Fuzzy Information and Engineering-2019 (Advances in Intelligent Systems and Computing, 1094) 9811524580, 9789811524585

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Advances in Intelligent Systems and Computing 1094

Bing-yuan Cao   Editor

Fuzzy Information and Engineering-2019

Advances in Intelligent Systems and Computing Volume 1094

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Advisory Editors Nikhil R. Pal, Indian Statistical Institute, Kolkata, India Rafael Bello Perez, Faculty of Mathematics, Physics and Computing, Universidad Central de Las Villas, Santa Clara, Cuba Emilio S. Corchado, University of Salamanca, Salamanca, Spain Hani Hagras, School of Computer Science and Electronic Engineering, University of Essex, Colchester, UK László T. Kóczy, Department of Automation, Széchenyi István University, Gyor, Hungary Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX, USA Chin-Teng Lin, Department of Electrical Engineering, National Chiao Tung University, Hsinchu, Taiwan Jie Lu, Faculty of Engineering and Information Technology, University of Technology Sydney, Sydney, NSW, Australia Patricia Melin, Graduate Program of Computer Science, Tijuana Institute of Technology, Tijuana, Mexico Nadia Nedjah, Department of Electronics Engineering, University of Rio de Janeiro, Rio de Janeiro, Brazil Ngoc Thanh Nguyen , Faculty of Computer Science and Management, Wrocław University of Technology, Wrocław, Poland Jun Wang, Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong

The series “Advances in Intelligent Systems and Computing” contains publications on theory, applications, and design methods of Intelligent Systems and Intelligent Computing. Virtually all disciplines such as engineering, natural sciences, computer and information science, ICT, economics, business, e-commerce, environment, healthcare, life science are covered. The list of topics spans all the areas of modern intelligent systems and computing such as: computational intelligence, soft computing including neural networks, fuzzy systems, evolutionary computing and the fusion of these paradigms, social intelligence, ambient intelligence, computational neuroscience, artificial life, virtual worlds and society, cognitive science and systems, Perception and Vision, DNA and immune based systems, self-organizing and adaptive systems, e-Learning and teaching, human-centered and human-centric computing, recommender systems, intelligent control, robotics and mechatronics including human-machine teaming, knowledge-based paradigms, learning paradigms, machine ethics, intelligent data analysis, knowledge management, intelligent agents, intelligent decision making and support, intelligent network security, trust management, interactive entertainment, Web intelligence and multimedia. The publications within “Advances in Intelligent Systems and Computing” are primarily proceedings of important conferences, symposia and congresses. They cover significant recent developments in the field, both of a foundational and applicable character. An important characteristic feature of the series is the short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. ** Indexing: The books of this series are submitted to ISI Proceedings, EI-Compendex, DBLP, SCOPUS, Google Scholar and Springerlink **

More information about this series at http://www.springer.com/series/11156

Bing-yuan Cao Editor

Fuzzy Information and Engineering-2019

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Editor Bing-yuan Cao Foshan University Foshan, Guangdong, China Guangzhou Vocational and Technical University of Science and Technology Guangzhou, Guangdong, China

ISSN 2194-5357 ISSN 2194-5365 (electronic) Advances in Intelligent Systems and Computing ISBN 978-981-15-2458-5 ISBN 978-981-15-2459-2 (eBook) https://doi.org/10.1007/978-981-15-2459-2 © Springer Nature Singapore Pte Ltd. 2020, corrected publication 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Organizing Committee

Program Committee of Satellite Conference in ICFIAE 2018 (December 27, 2018, Foshan, China) Conference General Chair: Bing-yuan Cao, China Co-chair: Zhi-feng Hao, China Secretary: Xue-hai Fan, China Steering Committee J. C. Bezdek, USA Z. Bien, Korea D. Dubois, France Gui-rong Guo, China M. M. Gupta, Canada Xin-gui He, China Abraham Kandel, Hungary J. Kacprzyk, Poland E. Mamdani, UK R. P. Nikhil, India M. Sugeno, Japan Hao Wang, China P. Z. Wang, USA Witold Pedrycz, Canada Jing-zhong Zhang, China H. J. Zimmermann, Germany

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Members R. Ameri, Iran K. Asai, Japan Shi-zhong Bai, China J. P. Barthelemy, France Rajabali Borzooei, Iran Tian-you Chai, China Guo-qing Chen, China Shui-li Chen, China Ovanes Chorayan, Russia Sen-lin Cheng, China He-pu Deng, Australia Ali Ebrahimnejad, Iran M. Fedrizzi, Italy Jia-li Feng, China Yin-jun Feng, China Si-cong Guo, China Ming-hu Ha, China Li-yan Han, China Cheng-ming Hu, USA Bao-qing Hu, China Zhe-xue Huang, Australia Hiroshi Inoue, Japan Li-min Jia, China X. Q. Jin, Macau Guy Jumarie, Canada Jim Keller, USA E. E. Kerre, Belgium K. H. Kim, USA N. Kuroki, Japan D. Lakov, Bulgaria Tsu-Tian Lee, China, Taiwan Dong-hui Li, China Hong-xing Li, China Jun Li, China Tai-fu Li, China Yu-cheng Li, China T. Y. Lin, USA Zhong-fei Li, China Bao-ding Liu, China Zhi-qiang Liu, Hong Kong Ming Ma, China Sheng-quan Ma, China

Organizing Committee

Organizing Committee

D. Dutta Majumder, Kolkata, India Hong-hai Mi, China M. Mizumoto, Japan Zhi-wen Mo, China J. Motiwalla, Singapore M. Mukaidono, Japan J. Mustonen, Finland Shohachiro Nakanishi, Japan Michael Ng, Hong Kong Jin-ping Ou, China H. Prade, France D. A. Ralescu, USA E. Sanchez, France Victor V. Senkevich, Russia Qiang Shen, UK Kai-quan Shi, China Zhen-ming Song, China Lan Su, China Enric Trillas, Spain P. Veeramani, India Jos Luis Verdegay, Spain Xi-zhao Wang, China Xue-ping Wang, China Zhen-yuan Wang, USA Fu-yi Wei, China Xin-jiang Wei, China Ber-lin Wu, China Taiwan Yun-dong Wu, China Yang Xu, China Ze-shui Xu, China R. R. Yager, USA T. Yamakawa, Japan Liang-zhong Yi, China Xue-hai Yuan, China Qiang Zhang, China Cheng-yi Zhang, China Yun-jie Zhang, China Shao-hui Zeng, Hongkong Kai-qi Zou, China

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Organizing Committee

Program Committee of ICFIAE (Kish Island, Iran) Conference General Chair: Bing-yuan Cao, China Scientific and Program Chair: Seyed Hadi Nasseri, Iran Executive Manager: Davood Darvishi Salookolaei Steering Committee Tian-you Chai, China Zhi-feng Hao, China Zeng-liang Liu, China Hao Wang, China Pei-zhuang Wang, China Reza Tavakkoli Moghaddam, Iran Mohammadreza Akbarzadeh, Iran Members José Luis Verdegay, Spain J. Vahidi, South Africa Bing-yuan Cao, China S. H. Nasseri, Iran R. Tavakkoli Moghadam, Iran Cornelis Roos, The Netherland G. Nakhaeizadeh, Germany Ming-hu Ha, China Nicu Bizon, Romania Nezam Mahdavi-Amiri, Iran Julian Vasilev, Bulgaria Peter Lohmander, Sweden Sifeng Liu, China N. Kazemi, USA Chefi Triki, Oman Djamel Slamina, Algeria Miroslaw Moroz, Poland Didier Dubois, France Z. Polkowski, Poland F. Hosseinzadeh Lotfi, Iran Palash Dutta, India A. Safaee Ghadikolaei, Iran M. M. Zahedi, Iran M. R. Safi, Canada R. Neisani, USA

Organizing Committee

S. Babaie-Kafaki, Iran Micheal Ng, Hong Kong B. Mirashrafi, Germany A. Tajdin, Iran S. Kordrostami, Iran B. Alizadeh, Iran I. Mahdavi, Iran T. Allahviranloo, Iran M. Taheri, Iran Cheng-yi Zhang, China H. A. Aghajani, Iran D. Darvishi, Iran A. Amirteimori, Iran Guo-qing Chen, China Bao-ding Liu, China M. R. Akbarzadeh-T, Iran M. Niksirat, Iran Ze-shui Xu, China Tai-fu Li, China M. M. Paydar, Iran Hong-hai Mi, China Nitul Dutta, India A. Pourdarvish, Iran Sheng-ming Hu, USA Xi-zhao Wang, China Executive Committee Salim Bavandi Gohar Shakouri Seyed Hadi Nasseri Davood Darvishi Salookolaei Morteza Goli Behdad Asadi Hanieh Mohammadi Hadi Zavieh Meysam Ranjbar Sara Sedighi Amirhossein Nafei Roghaye Chameh

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Preface

Approved by People’s Government of Guangdong Province, hosted by International Fuzzy Information and Engineering Association (chips), the Operations Research Society of Guangdong province and Foshan University, on December 27, 2018, the 9th International Satellite Conference of Fuzzy Information and Engineering (ICFIAE) was successfully held, among which, the 3rd Annual Meeting of the Operations Research Society of Guangdong Province and Guangdong, Hong Kong, Macao and Operational Research Society, (http://en.hudongxuetang.com/m/ ZNBGR), supported by one of the series of academic conferences celebrating Foshan University in 60th anniversary of the founding. The 9th International Conference of ICFIAE was held on February 13–15, 2019, in Kish Island, Persian Gulf, Iran (www.icodm.com), organized by International Fuzzy Information and Engineering Association (Chips) with the cooperation of Iranian Operations Research Society, Foshan University, Iranian Fuzzy Systems Society, University of Mazandaran, International Center of Optimization and Decision Making (ODM), The International Association of Grey Systems and Uncertainty Analysis (IAGSUA), Models of Decision and Optimization (MODO) Research Group, Iranian Data Envelopment Analysis Society and some scientific organizations from Iran, China, Spain and around the world. The current proceedings are a copy of the outcome effort from a number of people, who participated in the conference mentioned above. The committee in the two conferences received a total of 200 submissions, most of which were reviewed at least by two reviewers for the recommendation. The first conferences’ program was further enriched by eight keynote lectures: At the first meeting, the president of Foshan University, Prof. Zhi-feng Hao, “Data Analysis and Intelligent Optimization in Urban Computing—Bionics, Unique Decomposition Theorem, and Prime Number” and Prof. Hong-xing Li from Dalian University of Technology, the first person realized four inverted pendulums in the world, made a special report on entitled “Fuzzy System to Quantum Mechanics.” Professor Hadi Nasseri, the vice president of the Iranian Operations Research Society, Iran National Elite Fund Award, professor of the University of Mazandaran in Iran and distinguished professor of Foshan University, gave a report xi

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entitled “A multi-parametric approach for the flexible fuzzy mathematical models.” Professor Shui-li Chen of Jimei University presented a report on the key technology research of the fuzzy recognition system for abnormal human behavior. The topic entitled a Method to Mine Temporal Association Rules in Attributed Graph Sequence is a report by Prof. Fu-sheng Yu of Beijing Normal University. Professor Sheng-quan Ma from Hainan Normal University demonstrated a report on “The Humanoid Learning Model for Fuzzy Complex Value Information” was given by Dr. S. K. Hayat of the Pakistan University of Guangzhou who gave a report on Aggregation Operators GIFSSs. On the other conference, there were four keynote lectures: Prof. M. R. Akbarzadeh-T, Ferdowsi University of Mashhad, Iran, “On the Type II Equivalency of Bio-inspired Swarm Type I Fuzzy Systems”; Prof. Bing-yuan Cao, Foshan University, Guangzhou Vocational and Technical University of Science and Technology, and Guangzhou University, China, “Fuzzy Geometric Programming: Past, Present and Future”; Prof. R. Tavakkoli-Moghaddam, University of Tehran, Iran, “Multi-Criteria Decision-Making Problems and Methods under Uncertainty”; and Prof. S. H. Nasseri, University of Mazandaran, Iran and Foshan University, China, “A revised approach for flexible fuzzy linear programming problems.” And also one workshop is held by Dr. Seyed Bagher Mirashrafi, Karlsruhe Institute of Technology (KIT), Germany, and University of Mazandaran (UMZ), Babolsar, Iran, “Business Intelligence and Fuzzy Data Mining.” Besides the key speeches above, the proceedings of 22 pieces from the conference were accepted because only high-quality papers are included in the book, and its contents show as follows: I. II. III. IV. V.

Fuzzy Information and Intelligent System. Fuzzy Decision-Making and Programming. Fuzzy Engineering Model and Optimize. Fuzzy Algebra and Analysis. Others.

As for the above work, our warm thanks go to the speakers who kindly accepted to share their expertise with the attendees. We also would like to say that ICFIAE 2019 and Satellite Conference would not have been successful without the support of many people and organizations. We wish to thank the members of the ICFIAE Board for their invaluable support throughout the organization process. We are very grateful to the IFIEA (Chips), Operations Research Society of Guangdong Province, Foshan Science and Technology Association, ICODM, IORS, IFSS, IAGSUA members, to all the session organizers and to all the external reviewers for their precious support in providing a rigorous reviewing process and a rich scientific program. We are also grateful to Prof. Zhi-feng Hao, Prof. Hai-wu Rong, Ms. Pei-hua Wang, Prof. Jose Luis Verdegay, Prof. Shi-feng Liu, Prof. F. Hosseinzadeh Lotfi, Prof. S. Kordrostami, Prof. A. Amirteimori, Dr. Reza Neisani, Mr. Salim Bavandi, Dr. Davod Darvishi, Mr. Dong Wang and Dr. Min-fan He, who effectively helped the work in the

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organization process. Last but not least, we appreciate the institutions that hosted and supported the organization of the conference, especially Foshan University and its Mathematics and Big Data College, and the University of Mazandaran. Foshan, China September 2019

Prof. Bing-yuan Cao Chief Chairman of Conference Prof. Seyed Hadi Nasseri Scientific and Program Chair ICFIAE 2019 and Its Satellite Conference

Messages of the International Advisory Board

Dear Prof. Cao, Dear Prof. Nasseri, Dear Attendants and Dear Colleagues, It is a great honor and an authentic pleasure for me to accept the kind invitation made by Profs. Cao and Nasseri to participate by this means in this 9th International Conference of Fuzzy Information and Engineering (ICFIAE) held on February 13–15, 2019, at Kish Island, Iran, with the cooperation of International Fuzzy Information and Engineering Association (Chips), Iranian Operations Research Society, Foshan University, Iranian Fuzzy Systems Society, University of Mazandaran, International Center of Optimization and Decision Making (ODM) and other scientific organizations of Iran and around the world. The role played by decision-making in any area of Engineering in which real-world problems are solved is beyond doubt. Even beyond this, in our day to day we are constantly faced with decision-making, which can manifest itself in many ways, from checking our mobile phones to deciding that we are going to wear and even thinking that we are going to eat today or tomorrow, for example. Thus, while we know that we make an average of 35,000 decisions per day, a recent report by the Huawei company has shown that we are really aware of less than 1% of the decisions we make every day, in fact, we ignore a 99, 74% of the decisions we make. It is obvious that the relevant advances that have occurred in the last decade in machine learning have produced systems that compete with the behavior of people in situations that involve challenges, are ambiguously raised or require high doses of skill, such as recognition of the speech or images, games. Sometimes, these systems even surpass us. These systems, which we will generically call Autonomous Decision Systems (ADS) in the following when managed with intelligent automation techniques, can increase, and in some cases replace, through fully automatic systems, the capacity to act, that is, to make decisions, of human beings. The crucial fact is that this “substitution” of functions could produce, sooner rather than later, massive job losses, the disqualification of the people who performed them, unknown effects in the systems they manage, or contexts of unwanted ungovernability, and hence, it is patent that,

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Messages of the International Advisory Board

(a) the ethical issues related to the behavior of the ADS must be included in its technological design, to facilitate that rather than as “risk factors” or restrictions, they act as the driving forces of innovation, and (b) in the event of such substitution of functions, that they do not produce malfunctions, that is, that the corresponding ADS act exactly as the human decision maker in turn, reproducing and improving its behavior and trying to avoid the inescapable and unpredictable failures that the people we can have when making decisions, especially when they have to be taken in unknown environments. Aware of these two demands, on the ethical character and proper functioning of the ADS, multiple institutions around the world have begun to discuss the conditions under which these systems should perform, as well as the premises that should guide its design, construction and material location. There are many initiatives carried out so far. Some of them, probably due to the social interest and the transcendence of the subject they consider, somewhat hasty in their recommendations, or even biased by being driven by particular interests. But what is evident in any case is that to achieve ADS behaviors analogous to those of human beings, the Fuzzy Sets Theory, Computational Intelligence and the Computing With Words paradigm cannot be left out, because they are the best tools to reproduce the behaviors of human beings. From this point of view, the celebration of this 9th International Conference of Fuzzy Information and Engineering (ICFIAE) is as relevant as opportune, and in my opinion it constitutes a great opportunity to become a permanent forum for reflection and debate on the role of the Fuzzy Sets and Systems in the future of the Society and more particularly in the future of the Digital Society. With an endearing and emotional memory to our always beloved and admired Prof. Zadeh, I want to express my gratitude to the Profs. Hadi Nasseri and Bing-yuan Cao for allowing me to express my opinions on the topic that this important conference addresses. Granada, Spain January 2019

Prof. José Luis Verdegay

Contents

Fuzzy Information and Intelligent System Solving a Discounted Closed-Loop Supply Chain Network Design Problem by Recent Metaheuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atefeh Samadi, Mostafa Hajiaghaei-Keshteli, and Reza Tavakkoli-Moghaddam Intelligent Antenatal Fetal Monitoring Model Based on Adaptive Neuro-Fuzzy Inference System Through Cardiotocography . . . . . . . . . . Xiao-qian Huang, Li Li, Qin-qun Chen, Hang Wei, and Zhi-feng Hao Fighting Detection Based on Hybrid Features . . . . . . . . . . . . . . . . . . . . Shuili Chen, Tengfang Li, Yongli Niu, and Guorong Cai A Method for Mining Temporal Association Rules in Single-Attributed Graph Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . Qi-kai Guo and Fu-sheng Yu

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Fuzzy Decision-Making and Programming Optimal Decision Making in Fuzzy Stochastic Hybrid Uncertainty Environments and Their Application in Transportation Problems . . . . . S. Bavandi, S. H. Nasseri, and C. Triki A Double Interactive Alternative Reduction Approach for Probabilistic Linguistic Multi-criteria Decision-Making with Incomplete Criteria Weight Information . . . . . . . . . . . . . . . . . . . . Na Yue, Jialiang Xie, and Shuili Chen Research on Method of Complex Fuzzy-Valued Integral Classifier in Evaluation of Ecotourism Safety in Hainan Island . . . . . . . . . . . . . . . Jing Ma, Gen-nian Sun, and Sheng-quan Ma

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Contents

Multi-attribute Decision Making Based on the Choquet Integral Operator with Hesitant Fuzzy Linguistic Information . . . . . . . . . . . . . . 107 Xiuqin Xu, Jialiang Xie, and Shuili Chen Fuzzy Engineering Model and Optimize Design of Hierarchical Cone Fuzzy System for Nonlinear System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Ming-zuo Jiang, Xue-hai Yuan, and Jia-xia Wang GPU Local PSO Algorithm at Dimension Level-Based Medical Image Registration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Xicheng Fu, ShengQuan Ma, DaWei Yun, and JiaJing Cai Fuzzy Geometric Programming: Past, Present, and Future . . . . . . . . . . 145 Bing-yuan Cao and Pei-Hua Wang Fuzzy Clustering Analysis of Hotel Online Booking Marketing—A Case of eLong Net . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Nan Wu Fuzzy Algebra and Analysis Application of the Definite Integral of Fuzzy-Valued Function Generated Linearly by Structural Elements . . . . . . . . . . . . . . . . . . . . . . 171 Tian-jun Shu and Zhi-wen Mo Some Typical Fuzzy Complex Set-Valued Integrals . . . . . . . . . . . . . . . . 181 Xue-Ping Zhang, Zhuang-Jian Mo, Jun Shen, and Sheng-Quan Ma The Application of Fuzzy Relational Equations and Genetic Algorithm in Fault Diagnosis Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Hong Mai, Bing-yuan Cao, and Xue-Gang Zhou On Regular, Intra-regular Ordered and Fuzzy Ordered Hypersemigroups in Terms of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Xiang-yun Xie, Lian-fei Gao, and Min Li Semi-weakly ðx1 ; x2 Þ Continuity in Lx-Spaces . . . . . . . . . . . . . . . . . . . . 223 Shu-ting Chen, Shui-li Chen, Jia-liang Xie, and Wei-quan Liu Others Research on System Structure Weight and Error System Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Q. W. Guo, L. T. Zeng, and K. Z. Guo

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Discussion on the Mechanism of Influencing Tax Policy Innovation Based on SEM Model—Taking Guangdong Free Trade Zone for Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Ren-Shou Zhang, Yun-Hui Zhang, Tian-Lin Gu, and Xiao-Jun Huang Online Information Evaluation of High-Quality Hotels in Guangzhou by Comprehensive Factor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Xiang Li Research on System Element Weight and Error System Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Q. W. Guo, L. T. Zeng, and K. Z. Guo Correction to: Intelligent Antenatal Fetal Monitoring Model Based on Adaptive Neuro-Fuzzy Inference System Through Cardiotocography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xiao-qian Huang, Li Li, Qin-qun Chen, Hang Wei, and Zhi-feng Hao

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Fuzzy Information and Intelligent System

Solving a Discounted Closed-Loop Supply Chain Network Design Problem by Recent Metaheuristics Atefeh Samadi, Mostafa Hajiaghaei-Keshteli, and Reza Tavakkoli-Moghaddam

Abstract This paper proposes a discounted closed-loop supply chain (D-CLSC) among the first studies in the area of supply chain network design (SCND) to better focus on the economic performance. Accordingly, a new mixed integer nonlinear programming model is developed to formulate the discount supposition in the transportation costs. To solve the proposed problem, four metaheuristic approaches are tackled to address the problem. In this regard, not only genetic algorithm (GA) is a well-known approach in the field but also two recent nature-inspired algorithms including Keshtel algorithm (KA) and red deer algorithm (RDA) are employed to solve the problem. In addition to this development, the main innovation of this paper is to propose a novel hybrid metaheuristic algorithm based on the benefits of presented algorithms. As a comparative study, some numerical instances are defined and solved by the proposed algorithms and also validated by the outputs of exact solver. Finally, the results indicate that the proposed hybrid algorithm not only is more appropriate than the exact solver but it also outperforms the performance of individual ones particularly in medium- and large-size problems. Keywords Supply chain network design · Discount · Closed-loop supply chain · Metaheuristics · Hybrid methods

A. Samadi (B) Department of Industrial Engineering, Shomal University, Amol, Iran e-mail: [email protected] M. Hajiaghaei-Keshteli Department of Industrial Engineering, University of Science and Technology of Mazandaran, Behshahr, Iran e-mail: [email protected] R. Tavakkoli-Moghaddam School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran © Springer Nature Singapore Pte Ltd. 2020 B.-y. Cao (ed.), Fuzzy Information and Engineering-2019, Advances in Intelligent Systems and Computing 1094, https://doi.org/10.1007/978-981-15-2459-2_1

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1 Introduction and Literature Review Nowadays, supply chain (SC) is described as an integrated system of facilities and activities which involve and coordinate interrelated business functions of material procurement, material transformation to final products and also distribution of these products to the end users [29]. Supply chain management (SCM) is also defined as a management of all parties’ concerns in SC such as movement and replenishment of raw materials and finished goods alongside transportation decisions [28]. Utilizing this concept yields considerable competitive advantages for all partners of a chain [9]. Supply chain network design (SCND) provides strategic decisions playing an important role in the performance of a supply chain and its competitive advantages and also effects on other decisions in an operational and tactical level of supply chain over the time [13, 14]. Choosing good facilities from the possible locations among all potential ones, determining numbers and capacities of network facilities as well as the material flow through the network are the main concerns and decisions in SCND. These decisions are the most important ones in SCM because any minor or major changes in such crucially important decisions in each network influence both directly and indirectly other decisions in lower levels, and consequently cause many costs in whole chain (Devika et al. [6]). There is a gap between the previously published papers and the latest ones in the SCND. Old studies in this research zone mainly focused on the minimizing of the total cost or maximizing of profit, while recent works added the environmental impacts of products and operations as well as the health and safety of employees and whole society [1]. In this regard, activists like people worked as Non-Governmental Organizations (NGOs), and also media declare people’s desires to firms to take the responsibility of their own actions as well as other partners in the supply chains (Devika et al. 2014). Besides, governments and international committees have enacted legislations in favor of the environment such as regulations for greenhouse gases (GHGs) reduction in industrial and developing countries [20]. In a nutshell, supply chains are shifting toward sustainable supply chain management (SSCM) with various motivations such as satisfying activists’ requirements ([30], Yi et al. 2016), and maintaining customers for long term [3], gaining public image [16]. One of recent suppositions in the SCND is the closed-loop supply chain (CLSC) to focus on developing an integrated supply chain network to consider both forward and reverse flows [19]. A forward supply chain network can be considered as a set of levels including supplying, manufacturing and distributing activities [17, 35]. A reverse supply chain network includes some activities which are started with the used products collected by recovering centers [32]. Based on the quality of the collected products, remanufacturing, recycling and or disposal activities should be performed on these collected products [13, 10]. Last but not least, the CLSC not only considers supplying, manufacturing and distribution activities for a forward supply chain, but also collects, remanufactures, recycles and disposes sections for a reverse supply chain. The main benefits of CLSC are the added-value and the reuse of returned

Solving a Discounted Closed-Loop Supply Chain Network …

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products. There are many papers in the literature of forward, reverse and CLSC network design problems which have been classified and analyzed comprehensively these optimization problems in recent years (e.g., Devika et al. [6] and Govindan et al. [25, 20, 21] and also Sahebjamnia et al. [28]). Given CLSC networks tries to develop decision-making models based on the optimization to get closer the realworld applications of SCND. In this regard, the discount problem to be considered in a CLSC network is one of challengeable issues which show a great deal of attention for all governments and international comities [4]. One of important factors of SCND models is the transportation cost (TC). On the other hand, one of the real assumptions about the TC is the discount situation. For instance, Hajiaghaei-Keshteli and Fard [24] presented a SCND model by considering TC discount. They considered two types of the discount; using the distance between facilities and also considering the quantity of production to reduce the TC. They presented two heuristics to handle their proposed multi-objective sustainable CLSC. This paper is another discounted CLSC (D-CLSC) in this area to keep on this interesting research area. Finally, the CLSC network design is NP-hard problem [7, 27, 31]. This motivated several researchers to propose novel and strong approaches in metaheuristics to find robust near-optimal solutions. Generally, metaheuristics are the popular feasible alternatives to solve this complicated problem form the literature [22]. As one of the NP-hard problems, this chance even with low possibility always exists for a new metaheuristic to better solve such complicated model to get a near-optimal solution in less time [13]. This reason motivates several authors to employ different types of metaheuristics in this research area. To cover the limitations of several metaheuristics employed in the literature, this study not only utilizes genetic algorithm (GA) [26] as a well-known in the field and two recent metaheuristics, namely red deer algorithm (RDA) [8] and Keshtel algorithm (KA) [23], but also develops a new hybrid metaheuristic to consider the benefits of these individual algorithms. The reminder of this paper is organized as follows. Section 2 explains the proposed problem based on the details of mathematical formulation. Section 3 introduces and implements the proposed metaheuristics to solve the proposed problem. An extensive comparison and justifications are performed to show a comparative study and to find the best algorithm in Sect. 4. The conclusion and future remarks are presented in Sect. 5.

2 Proposed Problem An MINLP formulation for a CLSC network design problem with ten echelons is introduced. The developed model mainly focuses on the treatment process by taking into account four types of facilities called treatment centers in the reverse network: recovering, remanufacturing, recycling and disposal (Fig. 1). The developed model is founded on the following suppositions in this research area ([10–12, 24, 6]):

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Suppliers (i)

Manufacturing centers (j) Fixed location

Distribution centers (k)

Location to be selected

Forward flow Reverse flow

Costumer zones (l)

Collection/ Inspection centers (m)

Recovering centers (n)

Remanufactures centers (p)

Reuse market

Fig. 1 Graphical illustration of CLSC network

Recycling centers (r)

Disposal centers (s)

Solving a Discounted Closed-Loop Supply Chain Network …

• • • • •

7

The demand of each customer must be met. The number of facilities and their potential sites in each echelon is predefined. The maximum quantity of production flow to allocate the discount is predefined. No flow exists between the facilities of the same echelon. As a special supposition in closed-loop logistics suggested by van der Laan et al. [34], it is assumed that the number of the EOL products returned to the collection/inspection centers is a fraction of the customers’ demands. In addition, they are allocated to the treatment facilities based on their qualities.

This problem aims to determine the amount of products which have to be manufactured at each plant, the assignment of customers to distribution centers and collection/inspection centers, and also the flow of materials should be decided. Tables 1, 2 and 3 illustrate the notations of the proposed model. Table 1 Model notation Indices I

Set of suppliers: i ∈ {1, 2, . . . , I }

J

Set of potential manufacturing centers: j ∈ {1, 2, . . . , J }

K

Set of potential distribution centers: k ∈ {1, 2, . . . , K }

L

Set of customer zones: l ∈ {1, 2, . . . , L}

M

Set of collection/inspection centers: m ∈ {1, 2, . . . , M}

N

Set of manufacturing centers: n ∈ {1, 2, . . . , N }

P

Set of potential remanufacturing centers: p ∈ {1, 2, . . . , P}

R

Set of potential recycling centers: r ∈ {1, 2, . . . , R} Set of potential disposal centers: s ∈ {1, 2, . . . , S}

S e,

e

Set of echelons: e, e ∈ {i, j, k, l, m, n, p, r, s} 

Set of facilities in echelon e : f e , f e ∈ {1, . . . , Fe }

Fe

Table 2 Model notation Variables Continues variables X fe f 

Flow of products from facility f e to facility f e as illustrate in Fig. 1

Hj

Amount of products manufactured at manufacturing center j

e

Binary variables Y fe

1 if facility f e | e {k, m, n, p, r, s, j} is to be established, 0 otherwise

Zkl

1 if customer zone l is assigned to distribution center k, 0 otherwise

Zlm

1 if customer zone l is assigned to collection/inspection center m, 0 otherwise

dc fe f  e

1 if the volume of production is upper than mean fe f  , 0 otherwise e

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A. Samadi et al.

Table 3 Model notation Parameters pci

Cost of purchasing raw material from supplier i

fc fe

Fixed opening cost of facility f e |e ∈ {k, m, n, p, r, s, j}

mc j

Manufacturing cost of each unit product at manufacturing center j

vc fe

Per unit handling cost at facility f e |e ∈ {k, m, n, p, r, s}

tc1f

e f  e

Transportation cost per unit by considering discount from facility f e to facility f e corresponding to X fe f  illustrated in Fig. 1

e f  e

Transportation cost per unit without considering discount from facility f e to facility f e

e

tc2f

mean fe f 

Minimum of volume of production to assignment the discount corresponding to X fe f 

ackl

Per unit cost of assigning customer zone l to distribution center k

cclm

Per unit cost collecting EOL products from customer zone l and shipping to collection/inspection center m

pj

Capacity of manufacturing center j

pfe

Capacity of facility f e |e ∈ {k, m, n, p, r, s}

maxe

Maximum desired number of established sites in echelon e ∈ {k, m, n, p, r, s}

dl

Demand of customer zone l

bj

The fraction of broken products manufactured at manufacturing center j

e

e

∝l

The fraction of used products returned from customer zone l

βmn , βm , βmr and βms

p

The fraction of reusable, recoverable, recyclable and scrapped products p in collection/inspection center m, respectively (βmn + βm + βmr + βms = 1)

γn , γ p and γr

The fraction of products shipped from a recovering and remanufacturing and recycling center to used products market, respectively

scd , scm and scr

Per unit monetary saving resulted from using recovered, remanufactured and recycled EOL products, respectively

scu

Unit selling price of each EOL product in the reuse market

sed , sem and ser

Per unit environmental benefits resulted from using recovered, remanufactured and recycled EOL products, respectively

All in all, the formulations of the proposed CLSC network design problem by using the above notations are presented. These formulations are developed by extending the pervious works (mainly [6]). Min OBJ1 =

 e

+

fc f e Y f e +

fe

 i

j

pci X i j +



mc j H j

j

      tc1f f  e X f e f  , dc f e f  + 1 − dcf f  tc f e f  X f e f  e e  e e e e e

e

f e f  e

e

Solving a Discounted Closed-Loop Supply Chain Network …

+

 k

ackl dl Z kl +



l

9 ⎛

(cclm + vcm ) ∝l dl Z lm − scd ⎝

m

l

 n

⎞ X nk ⎠

k

⎛ ⎞ ⎛ ⎞ ⎛ ⎞    X pj ⎠ − scr ⎝ X ri ⎠ − scu ⎝ γe X m f ⎠ − scm ⎝ e

p

r

j

e

i

fe

(1)

m

The first objective minimizes the total costs of the network. In this regard, the first term is the fixed costs of opening facilities. The second to seventh summations are affiliated with purchasing, manufacturing, handling, transportation and assignment and collection costs. The last four terms stand for the savings resulted from reusing products at the manufacturing centers, from redistributing recovered or remanufactured products, and from selling products to reuse market. In addition, the fifth term of the first objective shows the discount in the TC. The following avouches ensure that the flow of products is maintained and the demands are convinced.  i

  fe

j

X m fe = βme



X jk ∀ j

(2)

dl Z kl ∀k

(3)

k

X jk = 

 l

αl dl Z lm ∀m, e ∈ {n, p, r, s}

l

 k



Xi j =

X nk = (1 − γn )



(4)

X mn ∀n

(5)

 X pj = 1 − γ p X mp ∀ p

(6)

m

m

j



X ri = (1 − γr )



X mr ∀r

(7)

m

i

The amount of products manufactured by each manufacturing center is computed by the following constraint: Hj =



Xi j ∀ j

(8)

i

Each customer zone should be assigned to only one distribution center and collection/inspection center:  k

Z kl =



Z lm = 1 ∀l

(9)

m

The amount of products procured from each supplier is restricted by its capacity:

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A. Samadi et al.



X i j ≤ pi

(10)

j

A manufacturing center can manufacture only when it is opened, and it has idle capacity: Hj ≤ pjYj ∀ j

(11)

The flow of products through a facility is allowed only when the respective facility is operating and has enough capacity as well: 

X jk ≤ pk Yk ∀k

(12)

∝l dl Z lm ≤ pm Ym ∀m

(13)

X m fe ≤ p fe Y fe ∀e ∈ {n, p, r, s}

(14)

 

j

l

m

The number of facilities in each echelon is restricted by a predefined number. 

Y fe ≤ max ∀e ∈ {k, m, n, p, r, s, j } e

fe

(15)

At the end, binary and positive variables are guaranteed. Y fe , Z kl , Z lm , dc fe fe ∈ {0, 1}

(16)

X fe fe , H j ≥ 0

(17)

3 Solution Approach As stated before, the CLSC models are non-polynomial time hard problems that are difficult to solve using different types of exact methods. In addition to its natural complexity, considering dynamic conditions increase its difficulty and combinatorial nature. Hence, metaheuristic approaches should be employed to obtain a satisfying solution in a reasonable time. Several algorithms have been applied in the context of a CLSC network design. One of the most popular algorithms is the genetic algorithm (GA). This motivated to use GA in this study based on previous works in the literature [5]. Due to a no free lunch theory, this chance for a new metaheuristic always exists to reveal a better output in comparison with other existing algorithms [14]. In regards

Solving a Discounted Closed-Loop Supply Chain Network …

11

to this theory, this study employs two recent nature-inspired metaheuristics including Keshtel algorithm (KA) and red deer algorithm (RDA). Last but not least, the main innovation of this study is to propose a novel hybrid metaheuristic based on the advantages of aforementioned algorithms. Here, first of all, the encoding plan of metaheuristics is illustrated and after that the procedures of metaheuristics utilized are addressed in the following subsections.

3.1 Encoding Plan and Initialization In the encoding scheme, a two-stage technique called random-key (RK) is utilized to solve developed discrete problem [14]. This technique helps us to solve our problem with various methods and operators. In RK, two phases are existed. At first, a solution is made by random numbers and then this solution is converted to a feasible discrete solution by a procedure. In this problem, the solutions are defined in two different sub-solutions. The illustration of encoding sub-solutions is depicted in Fig. 2. As shown in this figure, at first a matrix with |p| elements obtained by uniform distribution U(0, 1) is made (Fig. 2a). This sub-solution is used for binary variables such as selection the distribution centers, retailers, collection/inspection centers and recycling centers. Eventually, Fig. 2b determines the flow of products. In the other words, a |P2 |×|D|, |P2 |≤PMax random matrix is produced. Afterward, we normalize the columns of the second matrix specify how DCs supply their demands between different elected plants. To see more examples of the used procedure, Devika et al. [6] is ordered to study. The main phases of the random initialization process for metaheuristics are shown in Fig. 3. In an N dimensional hyperspace distributed by a uniform distribution U(L n , U n ), in which L n is defined the lower bounds and U n is limited the upper bound of variables, respectively, (Devika et al. [6, 24]). When the particles are generated, nine

1

2

0.34

0.57

0.25

0.68

0.92

0

1

0

1

1

Facilities selection sub-solution (a) 2

1 0.65

0.12

0.45

0.33

0.36

0.21

0.33

0.25

0.49

0.08

0.38

0.17

0.27

0.14

0.28

0.13

0.68

0.38

0.52

0.82

0.37

0.66

0.39

0.62

Shipment from plants to distribution centers (b)

Fig. 2 Graphical illustration of structure solution with two parts in a two-stage RK approach. a Facilities selection sub-solution. b Flow of products sub-solution

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A. Samadi et al.

