Table 3.

Amount of mandatory (related to intrinsic restrictions)) and optional (related to real-world BPP restrictions) used variables depending on the number of items and bins (see Table 1).

	n	$x_i,y_i,z_i$	$u_{i,j}$	$v_j$	$r_{i,k}$	$b_{i,k,q}$	Binary	Continuous
Mandatory	1	3m	–	–	$\kappa$	$6\left(\begin{array}{c}m\\ 2\end{array}\right)$	$6\left(\begin{array}{c}m\\ 2\end{array}\right)+\kappa$	3m
	$\ge 2$	3m	nm	n	$\kappa$	$6\left(\begin{array}{c}m\\ 2\end{array}\right)$	$6\left(\begin{array}{c}m\\ 2\end{array}\right)+n(m+1)+\kappa$	3m

	n	${\tilde{x}}_i,{\tilde{y}}_i$	(26) as HC	Binary	Continuous
Optional	1	2m	$-p^{-}-6p^{+}$	$-p^{-}-6p^{+}$	2m
	$\ge 2$	2m	$-p^{-}-6p^{+}$	$-p^{-}-6p^{+}$	2m