Post-Pareto optimality methods for the analysis of large Pareto sets in multi-objective optimization
Multiple objective optimization involves the simultaneous optimization of more than one, possibly conflicting, objectives. Multiple objective optimization problems arise in a variety of real-world applications. In general, the main difference between single and multi-objective optimization is that in multi-objective optimization there is usually no single optimal solution, but a set of equally good alternatives with different trade-offs, also known as Pareto-optimal solutions. There are two general approaches to solve multiple objective optimization problems: mathematical methods and meta-heuristic methods. The first approach involves the aggregation of the attributes into a linear combination of the objective functions, also known as scalarization. The second general approach involves populating a number of feasible solutions along the Pareto frontier, and the final solution is a set of non-dominated solutions, also called Pareto-optimal solutions. Once a Pareto-optimal set has been obtained, the decision-maker faces the challenge of analyzing a potentially large set of solutions, and selecting one solution over others can be quite a challenging task since the Pareto set can contain an unmanageable number of solutions. Therefore there exists a need for efficient methods that can reduce the size of the Pareto-optimal set to facilitate decision-making. This decision-making stage is usually known as the post-Pareto analysis stage and is the main focus of this work. ^ This work presents four different methods to perform post-Pareto analysis. The first method is the generalization of a method known as the non-numerical ranking preferences method. This method can help decision makers reduce the number of design possibilities to small subsets that clearly reflect the decision maker's objective function preferences without having to provide specific weight values. Previous research has only presented the application of the non-numerical ranking preferences 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 non-numerical ranking preferences method. The second method presented in this thesis uses a non-uniform weight generator method to reduce the size of the Pareto-optimal set. A third method, called sweeping cones technique, is introduced to reduce the size of the Pareto set. Geometrically speaking, this method projects all of the objective function values and weights into the space over a unit radius sphere, and then sweeping cones are used to capture the Pareto points that reflect decision-maker's preferences. The fourth and last method developed is called Orthogonal Search for post-Pareto optimality. This method generates a decreasing succession of mesh points guided by what is called an ideal direction. All methods have been tested on different problem instances to show their performance.^
Carrillo, Victor M, "Post-Pareto optimality methods for the analysis of large Pareto sets in multi-objective optimization" (2013). ETD Collection for University of Texas, El Paso. AAI3609477.