lane-exchange mechanisms for truckload carrier collaboration

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Lane-Exchange Mechanisms for Truckload CarrierCollaborationOkan Örsan Özener, Özlem Ergun, Martin Savelsbergh,

To cite this article:Okan Örsan Özener, Özlem Ergun, Martin Savelsbergh, (2011) Lane-Exchange Mechanisms for Truckload Carrier Collaboration.Transportation Science 45(1):1-17. http://dx.doi.org/10.1287/trsc.1100.0327

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Lane-Exchange Mechanisms for TruckloadCarrier Collaboration

Okan Örsan ÖzenerOzyegin University, 34662 Istanbul, Turkey, [email protected]

Özlem ErgunH. Milton Stewart School of Industrial & Systems Engineering, Georgia Institute of Technology,

Atlanta, Georgia 30332, [email protected]

Martin SavelsberghCSIRO Mathematics, Informatics, and Statistics, North Ryde, NSW 1670, Australia,

[email protected]

Because of historically high fuel prices, the trucking industry’s operating expenses are higher than ever andthus profit margins are lower than ever. To cut costs, the trucking industry is searching for and exploring

new ideas. We investigate the potential of collaborative opportunities in truckload transportation. When carriersserve transportation requests from many shippers, they may be able to reduce their repositioning costs byexchanging one or more of them. We develop optimization models to determine the maximum benefit thatcan be derived from collaborating. We also develop various exchange mechanisms which differ in terms ofinformation sharing requirements and side payment options that allow carriers to realize some or all of thecosts savings opportunities.

Key words : carrier collaboration; truckload shipping; lane-covering problem; lane exchangesHistory : Received: October 2008; revisions received: March 2009, September 2009, February 2010;accepted: March 2010. Published online in Articles in Advance July 7, 2010.

1. IntroductionTrucking is the backbone of U.S. freight move-ment. According to the American Trucking Associa-tion (ATA), the trucking industry’s share of the totalvolume of freight transported in the United Stateswas 68.9% in 2005. The U.S. trucking industry pro-duced an annual revenue of $623 billion by haul-ing 10.7 billion tons in 2005, which was 84.3% ofthe nation’s freight bill. Mostly because of historicallyhigh fuel prices, the trucking industry’s operatingexpenses are higher than ever. “Each penny increasein diesel costs the trucking industry $381 million overa full year” (ATA 2007). Eventually, these increasedoperating costs are reflected in the prices charged tothe shippers.We focus on the full truckload segment of the truck-

ing industry. Razor-thin profit margins have forcedfull truckload carriers to search for ways to cut costs.One way to do so is through mergers and acqui-sitions. There are economies of scale in the formof increased buying power and reduced marketingand administrative expenses, as well as economiesof scope in the form of reduced repositioning costs.When shipment density increases because of a mergeror acquisition, the likelihood of geographical syn-ergies between shipments increases, which in turnwill reduce costly empty travel between consecutive

shipments. However, mergers and acquisitions maybe beyond reach in many situations. In those cases,companies may consider strategic collaborations as ameans to achieve some economies of scale and scope.Strategic collaboration among full truckload carri-

ers may result in significant cost savings, but realizingthe full potential of such collaborations is challenging.Ideally, a centralized decision maker with completeinformation about all participating carriers deter-mines the optimal assignment of shipment requeststo the carriers and identifies the minimum cost routesfor them. However, in addition to the fact that acomplex optimization problem needs to be solved,companies may not be willing to share the neces-sary information. Trust is a central issue for selfishindividuals in collaborative relationships, especiallyin horizontal collaborations where competing entitiescollaborate.If centralized decision making is not viable, alter-

native mechanisms need to be explored. That is thefocus of our paper. We design and analyze rela-tively simple mechanisms that allow full truckloadcarriers to initiate and manage strategic collabora-tions. Because the carriers are guided by their ownself-interests, any proposed mechanism to managecollaboration activities has to yield collectively andindividually desirable solutions. Also, because of the

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Özener, Ergun, and Savelsbergh: Lane-Exchange Mechanisms for Truckload Carrier Collaboration2 Transportation Science 45(1), pp. 1–17, © 2011 INFORMS

trust issues associated with horizontal collaborations,proposed mechanisms cannot rely on the availabilityof full information. The challenge is to design mech-anisms that are simple and implementable yet effec-tive in terms of benefits to all participants. We willshow that bilateral lane exchanges satisfy all theserequirements. We will also show that when carriersshare information and allow side payments, most ofthe potential cost savings can be realized.The remainder of the paper is organized as fol-

lows. In §2, we illustrate economies of scope in truck-load transportation and discuss how collaborationsmay exploit economies of scope. In §3, we brieflyreview related work. In §4, we introduce and ana-lyze the multicarrier lane-covering problem, whichis the optimization problem that needs to be solvedto determine the maximum benefits of collaborat-ing. In §5, we present and investigate lane-exchangemechanisms for different collaborative environmentsin terms of information sharing and side paymentoptions.

2. Economies of Scope—An ExampleWe demonstrate the potential benefits of carrier col-laboration with a simple example. Consider a networkwith four cities and two carriers A and B. We assumethat the cost of traveling between two of the citiesis the same for both carriers and, for simplicity, thatthere is no difference in cost between traveling loadedor empty. We further assume that carrier A has toserve lanes (3�2) and (2�4) and that carrier B hasto serve lanes (1�2), (2�3), and (4�1). The minimumcost routes covering the lanes that have to be servedby both carriers are presented in Figure 1, where adashed line represents repositioning.If carriers A and B collaborate and exchange lanes

(2�4) and (2�3), then they both reduce their costs—from 3 to 2 for carrier A and from 4.73 to 3.73 for car-rier B. The minimum cost routes covering the lanes

1

2

3

4

Carrier B

(b)

1

1

1

1.73

1

2

3

4

Carrier A

(a)

11

1

Figure 1 Minimum Cost Routes

1

2

3

4

Carrier B

(b)

1

1

1

1.73

1

2

3

4

Carrier A

(a)

1

Figure 2 Minimum Cost Routes After the Lane Exchange

that have to be served by both carriers after the laneexchange are presented in Figure 2.

3. Literature ReviewIn this section, we briefly review related work. The lit-erature on truckload transportation procurement canbe roughly divided into two streams: centralized andauction-based approaches.Much of the research on centralized approaches

considers only single-carrier settings and focuses onoptimizing the routing decisions for those carriers,i.e., finding continuous-move paths and tours thatcover the lanes and minimize repositioning costs.Ergun, Kuyzu, and Savelsbergh (2007a) introducethe lane-covering problem (LCP), i.e., the problemof finding a minimum-cost set of cycles covering agiven set of lanes, and show that LCP is polynomi-ally solvable. Ergun, Kuyzu, and Savelsbergh (2007b)consider variants of LCP that include practically rel-evant side constraints such as dispatch windows anddriver restrictions and show that these variants areNP-hard. Their proposed heuristics appear to be effec-tive. Moore, Warmke, and Gorban (1991) develop amixed-integer programming model and a simulationtool to facilitate the centralized management of inter-state truckload shipments, which involves selectingcarriers and dispatching shipments.Özener and Ergun (2008) study the cost-allocation

problem of a collaborative transportation procure-ment network. In this setting, shippers are trying toachieve a common goal of minimizing the total trans-portation cost of the collaboration. After the opti-mal collaborative routes are determined, the costof these routes is to be allocated among the mem-bers of the collaboration. Özener and Ergun (2008)develop a framework to allocate costs and benefitsfairly among selfish collaborators to ensure the sus-tainability of the collaboration. Agarwal et al. (2009)

