The Stochastic Guaranteed Service Model with Recourse for MultiEchelon Warehouse Management


 Whitney Hood
 2 years ago
 Views:
Transcription
1 The Stochastc Guaranteed Servce Model wth Recourse for MultEchelon Warehouse Management Jörg Rambau, Konrad Schade 1 Lehrstuhl für Wrtschaftsmathematk Unverstät Bayreuth Bayreuth, Germany Abstract The Guaranteed Servce Model (GSM) computes optmal orderponts 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 realworld data of a large German car manufacturer show that polces based on the SGSM domnate GSMpolces. Keywords: Multechelon nventory control, guaranteed servce model, stochastc programmng, nteger lnear programmng, realworld 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 socalled 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 orderponts s for the wdely accepted (s, S)polces n multechelon 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 hgherdmensonal decsons n a blackbox. 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 multechelon 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 multechelon 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 nontrval complete recourse consstng of a transportaton opton besdes the penalty cost for nonsales,.e., requested parts that cannot be delvered n tme. We solve the SGSM by a combnaton of sample average approxmaton wth stateofthe 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 realworld data of a large German automoble manufacturer and Possondstrbuted demand wth realworld ntensty forecasts show that our nventory polces based on the SGSM domnate GSMpolces 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 nonsales 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 realworld 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 mxednteger 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 GuaranteedServceModel The GSM ILP follows the orgnal work n [3], except for the ntegralty of the orderponts, whch s mandatory n sparepart systems wth occasonally large, expensve parts at very small stocklevels. 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 endcustomers) 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 ) orderpont 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 pecewselnear modellng technques wth addtonal bnary varables, ths model can approxmately be transformed nto an ILP (see [3]) The Stochastc GuaranteedServceModel 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 realworld 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 nonsale or mssed lead tme. The hope s that the SGSM can balance nventory costs and nonsales 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 twostage 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 topayapenaltyforanonsaleftcannotdelvertheorderedparts. 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 wrapup the bascs about Sample AverageApproxmaton (SAA) Methods for general dscrete approxmatons of probablty dstrbutons n Subsecton SAAMethod 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 pmedan problem. In order to save computaton tme, we chose to apply the socalled 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 fastforward scenaro reducton,.e., t s approxmated by a dscrete dstrbuton over a subset of scenaros of prescrbed cardnalty, whch have, n general, nonunform 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 realworld data from our partner General Issues Before we report on our tests, we want to make some general remarks concernng some sdeeffects of modelngdecsons 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 straghtforward 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 starshaped twoechelon 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 pecebased recourse cost (socalled nonsales ). 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 multechelon 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 straghtforwardly 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, Inprocess 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. SmchLev, 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 multstage nventory systems, Internatonal Journal of Producton Economcs 24 (12) (1991) do: / (91)90157o. 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 multstage 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 multechelon nventory problem, Management Scence 6 (1960) [9] C. Sherbrooke, Metrc: A multechelon technque for recoverable tem control, Operatons Research 16 (1968) [10] A. Daz, M. C. Fu, Multechelon 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 onewarehouse multretaler 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 multechelon warehouse management, n: Proceedngs of the 20
PowerofTwo Policies for Single Warehouse MultiRetailer Inventory Systems with Order Frequency Discounts
Powerofwo Polces for Sngle Warehouse MultRetaler Inventory Systems wth Order Frequency Dscounts José A. Ventura Pennsylvana State Unversty (USA) Yale. Herer echnon Israel Insttute of echnology (Israel)
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationOn the Optimal Control of a Cascade of HydroElectric Power Stations
On the Optmal Control of a Cascade of HydroElectrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;
More informationProject Networks With MixedTime Constraints
Project Networs Wth MxedTme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa
More information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More informationCommunication Networks II Contents
8 / 1  Communcaton Networs II (Görg)  www.comnets.unbremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP
More information2008/8. An integrated model for warehouse and inventory planning. Géraldine Strack and Yves Pochet
2008/8 An ntegrated model for warehouse and nventory plannng Géraldne Strack and Yves Pochet CORE Voe du Roman Pays 34 B1348 LouvanlaNeuve, Belgum. Tel (32 10) 47 43 04 Fax (32 10) 47 43 01 Emal: corestatlbrary@uclouvan.be
More informationSimulation and optimization of supply chains: alternative or complementary approaches?
