Aczel–Alsina Power Aggregation Operators for Complex Picture Fuzzy (CPF) Sets with Application in CPF Multi-Attribute Decision Making
Abstract
:1. Introduction
- To explore the basic Aczel–Alsina operational law for complex picture fuzzy (CPF) values.
- To derive the CPF Aczel–Alsina power averaging (CPFAAP-A), CPF Aczel–Alsina weighted power averaging (CPFAAWP-A), CPF Aczel–Alsina power geometric (CPFAAP-G), and CPF Aczel–Alsina weighted power geometric (CPFAAWP-G) operators.
- To examine the three basic properties of the above operators, such as idempotency, monotonicity, and boundedness.
- To justify the above problem, we illustrate a procedure of decision making in the presence of the CPF values (CPFVs) and derive an algorithm to evaluate the MADM problems. Furthermore, these can be extended to CPF Maclaurin symmetric mean and power generalized Maclaurin symmetric mean operators.
- To illustrate a practical example of a decision-making procedure under the consideration of derived operators and compare their performance with various operators to show the supremacy and validity of the derived approaches.
2. Preliminaries
- (1)
- .
- (2)
- .
- (3)
- when .
- If ;
- If ;
- If ;
- (i)
- If ;
- (ii)
- If ;
- (iii)
- If .
- ;
- ;
- ;
- ;
- ;
- .
3. The Proposed Aczel-Alsina Power AOs for CPFVs
- By setting the value of in the invented theory, the invented theory will be reduced for CIFSs.
- By setting the value of in the invented theory, the invented theory will be reduced for CFSs.
- By setting the value of in the invented theory, the invented theory will be reduced for PFSs.
- By setting the value of in the invented theory, the invented theory will be reduced for IFSs.
- By setting the value of in the invented theory, the invented theory will be reduced for FSs.
- Furthermore, the Aczel–Alsina aggregation operators for FSs, IFSs, PFSs, CFSs, CIFSs, and CPFSs are the special cases of the proposed theory, when we remove the power aggregation operators.
- The power aggregation operators for FSs, IFSs, PFSs, CFSs, CIFSs, and CPFSs are the special cases of the proposed theory, when we remove the Aczel–Alsina information.
- The simple averaging and geometric aggregation operators for FSs, IFSs, PFSs, CFSs, CIFSs, and CPFSs are the special cases of the proposed theory, when we use the algebraic information instead of Aczel–Alsina and power aggregation operators.
4. Strategic CPF MADM Methods
5. Comparative Analysis
6. Conclusions
- We exposed the theory of Aczel–Alsina operational laws for the invented theory such as CPF information.
- We derived the CPFAAP-A, CPFAAWP-A, CPFAAP-G, and CPFAAWP-G operators and discover their properties, and then illustrated the procedure of the decision-making technique in the presence of the CPF values and derived an algorithm to evaluate the MADM problems.
- We give an example to illustrate the decision-making procedure based on the derived CPFAAP-A, CPFAAWP-A, CPFAAP-G, and CPFAAWP-G operators.
- We compared the derived theory with the existing operators, which showed the supremacy and validity of our proposed approaches.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Zadeh, L.A. Fuzzy sets. Inf. Control 1965, 8, 338–353. [Google Scholar] [CrossRef] [Green Version]
- Chaomurilige, C.; Yu, J.; Yang, M.S. Analysis of parameter selection for Gustafson-Kessel fuzzy clustering using Jacobian matrix. IEEE Trans. Fuzzy Syst. 2015, 23, 2329–2342. [Google Scholar] [CrossRef]
- Lu, K.P.; Chang, S.T.; Yang, M.S. Change-point detection for shifts in control charts using fuzzy shift change-point algorithms. Comput. Ind. Eng. 2016, 93, 12–27. [Google Scholar] [CrossRef]
