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
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