The effect of demand distributions on the performance of inventory policies

Transcription

1 DOI /LJ_Ref_Kuhn_en_ The effec of demand disribuions on he performance of invenory policies SONJA KUHNT & WIEBKE SIEBEN FAKULTÄT STATISTIK TECHNISCHE UNIVERSITÄT DORTMUND DORTMUND Invenory policies provide insrucions on how o calculae sock levels and order quaniies. This paper examines he effec of he demand disribuion on he performance of several well-known policies. Their performance is compared in erms of achieved service levels. As a conclusion from hese comparisons, a nonparameric demand model is proposed o be used in invenory policies. Keywords: Invenory policies, nonparameric demand model, applicaions in logisics 1. Inroducion Invenory conrol policies offer suppor in esablishing he amoun o be socked such ha fuure demand will be me bes. Financial aspecs as well as he necessiy of providing good service are relevan when choosing he opimal sock level and replenishmen quaniy. For he general conceps compare [Silv. 1998] or [Temp. 2006] and he lieraure menioned herein. We resric our invesigaions o he demand of one iem a a ime in a single-iem single-level periodic review seing wih order-up-o policies. This means ha he sock level is updaed a he end of each period and a decision is made, wheher he invenory should be refilled up o he order-up-o level S using an invenory policy (see Fig. 1). Figure 1: Seup Along wih a vas body of lieraure, here exis many differen invenory policies, some of hem being very specific and elaborae, aking ino accoun los of variables and condiions. Mos invenory policies make a leas some assumpions on he demand, for example he common assumpion of i.i.d. normal demand, bu hese assumpions are usually no checked in pracice. The aim of his sudy is o demonsrae he effec of valid and violaed demand model assumpions on he performance of invenory policies. To sudy pure demand model effecs, he lead ime is se zero, which means ha he invenory is refilled insananeously. This is of course no a realisic assumpion as replenishing akes some ime and is usually random bu i serves our purpose Logisics Journal : Referiere Veröffenlichungen ISSN Seie 1

2 DOI /LJ_Ref_Kuhn_en_ Firs sochasic approaches wih respec o he opimal sock level and he modelling of invenory siuaions were made in he Fifies of he las cenury. Arrow e al. [Arro. 1951] developed opimal (s,s)-policies for a saic and a dynamic invenory model. Boh models make assumpions concerning he demand disribuion and ake shorage coss ino accoun. In he dynamic model he developmen of he invenory level is regarded as a Markov chain. Shorly laer, in 1953 Dvorezky e al. [Dvor. 1953] modelled he invenory siuaion as a woperson zero-sum game wih loss funcions and decision rules. Regarding he opimaliy of invenory policies Scarf [Scar. 1960] poined ou, ha under linear holding and shorage coss an opimal policy is always of (s,s)- ype. Based on hese findings Iglehar [Igle. 1963] developed an approximaively opimal policy for he infinie horizon problem. Today, many publicaions focus on he opimaliy of ordering policies in invenory models. These models are ofen quie elaborae and specific. Ceinkaya and Parlar [Cei. 2004] propose he opimal saionary base-sock policy for a periodic review invenory model wih independen idenically disribued demand, non-linear shorage coss and finie planning horizon. Gallego and Hu [Gall. 2004] presen opimal policies for finie and infinie planning horizons in a periodic review invenory model wih finie capaciy, in which he demand and supply processes boh are regarded as Markov chains. Hollier, Mak und Lai [Holl. 2002] develop an opimal (s,s)-policy for a coninuous review invenory model wih compound Poisson disribued demand. If he demand exceeds some criical value and if he invenory level is below some criical value, he demand will be ransformed ino a replenishmen order direcly while all oher demands will be me from sock. Some oher approaches expand he deerminisic "economic order quaniy" (EOQ) model [Harr. 1915]. Yu [Yu 1995] develops a robus EOQ model conaining he demand, ordering coss and holding coss as random variables. Presman and Sehi [Pres. 2004] inroduce a policy for