Rectangle Packing Research Articles

Interpretability of rectangle packing solutions with Monte Carlo tree search

Packing problems have been studied for a long time and have great applications in real-world scenarios. In recent times, with problems in the industrial world increasing in size, exact algorithms are often not a viable option and faster approaches are needed. We study Monte Carlo tree search, a random sampling algorithm that has gained great importance in literature in the last few years. We propose three approaches based on MCTS and its integration with metaheuristic algorithms or deep learning models to obtain approximated solutions to packing problems that are also interpretable by means of MCTS exploration and from which knowledge can be extracted. We focus on two-dimensional rectangle packing problems in our experimentation and use several well known benchmarks from literature to compare our solutions with existing approaches and offer a view on the potential uses for knowledge extraction from our method. We manage to match the quality of state-of-the-art methods, with improvements in time with respect to some of them and greater interpretability.

Online Rectangle Packing Algorithm for Swapped Battery Charging Dispatch Model Considering Continuous Charging Power

The vigorous development of electric vehicles (EVs) is an important means of reducing carbon emissions and mitigating environmental problems such as the greenhouse effect. Battery swapping stations (BSSs) can both provide battery swapping services for large-scale EVs and charge batteries centrally. As the supply of fully charged batteries in the BSS shrinks, it becomes necessary to schedule the charging of the depleted batteries rapidly that users have swapped for fully-charged ones. The charging schedule for depleted batteries must be made without knowledge of future battery arrivals. In this context, this paper develops a mathematical model for online charging scheduling of BSSs, formulates the charging strategy as a two-dimensional rectangle packing problem, and quickly calculates the scheduling arrangement of batteries by partitioning the remaining available capacity of a BSS. Since there are limited battery types within the BSS which can provide battery replacement services, this paper supplements the proposed model with known battery types, which improves the utilization of the available capacity of BSSs. Finally, numerical results verify the effectiveness of the proposed model. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Electric vehicles (EVs) are becoming an alternative way to reduce carbon emissions in transportation systems. Herein, the optimal battery charging problem is the core problem when it comes to dispatching a huge number of EVs. Up to now, battery-swapping is widely used for EVs due to its simple, convenient way. Furthermore, a business model for the battery swapping stations (BSSs) is brought up, where EV users send their depleted batteries to the BSS and the BSS provides the users with a fully charged replacement battery from its warehouse, which only takes a few minutes. Since the maximum charging power of the BSS is limited by the capacity of the transformer connecting the BSS to the power grid, the BSS will adopt an optimal charging schedule that maximizes the charging benefit for large quantities of depleted batteries in the warehouse. However, the challenge is that the charging schedule for depleted batteries must be made without knowledge of future battery arrivals because the EV behaviors are difficult to predict. To address this problem, this paper developed an online charging scheduling algorithm, which formulates the charging strategy as a two-dimensional rectangle packing problem. The proposed method can provide battery replacement services in real-time and solve quickly without any information about incoming depleted EV batteries. The proposed model and method have been tested on the system with different numbers of batteries to show the effectiveness. Besides, the online two-dimensional rectangle packing problem can provide an online decision for BSSs.

Bounds on soft rectangle packing ratios

We examine rectangle packing problems where only the areas a1,…,an of the rectangles to be packed are given while their aspect ratios may be chosen from a given interval [1γ,γ]. In particular, we ask for the smallest possible size of a rectangle R such that, under these constraints, any collection a1,…,an of rectangle areas of total size 1 can be packed into R. As for standard square packing problems, which are contained as a special case for γ=1, this question leads us to three different answers, depending on whether the aspect ratio of R is given or whether we may choose it either with or without knowing the areas a1,…,an. Generalizing known results for square packing problems, we provide upper and lower bounds for the size of R with respect to all three variants of the problem, which are tight at least for larger values of γ. Moreover, we show how to improve these bounds on the size of R if we restrict ourselves to instances where the largest element in a1,…,an is bounded.

