The Can-Order Policy for One-Warehouse N-Retailer Inventory System: A Heuristic Approach

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1 Atcle Te Can-Ode Polcy fo One-Waeouse N-Retale Inventoy ystem: A Heustc Appoac Vaapon Pukcanon, Paveena Caovaltongse, and Naagan Pumcus Depatment of Industal Engneeng, Faculty of Engneeng, Culalongkon Unvesty, Bangkok 133, Taland E-mal: [email protected] Abstact. We study an applcaton of te can-ode polcy n one-aeouse n-etale nventoy systems, and popose a eustc appoac fo settng te appopate nventoy polcy. On te can-ode polcy, an ode s tggeed en a etale s nventoy poston eaces ts must-ode level. Ten ote etales ae examned ete te nventoy eaces te can-ode level, and f so tey ae flled by ts ode as ell. Waeouse fulflls all nvolved etales nventoy to te ode-up-to levels. Te can-ode polcy s not only able to save te total system-de cost fom jont eplensment, but t s also smple to use. Compute smulaton s utlzed to pelmnaly study and to detemne te best-knon soluton. We popose a eustc appoac utlzng te decomposton tecnque, teatve pocedue, and golden secton seac to obtan te satsfyng total system-de cost. Ts can save ou computatonal tme to fnd te appopate nventoy polcy settng fom te educed seac space. We found tat te poposed eustc appoac pefoms vey ell t te aveage cost gap of less tan 2% compang to te best-knon soluton. Tus, te can-ode polcy can be vey useful fo suc systems. Keyods: Can-ode polcy, jont eplensment, one-aeouse n-etale nventoy system, eustc appoac. ENGINEERING JOURNAL Volume 18 Issue 4 Receved 27 eptembe 213 Accepted 25 Mac 214 Publsed 16 Octobe 214 Onlne at ttp://.engj.og/ DOI:1.4186/ej

2 DOI:1.4186/ej Intoducton Ts pape focuses on te one-aeouse n-etale nventoy system (OWNR) c s a geneal patten of to-ecelon supply can. uc system confonts te uncetanty of demand n ealty. upply coodnaton called centalzed contol as been dely appled to educe te total system-de cost and te demand vaaton, as ell as to mpove supply pefomance ncludng nventoy plannng pocess [1-3]. In addton, vaous supply cans gve te attenton nto contnuous eplensment accodng to te esponsve nfomaton tecnology. Ts can not only educe te buffe stocks but also mpove te ente system s pefomance. Hence, e concentate on te nventoy polcy fo contollng suc system unde stocastc demand and contnuous eplensment. A numbe of eseaces on OWNR ave been conducted unde ete contnuous o peodc eplensment. Tey poposed matematcal models and soluton appoaces fo settng an appopate nventoy polcy. Most of te oks studed to majo types of te nventoy polces: Fxed-nteval odeup-to polces and tock-based batc-odeng polces, on dffeent condtons and elevant paametes. Fute detals can be seen n te eves of Axsäte et al. [4], Wang et al. [5], and cnede et al. [6]. Focusng on contnuous eplensment, most eseaces manage multple etales by ndvdual odeng decson. Factually, multple etales can coodnate te odeng decson to sae te odeng cost en an ode s tggeed. It ceates an oppotunty fo educng te total system-de cost. We found tat just a fe oks concened ts cost-savng oppotunty n te odeng decsons. Wt egad to coodnated odeng decson, most lteatues appled jont eplensment poblem (JRP) to OWNR due to te smlaty of cost functons and soluton pocedues [7, 8]. JRP s ognally developed fo te mult-poduct nventoy poblem t te eplensment coodnaton of a goup of tems jontly odeed fom te same supple. Ceung and Lee [9] studed te ( Q), polcy. Wen te cumulatve demands ove all etales eac a gven Q unts (.e. tuckload sze fo all etales n sngle tp), an ode s placed at te aeouse to eplens te etale to te ode-up-to level. At te aeouse, a tadtonal eode pont-fxed ode quantty polcy as employed. Özkaya [1] poposed analytcal models and eustc appoaces fo fou types of jont eplensment polcy at te etales, and utlzed a tadtonal eode pont-based stock polcy at te aeouse. Fou types of jont eplensment polcy ae ( Q), polcy, ( Q,, T ) polcy, ( Q, T ) polcy, and ( s, 1, ) polcy. Te ( Q), polcy of Ceung and Lee [9] and Özkaya [1] as studed on dffeent stuctues. Te fome sets te taget sevce level at te aeouse and te penalty cost at te etales; meanle te latte sets te taget sevce level only at te etales. Te ( Q,, T ) polcy s a ybd contnuous and peodc eplensments. An ode s placed at te aeouse ete en te cumulatve demands ove all etales eac Q unts o en at least one demand aves n T tme unts afte te last odeng nstance. Te ( Q, T ) polcy s a peodc eplensment polcy and te odeng decson ases evey T tme unts. At te decson epoc, f at least Q demands ave accumulated fo te etales snce te last odeng nstance, an ode s placed at te aeouse. Te ( s, 1, ) polcy s a contnuous eplensment polcy en an ode s tggeed and any etale s nventoy poston eaces ts must-ode level s. Ten ote etales n te system ll be also ncluded by ts ode f at least one demand aves to eac etale. All poposed polces commonly ave te etale s ode-up-to level to c te aeouse eplenses all etales nventoes. Özkaya [1] soed compaatve esults among tese polces tout compang to te loe bound o te best-knon soluton. Ten, e tested Özkaya [1] s esults n te case of cossdockng system (no nventoy on-and at te aeouse) n ode to develop a smple loe bound 1 fo compason. Te testng esults soed a vast amount of cost gap beteen te Özkaya [1] s best soluton and te loe bound (156% on aveage). Gou et al. [11] ntoduced a jont eplensment polcy ee te aeouse takes a tadtonal eode pont-based stock polcy and te etales utlze te can-ode ( s, c, ) polcy. Te can-ode polcy as te same mecansm as te ( s, 1, ) polcy except te can-ode level c. Wen an ode s tggeed by a etale, ote etales ose nventoy poston eaces ts can-ode level c ll be ncluded by ts ode 1 Te smple loe bound can be detemned by to steps. Te fst step s to fnd te ode quantty Q fo all etales by assumng tat tey ae eplensed at te same ode nteval. Te second step s to fnd te eode pont s espondng to te taget sevce level. Tus, te total system-de cost can be calculated fom suc to decson vaables. 54 ENGINEERING JOURNAL Volume 18 Issue 4, IN (ttp://.engj.og/)

