A heuristic approach for an inventory routing problem with backorder decisions

Transcription

1 Lecure Noes n Managemen Scence (2014) Vol. 6: h Inernaonal Conference on Appled Operaonal Research, Proceedngs Tadbr Operaonal Research Group Ld. All rghs reserved. ISSN (Prn), ISSN (Onlne) A heursc approach for an nvenory roung problem wh backorder decsons Sella Sofanopoulou Deparmen of Indusral Managemen & Technology, Unversy of Praeus, Greece sofanop@unp.gr Absrac. A mul-perod nvenory-roung problem s consdered where a vendor serves mulple geographcally dspersed cusomers who receve uns of a sngle produc from a depo, wh adequae supply, usng a capacaed vehcle. The class of problems arsng from he combnaon of roung and nvenory managemen decsons s known as he nvenory roung problem (IRP). In hs caegory of problems, he nvenory roung problem wh backorders (IRPB) deals wh deermnng nvenory levels when backorders are allowed. The am s o mnmze he oal cos for he plannng perod, comprsng of holdng cos, ransporaon and backorder penaly cos whle ensurng ha nvenory level capacy consrans are no volaed. An Ineger Programmng model s frs developed o provde an accurae descrpon of he problem and hen a Genec Algorhm (GA) wh suably desgned genec operaors s employed n order o oban near opmal soluons. Compuaonal resuls are presened o demonsrae he effecveness of he proposed procedure. Keywords: nvenory roung problem; backloggng; genec algorhms Inroducon Vendor Managed Invenory ogeher wh he underlyng Invenory Roung Problem resuls n sgnfcan cos savngs of nvenory handlng, ncreased effcency, beer performance, and commed delvery daes. More precsely, nsead of cusomers need o keep her socks and make sure s replenshed on a regular bass, he suppler s responsble for respondng o cusomer needs by smulaneously deermnng he mng and sze of delveres and schedules effecvely vehcles o mnmze he oal ransporaon and nvenory holdng coss. The class of problems arsng from he combnaon of nvenory conrol problems and NP-hard vehcle roung problems, whch deermnes delvery volumes

2 50 Lecure Noes n Managemen Scence Vol. 6: ICAOR 2014, Proceedngs o he cusomers ha he suppler serves n each perod, and vehcle roues o delver he volumes s known as "nvenory roung problems" (IRP). The nvenory roung problem wh backloggng (IRPB),.e. he nvenory roung problem when backorders are allowed, s an IRP varan wh he am o deermne nvenory levels, backloggng and vehcle roung decsons for a se of n cusomers on a number of me perods. A varey of ndusres have provded ferle ground for he developmen for he IRP and s varans. Researchers have generaed many mehods of solvng IRP and o a lesser exend IRPB. Mos of he proposed mehods use eher a "heorecal" or a more "praccal" approach. The heory concenraes manly o compue lower bounds of he problem, whle he praccal approaches employ heursc procedures o oban near opmal soluons. Mos researchers concenrae on he IRP where nvenory and roung decsons are combned n a cos opmzaon problem where backorders are no permed and mulple dsrbuon levels may occur (Janxang L, Feng Chu and Haoxun Chen, 2011). Dependng now on he approaches developed for he soluon of IRP, we wll adop he classfcaon nroduced by Bramel and Smch-Lev (1997). Accordng o hs, mos of he research on IRP s n one of hree drecons: sngle-day models usng deermnsc demand (Chen, Balakrshnan, and Wong, 1989; Berazz, Palea, and Speranza, 2002), mul-day models usng deermnsc demand (Bard, Huang, Jalle and Dror, 1998; Campbell and Savelsbergh, 2004), and nfne me horzon usually for long-erm plannng purposes (Anly and Bramel, 2004). Fnally, sochasc demand has also been consdered by several researchers (Kleyweg, Nor, and Savelsbergh, 2002). Concernng now he leraure ha refers o IRP models wh backloggng, recen works nclude a local search approach appled o he problem (Zacharads, Taranls, and Kranouds, 2009), where wo local search operaors for jonly dealng wh he nvenory and roung aspecs of he problem are employed ogeher wh a Tabu Search approach for ransporaon cos reducon. Also n (Abdelmaguda, Dessouky and Ordóñez, 2009) he auhors sudy an nvenory-roung problem n whch mul-perod nvenory holdng, backloggng, and vehcle roung decsons are o be aken usng consrucve and mprovemen heurscs. The nvenory roung problem wh backloggng (IRPB) s nroduced and solved n hs work. In hs paper a mul-realer sngle produc roung problem where backorders are permed s examned and revewed. In he followng secon he problem s formally saed, whle n he nex secon a Genec Algorhm s presened and relaed operaors are dscussed. Nex he applcaon of he algorhmc procedure o IRPB s mplemened, whle concludng remarks and suggesons for furher research are presened n he las secon. Problem Saemen IRPB s formally presened n hs secon. The problem s concerned wh he repeaed dsrbuon of a sngle produc, from a sngle facly,.e. a cenral depo, o a se of N={1,2,,n} dspersed realers over a gven plannng horzon of lengh T usng a sngle vehcle of gven capacy. For each realer N, we have a deermnsc demand d, a level of nvenory I (wh a maxmum capacy of C ) and a backloggng level B wh a maxmum level Bl per me perod. The amoun of delveryq, o cusomer n perod, s o be decded and based on he delvery amouns o oher realers n perod.