a coninuous review invenory model wih a demand disribuion wih a deerminisic and a sochasic componen. This policy reduces o he EOQ formula, if he sochasic componen is missing. Oher publicaions ake he difference beween demand forecass and acual demands ino accoun. Heah and Jackson [Hea. 1994] presen heir "Maringale Model of Forecas Evoluion". Tokay and Wein [Tok. 2001] ransfer his idea o an invenory model wih finie capaciy. Abels [Abel. 1999] suggess o supplemenary order he upper confidence boundary for he median of he forecas errors in addiion o he forecased demand. There is obviously a wide variey of invenory models and policies. In pracice and paricularly in large invenories wih many iems, model assumpions on he demand disribuion of every single iem canno be checked due o coss and effors his would cause. I is herefore of ousanding imporance o examine pracical invenory policies wih few model assumpions wih respec o heir goodness of service (achieved service level) and compare achieved service levels if even hese few assumpions are violaed. Overall an invenory policy is required ha copes well wih many poenially underlying demand disribuions. This paper compares invenory models and ordering policies wih respec o heir performance in simulaed and real invenory siuaions. Also, he impac of highly frequen zero-demand periods, which are a common problem in invenory pracice, on he resuling service level is invesigaed. The paper sars wih a discussion of service levels as measures of good service in Secion 2. Aferwards, in Secion 3, six invenory models and heir ordering policies are presened. In Secion 4 simulaed invenory siuaions and real daa wih a challengingly high proporion of zero-demand periods are described. In Secion 5, invenory policies are applied and differences beween arge and achieved service levels compared. Secion 6 provides a summary and discussion of he main resuls. 2. Service levels Comparing invenory policies by heir service performances in differen invenory siuaions requires he use of a service measure. The hree commonly used service measures are he α, β and γ service level. In general, service levels measure he abiliy o deliver demanded goods. They reflec he relaionship beween demand, me demand and unme demand and so hey are powerful ools when comparing invenory policies. The measures are esimaes for unknown heoreical probabiliies of sockous relaed o he assumed underlying invenory model. The achieved α service level is he empirical proporion of periods wih fully me demand compared o all periods. The achieved β service level is he empirical proporion of he demand me in he period of is occurrence o he oal demand. The β service level provides informaion on he amoun missing, while he achieved α service level jus noes wheher i came o shorages or no. For he γ service level, as a hird measure of service, differen definiions exis. Generally i measures no only he size of he amoun missing bu also he ime of being unable o deliver. In cases of no backorders bu los sales and a lead ime assumed o be zero, as we will have laer, he γ service level is he same as he β service level. For furher deails on hese service measures see [Schn. 1981] Logisics Journal : Referiere Veröffenlichungen ISSN Seie 2

3 DOI /LJ_Ref_Kuhn_en_ As every period wihou demand is a period wih fully me demand, an aricle wih 80 percen zero demand periods achieves an α service level of 80 percen, when i is no kep in sock, i.e. he sock level for his aricle is always zero. If he aricle is kep in sock, i.e. he sock level is greaer han zero, hen he achieved α service level will be a leas 80 percen. Given a arge α service level of 90 percen, he possible difference beween he achieved α service level and he arge α service level is smaller for an aricle wih many periods wihou demand han for an aricle wih less periods wihou demand. Neverheless, he α service level is an useful ool for judging he abiliy o deliver demanded aricles. An aricle demanded only once a year maybe needs no o be kep in sock bu can be ordered wih a cerain lead ime when requesed. The consequences of sockous need o be weighed up agains coss and effors of having he paricular aricle in sock. If he