A New Model for Location and Transportation Problem of Cross-Docks in Distribution Networks

Transcription

1 Internatonal Journal of Modelng and Optmzaton Vol 7 No 6 December 07 A New Model for Locaton and Transportaton of Cross-Docks n Dstrbuton Networks A Hasan Goodarz and H Zegord Abstract Ths paper consders a locaton and transportaton problem n a dstrbuton network wth a set of part supplers cross-dockng centers and assembly plants known as customers In usual studes nvestgatng crossdockng networks all vehcles are forced to pass through crossdocks even f they pck up and delver the same set of goods and products In order to elmnate unnecessary stops at crossdockng centers and reduce transportaton costs we address a dstrbuton network n whch shpments are allowed to be transferred from supplers to assembly plants drectly as well as through cross-docks (ndrect shpment) A mathematcal model s developed for ths problem n whch the best locaton for establshng cross-docks s determned whle smultaneously supplers and plants are allocated to cross-dockng centers n order to transport parts va two transportaton strateges: drect shpment and shpment through cross-docks In order to solve the model GAM software has been used and some numercal examples are performed n small and medum sze problem nstances The results are reported n terms of total cost of the network and computatonal tme Index Terms Cross-dockng drect shpment dstrbuton network locaton problem mathematcal model I INTRODUCTION Nowadays many supply chans and companes are tryng to mprove ther dstrbuton strateges to effectvely move products n the network snce about 0% of a product prce s ncurred n the dstrbuton process [] There are several strateges n dstrbuton networks ncludng drect shppng mlk runs cross-dockng and talored networks [] Crossdockng s known as a logstc technque mplemented to consoldate shpments from varous sources and sort them for dstrbuton and delver to dfferent fnal destnatons After collectng products from varous orgns (supplers) nbound vehcles carry shpments to the cross-docks (CDs) Then they are unloaded at the nbound doors sorted consoldated accordng to destnaton of products and reloaded nto outgong vehcles at the outbound doors wthn less than 6 hours Other handlng operatons ncludng weghng szng packagng prcng and product labelng can be done on shpments hppng small orders by ndvdual vehcles results n Less than Truck-Load (LTL) shpments n whch the entre vehcle capacty s not occuped whle n crossdockng strategy small orders are consoldated nto one bg shpment to use the whole vehcle capacty (Full-Truck- Load) Therefore ths approach reduces the numbers of Manuscrpt receved eptember 07; revsed December 07 A Hasan Goodarz and H Zegord are wth the Industral Engneerng n the Faculty of Industral & ystems Engneerng at Tarbat Modares Unversty Tehran Iran (e-mal: asefehhasan@modaresacr Zegord@modaresacr) utlzed vehcles and helps to make more frequent and economcal pckup and delveres by tryng to meet the entre vehcle capacty [] Apart from reducng transportaton cost and consoldaton of shpments some other mportant advantages of crossdockng are reducng nventory level and need for warehouse space decreasng labor cost mprovng servce level and balancng demand and supply Cross-dockng approach cannot be mplemented n any condton ome goods and products are more suted to mplement ths approach ncludng fast-movng products wth constant demand [] or low demand varance pershable tems needng mmedate shpment hgh qualty products that do not requre qualty nspectons durng recevng process products that are pre-tagged and ready for sale at the customer In the lterature on cross-dockng problems are defned n strategc and operatonal level The problems n strategc level are often related to decsons that are not made frequently for nstance the locaton of cross-docks and ther layout In the operatonal level decsons are made for short term (daly weekly or monthly) ncludng decsons about dock door assgnment vehcle routng and schedulng problems Recently cross-dockng approach has attracted strong attenton among researchers especally n the feld of vehcle routng schedulng problem The work of Lee et al [5] was seemngly the frst study that nvestgated vehcle routng schedulng n cross-dockng problem (VRPCD) They consdered a cross-dock n order to transport goods from a set of supplers to retalers The authors proposed a Tabu search (T) method to determne the number of vehcles and the optmal vehcle routng schedule at cross-dock Then Lao et al [6] presented a T algorthm to solve the model proposed by Lee et al [5] and compared the results wth those of Lee et al [5] Vahdan et al [7] also consdered the