In this thesis we address three related topic in the field of Operations Research. Firstly we discuss the problems and limitation of most common solvers for linear programming, precision. We then present a solver that generate rational optimal solutions to linear programming problems by solving a succession of (increasingly more precise) floating point approximations of the original rational problem until the rational optimality conditions are achieved. This method is shown to be (on average) only 20% slower than the common pure floating point approach, while returning true optimal solutions to the problems. Secondly we present an extension of the Local Cut procedure introduced by Applegate et al, 2001, for the Symmetric Traveling Salesman Problem (STSP), to the general setting of MIP problems. This extension also proves finiteness of the separation, facet and tilting procedures in the general MIP setting, and also provides conditions under which the separation procedure is guaranteed to generate cuts that separate the current fractional solution from the convex hull of the mixed-integer polyhedron. We then move on to explore some configurations for local cuts, realizing extensive testing on the instances from MIPLIB. Those results show that this technique may be useful in general MIP problems, while the experience of Applegate et al, shows that the ideas can be successfully applied to structures problems as well. Thirdly we present an extensive computational experiment on the TSP and Domino Parity inequalities as introduced by Letchford, 2000. This work also include a safe-shrinking theorem for domino parity inequalities, heuristics to apply the planar separation algorithm introduced by Letchford to instances where the planarity requirement does not hold, and several practical speed-ups. Our computational experience showed that this class of inequalities effectively improve the lower bounds from the best relaxations obtained with Concorde, which is one of the state of the art solvers for the STSP. As part of these experience, we solved to optimality the (up to now) largest two STSP instances, both of them belong to the TSPLIB set of instances and they have 18,520 and 33,810 cities respectively.

The classical multiple knapsack problem selects a set of items and assigns each to one of the knapsacks so as to maximize the total profit. The knapsacks have limited capacities. The cardinality constrained multiple knapsack problem assumes limits on the number of items that are to be put in each knapsack, as well. Despite many efforts on the classical multiple knapsack problem, the research on the cardinality constrained multiple knapsack problem is scarce. In this study we consider the cardinality constrained multiple knapsack problem. We propose heuristic and optimization procedures that rely on the optimal solutions of the linear programming relaxation problem. Our computational results on the large-sized problem instances have shown the satisfactory performances of our algorithms.

Management of public debt is crucial for every country. Public debt managers make efforts to both minimize the cost of borrowing and to keep debt stock at sustainable levels. However, due to competition for funds in the continuously changing and developing financial markets, new threats and opportunities appear constantly. Public debt managers construct borrowing policies in order to minimize the cost of borrowing and also to decrease risk by using various borrowing instruments. This thesis presents a mathematical model to determine the borrowing policy that minimizes the cost of borrowing in line with future projections and then seeks to extend it to construct risk sensitive policies that allow minimizing the effects of changes in the market on the cost of borrowing. The model&rsquo / s application results for determining the borrowing strategies of Turkish Treasury for 100 month horizon have been evaluated through the study.

In this thesis, we consider a Disassembly Line Balancing Problem (DLBP) with fixed number of workstations. We aim to maximize the total value of the recovered parts. We assume that there is a limited supply for the products to be disassembled. Different components can be obtained by disassembling different units of the product. Our aim is to assign the tasks to the workstations of the disassembly line so as to maximize the total value of the recovered parts. We present several upper and one lower bounding procedure. The results of our computational study have revealed the satisfactory behavior of our bounding mechanisms.

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Branch-price-and-cut algorithms are among the most successful exact optimization approaches for solving many routing and scheduling problems. This is due, in part, to the availability of extremely efficient and effective dynamic programming algorithms for solving the pricing problem, and the availability of efficient and effective branching schemes and cutting planes that drive integrality. In terms of branch-price-and-cut, two obstacles we face today are (1) being able to solve harder and larger pricing problems, and (2) solving mixed-integer column generation formulations that suffer from relatively weak LP bounds compared to the more traditional 0-1 set partitioning type. As part of the work presented in this thesis, we encounter column generation formulations motivated by real life problems that require overcoming both types of challenges. The first part of this thesis is dedicated to solving the resource constrained shortest path problem (RCSPP) arising in column generation pricing problems for formulations involving extremely large networks and a huge number of local resource constraints. We present a relaxation-based dynamic programming algorithm that alternates between a forward and a backward search. Each search employs bounds derived in the previous search to prune the search, and between consecutive searches, the relaxation is tightened over a set of critical resources and arcs. The second part of this thesis focuses in the fixed charge shortest path problem (FCSPP) in which the amount of resource consumed is itself a continuous bounded variable. By exploiting the structure of optimal solutions to FCSPP, we design and implement a solution approach that relies on solving multiple RCSPPs, and therefore can again make use of extremely efficient and effective dynamic programming algorithms. In the third and final part of this thesis, we present a branch-price-and-cut algorithm for the inventory routing problem (IRP). We extend a class of cuts known for the vehicle routing problem, and develop a new class of cuts specifically for IRP to tighten the formulation. Both the branching schemes and cuts preserve the structure of the pricing problem making them efficiently implementable within a branch-price-and-cut algorithm.

