One-dimensional Bin Packing Problem Research Articles

Lower and upper bounding procedures for the bin packing problem with concave loading cost

We address the one-dimensional bin packing problem with concave loading cost (BPPC), which commonly arises in less-than-truckload shipping services. Our contribution is twofold. First, we propose three lower bounds for this problem. The first one is the optimal solution of the continuous relaxation of the problem for which a closed form is proposed. The second one allows the splitting of items but not the fractioning of bins. The third one is based on a large-scale set partitioning formulation of the problem. In order to circumvent the challenges posed by the non-linearity of the objective function coefficients, we considered the inner-approximation of the concave load cost and derived a relaxed formulation that is solved by column generation. In addition, we propose two subset-sum-based heuristics. The first one is a constructive heuristic while the second one is a local search heuristic that iteratively attempts to improve the current solution by selecting pairs of bins and solving the corresponding subset sum-problem. We show that the worst-case performance of any BPPC heuristic and any concave loading cost function is bounded by 2. We present the results of an extensive computational study that was carried out on large set of benchmark instances. This study provides empirical evidence that the column generation-based lower bound and the local search heuristic consistently exhibit remarkable performance.

Optimization of one-dimensional bin packing problem using a hybrid flower pollination algorithm

Optimization of one-dimensional bin packing problem using a hybrid flower pollination algorithm

The 1-D bin packing problem optimisation using bees algorithm

The bin packing problem is a classic combinatorial optimisation problem that is widely used in various applications such as assembly line balancing, scheduling, and time-tabling. Metaheuristic algorithms can provide solutions to these problems faster than exact methods. Bees Algorithm, a metaheuristic algorithm inspired by the foraging activity of bees, is known for its performance in solving optimisation problems. To our best knowledge, this is the first use of the Bees Algorithm to solve a bin packing problem. In this paper, we use the basic Bees Algorithm to demonstrate near-optimal solutions and measure the accuracy of solutions to the one-dimensional bin packing problem. The algorithm procedure and parameter settings are set following the previous research. Benchmark datasets are used in the experiment, and accuracy is measured. The results indicate that the basic Bees Algorithm for bin packing problems and previous research on travelling salesman problems produce similar accuracy results.

Exact and approximate methods for the score-constrained packing problem

This paper investigates a packing problem related to the one-dimensional bin packing problem in which the order and orientation of items influences the feasibility of a solution. We give an exact polynomial-time algorithm for the Constrained Ordering Problem, explaining how it can be used to find a feasible packing of items in a single bin. We then introduce an evolutionary algorithm for the multi-bin version of the problem, which incorporates the exact algorithm along with a local search procedure and three recombination operators. The mechanisms and results produced by each of the recombination operators are compared, and we discuss the circumstances in which each approach proves most advantageous.

Two heuristics for one of Bin-Packing problems

In this article, a new practical problem is proposed, as well as algorithms for solving the one-dimensional bin packing problem. The generalization of this problem is one of the most fundamental problems of combinatorial optimization and has been widely studied for decades. Our formulation for this problem takes into account not only the different weights of products and separability but the difference in their types. An objective function is formulated that minimizes the sum of the components with weights responsible for different characteristics of product distribution. Parameter generation of the problem is based on data that approximate the real one. In order to compare the algorithms 168 test instances were generated. In addition they were solved optimally using the Gurobi solver with a time limit of 1 hour. The algorithms proposed are based on the separation of the set of subjects under consideration on the basis of divisibility. Also, the proposed algorithms have been tested on a large family of generated instances with the number of bins from 100 to 1000 ones.

Pool-based Evolutionary Algorithm for the Bin Packing Problem

Bin packing problem is one of the most important optimization problems from the literature. In this work, we propose a novel pool-based evolutionary algorithm for solving the one-dimensional bin packing problem. The algorithm exploits a pool-based crossover operator that aims to increase the search space of the problem and combine and remap as a local search technique that aims to improve the quality of the solution by considering underutilized bins in the offspring. The proposed method is applied to medium and hard instances taken from benchmark problem sets and is compared with six algorithms from the literature. Our experimental evolution indicates that the proposed algorithm outperforms these algorithms in most of the test cases provided.

A Flexible Reinforced Bin Packing Framework with Automatic Slack Selection

The slack-based algorithms are popular bin-focus heuristics for the bin packing problem (BPP). The selection of slacks in existing methods only consider predetermined policies, ignoring the dynamic exploration of the global data structure, which leads to nonfully utilization of the information in the data space. In this paper, we propose a novel slack-based flexible bin packing framework called reinforced bin packing framework (RBF) for the one-dimensional BPP. RBF considers the RL-system, the instance-eigenvalue mapping process, and the reinforced-MBS strategy simultaneously. In our work, the slack is generated with a reinforcement learning strategy, in which the performance-driven rewards are used to capture the intuition of learning the current state of the container space, the action is the choice of the packing container, and the state is the remaining capacity after packing. During the construction of the slack, an instance-eigenvalue mapping process is designed and utilized to generate the representative and classified validate set. Furthermore, the provision of the slack coefficient is integrated into MBS-based packing process. Experimental results show that, in comparison with fit algorithms, MBS and MBS’, RBF achieves state-of-the-art performance on BINDATA and SCH_WAE datasets. In particular, it outperforms its baseline MBS and MBS’, averaging the number increase of optimal solutions of 189.05% and 27.41%, respectively.