Smulaton and optmzaton of supply chans: alternatve or complementary approaches? Chrstan Almeder Margaretha Preusser Rchard F. Hartl Orgnally publshed n: OR Spectrum (2009) 31:95 119 DOI 10.1007/s002910070118z
More informationIn some supply chains, materials are ordered periodically according to local information. This paper investigates
MANUFACTURING & SRVIC OPRATIONS MANAGMNT Vol. 12, No. 3, Summer 2010, pp. 430 448 ssn 15234614 essn 15265498 10 1203 0430 nforms do 10.1287/msom.1090.0277 2010 INFORMS Improvng Supply Chan Performance:
More informationRobust Design of Public Storage Warehouses. Yeming (Yale) Gong EMLYON Business School
Robust Desgn of Publc Storage Warehouses Yemng (Yale) Gong EMLYON Busness School Rene de Koster Rotterdam school of management, Erasmus Unversty Abstract We apply robust optmzaton and revenue management
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationRetailers must constantly strive for excellence in operations; extremely narrow profit margins
Managng a Retaler s Shelf Space, Inventory, and Transportaton Gerard Cachon 300 SH/DH, The Wharton School, Unversty of Pennsylvana, Phladelpha, Pennsylvana 90 cachon@wharton.upenn.edu http://opm.wharton.upenn.edu/cachon/
More informationEuropean Journal of Operational Research
European Journal of Operatonal Research 221 (2012) 317 327 Contents lsts avalable at ScVerse ScenceDrect European Journal of Operatonal Research journal homepage: www.elsever.com/locate/ejor Producton,
More informationOutsourcing inventory management decisions in healthcare: Models and application
European Journal of Operatonal Research 154 (24) 271 29 O.R. Applcatons Outsourcng nventory management decsons n healthcare: Models and applcaton www.elsever.com/locate/dsw Lawrence Ncholson a, Asoo J.
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationBenefits of MultiEchelon Inventory Control
2012, Lund Lund Unversty Lund Insttute of Technology Dvson of Producton Management Department of Industral Management and Logstcs Benefts of MultEchelon Inventory Control A case study at Tetra Pak Author:
More informationCredit Limit Optimization (CLO) for Credit Cards
Credt Lmt Optmzaton (CLO) for Credt Cards Vay S. Desa CSCC IX, Ednburgh September 8, 2005 Copyrght 2003, SAS Insttute Inc. All rghts reserved. SAS Propretary Agenda Background Tradtonal approaches to credt
More informationThe Development of Web Log Mining Based on ImproveKMeans Clustering Analysis
The Development of Web Log Mnng Based on ImproveKMeans Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More information8 Algorithm for Binary Searching in Trees
8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the
More informationStochastic Inventory Management for Tactical Process Planning under Uncertainties: MINLP Models and Algorithms
Stochastc Inventory Management for Tactcal Process Plannng under Uncertantes: MINLP Models and Algorthms Fengq You, Ignaco E. Grossmann Department of Chemcal Engneerng, Carnege Mellon Unversty Pttsburgh,
More informationPreventive Maintenance and Replacement Scheduling: Models and Algorithms
Preventve Mantenance and Replacement Schedulng: Models and Algorthms By Kamran S. Moghaddam B.S. Unversty of Tehran 200 M.S. Tehran Polytechnc 2003 A Dssertaton Proposal Submtted to the Faculty of the
More informationCan Auto Liability Insurance Purchases Signal Risk Attitude?
Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? ChuShu L Department of Internatonal Busness, Asa Unversty, Tawan ShengChang
More informationA DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATIONBASED OPTIMIZATION. Michael E. Kuhl Radhamés A. TolentinoPeña
Proceedngs of the 2008 Wnter Smulaton Conference S. J. Mason, R. R. Hll, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. A DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATIONBASED OPTIMIZATION
More informationA hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm
Avalable onlne www.ocpr.com Journal of Chemcal and Pharmaceutcal Research, 2014, 6(7):18841889 Research Artcle ISSN : 09757384 CODEN(USA) : JCPRC5 A hybrd global optmzaton algorthm based on parallel
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a twostage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationTHE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek
HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo
More informationMany etailers providing attended home delivery, especially egrocers, offer narrow delivery time slots to
Vol. 45, No. 3, August 2011, pp. 435 449 ssn 00411655 essn 15265447 11 4503 0435 do 10.1287/trsc.1100.0346 2011 INFORMS Tme Slot Management n Attended Home Delvery Nels Agatz Department of Decson and
More informationEfficient Project Portfolio as a tool for Enterprise Risk Management
Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse
More informationFormulating & Solving Integer Problems Chapter 11 289
Formulatng & Solvng Integer Problems Chapter 11 289 The Optonal Stop TSP If we drop the requrement that every stop must be vsted, we then get the optonal stop TSP. Ths mght correspond to a ob sequencng
More informationAn MILP model for planning of batch plants operating in a campaignmode
An MILP model for plannng of batch plants operatng n a campagnmode Yanna Fumero Insttuto de Desarrollo y Dseño CONICET UTN yfumero@santafeconcet.gov.ar Gabrela Corsano Insttuto de Desarrollo y Dseño
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationJ. Parallel Distrib. Comput.
