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Oil Refinery Optimization
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A refinery has a distillation capacity of 100,000 barrels of crude/day
in its fractionater. Here crude oil is basically heated and as the
temperature increases different products called DN (distillation
naptha), DHO (distillation heating oil), DGO (distillation gas oil),
and P (Pitch), are given off in vapor form, and are collected at
various levels. The refinery gets crude oil from three different
countries, these are called crudes 1,2,3. All the crudes and the
various products are measured by volume in barrels. The output
statistics from the distillation of each of the available crudes are
given in Table \ref{tab:output}.
\begin{table}
\begin{center}
\begin{tabular}{c|ccc|}
Distillation & \multicolumn{3}{c}{Yield (barrels/barrel) from}\\
output & \multicolumn{3}{c}{distillation of}\\
\hline
&    Crude 1 & Crude 2 & Crude 3\\
\hline
DN &  0.19 &0.16 &0.02  \\
DHO &  0.27 &0.32 &0.24  \\
DGO &  0.38 &0.27 &0.26  \\
P &  0.05 &0.13 &0.39  \\
\hline
Price (\$/barrel) &  23.25& 22.00 &20.50\\
Available &  60,000& 90,000 &80,000\\
(barrels/day) \\
\hline
\end{tabular}
\caption{Distillation Output}
\label{tab:output}
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\end{table}
It can be verified that the total volume of outputs from the
distillation of one barrel of crude is $<1$. The loss is due to
evaporation and unusable heavy residuals. 

DHO can be sold directly as heating oil. DGO can be sold directly as
diesel fuel. Sale prices of these products are given in Table
\ref{tab:prices}.
\begin{table}
\begin{center}
\begin{tabular}{c|ccc|}
Catalytic cracker & \multicolumn{3}{c}{Output (barrels/barrel of feed)}\\
outputs &   \\
\hline
&    DHO normal  & DHO high & DGO normal\\
&    level  & severity level & level\\
\hline
Catalytic naptha (CN) &  0.18 &0.32 &0.48  \\
Catalytic heating oil (CHO) &  0.80 &0.69 &0.52  \\
Pitch (P) &  0.11 &0.10 &0.15  \\
\hline
\end{tabular}
\caption{Cracker Output}
\label{tab:coutput}
\end{center}
\end{table}
DHO and DGO can also be processed further in a catalytic cracker. The
catalytic cracker can either process a maximum of 100,000 barrels/day
of DHO, or a maximum of 50,000 barrels/day of DGO, or a combination of
these in proportion of these levels adding up to 1. Also, when
processing DHO the catalytic cracker can be run either at a normal
level or at a high severity level. The high severity level helps to
convert more of the DHO into naptha as seen from Table \ref{tab:coutput}.
In
processing DGO the catalytic cracker is run at normal level only and
never on high severity level.

The cracking process converts the feed into products whose density is
smaller than that of the feed, that's why the volume of outputs from
this process is greater than the feed volume.

The pitch (from fractionater and catalytic cracker) can be combined
with CHO (two parts of CHO to 17 parts of pitch) and sold as heavy
fuel oil. DN and CN can be combined (20 parts of DN with 17 parts or
greater of CN) and sold as gasoline. The quality of the gasoline
improves with the proportion of CN in this blend. Table
\ref{tab:prices} gives the  selling prices and demand for the various
final products. 
\begin{table}
\begin{center}
\begin{tabular}{c|c|c}
Final Product & Selling price  & Daily demand \\
                &  \$/barrel  \\
\hline
Gasoline  &  43.25   & Up to 40,000 barrels\\
Heating oil  &  39.75   & Up to 40,000 barrels\\
Diesel fuel  &  39.00   & Any amount\\
Heavy fuel oil  &  30.00   & Any amount\\
\hline
\end{tabular}
\caption{Final Prices}
\label{tab:prices}
\end{center}
\end{table}
The processing cost on the fractionater is estimated to be
\$0.60/barrel of crude processed. On the catalytic cracker the
processing costs are \$1.50/barrel of fresh feed at the high severity
level, and \$0.95/barrel of fresh feed at the normal level. Formulate
the problem of determining how much of each final product to produce
daily in order to maximize daily net profit, as an LP.

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