A BRANCH-AND-PRICE METHOD FOR THE VEHICLE ROUTING PROBLEM WITH CROSS-DOCKING AND TIME WINDOWS

Transcription

1 A BRANCH-AN-PRCE EHO FOR HE VEHCLE ROUNG PROBLE WH CROSS-OCKNG AN E WNOWS Rodolfo ondo ABSRAC: One mpoan faco n supply chan managemen s o effcenly conol he supply chan flows. ue o s mpoance, many companes ae yng o develop effcen mehods o ncease cusome sasfacon and educe coss. Coss-dockng s consdeed a good mehod o educe nvenoy and mpove esponsveness. he Vehcle Roung Poblem wh Coss-ockng and me Wndows (VRP-C-W) consss on desgnng he mnmumcos se of oues o seve a gven se of anspoaon equess whle especng consans on vehcles capacy, cusome me wndows and usng ansfes on a coss-dockng base. Each cusome mus be vsed us once and mxed ous compsng pck-up and delvey sops ae no allowed. Fo a gven vehcle, he desgned pck-up ou mus pecede s delvey ou. n hs wok, we model he VRP-C-W assumng ha all feasble odes ae known n advance. We pesen a new mxed nege pogam o model he VRP-C-W and efomulae va anzg Wolfe decomposon o lae develop a column geneaon pocedue. he poposed banch-and-pce algohm shows encouagng esuls on solvng some Solomon-based nsances. Keywods: Supply-chan managemen. Coss-dockng. Vehcle oung. Columns geneaon. NROUCON Coss-dockng has emeged as an mpoan and effcen goods anspoaon saegy. As a vaaon of he well-known vehcle oung poblem (VRP), he VRP wh Cossdockng (VRPC) ases n a numbe of logsc plannng conexs. A coss-dockng emnals ncomng delvees of nbound ucks ae unloaded, soed, moved acoss he dock and fnally loaded ono oubound ucks, whch mmedaely leave he emnal owads he nex desnaon n he dsbuon chan. he coss-dock s a consoldaon pon n a dsbuon newok, whee mulple smalle shpmens can be meged o full uck loads n ode o oban savngs n., en ngeneía Químca Pofeso Aduno, Unvesdad Naconal del Loal, nsuo de esaollo ecnológco paa la ndusa Químca NEC, Conseo Naconal de nvesgacones Cenífcas y écncas CONCE, Güemes 340 (3000) Sana Fe- Agenna. E-mal: dondo@sanafe-conce.gov.a.

2 anspoaon (BOSEN; FLENER, 200). he obecve of he VRP-C-W consss of desgnng he mnmum-cos se of oues o seve a gven se of anspoaon equess whle fulfllng consans on vehcles capacy and cusome me wndows and usng goods ansfes on a coss-dockng base. As coss-dockng s a compaavely new logsc saegy, hee s no ye a massve body of leaue on he subec. n fac, no eseach on he shoem uck-schedulng-poblem was publshed befoe 200. Lee a al. (2006) consdeed coss-dockng fom an opeaonal vewpon n ode o fnd he opmal vehcle oung schedule. hus, an negaed model consdeng boh cossdockng and vehcle oung schedulng was pesened. Snce he poblem s NP-had, a heusc algohm based on a aboo seach algohm was poposed. One of he obecves fo coss dockng sysems s how well he ucks can be scheduled a he dock and how he ems n nbound ucks can be allocaed o he oubound ucks o opmze on some measue of sysem pefomance (U; EGBELU, 2008). he auhos eseached on how o fnd he bes uck dockng o schedulng sequence fo boh nbound and oubound ucks o mnmze oal opeaon me when a soage buffe o hold ems empoaly s locaed a he shppng dock. he poduc assgnmen o ucks and he dockng sequences of he nbound and oubound ucks ae all deemned smulaneously. Wen e al. (2009) pesened a vey dealed mxed fomulaon fo he poblem and a aboo seach heusc embedded whn an adapve memoy pocedue. he pocedue was esed on daa ses unepoed n he pape ha wee povded by he ansh ansvson consulancy nvolvng up o 200 pa of nodes. Lao e al. (200) pesened a new aboo seach (S) algohm ha was poposed o oban a good feasble soluon fo he poblem. hough exensve compuaonal expemens, hey clam ha he poposed S algohm can acheve bee pefomance han a pevous S by Lee e al. algohm whle usng much less compuaon me. Boloo Aaban e al. (20) suded some mea-heuscs o fnd he bes sequence of nbound and oubound ucks, so ha he obecve, mnmzng he oal opeaon me o makespan, can be sasfed. n hs wok, we sudy he VRP-C-W and popose a banchand-pce algohm fo solvng such a poblem. n secon 2 we fomulae he poblem. n secon 3 we pesen and descbe he banch-and-pce algohm devsed o solve he CRP- C-W. Compuaonal esuls ae pesened and dscussed n secon and he conclusons ae oulned n secon 4. 7

