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Technical note: split algorithm in O(n) for the capacitated vehicle routing problem

COMPUTERS & OPERATIONS RESEARCH 2016年 69卷 40-47页

作者： Vidal, Thibaut Pontificia Univ Catolica Rio De Janeiro PUC Rio Dept Informat Rua Marques de Sao Vicente225 Gavea BR-22451900 Rio De Janeiro RJ Brazil

The split algorithm is an essential building block of route-first cluster-second heuristics and modern genetic algorithms for vehicle routing problems. The algorithm is used to partition a solution, represented as a giant tour without occurrences of the depot, into separate routes with minimum cost. As highlighted by the recent survey of Prins et al. [18], no less than 70 recent articles use this technique. In the vehicle routing literature, split is usually assimilated to the search for a shortest path in a directed acyclic graph g and solved in O(nB) using Bellman's algorithm, where n is the number of delivery points and B is the average number of feasible routes that start with a given customer in the giant tour. Some linear-time algorithms are also known for this problem as a consequence of a Monge property of g. In this paper, we highlight a stronger property of this graph, leading to a simple alternative algorithm in O(n). Experimentally, we observe that the approach is faster than the classical split for problem instances of practical size. We also extend the method to deal with a limited fleet and soft capacity constraints. (C) 2015 Elsevier Ltd. All rights reserved.

关键词： Vehicle routing problem Large neighborhood search split algorithm Cluster-first route-second heuristic

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A hybrid genetic search and dynamic programming-based split algorithm for the multi-trip time-dependent vehicle routing problem

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2024年第3期317卷 921-935页

作者： Zhao, Jingyi Poon, Mark Tan, Vincent Y. F. Zhang, Zhenzhen Ecole Polytech Montreal Dept Math & Ind Engn Montreal PQ H3T 1N8 Canada Amazon Seattle WA 98109 USA Natl Univ Singapore Dept Math Singapore 119077 Singapore Natl Univ Singapore Dept Elect & Comp Engn Singapore 119077 Singapore Tongji Univ Sch Econ & Management Shanghai 200092 Peoples R China

We design a hybrid algorithm for the multi -trip time -dependent vehicle routing problem (MT-TD-VRP). One of its components is the Time -Dependent split algorithm (TD -SPA), which is a dynamic programming -based algorithm specifically designed to handle both the multi -trip per vehicle and the time -dependent aspects of the problem. The hybrid algorithm combines the proposed TD -SPA, designed to efficiently split a giant tour into complete vehicle routes, with a genetic algorithm for generating these tours. We introduce a monotone queue optimization (MQO) technique to accelerate the TD -SPA. The effectiveness of MQO is evaluated by comparing computation times between the original split algorithm for the capacitated vehicle routing problem (CVRP) and its MQO-enhanced counterpart. Extensive numerical experiments with a real -world dataset from a Singapore food and beverage company are conducted to assess our algorithm's performance on various MTTD-VRP instances. The results indicate that our algorithm surpasses the performance of the commercial solver Gurobi, with an average improvement of 25.13% on the best solutions found within a prescribed duration. Our numerical simulations further reveal the algorithm's ability to efficiently solve both the capacitated vehicle routing problem (CVRP) and the multi -trip vehicle routing problem (MTVRP), consistently producing competitive solutions. Moreover, to highlight the importance of incorporating the time -dependent (TD) factor into our model and algorithm, we demonstrate a notable enhancement in performance-averaging at 7.47% for the best solutions under TD conditions for an MTVRP dataset.

关键词： Routing Time-dependent travel time Multi-trip split algorithm Monotone queue optimization

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Enhancing Data Security in Cloud Using split algorithm, Caesar Cipher, and Vigenere Cipher, Homomorphism Encryption Scheme

Enhancing Data Security in Cloud Using Split Algorithm, Caes...

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International Conference on Emerging Trends in Expert Applications and Security (ICETEAS)

作者： Singh, Abhishek Sharma, Shilpi Amity Univ Uttar Pradesh Amity Sch Engn & Technol Noida Uttar Pradesh India

ISBN: (纸本)9789811322853;9789811322846

Cloud computing declared among the fastest growing technology. The amount of users using this technology has exploded. Therefore, cloud users expect their data to be safe on the cloud. We can secure the data from being accessed illegally by encrypting it with a key with various methods likewise Advanced and Data encryption standard. However, the issue with these encryption methods is that they are very complex and have a lengthy computing time due to finite key space. To overcome these problems, Cloud Service Providers (CSP) which is becoming a great area to have researched for ensuring the security of data with minimum processing time. This research paper emphasizes various secured encryption techniques which have efficient computing time. This paper focus on the complexity, efficiency, and the security of the algorithm used.

