A tree search algorithm for the container loading problem

This paper presents a binary tree search algorithm for the three dimensional container loading problem (3D-CLP). The 3D-CLP is about how to load a subset of a given set of rectangular boxes into a rectangular container, such that the packing volume is maximized. In this algorithm, all the boxes are grouped into strips and layers while three constraints, i.e., full support constraint, orientation constraint and guillotine cutting constraint are satisfied. A binary tree is created where each tree node denotes a container loading plan. For a non-root each node, the layer set of its left (or right) child is obtained by inserting a directed layer into its layer set. A directed layer is parallel (or perpendicular) to the left side of the container. Each leaf node denotes a complete container loading plan. The solution is the layer set whose total volume of the boxes is the greatest among all tree nodes. The proposed algorithm achieves good results for the well-known 3D-CLP instances suggested by Bischoff and Ratcliff with reasonable computing time.     

A heuristic block-loading algorithm based on multi-layer search for the container loading problem

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.     

Lagrangean relaxation with clusters and column generation for the manufacturer's pallet loading problem

We consider in this paper a new lagrangean relaxation with clusters for the Manufacturer's Pallet Loading Problem (MPLP). The relaxation is based on the MPLP formulated as a Maximum Independent Set Problem (MISP) and represented in a conflict graph that can be partitioned in clusters. The edges inter clusters are relaxed in a lagrangean fashion. Computational tests attain the optimality for some instances considered difficult for a lagrangean relaxation. Our results show that this relaxation can be a successful approach for hard combinatorial problems modeled in conflict graphs. Moreover, we propose a column generation approach for the MPLP derived from the idea behind the lagrangean relaxation proposed.     

Multi-objective multi-population biased random-key genetic algorithm for the 3-D container loading problem

The container loading problem (CLP) has important industrial and commercial application for global logistics and supply chain. Many algorithms have been proposed for solving the 2D/3D container loading problem, yet most of them consider single objective optimization. In practice, container loading involves optimizing a number of objectives. This study aims to develop a multi-objective multi-population biased random-key genetic algorithm for the three-dimensional single container loading problem. In particular, the proposed genetic algorithm applied multi-population strategy and fuzzy logic controller (FLC) to improve efficiency and effectiveness. Indeed, the proposed approach maximizes the container space utilization and the value of total loaded boxes by employing Pareto approach and adaptive weights approach. Numerical experiments are designed to compare the results between the proposed approach and existing approaches in hard and weak heterogeneous cases to estimate the validity of this approach. The results have shown practical viability of this approach. This study concludes with discussions of contributions and future research directions.     

A beam search approach to the container loading problem

The single container loading problem is a three-dimensional packing problem in which a container has to be filled with a set of boxes. The objective is to maximize the space utilization of the container. This problem has wide applications in the logistics industry. In this work, a new constructive approach to this problem is introduced. The approach uses a beam search strategy. This strategy can be viewed as a variant of the branch-and-bound search that only expands the most promising nodes at each level of the search tree. The approach is compared with state-of-the-art algorithms using 16 well-known sets of benchmark instances. Results show that the new approach outperforms all the others for each set of instances.     

应用自适应遗传算法解决集装箱装载问题

集装箱配载是一个复杂的组合优化问题,其约束条件多,属于NP完全问题,并且求解难度大．在满足一定的约束条件下。提出一种3维集装箱装载的自适应遗传算法．算法中考虑了货物放置方向和装载容积等约束条件,给出了有效的解码算法．实例仿真结果表明了该算法的有效性和实用性．     

一种求解三维集装箱装箱问题的混合遗传算法

在遗传算法的基础上结合传统启发式装箱算法,设计了一个混合遗传算法,该算法既继承了遗传算法的全局搜索好的优点,也克服了遗传算法局部搜索能力差的缺点,能够较好地解决集装箱这类多目标多约束的空间三维分布的问题。     

Approximate algorithms for the container loading problem

In this paper we develop several algorithms for solving three–dimensional cutting/packing problems. We begin by proposing an adaptation of the approach proposed in Hifi and Ouafi (1997) for solving two–staged unconstrained two–dimensional cutting problems. We show how the algorithm can be polynomially solved for producing a constant approximation ratio. We then extend this algorithm for developing better approximate algorithms. By using hill–climbing strategies, we construct some heuristics which produce a good trade–off between the computational time and the solution quality. The performance of the proposed algorithms is evaluated on different problem instances of the literature, with different sizes and densities (a total of 144 problem instances).     

