6 An initial simplex tableau is shown below.
| \(P\) | \(x\) | \(y\) | \(s\) | \(t\) | \(u\) | RHS |
| 1 | 2 | -5 | 0 | 0 | 0 | 0 |
| 0 | 2 | 1 | 1 | 0 | 0 | 25.8 |
| 0 | -1 | 3 | 0 | 1 | 0 | 13.8 |
| 0 | 4 | -3 | 0 | 0 | 1 | 18.8 |
The variables \(s , t\) and \(u\) are slack variables.
- For the LP problem that this tableau represents, write down the following, in terms of \(x\) and \(y\) only.
- The objective function, \(P\), to be maximised.
- The constraints as inequalities.
The graph below shows the feasible region for the problem (as the unshaded region, and its boundaries), and objective lines \(P = 10\) and \(P = 20\) (shown as dotted lines).
\includegraphics[max width=\textwidth, alt={}, center]{133395d2-5020-4054-a229-70168f1d0f95-7_883_1043_1272_244}
The optimal solution is \(P = 23\), when \(x = 0\) and \(y = 4.6\). - Complete the first three rows of branch-and-bound in the Printed Answer Booklet, branching on \(y\) first, to determine an optimal solution when \(x\) and \(y\) are constrained to take integer values. In your working, you should show non-integer values to \(\mathbf { 2 }\) decimal places.
The tableau entry 18.8 is reduced to 0 .
- Describe carefully what changes, if any, this makes to the following.
- The graph of the feasible region.
- The optimal integer valued solution.