Vehicle Routing Problem with Simultaneous Pickup and Delivery in Cross-Docking Environment

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1 Vhicl Routing Problm with Simultanous Picup and Dlivry in Cross-Docing Environmnt Chiong Huang and Yun-Xi Liu Abstract This study will discuss th vhicl routing problm with simultanous picup and dlivry in a cross-docing nvironmnt. Th transportation systm includs thr s: (1) picup goods, (2) dlivry goods and picup rturnd goods, and (3) dlivry all rturnd goods. In th scond, som oprations of (3) may b accptd if tim and loading capacity ar availabl. W bliv this spcial dsign can furthr rduc th transportation cost. A mathmatical modl is dvlopd. Th objctiv of this modl is to minimiz ovrall transportation cost including th vhicl transportation cost and th fixd cost of ach rout. An appropriat huristic algorithm for solving th problm is also dvlopd by using Tabu logic. Rsult of numrical xampl indicats that th proposd thr- transportation systm and huristic algorithm can solv th vhicl routing problm ffctivly and fficintly. Indx Trms Cross-docing, picup and dlivry, vhicl routing problm. com bac to th sam ara gathring all rturnd goods. Aftr sorting th rturnd goods at th cross-docing ara, th third transportation starts at th sam tim. In this, all rturnd goods will b dlivrd bac to th supplirs through svral indpndnt vhicl routs. On spcial dsign in th scond is that w accpt som vhicls to dlivr th rturnd goods to thir final dstinations if tim and loading capacity ar prmittd. W bliv this optional tas, which originally should b xcutd in th third transportation, can furthr rduc th ovrall transportation cost. Th thr- transportation systm in this rsarch mas th solution procss mor difficult. Howvr, if th ovrall cost can b furthr rducd, it is still worth to do. Basd on this ida, a mathmatical modl will b dvlopd in sction III and a solution algorithm is also proposd in sction IV. I. INTRODUCTION In th xtrmly comptitiv and rapidly changing mart, improving th fficincy of logistics systm can not only rduc opration cost for ntrpriss but also improv customr satisfaction. A traditional transportation systm can ithr dlivr goods or picup goods through vhicl routings. Howvr, simultanous picup and dlivry within singl opration bcoms a nw trnd in Taiwan. In addition, th cross-docing concpt has bn introducd to logistic systm. This approach can ffctivly rduc th cost of warhous facilitis and incras opration fficincy. It is blivd that a cross-docing systm combind with simultanous picup and dlivry oprations will b a nw ida for dsign of logistic systm. This study proposs a nw concpt of transportation systm assuming simultanous picup and dlivry oprations in a cross-docing nvironmnt. Within a planning priod, this transportation systm aggrgats all dlivry and picup oprations in thr conscutiv transportation s. In th first, all goods ar picd up from th supplirs to th cross-docing ara by svral indpndnt vhicl routings. Th scond transportation, all goods ar dlivrd to th nd customrs. Simultanously, this prforms picup oprations for all rturnd goods from th nd customrs if thy call for. All vhicls in this dpart from th cross-docing ara and Manuscript rcivd August 25, 2013; rvisd Novmbr 6, This wor was supportd in part by th National Scinc Counsl, Taiwan ROC undr Grant NSC E Chiong Huang and Yun-Xi Liu ar with th National Yunlin Univrsity of Scinc and