The remanufacturing industry emerged as many countries and increasing number of environmentally conscious consumers come to realize the natural resources depletion and widely spread pollution. While commonly believed as a compromise to the strict government regulations, remanufacturing is actually becoming a profitable business model. Moreover, researchers have found that it may increase market share under right circumstances. In this thesis, we study one major problem in remanufacturing, namely, sorting policies that specify which returned items should be remanufactured and which should be scrapped. We examine a remanufacturer who acquires used products from third party brokers or directly from the market in both single and multiple periods. In single period setting, we examine the optimal policies when the acquisition cost is piecewise linear convex as well as linear. We show that a simple sorting policy exists when the acquisition cost is linear. We study the multi-period problem and prove that the unique optimal policies in multiple periods exist. While the remanufacturer may decide to keep inventory for final products only or he may decide to keep inventory for raw cores as well, we illustrate the algorithm to solve the optimization problem. For linear acquisition cost problems in both single and multiple period problems, we show that they can be solved as general inventory problems which may include setup costs, backlogging and uncertain demand.

The Goal Oriented Systems Approach to Building Fire Safety developed by the U.S. General Services Administration is presently the only probabilistic methodology for fire protection evaluation in use in the United States. This paper describes and analyzes the GSA approach and formulates a more scientific procedure by synthesizing GSA concepts with additional probability theory. Discussion of systems analysis and modeling concepts emphasizes the need for probabilistic considerations of fire safety. The revised model, identified by the hyphenated expression Goal-Oriented, simplifies data requirements through parameter estimation techniques. The new approach is consistant with the GSA model for several example cases. Facility for sensitivity analysis of alternative fire protection strategies is a demonstrated advantage of the new methodology.

In general, mixed integer linear programming models are known to be computationally intractable. Algorithmic procedures that provide optimal or near optimal solutions are usually specific to a particular model or class of models. In recent decades, the focus has shifted to developing, what are termed, tight formulations through the use of valid inequalities. Valid inequalities are typically specific to a class of optimization models and have been shown to be crucial to obtaining optimal solutions. Modeling the supply chain to determine efficient and effective policies across and within the echelons of a supply chain can be a formidable task. Supply chains are not only defined by their physical infrastructure of plants, warehouses, distribution centers, and transportation alternatives, but also by defining characteristics such as the number of products, product volumes, and their demand patterns over time. A model for a supply chain can defy solution for specific instances of both the product characteristics and the spatial characteristics of the infrastructure. The difference in two supply chain instances that render the model for one to be intractable, and the model for the other to yield instantaneously obtained optimal solutions, can be due to a combination of factors, such as cost structures, demand patterns, or spatial configurations. This dissertation examines various model instances of each of a suite of single and multi-period supply chain models to reveal the manner in which specific supply chain characteristics can lead to computational intractability. Valid inequalities, which are developed for each model studied, are shown to not only alleviate the model's computational intractability by orders of magnitude, but also afford identification of optimal solutions to a spectrum of model instances. This study of various problem instances reveals that particular patterns of transportation, inventory, and distribution are determined by the particular cost structures and cost tradeoffs. The studies reveal that computational intractability is, in fact, due to factors that include specific types of spatial configuration, cost structures of multiple modes of transportation, and temporal relationships between inventory and transportation.

Given a multi-story facility, the problem is how to design its evacuation system so that, when an emergency occurs, it will take the minimal clearance time to evacuate the occupant population from the facility. This research formulates the design problem of an evacuation planning system of a multi-story facility on a rectilinear metric and presents an approach to its solution based on a tri-partite series of optimization and stochastic models. In order to find the optimal locations of stairwells on each floor, a heuristic for equi-area partitioning for rectilinear simple polygons is developed. The facility evacuation system is studied as a state dependent stochastic model, and a simulation program based on this model is used to evaluate the efficiency of the evacuation system. This research also addresses the problem of how to optimally determine the width of the stairwells with the objective of obtaining the minimal clearance time of the evacuation. Finally, a case study is conducted based on a real world problem - the evaluation of the evacuation process of the New York International Gift Fair held at the Jacob Javits Convention Center. The problem is modeled with the state dependent stochastic model; and an evacuation is conducted and suggested solutions for improvement are recommended.

