# The Vehicle Routing Problem with Service Level Constraints

## TL;DR

This paper introduces a new vehicle routing problem with service level constraints, proposing algorithms that effectively balance costs and service levels, and demonstrating high-quality solutions and new optimal results through computational experiments.

## Contribution

It presents a novel formulation and algorithms for the vehicle routing problem with service level constraints, extending existing models and providing new optimal solutions.

## Key findings

- High-quality heuristic solutions for complex problem instances
- Matching all optimal solutions for small and medium benchmarks
- Improving algorithms for related vehicle routing problems

## Abstract

We consider a vehicle routing problem which seeks to minimize cost subject to service level constraints on several groups of deliveries. This problem captures some essential challenges faced by a logistics provider which operates transportation services for a limited number of partners and should respect contractual obligations on service levels. The problem also generalizes several important classes of vehicle routing problems with profits. To solve it, we propose a compact mathematical formulation, a branch-and-price algorithm, and a hybrid genetic algorithm with population management, which relies on problem-tailored solution representation, crossover and local search operators, as well as an adaptive penalization mechanism establishing a good balance between service levels and costs. Our computational experiments show that the proposed heuristic returns very high-quality solutions for this difficult problem, matches all optimal solutions found for small and medium-scale benchmark instances, and improves upon existing algorithms for two important special cases: the vehicle routing problem with private fleet and common carrier, and the capacitated profitable tour problem. The branch-and-price algorithm also produces new optimal solutions for all three problems.