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        riddles >> medium >> Taco stand
        
(Message started by: Arne on Jul 3rd, 2017, 2:47pm)
Title: Taco stand
Post by Arne on Jul 3rd, 2017, 2:47pm
A taco stand chain wants to build a new location in a city. Where is the best location for the new stand? To decrease competition the taco stand should be built in a location where the distance to the nearest taco stand is maximized. All other factors being equal.

Interesting problem from this video: youtu.be/b_uvofsYl9s?t=2m19s
Title: Re: Taco stand
Post by dudiobugtron on Jul 4th, 2017, 12:50pm
How high up does it stop being in the city?  I imagine a taco stand in orbit above the city would be far enough away from all of the other ones.
Title: Re: Taco stand
Post by towr on Jul 4th, 2017, 10:13pm
They wouldn't get many customers there, though. And the prices would be out of this world.
Title: Re: Taco stand
Post by Arne on Jul 5th, 2017, 1:01pm

on 07/04/17 at 12:50:54, dudiobugtron wrote:
How high up does it stop being in the city?  I imagine a taco stand in orbit above the city would be far enough away from all of the other ones.


If you already go up a dimension why not generalize all the way? Find the furthest point from all other points in a volume in n dimensions.
Title: Re: Taco stand
Post by Grimbal on Jul 6th, 2017, 8:21am
Go to the future.  Wait until the concept of a taco is long forgotten and historians debate about what kind of religious ritual it was to line up at a "taco" stand and deposit small pieces of metal.

Then in time and in space you have the largest distance to the nearest taco stand.  That is, if you really can't wait a bit more.
Title: Re: Taco stand
Post by rmsgrey on Jul 6th, 2017, 3:36pm
So, I can get as far as independently reinventing Voronoi cells but can't come up with anything cleverer than brute-force checking all the corners for finding the global optimum.
Title: Re: Taco stand
Post by Grimbal on Jul 7th, 2017, 12:53am
And there is the problem of defining exactly where you stop to be "in the city".
Title: Re: Taco stand
Post by towr on Jul 7th, 2017, 1:26am
Okay, how about this, you have a city which is a circle with radius 1, which is uniformly packed with customers with a willingness to visit a taco-shop inversely proportion to the square of their distance from it, who will always pick the closest taco shop if any. You incrementally add shops to your franchise in the city, optimizing for each single expansion (instead of for future expansion). Where do you put shops 1, 2, 3, etc?
Title: Re: Taco stand
Post by rmsgrey on Jul 9th, 2017, 1:12pm

on 07/07/17 at 01:26:11, towr wrote:
Okay, how about this, you have a city which is a circle with radius 1, which is uniformly packed with customers with a willingness to visit a taco-shop inversely proportion to the square of their distance from it, who will always pick the closest taco shop if any. You incrementally add shops to your franchise in the city, optimizing for each single expansion (instead of for future expansion). Where do you put shops 1, 2, 3, etc?

Shop 1 is trivial - it goes at the origin. Shop 2 goes anywhere on a circle with radius r (exact radius to be determined). Shop 3 then goes anywhere on an arc with the same radius, but roughly opposite Shop 2 - at the endpoints of the arc, the perpendicular bisectors of the radial lines to shops 2 and 3 meet at the circumference.

The angle between shops 2 and 3 determines where future shops go, possibly uniquely.
Title: Re: Taco stand
Post by Grimbal on Jul 10th, 2017, 4:11am
Sometimes the best move to not move it is.

At some point, the cost of installing and maintaining a taco stand is bound to fall below the expected revenue.  The optimal solution would stop after a finite time.

(rmsgrey already gave a perfectly good reply.  There is nothing left for me but going back and exploring another point of view)
Title: Re: Taco stand
Post by Hippo on Jul 11th, 2017, 5:56am

on 07/03/17 at 14:47:30, Arne wrote:
To decrease competition the taco stand should be built in a location where the distance to the nearest taco stand is maximized.


In that case r=1.

If the criteruim would be to split the customers to as equal sizes as possible, rmsgreys describtion would fail.
2nd stand would be infinitely close to the 1st. 3rd would be on their axis such that line distant r/2 from origin cuts 1/3 of disc. 4th would be opposite to 3, but further from center as it should cut ... what it should cut? Something between 2/9 and 1/4. And 5th should definitely cut mainly from 3rd ...
Title: Re: Taco stand
Post by Grimbal on Jul 13th, 2017, 8:25am

on 07/11/17 at 05:56:58, Hippo wrote:
In that case r=1.

If the criteruim would be to split the customers to as equal sizes as possible, rmsgreys describtion would fail.
2nd stand would be infinitely close to the 1st. 3rd would be on their axis such that line distant r/2 from origin cuts 1/3 of disc. 4th would be opposite to 3, but further from center as it should cut ... what it should cut? Something between 2/9 and 1/4. And 5th should definitely cut mainly from 3rd ...

Then we need to consider whether taco stands are fermions or bosons.

Towr gave a rule telling how many people are attracted by a stand.  That rule favors putting the next stand in a region far from any existing stand, where few people are willing to go to any stand yet.

But then towr's rule is not perfect.  For a small radius near the taco stand the willigness would exceed 1 i.e.100%.  I don't know how it translates into actual purchasing pattern.
Title: Re: Taco stand
Post by towr on Jul 13th, 2017, 10:17am

on 07/13/17 at 08:25:25, Grimbal wrote:
But then towr's rule is not perfect.
Just you wait till I've conquered the world, and try saying that then ;)


Quote:
For a small radius near the taco stand the willingness would exceed 1 i.e.100%.
Only if people can get within that circle. But customers aren't allowed in the kitchen ;)
Hmm, yeah, maybe we need a attraction function that goes back down to zero when you get really close, to simulate the effect of seeing what actually goes in the taco's and how the staff handles them.

Though, I suppose easiest would be just to pick a function that tops out at 1, like exp(-d2)
Title: Re: Taco stand
Post by dudiobugtron on Jul 13th, 2017, 8:49pm
Or 1/(1+x)2
Title: Re: Taco stand
Post by Grimbal on Jul 14th, 2017, 8:13am

on 07/13/17 at 10:17:44, towr wrote:
Just you wait till I've conquered the world, and try saying that then ;)

It will still not be perfect, I would just shut up about it.  But people would read about it on anonymous leaflets distributed all over the city during the night.
Title: Re: Taco stand
Post by Grimbal on Jul 14th, 2017, 8:17am

on 07/13/17 at 10:17:44, towr wrote:
Only if people can get within that circle. But customers aren't allowed in the kitchen ;)

No, no, no!  You said "which is uniformly packed with customers".  That means customers are everywhere.  How can we solve riddles if you keep changing the rules?
Title: Re: Taco stand
Post by towr on Jul 14th, 2017, 12:43pm
OMG you're right.. But that means.. The customers are also in the tacos!! It's soylent green all over again.


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