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Özener, Ergun, and Savelsbergh: Lane-Exchange Mechanisms for Truckload Carrier CollaborationTransportation Science 45(1), pp. 1–17, © 2011 INFORMS 3

review two forms of collaboration in cargo trans-portation: carrier alliances in sea and air cargo, andshipper collaborations in trucking. Agarwal et al.(2009) investigate such issues as the maximum ben-efit the collaboration can achieve and how to allo-cate benefits as well as how to develop implementablemechanisms to manage the collaboration structureseffectively. Similarly, Krajewska et al. (2008) consider acollaboration among freight carriers. They analyze theprofit margins resulting from horizontal cooperationamong the carriers, where the underlying optimiza-tion problem is a pick-up and delivery problem withtime windows, and then they discuss a mechanism(namely, Shapley value) to allocate the profit marginsamong the collaborators. Kuo et al. (2008) proposethree decision-making strategies for the collaborativeoperation of international rail-based intermodal ser-vices by multiple carriers: (a) train slot-cooperationtechniques, (b) train space-leasing techniques, and(c) train slot-swapping techniques. These techniquesrely on various mechanisms for collaboration amongcarriers. Kuo et al. (2008) show that all three strate-gies result in win-win situations for all collaboratingcarriers.Our perspective in this paper is quite differ-

ent from the centralized approaches studied in theabove-mentioned papers. Contrary to those studies,in our setting the carriers do not form collabora-tions to achieve common objectives and then sharethe resulting benefits. Instead, the carriers collaborateby defining the rules of a lane-exchange mechanism.Each carrier is selfish and pursues only a selfish goal,which is to maximize its individual benefit. The carri-ers are not trying to establish a long-term partnershipand are not trying to develop mechanisms to sustaina collaboration; they are just trying to identify sim-ple lane exchanges that will benefit both parties. As aresult, our methodology is based on noncollaborativegame theory, as opposed to collaborative game the-ory, and models individual strategies as opposed tocentralized decisions.The research on truckload transportation procure-

ment using auction-based methods mostly focuses onthe assignment of lanes from a single shipper to aset of carriers so as to minimize the total transporta-tion cost. Combinatorial auctions are most suitablefor transportation procurement auctions because theyallow the capturing of synergies between shipmentrequests through bundle bids. In this context, theshipper has to solve a winner-determination problemto assign lanes to the carriers. Sheffi (2004) providesa survey on combinatorial auctions in procurementof truckload transportation services. He observes thatsolving large instances is quite challenging becausereal problems include thousands of lanes, dozens orhundreds of carriers, and millions of combination

bids (see also De Vries and Vohra 2003, Caplice andSheffi 2003, and Song and Regan 2003). Elmaghrabyand Keskinocak (2003) give an overview of the chal-lenges in designing and implementing combinatorialauctions and present a case study on how HomeDepot utilizes combinatorial auctions in the procure-ment of transportation services. An, Elmaghraby, andKeskinocak (2005) investigate how the participantsin a combinatorial auction should select their bidsbecause evaluating the bids for all possible bundlesis not practical for both the bidders and the auction-eer. Ledyard et al. (2002) discuss the implementationof “combined-value auctions” for the purchasing oftransportation services by Sears Logistics Services.Chen et al. (2007) study a Vickrey-Clarke-Groves com-binatorial truckload procurement auction, where thecosts of the carriers are assumed to be coming from anetwork optimization problem solved by the carriers.Figliozzi (2006) proposes and analyzes dynamic col-

laborative mechanisms that are incentive compatible.In this setting, there is a set of carriers, each withtheir own set of customers generating a stream ofrandom transportation requests over time. He pro-poses a collaboration structure in which each carrierhas the incentive to submit any transportation requestreceived to the collaborative mechanism, which isbased on sequential second-price auctions. Althoughauction-based mechanisms have certain advantagessuch as truthfulness of the collaborators in second-price auction settings, there exist several practicalissues associated with these types of mechanisms.In a sequential auction structure, the collaboratorsbase their decisions on their current valuations, whichare likely to change in dynamic environments. Onthe other hand, the bundling of items (shipments)puts a computational burden on the participants todetermine the valuation of the bundles, handle anexponential number of bundles, and solve a winner-determination problem.Again, our perspective is quite different. We are

not interested in the transportation procurement pro-cess, i.e., in how shippers select carriers and assignlanes to carriers. Instead, we want to know howcarriers can reduce cost by collaborating, given theset of lanes they have to serve. Our goal is todesign lane-exchange processes to facilitate collabo-ration among carriers. We will measure the effective-ness of our lane-exchange mechanisms by comparingthe solutions they produce to a fully centralized set-ting (which is equivalent to assuming that the carri-ers have merged and have become a single entity).Furthermore, we will quantify the effects of moreadvanced mechanisms that require the exchange ofinformation and side payments.As is common practice in the truckload transporta-

tion procurement literature, we determine the syn-ergy between lanes purely based on the origin and

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destination locations. It should be recognized, though,that this introduces some uncertainty when it comesto achieving network balance because that dependson the loads being tendered on these lanes duringdaily execution.We assume that a carrier divides the lanes that

have to be served into (a) those the carrier wants orneeds to serve himself because of contractual obli-gations (subcontract prohibitions) or perhaps becausethe shippers on these lanes are responsible for a largeportion of the carrier’s business and the carrier wantsto ensure the highest level of service, and (b) thosethe carrier does not necessarily want or need to servehimself.Finally, we assume that the collaborating carriers

are truthful in sharing their cost and network infor-mation when information sharing is required by theagreed-upon lane-exchange setting. Note that, in real-ity, the carriers may lie to try to gain higher bene-fits from the exchange process. In most collaborativesettings, “untruthfulness” can threaten the collabora-tive structure. However, in our proposed exchangemechanisms, a lane exchange will only take placewhen it is profitable for both carriers. Our intuitionis that, therefore, the carriers will rarely benefit fromlying about their costs and their network information.If one of the carriers tries to manipulate the selec-tion of an exchange by lying and the resulting selec-tion is not acceptable to the other carrier, the latterwill not accept the exchange and both carriers willgain nothing from the collaboration. In summary, wepropose exchange mechanisms with voluntary par-ticipation and assumed truthfulness, because sharingincomplete and incorrect information usually resultsin unacceptable exchanges and benefits no one.

4. The MulticarrierLane-Covering Problem

To assess the maximum benefit of collaborating, theoptimal assignment of lanes to carriers has to bedetermined and the optimal set of cycles coveringthe lanes assigned to each carrier needs to be found.We refer to this optimization problem as the multicar-rier lane-covering problem (MCLCP). This setting isequivalent to assuming that the carriers have mergedinto a single carrier with centralized control. We studythis setting primarily to be able to assess the perfor-mance of the lane-exchange mechanisms that will beintroduced in §5.2.The total transportation costs break down into two

components: lane-covering costs and repositioningcosts. The cost of covering a lane depends only onthe carrier assigned to operate on that lane. There-fore, if the carriers have identical cost structures, thelane-covering costs are independent of the assign-ment and routing decisions. On the other hand, the

repositioning costs depend on the synergy among (orcomplementarity of) the set of lanes assigned to acarrier (irrespective of any differences in carrier coststructures) and thus on the assignment and routingdecisions.Our starting point is a set of carriers, each with

two sets of lanes: lanes that have to be served by thecarrier and lanes that can be served by any carrier.(The set of lanes that have to be served by the car-rier may be empty.) Of course, only the lanes thatcan be served by any carrier can be reassigned duringthe optimization. We further assume that the cost ofserving lane �i� j� is the same as the cost of servinglane �j� i� and that the cost of repositioning over �i� j�is a percentage of the cost-serving lane �i� j�. Thesecosts can be different for different carriers. There areno capacity restrictions; each carrier is able to handleall the assigned lanes. Because a lane �i� j� representsa commitment from a carrier to a shipper to providetruckload transportation services from origin i to des-tination j , we allow multiple lanes �i� j�.We model MCLCP on a complete graph G= �N�A�,

where N is the set of nodes 1� �n� and A is the setof arcs. Let K be the set of carriers offering transporta-tion services. Let ckij denote the cost of serving lane�i� j� for carrier k. The repositioning cost coefficientof carrier k is denoted by �k, where 0 < �k ≤ 1, andhence the repositioning cost along �i� j� ∈ A is equalto �kckij for carrier k. The set of lanes that can be cov-ered by any carrier is denoted by L⊆A, whereas theset of lanes that can only be covered by carrier k isdenoted by I k ⊆A. The objective is to cover the lanesat minimum cost.MCLCP can be formulated as an integer linear pro-

gram as follows:

zLP =min∑k∈K

∑�i� j�∈L

ckijxkij +

∑k∈K

�k∑

�i� j�∈Ackijz

kij � (1)

st∑j∈N

xkij −∑j∈N

xkji +∑j∈N

zkij −∑j∈N

zkji = 0�

∀ i ∈N� ∀k ∈K� (2)∑k∈K

xkij ≥ rij +∑k∈K

Ikij ∀ �i� j� ∈ L� (3)

xkij ≥ I kij ∀ �i� j� ∈ I k� ∀k ∈K� (4)

zkij ≥ 0 ∀ �i� j� ∈A� ∀k ∈K� (5)

xkij � zkij ∈� (6)

