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Resource constrained shortest path problems are usually solved thanks to asmart enumeration of all the non-dominated paths. Recent improvements of theseenumeration algorithms rely on the use of bounds on path resources to discardpartial solutions. The quality of the bounds determines the performance of thealgorithm. The main contribution of this paper is to introduce a standardprocedure to generate bounds on paths resources in a general setting whichcovers most resource constrained shortest path problems, among which stochasticversions.  In that purpose, we introduce a generalization of the resource constrainedshortest path problem where the resources are taken in a monoid. The resourceof a path is the monoid sum of the resources of its arcs. The problem consistsin finding a path whose resource minimizes a non-decreasing cost function ofthe path resource among the paths that respect a given constraint. Enumerationalgorithms are generalized to this framework. We use lattice theory to providepolynomial procedures to find good quality bounds. These procedures solve ageneralization of the algebraic path problem, where arc resources belong to alattice ordered monoid. The practical efficiency of the approach is provedthrough an extensive numerical study on some deterministic and stochasticresource constrained shortest path problems.

Algorithms for Non-Linear and Stochastic Resource Constrained Shortest Paths