Centralized auctions for the procurement of full truckload transportation services : impacts on carriers and shippers

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Centralized auctions for the procurement of full

truckload transportation services: impacts on carriers

and shippers

Thèse

Intissar Ben Othmane

Doctorat en sciences de l'administration

Philosophiæ doctor (Ph. D.)

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Centralized auctions for the procurement of full

truckload transportation services: impacts on

carriers and shippers

Thèse

Intissar Ben Othmane

Sous la direction de:

Sehl Mellouli, directeur de recherche Monia Rekik, codirectrice de recherche

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Résumé

Cette thèse porte sur l’approvisionnement des services de transport routier à charge pleine en utilisant le mécanisme d’enchères combinatoires. Traditionnellement, chaque expéditeur roule une enchère séparée pour satisfaire ses besoins internes de transport indépendamment des autres enchères roulées par d’autres expéditeurs. Étant donné que le processus de négociation pour l’établissement de contrats entre les expéditeurs et les transporteurs prend un à deux ans, les transporteurs sont amenés, dans certains cas, à soumettre des mises dans plus d’une enchère en même temps sans connaitre les résultats des autres enchères et savoir laquelle des offres qu’ils ont déjà soumises est effectivement gagnée. Pour contourner les inconvénients d’un tel mécanisme, un nouveau mécanisme d’approvisionnement des services de transport basé sur des enchères est proposé. Il offre aux transporteurs la possibilité de soumettre simultanément sur les contrats requis par différents expéditeurs dans une seule enchère combinatoire où les demandes des expéditeurs sont centralisées. Il s’agit d’une vente aux enchères unique dans laquelle plusieurs expéditeurs présentent dans une seule enchère leurs services de transport qu’ils veulent sous-traiter et les transporteurs soumettent des mises combinatoires couvrant un ensemble de contrats appartenant à différents expéditeurs. Cette recherche examine l’impact d’un tel mécanisme centralisé sur les transporteurs ainsi que sur les expéditeurs en le comparant au mécanisme décentralisé traditionnel et en tenant en considération différents comportements à risque des transporteurs.

La première partie de cette thèse présente une heuristique utilisée par les transporteurs pour construire des offres combinatoires dans les enchères dédiées à l’approvisionnement des ser-vices de transport routier à charge pleine. L’heurisque peut être utilisée par les transporteurs dans les enchères centralisées et décentralisées à la fois. On suppose que les transporteurs ont déjà des engagements avec d’autres expéditeurs pour servir leurs contrats avant de par-ticiper à l’enchère et que les routes qui sont déjà définies pour les contrats existants doivent être conservées. L’heuristique proposée identifie d’abord les nouveaux contrats rentables et les intègre efficacement dans les routes existantes des transporteurs. Ensuite, elle construit de nouvelles routes pour les véhicules inutilisés avec les nouveaux contrats restants, quand cela est rentable. L’heuristique offre au transporteur la possibilité de soumettre des mises simples ou combinatoires avec un intervalle de prix de vente rentable pour chaque mise. L’heuristique se révèle rapide et efficace: elle exploite bien le réseau existant des transporteurs, génère des

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offres intéressantes permettant un profit additionnel considérable et couvre un pourcentage important de nouveaux contrats. Comparée à une méthode de solution exacte pour le pro-blème de construction de mises, les résultats prouvent que l’heuristique identifie des solutions optimales ou quasi-optimales pour des instances de petite taille. Pour les grandes instances, la solution exacte soit elle identifie une solution pire que celle produite par notre heuristique ou échoue à identifier une solution réalisable.

Dans la deuxième partie de la thèse, nous étudions les avantages/inconvénients du mécanisme d’enchère centralisée sur les transporteurs, en le comparant aux enchères décentralisées consi-dérant différents comportements à risque des transporteurs. Lors de la construction des mises, les transporteurs peuvent miser en ayant un comportement trop risqué ou le contraire averse aux risques. Les résultats montrent que du point de vue du transporteur, les enchères cen-tralisées offrent le meilleur compromis entre profit, efficacité du réseau et diversification du marché.

Dans la troisième partie de la thèse, les avantages/inconvénients de l’enchère centralisée sur les expéditeurs sont étudiés en comparant les enchères centralisées aux décentralisées et en tenant compte des différents comportements à risque des transporteurs au moment de la construction des mises. Les résultats obtenus prouvent que du point de vue de l’expéditeur, les enchères centralisées amènent les expéditeurs à réaliser des économies de coûts considérables réalisées principalement grâce aux bas prix des mises proposées dans les enchères centralisées en les comparant avec les enchères décentralisées.

La dernière partie de la thèse modélise et résout le problème de détermination des mises gagnantes (DMG) basé sur la réputation pour les enchères de transport combinatoires et cen-tralisées. Dans ces enchères, le commissaire-priseur doit décider des mises gagnantes sachant qu’une même mise peut inclure des contrats de différents expéditeurs et que la réputation d’un transporteur peut différer d’un expéditeur à un autre. L’objectif est de trouver le meilleur compromis entre les prix offerts et le niveau de service des transporteurs. Le modèle DMG centralisé basé sur la réputation est résolu à l’optimalité et les résultats montrent que la prise en compte de la réputation dans un DMG centralisé peut entraîner des économies considé-rables sur les coûts totaux payés par tous les expéditeurs. De plus, les expéditeurs peuvent faire affaire avec des transporteurs moins privilégiés pour eux tout en réajustant leurs propres évaluations via cette collaboration.

Cette thèse est organisée comme suit. Après un chapitre d’introduction générale, nous présen-tons une approche heuristique pour le problème de construction des mises d’enchères combina-toires. Après cela, nous présentons dans les chapitres 3 et 4 ce que nous appelons les enchères centralisées et nous les comparons aux enchères décentralisées du côté des transporteurs ainsi que des expéditeurs. Le cinquième chapitre résout le modèle de problème de détermination des mises gagnantes basé sur la réputation pour les enchères de transport combinatoire

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cen-tralisé et la conclusion et les orientations pour les travaux futurs sont présentés dans le dernier chapitre.

Mots clés: enchères combinatoires, services de transport routier, construction de mises, heu-ristique, approvisionnement centralisé et décentralisé, comportement à risque du transporteur, détermination du gagnant, réputation du transporteur.

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Abstract

This thesis addresses the procurement of truckload services through combinatorial auctions. Traditionally, each shipper runs a separate auction to satisfy its proper transportation needs independently from other shippers in decentralized auctions. Since the negotiation process for establishing contracts between shippers and carriers takes one to two years, carriers, in some cases, are led to bid in more than one auction at the same time before knowing which of the bids they submitted are effectively won. To circumvent the inconvenient of such mechanism, a novel auction-based procurement mechanism is proposed. It offers to carriers the opportunity to bid on the contracts required by different shippers simultaneously in a single combinatorial auction where shippers’ requests are centralized. That is a single auction where multiple shippers present their transportation requests simultaneously, and carriers submit package bids covering a set of contracts belonging to different shippers. The research study investigates the impact of such a centralized mechanism on carriers as well as on shippers by comparing it to the traditional decentralized mechanism under different carriers’ risk behaviours.

