Dr. Lakmali Weerasena, Assistant Professor of Mathematics at the University of Tennessee at Chattanooga Department of Mathematics, has been awarded a $249,860 grant from the National Science Foundation for the project entitled “Advancing Approximation of Heterogeneous Multi-Objective Set Covering Problems with Modeling and Applications.”
This project aims to efficiently solve large-scale set covering problems (SCPs) associated with the decision-making in conservation planning and emergency medical service management through new multiobjective models, algorithms, and theory. The new models investigated in this project are complex covering models, and mathematically-driven approximations algorithms are necessary to solve them efficiently. Most of the covering models studied in conservation biology and emergency medical service management directly relate to the multiobjective SCP (MOSCP) or at least the MOSCP can be found as a sub-problem. A significant component of such algorithms is the predetermined quality measure. With that property, the optimal solution set is always guaranteed to be within the returned solution set’s predetermined error. This predetermined error is essential in applications to better understand the limits of approximating the desired optimization problem. Estimating the quality of the approximation beforehand is vital in establishing an actionable strategy in applications. Otherwise, the available resources will not be used efficiently, and future adverse conditions are less likely to be met successfully. The broad applicability of this project is to develop optimization tools to integrate voluminous spatially-explicit data to effectively allocate resources and analyze information such as species distribution, population parameters, the spatial configuration of reserves, land costs, and environmental dynamics. Integration of education and outreach activities in this project and broadening participation of underrepresented groups in STEM are among the highly impacted outcomes of this project. A significant societal impact of this project is that it involves developing systematic, structured mathematical models for locally pressing environmental problems and new theories to solve these models.
This project advances theory and applications in multiobjective optimization (MO), focusing on the generalized MOSCP framework. To fully employ MOSCP in real-life applications, it is required to develop systematic, structured mathematical models and computationally efficient algorithms with solid theoretical support to solve the models. MO problems typically do not have optimal solutions in the totally ordered sense of single objective optimization; instead, they have optimal solutions in a partially ordered sense appropriate in the presence of multiple conflicting objectives. The success of applying MO in practice depends mainly on the ability to compute the elements of the solution set (known as the efficient/Pareto set). This task is highly challenging when solving mixed-integer multiobjective optimization problems, which are the fundamental models developed in this project. For such models, some parts of the Pareto set are discrete and some parts continuous, and therefore far more challenging to solve. Further, determining the efficiency of a given solution is computationally challenging for most MO problems. Among MO problems, the generalized MOSCP used here emerges as a computationally-intensive problem with a pressing need for new theory and methods to approximate the Pareto set. Therefore, the technical theme of this project is the investigation of structural properties of the generalized MOSCPs and the development of theoretically-supported, computationally efficient methods for solving them. These methods are also motivated by the practical requirements of robustness against variable and parameter uncertainties and complexities while efficiently handling computationally intensive models. The proposed approximation methods and algorithms are developed using mathematical programming techniques. These mathematical findings contribute to the nascent literature of MOSCP and significantly contribute to the field of approximation in MO optimization. The expected outcome of the research is novel generalized MOSCPs, approximation algorithms with predetermined quality measures, and advanced theory to enable the solutions of novel models addressing decision making in conservation planning and emergency medical service management.