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Journal articles on the topic 'Einstein weighted averaging operator'

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Published: 5 June 2025

Last updated: 1 August 2025

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1

Rahman, Khaista, Ibrahim M. Hezam, Darko Božanić, Adis Puška, and Miloš Milovančević. "Some Logarithmic Intuitionistic Fuzzy Einstein Aggregation Operators under Confidence Level." Processes 11, no. 4 (2023): 1298. http://dx.doi.org/10.3390/pr11041298.

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The objective of this paper is to introduce some new logarithm operational laws for intuitionistic fuzzy sets. Some structure properties have been developed and based on these, various aggregation operators, namely confidence logarithmic intuitionistic fuzzy Einstein weighted geometric (CLIFEWG) operator, confidence logarithmic intuitionistic fuzzy Einstein ordered weighted geometric (CLIFEOWG) operator, confidence logarithmic intuitionistic fuzzy Einstein hybrid geometric (CLIFEHG) operator, confidence logarithmic intuitionistic fuzzy Einstein weighted averaging (CLIFEWA) operator, confidence

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Fahmi, Aliya, Fazli Amin, Florentin Smarandache, Madad Khan, and Nasruddin Hassan. "Triangular Cubic Hesitant Fuzzy Einstein Hybrid Weighted Averaging Operator and Its Application to Decision Making." Symmetry 10, no. 11 (2018): 658. http://dx.doi.org/10.3390/sym10110658.

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In this paper, triangular cubic hesitant fuzzy Einstein weighted averaging (TCHFEWA) operator, triangular cubic hesitant fuzzy Einstein ordered weighted averaging (TCHFEOWA) operator and triangular cubic hesitant fuzzy Einstein hybrid weighted averaging (TCHFEHWA) operator are proposed. An approach to multiple attribute group decision making with linguistic information is developed based on the TCHFEWA and the TCHFEHWA operators. Furthermore, we establish various properties of these operators and derive the relationship between the proposed operators and the existing aggregation operators. Fin

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Zhang, Wenkai, Xia Li, and Yanbing Ju. "Some Aggregation Operators Based on Einstein Operations under Interval-Valued Dual Hesitant Fuzzy Setting and Their Application." Mathematical Problems in Engineering 2014 (2014): 1–21. http://dx.doi.org/10.1155/2014/958927.

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We investigate the multiple attribute decision making (MADM) problems in which attribute values take the form of interval-valued dual hesitant fuzzy information. Firstly, some operational laws for interval-valued dual hesitation fuzzy elements (IVDHFEs) based on Einstein operations are developed. Then we develop some aggregation operators based on Einstein operations: the interval-valued dual hesitant fuzzy Einstein weighted averaging (IVDHFEWA) operator, interval-valued dual hesitant fuzzy Einstein ordered weighted averaging (IVDHFEOWA) operator, interval-valued dual hesitant fuzzy Einstein h

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Rahman, Khaista, Saleem Abdullah, Asad Ali, and Fazli Amin. "Pythagorean Fuzzy Einstein Hybrid Averaging Aggregation Operator and its Application to Multiple-Attribute Group Decision Making." Journal of Intelligent Systems 29, no. 1 (2018): 736–52. http://dx.doi.org/10.1515/jisys-2018-0071.

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Abstract Pythagorean fuzzy set is one of the successful extensions of the intuitionistic fuzzy set for handling uncertainties in information. Under this environment, in this paper, we introduce the notion of Pythagorean fuzzy Einstein hybrid averaging (PFEHA) aggregation operator along with some of its properties, namely idempotency, boundedness, and monotonicity. PFEHA aggregation operator is the generalization of Pythagorean fuzzy Einstein weighted averaging aggregation operator and Pythagorean fuzzy Einstein ordered weighted averaging aggregation operator. The operator proposed in this pape

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Khan, Arshad, Saleem Abdullah, Muhammad Shakeel, Faisal Khan, Noor Amin, and Jianchao Luo. "A New Ranking Methodology for Pythagorean Trapezoidal Uncertain Linguistic Fuzzy Sets Based on Einstein Operations." Symmetry 11, no. 3 (2019): 440. http://dx.doi.org/10.3390/sym11030440.

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In this article, we proposed new Pythagorean trapezoidal uncertain linguistic fuzzy aggregation information—namely, the Pythagorean trapezoidal uncertain linguistic fuzzy Einstein weighted averaging (PTULFEWA) operator, the Pythagorean trapezoidal uncertain linguistic fuzzy Einstein ordered weighted averaging (PTULFEOWA) operator, and the Pythagorean trapezoidal uncertain linguistic fuzzy Einstein hybrid weighted averaging (PTULFEHWA) operator—using the Einstein operational laws. We studied some important properties of the suggested aggregation operators and showed that the PTULFEHWA is more g

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Chen, Jinping. "An Approach to Multiple Attribute Decision Making with Triangular Intuitionistic Fuzzy Information." Journal of Computational and Theoretical Nanoscience 13, no. 10 (2016): 7285–88. http://dx.doi.org/10.1166/jctn.2016.5710.

