The Stochastic Guaranteed Service Model with Recourse for Multi-Echelon Warehouse Management

The S ochas ic Gua an eed Se ice Model wi h Recou se o Mul i-Echelon Wa ehouse Managemen J¨o g Rambau, Kon ad Schade1 Leh s uhl ¨u Wi scha sma hema ik Uni e si ¨a Bay eu h Bay eu h, Ge many Abs ac The Gua an eed Se ice Model (GSM) compu es op imal o de -poin s in mul i- echelon in en o y con ol unde he assump ions ha deli e y imes can be gua an eed and he demand is bounded. Ou new S ochas ic Gua an eed Se - ice Model (SGSM) wi h Recou se co e s also scena ios ha iola e hese as- sump ions. Simula ion expe imen s on eal-wo ld da a o a la ge Ge man ca manu ac u e show ha policies based on he SGSM domina e GSM-policies. Keywo ds: Mul i-echelon in en o y con ol, gua an eed se ice model, s ochas ic p og amming, in ege linea p og amming, eal-wo ld applica ion 1. In oduc ion In en o y con ol o a spa e pa dis ibu ion sys em ollows wo goals: deli e as p omp ly as possible o he end cus ome and minimize in en o y cos s. One op ion o deal wi h wo goals a he same ime is o impose a bound o one and op imize he o he . Fo example: minimize in en o y cos subjec o a gi en se ice le el, i.e., he ac ion o demands ha can be se ed immedia ely. This is he s a egy ha is used, e.g., by he so-called gua an eed- se ice model. See [1] which includes he idea o gua an eed se ice imes o he i s ime, [2] o an ex ension o a ee s uc u e ne wo k, and [3] whe e he model is ex ended o acyclic ne wo ks. In [4] he model was applied o he spa e pa dis ibu ion sys em o a la ge Ge man ca manu ac u e . See also he wo k o Inde u h [5, 6] and Minne [7]. The gua an eed se ice model cha ac e izes, o a gi en se ice le el, op imal o de -poin s s o he widely accep ed (s, S)-policies in mul i-echelon in en o y con ol (see [8] o he classical p oblem s a emen and he heo e ical mo i a ion o (s, S) policies). I can be conside ed as an ad an age o he GSM ha i Email add esses: joe g. ambau@uni-bay eu h.de (J¨o g Rambau), kon ad.schade@uni-bay eu h.de (Kon ad Schade) 1Suppo ed by a g an o “Eli ene zwe k Baye n” P ep in submi ed o Else ie No embe 14, 2011 only makes decisions on he sa e y s ock le el s o he p esc ibed (s, S)-policy: e en hough (s, S)-policies may be subop imal, hey a e anspa en o human ope a o s – i is much easie o make plausibili y checks o sa e y s ock le els han o models ha compu a ionally p oduce highe -dimensional decisions in a black-box. An addi ional ad an age is ha he GSM can be implemen ed and (app oxima ely) sol ed as an in ege linea p og am (see [3]). The GSM, howe e , can only handle bounded demands and de e minis ic deli e y imes in he ne wo k. Ex eme demands and missed in e nal deli e y imes p oduce si ua ions ha a e no cap u ed by he model, and hus he co e- sponding cos can no be accoun ed o by he GSM. The e a e, o cou se, o he policies o mul i-echelon in en o y con ol – including sophis ica ed s ochas- ic se ice models – wi h o he s eng hs and weaknesses (see, e.g., [9] o he METRIC sys em, [10] o a su ey, and [11] o a special e sion o a s ochas ic se ice model). In pa icula , in s ochas ic se ice models adding u he e- s ic ions, e.g., imposed by he business p ocesses o a company, can ende he me hod imp ac ical, whe e as adding es ic ions o he ILP model o he GSM o a ce ain ex en does no a ec he solu ion p ocedu e oo much. Ou con ibu ion: We in oduce he new s ochas ic gua an eed se ice model wi h ecou se (SGSM) and apply wo e sions o i o he in en o y con ol p oblem in a mul i-echelon wa ehouse sys em o a spa e pa dis ibu o . The model is a s ochas ic enhancemen o he gua an eed se ice model by a ecou se componen and demand scena io sampling, so ha all demand scena ios ha a e cap u ed by he