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.     

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.     

A hybrid heuristic algorithm for the 2D variable-sized bin packing problem

In this paper, we consider the two-dimensional variable-sized bin packing problem (2DVSBPP) with guillotine constraint. 2DVSBPP is a well-known NP-hard optimization problem which has several real applications. A mixed bin packing algorithm (MixPacking) which combines a heuristic packing algorithm with the Best Fit algorithm is proposed to solve the single bin problem, and then a backtracking algorithm which embeds MixPacking is developed to solve the 2DVSBPP. A hybrid heuristic algorithm based on iterative simulated annealing and binary search (named HHA) is then developed to further improve the results of our Backtracking algorithm. Computational experiments on the benchmark instances for 2DVSBPP show that HHA has achieved good results and outperforms existing algorithms.     

Dynamic multi-dimensional bin packing

A natural generalization of the classical online bin packing problem is the dynamic bin packing problem introduced by Coffman et al. (1983) [7]. In this formulation, items arrive and depart and the objective is to minimize the maximal number of bins ever used over all times. We study the oriented multi-dimensional dynamic bin packing problem for two dimensions, three dimensions and multiple dimensions. Specifically, we consider dynamic packing of squares and rectangles into unit squares and dynamic packing of three-dimensional cubes and boxes into unit cubes. We also study dynamic d-dimensional hypercube and hyperbox packing. For dynamic d-dimensional box packing we define and analyze the algorithm NFDH for the offline problem and present a dynamic version. This algorithm was studied before for rectangle packing and for square packing and was generalized only for multi-dimensional cubes. We present upper and lower bounds for each of these cases.     

A note on a variant of the online open end bin packing problem

We study the minimum variant of the online open end bin packing problem. Items are presented one by one, and an item can be packed into a bin while the resulting total size of items excluding the minimum size item of the bin will be below 1. We design an improved online algorithm whose asymptotic competitive ratio does not exceed 1.180952381, and we prove a new lower bound of 1.1666666 on the asymptotic competitive ratio of any online algorithm.     

A note on online hypercube packing

In this paper, we study an online multi-dimensional bin packing problem where all items are hypercubes. Hypercubes of different size arrive one by one, and we are asked to pack each of them without knowledge of the next pieces so that the number of bins used is minimized. Based on the techniques from one dimensional bin packing and specifically the algorithm Super Harmonic by Seiden (J ACM 49:640–671, 2002), we extend the framework for online bin packing problems developed by Seiden to the hypercube packing problem. To the best of our knowledge, this is the first paper to apply a version of Super Harmonic (and not of the Improved Harmonic algorithm) for online square packing, although the Super Harmonic has been already known before. Note that the best previous result was obtained by Epstein and van Stee (Acta Inform 41(9):595–606, 2005b) using an instance of Improved Harmonic. In this paper we show that Super Harmonic is more powerful than Improve Harmonic for online hypercube packing, and then we obtain better upper bounds on asymptotic competitive ratios. More precisely, we get an upper bound of 2.1187 for square packing and an upper bound of 2.6161 for cube packing, which improve upon the previous upper bounds 2.24437 and 2.9421 (Epstein and van Stee in Acta Inform 41(9):595–606, 2005b) for the two problems, respectively.     

Approximation algorithms for the oriented two-dimensional bin packing problem

Given a set of rectangular items which may not be rotated and an unlimited number of identical rectangular bins, we consider the problem of packing each item into a bin so that no two items overlap and the number of required bins is minimized. The problem is strongly NP-hard and finds practical applications in cutting and packing. We discuss a simple deterministic approximation algorithm which is used in the initialization of a tabu search approach. We then present a tabu search algorithm and analyze its average performance through extensive computational experiments.     

A hybrid GRASP/VND algorithm for two- and three-dimensional bin packing

The three-dimensional bin packing problem consists of packing a set of boxes into the minimum number of bins. In this paper we propose a new GRASP algorithm for solving three-dimensional bin packing problems which can also be directly applied to the two-dimensional case. The constructive phase is based on a maximal-space heuristic developed for the container loading problem. In the improvement phase, several new moves are designed and combined in a VND structure. The resulting hybrid GRASP/VND algorithm is simple and quite fast and the extensive computational results on test instances from the literature show that the quality of the solutions is equal to or better than that obtained by the best existing heuristic procedures.     

Bin packing with general cost structures

Following the work of Anily et?al., we consider a variant of bin packing called bin packing with general cost structures (GCBP) and design an asymptotic fully polynomial time approximation scheme (AFPTAS) for this problem. In the classic bin packing problem, a set of one-dimensional items is to be assigned to subsets of total size at most 1, that is, to be packed into unit sized bins. However, in GCBP, the cost of a bin is not 1 as in classic bin packing, but it is a non-decreasing and concave function of the number of items packed in it, where the cost of an empty bin is zero. The construction of the AFPTAS requires novel techniques for dealing with small items, which are developed in this work. In addition, we develop a fast approximation algorithm which acts identically for all non-decreasing and concave functions, and has an asymptotic approximation ratio of 1.5 for all functions simultaneously.     

