Optiml Control of Seril, MultiEchelon Inventory (E&I) & Mixed Erlng demnds


 Rhoda Quinn
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1 Optiml Control of Seril, MultiEchelon Inventory/Production Systems with Periodic Btching GeertJn vn Houtum Deprtment of Technology Mngement, Technische Universiteit Eindhoven, P.O. Box 513, 56 MB, Eindhoven, The Netherlnds, Aln SchellerWolf Grdute School of Industril Administrtion, Crnegie Mellon University, Pittsburgh, PA , Jinxin Yi SAS Institute Inc., Cry, NC 27513, December 8, 23 Abstrct We consider singleitem, periodicreview, seril, multiechelon inventory system, with liner inventory holding nd penlty costs. In order to fcilitte shipment consolidtion nd cpcity plnning, we ssume the system hs implemented periodic btching: ech stge is llowed to order t given equidistnt times. Further, for ech stge except the most downstrem one, the reorder intervl is ssumed to be n integer multiple of the reorder intervl of the next downstrem stge. This reflects the fct tht the further upstrem in supply chin, the higher setup times nd costs tend to be, nd thus stronger btching is desired. Our model with periodic btching is direct generliztion of the seril, multiechelon model of Clrk nd Scrf (196). For this generlized model, we prove the optimlity of bsestock policies, we derive Newsboytype chrcteriztions for the optiml bsestock levels, nd we describe n efficient exct solution procedure for the cse with mixed Erlng demnds. Finlly, we present extensions to ssembly systems nd to systems with modified fill rte constrint insted of bckorder costs. Subject clssifiction: Inventory/Production: Multiechelon, stochstic demnd, periodic btching, optiml policies.
2 1 Introduction Two min costs of supply chins consist of cpcity costs nd mteril costs; hence decisions ffecting ech of these should be mde tking decisions regrding the other into ccount. Typiclly cpcity decisions re mde for longer term (> 5 yers, sy) thn mteril decisions; thus cpcity decisions re often mde first, with mterils decisions following fterwrds (nd being revisited more often). Such mteril decisions concern btching rules (often used to fcilitte cpcity prtitioning mong different products), which my be reviewed nnully, nd reorder nd bsestock levels, which my even be dpted on dily or weekly bsis (e.g., when procedures like exponentil smoothing re used for demnd forecsting). These mterils decisions, constrined to ccommodte previous cpcity choices, re the focus of this pper. Specificlly we consider the setting of optiml reorder points nd bsestock levels in multiechelon supply chins utilizing periodic btching. With respect to btching, for singleitem, singlestge sitution, with either continuous or discrete time, we distinguish the following forms: (i) Periodic btching: Think of n (R, S)policy, where R represents the reorder intervl nd S the bsestock or orderupto level. Under this policy, every R time units (or periods) n order is plced to return the inventory position to S; (ii) Fixed btch sizes: Think of n (s, Q)policy, where s represents the reorder level nd Q the fixed btch size. Ech time tht the inventory position hs dropped to or below s, one or more btches of size Q re ordered to bring the inventory position up to or bove s; (iii) A combintion of both: Think of n (R, s, Q)policy with the review intervl R, fixed btch size Q, nd reorder level s s decision vribles. Under this policy, every R time units one is llowed to plce n order, nd if t such time epoch the inventory position hs dropped to or below s, then one or more btches of size Q re ordered to bring the inventory position up to or bove s. When looking t multiple items nd/or multiple stges, one lso hs these three forms, nd further mny different combintions re possible. Which form of btching is best is very hrd question, becuse spects like setups (setup times nd costs), cpcity constrints, cpcity flexibility, nd shipment consolidtion re 1
3 ffected. In fct, cler nswer is only known for periodicreview, singleitem, singlestge system with fixed ordering costs nd convex inventory holding nd penlty costs, nd for some vrints of this system. Then n (s, S)policy is optiml (cf. Scrf, 196, Zipkin, 2, nd Porteus, 22; notice tht n (s, S)policy is equivlent to n (s, Q)policy in certin cses, e.g. under continuous review nd Poisson demnd process). Further, Ro (23) compred n optiml (R, S)policy (where both R nd S re optimized) to n optiml (s, Q)policy for singleitem, singlestge system with fixed ordering costs nd liner inventory holding nd penlty costs. Computtionl results, for instnces with Poisson demnd processes, show tht the (s, Q)policy is superior but the reltive difference in costs is limited. In subsequent study, Feng nd Ro (23) compre nested multiechelon (R, S)policies to nested multiechelon (s, Q)policies for twoechelon seril system with fixed ordering costs t both stges. Bsed on series of instnces with Poisson demnd processes, they find tht the optiml multiechelon (s, Q)policy is better thn the optiml multiechelon (R, S)policy, but the difference is smll. And, thus, (R, S)policies esily my become more ttrctive when other fctors thn just fixed ordering costs re importnt. For ll other cses, not so much is known. Even for the reltively simple Stochstic Economic Lotsizing Problem (which is multiitem, singlestge), no cler nswers re vilble. Becuse of the complexity, for the mterils decisions in supply chins, we, s mny uthors before us, dvocte use of hierrchicl pproch with two decision levels: (i) n upper level to decide on the form of btching nd the btch sizes nd reorder intervls, where multiitem, multiechelon view is tken in order to del with setups, cpcity constrints, cpcity flexibilities, nd shipment consolidtion; nd (ii) lower level to decide on reorder nd bsestock levels, where the btching rule is given nd singleitem, multiechelon view cn be tken. This is in line with the seprtion between btching nd sfety stock decisions s dvocted by Grves (1996), nd with comments of severl other uthors; see e.g. Yno nd Crlson (1988), Zipkin (2, p. 235), nd Ro (23). Models for the upper level cn be of vrious types. If only setup costs ply role, then lotsizing models with deterministic demnd my be pproprite (see e.g. Roundy, 1986). However, for rellife problems, 2
