3 An initial simplex tableau is given below.
| \(P\) | \(x\) | \(y\) | \(z\) | \(s\) | \(t\) | RHS |
| 1 | - 2 | 3 | - 1 | 0 | 0 | 0 |
| 0 | 5 | - 4 | 1 | 1 | 0 | 20 |
| 0 | 2 | - 1 | 0 | 0 | 1 | 6 |
- Carry out two iterations of the simplex algorithm, choosing the first pivot from the \(x\) column.
After three iterations the resulting tableau is as follows.
| \(P\) | \(x\) | \(y\) | \(z\) | \(s\) | \(t\) | RHS |
| 1 | 3 | - 1 | 0 | 1 | 0 | 20 |
| 0 | 5 | - 4 | 1 | 1 | 0 | 20 |
| 0 | 2 | - 1 | 0 | 0 | 1 | 6 |
- State the values of \(P , x , y , z , s\) and \(t\) that result from these three iterations.
- Explain why no further iterations are possible.
The initial simplex tableau is changed to the following, where \(k\) is a positive real value.
| \(P\) | \(x\) | \(y\) | \(z\) | \(s\) | \(t\) | RHS |
| 1 | 2 | - 3 | 1 | 0 | 0 | 0 |
| 0 | 5 | \(k\) | 1 | 1 | 0 | 20 |
| 0 | 2 | - 1 | 0 | 0 | 1 | 6 |
- Deduce the value of \(k\).