A novel robustness measure for multi-objective optimization problems under interval uncertainty

In this paper, a novel robustness index is introduced to provide a measure of the robustness of a solution against variations in decision variables and parameters. Most of the proposed robustness measures in the literature consider only magnitude of variations in the objectives space and don’t take into account the direction, or in the other words, the type of variations. In this paper, two types of variation named dominating and Pareto variations are introduced and argued that the Pareto variations are more robust than the other one. An index is also proposed here to help measuring the proportion of dominating variations. We proved that this index is independent of magnitude of variations. A robustness index is developed based on these two measures. The robustness index is then used as an additional objective and constraint function so that the uncertain multi-objective optimization problem is transformed to a deterministic one. The resulting deterministic multi-objective optimization problem is solved by NSGA-III. Moreover, Mont Carlo simulation is used to evaluate solutions during the algorithm and compute the robustness index. Two test problems from the context of engineering design optimization are used to illustrate the applicability and efficiency of our proposed robustness index.