6 Consider the linear programming problem:
| maximise | \(P = 2 x - 5 y - z\), |
| subject to | \(5 x + 3 y - 5 z \leqslant 15\), |
| \(2 x + 6 y + 8 z \leqslant 24\), |
| and | \(x \geqslant 0 , y \geqslant 0 , z \geqslant 0\). |
- Using slack variables, \(s\) and \(t\), express the non-trivial constraints as two equations.
- Represent the problem as an initial Simplex tableau. Perform one iteration of the Simplex algorithm.
- Use the Simplex algorithm to find the values of \(x , y\) and \(z\) for which \(P\) is maximised, subject to the constraints above.
- The value 15 in the first constraint is increased to a new value \(k\). As a result the pivot for the first iteration changes. Show what effect this has on the final value of \(y\).