accommodaion of demand is considered compulsory, he α service level can be se so high, ha he arge α service level canno be achieved by periods wihou demand only. In conras, he β service level ignores periods wihou demand. Even so, i is no more suiable o capure he abiliy o mee he demand han he α service level. If an iem is ordered jus once a year and due o his no in sock, hen none of he demanded amoun can be me in he period when he demand occurred. In his example he achieved β service level for a whole year equals zero. I does no reflec, ha he invenory level of zero is appropriaely chosen mos of he ime. We herefore concenrae on he α service level as a measure for he abiliy o mee he demand. 3. Invenory Policies Invenory models are based on assumpions concerning he naure of he reaed invenory siuaion. Using hese models, invenory policies are insrucions on how o calculae he opimal sock level and order quaniy. Cos or service level consideraions can be used o find an opimal policy. These wo approaches are ofen equivalen and boh leave space for he users' subjeciveness, as ypically no all coss are known and need o be guessed and here is no objecive way o deermine he appropriae arge service level. We discuss six differen (1,S) invenory policies as special cases of (s,s) sraegies. Two of he six policies conain fixed safey and base sock levels up o which he invenory is replenished in he beginning of each period. The firs invenory policy "P1" is based on he assumpion of an i.i.d. normal demand. For he second invenory policy "P2" an i.i.d. gamma disribuion is assumed. Invenory policies "P3" and "P4" have fixed safey sock levels bu he base sock levels are updaed in each period by using pas and recen demand observaions in a one-sep-ahead predicion of an auoregressive inegraed moving average (ARIMA) model. The fixed safey sock levels are deermined via he variance of he whie noise process (P3) or he α quanile of he shorages ha occur when no safey sock is kep (P4) respecively. The remaining invenory policies "P5" and "P6" have safey and base sock levels, which are in each period calculaed from he las demand observaions. For invenory policy P5 again i.i.d. normal demand is assumed, bu in conras o P1 he safey sock and order-up-o level are recalculaed every period. Policy P6 uses he empirical disribuion funcion as esimae for he unknown demand disribuion, which is updaed every period. In all policies he lead ime is se zero and no coss bu service level consrains are considered. Permiing backorders or no herefore makes no difference Policies wih fixed base sock and safey sock levels Invenory policies P1 and P2 are parameric policies ha provide raher simple formulae for calculaing he opimal order-up-o level S*. Like in he saic AHM-model [Arro. 1951] i is assumed ha he demand x a any poin in ime is independen idenically disribued wih disribuion funcion F. Here, S* is no calculaed by minimizing coss bu by giving a arge α service level o be achieved. Thus, S*is he opimal order-up-o level ha saisfies he equaion P ( x S ) = F( S ) = α 2 In case of P1, normally disribued demand wih expecaion µ and variance σ is assumed, and so S* equals = Φ 1 S ( α ) σ + μ 2009 Logisics Journal : Referiere Veröffenlichungen ISSN Seie 3

4 DOI /LJ_Ref_Kuhn_en_ where Φ is he sandard normal disribuion funcion. This order-up-o level can be spli ino base sock level µ and safey sock level Φ 1 ( α) σ. This is he usual exbook formula. In case of P2 he disribuion funcion F is ha of a gamma disribuion and S* equals he α quanile of F. The order-up-o level S* can be spli ino a base sock level given by he median ~ x 0. 