A heuristic block-loading algor ithm based on multi-layer search for the
container loading problem
Defu Zhang
a
, Yu Peng
b,
n
, Stephen C.H. Leung
c
a
Department of Computer Science, Xiamen University, 361005, China
b
Department of Computer Science, Hong Kong University, Hong Kong
c
Department of Management Sciences, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
article info
Available online 25 October 2011
Keywords:
Container loading problem
Heuristic algorithm
Depth-first search
abstract
This paper presents an efficient heuristic block-loading algorithm based on multi-layer search for the
three-dimensional container loading problem. First, a basic heuristic block-loading algorithm is
introduced. This algorithm loads one block, determined by a block selecting algorithm, in one packing
phase, according to a fixed strategy, until no blocks are available. Second, the concept of composite
block is introduced, the difference between traditional block and composite block being that composite
block can contain multiple types of boxes in one block under some restrictions. Third, based on the
depth-first search algorithm, a multi-layer search algorithm is developed for determining the selected
block in each packing phase, and making this result closer to the optimal solution. Computational
results on a classic data set show that the proposed algorithm outperforms the best known algorithm in
almost all the test data.
& 2011 Elsevier Ltd. All rights reserved.
1. Introduction
In logistics and transportation industries, three-dimensional
packing problems often occur. Improving the efficiency of packing
(space utilization) has become a very important issue as the
global economy is becoming ever more competitive. An effective
method for solving three-dimensional packing problems is not
only of great economic importance in logistics and transportation
processes but also has environmental implications since lower
fuel consumption helps to reduce pollution.
The container loading problem is a typical three-dimensional
packing problem that involves several practical applications
having different optimization objectives and loading constraints.
Thus there exist many variants of the problem. Dyckhoff and
Finke [1] outline the different types of loading problems, and give
the corresponding classification, such as a single container load-
ing problem, multi-container loading problem, homogeneous
loading problem (containing only one type of box) and hetero-
geneous loading problems (including many types of boxes). The
single container loading problem considered by this paper can be
described as follows.
Assume a container (of volume V) and a series of boxes to be
packed into the container. The container and the boxes are of
cuboid shape. The objective is to determine a feasible loading plan
that meets the given loading constraints and maximizes utiliza-
tion of space in the container, i.e. the filling rate (S/V 100%) is as
large as possible, where S is the total volume of boxes loaded in
the container. A feasible loading plan must meet the following
criteria:
(1) Any loaded box cannot overlap with the container, and no two
boxes can overlap each other;
(2) The surface of the loaded boxes is parallel to the surface of the
container.
When solving real-world container loading problems, one has
to consider some practical constraints, such as orientation con-
straints, loading stability, load bearing strength of boxes, handling
constraints, container weight limit and so on [2]. This paper
considers the following two constraints:
(C1) Orientation constraint: for certain box types, up to five of
the maximum six possible orientations are prohibited.
(C2) Support constraint: in a given packing plan the area of
each box not placed on the floor of the container must be
supported completely (i.e. 100%) by other boxes.
The remainder of this paper is organized as follows. In the next
section an overview of literature is provided. In Section 3 main
parts of the proposed algorithm are introduced. In Section 4
computational experiments are described and analyzed. Finally,
in Section 5 conclusions are drawn.
Contents lists available at SciVerse ScienceDirect
journal homepage: www.elsevier.com/locate/caor
Computers & Operations Research
0305-0548/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.cor.2011.10.019
n
Corresponding author. Tel.: þ 852 28578465; fax: þ852 25598447.
E-mail address: ypeng@cs.hku.hk (Y. Peng).
Computers & Operations Research 39 (2012) 2267–2276