Robust Optimization for the Vehicle Routing Problem with Multiple Deliverymen

This paper studies the vehicle routing problem with time windows and multiple deliverymen in which customer demands are uncertain and belong to a predetermined polytope. In addition to the routing decisions, this problem aims to define the number of deliverymen used to provide the 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 of solving this formulation, we propose a constructive heuristic to generate a robust solution that is used as an initial solution 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 this static robust optimization to deal with trade-off between cost and risk. The results also reveal that the proposed approach provides good results even without having exact knowledge of some probabilistic measure of the customer demand.