Solving mixed Pareto-Lexicographic multi-objective optimization problems: The case of priority chains

This paper introduces a new class of optimization problems, called Mixed Pareto-Lexicographic Multi-objective Optimization Problems (MPL-MOPs), to provide a suitable model for scenarios where some objectives have priority over some others. Specifically, this work focuses on a relevant subclass of MPL-MOPs, namely problems involving Pareto optimization of two or more emph{priority chains}. A priority chain (PC) is a sequence of objectives lexicographically ordered by importance. After examining the main features of those problems, named PC-MPL-MOPs, we propose an innovative approach to deal with them, built upon the Grossone Methodology, a recent theory which enables handling the priority in an elegant and powerful way. The most interesting aspect of this technique is the possibility to seamlessly embed it in any existing evolutionary algorithm, without altering its logical structure. In order to provide concrete examples, we implemented it on top of the well-known NSGA-II and MOEA/D algorithms, calling these new generalized versions PC-NSGA-II and PC-MOEA/D, respectively. In the second part of this article, we test the strength of our strategy in solving multi- and even many-objective problems with priority chains, comparing it against the results achieved by standard priority-based and non-priority-based approaches. Experiments show that our algorithms are generally able to produce more solutions and of higher quality.