4. The tableau below is the initial tableau for a maximising linear programming problem in \(x , y\) and \(z\) which is to be solved.
| Basic variable | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value |
| \(r\) | 5 | \(\frac { 1 } { 2 }\) | 0 | 1 | 0 | 0 | 5 |
| \(s\) | 1 | -2 | 4 | 0 | 1 | 0 | 3 |
| \(t\) | 8 | 4 | 6 | 0 | 0 | 1 | 6 |
| \(P\) | -5 | -7 | -4 | 0 | 0 | 0 | 0 |
- Starting by increasing \(y\), perform one complete iteration of the simplex algorithm, to obtain tableau T. State the row operations you use.
- Write down the profit equation given by tableau T .
- Use the profit equation from part (b) to explain why tableau T is optimal.