5 and a safey sock level S ~ x Policies wih fixed safey sock levels and updaed base sock levels The assumpion of independen demand observaions migh, however, ofen be oo far from realiy. In such a case, correlaed demand, for example, can be modelled by an ARIMA model (P3 and P4). These policies, oo, are parameric. If he underlying process of correlaed demand is an ARIMA process, he demand x in period can be described by 1 p p j ϕ jb j= 1 j= 1 d j ( 1 B) ( x μ) = 1 θ B ε where θ,..., θq, ϕ,..., ϕ are consan, B he backshif operaor, µ he mean, and 1 1 p ε a whie noise process wih 2 variance σ. Under cerain assumpions he order-up-o level ε S = ˆ x + Φ 1 ( α) σ ε is opimal [John. 1975], wih x ˆ E( x x,..., x, ˆ,..., ε ) ˆ = 1 1 ε 1 1. Hence for invenory policy P3, he opimal order-up-o level S in period is given by he sum of he one sep ahead predicion xˆ and he α quanile of he disribuion of ε. Here S can be spli ino he base sock level xˆ and he safey sock level Φ 1 ( α ) σ. ε Invenory policy P4 has i's base sock level calculaed as policy P3 bu i's safey sock level is given by he empirical α quanile of he disribuion of shorages [Abel. 1999], ha occur when only he one-sep-ahead predicion is supplied. j 3.3. Policies wih updaed base and safey sock levels Policies wih updaed base and safey sock levels use all available informaion for he decision on he amoun o be ordered: in every period he esimaes are recalculaed using he laes demand observaions. For he following policies i is assumed ha he demand is independen and idenically disribued wih a disribuion funcion F, like for P1 and P2. The opimal order-up-o level S for a given arge service level α is hen again given by 1 ( x S ) = F( S ) = α S = F ( α ) P. Policy P5 is an updaed version of policy P1, based on he assumpion of normal demand. The base sock level μ in period is esimaed by x, he mean demand during periods 1 o -1. And he safey sock level is 1 esimaed by Φ ( α ) ˆ σ, which is he produc of he α quanile of a sandard normal disribuion and he updaed esimaed sandard deviaion which ogeher form he opimal order-up-o level S in period. As an alernaive o he previously described parameric invenory policies we inroduce a non-parameric policy (P6) resing on he assumpion of independen idenically disribued demand observaions. The unknown demand disribuion is esimaed by he empirical disribuion funcion. Is order-up-o level S in 2009 Logisics Journal : Referiere Veröffenlichungen ISSN Seie 4

5 DOI /LJ_Ref_Kuhn_en_ period can be spli ino base sock level ~ x 0.5,, he updaed median, and safey sock level ~ x, ~ α x0.5,, he difference beween S and he median. Wih hese wo policies he invenory will be replenished a he beginning of every period up o he opimal order-up-o level S, unless he invenory level a he end of he previous period is greaer han S. I is also possible o ake only a fixed number of recen demand observaions o esimae quanile of he empirical demand is calculaed in a moving window. 1 F, such ha he α 4. Invenory siuaions The inroduced invenory policies are based on assumpions wih respec o he demand disribuion. The effec of violaed disribuional assumpions in real invenory siuaions can no be deermined, as he rue disribuion of he demand is of course unknown. Applying he invenory policies o real and simulaed invenory siuaions and hen comparing he achieved service levels, however, can reveal how he policies reac o violaed model assumpions. We base our analysis on he following simulaed invenory siuaions. Firs, demand series are simulaed from normal and gamma disribuions, as hey are widely assumed in invenory pracice. Parameers for a simulaion of 500 series wih lengh 52 from he gamma disribuion were generaed by esimaion from real invenory siuaions. Generaed his way, he simulaed gamma disribued demand series will resemble real demand series, if he assumpion of gamma disribued demand is rue. Normal disribuions are symmeric around he expecaion and negaive realizaions always occur wih a cerain probabiliy. As demand can no be negaive, parameers for simulaing normal disribued demand are chosen in a way ha negaive demand has only a small probabiliy o be generaed. So, 500 series from normal disribuions are simulaed wih expecaion 1000 and random variance beween 1 and 250. In addiion o he simulaed invenory siuaions real invenory siuaions are considered. The daa comes from a coal-mining company. I consiss of invenory informaion on iems conaining, amongs ohers, he pronouncemen dae of demand, he desired delivery dae and he dae of he acual delivery. The iems are primarily spare pars, as for insance screws and ubes, pain and varnish, insrumens, machines and pars of hem, bu also work clohes, deergens, diesel fuel, saniary equipmen, fire exinguishers, Firs-Aid devices, personal hygiene