Variable xkij represents the number of times lane�i� j� ∈ L is assigned to carrier k and variable zkij repre-sents the number of times �i� j� ∈A is used for repo-sitioning by carrier k. Furthermore, I kij represents thenumber of times lane �i� j� ∈ L has to be assignedto carrier k and rij represents the number of times

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lane �i� j� ∈ L has to be assigned to any of the car-riers. The objective is to minimize the sum of thelane-covering costs of all the carriers plus the sum ofthe repositioning costs of all the carriers. Constraints(2) are flow-balance constraints for all the nodes inthe network and all the carriers offering service. Con-straints (3) ensure that the lanes that can be served byany carrier are assigned. Constraints (4) ensure thatthe lanes that can only be served by a specific carrierare assigned. A solution to MCLCP gives the optimalassignment of the lanes to the carriers and the optimalroutes for covering the lanes assigned to each carrier.The objective function represents the minimum costfor serving all lanes.We have presented MCLCP as the optimization

problem that carriers need to solve to determine themost cost-effective way to serve their combined lanesets. However, it is important to note that solvingMCLCP is also equivalent to solving the winner-determination problem in the context of a VCG com-binatorial transportation procurement auction see alsoChen et al. 2007.)Next, we examine the complexity of MCLCP. There

are a few characteristics that impact the complexity:the number of carriers, whether or not carriers havesets of lanes that they have to serve, and whether ornot the carriers have identical cost structures. We havethe following set of results.

Lemma 1. When carriers have no sets of lanes that onlythey can serve and when their cost structures are identical,then MCLCP can be solved in polynomial time for anynumber of carriers.

Proof. The proof follows from the fact that theunconstrained version of the single-carrier lane-covering problem is polynomially solvable (Ergun,Kuyzu, and Savelsbergh 2007a). �

Theorem 1. When there are three or more carriers withnonidentical cost structures, then MCLCP is NP-hard.

Proof. See the appendix. �

As MCLCP is NP-hard, the LP-relaxation of (2)–(6)may not yield an integral solution. Next, we presentan observation concerning the integrality gap.

Lemma 2. The integrality gap for MCLCP is less thanor equal to 100%.

Proof. Let z∗LP be the objective function value of theLP-relaxation. Then, we have

z∗LP ≥∑

�i� j�∈Lmink∈K

ckij rij +∑k∈K

∑�i� j�∈L

ckij Ikij

because the right-hand side represents the cost oftraversing each lane in L using the minimum costcarrier with no repositioning. Furthermore, traversingeach lane in L with its lowest-cost carrier option and

returning back empty along that lane is an integralfeasible solution to MCLCP. Because �k ≤ 1 ∀k ∈K, theobjective function value of this feasible solution—say,zIP—satisfies

zIP ≤ 2{ ∑

�i� j�∈Lmink∈K

ckij rij +∑k∈K

∑�i� j�∈L

ckij Ikij

}

Hence, the integrality gap is less than or equal to100%. �

Because the size of instances of MCLCP encoun-tered in different contexts may differ considerablyand may sometimes be large, we have developeda few heuristic approaches to solve MCLCP andcompared their performances to simply solving theinteger programming (IP) formulation using a com-mercial solver. The first heuristic approach relaxesthe precision of the IP solver by specifying a relativeoptimality stopping criterion. The second heuristicexploits the structure of a solution to the LP-relaxationof (2)–(6) by fixing integral flows along cycles in thesolution of the LP-relaxation at the root node (thusreducing the feasible solution space) before startingthe branch-and-bound process. We refer the readerto Özener (2008) for the details of these approaches.The computational study to evaluate the perfor-mance of these heuristics uses randomly generatedinstances. The results demonstrate that integer pro-gramming with relaxed precision as well as the cycle-fixing approach are effective heuristics for solvingMCLCP because the maximum optimality gap overall instances (of various sizes) is less than 0.04% and0.6%, respectively. The results also show that the com-putational advantages of these heuristic approachesbecomes apparent only for relatively large instances(with at least 300 nodes). Again, we refer the readerto Özener (2008) for details on the computationalstudy.

5. Carrier CollaborationWe consider settings in which several carriers collab-orate by means of bilateral lane exchanges, i.e., bytwo carriers exchanging shipment requests sometimesaccompanied by side payments. The goal is to reduceany geographic imbalances in the carriers’ networksso as to reduce repositioning costs. The carriers areselfish in the sense that their objective is to minimizetheir own costs, although they may be willing to sharea portion of the benefits of a lane exchange if theybelieve this is required for the exchange to happen.The carriers have agreed up front on the rules andregulations governing the lane-exchange process, pri-marily the level of information sharing and whetheror not side payments can be offered. We have cho-sen to focus on bilateral lane exchanges because theyrepresent the simplest setting for exchanges and thus

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facilitate the analysis. This does not prevent severalcarriers from collaborating. It just means that theydo so through a sequence of bilateral lane exchanges.Furthermore, the ideas discussed can be extended tolane exchanges involving more than two carriers.Given a lane-exchange mechanism, i.e., a set of

rules and regulations governing the lane-exchangeprocess, a carrier determines his own strategy. That is,a carrier decides himself on the lane to offer to theother carrier and on the side payment if it is allowedby the exchange mechanism. Depending, again, onthe exchange mechanism, a carrier may have only hisown individual cost and network information, or hemay have information about the other carrier’s costand network. In order for a lane exchange to hap-pen, both carriers must agree to the exchange; hence,the result of the exchange must be acceptable to bothcarriers. This property ensures that the carriers arealways better off participating in the collaboration,which is in line with the selfishness of the carriers.The level of information sharing between carriers

and the decision to allow or disallow side paymentsare the two primary factors defining a lane-exchangemechanism. Carriers may be hesitant to share infor-mation with their competitors and, therefore, maydecide not to do so. This, of course, forces them toselect the lane to offer and to decide on the sidepayment based solely on their own cost and net-work information and thus one somewhat they blind-folded; some beneficial lane-exchange opportunitieswill not be recognized or identified. Limited informa-tion sharing not only restricts the number of possibleexchanges considered but also decreases the possibil-ity of an exchange being accepted because a carrier ismore likely to offer a lane or a side payment that isunacceptable to the other carrier. With informationsharing carriers can estimate the counterstrategies oftheir collaborators, which makes it easier to identifythe best overall strategy. Hence, increasing the level ofinformation sharing usually increases the value of thecollaboration. However, as will be shown later, this isnot always the case.Side payments allow carriers to share some of their

benefits from an exchange with the other carrier.Because an exchange will only take place if both car-riers agree to it, a carrier has to make his offer attrac-tive to the other carrier so as to make it acceptable.Suppose that carrier A offers a lane with low syner-gies with the other lanes in his network. Because ofthe low synergies, the operating cost for this lane ishigh and if the offer is accepted, it will likely resultin considerable cost savings. Carrier B may be able tohandle this lane at a lower cost but, because of pos-sible synergies with the other lanes in his network,the exchange may be unacceptable for carrier B ashis costs may still increase. In this case, carrier A can

share some of his cost savings with carrier B via aside payment and make the exchange offer acceptableto carrier B. Hence, allowing side payments increasesthe number of acceptable exchanges.Designing an effective lane-exchange mechanism

presents several challenges. The mechanism shouldidentify exchanges that are acceptable to the individ-ual carriers while trying to reach a system-optimalsolution (i.e., an optimal solution to MCLCP). Fur-thermore, the mechanism should be computationallyviable and be able to determine a candidate exchangewithout having to enumerate all possible exchanges.Different levels of information sharing and whether

or not side payments are allowed present differentchallenges, all of which a mechanism should accom-modate. For example, without information sharingand side payments, the carriers are myopic whendeciding on a lane to offer and have no means tomake this offer acceptable to the other carrier. If sidepayments are allowed, determining a side paymentfor the offered lane is nontrivial because of the lackof information about the other carrier. With full infor-mation sharing, the number of alternative strategiesis quite large and makes the exchange process com-plicated, especially if side payments are allowed.A carrier has to evaluate all lanes and simultaneouslyevaluate all possible side payments because the ben-efits of a lane exchange depend on both the lane andthe associated side payment.

