The Optimality of Generalized (s, S) Policies under Uniform Demand Densities

This note considers the single product, single echelon, periodic review, stochastic, dynamic inventory model discussed recently [Porteus, E. L. 1971. On the optimality of generalized (s, S) policies. Management Sci. 17 411-426.], where the ordering cost function is concave increasing, rather than simply linear with a setup cost. We show that a generalized (s, S) policy will be optimal in a finite horizon problem when the probability densities of demand are uniform or convolutions of a finite number of uniform and/or one-sided Pólya densities. Such densities are not necessarily one-sided Pólya densities, for which this result has already been established. To prove the result here we need only show, roughly, that a certain subclass of the quasi-K-convex functions is closed under convolution with uniform densities which describe nonnegative random variables.