3 S Sofanopoulou 51 Each realer, ncurs an nvenory holdng cos of h and a shorage cos (penaly) of b per me perod per un. We assume ha he depo has a suffcen supply of ems o cover all cusomers demands hroughou he plannng horzon. Delveres can be carred ou a any me perod, usng a vehcle of lmed capacy. Any combnaon of realers can be vsed n a sngle delvery roue and he ransporaon cos c j,.e. from realer o realer j, s gven. The followng s an Ineger programmng formulaon of IRPB presened n he paper manly for accuraely descrbng he problem ackled: mn h I j j c w b B (1) Subjec o: 1 I I B B 1 q d, (2) I C, (3) B Bl j, (4) w 1, (5) j zj Mwj, j, (6) q VQ (7) j w w 0, (8) j l l l k l z z q, (9) k I,B,q,z j 0, w 0, 1 Μ: large number (10) j The oal cos comprses of he nvenory holdng cos, he ransporaon cos, and he shorage cos as depced n he objecve funcon (1). Consrans (2) are he nvenory balance equaons for cusomers. Consrans (3) and (4) lm he oal amoun of nvenory and backloggng o C and B respecvely. Consrans (5) ensure ha a vehcle wll no vs he locaon of a specfc realer more han once. w j s a bnary varable ha s equal o 1, f he vehcle vss realers a locaon and j successvely. Consrans (6) make sure ha f here sn a vehcle ravellng beween wo locaons (.e. realers) hen he amoun delvered beween hem wll be zero. z j s a connuous varable represenng maeral flow. Consrans (7) ensure ha he oal quany delvered o all cusomers durng perod does no exceed vehcle s capacy VQ. Consrans (8) are used n order o ensure roue connuy and consrans (9) reflec he equy beween maerals flow and he quany of producs o be delvered. Consrans (10) are doman consrans.

4 52 Lecure Noes n Managemen Scence Vol. 6: ICAOR 2014, Proceedngs Genec Algorhms The IRP s classfed as NP-hard problem snce subsumes he Vehcle Roung Problem (VRP). Ths fac led o he developmen of many heursc approaches, alhough a small number of exac mehods have been recenly nroduced (Arche e al. 2007; Solyalı and Süral 2011; Adulyasak, Cordeau, and Jans 2012; Coelho and Lapore 2013). Neverheless, here are lmaons n her use as benchmarks as here exs several IRP varans. Snce he IRPB s NP-hard, accurae soluon mehods are dffcul o adop, and heursc approaches are more promsng. Genec Algorhms (GAs) are effcen heursc procedures ofen surpassng he effecveness of more radonal algorhms. There are several reasons for ha. Frsly, GAs work wh a codng of he parameers whereas normal opmsaon and search procedures use he parameers hemselves. Second, GAs perform her search from a populaon of pons, no a sngle pon. Thrd, hey use payoff (objecve funcon) nformaon and do no rely on dervaves or oher auxlary nformaon. Fnally, GAs make use of probablsc ranson rules, no deermnsc ones. In genec algorhms, he erm chromosome ypcally refers o a canddae soluon o a problem, ofen encoded as a b srng. The "genes" are eher sngle bs or shor blocks of adjacen bs ha encode a parcular elemen of he canddae soluon (e.g., n he conex of mul-parameer funcon opmzaon he bs encodng a parcular parameer mgh be consdered o be a gene). An allele of a b srng s eher 0 or 1; for larger alphabes more alleles are possble a each locus. In correspondence o he erm genoype used n naural sysems, n arfcal sysems he erm srucure s employed n order o refer o he oal package of srngs. Moreover, he erm phenoype s analogous o a parameer se, or a soluon alernave ha s formed by he decodng of a srucure and he locus of a gene s analogous o he poson of a b n a srng. Genec operaors As menoned earler, GAs resemble naural sysems and bascally follow he same prncples. In GAs, durng each successve generaon, a proporon of he exsng populaon s seleced o breed a new generaon. Indvdual soluons (srngs) are seleced hrough a fness-based process. In GAs, as a number of genec operaors s employed (selecon, crossover, muaon) o ensure ha he average fness of he populaon of srngs wll connue o ncrease n successve generaons, good paral soluons combne o form even beer compose soluons. Genec operaors le a he core of GAs and are of crucal mporance o he ably of he algorhm o produce hgh qualy soluons. Selecon ensures he survval of he fes and resembles he naural selecon process. I deermnes durng each successve generaon, whch srngs wll reproduce and pass on her genes o he nex generaon, accordng o fness crera. Popular and well-suded selecon mehods nclude roulee wheel selecon and ournamen selecon. In roulee-wheel selecon, he fness funcon assgns a fness value o possble soluons (srngs). Afer selecng he srngs, he nex sep s o generae a new populaon of soluons from hose seleced hrough genec operaors: crossover (also called recombnaon), and/or muaon. For each new soluon o be produced, a par of "paren" soluons s

