ANALYSIS OF DYNAMIC PROPERTIES OF AN INVENTORY SYSTEM WITH SERVICE-SENSITIVE DEMAND USING SIMULATION Yuri Merkuryev and Julija Peuhova Janis Grabis Deparmen of Modelling and Simulaion Deparmen of Operaion Research Riga Technical Universiy Riga Technical Universiy Kalku Sree Kalku Sree LV-658 Riga, Lavia LV-658 Riga, Lavia E-Mail: merkur@il.ru.lv, julija@il.ru.lv E-Mail: grabis@il.ru.lv KEYWORDS Invenory managemen, service-sensiive demand, hybrid modelling. ABSTRACT Complexiy of many problems solvable by analyical mehods quickly increases under addiional assumpion. Invenory managemen under he service-sensiive demand is one of such pracical problems. This paper considers applicaion of he hybrid simulaion/analyical approach for dealing wih his problem. The appropriae closed loop model, ha incorporaes boh simulaion and analyical models, has been developed. I has been applied o sudy behaviour of he invenory sysem under he service-sensiive demand. The regression analysis conduced indicaes ha he service-sensiive demand causes subsanial deviaions of he provided service level from he arge service level. The arge service level, he demand variabiliy and he lead ime are facors subsanially influencing he difference beween he arge service level and he provided service level. The resuls obained are o be used for design of a mechanism for adjusing he parameers of he invenory sysem in order o mainain he arge service level. INTRODUCTION The comparaive advanages and disadvanages of analyical versus simulaion models are well known (for insance, see Nolan and Sovereign 972). Hybrid simulaion/analyical models are used o aain some of he advanages of boh ypes of models, while avoiding he disadvanages. Shanhikumar and Sargen (983) idenify four classes of hybrid simulaion/analyical models. Simulaion and analyical models are independen pars of a model for he firs class of hybrid models. These models are used sequenially. The second class includes hybrid models consising of simulaion and analyical models operaing in parallel. The hird class comprises hybrid models wih dominan analyical models, which use subordinae simulaion models for performing special asks. Finally, in he fourh class a simulaion model is a primary model of he sysem, and i uses inpus from one or more secondary analyical models during he modelling process. Such uilizaion of analyical models is frequenly encounered in complex simulaion models. For insance, analyical models are used o generae demand forecass (Bhaskaran 998) and for invenory managemen (Ganeshan e al. 200). In order o simplify usage of analyical models in simulaion, Baker (997) develops a mehodology for incorporaing classic algorihms of operaions research ino simulaion models. This paper considers a special case of he fourh class hybrid simulaion/analyical models. This case describes a siuaion, where a simulaion model is buil around an analyical model in order o exend funcionaliy of he analyical model. This ype of combinaion of analyical and simulaion models is used o evaluae analyical models under realisic condiions and o consider model and environmenal parameers no represened in he radiional formulaion of analyical models. Cerda (997) uses simulaion o selec he mos appropriae ordering opions for he complex muli-iem re-order poin invenory managemen policy. Clay & Grange (997) simulae he supply chain of auomoive service pars. The simulaion model is used o evaluae he impac of differen forecasing mehods on he supply chain performance. Such analysis provides means for direc evaluaion of forecasing mehods by evaluaing a resuled value of a specified goal funcion or variable (e.g., a service level) insead of evaluaion of forecasing mehods according o he forecasing accuracy crierion. Enns (2002) invesigaes he impac of forecasing accuracy on efficiency of maerials requiremens planning. In order o overcome limiaions of previous research in his area, he auhor develops a shop floor simulaion model for more realisic evaluaion of elaboraed producion schedules. The applicaion of he realisic producion schedule evaluaion procedure has enabled idenificaion of complex ineracion among properies of demand forecass and characerisics of demand process. Takakuwa & Fujii (999) develop a sandardized simulaion model for analysis of ranshipmen invenory sysems. The simulaion model is used o provide a more realisic represenaion of he ranshipmen problem comparaive o he radiional mahemaical programming represenaion. For model building purposes, he auhors idenify and sandardize modules defining he ranshipmen problem and parameers of hese modules.