3 2 PROBLE FORULAON Le G [; A] a deced gaph nvolvng he locaons se = {w + - } and he ne A = {a ' :, ' and '}. conans he he coss-dock-base w, he pck-up nodes + = {,..., n } and he delvey nodes - = {',..., ' n }. A eques = {, '} of a eques ls R = {,... } consss of a demand of a anspoaon sevce fom he ogn-node(s) + o he desnaon node(s) ' + fo cean load l. Each ac a ' A have an assocaed non-negave cos c and an assocaed non-negave avel-me. he sevce me on each node s defned by he paamee s. Requess mus be fulflled hough a homogeneous vehcles flee V = {v, v 2,..., v m }. he soluon consss of a fne se of sequences of acs, called oues, fo some vehcles v V such ha: () Each vehcle can pefom pck-up and/o delvey ous bu each ndvdual ou conans ehe pck-up o delvey locaons. No mxed oues wh boh ype of locaons ae allowed. () Fo each vehcle, he pck-up ou mus pecede s desgned delvey oue. () Each vehcle sas and ends he pck-up and/o delvey oues a he coss-dock base w. (v) Each pck-up/delvey se + - s assgned o exacly one oue. (v) he acual load caed by a vehcle v mus neve exceed s anspo capacy q v. (v) he sevce fo any pck-up/delvey node + - mus sa whn he me-wndow [a, b ]; (v) he whole duaon of he vehcle-v p, ncludng he pck-up ou, he ansfe opeaons and he delvey ou mus be shoe han a maxmum oung me v max v. (v) he poblem goal s o mnmze he oal cos fo povdng pckup/ delvey sevce o evey node COLUN GENERAON n hs secon, we noduce a se paonng fomulaon of he poblem above pesened o lae povde he fomulaons of he mase poblem and he pcng subpoblem. 3. he ase Poblem he defnon of he mase poblem eques he followng noaon: c + s he cos of he pck-up oue + and a denoes a bnay paamee equal o f he oue + pcks-up he cago of he eques R. 8

4 - n he same way, c s he cos of he delvey oue - whle b denoes a bnay paamee equal o f he oue - delves he cago of he eques R. Afe he noducon of he me-coodnaon consan (4), he mase poblem can be fomulaed as follows (Consan -4): nmze Sue o R c x R c x () R a x (2) R a end R x b x R b x sa, : (, ) R (3) (4) x 0, whee x + and x - ae bnay vaables assocaed especvely o pck-up oues + and -. he dual vaables assocaed o Consan 2 and 3 ae boh colleced n he veco π = [π,..., π,...] whle π = [π,..., π ] collec he dual values assocaed o coodnaon Consan 4. he obecve funcon () selecs he se of pck-up and delvey oues ha mnmzes he oal avelng cos. Consans 2 ensue ha all equess R ae pcked fom pck-up ses +, whle Consan 3 ensue ha all equess R ae delveed o delvey locaons -. Consan 4 coodnaes he pck-up and delvey ous ha move a eques fom s ogn o s desnaon. So, he end of he pck-up ou mus pecede he me-sa of he delvey ou. We emak ha Consan 2 and 3 need o be modeled as paonng consans n he VRP-C- W, unlke common efomulaons fo oung poblems ha geneally make use of coveng consans. 9

5 20 hs s due o he fac ha a lnea combnaon of acve pck-up and delvey oues mus lead o a soluon on whch he pck-up ou sas befoe he end of he delvey oue. As a consequence, a paonng soluon equvalen o he opmal coveng soluon may be nfeasble. 3.2 Pcng sub-poblem n a column geneaon scheme, gven a dual soluon of he (esced) mase poblem, he pcng sub-poblem denfes he oue * wh he mnmum educed cos: R R CV CV c ), :( * he sub-poblem fomulaon eles on he some nege and connuous vaables and can be wen as follows (Consan -23): nmze R R CV CV ), :( () Subec o: d 0 S d S d 2 2 :, C d CV 0 (6) (7.a) (7.b) (8) 0 S s S s 2 2 :, V (9) (0.a) (0.b) ()