A caving degree based flake arrangement approach for the container loading problem

A caving degree based flake arrangement (CDFA) approach for the NP-hard container loading problem is presented in this paper. Based on the caving degree approach, CDFA binds items in the same size into a larger flake whose binding number is 1 in one of its three dimensions, and then it tries to pack the flake into a corner nearest to the eight vertices of the container. Then, caving degree is redefined to evaluate different placements of flakes, such that an action is selected whose packing flake is as compact and close as possible with other flakes already in (the six surfaces of the container can be regarded as six special flakes). CDFA is extensively tested over two sets of famous benchmarks: 15 LN instances and 1500 BR instances. Experimental results show the high performance of this new approach. Especially for the LN set, CDFA improved current highest volume utilization by 1.3% and 0.5% for two difficult instances LN2 and LN6 respectively; at the same time it got optimal layouts for the other 13 instances, equalling current best records. This breaks current best record created and held by Bortfeldt and Gehring since 1998. Besides, CDFA achieved the second highest average volume utilization among several top efficient algorithms for the BR set.     

A parallel multi-population biased random-key genetic algorithm for a container loading problem

This paper presents a multi-population biased random-key genetic algorithm (BRKGA) for the single container loading problem (3D-CLP) where several rectangular boxes of different sizes are loaded into a single rectangular container. The approach uses a maximal-space representation to manage the free spaces in the container. The proposed algorithm hybridizes a novel placement procedure with a multi-population genetic algorithm based on random keys. The BRKGA is used to evolve the order in which the box types are loaded into the container and the corresponding type of layer used in the placement procedure. A heuristic is used to determine the maximal space where each box is placed. A novel procedure is developed for joining free spaces in the case where full support from below is required. The approach is extensively tested on the complete set of test problem instances of Bischoff and Ratcliff [1] and Davies and Bischoff [2] and is compared with 13 other approaches. The test set consists of 1500 instances from weakly to strongly heterogeneous cargo. The computational experiments demonstrate that not only the approach performs very well in all types of instance classes but also it obtains the best overall results when compared with other approaches published in the literature.     

An optimization model for the container pre-marshalling problem

In most container yards around the world, containers are stacked high to utilize yard space more efficiently. In these yards, one major factor that affects their operational efficiency is the need to re-shuffle containers when accessing a container that is buried beneath other containers. One way to achieve higher loading efficiency is to pre-marshal the containers in such a way that it fits the loading sequence. In this research, we present a mathematical model for the container pre-marshalling problem. With respect to a given yard layout and a given sequence that containers are loaded onto a ship, the model yields a plan to re-position the export containers within the yard, so that no extra re-handles will be needed during the loading operation. The optimization goal is to minimize the number of container movements during pre-marshalling. The resulting model is an integer programming model composed of a multi-commodity flow problem and a set of side constraints. Several possible variations of the model as well as a solution heuristic are also discussed. Computation results are provided.     

蚁群算法求解复杂集装箱装载问题

针对复杂集装箱装载问题(CLP),应用启发式信息与蚁群算法求解了最优装载方案。首先,建立了复杂集装箱装载问题的数学模型,利用蚁群算法对解空间的强搜索能力、潜在并行性及可扩充性,结合三空间分解策略将布局空间依次分割;然后,装入满足约束条件的最优货物块,完成不同大小三维矩形货物的装载布局。在此基础上,设计了基于空间划分策略的蚁群算法。最后以700件货物装入40尺(12.025m)高柜箱进行计算,结果表明该方法能提高集装箱的空间利用率,同时兼顾了多个装载约束条件,可应用性好。     

求解强异类集装箱装载问题的混合蚁群算法

针对强异类集装箱装载问题,设计了一种混合蚁群算法。算法中搜索空间分为货物摆放的优先序列和货物摆放的状态两部分;引入体积大的货物优先放入的启发式规则;将蚂蚁搜索得到的序列与历史最优序列进行交叉,取三者最优序列作为该蚂蚁的搜索路径;在更新信息素时,采取两种挥发系数更新信息素以避免信息素过快饱和,同时分析了算法的复杂度。通过三个强异类实例的测试,表明算法得到的装载方案有较高的空间利用率。     

New lower bounds based on column generation and constraint programming for the pattern minimization problem

The pattern minimization problem is a cutting and packing problem that consists in finding a cutting plan with the minimum number of different patterns. This objective may be relevant when changing from one pattern to another involves a cost for setting up the cutting machine. When the minimization of the number of different patterns is done by assuming that no more than the minimum number of rolls can be used, the problem is also referred to as the cutting stock problem with setup costs.     

