Given:

Let xi be the number of hours process which takes for “i” run per week where i=1,2,3. Let Oi be the number of barrels of oil and G2 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(2x3). Here, the variable for gas 2 is sold and the remaining gas will be used in producing gas 3.

Maximize z=9(2x1)+10G2+24(2x3)−5x1−4x2−x3−2O1−3O2 =13x1−4x2+47x3+10G2−2O1−3O2

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 x1, x2and x3 :

(2barrel of oil 1 used for process 1)+(1 barrel of oil 2 used for process 2)=O1 2x1+x2=O1 2x1+x2−O1=0

Expressing the constraint 2 in terms of x1,

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