4 more sophisticted lotsizing models re often needed. Such models should work in concert with pproprite models for the lower level, which re the focus of this work. For this lower level, models with given fixed btch sizes  multiechelon (s, Q)policies  re vilble, see e.g. DeBodt nd Grves (1985), Chen (2), nd the mny references therein. Generl multiechelon systems with periodic btching  multiechelon (R, S) policies , or combintion of fixed btch sizes nd periodic btching  multiechelon (R, s, Q)policies , lck such definitive results however. This is quite surprising, given the fct tht periodic btching ws lredy recognized s common prctice to fcilitte freight consolidtions nd logistics/production scheduling nerly ten yers go; see e.g. Grves (1996, p. 4) who reports on periodic replenishments in the context of lrge retil chins such s WlMrts. Grves lso comments on the expecttion tht the replenishment schedule would be nested, wht we will refer to s the integer rtio nd synchroniztion constrint. We do not wish to imply tht periodic btching models hve been ignored in the literture, s in fct some results for specil cses re known. For twoechelon seril systems, there is recent work by Feng nd Ro (23), s mentioned bove. For twostge ssembly system with periodic btching nd normlly distributed demnd, Yno nd Crlson (1988) developed n pproximte evlution nd optimiztion procedure for instlltionstock bsestock policies. Further, number of ppers hs been devoted to twoechelon distribution systems with periodic btching. Prtil chrcteriztions of optiml policies hve been derived by Jckson (1988), McGvin, Schwrtz, nd Wrd (1993), nd Güllü nd Erkip (1996). Severl computtionl experiments were executed by these uthors nd by Jönsson nd Silver (1987), Grves (1996), nd Vn der Heijden (1999), in order to gin further insights into pproprite control rules. In ddition, Grves (1996) provides n exct evlution (but not optimiztion) for bsestock policies in multiechelon distribution systems with periodic btching nd Poisson demnd processes. Conspicuously bsent from this literture is chrcteriztion of the exct form of the optiml policy; we demonstrte this in this pper. In this pper, we study singleitem, periodicreview, seril, multiechelon inventory system with periodic btching. We ssume tht for ech stge reorder intervl hs lredy been determined, nd for ech stge except the most downstrem one the reorder intervl is n integer multiple of the reorder intervl in the next downstrem stge. We cll this the integerrtio constrint. This constrint fcilittes synchroniztion within the chin nd reflects the fct tht the further upstrem we re in supply chin, the higher setup 3
5 times nd costs tend to be, nd thus stronger btching is desired. Further, we ssume the reorder epochs re timed such tht rriving mterils t one stockpoint cn be forwrded immeditely to the next stockpoint if desired. This is clled the synchroniztion constrint. (Together, these two constrints formlize the concept of nesting of Grves, 1996.) Our model is direct generliztion of the ClrkndScrf model (Clrk nd Scrf, 196), nd extends mny of the known Clrk nd Scrf results (see next prgrph) to the periodic btching domin. We lso complement the work by Chen (2), who generlized severl of the existing results for the ClrkndScrf model to the cse with given fixed btch size per stge. As we llow btch sizes to vry, nd insted fix reorder intervls, our model is nlogous to n (R, S)policy, wheres Chen (2) is nlogous to n (s, Q)policy. For the generl ClrkndScrf model, mny results re vilble. Clrk nd Scrf (196) proved the optimlity of bsestock policies in finitehorizon setting. Federgruen nd Zipkin (1984) extended this result to the infinitehorizon cse. Alterntive proofs were given by Lngenhoff nd Zijm (199) nd Chen nd Zheng (1994). Vn Houtum nd Zijm (1991, 1997) derived Newsboytype chrcteriztions for the optiml bsestock levels, nd proved the optimlity of bsestock policies under modified service level constrint (which is equivlent to n verge bcklog constrint). In ddition, they derived n efficient exct solution procedure for the cse with mixed Erlng demnds, nd n even fster pproximte solution procedure for the generl demnd cse (see lso Vn Houtum, Inderfurth, nd Zijm, 1996). As stted bove, Chen (2) generlized the min results for the cse with given fixed btch sizes. Further, Chen nd Song (21) considered the extension to Mrkov modulted demnd, nd Gllego nd Özer (23) to specific type of dvnce demnd informtion. In both cses, the optimlity of sttedependent bsestock policies ws derived, together with n efficient lgorithm for the determintion of the optiml bsestock levels. Finlly, bounds llowing simple spredsheet computtions were derived by Shng nd Song (23). The min contributions of this pper re s follows: First, this pper derives, for the first time, the optiml policy for generl multiechelon system with given periodic btching rule. Second, we generlize mny of the existing results for the ClrkndScrf model to the periodic btching domin  we prove the optimlity of bsestock policies, derive Newsboytype chrcteriztions for the optiml bsestock levels, nd describe n efficient exct solution procedure for the cse with mixed Erlng demnds. Third, for our proofs 4
6 we follow the line sketched in Vn Houtum et l. (1996), but obtin more cler formule nd simpler exct solution procedure thn hs been vilble so fr for the ClrkndScrf model. Forth, we discuss extensions to ssembly systems, systems with γservice level constrint (modified fill rte constrint), nd systems violting the integerrtio or synchroniztion constrint. This pper is orgnized s follows. In Section 2, we discuss the model. The complete nlysis is presented in Section 3, nd extensions re discussed in Section 4. Finlly, concluding remrks re mde in Section 5. Throughout the pper, n illustrtive exmple is used to support the presenttion. 2 Model In this section, we describe our model for seril, multiechelon inventory system with periodic btching. The ssumptions re given in Subsection 2.1, the nottion is listed in Subsection 2.2, nd the objective nd the concept of echelon costs re introduced in Subsection Assumptions The inventory/production system consists of number of stges in series. Inventory my be held in stock t the end of ech stge. Time is divided into periods of equl length (w.l.o.g., the length of ech period is ssumed to be equl to 1), nd the time horizon tht we consider is infinitely long. The periods re numbered, 1,... Externl demnd occurs t the most downstrem stge only. The demnds per period re i.i.d., strictly continuously distributed rndom vribles on (, ). Ledtimes re constnt. The most upstrem stge orders t n externl supplier, which cn lwys deliver. Demnd tht cnnot be stisfied from stock t the most downstrem stockpoint is bcklogged nd stisfied s soon s possible (in FCFS order). 5