- Atanassov, K.T. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20, 87–96. [Google Scholar] [CrossRef]
- Saqlain, M.; Riaz, M.; Saleem, M.A.; Yang, M.S. Distance and similarity measures for neutrosophic hypersoft set (NHSS) with construction of NHSS-TOPSIS and applications. IEEE Access 2021, 9, 30803–30816. [Google Scholar] [CrossRef]
- Cuong, B.C. Picture fuzzy sets. J. Comput. Sci. Cybern. 2014, 30, 409–420. [Google Scholar]
- Zhao, R.; Luo, M.; Li, S. A dynamic distance measure of picture fuzzy sets and its application. Symmetry 2021, 13, 436. [Google Scholar] [CrossRef]
- Ramot, D.; Milo, R.; Friedman, M.; Kandel, A. Complex fuzzy sets. IEEE Trans. Fuzzy Syst. 2002, 10, 171–186. [Google Scholar] [CrossRef]
- Yazdanbakhsh, O.; Dick, S. A systematic review of complex fuzzy sets and logic. Fuzzy Sets Syst. 2018, 338, 1–22. [Google Scholar] [CrossRef]
- Alkouri, A.M.D.J.S.; Salleh, A.R. Complex intuitionistic fuzzy sets. In AIP Conference Proceedings; American Institute of Physics: College Park, MD, USA, 2012; Volume 1482, pp. 464–470. [Google Scholar]
- Akram, M.; Bashir, A.; Garg, H. Decision-making model under complex picture fuzzy Hamacher aggregation operators. Comput. Appl. Math. 2020, 39, 1–38. [Google Scholar] [CrossRef]
- Aczél, J.; Alsina, C. Characterizations of some classes of quasilinear functions with applications to triangular norms and to synthesizing judgements. Aequ. Math. 1982, 25, 313–315. [Google Scholar] [CrossRef]
- Yager, R.R. The power average operator. IEEE Trans. Syst. Man Cybern-Part A Syst. Hum. 2001, 31, 724–731. [Google Scholar] [CrossRef]
- Senapati, T.; Chen, G.; Yager, R.R. Aczel–Alsina aggregation operators and their application to intuitionistic fuzzy multiple attribute decision making. Int. J. Intell. Syst. 2022, 37, 1529–1551. [Google Scholar] [CrossRef]
- Senapati, T.; Chen, G.; Mesiar, R.; Yager, R.R. Intuitionistic fuzzy geometric aggregation operators in the framework of Aczel-Alsina triangular norms and their application to multiple attribute decision making. Expert Syst. Appl. 2023, 212, 118832. [Google Scholar] [CrossRef]
- Sarfraz, M.; Ullah, K.; Akram, M.; Pamucar, D.; Božanić, D. Prioritized aggregation operators for intuitionistic fuzzy information based on Aczel–Alsina T-norm and T-conorm and their applications in group decision-making. Symmetry 2022, 14, 2655. [Google Scholar] [CrossRef]
- Hussain, A.; Ullah, K.; Alshahrani, M.N.; Yang, M.S.; Pamucar, D. Novel Aczel-Alsina operators for Pythagorean fuzzy sets with application in multi-attribute decision making. Symmetry 2022, 14, 940. [Google Scholar] [CrossRef]
- Jin, H.; Hussain, A.; Ullah, K.; Javed, A. Novel complex Pythagorean fuzzy sets under Aczel–Alsina operators and their application in multi-attribute decision making. Symmetry 2023, 15, 68. [Google Scholar] [CrossRef]
- Ye, J.; Du, S.; Yong, R. Aczel–Alsina weighted aggregation operators of neutrosophic Z-numbers and their multiple attribute decision-making method. Int. J. Fuzzy Syst. 2022, 24, 2397–2410. [Google Scholar] [CrossRef]
- Senapati, T. Approaches to multi-attribute decision-making based on picture fuzzy Aczel–Alsina average aggregation operators. Comput. Appl. Math. 2022, 41, 40. [Google Scholar] [CrossRef]
- Naeem, M.; Khan, Y.; Ashraf, S.; Weera, W.; Batool, B. A novel picture fuzzy Aczel-Alsina geometric aggregation information: Application to determining the factors affecting mango crops. AIMS Math. 2022, 7, 12264–12288. [Google Scholar] [CrossRef]
- Xu, Z. Approaches to multiple attribute group decision making based on intuitionistic fuzzy power aggregation operators. Knowl.-Based Syst. 2011, 24, 749–760. [Google Scholar] [CrossRef]
