# Simplex Method Of Solving Linear Programming Problem

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.George Dantzig worked on planning methods for the US Army Air Force during World War II using a desk calculator.

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.George Dantzig worked on planning methods for the US Army Air Force during World War II using a desk calculator.

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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.

The solution of a linear program is accomplished in two steps.

In the first step, known as Phase I, a starting extreme point is found.

On this example, we can see that on first iteration objective function value made no gains.

In general, there might be longer runs of degenerate pivot steps.

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.

It may even happen that some tableau is repeated in a sequence of degenerate pivot steps.

It may even happen that some tableau is repeated in a sequence of degenerate pivot steps, and so the algorithm might pass through an infinite sequence of tableau without any progress. A pivot rule is a rule for selecting the entering variable if there are several possibilities, which is usually the case(in our algorithm determine this element).