Rectangular Packing Problem Research Articles

A Genetic Algorithm with Lower Neighborhood Search for the Three-Dimensional Multiorder Open-Size Rectangular Packing Problem

This paper addresses the multiorder open-dimension three-dimensional rectangular packing problem (3D-MOSB-ODRPP), which involves packing rectangular items from multiple orders into a single, size-adjustable container. We propose a novel metaheuristic approach combining a genetic algorithm with the Gurobi solver. The algorithm incorporates a lower neighborhood search strategy and is underpinned by a mathematical model representing the multiorder open-dimension packing scenario. Extensive experiments validate the effectiveness of the proposed approach. The LNSGA algorithm outperforms Gurobi and the traditional genetic algorithm in solution quality and computational efficiency. For small-scale instances, LNSGA achieves optimal values in most cases. LNSGA demonstrates significant optimization improvements over Gurobi and the genetic algorithm for large-scale instances. The superior performance is attributed to the effective integration of the lower neighborhood search mechanism and the Gurobi solver. This study offers valuable insights for optimizing the packing process in e-commerce warehousing and logistics operations.

Solving the three-dimensional open-dimension rectangular packing problem: A constraint programming model

This paper addresses the three-dimensional open-dimension rectangular packing problem (3D-ODRPP). This problem addresses a set of rectangular boxes of given dimensions and a rectangular container of open dimensions. The objective is to pack all boxes orthogonally into the container while minimizing the container volume. Real-world applications of the 3D-ODRPP arise in production systems with operations of shipping or moving. The literature has presented mainly mixed-integer programming (MIP) formulations and their linearization techniques for the problem allied with general-purpose optimization solvers. To model and solve the 3D-ODRPP, we propose a constraint programming model based on a position-free modeling approach with logic operators. We ran computational experiments to assess the performance of the proposed model compared to the benchmark MIP models from instances of the literature. The results show our approach is competitive in different sets of problem instances in terms of reaching optimality as well as providing satisfactory feasible solutions quickly.

An efficient constructive heuristic for the rectangular packing problem with rotations.

The rectangular packing problem has been extensively studied over the years due to its wide application in industry. However, most of the research efforts are devoted to positioning techniques of the rectangles for various problem variants, the efficient implementation of the packing procedure is relatively less studied. In this paper, we propose an efficient constructive algorithm for the rectangular packing problem with rotations. We design a preprocess procedure with four data structures to store the information used for item selection. The gaps on the skyline are categorized into three types according to their associated edges for the placement procedure, during which the item is searched and packed in a descending order of the fitness value. The entire constructive phase takes a time complexity of O(nlogn). For the packing improvement phase, we optimize the packing through random perturbation on the sequence and orientation of the item. Three classes of stochastic problems are generated ranging from small-scale to extra-large-scale, the recorded running time confirms the efficiency of the proposed algorithm. We also test the proposed algorithm on the benchmark problem C21, N13, NT, Babu and CX, the computational results show that it delivers a good performance.

Optimizing e-commerce warehousing through open dimension management in a three-dimensional bin packing system.

In the field of e-commerce warehousing, maximizing the utilization of packing bins is a fundamental goal for all major logistics enterprises. However, determining the appropriate size of packing bins poses a practical challenge for many logistics companies. Given the limited research on the open-size 3D bin packing problem as well as the high complexity and lengthy computation time of existing models, this study focuses on optimizing multiple-bin sizes within the e-commerce context. Building upon existing research, we propose a hybrid integer programming model, denoted as the three dimensional multiple option dimensional rectangular packing problem (3D-MODRPP), to address the multiple-bin size 3D bin packing problem. Additionally, we leverage well-established hardware and software technologies to propose a 3D bin packing system capable of accommodating multiple bin types with open dimensions. To reduce the complexity of the model and the number of constraints, we introduce a novel assumption method for 0-1 integer variables in the overlap and rotation constraints. By applying this approach, we significantly streamline the computational complexity associated with the model calculations. Furthermore, we refine the dataset by developing a customized version based on the classical Three-Dimensional One-Size Dependent Rectangular Packing Problem (3D-ODRPP) dataset, leading to improved outcomes. Through comprehensive analysis of the research results, our model exhibits remarkable advancements in addressing the strong heterogeneous bin packing problem, the weak heterogeneous bin packing problem, the actual bin packing problem, and the bin packing problem with multiple bin types and open sizes. Specifically, it significantly reduces model complexity and computation time and increases space utilization. The system designed in this study paves the way for practical applications based on the proposed model, providing researchers with broader research prospects and directions to expand the scope of investigation in the field of 3D bin packing. Consequently, this system contributes to solving complex 3D packing problems, reducing space waste, and enhancing transportation efficiency.

