Explicit Solution of a Two-Dimensional Deterministic Inventory Problem

We determine the optimal ordering policy for impulse control of a deterministic two-product inventory system subject to constant demand rates, linear storage and shortage costs and economies of joint ordering. The optimal cost is explicitly obtained as the smoothest solution of a two-dimensional Quasi-Variational Inequality. Exact construction of the boundary of the continuation set leads to the optimal ordering policy. Extensions to independent products, average cost and n-product case n > 2 are also briefly considered.