Application of GRA Method in Multi-Attribute Decision Making Problem Base on Picture Fuzzy Choquet Integral Information

Muhammad Azam; Muhammad Zubair; Khalil Ullah; Muhammad Imtiaz Khan; Naseer Ullah; Ali Hassan

doi:doi:10.11648/j.ajmcm.20220701.12

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Application of GRA Method in Multi-Attribute Decision Making Problem Base on Picture Fuzzy Choquet Integral Information

Muhammad Azam, Muhammad Zubair, Khalil Ullah, Muhammad Imtiaz Khan, Naseer Ullah, Ali Hassan

Published in American Journal of Mathematical and Computer Modelling (Volume 7, Issue 1)

Received: 20 March 2022 Accepted: 7 April 2022 Published: 14 April 2022

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Abstract

Picture fuzzy set (PFS) is the generalize structure over existing structures of fuzzy sets to arrange uncertainty and imprecise information in the decision making problem. Picture fuzzy set is a direct extension of the intuitionistic fuzzy set (IFS) that can model uncertainty in situations with multiple types of answers, such as yes, no, abstain, and refuse. In this paper, we propose Choquet integral (CI) for the picture fuzzy set (PFS). Then, we defined some basic operational laws for picture fuzzy set. Also, based on the defined operational laws, we proposed picture fuzzy averaging (PFA) and picture fuzzy Choquet integral weighted averaging operator (PFCIWAO) along with their basic properties. Also, we propose the normalized Hamming distance and normalized Euclidean distance for the picture fuzzy numbers. Viewing the electiveness of the picture fuzzy set, we proposed a decision-making approach for the multi-criteria decision-making problems. We also propose the GRA method using Choquet integral for dealing uncertainty in decision making problems under picture fuzzy information. Lastly, we illustrate an example show the effectiveness and reliability of the proposed method.

Published in	American Journal of Mathematical and Computer Modelling (Volume 7, Issue 1)
DOI	10.11648/j.ajmcm.20220701.12
Page(s)	12-19
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

IFS, CI, PFS, GRA Method, PFA and PFCIWAO

References

[1]	K. Atanassov, Intuitionistic fuzzy sets: theory and applications, in: Studies in fuzziness and soft computing, Vol. 35, Physica-Verlag, Heidelberg, 1999.
[2]	B. C. Cuong, Picture fuzzy sets, J. Comput. Sci. Cybern. 30 (2014) 409ñ420.
[3]	B. C. Cuong, P. V. Hai, Some fuzzy logic operators for picture fuzzy sets, in: Seventh International Conference on Knowledge and Systems Engineering, 2015, pp. 132ñ137.
[4]	B. C. Cuong, V. Kreinovich, R. T. Ngan, A classiÖcation of representable t-norm operators for picture fuzzy sets, Departmental Technical Reports (CS), (2016) Paper 1047.
[5]	J. L. Deng, Introduction to grey system theory. Grey Systems; 1: 1-24, (1989).
[6]	J. L. Deng, Grey system theory. Press of Huazhong University of Science and Technology, Wuhan, (2002).
[7]	P. H. Phong, D. T. Hieu, R. T. H. Ngan, P. T. Them, Some compositions of picture fuzzy relations, in: Proceedings ofthe 7thNational Conference on Fundamental and Applied Information Technology Research, FAIRí7, Thai Nguyen, 2014, pp. 19ñ20.
[8]	P. Singh, Correlation coe¢ cients for picture fuzzy sets, J. Intell. Fuzzy Syst. 27 (2014) 2857ñ 2868.
[9]	L. H. Son, Generalized picture distance measure and applications to picture fuzzy clustering, Appl. Soft Comput. 46 (2016) 284ñ295.
[10]	L. H. Son, Measuring analogousness in picture fuzzy sets: from picture distance measures topicture association measures, Fuzzy Optim. Decis. Mak. 16 (3) (2017) 1ñ20.
[11]	G. W. Wei, Picture fuzzy cross-entropy for multiple attribute decision making problems, J. Bus. Econ. Manag. 17 (4) (2016) 491ñ502.
[12]	G. W. Wei, F. E. Alsaadi, T. Hayat, A. Alsaedi, Pro jection models for multiple attribute decision making with picture fuzzy information, Int. J. Mach. Learn. Cybern. 9 (4) (2018) 713ñ719.
[13]	G. W. Wei, H. Gao, The generalized dice similarity measures for picture fuzzy sets and their applications, Informatica 29 (1) (2018) 1ñ18.
[14]	G. W. Wei, Some similarity measures for picture fuzzy sets and their applications, Iran. J. Fuzzy Syst. 15 (1) (2018) 77ñ89.
[15]	Zadeh, L. A., 1965. Fuzzy sets. Information and control, 8 (3), pp. 338-353.

