Two-dimensional Packing Problem 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%.

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.

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

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.

Hybrid branch-and-price-and-cut algorithm for the two-dimensional vector packing problem with time windows

This paper introduces the two-dimensional vector packing problem with time windows (2DVPPTW). It packs all items into identical bins to minimize the number of used bins. Items are characterized by their weights, volumes, and time windows. Items have different time requirements for delivery in many practical settings. For example, items have their time windows for packing and delivery. The 2DVPPTW is subjected to three constraints: weight, volume, and time windows. We present an integer programming (IP) model for the proposed 2DVPPTW. The IP model is reformulated into the master problem and the subproblem (SP). A hybrid branch-and-price-and-cut (H-BPC) algorithm is developed to solve the 2DVPPTW. On the basis of extensive computational experiments, the proposed H-BPC is significantly effective in comparison with the commercial branch-and-bound/cut solvers, such as CPLEX, and two exact methods. The primary reason for the computational efficiency of the H-BPC is the development of a heuristic algorithm, namely, adaptive large neighborhood search (ALNS), to solve the SP, which is usually computationally expensive. Developing the heuristic algorithm extends the works on the solution to the SPs of bin packing problems and two-dimensional vector packing problems. Two dynamic programming (DP) algorithms are also proposed to solve the SP optimally, namely, the label-setting algorithm and the maximal clique-based DP (MC_DP) algorithm. Moreover, two cooperative schemes are proposed and compared. One is composed of the ALNS and label-setting algorithm, which has a shorter computational time, and the other is composed of the ALNS and MC_DP, which can obtain more optimal solutions. Furthermore, subset-row inequalities, rounded capacity inequalities, multidimensional dual-feasible function, and incompatibility preprocessing policy help to decrease the computational burden dramatically.

The generalized serial-lock scheduling problem on inland waterway: A novel decomposition-based solution framework and efficient heuristic approach

Serial locks with multiple chambers are common on major inland waterways worldwide. For the first time, this study designs a novel heuristic algorithmic structure for the generalized serial lock scheduling problem (GSLSP) from a batch-processing-based flexible job-shop scheduling perspective, which converts the GSLSP into a discrete optimization problem. Subsequently, the overall discrete optimization problem is decomposed into an assignment problem and two subproblems (a two-dimensional packing problem and the shortest path problem). They are solved by a tabu-based adaptive large neighborhood search algorithm (TALNS), a multi-order best-fit algorithm, and the Bellman–Ford algorithm, respectively. Additionally, a solution-space cutting strategy is proposed to accelerate the TALNS, which occupies most of the computational burden. Case studies are implemented based on practical historical data from the two-lock system in Yangtze River and the six-lock system along the Albertkanaal canal. Numerical results demonstrate that high-quality solutions can be obtained for large-scale GSLSPs within a reasonable time by the proposed hybrid algorithmic structure. After that, GSLSPs with simplified settings are solved with the proposed framework, and the computational results are compared with those obtained by state-of-the-art methods in the literature. Extensive numerical experiments and comparisons infer the effectiveness and versatility of the proposed approach for solving different variants of GSLSP and its superiority over existing methods. Furthermore, experimental results on GSLSPs with varying lock configurations show that the throughput capacity of locks in the middle of a serial-lock system has a more significant impact on the overall transportation efficiency of the serial lock system.

Two-dimensional irregular packing problems: A review

Two-dimensional (2D) irregular packing problems are widespread in manufacturing industries such as shipbuilding, metalworking, automotive production, aerospace, clothing and furniture manufacturing. Research on 2D irregular packing problems is essential for improving material utilization and industrial automation. Much research has been conducted on this problem with significant research results and certain algorithms. The work has made important contributions to solving practical problems. This paper reviews recent advances in the domain of 2D irregular packing problems based on a variety of research papers. We first introduce the basic concept and research background of 2D irregular packing problems and then summarize algorithms and strategies that have been proposed for the problems in recent years. Conclusion summarize development trends and research hotspots of typical 2D irregular shape packing problems. We hope that this review could provide guidance for researchers in the field of 2D irregular packing.

Covering, corner-searching and occupying: A three-stage intelligent algorithm for the 2d multishape part packing problem.

