.

Likewise, still comparing with a solution obtained by minimizing that total cost, there is a

23.28% increase in the costs by using a solution that minimizes emissions. The percentage

increase is calculated as 100× z(F )− z(TC)

z(F ), where z(F ) is the value of the total cost related

to the solution that minimizes function F (distance in the first column and emissions in the

second) and z(TC) is the total cost related to the solution that minimizes the total cost. A

similar procedure is followed for clusters 2 and 3. In cluster 2, the distance is evaluated for

87

Table 5.1: Performance summary for IRP

Instance Total cost Distance Emissions# retailers Gap (%) Time(s) Gap (%) Time(s) Gap (%) Time(s)

3-High

5 0.00 0 0.00 0 0.00 010 0.00 0 0.00 0 22.99 360015 0.00 0 0.00 0 31.62 360020 0.00 1 0.00 1 39.15 360025 0.00 1 0.00 1 43.61 360030 0.00 3 0.00 3 56.67 360035 0.00 1 0.00 1 57.98 360040 0.00 8 0.00 6 68.57 360045 0.00 10 0.00 8 70.19 360050 0.00 38 0.00 37 69.65 3600

Average 0.00 6.24 0.00 5.62 46.04 3240.16

3-Low

5 0.00 0 0.00 0 0.00 010 0.00 0 0.00 0 22.99 360015 0.00 0 0.00 0 31.47 360020 0.00 1 0.00 1 37.73 360025 0.00 1 0.00 1 43.05 360030 0.00 2 0.00 3 56.15 360035 0.00 1 0.00 1 59.58 360040 0.00 5 0.00 6 69.11 360045 0.00 13 0.00 8 70.95 360050 0.00 43 0.00 38 69.98 3600

Average 0.00 6.58 0.00 5.76 46.10 3240.12

6-High

5 0.00 0 0.00 0 0.74 184910 0.00 0 0.00 1 27.92 360015 0.00 2 0.00 2 34.04 360020 0.00 7 0.00 47 43.67 360025 0.00 12 0.00 43 53.94 360030 0.00 26 0.00 65 64.21 3600

Average 0.00 7.93 0.00 26.43 37.42 3308.37

6-High

5 0.00 0 0.00 0 0.66 180810 0.00 1 0.00 0 28.50 360015 0.00 2 0.00 2 33.94 360020 0.00 13 0.00 15 43.27 360025 0.00 24 0.00 17 55.07 360030 0.00 58 0.00 52 64.18 3600

Average 0.00 16.30 0.00 14.57 37.60 3301.40

solutions that minimize the total cost, first, and the emissions, second. For cluster 3, the

emissions are evaluated for solutions that minimize the total cost, first, and the distance,

second.

As shown in the table, when minimizing the distance we obtain a slight increase in total

cost. Instead, when we minimize emissions, the cost increases on average up to 50.42%.

Minimizing the total cost would almost always provide solutions coinciding with those obtained

when minimizing the distance. However, compared to the total cost minimization, when only

distances are minimized the emission level increases. Despite the large optimality gap obtained

for the emission objective function problems, they always yield the best solutions in terms of

emissions. As shown in Table 5.2 emission levels increase by at least 22.16% and at most

56.62% under any of the other objective functions.

88

Table 5.2: Comparison of solutions with different objective functions for the IRP

Instance From distance From emissions From total cost From emissions From total cost From distance# retailers To total cost To distance To emissions

3-High

5 0.35 23.28 0.00 35.95 42.13 43.8810 0.32 20.94 0.00 40.85 47.60 45.2415 0.27 18.72 0.00 39.97 56.49 55.5720 0.20 18.30 0.00 42.59 52.64 51.2925 0.21 19.10 0.00 47.49 53.12 51.1330 0.29 22.13 0.00 55.56 52.22 50.2035 0.42 30.15 0.00 65.56 48.03 45.9440 0.27 35.18 0.42 71.32 40.65 39.7745 0.35 35.60 0.00 73.41 35.00 32.3750 0.18 43.88 0.27 79.54 25.49 23.23

Average 0.28 26.73 0.07 55.22 45.34 43.86

3-Low

5 0.07 34.08 0.00 35.95 43.39 43.8810 0.08 37.26 0.00 40.85 47.60 45.2415 0.06 36.22 0.00 40.35 56.62 55.6920 0.05 36.99 0.00 42.01 53.21 52.1625 0.07 41.79 0.00 48.00 53.72 51.7430 0.09 47.28 0.00 54.70 52.85 50.5035 0.14 59.47 0.00 66.33 46.13 44.1240 0.08 65.50 0.00 72.08 40.57 39.1045 0.13 72.03 0.00 78.31 34.22 31.2350 0.05 73.56 0.00 79.56 23.91 22.16

Average 0.08 50.42 0.00 55.81 45.22 43.58

6-High

5 0.61 16.24 0.06 25.95 35.71 32.4210 0.37 17.25 0.00 29.58 40.22 40.0115 0.28 16.83 0.37 32.58 49.03 47.0520 0.39 18.84 0.29 36.66 38.77 39.3625 0.57 33.82 0.20 58.02 35.09 35.7930 0.59 35.05 0.41 63.37 23.49 24.30

Average 0.47 23.00 0.22 41.03 37.05 36.49

6-Low

5 0.10 24.46 0.00 25.95 36.19 32.4210 0.07 29.21 0.00 31.31 39.76 39.6115 0.04 27.55 0.00 30.29 47.37 47.1220 0.06 33.70 0.00 36.96 41.19 39.8525 0.11 53.36 0.00 57.29 35.55 34.1430 0.11 58.36 0.01 63.18 23.44 23.40

Average 0.08 37.77 0.00 40.83 37.25 36.09

5.3.2.2 Computational results for the PRP

Table 5.3 presents the performance summary for the PRP instances. Class I instances with

total cost and distance objective functions are solved to optimality within on average 2.2

and 25.11 seconds, Class II instances within 2.68 and 51.19, Class III within 5.04 and 22.04,

and finally Class IV within 4.98 and 42.51. Again, when considering the emission objective

function, the problem becomes more difficult to be solved to optimality within one hour of

computing time. The maximum gap belongs to Class II instances, for which, after 3404

seconds, the average gap is 11.36%. However, further analysis indicates that even with large

optimality gaps, still the emission objective function provides the best solution in terms of

emission level compared to the results obtained when minimizing distances or total costs.

For class I instances, presented in Table 5.4, we see that the increase in total cost when mini-

89

Table 5.3: Performance summary for PRP

Total cost Distance EmissionsInstance Gap (%) Time(s) Gap (%) Time(s) Gap (%) Time(s)Class I 0.00 2.20 0.00 25.11 10.79 3439.25Class II 0.00 2.68 0.00 51.19 11.36 3404.27Class III 0.00 5.04 0.00 22.04 9.49 3129.88Class IV 0.00 4.98 0.00 42.51 7.97 3058.49

mizing the distance is 5.33% while, on the contrary, the increase in distance when minimizing

the total cost is 8.60%. Thus, we can say that the two objectives obtain almost the same

solutions. On the other side, when considering emissions, we get a totally different picture.

In fact, when minimizing the emissions, the total cost increases by 11.28% and the distance

increases by 38.19%, thus leading to a substantial difference. In addition, when we minimize

both total cost and distance, we incur a large increase in emissions, of 41.83% and 39.51%,

respectively. A similar behavior is realized for the other classes of instances (Tables 5.5– 5.7)

with a lower difference between the solutions obtained when minimizing the total cost and the

distance.

