Bin Packing Research Articles

GPUTreeShap: massively parallel exact calculation of SHAP scores for tree ensembles

SHapley Additive exPlanation (SHAP) values (Lundberg & Lee, 2017) provide a game theoretic interpretation of the predictions of machine learning models based on Shapley values (Shapley, 1953). While exact calculation of SHAP values is computationally intractable in general, a recursive polynomial-time algorithm called TreeShap (Lundberg et al., 2020) is available for decision tree models. However, despite its polynomial time complexity, TreeShap can become a significant bottleneck in practical machine learning pipelines when applied to large decision tree ensembles. Unfortunately, the complicated TreeShap algorithm is difficult to map to hardware accelerators such as GPUs. In this work, we present GPUTreeShap, a reformulated TreeShap algorithm suitable for massively parallel computation on graphics processing units. Our approach first preprocesses each decision tree to isolate variable sized sub-problems from the original recursive algorithm, then solves a bin packing problem, and finally maps sub-problems to single-instruction, multiple-thread (SIMT) tasks for parallel execution with specialised hardware instructions. With a single NVIDIA Tesla V100-32 GPU, we achieve speedups of up to 19× for SHAP values, and speedups of up to 340× for SHAP interaction values, over a state-of-the-art multi-core CPU implementation executed on two 20-core Xeon E5-2698 v4 2.2 GHz CPUs. We also experiment with multi-GPU computing using eight V100 GPUs, demonstrating throughput of 1.2 M rows per second—equivalent CPU-based performance is estimated to require 6850 CPU cores.

Homogeneous grouping of non-prime steel products for online auctions: a case study

Not all products meet customers’ quality expectations after the steelmaking process. Some of them, labelled as ‘non-prime’ products, are sold in a periodic online auction. These products need to be grouped into the smallest feasible number of bundles as homogeneous as possible, as this increases the attractiveness of the bundles and hence their selling prices. This results in a highly complex optimisation problem, also conditioned by other requirements, with large economic implications. It may be interpreted as a variant of the well-known bin packing problem. In this article, we formalise it mathematically by studying the real problem faced by a multinational in the steel industry. We also propose a structured, three-stage solution procedure: (i) initial division of the products according to their characteristics; (ii) cluster analysis; and (iii) allocation of products to bundles via optimisation methods. In the last stage, we implement three heuristic algorithms: FIFO, greedy, and distance-based. Building on previous works, we develop 80 test instances, which we use to compare the heuristics. We observe that the greedy algorithm generally outperforms its competitors; however, the distance-based one proves to be more appropriate for large sets of products. Last, we apply the proposed solution procedure to real-world datasets and discuss the benefits obtained by the organisation.

Interactive Bin Packing: A Java Application for Learning Constructive Heuristics for Combinatorial Optimization

Cicirello, V. A., (2022). Interactive Bin Packing: A Java Application for Learning Constructive Heuristics for Combinatorial Optimization. Journal of Open Source Education, 5(49), 140, https://doi.org/10.21105/jose.00140

Variable neighborhood search for quadratic multiple constraint variable sized bin-packing problem

The bin-packing problem is well known in the field of combinatorial optimization. In this paper, we propose a quadratic multiple constraint variable sized bin-packing problem (QMC-VSBPP), which is a generalization of the variable sized bin-packing problem (VSBPP). In QMC-VSBPP, each bin type has multiple capacities, each item has multiple weights, and the items in different bins have joint costs. The objective of QMC-VSBPP is to minimize the sum of costs of used bins and the joint costs of items in different bins. QMC-VSBPP is a new combinatorial optimization problem. Herein, a variable neighborhood search (VNS) heuristic is proposed to solve QMC-VSBPP. The proposed VNS uses a special hierarchical clustering algorithm to generate the initial solution, in which three neighborhood operators are presented to expand solution space, and a local search algorithm based on a combination of two local operators is presented to find a local optimal solution. To evaluate the performance of the proposed VNS for QMC-VSBPP, a genetic algorithm and a variable neighborhood search for VSBPP were adapted for the state-of-the-art QMC-VSBPP, and the commercial solver CPLEX was also applied to QMC-VSBPP. Results of computational experiments on 96 test instances show that the proposed VNS could yield better solutions than those of the adapted genetic algorithm (GA), the adapted VNS, and CPLEX.