7 For ech stge, fixed reorder intervl is given. The reorder intervls re nondecresing from downstrem to upstrem in the chin, nd the reorder intervls stisfy n integerrtio constrint, i.e., for ech except the most downstrem stge, the reorder intervl is n integer multiple of the reorder intervl of the next downstrem stge. Further, the reorder time instnts re synchronized, i.e., ech stge, excluding the most upstrem stge, hs its order moments t the time instnts tht orders of the next upstrem stge rrive nd t equidistnt time instnts in between. In ech period the following events occur: (i) At ech stge, n order is plced if this is llowed in tht period; (ii) Arrivl of orders; (iii) Demnd occurs; (iv) Inventory holding nd penlty costs re chrged. The first two events re ssumed to tke plce t the beginning of the period, nd the order of these two events my be interchnged, except for the most downstrem stge when its ledtime equls. The lst event is ssumed to occur t the end of period. The third event, the demnd, my occur nywhere in between. 2.2 Nottion We define nottion below, illustrting it with n illustrtive exmple described below (see lso Figures 1 nd 2). Generl N: Number of stges of the seril system (N N, N 2). The stges (nd the corresponding stockpoints) re numbered from n = 1,..., N, where 1 denotes the most downstrem stge nd N the most upstrem stge. Ledtimes l n, n = 1,..., N: Fixed ledtime for the nth stge (l n N for n = 2,..., N, nd l 1 N := {, 1,...}). L n, n = 1,..., N: L n := N i=n l n denotes the cumultive ledtime from the externl supplier to the stockpoint t the end of stge n. For nottionl convenience, L N+1 :=. 6
8 Demnd D t1,t 2, t 1, t 2 N, t 2 t 1 : Rndom vrible which denotes the cumultive demnd over the periods t 1,..., t 2. F : Generic distribution function of the demnd D t,t in n rbitrry period t N. We ssume tht F is continuous distribution on (, ), with F () =. (Lter, when we consider the sme model with γservice level constrint, we in ddition ssume tht F hs compct support, i.e., tht F is strictly incresing from to 1 on n intervl [ n, b n ) with n < b n.) µ: Expected vlue of D t,t ; notice tht µ >. Reorder Intervls, n = 1,..., N: The given reorder intervl for the nth stge (exogenous vrible). The reorder intervls re ssumed to stisfy the integerrtio constrint, i.e., we ssume tht for ech n = 1,..., N 1, +1 / = r n N, or, equivlently = +1 /r n, where r n denotes the number of times tht stge n cn order per order of stge n + 1. Let r := R 1 (so, r denotes how often customer demnd occurs per order of stge 1) nd r N := 1. For ll n = 1,..., N, = n 1 i= r i. The r n re clled the reltive reorder frequencies. T n, n = 1,..., N: The set of periods t N in which stge n is llowed to plce n order. W.l.o.g., we ssume tht stge N plces its first order in period. Then T N := {kr N k N }, nd T n := {L n+1 + k t T N, k Z, t + L n+1 + k } for ll n = 1,..., N 1. Note tht the reorder epochs T n re offset by L n+1 to llow the lower stge to reorder from the upper t the exct moment n order rrives t the upper stge (nd equidistnt times therefter). This constrint is clled the synchroniztion constrint. Costs H n, n = 1,..., N: The inventory holding cost prmeters. A cost of H n, n = 2,..., N is chrged for ech unit tht is on stock in stockpoint n t the end of period nd for ech unit in the pipeline from the nth to the (n 1)th stockpoint. A cost of H 1 is chrged for ech unit tht is on stock in 7
9 L 2 = 1 L 1 = 2 Externl supplier l 2 = 1 l 1 = D t,t H 2 H 1, p Figure 1: The 2stge seril inventory/production system of Exmple 1. stockpoint 1 t the end of period. So, we pretend tht in ech stge the vlue of product chnges t the end of the stge only, i.e., when the product enters the stockpoint t the end of the stge. We ssume tht we hve nonnegtive dded vlue per stge, i.e., tht H 1 H 2... H N. For nottionl convenience, H N+1 :=. h n, n = 1,..., N: The dditionl inventory holding cost prmeters; h n := H n H n+1 for n = N,..., 1. Notice tht h n for ll n = 1,..., N. p: Penlty cost prmeter. A cost p is chrged per unit of bcklog t stockpoint 1 t the end of period. We ssume tht p >. Further nottion will be introduced during the nlysis s needed. Exmple 1 During this pper we use seril system consisting of N = 2 stges, with ledtimes l 1 = l 2 = 1 nd reorder intervls R 1 = 2 nd R 2 = 4, s n illustrtive exmple. This implies tht L 3 =, L 2 = 1, L 1 = 2, r 2 = 1, r 1 = 2, r = 2, T 2 = {, 4, 8,...}, T 1 = {1, 3, 5,...}. The other input vribles re specified when needed. See Figures 1 nd 2 for visuliztion of the input vribles for our exmple nd the time epochs t which orders re plced by stges 1 nd Objective Function nd Echelon Costs Let Π denote the set of ll fesible ordering policies, nd let G(π) denote the verge costs of ordering policy π for ll π Π. The objective is to find n ordering policy under which the verge costs per period re 8
10 t +1 t +2 t +3 t +4 T 2 2t t +5 2: order by stge 2 1: order by stge 1 Figure 2: Timing of orders plced by stges 1 nd 2 for the system of Exmple 1. minimized, or, equivlently, to solve: (P ) : Min G(π) s.t. π Π. Here, the verge costs consist of inventory holding costs nd penlty costs. In Section 4 we will show how the optiml ordering policy cn lso be found when the objective is to minimize the verge inventory holding costs subject to γservice level constrint. Before we strt with the nlysis we hve to introduce the concepts echelon stock nd echelon inventory position, s well s some relevnt cost functions. These re ll stndrd in multiechelon inventory theory, so we just summrize them here. For n explntion in greter depth, the reder is referred to Zipkin (2, p ). The portion of our seril supply chin from the most downstrem stockpoint 1 up to ny other stockpoint is clled n echelon. Echelons re numbered ccording to the highest stockpoint in tht echelon. The echelon stock, or echelon inventory level, of given echelon n denotes ll physicl stock t stockpoint n plus ll mterils in trnsit to or on hnd t ny stockpoint downstrem, minus possible bcklog t stockpoint 1. The echelon inventory position of n echelon n is defined s its echelon stock plus ll mterils which re in trnsit to stockpoint n. As we hve centrlized control, we my ssume w.l.o.g. tht stockpoint never orders more thn wht is vilble t the next upstrem stockpoint (cf. Chen nd Zheng, 1994). Hence, our definition of echelon inventory position is equivlent to defining the echelon inventory position s the echelon stock plus ll mterils which re on order. The echelon stock nd echelon inventory position of echelon n 9
