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 tableau is obtained after the first iteration.
| Basic Variable | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value |
| r | 0 | 5 | 2 | 1 | -3 | 0 | 10 |
| \(x\) | 1 | 2 | 3 | 0 | 1 | 0 | 18 |
| \(t\) | 0 | 1 | -1 | 0 | 4 | 1 | 3 |
| \(P\) | 0 | 3 | -4 | 0 | 1 | 0 | 7 |
- State which variable was increased first, giving a reason for your answer.
- Perform one complete iteration of the simplex algorithm, to obtain a new tableau, T. Make your method clear by stating the row operations you use.
- Write down the profit equation given by T .
- State whether T is optimal. You must use your answer to (c) to justify your answer.