T-norm Research Articles

Additive generators of fuzzy operations in the form of linear fractional functions

This paper is devoted to the analysis of additive generators in the form of linear fractional functions. The limitations on the coefficients of increasing and decreasing generators are defined. It is shown that the compositions of increasing and/or decreasing generators are also generators. For each case, the corresponding triangular norms or conorms are found. It is established that the triangular norms and conorms obtained on the basis of linear fractional functions, and which are also dual to them, have the same structure. In fact, a new family of dual triangular norms and conorms is proposed.

Fuzzy Phonetic Encoding of Speech Signals in Voice Processing Systems

In this paper, we studied the phonetic approach for voice processing. A method for automatic recognition of speech signals, in which each quasistationary segment is associated with a fuzzy set of phonemes, was developed. We proposed the operation of the probabilistic triangular norm for fuzzy sets corresponding to the input frame and the nearest reference phoneme. The developed method was experimentally shown to allow a 1.5–5% reduction in the probability of erroneous recognition in comparison with known analogues.

The cross-migrativity with respect to continuous triangular norms revisited

The structures of continuous cross-migrative triangular norms with respective to some continuous triangular norm have been studied, and only partial results were obtained. This paper is devoted to presenting a complete characterization of the class of continuous triangular norms, being cross-migrative with respect to some continuous triangular norm.

A new structure for uninorms on bounded lattices

This article investigates the construction of uninorms on bounded lattices. First, by using the fact that triangular norms (t-norms) and triangular conorms (t-conorms) on an arbitrary bounded lattice always exist, we introduce a new construction of uninorms on arbitrary bounded lattices with the neutral element. Then we illustrate how our new construction method is different from some existing methods for the construction of uninorms on bounded lattices. Some illustrative examples for the construction of uninorms on bounded lattices are provided.

Uninorms Having an Identity and an Annihilator on Bounded Lattices

Sınırlı kafesler üzerinde tanımlanan uninormlar, üçgensel normların ve üçgensel konormların önemli bir genelleştirmesidir ve bu operatörler, birimin sınırlı kafesin herhangi bir noktasında olmasına olanak sağlarlar. Bu çalışmada, sınırlı kafesler üzerinde uninormlar konusu üzerine çalışılmıştır. Sınırlı kafesler üzerinde bir birime ve sıfırlayana sahip uninormları karakterize etmek için bir metot önerildi ve bu şekildeki uninormların bazı temel özellikleri araştırıldı. Ayrıca, önerilen metotun uygulanabilirliğini göstermek için bir örnek verildi.

Analytical Structures of Takagi-Sugeno Fuzzy Two-Input Two-Output Proportional-Integral/Proportional-Derivative Controllers With Multiple Fuzzy Sets

In this paper, an attempt is made to generalize the analytical structures of Takagi-Sugeno (TS) fuzzy two-input two-output (TITO) proportional-integral (PI)/proportional-derivative (PD) controllers using multiple input fuzzy sets. Two models of fuzzy TITO PI/PD controllers are proposed based on two distinct control strategies. The inputs are fuzzified by multiple fuzzy sets with trapezoidal/triangular membership functions. The generalized rule base consists of nine control rules imbibing the complete control strategy and is closer in spirit to the original TS rule base. Algebraic product (AP) triangular norm, bounded sum (BS) triangular conorm, and center of gravity (CoG) defuzzifier are applied to derive the models. The models of the fuzzy TITO PI/PD controllers with multiple input fuzzy sets are (nonlinear) variable gain/structure controllers. Also, each output of the fuzzy controller is the sum of two nonlinear PI/PD controllers with variable gains. The gain variation and properties of the proposed controllers are studied. Two examples of nonlinear dynamic processes are considered to demonstrate the applicability of the proposed controllers.

Fuzzy [formula omitted]-covering based [formula omitted]-fuzzy rough set models and applications to multi-attribute decision-making

Multi-attribute decision-making (MADM) can be regarded as a process of selecting the optimal one from all alternatives. Traditional MADM problems with fuzzy information are mainly focused on a fundamental tool which is a fuzzy binary relation. However some complicated problems cannot be effectively solved by a fuzzy relation. For this reason, in order to solve these issues, we set forth two decision-making methods that are stated in terms of novel and flexible fuzzy rough set models. For the purpose of defining these models we employ a fuzzy implication operator I and a triangular norm T. With these adaptable tools we design four kinds of fuzzy β-coverings based (I,T)-fuzzy rough set models. The elements that make these models different are the combination of fuzzy β-neighborhoods that intervene in the definitions of the lower (upper) approximations. Then, we discuss the relationships among these given four types of models. Finally, we propose two novel methodologies to solve MADM problems with evaluation of fuzzy information, which rely on these models. Through the analysis of the ranking results of these two methods, we observe that the optimal selected alternative is the same, which means that these two decision-making methods are reasonable. In addition, by comparing the ranking results of these two methods and the existing traditional methods (WA operator and TOPSIS), we observe that our proposed methods can solve the ranking problems that traditional methods cannot solve, which means that our proposed methods are superior to traditional methods.

