Robust optimization for the vehicle routing problem with multiple deliverymen

Abstract

This paper addresses the vehicle routing problem with time windows and multiple deliverymen in which the customer demands are uncertain and belong to a predetermined polytope. In addition to the routing decisions, this problem attempts to define the number of deliverymen used to service to the customers on each route. A new mathematical formulation is presented for the deterministic counterpart based on auxiliary variables that define the assignment of customers to routes. Building on this formulation, we apply a static robust optimization approach to obtain a robust counterpart formulation that captures the random nature of customer demand. Due to the difficulty in solving this formulation, we propose a constructive heuristic to generate a robust solution, which is used as a starting point for solving the robust counterpart formulation. The heuristic is an extension of Solomon’s heuristic I1. Computational results using problem instances from the literature and risk analysis via Monte-Carlo simulation indicate the potential of static robust optimization to address the trade-off between cost and risk. The results also reveal that the proposed approach provides good results even without exact knowledge of some probabilistic measure of the customer demand.