Model and Algorithm for the Vendor-Warehouse Transportation and Inventory Problem in a Three-Level Distribution System

We consider the inventory-routing problem in a three-level distribution system with a single vendor, a single warehouse and many geographically dispersed retailers. In this problem, each retailer faces a demand at a deterministic, retailer-specific rate. The demand of each retailer is replenished either from the warehouse by a small vehicle or from the vendor bypassing the warehouse by a big vehicle. Inventories are kept not only at the retailers but also at the warehouse. The objective is to find a combined inventory policy and routing pattern minimizing a long-run average system-wide cost while meeting the demand of each retailer without shortage. We present an efficient solution approach based on a fixed partition policy where the retailers are partitioned into disjoint and collectively exhaustive sets and each set of retailers is served on a separate route. Given a fixed partition, the original problem is decomposed into three subproblems. In this paper, we focus on the modelling and resolution of the vendor-warehouse transportation and inventory subproblem. We demonstrate that the subproblem can be reduced to a C/C/C/Z capacitated dynamic lot sizing problem and there exists an algorithm to solve the reduced problem to optimality in O(T 2 ) time.