4 A linear programming problem involving variables \(x , y\) and \(z\) is to be solved. The objective function to be maximised is \(P = 2 x + 6 y + k z\), where \(k\) is a constant.
The initial Simplex tableau is given below.
| \(\boldsymbol { P }\) | \(x\) | \(y\) | \(\boldsymbol { Z }\) | \(\boldsymbol { s }\) | \(\boldsymbol { t }\) | \(\boldsymbol { u }\) | value |
| 1 | -2 | -6 | \(- k\) | 0 | 0 | 0 | 0 |
| 0 | 5 | 3 | 10 | 1 | 0 | 0 | 15 |
| 0 | 7 | 6 | 4 | 0 | 1 | 0 | 28 |
| 0 | 4 | 3 | 6 | 0 | 0 | 1 | 12 |
- 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 \(\boldsymbol { y }\)-column, perform one iteration of the Simplex method.
- Given that the optimal value has not been reached, find the possible values of \(k\).
- In the case when \(k = 20\) :
- perform one further iteration;
- interpret the final tableau and state the values of the slack variables.