The rst part of this thesis presents a heuristic approach used by carriers in centralized and decentralized auctions for constructing combinatorial bids in transportation auctions for the procurement of truckload transportation services. It assumes that carriers have already en-gaged on a set of transportation contracts before participating to the auction, and the routes already defined for the existing contracts must be kept. The proposed heuristic first identifies profitable new contracts and efficiently integrates them into a carrier’s existing routes. Then it builds new routes for unused vehicles with the remaining new contracts, when it is prof-itable. The heuristic offers the carrier the possibility to submit either a single bid or multiple bids with an interval of profitable ask prices for each bid. The heuristic is shown to be fast and efficient: It exploits well the carriers existing network, generates interesting bids enabling an additional potential profit and covers a large percentage of new contracts. Even when compared with an exact solution method, results prove that the heuristic identifies optimal or near-optimal solutions for small-sized instances. However, for larger instances, the exact method either identifies a solution that is worse than that output by our heuristic or fails in identifying a feasible one.

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auction on carriers by comparing it to decentralized auctions under different carriers risk-behaviour attitudes. When bidding, two types of carriers are considered: too risky carriers and averse-to-risk carriers. Results show that from the carriers perspective, centralized auctions offer the best compromise between profit, network efficiency and business diversification. In the third part of the thesis, the benefits/drawbacks of centralized auctions on shippers are studied by comparing them to decentralized auctions and considering different carrier’s risk-behaviour attitudes when bidding. Obtained results prove that from shippers perspectives, centralized auctions lead shippers to realize considerable cost savings achieved mainly due to attractive bids prices offered in centralized auctions.

The last part of the thesis considers a reputation-based winner determination problem (WDP) model for centralized combinatorial transportation auctions in which the auctioneer should decide on the winning bids knowing that a same bid may include shipping contracts requested by different shippers and a carrier reputation may differ from one shipper to another. The objective is to find the best trade-off between bid prices and carriers level of service. The centralized reputation-based WDP model is solved to optimality and results show that con-sidering reputation in a centralized WDP may lead to considerable savings in total costs paid by all shippers. Also, shippers may deal with less-preferred carriers while readjusting their own evaluations.

This thesis is organized as follows. After a general introduction chapter, we present a heuristic approach for combinatorial bid construction problem. After that, we present in chapters 3 and 4 what we call centralized auctions, and we compare them to decentralized auctions from the perspective of carriers and shippers. The fifth chapter solves the reputation-based winner determination problem model for centralized combinatorial transportation auctions and the conclusion and directions for future work are presented in the last chapter.

Keywords: combinatorial auctions, truckload transportation services, bid construction, heuris-tic, centralized and decentralized procurement, carrier’s risk behaviour, winner determination, carrier Reputation.

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List of Tables

1.1 Characteristics of the problem tests. . . 21

1.2 Solution times (in seconds) of the proposed heuristic . . . 22

1.3 Comparison with the B&B of CPLEX . . . 22

1.4 Quality of the solutions obtained with the proposed heuristic . . . 24

1.5 Results of the auction process simulation. . . 25

2.1 Description of the instances sets . . . 42

2.2 Cases for centralized and decentralized mechanisms’ comparison. . . 42

2.3 CP versus DPA: Average results for ∆Pda_{, ∆Q and ∆E} _{. . . .} ₄₃

2.4 CP versus DPR: Average results for ∆Pdr_{, ∆P}dr_{, ∆Q, ∆Q and ∆E} _{. . . .} ₄₆

3.1 Problem tests . . . 60

3.2 Centralized versus decentralized: Average results for ∆DC, ∆SC, ∆T C, ∆P CC 62 4.1 Description of the instances sets . . . 74

4.2 Impact of considering carriers’ reputation on direct, hidden and total costs . . . 75

4.3 Number of instance for which a proposed approach yielded the lowest cost . . . 76

4.4 Gap (%) between the hidden cost obtained with a proposed approach and the optimal hidden cost for s1 and s2 . . . 78

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List of Figures

2.1 Gain/loss in profits for CP versus DPA. . . 43

2.2 Variation in the percentage of contracts bid on in CP versus DPA . . . 44

2.3 Gain/loss in empty movement distance for CP versus DPA . . . 44

2.4 Market diversification: CP versus DPA . . . 45

2.5 Gain/loss in profits for CP versus DPR . . . 47

2.6 Variation in the percentage of contracts bid on in CP versus DPR . . . 47

2.7 Gain/loss in empty movement distance for CP versus DPR . . . 47

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List of Abbreviations

CA Combinatorial Auction BCP Bid Construction Problem WDP Winner Determination Problem CAP Cost Allocation Problem FTL Full TruckLoad

LTL Less Than TruckLoad CPU Central Processing Unit

ALNS Adapted Large Neighborhood Search LCP Lane Covering Problem

CCLCP Cardinality Constraint Lane Covering Problem LCLCP Length Constraint Lane Covering Problem TCLCP Time Constraint Lane Covering Problem VRP Vehicle Routing Problem

DSS Decision Support System MIP Mixed-Integer Programming B&B Branch And Bound

DPA Decentralized Procurement with a risk-Averse bidding DPR Decentralized Procurement with a Risky bidding CP Centralized Procurement

IBM International Business Machines Corporation

GO Giga Octet

RAM Random Access Memory

BP Bid Price

P Profit

BP Bid Price

SC Spot Cost

PCC Percentage of Covered Contracts

HC Hidden Cost

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To my amazing mom and dad... You will always be in my heart.

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If you can’t fly then run, if you can’t run then walk, if you can’t walk then crawl, but whatever you do you have to keep moving forward.

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Acknowledgments

First and foremost, I wish to present my thanks and admiration to my supervisors, Professor Sehl Mellouli and Professor Monia Rekik. Sehl gave me the opportunity to begin my doctoral studies at Laval University and supported me since my first day at my Ph.D. degree. He oered me the right assistance, shared his knowledge with me, and supported me both scientifically and professionally. Second, I would like to thank my co-supervisor, Monia who inspired me in many ways. I appreciated a lot her attention to details, her scientific rigour, and her non-stop quest for challenges. Thanks to both of you, I will never be able to thank you enough for your kindness, availability, fruitful discussions, guidance, feedback, and trust in my choices throughout this research work. This work would not have been possible without the two of you.

I wish to thank my committee members: Professors Adnène Hajji, Rohit Nishant and Jean-Marc Frayret for accepting to evaluate my research.

