Rectangular Packing Problem Research Articles

Parameter Analysis of Placement Function for the Rectangular Packing Problem Based on GA

Rectangle part is the foundation of irregular part layout, about which domestic and overseas scholars have studied a lot and have put forward many algorithms. Based on a careful study of these algorithms in the paper, it is determined to solve the rectangle packing problem with genetic algorithm analysis. Different from the formerly used the genetic algorithm optimization layout, the algorithm of this paper stresses optimization localization rule in order to solve layout problem with the advantages of global searching ability of genetic algorithm. This algorithm sets the utilization ratio of the maximum area of capacity as the goal, after confirming the priority of deposition sequence of rectangle, in view of the locating rule of rectangle packing, based on feasible region and by introducing the method of attractive factors, optimizing the calculation for each parameter of placement function by utilizing genetic algorithm.Positioning function of the structure of this paper by changing the parameter value can cover the previous golden horn strategy, the lower left corner strategy, down the steps such as positioning method. Use VC programming to realize automatic two dimensional rectangular layout systems, the algorithm and example verification, the precision of parameter on the result of layout, the number of attractor for layout results, and the influence of parameter values for different rectangular piece of regularity. According to different rectangular block configuration, layout scheme can be better and faster.

A quasi-human algorithm for the two dimensional rectangular strip packing problem: in memory of Prof. Wenqi Huang

This paper presents a quasi-human algorithm for the rectangular strip packing problem without guillotine constraint. The basic version of the algorithm works according to seven heuristic selection rules, which are designed to select a corner-occupying action. The strengthened version of the algorithm adopts a branching scheme in which the basic version of the algorithm is applied in a heuristic series of parallel branches. Computational tests carried on eight sets of well-known benchmark instances show that the algorithm is efficient for approximately solving the problem. In comparison with the best algorithms in the literature, the algorithm performs better for zero-waste instances and large scale non-zero-waste instances.

A priority heuristic for the guillotine rectangular packing problem

A new priority heuristic is presented for the guillotine rectangular packing problem. This heuristic first selects one available item for a given position by a priority strategy. Then it divides the remaining space into two rectangular bins and packs them recursively, and its worst-case time complexity is T(n)=O(n2). The proposed algorithm is a general, simple and efficient method, and can solve different packing problems. Computational results on a wide range of benchmark problems have shown that the proposed algorithm outperforms existing heuristics in the literature, on average.

Heuristics for the local grid scheduling problem with processing time constraints

In this paper, the local job scheduling problem, with processing time constraints (i.e., deadline, earliest start time, reservation) in a computational grid, is addressed. The problem under investigation is formulated as a rectangular packing problem and several heuristic approaches are developed for its solution. The performance of the proposed algorithms are evaluated under different scenarios. Extensive experimental tests demonstrate that the defined solution strategies outperform the state-of-art methods.

An action-space-based global optimization algorithm for packing circles into a square container

This paper proposes an action-space-based global optimization (ASGO) approach for the problem of packing unequal circles into a square container such that the size of the square is minimized. Starting from several random configurations, ASGO runs the following potential descent method and basin-hopping strategy iteratively. It finds configurations with the local minimum potential energy by the limited-memory BFGS (LBFGS) algorithm, then selects the circular items having the most deformations and moves them to some large vacant space or randomly chosen vacant space. By adapting the action space defined for the rectangular packing problem, we approximate each circular item as a rectangular item, thus making it much easier to find comparatively larger vacant spaces for any given configuration. The tabu strategy is used to prevent cycling and enhance the diversification during the search procedure. Several other strategies, such as swapping two similar circles or swapping two circles in different quadrants in the container, are combined to increase the diversity of the configurations. We compare the performance of ASGO on 68 benchmark instances at the Packomania website with the state-of-the-art results. ASGO obtains configurations with smaller square containers on 63 instances; at the same time it matches or approaches the current best results on the other five instances.

Investigation on the rectangular packing problem based on the ant colony algorithm

The large-scale rectangular packing problem is a typical combinatorial optimisation problem, which belongs to the most complicated NP-complete problem. Now, it is very difficult to obtain the best solution within a certain time by the commonly used theories and methods of calculation. The ant colony algorithm, developed in recent years, is very effective in solving combinatorial optimisation and NP-problems. Based on the analysis of ant colony algorithm and the rectangular packing problem, this paper combine the ant colony algorithm and rectangular packing problem, the best-worst ant model is built and then simulated, finally, an example is used to prove the effectiveness of the proposed algorithm.

