A Fixed Rate Production Problem with Poisson Demand and Lost Sales Penalties
Bruce Faaland, Mark McKay, Thomas Schmitt
Production and Operations Management · 2019 · Volume 28 · Issue 3
Abstract
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 levelQand the trigger levelRthat signals a setup for production. Our search over the (Q,R) array begins by findingQ0, the minimizing value ofQforR= 0. Total computation to solve the overall problem, measured in arithmetic operations, is quadratic inQ0. At most 3Q0cost 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 anN-policyM/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.
Keywords
production/inventory, lost sales, continuous review, make-to-stock and make-to-order, N-policyM/D/1 finite queue
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