The bin packing problem has a wide range of applications in industry. With the upgrade of the task difficulty, the traditional 2d rectangular layout algorithm can no longer meet the needs of modern industry, such as express packing task and exoplanet ore collection task. The express or ore samples come in heterogeneous shapes so they cannot all be treated as rectangular pieces. In this paper, we propose a three-stage method called covering, corner-searching and occupying (C,S&O) to solve the two-dimensional multishape part packing problem. The objective of the packing problem variant is to ensure maximum use of the raw material and minimize the trim loss. The algorithm cannot make use of information about the sequence of future objects that are going to arrive, only knowing the shape and size of the coming one, and the coming part must be packed into the bin immediately after its arrival without buffering or readjusting. The method of C,S&O hybridizes the idea of “gold corners, silver edges and grass belly” in the Chinese game Go and the method of finding picture corners in machine vision. In the first stage, the rectangular bin and the coming part are transformed into matrix representation, and generating the position matrix that indicates possible ways of packing the part into the bin. In the second stage, the suitable layout position of the coming part is obtained using machine vision image processing technology for reference. The third stage is calculating the environment matching degree to determine the current optimal placement orientation. In order to facilitate the display of the simulation results, only three shapes of parts are considered in the simulation, rectangle, circle and triangle. The experimental results show the effectiveness of this method. Consulting the literature, it is found that this paper is the first to propose a layout method for multishape manufacturing parts.

Minimizing makespan on parallel batch processing machines with two-dimensional rectangular jobs

This paper addresses the parallel batch processing machines scheduling problem with two-dimensional bin packing constraints. One typical application of this problem is last-mile truck delivery. A set of rectangular items (or jobs) that have delivery times are processed by a set of homogeneous vehicles (or machines). Each vehicle can deliver a batch of items at a trip following a fixed route as long as these items in the batch can be placed onto the vehicle without overlapping. The total delivery time of a batch is determined by the longest delivery time of the items in the batch. The problem is to allocate the items, which may be rotated, into batches and then assign the batches to the vehicles so as to minimize the makespan. A mixed integer programming model of the problem is formulated. A number of heuristic algorithms are proposed and compared. Some heuristics for two-dimensional packing problems are adapted to the problem. The performance of the algorithms is evaluated based on computational experiments. The experimental results show that the best-fit maximal rectangle (BFMR) heuristic is the best, in which the batch with the smallest remaining capacity is selected for an item, the empty maximal rectangle with the least space in the batch is chosen to place the item, and finally the generated batches are allocated to the vehicles according to the longest batch delivery time algorithm.

A cutting plane method and a parallel algorithm for packing rectangles in a circular container

We study a two-dimensional packing problem where rectangular items are placed into a circular container to maximize either the number or the total area of items packed. We adapt a mixed-integer linear programming model from the case with a rectangular container and design a cutting plane method to solve this problem by adding linear cuts to forbid items from being placed outside the circle. We show that this linear model allows us to prove optimality for instances larger than those solved using the state-of-the-art non-linear model for the same problem. We also propose a simple parallel algorithm that efficiently enumerates all non-dominated subsets of items and verifies whether pertinent subsets fit into the container using an adapted version of our linear model. Computational experiments using large benchmark instances attest that this enumerative algorithm generally provides better solutions than the best heuristics from the literature when maximizing the number of items packed. Instances with up to 30 items are now solved to optimality, against the eight-item instance previously solved.

A biased genetic algorithm hybridized with VNS for the two-dimensional knapsack packing problem with defects

This paper addresses a two-dimensional knapsack packing problem which packing a set of rectangles into a rectangular board to maximize the total value of the rectangles packed. The rectangle has finite types while its quantity is unlimited, and the board has unusable defects. A biased genetic algorithm hybridized with variable neighborhood search (VNS) is proposed to solve the problem as genetic algorithm can effectively solve the combinational optimization problem, has good searching performance, and is easy to implement. We adopt the replacing strategy for increasing the diversity of the population and avoiding converging too early. An improved placement procedure in charge of producing the layout is presented and four neighborhood structures are constructed. We conduct lots of numerical experiments using 5414 benchmark instances taken from the literature for evaluating our approach and comparing it to other excellent approaches. The experimental results show that the proposed algorithm gets many new best solutions in these benchmark instances and is very competitive.

Research on the Cutting of Rectangular Blanks to Solve the Two-Dimensional Problem

This paper introduces the research background of the two-dimensional packing problem, classifies the rectangular two-dimensional packing problem, then analyses the pattern scheme of the classic test questions, and finally, studies the three factors that affect the utilization rate of the plate.