Fig. 3 Pseudo-code of initialization procedure

metaheuristic methods are obtained to decompose the initial solutions and guide the search with creating new values of each particle in the feasible zone, and the new particles are evaluated.

3.2 Genetic Algorithm (GA) The GA proposed by Holland [26] is a well-known evolutionary algorithm by considering each solution as a chromosome and do the mutation and crossover operators to search the feasible area. When GA searches the solution space, the solutions with better quality and also with lower fitness are acceptable leading to escape from local optimum points. In this algorithm, design space should be converted to genetic space. The processing principle in this algorithm is random and guided to optimum place. GA varies with traditional optimization methods in many ways. Generally, the differences among GA and other optimization methods can be expressed as follows: • GA does not search the solution in a single point; however, it searches the solution in parallel. • GA does not use the deterministic rules but uses probabilistic rules. • GA is based on coded variables, unless in cases which variables are illustrated as real numbers. • GA does not require backup information. It only determines the members of objective function and the fitness of path in search space. To apply GA, the following steps are necessary to be taken into account: • Representing an appropriate solution structure. • Obtaining appropriate initial solutions in a population size. • Employing appropriate genetic operators (i.e., mutation and crossover) to obtain new solutions. • Selecting population of the next generation from parent and offspring chromosomes. • Chromosome evaluation measure (i.e., fitness function). • Specifying the stopping criteria. Crossover operator is applied on every pair of randomly selected chromosomes as parents and combines their information to generate two offspring [10]. In the

Solving a Discounted Closed-Loop Supply Chain Network …

13

proposed algorithm, crossover operator substitutes a part of a row from one parent with the same part of the same row of another parent in order to generate two offspring similar to both parents [25]. In this study, parameterized uniform crossover is employed. We choose the first parent among the best individuals in the population while the other one is chosen randomly from the whole of population. Then, a real random number row is produced for each row in the interval [0, 1]. If the random number is larger than a predetermined threshold value, called crossover probability (CProb), then the allele of the first parent is applied. After doing mutation and crossover and choosing the next generation of solutions, to stop GA and present a final solution, stopping criterion should be considered. The stopping criteria mainly defined as follows: • The maximum specified numbers of generation: If the number of generations passes the specified maximum numbers of generation, algorithm will end. • The convergence of population: In broad terms, GA attempts to converge the population to a single population. If the current population converges to a single solution, algorithm will end. • Reaching the specified solving time. In the proposed algorithm, the specified maximum numbers of the generations is used as the stopping criterion.

3.3 Keshtel Algorithm (KA) The Keshtel algorithm is a recent nature-inspired optimizer developed by HjiaghaeiKeshteli and Aminnayeri [23]. It is inspired by the feeding behavior of a dabbling dock, namely Keshtel, in Anas family. The bird inhabits in the north of Iran in cold seasons. They have an amazing behavior for searching foods in the shallow lakes. They migrate every year in cold seasons to the south. The anatomy of these docks helps them to search food carefully and find good foods. Keshtels move so fast in the lake, until they reach a good source food. When a Keshtel reaches a good source food, its neighbors approach to it and they swirl together in a circle. The other that have not found food, may fly to another lake and territory. Accordingly, the swirling process will be continued until the source food run out [13]. To explain the counterpart of proposed algorithm for solving a problem, the user generates the initial population called Keshtel and divides them in three types (i.e. N 1 , N 2 and N 3 ). N 1 includes some Keshtels which have found good foods in first time and called them lucky Keshtels. Also, N 3 includes of the worst solutions. The lucky Keshtels search more food around them. To find the nearest neighbor to lucky Keshtel, the distance between this lucky Kehstel and all Keshtels has to be computed. When better food is found in around a lucky Keshtel, a new lucky is replaced, if not, the swirling will be continued (there is also a parameter to specify the maximum number of swirling). For N 2 population, they move between the two other Keshtels. In addition, for N 3 population, they regenerated randomly for each generation [10, 11].

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A. Samadi et al.

Fig. 4 Pseudo-code of KA

In the KA, trade-off between the phases is very important and relies on the number of each kind of populations. For example, the lucky Keshtels (N 1 ) work for the intensification phase, whereas N 2 and N 3 are considered for the diversification phase. However, the steps in KA are developed in a way that the user can employ just one or two operators among the three ones. According to the best of our knowledge and previous works [13, 23, 24], there is no attempt to employ this metaheuristic in the literature of CMS scheduling problems. Therefore, this study proposes KA by forming the solution representation of proposed model. Given more details about this metaheuristic, the steps of KA are addressed in Fig. 4.

3.4 Red Deer Algorithm (RDA) There is a great deal of interest in developing new nature-inspired metaheuristics by focusing on the two important phases including exploration and exploitation, and their trade-offs. The red deer algorithm introduced by Fathollahi-Fard and Hajiaghaei-Keshteli [15] is one of first successful methods among recent metaheuristics to give the opportunity to a user to make a balance between the exploitation and exploration phases, easily [13]. This algorithm explored the red deer’s characteristics in breeding season and simulated their main behaviors in this specially time of year [18]. The Scottish red feer (Cervus elaphus scoticus) is a subspecies of red deer

Solving a Discounted Closed-Loop Supply Chain Network …

15

and lives in British Isles. The males roar loudly and repeatedly during the breeding season, and females prefer a high to a low roaring rate [28]. The males want to increase their territory and the number of hinds in their harems. So, the course of fight is unavoidable. Although it is possible that a male has no territory and harem, hence, they prefer to mate with a handy hind. In a nutshell, RDA starts with an initial population, called red deers (RD). They are divided into two types: hinds and male RDs. Besides, a harem is a group of female RDs, and the competition of male RDs to get the harem with more hinds via roaring and fighting, and their mating behavior is the basis of the proposed evolutionary algorithm. According to the best of our knowledge [8, 13, 18, 28], there is no tried to use this algorithm in the area of CMS scheduling problems. In this regard, this is the first attempt to employ this recent nature-inspired optimizer to solve the proposed problem. In continuous, the steps of the algorithm are detailed by considering a multi-objective version of RDA as seen in Fig. 5. In the red deer algorithm (RDA), the authors find a creative way to trade-off between the phases by three simple operators. RDA does local search by roaring males and also performs the exploitation phase by fighting between different types of males. Also, by mating in three ways to search the potential areas, diversification phase is done.

Fig. 5 Pseudo-code of RDA

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A. Samadi et al.

3.5 Proposed Novel Hybrid Metaheuristic (H-RDKGA) As indicated from previous discussion, the KA is very good at doing the exploitation action. It seems that the swirling process can be done instead of two processes including of roaring and fighting in RDA. Accordingly, for each male, the closest neighbor is specified and the swirling action is done. Due to mating process, the GA mechanism is considered in this regard. Having a brief illustration, the KA is chosen the intensification properties as well as the GA is measured the diversification phase. This opinion is employed to examine the proposed method with their individual methods and also other feasible alternatives for combinations. Given more details of proposed H-RDKGA, a pseudo-code is provided as seen in Fig. 6.

4 Experimental Results A comparative study is presented in this section. First of all, to enhance the performance of employed metaheuristics and having a fair comparison, a full factorial

Fig. 6 Pseudo-code of H-RDKGA

Solving a Discounted Closed-Loop Supply Chain Network …

17

design method is applied to tune the algorithms’ parameters properly. After that, an extensive comparison among metaheuristics based on different criteria is presented in the following subsection.

4.1 Tuning of Metaheuristics For tuning the parameters of the metaheuristic techniques, we use the design of experiment (DOE) method discussed in Bezerra et al. [2]. The reason why we chose to use DOE compared to more efficient calibration techniques is that the method is simple and coarse [10, 11], and hence, the direct impact of the algorithms on the problem solutions can be understood without the presence of a good calibration technique. As given in the solution algorithm, the main parameters under consideration for GA are the population size, maximum number of iterations, mutation and crossover rates. In the proposed KA, the population size, maximum number of iterations, the percentage of N 1 and N 2 , maximum number of swirling are the key parameters. As such, the main parameters of RDA are the population size, maximum number of iterations, number of males, and the rate of alpha, beta and gamma. At the last, the proposed parameters of H-RDKGA are only the population size, maximum number of iterations, the number of males and the maximum number of swirling. We use the full factorial design to evaluate the different sets of values considered for conducting the DOE (given in Table 4). These range of values are decided based on the parameter settings provided in the literature ([13, 14, 18, 28], [25]).

4.2 Comparison Among Employed Metaheuristics Here, an extensive comparison among employed metaheuristics is done. First of all, nine test problems with different complexity from small to large are conducted to evaluate the algorithms. Due to natural stochastic of metaheuristics, all of them in each size test problem are run for 10 times. In this regard, the best, the worst, the average and the standard deviation of solutions among all outputted ones are analyzed. The computational time of algorithms are also noted. To check the optimal results of algorithms, an exact solver implemented by GAMS software is also selected to solve the test problems. All results are given in Table 5. Regarding the table, it should be noted that the exact solver is not capable of finding feasible solutions for the large-sized problems even after 1 h. However, metaheuristics have solved these examples in less than 2 min. The behavior of algorithms based on the computational time is given in Fig. 7. The gap of the best metaheuristic solutions from the best solution ever found by the exact solver is depicted by Fig. 8. At the end, some statistical analyses are done to show the robustness of metaheuristics in comparison with each other as seen in Fig. 9.

18

A. Samadi et al.

Table 4 Tuning of metaheuristics Metaheuristic

Parameters

Levels

GA

Population size

100

150

200

200

Maximum number of iterations

300

500

700

500

Rate of mutation

0.05

0.15

0.25

0.15

KA

RDA

H-RDKGA

Tuned value

Rate of crossover

0.6

0.7

0.8

0.8

Population size

100

150

200

100

Maximum number of iterations

300

500

700

300

Percentage of N1

0.1

0.2

0.3

0.1

Percentage of N2

0.4

0.5

0.6

0.6

Maximum number of swirling

5

10

15

10

Population size

100

150

200

150

Maximum number of iterations

300

500

700

700

Number of males

15

25

30

25

Alpha

0.5

0.6

0.7

0.6

Beta

0.7

0.8

0.9

0.7

Gamma

0.8

0.9

1

0.8

Population size

100

150

200

150

Maximum number of iterations

300

500

700

500

Number of males

15

25

30

30

Maximum number of swirling

5

10

15

15

As indicated from Fig. 7, the behavior of algorithms is as the same in overall. The proposed hybrid algorithm and KA show a competitive results in this item. Generally, the best algorithm in this criterion is the KA. However, the worst behavior can be concluded from RDA in most of testes. What can be seen at the first glance of Fig. 8 reveals that there is a set of similarities between the gap behaviors of metaheuristics. However, the proposed hybrid H-RDKGA shows robust behavior and reaches a near-global solution in all of small and medium test problems. As can be seen from Fig. 9, there is a clear difference between the performances of metaheuristics. The best algorithm is clearly the proposed hybrid metaheuristic. Conversely, the worst metaheuristic is the GA in this comparison. There is a little difference between the KA and the RDA. However, the RDA is slightly better than the KA. In conclusion, by considering different measurements to analyze the performance of algorithms, it is evident that the proposed hybrid algorithm called as H-RDKGA is the most powerful metaheuristic in this comparative study.

RDA

KA

GA

EX

Algorithm

24,524

18

CPU

W

194

SD

24,283

24,428

B

24,525

22

CPU

A

4205

SD

W

24,283

A

24,283

27,925

B

24,283

W

18

CPU

B

24,283

A

P1

28,926

28,641

15

230

28,812

28,927

28,641

17

4960

28,641

32,937

28,641

64

28,641

P2

Test problems

Table 5 Comparison of algorithms

85,021

84,180

20

677

84,684

85,021

84,180

22

15,740

85,637

98,482

84,180

201

84,180

P3

122,617

119,046

28

3834

121,902

123,807

119,046

32

22,090

120,903

139,038

119,046

836

119,046

P4

226,445

219,850

38

7085

225,246

228,766

219,968

42

40,414

224,095

257,709

221,470

1872

218,907

P5

490,652

476,362

58

19,358

495,212

504,828

480,789

65

101,164

497,882

572,564

476,124

3315

476,036

P6

708,591

687,953

72

28,254

722,763

736,797

701,712

79

129,847

706,813

812,835

694,902

3600

–

P7

971,677

943,376

88

38,745

991,110

1,010,355

962,243

96

180,812

972,491

1,118,365

952,906

3600

–

P8

(continued)

122,459

118,893

92

4882

124,907

127,333

121,270

102

20,929

120,265

138,304

120,094

3600

–

P9

Solving a Discounted Closed-Loop Supply Chain Network … 19

24,283

0

22

A

SD

CPU

24,283

W

24

CPU

24,283

SD

B

24,451

160

A

P1

16

286

28,927

29,213

28,641

18

190

28,840

P2

Test problems

20

841

85,021

85,863

84,180

26

560

84,768

P3

26

1190

120,236

121,426

119,046

33

2382

121,545

P4

40

2189

221,096

223,285

218,907

43

4399

224,466

P5

62

4761

480,862

485,623

476,101

66

9532

486,365

P6

72

6810

687,814

694,624

681,004

78

13,767

702,399

P7

90

9338

943,185

952,523

933,847

95

18,879

963,186

P8

EX = exact solver; B = best, W = worst, A = average, SD = standard deviation, CPU = computational time based on the second

H-RDKGA

Algorithm

Table 5 (continued)

94

1176

118,868

120,045

117,692

106

2378

121,389

P9

20 A. Samadi et al.

Solving a Discounted Closed-Loop Supply Chain Network …

21

105 95

CPU (second)

85 75 65 55 45 35 25 15

Test problems GA

KA

RDA

H-RDKGA

Fig. 7 Behavior of algorithms in terms of computational time 0.014 0.012

Gap

0.01 0.008 0.006 0.004 0.002 0

Test problems GA

KA

RDA

H-RDKGA

Fig. 8 Gap behavior of algorithms

5 Conclusion and Future Works This paper developed a discounted closed-loop supply chain (D-CLSC) among the first studies in the area of supply chain network design (SCND) to better focus on the economic performance. Accordingly, a new mixed integer nonlinear programming model was propped to formulate the discount supposition in the transportation costs. To solve the proposed problem, four metaheuristic approaches were tackled to address the problem. In this regard, not only genetic algorithm (GA) as a well-known approach in the field but also two recent nature-inspired algorithms including Keshtel

22

A. Samadi et al. Interval Plot of H-RDKGA, RDA, KA, GA 95% CI for the Mean

1.0 0.8

Data

0.6 0.4 0.2 0.0 H-RDKGA

RDA

KA

GA

Fig. 9 Interval plots of algorithms based on the standard deviation of algorithms

algorithm (KA) and red deer algorithm (RDA) were employed to solve the problem. In addition to this development, the main innovation of this paper was to propose a novel hybrid metaheuristic algorithm based on the benefits of presented algorithms. As a comparative study, some numerical instances were defined and solved by the proposed algorithms and also validated by the outputs of exact solver. Finally, the results indicate that the proposed hybrid algorithm not only was more appropriate than the exact solver but it also outperformed the performance of individual ones particularly in medium- and large-size problems. There are several recommendations for future works. New hybrid and improved algorithms can be employed to solve the problem and compared with own results. More in-depth analyses such as sensitivity analyses can be done. Other real-life suppositions such routing operations and inventory policies can be applied to better simulate the CLSC network design problem. Uncertainty of parameters by using stochastic or fuzzy numbers, is another interesting continuation of this study. Acknowledgements Thanks to the supported by the 9th International Conference on Fuzzy Information and Engineering (Kish Island). Recommender: Thanks to Professor Hadi Nasseri’s recommendation of University of Mazandaran in Iran.

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Intelligent Antenatal Fetal Monitoring Model Based on Adaptive Neuro-Fuzzy Inference System Through Cardiotocography Xiao-qian Huang, Li Li, Qin-qun Chen, Hang Wei and Zhi-feng Hao

Abstract In non-stress tests (NST) of antenatal fetal monitoring, obstetricians usually interpret the cardiotocography (CTG), not only in line with fetal monitoring guidelines, but also in accordance with the knowledge gained from individual clinical experience, by which prognosis would be uncertain and ambiguous. Moreover, CTG contains considerable uncertainty and fuzziness, for it is a multi-component and nonlinear complex system which has large amount of information-doped noise. Therefore, an intelligent antenatal fetal monitoring model based on adaptive neural network fuzzy inference system (ANFIS) is presented in this paper. Nine important features were extracted from the CTG case data, the number of fuzzy rules was determined by subtractive clustering, after which the fuzzy system was initialized and adjusted through the self-learning mechanism of neural networks. The efficiency of the proposed model was tested on the antenatal CTG dataset from the UCI repository. The experimental results showed that the method outperformed the existing state-of-the-art antenatal fetal monitoring models and had a significant advantage on the gray stage of “suspicious” discrimination. It indicates that the proposed model has promising learning ability and adaptability for the uncertainty of antenatal fetal monitoring. Keywords Intelligent antenatal fetal monitoring · Cardiotocography (CTG) · Uncertainty · Adaptive neural network inference system (ANFIS) The original version of this chapter was revised: The author’s name was updated from Hag Wei to Hang Wei. The correction to this chapter is available at https://doi.org/10.1007/978-981-15-24592_22 X. Huang · Q. Chen · H. Wei (B) School of Medical Information Engineering, Guangzhou University of Chinese Medicine, Guangzhou, China e-mail: [email protected] L. Li First Affiliated Hospital of Jinan University, Tianhe District People’s Hospital, Guangzhou, China Guangzhou Sunray Medical Apparatus Co. Ltd, Guangzhou, China Z. Hao School of Mathematics and Big Data, Foshan University, Foshan, China © Springer Nature Singapore Pte Ltd. 2020, corrected publication 2020 B.-y. Cao (ed.), Fuzzy Information and Engineering-2019, Advances in Intelligent Systems and Computing 1094, https://doi.org/10.1007/978-981-15-2459-2_2

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1 Introduction Antenatal fetal monitoring is a key link in fetal health monitoring. It helps early diagnosis of abnormalities, such as congenital heart defects, fetal distress or hypoxia, by responding to which obstetricians can take measures to prevent the irreversible damage to the fetus [1]. Non-stress tests (NST) were widely introduced to antenatal fetal monitoring in the late 1960s and are still used extensively, for their low cost, easy operation, and non-invasion [2]. In the NST, the interpretation of cardiotocography (CTG), which includes signals of both fetal heart rate and uterine contraction, to date primarily depends on obstetricians. However, the demands of fetal monitoring have been dramatically increasing, while shortages of professional obstetricians remain acute and fetal monitoring levels in primary or rural hospitals are quite low [3]. Hence, intelligent antenatal fetal monitoring has been being intensively studied as the development of machine learning and artificial intelligence [4–10]. In antenatal fetal monitoring, obstetrician interpretations of CTG contain uncertainty and fuzziness. The studies by Gagnon et al. have found that even the same obstetrician has a different interpretation of the same CTG at a subsequent time, let alone by a different obstetrician [11]. Obstetricians usually interpret the CTG, not only according to fetal monitoring guidelines but also in line with the knowledge obtained from individual clinical experience [12–14]. Moreover, the outcomes of fetal status evaluation are labeled as dynamic ranking stages, taking “normal, suspicious, or pathological” patterns in the International Federation of Gynecology and Obstetrics (FIGO) guidelines. In addition, the CTG also shows considerable uncertainty and fuzziness because it is a multi-component and nonlinear complex system which contains large amount information-doped noise. So far, however, the studies on intelligent antenatal fetal monitoring have paid scant attention to the uncertainty and fuzziness of both obstetrician interpretations and CTG data. Although Hasan proposed an antenatal fetal evaluation model based on adaptive neuro-fuzzy inference system (ANFIS), merely the normal and pathological groups were taken into the study, which lacked the gray group of “suspicious” discrimination [15]. Fuzzy system theory devotes itself to the modeling, prediction, decision making, control and such issues of systems which have unclear patterns, incomplete behavioral information, and indistinct operation mechanisms [16]. Fuzzy system theory was introduced in this study to deal with the uncertainty and fuzziness in antenatal fetal monitoring; the ANFIS was applied in the intelligent discriminant model. The important features were extracted from the CTG case data, the number of fuzzy rules was determined by subtractive clustering, after which the fuzzy system was initialized and adjusted through the self-learning mechanism of neural networks. The experimental results of UCI repository showed that the ANFIS outperformed the existing state-of-the-art antenatal fetal monitoring models and had an outstanding advantage on the gray stage of “suspicious” recognition. By the proposed model, the accuracy values of the “normal, suspicious, and pathological” groups are 98.53%, 97.30%, and 96.00%, respectively. These results demonstrate that the proposed ANFIS is feasible

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in antenatal fetal monitoring, and has promising learning ability and adaptability for the uncertainty in evaluation of antenatal fetal health. The rest of this paper is organized as follows. Section 2 introduces the related concepts and principles. Materials and methods are illustrated in Sect. 3. To demonstrate the performance and applicability of the proposed method, a series of experiments were carried out; the numerical results and comparative analysis for these experiments are given in Sect. 4, and the conclusions are summarized in Sect. 5.

2 Basic Concepts and Principles In this section, the use of continuous cardiotocography (CTG) and its interpretative methods for antenatal fetal monitoring are introduced. Subsequently, the mathematical formulations of the fuzzy rules for antenatal CTG classification based on Takagi–Sugeno (T-S) fuzzy system are proposed.

2.1 CTG and Its Interpretations in Antenatal Fetal Monitoring At the present time, the most widely used electronic fetal monitoring (EFM) in clinical non-stress tests (NST) is continuous cardiotocography (CTG) [17]. When utilizing CTG monitoring, it is necessary to record both fetal heart rate (FHR) and tocography (TOCO) representing uterine pressure of pregnant women. Obstetricians interpret and evaluate the fetal status according to whether or not the acceleration criteria are met. Aside from acceleration, other parameters of electronic fetal heart assessment, including baseline rate, variability, and the presence or absence of decelerations, should also be assessed [18]. To the extent known by the authors of this paper, there are two CTG interpretative methods, presently in use for antenatal fetal monitoring in NST, namely “scoring” and “ranking”. The scoring methods for evaluating antenatal fetal status include NST, Krebs, Fischer, and improved Fischer [2, 19, 20]. The above scoring methods have been developed abroad and have the following shortcomings. Most notably among these: it is unclear for an untrained evaluator as to whether or not the fetal status is normal in corresponding to the given score. Secondly, the scales are not uniform. Krebs is on 12 scale, while others are on 10 scale. Thirdly, the parameters themselves are inconsistent with each other. Finally, the above methods were developed based on the particular population of people; therefore, they may not give the desired result when applied to another population of people, for example, Chinese people. With respect to the ranking methods for antenatal CTG, there have been a number of academic research projects with difference results carried out in various countries. At the present, the accepted and recognized antenatal fetal monitoring guidelines for

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ranking methods are those referenced by the Society of Obstetricians and Gynecologists of Canada (SOGC) [18], American College of Obstetrics and Gynecology (ACOG), National Institute of Health and Clinical Excellence (NICE) [21], the International Federation of Gynecology and Obstetrics (FIGO) [22, 23], and Expert Consensus in China (ECC) [24]. Among them, SOGC, ACOG, and FIGO guidelines adopt three-category classification, but NICE guidelines employ four-category classification [21] while ECC guidelines only classify antenatal CTG into two categories, namely “reactive” and “non-reactive” [24]. Generally speaking, although the above interpretation methods can illustrate the fetal status, high sensitivity and low specificity are common in clinical application [2, 22, 24, 25]. Particularly, when the recording lasts less than 40 min, a false positive is prone to occur, a finding which will lead to fetal distress over-treatment and result in unnecessary cesarean sections for pregnant women [25].

2.2 Takagi–Sugeno Fuzzy Inference System The ANFIS is one of fuzzy inference models based on the Takagi–Sugeno (T-S) fuzzy system [26]. In this paper, membership function and fuzzy rules of the proposed model were able to be developed by training antenatal CTG samples which have been evaluated by obstetrics experts. The authenticity and completeness of the CTG data help to improve the classification accuracy in fetal status evaluation. If antenatal CTG data have p features and q rules, the fuzzy rules are as follows: ⎧ i f x1 i s A1,1 , x2 i s A1,2 , . . . , x j i s A1, j , then f 1 = k1,1 x1 + k1,2 x2 + . . . + k1, j x j + r1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i f x1 i s A2,1 , x2 i s A2,2 , . . . , x j i s A2, j , then f 2 = k2,1 x1 + k2,2 x2 + . . . + k2, j x j + r2 ⎪ ⎪ ⎪ ⎪ ⎨ . . . ⎪ ⎪ . . . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ . . . ⎪ ⎪ ⎪ ⎩ i f x1 i s Ai,1 , x2 i s Ai,2 , . . . , x j i s Ai, j , then f i = ki,1 x1 + ki,2 x2 + . . . + ki, j x j + ri

(1)

where x j ( j = 1, 2, . . . , p) is the jth feature in antenatal CTG data; f i (i = 1, 2, . . . , q) is the ith rule; ri (i = 1, 2, . . . , q) is the constant term of the ith rule; Ai, j (i = 1, 2, . . . , q; j = 1, 2, . . . , p) is the fuzzification of the jth feature and rule i; pi, j (i = 1, 2, . . . , q; j = 1, 2, . . . , p) is the linear parameter of the jth feature in the ith rule. The ANFIS integrates the interpretability of fuzzy interference system and the self-adaptive learning of neural network system, thus it can more effectively solve the complex and nonlinear problems involving modeling and control systems [27, 28]. Meanwhile, antenatal CTG evaluation is a complex, multi-component, and nonlinear system with uncertainty and fuzziness.

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Fig. 1 CTG data distribution

3 Materials and Methods 3.1 Dataset Description The simulation experiments were carried out on a standard antenatal CTG dataset taken from UCI (Univ. of California at Irvine) Machine Learning Repository [29]. The CTG data, respecting gestational ages that ranged from 29 to 42 weeks, were collected in SisPorto 2.0 of Portugal [30]. The dataset consisted of 2126 instances with 21 attributes and the instances were categorized by the consensus of three expert obstetricians in line with FIGO guideline into three groups: 1655 normal, 295 suspicious, and 176 pathological datum. It can be seen in Fig. 1 that the CTG dataset had classification imbalance problem, which is generally more serious in clinical practice.

3.2 CTG Data Exploration, Feature Extraction, and Data Preprocessing To explore the important features, the distribution of each attribute and fetal status were depicted by pivot table (Fig. 2). For example, Fig. 2 indicated that the percentage of time with abnormal long-term variability (ALTV) and percentage of time with abnormal short-term variability (ASTV) had great impact on the fetal status evaluation. When their values increased, the possibilities of pathological pattern also rose.

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(a)

(b)

Fig. 2 a Distribution of ALTV and fetal status. b Distribution of ASTV and fetal status

Table 1 One-hot encoding for CTG categories

NSP

NSP_1

NSP_2

NSP_3

Normal

1

0

0

Suspicious

0

1

0

Abnormal

0

0

1

After the analysis of the correlation between each attribute and fetal status, the most influential CTG attributes, namely fetal movement (FM), prolonged deceleration (DP), abnormal long-term variation (ALTV), abnormal short-term variation (ASTV), mean, mode, MLTV, median and accelerated (AC), were found. The important attributes would be employed as the input features of the proposed ANFIS model. The input features had different measurements and their values varied greatly. In order to eliminate the internal influence, the input features were normalized. One-hot encoding was used to transform the categories into multiple patterns (Table 1).

3.3 ANFIS Model for Antenatal Fetal Monitoring The dataset was randomly divided into training set and test set by 3:1 ratio. The structure of ANFIS system was preliminarily determined by subtractive clustering. The parameters of subtractive clustering were set as follows: range of influence = 0.2, squash factor = 1.38, accept ratio = 0.5, and reject ratio = 0.15 according to the tests. The initial structure and parameters of ANFIS were trained by a hybrid learning algorithm in training set. Error tolerance was set to 0, the number of training was set to 500, and the terminal error was 0.26% (Fig. 3). Terminally, five fuzzy rules were obtained. The structure diagram of this new model ANFIS was shown in Fig. 4.

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Fig. 3 Training iteration process of the ANFIS model

Fig. 4 Structure diagram of the ANFIS model

4 Results and Discussion 4.1 The Experimental Results of the ANFIS Model The accuracy values of the proposed ANFIS model for antenatal fetal monitoring in training set and test set were 97.3% and 97.54%, respectively. The mixed efficiency matrix of test set was shown in Table 2. The accuracy values of normal, suspicious, and abnormal categories were 98.53%, 97.30%, and 96.00%, respectively.

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Table 2 Mixed efficiency matrix of the ANFIS model

Reality prediction

Normal

Suspicious

Abnormal

Normal

0.9853

0

0.0400

Suspicious

0.0049

0.9730

0

Abnormal

0.0098

0.0270

0.9600

4.2 Comparison with the Conventional Machine Learning Methods To demonstrate the performance of the proposed ANFIS model, a series of comparison experiments were carried out with conventional machine learning methods, including decision tree (DT), random forest (RF), support vector machine (SVM), and BP neural network (BP). The numerical outcomes were shown in Table 3. The proposed ANFIS, with the highest accuracy and the shortest running time, obviously outperformed the above four conventional machine learning methods. In clinical practice, there frequently exists serious classification imbalance in antenatal fetal monitoring. Contrasted with the normal, there are much fewer cases of the suspicious and abnormal categories, especially those of abnormal category. To illustrate the adaptability of the ANFIS to solving the classification imbalance problem, misclassification cobweb for six classifiers, namely ANFIS, DT, RT, BP, SVM, and chance, was graphed in Fig. 5. Among the six kinds of misclassification, the most dangerous one is to misclassify the abnormal or suspicious into the normal ones, because it will lead to missing the best treatment time and even endangering the fetal health. In addition, it is also dangerous that the normal one is evaluated as abnormal one, for it may result in over-treatment, excessive intervention, and unnecessary cesarean section. It showed that among the three most harmful misclassifications, the ANFIS model had the lowest misclassification rate in the comparison experiments. Furthermore, the highest misclassification rate of the ANFIS model was misjudging the abnormal ones as normal ones with a quite low probability of 0.04. To demonstrate how accurately the ANFIS model can discriminate the threecategory CTG data, the precision, recall, and F1-score values of the normal, suspicious, and abnormal categories by the above five models were displayed in Table 4. Table 3 Comparison of overall accuracy and efficiency between ANFIS and other models Model

Accuracy

Running time (s)

Prediction error

ANFIS

0.9727

0.017

0.0273

DT

0.9041

0.044

0.0959

RF

0.9323

1.199

0.0677

SVM

0.8726

1.1284

0.1274

BP

0.8383

9.1507

0.1617

Note The CPU configuration of the machine is Core (TM) i5-6300HQ, 2.3 GHz, 8 GB of memory, and the system is 64 bits in the Chinese version of Win10

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Fig. 5 Misclassification cobweb for six classifiers

Table 4 Classification evaluation tables for ANFIS and other machine learning methods Evaluating indicator

Precision

Recall

F1

ANFIS

Normal

0.9950

0.9853

0.9901

Suspicious

0.9474

0.9728

0.9599

Abnormal

0.9057

0.9600

0.9320

Normal

0.9442

0.9760

0.9598

Suspicious

0.8750

0.6622

0.7539

Abnormal

0.8913

0.9762

0.9318

Normal

0.9443

0.9375

0.9409

Suspicious

0.6923

0.7297

0.7105

Abnormal

0.9024

0.8810

0.8916

Normal

0.8998

0.9495

0.9240

Suspicious

0.5862

0.4595

0.5152

Abnormal

0.8788

0.6905

0.7734

Normal

0.8758

0.9495

0.9112

Suspicious

0.3846

0.0676

0.1150

Abnormal

0.6207

0.8571

0.7180

RF

DT

SVM

BP

Compared with the conventional machine learning methods, the ANFIS model greatly reduced the misclassification rate of suspicious class in the gray stage, and the recognition of abnormal class was also significantly improved. These results illustrate that the ANFIS is promising for dealing with the uncertainties and fuzziness in CTG interpretation, but also the class-imbalanced problems in three-category CTG data.

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4.3 Comparison with the Existing Antenatal Fetal Monitoring Models To illustrate the proposed ANFIS model, accuracy values and F1 scores of the existing antenatal fetal monitoring models were listed in Tables 5 and 6, respectively. The ANFIS was far superior on unbalanced CTG data classification and had the best classification performance. These results furthermore demonstrate that the ANFIS can effectively distinguish unbalanced CTG data and deal with the uncertainty and fuzziness in antenatal CTG interpretations. Table 5 Comparison of overall accuracy of CTG discriminant models at home and abroad Model

Accuracy (%)

Model

Accuracy (%)

BP [4]

91.30

DA [8]

82.03

GRNN [5]

91.86

DT [8]

86.31

PNN [5]

92.14

LS-SVM-PSO-BDT [9]

91.58

MLPNN [5]

90.36

DT-AdaBoost [10]

95.01

RF [6]

93.60

ANFIS [this paper]

97.27

IAGA [7]

93.89

Table 6 Comparison of classification performance of CTG discriminant models at home and abroad Model

Feature

Normal (%)

Suspicious (%)

Abnormal (%)

Average F1 (%)

BP [4]

22

97.84

45.14

97.24

80.07

GRNN [5]

21

95.70

73.92

84.88

84.83

PNN [5]

21

95.91

73.81

85.45

85.06

MLPNN [5]

21

95.00

68.43

80.85

81.31

RF [6]

21

96.40

79.60

91.20

89.07

6

96.83

79.15

89.41

88.46

DT [8]

10

97.1

83.69

92.84

91.23

DA [8]

10

89.69

58.50

65.58

71.26

LS-SVM-PSO-BOT [9]

21

96.02

72.98

79.18

82.73

DT-AdaBoost [10]

21

93.31

60.09

66.43

73.28

ANFIS [this paper]

9

99.01

95.99

93.20

96.07

IAGA [7]

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5 Conclusion Intelligent antenatal fetal monitoring through CTG is challenging. Both CTG and its interpretation embody uncertainty and fuzziness. At the same time, a serious class-imbalanced problem exists in prenatal fetal monitoring cases. Nine important features, such as baseline, long variation, and short variation, were selected and then an intelligent antenatal fetal monitoring based on ANFIS was established. The experimental results showed that the proposed model holds promise for solving not only the uncertainty problem but also the class-imbalanced problem as well. The data explorations for extracting important CTG features are the essential steps for the establishment of fuzzy logical model. Hence it would be desirable that future work investigates the applicable conditions of feature extracting methods in practical antenatal fetal monitoring. On the other side, the self-learning algorithm of neural networks in ANFIS needs to be improved, so that local optimum can be avoided and convergence can be seeped up even in the case of many input features, for example, beyond 15. Acknowledgements This work is supported by the Medical Scientific Research Foundation of Guangdong Province under Grant No. A2019428 and the Natural Science Foundation of Guangdong Province under Grant No. 2015A030310312. Recommender: This paper is recommended by Rui-chu Cai who is a professor of Guangdong University of Technology in China.

