Inventory decision with two substitutable products

In this paper, we consider a single period simplified version of substitution problem with one retailer and two substitutable products. The retailer retails the two substitutable products, one is the innovation type of the other, therefore, when there is a stock-out, the new type of product can substituted for the old type of product. Both of them must be ordered before the demand realized. How to manage inventories of products in such a system is of managerial interest. The substitution problem is notoriously difficult. Using a general model, we formulate the inventory management problem for two end-products with substitution as a two-stage stochastic nonlinear program. We solve the first stage of the problem to determine the optimal inventory levels of the two products, and solve the second stage problem, to determine the product allocation decisions when demands realized. At last, we give a numerical example.