Applications of accuracy certificates for problems with convex structure   This dissertation addresses the efficient generation and potential applications of accuracy certificates in the framework of “black-box-represented” convex optimization problems - convex problems where the objective and the constraints are represented by  “black boxes” which, given on input a value x of the argument, somehow (perhaps in a fashion unknown to the user) provide on output the values and the derivatives of the objective and the constraints at x. The main body of the dissertation can be split into three parts.  In the first part, we provide our background --- state of the art of the theory of accuracy certificates for black-box-represented convex optimization. In the second part, we extend the toolbox of black-box-oriented convex optimization algorithms with accuracy certificates by equipping with these certificates a state-of-the-art algorithm for large-scale nonsmooth black-box-represented problems with convex structure, specifically, the Non-Euclidean Restricted Memory Level (NERML) method. In the third part, we present several novel academic applications of accuracy certificates. The dissertation is organized as follows: In Chapter 1, we motivate our research goals and present a detailed summary of our results. In Chapter 2, we outline the relevant background, specifically, describe four generic black-box-represented generic problems with convex structure (Convex Minimization, Convex-Concave Saddle Point, Convex Nash Equilibrium, and Variational Inequality with Monotone Operator), and outline the existing theory of accuracy certificates for these problems. In Chapter 3, we develop techniques for equipping with on-line accuracy certificates the state-of-the-art NERML algorithm for large-scale nonsmooth problems with convex structure, both in the cases when the domain of the problem is a simple solid and in the case when the domain is given by Separation oracle. In Chapter 4, we develop  several novel academic applications of accuracy certificates, primarily to (a) efficient certifying emptiness of the intersection of finitely many solids given by Separation oracles, and (b) building efficient algorithms for convex minimization over solids given by Linear Optimization oracles (both precise and approximate). In Chapter 5, we apply accuracy certificates to efficient decomposition of “well structured” convex-concave saddle point problems, with applications to computationally attractive decomposition of a large-scale LP program with the constraint matrix which becomes block-diagonal after eliminating a relatively small number of possibly dense columns (corresponding to “linking variables”) and possibly dense rows (corresponding to “linking constraints”).

<p>This thesis consists of an introduction and sevenindependent, but closely related, papers which all deal withproblems in structural optimization. In particular, we considermodels and methods for global optimization of problems intopology design of discrete and continuum structures.</p><p>In the first four papers of the thesis the nonconvex problemof minimizing the weight of a truss structure subject to stressconstraints is considered. First itis shown that a certainsubclass of these problems can equivalently be cast as linearprograms and thus efficiently solved to global optimality.Thereafter, the behavior of a certain well-known perturbationtechnique is studied. It is concluded that, in practice, thistechnique can not guarantee that a global minimizer is found.Finally, a convergent continuous branch-and-bound method forglobal optimization of minimum weight problems with stress,displacement, and local buckling constraints is developed.Using this method, several problems taken from the literatureare solved with a proof of global optimality for the firsttime.</p><p>The last three papers of the thesis deal with topologyoptimization of discretized continuum structures. Theseproblems are usually modeled as mixed or pure nonlinear 0-1programs. First, the behavior of certain often usedpenalization methods for minimum compliance problems isstudied. It is concluded that these methods may fail to producea zero-one solution to the considered problem. To remedy this,a material interpolation scheme based on a rational functionsuch that compli- ance becomes a concave function is proposed.Finally, it is shown that a broad range of nonlinear 0-1topology optimization problems, including stress- anddisplacement-constrained minimum weight problems, canequivalently be modeled as linear mixed 0-1 programs. Thisresult implies that any of the standard methods available forgeneral linear integer programming can now be used on topologyoptimization problems.</p><p><b>Keywords:</b>topology optimization, global optimization,stress constraints, linear programming, mixed integerprogramming, branch-and-bound.</p>

This work integrates perennial feedstock yield modeling using climate model data from current and future climate scenarios, land use datasets, transportation network data sets, Geographic Information Systems (GIS) tools, and Mixed integer linear programming (MILP) optimization models to examine biorefinery network designs in the southeastern United States from an overall systems perspective. Both deterministic and stochastic cases are modeled. Findings indicate that the high transportation costs incurred by biorefinery networks resulting from the need to transport harvested biomass from harvest location to processing facilities can be mitigated by performing initial processing steps in small scale mobile units at the cost of increased unit production costs associated with operating at smaller scales. Indeed, it can be financially advantageous to move the processing units instead of the harvested biomass, particularly when considering a 10-year planning period (typical switchgrass stand life). In this case, the mobile processing supply chain configuration provides added flexibility to respond to year-to-year variation in the geographic distribution of switchgrass yields. In order to capture the effects of variation in switchgrass yields and incorporate it in optimization models, yield modeling was conducted for both current and future climate scenarios. (In general profits are lower in future climate scenarios). Thus, both the effects of annual variation in weather patterns and varying climate scenarios on optimization model decisions can be observed.

According to the Institute of Medicine (IOM), the U.S. health care delivery system does not provide consistent, high-quality medical care to all people all the time. As a significant component of the health care delivery system, the state of Graduate Medical Education in the United States has prompted much analysis in recent years due to the general view that desired and actual outcomes are increasingly at variance with each other. One area of focus has been the implications of change for provider credentialing and funding of graduate medical education. With this research we test the hypothesis that residents perform valuable work in the teaching hospitals where they undergo training, to inform the issue regarding provider credentialing for residents. We developed a framework to compare second-year residents (PGY2), physician assistants with one year of experience, and nurse practitioners with one year of experience to measurably address the interchangeability of providers. Data was collected by obtaining expert opinions on the proficiency of the three provider options (resident, physician assistant, nurse practitioner) in performing a set of tasks/procedures by surveying the program directors of Internal Medicine residency programs in the United States. The other residency programs at the University of South Floridas College of Medicine were also surveyed to obtain measurable performance on the service providers.Statistical tools were used to analyze the survey responses, aggregate patient data and salary data for each provider. The data analysis and summary indicated that residents displayed higher levels of proficiency than physician assistants and nurse practitioners for the tasks investigated.