Question: MaxZ=$3x+$9y,
What is the amount of slack associated with the second restraint? Subject to:
20x+32y<1600
4x+2y<240
y<40
x,y>0
Maximise z = 3x + 9y
Subject to
20x + 32y < 1600
4x + 2y < 240
y < 40
x,y > 0
Or,
20x + 32y +v1 = 1600
4x + 2y + v2 = 240
y + v3 = 40
x,y > 0
where v1, v2, v3 are the slack variables.
Graphical Solution
Draw the straight lines corresponding to the following constraints.
l1: 20x + 32y = 1600, or 5x + 8y = 400
l3: 4x + 2y = 240, or 2x + y = 120
l3: y = 40
The objective function z is Bottom of Form
on the line 3x + 9y = const.
The objective function is maximised when the line, 3x + 9y = const, is furthest from the origin and still within the feasible region.
This happens when the line 3x + 9y = const meets the point A, the intersection of the lines l1 and l3.
Intersection Point
l1: 5x + 8y = 400
l3: y = 40 => x = 16
Therefore z = 3x + 9y = 48 + 340 = 408.00
Maximisation of objective function is z = $408.00
The slack variable
The 2nd constraint is
4x + 2y + v2 = 240
Using x = 16, y = 40,
64 + 80 + v2 = 240
v2 = 240 – 144
v2 = 96