Mixed Integer Programming Model for open Vehicle Routing Problem with Fleet and driver Scheduling Considering Delivery and Pick-Up Simultaneously

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1 Inernaona Journa on Recen and Innovaon rends n Compung and Communcaon ISSN: Voume: 3 Issue: Mxed Ineger Programmng Mode for open Vehce Roung Probem wh Fee and drver Schedung Consderng Devery and Pck-Up Smuaneousy Hard ambunan Deparemen of Mahemacs Quay Unversy Medan Indonesa Herman Mawengkang Deparemen of Mahemacs Unversy of Suamera Uara Medan Indonesa hmawengkang@yahoo.com Absrac Vehce Roung Probem (VRP) s a key eemen of many ogsc sysems whch nvove roung and schedung of vehces from a depo o a se of cusomers node. hs s a combnaora opmzaon probem wh he obecve o fnd an opma se of roues used by a fee of vehces o serve a se of cusomers I s requred ha hese vehces reurn o he depo afer servng cusomers demand. hs paper nvesgaes a varan of VRP n whch he vehces do no need o reurn o he depo caed open vehce roung probem (OVRP). he probem ncorporaes me wndows fee and drver schedung pck-up and devery n he pannng horzon. he goa s o schedue he deveres accordng o feasbe combnaons of devery days and o deermne he schedung of fee and drver and roung poces of he vehces. he obecve s o mnmze he sum of he coss of a roues over he pannng horzon. We mode he probem as a near mxed neger program. We deveop a combnaon of heurscs and exac mehod for sovng he mode. Keywords- Logscs Ineger programmng Heurscs Neghborhood search ***** I. INRODUCION Vehce Roung Probem (VRP) s one of he mporan ssues ha exs n ransporaon sysem. hs s a we known combnaora opmzaon probem whch consss of a cusomer popuaon wh deermnsc demands and a cenra depo whch acs as he base of a homogeneous fee of vehces. he obecve s o desgn a se of Hamonan cyces (vehce roues) sarng and ermnang a he cenra depo such ha he demand of cusomers s oay sasfed each cusomer s vsed once by a snge vehce he oa demand of he cusomers assgned o a roue does no exceed vehce capacy and he overa rave cos akng no accoun varous operaona consrans. VRP was frs nroduced by [8]. Snce hen many researchers have been workng n hs area o dscover new mehodooges n seecng he bes roues n order o fnd he beer souons. here are a number of survey can be found n eraure for VRP such as ([6] [5] [1] [3]) and books ([2] [7]). Mahemacay VRP can be defned as foows: vehces wh a fxed capacy Q mus dever order quanes ( 1 n) of goods o n cusomers from a snge depo ( = ). Knowng he dsance d beween cusomers and ( 1 n ) he obecve of he probem s o mnmze he oa dsance raveed by he vehces n a way ha ony one vehce handes he deveres for a gven cusomer and he oa quany of goods ha a snge vehce devers s no arger han Q [18]. In some cases parcuary when he busness frms do no own a vehce fee or her prvae fee s nadequae for fuy sasfyng cusomer demand dsrbuon servces woud be carred ou by exerna conracors such as a hred vehce fee. herefore vehces are no requred o reurn o he cenra depo afer her deveres have been sasfed. hs dsrbuon mode s referred o as an open vehce roung probem (OVRP). Open vehce roung probem s an expanson q probem of he cassc vehce roung probem. he mos sgnfcan dfference beween OVRP and VRP s ha n he OVRP vehces do no reurn o he orgna depo afer servcng he as cusomer on he roue or f hey are requred hey reurn by raveng he same roue back. Open vehce roung probem s a key sep of ogscs opmzaon and he ndspensabe par of he ecommerce acves. Foowng he VRP whch beongs o an NP-hard probem he OVRP s aso an NP-hard; herefore o dea wh OVRP