Synergic ranking of fuzzy Z-numbers based on vectorial distance and spread for application in decision-making

Nik Muhammad Farhan Hakim Nik Badrul Alam; Ku Muhammad Naim Ku Khalif; Nor Izzati Jaini; Nik Muhammad Farhan Hakim Nik Badrul Alam; Ku Muhammad Naim Ku Khalif; Nor Izzati Jaini

doi:10.3934/math.2023560

AIMS Mathematics

2023, Volume 8, Issue 5: 11057-11083. doi: 10.3934/math.2023560

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Synergic ranking of fuzzy Z-numbers based on vectorial distance and spread for application in decision-making

Nik Muhammad Farhan Hakim Nik Badrul Alam ^1,2,
Ku Muhammad Naim Ku Khalif ^{1,3
,
,},
Nor Izzati Jaini ¹

1.
Centre for Mathematical Sciences, Universiti Malaysia Pahang, 26300 Gambang, Pahang, Malaysia
2.
Mathematical Sciences Studies, College of Computing, Informatics and Media, Universiti Teknologi MARA (UiTM) Pahang Branch, Jengka Campus, Bandar Tun Abdul Razak Jengka, Pahang, Malaysia
3.
Centre of Excellence for Artificial Intelligence and Data Science, Universiti Malaysia Pahang, 26300 Gambang, Pahang, Malaysia

Received: 15 November 2022 Revised: 19 January 2023 Accepted: 02 February 2023 Published: 08 March 2023
MSC : 91B06

Decision science has a wide range of applications in daily life. Decision information is usually incomplete and partially reliable. In the fuzzy set theory, Z-numbers are introduced to handle this situation because they contain the restriction and reliability components, which complement the impaired information. The ranking of Z-numbers is a challenging task since they are composed of pairs of fuzzy numbers. In this research, the vectorial distance and spread of Z-numbers were proposed synergically, in which the vectorial distance measures how much the fuzzy numbers are apart from the origin, which was set as a relative point, and their spreads over a horizontal axis. Furthermore, a ranking method based on the convex compound was proposed to combine the restriction and reliability components of Z-numbers. The proposed ranking method was validated using several empirical examples and a comparative analysis was conducted. The application of the proposed ranking method in decision-making was illustrated via the development of the Analytic Hierarchy Process-Weighted Aggregated Sum Product Assessment (AHP-WASPAS) model to solve the prioritization of public services for the implementation of Industry 4.0 tools. Sensitivity analysis was also conducted to evaluate the performance of the proposed model and the results showed that the proposed model has improved its consistency from 66.67% of the existing model to 83.33%. This research leads to a future direction of the application of ranking based on the vectorial distance and spread in multi-criteria decision-making methods, which use Z-numbers as linguistic values.
- Z-numbers,
- vectorial distance,
- spread,
- ranking,
- decision-making,
- AHP-WASPAS
Citation: Nik Muhammad Farhan Hakim Nik Badrul Alam, Ku Muhammad Naim Ku Khalif, Nor Izzati Jaini. Synergic ranking of fuzzy Z-numbers based on vectorial distance and spread for application in decision-making[J]. AIMS Mathematics, 2023, 8(5): 11057-11083. doi: 10.3934/math.2023560

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Abstract

Decision science has a wide range of applications in daily life. Decision information is usually incomplete and partially reliable. In the fuzzy set theory, Z-numbers are introduced to handle this situation because they contain the restriction and reliability components, which complement the impaired information. The ranking of Z-numbers is a challenging task since they are composed of pairs of fuzzy numbers. In this research, the vectorial distance and spread of Z-numbers were proposed synergically, in which the vectorial distance measures how much the fuzzy numbers are apart from the origin, which was set as a relative point, and their spreads over a horizontal axis. Furthermore, a ranking method based on the convex compound was proposed to combine the restriction and reliability components of Z-numbers. The proposed ranking method was validated using several empirical examples and a comparative analysis was conducted. The application of the proposed ranking method in decision-making was illustrated via the development of the Analytic Hierarchy Process-Weighted Aggregated Sum Product Assessment (AHP-WASPAS) model to solve the prioritization of public services for the implementation of Industry 4.0 tools. Sensitivity analysis was also conducted to evaluate the performance of the proposed model and the results showed that the proposed model has improved its consistency from 66.67% of the existing model to 83.33%. This research leads to a future direction of the application of ranking based on the vectorial distance and spread in multi-criteria decision-making methods, which use Z-numbers as linguistic values.

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