An Effective Scoring-Based Metaheuristic Algorithm for Rectilinear Block Placement

Due to the new packaging schemes such as multi-project wafer (MPW), the shape of integrated circuit blocks becomes more flexible and the demand for rectilinear block placement is rising. MPW can be modeled as the rectilinear block placement problem, which requires to place a set of L/T/U-shaped rectilinear blocks on a plane without overlapping. The objective is to minimize the area of the enveloping rectangle with the aspect ratio of the envelope close to 1:1. This paper presents an effective scoring-based metaheuristic algorithm (SMA) for the MPW problem. A skyline heuristic is proposed to find a suitable block for the lowest and leftmost space for the rectilinear block placement problem. Comparing to the previous algorithms for rectangle packing, our algorithm considers the characteristics of L/T/U-shaped blocks and different orientations in the scoring strategies for block selection. In order to improve the placement sequence, swapping and shifting operations are incorporated to iteratively change the current sequence. Computational experiments were conducted on ten benchmark instances from the Integrated Circuit EDA Elite Challenge 2020 (ICEEC 2020). The experimental results show that the average filling ratio of our algorithm is around 95% which demonstrates the effectiveness of SMA.

Rectangle packing with a recursive Pilot Method

We propose a deterministic tree search procedure for the Rectangle Packing Problem, where a set of rectangular items must be packed into a rectangular container such that the unoccupied area of the container is the minimum possible. The tree search is based on two literature methods whose strong points complement each other. First, a skyline-based heuristic strongly restricts the number of items that must be evaluated and the number of positions where they may be placed. Also, a generalization of the Pilot Method meta-heuristic expands the search tree gradually, iteratively increasing the tree size within a given time limit. We also devise tree pruning strategies to further reduce the exploration size. Empirical results surpass state-of-the-art solutions on most benchmark data sets, subject to computation time limits comparable to those of the literature, and we analyze how the quality of the solutions varies depending on the geometry of the items.

Hybrid Swapped Battery Charging and Logistics Dispatch Model in Continuous Time Domain

Electric vehicles (EVs) have attracted enormous attention in recent years due to their potentials in mitigating energy crisis and air pollutions. However, the long battery charging time and lack of sufficient charging infrastructure highly restrict the popularization of EVs. In this context, it is promising to establish battery charging and swapping systems (BCSSs) based on the concept of battery swapping services. To optimally achieve the combined operation of BCSSs, this paper proposes a hybrid swapped battery charging and logistics dispatch model in continuous time domain. Identifying the special structure of the mathematical models of the two problems, this paper innovatively formulated the swapped battery charging strategy as the rectangle packing problem and the battery logistics model as the vehicle routing problem. The two models are closely linked by the delivery time of transporting the well-charged batteries from battery charging stations to battery swapping stations. A hybrid optimal operation model of BCSSs is further formulated as a mixed-integer linear programming model by incorporating the interaction between the battery charging and battery logistics. Finally, case studies are conducted on several BCSSs and numerical results validate the effectiveness of the proposed model.

Online Routing and Charging Schedule of Electric Vehicles With Uninterrupted Charging Rates

With the development of electric vehicle technology and the Internet of Things (IoT) technology, IoT-Based electric vehicles (IoEVs) will become a main public transposition in smart city. The increasing penetration of IoEV will have a great impact on the transportation network and power grid. In this work, we aim to study the routing and charging scheduling problem of IoEVs with the constraint of power limit. This paper proposes a clustered rolling framework and a mathematical model for the IoEVs optimal routing and charging problem to minimize the waiting time of customers and get a good converge ratio of the energy block with power limit. This framework can effectively deal with the multi-step pickup and delivery of each vehicle over the time horizon in a dynamic environment. We give a criterion to judge whether pickup and delivery tasks can be assigned to the IoEV. Then, we design a new rectangle packing method for IoEV charging dispatch with the constraint of power limit. Moreover, we consider that the charging process of IoEVs cannot be interrupted before IoEVs meet the electricity demand. We propose the IoEV Routing and Charging Algorithm and analyze its computational complexity. Simulation results show that IoEVs can offer the shuttle service to customers with the minimum waiting time and IoEV charging energy block has a good coverage ratio of the total energy block. Simulation results can verify the effectiveness of our proposed framework, compared with two benchmark algorithms.

