Simplex Method and Linear Programming

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Simplex Method and Linear Programming The simplex method invented by George B. Dantzig in 1947 is one of the most impressive developments in computational optimization. Dantzig wrote “Linear programming is viewed as a revolutionary development giving man the ability to state general objectives and to find, by means of the simplex method, optimal policy decisions for a broad class of practical decision problems of great complexity. In the real world, planning tends to be ad hoc because of the many special-interest groups with their multiple objectives” (Dantzig, 1981). Dantzig describes this in his book Linear programming and extensions (1963) that “One of the first applications of the simplex algorithm was to the determination of an adequate diet that was of least cost” (Dantzig, 1963). The particular problem proposed was a solution based on the substitution of certain foods by others that gave more nutrition per dollar. Simplex Method The simplex method is composed of a matrix and row operations used to modify a linear programming problem. The simplex method can solve large linear programming problems without needing to solve all of the solutions that are obtainable. Constraints are the required numbers that the problem has to fit. Constraints are translated into linear equations with the use of slack variables. Slack variables absorb the "slack" between each side of the equation (Barnett, Ziegler, & Byleen, 2011). Simplex Problem To understand how the simplex method is used by a business to determine costs and efficiency of operation, the CEO of a dairy cooperative must determine the objective function and constraints of the problem. The dairy cooperative has 3 dairy farms that produce white milk and chocolate milk. Dairy farm C is able to produce 40 quarts of white milk and 20 quarts of chocolate milk per hour of operation. Dairy farm D is able

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