Metaheuristic algorithms for one-dimensional bin-packing problems: A survey of recent advances and applications

AbstractThe bin-packing problem (BPP) is an age-old NP-hard combinatorial optimization problem, which is defined as the placement of a set of different-sized items into identical bins such that the number of containers used is optimally minimized. Besides, different variations of the problem do exist in practice depending on the bins dimension, placement constraints, and priority. More so, there are several important real-world applications of the BPP, especially in cutting industries, transportation, warehousing, and supply chain management. Due to the practical relevance of this problem, researchers are consistently investigating new and improved techniques to solve the problem optimally. Nature-inspired metaheuristics are powerful algorithms that have proven their incredible capability of solving challenging and complex optimization problems, including several variants of BPPs. However, no comprehensive literature review exists on the applications of the metaheuristic approaches to solve the BPPs. Therefore, to fill this gap, this article presents a survey of the recent advances achieved for the one-dimensional BPP, with specific emphasis on population-based metaheuristic algorithms. We believe that this article can serve as a reference guide for researchers to explore and develop more robust state-of-the-art metaheuristics algorithms for solving the emerging variants of the bin-parking problems.

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.

Approximation Algorithms for a New Truck Loading Problem in Urban Freight Transportation

Motivated by urban freight transportation practices in China, we study an optimal truck loading problem in which a fixed cost and an additional cost that depends on the number of unloading points are associated with each truck used. The truck loading problem is modeled as a one-dimensional bin packing problem, where the cost of each bin is a convex fixed-plus-linear function of the number of items in the bin. The objective is to minimize the total cost of bins used. We develop an asymptotic polynomial time approximation scheme and an efficient approximation algorithm with an asymptotic worst-case performance ratio of 1.5 to tackle the problem. Our computational experiments indicate that the approximation algorithm performs well for certain order patterns, and also reveal several important insights on using the freight charge scheme.

Improvement Grouping Genetic Algorithm for Solving the Bin Packing Problem

An improved grouping genetic algorithm is proposed to solve the one-dimensional bin packing problem. In this algorithm, a new crossover operator is constructed according to the characteristics of the encoding. At the same time, a replacement insertion method is designed to improve the mutation operation. The new mutation operation prevents the algorithm from falling into a local optimum and improves the transmission speed of the best individual in the chromosome. The analysis and comparison of experimental results show that the proposed algorithm can effectively solve the one-dimensional bin packing problem and has reliable global convergence.

Metaheuristic Approaches for One-Dimensional Bin Packing Problem: A Comparative Performance Study

Nature-inspired metaheuristic algorithms have steadily gained popularity over the last two decades. They have been applied to a plethora of optimization problems both in continuous and combinatorial domains. In this paper, the one-dimensional bin packing problem is solved through the implementation of two underlying heuristics, namely, best fit and better fit, and four representative state-of-the-art global metaheuristic algorithms, namely, firefly algorithm, genetic algorithm, adaptive cuckoo search algorithm, and artificial bee colony algorithm. The underlying best fit and better fit heuristics are employed by the four aforementioned global metaheuristic algorithms as reordering heuristics and local search improvement mechanisms are used for generating good packing schema. Furthermore, these two local heuristics possess special characteristics which, when incorporated into the global metaheuristics, allow them to escape local optimums and avoid getting stuck, and more so, enables the algorithms to generate good quality solutions. The main focus of this paper is the presentation of a systematic performance evaluation study for the representative algorithms, with some initial computational results that show the effectiveness of the respective algorithms and their ability to achieve promising solutions. The experiments conducted here were carried out using three standard bin packing problem datasets categories with over 1,210 instances in total, each with differing capacities, number of items and distributions between item weights. The numerical results of the representative algorithms were compared with the solutions achieved with the underlying heuristic techniques, in this case, the best fit and better fit heuristics, respectively. Similarly, the analysis of the initial computational results obtained revealed the superior performance of the individual algorithm implementation. Moreover, performance was established by taking into account both the algorithms computational time and the solution quality. Overall, several observations regarding the solution and the underlying heuristics were made. It is worth noting that by utilizing best fit heuristic, the algorithms attained optimal solutions for instances of the easy dataset requiring smaller capacity bins. However, as the complexity of the instances increased, the ability to produce high-quality results decreased. Nevertheless, this heuristic can produce near-optimal solutions and a good packing schema for most instances. On the other hand, utilizing better fit as the underlying heuristic results in optimal solutions almost all the time, regardless of capacity and number of items.

