Simplex Method

Abstract

The general linear programming problem can be formulated as follows: minimize the function

$$ f(x) = {c^1}{x^1} + {c^2}{x^2} + ... + {c^n}{x^n} $$

((1.1.1))

under the conditions

$$ {x^k} \ge 0,k \in {I_+ } $$

((1.1.2))

,

$$ \left. \begin{array}{l}{a_{11}}{x^1} + {a_{12}}{x^2} + .... + {a_{1n}}{x^n} \le {b^1}, \\........................................... \\{a_{m1}}{x^1} + {a_{m2}}{x^2} + ... + {a_{mn}}{x^n} \le {b^m}, \\\end{array} \right\} $$

((1.1.3))

$$ \left. \begin{array}{l}{a_{m + 1,1}}{x^1} + {a_{m + 1,2}}{x^2} + ... + {a_{m + 1,n}}{x^n} \le {b^{m + 1}}, \\........................................................... \\{a_{s1}}{x^1} + {a_{s2}}{x^2} + ... + {a_{sn}}{x^n} \le {b^s} \\\end{array} \right\} $$

((1.1.4))

where ${c^j},{b^i},{a_{ij}},i = \overline {1,s} ,j = \overline {1,n}$ are given numbers, I ₊ is the given subset of indices from the set {1,2,... , n}. The function (1.1.1) is known as an objective function, conditions (1.1.3) are constraints of the type of inequalities, conditions (1.1.4) are constraints of the type of equalities. Conditions (1.1.2) of nonnegativity of the variables are, of course, also constraints of the type of inequalities, but it is customary to consider them separately. Problem (1.1.1)-(1.1.4) may include cases where ${I_ + } = \emptyset or{I_ + } = \{ 1,2,...,n\}$ it is also possible that problem (1.1.1)-(1.1.4) may not contain constraints of the type of inequalities or equalities. We call the point x = (x ¹, ..., x ⁿ) which satisfies all conditions (1.1.2)–(1.1.4) an admissible point of problem (1.1.1)–(1.1.4) or simply an admissible point.