A Matrix Representation of the Multiple Vehicle Routing Problem for Pickup and Delivery

This paper develops a computationally efficient algorithm for the Multiple Vehicle Pickup and Delivery Problem (MVPDP) with the objective of minimizing the tour cost incurred while completing the task of pickup and delivery of customers. To this end, this paper constructs a novel 0-1 Integer Quadratic Programming (IQP) problem to exactly solve the MVPDP. Compared to the state-of-the-art Mixed Integer Linear Programming (MILP) formulation of the problem, the one presented here requires fewer constraints and decision variables. To ensure that this IQP formulation of the MVPDP can be solved in a computationally efficient manner, this paper devises a set of sufficient conditions to ensure convexity of this formulation when the integer variables are relaxed. In addition, this paper describes a transformation to map any non-convex IQP formulation of the MVPDP into an equivalent convex one. The superior computational efficacy of this convex IQP method when compared to the state-of-the-art MILP formulation is demonstrated through extensive simulated and real-world experiments.