Question

Minimise Z = x + 2 y subject to .

Answer 1

The given constraints are,

2x+y≥3 x+2y≥6 x≥0 y≥0

The given objective function which needs to minimise is,

Z=x+2y

The line 2x+y≥3 gives the intersection point as,

x	0	3 2
y	3	0

Also, when x=0,y=0 for the line 2x+y≥3, then,

0+0≥3 0≥3

This is false, so the graph have the shaded region away from the origin.

The line x+2y≥6 gives the intersection point as,

x	0	6
y	3	0

Also, when x=0,y=0 for the line x+2y≥6, then,

0+0≥6 0≥6

This is false, so the graph have the shaded region away the origin.

By the substitution method, the intersection points of the lines 2x+y≥3 and x+2y≥6 is ( 0,3 )

Plot the points of all the constraint lines,

It can be observed that the corner points are A( 6,0 ),B( 0,3 ).

Substitute these points in the given objective function to find the minimum value of Z.

Corner points	Z=x+2y
A( 6,0 )	6
B( 0,3 )	6

This shows that the value of Z at points A and B are same.

Consider the line of equation x+2y=6, then, at x=2 and y=2, the value comes out to be Z=6.

It is observed that the minimum value of Z occurs at more than 2 points.

Therefore, the value of Z is minimum at every point on the line x+2y=6.