Question

Solve the following using graphical method : 

Minimize :Z=3x+5y

`2x+3x>=12` 

`-x+y<=3` 

`x<=4,y>=3,x>=0,y>=0` 

 

 

Answer 1

Min `Z = 3x + 5y`
S.t. `2x + 3y ≥ 12     `               …(i)
`– x + y ≤ 3          `                   …(ii)
`x ≤ 4, y ≥ 3, x ≥ 0, y ≥ 0` 

Taking eqn (i)
2x + 3y = 12
Putting x = 0, y = 4 Let the point is (0, 4)
Now putting y = 0, x = 6 Let the point is (6, 0)
Now taking eqn (ii)
– x + y = 3
Putting x = 0, y = 3 (0, 3)
Putting y = 0, x = – 3 (– 3, 0)
The graph is as follows 

ABCDA be the feasible region bounded by these
lines Now we find the coordinates of A, B, C and D
for A, Solving the eqns.

`2x+3y=12 and -xy=3` 

We get 

`x=(+3)/5  and  y=18/5`

coordinate of `A((+3)/5,18/5)` 

Now 

`Z=3xx(+3/5)+5xx18/5` 

=`(+9)/5+90/5=90/5`

For B, Solving the eqns 

`2x+3y=12 and y=3` 

We get `x=3/2,y=3`

∴Coordinate of B `(3/2,3)` 

Now` Z=3xx3/2+5xx3` 

=`9/2+15=39/2 `

For C. Solving the eqn x = 4 and y = 3  

∴ Coordinate of C (4, 3) 

Now `Z=3xx4+5xx3` 

=`12+1=27` 

For D, Solving the eqn 

– x + y = 3 and x = 4
We get  x = 4, y = 7

Now Z = 3 × 4 + 5 × 7
= 12 + 35 = 47 

Min Z = `39/2, "for" x=3/2,y=3`