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The aim of this paper is to investigate the multiple attribute decision making problems with triangular intuitionistic fuzzy information. Some operational laws of triangular intuitionistic fuzzy sets, score functions of triangular intuitionistic fuzzy sets are introduced. Based on these operational laws, some Einstein aggregation operators, including triangular intuitionistic fuzzy Einstein weighted averaging (TIFEWA) operator, triangular intuitionistic fuzzy Einstein ordered weighted averaging (TIFEOWA) operator and triangular intuitionistic fuzzy Einstein hybrid aggregation (TIFEHA) operator

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Riaz, Muhammad, Wojciech Sałabun, Hafiz Muhammad Athar Farid, Nawazish Ali, and Jarosław Wątróbski. "A Robust q-Rung Orthopair Fuzzy Information Aggregation Using Einstein Operations with Application to Sustainable Energy Planning Decision Management." Energies 13, no. 9 (2020): 2155. http://dx.doi.org/10.3390/en13092155.

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A q-rung orthopair fuzzy set (q-ROFS), an extension of the Pythagorean fuzzy set (PFS) and intuitionistic fuzzy set (IFS), is very helpful in representing vague information that occurs in real-world circumstances. The intention of this article is to introduce several aggregation operators in the framework of q-rung orthopair fuzzy numbers (q-ROFNs). The key feature of q-ROFNs is to deal with the situation when the sum of the qth powers of membership and non-membership grades of each alternative in the universe is less than one. The Einstein operators with their operational laws have excellent

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Rahman, Khaista, Saleem Abdullah, and Muhammad Sajjad Ali Khan. "Some Interval-Valued Pythagorean Fuzzy Einstein Weighted Averaging Aggregation Operators and Their Application to Group Decision Making." Journal of Intelligent Systems 29, no. 1 (2018): 393–408. http://dx.doi.org/10.1515/jisys-2017-0212.

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Abstract In this paper, we introduce the notion of Einstein aggregation operators, such as the interval-valued Pythagorean fuzzy Einstein weighted averaging aggregation operator and the interval-valued Pythagorean fuzzy Einstein ordered weighted averaging aggregation operator. We also discuss some desirable properties, such as idempotency, boundedness, commutativity, and monotonicity. The main advantage of using the proposed operators is that these operators give a more complete view of the problem to the decision makers. These operators provide more accurate and precise results as compared th

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9

Park, Jin, Yu Park, and Mi Son. "Hesitant Probabilistic Fuzzy Information Aggregation Using Einstein Operations." Information 9, no. 9 (2018): 226. http://dx.doi.org/10.3390/info9090226.

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In this paper, a hesitant probabilistic fuzzy multiple attribute group decision making is studied. First, some Einstein operations on hesitant probability fuzzy elements such as the Einstein sum, Einstein product, and Einstein scalar multiplication are presented and their properties are discussed. Then, several hesitant probabilistic fuzzy Einstein aggregation operators, including the hesitant probabilistic fuzzy Einstein weighted averaging operator and the hesitant probabilistic fuzzy Einstein weighted geometric operator and so on, are introduced. Moreover, some desirable properties and speci

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Rong, Yuan, Zheng Pei, and Yi Liu. "Linguistic Pythagorean Einstein Operators and Their Application to Decision Making." Information 11, no. 1 (2020): 46. http://dx.doi.org/10.3390/info11010046.

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Linguistic Pythagorean fuzzy (LPF) set is an efficacious technique to comprehensively represent uncertain assessment information by combining the Pythagorean fuzzy numbers and linguistic variables. In this paper, we define several novel essential operations of LPF numbers based upon Einstein operations and discuss several relations between these operations. For solving the LPF numbers fusion problem, several LPF aggregation operators, including LPF Einstein weighted averaging (LPFEWA) operator, LPF Einstein weighted geometric (LPFEWG) operator and LPF Einstein hybrid operator, are propounded;

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Agbodah, Kobina, and Adjei Peter Darko. "Probabilistic Linguistic Aggregation Operators Based on Einstein t-Norm and t-Conorm and Their Application in Multi-Criteria Group Decision Making." Symmetry 11, no. 1 (2019): 39. http://dx.doi.org/10.3390/sym11010039.

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One of the major problems of varied knowledge-based systems has to do with aggregation and fusion. Pang’s probabilistic linguistic term sets denotes aggregation of fuzzy information and it has attracted tremendous interest from researchers recently. The purpose of this article is to deal investigating methods of information aggregation under the probabilistic linguistic environment. In this situation we defined certain Einstein operational laws on probabilistic linguistic term elements (PLTESs) based on Einstein product and Einstein sum. Consequently, we develop some probabilistic linguistic a

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Nur Qafareeny Abdul Halim, Noor Azzah Awang, Siti Nurhidayah Yaacob, Hazwani Hashim, and Lazim Abdullah. "Rough Neutrosophic Shapley Weighted Einstein Averaging Aggregation Operator and its Application in Multi-Criteria Decision-Making Problem." Journal of Advanced Research in Applied Sciences and Engineering Technology 43, no. 2 (2024): 52–64. http://dx.doi.org/10.37934/araset.43.2.5264.

