Development of new mathematical methods for post-pareto optimality

Many real-world applications of multi-objective optimization involve a large number of objectives. A multi-objective optimization task involving multiple conflicting objectives ideally demands finding a multi-dimensional Pareto-optimal front. Although the classical methods have dealt with finding one preferred solution with the help of a decision-maker, evolutionary multi-objective optimization (EMO) methods have been attempted to find a representative set of solutions in the Pareto-optimal front. Multiple objective evolutionary algorithms (MOEAs), which are biologically-inspired optimization methods, have become popular approaches to solve problems with multiple objective functions. With the use of MOEAs, multiple objective optimization becomes a two-part problem. First, the multiple objective optimization problem needs to be formulated and successfully solved using an MOEA. Then, a non-dominated set—also known as efficient or Pareto frontier—needs to be analyzed to select a solution to the problem. This can represent a challenging task to the decision-maker because this set can contain a large number of solutions. This decision-making stage is usually known as the post-Pareto analysis stage. This thesis presents two different methods to perform post-Pareto analysis. The first method is the generalization of a method known as the non-numerical ranking preferences (NNRP) method. This method can help decision makers reduce the number of design possibilities to small subsets that clearly reflect their objective function preferences without having to provide specific weight values. Previous research has only presented the application of the NNRP method using three and four objective functions but had not been generalized to the case of n objective functions. The work presented in this thesis expands the NNRP method. The second method presented in this thesis uses a non-uniform weight generator method to reduce the size of the Pareto-optimal set. Both methods have been tested on different problem instances with successful results