The pivot row is the row that has the smallest non-negative ratio.If no non-negative ratios can be found, stop, the problem doesn't have a solution.That means that variable is exiting the set of basic variables and becoming non-basic. Now that we have a direction picked, we need to determine how far we should move in that direction.
If the column is cleared out and has only one non-zero element in it, then that variable is a basic variable.
If a column is not cleared out and has more than one non-zero element in it, that variable is non-basic and the value of that variable is zero.
What will happen if we apply the simplex algorithm for it?
On this example, we can see that on first iteration objective function value made no gains.
For the columns that are cleared out and have only one non-zero entry in them, you go down the column until you find the non-zero entry.
Each column will have it's non-zero element in a different row.Since we're trying to maximize the value of the objective function, that would be counter-productive. As the independent terms of all restrictions are positive no further action is required.In general, there might be longer runs of degenerate pivot steps.It may even happen that some tableau is repeated in a sequence of degenerate pivot steps.Now, think about how that 40 is represented in the objective function of the tableau.When we placed the objective function into the tableau, we moved the decision variables and their coefficients to the left hand side and made them negative.If we move any more than 8, we're leaving the feasible region.Therefore, we have to move the smallest distance possible to stay within the feasible region.Therefore, the most negative number in the bottom row corresponds to the most positive coefficient in the objective function and indicates the direction we should head.The pivot column is the column with the most negative number in its bottom row.