Carrier assignment models in transportation procurement

Transcription

1 Journal of he Operaional Research Sociey (26) 57, r 26 Operaional Research Sociey Ld. All righs reserved /6 $3. arrier assignmen models in ransporaion procuremen Y Guo 1, A Lim 2, B Rodrigues 3 * and Y Zhu 2 1 Naional Universiy of Singapore, Singapore; 2 Hong Kong Universiy of Science and Technology, Hong Kong; and 3 Singapore Managemen Universiy, Singapore This paper exends carrier assignmen models used in winner deerminaion aucions for ransporaion procuremen o include shipper non-price objecives and carrier ransi poin coss. The models are unlike radiional carrier assignmen models which incorporae only carrier lane bids, and differen from combinaorial aucion models which focus on packes of lanes wihou considering ransi poin coss. We develop soluions, including meaheurisics, for he new models and hrough compuaional experimenaion show ha he algorihms work well and can be easily implemened. Journal of he Operaional Research Sociey (26) 57, doi:1.157/palgrave.jors Published online 28 December 25 Keywords: ransporaion; procuremen; bid; meaheurisics 1. Inroducion In he huge rucking marke, valued a $6 billion in he Unied Saes alone (American Trucking Associaion, 22; aplice and Sheffi, 23) procuring ransporaion services is crucial o shippers who mus secure high-qualiy carrier services while conrolling supply chain coss. In buying ransporaion services, a shipper ypically enders a reques for quoes (RFQ) for a nework of lanes following a bid preparaion sage (Rhinehar, 1989; Foser and Srasser, 1991; Gibson e al, 1993; aplice and Sheffi, 23; Sheffi, 24), where a lane is a one-way movemen from an origin o a desinaion wih an associaed se of shipmens for he period covered by he RFQ (aplice and Sheffi, 23). Once bids are received, a bid-analysis exercise is used o allocae lanes o carriers. Foser and Srasser (1991) sudied RFQ aucions where he shipper provides a lis of lanes o carriers o bid for, and decides he winners using a single price crierion. RFQ aucions coninue o be used in ransporaion procuremen as repored by Sheffi (24). In combinaorial aucion (Sheffi, 24) mechanisms, used by many shippers and hirdpary-logisics providers o encourage more aggressive bidding, shippers reques bids for groups of lanes, in addiion o individual lanes. This allows carriers o form bid packages based on heir economics exising clien base, driver domiciles, mainenance neworks, ec and herefore cu coss and pass on par of he savings o he shipper. In deciding he winning bids for lanes, he shipper uses an opimizaion carrier assignmen model (AM) o minimize *orrespondence: B Rodrigues, School of Business, Singapore Managemen Universiy, 5 Samford Road, Singapore , Singapore. br@smu.edu.sg ransporaion coss while ensuring ha all lanes are covered. Se covering models, including hose wih combinaorial inpu, have been well sudied in he lieraure (Ledyard e al, 22; Song and Regan, 22; aplice and Sheffi, 23; Elmaghraby and Keskinocak, 23; Vohra and de Vries, 23; Sheffi, 24). Recenly, aplice and Sheffi (23) and Sheffi (24) sudied non-price and level-of-service facors in ransporaion procuremen. These include shipper resricions on he number of lanes a carrier can win, favouring incumbens, keeping specific carriers off cerain lanes, resricing carriers from serving pars of he nework, and incorporaing performance as a facor in carrier selecion. This has led o specialized RFQ aucions where winner deerminaion AMs are required o deal wih hese facors, oher han price alone. Sheffi (24) repors ha many leading companies, including olgae-palmolive, Ford Moors, Lucen Technologies, Procor and Gamble, and Wal-Mar Sores, have used combinaorial aucions successfully o obain lowransporaion coss and high levels of service. Here, a level-of-service performance requiremen can be announced by he shipper in his RFQ as a non-price aribue, and evaluaion crieria given for i. The carrier responds wih a descripion of his service qualiy level and a bid price. The shipper can hen se a value for his service level and can, for example, choose o selec he carrier wih he larges surplus margin beween service level and bid price. This illusraes he fac ha he RFQ process is a sealed bid aucion wih independen privae values. As deermining he winner is a combinaorial problem wih many objecives, a AM mus be used. In he example of service level inpu, a penaly cos can be modelled in he

2 Y Guo e al arrier assignmen models in ransporaion procuremen 1473 AM, inverse o he value se by he shipper for he carrier s level of service or he shipper can adjus he fees charged by he carrier o reflec he service level for he carrier on each lane (aplice and Sheffi, 23). Transporaion and logisics soluion providers such as Manugisics Inc. (hp:// offer elecronically disribued e-rfqs wih which shippers can uilize configurable algorihms o furher analyse and opimize carrier bids across a number of business consrains. Using online bidding plaforms, carriers are able o adjus o he shipper s requiremens. For example, if he shipper imposes a cap on he number of lanes he carrier can win, hen he carrier will mos likely respond by sraegically increasing is bids over he nework using many combinaorial bids (Sheffi, 24). If, for example, he carrier is aware ha his level of service is lowin viewof he shipper s crieria, i migh choose o reduce is bid price. In he case of shippers insising on lowpackage prices while awarding only a single lane or parial packages, a carrier will submi bids for single lanes and parial packages o proec iself (Sheffi, 