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 daily support. I would also like to thank all my professors in Operations and Decision Systems Department especially Dr. Jacques Renaud and Dr. Benoit Montreuil. I also thank my colleagues from the CIRRELT for the different moments shared throughout my university journey. I especially thank Antoine, Houcine Farouk, Anouar and Anas for all the effervescent exchanges we had, as well as Mariem, Lilia, Maha, Wiem, Sana & Sana, Salma & Salma, Amal & Amal, Zeineb, Ines, Chourouk, and Raja for their encouragement.

My deepest gratitude goes to my family for their patience and love throughout all of my studies and my life. Thanks to my wonderful parents Ali Ben Othmane and Khaouira Ben Jabeur, who always encouraged us to study and who made our achievements their priority in life. Thank you for your unconditional love, for your unlimited support, for your endless devotion through every step I did in this world, and for brightening my universe all the time. Thanks to my amazing sister Nawel and brothers Nidhal and Nefaa, who are always soothing me, caring about me, and reminding me how lucky and honored I am to have them in. I especially want to thank my mother in law Chadiya Neiliya for her help and support in

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achieving my goal.

A huge thanks are due to Marie-claire Moreau, Jean-Marc Massard and Nabil Djoumi, who believed in me and offered me the opportunity to join their team in the ministry of higher education. My great thanks to my colleagues Sana, Nessrine, Sahar and Abdou for their encouragement and support.

To my funny virtual friends Faten Fazaa, Maria Teresa and Mona Dachri residing respectively in Germany, Italy and Tunisia, thank you very much for making me happy always and even in the worst circumstances.

Last but not least, I would like to offer my heartfelt thanks to my companions along the way, Mariem and Lilia, for all their endless support, understanding and encouragement.

In the end, my great appreciation and thanks go to my dearest and wonderful husband Hamza Heni and my lovely children Molk and Ayoub-Sanad for their never-ending unconditional support and understanding during my busy and long doctorate journey. I promise I’ll be there with you!

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Preface

This thesis presents my work as a Ph.D. student developed 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. This thesis consists of four papers, each of which is written in collaboration with my directors Sehl Mellouli and Monia Rekik. The first paper is already published and the others are submitted. In all four papers, I remain the rst author and have played the major role of setting up and conducting the research, modelling and implementing the algorithms, analyzing the results, and preparing and writing the papers.

The first paper entitled A Profit-Maximization Heuristic for Combinatorial Bid Construction with Pre-existing Network Restrictions is written in collaboration with Monia Rekik and Sehl Mellouli. The paper is accepted by the Journal of the Operational Research Society in August 2018 and was published online in February 2019.

The second paper entitled Centralized auctions for the procurement of truckload transportation services: impacts on carriers is written in collaboration with Monia Rekik and Sehl Mellouli. The paper has been submitted for publication in Transportation Research Part E: Logistics and Transportation Review in December 2019.

The third paper entitled Centralized auctions for the procurement of truckload transportation services: impacts on shippers is written in collaboration with Monia Rekik and Sehl Mellouli. This paper will be submitted afterwards.

The fourth paper entitled Reputation-based winner determination problem in centralized com-binatorial auctions for the procurement of transportation services is written in collaboration with Sehl Mellouli and Monia Rekik. The paper has been submitted for publication in Trans-portation Research Part B: Methodological, in December 2019.

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Introduction

The trucking market generated in 2017 a total revenue of 38.951 _{billion Canadian dollars}

from almost 66.51 _{million shipments and transported cargo on around 323.5}1 _billion

tonne-kilometers. Trucking operations are either Full Truckload (FTL) or Less-Than Truckload (LTL). In FTL operations, shipments must be driven directly from pick- up to delivery loca-tions without any intermediate stop. However, LTL operaloca-tions allow consolidation of many smaller shipments on a single vehicle. Trucking procurement markets include two major ac-tors which are shippers (typically large manufacturing companies, retailers, distribuac-tors or large carrier organizations as sub-contracts) who have loads that need to be transported from origins to destinations, and carriers (which generally refer to transportation companies that own transportation means) who are the service providers. In order to connect shippers and carriers, auctions can be used. In an auction context, shippers present their requests and seek to outsource all or part of their FTL operations to external carriers. Shippers are often the auctioneers in the procurement of transportation services. Otherwise, in many cases, a third party, such as a 3rd Party Logistics Provider (3PL), consultant or software vendor, will run the auction and act as the auctioneer. Carriers act as bidders on shippers’ requests and compete by bidding on the shipper requests. In the following, we will use the term contract to refer to a shipper request. A contract, in its simple form, is defined by an origin-destination pair (also called a lane).

Several studies in auction design determinate the potential benefits of the so-called combinato-rial auctions, especially when some synergy exists between the traded items (egLedyard et al.

(2002), Peke and Rothkopf (2003), Rekik and Mellouli (2012)). In combinatorial auctions, carriers are allowed to bid on combinations of items rather than individual items. Hence, bidders are offered the possibility to express their valuations of any package of items they want to acquire Rekik and Mellouli (2012). In general, the package bidding is beneficial for bidders whose valuation of a collection of items is greater than the sum of the values of the items taken independently. These items are referred to as complementary items (eg de Vries and Vohra(2003),Peke and Rothkopf(2003),Abrache et al.(2007)). Such complementarities are intensively present in transportation service procurement. In fact, a significant portion of

1

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the trucking industry costs is due to the repositioning of empty vehicles from the destination of one load to the origin of a subsequent load. Hence, carriers aim to acquire adjacent lanes and/or lanes that form a closed-loop in order to minimize empty moves (Song and Regan 2005,Lee, Kwon, and Ma 2007). In consequence, carriers can make economies of scope since the cost of serving one lane depend more on whether another close lane is served than on the shipped volume Caplice and Sheffi (2003), Nandiraju (2006). In last decades, the use of combinatorial auctions for the procurement of FTL transportation services has achieved great success (Caplice and Sheffi 2006) and five software tools incorporating package bidding and dedicated to transport procurement were developed between 1997 to 2003 and used by more than one hundred companies realizing an average saving of 13% in transportation costs for these companies.

During an auction process, two main decisional problems are addressed: the bid construction problem (BCP), also called the bid generation problem and the winner determination prob-lem (WDP). The BCP is a probprob-lem faced by carriers and consists of determining the set of profitable contracts they are interested to acquire and the price they are asking for serving them. The WDP must rather be solved by the shipper (or more generaly the auctioneer) and consist of determining the set of winning bids that minimize the costs to the shipper. From a shipper perspective, the performance of a carrier may influence the effectiveness of its entire logistics. In fact, several studies show the importance of considering attributes other than the price to select carriers Sheffi(2004), Danielis, Marcucci, and Rotaris(2005),Caplice and Sheffi (2006). Consequently, shippers should manage a trade-off between a cost-minimization goal as well as the service quality level of the carriers with whom they will engage (Meixell and Norbis 2008).