Table 5.4: Comparison of solutions with different objective functions for PRP-Class I

From distance From emissions From total cost From emissions From total cost From distanceInstance To total cost To distance To emissions

Class I

1 9.69 12.73 14.60 43.53 51.69 50.782 5.83 12.28 27.00 39.96 48.57 40.403 9.35 11.05 17.57 31.39 37.61 26.374 4.86 8.80 3.45 44.80 49.71 50.605 3.37 8.33 1.93 40.16 43.38 40.376 4.57 7.42 4.16 29.81 27.69 28.007 5.18 13.24 1.40 44.80 51.59 50.608 3.39 12.20 1.93 40.16 43.38 40.379 4.19 9.93 4.16 29.81 27.74 28.0010 6.21 11.83 11.71 42.31 48.91 50.6411 3.95 10.80 8.40 39.49 39.96 40.6112 3.54 9.16 4.90 32.91 28.45 27.2313 7.73 14.99 9.20 41.29 51.34 50.2914 5.61 14.44 8.72 40.14 44.39 41.0315 5.47 12.03 9.87 32.24 33.09 27.37

Average 5.53 11.28 8.60 38.19 41.83 39.51

5.3.3 Key performance indicators

In order to compare solutions obtained from different objective functions and to better un-

derstand the trade-offs, we have defined a set of key performance indicators (KPIs). These

KPIs are categorized in three groups: inventory, delivery, and load. Inventory KPIs category

includes quantity and cost of inventory at retailers and at the depot. Delivery KPIs measure

the total number of deliveries, total quantity delivered to all retailers, and finally the average

90

Table 5.5: Comparison of solutions with different objective functions for PRP-Class II

From distance From emissions From total cost From emissions From total cost From distanceInstance To total cost To distance To emissions

Class II

16 4.43 10.32 0.53 41.30 47.97 50.1917 2.20 10.02 0.95 39.80 41.65 40.9418 3.30 8.64 1.74 30.45 28.77 27.5019 4.31 16.63 0.53 42.09 48.13 50.3520 2.11 16.02 0.95 40.74 41.67 40.9721 2.72 12.71 1.74 31.05 28.77 27.4922 5.50 13.25 7.75 41.30 48.61 50.1923 2.58 12.69 2.81 40.32 42.30 41.1324 2.98 10.12 3.00 31.08 27.43 27.4425 3.45 2.36 14.60 43.53 51.66 50.7826 1.48 2.26 25.11 38.68 49.57 40.7127 1.75 2.10 17.57 32.96 38.30 27.1728 2.50 1.24 0.87 42.07 51.19 50.7329 0.47 1.22 1.93 40.78 43.52 40.4930 0.65 1.14 4.16 32.91 27.06 27.2331 2.56 2.01 0.87 43.53 51.24 50.7832 0.48 1.89 1.93 40.46 43.69 40.6733 0.62 1.67 4.16 32.91 26.99 27.2334 2.63 1.55 11.71 41.68 48.96 50.6935 0.49 1.46 8.40 40.46 40.37 40.6736 0.45 1.20 4.90 32.05 28.50 27.3337 3.18 3.08 6.75 41.30 49.86 50.1938 1.07 3.06 7.51 41.85 43.90 40.9939 1.08 2.46 9.87 31.65 34.24 28.6240 2.48 1.59 0.53 41.56 47.70 49.9441 0.31 1.56 0.96 39.63 41.89 41.0742 0.48 1.36 1.74 31.65 29.81 28.6243 2.49 2.83 0.53 41.39 47.76 50.0044 0.33 2.75 0.96 39.63 41.89 41.0745 0.43 2.25 1.74 31.65 29.91 28.6246 2.58 1.87 7.75 41.59 48.35 49.9547 0.33 1.82 2.81 40.15 42.27 41.0948 0.39 1.43 3.00 31.65 28.58 28.62

Average 1.90 4.74 4.86 38.00 40.68 39.68

trip length. The average trip length is obtained by calculating the total distance traveled di-

vided by the number of arcs used. Finally, we define vehicle fill, average load, empty running,

logistic ratio, and distribution cost per delivery as the load KPIs. The vehicle fill is the total

load on the vehicle divided by its capacity. The average load is the ratio between the total

load and number of deliveries. The empty running KPI is the distance the vehicle travels

getting back to the depot, after the last delivery, as all vehicles come back empty to the depot.

For the emission problem in which the graph is directed, the empty running of each vehicle is

the distance the vehicle travels from the last visited customer to the depot. For the total cost

and distance minimization problems, as the edges are not directed, we use the average of the

distance from the depot to the first and last visited customer as the empty running KPI. Note

that, for the emission problem, the direction in which the route is traversed has an impact on

the emission value. Thus, we cannot consider the reverse direction. Finally, logistic ratio is

the distance traveled per quantity of items delivered, and the distribution cost per delivery,

91

Table 5.6: Comparison of solutions with different objective functions for PRP-Class III

From distance From emissions From total cost From emissions From total cost From distanceInstance To total cost To distance To emissions

Class III

49 4.58 27.07 0.13 42.35 50.35 50.5650 4.01 24.89 0.63 39.23 39.91 39.6051 3.58 20.52 1.38 32.00 30.02 27.3852 2.61 20.57 0.01 42.35 49.45 50.5653 2.53 19.14 0.44 39.23 39.34 39.6054 2.78 15.99 1.39 32.00 28.93 27.3855 1.89 29.87 0.01 42.35 49.46 50.5656 1.63 27.54 0.44 39.23 39.34 39.6057 1.30 22.30 13.19 31.86 26.64 27.3058 2.52 23.32 0.01 42.35 49.36 50.5659 2.42 21.66 0.60 39.23 39.95 39.6060 1.55 17.12 0.65 32.00 26.88 27.3861 3.01 32.91 0.10 42.55 50.25 49.7462 2.09 30.91 0.65 40.14 39.82 41.1563 1.94 23.59 0.52 30.75 28.02 28.6064 2.61 26.85 0.01 42.55 49.04 49.6965 1.32 26.17 0.46 41.26 39.30 41.1966 1.78 19.51 1.39 30.75 29.20 28.6067 1.64 33.96 0.01 41.49 49.41 49.9168 0.61 33.76 0.46 41.26 39.30 41.1969 0.65 25.50 0.52 31.34 26.94 28.5470 2.94 29.01 0.01 42.55 49.10 49.7471 1.27 28.10 0.56 41.26 41.11 41.1972 0.91 21.47 0.37 31.29 24.73 29.27

Average 2.17 25.07 1.00 37.97 38.99 39.54

Table 5.7: Comparison of solutions with different objective functions for PRP-Class IV

From distance From emissions From total cost From emissions From total cost From distanceInstance To total cost To distance To emissions

Class IV

73 5.42 16.74 7.73 43.54 45.16 48.2174 4.37 13.89 4.41 37.03 38.51 39.0075 2.23 11.76 6.97 34.96 33.45 34.3076 6.57 13.89 10.24 42.29 44.32 48.5577 5.24 11.57 7.89 37.03 40.27 39.0178 2.69 10.49 2.69 35.19 37.90 40.3079 3.19 19.34 1.42 41.15 44.02 46.3980 2.98 17.70 1.13 38.53 40.96 39.2381 4.77 14.78 4.01 34.15 28.38 36.8482 4.34 15.08 8.06 40.52 44.40 46.2783 3.81 13.18 6.08 37.88 38.50 39.1184 1.23 6.95 4.17 33.64 29.41 32.7085 2.53 2.28 7.73 42.29 45.40 48.5586 0.55 1.92 4.41 37.03 38.50 39.0087 0.27 1.54 6.97 34.47 33.68 34.3088 2.64 1.77 10.24 43.54 44.18 48.2189 0.62 1.45 7.89 37.03 40.09 39.0090 1.16 1.35 2.69 35.52 38.27 40.2691 0.43 3.07 1.42 40.85 43.59 46.6392 0.41 2.79 1.13 37.88 41.21 39.1193 4.17 2.19 4.01 34.15 29.46 36.8494 0.52 1.97 8.06 40.79 45.22 46.7695 0.46 1.76 6.08 37.88 38.61 39.1196 0.26 1.48 3.78 31.65 27.15 29.63

Average 2.54 7.87 5.38 37.88 38.78 40.72

92

as the name suggests, is the cost per units delivered.

We use the above mentioned KPIs mainly for two reasons. First, they are already established

in the literature, as they are well documented by Mckinnon and Hr [2007]. Second, they are

all based on measures that we are able to evaluate by running our models on benchmark IRP

and PRP instances, i.e., distance and vehicle load. In fact, one may argue that traffic and

congestion are also important factors that have an impact on emissions. However, they are

difficult to measure in general and not suited for the models and instances under study.

5.3.3.1 Analysis for the IRP

The average of inventory KPIs for the IRP are presented in Table 5.8. Inventory costs and

quantities remain roughly unchanged within the three different objective functions. This

leads to the conclusion that the difference in total cost observed when optimizing the different

objective functions, especially emissions, is mainly due to transportation cost (Table 5.2). The

inventory at the depot is the lowest under the total cost minimization and the highest under

the distance minimization. However, for the amount of inventory kept at the retailers we

observe the opposite.

