一种新的多约束尺寸可变的装箱问题

多约束尺寸可变的装箱问题作为经典装箱问题的扩展,具有极为广泛的应用背景。在以货车运输为主的物流公司的装载环节中,运输成本不仅仅由车厢的空间利用率决定。分析了该类装箱问题与传统的集装箱装载问题的区别,并据此给出了一种新的尺寸可变装箱问题的定义。除了经典装箱问题中物品体积这一参数,还引入了物品类型、箱子类型等参数,建立了数学模型,将经典的FFD（First Fit Decreasing）算法进行了推广,提出了新的算法MFFD,并分析了相关的算法复杂性。最后对FF、FFD以及MFFD算法进行了模拟实验,实验结果表明,在相关参数符合均匀分布的条件下,MFFD算法效果较好。     

New heuristics for one-dimensional bin-packing

Several new heuristics for solving the one-dimensional bin packing problem are presented. Some of these are based on the minimal bin slack (MBS) heuristic of Gupta and Ho. A different algorithm is one based on the variable neighbourhood search metaheuristic. The most effective algorithm turned out to be one based on running one of the former to provide an initial solution for the latter. When tested on 1370 benchmark test problem instances from two sources, this last hybrid algorithm proved capable of achieving the optimal solution for 1329, and could find for 4 instances solutions better than the best known. This is remarkable performance when set against other methods, both heuristic and optimum seeking.Scope and purposePacking items into boxes or bins is a task that occurs frequently in distribution and production. A large variety of different packing problems can be distinguished, depending on the size and shape of the items, as well as on the form and capacity of the bins (H. Dyckhoff and U. Finke, Cutting and Packing in Production and Distribution: a Typology and Bibliography, Springer, Berlin, 1992). Similar problems occur in minimising material wastage while cutting pieces into particular smaller ones and in the scheduling of identical processors in order to minimise total completion time. This work addresses the basic packing problem, known as the one-dimensional bin packing problem, where it is required to pack a number of items into the smallest possible number of bins of pre-specified equal capacity. Even though this problem is simple to state, it is NP hard, i.e., it is unlikely that there exists an algorithm that could solve every instance of it in polynomial time. Solution of more general realistic packing problems is probably contingent upon the availability of effective and computationally efficient solution procedures for the basic problem. In this work we present several heuristics capable of doing that. Extensive computational testing attests to the power of these heuristics, as well as to their computational efficiency.     

Bin Packing with Rejection Revisited

We consider the following generalization of bin packing. Each item has a size in (0,1] associated with it, as well as a rejection cost, that an algorithm must pay if it chooses not to pack this item. The cost of an algorithm is the sum of all rejection costs of rejected items plus the number of unit sized bins used for packing all other items. We first study the offline version of the problem and design an APTAS for it. This is a non-trivial generalization of the APTAS given by Fernandez de la Vega and Lueker for the standard bin packing problem. We further give an approximation algorithm of an absolute approximation ratio 3/2, where this value is best possible unless P=NP.     

A parallel multi-objective algorithm for two-dimensional bin packing with rotations and load balancing

Bin packing problems are NP-hard combinatorial optimization problems of fundamental importance in several fields, including computer science, engineering, economics, management, manufacturing, transportation, and logistics. In particular, the non-guillotine version of the single-objective two-dimensional bin packing problem with rotations is a highly complex scheduling problem that consists in packing a set of items into the minimum number of bins, where items can be rotated 90° and are characterized by having different heights and widths. Recently, some authors have proposed multi-objective formulations that also consider additional objectives, such as the balancing the bin load in order to increase its stability. The load imbalance minimization, which depends on the distribution of the items packed in them, is a critical point in many real applications. This paper analyzes how to solve two-dimensional bin packing problems with rotations and load balancing using parallel and multi-objective memetic algorithms that apply a set of search operators specifically designed to solve this problem. Results obtained using a set of test problems show the good performance of parallel and multi-objective memetic algorithms in comparison with other methods found in the literature.     

