Linear Programming: Graphical Method

GRAPHICAL METHOD

Learning Objectives

After reading this chapter the student must be able to

a. Plot the constraints of the LPP on a graph paper

b. Identify the feasible region in the graph

c. Identify the corner points of the region

d. Plot the objective function

e. Deduce the optimum solution of the problem

Basic terms

i. Solution

Values of the decision variable x;(i = 1,2,3, in) satisfying the constraints of a general linear programming model is known as the solution to that linear programming model.

ii. Feasible Solution

Out of the total available solution a solution that also satisfies the non-negativity restrictions of the linear programming problem is called a feasible solution.

iii. Basic Solution

For a set of simultaneous equations in Q unknowns (p Q) a solution obtained by setting (P - Q) of the variables equal to zero & solving the remaining P equation in P unknowns is known as a basic solution. The variables which take zero values at any solution are detained as non-basic variables & remaining are known as-basic variables, often called basic.

iv. Basic Feasible Solution

A feasible solution to a general linear programming problem which is also basic solution is called a basic feasible solution.

v. Optimal Feasible Solution

Any basic feasible solution which optimizes (i.e.; maximize or minimizes) the objective function of a linear programming modes known as the optimal feasible solution to that linear programming model

vi. Degenerate Solution

A basic solution to the system of equations is termed as degenerate if one or more of the basic variables become equal to zero.

I hope the concepts that we have so far discussed have been fully understood by all of you.

Friends, it is now the time to supplement our understanding with the help of examples.

Example-1 solve the following LPP graphically

                      

Solution:

  First consider the inequality constraints are to be equal

          

 Let we assume like following

      

      In the same way  

           

                        X₁=2

And   for third equation we can get (0,3) and (3,0) this can be plotted in the graph

 

In the above graph common area is between O,A,B and C points, these area is called Feasible area and   there fore

At O(0,0),A (2,0),B(2,1) and C(2/3,7/3)

Example-2 Solve graphically following LPP

             

Solution :

 

 in the above Graph it is noted that the feasible region is  unbounded, there fore solution is unbounded

Related Topics 
 
Linear Programming Problem
LINEAR PROGRAMMING –SIMPLEX METHOD