7. A maximisation linear programming problem in \(x , y\) and \(z\) is to be solved using the Simplex method.
The tableau after the 1st iteration is shown below.
| b.v. | \(x\) | \(y\) | \(z\) | \(s _ { 1 }\) | \(S _ { 2 }\) | \(S _ { 3 }\) | Value |
| \(s _ { 1 }\) | 0 | \(- \frac { 1 } { 2 }\) | \(\frac { 3 } { 2 }\) | 1 | \(- \frac { 1 } { 2 }\) | 0 | 30 |
| \(x\) | 1 | \(\frac { 1 } { 4 }\) | \(- \frac { 1 } { 4 }\) | 0 | \(\frac { 1 } { 4 }\) | 0 | 10 |
| \(S _ { 3 }\) | 0 | 1 | 1 | 0 | 0 | 1 | 26 |
| \(P\) | 0 | \(- \frac { 1 } { 4 }\) | \(- \frac { 11 } { 4 }\) | 0 | \(\frac { 3 } { 4 }\) | 0 | 30 |
- State the column that contains the pivot value for the 1st iteration. You must give a reason for your answer.
- By considering the equations represented in the above tableau, formulate the linear programming problem in \(x , y\) and \(z\) only. State the objective and list the constraints as inequalities with integer coefficients.
- Taking the most negative number in the profit row to indicate the pivot column, perform the 2nd iteration of the Simplex algorithm, to obtain a new tableau, T . Make your method clear by stating the row operations you use.
- Explain, using T, how you know that an optimal solution to the original linear programming problem has not been found after the 2nd iteration.
- State the values of the basic variables after the 2nd iteration.
A student attempts the 3rd iteration of the Simplex algorithm and obtains the tableau below.
| b.v. | \(x\) | \(y\) | \(z\) | \(s _ { 1 }\) | \(S _ { 2 }\) | \(\mathrm { S } _ { 3 }\) | Value |
| z | 0 | 0 | 1 | \(\frac { 1 } { 2 }\) | \(- \frac { 1 } { 4 }\) | \(\frac { 1 } { 4 }\) | \(\frac { 43 } { 2 }\) |
| \(x\) | 1 | 0 | 0 | \(\frac { 1 } { 4 }\) | \(\frac { 1 } { 8 }\) | \(- \frac { 1 } { 8 }\) | \(\frac { 57 } { 4 }\) |
| \(y\) | 0 | 1 | 0 | \(- \frac { 1 } { 2 }\) | \(\frac { 1 } { 4 }\) | \(\frac { 3 } { 4 }\) | \(\frac { 9 } { 2 }\) |
| \(P\) | 0 | 1 | 0 | \(\frac { 5 } { 4 }\) | \(\frac { 1 } { 8 }\) | \(\frac { 7 } { 8 }\) | \(\frac { 361 } { 4 }\) |
- Explain how you know that the student's attempt at the 3rd iteration is not correct.