5. The initial tableau for a linear programming problem in \(x , y\) and \(z\) is shown below. The objective function to be maximised is \(P = 4 x + 2 y + k z\), where \(k\) is a positive constant.
| Basic Variable | \(x\) | \(y\) | \(z\) | r | \(s\) | \(t\) | Value |
| \(r\) | -2 | -6 | 1 | 1 | 0 | 0 | 40 |
| \(s\) | 2 | 3 | 2 | 0 | 1 | 0 | 80 |
| \(t\) | 1 | 2 | 2 | 0 | 0 | 1 | 50 |
| \(P\) | -4 | -2 | -k | 0 | 0 | 0 | 0 |
- Using the information in the tableau, write down the three constraints as inequalities.
- By increasing \(x\), perform one complete iteration of the simplex algorithm to obtain tableau \(T _ { 1 }\) and state the row operations you use.
- Given that \(T _ { 1 }\) is not optimal, find an inequality for the value of \(k\).
- Perform a second complete iteration of the simplex algorithm to obtain tableau \(T _ { 2 }\) and state the row operations you use.
- Given that \(T _ { 2 }\) is optimal, find a second inequality for the value of \(k\).
- State the final value of each variable and give an expression for the final value of \(P\) in terms of \(k\).
- Hence find the range of possible values of \(P\).