b$)$ Your employer company has recently acquired a refinery which produces three types of gasoline by blending three crude oil distillates: A$,$ B and C$.$ Answer the questions below, stating and justifying any assumptions. Octane Number $($ON$),$ quality and market data are provided in Table Q$3$b$.$

Table Q$3$b: Refinery feed and product data $($bbl: barrel$).$

$\backslash$table$[[$Feeed$,$Name,ON$,$Availability $($bbl$/$day$),$Cost $(/$bbl$),],[$Distillate$,$A$,83,32,000,40],[$Distillate$,$B$,88,45,000,58,],[$Distillate$,$C$,93,35,000,68],[$Gasoline$,$Name,min. ON$,$max. A $(\%),$min. C $(\%),$Price $(/$bbl$)],[$Regular$,$R$,87,-,10,66],[$Mid$-$Grade,M$,89,22,37,68],[$Premium$,$P$,90,60,40,73]]$

$($i$)$ Express the revenue generated from selling each of these three fuel types $($at given prices$)$ as an explicit function of feed quantities blended.

$($ii$)$ Derive the objective function for gasoline blending profit maximisation. Define in detail all terms necessary and then obtain the simplest form.

$($iii$)$ Express all constraints on gasoline blending operation with regard to specifications and physical limitations$.$ Derive all in the simplest form.

For each of the following questions, use at most four $(4)$ sentences.

$($iv$)$ If the premium grade gasoline $ON$ specification is changed to $92,$ find and explain the maximum daily output $(P)$ attainable by this refinery.

$($v$)$ Could you draw a feasible region $($FR$)$ for the $[($ii$)-($iii$)]$ model on paper? Justify your response in detail, providing clear quantitative arguments.

$($vi$)$ State the key LP theorem. How can the optimal blending operating point be determined? How can we check on the FR that this is the optimum?