关键词： Cloud computing Encryption Data security split algorithm Caesar cipher Vigenere cipher

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A split algorithm of manufacturing stencil mask and dividing pattern for making holographic optical elements 8

A split algorithm of manufacturing stencil mask and dividing...

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Conference on Holography, Diffractive Optics, and Applications VIII

作者： Aoki, Yuta Sakamoto, Kunio Konan Univ Fac Intelligence & Informat Higashinada Ku 8-9-1 Okamoto Kobe Hyogo 6588501 Japan

ISBN: (数字)9781510622357

ISBN: (纸本)9781510622357

Suppose a donut-shaped pillow. We illustrate this pillow with inner and outer circles. Then we cut a donut-like shape along the lines of circles. We can only get a big circle, not a donut-like shape. It is inconvenient for manufacturers that the hole-shapes of a mask is different from what we want. Because the deformed donut-like shape is different from the original in having "bridges," the authors adopt a method of dividing the stencil into two in order to get what we want. In this paper, we propose mask split algorithm for a method of manufacturing stencil masks and dividing pattern. We can have made masks with numerous holes automatically and holographic optical elements with various shapes by using our proposed design algorithm.

关键词： holographic optical element holographic grating exposure recording photomask stencil template split algorithm

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Hybrid ANN-Based and Text Similarity Method for Automatic Short-Answer Grading in Polish

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APPLIED SCIENCES-BASEL 2025年第3期15卷 1605-1605页

作者： Saad, Marwah Bani Jackowska-Strumillo, Lidia Bieniecki, Wojciech Lodz Univ Technol Inst Appl Comp Sci Stefanowskiego 18 PL-90537 Lodz Poland

Computer-assisted grading plays an important role in an educational context, mainly by reducing the workload of teachers in manual scoring. While electronic choice surveys have long been used in many web applications, automatic scoring of open-ended responses remains an interesting research problem in natural language processing. In this article, we propose a new hybrid text-processing method for scoring students' responses based on word splitting and preprocessing, which will then combine textual algorithms with a set of artificial neural network classifiers and a set of heuristic decision rules. This concept has been implemented in the interactive e-test system operating in the local computer network of the Institute of Applied Computer Science at the Lodz University of Technology. The dataset is acquired as questions, reference answers, and students' answers generated on the basis of exams conducted at our institute in the years 2015-2022 for more than a thousand students. This article extends our previous research and includes comparative tests. The proposed method achieves excellent results and outperforms the previous approaches. The obtained precision is equal to 1, and the recall measure is 0.97 for the final results.

关键词： open question test automatic test assessment natural language processing split algorithm similarity measures machine learning artificial neural network hybrid text-processing method

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split algorithms for skewsymmetric Toeplitz matrices with arbitrary rank profile

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THEORETICAL COMPUTER SCIENCE 2004年第2-3期315卷 453-468页

作者： Heinig, G Rost, K Kuwait Univ Dept Math & Comp Sci Kuwait 13060 Kuwait Tech Univ Chemnitz Dept Math D-09107 Chemnitz Germany

split Levinson-type and Schur-type algorithms for the solutions of linear systems with a non-singular skewsymmetric Toeplitz matrix are designed. In contrast to previous ones, the algorithms work for any nonsingular skewsymmetric Toeplitz matrix. Moreover, generalizations of ZW- and WZ-factorizations of skewsymmetric Toeplitz: matrices related to the new split algorithms are presented. (C) 2004 Elsevier B.V. All rights reserved.

