Third Party Logistics Planning with Routing and Inventory Costs

Abstract

We address a scheduling and routing problem faced by a third-party logistics provider in planning its day-of-week delivery schedule and routes for a set of existing and/or prospective customers who need to make shipments to their customers (whom we call “end-customers”). The goal is to minimize the total cost of transportation and inventory while satisfying a customer service requirement that stipulates a minimum number of visits to each customer each week and satisfaction of time-varying demand at the end-customers. Explicit constraints on the minimum number of visits to each customer each week give rise to interdependencies that result in a dimension of problem difficulty not commonly found in models in the literature. Our model includes two other realistic factors that the third-party logistics provider needs to consider: the cost of holding inventory borne by end-customers if deliveries are not made “just-in-time” and the possibility of multiple vehicle visits to an end-customer in the same period (day).

We develop a solution procedure based on Lagrangian relaxation in which the particular form of the relaxation provides strong bounds. One of the subproblems that arises from the relaxation serves to integrate the impact of the timing of deliveries to the various end-customers with inventory decisions, which not only contributes to the strong lower bound that the relaxation provides, but also yields a mathematical structure with some unusual characteristics; we develop an optimal polynomial-time solution procedure for this subproblem. We also consider two variants of the original problem with more restrictive assumptions that are usually imposed implicitly in many vehicle routing problems. Computational results indicate that the Lagrangian procedure performs well for both the original problem and the variants. In many realistic cases, the imposition of the additional restrictive assumptions does not significantly affect the quality of the solutions but substantially reduces computational effort.