Production, Manufacturing and Logistics
A dynamic analysis of the single-item periodic stochastic inventory system with order capacity

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Abstract

Consider a single-item periodic review stochastic inventory system with positive setup cost and finite order capacity. Chen and Lambrecht [Operations Research 44 (1996) 1013] showed that the optimal policy has a systematic pattern called the XY band structure. However there is no clear pattern for inventory positions between X and Y. Some properties of the optimal order policy are provided when the inventory position falls between X and Y. As a consequence of the analysis, an efficient algorithm is provided to compute the optimal ordering policy parameters.

Section snippets

Literature review

Consider a single-item periodic review stochastic inventory system with positive setup cost and finite order capacity. The order cost includes a fixed setup cost and a variable cost. The single-period expected shortage and holding cost is a non-negative convex function. The demand in each period is a non-negative discrete random variable which is independent of those in other periods. If the lead time between the placement of an order and its receipt is a positive integral multiple of the

Notations, assumptions, and formulation

Assume that the demand D in each period is a non-negative discrete random variable which is independently and identically distributed from period to period. For the real application, the random demand is usually bounded. Even if it is unbounded, we may use some approximation to replace the unbounded random variable by a bounded one. Thus, we assume that D is bounded by some positive integer N. Let pj=Pr(D=j), j=0,1,…,N, where ∑j=0Npj=1. In each period, define the following respective variables

(α,β)-Convexity and its applications

The study of inventory systems with inventory capacity CI, order capacity CP and positive setup cost K is related to a generalized concept of convexity called (α,β)-convexity.

Definition 1

A function h(x) is said to be (α,β)-convex on a convex set C if h(λx+(1−λ)y)⩽λ(α+h(x))+(1−λ)h(y) for any x,yC and 0⩽λ⩽1 such that yxy+β.

Note that a (0,+∞)-convex function is a convex function. The function h(x) defined below is (3,2)-convex but not convex:h(x)=1,−∞<x⩽1,x,1<x⩽2,4−x,2<x⩽3,1,3<x<+∞.

The graph of the

Main results

In this section, we assume that K>0 and CP<∞. The main result is the characterization of the structure of the optimal ordering quantities for all inventory positions fall between X and Y, where X and Y are defined in Chen and Lambrecht (1996) and used in the definition of XY band structure. Before we present the main result, we first study an intuitively obvious fact about the cost function Hn+1(y) over the (n+1)-period planning horizon. The proof is given in Appendix C.

Lemma 3

Suppose that Fn(x) is (K

An algorithm and a numerical example

Using the results of Theorem 2, we can design a new algorithm that can greatly reduce the computation time compared with the traditional dynamic programming approach. The following algorithm provides an efficient procedure for the computation of the optimal order quantities On(x) given that Fn−1(x) is (K,CP)-convex.
Algorithm for finding the optimal order policy over n-period planning horizon:

  • Step 1. Compute the X-band and Y-band according to the method given in Chen and Lambrecht (1996).

  • Step 2.

A conjecture

First, we introduce the following lemma appeared in Chen and Lambrecht (1996) which shows that there exists an inventory position y1 such that all the inventory positions xy1 are the order levels over the n-period planning horizon, n=1,2,…

Lemma 4

Let y0 be a minimizer of the convex function ay+L(y) on (−∞,CI] and let y1<y0 be the largest inventory position such that L(y1)+ay1K+L(y0)+ay0. If CPy0y1, then every inventory position yy1 is an order level over the n-period planning horizon, n=1,2,…

We

Conclusion

The single-item periodic review stochastic inventory system with positive setup cost and finite order capacity is considered in this paper. Chen and Lambrecht (1996) showed that the optimal policy has a systematic pattern called the XY band structure. However there is no clear pattern for inventory positions between X and Y. We first provide some mathematical analysis about this inventory system and characterize the structure of the optimal ordering policy when the inventory positions between X

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