4 A linear programming problem consists of maximising an objective function \(P\) involving three variables \(x , y\) and \(z\). Slack variables \(s , t , u\) and \(v\) are introduced and the Simplex method is used to solve the problem. Several iterations of the method lead to the following tableau.
| \(\boldsymbol { P }\) | \(x\) | \(y\) | \(\boldsymbol { Z }\) | \(\boldsymbol { s }\) | \(\boldsymbol { t }\) | \(\boldsymbol { u }\) | \(v\) | value |
| 1 | 0 | -12 | 0 | 5 | -3 | 0 | 0 | 37 |
| 0 | 1 | -8 | 0 | 1 | 2 | 0 | 0 | 16 |
| 0 | 0 | 4 | 0 | 0 | 3 | 0 | 1 | 20 |
| 0 | 0 | 2 | 0 | -3 | 2 | 1 | 0 | 14 |
| 0 | 0 | 1 | 1 | 2 | 5 | 0 | 0 | 8 |
- The pivot for the next iteration is chosen from the \(\boldsymbol { y }\)-column. State which value should be chosen and explain the reason for your choice.
- Perform the next iteration of the Simplex method.
- Explain why your new tableau solves the original problem.
- State the maximum value of \(P\) and the values of \(x , y\) and \(z\) that produce this maximum value.
- State the values of the slack variables at the optimum point. Hence determine how many of the original inequalities still have some slack when the optimum is reached.