A Time-Efficient and Exploratory Algorithm for the Rectangle Packing Problem

Today, resource waste is considered as one of the most important challenges in different industries. In this regard, the Rectangle Packing Problem (RPP) can affect noticeably both time and design issues in businesses. In this study, the main objective is to create a set of non-overlapping rectangles so that they have specific dimensions within a rectangular plate with a specified width and an unlimited height. The ensued challenge is an NP-complete problem. NP-complete problem, any of a class of computational problems that still there are no efficient solution for them. Most substantial computer-science problems such as the traveling salesman problem, satisfiability problems (sometimes called propositional satisfiability problem and abbreviated SAT or B-SAT), and graph-covering problems are belong to this class. Essentially, it is complicated to spot the best arrangement with the highest rate of resource utilization by emphasizing the linear computation time. This study introduces a time-efficient and exploratory algorithm for the RPP, including the lowest front-line strategy and a Best-Fit algorithm. The obtained results confirmed that the proposed algorithm can lead to a good performance with simplicity and time efficiency. Our evaluation shows that the proposed model with utilization rate about 94.37% outperforms others with 87.75%, 50.54%, and 87.17% utilization rate, respectively. Consequently, the proposed method is capable to of achieving much better utilization rate in comparison with other mentioned algorithms in just 0.023 s running-time, which is much faster than others.

Which rectangle sets have perfect packings?

In the perfect rectangle packing problem, a set of rectangular items have to be placed inside a rectangular container without overlap or empty space. In this paper, we generate a large number of random instances and decide them all with an exact solving algorithm. Both an instance’s solution probability and its hardness measured in recursions or system time, seems to critically depend on tmax, a parameter in the generation procedure that assigns the maximally choosable random side lengths of items in the instance. We numerically characterize the solvability across instance sizes, and derive a rule for generating (un)solvable problem instances of arbitrary size.

Optimal Packing of High-Precision Rectangles

The rectangle-packing problem consists of finding an enclosing rectangle of smallest area that can contain a given set of rectangles without overlap. Our new benchmark includes rectangles of successively higher precision, a problem for the previous state-of-the-art, which enumerates all locations for placing rectangles. We instead limit these locations and bounding box dimensions to the set of subset sums of the rectangles' dimensions, allowing us to test 4,500 times fewer bounding boxes and solve N=9 over two orders of magnitude faster. Finally, on the open problem of the feasibility of packing a specific infinite series of rectangles into the unit square, we pack the first 50,000 such rectangles and conjecture that the entire infinite series can fit.

NHACR: A novel heuristic approach for 2D rectangle packing area minimization problem with central rectangle

As a special 2D rectangle packing problem, the 2D rectangle packing area minimization problem with central rectangle (CR-RPAMP) has one or more central rectangles among the candidate items. The central rectangles should be packed near the center of the final layout. At the same time, the length–width ratio of the final layout must be within a reasonable region. The CR-RPAMP has been applied in many areas such as the layout design of semi-submersible platform, farmland irrigation, urban planning. In this study, to improve the performance of heuristic approach for solving the CR-RPAMP, three novel strategies including strategy of filling inner space, strategy of generating inner space and strategy of simplifying packing process, are proposed and combined to form a novel heuristic approach NHACR (Novel Heuristic Approach for CR-RPAMP). The analyses of computational complexity of the proposed NHACR indicate that it has low computational complexity. Based on two famous sets of benchmark instances, the proposed NHACR shows better performance compared with several existing algorithms considering the success rate, the filling rate of final layout and the computing time. Finally, the NHACR is utilized to solve a specific application problem in oil and gas industry and its advantage is further confirmed.

Temporal and Spectral 2D Fragmentation-Aware RMSA Algorithm for Advance Reservation Requests in EONs

In this paper, we explore the problem of routing, modulation and spectrum assignment (RMSA) for advance reservation (AR) requests in elastic optical networks (EONs), with the objective to further reduce the blocking probability and improve the spectrum utilization. On the one hand, we propose a minimum resource consumption routing policy (MRC) for leaving more free resources for future requests. On the other hand, we introduce a two-dimensional (2D) path resource model, in which each AR request can be transformed into a rectangle block whose length and width are related to request bandwidth and request duration respectively. Therefore, the spectrum assignment problem of AR requests is simplified to the two-dimensional rectangle packing problem. Furthermore, we design several factors to evaluate the two-dimensional fragmentation caused by a rectangle packing operation, where we consider the fragmentation situation in the rectangle block neighborhood for the first time. Based on these factors, we propose a two-dimensional fragmentation-aware spectrum assignment strategy (2D-FA). Through the combination of MRC and 2D-FA, we propose a novel RMSA algorithm named MRC-2D-FA for dynamically provisioning AR requests in EONs. Simulation results show that MRC-2D-FA can achieve lower bandwidth blocking probability and higher spectrum utilization compared with several well-performed benchmark algorithms.

