Joint inventory control across products with supply uncertainty and responsive pricing

We study sourcing and responsive pricing decisions of a firm with two correlated products and price-dependent demand when supply capacities for these two products are uncertain. Cross-price effects exist between the two products, which means that the demand of each product depends on the prices of both products. First, we apply the tool of L♮$ L^{\natural } $ convexity and some recent results to establish the L♮$ L^{\natural } $ convexity of the value-to-go function during each period; then we use KKT conditions and some structural properties given by L♮$ L^{\natural } $ convexity to derive structural results of the optimal order policy for these two correlated products in a periodic review setting. Contrary to the classical “base-stock” policy derived in the literature for single product or multiple products without uncertain supply capacity, we show that the optimal order policy in our inventory control problem is complicated and it follows the “order-up-to” structural property only under some conditions. Numerical studies are carried out to present the structural properties of the optimal order policy and show the effect of the complementarity or substitutability on the profits.