3. A three-variable linear programming problem in \(x , y\) and \(z\) is to be solved. The objective is to maximise the profit, \(P\).
The following tableau is obtained.
| Basic variable | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value |
| \(r\) | \(- \frac { 1 } { 2 }\) | 0 | 2 | 1 | \(- \frac { 1 } { 2 }\) | 0 | 10 |
| \(y\) | \(\frac { 1 } { 2 }\) | 1 | \(\frac { 3 } { 4 }\) | 0 | \(\frac { 1 } { 4 }\) | 0 | 5 |
| \(t\) | \(\frac { 1 } { 2 }\) | 0 | 1 | 0 | \(- \frac { 1 } { 4 }\) | 1 | 4 |
| \(P\) | - 7 | 0 | 1 | 0 | 4 | 0 | 320 |
- Write down the profit equation represented in the tableau.
- 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.
- State the value of the objective function and of each variable.