A Fixed Rate Production Problem with Poisson Demand and Lost Sales Penalties

We solve a variation of a classic make-to-stock inventory problem introduced by Gavish and Graves. A machine is dedicated to a single product whose demand follows a stationary Poisson distribution. When the machine is on, items are produced one at a time at a fixed rate and placed into finished-goods inventory until they are sold. In addition, there is an expense for setting up the machine to begin a production run. Our departure from Gavish and Graves involves the handling of unsatisfied demand. Gavish and Graves assumed it is backordered, while we assume it is lost, with a unit penalty for each lost sale. We obtain an optimal solution, which involves a produce-up-to policy, and prove that the expected time-average cost function, which we derive explicitly, is quasi-convex separately in both the produce-up-to inventory level Q and the trigger level R that signals a setup for production. Our search over the (Q, R) array begins by finding Q(0), the minimizing value of Q for R = 0. Total computation to solve the overall problem, measured in arithmetic operations, is quadratic in Q(0). At most 3 Q(0) cost function evaluations are required. In addition, we derive closed-form expressions for the objective function of two related problems: one involving make-to-order production and another for control of an N-policy M/D/1 finite queue. Finally, we explore the possibility of solving the lost sales problem by applying the Gavish and Graves algorithm for the backorder problem.