A Novel Approached Based on T-Spherical Fuzzy Schweizer-Sklar Power Heronian Mean Operator for Evaluating Water Reuse Applications under Uncertainty
Abstract
:1. Introduction
2. Materials
2.1. Literature Review
2.2. Theoretical Fundamentals
- (1)
- In dealing with MADM challenges, T-SPHFNs outperform ITFS, PytFS, Q-RUOFS and, PIFS in displaying speculative data by detecting the positive MED, abstinence MD, and negative MED.
- (2)
- The SS operations are far more versatile and, by a variable parameter, superior to the previous procedures.
- (3)
- Fortunately, there are a variety of MADM issues in which the attributes are interrelated, and a number of current AGOs can only mitigate such situations when attributes have been in the form of actual numbers or in the form of other fuzzy structures.
- (4)
- There are currently no AGOs in place to handle the MADM issues under T-SPHF information provided on SSTNO and SSTCNO. We integrated the combination of PA and HM operators with SS operations to tackle MADM issues using T-SPF information in response to this constraint.
- (1)
- Initiating novel SS ALs for T-SPHFNs, discussing its basic properties, and deploying it on the SS ALs anticipating T-SPHFSS power Heronian mean operators, T-SPHFSS power geometric Heronian mean operators, and its weighted form.
- (2)
- Inspecting its basic properties and special cases of these initiating AOs.
- (3)
- Anticipating a MADM model deployed on these initiating AOs.
- (4)
- Applying a MADM model to assess water reuse applications.
- (5)
- Verifying the initiated approach’ effectiveness and practicality.
3. Methodology
3.1. The T-Spherical Fuzzy Set and Its Operational Laws
- If then is finer than and is designated by
- If then is superior to and is designated by
- If then is identical to and is designated by
3.2. The Power Average (POA) Operator
3.3. Heronian Mean (HEM) Operator
4. Results
4.1. Schweizer–Sklar ALs for T-SPHFNs
- (1)
- (2)
- (3)
- (4)
- (5)
- (6)
- .
4.2. The T-Spherical Fuzzy Schweizer-Sklar Power Heronian Mean Operators
The T-SPHFSSPOHEM and T-SPHFSSPOGHEM Operators
4.3. T-SPHFSSWPOHEM and T-SPHFSSWPOGHEM Operators
5. The Approach to Solve MADM Problems
6. Numerical Example
6.1. Effect of Parameters
6.1.1. Effect of the Parameters on Ranking Order Utilizing T-SPFSSWHM and T-SPFSSPWGHM Operators
6.1.2. Effect of the Parameter on Final Ranking Orders Utilizing T-SPFSSPWHM and T-SPFSSPWGHM Operators
6.1.3. Effect of the Parameter on Final Ranking Orders Utilizing T-SPFSSPWHM and T-SPFSSPWGHM Operators
6.1.4. Comparison with Existing Approaches
- (1)
- The anticipated MADM model is based on the newly initiated aggregation operators. That is, these aggregation operators are proposed utilizing SS ALs for T-SPFNs, which consist of general parameters that make the decision-making process more flexible. Meanwhile, the existing MADM models are based on the aggregation operators, which are initiated utilizing algebraic ALs.
- (2)
- The existing aggregation operators have the characteristic that they can only remove the effect of awkward data by utilizing power weight vector, while the anticipated aggregation operators have the ability to remove the effect of awkward data as well as consider the interrelationship among the input data at the same time.
- (3)
- The other advantage of the anticipated AOs is that it consists of general parameters, which make the decision-making process more flexible. Therefore, the initiated AOs are more practical and comparative in their utilization while solving MADM models under T-SPF information.
