Model Answer
0 min readIntroduction
Linear programming is a powerful mathematical technique used for optimizing resource allocation in situations where objectives are linear and constraints are also linear. It finds extensive applications in operations management, particularly in production planning, inventory control, and transportation. This problem presents a classic scenario where MNP Manufacturing aims to maximize its profit by deciding the optimal number of doors and windows to produce, given limited resources – cutting, sanding, and finishing capacities. Formulating and solving this problem using linear programming will provide MNP with a data-driven decision-making tool.
Formulation of the Linear Programming Model
Let:
- x = Number of doors produced
- y = Number of windows produced
Objective Function: Maximize Profit (Z)
Z = 500x + 400y
Constraints:
- Cutting Capacity: 1x + 0.5y ≤ 40
- Sanding Capacity: 0.5x + 0.75y ≤ 40
- Finishing Capacity: 0.5x + 1y ≤ 60
- Non-negativity: x ≥ 0, y ≥ 0
Graphical Solution
To solve this graphically, we first convert the inequalities into equations and plot them on a graph. The feasible region is the area that satisfies all the constraints simultaneously.
Step 1: Plotting the Constraints
- 1x + 0.5y = 40: When x=0, y=80; When y=0, x=40.
- 0.5x + 0.75y = 40: When x=0, y=53.33; When y=0, x=80.
- 0.5x + 1y = 60: When x=0, y=60; When y=0, x=120.
Step 2: Identifying the Feasible Region
The feasible region is the area bounded by the axes (x ≥ 0, y ≥ 0) and the constraint lines. We need to determine which side of each line satisfies the inequality.
Step 3: Finding the Corner Points
The optimal solution will lie at one of the corner points of the feasible region. The corner points are:
- (0, 0)
- (40, 0)
- (0, 60)
- Intersection of 1x + 0.5y = 40 and 0.5x + 0.75y = 40
- Intersection of 0.5x + 0.75y = 40 and 0.5x + 1y = 60
Let's calculate the intersection points:
- Intersection of 1x + 0.5y = 40 and 0.5x + 0.75y = 40: Multiplying the first equation by 0.5, we get 0.5x + 0.25y = 20. Subtracting this from the second equation, we get 0.5y = 20, so y = 40. Substituting y = 40 into the first equation, we get x + 0.5(40) = 40, so x = 20. Therefore, the intersection point is (20, 40).
- Intersection of 0.5x + 0.75y = 40 and 0.5x + 1y = 60: Subtracting the first equation from the second, we get 0.25y = 20, so y = 80. Substituting y = 80 into the second equation, we get 0.5x + 80 = 60, so 0.5x = -20, which gives x = -40. This point is not in the feasible region as x must be non-negative.
Step 4: Evaluating the Objective Function at Corner Points
| Corner Point (x, y) | Z = 500x + 400y |
|---|---|
| (0, 0) | Z = 0 |
| (40, 0) | Z = 500(40) + 400(0) = 20000 |
| (0, 60) | Z = 500(0) + 400(60) = 24000 |
| (20, 40) | Z = 500(20) + 400(40) = 10000 + 16000 = 26000 |
The maximum profit (Z) is ₹26,000, which occurs when x = 20 and y = 40.
Conclusion
Therefore, MNP Manufacturing should produce 20 doors and 40 windows to maximize its profit, given the available cutting, sanding, and finishing capacities. This solution utilizes all available resources efficiently. The graphical method provides a clear visual representation of the feasible region and helps in identifying the optimal production plan. Further analysis could involve sensitivity analysis to understand how changes in resource availability or profit margins would affect the optimal solution.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.