5 Consider the following LP problem.
$$\begin{aligned}
\text { Minimise } & 2 a - 3 b + c + 18 , \\
\text { subject to } & a + b - c \geqslant 14 , \\
& - 2 a + 3 c \leqslant 50 , \\
\text { and } & a \leqslant 4 a \leqslant 5 b , \\
& a \leqslant 20 , b \leqslant 10 , c \leqslant 8 .
\end{aligned}$$
- By replacing \(a\) by \(20 - x , b\) by \(10 - y\) and \(c\) by \(8 - z\), show that the problem can be expressed as follows.
$$\begin{aligned}
\text { Maximise } & 2 x - 3 y + z , \\
\text { subject to } & x + y - z \leqslant 8 , \\
& 2 x - 3 z \leqslant 66 , \\
& 4 x - 5 y \leqslant 40 , \\
\text { and } & x \geqslant 0 , y \geqslant 0 , z \geqslant 0 .
\end{aligned}$$
- Represent the problem as an initial Simplex tableau. Perform one iteration of the Simplex algorithm. Explain how the choice of pivot was made and show how each row was obtained. Write down the values of \(x , y\) and \(z\) at this stage. Hence write down the corresponding values of \(a , b\) and \(c\).
- If, additionally, the variables \(a , b\) and \(c\) are non-negative, what additional constraints are there on the values of \(x , y\) and \(z\) ?