11 re lso clled echelon stock n nd echelon inventory position n, respectively. We now define soclled costs ttched per echelon. Let x n denote echelon stock n t the end of period. Notice tht, by the bove definitions, it holds tht x n x n 1 for n = 2,..., N. Therefore we find tht the totl costs t the end of the period under considertion re equl to N H n (x n x n 1 ) + H 1 x px 1 n=2 N 1 = H N x N + (H n H n+1 )x n + (p + H 1 )x 1 = n=1 N h n x n + (p + H 1 )x 1, n=1 where x + = mx{, x} nd x = mx{, x} = min{, x} for ny x R. This formul shows tht the costs my be written s sum of cost terms per echelon. The costs h n x n re the costs ttched to echelon n (or the echelon n costs), n = 2,..., N, nd the costs h 1 x 1 + (p + H 1 )x 1 re the costs ttched to echelon 1. Notice tht the terms h n x n lwys pper, independent of the sign of x n. Nottionlly, for ech n = 1,..., N nd t N, we let IL t,n nd IP t,n denote echelon stock n (= echelon inventory level n) nd echelon inventory position n t the beginning of period t (just before the demnd occurs), nd we let C t,n denote the costs ttched to echelon n t the end of period t. 3 Anlysis We first show the reltionship between, nd direct impct of, ordering decisions of different stges, in Subsection 3.1. This constitutes the bsis for the nlysis of bsestock policies nd the derivtion of n optiml policy, in Subsections 3.2 nd 3.3, respectively. In Subsection 3.4, we describe dditionl results tht follow directly from the min results. Finlly, n exct solution procedure for the cse with mixed Erlng demnds is described in Subsection Setup of the Anlysis In this subsection, we describe the connection between ordering decisions t different stges nd which costs they ffect. 1
12 Let t be period in which stge N my plce n order; i.e., t T N. By this order, IP t,n is incresed to certin level z. We sy tht this order strts whole order cycle in the chin; the ordering decision for stge N in period t ffects whole tree of decisions. First, the decision directly ffects the ordering by stge N 1 in the periods τ k := t + l N + (k 1)R N 1, with k = 1,..., r N 1, by which echelon inventory level IL N 1 is incresed. Next, ech of the ordering decisions for stge N 1 directly ffects the ordering by stge N 2 in the periods τ k + l N 1 + (m 1)R N 2, with m = 1,..., r N 2 ; nd so on. To denote the whole tree of decisions, we introduce the following set of vectors, which hs direct correspondence with the decisions in the order cycle: A def = {(, 1,..., N ) N N+1 i = for i k 1, where k {,..., N}, nd i {1,..., r i } for k i N}. In ddition, for ech A, we define lev() s the index of the first nonzero component of, i.e., lev() def = min{i i > }. The vector (,...,, 1) is the only vector in A with lev() = N. This vector is used s the lbel for the ordering decision for echelon inventory position N t the beginning of period t. Next, the vectors (,...,, k, 1), k = 1,..., r N 1 (ech with level N 1) re used to denote the ordering decisions for stge N 1 in the periods t + l N + (k 1)R N 1 ; etceter. Define A n def = { A lev() = n}, n =,..., N. Thus the vectors A n, n = 1,..., N, denote the decisions t level n in the order cycle, with ech vector corresponding to decision epoch in specific wy. Specificlly, vector A n, n 1, denotes the ordering decision for stge n t the beginning of period t def = t + L n+1 + N i=n ( i 1)R i. The remining elements of A, i.e. the vectors A, re used to lbel the relevnt bcklogs s seen by the customers. Vector A is the lbel for the bcklog t stge 1 t the end of period t def = t + L 1 + N i= ( i 1)R i. Finlly, for ech A \ {(,...,, 1)}, we define pr() s the prent of, obtined by replcing the first nonzero component of by zero. For ech A \ A, the vectors ã A for which pr(ã) = re clled the children of. For ll A n, n 2, decision directly ffects the decisions ã A n 1 for which pr(ã) =. For ll A 1, decision directly ffects the bcklogs ã A for which pr(ã) =. 11
13 Exmple 1 (continued) For our 2stge exmple system, we hve: A = {(,, 1), (, 1, 1), (, 2, 1), (1, 1, 1), (2, 1, 1), (1, 2, 1), (2, 2, 1)}, lev((,, 1)) = 2, lev((, 1, 1)) = lev((, 2, 1)) = 1, lev((1, 1, 1)) = lev((2, 1, 1)) = lev((1, 2, 1)) = lev((2, 2, 1)) =, A 2 = {(,, 1)}, A 1 = {(, 1, 1), (, 2, 1)}, A = {(1, 1, 1), (2, 1, 1), (1, 2, 1), (2, 2, 1)}, t (,,1) = t, t (,1,1) = t + 1, t (,2,1) = t + 3, t (1,1,1) = t + 2, t (2,1,1) = t + 3, t (1,2,1) = t + 4, t (2,2,1) = t + 5, where t T 2, pr((, 1, 1)) = pr((, 2, 1)) = (,, 1), pr((1, 1, 1)) = pr((2, 1, 1)) = (, 1, 1), pr((1, 2, 1)) = pr((2, 2, 1)) = (, 2, 1). Here, e.g., the vector (, 2, 1) denotes the ordering decision tken t stge 1 t time t + 3, where t T 2. This decision is directly ffected by the prent decision (,, 1). The vector (2, 2, 1) is the lbel for the bcklog t the end of period t + 5. See Figure 3. We now describe which costs re directly ffected by the decisions A \ A, nd, while doing tht, we lso give more detiled description of the reltionship between these decisions A \ A,: Decision = (,...,, 1): This decision concerns the decision t the beginning of period t with respect to the order plced by stge N t the externl supplier. Suppose tht this order is such tht the echelon inventory position IP t,n becomes equl to some level z (,...,,1). First of ll, this decision directly ffects the echelon N costs t the end of the periods t + l N + k, k =,..., R N 1. The expected vlues of these costs re equl to E{C t +l N +k,n IP t,n = z (,...,,1) } = E{h N (z (,...,,1) D t,t +l N +k)} = h N (z (,...,,1) (l N + k + 1)µ), nd for the sum we find R N 1 k= [ E{C t+l N +k,n IP t,n = z (,...,,1) } = R N h N z (,...,,1) (l N + 1 ] 2 (R N + 1))µ. (1) 12