A beauty degree enumeration algorithm for the 2D rectangular packing problem

To address the 2D rectangular packing problem, this paper presents a basic algorithm based on corner areas, and presents a beauty degree enumeration algorithm. For two sets of representative benchmark instances c1-c21 and zdf1-zdf16, the algorithm outperforms the best algorithms in the literature. When the orientation of the rectangles is fixed, the algorithm finds better packing con gurations than the best reported results of four open benchmark instances zdf6-zdf9. The proposed algorithm finds the optimal solutions for zdf8 and zdf9; to the best of our knowledge, this is the first time this has been achieved.

Research on PSO algorithms for the rectangular packing problem

For packing problems that are complex optimisation problems, the solutions must rely on heuristics. Particle swarm optimisation PSO is an emerging technique that utilises heuristics. Because of its simplicity, PSO has been widely applied to a variety of fields, including applications to the packing problem. This paper researches the search efficiency of the PSO algorithm. Through fitting the data to the length of the solution interval and the number of iterations, a strong search capability for the PSO is proved. Then, the results of experiments with comparisons of many disturbance strategies show that a stochastic strategy is beneficial for raising the search capability of the algorithm.

An Algorithm for Automatic Photo Typesetting Based on Kinship

Photo typesetting belongs to the rectangular packing problems, currently the algorithms of rectangular packing problems can be divided into two categories: heuristic methods and modern intelligent based methods. For both of these methods, the similarity of contents do not be well considered, the packing order of the rectangular pieces are chaos. Therefore, this drawback will cause problems for photo typesetting. In this paper we propose a hybrid algorithm for the photo typesetting problem based on kinship. According to our algorithm, the kinship of difierent photos is calculated by their absolute paths, and then all photos are packed based on the distance of their kinship. Compared with the existing methods, according to the computation of difierent scales of tests, our algorithm shows faster typesetting speed and higher utilization ratios. This algorithm could solve the sort problem caused by chaos typesetting photos of difierent customers after lots of photos were typeset. Our algorithm could greatly improve the practical application of photo typesetting.

Hybrid algorithm for the two-dimensional rectangular layer-packing problem

In this paper, a rectangular layer-packing algorithm (RLPA) combined with modified genetic algorithm (GA) or particle swarm optimization (PSO) algorithm is developed to solve the problem with emerging restraints, which is raised from the two-dimensional rectangular packing problem with some small rectangles that need to be packed into a fixed rectangular object. RLPA is designed from the BL algorithm and lowest horizontal line algorithm. GA and PSO are also modified to satisfy the constraint conditions. Best GA or PSO parameters are obtained by conducting experiments on some typical instances. The results are also compared, which validate the quality of the solutions and show the effectiveness of the modified algorithm.

A block-based layer building approach for the 2D guillotine strip packing problem

We examine the 2D strip packing problems with guillotine-cut constraint, where the objective is to pack all rectangles into a strip with fixed width and minimize the total height of the strip. We combine three most successful ideas for the orthogonal rectangular packing problems into a single coherent algorithm: (1) packing a block of rectangles instead of a single rectangle in each step; (2) dividing the strip into layers and pack layer by layer; and (3) unrolling and repacking the top portion of the solutions where usually wasted space occurs. Computational experiments on benchmark test sets suggest that our approach rivals existing approaches.

A global optimization approach for solving three-dimensional open dimension rectangular packing problems

The three-dimensional open dimension rectangular packing problem (3D-ODRPP) aims to pack a set of given rectangular boxes into a large rectangular container of minimal volume. This problem is an important issue in the shipping and moving industries. All the boxes can be any rectangular stackable objects with different sizes and may be freely rotated. The 3D-ODRPP is usually formulated as a mixed-integer non-linear programming problem. Most existing packing optimization methods cannot guarantee to find a globally optimal solution or are computationally inefficient. Therefore, this paper proposes an efficient global optimization method that transforms a 3D-ODRPP as a mixed-integer linear program using fewer extra 0–1 variables and constraints compared to existing deterministic approaches. The reformulated model can be solved to obtain a global optimum. Experimental results demonstrate the computational efficiency of the proposed approach in globally solving 3D-ODRPPs drawn from the literature and the practical applications.

An Ordering Approach Based on Evaluative Function for Rectangular Packing Problem

The most of the ordering approaches are derived from the practice of the production and life, and these approaches ensure relatively good packing result. In view of rectangular packing problem, this paper presents an ordering approach which is based on evaluative function. By the analysis of this evaluative function, some properties of this ordering approach are obtained. And through contrast experiment, it is proved that this ordering approach is beneficial to improve the space utilization of packing result.

