THE LOAD PLANNING PROBLEM FOR LESS-THAN-TRUCKLOAD MOTOR CARRIERS AND A SOLUTION APPROACH. Professor Naoto Katayama* and Professor Shigeru Yurimoto*

Transcription

1 7th Internatonal Symposum on Logstcs THE LOAD PLAIG PROBLEM FOR LESS-THA-TRUCKLOAD MOTOR CARRIERS AD A SOLUTIO APPROACH Professor aoto Katayama* an Professor Shgeru Yurmoto* * Faculty of Dstrbuton an Logstcs Systems, Ryutsu Keza Unversty 10 Hrahata, Ryugasak, Ibarak , Japan ABSTRUCT The man stress n the loa plannng problem for less-than-truckloa (LTL) motor carrers falls on etermnng how to consolate freght on small-lot consgnment over a loa plannng network nclung break-bulk termnals. The goal of ths problem s to mnmze the total lne-haul cost uner the conton that the mnmum frequency of elvery per week between a par of termnals must satsfy a gven servce level. In ths stuy, we propose a loa plannng mel an new algorthm usng a Lagrangan relaaton meth. umercal eperments are presente to evaluate the effectveness of our Lagrangan relaaton meth. ITRODUCTIO Progress of the supply chan management an the current tren towar eregulaton of the Japanese truckng nustry places the freght motor carrers n a hghly compettve envronment. As a result of that, the carrers nee to conser strateges an tactcs that satsfy both cost mnmzaton an a efnte level of servce qualty. In general, a less-than-truckloa motor carrer hauls shpments n the range of 50kg to 5000kg. Snce a stanar traler can hol 10-ton to 30-ton of shpment, t s necessary for the LTL motor carrers to consolate the freght to make the best use of tralers. The freght orgnatng at an en-of-lne s loae onto a lne-haul truck, whch carres t to a break-bulk termnal. At ths termnal, the freght s unloae, sorte an reloae onto a traler, whch carres t to another termnal. One of the man problems face by LTL motor carrers s to etermne how freght shoul be route over the network. Ths problem s calle the loa plannng problem for LTL motor carrers. It can be formulate as a huge me nteger optmzaton problem. Prevous research on ths problem s lmte. Powell (1986) an Powell-Sheff (1989) propose heurstc approaches usng a-rop local search meths. Cranc-Roy (199) escrbes a set-coverng formulaton an a soluton meth for the loa plannng problem. Powell-Delorme (1989) presents so-calle ETPLA an Powell-Koskoss (199) uses a graent-base local search meth an the Lagrangan heurstc approach wth a relaaton of mnmum servce level constrants. Hoppe et al.(1999) proposes a heurstc approach usng the labelng algorthm an a-rop local search meths. Cranc (1999) surveys we varety of freght transportaton plannng problems. The prncpal concern n the loa plannng problem s about etermnng how to consolate freght on small-lot consgnment over a loa plannng network nclung break-bulk termnals n orer to mnmze the total lne-haul cost. Ths problem s approache as a two-tere problem: 1) between whch pars of termnals shoul rect servce be offere, ) gven a set of rect servces, how shoul the freght be route over the network. Ths problem s formulate as follows: 1) lne-haul costs between termnals shoul be mnmze, ) the mnmum frequency of elvery per week between a par of termnals must satsfy a gven servce level, 3) the paths from all orgn termnals nto a estnaton termnal form a tree, whch reflects that the freght at a termnal wth same estnaton shoul be loae onto a 1

2 7th Internatonal Symposum on Logstcs Destnaton Termnal truck heang for one termnal. Fgure 1 llustrates an eample of the loa plannng network. Fgure llustrates frequency of servce between a par of termnals. In ths stuy, we propose a loa plannng mel an new algorthm usng a Lagrangan relaaton meth. FORMULATIO FOR THE LOAD PLAIG PROBLEM The loa plannng problem can be formulate as a me nteger programmng problem. We use the followng notaton for our mel. s the set of nes, whch conssts of en-of-lne an break-bulk termnals. A s the set of all potental lnks for rect servces n the loa plannng network, A. f s the mnmum frequency of tralers per week on the lnk (,) from ne to, (,)A. e s the loa capacty of a traler on the lnk (,). a s the lne-haul cost per traler on the lnk (,). q s the total LTL freght eman of commty (o,) orgnatng at termnal o an estne for termnal per week, o. n s a constant, whch equals to 1 f ne n s estnaton, -1 f ne n s orgn o an 0 otherwse, n,o,. z s the total lne-haul cost on the lnk (,). s the total freght flow on Flow of Freght (Ton per Week) the lnk (,). s the bnary ecson varable, whch equals to 1 f the freght flow of commty (o,) s route on the lnk (,) an 0 f not. y s the bnary ecson varable, whch equals to 1 f a rect servce s beng offere on the lnk (,) an 0 f not. y s the bnary ecson varable, whch equals to 1 f the freght flow estne for termnal s route on the lnk(,) an 0 f not. The loa plannng problem can be state as follows. (LTL) mnmze z y, A subect to n q n o () o, (3) n n A, o (4) y y y, Drect Servce Break-Bulk Termnal En-of-Lne Termnal Fgure 1. Illustratve Loa Plannng etwork Frequency of Servce (Tralers per Week) Mnmum Frequency Freght Route Destne for a Termnal Fgure. Frequency of Drect Servce between a Par of Termnals (1), (5)

