A finite horizon lot-sizing problem with time-varying deterministic demand and waiting-time-dependent partial backlogging

This paper presents a general time-varying demand inventory lot-sizing model with waiting-time-dependent backlogging and a lot-size-dependent replenishment cost. It differs from many related trended inventory replenishment models in two directions. First, our model not only allows part of the backlogged demands to turn into lost sales, but this backlog-to-lost-sales conversion rate is modeled by a general continuously decreasing function of the remaining waiting time until the next replenishment delivery. Second, this paper considers the dependence of replenishment cost on lot size. We derive the model's cost function for a “shortages followed by inventory” replenishment policy. Some convenient mathematical properties of the cost function are identified, with which an effective numerical solution procedure is developed for determining the optimal replenishment policy. Numerical examples and some sensitive-analysis results are then reported.