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Miller, G.W. Integrated concepts in water reuse: Managing global water needs. Desalination 2006, 187, 65–75. [Google Scholar] [CrossRef]
- Zarghami, M.; Szidarovszky, F. Stochastic-fuzzy multi criteria decision making for robust water resources management. Stoch. Environ. Res. Risk Assess. 2009, 23, 329–339. [Google Scholar] [CrossRef]
- Zadeh, L.A. Fuzzy sets. Inf. Control. 1965, 8, 338–353. [Google Scholar] [CrossRef] [Green Version]
- Mahmood, T.; Ullah, K.; Khan, Q.; Jan, N. An approach toward decision-making and medical diagnosis problems using the concept of spherical fuzzy sets. Neural Comput. Appl. 2019, 31, 7041–7053. [Google Scholar] [CrossRef]
- Xu, Z. Intuitionistic fuzzy aggregation operators. IEEE Trans. Fuzzy Syst. 2007, 15, 1179–1187. [Google Scholar]
- Xu, Z.; Yager, R.R. Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Gener. Syst. 2006, 35, 417–433. [Google Scholar] [CrossRef]
- Liu, P.; Khan, Q.; Mahmood, T.; Hassan, N. T-spherical fuzzy power Muirhead mean operator based on novel operational laws and their application in multi-attribute group decision making. IEEE Access 2019, 7, 22613–22632. [Google Scholar] [CrossRef]
- Bonferroni, C. Sulle medie multiple di potenze. Boll. dell’Unione Mat. Ital. 1950, 5, 267–270. [Google Scholar]
- Sykora, S. Mathematical Means and Averages: Generalized Heronian Means. Stan’s Libr. Castano Primo 2009, 3. [Google Scholar] [CrossRef]
- Muirhead, R.F. Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters. Proc. Edinb. Math. Soc. 1902, 21, 144–162. [Google Scholar] [CrossRef] [Green Version]
- Maclaurin, C. A second letter to Martin Folkes, Esq.; concerning the roots of equations, with the demonstration of other rules of algebra. Philos. Trans. 1729, 36, 59–96. [Google Scholar]
- Pan, Q.; Chhipi-Shrestha, G.; Zhou, D.; Zhang, K.; Hewage, K.; Sadiq, R. Evaluating water reuse applications under uncertainty: Generalized intuitionistic fuzzy-based approach. Stoch. Environ. Res. Risk Assess. 2018, 32, 1099–1111. [Google Scholar] [CrossRef]
- Phochanikorn, P.; Tan, C. A new extension to a multi-criteria decision-making model for sustainable supplier selection under an intuitionistic fuzzy environment. Sustainability 2019, 11, 5413. [Google Scholar] [CrossRef] [Green Version]
- Tseng, M.L.; Tan, P.A.; Jeng, S.Y.; Lin, C.W.R.; Negash, Y.T.; Darsono, S.N.A.C. Sustainable investment: Interrelated among corporate governance, economic performance and market risks using investor preference approach. Sustainability 2019, 11, 2108. [Google Scholar] [CrossRef] [Green Version]
- Wang, R.; Nan, G.; Chen, L.; Li, M. Channel integration choices and pricing strategies for competing dual-channel retailers. IEEE Trans. Eng. Manag. 2020, 1–15. [Google Scholar] [CrossRef]
- Mousavi-Avval, S.H.; Rafiee, S.; Mohammadi, A. Development and Evaluation of Combined Adaptive Neuro-Fuzzy Inference System and Multi-Objective Genetic Algorithm in Energy, Economic and Environmental Life Cycle Assessments of Oilseed Production. Sustainability 2021, 13, 290. [Google Scholar] [CrossRef]
- Stekelorum, R.; Laguir, I.; Gupta, S.; Kumar, S. Green supply chain management practices and third-party logistics providers’ performances: A fuzzy-set approach. Int. J. Prod. Econ. 2021, 235, 108093. [Google Scholar] [CrossRef]
- Coppolino, L.; Romano, L.; Scaletti, A.; Sgaglione, L. Fuzzy set theory-based comparative evaluation of cloud service offerings: An agro-food supply chain case study. Technol. Anal. Strateg. Manag. 2020, 1–14. [Google Scholar] [CrossRef]
