Algorithms for Non-Linear and Stochastic Resource Constrained Shortest Paths

Resource constrained shortest path problems are usually solved by label algorithms, which consist in a smart enumeration of the non-dominated paths. Recent improvements of these algorithms rely on the use of bounds on path resources to discard partial solutions. The quality of the bounds determines the performance of the algorithm. The main contribution of this paper is to introduce a standard procedure to generate bounds on paths resources in a general setting which covers most resource constrained shortest path problems, among which stochastic versions. In that purpose, we introduce a generalization of the resource constrained shortest path problem where the resources are taken in a monoid. The resource of a path is the monoid sum of the resources of its arcs. The problem consists in finding a path whose resource minimizes a non-decreasing cost function of the path resource among the paths that respect a given constraint. Label algorithms are generalized to this framework. We use lattice theory to provide a polynomial procedure to find good quality bounds. The efficiency of the approach is proved through an extensive numerical study in the case of the Stochastic Resource Constrained Shortest Path problem.