J. Servce Scence & Management, 2010, 3: 143-149 do:10.4236/jssm.2010.31018 Publshed Onlne March 2010 (http://www.scrp.org/journal/jssm) 143 Allocatng Collaboratve Proft n Less-than-Truckload Carrer Allance Peng Lu 1, Yaohua Wu 1 *, Na Xu 2 1 School of Control Scence and Engneerng, Shandong Unversty, Jnan, Chna; 2 School of Busness, Shandong Jan-zhu Unversty, Jnan, Chna; *Correspondng Author. Emal: ken0211@gmal.com, mke.wu@263.net, xuna1011@hotmal.com Receved September 21 st, 2009; revsed November 5 th, 2009; accepted December 20 th, 2009. ABSTRACT Internatonal Fnancal Crss has made the less-than-truckload (LTL) ndustry face wth severe challenges of survval and development. More and more small and medum-szed LTL carrers choose to collaborate as the potental savngs are large, often n the range 5 15%. A key queston s how to dstrbute profts/savngs among the partcpants. Snce every LTL carrers are guded by ther own self-nterests and ther contrbutons to the collaboraton are qute dfferent, the proposed allocaton method should be a collectvely and ndvdually desrable soluton. In ths paper, we frstly analyze the proft opportuntes from collaboraton and present mechansms to realze these benefts by two llustratve examples. Based on the cooperatve game theory, we formulate the LTL collaboraton game and dscuss the well-known proft allocaton concepts ncludng Proportonal Allocaton, Shapley value and Nucleolus. We then propose a new al-locaton method named Weghted Relatve Savngs Model (WRSM) whch s n the core and mnmzes the maxmum dfference between weghted relatve savngs among the partcpants. Smulaton result for real-lfe nstances shows the effectveness of WRSM. Keywords: Cooperatve Game, Proft Allocaton, Collaboratve Transportaton 1. Introducton Internatonal Fnancal Crss causes a huge decrease n transportaton requests and has made less-than-truckload (LTL) segment of the truckng ndustry face wth severe challenges of survval and development. Under ths crcumstance, horzontal collaboraton becomes a good choce for small and medum-szed LTL carrers. In the collaboratve allance, a number of complementary transportaton resources from the partcpants could be ntegrated and thus more profts could be ganed for every partcpant compared wth ther stand-alone operaton. The potental cost savngs from collaboraton are often range from 5% to 15%. Although the benefts from collaboraton are appealng, the key queston s how to dstrbute the collaboratve profts among every partcpant to ensure the establshment and sustanablty of the allance and realze the potental of collaboraton. Snce every partcpant s guded by ther own self-nterests and ther contrbutons to the collaboraton are qute dfferent, the proposed allocaton method should be a collectvely and ndvdually desrable soluton [1]. The challenge s to desgn mechansms that are far, reasonable and easy- to-mplement. We wll show that the proposed Weghted Relatve Savngs Model (WRSM) satsfes all these requrements. The remander of the paper s organzed as follows. In Secton 2, we analyze the opportunty to ncrease every LTL carrer s proft through collaboraton and present two llustratve examples to demonstrate the mechansms to realze these benefts. In Secton 3, based on the cooperatve game theory, we formulate the LTL collaboratve game and dscuss the well-known proft allocaton concepts. We then propose a new soluton method called Weghted Relatve Savngs Model (WRSM) whch s n the core and mnmzes the maxmum dfference between the weghted relatve savngs among the partcpants. Smulaton result for real-lfe nstances s presented and analyzed n Secton 4 to show the effectveness of WRSM. 2. Proft Opportuntes from Collaboraton The constructon of LTL carrers allance wll enable the formulaton of collaboratve transportaton system. In ths secton, we wll analyze the proft opportuntes of ths system. Copyrght 2010 ScRes
