# Linear Programming Class 12 Notes Maths Chapter 12 - CBSE

## What are Linear Programming ?

A linear programming problem is one that is concerned with finding the optimal value of a linear function of several variables subject to the conditions that the variables are non- negative and satisfy a set of linear inequalities.

## Angle Between Two Lines

• Constraints : The linear inequalities or equations or restrictions on the variables of a linear programming problem are called constraints. The conditions x ≥ 0, y ≥ 0 are called non-negative restrictions.
• Objective Function : Linear function Z = ax + by, where a, b are constants, which has to be maximized or minimized is called a linear objective function.
• Optimisation : A problem which seeks to maximise or minimize a linear function subject to certain constraints as determined by a set of linear inequalities is called an optimization.
• Different Types Of Linear Programming Problems : A few important linear programming problems are: (i) Diet problems (ii) Manufacturing problems (iii) Transportation problems

## Graphical Method Of Solution For Problems In Two Variables

Step: 1→Formulate the LP problem

Step: 2→Construct a graph and plot the constraint lines.

Step: 3→Determine the valid side of each constraint line.

Step: 4→Identify the feasible solution region.

Step: 5→Plot the objective function on the graph.

Step: 6→Find optimum point.

## Feasible And Infeasible Regions

The common region determined by all the constraints including the non-negative constraints x ≥ 0, y ≥ 0 of a linear programming problem is called the feasible region. The region other than feasible region is called an infeasible region.

## Feasible And Infeasible Solutions

Points within and on the boundary of the feasible region represent feasible solutions of the constraint. Any point outside the feasible region is an infeasible solution.

## Optimal Feasible Solutions

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