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Unformatted text preview: Mathematical Programming: An Overview 1 Management science is characterized by a scientiﬁc approach to managerial decision making. It attempts to apply mathematical methods and the capabilities of modern computers to the difﬁcult and unstructured problems confronting modern managers. It is a young and novel discipline. Although its roots can be traced back to problems posed by early civilizations, it was not until World War II that it became identiﬁed as a respectable and well deﬁned body of knowledge. Since then, it has grown at an impressive pace, unprecedented for most scientiﬁc accomplishments; it is changing our attitudes toward decisionmaking, and inﬁltrating every conceivable area of application, covering a wide variety of business, industrial, military, and publicsector problems. Management science has been known by a variety of other names. In the United States, operations research has served as a synonym and it is used widely today, while in Britain operational research seems to be the more accepted name. Some people tend to identify the scientiﬁc approach to managerial problemsolving under such other names as systems analysis, cost–beneﬁt analysis, and costeffectiveness analysis. We will adhere to management science throughout this book. Mathematical programming, and especially linear programming, is one of the best developed and most used branches of management science. It concerns the optimum allocation of limited resources among competing activities, under a set of constraints imposed by the nature of the problem being studied. These constraints could reﬂect ﬁnancial, technological, marketing, organizational, or many other considerations. In broad terms, mathematical programming can be deﬁned as a mathematical representation aimed at programming or planning the best possible allocation of scarce resources. When the mathematical representation uses linear functions exclusively, we have a linearprogramming model. In 1947, George B. Dantzig, then part of a research group of the U.S. Air Force known as Project SCOOP (Scientiﬁc Computation Of Optimum Programs), developed the simplex method for solving the general linearprogramming problem. The extraordinary computational efﬁciency and robustness of the simplex method, together with the availability of highspeed digital computers, have made linear programming the most powerful optimization method ever designed and the most widely applied in the business environment. Since then, many additional techniques have been developed, which relax the assumptions of the linearprogramming model and broaden the applications of the mathematicalprogramming approach. It is this spectrum of techniques and their effective implementation in practice that are considered in this book....
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 Spring '09
 mathematicalprogrammibg
 Operations Research, Linear Programming, Optimization, objective function, optimal solution

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