3 A problem to maximise \(P\) as a function of \(x , y\) and \(z\) is represented by the initial Simplex tableau below.
| \(P\) | \(x\) | \(y\) | \(z\) | \(s\) | \(t\) | RHS |
| 1 | - 10 | 2 | 3 | 0 | 0 | 0 |
| 0 | 5 | 0 | - 5 | 1 | 0 | 60 |
| 0 | 4 | 3 | 0 | 0 | 1 | 100 |
- Write down \(P\) as a function of \(x , y\) and \(z\).
- Write down the constraints as inequalities involving \(x , y\) and \(z\).
- Carry out one iteration of the Simplex algorithm.
After a second iteration of the Simplex algorithm the tableau is as given below.
| \(P\) | \(x\) | \(y\) | \(z\) | \(s\) | \(t\) | RHS |
| 1 | 0 | 7.25 | 0 | 0.6 | 1.75 | 211 |
| 0 | 1 | 0.75 | 0 | 0 | 0.25 | 25 |
| 0 | 0 | 0.75 | 1 | - 0.2 | 0.25 | 13 |
- Explain how you know that the optimal solution has been achieved.
- Write down the values of \(x , y\) and \(z\) that maximise \(P\). Write down the optimal value of \(P\).