ornotes1 - Introduction to Mathematical Programming AGEC...

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Introduction to Mathematical Programming AGEC 7100 (Notes, Set 1) Mathematical programming problems generally involve the allocation of scarce resources to achieve some objective. Because economics is often defined in similar terms, mathematical programming models have been used extensively in economic applications, such as: 1) Maximizing the profit of a business 2) Minimizing the cost of producing a given level of output 3) Finding the "best" (least cost or least time) transportation model 4) Analyzing producer responses to policy incentives or constraints In general terms, mathematical programming models will include a set of 1) DECISION VARIABLES (variables whose values can change), 2) an OBJECTIVE FUNCTION , to be maximized or minimized by changing the level of the decision variables, and 3) a set of CONSTRAINTS that limit the decision process in some way. Operations Research involves a systematic and scientific approach to decision making. There are several different types of mathematical programming models including: Linear Programming Models, Integer Programming Models, Quadratic Programming Models, Nonlinear Programming Models, and Dynamic Programming Models. Each type of model involves different assumptions about the nature of the objective function, the decision variables, or the restrictions. Some History Although some of the mathematical theory underlying much of mathematical programming can be traced back to earlier origins, it is generally agreed that the modern approach to programming began during the World War II era. Hence, mathematical programming is a recent development in mathematics. Because of the complexity of problems during World War II and the pressing need to allocate scarce resources to various military operations in the most efficient way, teams of scientists and mathematicians from various backgrounds were brought together first in Great Britain and then in other countries.
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Because the focus of their work was on military operations, the emerging discipline became known as "operations research." The efforts of these teams were credited with the success of several important allied military campaigns. At the end of World War II, those involved in the industrial boom saw the possibility of using these military operations research methods for other purposes. Also, many of the scientists who started the work in this field during the war were interested in continuing their work. In particular, in 1947, George Dantzig developed the simplex method for solving linear programming models. Many of the standard tools of operations research were developed before the end of the 1950s. Without the advent of the "computer age" operations research applications would have remained limited in size and complexity. High speed computing has allowed us to solve larger, more complicated problems using the theories and techniques developed by Dantzig and his colleagues (much of the following material is modified from Dr. McCarl's lecture notes) Basic Optimization Problem, Expressed Mathematically Optimize F(X) Subject To (s.t.) G(X)
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