sampling p ocess a e handled inside he model. The bene i is ha se ice le els a e now an ou come o he model. The ad an age o he GSM ILP model ha can ake u he es ic ions is main ained. The d awback is ha ecou se cos da a o he cases o los demands ha e o be gi en. (See [12] o backg ound on s ochas ic p og amming.) The con ibu ion o his a icle goes beyond he con e ence p esen a ion [13] in he ollowing aspec s (among o he s): •We in oduce he new SGSM wi h a non- i ial comple e ecou se consis - ing o a anspo a ion op ion besides he penal y cos o non-sales, i.e., eques ed pa s ha canno be deli e ed in ime. •We sol e he SGSM by a combina ion o sample a e age app oxima ion wi h s a e-o - he a scena io educ ion echniques. This way, a be e co e age o unlikely bu expensi e scena ios is achie ed wi hou inc easing he compu a ion imes in he MILP sol e . Ou new asymme ic dis ance unc ion o he asymme ic scena io educ ion akes in o accoun he in luence o he scena io educ ion on he esul o he op imiza ion. To he bes o ou knowledge, his is new. •We p esen a mo e comp ehensi e documen a ion o ex ended compu a- ional esul s, including a new compa ison o one ep esen a i e [11] o he class o s ochas ic se ice models ha could be implemen ed o cope wi h ou es da a. 2 Simula ion esul s on eal-wo ld da a o a la ge Ge man au omobile manu ac- u e and Poisson-dis ibu ed demand wi h eal-wo ld in ensi y o ecas s show ha ou in en o y policies based on he SGSM domina e GSM-policies and yield be e esul s han he conside ed s ochas ic se ice ime model. One eason o his is, among o he s, ha he se ice le el gua an ees o he GSM do no ake in o accoun ha non-sales can ha e qui e di e en impac on he o al cos , which depends on he pa icula pa and on he numbe o pa s missing. I would be in e es ing om a heo e ical poin o iew o also check pe o mances on a i icial andomized da a. Fo his wo k, we ocussed on he p ac ical impac in eal-wo ld applica ions, o which andomized da a is a ely ep esen a i e. We emphasize ha , o his eason, ou simula ion es is comple ely indepen- den o he assump ions o he es ed models – i a he ep esen s ou pa ne ’s p ocess as closely as possible. In he ollowing sec ion we in oduce he modeling o he GSM and he SGSM be o e we show he me hods used o scena io gene a ion and scena io educ ion in sec ion 3. A e he desc ip ion o he simula ion me hod and some compu a ional esul s in sec ion 4 we end wi h some conclusions. 2. Modeling In his sec ion we i s gi e an in oduc ion o he GSM. We use he ILP modeling app oach as in [3]. Then we p esen he SGSM in wo di e en ways. Fi s , in 2.2 we in oduce he SGSM as a wo s age s ochas ic mixed-in ege linea p og am wi h simple ecou se. Second, in 2.3 we show an ex ension whe e he ecou se ac ion o he loca ions supplying he end cus ome s a e modeled as a anspo a ion p oblem. 2.1. The Gua an eed-Se ice-Model The GSM ILP ollows he o iginal wo k in [3], excep o he in eg ali y o he o de -poin s, which is manda o y in spa e-pa sys ems wi h occasionally la ge, expensi e pa s a e y small s ock-le els. Pa ame e s o he model GSM a e: Gdi ec ed g aph desc ibing he wa ehouse ne wo k Nnumbe o wa ehouses N(G) se o nodes in G A(G) se o a cs in G D(G) se o lea es in G(wa ehouses deli e ing o end-cus ome s) hiin en o y holding cos in loca ion i Lideli e y ime o loca ion i ¯sou igi en se ice ime o a lea i∈D(G) Φi(xi) uppe bound o he demand in i∈N(G) du ing he ime pe iod xi 3 The model GSM uses he ollowing a iables o wa ehouses i∈N(G): sin ise ice imes gua an eed by he p edecesso s o i sou ise ice imes gua an eed by i o i s successo s xi ime pe iod ha ineeds o b idge wi h i s in en o y (i.e., he ime be ween o de and deli e y o eplenishmen s om he p edecesso s o i) yio de -poin in i The model GSM now eads as ollows: min PN i=1 hiyi s. . xi≥sin i−sou i+Li∀i∈N(G) sin i≥sou j∀(j, i)∈A(G) sou i≤¯sou i∀i∈D(G) yi≥Φi(xi)∀i∈N(G) xi, sin i, sou i, yi≥0∀i∈N(G) yi∈Z∀i∈N(G) This is no qui e an ILP ye because o he uppe bound on he demand in he loca ion iwhich is deno ed by Φi(xi). Wi h s anda d piecewise-linea modelling echniques wi h addi ional bina y a iables, his model can app oxima ely be ans o med in o an ILP (see [3]). 