5.1. Lane Exchanges—Illustrative ExamplesWe demonstrate the challenges associated withdesigning lane-exchange mechanisms by means ofexamples. We consider a network with five cities andthree carriers, namely, A, B, and C. The cost of servinga lane is assumed to be the same for all carriers; thecosts are shown in Figure 3(a). For simplicity, we alsoassume that the cost of using a lane for reposition-ing is equal to the cost of serving the lane. The lanesof each carrier and the optimal way to cover themare shown in Figure 3(b), 3(c), and 3(d), respectively,where the dashed line represents repositioning. Thecorresponding costs are 3, 4.73, and 5.46, respectively.Consider a lane-exchange setting in which there is

no information sharing and there are no side pay-ments. In this setting, the only task of a carrier is toselect the lane to offer to the other carrier. Supposethat carriers A and B try to exchange lanes. Becausea carrier has no information about the other carrier,he is likely to choose a lane that is costly to servesuch as the highest marginal cost lane or the highestper-mile marginal cost lane in the hope that the lanehas a greater synergy within the other carrier’s net-work. Hence, carrier A may offer either lane (3�2) orlane (2�4); the lanes are equivalent from carrier A’spoint of view. Carrier B may offer lane (2�3). As a

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Costs

1

1

1

2

3

4

5

1.5

0.865

1.73

1

1

A

1

2

3

4

5

B

1

2

3

4

5

C

1

2

3

4

5

Cost = 3 Cost = 4.73 Cost = 5.46

(a) (b) (c) (d)

Figure 3 Collaboration Among Three Carriers

consequence, if carrier A decides to offer lane (3�2),the exchange will not be acceptable but if carrier Adecides to offer lane (2�4) instead, both carriers willagree on the exchange as it reduces their costs to 2and 3.73, respectively.Allowing side payments results in more acceptable

lane exchanges. Suppose that carriers B and C try toexchange lanes. Furthermore, let carrier B offer lane(2�3) because it is the highest marginal cost lane and,similarly, let carrier C offer lane (1�4). If the carri-ers exchange these lanes, then their costs become 5.46and 3, respectively. Hence, carrier B will not agree tothe exchange. However, if carrier C is willing to sharesome of his benefits of 546−3= 246 and offer a sidepayment of, say, one unit, then carrier B will acceptthe improved offer because it results in a cost of 546−1 = 446, which is less than 4.73. At the same time,carrier C still benefits because his costs are 3+ 1= 4,which is still less than 5.46. This shows that carrierscan benefit from allowing side payments. However,it is not obvious how a carrier should determine theamount of the side payment, especially without infor-mation about the other carrier’s network.Information sharing allows carriers to calculate the

value and outcome of any possible exchange. There-fore, carriers can initiate lane exchanges with higherbenefits and higher likelihoods of acceptance. Sup-pose that carriers A and B try to exchange lanes andshare network and cost information. Carrier B real-izes that if he offers any lane but (2�3), then car-rier A will reject the exchange regardless of the laneoffer by carrier A itself. Therefore, carrier B offers lane(2�3). Similarly, carrier A realizes that carrier B willoffer lane (2�3) and so he chooses to offer lane (2�4).The exchange will be accepted because both carriersreduce their costs (to 2 and 3.73, respectively).Not surprisingly, with information sharing, deter-

mining a strategy becomes considerably more in-volved. Evaluating each possible lane exchange is not

only challenging and time-consuming but also it maynot yield a dominant solution, i.e., a unique laneexchange that results in the largest cost savings forboth carriers. Consider a network with eight cities andtwo carriers, A and B. The cost of covering a lane isassumed to be the same for all carriers; the costs areshown in Figure 4(a). Furthermore, assume that thecost of using a lane for repositioning is equal to thecost of serving the lane. The networks of carriers Aand B are shown in Figure 4(b). The carriers sharenetwork and cost information. If carrier A offers lane(2�3), then his cost savings will be 0 in case carrier Boffers (3�2) or 2 if carrier B offers (7�6). On the otherhand, if carrier A offers lane (6�7), then his cost sav-ings will be 2 in case carrier B offers (3�2) or 0 ifcarrier B offers (7�6). The situation is identical for car-rier B. That is, there is no dominant solution and thecarriers are still uncertain about their strategies eventhough they have perfect information. The reason forthe uncertainty is that the carriers do not know thestrategy of the other carrier.

(b)

A

2

3 7

6

B

2

3 7

6

(c)

A

1

4

7

6

B

1

4

7

6

(a)

4

2

1

3

8

6

5

7

Network

1

1

1

1

1

1

8

Figure 4 Collaboration with Information Sharing

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A B

(b)(a)

Network

5

3

1

4

10

7

6

8

0.5

1

2 2

1

8

2

9

0.5

0.5

0.5

5

3

1

4

10

7

6

8

2

5

3

1

4

10

7

6

8

9

Figure 5 An Adversary Example for Information Sharing

Finally, although information sharing is generallyhelpful, it is easy to show that this is not alwaysthe case. Suppose the networks of carrier A and Bare as shown in Figure 5. Suppose that carrier A canoffer only lanes (4�1) and (8�6) and carrier B canoffer only lanes (5�3) and (10�7). Without informationsharing, the carriers will likely offer lanes with thehighest marginal costs, i.e., (4�1) and (10�7), respec-tively, which will result in cost savings of 1 for bothcarriers. With information sharing, if carrier A offers(4�1), then his cost savings will be 1 in case carrier Boffers (10�7) or 0 if carrier B offers (5�3). On the otherhand, if carrier A offers lane (8�6), then his cost sav-ings will be 2 in case carrier B offers (10�7) or 0 ifcarrier B offers (5�3). As the potential payoffs fromoffering lane (8�6) dominate the alternative, carrier Achooses to offer lane (8�6). Similarly, carrier B choosesto offer lane (5�3). Thus, their cost savings will be 0whereas without information sharing, their cost sav-ings would have been 1.Also, with information sharing, allowing side pay-

ments is generally beneficial because it increases thenumber of acceptable lane exchanges. However, deter-mining an appropriate side payment with an offeredlane is even more difficult in this setting. The cost sav-ings for a carrier depend on the lane offered by theother carrier so even if a carrier is willing to share thebenefits, he is uncertain about the actual cost savingsof a lane exchange until the moment of the exchange.Given a candidate lane to offer, calculating an appro-priate side payment for all possible counteroffers may

be too time-consuming. Determining an appropriateside payment is further complicated by the fact thatthe other carrier may also offer a side payment.

5.2. Lane-Exchange MechanismsWeproposeandanalyze lane-exchangemechanisms forfourdifferentcarrier-collaborationsettings:no informa-tion sharing and no side payments (NINS), no informa-tion sharing with side payments (NIWS), informationsharing without side payments (WINS), and informa-tion sharing with side payments (WIWS). We first dis-cuss some common properties of these lane-exchangemechanisms and then describe them in detail and dis-cuss their advantages and disadvantages.Let carriers A and B be the two carriers in a lane-

exchange process. Let LA and LB be the sets of lanesthe carriers can offer for exchange. (Recall that somelanes may have to be served by a carrier itself.) Let cAijrepresent the cost of serving lane �i� j� for carrier A,and let �cAij be the repositioning cost for carrier Aalong that same lane. Let z∗A�L

A� be the minimum costof covering all the lanes in the lane set LA by car-rier A, i.e., the optimal objective function value ofthe lane-covering problem (LCP) for carrier A. Themarginal cost MCAij �L

A� of lane �i� j� ∈ LA for carrier Ais z∗A�L

A� − z∗A�LA/�i� j��. Furthermore, let pAij denote

the side payment for lane �i� j� ∈ LA by carrier A,i.e., the amount that carrier A will pay to carrier Bif the lane exchange is accepted. The “payoff” of anexchange to a carrier refers to the cost savings forthe carrier if the lane exchange is accepted and per-formed. The payoff to carrier A is denoted by �Aij�uv,where �i� j� represents the lane offered by carrier Aand �u�v� the lane offered by carrier B. Similarly, thepayoff to carrier B is denoted by �Buv� ij . The payoffsare equal to

�Aij�uv =

MCAij �L

A�−MCAuv�LA\�i� j�∪ �u�v��− pAij + pBuvif exchange occurs�

0 otherwise

and

�Buv� ij =

MCBuv�L

B�−MCBij �LB\�u�v�∪ �i� j��− pBuv + pAijif exchange occurs�

0 otherwise

Note that the payoffs depend on both lanes. That is,the marginal cost of an offered lane depends on thelane that the other carrier offers. Therefore, strategiesthat ignore the other carrier’s actions are less likely tobe effective.Although any acceptable lane exchange results in

benefits for both carriers, a single exchange is not

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likely to capture the full potential of existing net-work synergies. Hence, our lane-exchange mecha-nisms work their way to a “perfect collaboration”through a sequence of bilateral exchanges. In eachiteration, the lane-exchange mechanisms greedilychoose an exchange with high probability of accep-tance and high combined benefits. Even though thelane-exchange mechanisms may not identify the bestpossible exchange in terms of combined payoffs in agiven iteration, our computational experiments showthat upon termination the lane-exchange mechanismsdo result in close to perfect collaborations.Another important point is that the lane mecha-

nisms do not guarantee truthfulness; hence, the carri-ers may lie in an attempt to gain higher benefits. Thiscan be seen as a potential weakness of the proposedlane-exchange mechanisms. However, the exchangemechanisms rely on voluntary participation and anexchange only takes place when the payoffs are posi-tive for both carriers. Therefore, it is in the best inter-est of each carrier to make the exchange appealingto the other carrier and it is not obvious that lyingis a good way to do so. Furthermore, it is extremelychallenging and may even be impossible to provethe truthfulness of the collaborators in this type ofnetwork optimization setting because of the complexinteractions.