A decomposition approach for the solution of the unit loading problem in hydroplants

This paper presents a decomposition approach for the solution of the dynamic programming formulation of the unit loading problem in hydroplant management. This decomposition approach allows the consideration of network and canal constraints without additional computational effort.     

A tree search procedure for the container relocation problem

In the container relocation problem (CRP) n items are given that belong to G different item groups (g=1,…,G). The items are piled up in up to S stacks with a maximum stack height H. A move can either shift one item from the top of a stack to the top of another one (relocation) or pick an item from the top of a stack and entirely remove it (remove). A move of the latter type is only feasible if the group index of the item is minimum compared to all remaining items in all stacks. A move sequence of minimum length has to be determined that removes all items from the stacks. The CRP occurs frequently in container terminals of seaports. It has to be solved when containers, piled up in stacks, need to be transported to a ship or to trucks in a predefined sequence. This article presents a heuristic tree search procedure for the CRP. The procedure is compared to all known solution approaches for the CRP and turns out to be very competitive.     

A parallel hybrid local search algorithm for the container loading problem

In this contribution, a parallel hybrid local search algorithm for the three‐dimensional container loading problem (CLP) is proposed. First a simulated annealing method for the CLP is developed, which is then combined with an existing tabu search algorithm to form a hybrid metaheuristic. Finally, parallel versions are introduced for these algorithms. The emphasis is on CLP instances with a weakly heterogeneous load. Numerical tests based on the well‐known 700 test instances from Bischoff and Ratcliff are performed, and the outcome is compared with methods from other authors. The results show a high solution quality obtained with reasonable computing time.     

Improved LP-based algorithms for the closest string problem

In a recent paper by Liu et al. [Exact algorithm and heuristic for the closest string problem, Computers & Operations Research 2011;38:1513-20], a polynomial time heuristic procedure is proposed for the closest string problem (CSP). Such heuristic called LDDA_LSS is a combination of a previously published approximation algorithm and local search strategies. This paper points out that an instant algorithm deriving a feasible solution directly from the continuous relaxation solution of a standard ILP formulation of CSP already strongly outperforms LDDA_LSS both in terms of solution quality and computing time. Two core based procedures are then proposed that further improve the results of the instant algorithm. Based on these results, we conclude that such LP-based approaches for their efficiency and simplicity should be used as a benchmark for future heuristics on CSP.     

An iterated construction approach with dynamic prioritization for solving the container loading problems

This paper addresses the single and multiple container loading problems, which forms the core engine of a warehouse management system we are contracted to implement for a Hong Kong logistics company. We propose to use dynamic prioritization to handle the awkward box types, whereas the box type with a higher priority is packed onto lower surfaces of the container for the single container case, or packed in earlier containers for the multiple container case. The solution found in one iteration of the algorithm is analyzed, and the priorities are updated and used in the next iteration. This approach provides very competitive results using standard benchmark data sets as compared with other methods. It helps to reduce the difficulty in system implementation and maintenance, because the algorithm is easy to understand for practitioners in the local industry, and it is applicable for both the single and multiple container loading problems at the same time. In addition, we find the existing test data for the multiple container loading problem to be deficient and supplement them by generating new test data consisting of 2800 test cases. Last but not least, our algorithm has been packaged into a software component with full graphical user interface and integrated into a warehouse management system for daily operations.     

集装箱装载问题的一种DNA遗传算法

三维集装箱装载是一个复杂的组合优化问题,约束条件多,属于NP完全问题,求解难度大.在考虑方向性约束和稳定性约束的情况下,提出了一种DNA遗传算法（DNA-GA）,给出了有效的编码和解码方法。实例计算结果表明,利用DNA-GA解决装箱问题是行之有效的一种方法,对推广DNA计算在求解NP难解问题中的应用具有一定的意义。