2.2. The S ochas ic Gua an eed-Se ice-Model wi h Simple Recou se We now add ess wo majo d awbacks o he GSM: he bounded demand (gi en by he p esc ibed se ice le el) and he gua an eed deli e y imes inside he ne wo k. Whene e one o hem happens o be iola ed, an ac ion has o be aken ha is no cap u ed by he model which incu s a cos ha is no aken in o accoun by he model. In o de o inco po a e he wo aspec s in o he model in he simples way, we in oduce simple comple e ecou se o bo h delays and unme demand. Tha is: •Whene e he gua an eed deli e y ime o a wa ehouse is missed, he e is some agen ha o some cos pe ime uni deli e s he pa in ime; his can also be in e p e ed as a penal y o pay o missed deadlines. •Whene e a wa ehouse can no deli e a piece, he e is some (o he ) agen ha deli e s he piece o he wa ehouse immedia ely; his can also be in e p e ed as a penal y o pay o unme demand. O cou se, in p ac ice, he ecou se may be comple e bu mos p obably no sim- ple. A eal-wo ld model o he ecou se p ocess in use depends on he pa icula applica ion and equi es da a abou he cos o cou ie se ices, he cos o a 4 damage in epu a ion, and he like. Howe e , ou i s goal was o in es iga e how he ecou se model as such would in luence he esul ing policy. And o his end, simple ecou se is al eady elling, as we will see. Fo mally, he SGSM has he ollowing addi ional scena io and ecou se pa- ame e s: Sse o scena ios psp obabili y o scena io s∈S icos o compensa e o one ime uni o la e deli e y cicos o compensa e o one piece o unme demand Ls iac ual deli e y ime o iin scena io s Ψs i(xi) ac ual demand in i, du ing ime pe iod xiin scena io s Following he idea o simple ecou se, he SGSM has he ollowing addi ional ecou se a iables: s i ecou se a iable o missed deadlines; “how many ime uni s should be compensa ed a a cos o ipe uni ?” qs i ecou se a iable o missed pieces; “how many pieces should be compensa ed a a cos o ipe uni ?” Since he e is no ob ious implemen a ion o ac ions in he eal wo ld ac- co ding o hese ecou se a iables, hey se e as penal ies o each non-sale o missed lead ime. The hope is ha he SGSM can balance in en o y cos s and non-sales in a mo e de ailed way han he GSM. A he same ime, we main ain he modelling powe o he MILP o mula ion: addi ional es ic ions can be easie inco po a ed han in s ochas ic se ice models we know o . The wo-s age s ochas ic model SGSM now eads as ollows: min PN i=1 hiyi+Ps∈Sps( i s i+ciqs i) s. . xi+ s i≥sin i−sou i+Ls i∀i∈N(G),∀s∈S sin i≥sou j∀(j, i)∈A(G) sou i≤¯sou i∀i∈D(G) yi+qs i≥Ψs i(xi)∀i∈N(G),∀s∈S xi, sin i, sou i, s i, qs i≥0∀i∈N(G),∀s∈S yi, qs i∈Z∀i∈N(G),∀s∈S Again, a linea iza ion o Ψ(xi) can be ca ied ou by s anda d piecewise- linea modelling wi h addi ional bina y a iables. 