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

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1 The Stochastc Guaranteed Servce Model wth Recourse for Mult-Echelon Warehouse Management Jörg Rambau, Konrad Schade 1 Lehrstuhl für Wrtschaftsmathematk Unverstät Bayreuth Bayreuth, Germany Abstract The Guaranteed Servce Model (GSM) computes optmal order-ponts n multechelon nventory control under the assumptons that delvery tmes can be guaranteed and the demand s bounded. Our new Stochastc Guaranteed Servce Model (SGSM) wth Recourse covers also scenaros that volate these assumptons. Smulaton experments on real-world data of a large German car manufacturer show that polces based on the SGSM domnate GSM-polces. Keywords: Mult-echelon nventory control, guaranteed servce model, stochastc programmng, nteger lnear programmng, real-world applcaton 1. Introducton Inventory control for a spare part dstrbuton system follows two goals: delver as promptly as possble to the end customer and mnmze nventory costs. One opton to deal wth two goals at the same tme s to mpose a bound for one and optmze the other. For example: mnmze nventory cost subject to a gven servce level,.e., the fracton of demands that can be served mmedately. Ths s the strategy that s used, e.g., by the so-called guaranteedservce model. See [1] whch ncludes the dea of guaranteed servce tmes for the frst tme, [2] for an extenson to a tree structure network, and [3] where the model s extended to acyclc networks. In [4] the model was appled to the spare part dstrbuton system of a large German car manufacturer. See also the work of Inderfurth [5, 6] and Mnner [7]. The guaranteed servce model characterzes, for a gven servce level, optmal order-ponts s for the wdely accepted (s, S)-polces n mult-echelon nventory control(see[8] for the classcal problem statement and the theoretcal motvaton for (s,s) polces). It can be consdered as an advantage of the GSM that t Emal addresses: (Jörg Rambau), (Konrad Schade) 1 Supported by a grant of Eltenetzwerk Bayern Preprnt submtted to Elsever November 14, 2011

2 only makes decsons on the safety stock level s for the prescrbed (s, S)-polcy: even though (s, S)-polces may be suboptmal, they are transparent to human operators t s much easer to make plausblty checks for safety stock levels than for models that computatonally produce hgher-dmensonal decsons n a black-box. An addtonal advantage s that the GSM can be mplemented and (approxmately) solved as an nteger lnear program (see [3]). The GSM, however, can only handle bounded demands and determnstc delvery tmes n the network. Extreme demands and mssed nternal delvery tmes produce stuatons that are not captured by the model, and thus the correspondng cost can not be accounted for by the GSM. There are, of course, other polces for mult-echelon nventory control ncludng sophstcated stochastc servce models wth other strengths and weaknesses (see, e.g., [9] for the METRIC system, [10] for a survey, and [11] for a specal verson of a stochastc servce model). In partcular, n stochastc servce models addng further restrctons, e.g., mposed by the busness processes of a company, can render the method mpractcal, where as addng restrctons to the ILP model of the GSM to a certan extent does not affect the soluton procedure too much. Our contrbuton: We ntroduce the new stochastc guaranteed servce model wth recourse (SGSM) and apply two versons of t to the nventory control problem n a mult-echelon warehouse system of a spare part dstrbutor. The model s a stochastc enhancement of the guaranteed servce model by a recourse component and demand scenaro samplng, so that all demand scenaros that are captured by the samplng process are handled nsde the model. The beneft s that servce levels are now an outcome of the model. The advantage of the GSM ILP model that can take further restrctons s mantaned. The drawback s that recourse cost data for the cases of lost demands have to be gven. (See [12] for background on stochastc programmng.) The contrbuton of ths artcle goes beyond the conference presentaton [13] n the followng aspects (among others): We ntroduce the new SGSM wth a non-trval complete recourse consstng of a transportaton opton besdes the penalty cost for non-sales,.e., requested parts that cannot be delvered n tme. We solve the SGSM by a combnaton of sample average approxmaton wth state-of-the art scenaro reducton technques. Ths way, a better coverage of unlkely but expensve scenaros s acheved wthout ncreasng the computaton tmes n the MILP solver. Our new asymmetrc dstance functon for the asymmetrc scenaro reducton takes nto account the nfluence of the scenaro reducton on the result of the optmzaton. To the best of our knowledge, ths s new. We present a more comprehensve documentaton of extended computatonal results, ncludng a new comparson to one representatve[11] of the class of stochastc servce models that could be mplemented to cope wth our test data. 2