24). Depending on specificaions of he RFQ provided by he shipper, carriers respond by adjusing heir bid sraegy. As mos shippers use only a single round in he process (Sheffi, 24), combinaorial bidding is imporan o carriers who canno use muliple rounds o signal each oher as o which lanes hey wan and mus hedge agains uncerainy in response o he bidding mehod used by he shipper (aplice and Sheffi, 23). In all cases, carriers have o make assumpions wheher hey will win lanes which inerac wih lanes in heir nework and esimae he probabiliy of achieving economies of scope arising from line inerdependencies ec, where he cos of hauling on one lane is affeced by oher serviced lanes. In his work, we focus on opimizaion models and exend AMs o address shipper s non-price business objecives menioned above. Furher o his, we develop a model ha incorporaes carrier ransi poin coss, in addiion o lane coss. This provision originaes from he auhors sudy wih Royal Philips Elecronics (a shipper) which used carriers ha incurred varying coss a ransi poins. These coss included sopover, parking/berhing, warehousing, axes and accommodaion coss. ombinaorial aucion models available do no address hese shipper and carrier consideraions since combinaions sudied are concerned wih packes of lanes and no wih he poins lanes ransi. In realiy, however, carrier quoes and shipper supply chain planning are conneced o boh ransporaion roues and ransi/erminal locaions, and inerdependencies of coss resuling from economies of scope (aplice and Sheffi, 23; Sheffi, 24) are no only derived from connecing served lanes bu also on he locaions ha connec lanes. The work is organized as follows: In he nex secion, he bid analysis process is described. In secion 3, wo new models ha incorporae shipper s objecives and ransi poin coss are given. Soluions for he models are hen provided. In secion 4, compuaional experimens o compare he soluion approaches are described. The work is concluded in secion Background 2.1. AMs in bid analysis In he bid analysis sage in he ransporaion procuremen process, a AM helps he shipper minimize oal coss while ensuring ha each lane is served and is required capaciy saisfied. Generalized AMs specify ha allocaed lanes and volumes are feasible for boh shippers and carriers. In AM models, decision variables are binary, whereas more general forms are mixed ineger programs (MIP) for which soluion approaches are available (Nemhauser and Wolsey, 1999; aplice and Sheffi, 23) using MIP solvers. There has been much ineres in combinaorial aucions used in bid analysis in ransporaion procuremen. In an early work, Moore e al (1991) employed MIP for carrier selecion wihou condiional bids, while, more recenly, Ledyard e al (22) allowed for condiional bids wihou consideraion of capaciy limiaions and performance facors. Shippers ofen use opimizaion models in Wha-if sensiiviy analysis (Gibson e al, 1993): Wha if I assign only incumbens and do no allownewcarriers ; Wha if I reduce he number of carriers servicing ciy X, ec. I is preferable herefore ha shippers can analyze coningencies direcly in AMs Shipper consideraions As poined ou, shipper inpu is usually absen from radiional AMs, including hose of a combinaorial naure. The inclusion of shipper non-price consideraions has been found o be one of he sronges added values o he whole process as repored recenly by aplice and Sheffi (23). The auhors lis he following as some of he consideraions ha enhance he pracical value of AMs: Seing minimum/maximum carrier numbers: Shippers would require ha no more and no less han a cerain number of carriers can win lanes o deermine opimal sizing of a carrier group. Favouring incumbens: Because of he addiional coss of newcarriers, he shipper can apply a penaly o nonincumbens (or reward incumbens). In sraegic supply managemen, shippers wan o ensure hey have he righ se of suppliers. Seing maximum/minimum coverage: Shippers may wish o resric he amoun of raffic a carrier can win on a lane or in he sysem. Resricing carriers: Shippers may wish o resric carriers or groups of carriers from serving par of he nework. For example, a shipper may wan o resric a carrier from serving cerain nodes in he nework in an inernaional

3 1474 Journal of he Operaional Research Sociey Vol. 57, No.12 nework for poliical reasons or may wan o penalize a shipper a a ciy where is saff are unfamiliar wih he carrier s operaions. In inernaional ransporaion, carriers foreign o counries hrough which lanes ransi may be less preferred. Facoring in performance facors: Level of service provided by carriers can be a facor leading o awarding lanes. One way o do his, suggesed by he auhors, is o modify cos coefficiens arrier consideraions arriers radiionally submi quoes only for lanes. I is common, however, ha carriers have srenghs (or weaknesses) in cerain ciies, regions or groups of ciies and incur varying coss by operaing hrough differen poins. For example, carriers can have hubs a cerain ciies hrough and from which hey can operae wih lower coss. In oher siuaions, carriers may have o inves in sar-up coss a ciies newo heir nework. Ye in oher siuaions, carriers may be liable for operaing coss, including ad hoc ones, for example, axes and levies in inernaional carriage a ransi poins. Take he case where a carrier wins a lane connecing A and B, and a lane connecing A and bu will incur separae coss a A, B and. I will be preferable o boh he carrier and he shipper if he quoe given for lanes AB and A did no include operaing coss due o A, B and, since, oherwise, hey would accrue wice from A, resuling in a higher oal