Mathematics Notes for Class 12 Linear Programming

Linear Programming

Welcome to the chapter on Linear Programming for Class 12. In this chapter, you will learn how to formulate and solve linear programming problems (LPPs), understand their applications, and use graphical methods to find optimal solutions. By the end of this chapter, you will be able to model real-life situations as LPPs and solve them using mathematical techniques.

Key Concepts

Linear Programming: A method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.
Objective Function: The function to be maximized or minimized (e.g., profit, cost).
Constraints: The restrictions or limitations on the variables, expressed as linear inequalities.
Feasible Region: The region in the graph that satisfies all the constraints.
Optimal Solution: The point(s) in the feasible region where the objective function reaches its maximum or minimum value.

Formulating a Linear Programming Problem

Identify the decision variables.
Write the objective function to be maximized or minimized.
Write the constraints as linear inequalities.
Include non-negativity restrictions (variables must be ≥ 0).

Example: A company makes two products, A and B. Each unit of A requires 2 hours of labor and 3 units of raw material. Each unit of B requires 4 hours of labor and 2 units of raw material. The company has 100 hours of labor and 90 units of raw material available. The profit on each unit of A is ₹30 and on B is ₹50. How many units of each should be produced to maximize profit?

Let x = number of units of A, y = number of units of B
Objective: Maximize Z = 30x + 50y
Constraints: 2x + 4y ≤ 100, 3x + 2y ≤ 90, x ≥ 0, y ≥ 0

Graphical Method for Solving LPP

Draw the constraint inequalities on the graph.
Identify the feasible region.
Find the corner points (vertices) of the feasible region.
Evaluate the objective function at each corner point.
The maximum or minimum value occurs at one of the corner points.

Applications of Linear Programming

Maximizing profit or minimizing cost in business.
Resource allocation (labor, materials, time).
Diet problems (minimizing cost while meeting nutritional requirements).
Transportation and assignment problems.

Practice Questions

Formulate an LPP to maximize profit for a factory making two products with given constraints on labor and material.
Solve graphically: Maximize Z = 4x + 3y, subject to x + 2y ≤ 8, 3x + y ≤ 9, x ≥ 0, y ≥ 0.
Explain what is meant by the feasible region in an LPP.
Why does the optimal solution of an LPP always occur at a corner point of the feasible region?
Give one real-life example where linear programming can be used.

Challenge Yourself

Draw the feasible region for the constraints: x + y ≤ 4, x ≥ 0, y ≥ 0. Find the maximum value of Z = 2x + 3y.
Formulate and solve a diet problem using linear programming.

Did You Know?

Linear programming was developed during World War II to solve military logistics problems.
The Simplex method is a popular algorithm for solving LPPs with more than two variables.

Glossary

Objective Function: The function to be maximized or minimized in an LPP.
Constraint: A condition that the solution must satisfy.
Feasible Region: The set of all possible points that satisfy the constraints.
Corner Point: A vertex of the feasible region where two or more constraints intersect.

Answers to Practice Questions

Let x and y be the number of units of two products. Write the objective function for profit and constraints for labor and material. (See example above.)
Draw the inequalities, find the feasible region, evaluate Z at each corner point to find the maximum value.
The feasible region is the area on the graph where all constraints overlap and are satisfied.
Because the objective function reaches its maximum or minimum at a vertex of the feasible region.
Resource allocation in factories, diet planning, transportation, etc.

Practice formulating and solving linear programming problems to master this powerful mathematical tool!

Mathematics Class 12 - Linear Programming Notes