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In the previous part we implemented and tested the simplex method on a simple example, and it has executed without any problems. In the first part, we have seen an example of the unbounded linear program.What will happen if we apply the simplex algorithm for it?Let’s list some of the common pivot rules: It may happen that for some linear programs the simplex method cycles and theoretically, this is the only possibility of how it may fail.
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.
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.
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 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.
There is a straightforward process to convert any linear program into one in standard form, so using this form of linear programs results in no loss of generality.
In geometric terms, the feasible region defined by all values of is a (possibly unbounded) convex polytope.