On a Better Understanding of Unique Identifiers of Pareto Solutions for Multi-criterion Optimization, Visualization, and Decision-making

A multi-criterion optimization task requires a decision-making activity during or after the Pareto solutions are found. A convenient and effective decision-making task can be achieved with well-represented Pareto solutions and by means of user-friendly and easily comprehensible visualization tools. To have a comprehensive idea of the spread of Pareto solutions obtained by evolutionary multi-criterion optimization (EMO) or other algorithms, each Pareto solution can be associated with a unique identifier - a vector of size of the dimension of the Pareto surface. While the Pareto solutions can lie on an arbitrary surface with non-domination properties but having its shape and size largely dependent on the problem being solved, these identifiers can have much simpler properties, such as lying on a unit simplex or a unit sphere, or directly relating to preference values. Once achieved, these unique identifiers can help EMO algorithms evaluate the extent of the spread of solutions during the optimization process and decision-makers to have a better understanding of trade-offs among solutions to make better decisions. Moreover, apparent gaps or other complexities of the obtained Pareto solutions can be more easily located in the identifier space. In this paper, we present and compare five identifiers - ideal point reference vectors (RV), nadir point RVs, projection RVs, pseudo-weights, and angle vectors - with respect to their advantages and disadvantages in decision-making and visualization purposes. We also demonstrate that, if desired by the decision maker, these alternative identifiers can be used during optimization to achieve a good distribution of solutions in this space.