Chapter 3.9, Problem 1P

Explanation of Solution

Given:

Let $x_i$ be the number of hours process which takes for “i” run per week where $i=1,2,3$. Let $O_i$ be the number of barrels of oil and $G_2$ be the barrels of gas 2 sold per week.

Objective function:

The processor 1 that runs for 1 hour requires 2 barrels of crude oil 1 and 3 barrels of crude oil 2 and the output of process 1 for one hour produces 2 barrels of gas 1 and 1 barrel of gas 2.

The processor 2 that runs for 1 hour requires 2 barrels of crude oil 1 and 3 barrels of crude oil 2 and the output of process 2 for one hour produces 3 barrels of gas 2.

The process 3 that runs for 1 hour requires 2 barrels of crude 2 and 2 barrels of gas 2 and the output of process 3 for one hour produces 2 barrels of gas 3.

The revenue from the gas 1 is sold for $9 and the revenue from gas 3 is sold for $24 is $24(2x_3)$. Here, the variable for gas 2 is sold and the remaining gas will be used in producing gas 3.

$\begin{array}{l}\text{Maximize}z=9(2x_1)+10G_2+24(2x_3)-5x_1-4x_2-x_3-2O_1-3O_2\\ =13x_1-4x_2+47x_3+10G_2-2O_1-3O_2\end{array}$

Considering the constraints,

Constraint 1: Total oil 1 used in process 1 and 2.

Constraint 2: Total oil 2 used in process 1,2 and 3.

Constraint 3 and 4: Each week, 200 barrels of crude oil 1 and 300 barrels of crude oil 2 must be purchased.

Constraint 4: Production gas 2 will be equal to the gas 2 sold and gas 2 used in 3 that have been produced by process 1 and 2 by using the amount of gas 2 produced and from each process.

Constraint 6: Each week, 100 hours of time catalytic crackers are available.

Expressing the constraint 1 in terms of $x_1$, $x_2$and $x_3$ :

$\begin{array}{l}{\text{(2barrel of oil 1 used for process 1)+(1 barrel of oil 2 used for process 2)=O}}_{\text{1}}\\ 2x_1+x_2=O_{\begin{array}{l}1\\ \end{array}}\\ 2x_1+x_2-O_1=0\\ \end{array}$

Expressing the constraint 2 in terms of $x_1$, $$