An exact algorithm for two-dimensional vector packing problem with volumetric weight and general costs

The volumetric weight of a package has become an essential factor in calculating the delivery cost of shipments in the international logistics market. In this work, we extend the two-dimensional vector packing problem by considering a more realistic cost structure, which is a general function of volumetric weight. The problem is to pack a set of different items into some identical bins without violating weight limits and volume capacities so that the total delivery cost is minimized. We develop an exact approach based on a branch-and-price algorithm and subset-row inequalities for the problem. To efficiently solve the pricing problem in column generation, a label-setting algorithm with an effective label dominance rule and a bounding procedure is presented. A stronger label dominance rule is derived for the case where the cost function is convex. The computational results show that the exact method is effective in solving the various test instances of the problem. If the volumetric weight is not considered, the exact method can be adapted to solve the two-dimensional vector packing problem with piecewise linear cost function and outperformed the existing exact algorithm by computing 27 optimal solutions for previously open instances.

Bin packing with lexicographic objectives for loading weight- and volume-constrained trucks in a direct-shipping system

We consider a packing problem that arises in a direct-shipping system in the food and beverage industry: Trucks are the containers, and products to be distributed are the items. The packing is constrained by two independent quantities, weight (e.g., measured in kg) and volume (number of pallets). Additionally, the products are grouped into the three categories: standard, cooled, and frozen (the latter two require refrigerated trucks). Products of different categories can be transported in one truck using separated zones, but the cost of a truck depends on the transported product categories. Moreover, splitting orders of a product should be avoided so that (un-)loading is simplified. As a result, we seek for a feasible packing optimizing the following objective functions in a strictly lexicographic sense: minimize the (1) total number of trucks; (2) number of refrigerated trucks; (3) number of refrigerated trucks which contain frozen products; (4) number of refrigerated trucks which also transport standard products; (5) and minimize splitting. This is a real-world application of a bin-packing problem with cardinality constraints a.k.a. the two-dimensional vector packing problem with additional constraints. We provide a heuristic and an exact solution approach. The heuristic meta-scheme considers the multi-compartment and item fragmentation features of the problem and applies various problem-specific heuristics. The exact solution algorithm covering all five stages is based on branch-and-price using stabilization techniques exploiting dual-optimal inequalities. Computational results on real-world and difficult self-generated instances prove the applicability of our approach.

An Improved Adaptive Genetic Algorithm for Two-Dimensional Rectangular Packing Problem

This paper proposes the hybrid adaptive genetic algorithm (HAGA) as an improved method for solving the NP-hard two-dimensional rectangular packing problem to maximize the filling rate of a rectangular sheet. The packing sequence and rotation state are encoded in a two-stage approach, and the initial population is constructed from random generation by a combination of sorting rules. After using the sort-based method as an improved selection operator for the hybrid adaptive genetic algorithm, the crossover probability and mutation probability are adjusted adaptively according to the joint action of individual fitness from the local perspective and the global perspective of population evolution. The approach not only can obtain differential performance for individuals but also deals with the impact of dynamic changes on population evolution to quickly find a further improved solution. The heuristic placement algorithm decodes the rectangular packing sequence and addresses the two-dimensional rectangular packing problem through continuous iterative optimization. The computational results of a wide range of benchmark instances from zero-waste to non-zero-waste problems show that the HAGA outperforms those of two adaptive genetic algorithms from the related literature. Compared with some recent algorithms, this algorithm, which can be increased by up to 1.6604% for the average filling rate, has great significance for improving the quality of work in fields such as packing and cutting.

Sequence Transfer-Based Particle Swarm Optimization Algorithm for Irregular Packing Problems

The two-dimensional (2D) irregular packing problem is a classical optimization problem with NP-hard characteristics and has high computational complexity. To date, packing problems have generally been solved by artificial experience and heuristic algorithms. However, these algorithms are not highly efficient and the excellent cases cannot be preserved, which both time and economic costs are high. Inspire by transfer learning and considering the characteristics of 2D irregular packing problems, we propose a sequence transfer-based particle swarm optimization algorithm (ST-PSO) to solve the multi-constraint packing problem. A piece-matching strategy based on an improved shape context algorithm, and a piece-sequencing generation strategy for transferring the packing sequence are developed for particle swarm optimization(PSO) initialization. In the process of PSO, an adaptive adjustment strategy is used with an improved positioning strategy to adjust the packing position of the pieces. The results indicate that this method can robustly, quickly, and efficiently achieve the packing of 2D irregular pieces. Compared with the data prior to transfer, the ST-PSO can inherit and transfer the historical packing sequence in less time and retain or exceed the actual packing data onto the samples. This algorithm could be applied to industrial applications to reduce waste, packing time, and production costs.

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.

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.