D-PHI: Desirability-Based Hypervolume Indicator for Interactive Multiobjective Optimization Using Aspiration and Reservation Levels as Preferences

To address problems with multiple conflicting objective functions while incorporating preference information from a decision maker (a domain expert), interactive evolutionary multiobjective optimization methods have been proposed and widely adopted. To systematically assess the performance of these methods, it is desirable to have indicators that not only rely on the Pareto dominance relation, where solutions with better objective function values are preferred, but also take the decision maker’s preferences into account. In this way, indicator values can meaningfully reflect the decision maker’s preferences and the quality of the solutions obtained. In this article, we propose the first performance indicator that incorporates preference information in the form of two reference points, specifically an aspiration point, consisting of aspiration levels that represent the decision maker’s full satisfaction, and a reservation point, consisting of reservation levels that represent dissatisfaction. The new indicator, termed D-PHI, translates information about reservation and aspiration points into a transformed hypervolume measure that assesses the performance of a solution set. We introduce D-PHI with an extended set of guidelines for preference-based performance evaluation, discuss its theoretical properties, and demonstrate its behavior through empirical examples. Moreover, we define a complementary indicator (CI) based on a so-called achievement scalarizing function. CI offers additional insight into whether aspiration and reservation levels have been achieved, and its usefulness is demonstrated through examples.