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The study of aggregation operators has played a crucial role in various decision-making methods. The primary function of the aggregation operator is to combine multiple numbers into a single value. While Einstein operators offer a compact notation and can handle complex and large datasets, they do not consider the interaction involved in determining the criteria weights or account for imprecise and indeterminate data. To overcome this limitation, this paper introduces an improved aggregation operator, the rough neutrosophic Shapley weighted Einstein averaging aggregation operator. The rough ne

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Bilal, Muhammad. "Spontaneous Symmetry Breaking in Group Decision-Making with Complex Polytopic Fuzzy System." Symmetry 17, no. 1 (2024): 34. https://doi.org/10.3390/sym17010034.

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Beginning with a symmetrical multiple-choice individual as the foundation, I develop a sociophysics model of decision-making. By simplifying the range of choices, the framework incorporates the complex Polytopic fuzzy model to capture nuanced dynamics. This approach enables a deeper analysis of decision-making processes within social systems. Decision-making problems commonly involve uncertainty and complexity, posing considerable challenges for organizations and individuals. Due to their structure and variable parameters, the Einstein t-norm (ETN) and t-conorm (ETCN) offer more elasticity tha

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Liu, Jia-Bao, Rashad Ismail, Muhammad Kamran, Esmail Hassan Abdullatif Al-Sabri, Shahzaib Ashraf, and Ismail Naci Cangul. "An optimization strategy with SV-neutrosophic quaternion information and probabilistic hesitant fuzzy rough Einstein aggregation operator." AIMS Mathematics 8, no. 9 (2023): 20612–53. http://dx.doi.org/10.3934/math.20231051.

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<abstract><p>The single valued neutrosophic probabilistic hesitant fuzzy rough Einstein aggregation operator (SV-NPHFRE-AO) is an extension of the neutrosophic probabilistic hesitant fuzzy rough set theory. It is a powerful decision-making tool that combines the concepts of neutrosophic logic, probability theory, hesitant fuzzy sets, rough sets, and Einstein aggregation operators. SV-NPHFRE-AO can be applied in many fields, including livestock decision making. Making judgments about a wide range of issues, including feed formulation, breeding program design, disease diagnostics, an

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Iampan, Aiyared, Gustavo Santos García, Muhammad Riaz, Hafiz Muhammad Athar Farid, and Ronnason Chinram. "Linear Diophantine Fuzzy Einstein Aggregation Operators for Multi-Criteria Decision-Making Problems." Journal of Mathematics 2021 (July 17, 2021): 1–31. http://dx.doi.org/10.1155/2021/5548033.

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The linear Diophantine fuzzy set (LDFS) has been proved to be an efficient tool in expressing decision maker (DM) evaluation values in multicriteria decision-making (MCDM) procedure. To more effectively represent DMs’ evaluation information in complicated MCDM process, this paper proposes a MCDM method based on proposed novel aggregation operators (AOs) under linear Diophantine fuzzy set (LDFS). A q -Rung orthopair fuzzy set ( q -ROFS), Pythagorean fuzzy set (PFS), and intuitionistic fuzzy set (IFS) are rudimentary concepts in computational intelligence, which have diverse applications in mode

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Akram, Muhammad, Gulfam Shahzadi, and Abdullah Ali H. Ahmadini. "Decision-Making Framework for an Effective Sanitizer to Reduce COVID-19 under Fermatean Fuzzy Environment." Journal of Mathematics 2020 (October 29, 2020): 1–19. http://dx.doi.org/10.1155/2020/3263407.

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The purpose of this article is to develop some general aggregation operators (AOs) based on Einstein’s norm operations, to cumulate the Fermatean fuzzy data in decision-making environments. A Fermatean fuzzy set (FFS), possessing the more flexible structure than the intuitionistic fuzzy set (IFS) and Pythagorean fuzzy set (PFS), is a competent tool to handle vague information in the decision-making process by the means of membership degree (MD) and nonmembership degree (NMD). Our target is to empower the AOs using the theoretical basis of Einstein norms for the FFS to establish some advantageo

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Du, Wen Sheng. "A further investigation on q-rung orthopair fuzzy Einstein aggregation operators." Journal of Intelligent & Fuzzy Systems 41, no. 6 (2021): 6655–73. http://dx.doi.org/10.3233/jifs-210548.

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Aggregation of q-rung orthopair fuzzy information serves as an important branch of the q-rung orthopair fuzzy set theory, where operations on q-rung orthopair fuzzy values (q-ROFVs) play a crucial role. Recently, aggregation operators on q-ROFVs were established by employing the Einstein operations rather than the algebraic operations. In this paper, we give a further investigation on operations and aggregation operators for q-ROFVs based on the Einstein operational laws. We present the operational principles of Einstein operations over q-ROFVs and compare them with those built on the algebrai

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Qiyas, Muhammad, Darjan Karabasevic, Neelam Khan, and Srdjan Maričić. "Einstein Exponential Operational Laws Based on Fractional Orthotriple Fuzzy Sets and Their Applications in Decision Making Problems." Mathematics 12, no. 20 (2024): 3216. http://dx.doi.org/10.3390/math12203216.