Traditionally, each shipper runs an individual auction to satisfy its proper transportation needs independently from other shippers. In the following, we refer to such auction as a "decentralized auction". Basu, Subramanian, and Cheikhrouhou (2015) reported that the negotiation process for establishing contracts between shippers and carriers takes one to two years. In some cases, carriers are led to bid in more than one auction at the same time before knowing which of the bids they submitted are effectively won. Hence, a carrier participating asynchronously into different decentralized auctions may face a problem similar to the exposure problem: when it is invited to participate into an auction run by a shipper s2, it probably

had already submitted bids to another auction previously run by another shipper s1 without

knowing yet if the bids it submitted to s1 has won or not. Consequently, it may happen that

when some bids submitted to the first auction are rejected by s1, the bids submitted to s2

are no more profitable. This is exactly the exposure problem initially criticized in single-bid auctions.

As already explained above, combinatorial auctions came to face the exposure problem by allowing a carrier to bid on a collection of contracts simultaneously. Our research extends the

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same idea by considering what we call a "centralized auction" in which carriers are the only bidders and have the opportunity to bid on the contracts required by the different shippers simultaneously. In this thesis, we propose a novel auction-based procurement mechanism in which multiple shippers present their transportation contracts simultaneously in a centralized auction and carriers submit package bids, including a set of contracts belonging to different shippers.

The remainder of this thesis is organized as follows. In Chapter 1 we propose a heuristic approach for constructing combinatorial bids in TL transportation services procurement auc-tions. It considers a particular context in which the carrier has already engaged on a set of transportation contracts before participating in the auction. The proposed heuristic identi-fies profitable new contracts and efficiently integrates them in the carrier exiting routes and where the routes already defined for existing contracts must be kept. The carrier also has the possibility to build new routes composed of only new contracts so that its unused vehicles could be operated, when profitable. In the two cases, the carrier’s objective is to maximize its profit. The heuristic is shown to be efficient for solving up to 300 existing contracts and 300 auctioned contracts in a short computational time. It exploits well the carriers existing network, generates interesting bids enabling an additional potential profit and covers a large percentage of new contracts. The heuristic is also compared with an exact solution method of

Hammami, Rekik, and Coelho (2019) and results prove that our heuristic identifies optimal or near-optimal solutions for small-sized instances. For larger instances, the exact method either identifies a solution that is worse than that output by our heuristic or fails in identifying a feasible one. This heuristic is used by carriers in centralized as well as decentralized auctions and offers the carrier the possibility to submit either a single bid or multiple bids with an ask price for each bid.

In chapter 2, we present centralized and decentralized procurement mechanisms and investi-gates the impact of such a centralized auction on carriers by comparing it to decentralized auctions under different carrier’s risk-behaviour attitudes. We distinguish two types of carriers risk-behaviour, which are a too risky carrier and an averse-to-risk carrier. A comparison be-tween centralized and decentralized auctions from the carrier’s perspective is made over a large set of generated instances by computing relevant performance measures. Results show that centralized procurement offers to participating carriers the best compromise between profit, network efficiency and business diversification.

Chapter 3 extends the study of centralized auctions by investigating its impact on shippers. The comparison between centralized and decentralized auctions from a shipper’s perspective is also performed over a large set of generated instances and by computing relevant performance indicators. Results show that in most cases, centralized auctions allow shippers to sell less contracts than decentralized ones. However, in all cases, shippers realize considerable total cost savings.

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In Chapter 4, we consider only a centralized auction in which shippers select carriers based not only on bid prices but also on the reputation of carriers which is translated into an unexpected hidden cost that shippers may incur when dealing with winning carriers. We consider a reputation-based WDP model for centralized combinatorial transportation auctions in which the auctioneer should decide on the winning bids knowing that: (1) the same bid may include shipping contracts requested by different shippers and (2) a carrier’s reputation may differ from one shipper to another. The objective is to find the best trade-off between bid prices and carriers level of service. Three methods are proposed to weight the different shippers valuations of carriers reputations, and the reputation-based WDP model is solved to optimality. We compare methods among others over a large set of generated instances and by computing relevant performance measures. Results show that considering reputation in a centralized WDP may lead to considerable savings in total costs paid by all shippers. Also, we observe, by comparing the three proposed approaches, that choosing the appropriate one depends closely on the shippers profiles, their weight in the auction and their perception of centralization.

Chapter 5 discusses the conclusions of this research as well as meaningful extensions to this thesis.

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Chapter 1

A Profit-Maximization Heuristic for

Combinatorial Bid Construction with

Pre-existing Network Restrictions

Résumé

Cet article propose une approche heuristique pour la construction de mises combinatoires dans les enchères d’approvisionnement des services de transport routier à charge pleine. Il considère le contexte où le transporteur a déjà un ensemble de contrats de transport qu’il doit servir à d’autres expéditeurs avant de participer à l’enchère. L’heuristique proposée identifie les nouveaux contrats profitables et les intègre efficacement dans les routes des transporteurs, et/ou construit de nouvelles routes pour les véhicules inutilisés. Les résultats expérimentaux prouvent l’efficacité de l’heuristique proposée en termes de temps de calcul et de qualité des solutions.

Chapter informationA research paper based on this chapter, named A Profit-Maximization Heuristic for Combinatorial Bid Construction with Pre-existing Network Restrictions, has been published in the Journal of Operational Research Society: Ben Othmane I., Rekik M., Mellouli S., 2019. A profit-maximization heuristic for combinatorial bid construction with pre-existing network restrictions. Journal of the Operational Research Society 70(12), 2097–2111. This research was the subject of one presentation: at the International Conference on Information Systems, Logistics and Supply Chain ILS in Quebec (Canada) 2012.

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Abstract

This paper proposes a heuristic approach for constructing combinatorial bids in TL trans-portation services procurement auctions. It considers the case where the carrier has already engaged on a set of transportation contracts before participating in the auction. The pro-posed heuristic identifies profitable new contracts and efficiently integrates them in the carrier exiting routes, and/or builds new routes for unused vehicles. Experimental results prove the efficiency of the proposed heuristic in terms of computing times and solutions quality.

Keywords: combinatorial auctions, TL operations, bid construction, heuristic.

