# A LP Model for producing oil (involving 3 processes)

A company produces two kinds of gasoline from the combination of two kinds of petroleum. For this objective, the company can use 3 kinds of process:

1. In an hour, with process 1, 2 barrels of petroleum kind 1 and 3 barrels of petroleum kind 2 are combined to produce 2 barrels of gasoline kind 1 and 2 barrels of gasoline kind 2.
2. In an hour, with process 2, 1 barrel of petroleum kind 1 and 3 barrels of petroleum kind 2 are combined to produce 3 barrels of gasoline kind 2.
3. In an hour, with process 3, 2 barrels of petroleum kind 1 and 3 barrels of petroleum kind 2 are combined to produce 2 barrels of gasoline kind 3.

1 hour of using process 1, 2 and 3 costs 5, 4 and 1 unit respectively. The company can buy at most 200 barrels of petroleum kind 1 with the price of 2 units per each barrel, and at most 300 barrels of petroleum kind 2 with the price of 3 units per each barrel.

Also, Each barrel of gasoline kind 1, 2 and 3 can be sold with the price of 9, 10 and 24 units respectively.

There is at most 100 hours of time available in each week.

Provide a LP Model for this problem.

Note : My problem is the decision variables. I've seen some similar questions. But in that questions, the decision variables were the amount of petroleum kind i used to produce gasoline kind j. Here, i don't know how to involve these three kinds of processes in the model.

Let $x_i$ be the duration of process $i$ measured in hours. And you have for instance this condition

The company can buy at most 200 barrels of petroleum kind 1 ...

Process $1$ needs $2$ barrels of petroleum kind 1, process $2$ needs $1$ barrel of petroleum kind $1$ and process $3$ needs $2$ barrels of petroleum kind $1$.

Therefore the inequality is

$2x_1+x_2+2x_3\leq 200$

Similar approach for the condition

at most 300 barrels of petroleum kind 2

The condition

There is at most 100 hours of time available in each week.

can be expressed as $x_1+x_2+x_3\leq 100$

If you have questions about other conditions or the objective function feel free to ask.

One way to approach a problem like this is to use the numbers of hours each process is run as decision variables. The numbers of barrels of each kind of oil and gasoline are easily expressed in terms of those hours as equality constraints.