devices, coffee, ea and biscuis. The available daa is recorded for one year wih respec o he dae of he acual delivery alhough he dae of he desired delivery comes closes o he demand, he variable of ineres. So all deliveries of one year are a hand bu no all desired deliveries. From he daa i canno be seen wheher he delivered amoun corresponds o he amoun demanded or wheher i has been spli up ino par-deliveries. However, poenial par-deliveries are merged by summing up all delivered amouns of one aricle wih he same desired delivery dae and so he daily demand for one aricle is formed. There is a large number of days wihou demand, especially mos Saurdays and Sundays. We se he period lengh o one week, which reduces he number of zero demand periods compared o a period lengh of one day. This choice appears o be reasonable as demand is rarely me in he week of pronouncemen and delays of up o wo years occur. However, despie his choice of period lengh he demand is sill inermien (see able 1). We group he iems wih respec o heir achieved α service levels when he sock level is always kep zero, which is jus he same as he zero demand frequency for each iem. Applying ordering policies o hese groups may reveal he influence of zero demand frequencies on achieved service levels. Iems wih heir firs quarile of zero demand frequency number of iems up o 10% % - 20% % - 30% % - 40% % - 50% % - 60% % - 70% % - 80% % - 90% % - 95% 2060 a leas 95% 3920 Table 1: Zero demand frequency in weekly demand daa 2009 Logisics Journal : Referiere Veröffenlichungen ISSN Seie 5

6 DOI /LJ_Ref_Kuhn_en_ demand greaer han zero have less han 25 percen periods wihou demand iems fall ino his caegory, hey form group 1. Group 2 consiss of all iems wih heir firs quarile equal o zero bu heir second quarile greaer han zero. I conains 897 iems. Analogical he 1785 iems are assigned o group 3. They have heir second quarile equal o zero and heir hird quarile greaer han zero. The remaining 8139 iems build group 4. All iems in his group have zero demand in a leas 75 percen of all periods. None of he invenory models presened in Chaper 3 can be regarded as solely underlying in he real invenory 2 demand series. χ -goodness-of-fi ess lead o he rejecion of he hypohesis of normal, exponenial, poisson and lognormal disribued demand for all or nearly all iems. Only for 588 iems from group 1 he hypohesis of gamma disribued demand could no be rejeced. Bu for mos of hese 588 iems ARIMA-processes could neiher be eliminaed as underlying processes. The plausibiliy of underlying ARIMA-processes has due o he vas amoun of daa only been checked by ess for correlaed residuals. Mos of he es resuls do no indicae inadequacy of he fied ARIMA models. 5. Performance of he invenory policies Mos of he described invenory policies conain unknown parameers which need o be esimaed before he policies can be applied o he invenory siuaions. To esimae he unknown parameers in P1 and P2, we cu he demand series ino wo halves. The required parameers are for each series esimaed from he firs half, and hen he policies are applied o he second half of he demand series. When applying invenory policies P3 and P4 o daa, ARIMA models need o be fi. The order of hese models, he parameers θ,..., θ q, ϕ,..., ϕ and he variance 1 1 p σ ε are unknown. For policy P3, order and variance are esimaed from he firs half of he demand series. The parameer esimaes are updaed in every period of he second half when applying he policies o he daa and he one sep ahead predicion for he nex period is calculaed. If he resuling order-up-o level is less han he sock level a he end of he previous period, nohing is replenished. For policy P4, model order and variance are esimaed from he firs quarer of he series. Then, in every period of he second quarer of he series, he parameer esimaes are updaed and he one-sep-ahead predicion is supplied in he following period. The empirical α quaniles of he shorages in he second quarers of he series form he safey sock levels. Invenory policy P4 is hen applied o he remaining second halves of he demand series. For policies P5 and P6 he demand series are halved o provide for comparabiliy alhough no parameers need o be esimaed from