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APA Style

Muhammad Azam, Muhammad Zubair, Khalil Ullah, Muhammad Imtiaz Khan, Naseer Ullah, et al. (2022). Application of GRA Method in Multi-Attribute Decision Making Problem Base on Picture Fuzzy Choquet Integral Information. American Journal of Mathematical and Computer Modelling, 7(1), 12-19. https://doi.org/10.11648/j.ajmcm.20220701.12

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ACS Style

Muhammad Azam; Muhammad Zubair; Khalil Ullah; Muhammad Imtiaz Khan; Naseer Ullah, et al. Application of GRA Method in Multi-Attribute Decision Making Problem Base on Picture Fuzzy Choquet Integral Information. Am. J. Math. Comput. Model. 2022, 7(1), 12-19. doi: 10.11648/j.ajmcm.20220701.12

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AMA Style

Muhammad Azam, Muhammad Zubair, Khalil Ullah, Muhammad Imtiaz Khan, Naseer Ullah, et al. Application of GRA Method in Multi-Attribute Decision Making Problem Base on Picture Fuzzy Choquet Integral Information. Am J Math Comput Model. 2022;7(1):12-19. doi: 10.11648/j.ajmcm.20220701.12

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@article{10.11648/j.ajmcm.20220701.12,
  author = {Muhammad Azam and Muhammad Zubair and Khalil Ullah and Muhammad Imtiaz Khan and Naseer Ullah and Ali Hassan},
  title = {Application of GRA Method in Multi-Attribute Decision Making Problem Base on Picture Fuzzy Choquet Integral Information},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {7},
  number = {1},
  pages = {12-19},
  doi = {10.11648/j.ajmcm.20220701.12},
  url = {https://doi.org/10.11648/j.ajmcm.20220701.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20220701.12},
  abstract = {Picture fuzzy set (PFS) is the generalize structure over existing structures of fuzzy sets to arrange uncertainty and imprecise information in the decision making problem. Picture fuzzy set is a direct extension of the intuitionistic fuzzy set (IFS) that can model uncertainty in situations with multiple types of answers, such as yes, no, abstain, and refuse. In this paper, we propose Choquet integral (CI) for the picture fuzzy set (PFS). Then, we defined some basic operational laws for picture fuzzy set. Also, based on the defined operational laws, we proposed picture fuzzy averaging (PFA) and picture fuzzy Choquet integral weighted averaging operator (PFCIWAO) along with their basic properties. Also, we propose the normalized Hamming distance and normalized Euclidean distance for the picture fuzzy numbers. Viewing the electiveness of the picture fuzzy set, we proposed a decision-making approach for the multi-criteria decision-making problems. We also propose the GRA method using Choquet integral for dealing uncertainty in decision making problems under picture fuzzy information. Lastly, we illustrate an example show the effectiveness and reliability of the proposed method.},
 year = {2022}
}

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TY  - JOUR
T1  - Application of GRA Method in Multi-Attribute Decision Making Problem Base on Picture Fuzzy Choquet Integral Information
AU  - Muhammad Azam
AU  - Muhammad Zubair
AU  - Khalil Ullah
AU  - Muhammad Imtiaz Khan
AU  - Naseer Ullah
AU  - Ali Hassan
Y1  - 2022/04/14
PY  - 2022
N1  - https://doi.org/10.11648/j.ajmcm.20220701.12
DO  - 10.11648/j.ajmcm.20220701.12
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 12
EP  - 19
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20220701.12
AB  - Picture fuzzy set (PFS) is the generalize structure over existing structures of fuzzy sets to arrange uncertainty and imprecise information in the decision making problem. Picture fuzzy set is a direct extension of the intuitionistic fuzzy set (IFS) that can model uncertainty in situations with multiple types of answers, such as yes, no, abstain, and refuse. In this paper, we propose Choquet integral (CI) for the picture fuzzy set (PFS). Then, we defined some basic operational laws for picture fuzzy set. Also, based on the defined operational laws, we proposed picture fuzzy averaging (PFA) and picture fuzzy Choquet integral weighted averaging operator (PFCIWAO) along with their basic properties. Also, we propose the normalized Hamming distance and normalized Euclidean distance for the picture fuzzy numbers. Viewing the electiveness of the picture fuzzy set, we proposed a decision-making approach for the multi-criteria decision-making problems. We also propose the GRA method using Choquet integral for dealing uncertainty in decision making problems under picture fuzzy information. Lastly, we illustrate an example show the effectiveness and reliability of the proposed method.
VL  - 7
IS  - 1
ER  -

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Author Information

Muhammad Azam

Department of Mathematics, Faculty of Applied Sciences, Beijing University of Technology, Beijing, China
Muhammad Zubair

Centre for Advanced Materials Application CEMEA, Slovak Academy of Sciences, Bratislava, Slovak Republic
Khalil Ullah

Department of Mathematics, Faculty of Physical and Numerical Sciences, Abdul Wali Khan University Mardan, KPK, Pakistan
Muhammad Imtiaz Khan