5.2.1. No Information Sharing and No Side Pay-ments (NINS). In this most basic lane-exchange set-ting, a carrier only needs to select the lane tooffer to the other carrier. In the following discus-sion about the lane-exchange mechanism we pro-pose, we assume that the carriers offer the laneswith the highest marginal costs. That is, carrier Aoffers argmin�i� j�∈LAMC

Aij �L

A�� and carrier B offersargmin�u�v�∈LBMC

Buv�L

B��. The selection strategy iscompletely based on a carrier’s individual networkand ignores the other carrier’s network. The motiva-tion for this selection strategy is that it maximizes theknown benefit from a potential exchange as it discardsthe lane with highest cost.An alternative selection strategy is to offer the lane

with the highest marginal cost per mile, which isan indicator for the repositioning requirement of thelane. The motivation for this selection strategy is thatthe highest marginal cost lane may be a long lanerather than a lane with low synergies. On the otherhand, the lane with the highest marginal cost per milehas limited or no synergies with the existing lanes ofthe carrier.The computational requirements of both selection

strategies are small because the exchange mecha-nism is simple and only requires the ranking of thelanes according to their marginal costs (or per milemarginal costs).

The following algorithm evaluates the benefits ofthis exchange mechanism for two carriers, A and B,where we assume that the carriers continue to ex-change lanes as long as beneficial lane exchanges areidentified.

NINSStep 1. Rank the lanes that can be offered according

to marginal costs (or per mile marginal costs) for bothcarriers and select the best one from each to form thelane-exchange pair.Step 2. Let ��i� j�� u�v�� be the selected pair of

lanes. Compute the payoffs for each carrier:

�Aij�uv =MCAij �LA�−MCAuv(LA\�i� j�∪ �u�v�

)�

�Buv� ij =MCBuv�LB�−MCBij(LB\�u�v�∪ �i� j�

)

Step 3. If the individual payoffs are nonnegative,i.e., �A ≥ 0 and �B ≥ 0, and the combined payoff isstrictly positive, i.e., �A + �B > �, then execute theexchange, update the lane sets of the carriers, andreturn to Step 1. Otherwise, discard the exchange andstop.This algorithm terminates after a finite number of

iterations because the combined payoff is greater than� at each iteration and the minimum total cost for thecarriers is positive. The only computationally inten-sive step in the algorithm is the ranking of the lanesaccording to their marginal costs. To compute themarginal cost of a lane, we have to solve two LCPs.Fortunately, LCP can be solved efficiently as a min-cost flow problem.

5.2.2. No Information Sharing with Side Pay-ments (NIWS). In this lane-exchange setting, acarrier needs to select a lane to offer and needsto determine an appropriate side payment for theoffered lane. Note that we assume the side paymentsfor lanes are set beforehand. In the lane mechanismwe propose, the carriers still offer the lanes with thehighest marginal costs (or highest marginal costs permile). However, because a lane exchange may resultin cost savings, the carriers have an incentive to makethe offer attractive in order to realize the cost savings.Side payments provide a mechanism for making anoffered lane more attractive. We propose a methodfor determining side payments that is based on thesynergies of a lane within the network of the carrier.We can assess the synergies of a lane by comparingthe marginal cost to the lane cost. For instance, whenthe marginal cost of lane �i� j� for carrier A is equal to�1+��cAij , the lane has no synergies within the networkof carrier A because the marginal cost value impliesthat carrier A has to reposition from j to i. On theother hand, when the marginal cost of lane �i� j� forcarrier A is equal to �1 − ��cAij , the lane has perfect

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synergies within the network of carrier A because itimplies that covering the lane also serves as a reposi-tioning opportunity for other lanes in the network ofcarrier A.As we increase the side payment, the likelihood of

acceptance increases. However, at the same time, thebenefits from the lane exchange decrease. As such,side payments determine how the cost savings thatresult from exchanging lanes are shared among thecarriers. As a carrier does not know these cost sav-ings in advance because they depend on the othercarrier’s action, an estimate has to be used. Oneoption is to use MCAij �L

A�− cAij as an estimate of thecost savings. When the marginal cost of a lane ishigh compared to the lane cost, the synergies withinthe network are low. Consequently, the other carriermay be able to cover this lane more cheaply becauseof potential synergies within his network. Sharingthe estimated cost savings equally provides a reason-able balance between reducing the risk of a rejec-tion by the other carrier and reducing the size of thebenefit of the lane exchange. Thus, the carrier mayput forth a side payment equal to half the differ-ence between the marginal cost and original lane cost:�MCAij �L

A�− cAij �/2.When the marginal cost of a lane is low compared

to the lane cost (but still greater than the marginalcost), there are synergies within the network. Conse-quently, it is less likely that the other carrier can coverthis lane more cheaply as a result of potential syner-gies within his network. Therefore, the carrier’s pri-ority is to make the exchange attractive to the othercarrier so as to avoid a rejection of the lane exchange.This means that a higher portion of the estimated costsavings has to be “transferred” to the other carrier toincrease the likelihood of acceptance. Thus, the car-rier may opt to offer a side payment that is more thanhalf the difference between the marginal cost and theoriginal cost. We have chosen to use the adjustmentfactor �1+ ��cAij /�MC

Aij �L

A��, which results in the fol-lowing rule for determining the side payment:

pAij =

�1+ ��cAij

MCAij �LA�× MC

Aij �L

A�− cAij

2if MCAij �L

A�≥ cAij �

0 otherwise

Note that the adjustment factor is a value between 1and 2 (because � ≤ 1) and that it gets larger when themarginal cost gets smaller.The following algorithm evaluates the benefits of

this exchange mechanism for two carriers, A and B,where we assume that the carriers continue to ex-change lanes as long as beneficial lane exchanges areidentified.

NIWSStep 1. Rank the lanes that can be offered according

to marginal costs (or per mile marginal costs) for bothcarriers and select the best one from each to form thelane-exchange pair.Step 2. Let ��i� j�� u�v�� be the selected pair of

lanes. Compute the side payments on the lanes:

pAij =

�1+ ��cAij

2×MCAij �LA��MCAij �L

A�− cAij �

if MCAij �LA� > cAij �

0 otherwise�

pBuv =

�1+ ��cBuv2×MCBuv�LB�

�MCBuv�LB�− cBuv�

if MCBuv�LB� > cBuv�

0 otherwise

Compute the payoffs for each carrier:

�Aij�uv =MCAij �LA�−MCAuv�LA\�i� j�∪ �u�v��− pAij + pBuv�

�Buv� ij =MCBuv�LB�−MCBij �LB\�u�v�∪ �i� j��− pBuv + pAij

Step 3. If the individual payoffs are nonnegative,i.e., �A ≥ 0 and �B ≥ 0, and the combined payoff isstrictly positive, i.e., �A + �B > �, then execute theexchange, update the lane sets of the carriers, andreturn to Step 1. Otherwise, discard the exchange andstop.As before, the algorithm terminates after a finite

number of iterations and the only computationallyintensive step is computing the marginal cost of thelanes.

