Performance Analysis and Improvement for Some Linear On-Line Bin-Packing Algorithms

The two-dimensional strip packing problem is a generalization of the classic one-dimensional bin packing problem. It has many important applications such as costume clipping, material cutting, real-world planning, packing, scheduling etc. Average-case performance analysis of approximation algorithms attracts a lot of attention because it plays a crucial role in selecting an appropriate algorithm for a given application. While approximation algorithms for two-dimensional packing are frequently presented, the results of their average-case performance analyses have seldom been reported due to intractability. In this paper, we analyze the average-case performance of Next Fit Decreasing Height (NFDH) algorithm, one of the first strip packing algorithms proposed by Coffman, Jr. in 1980. We prove that the expected height of packing with NFDH algorithm, when the heights and widths of the rectangle items are independent and both obey (0, 1] uniform distribution, is about n/3, where n is the number of rectangle items. We also validate the theoretical result with experiments.

In this paper, we analyze the average-case performance of the Modified Harmonic algorithm for bin packing. We first analyze the average-case performance for arbitrary distribution of item sizes over (0, 1]. This analysis is based on the following result. Let f1 and f2 be two linear combinations of random variables {ni} n=1 k , where the Nis have a joint multinomial distribution for each \(n = \sum\limits_{i = 1}^k {N_i }\). Let E(f1)≠0, and E(f2)↮0. Then \(\mathop {\lim }\limits_{n \to \infty }\)E(max(f1, f2))/n=\(\mathop {\lim }\limits_{n \to \infty }\)max(E(f1), E(f2))/n. We then consider the special case when the item sizes are uniformly distributed over (0, 1], and obtain optimal values for the parameters of the algorithm. For these values of the parameters, the average-case performance ratio is less than 1.19. This compares well with the performance ratio 1.2865... of the Harmonic algorithm.

In this paper we consider a one-dimensional bin packing problem where the bins do not have identical sizes, or a variable-sized bin packing problem, to minimize the bin consumption, i.e., the total size of the opened bins. The identical size problem has been extensively studied in the literature both for static and dynamic settings. The worst-case or average-case performance has been analyzed. Our problem setting particularly arises in metal cutting industries. Therefore, it presents a great practical relevance. Four greedy approximation algorithms based on a construction approach called largest object first with least absolute waste (LFLAW), largest object first with least relative waste (LFLRW), least absolute waste (LAW), and least relative waste (LRW) are examined. Their absolute worst-case performances are analyzed. The worst case bounds are 2 for LFLAW and LFLRW, 3 for LAW, and 2+ln 2 for LRW. We also show that these worst-case bounds are tight.

Multi-capacity bin-packing is a generalization of the classical one-dimensional bin-packing problem in which the bin capacity and the item sizes are represented by d-dimensional vectors. Previous work in d-capacity bin-packing algorithms analyzed variants of single capacity bin-packing, extended to deal with the d capacities. These extended algorithms are oblivious to the additional capacity information, however, and do not scale well with increasing d. We provide new packing algorithms which use the additional capacity information to provide better packing and show how these algorithms might lead to better multi-resource allocation and scheduling solutions.

This work aims to study and explore the use of Gene Expression Programming (GEP) in solving on-line Bin-Packing problem. The main idea is to show how GEP can automatically find acceptable heuristic rules to solve the problem efficiently and economically. One dimensional Bin-Packing problem is considered in the course of this work with the constraint of minimizing the number of bins filled with the given pieces. Experimental Data includes instances of benchmark test data taken from Falkenauer (1996) for One-dimensional Bin-Packing Problems. Results show that GEP can be used as a very powerful and flexible tool for finding interesting compact rules suited for the problem. The impact of functions is also investigated to show how they can affect and influence the success of rates when they appear in rules. High success rates are gained with smaller population size and fewer generations compared to a previous work performed using Genetic Programming.

Background: Logistics and transport, core of many business processes, are continuously optimized to improve efficiency and market competitiveness. The paper describes a modular coordination of vehicle routing and bin packing problems that enables independent instances of the problems to be joined together, with the aim that the vehicle routing solution satisfies all the constraints from real-world applications. Methods: The vehicle routing algorithm is based on an adaptive memory procedure that also incorporates a simple, one-dimensional bin packing problem. This preliminary packing solution is refined by a complex, three dimensional bin packing for each vehicle to identify the infeasible packages. The method iteratively adjusts virtual volumes until reaching near-optimal routes that respect bin-packing constraints. Results: The coordination enables independent applications of an adaptive memory procedure to vehicle routing and a genetic algorithm approach to bin packing while joining them in a computationally tractable way. Such a coordinated approach is applied to a frequently used public benchmark and proven to provide commensurate costs while significantly lowering algorithm complexity. Conclusions: The proposed method is further validated on a real industrial case study and provided additional savings of 14.48% in average daily distance traveled compared to the current industrial standard.