Table 5.8: Average inventory KPIs for IRP

Instance Objective Inventory Inventory cost Inventory Inventory costretailer retailer depot depot

3-HighTotal cost 4,530.32 1,350.01 1,8694.32 5,608.30Distance 4,370.66 1,330.00 1,8853.98 5,656.19Emissions 4,381.96 1,336.06 1,8842.68 5,652.80

3-LowTotal cost 4,511.68 134.88 18,712.96 561.39Distance 4,370.66 133.17 18,853.98 565.62Emissions 4,365.16 133.49 18,859.48 565.78

6-HighTotal cost 5,096.50 1,472.65 21,390.57 6,417.17Distance 4,750.57 1,443.13 21,736.50 6,520.95Emissions 4,907.47 1,494.70 21,579.60 6,473.88

6-LowTotal cost 5,016.87 147.72 2,1470.20 644.11Distance 4,750.57 142.93 2,1736.50 652.10Emissions 4,904.67 149.47 2,1582.40 647.47

AverageTotal cost 4,788.84 776.31 2,0067.01 3,307.74Distance 4,560.61 762.31 2,0295.24 3,348.71Emissions 4,639.81 778.43 2,0216.04 3,334.99

Table 5.9 shows the average delivery KPIs for the IRP. As indicated, the total number of

deliveries when minimizing the emissions is almost double the number of deliveries in total

cost and distance minimization problems. Total deliveries are similar in all three problems,

however, in the total cost minimization, more quantities are delivered to the retailers. The

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extra delivery is kept as the inventory at the retailers and, as shown before by inventory KPIs,

the total cost minimization problem tends to keep more inventory at the retailers. This is

due to the structure of inventory costs. While the average length of the trip does not vary in

total cost and distance minimization problems, they considerably increase with an emission

minimization objective function. This is due to the fact that, when minimizing the emissions,

we consider both distance and quantity delivered, so a higher number of deliveries is performed.

Table 5.9: Average delivery KPIs for IRP

Instance Objective # Deliveries Total delivery Trip length

3-HighTotal cost 29.60 2,257.98 91.74Distance 29.54 2,159.30 91.54Emissions 45.42 2,155.26 148.11

3-LowTotal cost 29.54 2,250.04 91.54Distance 29.54 2,159.30 91.54Emissions 45.84 2,155.26 150.34

6-HighTotal cost 49.57 4,380.37 115.13Distance 49.00 4,275.90 116.04Emissions 62.63 4,271.37 162.54

6-LowTotal cost 49.03 4,337.33 116.02Distance 49.00 4,275.90 116.04Emissions 61.93 4,271.37 163.48

AverageTotal cost 39.44 3,306.43 103.61Distance 39.27 3,217.60 103.79Emissions 53.96 3,213.31 156.12

As shown in Table 5.10, which reports the average load KPIs, the vehicle fill is the same

with all three objective functions, and, when minimizing the distance, the average load on

the vehicle tends to be the biggest among the three objective functions and significantly lower

with respect to the case in which we minimize the emissions. The differences between the

empty running KPIs obtained by three objective functions is quite notable. In the emission

minimization problem, the empty vehicles travel a much longer distance. This is mainly due

to the fact that we define emissions as the product of the load and the distance, therefore,

to reduce the emissions, the vehicle has to travel the longer way with no load. Logistic ratio

and distribution cost per delivery are also higher while minimizing emissions, as the number

of deliveries increase.

5.3.3.2 Analysis for the PRP

The inventory KPIs for the PRP are shown in Table 5.11. The table shows that to minimize

the total cost, one needs to reduce the inventory level at the depot, while to minimize the

emissions, the inventory at the depot should almost be doubled. Although almost the same

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Table 5.10: Average load KPIs for IRP

Instance Objective Vehicle Average Empty Logistic Distribution costfill load running ratio per delivery

3-HighTotal cost 0.35 73.79 141.91 1.34 91.74Distance 0.33 91.54 141.03 1.43 91.54Emissions 0.33 49.96 841.98 3.15 148.11

3-LowTotal cost 0.35 73.87 140.93 1.34 91.54Distance 0.33 91.54 141.03 1.43 91.54Emissions 0.33 49.72 887.38 3.23 150.34

6-HighTotal cost 0.51 86.55 447.57 1.37 115.13Distance 0.50 116.04 442.20 1.39 116.04Emissions 0.50 70.57 1706.07 2.35 162.54

6-LowTotal cost 0.51 86.64 444.33 1.38 116.02Distance 0.50 85.56 442.20 1.39 116.04Emissions 0.50 71.08 1703.17 2.35 163.48

AverageTotal cost 0.43 80.21 293.69 1.36 103.61Distance 0.42 96.17 291.62 1.41 103.79Emissions 0.42 60.33 1284.65 2.77 156.12

amount of inventory is kept at the retailers in both total cost and emission minimization, it

is quite lower when the distance is minimized.

Table 5.11: Average inventory KPIs for PRP

Instance Objective Inventory Inventory cost Inventory Inventory costretailer retailer depot depot

Class ITotal cost 1,729.23 8,308.91 651.45 2,939.69Distance 2,033.77 10,517.09 1,022.44 5,079.85Emissions 1,809.12 9,367.11 1,232.88 6,168.37

Class IITotal cost 1,855.67 7,801.08 642.73 3,364.03Distance 2,006.22 8,920.38 1,057.73 6,064.58Emissions 1,800.44 8,022.16 1,241.56 7,103.07

Class IIITotal cost 1,870.23 8,471.42 632.87 3,344.18Distance 2,046.60 9,571.50 997.40 5,485.70Emissions 1,796.50 8,385.93 1,245.50 6,856.50

Class IVTotal cost 2,045.13 0.00 479.93 2,590.55Distance 2,014.83 0.00 1,046.70 5,756.85Emissions 1,802.38 0.00 1,239.63 6,818.00

AverageTotal cost 1,875.07 6,145.35 601.75 3,059.61Distance 2,025.36 7,252.24 1,031.07 5,596.74Emissions 1,802.11 6,443.80 1,239.89 6,736.49

The delivery KPIs are shown in Table 5.12. It is interesting to note that with the distance

minimization objective function, the number of deliveries and the average length of the trip

95

are also minimized. The number of deliveries slightly increases when minimizing the emissions

rather than the total cost, while the total deliveries remain the same. As was the case with

IRP, the trip length of each vehicle increases when reducing the emissions with respect to the

case of minimizing the total cost or distance.

Table 5.12: Average delivery KPIs for PRP

Instance Objective # Deliveries Total Delivery Trip Length

Class ITotal cost 24.59 543.60 163.54Distance 22.72 548.28 161.18Emissions 26.03 543.60 227.78

Class IITotal cost 23.78 543.60 224.50Distance 22.69 548.31 220.08Emissions 26.10 543.60 308.54

Class IIITotal cost 22.43 543.60 2,044.86Distance 22.43 546.53 2,037.68Emissions 25.82 543.60 2,867.04

Class IVTotal cost 22.63 543.60 213.90Distance 22.70 548.30 201.67Emissions 25.85 543.60 285.53

AverageTotal cost 23.28 543.60 661.70Distance 22.64 547.86 655.15Emissions 25.95 543.60 922.29

Table 5.13 shows the load KPIs for PRP. Just like for the IRP, the vehicle fill does not

vary with respect to the objective function. When minimizing the emissions, the average

load tends to decrease while the empty running, logistic efficiency, and distribution cost per

delivery considerably increase. Concerning the total cost and distance minimization objective

functions, while the distribution cost per delivery remains almost the same, the average load

and empty running increase when minimizing the distance and the logistic ratio decreases.

5.4 Conclusions

In this paper we have studied what is the impact of minimizing emissions on classical integrated

logistics optimization problems. We have considered the IRP and the PRP, two classical

supply chain problems treated under a cost minimization objective, and we have assessed

their solutions under a more encompassing objective in which one aims to decrease emission

levels and be more environmentally friendly. To that end, we have focused on both minimizing

the total distance traveled to satisfy all the demand, and a more complex and richer function

serving as a proxy for true emissions. After solving sets of benchmark instances under these

three scenarios, we have evaluated the shape of the solutions in terms of different business

96

Table 5.13: Average load KPIs for PRP

Instance Objective Vehicle Average Empty Logistic Distribution costfill load running ratio per delivery

Class ITotal cost 0.41 22.38 608.76 7.44 163.54Distance 0.41 24.23 606.54 6.72 161.18Emissions 0.41 20.87 1,840.34 10.95 227.78

Class IITotal cost 0.41 23.34 804.90 9.66 224.50Distance 0.41 24.24 819.79 9.17 220.08Emissions 0.41 20.80 2,500.79 14.86 308.54

Class IIITotal cost 0.41 24.25 7,186.53 84.97 2,044.86Distance 0.41 24.39 7,855.75 84.18 2,037.68Emissions 0.41 21.04 2,2877.78 136.49 2,867.04

Class IVTotal cost 0.41 24.02 797.51 8.91 213.90Distance 0.41 24.24 753.15 8.40 201.67Emissions 0.41 21.02 2,293.50 13.62 285.78

AverageTotal cost 0.41 23.50 2,349.42 27.74 661.70Distance 0.41 24.28 2,508.81 27.12 655.15Emissions 0.41 20.93 7,378.10 43.98 922.29

KPIs designed to highlight the impact of the objective function chosen on costs, distances,

and emissions.