3 z 7th Internatonal Symposum on Logstcs y 1 \, (6) y 0 (7) a / e f fe, a f f fe,1, y A (8),1, o 0 (9) 0 A (10),1, y 0 (11) The obectve functon (1) s the total lne-haul cost an shoul be mnmze. Constrant () epresses the stanar flow conservaton. Constrant (3) shows the relatonshp between the total freght flow an the emans of commty (o,). Constrant (4) states that f the flow estne for s not route on the lnk(,), then the flow of every commty (o,) on the lnk (,) must be zero. Constrant (5) states that f the rect servce on the lnk (,) s not beng offere, then the flow estne for must not be route on the lnk (,). Constrants (6) an (7) nsure that the paths from all orgn termnals nto a estnaton termnal form a tree. Constrant (8) states that the mnmum frequency of elveres between a par of termnals must satsfy a gven servce level. Constrants (9), (10) an (11) are the bnary requrements. A LAGRAGIA RELAXATIO PROBLEM The Lagrangan relaaton s one of general soluton strateges for solvng mathematcal programmng problems that permt us to ecompose problems to eplot ther specal structure. When we use vectors of the Lagrange multplers v={v } relatve to constrant () an w={w } relatve to (6) an (7), an a them to the obectve functon (1), the followng Lagrangan relaaton problem LG can be forme. (LG) mnmze z y v v w y, A, A o v v w o o \ subect to (3)-(5) an (8)-(11) where w 0, \ {},. Gven the Lagrange multplers v an w, we can eal wth the thr an the fourth terms of the obectve functon (1) as constant terms. LG can be ecompose nto followng subproblem LG for each lnk (,). z y v v w y (13) (LG ) mnmze subect to z y a o q a f,1 o o (1) y o (14) y y (15) / e f f o o 0 (17) 0,1 0, (19) 1 q q f f e e (16) y (18) Furthermore, LG can be ecompose nto followng two subproblems, LG 1 an LG. a q / e y v v w y (0) (LG 1 ) mnmze o subect to q fe o (1) (14),(15) an (17)-(19) (LG ) mnmze f v v a w y () o 3

4 7th Internatonal Symposum on Logstcs subect to o q fe (3) (14),(15) an (17)-(19) When we gve the Lagrange multplers v an w, an solve the Lagrangan relaaton problem LG or further a relaaton problem optmally, a lower boun for LTL can be obtane. A OPTIMAL SOLUTIO FOR A LAGRAGIA RELAXATIO PROBLEM At frst, we assume that the Lagrange multplers v are gven an let w=0. Then LG 1 can be rewrtten as a smple problem. a q / e y v v (4) mnmze subect to o y o (5) (17), (18) an (1) We ecompose ths problem nto two subproblems n the case of y =0 an y =1. Obvously, when y =0, the optmal soluton s =0( o,) an the optmal value of equaton (4) s 0. When y =1, ths problem can be rewrtten as the followng 0-1 knapsack problem LG 11. (LG 11 ) mnmze a q / e v v (6) 1 o subect to (17) an (1) Ths problem relae 0-1 contons turn out to be the contnuous knapsack problem an can be smply solve by sortng, an then a lower boun an the relaaton soluton for LG 11 are easly obtane. Accorngly the lower boun for LG 1 s mn{0, 1 }, whch s the mnmum value of the optmum n the case of y =0 an y =1. As wth LG 1, LG can be ecompose nto two cases of y =0 an y =1. When y =1, ths problem can be rewrtten as the followng problem LG 1. (LG 1 ) a f v v (7) mnmze subect to (17) an (3) o Consequently the optmal value or the lower boun for LG s mn 0, 1, v v w. (8), A o o \ For (,) the optmal soluton s y =1 f 1 <0 or <0 an y =0 f not. For o (,) the optmal value of s X 1 f 1 an y =1, X f < 1 an y =1, an 0 otherwse, where X 1 s the soluton for LG 11 an X s the soluton for LG 1. Atonally, for (,) the optmal soluton s y =1 f some >0(o) an y =0 f not, because w=0. From these epressons, we can solve the Lagrangan relaaton problem LG an obtan the lower boun for the loa plannng problem LTL. A MULTIPLIER ADJUSTMET AD A SUBGRADIET METHOD We evelop the multpler austment meth for settng the value of w. Increasng the value of w from 0, whle w s feasble an the soluton an y for LG o not change, we coul also ncrease the lower boun for LTL. For LG 1 1, let mn{ mna q / e y v v 0, o maa q e y v v 0, o } /. (9) 4

