3 dimensional bin packing algorithms
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 !
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.
From wikipedia:
Although these simple strategies are often good enough, efficient approximation algorithms have been demonstrated that can solve the bin packing problem within any fixed percentage of the optimal solution for sufficiently large inputs
Here are the two sources they give for this:
- Approximation Algorithms
- Bin packing can be solved within 1 + ε in linear time
Best exact solver: Use dynamic programming.
State variables:
- Items you have packed and discarded.
- Space filled in the container.
If the container is a parallelepiped grid, and the items "fit" in exact cells of the grid, you can use a 3-dimensional array to represent state variable 2. Otherwise, you will have to use more complex data structures.
Best heuristic solvers
I don't know. Perhaps Variable Neighborhood Search. There are some similarities between your problem and the timetable construction problem (which I'm working on), so the same heuristic might be good for both.
Off-the-shelf solutions to conduct experiments
I'm sorry, I don't even have a clue.