Abstract
As we have shown, the simplex method refers to the so-called finite methods which allow one to find a solution for any problem of linear programming or to prove its unsolvability performing a finite number of elementary operations (addition, subtraction, multiplication, division, comparison of two real numbers). It stands to reason that the number of operations depends on the dimension (n, m) of the problem (n is the number of variables, m is the number of equality and inequality constraints). The practice of solving linear programming problems has shown that the simplex method and its modifications are very effective. It is accepted as a fact that in the majority of linear programming problems the number of elementary operations which are necessary for their solution is of the order O(n 2 m + m 2 n) [1, 91]. Here and in the sequel we denote by O(a) the quantities for which |O(a)| ≤ C(a), where C is a positive constant independent of a. We have found out that there exist “poor” linear programming problems in which the amount of elementary operations required for their solution by the simplex method is estimated by the number which exponentially depends on n and m.
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