An easy derivation of the order level optimality condition for inventory systems with backordering

doi:10.1016/j.ijpe.2008.01.009

International Journal of Production Economics

Volume 114, Issue 1, July 2008, Pages 201-204

https://doi.org/10.1016/j.ijpe.2008.01.009 Get rights and content

Abstract

We analyze the classical inventory model with backordering, where the inventory position is controlled by an order level, order quantity policy. The cost for a backorder contains a fixed and a time-proportional component. Demand can be driven by any discrete process. Order lead times may be stochastic and orders are allowed to cross. The optimality condition for the order-level, given some predetermined order quantity, is derived using an easy and insightful marginal cost analysis. It is further shown how this condition can easily be (approximately) rewritten in well-known forms for special cases.

Introduction

We analyze the classical continuous review inventory system with backordering, where the inventory position is controlled by an order level, order quantity policy, in a rather general setting. We derive the optimality condition for the order level r, given some predetermined order quantity Q. Other authors have done so in the past, but our marginal cost approach is easier and more insightful. The main advantage of this approach is that there is no need to find an expression for the average cost (per time unit) associated with an $(r, Q)$ policy. The analysis of Hadley and Whitin (1963, Section 4.7) shows how complicated that is, even for the relatively simple case where demand is driven by a unit Poisson process and the replenishment order lead time is deterministic. Furthermore, as we will show, the optimality condition that is obtained using the marginal approach can easily be (approximately) rewritten in well-known forms for special cases.

Together with a derivation of the EOQ formula, our derivation and rewriting of the optimality condition for r could be explained to managers and students with a limited mathematical background. This especially holds in combination with a simple algebraic derivation of the EOQ formula. Such a simple derivation, not requiring the use of differential calculus, was first presented for the standard EOQ model by Grubbström (1996). It was later generalized for cases with backlogging by Grubbström and Erdem (1999), and extended to the EPQ model by Cárdenas-Barrón (2001). Ronald et al. (2004) further simplify the derivations of both the EOQ and the EPQ formula.

Section snippets

System description

We study an inventory system with backordering, where the inventory position is controlled by an order level, order quantity $(r, Q)$ policy. Demand can be driven by any discrete process (e.g. unit Poisson or compound Poisson). Demand splitting is applied. That is, all available stock on hand is used to satisfy (backordered) demands, either fully or partially. Demands are satisfied on a first come first served (FCFS) basis. There are no assumptions about the process that generates replenishment

Derivation of the order level optimality condition

In this section, we use a marginal cost approach to derive an optimality condition for $r^{*}$ . We find an expression for the marginal cost $Δ$ , which is the difference in average cost between policies $π (r, Q)$ and $π (r - 1, Q)$ , by comparing those two policies in period $[0, \infty)$ . In doing so, we assume that the inventory position of policy $π (r, Q)$ is one item higher at time 0. It is further assumed that both policies have the same number of outstanding replenishment orders, and that the remaining lead times of

References (7)

L.E. Cárdenas-Barrón
The economic production quantity (EPQ) with shortage derived algebraically
International Journal of Production Economics
(2001)
R.W. Grubbström et al.
The EOQ with backlogging derived without derivatives
International Journal of Production Economics
(1999)
R. Ronald et al.
Technical note: The EOQ and EPQ models with shortages derived without derivatives
International Journal of Production Economics
(2004)

There are more references available in the full text version of this article.

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The generated demands are processed in order to be consistent with the underlying theory of the (r, Q) inventory model. Since the given demand generator drives demands in discrete units, it is possible to apply demand-splitting (Teunter and Dekker, 2008). In addition, the simulation model processes demands based on a first-come-first-served ordering.
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