# A Relax-and-Decomposition Algorithm for a p-Robust Hub Location Problem

## TL;DR

This paper introduces a novel relax-and-decomposition heuristic for solving a non-linear p-robust hub location problem under risk, incorporating network security, pollution, and congestion factors, validated through numerical tests.

## Contribution

It develops a new heuristic combining accelerated Benders decomposition and Lagrangian relaxation with multi-Pareto cuts for robust hub network design under risk.

## Key findings

- The model effectively designs robust networks considering multiple risk factors.
- The sample average approximation method is validated for accuracy.
- The proposed algorithm outperforms existing approaches in efficiency.

## Abstract

In this paper, a non-linear p-robust hub location problem is extended to a risky environment where augmented chance constraint with a min-max regret form is employed to consider network risk as one of the objectives. The model considers risk factors such as security, air pollution and congestion to design the robust hub network. A Monte-Carlo simulation based algorithm, namely, a sample average approximation scheme is applied to select a set of efficient scenarios. The problem is then solved using a novel relax-and-decomposition heuristic based on the coupling of an accelerated Benders decomposition with a Lagrangian relaxation method. To improve the decomposition mechanism, a multi-Pareto cut version is applied in the proposed algorithm. In our numerical tests a modification of the well-known CAB data set is used with different levels of parameters uncertainty. The results demonstrate the capability of the proposed model to design a robust network. We also verify the accuracy of the sample average approximation method. Finally, the results of the proposed algorithm for different instances were compared to other solution approaches which confirm the efficiency of the proposed solution method.