3 DOI:1.4186/ej as ell. Even toug zeo lead tme as assumed n te study, tey cannot povde an analytcal model due to te complcaton. Tus, compute smulaton as used nstead. Te esult soed tat about 5 to 2% of te cost can be saved as compang t te ndependent contolled polcy at te etales. Pukcanon et al. [12] also confmed te advantage of te can-ode polcy on OWNR by expementng on te boade anges of elevant factos. Te can-ode polcy can save te total system-de cost by ove 3% dependng on elevant factos. Neveteless, bot oks dd not povde a soluton appoac fo settng te appopate nventoy polcy. Tee ae ote eseaces on jont odeng decson conducted on dffeent cost stuctues. Coss-dockng systems ee caed out n Özkaya [8] and Gübüz [13]. Cetnkaya and Lee [14] studed vendo managed nventoy (VMI) system tout te oldng cost at etales. Axsäte and Zang [15] developed a jont odeng polcy by not concenng te saed odeng cost. Accodng to te exstng lteatues, coodnated odeng decson as been consdeed n vaous systems. It s nteestng to study nventoy polcy settng fo bot aeouse and all etales unde typcal cost stuctue contanng te odeng costs and te oldng costs at aeouse and all etales. nce te can-ode polcy not only pefoms ell as found n Gou et al. [11] and Pukcanon et al. [12] but also be stagtfoad and appealng to one s common sense [16]. Tus, n ts pape e focus on te can-ode polcy fo OWNR. Te can-ode polcy as fst ntoduced by Balnfy [17], and ten t as caed out by many eseaces n dffeent ays [18-24]. Te can-ode polcy as ntensvely studed on te mult-tem sngle-locaton nventoy system. Heetofoe fe of pevous eseaces focused on detemnng te appopate nventoy polcy settng fo te can-ode polcy n OWNR. Hence, ts pape s objectve s to popose a eustc appoac to detemne te appopate can-ode polcy n OWNR. We extend te knoledge of te can-ode polcy nto te to-ecelon nventoy system n ts study. Te pape s oganzed nto sx pats. ecton 2 descbes ou poblem t te elevant factos and assumptons. ecton 3 explans oveall metodology used n te eseac. ecton 4 poposes a eustc appoac t pelmnay analyss, concept, matematcal model and algotm. ecton 5 demonstates te expemental esults t analyss and dscusson. ecton 6 concludes all valuable fndngs and poposes deas to extend ts eseac on te can-ode polcy fo OWNR n futue studes. 2. Poblem Descpton Te system conssts of a aeouse and multple etales t sngle commodty. Let n denote te numbe of etales and denote te locaton ee te aeouse s set by = and te etale N, N {1, 2,..., n }. Waeouse s assgned n te fst ecelon called aeouse ecelon, and all etales ae assgned n te second ecelon called etale ecelon. Demands come fom eac etale s customes defned as end customes. A aeouse and multple etales ae coopeated as a sngle fm to concen te total systemde cost unde global nfomaton and centalzed contol. Te aeouse s avalable to old nventoes fo supplyng all etales odes. Inventoes at aeouse ae fulflled by an outsde supple ose ample stock s not consdeed n te poblem. Te aeouse dstbutes all equed tems to te etales n a sngle tp tout splttng lot. It s supposed tat uncapactated vecle s avalable to supply all equed tems n te ode. Multple etales ave te on nventoes to seve te custome demands. Posson demand s assumed to epesent te custome demands, denoted by c s a constant mean of custome demand at te etale. Regadng te can-ode ( s, c, ) polcy appled to ou system, t as to eode ponts: te mustode level s povdng nomal eplensment, and te can-ode level c makng specal eplensment. pecal eplensment s an oppotunty of a etale s jont eplensment en ote etales eac te must-ode levels. Wen te nventoy poston of any etale dops to o belo ts must-ode level s, an ode s tggeed to ceate nomal eplensment. Ten, ote etales n te system can also be ncluded by ts ode f te nventoy poston s at o belo ts can-ode level c ; a specal eplensment s occued. All te nvolved etales nventoes ae fulflled fom te aeouse to te on ode-up-to level. Consdeng sngle commodty, te aeouse modfes te can-ode polcy to a tadtonal ( s, ) polcy by settng ts can-ode level equals ts must-ode level. Te aeouse ssues an ode en ts nventoy poston eaces ts must-ode level s. Ten te outsde supple ll eplens te aeouse s nventoy to ts ode-up-to level. Te aeouse places an ode to te outsde supple f and only f etale ecelon tgges an ode to te aeouse. We dffeentate beteen ode cycle at ENGINEERING JOURNAL Volume 18 Issue 4, IN (ttp://.engj.og/) 55

4 DOI:1.4186/ej etale ecelon and ode cycle at aeouse ecelon by defnng dspatc cycle and eplensment cycle fo etale ecelon and aeouse ecelon, espectvely. Ou system consdes all nventoy costs at bot ecelons. Te nventoy costs ae composed of 1) Te oldng costs at te aeouse and all etales, 2) Te majo odeng costs fo aeouse ecelon and etale ecelon, and 3) Te mno odeng costs fo etale ecelon. Te oldng cost occus at eac locaton avng pyscal stock. Te total oldng cost ove te tme peod at locaton ( HC ) can be detemned fom te unt oldng cost ( ) and te accumulated nventoy ove te tme peod ( INV ). Te majo odeng cost s te fxed cost occung once an ode s tggeed. Ts cost ncludes admnstatve costs, mateal andlng costs, and tanspotaton costs c do not depended on te numbe of etales n te ode. o, te etales n te system can sae te majo odeng cost togete fo eplensng n one ound tp. Te total majo odeng cost ove te tme peod at etale ecelon ( MJ ) s te etales majo odeng cost pe ode ( K ) multpled by te numbe of dspatc cycle ( ND ). mlaly, te total majo odeng cost ove te tme peod at aeouse ecelon ( MJ ) s te multplcaton of te aeouse majo odeng cost pe ode ( K ) and te numbe of eplensment cycle ( NR ). Te mno odeng cost s an addtonal cost of eac etale en eplensng te nventoes, suc as addtonal tanspotaton cost elatng to dstance o ote cages. Ts cost depends on te numbe of nvolved etales n tat ode. Te total mno odeng cost ove te tme peod ( MN ) s accumulated fom te nvolved etales n eac ode multpled by ts mno odeng cost of etale ( ) ove te tme peod. Po oks on coodnated odeng decson gnoed ts addtonal cost n spte of te fact tat ts addtonal cost dectly affects te nventoy polcy settng [23, 24]. Te concept of te can-ode polcy s balancng among educed majo odeng costs, vaed mno odeng costs, and nceased oldng costs. Reduced majo odeng cost occus f a specal eplensment s ncluded n an ode. On te ote and, fom specal eplensment tee s a esdual stock [23] c s a stock left above te must-ode level at te ode-tggeed pont. Ten, te nvolved etales ave to old moe stock nceasng te oldng cost. Meanle, te mno odeng costs can be ete educed o nceased dependng on te ode fequency at eac etale. Hence, e ave to consoldate all elevant costs to detemne te appopate nventoy polcy settng unde te total system-de cost mnmzaton. It s, oeve, dffcult to deal t te poblem manly because of te demand uncetanty, vaaton of etales ode quantty, etale s to-ode pont settng, and ode tme synconzaton at all locatons. In ts pape e smplfy te poblem by assumng zeo lead tme. Retales ode s nstantly dspatced fom te aeouse. All etales must-ode levels ae ten equal to zeo ( s =, N). Te aeouse s ode s also eplensed fom te outsde supple mmedately. In ts case, aeouse s must-ode level s equal to -1 because te aeouse s alloed to old zeo nventoy level untl te next eplensment ll be ssued. Ts can elp te aeouse not to keep te excessve stock atng fo te next dspatc to etale ecelon. Teefoe, decson vaables ae c, and. Te notatons and poblem fomulaton ae demonstated as follos: n = Numbe of etales n te system = Index of locaton ; te aeouse = and te etale N T = Te tme peod consdeed n te poblem (tme unts) s = Te must-ode level at te aeouse (unts); (Assgn s = -1 fom te zeo-lead tme assumpton) = Te ode-up-to level at te aeouse (unts) s = Te must-ode level at etale (unts); (Assgn s = fom te zeo-lead tme assumpton) c = Te can-ode level at etale (unts) = Te ode-up-to level at etale (unts) = Demand ate of etale (unts/tme unt) = Te unt oldng cost pe unt tme at te aeouse ($/unt tme unt) = Te unt oldng cost pe unt tme at etale ($/unt tme unt) K = Te aeouse s majo odeng cost pe a eplensment cycle ($/tme) K = Te etales majo odeng cost pe a dspatc cycle ($/tme) 56 ENGINEERING JOURNAL Volume 18 Issue 4, IN (ttp://.engj.og/)