- Jiang, W.; Wei, B.; Liu, X.; Li, X.; Zheng, H. Intuitionistic fuzzy power aggregation operator based on entropy and its application in decision making. Int. J. Intell. Syst. 2018, 33, 49–67. [Google Scholar] [CrossRef]
- Rani, D.; Garg, H. Complex intuitionistic fuzzy power aggregation operators and their applications in multicriteria decision-making. Expert Syst. 2018, 35, e12325. [Google Scholar] [CrossRef]
- Liu, P.; Akram, M.; Bashir, A. Extensions of power aggregation operators for decision making based on complex picture fuzzy knowledge. J. Intell. Fuzzy Syst. 2021, 40, 1107–1128. [Google Scholar] [CrossRef]
- Mahmood, T.; ur Rehman, U.; Ali, Z. Analysis and Application of Aczel-Alsina Aggregation operators Based on Bipolar Complex Fuzzy Information in Multiple Attribute Decision Making. Inf. Sci. 2023, 619, 817–833. [Google Scholar] [CrossRef]
Organized Algorithm for Evaluating the Decision-Making Problem |
---|
Notice that here we construct a decision-making procedure based on the proposed theory, whose main steps are stated below: Step 1: During the construction of every decision matrix, experts have faced two types of information like cost and benefit. In this problem, if we have cost type of data, then normalize it, such as Equation (26) But, if we have a benefit type of data, then proceed with the procedure of decision-making application. Step 2: To use the idea of CPFAAP-A, CPFAAWP-A, CPFAAP-G, and CPFAAWP-G operators, we concentrate to aggregate the information matrix into a single set theory. Step 3: To use the idea of score and accuracy values, we focus to more evaluate the single aggregated values into real-valued information. Step 4: To use the score values, we examine the ranking values for evaluating the finest preferences from the collection of finite preferences. |
Alternatives/Attributes | ||
---|---|---|
Alternatives/Attributes | ||
Alternatives | CPFAAP-A Operator | CPFAAP-G Operator |
---|---|---|
Alternatives | CPFAAWP-A Operator | CPFAAWP-G Operator |
Alternatives | CPFAAP-A Operator | CPFAAP-G Operator | CPFAAWP-A Operator | CPFAAWP-G Operator |
---|---|---|---|---|
CPFAAP-A Operator | |
CPFAAP-G Operator | |
CPFAAWP-A Operator | |
CPFAAWP-G Operator |
Alternatives | CPFAAP-A Operator | CPFAAP-G Operator | CPFAAWP-A Operator | CPFAAWP-G Operator |
---|---|---|---|---|
−0.3625 | 0.2729 | −0.3616 | 0.273 | |
−0.4321 | 0.2112 | −0.4311 | 0.2113 | |
−0.2813 | 0.3288 | −0.2803 | 0.3289 | |
−0.4366 | 0.2026 | −0.4358 | 0.2026 | |
−0.3637 | 0.1429 | −0.3623 | 0.1425 |
CPFAAP-A Operator | |
CPFAAP-G Operator | |
CPFAAWP-A Operator | |
CPFAAWP-G Operator |
Paramete | Operato | Score Value | Ranking Value |
---|---|---|---|
CPFAAP-A | −0.4735, −0.6104, −0.3077, −0.6155, −0.5105 | ||
CPFAAP-G | 0.359, 0.259, 0.473, 0.2196, 0.2159 | ||
CPFAAWP-A | −0.4722, −0.6093, −0.3062, −0.6143, −0.5088 | ||
CPFAAWP-G | 0.3592,0.2591,0.4733,0.2196,0.2157 | ||
CPFAAP-A | −0.4727, −0.6097, −0.3066, −0.6147, −0.5093 | ||
CPFAAP-G | 0.3583, 0.2583, 0.4722, 0.2189, 0.2149 | ||
CPFAAWP-A | −0.4712, −0.6085, −0.305, −0.6134, −0.5075 | ||
CPFAAWP-G | 0.3584, 0.2584, 0.4723, 0.2188, 0.2145 | ||
CPFAAP-A | −0.4718, −0.609, −0.3056, −0.6139, −0.5082 | ||
CPFAAP-G | 0.3577, 0.2577, 0.4714, 0.2182, 0.2139 | ||
CPFAAWP-A | −0.4703, −0.6077, −0.3038, −0.6125, −0.5062 | ||
CPFAAWP-G | 0.3576, 0.2577, 0.4714, 0.218, 0.2133 | ||
CPFAAP-A | −0.4709, −0.6082, −0.3045, −0.6131, −0.507 | ||
CPFAAP-G | 0.357, 0.2571, 0.4707, 0.2176, 0.213 | ||
CPFAAWP-A | −0.4693, −0.6069, −0.3027, −0.6117, −0.5049 | ||
CPFAAWP-G | 0.3568, 0.257, 0.4705, 0.2172, 0.2122 | ||