1.1 Introduction

Transportation procurement markets include a set of shippers (the service askers) that de-cide to outsource all or a part of their transport operations to external carriers (the service providers). The latter compete on shippers’ transportation requests by submitting offers. Our paper particularly addresses Total TruckLoad (TL) operations for which shipments must be driven directly from pick- up to delivery locations without any intermediate stop. In the rest of the paper, we use the term transportation contract or simply contract to refer to a shipper request. The use of combinatorial auctions for the procurement of freight transportation ser-vices considerably increased throughout the 1990’s (Caplice and Sheffi 2003,Elmaghraby and Keskinocak 2004). This was strongly motivated by the significant cost savings achieved by such mechanisms (Caplice and Sheffi 2006). Combinatorial auctions are trading mechanisms where a carrier’s offer takes the form of a combinatorial bid in which the carrier expresses its interest to serve a package of contracts. If the carrier’s bid wins, then all the contracts submitted in the bid must be allocated to it. In general, bids on combinations of items are beneficial for bidders whose value for a combination of items is greater than the sum of the values of the items in the combination, taken separately (de Vries and Vohra 2003). Such complementarities are intensively present in transportation service procurement markets. This paper addresses an important decisional problem encountered by each carrier partici-pating in a transportation combinatorial auction: the Bid Construction Problem (BCP), also known as the bid generation problem. More precisely, Each carrier then solves its own BCP. A BCP mainly consists in answering two questions: Which contracts the carrier should put in a package bid? What price should be asked for serving this package?

After receiving all carriers’ bids, the shipper solves the Winner Determination Problem (WDP) to determine the winners. When combinatorial bidding is permitted, the WDP is an NP hard problem. It is generally formulated as a set covering or a set partitioning problem where the objective is to minimize the total price to be paid by the shipper so that each requested

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contract is served once (for a set partitioning formulation) or at least once (for a set covering formulation). A first-price auction implies that the shipper should pay a carrier exactly the price it asks in its winning bid. We refer the reader to the paper by Abrache et al.(2007) for more details on auctions taxonomy.

Our paper considers a new variant of the BCP where the carrier has already engaged on a set of transportation contracts (referred to in what follows as existing contracts) before participating in the auction and where the routes already defined for the existing contracts must be kept. Keeping existing routes is relevant in situations where the carrier has a residual capacity and aims to maximize its resources utilization by adding new profitable contracts without drastically changing its transportation network. This would be the case, for example, in spot markets where shippers communicate their urgent needs on a day-to-day basis and carriers have no time for re-planning all their shipments. The objective for the carrier is to identify profitable packages of new contracts and efficiently integrate them in its existing network. The carrier is also offered the possibility to define new routes composed of only new contracts so that its unused vehicles could be operated, when profitable.

A very efficient heuristic approach is proposed to solve the BCP in the particular context described above. The proposed heuristic can be easily implemented within a Decision Support System (DSS) to provide the carrier with different alternative bids. It particularly offers the carrier the possibility to submit either a single bid or multiple bids with an interval of profitable ask prices for each bid. Our experimental results show that the proposed heuristic is very fast and computationally robust. To the best of our knowledge, this paper is the first to propose an efficient heuristic for solving a BCP including up to 300 new contracts and 300 existing ones within less than 149 seconds. Short computational times offer the carrier the possibility to test different alternative bids under different bidding languages such as XOR, OR-of-XOR, XOR-of-OR, etc. We refer the reader to Abrache et al. (2007) for more details on bidding langages. They also enable them to respond quickly to requests in spot markets where shippers are generally looking out for short response times to their urgent requests.

An exhaustive experimental study is conducted to highlight the high performance of the pro-posed heuristic. First, our heuristic is compared to an exact solution method for small-sized instances where an optimal solution could be identified. The exact method uses the branch-and-bound procedure embedded in CPLEX 12.6.1 applied to an arc-based mathematical model recently proposed byHammami, Rekik, and Coelho(2018) for a standard BCP. This arc-based formulation is adapted to the new variant of BCP addressed in this paper. Our results prove that our heuristic identifies optimal and near-optimal solutions for small-sized instances. For larger instances, the exact method either identifies a solution that is worse than that output by our heuristic or fails in identifying a feasible one.

Moreover, to assess on the quality of the solutions obtained, a number of performance mea-sures are computed with regard to the percentage of new contracts covered, the fleet utilization

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and, the additional potential profit resulting from submitting the proposed bids. The reported results prove that our heuristic exploits well the carriers existing network and generates inter-esting bids enabling an additional potential profit of 31%, on average, if all the submitted bids win. The percentage of new contracts covered by the generated bids is also relatively large for the majority of the instances with an average coverage of 67%.

Finally, to measure the efficiency of our heuristic in practice, we simulate a first-price sealed-bid auction with 10 competing carriers, each using the proposed heuristic to generate sealed-bids. A WDP is then solved to optimality and the winning bids determined. The effective values of the different performance measures previously described are computed with respect to the bids effectively won by each carrier. Our results prove, once again, that our heuristic enables a carrier to realize higher profits.

The remainder of the paper is organized as follows. Section 1.2is a literature review recalling the main recent published papers dealing with BCP in TL transportation procurement auc-tions. Section1.3gives a formal description of the problem addressed. Section1.4details the proposed heuristic. In Section 2.5, we present the results of our experimental study. Section

1.6 concludes the paper and considers future research avenues.

1.2 Literature review

In this section, we discuss the published papers dealing with BCPs in combinatorial auctions for the procurement of TL services, which are more close to our problem setting. Hence, works addressing the BCP for less-than-truckload operations or for exchanges in a horizontal collaboration between carriers are not reported.

Song and Regan (2004) propose a greedy search heuristic to help the carrier construct desir-able package lanes (a lane is an origin-destination pair). Routes including at most three lanes are greedily generated by giving a higher priority to new lanes with a matching opportunity with current lanes. Song and Regan (2004) show, through an experimental study, that both shippers’ procurement costs and carriers’ average empty movement costs are lower in combi-natorial auction compared to traditional single-item auctions.

Later, Song and Regan (2005) propose approximation methods to solve the BCP under two scenarios: with and without pre-existent commitments. It is assumed that trucks are available at any location at the beginning of the auction and can reside in any destination of a lane. A two-phase solution approach is proposed. In the first phase, an exhaustive search algorithm enumerates all routes with respect to a number of routing constraints such as maximum route distance or driver work rules. A set covering model is solved in the second phase to select optimal routes (packages) that minimize empty movement costs for both scenarios. In case the carrier wants to integrate the new lanes into its current operations with pre-existing com-mitments, more candidate routes need to be generated at the first stage. The authors give no

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details on how these routes are generated. The authors discuss the benefits of using a set cov-ering model (rather than a set partitioning one) for which each new lane could be covered by more than one bid. In this case, substitutable bids (i.e. bids that include lanes in common) are possible and an augmentation step is proposed to strengthen the carrier competitiveness. A “modified" branch-and-bound algorithm is used to solve the proposed set covering formulation for which the MIP solver is forced to identify all optimal solutions.

Song and Regan(2005) compare the performance of the proposed bid construction algorithm to a complete enumeration approach through simulating a combinatorial procurement auction organized by a single shipper with two competing carriers. In a complete enumeration, every possible combination of new lanes is considered. The experimental study considers only small-sized instances where the number of new lanes ranges from 4 to 10. The performance of the proposed heuristic is evaluated by reporting the number of submitted bids, the percentage of new lanes won, and the ratio between the empty distance and the total distance traveled under both scenarios (with and without pre-existing commitments). The authors point out the large gap between the bid sizes versus a small variation in the percentage of new lanes won and in empty move distances of the“smart agent" using their proposed bid construction method compared to the agent using the enumeration approach. They report that their method was “able to routinely generate more than 1500 atomic bids for 100 new lanes in less than 3 min". They give no more details on the of the proposed heuristic for large instances.

