# Algorithms for non-linear and stochastic resource constrained shortest path

## Abstract

Resource constrained shortest path problems are usually solved thanks to a smart enumeration of all the non-dominated paths. Recent improvements of these enumeration 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. Enumeration algorithms are generalized to this framework. We use lattice theory to provide polynomial procedures to find good quality bounds. These procedures solve a generalization of the algebraic path problem, where arc resources belong to a lattice ordered monoid. The practical efficiency of the approach is proved through an extensive numerical study on some deterministic and stochastic resource constrained shortest path problems.

## Keywords

Resource constrained shortest path Stochastic shortest path Risk measures Lattice ordered monoid## Mathematics Subject Classification

90B99## Notes

### Acknowledgements

I greatly thank my Ph.D. advisor Frédéric Meunier for his numerous and deep remarks on the mathematics and the way to write this article. I am also grateful to the anonymous referees for their useful comments that clearly helped me improve this paper.

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