A branch-and-price method for the Vehicle Routing Problem with Cross-Docking and Time Windows

Transcription

1 A banch-and-pce method fo the Vehcle Routng Poblem wth Coss-ockng and me Wndows Rodolfo ondo INEC (UNL - CONICE) - Güemes Santa Fe ARGENINA Rodolfo ondo, dondo@santafe-concet.gov.a Abstact. One mpotant facto n supply chan management s to effcently contol the supply chan flows. ue to ts mpotance, many companes ae tyng to develop effcent methods to ncease custome satsfacton and educe costs. Coss-dockng s consdeed a good method to educe nventoy and mpove esponsveness. he Vehcle Routng Poblem wth Coss-ockng and me Wndows (VRP-C-W) conssts on desgnng the mnmum-cost set of outes to seve a gven set of tanspotaton equests whle espectng constants on vehcles capacty, custome tme wndows and usng tansfes on a coss-dockng base. Each custome must be vsted ust once and mxed tous compsng pck-up and delvey stops ae not allowed. Fo a gven vehcle, the desgned pck-up tou must pecede ts delvey tou. In ths wok, we model the VRP-C-W assumng that all feasble odes ae known n advance. We pesent a new mxed ntege pogam to model the VRP-C-W and efomulate t va antzg Wolfe decomposton to late develop a column geneaton pocedue. he poposed banch-and-pce algothm shows encouagng esults on solvng some Solomon-based nstances. Keywods: Supply-chan management, coss-dockng, vehcle outng, columns geneaton. Intoducton Coss-dockng has emeged as an mpotant and effcent goods tanspotaton stategy. As a vaaton of the well-known vehcle outng poblem (VRP), the VRP wth Coss-dockng (VRPC) ases n a numbe of logstc plannng contexts. At coss-dockng temnals ncomng delvees of nbound tucks ae unloaded, soted, moved acoss the dock and fnally loaded onto outbound tucks, whch mmedately leave the temnal towads the next destnaton n the dstbuton chan. he cossdock s a consoldaton pont n a dstbuton netwok, whee multple smalle shpments can be meged to full tuck loads n ode to obtan savngs n tanspotaton []. he obectve of the VRP-C-W conssts of desgnng the mnmum-cost set of outes to seve a gven set of tanspotaton equests whle fulfllng constants on vehcles capacty and custome tme wndows and usng goods tansfes on a coss-dockng base. As coss-dockng s a compaatvely new logstc stategy, thee s not yet a massve body of lteatue on the subect. In fact, no 42 JAIIO - SII ISSN: Page 73

2 eseach on the shot-tem tuck-schedulng-poblem was publshed befoe 200. Lee at al. [2] consdeed coss-dockng fom an opeatonal vewpont n ode to fnd the optmal vehcle outng schedule. hus, an ntegated model consdeng both cossdockng and vehcle outng schedulng was pesented. Snce the poblem s NP-had, a heustc algothm based on a taboo seach algothm was poposed. One of the obectves fo coss dockng systems s how well the tucks can be scheduled at the dock and how the tems n nbound tucks can be allocated to the outbound tucks to optmze on some measue of system pefomance [3]. he authos eseached on how to fnd the best tuck dockng o schedulng sequence fo both nbound and outbound tucks to mnmze total opeaton tme when a stoage buffe to hold tems tempoaly s located at the shppng dock. he poduct assgnment to tucks and the dockng sequences of the nbound and outbound tucks ae all detemned smultaneously. Wen et al. [4] pesented a vey detaled mxed fomulaton fo the poblem and a taboo seach heustc embedded wthn an adaptve memoy pocedue. he pocedue was tested on data sets unepoted n the pape that wee povded by the ansh ansvson consultancy nvolvng up to 200 pa of nodes. Lao et al. [] pesented a new taboo seach (S) algothm that was poposed to obtan a good feasble soluton fo the poblem. hough extensve computatonal expements, they clam that the poposed S algothm can acheve bette pefomance than a pevous S by Lee et al. algothm whle usng much less computaton tme. Boloo Aaban et al [6] studed some meta-heustcs to fnd the best sequence of nbound and outbound tucks, so that the obectve, mnmzng the total opeaton tme o makespan, can be satsfed. In ths wok, we study the VRP- C-W and popose a banch-and-pce algothm fo solvng such a poblem. In secton 2 we fomulate the poblem. In secton 3 we pesent and descbe the banchand-pce algothm devsed to solve the CRP-C-W. Computatonal esults ae pesented and dscussed n secton and the conclusons ae outlned n secton 4. 2 Poblem Fomulaton Let G[I; A] a dected gaph nvolvng the locatons set I = {w I I - } and the net A = {a ' :, ' and '}. I contans the the coss-dock-base w, the pck-up nodes I = {,..., n } and the delvey nodes I - = {',..., ' n }. A equest = {, '} of a equest lst R = {,... } conssts of a demand of a tanspotaton sevce fom the ogn-node(s) to the destnaton node(s) ' I fo cetan load l. Each ac a ' A have an assocated non-negatve cost c and an assocated non-negatve tavel-tme t. he sevce tme on each node s defned by the paamete st. Requests must be fulflled though a homogeneous vehcles fleet V = {v, v 2,..., v m }. he soluton conssts of a fnte set of sequences of acs, called outes, fo some vehcles v V such that: () Each vehcle can pefom pck-up and/o delvey tous but each ndvdual tou contans ethe pck-up o delvey locatons. No "mxed" outes wth both type of locatons ae allowed. () Fo each vehcle, the pck-up tou must pecede ts desgned delvey oute. () Each vehcle stats and ends the pck-up and/o delvey outes at the coss-dock base w. (v) Each pck-up/delvey ste I - s assgned 42 JAIIO - SII ISSN: Page 74

