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Radko Mesiar - One of the best experts on this subject based on the ideXlab platform.
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fuzzy number valued Triangular Norm based decomposable time stamped fuzzy measure and integration
Fuzzy Sets and Systems, 2021Co-Authors: Abbas Ghaffari, Reza Saadati, Radko MesiarAbstract:Abstract In this paper, we introduce a new concept of fuzzy measure and fuzzy integration which has a dynamic position and is different from previous approaches. Our definition of a new type of fuzzy measure deals with distance functions (special L-fuzzy numbers) and is based on continuous Triangular Norms. By this concept, we construct a new version of measure theory and integration which is more flexible since the measure of the set both depends on the set itself and on the other parameter named by time. Our approach is related to the idea of fuzzy metric spaces. We study some fuzzy measures induced by classical measures. An integral based on the introduced measures is proposed and studied, too. To complete our paper, we prove some limits and convergence theorems.
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inequalities in Triangular Norm based fuzzy l p spaces
Mathematics, 2020Co-Authors: Abbas Ghaffari, Reza Saadati, Radko MesiarAbstract:In this article, we introduce the ∗-fuzzy (L+)p spaces for 1≤p<∞ on Triangular Norm-based ∗-fuzzy measure spaces and show that they are complete ∗-fuzzy Normed space and investigate some properties in these space. Next, we prove Chebyshev’s inequality and Holder’s inequality in ∗-fuzzy (L+)p spaces.
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on the order of Triangular Norms comments on a Triangular Norm hierarchy by e cretu
Fuzzy Sets and Systems, 2002Co-Authors: Erich Peter Klement, Radko Mesiar, Endre PapAbstract:In Cretu (Fuzzy Sets and Systems 120 (2001) 371), the members of the families of Frank, Dubois-Prade, Yager, and Hamacher t-Norms, respectively, are compared (in a pointwise way) with the minimum, the product, and the Lukasiewicz t-Norm. All these results are well-known and trivial. Moreover, these families of t-Norms cannot only be compared with the three basic t-Norms above, but all these families are monotone with respect to their index, a fact which is also well known and straightforward to prove (with the exception of the family of Frank t-Norms whose monotonicity has been first proven in Butnariu and Klement, Triangular Norm-Based Measures and Games with Fuzzy Coalitions, Kluwer, Dordrecht, 1993).
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pseudo additive measures and Triangular Norm based conditioning
Annals of Mathematics and Artificial Intelligence, 2002Co-Authors: Pietro Benvenuti, Radko MesiarAbstract:Conditioning in the framework of fuzzy measures (monotone Normalized set functions vanishing in the empty set) is introduced. For every set i>B with non-null measure i>m(i>B) a conditional measure i>mi>B, based on a Triangular Norm i>T, is introduced. Universal conditioning preserving the lower semi-continuity is shown to be necessarily based on some strict Triangular Norm. Then also each conditional measure i>mi>B related to a pseudo-additive measure i>m is pseudo-additive. However, the pseudo-addition ⊕i>B operating on the measures i>mi>B is in general different from the pseudo-addition ⊕ operating on the measure i>m. Specific cases of universal conditioning preserving the pseudo-addition ⊕ are characterized. Classical probabilistic conditioning is shown to be a special case.
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operations fitting Triangular Norm based biresiduation
Fuzzy Sets and Systems, 1999Co-Authors: Radko Mesiar, Vilem NovakAbstract:Abstract Given a t-Norm T and the corresponding biresiduation ↔T. An operation K fits the latter (or shortly, is fitting) if ↔T is respected by K in all its arguments. In this paper, the structure of fitting operations is investigated. Special attention is paid to the case of basic t-Norms. The connection between the fitting property and the Lipschitz property is stressed. Some examples are given.
Erich Peter Klement - One of the best experts on this subject based on the ideXlab platform.
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on the order of Triangular Norms comments on a Triangular Norm hierarchy by e cretu
Fuzzy Sets and Systems, 2002Co-Authors: Erich Peter Klement, Radko Mesiar, Endre PapAbstract:In Cretu (Fuzzy Sets and Systems 120 (2001) 371), the members of the families of Frank, Dubois-Prade, Yager, and Hamacher t-Norms, respectively, are compared (in a pointwise way) with the minimum, the product, and the Lukasiewicz t-Norm. All these results are well-known and trivial. Moreover, these families of t-Norms cannot only be compared with the three basic t-Norms above, but all these families are monotone with respect to their index, a fact which is also well known and straightforward to prove (with the exception of the family of Frank t-Norms whose monotonicity has been first proven in Butnariu and Klement, Triangular Norm-Based Measures and Games with Fuzzy Coalitions, Kluwer, Dordrecht, 1993).