受启动空间约束的装箱问题

提出了一种带有启动空间的约束装箱问题(start-up bin packing problem,简称SBPP),即不同类型的物品放入同一箱子中需要一个启动空间.该问题在工作分配、任务调度和日常生活中的包装等问题中有着广泛的应用背景.给出了一个求解SBPP的线性脱线算法C-NF,其最坏情况渐近性能比为2,与启动空间的大小无关.对该算法的平均性能进行了实验分析.另外,还分析了SBPP的在线特性,指出大量的经典在线装箱算法应用于SBPP都不存在确定的最坏情况渐近性能比,也给出了一种具有确定的最坏情况渐近性能比的在线算法.     

Bin packing problems with rejection penalties and their dual problems

In this paper we consider the following problems: we are given a set of n items {u₁, … , u_n} and a number of unit-capacity bins. Each item u_i has a size w_i ∈ (0, 1] and a penalty p_i ⩾ 0. An item can be either rejected, in which case we pay its penalty, or put into one bin under the constraint that the total size of the items in the bin is no greater than 1. No item can be spread into more than one bin. The objective is to minimize the total number of used bins plus the total penalty paid for the rejected items. We call the problem bin packing with rejection penalties, and denote it as BPR. For the on-line BPR problem, we present an algorithm with an absolute competitive ratio of 2.618 while the lower bound is 2.343, and an algorithm with an asymptotic competitive ratio arbitrarily close to 1.75 while the lower bound is 1.540. For the off-line BPR problem, we present an algorithm with an absolute worst-case ratio of 2 while the lower bound is 1.5, and an algorithm with an asymptotic worst-case ratio of 1.5. We also study a closely related bin covering version of the problem. In this case p_i means some amount of profit. If an item is rejected, we get its profit, or it can be put into a bin in such a way that the total size of the items in the bin is no smaller than 1. The objective is to maximize the number of covered bins plus the total profit of all rejected items. We call this problem bin covering with rejection (BCR). For the on-line BCR problem, we show that no algorithm can have absolute competitive ratio greater than 0, and present an algorithm with asymptotic competitive ratio 1/2, which is the best possible. For the off-line BCR problem, we also present an algorithm with an absolute worst-case ratio of 1/2 which matches the lower bound.     

Approximation Algorithms for Extensible Bin Packing

In a variation of bin packing called extensible bin packing, the number of bins is specified as part of the input, and bins may be extended to hold more than the usual unit capacity. The cost of a bin is 1 if it is not extended, and the size if it is extended. The goal is to pack a set of items of given sizes into the specified number of bins so as to minimize the total cost. Adapting ideas Grötschel et al. (1981), Grötschel et al. (1988), Karmarkar and Karp (1982), Murgolo (1987), we give a fully polynomial time asymptotic approximation scheme (FPTAAS) for extensible bin packing. We close with comments on the complexity of obtaining stronger results.     

Two-dimensional online bin packing with rotation

In two-dimensional bin packing problems, the input items are rectangles which need to be packed in a non-overlapping manner. The goal is to assign the items into unit squares using an axis-parallel packing. Most previous work on online packing concentrated on items of fixed orientation, which must be assigned such that their bottom side is parallel to the bottom of the bin. In this paper we study the case of rotatable items, which can be rotated by ninety degrees. We give almost tight bounds on the (asymptotic) competitive ratio of bounded space bin packing of rotatable items, and introduce a new unbounded space algorithm. This improves the results of Fujita and Hada.     

Improved Results for a Memory Allocation Problem

We consider a memory allocation problem. This problem can be modeled as a version of bin packing where items may be split, but each bin may contain at most two (parts of) items. This problem was recently introduced by Chung et al. (Theory Comput. Syst. 39(6):829–849, 2006). We give a simple \frac32\frac{3}{2} -approximation algorithm for this problem which is in fact an online algorithm. This algorithm also has good performance for the more general case where each bin may contain at most k parts of items. We show that this general case is strongly NP-hard for any k≥3. Additionally, we design an efficient approximation algorithm, for which the approximation ratio can be made arbitrarily close to \frac75\frac{7}{5} .     

Variable neighbourhood search for the variable sized bin packing problem

The variable sized bin packing problem is a generalisation of the one-dimensional bin packing problem. Given is a set of weighted items, which must be packed into a minimum-cost set of bins of variable sizes and costs. This problem has practical applications, for example, in packing, transportation planning, and cutting. In this work we propose a variable neighbourhood search metaheuristic for tackling the variable sized bin packing problem. The presented algorithm can be seen as a hybrid metaheuristic, because it makes use of lower bounding techniques and dynamic programming in various algorithmic components. An extensive experimentation on a diverse set of problem instances shows that the proposed algorithm is very competitive with current state-of-the-art approaches.     