bid for lanes AB and A. In designing AM s, he use of explici ransi poin coss can herefore only benefi boh carriers and shippers. arriers can beer idenify opimal lane packages and provide more realisic bids when lane coss are clearly separaed from ransi poin coss, raher han subsumed ino one se of lane coss. 3. Exended carrier assignmen models Alhough i is impracical o include every shipper non-price and level-of-service facor in one model, we address some of hese facors here. To achieve his, we provide wo models. The firs addresses he issue of resricing he number of lanes a carrier can win. Alhough his is deermined by he shipper, carriers can sugges he maximum lane coverage hreshold hey wish o impose on hemselves. The model is an ineger program, for which a nework flow soluion is provided. The second model addresses oher facors which include favoring incumbens, resricing carriers o lanes and service performance facors. From he carrier s poin of view, he model allows for explici coss a ransi poins o be managed separaely from lane coss. We call hese separae coss, carrier ransi poin coss. This model is shown o be NPcomplee for which meaheurisic soluions are developed Model 1: a AM wih shipper s business consrains Shipper s perspecive. As we have seen, shippers may wish o have a maximum (minimum) number of carriers on each lane or wish o resric he number of carriers in pars or all of is nework. One way his can be achieved is by conrolling he number of lanes awarded o each carrier. From he shipper s poin of view, he number of lanes a carrier can ulimaely win is dependen on several facors; ypically, hese include he shipper s percepion of he carrier capabiliy, rack records, synergies wih he shipper and spread requiremens. This can be addressed in various ways. Enforcing a cap on he oal number of winnable lanes can ensure a beer spread and larger carrier paricipaion resuling in beer choices for he shipper. onversely, he shipper can wish o award a minimum number of lanes o a preferred carrier. arriers perspecive. In compeiion for conracs, small ransporaion companies ofen submi lower bids compared o heir larger counerpars. However, hese carriers ofen have smaller capabiliies and, as a resul, usually service only a limied number of lanes. Because of his, hey are resriced o bid for smaller numbers of lanes alhough hey would be beer off bidding for many packes of lanes o increase heir chances of winning roues ha maximize profis. A carrier wih a five-lane capabiliy will wan o bid for a number of five-lane packages hoping o win one wih he bes profi. The possibiliy, however, of winning more lanes han i can handle is a consequence ha he carrier may no be able o bear. arriers herefore will wan o limi he number of winnable conracs o be wihin heir capabiliies, bu oherwise aemp o bid for as many combinaions of lanes or packages as possible. The following shipper s ineger programming model addresses boh hese shipper and carrier objecives. As poined ou, lane caps can be shipper deermined, or provided o he shipper by he carrier. Parameers: L= he number of lanes, S= he number of carriers, M= a sufficienly large number, b kj =,1,2,3,y is he bid value carrier k places on lane j; b kj ¼ M if carrier k does no bid for lane j (1pkpS, 1pjpL), k min = he minimum oal number of lanes assignable o carrier k (1pkpS) deermined by he shipper (or he carrier if desired), k max =he maximum oal number of lanes assignable o carrier k (1pkpS) deermined by he shipper (or he carrier if desired). Decision variables: x kj ¼ 1 if lane j is assigned o carrier k (1pkpS, 1pjpL); oherwise

4 Y Guo e al arrier assignmen models in ransporaion procuremen 1475 Objecive: subjec o minimize ¼ XS X S k¼1 X L k¼1 j¼1 b kj x kj ð1þ x kj ¼ 1; 1pjpL ð2þ Source /2 /1 a b 2/1 3/1 5/1 9/1 4/1 d c /1 /1 /1 Sink X S b kj x kj pm 1; 1pjpL ð3þ 7/1 e k¼1 k min p XL j¼1 x kj pk max ; 1pkpS ð4þ onsrain (2) ensures ha each lane is assigned o exacly one carrier and consrain (3) ensures ha when lane j is assigned o carrier k, he bid b kj value canno be M. In (4), we noe ha when he maximum values k max are se o L (wih k min se o ), he model reduces o a basic AM where here is no cap on he number of lanes. In he model, greaer spread of lanes among carriers can be achieved by reducing k max, where in he exreme case k max ¼ 1 for all k. In he case ha carriers wish o resric he number of lanes ha can be won by any carrier in a paricular region, consrain (4) can be modified o have only he se of lanes presen in he region used and k resriced o ha paricular carrier. Here consrains (4) would be ransformed o: k min p P jarx kj pk max,wherer denoes he region or subse oflanesinquesion A nework maximum flow soluion. A minimum cos maximum flowsoluion is given for his problem. Wihou loss of generaliy, ake k min ¼ and apply a ransformaion o he graph which represens he problem. In he graph, edges are creaed for every lane bid a carrier submis. To illusrae he nework, we use a simple example wih wo carriers and hree lanes, and where B in Figure 1 denoes he corresponding carrier-lane bid marix. Nodes a and b represen carrier 1 and carrier 2, respecively, whereas nodes c, d and e represen he hree lanes. The pair i/j on each edge represens he cos i of a uni flowand he capaciy j of ha edge. arriers 1 and 2 are allowed o serve a mos 2 and 1 lanes, respecively. In consrucing he nework in Figure 1, wo addiional nodes are added: a source and a sink. Phase 1 consiss of he capaciy consrains of each carrier, where edges are consruced from he source node o he carrier nodes. Here, /2 indicaes ha carrier 