Efficient Algorithm Based on Action Space for Solving the 2D Rectangular Packing Problem

对于二维矩形Packing 这一典型的NP 难度问题,在黄文奇等人提出的拟人型穴度算法的基础上,通过定义动作空间来简化对不同放入动作的评价,使穴度的计算时间明显缩短,从而使算法能够快速地得到空间利用率较高的布局图案.实验测试了Hopper 和Turton 提出的21 个著名的二维矩形Packing 问题的实例.改进的算法对其中的每一个实例都得到了空间利用率为100%的最优布局,且在普通PC 机上的平均计算时间未超过7 分钟.实验结果表明,基于动作空间对拟人型穴度算法所进行的改进是明显而有效的.;This paper addresses a typical NP-hard problem, the two-dimensional (2D) rectangular packing problem. The study makes improvements on a quasi-human approach, a caving degree algorithm proposed by Huang Wen-Qi, et al., by defining the conception of action space such that the calculation of the caving degree is simplified. Therefore, the evaluation on different placements is reduced considerably, and good layouts could be obtained quickly. The experiments tested 21 famous instances of the 2D rectangular packing problem provided by Hopper and Turton. The improved algorithm achieved optimal layout with a space utilization of 100% for each instance, and the average computing time on a personal computer was within seven minutes. Computational results show that the improvement strategies on the quasi-human caving degree approach are evident and effective.

An efficient deterministic optimization approach for rectangular packing problems

The rectangular packing problem aims to seek the best way of placing a given set of rectangular pieces within a large rectangle of minimal area. Such a problem is often constructed as a quadratic mixed-integer program. To find the global optimum of a rectangular packing problem, this study transforms the original problem as a mixed-integer linear programming problem by logarithmic transformations and an efficient piecewise linearization approach that uses a number of binary variables and constraints logarithmic in the number of piecewise line segments. The reformulated problem can be solved to obtain an optimal solution within a tolerable error. Numerical examples demonstrate the computational efficiency of the proposed method in globally solving rectangular packing problems.

An efficient deterministic heuristic for two-dimensional rectangular packing

This paper proposes a deterministic heuristic, a best fit algorithm (BFA), for solving the NP-hard two-dimensional rectangular packing problem to maximize the filling rate of a rectangular sheet. There are two stages in this new approach: the constructive stage and the tree search stage. The former aims to rapidly generate an initial solution by employing the concepts of action space and fit degree in evaluating different placements. The latter seeks to further improve the solution and searches for promising placements by a partial tree search procedure. We then compare BFA with other approaches in terms of solution quality and computing time. We carry out computational experiments on two sets of well-known benchmark instances, C21 proposed by Hopper and Turton, and N13 proposed by Burke et al. BFA gained an average filling rate of 100% for the C21 instances within short times, indicating that all the layouts obtained are optimal. To the best of our knowledge, this is the first time that optimal layouts on all the 21 instances were obtained by a deterministic algorithm. As for the N13 instances, to date, researchers have found optimal solutions to the first three instances, whereas BFA solved seven, including the first three, within a reasonable period. An additional work is to adapt BFA to solve a relevant problem, the constrained two-dimensional cutting (or packing) problem (CTDC). Though BFA is not for the CTDC in the original design such that some specific characteristics of CTDC are not considered, the adapted algorithm still performed well on 21 public CTDC instances.

A skyline heuristic for the 2D rectangular packing and strip packing problems

In this paper, we propose a greedy heuristic for the 2D rectangular packing problem (2DRP) that represents packings using a skyline; the use of this heuristic in a simple tabu search approach outperforms the best existing approach for the 2DRP on benchmark test cases. We then make use of this 2DRP approach as a subroutine in an “iterative doubling” binary search on the height of the packing to solve the 2D rectangular strip packing problem (2DSP). This approach outperforms all existing approaches on standard benchmark test cases for the 2DSP.

A New Effective Hybrid Optimization Strategy for Rectangular Packing Problem

In this paper we address a rectangular packing problem (RPP), which is one of the most difficult NP-complete problems. First, greedy biggest space sequencing (GBSS) is presented as a new placement strategy, which is very essential to RPP. Then, borrowing from the respective advantages of the two algorithms, genetic algorithm (GA) and simulated annealing (SA), a hybrid optimization policy is developed. The hybrid GASA is subjected to a test using a set of benchmarks. Compared to other approaches from the literature the hybrid optimization strategy performs better.

Solving the two-dimensional packing problem with m-M calculus

This paper considers the two dimensional rectangular packing problem. The mathematical formulation is based on the optimization of a non-linear function with piecewise linear constraints with a small number of real variables. The presented method of m-M calculus finds all optimal solutions on small instances. Computational performance is good on smaller instances.

The Particle Swarm Optimization Algorithm for Solving Rectangular Packing Problem

Based on extensive researches on various heuristic algorithms, the particle swarm optimization algorithm was developed to solve the rectangular packing problems. The algorithm optimizes the parameter of dynamic attractive factors by updating the position and the velocity of the particles, and applies perturbation strategy to solve the matter that it is easy to stick at local optima. The experimental result shows that the algorithm can get a better packing result by less time.