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Idempotent Computing Rules and Novel Comparative Laws for Hesitant Fuzzy Cognitive Information and Their Application to Multiattribute Decision Making

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Abstract

Hesitant fuzzy cognitive information provides an effective and convenient form for expressing human subjective cognition about the research object, i.e., hesitant fuzzy sets (HFSs). Studies on decision making with HFSs have become an important branch in decision theory, in which operational laws of hesitant fuzzy elements (HFEs) play a core role in the solution. However, current HFE computational laws have the disadvantages of subjectivity and dimensional problems. Therefore, how to define an objective HFE operational rule without dimensional problems is an open issue. This paper introduces an idempotent HFE computing rule to overcome current disadvantages. The weighted mean of HFEs under the developed computing rule is further discussed. In addition, a novel comparison law between HFEs is proposed. The property of idempotence is introduced to provide an intuitive integrated result of HFEs. To decrease the integrated HFEs dimensions, the sliding window model is utilized. Fundamental mathematical properties of the developed operations are discussed. Furthermore, the normal weighted means of HFEs are extended by using the developed idempotent computing rules. Finally, a novel comparison law for comparing HFEs is designed, which is further used to provide a multiattribute decision procedure. Additive idempotent is developed as a special and intuitive property for the HFE additive operation. Normal weighted means of HFEs, including arithmetic and geometric means, are correspondingly derived. Numerical examples have shown that the proposed HFE operational laws are valid, which can effectively decrease the dimensions of integrated results. The developed idempotent computing rules provide a novel HFE algebra structure, which includes the additive operation, multiplicative operation, scalar multiplication and power operation. By using the sliding window model, the developed idempotent computing rules can effectively reduce the integrated HFEs dimensions. The strength of the developed computational model is that integrating two identical pieces of cognitive information produces the same result. In addition, the modified HFE comparison law can overcome the drawback of current comparison laws, and a much more reasonable comparison result can be obtained.

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Data Availability

No data were used to support the findings of the study performed.

Notes

  1. The composed HFE dimensions are not required to be the same.

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Acknowledgements

The authors first want to thank the Editors and anonymous reviewers for their constructive and valuable comments, which have greatly improved the paper. The authors also want to thank Dr. Shahid Hussain Gurmani for proofreading the revised manuscript.

Funding

The work was supported by the Humanities and Social Sciences Research Youth Project of the Ministry of Education of China (grant number 21YJCZH148), the Natural Science Foundation of Anhui Province (grant number 2108085MG239), the Humanities and Social Science Research Project of Universities in Anhui Province (grant number SK2020A0049), the National Natural Science Foundation of China (grant numbers 71701001, 71871001, 71901001, 71901088, 72071001 and 72001001) and the Provincial Natural Science Research Project of Anhui Colleges (grant number KJ2020A0004).

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Authors and Affiliations

  1. School of Economics, Anhui University, Anhui, Hefei, 230601, China

    Zhifu Tao

  2. Research Center for Finance and Statistics, Anhui University, Anhui, Hefei, 230601, China

    Zhifu Tao

  3. School of Mathematical Sciences, Anhui University, Anhui, Hefei, 230601, China

    Ligang Zhou & Huayou Chen

  4. Research Center for Applied Mathematics Research, Anhui University, Anhui, Hefei, 230601, China

    Ligang Zhou

  5. School of Business, Anhui University, Anhui, Hefei, 230601, China

    Jinpei Liu

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  1. Zhifu Tao
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Correspondence to Zhifu Tao.

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Tao, Z., Zhou, L., Liu, J. et al. Idempotent Computing Rules and Novel Comparative Laws for Hesitant Fuzzy Cognitive Information and Their Application to Multiattribute Decision Making. Cogn Comput 13, 1515–1529 (2021). https://doi.org/10.1007/s12559-021-09937-3

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