model of [5] and suggested a hybrd metaheurstc algorthm to solve t The algorthm combnes partcle swarm optmzaton (PO) varable neghborhood search (VN) and smulated annealng (A) n a populaton-based context Numercal results ndcated that the hybrd metaheurstc algorthm outperforms the T proposed by [5] Musa et al [8] addressed the transportaton problem n cross-dockng networks and assumed that the shpments could be transferred from supplers to customers drectly as well as va cross-docks They presented an Ant Colony Optmzaton (ACO) algorthm to solve the proposed model Dondo et al [9] consdered the vehcle routng and schedulng problem n a mult-echelon mult-tem crossdockng network wth tme wndows and presented a mxednteger lnear programmng (MILP) formulaton for t The objectve functon tres to meet the customer demands at DOI: 0776/IJMO07V7607 7

2 Internatonal Journal of Modelng and Optmzaton Vol 7 No 6 December 07 mnmum total transportaton cost Ma et al [0] nvestgated a consoldaton and transportaton problem n cross-dockng dstrbuton networks to fnd the trade-offs between transportaton cost nventory and schedulng requrements antos et al [] extended the usual VRP n crossdockng (VRPCD) n a way that a cost wll be added n the objectve functon whenever a good s moved from a vehcle to another one at the cross-dockng center Then antos et al [] extended ther prevous study by consderng a VRPCD n whch vehcles are permtted to avod the stop at the cross-dock after pckup process They presented an nteger programmng (IP) model for problem and appled a Branchand-prce method to solve t Konur and Golas [] formulated a b-objectve b-level optmzaton problem to determne a cost-stable schedulng at nbound doors of cross-dock that mnmzes the average of total servce costs They utlzed a genetc algorthm (GA) to fnd an effcent Pareto fronter Tarantls [] addressed a mult-source vehcle routng problem wth a sngle crossdockng center and employed a T algorthm n order to solve t Mousav and Tavakkol-Moghaddam [5] and Mousav et al [6] addressed strategc tactcal and operatonal decson levels by consderng the locaton problem of multple cross-dockng facltes and vehcle routng schedulng problem In Mousav and Tavakkol- Moghaddam [5] the problem has been made more realstc by consderng uncertanty n decson levels They developed a two-phase mxed-nteger lnear programmng (MILP) formulaton and ncorporated two types of uncertantes nto mathematcal formulaton by proposng a hybrd fuzzy possblstc stochastc approach Hasan Goodarz and Zegord [7] also nvestgated a locaton-routng problem (LRP) n a determnstc dstrbuton network as an ntegrated model They developed a mxed nteger non-lnear programng (MINLP) formulaton for the problem n whch the best locaton of cross-docks s determned and smultaneously a fleet of vehcles s utlzed to transport products from part supplers to customers As consdered n the current study two transportaton strateges nclude drect shpment and ndrect shpment (through cross-dock) In order to study more n ths scope the reader can fnd more nformaton on [8]-[0] Gong through the lterature shows that the locatonallocaton problem n cross-dockng requres some adoptons n order to be applcable n real world envronments In some cases the pcked up shpment from a typcal suppler may be near FTL or geographcal dstrbuton of nodes s n a way that passng through cross-docks s not cost effectve Therefore n dstrbuton networks wth cross-docks movng shpments through cross-dock and consoldaton process should be done when t s ratonale Usng these consderatons ths study tres to present a new formulaton A Assumptons II PROBLEM DEFINITION The dstrbuton network n ths study ncludes supplers cross-docks and assembly plants (as customers) The transportaton system for shppng loads to plants ncludes drect and ndrect shpments (va cross-docks) By usng cross-dockng strategy products n varous locatons are collected n the cross-dock pror to transportaton to ther fnal destnaton The loads sent to the cross-dock are consoldated accordng to ther fnal destnaton The demand of a typcal assembly plant served by drect shpment strategy usually s near FTL The model tres to fnd the best locaton of cross-dock(s) and dstrbuton plan such that the total transportaton cost s mnmzed Therefore the programmng model determnes the shpments to be sent drectly from supplers to plants and loads to be sent ndrectly through cross-dockng centers also the optmal locaton of cross-docks A plant may demand several products provded by several supplers All plants are drectly connected to one or more cross-docks It s not possble to have routes between nodes n the pck up or delvery process Fg shows the flow of products n the proposed