3 Smulaton results on real-world data of a large German automoble manufacturer and Posson-dstrbuted demand wth real-world ntensty forecasts show that our nventory polces based on the SGSM domnate GSM-polces and yeld better results than the consdered stochastc servce tme model. One reason for ths s, among others, that the servce level guarantees of the GSM do not take nto account that non-sales can have qute dfferent mpact on the total cost, whch depends on the partcular part and on the number of parts mssng. It would be nterestng from a theoretcal pont of vew to also check performances on artfcal randomzed data. For ths work, we focussed on the practcal mpact n real-world applcatons, for whch randomzed data s rarely representatve. We emphasze that, for ths reason, our smulaton test s completely ndependent of the assumptons of the tested models t rather represents our partner s process as closely as possble. In the followng secton we ntroduce the modelng of the GSM and the SGSM before we show the methods used of scenaro generaton and scenaro reducton n secton 3. After the descrpton of the smulaton method and some computatonal results n secton 4 we end wth some conclusons. 2. Modelng In ths secton we frst gve an ntroducton to the GSM. We use the ILP modelng approach as n [3]. Then we present the SGSM n two dfferent ways. Frst, n 2.2 we ntroduce the SGSM as a two stage stochastc mxed-nteger lnear program wth smple recourse. Second, n 2.3 we show an extenson where the recourse acton of the locatons supplyng the end customers are modeled as a transportaton problem The Guaranteed-Servce-Model The GSM ILP follows the orgnal work n [3], except for the ntegralty of the order-ponts, whch s mandatory n spare-part systems wth occasonally large, expensve parts at very small stock-levels. Parameters of the model GSM are: G N N(G) A(G) D(G) h L s out Φ (x ) drected graph descrbng the warehouse network number of warehouses set of nodes n G set of arcs n G set of leaves n G (warehouses delverng to end-customers) nventory holdng cost n locaton delvery tme to locaton gven servce tme for a leaf D(G) upper bound for the demand n N(G) durng the tme perod x 3

4 The model GSM uses the followng varables for warehouses N(G): s n s out x y servce tmes guaranteed by the predecessors of servce tmes guaranteed by for ts successors tme perod that needs to brdge wth ts nventory (.e., the tme between order and delvery of replenshments from the predecessors of ) order-pont n The model GSM now reads as follows: mn s.t. N =1 h y x s n s out +L N(G) s n s out j (j,) A(G) s out s out D(G) y Φ (x ) N(G) x, s n, sout, y 0 y Z N(G) N(G) ThssnotquteanILPyetbecauseoftheupperboundonthedemandnthe locaton whch s denoted by Φ (x ). Wth standard pecewse-lnear modellng technques wth addtonal bnary varables, ths model can approxmately be transformed nto an ILP (see [3]) The Stochastc Guaranteed-Servce-Model wth Smple Recourse We now address two major drawbacks of the GSM: the bounded demand (gven by the prescrbed servce level) and the guaranteed delvery tmes nsde the network. Whenever one of them happens to be volated, an acton has to be taken that s not captured by the model whch ncurs a cost that s not taken nto account by the model. In order to ncorporate the two aspects nto the model n the smplest way, we ntroduce smple complete recourse for both delays and unmet demand. That s: Whenever the guaranteed delvery tme of a warehouse s mssed, there s some agent that for some cost per tme unt delvers the part n tme; ths can also be nterpreted as a penalty to pay for mssed deadlnes. Whenever a warehouse can not delver a pece, there s some (other) agent that delvers the pece to the warehouse mmedately; ths can also be nterpreted as a penalty to pay for unmet demand. Of course, n practce, the recourse may be complete but most probably not smple. A real-world model of the recourse process n use depends on the partcular applcaton and requres data about the cost of courer servces, the cost of a 4

