SOLUTION: Minimize Z=2X+Y 3Y+4X≥24 3Y+2X≥16 Subject to2Y+2X≥14 X≥0 Y≥0 X= Y=

Question 1206461: Minimize Z=2X+Y
3Y+4X≥24
3Y+2X≥16
Subject to2Y+2X≥14
X≥0
Y≥0
X=
Y=

Found 3 solutions by Theo, greenestamps, ikleyn:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
i think you mean:

minimize z = 2x + y subject to
3y + 4x >= 24
3y + 2x >= 16
2y + 2x >= 14
x >= 0
y >= 0

using the desmos.com calculator, you would graph the opposite of the inequalities.
the area of the graph not shaded is the region of feasibility.
the corner points of the region of feasibility will contain the minimum value of the objective function.

the objective function is z = 2x + y.

the corner points of the feasible region are:

(0,8)
(3,4)
(5,2)
(8,0)

the minimum value of z = 2x + y is at the point (0,8), where z = 8.

all the constraints are satisfied.

you get:

3y + 4x >= 24 becomes 3*8 + 4*0 = 24 >= 24 which is true.
3y + 2x >= 16 becomes 3*8 + 2*0 = 24 >= 16 which is true.
2y + 2x >= 14 becomes 2*8 + 2*0 = 16 >= 14 which is true.
x >= 0 becomes 0 >= 0 which is true.
y >= 0 becomes 8 >= 0 which is true.

the value of z is minimum when x = 0 and y = 8.

here's what the graph looks like.

Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

The response from the other tutor shows a solution using the standard method shown in most references.

There is a refinement of that standard method that reduces the amount of effort needed to solve the problem.

Here is a graph of the constraint boundary lines:

You can determine where the objective function is minimized by comparing the slope of the objective function to the slopes of the constraint boundary lines.

The slopes of the constraint boundary lines are -4/3, -2/3, and -1; the slope of the objective function is -2.

The objective function will be minimized where a line with slope -2 just touches the feasibility region.

Imagine lines with slopes of -2 added to the graph above of the constraint boundary lines. Since the slope of the objective function is more negative than the slopes of all of the constraint boundary lines, the function will be minimized at the corner of the feasibility region where the constraint boundary line has a slope closest to -2. That slope is -4/3, which is the red line in the graph.

Here is a graph showing the constraint boundary lines along with a line with slope -2 that just touches the feasibility region:

All other lines with slope -2 would either not pass through any part of the feasibility region, or they would pass through the middle of the feasibility region, where the objective function would not be minimized.

So the objective function is minimized at (0,8). At that point the value of the objective function is 2x+y = 2(0)+8 = 8.

ANSWER: The objective function has a minimum value of 8 at (0,8).

Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!

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