Question

1) An oil company has three different processes that can be used to manufacture different types of gasoline. Each process involves blending oils in the company’s catalytic cracker. Running process 1 for an hour costs $5 and requires 2 barrels of crude oil 1 and 3 barrels of crude oil 2. The output from running process 1 for an hour is 2 barrels of gas 1 and 1 barrel of gas 2. Running process 2 for an hour costs $4 and requires 1 barrel of crude 1 and 3 barrels of crude 2. The output from running process 2 for an hour is 3 barrels of gas 2. Running process 3 for an hour costs $1 and requires 2 barrels of crude 2 and 3 barrels of gas 2. The output from running process 3 for an hour is 2 barrels of gas 3. Each week, 200 barrels of crude 1, at $2/barrel, and 300 barrels of crude 2, at $3/barrel, may be purchased. All gas produced can be sold at the following per-barrel prices: gas 1, $9; gas 2, $10; gas 3, $24. Assume that only 100 hours of time on the catalytic cracker are available each week.

a) Formulate an LP whose solution will maximize revenues less costs.

Answer 1

Explanation:

Let's define the decision variables:

x1 = number of hours to run process 1

x2 = number of hours to run process 2

x3 = number of hours to run process 3

The objective is to maximize the profit, which is the revenue minus the cost. The revenue comes from selling the gasoline, and the cost comes from the raw materials and the running costs of each process. The profit can be expressed as:

Profit = (92x1 + 104x2 + 242x3) - (5x1 + 4x2 + 1x3 + 22200 + 33*300)

The first term in the equation represents the revenue, which is calculated by multiplying the selling price of each type of gasoline by the volume produced by each process. The second term represents the cost, which is calculated by multiplying the running cost of each process by the number of hours used, and adding the cost of the raw materials.

The LP model can be formulated as follows:

Maximize:

Profit = 7x1 + 36x2 + 46x3 - 2200

Subject to:

2x1 + x2 <= 200 (crude 1 constraint)

3x1 + 3x2 + 2x3 <= 300 (crude 2 constraint)

2x1 + x3 <= 100 (time constraint)

x1, x2, x3 >= 0 (non-negativity constraint)

The first two constraints limit the amount of crude oil that can be used for each process, while the third constraint limits the total hours available on the catalytic cracker. The non-negativity constraint ensures that the decision variables cannot be negative.

Therefore, the complete LP model for maximizing revenues less costs is:

Maximize:

7x1 + 36x2 + 46x3 - 2200

Subject to:

2x1 + x2 <= 200

3x1 + 3x2 + 2x3 <= 300

2x1 + x3 <= 100

x1, x2, x3 >= 0