Two-dimensional Bin Packing Research Articles

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.

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

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

A Hybrid Discrete Grey Wolf Optimization Algorithm Imbalance-ness Aware for Solving Two-dimensional Bin-packing Problems

A Hybrid Discrete Grey Wolf Optimization Algorithm Imbalance-ness Aware for Solving Two-dimensional Bin-packing Problems

Real-time cyber/physical interplay in scheduling for peak load optimisation in Cyber–Physical Energy Systems

This paper is about reducing the power consumption of Cyber–Physical Energy Systems (CPESs) composed of many loads through the usage of a scheduling technique inspired by the real-time-computing domain; the electric-load coordination guarantees a more efficient operation of the entire system by avoiding unnecessary concurrent activation of loads and thus limiting the peak load. We represent the power loads themselves as “physical” components and the computing devices that coordinate them as “cyber” components and formally derive the relationship between the operations in cyber and physical domains as the interplay between the schedules enforced on each component: indeed, the schedule of the loads – generated by combining a two-dimensional bin-packing and an optimal multi-processor real-time scheduling algorithm – influences the timing of the processing tasks that are dedicated to the activation/deactivation of loads themselves. We also consider non-schedulable loads by introducing a policy to cope with the presence of such loads. Numerical simulations and experiments confirm the good performance of the proposed peak load reduction method. The usage of real-time scheduling in this context provides inherent resource optimisation by limiting the number of concurrent loads that are active at the same time, thus directly reducing the overall peak load; moreover, thanks to the limited computational complexity of the algorithms, it scales to large systems, overcoming the scalability issues of common optimisation methods. Numerical simulations and experiments confirm the good performance of the proposed peak load reduction method.

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.

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.

Parallel Batch Processing Machine Scheduling Under Two-Dimensional Bin-Packing Constraints

The parallel batch processing machine scheduling problem with two-dimensional bin packing constraints (PBS-2D) has appeared in many industrial environments, but the research on this problem is still not sufficient. In this article, we use the 2-D rectangular packing constraints to describe the spatial geometric layout of jobs more accurately, and propose a mixed integer linear programming (MILP) model to minimize the makespan of PBS-2D. According to the in-depth analysis of the problem properties, PBS-2D is separated into three subproblems, which are job-machine assignment, 2-D job placement, and job sequence optimization. An improved biased random key genetic algorithm (OBRKGA) is proposed to solve job-machine assignment and job sequence optimization. This algorithm generates the initial population based on the idea of orthogonal experimental design, so that the whole solution space can be scanned evenly when generating the initial solution, making the algorithm be more likely to find better solutions for subsequent evolution. At the same time, a bin packing algorithm based on best fit is designed to handle 2-D job placement problem. Finally, the algorithms are tested on a large number of randomly generated instances, the results show that OBRKGA outperformed the original BRKGA, the classical moth-flame optimization algorithm, particle swarm optimization algorithm, gravity search algorithm, and MILP.

Heuristics for the two-dimensional irregular bin packing problem with limited rotations

This paper studies a two-dimensional irregular bin packing problem with some specific degrees of rotations. Despite a large number of studies on the two-dimensional irregular strip packing problem, much less attention has been paid to the two-dimensional irregular bin packing problem (2DIBPP), and most of the current research on 2DIBPP is about free rotations. However, limited by the texture of the material, sometimes only some specific degrees of rotations are allowed. To pack a set of irregular pieces with some specific degrees into multi-stock sheets and maximize the utilization, we propose a heuristic algorithm that combines some pieces into one block for packing and uses residual spaces with reference lines to help select the next piece to be packed. In order to further increase the utilization of bins, we adopt fine-tuning, movement, filling, replacement and swap operations, as well as partial and full repack strategies. The computational results show that the proposed algorithm can obtain superior results on the instances with small number of pieces, relatively large number of types of pieces, large bins and more specific degrees of rotations.

A hybrid estimation of distribution algorithm for the offline 2D variable-sized bin packing problem

In this paper we present an evolutionary heuristic for the offline two-dimensional variable-sized bin packing problem. In this problem we have to pack a set of rectangles into two-dimensional variable-sized rectangular bins. The bins are divided into types, and the bins in different types have different sizes and possibly different weights (costs). There are (sufficiently) many bins from each type, and any rectangle fits into at least one bin-type. The goal is to pack the rectangles into the bins without overlap, parallel to the sides, so that the total area of the used bins (or total cost) is minimized. Our algorithm is a hybrid heuristic. It uses two different techniques to generate the descendants: either estimation of distribution algorithm and sampling the resulting probability model, or applying the usual operators of evolutionary algorithms (selection, mutation). To pack the rectangles into the bins the algorithm uses the strategy of randomly choosing one of two placement heuristics, that pack always only one group (one to three) of rectangles. It improves the quality of the solutions with three local search procedures. The algorithm has been tested on benchmark instances from the literature and has been compared with other heuristics and metaheuristics. Our algorithm outperformed the previously published results.