Analytical structures and stability analysis of the simplest Takagi-Sugeno fuzzy two-term controllers

This paper deals with the simplest Takagi-Sugeno fuzzy two-term controllers. The analytical structures of the simplest fuzzy controllers were developed using a modified rule base. The number of tuning parameters of the controllers is reduced by introducing a novel rule base consisting of two rules. These controllers are termed as 'the simplest' since minimal (two) number of fuzzy sets is used for fuzzification. Algebraic product (AP)/minimum (min) triangular norm, bounded sum (BS)/maximum (max) triangular co-norm, different universes of discourses (UoDs) of inputs, and centre of gravity (CoG) defuzzification method are chosen to derive the mathematical models of the fuzzy controllers. The simplest fuzzy controller with modified rule base is equivalent to a (nonlinear) variable gain/structure PI/PD controller. The BIBO stability of the closed loop control system is investigated using the small gain theorem. The gain of the controller either varies or remains constant in different regions of the input plane. The gain variations and the computational burden of the fuzzy controllers are also studied. Two examples of nonlinear dynamical systems are considered to validate the developed models of the fuzzy controllers.

Multiattribute Group Decision-Making Approach With Linguistic Pythagorean Fuzzy Information

The purpose of this study is to construct the multi-attribute group decision making (MAGDM) approach with linguistic Pythagorean fuzzy information (LPFI) based on generalized linguistic Pythagorean fuzzy aggregation operators (GLPFA). To begin with, we define the generalized indeterminacy degree-preference distance of linguistic Pythagorean fuzzy numbers (LPFNs), on the basis of it, we build a new approach for ranking the alternatives after analysing the existed comparison rule. In addition, we introduce the new version of t-norms (TNs) and t-conorms (TCs) named linguistic Pythagorean t-norms (LPTNs) and linguistic Pythagorean t-conorms (LPTCs), which can be used to handle the LPFI; some special cases for LPTNs and LPTCs are obtained and they can deal with Pythagorean fuzzy information (PFI). Thirdly, we introduce the generalized linguistic Pythagorean fuzzy average aggregation operator (GLPFAA) based on LPTN and LPTC along with their properties are also investigated, whilst, some special cases of GLPAA are obtained when LPTN and LPTC take some special TNs and TCs. Finally, a MAGDM approach based on some LPTNs and LPTCs is constructed to deal with some MAGDM problems with unknown attributes'weights and experts' weights, before building the MAGDM approach, we define new cross-entropy to fix the experts's weights and use the maximizing deviation to calculate the attributes' weights based on the proposed indeterminacy degree-preference distance. Consequently, an illustrative example is provided in order to show the effectiveness and advantages of the proposed method and some comparisons are also carried out.

A new perspective on traffic control management using triangular interval type-2 fuzzy sets and interval neutrosophic sets

Controlling traffic flow on roads is an important traffic management task necessary to ensure a peaceful and safe environment for people. The number of cars on roads at any given time is always unknown. Type-2 fuzzy sets and neutrosophic sets play a vital role in dealing efficiently with such uncertainty. In this paper, a triangular interval type-2 Schweizer and Sklar weighted arithmetic (TIT2SSWA) operator and a triangular interval type-2 Schweizer and Sklar weighted geometric (TIT2SSWG) operator based on Schweizer and Sklar triangular norms have been studied, and the validity of these operators has been checked using a numerical example and extended to an interval neutrosophic environment by proposing interval neutrosophic Schweizer and Sklar weighted arithmetic (INSSWA) and interval neutrosophic Schweizer and Sklar weighted geometric (INSSWG) operators. Furthermore, their properties have been examined; some of the more important properties are examined in detail. Moreover, we proposed an improved score function for interval neutrosophic numbers (INNs) to control traffic flow that has been analyzed by identifying the junction that has more vehicles. This improved score function uses score values of triangular interval type-2 fuzzy numbers (TIT2FNs) and interval neutrosophic numbers.

