Space defragmentation for packing problems

One of main difficulties of multi-dimensional packing problems is the fragmentation of free space into several unusable small parts after a few items are packed. This study proposes a defragmentation technique to combine the fragmented space into a continuous usable space, which potentially allows the packing of additional items. We illustrate the effectiveness of this technique using the two- and three-dimensional bin packing problem, where the aim is to load all given items (represented by rectangular boxes) into the minimum number of identical bins. Experimental results based on well-known 2D and 3D bin packing data sets show that our defragmentation technique alone is able to produce solutions approaching the quality of considerably more complex meta-heuristic approaches for the problem. In conjunction with a bin shuffling strategy for incremental improvement, our resultant algorithm outperforms all leading meta-heuristic approaches based on the commonly used benchmark data by a significant margin.     

Multiple-type,two-dimensional bin packing problems: Applications and algorithms

In this paper we consider a class of bin selection and packing problems (BPP) in which potential bins are of various types, have two resource constraints, and the resource requirement for each object differs for each bin type. The problem is to select bins and assign the objects to bins so as to minimize the sum of bin costs while meeting the two resource constraints. This problem represents an extension of the classical two-dimensional BPP in which bins are homogeneous. Typical applications of this research include computer storage device selection with file assignment, robot selection with work station assignment, and computer processor selection with task assignment. Three solution algorithms have been developed and tested: a simple greedy heuristic, a method based onsimulated annealing (SA) and an exact algorithm based onColumn Generation with Branch and Bound (CG). An LP-based method for generating tight lower bounds was also developed (LB). Several hundred test problems based on computer storage device selection and file assignment were generated and solved. The heuristic solved problems up to 100 objects in less than a second; average solution value was within about 3% of the optimum. SA improved solutions to an average gap of less than 1% but a significant increase in computing time. LB produced average lower bounds within 3% of optimum within a few seconds. CG is practical for small to moderately-sized problems — possibly as many as 50 objects.     

带冲突关系装箱问题的启发式求解算法

现实物流活动中大量存在的食品、药品和危险品等货物的分组包装问题属于带冲突关系的装箱问题(BPPC),其优化目标是在满足货物间冲突限制的前提下完成装箱操作,并最小化使用货箱的数量。本文从实际需求出发,基于货物之间的冲突关系、装箱顺序和货箱容量等约束建立相应的数学规划模型;随后设计了求解BPPC问题的启发式算法,算法通过迭代求解最大团结构实现货物间冲突关系的消去,根据当前货物最大团采用改进降序首次适应算法(FFD)完成货物装箱操作,并通过“洗牌”策略对已有装箱方案进行局部优化;最后,针对Iori算例数据,将以上算法与基于图着色的启发式算法进行比较分析,结果表明,本文算法是求解BPPC问题更为有效的方法。     

Selfish Square Packing

We consider a game-theoretical bin packing problem. The 1D (one dimensional) case has been treated in the literature as the ʼselfish bin packing problemʼ. We investigate a 2D version, in which the items to be packed are squares and the bins are unit squares. In this game, a set of items is packed into bins. Each player controls exactly one item and is charged with a cost defined as the ratio between the area of the item and the occupied area of the respective bin. One at a time, players selfishly move their items from one bin to another, in order to minimize the costs they are charged. At a Nash equilibrium, no player can reduce the cost he is charged by moving his item to a different bin. In the 2D case, to decide whether an item can be placed in another bin with other items is NP-complete, so we consider that players use a packing algorithm to make this decision. We show that this game converges to a Nash equilibrium, independently of the packing algorithm used. We prove that the price of anarchy is at least 2.27. We also prove that, using the NFDH packing algorithm, the asymptotic price of anarchy is at most 2.6875.     

经典一维装箱问题近似算法的研究进展

自20世纪70年代开始,随着计算复杂性理论的建立,近似算法逐渐成为组合优化的重要研究方向。作为第一批研究对象,装箱问题引起了组合优化领域学者的极大关注。装箱问题模型简单、拓展性强,广泛出现在各种带容量约束的资源分配问题中。除了在物流装载和材料切割等方面愈来愈重要的应用外,装箱算法的任何理论突破都关乎到整个组合优化领域的发展。直到今天,对装箱问题近似算法的研究仍如火如荼。本文主要针对一维模型,简述若干经典Fit算法的发展历程,分析基于线性规划松弛的近似方案的主要思路,总结当前的研究现状并对未来的研究提供一些参考建议。     

经典一维装箱问题近似算法的研究进展

自20世纪70年代开始,随着计算复杂性理论的建立,近似算法逐渐成为组合优化的重要研究方向。作为第一批研究对象,装箱问题引起了组合优化领域学者的极大关注。装箱问题模型简单、拓展性强,广泛出现在各种带容量约束的资源分配问题中。除了在物流装载和材料切割等方面愈来愈重要的应用外,装箱算法的任何理论突破都关乎到整个组合优化领域的发展。直到今天,对装箱问题近似算法的研究仍如火如荼。本文主要针对一维模型,简述若干经典Fit算法的发展历程,分析基于线性规划松弛的近似方案的主要思路,总结当前的研究现状并对未来的研究提供一些参考建议。     

