Elsevier

Operations Research Letters

Volume 29, Issue 4, November 2001, Pages 187-192
Operations Research Letters

Minimizing holding and ordering costs subject to a bound on backorders is as easy as solving a single backorder cost model

https://doi.org/10.1016/S0167-6377(01)00095-5Get rights and content

Abstract

Minimizing average ordering and holding cost subject to a constraint on expected backorders has required to iteratively solve the backorder cost model with different backorder penalty rates until the constraint is satisfied. Here we present a direct approach, that is as simple 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 asminQ,rc(Q,r)s.t.B(Q,r)⩽η,where 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: minQ[minrc(Q,r)]. Let r(Q) denote an optimal r for fixed Q. LetS(Q,r)=rr+QE[D−y]+dy−ηQ=Q(B(Q,r)−η).Notice that for fixed Q, c(Q,r) is increasing and B(Q,r) is decreasing in r. Therefore r(Q) is defined by the identityS(Q,r)≡0.

Let r′(Q) denote the derivative of r(Q).

Lemma 1

r(Q) is decreasing and r(Q)+Q is increasing in Q. Specifically,12⩽r′(Q)⩽0.

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(yD)++pE[Dy]+, 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 thatGT(rT(Q))=GT(rT(Q)+Q).As shown in Gallego [6], this condition is equivalent to1QrT(Q)rT

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

References (26)

  • H Tempelmeier

    Inventory control using a service constraint on the expected customer order waiting time

    European J. Oper. Res.

    (1985)
  • H.C Tijms et al.

    Simple approximations for the reorder point in periodic and continuous review (s,S) inventory systems with service level constraints

    European J. Oper. Res.

    (1984)
  • T. Boyaci, Supply chain coordination and service level management, Ph.D. Thesis, Department of IEOR, Columbia...
  • S Browne et al.

    Inventory models with continuous, stochastic demands

    Ann. Appl. Probab.

    (1991)
  • X. Chao, G. Gallego, Stochastic production systems with finite capacity: bounds, heuristics, and algorithms, Working...
  • M.A Cohen et al.

    Service constrained (s,S) inventory systems with priority demand classes and lost sales

    Management Sci.

    (1988)
  • A Federgruen et al.

    An efficient algorithm for computing an optimal (r,Q) policy in continuous review stochastic inventory systems

    Oper. Res.

    (1992)
  • G Gallego

    New bounds and heuristics for (Q,r) policies

    Management Sci.

    (1998)
  • H Galliher et al.

    Dynamics of two classes of continuous-review inventory systems

    Oper. Res.

    (1959)
  • G Hadley et al.

    Analysis of Inventory Systems

    (1963)
  • S Nahmias

    On the equivalence of three approximate continuous review inventory models

    Nav. Res. Logist. Quart.

    (1976)
  • S Nahmias

    Production and Operations Analysis

    (1993)
  • D.E Platt et al.

    Tractable (Q,r) heuristic models for constrained service levels

    Management Sci.

    (1997)
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