Abstract
This chapter describes theory of simplex methods. To develop the simplex method, one should proceed with the standard form of the linear programming problem. In the presence of degeneracy, one cannot be sure that the simplex procedure will necessarily terminate in a finite number of steps. If a basic feasible solution is degenerate, that is one or more of the basic variables are zero, then θ0 is zero and the new solution obtained is again degenerate with no improvement in the value of the objective function. This may occur in several successive iterations and it is possible to return to a basis already obtained and then the procedure may cycle indefinitely. It is therefore desirable to develop a procedure to avoid degeneracy in the problem. The theorems also indicate how to proceed step by step so that the procedure converges to an optimal solution to the linear programming problem or determines that the problem has no optimal solution.
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