144 Allocatng Collaboratve Proft n Less-than-Truckload Carrer Allance 2.1 Collaboratve Transportaton System As t s shown n Fgure 1, the collaboratve transportaton system s a knd of system n whch all partcpants share the network and transportaton resources. E denotes Termnal Pont (TP) whch s the boundary pont of the carrer s busness coverage. N denotes Swtch Pont (SP), through whch the cargos transport to the carrer s adjacent busness pont. W denotes Exchange Pont (EP) where two or more collaboratve carrers n the allance exchange ther cargos and transport the exchanged cargos to ther own busness pont. S de-notes Shared Pont (SDP) whch s shared by two or more collaboratve carrers n the allance. From the system wde pont of vew, transportaton network and re-sources are shared among all the LTL carrers n the allance through EP and SDP whch expand the busness scope of every partcpant. Resource sharng wll help to buld more reasonable transportaton plans to better utlze vehcles, reduce travel tme, unloaded dstance and lower the total transportaton cost effectvely. 2.2 Benefts of Collaboraton Crujssen and Salomon [2] analyze the effect of collaboraton for an entre coalton and show, usng a case study that cost savngs may range from 5 to 15% and can be even hgher. Ergun et al. [3] note that shppers can reduce ther hdden costs by cooperatng, partly due to hgher utlzaton of ther less-than-truckload loadng and asset repostonng capabltes. In the tme-constraned lane coverng problem, the savngs range s from about 5.5 percent to a lttle over 13 percent, where the savngs tend to be larger when the sze of the nstance s larger. [4] Krajewska and Kopfer [5] show that, usng a case study, cooperaton between the two carrers yelds a 10% reducton n the number of vehcles and a 12.46% reducton n routng cost. In practce, after formng collaboratve partnershps wth others n the Nstevo work, Georga- Pacfc s percentage of empty movements decreased from 18% to 3%, whch corresponds to $11,250,000 savngs yearly [6]. We demonstrate the potental benefts of LTL carrer collaboraton wth the followng two examples. 1) Backhaulng Consder a network wth three ctes and two carrers A and B. We assume that the cost of travelng between two ctes s the same for both carrers and, for smplcty, that there s no dfference n cost between travelng loaded or empty. We further assume that carrer A has a contract n place to serve lane (2, 1), (1, 3) and that carrer B has a contract n place to serve lane (3, 2). The cost C and freght F of each lane n the network and other relevant nformaton are gven n Fgure 2, where a dashed lne represents repostonng (or empty travel). Wthout collaboraton, carrer A and B operate ndvdually and the correspondng proft of them are Proft A = F21 + F13 C21 C13 C32 = 1300 Proft B = F32 C32 C23 = 400 As t s shown n Fgure 3, f carrer A and carrer B collaborate and carrer A serve lane (3, 2) nstead of carrer B, they sgnfcantly ncrease ther total proft to 2100 by reducng two empty trps. Assume that the proft allocaton rate s 0.75, then the new proft become 1575 for carrer A and 525 for carrer B. Carrer A and B ncrease ther profts by 21% and 31% respectvely. Through collaboraton, carrer A reduces ts empty trp and fully utlzes the truck whle carrer B does not need to transport the cargos. But they both gan more benefts snce the total repostonng cost s much lower. Fgure 2. work nformaton and transportaton requests Fgure 1. Collaboratve transportaton system Fgure 3. Collaboraton between carrer A and B Copyrght 2010 ScRes