5 2.3. Ex ension wi h Ex e nal Supplie s and Los Sales The model wi h simple ecou se om he p e ious sec ion can be ex ended by modelling an explici ecou se p ocess. We assume ha unme cus ome demands a e los . Howe e , in e nal o de s a e backlogged. The loca ions ha deli e pa s o he end cus ome s can o de pa s om ex e nal supplie s o p e en los sales. The ex e nal supplie s deli e he pa s di ec ly o he end cus ome s so ha he e is no delay in he deli e y. The cos s o an o de om an ex e nal supplie depends on he dis ance be ween he o de ing loca ion and he supplie . O cou se he supplie do no ha e unlimi ed s ock so ha capaci y cons ain s ha e o be aken in o accoun . To concen a e on hese ecou se ac ions we assume ha he deli e y imes in he sys em a e ix. An ex ension wi h deli e y ime unce ain ies would be s aigh o wa d. We need some mo e no a ion o model he new si ua ion Jse o ex e nal supplie s Cjcapaci y o he ex e nal supplie j qs ji ecou se a iable o pa s o de ed by loca ion ia supplie j cji cos s o loca ion i o o de one pa om supplie j This leads us o he ollowing model: min PN i=1 hiyi+Ps∈SpsPj∈Jcji qs ji s. . xi≥sin i−sou i+Li∀i∈N(G) sin i≥sou j∀(j, i)∈A(G) sou i≤¯sou i∀i∈D(G) yi+Pj∈Jqs ji ≥Ψs i∀i∈N(G),∀s∈S Pi∈D(G)qs ji ≤Cj∀j∈J, ∀s∈S xi, sin i, sou i, qs ji ≥0∀i∈N(G),∀j∈J, ∀s∈S yi, qs ji ∈Z∀i∈N(G),∀s∈S So a , his model does no ha e comple e ecou se. The e o e, we in oduce an o he ecou se a iable. As be o e, we enable o e e y loca ion he possibili y o pay a penal y o a non-sale i i can no deli e he o de ed pa s. Fo ins ance one can p o ide he cus ome wi h a eplacemen ehicle un il he spa e pa can be deli e ed and he cus ome ‘s ca is ixed. The co esponding penal y ecou se a iable is deno ed by qs i, as in he i s model, and he penal y cos s a e deno ed by ciagain. No e, ha by using he penal y ecou se a iables we o ce comple e ecou se bu accoun o ailu e by some cos . The compu a ional esul s in Sec ion 4.3 sugges ha he SGSM policies wi h he es ed penal y alues domina e GSM- policies in e ms o bo h in en o y and ecou se cos , no only o al cos . This means, he esul ing SGSM policy, in e nally using hose success ul penal y 6 alues, will pe o m be e han he co esponding GSM policies also o any o he penal y alues. We ob ain a wo s age s ochas ic model wi h comple e ecou se: min PN i=1 hiyi+Ps∈Spsciqs i+Pj∈Jcji qs ji s. . xi≥sin i−sou i+Li∀i∈N(G) sin i≥sou j∀(j, i)∈A(G) sou i≤¯sou i∀i∈D(G) yi+qs i+Pj∈Jqs ji ≥Ψs i∀i∈D(G),∀s∈S Pi∈D(G)qs ji ≤Cj∀j∈J, ∀s∈S yi+qs i≥Ψs i∀i∈N(G) D(G),∀s∈S xi, sin i, sou i, qs ji ≥0∀i∈N(G),∀j∈J, ∀s∈S yi, qs ji ∈Z∀i∈N(G),∀s∈S 3. Scena io Gene a ion and Reduc ion An app op ia e disc e e app oxima ion o he assumed dis ibu ion o he s ochas ic pa ame e s in he model o en needs many scena ios. The ex en- si e o m o he de e minis ic equi alen p oblem g ows qui e as wi h he numbe o scena ios. This is he eason why we employ scena io educ ion as desc ibed in Subsec ion 3.2. Bu i s we w ap-up he basics abou Sample- A e age-App oxima ion (SAA) Me hods o gene al disc e e app oxima ions o p obabili y dis ibu ions in Subsec ion 3.1. 3.1. SAA-Me hod o Scena io Gene a ion To app oxima e he dis ibu ions o he s ochas ic pa ame e s we gene a e andom numbe s acco ding o he assumed dis ibu ion. These andom num- be s build he scena ios in he disc e e dis ibu ion app oxima ing he eal dis- ibu ion o he s ochas ic pa ame e s. All samples a e assigned p obabili ies p opo ional o he numbe o imes hey we e gene a ed. Sampling echniques like his a e qui e common in s ochas ic p og amming. See o example [12]. The idea o sampling echniques is o app oxima e a s ochas ic p og am (x) = min x∈XcTx+Q(x, ξ).(1) He e Q(x, ξ) deno es he expec ed alue o he op imal solu ion o he second s age p oblem Q(x, ξ) depending on he ac ual ealiza ion ξo ξ. Assume he e is a possibili y o ge independen , iden ically dis ibu ed sam- ples {ξ1,...,ξS}o ξ. The p oblem ˆ (x) = min x∈X(cTx+ S X s=1 Q(x, ξs))(2) 7 can be sol ed concep ually easily and gi es us an unbiased es ima o o (x) he solu ion o he o iginal p oblem. Fu he in o ma ion o SAA can o example be ound in [14]. 