A modelling problem considered in his paper is invenory managemen under service-sensiive demand. The service-sensiive demand implies ha demand for fuure periods depends upon he service level observed a he curren period. A radiional formulaion of analyical invenory models does no consider such dependence. The exising research on invenory managemen under service-sensiive demand has been resriced o eiher siuaions wih deerminisic demand or woperiod problems (Baker and Urban 988; Erns and Powell 995). These limiaions can be explained by an explicily dynamic characer of he problem leading o a complicaed analyical analysis. Simulaion modelling allows analysing a muli-period problem under sochasic demand. The demand parameers change from one period o anoher in he case of he muliperiod problem. Such behaviour can be observed in highly dynamic and compeiive invenory sysems. For insance, wholesalers of compuer chips ofen are no able o mee demand due o insufficien supplies form upsream supply chain levels. In he case of he shorage, cusomers are likely o seek alernaive vendors and may choose o place orders o a newly seleced vendor for following periods as long as he service level is mainained. The cusomers may swich back o he iniial vendor, if he newly seleced vendor reduces is service level. Similar relaionships beween demand and performance of he invenory sysem are also observed in reail (Silver and Peerson 985). The research on service-sensiive demand also relaes o research on invenory level dependen demand (see Chung (2003) for a recen accoun). Main objecives of his research are o expand modelling of invenory sysems wih service-sensiive demand by considering muli-period sochasic problems, idenify properies of such sysems and o es abiliy of radiional invenory models o mee service level requiremens in he case of service-sensiive demand. The service level sensiive demand is modelled similarly o Erns and Powell (995). A simulaion model is buil around analyical models used for invenory managemen and updaing demand parameers according o he observed shor erm service level. Experimenal sudies are conduced wih he model, and he regression analysis is used o deermine impac of service-sensiive demand on performance of he invenory sysem. The remaining par of he paper discusses issues in invenory managemen under service-sensiive demand, describes a simulaion model for evaluaion of such invenory sysems and provides preliminary experimenal resuls. INVENTORY SYSTEM A single-iem, single-sage, muli-period invenory sysem is considered. The radiional re-order poin policy is used for invenory managemen. The order size is fixed independenly of he re-order poin level. The service level is measured using a proporion of demand saisfied direcly from he invenory. Any unsaisfied demand is los. Exernal demand is normally disribued wih mean D and he sandard deviaion σ, where,..., T,... =. The parameers of he exernal demand change dynamically according o shor-erm flucuaions of he service level provided. The invenory managemen objecive is o preserve he service level a a fixed level. Alernaively, one could ry o increase his or her marke share. However, he marke is assumed o be highly compeiive, and oher players are expeced o ake analogous acions prevening one player o achieve a permanen increase of he marke share. Invenory size Invenory parameer Service Level Figure : Ineracions among, Invenory Size and Service Level Relaionships beween he demand, service level and invenory parameers are shown in Figure. The exernal demand causes depleion of he invenory level. The invenory is replenished according o he re-order poin policy specified by a se of he invenory parameers (re-order poin, order size, mean demand, demand sandard deviaion, lead ime and arge service level). The service level achieved during a relaively shor ime period is observed. This service level is mos likely o differ from he arge, required service level. In he case of he service-sensiive demand, his causes changes of he demand parameers. A higher han arge service level causes increase of he mean demand and he sandard deviaion. A lower han arge service level