Multi-objective steel plate cutting optimization problem based on real number coding genetic algorithm

The rectangular packing problem is an NP-complete combinatorial optimization problem. This problem occurs widely in social production scenarios, with steel plate cutting being one example. The cutting scheme for the rectangular packing problem needs to be improved because, without the globally optimal solution, there are many unnecessary edges in the steel cutting process. Based on a practical roll-fed disc shearing steel plate optimization problem, this paper explores a generalized packing method for rectangles of special dimensions and abstractly condenses complex quantitative relationships to establish a multi-objective mixed-integer nonlinear programming model. An innovative algorithm design based on a genetic algorithm is established to plan the cutting scheme in a high-speed and efficient way. The outcome is a utilization rate of up to 92.73% for raw materials and a significant reduction in labor, providing a guide for practical production and processing tasks. The advantages and disadvantages of the model and algorithm are discussed, and it is concluded that this rectangular packing method has strong universality and generalization ability, allowing rectangular packing tasks with large data volumes to be completed within a short time.

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 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.

An improved priority heuristic for the fixed guillotine rectangular packing problem

An improved priority heuristic (IPH) is presented for the guillotine rectangular packing problem with a fixed orientation constraint. This algorithm can continuously divide the remaining space into blocks and recursively layout the entire space by filling each current block. IPH inherits the placement method of PH and adopts an improved space partitioning rule. The partitioning rule comprehensively considers the effects of the newly generated block on subsequent item placement. Thus, the remaining space can be filled as much as possible from a probability perspective. Given the introduction and utilization of different solving parameters, the worst-case time complexity is O(40n 2). Computational results on benchmark datasets show that IPH outperforms existing heuristics on average. For large-scale instances, several perfect layouts can be obtained for the first time.

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.

The Orthogonal Packing and Scheduling Problem: Model, Heuristic, and Benchmark

This paper addresses a new orthogonal packing and time scheduling problem—3D space-time optimization problem (3D-STO). Given a rectangular sheet and a set of rectangular items, each item needs to be continuously processed with a time length on the sheet, the 3D-STO consists in arranging each item’s loading time, location, and orientation for a certain period such that the total utilization time of the sheet, i.e., the makespan of the schedule, is minimized. The 3D-STO involves a series of 2D rectangular packing problem (2D-RPP) for the whole scheduling period. If the processing time of each item is simply regarded as a space dimension for the packing, then the 3D-STO can be reduced to an NP-hard packing problem—3D strip packing problem (3D-SPP). The 3D-STO differs from the 3D-SPP in that the position and orientation of items can be changed over the processing time (the third dimension) such that the 3D-STO has a much larger search space compared with the 3D-SPP. Hereby, the optimal solution of the 3D-STO is better than or equal to the optimal solution of the 3D-SPP, and a possibly better solution could be found for the 3D-STO model. As a new model, there is no algorithm or benchmark in the literature. Moreover, the 2D-RPP and 3D-SPP algorithms are not suitable for solving the 3D-STO which considers the packing and scheduling simultaneously. Hereby, we propose a caving-degree-based scheduling algorithm (CDS) for the 3D-STO. This is the first proposed 3D-STO algorithm. We also formalize the problem as a mixed integer programming model and solve it by ILOG CPLEX. For evaluation, we provide a synthesized method to generate a total of 195 various benchmark instances with guillotine and nonguillotine cut constraints. The comparative results show that CDS is more effective than the CPLEX solver for the 3D-STO. Also, when comparing CDS for the 3D-STO and the adapted CDS for the 3D-SPP, we see that the new 3D-STO model can enhance the flexibility of the item arrangement and make maximum utilization of the sheet and time.

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.