nsances of pracca sze researchers have focused her neres on he deveopmen of effecve heursc and meaheursc souon approaches. he arce of [13] whch cassfes he feaures encounered n pracca roung probems was he frs o dsngush beween cosed rps raveed by prvae vehces and open rps assgned o common carrer vehces. he frs souon approach for he OVRP s due o [12]. her paper deas wh a pracca roung probem faced by he arpane fee of FedEx. In specfc arpanes ayover a he end of her devery roues o aer perform her pck-up rps. hese devery roues can be seen as an appcaon of he OVRP n he sense ha arpanes do no reurn o he depo. her souon approach s a varan of he [21] agorhm adaped o he examned probem. Foowng he VRP whch beongs o an NP-hard probem he OVRP s aso an NP-hard; herefore o dea wh OVRP nsances of pracca sze researchers have focused her neres on he deveopmen of effecve heursc and meaheursc souon approaches. Reference [14] address a heursc mehod based on a mnmum spannng ree combned wh a penazaon procedure for sovng OVRP. Reference [11] presens a abu search procedure whch makes use of cusomer nseron and swap oca search operaors. Reference [15 16] have presened sudes of mea-heurscs on he OVRP whch beong o he hreshod accepng caegory of agorhms. he frs one [15] proposes an anneang based mehod ha uzes a backrackng pocy of he hreshod IJRICC Sepember 215 hp:// 558

2 Inernaona Journa on Recen and Innovaon rends n Compung and Communcaon ISSN: Voume: 3 Issue: vaue when no accepances of feasbe souons occur durng he search process whereas he second one [16] presens a snge-parameer meaheursc mehod ha expos a s of hreshod vaues o negeny gude an advanced oca search mehod. Reference [17] have aso pubshed an adapve memory approach for he OVRP. her approach nvoves a poo of roues ha beong o he hghes quay souons encounered hrough he search process. Sequences of cusomers are exraced from he adapve memory o form new para souons aer o be mproved by a abu search procedure. he roues of hese mproved souons are used o updae he adapve memory conens formng n hs way a cycc agorhm. Noe ha he aforemenoned hree works am a soey mnmzng he oa dsance of he open roues dsregardng he requred fee sze. se. Wh each cusomer verex s assocaed a non-negave known demand q whereas wh each arc (v v ) E s assocaed a cos c whch corresponds o he cos (rave me dsance) for ransng from v o v. As wh mos prevous OVRP approaches we consder ha he cos marx s obaned by cacuang he Eucdean dsances beween verex pars so ha c = c ( < n ). he obecve of he probem s o desgn he se of Hamonan pahs o serve a cusomers such ha: he number of vehces s mnmzed as we as o mnmze he oa cos of he generaed pahs. here are some resrcons whch mus be sasfed such as every pah orgnaes from he cenra depo v each cusomer verex s assgned o a snge pah and he oa demand of he cusomer se assgned o a snge pah does no exceed he maxmum carryng oad Q of he vehces (capacy consran). Reference [91] propose a meaheursc framework whch consrucs an na OVRP souon va a farhes-frs heursc. hs souon s hen mproved by a abu search mehod whch empoys he we-known reocaon swap and 2-op operaors. reference [18] have dea wh he OVRP by deveopng a oca search meaheursc agorhm whch uses he concep of record-o-record rave [19]. In her work hey nroduce egh arge-scae OVRP nsances whch have served as a comparson bass for he effecveness of recen OVRP mehodooges. hese works ncude he genera roung heursc of [2] whch has been effecvey apped o he OVRP varan. her approach nvoves he appcaon of he adapve arge neghborhood search framework. Hybrd genec