A goal-driven approach to the 2D bin packing and variable-sized bin packing problems

In this paper, we examine the two-dimensional variable-sized bin packing problem (2DVSBPP), where the task is to pack all given rectangles into bins of various sizes such that the total area of the used bins is minimized. We partition the search space of the 2DVSBPP into sets and impose an order on the sets, and then use a goal-driven approach to take advantage of the special structure of this partitioned solution space. Since the 2DVSBPP is a generalization of the two-dimensional bin packing problem (2DBPP), our approach can be adapted to the 2DBPP with minimal changes. Computational experiments on the standard benchmark data for both the 2DVSBPP and 2DBPP shows that our approach is more effective than existing approaches in literature.     

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.     

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.     

Resource augmentation for online bounded space bin packing

We study online bounded space bin packing in the resource augmentation model of competitive analysis. In this model, the online bounded space packing algorithm has to pack a list L of items in (0,1] into a small number of bins of size b1. Its performance is measured by comparing the produced packing against the optimal offline packing of the list L into bins of size 1.We present a complete solution to this problem: For every bin size b1, we design online bounded space bin packing algorithms whose worst case ratio in this model comes arbitrarily close to a certain bound ρ(b). Moreover, we prove that no online bounded space algorithm can perform better than ρ(b) in the worst case.     

A 3-approximation algorithm for two-dimensional bin packing

In the classical two-dimensional bin packing problem one is asked to pack a set of rectangular items, without overlap and without any rotation, into the minimum number of identical square bins. We give an approximation algorithm with absolute worst-case ratio of 3.     

On-line bin packing — A restricted survey

In the classical bin packing problem, one is asked to pack items of various sizes into the minimum number of equal-sized bins. In the on-line version of this problem, the packer is given the items one by one and must immediately and irrevocably assign every item to its bin, without knowing the future items. Beginning with the first results in the early 1970's, we survey — from the worst case point of view — the approximation results obtained for on-line bin packing, higher dimensional versions of the problem, lower bounds on worst case ratios and related results.This work was partially supported by the Christian Doppier Laboratorium für Diskrete Optimierung.     

Resource augmented semi-online bounded space bin packing

We study on-line bounded space bin-packing in the resource augmentation model of competitive analysis. In this model, the on-line bounded space packing algorithm has to pack a list L of items with sizes in (0, 1], into a minimum number of bins of size b, b≥1. A bounded space algorithm has the property that it only has a constant number of active bins available to accept items at any point during processing. The performance of the algorithm is measured by comparing the produced packing with an optimal offline packing of the list L into bins of size 1. The competitive ratio then becomes a function of the on-line bin size b. Csirik and Woeginger studied this problem in [J. Csirik, G.J. Woeginger, Resource augmentation for online bounded space bin packing, Journal of Algorithms 44(2) (2002) 308-320] and proved that no on-line bounded space algorithm can perform better than a certain bound ρ(b) in the worst case. We relax the on-line condition by allowing a complete repacking within the active bins, and show that the same lower bound holds for this problem as well, and repacking may only allow one to obtain the exact best possible competitive ratio of ρ(b) having a constant number of active bins, instead of achieving this bound in the limit. We design a polynomial time on-line algorithm that uses three active bins and achieves the exact best possible competitive ratio ρ(b) for the given problem.     

Online variable-sized bin packing with conflicts

We study a new kind of online bin packing with conflicts, motivated by a problem arising when scheduling jobs on the Grid. In this bin packing problem, the set of items is given at the beginning, together with a set of conflicts on pairs of items. A conflict on a pair of items implies that they cannot be assigned to a common bin. The online scenario is realized as follows. Variable-sized bins arrive one by one, and items need to be assigned to each bin before the next bin arrives. We analyze the online problem as well as semi-online versions of it, which are the variant where the sizes of the arriving bins are monotonically non-increasing as well as the variant where they are monotonically non-decreasing.     

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.     

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．     

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.     

A heuristic for solving large bin packing problems in two and three dimensions

The more-dimensional bin packing problem (BPP) considered here requires packing a set of rectangular-shaped items into a minimum number of identical rectangular-shaped bins. All items may be rotated and the guillotine cut constraint has to be respected. A straightforward heuristic is presented that is based on a method for the container loading problem following a wall-building approach and on a method for the one-dimensional BPP. 1,800 new benchmark instances are introduced for the two-dimensional and three-dimensional BPP. The instances include more than 1,500 items on average. Applied to these very large instances, the heuristic generates solutions of acceptable quality in short computation times. Moreover, the influence of different instance parameters on the solution quality is investigated by an extended computational study.     