5 damage n reputaton, and the lke. However, our frst goal was to nvestgate how the recourse model as such would nfluence the resultng polcy. And to ths end, smple recourse s already tellng, as we wll see. Formally, the SGSM has the followng addtonal scenaro and recourse parameters: S p s t c L s set of scenaros probablty of scenaro s S cost to compensate for one tme unt of late delvery cost to compensate for one pece of unmet demand actual delvery tme to n scenaro s Ψ s (x ) actual demand n, durng tme perod x n scenaro s Followng the dea of smple recourse, the SGSM has the followng addtonal recourse varables: r s q s recourse varable for mssed deadlnes; how many tme unts should be compensated at a cost of t per unt? recourse varable for mssed peces; how many peces should be compensated at a cost of t per unt? Snce there s no obvous mplementaton of actons n the real world accordng to these recourse varables, they serve as penaltes for each non-sale or mssed lead tme. The hope s that the SGSM can balance nventory costs and non-sales n a more detaled way than the GSM. At the same tme, we mantan the modellng power of the MILP formulaton: addtonal restrctons can be easer ncorporated than n stochastc servce models we know of. The two-stage stochastc model SGSM now reads as follows: mn N ( =1 h y + s S p s(t r s +c q s)) s.t. x +r s sn s out +L s N(G), s S s n s out j (j,) A(G) s out s out y +q s Ψs (x ) x, s n, sout, r s, qs 0 y, q s Z D(G) N(G), s S N(G), s S N(G), s S Agan, a lnearzaton of Ψ(x ) can be carred out by standard pecewselnear modellng wth addtonal bnary varables. 5

6 2.3. Extenson wth External Supplers and Lost Sales The model wth smple recourse from the prevous secton can be extended by modellng an explct recourse process. We assume that unmet customer demands are lost. However, nternal orders are backlogged. The locatons that delver parts to the end customers can order parts from external supplers to prevent lost sales. The external supplers delver the parts drectly to the end customers so that there s no delay n the delvery. The costs of an order from an external suppler depends on the dstance between the orderng locaton and the suppler. Of course the suppler do not have unlmted stock so that capacty constrants have to be taken nto account. To concentrate on these recourse actons we assume that the delvery tmes n the system are fx. An extenson wth delvery tme uncertantes would be straght forward. We need some more notaton to model the new stuaton J C j qj s c j set of external supplers capacty of the external suppler j recourse varable for parts ordered by locaton at suppler j costs for locaton to order one part from suppler j Ths leads us to the followng model: mn ( N =1 h y + s S p ) s j J c jqj s s.t. x s n s out +L N(G) s n s out j (j,) A(G) s out s out y + j J qs j Ψs D(G) qs j C j x, s n, sout, qj s 0 y, qj s Z D(G) N(G), s S j J, s S N(G), j J, s S N(G), s S So far, ths model does not have complete recourse. Therefore, we ntroduce an other recourse varable. As before, we enable for every locaton the possblty topayapenaltyforanon-saleftcannotdelvertheorderedparts. Fornstance one can provde the customer wth a replacement vehcle untl the spare part can be delvered and the customer s car s fxed. The correspondng penalty recourse varable s denoted by q s, as n the frst model, and the penalty costs are denoted by c agan. Note, that by usng the penalty recourse varables we force complete recourse but account for falure by some cost. The computatonal results n Secton 4.3 suggest that the SGSM polces wth the tested penalty values domnate GSMpolces n terms of both nventory and recourse cost, not only total cost. Ths means, the resultng SGSM polcy, nternally usng those successful penalty 6

7 values, wll perform better than the correspondng GSM polces also for any other penalty values. We obtan a two stage stochastc model wth complete recourse: mn N =1 s.t. ( h y + s S p s ( c q s + j J c jq s j )) x s n s out +L N(G) s n s out j (j,) A(G) s out s out y +q s + j J qs j Ψs D(G) qs j C j y +q s Ψs x, s n, sout, qj s 0 y, qj s Z D(G) D(G), s S j J, s S N(G)\D(G), s S N(G), j J, s S N(G), s S 3. Scenaro Generaton and Reducton An approprate dscrete approxmaton of the assumed dstrbuton of the stochastc parameters n the model often needs many scenaros. The extensve form of the determnstc equvalent problem grows qute fast wth the number of scenaros. Ths s the reason why we employ scenaro reducton as descrbed n Subsecton 3.2. But frst we wrap-up the bascs about Sample- Average-Approxmaton (SAA) Methods for general dscrete approxmatons of probablty dstrbutons n Subsecton SAA-Method for Scenaro Generaton To approxmate the dstrbutons of the stochastc parameters we generate random numbers accordng to the assumed dstrbuton. These random numbers buld the scenaros n the dscrete dstrbuton approxmatng the real dstrbuton of the stochastc parameters. All samples are assgned probabltes proportonal to the number of tmes they were generated. Samplng technques lke ths are qute common n stochastc programmng. See for example [12]. The dea of samplng technques s to approxmate a stochastc program f(x) = mn x X { c T x+q(x,ξ) }. (1) Here Q(x,ξ) denotes the expected value of the optmal soluton of the second stage problem Q(x,ξ) dependng on the actual realzaton ξ of ξ. Assume there s a possblty to get ndependent, dentcally dstrbuted samples {ξ 1,...,ξ S } of ξ. The problem ˆf(x) = mn x X { c T x+ } S Q(x,ξ s ) s=1 (2) 7