3.2. Scena io Reduc ion: The Fas Fo wa d Selec ion The goal o scena io educ ion is o app oxima e a disc e e dis ibu ion wi h many scena ios by ano he disc e e dis ibu ion wi h signi ican ly ewe scena - ios. The e a e se e al me hods o achie e his goal, usually based on a me ic on he space o all possible scena ios (see [15, 16, 17]). An exac app oach o ind he bes app oxima ion wi h a ixed numbe o scena ios is o model he app oxima ion p oblem as a p-median p oblem. In o de o sa e compu a ion ime, we chose o apply he so-called as o wa d selec ion, one o he heu is ics in oduced in [15, 16, 17]. The app oxima ion o he deli e y imes and demand dis ibu ions is spli in o wo pa s. Fi s , a numbe o samples S={ξ1,...,ξS}is gene a ed acco d- ing o he assumed dis ibu ion. These samples buil a i s disc e e app oxi- ma ion whe e e e y scena io ins ance occu s wi h equal p obabili y ps= 1/S. Second, he esul ing disc e e dis ibu ion is ed in o he as - o wa d scena io educ ion, i.e., i is app oxima ed by a disc e e dis ibu ion o e a subse o scena ios o p esc ibed ca dinali y, which ha e, in gene al, non-uni o m p oba- bili ies. Le us now ske ch he p inciple o scena io educ ion, since we ha e o make some choices. The app oach o educe he numbe o scena ios is based on a dis ance be ween wo scena ios deno ed by d(ξ1, ξ2), a quan i y ha we ha e o de ine. When he se o scena ios S′is de ined we add he p obabili y ps o all ξs∈S S′ o he scena io ξs′∈S′which has minimal dis ance o ξs. The as o wa d heu is ic wo ks as ollows. I uses he ac ha i is qui e easy o ind he scena io ξs′∈S o which he o al dis ance o all ξs∈S ξs′, which is X ξs∈S ξs′ psd(ξs, ξs′),(3) is minimal. As ps= 1/S o all scena ios i can be eplaced by a combina ion o he o he scena ios. I e a ing his un il he se S′includes he p ede ined numbe o scena ios is he idea o he as o wa d heu is ic. Gi en he gene a ed scena ios s∈S={ξ1,...,ξS}, he dis ances dbe ween he scena ios, and he ca dinali y o S′,|S′|=k he as o wa d selec ion wo ks as ollows: begin S0={1,...,S} ¯ d=d o i= 1,...,k do s′ i∈a gmins∈Si−1nPj∈Si−1 smini/∈Si−1 s¯ d(ξi, ξj)o Si=Si−1 s′ i 8 upda e(¯ d, s′ i) S′=S0 Sk o s′∈S′do p′ s′=1 S+Ps∈Sk|s′=a gmin{˜s∈S′}d(ξ˜s,ξs)1 S e u n S′and p′ end whe e upda e(¯ d, s′ i) is he ollowing unc ion: begin o i= 1,...,|S|do o j= 1, . . . , |S|do ¯ d(ξi, ξj) = min n¯ d(ξi, ξj),¯ d(ξi, ξs′ i)o end In ou compu a ional es s we use wo di e en kinds o dis ances be ween wo scena ios. The i s dis ance we will e e o as symme ic dis ance. Fo he lead ime we jus ake he euclidean dis ance d(L1 i, L2 i) = |L1 i−L2 i|.(4) Since a demand scena io consis s o di e en demand a es o e e y ime in- e al, we ha e o compa e piecewise linea unc ions. We de ine he dis ance be ween wo demand scena ios Ψ1 iand Ψ2 ias d(Ψ1 i,Ψ2 i) =  α1, i−α2, i 2  ,(5) whe e αs, ideno es he demand a e du ing he ime in e al a s. The e is ano he op ion ha leads o asymme ic dis ances. The idea is o an icipa e ha he app oxima ion is cons uc ed o he use in a s ochas ic op imiza ion p oblem. Thus, we would like o ind he app oxima ion ha yields he leas change in he esul o he op imiza ion. To decide which scena io is mo e impo an o op imiza ion, we need some in o ma ion abou he cos s ha occu in case o s ockholding and in case o s ockou . We ha e his in o ma ion gi en as pa ame e hi, cos s o holding one piece in s ock, and cicos s o ha ing a s ockou o one piece. This way, we can de ine he asymme ic dis ance be ween wo lead ime sce- na ios as d(L1 i, L2 i) = |L1 i−L2 i|ci hi (6) i L1 i> L2 i d(L1 i, L2 i) = |L1 i−L2 i|hi ci ,(7) o he wise. 9 Table 6: Compa ison o se ice le els (%) Wa ehouse (1) (2) (3) (4) (5) (6) (7) (8) (9) 0 84.9 87.2 85.0 87.6 88.2 88.2 88.0 88.7 88.8 1 94.2 94.3 94.2 94.3 94.5 94.5 94.6 94.6 94.6 2 94.1 94.1 94.0 94.1 94.3 94.3 94.3 94.3 94.4 3 94.4 94.5 94.4 94.5 94.7 94.7 94.7 94.7 94.7 4 93.9 94.0 93.9 94.0 