Approximation algorithms for a virtual machine allocation problem with finite types

We study the virtual machine allocation (VMA) problem with physical machines (PMs) of identical configuration and virtual machines (VMs) of a constant number of types. Only CPU and memory (MEM) resources are considered, so it is a special two-dimensional vector bin packing (2-DVBP) problem. Applying the best fit heuristic, we develop a fast online algorithm, which uses 209OPT(L)+1 PMs at most, where OPT(L) is the optimal value. According to the limited types of VMs, the progress of combining VMs and dividing the PMs' capacity contributes to a linear-time offline algorithm that uses 109OPT(L)+2 PMs. It outperforms the state-of-the-art (1.405+ϵ)-asymptotic approximation algorithm for the general 2-DVBP problem. These ideas can be extended to develop fast and practical algorithms.

Strip based compact formulation for two-dimensional guillotine cutting problems

In this paper, we present a new mixed integer programming formulation for two-dimensional guillotine cutting and packing problems based on a strip decomposition of the rectangular spaces. The formulation covers most of the main related problems in the literature by setting parameters accordingly. The strip concept commonly used for problems with a limited number of stages (two or three stages) has been, in this study, extended to a general concept that can cover an arbitrary number of stages. Due to the easy adaptation of the proposed formulation, which is presented in both heuristic and exact version, an extensive set of computational experiments was performed with instances for the two-dimensional guillotine knapsack, cutting stock and bin packing problems. The experiments, which involve existing benchmark instances and randomly generated instances from the literature, showed that our proposed formulation, considering its heuristic version, can output competitive results both in terms of computational time and percentage of optimally solved instances.

A hierarchical hyper-heuristic for the bin packing problem

AbstractThis paper addresses the two-dimensional irregular bin packing problem, whose main aim is to allocate a given set of irregular pieces to larger rectangular containers (bins), while minimizing the number of bins required to contain all pieces. To solve the problem under study a dynamic hierarchical hyper-heuristic approach is proposed. The main idea of the hyper-heuristics is to search the space of low-level heuristics for solving computationally difficult problems. The proposed approach is “dynamic” since the low-level heuristic to be executed is chosen on the basis of the main characteristics of the instance to be solved. The term “hierarchical” is used to indicate the fact that the main hyper-heuristic can execute either simple heuristics or can run in a “recursive fashion” a hyper-heuristic. The developed solution strategy is evaluated empirically by performing extensive experiments on irregular packing benchmark instances. A comparison with the state-of-the-art approaches is also carried out. The computational results are very encouraging.

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.

Optimization of two-dimensional irregular bin packing problem considering slit distance and free rotation of pieces

In this paper, we present a two-dimensional irregular bin packing problem (2DIBPP) that takes into account the slit distance and allows the pieces to rotate freely. The target is to arrange a specified collection of pieces with irregular shapes into a minimal number of bins. Firstly, we develop a mathematical model for the 2DIBPP that considers slit distance and free rotation of the pieces, and an equidistant edge expansion approach is then proposed to handle the slit distance. Secondly, a two-stage method is implemented to get a finite collection of promising rotation angles, effectively decreasing the search neighbourhood. Thirdly, we decompose the 2DIBPP into two sub-problems: piece assignment and packing. The Partial Bin Packing (PBP) strategy is employed in the allocation stage, and we adopt an overlap minimization method to pack the pieces into an individual bin. Finally, we use a local search (LS) algorithm to advance the quality of the solutions by adjusting the piece assignment across bins. Experimental evidence exhibits that our approach is competitive in most instances of the literature, with four better results in five benchmark instances.

A Reduced Variable Neighborhood Search Approach to the Heterogeneous Vector Bin Packing Problem

The two-dimensional heterogeneous vector bin packing problem (2DHet-VBPP) consists of packing the set of items into the set of various type bins, respecting their two resource limits. The problem is to minimize the total cost of all bins. The problem, known to be NP-hard, can be formulated as a pure integer linear program, but optimal solutions can be obtained by the CPLEX Optimizer engine only for small instances. This paper proposes a metaheuristic approach to the 2DHet-VBPP, based on Reduced variable neighborhood search (RVNS). All RVNS elements are adapted to the considered problem and many procedures are designed to improve efficiency of the method. As the Two-dimensional Homogeneous-VBPP (2DHom-VBPP) is more often treated, we considered also a special version of the RVNS algorithm to solve the 2DHom-VBPP. The results obtained and compared to both CPLEX results and results on benchmark instances from literature, justify the use of the RVNS algorithm to solve large instances of these optimization problems.

A goal-driven ruin and recreate heuristic for the 2D variable-sized bin packing problem with guillotine constraints

This paper addresses the two-dimensional bin packing problem with guillotine constraints. The problem requires a set of rectangular items to be cut from larger rectangles, known as bins, while only making use of edge-to-edge (guillotine) cuts. The goal is to minimize the total bin area needed to cut all required items. This paper also addresses variants of the problem which permit 90∘ rotation of items and/or a heterogeneous set of bins. A novel heuristic is introduced which is based on the ruin and recreate paradigm combined with a goal-driven approach. When applying the proposed heuristic to benchmark instances from the literature, it outperforms the current state-of-the-art algorithms in terms of solution quality for all variants of the problem considered.