cross-dockng network Other assumptons are as follows: Every suppler sends products drectly to assembly plants or thorough a cross-dock There are multple cross-docks wth lmted capacty n the dstrbuton network There s no nventory kept at the cross-docks whch means that the total quantty of products pcked up from supplers should be equal to total amount that s delvered to plants The load to be sent from each suppler to each plant s known (d ) If d s more than vehcle capacty or near FTL the soluton s trval and the vehcle s needed to go drectly for that flow (from suppler l to plant j) snce consoldaton s not possble n such a stuaton due to full truckload and t s more economcal to go drectly to destnaton All vehcles have the same capacty and fleet of vehcles s homogeneous The total load shpped by each vehcle cannot exceed the capacty B Mathematcal Model The sets parameters and varables used n the model are as follows: ets J The set of assembly plants {j= J} I The set of cross-docks {= I} L The set of supplers {l= L} K The set of vehcles {k= K} Parameters c j The transportaton cost from node to node j ( j (L I J)) g The fxed cost of establshng cross-dock v The varable cost per part unt at cross-dock V The maxmum capacty of cross-dock d Demand of plant j from suppler l Q The capacty of vehcles Decson varables U : f load from suppler l to plant j s sent drectly; 0: otherwse y : f cross-dock s open; 0: otherwse Z l : f load from suppler l s served by cross-dock ; 0: otherwse F j m l : f plant j s served by cross-dock ; 0: otherwse The number of vehcles from suppler l to cross-dock (nonnegatve nteger varable) 8

3 Internatonal Journal of Modelng and Optmzaton Vol 7 No 6 December 07 n j The number of vehcles from cross-dock to plant j (nonnegatve nteger varable) L I J plants Constrant () ensures that each customer s demand s suppled va cross-docks or drect transshpment Constrant () stpulates the maxmum vehcle capacty Equaton () determnes the load to be sent from cross-docks to assembly plants and also ensures that a flow enterng a cross-dock s equal to the flow extng t Constrant (5) lmts the flow through a cross-dock to ts capacty Constrants (6) and (7) ensure that when Σj R s postve then Z l should be equal to one otherwse Z l =0 These constrants stpulate that cross-dock gves servce to suppler l f R s equal to one at least for one assembly plant Constrants (8) and (9) are smlar to (6) and (7) but n delvery process Equaton (0) determnes the number of vehcles shpped from each cross-dock to each plant The types of decson varables are defned n ()-() Part supplers Cross-Dock Indrect flow through cross dock Drect flow plants Fg Consdered cross-dockng network when drect flow s allowed R : f demand of plant j from suppler l goes thorough crossdock ; 0: otherwse w j Quantty of parts shpped from cross-dock to plant j Wth the notatons ntroduced above the problem can be formulated as a mathematcal model as follows: Mn I I ll ll g y v { jj jj ll d ( U )} Z c U l l I ll j I jj U R mn{ d }; l j jj I R d ml Q; l c m l c n d ( U) Zl Fj wj 0; j jj ji ll Z w y V ; j R;l l ji R J Z ;l l Fj R; j ll R L Fj; j j () () () () (5) (6) (7) (8) (9) w n Q; j (0) j j y U R Z F {0} () w j l j 0;( I; j J) () m l n j {0} () The objectve functon tres to mnmze the total cost ncludng fxed cost of establshng cross-dock drect transshpment cost ndrect transportaton cost from supplers to cross-docks varable cross-dockng cost at recevng process and fnally delvery cost to assembly III EXPERIMENTAL TUDY In ths secton the tractablty of the proposed programmng model s evaluated n terms of objectve functon value and requred computatonal tme To do so we perform some numercal experments on a set of randomly generated problem nstances n small and medum szes The programmng model was mplemented n GAM 9 modelng language All experments were performed on a laptop wth a Core 5 Duo CPU processor and GB of RAM The problem sets are assumed to be as follows: et (0 ): 0 part supplers cross-docks and et (5 ): 0 part supplers cross-docks and et (5 5 ): 5 part supplers 5 cross-docks and et (5 5 ): 5 part supplers 5 cross-docks and plants The maxmum capacty of vehcles s assumed to be 00 The demand of plants from supplers s randomly generated n Unform (0 and 90) Maxmum capacty of cross-docks s set to 900 for set and and to 500 for set Other values of data are presented n Table I A A ample Instance For problem nstance - the dstrbuton of supplers cross-docks and plants are llustrated n Fg The vehcle capacty s 00 and varable cost at each cross-dock s set to 00 The wrtten numbers under each suppler node ndcates the demand of assembly plants from that suppler as s shown n Fg For ths nstance consder three possble scenaros: n the frst one all transshpments have to go drectly from supplers to plants and there s no cross-dockng center n the network In such a condton totally 0 shpments (equvalent to 0 vehcles) are requred to meet all the demand of two plants For ths scenaro cost terms of objectve functon are summarzed n Table In the second scenaro t s assumed that all shpments have to pass through cross-docks and no drect shpment s allowed In ths condton transportaton costs from supplers to plants (c ) drect costs of transportng parts are set to an extremely large number thus all shpment are 9

