Solving container loading problems by block arrangement

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Abstract

In order to solve heterogeneous single and multiple container loading problems, an algorithm is presented that builds homogeneous blocks of identically orientated items. First a greedy heuristic is presented that generates the desired block arrangements. Second the solutions provided by the greedy heuristic are improved by a tree search. Additional aspects such as load stability and weight distribution within the container are also taken into account. The test cases of Bischoff and Ratcliff are used for benchmarking purposes.

Introduction

The paper addresses the following packing problem: Several small, three-dimensional, rectangular items (e.g., boxes) are to be stowed in one or more containers in such a way that maximum use is made of the container volume. The items should be stowed completely in the container so that their edges lie parallel to the edges of the container and that no two items overlap. Items may be of different sizes. Those of identical size can be grouped into types. In contrast to the homogeneous problem with only one item type, this problem is often referred to as the heterogeneous problem or the container loading problem.

According to the classification system of Dyckhoff and Finke (1992), two problem areas in the heterogeneous problem can be distinguished. For the so called single container problem, the capacity of the container is not large enough to stow all given items and consequently the objective should be to minimize the unused container space. The quality of a solution is measured by volume utilization. This is defined by the ratio of the sum of the volume of the packed items to the total container volume. The second problem type is known as the multiple container problem. It aims to minimize the number of containers needed to stow all given items. A possible extension of the multiple container problem, not addressed in this paper, would be to consider containers of different sizes. Instead of minimizing the number of required containers a more appropriate objective would then be to minimize the total cost necessary to ship all items.

Packing problems contain knapsack problems. Therefore they are NP-hard. Consequently, finding the optimal arrangement in view of volume utilization is not a trivial task. But for many real world applications it is not sufficient to consider volume utilization as the sole objective. As argued by Bischoff and Ratcliff (1995a) a variety of additional factors have to be taken into account. Examples for this are considering load stability, catering for orientation constraints for the items or balancing the weight distribution within the container. Hence, approaches that neglect additional factors are likely to be of limited value.

In order to solve container loading problems, several heuristics, designed mainly for the single container issue, have been developed in recent years. They all address the question of how to find a solution that achieves high volume utilization, but some of them are designed to cater for additional aspects. In order to reduce the computing time of the algorithms and the material handling effort necessary to implement the generated solutions, most of the approaches generate solutions of a specific given arrangement, e.g., layers or towers. Bischoff et al. (1995) and Bortfeldt and Gehring (1998) have developed algorithms where the container is filled in layers. The first is a greedy heuristic, the latter a tabu search approach. Bischoff and Ratcliff's heuristic (1995a), which beside volume utilization also caters for multidrop situations, and Gehring and Bortfeldt's Genetic Algorithm (1997) solve the problem by arranging the items in stacks. Ngoi et al. (1994) have developed an algorithm that does not explicitly make use of a specified arrangement. Tree search algorithms were presented by Bortfeldt (1995) and by Morabito and Arenales (1994). Whereas the first approach was outperformed by Gehring and Bortfeldt's algorithm (1997), the latter one only considers arrangements that can be generated by applying guillotine cuts. The paper of Davies and Bischoff (1999) addresses the question of how to achieve an even weight distribution. Finally, the approach of Terno et al. (2000) was primarily designed for solving heterogeneous pallet loading problems by layer arrangements.

Not much research has been devoted to the multiple container problem to date. In the eighties Liu and Chen (1981) and Ivancic et al. (1989) published approaches on this issue. Bischoff and Ratcliff (1995b) extended the single container approach of Bischoff et al. (1995) to address the multiple container problem. Recently, Bortfeldt (2000) modified Bortfeldt and Gehring's tabu search approach (1998) to cater for the multiple container issue.

This paper presents a new approach for the container loading problem with a heterogeneous consignment. Instead of using wall, layer or tower arrangements, the container is filled by homogeneous blocks made up of identical items. All items within a block have the same orientation (cf. Fig. 1).

These homogeneous blocks offer a variety of advantages for many practical problems and allow considering additional factors:

  • Solving the container loading problem as a three-dimensional “puzzle” is not appropriate for practical reasons as it is often desirable that items of the same type be stowed in close proximity to one another, thus avoiding re-sorting the items after unloading. Homogeneous block arrangements satisfy this requirement.

  • Homogeneous blocks are easy to arrange and therefore enable a quicker loading time.

  • Stacking identical items is in view of their load bearing strength often less problematic than placing a heavy item with a small base area on top of a bigger item.

  • Items cannot easily slip within a block structure.


The paper is structured as follows. Section 2 presents a simple greedy heuristic for solving the single container problem using homogeneous blocks. An improved algorithm based on a tree search procedure is discussed in Section 3. In 4 Computational results for the single container problem, 5 Computational results for the multiple container problem, computational results for the single and the multiple container problem are presented with respect to different objectives such as volume utilization, load stability and weight distribution. Finally, Section 6 summarizes the conclusions reached and attempts to put them into perspective.

Section snippets

Greedy heuristic

In the first step a heuristic was implemented that implicitly generates the desired block arrangements. Starting with an empty container an additional item is stowed in every iteration of the heuristic. Pursuing a greedy approach, the items are sorted by volume with larger items being chosen first. All possible positions for stowing the additional item in the container are examined. These positions are called empty spaces and are stored in a list. The additional item is tentatively placed with

Improvement heuristic

In order to improve the solutions generated by the greedy heuristic, a tree search was implemented that allows consideration of different item loading sequences, alongside the volume determined sequence. Branching is not carried out for every different item, rather for different types, and the permitted item orientations within the considered type. The root node represents an empty container; each further node constitutes a (partially) filled container. If m is the number of different types,

Computational results for the single container problem

The algorithm presented above, hereafter known as Algorithm 1, was implemented in Borland Delphi 3.0 under Windows 95. Test runs were carried out on a Pentium PC with 200 MHz clock and 32 MB memory. The value of the breadth parameter was set to seven.

Standard test cases from the literature were used for benchmarking purposes. Bischoff and Ratcliff (1995a) generated seven test classes BR1–BR7 with 100 test cases each. Test cases in BR1 use three different types of items only. The number of types

Computational results for the multiple container problem

Benchmarking was carried out with test cases from the literature and with randomly generated test cases. Both strategies for the multiple container problem were tested. In the following tests the sequential approach is referred to as Algorithm 2, and the simultaneous approach as Algorithm 3.

Conclusion

A new algorithm based on the generation of homogeneous block arrangements was developed in order to solve single container problems. These homogeneous blocks have several advantages for practical problems. The algorithm was tested by using standard test cases from the literature. With respect to volume utilization, the algorithm competed very well compared to several benchmark approaches, in particular for stronger heterogeneous problems.

Many industrial packing problems involve multiple

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