b$)$ Two diesel feedstocks of Cetane Number $($CN$)$ of $20$ and $50$ are available at refinery inventory volumes of $50000$ and $40000$ litres, respectively. Blending optimisation requires you to formulate a Lidear Programming $($LP$)$ problem and solve it to obtain optimal volumes from each feed for mixing, to two target fuels: $CN=30$ and $40($respective prices: $0\mathrm{.}70,\frac{1\mathrm{.}20}{?}$ litre$).$ There is no volume contraction, and any leftover feedstock cannot be sold.

$($i$)$ Select, state and derive the objective function for this blending problem. Define in detail all terms necessary and then obtain the simplest form.

$($ii$)$ Express all constraints on diesel blending in the simplest form possible.

$(3)$

$($iii$)$ Draw and annotate fully the feasible region $($FR$)$ of the LP problem.

$($iv$)$ State the key LP theorem. Explain how a refinery can systematically determine and check the optimal blending operating point on the FR$.$

$($v$)$ Obtain, state and comment on the solution within two $(2)$ sentences.

$($vi$)$ If the high$-$grade diesel $CN$ specification is changed to $55,$ comment on and explain the resulting $(\%)$ effect on refinery output and performance.

$(3)$