Zadeh Operator Research Articles

A Lattice Structure on Hesitant Fuzzy Sets

In this paper we deal with the lattice-compatibility between several classes of extended fuzzy sets. Concretely, we treat the problem of finding a lattice structure on set-valued fuzzy sets (SVFSs) whose restriction to interval-valued fuzzy sets (IVFSs) and (type-1) fuzzy sets (FSs) match Zadeh's classical lattice operations. A prominent approach to this problem was given by Torra by means of the so-called hesitant fuzzy sets (HFSs). Nevertheless, despite their usefulness in group decision making problems, it is well-known that Torra's operations do not produce a lattice. Here, we mend partially this handicap by giving two lattice orders. Each of them preserves one of the Torra's operations and, additionally, reduces to Zadeh's orders on FSs and on IVFSs. As a counterpart, they cannot be defined on the whole class of HFSs, or SVFSs. Finally, we provide a full answer combining both orders. We define a partial order, that we call the symmetric order, on the whole class of non-empty subsets of [0,1]. This order extends the usual ones on [0,1] and on closed intervals of [0,1]. As a consequence, we find a lattice structure on HFSs whose restriction to FSs and IVFSs reduces to Zadeh's operations.

Fuzzy parallel dynamical systems on Zadeh operators

In this paper, we describe the orbital structure of homogeneous fuzzy parallel dynamical systems on Zadeh operators. In particular, we show that, when the (global) evolution operator is either the Zadeh conjunction operator AND or the Zadeh disjunction operator OR, all the points of the system are fixed points or eventually fixed points. Specifically, such systems present infinitely many fixed points. On the other hand, when the (global) evolution operator is either the Zadeh operator NOR or the Zadeh operator NAND, the orbital structure consists of an only fixed point and infinitely many 2‐periodic points.

AHP_FCE-based high-rise building construction risk assessment

There are many uncertain factors in the construction of high-rise buildings. By identifying the risk factors existing in the construction of high-rise buildings, this paper establishes a comprehensive evaluation index system for construction risks of high-rise buildings; and the method of combining analytic hierarchy process and fuzzy comprehensive evaluation is adopted, and the respective advantages are used to establish a construction risk evaluation model; The analytic hierarchy process was used to calculate the weights of the indicators at all levels with the help of MALAB software. The fuzzy comprehensive evaluation method was used to construct the risk evaluation matrix, and the Zadeh operator was used to calculate the evaluation indicators at various levels. The risk factors in the construction process of high-rise buildings are evaluated at the second level and passed. An example verifies the practicability of the above method.

Embeddings between lattices of fuzzy sets: An application of closed-valued fuzzy sets

In this paper we deal with the problem of extending Zadeh's operators on fuzzy sets (FSs) to interval-valued (IVFSs), set-valued (SVFSs) and type-2 (T2FSs) fuzzy sets. Namely, it is known that seeing FSs as SVFSs, or T2FSs, whose membership degrees are singletons is not order-preserving. We then describe a family of lattice embeddings from FSs to SVFSs. Alternatively, if the former singleton viewpoint is required, we reformulate the intersection on hesitant fuzzy sets and introduce what we have called closed-valued fuzzy sets. This new type of fuzzy sets extends standard union and intersection on FSs. In addition, it allows handling together membership degrees of different nature as, for instance, closed intervals and finite sets. Finally, all these constructions are viewed as T2FSs forming a chain of lattices.

The Membership Relation in Fuzzy Operators

Firstly the membership relation in fuzzy operators located outside Zadeh operators be discussed and given. Secondly the membership relation in fuzzy operators located within Zadeh operators be discussed and given.

A New Specific Property of Schweizer-Sklard Operators

A new specific property of Schweizer-Sklard operators is discovered that Schweizer-Sklard operations are continued fuzzy operators. The relation between Schweizer-Sklard operators and other Fuzzy operators is analysis. The conclusion that Schweizer-Sklard operations could partly replace other fuzzy operations most in use is given. At the same time the relation between Schweizer-Sklard operations and Zadeh operations as well as probability operators is given.

