Question

Solve the following LPP by using graphical method.

Maximize : Z = 6x + 4y

Subject to x ≤ 2, x + y ≤  3, -2x + y ≤  1, x ≥  0, y ≥ 0.

Also find maximum value of Z.

Answer 1

Inequalities	x ≤ 2	x+y≤3	-2x+y≤1
Equalities	x=2	x+y=3	-2x+y=1
Intercept form	`x/2=1`	`x/3+y/3=1`	`x/(-1/2)+y/1=1`
Origin Test	0≤2	0+0≤3	-2(0)+0≤1
	True Origin Side	True Origin Side	True Origin Side

Shaded portion OABC is the feasible region, Where O(0,0) A(2, 0) D(0, 1), B(2, 1)

For C :

   x + y = 3

– 2x + y = 1

   –   –      –

----------------------

   3x = 2

∴ x = 2/3

2/3+y=3 i.e y=7/3

`∴ c(2/3,7/3)`

Z = 6x + 4y

Z at O(0, 0) = 6(0) + 4(0) = 0

Z at A(2, 0) = 6(2) + 4(0) = 12

Z at B(2, 1) = 6(2) + 4(1) = 16

Z at `c(2/3,7/3)=6(2/3)+(7/3)4=40/3`

Z at D(0,1) = 6(0) + 4(1) = 4 Thus, Z is maximized at B(2, 1) and its maximum value is 16.