Wang and Xia (2005) propose two heuristics to solve a BCP where a pick-up time window is associated with each committed and each auctioned lane. The objective is to minimize the total expected empty travel distance given that not all the combinations of lanes are guaranteed to be won. The first heuristic is based on a fleet assignment model where a binary decision variable is assigned to each arc of the carrier network to indicate whether an assignment exists on that arc. The second heuristic is based on the nearest insertion method: lanes are gradually added to vehicle routes based on the extra empty travel distance they generate. These algorithms are tested on a set of instances including up to 10 vehicles and 20 auctioned lanes. The authors report that, on average, the fleet assignment based heuristic generates slightly better bids than the nearest insertion method.

Lee, Kwon, and Ma (2007) consider vehicle routing models to help bidders identify sets of origin-destination pairs that maximize their profits rather than just minimizing empty repo-sitioning costs. The fleet size is assumed limited and all trucks are identical. The proposed model is a quadratic integer program that simultaneously generates and selects profitable routes while taking into account existing commitments. Column generation and Lagrangian relaxation based techniques are combined for solving the problem. The results obtained show that the solution method yields relatively short computing times for relatively small instances with less than 100 contracts. However, larger instances involving up to 400 contracts required large computing times (up to 120 hours).

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InChang(2009), a so-called bidding advisor is proposed to help the carrier generate desirable bids in one-shot combinatorial auctions for TL spot markets. The spot market assumes a short planning horizon (one week) with known and estimated information on the number of available vehicles, the current locations of all vehicles, the booked, forecasted and auctioned loads, their pick-up and delivery times, their revenues, etc. The BCP addressed has the particularity of considering forecasted loads, in addition to booked (what we call existing) and new loads. Forecasted loads are loads with a high probability to be materialized based on the carrier’s historical data. Both booked and forecasted loads must be serviced and only profitable new loads are selected. The author claims that, “in truckload procurement, the bid generation problem is better formulated as a time-space network based fleet management problem instead of vehicle routing problems". He probably refers to papers that force all auctioned items to be served (Song and Regan 2005,Wang and Xia 2005) using VRP concepts. Chang(2009) model the BCP as a synergetic minimum-cost flow problem in the sense that the potential synergy effect between consecutive loads on the same vehicle tour is taken into account. A path-based formulation is proposed and solved by column generation in which a shortest path algorithm with synergy considerations is proposed to generate all optimal paths. The computational performance of the column generation solution approach is evaluated on large-sized instances with 100, 250, 500, 750, or 1000 vehicles “serving loads with a double size of corresponding fleet between 20, 40, or 60 regions" within a one-week planning horizon (Chang 2009). The CPU time varies between 13 and 522 minutes. Instances with 750 or 1000 vehicles serving loads between 60 regions could not be solved.

Triki et al.(2014) address the BCP in a single-round sealed-bid combinatorial auction for a TL spot market. They propose a probabilistic mixed integer programming model that incorporates the package bid price as a decision variable while defining the auction clearing prices as random variables. The clearing price associated with a package bid represents the lowest price offered by the competitors for that package. The probabilistic constraints guide the selection of the package whose price is guaranteed to be less than the corresponding clearing price by a certain probability threshold. The BCP is formulated as a fleet management problem based on a time-space network where the objective is to maximize the carrier’s profit. The authors report that the CPLEX branch-and-cut method required unreasonable computing times to solve the proposed model even for problems with 20 auctioned loads. They propose two heuristics: a sequential descending method and a sequential ascending method. Both heuristics generate a single package bid by evaluating incremental profits. The descending method starts with a bundle including all contracts and drops the load with the best incremental profit, one load at a time. Loads that yielded non-positive incremental profits are included in the package bid. The ascending method rather starts with an empty set and adds loads resulting in positive incremental profits, one at a time. The results obtained show that the ascending heuristic outperforms the another one for all tested instances. The computational time required by the ascending heuristic varies between 2261 and 6889 seconds for instances including 10 auctioned

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loads.

Recently, Hammami, Rekik, and Coelho (2018) propose an Adapted Large Neighborhood Search (ALNS) heuristic to solve a BCP with almost the same problem setting as in our paper. The main difference, is that the carrier pre-existing network could be literally changed. Their heuristic shows good computational performances when compared to the branch-and-bound procedure of CPLEX applied to an arc-based MIP model. Whereas the ALNS heuristic requires very short computing times (less than 15 seconds) for instances with up to 35 new and 182 existing contracts, the CPU time considerably increases (more than 4 hours) when the number of new and existing contracts exceeds 140 and 33, respectively.

Based on the literature review, our paper makes a number of interesting contributions to combinatorial bid generation in TL transportation procurement auctions. Mainly, it is the first to address a new variant of the BCP where the routes already defined for the existing contracts must be kept. As will be explained later, the way bids are generated with this variant ensures the carrier to make a non negative profit even if only a subset of the submitted bids are won. Unlike Song and Regan (2005) and Wang and Xia (2005), we do not force the carrier to generate bids covering all the new contracts. New contracts are selected only if they are profitable for the carrier given its existing transportation plans. Second, our paper proposes a very fast and efficient heuristic that solves instances including 300 new and 300 existing contracts in very short computing times. To the best of our knowledge, none of the solution approaches reported in the literature for similar problem settings show such a good computational performance. Moreover, our paper is the first to test the performance of the proposed heuristic in terms of bids quality through simulating a competitive auction process with 10 carriers with different risk behavior profiles. The risk behavior is modeled with the profit margin applied to the incremental cost resulting from serving the new contracts in each bid.

1.3 Problem description

We consider a TL transportation services procurement auction in which a shipper (the auc-tioneer) submits the transportation contracts -it decided to outsource- to a set of competing carriers (the bidders). Each carrier is then faced with the BCP and must determine: (1) the packages of contracts to bid on, and (2) the price (for each submitted package) that should be asked. These decisions must take into account the carrier constraints (existing contracts, fleet size, work rules) while maximizing its total profit. A transportation contract k is described by a tuple (ok, dk, Vk), where ok and dkrepresent the pick-up and delivery locations, respectively,

and Vk is the volume to be shipped directly from ok to dk. The set of existing contracts in the

carrier network is denoted Ke_{. The set of new contracts traded in the auction is denoted K}n_.