3 to exactly one oute. (v) he actual load caed by a vehcle v must neve exceed ts tanspot capacty q v. (v) he sevce fo any pck-up/delvey node I - must stat wthn the tme-wndow [a, b ]; (v) he whole duaton of the vehcle-v tp, ncludng the pck-up tou, the tansfe opeatons and the delvey tou must be shote than a maxmum outng tme tv v max. (v) he poblem goal s to mnmze the total cost fo povdng pckup/ delvey sevce to evey node I -. 3 Column Geneaton In ths secton we ntoduce a set pattonng fomulaton of the poblem above pesented to late povde the fomulatons of the maste poblem and the pcng subpoblem. 3. he Maste Poblem he defnton of the maste poblem eques the followng notaton: c s the cost of the pck-up oute and a denotes a bnay paamete equal to f the oute pcks-up the cago of the equest R. In the same way, c - s the cost of the delvey oute - whle b denotes a bnay paamete equal to f the oute - delves the cago of the equest R. Afte the ntoducton of the tme-coodnaton constant (4), the maste poblem can be fomulated as follows: Mnmze Suet to c x c x () a x = (2) b x = (3) (4) a t end x b t stat x x = { 0,}, : (, = R whee x and x - ae bnay vaables assocated espectvely to pck-up outes and -. he dual vaables assocated to constants (2) and (3) ae both collected n the vecto π = [π,..., π,...] whle π = [π,..., π ] collect the dual values assocated to 42 JAIIO - SII ISSN: Page 7

4 coodnaton constant (4). he obectve functon () selects the set of pck-up and delvey outes that mnmzes the total tavelng cost. Constants (2) ensue that all equests R ae pcked fom pck-up stes I, whle constant (3) ensue that all equests R ae delveed to delvey locatons I -. Constant (4) coodnates the pck-up and delvey tous that move a equest fom ts ogn to ts destnaton. So, the end of the pck-up tou must pecede the tme-stat of the delvey tou. We emak that constants (2) and (3) need to be modeled as pattonng constants n the VRP-C-W, unlke common efomulatons fo outng poblems that geneally make use of coveng constants. hs s due to the fact that a lnea combnaton of actve pck-up and delvey outes must lead to a soluton on whch the pck-up tou stats befoe the end of the delvey oute. As a consequence, a pattonng soluton equvalent to the optmal coveng soluton may be nfeasble. 3.2 Pcng sub-poblem In a column geneaton scheme, gven a dual soluton of the (estcted) maste poblem, the pcng sub-poblem dentfes the oute * wth the mnmum educed cost: c = CV CV Y * π π π :(, = he sub-poblem fomulaton eles on the some ntege and contnuous vaables and can be wtten as follows Mnmze CV Subect to: d 0 π Y CV π Y π YR () :(, = d M d M CV t0 0 C Y ( S ) ( ) M 2 Y Y ( ) S M 2 Y Y d M st t M st t M V M ( Y ) ( Y ), : < ( S ) ( ) M 2 Y Y ( ) S M 2 Y Y, I : < I YR (6) (7.a) (7.b) (8) (9) (0.a) (0.b) () 42 JAIIO - SII ISSN: Page 76

