ON THE (Q, R) POLICY IN PRODUCTION-INVENTORY SYSTEMS

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1 ON THE R POLICY IN PRODUCTION-INVENTORY SYSTEMS Saifallah Benjaafa and Joon-Seok Kim Depatment of Mechanical Engineeing Univesity of Minnesota Minneapolis MN Abstact We conside a poduction-inventoy system with finite capacity stochastic demand and stochastic pocessing times times. We examine the impact of explicitly modeling poduction lead times on inventoy costs ode eplenishment lead times and the optimal policy. We contast esults obtained using ou model with those obtained using the conventional o standad model. We show that fo systems with finite poduction capacity ode eplenishment lead times ae highly sensitive to loading and ode quantity. Consequently the choice of optimal ode quantity and optimal eode point can vay significantly fom those obtained unde the usual assumption of a load-independent lead time. Moe impotantly we show that fo a given policy the conventional model can gossly unde o ove-estimate the actual cost of the policy depending on the value of lead time used in the standad model. Equally significant we show that the ode quantity that minimizes ode eplenishment lead time is not necessaily the one that minimizes total cost. In cases whee a setup time as well as a setup cost is associated with placing a poduction ode the optimal policy deived fom the standad model can be in fact infeasible.. INTRODUCTION One of the most common methods fo managing inventoy systems with stochastic demand is the eode point/ode quantity policy also known as the policy. Unde a policy finished goods inventoy is continuously eviewed and a new poduction ode is placed each time inventoy position on-hand inventoy outstanding odes backodes falls to a eode point. Values fo and ae typically selected so that inventoy costs odeing costs holding costs backodeing costs ae minimized. Policies of the type have been widely studied with a ich body of liteatue on models dating back to the late 5 s see fo example Gallihe et al. [8] and Hadley and Whittin [9]. Fo single item inventoy systems unde the standad assumptions an optimal policy can in fact be shown to exist within the class of policies [6]. Recently the applicability of policies has been extended to inventoy systems with multiple items and multiple echelons see fo example Atkins and Iyogum [] and Chen and Zheng [7] and the efeences theein. In detemining optimal values fo and it is assumed in most of the existing liteatue that ode eplenishment lead times ae not sensitive to the loading of the poduction facility o to the ode quantity. In fact the most common assumption is a fixed positive lead time [4]. This assumption is ealistic when the inventoy system is fully decoupled fom the poduction system though lage inventoy holding at the poduction facility o at

2 subsequent stages of the supply chain fo example when a local etaile is eplenished fom a lage egional waehouse. It may also be ealistic when non-poduction lead time is long fo example when tanspotation lead times ae significantly longe than manufactuing lead times. Howeve fo most integated poduction-inventoy systems these assumptions aely hold. In fact with inceased emphasis on lean manufactuing finished goods inventoy at factoies has become much moe tightly contolled and mateial handling time between the poduction facility and the finished-goods waehouse is often minimal [4]. Simila pinciples ae being applied to the entie supply chain with much tighte coupling between etailes distibutos and supplies taking place [5]. This inceased coupling means that distibutos and etailes ae often immediately affected by congestion and delays on the factoy floo. In this pape we examine the impact of explicitly modeling poduction lead times on inventoy costs ode eplenishment lead times and the optimal policy. We contast esults obtained using ou model with those obtained using the conventional o standad model. We show that fo systems with finite poduction capacity ode eplenishment lead times ae highly sensitive to loading and ode quantity. Consequently the choice of optimal ode quantity and optimal eode point can vay significantly fom those obtained unde the usual assumption of a load-independent lead time. Moe impotantly we show that fo a given policy the conventional model can gossly unde o ove-estimate the actual cost of the policy depending on the value of lead time used in the standad model. Equally significant we show that the ode quantity that minimizes ode eplenishment lead time is not necessaily the one that minimizes total cost. In cases whee a setup time as well as a setup cost is associated with placing a poduction ode the optimal policy deived fom the standad model can be in fact infeasible. To illustate ou esults we estict ou discussion initially to the case whee demand is Poisson distibuted since this is one of the few cases fo which the optimal solution to the standad model can be easily obtained. A model fo systems with geneal demand distibutions is pesented late in the pape.. THE STANDARD R MODEL In this pape the standad model efes to a single item continuous eview inventoy model whee demand is stationay and occus one unit at a time with ate λ. An ode of fixed quantity is placed wheneve inventoy position dops to a fixed eode point. Odes ae deliveed afte a fixed positive lead time L. All stockouts ae backodeed. The elevant costs include a fixed setup cost K fo each placed ode a holding cost h pe unit of held inventoy pe unit time and a backodeing cost b pe unit backodeed pe unit time. The long un aveage cost C S unde the above conditions is given by the following see fo example Fedeguen and Zheng [6]: y CS = λ K { h y i p i b i y p i } / y= i= i= y whee p i is the pobability that total demand duing lead time is i. The above expession is exact when the inventoy position in steady state is unifomly distibuted on {... } and is independent of demand duing lead time. This condition is satisfied when λl i demand is Poisson in which case p i is given by pi = e λl / i!. Although closed fom expessions fo the optimal ode quantity and optimal eode point * * ae difficult to obtain an efficient algoithm fo calculating * and * has been ecently poposed by Fedeguen and Zhen [6].