Our findings indicate that an important factor in reducing the emissions is the empty running

of the vehicle.In fact, when minimizing emissions, a ‘lighter’ load is preferable, and, as a

consequence, empty running to come back to the depot tends to be longer. In addition, the

findings indicate that there needs to be a balance between the load on the vehicle and the

distance it travels. In order to reduce fuel consumption, it would be preferable to perform more

deliveries to the customers with respect to the number of deliveries obtained when optimizing

classical objective functions like total cost or distance. Finally, our computational experiments

corroborate the previous studies pointing out that to reduce emissions, it is important to

measure not only the distance traveled but also the load of the vehicles. This may lead to

counter-intuitive solutions in which the distance traveled increases and the average vehicle

load decreases. However, given that vehicles travel with a lower load, this may lead to benefits

in terms of emissions. Our evaluation of several KPIs has shed some light on the shape the

solutions should have in order to minimize the total costs or emission levels.

In conclusion, our work highlight that solutions may differ a lot when changing the objective

function. Hence, managers have to properly evaluate what is best for their company before

planning the corresponding operations.

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Chapter 6

Conclusion

Integration and coordination of decisions from different supply chain functions have become

a growing area of interest in both academia and practice. In this research we have described,

modeled and solved several integrated supply chain problems. The two main themes of this

thesis are value of integration and benefits of flexibility in integrated problems. We have

investigated the financial value of integrated approach in rich supply chain settings inspired

by practical problems. The trade-offs between financially good decisions and environmental

friendly ones are also investigated. The two sources of flexibility studied in this thesis stem

from delivery time windows and the supply chain network design.

In Chapter 1, after introducing well-known integrated supply chain problems, we have fo-

cused on the integrated production-distribution problems. First, we have clarified the several

common terms used interchangeably for the same problem and then classified the literature

that presents mathematical models to solve a single stage integrated optimization problem

considering the following criteria: the number of products (single or multiple), the number of

echelons (single or multiple), the number of plants (single or multiple), the number of periods

(single or multiple), whether production/inventory capacities exist, whether the demand is

deterministic or stochastic, whether the production setup cost is considered in the objective

function, and finally whether location decisions are addressed in the model. We have also

reviewed the solution algorithms and applications of the models. Our review of literature

showed that LSP with time windows, integration of location decisions with production and

distribution decisions, and environmental issues of the supply chains are among the recent hot

topics of interest that have not been well studied in the integrated optimization area.

In Chapter 2, we have introduced a real-case single-product, multi-plant, multi-period pro-

duction-distribution problem in which production, inventory, and distribution decisions were

simultaneously optimized. We considered production and inventory capacities as well as inter-

98

plant transfers in our model and solved it by the branch-and-bound method. Changing the

parameters of the model, we have conducted elaborate sensitivity analysis and compared

various production scenarios. Our results revealed that using an integrated approach toward

optimizing lot sizing and distribution decisions, companies could save on logistics costs and

also promise more competitive delivery time windows.

We have extended the problem in Chapter 3 by considering a multi-product setting, and adding

a new echelon for DCs. We have introduced the facility location decision to the model as the

location of the DCs had to be selected in each period. We have tackled the problem taking

both sequential and integrated approaches. For sequential approaches, we have used very

simple procedures inspired by production systems managed in silos. For the integrated one,

we have first solved the problem with exact methods, and then to overcome the limitation

of the exact methods in solving large instances, we have proposed a hybrid adaptive large

neighborhood search (ALNS) heuristic. By conducting extensive computational experiments

on instances with varying sizes and levels of complexity, we have highlighted the value of the

integrated approach compared to the sequential one. Showing the inefficiency of the exact

methods, we have drawn the attention to the need for more performant solution algorithms.

In both problems presented in Chapters 2 and 3, we deal with deliveries within time windows,

and particularly, in Chapter 3 we solve an integrated production-distribution problem with

flexible network design. The values of these two types of flexibility, the flexibility obtained

from delivery time windows and the one obtained from the network design, are investigated in

a richer model setting presented in Chapter 4. In order to have a better understanding on the

role of flexibility in reducing supply chain costs, we have described and modeled an integrated

problem in which location and routing decisions are simultaneously optimized. We propose a

mathematical formulation of the problem together with different classes of valid inequalities.

Extensive computational tests are made on randomly generated instances to show the value

of the two kinds of flexibility and provide managerial insights. Our results show that the

combined effect of the two kinds of flexibility leads to considerable cost savings.

In Chapter 5, we introduce the idea of assessing the impact of emission reduction on classical

integrated logistics optimization problems. Comparing the solutions obtained from total cost,

distance, and emission reduction objectives, we study the trade-offs between environmental

friendly solutions and the cost-efficient ones. Our further analysis reveals the differences in

the shape of the solutions, in terms of different business KPIs designed to highlight the impact

of the objective function chosen on costs, distances, and emissions.

This thesis contributes to the literature in a number of ways. Comparing the traditional

approach towards decisions making in supply chain planning with the integrated one, we have

99

highlighted the benefits of the integrated approach. We have shown how the cost reduction

benefits obtained by these methods compensate their complexity and time consuming solving

process. We have added the flexibility feature to our rich integrated problems. This flexibility

mainly stems from two sources of the flexibility in delivery due dates, and flexibility in the

network design. We have also incorporated the location decision into the integrated problem,

which due to its strategic nature is not jointly considered with the operational level decisions.

Finally, we have reassessed two classical integrated problems to shed light on the trade-offs

between cost reduction versus energy savings and pollutant emission diminution.

This thesis is also limited in a number of ways. Solving an integrated problem is a complex

task itself, adding several real-world features to the problem makes it even more difficult

to solve. In order to be able to have a tractable model and implementable solutions, we

have considered several simplifying assumptions. For instance, we only consider the demand

to be deterministic and all time delays in production and transportation are ignored. In this

thesis, we have used instances, either randomly generated or some already existing benchmarks

from the literature. Although all our instances are generated randomly, the findings of our

experimental studies are valid for the set of instances used. Finally, in this thesis, we take a

problem solving approach; we define the problems and try to solve them first only by means

of exact methods, and then later, if needed, a heuristic method is applied. Large instances

of our problems still remain very difficult to be solved to optimality either by using the exact

methods or our powerful heuristic.

As discussed in Chapter 1, research opportunities still exists on investigating stochastic pro-

duction-distribution integration problems, in the development of faster and more efficient exact

methods to deal with large instances of flexible integrated problems. An interesting extension

of this thesis would be to integrate location and procurement decisions of the supply chains into

the production-distribution models. In Chapter 2, we have provided a real-world application

for the integrated production-distribution model, having a wide applicability a future research

venue would be to adapt other theoretical integrated models to real business cases.

100

Bibliography

N. Absi, S. Kedad-Sidhoum, and S. Dauzère-Pérès. Uncapacitated lot-sizing problem with

production time windows, early productions, backlogs and lost sales. International Journal

of Production Research, 49(9):2551–2566, 2011.

N. Absi, S. Dauzère-Pérès, S. Kedad-Sidhoum, B. Penz, and C. Rapine. Lot sizing with carbon

emission constraints. European Journal of Operational Research, 227(1):55–61, 2013.

N. Absi, C. Archetti, S. Dauzère-Pérès, and D. Feillet. A two-phase iterative heuristic approach

for the production routing problem. Transportation Science, 49(4):784–795, 2014.

N. Absi, S. Dauzère-Pérès, S. Kedad-Sidhoum, B. Penz, and C. Rapine. The single-item green

lot-sizing problem with fixed carbon emissions. European Journal of Operational Research,

248(3):849–855, 2016.

Y. Acar and S.N. Atadeniz. Comparison of integrated and local planning approaches for

the supply network of a globally-dispersed enterprise. International Journal of Production

Economics, 167:204–219, 2015.

Y. Adulyasak, J.-F. Cordeau, and R. Jans. Optimization-based adaptive large neighborhood

search for the production routing problem. Transportation Science, 48(1):20–45, 2012.

Y. Adulyasak, J.-F. Cordeau, and R. Jans. Formulations and branch-and-cut algorithms

for multi-vehicle production and inventory routing problems. INFORMS Journal on

Computing, 26(1):103–120, 2014.

Y. Adulyasak, J.-F. Cordeau, and R. Jans. The production routing problem: A review of

formulations and solution algorithms. Computers & Operations Research, 55:141–152, 2015.

R.K. Ahuja, W. Huang, H.E. Romeijn, and D.R. Morales. A heuristic approach to the multi-

period single-sourcing problem with production and inventory capacities and perishability

constraints. INFORMS Journal on Computing, 19(1):14–26, 2007.

