A class of fuzzy multisets with a fixed number of memberships

A novel MADM algorithm for landfill site selection based on q-rung orthopair probabilistic hesitant fuzzy power Muirhead mean operator

With the rapid development of economy and the acceleration of urbanization, the garbage produced by urban residents also increases with the increase of population. In many big cities, the phenomenon of "garbage siege" has seriously affected the development of cities and the lives of residents. Sanitary landfill is an important way of municipal solid waste disposal. However, due to the restriction of social, environmental and economic conditions, landfill site selection has become a very challenging task. In addition, landfill site selection is full of uncertainty and complexity due to the lack of cognitive ability of decision-makers and the existence of uncertain information in the decision-making process. Therefore, a novel multi-attribute decision making method based on q-rung orthopair probabilistic hesitant fuzzy power weight Muirhead mean operator is proposed in this paper, which can solve the problem of landfill site selection well. This method uses probability to represent the hesitance of decision maker and retains decision information more comprehensively. The negative effect of abnormal data on the decision result is eliminated by using the power average operator. Muirhead mean operator is used to describe the correlation between attributes. Then, an example of landfill site selection is given to verify the effectiveness of the proposed method, and the advantages of the proposed method are illustrated by parameter analysis and comparative analysis. The results show that this method has a wider space for information expression, gives the decision maker a great degree of freedom in decision-making, and has robustness.

n-Dimensional (S,N)-implications

The n-dimensional fuzzy logic (n-DFL) has been contributed to overcome the insufficiency of traditional fuzzy logic in modeling imperfect and imprecise information, coming from different opinions of many experts by considering the possibility to model not only ordered but also repeated membership degrees. Thus, n-DFL provides a consolidated logical strategy for applied technologies since the ordered evaluations provided by decision makers impact not only by selecting the best solutions for a decision making problem, but also by enabling their comparisons. In such context, this paper studies the n-dimensional fuzzy implications (n-DI) following distinct approaches: (i) analytical studies, presenting the most desirable properties as neutrality, ordering, (contra-)symmetry, exchange and identity principles, discussing their interrelations and exemplifications; (ii) algebraic aspects mainly related to left- and right-continuity of representable n-dimensional fuzzy t-conorms; and (iii) generating n-DI from existing fuzzy implications. As the most relevant contribution, the prospective studies in the class of n-dimensional interval (S,N)-implications include results obtained from t-representable n-dimensional conorms and involutive n-dimensional fuzzy negations. And, these theoretical results are applied to model approximate reasoning of inference schemes, dealing with based-rule in n-dimensional interval fuzzy systems. A synthetic case-study illustrates the solution for a decision-making problem in medical diagnoses.

Hesitant Probabilistic Fuzzy Information Aggregation Using Einstein Operations

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 special cases are investigated. It is shown that some existing hesitant fuzzy aggregation operators and hesitant probabilistic fuzzy aggregation operators are special cases of the proposed operators. Further, based on the proposed operators, a new approach of hesitant probabilistic fuzzy multiple attribute decision making is developed. Finally, a practical example is provided to illustrate the developed approach.

L-Fuzzy Sets and Isomorphic Lattices: Are All the “New” Results Really New? †

We review several generalizations of the concept of fuzzy sets with two- or three-dimensional lattices of truth values and study their relationship. It turns out that, in the two-dimensional case, several of the lattices of truth values considered here are pairwise isomorphic, and so are the corresponding families of fuzzy sets. Therefore, each result for one of these types of fuzzy sets can be directly rewritten for each (isomorphic) type of fuzzy set. Finally we also discuss some questionable notations, in particular, those of “intuitionistic” and “Pythagorean” fuzzy sets.

Correlation in Interval-Valued Atanassov’s Intuitionistic Fuzzy Sets — Conjugate and Negation Operators

This paper studies the conjugate functions related to main connectives of the Intervalvalued Atanassov’s Intuitionistic Fuzzy Logic. The relationships among automorphism classes are formalized by the ϕ-representability theorem, passing from automorphisms to interval-valued intuitionistic automorphisms, also visiting other two ones, intuitionistic automorphisms and interval-valued automorphisms. Additionally, the ϕ-conjugate of an interval-valued Atanassov’s intuitionistic fuzzy negation can be obtained either from an interval-valued fuzzy negation or from an Atanassov’s intuitionistic fuzzy negation, including a discussion presenting such reverse constructions. The ϕ-conjugate of an interval-valued Atanassov’s intuitionistic fuzzy negation not only preserves the main properties of its corresponding fuzzy negation but also of two other ones, the intuitionistic fuzzy negation and interval-valued fuzzy negation. Moreover, an extension of the intuitionistic fuzzy index as well as the correlation coefficient is discussed in terms of fuzzy negations, by considering the Atanassov’s Intuitionistic Fuzzy Logic.