The Multi-objective Multi-vehicle Pickup and Delivery Problem with Time Windows

Abstract The Single Objective Single Vehicle Pickup and Delivery Problem ( SOSV-PDP ) is a Vehicle Routing Problem ( VRP ) in which the vehicle, based at the depot, has to visit exactly once a set of customers with known demand. Each request specifies two locations: an origin for the picking and one for the delivery. The vehicle must start and finish at the depot and the total handled demand must not exceed its capacity. Moreover, for each request, the origin must precede the destination ( precedence constraints ). In the SOSV-PDP with Time Windows ( SOSV-PDPTW ), each request specifies also a time window. Therefore, the vehicle has to serve the customer within the time window ( time window constraint ). The Single Objective Multiple Vehicle - PDPTW ( SOMV-PDPTW ) is an extension of SOSV-PDPTW where customers are served by a fleet (usually homogeneous) of vehicles. Therefore, together with the precedencies, for each request, the origin and the destination have to belong to the same route ( pairing constraints ). In the traditional SOMV-PDPTW , only one objective is optimized (usually, the total travel cost); while, in the literature, few multi-objective MOSV-PDPTW exist that optimize at most three criteria simultaneously. The contribution of this paper consists in addressing the MOMV-PDPTW from both a modeling and methodological point of view. In fact, the MOMV-PDPTW is firstly modeled with the aim of optimizing the number of vehicles, the total travel cost and the longest travel cost, simultaneously; then, a two-step solution approach is proposed. In particular, in the first step, a set of feasible routes is generated by properly adapting some meta-heuristics proposed in literature for the SOMV-PDPTW ; then, set partitioning optimization problems are solved within an є-constraint framework . More specifically, each set partitioning problem selects the routes from the feasible set, optimizing one criterion at time, constraining the remaining ones by appropriate upper bounds and satisfying customer requirements. Finally, the second step finds the set of efficient solutions for approximating the Pareto Fronts . Computational experiments, carried out on some instances generated in literature, show that our approach determines good quality Efficient Pareto Fronts (in terms of number of efficient solutions) and also provides well-diversified efficient sets. This last aspect is properly evaluated by computing the Spread metric on each of the instances.