5 DOI:1.4186/ej = Te mno odeng cost at etale ($) TC( c,, ) = Te total system-de cost pe unt tme ($/tme unt) HC = Te total oldng cost at locaton ove te tme T unts ($) MJ = Te total majo odeng cost at etale ecelon ove te tme T unts ($) MN = Te total mno odeng cost at etale ecelon ove te tme T unts ($) MJ = Te total majo odeng cost at aeouse ecelon ove te tme T unts ($) INV = Te accumulated nventoy ove tme peod at locaton (unt tme unt) ND = Te total numbe of dspatc cycle ove te tme T unts (tmes) NR = Te total numbe of eplensment cycle ove te tme T unts (tmes) j = An ndcato c equals 1 en etale s ncluded n te dspatc cycle j and equals (, ) otese Objectve functon: Mnmze TC( c,, ) n HC MJ MN MJ T (1) ee HC INV (2) MJ K ND (3) ND n MN (4) (, j) j1 1 MJ K NR (5) Te objectve functon of te poblem s to mnmze te total system-de cost pe unt tme. nce s and s can be gven by te zeo-lead tme assumpton, te total system-de cost pe unt tme can be a functon of only tee decson vaables: c,,. Ts enables us to smple manpulate te poblem. Hoeve, te poblem emans te complcatons, suc as demand uncetanty, vaaton of etales ode quantty, and ode-tme synconzaton at all locatons. 3. Metodology Dealng t te complcaton of ou poblem, te optmal soluton cannot be smply deved fom an analytcal appoac. Hence, e ntally study te can-ode polcy on OWNR by usng compute smulaton. Compute smulaton s an effcent appoac epesentng te nventoy pocess even n te complcated system. Te pelmnay study leads us to developng a eustc appoac. In addton, fom te smulaton e can detemne te best-knon soluton used to measue te poposed eustc appoac s pefomance Compute mulaton Te compute algotm epesentng te nventoy pocess s llustated n Fg. 1. Te nputs fo smulatng te system can be dvded nto tee goups as follos: ENGINEERING JOURNAL Volume 18 Issue 4, IN (ttp://.engj.og/) 57

6 DOI:1.4186/ej Input paametes Relevant Factos: n = Numbe of etales λ = Demand ate at etale = Unt oldng cost at te aeouse = Unt oldng cost at etale K = Waeouse s majo odeng cost K = Retales majo odeng cost к = Mno odeng cost at etale Decson vaables: Waeouse s = -1 = [mn,max] Retale s = c = [mn,max] = [mn,max] Expement settng: I () = Intal nventoy level at te aeouse; I () = I () = Intal nventoy level at etale ; I () = T = Tme peod; T = 1, eed numbe = [, 99] Output secton A epot of nventoy costs and tansactons TART et Dspatc cycle j = 1 Replensment cycle = 1 Fo eac etale, Geneate nte-aval tme of demands and sot all demands by aval tme No Monto demand aval of te system Is demand aval tme < T? Yes Fo etale o ons an aved demand, ubtact demand fom nventoy poston I (t) = I (t) - demand Is I (t) s? Yes No Recod dspatc event and set dspatc cycle j = j+1 Dspatc quantty = - I (t), set I (t) =, and set δ (,j) = 1 Compute algotm Fo eac etale k, Is I k(t) c k? Yes Dspatc quantty = k - I k(t), set I k(t) = k, and set δ (k,j) = 1 Fo aeouse, No Collect total dspatc quantty Dspatc quantty = and set δ (k,j) = ubtact total dspatc quantty I (t) = I (t) total dspatc quantty Is I (t) s? No Yes Recod eplensment event and set eplensment cycle = +1 Replensment quantty = I (t) and set I (t) = Calculate nventoy costs Calculate total system-de costs pe unt tme END Fg. 1. Te compute algotm fo smulaton 1) Decson vaables ( c,, ): Eac vaable s nputted as a ange of mnmum and maxmum values. A combnaton of ( c,, ) s called soluton. A soluton povdes a value of te total systemde cost and ts tansacton (e.g. numbe of dspatc cycles, numbe of eplensment cycles). 2) Relevant factos (.e. cost paametes, demand ates, and numbe of etales): We set a combnaton of elevant factos to scenao. A scenao contans dffeent solutons. Te best soluton povdng te mnmum total system-de cost s selected fo eac scenao. 3) Expement settng: Let I () t denote te nventoy level of locaton at tme t. At te begnnng of te unnng peod, all locatons ntal nventoy levels stat at zeo, I () =. We cose 1, unnng peods fo ou smulaton, snce ts unnng peod povdes te steady state fo te system. Addtonally, vaous seed numbes ae tested to vefy te solutons snce dffeent seed numbes geneate dffeent nte-aval tme sets. Fnally, e obtan a epot of te nventoy costs and ts tansacton. In consequence, e can fnd te mnmum total system-de cost fo eac ange of decson vaables nputted unde a gven scenao Te Best oluton Fndng Input paametes Fst of all, e andomly select a seed numbe beteen [, 99] to use fo te fst eplcaton (.e. a eplcaton comes fom a seed numbe). Decson vaables ae nputted as a ange of mnmum and maxmum values. Te ange s dynamc dependng on ou settng. In te expement, e set te dt of ange to be 5 unts fo c and and 2 unts fo. nce ove 5 unts of c and ceates multpled combnatons spendng moe unnng tme. Weeas te ange s lage because lnealy ceates combnatons. Te fst ange can be set fom te ntal pont of and calculated by 58 ENGINEERING JOURNAL Volume 18 Issue 4, IN (ttp://.engj.og/)