An iteratively doubling local search for the two-dimensional irregular bin packing problem with limited rotations

This paper proposes an iteratively doubling local search approach for the two-dimensional irregular bin packing problem (2DIRBPP) with limited rotations, whose objective is to pack a given set of irregular pieces into the minimum number of rectangular bins. The allowable angles of each piece are limited. To solve this problem, a waste least first decreasing strategy is introduced to assign the piece to the bins. A simple greedy local search approach by moving the pieces from one bin to another is utilized to improve the solution. We adapt an overlap minimization approach to solving the one bin placement problem. The classical bottom-left method is utilized to generate the initial position for each piece, and the random local search by exchanging the positions of two pieces is used to minimize the overlap. Moreover, a novel warm start and an iteratively doubling search strategy are proposed to speed up the search process. The standard benchmark results show that our approach improves the results for most of the instances in the literature.

A lookahead matheuristic for the unweighed variable-sized two-dimensional bin packing problem

The unweighed oriented variable-sized two-dimensional guillotine bin packing problem consists in packing without overlap small rectangular items into large non-identical rectangular bins, with the items obtained via guillotine cuts. It minimizes the waste of the used bins. It is herein approximately solved using a hybrid matheuristic, which applies a sequence of low-level mixed-integer programs that reserve space for unpacked items and that are guided by feasibility constraints and by upper bounds on the objective function. The embedded constraints constitute a lookahead mechanism that prohibits the investigation of infeasible directions and constrains the search to improving ones. The matheuristic further employs high-level diversification and intensification mechanisms. The diversification incorporates a sequential value correction algorithm that tags a pseudo-price to each item to govern the fitness functions of mixed integer programs and subsequently their solution construction process. The intensification is a local search that investigates the neighbourhood of promising solutions. The extensive computational experiments provide evidence of the good performance of the proposed matheuristic. For the variable-sized bin packing benchmark instances, the matheuristic matches and improves 90.8% of the upper bounds. For the single bin-size bin packing benchmark instances, the matheuristic further proves the optimality of 82.6% upper bounds while it matches 14.4% and improves 2.6% existing bounds.

An optimization model of variable size two-dimensional nesting with stacked sheets and usable leftovers for the CNC routing process

The research is motivated by a problem in the aviation industry, where nesting is performed to allocate and arrange parts in metal sheets for a CNC router machine. In addition to the usual objective to minimize material waste, the nesting problem has a few characteristics, i.e., variable sheet size, stacked sheet routing, and usable leftovers. The three aspects are integrated into an optimization model developed from the fundamental two-dimensional bin packing problem. A simple numerical example is given to show the capability of the model and the characteristics of the results.

Combinatorial Benders Decomposition for the Two-Dimensional Bin Packing Problem

The two-dimensional bin packing problem calls for packing a set of rectangular items into a minimal set of larger rectangular bins. Items must be packed with their edges parallel to the borders of the bins, cannot be rotated, and cannot overlap among them. The problem is of interest because it models many real-world applications, including production, warehouse management, and transportation. It is, unfortunately, very difficult, and instances with just 40 items are unsolved to proven optimality, despite many attempts, since the 1990s. In this paper, we solve the problem with a combinatorial Benders decomposition that is based on a simple model in which the two-dimensional items and bins are just represented by their areas, and infeasible packings are imposed by means of exponentially many no-good cuts. The basic decomposition scheme is quite naive, but we enrich it with a number of preprocessing techniques, valid inequalities, lower bounding methods, and enhanced algorithms to produce the strongest possible cuts. The resulting algorithm behaved very well on the benchmark sets of instances, improving on average on previous algorithms from the literature and solving for the first time a number of open instances. Summary of Contribution: We address the two-dimensional bin packing problem (2D-BPP), which calls for packing a set of rectangular items into a minimal set of larger rectangular bins. The 2D-BPP is a very difficult generalization of the standard one-dimensional bin packing problem, and it has been widely studied in the past because it models many real-world applications, including production, warehouse management, and transportation. We solve the 2D-BPP with a combinatorial Benders decomposition that is based on a model in which the two-dimensional items and bins are represented by their areas, and infeasible packings are imposed by means of exponentially many no-good cuts. The basic decomposition scheme is quite naive, but it is enriched with a number of preprocessing techniques, valid inequalities, lower bounding methods, and enhanced algorithms to produce the strongest possible cuts. The algorithm we developed has been extensively tested on the most well-known benchmark set from the literature, which contains 500 instances. It behaved very well, improving on average upon previous algorithms, and solving for the first time a number of open instances. We analyzed in detail several configurations before obtaining the best one and discussed several insights from this analysis in the manuscript.