SPARE PARS INVENORY SYSEMS UNDER AN INCREASING FAILURE RAE DEMAND INERVAL DISRIBUION Safa Saidane 1, M. Zied Babai 2, M. Salah Aguir 3, Ouajdi Korbaa 4 1 National School of Computer Sciences (unisia), 2 BEM-Bordeaux Management School (France), 3 Higher School of echnology and Computer Science (unisia), 4 ISICom, University of Sousse (unisia) safa_saidane6@yahoo.fr, mohamed-zied.babai@bem.edu, aguirms@yahoo.fr, Ouajdi.Korbaa@Centraliens-Lille.org Abstract: Managing spare parts inventories is a challenging task and can benefit considerably from any information on the failure rate of the parts. It has often been considered in the stock control literature that parts' failures are only random, caused by external events which results in the assumption of constant failure rate and therefore the consideration of the Poisson process to represent spare parts demand. his is obviously a restricted and an unrealistic modeling assumption. In fact, a spare part recently replaced will unlikely fail shortly again but generally the older it gets, the greater this probability. Moreover, it has been shown that models with a Poisson demand process assumption advise more stock than necessary. hat it is to say that it is more appropriate to model spare parts demand by a renewal process with an increasing failure rate demand interval distribution. For a continuous treatment of time, an Erlang distribution covers a large spectrum of spare parts demand interval distributions. In this paper, we develop a model for a spare part inventory system where demand is a renewal process and the supply lead-time is constant. Demand intervals follow an Erlang distribution and the demand sizes follow a Gamma distribution. he stock is controlled according to a base-stock policy which is often used in spare parts inventory control. We determine analytically the expected total inventory cost and the optimal base-stock level for a cost oriented inventory system where unfilled demands are backordered. A numerical investigation is also conducted to analyze the inventory system with respect to the different system parameters. Keywords: spare parts, base-stock inventory control, compound Erlang demand, increasing failure rate 1. Introduction Spare parts inventories are mostly managed according to the base-stock policy. For such slow moving stock keeping units, when time is treated as a continuous variable, most of the stock control research work (Archibald and Silver, 1978; Cheung, 1996; Zipkin, ; Babai et al., 11) makes the practical assumption that failures and consequently demands occur following a compound Poisson process. his assumption is attractive due to the memoryless propriety of the exponential distribution of the demand intervals. he Poisson assumption can be easily justified by Palm s theorem for complex systems (Gupta et al., ) but if we consider one single part, it is obviously not always true because this assumption means that the failure process does not have a memory. hat is to say that once replaced, the part will likely fail shortly again. Subsequently, we can easily understand that the Poisson demand process assumption advises excess stock (Smith and Dekker, 1997). Actually, a part's failure rate depends on time, whereas a Poisson failure process implies constant failure rate. In this paper, we consider Increasing Failure Rate (IFR) parts i.e. we treat the case when failure is caused by wear-out such for mechanical parts which more likely fail when they get older. We assume that the failure intervals follow an Erlang-k distribution which is an advantageous IFR distribution (Gupta et al., ). In fact, when the parameter k varies, the Erlang-k covers a wide range of probability distributions. Note that for k = 1, an Erlang distribution is reduced to an exponential distribution and when k increases the distribution becomes less variable, and the variance goes to zero (i.e. constant times between demands) as k goes to infinity. Despite the considerable amount of research contributions in the stock control field, periodic review policies have clearly dominated the academic literature due to their practical appeal (eunter et al., ; Sani and Kingsman, 1997). Continuous review policies may appear less feasible but actually, in 768
order to provide the same level of customer service, they require less safety stock than periodic review policies and then, they incur lower inventory holding costs so, lower total inventory costs. Moreover, most of the exact and approximate proposed methods for determining the optimal parameters of inventory control policies under stochastic demands are not so simple to implement and to use directly for practitioners. he failure intervals correspond in inventory language to the demand intervals and the demand sizes to the number of failed parts. In this paper, we develop a single echelon single item inventory system method to calculate the optimal base-stock level for a cost oriented continuous review inventory system where unfilled demands are backordered. he demand intervals are modeled by an Erlang-k distribution and the demand sizes follow an arbitrary non-negative probability distribution. he leadtime is assumed to be constant. We assume that there is no capacity constraint in the replenishment system i.e. a replenishment order is immediately generated after a failure occurrence without delay which is a little uneconomic (Kalpakam and Sapna, 1998). A numerical investigation is conducted to analyze the inventory system with respect to the different parameters of the demand process. We increase the number of Erlang stages k to observe our Erlangian model s behavior and to illustrate the advantages of using the Erlang-k distribution instead of the exponential distribution for the demand arrival process. he remainder of the paper is organized as follows. Section 2 describes the inventory system and presents the notations used in this paper. hen, the expected total cost and the expression of the optimal base-stock level are derived. In Section 3, we present the results and interpretations of the numerical investigation. In section 4, we conclude and we expose research perspectives. 