101

C.H. Aikens. Facility location models for distribution planning. European Journal of

Operational Research, 22(3):263–279, 1985.

A. Akbalik and B. Penz. Comparison of just-in-time and time window delivery policies for a

single-item capacitated lot sizing problem. International Journal of Production Research,

49(9):2567–2585, 2011.

B. Almada-Lobo and R.J.W. James. Neighbourhood search meta-heuristics for capacitated

lot-sizing with sequence-dependent setups. International Journal of Production Research,

48(3):861–878, 2010.

C. Almeder. A hybrid optimization approach for multi-level capacitated lot-sizing problems.

European Journal of Operational Research, 200(2):599–606, 2010.

A. Amiri. Designing a distribution network in a supply chain system: Formulation and efficient

solution procedure. European Journal of Operational Research, 171(2):567–576, 2006.

P. Amorim, H.-O. Günther, and B. Almada-Lobo. Multi-objective integrated production

and distribution planning of perishable products. International Journal of Production

Economics, 138(1):89–101, 2012.

H. Andersson, A. Hoff, M. Christiansen, G. Hasle, and A. Løkketangen. Industrial aspects and

literature survey: Combined inventory management and routing. Computers & Operations

Research, 37(9):1515–1536, 2010.

C. Archetti and M.G. Speranza. The inventory routing problem: the value of integration.

International Transactions in Operational Research, 23(3):393–407, 2016.

C. Archetti, L. Bertazzi, G. Laporte, and M.G. Speranza. A branch-and-cut algorithm for a

vendor-managed inventory-routing problem. Transportation Science, 41(3):382–391, 2007.

C. Archetti, L. Bertazzi, G. Paletta, and M.G. Speranza. Analysis of the maximum level policy

in a production-distribution system. Computers & Operations Research, 12(38):1731–1746,

2011.

C. Archetti, L. Bertazzi, A. Hertz, and M.G. Speranza. A hybrid heuristic for an inventory

routing problem. INFORMS Journal on Computing, 24(1):101–116, 2012.

C. Archetti, O. Jabali, and M.G. Speranza. Multi-period vehicle routing problem with due

dates. Computers & Operations Research, 61:122–134, 2015.

C. Archetti, E. Fernández, and D.L. Huerta-Muñoz. The flexible periodic vehicle routing

problem. Computers & Operations Research, 85:58–70, 2017.

102

V.A.L. Armentano, A.L. Shiguemoto, and A. Løkketangen. Tabu search with path relinking

for an integrated production–distribution problem. Computers & Operations Research, 38

(8):1199–1209, 2011.

S. Axsäter. Inventory Control, chapter Multi-Echelon Systems: Lot Sizing, pages 171–189.

Springer International Publishing, Cham, 2015.

A. Baboli, J. Fondrevelle, R. Tavakkoli-Moghaddam, and A. Mehrabi. A replenishment policy

based on joint optimization in a downstream pharmaceutical supply chain: centralized

vs. decentralized replenishment. The International Journal of Advanced Manufacturing

Technology, 57(1-4):367–378, 2011.

H.C. Bahl, L.P. Ritzman, and J.N.D. Gupta. OR practice – determining lot sizes and resource

requirements: A review. Operations Research, 35(3):329–345, 1987.

I. Barany, T.J. Van Roy, and L.A. Wolsey. Strong formulations for multi-item capacitated lot

sizing. Management Science, 30(10):1255–1261, 1984.

G. Barbarosoğlu and D. Özgür. Hierarchical design of an integrated production and 2-echelon

distribution system. European Journal of Operational Research, 118(3):464–484, 1999.

J.F. Bard and N. Nananukul. Heuristics for a multiperiod inventory routing problem with

production decisions. Computers & Industrial Engineering, 57(3):713–723, 2009a.

J.F. Bard and N. Nananukul. The integrated production–inventory–distribution–routing prob-

lem. Journal of Scheduling, 12(3):257–280, 2009b.

J.F. Bard and N. Nananukul. A branch-and-price algorithm for an integrated production and

inventory routing problem. Computers & Operations Research, 37(12):2202–2217, 2010.

T. Bektaş and G. Laporte. The pollution-routing problem. Transportation Research Part B:

Methodological, 45(8):1232–1250, 2011.

W.J. Bell, L.M. Dalberto, M.L. Fisher, A.J. Greenfield, R. Jaikumar, P. Kedia, Robert G.

Mack, and P.J. Prutzman. Improving the distribution of industrial gases with an on-line

computerized routing and scheduling optimizer. Interfaces, 13(6):4–23, 1983.

M.A.F. Belo-Filho, P. Amorim, and B. Almada-Lobo. An adaptive large neighbourhood search

for the operational integrated production and distribution problem of perishable products.

International Journal of Production Research, 53(20):6040–6058, 2015.

S. Benjaafar, Y. Li, and M. Daskin. Carbon footprint and the management of supply chains:

Insights from simple models. IEEE Transactions on Automation Science and Engineering,

10(1):99–116, 2013.

103

L. Bertazzi and M.G. Speranza. Inventory routing problems: An introduction. EURO Journal

on Transportation and Logistics, 1(4):307–326, 2012.

L. Bertazzi and M.G. Speranza. Inventory routing problems with multiple customers. EURO

Journal on Transportation and Logistics, 2(3):255–275, 2013.

L. Bertazzi, G. Paletta, and M.G. Speranza. Deterministic order-up-to level policies in an

inventory routing problem. Transportation Science, 36(1):119–132, 2002.

M. Boudia and C. Prins. A memetic algorithm with dynamic population management for

an integrated production-distribution problem. European Journal of Operational Research,

195(3):703–715, 2009.

M. Boudia, M.A.O. Louly, and C. Prins. A reactive GRASP and path relinking for a combined

production-distribution problem. Computers & Operations Research, 34(11):3402–3419,

2007.

M. Boudia, M.A.O. Louly, and C. Prins. Fast heuristics for a combined production planning

and vehicle routing problem. Production Planning and Control, 19(2):85–96, 2008.

E. Boventer. The relationship between transportation costs and location rent in transportation

problems. Journal of Regional Science, 3(2):27–40, 1961.

N. Brahimi and T. Aouam. Multi-item production routing problem with backordering: a

MILP approach. International Journal of Production Research, 54(4):1076–1093, 2016.

N. Brahimi, S. Dauzère-Pérès, and N.M. Najid. Capacitated multi-item lot-sizing problems

with time windows. Operations Research, 54(5):951–967, 2006a.

N. Brahimi, S. Dauzère-Pérès, N.M. Najid, and A. Nordli. Single item lot sizing problems.

European Journal of Operational Research, 168(1):1–16, 2006b.

M.L. Brandeau and S.S. Chiu. An overview of representative problems in location research.

Management Science, 35(6):645–674, 1989.

G. Brown, J. Keegan, B. Vigus, and K. Wood. The Kellogg company optimizes production,

inventory, and distribution. Interfaces, 31(6):1–15, 2001.

G. Bruno, A. Genovese, and C. Piccolo. The capacitated lot sizing model: A powerful tool

for logistics decision making. International Journal of Production Economics, 155:380–390,

2014.

104

L. Buschkühl, F. Sahling, S. Helber, and H. Tempelmeier. Dynamic capacitated lot-sizing

problems: a classification and review of solution approaches. OR Spectrum, 32(2):231–261,

2010.

J.-F. Camacho-Vallejo, R. Muñoz-Sánchez, and J.L. González-Velarde. A heuristic algorithm

for a supply chain’s production-distribution planning. Computers & Operations Research,

61:110–121, 2015.

A. Campbell, L. Clarke, A. Kleywegt, and M.W.P. Savelsbergh. The inventory routing prob-

lem. In T.G. Crainic, editor, Fleet Management and Logistics, pages 95–113. Springer,

1998.

L.E. Cárdenas-Barrón, J.L. González-Velarde, and G. Treviño-Garza. A new approach to

solve the multi-product multi-period inventory lot sizing with supplier selection problem.

Computers & Operations Research, 64:225–232, 2015.

C.R. Carter and D.S. Rogers. A framework of sustainable supply chain management: moving

toward new theory. International Journal of Physical Distribution & Logistics Management,

38(5):360–387, 2008.

S. Çetinkaya, H. Üster, G. Easwaran, and B.B. Keskin. An integrated outbound logistics

model for Frito-Lay: Coordinating aggregate-level production and distribution decisions.

Interfaces, 39(5):460–475, 2009.

P. Chandra. A dynamic distribution model with warehouse and customer replenishment re-

quirements. Journal of the Operational Research Society, 44(7):681–692, 1993.

