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?     use greater then and less then signs to formulate eqautions then solve for 15 marks

The problem at hand is a linear programming optimization problem. We need to determine the amounts of light and heavy crude the refinery should purchase in order to meet its production requirements at the minimum cost.

### Variables:
Let:
- $ x_1 $ = barrels of light crude to be purchased
- $ x_2 $ = barrels of heavy crude to be purchased

### Objective:
The cost function we need to minimize is the total cost of purchasing the crude oil:
\[
\text{Minimize } Z = 35x_1 + 30x_2
\]

### Constraints:
From the table, the refinery produces different amounts of gasoline, heating oil, and jet fuel per barrel of crude oil.

**Gasoline constraint:**
\[
0.3x_1 + 0.3x_2 \geq 900,000
\]
This ensures that the combined gasoline produced from both light and heavy crude is at least 900,000 barrels.

**Heating oil constraint:**
\[
0.2x_1 + 0.4x_2 \geq 800,000
\]
This ensures that the combined heating oil produced from both light and heavy crude is at least 800,000 barrels.

**Jet fuel constraint:**
\[
0.3x_1 + 0.2x_2 \geq 500,000
\]
This ensures that the combined jet fuel produced from both light and heavy crude is at least 500,000 barrels.

### Non-negativity constraint:
Both $ x_1 $ and $ x_2 $ represent quantities of crude oil, so they must be non-negative:
\[
x_1 \geq 0, \quad x_2 \geq 0
\]

### Formulation Summary:
Minimize:
\[
Z = 35x_1 + 30x_2
\]
Subject to the constraints:
\[
0.3x_1 + 0.3x_2 \geq 900,000
\]
\[
0.2x_1 + 0.4x_2 \geq 800,000
\]
\[
0.3x_1 + 0.2x_2 \geq 500,000
\]
\[
x_1 \geq 0, \quad x_2 \geq 0
\]

### Solving the system:
We can solve this linear programming problem using the simplex method. Let me proceed with the calculation.The refinery should purchase:
- $ x_1 = 0 $ barrels of light crude
- $ x_2 = 3,000,000 $ barrels of heavy crude

The minimum cost for the refinery to meet its production obligations is **$90,000,000**.

This result implies that only heavy crude is required to meet the gasoline, heating oil, and jet fuel demands at the minimum cost.

Would you like more details on the solution process, or have any further questions?

Here are 5 related questions you can explore:
1. How would the result change if the cost of heavy crude increased?
2. What if the refinery needed to supply additional products like diesel or kerosene?
3. How does the method of linear programming compare to other optimization techniques?
4. Can linear programming handle more complex refinery operations with variable product yields?
5. What if there were additional constraints on the maximum amount of crude that could be purchased?

**Tip**: In optimization problems, understanding constraints and objective functions is key to effectively setting up and solving the problem.

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 the oil refinery problem where light crude costs $35 per barrel and heavy crude costs $30 per barrel. The refinery must produce 900,000 barrels of gasoline, 800,000 barrels of heating oil, and 500,000 barrels of jet fuel at minimum cost. Learn how to set up and solve this linear programming problem using constraints and the simplex method. Suitable for advanced high school or university students.

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