5 l Y Q YR Y YR Y YR Y t t end stat V t end d 0 Y d M d M CV 0 st Y st :(, = 0 st Y st :(, = 0 YR YR ( S ) ( ) M 2 Y Y ( ) S M 2 Y Y d M t stat t0 st t M st t M V M l Y Q C ( Y ), : < ( S ) ( ) M 2 Y Y ( ) S M 2 Y Y ( Y ), I : < V t a b (2) (3.a) (3.b) (3.c) (4.a) (4.b) () (6.a) (6.b) (7) (8) (9.a) (9.b) (20) (2) max (22) he obectve () dentfes the oute wth mnmum educed cost to pefom by a sngle vehcle. Pces values ae obtaned fom the dual values fo constants (3), (4) and (). he cost constants of the pck-up tou ae gven by eqs (6), (7) and (8). hey compute the dstances tavelled to each the vsted stes and the total cost of the geneated oute espectvely. So, eqs. (7) fx the accumulated dstance up to each vsted ste. I.e. f nodes and ae allocated to the geneated pck-up oute (Y = Y = ), the vstng odeng fo both stes s detemned by the value of the sequencng vaable S. If locaton s vsted befoe (S = ), accodng constants (7.a), the tavelled dstance up to the locaton ( ) must be lage than by at least d. In case node s vsted eale, (S = 0), the evese statement holds and constant (7.b) becomes actve. If one o both nodes ae not allocated to the tou, eqs (7.a)-(7.b) (23) 42 JAIIO - SII ISSN: Page 77

6 become edundant. M s an uppe bound fo vaables. Eq (8) tansfoms the total tavelled dstance nto the oute-cost CV. M C s an uppe bound fo the vaable CV. he tmng constants stated by eqs. (9), (0) and () defne vstng-tme constants that ae smla to constants (6), (7) and (9) but apply to the tme dmenson. M s an uppe bound fo the tmes spent to each the nodes. Eq. (2) s the cago-capacty constant elated to the pck-up tou. Eqs. (3) actvate the vaables YR ndcatng that the cago s pcked-up and delveed by the same vehcle. In such a case, the equest cago wll eman on the tuck and no dopoff/pck-up actvtes ae ncued. Constant (4.a) states that the tme-end of the pck-up phase must be the sum of the tme spent completng the pck-up tou plus the tme ncued unloadng cagos at the coss-dock base. Convesely, eq. (4.b) states that the delvey stage must stat afte the sum of tme t end and the tme ncued n loadng goods to be late delveed. Constants ()-(7) ae cost constants elated to the delvey tou whle eqs. (8)-(20) defne tme constants elated to ths delvey tou. Eq. (2) apply the vehcle capacty constant to the delvey tou and eq. (22) foces the sevce tme at any pck-up/delvey ste to stat at a tme bounded by the tme nteval [a, b ]. Fnally, the eq. (23) mposes the uppe bound t v max to the total tou-tme. 3.3 Banch and Pce Implementaton he Foste and Ryan banchng ule s vey favouable fo banch and pce applcatons and fts easly wth the VRP-C-W pcng poblem snce t s equvalent to banch on assgnment decsons Y, fo all I -. hs banchng ule s mplemented as follows: Routes wthout any banchng constant ae geneated n the oot node untl no moe columns can be obtaned. Afte the bounds compason n ths node, a locaton s chosen based on a usual banchng cteon (.e. best fst seach) to late geneate two sub-poblems n the second tee-level. he fst sub-poblem geneates outes that nclude a second chosen locaton whle the second sub-poblem geneates a oute wthout t. In the thd level anothe locaton s ntoduced sng to (2 n- ) = 3 the numbe of nodes of the banch-and-pce tee. he wost case popagaton nvolves (2 n- ) nodes at the n level of the tee. o pune ths tee, the global uppe bound (GUB) and the local lowe bound (LLB) ae computed on each tee-node. he lowe bound wthn a banch-and-pce node s found by solvng the lnea poblem defned by eqs. ()-(4) on the subset of columns that fulfl the fozen assgnment decsons of the node. hat s easy because lnea RMP nvolvng seveal thousand columns ae solved wthn a small facton of a second usng moden CPLEX codes n standad PCs. he GUB s moe dffcult to compute because nvolves the esoluton of an ntege RMP on all geneated columns. Seveal computaton technques amed at avodng the complete poblem esoluton have been poposed. Nevetheless, the best uppe bound can be computed by solvng the ntege RMP, banchng on the column selecton vaables x, because f the columns set emans n modeate szes (.e. a few thousand columns) ths poblem can be solved n a few seconds wth state of the at banch-and-cut solves. If the sze of the columns poll gows above ths theshold, some columns can be gnoed accodng a flteng cteon. he flteng leads to a slmmed ntege poblem that povdes a 42 JAIIO - SII ISSN: Page 78

