Economic production with Poisson demand, lost sales, fixed-rate discrete replenishment, and a constant setup time

Abstract

We address a production/inventory problem for a single product and machine where demand is Poisson distributed, and the times for unit production and setup are constant. Demand not in stock is lost. We derive a solution for a produce-up-to policy that minimizes average cost per unit time, including costs of setup, inventory carrying, and lost sales. The machine is stopped periodically, possibly rendered idle, set up for a fixed period, and then restarted. The average cost function, which we derive explicitly, is quasi-convex sparately in the produce-up-to levelQ, the low-levelRthat prompts a setup, and jointly inRequalsQ. We start by finding the minimizing value ofQwhereRequals 0, and then extend the search over largerRvalues. The discrete search may end withRless thanQ, or on the matrix diagonal whereRequalsQ, depending on the problem parameters. Idle time disappears in the cycle whenRequalsQ, and the two-parameter system folds into one. This hybrid policy is novel in make-to-stock problems with a setup time. The number of arithmetic operations to calculate costs in the (Q,R) matrix depends on a vector search overQ. The computation of the algorithm is bounded by a quadratic function of the minimizing value ofQ. The storage requirements and number of cells visited are proportional to it.