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chapter 23 Triangular Norm based measures
Handbook of Measure Theory, 2002Co-Authors: Dan Butnariu, Erich Peter KlementAbstract:This chapter describes the various aspects of Triangular Norm-based measures. Triangular Norm-based measures (T-measures) are special real valuations defined on T-tribes, the latter being classes of fuzzy sets based on a Triangular Norm. The necessary preliminaries about Triangular Norms and fuzzy sets, including concepts of disjointness and emphasizing the special role of Frank t-Norms are presented in the chapter. T-tribes are introduced and their most prominent properties are listed in the chapter. Three different representations for T-measures as integral with respect to a suitable Markov kernel, as well as a decomposition of monotone measures are presented in the chapter. The chapter discusses measures with respect to the Lukasiewicz t-Norm and some of their most important properties, which include Jordan decomposition, absolute continuity, Darboux property, and nonatomicity. The Liapounoff type theorem concerning the compactness and convexity of the range of vector measures is presented in the chapter. The existence of an Aumann–Shapley value on the space of games with fuzzy coalitions spanned by positive integer powers of monotone TL-measures is also shown in the chapter.
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a survey on different Triangular Norm based fuzzy logics
Fuzzy Sets and Systems, 1999Co-Authors: Erich Peter Klement, Mirko NavaraAbstract:Among various approaches to fuzzy logics, we have chosen two of them, which are built up in a similar way. Although starting from different basic logical connectives, they both use interpretations based on Frank t-Norms. Different interpretations of the implication lead to different axiomatizations, but most logics studied here are complete. We compare the properties, advantages and disadvantages of the two approaches.
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on some geometric transformation of t Norms
Soft Computing, 1998Co-Authors: Erich Peter Klement, Radko Mesiar, Endre PapAbstract:Given a Triangular Norm $T$, its $t$-reverse $T^*$, introduced by C. Kimberling ({\it Publ. Math. Debrecen} 20, 21-39, 1973) under the name invert, is studied. The question under which conditions we have $ T^{**} = T$ is completely solved. The $t$-reverses of ordinal sums of $t$-Norms are investigated and a complete description of continuous, self-reverse $t$-Norms is given, leading to a new characterization of the continuous $t$-Norms $T$ such that the function $ G(x,y) = x + y - T(x,y)$ is a $t$-coNorm, a problem originally studied by M.J. Frank ({\it Aequationes Math.} 19, 194-226, 1979). Finally, some open problems are formulated.
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on Triangular Norm based propositional fuzzy logics
Fuzzy Sets and Systems, 1995Co-Authors: Dan Butnariu, Erich Peter Klement, Samy ZafranyAbstract:Abstract Fuzzy logics based on Triangular Norms and their corresponding coNorms are investigated. An affirmative answer to the question whether in such logics a specific level of satisfiability of a set of formulas can be characterized by the same level of satisfiability of its finite subsets is given. Tautologies, contradictions and contigencies with respect to such fuzzy logics are studied, in particular for the important cases of min-max and Łukasiewicz logics. Finally, fundamental t-Norm-based fuzzy logics are shown to provide a gradual transition between min-max and Łukasiewicz logics.
Z Valizadeh - One of the best experts on this subject based on the ideXlab platform.
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linear fractional multi objective optimization problems subject to fuzzy relational equations with a continuous archimedean Triangular Norm
Information Sciences, 2014Co-Authors: Esmaile Khorram, Reza Ezzati, Z ValizadehAbstract:In this paper, linear fractional multi-objective optimization problems subject to a system of fuzzy relational equations (FREs) using the max-Archimedean Triangular Norm composition are considered. First, some theorems and results are presented to simplify systems of fuzzy relational equations. Next, the feasible set of the optimization problem is reduced. Then, the linear fractional multi-objective optimization problem is converted to a linear one using Nykowski and Zolkiewski's approach. Furthermore, the efficient solutions are obtained by applying the improved @e-constraint method. In addition, we design an algorithm to generate consistent systems of fuzzy relational equations, randomly, using the covering problem. Finally, the proposed method is effectively tested by solving a randomly generated consistent test problem.
