4 A linear programming problem involving the variables \(x , y\) and \(z\) is to be solved. The objective function to be maximised is \(P = 2 x + 3 y + 5 z\). The initial Simplex tableau is given below.
| \(\boldsymbol { P }\) | \(\boldsymbol { x }\) | \(\boldsymbol { y }\) | \(\boldsymbol { z }\) | \(s\) | \(\boldsymbol { t }\) | \(\boldsymbol { u }\) | value |
| 1 | -2 | -3 | -5 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 1 | 0 | 0 | 9 |
| 0 | 2 | 1 | 4 | 0 | 1 | 0 | 40 |
| 0 | 4 | 2 | 3 | 0 | 0 | 1 | 33 |
- In addition to \(x \geqslant 0 , y \geqslant 0 , z \geqslant 0\), write down three inequalities involving \(x , y\) and \(z\) for this problem.
- By choosing the first pivot from the \(z\)-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.