受位置约束的有色装箱问题

作为对有色装箱问题的推广,提出了一种受位置约束的有色装箱问题（longest item at the bottom coloring bin packing problem,LIBCBPP）,即在有色物品的装箱过程中,要求重（长）的物品置于轻（短）的物品下方.该问题在任务调度和日常生活中的运输等问题中有着广泛的应用背景.给出了一个求解该问题的近似KC-LIBFF算法,分析其最坏情况渐进性能比为2,并给出了相应的实验结果.     

On Dynamic Bin Packing: An Improved Lower Bound and Resource Augmentation Analysis

We study the dynamic bin packing problem introduced by Coffman, Garey and Johnson. This problem is a generalization of the bin packing problem in which items may arrive and depart from the packing dynamically. The main result in this paper is a lower bound of 2.5 on the achievable competitive ratio, improving the best known 2.428 lower bound, and revealing that packing items of restricted form like unit fractions (i.e., of size 1/k for some integer k), for which a 2.4985-competitive algorithm is known, is indeed easier. We also investigate the resource augmentation version of the problem where the on-line algorithm can use bins of size b (>1) times that of the optimal off-line algorithm. An interesting result is that we prove b=2 is both necessary and sufficient for the on-line algorithm to match the performance of the optimal off-line algorithm, i.e., achieve 1-competitiveness. Further analysis gives a trade-off between the bin size multiplier 1<b≤2 and the achievable competitive ratio. J.W.-T. Chan’s research is partly supported by Hong Kong RGC Grant HKU5172/03E when the author was with the Department of Computer Science, University of Hong Kong, Hong Kong. P.W.H. Wong’s research is partly supported by Nuffield Foundation Grant NAL/01004/G.     

Approximation Schemes for Packing Splittable Items with Cardinality Constraints

We continue the study of bin packing with splittable items and cardinality constraints. In this problem, a set of n items must be packed into as few bins as possible. Items may be split, but each bin may contain at most?k (parts of) items, where k is some given parameter. Complicating the problem further is the fact that items may be larger than?1, which is the size of a bin. The problem is known to be strongly NP-hard for any fixed value of?k. We essentially close this problem by providing an efficient polynomial-time approximation scheme (EPTAS) for most of its versions. Namely, we present an efficient polynomial time approximation scheme for k=o(n). A?PTAS for k=Θ(n) does not exist unless P = NP. Additionally, we present dual approximation schemes for k=2 and for constant values of?k. Thus we show that for any ε>0, it is possible to pack the items into the optimal number of bins in polynomial time, if the algorithm may use bins of size 1+ε.     

The load-balanced multi-dimensional bin-packing problem

The bin-packing problem is one of the most investigated and applicable combinatorial optimization problems. In this paper we consider its multi-dimensional version with the practical extension of load balancing, i.e. to find the packing requiring the minimum number of bins while ensuring that the average center of mass of the loaded bins falls as close as possible to an ideal point, for instance, the center of the bin. We formally describe the problem using mixed-integer linear programming models, from the simple case where we want to optimally balance a set of items already assigned to a single bin, to the general balanced bin-packing problem. Given the difficulty for standard solvers to deal even with small size instances, a multi-level local search heuristic is presented. The algorithm takes advantage of the Fekete–Schepers representation of feasible packings in terms of particular classes of interval graphs, and iteratively improves the load balancing of a bin-packing solution using different search levels. The first level explores the space of transitive orientations of the complement graphs associated with the packing, the second modifies the structure itself of the interval graphs, the third exchanges items between bins repacking proper n-tuples of weakly balanced bins. Computational experiments show very promising results on a set of 3D bin-packing instances from the literature.     

