New cluster mapping tools for the graphical assessment of non-dominated solutions in multi-objective optimization

Two new graphical tools for the interpretation of Pareto fronts and the selection of non-dominated solutions produced in multi-objective optimization processes (MOOPs), are presented. The first is a version of the parallel coordinates plots (PCP), modified by combining the PCP with the dendrogram representing the cluster analysis of non-dominated solutions in the decision variable space or in the objective space. A correspondence plot that simplifies interpretation of the above plots has also been developed. The second graphical tool is a cluster map (PFCM), produced by combining the information provided by the dendrograms calculated in the decision and the objective spaces, to provide a two-dimensional plot in which the non-dominated solutions are organized according to both dendrograms; the plot is colored on the basis of any of the objectives or a combination of these objectives when convenient. Two derived graphic tools consisting of a combination of the decision variables and the objectives and the dendrograms produced in the decision and the objective spaces have also been developed. All of these graphical tools are demonstrated with several mathematical functions available in the MOOP-related literature and with a real-world optimization process consisting of the computer-assisted method development of high-performance liquid chromatography.

► Dendrogram-assisted parallel coordinates plots allow interpretation of pareto-fronts in MOOP.
► Pareto-front cluster maps provide a global view of groupings in Pareto-fronts.
► These graphical tools help in decision making at final stage of any multi-objective optimization.