Optimality and Computation of (σ, S) Policies in the Multi-Item Infinite Horizon Inventory Problem

A multi-item inventory system with periodic review, single set-up cost K plus linear ordering cost c for changing stock levels, and holding and shortage cost l(x) for being in stock position x at the beginning of a period is considered. Demand in any period is assumed to be ξ with probability ϕ(x,ξ), where x is the stock level at the beginning of the period. The infinite horizon optimum policies found consist of ordering up to S for any point x in σ, the reorder region, and not ordering at x not in σ. A computational procedure is given, bounds are derived, and, for further assumptions on l, σ is further characterized.