Scheduling a single batch processing machine with non-identical two-dimensional job sizes

This paper considers a single batch processing machine scheduling problem with non-identical two-dimensional job sizes. Jobs may be rotated but must be placed orthogonally without overlapping in the machine. The machine can simultaneously process a batch of jobs as long as its two-dimensional capacity is not exceeded. The processing time of a batch is defined by the longest processing time of all jobs in the batch. The objective is to assign jobs to batches considering the machine capacity constraint such that the makespan is minimized. An improved mixed integer programming formulation is proposed to reduce the solution space. 16 heuristics, combining four job sequencing, two batch selection, and two job placement alternatives, are investigated. The effectiveness of the model and heuristics is analyzed through a comprehensive experimental study. The results show that the LPTBFSA heuristic outperforms the others, in which jobs are sorted in decreasing processing time, a batch is selected for each job based on the best-fit algorithm, and a job is placed in the selected batch following the “as compact as possible” packing principle. The LPTFFSA heuristic also works well, in which the first-fit algorithm is used to select a batch for each job.

Multi-objective optimization of the 3D container stowage planning problem in a barge convoy system

Mainly motivated by the massification of flows in inland shipping, where several barges can be assembled in convoys to transport a large number of containers to and from the hinterland, we introduce an original extension of the Container Stowage Planning Problem (CSPP) (which we denote the 3D – barge Convoy Container Stowage Planning Problem (3D –CCSPP)). The requirements for stowing all containers in a barge convoy entail additional constraints related to both the containers and the convoy. In this paper, the 3D –CCSPP is dealt with, on the basis of its relation to the 3D Bin-Packing Problem (3D-BPP), in which the filled objects represent the containers, and the bins indicate the convoy barges. We consider the multi-objective aspect of the problem with three realistic functions to be optimized: shifting movements, the convoy stability and the number of real-used barges. This is a new and effective aspect considering the state of the art of CSPP in inland shipping. We develop a multi-objective mathematical formulation of the 3D-CCSPP and propose a novel adaptation of the multi-objective evolutionary algorithm NSGA-II (Non-Dominated Sorting Genetic Algorithm-II) based on a set of heuristics introduced by the BPP resolution methods. Specific encoding method, generation of the initial population and genetic operators are designed based on the mathematical description of the problem. Extensive experiments are then carried out on a varied set of instances. The computational results demonstrate the effectiveness of the proposed approaches, they are evaluated using CPLEX solver and their performance is verified by a set of measurements which are adapted to the problem in real-world applications in relation to each objective function. Additional studies are carried out in sensitivity analysis to provide decision makers with a better knowledge of the problem. They can then choose, from the set of compromise solutions of the Pareto front, the most promising one for practical implementation based on their experience and preferences.

Scheduling a single parallel-batching machine with non-identical job sizes and incompatible job families

We study the scheduling of jobs on a single parallel-batching machine with non-identical job sizes and incompatible job families. Jobs from the same family have the same processing time and can be loaded into a batch, as long as the batch size respects the machine capacity. The objective is to minimize the total weighted completion time; this common scheduling objective is equivalent with the minimization of in-process inventory when all jobs have the same release date. Our problem combines two classic combinatorial problems, namely bin packing and single machine scheduling. We develop three new mixed-integer linear-programming formulations, namely an assignment-based formulation, a time-indexed formulation (TIF), and a set-partitioning formulation (SPF). We also propose a column generation (CG) algorithm for the SPF, a branch-and-price (B&P) algorithm and a CG-based heuristic. A heuristic framework based on proximity search is also developed using the TIF. We examine how to reduce the size (number of variables) of the formulations in a preprocessing step. The SPF and B&P can solve instances with non-unit and unit job durations to optimality with up to 80 and 150 jobs within reasonable runtime limits, respectively. The proposed heuristics perform better than the methods available in the literature for the same problem.

