An oil refinery has two sources of crude oil: a light crude that costs $ 35/barrel and a 
heavy crude that costs $ 30/barrel. The refinery produces gasoline, heating oil, and jet 
fuel from crude in the amounts per barrel indicated in the following table:
Gasoline Heating Oil Jet Fuel
Light Crude 0.3 0.2 0.3
Heavy Crude 0.3 0.4 0.2
The refinery has been contracted to supply 900,000 barrels of gasoline, 800,000 barrels 
of heating oil, and 500,000 barrels of jet fuel. Find the amounts of light and heavy crude 
that the refinery has to purchase so as to be able to meet its obligations at minimum 
cost. What is the minimum cost?

This problem can be formulated as a linear programming problem, where the objective is to minimize the cost of purchasing crude oil, while the constraints ensure that the refinery meets the required quantities of gasoline, heating oil, and jet fuel.

### Variables:
Let:
- $ x_1 $ = number of barrels of light crude to purchase
- $ x_2 $ = number of barrels of heavy crude to purchase

### Objective:
We want to minimize the total cost:
\[
\text{Cost} = 35x_1 + 30x_2
\]

### Constraints:
We have to satisfy the demands for gasoline, heating oil, and jet fuel.

- **Gasoline constraint**: 
  - From light crude, we get 0.3 barrels of gasoline per barrel of crude.
  - From heavy crude, we also get 0.3 barrels of gasoline per barrel of crude.
  \[
  0.3x_1 + 0.3x_2 \geq 900,000
  \]

- **Heating oil constraint**:
  - From light crude, we get 0.2 barrels of heating oil per barrel of crude.
  - From heavy crude, we get 0.4 barrels of heating oil per barrel of crude.
  \[
  0.2x_1 + 0.4x_2 \geq 800,000
  \]

- **Jet fuel constraint**:
  - From light crude, we get 0.3 barrels of jet fuel per barrel of crude.
  - From heavy crude, we get 0.2 barrels of jet fuel per barrel of crude.
  \[
  0.3x_1 + 0.2x_2 \geq 500,000
  \]

Additionally, we assume that $ x_1 \geq 0 $ and $ x_2 \geq 0 $, as we can't purchase negative amounts of crude oil.

### Problem Summary:
We need to solve the following linear programming problem:
\[
\min 35x_1 + 30x_2
\]
subject to:
\[
\begin{aligned}
0.3x_1 + 0.3x_2 &\geq 900,000 \\
0.2x_1 + 0.4x_2 &\geq 800,000 \\
0.3x_1 + 0.2x_2 &\geq 500,000 \\
x_1, x_2 &\geq 0
\]

### Solution:
To solve this linear programming problem, we can use the **Simplex method** or other LP solvers. I will now calculate the solution.

Let me compute it.The refinery should purchase the following amounts of crude oil to minimize costs while meeting its obligations:

- **Light crude**: 0 barrels
- **Heavy crude**: 3,000,000 barrels

The **minimum cost** of purchasing the crude oil is $90,000,000.

Would you like more details on how this solution was derived, or do you have any further questions?

Here are 5 related questions for further exploration:
1. What happens if the cost of light crude changes? How does that affect the optimal solution?
2. Could the refinery use a mix of light and heavy crude if constraints were adjusted?
3. How would the solution change if the refinery needed to produce different amounts of fuel?
4. Can the refinery benefit from increasing the efficiency of producing fuel from crude oil?
5. What if the refinery had a limit on the maximum number of barrels it could process?

**Tip:** In optimization problems, constraints play a crucial role in shaping the solution. Even if one input is cheaper, constraints can force the use of more expensive options.

An oil refinery has two sources of crude oil: a light crude that costs $35/barrel and a heavy crude that costs $30/barrel. The refinery produces gasoline, heating oil, and jet fuel from crude in the amounts per barrel indicated in the following table:

|           | Gasoline | Heating Oil | Jet Fuel |
|-----------|----------|-------------|----------|
| Light Crude | 0.3      | 0.2         | 0.3      |
| Heavy Crude | 0.3      | 0.4         | 0.2      |

The refinery has been contracted to supply 900,000 barrels of gasoline, 800,000 barrels of heating oil, and 500,000 barrels of jet fuel. Find the amounts of light and heavy crude that the refinery has to purchase so as to be able to meet its obligations at minimum cost. What is the minimum cost?

Solve a linear programming problem to minimize the cost of purchasing light and heavy crude oil for a refinery. Learn how to formulate and solve constraints to meet fuel production requirements for gasoline, heating oil, and jet fuel. Ideal for students in Grades 11-12 and above.

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