Stable Limited manufactures two models of refrigerators. Deluxe Model requires 14 hours of Department A and 40 hours of Department B. Standard Model requires 7 hours of Department A and 30 hours of Department B. Each month, a maximum of 2100 hours are available in Department A and 8400 hours in Department B. The company makes a profit of Rs 5,000 on each Deluxe Model and Rs 4,000 on each Standard Model.

) Construct the set of constraints and the objective function for profit maximization.

(a (b) Draw the graph and identify the feasible region, clearly indicating its boundaries.

(c) How many refrigerators of each model should the company manufacture per day to maximize its profit

Let's solve this problem step by step.

### (a) Construct the set of constraints and the objective function for profit maximization.

Let:
- \( x_1 \) be the number of Deluxe Models produced per month.
- \( x_2 \) be the number of Standard Models produced per month.

#### Objective Function:
The company wants to maximize its profit. The profit from Deluxe Models is Rs 5,000 per unit, and the profit from Standard Models is Rs 4,000 per unit. Therefore, the objective function is:

\[
\text{Maximize } Z = 5000x_1 + 4000x_2
\]

#### Constraints:

1. **Department A Constraint**:
   - Each Deluxe Model requires 14 hours.
   - Each Standard Model requires 7 hours.
   - A maximum of 2100 hours are available per month in Department A.

   This gives the constraint:
   \[
   14x_1 + 7x_2 \leq 2100
   \]

2. **Department B Constraint**:
   - Each Deluxe Model requires 40 hours.
   - Each Standard Model requires 30 hours.
   - A maximum of 8400 hours are available per month in Department B.

   This gives the constraint:
   \[
   40x_1 + 30x_2 \leq 8400
   \]

3. **Non-negativity Constraints**:
   - The number of refrigerators produced cannot be negative:
   \[
   x_1 \geq 0, \quad x_2 \geq 0
   \]

### Summary of the Mathematical Model:

Objective Function:
\[
\text{Maximize } Z = 5000x_1 + 4000x_2
\]

Subject to:
\[
14x_1 + 7x_2 \leq 2100
\]
\[
40x_1 + 30x_2 \leq 8400
\]
\[
x_1 \geq 0, \quad x_2 \geq 0
\]

### (b) Draw the graph and identify the feasible region, clearly indicating its boundaries.

Let's solve the constraints graphically. First, we'll find the intercepts of each constraint.

#### Constraint 1: \( 14x_1 + 7x_2 \leq 2100 \)

- For \( x_1 = 0 \):
  \[
  7x_2 = 2100 \quad \Rightarrow \quad x_2 = 300
  \]

- For \( x_2 = 0 \):
  \[
  14x_1 = 2100 \quad \Rightarrow \quad x_1 = 150
  \]

#### Constraint 2: \( 40x_1 + 30x_2 \leq 8400 \)

- For \( x_1 = 0 \):
  \[


Learn how to maximize profit using linear programming with Deluxe and Standard refrigerator models. Understand the constraints and objective function for profit maximization. Suitable for college-level students and professionals.

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