Model Answer
0 min readIntroduction
Operations Research, a discipline that applies advanced analytical methods to help make better decisions, is increasingly vital in modern management. Linear Programming (LP) is a core technique within Operations Research, used to optimize resource allocation subject to constraints. This problem exemplifies a typical LP scenario where a company aims to maximize profit by determining the optimal production levels of different products, given limited resources like processing time. The efficient allocation of resources to maximize profitability is a fundamental challenge for businesses, and LP provides a structured approach to address it.
Problem Formulation
Let:
- x = Number of units of product A to be produced
- y = Number of units of product B to be produced
Objective Function: Maximize the total net profit (Z)
Z = 3.50x + 5.00y
Constraints:
- Processing time in section L: 2x + y ≤ 6000
- Processing time in section M: x + 4y ≤ 10000
- Non-negativity constraints: x ≥ 0, y ≥ 0
Graphical Solution
To solve this problem graphically, we first convert the inequalities into equations and plot them on a graph:
- 2x + y = 6000 => y = 6000 - 2x
- x + 4y = 10000 => y = 2500 - 0.25x
The feasible region is the area that satisfies all the constraints. The corner points of the feasible region are:
- (0, 0)
- (3000, 0)
- (0, 2500)
- Intersection of 2x + y = 6000 and x + 4y = 10000
Finding the Intersection Point
Solving the system of equations:
2x + y = 6000
x + 4y = 10000
Multiply the first equation by 4: 8x + 4y = 24000
Subtract the second equation from the modified first equation: 7x = 14000 => x = 2000
Substitute x = 2000 into the first equation: 2(2000) + y = 6000 => y = 2000
Therefore, the intersection point is (2000, 2000).
Evaluating the Objective Function at Corner Points
| Corner Point (x, y) | Z = 3.50x + 5.00y |
|---|---|
| (0, 0) | Z = 0 |
| (3000, 0) | Z = 3.50(3000) + 5.00(0) = 10500 |
| (0, 2500) | Z = 3.50(0) + 5.00(2500) = 12500 |
| (2000, 2000) | Z = 3.50(2000) + 5.00(2000) = 7000 + 10000 = 17000 |
The maximum profit (Z) is ₹ 17000, which occurs when x = 2000 and y = 2000.
Conclusion
Therefore, the company should produce 2000 units of product A and 2000 units of product B to maximize its total net profit, which will be ₹ 17000. This solution demonstrates the power of linear programming in optimizing resource allocation and maximizing profitability in a manufacturing setting.
Conclusion
In conclusion, this linear programming problem highlights the importance of mathematical modeling in managerial decision-making. By formulating the problem correctly and applying appropriate solution techniques, the company can achieve optimal production levels and maximize its profits. The graphical method, while effective for two-variable problems, can be extended to more complex scenarios using the simplex method and specialized software. Continuous monitoring of resource availability and market demand is crucial to maintain the effectiveness of this optimization strategy.
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.