Min-Max-Min Robustness for Combinatorial Problems with Discrete Budgeted Uncertainty

TL;DR

This paper studies a robust optimization approach where decision makers prepare multiple solutions in advance to handle uncertain costs, introducing new formulations and heuristics for solving complex combinatorial problems under budgeted uncertainty.

Contribution

It introduces an integer programming formulation and heuristics for min-max-min robustness in combinatorial problems with budgeted uncertainty, addressing computational challenges.

Findings

Heuristics produce high-quality solutions quickly.

Exact algorithms are limited to small instances.

Heuristics outperform exact methods in larger problems.

Abstract

We consider robust combinatorial optimization problems with cost uncertainty where the decision maker can prepare K solutions beforehand and chooses the best of them once the true cost is revealed. Also known as min-max-min robustness (a special case of K-adaptability), it is a viable alternative to otherwise intractable two-stage problems. The uncertainty set assumed in this paper considers that in any scenario, at most Gamma of the components of the cost vectors will be higher than expected, which corresponds to the extreme points of the budgeted uncertainty set. While the classical min-max problem with budgeted uncertainty is essentially as easy as the underlying deterministic problem, it turns out that the min-max-min problem is NPhard for many easy combinatorial optimization problems, and not approximable in general. We thus present an integer programming formulation for solving the problem through a row-and-column generation algorithm. While exact, this algorithm can only cope with small problems, so we present two additional heuristics leveraging the structure of budgeted uncertainty. We compare our row-and-column generation algorithm and our heuristics on knapsack and shortest path instances previously used in the scientific literature and find that the heuristics obtain good quality solutions in short computational times.