GENERALIZED FIRST-FIT ALGORITHMS IN TWO AND THREE DIMENSIONS

We investigate the two and three dimensional bin packing problems, i.e., packing a list of rectangles (boxes) into unit square (cube) bins so that the number of bins used is a minimum. A simple on-line packing algorithm for the one dimensional bin packing problem, the First-Fit algorithm, is generalized to two and three dimensions. We first give an algorithm for the two dimensional case and show that its asymptotic worse case performance ratio is [Formula: see text]. The algorithm is then generalized to the three dimensional case and its performance ratio [Formula: see text]. The second algorithm takes a parameter and we prove that by choosing the parameter properly, it has an asymptotic worst case performance bound which can be made as close as desired to 1.72=2.89 and 1.73=4.913 respectively in two and three dimensions.

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Erratum to “The Three-Dimensional Bin Packing Problem”: Robot-Packable and Orthogonal Variants of Packing Problems

Operations Research ◽

10.1287/opre.1050.0210 ◽

2005 ◽

Vol 53 (4) ◽

pp. 735-736 ◽

Cited By ~ 27

Author(s):

Edgar den Boef ◽

Jan Korst ◽

Silvano Martello ◽

David Pisinger ◽

Daniele Vigo

Keyword(s):

Bin Packing ◽

Three Dimensional ◽

Packing Problem ◽

Packing Problems ◽

Bin Packing Problem

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Metaheuristic algorithms for one-dimensional bin-packing problems: A survey of recent advances and applications

Journal of Intelligent Systems ◽

10.1515/jisys-2020-0117 ◽

2021 ◽

Vol 30 (1) ◽

pp. 636-663

Author(s):

Chanaleä Munien ◽

Absalom E. Ezugwu

Keyword(s):

Bin Packing ◽

Optimization Problems ◽

Packing Problem ◽

Population Based ◽

Metaheuristic Algorithms ◽

Packing Problems ◽

One Dimensional ◽

Reference Guide ◽

Recent Advances ◽

The One

Abstract The bin-packing problem (BPP) is an age-old NP-hard combinatorial optimization problem, which is defined as the placement of a set of different-sized items into identical bins such that the number of containers used is optimally minimized. Besides, different variations of the problem do exist in practice depending on the bins dimension, placement constraints, and priority. More so, there are several important real-world applications of the BPP, especially in cutting industries, transportation, warehousing, and supply chain management. Due to the practical relevance of this problem, researchers are consistently investigating new and improved techniques to solve the problem optimally. Nature-inspired metaheuristics are powerful algorithms that have proven their incredible capability of solving challenging and complex optimization problems, including several variants of BPPs. However, no comprehensive literature review exists on the applications of the metaheuristic approaches to solve the BPPs. Therefore, to fill this gap, this article presents a survey of the recent advances achieved for the one-dimensional BPP, with specific emphasis on population-based metaheuristic algorithms. We believe that this article can serve as a reference guide for researchers to explore and develop more robust state-of-the-art metaheuristics algorithms for solving the emerging variants of the bin-parking problems.

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Analysis of an Approximation Algorithm for Scheduling Independent Parallel Tasks

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.262 ◽

1999 ◽

Vol Vol. 3 no. 4 ◽

Author(s):

Keqin Li

Keyword(s):

Approximation Algorithm ◽

Bin Packing ◽

Random Variables ◽

Packing Problem ◽

Performance Ratio ◽

Worst Case ◽

Average Case ◽

Parallel Tasks ◽

International Audience ◽

Worst Case Performance Ratio

International audience In this paper, we consider the problem of scheduling independent parallel tasks in parallel systems with identical processors. The problem is NP-hard, since it includes the bin packing problem as a special case when all tasks have unit execution time. We propose and analyze a simple approximation algorithm called H_m, where m is a positive integer. Algorithm H_m has a moderate asymptotic worst-case performance ratio in the range [4/3 ... 31/18] for all m≥ 6; but the algorithm has a small asymptotic worst-case performance ratio in the range [1+1/(r+1)..1+1/r], when task sizes do not exceed 1/r of the total available processors, where r>1 is an integer. Furthermore, we show that if the task sizes are independent, identically distributed (i.i.d.) uniform random variables, and task execution times are i.i.d. random variables with finite mean and variance, then the average-case performance ratio of algorithm H_m is no larger than 1.2898680..., and for an exponential distribution of task sizes, it does not exceed 1.2898305.... As demonstrated by our analytical as well as numerical results, the average-case performance ratio improves significantly when tasks request for smaller numbers of processors.