关键词： Toeplitz matrix skewsymmetric matrix split algorithm Levinson algorithm Schur algorithm WZ-factorization

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split algorithms for hermitian Toeplitz matrices with arbitrary rank profile

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LINEAR ALGEBRA AND ITS APPLICATIONS 2004年第1-3期392卷 235-253页

作者： Heinig, G Rost, K Kuwait Univ Dept Math & Comp Sci Safat 13060 Kuwait Chemnitz Univ Dept Math D-09107 Chemnitz Germany

split algorithms for Toeplitz matrices exploit besides the Toeplitz structure additional symmetry properties to reduce the number of operations. In this paper split Levinson and Schur algorithms for hermitian Toeplitz matrices are presented that work, in contrast to previous algorithms, without additional conditions like strong nonsingularity. The main contribution is the generalization of the split Levinson-type algorithms of B. Krishna/H. Krishna and H. Krishna/S. Morgera to general nonsingular hermitian Toeplitz matrices. Furthermore, a Schur-type counterpart of this algorithm is presented that is also new in the strongly nonsingular case. Some auxiliary considerations concerning the kernel structure of hermitian Toeplitz matrices might be of independent interest. (C) 2004 Elsevier Inc. All rights reserved.

关键词： hermitian Toeplitz matrix split algorithm Levinson algorithm Schur algorithm

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split algorithms for symmetric Toeplitz matrices with arbitrary rank profile

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2005年第2-3期12卷 141-151页

作者： Heinig, G Rost, K Kuwait Univ Dept Math & Comp Sci Safat 13060 Kuwait Tech Univ Chemnitz Dept Math D-09107 Chemnitz Germany

split Levinson and Schur algorithms for the inversion of symmetric Toeplitz matrices are presented that work, in contrast to previous algorithms, without additional conditions like strong non-singularity. Copyright (c... 详细信息

关键词： symmetric Toeplitz matrix Levinson algorithm Schur algorithm split algorithm

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Advances in FE explicit formulation for simulation of metalforming processes

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JOURNAL OF MATERIALS PROCESSING TECHNOLOGY 2001年第1-3期119卷 41-47页

作者： Rojek, J Zienkiewicz, OC Oñate, E Postek, E Polish Acad Sci Inst Fundamental Technol Res PL-00049 Warsaw Poland Univ Coll Swansea Swansea W Glam Wales Int Ctr Numer Methods Engn Barcelona Spain

This paper presents some advances of finite element explicit formulation for simulation of metal forming processes. Because of their computational efficiency, finite element programs based on the explicit dynamic formulation proved to be a very attractive tool for the simulation of metal forming processes. The use of explicit programs in the sheet forming simulation is quite common, the possibilities of these codes in bulk forming simulation in our opinion are still not exploited sufficiently. In our paper, we will consider aspects of bulk forming simulation. We will present new formulations and algorithms developed for bulk metal forming within the explicit formulation. An extension of a finite element code for the thermomechanical coupled analysis is discussed. A new thermomechanical constitutive model developed by the authors and implemented in the program is presented. A new formulation based on the so-called split algorithm allows us to use linear triangular and tetrahedral elements in the analysis of large plastic deformations characteristic to forming processes. Linear triangles and tetrahedra have many advantages over quadrilateral and hexahedral elements. Linear triangles and tetrahedra based on the standard formulations exhibit volumetric locking and are not suitable for large plastic strain simulation. The new formulation allows to overcome this difficulty. New formulations and algorithms have been implemented in the finite element code Stampack developed at the International Centre for Numerical Methods in Engineering in Barcelona. Numerical examples illustrate some of the possibilities of the finite element code developed and validate new algorithms. (C) 2001 Elsevier Science B.V. All rights reserved.

关键词： explicit formulation forming simulation split algorithm

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Linear elements for metal-forming problems within the flow approach

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2000年第5-7期190卷 783-801页

作者： Antunez, HJ Polish Acad Sci Inst Fundamental Technol Res PL-00049 Warsaw Poland

Simulation of transient and stationary metal-forming processes is presented in the context of a mixed velocity-pressure formulation with a rigid-viscoplastic material model. A fractional step technique is used to obtain in a natural way a discrete Laplacian in the pressure-pressure entries of the system matrix which, as is known, introduces the stabilizing effect and allows for the use of equal order interpolating functions for both discretized variables, in spite of the Babuska-Brezzi condition. The method is suitable for both implicit and explicit time integration schemes. In particular, the explicit version (forward Euler with mass lumping) is developed. The discretization is carried out for linear triangles and tetrahedra which most easily can make use of automatic mesh generators and result in a simple code. Attention is focused on the stabilizing effect as a function of material and numerical parameters. The technique is illustrated with extrusion and upsetting simulation and compared with a known pressure stabilization method. (C) 2000 Elsevier Science S.A. All rights reserved.

关键词： metal-forming flow approach explicit split algorithm

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