Enhancment of Inventory Management Approaches in Vehicle Routing-Cross Docking Problems

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1 Enhanment of Inventory Management Approahes n Vehle Routng-Cross Dong Problems Mahd Alnaghan*, Hamed Amanpour*, Erfan Babaee rolaee* *Department of Industral and Systems Engneerng, Isfahan Unversty of ehnology, Isfahan, Iran. E-mal:alnaghan@.ut.a.r Abstrat A ross do s the onsoldaton pont n a dstrbuton networ, where multple smaller shpments an be merged to full tru loads n order to realse eonomes n transportaton. he fous of ths study s to evaluate the possblty of onsderng nventory management strutures n order to enhane the qualty of solutons aheved by the lassal VRP-Cross Do (VRPCD) modelng. A mxed-nteger programmng s taen nto aount for loatng ross dos and alloatng Pup and Delvery Nodes to them by onsderng eah node s tme wndow n order to mnmse the total transportaton osts and nventory holdng osts nsde eah ross do whh s the obetve funton of the modelng. Dret Shpment s used to attah Delvery and P-up nodes to ross do faltes. Determnng the neessty and amount of goods transshpment among ross dos s a ey pont n ths modelng whh auses a Sem-VMI behavour by onsderng ross do faltes as the oordnator base of the dstrbuton networ. he enhanement of the solutons by addng the ablty of transshpment between ross dos s proved. Loatng ross do faltes (Strateg deson) whle onsderng other tehnal ssues (Operatonal desons) s one of the maor ontrbutons of ths study. Keywords: Cross Dong, Inventory Management, ransportaton, Mxed-Integer Programmng, Falty Loaton Alloaton Introduton Cross dong studes have mostly been onerned wth the physal layout of a ross do or tehnal haratersts of these faltes. Loatonng, Door Assgnment, Vehle Routng, ru Shedulng, Resoure Shedulng and Pang-Unpang desons are the most popular desons made n a dstrbuton networ ontanng ross do faltes. Among those studes whh addressed operatonal haratersts le those onsderng vehle routng and ross dong smultaneously, few of them pad attenton to the osts nurred by mantanng nventores nsde ross dos. he reason may be that the lassal defnton of a ross do emphaszes that these faltes are ust onsoldaton ponts so that the mum tme that nventores an tae n are fewer than 24 hours, but that s not pragmat n real world. Although ths fat must not be a hndrane n onsderaton of nventory osts and a good reason for smplfyng models by gnorng the ross dos nventory levels at eah tme, beause besde the holdng osts themselves are negatve ponts for the whole dstrbuton networ even for suh a short tme, mplementng nventory management for goods nsde ross dos may lead to a total better performane of the whole supply han, by tang better tmng and alloatng desons (oordnaton). he fous of ths study s on mplementng some rual mprovements on one of the rare proposed models whh onsdered nventory osts nsde eah ross do. Whle the whole struture of the modelng s mantaned, many dfferent assumptons and onstrants are devoted n order to aheve real world exeutable solutons, whle senstvty analyss wll prove the applablty of the new modelng. Due to the nature of ths study we must frst have a qu survey on the prevous models n ths area. A lassfaton and revew on the reent lteratures ould be helpful to whh the thrd seton s devoted. In the fourth seton, we espeally fous on the three manly avalable models n ths area whh were from the frst seton, ombnng vehle routng problems and problems onsderng the usage of ross do faltes. In the next seton, an extenson for one of the above models whh s the most sutable for onsderng nventory management n VRPross do problems, proposed by Chen et al. (2006) s