7 DOI:1.4186/ej K and 2K due to te zeo-lead tme assumpton and te concept of N economc ode quantty. Fo example, ntal = 45 and ntal = 14, te fst anges ae dentfed as [41, 6], [11, 15], and c [1, 14]. Te next step s te pocess of movng te anges untl te soluton seems to be ose contnuously. Te ange s moved upad and donad by fxng te ange at all etales. Ten, e fnd te c and anges at te etale by keepng te same ange of and te c j and j anges at te etale j., c and anges ae canged epeatedly. We select te best soluton povdng te mnmzed total system-de cost fo te fst eplcaton. Afte tat, te valdaton pocess son n te next pat s utlzed to get te typcal best soluton Output valdaton Te typcal best soluton s a epesentatve of te best solutons fom vaous eplcatons. We defne te typcal best soluton as te best-knon soluton to geneally use n late sectons. nce abundant combnatons ae un n te fst eplcaton, n ts pocess e can educe unnecessay anges by statng at te best soluton s ange fom te fst eplcaton. By ts pocess, e can fnd te best soluton fo ote eplcatons faste. If tee s an eo fom te fst eplcaton, coss-ceckng s occued. In te plot testng (1 scenaos), e tested on ten andom seed numbes to detemne te best soluton fo eac seed numbe. We found tat te best-knon soluton appeaed snce te fst tee andom seed numbes ee conducted. Tus, nstead of a numbe of te expements e could save te computatonal tme on fve andom seed numbes fo detemnng eac seed numbe s best soluton. Consequently, e test anote fou eplcatons on dffeent andom seed numbes (afte te fst eplcaton as been done pevously). Most eplcatons povde te same best soluton; oeve, some dffeent solutons can appea. Ten, fo eac best soluton e detemne te aveage total system-de cost by addtonal 1 andom seed numbes. Te best-knon soluton s povded by te best soluton t te mnmum aveage total system-de cost Pefomance Measuement nce ts pape s objectve s to popose a eustc appoac fo settng te appopate can-ode polcy, te best-knon soluton s utlzed to compae t te eustc s best soluton. Heustc s pefomance s measued n tems of te cost gap calculated fom te follong equaton. ( HRT ) ( B ) ( TC TC ) 1 Cost Gap ( C. G.) (6) ( B ) TC ( HRT ) ( B ) ee TC and TC ae te aveage total system-de cost pe unt tme of te eustc appoac and te aveage total system-de cost pe unt tme of te best-knon soluton, espectvely. 4. Heustc Appoac Ts secton demonstates pelmnay analyss c s te man fndngs fom te compute smulaton leadng us to developng a eustc appoac, as ell as e smplfy te complcated model and popose an algotm to detemne te nventoy polcy settng Pelmnay Analyss In te pelmnay study, te expement as conducted fo 28 scenaos (see Appendx A). Identcal etales ae consdeed n te expement to study te effect of elevant factos. gnfcant fndngs ae demonstated as follos: ENGINEERING JOURNAL Volume 18 Issue 4, IN (ttp://.engj.og/) 59

8 Total ystem-wde Cost pe Unt Tme Total ystem-wde Cost pe Unt Tme DOI:1.4186/ej Te ode-up-to level at te aeouse Fo a gven, e can fnd te soluton of ( c, ) povdng te mnmum aveage total system-de cost as llustated n Fg. 2. Tee ae to local mnmum solutons located n to anges: Range I te soluton occus at = and Range II t occus at >. Fo Range I, stats fom zeo and ten nceases to eac te last value befoe te cost lne tuns to a convex functon. Fo Range II, t s defned afte tat last value to postve nfnty. Te best-knon soluton (global mnmum soluton) defntely occus n ete Range I o Range II Range I Range II Te best-knon soluton Range I Range II Te best-knon soluton Mnmum oluton at Eac (Te value on te ozontal lne s -c - ) Mnmum oluton at Eac (Te value on te ozontal lne s -c - ) Fg. 2. (a) (b) To anges of te best-knon soluton: (a) te best-knon soluton occued n Range I; and (b) te best-knon soluton occued n Range II. Fo Range I, none of oldng stock at te aeouse povdes te loest total system-de cost snce te nceasng ceates te excessve stock. Weneve etale ecelon tgges an ode all excessve stock s consumed and te aeouse s must-ode level s alays eaced. Te aeouse s eplensed evey dspatc cycle; teefoe, t s not necessay to keep stock atng fo te next dspatc cycle. Fo Range II, a tade-off beteen te nceasng oldng costs and te educed odeng costs fo an nceasng s occued as found n te economc ode quantty. We can set = fo a g / ato, snce moe stock ceates moe nventoy cost (.e. te nceased oldng cost s lage tan te educed odeng cost). Hoeve, tee s a possblty tat te best-knon soluton can move fom Range I to Range II en a elevant facto s canged, suc as smalle / ato, ge K, o ge numbe of etales snce suc stuatons affect te aeouse to old nventoes so as to educe te fequency of eplensment Te can-ode level at te etales Fom te exstng lteatues, te ato of te majo odeng cost and te mno odeng cost s one of te most sgnfcant factos fo te can-ode polcy s pefomance, snce suc ato affects te can-ode level c to ceate a combnaton of etales n an ode [23, 24]. Teefoe, e consde te expements n case of zeo mno odeng cost and non-zeo mno odeng cost. Regadng te case of zeo mno odeng cost (154 scenaos), a esult demonstates tat fo 87.66% of all scenaos (135 scenaos) te value c = - 1, ee c and denote te optmal can-ode level and te optmal ode-up-to level of etale. Ts esult s consstent t te study of van Ejs [23] c soed tat en K / ato s appoacng nfnty, ten c = - 1 fo all tems. It mples tat all tems ae jontly eplensed as soon as an tem tgges an ode. Ote tems ae not odeed f tee as been no demand afte te pecedng ode. Ts concept s pupose s to mostly educe te odeng cost fom jontly eplensng all tems n te ode. Fo ote 19 scenaos occung te best soluton at c - 1, te esult ndcates tat TC s geate tan TC.1% on aveage t a standad devaton of ( 1) 6 ENGINEERING JOURNAL Volume 18 Issue 4, IN (ttp://.engj.og/)