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The fractional orthotriple fuzzy set (FOFS) model is a recently created extension of fuzzy sets (FS) for coping with ambiguity in DM. The purpose of this study is to define new exponential and Einstein exponential operational (EO) laws for fractional orthotriple fuzzy sets and the aggregation procedures that accompany them. We present the operational laws for exponential and Einstein exponential FOFSs which have crisp numbers as base values and fractional orthotriple fuzzy numbers as exponents (weights). The proposed operations’ qualities and characteristics are then explored. Based on the def

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Fahmi, Aliya, Fazli Amin, Madad Khan, and Florentin Smarandache. "Group Decision Making Based on Triangular Neutrosophic Cubic Fuzzy Einstein Hybrid Weighted Averaging Operators." Symmetry 11, no. 2 (2019): 180. http://dx.doi.org/10.3390/sym11020180.

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In this paper, a new concept of the triangular neutrosophic cubic fuzzy numbers (TNCFNs), their score and accuracy functions are introduced. Based on TNCFNs, some new Einstein aggregation operators, such as the triangular neutrosophic cubic fuzzy Einstein weighted averaging (TNCFEWA), triangular neutrosophic cubic fuzzy Einstein ordered weighted averaging (TNCFEOWA) and triangular neutrosophic cubic fuzzy Einstein hybrid weighted averaging (TNCFEHWA) operators are developed. Furthermore, their application to multiple-attribute decision-making with triangular neutrosophic cubic fuzzy (TNCF) inf

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Riaz, Muhammad, Hafiz Muhammad Athar Farid, Humaira Kalsoom, Dragan Pamučar, and Yu-Ming Chu. "A Robust q-Rung Orthopair Fuzzy Einstein Prioritized Aggregation Operators with Application towards MCGDM." Symmetry 12, no. 6 (2020): 1058. http://dx.doi.org/10.3390/sym12061058.

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A q-rung orthopair fuzzy set (q-ROFS) provides a significant mechanism for managing symmetrical aspects in real life circumstances. The renowned distinguishing feature of q-ROFS is that the sum of the qth powers to each membership degree (MD) and non-membership degree (NMD) is less than or equal 1, and therefore the comprehensive uncertain space for q-ROF information is broader. Numerous researchers have suggested several aggregation operators based on q-ROFSs. In order to discuss prioritized relationship in the criterion and a smooth approximation of q-ROF information, we introduced q-rung or

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Kamran, Muhammad, Shahzaib Ashraf, Nadeem Salamat, Muhammad Naeem, and Thongchai Botmart. "Cyber security control selection based decision support algorithm under single valued neutrosophic hesitant fuzzy Einstein aggregation information." AIMS Mathematics 8, no. 3 (2022): 5551–73. http://dx.doi.org/10.3934/math.2023280.

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<abstract><p>The single-valued neutrosophic hesitant fuzzy set (SV-NHFS) is a hybrid structure of the single-valued neutrosophic set and the hesitant fuzzy set that is designed for some incomplete, uncertain, and inconsistent situations in which each element has a few different values designed by the truth membership hesitant function, indeterminacy membership hesitant function, and falsity membership hesitant function. A strategic decision-making technique can help the decision-maker accomplish and analyze the information in an efficient manner. However, in our real lives, uncerta

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Ali, Zeeshan, Tahir Mahmood, Kifayat Ullah, Dragan Pamucar, and Goran Cirovic. "Power Aggregation Operators Based on t-Norm and t-Conorm under the Complex Intuitionistic Fuzzy Soft Settings and Their Application in Multi-Attribute Decision Making." Symmetry 13, no. 11 (2021): 1986. http://dx.doi.org/10.3390/sym13111986.

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Multi-attribute decision-making (MADM) is commonly used to investigate fuzzy information effectively. However, selecting the best alternative information is not always symmetric because the alternatives do not have complete information, so asymmetric information is often involved. In this analysis, we use the massive dominant and more consistent principle of power aggregation operators (PAOs) based on general t-norm and t-conorm, which manage awkward and inconsistent data in real-world dilemmas such as medical diagnosis, pattern recognition, cleaner production evaluation in gold mines, the ana

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Zulqarnain, Rana Muhammad, Imran Siddique, Fahd Jarad, Y. S. Hamed, Khadijah M. Abualnaja, and Aiyared Iampan. "Einstein Aggregation Operators for Pythagorean Fuzzy Soft Sets with Their Application in Multiattribute Group Decision-Making." Journal of Function Spaces 2022 (March 31, 2022): 1–21. http://dx.doi.org/10.1155/2022/1358675.

