Schweizer-Sklar T-norms Research Articles

Waste Reduction and Recycling: Schweizer-Sklar Aggregation Operators Based on Neutrosophic Fuzzy Rough Sets and Their Application in Green Supply Chain Management

Green supply chain management (GSCM) is a valuable application that is used to reduce the overall environmental impact of the supply chain. Waste reduction and recycling are crucial components of sustainable technique that aims to reduce ecological impact and encourage reserve effectiveness. In this manuscript, we initiate the technique of Schweizer-Sklar (SS) operational laws based on neutrosophic fuzzy rough (NFR) values for SS t-norm (SSTN) and SS t-conorm (SSTCN). Further, we derive the NFR SS weighted averaging (NFRSSWA) operator and the NFR SS weighted geometric (NFRSSWG) operator. Some basic properties for the above-initiated techniques are derived. Additionally, we describe the application in green supply chain management, called waste reeducation and recycling based on initiated operators in multi-attribute decision-making (MADM) problems. Finally, we illustrate an example for comparing the ranking values of the proposed techniques with the ranking values of the existing technique to enhance the worth of the derived theory.

Schweizer-Sklar power aggregation operators based on complex intuitionistic fuzzy information and their application in decision-making

In 1960, Schweizer and Sklar introduced the novel Schweizer-Sklar t-norm and t-conorm which is used in the construction of aggregation operators. Schweizer-Sklar norms are more general than algebraic norms and Einstein norms. Additionally, computing the power operators based on the Schweizer-Sklar norms for complex Atanassov intuitionistic fuzzy (CA-IF) set is very awkward and complicated. In this manuscript, firstly, we propose the Schweizer-Sklar operational laws for CA-IF values, and secondly, we develop the CA-IF Schweizer-Sklar power averaging (CA-IFSSPA) operator, CA-IF Schweizer-Sklar power ordered averaging (CA-IFSSPOA) operator, CA-IF Schweizer-Sklar power geometric (CA-IFSSPG) operator, and CA-IF Schweizer-Sklar power ordered geometric (CA-IFSSPOG) operator. Some suitable and dominant properties for the above operators are also discussed. Furthermore, to simplify the above operators, we develop the procedure of decision-making technique, called multi-attribute decision-making (MADM) methods based on the proposed operators based on CA-IF values. Finally, we compare the proposed methods with some existing methods to describe the efficiency and capability of the discovered approaches by some examples.

T-Spherical Fuzzy Information and Shweizer-Sklar Operations Based Maclaurin Symmetric Mean Operator and Their Applications

T-spherical fuzzy set (TSFS) is the generalization of the fuzzy set (FS) which extracts the information from the real-life scenario with certainty. Aside from the remarkable advantage of being able to account for the connections among the multi-input considerations, such as multi-attributes or multi-experts in the multi-attribute group decision-making (MAGDM), the Maclaurin symmetric mean operator (MSMO) is also the generalization of several different existing operators. Moreover, one important class of T-norms (TN) and T-conorms (TC) is the Schweizer-Sklar TN (SSTN) and TC (SSTC). In this article, the operational laws for TSFS based on SSTN and SSTC are introduced first. Then the introduced operations are used to develop a class of aggregation operators (AOs) to aggregate the information in the form of the T-spherical fuzzy values (TSFVs). The introduced operators in this article are the T-spherical fuzzy Schweizer-Sklar MSMO (TSFSSMSMO) and the T-spherical fuzzy Schweizer-Sklar weighted MSMO (TSFSSWMSMO). Further, the TSFSSMSMO and TSFWMSMO are applied to a specific multi-attribute group decision-making (MAGDM) problem to show the significance of the developed operators.

