5. A three-variable linear programming problem in \(x , y\) and \(z\) is to be solved. The objective is to maximise the profit, \(P\).
The following tableau is obtained.
| Basic variable | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value |
| \(r\) | \(\frac { 1 } { 2 }\) | \(- \frac { 1 } { 2 }\) | 0 | 1 | 0 | \(- \frac { 1 } { 2 }\) | 10 |
| \(s\) | \(1 \frac { 1 } { 2 }\) | \(2 \frac { 1 } { 2 }\) | 0 | 0 | 1 | \(- \frac { 1 } { 2 }\) | 5 |
| \(z\) | \(\frac { 1 } { 2 }\) | \(\frac { 1 } { 2 }\) | 1 | 0 | 0 | \(\frac { 1 } { 2 }\) | 5 |
| \(P\) | -5 | -10 | 0 | 0 | 0 | 20 | 220 |
- Starting by increasing \(y\), perform one complete iteration of the Simplex algorithm, to obtain a new tableau, T. State the row operations you use.
- Write down the profit equation given by T .
- Use the profit equation from part (b) to explain why T is optimal.