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The Pythagorean fuzzy soft set (PFSS) is the most proficient and manipulative leeway of the Pythagorean fuzzy set (PFS), which contracts with parameterized values of the alternatives. It is a generalized form of the intuitionistic fuzzy soft set (IFSS), which provides healthier and more accurate evaluations through decision-making (DM). The main determination of this research is to prolong the idea of Einstein’s aggregation operators for PFSS. We introduce the Einstein operational laws for Pythagorean fuzzy soft numbers (PFSNs). Based on Einstein operational laws, we construct two novel aggreg

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Farid, Hafiz Muhammad Athar, Muhammad Riaz, Vladimir Simic, and Xindong Peng. "q-Rung orthopair fuzzy dynamic aggregation operators with time sequence preference for dynamic decision-making." PeerJ Computer Science 10 (January 31, 2024): e1742. http://dx.doi.org/10.7717/peerj-cs.1742.

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The q-rung orthopair fuzzy set (q-ROPFS) is a kind of fuzzy framework that is capable of introducing significantly more fuzzy information than other fuzzy frameworks. The concept of combining information and aggregating it plays a significant part in the multi-criteria decision-making method. However, this new branch has recently attracted scholars from several domains. The goal of this study is to introduce some dynamic q-rung orthopair fuzzy aggregation operators (AOs) for solving multi-period decision-making issues in which all decision information is given by decision makers in the form of

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Veeramachaneni, Sireesha, and Anusha V. "Selection of Optimal E-Learning Tool with Type-2 Intuitionistic Fuzzy Einstein Interactive Weighted Aggregation Operator." International Journal of Fuzzy System Applications 11, no. 1 (2022): 1–17. http://dx.doi.org/10.4018/ijfsa.312242.

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Not only are daily life activities being disrupted by the COVID-19 pandemic, but so are educational systems. To some extent, encouraging the use of e-learning technology has helped to stabilize the situation. The suitable selection of the appropriate e-learning platform for the institution depends upon different criteria with uncertain information. As type 2 intuitionistic fuzzy (T2IF) sets are conceptually intriguing and they provide a lot of expressive potential for dealing with uncertainty in expert knowledge, this work investigates the best e-learning tool for higher education in a type 2

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Attaullah, Shahzaib Ashraf, Noor Rehman, Hussain AlSalman, and Abdu H. Gumaei. "A Decision-Making Framework Using q-Rung Orthopair Probabilistic Hesitant Fuzzy Rough Aggregation Information for the Drug Selection to Treat COVID-19." Complexity 2022 (February 4, 2022): 1–37. http://dx.doi.org/10.1155/2022/5556309.

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In our current era, a new rapidly spreading pandemic disease called coronavirus disease (COVID-19), caused by a virus identified as a novel coronavirus (SARS-CoV-2), is becoming a crucial threat for the whole world. Currently, the number of patients infected by the virus is expanding exponentially, but there is no commercially available COVID-19 medication for this pandemic. However, numerous antiviral drugs are utilized for the treatment of the COVID-19 disease. Identification of the appropriate antivirus medicine to treat the infection of COVID-19 is still a complicated and uncertain decisio

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Mahmood, Tahir, Izatmand, Zeeshan Ali, et al. "Linear Diophantine Uncertain Linguistic Power Einstein Aggregation Operators and Their Applications to Multiattribute Decision Making." Complexity 2021 (August 5, 2021): 1–25. http://dx.doi.org/10.1155/2021/4168124.

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Linear Diophantine uncertain linguistic set (LDULS) is a modified variety of the fuzzy set (FS) to manage problematic and inconsistent information in actual life troubles. LDULS covers the grade of truth, grade of falsity, and their reference parameters with the uncertain linguistic term (ULT) with a rule 0 ≤ α AMG u AMG x + β ANG v AMG x ≤ 1 , where 0 ≤ α AMG + β ANG ≤ 1 . In this study, the principle of LDULS and their useful laws are elaborated. Additionally, the power Einstein (PE) aggregation operator (AO) is a conventional sort of AO utilized in innovative decision-making troubles, which

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Zulqarnain, Rana Muhammad, Imran Siddique, Shahzad Ahmad, et al. "Pythagorean Fuzzy Soft Einstein Ordered Weighted Average Operator in Sustainable Supplier Selection Problem." Mathematical Problems in Engineering 2021 (November 30, 2021): 1–16. http://dx.doi.org/10.1155/2021/2559979.

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Pythagorean fuzzy soft set (PFSS) is the most influential and operative extension of the Pythagorean fuzzy set (PFS), which contracts with the parametrized standards of the substitutes. It is also a generalized form of the intuitionistic fuzzy soft set (IFSS) and delivers a well and accurate estimation in the decision-making (DM) procedure. The primary purpose is to prolong and propose ideas related to Einstein’s ordered weighted aggregation operator from fuzzy to PFSS, comforting the condition that the sum of the degrees of membership function and nonmembership function is less than one and t

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Kausar, Rukhsana, Hafiz Muhammad Athar Farid, Muhammad Riaz, and Darko Božanić. "Cancer Therapy Assessment Accounting for Heterogeneity Using q-Rung Picture Fuzzy Dynamic Aggregation Approach." Symmetry 14, no. 12 (2022): 2538. http://dx.doi.org/10.3390/sym14122538.