8 canbe solvedconceptuallyeaslyand gvesusanunbasedestmatorforf(x) the soluton of the orgnal problem. Further nformaton to SAA can for example be found n [14] Scenaro Reducton: The Fast Forward Selecton The goal of scenaro reducton s to approxmate a dscrete dstrbuton wth many scenaros by another dscrete dstrbuton wth sgnfcantly fewer scenaros. There are several methods to acheve ths goal, usually based on a metrc on the space of all possble scenaros (see [15, 16, 17]). An exact approach to fnd the best approxmaton wth a fxed number of scenaros s to model the approxmaton problem as a p-medan problem. In order to save computaton tme, we chose to apply the so-called fast forward selecton, one of the heurstcs ntroduced n [15, 16, 17]. The approxmaton of the delvery tmes and demand dstrbutons s splt nto two parts. Frst, a number ofsamples S = {ξ 1,...,ξ S } s generated accordng to the assumed dstrbuton. These samples bult a frst dscrete approxmaton where every scenaro nstance occurs wth equal probablty p s = 1/S. Second, the resultng dscrete dstrbuton s fed nto the fast-forward scenaro reducton,.e., t s approxmated by a dscrete dstrbuton over a subset of scenaros of prescrbed cardnalty, whch have, n general, non-unform probabltes. Let us now sketch the prncple of scenaro reducton, snce we have to make some choces. The approach to reduce the number of scenaros s based on a dstance between two scenaros denoted by d(ξ 1,ξ 2 ), a quantty that we have to defne. WhenthesetofscenarosS sdefnedweaddtheprobabltyp s forallξ s S\S to the scenaro ξ s S whch has mnmal dstance to ξ s. The fast forward heurstc works as follows. It uses the fact that t s qute easy to fnd the scenaro ξ s S for whch the total dstance to all ξ s S\ξ s, whch s p s d(ξ s,ξ s ), (3) ξ s S\ξ s s mnmal. As p s = 1/S for all scenaros t can be replaced by a combnaton of the other scenaros. Iteratng ths untl the set S ncludes the predefned number of scenaros s the dea of the fast forward heurstc. Gven the generated scenaros s S = {ξ 1,...,ξ S }, the dstances d between the scenaros, and the cardnalty of S, S = k the fast forward selecton works as follows: begn S 0 = {1,...,S} d = d for = 1,...,k do { s argmn s S 1 j S 1 \s mn / S 1 \s { d(ξ,ξ j ) }} S = S 1 \s 8

9 update( d,s ) S = S 0 \S k for s S do p s = 1 S + s S k s =argmn { s S } d(ξ s,ξ s ) 1 S return S and p end where update( d,s ) s the followng functon: begn for = 1,..., S do for j = 1,..., S do d(ξ,ξ j ) = mn { d(ξ,ξ j ), d(ξ },ξ s ) end In our computatonal tests we use two dfferent knds of dstances between two scenaros. The frst dstance we wll refer to as symmetrc dstance. For the lead tme we just take the eucldean dstance d(l 1, L2 ) = L1 L2. (4) Snce a demand scenaro conssts of dfferent demand rates for every tme nterval, we have to compare pecewse lnear functons. We defne the dstance between two demand scenaros Ψ 1 and Ψ2 as d(ψ 1, Ψ 2 α 1,r α 2,r ) =, (5) where α s,r denotes the demand rate durng the tme nterval r at s. There s another opton that leads to asymmetrc dstances. The dea s to antcpate that the approxmaton s constructed for the use n a stochastc optmzaton problem. Thus, we would lke to fnd the approxmaton that yelds the least change n the result of the optmzaton. To decde whch scenaro s more mportant for optmzaton, we need some nformaton about the costs that occur n case of stockholdng and n case of stockout. We have ths nformaton gven as parameter h, costs for holdng one pece n stock, and c costs for havng a stockout of one pece. Ths way, we can defne the asymmetrc dstance between two lead tme scenaros as d(l 1, L2 ) = L1 L2 c h (6) 2 r f L 1 > L2 otherwse. d(l 1, L2 ) = L1 L2 h c, (7) 9