Tchnology, Yunlin 640, Taiwan (-mail: huangc@yuntch.du.tw, m @yuntch.du.tw). II. LITERATURE REVIEW A. Cross-Docing Transportation Th cross-docing transportation is on stratgy of warhous dsign. Goods can stay in cross-docing ara for sorting and r-loading vhicl in a short priod of tim. Goods will b dlivrd to nxt dstination by anothr flt which usually dparts th cross-docing ara at th sam tom. Du to th short tim storag and minimum storag facilitis rquird, it is blivd that th cross-docing dsign can ffctivly rduc transportation cost and minimiz invntory cost [1]. Rfrnc [2] indicats that this approach can also incras customr satisfaction, rduc oprating tim, and rduc lad tim of dlivry. In th ral-world application of cross-docing dsign, both Wal-Mart and Toyota show bnfits on cost-ffctiv. Timing is an important dsign factor, spcially for vhicl picup and dlivry systm [3]. This papr will try to ta ths advantags in dvloping a nw vhicl transportation systm. B. Vhicl Routing Problms Th vhicl routing problm (VRP) is first discussd by Dantzing & Ramsr [4] on th truc dispatching problm (TDP). Traditional VRP is focusd on dispatching flt on xcution th dlivry srvic givn a singl dpot, srvic nods and quantitis rquird. Th objctiv is to minimiz travl distanc or dlivry cost. In th rcnt yars, svral ral-world constraints hav bn considrd to a traditional vhicl routing problm, such as th capacitatd VRP, th multi-dpot VRP, th VRP with tim windows, th pic-up and dlivry VRP. Ths additional conditions ma th problm mor ralistic but mor difficult to solv. DOI: /JOEBM.2015.V

2 Sinc th cross-docing oprations will connct two conscutiv dlivris, thrfor, th VRP with hard tim windows will applis in this rsarch. In addition, a fixd and givn loading capacity for ach vhicl and simultanous picup and dlivry oprations in th scond ar all ncssary assumptions. C. Huristic Algorithms In th rcnt yars, svral mta-huristic algorithms hav bn wll dvlopd for solving th NP-hard vhicl routing problms, such as Simulatd Annaling algorithms [5], Tabu Sarchs, Gntic Algorithms [6]. For ffctivly us of ach mta-huristic, a st of dsign paramtrs should b tstd and fin-tund. This study will apply th logic of Tabu sarch. Th dsign paramtrs includ lngth of Tabu list, numbr of itrations, and probabilitis of nighborhood sarching approachs. Th Taguchi approach will b usd in this study to dcid suitabl dsign paramtrs for Tabu sarch. III. PROBLEM ASSUMPTIONS AND MODEL CONSTRUCTION A. Problm Statmnt and Assumptions Th proposd thr- transportation systm will b modld in this sction and it will also b dscribd in dtails. Oprations in on transportation ar indpndnt to th oprations in othr s. All transportation oprations should b ndd within a givn planning priod or woring priod. Th first transportation prforms all picup oprations from th supplirs. In this, all vhicls dpart th cross-docing ara at th sam tim and thy will rturn to th cross-docing ara within a givn tim window. Aftr sorting and arranging goods at th cross-docing ara, th scond can b startd. Th scond transportation xcuts two tass for nd customrs: (1) goods dlivry to nd customrs and (2) picup rturnd goods from nd customrs. In this, all vhicls dpart th cross-docing ara at th sam tim and thy will rturn to th cross-docing ara within a givn tim window. Aftr sorting and arranging th rturnd goods at th cross-docing ara, th third transportation can b startd. Th third