( iii) The polygonal region so obtained, satisfying all the constraints and the non-negative restrictions is the convex set of all feasible solutions of the given LPP, which is also known as feasible region.
( iv) Determine the extreme points of the feasible region.
( v) Give some convenient value k to the objective function Z and draw the corresponding straight line in the xy-plane.
( vi) If the problem is of maximization, then draw lines parallel to the line Z = k and obtain a line which is farthest from the origin and has atleast one point common to the feasible region. If the problem is of minimization, then draw lines parallel to the line Z = k and obtain a line, which is nearest to the origin and has atleast one point common to the feasible region.
( vii) The common point so obtained is the optimal solution of the given LPP.
Working Rule for Marking Feasible Region Consider the constraint ax + by ≤ c, where c > 0.
First draw the straight line ax + by = c by joining any two points on it. For this find two convenient points satisfying this equation.
This straight line divides the xy-plane in two parts. The inequation ax + by c will represent that part of the xy-plane which lies to that side of the line ax + by = c in which the origin lies.
Again, consider the constraint ax + by ≥ c, where c > 0. Draw the straight line ax + by = c by joining any two points on it.
This straight line divides the xy-plane in two parts. The inequation ax + by ≥ c will represent that part of the xy-plane, which lies to that side of the line ax + by = c in which the origin does not lie.
Important Points to be Remembered
( i) Basic Feasible Solution A BFS is a basic solution which also satisfies the non-negativity restrictions.
( ii) Optimum Basic Feasible Solution A BFS is said to be optimum, if it also optimizes( Max or min) the objective function.