The recent theories of scale-free and small-world networks have significantly enhanced our understanding of the behavior as well as the vulnerability of many real-world networks. However, the majority of network vulnerability studies focus solely on the topological characteristics. Although the topological structure of a network provides crucial information regarding network vulnerability, the flow on a network is also an important indicator, as are the flow-induced costs and the behavior of the users. Latora and Marchiori (2001, 2002, 2004) proposed a network efficiency measure that is shown to have advantages over several existing network measures. Nevertheless, their measure only considers geodesic information and, therefore, ignores important factors such as flows, costs, and behaviors. The first objective of this dissertation is to construct a network efficiency/performance measure that extends the Latora-Marchiori measure to incorporate such important network factors as flows, costs, and behaviors in order to assess the importance of network components. It is shown that the new network measure has advantages over several existing network measures. Furthermore, the measure is able to handle both fixed and elastic demands as well as static and dynamic networks, with the latter of particular relevance to the Internet. Moreover, it enables a ranking of the importance of network components. In addition, instead of looking at the situation where a network component is completely disrupted, network robustness, another important aspect of the network vulnerability, investigates cases in which network resources are reduced in stressful environments. The second goal of this dissertation is to study transportation network robustness based on the new network efficiency/performance measure in order to investigate the network functionality when the links are partially degraded. I also evaluate transportation network robustness under different user behaviors and with environmental concerns. Furthermore, based on the recent results regarding the supernetwork equivalence between transportation networks and supply chain networks as well as financial networks (Nagurney (2006a) and Liu and Nagurney (2007)), I apply the new network measure to study multitiered financial networks with intermediation. I also propose a novel supply chain model with disruption risks and uncertain demands and define a weighted supply chain network performance measure.

Recently, Stochastic Gradient Descent (SGD) and its variants have become the dominant methods in the large-scale optimization of machine learning problems. A variety of strategies have been proposed for tuning the step sizes, ranging from adaptive step sizes (e.g., AdaGrad) to heuristic methods to change the step size in each iteration. Also, momentum has been widely employed in machine learning tasks to accelerate the training process. Yet, there is a gap in our theoretical understanding of them. In this work, we start to close this gap by providing formal guarantees to a few heuristic optimization methods and proposing improved algorithms if the theoretical results are suboptimal. First, we analyze a generalized version of the AdaGrad (Delayed AdaGrad) step sizes in both convex and non-convex settings, showing that these step sizes allow the algorithms to automatically adapt to the level of noise of the stochastic gradients. We show sufficient conditions for Delayed AdaGrad to achieve almost sure convergence of the gradients to zero, which is the first guarantee for Delayed AdaGrad in the non-convex setting. Moreover, we present a high probability analysis for Delayed AdaGrad and its momentum variant in the non-convex setting. Second, we present an analysis of SGD with exponential and cosine step sizes, which are simple-to-use, empirically successful but lack of theoretical support. We provide the very first convergence guarantees for them in the smooth and non-convex setting, with and without the Polyak-Łojasiewicz (PL) condition. We show that these two strategies also have the good property of adaptivity to noise under PL condition. Third, we study the last iterate of momentum methods. We prove the first lower bound in the general convex setting for the last iterate of SGD with constant momentum. Based on the fact that the lower bound is suboptimal, we investigate a class of (both adaptive and non-adaptive) Follow-The-Regularized-Leader-based momentum algorithms (FTRL-based SGDM) with increasing momentum and shrinking updates. We show that their last iterate has optimal convergence for unconstrained convex stochastic optimization problems without projections onto bounded domains nor knowledge of the number of iterations.

Facility layout and location, land use planning and other problems that concern the optimal assignment of activities to locations are complex combinatorial optimization problems. Such problems can be represented by the Quadratic Set Packing (QSP) model, which offers distinct computational advantages over the more general Quadratic Assignment Problem formulation. Further, a lagrangian relaxation of the QSP is a simpler relaxed assignment problem that can be solved efficiently with an exact branch and bound algorithm embedded in a program called MAFLAD (Smith and Macleod, 1988). MAFLAD is based on the assumption that the study region is tesselated by a cartesian grid, and that clusters of grid cells representing alternate locations for each activity are input by the user. The first section of this thesis addresses the topological clustering problem, that is the problem of generating clusters of grid cells for input to MAFLAD. This is a bi-criterion problem in which both attribute data and spatial location impact the optimal grouping. Initially, this bi-criterion problem is decomposed and formulated as two non-linear zero-one integer programming problems that are solved successively to identify the optimal grouping. Due to the complexity of this approach, the topological clustering problem is reformulated as a QSP problem which is optimally solved with the branch and bound algorithm embedded in MAFLAD. The efficiency of this solution method is enhanced via data reduction techniques that exploit the geometry of the topological clustering problem to reduce the depth and breadth of the branch and bound tree. The second section of this thesis presents a parallel version of MAFLAD designed for an implemented on a Sequent Symmetry S-81 multiprocessor. This is an asynchronous algorithm in which multiple processors execute the branch and bound procedure on private data while sharing the best bound. This algorithm avoids problems associated with data dependency, minimizes overhead, and exploits the tightly coupled/shared memory architecture of the Sequent multiprocessor. A heuristic for ordering the sequence of assignments in the branch and bound procedure and a data partitioning algorithm to generate input for the parallel version of MAFLAD are presented, along with the results and analysis of experiments.

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