SOLUTION: Solve the following linear programming problem. Restrict x ≥ 0 and y ≥ 0. Minimize g = 9x + 8y subject to the following. 5x + 2y ≥ 16 3x + 7y ≥ 27

Question 1148028: Solve the following linear programming problem. Restrict
x ≥ 0
and
y ≥ 0.
Minimize
g = 9x + 8y
subject to the following.
5x + 2y ≥ 16
3x + 7y ≥ 27
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

For finding x- and y-intercepts of the constraint boundary lines, the equations in the given form are useful.

5x + 2y = 16 --> (0,8) and (16/5,0)
3x + 7y = 27 --> (0,27/7) and (9,0)

There are many ways to find the point of intersection of the two boundary lines. I won't specify any particular method; the intersection point is (2,3).

Since we are minimizing the objective function, the corner points of our (open) feasibility region are (0,8), (2,3), and (9,0).

Virtually all references will say that you need to evaluate the objective function at all corners of the feasibility region. But that is not true.

You can determine which corner of the feasibility region will yield the minimum value of the objective function by comparing the slopes of the objective function and the constraint boundary lines.

The slope of the first equation is -5/2; the slope of the second is -3/7. The slope of the objective function is -9/8. Since the slope of the objective function is between the slopes of the two constraint lines, the minimum value of the objective function will be obtained at the intersection of the two constraint lines.

So the minimum value of the objective function is 9(2)+8(3) = 18+24 = 42.

We can go ahead and evaluate the objective function at the other two corners to verify that our result is correct:
(0,8) --> 9(0)+8(8) = 0+64 = 64
(9,0) --> 9(9)+8(0) = 81+0 = 81

Yes, the minimum value is at (2,3).

Here is a graph of the two constraint boundary lines:

To visualize why we don't have to evaluate the objective function at every corner of the feasibility region, imagine the family of all functions with the same slope of the objective function. To minimize the value of the objective function, we want the function from that family that just touches a corner of the feasibility region, rather than passing through it.

And in this case, since the slope of the objective function is between the slopes of the two constraint lines, the corner we want is where the two constraint lines intersect.

Answer by ikleyn(52786)   (Show Source): You can put this solution on YOUR website!
.

To see similar problems solved by the Linear Programming method and to learn how the method works, look into the lesson
    - Solving minimax problems by the Linear Programming method
in this site.

Also, look into the solutions of other similar problems in the archive to this forum under the links

https://www.algebra.com/algebra/homework/Inequalities/Inequalities.faq.question.1142000.html
https://www.algebra.com/algebra/homework/Sequences-and-series/Sequences-and-series.faq.question.1137172.html
https://www.algebra.com/algebra/homework/Graphs/Graphs.faq.question.1136382.html
https://www.algebra.com/algebra/homework/Coordinate-system/Coordinate-system.faq.question.1134444.html
https://www.algebra.com/algebra/homework/Graphs/Graphs.faq.question.1131906.html
https://www.algebra.com/algebra/homework/coordinate/word/Linear_Equations_And_Systems_Word_Problems.faq.question.1131043.html
https://www.algebra.com/algebra/homework/word/finance/Money_Word_Problems.faq.question.1129285.html
https://www.algebra.com/algebra/homework/playground/test.faq.question.1112482.html
https://www.algebra.com/algebra/homework/Finance/Finance.faq.question.1102103.html

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