14 Second, decision (,...,, 1) ffects the decisions A N 1. At the beginning of period τ k = t + l N + (k 1)R N 1, k = 1,..., r N 1, echelon stock N becomes equl to IL τk,n = z (,...,,1) D t,τ k 1, nd this limits the level to which IP τk,n 1 cn be incresed t the beginning of tht period τ k. These decisions t level N 1 re the next decisions to consider. Decisions A n, for n = N 1,..., 2 (this rnge of vlues for n is empty when N = 2): Let n {2,..., N 1} nd A n. Decision concerns the decision with respect to the order plced by stge n t the beginning of period t = t + L n+1 + N i=n ( i 1)R i. Suppose tht by this order IP t,n becomes equl to some level z. First of ll, this decision directly ffects the echelon n costs t the end of the periods t + l n + k, k =,..., 1. The expected vlues of these costs re equl to E{C t+l n+k,n IP t,n = z } = E{h n (z D t,t +l n+k)} = h n (z (l n + k + 1)µ), nd for the sum we find R n 1 k= E{C t +l n +k,n IP t,n = z } = h n [z (l n + 1 ] 2 ( + 1))µ. (2) Second, decision ffects the decisions ã A n 1 for which pr(ã) =. At the beginning of period τ m := t +l n +(m 1) 1, m = 1,..., r n 1, echelon stock n becomes equl to IL τm,n = z D t,τ m 1, nd this limits the level to which IP τm,n 1 cn be incresed t the beginning of period τ m. Decisions A 1 : Let A 1. Decision concerns the decision with respect to the order plced by stge 1 t the beginning of period t = t + L 2 + N i=1 ( i 1)R i. Suppose tht by this order IP t,1 becomes equl to some level z. This decision directly ffects the echelon 1 costs t the end of the periods t + l 1 + k, k =,..., R 1 1. The expected vlues of these costs re equl to E{C t +l 1 +k,1 IP t,1 = z } = E{h 1 (z D t,t +l 1 +k) + (p + H 1 )(D t,t +l 1 +k z ) + } = h 1 (z (l 1 + k + 1)µ) + (p + H 1 )E{(D t,t +l 1+k z ) + }. 13
15 Strt of n ordering cycle Strt of next ordering cycle t t +1 t +2 t +3 t +4 t +5 Decision (,,1): IP t,2 is incresed up to z (,,1) Costs C t +k,2, k =1,2,3,4 Costs: C t +1,2 Decision (,1,1): is incresed IP t +1,1 Costs: C t +2,2 C t +2,1 Costs: C t +3,2 C t +3,1 Decision (,2,1): is incresed IP t +3,1 Costs: C t +4,2 C t +4,1 Costs: C t +5,1 up to z (,1,1) ( z (,,1) D t, t ) Costs C t +2,1 nd C t +3,1 up to z (,2,1) ( z (,,1) D t, t +2 ) Costs C t +4,1 nd C t +5,1 Figure 3: The reltionship between nd the consequences of the decisions A \ A for the system of Exmple 1. The sum of these expected costs equls R 1 1 k= E{C t+l 1+k,1 IP t,1 = z } = R 1 h 1 [z (l ] 2 (R 1 + 1))µ R 1 1 +(p + H 1 ) E{(D t,t +l 1+k z ) + }. (3) k= Figure 3 illustrtes the wy in which the bove decisions ffect ech other nd which costs re determined by them for Exmple 1. In the description bove, we hve explicitly described for ech decision N 1 n=1 A n how the level z to which IP t,lev() is incresed, is bounded from bove. We will need this in the nlysis below. Obviously, for ech decision N 1 n=1 A n, it lso holds tht the level z to which IP t,lev() is incresed, is bounded from below (by the level tht one lredy hs for echelon inventory position lev() just before the new order is plced). In the nlysis below, this is tken into ccount too. But, for the nlysis the bounding from below will pper to be less importnt. 14
16 The tree of decisions A \ A strts with decision (,...,, 1) tken in period t T N. It determines the costs C t over the corresponding cycle: C t def = N n=1 R N 1 k= C t +L n +k,n. These costs re defined for ech period t T N, nd we cll them the totl costs ttched to cycle t. Notice tht C t contins costs over different shifted time intervls for different echelons. It is esily verified tht C t lso my be written s C t = N R n 1 n=1 A n k= C t+l n+k,n. Exmple 1 (continued) Let us continue with the illustrtive exmple in order to explin the expressions for C t. Then for ech t T 2 : C t = (C t +1,2 + C t +2,2 + C t +3,2 + C t +4,2) = +(C t +2,1 + C t +3,1) + (C t +4,1 + C t +5,1) 2 R n 1 n=1 A n k= C t +l n +k,n. For ech positive recurrent policy π Π, the verge costs re equl to the verge vlue of the costs C t over ll cycles t T N divided by the cycle length R N : { T 1 1 G(π) = lim T T E t= = lim k = lim k N n=1 1 k 1 N E C jrn + kr N j= } { krn 1 1 C t,n = lim E k kr N n=1 L n 1 t= C t,n t= N n=1 } N C t,n n=1 kr N +L n 1 t=kr N C t,n k 1 1 EC jrn. (4) kr N j= The bove expression requires tht the expecttions exist nd be finite. While this need not be true for generl inventory policies (in prticulr those tht do not order sufficiently to stisfy demnd), ny policy tht is positive recurrent will meet this requirement. The clss we consider below, bsestock policies, re well known to be positive recurrent. 15
17 3.2 Bsestock Policies A relevnt clss of ordering policies is constituted by the clss of bsestock policies. A bsestock policy is denoted by tuple (y 1,..., y N ), where y n R denotes the desired orderupto level for the echelon inventory position n. Under bsestock policy (y 1,..., y N ), the ordering decisions re tken s follows: t the beginning of ech period t T N, echelon inventory position N is incresed to y N. For ech n = N 1,..., 1, t the beginning of ech period t T n, echelon inventory position n is incresed to the minimum of y n nd the ctul echelon stock of echelon n + 1 (the strt up phenomen, occurring in cse the initil echelon inventory positions re lrger thn the desired levels, re ignored, since the long run verge costs re not ffected by these). Notice tht we do not require tht the bsestock levels be nondecresing. The verge costs for bsestock policy (y 1,..., y N ) re denoted by G(y 1,..., y N ). It is esily seen tht G(y 1,..., y N ) = = 1 EC t R N 1 { N E R N R n 1 n=1 A n k= C t+l n+k,n z (,...,,1) = y N, z = min{il t,n+1, y n } for ll n = N 1,..., 1 nd A n }, (5) where the tree of decisions A \ A strts with decision (,...,, 1) t the beginning of some period t (,...,,1) = t T N (s described in the previous subsection). We now nlyze the sums 1 k= C t+l n+k,n for n = N,..., 1 nd A n, referring to formule (1) (3). The term for n = N nd = (,...,, 1) is the simplest one. The expected vlue of the costs RN 1 k= C t+l N +k,n equls (cf. (1)) R N 1 k= EC t+l N +k,n = R N h N [y N (l N + 1 ] 2 (R N + 1))µ. Next, we consider 1 k= C t +l n +k,n for n = N 1 nd A N 1. The expected vlue of this sum is equl to (cf. (2)) R N 1 1 k= EC t +l N 1 +k,n 1 = R N 1 h N 1 [E{z } (l N ] 2 (R N 1 + 1)µ, with z = min{y N D t,t 1, y N 1 }. This holds if N 3; below we describe the formule for the generl cse with N 2. The level z denotes the ctul level to which IP t,n 1 is incresed. The difference between this nd the desired level y N 1 is clled the shortfll, which cn lso be seen s bcklog t 16
