Date of Award
5-2025
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
School of Mathematical and Statistical Sciences
Committee Chair/Advisor
Margaret M. Wiecek
Committee Member
James B. Coykendall
Committee Member
Matthew J. Saltzman
Committee Member
Cameron Turner
Abstract
In this work, we consider finding Pareto efficient solutions for complex multiobjective optimization problems (MOPs). Complex MOPs are unique in the literature because they have many more objective functions than is typically considered. In fact, such complex MOPs will have 30+ objective functions. This large problem size presents computational and coginitive difficulties. Computationally, standard techniques for solving MOPs are often ineffective and cognitively it is difficult for a decision maker (DM) to handle all of the information provided in such a large problem. To address these challenges, we develop a decomposition and coordination framework. This framework will allow us to decompose the complex MOP into a set of subproblems, each of which are multiobjective but with fewer objective functions. Given efficient solutions to these subproblems, efficient solutions for the original complex MOP may be constructed. We develop this approach in four ways.
First, we make theoretical extensions to Achievement Scalarizing Functions, a well-established technique for transforming an MOP into a single-objective optimization problem, which uses a predetermined reference point that represents the DM's aspirations. Our extension, which we call Bivariate Achievement Scalarizing Functions (BASFs), allow the reference point to itself be an optimization variable. In so doing, we take into account the DM's uncertainty in her aspirations. This development is used to great effect in this work.
Secondly, we describe two decomposition and coordination methodologies. The first considers the case of a complex MOP with global and local variables. The second considers a complex MOP with global, quasi-global, and local variables. In the global, quasi-global, and local variables case, during the decomposition phase, we present results on constructing efficient solutions for the complex MOP given an efficient solution to a subproblem. However, the coordination stage ensures that the subproblems may work together to construct an efficient solution for the complex MOP. In particular, we use BASFs to define subproblem tradeoffs, which allows a DM to directly measure the conflict between entire subproblems, a feature that is not currently available in the literature. Using these subproblems, we construct three types of coordination. First, autonomous, which is independent of the DM's involvement; second, hierarchical coordination, which is entirely dependent on the DM ordering subproblems, selecting anchor points, and defining relaxations; third, hybrid coordination, which combines autonomous and hierarchical coordinations to create a robust decision making tool. We demonstrate these results on a case study of delivering humanitarian aid in the event of a natural disaster.
Thirdly, we focus our attention on the problem of actually finding efficient solutions of a subproblem MOP. We consider the case of a biobjective mixed-integer problem (BOMIP) and develop an algorithm, Pareto Leap, which exactly identifies all slices of the outcome set which contribute to the Pareto set. It is recognized in the literature that the problem of identifying Pareto slices is a difficult problem. Nonetheless, Pareto Leap is able to find all Pareto slices using tabu constraints and computationally it is only restrained by access to a sufficiently powerful solver that can handle integer variables and the underlying continuous problem. We provide the requisite theoretical results, present the algorithm, and test it on problems from the literature.
Finally, we consider the problem of coordinating a DM's preferences between efficient solutions of a complex MOP. We note that this is precisely equivalent to the problem of aggregating the preferences of a committee into a single group preference. We formulate this problem of group preference aggregation as a multiobjective optimization problem over binary relations. We further provide an initial result which shows that dictatorships provide Pareto efficient solutions to the group preference aggregation problem. This leads us to conjecture that efficient solutions to the group preference aggregation problem may be written as a "convex" combination of each individual's preferences. We conclude by discussing an example which demonstrates this conjecture.
Recommended Citation
de Castro, Philip J., "Decomposition and Coordination for Multiobjective Optimization: A Framework and Methodology" (2025). All Dissertations. 3939.
https://open.clemson.edu/all_dissertations/3939
Author ORCID Identifier
0009-0003-7306-9324