4 The Simplex method is to be used to maximise \(P = 3 x + 2 y + z\) subject to the constraints
$$\begin{aligned}
- x + y + z & \leqslant 4
2 x + y + 4 z & \leqslant 10
4 x + 2 y + 3 z & \leqslant 21
\end{aligned}$$
The initial Simplex tableau is given below.
| \(\boldsymbol { P }\) | \(\boldsymbol { x }\) | \(\boldsymbol { y }\) | \(\boldsymbol { z }\) | \(s\) | \(t\) | \(\boldsymbol { u }\) | value |
| 1 | -3 | -2 | -1 | 0 | 0 | 0 | 0 |
| 0 | -1 | 1 | 1 | 1 | 0 | 0 | 4 |
| 0 | 2 | 1 | 4 | 0 | 1 | 0 | 10 |
| 0 | 4 | 2 | 3 | 0 | 0 | 1 | 21 |
- The first pivot is to be chosen from the \(x\)-column. Identify the pivot and explain why this particular value is chosen.
- Perform one iteration of the Simplex method and explain how you know that the optimal value has not been reached.
- Perform one further iteration.
- Interpret the final tableau and write down the initial inequality that still has slack.
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