- Turksen, I.B. Interval valued fuzzy sets based on normal forms. Fuzzy Sets Syst. 1986, 20, 191–210. [Google Scholar] [CrossRef]
- Atanassov, K. Intuitionistic fuzzy sets. Fuzzy sets syst. 1986, 20, 87–96. [Google Scholar] [CrossRef]
- Yager, R.R. Pythagorean fuzzy subsets. In Proceedings of the 2013 Joint IFSA World Congress and NAFIPS Annual Meeting, Edmonton, AB, Canada, 24–28 June 2013; pp. 57–61. [Google Scholar]
- Yager, R.R.; Abbasov, A.M. Pythagorean membership grades, complex numbers, and decision making. Int. J. Intell. Syst. 2013, 28, 436–452. [Google Scholar] [CrossRef]
- Yager, R.R. Pythagorean membership grades in multicriteria decision making. IEEE Trans. Fuzzy Syst. 2014, 22, 958–965. [Google Scholar] [CrossRef]
- Wei, G.; Wei, Y. Similarity measures of Pythagorean fuzzy sets based on the cosine function and their applications. Int. J. Intell. Syst. 2018, 33, 634–652. [Google Scholar] [CrossRef]
- Peng, X.; Dai, J. Approaches to Pythagorean fuzzy stochastic multi-criteria decision making based on prospect theory and regret theory with new distance measure and score function. Int. J. Intell. Syst. 2017, 3, 1187–1214. [Google Scholar] [CrossRef]
- Chen, T.Y. Remoteness index-based Pythagorean fuzzy VIKOR methods with a generalized distance measure for multiple criteria decision analysis. Inf. Fusion 2018, 41, 129–150. [Google Scholar] [CrossRef]
- Zhang, X. A novel approach based on similarity measure for Pythagorean fuzzy multiple criteria group decision making. Int. J. Intell. Syst. 2016, 31, 593–611. [Google Scholar] [CrossRef]
- Zulqarnain, R.M.; Xin, X.L.; Siddique, I.; Khan, W.A.; Yousif, M.A. TOPSIS method based on correlation coefficient under pythagorean fuzzy soft environment and its application towards green supply chain management. Sustainability 2021, 13, 1642. [Google Scholar] [CrossRef]
- Garg, H. A novel correlation coefficient between Pythagorean fuzzy sets and its applications to decision-making processes. Int. J. Intell. Syst. 2016, 31, 1234–1252. [Google Scholar] [CrossRef]
- Zhang, X.; Xu, Z. Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets. Int. J. Intell. Syst. 2014, 29, 1061–1078. [Google Scholar] [CrossRef]
- Pérez-Dominguez, L.; Durán, S.N.A.; López, R.R.; Pérez-Olguin, I.J.C.; Luviano-Cruz, D.; Gómez, J.A.H. Assessment urban transport service and Pythagorean Fuzzy Sets CODAS method: A case of study of Ciudad Juárez. Sustainability 2021, 13, 1281. [Google Scholar] [CrossRef]
- Li, X.H.; Huang, L.; Li, Q.; Liu, H.C. Passenger satisfaction evaluation of public transportation using pythagorean fuzzy MULTIMOORA method under large group environment. Sustainability 2020, 12, 4996. [Google Scholar] [CrossRef]
- Rani, P.; Mishra, A.R.; Mardani, A.; Cavallaro, F.; Štreimikienė, D.; Khan, S.A.R. Pythagorean fuzzy SWARA–VIKOR framework for performance evaluation of solar panel selection. Sustainability 2020, 12, 4278. [Google Scholar] [CrossRef]
- Yager, R.R. Generalized orthopair fuzzy sets. IEEE Trans. Fuzzy Syst. 2016, 25, 1222–1230. [Google Scholar] [CrossRef]
- Cuong, B.C.; Kreinovich, V. Picture Fuzzy Sets-a new concept for computational intelligence problems. In Proceedings of the 2013 Third World Congress on Information and Communication Technologies, WICT, Hanoi, Vietnam, 15–18 December 2013; pp. 1–6. [Google Scholar]
- Yager, R.R. The power average operator. IEEE Trans. Syst. Man Cybern. Part A Syst. Hum. 2001, 31, 724–731. [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]
- Garg, H.; Ullah, K.; Mahmood, T.; Hassan, N.; Jan, N. T-spherical fuzzy power aggregation operators and their applications in multi-attribute decision making. J. Ambient Intell. Humaniz. Comput. 2021, 1–14. [Google Scholar] [CrossRef]