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M of vehicles. This set can be partitioned into two subsets: A subset Me _{of vehicles that are}

already assigned to serve existing contracts and a set Mu _{of unused vehicles. All the vehicles}

are located in a single depot s and are assumed to start and finish their trip, if any, at s. The capacity of a vehicle m ∈ M is denoted Qm_{. To alleviate the presentation, it is assumed}

that all the vehicles have the capacity required to serve all the new contracts. In other words, ∀k ∈ Kn_{, ∀m ∈ M, Q}m _{≥ V}

k. Observe however that the proposed heuristic could be adapted

to handle the case where the latter assumption does not hold. This will be discussed in more detail in Section 1.4. For all the vehicles, the route duration must not exceed a pre-specified maximum duration TM ax_.

The carrier’s existing network is represented by a graph ˜Ge = ( Ñe, Ãe), where the set Ñe of nodes consists of the depot s and all the origin and destination locations of the existing contracts (i.e., Ñe= {s} ∪ {ok, k ∈ Ke} ∪ {dk, k ∈ Ke}), and Ãe is the set of arcs. For each arc

(i, j) ∈ ˜Ae, we associate a cost ˜cij as well as a time ˜tij required for traversing it. Note that for

the arcs (i, j) associated with contracts (i.e., (i, j) = (ok, dk), k ∈ Ke) loading and unloading

times (if not negligible) are included in ˜tij. For these particular arcs, we also define a price

pk= ˜pij that represents the price the carrier will receive for serving the corresponding contract

k. These prices are assumed to be known in advance. More precisely, when a single-round sealed bid auction is used -which is the context we are considering- such a price is estimated based on the carrier experience and knowledge of the shipper and of the other potentially competing carriers. When multiple-round, also called iterative, auctions are used, both the WDP and the BCP must be solved at each round. Some information on winning bids are revealed to bidders and could be used to estimate the approximate price of each auctioned contract before solving the BCP at each round (More details on the way price pk could be

determined are given in Section1.4.3). Hence, our heuristic could be used for iterative auctions as well. The only change concerns the value of the price to consider for each new contract -which is an input parameter- that must be updated at each round based on the information obtained from the previous round.

Because of the TL context, the graph ˜Ge = ( ˜Ne, ˜Ae) can be alleviated by replacing each arc (ok, dk), k ∈ Kewith a single node ik, referred to as a contract node. Hence, the carrier existing

network can be represented by a complete graph Ge _{= (N}e_{, A}e₎_{where N}e_{= {s}∪{i}

k, k ∈ Ke},

and Ae_{is the new sets of arcs linking either two contracts nodes or a contract node to the depot.}

The cost cij and the traveling time tij associated with an arc (i, j) in Ae are deduced from

the graph ˜G as follows: (i) If the nodes i and j (in Ne) correspond respectively to contracts k and k0, then cij = ˜cokdk+ ˜cdkok0 and tij = ˜tokdk + ˜tdkok0.; (ii) If node i corresponds to the

depot s and node j to contract k0_{, then c}

sj = ˜csok0 and tsj = ˜tsok0.; (iii) If node i corresponds

to contract k and node j to the depot s, then cis= ˜cokdk + ˜cdks and tis= ˜tokdk+ ˜tdks.

Let Re_{be the set of routes serving existing contracts in G}e_{= (N}e_{, A}e₎_{. This set of routes is}

as-sumed to be known in advance. A route re_{∈ R}e_{is defined by a sequence (s, i}

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where {k(w), w = 1, ..., W (re_)}_{represents a subset of existing contracts, W (r}e₎_{is the number}

of contracts served by route re_{and k(w) identifies the contract having the w}th _{position in the}

route. The duration Tre of route re is computed as T_re = t_(s,i k(1))+

PW (re)−1

w=1 t(ik(w),ik(w+1))+

t_(i_{k(W (re))}_,s). Observe that given our representation of the carrier network with contract nodes, all arcs linking two successive nodes in re _{are empty move arcs.}

The BCP addressed in this paper consists in determining the set of new contracts (in Kn_{) that}

the carrier should bid on to maximize its profit without altering its existing network. More precisely, new contracts, if admissible, must: (1) either be inserted within deadhead moves (in the transformed graph) present in the existing routes re _{∈ R}e_{, (2) or form new routes with}

unused vehicles.

1.4 Solution Approach

Generating a combinatorial bid consists in identifying one or many packages of new admissible contracts that are profitable for the carrier. A three-stage heuristic is proposed. The first stage aims at identifying the admissible new contracts that generate additional profits when integrated into the carrier existing routes. New contracts are added one at a time, the carrier network is updated (the newly added contract is considered as an existing one), and the process iterates until no new contracts can be added. In the second stage, the new contracts that were not added in the first stage, are examined and new routes including all or a subset of the remaining new contracts are built for the unused vehicles. The third stage determines the interval of prices (for each generated bid) that the carrier should ask.

1.4.1 First stage: New contracts insertion into existing routes

Algorithm1.1presents the main steps of the first stage. Details on each step are given in the following subsections. In sum, three types of subsets are used and updated at each iteration of Algorithm 1.1: (1) A set Kp _{including the new contracts that have not yet been inserted}

into the carrier existing network at the current iteration, (2) A set Kc _{of contracts that are}

admissible in the current iteration, and (3) Sets Bre

, one for each existing route re _{∈ R}e_,

including possibly the new contracts that have been inserted into the existing route re _up

to the current iteration. At the end of the algorithm, each non-empty subset Bre

, re ∈ Re

represents a package of new contracts that is profitable for the carrier to bid on.

Admissible new contracts

A new contract k ∈ Kn _{is admissible for a route r}e _{∈ R}e _{if, when inserted in an}

empty-move of re_{, it generates a positive profit without violating the time capacity constraints.}

Formally, k ∈ Kn _{is admissible for r}e _{= (s, i}

k(1), ik(2), ...., ik(W (re₎₎, s) if there exists an arc

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Algorithm 1.1 New contracts insertion into existing routes Step 0: Initialization

Set Kp_=Kn_{, B}re

=∅, ∀re_{∈ R}e_{, K}c_{= ∅}_{. Go to Step 1.}

Step 1: Admissible new contracts(see Section 1.4.1) For each k ∈ Kp_,

Let Re_{(k) = {(r}e_{, w), r}e_{∈ R}e_{, w ∈ W (r}e_{), k} _{is admissible for (r}e_{, w)}.}

If Re_{(k) 6= ∅}_{, then K}c_{= K}c_{∪ {k}}_. if Kc6= ∅ then Go to Step 2 else return Bre, re∈ Re end if

Step 2: Optimal position for admissible contracts(see Section 1.4.1) For each k ∈ Kc_,

Determine the optimal (route, position) pair (re∗_{(k), w}∗_(k))_.

Determine the maximum profit g∗ k.

Go to Step 3.

Step 3: Contract selection (see Section1.4.1)

Let kmax _{be the maximum element in K}c _{with respect to the total order relationship ≺.}

Bre = Bre∪ {kmax_}_.

Kc_{= K}c_{\ {k}max_}_{; K}p _{= K}p_{\ {k}max_}_.

