NCERT Solutions for Class 12 Maths Chapter 12 – Linear Programming

Introduction

Linear Programming is a crucial topic in Class 12 Mathematics, introduced in Chapter 12 of the NCERT syllabus. It is an area of optimization concerned with maximizing or minimizing a linear objective function subject to a set of linear constraints. The practical applications of Linear Programming are vast, ranging from economics and business to engineering and logistics. This article provides an in-depth exploration of NCERT Solutions for Class 12 Maths Chapter 12 – Linear Programming, including key concepts, problems, and solutions to help students master this fundamental topic.

Understanding Linear Programming

Linear Programming involves the formulation of mathematical models to represent real-world problems and finding optimal solutions within given constraints. The general structure of a linear programming problem includes:

1. Objective Function: A linear function that needs to be maximized or minimized.
2. Constraints: Linear inequalities that restrict the feasible region in which the solution lies.
3. Non-negativity Constraints: Constraints that require decision variables to be non-negative.

Key Concepts in Linear Programming

1. Formulation of Linear Programming Problems: This involves translating a real-world problem into a mathematical model consisting of an objective function and constraints.

2. Graphical Method: For problems involving two variables, the graphical method is used to find the feasible region and the optimal solution by plotting constraints on a graph.

3. Simplex Method: For problems involving more than two variables, the simplex method is employed. This algorithm iterates through feasible solutions to find the optimal one.

4. Duality: Every linear programming problem has a corresponding dual problem. The solutions to the primal and dual problems provide insights into the problem’s optimality and feasibility.

5. Sensitivity Analysis: This involves studying how changes in the coefficients of the objective function and constraints affect the optimal solution.

NCERT Solutions for Class 12 Maths Chapter 12

The NCERT textbook provides a comprehensive approach to Linear Programming with detailed explanations and step-by-step solutions. Here, we’ll discuss the main types of problems and solutions provided in the chapter.

1. Formulation of Linear Programming Problems

Example Problem:

A company produces two products, A and B. The profit for each unit of product A is Rs 5, and for product B, it is Rs 3. The production constraints are as follows:

• Product A requires 2 hours of labor and 1 unit of raw material per unit.
• Product B requires 1 hour of labor and 2 units of raw material per unit.
• The total available labor hours are 8, and the total available raw material units are 10.

Formulate the Linear Programming problem to maximize the profit.

Solution:

1. Define Variables:

   • Let $x$ be the number of units of product A.
   • Let $y$ be the number of units of product B.

2. Objective Function:

   • Maximize Profit $Z = 5x + 3y$.

3. Constraints:

   • Labor constraint: $2x + y \leq 8$.
   • Raw material constraint: $x + 2y \leq 10$.
   • Non-negativity constraints: $x \geq 0$ and $y \geq 0$.

2. Graphical Method

Example Problem:

Solve the following linear programming problem graphically:

• Maximize $Z = 3x + 2y$
• Subject to:
  • $x + y \leq 4$
  • $x \leq 2$
  • $y \leq 3$
  • $x \geq 0$ and $y \geq 0$

Solution:

1. Plot the Constraints:

   • Draw the lines representing the constraints on a graph.
   • Identify the feasible region that satisfies all the constraints.

2. Find the Corner Points:

   • Determine the coordinates of the corner points of the feasible region by solving the system of linear equations given by the intersection of constraints.

3. Evaluate the Objective Function:

   • Calculate the value of the objective function $Z = 3x + 2y$ at each corner point.
   • The point where $Z$ is maximized is the optimal solution.

3. Simplex Method

Example Problem:

Solve the following linear programming problem using the simplex method:

• Maximize $Z = 4x_1 + 3x_2$
• Subject to:
  • $2x_1 + x_2 \leq 8$
  • $x_1 + 2x_2 \leq 6$
  • $x_1, x_2 \geq 0$

Solution:

1. Convert to Standard Form:

   • Introduce slack variables to convert inequalities into equalities.
   • The constraints become:
     • $2x_1 + x_2 + s_1 = 8$
     • $x_1 + 2x_2 + s_2 = 6$
   • Where $s_1$ and $s_2$ are slack variables.

2. Set Up the Initial Simplex Tableau:

   • Create the initial tableau with the objective function and constraints.

3. Perform Simplex Iterations:

   • Apply the simplex algorithm to iterate and update the tableau until an optimal solution is reached.

4. Extract the Solution:

   • The optimal values of $x_1$ and $x_2$ can be found in the final tableau.

4. Duality in Linear Programming

Example Problem:

Given the primal problem:

• Maximize $Z = 2x_1 + 3x_2$
• Subject to:
  • $x_1 + 2x_2 \leq 8$
  • $3x_1 + x_2 \leq 12$
  • $x_1, x_2 \geq 0$

Formulate the dual problem.

Solution:

1. Formulate the Dual Problem:

   • The dual problem involves minimizing the dual objective function subject to dual constraints.
   • For the given primal problem, the dual problem is:
     • Minimize $W = 8y_1 + 12y_2$
     • Subject to:
       • $y_1 + 3y_2 \geq 2$
       • $2y_1 + y_2 \geq 3$
       • $y_1, y_2 \geq 0$

2. Solve the Dual Problem:

   • The optimal solution to the dual problem provides insight into the primal problem's optimality.

5. Sensitivity Analysis

Example Problem:

Given the linear programming problem:

• Maximize $Z = 4x_1 + 5x_2$
• Subject to:
  • $2x_1 + 3x_2 \leq 12$
  • $4x_1 + x_2 \leq 8$
  • $x_1, x_2 \geq 0$

Suppose the constraint coefficients are changed. Perform a sensitivity analysis to determine how these changes affect the optimal solution.

Solution:

1. Recalculate the Optimal Solution:

   • Solve the revised linear programming problem with the new coefficients.

2. Analyze the Impact:

   • Compare the new optimal solution with the original to understand the impact of the changes.

Conclusion

Linear Programming is a vital area in Class 12 Mathematics, offering students insights into optimization and problem-solving techniques. The NCERT Solutions for Class 12 Maths Chapter 12 provide a thorough understanding of linear programming, covering the formulation of problems, graphical and simplex methods, duality, and sensitivity analysis. Mastering these concepts not only helps in academic success but also equips students with practical skills applicable in various fields. By working through the problems and solutions provided in the NCERT textbook, students can build a solid foundation in Linear Programming and prepare effectively for their examinations.