3 3. THE PRODUCTION-INVENTORY R MODEL In contast to the standad model we conside an integated poduction-inventoy system whee inventoy is eplenished fom a poduction facility with finite capacity. Inventoy is managed accoding to a policy simila to the standad policy. That is a poduction ode of size is placed each time inventoy position dops to the eode point. As in the standad model the ode is deliveed once all units have been poduced it is possible to extend the model to allow fo unit by unit eplenishment. As in the standad model we assume that demand occus one unit at a time accoding to a Poisson pocess with ate λ. Although custome odes aive individually a poduction ode is placed only afte custome odes ae eceived. Theefoe the inte-aival time of odes to the poduction system is Elang with phase. In contast to the standad model we let poduction capacity be finite with a positive pocessing ate µ. We conside the case whee unit pocessing times ae independent identical and exponentially distibuted andom vaiables with mean /µ. Since the pocessing time of an ode of size is the sum of independent identical and exponentially distibuted andom vaiables ode pocessing time is Elang with phase. Thus the poduction facility can be viewed as an E /E / queueing system i.e. a singleseve queueing system with Elang inte-aival and pocessing times. If we let i efe to inventoy level n to the numbe of odes in the poduction system each ode is of size z to the numbe of custome unit odes that have yet to be odeed an ode is placed only when inventoy position eaches the eode point and b to the numbe of backodes then it is easy to see that b = {n z } and i = { n z}. Noting that z is unifomly distibuted on { } this follows fom the fact that inventoy position is unifomly distibuted on { } it is staightfowad to show that: i P i = P N i =... I and b P B b = P N b = whee P I P B and P N efe espectively to the steady state pobability distibution of inventoy level numbe of backodes and numbe of odes in the poduction facility in queue in sevice. Fo the steady state pobabilities to exist we equie the stability condition ρ = λ/µ < whee ρ efes to the utilization of the poduction facility. Fom and 3 we can obtain aveage inventoy I aveage numbe of backodes B and aveage ode eplenishment lead time L as follows: I = ip I i 4 i = B = ip B i and 5 i= 3

4 L = ip N i / λ. 6 i= A closed fom expession fo the pobability distibution of the numbe of customes in a E /E / system is difficult to obtain. Howeve the pobabilities can be computed numeically see fo example Neuts [3] fo a matix-geometic appoach. Noting that P N i conveges to zeo as i appoaches infinity I B and L can be computed to desied accuacy. Ou cost function can now be witten as: i i i i C p = h P c P N kλ/ 7 i = N i = We can show that C P is convex in see Appendix. Theefoe fo each ode size the optimal eode point can be easily obtained using standad convex optimization []. The optimal ode quantity can be obtained by an exhaustive seach ove the ange of feasible ode sizes although numeical esults suggest that C P is jointly convex in and showing joint convexity analytically is difficult. 4. COMPARISONS AND ANALYSIS Since the ode quantity affects both the distibution of the aival pocess to and the pocessing time at the poduction facility it easy to show that lead time is sensitive to ou choice of ode quantity see expession 6. Also since congestion at the poduction facility is affected by both the available capacity and demand level lead time is affected by the utilization of the poduction facility. The behavio of aveage lead time fo vaying ode quantities and utilization levels is illustated in Figue. As we can see lead time is inceasing in both and ρ. Because lead time is affected by the choice of which in tun affects ou choice of the optimal values of and obtained espectively by the standad and the poductioninventoy models ae geneally diffeent. Depending on the value of lead time used in the standad model the diffeence in these values can be significant. This is illustated in Figues and 3 whee values of * and * fo both models ae shown fo vaying assumptions of fixed lead time we use the notation * s * * * s and p p to diffeentiate espectively between the optimal solution to the standad model and the poduction-inventoy model. We should note that even in cases whee the optimal ode quantities 4