10 The dstance d(ξ 1,ξ 2 ) descrbes the costs of deletng scenaro ξ 1 and addng ts probablty p 1 to p 2, the probablty of ξ 2. The defnton of asymmetrc dstance between two demand scenaros s based on the same dea but the order Ψ 1 (x ) > Ψ 2 (x ) depends on the x as Ψ s (x ) s pecewse lnear. That s why we look at the values of Ψ s (x ) where x equals the expected value of the delvery tme to locaton, L. So we defne the followng asymmetrc dstance between demand scenaros: f Ψ 1 (L ) > Ψ 2 (L ) d(ψ 1, Ψ 2 ) = d(ψ 2, Ψ1 ) = α 1,r α 1,r α 2,r 2 r α 2,r 2 r c h, (8) h c, (9) otherwse. We use these dstances n the fast forward selecton to determne the scenaros s S and ther new probabltes p s. The asymmetrc reducton does not approxmate the dstrbuton tself as fathfully as the reducton technque based on symmetrc dstances. We get a bas n our approxmaton that depends on the fracton of h and c. It wll be shown n the next secton that ths based reducton ndeed approxmates better the solutons to the optmzaton problems because t takes nto account the cost of endng up n a certan scenaro. To the best of our knowledge, ths s not yet standard n the Stochastc Programmng lterature. 4. Smulaton We performed comprehensve computatonal tests on real-world data from our partner General Issues Before we report on our tests, we want to make some general remarks concernng some sde-effects of modelng-decsons of the SGSM. Frst, the SGSM can only take fnte dscrete dstrbutons of demands and lead tmes. Second, all scenaros of the demand dstrbutons must be represented by pecewse lnear approxmatons n order to obtan an MILP formulaton for the SGSM. Our partner forecasts the demand for one month. The data nclude the expected total demand n the actual month, the expected total demand n the comng month and so on. Thus, a straght-forward approach would be to approxmate the demand lnearly durng one month. However: If we smply assume lnearty of the demand durng one month, then the rough dscretzaton of tme nto months leads to demand scenaros wth too lttle varaton over tme. We can, of course, choose a fner dscretzaton of tme n weeks or days. The fner the dscretzaton s the more realstc becomes the demand functon. 10

11 In order to get a feelng for ths nfluence, we generated stochastc numbers denotng the demand over one month or one week. Fgure 1 shows an example of dfferences n the scenaros for dscretzaton n months and n weeks. demand (cum) Scenaro 1 Scenaro 2 Scenaro 3 Scenaro tme (month) Fgure 1: Dfferent demand scenaros wth dscretzaton of tme n month (dashed lnes) and n weeks (sold lnes) A problem arses f the dscretzaton of tme becomes too small. The shorter the lnear peces n the demand functons, the more varables and constrants n the resultng MILP. Ths s the reason why the results n secton 4.3 are all based on dscretzaton n months or weeks. The dscretzaton n days also does not lead to hgh savngs compared to the one n weeks. Besdes the tme dscretzaton, the number of scenaros ncluded n the model s the other quantty that s crtcal for the mere sze and therefore to the computng tme of the SGSM. Therefore, we check the effectvty of SAA wth scenaro reducton n our tests. 11

12 4.2. Test Data We checked the SGSM on real data (nventory costs and demand ntenstes for 1127 spare parts) from our partner, a large German car manufacturer wth a star-shaped two-echelon spare part dstrbuton wth one master warehouse (no. 0) and seven warehouses (nos. 1 7) for end customer servce provders n the US. The model SGSM s not restrcted to ths specal structure; t can be appled to any acyclc network structure. The tme horzon was chosen to be 25 months. The demand n the leaves of the network was generated randomly accordng to Posson dstrbutons wth the gven ntenstes from our hstorcal data. Devatons of the delvery tmes up to 20% were randomly generated. Stochastc data was dentcally reproduced for all polces under consderaton. Replenshment orders n the smulaton are trggered by (s,s)-polces, where the values for s are chosen by the models under consderaton. Note: The expected servce tmes n the smulaton are always equal to the servce tmes computed n the respectve models,.e., on average there are no early delveres(ths s debatable; other optons are work n progress). Moreover, the nventory costs n both the GSM and the SGSM are only approxmatons of the actual nventory costs. The smulaton reports the actual (lnear) nventory costs. The SGSM produced scenaros by samplng from the Posson dstrbuton and was solved by Sample Average Approxmaton (SAA) (see Secton 3.1. We tred a varyng number of samples and scenaro reducton as descrbed n Secton 3.2. The network topology was easy enough for all nstances to solve n less than an hour for the testassortment of 1127 parts n the MILP solver gurob 3.0 up to an optmalty gap of fve percent. The calculaton was carred out on a standard PC (CPU: Intel(R) Core(TM) 2 Quad CPU 2.83 GHz, Mem: 8GB RAM) usng ubuntu In order to fnd out whether stochastc modellng as such has a postve mpact on the result, we tred dfferent parameter settngs n the smulaton experments. The GSM s parametrzed by the prescrbed servce level: we nvestgated the GSM wth 90% and wth 96% servce level called GSM(90%) and GSM(96%). Moreover, n order to substantate the beneft of a network model as opposed to a decentralzed optmzaton of each separate warehouse, we gve results for the decentralzed polces DEZ(90%) and DEZ(96%) for a servce level of 90% and 96%, respectvely. In these models each locaton tres to reach the gven servce level target. The cost coeffcents are taken from cost estmates of our partner for nventory cost and the pece-based recourse cost (so-called non-sales ). These coeffcents are part and warehouse dependent and cannot be lsted here Computatonal Results Table 1 shows the benefts of samplng ffty scenaros followed by a reducton to three compared to samplng three scenaros. The results presented n ths 12