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A New Branch-and-Price-and-Cut Algorithm for One-Dimensional Bin-Packing Problems

INFORMS Journal on Computing ◽

10.1287/ijoc.2018.0867 ◽

2020 ◽

Vol 32 (2) ◽

pp. 428-443

Author(s):

Lijun Wei ◽

Zhixing Luo ◽

Roberto Baldacci ◽

Andrew Lim

Keyword(s):

Bin Packing ◽

Large Family ◽

Exact Algorithm ◽

Packing Problem ◽

Set Partitioning ◽

Branch And Price ◽

Packing Problems ◽

One Dimensional ◽

Bin Packing Problem ◽

The One

In this paper, a new branch-and-price-and-cut algorithm is proposed to solve the one-dimensional bin-packing problem (1D-BPP). The 1D-BPP is one of the most fundamental problems in combinatorial optimization and has been extensively studied for decades. Recently, a set of new 500 test instances were proposed for the 1D-BPP, and the best exact algorithm proposed in the literature can optimally solve 167 of these new instances, with a time limit of 1 hour imposed on each execution of the algorithm. The exact algorithm proposed in this paper is based on the classical set-partitioning model for the 1DBPPs and the subset row inequalities. We describe an ad hoc label-setting algorithm to solve the pricing problem, dominance, and fathoming rules to speed up its computation and a new primal heuristic. The exact algorithm can easily handle some practical constraints, such as the incompatibility between the items, and therefore, we also apply it to solve the one-dimensional bin-packing problem with conflicts (1D-BPPC). The proposed method is tested on a large family of 1D-BPP and 1D-BPPC classes of instances. For the 1D-BPP, the proposed method can optimally solve 237 instances of the new set of difficult instances; the largest instance involves 1,003 items and bins of capacity 80,000. For the 1D-BPPC, the experiments show that the method is highly competitive with state-of-the-art methods and that it successfully closed several open 1D-BPPC instances.

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The Improvement of a Hybrid Genetic Algorithm for Three Dimensional Bin-Packing Problems

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.200.470 ◽

2012 ◽

Vol 200 ◽

pp. 470-473

Author(s):

Zhen Zhai ◽

Li Chen ◽

Xiao Min Han

Keyword(s):

Genetic Algorithm ◽

Bin Packing ◽

Hybrid Genetic Algorithm ◽

Three Dimensional ◽

Packing Problem ◽

Flow Process ◽

Packing Problems ◽

Loading Problem ◽

The Cost ◽

Process Diagrams

The multi-constrained bi-objective bin packing problem has many extensive applications. In the loading section of logistics it has mainly been transported by truck. The cost of transportation is not only determined by the bin space utilization, but also by the number of vehicles in transporta¬tion utilization. The type of items and bins is introduced in the mathematical model, as well as the volume of the items. In this paper, the hybrid genetic algorithm which tabu and simulated annealed rules are added for complex container-loading problem is studied. The effective coding and decod-ing method together with flow process diagrams are given.

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Analysis of the robustness of a steel frame structure with composite floors subject to multiple fire scenarios

Advances in Structural Engineering ◽

10.1177/1369433221992494 ◽

2021 ◽

pp. 136943322199249

Author(s):

Riza Suwondo ◽

Lee Cunningham ◽

Martin Gillie ◽

Colin Bailey

Keyword(s):

Progressive Collapse ◽

Three Dimensional ◽

Steel Structure ◽

Frame Structure ◽

Worst Case ◽

Collapse Resistance ◽

Fire Scenarios ◽

Element Approach ◽

The One ◽

Steel Frame Structure

This study presents robustness analyses of a three-dimensional multi-storey composite steel structure under the action of multiple fire scenarios. The main objective of the work is to improve current understanding of the collapse resistance of this type of building under different fire situations. A finite element approach was adopted with the model being firstly validated against previous studies available in the literature. The modelling approach was then used to investigate the collapse resistance of the structure for the various fire scenarios examined. Different sizes of fire compartment are considered in this study, starting from one bay, three bays and lastly the whole ground floor as the fire compartment. The investigation allows a fundamental understanding of load redistribution paths and member interactions when local failure occurs. It is concluded that the robustness of the focussed building in a fire is considerably affected by the size of fire compartments as well as fire location. The subject building can resist progressive collapse when the fire occurs only in the one-bay compartment. On the other hand, total collapse occurs when fire is located in the edge three-bay case. This shows that more than one fire scenario needs to be taken into consideration to ensure that a structure of this type can survive from collapse in the worst-case situation.

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