Jeeps in the Desert

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Jeeps in the Desert

Postby David P Bird » Fri Oct 05, 2012 10:29 am

In the middle of the desert you have a fuel depot and some jeeps each of which can travel 100 miles on a tank of fuel. The usual problem is to find out how far away from base it is possible to explore which involves building up intermediate fuel dumps in jerry cans. In this version however this is impossible (perhaps there are no jerry cans) and the only way a jeep can get extra fuel away from base is to transfer some from the tank of another.

If two jeeps go out together after 33.33 miles each will have used 1/3rd of a tank, the support jeep transfers 1/3rd to the scout jeep, returns to base, where it refuels and returns to the same point to wait for the scout jeep to return. It can then give it enough fuel for both to get back to base. This gives two jeeps a range of 33.33 + 50 = 83.333 miles.

What is the range for N jeeps for N = 3 to 7?

If fuel dumps are possible, there is an equation to calculate the range, but I don't think this is true for this version.

I tackled this problem many years ago and can't remember the answers I got so don't expect me to be able to confirm any results!
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Re: Jeeps in the Desert

Postby civiliza » Fri Oct 05, 2012 1:36 pm

Logically, the outermost Jeep (Jeep J) always has a range of 50 miles.

The staging post before that can hold 1 or more jeeps,

if more than one jeep is waiting at the staging post, then the maximum distance n to the previous staging post is:

n = 100 * (J-1) / (2J-1)

ie 2 jeeps (1 stager plus 1 outermost) can return 33.3 miles
3 jeeps (2 stagers plus 1 outermost) can return 40 miles
4 jeeps 42.9 miles
etc.

By contrast, with individual jeeps stationed every 33.3 miles, each one can be fully refueled from empty by three runs from the station(s) before. So:
1 Jeep gives range of 50 miles
2 Jeeps gives range of 33.3 + 50 miles
3 Jeeps gives range of 33.3 + 33.3 + 50 miles (ie 116.6 miles if chained vs 90 if paired)
n Jeeps give range of 50 + 33.3 * (n-1)
Last edited by civiliza on Fri Oct 05, 2012 2:24 pm, edited 1 time in total.
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Re: Jeeps in the Desert

Postby civiliza » Fri Oct 05, 2012 2:21 pm

Oops, I think that I became fixated with the 33.3 mile number. If every jeep but the last two was prepared to make 50 * 49 mile journeys to do one top up, the range would go up to:

50 + 33.3 + 49 * (J-2).

Theoretically, the maximum range would approach 50 + 33.3 + 50 * (J-2), but the number of journeys each jeep (and especially the one nearest the fuel depot) would do would get ridiculous.
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Re: Jeeps in the Desert

Postby rjamil » Sun Aug 05, 2018 1:19 am

Although, this thread is going to be 6 years old, but this might gives solution to the original problem.
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Re: Jeeps in the Desert

Postby David P Bird » Sun Aug 05, 2018 6:03 am

rjamil wrote:Although, this thread is going to be 6 years old, but this might gives solution to the original problem.

Oh! I had forgotten all about posting on this old problem before. Your reference paper is well found. Its references includes the flight of planes problem I mentioned in your camel & banana thread. That problem is a lot more awkward as support aircraft can't create fuel dumps but must rendezvous at a precise times.

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Re: Jeeps in the Desert

Postby tarek » Sun Aug 05, 2018 9:56 am

I was happy with the Wikipedia article explaining the basic camel and banana problem. The solution is 533 1/3 assuming the camel can eat part bananas :D

I have to thank you both for bringing this to my attention. It is a lovely brain teaser. And even getting close to the correct answer brings some satisfaction to the non mathematicians here!!!

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