agorhm was used by [22] o sove a varan of OVRP whch nvove snge and mxed fee sraegy. Reference [23] propose a hybrd an coony meaheursc approach for OVRP. hey presen a new ranson rue an effcen canddae s severa effecve oca search echnques and a new pheromone updang rue n a way o acheve beer souon. From eraure survey menoned before mos of he research are on how o sove OVRP usng meaheursc. In reay here some varan for hs OVRP hs paper concerns wh a comprehensve mode for he varan of OVRP whch ncorporaed me wndows fee and drver schedung pckup and devery (OVRFDPDWP). he basc framework of he vehce roung par can be vewed as a Heerogeneous Vehce Roung Probem wh me Wndows (HVRPW) n whch a med number of heerogeneous vehces characerzed by dfferen capaces are avaabe and he cusomers have a specfed me wndows for servces. We propose a mxed neger programmng formuaon o mode he probem. A feasbe neghbourhood heursc search s addressed o ge he neger feasbe souon afer sovng he connuous mode of he probem. II. MAHEMAICAL FORMULAION OF OVRDPDWP Usng graph OVRP can be defned as foows. Le a graph G = (V E) where V = {v v 1 v n } s he verex se and E = {(v v ): v v V } s he arc se. Verex v represens he cenra depo where a feeof vehces s ocaed each of hem wh maxmum carryng oad equa o Q. he remanng n verces of V \ {v } represen he cusomer o formuae he mode frsy we denoe as he pannng horzon and D as he se of drvers. he se of workdays for drver D s denoed by. he sar workng me and aes endng me for drver D on day are gven by g and h respecvey. For each drver D e H denoe he maxmum weeky workng duraon. We denoe he maxmum eapsed drvng me whou break by F and he duraon of a break by G. Le K denoe he se of vehces. For each vehce k K e Q k and P k denoe he capacy n wegh and n voume respecvey. We assume he number of vehces equas o he number of drvers. Denoe he se of n cusomers (/nodes) by N 12 n. Denoe he depo by n 1 vehce sars from and ermnaes a n 1. Each. Each cusomer N specfes a se of days o be vsed denoed by. On each day cusomer N requess servce wh demand of servce duraon he depo n 1 q n wegh and d and me wndow p n voume a b. Noe ha for on day we se q p d. Denoe he se of preferabe vehces for vsng cusomer by K ( K K ) and he exra servce me per pae by e f a cusomer s no vsed by a preferabe vehce. he rave me beween cusomer and s gven by c. Denoe he cos coeffcens of he rave me of he nerna drvers by A and he workng duraon of he exerna drvers by B. We defne bnary varabe x o be 1 f vehce k raves from node o on day bnary varabe w o be 1 f cusomer s no vsed by a preferred vehce on day. Varabe v s he me ha vehce k vss node on day. k Bnary varabe z k ndcaes wheher vehce k akes a break afer servng cusomer on day. Varabe u k s he eapsed IJRICC Sepember 215 hp:// 559

3 Inernaona Journa on Recen and Innovaon rends n Compung and Communcaon ISSN: Voume: 3 Issue: drvng me for vehce k a cusomer afer he prevous x k Bnary varabe ndcang wheher vehce break on day. Bnary varabe y k s se o 1 f vehce k s k K raves from node N o N assgned o drver on day. Varabes r and s are he oa on day workng duraon and he oa rave me for drver on day w respecvey. Bnary varabe ndcang wheher cusomer N s vsed by a non-preferabe vehce on day Noaons used are defned as foows. Se: he se of workdays n he pannng horzon D he se of drvers D = D I D E he se of workdays for drver D K he se of vehces N he se of cusomers N he se of cusomers and depo N = { n + 1} N K he se of preferabe vehces for cusomer N he se of days on whch cusomer N Parameer: orders Q k he wegh capacy of vehce k K P k he voume