The oriented non-guillotine two-dimensional strip packing problem is investigated. Several 1D relaxations are considered and compared. Moreover, an ILP formulation of the 1D contiguous relaxation is given. The appropriate combination of some of the relaxations enables us the formulate a new ILP model of the strip packing problem.

We study the problem of scheduling independent jobs in a hypercube where jobs are executed in subcubes of various dimensions. The problem being NP-complete, several approximation algorithms based on list scheduling have been proposed, having approximation ratio of order of 2. In this paper, a linear time &-approximation algorithm for the problem is provided when the size of the hypercube is fix...

In the three dimensional strip packing problem, we are given a set of three dimensional rectangular items and a three dimensional box B. The goal is to pack all the items in B such that the height of the packing is minimized. We consider the most basic version of the problem, where the items must be packed with their edges parallel to the edges of B and cannot be rotated. Building upon Caprara’...

The oriented strip packing problem is very important to manufacturing industries: given a strip of xed width and a set of many (> 100) nonconvex polygons with 1, 2, 4, or 8 orientations permitted for each polygon, nd a set of translations and orientations for the polygons that places them without overlapping into the strip of minimum length. Heuristics are given for two versions of strip packin...

Given a set of irregular shapes, the strip nesting problem is the problem of packing the shapes within a rectangular strip of material such that the strip length is minimized, or equivalently the utilization of material is maximized. If the packing found is to be repeated, e.g., on a roll of fabric or a coil of metal, then the separation between repeats is going to be a straight line. This cons...

In this paper we consider a two dimensional strip packing problem. The problem consists of packing a set of rectangular items in one strip of width W and infinite height. They must be packed without overlapping, parallel to the edge of the strip and we assume that the items are oriented, i.e. they cannot be rotated. To solve this problem, we use three exact methods: a branch and bound method, a...

Bansal and Sviridenko [4] proved that there is no asymptotic PTAS for 2-dimensional Orthogonal Bin Packing (without rotations), unless P = NP. We show that similar approximation hardness results hold for several 2and 3-dimensional rectangle packing and covering problems even if rotations by ninety degrees are allowed. Moreover, for some of these problems we provide explicit lower bounds on asym...

When handling 2D packing problems, numerous incomplete and complete algorithms maintain a so-called bottomleft (BL) property: every rectangle placed in a container is propped up bottom and left. While it is easy to make a rectangle BL when it is is added in a container, it is more expensive to maintain all the placed pieces BL when a rectangle is removed. This prevents researchers from designin...

The rectangular Strip Packing Problem (SPP) consists of packing a given set of rectangles in a rectangular strip with fixed width w and variable height minimizing the height of the packing. In this work we propose a ”Greedy Randomized and Adaptive Constructive” heuristic, a GRASP metaheuristic without the post-processing phase, for the SPP in the version where the rectangles can be freely rotat...