A novel multiple attribute decision-making method based on Schweizer-Sklar t-norm and t-conormwith q-rung dual hesitant fuzzy information

A novel multiple attribute decision-making method based on Schweizer-Sklar t-norm and t-conormwith q-rung dual hesitant fuzzy information

Prioritized aggregation operators based on Schweizer-Sklar t-norm for linear Diophantine fuzzy sets and their application in green sustainable chain

Schweizer and Sklar invented the idea of the Schweizer-Sklar t-norm and Schweizer-Sklar t-conorm in 1960. Additionally, prioritized aggregation operators were derived by Yager in 2008. Further, linear Diophantine fuzzy (LDF) information is a most recent generalization of q-rung orthopair, Pythagorean, and intuitionistic fuzzy sets. In this article, we invent the Schweizer-Sklar operational laws for LDF set based on the Schweizer-Sklar t-norm and t-conorm. Furthermore, we expose the LDF Schweizer-Sklar prioritized averaging (LDFSSPoA) operator, LDF Schweizer-Sklar weighted prioritized averaging (LDFSSWPoA) operator, LDF Schweizer-Sklar prioritized geometric (LDFSSPoG) operator, LDF Schweizer-Sklar weighted prioritized geometric (LDFSSWPoG) operator and evaluate their fundamental properties and results. Moreover, after evaluation of these operators, we evaluate the problem of the green sustainable chain to enhance the worth of the invented theory. Finally, we show the supremacy and accuracy of the proposed theory with the help of a comparative analysis between the proposed techniques with some existing techniques.

Measuring efficiency of retrieval algorithms with Schweizer-Sklar information aggregation

The task of an information retrieval system (IRS) is to retrieve relevant information from large datasets efficiently. However, selecting the best algorithm for an IRS is a complex decision support system (DSS) problem. The cubic m-polar fuzzy set (CmPFS) model is proposed to address the multi-polarity issue during algorithm selection and to express multi-polar information under m-fuzzy intervals. In this study, we introduce the Schweizer-Sklar (SS) aggregation operator based on CmPFS and develop various new aggregation operators using CmPFS and SS t-norm and t-conorm. New operations such as addition, usual multiplication, and power of two or more Cubic m-polar fuzzy Numbers (CmPFNs) based on SS t-norm and t-conorm are also defined. We also define score and accuracy functions to determine the priorities of alternatives. Four operators are proposed as follows: Cubic m-polar fuzzy Schweizer-Sklar weighted average (CmPFSSWA), Cubic m-polar fuzzy Schweizer-Sklar ordered weighted average (CmPOWA), Cubic m-polar fuzzy Schweizer-Sklar weighted geometric (CmPFSSWG), and Cubic m-polar fuzzy Schweizer-Sklar ordered weighted geometric (CmPFSSOWG). The desired qualities of these operators are explored and used to solve the DSS problem based on CmPFS. Finally, a practical application of an IRS is presented to demonstrate the effectiveness and reliability of the proposed operators.

Digital technology implementation and impact of artificial intelligence based on bipolar complex fuzzy Schweizer–Sklar power aggregation operators

Digital technology refers to any technology that uses digital signals or electronic data to process, store, and transmit information. Some examples of digital technologies include social media platforms, cloud computing, artificial intelligence, virtual and augmented reality, and blockchain technology. Digital technology has the potential to play a significant role in achieving sustainable development goals by providing solutions for a wide range of environmental, social, and economic challenges. In this manuscript, we investigate digital technology implementation under sustainable development and would find which area of sustainable development is most in need of digital technology. Further, we investigate the operational laws based on Schweizer–Sklar t-norm and t-conorm and originate aggregation operators based on these deduced operational laws under the environment of bipolar complex fuzzy set that is bipolar complex fuzzy Schweizer–Sklar power averaging, bipolar complex fuzzy Schweizer–Sklar power weighted averaging, bipolar complex fuzzy Schweizer–Sklar power geometric and bipolar complex fuzzy Schweizer–Sklar power weighted geometric operators and then we deduce techniques of decision-making utilizing these originated operators. Afterward, we tackle a numerical example related to the digital technology implementation under sustainable development by considering artificial data and finding the area of sustainable development which is most in need of digital technology. Moreover, we reveal the impact of one of the digital technologies that are artificial intelligence in the field of healthcare and study a numerical example by considering hypothetical data by employing the originated technique of decision-making. At the last, we do a comparison of the deduced operators with numerous current operators to reveal the superiority and benefits of the deduced operators.

