4 A linear programming problem involving variables \(x , y\) and \(z\) is to be solved. The objective function to be maximised is \(P = 2 x + 4 y + 3 z\). The initial Simplex tableau is given below.
| \(\boldsymbol { P }\) | \(\boldsymbol { x }\) | \(\boldsymbol { y }\) | \(\boldsymbol { Z }\) | \(s\) | \(t\) | \(\boldsymbol { u }\) | value |
| 1 | -2 | -4 | -3 | 0 | 0 | 0 | 0 |
| 0 | 2 | 2 | 1 | 1 | 0 | 0 | 14 |
| 0 | -1 | 1 | 2 | 0 | 1 | 0 | 6 |
| 0 | 4 | 4 | 3 | 0 | 0 | 1 | 29 |
- What name is given to the variables \(s , t\) and \(u\) ?
- Write down an equation involving \(x , y , z\) and \(s\) 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, stating the values of \(P , x , y\) and \(z\).