Identifier spaces for representing Pareto-optimal solutions in multi-objective optimization and decision-making

A multi-objective optimization task is not complete without a decision-making activity before, during, or after Pareto-optimal (PO) solutions are found. Evolutionary multi-objective optimization (EMO) algorithms, proposed in early nineties, are able to find multiple well-diversified near-PO solutions. Recent evolutionary many-objective optimization (EMaO) studies solve problems having more than three conflicting objectives using ideal-point-based reference vectors (RVs) on the objective space to guide the search and find diverse PO solutions for two- to 20-objective problems. In certain nonlinear problems, PO points guided by ideal-point-based RVs are not uniform on the PO front. This has caused EMaO researchers to develop other RVs suitable for finding a better distribution. We further argue that decision-makers (DMs) may want to visualize a well-distributed set of PO solutions on a different decision-making (identifier) space, other than the objective space, convenient to their decision-making preference. This article presents and compares six different identifier spaces, motivated from a decision-making perspective, based on ideal-point RVs, nadir-point RVs, projection RVs, pseudo-weight vectors, angle vectors, and RadViz coordinates. Advantages and disadvantages of these identifier spaces are laid out for optimization and decision-making purposes by implementing each on a suitable EMaO algorithm and applying them to standard test problems. The use of multiple identifier spaces simultaneously on a single EMO simulation is also executed for DMs to have a compromise distribution of PO solutions. The choice of one or more identifier spaces within EMO algorithms allows a generic, flexible and practical for addressing optimization and decision-making tasks together.