Department of Physics, Abbottabad University of Science and Technology, Hevellian, Khyber Pakhtunkhwa, Pakistan
Naseer Ullah

Department of Mathematics, Faculty of Applied Sciences, Beijing University of Technology, Beijing, China
Ali Hassan

Internation Science & Technology Cooperation Base for Laser Robots, Wenzhou University, Wenzhou, China

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APA Style

Muhammad Azam, Muhammad Zubair, Khalil Ullah, Muhammad Imtiaz Khan, Naseer Ullah, et al. (2022). Application of GRA Method in Multi-Attribute Decision Making Problem Base on Picture Fuzzy Choquet Integral Information. American Journal of Mathematical and Computer Modelling, 7(1), 12-19. https://doi.org/10.11648/j.ajmcm.20220701.12

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ACS Style

Muhammad Azam; Muhammad Zubair; Khalil Ullah; Muhammad Imtiaz Khan; Naseer Ullah, et al. Application of GRA Method in Multi-Attribute Decision Making Problem Base on Picture Fuzzy Choquet Integral Information. Am. J. Math. Comput. Model. 2022, 7(1), 12-19. doi: 10.11648/j.ajmcm.20220701.12

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AMA Style

Muhammad Azam, Muhammad Zubair, Khalil Ullah, Muhammad Imtiaz Khan, Naseer Ullah, et al. Application of GRA Method in Multi-Attribute Decision Making Problem Base on Picture Fuzzy Choquet Integral Information. Am J Math Comput Model. 2022;7(1):12-19. doi: 10.11648/j.ajmcm.20220701.12

Copy | Download

@article{10.11648/j.ajmcm.20220701.12,
  author = {Muhammad Azam and Muhammad Zubair and Khalil Ullah and Muhammad Imtiaz Khan and Naseer Ullah and Ali Hassan},
  title = {Application of GRA Method in Multi-Attribute Decision Making Problem Base on Picture Fuzzy Choquet Integral Information},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {7},
  number = {1},
  pages = {12-19},
  doi = {10.11648/j.ajmcm.20220701.12},
  url = {https://doi.org/10.11648/j.ajmcm.20220701.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20220701.12},
  abstract = {Picture fuzzy set (PFS) is the generalize structure over existing structures of fuzzy sets to arrange uncertainty and imprecise information in the decision making problem. Picture fuzzy set is a direct extension of the intuitionistic fuzzy set (IFS) that can model uncertainty in situations with multiple types of answers, such as yes, no, abstain, and refuse. In this paper, we propose Choquet integral (CI) for the picture fuzzy set (PFS). Then, we defined some basic operational laws for picture fuzzy set. Also, based on the defined operational laws, we proposed picture fuzzy averaging (PFA) and picture fuzzy Choquet integral weighted averaging operator (PFCIWAO) along with their basic properties. Also, we propose the normalized Hamming distance and normalized Euclidean distance for the picture fuzzy numbers. Viewing the electiveness of the picture fuzzy set, we proposed a decision-making approach for the multi-criteria decision-making problems. We also propose the GRA method using Choquet integral for dealing uncertainty in decision making problems under picture fuzzy information. Lastly, we illustrate an example show the effectiveness and reliability of the proposed method.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Application of GRA Method in Multi-Attribute Decision Making Problem Base on Picture Fuzzy Choquet Integral Information
AU  - Muhammad Azam
AU  - Muhammad Zubair
AU  - Khalil Ullah
AU  - Muhammad Imtiaz Khan
AU  - Naseer Ullah
AU  - Ali Hassan
Y1  - 2022/04/14
PY  - 2022
N1  - https://doi.org/10.11648/j.ajmcm.20220701.12
DO  - 10.11648/j.ajmcm.20220701.12
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 12
EP  - 19
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20220701.12
AB  - Picture fuzzy set (PFS) is the generalize structure over existing structures of fuzzy sets to arrange uncertainty and imprecise information in the decision making problem. Picture fuzzy set is a direct extension of the intuitionistic fuzzy set (IFS) that can model uncertainty in situations with multiple types of answers, such as yes, no, abstain, and refuse. In this paper, we propose Choquet integral (CI) for the picture fuzzy set (PFS). Then, we defined some basic operational laws for picture fuzzy set. Also, based on the defined operational laws, we proposed picture fuzzy averaging (PFA) and picture fuzzy Choquet integral weighted averaging operator (PFCIWAO) along with their basic properties. Also, we propose the normalized Hamming distance and normalized Euclidean distance for the picture fuzzy numbers. Viewing the electiveness of the picture fuzzy set, we proposed a decision-making approach for the multi-criteria decision-making problems. We also propose the GRA method using Choquet integral for dealing uncertainty in decision making problems under picture fuzzy information. Lastly, we illustrate an example show the effectiveness and reliability of the proposed method.
VL  - 7
IS  - 1
ER  -

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