Pivot Operations
The geometrical operation of moving from a basic feasible solution to an adjacent basic feasible solution is implemented as a pivot operation. First, a nonzero pivot element is selected in a nonbasic column. The row containing this element is multiplied by its reciprocal to change this element to 1, and then multiples of the row are added to the other rows to change the other entries in the column to 0. The result is that, if the pivot element is in row r, then the column becomes the r-th column of the identity matrix. The variable for this column is now a basic variable, replacing the variable which corresponded to the r-th column of the identity matrix before the operation. In effect, the variable corresponding to the pivot column enters the set of basic variables and is called the entering variable, and the variable being replaced leaves the set of basic variables and is called the leaving variable. The tableau is still in canonical form but with the set of basic variables changed by one element.
Read more about this topic: Simplex Algorithm
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