7 heustc uppe bound that may concde wth the optmal bound. hs fequently occus f the columns selecton s coectly caed out. 4 Computatonal Results he algothm was coded and compled wth Gams and the poblems un unde a Wndows 7 opeatng system on a 2 GHz 6 GB RAM PC. o the best of ou knowledge thee ae no standad datasets used to evaluate VRP-C-W soluton algothms. Some cted nstances as those povded by the ansh ansvson Goup [4] ae not openly avalable. Consequently, we geneated ou test bed fom the wellknown VRPW Solomon s test-bed [7]. o geneate VRP-C-W nstances, we took the fst n = 0, 20, 30, 40 and 0 customes of the Solomon s R-class poblems. Fo each nstance, the fst half of them ae consdeed pck-up stes whle the second half ae egaded as delvey locatons. he sevce tmes at pck-up and delvey nodes ae constants kept at the value st = 0 taken fom the Solomon nstances. Also, the maxmum outng tme s kept at the value t v max = 230. In addton, dffeent tme-wndows wthn wth the pckup and delvey sevces must stat, ae also consdeed. hey ae taken fom the Solomon nstances R0, R0 and R09. As pck-up tous must pecede delvey tous, some tme-wndows wll be nconsstent and must be elmnated. In such a case, no tme wndows must be consdeed on the delvey locatons. he able pesents a summay of the nstances solved by the banch-and pce pocedue. able. Summay of the solutons found by the Banch-and-pce pocedue on the geneated. Nodes me Wndows Intege Soluton Lnea Soluton % Gap Columns B & B nodes otal CPU tme JAIIO - SII ISSN: Page 79

8 Conclusons In ths wok we pesented a banch-and-pce algothm fo solvng the VRP-C-W. he banch-and-pce algothm has been mplemented by adaptng state-of-the-at technques to the specfc stuctue and popetes of the poblem. Computatonal esults have shown that ou algothm s qute effcent n solvng modeate-sze nstances. By analyzng the computatonal esults, we can conclude that the poblem s qute complex and theefoe, had to solve. Nevetheless, we managed to solve nstances wth up to 0 customes n easonable CPU tmes. Although moe effcent soluton technques could be exploed, we consde these esults satsfactoy and a good statng pont fo nvestgatng moe sophstcated appoaches n the futue. Refeences. Boysen N. and M. Fledne. Coss-dock schedulng: Classfcaton, lteatue evew and Reseach Agenda. Omega 38, (200). 2. Lee Y., Jung W.J., and K.M. Lee. Vehcle Routng Schedulng fo Coss-ockng n the Supply Chan. Comp. and Ind. Eng.,, (2006). 3. Yu W. and P. Egbelu. Schedulng of Inbound and Outbound ucks n Coss-ockng Systems wth empoay Stoage. Eu. J. of Op. Res. 84, (2008). 4. Wen, M., Lasen, J., Clausen, J., Codeau, J.-F., & Lapote, G. Vehcle outng wth coss-dockng. Jounal of the Opeatonal Reseach Socety, 60, (2009).. Lao, Ch.-J., Ln, Y., & Shh, S. C. Vehcle outng wth coss-dockng n the supply chan. Expet Systems wth Applcatons, 37, (200). 6. Boloo Aaban A., Fatem Ghom S. and M. Zandeh. Meta-heustcs mplementaton fo schedulng of tucks n a coss-dockng system wth tempoay stoage. Exp. Sys. wth Appl. 38, (20). 7. Solomon, M. Algothms fo the vehcle outng and schedulng poblem wth tme wndow constants. Opeatons Reseach, 32, (987). 42 JAIIO - SII ISSN: Page 80