Sequence based heuristics for two-dimensional bin packing problems

A Computational Study of Lower Bounds for the Two Dimensional Bin Packing Problem

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

We address the two-dimensional bin packing problem with fixed orientation. This problem requires packing a set of small rectangular items into a minimum number of standard two-dimensional bins. It is a notoriously intractable combinatorial optimization problem and has numerous applications in packing and cutting. The contribution of this paper is twofold. First, we propose a comprehensive theoretical analysis of lower bounds and we elucidate dominance relationships. We show that a previously presented dominance result is incorrect. Second, we present the results of an extensive computational study that was carried out, on a large set of 500 benchmark instances, to assess the empirical performance of the lower bounds. We found that the so-called Carlier-Clautiaux-Moukrim lower bounds exhibits an excellent relative performance and yields the tightest value for all of the benchmark instances.

We consider the two-dimensional bin packing and strip packing problem, where a list of rectangles has to be packed into a minimal number of rectangular bins or a strip of minimal height, respectively. All packings have to be non-overlapping and orthogonal, i.e., axis-parallel. Our algorithm for strip packing has an absolute approximation ratio of 1.9396 and is the first algorithm to break the approximation ratio of 2 which was established more than a decade ago. Moreover, we present a polynomial time approximation scheme ($\mathcal{PTAS}$) for strip packing where rotations by 90 degrees are permitted and an algorithm for two-dimensional bin packing with an absolute worst-case ratio of 2, which is optimal provided $\mathcal{P} \not= \mathcal{NP}$.

The following two-dimensional bin packing problems are considered. (1) Find a way to pack an arbitrary collection of rectangular pieces into an open-ended, rectangular bin so as to minimize the height to which the pieces fill the bin. (2) Given rectangular sheets, i.e. bins of fixed width and height, allocate the rectangular pieces, the object being to obtain an arrangement that minimizes the number of sheets needed. Approximation algorithms are given where the solutions are achieved with heuristic search methods. Test results are presented to support the feasibility of the methods.

This paper reconsidered the Two-Dimensional Bin Packing Problem from the perspective of practical application based on actual needs of small furniture companies in China. Although there have been many researches on this problem for many years, most of the existed researches put their foremost attention on developing an algorithm with low complexity or approximately best performance which means that it can nearly work out the best solution most of the time. This is the common situation when we do research on such problems, but unfortunately it will leave out abundant practical needs. For example, the computational resources of those small companies are seldom, which makes it impossible to use algorithms that almost gets the best solution but takes much time. The significant goal is to find out a solution satisfactory enough within a short time, so some complex algorithms which aim to get a nearly best result aren’t suitable here. Meanwhile, the cutting times are also a rather critical matter for them. Cutting too many times will cause high wastage to machines and is therefore unaffordable for small companies, but this was often ignored in previous researches especially in classic meta-heuristic algorithms since they frequently lead to complicated packing patterns with massive cutting times. This paper reviewed previous related researches and summarized their shortages and some necessary characteristics of a practically efficient algorithm. Based on classic algorithms, it gained a two-stage heuristic algorithm which performed better than current used ones in those small companies, as well as meets their special needs stated in this paper which is a rather important issue. And this algorithm itself may be used to meet needs of those factories or be used as a starter of the further optimization search.

Weight annealing is a metaheuristic that has been recently proposed in the physics literature. We develop a weight annealing-based algorithm for solving four variants of the two-dimensional bin packing problem. We apply our algorithm to 500 benchmark instances and find that it quickly produces very high-quality solutions that are comparable to the best published results.

Two-dimensional online bin packing with rotation

This paper looks into the steel mother plate design problem. A slab, which is an intermediate work in process, is subsequently rolled into a mother plate with the specific dimensions of thickness, length, and width. The mother plate is then cut into customer order plates. As a slab is rolled into a mother plate through a series of horizontal and vertical rolling processes, different-sized mother plates can be generated from a single-slab type. This flexibility allows for the size of a mother plate to be determined according to the order plates assigned to it. Furthermore, when the order plates are cut from a mother plate, a guillotine cut is required to reduce the production cost. The steel mother plate design problem involves the placing of order plates on the mother plates in a guillotine cut pattern and determining the sizes of the mother plates with the objective of minimising the number of slabs; thus it may be considered as a two-staged guillotine cut, two-dimensional bin packing problem with flexible bin size. This paper introduces the problem, presents several mathematical models, and proposes an iterative two-phase heuristic method consisting of several algorithms to solve the problem. Computational results for the benchmark problems show the effectiveness of the proposed method.

The vehicle routing problem with two-dimensional loading constraints (2L-CVRP) and the last-in-first-out (LIFO) rule presents significant practical and algorithmic challenges. While numerous heuristic approaches have been proposed to address its complexity, stemming from two NP-hard problems: the vehicle routing problem (VRP) and the two-dimensional bin packing problem (2D-BPP), less attention has been paid to developing exact algorithms. Bridging this gap, this article presents an exact algorithm that integrates advanced machine learning techniques, specifically a novel combination of attention and recurrence mechanisms. This integration accelerates the state-of-the-art exact algorithm by a median of 29.79% across various problem instances. Moreover, the proposed algorithm successfully resolves an open instance in the standard test-bed, demonstrating significant improvements brought about by the incorporation of machine learning models. Code is available at https://github.com/xyfffff/NCG-for-2L-CVRP.

