مدل سازی کنترل موجودی با فرآیندهای احیا کننده

In this paper we will discuss a general framework for single-item inventory models based on the theory of regenerative processes. After presenting without proof the main theorems for regenerative processes we analyze in detail how the different single-item models can be embedded within this general theory. This facilitates to write down the expressions for the average cost associated with an arbitrary costrate function f. Since these expressions are still complicated, involving convolutions, we use a recently developed numerically stable Laplace inversion algorithm to compute these objective functions in MATLAB©. This enables us to compute the costs instead of using approximations.

Ford Harris’ famous paper on the EOQ model in 1913 (cf. [1]) was the first of many publications on inventory theory. At present, thousands of papers have appeared in the management science and operations research literature. One may wonder why so much research is done on inventory models. The explanation is simply that, in practice, one encounters many different situations and each one requires a tailor-made analysis. For example, there may be differences with respect to the following aspects: number of locations and echelons, number of products, demand process, cost structure, service requirements and measurement, possible moments of placing a replenishment order, the way a stockout is handled, and the lead time of replenishment orders. Since so many different situations can be analyzed, we feel that there is a need to develop a general framework. Such a framework will help to improve the understanding of the models that appeared in the literature. In this paper the average cost for a number of basic inventory models will be derived using a general framework which is presented in the next section. This framework is based on the theory of regenerative processes. It can be shown that most inventory models satisfy the so-called regenerative property which allows for a nice derivation of the average cost. Moreover, the newly developed algorithm for numerically inverting Laplace transforms (cf. [2]) enables us to calculate the exact costs, instead of using approximations. We will restrict ourselves to inventory systems with a single product, a single location, backordering of stockouts and deterministic lead times. In most of the inventory literature such a system is considered and for an overview the reader is referred to Chikán [3]. A more recent discussion of those models is given by de Kok [4], usually the analysis of those models serve as a basis for multi-location, multi-echelon and multi-item systems. The theory of regenerative processes is briefly discussed in Section 2 and its application to inventory models is presented in Section 3. Section 4 deals with a more detailed discussion of the different classical single-item inventory models. In this section the exact expression of the average cost for each model is presented. In Section 5 the algorithm for numerically inverting Laplace transforms is applied to a numerical example of the so-called (R,S) inventory model. This is the simplest of the considered models and in a forthcoming paper we will show how to apply this algorithm to the other more complicated models. Finally, in Section 6 some conclusions are presented.

The general framework of regenerative processes enables us to derive the average cost for any of the classical single-item inventory control models in an easy and efficient way. Moreover, the main advantage of this approach is that presents a uniform analysis of these inventory control models, which helps improving the understanding of the theory, offers an insight into the differences and similarities among these different single-item models. The special convolution structure of the expressions for the average costs permits an easy derivation of the Laplace transforms. Applying the newly developed Laplace inversion method one can calculate these expressions with machine precision, therefore there is no need to use approximations (cf. [7]). Plotting the expected average cost function gives an insight to the behavior and sensitivity of the costs for the different models.