Minimizing holding and ordering costs subject to a bound on backorders is as easy as solving a single backorder cost model
Section snippets
Introduction and motivation
The traditional backorder cost (BC) model minimizes the sum of long-run average ordering, inventory holding, and backorder costs per unit time. Both inventory holding and backorder costs are traditionally charged at constant rates, proportional to the number of inventory and backorders, i.e. ($/unit/time). In practice, the backorder costs charged by the BC model are rarely out-of-pocket and therefore are difficult to specify. The use of proportional backorder penalty costs is justified by
Literature review
Initial treatment of the BC (Q,r) model dates back to late 1950s (see Whitin [23]). An exact formulation under the assumptions of Poisson demand and positive reorder levels was developed and analyzed by Galliher et al. [7] and by Hadley and Whitin [8]. Since then, substantial research has been done on (Q,r) and (s,S) policies. Various authors have analyzed the operating characteristics and the objective functions of (Q,r) and (s,S) policies under different system settings: Browne and Zipkin [2]
Preliminaries
For a given continuous review (Q,r) policy, let c(Q,r) denote the long-run average ordering and inventory holding cost, and let B(Q,r) denote the expected number of backorders. Then the model can be formulated aswhere η is the predetermined maximum tolerance for backorders.
Let λ denote the demand rate (the average demand per unit time) and D the sum of the demands that occur over the fixed leadtime interval (t,t+L] in steady state. We assume that D⩾0. Let F be the cdf
Analysis and algorithm
Our analysis is largely inspired by Zheng [25]. The two-dimensional service-constrained minimization problem minQ,r c(Q,r) can be carried out sequentially: . Let r(Q) denote an optimal r for fixed Q. LetNotice that for fixed Q, c(Q,r) is increasing and B(Q,r) is decreasing in r. Therefore r(Q) is defined by the identity
Let r′(Q) denote the derivative of r(Q). Lemma 1 r(Q) is decreasing and r(Q)+Q is increasing in Q. Specifically, Proof
Imputed backorder penalty rate
Here we show that it is possible to easily obtain the imputed backorder penalty cost as well as the fill-rate at the end of the algorithm. To see this, recall that the traditional BC model minimizes (4) with G(y) replaced by GT(y)=h(y−D)++pE[D−y]+, where p is the linear backorder penalty rate. It is shown in Zheng [25] that for fixed Q, the optimal reorder level rT(Q) for the BC model is selected such thatAs shown in Gallego [6], this condition is equivalent to
Numerical example
The purpose of this section is to illustrate the results of our model and to demonstrate the performance of the algorithm presented in Section 4. The following parameters are used in the example: {L=1,K=25,h=10}. The leadtime demand is assumed to be normally distributed with mean μ and standard deviation σ. Two distributions are tested: (μ,σ)∈{(10,2.5),(100,25)}. The tolerance limit for backorders is set as η=1, corresponding to 10% (resp. 1%) of mean leadtime demand in the case μ=10, (resp. μ
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2019, Operations Research PerspectivesCitation Excerpt :Such an exchange curve illustrated the trade-off between aggregate inventory costs versus the costs for maximum allowed backorder delay. Boyaci and Gallego [6] investigated a stock refilling problem with the aim of minimizing ordering and holding costs subject to a constraint on the expected level of backorders. A algorithm was developed to compute an optimal policy, the imputed penalty cost rate, and the corresponding fill-rate.
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