Multi-attribute decision-making method based on q-rung orthopair probabilistic hesitant fuzzy schweizer-sklar power weighted hamy mean operator.

In order to further improve the computing power of the information aggregation operator in the q-rung orthopair probabilistic hesitant fuzzy environment, this paper proposes a multi-attribute decision-making method based on the q-rung orthopair probabilistic hesitant fuzzy Schweizer-Sklar power weighted Hamy mean operator. Firstly, the algorithm of q-rung orthopair probabilistic hesitant fuzzy set is improved based on the Schweizer-Sklar T-norm. In order to better reflect the degree of hesitation of decision-making experts, a new q-rung orthopair probabilistic hesitant fuzzy distance measure is proposed, which provides a basis for subsequent power weighted calculations. Furthermore, considering the correlation between attributes and the influence of data extremes, some information aggregation operators and their power weighted forms are proposed. Finally, a multi-attribute decision-making model based on the q-rung orthopair probabilistic hesitant fuzzy Schweizer-Sklar power weighted Hamy mean operator is established, and the reliability and validity of the research content in this paper are verified through decision-making examples and comparative analysis.

Multiple attribute decision-making based on Fermatean fuzzy number

<abstract> <p>Multiple attribute decision-making concerns with production significant in our everyday life. To resolve the problems that decision makers might feel uncertain to choose the suitable assessment values among several conceivable ideals in the procedure. Fuzzy model, and its extensions are extensively applied to MADM problems. In this study, we proposed an innovative Schweizer-Sklar t-norm and t-conorm operation of FFNs, Fermatean fuzzy Schweizer-Sklar operators. They were used as a framework for the development of an MCDM method, which was illustrated by an example to demonstrate its effectiveness and applicability. Finally, a complete limitation study, rational examination, and comparative analysis of the presented approaches has been exhibited, we originate that our technique is superior in offering DMs a better decision-making choice and reducing the restrictions on stating individual partialities.</p> </abstract>

A Novel MAGDM Method Based on Hesitant Picture Fuzzy Schweizer-Sklar Maclaurin Symmetric Mean Operators and Their Application.

Multiple attribute group decision making (MAGDM) issues play important roles in our daily life. In order to solve the problem that decision makers (DMs) may feel hesitant to select the appropriate evaluation values from several possible values in the process of providing evaluations, fuzzy theory and its extensions are widely applied in MAGDM problems. In this study, we first proposed hesitant picture fuzzy sets (HPFSs), which is a combination of the hesitant fuzzy set and picture fuzzy set. Subsequently, we introduced a novel Schweizer–Sklar t-norm and t-conorm operation rules of HPFSs and proposed a family of hesitant picture fuzzy Schweizer–Sklar Maclaurin symmetric mean operators. To show the application procedure of the proposed method to practical MAGDM issues, a numerical example about enterprise informatization level evaluation was employed to elaborate the calculation process with the proposed method. Finally, through the parameter analysis, validity analysis, and comparative analysis with some existing methods, we found that our method is more superior in providing DMs a greater decision-making freedom and relaxing the constraints on expressing personal preferences. This study provides a general framework of the proposed method to MAGDM problems under hesitant picture fuzzy environment, which enriches the fuzzy theory and its applications.

Applications of Cubic Schweizer–Sklar Power Heronian Mean to Multiple Attribute Decision‐Making

The cubic set (CS) is a basic simplification of several fuzzy notions, including fuzzy set (FS), interval‐valued FS (IVFS), and intuitionistic FS (IFS). By the degrees of IVFS and FS, CS exposes fuzzy judgement, and this is a much more potent mathematical approach for dealing with information that is unclear, ambiguous, or indistinguishable. The article provides many innovative operational laws for cubic numbers (CNs) drawn on the Schweizer–Sklar (SS) t‐norm (SSTN), and the SS t‐conorm (SSTCN), as well as several desired properties of these operational laws. We also plan to emphasise on the cubic Schweizer–Sklar power Heronian mean (CSSPHM) operator, as well as the cubic Schweizer–Sklar power geometric Heronian mean (CSSPGHM) operator, in order to maintain the supremacy of the power aggregation (PA) operators that seize the complications of the unsuitable information and Heronian mean (HM) operators that contemplate the interrelationship between the input data being aggregated. A novel multiple attribute decision‐making (MADM) model is anticipated for these freshly launched aggregation operators (AOs). Finally, a numerical example of enterprise resource planning is used to validate the approach’s relevance and usefulness. There is also a comparison with existing decision‐making models.