Allocatng Collaboratve Proft n Less-than-Truckload Carrer Allance 145 2) Lane/Request Exchangng Consder a network wth four ctes and two carrers A and B. We assume that the cost of travelng between two ctes s the same for both carrers and, for smplcty, that there s no dfference n cost between travelng loaded or empty. We further assume that carrer A has a contract n place to serve lane (2, 1), (1, 3), (3, 4) and that carrer B has a contract n place to serve lane (4, 3), (3, 2). The cost C and freght F of each lane n the net-work and other relevant nformaton are gven n Fgure 4, where a dashed lne represents repostonng (or empty travel). Wthout collaboraton, carrer A and B operate ndvdually and the correspondng profts are 1900 for carrer A and 1000 for carrer B. We assume that the exstng contracts are not long term contractual agreements so can potentally be exchanged between the carrers [1]. As t s shown n Fgure 5, f carrer A and carrer B collaborate and exchange lanes (3, 4) and (3, 2), the correspondng profts are 2100 for carrer A and 1200 for carrer B. Carrer A and B ncrease ther profts by 10.5% and 20% respectvely. Through collaboraton, the optmal set of cycles coverng the contract lanes are assgned to each carrer. Empty travels are greatly reduced and total profts are redstrbuted between carrer A and carrer B. Fgure 4. work nformaton and transportaton requests Fgure 5. Collaboraton between carrer A and B 3. Proft Allocaton Problem Cooperatve game theory provdes a natural framework for the proft allocaton. There are a set of papers that jon the transportaton related proft or cost allocaton problems and cooperatve game theory. Sakawa et al. [7] dscuss the producton and transportaton proft and cost allocaton based on nucleolus n the fuzzy envronment and shows, usng actual data, the usefulness of fuzzy programmng and the effectveness nucleolus allocaton. Sanchez-Sorano et al. [8] study the core of the transportaton games, prove the nonemptness of the core for these games and provde some results about the relatonshp between the core and the dual optmal solutons of the underlyng transportaton problem. Sanchez-Sorano et al. [9] study the cost allocaton of the ntegrated transportaton servces provded by Alacant Unversty for students, formulate the problem as tree buses game, propose the aggregated egaltaran soluton concept and show t s the core of the game. Engevall et al. [10] formulate the travelng salesman game and vehcle routng game, dscuss nucleolus, TSP nucleolus, TSP demand weghted nucleolus, Shapley value and τ value respectvely. Matsubayash et al. [11] study a cost allocaton problem arsng from hub-spoke network systems and show that, f the demand across the system has a block structure and the fxed cost s hgh, allocatng the cost proportonal to the flow that an agent generates belongs to the core. Ozener [1] study the cost allocaton n the collaboratve transportaton procurement network and dscuss the truckload carrer s collaboraton. Krajewska et al. [5] study the proft sharng problem among carrers n the horzontal collaboraton, dscuss the possbltes of sharng these proft margns farly among the partners, apply the Shapley value to determne a far allocaton of the problem and present numercal results for real-lfe and artfcal nstances. These papers n general study the exstng proft or cost allocaton methods wth well-studed propertes from cooperatve game theory and present the computatonal results for such allocatons. However, to the best of our knowledge, there s no lterature on proft allocaton for LTL collaboratve transportaton problem that consders both the relatve cost savngs and contrbuton dfferences, whch are very mportant n the contractual agreement negotaton of the collaboraton. In ths secton, we wll search for a new proft allocaton method that satsfes these requrements based on the well-known soluton concepts from cooperatve game theory. 3.1 Problem Defntons and Assumptons We formulate the proft allocaton for LTL collaboratve transportaton problem as a co-operatve game Nv,. Copyrght 2010 ScRes