A product arrangement optimization method to reduce packaging environmental impacts

The Three-dimensional Open Dimension Rectangular Packing Problem (3D-ODRPP) is one of the most important optimization problems arise in reducing waste and shipping cost of packing and shipping industries. The 3D-ODRPP aims at seeking the length, width and height of a rectangular box of minimal volume that can pack a given rectangular products. Most 3D-ODRPP models in the literature use too many extra binary variables and don’t consider the equilibrium of product placement. This study presents an improved mathematical model for the 3D-ODRPP. The proposed model uses fewer decision variables and constraints to define product orientation. It also adopts the full base support (FBS) constraint as a product supporting condition to guarantee packaging stability and avoid infeasible arrangements. Literature instance tests show the improvement in packaging stability of the proposed method compared with existing methods. Industrial benchmark demonstrates that, by solving the 3D-ODRPP, the proposed method can reduce package volume and create economic and ecologic gains.

An efficient deterministic heuristic algorithm for the rectangular packing problem

This paper presents a deterministic heuristic algorithm for solving the NP-hard two-dimensional rectangular packing problem with the objective of maximizing the filling rate of a rectangular sheet. The key component of the proposed algorithm is a best-fit constructive procedure, according to which, the rectangles are packed into the sheet one by one and each rectangle is packed into the sheet by an angle-occupying placement with maximum fit degree. To further improve the algorithm’s searching ability, a look-ahead strategy and a multistart method are introduced. The proposed algorithm is evaluated on five sets of 112 well-known test instances, and the computational results disclose that the proposed algorithm is competitive with the current state-of-the-art algorithms. The effects of the essential components of the proposed algorithm are investigated by a series of experimental analysis. Additionally, we adapt the proposed packing strategy to solve a variant of 2DRP, the constrained two-dimensional cutting (or packing) (CTDC) problem. Computational experiments on 21 classical CTDC problem instances and comparisons with two state-of-the-art algorithms verifies the effectiveness and efficiency of the adapted algorithm.

Integrated Optimization of Mixed Cargo Packing and Cargo Location Assignment in Automated Storage and Retrieval Systems

To improve the delivery efficiency of automated storage and retrieval system, the problem of the integrated optimization of mixed cargo packing and cargo location assignment is addressed. An integrated optimization model of mixed cargo packing and location assignments with the shortest time for the stacker in a certain historical period is established and is transformed into a conditional packing problem. An improved hybrid genetic algorithm based on a group coding method is designed to solve the problem. When the initial population is generated, a new heuristic algorithm is designed to improve the convergence speed of the genetic algorithm considering the correlation and frequency of the goods outbound. A heuristic algorithm for a two-dimensional rectangular-packing problem is designed to determine whether a variety of goods can be mixed in packing. Taking actual data from an automated storage and retrieval system for an aviation food company as an example, the established model and design algorithm are verified and the influence of changes in the outbound delivery orders on the optimization result is analyzed. The results show that compared to the method of separate storage of goods based on cube-per-order index rules and a phased optimization method of mixed storage of goods, an integrated optimization method of mixed cargo packing and location assignment can improve the outbound delivery efficiency of the stacking machine by 11.43–25.98% and 1.73–5.51%, respectively, and reduce the cargo location used by 50–55% and 0–10%, respectively. The stronger the correlation of the goods leaving a warehouse, the greater the potential of the design method in this paper to improve the efficiency of the stacker.

An efficient heuristic algorithm for two-dimensional rectangular packing problem with central rectangle

<p style='text-indent:20px;'>This paper presents a heuristic algorithm for solving a specific NP-hard 2D rectangular packing problem in which a rectangle called central rectangle is required to be placed in the center of the final layout, and the aspect ratio of the container is also required to be in a given range. The key component of the proposed algorithm is a greedy constructive procedure, according to which, the rectangles are packed into the container one by one and each rectangle is packed into the container by an angle-occupying placement with maximum fit degree. The proposed algorithm is evaluated on two groups of 35 well-known benchmark instances. Computational results disclose that the proposed algorithm outperforms the previous algorithm for the packing problem. For the first group of test instances, solutions with average filling rate 99.31% can be obtained; for the real-world layout problem in the second group, the filling rate of the solution is 94.75%.