2 28 Journal of Supply Chan Management Systems Volume 3 Issue 3 July 204 prepared. he next seton s exlusvely foused on three numeral examples and analyss of the results obtaned by the extended model. Vablty of suh mehansm n real world s hallenged based on the aheved results. In the seventh seton, a bref onluson and dsusson wll be an end for ths study. Lt r tur R A ross dong falty or termnal s one of the dstrbuton networ nodes, whh s partularly dedated to transshpment of tru loads. he prmary purpose of a ross do s to enable a onsoldaton of dfferently szed shpments wth the same destnaton to full tru loads, so that eonomes n transportaton osts an be realzed (Apte & Vswanathan, 2000). In some lterature ross dong and short-term tru shedulng are onsdered as the same ategores due to the nature of a ross do falty to whh tmng and routng desons are the most mportant desons to be made (MWllams et al., 2005). Boysen et al. (2009) made a thorough and n-depth lassfaton of ross dong problems by utlsaton of a tuple ( α, β, γ) whh represents Door Envronment, Operatonal Charatersts and Obetves, respetvely whh dstnt eah problem s orgn (Boysen et al., 2009). hey mpled that there are seven lasses of desons to be made n a ross dong problem whh are: -Loaton of ross dong termnals, 2-Layout of the termnals, 3-Assgnment of destnatons to do doors, 4-Vehle Routng, 5-ru Shedulng, 6-Resoure Shedulng nsde the termnals, and 7- (un-)pang loads nto (from) trus. Loaton of ross dos (atal Deson) and vehle routng (Operatonal Desons) are foused n ths paper. Investgaton of a real world post offe where delvered parels are forwarded to outbound trus by a system of nteronneted onveyer belt s worth to menton (Apte & Vswanathan, 2000). Boysen (200) dsussed the omplexty of dstrbutng pershable goods whle onsderng tehnal ssues and routng desons. he wor of Lee et al. (2006) may be the frst that taes both VRP and ross dong nto onsderaton. he purpose of the modelng was to determne f vehle moves from node to node or not, whle onsderaton of only one ross do and zero nventory poly n all nodes besde p-up and delvery struture wth tour networ arethe man haratersts of the formulaton. Another effort n manpulatng ross dong approah and vehle routng smultaneously was made by Wen et al. (2009) whh had nherted the struture from the frst mentoned modelng wth some ustfatons le elmnaton of smultaneous arrvals onstrant and onsderng tehnal ssues nsde ross dos le load/unload order desons. Whle n none of the modelng, nventory osts were taen nto aount, Chen et al. (2006) proposed a new formulaton of vehle routng and ross dong whh onsderaton and mnmsaton of the nventory osts were the man dea of ther mxed-nteger modelng. C n Pr ou VRPCD In ths seton we provde a lassfaton for the three vehle routng- ross dong modelng. Smlartes and dfferenes as prelmnares for ntrodung the new approah are dsussed. In lterature we an fnd two dfferent types of ross dong strutures: -One-ouh Struture, and 2- Post Offe Struture. In the frst one, the smplfyng level of the modelng s at the hghest beause one maor assumpton s that goods are unloaded, onsoldated and loaded nto trus n a very short tme and therefore there wll be no nventores nsde ross dos at any pont of tme. But n the seond struture, whh was frstly ntrodued by Donaldson et al. as a proedure to onquer the shedulng omplexty of mal transshpment n U.S. post offes, the smplfyng level s not that hgh, so the onsoldaton proess le ts ounterpart n real world, taes tme and therefore aggregated nventores and the shedulng (tmng) are onsdered n the formulaton. Another haraterst of the ross dong problems s the absene or presene of nformaton flow n the dstrbuton networ. In the frst lass of models there n n = = are onstrants suh as P = D whh mples the equalty of produton and demand among all produton and demand nodes, whh s applable n modelng dary, flowers, and pershable goods dstrbuton networ. Consder that ths leads to zero nventory poly n ross do faltes. Another mportant haraterst of a VRPCD s Networ strutures. Dret shpments are used when produton or demand n the orrespondng nodes are equal or near to the mum tru apaty whle tour strutures are used when ths assumpton s not vald and therefore routng desons expand omplexty of the problem. Produt varety and Number of ross dos are other noteable haratersts of a VRPCD problem. able presents a lassfaton of the three bas models based on the above-mentoned haratersts:

3 Enhanment of Inventory Management Approahes n Vehle Routng-Cross Dong Problems 29 able : Classfaton of hree Manly VRPCDs Modelng Lee et al. (2006) Wen et al. (2009) Chen et al. (2006) Inv. Insde CDs Zero Inv. (One ouh) Zero Inv. (One ouh) Yes (Post Offe) Info. Flow Yes Yes No Networ Struture ours ours Dret Shpments Produt Varety Sngle Multple Multple Number of CDs One One wo Chen et al. formulaton After a bref revew of Chen et al. modelng, the rest of ths paper s dedated to mplement some mprovements on the above formulaton n order to enhane ts real world applablty. Let s defne the dfferent attrbutes: where COS ransportaton m = = t= = x DD + y PD mn n t,,, t,,, = = t= mn DP DA DD DS DE CAP COS mn, able 2a: Inputs D Set of m delveres, ndexed by P Set of n pups, ndexed by C Set of ross dos, ndexed by G Set of d produts, ndexed by r Set of tmes, ndexed by t able 2b: Parameters Bnary ndene matrx, f produt r s delvered by delvery and 0 o.w Vetor where DA s the amount delvered by delvery Matrx where DD, s the dstane from delvery I to ross do Vetor where DS s startng tme of delvery Vetor where DE s the endng tme of delvery (defne PP, P, PD, PS, PE for pup node n the same way) Vetor where CAP s the apaty of ross do Vetor where COS s the ost of handlng a unt produt for a unt tme at ross do Mnmum and mum tmes defnng the tme horzon able 2: Deson Varables d r= t= COS nventory = COS z S.t: PE = 2) x ( " ) t= PS PE t 3) y = ( " ) t= PS t mn r, t, 4) zrt CAP r (" rt,, ), t 5)z = 0( " r,,, t ), t, mn rt,, - mn mn m n 6) zrt,, = zr, t, - + xt,, DP, rda - y, t, PPr, PA = = all r, and mn 7 z r - = 0 ),, mn t for he output s to determne when and to whh ross do eah delvery node must transfer ts goods (and as the same for p up or demand nodes). Pro o d Mod X,, t Y,, t Z r,, t Bnary, f delvery s bound for ross do at tme t, 0 o.w Bnary, f pup goes to ross do at tme t, 0 o.w Integer, and s the amount of produt r at ross do at tme t And the modelng s as below: Mnmze (COS ransportaton +COS nventory ) By mplementng maor ustfatons on the ntrodued modelng and onsderng many dfferent onstrans and addng one mportant ablty to the formulaton, we an defne ontrbutons of our modelng as below: Consderng produton of dfferent bathes of dfferent nds of good n delvery nodes. Consderng dfferent bathes of dfferent nds of goods demand n p up nodes.

4 30 Journal of Supply Chan Management Systems Volume 3 Issue 3 July 204 Dsrete loatng of ross dos from a set of anddate loatons and alloatng delvery and p up nodes to them. Consderaton of dfferent apates and holdng osts n eah ross do for eah of the goods due to tehnal ssues. Consderng the ablty of transferrng goods between ross dos (whh leads to a sem-vmi behavour) by utlzng revsed nventory balane onstrant. Let s defne the dfferent attrbutes: able 3a: Parameters DP, r Bnary ndene matrx, f produt r s produed at delvery node, 0 ow (defne PP, r for the pup node n the same way). DA, r Vetor where DA, r s the amount of produt r delvered by delvery I (defne PA, r for the pup node n the same way). DD, Matrx, Dstane between delvery node and ross do. (defne PD, for the pup node n the same way). CAP, r Matrx, Capaty of ross do for produt r. COS, r Matrx, Holdng ost of Cross do for produt r. D, Matrx, Dstane between ross dos. F Fxed ost nurred f ross do s open. hh r Unt ost nurred f one amount of produt r s transferred between ross dos. YY RE r,,, t able 3b: Varables f ross do s open, 0 o.w. Integer, the amount of produt r, transferred from ross do to ross do at tme t. he proposed formulaton s as below: Obetve funton: Mnmze (COS ransportaton +COS nventory +COS CD open ) Where: COS ransportaton m n = xt,, DD, + yt,, PD, = = t= = = t= mn COS nventory mn = d COS zr, t, + = d = = r= t= r= t= mn mn RE COS CD open = F* yy = * hh( r)*d r,,, t, Note: we assumed that ost nurred by transferrng goods between two ross dos are proportonal to destnaton between them (D, ). s.t: 2) x ( " ) 3) y = ( " ) 4) z PE t= PS PE t= PS rt t t CAP (" r, t,), t r, mn 5)z rt,, - = 0( " r,,, tmn),mn t 6) zrt,, = zr, t, - + REr,,, t - REr,,,t + 7) yy ( " ) m π m 8) x yy (" t,) n t 9) y yy (",) t d t π x DPDA - y PP PA n t r r t r r 0 ) RE yy * M (" π ) r t= mn rt,,, )yy p(, ) yy ( " π ) 2) yy p(, ) yy (" π ) 3) yy + yy - yy (, )(" π ) p p Constrant (2) stpulates that t s not neessary to p up goods from eah delvery node whle (3) shows the neessty of servng every p up node n the dstrbuton networ. It means that the model s only oblgated to serve all p up nodes but not all delveres and as long as we an satsfy demand of all p up nodes wth some of the delveres, the onstrants are satsfed too. Constrant (4) s revsed by onsderng the CAP r, whh means the dfferent apaty for holdng dfferent goods

