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2.Mathematical Formulation
Class 12 • Mathematics • NCERT
Mathematical Formulation of Linear Programming Problem
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This lesson explains how to translate a real-life situation into a Linear Programming Problem by identifying variables, objective function, and constraints, as prescribed in NCERT Class 12.
Lesson Objectives
- Understand the meaning of mathematical formulation.
- Identify decision variables from word problems.
- Form objective function and constraints.
- Write a complete Linear Programming Problem.
1. Meaning of Mathematical Formulation
Mathematical formulation of an LPP means expressing a given real-life problem in mathematical terms using linear equations or inequalities.
It converts words into mathematical expressions.
[ Image Placeholder: Real-Life Situation to Mathematical Model ]
2. Decision Variables
Decision variables represent the quantities whose values are to be determined to achieve optimization.
Let x and y be the number of units of two products.
[ Image Placeholder: Identifying Decision Variables ]
3. Objective Function
The objective function is a linear function of decision variables that is to be maximized or minimized.
Example: Maximize Z = 3x + 5y
[ Image Placeholder: Objective Function Line ]
4. Constraints
Constraints are linear inequalities or equations that restrict the values of decision variables.
Example:
x + y ≤ 10
2x + y ≤ 15
x + y ≤ 10
2x + y ≤ 15
[ Image Placeholder: Graph of Constraints ]
5. Non-Negativity Conditions
Decision variables cannot take negative values in practical problems.
x ≥ 0, y ≥ 0
[ Image Placeholder: First Quadrant Representation ]
6. Steps to Formulate an LPP (NCERT)
• Identify decision variables
• Write objective function
• Write constraints
• Add non-negativity conditions
• Write objective function
• Write constraints
• Add non-negativity conditions
Practice Questions (NCERT)
- What is meant by mathematical formulation?
- What are decision variables?
- Define objective function.
- What are constraints?
- Why non-negativity conditions are required?
- Form objective function if profit is ₹4 on x and ₹3 on y.
- Write constraints if x + y ≤ 20.
- Is quadratic objective function allowed?
- Which quadrant is considered in LPP?
- Is this chapter important for board exams?
✅ Show Answer Key
- Converting problem into mathematical form
- Variables representing decisions
- Function to maximize or minimize
- Restrictions on variables
- Variables cannot be negative
- Z = 4x + 3y
- x + y ≤ 20
- No
- First quadrant
- Yes
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