Computationally Efficient Approaches for Optimization under Uncertainty and its Applications

Abstract

Uncertainty poses a significant challenge to decision making in many real-world problems, especially when it is high-dimensional. For example, disruptive events such as the COVID-19 pandemic have caused significant uncertainties in the supply and demand for many important products such as Neodymium-iron-boron (NdFeB) magnets. To overcome such challenge, advanced optimization approaches such as stochastic programming (SP), robust optimization (RO), and distributionally robust optimization (DRO) have been developed in order to enable decision makers to find an optimal trade-off between risk and reward by including some knowledge of the uncertainty into their decision-making process. In this dissertation, we study computationally efficient approaches for SP, RO, and DRO and their applications. First, we propose computationally efficient inner and outer approximations for DRO problems with a moment-based ambiguity set and a combined ambiguity set including Wasserstein distance and moment information. In these approximations, we split a random vector into smaller pieces, leading to smaller matrix constraints. In addition, we use principal component analysis (PCA) to shrink uncertainty space dimensionality. We quantify the quality of the developed approximations by deriving theoretical bounds on their optimality gap. We display the practical applicability of the proposed approximations in production-transportation and multi-product newsvendor problems. The results demonstrate that these approximations dramatically reduce computational time while maintaining high solution quality. The approximations also help construct an interval that includes the (unknown) optimal value for a large-scale DRO problem. Next, we propose a resilient reverse supply chain and logistics network design for recycling NdFeB magnets as a means to diversify the supply and alleviate the risks. We develop scenarios to model the unique impact of the COVID-19 pandemic on the proposed business, incorporating both disruption intensity and recovery rate. We formulate a chance-constrained two-stage SP model to maximize the profit while guaranteeing the network resiliency against disruption risks. The decision variables include facility locations, processing capacities, inventory levels, and material flows. To solve the problem in large-scale instances, we develop an efficient Benders decomposition algorithm. Finally, we apply the model to the United States to demonstrate the practical applicability of the proposed model and algorithm. Lastly, we propose a systematic approach to develop data-driven polyhedral uncertainty sets that mitigates these drawbacks. The proposed uncertainty sets are polytopes induced by a given set of scenarios, capture correlation information between uncertain parameters, and allows for direct trade-offs between traceability and conservativeness issue of conventional polyhedral uncertainty sets. To develop these uncertainty sets, we use principal component analysis (PCA) to transform the correlated scenarios into their uncorrelated principal components and to shrink theuncertainty space dimensionality. Thus, decision-makers can use the number of the leading principal components as a tool to trade-off tractability, conservativeness, and robustness of RO models. We quantify the quality of the lower bound of a static RO problem with a scenario-induced uncertainty set by deriving a theoretical bound on the gap between the optimal value of this problem and that of its lower bound. Additionally, we derive probabilistic guarantees for the performance of the proposed scenario-induced uncertainty sets by developing explicit lower bounds on the number of scenarios required to obtain the desired guarantees. Finally, we demonstrate the practical applicability of the proposed uncertainty sets to trade-off tractability, robustness, and conservativeness by examining a range of knapsack and power grid problems.