Re: [math-fun] length + width + height ??
Let's assume that these services are motivated by profits, and that there is enough competition to ensure that the prices track carefully to the costs. Let's further assume (for the moment) that weight doesn't matter; that the real problem is merely fitting the packages into the container for shipping. Let f(L,W,H) be a function that approximates the cost of shipping, and we will price based on some "cost plus" model. What would be a decent function to use that would maximize the value that one could carry? Clearly, if the standard container is 20'x8'x8', we can completely fill it with 4'x4'x4' cubes. But if we can charge more (per unit volume) for non-cubical shapes, we could still maximize the shipping value of the container, even though the container still might have some empty spaces. So what would an "optimal" f(L,W,H) look like? I would guess that f() is symmetric in all of its arguments, but perhaps that isn't even obvious. f() would somehow charge more per unit volume, the "odder" the shape is, because I assume that the statistics of the types of objects would make it less and less likely to be able to pack the container completely full. Note that mere simplicity of computation isn't all that important if the shipping costs are high enough; one would quickly implement a more sophisticated pricing function if there was significantly more profit in it. So, given these constraints, is f(L,W,H)=C*(L+W+H) even close to optimal? At 11:48 AM 3/4/2012, Robert Munafo wrote:
I think it's a lot simpler than that.
They're trying to give a "volume discount" for customers with lage packages. To some extent, if the customer had to pay 8 times as much for something that has 8 times the volume (2x in each direction), they'd probably consider it overpriced.
There is also an intentional surcharge for oddly-shaped packages, for the packing reason you mentioned and for other reasons.
"length + width + height" fairly neatly encapsulates both of these objectives.
On Sun, Mar 4, 2012 at 11:09, Henry Baker <hbaker1@pipeline.com> wrote:
Some airlines & postal systems charge by the curious measure "length + width + height".
1. Is there a standard name for this measure?
2. What is the scientific/mathematical rationale for using this measure for charging?
--
If there were such a mathematical rationale, I would think that it would somehow be based upon the statistics of packing many dissimilar boxes into a standard box -- e.g., a standard shipping container or a UPS truck.
What do we know about the statistics of packing various numbers of differently shaped boxes into a large cubical box?
[...]
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"HB" == Henry Baker <hbaker1@pipeline.com> writes:
HB> So, given these constraints, is f(L,W,H)=C*(L+W+H) even close to optimal? Probably not. The package shipping companies (USP, USPS, FedEx, et al) tend to use lenght * girth, making that f(L,W,H) = C * L*(W+H). I've heard -- but cannot confirm -- that L*G traces back to nautical shipping. I can confirm that USP and USPS, at least, were already using L*G back in the '70s. -JimC -- James Cloos <cloos@jhcloos.com> OpenPGP: 1024D/ED7DAEA6
Very interesting! Thanks! I wonder if "length" is treated differently because of a preferred direction ("This Side Up"). Or perhaps ships & airplanes are long & thin, so "length" is treated differently for those reasons. Although, the airplane shipping containers that I have seen look like 1/2 of a sliced cylinder -- probably because they are loaded below the "deck". These airline shipping containers are also not very long -- less than 6' -- so their length is less than their radius. I'll have to do some research to see how far back L*G goes in the shipping biz. So for 2 dimensions we have "perimeter" and "girth" (presumably the perimeter of the convex hull); there doesn't seem to be a name for L+W+H in 3 dimensions. Perhaps there is a name for supremum girth, over all 2-D projections of the object. At 04:12 PM 3/5/2012, James Cloos wrote:
"HB" == Henry Baker <hbaker1@pipeline.com> writes:
HB> So, given these constraints, is f(L,W,H)=C*(L+W+H) even close to optimal?
Probably not.
The package shipping companies (USP, USPS, FedEx, et al) tend to use lenght * girth, making that f(L,W,H) = C * L*(W+H). I've heard -- but cannot confirm -- that L*G traces back to nautical shipping.
I can confirm that USP and USPS, at least, were already using L*G back in the '70s.
-JimC -- James Cloos <cloos@jhcloos.com> OpenPGP: 1024D/ED7DAEA6
This has nothing to do with justifying its use in shipping, but the function L+W+H on boxes is a pretty important one. I thought it was called something like "linear girth", but I can't find any justification for that online, so seems likely that my memory is faulty. But I'll use that here since I need some name for it. The key theorem, from the domain of geometric probability, is that the space of continuous invariant measures on polyconvex sets is spanned by only a few basis elements -- in dimension d, they are the d+1 measures that take a box to one of the d+1 elementary symmetric functions of its edge lengths. You extend the measures by additivity on disjoint sets, and so by mu(A U B) = mu(A) + mu(B) - mu(A ^ B) on arbitrary sets. (Using ^ for intersection here.) Taking (L,W,H) to L*W*H gives you volume, of course. Taking (L,W,H) to LW+WH+HL gives you surface area (up to a scalar), and here you need to think about the mu(A ^ B) term: a unit cube gets mu=3, while a 1x1x2 block gets mu=5 because you can break it into two unit cubes that intersect on a 1x1x0 square with valuation 1. Taking (L,W,H) to L+W+H gives you the linear girth thing under discussion here. Taking (L,W,H) to 1 gives you Euler characteristic; you should work through the inclusion-exclusion yourself if you haven't seen this before. There are nice geometric probability interpretations of the measures, too. Suppose A and B are nice convex shapes and A fits inside of B. Then the ratio of volumes is the probability that a randomly-chosen point in B is also in A. The ratio of surface areas is the probability that a randomly-chosen line that passes through B also passes through A, and the ratio of linear girths is the probability that a randomly-chosen plane passing through B also passes through A. --Michael On Mon, Mar 5, 2012 at 7:43 PM, Henry Baker <hbaker1@pipeline.com> wrote:
Very interesting! Thanks!
I wonder if "length" is treated differently because of a preferred direction ("This Side Up").
Or perhaps ships & airplanes are long & thin, so "length" is treated differently for those reasons.
Although, the airplane shipping containers that I have seen look like 1/2 of a sliced cylinder -- probably because they are loaded below the "deck". These airline shipping containers are also not very long -- less than 6' -- so their length is less than their radius.
I'll have to do some research to see how far back L*G goes in the shipping biz.
So for 2 dimensions we have "perimeter" and "girth" (presumably the perimeter of the convex hull); there doesn't seem to be a name for L+W+H in 3 dimensions. Perhaps there is a name for supremum girth, over all 2-D projections of the object.
At 04:12 PM 3/5/2012, James Cloos wrote:
> "HB" == Henry Baker <hbaker1@pipeline.com> writes:
HB> So, given these constraints, is f(L,W,H)=C*(L+W+H) even close to optimal?
Probably not.
The package shipping companies (USP, USPS, FedEx, et al) tend to use lenght * girth, making that f(L,W,H) = C * L*(W+H). I've heard -- but cannot confirm -- that L*G traces back to nautical shipping.
I can confirm that USP and USPS, at least, were already using L*G back in the '70s.
-JimC -- James Cloos <cloos@jhcloos.com> OpenPGP: 1024D/ED7DAEA6
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On Tuesday 06 March 2012 15:43:05 Michael Kleber wrote:
This has nothing to do with justifying its use in shipping, but the function L+W+H on boxes is a pretty important one. ... The key theorem, from the domain of geometric probability, is that the space of continuous invariant measures on polyconvex sets is spanned by only a few basis elements -- in dimension d, they are the d+1 measures that take a box to one of the d+1 elementary symmetric functions of its edge lengths. ... Taking (L,W,H) to L*W*H gives you volume, of course. ... Taking (L,W,H) to 1 gives you Euler characteristic; you should work through the inclusion-exclusion yourself if you haven't seen this before.
There are nice geometric probability interpretations of the measures, too. Suppose A and B are nice convex shapes and A fits inside of B. Then the ratio of volumes is the probability that a randomly-chosen point in B is also in A. The ratio of surface areas is the probability that a randomly-chosen line that passes through B also passes through A, and the ratio of linear girths is the probability that a randomly-chosen plane passing through B also passes through A.
... And, indeed, the ratio of Euler characteristics is the probability that a randomly-chosen all-of-R^n passing through B also passes through A, since any convex set[1] has Euler characteristic 1. :-) [1] Er, I think. Certainly any compact convex set, which I think is the correct condition for the conditional-probability theorem to apply. -- g
participants (4)
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Gareth McCaughan -
Henry Baker -
James Cloos -
Michael Kleber