P. Chandra and M.L. Fisher. Coordination of production and distribution planning. European

Journal of Operational Research, 72(3):503–517, 1994.

H. Chen. Fix-and-optimize and variable neighborhood search approaches for multi-level ca-

pacitated lot sizing problems. Omega, 56:25–36, 2015.

H.-D. Chen, D.W. Hearn, and C.-Y. Lee. A new dynamic programming algorithm for the

single item capacitated dynamic lot size model. Journal of Global Optimization, 4(3):285–

300, 1994.

Z.-L. Chen. Integrated production and distribution operations. In D. Simchi-Levi, S. D. Wu,

and Z.-J. Shen, editors, Handbook of Quantitative Supply Chain Analysis, pages 711–745.

Springer US, Boston, MA, 2004.

Z.-L. Chen. Integrated production and outbound distribution scheduling: review and exten-

sions. Operations Research, 58(1):130–148, 2010.

105

M. Christopher. Logistics & Supply Chain Management. Financial Times/Pearson Education,

Harlow, 2016.

M. Christopher and D.R. Towill. An integrated model for the design of agile supply chains.

International Journal of Physical Distribution & Logistics Management, 31(4):235–246,

2001.

G.D.H. Claassen, J.C. Gerdessen, E.M.T. Hendrix, and J.G.A.J. van der Vorst. On production

planning and scheduling in food processing industry: Modelling non-triangular setups and

product decay. Computers & Operations Research, 76:147–154, 2016.

A. Clark, B. Almada-Lobo, and C. Almeder. Lot sizing and scheduling: industrial extensions

and research opportunities. International Journal of Production Research, 49(9):2457–2461,

2011.

L.C. Coelho and G. Laporte. A branch-and-cut algorithm for the multi-product multi-vehicle

inventory-routing problem. International Journal of Production Research, 51(23-24):7156–

7169, 2013a.

L.C. Coelho and G. Laporte. The exact solution of several classes of inventory-routing prob-

lems. Computers & Operations Research, 40(2):558–565, 2013b.

L.C. Coelho and G. Laporte. Improved solutions for inventory-routing problems through valid

inequalities and input ordering. International Journal of Production Economics, 155:391–

397, 2014.

L.C. Coelho, J.-F. Cordeau, and G. Laporte. The inventory-routing problem with transship-

ment. Computers & Operations Research, 39(11):2537–2548, 2012.

L.C. Coelho, J.-F. Cordeau, and G. Laporte. Thirty years of inventory routing. Transportation

Science, 48(1):1–19, 2014.

L.C. Coelho, J. Renaud, and G. Laporte. Road-based goods transportation: a survey of

real-world logistics applications from 2000 to 2015. INFOR: Information Systems and

Operational Research, 54(2):79–96, 2016.

M.A. Cohen and H.L. Lee. Strategic analysis of integrated production-distribution systems:

Models and methods. Operations Research, 36(2):216–228, 1988.

M.C. Cooper, D.M. Lambert, and J.D. Pagh. Supply chain management: more than a new

name for logistics. The International Journal of Logistics Management, 8(1):1–14, 1997.

106

J.-F. Cordeau, M. Gendreau, and G. Laporte. A tabu search heuristic for periodic and multi-

depot vehicle routing problems. Networks, 30(2):105–119, 1997.

I. Correia, T. Melo, and F. Saldanha da Gama. Comparing classical performance measures

for a multi-period, two-echelon supply chain network design problem with sizing decisions.

Computers & Industrial Engineering, 64(1):366–380, 2013.

R. Cuda, G. Guastaroba, and M.G. Speranza. A survey on two-echelon routing problems.

Computers & Operations Research, 55:185–199, 2015.

J.O. Cunha and R.A. Melo. On reformulations for the one-warehouse multi-retailer problem.

Annals of Operations Research, 238(1-2):99–122, 2016.

S. Dabia, E. Demir, and T. Van Woensel. An exact approach for a variant of the pollution-

routing problem. Transportation Science, 51(2):607–628, 2017.

M. Darvish, H. Larrain, and L.C. Coelho. A dynamic multi-plant lot-sizing and distribution

problem. International Journal of Production Research, 54(22):6707–6717, 2016.

M.S. Daskin, C.R. Coullard, and Z.-J.M. Shen. An inventory-location model: Formulation,

solution algorithm and computational results. Annals of Operations Research, 110(1–4):

83–106, 2002.

R.E. De Matta, V.N. Hsu, and C.-L. Li. Coordinated production and delivery for an exporter.

IIE Transactions, 47(4):373–391, 2015.

A. Degbotse, B.T. Denton, K. Fordyce, R.J. Milne, R. Orzell, and C.T. Wang. IBM blends

heuristics and optimization to plan its semiconductor supply chain. Interfaces, 43(2):130–

141, 2013.

E. Demir, T. Bektaş, and G. Laporte. A comparative analysis of several vehicle emission

models for road freight transportation. Transportation Research Part D: Transport and

Environment, 16(5):347–357, 2011.

E. Demir, T. Bektaş, and G. Laporte. A review of recent research on green road freight

transportation. European Journal of Operational Research, 237(3):775–793, 2014.

G. Desaulniers, J.G. Rakke, and L.C. Coelho. A branch-price-and-cut algorithm for the

inventory-routing problem. Transportation Science, 50(3):1060–1076, 2015.

M. Desrochers, J. Desrosiers, and M. Solomon. A new optimization algorithm for the vehicle

routing problem with time windows. Operations Research, 40(2):342–354, 1992.

107

M. Díaz-Madroñero, D. Peidro, and J. Mula. A review of tactical optimization models for

integrated production and transport routing planning decisions. Computers & Industrial

Engineering, 88:518–535, 2015.

M. Drexl and M. Schneider. A survey of variants and extensions of the location-routing

problem. European Journal of Operational Research, 241(2):283–308, 2015.

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

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

S.D. Ekşioğlu, H.E. Romeijn, and P.M. Pardalos. Cross-facility management of production

and transportation planning problem. Computers & Operations Research, 33(11):3231–

3251, 2006.

S.D. Ekşioğlu, B. Ekşioğlu, and H.E. Romeijn. A lagrangean heuristic for integrated production

and transportation planning problems in a dynamic, multi-item, two-layer supply chain. IIE

Transactions, 39(2):191–201, 2007.

S. Elhedhli and J.-L. Goffin. Efficient production-distribution system design. Management

Science, 51(7):1151–1164, 2005.

L.M. Ellram. Supply-chain management: the industrial organisation perspective. International

Journal of Physical Distribution & Logistics Management, 21(1):13–22, 1991.

S.J. Erlebacher and R.D. Meller. The interaction of location and inventory in designing

distribution systems. IIE Transactions, 32(2):155–166, 2000.

D. Erlenkotter. A dual-based procedure for uncapacitated facility location. Operations

Research, 26(6):992–1009, 1978.

B. Fahimnia, R.Z. Farahani, R. Marian, and L. Luong. A review and critique on inte-

grated production–distribution planning models and techniques. Journal of Manufacturing

Systems, 32(1):1–19, 2013.

R.Z. Farahani, N. Asgari, N. Heidari, M. Hosseininia, and M. Goh. Covering problems in

facility location: A review. Computers & Industrial Engineering, 62(1):368–407, 2012.

R.Z. Farahani, M. Hekmatfar, B. Fahimnia, and N. Kazemzadeh. Hierarchical facility location

problem: Models, classifications, techniques, and applications. Computers & Industrial

Engineering, 68:104–117, 2014.

R.Z. Farahani, H. Rashidi Bajgan, B. Fahimnia, and M. Kaviani. Location-inventory problem

in supply chains: a modelling review. International Journal of Production Research, 53(12):

3769–3788, 2015.

108

M. Florian, J.K. Lenstra, and A.H.G. Rinnooy Kan. Deterministic production planning:

Algorithms and complexity. Management Science, 26(7):669–679, 1980.

A. Franceschetti, E. Demir, D. Honhon, T. Van Woensel, G. Laporte, and M. Stobbe. A

metaheuristic for the time-dependent pollution-routing problem. European Journal of

Operational Research, 259(3):972–991, 2017.

R.L. Francis, L.F. McGinnis, and J.A. White. Locational analysis. European Journal of

Operational Research, 12(3):220–252, 1983.

F. Fumero and C. Vercellis. Synchronized development of production, inventory and distribu-

tion schedules. Transportation Science, 33(3):330–340, 1999.

W.M. Garvin, H.W. Crandall, J.B. John, and R.A. Spellman. Applications of linear program-

ming in the oil industry. Management Science, 3(4):407–430, 1957.