13 table are the average costs of ten calculatons wth the gven number of scenaros generated. The demands and delvery tmes are dentcal n all the smulatons. Table 1: results applyng the SGSM wth dfferent scenaro reducton technques Reducton Inventory Cost Recourse Cost Total Costs no symmetrc asymmetrc no We can see an enormous reducton n the total costs by applyng the reducton technques ntroduced n secton 3. In the case of generatng only three scenaros we observe a very hgh varablty n the costs over the ten smulaton runs. Durng ten smulatons, the mnmal total costs were , and the maxmal total costs were Applyng the symmetrc/asymmetrc reducton technque the mnmal total costs were / and the maxmal total costs were / , respectvely. The costs occurrng n the sngle smulaton runs are lsted n Appendx A. These results show that applyng scenaro reducton leads to a much lower varablty n the costs because also scenaros wth small probablty are taken nto account. We can see that the results for the asymmetrc reducton are qute close to those where all the ffty generated scenaros are ncluded n the model. Table 2 ncludes the servce levels n the dfferent locatons durng the frst of the ten smulaton runs. Table 2: Comparson of servce levels (%) Warehouse sym 50 3 asym The servce levels n table 2 show the dfference between the symmetrc and the (new) asymmetrc reducton technque. The asymmetrc technque takes nto account that for many parts the quotent h /c s greater for the leaf warehouses than for the master warehouse. Therefore, for the symmetrc technque we get a hgher servce level n the master warehouse (no. 0), but lower servce levels n the warehouses (nos. 1 7). 13

14 Smulatng the stuaton modeled n the SGSM wth smple recourse leads to the results lsted n table 3. Table 3: Results of smulaton wth posson dstrbuted demand and equal dstrbuted delvery tme Model Inventory Cost Recourse Cost Total Cost (1) DEZ 90% (2) DEZ 96% (3) GSM 90% (4) GSM 96% (5) SGSM 50, months , (6) SGSM , months, sym , ,46 (7) SGSM , months, asym (8) SGSM 200 1, months (9) SGSM , weeks, sym (10) SGSM , weeks, asym Ths table ncludes the average costs of the dfferent approaches. Here we calculated the orderponts s usng all the dfferent methods and run the smulaton ten tmes wth dfferent demand and delvery tme. For all dfferent approaches the demand and delvery tme n the smulaton s dentcal. The results for the decentralzed method are a bt worse than the results, when the orderponts are calculated by the GSM. Usng one of the lsted SGSM approaches leads to a cost reducton of 30% and more. Agan, the asymmetrc scenaro reducton domnates the symmetrc one. Another mportant aspect to notce s that the results usng a dscretzaton of tme n weeks are remarkably better than results usng a dscretzaton n month. Results for each of the ten smulaton runs for Model (4) and (10) can be found n Appendx A. In Method (8) a specal heurstc s appled (dfferent from the fast forward reducton) that tres to fnd a crtcal scenaro of the delvery tme and the demand for every locaton. Ths shows that much of the problem s structure can be encoded nto a sngle scenaro. Ths heurstc works properly for the dscretzaton n months and may be extended to fner dscretzaton. Ths s work n progress. The resultng servce levels for the dfferent methods n the frst smulatons are shown n table 4. Table 4: Comparson of servce levels (%) Warehouse (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