5 Aveage lead time ρ =.8 ρ =.7 ρ =.6 ρ = Ode quantity Figue. The effect of ode quantity on aveage lead time λ = µ = h =. c =. K =. 8 6 Optimal ode quantity * Standad model Poduction-Inventoy model Lead time used in the standad model Figue. The effect of lead time on optimal ode quantity λ = µ =.5 h =. c =. K =. 5

6 Optimal eode point * Standad model Poduction-Inventoy model Lead time Figue 3. The effect of lead time on optimal eode point λ = µ =.5 h =. c =. K =. happen to be the same the eode points ae geneally diffeent. Moe impotantly the value of fixed lead time fo which the two optimal ode quantities ae the same does not geneally coespond to the actual lead time expeienced by the poduction-inventoy system. The fact that * and * as well as lead time ae diffeent unde the two models means that aveage inventoy and aveage numbe of backodes ae also diffeent. In tun this means that the estimated costs of the optimal policies unde the two models can also be vey diffeent. In fact depending on the value of lead time used in the standad model this diffeence can be quite significant. This is illustated in Figue 4. Note that thee may exist a fixed lead time fo which the pedicted optimal costs ae the same fo the two models. Howeve the coesponding values of * and * ae not necessaily the same and the fixed lead time does not usually coespond to the actual lead time in the poduction inventoy system. 4 Aveage inventoy cost Standad model Poduction-Inventoy model Lead time used in standad model Figue 4. The effect of lead time on inventoy cost λ = µ =.5 h =. c =. K =. 6

7 Moe impotantly the standad model can significantly ove o unde-estimate the tue costs of implementing the coesponding optimal policy o in fact any policy. In Figue 5 we show the diffeence between the optimal costs as estimated by the standad model and the tue costs of implementing this policy as obtained fom the poduction inventoy model. Again we can see that this diffeence is dependent on the choice of lead time in the standad model and can be quite lage. 8 6 Aveage inventoy cost Estimated cost by standad model Actual cost Lead time used in standad model Figue 5. The impact of lead time on estimated and actual inventoy costs λ = µ =.5 h =. c =. K =. In addition to inducing eos in estimating inventoy holding costs backodeing costs and odeing costs the standad model can lead to eos in estimating othe pefomance metics such as custome ode fulfillment time fill ate o the pobability of stockout. Fo example custome ode fulfillment time time fom when a custome places an ode to the time when the ode is shipped is given espectively fo the standad and poduction-inventoy models by the following: B F s = = i y p i }/ λ 8 λ y= i= y F p B = = λ i i = λ P N i Since aveage ode fulfillment time is linealy inceasing in the aveage numbe of backodes ode fulfillment is similaly sensitive to the choice of ode quantity. 5. THE IMPACT OF SETUP TIMES In many poduction systems a setup time is equied pio to the initiation of an ode. A non-zeo setup time inceases ode-pocessing time and theefoe inceases ode lead time. Since both aveage inventoy and numbe of backodes ae sensitive to lead time the intoduction of setup time affects total cost. In tun this affects the value of the optimal ode quantity and the optimal eode point. Moe impotantly a non-zeo setup time affects the stability condition of the poduction system and places a minimum equiement on ode 9 7

