An algorithm for the resource constrained shortest path problem

Abstract

In this paper we examine an integer programming formulation of the resource constrained shortest path problem. This is the problem of a traveller with a budget of various resources who has to reach a given destination as quickly as possible within the resource constraints imposed by his budget. A lagrangean relaxation of the integer programming formulation of the problem into a minimum cost network flow problem (which in certain circumstances reduces to an unconstrained shortest path problem) is developed which provides a lower bound for use in a tree search procedure. Problem reduction tests based on both the original problem and this lagrangean relaxation are given. Computational results are presented for the solution of problems involving up to 500 vertices, 5000 arcs, and 10 resources.

Solution approaches for determining user-oriented paths on dynamic networks

“…Several solution approaches have been defined for solving to optimality the RCSPP. The main strategies are based on dynamic-programming (Beasley and Christofides, 1989;Mehlhorn and Ziegelmann 2000;Dumitrescu and Boland, 2003), path ranking (Santos et al, 2007;Di Puglia Pugliese and Guerriero, 2013a) and branch and bound (Carlyle et al, 2008;Muhandiramge and Boland, 2009) procedures. When negative cost cycles are present, elementary requirements must be explicitly introduced.…”

Vehicle routing with soft time windows and stochastic travel times: A column generation and branch-and-price solution approach

“…Feillet et al [13] extend this classical label correcting algorithm, which had been developed for the non-elementary shortest path problem with resource constraints, by including node resources to solve the ESPPRC optimally. Beasley and Christofides [2] propose the idea of adding a binary resource for each node in the graph but they do not conduct any computational experiments for that. Feillet et al [13] solve the ESPPRC on a full-dimensional state space, and thus they apply the idea of using unreachable nodes in the labels to have an efficient algorithm.…”

Optimal Routing of Traffic Flows with Length Restrictions in Networks with Congestion

“…Two algorithms for the constrained shortest path problem have been implemented for the diploma thesis: A branch-and-bound scheme with Lagrangian relaxation by Beasley and Christofides [3], and a labeling algorithm with lists of labels for each node by Aneja, Aggarwal, and Nair [2].…”