146 Allocatng Collaboratve Proft n Less-than-Truckload Carrer Allance 1, 2,..., 2 coalton S. N n n s called the grand coalton whch denotes all collaboratve carrers. vs s the characterstc functon whch assgns to each possble coalton of carrers SS N a numercal value to be nterpreted as the cost savngs realzed by the carrers n y N s the proft/cost savngs allocated to carrer. Y y1, y2,..., yn s the proft allocaton. It s assumed that all carrers have the opportunty to form and cooperate n coalton. When coalton S cooperates, the total cost cs s generated and we have, v S c c S S N (1) S Below we dscuss some of the most commonly used proft allocaton propertes from cooperatve game theory. A proft allocaton method that splts the total proft vnamong the carrers N s sad to be effcent or budget balanced, that s y v( N). N A proft allocaton s sad to be ndvdual ratonal f no carrer gans less than ts stand alone proft/cost savng, whch equals to zero. Mathematcally, ths property s expressed as y v({ }), N. The core of the game s defned as those proft alloca- Y y y y that satsfy the condtons tons 1, 2,..., n y v( S), S N (2) S y v( N) (3) N That s, no sngle carrer or coalton of carrers would be better off f they decde to opt out and collaborate only among themselves. A proft allocaton n the core s sad to be stable. For each coalton S and a gven proft allocaton 1, 2,..., n Y y y y, we can compute the excess ey (, S) y vs ( ), S N (4) S whch expresses the dfference between the sum of the profts allocated to ts members and the total proft of a coalton. For a gven proft allocaton, the vector of all excesses can be thought of as a measure of how far the proft allocaton s from the core. If a proft allocaton s not n the core, at least one excess s negatve. 3.2 Well-known Proft Allocaton Concepts 3.2.1 Proportonal Allocaton In practce, the most commonly used soluton s to dstrbute the collaboratve proft/cost savngs of the grand allance v N among the carrers equally, weghted wth each carrer s stand alone cost. Ths s expressed as y r v N, N (5) where r s equal to c c. N Although ths method s easy to understand, easy to show and easy to compute, t s not stable from a cooperatve game theoretc pont of vew snce a partcpant wll pay, possbly, more than when operatng alone [1]. 3.2.2 Shapley Value A well-known cost allocaton method s the Shapley Value, whch s defned for each player as the weghted average of the player s margnal contrbuton to each subset of the collaboraton [12]. Shapley Value can be nterpreted as the average margnal contrbuton each member would make to the grand coalton f t were to form one member at a tme [13]. Mathematcally, Shapley Value s expressed as ns! s1! y vsvs \, N S n! where s denotes the number of carrers n coalton S. Shapley Value s the unque allocaton method to satsfy three axoms: dummy, addtvty and equal treatment of equals. Although Shapley Value may return cost allocatons n the core for some nstances, there are many nstances where allocatons based on Shapley Value are not stable [1]. 3.2.3 Nucleolus Nucleolus, ntroduced by Schmedler [14], s the cost allocaton that lexcographcally mnmzes the maxmal excess, the dfference between the total allocated proft to a subset and the stand alone cost of that subset, over all the subsets of the collaboraton. Mathematcally, t s expressed as Mnmze st.. y v( S) S S N, S S N y v({ }) N y v( N) The nucleolus exsts and s unque. However t does not take nto account each carrer s contrbutons to the coalton and the relatve cost savngs. 3.3 Weghted Relatve Savngs Model As dscussed above, the exstng solutons are not always stable, whch keeps the sustanablty of the LTL collaboraton, and dfferent to show that some partcpants can gan more f they contrbute more and all partcpants have a smlar relatve proft or cost savngs. In a nego- Copyrght 2010 ScRes
Allocatng Collaboratve Proft n Less-than-Truckload Carrer Allance 147 taton stuaton t would be benefcal to have an ntal allocaton where the relatve savngs are as smlar as possble for all partcpants. We therefore propose the Weghted Relatve Savngs Model (WRSM) whch s completely new and motvated by fndng a stable allocaton that mnmzes the maxmum dfference between relatve savngs among the partcpants and also reflects the contrbuton dfference. The relatve savngs of carrer s expressed as y c. Thus, the dfference n relatve savngs between two partcpants and j s equal to y y j (6) c c j The contrbuton to the collaboraton depends on the dstrbuton of power among freght carrers, on ther