5.2.3. With Information Sharing and No SidePayments (WINS). In this lane-exchange setting, thecarriers share information to be better able to identifylane-exchange opportunities. More specifically, theyshare their cost structure as well as the lanes theyneed to serve. With information about the other car-rier, a carrier can consider and analyze potential coun-teroffers and estimate the outcome of possible laneexchanges, and then choose a strategy accordingly toincrease the value of the benefits as well as the prob-ability of acceptance of a lane exchange.A carrier only needs to select a lane to offer to the

other carrier. However, this selection should be basednot only on information about their own networkbut also on information about the other carrier’s net-work and thus expectations about the other carrier’sstrategy. We propose to model the selection prob-lem as a two-person nonzero sum game, where therows correspond to lanes from carrier A and columnscorrespond to lanes from carrier B. As before, the

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payoff for a carrier for a given lane-exchange pairis the value of the cost savings resulting from theexchange. Both carriers can enumerate all possiblelane-exchange pairs and compute the correspondingpayoffs (thus constructing the payoff matrix for thegame). Then, based on this information, the carrierscan choose the strategy that maximizes the minimumpayoff to themselves, which corresponds to findinga Nash equilibrium of the two-person nonzero sumgame.We have the following results.

Lemma 3. A mixed-strategy Nash equilibrium pointalways exists for the lane-exchange game.

Proof. The proof follows directly from the well-known Nash theorem “every finite strategic game hasa mixed-strategy Nash equilibrium” (Osborne andRubinstein 1994). �

Lemma 4. A pure-strategy Nash equilibrium point maynot exist for the lane-exchange game.

Proof. Consider the example in Figure 6. Two car-riers (namely, I and II) have four lanes each (A, B,E, F} and {C, D, G, H}, respectively) and identical coststructures on this network (cost figures are shown inFigure 6). Suppose that carrier I can offer either lane Aor B for an exchange and carrier II can offer eitherlane C or D. Hence, the resulting payoff matrix is asfollows: [

�25�107� �107�25�

�107�25� �25�107�

]

C

B

14.1

10

14.1

20

22.4

A

D

E

F

G

H

Figure 6 An Instance with No Pure-Strategy Nash Equilibrium

Consequently, this instance has no pure-strategy Nashequilibrium point. �

Based on these results, we first check whether thereexists a pure-strategy Nash equilibrium point for thelane-exchange game. If one or more pure-strategyNash equilibria exist, then we select the one with thehighest combined benefit to both carriers. When apure-strategy Nash equilibrium point does not exist,we find the set of mixed-strategy equilibria and selectthe one with the highest combined benefit to bothcarriers. For that mixed-equilibrium point, we selectthe row and column with the highest probabilitiesas the strategies for the carriers. Note that regard-less of the type of the equilibrium, this method maynot return the best lane-exchange pair from a system-wide perspective. This is a result of the selfishness ofthe carriers. Intuitively, though, this method shouldbe quite effective in terms of the resulting benefits.With information sharing, the carriers can evaluatethe response of the other carrier and thus choose astrategy that maximizes the expected cost savings.The following algorithm evaluates the benefits of


WINSStep 1. For every possible pair of lanes ��i� j��

�u�v��, compute the payoffs for each carrier:

�Aij�uv =MCAij �LA�−MCAuv�LA\�i� j�∪ �u�v��

�Buv� ij =MCBuv�LB�−MCBij �LB\�u�v�∪ �i� j��

�Aij�uv ={ �Aij�uv �Aij�uv� �Buv� ij ≥ 0, �Aij�uv + �Buv� ij > 0�0 otherwise�

�Buv� ij ={ �Buv� ij �Aij�uv� �Buv� ij ≥ 0, �Aij�uv + �Buv� ij > 0�0 otherwise

Step 2. Construct the payoff matrix using �Asand �Bs.Step 3. Solve the two-person nonzero sum game

to find the pure-strategy Nash equilibrium points.If multiple pure-strategy Nash equilibria exist, thenselect the one with the highest combined benefits��A+�B�. If no pure-strategy Nash equilibrium pointexists, solve the game to find the mixed-strategy Nashequilibrium points. If multiple mixed Nash equilibriaexist, then select the one with the highest combinedbenefits ��A + �B�. For the selected mixed-strategyNash equilibrium point, select the row and columnstrategies with highest probabilities and take corre-sponding payoffs as the payoffs of the carriers.

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Step 4. If the individual payoffs are nonnegative,i.e., �A ≥ 0 and �B ≥ 0, and the combined payoff isstrictly positive, i.e., �A + �B > �, then execute theexchange, update the lane sets of the carriers, andreturn to Step 1. Otherwise, discard the exchange andstop.The algorithm terminates after a finite number of

iterations. This algorithm is much more computation-ally intensive. To construct the payoff matrix, we haveto compute the payoffs for all possible lane-exchangepairs. For a given lane-exchange pair, this involvessolving four LCPs (two for each marginal cost calcu-lation). After constructing the payoff matrix, we haveto find the Nash equilibria of the game.

5.2.4. With Information and with Side Payments(WIWS). As mentioned before, by allowing side pay-ments, the number of lane-exchange opportunitiesincreases. Observe that with information sharing butwithout side payoffs, if for a given lane-exchange pairone of the payoffs is negative, the exchange will not beacceptable. On the other hand, the payoff for the offer-ing carrier may be quite high because of the fact thathe is giving away a high-cost lane. A side paymentmay result in nonnegative payoffs for both carriers.In this lane-exchange setting, a carrier needs to

select a lane to offer and needs to determine an appro-priate side payment for the offered lane. Again, wepropose to model the decision process as a two-personnonzero sum game, where the rows correspond tolanes from carrier A and the columns correspond tolanes from carrier B. The only difference is that thepayoff values have to include any side payments.Determining appropriate side payments, however, isquite complicated. The basic idea of a side payment isto make an offer attractive to the other carrier. How-ever, when a carrier includes a side payment with alane, it affects the payoffs of all the lane-exchange pairsthat involve the lane. Furthermore, the other carriermay include a side payment with the offered lane aswell, in effect returning a portion of the side payment.We propose a “conservative approach” for determin-ing side payments. If carrier A offers lane �i� j� andcarrier B would choose �u�v� as the counteroffer tomaximize his payoff, i.e., �u�v�= argmax�u�v�∈LB �Buv� ijbut �Buv� ij < 0, then carrier A will offer side paymentpAij = � − �Buv� ij , which will result in a positive payofffor carrier B. Of course, carrier A will do this onlyif �Aij�uv + �Buv� ij > �; otherwise, his own payoff wouldbecome negative.The following algorithm evaluates the benefits of


WIWSStep 1. For every possible lane-exchange pair ��i� j��

��u�v��, compute the tentative payoffs for eachcarrier:

�Aij�uv =MCAij �LA�−MCAuv�LA\�i� j�∪ �u�v��

�Buv� ij =MCBuv�LB�−MCBij �LB\�u�v�∪ �i� j��

Step 2. For every lane �i� j� of carrier A, determine�u�v�= argmax�u�v�∈LB �Buv� ij and

pAij ={�− �Buv� ij �Buv� ij < 0, �

Aij�uv + �Buv� ij > ��

0 otherwise

For every lane �u�v� of carrier B, determine �i� j� =argmax�i� j�∈LA �Aij�uv and

pBuv ={�− �Aij�uv �Aij�uv < 0, �

Aij�uv + �Buv� ij > ��

0 otherwise

Update the tentative payoff values as

�Aij�uv = �Aij�uv − pAij + pBuv�

�Buv� ij = �Buv� ij + pAij − pBuv

For every possible lane-exchange pair ��i� j�� u�v��,compute the actual payoffs

�Aij�uv ={ �Aij�uv �Aij�uv� �Buv� ij ≥ 0, �Aij�uv + �Buv� ij > 0�0 otherwise�

�Buv� ij ={ �Buv� ij �Aij�uv� �Buv� ij ≥ 0, �Aij�uv + �Buv� ij > 0�0 otherwise

Step 3. Construct the payoff matrix using �Asand �Bs.Step 4. Solve the two-person nonzero sum game

to find the pure-strategy Nash equilibrium points.If multiple pure-strategy Nash equilibria exist, thenselect the one with the highest combined benefits��A+�B�. If no pure-strategy Nash equilibria exists,solve the game to find the mixed-strategy Nash equi-librium points. If multiple mixed-strategy Nash equili-bria exist, then select the one with the highest benefits��A + �B�. For the select mixed-strategy Nash equili-bria, select the row and column strategies with high-est probabilities and take corresponding payoffs as thepayoffs of the carriers.Step 5. If the individual payoffs are nonnegative,

i.e., �A ≥ 0 and �B ≥ 0, and the combined payoff isstrictly positive, i.e., �A + �B > �, then execute theexchange, update the lane sets of the carriers, andreturn to Step 1. Otherwise, discard the exchangeand stop.