transportation dlivrs all rturnd goods bac to th supplirs or rturn points. All vhicls in th third dpart th cross-docing ara at th sam tim and thy should rturn to th cross-docing ara bfor th nd of planning priod, i.. th nd of woring priod. Bcaus th singl cross-docing ara plays an important rol for goods sorting, vhicl dispatching, and vhicl r-loading. Thrfor, a hard tim window is absolutly ncssary spcially for th tim priod btwn th 1 st and 2 nd s and th tim priod btwn th 2 nd and 3 rd s. Svral assumptions and limitations should b clarly dfind bfor th modl construction. Th basic assumptions applid for all thr s ar listd as follows. Thr is a singl cross-docing ara, location is fixd, givn, and nown. Location and quantity for ach supplir and nd customr ar fixd, givn, and nown. In ach, ach nod can b srvd by on vhicl and b srvd onc only. Only on vhicl typ, loading capacity and vhicl spd ar fixd, givn, and nown. Transportation cost pr unit distanc and fixd cost pr rout ar fixd, givn, and nown. Dlivry quantity for ach nod should b lss than th vhicl loading capacity. No invntory and no dfct itm occur in all s. All goods and rturnd goods ar balancd in th systm. Tim for goods sorting, vhicl dispatching, and vhicl r-loading at th cross-docing ara is fixd, givn, and nown. No invntory cost is considrd in this study. All oprations should b finishd within th planning priod. Shipping quantitis of th planning priod should mt th quantitis rquird by nd customrs. Th basic assumptions for th first transportation ar listd as follows: 1) Only picup oprations ar xcutd in this. 2) All vhicls dpart from th cross-docing ara at th sam tim with mpty loading. 3) All vhicls rturn to th cross-docing ara within a givn tim window. Th basic assumptions for th scond transportation ar listd as follows: 1) Assum thr typs of dlivry rquirmnt for th nd customr: (1) dlivry goods only, (2) rturnd goods only, (3) both dlivry goods and rturnd goods at th sam tim. 2) Vhicl can snd th rturnd goods dirctly to th final dstinations, i.. supplirs, if th tim and loading capacity ar tolrabl. 3) All vhicls dpart from th cross-docing ara at th sam tim which is stimatd by th latst vhicl rturn tim in th first plus th fixd sorting tim spnt in th cross-docing ara. 4) All vhicls should rturn to th cross-docing ara within a givn tim window. Th basic assumptions for th third transportation ar listd as follows. 1) Only th rturnd goods will b srvd in this. 2) All vhicls dpart from th cross-docing ara at th sam tim which is calculatd by th latst vhicl rturn tim in th scond plus th fixd sorting tim spnt in th cross-docing ara. 3) All mpty vhicls should rturn to th cross-docing ara bfor th nd of planning priod. B. Dfinitions of Variabls and Notations Th following dfinition for notations, sts, paramtrs, and dcision variabls will b usd in th mathmatical modl. Notations: i, j, h: Indx for supplirs, nd customrs, and rturn points (i.. supplirs) : Indx for vhicls Sts: N: Th st of all nods which includ supplirs (n), nd customrs (m), and rturn points (n) N 0 : Th st of all nods including cross-docing point p: Th st of all supplirs, p = 1, 2,, n 61