Research on Four Type-2 Fuzzy Reasoning Models

The previous discussion about type-2 fuzzy reasoning mechanisms just paid attentions to Mamdani models based on Mamdani implication operator. There is no doubt that the operator is justifiable, since such an operator possesses characteristic clipping effect, i.e. limiting the consequence membership function with the truth value of the antecedent. However, Mamdani implication operator does not uphold classical logic, i.e. it is in no agreement with the classic binary interpretation of implication. All of the Kleene-Dienes operator, Lukasiewicz operator, Zadeh operator and Reichenbach operator, which derive from equivalencies of the classic binary implication, have in common the feature of generalizing classic logic. This paper investigates type-2 fuzzy reasoning models and algorithms based on these four implication operators, respectively. In the view upholding classical logic, proposed models and algorithms seem to be more natural than of the Mamdani type.

ON SOME STRUCTURES OF FUZZY NUMBERS

The operations in the set of fuzzy numbers are usually obtained by the Zadeh extension principle. But these definitions can have some disadvan- tages for the applications both by an algebraic point of view and by practical aspects. In fact the Zadeh multiplication is not distributive with respect to the addition, the shape of fuzzy numbers is not preserved by multiplication, the indeterminateness of the sum is too increasing. Then, for the applications in the Natural and Social Sciences it is important to individuate some suitable variants of the classical addition and multiplication of fuzzy numbers that have not the previous disadvantage. Here, some possible alternatives to the Zadeh operations are studied.

Fuzzy sets and sheaves. Part II:: Sheaf-theoretic foundations of fuzzy set theory with applications to algebra and topology

This essay shows that large parts of fuzzy set theory are actually subfields of sheaf theory, resp. of the theory of complete Ω -valued sets. Hence fuzzy set theory is closer to the mainstream in mathematics than many people would expect. Part II of this series of two papers explains the sheaf-theoretic basis of Zadeh's operations on fuzzy sets (with the exception of the complementation). Further, the quotient problem w.r.t. similarity relations and the quotient problem of fuzzy groups w.r.t. fuzzy congruence relations are solved. Finally, stratified fuzzy topologies are identified with internal topologies on constant sheaves.

Distributivity and Zadeh's operators

PurposeThe purpose of this paper is to study the relationship between the property of distributivity and Zadeh's operators.Design/methodology/approachThe results are established by studying the relationship between the condition of distributivity and Zadeh's operations.FindingsWhen the t‐conorm T* and t‐norm T are Zadeh's “intersection” and “union” operations, respectively, or when the pseudo‐addition ⊕ and pseudo‐multiplication ⊗ are Zadeh's “intersection” and “union” operations, respectively, a series of sufficient and necessary conditions for T* and T (or ⊕ and ⊗) to satisfy a distributivity property are obtained. These results are readily applied to semirings.Originality/valueNew theoretical results are discovered and are shown to be useful to the study of triangular operators and applications in intelligent systems.

A new t-norm

We define a new t-norm, and t-conorm, whose domain is usually a restricted subset of [0, 1] x [0, 1], with one parameter, the correlation coefficient between the membership values, which ranges from −1 to 1. We then investigate their properties showing at one extreme they equal the Zadeh operators (max, min), and at the other extreme they produce the Lukasiewicz operators (max( a + b - l, 0), min( a + b,1)).

On a possible generalization of the Bayes method of inference

The main purpose of this paper is investigation of the Bayes Formula problem for Klement's fuzzy probability measure. This measure is defined here on a family of fuzzy subsets which is, among other things, closed under complement given as an involution satisfying Zadeh's Extension Principle. Due to Ovchinnikov's result [6], we express this complement as a variable pointwise involution. Next we prove that the Bayes Formula holds for fuzzy probability measure iff it is a fuzzy P-measure, more generally defined here. Moreover, we show that there exists an automorphism between fuzzy probability space with the usual Zadeh operator of complement and fuzzy probability space with Ovchinnikov's operator of complement. So, using the Bayes method of inference, we can always replace the generalized operator of complement by Zadeh's one.