5 S Sofanopoulou 53 seleced for breedng from he pool seleced prevously. By producng a "chld" soluon usng he mehods of crossover and muaon, a new soluon s creaed whch ypcally shares many of he characerscs of s "parens". New parens are seleced for each chld, and he process connues unl a new populaon of soluons of approprae sze s generaed. These processes ulmaely resul n he nex generaon populaon of chromosomes ha s dfferen from he nal generaon. Generally he average fness wll have ncreased by hs procedure for he populaon, snce only he bes organsms from he frs generaon are seleced for breedng, along wh a small proporon of less f soluons, for reasons already menoned above. Analycally, durng he process of crossover, wo ndvduals are seleced from he populaon and a crossover se (a poson) along he b srngs s randomly chosen. Then, subsrngs from correspondng posons whn he ndvduals are exchanged. One or boh of he new ndvduals are nsered no he populaon a he nex generaon. For example f S 1 = and S 2 = are he paren srngs and he crossover pon s 2 hen S 1 '= and S 2 '= Muaon helps he GA procedure o manan genec dversy from one generaon of a populaon of chromosomes o he nex. A common mehod of mplemenng he muaon operaor nvolves generang a random varable for each b n a sequence. Ths random varable ells wheher or no a parcular b wll be modfed. Srucure of he algorhm The seps for mplemenng a GA are he followng: Randomly creae an nal populaon (generaon 0) Ieravely perform he followng sub-seps on he populaon unl he ermnaon creron (.e. a sasfacory fness level or a maxmum number of generaons), s sasfed: For each srng n he populaon calculae s fness. Selec one or wo ndvdual(s) from he populaon wh a probably based on fness o parcpae n he genec operaons n (c). Creae new ndvdual for he populaon by applyng he followng genec operaons wh specfed probables: Reproducon: Copy he seleced ndvdual o he new populaon. Crossover: Creae new offsprng(s) for he new populaon by recombnng randomly chosen pars from wo ndvduals. Muaon: Creae one new offsprng for he new populaon by randomly muang a randomly chosen par of one seleced ndvdual. Afer he ermnaon creron s sasfed, he sngle bes ndvdual (srng) n he populaon produced durng he run (he bes-so-far ndvdual) s harvesed and desgnaed as he resul of he run. If he run s successful, he resul may be a soluon o he problem.