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This algorithm terminates after a finite number ofiterations and has the same computational challengesas the algorithm for WINS.Computational Efficiency. All proposed algorithms

require the solution of a large number of LCPs to com-pute the marginal costs of the lanes of the carriers.The following property concerning the marginal costof a lane allows us to significantly reduce the compu-tational time of our algorithms.

Theorem 2. The marginal cost of a lane is less than orequal to the value of the corresponding dual variable in anoptimal solution to the dual of the lane-covering problem.

Proof. Let Lk be the lane set of carrier k and rkij bethe number of times lane �i� j� ∈ Lk is required to becovered. Hence, for a carrier k, the dual of the lane-covering problem is as follows:

DLCPk

z�Lk�=max ∑�i� j�∈Lk

rkij akij � (7)

st akij + yki − yk

j = ckij ∀ �i� j� ∈ Lk� (8)

yki − yk

j ≤ �ckij ∀ �i� j� ∈A� (9)

akij ≥ 0 ∀ �i� j� ∈ Lk (10)

Let �ak� yk� represent the optimal solution to DLCPk.Because the optimal objective function value of thedual LCP is equal to the total cost of covering all thelanes by carrier k, the marginal cost of lane �u�v� tocarrier k’s network is equal to

MCkuv�L

k�= z∗�Lk ∪ �u�v��− z∗�Lk��

where z∗� · � represents the optimal objective functionvalue of the associated dual LP. After adding lane�u�v� to the network, the resulting cost of carrier k canbe determined by solving the modified dual problem

DLCPk

z�Lk∪�u�v��=max ∑�i�j�∈Lk

rkij akij+akuv� (11)

st akij+yki −yk

j =ckij

∀�i�j�∈Lk� (12)

akuv+yku−yk

v=ckuv� (13)

yki −yk

j ≤�ckij ∀�i�j�∈A� (14)

akij ≥0 ∀�i�j�∈Lk∪�u�v� (15)

Let �ak� yk� represent the optimal solution to DLCPk.

Because �ak� yk� is feasible for the original dual prob-lem DLCPk, we have∑

�i� j�∈Lkrkij aij ≤

∑�i� j�∈Lk

rkij aij

because DLCPk is a maximization problem. Combin-ing these two results, we obtain the following prop-erty for the marginal costs:

MCkuv�L

k� = z∗�Lk ∪ �u�v��− z∗�Lk�

= ∑�i� j�∈Lk

rkij aij + akij −∑

�i� j�∈Lkrkij aij ≤ akij �

The optimal values of the dual variables can eas-ily be found using linear programming technology.Consequently, we have an efficient mechanism forobtaining upper bounds on the marginal costs of thelanes, which we will use instead of the marginal costsunless these values are insufficient. For example, thelane-exchange mechanism without information shar-ing and without side payments requires the identi-fication of the lane with the highest marginal cost.We sort the lanes based on the upper bounds onthe marginal cost and compute the actual marginalcost for the lane with the largest upper bound, whichprovides a lower bound on the value of the highestmarginal cost. For the remaining lanes, if the upperbound is less than or equal to the lower bound onthe value of the highest marginal cost, we simply skipthe computation of the true marginal cost. Otherwise,we compute the true marginal cost of the lane andupdate the lower bound if necessary. Computationalexperience shows that this simple procedure signifi-cantly reduces computation times. The lane-exchangemechanism with information sharing but without sidepayments can take advantage of Theorem 2. The com-putation of the tentative payoffs ( �s) involves twomarginal costs. One of these can be replaced with itsupper bound to get an upper bound on the payoff.Only if the upper bound on the payoff is positive dowe compute the actual marginal cost of the lane forwhich we have used the upper bound. Let aAij andaBuv represent the relevant upper bounds. Then, weexamine

�Aij�uv = aAij −MCAuv�LA ∪ �u�v��

and�Buv� ij = aBuv −MCBij �LB ∪ �i� j��

to see if true payoff values need to be computed. Thesame idea can be applied if there is information shar-ing and if there are side payments. Computationalexperience shows significant reductions in computa-tion times.

5.3. Computational StudyWe have carried out a computational study to eval-uate the practical performance of the proposed lane-exchange mechanisms. (Note that when there is noinformation sharing and there are no side payments,it is easy to construct examples in which each carrier

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initially serves all of his lanes by out-and-back toursand there exist lane exchanges that completely elim-inate the repositioning but where the lane-exchangemechanisms fail to identify any profitable exchangesso that there is an optimality gap of 100%. Witha little more effort, examples with similar optimal-ity gaps can also be constructed for the more com-plex settings.)We generated a large set of instances in a 1,000-

mile × 1,000-mile square region by varying thenumber of locations, the average number of lanes inci-dent to a location, the number of clusters, and theratio of cluster locations to total locations. Lanes arecreated by randomly picking an origin and a desti-nation, where we ensure that no lanes are generatedbetween two locations in the same cluster. The costof traveling between locations with a full truckload isequal to the Euclidean distance between the locationsand the repositioning cost coefficient for all carriersis equal to 0.75. We generated 270 instances with 100,200, and 300 locations, with 2, 4, and 6 as the averagenumbers of lanes incident to a location, with 4 clus-ters, and with 45% of the locations within clusters. Tobe able to investigate the impact of lanes that haveto be served by the carrier, we created three addi-tional instances for every instance by adding lanesthat have to be served by a carrier. (Thus, the numberof lanes that can be exchanged is the same for the fourinstances.) The 3 additional instances are created byadding 0.5, 1, and 2 times the number of lanes that canbe exchanged. That means that if the original instancehas 120 lanes, the variant with 0.5 times the lanes thathave to be served by a carrier added has a total of180 lanes, i.e., 120 that can be exchanged (assignedrandomly to the two carriers) and 60 that cannotbe exchanged (assigned randomly and evenly among

Table 1 Optimality Gaps for the Proposed Lane-Exchange Mechanisms

Random NINS cpu NINS NIWS cpu NIWS WINS cpu WINS WIWS cpu WIWS

40 lanes0 5�66 3�55 3�20 3�14 2�00 0�37 74�60 0�16 268�000.5 8�07 4�76 3�00 3�81 2�40 0�92 69�40 0�36 302�201 5�45 4�50 2�80 4�69 1�80 0�77 56�80 0�26 239�602 11�12 8�03 2�60 7�48 1�80 1�97 82�00 0�73 352�20

80 lanes0 3�34 2�28 3�80 2�29 2�40 0�11 233�20 0�07 1�034�000.5 5�31 4�52 3�20 4�07 1�60 0�34 277�40 0�20 1�345�401 5�48 4�79 3�00 4�71 1�40 0�70 270�60 0�55 1�233�002 4�42 4�42 2�20 4�21 1�20 0�69 224�40 0�31 1�192�60

120 lanes0 3�27 2�61 3�20 2�61 1�40 0�13 697�40 0�03 3�042�600.5 3�86 2�03 3�80 1�85 2�60 0�12 596�60 0�13 2�747�401 4�96 3�59 3�40 3�20 2�40 0�13 879�00 0�09 3�714�802 5�61 4�31 3�20 4�54 1�60 0�40 817�20 0�29 4�360�00

Average 5�54 4�12 3�12 3�88 1�88 0�55 356�55 0�27 1�652�65

the carriers). All the experiments assume two carri-ers with identical cost structures, which is the settingwith the least potential for cost-savings opportunities.Our goal is to show that bilateral lane-exchange

mechanisms are capable of identifying and exploitingthe synergies that exist among the lanes of the twocarriers. For a given random assignment of lanes tocarriers A and B, we determine the maximum pos-sible savings by computing the minimum cost forcarrier A (obtained by solving LCP over the lanesassigned to carrier A) plus the minimum cost forcarrier B (obtained by solving LCP over the lanesassigned to carrier B) minus the cost of a centralizedsolution (obtained by solving MCLCP over all lanes).We use the term optimality gap to refer to these max-imum possible savings as a percentage of the cost ofthe perfect collaboration.We evaluate each of the four lane-exchange

mechanisms (NINS, NIWS, WINS, and WIWS) bycomputing the optimality gap after the lane-exchangemechanism has been applied. We solve the two-person nonzero sum games in mechanisms WINS andWIWS using the Gambit software package (Ruchira2003). Table 1 summarizes the performance of theexchange methods. We report the optimality gapfor the random assignment (“Random”), the aver-age optimality gap for mechanism NINS (“NINS”),the average computation time for mechanism NINSin CPU seconds (“cpu NINS”), the average optimal-ity gap for mechanism NIWS (“NIWS”), the aver-age computation time for mechanism NIWS (“cpuNIWS”), the average optimality gap for mechanismWINS (“WINS”), the average computation time formechanism WINS (“cpu WINS”), the average opti-mality gap for mechanism WIWS (“WIWS”), andthe average computation time for mechanism WIWS(“cpu WIWS”). For each instance size, we present

