Solving Mixed Pareto-Lexicographic Multi-Objective Optimization Problems: The Case of Priority Levels

This paper concerns the study of Mixed Pareto-Lexicographic Multi-objective Optimization Problems where the objectives must be partitioned in multiple priority levels. A priority level (PL) is a group of objectives having the same importance in terms of optimization and subsequent decision-making, while between PLs a lexicographic ordering exists. A naive approach would be to define a multi-level dominance relationship and apply a standard EMO/EMaO algorithm, but the concept does not conform to a stable optimization process as the resulting dominance relationship violates the transitive property needed to achieve consistent comparisons. To overcome this, we present a novel approach which merges a custom non-dominance relation with the Grossone methodology, a mathematical framework to handle infinite and infinitesimal quantities. The proposed method is implemented on a popular multi-objective optimization algorithm (NSGA-II), deriving a generalization of it called by us PL-NSGA-II. We also demonstrate the usability of our strategy by quantitatively comparing the results obtained by PL-NSGA-II against other priority and non-priority-based approaches. Among the test cases, we include two real-world applications: one 10-objective aircraft design problem and one 3-objective crash safety vehicle design task. The obtained results show that PL-NSGA-II is more suited to solve lexicographical many-objective problems than the general purpose EMaO algorithms.