Question 1137424: Solve the following LLP graphically using the iso-profit method. Maximize Z=100+100y
subject to constraints:
10x+5y≤80
6x+6y≤66
4x+8y≥24
5x+6y≤90
X≥0,Y≥0
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
Here is a graph of the constraint boundary lines:
red 10x+5y <= 80 (y = -2x+16)
green 6x+6y <= 66 (y = -x+11)
blue 4x+8y >= 24 (y = -.5x+3)
purple 5x+6y<= 90 (y = -(5/6)x+15)
The feasibility region is in the first quadrant, above the blue line and below the red and green lines. Note the purple line has nothing to do with the solution of the problem. The corners of the feasibility region are...
(0,3) (y-intercept of blue line)
(0,11) (y-intercept of green line)
(5,6) (intersection of red and green lines)
(8,0) (x-intercept of red line)
(6,0) (x-intercept of blue line)
Evaluate the objective function z = 100x+100y at each corner of the feasibility region:
(0,3): z = 300
(0,11): z = 1100
(5,6): z = 1100
(8,0): z = 800
(6,0): z = 600
The objective function is maximized at both (0,11) and (5,6). That means the maximum value of the objective function is anywhere along the boundary of the feasibility region between (0,11) and (5,6).
NOTE: It is NOT necessary to evaluate the objective function at every corner of the feasibility region. You can tell where the objective function will be maximized by comparing the slope of the objective function to the slopes of the constraint boundary lines.
In this example, the slope of the objective function z=100x+100y is -1, which is the SAME as the slope of the constraint boundary line 6x+6y=66. That means the objective function will be maximized at any point on the portion of the constraint boundary line 6x+6y=66 that is a border of the feasibility region.
Answer by ikleyn(52800)  (Show Source):
You can put this solution on YOUR website! .
The problem asks to maximize Z = 100 + 100y.
@greenestamps solves another problem maximizing 100x + 100y.
I understand that with high probability there is A MISTAKE in the problem' formulation,
and in reality Z = 100x + 100y should be there.
But in such cases it should be noted in solution, isn't it ?
In such cases, I always do it placing warning notices before the solution . . .
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