Two things to get you started: (1) Choose your coordinates for the different cities. It will be enough to assume the Earth is flat, so you can use an XY-plane. (2) You will need the latitude and longitude of each city. Try www.indo.com/distance/ for starters.

In order to present a kinder, more capitalist image of extraterrestrials than what is usually shown in the media, I have decided to open up a business in your prime locality. I came to this decision after seeing all the hoopla in a town to your west called Roswell. (Evidently something really strange happened there. I don't recall any of my friends dive-bombing Earth in quite some time, but I digress......) I am starting Space Modulators Incorporated with a grant from the Small Business Administration for legal aliens, and I plan to open several retail outlets to sell my Illudium Q-36 Explosive Space Modulator, since the Earth blocks my view of Venus. (Don't worry, I'll set the phasers to stun, not kill). I need the help of your company to determine the best location for my central distribution site. I plan to open one store in Washington, D.C., one store in San Fransisco, CA, and the third store in College Station, TX. I plan on having just one central distribution site, and will make weekly deliveries to each of the stores. I plan on making three trips per week to San Fransisco, CA, only two per week to Washington, D.C., and five per week to College Station, TX. (They seem to really be into destroying orange and white things in that town; any idea why?) Anyway, I want to minimize the total flight distance, to keep my costs down. If the market is as strong as I anticipate, I plan to open another outlet store in Key West, Florida. I expect the Illudium Q-36 to be so popular there that I will have to make six trips per week there. Before I commit to opening a fourth outlet, I would like to know what effect this will have on the location of the central distribution site for the three stores. I also need to know the optimal site for the distribution site if all four stores are open, and I would like your recommendation on which of the two locations is better.

This forum on the-w is for this type of activity and I am sure you'll get 30-40 answers in a day or so. By the way, don't worry about registration or spam or anything. The forum Admin is a pal of mine and he's the best. No data gets out.

In case you were wondering why we are getting homework requests.

I believe in this case, he was overly optimistic with the potential response. ;-) This falls into an area of math I sincerely tried to avoid while I was in college.

Originally posted by arkansasst555I need the help of your company to determine the best location for my central distribution site. I plan to open one store in Washington, D.C., one store in San Fransisco, CA, and the third store in College Station, TX. I plan on having just one central distribution site, and will make weekly deliveries to each of the stores. I plan on making three trips per week to San Fransisco, CA, only two per week to Washington, D.C., and five per week to College Station, TX.

Isn't the answer to this part "College Station, TX"? Why wouldn't you put the distribution center THERE so you would make ZERO trips per week to College Station. Then you would have only three trips to San Fransisco (sic) at 1576 miles a pop and two to Washingotn at 1235 miles each, for a total if 7198 miles/week.

Even if you put the place EXACTLY halfway between SF and DC - they are 2449 miles apart, so you would have to go 1224.5 miles 5 times - that's 6122.5 miles without even including the five weekly trips to College Station.

The halfway point between SF and DC is roughly Wichita, which is 500 miles away from College Station - so five trips would add over 2500 more miles. It's been a while since I was in school but I'm pretty sure 6122.5 + 2500 > 7198.

Holy fuck shit motherfucker shit. Read comics. Fuck shit shit fuck shit I sold out when I did my job. Fuck fuck fuck shit fuck. Sorry had to do it....

*snip*

Revenge of the Sith = one thumb up from me. Fuck shit. I want to tittie fuck your ass. -- The Guinness. to Cerebus

I was in 3rd semester calculus when I dropped out of college but after 5+ years of not using it I've forgotten a lot.

I think I can get you started. I think this is going to end up involving multivariable function, which was the class I dropped out of. Have you used multivariable graphing programs like MAPLE?

Let's start with the numbers of those locations.

Let X be the longitude and Y be the latitude of each location.

Washington DC is at (77.016, 38.905) San Francisco is at (122.555, 37.793) College Station is at (96.312, 30.601) Key West (for the second part) is at (81.853, 24.563)

We can make all the longitudes negative if we want, but that's going to end up with the same result.

We are trying to find the point (X,Y) that will minimize the distance traveled in a week.

The distance required for a one-way trip from Washington DC to the distribution center (X,Y) can be found using the pythagorean theorem. It is sqrt((77.016-x)^2+(38.905-y)^2. The distance for a one-way trip to the other locations are similarly calculated.

Then we consider how many times we are traveling to and from each location.

The Weekly Distance traveled with the 3 locations is f(x,y) = 4*sqrt((77.016-x)^2+(38.905-y)^2)+6*sqrt((122.555-x)^2+(37.793-y)^2)+10*sqrt((96.312-x)^2+(30.601-y)^2)

Now at this point you need to find some way to figure out the minimum of that equation. That means finding the appropriate critical point of the equation such that the determinant equals whatever it's supposed to equal for a minimum point.

I made a very rough estimate using a spreadsheet, and if I did that formula right, the best distribution site with the 3 locations is right in College Station. This seems to make sense as it is the point that is traveled to the most, and it is already pretty centrally located between the other two points.

Then in the second part you just add the distance to Key West in to the equation and find the new minimum. It would definitely seem that the distribution center would move closer to Key West, but I can't say how much.

EDIT: Screw it, I'm still board. Adding Key West to the mix actually does almost nothing to change the minimum. Using full degree increments in a spreadsheet once again, the minimum moves from (96,31) to (95,30).

A very nice job by the teacher finding cities which give such a clean answer.