Quality & Quantity 38: 771–786, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Extend (r, Q) Inventory Model Under Lead Time
and Ordering Cost Reductions When the Receiving
Quantity is Different from the Ordered Quantity
and I-CHUAN LIN
Department of Business Administration, Tamkang University, Tamsui, Taipei 251, Taiwan;
Graduate of Management Sciences, Aletheia University, Tamsui, Taipei 251, Taiwan
Abstract. This paper investigates the continuous review inventory model involving variable lead
time with partial backorders, where the amount received is uncertain. The options of investing in
ordering cost reduction is included, and lead time can be shortened at an extra crashing cost. The
objective of this article is to simultaneously optimize the order quantity, reorder point, ordering cost
and lead time. We ﬁrst assume that the lead time demand follows a normal distribution and develop
an algorithm to ﬁnd the optimal solution. Then, we relax the assumption of normality to consider a
distribution free case where only the mean and standard deviation of lead time demand are known.
We apply the minimax distribution free procedure to solve this problem. For both cases, we also
show that the objective cost function to be minimized is jointly convex in the decision variables.
Furthermore, two numerical examples are given to illustrate the results.
Key words: inventory, ordering cost reduction, lead time, minimax distribution free.
Among the modern production management, the Japanese successful experiences
of using Just-In-Time (JIT) production show that the advantages and beneﬁts asso-
ciated with the efforts to control the lead-time can be clearly perceived. The goal of
JIT inventory management philosophies is the focus that emphasizes high quality,
keeps low inventory level and lead-time to a practical minimum. Shortening the
lead time is recognized as the feasible and effective way to achieve the goal of JIT.
In traditional, most deterministic and stochastic inventory models assume that
the lead-time is a given parameter or a random variable (therefore it is uncontrol-
lable), and determines the optimal operating policy on the basis of this unrealistic
assumption (Naddor, 1966; Silver and Peterson, 1998). In fact, in many practical
situations, lead-time is not a given parameter or a random variable; it can be con-
trolled and reduced at an added cost. Recently, some models considering lead-time
as a decision variable have been developed. Liao and Shyu (1991) have initiated
a study on lead-time reduction by presenting an inventory model in which lead-
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