Two Combinatorial Optimization Problems at the Interface of Computer Science and Operations Research

Solving large combinatorial optimization problems is a ubiquitous task across multiple disciplines. Developing efficient procedures for solving these problems has been of great interest to both researchers and practitioners. Over the last half century, vast amounts of research have been devoted to studying various methods in tackling these problems. These methods can be divided into two categories, heuristic methods and exact algorithms. Heuristic methods can often lead to near optimal solutions in a relatively time efficient manner, but provide no guarantees on optimality. Exact algorithms guarantee optimality, but are often very time consuming. This dissertation focuses on designing efficient exact algorithms that can solve larger problem instances with faster computational time. A general framework for an exact algorithm, called the Branch, Bound, and Remember algorithm, is proposed in this dissertation. Three variations of single machine scheduling problems are presented and used to evaluate the efficiency of the Branch, Bound, and Remember algorithm. The computational results show that the Branch, Bound, and Remember algorithms outperforms the best known algorithms in the literature. While the Branch, Bound, and Remember algorithm can be used for solving combinatorial optimization problems, it does not address the subject of post-optimality selection after the combinatorial optimization problem is solved. Post-optimality selection is a common problem in multi-objective combinatorial optimization problems where there exists a set of optimal solutions called Pareto optimal (non-dominated) solutions. Post-optimality selection is the process of selecting the best solutions within the Pareto optimal solution set. In many real-world applications, a Pareto solution set (either optimal or near-optimal) can be extremely large, and can be very challenging for a decision maker to evaluate and select the best solution. To address the post-optimality selection problem, this dissertation also proposes a new discrete optimization problem to help the decision-maker to obtain an optimal preferred subset of Pareto optimal solutions. This discrete optimization problem is proven to be NP-hard. To solve this problem, exact algorithms and heuristic methods are presented. Different multi-objective problems with various numbers of objectives and constraints are used to compare the performances of the proposed algorithms and heuristics

Biobjective Optimization over the Efficient Set Methodology for Pareto Set Reduction in Multiobjective Decision Making: Theory and Application

A large number of available solutions to choose from poses a significant challenge for multiple criteria decision making. This research develops a methodology that reduces the set of efficient solutions under consideration. This dissertation is composed of three major parts: (i) the formalization of a theoretical framework; (ii) the development of a solution approach; and (iii) a case study application of the methodology. In the first part, the problem is posed as a multiobjective optimization over the efficient set and considers secondary robustness criteria when the exact values of decision variables are subjected to uncertainties during implementation. The contributions are centered at the modeling of uncertainty directly affecting decision variables, the use of robustness to provide additional trade-off analysis, the study of theoretical bounds on the measures of robustness, and properties to ensure that fewer solutions are identified. In the second part, the problem is reformulated as a biobjective mixed binary program and the secondary criteria are generalized to any convenient linear functions. A solution approach is devised in which an auxiliary mixed binary program searches for unsupported Pareto outcomes and a novel linear programming filtering excludes any dominated solutions in the space of the secondary criteria. Experiments show that the algorithm tends to run faster than existing approaches for mixed binary programs. The algorithm enables dealing with continuous Pareto sets, avoiding discretization procedures common to the related literature. In the last part, the methodology is applied in a case study regarding the electricity generation capacity expansion problem in Texas. While water and energy are interconnected issues, to the best of our knowledge, this is the first study to consider both water and cost objectives. Experiments illustrate how the methodology can facilitate decision making and be used to answer strategic questions pertaining to the trade-off among different generation technologies, power plant locations, and the effect of uncertainty. A simulation shows that robust solutions tend to maintain feasibility and stability of objective values when power plant design capacity values are perturbed