# Mathematical Programming

**Topics:**Optimization, Linear programming, Operations research

**Pages:**55 (18473 words)

**Published:**December 14, 2011

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 decision-making, and inﬁltrating every conceivable area of application, covering a wide variety of business, industrial, military, and public-sector 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 cost-effectiveness 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 linear-programming 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 linear-programming problem. The extraordinary computational efﬁciency and robustness of the simplex method, together with the availability of high-speed 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 mathematical-programming approach. It is this spectrum of techniques and their effective implementation in practice that are considered in this book. 1.1 AN INTRODUCTION TO MANAGEMENT SCIENCE

Since mathematical programming is only a tool of the broad discipline known as management science, let us ﬁrst attempt to understand the management-science approach and identify the role of mathematical programming within that approach. 1

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Mathematical Programming: An Overview

1.2

It is hard to give a noncontroversial deﬁnition of management science. As we have indicated before, this is a rather new ﬁeld that is renewing itself and changing constantly. It has beneﬁted from contributions originating in the social and natural sciences, econometrics, and mathematics, much of which escape the rigidity of a deﬁnition. Nonetheless it is possible to provide a general statement about the basic elements of the management-science approach. Management science is characterized by the use of mathematical models in providing guidelines to managers for making effective decisions within the state of the current information, or in seeking further information if current knowledge is insufﬁcient to reach a proper decision. There are several elements of this statement that are deserving of emphasis. First, the essence of management science is the...

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