3 P: Th st of all supplirs including cross-docing point, P = 0, 1, 2,, n d: Th st of all nd customrs, d = n + 1, n + 2,, n + m D: Th st of all nd customrs including cross-docing point, D = 0, n + 1, n + 2,, n + m z: Th st of all rturn points, z = p Z: Th st of all rturn points including cross-docing point, Z = P K: Th st of all vhicls Paramtrs: C: Transportation cost pr unit distanc FC: Fixd cost pr rout Q: Loading capacity of vhicl Dq1 0 : In th 1 st, loading quantity of vhicl whn it dparts from cross-docing point Dq1 i : In th 1 st, loading quantity of vhicl whn it dparts from nod i Dq2 0 : In th 2 nd, loading quantity of vhicl whn it dparts from cross-docing point Dq2 i : In th 2 nd, loading quantity of vhicl whn it dparts from nod i Dq3 0 : In th 3 rd, loading quantity of vhicl whn it dparts from cross-docing point Dq3 i : In th 3 rd, loading quantity of vhicl whn it dparts from nod i p i : Picup quantity at supplir i d i : Dlivry quantity at nd customr i dp i : Picup quantity for rturnd goods at nd customr i z i : Quantity of rturnd goods at rturn point (supplir) i d ij : Distanc from nod i to nod j SP: Vhicl spd C ij : Transportation cost from nod i to nod j, C ij = C d ij t ij : Vhicl travl tim from nod i to nod j, t ij = d ij SP UP1: Aftr nd of 1 st, tim for sorting and rloading vhicl at cross-docing point UP2: Aftr nd of 2 nd, tim for sorting and rloading vhicl at cross-docing point s i : Srvic tim at nod i DT1 i : In th 1 st, dpartur tim of vhicl at nod i AT1 i : In th 1 st, arrival tim of vhicl at nod i DT2 i : In th 2 nd, dpartur tim of vhicl at nod i AT2 i : In th 2 nd, arrival tim of vhicl at nod i DT3 i : In th 3 rd, dpartur tim of vhicl at nod i AT3 i : In th 3 rd, arrival tim of vhicl at nod i ATP max : In th 1 st, th latst vhicl rturning tim at cross-docing point ATD max : In th 2 nd, th latst vhicl rturning tim at cross-docing point DDT: In th 2 nd, dpartur tim for all vhicls at cross-docing point, DDT = ATP max + UP1 ZDT: In th 3 rd, dpartur tim for all vhicls at cross-docing point, ZDT = ATD max + UP2 T p : Srvic tim (uppr limit) for th 1 st T d : Srvic tim (uppr limit) for th 1 st plus th 2 nd T: Srvic tim (uppr limit) for all thr s, tim for on planning priod X ij : Y ij : Z ij : U i : M: a maximal positiv numbr Dcision variabls: V i : B i : 1, If vhicl srv th path btwn nod i and nod j in th first 1, If vhicl srv th path btwn nod i and nod j in th scond 1, If vhicl srv th path btwn nod i and nod j in th third 1, If vhcl srv nod i in th first 1, If vhcl srv nod i in th scond 1, If vhcl srv nod i in th third C. Modl Construction Th mathmatical modl in this sction intgrats thr transportation s and it bcoms a complicat and larg-scal vhicl routing problm. Minimiz Z = Subjct to: i P j P K C ij X ij + j N 0 K C ij Y ij i Z j Z K C ij Z ij + K FC X 0j j N K FC Y 0j + K FC Z 0j i N 0 + j p + j z (1) i P K X ij = 1 j = 1,, n (2) j P K X ij = 1 i = 1,, n (3) i N K Y ij = 1 j = 1,, n (4) j N K Y ij = 1 i = 1,, n (5) i Z K Z ij 1 j = 1,, n (6) j Z K Z ij 1 i = 1,, n (7) i P X ij i P X ji = 0 j P K i j (8) i N 0 Y ij i N 0 Y ji = 0 j N K i j (9) i Z Z ij i Z Z ji = 0 j Z K i j (10) j p X oj = 1 K (11) X io i p = 1 K (12) Y oj j d = 1 K (13) Y io i N = 1 K (14) j z Z oj 1 K (15) Z io i z 1 K (16) Dq1 0 = 0 K (17) Dq1 j Dq 0 + p j 1 X 0j M jεp εk (18) Dq1 j Dq1 i + p j 1 X ij M i, jεp εk (19) Dq2 0 = i N 0 j N d j Y ij K (20) 62