8 size. This can be seen by noting that with non-zeo setup times the stability condition is given by the following: λs/ λ/µ < which can be ewitten as: > λs/ λ/µ whee S is aveage setup time. The ight-hand side of epesents fo a given aveage setup time the minimum feasible ode size i.e. a smalle ode size would esult in infinitely long lead times. We should note that although a setup cost is included in the standad model the model does not account fo setup time no does it account fo the elationship between ode size and fequency of setups and thei joint effect on poduction capacity. Theefoe the standad model could lead to the choice of an infeasible ode quantity. In ode to examine the effect of setup time on lead time optimal ode quantity and optimal eode point we consideed the case whee setup times ae exponentially distibuted. In this case ode pocessing time can be epesented by a genealized Elang distibution with phases whee phase epesents setup time. A simila appoach to the one descibed in section 3 can be used to compute aveage inventoy level aveage numbe of backodes and aveage lead time. These can then be used to detemine optimal ode quantity and eode point. In figue 6 we show numeical esults that illustate the effect of ode quantity on lead time fo vaying values of setup time. It is inteesting to note that in contast to the case with zeo setup time the effect of ode quantity on lead time is not monotonic. Initial inceases in educe lead time by educing the fequency of setups. Howeve futhe inceases in esult in sufficient inceases in the ode pocessing time leading to an oveall incease in lead time. Simila obsevations wee made by Kamaka [] and Benjaafa and Sheikhzadeh [] fo make to ode systems. In Figues 7 and 8 we illustate the impact of setup time on the value of the optimal ode quantity and eode point. As we can see both quantities ae highly sensitive to setup time. The effect of setup time is paticulaly ponounced when utilization is high. We should note that since these effects ae ignoed by the standad model the values of * and * obtained fom the standad model could be again vey diffeent fom those we obtain using the poduction-inventoy model. 6. EXTENSIONS We have so fa limited ou discussion to the case whee demand is Poisson and pocessing and setup times ae exponentially distibuted. These cases allowed us to benchmak ou esults against those obtained by the standad 8

9 6 4 Aveage lead time S =. S =.3 S = Ode uantity Figue 6. The effect of ode quantity on aveage lead time λ = µ =.5 h =. c =. K = Optimal ode quantity * ρ =.8 ρ =.7 ρ = Setup time S Figue 7. The effect of setup time on optimal ode quantity λ = µ = h =. c =. K =. 9

10 5 Optimal eode point * Setup time S Figue 8. The effect of setup time on optimal eode point λ = µ = h =. c =. K =. model unde simila assumptions using exact analysis. Howeve fo many poductioninventoy systems these distibutional assumptions do not hold. Unfotunately exact analysis of queueing and inventoy systems with geneal distibutions is difficult. In this section we pesent an appoach based on appoximations to model systems with geneal demand pocessing time and setup time distibutions. This appoach allows us to obtain closed fom appoximations fo aveage inventoy aveage numbe of backodes and aveage lead time. These appoximations ae useful even fo the case of Poisson demand and exponential pocessing times whee the computational effot needed fo the exact analysis could be significant fo lage. In geneal if custome odes fom a enewal pocess and the pocessing times ae independent and identically distibuted we can model the poduction system as a GI/G/ queue. The pobability distibution of the numbe of customes in a GI/G/ queue can be appoximated as follows see Buzacott and Shanthikuma [4]: ρ i = P N i i ρ σ σ i =... whee σ = N ρ / N and N is the appoximation fo the numbe of customes in a GI/G/ queue. By vitue of Little s law we have: N = λ W ρ 3 whee Wˆ is the appoximate waiting time in a GI/G/ system. Aveage waiting time can be appoximated as follows [4]: ρ C s C a ρ C s W = 4 ρ C λ ρ s whee C a and C s efe espectively to the squaed coefficient of vaiation atio of the vaiance ove the squaed mean in ode inte-aival and pocessing times at the poduction system altenative appoximations can be found in [3] and [3]. The pobability distibution of inventoy level and numbe of back odes can be obtained using expessions and 3 fom which aveage inventoy aveage numbe of backodes and aveage ode lead time can be calculated as follows: ρ =.8 ρ =.7 ρ =.6