Bin Packing Problem with Time Lags

We introduce and motivate several variants of the bin packing problem where bins are assigned to time slots, and minimum and maximum lags are required between some pairs of items. We suggest two integer programming formulations for the general problem: a compact one and a stronger formulation with an exponential number of variables and constraints. We propose a branch-cut-and-price approach that exploits the latter formulation. For this purpose, we devise separation algorithms based on a mathematical characterization of feasible assignments for two important special cases of the problem: when the number of possible bins available at each period is infinite and when this number is limited to one and time lags are nonnegative. Computational experiments are reported for instances inspired from a real-case application of chemical treatment planning in vineyards, as well as for literature instances for special cases of the problem. The experimental results show the efficiency of our branch-cut-and-price approach, as it outperforms the compact formulation on newly proposed instances and is able to obtain improved lower and upper bounds for literature instances. Summary of Contribution: The paper considers a new variant of the bin packing problem, which is one of the most important problems in operations research. A motivation for introducing this variant is given, as well as a real-life application. We present a novel and original exact branch-cut-and-price algorithm for the problem. We implement this algorithm, and we present the results of extensive computational experiments. The results show a very good performance of our algorithm. We give several research directions that can be followed by subsequent researchers to extend our contribution to more complex and generic problems.

Fast Multi-UAV Path Planning for Optimal Area Coverage in Aerial Sensing Applications.

This paper deals with the problems and the solutions of fast coverage path planning (CPP) for multiple UAVs. Through this research, the problem is solved and analyzed with both a software framework and algorithm. The implemented algorithm generates a back-and-forth path based on the onboard sensor footprint. In addition, three methods are proposed for the individual path assignment: simple bin packing trajectory planner (SIMPLE-BINPAT); bin packing trajectory planner (BINPAT); and Powell optimized bin packing trajectory planner (POWELL-BINPAT). The three methods use heuristic algorithms, linear sum assignment, and minimization techniques to optimize the planning task. Furthermore, this approach is implemented with applicable software to be easily used by first responders such as police and firefighters. In addition, simulation and real-world experiments were performed using UAVs with RGB and thermal cameras. The results show that POWELL-BINPAT generates optimal UAV paths to complete the entire mission in minimum time. Furthermore, the computation time for the trajectory generation task decreases compared to other techniques in the literature. This research is part of a real project funded by the H2020 FASTER Project, with grant ID: 833507.

A cluster-based yard template design integrated with yard crane deployment using a placement heuristic

The increasing demand for container transportation calls for higher terminal storage capacity and greater handling efficiency. A good yard template design will facilitate both space and yard crane utilization. However, most of the existing research designs the template based on the parallel layout and caters only to destination vessels. This study proposes a cluster-based yard template design method that reserves storage space based on the container destination and origin, which helps avoid yard crane conflict and is therefore applicable to various yard designs. A mixed integer programming model is proposed to assign clusters considering yard crane deployment, with the objective to ensure the yard space is used intensively most of the time. To solve this problem in large-scale cases, three heuristics are developed based on the best-fit method in solving the bin packing problem. Numerical experiments under different yard layouts are conducted to validate the efficiency and effectiveness of the model and algorithms.

Integrated bin packing and lot-sizing problem considering the configuration-dependent bin packing process

This paper addresses a practical production problem in an aviation manufacturing factory that focuses on the production of composite aeronautic products, where packing and assembling are two major operations conducted in the production process. We extract a novel integer programming model that integrates bin packing with the multi-level lot-sizing problem, which is characterized by re-configurable bins and configuration-dependent processing time. To obtain tighter lower bounds, we decompose the original model into a master problem and several sub-problems based on the Dantzig–Wolfe framework. To accelerate the column generation process, we develop a bi-level dynamic programming algorithm to solve the sub-problems exactly. A hybrid algorithm that combines the column generation and the fix-and-optimize heuristic is proposed to generate high-quality upper bounds. We first evaluate the quality of the lower bounds obtained from the linearly relaxed restricted master problem and then demonstrate the competitiveness of the proposed heuristic algorithm in terms of the solution quality and computational efficiency by comparing it with CPLEX, a commercial solver. In addition, we conduct sensitivity analysis of the bi-level dynamic programming to investigate the impact of the processing time required by the configuration-based bins. Moreover, a case study was conducted, which demonstrated the effect on cost reduction by shortening the length of unit time period in the planning horizon.

