Need help with Linear Programming

[seangman]

I am trying to figure out these problems, but the book isn't to clear with its examples so I am stuck. I was never good with math (english major yay), and I hope someone can help me out and what I need to do to solve these. Thanks for your time and help in advance for anyone who helps me out.

1. Maximize the aim function 7x+3y subject to the constraints:

3x+2y<36, x+4y<32, x>4, y>0


2. Maximize the aim function 2/3x+3/4y subject to the constraints:

2x+y<8, 3x+5y<16, x>0, y>3


3. A farmer has 100 acres on which to plant oats or corn. Each acre of oats requires $18 capital and 2 hours of labor. Each acre of corn requires $36 capital and 6 hours of labor. The farmer has $2100 available for capital and 360 hours for labor. If the revenue is $55 from each acre of oats and $125 from each acre of corn, what planting combination will produce the greatest revenue?[/u]

 

[tkhunny]

Could you suggest a methodology? Are we just graphing? Since there are only two variables, it should be possible.

Graph all your inequalities, marking the acceptable region for each. In this case, graph:

3x + 2y <= 36
y <= 18 - (3/2)x

x+4y <= 32
y <= 8 - (1/4)x

x >= 4

y >= 0

The cool thing is, you have to check ONLY the corners. You WILL find the maximum value at one of the corners.

 

[seangman]

Here is what we have to do:

Step 1- Make a table

Step 2 - Write aim function & constraint inequalities

Step 3 - Fine feasible set by drawing lines

Step 4 - Find corner point & plug in aim function

Step 5 - Give answer.

Sorry for being vague, I hope that is what you were asking for.

 

[tkhunny]

That sounds essentially as I described it above.

The only point of caution is to encourage you to find the EXAACT points of intersection. Your graph may not be good enough to provide sufficient information. The algebraic solution should rule.

Well, graph away and find the corners.

 

[seangman]

All right, I got the first two easy enough thanks to the corner points. Can you give me some hints on the set up of the third problem? I don't know where to start really on that one, but I will keep trying until I figure something out.

 

[tkhunny]

3. A farmer has 100 acres on which to plant oats or corn. Each acre of oats requires $18 capital and 2 hours of labor. Each acre of corn requires $36 capital and 6 hours of labor. The farmer has $2100 available for capital and 360 hours for labor. If the revenue is $55 from each acre of oats and $125 from each acre of corn, what planting combination will produce the greatest revenue?

100 Acres Total
$2100 Capital Total
360 hours Labor Total

Oats Cost: $18 + 2 hours
Corn Cost: $36 + 6 hours

Revenue: $55*Oats + $125*Corn

OT is the numer of acres of Oats
CR is the number of acres of Corn

From each limited resource, we build a constraint.

100 Acres Total

OT + CR <= 100

$2100 Capital Total
OT*18 + CR*36 <= 2100

360 hours Labor Total
OT*2 + CR*6 <= 360

Add some practical considerations.
OT >= 0
CR >= 0

And you're on your way.

Note: You should observe that one of the constraints is redundant. Making it otherwise would require what?