SISO Intuitionistic Fuzzy Systems: IF-${t}$ -Norm, IF-${R}$ -Implication, and Universal Approximators

Fuzzy system, as the main method of approximate reasoning, has been widely used to solve practical problems. The existing results on fuzzy systems were based on fuzzy sets, fuzzy if-then rule, and fuzzy inference. In this work, we investigate the component of the intuitionistic fuzzy systems, such as the intuitionistic fuzzy triangular norms (IF-t-norm), the intuitionistic fuzzy residual implications (IF-R-implication), the intuitionistic fuzzy if-then rule, the intuitionistic fuzzy inference, and the intuitionistic defuzzification. Accordingly, the single input single output (SISO) intuitionistic fuzzy systems are built. Some examples are paid to illustrate the methods and their application. Moreover, the approximation properties of the intuitionistic Mamdani, the intuitionistic Larsen, the IF-t-norm, and the intuitionistic triple-I fuzzy systems are obtained, which show that the SISO intuitionistic fuzzy systems are the universal approximators.

Intuitionistic Fuzzy Measures of Correlation Coefficient of Intuitionistic Fuzzy Numbers Under Weakest Triangular Norm

The correlation coefficient of variables has wide applications in statistics and is often calculated in crisp or fuzzy environment. This article extends the application of correlation coefficient to intuitionistic fuzzy environment. In this article, a new method is proposed to measure the correlation coefficient of intuitionistic fuzzy numbers using weakest triangular norm based intuitionistic fuzzy arithmetic operations. Different from previous studies, the correlation coefficient computed in this article is an intuitionistic fuzzy number rather than a crisp or fuzzy number. It is well known that the weakest t-norm arithmetic operations effectively reduce fuzzy spreads (fuzzy intervals) and provide more exact results. Therefore, a simplified, effective and exact method based on weakest t-norm arithmetic operations is presented to compute the correlation coefficient of intuitionistic fuzzy numbers. To illustrate the proposed method, the correlation coefficient between the technology level and management achievement from a sample of 15 machinery firms in Taiwan is calculated using proposed approach.

Some operations of fuzzy sets in UP-algebras with respect to a triangular norm

This paper aim is to apply the notions of the intersection and the union of any fuzzy set to UP-algebras. We investigate properties of the intersection and the union of T-fuzzy UP-subalgebras, T-fuzzy near UP-filters, T-fuzzy UP-filters, T-fuzzy UPideals, T-fuzzy strongly UP-ideals, anti-T-fuzzy UP-subalgebras, and anti-T-fuzzy near UP-filters of UP-algebras.

Prediction of cancer using customised fuzzy rough machine learning approaches.

This Letter proposes a customised approach for attribute selection applied to the fuzzy rough quick reduct algorithm. The unbalanced data is balanced using synthetic minority oversampling technique. The huge dimensionality of the cancer data is reduced using a correlation-based filter. The dimensionality reduced balanced attribute gene subset is used to compute the final minimal reduct set using a customised fuzzy triangular norm operator on the fuzzy rough quick reduct algorithm. The customised fuzzy triangular norm operator is used with a Lukasiewicz fuzzy implicator to compute the fuzzy approximation. The customised operator selects the least number of informative feature genes from the dimensionality reduced datasets. Classification accuracy using leave-one-out cross validation of 94.85, 76.54, 98.11, and 99.13% is obtained using a customised function for Lukasiewicz triangular norm operator on leukemia, central nervous system, lung, and ovarian datasets, respectively. Performance analysis of the conventional fuzzy rough quick reduct and the proposed method are performed using parameters such as classification accuracy, precision, recall, F-measure, scatter plots, receiver operating characteristic area, McNemar test, chi-squared test, Matthew's correlation coefficient and false discovery rate that are used to prove that the proposed approach performs better than available methods in the literature.

Sufficient Conditions for Triangular Norms Preserving ⊗-Convexity

The convexity in triangular norm (for short, ⊗−convexity) is a generalization of Zadeh’s quasiconvexity. The aggregation of two ⊗−convex sets is under the aggregation operator ⊗ is also ⊗−convex, but the aggregation operator ⊗ is not unique. To solve it in complexity, in the present paper, we give some sufficient conditions for aggregation operators preserve ⊗−convexity. In particular, when aggregation operators are triangular norms, we have that several results such as arbitrary triangular norm preserve ⊗ D − convexity and ⊗ a − convexity on bounded lattices, ⊗ M preserves ⊗ H − convexity in the real unite interval [ 0 , 1 ] .