1 can serve a maximum of wo lanes. In phase 2, edges beween carrier nodes {a, b) and lane nodes {c, d, e} are consruced. Here, 3/1 indicaes he cos o carrier a o cover roue c is 3 and flowcapaciy 1 for consisency beween phases 1 and 3. In phase 3, edges Phase 1 Phase 2 Phase 3 B = Figure connecing lane nodes o he sink are consruced wih cos and capaciy 1 o ensure each edge is served by a carrier only once. I is noweasy o see ha he minimum cos maximum flowin he nework solves he problem. Firs, edge capaciies in phase 1 ensure ha no carrier is assigned more han he number lanes allowed and flow in phase 3 ensures every edge is served by a carrier once. Nex, any soluion wih cos less han he minimum cos maximum flow, can be ransformed o a nework flow as described implying a cos lower han he minimum cos maximum flow, which is a conradicion. Hence, he minimum cos maximum flowmus solve he problem. Algorihms for finding he minimum cos maximum flow in a nework have been well-sudied (Oldham, 21). By applying he ransformaion above, he problem can be solved in O((L þ S) 3 * L)ime Model 2: a AM wih penaly and ransi poin coss (AMP) Shipper s perspecive. Shipper non-price business consideraions include favouring incumbens, resricing carriers o lanes, excluding or penalizing carriers a ransi poins and including carrier performance facors, such as level of service. To deal wih hese objecives, we propose a AM ha includes penaly coss. In he case where an aribue is desirable, as wih level of service, penaly cos can be aken as he inverse value of he level-of-service price assigned o he carrier by he shipper. The model can easily be exended o have more han one penaly cos. Alernaively, he model can be used o achieve each of hese objecives separaely. For example, if he shipper wished o exclude a carrier from a lane, hen high penaly coss can be assigned o he carrier Nework flow example.

5 1476 Journal of he Operaional Research Sociey Vol. 57, No.12 a locaions adjacen o he lane. If he shipper wished o resric or exclude a carrier a a ransi locaion, he would assign a high penaly o he carrier a ha poin. By adjusing penaly coss, he shipper can deermine he carrier allocaion which is bes suied o his business objecives. arrier s perspecive. arriers and shippers will benefi if lane bids can be separaed from ransi poin coss which provides for more realisic bidding. arriers can adjus bids according o varying coss incurred a differen ransi locaions. A model (AMP) which addresses hese shipper and carrier objecives is given in he ineger program: Parameers: n= he number of nodes L= he number of lanes S= he number of carriers M= a sufficienly large number a ij = 1 if and only if node i is adjacen o lane j (1pipn, 1pjpL); oherwise b kj =, 1, 2,y is he value carrier k bids for lane j; b kj ¼ M if carrier k does no bid for lane j (1pkpS, 1pjpL) K b ki =, 1, 2,. is he carrier k s bid cos a ransi node i (1pkpS, 1pipn) p ki =,1,2,y is he shipper s penaly cos assigned o carrier k a node i (lpkps, 1pipn) Decision variables: x kj ¼ 1, if lane j is assigned o carrier k (1pkpS, 1pjpL); oherwise, y ki ¼ 1, if carrier k wins a lane adjacen o node i (1pkpS, 1pipn); oherwise Objecive: subjec o minimize ¼ XS X L j¼1 X S k¼1 X S k¼1 X L k¼1 j¼1 þ XS k¼1 i¼1 b kj x kj þ XS X n p ki y ki X n k¼1 i¼1 b kj y ki ð5þ x kj ¼ 1; 1pjpL ð6þ b kj x kj pm 1; 1pjpL ð7þ a ij x kj Xy ki ; 1pipn; 1pkpS ð8þ X L j¼1 a ij x kj pmy ki ; 1pipn; 1pkpS ð9þ onsrain (6) ensures ha each lane is assigned o exacly one carrier and consrain (7) ensures ha when lane j is assigned o carrier k, he bid b kj is no M. onsrain (8) ensures ha when y ki is 1, here is a leas one edge j connecing node i o carrier k and (9) ensures ha when y ki is, no edge connecing node i is assigned o carrier k. The AMP is NP-complee and a proof of his can be found in he Appendix Benchmarking he APMP using branch-andbound soluions. A branch-and-bound (B&B) complee search can be used o examine all possible assignmens of he lanes o carriers for small es sizes, where he ime performance of he algorihm depends largely on he bounding funcion used (Viswanahkumar and Srinivasan, 22). The B&B algorihm begins wih an empy soluion se and divides he problem ino L recursive sages, where a lane is assigned o a carrier in each sage. A sage k, when lanes 1 o k 1 have been assigned, carriers who bid for lane k are examined. Given a carrier s from his se, lane k is assigned o s if he curren bes soluion is larger han he lower bound (see below); oherwise, he search branch is discarded and he search moves o consider he nex carrier. The process moves recursively o sage k þ 1oassignacarriero lane k þ 1. To calculae he lower bound: Le s(i) (1pipL) behe carrier ha edge i is assigned o. If s{i) isdeerminedfor 1pipk and undeermined for k þ 1pipL, he lower bound for he oal cos is he sum of hree componens: L 1, L 2 and L 3,whereL 1 is he bid cos for edges 1 o k, already assigned o some carrier, L 2 is he bid lower bound for edges k þ 1o L, ye o be assigned o any carriers, and L 3 is he leas possible oal penaly cos. Leing B denoe he S L carrier bid marix, we have L 1 ¼ Xk L 2 ¼ Xk i¼k þ 1 B sðiþ;i i¼1 minfb j;i 1pjpSg L 3 ¼ Xn p i i¼1 where p i is he penaly cos incurred a node i. If here is an edge j wih 1pjpk which is conneced o node i, p i is he sum of penaly coss assigned o carriers who serve some edge beween 1 and k; ifhereisnoedgej wih 1pjpk conneced o node i,henp i is he minimum penaly assigned o any carrier ha can cover any edge conneced o node i. Thus, a any poin in he B&B process, he bounding funcion used is L 1 þ L 2 þ L 3.