18 stge N (it would be the bcklog t stge N if stge N 1 would order such tht IP t,n 1 is incresed up to y N 1, without tking into ccount how much is vilble t stge N). We denote this shortfll by B = y N 1 z = y N 1 min{y N D t,t 1, y N 1 } = (D t,t 1 (y N y N 1 )) +. (Notice tht this shortfll is if nd only if y N y N 1 nd D t,t 1 y N y N 1 ; if y N < y N 1, then this shortfll is positive by definition.) We now find tht R N 1 1 k= EC t+l N 1 +k,n 1 = R N 1 h N 1 [ y N 1 (l N (R N 1 + 1)µ EB ]. A similr expression holds for the sums 1 k= C t+l n+k,n with n N 2 nd A n. To find the generl expressions for the expected vlues of the sums 1 k= C t+l n+k,n, for the generl cse with N 2, we define B = for = (,...,, 1), (6) B = (B pr() + D tpr(),t 1 (y n+1 y n )) + for ll n = N 1,..., 1, A n, (7) B = (B pr() + D tpr(),t y 1 ) + for ll A. (8) For ech A\A, B denotes the shortfll when decision is tken, i.e., the shortfll t stge lev()+1 (red externl supplier when lev() = N) t the beginning of period t. For ech A, the rndom vrible B denotes the bcklog t stge 1 t the end of period t. Then, using (2) nd (3), it cn be shown tht R n 1 k= R 1 1 k= EC t+l n+k,n = h n [ y n (l n ( + 1))µ EB ] EC t+l 1+k,1 = R 1 h 1 [ y 1 (l (R 1 + 1))µ EB ] +(p + H 1 ) ã A,pr(ã)= EBã, A 1. By substitution of these formule into (5), we obtin the following theorem., n = N,..., 2, A n, Theorem 2 The verge costs of bsestock policy (y 1,..., y N ), with y n R for ll n = 1,..., N, re equl to G(y 1,..., y N ) = N { h n y n (l n ( + 1))µ R n=1 N +(p + H 1 ) 1 EB, R N A A n EB } where the rndom vribles B re given by (6)(8). 17
19 Proof : Strting with the substitution of the expressions given just before the theorem into eqution (5), we obtin: G(y 1,..., y N ) = = = 1 [ N R N N [y n (l n ( + 1))µ EB ] h n n=2 A n + {R 1 h 1 [y 1 (l ] 2 (R 1 + 1))µ EB n=1 A n A 1 +(p + H 1 ) ã A,pr(ã)= }] EBã R [ n h n y n (l n + 1 ] R N 2 ( + 1))µ EB +(p + H 1 ) 1 R N A 1 ã A,pr(ã)= N { h n y n (l n ( + 1))µ R n=1 N +(p + H 1 ) 1 EB, R N A EBã A n EB } where in the lst step we use tht the number of elements of A n is equl to A n = N i=n r i = R N. Note tht s the EB depend on, we cnnot mke this substitution for the remining sums A n EB in the lst expression. Theorem 2 gives very simple expression for the verge costs, subject to the evlution of the verge shortflls/bcklogs EB, A. This ide will be importnt in Subsection 3.5 Exmple 1 (continued) For our illustrtive exmple, we find tht the verge costs of bsestock policy (y 1, y 2 ), y 1, y 2 R, re equl to G(y 1, y 2 ) = h 2 {y µ} + h 1{y µ 1 2 (EB (,1,1) + EB (,2,1) )} +(p + H 1 ) 1 4 (EB (1,1,1) + EB (2,1,1) + EB (1,2,1) + EB (2,2,1) ), (9) 18
20 where B (,1,1) = (D t,t (y 2 y 1 )) +, B (,2,1) = (D t,t +2 (y 2 y 1 )) +, B (1,1,1) = (B (,1,1) + D t+1,t +2 y 1 ) +, B (2,1,1) = (B (,1,1) + D t+1,t +3 y 1 ) +, B (1,2,1) = (B (,2,1) + D t+3,t +4 y 1 ) +, B (2,2,1) = (B (,2,1) + D t+3,t +5 y 1 ) +. For the ske of the nlysis below, we now introduce cost functions G n (y 1,..., y n ), with n = 1,..., N, nd y i R for ll i = 1,..., n. The function G n (y 1,..., y n ) is defined s the verge costs ttched to the echelons 1,..., n when ech of the stges 1,..., n pplies bsestock policy with bsestock level y i nd when stge n+1 cn lwys deliver. Obviously, G N (y 1,..., y N ) = G(y 1,..., y N ). For the functions G n (y 1,..., y n ) we cn derive expressions similr to G(y 1,..., y N ). We first define A (n) def = {(, 1,..., n ) N n+1 i = for i k 1, where k {,..., n}, i {1,..., r i } for k i n 1, n = 1}, nd then define lev(), A (n) i nd pr() respective to A (n) in n nlogous mnner s we did for the set A. Then it is esily verified tht (cf. Eqution (5)) G n (y 1,..., y n ) = 1 { n E i=1 i R i 1 k= C t +l i +k,i z (,...,,1) = y n, z = min{il t,i+1, y i } for ll 1 i n 1 nd A (n) i }, (1) where the tree of decisions A (n) \ A (n) strts with decision (,...,, 1) t the beginning of some period t (,...,,1) = t T n, nd t def = t + n j=i+1 l j + n j=i ( j 1)R j, i =,..., n, A (n) i. Next, long the sme lines s Theorem 2, we find the following result. Lemm 3 For n = 1,..., N, nd y i R for ll i = 1,..., n, G n (y 1,..., y n ) = n i=1 h i { y i (l i (R i + 1))µ R i +(p + H 1 ) 1 EB (n), i } EB (n) 19
21 where the rndom vribles B (n) re defined by B (n) = for = (,...,, 1), B (n) = (B (n) pr() + D t pr(),t 1 (y i+1 y i )) + for ll 1 i n 1, A (n) i, B (n) = (B (n) pr() + D t pr(),t y 1 ) + for ll A (n). Notice tht B (N) for n = N), the B (n) = B for ll A (N) = A. For n = 1,..., N 1 (the formultion tht follows lso holds re relted to the B s follows. Let n {1,..., N 1} nd let ã A n. Then B (n) d = (B (,..., n 1,ã n,...,ã N ) Bã = ) for ll A (n), where d = denotes equlity in distribution. In other words, B (n) is equl in distribution to the shortfll of ny full Nvector on the sme decision epoch under the condition tht there is no shortfll upstrem of stge n in the Nvector. Similrly, the B (k) re relted to the B (n) for some k < n s follows. Let k, n {1,..., N} with k < n nd let ã A (n) k. Then B (k) d = (B (n) (,..., k 1,ã k,...,ã n ) B(n) ã = ) for ll A (k). (11) For ech of the functions G n (y 1,..., y n ), we lso need the prtil derivtive with respect to the lst component y n. Hence, for n = 1,..., N, we define g n (y 1,..., y n ) def = δ δy n {G n (y 1,..., y n )}, y i R for ll i = 1,..., n. For the prtil derivtives, the following result holds. Lemm 4 For n = 1,..., N, nd y i R for ll i = 1,..., n, g n (y 1,..., y n ) = n i=1 n 1 h i (p + H 1 ) 1 R i R i=1 n i > } = }g i (y 1,..., y i ) (with the convention tht the lst sum on the righthnd side is equl to when n = 1). The proof of this lemm is given in the ppendix. 2