J. Parallel Dstrb. Comput. 71 (2011) 62 76 Contents lsts avalable at ScenceDrect J. Parallel Dstrb. Comput. journal homepage: www.elsever.com/locate/jpdc Optmzng server placement n dstrbuted systems n
More informationNONCONSTANT SUM REDANDBLACK GAMES WITH BETDEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia
To appear n Journal o Appled Probablty June 2007 OCOSTAT SUM REDADBLACK GAMES WITH BETDEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate
More informationOptimal Joint Replenishment, Delivery and Inventory Management Policies for Perishable Products
Optmal Jont Replenshment, Delvery and Inventory Management Polces for Pershable Products Leandro C. Coelho Glbert Laporte May 2013 CIRRELT201332 Bureaux de Montréal : Bureaux de Québec : Unversté de
More informationSIMULATION OF INVENTORY CONTROL SYSTEM FOR SUPPLY CHAIN PRODUCER WHOLESALER CLIENT IN EXTENDSIM ENVIRONMENT
SIMULATION OF INVENTOY CONTOL SYSTEM FO SUPPLY CHAIN PODUCE WHOLESALE CLIENT IN EXTENDSIM ENVIONMENT Eugene Kopytov and Avars Muravjovs Transport and Telecommuncaton Insttute, Lomonosov Street, ga, LV09,
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 /  Communcaton Networks II (Görg) SS20  www.comnets.unbremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More informationForecasting the Direction and Strength of Stock Market Movement
Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract  Stock market s one of the most complcated systems
More informationOptimization of network mesh topologies and link capacities for congestion relief
Optmzaton of networ mesh topologes and ln capactes for congeston relef D. de Vllers * J.M. Hattngh School of Computer, Statstcal and Mathematcal Scences Potchefstroom Unversty for CHE * Emal: rwddv@pu.ac.za
More informationFeature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College
Feature selecton for ntruson detecton Slobodan Petrovć NISlab, Gjøvk Unversty College Contents The feature selecton problem Intruson detecton Traffc features relevant for IDS The CFS measure The mrmr measure
More informationOmega 39 (2011) 313 322. Contents lists available at ScienceDirect. Omega. journal homepage: www.elsevier.com/locate/omega
Omega 39 (2011) 313 322 Contents lsts avalable at ScenceDrect Omega journal homepage: www.elsever.com/locate/omega Supply chan confguraton for dffuson of new products: An ntegrated optmzaton approach Mehd
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMISP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationDEVELOPMENT & IMPLEMENTATION OF SUPPLY CHAIN MANAGEMENT FOCUSING ON PROCUREMENT PROCESSES & SUPPLIERS 1.
