4 A linear programming problem involving variables \(x , y\) and \(z\) is to be solved. The objective function to be maximised is \(P = 4 x + y + k z\), where \(k\) is a constant. The initial Simplex tableau is given below.
| \(\boldsymbol { P }\) | \(\boldsymbol { x }\) | \(\boldsymbol { y }\) | \(\boldsymbol { z }\) | \(s\) | \(\boldsymbol { t }\) | value |
| 1 | -4 | -1 | \(- k\) | 0 | 0 | 0 |
| 0 | 1 | 2 | 3 | 1 | 0 | 7 |
| 0 | 2 | 1 | 4 | 0 | 1 | 10 |
- In addition to \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\), write down two inequalities involving \(x , y\) and \(z\) for this problem.
- The first pivot is chosen from the \(\boldsymbol { x }\)-column. Identify the pivot and perform one iteration of the Simplex method.
- Given that the optimal value of \(P\) has not been reached after this first iteration, find the possible values of \(k\).
- Given that \(k = 10\) :
- perform one further iteration of the Simplex method;
- interpret the final tableau.