Frank Prioritized Aggregation Operators and WASPAS Method Based on Complex Intuitionistic Fuzzy Sets and Their Application in Multi-Attribute Decision-Making
Abstract
:1. Introduction
- To discover Frank operational laws for managing the theory of CIF information;
- To derive the CIF Frank prioritized averaging (CIFFPA) operator, the CIF Frank prioritized ordered averaging (CIFFPOA) operator, the CIF Frank prioritized geometric (CIFFPG) operator, and the CIF Frank prioritized ordered geometric (CIFFPOG) operator with their properties;
- To expose the idea of the weighted aggregates sum product assessment (WASPAS) procedure under the consideration or presence of the CIF information and try to simplify it with the help of a suitable example;
- To demonstrate an example in the presence of the MADM procedures for evaluating the comparison between the proposed operators with some well-known existing operators to show the validity and worth of the discovered approaches.
2. Preliminaries
2.1. WASPAS Method for Classical Set Theory
- When , we obtain the data in Equation (2);
- When , we obtain the data in Equation (3).
2.2. Existing Ideas
3. CIF Frank Prioritized Aggregation Operators
4. CIF WASPAS Procedures
- When , we obtain the data in Equation (40);
- When , we obtain the data in Equation (41).
5. Application in MADM Method
- Step 1: Before evaluating the normalization, we arrange a collection of CIF data which may be of a benefit type or cost type. If the data are of a benefit type, then good, otherwise, using the below theory, we normalize the information, such as:
- Step 2: After performing the above evaluation, we calculate the CIFFPA operator and CIFFPG operator with the help of the derived theory.
- Step 3: Evaluate the score or accuracy values of the aggregated information.
- Step 4: Examine the ranking values in the presence of the score information.
6. Comparative Analysis
7. Conclusions
- We evaluated the Frank operational laws for the theory of CIF information;
- We examined the theory of the CIFFPA, CIFFPOA, CIFFPG, and CIFFPOG operators, and their properties of idempotency, monotonicity, and boundedness;
- We derived the WASPAS under the presence of the CIFFPA and CIFFPG operators;
- We demonstrated the MADM procedures based on the invented theory for CIF information;
- We compared the derived theory with various existing information to show the validity and worth of the discovered approaches.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Symbols | Meanings | Symbols | Meanings | Symbols | Meanings |
---|---|---|---|---|---|
Real part of membership grade | Real part of the non-membership grade | Universal set | |||
Imaginary part of membership grade | Imaginary part of the non-membership grade | Element of the universal set | |||
Real part of the refusal grade | Imaginary part of refusal grade | Refusal grade | |||
Complex intuitionistic fuzzy set | Complex intuitionistic fuzzy value | Score value | |||
Accuracy value | Weighted vector | Scaler | |||
t-norm | t-conorm | Scaler |
Methods | |
Methods | |
Methods | Score Information | Ranking Information |
---|---|---|
Xu [23] | ||
Xu and Yager [24] | ||
Yahya, et al. [26] | ||
Yu [31] | ||
Lin, et al. [32] | ||
Garg and Rani [33] | 0.1506, 0.5008, 0.0506, 0.3005, 0.6010 | |
Garg and Rani [34] | 0.1497, 0.4998, 0.0496, 0.2997, 0.5998 | |
Mahmood, et al. [35] | 0.1506, 0.5007, 0.0505, 0.3005, 0.6009 | |
CIFFPA | ||
CIFFPG |
Methods | Score Information | Ranking Information |
---|---|---|
Xu [23] | ||
Xu and Yager [24] | ||
Yahya, et al. [26] | ||
Yu [31] | ||
Lin, et al. [32] | ||
Garg and Rani [33] | 0.1502, 0.2002, 0.0001, 0.2502, 0.3003 | |
Garg and Rani [34] | 0.1499, 0.1999, 0.0001, 0.2499, 0.2999 | |
Mahmood, et al. [35] | 0.1502, 0.2001, 0.00009, 0.2502, 0.3003 | |
CIFFPA | ||
CIFFPG |
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Ali, Z.; Mahmood, T.; Yang, M.-S. Frank Prioritized Aggregation Operators and WASPAS Method Based on Complex Intuitionistic Fuzzy Sets and Their Application in Multi-Attribute Decision-Making. Mathematics 2023, 11, 2058. https://doi.org/10.3390/math11092058
Ali Z, Mahmood T, Yang M-S. Frank Prioritized Aggregation Operators and WASPAS Method Based on Complex Intuitionistic Fuzzy Sets and Their Application in Multi-Attribute Decision-Making. Mathematics. 2023; 11(9):2058. https://doi.org/10.3390/math11092058
Chicago/Turabian StyleAli, Zeeshan, Tahir Mahmood, and Miin-Shen Yang. 2023. "Frank Prioritized Aggregation Operators and WASPAS Method Based on Complex Intuitionistic Fuzzy Sets and Their Application in Multi-Attribute Decision-Making" Mathematics 11, no. 9: 2058. https://doi.org/10.3390/math11092058
APA StyleAli, Z., Mahmood, T., & Yang, M.-S. (2023). Frank Prioritized Aggregation Operators and WASPAS Method Based on Complex Intuitionistic Fuzzy Sets and Their Application in Multi-Attribute Decision-Making. Mathematics, 11(9), 2058. https://doi.org/10.3390/math11092058