Production, Manufacturing and LogisticsA dynamic analysis of the single-item periodic stochastic inventory system with order capacity
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,y∈C and 0⩽λ⩽1 such that y⩽x⩽y+β.
Note that a (0,+∞)-convex function is a convex function. The function h(x) defined below is (3,2)-convex but not convex:
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 X–Y 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 x⩽y1 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)+ay1⩾K+L(y0)+ay0. If CP⩾y0−y1, then every inventory position y⩽y1 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 X–Y 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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2015, Computers and Industrial EngineeringCitation Excerpt :Over the last half a century, one of the most commonly studied research topics, in applied operations research and industrial engineering, is the analysis of production inventory system. A few examples of such studies include: a single item inventory system with a non-stationary demand process in a single stage production inventory system (Graves, 1999), a periodic review stochastic inventory system (Chan & Song, 2003), determination of lot size and order level for a single item, single stage inventory model with a deterministic time-dependent demand (Dave, 1989), a single item and single stage inventory system with stochastic demand in a periodic review where the system must order either none, or at least as much, as a minimum order quantity (Kiesmüller, De Kok, & Dabia, 2011) and to analyse three inventory models; a replenishment batching policy, a production batching policy, and an integrated replenishment/production batching policy (Rau & OuYang, 2007). Recently, a few studies have considered some important practical issues while analysing production–inventory system, such as: machine breakdown, process interruptions and supplier reliability.
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2015, Computers and Operations ResearchCitation Excerpt :Goyal [9] also applied the basic concept of the EPQ model to determine optimal lot sizes in a two-stage production system that minimized the sum of all costs. Other such extensions of EPQ models in single-stage production-inventory systems were developed by Ishii and Imori [16], Graves [11], Biskup et al. [2], Chan and Song [3], Dave [6], Chiu et al. [5], Pentico et al. [29], Paul et al. [24], Sarkar and Moon [31], and Kiesmüller et al. [17]. However, the above studies focused mainly on either single- or two-stage production systems whereas, on many production lines, products are processed in multiple stages [10].
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