Essays on Logic-Based Benders Decomposition, Portfolio Optimization, and Fair Allocation of Resources

This thesis offers methodological and computational contributions to several fields of operations research including stochastic programming, decomposition-based methods, robust optimization, and fairness in resource allocation. In the first chapter, we introduce the stochastic planning and scheduling problem and formulate it as a twostage stochastic program. We devise a logic-based Benders decomposition algorithm that can solve this problem exactly. We present an extensive numerical analysis on the effectiveness of the proposed solution algorithm. In the second chapter, we extend our analysis on the planning and scheduling problem. We introduce new Benders cuts and improve some of the cuts proposed in the literature. We then focus on a class of sequence-dependent scheduling problems. We provide an exact solution method for this problem by deriving novel logic-based Benders cuts. The cuts we propose generalize some of the other well-known cuts in the literature. The numerical experiments show that the proposed method outperforms the benchmark. In the third chapter, we study the classical Markowitz model for portfolio optimization. We focus on a robust portfolio optimization model that attempts to address the uncertainty in the expected returns. We provide a theoretical analysis on the selection of the error covariance matrix that is used to define the uncertainty set.Our results show that the class of diagonal estimation-error matrices can achieve an arbitrarily small loss in the expected portfolio return as compared to the optimum. The computational experiments we perform show that even using the identity matrix as the error covariance matrix outperforms the classical Markowitz model. In the fourth and the last chapter, we focus on the use of optimization models for fair allocation of scarce resources. We study several social welfare functions that are used in optimization models to balance efficiency and equity in resource distribution. We analyze the structural properties of the socially optimal distributions based on each social welfare function. We discuss the implications of selecting a social welfare

function with respect to the incentives it creates for both the players and the social planner. We then extend our analysis to hierarchical networks. Our analysis offers a novel approach to evaluating the adequacy of well-known social welfare functions for distribution of scarce resources.