A Novel Approached Based on T-Spherical Fuzzy Schweizer-Sklar Power Heronian Mean Operator for Evaluating Water Reuse Applications under Uncertainty

The T-Spherical Fuzzy set (T-SPHFS) is one of the core simplifications of quite a lot of fuzzy concepts such as fuzzy set (FS), intuitionistic fuzzy set (ITFS), picture fuzzy set (PIFS), Q-rung orthopair fuzzy set (Q-RUOFS), etc. T-SPHFS reveals fuzzy judgment by the degree of positive membership, degree of abstinence, degree of negative membership, and degree of refusal with relaxed conditions, and this is a more powerful mathematical tool to pair with inconsistent, indecisive, and indistinguishable information. In this article, several novel operational laws for T-SPFNs based on the Schweizer–Sklar t-norm (SSTN) and the Schweizer–Sklar t-conorm (SSTCN) are initiated, and some desirable characteristics of these operational laws are investigated. Further, maintaining the dominance of the power aggregation (POA) operators that confiscate the ramifications of the inappropriate data and Heronian mean (HEM) operators that consider the interrelationship among the input information being aggregated, we intend to focus on the T-Spherical fuzzy Schweizer–Sklar power Heronian mean (T-SPHFSSPHEM) operator, the T-Spherical fuzzy Schweizer–Sklar power geometric Heronian mean (T-SPHFSSPGHEM) operator, the T-Spherical fuzzy Schweizer–Sklar power weighted Heronian mean (T-SPHFSSPWHEM) operator, the T-Spherical fuzzy Schweizer–Sklar power weighted geometric Heronian mean (T-SPHFSSPWGHEM) operator, and their core properties and exceptional cases in connection with the parameters. Additionally, deployed on these newly initiated aggregation operators (AOs), a novel multiple attribute decision making (MADM) model is proposed. Then, the initiated model is applied to the City of Penticton (British Columbia, Canada) to select the best choice among the accessible seven water reuse choices to manifest the practicality and potency of the preferred model and a comparison with the proffered models is also particularized.

The q-Rung Orthopair Hesitant Fuzzy Uncertain Linguistic Aggregation Operators and Their Application in Multi-Attribute Decision Making

This paper combines the q-rung orthopair hesitant fuzzy sets (q-ROHFSs) with the uncertain linguistic variables, and proposes the q-rung orthopair hesitant fuzzy uncertain linguistic sets (q-ROHFULSs). In addition, the Schweizer-Sklar T-norm is introduced, and a multi-attribute decision-making method based on the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar aggregation operators is established. Firstly, based on the Schweizer-Sklar T-norm, the operational properties of q-rung orthopair hesitant fuzzy uncertain linguistic elements are defined, and the score function, accuracy function and ranking method of the q-rung orthopair hesitant fuzzy uncertain linguistic elements are proposed. Secondly, the q-rung orthopair hesitant fuzzy uncertain linguistic Schweizer-Sklar Bonferroni mean (BM) operator and geometric Bonferroni mean (GBM) operator, Maclaurin symmetric mean (MSM) operator and dual Maclaurin symmetric mean (DMSM) operator, Maclaurin mean (MM) operator and dual Maclaurin mean (DMM) operator are defined. The calculation formulas of the operators are given, the related properties are studied, and the special forms of the operators are discussed. Finally, a multi-attribute decision-making model based on the q-rung orthopair hesitant fuzzy uncertain linguistic aggregation operators is established, and the feasibility and effectiveness of the decision-making method are demonstrated through calculation examples and comparative analyses.