6 Y Guo e al arrier assignmen models in ransporaion procuremen Using meaheurisics o solve he AMP. Heurisics have been used for combinaorial aucions problems (Sandholm e al, 22; Sandholm, 22; Vohra and de Vries, 23) and for oher difficul combinaorial opimizaion problems (eg Foser and Srasser, 1991; Lim e al, 24). As he AMP is an NP-complee problem, meaheurisic soluions are developed based on widely used geneic algorihm and abu search echniques, which have been successful in oher comparable applicaions. A hybrid of hese is hen consruced which provides a hird heurisic approach o he problem. A geneic algorihm Geneic algorihms have been widely used for combinaorial opimizaion problems (Dowsland, 1996); for example, hey have been applied o ask allocaion problems (Song and Regan, 22; Wen and David, 21). Here, a geneic algorihm (GA) is used for he AMP which is described as follows: Ouline. For each disinc chromosome pair in a subse of a randomly generaed iniial populaion wih size pop_size, perform crossover operaions and muae newly generaed chromosomes according o a muaion probabiliy. By evaluaing he objecive funcion value of he newand old chromosomes, reain pop_size bes chromosomes. In he implemenaion, erminaion is effeced when he bes soluion does no improve in a given number of ieraions or if a maximum number of ieraions is reached. hromosome represenaion. Soluions are encoded as chromosome srings s ¼ (s(l), s(2), ys(l)) conaining L inegers, where L is he number of edges o be covered. Each s(i) (1pipL) is an ineger which represens he carrier index ha is assigned o edge i, soha1ps(i)ps. Iniial soluions are generaed so ha b s(i),i on is ensured. rossover operaion. For chromosomes s and s, a random number k is generaed beween 1 and L 1 and he genes are cu a posiion k. The subsequence (s(k þ 1), s(k þ 2), y, s(l)) is placed afer (s (l), s (2), y, s (k)) and (s (k þ 1), s (k þ 2), y, s(l)) is placed afer (s(l), s(2), y, s(k)) o generae wo new chromosomes. Wih his operaion, newly formed chromosomes are always feasible, given heir parens are feasible since he posiions of carrier assignmens are no shifed when performing he crossover. Muaion operaion. For each chromosome in he offspring generaion, a muaion probabiliy q is used. Generaing a random number in ra[, 1] for each i, if roq, s(i) is changed randomly o anoher value hus changing he carrier assigned o edge i. In implemenaion, he following parameers were used: pop_size ¼ 1, q ¼.5. The convergence crieria were 5 generaions reached or more han 2 generaions wih no improvemen o curren bes soluion. A abusearch. Tabu search is a search sraegy which moves ieraively from one soluion o anoher in a neighbourhood search space using an adapive memory. The mehod declares abu, soluions wih aribue changes recorded in he shor-erm memory from being reused, where he ime a resricion is in effec depends on a abuenure parameer (Glover and Laguna, 1997). Tabu search is applied o he AMP, wih he soluion represenaion used in GA, ha is, a sring of inegers. A neighbourhood move is denned as a change in carrier-edge assignmen in he soluion (similar o he muaion operaor in GA) and, o avoid recycling, abu liss consis of he recen abuenure soluions. The heurisic (TS) is oulined as follows: Find an iniial soluion x now.sex bes ¼ x now. If he erminaion condiion is saisfied, qui wih x bes. The erminaion condiion is saisfied if eiher more han 2 ieraions have been execued or here is no improvemen in he curren bes soluion in he mos recen 1 ieraions. Oherwise, generae he neighbourhood N(x now )ofx now wih 1 new neighbourhood soluions by randomly selecing hree lanes among he L lanes in x now and reassigning hem o oher feasible carriers randomly. Evaluae he cos of x rial for each newcandidae rial soluion x rial.selecx nex ¼ min XriaieN ( Xnow )os(x ria l)forx ria i no in he abu lis. Updae he abu memory by adding he newcurren soluion x nex ino he abu lis. Se x now ¼ x nex.ifcos(x now )ocos(x bes ), se cos(x bes ) ¼ cos(x now ). In implemenaion, 1 recen soluions were mainained in he abu lis. A geneic algorihm wih abusearch (GA þ TS) As iniial soluions can conribue o he qualiy of soluions provided by he meaheurisics, we developed a hird heurisic using iniial soluions from GA in TS. 4. ompuaional experimens for he AMP In order o gauge he performance of he soluion approaches, a series of experimens were conduced. The B&B mehod and hree heurisics (GA, TS, GA þ TS) were coded using þþ and run on a Penium IV 1.4G P wih 256 Mb of memory. The experimens consised of wo pars. The firs se of experimens used small-size es insances as opimal soluions could be found wih he exac enumeraive B&B mehod for hese sizes. Soluions obained by he heurisic approaches were compared wih he opimal soluions found. The second se of experimens compared he heurisics for larger-size problems agains he bes soluions obained from he heurisics. Tes insances were generaed using he following seps: Sep 1: Sep 2: Given he inpu daa: n nodes, S carriers and L edges, consruc a graph from an iniial graph wih n nodes and no edges by connecing randomly seleced nodes wih no edge beween hem. Repea his unil all L edges are added o he graph. Assign carrier coss o nodes: For each carrier and each node, pick a number from {1, 2, 3} randomly

