Single Period Multi-Product Inventory Models with Substitution
Y. Bassok, R. Anupindi, and R. Akella
*** Abstract ****
We study a single period multi-product inventory problem with substitution and
proportional costs and revenues. We consider $n$ products and $n$ demand classes
with full downward substitution, i.e. excess demand for class $i$ can
be satisfied using product $j$ for $j > i$. We first discuss a two--stage profit
maximization formulation for the multiproduct substitution problem. We use an
allocation policy that is (myopically) optimal to write the expected profits
and it's first partials explicitly. This in turn enables us to prove additional
properties of the profit function and several interesting properties of the
optimal solution. In a limited computational study using two products, we illustrate
the benefits of solving for the optimal quantities when substitution is considered
at the ordering stage over similar computations without considering substitution
while ordering. Specifically we show that the benefits are higher with high
demand variability, low substitution cost, low profit margins (or low price
to cost ratio), high salvage values, and similarity of products in terms of
prices and costs.