References 1. Haran, S.S., Everett, T.R.: Antenatal fetal wellbeing. Obstet. Gynaecol. Reprod. Med. 27(2), 44–49 (2017) 2. Grivell, R.M., Alfirevic, Z., Gyte, G.M.L., et al.: Antenatal cardiotocography for fetal assessment. The Cochrane Library (2015) 3. State Statistical Bureau. Statistical Yearbook of China. China Statistics Press, Beijing (2017) 4. Sundar, C., Chitradevi, M., Geetharamani, G.: Classification of cardiotocogram data using neural network based machine learning technique. Int. J. Comput. Appl. 47(14), 19–25 (2013) 5. Yılmaz, E.: Fetal state assessment from cardiotocogram data using artificial neural networks. J. Med. Biol. Eng. 36(6), 820–832 (2016) 6. Arif, M.: Classification of cardiotocograms using random forest classifier and selection of important features from cardiotocogram signal. Biomater. Biomech. Bioeng. 2(3), 173–183 (2015) 7. Ravindran, S., Jambek, A.B., Muthusamy, H., et al.: A novel clinical decision support system using improved adaptive genetic algorithm for the assessment of fetal well-being. Comput. Math. Methods Med 2015(283532) 8. Huang, M.L., Hsu, Y.: Fetal distress prediction using discriminant analysis, decision tree, and artificial neural network. J. Biomed. Sci. Eng. 5(9) (2012) 9. Yılmaz, E., Kılıkçıer, Ç.: Determination of fetal state from cardiotocogram using LS-SVM with particle swarm optimization and binary decision tree. Comput. Math. Methods Med. 2, 487179 (2013)

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10. Karabulut, E.M., Ibrikci, T.: Analysis of cardiotocogram data for fetal distress determination by decision tree based adaptive boosting approach. J. Comput. Commun. 02(9), 32–37 (2014) 11. Gagnon, R., Campbell, M.K., Hunse, C.: A comparison between visual and computer analysis of antepartum fetal heart rate tracings. Am. J. Obstet. Gynecol. 168(3 Pt 1), 842 (1993) 12. Goddard, R.: Electronic fetal monitoring: is not necessary for low risk labours. BMJ Br. Med. J. 322(7300), 1436 (2001) 13. Rooth, G., Huch, A., Huch, R.: Guidelines for the use of fetal monitoring. Int. J. Gynaecol. Obstet. 25, 159–167 (1987) 14. Bernardes, J., Costa-Pereira, A., Ayres-de-Campos, D., et al.: Evaluation of interobserver agreement of cardiotocograms. Int. J. Gynecol. Obstet. 57(1), 33–37 (1997) 15. Ocak, H., Ertunc, H.M.: Prediction of fetal state from the cardiotocogram recordings using adaptive neuro -fuzzy inference systems. Neural Comput. Appl. 23(6), 1583–1589 (2013) 16. Mendel, J.M.: Uncertain Rule-Based Fuzzy Systems. Springer, Cham CrossRef MATH Google Scholar (2017) 17. Alfirevic, Z., Devane, D., Gyte, G.M.L., et al.: Continuous cardiotocography (CTG) as a form of electronic fetal monitoring (EFM) for fetal assessment during labour. The Cochrane Library (2017) 18. Liston, R., Sawchuck, D., Young, D., et al.: Fetal health surveillance: antepartum and intrapartum consensus guideline. J. Obstet. Gynaecol. Can. 29(9), S3–S4 (2007) 19. Lyons, E.R., Bylsma-Howell, M., Shamsi, S., et al.: A scoring system for nonstressed antepartum fetal heart rate monitoring. Am. J. Obstet. Gynecol. 133(3), 242–246 (1979) 20. Caixuan, T.: The dependability analysis between the evaluates on the fetus electron monitoring and the birth status of infant by NST score and reforming Fischer score. China Mod. Med. 16(22), 28–29 (2009) 21. National Collaborating Centre for Women’s and Children’s Health (UK): Intrapartum Care: Care of Healthy Women and Their Babies During Childbirth (2014) 22. Santo, S., Ayres-de-Campos, D., Costa-Santos, C., et al.: Agreement and accuracy using the FIGO, ACOG and NICE cardiotocography interpretation guidelines. Acta Obstet. Gynecol. Scand. 96(2), 166–175 (2017) 23. Ayres-de-Campos, D., Spong, C.Y., Chandraharan, E.: FIGO consensus guidelines on intrapartum fetal monitoring: cardiotocography. Int. J. Gynecol. Obstet. 131(1), 13–24 (2015) 24. Chinese Society of Perinatal Medicine: Expert consensus on the electronic fetal monitoring’s application. Chin. J. Perinat. Med. 7, 486–490 (2015) 25. Huiping, Zhang: Analysis and management on 368 abnormal fetal monitoring in non-stress test. Chin. Gen. Pract. Nurs. 14(36), 3837–3838 (2016) 26. Li, L., Chadli, M., Ding, S.X., et al.: Diagnostic observer design for T-S fuzzy systems: application to real-time-weighted fault-detection approach. IEEE Trans. Fuzzy Syst. 26(2), 805–816 (2018) 27. Karaboga, D., Kaya, E.: Adaptive network based fuzzy inference system (ANFIS) training approaches: a comprehensive survey. Artif. Intell. Rev. 1–31 (2018) 28. Mlaki´c, D., Baghaee, H.R., Nikolovski, S.: A novel ANFIS-based islanding detection for inverter–interfaced microgrids. IEEE Trans. Smart Grid 10, 4411–4424 (2018) 29. Asuncion, A., Newman, D.: UCI Machine Learning Repository (2007) 30. Ayresde, C.D., Bernardes, J., Garrido, A., et al.: SisPorto 2.0: a program for automated analysis of cardiotocograms. J. Maternal-Fetal Med 9(5), 311–318 (2000)

Fighting Detection Based on Hybrid Features Shuili Chen, Tengfang Li, Yongli Niu, and Guorong Cai

Abstract This paper presents a fighting detection algorithm based on hybrid features. An effective coding scheme called locality-constrained linear coding (LLC) is used for the features by incorporating histograms of quadruples optical flow gradient (HQOFG) into local spatio-temporal features. Our results confirm that the fighting detector only with a linear support vector machine for sequence classification performs well. Compared to the fighting detector using bag-of-words (BOW) framework and local spatio-temporal features, the detector proposed in the paper improves the recognition accuracy. Keywords Fighting detection · Global feature · Spatio-temporal features · LLC

1 Introduction Human action recognition is a hot research topic in computer vision area [1–3], which can be applied to abnormal human behavior recognition [4] and user identification [5], etc. Especially, detecting fighting behavior in video monitoring scenes in order to stop violence is very important to public safety. Compared with traditional action recognition, fighting behavior is more difficult because it is related to the interaction between people, occlusion and irregularity. Datta proposes an in-depth hierarchical approach [6] which computes information regarding the motion trajectory of image structures for detecting violent actions between two people. Giannakopoulos developed the violent detector [7] by using S. Chen (B) Chengyi University College, Jimei University, 361021 Xiamen, Fujian, China e-mail: [email protected] T. Li Zhangzhou Branch of China Mobile, 363000 Zhangzhou, Fujian, China Y. Niu Department of Information Engineering, 239000 Chuzhou, Anhui, China G. Cai School of Computer Engineering, Jimei University, 361021 Xiamen, China © Springer Nature Singapore Pte Ltd. 2020 B.-y. Cao (ed.), Fuzzy Information and Engineering-2019, Advances in Intelligent Systems and Computing 1094, https://doi.org/10.1007/978-981-15-2459-2_3

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the audio features. They showed the audio signals are self-sufficient resources for detecting violence in video and can rate the movies containing the violent content. However, the audio features in monitoring scenes and movies are quite different and many audio features are not related to the violence, which may lead to the misclassification. As audio signal is an effective way of supplementing measurement of human actions, some violent detectors combine video and audio features for violence recognition [8, 9]. But this method requires audio signal, which is always not provided in the monitoring scenes. Clarin presented a violence detector [10] consisting of four modules. They can determine whether the scene contains violence or not through retrieving the skin and blood data, and then computing the motion intensities of these candidate regions. However, the color information may not be provided in the monitoring scenes, so the skin and blood data cannot be retrieved. Most of the previous work on simple human action recognition is based on wellknown local spatio-temporal features [11–16]. It has been applied to the detection of violence [17, 18]. These methods are built on the local spatio-temporal features with bags of visual words using a linear support vector machine. The results obtained demonstrated that motion patterns are crucial to distinguishing violence from regular activities in comparison with visual descriptors that rely solely on the space domain. Although the action recognition method based on local spatio-temporal features provides a practical strategy, it ignores the global information of movement which often reflects the macroscopic state of motion and describes the trend of motion. On the other hand, the bag-of-word (BOW) disregards the information about the spatial layout of features and usually needs classifiers with nonlinear to achieve good performance. Accordingly, the nonlinear classifiers often have limitations, such as more time-consuming or taking up more memory in computation; therefore, it is impractical to apply them to real-world applications. In this paper, we propose a global descriptor that is based on the gradient of optical flow [19]. Then the global feature describing macroscopic motion is combined with local spatio-temporal features. The global feature and local feature separately are coded by LLC and concatenated into vectors. Finally, the vectors can be classified using linear support vector machine (SVM). For the purpose of evaluation on fighting detection, we use two databases to test and analyze the time of feature extraction. The results obtained demonstrate that the fighting detector can achieve impressive fighting classification accuracy with only a linear SVM classifier and the global feature only adds a little computational time.

2 The Descriptor-Based Gradient Optical Flow for Global Feature Extraction and Description As we all know, the motion state of fighting behavior is changing rapidly. The changes include moving direction and speed. To describe the motion state, the most representative is optical flow. So, we propose a global descriptor, histograms of quadruples

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optical flow gradient (HQOFG), which is based on the optical flow. Our descriptor can not only describe the motion but also cancel out most of the effects of camera motion. First given a sequence of frames, V, we calculate each pixel (u, v) optical flow of each frame Ft by the method of Horn–Schunck [19] or Lucas–Kanade [20], where t is the frame index, and vector (u, v) denote the x (horizontal) and y (vertical) components of optical flow. Then we take their horizontal and vertical gradients separately: u x = u(x + 1, y) − u(x − 1, y) u y = u(x, y + 1) − u(x, y − 1) vx = v(x + 1, y) − v(x − 1, y) v y = v(x, y + 1) − v(x, y − 1)

(1)

where u x , u y , vx , v y denote the corresponding horizontal and vertical gradients of of flow, e.g., u x is the horizontal gradient of the    optical  flow.    x component , v , u , u , v , v Then we use the gradients to assemble u x y x y x y and vx , u y     as pairs. We use u x , u y and vx , v y to capture the relative movements of limb and body edges because the gradient of optical flow cancels out most of the effects of camera translation. According to themethod for the human  thatAli S proposed  action behave recognition [14], we use u x , v y and vx , u y to describe the motion on ration, expansion and vorticity. The large gradients of optical flow computed for two frames are shown in Fig. 1. The four pairs of gradients can describe the motion boundaries and reflect the motion state. Finally, we built histograms for the four pairs extracted from each frame. These feature points are distributed based on the magnitude and orientation together. The histograms are three-dimensional and not the same as HOG. The HOG method divides an image into a dense grid of cells. Each pixel of the image gradient vector is calculated and converted to an angle, voting into the corresponding orientation bin with a vote weighted by the gradient magnitude. The cells are grouped into blocks. All of the blocks are concatenated into a vector. Therefore, the highlighted feature

Fig. 1 Top: optical flow gradients of fighting behavior.  Bottom:   optical   flow gradients   of walk  action. From left to right: frame from input sequence, u x , u y , vx , v y , u x , v y and vx , u y

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will be lost when the blocks are sparse and the dimension of vector will be large when the blocks are dense. We utilize the three-dimensional histogram which can retain more highlighted features when the blocks are dense. Note that in the papers, we only divide each frame  into a block. For the pairs of optical flow gradient at each pixel, such as given the u x , u y , it can be calculated  into (r, θ ), where r represents  the magnitude of u x , u y and θ is the angle of u x , u y . The magnitudes ranging from Mmin to Mmax are evenly divided into p magnitude sets. In a similar way, the angles ranging from 0 to 360 are evenly divided into q angle sets. And, the histogram is calculated by the corresponding magnitude and orientation bin with a vote, hist(ir , jθ ) = hist(ir , jθ ) + 1 r ∈ ir , θ ∈ jθ

(2)

Therefore, we can build four histograms for the four pairs of optical flow gradients, respectively. We define the four histograms as H1 , H2 , H3 and H4 for each frame F. The four histograms are normalized and concatenated into a vector Ft , respectively, where Ft = [H1 , H2 , H3 , H4 ] and t is the frame index. So, each video V is composed by the vectors and denoted by V = {F1 , F2 , F3 , F4 , . . . , Fm }, where m is the total number of frames that compose V.

3 LLC Feature Representation To achieve better performance, several feature representation methods have been proposed, such as VQ, SC [21], LLC [22]. These methods have achieved excellent performance in image filed. The BOW utilizes VQ. The VQ uses a single basis in the codebook which is the most similar to the feature and always ignores the relationships between different bases. Hence, it requires nonlinear kernel to make up such information loss. The SC might select quite different bases for similar patches to favor sparsity, thus losing correlations between codes. Wang J has confirmed that the LLC is easy to compute and has better performance than many representation methods. Therefore, we adopt the LLC feature representation for the global features and local spatio-temporal features. The flowchart of LLC coding in this paper is shown in Fig. 2. The global features and local spatio-temporal features are extracted from the input video. Assume the X = [x1 , x2 , . . . , x N ] ∈ R D×N denotes either the global features or local spatiotemporal features from each video. The codebook B(B = [b1 , b2 , . . . , b K ] ∈ R D×K ) is generated by the features. For each feature xi , find the nearest neighbors of xi . They are used to reconstruct the locality-constrained liner code. The LLC uses the following criteria: min

N i=1

xi − Bci 2 + λdi  ci 2

s.t. 1T ci = 1, ∀i

(3)

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Fig. 2 Flowchart of LLC





dist(xi ,B) σ  T

∈ R K is the  locality adaptor. dist(xi , B) = [dist(xi , b1 ), . . . , dist(xi , b K )] , dist xi , b j is the Euclidean distance between xi and b j . For each video, the codes are pooled together to get the corresponding pooled feature. Specifically, two pooling methods have been used [21, 23]

where  denotes the element-wise multiplication, di = exp

sum pooling: cout = cin1 +, . . . , +cink

(4)

max pooling: cout = max(cin1 , . . . , cink )

(5)

4 Considering Temporal Information of the Global Feature To remedy the neglecting the temporal information, which is significant in fighting recognition, we concatenate the feature vectors of two frames. The global feature vector of each frame is Ft = [H1 , H2 , H3 , H4 ], define the new vector Pt , the new vector consider temporal information. Pt is written as   Pt = Ft−Δt , Ft , t ≥ Δt + 1

(6)

Δt is the interval. Pt is concatenated by Ft−Δt and Ft , retain the temporal information underlying the action and enhance the feature more distinguishable.

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5 The Process of Algorithm Step 1: Input the sets of videos SV = {V1 , V2 , . . . , Vn }. Each of videos Vi =

f i,1 , f i,2 , . . . , f i,m , where f i, j is the jth frame and m is the total number of frames. Step 2: Each of videos Vi submitted to the feature extraction process using HQOFG descriptor and spatio-temporal descriptor. The

extraction features are F G i = Pi,Δt+1 , Pi,Δt+2 , . . . , Pi,m and F L i = Si,1 , Si,2 , . . . , Si,n , where F G i denotes the global features and F L i denotes the spatio-temporal features, m is the total number of frames and n is the total number of points of interest. Step 3: The extraction features F G i and F L i , respectively, use the LLC feature representation by the corresponding codebook. Note that there are two codebooks generated by the global features and spatio-temporal features, respecg l , concatenate tively. The F G i is coded as Cout,i and F L i is coded as Cout,i g l the Cout,i and Cout,i to a vector. Step 4: The video vectors are fed into the SVM; the SVM learns from the training data and generates vectors for determining whether the video includes fighting behavior. The process of algorithmic is shown in Fig. 3.

Input videos

Generate Visual Codebook

Local Spatio-tempral Feature Extraction

The Global Feature Extraction

Is Training Dataset?

Is Training Dataset?

LLC Code

LLC Code

Concatenate the Global and Local Feature

Is Training Dataset?

Video Classification

Fig. 3 Process of algorithmic

Classifier Learning

Generate Visual Codebook

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6 Experimental Results and Analysis In this section, extensive experiments are carried out on two databases to evaluate the performance of the proposed algorithm. Our method was implemented in Visual Studio 2010. The experimental results are also compared with the fighting detector using bag-of-words (BOW) framework and local spatio-temporal features [12, 13]. The results are evaluated by accuracy of classification, accuracy of detection and ROC. The spatio-temporal descriptor uses Space-Time Interest Points (STIP) [6] and extracts HOG [24], HOG\HOF [8] feature vectors. A. Dataset In BEHAVE [25], the scenes are closely similar to the monitoring scenes of daily life including standing, meeting, shaking hands, walking together, fighting, escaping as well as running, and therefore are suitable for evaluating our algorithm. Some samples are shown in Fig. 4. To evaluate classification, we collect a dataset from the BEHAVE dataset. The dataset consists of 50 clips of fighting events and 50 clips of normal events. The clips only contain one group and most of background has been wiped off. The length of clips is range from 1 to 4 s. Some samples of the dataset are shown in Fig. 5. To evaluate performance of classification, we also use a publicly available violence dataset NHL [13], as shown in Fig. 6. The dataset can be used to measure the performance of violence recognition approaches. The dataset contains 1000 clips of actions from hockey games and each clip is labeled as fight or non-fight. The camera is rotated, and the background and focal distance can change. B. Experimental results on BEHAVE a. Classification The section utilizes the dataset built from BEHAVE. The results are compared with the violence detection [12] using spatio-temporal features with bags of visual

Fig. 4 BEHAVE dataset. Top: samples from the fighting; bottom row: samples from the normal events

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Fig. 5 Dataset. Top: samples from the fighting; bottom row: samples from the normal events

Fig. 6 NHL dataset. Top: samples from the fighting; bottom row: samples from the normal events

words. The classification tests use a 10-fold cross-validation test and calculate the mean accuracy for repeat five times. The main parameters in the experiment are listed in Table 1. The linear SVM and the sum pooling of LLC are adopted in this Table 1 Parameters setting Our method

Compared method

Magnitude sets

Angle sets

Interval time

Vocabularies (global)

Vocabularies (local)

Neighbors (global)

Neighbors (local)

Vocabularies

p

q

8

8

t

vac

vac

K

K

vac

2

100

1000

7

5

1000

Fighting Detection Based on Hybrid Features Table 2 Classification results of the methods

45

Method

Accuracy (%)

STIP(HOG) + BOW

83.2

STIP(HOF) + BOW

84.4

STIP(HOG/HOF) + BOW

88.2

HQOFG + LLC

90.2

STIP(HOG) + LLC

90.6

STIP(HOF) + LLC

89.4

STIP(HOG/HOF) + LLC

90.8

HQOFG + STIP(HOG) + LLC

94.2

HQOFG + STIP(HOF) + LLC

92.4

HQOFG + STIP(HOG/HOF) + LLC

93.2

method. In [12], the author compared with the traditional nonlinear kernel and the linear kernel performance best. So, the compared method also used linear SVM. Table 2 presents the accuracy of fighting classification. Table 2 shows that the LLC method can achieve better performance than BOW by using the linear SVM. The HQOFG descriptor proposed in the paper performed well and it is shown that the HQOFG descriptor can distinguish the fighting and normal behavior. The hybrid features proposed in our paper improve the classification accuracy by 2–4% and the best performance can achieve 94.2% by HOG descriptor and HQOFG descriptor. The results illustrate that the method of hybrid features proposed in this paper outperforms the method only using spatio-temporal features and indicate the effectiveness of our fighting detector. b. Detection In this section, we detect some videos with static background on BEHAVE dataset. The videos include fighting and normal behavior. The videos are divided into segments of 50 s. We detect the person by the method of frame difference and label the same group with block. To make the scene of block similar to the scene that we train in section a, the block is the area where the persons move in and the block is fixed within 50 s. Some samples are shown in Fig. 7. The segment extracted from the block within 50 s is a test sample. We can detect whether the input video includes fighting based on the test samples. We utilize the model that we train in the section with the HQOFG detector and STIP (HOG) detector for the detection. The results are shown in Table 3. We can see from the table that the detector is effective for detecting fighting. C. Experimental results on NHL To confirm the performance of our method, we experiment on NHL which is a publicly available violence dataset. The compared methods use spatio-temporal features with bags of visual words [12, 13]. However in [13], the authors use the HIK SVM, while in other methods, linear SVM is used in this section. The classification

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Fig. 7 Some samples of detect group from the BEHAVE

Table 3 Detection results The number of the video segments

The number of fighting segments

True fighting (%)

Fale fighting (%)

235(11,711 frames)

14

85.8

4.9

Table 4 Parameters setting Our method

Compared method

Magnitude sets

Angle sets

Interval time

Vocabularies (global)

Vocabularies (local)

Neighbors (global)

Neighbors (local)

Vocabularies

p

q

5

8

Δt

vac

vac

K

K

vac

1

1000

1000

5

5

1000

tests use a 5-fold cross-validation test and calculate the mean accuracy from five repetitive tests. The max pooling of LLC is adopted. The main parameters in the experiment are listed in Table 4. Table 5 illustrates that our method performed a little better than other methods. The methods using STIP with HIK kernel are obvious advantageous over the STIP with the linear kernel. However, our method using hybrid features and LLC with linear kernel achieves 1–2% over the STIP with HIK kernel. The hybrid features really outperform the methods only using single features. The STIP(HOG/HOF) detector and HQOFG detector with LLC achieve the best accuracy of 92.7%. The ROC curve for one of those runs is shown in Fig. 8. The results imply that our hybrid features worked well for fighting recognition, and incorporating the hybrid features with other feature representation processes and kernels may perform better, but the other kernels, especially nonlinear kernel, usually take more time in training and prediction. The result of MoSIFT [26] is also not bad. Therefore, using other descriptors in our method may also improve the performance. Overall, our method is effective for recognizing fighting. D. The time of feature extraction analysis

Fighting Detection Based on Hybrid Features Table 5 Classification results of the methods

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Method

Accuracy (%)

STIP(HOG) + BOW HIK

91.7

STIP(HOF) + BOW HIK

88.6

MoSIFT + BOW HIK

90.9

STIP(HOG) + BOW

86.0

STIP(HOF) + BOW

81.6

STIP(HOG/HOF) + BOW

85.0

HQOFG + LLC

88.3

STIP(HOG) + LLC

88.7

STIP(HOF) + LLC

84.1

STIP(HOG/HOF) + LLC

88.0

HQOFG + STIP(HOG) + LLC

92.4

HQOFG + STIP(HOF) + LLC

92.3

HQOFG + STIP(HOG/HOF) + LLC

92.7

Fig. 8 ROC curve on NHL database

48 Table 6 Time of feature extraction

S. Chen et al. Method

Frame/s

Harris3D + HOG/HOF

1.6

HESSIAN + ESURF

4.6

Cuboid Detector + Descriptor

0.9

Dense + HOG3D

0.8

Dense + HOG/HOF

1.2

HQOFG

0.14

We simply analyze the time of the global features and spatio-temporal feature extraction in this section. In [27], the author listed the extraction time for different features by various descriptors. The experiment in [27] was performed on a set of videos from Hollywood2 with spatial resolution of 360 × 288 pixels and the computation was performed on a Dell Precision T3400 Dual core PC with 2.66 GHz processors and 4 GB RAM. Because the global feature extraction is only related to the resolution, we also use video with 360 × 288 pixels from NHL. Our environment is on Intel Dual core PC with 2.80 GHz processors and 3 GB RAM. Although the core of our environment is a little faster than the 2.66 GHz, the result in this paper did not increase the computational speed by more than 10% and our codes of the algorithm still have room for optimization. The time for global features and local spatio-temporal features extraction is shown in Table 6. The time for global features extraction is far less than spatio-temporal features. Due to more points detected in Hollywood2, the extraction of spatio-temporal features takes more time. In a simple situation, the extraction of spatio-temporal features could take less time but will still be more than the extraction in our HQOFG descriptor method. Therefore, adding the global feature does not add computational time. If using parallel computing, the extraction time will depend on the spatio-temporal features.

7 Conclusions In this paper, aiming at solving the problem of local spatio-temporal features, we propose a global feature and combine the global feature with local spatio-temporal features. Then adopt the LLC for feature representation and concatenate the two features, respectively. The linear SVM is used in classification and prediction. Our method is confirmed on two databases and the results show that our method improved the effect for the fighting detection. Acknowledgements This research was supported by the Natural Science Foundation of Fujian Province (2013J01245), the Major Program of Industrial Collaboration of Science and Technology Department of Fujian Province (2017H6015) and the projects of Science and Technology Department of Xiamen City (3502Z20123022).

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A Method for Mining Temporal Association Rules in Single-Attributed Graph Sequence Qi-kai Guo and Fu-sheng Yu

Abstract This paper focuses on how to define and how to mine temporal association rules of an attributed graph sequence. Different from other temporal data, an attributed graph sequence contains edge-oriented structural data and vertex-oriented attribute data both of which vary with time. By reviewing the different forms of temporal association rules in the literature of related date types, we propose a definition of temporal association rule of an attributed graph sequence, which is a generalization of existing temporal association rules. In this definition, a two-tuple is used to describe the antecedent and consequent of a temporal association rule, which contains both structural information and attribute information. From this definition, a method for mining temporal association rules of an attributed graph sequence is given, and an Apriori algorithm-based mining algorithm is then designed. In the new mining method, two stages are given: the first one is for mining temporal structural association rules, while the second one is for mining temporal attribute association rules based on the result obtained in the first stage. This mining method exhibits good performance and high efficiency which was validated by the experiments carried on some datasets presented in this paper. Keywords Attributed graph sequence · Temporal association rule · Apriori algorithm

1 Introduction In recent years, numerous attributed graph sequences, which reflect characteristics of objects and relationships between objects changing over time, quickly arise in many areas, such as finance, environment, traffic and so on. Consider five cities which are directly connected by roads, railways or airlines. Each city is characterized by monthly consumption level. At a moment, the five cities construct an attributed graph with five vertexes where each vertex is described by an attribute, and each edge is a road, a railway or an airline. With the change of time, the attributed graphs at different Q. Guo · F. Yu (B) School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2020 B.-y. Cao (ed.), Fuzzy Information and Engineering-2019, Advances in Intelligent Systems and Computing 1094, https://doi.org/10.1007/978-981-15-2459-2_4

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moments will form an attributed graph sequence. Facing such kind of data, people are interested in the valuable information hiding there. For example, one maybe wanted to know: whether there is a correlation between the monthly consumption level of one city at a certain time and that of another city after a certain period of time. Besides, one may also want to know the correlation between a group of cities and another group of cities in the monthly consumption level at different times, and what is the influence of the connections of the cities on the correlations, etc. How to obtain the valuable information in an attributed graph sequence will be discussed in this paper. However, there is no literature on definition and mining method of temporal association rule of an attributed graph sequence. Some researches focus on mining frequent patterns in an attributed graph sequences [1, 2], which are different from temporal association rule of attributed graph sequences. Besides, researchers have paid attentions to patterns mining in non-attributed graph sequence including frequent subgraphs [3–5], frequent subsequences [6], triggering patterns [7] and weight pattern of edges [8]. So, we propose a definition and mining method of temporal association rules of an attributed graph sequence to meet the needs of new research and practical problems. The definition of association rule was first proposed by Agrawal [9] for describing the associations between different products in sale transactions and the patterns of customers’ purchase behaviors. By investigating the customers’ purchase behaviors in a period of time, the temporal association rule was proposed to reflect customers’ multiple shopping behaviors at different times [10]. With the development of the age, temporal non-attributed graph data appear more and more in reality, and the temporal association rule of non-attributed graph sequence is defined to reflect correlation between subgraphs over time. In short, several definitions of temporal association rules were put forward for temporal transactional association rule, temporal numerical association rule [11] and temporal association rule of non-attributed graph sequence. In this paper, we focus on the definition of temporal association rules of temporal single-attributed graphs or single-attributed graph sequences which also appear more and more in real world. The proposed definition of temporal association rule in this paper is a general version of the definitions of temporal transactional association rules and temporal association rules of non-attributed graph sequences. Based on the proposed definition, we design a method for mining the association rules of attributed graph sequences. This method first mines the structural association rules of structure information and then mines the association rules in the attribute values of the vertexes involved in the obtained structural association rules. This greatly reduces the search space resulting in decreasing searching time. Three experiments presented in this paper show the feasibility and effectiveness of the proposed method. The reminder of this paper is organized as follows: Section 2 presents a new definition of temporal association rules of an attributed graph sequence whose rationality is also explained. Section 3 presents the proposed Apriori algorithm-based method for mining temporal association rules of an attributed graph sequence. Experiments are shown in Sect. 4. Conclusions are made in Sect. 5.

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2 Definition of Association Rule of Attributed Graph Sequence An attributed graph sequence that reflects characteristics of objects and relationships between objects changing over time is a temporal dataset. The temporal data types related to an attributed graph sequence are time series and non-attributed graph sequence. The relationship between an attributed graph sequence, a non-attributed graph sequence and a time series is shown in Fig. 1. An attributed graph sequence will become a non-attributed graph sequence when ignoring the attributes of each vertex in every attributed graph of an attributed graph sequence. If ignoring all the edges in an attributed graph sequence, then the attribute values of each vertex in every attributed graph will form a numerical time series. As for a non-attributed graph sequence, each vertex can be regarded as a transactional time series when ignoring all attributes of each vertex. In order to simplify the discussion, we first give the definition of an attributed graph sequence. Definition 1 (Attributed graph sequence) G = {G 1 , G  2 , . . . , G n } is called an attributed graph sequence, where G k = V k , E k , Λ is an attributed graph (1 ≤ k ≤ n). V k and E k are the vertex set and edge set of graph G k , respectively. used for describing each vertex. For Λ = {ai |i = 1, 2, . . . , m}   is the set of attributes li 1 2 any ai ∈ Λ, G D(ai ) = ai , ai , . . . , ai (1 ≤ i ≤ m) denotes the domain of its all attribute values. As is shown in Definition 2, an association rule that describes the potential relations between transactional items in dataset is an important concept in data mining, which was proposed by R. Agrawal. An association rule with antecedent and consequent at different times is called temporal association rule. For the temporal datasets mentioned above, the existing definitions of association rule are listed below. Definition 2 (Association rule of transactions) Let J = {I1 , I2 , . . . , Im } be a set of itemset, D = {S1 , S2 , . . . , Sn } be a transactional database, where Si (⊆ J ) is called a transaction. An association rule has the form of X ⇒ Y , where X ⊆ J , Y ⊆ J , X ∩ Y = ∅. X and Y are named antecedent and consequent of the association rule. Definition 3 (Temporal association rule of temporal transaction) Let J = {I1 , I2 , . . . , Im } be a set of itemset, D = {(S1 , t1 ), (S2 , t2 ), . . . , (Sn , tn )} be a temporal transactional database, where Si (⊆ J ) is called a temporal transaction, ti is the Attributed graph sequence

Ignoring edges

Ignoring attributes

Numerical time series Ignoring attributes

Ignoring edges Non-attributed graph sequence Fig. 1 Relationships between different temporal data types

Transaction time series

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happening time of Si . A temporal association rule has the form of X ⇒ Y , where X ⊆ J , Y ⊆ J , T > 0. X and Y are, respectively, named antecedent and consequent. Definition 4 (Temporal association rule of non-attributed graph Let G =   sequence) {G 1 , G 2 , . . . , G n } be a non-attributed graph sequence, G k = V k , E k (1 ≤ k ≤ n) be a graph.V = nk=1 V k , J = V × V is a set of itemset. A temporal transactional database is D = {(E 1 , t1 ), (E 2 , t2 ), . . . , (E n , tn )} where E i ⊆ J, ti is the happening T time of E i . A temporal association rule has the form of X ⇒ Y , where X ⊆ J , Y ⊆ J , T > 0. X and Y are named antecedent and consequent of temporal association rule. In an attributed graph sequence, what varies with time are the numbers of vertexes and edges as well as the attributes of vertexes. Thus, the temporal association rule to be defined of an attributed graph sequence should reflect the association of not only vertexes but also their attribute values. That is, a temporal association rule of an attributed graph sequence should include structural information and attribute information. So, we use a two-tuple to represent the antecedent and consequence. Definition 5 (Temporal association rule of an attributed graphsequence) Let n k G = {G 1 , G 2 , . . . , G n } be an attributed graph sequence, V = k=1 V , L = m |V | k J1 = V ∪ (V × V) and J2 = k=1 i=1 G D(ai ),  L . Let J = J1 × J2 be a set of itemset, D = S 1 , P 1 , S 2 , P 2 , . . . , (S n , P n ) (Si ⊆ J1 , Pi ⊆ J2 , 1 ≤ i ≤ n) be a temporal dataset. A temporal association rule of attributed graph sequence G T has the form of (X, A) ⇒(Y, B), where X, Y ⊆ J1 , A, B ⊆ J2 , T is a time span. It means that (Y, B) will happen at the T th time moment after (X, A) happens. (X, A) and (Y, B) are called antecedent and consequent of the temporal association rule. T

The support and confidence of temporal association rule (X, A) ⇒(Y, B) are, respectively, defined as

T



  ((X, A), (Y, B))|X ⊆ S i , A ⊆ P i , Y ⊆ S i+T , B P i+T

sup (X, A) ⇒(Y, B) = n−T  

 ((X, A), (Y, B))|X ⊆ S i , A ⊆ P i , Y ⊆ S i+T , B ⊆ P i+T T  conf (X, A) ⇒(Y, B) = (X, A)|X ⊆ S i , A ⊆ P i , i ≤ n − T T

For a temporal association rule (X, A) ⇒(Y, B), if ignoring the attribute items T A, B, (X, A) ⇒(Y, B) will degenerate to a temporal association rule of nonT attributed graph sequence X ⇒ Y that is called structure association rule of attributed T

graph sequence, and if ignoring the edges items X, Y , (X, A) ⇒(Y, B) will T degenerate to temporal association rule of transactions A ⇒ B. For example, given an attributed graph sequence G = {G 1 , G 2 , G 3 , G 4 }, where V 3 = {v1 , vl 2 , v3 }, L = {a, b}. J1 = {v1 , v2 , v3 , (v1 , v2 ), (v1 , v3 ), (v2 , v3 )}, J2 = l=1 {a, b} . Setting the minimum support smin = 0.5 and minimum confidence

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Fig. 2 Temporal association rule of the given attributed graph sequence 1

cmin = 0.7, an association rule ((v1 , v2 ), (a, a)) ⇒((v1 , v3 ), (a, b)) can be easily found as shown in Fig. 2, whose antecedent is marked in red and consequent is marked in blue.

3 Principles of the Proposed Mining Method In our proposed method, the given attributed graph sequence will be first formalized so as to take means by the Apriori algorithm to implement the task of mining association rules. As is known, the size of the itemset from where the antecedent and consequent of an association rule come will heavily influence the efficiency of the mining algorithm. For a given attributed graph sequence, the itemset constructed for the mining task is usually of big size. Thus, enumerating all possible antecedents and consequents in an attributed graph sequence and calculating the support and confidence of candidate temporal association rules will take much more time. Therefore, deleting some candidate temporal association rules is a necessary processing step. Theorem 1 provides a theoretical basis for this step. In short words, for a given attributed graph sequence G, if a structure temporal association rule T X ⇒ Y of an attributed graph sequence is not true, then the temporal association T

rules (X, A) ⇒(Y, B) are also not true, which is beneficial for us to delete many candidate association rules in searching space. T

Theorem 1 If (X, A) ⇒(Y, B) is a temporal association rule of attributed graph T sequence G, then X ⇒ Y is a structure temporal association rule of G. For a given attributed graph sequence, mining temporal association rules is generally divided into two stages according to Theorem 1. The first one is for mining all structure association rules. The second one is for mining temporal association rules that meets the support and confidence requirements based on first stage. In the first stage, the improved Apriori algorithm is suitable to mine all frequent subgraphs when ignoring attribute of each vertex in a given attributed graph sequence. T Next, structure association rules of the attributed graph sequence X ⇒ Y that meets the confidence requirements from frequent subgraphs are mined. Once structure X, Y

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is determined from the frequent subgraphs, corresponding vertexes in the attributed graph sequence can also be determined. In the second stage, only the attribute values of those vertexes in the first stage are taken into consideration. Enumerate all possible attributes based on the structure X, Y and count support and confidence degree to mine temporal association rules. T If (X, A) ⇒(Y, B) satisfy the threshold of support and confidence, it is a temporal association rule of the attributed graph sequence. According to the above analysis, we list below the proposed algorithm for mining temporal association rules of an attributed graph sequence: Step1. Data preprocessing: process data and set appropriate parameters. T Step2. Mine structure association rule X ⇒ Y of the attributed graph sequence. T

Step3. Enumerate all possible temporal association rules (X, A) ⇒(Y, B) for each T structure association rule X ⇒ Y in Step 2. Step4. Give mining result by checking support and confidence of the temporal association rules obtained in Step 3.

4 Experiment Studies In this section, experiments on artificial and real datasets are designed to show the superiority as well as the effectiveness of our proposed method. Experiment A shows that the proposed method works well in a small-scale database. Experiment B shows the effectiveness of the proposed method by comparing with brute force method. It should be emphasized that the proposed method can reduce the searching space effectively. The feasibility of the proposed method in terms of accuracy of mining is illustrated by Experiment C which uses the data from China Meteorological Administration.

