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It can be shown that for a linear program in standard form, if the objective function has a maximum value on the feasible region, then it has this value on at least one of the extreme points.
The simplex algorithm applies this insight by walking along edges of the polytope to extreme points with greater and greater objective values. This continues until the maximum value is reached, or an unbounded edge is visited concluding that the problem has no solution. 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 functionwe hope that the number of vertices visited will be small.
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.
The possible results of Phase I are either that a basic feasible solution is found or that the feasible region is empty. In the latter case the linear program is called infeasible.
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 below.
During his colleague challenged him to mechanize the planning process to distract him from taking another job.
Dantzig formulated the problem as linear inequalities inspired by the work of Wassily Leontiefhowever, at that time he didn't include an objective as part of his formulation.
Without an objective, a vast number of solutions can be feasible, and therefore to find the "best" feasible solution, military-specified "ground rules" must be used that describe how goals can be achieved as opposed to specifying a goal itself.
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.
Dantzig realized that one of the unsolved problems that he had mistaken as homework in his professor Jerzy Neyman 's class and actually later solvedwas 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.
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 original variable can then be eliminated by substitution.
For example, given the constraint x.Since there are only two variables, we can solve this problem by graphing the set of points in the plane that satisﬁes all the constraints (called the constraint set) and then ﬁnding which point of this set maximizes the value of the objective function.
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Usually I have been asked to write problems in standard form that have inequalities involved. However, this problem has none and I was wondering if anyone had insight on how to go about solving it.
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