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 Q0, the minimizing value of Q for R = 0. Total computation to solve the overall problem, measured in arithmetic operations, is quadratic in Q0. At most 3 Q0 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.