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

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 level Q, the low‐level R that prompts a setup, and jointly in R equals Q. We start by finding the minimizing value of Q where R equals 0, and then extend the search over larger R values. The discrete search may end with R less than Q, or on the matrix diagonal where R equals Q, depending on the problem parameters. Idle time disappears in the cycle when R equals Q, 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 over Q. The computation of the algorithm is bounded by a quadratic function of the minimizing value of Q. The storage requirements and number of cells visited are proportional to it.