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Table 2 Optimality Gaps for the Proposed Lane-Exchange Mechanisms When the Carriers Have Geographically Separated Sets of Lanes That TheyHave to Serve

Random NINS cpu NINS NIWS cpu NIWS WINS cpu WINS WIWS cpu WIWS

40 lanes0.5 5�23 4�10 3�20 4�10 1�20 0�70 64�00 0�34 277�401 6�64 4�63 2�80 4�37 2�00 1�32 60�20 0�54 229�402 10�15 6�91 3�00 5�70 2�00 2�29 63�40 0�59 236�60

80 lanes0.5 10�55 4�47 4�60 4�12 3�00 0�24 543�40 0�27 1�651�001 11�27 5�04 4�40 3�93 4�40 0�41 503�00 0�32 1�549�802 7�37 5�25 3�40 4�28 2�80 0�63 368�60 0�43 1�606�00

120 lanes0.5 5�81 2�87 5�40 2�56 4�00 0�19 1�094�20 0�10 3�677�801 4�19 3�21 4�00 2�98 2�40 0�29 720�60 0�07 3�864�202 4�67 3�39 3�40 3�26 2�00 0�40 690�60 0�24 3�987�00

Average 7�32 4�43 3�80 3�92 2�64 0�72 456�44 0�32 1�897�69

summary statistics for the different levels of lanes thathave to be served by the carrier itself.The computational experiments reveal that the pro-

posed lane-exchange mechanisms perform quite welland are capable of identifying and exploiting the syn-ergies among the lanes of the two carriers. The aver-age optimality gap for NINS is 4.12% and the averageoptimality gap for of NIWS is 3.88%, which impliesthat 26% and 30% of the possible cost savings hasbeen realized. Although NIWS outperforms NINS onaverage, for some instances NINS has a lower opti-mality gap than NIWS. As we mentioned earlier,allowing side payments is not always beneficial andhere we observe this phenomenon in our experiments.It is also clear that lane-exchange mechanisms

with information sharing perform considerably betterthan those without information sharing. The aver-age optimality gap for WINS is 0.55% and the aver-age optimality gap for WIWS is only 0.27%, whichimplies that 90% and 95% of the possible cost sav-ings have been realized. In fact, in most of theinstances WIWS achieves the optimal solution (i.e.,the centralized solution representing perfect collabo-ration). Information sharing allows carriers to selecttheir strategies based on the anticipated counterof-fers and thus allows carriers to identify the best pos-sible lane exchange. Allowing side payments helps,too, but that effect is much less significant than shar-ing information.Next, we generate 60 instances in which the sets

of lanes that have to be served by a carrier aregeographically clustered in different regions. This isaccomplished by dividing the square region into anupper triangular area and a complementing lowertriangular area, and generating the lanes that haveto be served by one carrier in the upper triangu-lar region and the lanes that have to be served bythe other carrier in the lower triangular region. The

results can be found in Table 2. For these instances,the average optimality gap for NINS and NIWS is4.43% and 3.92%, respectively. These results implythat 40% and 47% of the possible cost savings havebeen realized; hence, the performance is better thanwhen the lanes that have to be served by the carri-ers are randomly assigned to carriers and have nospecial geographical properties. The average optimal-ity gap for WINS and WIWS are 0.72% and 0.32%,respectively, and these results imply that 90% and96% of the possible cost savings have been realized.Note that the nature of the geographically separatedinstances is such that the carriers need to have pre-existing lane sets that cannot be exchanged. Hence,in Table 2, there are no instances without a preexist-ing network (i.e., no zero instances as in Table 1).Therefore, for a fair comparison of the performance ofthe lane-exchange mechanisms on random and geo-graphically separated instances, we should excludethe instances without a preexisting network from theset of random instances. The adjusted average opti-mality gaps for random assignment, NINS, NIWS,WINS, and WIWS for the remaining instances become6.03%, 4.55%, 4.28%, 0.67%, and 0.32%, respectively.A comparison of the average optimality gaps for ran-dom and geographically separated instances suggeststhat the performance of the lane exchange mecha-nisms is slightly better when the carriers have a “clearidentity” in the form of the geographically separatedand clustered sets of lanes they have to serve. This isprobably because of the fact that fewer synergies arisein such situations.The computation times of the exchange mecha-

nisms are within acceptable limits. Although the aver-age computation times of the proposed mechanismsincrease as the mechanisms becomes more complex(and more effective), the highest computation time isstill less than 4,500 seconds.

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AcknowledgmentsÖzlem Ergun and Martin Savelsbergh were partially sup-ported by NSF Grant DMI-0427446.

Appendix. Complexity Proof

Theorem 3. When there are three or more carriers with non-identical cost structures, then MCLCP is NP-hard.

Proof.We prove theorem 3 by a reduction from 3-SAT. In3-SAT, we have a set of Boolean variables �= B1� �Bn�and a set of clauses � = CLA1� �CLAm�. Each clausehas exactly three literals, where a literal corresponds to aBoolean variable Bi or its complement Bi. A clause is sat-isfied when it contains at least one true literal. The ques-tion is whether every clause can be satisfied by setting eachBoolean variable to either true or false.We give a polynomial reduction from 3-SAT to MCLCP

with three or more carriers and nonidentical cost structures(3+ −MCLCP).Each clause is converted to an arc in such a way that these

arcs form a path, the so-called clause path (see Figure A1).For each Boolean variable Bi, we construct a network withan upper path corresponding to the variable itself and alower path corresponding to its complement. An exampleis depicted in Figure A2, where the upper path is shownin solid lines and the lower path is shown in dashed lines.Each of these paths intersect the clause path whenever thevariable appears in the clause. In the example, the upperpath intersects with the clause path twice, in arcs �2�3� and�4�5�, which indicates that variable Bi is included in bothCLA2 and CLA4. Similarly, the lower path intersects withthe clause path once, in arc �3�4�, which indicates that Bi

is only included in CLA3. The network is completed with

1 2CLA1

3 4 5 mCLA2 CLA3 CLA4

m +1CLAm

Figure A1 The Clause Path

1 2CLA1

CLA2 CLA3 CLA4 CLAm

Upper path

Lower path

Shared arc

Returning arc

3 4 5 m m +1

Figure A2 Network Corresponding to a Boolean Variable

two more arcs: a shared arc and a returning arc (also shownin Figure A2). The lane set L consists of the arcs in theclause path and the shared arcs in each of the networksassociated with the Boolean variables. We set the rij valuefor each clause arc and for each shared arc to one. The latterguarantees that there is at least a unit flow on either theupper or the lower path of every Boolean variable.Next, we create a carrier for each Boolean variable in

the sense that a carrier can only cover arcs in the networkassociated with the Boolean variable. This can be done bysetting the cost of all other arcs to infinity for that carrier.This guarantees that a flow in the network associated witha Boolean variable stays on the lower or upper path. Forthe arcs in the network associated with a Boolean variable,we set the costs to zero except for the shared arc, which getsa cost of one. Repositioning costs are assumed to be equalto the original lane costs for every carrier.Note that each clause arc appears in exactly three Boolean

variable paths (either an upper or a lower path). A clause issatisfied if and only if the corresponding clause arc is cov-ered by at least one of the three carriers associated with theBoolean variables. Note that, by setting rij equal to one forall the clause arcs, we ensure that each clause in � is sat-isfied. The constructed instance of 3+ −MCLCP is alwaysfeasible because by sending a unit flow on all upper andlower paths corresponding to the Boolean variables, we willsatisfy every clause. In that case, the cost for each carrier isequal to two because the cost of the shared arc is equal toone. Thus, the total cost of the carriers is equal to 2×�n� and,hence, an instance of 3-SAT is feasible if and only if the cor-responding 3+ −MCLCP instance has an optimal solutionwith a value of �n�. Any feasible solution with cost strictlygreater than �n� will have at least one Boolean variable with

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a flow of one on both the upper and lower path, whichimplies that the 3-SAT instance is not feasible.We conclude that MCLCP with three or more carriers and

nonidentical cost structures is NP-hard. �

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lane-exchange mechanisms for truckload carrier collaboration

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