4.1 Experiment A As shown in Fig. 3, the dataset of Experiment A is an attributed graph sequence consisting of ten attributed graphs. In each attributed graph, there are four vertexes with attribute value in {large, small} and some edges between vertexes. In this dataset, a temporal association rule whose antecedent and consequent are marked in red and blue, respectively, can be easily found. Edge (v3 , v4 ) with attribute value (large, large) happens at time t = 1, and edge (v2 , v4 ) with attribute value (small, large) happens at time t = 4, which is listed in Table 2 marked with (*). The method can find all temporal association rules of the attributed graph sequence. As shown in Table 1, different time constraints T, minimum support and minimum confidence will result in different numbers of temporal association rules. When T =

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Fig. 3 Dataset of Experiment A and a temporal association rule

Table 1 Temporal association rules of different parameters of Experiment A Time

smin

cmin

Example

1

0.7

0.7

1

0.7

0.4

1

0.5

0.7

((v2 , v4 ), (small, large)) ⇒((v3 , v4 ), (large, large))

3

0.7

0.7

((v3 , v4 ), (large, large)) ⇒((v2 , v4 ), (small, large))

3

0.7

0.4

((v3 , v4 ), (large, large)) ⇒((v2 , v4 )(v3 , v4 ), (small, large, large))

3

0.5

0.7

((v2 , v4 )(v3 , v4 ), (small, large, large)) ⇒((v3 , v4 ), (large, large))

1

3 3

3

1, smin = 0.7, cmin = 0.7, there are no temporal association rules in this attributed graph sequence; however, three temporal association rules can be found if decreasing minimum support to smin = 0.5. Table 2 shows all the obtained temporal association rules when T = 1, smin = 0.5, cmin = 0.7, and T = 3, smin = 0.7, cmin = 0.7.

4.2 Experiment B In this experiment, the dataset is larger than that in experiment A. The length of the attributed graph sequence is thirty. There are five vertexes and some edges in each attributed graph. Table 3 shows part of the attribute values of the vertexes in the dataset. Table 4 presents the connections between vertexes at different times. If there is an edge between vertexes vi and v j , then the value of (vi , v j ) is 1, otherwise is 0. After setting T = 1, smin = 0.5, cmin = 0.5, three temporal association rules of the given attributed graph sequence can be found, such as

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Table 2 Result of temporal association rules at specific parameter of Experiment A Parameter

Association rules

T =1 smin = 0.5 cmin = 0.7

((v2 , v4 ), (small, large)) ⇒((v3 , v4 ), (large, large))

1

1

((v3 , v4 ), (large, large)) ⇒((v3 , v4 ), (large, large)) 1

((v2 , v4 )(v3 , v4 ), (small, large, large)) ⇒((v3 , v4 ), (large, large)) 3

((v2 , v4 ), (small, large)) ⇒((v3 , v4 ), (large, large))

T =3 smin = 0.7

3

((v3 , v4 ), (large, large)) ⇒((v2 , v4 ), (small, large))(∗) 3

((v3 , v4 ), (large, large)) ⇒((v3 , v4 ), (large, large)) 3

((v3 , v4 ), (large, large)) ⇒((v2 , v4 )(v3 , v4 ), (small, large, large)) 3

((v2 , v4 )(v3 , v4 ), (small, large, large)) ⇒((v3 , v4 ), (large, large))

Table 3 Attribute value of vertexes of dataset in Experiment B Time

Attribute value of v1

Attribute value of v2

Attribute value of v3

Attribute value of v4

Attribute value of v5

t =1

10

22

15

30

t =2

13

21

12

32

8

t =3

15

23

15

26

11

t =4

12

21

18

28

14

9

Table 4 Dataset about edges of Experiment B Time

(v1 , v2 )

(v1 , v3 )

(v1 , v4 )

…

(v3 , v4 )

(v3 , v5 )

(v4 , v5 )

t =1

1

1

1

…

1

0

0

t =2

1

1

1

…

1

0

0

t =3

0

0

1

…

0

1

1

t =4

1

1

1

…

1

1

1

1

((v2 , v4 ), (large, large)) ⇒((v3 , v4 ), (large,large)). It means that: if “edge (v2 , v4 ) appears, both v2 and v4 take large value” happens, then “edge (v3 , v4 ) appears, both v3 and v4 take large value” will happen after one time span. The part results are shown in Table 5. The classical method of mining temporal association rules is brute force method that is divided into three steps. Enumerating all possible antecedent and consequent in a given attributed graph sequence to form a candidate set is the first step. In the second step, the candidate temporal association rules are established by selecting antecedents and consequents from the candidate set. In the third step, those candidate temporal association rules whose support and confidence bigger than a given threshold will be

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Table 5 Temporal association rules of Experiment B Parameter

Association rules

T =1 smin = 0.5 cmin = 0.5

((v2 , v4 ), (large, large)) ⇒((v3 , v4 ), (large, large))

1

1

((v3 , v4 ), (large, large)) ⇒((v2 , v3 ), (large, large)) 1

((v3 , v4 ), (large, large)) ⇒((v2 , v4 ), (large, large)) 3

((v2 , v4 ), (large, large)) ⇒((v1 , v2 ), (large, large))

T =3 smin = 0.5 cmin = 0.5

3

((v2 , v3 ), (large, large)) ⇒((v1 , v4 ), (large, large)) 3

((v3 , v4 ), (large, large)) ⇒((v2 , v4 ), (large, large)) 3

((v3 , v4 ), (large, large)) ⇒((v3 , v4 ), (large, large))

Table 6 Comparison of efficiency of the brute force method and our method Method

Time (s)

Brute force method

200

Proposed method

80

mined. Comparing the result of the brute force method with our method, we find that the two methods produce all the temporal association rules without missing. But the proposed method achieves a marked improvement in efficiency (Seen as in Table 6).

4.3 Experiment C The dataset comes from China Meteorological Administration, which contains 82 records of daily air quality index(AQI) and nitrogen dioxide index of Beijing(B), Tianjin(T) and Taiyuan(t). Some of data are shown in Table 7. Take the three cities to be the vertexes of an attributed graph sequence, AQI and NO2 to be the attributes of the vertexes. The NO2 attribute values are used to create edges between vertexes. Table 7 Comparison of efficiency of the brute force method and our method Times

AQI(B)

NO2 (B)

AQI(T)

NO2 (T)

AQI(t)

NO2 (t)

t =1

159

41

107

41

120

30

t =2

48

21

58

21

66

11

t =3

89

43

97

36

75

21

t =4

55

17

72

20

76

18

t =5

51

36

60

40

76

40

t =6

99

61

95

58

105

52

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Fig. 4 The attributed graph sequence of Experiment C

Table 8 Results of Experiment C when T = 12, smin = 0.1, cmin = 0.1 Rules 12

((B, T ), (2, 2)) ⇒((B, t), (2, 2)) 12

((B, t), (2, 2)) ⇒((T, t), (2, 2)) 12

((B, t), (2, 2)) ⇒((B, t)(T, t), (2, 2, 2)) 12

((B, t)(t, T ), (2, 2, 2)) ⇒((T, t), (2, 2))

Support

Confidence

0.11

0.33

0.18

0.52

0.11

0.25

0.13

0.47

If the absolute value of the difference between two cities is less than 10, an edge is added. After that, an attributed graph sequence is formed as shown in Fig. 4. By setting T = 12, smin = 0.1, cmin = 0.1, 25 temporal association rules are obtained. No new temporal association rule can be found when increasing the minimum support. Table 8 shows some typical temporal association rules.

4.4 Summary of Experimental Studies From the results in the experiments, we can draw the conclusions: Once the minimum support, minimum confidence and time constrain T are given, the proposed method can find all the temporal association rules hiding in a given attributed graph sequence. Compared with the brute force method, the proposed method achieves better results in efficiency of mining temporal association rules of attributed graph sequence.

5 Conclusions In this paper, we proposed a definition of temporal association rule of an attributed graph sequence, which is the general version of temporal association rule of transactions, numerical transactions and non-attributed graph sequence. Based on these definitions, we proposed a method to mine temporal association rules in an attributed graph sequence. We proved a theorem to emphasize that giving priority to structure information can reduce search space. According to this, mining temporal association

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rules are divided into two stages: the first one is mining temporal structural association rules, while the second one is mining temporal attribute association rules based on the result obtained in the first stage. The result of experiments shows that our method works accurately and effectively. In the study of this paper, we considered only the case where the domain of each attribute is discrete in an attributed graph sequence. How to deal with the continuous domains is worth investigating in future work. Acknowledgements This work was supported by National Natural Science Foundation of China (No. 11971065, No. 11571001, No. 11701338). Recommender This paper is recommended by W. Y. Zeng who is a professor of Beijing Normal University in China.

References 1. Desmier, E., Plantevit, M.: Trend mining in dynamic attributed graphs. In: European Conference on Machine Learning and Knowledge Discovery in Databases. pp. 654–669. Springer, New York (2013) 2. Desmier, E., Plantevit, M., Robardet, C.: Cohesive co-evolution patterns in dynamic attributed graphs. In: International Conference on Discovery Science, pp. 110–124 (2012) 3. Wang, W., Zhou, H., Yuan, Q.: Frequent pattern mining based on graph theory. Comput. Res. Dev. 42(2), 230–235 (2005) 4. Robardet, L.: Constraint-based pattern mining in dynamic graphs. In: Ninth IEEE International Conference on Data Mining, pp. 950–955. IEEE Computer Society (2009) 5. Lee, J., Park, K., Prabhakar, S.: Mining statistically significant attribute associations in attributed graphs. In: International Conference on Data Mining (2017) 6. Inokuchi, A., Washio, T.: A fast method to mine frequent subsequences from graph sequence data. In: Eighth IEEE International Conference on Data Mining, pp. 303–312. IEEE Computer Society (2008) 7. Kaytoue, M., Pitarch, Y., Plantevit, M.: Triggering patterns of topology changes in dynamic graphs. In: International Conference on Advances in Social Networks Analysis and Mining, pp. 158–165. IEEE (2014) 8. Gupta, A., Thakur, H., Gundherva, N.: Mining regular pattern in edge labeled dynamic graph. In: International Conference on Industrial and Information Systems. pp. 1–6. IEEE (2015) 9. Agrawal, R., Imielinski, T., Swami, A.: Mining association rules between sets of items in large databases, pp. 207–216. ACM (1993) 10. Zhan, L., Yu, F., Zhang, H.: A fast algorithm for mining temporal association rules based on a new definition. In: International Conference on Natural Computation, IEEE (2018) 11. Zhan, L.: Mining of temporal association rules. Beijing Normal University (2018)

Fuzzy Decision-Making and Programming

Optimal Decision Making in Fuzzy Stochastic Hybrid Uncertainty Environments and Their Application in Transportation Problems S. Bavandi, S. H. Nasseri, and C. Triki

Abstract Decision making plays an important role in economic, management, business, marketing, psychology, philosophy, mathematics, statistics, and many other fields. In each field, decision making consists of identifying the values, uncertainties, and other issues that define the decision. Randomness and fuzziness or vagueness are two major sources of uncertainty in the real world. Practical applications in areas of industrial engineering, management, and economics, are such that decision-makers are being confronted with information that is simultaneously probabilistically uncertain and fuzzily imprecise, and a decision making has to be performed under such a twofold uncertain environment of co-occurrence of randomness and fuzziness. This paper presents an application to the transportation problems in fuzzy stochastic hybrid uncertainty environments. In this paper, we focus on our attention on unbalanced transportation problems in the fuzzy stochastic environment. Keywords Decision making · Uncertainty · Fuzzy random variable · Transportation problem

1 Introduction All of us take different decisions in our individual and organizational lives, knowingly and unknowingly; Namely, we choose a solution from among several solutions. In fact, every day of life can be depicted in the form of a set of microdecisions such as coverage, transportation vehicle, and the route to decide on participation in a particular activity. The basic point in this regard is to have methods that help us to make S. Bavandi (B) · S. H. Nasseri Department of Mathematics, University of Mazandaran, Babolsar, Iran e-mail: [email protected] S. H. Nasseri Department of Mathematics and Big Data, Foshan University, Foshan, China C. Triki Department of Industrial Engineering, Sulan Qaboos University, Muscat, Oman © Springer Nature Singapore Pte Ltd. 2020 B.-y. Cao (ed.), Fuzzy Information and Engineering-2019, Advances in Intelligent Systems and Computing 1094, https://doi.org/10.1007/978-981-15-2459-2_5

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the right decision in real-life situations. On the other hand, with the development of the human sciences and knowledge, decision making has also undergone many changes as a scientific field, and this process continues. So that different standpoints about systematic and specific analytical methods have been developed in the face of complex and inadequate information and have led to the development of methods for solving decision-making problems in conditions of uncertainty. The uncertainty handling has been one of the main concerns of the decision makers (including governors, engineers, managers, and scientists) for many years [1]. To deal with various uncertainties in real-world problems through mathematical programming, decision scientists, mathematicians, and economists have written hundreds of books and thousands of articles on the topic. Uncertainty can be described in several ways, depending on the information at hand. Among mathematical tools for coping with uncertainty, we mention worst-case scenario analysis, evidence theory, probability theory and fuzzy sets theory, etc. Accessible accounts of these tools may be found in [15–17] and in references therein. While research has progressed at a steady pace in the fields of stochastic optimization and fuzzy mathematical programming, the past decade, in particular, has witnessed a developing interest in situations where fuzziness and randomness are under one roof? in an optimization framework. Uncertainty in transportation problem (TP) is a well-established phenomena. In fact, uncertainty exists everywhere in practical problems. It can be mainly classified in two senses— stochastic and fuzzy. Uncertainty can be mainly classified into two types—stochastic and fuzzy. Several researchers considered TP in stochastic [3, 4, 12] and fuzzy [6, 7, 11, 13] environments. Again, data/parameters imprecise in both fuzzy and stochastic senses are called fuzzy random parameters. The concept of fuzzy random variable was introduced by Kwakernaak [8, 9], Puri and Ralescu [14]. The occurrence of fuzzy random variable/parameter makes the combination of randomness and fuzziness more persuasive. Though in the literature, these are some decision-making problems formulated and solved with fuzzy random parameters/variables, till now, to the best of our knowledge, very few TPs have been formulated and solved with fuzzy random costs/resources. Recently, Xu et al. [19] pointed out that the time windows in a Vehicle Routing Problem with Soft Time Windows contains both fuzzy and stochastic information together. This paper focuses on a fuzzy stochastic transportation problem with an application view under uncertainty conditions. In Sect. 2, we describe some of the underlying concepts. The fuzzy stochastic transportation model is presented in Sect. 3. In Sect. 4, a method is presented to solve the unbalanced TPs. This paper presents an application to the transportation problems in fuzzy stochastic hybrid uncertainty environments. In this paper, we focus on our attention on unbalanced transportation problems in the fuzzy stochastic environment.

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2 Basic Preliminaries In this section, we describe some of the underlying concepts. Definition 2.1 A fuzzy number A˜ is a convex normalized fuzzy set A˜ of the real line R , with membership function A˜ : R → [0, 1], satisfying the following conditions: 1. There exist unique interval J ∈ R such that μ A (x) = 1 : x ∈ J 2. The membership function μ A is piecewise continuous. Definition 2.2 [10] A fuzzy number A˜ = [a1 ; a2 ; a3 ] is said to be Triangular Fuzzy Number (TFN) if its membership function is strictly increasing in the interval (a1 , a2 ) and strictly decreasing (a2 , a3 ) and μ B (a2 ) = 1 , where a2 is core, (a2 − a1 ) is left ˜ spread and (a3 − a2 ) is right spread of the fuzzy number A. Definition 2.3 [2] A cut of the fuzzy number A˜ is the set {x|μ A (x) ≥ α} for 0 ≤ α < 1 and denoted by A˜ [α]. According to Kaufmann and Gupta [5], the approximated value of TFN A˜ = [a1 ; a2 ; a3 ] is given by A˜ = a1 +2a42 +a3 . Definition 2.4 Let A˜ = [a1 ; a2 ; a3 ] and B˜ = [b1 ; b2 ; b3 ] are two TFNs. Using ˜ ∀x, y, z ∈ X maxmin convolution on fuzzy sets A˜ and B,   μ A+ ˜ B˜ (z) = ∨ μ A˜ (x) ∧ μ B˜ (y) : z = x + y

(1)

therefore, A˜ + B˜ = [a1 ; a2 ; a3 ] ⊕ [b1 ; b2 ; b3 ] = [a1 + b1 ; a2 + b2 ; a3 + b3 ] A˜ − B˜ = [a1 ; a2 ; a3 ] [b1 ; b2 ; b3 ] = [a1 − b3 ; a2 − b2 ; a3 − b1 ]   Scalar multiplication: Using μk A˜ (z) = ∨ μ A˜ (x) ∧ μk (y) : z = kx k A˜ =



[ka1 ; ka2 ; ka3 ] for k ≥ 0; [ka3 ; ka2 ; ka1 ] for k < 0;

    Definition 2.5 Let a˜ and b˜ be represented by a˜ = α∈[0,1] aαL , aαR and b˜ = α∈[0,1]  L R bα , bα , and then, the signed distance of a˜ and b˜ is the distance between the mid         points M = = 21 aαL , aαR of aαL , aαR and M (b (α)) = = 21 bαL , bαR of  L R  (a (α)) bα , bα over α in [0, 1], which is as follows

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1 1 [M (a (α)) − M (b (α))] dα 1−0 0  1 1 L = aα + aαR − bαL − bαR dα 2 0

 d a, ˜ b˜ =

(2)

A fuzzy random variable (FRV) is a random variable and a Borel measurable function whose actual value is a fuzzy number [14].   Definition 2.6 [18] If X˜ be a FRV, then an α-cut X α (ω) = t ∈ |μ X (ω) (t) ≥ α =   L X α (ω) , X αR (ω) is a random interval for every α ∈ (0, 1]. According to the recent definition, X αL and X αR are random variables for all α ∈ (0, 1], having the same continuous probability density function f (x), x ∈ R. For  any fuzzy observation x˜ of fuzzy random variable X˜ , the α-level set x˜α = xαL , xαR , where xαL and xαR arethe observations of X αL and X αL , respectively. For any x ∈ xαL , xαR , there exists β ≥ α such that x = xβL or x = xβR . So for any   x ∈ xαL , xαR , we can associate a probability density function f (x) with x. Then, by extension principle, the membership function of fuzzy probability density of x, ˜ denoted by f (x), ˜ is defined by μ f (x) ˜ (z) =

sup

w∈{x: f (x)=z}

[μx˜ (w)]

(3)

˜ where μx˜ (w) is the membership function of fuzzy random variable x. ˜ Definition

 2.7 The expectation of fuzzy random variable X is an unique fuzzy number E X˜ which is defined by



    E X αL , E X αR E X˜ = α∈[0,1]

=

⎡ ⎣

∞

∞ xαL f αL (x) d x,

α∈[0,1] −∞

⎤ xαR f αR (x) d x ⎦

−∞

3 Fuzzy Stochastic TP Model Suppose a two-dimensional imbalance TP is considered with sources M and n, and limitation on the total cost at different destinations is imposed in the form of budget constraints. The unit transportation costs and the available budgets at destinations are fuzzy stochastic in nature. Also, in the unbalanced TP, available total resources are more than total requirements. We consider an unbalanced TP with fuzzy stochastic costs where total availabilities are more than the total demands with the following notations

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m: the number of sources. n: the number of demands. ai : amount of a homogeneous product available at i-th origin. b j : the quantity of the commodity demanded at the j-th destination Cˆ˜ i j : fuzzy random unit transportation cost from i-th origin to j-th destination. B j : the budget allocated for transportation for each j = 1, 2, . . . , n which is less than the minimum total cost of transportation. xi j : the amount to be transported from the i-th origin to the j-th destination (decision variables). The mathematical model of the transportation problem is presented as follows: m  n  Cˆ˜ i j xi j

Min Z =

(4)

i=1 j=1

s.t.

m  C˜ˆ i j xi j ≤ B j , ∀ j

(5)

i=1 n 

xi j ≤ ai , ∀i

(6)

xi j ≥ b j , ∀ j

(7)

b j xi j ≥ 0, for all i, j

(8)

j=1 m  i=1

m 

ai ≥

i=1

n  j=1

4 Crisp Transformation of Model In this section, a method is presented to solve the unbalanced TPs. First, we consider the fuzzy nature of the parameter and resolve it taking the α-cut of the parameter with corresponding probability distribution. Then, the expectation is taken to reduce it to the corresponding crisp parameter. So, the objective function (4) reduces to 

Min E Z

∗



=

m  n  i=1

1 xi j . 2 j=1

0

1

  

  E CiLj α + E CiRj α dα



 where CiLj = Ci j − (1 − α) θ1 and CiRj = Ci j + (1 − α) θ2 . α

α

(9)

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In the next step, we will go to the constraints of the problem. The chanceconstrained programming technique can be used to solve problems involving chance constraints. If ξ j is the probability of non-violation of the constraint (5), then the constraint can be written as   (10) Pr Pˆ j ≤ A j ≥ ξ j where Pˆ j = m B j − i=1 m i j xi j m 2 2 i=1 σi j x i j

m i=1

m m i j xi j C˜ isj xi j − i=1 m 2 2 i=1 σi j x i j

Bj ≥

are the standard normal variant and A j =

m 

m i j xi j + λ j

i=1

m 

σi2j xi2j

i=1

where λ j are the real numbers such that   Pr P j ≤ λ j = ξ j In particular, let us consider the normal fuzzy probability density function 

 − 1 f C˜ i j = √ e 2π σi j

C˜ i j −Mi j √ 2πσi j

2

, −∞ < C˜ i j < ∞

Then,

 f C˜ iLj = α

√ 1 2π σi j

e

α

√ 1 2πσi j

e

 f C˜ iRj =

2  C −M − √i j2πσ i j ij

2  C −M − √i j2πσ i j ij

, Ci j ≥ (1 − α) θ1 , Ci j ≥ − (1 − α) θ2

So, the Eq. (9) reduces ∗

Min Z =

m  n 



σi j Mi j θ1 − θ2 1 − + √ xi j . √ + 2 8 2π 12 2πσ ij i=1 j=1   2 6Mi j − 3Mi j (θ1 − θ2 ) + θ12 + θ22

Therefore, the final model of the problem is obtained as follows:

Optimal Decision Making in Fuzzy Stochastic Hybrid Uncertainty Environments … ∗

Min Z =



m  n  i=1

71

σi j Mi j θ1 − θ2 − xi j . √ + 2 8 2π j=1 +

s.t. B j ≥

m 

√ 12 2π σi j

m i j xi j + λ j

i=1 n 

1

m 



6Mi2j

− 3Mi j (θ1 − θ2 ) +

θ12

+

θ22



σi2j xi2j

i=1

xi j ≤ ai , for all i

(11)

j=1 m  i=1 m  i=1

xi j ≥ b j , for all j ai ≥

n 

bj

j=1

xi j ≥ 0 i = 1, . . . , m and j = 1, . . . , n. The recent problem can be solved using genetic algorithm. The fuzzy stochastic transportation problem coefficients solving process can be summarized as follows: Step 1: Consider the TP with fuzzy stochastic coefficients and get the α-cut of C˜i j  and f (C i j ). Then, by using signed distance method, reduce the objective function (4) as (9). Step 2: Consider ξ j the probability of non-violation of the constraint (5), then convert the constraints of TP to the corpse form using the chance-constrained programming technique by the process explained in Sect. 4. Step 3: Solve problem (11) using a genetic algorithm.

5 Conclusion Significant uncertain information is involved in environmental decision making due to the complexities of natural systems and lack of sufficient data. Different types of uncertain information are identified as aleatory and epistemic information which are better represented using probability theory and fuzzy set theory and its extension possibility theory. The fuzzy stochastic parameters were first introduced in the literature in 1978, and consideration of this type of parameters in the decision-making problems is in the developing stage. But, in some real-life problems, available data are fuzzy stochastic. Hence, decision-making problems with fuzzy stochastic data are of great importance though there are very few such models in the literature. Here, a constrained unbalanced TP is formulated with fuzzy stochastic costs and resources

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and reduced to corresponding crisp ones. The applied procedure is quite general, and it can be used for other types of TPs with the types of constraints such as fuzzy stochastic TPs for breakable items, fuzzy stochastic solid TPs, with fuzzy stochastic space constraint, etc. Acknowledgements Thanks to the supported by the 9th International Conference on Fuzzy Information and Engineering (Kish island).

References 1. Attoh-Okine, N., Ayyub, B.: Applied Research in Uncertainty Modeling and Analysis, vol. 20 (2005) 2. Buckley, J.J.: Fuzzy Probabilities: New Approach and Applications, vol. 115. Springer Science and Business Media, Berlin (2005) 3. Chalam, G.A.: Fuzzy goal programming (FGP) approach to a stochastic transportation problem under budgetary constraint. Fuzzy Sets Syst. 66, 293–299 (1994) 4. Copper, L.: The stochastic transportation-location problem. Comput. Math. Appl. 4, 265–275 (1978) 5. Kaufman, A., Gupta, M.M.: Fuzzy Mathematical Models in Engineering and Management Sciences. North-Holland (1988) 6. Kaur, A., Kumar, A.: A new approach for solving fuzzy transportation problems using generalized trapezoidal fuzzy numbers. Manage. Sci. 22, 1116–1126 (2012) 7. Keshavarz, E., Khorram, E.: A fuzzy bi-criteria transportation problem. Comput. Ind. Eng. 61, 947–957 (2011) 8. Kwakernaak, H.: Fuzzy random variables - I. Definitions and theorems. Inf. Sci. 15, 1–19 (1978) 9. Kwakernaak, H.: Fuzzy random variables -II, algorithms and examples for the discrete case. Inf. Sci. 17, 253–278 (1979) 10. Lai, Y.J., Hwang, C.L.: A new approach to some possibilistic linear programming problems. Fuzzy Sets Syst. 49(2), 121–33 (1992) 11. Liu, S.L., Kao, C.: Solving fuzzy transportation problems based on extension principle. Euro. J. Oper. Res. 153, 661–674 (2004) 12. Ojha, A., Das, B., Mondal, S.K., Maiti, M.: A stochastic discounted multiobjective solid transportation problem for breakable items using analytical hierarchy process. Appl. Math. Model. 34, 2256–2271 (2010) 13. Saad, Omar M., Abass, Samir A.: A Parametric study on transportation problem under fuzzy environment. J. Fuzzy Math. 11(1), 115–124 (2003) 14. Puri, M.L., Ralescu, D.: Fuzzy random variables. J. Math. Anal. Appl. 114, 409–422 (1986) 15. Sakawa, M.: Fuzzy sets and Interactive Optimization. Plenum Press, New York (1993) 16. Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton, NJ (1976) 17. Shiryaer, A.: Probability, 2nd edn. Springer, Berlin (1996) 18. Wang, G.-Y., Zang, Y.: The theory of fuzzy stochastic processes. Fuzzy Sets Syst. 51(2), 161–78 (1992) 19. Xu, J., Yan, F., Li, S.: Vehicle routing optimization with soft time windows in a fuzzy random environment. Transp. Res. Part E 47, 1075–1091 (2011)

A Double Interactive Alternative Reduction Approach for Probabilistic Linguistic Multi-criteria Decision-Making with Incomplete Criteria Weight Information Na Yue, Jialiang Xie, and Shuili Chen Abstract The paper studies a double interactive search over alternative reduction for probabilistic linguistic multi-criteria decision-making problems, where the criteria weight values are incomplete known but the ranges are available. The method transforms the probabilistic linguistic term set into its associated vector, then defines the achievement scale, satisfactory degree and dominated alternative under the probabilistic linguistic circumstance. Next, with the expected levels of all alternatives proposed by decision makers, the algorithm of the probabilistic linguistic double interactive alternative reduction approach is presented. After that, decision makers communicate with each other to determine the optimal alternative. Finally, a practical example is used to illustrate the applicability and validity of the described method. Keywords Double interactive alternative reduction · Achievement scale · Satisfactory degree · Dominated alternative · Probabilistic linguistic multi-criteria decision-making

1 Introduction With the introduction of the probabilistic linguistic term set (PLTS) [7], the study of the multi-criteria decision-making (MCDM) problems under the probabilistic linguistic circumstance has caused widespread concern, and a lot of remarkable results have achieved [2–6, 10, 11]. However, in the practical probabilistic linguistic MCDM (PL-MCDM) problems, the criteria weight information we obtained may be uncertain since the complexity N. Yue · J. Xie (B) College of Science, Jimei University, Xiamen 361021, Fujian, China e-mail: [email protected] N. Yue e-mail: [email protected] S. Chen Chengyi University College, Jimei University, Xiamen 361021, Fujian, China e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2020 B.-y. Cao (ed.), Fuzzy Information and Engineering-2019, Advances in Intelligent Systems and Computing 1094, https://doi.org/10.1007/978-981-15-2459-2_6

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of the decision-making problems. In these uncertain situations, one of the cases is that the criteria weight values are incomplete known but the ranges are available. There are several methods to address this topic, for instance, the single-objective optimization model [1], the maximum deviation method [13], the classical two-step method [14], optimization model [16] and so on. But these methods have certain limitation for determining the optimal solution, such as these methods can only use known objective information and cannot fully exert the subjective initiative of decision makers (DMs). As the interaction requirements of DMs are raised, up to now, there have been just a few studies considering incomplete criteria weight information with interactive. Wang [9] introduced the interactive decision-making ideas in the multi-objective decision-making domain to the MCDM problems to solve the decision-making problems with incomplete criteria weight information. Xu and Chen [15] proposed an interactive method for fuzzy multiple attribute group decision-making, where the attribute weights are partly known. Based on the alternative achievement scale and alternative comprehensive scale, Xu [12] introduced an interactive method to solve the MCDM problems under the fuzzy circumstance. For the incomplete criteria weight information appearing in the probabilistic linguistic circumstance, since the PLTS can accurately express the true probability of each linguistic term, the interactive idea is used to solve the PL-MCDM problems with incomplete criteria weight information will be an important topic. To cope with the requirements of DMs under the probabilistic linguistic circumstance, we firstly give the equivalent expression form of the PLTS and the equivalent transformation functions between the PLTS and its associated vector. Secondly, to quantify the coordination of DMs and give full play to the subjective initiative of DMs, the achievement scale, satisfactory degree and dominated alternative under the probabilistic linguistic circumstance are proposed. Third, we present the probabilistic linguistic double interactive alternative reduction approach to cope with the PL-MCDM problems with the criteria weight values are incomplete known but the ranges are available. Finally, a practical case is used to illustrate the applicability and validity of the described method. The optimal alternative obtained by this method can achieve its good state as far as possible while meeting the requirements of DMs. Moreover, the amount of calculation in the decision-making process is reduced. The rest of this paper is organized as follows. In Sect. 2, we look back to some basic concepts that need to be used. Sect. 3 give the equivalent expression form of the PLTS and the equivalent transformation functions between the PLTS and its associated vector, then define the achievement scale, satisfactory degree and dominated alternative under the probabilistic linguistic circumstance and finally present the algorithm of the probabilistic linguistic double interactive alternative reduction approach. A case about the performance assessment of the big data analysis tools is carried out to illustrate the applicability and validity of our proposed method in Sect. 4. The paper finishes in Sect. 5 with some conclusions and the directions for future studies.

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2 Preparation In this section, we review some basic concepts related to the PLTS and the comprehensive criteria value of the alternative that needs to be used.

2.1 The Probabilistic Linguistic Term Set To express the true probability of each possible linguistic term, the probabilistic linguistic term set (PLTS) [7] based on the hesitant fuzzy linguistic term set (HFLTS) [8] is as follows. Definition 1 [7] The  probabilistic  linguistic term set (PLTS) PL(p) on linguistic term set (LTS) L = l0 , l1 , . . . , lδ can be defined as   #PL(p)    PL(p) = l(k) (p(k) )  l(k) ∈ L, p(k) ≥ 0, k = 1, 2, . . . , #PL(p), p(k) ≤ 1 k=1

where l(k) (p(k) ) is made up of the linguistic term l(k) and its associated probability p(k) , and #PL(p) is the cardinality of the PLTS PL(p).

2.2 The Comprehensive Criteria Value of the Alternative   Suppose that the MCDM a set of alternatives Ψ = ψ1 , ψ2 , . . . , ψn ,   problems contain a set of criteria Θ = θ1 , θ2 , . . . , θm with the weight vector Ω = (ω1 , ω2 , . . . , ωm ), where ωi ∈ [αi , βi ] ⊆ [0, 1], m i=1 ωi = 1. And the assessment of the alternative ψj with respect to a criterion θi is denoted as rij (i = 1, 2, . . . , m; j = 1, 2, . . . , n). Thenthe comprehensive criteria value of the alternative ψj (j = 1, 2, . . . , n) is zj = m i=1 ωi rij . In addition, since the criteria weight values are given in the form of intervals, the alternative ψj (j = 1, 2, . . . , n) has the minimum comprehensive criteria value zjmin (j = 1, 2, . . . , n) and the maximum comprehensive criteria value zjmax (j = 1, 2, . . . , n), which can be obtained by the following model (M.1). (M.1) min/max zj =

m 

ωi rij

i=1

⎧ ⎨ ωi ∈ [αi , βi ] ⊂ [0, 1] m s.t.  ⎩ ωi = 1 i=1

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3 The Probabilistic Linguistic Double Interactive Alternative Reduction Approach Before discussing the achievement scale, satisfactory degree and dominated alternative under the probabilistic linguistic circumstance, the equivalent transformation functions of the PLTS are first given to make sure that the cardinalities of all PLTSs are equal and easy to operate.   Definition 2 Let PL(p) be a PLTS on LTS L = l0 , l1 , . . . , lδ , then its equivalent expression form is defined as   δ+1    p(k) ≥ 0, k = 1, 2, . . . , δ + 1, p(k) ) 

p(k) ≤ 1 PL(p) = l(k−1) ( k=1

where p(k) = p(k) if l(k−1) (p(k) ) ∈ PL(p), otherwise, p(k) = 0. With the equivalent expression form of the PLTS, the equivalent transformation functions are proposed to implement the conversion between the PLTS and its associated vector v.    p(k) ≥ 0, k = 1, 2, . . . , δ + 1, δ+1 Definition 3 Suppose PL(p) = l(k−1) ( p(k) )  k=1 

p(k) ≤ 1 is the equivalent expression form of the PLTS PL(p) on LTS L =   l0 , l1 , . . . , lδ , and the associated vector of PL(p) is v = (v1 , v2 , . . . , vδ+1 ). Then p(k) ) and vk can be transformed into each other by the functions σ and σ −1 l(k−1) ( given as σ : PL(p) →v   k

p(k) = vk p(k) ) → σ l(k−1) ( p(k) ) = l(k−1) ( δ+1 σ −1 : v → PL(p) δ + 1  vk vk → σ −1 (vk ) = l(k−1) k Thus, vector of the PLTS  PL(p) on LTS L can be denoted as  1 the associated 2 δ v = δ+1

p(1) , δ+1

p(2) , . . . , δ+1

p(δ) , p(δ+1) . Therefore, for the PL-MCDM problems which contain a set of alternatives  θm with the weight vecΨ = ψ1 , ψ2 , . . . , ψn , a set of criteria Θ = θ1 , θ2 , . . . , tor Ω = (ω1 , ω2 , . . . , ωm ), where ωi ∈ [αi , βi ] ⊆ [0, 1], m i=1 ωi = 1. And the asas a PLTS sessment of the alternative ψj with respect to a criterion θi is denoted  PLij (p)(i = 1, 2, . . . , m; j = 1, 2, . . . , n) via using the LTS L = l0 , l1 , . . . , lδ , and the probabilistic linguistic decision matrix is PLm×n = (PLij (p))m×n . Let rij = |vij |(i = 1, 2, . . . , m; j = 1, 2, . . . , n), where vij is the associated vector of PLij (p), then the comprehensive criteriavalue of the alternative ψj under the probabilistic linguistic circumstance is zj = m i=1 ωi rij (j = 1, 2, . . . , n).