An Adaptive Fitness-Dependent Optimizer for the One-Dimensional Bin Packing Problem

In recent years, the one-dimensional bin packing problem (1D-BPP) has become one of the most famous combinatorial optimization problems. The 1D-BPP is a robust NP-hard problem that can be solved through optimization algorithms. This paper proposes an adaptive procedure using a recently optimized swarm algorithm and fitness-dependent optimizer (FDO), named the AFDO, to solve the BPP. The proposed algorithm is based on the generation of a feasible initial population through a modified well-known first fit (FF) heuristic approach. To obtain a final optimized solution, the most critical parameters of the algorithm are adapted for the problem. To the best of our knowledge, this is the first study to apply the FDO algorithm in a discrete optimization problem, especially for solving the BPP. The adaptive algorithm was tested on 30 instances obtained from benchmark datasets. The performance and evaluation results of this algorithm were compared with those of other popular algorithms, such as the particle swarm optimization (PSO) algorithm, crow search algorithm (CSA), and Jaya algorithm. The AFDO algorithm obtained the smallest fitness values and outperformed the PSO, CS, and Jaya algorithms by 16%, 17%, and 11%, respectively. Moreover, the AFDO shows superiority in terms of execution time with improvements over the execution times of the PSO, CS, and Jaya algorithms by up to 46%, 54%, and 43%, respectively. The experimental results illustrate the effectiveness of the proposed adaptive algorithm for solving the 1D-BPP.

A New Branch-and-Price-and-Cut Algorithm for One-Dimensional Bin-Packing Problems

In this paper, a new branch-and-price-and-cut algorithm is proposed to solve the one-dimensional bin-packing problem (1D-BPP). The 1D-BPP is one of the most fundamental problems in combinatorial optimization and has been extensively studied for decades. Recently, a set of new 500 test instances were proposed for the 1D-BPP, and the best exact algorithm proposed in the literature can optimally solve 167 of these new instances, with a time limit of 1 hour imposed on each execution of the algorithm. The exact algorithm proposed in this paper is based on the classical set-partitioning model for the 1DBPPs and the subset row inequalities. We describe an ad hoc label-setting algorithm to solve the pricing problem, dominance, and fathoming rules to speed up its computation and a new primal heuristic. The exact algorithm can easily handle some practical constraints, such as the incompatibility between the items, and therefore, we also apply it to solve the one-dimensional bin-packing problem with conflicts (1D-BPPC). The proposed method is tested on a large family of 1D-BPP and 1D-BPPC classes of instances. For the 1D-BPP, the proposed method can optimally solve 237 instances of the new set of difficult instances; the largest instance involves 1,003 items and bins of capacity 80,000. For the 1D-BPPC, the experiments show that the method is highly competitive with state-of-the-art methods and that it successfully closed several open 1D-BPPC instances.

A modified squirrel search algorithm based on improved best fit heuristic and operator strategy for bin packing problem

Bin Packing Problem (BPP) is one of the main optimization problems in which a set of items of known sizes should be packed into a minimum number of bins. In which the total size of items in each bin is not exceeding the bin capacity. Several approximation algorithms have been suggested that provide an approximate solution to the BPP. Nevertheless, they are not applicable to solve large-scale instances within a reasonable time or obtain an optimal number of bins. In this paper, we propose a modified version of the most recent optimization algorithm squirrel search algorithm (SSA) for solving the one-dimensional bin packing problem (MSBPP). The proposed algorithm is based on an improved best fit heuristic for generating a feasible initial packing of items into bins. During improving the solution, the operator’s strategy can take place to obtain an optimized solution for the BPP. To the best of our knowledge, this paper is the first work of applying squirrel search algorithm in a bin packing problem as an example of the optimization problem. The modified algorithm has been tested on various benchmark datasets ranged from easy to hard class dataset. As well as, we compare it against other popular algorithms such as Particle swarm optimization (PSO), African buffalo optimization (ABO) and Crow search algorithm (CSA) to demonstrate its efficiency and accuracy. According to the results of an easy class, the proposed algorithm has achieved an improvement ranged from 16% to 32% for N2C1W1_A instance when compared with other algorithms to reach an optimal result with the least number of iterations. In addition, the average execution time of it beats other compared algorithms with improvement reached up to 91% faster than PSO, 98% faster than ABO and 94% faster than CSA. For medium class, the bin efficiency of the proposed algorithm has been improved by 26% during iterations for N1W1B2R9 instance with minimum number of iterations and least execution time while for hard class, it has obtained an average efficiency 0.175 which is lower than other algorithms and average execution time 99.8 as a result of HARD3 instance. The experimental results validate the superiority of the proposed MSBPP in terms of not only the solution accuracy but also both the execution time and the number of iterations.