1. Claudu V. KIFOR, 2. Amela BUCUR, 3. Muhammad Arsalan FAROOQ DEVELOPMENT & IMPLEMENTATION OF SUPPLY CHAIN MANAGEMENT FOCUSING ON PROCUREMENT PROCESSES & SUPPLIERS 1. LUCIAN BLAGA UNIVERSITY SIBIU, RESEARCH
More informationFault tolerance in cloud technologies presented as a service
Internatonal Scentfc Conference Computer Scence 2015 Pavel Dzhunev, PhD student Fault tolerance n cloud technologes presented as a servce INTRODUCTION Improvements n technques for vrtualzaton and performance
More informationAn Analysis of Central Processor Scheduling in Multiprogrammed Computer Systems
STANCS73355 I SUSE73013 An Analyss of Central Processor Schedulng n Multprogrammed Computer Systems (Dgest Edton) by Thomas G. Prce October 1972 Techncal Report No. 57 Reproducton n whole or n part
More informationExtending Probabilistic Dynamic Epistemic Logic
Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σalgebra: a set
More informationAn InterestOriented Network Evolution Mechanism for Online Communities
An InterestOrented Network Evoluton Mechansm for Onlne Communtes Cahong Sun and Xaopng Yang School of Informaton, Renmn Unversty of Chna, Bejng 100872, P.R. Chna {chsun,yang}@ruc.edu.cn Abstract. Onlne
More informationTraffic State Estimation in the Traffic Management Center of Berlin
Traffc State Estmaton n the Traffc Management Center of Berln Authors: Peter Vortsch, PTV AG, Stumpfstrasse, D763 Karlsruhe, Germany phone ++49/72/965/35, emal peter.vortsch@ptv.de Peter Möhl, PTV AG,
More informationDynamic Pricing for Smart Grid with Reinforcement Learning
Dynamc Prcng for Smart Grd wth Renforcement Learnng ByungGook Km, Yu Zhang, Mhaela van der Schaar, and JangWon Lee Samsung Electroncs, Suwon, Korea Department of Electrcal Engneerng, UCLA, Los Angeles,
More informationSCHEDULING OF CONSTRUCTION PROJECTS BY MEANS OF EVOLUTIONARY ALGORITHMS
SCHEDULING OF CONSTRUCTION PROJECTS BY MEANS OF EVOLUTIONARY ALGORITHMS Magdalena Rogalska 1, Wocech Bożeko 2,Zdzsław Heduck 3, 1 Lubln Unversty of Technology, 2 Lubln, Nadbystrzycka 4., Poland. Emal:rogalska@akropols.pol.lubln.pl
More informationRiskbased Fatigue Estimate of Deep Water Risers  Course Project for EM388F: Fracture Mechanics, Spring 2008
Rskbased Fatgue Estmate of Deep Water Rsers  Course Project for EM388F: Fracture Mechancs, Sprng 2008 Chen Sh Department of Cvl, Archtectural, and Envronmental Engneerng The Unversty of Texas at Austn
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationDynamic optimization of the LNG value chain
Proceedngs of the 1 st Annual Gas Processng Symposum H. Alfadala, G.V. Rex Reklats and M.M. ElHalwag (Edtors) 2009 Elsever B.V. All rghts reserved. 1 Dynamc optmzaton of the LNG value chan Bjarne A. Foss
More informationRealistic Image Synthesis
Realstc Image Synthess  Combned Samplng and Path Tracng  Phlpp Slusallek Karol Myszkowsk Vncent Pegoraro Overvew: Today Combned Samplng (Multple Importance Samplng) Renderng and Measurng Equaton Random
More informationA method for a robust optimization of joint product and supply chain design
DOI 10.1007/s1084501409085 A method for a robust optmzaton of jont product and supply chan desgn Bertrand BaudLavgne Samuel Bassetto Bruno Agard Receved: 10 September 2013 / Accepted: 21 March 2014
More informationInequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.
Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.
More informationJoint Scheduling of Processing and Shuffle Phases in MapReduce Systems
Jont Schedulng of Processng and Shuffle Phases n MapReduce Systems Fangfe Chen, Mural Kodalam, T. V. Lakshman Department of Computer Scence and Engneerng, The Penn State Unversty Bell Laboratores, AlcatelLucent
More informationAnalysis of Premium Liabilities for Australian Lines of Business
Summary of Analyss of Premum Labltes for Australan Lnes of Busness Emly Tao Honours Research Paper, The Unversty of Melbourne Emly Tao Acknowledgements I am grateful to the Australan Prudental Regulaton
More informationAn Evaluation of the Extended Logistic, Simple Logistic, and Gompertz Models for Forecasting Short Lifecycle Products and Services
An Evaluaton of the Extended Logstc, Smple Logstc, and Gompertz Models for Forecastng Short Lfecycle Products and Servces Charles V. Trappey a,1, Hsnyng Wu b a Professor (Management Scence), Natonal Chao
More informationOptimization under uncertainty. Antonio J. Conejo The Ohio State University 2014
Optmzaton under uncertant Antono J. Conejo The Oho State Unverst 2014 Contents Stochastc programmng (SP) Robust optmzaton (RO) Power sstem applcatons A. J. Conejo The Oho State Unverst 2 Stochastc Programmng
