A two-demand-class inventory system with lost-sales and backorders

doi:10.1016/j.orl.2010.03.006

Operations Research Letters

Volume 38, Issue 4, July 2010, Pages 261-266

https://doi.org/10.1016/j.orl.2010.03.006 Get rights and content

Abstract

A periodic review inventory system serves two demand classes with different priorities. Unsatisfied demands in the high-priority class are lost, whereas those in the low-priority class are backlogged. We formulate the problem as a dynamic programming model and characterize the structure of the optimal replenishment policy.

Introduction

In the context of inventory theory, models typically assume that unsatisfied demands (or shortages) are either totally lost or fully backlogged. A comparison between these two types of inventory models (lost-sales and backorders) is provided in [9]. Many real inventory systems serve multiple demand classes with different priorities, in which on-hand inventories are allocated to classes for satisfying demands. In the literature, most researchers assume that shortages for all demand classes are treated identically, as either backorders or lost-sales. Papers focusing on backorders include [1], [2], [3], [8], [10], [13], [14], [16], [18], [19], [20], [21], and papers dealing with lost-sales include [5], [6], [7], [11], [12].

In practice, there are many multiple-demand-class inventory systems with both lost-sales and backorders. Depending on the business scenario, high priorities can be assigned to classes with lost-sales or to classes with backorders. Only a few papers analyze related models for optimal decisions [4], [17]. The authors in [4] discuss a two-class system with supply capacity, in which high priority is assigned to the class with lost-sales, whereas for the low-priority class part of the unsatisfied demands are lost and the rest are backlogged for one period. They assume that at the beginning of each period the replenishment decision must first fully satisfy all previous backorders, with the remaining on-hand inventories used to satisfy new demands. The authors in [17] address a two-class model with high priority given to the backorder demand class and low priority to the lost-sales demand class.

In this paper, we study a periodic review inventory system serving two demand classes, in which high priority is assigned to the class with lost-sales, while low priority is assigned to the class with backorders. Compared with [4], our paper assumes backorders to be satisfied in any later period, which is common in many real inventory systems, rather than backlogging part of them only for one period and losing the rest. We find new results that differ from existing results. In fact, the optimal policies in [4], [17] are of base-stock (or modified base-stock) type. However, as we show in Section 3 for our model, the base-stock structure is no longer optimal for negative inventory levels.

In the subsequent sections, we formulate the problem as a dynamic programming model and characterize the structure of the optimal replenishment policy.

Section snippets

Assumptions

Consider an inventory system that operates for a finite horizon of $N$ periods and serves two stochastic demand classes denoted by class $L$ and class $B$ . Unmet demands for class $L$ are lost, whereas those for class $B$ are backlogged. When satisfying demands using on-hand inventories, class $L$ has higher priority than class $B$ . Demands are independent across periods, while demands between class $L$ and class $B$ in a given period may be dependent. The system is replenished from a supplier with unlimited

Structure of the optimal policy

The following theorem presents the main result of this paper, where $Φ_{R} (z)$ is the cumulative distribution function of random variable $R$ and $Φ_{R}^{- 1} (u)$ denotes its inverse.

Theorem 3.1

If $x < 0$ , it holds that $r_{n} (x) \leq r_{n} (x + δ) \leq r_{n} (x) + δ$ , $\forall δ > 0$ satisfying $x + δ \leq 0$ . If $x \geq 0$ , there exists a critical level $S_{n} \in [0, Φ_{D^{L} + D^{B}}^{- 1} (\frac{p^{L} - c}{p^{L} + h - β c})]$ such that $r_{n} (x) = max {S_{n}, x}$ .

The proof of the above theorem is given in Section 3.2.

Theorem 3.1 implies that the optimal decision follows a base-stock policy if the inventory level is nonnegative.

Extensions and discussions

Based on Proposition 4.3.1 in [15], our model can be extended to the infinite-horizon discounted-cost case. A stationary policy $r (x)$ with the following properties is optimal: if $x < 0$ , then $0 \leq r (x + δ) - r (x) \leq δ$ for $\forall δ > 0$ such that $x + δ \leq 0$ ; if $x \geq 0$ , then the policy possesses a base-stock form, i.e., $r (x) = max {S, x}$ for an $S \in [0, Φ_{D^{B} + D^{L}}^{- 1} (\frac{p^{L} - c}{p^{L} + h - β c})]$ .

On the other hand, parallel results on the optimal policy (including for the infinite-horizon discounted-cost case) can be obtained if we consider integer-valued

Acknowledgements

This research is supported by NSF of China under Grant 70871066, and is partially supported by NSF of China under grant 70971072. The authors are grateful to Dr. David Robb for his kind help in improving the English of this paper. Also, the authors would like to thank the associate editor and the area editor for their helpful comments.

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View full text

A two-demand-class inventory system with lost-sales and backorders

Abstract

Introduction

Section snippets

Assumptions

Structure of the optimal policy

Extensions and discussions

Acknowledgements

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IIE Transactions

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Management Science