2. Inventory system description and analysis 2.1 System description and assumptions We consider a single echelon single item inventory system where an inventory holding cost h is incurred for each unit kept in stock and a demand excess is backordered with a penalty cost b, both costs are per unit per unit of time. We assume that the stock level is controlled according to a continuous review base-stock policy. Each replenishment order to the base-stock level is triggered immediately by only one separate demand arrival and has an assumed constant lead-time L. he Demand process is modeled as a compound Erlang process; demand inter-arrival times are i.i.d. according to an Erlang distribution with k identical phases, mean (rate λ, i.e. =1/λ) and CV² equal to 1/k. his implies that the duration of each phase is exponentially distributed with mean /k (rate kλ). he demand sizes X are i.i.d. following an arbitrary non-negative probability distribution G with mean µ X and standard deviation σ X. 2.2. Inventory system analysis he steady-state probability, P m (L), of the number m of Erlang-k demand arrivals during the lead-time L, following from (Kleinrock, 1975; Larsen and horstenson, 8) based on the renewal theory, is simplified to hen, the stock level decreases by a quantity X m that represents the sum of the m demand arrivals, each with size X. he probability distribution of X m is the m-fold iterated convolution of G, say G (m). Note that if the demand size distribution is regenerative so, the probability distributions G (m) and G are of the same type. herefore, the parameters of G (m) can be directly calculated based on µ X and σ X. he expected total cost, including holding and backordering costs, is given by 769
he optimal base-stock level S * corresponds to the value of S that gives the minimal expected total cost. Hence, in order to determine S *, one should study the convexity of the function with respect to S. By making the derivative of the expected total cost with respect to S, we obtain is increasing in S, which implies that the derivative of the total expected cost is also increasing in S. If a solution of a null derivative exists this means that is convex in S and this solution is necessary the optimal base-stock level S * which leads to the following equation that can be used to determine the optimal base-stock level S * Due to the infinite sum in (5), a truncation should be performed to approximate the optimal base-stock level. Bounds, with finite sums of an order n, can be used to approximate S *. We mainly refer to the method proposed by Babai et al. (11) to determine the expressions of the upper bound S U and the lower bound S L such that. For the proof, please refer to the appendix C in Babai et al. (11). he advantage of this method consists in the convergence of both to as n increases. So, we can fix the degree of accuracy of S * as we desire it. In our case, where is the solution of and is the solution of Once S * is determined numerically, the expected cost can be easily calculated. In fact, as mentioned above, for regenerative demand size distributions, (e.g. Normal or Gamma), our model becomes easy to implement in an Excel sheet. For the purpose of our numerical investigation, Gamma distribution is considered since it covers a large spectrum of probability distributions and it is positive. Please note that under the assumption of Gamma distributed demand sizes, the minimal expected total cost expression is 3. Numerical analysis In this paper, we report numerical results for L = 1, h = 1 and b =. We consider values of in the interval [.1, ]. Note that demand with < 1 will be referred in this paper as demand of fast moving items and demand with > 1 will be referred as demand of slow moving items. Demand sizes X are 77
distributed following a Gamma function with coefficient of variability CV² X =.9 and 4 (i.e. µ X = and σ X = 3 and ). Please note that we have considered other values of CV² X in order to calculate the optimal base-stock level and the expected minimal total cost but due to space constraints, we only report results for these mentioned values. First, we start by analyzing the order of convergence of the model and the evolution of and values, according to the system parameters, i.e. average demand inter-arrivals times, the variability of the demand sizes X and particularly number of Erlang phases k. For the considered demand sizes distributions, we retain a gap value of -2 between the lower bound and the upper bound to ensure convergence for an order n. In fact, we consider that and values are sufficiently high to retain this stopping convergence criterion. We denote by n * the smallest order corresponding to the convergence of and for any value. We present in able 1 the variation of the convergence order with respect to k and and for demand sizes with CV² X =.9. able 1. n * for CV² X =.9 k\.1.5 1 1.5 2 4 6 8 1 23 8 6 4 4 3 2 2 2 6 14 4 2 2 1 1 1 1 1 15 12 3 1 1 1 1 1 1 1 able 1 shows that the convergence order is really low which confirms what has been shown by Babai et al. (11) under the compound Poisson demand and shows that this method remains robust under the Erlang assumption as well. For fast moving items, it is obvious that a higher convergence order is required than for slow moving items. able 1 also shows that the convergence order decreases as the degree of uncertainty of the system decreases i.e. the number of Erlang phases k raises and/or the variability of demand sizes reduces. Next, we analyze the variation of with respect to. Figures 1 and 2 show the curves for and CV² X =.9 and 4 respectively. Figures 3 and 4 show the curves for and CV² X =.9 and 4 respectively. 1 1 Figure 1. S * for CV² X =.9 and < 1 Figure 2. S * for CV² X = 4 and < 1 3 3 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 Figure 3. S * for CV² X =.9 and Figure 4. S * for CV² X = 4 and 771
