4 Consider the following LP problem.
| Maximise | \(P = - 3 w + 5 x - 7 y + 2 z\), |
| subject to | \(w + 2 x - 2 y - z \leqslant 10\), |
| \(2 w + 3 y - 4 z \leqslant 12\), |
| and | \(4 w + 5 x + y \leqslant 30\), |
| \(w \geqslant 0 , x \geqslant 0 , y \geqslant 0 , z \geqslant 0\). |
- Represent the problem as an initial Simplex tableau. Explain why the pivot can only be chosen from the \(x\) column.
- Perform one iteration of the Simplex algorithm. Show how each row was obtained and write down the values of \(w , x , y , z\) and \(P\) at this stage.
- Perform a second iteration of the Simplex algorithm. Write down the values of \(w , x , y , z\) and \(P\) at this stage and explain how you can tell from this tableau that \(P\) can be increased without limit. How could you have known from the LP formulation above that \(P\) could be increased without limit?