9 Aveage Level Gap Aveage Pecentage of Cost Gap DOI:1.4186/ej % ee TC s te optmal aveage total system-de cost and system-de cost of te soluton at c = - 1. In case of non-zeo mno odeng cost (54 scenaos), smalle TC s te mnmum aveage total ( 1) K / ato nfluences a lage dffeence beteen c and as son n Fg. 3(a). nce suc dffeence can educe te numbe of nvolved etales n te ode and dspatc quantty, but ncease dspatc fequency. In te mult-tem sngle locaton poblem, van Ejs [23] uled tat f K / ato s less tan 5, te can-ode polcy mgt not appen to be Compang TC c = and ( 1) - 1. Addtonally, g demand ate affects a ge level gap beteen TC, te esult ndcates tat TC s geate tan ( 1) c and. TC by.91% on aveage t a standad devaton of 1.85%. malle K / ato nceases cost gap as son n Fg. 3(b). ettng c nea nceases te total odeng cost because of too many etales ncluded n an ode (a) (b) Fg. 3. Te effect of K / ato on te can-ode level at te etales: (a) Aveage level gap beteen and K / ato Demand ate, λ 2, and (b) Aveage pecentage of cost gap beteen TC Te ode-up-to level at te etales 5 ( 1) and TC. Wen e fx te nventoy polcy at te aeouse, te aveage total system-de cost at etale s a convex (unmodal) functon of as son n Fg. 4. Fgue 4(a) and Fgue 4(b) llustate dffeent scenaos but povde te same patten. Te convex functon occus fom a tade-off beteen te nceasng oldng costs and te educed odeng costs fo an nceasng, ten te economc ode quantty s detemned. 1.8% 1.6% 1.4% 1.2% 1.%.8%.6%.4%.2%.% K / ato c ENGINEERING JOURNAL Volume 18 Issue 4, IN (ttp://.engj.og/) 61

10 Aveage Total ystem-wde Cost pe Unt Tme Aveage Total ystem-wde Cost pe Unt Tme DOI:1.4186/ej Fg (a) Convex functon of on gven : (a) cenao at = 2, n = 2, = 48; and (b) cenao at n = 4, = 27. K = 1, K = 1, (b) K = 5, =, K = 5, = 25, = 2, = 2.5, = 25, = 1, = 5, As te above esults, e can smplfy te matematcal model by usng te can-ode level c = - 1 snce small aveage cost gap beteen TC and TC s occued. Addtonally, a convex functon of ( 1) enables us to develop a eustc appoac at ease t one-dmensonal seac Matematcal Model level Ou pupose of developng a eustc appoac s to povde an appopate nventoy polcy ( c,, ). Te total system-de cost of matematcal model s able to be appoxmated as long as te acceptable soluton s povded. Ts can educe te complexty of ou model. Hence, elatng to te pelmnay analyss ou matematcal model utlzes te can-ode level at c = - 1. Ten, tee exsts only to decson vaables (, ) concened n te matematcal model. Van Ejs [23] developed exact equatons by usng c = - 1 fo non-dentcal tems on sngle locaton. Hs model used te exact pobablty of te specal eplensment, unlke ote models assumng Posson dstbutons. It pefomed vey ell en te K / ato s moe tan 5. Teefoe, e adapt s ok nto ou consdeaton. Based on van Ejs [23], e can calculate te nventoy cost at te etale ecelon close to te exact value. Hoeve, detemnaton of nventoy cost at aeouse s anote dffcult pat. Te aeouse s nventoy level s consumed by an uncetan lot-szng ode fom etale ecelon. Fom pelmnay testng, e detemne te expected dspatc quantty at etale ecelon by usng te exact model of van Ejs [23]. We found tat te expected dspatc quantty pe dspatc cycle s alays equal to te cumulatve demand fom all etales. Tus, e smplfy ts pat by assumng tat te aeouse s nventoy level s consumed contnuously follong te total Posson demand cumulated fom all etales,. By ts assumpton, aeouse ecelon and etale ecelon ae ndependent to fnd te mnmum nventoy costs at eac ecelon. Even toug te assumpton povdes te appoxmate aeouse s nventoy cost ge tan te aeouse s actual nventoy cost, e compensate te appoxmate value by utlzng te mnmum nventoy cost at etale ecelon. Te cost model can be fomulated fo a gven (, ) polcy. It follos tat, level N K 1 ( ) ) E[ H ] N K E[ H] TC(, ) E[ DT ] E[ RT ] (7) TC(, ) = Te long-un aveage total system-de cost pe unt tme ($/tme unt) ( ) = Te pobablty tat no demand aves fo etale dung a dspatc cycle EH [ ] = Te expected oldng cost of etale dung a dspatc cycle ($) 62 ENGINEERING JOURNAL Volume 18 Issue 4, IN (ttp://.engj.og/)

11 DOI:1.4186/ej EH [ ] = Te expected oldng cost of te aeouse dung a eplensment cycle ($) E[ DT ] = Te expected lengt of a dspatc cycle (tme unt) E[ RT ] = Te expected lengt of a eplensment cycle (tme unt) Accodng to te equaton (7), e consde te pobablty tat at least one demand aves fo etale dung a dspatc cycle to be consstent t te value c = - 1. uc pobablty affects te occuence of te mno odeng cost. Retale Ecelon Te model s developed accodng to te ndependent Posson pocess of demands fo ndvdual etales, so nte-aval tmes of demands ae exponentally dstbuted. uppose a dspatc cycle stats at tme. We defne te follong vaables accodng to stocastc pocess: DT = Tme untl etale tgges an ode to te aeouse (tme unt) DT = Tme untl any etale tgges an ode to te aeouse; DT mn( DT ) (tme unt) f () t = Pobablty densty functon of DT F () t = Dstbuton functon of DT f() t = Pobablty densty functon of DT Ft () = Dstbuton functon of DT Retale ll tgge an ode f te total demand fo etale fom tme equals. Tus, accodng to te exponental dstbuton of nte-aval tmes of demands, DT follos Elang dstbuton t paametes and. Te value of f () t and F () t ae detemned by te geneal fomula of Elang dstbuton [25]. Ten, te pobablty densty functon and dstbuton functon of DT can be calculated by f ( t) f ( t) 1 F ( t ) (8) N Tus, te expected lengt of a dspatc cycle s j F( t) 1 1 F ( t ) (9) N E[ DT ] tf ( t) dt 1 F( t) dt 1 F ( t) dt (1) t t t N Te expected oldng cost of etale dung a dspatc cycle s assocated t te etale s nventoy on and at te begnnng and at te end of te dspatc cycle. At te begnnng of te cycle, settng c = - 1 makes all etales nventoy on and equal. At te end of te cycle, te nventoy on and depends on te esdual stock level, c s a stock above te must-ode level en an ode s tggeed. Tus, e defne ( x ) as te pobablty tat at tme DT te esdual stock of etale equals x. Tee ae to cases fo detemnng ( x ). Te fst case s en te esdual stock level of etale s equal to zeo; only etale tgges an ode. Te second case s en te esdual stock level of etale s postve. o, an ode s tggeed by etale j. Tus, te value of ( x) can be calculated by te follong expessons: ENGINEERING JOURNAL Volume 18 Issue 4, IN (ttp://.engj.og/) 63

