 6. An oil refinery wishes its product to have at least minimum amount of 3 components: 10%
 of A, 20% of B and 12% of C. It has available three different grades of crude oil: x, y, and z.
 Grade x contains: 15% of A, 10% of B and 9% of C and costs Rs. 200/– per barrel.
 Grade y contains: 18% of A, 25% of B and 3% of C and costs Rs. 250/– per barrel.
 Grade z contains 10% of A, 15% of B and 30% of C and costs Rs. 180/– per barrel....formulate lpp

To formulate this problem as a Linear Programming Problem (LPP), we need to define the decision variables, objective function, and constraints.

### **Step 1: Define the Decision Variables**
Let's define the quantities of each grade of crude oil that the refinery should purchase:
- Let \( x_1 \) be the number of barrels of grade \( x \) to purchase.
- Let \( x_2 \) be the number of barrels of grade \( y \) to purchase.
- Let \( x_3 \) be the number of barrels of grade \( z \) to purchase.

### **Step 2: Define the Objective Function**
The refinery wants to minimize the cost of purchasing these barrels. The cost function can be expressed as:

\[
\text{Minimize } Z = 200x_1 + 250x_2 + 180x_3
\]

### **Step 3: Define the Constraints**
The refinery needs to ensure that the mix of crude oil purchased contains at least the minimum required percentages of components A, B, and C.

#### **1. For Component A:**
The total amount of component A should be at least 10% of the total mixture.

\[
0.15x_1 + 0.18x_2 + 0.10x_3 \geq 0.10(x_1 + x_2 + x_3)
\]

This simplifies to:

\[
0.05x_1 + 0.08x_2 - 0.00x_3 \geq 0
\]

#### **2. For Component B:**
The total amount of component B should be at least 20% of the total mixture.

\[
0.10x_1 + 0.25x_2 + 0.15x_3 \geq 0.20(x_1 + x_2 + x_3)
\]

This simplifies to:

\[
-0.10x_1 + 0.05x_2 - 0.05x_3 \geq 0
\]

#### **3. For Component C:**
The total amount of component C should be at least 12% of the total mixture.

\[
0.09x_1 + 0.03x_2 + 0.30x_3 \geq 0.12(x_1 + x_2 + x_3)
\]

This simplifies to:

\[
-0.03x_1 - 0.09x_2 + 0.18x_3 \geq 0
\]

### **Step 4: Non-Negativity Constraints**
Finally, we include the non-negativity constraints, as the quantities purchased cannot be negative:

\[
x_1 \geq 0, \quad x_2 \geq 0, \quad x_3 \geq 0
\]

### **Summary of the LPP:**

\[
\text{Minimize } Z = 200x_1 + 250x_2 + 180x_3
\]

Subject to:

\[
\begin{aligned}
0.05x_1 + 0.08x_2 - 0.00x_3 & \geq 0, \\
-0.10x_1 + 0.05x_2 - 0.05x_3 & \geq 0, \\
-0.03x_1 - 0.09x_2 + 0.18x_3 & \geq 0, \\
x_1, x_2, x_3 & \geq 0.
\end{aligned}
\]

This Linear Programming Problem can be solved using standard methods such as the Simplex method or using software tools like MATLAB, Excel Solver, or other optimization software.

Would you like a detailed explanation of the simplex method to solve this problem, or do you have any other questions? 

Here are 5 related questions for further exploration:
1. How does the Simplex method solve linear programming problems?
2. What are some real-life applications of linear programming in industries?
3. How would the formulation change if there were more grades of crude oil available?
4. What happens if the refinery wants to maximize quality instead of minimizing cost?
5. How does sensitivity analysis work in linear programming?

**Tip:** In linear programming, always ensure your constraints are correctly derived and reflect the real-world requirements of the problem.

Learn how to formulate a Linear Programming Problem (LPP) for optimizing crude oil purchases at an oil refinery. This problem involves minimizing costs while meeting minimum component requirements of crude oils x, y, and z. Suitable for advanced level students and professionals.

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