Two-dimensional Packing Research Articles

Research on irregular sponge layout system based on heuristic algorithms

Abstract A layout optimization method that integrates No-Fit Polygons (NFP) positioning and a genetic algorithm is proposed for the two-dimensional irregular sponge packing problem. Firstly, according to the geometric characteristics of the parts, they are classified and encoded to calculate the NFP between part classes, avoiding redundant NFP calculations. Simultaneously, heuristic rules aimed at minimizing the packing height are introduced to provide a comprehensive evaluation function for part positioning strategies, thus improving packing efficiency. Secondly, in the optimization of part sorting, a genetic algorithm is introduced to optimize the sponge packing algorithm. Finally, several standard cases provided by the European Special Interest Group on Cutting and Packing (ESICUP) are used to test the proposed packing method. The results show that compared to the traditional Bottom-Left algorithm, the computational efficiency of the proposed packing method is improved by 4.20% to 22.73%.

Task scheduling for control system based on deep reinforcement learning

We investigate the control system’s computational task scheduling problem within limited time and with limited CPU cores in the cloud server. We employ a neural network model to estimate the runtime consumption of linear quadratic regulators (LQR) under varying numbers of CPU cores. Building upon this, we model the task scheduling problem as a two-dimensional bin packing problem (2D BPP) and formulate the BPP as a Markov Decision Process (MDP). By studying the characteristics of the MDP, we simplify the action space, design an efficient reward function, and propose a Double DQN-based algorithm with a simplified action space. Experimental results demonstrate that the proposed approach has improved training efficiency and learning performance compared to other packing algorithms, effectively addressing the challenges of task scheduling in the context of the control system.

Improved Best Fit Heuristics for Offline Not Oriented Two-dimensional Rectangular Strip Packing Problem

Improved Best Fit Heuristics for Offline Not Oriented Two-dimensional Rectangular Strip Packing Problem

Optimizing Two-Dimensional Irregular Pattern Packing with Advanced Overlap Optimization Techniques

This research introduces the Iterative Overlap Optimization Placement (IOOP) method, a novel approach designed to enhance the efficiency of irregular pattern packing by dynamically optimizing overlap ratios and pattern placements. Utilizing a modified genetic algorithm, IOOP addresses the complexities of arranging irregular patterns in a given space, focusing on improving spatial and material efficiency. This study demonstrates the method’s superiority over the traditional Size-First Non-Iterative Overlap Optimization Placement technique through comparative analysis, highlighting significant improvements in spatial utilization, flexibility, and material conservation. The effectiveness of IOOP is further validated by its robustness in handling diverse pattern groups and its adaptability in adjusting pattern placements iteratively. This research not only showcases the potential of IOOP in manufacturing and design processes requiring precise spatial planning but also opens avenues for its application across various industries, underscoring the need for further exploration into advanced technological integrations for tackling complex spatial optimization challenges.

The Normalized Direct Trigonometry Model for the Two-Dimensional Irregular Strip Packing Problem

Background: The Irregular Strip Packing Problem (ISPP) involves packing a set of irregularly shaped items within a strip while minimizing its length. Methods: This study introduces the Normalized Direct Trigonometry Model (NDTM), an innovative enhancement of the Direct Trigonometry Model (DTM). The NDTM incorporates a distance function that supports the integration of the separation constraint, which mandates a minimum separation distance between items. Additionally, the paper proposes a new set of constraints based on the bounding boxes of the pieces aimed at improving the non-overlapping condition. Results: Comparative computational experiments were performed using a comprehensive set of 90 instances. Results show that the NDTM finds more feasible and optimal solutions than the DTM. While the NDTM allows for the implementation of the separation constraint, the number of feasible and optimal solutions tends to decrease as more separation among the items is considered, despite not increasing the number of variables or constraints. Conclusions: The NDTM outperforms the DTM. Moreover, the results indicate that the new set of non-overlapping constraints facilitates the exploration of feasible solutions at the expense of optimality in some cases.