level of nterdependency and wllngness to make compromses, and on the market wthn whch the freght carrers operate [5]. Followng the deas of the Shapley Value, we defne the contrbuton of carrer to the grand coalton as vsvs \, N (7) S In order to reflect the contrbuton dfference, we modfy the relatve savngs by addng the contrbuton rato weght whch s expressed as 1 S N S \ vs vs \ v S v S The weghted relatve savngs of carrer s then equal to y c and the dfference n relatve savngs between two partcpants and j s equal to (8) y j y j (9) c c j The Weghted Relatve Savngs Model (WRSM) s the followng LP problem whch we need to solve to fnd the allocaton. Mnmze f.. y jy j s t f, jn c c j y v( S) S N S N y v( N) The frst constrant set s to measure the dfference between all partcpants weghted relatve savngs. The varable f s used n the objectve functon to mnmze the maxmum dfference. The other two constrant sets ensure that the allocaton s n the core and thus stable. We add a mnmum penalzed slack n the constrants defnng the core. In the case the core s empty we propose to use the epslon-core or alternatvely seek the maxmal number of players present n a game for whch the core exsts. However, how ths subgroup of players should be selected remans to be studed n future research. Compared wth the Proportonal Allocaton and the Shapley Value, ths allocaton s stable. Snce the objectve s a combnaton between partcpants and consders the relatve savngs and the contrbuton dfference, ths model s not a weghted nucleolus. In the lterature of ths feld, we have not been able to fnd an allocaton method wth smlar objectve. Therefore, to the best of our knowledge, ths allocaton concept s new. 4. Smulaton Result and Analyss In order to show the effectveness of the method we propose, we compare the Weghted Relatve Savngs Model (WRSM) wth Proportonal Allocaton, Shapley Value and Nucleolus based on the exstng test nstances n [5]. Table 1 presents the nstances used n our test and related calculatons. There are three carrers n the grand coalton and the optmal number of vehcles and cost of each subset of the grand coalton s calculated accordng to the transportaton requests n the sub-coalton [5]. Savngs of Coalton s calculated usng (1). Contrbuton to the Grand Coalton s calculated usng (7) and Contrbuton Rato Weght s calculated usng (8) respectvely. Table 2 shows the results for test nstances and the comparson among Proportonal Allocaton, Shapley Value, Nucleolus and WRSM. For each allocaton concept, Savngs allocated to carrer s calculated accordng to the related defntons and algorthms dscussed above. equals to Stand-alone mnus Savngs. These results show clearly that t s ndeed worth poolng the LTL carrers transportaton resources through collaboraton to serve customer requests. The cost savngs s range from 7.3% to 18.7%. Although the Proportonal Allocaton and Shapley Value s stable usng our test nstances, carrer C wll not agree wth those allocaton methods snce he contrbutes more to the grand coalton but gans the same relatve savngs as carrer A and B n Proportonal Allocaton and the smallest savngs n Shapley Value allocaton. The Nucleolus, whch dvdes the cost savngs equally among three carrers, does not take nto account the contrbuton dfference among the three and may be rejected by any of them. WRSM whch s n the core and consders both relatve savngs and contrbuton dfference makes the Copyrght 2010 ScRes
