Model Answer
0 min readIntroduction
Linear Programming (LP) is a mathematical method for achieving the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. It’s a powerful tool used in resource allocation, production planning, and various optimization problems. The core idea is to maximize or minimize a linear objective function, subject to linear equality and inequality constraints. Formulating a problem as an LP model is the first and most critical step in applying this technique to real-world scenarios. This answer will demonstrate this process with a hypothetical manufacturing example.
Problem Formulation: A Manufacturing Scenario
Let's consider a company that manufactures two products: chairs and tables. The company wants to determine the optimal production quantities of each product to maximize its profit.
1. Decision Variables
First, we need to define the decision variables. These are the variables that we can control to achieve our objective.
- Let x represent the number of chairs produced.
- Let y represent the number of tables produced.
2. Objective Function
The objective function represents the quantity we want to maximize or minimize. In this case, we want to maximize profit.
Assume that the profit from selling one chair is ₹50 and the profit from selling one table is ₹80. Therefore, the objective function is:
Maximize Z = 50x + 80y
3. Constraints
Constraints are limitations or restrictions on the decision variables. These constraints represent the available resources or other limitations.
a) Resource Constraints
Assume the company has limited resources, specifically labor hours and raw materials.
- Labor Constraint: Each chair requires 2 labor hours, and each table requires 4 labor hours. The company has a total of 120 labor hours available. This can be expressed as: 2x + 4y ≤ 120
- Raw Material Constraint: Each chair requires 3 units of raw material, and each table requires 5 units of raw material. The company has 150 units of raw material available. This can be expressed as: 3x + 5y ≤ 150
b) Non-Negativity Constraints
The number of chairs and tables produced cannot be negative.
- x ≥ 0
- y ≥ 0
4. Complete Linear Programming Formulation
Combining all the above, the complete linear programming formulation is:
Maximize Z = 50x + 80y
Subject to:
- 2x + 4y ≤ 120
- 3x + 5y ≤ 150
- x ≥ 0
- y ≥ 0
5. Graphical Representation (Conceptual)
While not explicitly asked for, understanding that this formulation can be graphically represented is important. The feasible region is defined by the constraints, and the optimal solution lies at a corner point of this region.
| Component | Description | Mathematical Representation |
|---|---|---|
| Decision Variables | Quantities to be determined | x = number of chairs, y = number of tables |
| Objective Function | Quantity to be maximized/minimized | Maximize Z = 50x + 80y |
| Labor Constraint | Limitation on labor hours | 2x + 4y ≤ 120 |
| Raw Material Constraint | Limitation on raw material | 3x + 5y ≤ 150 |
| Non-Negativity Constraints | Variables cannot be negative | x ≥ 0, y ≥ 0 |
Conclusion
In conclusion, we have successfully formulated a manufacturing problem as a linear programming model. This involves defining decision variables, formulating an objective function to maximize profit, and establishing constraints based on resource limitations and non-negativity requirements. This formulation provides a solid foundation for solving the problem using various LP techniques, such as the simplex method or graphical analysis, to determine the optimal production quantities of chairs and tables. The ability to accurately translate a real-world problem into a mathematical LP model is a crucial skill in operations research and management science.
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.