- Liu, P.; You, X. Interval neutrosophic Muirhead mean operators and their applications in multiple-attribute group decision making. Int. J. Uncertain. Quant. 2017, 7, 303–334. [Google Scholar] [CrossRef]
- Liu, P.; Khan, Q.; Mahmood, T. Some single-valued neutrosophic power muirhead mean operators and their application to group decision making. J. Intell. Fuzzy Syst. 2019, 37, 2515–2537. [Google Scholar] [CrossRef]
- Li, Y.; Liu, P.; Chen, Y. Some single valued neutrosophic number Heronian mean operators and their application in multiple attribute group decision making. Informatica 2016, 27, 85–110. [Google Scholar] [CrossRef] [Green Version]
- Xu, Z.; Yager, R.R. Intuitionistic fuzzy Bonferroni means. IEEE Trans. Syst. Man Cybern. Part B 2011, 41, 568–578. [Google Scholar]
- Zhou, W.; He, J.M. Intuitionistic fuzzy geometric Bonferroni means and their application in multi-criteria decision making. Int. J. Intell. Syst. 2012, 27, 995–1019. [Google Scholar] [CrossRef]
- Ashraf, S.; Abdullah, S.; Mahmood, T. Spherical fuzzy Dombi aggregation operators and their application in group decision making problems. J. Ambient Intell. Humaniz. Comput. 2019, 1–19. [Google Scholar] [CrossRef]
- Dombi, J. A general class of fuzzy operators, the DeMorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators. Fuzzy Sets Syst. 1982, 8, 149–163. [Google Scholar] [CrossRef]
- Deschrijver, G.; Kerre, E.E. A generalization of operators on intuitionistic fuzzy sets using triangular norms and conorms. Notes Intuit. Fuzzy Sets 2002, 8, 19–27. [Google Scholar]
- Deschrijver, G. Generalized arithmetic operators and their relationship to t-norms in interval-valued fuzzy set theory. Fuzzy Sets Syst. 2009, 160, 3080–3102. [Google Scholar] [CrossRef] [Green Version]
- Zhang, X.; He, H.; Xu, Y. A fuzzy logic system based on Schweizer-Sklar t-norm. Sci. China Ser. F Inf. Sci. 2006, 49, 175–188. [Google Scholar] [CrossRef]
- Liu, P.; Wang, P. Some interval-valued intuitionistic fuzzy Schweizer–Sklar power aggregation operators and their application to supplier selection. Int. J. Syst. Sci. 2018, 49, 1188–1211. [Google Scholar] [CrossRef]
- Zhang, L. Intuitionistic fuzzy averaging Schweizer-Sklar operators based on interval-valued intuitionistic fuzzy numbers and its applications. In Proceedings of the 2018 Chinese Control and Decision Conference (CCDC), Shenyang, China, 9–11 June 2018; pp. 2194–2197. [Google Scholar]
- Wang, P.; Liu, P. Some Maclaurin symmetric mean aggregation operators based on Schweizer-Sklar operations for intuitionistic fuzzy numbers and their application to decision making. J. Intell. Fuzzy Syst. 2019, 36, 3801–3824. [Google Scholar] [CrossRef]
- Liu, P.; Khan, Q.; Mahmood, T. Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator. Cogn. Syst. Res. 2019, 57, 175–196. [Google Scholar] [CrossRef] [Green Version]
- Zhang, H.; Wang, F.; Geng, Y. Multi-Criteria Decision-Making Method Based on Single-Valued Neutrosophic Schweizer–Sklar Muirhead Mean Aggregation Operators. Symmetry 2019, 11, 152. [Google Scholar] [CrossRef] [Green Version]
- Nagarajan, D.; LathaMaheswari, M.; Broumi, S.; Kavikumar, J. A new perspective on traffic control management using triangular interval type-2 fuzzy sets and interval neutrosophic sets. Oper. Res. Perspect. 2019, 6, 100099. [Google Scholar] [CrossRef]
- Rong, Y.; Li, Q.; Pei, Z. A Novel Q-rung Orthopair Fuzzy Multi-attribute Group Decision-making Approach Based on Schweizer-sklar Operations and Improved COPRAS Method. In Proceedings of the 4th International Conference on Computer Science and Application Engineering, Sanya, China, 20–22 October 2020; pp. 1–6; Association for Computing Machinery: New York, NY, USA, 2020. [Google Scholar]
