Question

I need some QUICK help here please...

What is the minimum value for.....

Answer 1

z = -x + 3y over the feasibility region defined by the constraints shown above?

Answer 2

I'm kinda in a rush to get his done since I am quite behind so can somebody do a quick help session with me please?

Answer 3

A. 5
B. 4
C. -4
D. -1

Answer 4

You haven't posted the constraints.

Answer 5

oh oops sorry !!!

Answer 6

Which variable has the most impact on the value of \(z\)?

Answer 7

y?

Answer 8

Yes, a 1 unit change in y affects the value of z by 3, whereas a 1 unit change in x affects it by 1.

Do we want y to be large or small when optimizing z here?

Answer 9

large I think

Answer 10

Isn't the coefficient of y positive? Increasing y will increase z, and we want the minimum value of z possible, correct?

Answer 11

Oh ok

Answer 12

Do you agree that our maximum and minimum points are going to be located on vertices of this figure?

Answer 13

So when optimizing the z we want it smaller, how do we get that?

Answer 14

yes

Answer 15

Well, we look for the minimum values of the input variables (x,y) with positive coefficients, and maximum values of the input variables with negative coefficients. That will add the least positive contribution to the value of z, and the greatest negative contribution.

Answer 16

Looking at the contribution of (-x) to the sum, which side of the figure should we investigate?

Answer 17

x=1, or x=7?

Answer 18

x = 1

Answer 19

wait I also think it might be -7 since the way the graph is shaped... I'm not sure

Answer 20

Can you tell me the coordinates of the 4 vertices of this figure?

Answer 21

Yep,

(1, 5.5)
(1, 2)
(7, 3.5)
(7, 2)

Answer 22

right?

Answer 23

Close, but the ones involving a fraction aren't quite right, because the slope of the line is -1/3, not -1/2. (1, 5 2/3) would be correct, as well as (7, 3 2/3)

Now, what do we get for z = -x + 3y at each of those points?

Answer 24

(1, 5 2/3) -> z = -1 + 3*(5 2/3) = -1 + 15 + 2 = 16

Answer 25

oh ok I seek what you did, but now what do we do with that 16?
Also you should know that I am very bad at fractions....in other words, I have no idea how to multiply them but I can do the whole numbers just fine :)

Answer 26

see*

Answer 27

well, that 16 is the value of z at that corner of the figure. We're trying to find the point that gives us the smallest value of z possible while staying on or in that figure. The smallest value of z is our answer.

Answer 28

So now should we try point (7, 3 2/3)?

Answer 29

I'll do the other one involving a fraction if you promise to learn how to multiply fractions!

(7, 3 2/3) -> -7 + 3*(3 2/3) = -7 + 3*3 + 3*(2/3) = -7 + 9 + 2 = 4

Clearly that is a much smaller value of z! But is it the smallest possible?

Answer 30

Ok thanks I'll do the rest right now and then post them :) So, so far we have 16 and 4

Answer 31

(7, 2) -> -7 + 3*(2) = -1?

Answer 32

Is that right?

Answer 33

(1, 2) -> -1 + 3* (2) = 5?

Answer 34

@whpalmer4

Answer 35

Yes, those appear to be correct. So what is the minimum value of z of those four points we tried?

Answer 36

-1!

Answer 37

That's correct.

Answer 38

Thanks can you help me with more please?

Answer 39

But we should convince ourselves that there isn't some point somewhere in the interior of the figure that might have an even smaller value...

Answer 40

but we did all four vertices ?

Answer 41

The point we found our minimum value of z at was (7,2), right?

Answer 42

yes

Answer 43

Yes, we did all 4 vertices. But let's say you need to convince me that there isn't a point right in the middle that has an even smaller value of z associated with it. How would you do that? By doing this, you'll solidify your understanding of this process, and see how you could have come to the correct answer much more quickly than by trying all of the vertices...

Answer 44

ok

Answer 45

so what do I do..... remember I'm kinda In a rush....my spring break is almost over!

Answer 46

Let's look at our equation for \(z\):\[z = -x + 3y\]

What if we hold \(y\) constant, and only vary \(x\)? Starting at the lower left corner, \(y = 2\), so \(z = -x + 3(2) = -x + 6\)

As we move to the right, \(x\) increases and \(z\) steadily decreases at the same rate that \(x\) increases, right?

What that tells us is moving along the x-axis to the right decreases \(z\) (which is what we want). Any point in the middle which has a point at the same value of \(y\) but further to the right on the \(x\) axis is not going to be the optimum value of \(z\). Agreed?

Answer 47

I understand...

Answer 48

Now, what about moving up and down the line at a fixed value of \(x\)?

\[z = -x + 3y\]Let's say \(x = 4\), just to pick something in the middle:
\[z = -4 + 3y\]

To minimize \(z\), we'll want something as close to the x-axis as possible in this figure, right?

And again, for any point in the middle, if there's another point at the same value of \(x\) but having a smaller value of \(y\), then our point in the middle won't be the optimum value of \(z\).

Answer 49

okay

Answer 50

So with our sides all being straight lines like they are, the result is we can just look at our formula: \(z = -x + 3y\) and see that if we want to minimize \(z\), we take the smallest available value of \(y\) (which is 2) and the largest available value of \(x\), because \(y\) adds to \(z\)'s value, and \(x\) subtracts from it. If we have to choose between a couple of points (because the side is slanted, perhaps), then we compare the relative impact of the coefficients, but we don't have that complication here.

Answer 51

So, quickly, without looking back at the values we found, which corner is going to give the maximum value of z?