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Implied backorder costs-DERIVATIONS

Implied backorder costs-DERIVATIONS - J OURNAL OF...

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 97, No. 1, pp. 71-92, APRIL 1998 Nonlinear Programming Analysis to Estimate Implicit Inventory Backorder Costs 1 S. CETINKAYA 2 AND M. PARLAR 3 Communicated by D. G. Luenberger Abstract. In this paper, we use nonlinear programming to provide an alternative treatment of the economic order quantity problem with planned backorders. Many businesses, such as capital-goods firms that deal with expensive products and some service industries that cannot store their services, operate with substantial backlogs. In practical prob- lems, it is usually very difficult to estimate accurately the values of the two types of backorder costs, i.e., the time-dependent unit backorder cost and the unit backorder cost. We redefine the original problem without including these backorder costs and construct a nonlinear pro- gramming problem with two service measure constraints which may be easier to specify than the backorder costs. We find that, with this differ- ent formulation of our new problem, we obtain results which give implicit estimates of the backorder costs. The alternative formulation provides an easier-to-use model and managerially meaningful results. Next, we show that, for a wide range of parameter values, it usually suffices to consider only one type of backorder cost, or equivalently, only one type of service measure constraint. Finally, we develop expres- sions which bracket the optimal values of the decision variables in a narrow range and provide a simple method for computing the optimal solution. In the most complicated case, this method requires finding the unique root of a polynomial. Key Words. Nonlinear programming, inventory theory, backorders costs. 1. Introduction One of the well-known generalizations of the economic order quantity (EOQ) model of inventory theory is the case where backorders are allowed This research was conducted while Dr. Cetinkaya was PhD Student in Management Science/ Systems at McMaster University, Hamilton, Ontario, Canada. 2 Assistant Professor, Department of Industrial Engineering, Texas A&M University, College Station, Texas. 3 Professor. DeGroote School of Business, McMaster University, Hamilton, Ontario, Canada. 71 0022-3239/98/0400-001IJ15.00/0 © 1998 Plenum Publishing Corporation
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(Refs. 1-7). The optimal solution is determined by minimizing the average annual cost function, which is the sum of three terms: (i) ordering cost, (ii) holding cost, and (iii) two types of backorder costs. It is a long established fact (Hadley and Whitin, Ref. 2, pp. 18-19) that in practice the backorder costs are very difficult to specify, since they usually represent intangible factors such as the loss of goodwill resulting from a delay in meeting demands. However, for those cases where backorder costs can be accurately estimated, Hadley and Whitin (Ref. 2) use a nondecreasing function of time n(t) to represent the cost of each unit backordered for which the backorder remained on the books. They specialize the function n(t) to the linear form ;r(0 = nt + n, where n is the time-dependent per
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