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Non-stationary stochastic demands are very common in industrial settings with seasonal patterns, trends, business cycles, and limited-life items. In such cases, the optimal inventory control policies are also non-stationary. However, due to high computational complexity, non-stationary inventory policies are not usually preferred in real-life applications. In this paper, we investigate the cost of using a stationary policy as an approximation to the optimal non-stationary one. Our numerical study points to two important results: (i) Using stationary policies can be very expensive depending on the magnitude of demand variability. (ii) Stationary policies may be efficient approximations to optimal non-stationary policies when demand information contains high uncertainty, setup costs are high and penalty costs are low.

Stochastic inventory control systems have been studied extensively under various assumptions on demand. Nevertheless, the literature reflects a clear dichotomy between inventory models with stationary and non-stationary demands. The former assumes a steady demand process, whereas the latter assumes a demand process that varies in time. Strictly speaking, most practical demand patterns are non-stationary [1]. Furthermore, as product life cycles are becoming shorter, demand that evolves over the life of the product never follows stationary patterns [2]. For instance, electronic products, which have relatively short life cycle, generally follow non-stationary demand patterns (see e.g. [2] and [3]). Moreover, many authors have reported that providers of components and subassemblies often face unstable customer orders (see e.g. [4] and [5]). One major theme in the continuing development of inventory theory is the incorporation of more realistic demand assumptions into inventory models. Consequently, one would expect increasing number of studies concerned with non-stationary inventory models. However, the literature on non-stationary demand is rather limited, whereas it is vast for stationary demand. A topic search (title, abstract and keywords) on the ISI Web of Knowledge, since the year 2000, using the terms stationary and inventory gives 221 published papers, whilst this figure is only 29 for the terms non-stationary and inventory. It is obvious that, there is also a large number of papers assuming stationary demand without using the term stationary. This disparity is mainly due to the ill structure of non-stationary problems from a theoretical point of view and the complexity inherent in non-stationary models from a computational point of view. Silver et al. [6] point out that non-stationary demand is too complicated for routine use in practice. Furthermore, as Kurawarwala and Matsuo [7] stated, the unique characteristics of non-stationary demand preclude the use of traditional forecasting methods not designed for this environment and raise a need for tailor-made forecasting methods. Consequently, stationary policies have always been preferred to non-stationary policies in many real-life applications for the sake of their relative simplicity even if the underlying actual demand is non-stationary. In spite of all the above mentioned issues related to non-stationary inventory policies, when demand is non-stationary, a stationary policy is an approximation to the optimal non-stationary one, and hence, is sub-optimal with respect to total expected cost. This research investigates the magnitude of this sub-optimality under various settings. To the best of our knowledge, no work has been done that can be used as a guideline to compute the cost of using stationary policies when demand is non-stationary. We establish our analysis by using the (s,S) inventory control policy. The (s,S) policy is proven to be optimal both in stationary and non-stationary demand cases, and therefore, constitutes an inherent frame of reference. Our contribution is two-fold. First, we show that using stationary policies can be very expensive depending on the extent of demand variability as well as other factors. Second, we provide some insight on cases where stationary models provide good approximations to non-stationary models. In the remainder of this section, we concisely review related literature. In Section 2, we give the key assumptions of the inventory problem considered, and present algorithms used to compute the stationary and the non-stationary (s,S) policies. In Section 3, we present the experimental design and computational results. Finally, in Section 4, we draw general conclusions and provide some managerial insights. Most of the research in inventory literature assumes either a stationary or a non-stationary demand, and develop models and policies accordingly. Therefore, it is difficult to refer to any research addressing the cost performance of stationary policies when demand is non-stationary. However, we believe that it is necessary to briefly discuss the key literature in order to ease the exposition of the remaining sections. One of the most exciting developments in the inventory theory is Scarf's [8] proof of the optimality of (s,S) policies. (s,S) policies are characterized by two