BG Maths Contest !

[LinktheDeme]

Well I got a homework problem that is about maximizing and I know shit about it so I just did trial and error and compared with my friends what their total profit was.

My homework is to optimize the profit per box sold. The highest I got out of 10 tries was $72. Problem involved more math but I did some of it since typing out everything else would be meh.

Rules

-No more than 10 of the same nail in a box.
-No less than 5, of each of at least 3 types of nails in a box
(meaning they're has to be three different types of nails that have 5 in them) (example: five 4-inch nails, five 3.5 inch nails, five 3 inch nails and the rest of the 15 can be distributed assuming the other rules arent broken)
-The box cannot weigh more than 20 ounces total
-Their cannot be more than 30 nails in a box

4 inch nails = 1 ounce and $4 profit per nail
3.5 inch nails = .85 and $2.5 profit per nail
3 inch nails = .7 and $2 profit per nail
2.5 inch nails = .5 and $1.5 profit per nail
2 inch nails = .25 and $1.0 profit per nail
1.5 inch nails = .1 and $0.5 profit per nail

Whoever beats $72 per box and has the highest one by 11 PM EST Tonight. I'll paypal five real dollars to you (not much but its incentive if your doing this for fun) if you tell me where you allocated each nail and it's within the rules. If multiple people PM me with one higher than $72 per box than the person who PMed me first wins.

An example of one that I did that works is

4 inch - 10
3.5 inch - 10
3 inch - 0
2.5 inch - 0
2 inch - 3
1.5 inch - 7

This results in 19.95 which is no more than 20 (20 ounces is the max) so that rule is fine

30 nails so that rule is fine

and 4 inch, 3.5 inch, and 1.5 inch are 3 different nails that have at least 5 in them so this rule is fine

Lastly no more of 10 nails were in any category so that rule is fine.

edit: Resulting Total Profit in example was 4(10) + 2.5(10) + 1(3) + .5(7) = $71.5

[Mizango]

Nice try. I saw the word "contest" and got excited, then I saw it was homework and changed my mind.

Spoiler: show

http://i5.photobucket.com/albums/y15...aReaction2.png

[LinktheDeme]

Nice try. I saw the word "contest" and got excited, then I saw it was homework and changed my mind.

Spoiler: show

http://i5.photobucket.com/albums/y15...aReaction2.png

Still a contest because you can wins and I'll give you moneyz >.>

[Eliseos]

Nice try. I saw the word "contest" and got excited, then I saw it was homework and changed my mind.

Spoiler: show

http://i5.photobucket.com/albums/y15...aReaction2.png

We should report him to his teacher, saying that he was attempting to pay us to do his work.

[Mizango]

lol

[kuronosan]

E-mailing teacher now.

[Cooleko]

I'm not sure what your lvl of math is, but the general way to solve an optimization problem is:

Make base equation to be optimized (in this case your box total)
Make an equation for each constraint
apply first and second derivatives
???
Profit!

this will always give you the optimal solution to any bounds/optimization problem

[Woozie]

Since it's homework, we're obviously not going to just solve the problem and post the solution. Try what Cooleko said, or try something else. If you get stuck, tell us what you've done so far, and tell us what math class this is for (we need to know what class this is in order to know what level you're on and how your teacher expects you to solve), show some of your work, and we'll guide you from wherever you're stuck, to a solution.

[Titanss]

right now the op is fucking doing a

http://i57.photobucket.com/albums/g2...m/Faceslap.jpg

[Blubbartron]

This is the knapsack problem with some extra rules, if anybody cares. It seems to be the right answer to me so far. Grab the most valuable item with the highest value:weight ratio until you can't grab any more, then the next, then the next, then the next.

[Khamsin]

Simple.

6 types of nails = a,b,c,d,e,f
total profit = y

Graph the following on 7-dimensional graph paper:

y = 4a + 2.5b + 2c + 1.5d + e + 0.5f

Then draw a line at each boundary, namely where:

a + b + c + d + e + f = 30
a + .85b + .7c + .5d + .25e + .1f = 20
a = 10
b = 10
c = 10
d = 10
e = 10
f = 10

You'll have a 7-dimensional shape with a bunch of lines crisscrossing it. See how far up the y-direction you can go, within that shape and within those lines, and still meet the 5 minimum of 3 types of nails requirement.