7 1478 Journal of he Operaional Research Sociey Vol. 57, No.12 Sep 3: o indicae he range he carrier cos will be in 1 for a lowrange, 2 for a middle range, and 3 for a high range. Take each cos range o be given by an inerval. Wih his, assign a cos value randomly from he range. Assign carrier coss o edges: Assign coss as in Sep omparisons wih branch-and-bound In oal, 3 small size insances were generaed wih he number of edges ranging from 3 o 21. The exac opimal cos value was found by B&B, and he running ime (in seconds) calculaed for each algorihm, for each insance. From he experimens, we found ha he heurisics performed well when insances were small, which is expeced of a good heurisic. All hree mehods provided opimal soluions for mos of he 3 insances. The mean values of he percenage difference beween each heurisic soluion and he exac soluion found by B&B were very close o for all he heurisics. The mean values and sandard deviaions are provided in Table 1. Mean values were.82 for GA, and.4 for TS and GA þ TS, and he sandard deviaions were for GA, 1.56 for TS and 1.56 for GA þ TS. The ime required for B&B varied wih size and among he heurisics, he GA algorihm consumed more ime han he oher wo heurisics for mos insances since i depends on he populaion size of each generaion. The imes required for B&B were large since is performance depended on inpu size and graph srucure. In conras, he heurisics were more likely o be independen of hese inpus and had more sable running imes han B&B. Running ime saisics are provided in Table 2. In he able, m B&B, m GA, m TS, m GA þ TS is he mean running ime for B&B, GA, TS and GA þ TS respecively, and s is he ime sandard deviaion in each group. From he able, B&B required considerably more ime han he oher heurisics when he number of edges was 13 and higher. Moreover, B&B had a large deviaion for running imes. Alhough he heurisics performed well for small cases, his is no sufficien o guaranee good performance for larger-size cases. In he nex secion, we provide a bes case analysis for he heurisics for larger cases omparisons beween he heurisics To deermine he performance of he heurisics for large es sizes, a oal of 25 insances were esed in five groups which were generaed wih sizes of up o 5 edges. Saisics for he experimens are provided in Tables 3 and 4. In Table 3, he insance size is in he form: number of edges_number of nodes_number of carriers, and m GA þ TS, m TS and m GA denoe he soluion objecive funcion mean value obained using GA þ TS, TS and GA, respecively, whereas m GA, m TS and m GA þ TS is he mean ime required by Table 1 Performance of heurisics for small es sizes No. of edges No. of insances m B&B s B&B m GA s GA m TS s TS m GA þ TS s GA þ TS Table 2 Performance of heurisics for small es sizes ime saisics No. of edges No. of insances m B&B s B&B m GA s GA m TS s TS m GA þ TS s GA þ TS Table 3 Lane experimens Size m GA þ TS m GA þ TS m TS m TS d 1 m GA m GA d 2 1_5_ _7_ _8_ _9_ _1_

8 Y Guo e al arrier assignmen models in ransporaion procuremen 1479 he algorihms o find soluions. The values d 1 and d 2 denoe he raio of difference beween m TS and m GA wih m GA þ TS as a percenage. From his able, GA þ TS and TS can be seen o perform equally well, wih resuls differing by no more han 1%, all of which were found wihin 9 s. On he oher hand, he GA algorihm did no perform well when compared o is performance for smaller es cases. The relaive performance difference beween GA and GA þ TS increased from abou 4% o 43% as size increased. In addiion, GA required 8 1 imes he running ime required by GA þ TS. Table 4 provides addiional saisical informaion on hese experimens. In he able, b GA þ TS, b TS, and b GA, denoe he number of bes soluions obained in each group by he algorihms GA þ TS, TS and GA, respecively, and s GA þ TS and s GA þ TS, ec, is he sandard deviaion of he soluions and running imes, respecively. Again, here was a marginal advanage of GA þ TS over TS in he number of bes soluions obained, while GA failed o provide resuls ha are compeiive wih he oher heurisics. Boh GA þ TS and TS soluions had a small sandard deviaion