5 Consider the linear programming problem:
| maximise | \(P = x - 2 y - 3 z\), |
| subject to | \(2 x - 5 y + 2 z \leqslant 10\), |
| \(2 x \quad + 3 z \leqslant 30\), |
| and | \(x \geqslant 0 , y \geqslant 0 , z \geqslant 0\). |
- Using slack variables, \(s \geqslant 0\) and \(t \geqslant 0\), express the two non-trivial constraints as equations.
- Represent the problem as an initial Simplex tableau.
- Explain why the pivot element must be chosen from the \(x\)-column and show the calculations that are used to choose the pivot.
- Perform one iteration of the Simplex algorithm. Show how you obtained each row of your tableau and write down the values of \(x , y , z\) and \(P\) that result from this iteration. State whether or not this is the maximum feasible value of \(P\) and describe how you know this from the values in the tableau.