Table 1.

Parameters and variables used in our formulation.

Parameters
$I,J,K_i,Q$	Sets of items, bins, orientations of i-th item (see Fig. 1) and relative positions between items (see Fig. 2)
m, n	Number of items and bins
$l_i,w_i,h_i,{\mu }_i$	Length, width, height and weight of item $i\in I$
L, W, H, M	Length, width, height and maximum capacity (optional) of bins
$A^{\text{pos}},A^{\text{neg}}$	Sets of positive and negative affinities (incompatibilities) between items of type $\alpha$ and $\beta$ (optional)
$\eta$	Maximum mass ratio between two items where one is placed at the top of the other with $\eta >1$ (optional)
$(\tilde{L},\tilde{W})$	Target center of mass of the resultant packings (optional)
Variables
$v_j$	Binary variable that represents if bin $j\in J$ is used
$u_{i,j}$	Binary variable that represents if item i is added to bin j
$r_{i,k}$	Binary variable that represents if the orientation $k\in K_i$ is applied to the item i. They are used to compute the effective length, width and height $(x_i^{\prime },y_i^{\prime },z_i^{\prime })$ of the i-th item (see (6)–(8))
$x_i,y_i,z_i$	Continuous variables that return the location of the back lower left corner of item i along x, y and z axes
${\tilde{x}}_i,{\tilde{y}}_i$	Continuous variables that account the relative distance between item i and $(\tilde{L},\tilde{W})$ along x and y axes. Both are used if $(\tilde{L},\tilde{W})$ is defined previously
$b_{i,k,q}$	binary variable that returns the relative position $q\in Q$ between items $i,k\in I$. See Fig. 2