4. A three-variable linear programming problem in \(x , y\) and \(z\) is to be solved. The objective is to maximise the profit \(P\). The following initial tableau was obtained.
| Basic variable | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | Value |
| \(r\) | 2 | 0 | 4 | 1 | 0 | 80 |
| \(s\) | 1 | 4 | 2 | 0 | 1 | 160 |
| \(P\) | - 2 | - 8 | - 20 | 0 | 0 | 0 |
- Taking the most negative number in the profit row to indicate the pivot column, perform one complete iteration of the simplex algorithm, to obtain tableau \(T\). State the row operations that you use.
- Write down the profit equation shown in tableau \(T\).
- State whether tableau \(T\) is optimal. Give a reason for your answer.