一种装箱问题的高效定位启发式算法

An efficient placement heuristic for three-dimensional rectangular packing

Kun He,Wenqi HuangÃ

School of Computer Science and Technology,Huazhong University of Science and Technology,Wuhan430074,China

a r t i c l e i n f o

Available online28April2010

Keywords:

Cutting and packing

Three-dimension

Rectangular packing

Container loading

Heuristic

a b s t r a c t

By embodying the spirit of‘‘gold corner,silver side and strawy void’’directly on the candidate packing

place such that the searching space is reduced considerably,and by utilizing the characteristic of

weakly heterogeneous problems that many items are in the same size,afit degree algorithm(FDA)is

proposed for solving a classical3D rectangular packing problem,container loading problem.

Experiments show that FDA works well on the complete set of1500instances proposed by Bischoff,

Ratcliff and Davies.Especially for the800difficult strongly heterogeneous instances among them,FDA

outperforms other algorithms with an average volume utilization of91.91%,which to our knowledge is

0.45%higher than current best result just reported in2010.

&2010Elsevier Ltd.All rights reserved.

1.Introduction

Cutting and packing problems are representative combinational

optimization problems with many important applications in the

industry.This paper addresses the problem of loading a subset of

three-dimensional(3D)rectangular items into a3D rectangular

container,such that the total volume of the packed items is maxi-

mized,i.e.the container’s wasted volume is minimized.A layout is

called feasible,if each packed item is located orthogonally and

completely in the container and there is no overlapping between

any two items.This problem is an NP-hard problem,whose1D

degradation,the0–1knapsack problem,is still NP-hard.

This3D rectangular packing problem is also called the

container loading problem,because the most common and

important application of this problem is to load rectangular

cargoes into containers,vehicles or ships in the transportation

industry.There are some additional considerations that would be

taken into account in the real world[1,2],among which the

orientation constraint and the stability constraint are the most

important ones.In our opinion,if there is an efficient approach to

solve the basic problem that has no additional constraints,then it

is not difficult to make the approach adapted to problems

considering some additional ones.

Since the orientation constraint has been widely considered by

the researchers,and it has been accepted by the famous BR

benchmarks[1],we take the orientation constraint into account

that one or two sides of the items may not be used as the height.

This situation usually happens when cargoes are boxes full of oil

or wine.We do not concern the stability constraint for the

following reasons:(1)Stability constraint is not considered as

widely as the orientation one and the stability criteria is

inconsistent in the literature.In some cases it requires that each

item is fully supported,or partially supported with at least a given

percentage;in other cases it requires that the gravity center of

each item falls over an underlying item or over the bottom of the

container.(2)Stability could become a consequence of the high

cargo compactness when the container’s volume utilization is

high enough.(3)Foam rubber or other stuffing could be used to

fill the small empty spaces left,as what has been done in some

freight companies.

Many efficient algorithms have been proposed for solving this

classical3D packing problem.The most prevalent approach is

wall building or layering[1,3–9],first proposed by George and

Robinson in1980[3].In the past thousand years,people usually

start packing goods from the bottom and build up the packing

in layers,inserting each goods such that it is contiguous with

what is already packed.Inspired by these human’s experience,the

wall building or layering method usually opens a new layer

or wall with a width equals to some item dimension,then each

layer isfilled up by a number of horizontal strips,and each strip

is packed in a greedy way.Another efficient approach widely

adopted by the researchers is block arrangement[10–12],first

proposed by Bortfeldt and Gehring in1998[10],which binds

items of the same or similar size into a larger rectangular block to

do the tentative packing.By utilizing the block arrangement

method,Parren˜o proposed an approach that always places a

column or layer built by same size items into a maximal space,the

largest empty parallelepiped spaces[13,14].Above approaches all

utilized the characteristic of the weakly heterogeneous instances

that many items are in the same size.Therefore,the solution

qualities are in a downtrend when the problems become more

Contents lists available at ScienceDirect

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Computers&Operations Research

0305-0548/$-see front matter&2010Elsevier Ltd.All rights reserved.

doi:10.1016/j.cor.2010.04.015

ÃCorresponding author.Tel.:+862787543018;fax:+862787545004.

E-mail addresses:brooklet60@(K.He),wqhuangwh@

(W.Huang).

Computers&Operations Research38(2011)227–233