A fix-and-optimize heuristic for the integrated fleet sizing and replenishment planning problem with predetermined delivery frequencies

We tackled an integrated fleet sizing and replenishment planning problem in a vendor managed inventory system. There is a set of customers which must be replenished based on a given set of predetermined frequencies. The vehicle fleet consists of multiple types of heterogeneous vehicles which differ in carrying capacity, cost per kilometer, and ownership costs. Customer demands are taken as deterministic values. The main decision we make in this problem is the triple assignment of vehicle-frequency-customer. As a result of these assignment decisions, we obtain an annual costs consisting of vehicle ownership cost, routing cost, inventory holding and fixed replenishment costs. A key simplification in the model is the use of linear approximation for the routing cost based on the number of customers visited in a tour. The developed model, which is new in the literature, integrates fleet sizing and replenishment planning decisions. Our problem is NP-hard since it can be shown that a special case of our problem is a bin packing problem. In order to solve large problems efficiently, we suggested and applied a fix and optimize heuristic as a solution procedure. This fix and optimize heuristic divides the problem into smaller problems in which some variables are binaries and the others are linearly relaxed, and it fixes the linear decision variable iteratively. We also showed the effectiveness of the suggested heuristic solution procedure on a large set of randomly generated problems.