## Chapter:12

# 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.