94.1 94.1 94.1 94.2 94.2 5 94.3 94.5 94.3 94.5 94.7 94.8 94.8 94.8 94.8 6 93.6 94.0 93.6 93.9 94.0 94.0 94.0 94.0 94.0 7 94.1 94.2 94.1 94.3 94.4 94.5 94.4 94.5 94.5 96%. The cos s in able 5 ell us ha he SGSM ea s di e en pa s di e en ly, while he GSM and he decen alized model co e 96% o he demand o e e y pa , no ma e wha he cos s hiand cji a e. This is he eason why he o de poin s calcula ed by he SGSM can lead o cheape in en o y cos s and ecou se cos s a he same ime. Las we wan o compa e ou model o a model ha was in oduced by Doˇg u, de Kok and an Hou um, see [11]. In he ollowing we will e e o his model as DoKoHo. In he simula ion we need o apply ix lead imes as his is one assump ion o he DoKoHo model. We simula e a si ua ion ha i s o he DoKoHo assump ions whe e demand is backlocked and he e a e penal y cos s i a loca ion is no able o deli e as demanded. Table 7 shows he esul s o he simula ion o he GSM, he SGSM and DoKoHo. The e a e some di e en pa ame e se ings o DoKoHo whe e he penal y cos s used in he model a e mul iplied by a ac o (γ). Table 7: Resul s o simula ion wi h poisson dis ibu ed demand and ix lead ime Model In en o y Cos Recou se Cos To al Cos (1) DoKoHo (γ= 1)1 450 564.94 2 510 522.12 3 961 087.06 (2) DoKoHo (γ= 5)1 816 753.33 1 534 991.82 3 351 745.15 (3) DoKoHo (γ= 10)1 954 727.95 1 387 486.39 3 342 214.34 (4) GSM 96% 1 834 581.46 1 980 314.45 3 814 895.91 (5) SGSM 300 →75, weeks, asym 1 058 309.50 1 629 832.65 2 688 142.15 The DoKoHo model ou pe o ms he GSM bu causes highe cos s han he SGSM. The simula ion o he di e en models lead o he se ice le els ha a e shown in able 8. The eason o he e y low se ice le els a he mas e wa ehouse using he DoKoHo model compa ed o he ones using GSM o SGSM can be explained easily. In he DoKoHo model he e a e no explici se ice imes gua an eed o he successo s. Bu he lowe pe o mance o he mas e wa ehouse is conside ed when he successo ‘s sa e y s ock is calcula ed. In he simula ion we use a se ice ime o ze o o his case so he deli e y o mas e wa ehouse is o en la e. 16 Table 8: Compa ison o se ice le els (%) Wa ehouse (1) (2) (3) (4) (5) 0 48.0 53.7 55.4 91.9 83.3 1 96.8 98.7 99.0 98.1 97.4 2 96.4 98.4 98.6 97.8 97.1 3 96.6 98.7 98.9 97.9 97.4 4 96.3 98.2 98.4 97.6 96.5 5 97.0 98.9 99.1 98.7 97.6 6 96.7 98.1 98.2 97.8 96.6 7 96.6 98.5 98.7 97.9 97.5 As we do no conside penal y cos s o mas e wa ehouse in he simula ion, his does no e ec he cos s ha we ge applying he DoKoHo models. 5. Conclusion We ha e in oduced he S ochas ic Gua an eed Se ice Model (SGSM), a s ochas ic p og amming e sion o he Gua an eed Se ice Model (GSM) o he compu a ion o sa e y s ock le els in a mul i-echelon spa e pa dis ibu ion sys em o a la ge Ge man ca manu ac u e . Whe eas he GSM makes assump ions ha equi e ex eme demand scena - ios and missed deli e y da es o be handled ou side he model, he SGSM is capable o inco po a ing hese ola ili ies inside he model, he eby accoun ing o he co esponding cos . The s ochas ici y needs o be cap u ed by su icien ly la ge sample sizes: in ou example we gene a ed 200 scena ios mos o he ime and educed hem o 50 applying modi ied scena io educ ion echniques. The esul ing MILP models could be sol ed s aigh - o wa dly in ou example. The SGSM makes some assump ions ha a e only app oxima ions o eali y (comple e ecou se, piecewise linea demand). Howe e , ou simula ion was no es ic ed by hese assump ions; i only checked he esul ing policies, no ma e wha hey assumed, and accoun ed o all he occu ing cos s. And in his qui e ealis ic simula ion expe imen , he policies calcula ed wi h he SGSM pe o med ex emely well. One eason o his is ha he SGSM can ha e s uc u ally di e en op imal solu ions han he GSM: no all op imal SGSM solu ions a e ex eme in he space o a iables o he GSM. Thus he SGSM some imes inds solu ions ha he GSM can ne e p o ide, no ma e which pa ame e se ing. And such solu ions domina ed GSM solu ions in ou simula ions. We he e o e hink ha he SGSM can be applied ou inely in spa e pa dis ibu ion sys ems like he one o ou pa ne . Nex , we will model he eal- wo ld ecou se ac ions in mo e de ail in o de o ind mo e ealis ic ecou se cos alues. 