Minimising the maximum relative regret for linear programs with interval objective function coefficients

The minimax relative regret solution to a linear program with interval objective function coefficients can be found using an algorithm that, at each iteration, solves a linear program to generate a candidate solution and a mixed integer program (MIP) to find the corresponding maximum regret. This paper first shows that there exists a regret-maximising solution in which all uncertain costs are at a bound, and then uses this to derive an MIP formulation that maximises the regret of a candidate solution. Computational experiments demonstrate that this approach is effective for problems with up to 50 uncertain objective function coefficients, significantly improving upon the existing enumerative method.