Verified Approximation Algorithms

We present the first formal verification of approximation algorithms for NP-complete optimization problems: vertex cover, independent set, set cover, center selection, load balancing, and bin packing. We uncover incompletenesses in existing proofs and improve the approximation ratio in one case. All proofs are uniformly invariant based.

Decomposition of the Knapsack Problem for Increasing the Capacity of Operating Rooms

This paper is aimed at the problem of scheduling surgeries in operating rooms. To solve this problem, we suggest using some variation of the bin packing problem. The model is based on the actual operation of 10 operating rooms, each of which belongs to a specific department of the hospital. Departments are unevenly loaded, so operations can be moved to operating rooms in other departments. The main goal is to increase patient throughput. It is also necessary to measure how many operations take place in other departments with the proposed solution. The preferred solution is a solution with fewer such operations, all other things being equal. Due to the fact that the mixed-integer linear programming model turned out to be computationally complex, two approximation algorithms were also proposed. They are based on decomposition. The complexity of the proposed algorithms is estimated, and arguments are made regarding their accuracy from a theoretical point of view. To assess the practical accuracy of the algorithms, the Gurobi solver is used. Experiments were conducted on real historical data on surgeries obtained from the Burdenko Neurosurgical Center. Two decomposition algorithms were constructed and a comparative analysis was performed for 10 operating rooms based on real data.

Adaptive Bin Packing with Overflow

Motivated by bursty bandwidth allocation and by the allocation of virtual machines to servers in the cloud, we consider the online problem of packing items with random sizes into unit-capacity bins. Items arrive sequentially, but on arrival, an item’s actual size is unknown; only its probabilistic information is available to the decision maker. Without knowing this size, the decision maker must irrevocably pack the item into an available bin or place it in a new bin. Once packed in a bin, the decision maker observes the item’s actual size, and overflowing the bin is a possibility. An overflow incurs a large penalty cost, and the corresponding bin is unusable for the rest of the process. In practical terms, this overflow models delayed services, failure of servers, and/or loss of end-user goodwill. The objective is to minimize the total expected cost given by the sum of the number of opened bins and the overflow penalty cost. We present an online algorithm with expected cost at most a constant factor times the cost incurred by the optimal packing policy when item sizes are drawn from an independent and identically distributed (i.i.d.) sequence of unknown length. We give a similar result when item size distributions are exponential with arbitrary rates. We also study the offline model, where distributions are known in advance but must be packed sequentially. We construct a soft-capacity polynomial-time approximation scheme for this problem and show that the complexity of computing the optimal offline cost is [Formula: see text]-hard. Finally, we provide an empirical study of our online algorithm’s performance.

A MILP model and two heuristics for the Bin Packing Problem with Conflicts and Item Fragmentation

Bin Packing Problem with Conflicts and Item Fragmentation (BPPC-IF) is a variant of the classical bin packing problem in which fragments of the same item can be packed in different bins (item fragmentation), and some pairs of items cannot be packed in the same bin (item conflicts). To solve the problem, we propose a new mixed-integer linear programming (MILP) model and two heuristics. The MILP model represents BPPC-IF as a generalized transportation problem in which sources correspond to items, and sinks are defined for all maximal independent sets in the graph representing conflicts between items. The proposed heuristics, called Sequential Packing Heuristics 1 and 2 (SPH1 and SPH2, respectively) are modifications of the previously proposed Sequential Maximum Degree Packing Heuristic (SMDPH) for BPPC-IF, which fills bins by solving a series of MILP subproblems. Both new heuristics use improved criteria for selecting items to pack. Additionally, in SPH1, bins are filled heuristically without solving any MILP subproblems. Computational experiments carried out on a large set of benchmark problem instances from the literature show that (i) SPH1 obtains good quality solutions and is orders of magnitude faster than the MILP-based heuristics, (ii) compound heuristic SPH1+SPH2 finds on average better solutions than any other heuristic and is on average faster than any other MILP-based heuristic, and (iii) the proposed MILP model greatly outperforms the best previously proposed MILP model.

