Step 1: Define Variables

Let:

• $x$ be the number of units of product $P$.
• $y$ be the number of units of product $Q$.

Step 2: Objective Function

The company wants to maximize profit, which is given by:

$Z = 60x + 50y$

Step 3: Constraints

1. Labor Constraint: Each unit of $P$ requires 4 hours, and each unit of $Q$ requires 3 hours. The company has 24 hours available:

   $4x + 3y \leq 24$

2. Production Limits:

   • At most 6 units of $P$:

     $x \leq 6$

   • At most 8 units of $Q$:

     $y \leq 8$

3. Non-Negativity Constraints: Since the number of units cannot be negative,

   $x \geq 0, \quad y \geq 0$

Step 4: Graphical Solution

We graph the feasible region by plotting the constraints.

Plotting the Constraint $4x + 3y \leq 24$

• Find intercepts:

  • Set $x = 0$: $3y = 24 \Rightarrow y = 8$ → Point $(0,8)$
  • Set $y = 0$: $4x = 24 \Rightarrow x = 6$ → Point $(6,0)$

• Other Constraints:

  • The vertical line $x = 6$
  • The horizontal line $y = 8$
  • The region is bounded in the first quadrant.

Finding Corner Points

The feasible region is bounded by:

1. (0,0) – No production.

2. (6,0) – Producing only P.

3. (0,8) – Producing only Q.

4. Intersection of $4x + 3y = 24$ with $x = 6$

   • Substituting $x = 6$ in $4(6) + 3y = 24$:

     $24 + 3y = 24 \Rightarrow y = 0$

     So, the point is $(6,0)$, which was already considered.

5. Intersection of $4x + 3y = 24$ with $y = 8$

   • Substituting $y = 8$ in $4x + 3(8) = 24$:

     $4x + 24 = 24 \Rightarrow 4x = 0 \Rightarrow x = 0$

     So, the point is $(0,8)$, which was already considered.

Thus, the corner points are: $(0,0)$, $(6,0)$, and $(0,8)$.

Step 5: Calculate Profit at Each Corner

• At (0,0): $Z = 60(0) + 50(0) = 0$
• At (6,0): $Z = 60(6) + 50(0) = 360$
• At (0,8): $Z = 60(0) + 50(8) = 400$

Step 6: Conclusion

The maximum profit is ₹400, achieved by producing 0 units of P and 8 units of Q.

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Would you like a graphical representation?

Here are some related questions:

1. What happens if the labor hours increase from 24 to 30?
2. How would the solution change if profit per unit of Q increased to ₹60?
3. What if an additional constraint, like a budget limit, is introduced?
4. How do we solve this problem using the Simplex Method instead of graphically?
5. How would we interpret the shadow price of labor in this case?

💡 Tip: In graphical solutions, always check the feasible region before evaluating corner points.