# Multi-objective minmax robust combinatorial optimization with cardinality-constrained uncertainty

## TL;DR

This paper presents two novel methods for solving multi-objective minmax robust combinatorial optimization problems with cardinality-constrained uncertainty, including algorithmic enhancements and a new deterministic approach, demonstrated on hazardous material transportation instances.

## Contribution

It extends existing algorithms to multi-objective cases, introduces a deterministic problem formulation, and develops specialized algorithms for robust shortest path problems.

## Key findings

- Enhanced algorithms outperform previous methods in speed.
- Deterministic approach provides a comprehensive solution set.
- Effective on hazardous material transportation instances.

## Abstract

In this paper we develop two approaches to find minmax robust efficient solutions for multi-objective combinatorial optimization problems with cardinality-constrained uncertainty. First, we extend an algorithm of Bertsimas and Sim (2003) for the single-objective problem to multi-objective optimization. We propose also an enhancement to accelerate the algorithm, even for the single-objective case, and we develop a faster version for special multi-objective instances. Second, we introduce a deterministic multi-objective problem with sum and bottleneck functions, which provides a superset of the robust efficient solutions. Based on this, we develop a label setting algorithm to solve the multi-objective uncertain shortest path problem. We compare both approaches on instances of the multi-objective uncertain shortest path problem originating from hazardous material transportation.