More informationActivity Scheduling for CostTime Investment Optimization in Project Management
PROJECT MANAGEMENT 4 th Internatonal Conference on Industral Engneerng and Industral Management XIV Congreso de Ingenería de Organzacón Donosta San Sebastán, September 8 th 10 th 010 Actvty Schedulng
More informationHow Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence
1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh
More informationData Broadcast on a MultiSystem Heterogeneous Overlayed Wireless Network *
JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 24, 819840 (2008) Data Broadcast on a MultSystem Heterogeneous Overlayed Wreless Network * Department of Computer Scence Natonal Chao Tung Unversty Hsnchu,
More informationMarginal RevenueBased Capacity Management Models and Benchmark 1
Margnal RevenueBased Capacty Management Models and Benchmark 1 Qwen Wang 2 Guanghua School of Management, Pekng Unversty Sherry Xaoyun Sun 3 Ctgroup ABSTRACT To effcently meet customer requrements, a
More informationMinimal Coding Network With Combinatorial Structure For Instantaneous Recovery From Edge Failures
Mnmal Codng Network Wth Combnatoral Structure For Instantaneous Recovery From Edge Falures Ashly Joseph 1, Mr.M.Sadsh Sendl 2, Dr.S.Karthk 3 1 Fnal Year ME CSE Student Department of Computer Scence Engneerng
More informationConversion between the vector and raster data structures using Fuzzy Geographical Entities
Converson between the vector and raster data structures usng Fuzzy Geographcal Enttes Cdála Fonte Department of Mathematcs Faculty of Scences and Technology Unversty of Combra, Apartado 38, 3 454 Combra,
More informationNonlinear data mapping by neural networks
Nonlnear data mappng by neural networks R.P.W. Dun Delft Unversty of Technology, Netherlands Abstract A revew s gven of the use of neural networks for nonlnear mappng of hgh dmensonal data on lower dmensonal
More informationEffective Network Defense Strategies against Malicious Attacks with Various Defense Mechanisms under Quality of Service Constraints
Effectve Network Defense Strateges aganst Malcous Attacks wth Varous Defense Mechansms under Qualty of Servce Constrants Frank YeongSung Ln Department of Informaton Natonal Tawan Unversty Tape, Tawan,
More informationThis paper concerns the evaluation and analysis of order
ORDERFULFILLMENT PERFORMANCE MEASURES IN AN ASSEMBLE TOORDER SYSTEM WITH STOCHASTIC LEADTIMES JINGSHENG SONG Unversty of Calforna, Irvne, Calforna SUSAN H. XU Penn State Unversty, Unversty Park, Pennsylvana
More informationInformation Sciences
Informaton Scences 0 (013) 45 441 Contents lsts avalable at ScVerse ScenceDrect Informaton Scences journal homepage: www.elsever.com/locate/ns A hybrd method of fuzzy smulaton and genetc algorthm to optmze
More informationSurvey on Virtual Machine Placement Techniques in Cloud Computing Environment
Survey on Vrtual Machne Placement Technques n Cloud Computng Envronment Rajeev Kumar Gupta and R. K. Paterya Department of Computer Scence & Engneerng, MANIT, Bhopal, Inda ABSTRACT In tradtonal data center
More informationx f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60
BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true
More informationA Binary Particle Swarm Optimization Algorithm for Lot Sizing Problem
Journal o Economc and Socal Research 5 (2), 2 A Bnary Partcle Swarm Optmzaton Algorthm or Lot Szng Problem M. Fath Taşgetren & YunCha Lang Abstract. Ths paper presents a bnary partcle swarm optmzaton
More informationOptimal Bidding Strategies for Generation Companies in a DayAhead Electricity Market with Risk Management Taken into Account
Amercan J. of Engneerng and Appled Scences (): 86, 009 ISSN 94700 009 Scence Publcatons Optmal Bddng Strateges for Generaton Companes n a DayAhead Electrcty Market wth Rsk Management Taken nto Account
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationThe Application of Fractional Brownian Motion in Option Pricing
Vol. 0, No. (05), pp. 738 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qngxn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes causeandeffect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationTime Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University
Tme Seres Analyss n Studes of AGN Varablty Bradley M. Peterson The Oho State Unversty 1 Lnear Correlaton Degree to whch two parameters are lnearly correlated can be expressed n terms of the lnear correlaton
More informationDynamic Fleet Management for Cybercars
Proceedngs of the IEEE ITSC 2006 2006 IEEE Intellgent Transportaton Systems Conference Toronto, Canada, September 1720, 2006 TC7.5 Dynamc Fleet Management for Cybercars Fenghu. Wang, Mng. Yang, Ruqng.