超尺寸物品装箱问题及其算法

本文探讨一类新装箱问题-超尺寸物品装箱问题。针对实际解决该问题的两涉法，我们提出了一个评价效率更高的目标函数，证明了在此目标函数下两步法的渐近最坏比不小于2，并给出了渐近量坏比与拆分次数的关系。最后本文提出了一种不同于两步法的新在线算法MA，证明了在新目标函数下其渐近最坏比不超过7/4。     

A note on the minimum bounded edge-partition of a tree

Minimum bounded edge-partition divides the edge set of a tree into the minimum number of disjoint connected components given a maximum weight for any component. It is an adaptation of the uniform edge-partition of a tree. An optimization algorithm is developed for this NP-hard problem, based on repeated bin packing of inter-related instances. The algorithm has linear running time for the class of ‘balanced trees’ common for the stochastic programming application which motivated investigation of this problem.Fast 2-approximation algorithms are formed for general instances by replacing the optimal bin packing with almost any bin packing heuristic. The asymptotic worst-case ratio of these approximation algorithms is never better than the absolute worst-case ratio of the bin packing heuristic used.     

Bin packing with divisible item sizes

We study a variety of NP-hard bin packing problems under a divisibility constraint that generalizes the often encountered situation in which all item sizes are powers of 2. For ordinary one-dimensional bin packing, we show that First Fit Decreasing produces optimal packings under this restriction, and that if in addition the largest item size divides the bin capacity, then even the less powerful First Fit algorithm is optimal. Similar results are obtained for two-dimensional bin packing and multiprocessor scheduling, along with several other simple variants. For more complicated problems, like vector packing and dynamic bin packing, the improvement is less substantial, and we indicate why.     

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.     

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.     

A hybrid algorithm for constrained order packing

Constraint order packing, which is an extension to the classical two-dimensional bin packing, adds an additional layer of complexity to known bin packing problems by new additional placement and order constraints. While existing meta heuristics usually produce good results for common bin packing problems in any dimension, they are not able to take advantage of special structures resulting from these constraints in this particular two-dimensional prolbem type. We introduce a hybrid algorithm that is based on greedy search and is nested within a network search algorithm with dynamic node expansion and meta logic, inspired by human intuition, to overrule decisions implied by the greedy search. Due to the design of this algorithm we can control the performance characteristics to lie anywhere between classical network search algorithms and local greedy search. We will present the algorithm, discuss bounds and show that their performance outperforms common approaches on a variety of data sets based on industrial applications. Furthermore, we discuss time complexity and show some ideas to speed up calculations and improve the quality of results.     

Fuzzy bin packing problem

This paper deals with the fuzzy bin packing problem that is a packing problem of non-rigid rectangles into an open rectangular bin. This problem is different from the conventional bin packing problem, which considers only rigid rectangles. The goal of the fuzzy bin packing problem is to minimize both the height of a packing and the extra cost due to the reduction of each piece. The total cost of the problem is represented as the sum of the height cost and the extra cost due to reductions of the pieces, which is called reduction cost. Because the conventional bin packing problem itself is an NP-hard problem, the presented optimization method assumes that an initial packing for non-reduced pieces has already been found. A closed form solution is presented for fuzzy bin packing problems, in which fuzzy numbers are triangular and the reduction cost is given by a quadratic function.     

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.     

THE FFD ALGORITHM FOR THE BIN PACKING PROBLEM WITH KERNEL ITEMS

The FFD algorithm is one of the most famous algorithms for the classical bin packing problem. In this paper,some versions of the FFD algorithm are considered in several bin packing problems. Especially,two of them applied to the bin packing problem with kernel items are analyzed. Tight worst-case performance ratios are obtained.     

Bin packing: On optimizing the number of pieces packed

We consider bin-packing variations related to the well-studied problem of maximizing the total number of pieces packed into a fixed set of bins. We show that, when the objective is to minimize the total number of pieces packed subject to the constraint that no unpacked piece will fit, no polynomial-time relative approximation algorithm exists (unless, of course,P=NP). That is, we prove that it isNP-hard to guarantee packing no more than a constant multiple of the optimal number of pieces, for any constant. This appears to be the first bin-packing problem for which this property has been demonstrated. Furthermore, this result also holds for the allied packing variant which seeks to minimize the maximum number of pieces packed in any single bin. We find the situation to be markedly better for the problem of maximizing the minimum number of pieces in any bin. If all bins possess the same capacity, we prove that the familiar SPF rule is an absolute approximation algorithm with additive constant 1, and can therefore be regarded as a best possible heuristic. For the more general and difficult case in which bin capacities may differ, it turns out that SPF fails to qualify as even a relative approximation algorithm. However, we devise an implementation of the well-known FFD heuristic, which we show to be a relative approximation algorithm, yielding a worst-case performance ratio of 1/2 with additive constant 0. Moreover, we prove that (unlessP=NP) no polynomial-time algorithm can guarantee a higher ratio without sacrificing the additive constant.This author's research is supported in part by the National Science Foundation under grants ECS-8403859 and MIP-8603879.     

New lower bounds for the three-dimensional orthogonal bin packing problem

In this paper, we consider the three-dimensional orthogonal bin packing problem, which is a generalization of the well-known bin packing problem. We present new lower bounds for the problem from a combinatorial point of view and demonstrate that they theoretically dominate all previous results from the literature. The comparison is also done concerning asymptotic worst-case performance ratios. The new lower bounds can be more efficiently computed in polynomial time. In addition, we study the non-oriented model, which allows items to be rotated.