Solving Linear Programming Problems Using Simplex Method

Solving Linear Programming Problems Using Simplex Method-72
Dantzig later published his "homework" as a thesis to earn his doctorate.The column geometry used in this thesis gave Dantzig insight that made him believe that the Simplex method would be very efficient.The algorithm always terminates because the number of vertices in the polytope is finite; moreover since we jump between vertices always in the same direction (that of the objective function), we hope that the number of vertices visited will be small.

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It can also be shown that, if an extreme point is not a maximum point of the objective function, then there is an edge containing the point so that the objective function is strictly increasing on the edge moving away from the point.

If the edge is finite, then the edge connects to another extreme point where the objective function has a greater value, otherwise the objective function is unbounded above on the edge and the linear program has no solution.

In geometric terms, the feasible region defined by all values of is a (possibly unbounded) convex polytope.

An extreme point or vertex of this polytope is known as basic feasible solution (BFS).

Dantzig's core insight was to realize that most such ground rules can be translated into a linear objective function that needs to be maximized.

After Dantzig included an objective function as part of his formulation during mid-1947, the problem was mathematically more tractable.The shape of this polytope is defined by the constraints applied to the objective function.A system of linear inequalities defines a polytope as a feasible region.In the first step, known as Phase I, a starting extreme point is found.Depending on the nature of the program this may be trivial, but in general it can be solved by applying the simplex algorithm to a modified version of the original program.In the second step, Phase II, the simplex algorithm is applied using the basic feasible solution found in Phase I as a starting point.The possible results from Phase II are either an optimum basic feasible solution or an infinite edge on which the objective function is unbounded above.By continuing to use this site, you consent to the use of cookies.We use cookies to offer you a better experience, personalize content, tailor advertising, provide social media features, and better understand the use of our services.Dantzig realized that one of the unsolved problems that he had mistaken as homework in his professor Jerzy Neyman's class (and actually later solved), was applicable to finding an algorithm for linear programs.This problem involved finding the existence of Lagrange multipliers for general linear programs over a continuum of variables, each bounded between zero and one, and satisfying linear constraints expressed in the form of Lebesgue integrals.

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