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   Author  Topic: Taco stand  (Read 580 times)
Arne
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Taco stand  
« on: Jul 3rd, 2017, 2:47pm »
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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
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dudiobugtron
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Re: Taco stand  
« Reply #1 on: Jul 4th, 2017, 12:50pm »
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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.
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towr
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Re: Taco stand  
« Reply #2 on: Jul 4th, 2017, 10:13pm »
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They wouldn't get many customers there, though. And the prices would be out of this world.
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Arne
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Re: Taco stand  
« Reply #3 on: Jul 5th, 2017, 1:01pm »
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on Jul 4th, 2017, 12:50pm, 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.
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Grimbal
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Re: Taco stand  
« Reply #4 on: Jul 6th, 2017, 8:21am »
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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.
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rmsgrey
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Re: Taco stand  
« Reply #5 on: Jul 6th, 2017, 3:36pm »
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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.
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Re: Taco stand  
« Reply #6 on: Jul 7th, 2017, 12:53am »
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And there is the problem of defining exactly where you stop to be "in the city".
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towr
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Re: Taco stand  
« Reply #7 on: Jul 7th, 2017, 1:26am »
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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?
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Re: Taco stand  
« Reply #8 on: Jul 9th, 2017, 1:12pm »
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on Jul 7th, 2017, 1:26am, 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.
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Re: Taco stand  
« Reply #9 on: Jul 10th, 2017, 4:11am »
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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)
« Last Edit: Jul 10th, 2017, 4:12am by Grimbal » IP Logged
Hippo
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Re: Taco stand  
« Reply #10 on: Jul 11th, 2017, 5:56am »
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on Jul 3rd, 2017, 2:47pm, 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 ...
« Last Edit: Jul 11th, 2017, 5:58am by Hippo » IP Logged
Grimbal
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Re: Taco stand  
« Reply #11 on: Jul 13th, 2017, 8:25am »
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on Jul 11th, 2017, 5:56am, 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.
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towr
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Re: Taco stand  
« Reply #12 on: Jul 13th, 2017, 10:17am »
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on Jul 13th, 2017, 8:25am, Grimbal wrote:
But then towr's rule is not perfect.
Just you wait till I've conquered the world, and try saying that then Wink
 
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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 Wink
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)
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dudiobugtron
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Re: Taco stand  
« Reply #13 on: Jul 13th, 2017, 8:49pm »
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Or 1/(1+x)2
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Grimbal
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Re: Taco stand  
« Reply #14 on: Jul 14th, 2017, 8:13am »
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on Jul 13th, 2017, 10:17am, towr wrote:
Just you wait till I've conquered the world, and try saying that then Wink

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.
« Last Edit: Jul 14th, 2017, 12:52pm by Grimbal » IP Logged
Grimbal
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Re: Taco stand  
« Reply #15 on: Jul 14th, 2017, 8:17am »
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on Jul 13th, 2017, 10:17am, towr wrote:
Only if people can get within that circle. But customers aren't allowed in the kitchen Wink

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?
« Last Edit: Jul 14th, 2017, 8:18am by Grimbal » IP Logged
towr
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Re: Taco stand  
« Reply #16 on: Jul 14th, 2017, 12:43pm »
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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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