ADAPTIVE HARMONY SEARCH FOR OPTIMIZING CONSTRAINED RESOURCE ALLOCATION PROBLEM

Resource Allocation problem is finding the optimal assignment of finite available resources to tasks or users. Resource allocation problems refer to a wide range of applications such as production, supply chain management, transportation, ICT technologies, etc. Resource allocation problems are NP-hard in nature where the objective is to find the optimal allocations satisfying given constraints. Harmony search (HS) algorithm is a meta-heuristic population based algorithm found good for solving different optimization problems. This paper presents adaptive harmony search (AHS) for solving one-dimensional bin packing problem (BPP) and multi-objective virtual machine placement problem (VMP). The proposed real coded solution representation supports partial constraint satisfaction. Adaptive pitch adjustment rate (PAR) based on population diversity improves the performance of harmony search algorithm. Results show that proposed HS gives optimal solution for 50 BPP instances with 100 % success rate. The performance reduced for large instances of BPP. The proposed weighted AHS for multi objective VMP problem gives better results than genetic algorithm.

Lower bounds for a bin packing problem with linear usage cost

In this paper, we address a one-dimensional bin packing problem with bin-specific usage cost. The cost coefficients have a direct linear relationship to the bin index, favoring packings with higher loads in “earlier” bins. We show how lower bounding schemes known from standard bin packing can be adapted to fit this objective function and conduct a worst-case performance analysis. The contribution also covers a conceptually new lower bound for the problem at hand. Computational experience based on randomly generated instances and benchmark libraries provides strong evidence for high quality bounds achievable with low computational effort. This observation is further underpinned by a successful embedding of the lower bounds into a branch-and-bound approach as a computational framework. Clear advantages over a state-of-the-art mixed-integer programming formulation can be obtained for particular problem settings.

Cooperative parallel grouping genetic algorithm for the one-dimensional bin packing problem

Evolutionary algorithms have been reported to be efficient metaheuristics for the optimization of several NP-Hard combinatorial optimization problems. In addition to their ability to solve difficult and complex problems in reasonable execution times, parallelized versions of evolutionary algorithms are reported to explore and exploit the problem search space more effectively than their sequential counterparts. The Island Model, where the population of a given run is divided into semi isolated subpopulations, is a popular parallelization approach for evolutionary algorithms such as Grouping Genetic Algorithms (GGA). Although the nature of GGAs is very suitable for coarse-grained parallel processing, designing an Island-parallel model for them is not a straightforward task. Selecting the communication topology, deciding migration and assimilation strategies, adjusting the migration rate and frequency, and using efficient diversification techniques are some of the important issues that needs to be covered in a successful Island-parallel Model. In this study, we propose a novel, scalable Island-parallel GGA (IPGGA) for the well-known combinatorial optimization Problem 1D Bin-Packing (1DBPP). We provide a thorough experimental evaluation of the parallel model and report significant improvements on the Hard28 problem instances by outperforming the state-of-the-art genetic algorithms. Additionally, we analyze and evaluate the parallelization parameters of IPGGA with an emphasis on problem search-space diversity and report several interesting results.

Peak demand shaving and load‐levelling using a combination of bin packing and subset sum algorithms for electrical energy storage system scheduling

An algorithm that uses demand profile information and a minimal set of energy storage system (ESS) parameters is formulated in this study for obtaining ESS operation schedules to achieve peak demand shaving and load-levelling. The algorithm is developed by formulating an electricity demand profile and ESS as a combination of the one-dimensional bin packing problem and the subset sum problem using sets of mathematical equations. The electrical system undergoes specific transformations in the charging and discharging phases of the ESS and heuristic algorithms for solving these problems are applied to obtain a viable operation schedule to meet these objectives. A comparative analysis with another algorithm is presented for verification. Results for a case study are also presented alongside a discussion to show a practical application of the algorithm.

Maximum lateness minimization in one-dimensional bin packing

In the One-dimensional Bin Packing problem (1-BP) items of different lengths must be assigned to a minimum number of bins of unit length. Regarding each item as a job that requires unit time and some resource amount, and each bin as the total (discrete) resource available per time unit, the 1-BP objective is the minimization of the makespan Cmax=maxj{Cj}. We here generalize the problem to the case in which each item j is due by some date dj: our objective is to minimize a convex combination of Cmax and Lmax=maxj{Cj−dj}. For this problem we propose a time-indexed Mixed Integer Linear Programming formulation. The formulation can be decomposed and solved by column generation relegating single-bin packing to a pricing problem to be solved dynamically. We use bounds to (individual terms of) the objective function to address the oddity of activation constraints. In this way, we get very good gaps for instances that are considered difficult for the 1-BP.