Three-dimensional Strip Packing Problem Research Articles

A reference length approach for the 3D strip packing problem

In the three-dimensional strip packing problem (3DSP), we are given a container with an open dimension and a set of rectangular cuboids (boxes) and the task is to orthogonally pack all the boxes into the container such that the magnitude of the open dimension is minimized. We propose a block building heuristic based on extreme points for this problem that uses a reference length to guide its solution. Our 3DSP approach employs this heuristic in a one-step lookahead tree search algorithm using an iterative construction strategy. We tested our approach on standard 3DSP benchmark test data; the results show that our approach produces better solutions on average than all other approaches in literature for the majority of these data sets using comparable computation time.

A global search framework for practical three-dimensional packing with variable carton orientations

This article aims to tackle a practical three-dimensional packing problem, where a number of cartons of diverse sizes are to be packed into a bin with fixed length and width but open height. Each carton is allowed to be packed in any one of the six orientations, and the carton edges are parallel to the bin edges. The allowance of variable carton orientations exponentially increases the solution space and makes the problem very challenging to solve. This study first elaborately devises the packing procedure, which converts an arbitrary sequence of cartons into a compact packing solution and subsequently develops an improved genetic algorithm (IGA) to evolve a set of solutions. Moreover, a novel global search framework (GSF), utilizing the concept of evolutionary gradient, is proposed to further improve the solution quality. Numerical experiments indicate that IGA provides faster and better results and GSF demonstrates its superior performance, especially in solving relative large-size and heterogeneous instances. Applying the proposed algorithms to some benchmarking cases of the three-dimensional strip packing problem also indicates that the algorithms are robust and effective compared to existing methods in the literature.

GREEDY ALGORITHMS FOR PACKING UNEQUAL SPHERES INTO A CUBOIDAL STRIP OR A CUBOID

Given a finite set of spheres of different sizes, we study the three-dimensional Strip Packing Problem (3D-SPP) as well as the three-dimensional Knapsack Problem (3D-KP). The 3D-SPP asks for a placement of all spheres within a cuboidal strip of fixed width and height so that the variable length of the cuboidal strip is minimized. The 3D-KP requires packing of a subset of the spheres in a given cuboid so that the wasted space is minimized. To solve these problems two greedy algorithms were developed which adapt the algorithms proposed by Huang et al. (2005) to the 3D case with some important enhancements. The resulting methods were tested using the instances provided by Stoyan et al. (2003). Additionally, two series of 12 instances each for the 3D-SPP and for the 3D-KP are introduced and results for these new instances are also reported.

A hybrid placement strategy for the three-dimensional strip packing problem

This paper presents a hybrid placement strategy for the three-dimensional strip packing problem which involves packing a set of cuboids (‘boxes’) into a three-dimensional bin (parallelepiped) of fixed width and height but unconstrained length (the ‘container’). The goal is to pack all of the boxes into the container, minimising its resulting length. This problem has potential industry application in stock cutting (wood, polystyrene, etc. – minimising wastage) and also cargo loading, as well as other applications in areas such as multi-dimensional resource scheduling. In addition to the proposed strategy a number of test results on available literature benchmark problems are presented and analysed. The results of empirical testing of the algorithm show that it out-performs other methods from the literature, consistently in terms of speed and solution quality-producing 28 best known results from 35 test cases.

Three-dimensional packings with rotations

We present approximation algorithms for the three-dimensional strip packing problem, and the three-dimensional bin packing problem. We consider orthogonal packings where 90 ∘ rotations are allowed. The algorithms we show for these problems have asymptotic performance bounds 2.64, and 4.89, respectively. These algorithms are for the more general case in which the bounded dimensions of the bin given in the input are not necessarily equal (that is, we consider bins for which the length, the width and the height are not necessarily equal). Moreover, we show that these problems—in the general version—are as hard to approximate as the corresponding oriented version.

A heuristic for the three-dimensional strip packing problem

The contribution presents a heuristic for the three-dimensional strip packing problem (3D-SPP) with rectangular pieces (boxes). The considered 3D-SPP can be formulated as follows: for a given set of boxes and a given longitudinal open container, determine an arrangement of all boxes within the container so that the required container length is minimized. The presented heuristic was derived from a branch-and-bound approach for the container loading problem (CLP) that was recently proposed by Pisinger. Two approaches are investigated to adapt the CLP method to the 3D-SPP. The first approach looks directly at a container of practically infinite length. The second one solves an SPP instance by computing a series of CLP instances with descending container lengths. The best parts of previously best solutions are reused systematically within the second approach that proves to be more successful. The method is tested by means of 800 benchmark instances with up to 1000 boxes. A comparison to other methods from the literature shows the high performance of the 3D-SPP heuristic.

This side up!

We consider two- and three-dimensional bin-packing problems where 90° rotations are allowed. We improve all known asymptotic performance bounds for these problems. In particular, we show how to combine ideas from strip packing and two-dimensional bin packing to give a new algorithm for the three-dimensional strip packing problem where boxes can only be rotated sideways. We propose to call this problem “This side up”. Our algorithm has an asymptotic performance bound of 9/4.

Two- and three-dimensional parametric packing

We present approximation algorithms for the two- and three-dimensional bin packing problems and the three-dimensional strip packing problem. We consider the special case of these problems in which a parameter m (a positive integer) is given, indicating that each of the dimensions of the items to be packed is at most 1 / m of the corresponding dimension of the recipient. We analyze the asymptotic performance of these algorithms and exhibit bounds that, to our knowledge, are the best known for this special case.

Two- and three-dimensional parametric packing

We present approximation algorithms for the two- and three-dimensional bin packing problems and the three-dimensional strip packing problem. We consider the special case of these problems in which a parameter m (a positive integer) is given, indicating that each of the dimensions of the items to be packed is at most 1/m of the corresponding dimension of the recipient. We analyze the asymptotic performance of these algorithms and exhibit bounds that, to our knowledge, are the best known for this special case.