Optimal (r,nQ,T) batch ordering with quantized supplies

We consider the control of a single-echelon inventory installation under the (r,nQ,T) batch ordering policy. Demand follows a stationary stochastic process and, when unsatisfied, is backordered. The supply process is quantized; so the policy base batch Q needs to satisfy the constraint Q=kg, where k is an integer and q an exogenous fixed supply lot, often reflecting physical supply limitations. Assuming continuous time and a standard cost structure, we determine the (r,nQ,T) policy variables that minimize total average cost per unit time subject to the supply lot constraint. While total average cost is not convex, we show that average holding and backorders costs are jointly convex in all policy variables. This helps establish optimality properties and convex bounds that allow developing an algorithm to exactly solve the quantized supplies optimization problem. Computations reveal that, depending on the supply lot size, quantized supplies may cause serious cost increase.