A multi-period location model with transportation economies-of-scale and perishable inventory

In this paper, we propose a multi-period location model with transportation economies-of-scale that distributes a single perishable product. The problem involves a single supplier, a layer of potential facility locations, and a layer of retailers. Each facility is assumed to practice cross-docking and each retailer faces a constant demand rate in each planning period. A Zero-Inventory-Ordering (ZIO) inventory policy is assumed to be adopted at each retailer. The demand at each retailer must be satisfied by the end of the planning horizon, although backlogging is allowed in the intermediate periods. The decisions to be made comprise the location of facilities, the allocation of retailers to the open facilities with single-sourcing, and the logistics shipping plan over time for the open facilities and the retailers allocated to them. The goal is to minimize the total cost that includes (1) the fixed setup cost for locating facilities, (2) the inbound cost at the open facilities, (3) the transportation cost from the facilities to the retailers (which is assumed to be an economies-of-scale function), and (4) the inventory cost at the retailers. We first formulate the problem as a mixed integer nonlinear programming model. By effectively utilizing the structural properties of the cost functions and combining with the ZIO, we show how to linearize the nonlinear cost components. The results of a set of numerical experiments performed on randomly generated medium-sized problem instances are reported.