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# bin_packing_md(+Items, ++ItemMDSizes, +BinMDLoads)

The multi-dimensional bin packing constraint with loads: packing M L-Dimensional items into N L-Dimensional bins, each bin having a load in each dimension
*Items*
- A collection of M (domain) variables or integers (domain/value between 1 and N)
*ItemMDSizes*
- A 2-D collection of M*L non-negative integers
*BinMDLoads*
- A 2-D collection of N*L (domai) variables or non-negative integers

## Description

This constraint is for multi-dimensional bin-packing, that is, to
pack M L-dimensional items with individual sizes in each dimension
into N L-dimensional bins, such that the sum of sizes of items in
each dimension of each bin equals the load for that dimension of
that bin, as specified in BinMDLoads. The constraint is logically
equivalent to posting L 1-dimensional bin_packing constraints, one
for each dimension, with the additional constraint that the items
are placed in the same bin for all the constraints.
Items and Bins are L-dimensional, i.e. each Item/Bin has a
size/load in each dimension. Thus,ItemMDSizes and BinMDLoads are
2-D collections, i.e. a 2-D matrix or a list of lists, such that
each element is itself a collection of L sub-elements.

Each element of Items and its corresponding element in ItemMDSizes
represents an item, such that the i'th element of ItemMDSizes is
the size of the i'th item, and the i'th element of Item is the bin
this item is packed into. BinMDLoads represent the loads of each
bin, i.e. the sum of the sizes of items (in each dimension)
assigned to that bin, with the j'th element representing the load
for bin j. An (integer finite domain) variable for the load in each
dimension allows a constraint on the load to be specified, such as
a minimum and/or maximum load for the bin in that dimension.

Note that the Gecode implementation of this constraint has index
starting from 0, i.e. the numbering for the bins starts from
0. These native indices are mapped to the ECLiPSe indices starting
from 1 with an additional dummy zero'th bin that is assigned a
dummy item 0.

This constraint is implemented using Gecode's multi-dimensional
variant of binpacking() constraint, which requires both the maximum
bin size (as in bin_packing_md/4) and all the bin loads to be specified.
This gfd version of the constraint deduces the maximum bin size from
BinMDLoads.

## Examples

[eclipse 2]: bin_packing_md([3,1,3], [[4,2], [3,0], [1,3]], [[L11,L12],[L21,L22],[L31,L32]]).
L11 = 3
L12 = 0
L21 = 0
L22 = 0
L31 = 5
L32 = 5

## See Also

bin_packing_md / 4, bin_packing / 4, bin_packing / 3