Prove that in the simplex method a variable which has just l

Prove that in the simplex method a variable which has just left the basis cannot re-enter the basis in the following iteration.

Solution

The simplex method is mainly concentrating on finding the minimum function or the minimizing the value. Let us suppose we enter a variable in the objective row that make an objective row of the function to be negative (<0) in order to minimize the function value and hence giving an optimum result. Now as soon as the variable will leave the function or its basis, the coefficient of the objective function in that particular row will become non-negative, hence in the next iteration the only permissible value will be the one which will lead to negative function objective value, hence the variable just left the basis cannot re-enter the basis otherwise it will not decrease the value but the increase the objective function of the row