一种自适应优化松弛量的装箱算法

装箱问题是物流系统和生产系统中的一个经典而重要的数学优化问题。装箱指把一系列物品按照一定顺序放进具有固定容量的箱子中,并最小化所使用的箱子数量,以最大限度地获取装箱问题的近似最优解。然而,现有的装箱算法存在明显的缺陷。遗传算法计算量过大,甚至无法求出所需解,启发式算法无法处理极端值问题,而现有的改进算法即使在引入松弛量的情况下,也极易陷入局部最小值。文中提出的Adaptive-MBS算法采用自适应权重来改进原有方法,即允许方法有一定的松弛量,并具有捕捉物体样本空间随时间变化的直觉,以使用更好的松弛量策略来装箱。Adaptive-MBS算法首先以当前箱子为中心,使用Adaptive_Search搜索算法迭代找到适合箱子容量的集合中所有物体的子集,Adaptive_Search搜索算法不要求完全装满箱子,而是允许箱子具有一定的松弛量,在训练过程中根据当前状态的变化,实现自动地调整松弛量,在找到完全填满箱子的子集后迭代至下轮搜索直至遍历完成。该方法不易陷入局部最优,具有较强的发现全局最优解的能力。文中使用装箱问题中经典的BINDATA和SCH_WAE数据集进行实验,结果表明,数据集中多达991例问题可以通过Adaptive-MBS算法得到最优解。在没有求解出最优解的实例上,所提算法也在所有对比算法上具有最低的相对偏移量百分比。数值实验结果表明,相较于其他经典的装箱算法,Adaptive-MBS算法有更好的效果,其收敛速度也显著优于其他算法。     

Dynamic bin packing with unit fraction items revisited

In this paper, we will study the problem of dynamic bin packing with unit fraction items. We focus on analyzing the First Fit (FF) algorithm on this problem. There are two main results: i) we give the first bound for the FF algorithm on cases when the largest item is at most 1/k; ii) we generalize the previous framework for analyzing FF and get an improved upper bound.     

Asymptotic fully polynomial approximation schemes for variants of open-end bin packing

We consider three variants of the open-end bin packing problem. Such variants of bin packing allow the total size of items packed into a bin to exceed the capacity of a bin, provided that a removal of the last item assigned to a bin would bring the contents of the bin below the capacity. In the first variant, this last item is the minimum sized item in the bin, that is, each bin must satisfy the property that the removal of any item should bring the total size of items in the bin below 1. The next variant (which is also known as lazy bin covering is similar to the first one, but in addition to the first condition, all bins (expect for possibly one bin) must contain a total size of items of at least 1. We show that these two problems admit asymptotic fully polynomial time approximation schemes (AFPTAS). Moreover, they turn out to be equivalent. We briefly discuss a third variant, where the input items are totally ordered, and the removal of the maximum indexed item should bring the total size of items in the bin below 1, and show that this variant is strongly NP-hard.     

Tree-decomposition based heuristics for the two-dimensional bin packing problem with conflicts

This paper deals with the two-dimensional bin packing problem with conflicts (BPC-2D). Given a finite set of rectangular items, an unlimited number of rectangular bins and a conflict graph, the goal is to find a conflict-free packing of the items minimizing the number of bins used. In this paper, we propose a new framework based on a tree-decomposition for solving this problem. It proceeds by decomposing a BPC-2D instance into subproblems to be solved independently. Applying this decomposition method is not straightforward, since merging partial solutions is hard. Several heuristic strategies are proposed to make an effective use of the decomposition. Computational experiments show the practical effectiveness of our approach.     

Batched bin packing revisited

We revisit the batched bin packing problem. In this model, items come in K consecutive batches, and the items of the earlier batches must be packed without any knowledge of later batches. We give the first approximation algorithm for the case \(K=2\), with tight asymptotic approximation ratio of 1.5833, while the known lower bound of the model is 1.378. With the application of this result, we are also able to provide an improved algorithm for the recently defined graph-bin packing problem in a special case, where we improve the upper bound from 3 to 2.5833.     

多约束的箱子大小可变的装箱问题

作为经典装箱问题的扩展,带有约束条件的装箱问题有着很强的应用背景。从钢铁公司板坯入库引发的装箱问题中,物品除了重量这一参数,还引入了特性和种类等参数。将板坯入库归结为多约束的箱子大小可变的装箱问题后,建立了数学模型,将经典装箱算法进行了推广,构建了新的启发式算法A,它对输入物品按特性进行预处理,再进行装箱。理论分析了NFD,FFD,算法A的最坏情况的性能,并对这三种算法进行了数值模拟,得出在相关特性均匀分布下,算法A的性能最好。