4 Dq2 j Dq2 0 d j + dp j 1 Y 0j M jεn, εk (21) Dq2 j Dq2 i d j + dp j z j 1 Y ij M i, jεn, εk (22) Dq3 0 = i Z j z z j Z ij K (23) Dq3 j Dq3 0 z j 1 Z 0j M jεz εk (24) Dq3 j Dq3 i z j 1 Z ij M i, jεz εk (25) Dq1 j Q + 1 Dq2 j Q + 1 Dq3 j Q + 1 Dq1 0 Q K (26) Dq2 0 Q K (27) Dq3 0 Q K (28) X ij i p M j p K (29) Y ij i p M j p K (30) Z ij i p M j p K (31) i P p i U i = i D d i V i K (32) i D dp i V i = i P z i V i + B i K (33) i P j P K s i X ij + i P j P t ij X K ij T p (34) i P j P K s i X ij + i P j P K t ij X ij + i D j N K s i Y ij + i D j N K t ij Y ij T d (35) i P j P K s i X ij + i P j P K t ij X ij + i D j N K s i Y ij + i P j P K t ij Y ij + i P j P K s i Z ij + i P j P K t ij Z ij T (36) DT1 j t ij + DT i + s j (1 X ij ) M i, j P K (37) DT2 j t ij + DT i + s j (1 Y ij ) M i, j N 0 K (38) DT3 j t ij + DT i + s j (1 Z ij ) M i, j Z K (39) AT1 j t ij + DT i (1 X ij ) M i, j P K (40) AT2 j t ij + DT i (1 Y ij ) M i, j N 0 K (41) AT3 j t ij + DT i (1 Z ij ) M i, j Z K (42) AT1 max = Max{AT1 0 X i0 i P K } (43) AT2 max = Max AT2 0 Y i0 i N 0 K (44) DDT AT1 max + UP1 1 K X i0 M Mi P K (45) ZDT AT2 max + UP2 1 K Y i0 M i N K (46) X ij j P = U i i p i j K (47) Y ij j N 0 = V i i N i j K (48) Z ij j Z = B i i z i j K (49) U i K = 1 i p (50) V i K = 1 i d (51) K V i + K B i = 1 i z (52) X ij 0,1 i, j P K (53) Y ij 0,1 i, j N 0 K (54) Z ij 0,1 i, j Z K (55) U i 0,1 i P K (56) V i 0,1 i N 0 K (57) B i 0,1 i Z K (58) Th objctiv function in quation (1) is to minimiz ovrall cost, which includ transportation costs in thr s and fixd costs of vhicl routing in thr s. Equation (2) to (7) mas sur that ach nod in ach is srvd onc by on vhicl. Equation (8) to (10) indicats a balancd flow quantity in ach. Equation (11) to (16) confirms that ach rout starts from th cross-docing point. Vhicl loading constraints and limitations ar dscribd from quation (17) to (31). Equation (17) mas sur th vhicl in th 1 st should b mpty whn dparting from th cross-docing point. For ach transportation, quation (18)-(19), (20)-(22), and (23)-(25) show th variation of loading quantity btwn two succssiv nods in a rout. Equation (26) to (31) confirms that no ovrloading is accptabl. Equation (32) mas sur that total picup quantity in th 1 st quals to total dlivry quantity in th 2 nd. For th rturnd goods, quation (33) indicats th picup quantity is th sam as th dlivry quantity. Th constraints of opration tim and tim fnc ar dscribd from quation (34) to (46). Equation (34) to (36) indicats that all oprations in ach should b finishd within tim window. Equation (37) to (42) shows how to accumulat dpartur tim and arrival tim btwn two succssiv nods. Equation (43) to (44) dfins th latst rturning tim for th 1 st and th 2 nd. Equation (45) to (46) dfins th dpartur tim for th 2 nd and th 3 rd. Equation (47) to (49) dfins th rlationship btwn two dcision variabls. Equation (50) mas sur that ach supplir should b srvd by on vhicl only. Equation (51) mas sur that ach nd customr should b srvd by on vhicl only. Equation (52) mas sur that ach rturn point should b srvd by on vhicl only ithr in th 2 nd or in th 3 rd. Finally, quation (53) to (58) forcs th dcision variabls to b an intgr, ithr 0 or 1. IV. CONCEPT OF SOLUTION ALGORITHM A huristic solution algorithm on th basis of Tabu logic is dvlopd for finding all thr- vhicl routings in dtails. Th output of this solution algorithm should answr th following qustions for managmnt lvl: how many routs rquird in ach, how many vhicl is ncssary to xcut all oprations in th planning priod, th dtaild vhicl travl plan for ach individual routing. Th most important information is th ovrall cost of this transportation systm, which includs th dlivry costs and th fixd costs 63