in he objecive funcion values and running imes. The GA algorihm, on he oher hand, had sandard deviaions significanly higher han GA þ TS and TS. The heurisics were furher esed using insances wih differen lane densiies. Fixing he number of lanes o be 2 and number of carriers o be 15, 3 insances were generaed in six groups according o he number of nodes. Saisics from he experimens are given in Table 5 and Table 6. The aribues used in he ables are similar o hose used for Tables 3 and 4. In Table 5, l is he lane densiy calculaed by L/((n/2) (n 1)), where he denominaor is he maximum possible number of lanes in a graph wih n nodes. From Table 5, i can be seen ha he performance of he heurisics mehods was no sensiive o he lane densiy, as he soluion qualiy and ime requiremens of each mehod were similar. TS resuls were very close o GA þ TS wih no more han 1% difference on average, whereas he GA algorihm soluions deviaed from 17 o 22% from he oher algorihms. In Table 6, addiional saisics on he lane densiy experimens is provided, similar o hose given in Table 4. Table 4 Lane experimens saisics Size b GA þ TS b TS b GA s GA þ TS s TS s GA s GA þ TS s TS s GA 1_5_ _7_ _8_ _9_ _1_ Table 5 Lane densiy experimens n l m GA þ TS m GA þ TS m TS m TS d 1 m GA m GA d Table 6 Lane densiy experimens saisics n b GA þ TS b TS b GA s GA þ TS s TS s GA s GA þ TS s TS s TS

9 148 Journal of he Operaional Research Sociey Vol. 57, No onclusions In his paper, opimizaion models used in winner deerminaion processes in ransporaion procuremen were inroduced. Alhough radiional combinaorial aucion models focus on carrier inpu using lane bids, hese models deal wih shipper non-price business consideraions such as carrier spread, and allowed for he added benefi of including ransi poin coss. In one model, he shipper is able o cap he number of lanes any one carrier can win. This ineger program was solved as a nework flow problem and a polynomial-ime algorihm provided. In a second model, a AM wih penaly and ransi poin coss was given o encourage more realisic bidding by carriers. The problem was shown o be NP-complee, and branch-and-bound and heurisics were developed o find soluions. ompuaional experimens were conduced o evaluae he algorihms on a range of es insances. I was found ha among he heurisics, a hybrid geneic algorihm wih abu search provided he bes soluions. This work provides a basis for he design and developmen of models ha address shipper non-price aribues and sysem consrains in AMs, and which include carrier cos inpu. The use of meaheurisics, paricularly a geneic algorihm wih abu search, has been shown o be effecive for hese problems and could be useful in oher similar opimizaion models, especially when mixed ineger programming commercial solvers canno be applied. Moore EW, Warmke JM and Gorban L (1991). The indispensable role of managemen science in cenralizing freigh operaions a Reynolds Meals ompany. Inerfaces 21(1): Nemhauser GL and Wolsey LA (1999). Ineger and ombinaorial Opimizaion. Wiley: NewYork. Oldham J (21). ombinaorial approximaion algorihms for generalized flowproblems. JAlgorihms38: Rhinehar LM (1989). Organizaional and personal facors influencing he negoiaion of moor carrier conracs: a survey of shippers and moor carriers. Transpor J 29: Sandholm T, Suri S, Gilpin A and Levine D (22). Winner deerminaion in combinaorial aucion generalizaions. In: aselfranchi and Johnson WL (eds). The Proceedings of The Firs Inernaional Join onference on Auonomous Agens and Muliagen Sysems, AAMAS 22, Bologna, Ialy, Vol. 1. AM Press, NewYork, USA, pp Sandholm T (22). Algorihms for opimal winner deerminaion in combinaorial aucions. Arificial Inelligence 135: Sheffi Y (24). ombinaorial aucions in he procuremen of ransporaion services. Inerfaces 34(4): Song J and Regan A (22). ombinaorial aucions for ransporaion service procuremen: he carrier perspecive. Working paper UI-ITS-LI-WP-2-9, Insiue of Transporaion Sudies, Universiy of alifornia, Irvine, USA. Viswanahkumar G and Srinivasan G (22). A branch and bound algorihm o minimize compleion ime variance on a single processor. ompu Opns Res 3: Vohra R and de Vries S (23). ombinaorial aucions: a survey. INFORMS J ompu 15: Wen R and David AK (21). A geneic algorihm based mehod for bidding sraegy coordinaion in energy and spinning reserve markes. Arificial Inelligence in Engineering 15: Acknowledgemens We hank he anonymous referees for suggesions which have helped improve his work. Appendix Theorem The AMP is NP-complee. References American Trucking Associaions (22). American Trucking Trends 22. American Trucking Associaions: Alexandria, Virginia. aplice and Sheffi Y (23). Opimizaion based procuremen for ransporaion