critical numbers sn and Sn for each period n, such that, the inventory position is replenished up to a target level Sn whenever the inventory position at the beginning of the period is lower than (or equal to) a re-order level sn. Scarf [8] showed the optimal value function satisfies a condition, which he called K-convexity, and provided a procedure for establishing the optimal policy parameters via a recursive function. Scarf's formulation required extensive computational power beyond the limitations of its time. As a matter of fact, there was no known way of computing policy parameters at that time [9]. Following Scarf [8], Iglehart [10] demonstrated the optimality of (s,S) policies in infinite horizon inventory problems with stationary demand. He showed that optimal policy parameters converge to two limit values s and S in this case. Iglehart's work has been followed by a large number of researchers (see e.g. [11], [12], [13], [14], [15], [16] and [17]) aiming at efficiently computing optimal policy parameters using the stationary analysis approach. However, not much work has been done for computing non-stationary (s,S) policies. A few authors addressed the inventory problem with non-stationary demands. Some of these work focused on alternative inventory control policies (see e.g. [18], [19], [20] and [21]), whereas some others proposed heuristics for computing near-optimal (s,S) parameters (see e.g. [22]). In this paper, we consider the inventory problem addressed in Scarf [8] and investigate the cost efficiency of stationary and non-stationary inventory policies.

As product life-cycles are getting shorter most of the real-life problems exhibit non-stationary demand patterns. However, non-stationary inventory policies have not been widespread among neither practitioners nor academics due to their complexity in computation and application. When demand is non-stationary, a stationary policy is only an approximation to the optimal non-stationary one, and hence, is sub-optimal with respect to the total expected cost. In this paper we analyzed the cost efficiency of using stationary inventory policies when demand is non-stationary. We used (s,S) policy as a frame of reference, and compared the optimal non-stationary (s,S) policy with the best possible stationary (s,S) policy in terms of cost performance. We conducted an extensive numerical study considering a variety of cases to examine the effects of various organizational parameters and showed that cost of neglecting the non-stationarity of demand is significantly high for the majority of cases. Our numerical study reveals that, the magnitude of the sub-optimality of stationary policies depends heavily on the variation of the demand pattern, i.e. the non-stationarity of demand, among other factors, such as, the stochasticity of demand, and cost parameters. Our study also provides some insight on cases where stationary models provide good approximations to non-stationary models. More specifically, when demands follow a rather stable pattern with high uncertainty, stationary policies may be a reasonable substitute for the optimal non-stationary policy. That is to say, higher coefficient of variation, which reflects the uncertainty of demand, resulted in better stationary policy performance. Moreover, the cost performance of stationary policies also improves when setup costs are high and penalty costs for stockouts are low. In other words, stationary policies may be efficient approximations to optimal non-stationary policies when demand information contains high uncertainty, setup costs are high and penalty costs are low. Our study underlines the need for careful evaluation when assuming stationary demands. We argue and provide some evidence that it may be very expensive to use stationary policies when actual demand is non-stationary. The recent literature is not sufficient to embed non-stationary models into real-life problems. Consequently, non-stationary stochastic demands have not yet been addressed in MRP/ERP environments. This observation points out the need for investigating and employing novel non-stationary models which are easy to understand and implement in practice. For instance, when demand rate changes rather slowly, even basic approaches such as periodically updating policy parameters could be satisfactory. Alternatively, reorder cycle policies, which are proven to be very flexible in modeling non-stationary demands can be of great use (see e.g. [21]). There are many other avenues for further research. One possibility would be to look at the effect of lead time on the sub-optimality of stationary policies. One would expect that the effect of a longer lead time is comparable to the effect of a higher uncertainty on demand. Nevertheless, it would be interesting to look at the magnitude of this effect. In this paper we investigated the cost of ignoring the non-stationarity of demand. Another interesting direction for further research is to investigate the cost of ignoring non-stationarity of other parameters such as unit, ordering, holding, and penalty costs. Finally, the effect of neglecting correlated demand, which is also very common in practice, is also worth exploring.