22 3.3 Derivtion of n Optiml Policy Consider gin the tree of decisions A \ A which strts with decision (,...,, 1) in some period t T N ; see the description in Subsection 3.1. We first consider how these decisions cn be tken such tht the expected totl costs ttched to cycle t (= EC t ) re minimized. Ech decision A n, with n = 1,..., N, is described by the level z, to which echelon inventory position n is incresed t the beginning of period t. The choice for the level z is limited from bove by wht is vilble t the next upstrem stge nd from below by the vlue of echelon inventory position n just before the order is plced. For the moment, we neglect the bounding from below, nd we consider the following relxed problem (RP (t )): Min EC t = s.t. R 1 1 k= R n 1 k= N R n 1 n=1 A n k= EC t +l n +k,n EC t+l 1+k,1 = R 1 h 1 [z (l ] 2 (R 1 + 1))µ R 1 1 +(p + H 1 ) E{(D t,t +l 1+k z ) + }, A 1, k= EC t+l n+k,n = h n [z (l n + 1 ] 2 ( + 1))µ, n = 2,..., N, A n, z IL t,n+1, n = 1,..., N 1, A n, IL t,n+1 = z pr() D tpr(),t 1, n = 1,..., N 1, A n. The solution of this Nstge stochstic progrmming problem follows from the following lemm. Lemm 5 For n equl to successively 1,..., N: (i) If n = 1, then with g 1 (y 1 ) = h 1 (p + H 1 ) 1 R 1 A (1) P{B (1) > }, y 1 R, B (1) = (D tpr(),t y 1 ) + for ll A (1). If n {2,..., N}, then g n (S 1,..., S n 1, y n ) = n i=1 h i (p + H 1 ) 1 > }, y n R, 21
23 with B (n) = for = (,...,, 1), B (n) = (B (n) pr() + D t pr(),t 1 (y n S n 1 )) + B (n) = (B (n) pr() + D t pr(),t 1 (S i+1 S i )) + for ll A (n) n 1, for ll i = n 2,..., 1, A (n) i, B (n) = (B (n) pr() + D t pr(),t S 1 ) + for ll A (n). (if one or more of the S i re equl to infinity, then in these formule the S i hve to be red s if they re equl to very lrge finite constnt). (ii) g n (S 1,..., S n 1, y n ) is continuous nd nondecresing s function of y n. In prticulr, g n (S 1,..., S n 1, y n ) = (p + H n+1 ) (< ) for ll y n nd g n (S 1,..., S n 1, y n ) h n ( ) s y n. (iii) G n (S 1,..., S n 1, y n ) is convex s function of y n. (iv) Let S n ( R { }) be chosen such tht S n def = rgmin yn R G n (S 1,..., S n 1, y n ) Then S n is such tht g n (S 1,..., S n 1, S n ) =. In prticulr, S n is positive nd finite if h n > ; S n = if h n = nd F hs infinite support; S n is positive nd my be finite s well s infinite if h n = nd F hs finite support. (v) For the problem (RP(t )), it is optiml to choose ech of the levels z, A n, equl to S n, or s high s possible if this level cn not be reched. The proof of this lemm is given in the ppendix. By, Lemm 5, bsestock policy (S 1,..., S N ) is optiml for the relxed problem RP(t ). The problem ws obtined by neglecting the bounding from below when plcing orders. However, the optimlity of bsestock policy (S 1,..., S N ) holds for ech cycle t T N. If this bsestock policy is used in ll cycles, then these lower bounds t most constitute limittion during trnsient period (when the echelon inventory positions my be bove the S n, nd nothing should be ordered). Hence, in the long run bsestock policy (S 1,..., S N ) is lso fesible for the unrelxed version of RP(t ), nd hence lso optiml for this problem. Thus bsestock policy (S 1,..., S N ) is lso optiml for problem (P). 22
24 Theorem 6 Bsestock policy (S 1,..., S N ) with the S n s defined in Lemm 5 is optiml for problem (P). The optiml bsestock levels (S 1,..., S N ) stisfy the Newsboytype chrcteriztions listed in the following corollry, which immeditely follows from the prts (i)(iv) of Lemm 5. Corollry 7 The optiml bsestock levels S 1,..., S N re such tht for ech n = 1,..., N, with 1 = } = p + H n+1 p + H 1, B (n) = for = (,...,, 1), B (n) = (B (n) pr() + D t pr(),t 1 (S i+1 S i )) + for ll i = n 1,..., 1, A (n) i, B (n) = (B (n) pr() + D t pr(),t S 1 ) + for ll A (n). This corollry sys tht, when S n is determined, it is imgined tht stge n + 1 cn lwys deliver (i.e., the nlysis is limited to the chin consisting of the stges n,..., 1) nd the vlue for S n is chosen such tht the verge probbility for nonnegtive stock t the most downstrem stge 1 is equl to p+hn+1 p+h 1 (notice tht the nonstockout probbility in ny period hs cyclic pttern). Exmple 1 (continued) For our illustrtive exmple, n optiml policy (S 1, S 2 ) is s follows. By Corollry 7 nd some lgebr, we find tht bsestock level S 1 stisfies 1 { } P{D t,t 2 +1 S 1 } + P{D t,t +2 S 1 } = p + H 2, (12) p + H 1 where t is n rbitrry period in which stge 1 orders (i.e., t T 1 ). So, under the ssumption tht stge 1 never experiences shortge t stge 2, S 1 is such tht the verge nonstockout probbility per cycle of 2 periods is equl to p+h2 p+h 1. Bsestock level S 2 stisfies 1 { P{(D t,t 4 (S 2 S 1 )) + + D t +1,t +2 S 1 } + P{(D t,t (S 2 S 1 )) + + D t +1,t +3 S 1 } + P{(D t,t +2 (S 2 S 1 )) + + D t+3,t +4 S 1 } } + P{(D t,t +2 (S 2 S 1 )) + p + D t+3,t +5 S 1 } =, (13) p + H 1 23
25 where t is n rbitrry period in which stge 2 orders (i.e., t T 2 ). S 2 is such tht the verge nonstockout probbility per cycle of 4 periods is equl to (S 1, S 2 ) cn be chosen to stisfy these equtions. p p+h 1. Note tht since the demnd distribution F is continuous, 3.4 Additionl Results The following corollry sys tht it is sufficient to hold stocks t the end of the most downstrem stge 1 nd in front of stges where vlue is dded to the product; i.e., it is not necessry to hold stock in front of stges n (< N) with h n =. Corollry 8 There exists n optiml bsestock policy under which no sfety stocks re held in front of stges where zero vlue is dded. Proof : Suppose tht h n = for some stge n = 1,..., N 1. Then, by prt (iv) of Lemm 5, S n my be chosen equl to S n =. This implies tht in ech period ll goods rriving in stockpoint n + 1 re immeditely forwrded to stockpoint n. This mens tht there is never stock present in stockpoint n + 1 t the end of period. For the bsestock policies we hve not ssumed tht they hve nondecresing bsestock levels. In fct, such n ssumption would hve complicted our nlysis. The following corollry reltes generl bsestock policy to bsestock policy with nondecresing bsestock levels. Corollry 9 Let y n R for n = 1,..., N, nd define ỹ n def = min{y n,..., y N } for n = 1,..., N. Then G(ỹ 1,..., ỹ N ) = G(y 1,..., y N ). This corollry follows directly from Theorem 2. In fct, it is esily verified tht under bsestock policy (ỹ 1,..., ỹ N ) the orders re identicl to the orders generted under bsestock policy (y 1,..., y N ) (t lest in the long run; in the first periods of the horizon there my be differences). 24