4 Internatonal Journal of Modelng and Optmzaton Vol 7 No 6 December 07 forced to pass through cross-docks Cost terms of objectve functon n ths scenaro are summarzed n Table III TABLE I: THE CHARACTERITIC AND INTERVAL OF INTANCE Parameters Intervals Transportaton cost between each couple of nodes (c j) U (0 00) Fxed cost of establshng cross-docks (g ) U (00 500) Varable cost per commodty unt at cross-dock (v ) U (00 ) TABLE III: COT TERM OF OBJECTIVE FUNCTION FOR PROBLEM INTANCE - (ECOND CENARIO) Fxed cost of establshng cross-dock 00 Drect transshpment cost 0 Indrect transportaton cost from supplers to CD (pck up cost) 85 Varable cross-dockng cost at recevng process 78 Indrect transportaton cost from CDs to plants (delvery cost) 57 Total cost plant CD CD plant CD Fg The dstrbuton of nodes n the smple nstance plant 9 5 V 0 CD 6 V plant Drect flow flow from supplers to cross-dock flow from cross-dock to plant Fg The optmal soluton of nstance - solved by GAM In the thrd scenaro the optmal soluton s found by GAM software when both transportaton strateges are allowed In ths condton the objectve value of problem nstance s 987 whch s 799 % and 9% better than cost of scenaro and scenaro respectvely The cost terms of optmum problem are summarzed n Table The optmal soluton has been found by GAM after 87 seconds In ths soluton cross-dock s opened Totally 59 commodty unts pass ths cross-dock therefore f the model determnes the best capacty of ths cross-dock t wll be 59 unts Fg shows the optmal soluton for ths nstance Ths smple nstance shows that when more than one transportaton strategy s utlzed the dstrbuton cost of network may decrease There are also condtons n whch consderng dfferent transportaton strateges does not necessarly have effect on cost TABLE II: COT TERM OF OBJECTIVE FUNCTION FOR PROBLEM INTANCE - (FIRT CENARIO) Fxed cost of establshng cross-dock 0 Drect transshpment cost 08 Indrect transportaton cost from supplers to CDs (pck up cost) 0 Varable cross-dockng cost at recevng process 0 Indrect transportaton cost from CDs to plants (delvery cost) 0 Total cost 08 7 TABLE IV: COT TERM OF OBJECTIVE FUNCTION FOR PROBLEM INTANCE - (THIRD CENARIO) Fxed cost of establshng cross-dock 00 Drect transshpment cost 7 Indrect transportaton cost from supplers to CDs (pck up cost) 5 Varable cross-dockng cost at recevng process 87 Indrect transportaton cost from CDs to plants (delvery cost) 85 Total cost 987 B Expermental Result Table V-VIII summarze the results for 0 problem nstances For each nstance t reports the total cost of dstrbuton parts and computatonal tme of executng shows the optmum number of opened cross-docks It s clear that the computatonal tme ncreases by ncreasng the number of supplers and plants The results are presented for problems wth up to 5 supplers TABLE V: COMPUTATIONAL REULT FOR PROBLEM ET GAM nstance Objectve CPU TABLE VI: COMPUTATIONAL REULT FOR PROBLEM ET GAM nstance Objectve CPU TABLE VII: COMPUTATIONAL REULT FOR PROBLEM ET GAM nstance Objectve CPU TABLE VIII: COMPUTATIONAL REULT FOR PROBLEM ET GAM nstance Objectve CPU

5 Internatonal Journal of Modelng and Optmzaton Vol 7 No 6 December IV CONCLUION Ths paper addresses the transportaton and locaton problem of cross-dockng network where the shpments are allowed to be transferred from supplers to plants drectly as well as va cross-dockng approach Cross-dockng s a logstcs strategy wth the am of reducng nventory and transportaton costs order pckng and delvery tme of demands n the supply networks Ths strategy smply deals wth supplyng demand of multple fnal destnatons from varous orgns by consoldatng shpments The objectve functon tres to fnd the best locaton of cross-docks and dstrbuton plan such that the total transportaton cost s mnmzed In order to solve the proposed model totally 0 nstances are defned n four problem sets and solved by GAM software The results are reported for each problem nstance n term of objectve functon value and computatonal tme As some drectons for future studes a heurstc algorthm can be used to solve large szed