causes decrease of he mean demand and he sandard deviaion. The increase of he demand parameers may resuls in a declining service level in forhcoming periods unless he invenory parameers are properly adjused. The decrease of he demand parameers may cause oversocking unless he invenory parameers are properly adjused. Therefore, a link represening updaing of he invenory parameers according o he observed shor-erm service level should be esablished. SIMULATION MODEL The invenory sysem described above has an explicily dynamic characer. Simulaion is used o capure his behaviour of he sysem. A simulaion model developed describes he invenory sysem and incorporaes an analyical model for implemening a feedback beween he simulaed shor-erm flucuaions in he service level
and he cusomer demand parameers. A srucure of he considered hybrid simulaion/analyical model is given in Figure 2. Iniial Parameers Parameers Invenory Parameers Simulaion Model Analyical Model Service Level Figure 2: Srucure of he Hybrid Simulaion/Analyical Model Simulaion is used for analysis of properies of he sysem. Resuls of he analysis are expeced o creae a basis for developing a mechanism for updaing of he invenory parameers in order o mainain he service a where is a curren ime period, SO is an unsaisfied demand in period, D is an observed acual demand in period. An impac of he cusomer service level on a fuure cusomer demand is quanified similar o Erns and Powell (995). In his approach, a linear relaionship beween he service level and he demand parameers is assumed. This dependence is evaluaed based on parameers esimaed by expers. This means ha he mean demand increases/decreases by α poins if he change in he service level doesn exceed a cerain hreshold: D = ( + α *( SL SL ))* D, (3) where SL is he shor erm service level in he previous ime period, D is he mean demand of he previous ime period, α is a coefficien of he change in mean demand wih increased/decreased service level. The sandard deviaion of he demand for he new demand level is expressed as a funcion of he parameers α, β and he sandard deviaion σ from he previous ime period: he arge level. 2 = [ + β α( SL SL ) ] 2 σ Main seps of he simulaion analysis being performed are as follows:. Iniialise he inpu daa module, including he iniial (received by radiional forecasing echniques) forecas of he cusomer demand; 2. Perform simulaion of invenory conrol processes; 3. Calculae he observed service level for a curren ime period; 4. Calculae he cusomer demand disribuion parameers based on he observed service level; 5. Updae he demand parameers in he simulaion model; 6. Go o sep 2 unil simulaion is compleed. Invenory managemen is based on he re-order poin policy, where he order size Q is fixed independenly of he re-order poin level. The re-order poin level is calculaed using a formula: ROP = LT * D0 + z * LT * σ 0, () where LT is he lead ime, D 0 is an iniial mean demand during one period, σ 0 is an iniial sandard deviaion of he demand during one period, z is a safey facor ha depends on a specified arge service level. The shor erm service level is calculaed each period using a formula: SL SO =, (2) D σ, (4) where β is a coefficien of he change in sandard deviaion of demand wih changed service level. In case if he increase/decrease in he service level exceed a resriced consan he mean demand is calculaed by a formula: D = ( + α *( SL SL )* MaxChange) * D, (5) where MaxChange is a consan of he maximal change in he service level. The sandard deviaion of he demand in his case is found by a formula: [ ] 2 + β α( SL SL )* MaxChange 2 σ σ (6) = If he shor erm service level is equal o one for wo consecuive periods and he demand parameers do no exceed heir iniial values, he demand parameers are updaed using he following expressions: α D = ( + )* D, (7) 0 2 β * α 2 σ = + σ. (8) 0 If his resricion is no imposed, he sysem may sele for providing a high service level on expense of carrying excessive invenory. The simulaion model is developed using he ARENA simulaion modelling environmen. Evaluaion of he service level and updaing of he demand parameers are implemened using Visual Basic.