6 54 Lecure Noes n Managemen Scence Vol. 6: ICAOR 2014, Proceedngs A Genec Algorhm Adaped o IRPB In order o use GAs for solvng a problem, he encodng of canddae soluons s necessary. Referrng o IRPB, a represenaon of a soluon s a wo-dmensonal marx (Abdelmagud and Dessouky, 2006), wh each cell conanng he quany of produc delvered o cusomer durng me perod, each row correspondng o a specfc cusomer and each column correspondng o a me perod. I s worh nong ha delvery amouns are se o be negers. Generally, realers are served f her nvenory level n perod canno cope wh he demand of ha perod. When realers are served a a ceran me perod and quany delvered s no adequae o cover demand, backloggng s assumed. In pracce, backloggng decsons are generally jusfed n wo cases. The frs s when here s a ransporaon cos savng ha s hgher han he ncurred shorage cos by a cusomer. The second case s when here s a holdng cos savng ha s hgher han he ncurred shorage cos by a cusomer. Then care s aken ha backorders are sasfed by he quany shpped n nex perod. An nal populaon s requred. The geness of hs populaon s conduced usng a wo-sep heursc algorhm. In phase one, he heursc procedure examnes whch cusomers are n need of a delvery. The heursc decdes whch cusomers are n mmedae need of a delvery, smply by usng a funcon ha calculaes he nvenory level of cusomer, n perod. In phase wo, a nework of he cusomers who are o be servced s generaed and he ransporaon cos, for supplyng each realer n he nework s mnmsed wh he use of a modfed Djksra algorhm. The amoun of produc o delver o each realer n he nework s q d I 1 B 1. Ths procedure s repeaed for each of he me perods under examnaon. The heursc also makes sure ha none of he consrans of he IRPB model s volaed by he proposed soluon. Genec operaors The selecon of ndvduals o produce successve generaons plays an exremely mporan role n GAs. The basc mehods we have chosen o apply are selecon based on roulee wheel and elsm. Our selecon operaon a frs locaes he ele chromosome ha s he soluon wh he leas oal cos. Ths soluon s auomacally passed on o he new populaon n order o safeguard ha wll no be elmnaed durng he selecon process. Nex, he fness of each ndvdual s compued. As we are dealng wh a mnmzaon problem n ha our goal s o mnmze he sum of he nvenory, backorder and roung coss for a number of perods, s val o map he underlyng naural objecve funcon o a fness funcon. Ths ransformaon s performed usng he mehod proposed by Goldberg (1989), accordng o whch: f ( x) Cmax g( x) when g(x)<c max (11) f ( x) 0 oherwse (12)

7 S Sofanopoulou 55 where g(x) s he cos mnmzaon funcon, f(x) s he fness funcon and C max he larges g value n he curren populaon. The new populaon s hen creaed usng he fness values compued earler and a roulee wheel mechansm. Gven he naure of he problem and he encodng seleced, crossover s he mos nrgung operaor. Crossover n our case can be performed eher horzonally or vercally. Whle horzonal crossover, ha s random exchange of delvery schedules (rows) of ceran cusomers beween he paren soluons, poses no problem, vercal crossover may cause a volaon of he consran concernng he capacy of each realer. Vercal crossover.e. exchange of rows beween paren soluons corresponds o he possbly of havng excessve amouns of produc whch n urn resuls o poor fness -as nvenory holdng cos are oo grea. Consequenly, he laer mehod s no gven any possbly o occur. Followng he crossover operaon, muaon s appled o each offsprng generaed wh a probably equal o he muaon rae. Specfcally he amoun of produc o be delvered s muaed (decreased or ncreased) randomly by a small amoun; and consequenly new soluons are creaed. As wh he prevous operaors, volaons of consrans by he muaed soluons are no allowed. Algorhm Tesng The heursc mehod has been coded and compled n MATLAB (verson R2008b) whle compuaonal esng was performed on a 2.17 GHz Inel Core 2 Duo processor wh 4 GB of RAM. For he evaluaon of he proposed approach we have used he nsances suggesed by Abdelmagud e al. (2009) for he Invenory Roung Problem. In hs laer work nsances for IRP were esed boh for he IRP wh backorders and whou. Those IRP wh backorders nsances were adaped o our IRPB. Abdelmagud e al. (2009) consder hree scenaros n her problem nsances, where he second scenaro s desgned usng such parameers so o provde condons n whch backorder decsons are economcal o be aken. We have herefore chosen o run nsances wh 5 cusomers, 5 me perods and one vehcle wh lmed capacy of 150, snce only one vehcle s consdered n our work. I should be also noed ha no fxed usage cos per vehcle ravellng per perod was consdered n our work. Followng he name convenon adoped by Abdelmagud e al. (2009) he nsances seleced o run are denoed by x-cc--v- where x sands for he scenaro (we have seleced only second scenaro nsances), cc sands for he number of cusomers consdered (fve cusomers nsances were seleced), for he number of me perods of he plannng horzon (5 perods), v for he number of vehcles ulzed and fnally sands for he nsance number. I should be menoned a hs pon ha hese es cases used provde ferle ground for a prelmnary esng of he algorhmc procedure descrbed and furher ess should be performed on larger problems. In Table 1 he resuls for hose nsances are repored when backorders are allowed. Table 1 presens he cos componens of soluons obaned n each case. The opmum (obaned hrough a MPI solver) s lsed n he second column whle TR, H and B are he ransporaon, holdng and backorder coss respecvely for each soluon obaned by our heursc procedure whle he oal cos, calculaed as he sum of all cos componens, s lsed n he las column.