26 3.5 Exct Solution Procedure for Mixed Erlng Demnds In generl, n optiml bsestock policy (S 1,..., S N ) nd the corresponding verge costs cn be computed s follows. First, for n = 1,..., N, S n my be computed by the Newsboytype chrcteriztion in Corollry 7, where in ech step bisection is pplied. Next, the optiml verge costs G(S 1,..., S N ) follow from Theorem 2. For both, it is required tht for given bsestock policy we re ble to evlute the shortflls/bcklogs B (n) s defined in Corollry 7 nd the B given by (6)(8). These shortflls/bcklogs my be determined recursively fter sufficiently fine discretiztion of the oneperiod demnd distribution F, lthough tht my be computtionlly inefficient, in prticulr s N grows lrge. We describe n lterntive, efficient procedure tht is pplicble when the oneperiod demnd is mixture of Erlng distributions with the sme scle prmeter. This is described below. Modeling demnd s mixture of Erlngs with the sme scle prmeter is motivted by the fct tht this clss of distributions is dense in the clss of ll distributions on [, ). We ssume tht the oneperiod demnd is simply given s such distribution. In prctice, however, often only the first two moments of the oneperiod demnd re given, nd then twomoment fit my be pplied first: soclled Erlng(k 1, k) distribution cn be fitted if the coefficient of vrition of the demnd is smller thn or equl to 1, nd soclled Erlng(1, k) distribution otherwise. More moments my be fit s desired, yielding lrger mixture. Let us ssume tht the demnd distribution F is mixture of Erlng distributions with the sme scle prmeter; i.e., there is discrete distribution on N, {q k } k N, nd λ > such tht F (x) = k=1 q ke k,λ (x), x R, where E k,λ denotes the Erlng distribution with k N phses nd scle prmeter λ >. We describe the evlution of the B (n) s defined in Corollry 7, for n 3. The evlution for ll other cses is similr. For = (,...,, 1), which is the only element of A (n) n, it is given tht B (n) = (which my be seen s mixture of E k,λ with ll probbility mss in. Next, we cn determine the B (n) for ech A (n) n 1. In this cse, B (n) = (D tpr(),t 1 (S n S n 1 )) +. The rndom vrible D tpr(),t 1 is the (t t pr() )fold convolution of the oneperiod demnd, nd thus is lso mixture of Erlng distributions with scle prmeter λ. The mixture probbilities re equl to the (t t pr() )fold convolution of {q k } k N. Next, if S n > S n 1, then we shift the distribution of D tpr(),t 1 to the left over distnce S n S n 1, while the probbility mss tht would rrive in the negtive rnge is bsorbed in. By tht opertion, we obtin the distribution 25
27 of (D tpr(),t 1 (S n S n 1 )) +, which is mixture of E k,λ distributions with positive probbility mss in (for the computtion of the new mixture probbilities, see SchellerWolf et l., 23). If S n S n 1, then we shift the distribution of D tpr(),t 1 to the right over distnce S n 1 S n. Then, the rndom vrible (D tpr(),t 1 (S n S n 1 )) + consists of deterministic prt plus mixture of E k,λ distributions. Next, the B (n) = (B (n) pr() + D t pr(),t 1 (S n 1 S n 2 )) + for ech A (n) n 2 my be determined, nd so on. The procedure is the sme then, except tht ech time first the convolution of D tpr(),t 1 with B (n) pr() hs to be determined. This gin will be mixture of Erlng distributions with the sme scle prmeter λ, possibly plus deterministic prt (see SchellerWolf et l., 23). The reson tht the bove procedure works is tht the clss of Erlng distributions with the sme scle prmeter, with possibly positive probbility mss in or deterministic prt being dded, is closed under the shift opertion s described bove nd when convolutions re tken. The bove procedure is simpler to implement thn the exct procedure described in Vn Houtum nd Zijm (1997) for the stndrd ClrkndScrf model. Finlly, the generl clss of phsetype distributions is likewise closed under the convolution nd shift opertion. So, n exct procedure cn lso be derived for phsetype distributions, lthough the different steps my become more complicted. If one is stisfied with ccurte pproximtions, then one my use the simple pproximte procedure bsed on twomoment fits s described in Vn Houtum nd Zijm (1991). Exmple 1 (continued) For our illustrtive exmple, we hve the formule (9), (12), nd (13) vilble. Assume tht H 2 =.5, H 1 = 1, p = 2 nd tht oneperiod demnd is exponentil with men µ = 1. Then, by the exct solution procedure, we find unique optiml bsestock levels S 1 = 6.67 nd S 2 = 9.9, nd optiml costs G(S 1, S 2 ) = The corresponding computtion time on regulr Pentium III PC is negligibly smll (only flsh). 26
28 4 Extensions 4.1 Service Level Problem We now consider our model with γservice level constrint (which is equivlent to n verge bcklog constrint) insted of penlty costs. The γservice level is lso known s modified fill rte, nd is closely relted to the regulr fill rte. For high service levels (more precisely, s long s ny demnd is not bckordered for more thn one period), both mesures re identicl (see Vn Houtum et l., 1996). Let γ be the trget γservice level. We ssume tht the demnd distribution F hs compct support. Then n optiml policy is obtined s described below. First, by Theorem 2, we find tht the γservice level of bsestock policy (y 1,..., u N ) equls γ(y 1,..., y N ) = 1 1 µ 1 R N A EB, where the rndom vribles B re given by (6)(8). Second, if for the penlty cost model with given penlty cost prmeter p, the resulting optiml policy (S 1,..., S N ) hs γservice level γ(s 1,..., S N ) = γ(p) tht is precisely equl γ, then (S 1,..., S N ) is lso optiml for the service level problem with trget service level γ (see Theorem 1 of Vn Houtum nd Zijm, 2). Third, the function γ(p) is nondecresing s function of p (see Theorem 2 of Vn Houtum nd Zijm, 2). Forth, under the ssumption tht F hs compct support, one cn show tht the optiml bsestock levels S 1,..., S N re continuous s function of p; thus, lso γ(p) is continuous s function of p, nd further γ(p) 1 s p. The bove shows tht the service level problem with given trget level γ < 1 my be solved by solving the penlty cost problem nd tuning the penlty cost prmeter p such tht the γservice level γ(p) of the optiml policy equls γ. Also, this implies tht the clss of bsestock policies is lso optiml for the service level problem with γservice level constrint. 4.2 NonSynchronized Reorder Epochs We hve ssumed tht the synchroniztion constrint is stisfied for the reorder epochs T n of the stges n = 1,..., N. If this constrint is not stisfied, then the results still hold but for system with dpted ledtimes. This is illustrted for the system of Exmple 1. Assume tht for this exmple system everything 27
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