- Kapustina, I.V.; Sergeev, S.M.; Kalinina, O.V.; Vilken, V.V.; Putikhin, Y.Y.; Volkova, L.V. Developing the physical distribution digital twin model within the trade network. Acad. Strateg. Manag. J. 2021, 20, 1–24. [Google Scholar]
Authors | Different Structures of Fuzzy MADM AOs Based on SS ALs |
---|---|
“Liu and Wang” [49] (2018) | IVIFSSPWA and IVIFSSPWG operators |
“Zhang” [50] (2018) | IFSSWA operator |
“Wang and Liu” [51] (2019) | IFSSMSM operator |
“Liu et al.” [52] (2019) | SVNSSPrWA and SVNSSPrWG operators |
“Zhang et al.” [53] (2019) | SVNSSMM and SVNSSDMM operators |
“Nagarajan et al.” [54] (2019) | INSSWA and INSSWG operators |
“Rong et al.” [55] (2020) | Improved COPRAS method |
Alternatives\Attributes | ||||||
---|---|---|---|---|---|---|
Agree | Neutrality | Disagree | Low | Medium | High | |
80 | 9 | 11 | 428.8 | 536 | 643.2 | |
63.5 | 13 | 23.5 | 2624.8 | 3281 | 3937.2 | |
84.5 | 10 | 5.5 | 3192.5 | 3990.6 | 4788.8 | |
74.5 | 10 | 15.5 | 3192.5 | 3990.6 | 4788.8 | |
85.5 | 8 | 6.5 | 886.3 | 1107.9 | 1329.5 | |
88.5 | 7 | 4.5 | 361.8 | 452.3 | 542.7 | |
24 | 14 | 62 | 3192.5 | 3990.6 | 4788.8 |
Alternatives\Attributes | |||||||
---|---|---|---|---|---|---|---|
Agree | Neutrality | Disagree | Low | Medium | High | ||
1,555,358 | 1,944,198 | 2,333,038 | 7.1 × 10−12 | 7.51 × 10−12 | 8.30 × 10−12 | ||
1,637,219 | 20,46,524 | 2,455,829 | 1.83 × 10−11 | 1.89 × 10−11 | 2.03 × 10−11 | ||
834,019 | 1,042,524 | 1,251,028 | 1.78 × 10−11 | 1.84 × 10−11 | 1.99 × 10−11 | ||
146,660 | 183,326 | 219,991 | 9.07 × 10−12 | 1.0 × 10−11 | 1.26 × 10−11 | ||
635,529 | 794,411 | 953,293 | 9.34 × 10−12 | 9.77 × 10−12 | 1.07 × 10−11 | ||
78,219 | 97,774 | 117,328 | 8.43 × 10−12 | 8.87 × 10−12 | 9.83 × 10−12 | ||
1,197,674 | 1,497,092 | 1,796,511 | 2.76 × 10-8 | 4.01 × 10−8 | 1.00 × 10−7 |
Alternatives\Attributes | |||
---|---|---|---|
Alternatives\Attributes | ||
---|---|---|
Parameters | Score Values | Ranking Order |
---|---|---|
Parameters | Score Values | Ranking Order |
---|---|---|
Parameter | Score Values | Ranking Order |
---|---|---|
Parameter | Score Values | Ranking Order |
---|---|---|
Parameter | Score Values Utilizing T-SPFSSPWHM Operator | Score Values Utilizing T-SPFSSPWHM Operator | Ranking Orders |
---|---|---|---|
and | |||
and | |||
and | |||
and | |||
and |
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 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
Khan, Q.; Gwak, J.; Shahzad, M.; Alam, M.K. A Novel Approached Based on T-Spherical Fuzzy Schweizer-Sklar Power Heronian Mean Operator for Evaluating Water Reuse Applications under Uncertainty. Sustainability 2021, 13, 7108. https://doi.org/10.3390/su13137108
Khan Q, Gwak J, Shahzad M, Alam MK. A Novel Approached Based on T-Spherical Fuzzy Schweizer-Sklar Power Heronian Mean Operator for Evaluating Water Reuse Applications under Uncertainty. Sustainability. 2021; 13(13):7108. https://doi.org/10.3390/su13137108
Chicago/Turabian StyleKhan, Qaisar, Jeonghwan Gwak, Muhammad Shahzad, and Muhammad Kamran Alam. 2021. "A Novel Approached Based on T-Spherical Fuzzy Schweizer-Sklar Power Heronian Mean Operator for Evaluating Water Reuse Applications under Uncertainty" Sustainability 13, no. 13: 7108. https://doi.org/10.3390/su13137108
APA StyleKhan, Q., Gwak, J., Shahzad, M., & Alam, M. K. (2021). A Novel Approached Based on T-Spherical Fuzzy Schweizer-Sklar Power Heronian Mean Operator for Evaluating Water Reuse Applications under Uncertainty. Sustainability, 13(13), 7108. https://doi.org/10.3390/su13137108