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Due to the fact that there is no symmetry in the division of cancer cells, it is important to consider this asymmetrical behavior. Because of this heterogeneity during any therapy, not every cancer cell that is killed only is abolished, which is sensitive to the particular treatment chosen. Mathematical models that describe these pathways are critical for predicting cancer cell proliferation behavior. The literature on the mathematical modeling of cancer onset, growth, and metastasis is extensive. Both deterministic and stochastic factors were used to develop mathematical models to mimic the d

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Quek, Selvachandran, Munir, et al. "Multi-Attribute Multi-Perception Decision-Making Based on Generalized T-Spherical Fuzzy Weighted Aggregation Operators on Neutrosophic Sets." Mathematics 7, no. 9 (2019): 780. http://dx.doi.org/10.3390/math7090780.

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The framework of the T-spherical fuzzy set is a recent development in fuzzy set theory that can describe imprecise events using four types of membership grades with no restrictions. The purpose of this manuscript is to point out the limitations of the existing intuitionistic fuzzy Einstein averaging and geometric operators and to develop some improved Einstein aggregation operators. To do so, first some new operational laws were developed for T-spherical fuzzy sets and their properties were investigated. Based on these new operations, two types of Einstein aggregation operators are proposed na

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Aruchsamy, Revathy, Inthumathi Velusamy, Prasantha Bharathi Dhandapani, and Taha Radwan. "Einstein Aggregation Operator Technique in Circular Fermatean Fuzzy Environment for MCDM." Symmetry 16, no. 9 (2024): 1243. http://dx.doi.org/10.3390/sym16091243.

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An Ethernet cable enables users to connect their electronic devices, such as smartphones, computers, routers, laptops, etc., to a network that permits them to utilize the internet. Additionally, it transfers broadband signals among connected devices. Wi-Fi is tremendously helpful with small, handheld gadgets, but if capacity is required, cable Ethernet connectivity cannot be surpassed. Ethernet connections typically work faster than Wi-Fi connections; they also tend to be more flexible, have fewer interruptions, can handle problems rapidly, and have a cleaner appearance. However, it becomes co

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Al-Sabri, Esmail Hassan Abdullatif, Muhammad Rahim, Fazli Amin, et al. "Multi-criteria decision-making based on Pythagorean cubic fuzzy Einstein aggregation operators for investment management." AIMS Mathematics 8, no. 7 (2023): 16961–88. http://dx.doi.org/10.3934/math.2023866.

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<abstract> <p>Pythagorean cubic fuzzy sets (PCFSs) are a more advanced version of interval-valued Pythagorean fuzzy sets where membership and non-membership are depicted using cubic sets. These sets offer a greater amount of data to handle uncertainties in the information. However, there has been no previous research on the use of Einstein operations for aggregating PCFSs. This study proposes two new aggregator operators, namely, Pythagorean cubic fuzzy Einstein weighted averaging (PCFEWA) and Pythagorean cubic fuzzy Einstein ordered weighted averaging (PCFEOWA), which extend the c

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Abdullah, Saleem, Alaa O. Almagrabi, and Ihsan Ullah. "A New Approach to Artificial Intelligent Based Three-Way Decision Making and Analyzing S-Box Image Encryption Using TOPSIS Method." Mathematics 11, no. 6 (2023): 1559. http://dx.doi.org/10.3390/math11061559.

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In fuzzy artificial intelligent decision support systems, three-way intelligent-decision making (TWIDM) has played a very important role in ranking objects under the double hierarchy linguistic variable (DHLV). The 8 × 8 S-boxes are very important for image encryption in secure communication. Therefore, the aim of the present study is to develop a new approach to artificial intelligent three-way decision making via DHLV and apply it to S-box image encryption. Artificial intelligent based three-way decision-making problems with double hierarchy hesitant linguistic terms are developed. The first

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Rahman, K., A. Ali, S. Abdullah, and F. Amin. "Approaches to Multi-Attribute Group Decision Making Based on Induced Interval-Valued Pythagorean Fuzzy Einstein Aggregation Operator." New Mathematics and Natural Computation 14, no. 03 (2018): 343–61. http://dx.doi.org/10.1142/s1793005718500217.

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Interval-valued Pythagorean fuzzy set is one of the successful extensions of the interval-valued intuitionistic fuzzy set for handling the uncertainties in the data. Under this environment, in this paper, we introduce the notion of induced interval-valued Pythagorean fuzzy Einstein ordered weighted averaging (I-IVPFEOWA) aggregation operator. Some of its desirable properties namely, idempotency, boundedness, commutatively, monotonicity have also been proved. The main advantage of using the proposed operator is that this operator gives a more complete view of the problem to the decision-makers.

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Aslam, Muhammad, and Aliya Fahmi. "New work of trapezoidal cubic linguistic uncertain fuzzy Einstein hybrid weighted averaging operator and decision making." Soft Computing 24, no. 5 (2019): 3331–54. http://dx.doi.org/10.1007/s00500-019-04096-y.

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Kumar, Kamal, and Shyi-Ming Chen. "Multiattribute decision making based on the improved intuitionistic fuzzy Einstein weighted averaging operator of intuitionistic fuzzy values." Information Sciences 568 (August 2021): 369–83. http://dx.doi.org/10.1016/j.ins.2021.03.020.

