A company produces three products A, B and C by using two raw materials X and Y. 4000 units of X and 6000 units of Z are available for production. The requirement of raw materials by each product is given below: Raw material Requirement per unit of product A B C X Y 2 3 5 4 2 7 The labour time for each unit of product A is twice that of product B and three times that of product C. The entire labour force of the company can produce the equivalent of 2500 units of product A. A market survey indicates the minimum demand of the three products are 500, 500 and 375 respectively for A, B and C. However, their ratio of number of units produced must be equal to 3: 2: 5. Assume that the profit per units of product A, B and C are Rupees 60/–, 40/– and 100 respectively. Formulate the L.P.P. for maximizing the profit.

To formulate the Linear Programming Problem (LPP) for maximizing profit, we need to define decision variables, the objective function, and the constraints based on the given information.

### 1. **Decision Variables:**
Let's denote:
- \(x\): Number of units of product A to be produced
- \(y\): Number of units of product B to be produced
- \(z\): Number of units of product C to be produced

### 2. **Objective Function:**
The objective is to maximize the profit. The profit contributions for each product are given as:
- Profit for product A = 60 Rupees per unit
- Profit for product B = 40 Rupees per unit
- Profit for product C = 100 Rupees per unit

The objective function to maximize the total profit \(P\) is:
\[
\text{Maximize } P = 60x + 40y + 100z
\]

### 3. **Constraints:**

#### a) **Raw Material Constraints:**

- **Material X:**  
  Each unit of A requires 2 units of X, each unit of B requires 3 units of X, and each unit of C requires 5 units of X. The total availability of X is 4000 units.
  \[
  2x + 3y + 5z \leq 4000
  \]

- **Material Y:**  
  Each unit of A requires 4 units of Y, each unit of B requires 2 units of Y, and each unit of C requires 7 units of Y. The total availability of Y is 6000 units.
  \[
  4x + 2y + 7z \leq 6000
  \]

#### b) **Labour Time Constraints:**

Let the labour time required for each unit of product B be \(t\). Then, the labour time required for each unit of:
- Product A is \(2t\) (twice that of B),
- Product C is \(\frac{t}{3}\) (one-third that of A).

Since the labour force can produce the equivalent of 2500 units of product A, the total labour time constraint can be formulated as:
\[
2tx + ty + \frac{t}{3}z \leq 2500 \times 2t
\]

Dividing the entire equation by \(t\), we get:
\[
2x + y + \frac{z}{3} \leq 5000
\]

#### c) **Market Demand Constraints:**

The minimum market demand for the products is given as:
- \(x \geq 500\) (for Product A)
- \(y \geq 500\) (for Product B)
- \(z \geq 375\) (for Product C)

#### d) **Production Ratio Constraint:**

The ratio of units produced must be in the ratio 3:2:5, i.e., 
\[
\frac{x}{3} = \frac{y}{2} = \frac{z}{5} \implies 2x = 3y \quad \text{and} \quad 5x = 3z
\]

Thus, we have two additional constraints:
\[
2x - 3y = 0
\]
\[
5x - 3z = 0
\]

### 4. **Non-negativity Constraints:**
\[
x \geq 0, \; y \geq 0, \; z \geq 0
\]

### 5. **Final LPP Formulation:**

\[
\text{Maximize } P = 60x + 40y + 100z
\]

**Subject to:**

1. \(2x + 3y + 5z \leq 4000\)
2. \(4x + 2y + 7z \leq 6000\)
3. \(2x + y + \frac{z}{3} \leq 5000\)
4. \(x \geq 500\)
5. \(y \geq 500\)
6. \(z \geq 375\)
7. \(2x - 3y = 0\)
8. \(5x - 3z = 0\)
9. \(x, y, z \geq 0\)

This formulation is now ready to be solved using a Linear Programming method like the Simplex Method to determine the optimal production quantities for maximizing profit.

Would you like a more detailed solution or explanation on any specific part? 

### Related Questions:
1. How do you solve this LPP using the Simplex Method?
2. What happens if the constraints on raw materials change?
3. How can we interpret the shadow prices in the context of this LPP?
4. What is the significance of the production ratio constraint in this problem?
5. How would the solution change if the profit per unit of each product were different?

**Tip:** When setting up an LPP, carefully check the units and ratios involved in constraints to ensure they are consistent across all equations.

Learn how to maximize profit using Linear Programming (LPP) with a detailed formulation and constraints. Solve for optimal production quantities of products A, B, and C given raw material limitations and market demands. Suitable for advanced college students and professionals.

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