# Exact algorithms for the chance-constrained vehicle routing problem

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## Abstract

We study the chance-constrained vehicle routing problem (CCVRP), a version of the vehicle routing problem (VRP) with stochastic demands, where a limit is imposed on the probability that each vehicle’s capacity is exceeded. A distinguishing feature of our proposed methodologies is that they allow correlation between random demands, whereas nearly all existing exact methods for the VRP with stochastic demands require independent demands. We first study an edge-based formulation for the CCVRP, in particular addressing the challenge of how to determine a lower bound on the number of vehicles required to serve a subset of customers. We then investigate the use of a branch-and-cut-and-price (BCP) algorithm. While BCP algorithms have been considered the state of the art in solving the deterministic VRP, few attempts have been made to extend this framework to the VRP with stochastic demands. In contrast to the deterministic VRP, we find that the pricing problem for the CCVRP problem is strongly \(\mathcal {NP}\)-hard, even when the routes being priced are allowed to have cycles. We therefore propose a further relaxation of the routes that enables pricing via dynamic programming. We also demonstrate how our proposed methodologies can be adapted to solve a distributionally robust CCVRP problem. Numerical results indicate that the proposed methods can solve instances of CCVRP having up to 55 vertices.

## Keywords

Stochastic vehicle routing Chance constraint Branch-and-cut-and-price## Mathematics Subject Classification

90C10 90C15 90B06## References

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