15 The dfferences n the servce levels of the symmetrc and the asymmetrc reducton are no longer substantal. The reason s that now the number of scenaros n the set S s much hgher; thus, both approaches lead to a good approxmaton of the dstrbuton and ts mpact on resultng servce levels. As table 3 shows, the dfferences n the resultng costs are stll remarkable. Ths s due to more scenaros n the more relevant parts of the dstrbuton n the asymmetrc reducton (hgh demand and delvery tme f h /c s low and vce versa). The results of smulatons of the SGSM wth external supplers from whch mssng parts can be ordered and lost sales (ntroduced n subsecton 2.3) are lsted n Table 5: Table 5: Results of smulaton wth external supplers Method Inventory costs Recourse Costs Total Costs (1) DEZ 90% (2) DEZ 96% (3) GSM 90% (4) GSM 96% (5) SGSM 100, weeks (6) SGSM , weeks, sym (7) SGSM , weeks, asym (8) SGSM , weeks, sym (9) SGSM , weeks, asym The smulaton works a lttle bt dfferent to the one appled n Tables 1 4. Here the demand that can not be delvered mmedately from the warehouses (nos. 1 7) to the end customers s lost. If the warehouses have not enough stock to delver the ordered parts, there s the possblty to buy these parts from an external suppler. Ths recourse acton causes costs dependng on the dstance between the warehouse and the external suppler. The suppler tself has lmted stock so that the warehouses are not able to order any amount from them. If a demand at a warehouse can be nether delvered from stock nor ordered from an external suppler, the demand s lost. Internal orders (from a warehouse to the master warehouse) are stll backlogged, and the master warehouse delvers the demand as soon as possble to the orderng warehouse. The orderng costs and the capactes of the external supplers are not ncluded n the data of our partner, so we had to set them artfcally. As we can see n the results oftable 5, the decentralzedmodel and the GSM perform much better n the case wth only one knd of uncertanty (demand uncertanty) than n the case of both, demand and delvery tme uncertanty. The SGSM stll outperforms the determnstc models achevng 10 20% of cost savngs. Table 6 show the resultng servce levels of the dfferent methods. Here the servce levels of the SGSM approaches are very smlar to these of the decentralzed model and the GSM, both wth a prescrbed servce level of 15

16 Table 6: Comparson of servce levels (%) Warehouse (1) (2) (3) (4) (5) (6) (7) (8) (9) %. The costsn table 5tell us that the SGSM treatsdfferent partsdfferently, whle the GSM and the decentralzed model cover 96% of the demand for every part, no matter what the costs h and c j are. Ths s the reason why the orderponts calculated by the SGSM can lead to cheaper nventory costs and recourse costs at the same tme. Last we want to compare our model to a model that was ntroduced by Doǧru, de Kok and van Houtum, see [11]. In the followng we wll refer to ths model as DoKoHo. In the smulaton we need to apply fx lead tmes as ths s one assumpton of the DoKoHo model. We smulate a stuaton that fts to the DoKoHo assumptons where demand s backlocked and there are penalty costs f a locaton s not able to delver as demanded. Table 7 shows the results of the smulaton for the GSM, the SGSM and DoKoHo. There are some dfferent parameter settngs for DoKoHo where the penalty costs used n the model are multpled by a factor (γ). Table 7: Results of smulaton wth posson dstrbuted demand and fx lead tme Model Inventory Cost Recourse Cost Total Cost (1) DoKoHo (γ = 1) (2) DoKoHo (γ = 5) (3) DoKoHo (γ = 10) (4) GSM 96% (5) SGSM , weeks, asym The DoKoHo model outperforms the GSM but causes hgher costs than the SGSM. The smulaton of the dfferent models lead to the servce levels that are shown n table 8. The reason for the very low servce levels at the master warehouse usng the DoKoHo model compared to the ones usng GSM or SGSM can be explaned easly. In the DoKoHo model there are no explct servce tmes guaranteed to the successors. But the lower performance of the master warehouse s consdered when the successor s safety stock s calculated. In the smulaton we use a servce tme of zero for ths case so the delvery of master warehouse s often late. 16