Website Optimization Design to Cut Stock Problems in 1 and 2 Dimensions with Genetic Algorithms

Many industries produce waste that cannot be reused, especially in the process of cutting raw materials. The use of raw materials that are not optimal indicates that the pattern of item placement on raw materials produces waste which cannot be used for other needs. This problem is known as cutting stock problem. This study aims to reduce the waste that arises when cutting raw material sheets into small pieces needed in the industry by developing a website as a tool in completing the placement of cutting patterns on raw materials. The developed website describes the results of optimal placement patterns in the form of images for 1 and 2 dimensions with the genetic algorithm method as an optimization algorithm using rectangle packing sequence in codification, cornerset modification algorithm on decoding, parameterized uniform crossover on crossover, roulette wheel on selection, and random generated mutations. Website development is made with hypertext markup language to display various information needed, cascading style sheets to arrange components in structured and uniform websites, and javascript to create genetic algorithm logic. This research resulted in the placement of cutting patterns on sheets of raw material on the canvas by labeling the names of each item.

Optimization Method for Guillotine Packing of Rectangular Items within an Irregular and Defective Slate

Research on the rectangle packing problems has mainly focused on rectangular raw material sheets without defects, while natural slate has irregular and defective characteristics, and the existing packing method adopts manual packing, which wastes material and is inefficient. In this work, we propose an effective packing optimization method for nature slate; to the best of our knowledge, this is the first attempt to solve the guillotine packing problem of rectangular items in a single irregular and defective slate. This method is modeled by the permutation model, uses the horizontal level (HL) heuristic proposed in this paper to obtain feasible solutions, and then applies the genetic algorithm to optimize the quality of solutions further. The HL heuristic is constructed on the basis of computational geometry and level packing. This heuristic aims to divide the irregular plate into multiple subplates horizontally, calculates the movable positions of the rectangle in the subplates, determines whether or not the rectangle can be packed in the movable positions through computational geometry, and fills the scraps appropriately. Theoretical analysis confirms that the rectangles obtained through the HL heuristic are inside the plate and do not overlap with the defects. In addition, the packed rectangles do not overlap each other and satisfy the guillotine constraint. Accordingly, the packing problem can be solved. Experiments on irregular slates with defects show that the slate utilization through our method is between 89% and 95%. This result is better than manual packing and can satisfy actual production requirements.

An effective approach to the two-dimensional rectangular packing problem in the manufacturing industry

The rectangle packing area minimization problem (RPAMP) has a wide range of applications in industry. The purpose of RPAMP is to ensure maximum use of the materials used in industries and minimize the trim loss. The yield from materials is significant issue. Because a small change in placement can result in a significant reduction in production costs and material savings. In this study, 2D-RPAMP, which is widely seen in the manufacturing industry, has been investigated. The solution approaches have been developed by using heuristic methods such as no-fit polygon and bottom-left fill and metaheuristic method such as Simulated Annealing (SA). In experimental studies, the bottom-left fill method is used for the placement process, no-fit polygon method is used in order to avoid overlapping between the parts placed, and the SA algorithm is employed for the faster and efficient packing process. As a result of experiments using the hybrid methodology consisting of the SA algorithm can reach reasonable level of the trim loss varies that is between 0,21% and 9,1% for most industry. The experimental results show the efficiency of the packing process with hybrid methodology.

