مقایسه (S، S) و (NQ) قوانین کنترل موجودی با توجه به ثبات برنامه ریزی

The application of a rolling horizon planning framework in inventory control causes different order release decisions in successive planning cycles. These lead to the so-called planning system nervousness. In this paper for a single-stage inventory system with arbitrary stochastic demand it is shown analytically how the setup-oriented planning stability concerning deviations of planned orders in all periods of a given stability horizon is influenced by the use of (s,S) and (s,nQ) control rules. It turns out that the reorder point s does not affect stability whereas the lot size determining parameters S−s and Q as well as the level of uncertainty have a considerable impact. The paper is concluded with directions for further research.

In the context of inventory control usually simple policies like (R,S), (s,nQ) or (s,S) policies are applied. These policies are investigated with respect to the optimal determination of their parameters according to their performance in terms of costs and service. Since in most practical applications inventory policies are used in connection with a rolling horizon planning framework, a third performance criterion implicitly exists: the measure of planning stability. It is worth noting that replenishment decisions are usually determined on the basis of quasi-deterministic models of the stochastic environment connected with a regular updating of all relevant parameters and succeeding replanning activities. This situation is typical for the wide-spread MRP applications whereas the above-mentioned procedure leads to so-called period order quantity (POQ) planning or fixed order quantity (FOQ) planning at each level of the production structure depending on the lotsizing procedure. The use of these planning rules can be interpreted as employing an (s,S) rule or an (s,Q) policy in a multi-period deterministic environment (see [1]) which evolves by using forecasts instead of stochastic data. The rolling horizon planning leads to replanning activities caused by permanent processing of new information in successive planning cycles. As a result formerly fixed order decisions are replanned in later periods. This discontinuity in maintaining former order releases is known as the nervousness syndrome (see [2]). The lack of planning stability can turn out to be a significant problem because it often generates a considerable amount of short- and medium-term adjustment efforts as well as a general loss in planning confidence. Especially in a multi-stage MRP production environment, nervousness on the top level (MPS level) is propagating throughout the system. In fact, due to MRP time-phasing, nervousness in future periods on the MPS-level may influence planning stability at the beginning of the planning process on lower stages in an MRP system. In many cases the consequences of replanning activities cannot be valued in terms of costs, therefore we will treat planning stability as an independent attribute for assessing an inventory control system (similar to the attribute of customer service). Defining a nervousness measure, we can distinguish between short-term planning instability (which measures nervousness with respect to only the first period's order decision) and long-term planning instability (which considers the entire planning horizon). Moreover we may differentiate between quantity adjustments, which are denoted by quantity-oriented instability, and pure changes in order setups, i.e. if in a new planning cycle a new setup is scheduled or, vice versa, if a formerly planned setup is canceled. This latter type of nervousness is described by setup-oriented instability (see, e.g., [3]). In order to treat planning stability as a specific performance criterion, we have to define a general numerical stability measure. In simulation studies which examine the influence of different planning parameters on nervousness only ad-hoc measures of planning stability are used (see, e.g. [4], [5], [6] and [7]). A systematic development of nervousness measures is given in [3], [8], [9], [10], [11], [12], [13] and [14]. In this paper we will refer to the measures described in [3], [8], [9], [10], [11], [12] and [15]. The above-mentioned simulation studies do not give a systematic insight into the relationship between planning stability and inventory policies. In [13] and [12] a comprehensive investigation of reorder-point lot-sizing policies is given, but its informative value is limited, since the simulation approach cannot provide an analytical description of the dependence of nervousness on inventory control rules. Analytical results are presented in [10] where the performance of the (s,S) and (s,nQ) policy for uniformly and exponentially distributed demand with respect to a short-term setup-oriented planning stability is analyzed. In [8], for the same short-term consideration as in [10], setup- as well as quantity-oriented stability of orders for more general demand distributions are examined for (s,S), (s,nQ) and (R,S) policies. In [9] the long-term setup-oriented stability performance of an (s,S) policy is analyzed. In this paper the same analysis is provided for an (s,nQ) policy. We will show the main analytical results and we will discuss the priority of both inventory control rules under the aspect of nervousness. For a more detailed discussion of the results presented in this paper the reader is referred to [15]. Here, we consider arbitrary stochastic demand to show analytically the influence of the demand distribution on our stability measure. In our analysis, differing with [8] and [10], we treat the long-term planning stability in order to capture its effect at all levels of the system (see above). In this context, we consider all periods within a so-called stability horizon which contains those planning periods where deviations in planned orders are perceived to be disadvantageous. This implies that plan changes in periods which exceed the stability horizon are not relevant with respect to planning stability. Since in some cases it seems to be reasonable to assume that replanning activities in periods near the beginning of the stability horizon are more critically than those in future periods (see, e.g. [13]), we modify the measure of planning stability proposed in [3], [8], [9], [10], [11] and [12] by introducing a weight parameter which reflects the (possibly) decreasing relevance of plan changes with progressing number of periods within the stability horizon. With suitable values for the weight parameter we can describe the short-term consideration as well as the case that all periods are weighted equally. This facilitates a comparison of our results to the outcome of [8] and [10] with respect to an (s,S) and (s,nQ) policy. Moreover, we consider the setup-oriented planning stability, because in many practical situations the fixed effort connected with replanning an order release, independently of the exact amount of adjustment, is mainly annoying in the execution of a planning process. The remainder of this paper is organized as follows. In Section 2 a measurable formalization of nervousness under rolling planning conditions is given. Section 3 contains a steady-state analysis for examining the impact of the (s,nQ) policy on the stability measure. In Section 4 analytical expression for the setup stability for the (s,S) and (s,nQ) policies are derived. In Section 5 the influence of the lot size parameters S−s, Q and the choice of the demand distribution as well as the impact of the length of the stability horizon and the accuracy of demand forecasts on the level of nervousness are shown. Finally, we draw the conclusions and provide guidelines for further research.

