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The standard inventory problems of the multi-period have been modeled under different situations. Specifically we have considered the demand subjects of a continuous distribution and a discrete distribution, and whether the demand of each period is unchanged or not. A method to get an economic order quantity in inventory systems with discrete and unchanged demand was presented in a previous paper, and this method has been generalized to an inventory model with varying continuous demand. However, it was not achieved due to there being many classified cases in the general situations. In this article the above method is discussed in the case discrete demand to determine whether it increases or decreases from period to period. A theoretical method is presented by using previous results and some examples are given which suggest how the concept can handle on inventory system. In order to make the decision, an algorithm is also presented under some conditions, and examples are shown by using the computer software program, Mathematica, which helps to explain the findings. In general cases, we view the optimal policy in the inventory problems in only a few periods.

Probabilistic inventory models of the multi-period have been studied in which some conditions are searched to help obtain an optimal policy, providing that the total cost function of a single period is known extensively (Sakaguchi and Kodama, 2002). In those models, demand of each period is assumed to be unchanged for simplicity. These facts were applied to the case varying demand. However, it remains difficult to get the precise economic order quantity. Moreover, it was not achieved because there were many cases in the general situation. Therefore, the study initially researched the inventory model in the restricted case where demand decreased over time (Sakaguchi and Kodama, 2005). A model with exponential demand is researched in Sakaguchi (2007a) since it is easier when demand is subjected to an exponential distribution. In this paper we will continue to develop the method stated above where the case demand is discrete. The results in Sakaguchi and Kodama (to appear) inspire us to consider some examples in Sakaguchi (2007b) and subsequently generate a lot of demanding calculations. The model investigated in this article is as follows. Let h and p be the holding cost and the shortage costs per unit per period, respectively, and let c be the purchasing cost per unit. Let us denote by z the amount on hand in initial period after a regular order is received and assume that demand in a single period be a discrete random variable. The decision criterion of single period is the minimization of the expected cost which includes the purchasing, holding and shortage costs. That is, the expectation E{C(b,z)}E{C(b,z)} of the total cost C(b,z)C(b,z) is View the MathML sourceE{C(b,z)}=c(purchasingquantity)+hE{holdingquantity}+pE{shortagequantity}. Turn MathJax on Our objective is to obtain a value z at which E{C(b,z)}E{C(b,z)} is minimized. Then, when the initial inventory quantity is x, the amount of the replenishment quantity is 0 or z−x. The basic facts about our model are stated in Sakaguchi (2007a) and we refer the general notion of the inventory management to Silver et al. (1998). Let N be the number of periods in the inventory model. In order to analyze the structure of our inventory models, many kinds of the functions are used in this article. The fundamental ones are Hi(z)Hi(z) and View the MathML sourcewi(z)(z=1,2,…,N) which are defined by the equations E{C(bi,z)}=-cx+Hi(z),E{C(bi,z)}=-cx+Hi(z), Turn MathJax on ΔE{C(bi,z)}=c-p+(h+p)wi(z),ΔE{C(bi,z)}=c-p+(h+p)wi(z), Turn MathJax on where bibi is a random variable of demand quantity in the i th period. Two conditions: D1 and D2 on the functions Hi(z)Hi(z) are assumed to get the fundamental Theorem 4 and it leads us to an algorithm of seeking an optimal policy. Two distributions, a uniform distribution and a Poisson distribution, are considered in this paper as making inventory examples. The statue of time-varying demand materializes by changing a parameter of distribution. We denote by DEC a decreasing demand from period to period and denote by INC an increasing one. Though we deal only with the case demand subjects to a discrete distribution, the model is built by using a continuous function g(x)g(x) that indicates occurrence of a demand quantity along time in one period. The functions View the MathML sourcewni(z)(n=1,2,…,N-1,i=1,2,…,N-n+1) Turn MathJax on are constructed inductively. Put View the MathML sourceκ=p-ch+p. Turn MathJax on In the case of DEC, a method, used to obtain the economic order quantity from period i through period i+n-1i+n-1 is achieved by solving an inequality View the MathML sourcewni(z)≥κ+αch+p, Turn MathJax on where α is a discount factor. In the case of INC, it is too complicated and consequently we view only problems of a few periods. Finally, a lot of examples are shown that are computed by making use of the computer software Mathematica.