Update existing route re∗_(kmax₎ _{by inserting contract k}max _{at position w}∗_(kmax₎_.

if Kp 6= ∅ then Go to Step 1. else return Bre, re∈ Re_. end if=0 1) t(ik(w),ik)+ t(ik,ik(w+1))− t(ik(w),ik(w+1)) ≤ T M ax_{− T} re and (Condition 2) gr e_,(w,w+1) k = pk+ c_(i_k(w)_,i_k(w+1)₎− c_(i k(w),ik)− c(ik,ik(w+1))> 0.

For Condition 1 , the left-hand side represents the additional time required when new contract kis added to route rebetween contract nodes ik(w)and ik(w+1). The right-hand side represents

the time margin left in existing route re _{with regard to the maximum route duration T}max_.

For Condition 2, gre_,(w,w+1)

k represents the additional profit (or loss) obtained by inserting

contract k between the wth _{and the (w + 1)}th _{positions of existing route r}e_{. This additional}

profit is computed as the difference between the amount of money earned (the price of the new contract) or saved (no empty move exists between contract nodes ik(w) and ik(w+1) after

the new contract insertion) and the amount of money spent for serving the new contracts and adding the two empty moves between (ik(w) and ik) and (ik and ik(w+1)). Observe that in

case our assumption on vehicles capacity does not hold, a new contract k admissibility for an existing route re_{must take into account the capacity of the vehicle assigned to route r}e_{. That}

is, if the vehicle m performing route re _{is such that Q}m_{< V}

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for k.

Definition 1: Admissibility of a new contract for an existing route

If conditions 1 and 2 are satisfied for a new contract k ∈ Kn _{and an existing route r}e_{, then}

contract k is said to be admissible for re_{at position w, or simply that k is admissible for (r}e_{, w).}

A contract k ∈ Kn_{is admissible for a route r}e _{if there exists a position w = 0, ..., W (r}e₎_such

that k is admissible for (re_{, w)}_.

Definition 2: Admissibility of a new contract

A new contract k ∈ Kn _{is said to be admissible if there exists a route r}e_{∈ R}e _{such that k}

is admissible for re_.

Optimal position for admissible new contracts

Consider a new contract k ∈ Kn_{and assume it is admissible (as defined in Section}_1.4.1_{). Let}

Re(k) be the set including all the pairs (re, w), re ∈ Re_{, w = 0, ..., W (r}e₎ _{for which contract}

k is admissible. Notice that since contract k is admissible, Re(k) is a non-empty set. In case, set Re_(k) _{includes more than one element, one should decide on the best pair (r}e_{, w)}

to choose for contract k, denoted (re∗_{(k), w}∗_{(k)). We propose to select the (route, position)}

pair inducing the maximum profit. The profit generated in this case is denoted g∗

k. Formally,

(re∗(k), w∗(k)) = argmax{g_kre,(w,w+1), re ∈ Re_{(k), w = 0, ..., W (r}e_)}_{. Observe that when two}

positions give the same maximum profit, we arbitrarily chose the pair (re_{, w)}_{with the smallest}

w value.

Contracts selection

Let Kc _{denote the set of admissible new contracts at a given iteration of Algorithm} _1.1_{. A}

total order relation is defined on set Kc_{as follows: a contract k ∈ K}c _{is less than a contract}

k0 ∈ Kc_{(denoted k ≺ k}0_{) if the maximum profit g}∗

k generated by k is less than the maximum

profit g∗

k0 generated by k0 (g_k∗ < g_k∗0) or g∗_k = g∗_k0 and k > k0. The new admissible contract

to be first selected is the maximum element, denoted kmax_{, in K}c _{with respect to the total}

order relation ≺. In other words, kmax _{is the contract inducing the maximum profit among}

all the admissible new contracts of the current iteration. Once a contract kmax _{is selected, it}

is inserted at the optimal position w∗_(kmax₎_{of its associated optimal existing route r}e∗_(kmax₎

and the process is reiterated. That is, contract kmax _{is considered as an existing contract.}

The carrier existing routes are updated (indeed, a change is made only for existing route re∗_(kmax₎_{). The set of admissible new contracts and their optimal positions in the existing}

routes are also updated. Set Kc _{is updated and ordered and its first element (contract), if}

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Stopping criteria

The process iterates until either : (1) set Kc _{becomes empty (Step 2) implying that there is}

no new admissible and profitable contract that could be inserted into the existing routes or (2) set Kp _{becomes empty (step 3) implying that all new contracts have been inserted. At}

the end of the algorithm, each non-empty subset Bre

, re ∈ Re _{represents a package of new}

contracts that is profitable for the carrier to bid on.

Diversification

To improve new contracts’ insertion in existing routes and alleviate the greediness of the proposed heuristic, we propose to run Algorithm 1.1 Ndiv times by diversifying the identity

of the contracts that are selected at Step 3; Ndiv being a parameter to be set in advance.

The solution resulting in the best profit over the Ndiv runs is passed to the second stage.

More specifically, at each run i = 1 . . . Ndiv, instead of selecting -at the first iteration- the new

contract inducing the maximum profit in Step 3, we rather select the contract resulting in the best ith _{profit. Furthermore, to intensify the diversification, contracts that were selected at}

the first iteration in the previous runs are tentatively discarded. They are reconsidered for insertion in the final iterations if they are the unique remaining admissible contracts. Observe that assigning large values to the parameter Ndiv increases the chance to improve the solution

output by the heuristic but at the expense of an increase in solution times. Hence, a trade off must be managed between solution quality and computational times when fixing the value of Ndiv. Based on a series of preliminary tests, we found that the value offering the best

compromise for the considered instances is Ndiv = 10.

1.4.2 Second stage: New routes for unused vehicles

Let Ku _{denote the set of new contracts that were not inserted in the carrier existing routes}

at the first stage. Formally, Ku _{= K}n_{\ ∪}

re_∈ReBr e

. In the second stage, contracts in Ku _are

examined and new routes are generated for the unused vehicles Mu _{to serve all or some of}

these contracts. Algorithm 1.2presents the main steps of the second stage.

A constructive heuristic is used to generate at maximum |Mu_|_{new routes that are admissible}

with respect to the maximum duration restriction Tmax_{and yield a non-negative profit for the}

carrier. The first stage in Algorithm 1.2is inspired by the Clark & Wright saving algorithm (Clarke and Wright 1964) used to solve Vehicle Routing Problems (VRP). It first starts by generating |Ku_|_{round trips, each visiting a single new contract in K}u _{and then returning to}

the depot. The feasibility of each round trip with respect to Tmax _{is tested and its profit}

computed. If there exist |Mu_|_{admissible round trips that generate a non-negative profit, the}

first stage of Algorithm 1.2 terminates. Otherwise, admissible round trips are merged using the Clark & Wright saving principle. The process is reiterated until either |Mu_| _admissible