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1.1. Introduction Solving systems of nonlinear equations has since long been of great interest to researchers in the field of economics, mathematics, en gineering, and many other professions. Many problems such as finding an equilibrium, a zero point, or a fixed point, can be formulated as the problem of finding a solution to a system of nonlinear equations. There are many methods to solve the nonlinear system such as Newton's method, the homotopy method, and the simplicial method. In this monograph we mainly consider the simplicial method. Traditionally, the zero point and fixed point problem have been solved by iterative methods such as Newton's method and modifications thereof. Among the difficulties which may cause an iterative method to perform inefficiently or even fail are: the lack of good starting points, slow convergence, and the lack of smoothness of the underlying function. These difficulties have been partly overcome by the introduction of homo topy methods.
In the last 25 years, the fuzzy set theory has been applied in many disciplines such as operations research, management science, control theory,artificial intelligence/expert system, etc. In this volume, methods and applications of fuzzy mathematical programming and possibilistic mathematical programming are first systematically and thoroughly reviewed and classified. This state-of-the-art survey provides readers with a capsule look into the existing methods, and their characteristics and applicability to analysis of fuzzy and possibilistic programming problems. To realize practical fuzzy modelling, we present solutions for real-world problems including production/manufacturing, transportation, assignment, game, environmental management, resource allocation, project investment, banking/finance, and agricultural economics. To improve flexibility and robustness of fuzzy mathematical programming techniques, we also present our expert decision-making support system IFLP which considers and solves all possibilities of a specific domain of (fuzzy) linear programming problems. Basic fuzzy set theories, membership functions, fuzzy decisions, operators and fuzzy arithmetic are introduced with simple numerical examples in aneasy-to-read and easy-to-follow manner. An updated bibliographical listing of 60 books, monographs or conference proceedings, and about 300 selected papers, reports or theses is presented in the end of this study.
Stochastic Programming offers models and methods for decision problems wheresome of the data are uncertain. These models have features and structural properties which are preferably exploited by SP methods within the solution process. This work contributes to the methodology for two-stagemodels. In these models the objective function is given as an integral, whose integrand depends on a random vector, on its probability measure and on a decision. The main results of this work have been derived with the intention to ease these difficulties: After investigating duality relations for convex optimization problems with supply/demand and prices being treated as parameters, a stability criterion is stated and proves subdifferentiability of the value function. This criterion is employed for proving the existence of bilinear functions, which minorize/majorize the integrand. Additionally, these minorants/majorants support the integrand on generalized barycenters of simplicial faces of specially shaped polytopes and amount to an approach which is denoted barycentric approximation scheme.
In this book, two seemingly unrelated fields -- algebraic topology and robust control -- are brought together. The book develops algebraic/differential topology from an application-oriented point of view. The book takes the reader on a path starting from a well-motivated robust stability problem, showing the relevance of the simplicial approximation theorem and how it can be efficiently implemented using computational geometry. The simplicial approximation theorem serves as a primer to more serious topological issues such as the obstruction to extending the Nyquist map, K-theory of robust stabilization, and eventually the differential topology of the Nyquist map, culminating in the explanation of the lack of continuity of the stability margin relative to rounding errors. The book is suitable for graduate students in engineering and/or applied mathematics, academic researchers and governmental laboratories.
This special volume contains the Proceedings of a Workshop on "Parallel Algorithms and Transputers for Optimization" which was held at the University of Siegen, on November 9, 1990. The purpose of the Workshop was to bring together those doing research on 2.lgorithms for parallel and distributed optimization and those representatives from industry and business who have an increasing demand for computing power and who may be the potential users of nonsequential approaches. In contrast to many other conferences, especially North-American, on parallel processing and supercomputers the main focus of the contributions and discussion was "problem oriented". This view reflects the following philosophy: How can the existing computing infrastructure (PC's, workstations, local area networks) of an institution or a company be used for parallel and/or distributed problem solution in optimization. This volume of the LECfURE NOTES ON ECONOMICS AND MA THEMA TICAL SYSTEMS contains most of the papers presented at the workshop, plus some additional invited papers covering other important topics related to this workshop. The papers appear here grouped according to four general areas. (1) Solution of optimization problems using massive parallel systems (data parallelism). The authors of these papers are: Lootsma; Gehne. (II) Solution of optimization problems using coarse-grained parallel approaches on multiprocessor systems (control parallelism). The authors of these papers are: Bierwirth, Mattfeld, and Stoppler; Schwartz; Boden, Gehne, and Grauer; and Taudes and Netousek.
This book is a collection of 34 papers presented by leading researchers at the International Workshop on Robust Control held in San Antonio, Texas in March 1991. The common theme tying these papers together is the analysis, synthesis, and design of control systems subject to various uncertainties. The papers describe the latest results in parametric understanding, H8 uncertainty, l1 optical control, and Quantitative Feedback Theory (QFT). The book is the first to bring together all the diverse points of view addressing the robust control problem and should strongly influence development in the robust control field for years to come. For this reason, control theorists, engineers, and applied mathematicians should consider it a crucial acquisition for their libraries.
In this book quantitative approaches are proposed for production planning problems in automated manufacturing. In particular techniques from operations research/combinatorial optimization provide ways to tackle these problems. Special attention is devoted to the efficient use of tools in production planning for automated manufacturing systems. The book presents models and tests solution strategies for different kinds of production decisions. A case study in the manufacturing of printed circuit boards highlights the methodology. This book will help understand the nature of production planning problems emerging in automated manufacturing and show how techniques from operations research may contribute to their solution.
Within a project human and non-human resources are pulled together in a tempo raray organization in order to achieve a predefined goal (d. [20], p. 187). That is, in contrast to manufacturing management, project management is directed to an end. One major function of project management is the scheduling of the project. Project scheduling is the time-based arrangement of the activities comprising the project subject to precedence-, time-and resource-constraints (d. [4], p. 170). In the 1950's the standard methods MPM (Metra Potential Method) and CPM (Cri tical Path Method) were developed. Given deterministic durations and precedence constraints the minimum project length, time windows for the start times and critical paths can be calculated. At the same time another group of researchers developed the Program Evaluation and Review Technique (PERT) (d. [19], [73] and [90]). In contrast to MPM and CPM, random variables describe the activity durations. Based on the optimistic, most likely and pessimistic estimations of the activity durations an assumed Beta distribution is derived in order to calculate the distribution of the project duration, the critical events, the distribution of earliest and latest occurence of an event, the distribution of the slack of the events and the probability of exceeding a date. By the time the estimates of the distributions have been improved (d. e.g. [52] and [56]). Nevertheless, there are some points of critique concerning the estimation of the resulting distributions and probabilities (d. e.g. [48], [49] and [50]).
Many problems in economics can be formulated as linearly constrained mathematical optimization problems, where the feasible solution set X represents a convex polyhedral set. In practice, the set X frequently contains degenerate verti- ces, yielding diverse problems in the determination of an optimal solution as well as in postoptimal analysis.The so- called degeneracy graphs represent a useful tool for des- cribing and solving degeneracy problems. The study of dege- neracy graphs opens a new field of research with many theo- retical aspects and practical applications. The present pu- blication pursues two aims. On the one hand the theory of degeneracy graphs is developed generally, which will serve as a basis for further applications. On the other hand dege- neracy graphs will be used to explain simplex cycling, i.e. necessary and sufficient conditions for cycling will be de- rived.