17 Table 8: Comparson of servce levels (%) Warehouse (1) (2) (3) (4) (5) As we do not consder penalty costs for master warehouse n the smulaton, ths does not effect the costs that we get applyng the DoKoHo models. 5. Concluson We have ntroduced the Stochastc Guaranteed Servce Model (SGSM), a stochastc programmng verson of the Guaranteed Servce Model (GSM) for the computaton of safety stock levels n a mult-echelon spare part dstrbuton system of a large German car manufacturer. Whereas the GSM makes assumptons that requre extreme demand scenaros and mssed delvery dates to be handled outsde the model, the SGSM s capable of ncorporatng these volatltes nsde the model, thereby accountng for the correspondng cost. The stochastcty needs to be captured by suffcently large sample szes: n our example we generated 200 scenaros most of the tme and reduced them to 50 applyng modfed scenaro reducton technques. The resultng MILP models could be solved straght-forwardly n our example. The SGSM makes some assumptons that are only approxmatons of realty (complete recourse, pecewse lnear demand). However, our smulaton was not restrcted by these assumptons; t only checked the resultng polces, no matter what they assumed, and accounted for all the occurrng costs. And n ths qute realstc smulaton experment, the polces calculated wth the SGSM performed extremely well. One reason for ths s that the SGSM can have structurally dfferent optmal solutons than the GSM: not all optmal SGSM solutons are extreme n the space of varables of the GSM. Thus the SGSM sometmes fnds solutons that the GSM can never provde, no matter whch parameter settng. And such solutons domnated GSM solutons n our smulatons. We therefore thnk that the SGSM can be appled routnely n spare part dstrbuton systems lke the one of our partner. Next, we wll model the realworld recourse actons n more detal n order to fnd more realstc recourse cost values. 17

18 Appendx A. Results of the smulaton runs In ths secton we show some of the numercal results n detal. The average costs of the ten smulaton runs lsted here are gven n tables 1 and 3. The tables A.9 A.12 nclude the results that lead to the average costs of table 2. Table A.9: No reducton (3 3) Run Inventory Costs Recourse Costs Total Costs Average Table A.10: Symmetrc reducton (50 3) Run Inventory Costs Recourse Costs Total Costs Average Table A.13 and A.14 nclude the results for the sngle runs of GSM 96% (4) and SGSM , weeks, asym (10) of table 3. References [1] Smpson, In-process nventory, Operatons Research 6 (1958) [2] S. Graves, S. Wllems, Optmzng strategc safety stock placement n supply chans, Manufacturng & Servce Operatons Management 2 (1) (2000)

19 Table A.11: Asymmetrc reducton (50 3) Run Inventory Costs Recourse Costs Total Costs Average Table A.12: No Reducton (50 50) Run Inventory Costs Recourse Costs Total Costs Average [3] T. Magnant, Z.-J. Shen, J. Shu, D. Smch-Lev, C.-P. Teo, Inventory placement n acyclc supply chan networks, Operatons Research Letters 34 (2006) [4] K. Schade, Lagerhaltungsstratege für mehrstufge Lagerhaltung n der Automoblndustre, Master s thess, Unverstät Bayreuth (2008). [5] K. Inderfurth, Safety stock optmzaton n mult-stage nventory systems, Internatonal Journal of Producton Economcs 24 (1-2) (1991) do: / (91)90157-o. URL [6] K. Inderfurth, Safety stocks n multstage dvergent nventory systems: A survey, nternatonal journal of producton economcs 35 (1994) [7] S. Mnner, Dynamc programmng algorthms for mult-stage safety stock optmzaton, OR Spectrum 19 (1997) , /BF URL 19

20 Table A.13: GSM wth a prescrbed servce level of 96% Run Inventory Costs Recourse Costs Total Costs Average Table A.14: SGSM asymmetrc reducton wth tme dscretzaton n weeks Run Inventory Costs Recourse Costs Total Costs Average [8] A. Clark, H. Scarf, Optmal polces for a mult-echelon nventory problem, Management Scence 6 (1960) [9] C. Sherbrooke, Metrc: A mult-echelon technque for recoverable tem control, Operatons Research 16 (1968) [10] A. Daz, M. C. Fu, Mult-echelon models for reparable tems: A revew, Document n Decson, Operatons & Informaton Technologes Research Works Unversty of Maryland (2005). URL [11] M. Doǧru, A. de Kok, G. van Houtum, Optmal control of one-warehouse mult-retaler systems wth dscrete demand, Workng paper (2005). [12] J. R. Brge, F. Louveaux, Introducton to Stochastc Programmng, Sprnger, [13] J. Rambau, K. Schade, The stochastc guaranteed servce model wth recourse for mult-echelon warehouse management, n: Proceedngs of the 20

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