he firs halves. Like all oher policies hey are hen applied o he second halves of he series. The invenory policies are applied o he simulaed demand series and he demand series from groups 1 o 4 (of size 500 each) wih arge service levels of 50, 70 and 90 percen. Invenory policies P3 and P4 differ in he calculaion of he safey sock level bu are he same in he calculaion of he base sock level. Comparing he service levels achieved wih P3 and P4 les us exclude P4 from furher examinaions, because is performance is even worse han P3. The differences beween he achieved and arge service levels are displayed by boxplos (see Fig.2, Fig.3, Fig.4). Ideally he median of such a performance boxplo should be zero and he range should be small. When analyzing hese figures, one needs o keep in mind, ha he policies are applied o only half of he demand series and ha no every second half of he demand series of an iem in group 4 for example has a zero demand proporion of a leas 75 percen as he whole series has Logisics Journal : Referiere Veröffenlichungen ISSN Seie 6

7 DOI /LJ_Ref_Kuhn_en_ Figure 2: Differences beween arge and achieved service level, gamma and normal demand The comparison of arge and achieved α service and safey sock levels under he differen invenory policies leads o several conclusions. Firsly, we consider he impac of model or disribuion assumpions on achieved service levels. Neiher valid nor violaed demand model assumpions imply beer or worse performance. Policies wih differen disribuion or model assumpions lead o differen achieved service levels. Bu surprisingly, violaed assumpions do no necessarily lead o higher absolue differences from he achieved o he arge service levels. This can be seen, when comparing he performance of P1 and P2 on he simulaed normal daa. Alhough policy P2 has a gamma demand assumpion which is violaed i performs beer han policy P1 which has he valid normal demand assumpion. On he simulaed gamma daa policy P2 again performs beer han policy P1. In his case he policy wih valid demand model assumpion leads o beer performance han P1 wih violaed demand model assumpion Logisics Journal : Referiere Veröffenlichungen ISSN Seie 7

8 DOI /LJ_Ref_Kuhn_en_ Figure 3: Differences beween arge and achieved service level, group 1 and group 2 Secondly, he performance ges beer wih rising arge service level, he impac of he arge service level. Along wih rising α service levels, he differences beween invenory policies wih respec o meeing he given service level become more and more negligible. And so does he choice of an invenory policy. And hirdly, here is an impac of zero demand proporions on achieved service levels. If he proporion of zero demand periods exceeds he arge α service level, he achieved service level will exceed he arge service level. Besides his, invenory policy P1 (assuming normal disribuion) leads o rising achieved α service levels wih rising zero demand proporions even when he zero demand proporions do no exceed he arge service levels. So if he pas rae of zero demand of an iem is higher han he desired service level or close o i, he arge service level is already achieved by zero demand periods and i migh be unnecessary o sore i in he fuure. These findings sugges o firs deermine he pas rae of zero demand and hen decide on he necessiy of an invenory policy Logisics Journal : Referiere Veröffenlichungen ISSN Seie 8

9 DOI /LJ_Ref_Kuhn_en_ Figure 4: Differences beween arge and achieved service level, group 3 and group 4 For groups 1 o 4 of he real invenory siuaion he policies lised in able 2 urned ou o mee he arge α service levels bes. If jus one policy is o be chosen for all iems in an invenory, hen he nonparameric policy P6 can be recommended. arge α service level demand 50% 70% 90% group 1 P6 P6 P6 group 2 P6 P6 P6 group 3 P2 oder P6 P2 oder P6 P3 oder P6 group 4 P1 oder P6 P1 oder P6 P2 oder P6 Table 2: Policies wih bes achieved α service levels 2009 Logisics Journal : Referiere Veröffenlichungen ISSN Seie 9

