بررسی بی ثباتی راه اندازی در سیستم های موجودی تصادفی غیر ثابت

In stochastic inventory systems unfolding uncertainties in demand lead to the revision of earlier replenishment plans which in turn results in an instability or so-called system nervousness. In this paper, we provide the grounds for measuring system nervousness in non-stationary demand environments, and gauge the stability and the cost performances of (R,S) and (s,S) inventory policies. Our results reveal that, both the stability and the cost performance of inventory policies are affected by the demand pattern as well as the cost parameters, and the (R,S) policy has the potential to replace the cost-optimal (s,S) policy for systems with limited flexibility.

In inventory planning systems, inventory replenishment plans, i.e. the timing and the size of replenishments, are often revised in response to realized demand. In practice, when this is the case, the replenishment plan is regenerated for the rest of the planning horizon leading to a planning instability or so-called system nervousness ( Vollmann et al., 1988). There is a large variety of inventory policies applied in inventory management practices (see e.g. Silver et al., 1998). These policies are extensively investigated in terms of their cost performance. However, in systems with low degrees of flexibility, the cost of implementing revisions in replenishment decisions may overcome the advantage of using the cost-efficient technique. In this context inventory control rules show different levels of instability. Thus system nervousness, as a performance criterion, can be of high importance in assessing inventory control rules. Omitting the planning instability can turn out to be a serious problem because it gives rise to a considerable amount of alteration efforts (Heisig, 2001). The (s,S) policy has been shown to be cost-optimal under very relaxed assumptions in both stationary and non-stationary demand cases (see Scarf, 1960 and Iglehart, 1963). Heisig, 1998 and Heisig, 2001, and de Kok and Inderfurth (1997) have questioned the performance of the (s,S) policy with respect to the nervousness criterion in the stationary demand case. Their research reveals the trade-off between cost effectiveness and nervousness and shows that the (s,S) policy exhibits the worst stability performance among a number of policies considered. Different strategies for dealing with the problem of nervousness are examined by Blackburn et al. (1986). They suggest an effective strategy based on freezing certain orders so they cannot be changed. In this regard, the (R,S) policy, in which the timing of future orders are fixed, provides a means of dampening the setup instability. Silver et al. (1998) points out that the (R,S) policy, which provides a rhythmic rather than a random replenishment pattern, is usually appealing from a practitioners point of view. One major difficulty in the continuing development of inventory theory is to incorporate more realistic assumptions about demand into inventory models. In many stable environments it is an adequate approximation to treat period demands as identically distributed random variables. However, demand patterns are often heavily seasonal or possess significant trends especially in industrial settings with business cycles. Furthermore, as product life cycles get shorter, the randomness and the unpredictability of demand increases. The essence of such situations can only be captured by means of finite horizon non-stationary inventory models. Literature provides guidelines about the stability performance of inventory policies in stationary systems. However, those may not be directly generalized to non-stationary systems. In stationary systems policy parameters are also stationary, and therefore, the measure of stability of the whole system can be determined by means of observing two arbitrary consecutive planning cycles. However, in non-stationary systems policy parameters are determined in connection with each and every period through the planning horizon, and consequently, stability is a function of the demand pattern (Blackburn et al., 1986). To the best of our knowledge, no work has been done on the measures of nervousness in non-stationary systems. In this paper, we aim to fill in this gap by investigating the system nervousness under non-stationary stochastic demand. Our contribution is two-fold. First, we propose a method for measuring system nervousness in non-stationary demand environments. Secondly, we gauge the setup stability of (R,S) and (s,S) inventory policies and demonstrate that the (R,S) policy has the potential to replace the cost-optimal (s,S) policy, especially for systems characterized by a low degree of flexibility to setup changes. The remainder of this paper is organized as follows. Section 2 investigates the related literature and positions the current work. Section 3 defines the inventory system addressed. Section 4 provides the grounds for computing setup instability in non-stationary environments and proposes a method for the computation thereof. Section 5 introduces the models computing (s,S) and (R,S) policies. Section 6 presents the computational experiments. Section 7 concludes and sketches some likely extensions of the study.

In this study, we extended the system nervousness definitions in the literature to cover non-stationary stochastic demand, and we investigated the cost and stability performances of (R,S) and cost-optimal (s,S) policies in terms of system nervousness. In contrast to previous studies using a rolling horizon framework which itself is a source of nervousness, we employed a re-planning approach and we analyzed the nervousness resulting from pure demand uncertainty. We proposed a strategy to obtain the expected cost and the instability performances of a given inventory problem simultaneously. We characterized the effects of setup frequency, demand variability, and various demand patterns on the cost and the instability performances of inventory policies. We showed that, although setup frequency and demand variability effect both the cost and the stability performance, they do not play a significant role in characterizing the trade-off between those. While on the contrary, we observed that the trade-off between the cost and the stability performance of inventory policies is a function of the demand pattern. These findings can be used to assess inventory control policies under different system settings and can be of help to choose the most efficient inventory policy depending on the flexibility of the system to setup changes and the importance of cost. In our numerical study, we showed that (R,S) is clearly dominant to (s,S) in terms of stability performance even when the inventory plan is regenerated through the planning horizon, and the cost penalty of using (R,S), rather than cost-optimal (s,S), is fairly small under general settings. These all together show that (R,S) can be a strong alternative to (s,S) with its superior stability performance, especially for systems characterized by a low degree of flexibility to setup changes. One possible direction for further research is to consider the case where the importance of the instabilities diminishes in time. This could be implemented by using a discount factor. Another potential extension is to investigate the cost and instability performance of different inventory policies.