capacy of vehce k K c he rave me from node N o node N [a b ] he eares and he aes vs me a node N N d he servce me of node on day q he wegh demand of node p he voume demand of node N on day N on day e he exra servce me per pae when a nonpreferabe vehce s used [ g h ] he sar me and he aes endng me of drver D on day Pck up quany for cusomer on day Devery quany for cusomer I on day H he maxmum workng duraon for each nerna drver over he pannng horzon F he maxmum eapsed drvng me whou break G he duraon of he break for drvers A he cos facor on he oa rave me Varabes: v he me a whch vehce k K k z k a node N on day sars servce Bnary varabe ndcang wheher vehce k K akes break afer servng node N on day u he eapsed drvng me of vehce k K a k y k node N afer he prevous break on day Bnary varabe ndcang wheher vehce k K s assgned o drver D r he oa workng duraon of drver D day s he oa rave dsance of drver D on day on on day Number of pck up demand of cusomer served by vehce kkon day Number of devery demands of cusomer served by vehce k Kon day he probem can be presened as a mxed neger near programmng mode. he obecve of he probem s o mnmze cos. Frsy we sum up he oa rave me n he pannng horzon and hen we mupy he resu wh cos facor A. herefore he obecve can be expressed as foows. Mnmze A( c x c ) N kk N N kk Subec o: x d d D (2) N (1) IJRICC Sepember 215 hp:// 551

4 Inernaona Journa on Recen and Innovaon rends n Compung and Communcaon ISSN: Voume: 3 Issue: x 1 N (3) {12...} N N x 1 N (4) kk \ K N x w N (5) N (5) q x Qk k K (6) k K (6) N N p x Pk k K (7) N N u u c M(1 x ) Mz k k Consrans (8-9) defne he eapsed drvng me. More N k K (8) specfcay for he vehce (8) (k) raveng from cusomer o on day he eapsed drvng me a equas he eapsed drvng me a pus he drvng me from o (.e. u u c M(1 x ) N k K (9) u c x F Mz k k N N k K (1) v v d ep w c Gz M(1 x ) k k k u + c ) f N he k vehce K does no ake (9) a break a cusomer cusomer (.e. z k = 1) he eapsed drvng me a w be consraned Nby (1) k whch K make sure (1) s greaer han or equa o he rave me beween and (.e. u c ). Consrans N k K (11) (1) guaranee N ha k K he eapsed drvng (11) me never exceeds an upper m F by mposng a break a cusomer (.e. z = 1) b v a N k K (12) f drvng from cusomer o s successor resus n a eapsed drvng me N greaer k K han F. (12) k K (13) Consrans (11) deermne he me o sar he servce a D each cusomer. If s vsed mmedaey afer he me k K (13) v k K (14) o sar he servce a shoud be greaer han or equa o k K (14) D he servce sarng me v k a pus s servce duraon d v k ( g yk ) vn 1 k ( h yk ) s c x M (1 y ) k N N D k K (15) vehce (.e. D I w k K = 1) he rave (15) me beween he wo r vn 1 k g M(1 yk ) D k K (16) afer servng D I k (.e. K k (16) r H D (17) D I (17) kk kk x w zk yk {1} v u r s k k N (18) N (19) N D k K (2) N D k K (21) N k K (22) As hs s an OVRP he vehces used are ony dspached from depo. Consran (2) s o make sure ha he number of vehce dspaches from depo shoud be he same as he number of drver. Consran (3) expresses ha a each node besde depo here shoud be exacy one enerng arc comng from a cusomer node or from he depo. he oher way around s expressed n Consran (4). Consrans (5) defne wheher each cusomer s vsed by a preferabe vehce. Consrans (6-7) guaranee ha he vehce capaces are respeced n boh wegh and voume. k K (7) k (.e. z k = ); Oherwse f he vehce akes a break a he exra servce me k e p f s vsed by an napproprae cusomers c and he break me G f he drver akes a break z = 1). Consrans (12) make sure he servces sar whn he cusomers me wndow. Consrans (13-14) ensure ha he sarng me