Optimality of Four-Threshold Policies in Inventory Systems with Customer Returns and Borrowing/Storage Options

Mark E. Lewis
Industrial and Operations Engineering Department
University of Michigan
1205 Beal Avenue
Ann Arbor, MI 48109-2117

Eugene A. Feinberg
Department of Applied Mathematics and Statistics
State University of New York at Stony Brook
Stony Brook, NY 11794-3600


This paper studies an extension of the basic inventory control and
cash management models so as to capture the availability of
one-period storage and borrowing. For models with zero setup costs
and possible negative demand, we show that if the inventory
position is too high, the optimal decision is to reduce the
inventory, but only to a certain point after which one should
store some of the commodity to meet future demand. Analogously, if
the inventory position is too low, the decision-maker should order
up to a certain level and then borrow from the secondary source to
meet potential demand. This leads to four thresholds as opposed to
the two threshold result that has been established in the cash
management literature. Moreover, the options to store or borrow
are only used a secondary options, so that the thresholds are
ordered. These facts hold under the finite and infinite horizon
discounted expected cost and the average cost criteria. We also
describe sufficient conditions when the borrowing and storage
options should not be used and provide numerical examples.