12 DOI:1.4186/ej j t j ( x) ( ) Pos(, x) f ( t) dt f x t f ( t) 1 F ( t) dt f x, m e Pos(, m) m! (11) (12) ( ) f ( t) f j( t) 1 Fk( t ) (13) j k j, ( ee Pos(, m) s te pobablty densty functon of Posson demand at etale, and f ) () t s te pobablty densty functon tat at tme t any etale j tgges an ode. Tus, ( ) llustated n equaton (7) can be calculated by usng equaton (11) as ell. Te expected oldng cost of etale dung a dspatc cycle s ten gven by ( ) x t E[ H ] ( x) f ( t) dt (14) x 2 t Accodng to te equaton (1) and (14), e tansfom te expesson to detemne te expected oldng cost of etale pe unt tme nstead. Tus, E[ H ] ( ) x ( x) E[ DT ] x 2 (15) Waeouse Ecelon To smplfy ts pat, e assume tat te aeouse s nventoy level s consumed contnuously by all etales Posson demands t ate. Inte-aval tmes of demands ae exponentally dstbuted, and ten te dstbuton of tme untl aeouse tgges an ode to an outsde supple s Elang, smla to te etale ecelon. Let RT denote te tme untl aeouse tgges an ode to an outsde supple. Te aeouse ll tgge an ode f te total demand fom tme equals, so te dstbuton of RT s Elang t paametes and. Te expected lengt of a eplensment cycle s te mean of Elang dstbuton. Tus, E[ RT ]. In case of oldng nventoy at te aeouse, te expected oldng cost of te aeouse dung a eplensment cycle s estmated follong te contnuous demands fom te etale ecelon. Ten, e can E[ H] detemne te expected oldng cost of te aeouse pe unt tme by. Accodng to te E[ RT ] 2 fom te can be easly calculated fom EOQ expesson at te aeouse, e can fnd te optmal ode-up-to level at te aeouse devatve of te cost functon t espect to. We found tat fomula. Ten, 2K Consequently, e can fgue out te long-un aveage total system-de cost pe unt tme fo a gven (, ) polcy. Ten, te next secton ll demonstate te algotm of eustc appoac to detemne te appopate decson vaables by usng te cost model Te Algotm of Heustc Appoac Wt egad to te pelmnay analyss and te matematcal model, te follong analyses demonstate ou concept fo developng te eustc appoac. 64 ENGINEERING JOURNAL Volume 18 Issue 4, IN (ttp://.engj.og/)

13 DOI:1.4186/ej ) Accodng to to local mnmum solutons located nto to anges, e can dentfy te value of to = fo Range I and 2K fo Range II. 2) To develop an ntal soluton at etale ecelon by assumng c = - 1, e can use detemnstc model to fnd economcal jont odeng tme en evey etales s eplensed n an ode. 3) Fxng nventoy polcy at etale j and at te aeouse, te total nventoy cost at etale s a convex functon of. We can fnd te local mnmum TC(, ) at te gven j and. Teefoe, te decomposton tecnque and teatve pocedue can be appled to beak multple locatons nto a sngle locaton and to ecuently fnd te mnmum soluton as fa as te best soluton as been found. Bot tecnques ave been ntensvely used n JRP [19, 22-24, 26-28]. 4) nce te total nventoy cost at etale s a unmodal functon unde one-dmensonal unconstaned poblem. We apply te lne seac called golden secton seac c s a smple and effcent metod fo fndng te extemum of a unmodal functon [23, 29, 3]. Te golden secton seac s sutable fo te case of non-devatve functon, lke ou model, by successvely naong te ange of seac space untl te desed accuacy n te mnmum value of te objectve functon s aceved. A golden ato, c s a constant educton facto fo te sze of te nteval, s utlzed to mantan te successve ange of dynamc tples of ponts (.e. uppe pont, mddle pont, and loe pont). Advantageously, eac successve ange e only ant to pefom one ne functon evaluaton. Fom ts tecnque, e can detemne te optmal fo te gven j and and save computatonal tme. Te golden secton seac as vefed to effcently use t te can-ode polcy n van Ejs [23] s ok. Hence, te eustc appoac s outlned n te follong algotm llustated n Fg. 5. In step 1 detemnaton of te ntal soluton, e calculate te jont dspatcng tme ( T d ) by detemnstc model accodng to te follong expesson: T d 2( K ) N N (16) Ten, te ntal fo etale s detemned by adaptng Love [16] s metod. It s selectng c povdes te mnmum gap beteen to pobabltes: 1) te pobablty tat te demand fo etale dung tme T d s less tan o equal to suc and 2) te pobablty tat an ode s tggeed by any etale (.e. ncludng nomal eplensment and specal eplensment). Tus, n n m f Pos( Td, m 1) Pos( Td, m) n1 n1 1 m Otese (17) Te ntal fom equaton (17) s close to te optmal soluton tan obtaned fom T. d ENGINEERING JOURNAL Volume 18 Issue 4, IN (ttp://.engj.og/) 65

14 DOI:1.4186/ej TART tep 1: Detemne ntal oluton at etale ecelon tep 2: Detemne te local optmal soluton fo eac ange R k: k = {1,2} tep 3: elect te best soluton at TC mn(, )= mn{tc mnr1(, ), TC mnr2(, )} END Output: Output: TC mnrk(, ) tep 1 tep 1.1: Calculate jont dspatcng tme (T d) tep 2 tep 2.1: (A) et values at te aeouse fo ange R k: k = {1,2} Fo ange R 1, set = and E[RT] = E[DT] (B) Calculate TC ntal(, ) tep 1.2: Fnd ntal fom T d by usng Posson pobablty functon Fo ange R 2, set = 2K λ / and E[RT] = / λ tep 2.2: Iteatve pocedue fo detemnng te local optmal soluton ( ) fo eac ange R k (A) et ntal value: - et loop y =, teaton m =, and assgn TC mnrk(, ) = TC ntal(, ) fo suc ntal value - Assgn etale = (B) et etale = +1, fx j and (C) Use golden secton seac fo detemnng te optmal unde gven j and (D) Update TC mnrk(, ) and f te bette soluton as been found (E) Count teaton m = m +1, go back to step (B) untl = n If = n, count loop y = y +1 (F) top f - fo ={1,...,n} does not cange n teatons n a o, o - TC mnrk(, ) of loop y and TC mnrk(, ) of loop y - 1 does not decease by moe tan ε% Otese go back to step (B) Fg. 5. Te algotm of eustc appoac. tep 2 s te most mpotant pocedue fo te eustc n ode to detemne te optmal fo eac ange of ange R 1 and R 2 (note tat fo ange R 1, te local optmal soluton occus at = and E[ RT ] E[ DT ], and fo ange R 2, t occus at 2K and E[ RT ] ). We use and te ntal fom step 1 to calculate te ntal long-un aveage total system-de cost pe unt tme, TC (, ). Te next step (2.2) s an teatve pocedue contanng steps (A) to (F). Fo eac teaton, ntal a golden secton seac s caed out fo etale : vay and fx j gven fom te pevous teaton. TC(, ) s an objectve functon fo golden secton seac. Te teatve pocess temnates as soon as evey does not cange n teatons n a o, o te mnmum long-un aveage total system-de cost pe unt tme, TCmn Rk (, ), fom te cuent loop does not decease fom te pevous loop by moe tan % (.e. en all etales ave been un, one loop s counted). Fom step 2, e obtan te local mnmum cost TCmn Rk (, ) fo k {1, 2}. Lastly, te compason of TCmn Rk (, ) fo k {1, 2} s caed out n step 3. Te mnmum long-un mn TC (, ), TC (, ). aveage total system-de cost pe unt tme s equal to 5. Expemental Results mn R1 mn R2 Te eustc appoac as been tested on vaous scenaos. Te expements on dentcal etales ae analyzed, specfcally n te case of zeo mno odeng cost and non-zeo mno odeng cost. nce bot cases affects te can-ode polcy at gven c = - 1 on dffeent esults as son n te pelmnay analyss. In addton, te expement on non-dentcal etales s also conducted to measue te eustc s pefomance on te dssmla stuaton. 66 ENGINEERING JOURNAL Volume 18 Issue 4, IN (ttp://.engj.og/)