Models for two-dimensional bin packing problems with customer order spread

Models for two-dimensional bin packing problems with customer order spread

Structural color in the bacterial domain: The ecogenomics of a 2-dimensional optical phenotype

Structural color is an optical phenomenon resulting from light interacting with nanostructured materials. Although structural color (SC) is widespread in the tree of life, the underlying genetics and genomics are not well understood. Here, we collected and sequenced a set of 87 structurally colored bacterial isolates and 30 related strains lacking SC. Optical analysis of colonies indicated that diverse bacteria from at least two different phyla (Bacteroidetes and Proteobacteria) can create two-dimensional packing of cells capable of producing SC. A pan-genome-wide association approach was used to identify genes associated with SC. The biosynthesis of uroporphyrin and pterins, as well as carbohydrate utilization and metabolism, was found to be involved. Using this information, we constructed a classifier to predict SC directly from bacterial genome sequences and validated it by cultivating and scoring 100 strains that were not part of the training set. We predicted that SCr is widely distributed within gram-negative bacteria. Analysis of over 13,000 assembled metagenomes suggested that SC is nearly absent from most habitats associated with multicellular organisms except macroalgae and is abundant in marine waters and surface/air interfaces. This work provides a large-scale ecogenomics view of SC in bacteria and identifies microbial pathways and evolutionary relationships that underlie this optical phenomenon.

Scheduling for additive manufacturing with two-dimensional packing and incompatible items

Additive manufacturing technology can enable low-cost, efficient production of low-demand, highly-complex customized items with reduced lead time if production is judiciously planned. This paper addresses the assignment of requested items into batches and the scheduling of the batches onto 3D printers. The objective is to minimize the manufacturing makespan while satisfying items’ compatibility, two-dimensional no-overlap and containment packing constraints within a batch, and machine’s capability to manufacture a batch. The problem is modeled as a mixed integer linear program (MIP) that solves instances up to 100 items. For hard and large instances, this paper proposes a matheuristic that fathoms packings using a step-wise check procedure. Computational results reveal that the proposed heuristic improves the makespan of MIP solutions of hard instances by 12% on average, with improvements reaching up to 72% for instances with 150 items. They further show that the proposed heuristic finds the best makespan for 88% of all cases. Finally, we provide useful managerial insights for production flexibility and scheduling policies.

Combinatorial Benders decomposition for single machine scheduling in additive manufacturing with two-dimensional packing constraints

This paper addresses a single-machine scheduling problem in additive manufacturing (AM). We focus on the direct metal laser sintering (DMLS) technology of an AM process, where parts can be produced simultaneously in a batch. In this problem, we consider simultaneously the assignment of parts to batches and the packing of parts onto two-dimensional surfaces of the build chamber to minimize the makespan objective. To solve this problem, a mixed integer linear programming model is formulated, and an approximation algorithm is proposed, followed by an exact algorithm based on combinatorial Benders decomposition method with various Benders cuts and acceleration strategies (Algorithm CBD). Finally, we conduct a comprehensive computational study to verify the efficiency of our proposed Algorithm CBD. The computational results show that our proposed Algorithm CBD can effectively solve the test instances with different sizes.

Framework for $\exists \mathbb{R}$-Completeness of Two-Dimensional Packing Problems

The aim in packing problems is to decide if a given set of pieces can be placed inside a given container. A packing problem is defined by the types of pieces and containers to be handled, and the motions that are allowed to move the pieces. The pieces must be placed so that in the resulting placement, they are pairwise interior-disjoint. We establish a framework which enables us to show that for many combinations of allowed pieces, containers and motions, the resulting problem is $\exists \mathbb{R}$-complete. This means that the problem is equivalent (under polynomial time reductions) to deciding whether a given system of polynomial equations and inequalities with integer coefficients has a real solution. We consider packing problems where only translations are allowed as the motions, and problems where arbitrary rigid motions are allowed, i.e., both translations and rotations. When rotations are allowed, we show that it is an $\exists \mathbb{R}$-complete problem to decide if a set of convex polygons, each of which has at most $7$ corners, can be packed into a square. Restricted to translations, we show that the following problems are $\exists \mathbb{R}$-complete: (i) pieces bounded by segments and hyperbolic curves to be packed in a square, and (ii) convex polygons to be packed in a container bounded by segments and hyperbolic curves.