5 Enhanment of Inventory Management Approahes n Vehle Routng-Cross Dong Problems 3 are taen nto aount, therefore we wll have r tmes more of ths nd of onstrant ompared to ts orgnal modelng ounterpart onstrant. Constrant (5) s an assumpton whh mples that at the begnnng of tme horzon of the problem we assume that all ross dos are empty and hold no nventores. One of the man deas of ths formulaton s present n onstrant (6) or revsed nventory balane onstrant, whh stpulates that at any gven tme (meanwhle the defned tme horzon), at eah of the avalable ross dos and for all nds of goods, the nventores must be balaned, whle onsderng for every ertan ross do ()the amount of a ertan good (r) browed from other () ross dos ( RE r,,, t π ) and the amount of a ertan good (r) lend to other ross dos ( RE rt,,, )at any gven pont of tme (t), must be π equal to the total amount of that ertan produt, for that ertan ross do () and n that ertan gven tme (t). Note that for means of smplfaton, we assume that transfer between ross dos ours mmedately. Constrants 8-3 are mentoned to manpulate the dsrete ross do loatng from a set of anddate loatons. Nu r E r nt hs seton s devoted to show the effetveness of our modelng by ntrodung and exeutng an example ontanng 5 possble ross dos, 2 dfferent nds of goods and 3 delvery and p up nodes wth ther ertan tme wndows. hese results were obtaned usng GAMS software verson, exeuted by a laptop runnng on wndows 7 operatng system, havng 2 GBs of RAM and Intel Core 5 CPU runnng at 2.4 GHz frequeny. ables 4-6 desrbe the parameters of the hypostatal example (). And onsder hh =80 and hh 2= 90 whle F =0, F 2 =, F 3 =0, F 4 =9 and F 5 =0. Fg. llustrates the obtaned results whh an be desrbed as below: x,, = x2, 47, = x32,, = x348,, = Whh means that for example delvery node, at tme onnets to the ross do (and so on), and for pups y48,, = y2, 47, = y3,, = y32,, = wth the same defnton as for delvery nodes. able 4: Parameters Related to Delvery Nodes DP, DP,2 DA, DA,2 DD, DD,2 DD,3 DD,4 DD,5 = = = able 5: Parameters Related to P up Nodes PP, PP,2 PA, PA,2 PD, PD,2 PD,3 PD,4 PD,5 = = = able 6: Charatersts of Cross Dos CAP, CAP,2 COS, COS,2 D, D,2 D,3 D,4 D,5 = = = = =

6 32 Journal of Supply Chan Management Systems Volume 3 Issue 3 July 204 able 7: me Wndows of Eah Delvery and P up Node Fg. 2: Aggregated Inventores at Cross Do DS DE PS PE = = = = = = Fg. : Obtaned Results Fg. 3: Aggregated Inventores at Cross Do 4 he results only show the neessty of openng ross dos and 4 ( yy = yy4 = ). he other lasses of varables are Z r,, t or the amount of aggregated nventores from every nd of produts at eah of the ross dos and any pont of tme whh has been olleted n able 8. able 8: Inventory Level of Dfferent Produts at Dfferent Cross Dos at Dfferent Ponts of mes Fgures 2 and 3 show the amount of aggregated nventores n avalable ross dos ( and 4 are deded to be open) durng the tme. Another nd of varables are RE r,,, t (transshpments between ross dos) whh at ths run are all equal to 0 and therefore any transshpment dd not our between ross dos (beause t was not eonomal or neessary ). Consder that the optmal obetve funton value for example () s Z * = Senstvty Analyss In order to expose the applablty and effetveness of the ablty to transfer goods between ross dos, we need to another example. We do so by mplementng some hanges n parameters of the orgnal example. Before that, let s defne the stuatons n whh, transshpment between ross dos beomes eonomal or neessary : he ost of transferrng goods between two ross dos s less than the holdng ost of goods n the orgn ross do. here are apaty lmtatons for holdng goods n the orgn ross do. ransshpment between ross dos by means of mang omplete demand bathes and mang ross dos empty whh leads to a Sem-VMI behavour of the modelng.