We also analyze the variation of with respect to. Figures 5 and 6 show the curves for and CV² X =.9 and 4 respectively. Figures 7 and 8 show the curves for and CV² X =.9 and 4 respectively. E[C()] 1 E[C()] 1 Figure 5. for CV² X =.9 and < 1 Figure 6. for CV² X = 4 and < 1 E[C()] 6 5 3 1 2 3 4 5 6 7 8 9 E[C()] 6 5 3 1 2 3 4 5 6 7 8 9 Figure 7. for CV² X =.9 and Figure 8. for CV² X = 4 and Figures 1-4 show that decreases in. As the average demand interval increases i.e. degree of intermittence increases, the advised stock level reduces. As expected and as it is shown by Figures 5-8, when demand gets more intermittent, the expected holding cost increases because the stock holding duration is longer and the expected backordering cost decreases because backorders are likely less frequent. In this case, it is clear that it is more economic to fall in stockout than to keep items in stock. Note also that for fast moving items, the optimal base-stock level is much higher than the one for slow moving items. In fact, for fast moving items, if backordering costs are high, when there is no sufficient stock it is more economically discerning to stock important quantities. As increases, the expected backordering costs reduce considerably then, the stocking is no longer necessary and optimal can be lowered. Figures 3 and 4 show that from some values,, i.e. it is not any more economic to keep in stock, the demand is sufficiently intermittent to satisfy it only with immediate replenishment orders. which are restrained to backorders are decreasing in (Figures 7 and 8). As explained above, demand intervals tend to be deterministic as k increases i.e. the randomness degree on demand arrivals decreases in k. hus, as shown by the S * curves, for all demand intervals, decreases in k (by approximately 3% for < 1 and by up to % for > 1) and consequently corresponding decreases in k. Furthermore, Figures 1-8 show that the more demand sizes are variable, the higher optimal base-stock level to enface this variability which generates higher. By comparing the shapes of these curves, we also conclude that the optimal base-stock level decreases to zero faster in if the size variability is high. his is expected since lumpy demand is risky and can imply important holding and backordering costs so, when the demand interval increases, it is economically safer to eliminate the stock and to rely more on replenishments. his result is not intuitive under customer service optimization reasoning. 772
4. Conclusion and future research We developed a method that can be used to determine numerically the optimal base-stock level for a single echelon single item inventory system. he demand intervals are modeled by an Erlang-k distribution and the demand sizes by a Gamma distribution. he Erlang-k assumption was motivated by our aspiration to propose a more suitable and more economic method than that under the compound Poisson demand, which is of a great utility in existing stock control systems, and particularly for wearing-out spare parts. he lead-time was assumed to be constant and unfilled demands are backordered. he optimal base-stock level is approximated by upper and lower bounds that converge after a relatively low number of iterations. his approximation could be as accurate as we decide it. When increasing the number of Erlang phases k and decreasing the variability of the demand sizes, we showed in this paper that the optimal base-stock level decreases as it is expected and it keeps decreasing in the average demands inter-arrivals. One of the most interesting results for us is the fact that in contrast with a customer service level oriented inventory control system which advises a bigger base-stock level as demand sizes variability increases, a cost oriented inventory control system advises to reduce this base-stock level because it costs cheaper to fall in stockout than providing stock to enface a very variable demand size. Immediate possible way to extend this research consists to conduct an empirical investigation on this Erlangian model to corroborate the benefits of applying it in a real industrial context. By the same way, we can assess the goodness of fit of the Erlang hypothesis on real demand sequences. Further research consists certainly to consider a stochastic lead-time. References: Archibald, B.C., Silver, E.A., 1978. (s,s) Policies under Continuous Review and Discrete Compound Poisson Demand. Management Science, 24, 899-99. Babai, M.Z., Jemai, Z., Dallery, Y., 11. Analysis of order-up-to-level inventory systems with compound Poisson demand, European Journal of Operational Research, 2, 552 558. Cheung, K.L., 1996. On the (S-1, S) Inventory Model under Compound Poisson Demands and i.i.d. Unit Resupply imes, Naval Research Logistics, 43, 563-572. Gupta, A. K., Zeng, W. B., Wu, Y.,. Probability and Statistical Models, Foundations for Problems in Reliabilityand Financial Mathematics. Springer Science, Business Media, LLC, NY. Kalpakam, S., Sapna, K.P., 1998. Optimum Ordering Policies For Expensive Slow Moving Items. Applied Mathematics Letters. Vol. 11, No. 3, pp. 95-99. Kleinrock, L., 1975. Queuing systems, volume 1: theory. John Wiley & sons, Canada. Larsen, C., horstenson, A., 8. A comparison between the order and the volume fill rate for a basestock inventory control system under a compound renewal demand process. Journal of the Operational Research Society 59, 798 4. Sani, B., Kingsman, B.G., 1997. Selecting the best periodic inventory control and demand forecasting methods for low demand items. Journal of the Operational Research Society 48, 7 713. Smith, M.A.J, Dekker, R., 1997. On the (S-1,S) stock model for renewal demand processes: Poisson s poison. Probability in the Engineering and informational Sciences 11, 375-386. eunter, R., Syntetos, A.A. and Babai, M.Z.,. Determining Order-Up-o Levels under Periodic Review for Compound Binomial (Intermittent) demand, European Journal of Operational Research, 3, 619-624. Zipkin, P.H.,. Fundamentals of queuing theory. McGraw-Hill, Boston. 773