7. A maximisation linear programming problem in \(x , y\) and \(z\) is to be solved using the two-stage simplex method.
The partially completed initial tableau is shown below.
| Basic variable | \(x\) | \(y\) | \(z\) | \(S _ { 1 }\) | \(S _ { 2 }\) | \(S _ { 3 }\) | \(a _ { 1 }\) | \(a _ { 2 }\) | Value |
| \(S _ { 1 }\) | 1 | 2 | 3 | 1 | 0 | 0 | 0 | 0 | 45 |
| \(a _ { 1 }\) | 3 | 2 | 0 | 0 | -1 | 0 | 1 | 0 | 9 |
| \(a _ { 2 }\) | -1 | 0 | 4 | 0 | 0 | -1 | 0 | 1 | 4 |
| \(P\) | -2 | -1 | -3 | 0 | 0 | 0 | 0 | 0 | 0 |
| A | | | | | | | | | |
- Using the information in the above tableau, formulate the linear programming problem. State the objective and list the constraints as inequalities.
- Complete the bottom row of Table 1 in the answer book. You should make your method and working clear.
The following tableau is obtained after two iterations of the first stage of the two-stage simplex method.
| Basic variable | \(x\) | \(y\) | \(z\) | \(S _ { 1 }\) | \(S _ { 2 }\) | \(S _ { 3 }\) | \(a _ { 1 }\) | \(a _ { 2 }\) | Value |
| \(S _ { 1 }\) | 0 | \(\frac { 5 } { 6 }\) | 0 | 1 | \(\frac { 7 } { 12 }\) | \(\frac { 3 } { 4 }\) | \(- \frac { 7 } { 12 }\) | \(- \frac { 3 } { 4 }\) | \(\frac { 147 } { 4 }\) |
| \(x\) | 1 | \(\frac { 2 } { 3 }\) | 0 | 0 | \(- \frac { 1 } { 3 }\) | 0 | \(\frac { 1 } { 3 }\) | 0 | 3 |
| \(z\) | 0 | \(\frac { 1 } { 6 }\) | 1 | 0 | \(- \frac { 1 } { 12 }\) | \(- \frac { 1 } { 4 }\) | \(\frac { 1 } { 12 }\) | \(\frac { 1 } { 4 }\) | \(\frac { 7 } { 4 }\) |
| \(P\) | 0 | \(\frac { 5 } { 6 }\) | 0 | 0 | \(- \frac { 11 } { 12 }\) | \(- \frac { 3 } { 4 }\) | \(\frac { 11 } { 12 }\) | \(\frac { 3 } { 4 }\) | \(\frac { 45 } { 4 }\) |
| A | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
- Explain how the above tableau shows that a basic feasible solution has been found for the original linear programming problem.
- Write down the basic feasible solution for the second stage.
- Taking the most negative number in the profit row to indicate the pivot column, perform one complete iteration of the second stage of the two-stage simplex method, to obtain a new tableau, \(T\). Make your method clear by stating the row operations you use.
- Explain, using \(T\), whether or not an optimal solution to the original linear programming problem has been found.
- Write down the value of the objective function.
- State the values of the basic variables.