Quality & Quantity 38: 457–473, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
A Note on EOQ Model for Items with Mixtures of
Exponential Distribution Deterioration, Shortages
and Time-varying Demand
and JONG-WUU WU
Department of Business Administration, Ming Chuan University, Taipei, Taiwan, R.O.C.
Department of Statistics, Tamkang University, Tamsui, Taipei, Taiwan, R.O.C.
Abstract. An inventory model is considered in which inventory is depleted not only by demand, but
also by deterioration. In this paper, we derive the EOQ model for inventory of items that deteriorates
at a mixtures of exponential distributed rate, assuming the demand rate with a continuous function
of time. Moreover, the proposed model cannot be solved directly in a closed form, thus we used the
computer software IMSL MATH/LIBRARY (1989) to ﬁnd the optimal reorder time Further, we also
ﬁnd that the optimal procedure is independent of the form of the demand rate. Finally, we also assume
that the holding cost is a continuous, nonnegative and non-decreasing function of time in order to
generalize EOQ model. Moreover, four numerical examples and sensitivity analyses are provided to
assess the solution procedure.
Key words: inventory, deterioration, exponential distribution, reorder time, differential equation
In many inventory models, the effect of deterioration is very important. Deteriora-
tion is deﬁned as decay, change or spoilage that prevents the items from being used
for its original purpose. Food items, drugs, pharmaceuticals, photographic ﬁlm,
electronic components, chemicals and radioactive substances are few examples of
items in which appreciable deterioration can take place during the normal stor-
age period of the units and consequently this loss must be taken into account
when analyzing the model. Hence, many authors have considered Economic Order
Quantity (EOQ) models for deteriorating items. An EOQ model for an inventory
with exponential decay of items (i.e., constant rate of deterioration over time) or
variable proportion of the on-hand inventory gets deteriorated per unit of time have
been developed by Ghare and Schrader (1963), Covert and Philip (1973), Misra
(1975), Shah (1977), Tadikamalla (1978), etc. In their investigations, the market
demand for the item is considered to be constant. As time progressed, several other
researchers developed inventory models with deteriorating items with time depend-
ent demand rates. In this connection, the works done by Donaldson (1977), Silver