M. Gendreau, A. Hertz, and G. Laporte. A tabu search heuristic for the vehicle routing

problem. Management Science, 40(10):1276–1290, 1994.

B. Gendron and T.G. Crainic. Relaxations for multicommodity capacitated network design

problems. Technical Report CRT-965, Centre de Recherche sur les Transports, Montreal,

1994.

C.H. Glock, E.H. Grosse, and J. Ries. The lot sizing problem: A tertiary study. International

Journal of Production Economics, 155:39–51, 2014.

H.G. Goren, S. Tunali, and R. Jans. A hybrid approach for the capacitated lot sizing problem

with setup carryover. International Journal of Production Research, 50(6):1582–1597, 2012.

G. Guastaroba, M.G. Speranza, and D. Vigo. Intermediate facilities in freight transportation

planning: A survey. Transportation Science, 50(3):763–789, 2016.

F. Hein and C. Almeder. Quantitative insights into the integrated supply vehicle routing and

production planning problem. International Journal of Production Economics, 177:66–76,

2016.

M.J.R. Helmrich, R. Jans, W. van den Heuvel, and A.P.M. Wagelmans. The economic lot-

sizing problem with an emission capacity constraint. European Journal of Operational

Research, 241(1):50–62, 2015.

M.H. Hugos. Essentials of supply chain management. John Wiley & Sons, 2011.

H.-C. Hwang. Economic lot-sizing for integrated production and transportation. Operations

Research, 58(2):428–444, 2010.

109

H.-C. Hwang and J. Kang. Two-phase algorithm for the lot-sizing problem with backlogging

for stepwise transportation cost without speculative motives. Omega, 59:238–250, 2016.

H.-C. Hwang, H.-S. Ahn, and P. Kaminsky. Algorithms for the two-stage production-

capacitated lot-sizing problem. Journal of Global Optimization, 65(4):777–799, 2016.

O. Jabali, T. Van Woensel, and A.G. de Kok. Analysis of travel times and CO2 emissions in

time-dependent vehicle routing. Production and Operations Management, 21(6):1060–1074,

2012.

R. Jans and Z. Degraeve. Meta-heuristics for dynamic lot sizing: A review and comparison of

solution approaches. European Journal of Operational Research, 177(3):1855–1875, 2007.

R. Jans and Z. Degraeve. Modeling industrial lot sizing problems: a review. International

Journal of Production Research, 46(6):1619–1643, 2008.

V. Jayaraman and H. Pirkul. Planning and coordination of production and distribution facil-

ities for multiple commodities. European Journal of Operational Research, 133(2):394–408,

2001.

S.D. Jena, J.-F. Cordeau, and B. Gendron. Modeling and solving a logging camp location

problem. Annals of Operations Research, 232(1):151–177, 2015a.

S.D. Jena, J.-F. Cordeau, and B. Gendron. Dynamic facility location with generalized modular

capacities. Transportation Science, 49(3):484–499, 2015b.

J.K. Jolayemi and F.O. Olorunniwo. A deterministic model for planning production quantities

in a multi-plant, multi-warehouse environment with extensible capacities. International

Journal of Production Economics, 87(2):99–113, 2004.

İ. Kara, B.Y. Kara, and M.K. Yetis. Energy minimizing vehicle routing problem. In A. Dress,

Y. Xu, and B. Zhu, editors, Combinatorial Optimization and Applications, volume 4616 of

Lecture Notes in Computer Science,. Springer, Berlin/Heidelberg, 2007.

İ. Karaoğlan and S.E. Kesen. The coordinated production and transportation scheduling

problem with a time-sensitive product: a branch-and-cut algorithm. International Journal

of Production Research, 55(2):536–557, 2017.

B. Karimi, S.M.T.F. Ghomi, and J.M. Wilson. The capacitated lot sizing problem: a review

of models and algorithms. Omega, 31(5):365–378, 2003.

J.P. Katz, M.D. Pagell, and J.M. Bloodgood. Strategies of supply communities. Supply Chain

Management: An International Journal, 8(4):291–302, 2003.

110

A. Klose and A. Drexl. Facility location models for distribution system design. European

Journal of Operational Research, 162(1):4–29, 2005.

Ç. Koç, T. Bektaş, O. Jabali, and G. Laporte. Thirty years of heterogeneous vehicle routing.

European Journal of Operational Research, 249(1):1–29, 2016a.

Ç. Koç, T. Bektaş, O. Jabali, and G. Laporte. The fleet size and mix location-routing prob-

lem with time windows: Formulations and a heuristic algorithm. European Journal of

Operational Research, 248(1):33–51, 2016b.

H.W. Kopfer, J. Schönberger, and H. Kopfer. Reducing greenhouse gas emissions of a hetero-

geneous vehicle fleet. Flexible Services and Manufacturing Journal, 26(1-2):221, 2014.

R.S. Kumar, K. Kondapaneni, V. Dixit, A. Goswami, L.S. Thakur, and M.K. Tiwari. Multi-

objective modeling of production and pollution routing problem with time window: A

self-learning particle swarm optimization approach. Computers & Industrial Engineering,

99:29–40, 2016.

R. Lahyani, L.C. Coelho, M. Khemakhem, G. Laporte, and F. Semet. A multi-compartment

vehicle routing problem arising in the collection of olive oil in Tunisia. Omega, 51(1):1–10,

2015a.

R. Lahyani, L.C. Coelho, and J. Renaud. Alternative formulations and improved bounds for

the multi-depot fleet size and mix vehicle routing. Technical Report 2015-36, CIRRELT,

Québec, Canada, 2015b.

R. Lahyani, M. Khemakhem, and F. Semet. Rich vehicle routing problems: From a taxonomy

to a definition. European Journal of Operational Research, 241(1):1–14, 2015c.

G. Laporte. The vehicle routing problem: An overview of exact and approximate algorithms.

European Journal of Operational Research, 59(3):345–358, 1992.

G. Laporte. Fifty years of vehicle routing. Transportation Science, 43(4):408–416, 2009.

C.-Y. Lee, S. Çetinkaya, and A.P.M. Wagelmans. A dynamic lot-sizing model with demand

time windows. Management Science, 47(10):1384–1395, 2001.

L. Lei, S. Liu, A.J. Ruszczynski, and S. Park. On the integrated production, inventory, and

distribution routing problem. IIE Transactions, 38(11):955–970, 2006.

Y. Li, Y. Tao, and F. Wang. An effective approach to multi-item capacitated dynamic lot-

sizing problems. International Journal of Production Research, 50(19):5348–5362, 2012.

111

Z. Liang, Y. He, T. Wu, and C. Zhang. An informative column generation and decomposition

method for a production planning and facility location problem. International Journal of

Production Economics, 170:88–96, 2015.

J. Maes, J.O. McClain, and L.N. Van Wassenhove. Multilevel capacitated lotsizing complexity

and LP-based heuristics. European Journal of Operational Research, 53(2):131–148, 1991.

R.P. Manatkar, K. Karthik, S.K. Kumar, and M.K. Tiwari. An integrated inventory opti-

mization model for facility location-allocation problem. International Journal of Production

Research, 54(12):3640–3658, 2016.

A.C. Mckinnon and E. Hr. Synchronised auditing of truck utilisation and energy efficiency: a

review of the British Government’s transport KPI programme. In Proceedings of the 11 th

World Conference on Transport Research, 2007.

C. Meinecke and B. Scholz-Reiter. A representation scheme for integrated production and

outbound distribution models. International Journal of Logistics Systems and Management,

18(3):283–301, 2014.

F. Meisel, T. Kirschstein, and C. Bierwirth. Integrated production and intermodal transporta-

tion planning in large scale production–distribution-networks. Transportation Research Part

E: Logistics and Transportation Review, 60:62–78, 2013.

M.T. Melo, S. Nickel, and F. Saldanha da Gama. Dynamic multi-commodity capacitated

facility location: a mathematical modeling framework for strategic supply chain planning.

Computers & Operations Research, 33(1):181–208, 2006.

M.T. Melo, S. Nickel, and F. Saldanha da Gama. Facility location and supply chain

management–a review. European Journal of Operational Research, 196(2):401–412, 2009.

R.A. Melo and L.A. Wolsey. MIP formulations and heuristics for two-level production-

transportation problems. Computers & Operations Research, 39(11):2776–2786, 2012.

N.H. Moin and S. Salhi. Inventory routing problems: a logistical overview. Journal of the

Operational Research Society, 58(9):1185–1194, 2007.

B. Montreuil. Toward a Physical Internet: meeting the global logistics sustainability grand

challenge. Logistics Research, 3(2-3):71–87, 2011.

