4 A linear programming problem involving variables \(x\) and \(y\) is to be solved. The objective function to be maximised is \(P = 3 x + 5 y\). The initial Simplex tableau is given below.
| \(\boldsymbol { P }\) | \(\boldsymbol { x }\) | \(\boldsymbol { y }\) | \(\boldsymbol { s }\) | \(\boldsymbol { t }\) | \(\boldsymbol { u }\) | value |
| 1 | - 3 | - 5 | 0 | 0 | 0 | 0 |
| 0 | 1 | 2 | 1 | 0 | 0 | 36 |
| 0 | 1 | 1 | 0 | 1 | 0 | 20 |
| 0 | 4 | 1 | 0 | 0 | 1 | 39 |
- In addition to \(x \geqslant 0 , y \geqslant 0\), write down three inequalities involving \(x\) and \(y\) for this problem.
- By choosing the first pivot from the \(\boldsymbol { y }\)-column, perform one iteration of the Simplex method.
- Explain how you know that the optimal value has not been reached.
- Perform one further iteration.
- Interpret the final tableau and state the values of the slack variables.