In this paper, our objective is to develop a mathematical formulation of solving the Bin Packing Problem (BPP) with different sizes of box. It has become one of the most important mathematical applications throughout the time. In our study, Modified Branch and Bound Algorithm (MBBA) is developed to generate all the feasible packing patterns of given boxes to required containers for One-Dimensional BPP. Further development of algorithms was made to ascertain the locations of each box within the containers by using Cartesian coordinate system. Developed algorithms are coded and programmed in the Python programming environment to generate feasible packing patterns.

Approximation and online algorithms for multidimensional bin packing: A survey

Best-fit is the best known algorithm for on-line binpacking, in the sense that no algorithm is known to behave better both in the worst case (when Best-fit has performance ratio 1.7) and in the average uniform case, with items drawn uniformly in the interval [0, l] (then Best-fit has expected wasted space O(n’/2(log n)“/“)). In practical applications, Best-fit appears to perform within a few percent of optimal. In this paper, in the spirit of previous work in computational geometry, we study the expected performance ratio, taking the worst-case multiset of items L, and assuming that the elements of L are inserted in random order, with all permutations equally likely. We show a lower bound of 1.08 . . . and an upper bound of 1.5 on the random order performance ratio of Best-fit. The upper bound contrasts with the result that in the worst case, any (deterministic or randomized) on-line bin-packing algorithm has performance ratio at least 1.54. . . .

The author presents a parallel algorithm for packing N>or=n/sup 2/ pieces in a massively parallel computing environment of n/sup 2/ processors operating in single-instruction multiple-data (SIMD) mode. The heuristic approach that is developed utilizes a grid topology that structures the input to increase packing efficiency without sorting the data set. The results of implementing this algorithm on a Connection Machine 2 system are compared with the results obtained from parallel versions of two traditional bin packing algorithms. It is concluded that the use of data partitioning to initialize the packing appears to be a practical method of allocating the packing workload to a set of parallel processors while maintaining the integrity of the packing algorithm. The results support the contention that parallelism offers opportunity for improvement in the efficiency of packing algorithms and that the SIMD model of computation with its data parallel programming approach is appropriate for bin packing problems. >

Construction heuristics for two-dimensional irregular shape bin packing with guillotine constraints

In this paper, we study extensions to bin packing algorithms to pack software modules into processors in real-time systems. We refer to this approach as Partitioning Bin-Packing. In this study, we analytically show that with partitioning bin-packing techniques the number of bins required by traditional bin packing can be reduced. We also evaluate heuristics to minimise both the number of processors (bins) needed and the network bandwidth required by communicating software modules that are partitioned across different processors. We find that a significant reduction in the number of bins is possible. Finally, different heuristics lead to different tradeoffs in processing vs. network needs.

A two-dimensional bin packing model was established to solve the problem of goods placement in the material box, which was optimized to improve space utilization. By discretization of the model, a bin packing algorithm with left-bottom priority is designed. By adopting the improved genetic algorithm with the optimal individual retention strategy, the optimal sequence of various kinds of goods in the material box is obtained. Two types of experiments of multi-algorithm comparison simulation and multi-solution simulation are designed. The first type of experimental results show that the algorithm has some advantages in fitness and convergence speed, and the second type of experimental results show that it can provide multiple optimal solutions for solving small-scale bin packing problems.

We consider the two-dimensional bin packing and strip packing problem, where a list of rectangles has to be packed into a minimal number of rectangular bins or a strip of minimal height, respectively. All packings have to be non-overlapping and orthogonal, i.e., axis-parallel. Our algorithm for strip packing has an absolute approximation ratio of 1.9396 and is the first algorithm to break the approximation ratio of 2 which was established more than a decade ago. Moreover, we present a polynomial time approximation scheme ($\mathcal{PTAS}$) for strip packing where rotations by 90 degrees are permitted and an algorithm for two-dimensional bin packing with an absolute worst-case ratio of 2, which is optimal provided $\mathcal{P} \not= \mathcal{NP}$.

Article Free Access Share on Tight bounds for minimax grid matching, with applications to the average case analysis of algorithms Authors: F T Leighton Mathematics Department and Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge, MA Mathematics Department and Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge, MAView Profile , P Shor Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CAView Profile Authors Info & Claims STOC '86: Proceedings of the eighteenth annual ACM symposium on Theory of computingNovember 1986Pages 91–103https://doi.org/10.1145/12130.12140Published:01 November 1986Publication History 27citation565DownloadsMetricsTotal Citations27Total Downloads565Last 12 Months73Last 6 weeks11 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF