A New replenishment Policy in a Two-echelon Inventory System with Stochastic Demand

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1 A ew eplenshment Polcy n a wo-echelon Inventoy System wth Stochastc Demand Rasoul Haj, Mohammadal Payesh eghab 2, Amand Babol 3,2 Industal Engneeng Dept, Shaf Unvesty of echnology, ehan, Ian ([email protected], [email protected], el: Lab. PRISMa, ISA de Lyon, 9 avenue Jean Capelle, 6962Vlleubanne Cedex, Fance ABSRAC In ths pape we consde a two-echelon nventoy system consstng of one cental waehouse and a numbe of nondentcal etales. he waehouse uses a egula one-fo-one polcy to eplensh ts nventoy, but the etales apply a new polcy that s each etale odes one unt to cental waehouse n a pe-detemned tme nteval. he most advantage of ths polcy s that the etales odes, whch consttute waehouse demand, ae detemnstc. Fo ths system we show how the nventoy costs can be evaluated. By the numecal examples we compae ou polcy wth onefo-one polcy. Keywods: nventoy contol, eplenshment polcy, two-echelon, stochastc demand - IRODUCIO We consde a two-echelon nventoy system consstng of one cental waehouse and a numbe of non-dentcal etales. We assume the etales face ndependent Posson demand and unsatsfed demand wll be lost. he tanspotaton tme fo an ode to ave at a etale fom the waehouse s assumed to be constant. he waehouse odes to an extenal supple. he lead tme fo an ode to ave at the waehouse s assumed to be constant. he cental waehouse uses a egula one-fo-one polcy, but the etales apply a new polcy whch s dffeent fom the nventoy polces used n the lteatue of nventoy and poducton contol systems. In ths system, each etale odes a demand to cental waehouse n a pe-detemned tme nteval. In othe wod, the tme nteval between any two consecutve odes of each etale s fxed and the quantty of each ode s one. he most advantage of ths polcy s that the etales odes, whch consttute waehouse demand, ae detemnstc. he detemnstc demand fo the waehouse leads to nventoy contol n waehouse s smplfed fo nstance, safety stock n waehouse s elmnated. We evaluate the total nventoy system cost n steady state consstng of holdng and shotage costs of etales and holdng cost of the waehouse. he pupose of ths study s to detemne the optmal tme nteval between /6/$2. 26 IEEE two odes of each etale whch mnmzes the total nventoy system cost. By the numecal examples we compae ou polcy wth one-fo-one polcy. he numecal esults ndcate that thee ae some cases whch make savngs by usng ou polcy nstead of a one-fo-one polcy. hs pape s oganzed as follow. Secton 2 we povde a lteatue evew of nventoy contol polcy fo the tow-echelon nventoy system. Secton 3 pesents cost evaluaton n the steady stats. Secton 4 gves some numecal examples and secton 5 pesents the conclusons and suggestons fo futhe eseach n ths aea. 2- IVEORY COROL POLICY FOR HE WO-ECHELO IVEORY SYSEM wo man polces appled n two-echelon nventoy models ae contnues evew and peodc evew. Most eseaches consde contnues evew polcy. Gaves [] consdes a mult-echelon nventoy model fo a epaable tem wth one-fo-one polcy. He pesents an exact model fo fndng the steady state dstbuton of the net nventoy level. Axsäte [2] nvestgates a twoechelon nventoy system n whch the nventoy polcy of each echelon s (, Q. Axsäte et al. [3] geneate an appoxmate method fo nventoy costs n a twoechelon nventoy system wth compound Posson demand, n whch the nventoy polcy of each echelon s ode up to S. Fosbeg [4] consdes an exact evaluaton of (, Q polces fo two-level nventoy systems wth Posson demand. Axsäte and Zhang [5] consde a two-level nventoy system wth a cental 246

2 waehouse and a numbe of dentcal etales. he waehouse uses a egula nstallaton stock batchodeng polcy, but the etales apply a dffeent type of polcy. When the sum of the etales nventoy postons declnes to a cetan jont eode pont, the etale wth the lowest nventoy poston places a batch quantty ode. Andesson and Melchos [6] popose an appoxmate method to evaluate nventoy costs n a two-echelon nventoy system wth one waehouse and multple etales. All nstallatons use (S-, S polcy. Seo et al. [7] develop an optmal eode polcy fo a two-echelon dstbuton system wth one cental waehouse and multple etales. Each faclty uses contnuous evew batch odeng polcy. Axsäte [8] n a two-echelon dstbuton nventoy system pesents a smple technque fo appoxmate optmzaton of the eode ponts. he system s contolled by contnuous evew nstallaton stock (, Q polces. Sefbagh and Akba [9] nvestgate the nventoy system consstng of one cental waehouse and many dentcal etales contolled by contnuous evew polcy (, Q. he demands of etales ae ndependent Posson and stockouts n etales ae lost sales. hee ae a few models n two-echelon nventoy whch consde peodc polcy. Axsäte [] nvestgates a two-echelon nventoy system appled ode-up-to-s polcy wth peodc evew. Matta and Snha [] nvestgate a twoechelon nventoy system conssts of a cental waehouse and a numbe of etales. Each etale apples (, S nventoy polcy wth an dentcal evew nteval and dffeent maxmum nventoy level S. the cental waehouse apples (, s, S polcy, whee s the same evew nteval as that of etales; s s ts eode ponts, and S s ts desed maxmum nventoy level. Kanchanasunton and echantsawad [2] nvestgate a peodc nventoy-dstbuton model base on Matta and Snha model [] to deal wth the case of fxed-lfe peshable poduct and lost sales at etales. 3- COS EVALUAIO We consde a two-echelon nventoy system consstng of one cental waehouse and a numbe of non-dentcal etales. he waehouse uses a egula one-fo-one polcy to eplensh ts nventoy, but the etales apply a new polcy that s each etale odes one unt to cental waehouse n a pe-detemned tme nteval. We evaluate the total nventoy system cost n steady state consstng of holdng and shotage costs of etales and holdng cost of the waehouse. In secton 4., we nvestgate fomulaton of etale costs. In secton 4.2 we study waehouse cost accodngly etale odes. Fnally, n secton 4.3, we demonstate total nventoy system cost. Assumptons: - he etales face ndependent Posson demand. 2- Unsatsfed demand n etales wll be lost. 3- he tanspotaton tme fo an ode to ave at a etale fom the waehouse s constant. 4- he waehouse odes to an extenal supple wth nfnte capacty. 5- he lead tme fo an ode to ave at the waehouse s constant. 6- At waehouse and etales, the eplenshment cost s assumed to be zeo o neglgble. otaton: : numbe of etales. : demand ntensty at etale,,2,,. s : penalty cost fo a lost sale at etale,,2,,. h : holdng cost ate at etale,,2,,. h w : holdng cost ate at cental waehouse. : tme nteval between any two consecutve odes of etale,,2,,. S : ode-up-to level at waehouse, I : nventoy level at etale n steady state,,2,,. H : aveage of nventoy level at etale n steady state,,2,,. Ch : holdng cost at etale n steady state,,2,,. CS: shotage cost at etale n steady state,,2,,. C ( : total cost at etale n steady state, (C ( Ch + CS,, 2,,. ISC: total nventoy system cost n steady state. 3.- FORMULAIO OF REAILER COSS he etales ae ndependent so that we evaluate holdng and lost shotage costs fo a etale (.e., etale and the cost evaluaton fo othe etales s smla to ths one. Retale odes one unt to cental waehouse n a pe-detemned tme nteval. In secton 3.2 we descbe applyng ths polcy amounts to a detemnstc demand fo the cental waehouse so that waehouse could plane ts nventoy n whch t doesn t face any shotages. Consdeng ths pont, the lead tme fo etale fom waehouse s equvalent to the tanspotaton tme. hus, we obtan a pobablty dstbuton of nventoy level n steady state wth ntenton to evaluate the aveage nventoy level at etale. We defne notaton n, pobablty dstbuton of nventoy level at etale n steady state, such as: o evaluate n n P( I n, n we use a queung system. We suppose the nventoy of etale such as the custome n a queung system. In ths concept, sevce to a costume 247

3 means watng to ave a demand. In the othe wod, a demand to etale s the same as the end of sevce n a queung system. Aval an ode to etale fom waehouse can be consdeed such as aval a custome to the queue. he queung system of nteest s D/M/ queue. he aval ate an ode to etale s equal and the demand ate to etale s equal. Pobablty dstbuton of nventoy level at etale n steady state s [3]: n ρ ρ ( x x n ; n ρ s ato the aval ate to the demand ate so, ρ x s a numbe between to that satsfes ths equaton: ( x x e he aveage nventoy level at etale n steady state s: ρ H n. n n x heefoe the holdng cost at etale n steady state s: ρh h Ch H. h Ch x ( x Inventoy level of etale s stable wheneve ρ <, so we consde ths constant n the model: ρ < s the pobablty of stock out at etale. he amount of demand that s lost n steady state s. Lost sale cost at etale n steady state s: CS s CS s ( he total cost at etale n steady state s: C ( Ch + CS C ( s. t x < x e ( x h ( x < + s ( 3.2. WAREHOUSE IVEORY AALYSIS We assumed that the nventoy polcy of waehouse s (S -, S. It means waehouse ode to the extenal supple as soon as the ecepton of a etale odes. he etales odes have detemnstc tme and quantty. Moeove, the lead tme to waehouse fom the extenal supple s constant, so that waehouse could plane the aval of an ode fom the extenal supple and delvey t to the etale smultaneously. hus, etale odes a unt to the waehouse and waehouse to the extenal supple evey unt of tme. In the othe wod, the optmal ode up-to-level s zeo, S,.heefoe, we tansfom the waehouse nto depot and ts holdng cost s zeo OAL IVEORY SYSEM COS he total nventoy system cost n steady state conssts of holdng and shotage costs of etales and holdng cost of the waehouse. It s descbed n secton 3.2; the holdng cost of the waehouse s zeo so the total nventoy system cost ncludes just the holdng and the shotage costs of etales. Basng ou fomulaton of etale costs (secton 3., the total nventoy system cost can be calculated as follow: ISC C ( he objectve s to detemne the optmal value of (,.., whch mnmze the total nventoy system cost, so the objectve functon s as follow: h ICS mn + s ( (2 ( x s. t x < x e ( x <,,,,..,,...,,.., 4- UMERICAL RESULS In ths secton we compae ou polcy wth common polcy one-fo-one by some numecal examples. he esults of numecal examples fo one-fo-one polcy ae calculated accodng to Andesson and Melchos method [6]. We obtan the optmal value of n functon (2 by MALAB softwae. In ou examples, we suppose that all etales ae dentcal. 248

4 able contans the esults of senstvty analyss on the ate of lost sale cost at etales. Fgue shows the compason between the total system costs of ou polcy wth one-fo-one polcy. In ths numecal example, f the ate of lost sale cost s less than 5, applyng ou polcy s bette. umbe 5 s equvalent to the ate of holdng cost at etales. Regads ths esult, we popose to use ou polcy when the ate of lost sale cost s less than the ate of holdng cost at etales. We test ths hypothess by anothe numecal example, table 2 and fgue 2 show the esults of ths numecal example. he esult of ths numecal example confms the hypothess. total Cost able. Senstvty analyss on the ate of lost sale cost 5,, L, h, h s (s-,s Polcy Ou Polcy (s-,s Polcy Ou Polcy Rate of lost sale cost Fgue. Senstvty analyss on the ate of lost sale cost able2. Senstvty analyss on ate of holdng cost at etales 5, 2, L, h, L, s h (s-,s Polcy Ou Polcy otal Cost (s-,s Polcy O u Po lc y Rate of holdng cost of etales Fgue2. Senstvty analyss on ate of holdng cost at etales 249

5 5- COCLUSIOS AD SUGGESIOS FOR FURHER RESEARCH In ths pape we ntoduced a new nventoy contol polcy and analyzed the applcaton of ths polcy n a tow-echelon nventoy system. he most advantage of ou polcy s to change etales odes n one-fo-one polcy nto the detemnstc odes. hs advantage facltates the nventoy plannng and lead to elmnate holng nventoy at the waehouse. he numecal esults llustate that n some cases ou polcy s bette than one-fo-one polcy. Fo futue eseach we can consde some othe ctea such as pce dscount o odeng cost dscount nto the tow-echelon systems. hs model can be completed by ncludng tanspotaton costs. We guess that ou polcy educes complexty of logstc plannng and deceases tanspotaton costs. hs polcy can be genealzed by consdeaton the batch sze odeng. [] Axsäte, S., Optmzaton of ode-up-to-s polces n two-echelon/nventoy systems wth peodc evew, aval Reseach Logstcs, 4, pp , 993 [] Matta, K.F., Snha, D., Polcy and cost appoxmatons of two-echelon dstbuton systems wth a pocuement cost at the hghe echelon, IIE ansacton, 27, pp , 995 [2] Kanchanasunton, K., echantsawad, A., An appoxmate peodc model fo fxed-lfe peshable poducts n a two-echelon nventoy-dstbuton system, Intenatonal Jounal of Poducton Economcs,, pp-5, 26 [3] Medh, J., Stochastc Models n Queueng heoy, Academc Pess, pp 39-39, 99 REFRECES [] Gaves, S., A mult-echelon nventoy model fo a epaable tem wth one fo-one eplenshment, Management Scence, Vol. 3, o., pp , 985. [2] Axsäte, S., Exact and appoxmate evaluaton of batch-odeng polces fo two-level nventoy systems, Opeatons.Reseach, Vol. 4, o. 4, pp , 993 [3] Axsäte, S., Fosbeg, R., Appoxmatng geneal mult-echelon nventoy systems by Posson models, Intenatonal Jounal of Poducton Economcs, 35, pp2-26, 994. [4] Fosbeg, F., Exact evaluaton of (R,Q polces fo two-level nventoy systems wth Posson demand, Euopean Jounal of Opeatonal Reseach, 96, pp3-38, 996 [5] Axsäte, S., Zhang, W.F., A jont eplenshment polcy fo mult-echelon nventoy contol, Intenatonal Jounal of Poducton Economcs, 59, pp243-25, 999 [6] Andesson, J., Melchos, Ph., A two-echelon nventoy model wth lost sales, Intenatonal Jounal of Poducton Economcs, 69, pp37-35, 2 [7] Seo, Y., Jung, S,. Hahm, J., Optmal eode decson utlzng centalzed stock nfomaton n a twoechelon dstbuton system, Computes & Opeaton Reseach, 29, pp7-93, 22 [8] Axsäte, S., Appoxmate optmzaton of a twolevel dstbuton nventoy system, Intenatonal Jounal of Poducton Economcs, 8-82, pp , 23 [9] Sefbaghy, M., Akba, M.R., Cost evaluaton of a two-echelon nventoy system wth lost sales and appoxmately Posson demand, Intenatonal Jounal of Poducton Economcs, Atcle n Pess, 26 25