and endng me of each roue mus e beween he sar workng me and aes endng me of he assgned drver. Consrans (15) cacuae he oa rave me for each nerna drver. Consrans (16) defne he workng duraon for each drver on every workday whch equas he me he drver reurns o he depo mnus he me he/she sars work. Consrans (17) make sure ha he nerna drvers work for no more han a maxmum weeky workng duraon referred o as 37 weekhour consrans. Consrans (18 19) defne he pck up and devery for each cusomer. Consrans (2-21) defne he bnary and posve varabes used n hs formuaon. IJRICC Sepember 215 hp:// 5511

5 Inernaona Journa on Recen and Innovaon rends n Compung and Communcaon ISSN: Voume: 3 Issue: he probem expressed as a mxed neger programmng Sep 1. Ge row * he smaes neger nfeasby such ha mode conans a arge number of varabes. mn{ f 1 f } * III. NEIGHBORHOOD SEARCH I shoud be noed ha generay n neger programmng he reduced graden vecor whch s normay used o deec an opmay condon s no avaabe even hough he probems are convex. hus we need o mpose a ceran condon for he oca esng search procedure n order o assure ha we have obaned he bes subopma neger feasbe souon. Furher n [4] has proposed a quany es o repace he prcng es for opmay n he neger programmng probem. he es s conduced by a search hrough he neghbors of a proposed feasbe pon o see wheher a nearby pon s aso feasbe and yeds an mprovemen o he obecve funcon. Le [] k be an neger pon beongs o a fne se of neghborhood N([] k ) We defne a neghborhood sysem assocaed wh [] k ha s f such an neger pon sasfes he foowng wo requremens 1. f [] N([] k ) hen [] k [] k. 2. N([] k ) = [] k + N() Wh respec o he neghborhood sysem menoned above he proposed negerzng sraegy can be descrbed as foows. Gven a non-neger componen x k of an opma vecor x B. he adacen pons of x k beng consdered are [x k ] dan [x k ] + 1. If one of hese pons sasfes he consrans and yeds a mnmum deeroraon of he opma obecve vaue we move o anoher componen f no we have neger-feasbe souon. Le [x k ] be he neger feasbe pon whch sasfes he above condons. We coud hen say f [x k ] + 1 N([x k ]) mpes ha he pon [x k ] + 1 s eher nfeasbe or yeds an nferor vaue o he obecve funcon obaned wh respec o [x k ]. In hs case [x k ] s sad o be an opma neger feasbe souon o he neger programmng probem. Obvousy n our case a negbourhood search s conduced hrough proposed feasbe pons such ha he neger feasbe souon woud be a he eas dsance from he opma connuous souon. IV. HE ALGORIHM We combne exac mehod and heurscs for sovng hs arge scae mxed neger programmng probem. Frsy we sove he near programmng par afer reaxng he neger resrcon. hen we usng he foowng heurscs for searchng a subopma bu neger-feasbe souon. Le x = [x] + f f 1 be he (connuous) souon of he reaxed probem [x] s he neger componen of non-neger varabe x and f s he fracona componen. Sage 1. (hs choce s movaed by he desre for mnma deeroraon n he obecve funcon and ceary corresponds o he neger basc wh smaes neger nfeasby). Sep 2. Do a prcng operaon * * IJRICC Sepember 215 hp:// v e B 1 Sep 3. Cacuae v * Wh corresponds o d mn Cacuae he maxmum movemen of nonbasc a ower bound and upper bound. Oherwse go o nex non-neger nonbasc or superbasc (f avaabe). Evenuay he coumn * s o be ncreased form LB or decreased from UB. If none go o nex *. Sep 4. Sove B * = * for * Sep 5. Do rao es for he basc varabes n order o say feasbe due o he reeasng of nonbasc * from s bounds. Sep 6. Exchange bass Sep 7. If row * = {} go o Sage 2 oherwse Repea from sep 1. Sage 2. Pass1 : adus neger nfeasbe superbascs by fracona seps o reach