5 of all routs in ach transportation. Th ovrall dsign logic of this proposd solution algorithm is prsntd in Fig. 1. Start Construct Initial Routings for th 1 st Stag Construct Initial Routings for th 2 nd Stag Construct Initial Routings for th 3 rd Stag Stup Paramtrs for th Tabu Sarch Improv Routings for All Stags Chc th Stopping Critria Output Final Rsults and Stop Fig. 1. Dsign logic for th solution algorithm. Thr ar two phass in th dsign logic. Th first phas is to construct initial vhicl routings starting from th 1 st transportation to th 3 rd transportation in squnc. Th basic ida for constructing an initial routing is using th nighborhood sarching approach starting from th dpot point, i.. th cross-docing point in this rsarch. Onc a narst nw srvic nod is found, svral chcing critria should b considrd, such as th accumulatd loading quantity, th accumulatd srvic tim, tim lft to rturn to th dpot point. In ach transportation, ths itrations will b rpatdly xcutd until all srvic points ar arrangd in a routing plan. Th scond phas of th dsign logic is to improv th initial vhicl routings in th systm. This phas applis th Tabu logic for routing improvmnt and intgrats all thr transportation s in on improving procss. Th dsign paramtrs of this Tabu sarch includ th lngth of Tabu list, numbr of itration to nd th sarching procss, probabilitis for slction of nighborhood xchang approachs. Thr ar four possibl nighborhood xchang approachs dvlopd in ach transportation. For th first transportation, th improvmnt approachs includ: (1) xtrnal rout 1-0 nod chang, (2) xtrnal rout 1-1 nod xchang, (3) intrnal rout 2-opt nod xchang, and (4) intrnal rout or-opt nod xchang. Th improvmnt approachs for th scond and third ar: (1) xtrnal rout 1-0 nod chang, (2) xtrnal rout 2-xchang, (3) intrnal rout 2-opt nod xchang, and (4) intrnal rout or-opt nod xchang. In addition, th Taguchi xprimnt approach is usd for fin tuning th dsign paramtrs of Tabu sarch. For solving th ral-world and larg-scal problm, th solution algorithm dvlopd in this rsarch is furthr codd by th VBA languag and running on Windows 7 systm. V. DEMONSTRATION EXAMPLE AND DISCUSSION A. Basic Data of th Illustration Exampl A numrical xampl with on cross-docing point is dvlopd in this sction to dmonstrat ffctivnss of th mathmatical modl and th fficincy of th solution algorithm proposd in this study. In this xampl, thr ar 50 nods within a squar ara, i by 100 ilomtr. 20 nods out of 50 ar supplirs which also rcognizd as th rturn points. Th rst of thm ar th nd customrs. Fig. 2 indicats all locations including: th cross-docing point (yllow squar), all supplirs (blu triangls), and all nd customrs (rd circls). Dlivry quantitis ar gnratd by a uniform distribution btwn 10 and 50. Quantitis of rturnd goods ar also uniformly distributd btwn 0 and Fig. 2. Locations of cross-docing point and 50 nods. Th loading capacity of ach vhicl is 150 units and th avrag vhicl spd is 60 ilomtr pr hour. Th fixd cost for ach vhicl rout is $500. Srvic tim at ach supplir or nd customr is 0.1 minut pr unit. Th opration tim for all thr transportation s is 480 minuts which is also rcognizd as on planning priod. Th tim priod for th first opration is 160 minuts. Aftr sorting goods and rloading truc, all vhicls should lav th cross-docing ara and start th scond transportation. Th opration tim for th scond