5 7th Internatonal Symposum on Logstcs For LG, let mn{ mnv v 0, o mav v 0, o }. (30) Then we set the ncrement value of w as 1 1 w : ma0, mn,,, \, f K, (31) where 1 mn a q / e y v v w (3) o \ mn a f v v w o \ number of 0, o (33) K the. (34) When the values of w ascen up to these values, w s feasble because w 0, an the optmal solutons for LG stll o not change. Then the lower boun can be ncrease as much as w K 1. (35) \ The frst term w K s the ncrement value of the secon term n (1) an the secon term -w s the ecrement value of the fourth term n (1). For settng the values of v appromately, we apply the stanar subgraent optmzaton proceure (Fsher,1981). Ths s an teratve proceure, whch uses the current multplers v, the current lower boun an an upper boun, n orer to compute the new multplers v use n the net teraton. Subgraent g of v can be efne follows, g n o. (36) n n n n Then, usng a step sze s t n teraton t, the new set of multplers are gven by t vn : vn s gn n o. (37) It can be shown for a fnte carnalty, f the step sze s t s selecte so that 0 t t lm t s 0,whle s, then the sequence v converges to the optmal value. We use t the step sze as s t p t the best known upperboun- the current lower boun / g (38) where p t s a scalar whch s ntally equal to 1 an s reuce every some teraton number. UMERICAL EXPERIMETS In orer to test the performance of our Lagrangan relaaton meth, a set of numercal eperments s carre out usng IBM compatble computer wth PETIUM4 1.7GHz, memory 56Mb an OS Wnows 000. Ths soluton meth s ce n COMPAQ VISUAL FORTRA Ver.6. The problem ata use n these eperments s ranomly generate up to 50 nes. the set of nes presente en-of-lne an break-bulk termnals s rawn from a unform strbuton over a rectangle measurng 100 by 100. the set of all potental lnks of rect servces s. The lne-haul cost per traler on the lnk s n Table 1. etwork Dmensons an Gaps A Commty Gap(%) % % % % % proporton to the Euclean stance between nes. Each of LTL freght eman s 1, the mnmum frequency s 1 an the loa capacty s. Obtanng for an upper boun an appromate solutons, we use three kns of Lagrangan heurstc algorthms (Katayama,00), whch 5

6 140% 130% 10% 110% 100% 90% 80% 70% 60% 50% 40% 30% 0% 10% 0% Upper Boun Lower Boun Gap o. of teraton Fgure 3. Convergence of meth for 30 nes 7th Internatonal Symposum on Logstcs are a lnk elete heurstc, a successor matr mfcaton heurstc an a tabu search meth. Table 1 brefly summarzes the effectveness of the Lagrangan relaaton meth. It shows the number of nes, potental rect servces, commty an the percentage gap between the best upper boun an the best lower boun. The percentage gaps range from 1.90% to 6.3%. Fgure 3 shows the rate of convergence for the problem wth 30 nes. The subgraent optmzaton algorthm ehbte the fastest rate of convergence. COCLUSIOS In ths paper, we evelope the loa plannng problem for LTL motor carrers an ts soluton meth usng the Lagrangan relaaton. The result of the eperments suggest that our Lagrangan relaaton problem an soluton approach can perform a go ob of entfyng a lower boun of the loa plannng problem for LTL motor carrers. Ths research s unerway to aapt solutons to the real worl problems, such as the empty traler balancng, the transt tme an the number of transshpment, etc. REFERECES Cranc T G an Roy J (199) Desgn of regular ntercty rver routes for the LTL motor carrer nustry, Transportaton Scence, 6, Cranc T G (1999) Long-Haul Freght Transportaton, n Hanbook of Transportaton Scence, , Kluwer Acaemc Publshers, Boston. Fsher M L (1981) The Lagrangan relaaton meth for solvng nteger programmng problem, Management Scence, 7, Hoppe B, Klampfl E Z, McZeal C an Rch J (1999) Strategc loa-plannng for less-than-truckloa truckng, CRPC-TR9981-S, Center for Research on Parallel Computaton, Rce Unversty. Katayama (00) Heurstcs for the less-than-truckloa plannng problem, Workng Paper, Ryutsu Keza Unversty (n Japanese). Powell W B (1986) A local mprovement heurstc for the esgn of less-than-truckloa motor carrer networks, Transportaton Scence, 0, Powell W B an Koskoss I A (199) Shpment routng algorthms wth tree constrants, Transportaton Scence, 6, Powell W B an Sheff Y (1989) Desgn an mplementaton of an nteractve optmzaton system for network esgn n the motor carrer nustry, Operatons Research, 37, 1-9. Roy J an Delorme L (1989) ETPLA:A network optmzaton mel for tactcal plannng n the less-than-truckloa motor-carrer nustry, IFOR, 7, -35. Roy J an Cranc T G (199) Improvng ntercty freght routng wth a tactcal plannng mel, Interfaces,,