Complex shadowed set theory and its application in decision-making problems

Doaa Alsharo; Eman Abuteen; Abd Ulazeez M. J. S. Alkouri; Mutasem Alkhasawneh; Fadi M. A. Al-Zubi; Doaa Alsharo; Eman Abuteen; Abd Ulazeez M. J. S. Alkouri; Mutasem Alkhasawneh; Fadi M. A. Al-Zubi

doi:10.3934/math.2024815

AIMS Mathematics

2024, Volume 9, Issue 6: 16810-16825. doi: 10.3934/math.2024815

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Research article

Complex shadowed set theory and its application in decision-making problems

1.
Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan, dmalsharo@bau.edu.jo
2.
Department of Physics Basic Science, Faculty of Engineering Technology, Al-Balqa Applied University, Amman, Jordan, Dr.eman.abuteen@bau.edu.jo
3.
Department of Mathematics, Faculty of Science, Ajloun National University, P.O. 43, Ajloun, 26810, Jordan, alkouriabdulazeez@anu.edu.jo
4.
Department of software engineering, Faculty of Science, Ajloun National University, P.O. 43, Ajloun, 26810, Jordan, mutasimkhasawneh@anu.edu.jo
5.
College of Natural and Health Sciences, Zayed University, Abu Dhabi, United Arab Emirates, Fadi.al-Zubi@zu.ac.ae
6.
School of Mathematical Sciences, University Kebangsaan Malaysia, 43600 UKM Bangi, Selangor DE, Malaysia, P115916@SISWA.UKM.EDU.MY

Received: 14 March 2024 Revised: 22 April 2024 Accepted: 30 April 2024 Published: 14 May 2024
MSC : 03E72, 47S40, 90B50

Modern technology makes it easier to store datasets, but extracting and isolating useful information with its full meaning from this data is crucial and hard. Recently, several algorithms for clustering data have used complex fuzzy sets (CFS) to improve clustering performance. Thus, adding a second dimension (phase term) to the range of membership avoids the problem of losing the full meaning of complicated information during the decision-making process. In this research, the notion of the complex shadowed set (CSHS) was introduced and considered as an example of the three region approximations method simplifying processing with the support of CFS and improving the representation of results attained within. This notion can be founded by extending the shadowed set codomain from $ \left\{0, \left[0, 1\right], 1\right\} $ into $ \left\{0{e}^{i\theta }, \left[0, 1\right]{e}^{i\theta }, 1{e}^{i\theta }\right\} $. The significance of CSHS was illustrated by giving an example. Additionally, some properties of the CSHS were examined. The basic CSHS operations, complement, union, and intersection were investigated with their properties. Finally, an application in decision-making was illuminated to support the present notion.
- fuzzy sets,
- shadowed sets,
- complex fuzzy sets,
- complex shadowed sets,
- thresholds approximation
Citation: Doaa Alsharo, Eman Abuteen, Abd Ulazeez M. J. S. Alkouri, Mutasem Alkhasawneh, Fadi M. A. Al-Zubi. Complex shadowed set theory and its application in decision-making problems[J]. AIMS Mathematics, 2024, 9(6): 16810-16825. doi: 10.3934/math.2024815

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Abstract

Modern technology makes it easier to store datasets, but extracting and isolating useful information with its full meaning from this data is crucial and hard. Recently, several algorithms for clustering data have used complex fuzzy sets (CFS) to improve clustering performance. Thus, adding a second dimension (phase term) to the range of membership avoids the problem of losing the full meaning of complicated information during the decision-making process. In this research, the notion of the complex shadowed set (CSHS) was introduced and considered as an example of the three region approximations method simplifying processing with the support of CFS and improving the representation of results attained within. This notion can be founded by extending the shadowed set codomain from $ \left\{0, \left[0, 1\right], 1\right\} $ into $ \left\{0{e}^{i\theta }, \left[0, 1\right]{e}^{i\theta }, 1{e}^{i\theta }\right\} $. The significance of CSHS was illustrated by giving an example. Additionally, some properties of the CSHS were examined. The basic CSHS operations, complement, union, and intersection were investigated with their properties. Finally, an application in decision-making was illuminated to support the present notion.

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