A continuous approximation location-inventory model with exact inventory costs and nonlinear delivery lead time penalties

Abstract

Most existing location-inventory models use exact locations (coordinates) for warehouses and customers but use simple approximations for inventory costs. A different approach is presented in this paper. While the locations are loosely modeled using a continuous approximation approach, a reorder point, order quantity inventory model with exact costs is used. Exact expressions for the transport time distribution and backorder time distribution (for Poisson demand) are combined with an exogenously given warehouse fulfillment time and form the delivery lead time. Customers perceive a long delivery lead time negatively. This creates a dynamic between the number of stocking locations used (which affects the transport time and cost) and the inventory policy (which affects the backorder time and inventory costs) because both influence the delivery lead time. Furthermore, we are able to model a nonlinear customer perception of the delivery lead time. Based on the presented model, the general characteristics and mechanisms of such a location-inventory problem are investigated. Among other things, we show for many probability distributions that the average backorder time per backorder is convex and that our model is quasi-convex, which allows for a wide range of optimization methods.