The hyper-radial visualization (HRV) method for visualization of the hyperspace pareto frontier for multi-objective optimization problems
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Most engineering design problems are characterized by more than two objective functions, and these problems are termed Multi-objective Optimization Problems (MOPs). MOPs always yield multiple Pareto optimum solutions, which are not dominated by any other feasible solutions. The region of the Pareto set is the Pareto frontier. When MOPs have more than three objectives, the resulting hyperspace surface is called the Hyperspace Pareto Frontier (HPF). In practice, only a few of these Pareto solutions hold any application value for designers. The true challenge with MOPs is to be able to identify the most desirable solution(s) from amongst the Pareto set. Visualization can provide decision makers with a way to see large complex data sets efficiently, thereby improving the solution selection process. However, most real world engineering design problems have more than a three-dimensional design space, and the visualization of such cases is limited by our three dimensional world. In this work, a visualization methodology is developed in which an HPF can be visualized in a two-dimensional space. The new approach is termed the Hyper-Radial Visualization (HRV) Method. By using this approach, the relationships of high dimensional Pareto sets can be viewed in an intuitive and straightforward manner. This dissertation includes several research issues pertaining to the development of the HRV Method. First, the concept of the HRV method is discussed. The HRV method can intuitively represent the HPF in a lossless way while maintaining the neighborhood relationships between each Pareto point. In HRV representations, designers can apply the HRV radial value for every Pareto point to select their final solution(s). Second, since every member of a Pareto frontier is mathematically equal, designers must consider additional preference information in order to distinguish between the Pareto solutions. There are two preference incorporation approaches proposed in this work. The weighting preference technology is proposed and tested in Chapter 4, and the range-based preference approach is introduced in Chapter 5. Both of these two preference structures are very powerful methods that enable filtering the final design(s), but the sorting efficiency of the range-based preference structure decreases seriously when applied in high dimensional problems. Hence, the final part of this work is to improve the filtering efficiency of the range-based preference structure. In Chapter 6, this work extends the concept of equal-range preference structures to a variable range structure. Moreover, three test problems are used to show that the variable range preference method can overcome the loss of sorting efficiency when studying high dimensional multiobjective optimization problems.