4 A linear programming problem consists of maximising an objective function \(P\) involving three variables, \(x , y\) and \(z\), subject to constraints given by three inequalities other than \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\). Slack variables \(s , t\) and \(u\) are introduced and the Simplex method is used to solve the problem. One iteration of the method leads to the following tableau.
| \(\boldsymbol { P }\) | \(\boldsymbol { x }\) | \(\boldsymbol { y }\) | \(\boldsymbol { Z }\) | \(\boldsymbol { s }\) | \(\boldsymbol { t }\) | \(\boldsymbol { u }\) | value |
| 1 | -2 | 11 | 0 | 3 | 0 | 0 | 6 |
| 0 | 2 | 3 | 1 | 1 | 0 | 0 | 2 |
| 0 | 6 | -30 | 0 | -6 | 1 | 0 | 3 |
| 0 | -1 | -9 | 0 | -3 | 0 | 1 | 4 |
- State the column from which the pivot for the next iteration should be chosen. Identify this pivot and explain the reason for your choice.
- Perform the next iteration of the Simplex method.
- Explain why you know that the maximum value of \(P\) has been achieved.
- State how many of the three original inequalities still have slack.
- State the maximum value of \(P\) and the values of \(x , y\) and \(z\) that produce this maximum value.
- The objective function for this problem is \(P = k x - 2 y + 3 z\), where \(k\) is a constant. Find the value of \(k\).
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