15 DOI:1.4186/ej Identcal Retales Zeo mno odeng cost Accodng to tee elevant factos (.e. cost paametes, demand ates, and numbe of etales), tey ae desgned to examne te eustc s pefomance unde 154 scenaos (see Appendx A). Table 1 sos some numecal examples elatng to te best-knon soluton of te system and te best soluton fom te eustc appoac. We found tat te pefomance of eustc appoac depends on all elevant factos. It povdes an aveage cost gap at 1.5% t standad devaton of 1.11% ove vaous scenaos. Ou appoac pefoms ell fo g numbe of etales, g K / K ato, and g / ato. Let ( B ) and ( B ) denote te best-knon ode-up-to level at etale and at te aeouse ( ) detemned fom te compute smulaton. Let HRT ( HRT ) and denote te best ode-up-to level at etale and at te aeouse and tey ae calculated by te eustc appoac. Teoetcally, a lage numbe of etales nceases te jont eplensment oppotunty fom specal eplensment, ts can ( ) educe. Tus, a ge numbe of etales educes B ( HRT ) ( B ) to be close to and also nceases ( HRT ) to be close to. Teefoe, te cost gap can educe. Fo ge K / K ato, and ae affected n a smla patten. Regadng te / ato, ge ato nfluences te aeouse s stock equal to zeo. Consequently, te nventoy cost at etale ecelon becomes te man pat of te system. Ou matematcal model povdes cost expesson at etale ecelon nea te exact value and eustc appoac can detemne te mnmum soluton at etale ecelon. Ten, te eustc appoac povdes te (nea) best-knon soluton. Table 3. Numecal examples fo compason of te best-kno soluton and te eustc s best soluton unde dentcal etales tout mno odeng cost. Relevant factos Best-knon oluton (B) Heustc Appoac Instance K K n,, ( B ) c TC,, c CG ,3,4 1, ,3,4.52% ,5,6 1,42.94,5,6.% ,9, ,1,11 2.3% ,12, ,18, % ,4, ,4,5.5% ,15, ,14, % ,12, ,1,11 2.5% ,11, ,1,11.98% ,6, ,5,6.36% ,4, ,4,5.11% ,4, ,4,5.15% ,8, ,8,9 1.3% ,6, ,6,7.29% ,5,6 1, ,5,6.17% Non-zeo mno odeng cost Altoug te can-ode level s not necessay to be equal to - 1 en tee s a mno odeng cost, ou eustc appoac can be appled nto ts poblem n some stuatons. To dentfy suc a stuaton, e tested on 54 scenaos (see Appendx A) by manly vayng te mno odeng cost. Te value of K / ato ae dentfed follong van Ejs s ok [23]. Te expemental esults ae depcted n Fg. 6. ENGINEERING JOURNAL Volume 18 Issue 4, IN (ttp://.engj.og/) 67

16 Aveage Pecentage of Cost Gap Aveage Pecentage of Cost Gap DOI:1.4186/ej % 2.5% 2.% 1.5% 1.%.5%.% K / ato (a) (b) Fg. 6. Te effect of K / ato on te can-ode level at te etales: (a) Heustc s pefomance and smulaton s pefomance en fxng c = - 1, and (b) Te effect of / ato. We found tat te eustc appoac povdes an aveage cost gap at 1.64% t standad devaton of 2.3% ove vaous scenaos. Heustc s pefomance s assocated t to easons. Fstly, ou eustc assumes c = - 1. As son n Fg. 6(a). Relatng to compute smulaton, e compae te best-knon soluton t te best soluton fxng c = - 1. Aveage pecentage of cost gap povdes n smulaton s lne. A smalle K / ato povdes a lage cost gap n smulaton s lne, consequently ou eustc also pefoms n te same ay. econdly, te nventoy cost at te aeouse s appoxmate. Cost gap of te eustc s lne s also added fom te smulaton s lne. Consdeng te / ato, a ge ato ( / s.4 and.6) nfluences te aeouse s stock equal to zeo. Ten, te eustc appoac povdes te (nea) best-knon soluton. On te ote and, a ge cost gap at te loe / ato comes fom an appoxmate nventoy cost at te aeouse, especally fo a small demand ate and g numbe of etales by te eason tat ou eustc gets = eeas te best-knon soluton s >. Te dffeence of soluton ceates a lage cost gap Non-Identcal Retales mulaton Heutstc To extend te expement on non-dentcal etales, e am at studyng te can-ode polcy on te etales dffeent demand ates because n ealty e fequently encounte suc stuaton. In addton, non-dentcal demands can ceate te dffeent dscount oppotuntes fom te saed odeng cost. Hence, t s nteestng to nvestgate and ts nquy as not been studed n te exstng lteatues. We tested on toetale scenaos and tee-etale scenaos (see Appendx B). Fgue 7 depcts te cost gap fom ou eustc appoac, as compaed to te best-knon solutons. Te eustc appoac povdes an aveage cost gap at 2.18% t standad devaton of.82% fo to-etale scenaos, and an aveage cost gap at 1.8% t standad devaton of.51% fo tee-etale scenaos. At small demand ate ato te eustc appoac pefoms ell because ode cycle of eac etale s not qute dffeent. o, te etales odeng cost can be moe saed t te balancng oldng costs. Hoeve, at a ge demand ate ato eustc s pefomance does not depend on te dffeent demand ates (.e. tee s no tend of te cost gap follong te demand ate ato). 5.% 4.5% 4.% 3.5% 3.% 2.5% 2.% 1.5% 1.%.5%.% K / ato Te / ato ENGINEERING JOURNAL Volume 18 Issue 4, IN (ttp://.engj.og/)