is 200 minuts and it is followd by th tim for goods sorting and truc rloading. All oprations should b ndd by 480 minuts. B. Dsign Paramtrs of Tabu Sarch Whn th solution program and basic data of xampl ar rady, a Taguchi xprimnt is conductd to st up appropriat dsign paramtrs for Tabu sarch. As indicatd in sction IV, th dsign paramtrs includ: lngth of Tabu list, maximal itration numbr, and probabilitis for slcting improvmnt approachs. Sinc ach paramtr is dfind in thr lvls, th L 9 orthogonal tabl is suitabl in this cas. Th final rsults of Taguchi xprimnt ar summarizd in Tabl I and Tabl II for all thr s. TABLE I: THE DESIGN PARAMETERS FOR THE FIRST STAGE Paramtr Lvl Valus Rmars Lngth of Tabu list 2 7 Maximal itration numbr /6 Extrnal 1-0 chang Probabilitis for 2/6 Extrnal 1-1 xchang improvmnt 3 1/6 Intrnal 2-opt xchang approachs 1/6 Intrnal or-opt xchang 64

6 TABLE II: THE DESIGN PARAMETERS FOR THE SECOND AND THIRD STAGE Paramtr Lvl Valus Rmars Lngth of Tabu list 2 7 Maximal itration numbr /6 Extrnal 1-0 chang Probabilitis for 1/6 Extrnal 2-xchang improvmnt 2 2/6 Intrnal 2-opt xchang approachs 2/6 Intrnal or-opt xchang C. Solution of th Exampl and Discussion Basd on th solution algorithm dscribd in sction IV, a computr program is thn dvlopd to solv this numrical xampl. For dmonstration and comparison purpos, Tabl III and Tabl IV show th initial solution and th final solution, rspctivly. Tabl V compars th initial and final solutions on th basis of total cost, travl tims, and numbr of routings usd in ach. Fig. 3 shows th final vhicl routings for all s in dtails. By comparing th initial solution and th final solution, th total cost can b rducd 14.69% and th numbr of routing can b ffctivly rducd up to 20%. Rduction of routings, i.. 3 routing in this xampl, may rprsnt a bttr utilization of vhicl and fully utilizing th tim of panning priod. Ths rsults indicat that th proposd solution algorithm can ffctivly rduc th cost and incras th fficincy of th transportation systm. Thr ar 135 rturnd goods should b dlivrd bac to supplirs starting from nd customrs. Data from th initial solution indicats that 35 rturnd goods, i %, ar dlivrd to thir dstination arlir in th scond. In th final solution, howvr, this quantity incrass to 70, i %. TABLE IV: THE FINAL SOLUTION OF THE ILLUSTRATION EXAMPLE Arri. Dpt. Vh Routing Tim tim Cost ID (min.) (min.) 1 st 2 nd 3 rd * * * * * * * -9 * * * * * -14 * Total cost Total travl tim Rmars: * Earlir dlivry in th scond TABLE V: COMPARISON OF THE INITIAL AND FINAL SOLUTIONS Initial solution (A) Final solution (B) Dviation * (C) Improvmnt prcntag ** (D) Total cost % Total travl tim % 1 st numbr of rout 2 nd numbr of rout 3 rd numbr of rout % % % Total numbr of rout % Rmars: * (C) = (A) - (B) ** (D) = (C) / (A) TABLE III: THE INITIAL SOLUTION OF THE ILLUSTRATION EXAMPLE 1 st 2 nd 3 rd Vh. ID 1 Routing Arri. tim (min.) Dpt. tim (min.) * * * * * * * * Total cost Total travl tim Rmars: * Earlir dlivry in th scond Cost Fig. 3. Final vhicl routings for all s. 1 st 2 nd 3 rd Additional xprimnt has bn conduct to compar th systm with pur indpndnt thr- (i.. no arlir dlivry in th scond ) and th systm with som flxibility in th scond (i.. accpt arlir dlivry in th scond ) as proposd in this rsarch. Tabl VI indicats th diffrncs basd on cost, tim, and numbr of rout. It is obviously that th arlir dlivry policy is bttr than no arlir dlivry policy. Th total cost will furthr rduc up to 9.40%. This phnomnon may imply that th arlir dlivry policy suggstd in this rsarch is worth to implmnt in ral-world cass. 65