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.

Developing a Maximum Inscribed Rectangle Heuristic to Satisfy Rush Orders for Heavy Plate Steel

Steel service centers receive rush orders that must be fulfilled on very short notice. Each order only consumes a portion of one steel plate, so plate selection and job placement are the critical factors that affect the service center’s primary performance metric: plate yield. In conjunction with a steel service center, Artco Steel, we model this problem as a two-dimensional online bin-packing algorithm. Unique in the online bin-packing literature, we calculate the maximum inscribed rectangle (MIR) before and after job placement as the basis for heuristics that assign each job to a plate and position the job on the plate. Our work is the first paper to extend the online two-dimensional bin-packing problem to incorporate scrap, rectangular bin sizes, and a finite number of bins. The MIR procedure significantly outperformed Artco’s existing practice of giving priority to the most recently used plate, and the heuristic’s straightforward nature allowed easy adoption in 2010.

Recursive first fit: a highly parallel optimal solution to spectrum allocation

We revisit the classical spectrum allocation (SA) problem, a fundamental subproblem in optical network design, and make three contributions. First, we show how some SA problem instances may be decomposed into smaller instances that may be solved independently without loss of optimality. Second, we prove an optimality property of the well-known first-fit (FF) heuristic. Finally, we leverage this property to develop a recursive and parallel algorithm that applies the FF heuristic to find an optimal solution efficiently. This recursive FF algorithm is highly scalable because of two unique properties: (1) it completely sidesteps the symmetry inherent in SA and hence drastically reduces the solution space compared to typical integer linear programming formulations, and (2) the solution space can be naturally decomposed in non-overlapping subtrees that may be explored in parallel almost independently of each other, resulting in faster than linear speedup.

Efficient Dynamic Deployment of Simulation Tasks in Collaborative Cloud and Edge Environments

Cloud computing has been studied and used extensively in many scenarios for its nearly unlimited resources and X as a service model. To reduce the latency for accessing the remote cloud data centers, small data centers or cloudlets are deployed near end-users, which is also called edge computing. In this paper, we mainly focus on the efficient scheduling of distributed simulation tasks in collaborative cloud and edge environments. Since simulation tasks are usually tightly coupled with each other by sending many messages and the status of tasks and hosts may also change frequently, it is essentially a dynamic bin-packing problem. Unfortunately, popular methods, such as meta-heuristics, and accurate algorithms are time-consuming and cannot deal with the dynamic changes of tasks and hosts efficiently. In this paper, we present Pool, an incremental flow-based scheduler, to minimize the overall communication cost of all tasks in a reasonable time span with the consideration of migration cost of task. After formulating such a scheduling problem as a min-cost max-flow (MCMF) problem, incremental MCMF algorithms are adopted to accelerate the procedure of calculating an optimal flow and heuristic scheduling algorithm, with the awareness of task migration cost, designed to assign tasks. Simulation experiments on Alibaba cluster trace show that Pool can schedule all of the tasks efficiently and is almost 5.8 times faster than the baseline method when few tasks and hosts change in the small problem scale.

Fast Non-dominated Sorting in Multi Objective Genetic Algorithm for Bin Packing Problem

The bin packing problem is a problem where goods with different volumes and dimensions are put into a container so that the volume of goods inserted is maximized. The problem of multi-objective bin packing is a problem that is more commonly found in everyday life, because what is considered in packing is usually not only volume.In this research, a multi-objective genetic algorithm is proposed to solve the multi-objective bin packing problem. The proposed genetic algorithm uses non-dominated sorting and crowding distance methods to get the best solution for each objective and to avoid bias. The algorithm is then tested with several test classes that represent different combinations of item and container sizes.From the results of the tests carried out, it was found that the proposed algorithm can find several solutions which are the best candidate solutions for each objective. Also found how the correlation of each objective in the population.