The Multiple Choice Elementary Constrained Shortest Path Problem

This paper considers a variation of the Elementary Constrained Shortest Path Problem in which nodes are separated into subsets and a feasible path through the network may visit at most one node from each subset. We refer to this problem as the Multiple Choice Elementary Constrained Shortest Path Problem (MC-ECSPP). The MC-ECSPP arises as a subproblem in branch-and-price approaches for variations of the vehicle routing problem in which the nodes to be visited are chosen among subsets. We present methods to obtain bounds and feasible solutions for the MC-ECSPP. Further, we incorporate these methods into a branch-and-price approach to solve a variation of the vehicle routing problem. This paper introduces a variation of the Elementary Constrained Shortest Path Problem (ECSPP), also known as the Elementary Shortest Path Problem with Resource Constraints or Time Windows. The ECSPP ̄nds a path of minimum cost between a source node and a sink node, visiting each node in the network at most once. The path may be constrained by the travel time of the path and time windows on the nodes. We consider a variation of the ECSPP in which nodes are separated into subsets and a feasible path through the network may visit at most one node from each subset. We refer to this problem as the Multiple Choice Elementary Constrained Shortest Path Problem (MC-ECSPP). The MC-ECSPP is de ̄ned on a directed network of nodes and arcs, with a cost and a non-negative travel time associated with each arc. The set of nodes is divided into subsets. The MC-ECSPP ̄nds a path with the minimum total cost that visits at most one node from each subset, constrained by the travel time of the path and time windows on nodes. In this paper, we analyze the MC-ECSPP and develop methods to obtain bounds and feasible solutions, making use of recent advances in solution methods for the ECSPP. As shown in this paper, the MC-ECSPP is important because it arises as a subproblem in a branch-and-price method for a variation of the vehicle routing problem, known as the Multi-Resource Routing Problem (MRRP). Therefore, the second part of the paper incorporates the solution and bounding methods for the MC-ECSPP in a branch-and-price approach for the MRRP. We show that these methods can improve the implementation of a branch-and-price method for the MRRP.