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

• 成果类型: Article
• 署名作者: Schmitt, Thomas; Faaland, Bruce; Mckay, Mark
• 署名单位: University of Washington; University of Washington Seattle; Trinity Western University; University of Washington; University of Washington Seattle
• 刊物名称: PRODUCTION AND OPERATIONS MANAGEMENT
• ISSN/ISSBN: 1059-1478
• DOI: 10.1111/poms.14072
• 发表日期: 2023
• 页码: 3968-3985
• 关键词: closed-form quasi-convex cost; discrete replenishment; lost sales; M/D/1 finite system; setup time

摘要

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.