Optimising Fuel Production From Crude Oil
How many units of each are needed from crude oil to satisfy the orders?
We can represent the conversion from one fuel to another in a table.
Consumed\Product | Petrol | Oil | Gas |
Petrol | 0 | 0 | 1/5 |
Oil | 1 | 1/5 | 2/5 |
Gas | 1 | 2/5 | 1/5 |
\[C =\left( \begin{array}{ccc} 0 & 0 & 1/5 \\ 1 & 1/5 & 2/5 \\ 1 & 2/5 & 1/5 \end{array} \right)\]
.Let
\[P, \: O, \: G\]
represent the production of petrol, oil and gas respectively. We can represent conversion between fuels by \[\left( \begin{array}{ccc} 0 & 0 & 1/5 \\ 1 & 1/5 & 2/5 \\ 1 & 2/5 & 1/5 \end{array} \right) \mathbf{Y}=\begin{pmatrix}P\\O\\G\end{pmatrix}\]
.Let
\[\mathbf{D}=\begin{pmatrix}100\\100\\100\end{pmatrix}\]
be the demand vector, then \[\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right) \begin{pmatrix}P\\O\\G\end{pmatrix} - \left( \begin{array}{ccc} 0 & 0 & 1/5 \\ 1 & 1/5 & 2/5 \\ 1 & 2/5 & 1/5 \end{array} \right) \begin{pmatrix}P\\O\\G\end{pmatrix}=\begin{pmatrix}100\\100\\100\end{pmatrix}\]
\[\left( \begin{array}{ccc} 1 & 0 & -1/5 \\ -1 & 4/5 & 3/5 \\ -1 & -2/5 & 4/5 \end{array} \right) \begin{pmatrix}P\\O\\G\end{pmatrix}=\begin{pmatrix}100\\100\\100\end{pmatrix}\]
\[\begin{pmatrix}P\\O\\G\end{pmatrix}={\left( \begin{array}{ccc} 1 & 0 & -1/5 \\ -1 & 4/5 & 3/5 \\ -1 & -2/5 & 4/5 \end{array} \right)}^{-1}\begin{pmatrix}100\\100\\100\end{pmatrix}=\left( \begin{array}{ccc} 2 & 1/3 & 2/3 \\ 5 & 5/2 & 5/2 \\ 5 & 5/3 & 10/3 \end{array} \right) \begin{pmatrix}100\\100\\100\end{pmatrix}=\begin{pmatrix}300\\1000\\1000\end{pmatrix}\]