Question 1118359: Please help me to understand how to do "optimization"
Objective function: C = 8x+10y Maximize
Constraints
2x+y _< 12
x+3y _< 21
X_>0, y_>0
Then I need to graph it?
Found 3 solutions by ankor@dixie-net.com, greenestamps, ikleyn:
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! Objective function: C = 8x+10y Maximize
Constraints
2x+y _< 12
x+3y _< 21
X_>0, y_>0
Graph this first, it will be easier to understand
We need the 1st two equations to be changed to the slope intercept
2x + y = 12
y = -2x + 12
and
x + 3y = 21
3y = -x + 21
y = - x + 7
Graph these equations, the area of interest is below the lowest line
Note that only positive solutions are considered
Find the objective solution at the point of intersection which is 3,6
C = 8x + 10y
C = 8(3) + 10(6)
C = 24 + 60
C = 84
Find the other two objective solutions
x=0, y=7
C = 8(0) + 10(7)
C = 70
and
x=6, y=0
C = 8(6) + 10(0)
C = 48
:
Obviously the optimum (max) objective solution is x=3, y=6, which is 84
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
Every resource I have ever seen explaining how to solve this kind of problem says that you have to find the value of the objective function at every corner of the feasibility region. That is simply not true.
Let's look again at your problem, where the constraint functions are  and  .
The objective function is  .
In slope-intercept form, that is  .
We don't know the y-intercept, because we don't know the value of C. But we know the slope of the objective function is -8/10, or -4/5.
Then to find the maximum value of the objective function you only need to determine which corner point of the feasibility region will be just touched by a line with slope -4/5. And since that slope -4/5 is between the slopes of the two constraint functions (-2 and -1/3), the corner point you want is at the intersection of the two constraint lines.
After that, you find that the intersection point is (3,6), and you evaluate the objective function at that point only, finding that the maximum value of the objective function is 10(3)+8(6) = 30+48 = 76.
If the slope of the objective function had been -3, which is smaller than -2, then the maximum value would have been obtained at (6,0); if the slope of the objective function had been -1/5, which is larger than -1/3, the maximum value would have been obtained at (0,12).
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website!
1. The notice that @greenestamps made in his post, is very well known fact,
and it is very well known to those who is familiar with more advanced courses in Linear Programming.
2. Since you, the student, are in the very beginning stage of learning this method, you may follow the method
presented by @ankor@dixie-net.com
I also recommend you to read my introductory lesson
Solving minimax problems by the Linear Programming method
in this site, which is a standard exposition of the method in all details for the beginner student.
It is the best exposition to start with . . .
3. For @greenestapms info, the college textbook, containing the conception he refers to, is (for example) this one:
"College Mathematics for management, life and social science"
3rd edition, R.A.Barnett and M.R.Ziegler, 1984, San Francisco, ISBN 0-02-306220-7.
4. Linear Programming method is very advanced and very well developed industrial optimization method now.
It works in 2D (in plane, and students learn it in introductory geometric version, as it is presented in this post),
in 3D, where visual interpretation is not so easy, and then as a quite abstract method in multi-dimensional spaces
(with 100, 1000 or even 10000 and more dimensions . . . ), where geometric intuition does not work at all.
For each of these stages, the interested reader may find textbooks of relevant levels.
5. For those who knows the method in all details, it might be the way to earn money for life.
6. For the history of the method and many interesting details/applications, see this Wikipedia article
https://en.wikipedia.org/wiki/Linear_programming
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