Combinatorial Benders' decomposition for the constrained two-dimensional non-guillotine cutting problem with defects

This paper studies the constrained two-dimensional non-guillotine cutting problem with defects, in which a set of items of a specific size is cut from a large rectangular sheet with defective areas, with the number of each type of cut item cannot exceed a specified quantity. The objective is to maximise the total value of the cut items. We propose a decomposition approach to address the problem. The process involves decomposing the original problem into a master problem and a subproblem. The master problem is formulated as a one-dimensional contiguous bin packing problem, while the subproblem is an x-Check problem to identify a two-dimensional packing that does not lead to any overlap. The x-Check problem is effectively addressed by using an integer linear programming model. When the x-Check problem proves infeasible, cuts are added to the master problem, and the iteration is repeated until the x-Check finds a feasible solution. Furthermore, we introduce several novel techniques, including valid inequalities, preprocessing techniques, and lifting the cut methods to improve the performance of the algorithm. Extensive computational results show that our method can quickly find the optimal solution for the 5450 instances in the literature.

High Open-Circuit Voltage Organic Solar Cells with 19.2% Efficiency Enabled by Synergistic Side-Chain Engineering.

Restricted by the energy-gap law, state-of-the-art organic solar cells (OSCs) exhibit relatively low open-circuit voltage (VOC) because of large nonradiative energy losses (ΔEnonrad). Moreover, the trade-off between VOC and external quantum efficiency (EQE) of OSCs is more distinctive; the power conversion efficiencies (PCEs) of OSCs are still <15% with VOCs of >1.0V. Herein, the electronic properties and aggregation behaviors of non-fullerene acceptors (NFAs) are carefully considered and then a new NFA (Z19) is delicately designed by simultaneously introducing alkoxy and phenyl-substituted alkyl chains to the conjugated backbone. Z19 exhibits a hypochromatic-shifted absorption spectrum, high-lying lowest unoccupied molecular orbital energy level and ordered 2D packing mode. The D18:Z19-based blend film exhibits favorable phase separation with face-on dominated molecular orientation, facilitating charge transport properties. Consequently, D18:Z19 binary devices afford an exciting PCE of 19.2% with a high VOC of 1.002V, surpassing Y6-2O-based devices. The former is the highest PCE reported to date for OSCs with VOCs of >1.0V. Moreover, theΔEnonrad of Z19- (0.200eV) and Y6-2O-based (0.155eV) devices are lower than that of Y6-based (0.239eV) devices. Indications are that the design of such NFA, considering the energy-gap law, could promote a new breakthrough in OSCs.

Percolation connectivity in deposits obtained usingcompetitive random sequential adsorption of binarydisk mixtures

Connectedness percolation phenomena in the two-dimensional (2D) packing of binary mixtures of disks with different diameters were studied numerically. The packings were produced using random sequential adsorption (RSA) model with simultaneous deposition of disks. The ratio of the particle diameters was varied within the range D=1-10, and the selection probability of the small disks was varied within the range 0-1. A core-shell structure of the particles was assumed for the analysis of connectivity. The packing coverages in a jamming state for different components, connectivities through small, large and both types of disks, the behavior of electrical conductivity were analyzed. The observed complex effects were explained accounting for the formation of conductive "bridges" from small disks in pores between large disks.

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.

An Iterative Compression Method for the Two-Dimensional Irregular Packing Problem With Lead Lines

An Iterative Compression Method for the Two-Dimensional Irregular Packing Problem With Lead Lines

Solving a 3D bin packing problem with stacking constraints

We consider a novel variant of the three-dimensional bin packing problem (3DBPP) arising from a maritime shipping port. Unlike the classical bin packing problem, our 3DBPP requires all items to be stacked vertically during the packing process. To tackle this problem effectively, we divide it into two subproblems: the stack packing problem (SPP) and the two-dimensional bin packing problem (2DBPP). We first formulate the SPP as an integer programming model to minimize the total bottom area of the stacks, which is further solved by a branch-and-price method. Then, in the 2DBPP, to pack all stacks into a minimum number of bins, we follow the left-down most extreme point placement strategy. An encoder–decoder model was developed to optimize the packing sequence and orientation of the stacks simultaneously. Industrial instances from shipping ports were collected and extended. The computational results show that our two-stage algorithm performs well in solving the stacking-constrained 3DBPP.