8 56 Lecure Noes n Managemen Scence Vol. 6: ICAOR 2014, Proceedngs Table 1. Resuls for he IRP wh backorders Problem Opmum TR H B Toal Cos , ,78 223,07 740, , ,72 53,22 542, , ,66 53,04 418, , ,95 110,97 513, , ,62 46,06 501,68 Concludng Remarks The purpose of hs work s o address he Invenory Roung Problem when backorder decsons are allowed (IRPB). We presen a smple GA approach for solvng he problem. The paper consss of hree pars. Frsly, IRPB s nroduced, followed by a formal problem saemen where a descrpon of he problem, usng a model of Mxed-Ineger Programmng s gven. Second, some basc heorecal foundaons for Genec Algorhms are presened. The soluon for he IRPB s demonsraed n he hrd par hrough some basc esng usng nsances of he problem presened n he relaed leraure. Furher expermenaon wll make possble for he researchers o deermne he bes combnaon of algorhmc feaures requred o mprove he algorhmc procedure or o ncorporae addonal feaures o he problem examned. References Abdelmagud, T.F. and Dessouky, M.M. (2006). A genec algorhm o he negraed nvenorydsrbuon problem, Inernaonal Journal of Producon Research, 44: Abdelmagud, T.F., Dessouky, M.M. and Ordóñez, F. (2009). Heursc approaches for he nvenoryroung problem wh backloggng, Compuers & Indusral Engneerng, 56: Adulyasak, Y., Cordeau, J.-F., and Jans, R., Formulaons and branch-and-cu algorhms for mul-vehcle producon and nvenory roung problems. Techncal Repor G , GERAD, Monreal. Anly, S. and Bramel, J. (2004). A Probablsc Analyss of a Fxed Paron Polcy for he Invenory- Roung Problem, Naval Research Logscs, 51: Arche, C., Berazz, L., Lapore, G., Speranza, M.G., A branch-and-cu algorhm for a vendor-managed nvenory-roung problem. Transporaon Scence, 41 (3), Bard, J., L. Huang, P. Jalle and M. Dror, A. (1998). Decomposon Approach o he Invenory Roung Problem wh Saelle Facles, Transporaon Scence, 32: Berazz, L, Palea, G. and Speranza, M. (2002). Deermnsc order-up-o level polces n an nvenory roung problem, Transporaon Scence, 36: Bramel, J. and Smch-Lev, D. (1997). The Logc of Logscs. Sprnger: New York. Campbell A and Savelsbergh MWP (2004). A decomposon approach for he nvenory-roung problem, Transporaon Scence, 38:

9 S Sofanopoulou 57 Chen, W., Balakrshnan, A. and Wong, R. (1989). An negraed nvenory allocaon and vehcle roung problem, Transporaon Scence, 23: Coelho, C. L., Cordeau, J-F and Lapore G. (2012). The nvenory-roung problem wh ransshpmen, Compuers & amp; Operaons Research, 39 (1): Coelho, L.C. and Lapore, G., The exac soluon of several classes of nvenory-roung problems. Compuers & Operaons Research, 40 (2), Goldberg D. (1989). Genec Algorhms n Search, Opmzaon, and Machne Learnng, Addson Wesley Professonal. Janxang L, Feng Chu, and Haoxun Chen. (2011). A soluon approach o he nvenory roung problem n a hree-level dsrbuon sysem, European Journal of Operaonal Research, 210: Kleyweg, J., Nor, V. S. and Savelsbergh, M. W. P. (2002). The Sochasc Invenory Roung Problem wh Drec Delveres, Transporaon Scence, 36: Solyalı O and Süral H (2011). A branch-and-cu algorhm usng a srong formulaon and an a pror our based heursc for an nvenory-roung problem. Transporaon Scence 45, Zacharads, E., Taranls, C. and Kranouds, C.T. (2009). An negraed local search mehod for nvenory and roung decsons, Exper Sysems wh Applcaons, 36: