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I'm faced with a 3 dimensional bin packing problem and am currently conducting some preliminary research as to which algorithms/heuristics are currently yielding the best results. Since the problem is NP hard I do not expect to find the optimal solution in every case, but I was wondering:
1) what are the best exact solvers? Branch and Bound? What problem instance sizes can I expect to solve with reasonable computing resources?
2) what are the best heuristic solvers?
3) What off-the-shelf solutions exist to conduct some experiments with?
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Best-fit is an online algorithm for bin packing. Its input is a list of items of different sizes. Its output is a packing - a partition of the items into bins of fixed capacity, such that the sum of sizes of items in each bin is at most the capacity.
Given n items of different weights and bins each of capacity c, assign each item to a bin such that number of total used bins is minimized. It may be assumed that all items have weights smaller than bin capacity.
The full bin packing algorithm is more likely to produce an optimal solution – using the least possible number of bins – than the first fit decreasing and first fit algorithms. It works by matching object so as to fill as many bins as possible.
The bin packing problem is an optimization problem, in which items of different sizes must be packed into a finite number of bins or containers, each of a fixed given capacity, in a way that minimizes the number of bins used.
As far as off the shelf solutions, check out MAXLOADPRO for loading trucks. It may be able to be configured to load any rectangular volume, but I haven't tried that yet. In general 3d bin-packing problems have the added complication that the objects can be rotated into different positions so for any object with a given length, width and height, you effectively have to create three variables representing each position, but you only use one in the solution.
In general, stand-alone MIP formulations (or branch and bound) don't work well for the 2d or 3d problem but constraint programming has met with some success producing exact solutions for the 2d problem. Check out this abstract. Without looking at the paper, I like the decomposition approach for the problem where you're trying to minimize the number of same-sized bins. I haven't seen as many results for the 3d problem, but let us know if you find any that are implementable.
Good luck !
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I've written a program which tests three various algorithms. Also this is a good source of information: A Thousand Ways to Pack the Bin - A Practical Approach to Two-Dimensional Rectangle Bin Packing. It is for two-dimensional rectangle bin, but you can always transform it to 3D.
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