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The Simplex Algorithm - Linear programming - Mathstools. The simplex algorithm seeks a solution between feasible region extreme points in linear programming problems which satisfies the optimality criterion. Simplex algorithm is based in an operation called pivots the matrix what it is precisely this iteration between the set of extreme points. The Simplex Algorithm output reduced to one of these 4 cases: unique finite optimal solution, unbounded solution, solution infinite and inconsistent restrictions.