In this paper it is shown that for basic inventory control rules like (s,nQ) and (s,S) policies analytical insights in the effects on setup planning stability can be given. Nervousness is not affected by the reorder point s, but the choice of the lot size Q and the minimum reorder quantity S−s, respectively, and the length of the stability horizon T as well as the weighting of the periods within T have a significant impact on the planning stability. Moreover, bad demand forecasts may cause a poor stability performance of the inventory control system. For both policies a high level of nervousness is caused by setting the lot size approximately equal to the forecasted demand per period. Furthermore, for an (s,S) policy a low level of stability is reached if the lot size is set to a multiple of the forecasted demand per period. In particular minimum stability may be attained for reorder quantities which are twice as large as the forecasted demand per period. Additionally, for both policies a change of the length of the stability horizon lead to a change of the level of planning stability. For reason of analytical tractability some specific assumptions concerning the distribution of the demand have been made. By simulation studies considering the impact of the lot size and the influence of the demand uncertainty it can be shown (see [3]) that our results also hold for more global situations. Summarizing it can be stated that, if nervousness plays a role, inventory control rules and control parameters should be determined under additional consideration of their stability effects. In particular, the choice of the lot size has a considerable impact on stability. Similar to service level constraints which ensure a certain level of customer service with respect to cost considerations, in inventory control the additional factor of planning stability can be included by using analogous stability constraints. The results given in this paper give some insights how lotsizing has to be restricted if such constraints with respect to a specified amount of nervousness are given by the management. Moreover, planning stability can be improved by modifying planning procedures and applying more complex control rules employing additional parameters (see [2], [3], [12] and [17]). The analysis presented here can be extended in several aspects. In a next step the analysis might be extended to the quantity-oriented long-term planning stability for the basic inventory control rules like the (R,S), (s,nQ) and (s,S) policy (see [8] for a comparison of basic inventory control rules with respect to their performance in short-term quantity-oriented planning stability). Moreover, an analytical study of the improvement of the planning stability by introducing a stabilization parameter seems to be promising (see, e.g., [3]). Finally, the planning stability of multi-stage inventory systems should be considered. There are first simulation experiments for simple systems, e.g., in [3] a two-stage serial as well as a divergent system are considered (see [3, p. 219]). These studies indicate that the control rule used at the end-item level has a major influence on the stability performance of the entire system. If in a multi-stage system the end-item level is controlled by a reorder point policy, and a lot-for-lot policy is used on upstream stages, then our results can directly be applied to such a multi-stage system. Since orders are placed in the same period at all stages, only the size of the order at each level may be altered, but there are no new, previously unplanned orders. Consequently, setup stability at the final product stage represents the planning stability of the entire system (see also [4] and [17] where, besides other strategies, the application of a lot-for-lot policy on upstream stages is suggested to stabilize production schedules).