EXPERIMENTAL EVALUATION Experimenal Design Objecive of experimenal sudies is o deermine he shor erm cusomer service level, o idenify parameers of he invenory sysem influencing disagreemen beween he arge and observed service levels and o evaluae changes of he cusomer demand parameers. Therefore, a se of experimens wih a feedback from he simulaion model o he analyical model, when he demand parameers are updaed aking ino consideraion he observed service level (servicesensiive demand), is performed. Performance of he invenory sysem is evaluaed under various facors such as iniial end cusomer mean demand, signal o noise raio, arge service level, lead ime, and order size coefficien (Table ). Table : Experimenal Design Facors D 0 Signal o Noise Targe Service Level LT Values Min 50 2 0.9 2 xlt Max 250 0 0.99 6 2xLT The Signal o Noise facor describes variabiliy of he demand process. A value of his facor normally should be in range beween and D 0. Given he value of D 0, he iniial sandard deviaion ( formula: σ 0 Q ) is found by a D0 σ 0 =, (9) Signal o Noise where D 0 is he iniial mean demand. The Targe Service Level facor describes he required service level of he invenory sysem considered. The LT facor value corresponds o ime beween he order placemen ime and he order arrival. I is measured in days. Two values of he fixed order size are considered. The minimum value is equal o he iniial mean lead ime demand, he maximum value is equal o he double iniial mean lead ime demand. The shor erm service level is observed every 5 days. The demand parameers are re-evaluaed a he same ime inerval. The difference beween he arge (SL Targe ) and he provided (SL Provided ) service level is he main performance measure. The provided service level represens he overall sysem service level and is calculaed a he end of each run by a formula: T SO = SLProvided =, (0) T D = where T is a replicaion lengh. Experimens are conduced according o a facorial experimenal design wih resoluion IV. This design consiss of 6 experimenal cells. The model was run for 5 replicaions. Each replicaion lengh is defined as 250 weeks and a warm-up period is 20 weeks. Thus, simulaion resuls are independen from he empy-andidle iniial sae; here is no predeermined saring and finishing poin for a simulaion run. Experimenal Resuls The simulaion resuls are summarized using he regression analysis. We analyse how a difference beween arge and provided service levels is affeced by values of he experimenal facors considered. The dependen variable Y is defined as (SL Targe -SL Provided ). Esimaed coefficiens of he regression equaion and associaed p-values are repored in Table 2. Table 2: Resuls of he Regression Analysis Independen variables Coefficiens p-value Consan -0.795 0.00 D 0 0.00002 0.05 Signal o Noise -0.004 0.00 LT 0.006 0.00 Order Size Coefficien 0.003 0.5 Targe Service Level 0.848 0.00 Ideally, he difference beween SL Targe and SL Provided should be equal o zero. However, he service-sensiive demand causes deviaion of he provided service level from he arge service level. Values larger han zero indicae ha he required, arge service level is no reached. Values smaller han zero mean ha reached service level is higher han expeced. The regression equaion suggess ha he Signal o Noise facor, LT and Targe Service Level have he mos significan impac on he difference beween he arge and provided service levels. The order size does no have he significan impac on his difference. The provided service level is likely o be smaller han he arge service level, if he arge service level is high, he lead ime is long and he demand is highly variable. Wih a high degree of confidence (95%) he observed service level averaged over all experimenal cells differs from he required service level by 5%, and he mean cusomer demand differs from he iniial mean demand on average by 3%. Dynamic behaviour of he mean demand and he observed shor erm service level during simulaion is shown in Figures 3, 4, and 5. The resuls are obained for differen values of he arge service level. Values of he iniial mean demand, he signal o noise raio, he lead ime and he order size coefficien are fixed a 250, 2, 2,, respecively.
Targe SL=0.9 piece/week 300 250 200 50 00 50.05 0.95 0.9 0.85 0.8 0.75 service level raio 0 0.7 0 50 00 50 200 250 300 weeks Mean Observed SL Figure 3: Mean wih Targe Service Level 0.9 300 Targe SL=0.95.05 piece/week 250 200 50 00 50 0 0 50 00 50 200 250 300 0.95 0.9 0.85 0.8 0.75 0.7 service level raio weeks Mean Observed SL Figure 4: Mean wih Targe Service Level 0.95 Targe SL=0.99 piece/week 350 300 250 200 50 00 50 0 0 50 00 50 200 250 300.05 0.95 0.9 0.85 0.8 0.75 0.7 service level raio weeks Mean Observed SL Figure 5: Mean wih Targe Service Level 0.99
