Data Envelopment Analysis (DEA) is a multi-criteria data analysis methodology introduced by Charnes, Cooper, and Rhodes in 1978. Since that time, it has proven to be a valuable analysis tool for strategic, policy, and operational decision problems. Its primary use is to conduct performance evaluations of technical, scale, and managerial efficiency. Since DEA generalizes the single-dimensional engineering and economic efficiency measure into a multi-dimensional measure, it has useful applications in engineering and economic studies. This dissertation addresses several aspects of the DEA methodology and presents original research results of both a theoretical and applied nature. Topics of the early chapters provide the reader with an intuitive understanding of DEA in addition to a finely-tuned technical understanding of the method. The later chapters build on this understanding through new theoretical results which contribute to a unifying DEA theory and through an empirical study of resource use efficiency in manufacturing. The theoretical research results give a thorough examination and specification of relationships between the economic concept of returns to scale enforced by different DEA models and variable set dimensionality. The relationships become apparent by examining properties of the set of units classified as efficient by each DEA model. These relationships are delineated, in set-theoretic terms, in a sequence of theorems with proofs. The applied research is an empirical DEA study of global resource use efficiency in international manufacturing using actual data obtained from the United Nations. By using the aggregate measure of efficiency which DEA provides, this research links multiple manufacturing outputs to consumption levels of multiple resources thereby incorporating the complexities of manufacturing environments which prior, simpler productivity analyses have been unable to capture. In particular, we analyze and interpret relationships between resource use and manufacturing efficiency. We compare performance of the manufacturing sectors in nations around the globe detecting temporal trends in efficiency, including differences in performance by economy type and by geographic location. Both the theoretical and the applied contributions presented in this dissertation are springboards to areas of future research. This dissertation concludes with mention of such possible extensions and follow-on studies.

The goal of this dissertation is to provide a series of static and dynamic models of competitive multi-instrument, multi-sector, and multi-currency financial equilibrium which will yield the optimal composition of assets and liabilities in the portfolio of every sector of each country. The equilibrium market prices of every instrument in each currency, as well as the equilibrium exchange rate prices for each currency are also obtained. In addition, market imperfections such as taxes, transaction costs, price policy interventions, and the presence of financial hedging instruments, are taken into consideration. The models presented here are based on the fundamental economic theory of finance, and relax many of the assumptions that much of the literature is based upon. For example, there is no need for a risk free instrument or a global portfolio, and all sectors in the economy do not have to share homogeneous expectations on prices. In the contrary, heterogeneity of opinions plays a critical role on the determination of the asset allocation as well as in the price derivation. Moreover, sectors do not hold the same amount of capital, and are not subjected to the same type of transaction costs and taxes, since the models under consideration have the ability to impose taxes and transaction costs that depend both on the identity of a sector and on the type of an instrument. Moreover, the monetary authorities of each country (or currency) have the ability to apply different price floors and ceilings on every instrument so that they can control the market according to their strategies. All the models as well as the computational methods suggested here are based on the methodologies of finite-dimensional variational inequality theory for the exploration of statics and equilibrium states, and on projected dynamical systems theory for the study of dynamics and disequilibrium behavior. Simultaneously, visualization and formulation of financial problems as network flow problems provide one with the opportunity of applying network-based algorithms, coupled with the aforementioned methodologies, for computational purposes. The models presented here are accompanied by a detailed qualitative analysis that provides conditions of existence and uniqueness of equilibrium patterns as well as general sensitivity analysis results.

The difficulty of straight edge and compass solutions to the Euclidean Steiner Minimal Tree Problem for more than three vertices, has been known for at least three centuries. Analytic geometry methods, in addition to these tools, use Algebra and Cartesian frames of reference. In E2, optimal solutions can be achieved for 10,000 points and more. For more than ten vertices in E 3 or higher dimensions, these exact formulations have proved difficult and cumbersome from the point of view of an algorithmic solution. A discrete version of the problem was shown to be NP-complete in 1977. Decomposition heuristics based on Computational Geometry were suggested, for these larger point sets. The thesis features a literature review of the considerable research efforts on the Steiner problem in two and three dimensional space, with the Euclidean metric. Heuristics of polynomial complexity, that have proven satisfactory for large point sets are considered after the O.R. methods that are of exponential order. The Steiner Ratio ρ of a vertex set is the length of the Steiner Minimal Tree (SMT), divided by the length of the Minimum Spanning Tree (MST), and in addition to execution time, is a key measure of performance, of algorithms and heuristics devoted to this problem. The consideration and comparison of the performance of algorithms, leads to the issue of the best Steiner ratio for a particular space. The d-Sausage, an unending geometric arrangement of regular simplices, has yielded the best Steiner Ratio in three and higher dimensional Euclidean space. A particular Full Steiner Tree topology, which we refer to as the path topology, is proven to be the optimal topology, for the d-Sausage when d = 1 or 2. Other structural properties of the flat sausage and the [special characters omitted]-Sausage, as these two instances of the d-Sausage are referred to, are proved as lemmas and theorems. This theoretical framework serves as a foundation for a heuristic for finding SMTs for the very large point sets. The sausage has been shown to have a superior Steiner ratio compared to a simplex. For this reason it is the preferred primitive for a decomposition technique. Finally, the ties between the Steiner Minimal Tree Problem, and the Euclidean Graph Embedding Problem, are explored in the light of the Minimum Energy Configuration of molecules, and Maxwell's theorem.