CPFAAP-A | −0.4697, −0.6072, −0.303, −0.612, −0.5054 | ||
CPFAAP-G | 0.3561, 0.2563, 0.4696, 0.2166, 0.2116 | ||
CPFAAWP-A | −0.4679, −0.6057, −0.301, −0.6104, −0.5031 | ||
CPFAAWP-G | 0.3557, 0.2559, 0.4693, 0.2161, 0.2107 |
Paramete | Operato | Score Value | Ranking Value |
---|---|---|---|
CPFAAP-A | −0.3628, −0.4324, −0.2816, −0.4369, −0.3642 | ||
CPFAAP-G | 0.2731, 0.2115, 0.329, 0.2028, 0.1433 | ||
CPFAAWP-A | −0.3619, −0.4315, −0.2807, −0.4361, −0.3629 | ||
CPFAAWP-G | 0.2733, 0.2116, 0.3292, 0.2029, 0.143 | ||
CPFAAP-A | −0.3622, −0.4318, −0.2809, −0.4364, −0.3632 | ||
CPFAAP-G | 0.2726, 0.211, 0.3285, 0.2024, 0.1424 | ||
CPFAAWP-A | −0.3612, −0.4308, −0.2799, −0.4355, −0.3617 | ||
CPFAAWP-G | 0.2727, 0.211, 0.3286, 0.2024, 0.142 | ||
CPFAAP-A | −0.3616, −0.4311, −0.2803, −0.4358, −0.3622 | ||
CPFAAP-G | 0.2721, 0.2105, 0.328, 0.2019, 0.1416 | ||
CPFAAWP-A | −0.3605, −0.4301, −0.2791, −0.4349, −0.3606 | ||
CPFAAWP-G | 0.2721, 0.2105, 0.328, 0.2018, 0.1409 | ||
CPFAAP-A | −0.361, −0.4305, −0.2796, −0.4353, −0.3613 | ||
CPFAAP-G | 0.2716, 0.21, 0.3274, 0.2014, 0.1407 | ||
CPFAAWP-A | −0.3598, −0.4294, −0.2783, −0.4343, −0.3595 | ||
CPFAAWP-G | 0.2716, 0.2099, 0.3274, 0.2013, 0.14 | ||
CPFAAP-A | −0.36, −0.4297, −0.2786, −0.4346, −0.3599 | ||
CPFAAP-G | 0.271, 0.2093, 0.3267, 0.2008, 0.1395 | ||
CPFAAWP-A | −0.3588, −0.4285, −0.2772, −0.4334, −0.358 | ||
CPFAAWP-G | 0.2708, 0.2091, 0.3265, 0.2005, 0.1386 |
Method | Score Value | Ranking Value |
---|---|---|
Senapati et al. [14] | ||
Senapati et al. [15] | ||
Sarfraz et al. [16] | ||
Senapati [19] | ||
Naeem et al. [20] | ||
Mahmood et al. [21] | ||
Xu [22] | ||
Jiang et al. [23] | ||
Rani and Garg [24] | ||
Liu et al. [25] | 0.0573, −0.1095, 0.2574, −0.1429, −0.076 | |
CPFAAP-A | −0.4731, −0.61, −0.3071, −0.6151, −0.5099 | |
CPFAAP-G | 0.3587, 0.2586, 0.4726, 0.2193, 0.2154 | |
CPFAAWP-A | −0.4717, −0.6089, −0.3056, −0.6139, −0.5082 | |
CPFAAWP-G | 0.3588, 0.2587, 0.4728, 0.2192, 0.2151 |
Methods | Score Values | Ranking Values |
---|---|---|
Senapati et al. [14] | ||
Senapati et al. [15] | ||
Sarfraz et al. [16] | ||
Senapati [19] | −0.3588,−0.4285,−0.2772,−0.4334,−0.358 | |
Naeem et al. [20] | −0.3588,−0.4285,−0.2772,−0.4334,−0.358 | |
Mahmood et al. [21] | ||
Xu [22] | ||
Jiang et al. [23] | ||
Rani and Garg [24] | ||
Liu et al. [25] | 0.0429, −0.0571, 0.143, −0.0572, −0.0569 | |
CPFAAP-A | −0.3625, −0.4321, −0.2813, −0.4366, −0.3637 | |
CPFAAP-G | 0.2729, 0.2112, 0.3288, 0.2026, 0.1429 | |
CPFAAWP-A | −0.3616, −0.4311, −0.2803, −0.4358, −0.3623 | |
CPFAAWP-G | 0.273, 0.2113, 0.3289, 0.2026, 0.1425 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Ali, Z.; Mahmood, T.; Yang, M.-S. Aczel–Alsina Power Aggregation Operators for Complex Picture Fuzzy (CPF) Sets with Application in CPF Multi-Attribute Decision Making. Symmetry 2023, 15, 651. https://doi.org/10.3390/sym15030651
Ali Z, Mahmood T, Yang M-S. Aczel–Alsina Power Aggregation Operators for Complex Picture Fuzzy (CPF) Sets with Application in CPF Multi-Attribute Decision Making. Symmetry. 2023; 15(3):651. https://doi.org/10.3390/sym15030651
Chicago/Turabian StyleAli, Zeeshan, Tahir Mahmood, and Miin-Shen Yang. 2023. "Aczel–Alsina Power Aggregation Operators for Complex Picture Fuzzy (CPF) Sets with Application in CPF Multi-Attribute Decision Making" Symmetry 15, no. 3: 651. https://doi.org/10.3390/sym15030651
APA StyleAli, Z., Mahmood, T., & Yang, M.-S. (2023). Aczel–Alsina Power Aggregation Operators for Complex Picture Fuzzy (CPF) Sets with Application in CPF Multi-Attribute Decision Making. Symmetry, 15(3), 651. https://doi.org/10.3390/sym15030651