Global optimization for the three-dimensional open-dimension rectangular packing problem

ABSTRACTThis article addresses the three-dimensional open-dimension rectangular packing problem (3D-ODRPP), which aims to pack a given set of unequal-size rectangular boxes within an enveloping rectangular space such that the volume of the occupied space is minimized. Even though the studied 3D-ODRPP is NP hard, the development of sophisticated global optimization methods has been stimulated. The mathematical programming formulation for the 3D-ODRPP has evolved into an effective and efficient mixed-integer linear programming (MILP) model. This study proposes an advanced exact scheme yielding a guaranteed global optimal solution given that all the instance data are non-negative rational numbers. The developed MILP retains not only fewer variables but also fewer constraints than the state-of-the-art models. The superior effectiveness and efficiency of the developed scheme are demonstrated with numerical experiments, where two sets of benchmark instances from references, real-world instances and instances with rational data are included.

An iterative bidirectional heuristic placement algorithm for solving the two-dimensional knapsack packing problem

ABSTRACTThis article presents an efficient heuristic placement algorithm, namely, a bidirectional heuristic placement, for solving the two-dimensional rectangular knapsack packing problem. The heuristic demonstrates ways to maximize space utilization by fitting the appropriate rectangle from both sides of the wall of the current residual space layer by layer. The iterative local search along with a shift strategy is developed and applied to the heuristic to balance the exploitation and exploration tasks in the solution space without the tuning of any parameters. The experimental results on many scales of packing problems show that this approach can produce high-quality solutions for most of the benchmark datasets, especially for large-scale problems, within a reasonable duration of computational time.

A quasi-human strategy-based improved basin filling algorithm for the orthogonal rectangular packing problem with mass balance constraint

Under the background of layout optimization of the satellite module, we study the orthogonal rectangle packing problem (ORPP) with mass balance constraint, which is an NP-hard problem. Based on the quasi-physical strategy, we convert the problem into an unconstrained optimization problem. The major challenge of solving this problem is that the objective function being optimized is characterized by a multitude of local minima separated by high-energy barriers. Basin filling (BF) algorithm is a new heuristic global optimization algorithm, which combines the energy landscape paving (ELP), based on Monte Carlo sampling, and local search, based on the gradient method. We use the improved basin filling (IBF) algorithm to solve the ORPP with mass balance constraint. In the IBF algorithm, in order to avoid the ELP falling into narrow and deep valleys of energy landscape, a new update mechanism of the histogram function in the ELP is proposed. In addition, a quasi-human corner-occupying strategy and a local movement strategy, based on the adaptive gradient method with retreat and acceleration, are used to update the layouts. Experimental results show that the proposed algorithm is an effective method for solving the ORPP with mass balance constraint.

A superior linearization method for signomial discrete functions in solving three-dimensional open-dimension rectangular packing problems

ABSTRACTThis article studies the three-dimensional open-dimension rectangular packing problem (3D-ODRPP) in which a set of given rectangular boxes is packed into a large container of minimal volume. This problem is usually formulated as a mixed-integer nonlinear programming problem with a signomial term in the objective. Existing exact methods experience difficulty in solving large-scale problems within a reasonable amount of time. This study reformulates the original problem as a mixed-integer linear programming problem by a novel method that reduces the number of constraints in linearizing the signomial term with discrete variables. In addition, the range reduction method is used to tighten variable bounds for further reducing the number of variables and constraints in problem transformation. Numerical experiments are presented to demonstrate that the computational efficiency of the proposed method is superior to existing methods in obtaining the global optimal solution of the 3D-ODRPP.

On solving three-dimensional open-dimension rectangular packing problems

ABSTRACTIn this article, a recently proposed three-dimensional open-dimension rectangular packing problem is considered, in which the objective is to find a minimal volume rectangular container that packs a set of rectangular boxes. The literature has tackled small-sized instances of this problem by means of optimization solvers, position-free mixed-integer programming (MIP) formulations and piecewise linearization approaches. In this study, the problem is alternatively addressed by means of grid-based position MIP formulations, whereas still considering optimization solvers and the same piecewise linearization techniques. A comparison of the computational performance of both models is then presented, when tested with benchmark problem instances and with new instances, and it is shown that the grid-based position MIP formulation can be competitive, depending on the characteristics of the instances. The grid-based position MIP formulation is also embedded with real-world practical constraints, such as cargo stability, and results are additionally presented.