Optimizing Two-Dimensional Irregular Packing: A Hybrid Approach of Genetic Algorithm and Linear Programming

The problem of two-dimensional irregular packing involves the arrangement of objects with diverse shapes and sizes within a given area. This challenge arises across various industrial sectors, where effective packing optimization can yield cost savings, enhanced productivity, and reduced material waste. Existing methods for addressing the two-dimensional irregular packing problem encounter several challenges, such as limited computing resources, a prolonged solving time, and the propensity to converge to local optima. To address this issue, this study proposes a hybrid algorithm called the GA-LP algorithm to optimize the two-dimensional irregular packing problem in the manufacturing industry. The algorithm combines the global search capability of a genetic algorithm with the precise solving characteristics of linear programming. Matheuristics merges the advantages of metaheuristics, such as genetic algorithms, and mathematical programming, such as linear programming. The algorithm employs the no-fit-polygon technique along with the bottom-left and lowest-gravity center mixing placement strategies to acquire an initial solution via the utilization of a genetic algorithm. The algorithm then optimizes the solution obtained by the genetic algorithm using linear programming to obtain the final packing result. Experimental results, drawn from a real case involving the European Special Interest Group on Cutting and Packing (ESICUP) demonstrate that the GA-LP algorithm outperforms two hybrid algorithms from the relevant literature. Compared with recent methods, this algorithm can increase the best and average utilization rates by up to 5.89% and 4.02%, respectively, with important implications for improving work quality in areas such as packing and cutting.

An iteratively doubling binary search for the two-dimensional irregular multiple-size bin packing problem raised in the steel industry

This paper examines the two-dimensional irregular multiple-size bin packing problem, where the goal is to pack all the given irregular pieces into bins of various sizes such that the total area of the used bins is minimized. Meanwhile, the irregular pieces include holes, and the bins can be irregular, also. This problem is raised in the steel industry considering the reuse of leftover material. An iteratively doubling binary search is proposed to solve this problem. Moreover, a binary search strategy is introduced to search the bin combination with minimum area, and an iteratively doubling strategy is utilized to control the search effort on each bin combination. Once the bins are identified, a first-fit bottom-left method is utilized to generate the initial position for each piece. An overlap minimization approach, which includes a local search by exchanging the positions of two pieces, is adapted to minimize the overlap in the initial solution. Experiment results on existing instances show that our approach could find a better solution than existing methods. Several instances representing different application scenarios are generated, and the results show our approach’s effectiveness and generality.

Unrelated parallel batch processing machine scheduling with time requirements and two-dimensional packing constraints

In various complex manufacturing systems such as wafer fabrication, ceramic processes, and additive manufacturing, a single machine often performs multiple distinct jobs simultaneously, requiring non-overlapping positions in the 2D-plane. As the demand for manufacturing with two-dimensional (2D) packing constraints continues to rise, there is an urgent need to address the integrated scheduling of batch processing machines (BPMs) and 2D packing. However, only a limited number of studies have explored BPM scheduling problems with 2D packing constraints. In this study, we focus on the scheduling of unrelated parallel BPMs to process jobs with non-identical sizes, unequal release dates, and due dates. To model the machine’s capacity, we move beyond the traditional one-dimensional knapsack approach and consider it as a 2D rectangle, taking into account the 2D packing constraints. It is crucial to ensure that no overlapping or stacking occurs within the machine’s capacity under these constraints. To address this problem, we propose a mixed integer linear programming model as the initial approach. Additionally, we develop an adaptive large neighborhood search (ALNS) heuristic specifically designed for handling large instances. The ALNS incorporates eleven removal operators and seven insertion operators, offering a comprehensive exploration and improvement of solutions. To tackle the packing procedure, we adopt two distinct packing patterns, namely Skyline and Open Space. These patterns play a significant role in optimizing the packing of jobs within the machine’s capacity. Furthermore, we conduct extensive computational experiments, generating 48 categories of instances with varying job attributes. These experiments serve to compare the performance of the two packing patterns and validate the effectiveness of our ALNS approach. In conclusion, our research addresses the critical gap in BPM scheduling problems with 2D packing constraints. By introducing a mixed integer linear programming model and developing the ALNS heuristic with diverse operators, coupled with the adoption of different packing patterns, we present a comprehensive approach to tackle this complex problem. Computational experiments are conducted to compare packing patterns and verify the performance of our ALNS method.