services. J Bus Logis 24: ormen Y, Leiserson E, Rives RL and Sein (21). Inroducion o Algorihms, 2nd edn. MIT Press: ambridge, MA, USA. Dowsland KA (1996). Geneic algorihms a ool for OR. J Opnl Res Soc 47: Elmaghraby W and Keskinocak P (23). ombinaorial aucions in procuremen. In: Billingon, Harrison T, Lee H, Neale J (eds). The Pracice of Supply hain Managemen. Kluw er Academic Publishers, Norwell, MA, USA, pp Foser JR and Srasser S (1991). arrier/modal selecion facors: he shipper/carrier paradox. Transporaion Research Forum 31: Gibson BJ, Sink H and Mundy R (1993). Shipper-arrier relaionship and carrier selecion crieria. Logis Transpor Rev 29: Glover F and Laguna M (1997). TabuSearch. Kluw er Academic Publishers: Norwell, MA, USA. Ledyard J e al (22). The firs use of a combined value aucion for ransporaion services. Inerfaces 32(5): Lim A, Rodrigues B and Song L (24). Manpower allocaion wih ime windows. J Opnl Res Soc 55: Proof Transi poin coss can be negleced and aken wih penaly coss in he objecive funcion. In order o showha he AMP is NP-complee, we show ha he decision form of he problem is NP-complee. The decision form can be saed as: Given S carriers, a bid cos marix B, apenalycos marix, and an ineger k, can we find a carrier-edge assignmen in he represenaive graph wih oal cos k? In order o prove his problem is NP-complee, i suffices o prove he problem is in NP and i is NP-hard. Obviously, given a carrier-edge assignmen, i is possible o deermine feasibiliy in polynomial ime, so he problems in NP. Proof ha he AMP is NP-hard: To showhe problem is NP-hard, we reduce he verex-cover problem (VP) a well-known NP-complee problem o he AMP. A verex cover of an undireced graph G ¼ (V, E) isasubse V DV such ha if (u, v) is an edge of G, hen eiher uav or vav (or boh). The VP is o find a subse V wih he minimal cardinaliy (ormen e al, 21). LeG(V, E) bean insance of he VP. We consruc an insance of AMP in polynomial ime. For he AMP, an insance consiss of agraphg (V, E ) of node-edge relaionships, a marix for he carrier-edge bids coss, and a marix of penaly coss.

10 Y Guo e al arrier assignmen models in ransporaion procuremen 1481 We consruc he inpu from G(V, E) as follows: Le G (V, E )DG(V, E) (ie V ¼ V and E ¼ E ), and n he number of nodes, L he number of edges and S he number of carriers. We have n ¼ V andl ¼ E. Leing S ¼ n, consruc as L carrier-edge bid cos marix B wih b ij ¼ if node i is adjacen o edge j, andb ij ¼ N oherwise (1pipS, 1pjpL). onsruc a S n carrier-node penaly cos marix ¼ [p ij ]wihp ij ¼ ifiaj, and p ij ¼ 1 oherwise (1pipS, 1pJpn). As S ¼ n, he marix is he uni square marix. An example is given in Figure 2: These can be compleed wihin polynomial ime. Nex we showha he VP has a soluion wih k verices if and only if he AMP has a soluion wih cos k. Firs, we prove ha, if he verex-cover problem has a feasible soluion of k verices, hen he AMP has a feasible soluion wih cos k. Le he se of chosen verices for he verex-cover problem be V 1,soha V 1 ¼ k and le h(i) (1pipk) beheindexofhe ih node in V 1. In he AMP, we choose k carriers s(1)ys(k) o be he edges, where s(i) ¼ h(i)(1pipk) whichis possible as S ¼ n. By he definiion of he VP, for any edge (u, v)ing(v, E), here is a node h i in V 1 which is conneced o (u, v); similarly, in he AMP, an edge (u, v) ing (V, E ) can be assigned o he carrier wih index s(i). This is a feasible soluion if we se he bid cos o be for carrier s(i) and edge (u, v) when node h(i) is adjacen o (u, v). Furhermore, he penaly cos assigned o carrier s(i) a node h(i) inmarix is 1 since s(i) ¼ h(i). Thus, his feasible soluion has cos k since each of he k carriers can only incur he cos of 1 and no oher cos is involved. onversely, we prove ha if he AMP has a feasible soluion of cos k, he VP also has a feasible soluion wih k verices. From he definiion of he AMP, he only way o obain he feasible soluion wih cos k is o choose k pairs of b 1 a 2 c d 3 e 4 Figure 2 B= = elemens (h(i), h(i)) (1pipk) in he diagonal of P. Inhe VP, choose k verices o be h(i) for 1pipk as in he AMP wih s(i) ¼ h(i). In he AMP, each edge is assigned o one of he k carriers s(i)ys(k), and if edge (u, v) is assigned o carrier s(j), hen edge (u, v) is conneced o node h(j) whereh(j) ¼ s(j) because he bid marix B mus be ; oherwise he feasible soluion for he penaly cos problem wih cos k is no possible. Thus, if he se of carriers s(1)ys(k) can serve he edges wih a cos k in he AMP hen he se of nodes h(1)yh(k) formaverex cover wih k verices in he VP. Hence, we have shown ha he VP can be reduced o he AMP by a polynomial-ime ransformaion so ha he AMP is NP-hard. & 1 Received June 24; acceped Sepember 25 afer four revisions 1 1 Reducion example.