6 l Q R R R end sa V end d0 0 s s R :(, ) R 0 s d d s R :(, ) R S 2 S 2 (2) (3.a) (3.b) (3.c) (4.a) (4.b) (), : CV d sa 0 0 s s V C S 2 S 2, : (6.a) (6.b) (7) (8) (9.a) (9.b) l Q (20) (2) max V (22) a b (23) he obecve denfes he oue wh mnmum educed cos o pefom by a sngle vehcle. Pces values ae obaned fom he dual values fo Consans 3, 4 and. he cos 2

7 consans of he pck-up ou ae gven by Equaon 6, 7 and 8. hey compue he dsances avelled o each he vsed ses + and he oal cos of he geneaed oue especvely. So, Equaon 7 fx he accumulaed dsance up o each vsed se..e. f nodes and ae allocaed o he geneaed pck-up oue ( = = ), he vsng odeng fo boh ses s deemned by he value of he sequencng vaable S. f locaon s vsed befoe (S = ), accodng Consan 7a, he avelled dsance up o he locaon ( ) mus be lage han by a leas d. n case node s vsed eale, (S = 0), he evese saemen holds and consan (7.b) becomes acve. f one o boh nodes ae no allocaed o he ou, Equaon 7a and 7b become edundan. s an uppe bound fo vaables. Equaon 8 ansfoms he oal avelled dsance no he oue-cos CV. C s an uppe bound fo he vaable CV. he mng consans saed by Equaon 9, 0 and defne vsng-me consans ha ae smla o Consans 6, 7 and 9 bu apply o he me dmenson. s an uppe bound fo he mes spen o each he nodes. Equaon 2 s he cago-capacy consan elaed o he pck-up ou. Equaons 3 acvae he vaables R ndcang ha he cago s pcked-up and delveed by he same vehcle. n such a case, he eques cago wll eman on he uck and no dop-off/pck-up acves ae ncued. Consan (4a) saes ha he me-end of he pck-up phase mus be he sum of he me spen compleng he pck-up ou plus he me ncued unloadng cagos a he cossdock base. Convesely, Equaon 4b saes ha he delvey sage mus sa afe he sum of me + end and he me ncued n loadng goods o be lae delveed. Consans -7 ae cos consans elaed o he delvey ou whle Equaon 8-20 defne me consans elaed o hs delvey ou. Eq. (2) apply he vehcle capacy consan o he delvey ou and Equaon 22 foces he sevce me a any pck-up/delvey se o sa a a me bounded by he me neval [a, b ]. Fnally, he Equaon 23 mposes he uppe bound max v o he oal ou-me. 3.3 Banch and Pce mplemenaon he Fose and Ryan banchng ule s vey favouable fo banch and pce applcaons and fs easly wh he VRP-C-W pcng poblem snce s equvalen o banch on assgnmen decsons, fo all