7 Enhanment of Inventory Management Approahes n Vehle Routng-Cross Dong Problems 33 Now we assume that the holdng ost n every ross do hasa 300 perent nrease (example 2). herefore the ustfed COS, r matrx beomes as shown n able 9. able 9: Justfed COS, r matrx (example 2) After runnng example (3), we see that there s not any transshpments between ross do ( RE = 0 for all r,,, t r,,, t) and the optmum value of the obetve funton nreases to hs fat proves the effeny of addng transshpment between ross dos ablty to the modelng. Fg.4 llustrates the nrease of the total osts n example 3 ompared to example 2. Fg. 4: Inreasng n otal Cost by Elmnatng ransshpment Between Cross Dos Capablty he obtaned results are desrbed here: x = x = x = x = x =,4,7 2,4,0 3,4,9 3,4,0 3,4, y,4,8 = y,4,9 = y,4,0 = y,4, = y2,4,7 = y3,4,0 = RE = RE = r,,, t 2,,4, We an see that due to a huge nrease n holdng osts n ross dos, model tres to empty ross dos at the end of plannng horzon whh leads to transferrng 0 unts of produt 2, from ross do =4 to ross do = at tme 7. Another mportant thng s the sem-vmi behavour whh means that n order to mae omplete bathes of demand and therefore enablng to mae the ross do empty at the end of the plannng horzon, we must have more delveres from delvery nodes (for example 3 tmes n a row, delveres from node =3 to ross do =4 at tme 9, 0 and are delvered) and onsequently more pups are sheduled than prevous. Consder that the optmum value of the obetve funton n example (2) beomes By elmnatng the ablty of transshpments between ross dos (n example (3)), we wll show the effetveness of addng suh apablty to the model. hs s done by a tremendous nrease n transshpment unt ost between ross dos or hh r whh has been shown n able 0. able 0: remendous Inrease n ransshpment Unt Cost Between Cross Dos hh r r= 8000 r= Con u on Cross dong s onsdered as an effent method to ontrol the nventory flow, whh s essental n a dstrbuton networ. he man dfferene between a ross dong falty and a dstrbuton enter s the duraton of holdng goods n these faltes. he defnton of ross do (holds goods for less than 24 hours) led to a gap n onsderaton of nventores osts n VRPCD lterature, but the truth s by gnorng osts nurred by aggregaton of nventores n ross dos, even for a very short tme, the opportunty for mang better shedulng and alloaton desons wll be lost. o onquer the dffultes made by ths level of smplfaton, our new model ontrbutes fve maor approahes whh are: - Dsreet loatonng of ross dos, 2- Alloatng PUp and Delvery demands to dfferent ross dos, 3- he ablty to transfer goods between ross dos, 4- Implementng a Sem-VMI struture (as a result), and 5- onsderng dfferent apaty/osts for holdng dfferent goods at eah of the ross dos. By utlsaton of ths new mxednteger modelng, the numeral experments showed about 2% derease n the total ost by onsderng a muh more ntellgent nventory management approah nsde ross dos. Also, mang Strateg desons suh as ross do loatonng whle onsderng Operatonal desons suh as alloatng demands to ross dos and transferrng goods between them are other ontrbutons of our study. By mplementng aheved solutons n

8 34 Journal of Supply Chan Management Systems Volume 3 Issue 3 July 204 prate, the management beomes aware f potental further dspathes to demand nodes (Sem- VMI Struture) are eonom or not. hs s done by tang optmzed desons about when and to (from) whh ross do olletng (dspathng) goods must tae plae. Future studes on ths area an be foused on expandng the proposed struture by onsder routng desons where dret shpment s no more the way of olletng and dspathng goods. Another trend an be defned by onsderaton of Soft me Wndows, where some penaltes may ompensate the osts assoated wth the resulted flexblty whh maes the modelng muh more realst. Revsng ths struture for modelng dstrbuton networ of pershable goods s another probable applaton n ths area. R r n Apte, U. M., & Vswanathan, S. (2000). Effetve ross dong for mprovng dstrbuton effenes. Internatonal Journal of Logsts: Researh and Applatons, 3, Boysen, N., & Fledner, M. (2009). Cross do shedulng: Classfaton, lterature revew and researh agenda. he Internatonal Journal of Management Sene, Boysen, N. (200). ru shedulng at zero-nventory ross dong termnals. Computers & Operatons Researh, 37, Chen, P., Guo, Y., & Lm, A., (2006). Rodrgues B. Multple ross dos wth nventory and tme wndows. Computers & Operatons Researh, 33, Lee, Y. H., Jung, J. W., & Lee, K. M. (2006).Vehle routng shedulng for ross-dong n the supply han. Computers & Industral Engneerng, MWllams, D. L., Stanfeld, P. M., & Geger C. D. (2005). he parel hub shedulng problem: A smulaton-based soluton approah. Computers & Industral Engneerng, 49, Wen, M., Larsen, J., Clausen., J, Cordeau, J. F., & Laporte, G. (2007). Vehle routng wth ross dong. Interunversty researh enter on enterprse networ, Logests and ransportaton.