The hyperspace Pareto frontier for intuitive visualization of multiobjective optimization problems

Almost all real world engineering design problems are characterized by the presence of several conflicting and/or cooperating objectives. The term Multiobjective Optimization Problems (MOPS) is used to broadly classify problems with more than one objective functions, which are optimized deterministically. As a solution strategy, multiobjective problems are often aggregated to form a single objective function leading to a single solution for the aggregated function. The solution found using this approach is strongly dependent on the way the objectives have been aggregated. A rather practical approach for dealing with multiobjective problems is to find a set of solutions, termed Pareto Optimal Solutions or the Pareto Set, instead of finding a single aggregated objective-dependent global optimum. All of the Pareto optimal solutions are equally important and all are global optimal solutions. Any final design solution is chosen from the Pareto Set as there would be no solution outside this set that would be better in all the objectives. Such a choice amongst these solutions would be made according to designer preferences.; The concept of Pareto optimality has been widely used in industry to aid designers in their decision-making processes. The decision-maker articulates his preference pertaining to the different objectives once he has knowledge of the Pareto set. The approach of visualizing the Pareto Set has been extensively used in decision-making for two and three objective problems, since the solutions can be readily visualized using traditional 2-D and 3-D graphical means. However, when the problem size is large (i.e. for more than three objectives), there is no easy or intuitive method to visually represent the Pareto frontier.; In this research, a new method for 'lossless' dimension blending is presented to enable development of an intuitive visualization capability for representing the Pareto frontier in multidimensional performance space for multiobjective optimization problems. The method presented is termed the Hyper-Space Diagonal Counting (HSDC) method for multidimensional visualization and the resulting visual representation of the multidimensional performance space is termed the Hyperspace Pareto Frontier (HPF). The HSDC method has been implemented on several test problems to examine its viability and demonstrate the potential of the technique.