17 Aveage Pecentage of Cost Gap Aveage Pecentage of Cost Gap DOI:1.4186/ej % 2.6% 2.4% 2.2% 2.% 1.8% 1.6% 1.4% 1.2% 1.% Aveage cost eo at 2.18% t standad devaton at.82% Demand Rate Rato 2.5% 2.% 1.5% 1.%.5%.% Aveage cost eo at 1.8% t standad devaton at.51% Demand Rate Rato Fg. 7. (a) (b) Heustc s pefomance unde non-dentcal etales: (a) To-etale scenaos, and (b) Tee-etale scenaos. As te expemental esults n vaous scenaos, te eustc appoac povdes te best solutons at a small aveage cost gap compang to te best-knon soluton. Moeove, te eustc appoac s computatonal tme can be saved fom te educed seac space as compang to te compute smulaton s computatonal tme. nce te golden secton seac can save te computatonal tme by educng te seacng ponts. It s a satsfactoy appoac to use fo te can-ode polcy settng unde OWNR. 6. Conclusons Ts pape poposed a eustc appoac fo detemnng on appopate can-ode polcy nto oneaeouse n-etale nventoy system. Dealng t te complcaton of ou poblem, compute smulaton as employed to exploe nsgts nto te can-ode polcy and to detemne te best-knon soluton. Te nsgts led us to developng te matematcal model and te algotm of te eustc appoac. Te man fndngs soed fom compute smulaton tat te aveage total system-de cost s a unmodal functon of te etale s ode-up-to level, en gven j and ae fxed. Decomposton tecnque and teatve pocedue can be appled to beak multple locatons nto a sngle locaton and to successvely fnd te mnmum as fa as te best soluton as been found. nce ou matematcal model s a non-devatve functon, e utlzed golden secton seac fo fndng te mnmum of a unmodal functon. Ts can save ou computatonal tme to fnd te appopate nventoy polcy settng. Te eustc appoac unde smplfed matematcal model and fxed c 1 pefoms vey ell, especally n case of g K / ato. Oveall, te expements tested on te de ange of data povded te cost gap of eustc appoac of less tan 2% on aveage. Wt satsfactoy computatonal tme and small cost gap, te eustc appoac s ell ot usng fo te can-ode polcy settng unde te oneaeouse n-etale nventoy system. In ts eseac, to man contbutons ee ganed. Fstly, te zeo lead tme assumpton can be ntepeted and appled n te stuaton en te ato of lead tme to ode cycle duaton s vey small. econdly, ou study can be used as te base case en e extend nto non-zeo lead tme. Futemoe, multple tems sould be concened to totally utlze te can-ode polcy on te aeouse. ENGINEERING JOURNAL Volume 18 Issue 4, IN (ttp://.engj.og/) 69

18 DOI:1.4186/ej Appendx A: Pelmnay Expement In te pelmnay study, te expement s desgned on te dentcal etales. Te follong table sos 28 scenaos expemented n sequence. Te astesk () n te table means tat paamete s vaed. Table A1. Numecal nput fo pelmnay expement unde te dentcal etales Fxed Paametes cenao No. K K n Vaed Paametes 1) Relatonsp beteen and (8 scenaos) {1, 25, 5, 1, 25}; {.1,.2,...,1} {.1,.5, 1, 2.5, 5}; {.1,.3,.5,.7,.9, 1} 2) Relatonsp beteen, and K (2 scenaos) K {1, 9}; {.1,.3,.5,.7,.9, 1} K {1, 9}; {.2,.4,.6,.8} 3) Relatonsp beteen, and K (2 scenaos) K {75, 2}; {.1,.3,.5,.7,.9, 1} K {125, 25}; {.2,.4,.6,.8} 4) Relatonsp beteen,, and K / K (14 scenaos) K / K {1.5, 3, 4, 5, 1, 1, 15}; {1, 25} 5) Relatonsp beteen,, and (1 scenaos) {.5, 1, 3, 5, 1, 4, 1, 5} {.5, 1} 6) Relatonsp beteen,, and n (1 scenaos) n {4, 8, 12, 2} {.5, 1}; n {4, 8, 12} 7) Te effect of (54 scenaos) {5, 1, 25}; {.5, 2}; {.2,.4,.6}; n{2, 4, 8} 7 ENGINEERING JOURNAL Volume 18 Issue 4, IN (ttp://.engj.og/)

19 cenao No. cenao No. cenao No. DOI:1.4186/ej Appendx B: Te Expement on Non-Identcal Retales Te follong table sos 45 scenaos on to-etale poblem and tee-etale poblem. All scenaos set dentcal cost components by K = 1, K = 5, =, = 2, and = 1. Table B1. Numecal nput fo te expement on non-dentcal etales: Demand Rate Demand Rate Demand Rate Demand Rate Rato Demand Rate Rato Demand Rate Rato , , 2, , , 2, , 1, , 1, 4 Acknoledgement Te colasp fom te Gaduate cool, Culalongkon Unvesty to commemoate te 72nd annvesay of s Majesty Kng Bumbala Aduladeja s gatefully acknoledged. Te autos ould lke tank te anonymous assocate edto, and efeees fo te elpful and valuable comments. A poton of te ok as pesented at 17 t Intenatonal Confeence on Industal Engneeng Teoy, Applcatons and Pactce (IJIE 213). Refeences [1] B. D. Wllams and T. Toka, A eve of nventoy management eseac n majo logstcs jounals: Temes and futue dectons, te Intenatonal Jounal of Logstcs Management, vol. 19, no. 2, pp , 28. [2] P. Kelle, H. cnede,. Wley-Patton, and J. Woosley, Healtcae supply can management, n Inventoy Management: Non-classcal ves. M. Y. Jabe, Ed., Baco Raton: CRC Pess, 29, pp [3] K. Asnde, A. Kanda, and. G. Desmuk, A Reve on supply can coodnaton: Coodnaton mecansms, managng uncetanty and eseac dectons, upply Can Coodnaton unde Uncetanty. Beln Hedelbeg: pnge, 211, pp [4]. Axsäte,. C. Gaves, and A. G. d. Kok, upply can opeatons: eal and dstbuton nventoy systems, Handbooks n Opeatons Reseac and Management cence, Elseve, 23, pp [5] Q. Wang, T.-M. Co, and T. C. E. Ceng, Contol polces fo mult-ecelon nventoy systems t stocastc demand, n upply Can Coodnaton unde Uncetanty. Beln Hedelbeg: pnge, 211, pp [6] H. cnede, D. B. Rnks, and P. Kelle, Poe appoxmatons fo a to-ecelon nventoy system usng sevce levels, Poducton and Opeatons Management, vol. 4, no. 4, pp , ENGINEERING JOURNAL Volume 18 Issue 4, IN (ttp://.engj.og/) 71

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