11 and i I = ip I i = i= i= = iρ ρ σ i = ρ σ ρ iσ i= i ρ σ B = = ip B i iσ i= i= i i σ 5 6 N ρ L = = ip N i =. 7 λ λ i= λ σ Although these esults ae based on a seies of appoximations compaisons with exact esults fo the E /E / model show the diffeence between the exact and appoximate estimates of aveage inventoy and aveage numbe of backodes not to exceed % fo a wide ange of paamete values. In most cases we found the optimal ode quantities and eode points to be the same as those obtained using the exact model. A detailed discussion of these appoximations and extensive numeical esults and compaisons can be found in []. 7. Conclusion In this pape we demonstated the impotance of modeling the dependency of lead time on system loading in poduction-inventoy systems. Specifically we showed that the optimal ode quantity and optimal eode point could vay significantly fom those obtained unde the usual assumption of a fixed lead time. We also showed that the costs estimated using a fixed lead time could be significantly diffeent fom those actually expeienced by the poduction-inventoy system. Fo systems whee the initiation of a poduction ode is peceded by a setup we found that system stability is dependent on ode size. Theefoe ignoing this dependency could lead to system instability. These esults highlight the fact that poduction and inventoy systems cannot be managed o modeled sepaately. Unfotunately the inteaction between poduction and inventoy as noted ecently by Buzacott and Shanthikuma [4] has gone lagely unde-studied. Theefoe thee is an oppotunity to extending the existing liteatue on inventoy theoy to include this impotant inteaction. Although we have focused on the policy in this pape it is wothwhile to cay out simila analysis with espect to othe common policies such as peiodic eview policies. It is also wothwhile to extend the analysis to systems with multiple echelons and multiple items. In that case captuing the inteaction between inventoy and poduction on one hand and inventoy and tanspotation on the othe becomes impotant. Acknowledgements: This eseach is funded by NSF unde gant DMII

12 Refeences [] Atkins D. and P. Iyogun Peiodic vesus Can-Ode Policies fo Coodinated Multiitem Inventoy Systems Management Science [] Benjaafa S. and M. Sheikhzadeh "Scheduling Policies Batch Sizes and Manufactuing Lead Times" IIE Tansactions [3] Bitan G. R. and D. Tiupati Multi-Poduct ueueing Netwoks with Deteministic Routing Decomposition Appoach and the Notion of Intefeence Management Science [4] Buzacott J. and G. Shanthikuma Stochastic Models of Manufactuing Systems Pentice Hall Englewood Cliffs NJ 99. [5] Cohen M. A. and H. L. Lee Stategic Analysis of Integated Poduction-Distibuted Systems: Models and Methods Opeations Reseach [6] Fedeguen A. and Y. S. Zheng An Efficient Algoithm fo Computing an Optimal Policy in Continuous Review Stochastic Inventoy Systems Opeations Reseach [7] Chen F. and Y. S. Zheng Evaluating Echelon Stock R n Policies in Seial Poduction/Inventoy Systems with Stochastic Demand Management Science [8] Gallihe H. P. P. M. Mose and M. Somond Dynamics of Two Classes of Continuous Review Inventoy Systems Opeations Reseach [9] Hadley G. and T. Whitin Analysis of Inventoy Systems Pentice-Hall Englewood Cliffs NJ 963. [] Kamaka U. S. Lot Sizes Lead Times and In-Pocess Inventoies Management Science [] Kim J. S. and S. Benjaafa On the Effectiveness of Policies in Poduction- Inventoy Systems Woking Pape Depatment of Mechanical Engineeing Univesity of Minnesota. [] Luenbege D. G. Linea and Nonlinea Pogamming Addison-Wesley Publishing Company Reading MA 98. [3] Neuts M. F. Matix-Geometic Solutions in Stochastic Models: An Algoithmic Appoach The Johns Hopkins Univesity Pess 98. [4] Silve E. A. D. F. Pyke and R. Peteson Inventoy Management and Poduction Planning and Scheduling John Wiley New Yok NY 998. [5] Whitt W. The ueueing Netwok Analyze Bell Systems Technical Jounal Appendix Lemma : Aveage inventoy cost C P is convex in. Poof: In ode to show that C P is convex in it is sufficient to show that aveage inventoy and aveage numbe of backodes ae convex in. Let I efe to aveage inventoy fo given and. In ode to show that aveage inventoy is convex in we need to show that { I I }/ I. Noting that

13 3 } { = P N P N P N P N I Κ it is not too difficult to show that. = P N I I I Hence aveage inventoy is convex in. Similaly we can show that. = P N B B B whee B is the aveage numbe of backodes. Thus B is also convex in which completes ou poof. #

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