17 Appendix A. Resul s o he simula ion uns In his sec ion we show some o he nume ical esul s in de ail. The a e age cos s o he en simula ion uns lis ed he e a e gi en in ables 1 and 3. The ables A.9–A.12 include he esul s ha lead o he a e age cos s o able 2. Table A.9: No educ ion (3 →3) Run In en o y Cos s Recou se Cos s To al Cos s 1 1 262 180.67 20 904 545.37 22 166 726.04 2 1 227 922.08 17 572 609.36 18 800 531.44 3 1 238 190.82 22 341 954.25 23 580 145.07 4 1 294 183.52 19 942 497.73 21 236 681.25 5 1 310 570.53 21 047 322.98 22 357 893.51 6 1 256 992.73 20 860 600.50 22 117 593.23 7 1 325 988.56 15 528 825.97 16 378 814.53 8 1 262 180.67 20 904 545.37 22 166 726.04 9 1 241 824.84 32 304 152.43 33 545 977.27 10 1 311 050.20 18 169 764.58 19 480 814.58 A e age 1 273 108.46 1 273 108.46 1 273 108.46 20 957 681.85 20 957 681.85 20 957 681.85 22 183 190.31 22 183 190.31 22 183 190.31 Table A.10: Symme ic educ ion (50 →3) Run In en o y Cos s Recou se Cos s To al Cos s 1 1 344 456.16 5 885 899.06 7 230 355.22 2 1 374 833.68 6 231 607.72 7 606 441.40 3 1 357 439.20 5 264 175.09 6 621 614.29 4 1 358 487.22 5 693 121.60 7 051 608.82 5 1 377 759.11 6 037 579.32 7 415 338.43 6 1 356 518.24 5 469 194.65 6 825 712.89 7 1 376 288.85 5 947 061.26 7 323 350.11 8 1 401 105.86 5 516 129.63 6 917 235.49 9 1 358 232.81 5 828 321.03 7 186 553.84 10 1 378 756.13 6 118 036.79 7 496 792.92 A e age 1 368 387.73 1 368 387.73 1 368 387.73 5 799 112.62 5 799 112.62 5 799 112.62 7 167 500.35 7 167 500.35 7 167 500.35 Table A.13 and A.14 include he esul s o he single uns o GSM 96% (4) and SGSM 200 →50, weeks, asym (10) o able 3. Re e ences [1] Simpson, In-p ocess in en o y, Ope a ions Resea ch 6 (1958) 863–873. [2] S. G a es, S. Willems, Op imizing s a egic sa e y s ock placemen in sup- ply chains, Manu ac u ing & Se ice Ope a ions Managemen 2 (1) (2000) 68–83. 18 Table A.11: Asymme ic educ ion (50 →3) Run In en o y Cos s Recou se Cos s To al Cos s 1 1 485 860.93 1 980 706.25 3 466 567.18 2 1 478 170.53 1 767 860.65 3 246 031.18 3 1 476 507.35 1 833 396.79 3 309 904.14 4 1 508 549.33 1 743 703.09 3 252 252.42 5 1 509 959.07 1 860 989.82 3 370 580.89 6 1 475 040.72 1 865 894.95 3 340 935.67 7 1 502 305.31 1 888 979.81 3 391 285.12 8 1 495 706.29 1 910 136.29 3 405 842.58 9 1 480 737.28 1 826 744.86 3 307 482.14 10 1 457 633.50 2 088 677.41 3 546 310.91 A e age 1 487 047.03 1 487 047.03 1 487 047.03 1 876 708.99 1 876 708.99 1 876 708.99 3 363 719.22 3 363 719.22 3 363 719.22 Table A.12: No Reduc ion (50 →50) Run In en o y Cos s Recou se Cos s To al Cos s 1 1 650 596.38 1 531 849.61 3 182 445.99 2 1 657 496.08 1 391 982.95 3 049 479.03 3 1 655 262.68 1 520 144.48 3 175 407.16 4 1 652 312.18 1 512 184.03 3 164 496.21 5 1 660 352.51 1 511 081.84 3 171 434.35 6 1 657 754.36 1 577 041.20 3 234 795.56 7 1 663 863.19 1 561 981.43 3 225 844.62 8 1 673 298.06 1 509 069.42 3 182 367.48 9 1 669 855.12 1 525 359.34 3 195 214.46 10 1 655 235.31 1 642 187.46 3 297 422.77 A e age 1 659 602.59 1 659 602.59 1 659 602.59 1 528 288.18 1 528 288.18 1 528 288.18 3 187 890.77 3 187 890.77 3 187 890.77 [3] T. Magnan i, Z.-J. Shen, J. Shu, D. Simchi-Le i, C.