Simple but effective heuristics for the 2-constraint bin packing problem

The 2-constraint bin packing problem consists in packing a given number of items, each one characterised by two different but not related dimensions, into the minimum number of bins in such a way to do not exceed the capacity of the bins in either dimension. The development of the heuristics for this problem is challenged by the need of a proper definition of the criterion for evaluating the feasibility of the two capacity constraints on the two different dimensions. In this paper, we propose a computational evaluation of several criteria, and two simple but effective algorithms—a greedy and neighbourhood search algorithms—for solving the 2-constraint bin packing problem. An extensive computational analysis supports our main claim.     

箱子是在线到达的带核元变尺寸装箱问题

本文考虑了一类箱子在线到达的带核元变尺寸装箱问题．假定箱子的尺寸可以是不同的．箱子是在线到达的，仅当箱子到达后其尺寸才知道．给定一个带有核元的物件表，目标是要将表中元素装入到达的箱子中，使得所用的箱子总长最小．我们首先证明了该问题是强NP—Hard，其次分析了经典算法NF(D)和FF(D)的性能界，指出NF(D)和FF(D)算法的性能界可以任意大．最后我们给出了相应的修改算法MNF(D)和MFF(D)，证明了它们的性能界都是3，此界是紧的．     

Constrained order packing: comparison of heuristic approaches for a new bin packing problem

Bin packing problems consist of allocating a set of small parts to a set of large bins by minimizing the number of used bins. Although several boundary conditions have been stated in the past, for example conflicts or restrictions on cutting and rotations, we introduce a set of constraints, which lead to a new problem structure. These constraints are motivated by the precast-concrete-part industry and represent requirements on the ordering of parts and their positions, machinery restrictions and due dates. Furthermore, we solve the problem using several heuristic approaches that are based on algorithms for the standard bin packing problem. Therefore, existing concepts are classified and adapted to fit the new problem, including Simulated Annealing, Genetic Algorithms, Tabu Search methods and SubSetSum based search routines. Finally, all proposed algorithms are tested and obtained results are discussed.     

Multiple reduced gradient method for multiobjective optimization problems

Multiobjective optimization has a large number of real-life applications. Under this motivation, in this paper, we present a new method for solving multiobjective optimization problems with both linear constraints and bound constraints on the variables. This method extends, to the multiobjective setting, the classical reduced gradient method for scalar-valued optimization. The proposed algorithm generates a feasible descent direction by solving an appropriate quadratic subproblem, without the use of any scalarization approaches. We prove that the sequence generated by the algorithm converges to Pareto-critical points of the problem. We also present some numerical results to show the efficiency of the proposed method.     

New and improved level heuristics for the rectangular strip packing and variable-sized bin packing problems

We consider two types of orthogonal, oriented, rectangular, two-dimensional packing problems. The first is the strip packing problem, for which four new and improved level-packing algorithms are presented. Two of these algorithms guarantee a packing that may be disentangled by guillotine cuts. These are combined with a two-stage heuristic designed to find a solution to the variable-sized bin packing problem, where the aim is to pack all items into bins so as to minimise the packing area. This heuristic packs the levels of a solution to the strip packing problem into large bins and then attempts to repack the items in those bins into smaller bins in order to reduce wasted space. The results of the algorithms are compared to those of seven level-packing heuristics from the literature by means of a large number of strip-packing benchmark instances. It is found that the new algorithms are an improvement over known level-packing heuristics for the strip packing problem. The advancements made by the new and improved algorithms are limited in terms of utilised space when applied to the variable-sized bin packing problem. However, they do provide results faster than many existing algorithms.     

An agent-based approach to the two-dimensional guillotine bin packing problem

The two-dimensional guillotine bin packing problem consists of packing, without overlap, small rectangular items into the smallest number of large rectangular bins where items are obtained via guillotine cuts. This problem is solved using a new guillotine bottom left (GBL) constructive heuristic and its agent-based (A–B) implementation. GBL, which is sequential, successively packs items into a bin and creates a new bin every time it can no longer fit any unpacked item into the current one. A–B, which is pseudo-parallel, uses the simplest system of artificial life. This system consists of active agents dynamically interacting in real time to jointly fill the bins while each agent is driven by its own parameters, decision process, and fitness assessment. A–B is particularly fast and yields near-optimal solutions. Its modularity makes it easily adaptable to knapsack related problems.     