On Performance of Binary Flower Pollination Algorithm for Rectangular Packing Problem

Background: Metaheuristic algorithms are optimization algorithms capable of finding near-optimal solutions for real world problems. Rectangle Packing Problem is a widely used industrial problem in which a number of small rectangles are placed into a large rectangular sheet to maximize the total area usage of the rectangular sheet. Metaheuristics have been widely used to solve the Rectangle Packing Problem. Objective: A recent metaheuristic approach, Binary Flower Pollination Algorithm, has been used to solve for rectangle packing optimization problem and its performance has been assessed. Methods: A heuristic placement strategy has been used for rectangle placement. Then, the Binary Flower Pollination Algorithm searches the optimal placement order and optimal layout. Results: Benchmark datasets have been used for experimentation to test the efficacy of Binary Flower Pollination Algorithm on the basis of utilization factor and number of bins used. The simulation results obtained show that the Binary Flower Pollination Algorithm outperforms in comparison to the other well-known algorithms. Conclusion: BFPA gave superior results and outperformed the existing state-of-the-art algorithms in many instances. Thus, the potential of a new nature based metaheuristic technique has been discovered.

Application of Nature Inspired Algorithms to Optimize Multi-objective Two-Dimensional Rectangle Packing Problem

This paper considers two-dimensional non-guillotine rectangular bin packing problem with multiple objectives in which small rectangular parts are to be arranged optimally on a large rectangular sheet. The optimization of rectangular parts is attained with respect to three objectives involving maximization of (1) utilization factor, minimization of (2) due dates of rectangles and (3) number of cuts. Three nature based metaheuristic algorithms — Cuckoo Search, Bat Algorithm and Flower Pollination Algorithm — have been used to solve the multi-objective packing problem. The purpose of this work is to consider multiple industrial objectives for improving the overall production process and to explore the potential of the recent metaheuristic techniques. Benchmark test data compare the performance of recent approaches with the popular approaches and also of the different objectives used. Different performance metrics analyze the behavior/performance of the proposed technique. Experimental results obtained in this work prove the effectiveness of the recent metaheuristic techniques used. Also, it was observed that considering multiple and independent factors as objectives for the production process does not degrade the overall performance and they do not necessarily conflict with each other.

Atlas refinement with bounded packing efficiency

We present a novel algorithm to refine an input atlas with bounded packing efficiency. Central to this method is the use of the axis-aligned structure that converts the general polygon packing problem to a rectangle packing problem, which is easier to achieve high packing efficiency. Given a parameterized mesh with no flipped triangles, we propose a new angle-driven deformation strategy to transform it into a set of axis-aligned charts, which can be decomposed into rectangles by the motorcycle graph algorithm. Since motorcycle graphs are not unique, we select the one balancing the trade-off between the packing efficiency and chart boundary length, while maintaining bounded packing efficiency. The axis-aligned chart often contains greater distortion than the input, so we try to reduce the distortion while bounding the packing efficiency and retaining bijection. We demonstrate the efficacy of our method on a data set containing over five thousand complex models. For all models, our method is able to produce packed atlases with bounded packing efficiency; for example, when the packing efficiency bound is set to 80%, we elongate the boundary length by an average of 78.7% and increase the distortion by an average of 0.0533%. Compared to state-of-the-art methods, our method is much faster and achieves greater packing efficiency.

Optimal charging strategy for electric vehicles using symbolic–graphic combination principle

To provide reliable technical support for the power grid to accommodate the increasing electric vehicle (EV) penetration, various charging strategies have been studied in the literature. Considering the difficulties of the device-level implementation and the limitations of the current EV charger technology, this study focuses on co-ordinating the charging start time (i.e. the time instant when each EV starts charging). To avoid excessive reduction of the adjustable range of the charging start time, this study applies a symbolic–graphic combination principle to address this problem. This involves formulating the valley-filling charging problem as a rectangle packing problem, which is equivalent to divide the valley-filling problem into two sub-problems: filling the load valley and optimising the valley-filling effect. Furthermore, a Zero-Gap Lowest Horizontal line (ZGLH) algorithm is proposed to fill the load valley without producing any undesirable gaps. In addition, a genetic algorithm (GA) is employed to further optimise the valley-filling effect by regulating the scheduled sequence. The proposed ZGLH algorithm can guarantee the charging start time to be continuously regulated within the controllable range rather than discretely regulated as in other works, and it does not add extra computational overhead. The effectiveness of the optimal charging strategy was quantitatively verified by several simulation cases.

A lower bound for online rectangle packing

We slightly improve the known lower bound on the asymptotic competitive ratio for online bin packing of rectangles. We present a complete proof for the new lower bound, whose value is above 1.91.