A 17/10-approximation algorithm fork-bounded space on-line variable-sized bin packing

A version of thek-bounded space on-line bin packing problem, where a fixed collection of bin sizes is allowed, is considered. By packing large items into appropriate bins and closing appropriate bins, we can derive an algorithm with worst-case performance bound 1.7 fork≥3. This research is supported by the Science Foundation under State Education Committee of China. The earlier version was done in Institute of Applied Mathematics, Academia Sinica.     

Two-dimensional packing problems: A survey

We consider problems requiring to allocate a set of rectangular items to larger rectangular standardized units by minimizing the waste. In two-dimensional bin packing problems these units are finite rectangles, and the objective is to pack all the items into the minimum number of units, while in two-dimensional strip packing problems there is a single standardized unit of given width, and the objective is to pack all the items within the minimum height. We discuss mathematical models, and survey lower bounds, classical approximation algorithms, recent heuristic and metaheuristic methods and exact enumerative approaches. The relevant special cases where the items have to be packed into rows forming levels are also discussed in detail.     

Three-dimensional bin packing problem with variable bin height

This paper studies a variant of the three-dimensional bin packing problem (3D-BPP), where the bin height can be adjusted to the cartons it packs. The bins and cartons to be packed are assumed rectangular in shape. The cartons are allowed to be rotated into any one of the six positions that keep the carton edges parallel to the bin edges. This greatly increases the difficulty of finding a good solution since the search space expands significantly comparing to the 3D-BPP where the cartons have fixed orientations. A mathematical (mixed integer programming) approach is modified based on [Chen, C. S., Lee, S. M., Shen, Q. S., 1995. An analytical model for the container loading problem. European Journal of Operational Research 80 (1), 68–76] and numerical experiments indicate that the mathematical approach is not suitable for the variable bin height 3D-BPP. A special bin packing algorithm based on packing index is designed to utilize the special problem feature and is used as a building block for a genetic algorithm designed for the 3D-BPP. The paper also investigates the situation where more than one type of bin are used and provides a heuristic for packing a batch of cartons using the genetic algorithm. Numerical experiments show that our proposed method yields quick and satisfactory results when benchmarked against the actual packing practice and the MIP model with the latest version of CPLEX.     

Exact algorithms for the bin packing problem with fragile objects

We are given a set of objects, each characterized by a weight and a fragility, and a large number of uncapacitated bins. Our aim is to find the minimum number of bins needed to pack all objects, in such a way that in each bin the sum of the object weights is less than or equal to the smallest fragility of an object in the bin. The problem is known in the literature as the Bin Packing Problem with Fragile Objects, and appears in the telecommunication field, when one has to assign cellular calls to available channels by ensuring that the total noise in a channel does not exceed the noise acceptance limit of a call.We propose a branch-and-bound and several branch-and-price algorithms for the exact solution of the problem, and improve their performance by the use of lower bounds and tailored optimization techniques. In addition we also develop algorithms for the optimal solution of the related knapsack problem with fragile objects. We conduct an extensive computational evaluation on the benchmark set of instances, and show that the proposed algorithms perform very well.     

Routing and wavelength assignment in optical networks using bin packing based algorithms

This paper addresses the problem of routing and wavelength assignment (RWA) of static lightpath requests in wavelength routed optical networks. The objective is to minimize the number of wavelengths used. This problem has been shown to be NP-complete and several heuristic algorithms have been developed to solve it. We suggest very efficient, yet simple, heuristic algorithms for the RWA problem developed by applying classical bin packing algorithms. The heuristics were tested on a series of large random networks and compared with an efficient existing algorithm for the same problem. Results indicate that the proposed algorithms yield solutions significantly superior in quality, not only with respect to the number of wavelength used, but also with respect to the physical length of the established lightpaths. Comparison with lower bounds shows that the proposed heuristics obtain optimal or near optimal solutions in many cases.     

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.     

New lower bounds for bin packing problems with conflicts

The bin packing problem with conflicts (BPC) consists of minimizing the number of bins used to pack a set of items, where some items cannot be packed together in the same bin due to compatibility restrictions. The concepts of dual-feasible functions (DFF) and data-dependent dual-feasible functions (DDFF) have been used in the literature to improve the resolution of several cutting and packing problems. In this paper, we propose a general framework for deriving new DDFF as well as a new concept of generalized data-dependent dual-feasible functions (GDDFF), a conflict generalization of DDFF. The GDDFF take into account the structure of the conflict graph using the techniques of graph triangulation and tree-decomposition. Then we show how these techniques can be used in order to improve the existing lower bounds.