A hybrid grouping genetic algorithm for bin packing

The grouping genetic algorithm (GGA) is a genetic algorithm heavily modified to suit the structure of grouping problems. Those are the problems where the aim is to find a good partition of a set or to group together the members of the set. The bin packing problem (BPP) is a well known NP-hard grouping problem: items of various sizes have to be grouped inside bins of fixed capacity. On the other hand, the reduction method of Martello and Toth, based on their dominance criterion, constitutes one of the best OR techniques for optimization of the BPP to date.In this article, we first describe the GGA paradigm as compared to the classic Holland-style GA and the ordering GA. We then show how the bin packing GGA can be enhanced with a local optimization inspired by the dominance criterion. An extensive experimental comparison shows that the combination yields an algorithm superior to either of its components.     

基于图着色模型的冲突装箱问题启发式算法

带有冲突关系装箱问题的优化目标是在满足货物冲突关系的前提下,使用数量最少的货箱完成货物装箱的目的。本文分析了冲突装箱问题的数学模型,提出了基于图着色模型的启发式算法进行求解。首先,使用冲突图来描述货物之间的冲突关系;其次,基于冲突图,采取图着色的方式将货物进行分组,并且组内的货物之间不存在冲突关系;最后,采取改进FFD算法对每组的货物进行装箱操作。实验表明,本文提出的启发式算法能够快速有效地找到问题的可行解,为此类装箱问题的求解提供了新思路。     

An exact algorithm for the type-constrained and variable sized bin packing problem

In this paper, we introduce an additional constraint to the one-dimensional variable sized bin packing problem. Practically, some of items have to be packed separately in different bins due to their specific requirement. Therefore, these items are labelled as different types. The bins can be used to pack either any type of items if they are empty originally or the same type of items as what they already have. We model the problem as a type-constrained and variable sized bin packing problem (TVSBPP), and solve it via a branch and bound method. An efficient backtracking procedure is proposed to improve the efficiency of the algorithm.     

Duality for multiobjective optimization problems with convex objective functions and D.C. constraints

In this paper we provide a duality theory for multiobjective optimization problems with convex objective functions and finitely many D.C. constraints. In order to do this, we study first the duality for a scalar convex optimization problem with inequality constraints defined by extended real-valued convex functions. For a family of multiobjective problems associated to the initial one we determine then, by means of the scalar duality results, their multiobjective dual problems. Finally, we consider as a special case the duality for the convex multiobjective optimization problem with convex constraints.     

The stochastic generalized bin packing problem

The Generalized Bin Packing Problem (GBPP) is a recently introduced packing problem where, given a set of bins characterized by volume and cost and a set of items characterized by volume and profit (which also depends on bins), we want to select a subset of items to be loaded into a subset of bins which maximizes the total net profit, while satisfying the volume and bin availability constraints. The total net profit is given by the difference between the total profit of the loaded items and the total cost of the used bins. In this paper, we consider the stochastic version of the GBPP (S-GBPP), where the item profits are random variables to take into account the profit oscillations due to the handling operations for bin loading. The probability distribution of these random variables is assumed to be unknown. By using the asymptotic theory of extreme values a deterministic approximation for the S-GBPP is derived.     

一类带多重核元集在线装箱问题

本文讨论了一类在线变尺寸装箱问题,假定箱子的尺寸可以是不同的.箱子是在线到达的,仅当箱子到达后其尺寸才知道.给定一个带有核元的物品表及其上的核元关系图.我们的目标是要将表中元素装入到达的箱子中,保证任何箱子所装物品不互为核元,即所装物品对应的点所导出的子图是个空图,并使得所用的箱子总长最小.我们证明了该问题是NPHard的,并给出了基于图的点染色、图的团分解和基于背包问题的近似算法,给出了算法的时间复杂度和性能界.     

Multiobjective optimization problems with equilibrium constraints

The paper is devoted to new applications of advanced tools of modern variational analysis and generalized differentiation to the study of broad classes of multiobjective optimization problems subject to equilibrium constraints in both finite-dimensional and infinite-dimensional settings. Performance criteria in multiobjective/vector optimization are defined by general preference relationships satisfying natural requirements, while equilibrium constraints are described by parameterized generalized equations/variational conditions in the sense of Robinson. Such problems are intrinsically nonsmooth and are handled in this paper via appropriate normal/coderivative/subdifferential constructions that exhibit full calculi. Most of the results obtained are new even in finite dimensions, while the case of infinite-dimensional spaces is significantly more involved requiring in addition certain “sequential normal compactness” properties of sets and mappings that are preserved under a broad spectrum of operations.     

A branch-and-price algorithm for the variable size bin packing problem with minimum filling constraint

In this paper we consider a variation of the bin packing problem in which bins of different types have different costs and capacities. Furthermore, each bin has to be filled at least to a certain level, which depends on its type. We present a set partitioning formulation and an exact optimization algorithm which exploits column generation and specialized heuristics. We compare our algorithm with the general purpose solver ILOG CPLEX, running on two compact ILP formulations and we report on experimental results on instances we have generated from data-sets for the variable size bin packing problem.