8 hs banchng ule s mplemened as follows: Roues whou any banchng consan ae geneaed n he oo node unl no moe columns can be obaned. Afe he bounds compason n hs node, a locaon s chosen based on a usual banchng ceon (.e. bes fs seach) o lae geneae wo sub-poblems n he second ee-level. he fs sub-poblem geneaes oues ha nclude a second chosen locaon whle he second sub-poblem geneaes a oue whou. n he hd level anohe locaon s noduced sng o (2 n- ) = 3 he numbe of nodes of he banch-and-pce ee. he wos case popagaon nvolves (2 n- ) nodes a he n level of he ee. o pune hs ee, he global uppe bound (GUB) and he local lowe bound (LLB) ae compued on each ee-node. he lowe bound whn a banch-and-pce node s found by solvng he lnea poblem defned by Equaon -4 on he subse of columns ha fulfl he fozen assgnmen decsons of he node. ha s easy because lnea RP nvolvng seveal housand columns ae solved whn a small facon of a second usng moden CPLEX codes n sandad PCs. he GUB s moe dffcul o compue because nvolves he esoluon of an nege RP on all geneaed columns. Seveal compuaon echnques amed a avodng he complee poblem esoluon have been poposed. Neveheless, he bes uppe bound can be compued by solvng he nege RP, banchng on he column selecon vaables x, because f he columns se emans n modeae szes (.e. a few housand columns) hs poblem can be solved n a few seconds wh sae of he a banch-and-cu solves. f he sze of he columns poll gows above hs heshold, some columns can be gnoed accodng a fleng ceon. he fleng leads o a slmmed nege poblem ha povdes a heusc uppe bound ha may concde wh he opmal bound. hs fequenly occus f he columns selecon s coecly caed ou. 4 COPUAONAL RESULS he algohm was coded and compled wh Gams and he poblems un unde a Wndows 7 opeang sysem on a 2 GHz 6 GB RA PC. o he bes of ou knowledge hee ae no sandad daases used o evaluae VRP-C-W soluon algohms. Some ced nsances as hose povded by he ansh ansvson Goup (WEN e al., 2009) ae no openly avalable. Consequenly, we geneaed ou es bed fom he well-known VRPW Solomon s es-bed (SOLOON, 987). 23

9 o geneae VRP-C-W nsances, we ook he fs n = 0, 20, 30, 40 and 0 cusomes of he Solomon s R-class poblems. Fo each nsance, he fs half of hem ae consdeed pck-up ses whle he second half ae egaded as delvey locaons. he sevce mes a pck-up and delvey nodes ae consans kep a he value s = 0 aken fom he Solomon nsances. Also, he maxmum oung me s kep a he value v max = 230. n addon, dffeen me-wndows whn wh he pck-up and delvey sevces mus sa, ae also consdeed. hey ae aken fom he Solomon nsances R0, R0 and R09. As pck-up ous mus pecede delvey ous, someme-wndows wll be nconssen and mus be elmnaed. n such a case, no me wndows mus be consdeed on he delvey locaons. he able pesens a summay of he nsances solved by he banch-and pce pocedue. able Summay of he soluons found by he Banch-and-pce pocedue on he geneaed Nodes me Wndows nege Soluon Lnea Soluon % Gap Columns B & B nodes oal CPU me CONCLUSONS n hs wok, we pesened a banch-and-pce algohm fo solvng he VRP-C-W. he banch-and-pce algohm has been mplemened by adapng sae-of-he-a echnques o he specfc sucue and popees of he poblem. 24

10 Compuaonal esuls have shown ha ou algohm s que effcen n solvng modeae-sze nsances. By analyzng he compuaonal esuls, we can conclude ha he poblem s que complex and heefoe, had o solve. Neveheless, we managed o solve nsances wh up o 0 cusomes n easonable CPU mes. Alhough moe effcen soluon echnques could be exploed, we consde hese esuls sasfacoy and a good sang pon fo nvesgang moe sophscaed appoaches n he fuue. REFERENCES BOSEN, N.; FLENER,. Coss-dock schedulng: classfcaon, leaue evew and eseach agenda. Omega, v. 38, p , 200. LEE,.; JUNG, W.J.; LEE, K.. Vehcle Roung Schedulng fo Coss-ockng n he Supply Chan. Compues & ndusal Engneeng, v., p , U, W.; EGBELU, P. Schedulng of nbound and Oubound ucks n Coss-ockng Sysems wh empoay Soage. Euopean Jounal of Opeaonal Reseach, v. 84, p , WEN,.; LARSEN, J.; CLAUSEN, J.; COREAU, J.-F.; LAPORE, G. Vehcle oung wh coss-dockng. Jounal of he Opeaonal Reseach Socey, v. 60, p , LAO, CH.-J.; LN,.; SHH, S.C. Vehcle oung wh coss-dockng n he supply chan. Expe Sysems wh Applcaons, v. 37, p , 200. BOLOOR ARABAN A., FAE GHO S. AN. ZANEH. ea-heuscs mplemenaon fo schedulng of ucks n a coss-dockng sysem wh empoay soage. Expe Sysems wh Applcaons, v. 38, p , 20. SOLOON,. Algohms fo he vehcle oung and schedulng poblem wh me wndow consans. Opeaons Reseach, v. 32, p , 987. Ognas ecebdos em: 26/0/203 Aceo paa publcação em: /04/204 2