Approximation Algorithms for the Orthogonal Z-Oriented Three-Dimensional Packing Problem

Abstract

We present approximation algorithms for the orthogonal z-oriented three-dimensional packing problem (TPPz) and analyze their asymptotic performance bound. This problem consists in packing a list of rectangular boxes L=(b1 ,b2 ,. . . ,bn ) into a rectangular box B=(l,w,\infty)$, orthogonally and oriented in the z-axis, in such a way that the height of thepacking is minimized. We say that a packing is oriented in the z-axis when the boxes in L are allowed to be rotated (by ninety degrees) around the z-axis. This problem has some nice applications but has been less investigated than the well-known variant of it---denoted by TPP (three-dimensional orthogonal packing problem)---in which rotations of the boxes are not allowed. The problem TPP can be reduced to TPPz. Given an algorithm for TPPz, we can obtain an algorithm for TPP with the same asymptotic bound. We present an algorithm for TPPz, called R, and three other algorithms, called LS, BS, and SS, for special cases of this problem in which the instances are more restricted. The algorithm LS is for the case in which all boxes in L have square bottoms; BS is for the case in which the box B has a square bottom, and SS is for the case in which the box B and all boxes in L have square bottoms. For an algorithm $\wa$, we denote by $r(\wa)$ the asymptotic performance bound of $\wa$. We show that $2.5\leq r(R) < 2.67$, $\ 2.5\leq r(LS)\leq 2.528$,$\ 2.5\leq r(BS)\leq 2.543$, and $\ 2.333\leq r(SS)\leq 2.361$. The algorithms presented here have the same complexity ${\cal O}(n\log n)$ as the other known algorithms for these problems, but they have better asymptotic performance bounds.