Interval-valued fuzzy reasoning algorithms based on Schweizer–Sklar t-norms and its application

Based on the normalized Minkowski distance in Hausdorff metrics, we study the sensitivity of interval-valued Schweizer-Sklar t-norms and their corresponding residual implications. Moreover, we investigate the robustness of interval-valued fuzzy reasoning triple I algorithms based on Schweizer–Sklar operators and illustrate the feasibility of the algorithms by a numerical example. Finally, the interval-valued fuzzy reasoning triple I algorithms are applied to medical diagnosis.

Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator

Single-valued neutrosophic (SVN) sets can successfully describe the uncertainty problems, and Schweizer-Sklar (SS) t-norm (TN) and t-conorm (TCN) can build the information aggregation process more flexible by a parameter. To fully consider the advantages of SVNS and SS operations, in this article, we extend the SS TN and TCN to single-valued neutrosophic numbers (SVNN) and propose the SS operational laws for SVNNs. Then, we merge the prioritized aggregation (PRA) operator with SS operations, and develop the single-valued neutrosophic Schweizer-Sklar prioritized weighted averaging (SVNSSPRWA) operator, single-valued neutrosophic Schweizer-Sklar prioritized ordered weighted averaging (SVNSSPROWA) operator, single-valued neutrosophic Schweizer-Sklar prioritized weighted geometric (SVNSSPRWG) operator, and single-valued neutrosophic Schweizer-Sklar prioritized ordered weighted geometric (SVNSSPROWG) operator. Moreover, we study some useful characteristics of these proposed aggregation operators (AOs) and propose two decision making models to deal with multiple-attribute decision making (MADM) problems under SVN information based on the SVNSSPRWA and SVNSSPRWG operators. Lastly, an illustrative example about talent introduction is given to testify the effectiveness of the developed methods.

Parameterized continuous models of fuzzy reasoning

In this paper, the ideas of universal logic is introduced into fuzzy systems. After giving the definitions of the softened fuzzy reasoning models based on Schweizer-Sklar t-norms and Schweizer-Sklar implications, i.e., α-models and β-models, we give the sufficient and necessary conditions for these models to be continuous, and discuss the continuity of some commonly used models. We also prove that when an α-model or a β-model is used as a fuzzy controller, it has universal property with respect to function approximation. The results we obtained show that α-models and β-models are more flexible than the existing models in applications.

Differential inequality conditions for dominance between continuous Archimedean t-norms

Dominance between triangular norms (t-norms) is a versatile relationship. For continuous Archimedean t-norms, dominance can be verified by checking one of many sufficient conditions derived from a generalization of the Mulholland inequality. These conditions pertain to various convexity properties of compositions of additive generators and their inverses. In this paper, assuming differentiability of these additive generators, we propose equivalent sufficient conditions that can be expressed as inequalities involving derivatives of the additive generators, avoiding the need of composing them. We demonstrate the powerfulness of the results by the straightforward rediscovery of dominance relationships in the Schweizer-Sklar t-norm family, as well as by unveiling some formerly unknown dominance relationships in the Sugeno-Weber t-norm family. Finally, we illustrate that the results can also be applied to members of different parametric families of t-norm.

A fuzzy logic system based on Schweizer-Sklar t-norm

Based on the Schweizer-Sklar t-norm, a fuzzy logic system UL* is established, and its soundness theorem and completeness theorem are proved. The following facts are pointed out: the well-known formal system SBL∼ is a semantic extension of UL*; the fuzzy logic system IMTLΔ is a special case of UL* when two negations in UL* coincide. Moreover, the connections between the system UL* and some fuzzy logic formal systems are investigated. Finally, starting from the concepts of “the strength of an ‘AND’ operator” by R.R. Yager and “the strength of fuzzy rule interaction” by T. Whalen, the essential meaning of a parameter p in UL* is explained and the use of fuzzy logic system UL* in approximate reasoning is presented.

A limit theorem for triangle functions

A wide class of triangle functions that can be interpreted as generalized pseudo-convolutions of distribution functions has been considered. In order to find out whether a nontrivial limit for the sequence of the pseudo-convolutions of distribution functions exists or not, some pseudo-Laplace-type transformations have been introduced. The main result is illustrated with a family of pseudo-convolutions based on a family of Schweizer–Sklar t-norms and another example is given with a calculation of the analytical form of the limit function.