A novel hybrid tabu search approach to container loading

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Abstract

The container loading problem, which is significant for a number of industrial sectors, aims to obtain a high space utilisation in the container while satisfying practical constraints. This paper presents a novel hybrid tabu search approach to the container loading problem. A loading heuristic is devised to incorporate heuristic strategies with a handling method for remaining spaces to generate optimal loading arrangements of boxes with stability considered. The tabu search technique, which covers the encoding, evaluation criteria and configuration of neighbourhood and candidate solutions, is used to improve the performance of the loading heuristic. Experimental results with benchmark data show that the hybrid approach provides a better space utilisation than the published approaches under the condition of all loaded boxes with one hundred percent support from below. Moreover, it is shown that the hybrid tabu search can solve problems with the constraints of weight limit and weight distribution with real world data.

Introduction

The three-dimensional container loading problem addresses the issue of how to load three-dimensional, rectangular items (e.g. boxes) in one or more containers in such a way that the maximum utilisation is made of the container space. The problem is significant for a number of industrial sectors where optimal loading is needed to place cargo effectively into aeroplanes, ships, trailers or trucks in order to obtain a high space utilisation. In the real world, the objective for container loading is not only to obtain high space utilisation, which is an optimal geometric combination of small items suitable for the loading process of the container, but also to satisfy practical constraints, e.g. orientation, stability, weight limits and load-bearing strength. The issue of stability is paramount in a number of real-world applications and to ensure high load stability each stowed box should be completely supported from below. The consideration of both space utilisation and constraints makes the problem difficult to solve by using a pure mathematical method. Hence, approximation approaches are used to explore solutions to the problem.

According to Dyckhoff's typology of cutting and packing [1] and the improved typology by Wäscher et al. [2], the problem is classified as 3DSLOPP, namely three-dimensional single large object placement problem. The last decade has seen significant progress in tackling container loading by using a wide variety of techniques. Several human knowledge-based heuristic approaches have been developed for specific arrangements, e.g. wall-building [3], [4], cuboid (block) arrangement [5], [6], [7], stack building [8], [9] and guillotine-cutting [10]. Metaheuristics with intelligent search have been exploited, e.g. genetic algorithms [9], [10], [11], [12], a co-evolutionary genetic algorithm [13], simulated annealing [14] and tabu search [15], [16]. Hybrid approaches have been proposed, e.g. tree search incorporated in the constructive heuristic [17], greedy randomized adaptive search procedure combined with the wall-building constructive heuristic [18]. Exact algorithms have also been proposed, e.g. a mixed-integer programming [19], [20]. Some of the published approaches are devoted to the maximum space utilisation of a single container considering the constraints of stability, loading orientations and weight distribution, based on the arrangements of blocks and walls. For example, the stability of goods has to be considered in the transportation [3], [5], [6], [7], [16], [21]. The weight distribution of cargos has to be considered in air freight [21], [22]. The load bearing strength is tackled by Bischoff [23]. In fact, all solution approaches face the issue of working within practical constraints in addition to the primary problem of maximum space utilisation. The consideration of each practical constraint further complicates the problem and increases the difficulty of the solution approach. This paper considers those situations in which it imperative that each box is completely supported from below. Real world scenarios in which this is important include fragile loads and loosely packed boxes in which the cargo inside can shift. Researchers have also remarked on the requirement for complete support, such as Bortfeldt and Gehring [24] who state “in the interests of high load stability it is required that each stowed box is completely supported by the container floor or the tops of other boxes.” Having developed novel heuristics for the container loading problem, Pisinger [4], noted the requirement for a postprocessing algorithm to improve the support of the boxes.

Based on a survey of the recent approaches, metaheuristics can obtain a high space utilisation but be ineffective to tackle the practical constraints, whereas the heuristics with the human knowledge is superior to metaheuristics for tackling the practical constraints, particularly for the stability and load bearing strength. However, heuristics result in a lower space utilisation compared to metaheuristics. Considering the trade-off between volume utilisation and practical constraints, this paper puts forward a novel hybrid approach to the container loading problem with strongly heterogeneous box types where the practical constraints are considered. The hybrid approach incorporates a loading heuristic and a handling method for remaining spaces with tabu search techniques. The paper is divided into seven sections. The next section describes the container loading problem and definition of the three-dimensional space. Section 3 describes the handling method for remaining spaces. Section 4 presents the heuristic method for the loading arrangement of boxes. The framework of hybrid tabu search and configuration of the tabu search are dealt with in Section 5. Comparisons using benchmark data and experiments using real world data are reported in Section 6. Finally, a summary of the approach and further research are drawn in Section 7.

Section snippets

The problem and constraints

There is a given set of n types of small, three-dimensional, rectangle-shaped items, called boxes, B={b1, b2, …, bn}, of which each box type is characterised by its length li, width wi, height hi and quantity mi, i={1, 2, …, n}. The boxes are loaded into a rectangular container with length L, width W and height H. Suppose that the right-hand face of the container is opened. Let a container be located in the first octant of a three-dimensional Cartesian coordinate system with the front-left-bottom

Partitioning remaining spaces

The handling of remaining space in the container is an important issue which directly influences volume utilisation and constraints fulfilment. Non-rectangular remaining spaces are generated when boxes are loaded. To ensure each remaining space to be rectangular shape, partitioning of the remaining spaces is necessary. After a box is loaded into the remaining rectangular space, the space is partitioned into three new rectangular remaining spaces, i.e. the right space to the right of the loaded

The loading heuristic

The published approaches load the boxes into the container one by one [10] or in a local arrangement into a given remaining space [9], [11], [15]. The one-by-one loading results in more fragmented remaining spaces. Thus merging remaining spaces must be considered, or low space utilisation will result. Gering and Bortfeldt [9] proposed another loading method implemented by yielding different local arrangements, which are composed of towers or layers. The towers or layers are constructed by

Hybrid tabu search with the loading heuristic

A novel hybrid tabu search method has been developed that encompasses both a tabu search and loading heuristics iteratively. The method has the following steps:

  • (1)

    An initial solution to the problem is generated by using the aforementioned loading heuristic. This initial solution is then transformed into a representation to be used in a tabu search. A number of feasible solutions are generated using the tabu search method.

  • (2)

    Each of the feasible solutions generated by the tabu search is transferred

Experiments with the hybrid tabu search

The hybrid tabu search approach presented, or HTS for short, is implemented in Visual C++ under Windows XP. All computational experiments are carried out on a laptop with Intel Centrino Duo CPU 1.66 GHz and RAM 1 GB.

Conclusion

The novel hybrid tabu search approach investigated combines the tabu search techniques, including: (1) an efficient encoding to overcome the expensive computational cost associated with large scale problems, (2) an evaluation criterion to satisfy the constraints of weight limit and weight distribution of boxes and (3) the loading heuristic to generate the loading arrangements of boxes by constructing blocks along with the handling method for remaining spaces while ensuring the constraints.

The

Acknowledgements

The authors would like to thank the participants and the anonymous reviewers for their comments and feedback. This research was partially supported by the Education Department of Liaoning Province in China (Grant no. 2004F043).

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