7 TABLE VI: COMPARISON OF THE EARLIER DELIVERY POLICY No Earlir Dviation arlir dlivry Improvmnt dlivry (this prcntag ** study) (A) (B) (C) (D) Total cost % Total travl tim % 1st numbr of % rout 2 nd numbr of % rout 3 rd numbr of % rout Total numbr of rout % Rmars: * (C) = (A) - (B) ** (D) = (C) / (A) VI. CONCLUSION This papr proposs a nw concpt for vhicl transportation systm. Th mathmatical modl and th associatd solution algorithm ar also prsntd in dtails. Finally, th illustration xampl furthr confirms that th thr- vhicl routing systm proposd in this papr is an fficint transportation systm which is worth to implmnt in ral world cass. ACKNOWLEDGMENT This rsarch is supportd by th National Scinc Counsl, Taiwan undr grant: NSC E REFERENCES [1] U. M. Apt and S. Viswanathan, Effctiv cross docing for improving distribution fficincis, Intrnational Journal of Logistics, vol. 3, no. 3, pp , [2] A. B. Arabania, M. Zandihb, and F. S. M. T. Ghomi, Multi-objctiv gntic-basd algorithms for a cross-docing schduling problm, Applid Soft Computing, vol. 11, no. 8, pp , [3] Y. H. L, W. J. Jung, and K. M. L, Vhicl routing schduling for cross-docing in th supply chain, Computrs and Industrial Enginring, vol. 51, no. 2, pp , [4] G. B. Dantzig and J. H. Rmsr, Th truc dispatching problm, Managmnt Scinc, vol. 6, no. 1, pp , 1959 [5] S. Kirpatric, C. D. Glatt, and M. P. Vcchi, Optimization by simulatd annaling, Scinc, vol. 220, no. 4598, pp , 1983 [6] H. F. Wang and Y. Y. Chn, A gntic algorithm for th simultanous dlivry and picup problms, Computrs and Industrial Enginring, vol. 62, no. 1, pp , [7] C. J. Liao and Y. Lin, Vhicl routing with cross-docing in th supply chain, Exprt Systms with Applications, vol. 37, no. 10, pp , [8] G. Clar and J. G. Wright, Schduling of vhicls from a cntral dpot to a numbr of dlivry points, Oprational Rsarch, vol. 12, no. 4, pp , Chiong Huang was born in Taiwan on July H rcivd his Bachlor dgr on industrial nginring from Tunghai Univrsity, Taichung, Taiwan ROC in In 1987, h rcivd his Mastr dgr on industrial nginring from th Univrsity of Txas at Arlington, Arlington, Txas USA. H rcivd his Ph.D. dgr on industrial nginring from th Univrsity of Txas at Arlington, Txas USA in H is a Profssor in th Dpartmnt of Industrial Enginring and Managmnt, National Yunlin Univrsity of Scinc & Tchnology, Touliu, Yunlin, Taiwan ROC sinc August H has bn word as an Industrial Enginr in th Organization & Efficincy Dpartmnt of Philips Taiwan Ltd., Chupi Factory, Taiwan ROC from March 1984 to July H also word as a Production Plannr in th Production Planning Dpartmnt of th Chin-Fong Machins Ltd., Changhua, Taiwan ROC from Sptmbr 1983 to Fbruary Currntly, his rsarch intrsts ar facilitis planning & dsign, production & oprations managmnt, and logistic planning & dsign. Prof. Huang is a mmbr of CIIE, Taiwan ROC. Yun-Xi Liu was born in Taiwan on August Sh rcivd hr Mastr dgr on industrial nginring & managmnt from th National Yunlin Univrsity of Scinc & Tchnology, Touliu, Yunlin, Taiwan ROC in Sh rcivd hr Bachlor dgr on industrial nginring & managmnt from th National Kaohsiung Univrsity of Applid Scincs in Hr rsarch intrsts ar logistics systm dsign, transportation systm dsign, and logic of mta-huristic. 66