Consider the linear programming problem $P$ :

$$\text { minimise } c^{T} x \text { subject to } A x \geqslant b, x \geqslant 0,$$

where $x$ and $c$ are in $\mathbb{R}^{n}, A$ is a real $m \times n$ matrix, $b$ is in $\mathbb{R}^{m}$ and ${ }^{T}$ denotes transpose. Derive the dual linear programming problem $D$. Show from first principles that the dual of $D$ is $P$.

Suppose that $c^{T}=(6,10,11), b^{T}=(1,1,3)$ and $A=\left(\begin{array}{lll}1 & 3 & 8 \\ 1 & 1 & 2 \\ 2 & 4 & 4\end{array}\right)$. Write down the dual $D$ and find the optimal solution of the dual using the simplex algorithm. Hence, or otherwise, find the optimal solution $x^{*}=\left(x_{1}^{*}, x_{2}^{*}, x_{3}^{*}\right)$ of $P$.

Suppose that $c$ is changed to $\tilde{c}=\left(6+\varepsilon_{1}, 10+\varepsilon_{2}, 11+\varepsilon_{3}\right)$. Give necessary and sufficient conditions for $x^{*}$ still to be the optimal solution of $P$. If $\varepsilon_{1}=\varepsilon_{2}=0$, find the range of values for $\varepsilon_{3}$ for which $x^{*}$ is still the optimal solution of $P$.