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With the comprehensive criteria value of the alternative under the probabilistic linguistic circumstance, the achievement scale and the satisfactory degree of the alternative in the PL-MCDM problems are given as follows. Definition 4 The achievement scale of the alternative ψj (j = 1, 2, . . . , n) is determined as zj − zjmin ϕ(zj ) = z j − zjmin and the satisfactory degree of the alternative ψj (j = 1, 2, . . . , n) is defined by ρ(zj ) =

zj − zjmin zjmax − zjmin

where zjmin ≤ z j ≤ zjmax is the expected level of the alternative ψj (j = 1, 2, . . . , n), which can be given by DMs. From Definition 4, we find that the achievement scale of the alternative reflects the achievement of each alternative and the satisfactory degree of the alternative reflects the satisfaction of DMs for this alternative. In what follows, when the sum of the achievement scales of all alternatives reaches a maximum, the criteria weight vector can be determined via model (M.2). (M.2) max ϕ =

n 

ϕ(zj )

j=1

⎧ ⎨ ωi ∈ [αi , βi ] ⊂ [0, 1] m s.t.  ⎩ ωi = 1 i=1

  By solving the model (M.2), the criteria weight vector is Ω 0 = ω10 , ω20 , . . . , ωm0 . Then the comprehensive criteria value and the satisfactory degree of the alternative ψj (j = 1, 2, . . . , n) can be calculated and denoted as zj (Ω 0 ), ρ(zj (Ω 0 ))(j = 1, 2, . . . , n) respectively. Based on the results obtained from model (M.2), we use model (M.3) to calculate the comprehensive criteria value of each alternative with the sum of the satisfactory degrees of all alternatives is also achieved a maximum. (M.3) max ρ =

n  j=1

ρ(zj )

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⎧ 0 ⎪ ⎪ ρ(zj ) ≥ ρ(zj (Ω )) ⎨ ωi ∈ [αi , βi ] ⊂ [0, 1] s.t.  m ⎪ ⎪ ωi = 1 ⎩ i=1

Solving the model (M.3) via using the software  MATLAB R2014b, we obtain  the criteria weight vector is Ω  = ω1 , ω2 , . . . , ωm , and the comprehensive criteria value of each alternative is zj (Ω  )(j = 1, 2, . . . , n). In the following, we remove the alternatives that do not need to be compared in the decision-making process according to the achievement scale and the satisfactory degree of each alternative. If zj (Ω  ) ≥ zj (Ω 0 ), the alternative ψj reserved, otherwise, delete the alternative ψj in Ψ . That is to say, the main idea of the interactive alternative reduction by using the achievement scale and satisfactory degree is to select alternatives with sufficient satisfactory degree under the achievement scale based on the expected level given by DMs. After that, the updated set of alternatives Ψ is obtained, and in this way, the amount of calculation for the alternatives can be reduced as the number of the alternative in Ψ is decreased. Furthermore, for the updated set of alternatives Ψ , we reduce the number of the alternatives in Ψ according to the dominated alternative and the non-dominated alternative to reduce the computation and complexity of the decision-making process. And, the concepts of the dominated alternative and the non-dominated alternative under the probabilistic linguistic circumstance are given as follows. Definition 5 The alternative ψj ∈ Ψ (j = 1, 2, . . . , n ; n ≤ n) is the dominated if there exists an alternative ψk ∈ Ψ such that zk > zj , where the updated set of alternatives Ψ consists n alternatives, zj and zk are the comprehensive criteria values of ψj and ψk respectively. Otherwise, alternative ψj is non-dominated. From Definition 5, we obtain the following theorem to determine whether the alternative ψj (j = 1, 2, . . . , n ) in Ψ is dominated or non-dominated.   Theorem 1 Given the criteria weight vector Ω = ω1 , ω2 , . . . , ωm , and the associated vector of the assessment of the alternative ψj ∈ Ψ (j = 1, 2, . . . , n ) with respect to a criterion θi ∈ Θ(i = 1, 2, . . . , m) is vij . Then ψj is the dominated alternative if and only if there is at least one alternative ψk ∈ Ψ such that − zkmax < 0, where rij = |vij |, rik = |vik |. δj k = zjmax Proof It is easy to verify, so it is omitted. Since the larger the comprehensive criteria value in the PL-MCDM problems and the better the alternative will be, the dominated alternatives determined by Theorem 1 in the updated set of alternatives Ψ can be deleted in the decision-making process. Based on the above three measures proposed under the probabilistic linguistic circumstance, the algorithm of the double interactive alternative reduction approach for PL-MCDM problems with incomplete criteria weight information is presented as follows.

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Algorithm 1 (The probabilistic linguistic double interactive alternative reduction approach) Step 1. Normalize the probabilistic linguistic decision matrix PLm×n via solving the normalization method presented in [7], and the normalized probabilistic linguistic ∗ = (PL∗ij (p))m×n ; decision matrix denoted as PLm×n Step 2. Determine the probabilistic linguistic real value matrix R = (rij )m×n , where rij = |vij∗ |, and vij∗ is the associated vector of the normalized PLTS PL∗ij (p)(i = 1, 2, . . . , m; j = 1, 2, . . . , n) in PL∗ ; Step 3. Calculate the minimum comprehensive criteria value zjmin (j = 1, 2, . . . , n) and the maximum comprehensive criteria value zjmax (j = 1, 2, . . . , n) by using model (M.1), then DMs propose the expected level z j (j = 1, 2, . . . , n); Step 4. Solve the models (M.2) and (M.3) to obtain the updated set of alternatives Ψ ; Step 5. Use Theorem 1 to delete the dominated alternatives in Ψ , a non-dominated set of alternatives Ψ is obtained; Step 6. If there is only one alternative in Ψ , the optimal alternative is determined, otherwise, DMs communicate with each other and select the optimal alternative from Ψ; Step 7. End.

4 Illustrative Example In this section, we use a case about the performance assessment of the big data analysis tools to illustrate the applicability and validity of the probabilistic linguistic double interactive alternative reduction approach. Suppose a big data analysis research team intends to choose the best from the following fifteen big data analysis tools: Hadoop (ψ1 ), MongoDB (ψ2 ), AWS Data Pipeline (ψ3 ), Snaplogic (ψ4 ), Treasuredata (ψ5 ), Striim (ψ6 ), HPCC (ψ7 ), Presto (ψ8 ), Oracle (ψ9 ), Pandas (ψ10 ), Lumify (ψ11 ), Disco (ψ12 ), Cloudera (ψ  13 ), Metabase (ψ14 ) and Statista (ψ15 ). The core members of the team use LTS L = l0 = terrible, l1 = very bad , l2 = bad , l3 = medium, l4 = good , l5 = very good , l6 = fantastic to evaluate the five basic aspects of big data analysis: analytic visualizations (θ1 ), data mining algorithms (θ2 ), predictive analytic capabilities (θ3 ), semantic engines (θ4 ),data quality and master data management (θ5 ). And the criteria weight vector Ω = (ω1 , ω2 , ω3 , ω4 , ω5 ) | 0.2 ≤ ω1 ≤ 0.4, 0.15 ≤ ω2 ≤ 0.35, 0.1 ≤ ω3 ≤   0.3, 0.2 ≤ ω4 ≤ 0.4, 0.01 ≤ ω5 ≤ 0.1, 5i=1 ωi = 1 . To reduce the length of the paper, we only give the normalized probabilistic linguistic decision matrix PL∗ 15×5 as shown in Table 1. Step 2. Determine the probabilistic linguistic real value matrix R 5×15 as shown in Table 2.

ψ1 ψ2 ψ3 ψ4 ψ5 ψ6 ψ7 ψ8 ψ9 ψ10 ψ11 ψ12 ψ13 ψ14 ψ15

{l5 (1)} {l3 (0.2), l5 (0.8)} {l2 (0.1), l3 (0.9)} {l3 (1)} {l2 (0.6), l3 (0.4)} {l4 (0.1), l5 (0.9)} {l3 (0.7), l4 (0.3)} {l4 (1)} {l2 (0.2), l4 (0.8)} {l3 (1)} {l3 (0.2), l5 (0.8)} {l4 (0.6), l5 (0.4)} {l3 (0.4), l6 (0.6)} {l2 (1)} {l5 (0.4), l6 (0.6)}

θ1

{l4 (0.2), l5 (0.8)} {l3 (0.9), l4 (0.1)} {l0 (0.1), l2 (0.8), l3 (0.1)} {l2 (0.2), l3 (0.2), l4 (0.6)} {l3 (0.9), l4 (0.1)} {l5 (1)} {l4 (0.4), l6 (0.6)} {l3 (0.1), l4 (0.8), l5 (0.1)} {l3 (0.7), l4 (0.3)} {l2 (0.6), l3 (0.4)} {l0 (0.1), l2 (0.9)} {l3 (0.1), l5 (0.4), l6 (0.5)} {l5 (0.9), l6 (0.1)} {l2 (0.5), l3 (0.2), l4 (0.3)} {l2 (0.8), l4 (0.2)}

θ2 {l5 (0.6), l6 (0.4)} {l5 (1)} {l3 (0.7), l4 (0.3)} {l2 (0.4), l4 (0.6)} {l3 (1)} {l3 (0.7), l4 (0.3)} {l4 (0.7), l5 (0.3)} {l3 (0.7), l4 (0.3)} {l6 (1)} {l0 (0.7), l1 (0.3)} {l3 (0.5), l4 (0.5)} {l4 (1)} {l2 (0.5), l4 (0.5)} {l3 (1)} {l2 (0.8), l3 (0.2)}

θ3

Table 1 The normalized probabilistic linguistic decision matrix PL∗ 15×5 {l5 (1)} {l3 (0.8), l4 (0.2)} {l2 (0.8), l4 (0.2)} {l3 (1)} {l1 (0.7), l3 (0.3)} {l3 (0.9), l4 (0.1)} {l2 (0.7), l3 (0.3)} {l4 (1)} {l3 (0.4), l4 (0.6)} {l2 (1)} {l3 (0.5), l4 (0.5)} {l5 (1)} {l2 (0.1), l3 (0.9)} {l3 (0.5), l4 (0.5)} {l4 (1)}

θ4

{l3 (0.4), l4 (0.6)} {l2 (0.1), l3 (0.6), l4 (0.3)} {l1 (0.3), l2 (0.3), l4 (0.4)} {l3 (0.3), l4 (0.7)} {l2 (0.8), l3 (0.2)} {l2 (0.4), l4 (0.1), l5 (0.5)} {l4 (0.6), l5 (0.4)} {l4 (0.5), l5 (0.5)} {l3 (1)} {l3 (0.1), l4 (0.9)} {l2 (0.1), l3 (0.9}) {l3 (0.1), l4 (0.8), l5 (0.1)} {l4 (0.2), l5 (0.8)} {l2 (0.6), l3 (0.4)} {l2 (1)}

θ5

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θ1 θ2 θ3 θ4 θ5

0.857 0.700 0.652 0.857 0.486

ψ1

0.695 0.519 0.857 0.479 0.407

ψ2

0.516 0.348 0.454 0.371 0.325

ψ3

0.571 0.452 0.462 0.571 0.529

ψ4 0.344 0.519 0.571 0.263 0.361

ψ5 0.775 0.857 0.454 0.519 0.467

ψ6

Table 2 The probabilistic linguistic real value matrix R 5×15 ψ7 0.454 0.665 0.562 0.346 0.549

ψ8 0.714 0.581 0.454 0.714 0.558

ψ9 0.578 0.454 1.000 0.486 0.571

ψ10 0.571 0.344 0.132 0.429 0.645

ψ11 0.695 0.386 0.457 0.457 0.516

ψ12 0.549 0.609 0.714 0.857 0.581

ψ13 0.642 0.778 0.417 0.516 0.700

ψ14 0.429 0.324 0.571 0.457 0.344

ψ15 0.691 0.371 0.361 0.714 0.429

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Min 0.727 Max 0.809 Expected level 0.755

z1

0.561 0.671 0.650

z2

0.396 0.445 0.420

z3 0.503 0.542 0.510

z4 0.361 0.447 0.435

z5

Table 3 Three comprehensive criteria values of each alternative 0.775 0.857 0.600

z6 0.596 0.717 0.510

z7 0.454 0.533 0.605

z8 0.546 0.668 0.640

z9 0.346 0.465 0.350

z10

0.480 0.547 0.485

z11

0.642 0.721 0.650

z12

0.553 0.648 0.560

z13

0.404 0.465 0.440

z14

0.501 0.617 0.530

z15

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zj ρ(zj (Ω 0 )) zj (Ω  )

(Ω 0 )

0.787 0.735 0.809

ψ1

0.614 0.483 0.608

ψ2

0.430 0.697 0.433

ψ3

0.525 0.568 0.542

ψ4 0.401 0.470 0.366

ψ5 0.712 0.962 0.665

ψ6 0.505 0.650 0.461

ψ7 0.648 0.738 0.666

ψ8 0.566 0.162 0.570

ψ9 0.434 0.736 0.446

ψ10 0.532 0.779 0.542

ψ11

0.645 0.036 0.680

ψ12

Table 4 The comprehensive criteria values and satisfactory degrees of all alternatives with different criteria weight vectors 0.634 0.856 0.598

ψ13

0.418 0.221 0.436

ψ14

0.567 0.571 0.615

ψ15

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Step 3. Calculate zjmin and zjmax (j = 1, 2, . . . , n) by using the model (M.1), then DMs propose the expected levels of all alternatives. And the results are listed in Table 3. Step 4. Solve the models (M.2) and (M.3) in turn, the criteria weight vectors we obtain are Ω 0 = (0.400, 0.290, 0.100, 0.200, 0.010), Ω  = (0.400, 0.150, 0.100, 0.340, 0.010). And, the comprehensive criteria values and satisfactory degrees of all alternatives concerning the criteria weight vectors Ω 0 and Ω  are shown in Table 4. Step 5. From Table 4, we have zj (Ω  ) < zj (Ω 0 )(j = 2, 5, 6, 7, 13), so the alternatives ψ2 , ψ5 , ψ6 , ψ7 , ψ13 should be deleted. Thus the updated set of alternatives is Ψ = {ψ1 , ψ3 , ψ4 , ψ8 , ψ9 , ψ10 , ψ11 , ψ12 , ψ14 , ψ15 }. And then by Theorem 1, we obtain the non-dominated set of alternatives is Ψ = {ψ1 }. Therefore, the optimal big data analysis tool is Hadoop (ψ1 ).

5 Conclusion In the process of decision-making, the probabilistic linguistic double interactive alternative reduction approach proposed in this paper can not only make full use of the known objective information but also maximize the subjective initiative of DMs. Through these two interactions between DMs, the optimal alternative obtained is more reasonable. Moreover, the probabilistic linguistic double interactive alternative reduction approach also provides a new way to solve the PL-MCDM problems with incomplete criteria weight information. However, only the optimal alternative can be obtained, and the order of the alternatives cannot be obtained. In the future, we will further study the interactive methods for decision-making problems with uncertain big data.

Acknowledgements This work is supported in part by the National Natural Science Foundation of China (No. 11371130), the Natural Science Foundation of Fujian Province (No. 2017J01558), Key Laboratory of Applied Mathematics of Fujian Province University (Putian University) (No. SX201906) and Pre-Research Fund of Jimei University.

References 1. Da, Q.L., Xu, Z.S.: Single-objective optimization model in uncertain multi-attribute decisionmaking. J. Syst. Eng. 17(1), 50–55 (2002) 2. Liang, D.C., Kobina, A., Quan, W.: Grey relational analysis method for probabilistic linguistic multi-criteria group decision-making based on Geometric Bonferroni mean. Int. J. Fuzzy Syst. 20(7), 2234–2244 (2018)

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3. Liao, H.C., Jiang, L.S., Xu, Z.S., Xu, J.P., Herrera, F.: A linear programming method for multiple criteria decision making with probabilistic linguistic information. Inf. Sci. 415–416, 341–355 (2017) 4. Lin, M.W., Xu, Z.S.: Probabilistic linguistic distance measures and their applications in multicriteria group decision making. Stud. Fuzziness Soft Comput. 357, 411–440 (2018) 5. Liu, P.D., Teng, F.: Some Muirhead mean operators for probabilistic linguistic term sets and their applications to multiple attribute decision-making. Appl. Soft Comput. 68, 396–431 (2018) 6. Liu, P.D., You, X.L.: Probabilistic linguistic TODIM approach for multiple attribute decisionmaking. Granular Comput. 2(4), 333–342 (2017) 7. Pang, Q., Wang, H., Xu, Z.S.: Probabilistic linguistic term sets in multi-attribute group decision making. Inf. Sci. 369, 128–143 (2016) 8. Rodríguez, R.M., Martínez, L., Herrera, F.: Hesitant fuzzy linguistic terms sets for decision making. IEEE Trans. Fuzzy Syst. 20(1), 109–119 (2012) 9. Wang, Y.B.: An interactive multi-attribute decision making method based on alternative set reduction strategy. Stat. Decis. 2, 68–70 (2016) 10. Wu, X.L., Liao, H.C.: An approach to quality function deployment based on probabilistic linguistic term sets and ORESTE method for multi-expert multi-criteria decision making. Inf. Fusion 43, 13–26 (2018) 11. Wu, X.L., Liao, H.C., Hafezalkotob, A., Herrera, F.: Probabilistic linguistic MULTIMOORA: a multi-criteria decision making method based on the probabilistic linguistic expectation function and the improved Borda rule. IEEE Trans. Fuzzy Syst. 26(6), 3688–3702 (2018) 12. Xu, Z.S.: Interactive method based on alternative achievement scale and alternative comprehensive scale for multi-attribute decision making problems. Control Decis. 17(4), 435–438 (2002) 13. Xu, Z.S.: Maximum deviation method based on deviation degree and possibility degree for uncertain multi-attribute decision making. Control Decis. 16, 818–821 (2001) 14. Xu, Z.S.: On method for multi-objective decision-making with partial weight information. Syst. Eng. Theor. Pract. 22(1), 43–47 (2002) 15. Xu, Z.S., Chen, J.: An interactive method for fuzzy multiple attribute group decision making. Inf. Sci. 117, 248–263 (2007) 16. Xu, Z.S., Zhang, X.L.: Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information. Knowl.-Based Syst. 52, 53–64 (2013)

Research on Method of Complex Fuzzy-Valued Integral Classifier in Evaluation of Ecotourism Safety in Hainan Island Jing Ma, Gen-nian Sun, and Sheng-quan Ma

Abstract This paper attempts to discuss the construction of tourism security evaluation model based on the new complex fuzzy-valued integral algorithm theoretically. It is the first time to study the application of this new classifier algorithm to the evaluation of tourism ecological security in Hainan Island. According to the actual situation of Hainan, this paper preliminarily establishes the early warning index system of ecological security in Hainan Province, also points out the technical problems, processes, and implementation steps in the application, etc., and it provides a new angle for theoretical research of tourism safety evaluation. Keywords Complex fuzzy-valued integral · Classifier · Tourism safety · Evaluation

1 Introduction Fuzzy integral classifier is a new and effective classifier training algorithm which emerged in recent ten years. It is especially suitable for the interaction between attributes. Classification as a technology, it purposes to construct a classification function or classification model (classifier) according to the characteristics of data sets to map unknown category samples to a given category. It is an important research field in data mining, machine learning, pattern recognition, and also the core issue of knowledge processing. Classification research has been studied for more than a century, and a large number of classification algorithms have been published. Common classification methods include Bayesian classification, neural network classification, linear classification, nearest neighbor classification, and so on. Since classification research is a very complex function extension study in theory, there is no optimal algorithm to adapt to various situations, so there are still many classification algorithms that appear one after another. The emergence of fuzzy sets makes the fuzzy J. Ma · G.-n. Sun School of Geography and Tourism, Shaanxi Normal University, Shaanxi 710119, China S.-q. Ma (B) School of Information Science and Technology, Hainan Normal University, Hainan 571158, China e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2020 B.-y. Cao (ed.), Fuzzy Information and Engineering-2019, Advances in Intelligent Systems and Computing 1094, https://doi.org/10.1007/978-981-15-2459-2_7

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technology combine with various classification algorithms to deal with the cognitive problems that classical classification algorithms cannot be addressed in the process of classification. Wang [1] and others used fuzzy real value measure and fuzzy real value integral as a classification tool in classifier design and fusion and achieved good results. Since Buckley [2] of Alabama University presented the concept of fuzzy complex number in 1989, many scholars have studied its theory and practical application. Some of them have studied its theoretical problems (measurement of complex fuzzy numerical function, measurable function of complex fuzzy numerical value, and complex fuzzy numerical integration (see [3–7])). Others have introduced a series of problems theoretically (complex fuzzy numerical measure, complex fuzzy measurable function, and complex fuzzy numerical integral) [8–11], and Sugeno-type complex fuzzy numerical integral is used as a new tool in classification fusion technology and other application problems. Since most of the learning and searching problems in classification are NP-hard and their exact solutions are still not yet achieved, it is still a meaningful work to explore various heuristic algorithms to approximate large-scale complex problems. In [12], it has studied the series of problems of complex fuzzy-valued integral (CFI for short). The complex fuzzy-valued integral classifier (CFIC for short) given in the later research is a new model and method to evaluate the recognition information by using the complex fuzzy-valued integral operator. As one of the hot spots of modern tourism research and development, ecotourism has become the fastest growing part of tourism. Its annual growth rate is about 20– 25%. Ecotourism pays attention to environmental protection and community development, which is an important form of sustainable tourism development. From the perspective of tourism, the potential impact of the number of tourists, tourism infrastructure, and different types of tourism activities on tourist destinations (regions) is analyzed. In order to realize the sustainable development of tourism industry, it is necessary to determine the tourism capacity, the lowest pollution, the ecological security and the protection of ecological diversity of tourist destinations (areas) based on the experience of tourists, the capacity of tourism space, the capacity of tourism heart, and the capacity of tourism ecology, without breaking through the scope of environmental carrying capacity. It is necessary to calculate the tourism spatial capacity, tourism psychological capacity, and tourism ecological capacity of scenic spots. In nowadays, people pay more attention to the protection of ecological environment and realize the importance of coexistence with the natural ecosystem for sustainable development of human beings (there are many complex links between human and natural system, and water is the most active and bonded.). Ecotourism is a term coined in the mid-1980s [13, 14]. Its two main purposes are to achieve sustainable management, to pursue natural resource protection and to satisfy tourists’ experience, that is, to pursue the maintaining and environment development and satisfying tourists’ needs. The concept of ecological security was put forward in the 1990s after the ecological problems directly and generally threatened the survival and safety of human beings on a large scale, and it was developed from the analysis of ecological risks. Its emergence has adapted to the change of environmental management objectives

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and environmental management concepts in the 1980s [15]. The concept of ecosystem and the rapid development of related theories provide a theoretical basis for people to understand the structure and function of ecosystem and the research and development of ecological security. Some researchers held that although the current research on tourism safety evaluation has taken shape, its main research fields are mainly focused on two aspects: tourism destination safety evaluation and tourism ecological safety evaluation, and its theoretical methods are mainly focused on fuzzy mathematics method and comprehensive index methods, such as mark method, neural network method, analytic hierarchy process, statistical analysis method, matter element extension method and so on [16]. In view of these developments, this paper attempts to use a new mathematical method—CFIC algorithm to study the evaluation of tourism safety. It can not only evaluate the safety of tourist destinations but also the ecological safety of tourism. And eco-safety management is a huge systematic project which integrates ecological safety into the framework of national safety management, and it is conducive to integrating resources development and utilization, environmental management, ecological protection, and many other fields, coordinating the responsibilities and interests of the competent departments, establishing a clear division of labor, coordinated and unified national ecological governance system, and promoting the modernization of ecological governance. Attention should be paid to the construction of a security assessment and early warning system, the establishment of a platform for police assessment, issuance and response, and the full protection of national ecological security. With the in-depth implementation of the strategy of “Developing Hainan International Tourism Island,” Hainan has further increased the pace of urbanization and industrialization. The influx of a large number of foreign tourists has brought significant negative impacts on the local ecological environment. The facing problem is how to rationally exploit and utilize natural resources in the new situation and reduce tremendous pressure of ecological security from human activities on regional livelihoods, fully bring out latent potentialities of existing resources, ensure ecological security, and promote the sustainable, healthy, and coordinated development of regional economy are important research topics in the process of regional development, as well as key contents in the construction of ecological-based area. To study the status of ecological security in Hainan Province, it is particularly important to evaluate its ecological security, which has significance to ensure the sustainable development of Hainan’s economy and society. This paper attempts to apply the CFIC classifier algorithm to the study of tourism safety. It is proposed to adopt the complex fuzzy-valued integral classifier algorithm to give a scientific evaluation method and theory for the evaluation of ecological security in Hainan and to explore the specific parts (key technologies, steps, processes, etc.) in the implementation of this problem. The rest of the paper is organized as follows. In Sect. 2, this paper introduces the concept and basic properties of the CFIC, especially for the discrete occasions in practical application, and it focuses on the calculation of Sugeno-type complex fuzzy-valued integral. And then, this part also introduces the complex fuzzy-valued measure to solve the problem of classifier calculation and gives the formula of the

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fuzzy density calculation, which lays a foundation for the application of the CFIC algorithm. And the key technology of data processing which will be obtained in actual case application is introduced, according to the key technology problems in the classifier algorithm and process. In Sect. 3, according to the actual situation of Hainan Island, this paper provides the construction of the evaluation index system of Hainan ecotourism security and the principle of comprehensive evaluation, constructs the evaluation index system of Hainan ecotourism security suitable for the situation of Hainan Province, discusses the technical methods to be used in the evaluation, and establishes comprehensive evaluation model.

2 Basic Concepts of Complex Fuzzy-Valued Integral Classifier (CFIC) The main concepts and related theoretical problems of the complex fuzzy-valued integral classifier algorithm are briefly introduced below. And the details can be found in Ref. [12].

2.1 Concept of Fuzzy-Valued Integral (CFI) Definition 1.1 Let (Z , F(Z ), μ) ˜ be the fuzzy measure space of CFI, E˜ ∈ F(Z ), and the complex fuzzy-valued integral of μ˜ of f˜ on E˜ is defined as: ⎞ ⎛   ⎟ ⎜ f˜dμ  ⎝ Re f˜dμ R , Im f˜dμ I ⎠,

 E˜

E˜

E˜

where 

⎡ Re f˜dμ R =

E˜

∪

λ∈(0,1]

⎢ λ⎣ 

=

∪

λ∈(0,1]

 = E˜

λ





Re f λ− dμ˜ R ,

E˜

⎤ ⎥ Re f λ+ dμ˜ R ⎦

E˜

sup

α∈[0,∞)

Im f˜dμ˜ I =



˜ (α ∧ Re μ− ), − λ ( A ∩ χ Fλ,α,1

∪

λ∈(0,1]

⎡

⎢ λ⎣



E˜

Im f λ− dμ˜ I ,

 E˜

˜ sup (α ∧ Re μ+ ) + λ ( A ∩ χ Fλ,α,1 α∈[0,∞) ⎤

⎥ Im f λ+ dμ˜ I ⎦,

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where    +    − = z Re f λ− (z) ≥ α , Fλ,α,1 = z Re f λ+ (z) ≥ α , Fλ,α,1    +    − Fλ,α,2 = z Im f λ− (z) ≥ α , Fλ,α,2 = z Im f λ+ (z) ≥ α . At this point, we define the complex fuzzy of μ of f˜ on E˜ is integrable. The properties of CFI are shown in Ref. [12]. Definition 1.2 Let f˜: Z → F0 (K + ) be a fuzzy measurable function of complex fuzzy value on complex fuzzy measure space (Z , F(Z ), μ), ˜ E˜ ∈ F(Z ). It is called  (s) E˜

˜ + i sup (α ∧ Im μ[( ˜ ˜ f˜1 )α ∩ E]) ˜ f˜2 )α ∩ E] f˜dμ˜ = sup (α ∧ Re μ[( α∈[0,∞)

α∈[0,∞)

is a Sugeno-type complex fuzzy-valued integral of μ˜ of f˜ on E˜ (S-type CFI), in which,         ( f˜1 )α = x Re f˜(x) ≥ α , ( f˜2 )α = x Im f˜(x) ≥ α , α ∈ (0, ∞). For the convenience of practical application, when X is a finite set (i.e., X = {x1 , x2 , . . . , xn }), and both real and imaginary parts of μ˜ are regular fuzzy measures, complex fuzzy functions are recorded as f˜(x) = a j = Re a j + iIm a j . Without√the loss of generality, supposing 0 + i0 ≤ a1 ≤ a2 ≤ · · · ≤ an ≤ 1 + i, i = −1 (follows the rules of plural sorting from Ref. [1], the X superscript can be rearranged to satisfy the relational expression), Sugeno-type CFIC can be simplified as  (s) E˜

˜ + i sup (α ∧ Im μ[( ˜ ˜ f˜1 )α ∩ E]) ˜ f˜2 )α ∩ E] f˜dμ˜ = sup (α ∧ Re μ[( α∈[0,∞)

α∈[0,∞)

   n  = ∨ (Re a j ∧ Re μ˜ x Re f˜(x) ≥ a j j=1    n  + i ∨ (Im a j ∧ Im μ˜ x Im f˜(x) ≥ Im a j j=1

n

n

j=1

j=1

˜ j )) + i ∨ (Im a j ∧ Im μ(A ˜ j )) = ∨ (Re a j ∧ Re μ(A     where A j = x1 , x2 , . . . , x j , A j = x j , x j+1 , . . . , xn .

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2.2 Basic Properties of Sugeno-Type CFI For the convenience of application, we summarize the most basic properties of Stype complex fuzzy-valued integrals as follows. The proofs of these properties are detailed in Ref. [12], which will not be rephrased here.     ˜ (c) f˜1 dμ˜ ≤ (c) f˜2 dμ; ˜ Theorem 1 If f˜1 ≤ f˜2 , then (s) A f˜1 dμ˜ ≤ (s) A f˜2 dμ,  (1) If μ(A) ˜ = 0, then (s) A f˜dμ˜ = 0;   / N ), then (c) f˜1 dμ˜ = (c) f˜2 dμ; ˜ (2) If N is a zero set, and f˜1 (x) = f˜2 (x)(∀x ∈    (3) (s) A ( f˜1 ∨ f˜2 )dμ˜ ≥ (s) A f˜1 dμ˜ ∨ (s) A f˜2 dμ; ˜   (4) If A ⊆ B, then (s) A f˜dμ˜ ≤ (s) B f˜dμ; ˜    (5) (s) A f˜dμ˜ = (s) X f˜χ A dμ; ˜ (c) χ A dμ˜ =μ(A); ˜   ˜ (6) If c is a nonnegative complex constant, then (s) A cd μ˜ = (s) A cd μ˜ ∧ μ(A); (7) If a is a nonnegative real number, b is a real number, then  (c)

(a f˜ + b)dμ˜ = a(c)

A

If a is a real number, then (c)



f˜dμ˜ + bμ(X ˜ )

A

 A

f˜d(a μ) ˜ = a(c)

 A

f˜dμ. ˜

2.3 Complex Fuzzy-Valued Integral Classifier (CFIC) 2.3.1

Complex Fuzzy Value gλ Measure

It is necessary to solve the problem of measuring complex fuzzy value in classification first, then the complex fuzzy-valued integral is used as a fusion operator to evaluate and classify. Because the fuzzy measure does not satisfy additivity, it is necessary to determine all subsets of the set, and it will cause massive calculation work. In order to facilitate practical application, we introduce the gλ complex fuzzy-valued measure. Definition 1.3 Let (Z , F(Z ), μ) ˜ be a complex fuzzy-valued measure space, if A, B ⊂ F(Z ), A ∩ B = φ, has μ(A ˜ ∪ B) = μ(A) ˜ + μ(B) ˜ + λμ(A) ˜ μ(B), ˜ where λ = Re λ + Im λ, Re λ, Im λ ∈ (−1, 0) ∪ (0, +∞), then call μ˜ complex fuzzyvalued measure as λ-complex fuzzy-valued measure. We still use gλ to represent a λ-complex fuzzy-valued measure. In practical applications, we consider X as limited. Let X = {x1 , x2 , . . . , xn } be a finite set and f˜ X → [0, 1]×[0, 1] be a complex function, with f˜(x j ) = a j (complex value). The complex values are sorted according to their real and imaginary parts (see

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Ref. [12]): If the real part values are large and the imaginary part values are large, the complex values will be large. These complex values are ordered as follows: a1 ≥ a2 ≥ · · · ≥ an , μ˜ is a gλ complex fuzzy-valued measure. In practical application, the real part Re μ˜ and imaginary part Im μ˜ of μ˜ complex fuzzy-valued measure can be appropriately constrained, such as “support degree” and “non-support degree,” “positive evaluation” and “negative evaluation,” “importance degree” and “non-importance degree,” which can be understood in combination with its actual meaning. Words should be ˜ i }) = Re μ˜ j ; satisfied with 0 < Re μ˜ + Im μ˜ ≤ 1; Im μ˜ j = 1 − Re μ˜ j ; Re μ({x ˜ j−1 ) + (Re λ)Re μ˜ j Re μ(A ˜ j−1 ), Re μ(A ˜ j ) = Re μ˜ j + Re μ(A   1 ≤ j ≤ n, A j = x1 , x2 , . . . , x j ; Re λ + 1 =

n 

(1 + Re λRe μ˜ j ); Im λ + 1 =

j=1

n 

(1 + Im λIm μ˜ j )

j=1

The existence and uniqueness of Re λ and Im λ in the upper formula are guaranteed by the following theorem: Theorem 2 For the fixed real number set {g j }, 1 ≤ j ≤ m, there is a unique real number λ ∈ (−1, +∞), λ = 0, which satisfies λ+1=

n 

(1 + λg j ),

i=1

Re n (l) jk

Re μ =  M l

j=1

Re n (l) jk

, k < M + 1,

Im n (l) jk

Im μ =  M l

j=1

Im n (l) jk

, k < M +1

where Re n (l) jk represents an output sample belonging to class j which is judged to be class k by the sub-classifier, and Im n (l) jk represents an output sample not belonging to class j which also is judged to be class k by the sub-classifier. When the complex fuzzy-valued measure is used to calculate the complex fuzzyvalued integral, only the complex fuzzy-valued measure needs to be known. The real part Re μ j of the jth density value μ j can be interpreted as the importance of information x j .

2.3.2

Complex Fuzzy-Valued Integral Calculation

At present, there are many forms of fuzzy complex-valued integral, such as C-type, Stype, and generalized CFI (see [1]), which have different effects on different practical

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application backgrounds. In this part, we will take the S-type CFI as an example to illustrate the calculation problem. According to the previous definition, the discrete formula for the corresponding S-type CFI is as follows:  (s)

n

n

j=1

j=1

˜ j )) + i ∨ (Im a j ∧ Im μ(A ˜ j )) f˜dμ˜ = ∨ (Re a j ∧ Re μ(A

(1)

E˜

    where A j = x1 , x2 , . . . , x j , A j = x j , x j+1 , . . . , xn . Special note: When X is a σ -algebra and gλ (X ) = 1, gλ is called Sugeno-type complex fuzzy-valued measure. Let X = {x1 , x2 , . . . , xn } be a finite set, f X → [0, 1] × [0, 1] be a complex function, and f (xi ) = ai , and a1 ≥ a2 ≥ · · · ≥ an , μ be a gλ complex fuzzy-valued measure. Im μ = 1 − Re μ, the value of Re μ can be calculated by following equation: Re μ({xi }) = Re μi Re μAi = Re μi + Re μ(Ai−1 ) + (Re λ)Re μi Re μ(Ai−1 ), 1 < i ≤ n

(2)

where Ai = {x1 , x2 , . . . , xi }. Re λ is given by the following equation Re λ + 1 =

n 

(1 + Re λRe μi )

(3)

i=1

2.3.3

Determination of Fuzzy Density

There are many methods to determine the fuzzy density (μi ), and the simplest of which is to take the recognition rate of each classifier as the value of the fuzzy density. In this paper, the complex fuzzy density of real part and imaginary part is determined according to the formula put forward by Yao Ming et al. (see [1]). Re n i(k) j

Re μ =  M k

i=1

Re n i(k) j

,

j < M +1

(4)

where Re n i(k) j represents an output sample belonging to class i, which is judged by the sub-classifier as class j, and Im n i(k) j represents an output sample not belonging to class i, which is judged by the sub-classifier as class J.

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Output of CFIC

For a classification of class c, let X = {x1 , x2 , . . . , xn } denote the set of attributes and f = ( f (x1 ), f (x2 ), . . . , f (xn )) denote the real value function defined on X , that is, the positive evaluation of each attribute is between 0 and 1, then 1 − f = (1 − f (x1 ), 1 − f (x2 ), . . . , 1 − f (xn )) is a negative evaluation of each attribute.  Let ai = x f dμi , bi = x (1 − f )dμi , here, μi is a complex fuzzy-valued measure corresponding to class i. The output of the integrator classifier is a set of two-dimensional vectors ϕ i ( f ) = ((a1 , b1 ), (a2 , b2 ), . . . , (ae , be )), k = 1, 2, . . . , c, (ak , bk ) represent the positive and negative membership evaluation of the example belonging to the class k. Let dk = ak − bk , k = 1, 2, . . . , c, the final classification can be given according to the principle of maximum a membership. The output of attributes can also be regarded as a complex value function, in which the real part represents the positive evaluation of the attributes and the imaginary part represents the negative evaluation of the attributes. Then, using the complex fuzzyvalued integral, examples can be used to evaluate the membership of each class, and the value of which is also a complex number and the real part generation. The real part represents the positive subordinate evaluation, and the imaginary part represents the negative one. Note: This classifier is still an integral-based classifier, but it is an improvement of the traditional integral classifier. It introduces negative membership evaluation, which makes the description of classification more comprehensive. Usually, using dk = αak − bk to describe the final membership evaluation value is also very rough. We suggest the following two ways: (1) Optimistic–pessimistic factor method Let dk = αak − (1 − α)bk .

(5)

(2) “Positive + negative” joint evaluation procedure: Step 1: Set up the fuzzy set of positive and negative subordinate evaluation. Step 2: Establishes the fuzzy rule base of positive and negative subordinate evaluation and final subordinate evaluation. Step 3: Input positive and negative membership evaluation values. The fuzzy logic toolbox in MATLAB is used to solve the problem. Steps for classification implementation: Taking the Sugeno-type CFI as the fusion operator, the implementation steps are as follows: Step 1: Calculate μi and λ for each classifier according to the above formula (3, 4). Calculate the value of complex fuzzy value according to Eq. (2).

