Approach for packing 2D shapes while minimizing total enclosing area
- by Dennis
Not sure on my tags for this question, but in short ....
I need to solve a problem of packing industrial parts into crates while minimizing total containing area. These parts are motors, or pumps, or custom-made components, and they have quite unusual shapes. For some, it may be possible to assume that a part === rectangular cuboid, but some are not so simple, i.e. they assume a shape more of that of a hammer or letter T. With those, (assuming 2D shape), by alternating direction of top & bottom, one can pack more objects into the same space, than if all tops were in the same direction. Crude example below with letter "T"-shaped parts:
***** xxxxx ***** x ***** *** ooo
* x vs * x vs * x vs * x o
* x * xxxxx * x * x o
xxxxx xxx
Right now we are solving the problem by something like this:
using CAD software, make actual models of how things fit in crate boxes
make estimates of actual crate dimensions & write them into Excel file
(1) is crazy amount of work and as the result we have just a limited amount of possible entries in (2), the Excel file.
The good things is that programming this is relatively easy. Given a combination of products to go into crates, we do a lookup, and if entry exists in the Excel (or Database), we bring it out. If it doesn't, we say "sorry, no data!".
I don't necessarily want to go full force on making up some crazy algorithm that given geometrical part description can align, rotate, and figure out best part packing into a crate, given its shape, but maybe I do..
Question
Well, here is my question: assuming that I can represent my parts as 2D (to be determined how), and that some parts look like letter T, and some parts look like rectangles, which algorithm can I use to give me a good estimate on the dimensions of the encompassing area, while ensuring that the parts are packed in a minimal possible area, to minimize crating/shipping costs?
Are there approximation algorithms?
Seeing how this can get complex, is there an existing library I could use?
My thought / Approach
My naive approach would be to define a way to describe position of parts, and place the first part, compute total enclosing area & dimensions. Then place 2nd part in 0 degree orientation, repeat, place it at 180 degree orientation, repeat (for my case I don't think 90 degree rotations will be meaningful due to long lengths of parts). Proceed using brute force "tacking on" other parts to the enclosing area until all parts are processed.
I may have to shift some parts a tad (see 3rd pictorial example above with letters T). This adds a layer of 2D complexity rather than 1D. I am not sure how to approach this. One idea I have is genetic algorithms, but I think those will take up too much processing power and time. I will need to look out for shape collisions, as well as adding extra padding space, since we are talking about real parts with irregularities rather than perfect imaginary blocks. I'm afraid this can get geometrically messy fairly fast, and I'd rather keep things simple, if I can.
But what if the best (practical) solution is to pack things into different crate boxes rather than just one? This can get a bit more tricky. There is human element involved as well, i.e. like parts can go into same box and are thus a constraint to be considered. Some parts that are not the same are sometimes grouped together for shipping and can be considered as a common grouped item. Sometimes customers want things shipped their way, which adds human element to constraints. so there will have to be some customization.