10 DOI /LJ_Ref_Kuhn_en_ Summary In pracical applicaions invenory policies are needed for auomaed decisions on opimal invenory levels and order quaniies. The majoriy of invenories conains large numbers of iems, making exensive individual decision processes impossible. Differen invenory policies have been proposed in he lieraure, varying wih he relaed invenory model and model assumpions. In mos invenories, however, he validiy of model assumpions canno be checked individually for every iem. Therefore invenory policies are invesigaed in his sudy wih respec o heir performance in simulaed siuaions wih valid and violaed demand model assumpions. Addiionally he adequacy of he policies is examined in a real invenory siuaions. The invesigaed real invenory daa show a high proporion of zero demand periods. The invenory models compared in his sudy cover policies wih i.i.d. normal and gamma demand disribuions auoregressive inegraed moving average modelled demand. The differen model and disribuional assumpions led o changes in he achieved service levels, bu violaed assumpions did no necessarily lead o bigger differences beween achieved and arge service levels. I urned ou, ha achieved service levels were influenced by zero demand proporions. The achieved service level canno be smaller han he zero demand proporion and so he achieved service level of rarely demanded iems can exceed he arge service level wihou being kep in sock. As a resul of his sudy, none of he demand models could be regarded as appropriae for all considered real invenory siuaions. The updaed nonparameric policy performs bes on he whole. Furher, neiher valid nor violaed demand model assumpions lead o beer or worse performance. Consequenly, we propose ha updaed disribuion free demand esimaes should be used for invenory managemen. Bibliography [Abel. 1999] Abels, H.: DISKOVER II, Ganzheiliche Besimmung von Sicherheisbesänden. Berlin Heidelberg New York: Springer. [Arro. 1951] Arrow, K.J., Harris, T., Marschak, J.: Opimal invenory policy. Economerica 19, [Cei. 2004] Ceinkaya, S., Parlar, M.: Compuing a saionary base-sock policy for a finie horizon sochasic invenory problem wih non-linear shorage coss. Sochasic Analysis and Applicaions 22, [Dvor. 1953] Dvorezky, A., Kiefer, J., Wolfowiz, J.: The invenory problem: II. case of unknown disribuions of demand. Economerica 21, [Gall. 2004] Gallego, G., Hu, H.C.: Opimal policies for producion/invenory sysems wih finie capaciy and Markov-modulaed demand and supply processes. Annals of Operaions Research 126, [Harr. 1915] [Hea. 1994] [Holl. 2002] [Igle. 1963] [John. 1975] [Pres. 2004] [Scar. 1960] [Schn. 1981] Harris, F. W.: Operaions Cos. Facory Managemen Series, Chicago: Shaw. Heah, D.C., Jackson, P.L.: Modelling he evoluion of demand forecass wih applicaion o safey sock analysis in producion/disribuion sysems. IIE Transacions 6, Hollier, R.H., Mak, K.L., Lai, K.K.: Compuing opimal (s,s) policies for invenory sysems wih a cu-off ransacion size and opion of join replenishmen. Inernaional Journal of Producion Research 40, Iglehar, D.L.: Opimaliy of (s,s) policies in he infinie horizon dynamic invenory problem. Managemen Science 9, Johnson, G.D., Thompson, H.E.: Opimaliy of myopic invenory policies for cerain dependen demand processes. Managemen Science 21, Presman, E., Sehi, S.P.: Sochasic invenory models wih coninuous and poisson demands and discouned and average coss. Working Paper, The Universiy of Texas a Dallas, Richardson, TX. Scarf, H.: Opimaliy of (s,s) policies in he dynamic invenory problem. Mahemaical Mehods in he Social Sciences. Sanford, California: Sanford Universiy Press, K.J. Arrow, K. Karlin, P. Suppes, eds. Schneider, H.: Effec of service-levels on order-poins or order-levels in invenory models. Inernaional Journal of Producion Research 19, Logisics Journal : Referiere Veröffenlichungen ISSN Seie 10

11 DOI /LJ_Ref_Kuhn_en_ [Silv. 1998] [Temp. 2006] [Tok. 2001] [Yu 1995] Silver, E. A., Pyke, D. F., Peerson, R.: Invenory Managemen and Producion Planning and Scheduling. New York: Wiley. Tempelmeier, H.: Invenory Managemen in Supply Neworks. Nordersed: Books on Demand. Tokay, L.B., Wein, L.M.: Analysis of a forecasing-producion-invenory sysem wih saionary demand. Managemen Science 47, Yu, G.: Robus economic order quaniy models. European Journal of Operaions Research 100, Logisics Journal : Referiere Veröffenlichungen ISSN Seie 11