problems In addton tme wndows can be consdered at suppler and plant locatons n order to mnmze transportaton tme Moreover the capacty of cross-dock can be optmzed by the model nstead of beng a constrant [] F A antos G R Mateus and A da Cunha The pckup and delvery problem wth cross-dockng Computers and Operatons Research vol 0 pp [] F A antos G R Mateus and A da Cunha A branch-and-prce algorthm for a vehcle routng problem wth cross dockng Electronc Notes n Dscrete Mathematcs vol 7 pp [] D Konur and M M Golas Cost-stable truck schedulng at a crossdock faclty wth unknown truck arrvals: A meta-heurstc approach Transportaton Research Part E vol 9 pp [] C D Tarantls Adaptve mult-restart tabu search algorthm for the vehcle routng problem wth cross-dockng Optm Lett vol 7 pp [5] M Mousav and R Tavakkol-Moghaddam A hybrd smulated annealng algorthm for locaton and routng schedulng problems wth cross-dockng n the supply chan Journal of Manufacturng ystems vol pp [6] M Mousav B Vahdan R Tavakkol-Moghaddam and H Hashem Locaton of cross-dockng centers and vehcle routng schedulng under uncertanty: A fuzzy possblstc stochastc programmng model Appled Mathematcal Modellng vol 8 pp [7] A H Goodarz and H Zegord A locaton- routng problem for cross-dockng networks: a bogeography-based optmzaton algorthm Computers and Industral Engneerng vol 0 pp 6 06 [8] D Agustna C K M Lee and R Pplan A revew: Mathematcal models for cross dockng plannng Internatonal Journal of Engneerng Busness Management vol pp [9] J Van Belle P Valckenaers and D Cattrysse Cross-dockng: tate of the art Omega vol 0 pp [0] P Bujs I F A Vs and H J Carlo ynchronzaton n crossdockng networks: A research classfcaton and framework European Journal of Operatonal Research vol 9 pp REFERENCE [] U M Apte and Vswanathan Effectve cross dockng for mprovng dstrbuton effcences Internatonal Journal of Logstcs vol pp [] C Chopra and P Mendl uppler Chan Management trateges Plannng and Operaton Prentce Hall Upper addle Rver New Jersey pp 00 [] D Vasevc M tepanovc and O Manoovc Cross dockng mplementaton n dstrbuton of food products Economcs of Agrculture vol 60 pp [] A Ross and V Jayaraman An evaluaton of new heurstcs for the locaton of cross-docks dstrbuton centers n supply chan network desgn Computers and Industral Engneerng vol 55 pp [5] Y H Lee J W Jung and K M Lee Vehcle routng schedulng for cross-dockng n the supply chan Computers and Industral Engneerng vol 5 pp [6] C J Lao Y Ln and C hh Vehcle routng wth crossdockng n the supply chan Expert ystems wth Applcatons vol 7 pp [7] B Vahdan R Tavakkol-Moghaddam M Zandeh and J Razm Vehcle routng schedulng usng an enhanced hybrd optmzaton approach Journal of Intellgent Manufacturng vol pp [8] R Musa J P Arnaout and H Jung Ant colony optmzaton algorthm to solve for the transportaton problem of cross-dockng network Computers and Industral Engneerng vol 59 pp [9] R Dondo C A Méndez and J Cerdá The mult-echelon vehcle routng problem wth cross dockng n supply chan management Computers and Chemcal Engneerng vol 5 pp [0] H Ma Z Mao A Lm and B Rodrgues Cross dockng dstrbuton networks wth setup cost and tme wndow constrant Omega vol 9 pp A Hasan Goodarz s a PhD canddate on Industral Engneerng n the faculty of Industral & ystems Engneerng at Tarbat Modares Unversty Tehran Iran he recently was a vstor at Grenoble INP Unversty Grenoble France and passed a sabbatcal n G-COP laboratory he receved her Msc from Department of Industral Engneerng at Unversty of Tehran Iran n 0 he holds a Bc n Industral Engneerng from Isfahan Unversty of Technology Iran Her man areas of research nterests nclude supply chan management operaton research mult-objectve optmzaton and combnatoral optmzaton he has publshed artcles n nternatonal conferences and academc journals ncludng Computers and Industral Engneerng and Appled Mathematcal Modelng H Zegord s an assocate professor of ndustral engneerng n the chool of Engneerng at Tarbat Modares Unversty Iran He receved hs PhD from Department of Industral Engneerng and management at Tokyo Insttute of Technology Japan n 99 He holds an Mc n ndustral engneerng and systems from harf Unversty of Technology Iran and a Bc n ndustral engneerng from Isfahan Unversty of Technology Iran Hs man areas of teachng and research nterests nclude producton plannng and schedulng mult-objectve optmzaton problems meta-heurstcs qualty management and productvty He has publshed several artcles n nternatonal conferences and academc journals ncludng European Journal of Operatonal Research Internatonal Journal of Producton Research Journal of Operatonal Research ocety of Japan Computers & Industral Engneerng and Amrkabr Journal of cence and Engneerng