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Step 2: Calculate the integral according to Eq. (1) ei (s) = (Re (ei (s)), Im(ei (s)) (i = 1, 2, . . . , n), n (Re ai ∧ Re μ{x|Re f (xi ) ≥ Re ai }) where Re (ei (s)) = ∨i=1 n

Im(ei (s)) = ∧ (Im ai ∨ Im μ{x|Im f (xi ) ≥ Im ai }). i=1

Step 3: Determine e+ (s) = (maxi Re(ei (s)), mini Im(ei (s))). Step 4: Calculate closeness degree N (e+ (s), e(s)).    1 N (e+ (s)) = 1 − (|Re (ei (s)) − Re(e+ (s)) + Im(ei (s)) − Im(e+ (s))) 2 Step 5: Use N (e+ (s), e(s)) to obtain the classification result and classify the sample x as the nearest class.

2.3.5

Flowchart of CFIC

See Fig. 1. Attention should be paid to the selection of Mamdani implication operator. The smaller the t-norm is, the higher the rank is.

3 Evaluation of Ecological Security System in Hainan Comprehensive evaluation of ecological security in Hainan Province involves natural resources, ecological environment, environmental pollution, and economic and social development. In choosing statistical units, it is difficult to obtain comprehensive data due to the large research area with numerous small villages, and most of the statistical data in experience are based on county-level administrative regions. Therefore, in order to facilitate the investigation and collection of data and minimize statistical errors, the county (city) region is chosen as one unit. At the same time, the county is a link between the cities and villages in the administrative division, and it is also the basic implementation unit of various ecological environment protection and control measures. The ecological environment condition of a county can also reflect or represent the ecological security of its region to a certain extent.

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Fig. 1 Flowchart of complex fuzzy-valued integral classifier

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Fig. 2 Hainan ecological security system framework

3.1 Index System of Ecological Security System in Hainan Here takes 19 counties (cities) in Hainan Province (Haikou City, Sanya City, Sansha City, Wuzhishan City, Wenchang City, Qionghai City, Wanning City, Danzhou City, Dongfang City, Dingan County, Tunchang County, Chengmai County, Lingao County, Baishali Autonomous County, Changjiang Li Autonomous County, Ledong Li Autonomous County, Lingshui Li Autonomous County, Baoting Li Autonomous County) as the basic unit of ecological security evaluation, based on the administrative regions at the level of Miao Autonomous County and Qiongzhong Li Miao Autonomous County, we can give the structure of Hainan ecological security system as follows (Fig. 2, Table 1).

3.2 Preliminary Study on Evaluation Method of Ecological Security in Hainan This paper tentatively attempts to apply the aforementioned algorithm based on the complex fuzzy-valued integral classifier to the evaluation of ecological security in Hainan Province. The specific implementation steps are as follows: (1) Firstly, we should investigate the ecological security situation in Hainan Island according to the index system before. (2) Secondly, according to the designed indicators, it is necessary to obtain the above-designed indicators data for the ecological security status of Hainan Province which can be obtained from the Hainan Tourism Bureau (3–5 years).

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Table 1 Hainan Tourism Administration on the investigation of ecological security Object layer

First-order indexes

Second-order indexes

Unit

Comprehensive assessment indicators of low-carbon society

Economic indicators

GDP per capita

Yuan

Growth rate of population

%

Density of population

People per square kilometer

The tertiary industry’s added value as percentage of GDP

%

GDP growth rate

%

Green GDP per capita

Yuan

Contribution rate of technological progress to GDP

%

Three industries structure

Social indicators

Domestic investment growth rate

%

Foreign investment in proportion to the social capital assets

%

Rural net incomes

Yuan

Urban disposable income

Yuan

Average financial income

Yuan

Employment rate

%

Engel coefficient Urbanization rate

%

Illiteracy rate among people over 15 years

%

Natural population growth rate

%

Average life expectancy

Year

Total dependency ratio

%

Social security coverage

% (continued)

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Table 1 (continued) Object layer

First-order indexes

Second-order indexes

Unit

Resources and environment indicators

GDP output ratio of major minerals

%

Low-carbon or new energy accounts for the proportion of total energy

%

Innocent treatment rate of domestic refuse

%

Urban air quality

%

Comprehensive treatment rate of industrial “three wastes”

%

Environmental investment as a percentage of GDP

%

Input intensity of ecological construction Input intensity of pollution control Popularization of environmental protection education Technical indicators

GDP energy intensity Carbon dioxide emissions per unit of GDP Renewable energy and new energy technology CO2 capture and storage technology Resource recycling efficiency

%

Elasticity coefficient of new energy R&D investment as a percentage of GDP

%

Environmental protection input as a percentage of GDP

%

(continued)

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Table 1 (continued) Object layer

First-order indexes

Second-order indexes

Low-carbon indicators

Ratio of low-carbon energy consumption building

Unit

Utilization rate of thermal insulation building materials CDM project approvals account for world total Carbon finance market development level Total carbon dioxide emissions

Ton

Carbon dioxide emission amount per capita

Ton

Output value of low-carbon products

Yuan

Development level of low-carbon agriculture Recognition of low-carbon consciousness Publicity of low-carbon concept Institutional indicators

System and policy of ecological security Perfection of ecological early warning mechanism Ecological safety regulations Effect of ecological safety system construction Openness of government affairs Administrative supervision (continued)

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Table 1 (continued) Object layer

First-order indexes

Second-order indexes

Unit

Enterprise social responsibility assessment Safety of territorial resources

Cultivated land security

Forest security

Land pollution

Others

Safety of water resource

Total cultivated land area

Thousand hectares

Per capita cultivated land

Hectares

Cultivated land quality index

%

Farmland drought and flood protection rate

%

Forest coverage

%

Reduction of forest coverage

%

Ecological forest area ratio

%

Coverage rate of garden network

%

Land pollution rate

%

“Three wastes” load of land and industry in unit area

Tons per square kilometer

Farmland chemical fertilizer and pesticide loading in unit area

Tons per square kilometer

Water loss and soil erosion

%

Population carrying rate

%

Land reserve resources

%

Soil incubation

%

Land barren rate

%

Coordination of soil and water conservation

%

Urban per capita green area

Square kilometer

Total water resources per capita

Cubic meter (continued)

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Table 1 (continued) Object layer

Safety of atmosphere resource

Safety of biology species

First-order indexes

Second-order indexes

Unit

Freshwater resources per capita

Cubic meter

Industrial wastewater discharge

Million tons

Unit water resource industrial wastewater load

Tons/cubic meter

Surface water quality index

%

Urban access to safe drinking water

%

The proportion of safe drinking water in rural areas

%

Annual freshwater extraction accounts for the total amount of water resources

%

Irrigated land to farmland rate

%

Sulfur dioxide emissions

Ten thousand tons

Sulfur dioxide emissions per capita

Ton

Discharge of industrial waste

Ten thousand tons

Air quality index

%

The proportion of electricity production from fossil fuels

%

The proportion of endangered species of mammals and birds

%

The proportion of endangered species in higher plants

%

National nature reserves account for land area

%

The indicator system is constructed according to the contents of natural resources, ecological environment, environmental pollution and economic and social development involved in the comprehensive evaluation of ecological security

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(3) Thirdly, according to the algorithm flow of the CFIC is given above, each evaluation value and final evaluation are obtained by evaluating the 1–3 level index values, respectively. In accordance with this step, we need to quantify the index factors, and the method of “Fuzzy Mathematics Thought + Expert Group Scoring” can be used. The train of thought is as follows: The range of index score is expressed by μ and real number interval [0, 1]. The highest level μ = 1(e.g., the most beautiful level), the lowest level μ = 0(e.g., the less beautiful level), and the real number in the interval correspond to the index factor of a certain level (see Table 1); we can give the range of initial evaluation value by 0–1 line segment scoring method. 0

×

0.5

×

1

Each index factor is evaluated by each member of the evaluation expert group according to the evaluation grade range determined by himself. The easiest way to deal with general problems is to evaluate the superiority of each indicator by means of arithmetic mean, which is as follows: μj =

n 1 μi n i=1

(6)

where μi is the evaluation value of each expert, i = 1, 2, . . . , n, and μ j is the relative membership degree of each index, j = 1, 2, . . . , m. We can also adopt more advanced methods: Let experts “score intervals” instead of “score.” Each expert gives a minimum value m and a maximum value M for a certain index, forming an interval [m, M]. We can get the final evaluation results by interval value statistics (interval overlay chart method) for each interval of the expert group. At this time, the result is still an interval, which means that the evaluation value of the index is between “minimum value and maximum value.”

3.2.1

Establishment of Comprehensive Evaluation Model

Commentary set is a description of the evaluation of indicators at all levels (quality, strength, etc.). We can generally divide the evaluation results into five levels, V(excellent), IV(good), III(medium), II(poor), and I(extremely poor), recorded as P = {V, IV, III, II, I}. Commentary set is composed of evaluation factors listed in the index system (see Table 1), generally recorded as U , A denotes the proportion or set of weights among index elements, which can give a comprehensive evaluation model by using the theory of fuzzy mathematics: B = A◦R

(7)

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μ : U → P; u → μ(u) where, u ∈ U, μ(u) ∈ [0, 1], the value of μ(u) corresponds to a certain level in the comment set P. Generally, we can get generally accepted criteria of comment set through actual investigation and analysis, which embodies that in the value of comment set at all levels (first-level indicators are divided into low-carbon social comprehensive evaluation indicators, land and resources security, water resources security, atmospheric resources security, biological species.). Generally, some people often choose B ∈ [0.0.45): corresponding to the level I of comment set (worst); B ∈ [0.45, 0.6) corresponding to the level II; B ∈ [0.6, 0.75) corresponding to the level III; B ∈ [0.75, 0.9) corresponding to the level IV; B ∈ [0.9, 1] corresponding to the level V. In the evaluation, we should evaluate each index by using multi-level fuzzy evaluation model and weighted average calculation. The equation is as follows: bi =

n 

ai ri j + θi

(8)

i=1

where ai —weight coefficient of ith elements bi —heavy load evaluation value of ith elements ri j —evaluation of jth and ith elements θi —adjustment added value. The Principle of Comprehensive Evaluation: Comprehensive Evaluation of Ecology (Tourism Safety) still adopts the “Maximum Subordination Principle.”

4 Conclusion This paper attempts to construct the early warning evaluation index system of ecological security in Hainan Island and to discuss the problems from the aspects of comprehensive evaluation index of low-carbon society, land and resources security, water resources security, atmospheric resources security, biological species security, and so on. The construction of early warning evaluation index system of ecological security must be centered on economic construction and sustainable. For the purpose of sustainable development, it embodies the contents and details of economy, society, resources and environment, technology, low carbon and system, establishes a set of feasible ecological security early warning index system, provides scientific basis for promoting sustainable development of Hainan International Tourism Island, and brings value to the construction of Hainan Free Trade Port. This paper tries to discuss the evaluation method based on the complex fuzzy-valued integral classifier. Using

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the method of fuzzy reasoning to deal with the output of classification is a more flexible and more intelligent method to evaluate and is a good value in practice in the evaluation of tourism studies. Acknowledgements Thanks to the support by The Ministry of Science and Technology of People’s Republic of China program (No. 2012DFA11270). And special thanks go to my supervisor, Gen-nian Sun.

References 1. Wang, Z.X.: Fuzzy measure and fuzzy integral and its application in classification technology. The Science Publishing Company, Beijing (2008) 2. Buckley, J.J.: Fuzzy complex numbers. Fuzzy Sets Syst. 33, 333–345 (1989) 3. Zhang, G.: The convergence for a sequence of fuzzy integrals of fuzzy number-valued function on the fuzzy set. Fuzzy Sets Syst. 59, 43–57 (1993) 4. Ma, S., Chen, F., Wang, Q.: Fuzzy complex-valued integral and its convergence. Fuzzy Eng. Oper. Res. AISC 147, 265–273 (2012) 5. Zhao, Z., Ma, S.: Fuzzy complex-valued fuzzy measure base on fuzzy complex sets. Fuzzy Eng. Oper. Res. AISC 147, 207–212 (2012) 6. Ma, S., Wang, Q.: The characters of the complex number-valued fuzzy measurable function. Adv. Intell. Soft Comput. 78, 49–54 (2010) 7. Ma, S.Q., Zhao, H., Li, S.G.: Measurable functions and properties on complex fuzzy-valued measure spaces. Fuzzy Syst. Math. 28(6), 73–79 (2014) 8. Ma, S.Q., Li, S.G., Zhao, H., Zhou, M.: Complex fuzzy-valued integrals and properties. J. Lanzhou Univ. (Nat. Sci.) 51(1), 109–114 (2015) 9. Ma, S., Chen, F., Wang, Q., Zhao, Z.: Sugeno type fuzzy complex-value integral and its application in classification. Procedia Eng. 29, 4140–4151 (2012) 10. Song, C., Ma, S.: An information fusion algorithm based on Sugeno fuzzy complex-valued integral. J. Comput. Inf. Syst. 7, 2166–2171 (2011) 11. Ma, S.-q., Chen, F.-c., Wang, Q., Zhao, Z.-q.: Sugeno type fuzzy complex-value integral and its application in classification. Procedia Eng. 29, 4140–4151 (2012) 12. Ma, S.Q.: Integral and application of complex fuzzy-valued functions. Doctoral dissertation of Shaanxi Normal University (2015) 13. Jin, B., Wang, R.Y., Cai, Y.L.: The concept of ecotourism and its application in China. Chin. J. Ecol. 20(3), 56–59 (2001) 14. Cui, S.H., Wang, J.: Non sustainable characteristics of ecotourism. Environ. Prot. 6, 29–31 (1999) 15. Fu, Z.Y., Xu, X.G.: Regional ecological risk assessment. Adv. Earth Sci. 16(2), 267–271 (2001) 16. Luo, J.F.: Current situation and prospect of tourism safety evaluation in China. J. Inst. Disaster Prev. Sci. Technol. 4, 51–59 (2014)

Multi-attribute Decision Making Based on the Choquet Integral Operator with Hesitant Fuzzy Linguistic Information Xiuqin Xu, Jialiang Xie, and Shuili Chen

Abstract In this paper, we investigate the multiple attribute decision making (MADM) problems under hesitant fuzzy linguistic environment. Firstly, by using Choquet integral, the hesitant fuzzy linguistic Choquet integral operator is developed for aggregating hesitant fuzzy linguistic information. The properties of the proposed operator are also investigated, such as idempotence, monotonicity, boundedness and exchangeability. Next, we apply the proposed operator to deal with MADM problems under hesitant fuzzy linguistic environment. Finally, a numerical example of information security risk assessment brought by paperless office is given to demonstrate the proposed method. This aggregation method can not only better solve the interaction between attributes but also reflect the practicability and effectiveness of decision making. Keywords The Choquet integral · Hesitant fuzzy linguistic information · Multi-attribute decision making · Non-additive measure

1 Introduction In many multi-attribute decision making (MADM) problems, still many challenges are faced. Due to the ambiguity and complexity of people’s cognition, sometimes experts cannot accurately express their views. Consequently, the fuzzy set (FS) proposed by Zadeh [1] can be used to represent uncertain and fuzzy information. Considering that the FS only includes the membership information, and there does not exist X. Xu · J. Xie College of Science, Jimei University, Xiamen 361021, Fujian, China e-mail: [email protected] J. Xie e-mail: [email protected] S. Chen (B) Chengyi University College, Jimei University, Xiamen 361021, Fujian, China e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2020 B.-y. Cao (ed.), Fuzzy Information and Engineering-2019, Advances in Intelligent Systems and Computing 1094, https://doi.org/10.1007/978-981-15-2459-2_8

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a clear approach or method for us to allocate one element to the membership degree of one set, the FS has been extended into various forms from different angles [2], such as intuitionistic fuzzy sets (IFSs), interval-valued intuitionistic fuzzy sets (IVIFSs), type-2 fuzzy sets, and hesitant fuzzy sets (HFSs). The linguistic terms, however, are much closer to people’s expression habits than the quantitative membership degrees [3]. Thus, Rodríguez et al. [3, 4] extended the HFS to linguistic context and introduced the hesitant fuzzy linguistic term sets (HFLTSs), which allow attributes of an object to be described in multiple linguistic terms, improving the flexibility of evaluating attributes. Since then, the application of HFLTSs in the field of the MADM problem has inspired many scholars’ interest, especially the research of information aggregation operators has attracted the attention of scholars [3, 5–7]. Motivated by the pessimistic and optimistic rules, Rodríguez [3] defines the min-upper operator and the max-lower operator to assemble the hesitant fuzzy linguistic terminology information and then provides a method to solve the MADM problem based on hesitant fuzzy linguistic evaluation information. Based on the convex combination of two linguistic terms, Wei [8] introduced the convex combination of two hesitant fuzzy linguistic elements (HFLEs), based on which, the hesitant fuzzy linguistic weighted averaging (HFLWA) operator and the hesitant fuzzy linguistic ordered weighted averaging (HFLOWA) operator were proposed. They then applied these two operators to multiple criteria decision making (MCDM) and group decision making (GDM), respectively. Zhang [9] introduced some aggregation operators for HFLTSs, such as the hesitant fuzzy linguistic averaging (HFLA) operator and the hesitant fuzzy linguistic geometric (HFLG) operator. All above-mentioned hesitant fuzzy linguistic aggregation operators only consider situations where all the elements are independent, that is, the attribute weights are additive. However, in many practical situations, the elements are usually correlative. In response to this situation, the MADM method with nonlinear integral as the integrated function can not only consider the relative importance of decision attributes but also flexibly describe and handle the interaction between decision attributes [10, 11]. It has been applied widely in MADM under different uncertain circumstances [12–16]. Motivated by the above analysis, it is natural to combine the Choquet integral and the HFLTS theory in decision making. In this paper, we proposed the hesitant fuzzy linguistic Choquet integral (HFLCI) operator under hesitant fuzzy linguistic information and discuss the operational properties. In addition, we apply the developed aggregation operator to solve MADM problems under hesitant fuzzy linguistic environment. Our results suggest that HFLCI operator can not only consider the interaction between attributes but also deal with inaccurate and uncertain decision information very well. The paper is organized as follows. After the introduction, we review some basic concepts related to hesitant fuzzy linguistic term set and the Choquet integral. In Sect. 3, we introduce the hesitant fuzzy linguistic Choquet integral (HFLCI) operator and discuss their basic properties. In Sect. 4, we develop a method based on the new operator under hesitant fuzzy linguistic environment to solve MADM problems.

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Meanwhile, an example of information security risk assessment is also given to show the effectiveness and accuracy of the developed approach. In Sect. 5, we give a conclusion.

2 Preliminaries In this section, we first clarify the definition of hesitant fuzzy linguistic term set (HFLTS). As the basis of nonlinear integral, integration function handles the importance and interactivity between decision attributes, and we also review the Choquet integral. Definition 1 [3] Let S = {s0 , s1 , . . . , sτ } be a linguistic term set. A hesitant fuzzy linguistic term set (HFLTS), HS , is an order finite subset of the consecutive linguistic terms of S. Liao et al. [17] redefined and formalized the HFLTS mathematically as follows. Definition 2 [17] Let xi ∈ X , i = 1, 2, . . . , N , be fixed and S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ } be a linguistic term set. A hesitant fuzzy linguistic term set (HFLTS) on X , HS , is in mathematical terms of HS = {< xi , hS (xi ) > |xi ∈ X },

(1)

where hS (xi )is a set of some values in the linguistic term set S and can be expressed  as hS (xi ) = sφl (xi )|sφl (xi ) ∈ S; l = 1, 2, . . . , L; φl ∈ {−τ , . . . , −1, 0, 1, τ } with L being the number of linguistic terms in hS (xi ). hS (xi ) denotes the possible degree of the linguistic variable xi to the linguistic term set S. For convenience, hS (xi ) is called the hesitant linguistic element (HFLE), and hS is the set of all HFLEs. Definition 3 [18] Let P(X ) be the power set of X = {x1 , x2 , . . . , xn }, μ : P(X ) → [0, 1], if the following conditions are met, then μ is called the fuzzy measure defined on X : (1) μ(φ) = 0, μ(X ) = 1; (2) B ⊂ C implies μ(B) ≤ μ(C), for all B, C ∈ X . Definition 4 [14] Let f be a non-negative function defined on X = {x1 , x2 , . . . , xn }, μ is the fuzzy measure defined on X , then the discrete Choquet integral of the fuzzy measure μ is defined as  (C)

fd μ =

n  (f (xσ(i) )[μ(Bσ(i) ) − μ(Bσ(i−1) )]), i=1

where (σ(1), σ(2), . . . , σ(n)) is a permutation of (1, 2, . . . , n), and satisfied f (xσ(1) )≥ f (xσ(2) ) ≥ · · · ≥ f (xσ(n) ); Bσ(i) = {xσ(1) , xσ(2) , . . . , xσ(n) }, i = 1, 2, . . . , n, Bσ(0) = φ.

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3 Hesitant Fuzzy Linguistic Choquet Integral Operator In this section, for achieving mutual conversion between hesitant fuzzy linguistic element (HFLE) and hesitant fuzzy element (HFE), we firstly introduce equivalent transformation functions. Then, we define the hesitant fuzzy linguistic Choquet integral operator and discuss its related properties. Definition 5 [19] Let S = {st |t = −τ , . . . , −1, 0, 1, . . . , τ } be a finite and totally ordered discrete linguistic term set hS = {st |t ∈ [−τ , τ ]} be a HFLE, and H = {γ|γ ∈ [0, 1]} be a HFS. Then, we can get this function as:   1 t + |t ∈ [−τ , τ ] = hγ . g : [−τ , τ ] → [0, 1], g(hS ) = g(st ) = 2τ 2   g −1 : [0, 1] → [−τ , τ ], g −1 (hγ ) = g −1 (γ) = s(2γ−1)τ |γ ∈ [0, 1] = hS . Furthermore, in order to compare any two HFLEs, the corresponding score function based on the transformation functions was proposed in Definition 6. Definition 6 [19] Let hS = {st |t ∈ [−τ , τ ]} be a HFLE, then 1 g(st ), l i=1 l

s(hS ) =

can be called the score function of hS , where st ∈ hS and l is the number of the elements of hS . Therefore, 1. If s(hs1 ) < s(hs2 ), then hs1 is smaller than hs2 , denoted by hs1 ≺ hs2 . 2. If s(hs1 ) = s(hs2 ), then hs1 is equal to hs2 , denoted by hs1 = hs2 . Based on these two equivalent transformation functions described in Definition 5, we can develop an aggregation operator for hesitant fuzzy linguistic information. Definition 7 Let hSi (i = 1, 2, . . . , n) be a collection of HFLEs, μ is the fuzzy measure defined on X , and μ(BS(i) ), μ(BS(i−1) ) > 0, then we call  (C1 )

hs d μ = HFLCI(hS1 , hS2 , . . . , hSn ) = g −1

n  

[μ(BS(i) ) − μ(BS(i−1) )]g(hS(i) ) i=1

n  

[μ(BS(i) ) − μ(BS(i−1) )](γi ) , = g −1 i=1

the  hesitant fuzzy linguistic Choquet integral operator (HFLCI), where (C1 ) hs d μ denoted Choquet integral, (S(1), S(2), . . . , S(n)) is a permutation of (1, 2, …, n), and satisfied g(hS(1) ) ≥ g(hS(2) ) ≥ · · · ≥ g(hS(n) ); BS(i) = {hS(1) , hS(2) , . . . , hS(n) |i = 1, 2, . . . , n}, Bσ(0) = φ.

Multi-attribute Decision Making Based on the Choquet Integral …

Obviously,

111

 n 

[μ(BS(i) ) − μ(BS(i−1) )](γi ) is the HFCI operator, which is a i=1

HFS by the operations on operations on HFSs. Finally, we use the other equivalent transformation function g −1 to transform the HFS into the HFLE. We can get the following theorem. Theorem 1 Let μ(BS(i) ), μ(BS(i−1) ) > 0, and hSi (i = 1, 2, . . . , n) be a collection of HFLEs, then the aggregated value by using the HFLCI is still HFLEs, and



HFLCI(hS1 , hS2 , . . . , hSn ) =

n    g −1 1 − (1 − γi )μ(BS(i) )−μ(BS(i−1) ) .

γi ∈h(i)

(2)

i=1

Proof By using mathematical inductive method, we prove Theorem 1 as follows. For n = 2, base on Definitions 4 and 7 in [20], we have HFLCI(hS1 , hS2 ) = g −1

2  

[μ(BS(i) ) − μ(BS(i−1) )](γi ) i=1

=g

  −1

2  

(1 − γi )μ(BS(i) )−μ(BS(i−1) )

1−

γi ∈h(i)

=

i=1

2

    g −1 1 − (1 − γi )μ(BS(i) )−μ(BS(i−1) ) . γi ∈h(i)

i=1

If Theorem 1 holds for n = k, that is, HFLCI(hS1 , hS2 , . . . , hSk ) =



k    g −1 1 − (1 − γi )μ(BS(i) )−μ(BS(i−1) ) .

γi ∈h(i)

i=1

Then, when n = k + 1, we have, HFLCI(hS1 , hS2 , . . . , hSk , hSk+1 ) = g −1

k+1 

 [μ(BS(i) ) − μ(BS(i−1) )](hi ) i=1

=g

k   

1 − (1 − γi )μ(BS(i) )−μ(BS(i−1) ) 1 − (1 − γk+1 )μ(BS(k+1) )−μ(BS(k) )

  −1 γi ∈h(i)

=

 γi ∈h(i)

 g −1 1 −

i=1 k+1 

(1 − γi )μ(BS(i) )−μ(BS(i−1) )



.

i=1

Obviously, Theorem 1 holds for n = k + 1, according to Definition 7, we can get the aggregated result of HFLCI(hS(1) , hS(2) , . . . , hS(n) ) is still a HFLE, which complete the proof. 

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Remark 1 If μ(xS(i) ) = μ(BS(i) ) − μ(BS(i−1) ), i = 1, 2, . . . , n, then the hesitant fuzzy linguistic Choquet integral (HFLCI) operator can be reduced into the hesitant fuzzy linguistic weighted average (HFLWA) operator [21].  |B| Remark 2 If μ(B) = i=1 ωi , where |B| represents the cardinality of set B, then i μ(BS(i) ) = k=1 ωk , so we can get ωi = μ(BS(i) ) − μ(BS(i−1) ), i = 1, 2, . . . , n, where ω = (ω1 , ω2 , . . . , ωn )T , ωi ≥ 0, i = 1, 2, . . . , n, ni=1 ωi = 1, then the hesitant fuzzy linguistic Choquet integral (HFLCI) operator can be reduced into the hesitant fuzzy linguistic ordered weighted average (HFLOWA) operator [21].  By Theorem 1, we can prove that the HFLCI operator has the following properties. Theorem 2 (Idempotency) Let X is a nonempty finite set and hS , hSi (i = 1, 2, . . . , n) be the hesitant fuzzy linguistic element on X , if hS = hSi (i = 1, 2, . . . , n), then we can get HFLCI(hS1 , hS2 , . . . , hSn ) = hS . Proof Let hS1 = hS2 = · · · = hSn = hS , (S1 , S2 , S3 , . . . , Sn ) is a permutation of (1, 2, . . . , n). By Theorem 1, we obtain HFLCI(hS1 , hS2 , . . . , hSn ) =



n    g −1 1 − (1 − γi )μ(BS(i) )−μ(BS(i−1) ) .

γi ∈h(i)

Since

n

i=1 (μ(BS(i) )

i=1

− μ(BS(i−1) )) = 1, and Definition 5, we have HFLCI(hS1 , hS2 , . . . , hSn ) = hS , 

which completes the proof.

Theorem 3 (Monotonicity) Let hSa = {hSa1 , hSa2 , . . . , hSan } and hSb = {hSb1 , hSb2 , . . . , hSbn } be two collections of HFLEs. IF for any Stai ∈ hSai and Stbi ∈ hSbi , we have Stai ≤ Stbi for any i, then HFLCI(hSa1 , hSa2 , . . . , hSan ) ≤ HFLCI(hSb1 , hSb2 , . . . , hSbn ). Proof Based on Definition 5, we can transform all HFLEs into HFSs, namely g : [−τ , τ ] → [0, 1], g(hSθi ) = {g(Stθi ) = γθi = where i = 1, 2, . . . , n and θ = a or θ = b.

1 tθ i + |tθi ∈ [−τ , τ ]} = hθi , 2τ 2

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113

Then, for any γθi ∈ hθi , we obtain γai ≤ γbi , so for any γai ∈ hγai and γbi ∈ hγbi , it follows n n   μ(BS(i) )−μ(BS(i−1) ) ) ≤ (1 − (1 − γbi )μ(BS(i) )−μ(BS(i−1) ) ). (1 − (1 − γai ) i=1

i=1

Based on Definition 7 and Theorem 1, we can get n

    g −1 1 − (1 − γai )μ(BS(i) )−μ(BS(i−1) )

HFLCI(hSa1 , hSa2 , . . . , hSan ) =

γai ∈hai

≤

  g −1 1 − γbi ∈hbi

n 

(1 − γbi )μ(BS(i) )−μ(BS(i−1) )

i=1



= HFLCI(hSb1 , hSb2 , . . . , hSbn ),

i=1



which completes the proof.

Theorem 4 (bouncy) Let hSi (i = 1, 2, . . . , n) be a collection of HFLEs, h+ Si =   − + + − − si ∈hS max{si } and hSi = si ∈hS min{si }, si ∈ hSi , si ∈ hSi , then i

i

+ h− Si ≤ HFLCI(hS1 , hS2 , . . . , hSn ) ≤ hSi

Proof Based on Definition 5, we can transform all HFLEs into HFSs, namely g : [−τ , τ ] → [0, 1], g(hsi ) = {g(si ) = γi =

1 i + |i ∈ [−τ , τ ]} = hi . 2τ 2

− − − + where γi ∈ hi (i = 1, 2, . . . , n), and there are γi+ ∈ h+ i , γi ∈ hi and γi ≤ γi ≤ γi , for all i. So, for any γi− ≤ γi and γi ≤ γi+ , we have n n n    (1 − γi+ )μ(BS(i) )−μ(BS(i−1) ) ≤ (1 − γi )μ(BS(i) )−μ(BS(i−1) ) ≤ (1 − γi− )μ(BS(i) )−μ(BS(i−1) ) i=1

i=1

i=1

Therefore, we can obtain, γi− ≤ 1 −

n  i=1

(1 − γi )μ(BS(i) )−μ(BS(i−1) ) ≤ γi+

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Additionally, from Definition 7 and Theorem 1, we can get h− Si =



 g −1 (γi− ) ≤ HFLCI(hS1 , hS2 , . . . , hSn )

γi− ∈h− (i)

=

 γi ∈h(i)

≤

n    g −1 1 − (1 − γi )μ(BS(i) )−μ(BS(i−1) )



i=1



g −1 (γi+ ) = h+ Si ,

γi+ ∈h+ (i) + then, h− Si ≤ HFLCI(hS(1) , hS(2) , . . . , hS(n ) ) ≤ hSi , which completes the proof.



Theorem 5 (Commutativity) Let hSi (i = 1, 2, . . . , n) be a collection of HFLEs, and (h(S1 ) , h(S2 ) , . . . , h(Sn ) ) be any permutation of (hS1 , hS2 , . . . , hSn ), then HFLCI(hS1 , hS2 , . . . , hSn ) = HFLCI(h(S1 ) , h(S2 ) , . . . , h(Sn ) ). Proof The commutativity of the HFLCI operator is obvious.

4 The Application of HFLCI Operator Based on the HFLCI operator, in the following, an approach is developed to deal with the MADM problems with hesitant fuzzy linguistic information. For MADM problems, let A = {a1 , a2 , . . . , am } be a set of m alternatives, and C = {c1 , c2 , . . . , cn } be a set of n attributes, DMs evaluate alternative ai (i = 1, 2, . . . , m) according to the attribute ci (i = 1, 2, . . . , n). During the process of decision making, the experts provide all their evaluation values that the alternative ai (i = 1, 2, . . . , m) satisfies the criterion cj (j = 1, 2, . . . , n) represented by the HFLEs hSi,j =

sij (i = 1, 2, . . . , m; j = 1, 2, . . . , n).

sij ∈hSij

All the HFLEs are contained in the hesitant fuzzy linguistic decision matrix H = (hSij )m×n shown as in Table 1. Step 1. Given the hesitant fuzzy linguistic matrix H = (hSij )m×n , for the decisionmaking problem with n attributes, the measures of the other 2n − 2 attribute subsets are further assigned, except the null set and the universal set are 0 and 1, respectively. Step 2. Constructing the hesitant fuzzy linguistic score matrix, sort the attributes in each alternative separately based on the score value.

Multi-attribute Decision Making Based on the Choquet Integral … Table 1 Hesitant fuzzy linguistic decision matrix C1 C2 a1 a2 ... am

hS11 hS21 ... hSm1

hS12 hS22 ... hSm2

115

...

Cn

... ... ... ...

hS1n hS2n ... hSmn

Step 3. Utilize the HFLCI operator to aggregate all HFLEs hSij (j = 1, 2, . . . , n) in the ith line of the normalized matrix H = (hSij )m×n , then, we can get the overall performance values hi corresponding to the alternative ai , hi = HFLCI(hSi1 , hSi2 , . . . , hSin ). Step 4. Calculate the score values s(hi ) (i = 1, 2, . . . , n) of all the performance values hi (i = 1, 2, . . . , n) based on Definition 6. Then, the ranking order of all the score values can be obtained, and the optimal alternative can also be selected. Example With the development of modern information technology, the trend of the paperless office is widely favored and promoted by all countries. However, the information security risks brought by it are indeed concerned and worried by all countries in the world. School is a large paperless office. School’s information security is worth our consideration, including a large number of students and teachers’ personal information and school confidential documents and other information security. To do so, we should consider three main attributes to determine the high security risks of information security in a certain department of the school. Firstly, we should consider the security threat factor, denoted by C1 , which includes four sub-criteria: software and hardware failure (C11 ), physical environment impact (C12 ), operational error (C13 ), external attack threat (C14 ). The second criterion is information vulnerability factors, denoted by C2 , which includes four sub-criteria: technical management (C21 ), the network structure (C22 ), the system software (C23 ), application middleware (C24 ). The next criterion is asset security factor, denoted by C3 , which includes four sub-criteria: confidentiality (C31 ), integrity (C32 ), usability (C33 ), importance (C34 ). The final criteria criterion is safety management factor, denoted by C4 , which includes four sub-criteria: accident coordination and control (C41 ), safety management guidelines and systems (C42 ), risk analysis (C43 ), accident prevention measures (C44 ). From above, we can see that the threat exploits the vulnerability to damage the assets and then put information at risk. Step 1. The expert gives evaluations about the information security risk for four departments of school (A, B, C, D), which constitute the hesitant fuzzy linguistic decision matrix, shown as in Table 2 (Here, we defined S = {s−3 = Extremely dangerous, s−2 = Highly dangerous, s−1 = Moderate risk, s0 = General danger, s1 = Safety, s2 = Highly secure, s3 = Extremely secure}).

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Table 2 Hesitant fuzzy linguistic decision matrix C1 C2 {s0 , s1 } {s1 } {s1 } {s2 }

A B C D

{s2 } {s−1 , s0 , s1 } {s1 , s2 } {s0 , s1 }

Table 3 The μ values of each attribute subset C μ(C) C μ(C) μ(C1 ) μ(C2 ) μ(C3 ) μ(C4 ) μ(C1 , C2 )

μ(C1 , C3 ) μ(C1 , C4 ) μ(C2 , C3 ) μ(C2 , C4 ) μ(C3 , C4 )

0.35 0.25 0.20 0.18 0.63

0.56 0.50 0.58 0.43 0.48

Table 4 Hesitant fuzzy linguistic score matrix C1 C2 A B C D

0.5833 0.6667 0.6667 0.8333

0.8333 0.7500 0.7500 0.5833

C3

C4

{s−1 , s0 } {s2 } {s2 , s3 } {s1 }

{s1 , s2 , s3 } {s0 } {s2 } {s1 , s2 }

C

μ(C)

μ(C1 , C2 , C3 ) μ(C1 , C2 , C4 ) μ(C1 , C3 , C4 ) μ(C2 , C3 , C4 ) μ(C1 , C2 , C3 , C4 )

0.75 0.70 0.78 0.74 1

C3

C4

0.8333 0.8333 0.9167 0.6667

0.7500 0.5000 0.8333 0.7500

Suppose the μ measure value of any attribute subset on attribute set C, shown as in Table 3. Step 2. Constructing the hesitant fuzzy linguistic score matrix, sort the attributes in each alternative separately based on score value, shown as in Table 4. Step 3. Utilize the HFLCI operator to aggregate all HFLEs hSij (j = 1, 2, . . . , n) in the ith line of the normalized matrix H = (hSij )m×n ; then, we can get the overall performance values hi corresponding to the alternative ai : hi = HFLCI(hSi1 , hSi2 , . . . , hSin ). Step 4. Calculate the score values s(hi ) (i = 1, 2, . . . , n) of all the performance values hi (i = 1, 2, . . . , n) based on Definition 6. Then, the ranking order of all the score values can be obtained, and the optimal alternative can also be selected. The following is the hesitant fuzzy linguistic evaluation of aggregation department A. It can be obtained from Tables 2 and 3. C(1) = {C2 , C3 , C1 , C4 }, C(2) = {C3 , C1 , C4 }, C(3) = {C1 , C4 }, C(1) = {C4 }, hS(1) = {s2 } = {5/6}, hS(2) = {s−1 , s0 } = {1/3, 1/2}, hS(3) = {s0 , s1 } = {1/2, 2/3}, hS(4) = {s1 , s2 } = {2/3, 5/6}.