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Rahman, Khaista, Saleem Abdullah, Asad Ali, and Fazli Amin. "Interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging aggregation operator and their application to group decision making." Complex & Intelligent Systems 5, no. 1 (2018): 41–52. http://dx.doi.org/10.1007/s40747-018-0076-x.

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Jamil, Muhammad, Saleem Abdullah, Muhammad Yaqub Khan, Florentin Smarandache, and Fazal Ghani. "Application of the Bipolar Neutrosophic Hamacher Averaging Aggregation Operators to Group Decision Making: An Illustrative Example." Symmetry 11, no. 5 (2019): 698. http://dx.doi.org/10.3390/sym11050698.

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The present study aims to introduce the notion of bipolar neutrosophic Hamacher aggregation operators and to also provide its application in real life. Then neutrosophic set (NS) can elaborate the incomplete, inconsistent, and indeterminate information, Hamacher aggregation operators, and extended Einstein aggregation operators to the arithmetic and geometric aggregation operators. First, we give the fundamental definition and operations of the neutrosophic set and the bipolar neutrosophic set. Our main focus is on the Hamacher aggregation operators of bipolar neutrosophic, namely, bipolar neu

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Seikh, Mijanur Rahaman, and Utpal Mandal. "q-Rung Orthopair Fuzzy Archimedean Aggregation Operators: Application in the Site Selection for Software Operating Units." Symmetry 15, no. 9 (2023): 1680. http://dx.doi.org/10.3390/sym15091680.

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The q-rung orthopair fuzzy (q-ROF) set is an efficient tool for dealing with uncertain and inaccurate data in real-world multi-attribute decision-making (MADM). In MADM, aggregation operators play a significant role. The majority of well-known aggregation operators are formed using algebraic, Einstein, Hamacher, Frank, and Yager t-conorms and t-norms. These existing t-conorms and t-norms are some special cases of Archimedean t-conorms (ATCNs) and Archimedean t-norms (ATNs). Therefore, this article aims to extend the ATCN and ATN operations under the q-ROF environment. In this paper, firstly, w

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ABDULLAH, SITI ROHANA GOH, and Muhammad Zaini Ahmad. "Multi-Attribute Decision-Making Based on Picture Fuzzy Einstein Operator and The TOPSIS Method." Applied Mathematics and Computational Intelligence (AMCI) 12, no. 4 (2023): 122–39. http://dx.doi.org/10.58915/amci.v12i4.314.

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Picture Fuzzy Sets (PFSs) denote the extension of conventional fuzzy sets, which capture a broader spectrum of human opinions, encompassing responses such as acceptance, neutrality, rejection, and hesitation. This wider range of responses cannot be accurately accommodated within fuzzy sets as well as intuitionistic fuzzy sets framework. In the realm of Multiple Attribute Group Decision-Making (MAGDM) methods, attributes frequently exhibit conflicts, uncertainties, imprecisions, as well as a lack of commensurability. To tackle the complexities inherent in MAGDM, the Technique for Order of Prefe

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Wang, Weize, and Xinwang Liu. "Interval-valued intuitionistic fuzzy hybrid weighted averaging operator based on Einstein operation and its application to decision making." Journal of Intelligent & Fuzzy Systems 25, no. 2 (2013): 279–90. http://dx.doi.org/10.3233/ifs-120635.

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Riaz, Muhammad, Hafiz Muhammad Athar Farid, Weiwei Wang, and Dragan Pamucar. "Interval-Valued Linear Diophantine Fuzzy Frank Aggregation Operators with Multi-Criteria Decision-Making." Mathematics 10, no. 11 (2022): 1811. http://dx.doi.org/10.3390/math10111811.

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We introduce the notion of the interval-valued linear Diophantine fuzzy set, which is a generalized fuzzy model for providing more accurate information, particularly in emergency decision-making, with the help of intervals of membership grades and non-membership grades, as well as reference parameters that provide freedom to the decision makers to analyze multiple objects and alternatives in the universe. The accuracy of interval-valued linear Diophantine fuzzy numbers is analyzed using Frank operations. We first extend the Frank t-conorm and t-norm (FTcTn) to interval-valued linear Diophantin

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Ali, Zeeshan, Tahir Mahmood, and Gustavo Santos-García. "Heronian Mean Operators Based on Novel Complex Linear Diophantine Uncertain Linguistic Variables and Their Applications in Multi-Attribute Decision Making." Mathematics 9, no. 21 (2021): 2730. http://dx.doi.org/10.3390/math9212730.