More informationMultiPeriod Resource Allocation for Estimating Project Costs in Competitive Bidding
Department of Industral Engneerng and Management Techncall Report No. 20146 MultPerod Resource Allocaton for Estmatng Project Costs n Compettve dng Yuch Takano, Nobuak Ish, and Masaak Murak September,
More informationANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING
ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 6105194390,
More informationCalculating the high frequency transmission line parameters of power cables
< ' Calculatng the hgh frequency transmsson lne parameters of power cables Authors: Dr. John Dcknson, Laboratory Servces Manager, N 0 RW E B Communcatons Mr. Peter J. Ncholson, Project Assgnment Manager,
More informationCALL ADMISSION CONTROL IN WIRELESS MULTIMEDIA NETWORKS
CALL ADMISSION CONTROL IN WIRELESS MULTIMEDIA NETWORKS Novella Bartoln 1, Imrch Chlamtac 2 1 Dpartmento d Informatca, Unverstà d Roma La Sapenza, Roma, Italy novella@ds.unroma1.t 2 Center for Advanced
More informationMAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date
Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPPATBDClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller
More informationDynamic Resource Allocation and Power Management in Virtualized Data Centers
Dynamc Resource Allocaton and Power Management n Vrtualzed Data Centers Rahul Urgaonkar, Ulas C. Kozat, Ken Igarash, Mchael J. Neely urgaonka@usc.edu, {kozat, garash}@docomolabsusa.com, mjneely@usc.edu
More informationPerformance Analysis of Energy Consumption of Smartphone Running Mobile Hotspot Application
Internatonal Journal of mart Grd and lean Energy Performance Analyss of Energy onsumpton of martphone Runnng Moble Hotspot Applcaton Yun on hung a chool of Electronc Engneerng, oongsl Unversty, 511 angdodong,
More informationErrorPropagation.nb 1. Error Propagation
ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then
More informationEVERY year, seasonal hurricanes threaten coastal areas.
1 Strategc Stockplng of Power System Supples for Dsaster Recovery Carleton Coffrn, Pascal Van Hentenryck, and Russell Bent Abstract Ths paper studes the Power System Stochastc Storage Problem (PSSSP),
More informationNumber of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000
Problem Set 5 Solutons 1 MIT s consderng buldng a new car park near Kendall Square. o unversty funds are avalable (overhead rates are under pressure and the new faclty would have to pay for tself from
More informationPerformance Analysis of Order Fulfillment for Low Demand Items in Etailing
Performance Analyss of Order Fulfllment for Low Demand Items n Etalng Pallav Chhaochhra, Stephen C Graves Massachusetts Insttute of Technology Abstract We study nventory allocaton and order fulfllment
More informationTo Fill or not to Fill: The Gas Station Problem
To Fll or not to Fll: The Gas Staton Problem Samr Khuller Azarakhsh Malekan Julán Mestre Abstract In ths paper we study several routng problems that generalze shortest paths and the Travelng Salesman Problem.
More informationAPPLICATION OF PROBE DATA COLLECTED VIA INFRARED BEACONS TO TRAFFIC MANEGEMENT
APPLICATION OF PROBE DATA COLLECTED VIA INFRARED BEACONS TO TRAFFIC MANEGEMENT Toshhko Oda (1), Kochro Iwaoka (2) (1), (2) Infrastructure Systems Busness Unt, Panasonc System Networks Co., Ltd. Saedocho
More informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy Scurve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy Scurve Regresson ChengWu Chen, Morrs H. L. Wang and TngYa Hseh Department of Cvl Engneerng, Natonal Central Unversty,
More informationgreatest common divisor
4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no
More informationStudy on Model of Risks Assessment of Standard Operation in Rural Power Network
Study on Model of Rsks Assessment of Standard Operaton n Rural Power Network Qngj L 1, Tao Yang 2 1 Qngj L, College of Informaton and Electrcal Engneerng, Shenyang Agrculture Unversty, Shenyang 110866,
More informationSUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW.
SUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW. Lucía Isabel García Cebrán Departamento de Economía y Dreccón de Empresas Unversdad de Zaragoza Gran Vía, 2 50.005 Zaragoza (Span) Phone: 976761000
More informationAnswer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 MultpleChoce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multplechoce questons. For each queston, only one of the answers s correct.
More informationSoftware project management with GAs
Informaton Scences 177 (27) 238 241 www.elsever.com/locate/ns Software project management wth GAs Enrque Alba *, J. Francsco Chcano Unversty of Málaga, Grupo GISUM, Departamento de Lenguajes y Cencas de
More information