This paper presents a finite-horizon dynamic Economic Lot
Size (ELS) model for perishable inventory with backorders allowed. Our model
generalizes a recent study by Hsu (2000a) on a similar ELS model but without
backordering. Hsu (2000a) points out that the inventory cost structure in the
traditional ELS models is reasonable for non-perishable products, but is not
applicable to perishable products. He proposes an ELS model without backordering
that includes stock deterioration which depends on the inventory's age (i.e.,
the time period in which the inventory
is
held). His model also defines multiple age-dependent inventory cost functions in
each period, one for each batch of stock with distinctive age.
In our generalized ELS model, in addition to the stock
deterioration and age-dependent inventory costs as in Hsu (2000a), we define
backordering cost functions in every period, one for each backorder with
distinctive age (i.e., the time period in which the backorder remains unfilled).
The proposed age-dependent backordering cost structure is a departure from the
traditional ELS model where backorders in each period are treated as the same
regardless when they are placed.
In another recent study, Hsu and Lowe (2001) discuss
situations in real world application where backordering costs (inventory costs)
may depend on the size of the backorder (inventory) and the time the backorder
is placed (inventory is produced) and/or filled (used). They propose so-called
period-pair-dependent backordering (inventory) cost functions which are defined
for pairs of periods in which the backorder is placed and filled (inventory is
produced and used). The difference between our model and that of Hsu and Lowe
(2001) are: (i) while Hsu and Lowe's cost functions are defined for every pair
of periods, the age-dependent inventory and backordering cost functions in our
model are defined for every single period, i.e., the inventory and backorder
costs are accounted for in a period-by-period fashion (see more discussions on
this difference in the next section); (ii) Hsu and Lowe's model does not
consider explicitly the stock deterioration, i.e., the possible inventory lost
from period to period.
In the remainder of the paper, we will present our model
in Section 2 and show that the general model with concave cost functions may be
more difficult to solve than its no backordering special case in Hsu (2000a). We
then propose two important special instances of the model, one with
non-decreasing demands and the other with non-decreasing marginal backordering
cost with respect to the age of backorders. In Section 3, we will establish some
structural properties for the optimal solutions to the two special instances and
develop a polynomial-time Dynamic Programming (DP) algorithm to solve the
problems. We also discuss a few more restricted instances with reduced
computational complexity. We conclude the paper in Section 4, where we provide a
table summarizing the results obtained in this and two other (Hsu, 2000a; Hsu
and Lowe, 2001) related papers