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117

By Eq. (2), we can get h1 = HFLCI(hS1 , hS2 , hS3 , hS4 ) =

γi ∈hS(i)

−1

{g (1 −

4 

(1 − γi )μ(BS(i) )−μ(BS(i−1) ) )}

i=1

−1

= g {0.6855, 0.6650, 0.6477, 0.6247, 0.6075, 0.5818, 0.5603, 0.5315} = {s1.11 , s0.99 , s0.87 , s0.75 , s0.65 , s0.49 , s0.36 , s0.19 }. Then, by Definition 6, we have s(h1 ) = 0.613. Similarly, we can calculate the score values of hi (i = 1, 2, 3, 4), which are s(h1 ) = 0.613, s(h2 ) = 0.6715, s(h3 ) = 0.7853, s(h4 ) = 0.6087. Then, the ranking order is s(h3 ) > s(h2 ) > s(h1 ) > s(h4 ) , so the department with the highest risk of information security is D.

5 Conclusion In this paper, under the background of uncertain evaluation information of hesitating fuzzy language, the multi-attribute decision making method of interrelation and interaction among attributes is given by constructing the hesitant fuzzy linguistic operator in the form of Choquet integral. The hesitant fuzzy linguistic Choquet integral (HFLCI) operator can be better applied to the practical problems of multi-attribute decision making. The content of this paper enriches the hesitant fuzzy linguistic operator theory, which provides a more choice space for people to make decisions. Acknowledgements This work is supported in part by the National Natural Science Foundation of China (No. 11371130), the Natural Science Foundation of Fujian Province (No. 2017J01558), Key Laboratory of Applied Mathematics of Fujian Province University (Putian University) (No. SX201906), and Pre-Research Fund of Jimei University.

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Fuzzy Engineering Model and Optimize

Design of Hierarchical Cone Fuzzy System for Nonlinear System Modeling Ming-zuo Jiang, Xue-hai Yuan, and Jia-xia Wang

Abstract The use of hierarchical fuzzy systems (HFS) is an effective way to deal with the curse of dimensionality which is the main drawback for the application of fuzzy models in the modeling and control of large-scale systems. This paper proposes the design of HFS which implements T-S type cone fuzzy system (CFS). The performance of the hierarchical fuzzy system is evaluated through time series prediction and function approximation, which demonstrate that the proposed HFS working together with the optimization of parameters by genetic algorithm (GA) and k-means clustering in fuzzy partition provides structurally simple and accurate fuzzy models. Keywords Takagi–Sugeno fuzzy system · Hierarchical fuzzy system · Cone fuzzy system · Genetic algorithm

1 Introduction Fuzzy system has been utilized to solve a wide range of problems that are ambiguous, uncertain, inaccurate, or noisy. The advantage of solving complex nonlinear problems by employing fuzzy system is that the experts’ experience and knowledge described as the fuzzy rule base can be directly embedded into the systems.

M. Jiang (B) School of Electronic Information, Jiangsu University of Science and Technology, Zhenjiang 212003, China e-mail: [email protected] X. Yuan The Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, China e-mail: [email protected] J. Wang School of Naval Architecture and Ocean Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, Jiangsu, China © Springer Nature Singapore Pte Ltd. 2020 B.-y. Cao (ed.), Fuzzy Information and Engineering-2019, Advances in Intelligent Systems and Computing 1094, https://doi.org/10.1007/978-981-15-2459-2_9

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Generally, in a conventional fuzzy system, to achieve the required accuracy, numerous antecedent membership functions must be defined. If we define m membership functions for n input variables, then mn rules are required. Hence, the number of fuzzy rules increases exponentially with the number of input variables. This phenomenon is called curse of dimensionality. A large number of fuzzy rules would overload the memory and make the fuzzy system very hard to implement. Consequently, how to reduce the total number of involved rules and their corresponding computation requirements is one of the important issues in fuzzy system. A compromise between the system computational complexity and accuracy should be found. In order to achieve a reasonable trade-off between model accuracy and complexity, many researches focus on automatically finding the appropriate structures and parameters of fuzzy system. The singular value decomposition (SVD)-based reduction method offers a way to automatically eliminate the redundant fuzzy rules [1–4]. The Higher Order Singular Value Decomposition (HOSVD)-based reduction method is a useful tool for handling this problem [3, 4]. Some design methods use GA to elicit the optimal number of rules [5–7]. Hierarchical fuzzy system which is made up of a number of low-dimensional fuzzy systems in a hierarchical form was first proposed by Raju and Zhou [8] to overcome curse of dimensionality, and its number of rules grows in a linear way with the number of input variables. It has been proven that HFSs are universal estimators [9]. There is no general conclusion about which input is more influential to the output of HFS, although rough observations were drawn for some particular fuzzy systems. To solve the problem that the defuzzification steps performed in the intermediate layers of HFS degenerate the fuzziness level of information, the defuzzification-free HFS is proposed in which the defuzzification steps are eliminated from HFS [10]. For a set of high-dimensional training data, it is not easy to determine an appropriate hierarchical structure of HFS. Several researchers focus on the optimization of structures and parameters by using various learning methods including derivative-free [11, 12] and derivative-based algorithms [13]. The T–S fuzzy model is composed of linear sub-models, and each sub-model is associated with a group of input–output data which is divided by the clustering algorithm. The division of the input space is one of the most important aspects of fuzzy modelling. This division is a coarse setting of model complexity, thus determining the performance of the model fine-tuning. One of the most widely used approaches in fuzzy space partition is fuzzy c-means algorithm. To improve the precision and simplify the structure of fuzzy system, CFS that using a proposed pyramid membership function (PMF) is constructed [14]. It is proved that CFS is a universal approximator, and it achieves superior approximation accuracy [14]. The following are the necessary factors to consider for the design of a fuzzy system: (1) input space partition; (2) fuzzy system structure representation; (3) rule formation; (4) rule tuning; (5) improving model’s accuracy and minimizing its complexity. In this work, hierarchical cone fuzzy system (HCFS) is proposed, which considers all these necessary fuzzy system design factors. The parameter tuning of the HCFS is performed by the GA, which is a metaheuristic algorithm that can solve complex optimization problems. Hence, they are useful in finding the appropriate

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parameter values for HCFS. K-means clustering is used to produce a fuzzy partition of the input space. It has been found that HCFS provides less complex and highly accurate models compared to the models produced by other methods.

2 T-S Type Cone Fuzzy System A T-S fuzzy system is described by a set of fuzzy if–then rules that locally represent the linear relations of input–output of a nonlinear system. The ijth rule Rij of T-S type CFS in [14] is represented in the following form: ij

ij

ij

Ri j : If (x, y) is Pi j , then z = c0 + c1 x + c2 y, where i = 1, 2, . . . , M, j = 1, 2, . . . , N ; (x, y) is the input vector, z is the output variable; Pij (x, y) is the pyramid membership function of the fuzzy set Pij defined in ij ij ij the space of input vector; c0 , c1 and c2 are free parameters. Similar with the conventional T-S fuzzy system, T-S type CFS is composed of fuzzification, fuzzy inference mechanism, fuzzy rule base, and defuzzification. The fuzzification block converts the crisp inputs into fuzzy sets. The fuzzy inference mechanism uses the rules in the fuzzy rule base to convert these fuzzy sets into other fuzzy sets that are representative of the recommendations of the various rules in the rule base. The defuzzification block combines these fuzzy recommendations to give a crisp output. The overall output of a T-S type CFS can be obtained by: (1) For a T-S type CFS with rectangular meshes, in a rectangular mesh [xi , xi+1 ] × [y j , y j+1 ], the expression is determined as follows [14]: ij

ij

ij

i+1, j

S(x, y) = [Pi j (x, y)(c0 + c1 x + c2 y) + Pi+1, j (x, y)(c0 +

i+1, j

+ c1

x

i+1, j y) c2

i+1, j+1 i+1, j+1 i+1 j+1 + Pi+1, j+1 (x, y)(c0 + c1 x + c2 y) i, j+1 i, j+1 i, j+1 Pi, j+1 (x, y)(c0 + c1 x + c2 y)]/[Pi j (x, y)

+ + Pi+1, j (x, y) + Pi+1, j+1 (x, y) + Pi, j+1 (x, y)],

where Pi j (x, y), Pi+1, j (x, y), Pi, j+1 (x, y) and Pi+1, j+1 (x, y) are pyramid membership functions with (xi , y j ), (xi , y j+1 ), (xi+1 , y j+1 ) and (xi+1 , y j ) as peak points, respectively. (2) For a T-S type CFS with triangular meshes, in a triangular mesh, the expression is determined as follows [14]: S(x, y) = [P1 (x, y)(c01 + c11 x + c21 y) + P2 (x, y)(c02 + c12 x + c22 y) + P3 (x, y)(c03 + c13 x + c23 y)]/[P1 (x, y) + P2 (x, y) + P3 (x, y)],

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where P1 (x, y), P2 (x, y), and P3 (x, y) are pyramid membership functions with (x 1 , y1 ), (x 2 , y2 ), and (x 3 , y3 ) as peak points, respectively.

3 Hierarchical Cone Fuzzy System Since a multiple-input multiple-output (MIMO) system can be divided into multipleinput single-output (MISO) systems. Throughout the rest of the paper, it is assumed that the system or function to be modeled or approximated is a MISO function defined on discrete spaces. HFS is organized in layers. We may distinguish between three types of modules of HCFS: DISO, MISO, and MIMO. There are different classifications of HFS. According to the most general classification, HCFS is divided into the following categories: (1) Serial HCFS (Fig. 1a): The input of a module is the output of the previous ones, along with system input variables. (2) Parallel HCFS (Fig. 1b): The first layer is made up of a set of modules receiving the system input variables. The outputs of the modules in the first layer are the inputs of the modules in the next layer, and the input of a module is the output of the previous ones. (3) Hybrid HCFS: This type of HCFS is a mixture of the two previous ones. Let x1 , x2 , . . . , xn+1 be the input variables of serial HCFS, then the construction process of serial HCFS (Fig. 1a) with T-S type CFS as its modules is conducted as follows: First layer: The T-S type CFS with the input vector (x 1 , x 2 ) has the following form of fuzzy rules (x1, x2)

FS

y1 (x , y ) 3 1

y2

FS

x3

yn-1

xn+1

(xn+1, yn-1)

FS

(a) Serial HCFS (x1, x2) (x3, x4)

y1,1 FS y1,2

FS (y1,1, y1,2)

FS

ym-1,1 FS

(x2n-3, x2n-2) (x2n-1, x2n)

FS FS

y1,n-1 y1,n

ym-1,2 FS 1,n-1

(y

,y ) 1,n

(b) Parallel HCFS Fig. 1 Types of HCFS

ym

yn

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If (x1 , x2 ) is Pk1 , then y1 = ak1 + bk1 x1 + ck1 x2 , where k = 1, 2, . . . , N1 , Pk1 (x1 , x2 ) is the pyramid membership functions. Then the output of the module in the first layer can be obtained by:  N1 y1 =

k=1

Pk1 (x1 , x2 )(ak1 + bk1 x1 + ck1 x2 ) .  N1 1 k=1 Pk (x 1 , x 2 )

Layer i (1 < i < n): The T-S type CFS with the input vector (x i+1 , yi−1 ) has the following form of fuzzy rules If (xi+1 , yi−1 ) is Pki , then yi = aki + bki xi+1 + cki yi−1 , where k = 1, 2, . . . , Ni , Pki (x1 , x2 ) is the pyramid membership functions. Then the output of the T-S type CFS in the ith layer can be obtained by:  Ni yi =

k=1

Pki (xi+1 , yi−1 )(aki + bki xi+1 + cki yi−1 ) .  Ni i k=1 Pk (x i+1 , yi−1 )

nth layer: The T-S type CFS with the input vector (x n+1 , yn−1 ) has the following form of fuzzy rules If (xn+1 , yn−1 ) is Pkn , then yn = akn + bkn xn+1 + ckn yn−1 , where k = 1, 2, . . . , Nn , Pkn (x1 , x2 ) is the pyramid membership functions. Then the overall output of the fuzzy model can be obtained by:  Nn yn =

k=1

Pkn (xn+1 , yn−1 )(ak1 + bk1 xn+1 + ck1 yn−1 ) .  Nn n k=1 Pk (x n+1 , yn−1 )

Parallel HCFS (Fig. 1b) with 2n inputs variables x1 , x2 , . . . , x2n is derived as follows: First layer: The first layer is made up of n T-S type CFSs with (x1 , x2 ), (x3 , x4 ), . . . , (x2n−1 , x2n ) as input vectors, respectively, then their rules are, respectively, as the following forms: If (x1 , x2 ) is Pk11 , then y 11 = ak11 + bk11 x1 + ck11 x2 (k = 1, 2, . . . , N 11 ). If (x3 , x4 ) is Pk12 , then y 12 = ak12 + bk12 x3 + ck12 x4 (k = 1, 2, . . . , N 12 ). .. .

If (x2n−1 , x2n ) is Pk1n , then y 1n = ak1n + bk1n x2n−1 + ck1n x2n (k = 1, 2, . . . , N 1n ). The outputs of n modules in the first layer can be, respectively, obtained by:

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y 11 = y 12 =

y 1n =

 N 11 k=1

 N 12

 N 1n k=1

k=1

Pk11 (x1 ,x2 )(ak11 +bk11 x1 +ck11 x2 ) ,  N1 11 k=1 Pk (x 1 ,x 2 ) Pk12 (x3 ,x4 )(ak12 +bk12 x3 +ck12 x4 ) ,  N 12 12 k=1 Pk (x 3 ,x 4 )

.. .

Pk1n (x2n−1 ,x2n )(ak1n +bk1n x2n−1 +ck1n x2n ) .  N 1n 1n k=1 Pk (x 2n−1 ,x 2n )

mth layer: The T-S type CFS module with the input vector (y m−1,1 , y m−1,2 ) has the following form of fuzzy rules: If (y m−1,1 , y m−1,2 ) is Pkm , then y = akm1 + bkm1 y m−1,1 + ckm1 y m−1,2 , where k = 1, 2, . . . , N m1 . Then the output of mth layer T-S type CFS can be calculated by:  N m1 ym =

k=1

Pkm (y m−1,1 , y m−1,2 )(akm1 + bkm1 y m−1,1 + ckm1 y m−1,2 ) .  Nn m m−1,1 m−1,2 ,y ) k=1 Pk (y

4 Identification of HCFS In the optimization of fuzzy system, the structure and parameters can be optimized simultaneously or in two stages, respectively. Since the membership function and fuzzy rule consequence in fuzzy system are interactive, in this paper, GA is used to simultaneously optimize the parameters of the membership functions and rule consequence. T-S type rectangular pyramid fuzzy system (RPFS), circular cone fuzzy system (CCFS), and triangular pyramid fuzzy system (TPFS) are proposed in [14]. The module of HCFS is selected as T-S type CFS. In the structure identification phase of HCFS, k-means algorithm is employed to divide the input space. In the parameter tuning phase of HCFS, a genotype representing the parameters of pyramid membership functions and fuzzy rule is optimized by using GA. The parameters of HCFS can be coded as a string of integer or real numbers. If there are N rules in a HCFS, a HCFS is coded by integer numbers as Fig. 2 shown. The selection of

rule 1 antecedent parameters

consequent parameters

Fig. 2 Encoding method of GA

rule i antecedent parameters

consequent parameters

rule N antecedent parameters

consequent parameters

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the best genotype in a single objective training GA is based on a comparison of the RMSE:   m  (yi − yi )2 . RMSE =  m i=1

5 Simulation Experiments In order to compare our method with other approaches, we performed two experimental cases, namely, cases 1 and 2. Case 1: The first example concerns the application of HCFS to the Mackey–Glass time series which is generated by the following differential equation: x(t + 1) = 0.9x(t) +

0.2x(t − 17) . 1 + x 10 (t − 17)

In this case, x(t − 1), x(t − 2), x(t − 3), x(t − 4) are chosen to predict x(t + 1). The first 500 data of x(101) to x(1100) are used as training data, and the last 500 data are used as testing data to validate the performance of the proposed model. All input–output data are normalized to the range [−1, 1] × [−1, 1]. The structure of the parallel hierarchical RPFS used in this case is shown in Fig. 3, then the output is x(t + 1) = g1 (g21 (x(t − 1), x(t − 2)), g22 (x(t − 3), x(t − 4))). The rule number of the module g1 in the first layer is r 1 = 2. The rule numbers of the modules g21 and g22 in the second layer are r 21 = 2 and r 22 = 2, respectively. The radius-scale parameters and consequent parameters are tuned by GA. The number of individuals in GA is 100, and the maximum generation is 300. The RMSE performance index is utilized to check the effectiveness of the proposed modeling approach. The comparative performance of HCFS and other models available in literatures are listed in Table 1. The RMSEs of the training and testing data of our model are 0.0011 and 0.0031, respectively. It is noted that the proposed Fig. 3 Structure of parallel hierarchical RPFS

(x(t−1), x(t−2))

g21 g1

(x(t−3), x(t−4))

g22

x(t+1)

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Number of inputs

Number of rules

RMSE

Tsekouras [15]

4

6

0.0041

Kukolj [15]

4

9

0.0061

Kim [15]

4

9

0.0264

ANFIS [16]

4

16

0.0016

HCFS

4

6

0.0031

error

Table 1 RMSE comparison of different fuzzy identification methods

data Fig. 4 Error curve for testing data (Mackey-Glass data)

model gives a more accurate result compared to other existing models, though model 2, model 3 and model 4 in Table 1 use much higher number of rules than HCFS. The error curve of the proposed method for testing data is shown in Fig. 4. Case 2: Consider the following nonlinear system   y(k − 1) + y(k − 2) 2.5y(k − 1)y(k − 2) + 0.3 cos y(k) = 1 + y 2 (k − 1) + y 2 (k − 2) 2 + 1.2u(k − 1) + e(k). The system output u(k) = (1/2)(sin(kπ/20) + sin(kπ/50)), e(k) is the gaussian white noise N(0, 0.3162 ). The initial condition is y(0) = y(−1) = 0. y(k − 1), y(k − 2), and u(k − 1) are selected as the input variables, and y(k) is the output variable. Choose 500 data of 1000 input–output pairs as testing data, and the last 500 data pairs are chosen as testing data. The structure of the serial hierarchical TPFS used in this case is shown in Fig. 5, then the output y(k) = g1 (g2 (y(k − 1), y(k − 2)), u(k − 1)).

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g2

(y(k−1), y(k−2))

g1

y(k)

u(k−1)

Fig. 5 Structure of serial hierarchical TPFS 2.5

2

2

1.5

1.5

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0.5

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1000

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Fig. 6 Model output

In the identification process of serial hierarchical TPFS, GA is used to adjust the radius-scale parameter of the pyramid membership functions and the consequent parameters. The number of individuals in GA is 100, and the maximum generation is 200. In the simulation experiments, firstly, the noisy samples are used for training and testing, and the results of the proposed model are compared with the corresponding results of HFS in [17], from the model precision and complexity (number of fuzzy rules). The parameters, in addition to the number of rules, the input variables, the training data, and testing data are the same as [17]. When the same white noise is taken into consideration, Salgado [17] adopted 13 rules with the MSE being 0.007902, and the serial hierarchical TPFS utilizes 10 rules with the MSE being 0.0053. Therefore, serial hierarchical TPFS can achieve higher precision with simpler model structure. When the gaussian white noise is removed, the MSE of serial hierarchical TPFS is 0.0036. The noisy and noisy-free model outputs are shown in Fig. 6.

6 Conclusion In order to achieve an accurate modeling method of the high-dimensional complex nonlinear systems, in this paper, we propose a new kind of HFS with T-S type CFS.

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A major characteristic of such models is that the antecedent of a rule uses pyramid membership functions. HFS can be constructed with two different hierarchical architectures, namely serial HCFS and parallel HCFS. The design of HCFS starts by producing a number of clusters in the multidimensional input space so that all the training examples belong to at least one cluster. GA is utilized for parameters learning. The proposed fuzzy model framework has been validated on the complex nonlinear systems, which demonstrate that HCFS provides competitive approximation performance compared to other algorithms and simultaneously it produces less complex models. Acknowledgements Thanks to the support by National Natural Science Foundation of China (Nos. 51609110, 51779110 and 51809122).

References 1. Yam, Y., Baranyi, P., Yang, C.T.: Reduction of fuzzy rule base via singular value decomposition. IEEE Trans. Fuzzy Syst. 7(2), 120–132 (1999) 2. Takacs, O., Varkonyi-Koczy, A.R.: SVD-based complexity reduction of rule-bases with nonlinear antecedent fuzzy sets. IEEE Trans. Instrum. Meas. 51(2), 217–221 (2002) 3. Tóth-Laufer, E., Várkonyi-Kóczy, A.R.: Anytime sport activity risk level calculation using HOSVD based hierarchical fuzzy models. In: IEEE International Symposium on Medical Measurements & Applications (IEEE, 2015) 4. Tóth-Laufer, E., Rövid, A., Takács, M.: Error calculation of the HOSVD-based rule base reduction in hierarchical fuzzy systems. Fuzzy Sets Syst. 307, 67–82 (2016) 5. Dennis, B., Muthukrishnan, S.: AGFS: adaptive genetic fuzzy system for medical data classification. Appl. Soft Comput. 25, 242–252 (2014) 6. Rey, M.I., Galende, M., Fuente, M.J., et al.: Multi-objective based fuzzy rule based systems (FRBSs) for trade-off improvement in accuracy and interpretability: a rule relevance point of view. Knowl.-Based Syst. 127, 67–84 (2017) 7. Lahsasna, A., Seng, W.C.: An improved genetic-fuzzy system for classification and data analysis. Expert Syst. Appl. 83, 49–62 (2017) 8. Raju, G.V.S., Zhou, J.: Adaptive hierarchical fuzzy controller. IEEE Trans. Syst. Man Cybern. 23(4), 973–980 (1993) 9. Wang, L.X.: Universal approximation by hierarchical fuzzy systems. Fuzzy Sets Syst. 93(2), 223–230 (1998) 10. Mutlu, B., Sezer, E.A., Nefeslioglu, H.A.: A defuzzification-free hierarchical fuzzy system (DF-HFS): rock mass rating prediction. Fuzzy Sets Syst. 2017(307), 50–66 (2017) 11. Ojha, V.K., Snasel, V., Abraham, A.: Multiobjective programming for type-2 hierarchical fuzzy inference trees. IEEE Trans. Fuzzy Syst. 26(2), 915–936 (2017) 12. Fernández, A., Jesus, M.J.D., Herrera, F.: Hierarchical fuzzy rule based classification systems with genetic rule selection for imbalanced data-sets. Int. J. Approx. Reason. 50(3), 561–577 (2009) 13. Sharifi, A., Aliyarishoorehdeli, M., Teshnehlab, M.: Hierarchical wavelet packet fuzzy inference system for pattern classification and system identification. Int. J. Syst. Sci. 44(1), 18 (2013) 14. Jiang, M.Z., Yuan, X.H.: A new type of fuzzy systems using pyramid membership functions (PMFs) and approximation properties. Soft Comput. 22, 7103–7118 (2018) 15. Tsekouras, G., Sarimveis, H., Kavakli, E., et al.: A hierarchical fuzzy-clustering approach to fuzzy modeling. Fuzzy Sets Syst. 150(2), 245–266 (2005)

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16. Kroll, A.: Identification of functional fuzzy models using multidimensional reference fuzzy sets. Fuzzy Sets Syst. 80(2), 149–158 (1996) 17. Salgado, P.: Rule generation for hierarchical collaborative fuzzy system. Appl. Math. Model. 32(7), 1159–1178 (2008)

GPU Local PSO Algorithm at Dimension Level-Based Medical Image Registration Xicheng Fu, ShengQuan Ma, DaWei Yun, and JiaJing Cai

Abstract For high dimension optimization problems, the performance of sequential particle swarm optimization is time-consuming. This paper presented a GPU-based parallel local particle swarm optimization algorithm at dimension level for medical image registration. Specially, a thread block represents a particle, while a thread in a block represents one dimension of an optimization function. Furthermore, the stop criterion is set as an acceptable range of derivation. In the experiment, for the high dimension optimization medical image registration problems, the speedup of GPU-based implementation can reach up to 95. The overall average difference of iterations to stop between the sequential algorithm and GPU-based algorithm is 17. Keywords Medical image registration · High dimension optimization problem · Local particle swarm optimization algorithm · Dimension-level parallelism · Graphical processing unit

1 Introduction 1.1 Medical Image Registration Medical image registration is an important technology for medical image processing in nearly 20 years ago. It analyzes quantitatively the matching problem of medical images in specific spatial position. When analyzing medical images, the mapping relation T will be determined in multiple images generated at different times and

X. Fu · D. Yun (B) · J. Cai Hainan Institution of Science and Technology, 18 Qiongshan Avenue, Meilan, Haikou 571126, Hainan, China e-mail: [email protected] X. Fu e-mail: [email protected] S. Ma Hainan Normal University, Haikou 571158, Hainan, China © Springer Nature Singapore Pte Ltd. 2020 B.-y. Cao (ed.), Fuzzy Information and Engineering-2019, Advances in Intelligent Systems and Computing 1094, https://doi.org/10.1007/978-981-15-2459-2_10

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Fig. 1 The transformation of image matching

conditions, which causes a pixel of image A correspond to the only pixel in medical image B. Then, more comprehensive medical information of patients would be obtained, and thereby, the level of diagnosis and treatment is improved as shown in Fig. 1.

1.2 Particle Swarm Optimization Algorithm By simulating the self-organizing movement of birds and fish, particle swarm optimization algorithm (PSOA) is defined as a simplified social model based on selfcognition and social influence [1, 2]. PSOA has been widely concerned since it was proposed because of the advantages such as simplistic concept, easy programming, and efficient search. It is one of the important branches in evolutionary computing [3]. On the other hand, NVIDIA Corporation introduced a GPU to support Computing Unified Device Architecture (CUDA) in 2007 applied in the field of general computing. The application after GPU acceleration could obtain considerable acceleration ratio [4]. In the analyzing work of accelerated PSOA, some papers represent that executing the PSOA in GPU can have better performance by compared with the serial implementation of CPU [5, 6]. Most of the work, however, is focused on parallelization at the particle level [7–11].

2 The CUDA Programming Model CPU is used as the host, and GPU can serve as the coprocessor in CUDA. While the CPU is responsible for logical transaction processing and serial computing, the GPU is focused on performing highly threaded parallel processing tasks. The process of CUDA computing usually includes three steps: data transfer from CPU to GPU, Kernel function execution [12], and data transfer from GPU to CPU. The specific process is shown in Fig. 2. CUDA adopts Single Instruction Multiple Thread (SIMT) execution mode. Global memory can be bound to texture memory, mainly used in the applications of graphics images. The CUDA programming model is displayed in Fig. 3.

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Fig. 2 CPU-GPU computing pattern

Fig. 3 CUDA programming model

3 Local PSO Execution Based on GPU From the topological structure of particles, PSO is mainly divided into global PSO and local PSO. Among them, global PSO regards the whole particle swarm as a “society”. The social influence that the particle receives is derived from the information of all particles during flying. In local PSO, however, a particle receives social influence from its surroundings. In terms of solving quality, global PSO converges rapidly, but it is easy to fall into local optimum, which is not suitable for solving multi-peak problem. Although the local PSO could settle the problem because it is not easy

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to fall into local optimum, the convergence speed is slower than global PSO when solving single-peak problem [3].

3.1 Sequential Execution of Local PSO Local PSO algorithm can be separated into many types. This paper mainly discusses the local PSO algorithm of ring topology. Specifically, the social impact of each particle is limited to one adjacent particle. During the execution of the algorithm, the future motion of each particle in the search space is determined by speed and orientation. At the same time, each particle also needs to record the best historical position which it has been searched for. At each iteration, the particle moves in the next coordinate direction by changing the velocity during the searching. The formulas which involve the velocity and position of the particle are shown in (1) and (2).     = wvi,(t)j + c1r1 yi, j − xi,(t)j + c2 r2 li, j − xi,(t)j vi,(t+1) j

(1)

xi,(t+1) = vi,(t+1) + xi,(t)j j j

(2)

where vi,j represents the velocity components of particle i in the j dimension. x i,j represents the search space coordinates of particle i in the j dimension; li,j represents the components in the j dimension of the coordinates of the best search space of particle I in its own history; L i,j represents the component of the j dimension of the coordinate of the best search space in the neighborhood of particle i. In addition, the parameters w represent inertial weight, c1 and c2 represent learning factors, and r 1 and r 2 represent random numbers in the interval [0, 1]. PSO can approach the location of global optimal solution in the search space by adaptive regulation of velocity. Following the above description, the sequential execution of pseudo-code of local PSO algorithm is shown in Algorithm 1: Algorithm 1. Local PSO

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In addition, in some optimization goals with large computational demands, the objective function generally exhibits the characteristics of high dimensions [13]. When the dimension of the objective function exceeds 30 dimensions, the execution efficiency of the serial local PSO decreases sharply. Therefore, it is necessary to propose an appropriate GPU parallel execution scheme according to the characteristics of the high-dimensional optimization target.

3.2 Implementation Framework of Local PSO Based on GPU The method in this paper has the characteristics of fine granularity parallelism, which is mainly due to the use of dimensional parallelism of the objective function. Thus, the thread blocks are corresponding to the particles in the cluster, while the threads in the block are corresponding to the dimensions in the target function. Logically, the size of the thread grid defines the number of particles, while the size of the thread block defines the dimension of the target function. The proposed local PSO that is parallel on the dimension level based on GPU is shown in Algorithm 2, and the GPU mainly includes four kernels. In the algorithm, ‘n’ represents the number of particles in the algorithm, and ‘d’ expresses the number of particles in each group. Algorithm 2. CPPSO

The first kernel is used to initialize the basic information about the position and velocity of each particle in the particle population. The best neighborhood particle information is initialized by the previous best location (Pbestx). Since a thread block corresponds to a particle, one thread in a thread block corresponds to a dimension of the target function. Therefore, the kernel starts n * d threads. In the second kernel, the fitness function of each particle is calculated firstly. It is then decided whether

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to update the Pbestx, and the kernel similarly enables n * d threads. In addition to updating the best neighborhood of each particle according to the ring topology among particles, the third kernel also needs to select the best local particle information. The last kernel starts n * d threads and updates the speed and location of each particle in the next iteration. After each iteration, the information of the best local particle is transferred to the host side to update the information of the best global particle. Finally, the stop condition of the algorithm is determined by judging whether the value of the target function reaches the specified precision. In each iteration, the algorithm updates the particle velocity according to random number. Thus, the CURAND library is used in this paper for generating random numbers in GPU. The library function can not only set random number seed but also generate pseudo-random sequence. Figure 4 displays an algorithm flowchart of local PSO parallel on dimension level based on GPU.

Fig. 4 Algorithm flowchart

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3.3 Specific Execution of Local PSO Based on GPU Algorithm 3. Calculate fitness and optimal particle

Algorithm 3 is the pseudo-code which helps to select kernel function for fitness calculation and optimal particle. Specifically, the half of the threads run simultaneously, and the second half suspends. The hypothesis is that the dimension of thread block is d, the statute process is performed by the previous d/2 threads at first, and next by the previous d/4 threads execute. That is, the number of threads that makes the protocol during each iteration is successively halved until only one thread gets the result of the target function. The schematic diagram of the statute process is shown in Fig. 5. Algorithm 4 represents the process of calculating local best(LBest). The topological relation between particles of the local PSOA in this paper is ring topology. Therefore, given that the quantity of particles is n, the calculation methods for the two neighborhood particles of particle i are (i + 1) mod n and (i − 1)mod n. In this

Fig. 5 Parallel reduction

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algorithm, each particle first updates its own best neighborhood particle information in parallel, and then, it selects the global best particle through only one thread block. The process of selecting the best global optimum particle is shown in Algorithm 5. Algorithm 4. Local optimal particle computing kernel

Similar to the process of calculating fitness in Algorithm 3, the parallel protocol process is used to find local optimum particles. Different from the addition operation used in fitness calculation, the comparison operation is adopted in this specification. In addition, the implementation of this part is executed directly in the GPU’s global memory, and there is no need to use synchronous statements. Algorithm 5. Calculate the best particle

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4 Experiment and Analysis In this paper, three benchmark functions which come from medical image registration are used to test the performance of the algorithm as shown in Table 1. To verify the effectiveness of the method, the same quantity of particles and the identical dimension of medical image registration objective function are used for serial and parallel execution of the algorithm. Due to the randomness of particle swarm optimization, the result of the algorithm derives from taking the average value after 20 times of execution. In this experiment, when the error of medical image registration objective function is less than 10−4 , the algorithm terminates. The differences between the efficiency of the serial implementation and parallel implementation in the local PSOA mainly include two aspects, that is, the running time of the whole algorithm (unit: ms) and the number of iterations required to reach the specified precision. Table 2 figures the operation results after the serial local PSOA- and GPU-based local PSO algorithm perform the optimized medical image registration objective function f 1 . Table 3 shows the running results after executing the optimized medical image registration objective function f 2 by the serial local PSO algorithm and the local PSO algorithm based on GPU. Table 4 shows the operating results after the serial local PSO algorithm and the local PSO algorithm based on GPU manipulates optimized medical image registration objective function f 3 . By analyzing the operation results, it seems that the execution time of the serial algorithm is proportional to the number of particles and the dimension of the objective function. The GPU implementation of the algorithm has obvious advantages in solving the objective function of high-dimensional medical image registration. The serial performance of the algorithm is better when optimizing the objective function Table 1 Target functions

Optimization function d f 1 (x) = i=1 x2 2  i f 2 (x) = 100 xi−1 − xi2 + (xi−1 − 1)2 √  d f 3 (x) = 418.9829σ − i=1 xi sin |xi |

Domain [−100, 100]d [−16, 16]d [−500, 500]d

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Table 2 Running time and number of iteration for f 1 Particle n

Dimension

Serial algorithm Running time

GPU algorithm Iterations

Running time

Iterations

8

2

2.8

640

39.4

647

16

4

15.4

834

50.2

827

32

8

70.5

1041

65.4

1037

64

16

360.2

1301

90.2

1297

128

32

1845

1652

135

1647

256

64

9452

2125

248.2

2119

512

128

52,102

2839

780

2837

1024

256

301,475

4168

3824.4

4164

Table 3 Running time and number of iteration for f 2 Particle n

Dimension

Serial algorithm Running time

GPU algorithm Iterations

Running time

Iterations

8

2

2.4

482

27.7

450

16

4

17.2

758

45.2

741

32

8

85.1

978

67.1

965

64

16

482.5

1215

92.1

1220

128

32

2641

1652

134.5

1665

256

64

17,054

2507

290

2492

512

128

130,054

4502

1140

4487

1024

256

672,471

5988

4857.4

5986

Table 4 Running time and number of iteration for f 3 Particle n

Dimension

Serial algorithm

GPU algorithm

Running time

Iterations

Running time

Iterations

8

2

7.8

825

58.4

879

16

4

28

824

68.4

872

32

8

118.4

853

62.1

892

64

16

498

870

64.8

915

128

32

2001

891

78.5

921

256

64

8320

914

168.1

947

512

128

34,001

935

270.4

957

1024

256

127,954

924

820

957

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of low dimensional medical image registration. It can be seen from the acceleration ratio of Fig. 6 that the difference between the execution of serial algorithm and parallel algorithm in different dimensions and the number of particles. Finally, Fig. 7 shows the difference in the number of iterations between the serial implementation and the parallel implementation as they converge on the three medical image registration objective functions. The number of random number generators that GPU performs in parallel is greater than the number of sequential executions of the algorithm. It can be seen that, in terms of the number of iterations, the serial execution of the algorithm and the GPU execution are generally close. Although the iterative difference of the algorithm is obvious when optimizing the function f3, the maximum difference is no more than 40. It is acceptable that the algorithm needs nearly 1000 iterations to converge.

Fig. 6 Speed-up ratio

Fig. 7 Derivation of number of iterations after convergence

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5 Conclusion In this paper, a local PSOA based on medical image registration GPU is proposed. The particle corresponds to the thread block, and the thread in the thread block corresponds to the dimension of the medical image registration objective function. The method in this paper has obvious advantages in the optimization of high-dimensional objective function, with the maximum acceleration ratio reaching 95 times. The total average difference in the number of iterations when the serial implementation and parallel implementation of the algorithm stop running under the premise of satisfying the accuracy is 17 times. In the future work, the proposed algorithm will be applied to the practical engineering of medical image registration. Acknowledgements This work was financially supported by Hainan Social Science Federation Fund (HNSK (2C) 18-22). Recommender This paper is recommended by Chao-hui Lu who is a professor of Hainan Normal University in China.

