Mixed Integer Programming Approaches for Group Decision Making

Group decision making problems are everywhere in our day-to-day lives and have great influence on the daily operation of companies and institutions. With the recent advances in computational technology, it's not surprising that some companies would want to harvest that power to aid their decision-making procedures. Ethelo, the company that we partnered with in this project, developed an online platform that aids decision-making procedures by formulating the decision-making problem as a mixed integer nonlinear program (MINLP), providing feedback by solving the MINLP in real-time, and allowing the general public to contribute their opinions. Since an interactive component is involved, it is the goal of this thesis to attempt to reduce the solve time of their MINLP by applying tools from Operational Research. The main contribution in this thesis is threefold: first, we noticed that a big proportion of the MINLPs can be easily reposed as linear integer programs, and that a runtime reduction of at least 87.9\% can be achieved by simply redirecting them to a linear solver. Second, we identified a knapsack-like polyhedral structure that, to the best of our knowledge, has not been studied before, and derived a sufficient condition to identify the cases for which all valid cuts can be derived by considering other knapsack or covering problems. Finally, for the more general case where the objective function is nonlinear and not continuous, we derived a few different formulations to get to different approximations of the nonlinear model, and tested all of the approximations computationally