Covering-based generalized IF rough sets with applications to multi-attribute decision-making

Multi-attribute decision-making (MADM) can be regarded as a process of selecting the optimal one from all objects. Traditional MADM problems with intuitionistic fuzzy (IF) information are mainly focused on an IF binary relation. However, some complicated problems can not be effectively solved by an IF relation. In order to solve these issues, we set forth two novel decision-making methods that are stated in terms of novel and flexible generalized IF rough set models. For defining these models, four types of IF neighborhoods based on an IF implicator I and an IF triangular norm T are firstly proposed. Secondly, by means of four types of IF neighborhoods, four types of coverings-based generalized IF rough set models are proposed. Furthermore, the relationships among the four types of coverings-based generalized IF rough set models and other types of IF rough set models are also discussed. By means of the principle of the IF-TOPSIS methods, MADM with IF information based on covering-based generalized IF rough sets or based on covering-based generalized fuzzy rough sets is put forward. Finally, we solve MADM problems with the evaluation of IF information based on covering-based generalized IF rough set models. By comparative analysis, we find that the results of this method based on covering-based generalized IF rough set models and based on covering-based generalized fuzzy rough set models are highly consistent. In particular, this method based on covering-based generalized IF rough set models is more effective to deal with MADM than that based on covering-based generalized fuzzy rough set models.

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.

Spherical aggregation operators and their application in multiattribute group decision-making

Spherical fuzzy sets (SFSs) are a new extension of Cuong's picture fuzzy sets (PFSs). In SFSs, membership degrees satisfy the condition instead of as is in PFSs. In the present work, we extend different strict archimedean triangular norm and conorm to aggregate spherical fuzzy information. Firstly, we define the SFS and discuss some operational rules. Generalized spherical aggregation operators for spherical fuzzy numbers utilizing these strict Archimedean t-norm and t-conorm are proposed. Finally, based on these operators, a decision-making method has been established for ranking the alternatives by utilizing a spherical fuzzy environment. The suggested technique has been demonstrated with a descriptive example for viewing their effectiveness as well as reliability. A test checking the reliability and validity has also been conducted for viewing the supremacy of the suggested technique.

On some equation related to the distributivity laws of fuzzy implications. Jensen equation extended to the infinity

Recently, in several articles related to the distributivity laws of fuzzy implications over triangular norms and conorms, the following functional equation appeared f(min⁡(x+y,a))=min⁡(f(x)+f(y),b), where a,b are finite or infinite nonnegative constants. In our earlier papers we have considered a generalized version of this equation, i.e., the equation f(m1(x+y))=m2(f(x)+f(y)). Firstly, we analyzed the situation when both functions m1, m2 are defined on some finite intervals of R. We also investigated the situation when both functions m1, m2 have finite or infinite domains and codomains, but they satisfy several additional assumptions. In this article we consider the above equation when m1, m2 are defined on some finite or infinite intervals and satisfy only one additional assumption: m2 is injective. Our proofs are based on the Jensen equation, therefore we also present the detailed analysis of this functional equation when the domain or codomain are extended to the infinity and/or are bounded.

Some Interval Neutrosophic Dombi Power Bonferroni Mean Operators and Their Application in Multi–Attribute Decision–Making

The power Bonferroni mean (PBM) operator is a hybrid structure and can take the advantage of a power average (PA) operator, which can reduce the impact of inappropriate data given by the prejudiced decision makers (DMs) and Bonferroni mean (BM) operator, which can take into account the correlation between two attributes. In recent years, many researchers have extended the PBM operator to handle fuzzy information. The Dombi operations of T-conorm (TCN) and T-norm (TN), proposed by Dombi, have the supremacy of outstanding flexibility with general parameters. However, in the existing literature, PBM and the Dombi operations have not been combined for the above advantages for interval-neutrosophic sets (INSs). In this article, we first define some operational laws for interval neutrosophic numbers (INNs) based on Dombi TN and TCN and discuss several desirable properties of these operational rules. Secondly, we extend the PBM operator based on Dombi operations to develop an interval-neutrosophic Dombi PBM (INDPBM) operator, an interval-neutrosophic weighted Dombi PBM (INWDPBM) operator, an interval-neutrosophic Dombi power geometric Bonferroni mean (INDPGBM) operator and an interval-neutrosophic weighted Dombi power geometric Bonferroni mean (INWDPGBM) operator, and discuss several properties of these aggregation operators. Then we develop a multi-attribute decision-making (MADM) method, based on these proposed aggregation operators, to deal with interval neutrosophic (IN) information. Lastly, an illustrative example is provided to show the usefulness and realism of the proposed MADM method. The developed aggregation operators are very practical for solving MADM problems, as it considers the interaction among two input arguments and removes the influence of awkward data in the decision-making process at the same time. The other advantage of the proposed aggregation operators is that they are flexible due to general parameter.