-P. Teo, In en o y placemen in acyclic supply chain ne wo ks, Ope a ions Resea ch Le e s 34 (2006) 228–238. [4] K. Schade, Lage hal ungss a egie ¨u meh s u ige Lage hal ung in de Au- omobilindus ie, Mas e ’s hesis, Uni e si ¨a Bay eu h (2008). [5] K. Inde u h, Sa e y s ock op imiza ion in mul i-s age in en o y sys ems, In e na ional Jou nal o P oduc ion Economics 24 (1-2) (1991) 103 – 113. doi:10.1016/0925-5273(91)90157-O. URL h p://www.sciencedi ec .com/science/a icle/pii/092552739190157O [6] K. Inde u h, Sa e y s ocks in mul is age di e gen in en o y sys ems: A su ey, in e na ional jou nal o p oduc ion economics 35 (1994) 321–329. [7] S. Minne , Dynamic p og amming algo i hms o mul i-s age sa e y s ock op imiza ion, OR Spec um 19 (1997) 261–271, 10.1007/BF01539783. URL h p://dx.doi.o g/10.1007/BF01539783 19 Table A.13: GSM wi h a p esc ibed se ice le el o 96% Run In en o y Cos s Recou se Cos s To al Cos s 1 2 977 194.04 953 576.93 3 930 770.97 2 2 985 082.45 957 196.39 3 942 278.84 3 2 981 576.40 1 017 176.75 3 998 753.15 4 2 988 402.12 945 210.41 3 933 612.53 5 2 983 057.11 1 085 943.06 4 069 000.17 6 2 992 958.19 914 166.88 3 907 125.07 7 2 997 632.67 881 167.01 3 878 799.68 8 2 974 985.18 962 779.43 3 937 764.61 9 2 971 511.93 927 915.91 3 899 427.84 10 2 978 386.83 985 449.47 3 963 836.30 A e age 2 983 078.69 2 983 078.69 2 983 078.69 963 058.22 963 058.22 963 058.22 3 946 136.91 3 946 136.91 3 946 136.91 Table A.14: SGSM 200 →3 asymme ic educ ion wi h ime disc e iza ion in weeks Run In en o y Cos s Recou se Cos s To al Cos s 1 1 869 685.34 794 347.91 2 664 033.25 2 1 894 667.10 742 510.22 2 637 177.32 3 1 892 332.95 834 364.91 2 726 697.86 4 1 874 564.45 764 024.10 2 638 588.55 5 1 875 607.36 997 203.15 2 872 810.51 6 1 883 575.61 772 162.01 2 655 737.62 7 1 879 543.93 812 130.95 2 691 674.88 8 1 894 412.96 808 671.05 2 703 084.01 9 1 885 131.65 767 510.04 2 652 641.69 10 1 889 852.08 787 893.38 2 677 745.46 A e age 1 883 937.34 1 883 937.34 1 883 937.34 808 081.77 808 081.77 808 081.77 2 692 019.11 2 692 019.11 2 692 019.11 [8] A. Cla k, H. Sca , Op imal policies o a mul i-echelon in en o y p oblem, Managemen Science 6 (1960) 475–490. [9] C. She b ooke, Me ic: A mul i-echelon echnique o eco e able i em con- ol, Ope a ions Resea ch 16 (1968) 122–141. [10] A. Diaz, M. C. Fu, Mul i-echelon models o epai able i ems: A e iew, Documen in Decision, Ope a ions & In o ma ion Technologies Resea ch Wo ks h p://hdl.handle.ne /1903/2300, Uni e si y o Ma yland (2005). URL h p://hdl.handle.ne /1903/2300 [11] M. Doˇg u, A. de Kok, G. an Hou um, Op imal con ol o one-wa ehouse mul i- e aile sys ems wi h disc e e demand, Wo king pape (2005). [12] J. R. Bi ge, F. Lou eaux, In oduc ion o S ochas ic P og amming, Sp inge , 1997. [13] J. Rambau, K. Schade, The s ochas ic gua an eed se ice model wi h e- cou se o mul i-echelon wa ehouse managemen , in: P oceedings o he 20 In e na ional Symposium on Combina o ial Op imiza ion (ISCO 2010), Vol. 36 o Elec onic No es in Disc e e Ma hema ics, Else ie , 2010, pp. 783–790, o appea . [14] A. Shapi o, Mon e ca lo sampling me hods, in: A. Ruszczynski, A. Shapi o (Eds.), S ochas ic P og amming, Vol. 10 o Handbooks in Ope a ions Re- sea ch and Managemen Science, Else ie , 2003, pp. 353 – 425. doi:DOI: 10.1016/S0927-0507(03)10006-0. [15] H. Hei sch, S abili ¨a und app oxima ion s ochas ische op imie ungsp ob- leme, PhD disse a ion, Humbold-Uni e si ¨a zu Be lin (No . 2007). [16] H. Hei sch, W. R¨omisch, Scena io educ ion algo i hms in s ochas ic p o- g amming, Compu a ional Op imiza ion and Applica ions 24 (2003) 187– 206. [17] R. Hen ion, C. K¨uchle , W. R¨omisch, Disc epancy dis ances and scena io educ ion in wo-s age s ochas ic mixed-in ege p og amming, JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION 4 (2) (2008) 363–384. 21