Linear Programming Class 12th Notes - Free NCERT Class 12 Maths Chapter 12 Notes - Download PDF

Ever wondered how delivery companies know the shortest path? How do airlines make their flight schedule work? Or how can any business company minimize the cost while maximising the profits? The answer to these real-world problems lies in Linear Programming, an interesting mathematical method for finding the optimum solution for various problems. From NCERT Class 12 Maths, the chapter Linear Programming contains the concepts of Objective Function, Decision Variables, Constraints, Feasible Region, Optimal Solutions, etc. Understanding these concepts will help the students grasp Linear Programming easily and enhance their problem-solving ability in real-world applications.

This article on NCERT notes Class 12 Maths Chapter 12 Linear Programming offers well-structured NCERT notes to help the students grasp the concepts of Linear Programming easily. Students who want to revise the key topics of Linear Programming quickly will find this article very useful. It will also boost the exam preparation of the students by many folds. These notes of Class 12 Maths Chapter 12 Linear Programming are made by the Subject Matter Experts according to the latest CBSE syllabus, ensuring that students can grasp the basic concepts effectively. NCERT solutions for class 12 maths and NCERT solutions for other subjects and classes can be downloaded from the NCERT Solutions.

NCERT Class 12 Maths Chapter 12 Notes

Linear programming is generally defined as the technique for maximizing or minimizing a linear function of several variables, like input or output cost.

Linear Programming Problem And Its Mathematical Formula

A Mountaineering equipment dealer deals in only two items–tents and rucksacks. He has Rs 50,000 for investment and has storage for compiling a maximum of 60 pieces. A tent is priced at Rs 2500 and a rucksack at Rs 500. He estimates that by selling one tent he can get a profit of Rs 250 and that from the. The sale of one rucksack earns a profit of Rs 75. So, how many tents and rucksacks should he buy with the money to maximize his total profit, assuming he can sell all the products he buys?

Here we can observe

• The dealer can invest his money in buying a tent or rucksacks, or a combination thereof. Later, he would earn different ways of profits by following different investment strategies.
• There are a lot of conditions, i.e., His investment is a maximum of Rs 50,000, and he has storage of a maximum of 60 pieces.

Let us assume that he decides to buy a tent only and no rucksacks, so he can buy 50000 ÷ 2500, i.e., 20 tents. Thus, his profit can be written as: Rs (250 × 20), i.e., Rs 5000.

Suppose he chooses to buy rucksacks only and no tent. With the capital of Rs 50,000, he can buy 50000 ÷ 500, i.e., 100 rucksacks. But he can only store 60 pieces. Therefore, he can only buy 60 rucksacks, which will give a profit of Rs (60 × 75), i.e., Rs 4500.

There are other possibilities too, like he may choose to buy 10 tents and 50 rucksacks, as his storage is limited to only 60 pieces. The total profit would be calculated as Rs (10 × 250 + 50 × 75), i.e., Rs 6250, and so on.

Thus, we find that the dealer can invest his money in many different ways so that he would earn different profits by the given different investment strategies.

Now the question arises, how should he invest to get the maximum profit? For this, let us see the process mathematically.

Mathematical Formulation Of The Problem

Let x be the number of tents and y be the number of rucksacks that the dealer buys. Here, x and y must be non-negative, i.e.,

$\begin{array}{rl}& x⩾0\dots \dots \dots \dots .1\\ & y⩾0\dots \dots \dots \dots .2\text{ (Non-negative constraints) }\end{array}$

Mathematically,

2500x + 500y ≤ 50000 (investment)

or 5x + y ≤ 100 ... (3)

and x + y ≤ 60 (storage) ... (4)

Z = 250x + 75y ( objective function) ... (5)

Now, the given problems now reduce to:

Maximize Z = 250x + 75y

5x + y ≤ 100

x + y ≤ 60

x ≥ 0, y ≥ 0

Such a problem is called linear programming.

Some Basic Terminology

Objective function :

Let Z = ax + by be a linear function, where a, b are the constants. Linear objective function goes by the computation of the maximum or the minimum of X

Let Z = ax + by be a linear objective function. Then the variables x and y are called decision variables.

Constraints :

Constraints are the limitations or restrictions imposed on the decision variables.

Optimization problem:

A problem that asks to maximize or minimize a linear function limited to certain constraints.

Graphical Method Of Solving Linear Programming Problems

Let us see a graph

5x + y ≤ 100 ... (1)

x + y ≤ 60 ... (2)

x ≥ 0 ... (3)

y ≥ 0 ... (4)

Each point in the shaded region represents a feasible choice; therefore, it is called the feasible region.

Any points outside feasible points are called infeasible points.

Optimal (feasible) solution:

Any point in the feasible region that gives the maximum or minimum value of the objective function is called an optimal solution.

Methods Of Solving A Linear Problem

1. Find the feasible region of the problem and find the vertices.

2. Find the objective function Z = ax + by. Let M and m be the largest and the smallest points of the problem

3. i) When the area is bounded. "M" and "m" are maximum and minimum values. If a feasible area is unbounded then

• ax + by > M, no common points with the feasible region.

• ax + by < m, no common points with the feasible region.

Different Types Of Linear Programming Problems

• Manufacturing Problem: Here we describe the different types of units that should be produced in a firm

• Diet Problems: Here we discuss the requirements of essential nutrients for our body.

• Transportation Problem: Here we discuss the cheapest way of transporting materials from factories to different locations

This brings us to the end of the chapter.

NCERT Class 12 Notes Chapter Wise.

Subject Wise NCERT Exemplar Solutions

After finishing the textbook exercises, students can use the following links to check the NCERT exemplar solutions for a better understanding of the concepts.

• NCERT Exemplar Class 12 Solutions
• NCERT Exemplar Class 12 Maths
• NCERT Exemplar Class 12 Physics
• NCERT Exemplar Class 12 Chemistry
• NCERT Exemplar Class 12 Biology

Subject Wise NCERT Solutions

Students can also check these well-structured, subject-wise solutions.

• NCERT Solutions for Class 12 Mathematics
• NCERT Solutions for Class 12 Chemistry
• NCERT Solutions for Class 12 Physics
• NCERT Solutions for Class 12 Biology

NCERT Books and Syllabus

Students should always analyze the latest CBSE syllabus before making a study routine. The following links will help them check the syllabus. Also, here is access to more reference books.

• NCERT Book for Class 12
• NCERT Syllabus for Class 12

Important points to note:

• NCERT problems are very important in order to perform well in the exams. Students must try to solve all the NCERT problems, including miscellaneous exercises, and if needed, refer to the NCERT solutions for class 12 Maths chapter 12 Linear Programming.
• Students are advised to go through the NCERT Class 12 Maths Chapter 3 Notes before solving the questions.
• To boost your exam preparation as well as for a quick revision, these NCERT notes are very useful.

Happy learning !!!