Coordinated home and locker deliveries: An exact approach for the urban delivery problem with conflicting time windows

This paper proposes an urban delivery problem with conflicting time windows arising in the urban last-mile delivery. The problem is to find a minimum cost tour over a set of customers and facilities in which the customer not visited on the tour is assigned to a feasible facility on the tour. The objective is to minimize the delivery and self-pick-up costs with the constraints of conflicting time windows provided by customers. This work puts forward a mixed-integer linear programming model to formulate the problem and provides a branch-and-cut algorithm to solve the problem. Several valid inequalities are developed and separated to improve the convergence of the algorithm. Sets of instances are generated, and the number of available facilities, the coverage radius of self-pick-up service are varied, to assess the effectiveness of the proposed formulations and algorithm. Experimental results show that instances involving up to 150 vertices can be solved optimally within one hour.