6. The tableau below is the initial tableau for a linear programming problem in \(x , y\) and \(z\). The objective is to maximise the profit, \(P\).
| Basic Variable | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value |
| \(r\) | 0 | 1 | 2 | 1 | 0 | 0 | 24 |
| \(s\) | 2 | 1 | 4 | 0 | 1 | 0 | 28 |
| \(t\) | -1 | \(\frac { 1 } { 2 }\) | 3 | 0 | 0 | 1 | 22 |
| \(P\) | -1 | -2 | -6 | 0 | 0 | 0 | 0 |
- Write down the profit equation represented in the initial tableau.
(1) - Taking the most negative number in the profit row to indicate the pivot column at each stage, solve this linear programming problem. Make your method clear by stating the row operations you use.
- State the final value of the objective function and of each variable.
(3)