Robustness Against the Decision-Maker's Attitude to Risk in Problems With Conflicting Objectives

In multiobjective optimization problems (MOPs), the Pareto set consists of efficient solutions that represent the best trade-offs between the conflicting objectives. Many forms of uncertainty affect the MOP, including uncertainty in the decision variables, parameters or objectives. A source of uncertainty that is not studied in the evolutionary multiobjective optimization (EMO) literature is the decision-maker's attitude to risk (DMAR) even though it has great significance in real-world applications. Often the decision-makers change over the course of the decision-making process and thus, some relevant information about preferences of future decision-makers is unknown at the time a decision is made. This poses a major risk to organizations because a new decision-maker may simply reject a decision that has been made previously. When an EMO technique attempts to generate the set of nondominated solutions for a problem, then DMAR-related uncertainty needs to be reduced. Solutions generated by an EMO technique should be robust against perturbations caused by the DMAR. In this paper, we focus on the DMAR as a source of uncertainty and present two new types of robustness in MOP. In the first type, dominance robustness (DR), the robust Pareto solutions are those which, if perturbed, would have a high chance to move to another Pareto solution. In the second type, preference robustness (PR), the robust Pareto solutions are those that are close to each other in configuration space. Dominance robustness captures the ability of a solution to move along the Pareto optimal front under some perturbative variation in the decision space, while PR captures the ability of a solution to produce a smooth transition (in the decision variable space) to its neighbors (defined in the objective space). We propose methods to quantify these robustness concepts, modify existing EMO techniques to capture robustness against the DMAR, and present test problems to examine both DR and PR.