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Solving nonlinear multi-objective optimization problems with fuzzy relation inequality constraints regarding Archimedean Triangular Norm compositions
Fuzzy Optimization and Decision Making, 2012Co-Authors: Esmaile Khorram, Reza Ezzati, Z ValizadehAbstract:We propose an approach to solve a nonlinear multi-objective problem subject to fuzzy relation inequalities with max-Archimedean-t-Norm composition by a genetic algorithm. The additive generator of Archimedean t-Norms is utilized to reform the existent genetic algorithm to solve the constrained nonlinear multi-objective optimization problems. We consider thoroughly the feasible set of the fuzzy relation inequality systems in three possible cases, namely “≤”, “≥” and the combination of them. In general, their feasible sets are nonconvex which are completely determined by one vector as their maximum solution and a finite number of minimal solutions. The maximum and minimal solutions are formulated by using the additive generator. Additionally, we present some conditions for each case under which the problem can be reduced. Finally, each reduced problem is solved by the genetic algorithm and the efficiency of the proposed method is shown by some numerical examples.
Esmaile Khorram - One of the best experts on this subject based on the ideXlab platform.
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linear fractional multi objective optimization problems subject to fuzzy relational equations with a continuous archimedean Triangular Norm
Information Sciences, 2014Co-Authors: Esmaile Khorram, Reza Ezzati, Z ValizadehAbstract:In this paper, linear fractional multi-objective optimization problems subject to a system of fuzzy relational equations (FREs) using the max-Archimedean Triangular Norm composition are considered. First, some theorems and results are presented to simplify systems of fuzzy relational equations. Next, the feasible set of the optimization problem is reduced. Then, the linear fractional multi-objective optimization problem is converted to a linear one using Nykowski and Zolkiewski's approach. Furthermore, the efficient solutions are obtained by applying the improved @e-constraint method. In addition, we design an algorithm to generate consistent systems of fuzzy relational equations, randomly, using the covering problem. Finally, the proposed method is effectively tested by solving a randomly generated consistent test problem.
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Solving nonlinear multi-objective optimization problems with fuzzy relation inequality constraints regarding Archimedean Triangular Norm compositions
Fuzzy Optimization and Decision Making, 2012Co-Authors: Esmaile Khorram, Reza Ezzati, Z ValizadehAbstract:We propose an approach to solve a nonlinear multi-objective problem subject to fuzzy relation inequalities with max-Archimedean-t-Norm composition by a genetic algorithm. The additive generator of Archimedean t-Norms is utilized to reform the existent genetic algorithm to solve the constrained nonlinear multi-objective optimization problems. We consider thoroughly the feasible set of the fuzzy relation inequality systems in three possible cases, namely “≤”, “≥” and the combination of them. In general, their feasible sets are nonconvex which are completely determined by one vector as their maximum solution and a finite number of minimal solutions. The maximum and minimal solutions are formulated by using the additive generator. Additionally, we present some conditions for each case under which the problem can be reduced. Finally, each reduced problem is solved by the genetic algorithm and the efficiency of the proposed method is shown by some numerical examples.
Shucherng Fang - One of the best experts on this subject based on the ideXlab platform.
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non l r type fuzzy parameters in mathematical programming problems
IEEE Transactions on Fuzzy Systems, 2014Co-Authors: Murat Adivar, Shucherng FangAbstract:The Triangular Norm-based operations in fuzzy logic usually lead to non-L-R type fuzzy sets. This study considers mathematical programming problems with non-L-R type fuzzy parameters. It shows that the fuzzy solutions to such problems can be obtained by solving an optimization problem on a mixed domain. The necessary and sufficient conditions for solving the resulting optimization problems are investigated by employing the theory of convex optimization on mixed domains. This is the first attempt to solve the fuzzy optimization problem with non-L-R type membership functions in view of optimization problems on a mixed domain.
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minimizing a linear fractional function subject to a system of sup t equations with a continuous archimedean Triangular Norm
Journal of Systems Science & Complexity, 2009Co-Authors: Pingke Li, Shucherng FangAbstract:This paper shows that the problem of minimizing a linear fractional function subject to a system of sup-T equations with a continuous Archimedean Triangular Norm T can be reduced to a 0-1 linear fractional optimization problem in polynomial time. Consequently, parametrization techniques, e.g., Dinkelbach’s algorithm, can be applied by solving a classical set covering problem in each iteration. Similar reduction can also be performed on the sup-T equation constrained optimization problems with an objective function being monotone in each variable separately. This method could be extended as well to the case in which the Triangular Norm is non-Archimedean.
