4. The tableau below is the initial tableau for a three-variable linear programming problem in \(x , y\) and \(z\). The objective is to maximise the profit, \(P\).
| Basic Variable | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value |
| \(r\) | 4 | 3 | \(\frac { 5 } { 2 }\) | 1 | 0 | 0 | 50 |
| \(s\) | 1 | 2 | 1 | 0 | 1 | 0 | 30 |
| \(t\) | 0 | 5 | 1 | 0 | 0 | 1 | 80 |
| \(P\) | - 25 | - 40 | - 35 | 0 | 0 | 0 | 0 |
- Taking the most negative number in the profit row to indicate the pivot column at each stage, perform two complete iterations of the simplex algorithm to obtain tableau T. Make your method clear by stating the row operations you use.
- Write down the profit equation given by T .
- Use your answer to (b) to determine whether T is optimal, justifying your answer.