References 1. Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of the IEEE International Conference on Neural Networks, vol. 4, Perth, WA, pp. 1942–1948 2. Poli, R., Kennedy, J., Blackwell, T.: Particle swarm optimization: an overview. Swarm Intell. 1(1), 33–57 (2007) 3. Zhang, X., Hu, X., Lin, Y.: Comparisons of genetic algorithm and particle swarm optimization. J. Front. Comput. Sci. Technol. 8(1), 90–102 (2014) 4. Owens, J.D., et al.: GPU computing. Proc. IEEE 96(5), 879–899 (2008) 5. Mussi, L., Daolio, F., Cagnoni, S.: Evaluation of parallel particle swarm optimization algorithms within the CUDA architecture. Inf. Sci. 181(20), 4642–4657 (2010) 6. Mussi, L., Naahed, Y.S.G., Cagnoni, S.: GPU-based asynchronous particle swarm optimization. In: Proceedings of the 13th Annual Conference on Genetic and Evolutionary Computation, Dublin, Ireland, pp. 1555–1562 7. Zhou, Y., Tan, Y.: GPU-based parallel particle swarm optimization. In: Proceedings of the IEEE Congress on Evolutionary Computation. Trondheim, pp. 1493–1500 (2009) 8. Veronese, L., Krohling, R.: Swarm’s flight: accelerating the particles using C-CUDA. In: Proceedings of the IEEE Congress on Evolutionary Computation, Trondheim, pp. 3264–3270 (2009) 9. Zhang Q.-K., Yang, B., Wang, L., Zhu, F.-X.: Research on parallel modern optimization algorithms using GPU. Comput. Sci. 39(4), 304–311 (2012) 10. Cai, Y., Li, G.-Y., Wang, H.: Research and implementation of parallel particle swarm optimization based on CUDA. Appl. Res. Comput. 30(8), 2415–2418 (2013) 11. Chen, F., Tian, Y.-B., Yang, M.: Research and design of parallel particle swarm optimization algorithm based on CUDA. Comput. Sci. 41(9), 263–268 (2014) 12. Kirk, D.B., Wen-mei, W.H.: Programming massively parallel processors: a hands-on approach. In: 2012: Newnes 13. Ivekoviˇc, Š., Trucco, E., Petillot, Y.: Human body pose estimation with particle swarm optimization. Evol. Comput. 16(4), 509–528 (2008)

Fuzzy Geometric Programming: Past, Present, and Future Bing-yuan Cao and Pei-Hua Wang

Abstract Cao first proposed the topic of fuzzy geometry programming since 1987, with its greatly developed in this field for 32 years. According to his previous researches, in the paper, he introduces its development process, aiming to promote this new branch, attracting scholars home and abroad to join in ranks of the research, and helping to solve three conjectures of fuzzy geometric programming. Keywords Fuzzy geometric programming · Origin and development · Past, present, and future · Conjecture

1 The Past of Geometric Programming The fuzzy positive geometric programming was first proposed by B.-y. Cao at the International Fuzzy Systems Association (IFSA) Conference in Tokyo, Japan, in 1987 [1, 2]. Its general form has the following expression. (G P)

min g˜ 0 (x) s.t. g˜ i (x)  1 (1  i  p  ), g˜ i (x)  1 ( p  + 1  i  p), x > 0,

it is a reversed posynomial GP, where  Ji all of gi (x)(0  i  p) are posynomials function of variable x with gi (x) = k=1 vik (x), here B.-y. Cao · P.-H. Wang (B) Guangzhou Vocational and Technical University of Science and Technology, Guangzhou, Guangdong 510550, China e-mail: [email protected] B.-y. Cao e-mail: [email protected] B.-y. Cao Foshan University, Foshan, Guangdong 528000, China © Springer Nature Singapore Pte Ltd. 2020 B.-y. Cao (ed.), Fuzzy Information and Engineering-2019, Advances in Intelligent Systems and Computing 1094, https://doi.org/10.1007/978-981-15-2459-2_11

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vik (x) =

⎧ m  γ ⎪ ⎪ xl ikl , (1  k  Ji ; 0  i  p  ) ⎨ cik l=1

m  −γ ⎪ ⎪ xl ikl , (1  k  Ji ; p  + 1  i  p) ⎩ cik

(1)

l=1

is a monomial function of variable x, variable x = (x1 , x2 , . . . , xn )T , and coefficients cik > 0, exponents γikl (1 ≤ k ≤ Ji , 0 ≤ i ≤ p, 1 ≤ l ≤ n) is arbitrary real number. We have done a lot of work in the research of fuzzy GP [3–12], putting forward its original algorithm, and using fuzzy arithmetic geometric inequality and setting value theory to obtain its dual form and dual algorithm. Then we proposed the geometric programming with fuzzy coefficient and fuzzy variable, and a multi-objective fuzzy geometric programming model [13–21]. At the same time, it is applied into power system, management optimization, and decision making [22–27]. Based on the published fuzzy GP, another monograph of Optimal Models and Methods with Fuzzy Quantities was published in Springer in 2010, in which he introduced some researches, such as GP models and algorithms with fuzzy coefficients and fuzzy variables [28, 29]. The author has been awarded the National Natural Science Foundation of China for three consecutive times: (1) Fuzzy generalized GP decision model and its method, 2008–2010. (2) Universal fuzzy GP and optimization techniques in management, 2003–2005. (3) Theory and method of fuzzy GP in the power system management, 1997–1999, for which he has achieved lots of results. In 2002, Kluwer Academic Publishers published a series of fuzzy GP in the Applied Mathematics series. In 2005, Fuzzy GP was awarded the third prize of Guangdong Science and Technology Award. The American Mathematical Reviews-Math SciNet and the Mathematical Digest Zentralblatt MATHcommented on its original research. In 1993, Liu Yingming, an academician of the Chinese Academy of Sciences, called this work an International Frontier. In 2004, Academician Wang Zikun, another academician of the Chinese Academy of Sciences, wrote an article in the Chinese Science Bulletin, saying that the monograph was at a high level in academic value with his identification of fuzzy GP, reaching an international advanced level. In 2007, Guo Bolin, a third academician of the Chinese Academy of Sciences, who served as the director of the appraisal committee, identified this research as an advanced level of the International Frontier, saying that it partially reached the leading level. While studying the theory and application of fuzzy GP, Cao and others are actively expanding it in other direction: In 2001, he proposed the extension GP. In 2005, the author and his doctoral student Yang Jihui, in the IEEE Fuzzy System Annual Meeting held in the USA, first proposed the fuzzy relation GP, and its form was described as follows. (PGPF)

min z(x) =

p  k=1

s.t. A ◦ x = b,

ck

n  j=1

γk j

xj

(2)

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where A = (ai j )m×n , x = (x j )n×1 , b = (bi )m×1 , ai j , x j , bi ∈ [0, 1], ck , γk j ∈ R, ck > 0, i ∈ I = {1, 2, . . . , m}, j ∈ J = {1, 2, . . . , n}, k ∈ K = {1, 2, . . . , p}, and for given j ∈ J , γk j (k ∈ K ) represent either all non-positive real numbers or all nonnegative real ones. Without loss of generality, we assume that problem (2) satisfies the following inequalities: 1  b1  b2  · · ·  bm  0. Otherwise, rearrange the components of b in decreasing order and adjust the rows of A accordingly b [36–44]. Then in 2005, Luo Dang proposed the gray positive GP. After that, in 2009, author proposed the rough posynomial GP, while in 2014, Cao’s doctoral student Zeinab Kheiri wrote the doctoral thesis with intuitionistic fuzzy posynomial GP [30–35]. So far, research in this direction has begun to heat up.

2 The Present of Geometric Programming From July 30 to August 1, 2016, Cao Bingyuans students, family, and relatives gathered in the Applied Mathematics Conference Room on the 7th floor of the Computer Experiment Building of Guangzhou University to celebrate his 30 years of fuzzy geometry programming and 40 years of his teaching job, and published a conference proceedings by Chinese Science and Education Press in 2019. The book, containing main articles of Cao and his students, has been published in the fuzzy GP for 30 years, reporting their projects, achievements, and rewards. At present, the research on fuzzy relational linear programming and fuzzy relational GP is becoming a hot topic. Nearly 100 papers have been published in journals, and recently, they have headed in some new research directions [45–48], and Cao’s book of fuzzy relational mathematical programming as well as the book of Indian scholars’ fuzzy relational GP and its application, will be published by Springer Nature. Now Cao’s colleagues and he are already prepared to collaborate with teams on optimized secured sharing of documents,which was proposed in Jana Wy˙zykowskiego University, Poland, on fuzzy relationships and their programming. At the same time, the three conjectures of fuzzy GP proposed in 2012 [49] still remain to be solved. The teams are expanding their propaganda and taking strong measures to attract more scholars to discuss the three. And they will use the 9th International Information and Engineering Conference of Kish Island to prepare for the establishment of the International Fuzzy Information and Engineering Association. We will use the EI collection and ISC conference proceedings and the magazine “Operational Management and Fuzzy Mathematics” to exchange ideas. Besides, the excellently chosen papers in this direction will be published in the journal of Fuzzy Information and Engineering, which is included in the Web of Sciences.

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3 The Future of Geometric Programming Thirty-two years witness the development of fuzzy GP. At present, its research has entered a critical phrase. More research scholars are involved in it and infiltration fields are constantly expanding, and its research needs more talents to participate in and support for. The three conjectures we have to solve are: (i) The local optimal (satisfactory) solution to fuzzy GP is still its global optimal (satisfactory) solution. (ii) After replacing the operator (+, ·) in (1) with other logical operators, the fuzzy relational GP still holds with its dual programming (2) established. (iii) Confirm the existence of fuzzy GP taxonomy and identification. As for the three above, they will continue to organize teams to deal with them, strive to make a breakthrough in theory, further find the background in the application, and make arduous efforts to establish a branch of fuzzy geometric programming. The fuzzy GP will attract all of us to further research because many aspects remain untouched. In the basic field, we shall consider the following topics. 1. Fuzzy reverse GP, including a GP problem with mixed sign-terms, is much more complex than the fuzzy convex (resp. concave) GP and fuzzy posynomial GP, so we want continuously explore their properties. 2. Fuzzy fractionation, extension, gray, and rough GP still need to be studied. 3. GPs with intuitionistic fuzzy coefficients and fuzzy variables have yet to be further refined and expanded. 4. Further solve real-world problems paradox with fuzzy GP. 5. Solve fuzzy relation GP. 6. Explore fuzzy GP with discrete variables and coefficients. 7. Fuzzy GP in application in BitTorrent-like Peer-to-Peer file sharing system. 8. GP problem subject to max-product fuzzy relation inequalities. The local optimal solution to the GP problem subject to fuzzy relation inequalities is also its global optimal solution. 9. Study fuzzy GP classification. 10. Gain an access to fuzzy GP’s genetic algorithm. Colleagues are welcome to participate and work hard to establish a new branch of fuzzy GP. Acknowledgements The work was supported by the Natural Science Foundation of Guangdong Province (No. 2016A030313552), the Innovation and Building Strong School Project of Colleges of Guangdong Province (2015KQNCX094) and Guangzhou Vocational College of Science and Technology (2016TD03).

References 1. Cao, B.-Y.: Solution and theory of question for a kind of fuzzy positive geometric program. In: Proceedings of the 2nd IFSA Conference, Tokyo, Japan, vol. 1, pp. 205–208 (1987) 2. Cao, B.-Y.: Fuzzy geometric programming (I). Int. J. Fuzzy Sets Syst. 53(2), 135–154 (1993)

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3. Cao, B.-Y.: Classification of fuzzy posynomial geometric programming and corresponding class properties. J. Fuzzy Syst. Math. 9(4), 60–64 (1995) 4. Cao, B.-Y.: Fuzzy geometric programming (II). J. Fuzzy Math. 4(1), 119–129 (1996) 5. Cao, B.-Y.: Primal algorithm of fuzzy posynomial geometric programming. In: Annual Conference of the North American Fuzzy Information Processing Society—NAFIPS, vol. 1, pp. 31–34 (2001) 6. Cao, B.-Y.: Antinomy in posynomial geometric programming. Adv. Syst. Sci. Appl. U.S.A. 4(1), 7–12 (2004) 7. Cao, B.-Y.: Lagrange problem in fuzzy reversed posynomial geometric programming. In: 2006, Fuzzy Systems and Knowledge Discovery—Second International Conference, FSKD 2005, Proceedings. Lecture Notes in Computer Science, LNAI, vol. 3614, pp. 546–550 (2005) 8. Yang, J.H., Cao, B.-Y.: The origin and its application of geometric programming. In: Proceedings of the Eighth National Conference of Operations Research Society of China, pp. 358–363. Global-Link Publishing Company, Hong Kong (2006). ISBN: 962-8286-09-9 9. Cao, B.-Y.: New proof to first dual theorem on fuzzy posynomial geometric programming. J. Fuzzy Math. 14(1), 1–14 (2006) 10. Cao, B.-Y.: Dual method to geometric programming with fuzzy variables. In: Advances in Intelligent and Soft Computing, vol. 62, pp. 1293–1301. Springer, Berlin (2009) 11. Cao, B.-Y.: The more-for-less paradox in fuzzy posynomial geometric programming. Inf. Sci. 211, 81–92 (2012) 12. Zahmatkesh, F., Cao, B.-Y.: On the fuzzy fractional posynomial geometric programming problems. In: Advances in Intelligent Systems and Computing, vol. 367, pp. 97–108 (2016) 13. Cao, B.-Y.: Posynomial geometry programming with L-R fuzzy coefficients. Int. J. Fuzzy Sets Syst. 67(3), 267–276 (1994) 14. Cao, B.-Y.: Research for a geometric programming model with T-fuzzy variations. J. Fuzzy Math. 5(3), 625–632 (1997) 15. Cao, B.-Y.: Multi-objective geometric programming with T-fuzzy variables. In: 22nd International Conference of NAFIPS Proceedings, pp. 456–461 (2003) 16. Cao, B.-Y.: Geometric programming with trapezoidal fuzzy variables. In: Annual Conference of the North American Fuzzy Information Processing Society-NAFIPS, vol. 2, pp. 826–831 (2004) 17. Cao, B.-Y.: Extension posynomial geometric programming. J. Guangdong Univ. Technol. 18(1), 61–64 (2001) 18. Cao, B.-Y.: Types of non-distinct multiobjective geometric programming. Hunan Ann. Math. 15(1), 99–106 (1995) 19. Verma, R.K.: Fuzzy geometric programming with several objective functions. Fuzzy Sets Syst. 35(1), 115–120 (1990) 20. Biswal, M.P.: Fuzzy programming technique to solve multi-objective geometric programming problems. Fuzzy Sets Syst. 51(1), 67–71 (1992) 21. Mandal, N.K., Roy, T.K., Maiti, M.: Multi-objective fuzzy inventory model with three constraints: a geometric programming approach. Fuzzy Sets Syst. 150(1), 87–106 (2005) 22. Cao, B.-Y.: Fuzzy geometric programming optimum seeking of scheme for waste-water disposal in power plant. In: Proceedings of the Fuzz-IEEE /IFES95 Conference, Yokohama, Japan, vol. 5, pp. 793–798 (1995) 23. Cao, B.-Y.: Fuzzy geometric programming optimum seeking in power supply radius of transformer substation. In: Proceedings of the Fuzz-IEEE99 Conference, Seoul, Korea, vol. 3, pp. 1749–1753 (1999) 24. Cao, B.-Y.: Reverse geometric programming with fuzzy coefficient and its application in chemical industry production cost analysis. In: IEEE International Conference on Fuzzy Systems, vol. 2, pp. 1311–1316 (2003) 25. Liu, S.-T.: Fuzzy geometric programming approach to a fuzzy machining economics model. Int. J. Prod. Res. 42(16), 3253–3269 (2004) 26. Islam, S., Kumar, R.T.: A new fuzzy multi-objective programming: entropy based geometric programming and its application of transportation problems. Eur. J. Oper. Res. 173(2), 387–404 (2006)

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27. Cao, B.-Y.: Power supply radius optimized with fuzzy geometric program in substation. Fuzzy Optim. Decis. Making 5(2), 123–139 (2006) 28. Cao, B.-Y.: Fuzzy Geometric Programming. Kluwer Academic Publishers, Dordrecht (2001) 29. Cao, B.-Y.: Optimal Models and Methods with Fuzzy Quantity. Springer, Berlin (2010) 30. Cao, B.-Y.: Extensional positive geometric programming. J. Guangdong Univ. Technol. 18(1), 61–64 (2001) 31. Cao, B.-Y.: Multi-objective geometric programming with T-fuzzy variables. In: Annual Conference of the North American Fuzzy Information Processing Society—NAFIPS, vol. 2003, pp. 456–461, Jan 2003 32. Dang, L.: Study on the gery posynomial geometric programming. Chin. Q. J. Math. 20(1), 34–41 (2005) 33. Cao, B.-Y.: Rough posynomial geometric programming. Fuzzy Inf. Eng. 1(1), 37–57 (2009) 34. Kheiri, Z., Zahmatkesh, F., Cao, B.-Y.: A new ranking approach to fuzzy posynomial geometric programming with trapezoidal fuzzy number. In: Advances in Intelligent and Soft Computing. AISC, vol. 147, pp. 517–523 (2012) 35. Zahmatkesh, F., Cao, B.-Y.: On the solution of fractional geometric programming problem with fuzzy coefficient. In: 4th Iranian Joint Congress on Fuzzy and Intelligent Systems, CFIS 2015 (2015) 36. Burnwal, A.P., Mukherjee, S.N., Singh, D.: Fuzzy geometric programming with nonequivalent objectives. Ranchi Univ. Math. J. 27, 53–58 (1996) 37. Yang, J.H., Cao, B.-Y.: Geometric programming with fuzzy relation equation constraints. In: Proceedings of the IEEE International Conference on Fuzzy Systems, pp. 557–560 (2005) 38. Yang, J.-H., Cao, B.-Y.: Posynomial fuzzy relation geometric programming. In: 2007, IFSA 2007, Proceedings. Lecture Notes in Computer Science. LNAI, vol. 4529, pp. 563–572 (2007) 39. Yang, J., Cao, B.-Y.: Monomial geometric programming with fuzzy relation equation constraints. Fuzzy Optim. Decis. Making 6(4), 337–349 (2007) 40. Zhou, X., Cao, B.-Y.: Optimizing the geometric programming problem with single-term exponents subject to max-product fuzzy relational equation. In: Proceedings—5th International Conference on Fuzzy Systems and Knowledge Discovery, FSKD 2008, vol. 1, pp. 621–625 (2008) 41. Yang, X.-P., Zhou, X.-G., Cao, B.-Y.: Single-variable term semi-latticized fuzzy relation geometric programming with max-product operator. Inf. Sci. 325, 271–287 (2015) 42. Zhou, X.-G., Yang, X.-P., Cao, B.-Y.: Posynomial geometric programming problem subject to max-min fuzzy relation equations. Inf. Sci. 328, 15–25 (2016) 43. Yang, X.-P., Zhou, X.-G., Cao, B.-Y.: Min-max programming problem subject to addition-min fuzzy relation inequalities. IEEE Trans. Fuzzy Syst. 24(1), 111–119 (2016) 44. Zhou, X.-G., Cao, B.-Y., Yang, X.-P.: The set of optimal solutions of geometric programming problem with max-product fuzzy relational equations constraints. Int. J. Fuzzy Syst. 18(3), 436–447 (2016) 45. Zeinab, K., Cao, B.-Y.: Posynomial geometric programming with intuitionistic fuzzy coefficients. In: Advances in Intelligent Systems and Computing, vol. 367, pp. 15–30 (2016) 46. Qin, Z., Cao, B.-Y., Fang, S.-C.: Geometric programming with discrete variables subject to max-product fuzzy relation constraints. Discrete Dyn. Nat. Soc. 1610349 (2018) 47. Yang, X.-P., Lin, H.-T., Zhou, X.-G., Cao, B.-Y.: Addition-min fuzzy relation inequalities with application in BitTorrent-like Peer-to-Peer file sharing system. Fuzzy Sets Syst. 343, 126–140 (2018) 48. Yang, X.-P., Yuan, D.-H., Cao, B.-Y.: Lexicographic optimal solution of the multi-objective programming problem subject to max-product fuzzy relation inequalities. Fuzzy Sets Syst. 341, 92–112 (2018) 49. Cao, B.-Y.: Three guess of fuzzy geometric programming. In: Advances in Intelligent and Soft Computing, vol. 147, pp. 591–594. Springer, Berlin (2012)

Fuzzy Clustering Analysis of Hotel Online Booking Marketing—A Case of eLong Net Nan Wu

Abstract In order to improve the network for hotel marketing quality of e-commerce and meet customers’ preferences, we, in the paper, propose a classification of hotel marketing quality on the basis of fuzzy clustering. The information of different types of 1579 hotels in Guangzhou was manually collected from the eLong net, including room characteristics, customer reviews, hotel service information, and reservation platform recommendation. Based on the indicator screening method to information sensitivity, indicators are constructed for 14 precision marketing evaluations. These 14 marketing evaluation indicators, affecting the marketing quality of different types of hotels, are refined through factor analysis into several key hotel marketing quality indicators. The extracted key indicators are clustered through a fuzzy C-means clustering method; based on the clustering results, the precise marketing differences are compared within different types of hotels. After that, by applying the entropy value method we identify the key factor affecting marketing quality in user ratings. Thus, for hotel precision marketing, it is suggested that an online booking marketing platform should be reconstructed with more friendship, stickiness, and valuableness as well. Keywords Fuzzy clustering analysis · Data mining · Hotel precision marketing

1 Introduction The Internet strongly transforms the hotel industry chain. As the network penetration and influence continue to expand, the scale data of hotel online booking continues to expand with the dimension also increasing and the speed of circulation accelerating. The data types are complex and challengeable to ordinary manual data analysis, variating from hotel rooms’ basic information, to customer reviews and booking platform recommendations that are generally random, unstructured, and non-procedural. N. Wu (B) Guangzhou Vocational and Technical University of Science and Technology, Guangzhou 510000, China e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2020 B.-y. Cao (ed.), Fuzzy Information and Engineering-2019, Advances in Intelligent Systems and Computing 1094, https://doi.org/10.1007/978-981-15-2459-2_12

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Therefore, the data mining technology of fuzzy clustering algorithm is used to analyze hotel online booking data processing, maximize the value information, and highlight the factors in the hotel reservation platform. Then the accurate marketing of hotel online booking can be realized to help customers to make the best purchase.

2 Research Status Home and Abroad Hotel is particularly suitable for online sales as a typical experiential service product [1], and the current research focuses on the impact of hotel online reviews on consumer purchase decisions, such as Dickinger & Mazanec think online reviews can influence consumer decisions [2], Gretzel & Yoo regard online reviews have the greatest impact on consumer decision-making [3], Deloitte Consulting considers the online reviews which were the key factors in influencing consumer’s decision-making [4, 5], Qiang & Yue analyzed consumer information perception from the length of online reviews and emotional polarity [6], Sparks & Browning built consumer’s high trust from positive online reviews [7], Vermeulen & Seegers increased consumer willingness to buy from positive online reviews [8]. The above researches focused only on one dimension of online information, i.e., the impact of online reviews on consumer purchase decisions. Only a handful of studies involve the impact on other different dimensions of online information for consumers to make purchase decisions. Chen pointed out the booking platform recommendations [9], and Chevalier and Mayzlin found book feature information [10], both of which affected the book’s online sales significantly. However, the study of the effects is rarely involved from different dimensions of online information on sales. Only Haul [11] and Zhang [12] pointed out that the different information presented on the online booking platform has different influence on consumers’ purchase, but scholars home and abroad have not made further study to form a good interaction with hotel industry. Accordingly, in order to overcome the diversity and complexity of the evaluation content of hotel online booking precision marketing, now we use both factor analysis and dynamic fuzzy clustering to compare and analyze the changing trend of hotel precision marketing difference. In addition, the entropy value method is used to find out the key factors that affecting the hotel precision marketing difference to provide marketing advice to the platform.

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3 Statistical Analysis Methods, Research Data Sources, and Indicator Selection 3.1 Statistical Analysis Methods Indicator Selection Method to Information Sensitivity. As the actual meaning is not clear of the main component reduced by using the principal component, which is not conducive to further study, the index screening model based on index information sensitivity proposed by Guotai [13] is used to screen the index and construct the index system. Factor Analysis Model. The purpose of factor analysis aims to seek the basic structure of variables, simplify the observation system, and the whole problem can be explained with a few key variables. Therefore, comprehensive evaluation of a precision marketing among each hotel can be carried on to observe the change of its precision marketing difference, helping the following analysis on hotel precision marketing division through dynamic fuzzy clustering. Dynamic Fuzzy Clustering Model. Clustering analysis means a method simplifying data by data modeling, while a dynamic fuzzy clustering method is a mathematical method to classify hotel precision marketing from dimensions of different characteristics, affinity and similarity by establishing fuzzy similarity relationship. It is more reasonable to do the classification by comparing the results of factor analysis. The main steps of the model refer to the regularization of data, the construction of fuzzy similarity matrix, the solution of transfer closure, the solution of truncated matrix [14], and so on. Entropy Value Method. The dispersion degree of a stochastic and disordered index can be judged by calculating an entropy value; the larger the value, the greater its degree of dispersion, and the greater its impact on the comprehensive evaluation. Therefore, this paper selects an index with the largest entropy value as a key index, which is a basis of the following study by using gradient adjustment method to simulate the influence of an index change on the difference of hotel precision marketing.

3.2 Data Acquisition and Processing Taking eLong net platform, for example, the manually collected data come from August 24, 2018, to August 30, 2018, 7 days in total. A total of 1579 Guangzhou hotels, including 108 five-star/luxury hotels, 378 four-star/high-class hotels, 510 three-star/comfort hotels, 583 economy hotels/Inns, were collected at random as online booking information data for research samples.

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3.3 Establishment of Index System A complex system results from hotel precision marketing (HPM) difference n affected f (xi) by a variety of factors. The mathematical model is expressed as: d = i=1 “d” is the hotel precision marketing effect. “x” refers to the hotel precision marketing effect factors. “i” means the number of affecting factors. Considering the reliability of the data, the variable selection is based on the online information of platforms. The information sensitivity index screening method is used to select online information influencing consumers, including the hotel room feature information, the recommended information provided by the booking platform and reviews from consumers [15]. The room feature information includes prices, room types, facilities, etc. Because it is difficult to compare the hotel facilities; room types are compared, together with the lowest price, the highest price, and the photographs uploaded by hotels. The recommended information provided by the platform serves as the hotel service index, timely confirmation rate, booking success rate, browsing and collection index, user rating, and so on. Hotel service index is an unique data to the eLong net, which is not available for other similar platforms. It includes the timely confirmation rate, booking success rate, user complaints rate, and the rankings of customer service efficiency of hotels in the same city. Browsing and collection index are the popularity of hotels. User ratings are based on customer reviews, the assessing stars from the National Tourism Administration, hotel facilities and equipment, the environment around, service quality, social reputation, brand awareness, customer integrity, and so on. The eLong platform rated hotels into four different types, including five-star/luxury hotel, four-star/high-class hotel, three-star/comfort hotel, and economy hotel/Inn. The consumer reviews index is combined with the number of comments, good review rate, recommendation rate, non-recommended rate, photographs uploaded by consumers. The number of comments reflects that of room consumption. Good review rate is the comprehensive impression of hotels, including location, facilities, services, sanitation, and performance-to-price ratio. If there are no complaints or comments, the system will regard it as good review, so the good review rate is the proportion of the total review. The recommendation rate is got by the proportion of recommendations, and the non-commendation rate can be got by using 1 to minus the recommendation rate, because the negative evaluations of consumers have greater influence on consumers’ choice than positive evaluations do [16]. The uploaded photographs are those taken and uploaded by hotel customers to share their feelings of staying in hotels, which are of high credibility (Table 1).

4 Analysis of HPM Difference Based on Factor Analysis A regression method is used to calculate the score of 4 public factors, also known as Thompson factor analysis. Its score is the score from 4 public factors on each sample point, which can represent the weight of each variable in a linear calculation

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Table 1 Hotel online booking accurate marketing evaluation index system Standard layer

Indicator layer

Description

Room features information

X 1 types of rooms

Types of room for booking offered by hotels

X 2 lowest price (RMB)

Lowest room rates offered by hotels

X 3 highest price (RMB)

Highest room rates offered by hotels

X 4 amount of hotel photographs

Amount of hotel photographs provided by the hotel itself

X 5 amount of reviews

Total number of customer reviews until August of 2018

X 6 positive review rate (%)

Rate of customer positive review for the hotel

X 7 recommendation rate (%)

Rate of customers’ recommendation for the hotel

X 8 non-recommendation rate (%)

Rate of customers’ non-recommendation for the hotel

X 9 amount of photographs uploaded by consumers

Amount of hotel photographs uploaded by hotel consumers

X 10 service index (0–5)

Timely confirmation rate, booking success rate, user complaints rate and the rankings of customer service efficiency of hotels in the same city

X 11 timely confirmation rate (%)

Ratio that hotel confirm customers’ reservation in time

X 12 booking success rate (%)

Ratio of successful consumption of the hotel products after customers’ booking

X 13 browsing and collection index

Number of customers’ browsing the information pages of hotels in the booking platform

X 14 user ratings (1 and 2)

Rating and classification of hotels by the booking platform

Customer comment information

Hotel service information

Booking platform referral information

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formula, so that the public factor is expressed as a linear combination of variables. β is weight of each  index variable, while Y is a new variable produced by the index 13 βiY i. assimilation Fn = i=1 We use a statistical product and service solution (SPSS) 20 to analyze 13 continuous variables of online information of four different types of hotel in Guangzhou. Using Kaiser-Meyer-Olkin (KMO) and Bartlett’s test, the KMO value of four models is greater than 0.5, probability value 0.000 < 0.01, reaching significant level, suitable for the factor analysis. According to the principle, the eigenvalue value is greater than 1; the cumulative contribution rate of the top 4 variance of four models is more than 60%, explaining most of the information and extracts the first four principal components. The original factor load matrix is established by using a maximum variance orthogonal rotation method, and the principal component is named. The main factors load larger in the “service index” “timely confirmation rate” “booking success rate,” named “hotel service index.” The main factors load larger in the “amount of reviews,” “browsing and collection index,” “photographs uploaded by consumers,” named “popularity index.” The main factors load larger in the “positive review rate,” “non-recommendation rate,” “recommendation rate,” named “evaluation index.” The main factors load larger in “room highest price,” “room lowest price,” “hotel photograph,” “room type,” named “room features.” The hotel factors for different hotel models have different connotations, and the details can be found in Table 2. As Table 3 shows, we use factor analysis to refine 13 online information of four different types of 1579 hotels in Guangzhou into four key factors F 1 , F 2 , F 3 , F 4 . The corresponding variance contribution rate is named λ1 , λ2 , λ3 , λ4 , and the cumulative variance contribution rate is named λ. According to the explanation of the amount of data from multi-oligopoly, different types of hotel factors mean different connotations. The comprehensive evaluation score of the hotel is calculated by using the variance contribution rate of each principal component as the weight: F = (λ1 ∗ F1 + λ2 ∗ F2 + λ3 ∗ F3 + λ4 ∗ F4 )/(λ1 + λ2 + λ3 + λ4 ) Model 1: F = (21.300F 1 + 18.981F 2 + 17.593F 3 + 9.482F 4 )/67.356. Model 2: F = (18.897F 1 + 18.654F 2 + 18.338F 3 + 10.971F4 )/66.860. Table 2 Summary of hotel factors of four hotel models F 1, F 2, F 3, F 4 Model

Model 1

Model 2

Model 3

Model 4

Five-star/luxury

Four-star/high-class

Three-star/comfort

Economy hotel/inn

F1

Hotel service

Hotel service

Evaluation

Evaluation

F2

Popularity index

Evaluation index

Popularity index

Popularity index

F3

Evaluation index

Popularity index

Room features

Hotel service

F4

Room features

Room features

Hotel service

Room features

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Table 3 Four hotel model results summary Model

Model 1

Model 2

Model 3

Model 4

Five-star/luxury

Four-star/high-class

Three-star/comfort

Economy hotel/inn

λ1 = F 1 variance contribution rate (%)

Hotel service index 21.300

Hotel service index 18.897

Evaluation index 21.735

Evaluation index 21.553

λ2 = F 2 variance contribution rate (%)

Popularity index 18.981

Evaluation index 18.654

Popularity index 19.507

Popularity index 17.064

λ3 = F 3 variance contribution rate (%)

Evaluation index 17.593

Popularity index 18.338

Room features 13.427

Hotel service index 14.594

λ4 = F 4 variance contribution rate (%)

Room features 9.482

Room features 10.971

Hotel service index 8.366

Room features 13.145

λ = cumulative variance contribution rate (%)

67.356

66.860

63.034

66.357

KMO

0.496

0.590

0.735

0.680

Sig

0.000

0.000

0.000

0.000

Model 3: F = (21.735F 1 + 19.507F 2 + 13.427F 3 + 8.366F 4 )/63.034. Model 4: F = (21.553F 1 + 17.064F 2 + 14.594F3 + 13.145F 4 )/66.357. We calculate the factor scores, comprehensive score and make ranking accordingly, but only list the calculations of model 1 as example, without the results of the other 3 listed here (Table 4). From the above analysis results: In Table 4, the hotel precision marketing score “hotel service index” standard deviation is the smallest, while the “evaluation index” standard deviation is the largest, indicating that the five-star/luxury hotel precision marketing difference is the “evaluation index,” so that hotel precision marketing has huge room for improvement.

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Table 4 Evaluations of F scores of five-star/luxury hotels for precision marketing (Excerpt) Hotel name

F score

F 1 score

F 2 score

F 3 score

F 4 score

Comprehensive evaluation

Hotel service index

Popularity index

Evaluation index

Room features

Guangzhou Changlong hotel

1.44532

−1.47274

6.99315

−0.25631

0.05199

1

Guangzhou Xiangxue International hotel apartments

1.13511

1.16931

0.25369

1.09699

2.89345

2

Guangzhou Baiyun hotel

0.9074

0.82184

2.60864

−0.35002

0.02714

3

Grand hotel Guangzhou Sofitel

0.87362

0.76056

1.11147

0.58022

1.19588

4

Guangzhou Berber international hotel

0.78064

0.96696

1.50502

−0.24577

0.81647

5

Guangzhou Jia Run Yunque hotel

0.74226

1.38536

0.06307

1.18407

Guangzhou zengcheng Wanda Ka Wah hotel

0.73219

1.22439

−0.47041

1.38239

0.82756

7

Nine Longhu Princess hotel Guangzhou

0.73007

0.56967

0.82619

0.1885

1.90287

8

Victoria Hotel Guangzhou

0.72841

0.41509

1.40748

0.67044

0.18046

9

Leiden hotel Guangzhou

0.71155

0.61156

1.83414

−0.17995

0.3431

10

Iste International Apartment-Guangzhou Poly Zhonghui

0.70275

1.30611

1.36009

−0.06367

−0.54643

11

Sheraton Guangzhou Guangdong hotel

0.69889

−0.10338

1.70557

1.00376

−0.07969

12

Guangzhou Blue Finch Boutique Art hotel

0.6798

0.48058

1.14469

1.0608

−0.51017

13

Sunshine hotel Guangzhou

0.61889

0.76499

−0.06591

1.2732

0.44753

14

Parkrow Business hotel

0.60552

1.28742

−0.18941

0.65059

0.58139

15

Standard deviation

2.6371

0.00428

4.2575

18.4494

−0.1625

13.9912

Rank

6

16

5 Analysis of Hotel Precision Marketing Difference Based on Fuzzy Clustering Analysis Based on the above refined four key factors, we, inn the paper, make a fuzzy clustering analysis of the precision marketing about four different types of 1579 hotels in

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Guangzhou. Contrast to the factor analysis above, compare and analyze the present situation and existing problems of precision marketing in the hotel. Fuzzy C-means cluster (FCM), given the number of categories c, the data group containing n samples is divided into c fuzzy classes. With the class center Vj of each class representing the class, through repeated iteration situation, the error value of the target function is gradually reduced until the target function converges, then the cluster is completed. c n m 2 Target function: min JFCM (U, P) = i=1 j=1 µi j di j c Constraints: i=1 u i j = 1 uij ∈ 0, 1, ∀i, j First, the data transformation is carried out through Excel, with the normalized matrix obtained. Secondly, the angle residual Yu Yufa is used to transform into fuzzy similarity matrix through MATLAB program. Then the fuzzy equivalence matrix is constructed by MATLAB program by using the transfer closure method. Finally, the elements are arranged in order from large to small, getting the truncated matrix corresponding to the confidence level. Figure 1 is the empirical result of hotel classification. Combined with the results of each hotel factor analysis above, the classification results are reasonable and in line with the actual situation of different types of hotel precision marketing differences in Guangzhou. Fig. 1 Classification results of fuzzy clustering analysis of each model hotel

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Program 1 funcƟon[center,U,obj_fun]=FCMCluster(data,n,opƟons) if nargin~=2 && nargin~=3 error('Too many or too few input arguments'); end data_n=size(data,1); in_n=size(data,2); default_opƟons=[2;100;1e-5;1]; if nargin==2 opƟons=default_opƟons; else ifl ength(opƟons)