Experimenal resuls show ha he observed service level achieved each week ofen differs from he arge service level. This leads o changes in he end cusomer demand. The demand volume changes are he main cause of he decreasing/increasing of he service level, because of reaching a high service level he demand volume become large and invenory conrol sysem is no able o adap o a new environmen during he shor ime. This leads o a lower service level in he nex period and he demand volume becomes smaller. Tha increases he service level in he nex period. This sequence is kep during a simulaion run. The arge service level has an impac on he demand parameers. The demand parameers increase, if he arge service level is high. The demand parameers decrease, if he arge service is low. The increase of he demand parameers is observed because, in he case of he high arge service level, he observed shor erm service level is ofen equal o one, shorages occur less frequenly causing fewer possibiliies for he demand parameers o decline. CONCLUSION The analysis of re-order poin invenory sysems under he service-sensiive demand has been exended o muli-period, sochasic demand siuaion. The hybrid simulaion/analyical modelling approach has been advocaed as an appropriae echnique for conducing his analysis. The appropriae simulaion model, which incorporaes analyical models for invenory managemen and modelling of he service-sensiive demand, has been developed. I has been applied o sudy behaviour of he invenory sysem under he service-sensiive demand. Analyical models provide a well-defined mechanism for invenory managemen. However hese models no give an impression of he sysem operaion over he ime. Therefore, a simulaion echnique is used o perform he analysis of he sysem dynamic behaviour. The regression analysis conduced indicaes ha he service-sensiive demand causes subsanial deviaions of he observed service level from he required, arge one. Addiionally, he observed service level is lower han he arge level, if laer is larger, while he observed service level is higher han he arge one, if laer is smaller. The demand variabiliy and he lead ime are oher facors subsanially influencing he difference beween he arge and he provided service levels. The resuls obained are o be used for design of a mechanism for adjusing he parameers of he invenory sysem in order o mainain he arge service level. The simples mechanism for achieving his objecive is recalculaing of he invenory parameers according o new values of he demand parameers. However, preliminary sudies of his mechanism sugges ha a more complex preempive approach is needed. REFERENCES Baker, G.S. 997. Taking The Work Ou Of Simulaion Modeling: An Applicaion Of Technology Inegraion. In Proceedings of he 997 Winer Simulaion Conference, 345-35. Baker, R.C.; and Urban, T.L. 988. A deerminisic invenory sysem wih an invenory-level-dependen demand rae. Journal of he Operaional Research Sociey 39, 9, 823-83. Bhaskaran, S. 998. Simulaion analysis of a manufacuring supply chain. Decision Sciences 29, 3, 633-657. Cerda C.B.R.; and de los Moneros, A.J.E. 997. Evaluaion of a (R,S,Q,C) Muli-Iem Invenory Replenishmen Policy Through Simulaion. In Proceedings of he 999 Winer Simulaion Conference, 825-83. Chung, K.-J. (2003), An algorihm for an invenory model wih invenory-level-dependen demand rae, Compuers & Operaions Research 30, 9, 3-37. Clay, G.R.; and Grange, F. 997. Evaluaing Forecasing Algorihms And Socking Level Sraegies Using Discree-Even Simulaion, In Proceedings of he 997 Winer Simulaion Conference, 87-824. Enns, S. T. 2002. MRP performance effecs due o forecasing bias and demand uncerainy. European Journal of Operaional Research 38, 87-02. Erns, R. and S. G. Powell (995), Opimal invenory policies under service-sensiive demand, European Journal of Operaional Research 87, 36-327. Erns, R. and S. G. Powell (998), Manufacurer incenives o improve reail service level, European Journal of Operaional Research 04, 437-450. Ganeshan, R., T. Boone and A. J. Senger (200), The impac of invenory and flow planning parameers on supply chain performance: An exploraory sudy, Inernaional Journal of Producion Economics 7, - 3, -8. Ho, P.-K. and J. Perl (995), Warehouse locaion under service-sensiive demand, Journal of Business Logisics 6,, 33-62. Nolan, R. L. and M. G. Sovereign (972), A recursive opimizaion and simulaion approach o analysis wih an applicaion o ransporaion sysems, Managemen Science 8, 2, 676-690. Shanhikumar, J. G. and R. G. Sargen (983), A Unifying View of Hybrid Simulaion/Analyic Models and Modeling, Operaions Research 3, 6, 030-052. Silver, E.A.; and Peerson R. 985. Decision sysems for invenory managemen and producion planning, 2nd ed. New York: Wiley. Takakuwa, S.; and Fujii, T. 999. A Pracical Module-Based Simulaion Model for Transshipmen-Invenory Sysems. In Proceedings of he 999 Winer Simulaion Conference, 324-332.