The two-dimensional heterogeneous vector bin packing problem (2DHet-VBPP) consists of packing the set of items into the set of various type bins, respecting their two resource limits. The problem is to minimize the total cost of all bins. The problem, known to be NP-hard, can be formulated as a pure integer linear program, but optimal solutions can be obtained by the CPLEX Optimizer engine only for small instances. This paper proposes a metaheuristic approach to the 2DHet-VBPP, based on Reduced variable neighborhood search (RVNS). All RVNS elements are adapted to the considered problem and many procedures are designed to improve efficiency of the method. As the Two-dimensional Homogeneous-VBPP (2DHom-VBPP) is more often treated, we considered also a special version of the RVNS algorithm to solve the 2DHom-VBPP. The results obtained and compared to both CPLEX results and results on benchmark instances from literature, justify the use of the RVNS algorithm to solve large instances of these optimization problems.

We propose an SOC test scheduling method based on simulated annealing. In our method, the test scheduling is formulated as a two-dimensional bin packing problem (rectangle packing) and a data structure called a sequence pair is used to represent the placement of the rectangles. Simulated annealing is used to find the optimal test schedule by altering an initial sequence pair and changing the width of the core wrapper. We also propose a method of wrapper design for cores without internal scan chains. Experiments are conducted on ITC'02 benchmarks, showing that overall the proposed method provides better solutions compared to earlier methods.

The automated design of search techniques is a recent trend in artificial intelligence research. Unfortunately, the majority of the automated design approaches are developed using trial and error which fails to justify or at least explain why some design decisions succeed while others fail. This approach is a host of evils as it has resulted in poorly understood systems for poorly understood problems. This study is an attempt to improve our understanding of the automated design of hybrid metaheuristics by utilizing fitness landscape analysis to reveal the topological characteristics that can be exploited to design better automated approaches. We consider the sequential hybridization, including algorithm configuration and parameter tuning, of single-point and multi-point metaheuristics and three optimization problems which are the earth-observing satellite scheduling problem, the aircraft landing problem and the two-dimensional bin packing problem. Interestingly, the design space exhibits similar trends regardless of the underlining optimization problem. The design space is found to be rugged, multimodal, moderately searchable, has multiple funnels, and almost no plateau. Based on these findings, deeper insights are provided to guide the development of future automated approaches instead of blindly trying different options.

2차원 저장소 적재는 NP-hard 문제로서 그 문제의 정확한 해를 구하는 것이 어려운 것으로 알려져 있으며, 이의 더 좋은 해를 얻기 위해 유전자(genetic) 알고리즘, 시뮬레이티드 어닐링(simulated annealing), 타부서치(tabu search)등과 같은 근사적 접근법이 제안되어 왔다. 하지만 분지한계(branch-and-bound)나 타부서치 기법들을 이용한 기존의 대표적인 근사 알고리즘들은 휴리스틱 알고리즘의 해에 기반을 둠으로 효율성이 낮고 반복수행에 의한 계산시간이 길다. 따라서 본 논문에서는 이러한 근사 알고리즘의 복잡성을 간소화하고, 알고리즘의 효율성을 높이기 위해 적재가능성을 판단하는 적합성 함수(fitness function)를 정의하고 이를 이용하여 어떤 특정 개체의 적재영역을 판단하는데 영향을 주는 적재영역의 수를 계산한다. 또한, 이들을 이용한 새로운 휴리스틱 알고리즘을 제안하였다. 끝으로 기존의 휴리스틱 또는 메타휴리스틱 기법과의 비교실험을 통해 기존의 휴리스틱 알고리즘인 FFF와 FBS에 비해 97%의 결과가 같거나 우수하였으며, 타부서치 알고리즘에 비해 86%의 결과가 같거나 우수한 것으로 나타났다. The two-dimensional bin packing problem(2D-BPP) has been known to be NP-hard, and it is difficult to solve the problem exactly. Many approximation methods, such as genetic algorithm, simulated annealing and tabu search etc, have been also proposed to gain better solutions. However, the existing approximation algorithms, such as branch-and-bound and tabu search, have shown the low efficiency and the long execution time due to a large of iterations. To solve these problems, we first define the fitness function to simplify and increase the utility of algorithm. The function decides whether an item is packed into a given area, and as an important information for a packing strategy, the number of subarea that can accommodate a given item is obtained from the variant of the fitness function. Then we present a heuristic algorithm BF for 2D bin packing, constructed by the fitness function and subarea. Finally, the effectiveness of the proposed algorithm will be expressed by the comparison experiments with the heuristic and the metaheuristic of the literatures. As comparing with existing heuristic algorithms and metaheuristic algorithms, it has been found that the packing rate of algorithm BP is the same as 97% as existing heuristic algorithms, FFF and FBS, or better than them. Also, it has been shown the same as 86% as tabu search algorithm or better.

We study the two-dimensional bin packing problem with and without rotations. Here we are given a set of two-dimensional rectangular items I and the goal is to pack these into a minimum number of unit square bins. We consider the orthogonal packing case where the edges of the items must be aligned parallel to the edges of the bin. Our main result is a 1.405-approximation for two-dimensional bin packing with and without rotation, which improves upon a recent 1.5 approximation due to Jansen and Pradel. We also show that a wide class of rounding based algorithms cannot improve upon the factor of 1.5.