Class 12 Maths Chapter 12 Linear Programming

Linear Programming sounds fancy, but once you break it down, it’s actually one of the most logical and practical topics you’ll study in Class 12 Mathematics.

These notes are carefully designed to explain every concept step by step, just like a teacher sitting beside you. No jargon, no unnecessary complexity, just clarity. By the end, you’ll see how simple and fun Linear Programming can actually be.

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Linear Programming Notes Class 12

Linear Programming (commonly abbreviated as LPP) is a mathematical technique used for finding the best possible outcome in a given situation where resources are limited. It is widely applied in real-life situations such as maximizing profit, minimizing cost, optimal allocation of resources, production scheduling, diet planning and transportation problems.

In Class 12 Mathematics (NCERT), we deal mainly with Linear Programming Problems involving two variables, as these can be conveniently represented and solved graphically. Although higher-level mathematics deals with more than two variables using advanced methods like the simplex method, at this stage the focus is only on the graphical method and its applications.

Basic Terms

Linear Programming Problem (LPP)

A Linear Programming Problem is one that seeks to optimize (maximize or minimize) a linear objective function subject to a set of linear inequalities (constraints).

Decision Variables

The variables whose values are to be determined are called decision variables. Usually denoted by x and y.

Objective Function

The function which needs to be optimized (maximized or minimized) is called the objective function.

It is usually expressed in the form:

Z = ax + by

where a and b are constants.

Constraints

The restrictions or limitations imposed on the decision variables are known as constraints. They are expressed as a system of linear inequalities such as:

x + y ≤ 10, x ≥ 0 , y ≥ 0

The region that satisfies all the given constraints simultaneously is called the feasible region. It is represented graphically as a polygonal (bounded) or open (unbounded) region on the coordinate plane.

Feasible Solutions

Every point within or on the boundary of the feasible region represents a feasible solution of the LPP.

Optimal Solution

Among all feasible solutions, the solution that optimizes the objective function (maximizes or minimizes it) is called the optimal solution.

Corner Point Theorem

If an LPP has an optimal solution, it must occur at one of the corner points (vertices) of the feasible region.

Formulation of a Linear Programming Problem - Linear Programming Class 12 Notes PDF

To formulate an LPP from a real-life situation:

• Identify the decision variables.
• Express the objective function (maximize profit/minimize cost).
• Write down the constraints based on the conditions of the problem.
• Represent non-negativity restrictions (x≥0,y≥0x \geq 0, y \geq 0).

Example:

A factory makes two types of products A and B. Each unit of A requires 2 hours of labour and 1 unit of raw material. Each unit of B requires 1 hour of labour and 2 units of raw material. The factory has 100 hours of labour and 120 units of raw material available. Profit is ₹40 on product A and ₹50 on product B. Formulate an LPP to maximize profit.

Solution (Formulation):

Let x = number of units of A, y = number of units of B.

Objective function: Maximize Z = 40x + 50y 

Constraints:

• Labour: 2x + y ≤ 100
• Raw material: x + 2y ≤ 120
• Non-negativity: x , y ≥ 0

Graphical Method of Solving LPP

The graphical method is the standard approach for solving LPPs with two variables.

Steps:

• Represent each inequality constraint as an equation of a straight line.
• Identify the feasible region by shading or testing points.
• Mark the corner points (vertices) of the feasible region.
• Evaluate the objective function at each corner point.
• The point giving the maximum or minimum value of the objective function is the optimal solution.

Types of Solutions

Feasible Solution

Any solution lying in the feasible region that satisfies all constraints.

Infeasible Solution: If no common region satisfies all the constraints, then no feasible solution exists and the problem is said to be infeasible.

Bounded Solution: If the feasible region is enclosed, it is bounded. The optimal solution always exists at a corner point.

Unbounded Solution: If the feasible region is open, it is unbounded. In this case, the objective function may or may not have an optimal solution. To check, we extend the objective line and see whether the function can increase/decrease indefinitely.

Multiple Solutions: If the objective function takes the same value at more than one corner point, then all points on the line segment joining these vertices are optimal solutions.

Important Applications of LPP

Linear Programming can be applied in:

• Manufacturing and production mix problems.
• Transportation and assignment problems.
• Diet problems (minimizing nutrition cost).
• Resource allocation.
• Scheduling tasks.

Example Problem

Q. Solve the following Linear Programming Problem graphically:

Maximize

=Z = 5x + 3y

Subject to the constraints:

x + 2y ≤ 10, x + y ≤ 6, x ≥ 0,y ≥ 0

Solution:

Step 1: Plot the constraints

• For x+2y≤10, the line x+2y=10 cuts the x-axis at (10, 0) and y-axis at (0, 5).
• For x+y≤6x + y \leq 6, the line x+y=6x + y = 6 cuts the x-axis at (6, 0) and y-axis at (0, 6).
• Non-negativity conditions x≥0,y≥0 restrict the feasible region to the first quadrant.

On plotting, the common shaded region forms a quadrilateral OBCD, where

O(0,0),  B(0,5),  C(2,4),  D(6,0) are the corner points.

Step 2: Evaluate the objective function

We calculate Z = 5x + 3y at each corner point:

• At O(0,0):  Z = 5(0) + 3(0) = 0
• At B(0,5):  Z = 5(0) + 3(5) = 15
• At C(2,4): Z = 5(2) + 3(4) = 10 + 12 = 22
• At D(6,0):  Z = 5(6) + 3(0) = 30

Step 3: Find the optimum value

• The maximum value of Z is 30 at (6, 0).
• The minimum value of Z is 0 at (0, 0).

The maximum value of the objective function is

Z = 30 at (x, y) = (6, 0).

The minimum value of the objective function is

Z = 0 at (x, y) = (0, 0).

Conclusion

Well done! You’ve reached the end of these Linear Programming notes and that already proves how determined you are. Feeling better now? Hopefully, the concepts of objective functions, constraints, feasible regions and corner point methods don’t seem so scary anymore. 

Remember, these notes aren’t just about passing exams, they’re about giving you confidence. Each problem you solve makes you sharper and every step you practice takes you closer to becoming a pro at optimization. 

So instead of worrying, download linear programming class 12 notes pdf. Believe in yourself and take it one problem at a time. You’re much more capable than you think. With these notes in your toolkit, Linear Programming will no longer feel like a burden but like a smart strategy game you can ace!

FAQs

Q1. What is Linear Programming in simple words?

Ans: Linear Programming is a method to find the best possible solution (maximum or minimum) for a problem by optimizing a linear objective function under given constraints.

Q2. Why do we only consider corner points in LPP?

Ans: According to the Corner Point Theorem, the optimal solution of an LPP (if it exists) always lies at one of the vertices of the feasible region.

Q3. What if the feasible region is unbounded?

Ans. If the region is unbounded, the solution may not exist. In such cases, we must check if the objective function can be increased or decreased indefinitely.

Q4. Can an LPP have multiple solutions?

Ans. Yes! If the objective function has the same value at two or more corner points, then every point on the line segment joining them gives optimal solutions.

Q5. How can I avoid stress while solving LPP questions?

Ans. Stay calm and follow steps: write constraints clearly, plot accurately, find corner points and test the objective function. With practice, it feels like solving a logical puzzle!