B. Montreuil, R.D. Meller, and E. Ballot. Physical internet foundations. In T. Borangiu,

A. Thomas, and D. Trentesaux, editors, Service Orientation in Holonic and Multi Agent

Manufacturing and Robotics, pages 151–166. Springer, Berlin/Heidelberg, 2013.

112

J. Mula, D. Peidro, M. Díaz-Madroñero, and E. Vicens. Mathematical programming models

for supply chain production and transport planning. European Journal of Operational

Research, 204(3):377–390, 2010.

M. Mustafa Kamal and Z. Irani. Analysing supply chain integration through a systematic

literature review: a normative perspective. Supply Chain Management: An International

Journal, 19(5/6):523–557, 2014.

G. Nagy and S. Salhi. Location-routing: Issues, models and methods. European Journal of

Operational Research, 177(2):649–672, 2007.

A.M. Nezhad, H. Manzour, and S. Salhi. Lagrangian relaxation heuristics for the uncapacitated

single-source multi-product facility location problem. International Journal of Production

Economics, 145(2):713–723, 2013.

S.H. Owen and M.S. Daskin. Strategic facility location: A review. European Journal of

Operational Research, 111(3):423–447, 1998.

Y.B. Park. An integrated approach for production and distribution planning in supply chain

management. International Journal of Production Research, 43(6):1205–1224, 2005.

M.A. Pereira, L.C. Coelho, L.A.N. Lorena, and L.C. De Souza. A hybrid method for the prob-

abilistic maximal covering location-allocation problem. Computers & Operations Research,

57:51–59, 2015.

V. Pillac, M. Gendreau, C. Guéret, and A.L. Medaglia. A review of dynamic vehicle routing

problems. European Journal of Operational Research, 225(1):1–11, 2013.

H. Pirkul and V. Jayaraman. Production, transportation, and distribution planning in a

multi-commodity tri-echelon system. Transportation Science, 30(4):291–302, 1996.

J. Pooley. Integrated production and distribution facility planning at Ault Foods. Interfaces,

24(4):113–121, 1994.

D. Power. Supply chain management integration and implementation: a literature review.

Supply Chain Management: An International Journal, 10(4):252–263, 2005.

C. Prodhon and C. Prins. A survey of recent research on location-routing problems. European

Journal of Operational Research, 238(1):1–17, 2014.

H.N. Psaraftis, M. Wen, and C.A. Kontovas. Dynamic vehicle routing problems: Three decades

and counting. Networks, 67(1):3–31, 2016.

113

D.F. Pyke and M.A. Cohen. Multiproduct integrated production—distribution systems.

European Journal of Operational Research, 74(1):18–49, 1994.

Y. Qiu, J. Qiao, and P.M. Pardalos. A branch-and-price algorithm for production routing

problems with carbon cap-and-trade. Omega, 68:49–61, 2017.

J. Renaud, G. Laporte, and F.F. Boctor. A tabu search heuristic for the multi-depot vehicle

routing problem. Computers & Operations Research, 23(3):229–235, 1996.

C.S. ReVelle and G. Laporte. The plant location problem: new models and research prospects.

Operations Research, 44(6):864–874, 1996.

P.L. Robinson, A. Narayanan, and F. Sahin. Coordinated deterministic dynamic demand

lot-sizing problem: A review of models and algorithms. Omega, 37(1):3–15, 2009.

H.E. Romeijn, T.C. Sharkey, Z.-J. M. Shen, and J. Zhang. Integrating facility location and

production planning decisions. Networks, 55(2):78–89, 2010.

S. Ropke and D. Pisinger. An adaptive large neighborghood search heuristic for the pickup

and delivery problem with time windows. Transportation Science, 40(4):455–472, 2006.

M. Ruokokoski, O. Solyalı, J.-F. Cordeau, R. Jans, and H. Süral. Efficient formulations and

a branch-and-cut algorithm for a production-routing problem. Les Cahiers du GERAD

G–2010–66, Tech. Rep., 2010.

G. Şahin and H. Süral. A review of hierarchical facility location models. Computers &

Operations Research, 34(8):2310–2331, 2007.

F. Sahling, L. Buschkühl, H. Tempelmeier, and S. Helber. Solving a multi-level capaci-

tated lot sizing problem with multi-period setup carry-over via a fix-and-optimize heuristic.

Computers & Operations Research, 36(9):2546–2553, 2009.

S. Salhi, A. Imran, and N.A. Wassan. The multi-depot vehicle routing problem with het-

erogeneous vehicle fleet: Formulation and a variable neighborhood search implementation.

Computers & Operations Research, 52:315–325, 2014.

M. Sambasivan and C.P. Schmidt. A heuristic procedure for solving multi-plant, mutli-item,

mutli-period capacitated lot-sizing problems. Asia-Pacific Journal of Operational Research,

19(1):87–105, 2002.

M. Sambasivan and S. Yahya. A Lagrangean-based heuristic for multi-plant, multi-item, multi-

period capacitated lot-sizing problems with inter-plant transfers. Computers & Operations

Research, 32(3):537–555, 2005.

114

A.M. Sarmiento and R. Nagi. A review of integrated analysis of production–distribution

systems. IIE Transactions, 31(11):1061–1074, 1999.

R. Sarraj, E. Ballot, S. Pan, D. Hakimi, and B. Montreuil. Interconnected logistic networks

and protocols: simulation-based efficiency assessment. International Journal of Production

Research, 52(11):3185–3208, 2014.

M.W.P. Savelsbergh and T. Van Woensel. City logistics: Challenges and opportunities.

Transportation Science, 50(2):579–590, 2016.

T.C. Sharkey, J. Geunes, Edwin R.H., and Z.-J.M. Shen. Exact algorithms for integrated

facility location and production planning problems. Naval Research Logistics, 58(5):419–

436, 2011.

D.X. Shaw and A.P.M. Wagelmans. An algorithm for single-item capacitated economic lot

sizing with piecewise linear production costs and general holding costs. Management Science,

44(6):831–838, 1998.

Z.-J.M. Shen, C. Coullard, and M.S. Daskin. A joint location-inventory model. Transportation

Science, 37(1):40–55, 2003.

L.V. Snyder. Facility location under uncertainty: a review. IIE Transactions, 38(7):547–564,

2006.

O. Solyalı and H. Süral. A multi-phase heuristic for the production routing problem.

Computers & Operations Research, 87:114 – 124, 2017.

M. Soysal, J.M. Bloemhof-Ruwaard, R. Haijema, and J.G.A.J. van der Vorst. Modeling an

inventory routing problem for perishable products with environmental considerations and

demand uncertainty. International Journal of Production Economics, 164:118–133, 2015.

M. Soysal, J.M. Bloemhof-Ruwaard, R. Haijema, and J.G.A.J. van der Vorst. Modeling a green

inventory routing problem for perishable products with horizontal collaboration. Computers

& Operations Research, 2016.

M.G. Speranza. Trends in transportation and logistics. European Journal of Operational

Research, 264(3):830–836, 2018.

H. Stadtler. Supply chain management and advanced planning–basics, overview and chal-

lenges. European Journal of Operational Research, 163(3):575–588, 2005.

M. Steinrücke. Integrated production, distribution and scheduling in the aluminium indus-

try: a continuous-time MILP model and decomposition method. International Journal of

Production Research, 53(19):5912–5930, 2015.

115

G.C. Stevens. Integrating the supply chain. International Journal of Physical Distribution &

Materials Management, 19(8):3–8, 1989.

Y. Suzuki. A new truck-routing approach for reducing fuel consumption and pollutants emis-

sion. Transportation Research Part D: Transport and Environment, 16(1):73–77, 2011.

É. Taillard, P. Badeau, M. Gendreau, F. Guertin, and J.-Y. Potvin. A tabu search heuristic for

the vehicle routing problem with soft time windows. Transportation Science, 31(2):170–186,

1997.

P. Toth and D. Vigo, editors. Vehicle Routing. Monographs on Discrete Mathematics and

Applications. MOS-SIAM Series on Optimization, Philadelphia, 2014.

S. van Hoesel, H.E. Romeijn, D.R. Morales, and A.P.M. Wagelmans. Integrated lot sizing

in serial supply chains with production capacities. Management Science, 51(11):1706–1719,

2005.

T. Vogel, B. Almada-Lobo, and C. Almeder. Integrated versus hierarchical approach to aggre-

gate production planning and master production scheduling. OR Spectrum, 39(1):193–229,

2017.

H.M. Wagner and T.M. Whitin. Dynamic version of the economic lot size model. Management

Science, 5(1):89–96, 1958.

Y. Xiao, Q. Zhao, I. Kaku, and Y. Xu. Development of a fuel consumption optimization

model for the capacitated vehicle routing problem. Computers & Operations Research, 39

(7):1419–1431, 2012.

116