148 Allocatng Collaboratve Proft n Less-than-Truckload Carrer Allance Carrers n Coalton Table 1. Test nstances and related calculatons # Requests # Vehcles Savngs of Coalton Contrbuton to the Grand Coalton Contrbuton Rato Weght A 61 13 16512.6 0.0 13216.5 0.64 B 96 11 17876.0 0.0 8463.7 0.77 C 100 28 38585.4 0.0 14575.7 0.60 A B 157 24 31961.6 2427.0 A C 161 36 49615.0 5483.0 B C 196 32 53354.8 3106.6 A B C 257 38 64560.9 8413.1 Carrer Stand-alone Table 2. Results for test nstances Proportonal Allocaton Shapley Value Nucleolus WRSM Savngs Savngs Rato Savngs Savngs Rato Savngs Savngs Rato Savngs Savngs Rato A 16512.6 1903.7 14608.9 11.5% 3087.2 13425.4 18.7% 2804.4 13708.2 17.0% 1920.5 14592.1 11.6% B 17876.0 2060.9 15815.1 11.5% 1899.0 15977.0 10.6% 2804.4 15071.6 15.7% 1723.5 16152.5 9.6% C 38585.4 4448.5 34136.9 11.5% 3427.0 35158.4 8.9% 2804.4 35781.0 7.3% 4769.1 33816.3 12.4% SUM 72974.0 8413.1 64560.9 8413.1 64560.9 8413.1 64560.9 8413.1 64560.9 weghted relatve savngs as smlar as possble among dfferent partcpants. It can be accepted by all the carrers and makes the collaboraton sustanable. 5. Conclusons Collaboraton s a good choce for small and medumszed LTL carrers under the background of the nternatonal fnancal crss. Potental cost savngs of the collaboratve allance s large and every partcpant can gan more profts comparng wth stand-alone operaton. In order to realze the benefts, collaboratve proft allocaton mechansm must be able to construct the allance and make t sustanable. The underlyng proft allocaton problem s dscussed n ths paper. We have demonstrated that collaboraton can yeld a consderable cost decrease and proposed a new proft allocaton method named Weghted Relatve Savngs Model (WRSM) based on the cooperatve game theory. Smulaton result for real-lfe nstances shows the effectveness of the proposed model. The truck transportaton ndustry has not yet adopted horzontal cooperaton on a large scale [5]. So the key challenge n terms of future developments s to adapt the proposed method for practcal use so that not all possble coaltons need to be analyzed. 6. Acknowledgements Ths research s supported by Natonal Natural Scence Foundaton of Chna (No. 50175064). The authors are also grateful to anonymous referees for ther helpful comments and nsghts. REFERENCES [1] O. O. Ozener, Collaboraton n transportaton, PhD thess, Georga Insttute of Technology, Atlanta, GA, USA, 2008. [2] F. Crujssen and M. Salomon, Emprcal study: Order sharng between transportaton companes may result n cost reductons between 5 to 15 percent, CentER Dscusson Paper, 2004, http://deas.repec.org/p/dgr/kubcen/200480.html. [3] O. Ergun, G. Kuyzu, and M. Savelsbergh, Shpper Collaboraton, Computers & Operatons Research, No. 34, pp. 1551 1560, 2007. [4] O. Ergun, G. Kuyzu, and M. Savelsbergh, Reducng truckload transportaton costs through collaboraton, Transportaton Scence, No. 41, pp. 206 221, 2007. [5] M. Krajewska and H. Kopfer, Horzontal cooperaton among freght carrers: Request allocaton and proft Sharng, Journal of the Operatonal Research Socety, No. 59, pp. 1483 1491, 2008. [6] P. Stroznak, Collaboratve logstcs: Overcomng ts challenges can lower transportaton and nventory costs and reduce stockouts, Frontlne Solutons, 2003, http://www.frontlnetoday.com. [7] M. Sakawa, I. Nshzak, and Y. Uemura, Fuzzy programmng and proft and cost allocaton for a producton and transportaton problem, European Journal of Operatonal Research, No. 131, pp. 1 15, 2001. [8] J. Sanchez-Sorano, M. A. Lopez, and I. Garca-Jurado, On the core of transportaton games, Mathematcal Socal Scences, No. 41, pp. 215 225, 2001. Copyrght 2010 ScRes
Allocatng Collaboratve Proft n Less-than-Truckload Carrer Allance 149 [9] J. Sanchez-Sorano, N. Llorce, A. Meca, and E. Molna, An ntegrated transport system for Alacant s students, Annals of Operatons Research, No. 109, pp. 41 60, 2002. [10] S. Engevall, M. Gothe-Lundgren, and P. Varbrand, The heterogeneous vehcle-routng game, Transportaton Scence, No. 38, pp. 71 85, 2004. [11] N. Matsubayash, M. Umezawa, Y. Masuda, and H. Nshno, A cost allocaton problem arsng n hub spoke network systems, European Journal of Operatonal Research, No. 160, pp. 821 838, 2005. [12] L. S. Shapley, A value for n-person games, Annals of Mathematcal Studes, No. 28, pp. 307 317, 1953. [13] H. P. Young, allocaton: Methods, prncples, applcatons, European Journal of Operatonal Research, No. 27, pp. 254 255, 1986. [14] D. Schmedler, Nucleolus of a characterstc functon game, Sam Journal on Appled Mathematcs, No. 17, pp. 1163 1170, 1969. Copyrght 2010 ScRes