[Bugpop]

Did your teacher say there was a larger answer than $72?

For my optimal solution, I get
4.0" - 10
3.5" - 8
2.5" - 1
2.0" - 10
1.5" - 1

profit = $72

[Conformity]

gonna cost you more than 5 dollas bub

[LinktheDeme]

Did your teacher say there was a larger answer than $72?

For my optimal solution, I get
4.0" - 10
3.5" - 8
2.5" - 1
2.0" - 10
1.5" - 1

profit = $72

Nope, didn't say what it was. No one beat it though so no winners

[Bugpop]

gonna cost you more than 5 dollas bub

The highest I can get is $72.33 using fractions of nails.

                             MOR/LP

Date: 10/18/09                                   Time: 23:00:33
                         C:\MOR\PROBLEM

                        MODEL DESCRIPTION
      ****************************************************


Max z = 4a+2.5b+2c+1.5d+e+0.5f
ST a+0.85b+0.7c+0.5d+0.25e+0.1f<=20
a+b+c+d+e+f>=15
a+b+c+d+e+f<=30
a<=10
b<=10 
c<=10
d<=10
e<=10
f<=10


      ****************************************************


Simplex Method





                            ^^^ Optimal Solution ^^^

Z               =     72.3333
A               =     10.0000
B               =      8.6667
C               =      0.0000
D               =      0.0000
E               =     10.0000
F               =      1.3333
 Surpl2         =     15.0000
 Slack5         =      1.3333
 Slack6         =     10.0000
 Slack7         =     10.0000
 Slack9         =      8.6667


Compile time:     0.00 (Secs.)
Run time    :     0.00 (Secs.)

[Khamsin]

Speaking of which, if I can get a $4 profit just by selling a single 4" nail, I'm in the wrong business.

[Plow]

$4 profit per nail

AFK getting into the nail business.

[Nyima]

Well there wasn't a constraint that you had to use whole nails was there?

[LinktheDeme]

Well there wasn't a constraint that you had to use whole nails was there?

Well technically no but realistically yes as you can't sell someone half a nail or it'll cease to be a nail lol. Regardless though it was a good effort but 10 minutes after the deadline I would have given it if otherwise though since I didn't list it.

[Bugpop]

Well technically no but realistically yes as you can't sell someone half a nail or it'll cease to be a nail lol. Regardless though it was a good effort but 10 minutes after the deadline I would have given it if otherwise though since I didn't list it.

I had to modify the problem a bit to get integer results. I took it for granted that whole nails were desired.

I can't see how 3 sizes had to have a minimum of 5 nails was a necessary constraint. At best it served as a hint for people using brute force methods. Are you covering linear programming in school?

                             MOR/LP

Date: 10/19/09                                   Time: 21:52:57
                         C:\MOR\problem3

                        MODEL DESCRIPTION
      ****************************************************


Max z = 400a+250b+200c+150d+100e+50f
ST 100a+85b+70c+50d+25e+10f<=2000
a+b+c+d+e+f>=15
a+b+c+d+e+f<=30
a<=10
b<=10 
c<=10
d<=10
e<=10
f<=10
integer(a,b,c,d,e,f)

      ****************************************************


Gomory Cut - Mixed Integer Linear Programming





                            ^^^ Optimal Solution ^^^



Integer solution has been obtained
Z               =   7200.0000
A               =     10.0000
B               =      8.0000
C               =      0.0000
D               =      1.0000
E               =     10.0000
F               =      1.0000
 Slack1         =     10.0000
 Surpl2         =     15.0000
 Slack5         =      2.0000
 Slack6         =     10.0000
 Slack7         =      9.0000
 Slack9         =      9.0000


Compile time:     0.00 (Secs.)
Run time    :     0.00 (Secs.)