compee neger feasby. Pass2 : adus neger feasbe superbascs. he obecve of hs phase s o conduc a hghy ocazed neghborhood search o verfy oca opmay. V. CONCLUSIONS hs paper s nended o deveop effcen echnque for sovng one of he mos economc mporance probems n opmzng ransporaon and dsrbuon sysems. he am of hs paper s o deveop a mode of open vehce roung wh me Wndows Fee and Drver Schedung Pck-up and Devery Probem hs probem has addona consran whch s he maon n he number of vehces. he proposed agorhm empoys neares neghbor heursc agorhm for sovng he mode. REFERENCES [1] B. L. Goden A. A. Assad and E. A. Was Roung Vehces n he rea word: Appcaons n he sod wase beverage food dary and newspaper ndusres. In P. oh and D. Vgo edors he Vehce Roung Probem pages SIAM Phadepha PA 22. [2] B. L. Goden S. Raghavan and E. A. Was he vehce roung probem: aes advances and new chaenges. Sprnger 28. [3] G. Lapore and F. Semen. Cassca Heurscs for he Vehce Roung Probem. In oh P. and Vgo D. edors he vehce Roung Probem SIAM Monographs on Dscree Maheamcs and Appcaons pages SIAM Phadepha PA

6 Inernaona Journa on Recen and Innovaon rends n Compung and Communcaon ISSN: Voume: 3 Issue: [4] H. E Scarf. esng for opmay n he absence of convexy n Waer P Heer Ross M Sarr and Davd A Sarre (Eds) (Cambrdge Unversy Press) pp [5] J.-F. Cordeau G. Lapore M.W.F. Savesbergh and D. Vgo. Vehve Roung. In Barnhar C. and Lapore G. edors ransporaon handbooks n Operaon Research and Managemen Scence Pages Norh-Hoand Amserdam 27. [6] O. Br ysy and M. Gendreau. Vehce Roung Probem wh mes Wndows Par II: Meaheurscs. ransporaon Scence 39(1): [7] P. oh and D. Vgo. he Vehce Roung Probem Socey for Indusra and Apped Maheamcs Phadepha PA 22. [8] G.B. Danzg and J.H. Ramser he ruck dspachng probem Managemen Scence Vo [9] Fu Z Egese RW L LYO. A new abu search heursc for he open vehce roung probem. J Oper Res Soc 25;56: [1] Fu Z Egese RW L LYO. Corrgendum. A new abu search heursc for he open vehce roung probem. J Oper Res Soc 26;57:118. [11] J. Brandão. A abu search agorhm for he open vehce roung probem. Eur J Oper Res 24; 157(3): [12] L. Bodn Goden B Assad A Ba M. Roung and schedung of vehces and crews : he sae of he ar. Compu Oper Res 1983;1(2): [13] L. Schrage. Formuaon and srucure of more compex/reasc roung and schedung probems. Neworks 1981;11(2): [14] D. Sarks Powe S. A heursc mehod for he open vehce roung probem. J Oper Res Soc 2;51(5): [15] C.D. arans Ioannou G Kranouds C Prasacos GP. A hreshod accepng approach o he open vehce roung probem. RAIRO 24;38: [16] C.D. arans Kranouds C Ioannou G Prasacos GP. Sovng he open vehce roung probem va a snge parameer meaheursc agorhm. J Oper Res Soc 25;56: [17] C.D. arans Dakouak D Kranouds C. Combnaon of geographca nformaon sysem and effecve roung agorhms for rea fe dsrbuon operaons. Eur J Oper Res 24;152: [18] L F Goden B Was E. he open vehce roung probem: Agorhms argescae es probems and compuaona resus. Compu Oper Res 24;34(1): [19] Dueck G. New opmzaon heurscs: he grea deuge agorhm and he recordo-record rave. J Compu Phys 1993;14: [2] D. Psnger Ropke S. A genera heursc for vehce roung probems. Compu Oper Res 27;34(8): Mah Mode Agor 24;3(2): [21] G. Carke Wrgh JW. Schedung of vehces from a cenra depo o a number of devery pons. Oper Res 1964; 12: [22] C. Ren. Research on Snge and Mxed Fee Sraegy for Open Vehce Roung Probem. Journa of Sofware 211;6(1): [23] M. Sedghpour V. Ahmad M. Yousefkhoshbakh F. Ddehvar F. Rahma. Sovng he Open Vehce Roung Probem by a Hybrd An Coony Opmzaon. Kuwa J. Sc. 214;41(3): IJRICC Sepember 215 hp:// 5513