8. The tableau below is the initial tableau for a linear programming problem in \(x , y\) and \(z\). The objective is to maximise the profit, \(P\).
| basic variable | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value |
| \(r\) | 12 | 4 | 5 | 1 | 0 | 0 | 246 |
| \(s\) | 9 | 6 | 3 | 0 | 1 | 0 | 153 |
| \(t\) | 5 | 2 | - 2 | 0 | 0 | 1 | 171 |
| \(P\) | - 2 | - 4 | - 3 | 0 | 0 | 0 | 0 |
Using the information in the tableau, write down
- the objective function,
- the three constraints as inequalities with integer coefficients.
Taking the most negative number in the profit row to indicate the pivot column at each stage,
- solve this linear programming problem. Make your method clear by stating the row operations you use.
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| b.v. | x | y | z | r | s | t | Value | Row operations |
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| b.v. | x | y | z | r | s | t | Value | Row operations |
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- State the final values of the objective function and each variable.
- One of the constraints is not at capacity. Explain how it can be identified.