Expert: Steve Holleran - 4/17/2007

Question
I have a problem that I have the answer to, but was hoping you could tell me how
I get the points?  Here is the problem:



Maximize Z= 4x + y subject constraints -x + y <2,



3x + y < 18, x > 0, y > 0, using the linear programing, that is graphing
techniques.



I get the 4 points are (0,0), (0,2), (4,6), (6,0).  I am just unsure without a
graphing calculator how to get these numbers, how do you take the contraints and
get the points out of it for the corners?



It also asks me to take this same problem and use the simplex method, if you
could show me how to do that, it would greatly help as well, which I know matrix
is used in this.  If you can't that is ok, I just really would like to know how
to get the points and how to do the graph.



If you could help me it would be a great help!!  I have asked this to three others and they could not help, so I am hoping you can....:)







Thank you,
Andy  

Answer
Hi Andy,

Well, I'm about to join the rest of the others--I can't ever recall using the simplex method to do this type of problem.  If I studied up on it, I may be able to help with it, but I just don't have any experience there.


On the first part of the question, though, I can be of some assistance:

You want to graph the lines represented by each of the constraints:

-x + y < 2 ----------->  y < x + 2   (y-int 2, slope 1)

3x + y < 18 ---------->  y < -3x + 18 (y-int 18, slope -3)

x > 0 ----------------> to right of y-axis

y > 0 ----------------> above x-axi

The point (0,0) is where x=0 and y=0 intersect,
The point (0,2) is where x = 0 and Y = x+2 intersect
The point (6,0) is where y = 0 and 3x+y = 18 intersect,
and to find the other point, take

y = -3x + 18 and y = x + 2 and solve simultaneously:

             -3x + 18 = x + 2

                   - 4x = - 16

                     x = 4  and then replace to get y = 6.

If you put each of the points into the objective function,
I think you get a maximum of 24 for Z at (6,0).

Sorry I can't help out on the simplex thing.

Steve Holleran