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In this manuscript, we combine the notion of linear Diophantine fuzzy set (LDFS), uncertain linguistic set (ULS), and complex fuzzy set (CFS) to elaborate the notion of complex linear Diophantine uncertain linguistic set (CLDULS). CLDULS refers to truth, falsity, reference parameters, and their uncertain linguistic terms to handle problematic and challenging data in factual life impasses. By using the elaborated CLDULSs, some operational laws are also settled. Furthermore, by using the power Einstein (PE) aggregation operators based on CLDULS: the complex linear Diophantine uncertain linguisti

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Fahmi, Aliya, Saleem Abdullah, Fazli Amin, and M. Sajjad Ali Khan. "Trapezoidal cubic fuzzy number Einstein hybrid weighted averaging operators and its application to decision making." Soft Computing 23, no. 14 (2018): 5753–83. http://dx.doi.org/10.1007/s00500-018-3242-6.

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Yasin, Yasir, Muhammad Riaz, and Kholood Alsager. "Synergy of machine learning and the Einstein Choquet integral with LOPCOW and fuzzy measures for sustainable solid waste management." AIMS Mathematics 10, no. 1 (2025): 460–98. https://doi.org/10.3934/math.2025022.

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<p>Solid waste management (SWM) protects public health, the environment, and limited resources in densely populated and urbanized countries such as Singapore. This work presents an advanced framework for optimizing SWM using advanced mathematical models and decision-making techniques, including the circular $ q $-rung orthopair fuzzy set (C$ q $-ROFS) for data, combined with the Choquet integral (CI) and logarithmic percentage change-driven objective weighting (LOPCOW) methods, enhanced by the aggregation operators (AOs) circular $ q $-rung orthopair fuzzy Einstein Choquet integral weigh

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Garg, Harish, and Gagandeep Kaur. "Algorithm for Probabilistic Dual Hesitant Fuzzy Multi-Criteria Decision-Making Based on Aggregation Operators With New Distance Measures." Mathematics 6, no. 12 (2018): 280. http://dx.doi.org/10.3390/math6120280.

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Probabilistic dual hesitant fuzzy set (PDHFS) is an enhanced version of a dual hesitant fuzzy set (DHFS) in which each membership and non-membership hesitant value is considered along with its occurrence probability. These assigned probabilities give more details about the level of agreeness or disagreeness. By emphasizing the advantages of the PDHFS and the aggregation operators, in this manuscript, we have proposed several weighted and ordered weighted averaging and geometric aggregation operators by using Einstein norm operations, where the preferences related to each object is taken in ter

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Sen, Supriyan, Laxminarayan Sahoo, and Sumanta Lal Ghosh. "Lifetime Extension of Wireless Sensor Networks by Perceptive Selection of Cluster Head Using K-Means and Einstein Weighted Averaging Aggregation Operator under Uncertainty." Journal of Industrial Intelligence 2, no. 1 (2024): 54–62. http://dx.doi.org/10.56578/jii020105.

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Ashraf, Shahzaib, Noor Rehman, Azmat Hussain, Hussain AlSalman, and Abdu H. Gumaei. "q-Rung Orthopair Fuzzy Rough Einstein Aggregation Information-Based EDAS Method: Applications in Robotic Agrifarming." Computational Intelligence and Neuroscience 2021 (October 30, 2021): 1–27. http://dx.doi.org/10.1155/2021/5520264.

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The main purpose of this manuscript is to present a novel idea on the q-rung orthopair fuzzy rough set (q-ROFRS) by the hybridized notion of q-ROFRSs and rough sets (RSs) and discuss its basic operations. Furthermore, by utilizing the developed concept, a list of q-ROFR Einstein weighted averaging and geometric aggregation operators are presented which are based on algebraic and Einstein norms. Similarly, some interesting characteristics of these operators are initiated. Moreover, the concept of the entropy and distance measures is presented to utilize the decision makers’ unknown weights as w

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Abosuliman, Shougi S., Abbas Qadir, and Saleem Abdullah. "Multi criteria group decision (MCGDM) for selecting third-party logistics provider (3PL) under Pythagorean fuzzy rough Einstein aggregators and entropy measures." AIMS Mathematics 8, no. 8 (2023): 18040–65. http://dx.doi.org/10.3934/math.2023917.

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<abstract> <p>In real life, with the trend of outsourcing logistics activities, choosing a third-party logistics (3PL) provider has become an inevitable choice for shippers. One of the most difficult decisions logistics consumers are facing the selecting the 3PL provider that best meets their needs. Decision making (DM) is an important in dealing with such situations because it allows them to make reliable decisions in a short period of time, as incorrect decisions can result in huge financial losses. In this regard, this article provides a new multi criteria group decision making

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PEI, ZHENG, LI ZOU, and LIANGZHONG YI. "A LINGUISTIC AGGREGATION OPERATOR INCLUDING WEIGHTS FOR LINGUISTIC VALUES AND EXPERTS IN GROUP DECISION MAKING." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 21, no. 06 (2013): 927–43. http://dx.doi.org/10.1142/s0218488513500426.

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Different linguistic aggregation methods have been proposed and applied in the linguistic decision making problems. Generally, weights for experts or criteria are considered in linguistic aggregation processes. In this paper, we provide a method to discovery new forms to compute weights and new interpretations in the linguistic ordered weighted averaging operator. In linguistic decision analysis, it can be noticed that some of initial linguistic values used by experts have priority over others linguistic values in evaluation processes. We formalize the priority over initial linguistic values a

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