Fuzzy Multisets Research Articles

Rough approximations of multi-fuzzy sets

Rough sets have been invented to cope with uncertainty and to deal with intelligent systems characterised by insufficient and incomplete information. This paper proposes a general framework for the study of relation-based multi-fuzzy rough approximation operators. We define rough multi-fuzzy sets and study their properties. An arbitrary multi-fuzzy relation is used to approximate multi-fuzzy sets. A pair of lower and upper rough approximation operators is hence obtained and basic properties of these approximation operators will be examined. By introducing the cut sets of multi-fuzzy sets, we study the connections between multi-fuzzy rough approximation operators and crisp approximation operators. Finally, the multi-fuzzy approximation operators will be defined by axioms.

Correlation for Dual Hesitant Fuzzy Sets and Dual Interval-Valued Hesitant Fuzzy Sets

Ever since fuzzy set has been introduced, several extensions have been established, such as interval-valued fuzzy sets, Atanassov's intuitionistic fuzzy sets, interval-valued Atanassov's intuitionistic fuzzy sets, fuzzy multisets, hesitant fuzzy sets, interval-valued hesitant fuzzy sets, and dual hesitant fuzzy sets. In this contribution, we propose dual interval-valued hesitant fuzzy sets, which encompass fuzzy sets and its aforementioned extensions as special cases. Because of the importance of correlation measure in data analysis, we propose an approach for deriving the correlation coefficient of dual hesitant fuzzy sets, and then extend the approach to the dual interval-valued hesitant fuzzy set theory. We also put forward some formulas to create new correlation coefficients for fuzzy sets and its extensions in a general way. In addition, we give a practical example to illustrate the application of correlation coefficient for dual hesitant fuzzy sets in medical diagnosis.

Consensus image method for unknown noise removal

Noise removal has been, and it is nowadays, an important task in computer vision. Usually, it is a previous task preceding other tasks, as segmentation or reconstruction. However, for most existing denoising algorithms the noise model has to be known in advance. In this paper, we introduce a new approach based on consensus to deal with unknown noise models. To do this, different filtered images are obtained, then combined using multifuzzy sets and averaging aggregation functions. The final decision is made by using a penalty function to deliver the compromised image. Results show that this approach is consistent and provides a good compromise between filters.

Group multi-criteria decision making using intuitionistic multi-fuzzy sets

AbstractIn this paper we propose an efficient approach for group multi-criteria decision making (MCDM) based on intuitionistic multi-fuzzy set (IMFS). First we construct intuitionistic multi-fuzzy matrices for decision makers with respect to the criteria (attributes) of the alternatives. Based on intuitionistic multi-fuzzy matrices, we construct the aggregated intuitionistic multi-fuzzy matrix using the proposed intuitionistic multi-fuzzy weighted averaging (IMFWA) operator. Then we use Hamming distance and Euclidean distance measurements in the context of IMFS between the aggregated matrix and the specified sample matrix to reach the optimal decision. This paper also presents score function and accuracy function of IMFS with an application to MCDM. Finally, a real-life case study related to heart disease diagnosis problem is provided to illustrate the advantage of the proposed approach.

A novel fuzzy-based expert system for RET selection

The aim of this work is to demonstrate a novel fuzzy-based expert system for selecting renewable energy technologies RET. Fuzzy multi-rules and fuzzy multi-sets are used to evaluate the main operational characteristics of six types of RET fuelled by biogas from municipal solid waste MSW landfills. The construction of the fuzzy multi-rules and fuzzy multi-sets is based on the following method: Mamdani controller using the Max-Min inference process and Center of Gravity defuzzification process. Several criteria are used for the investigation: costs, efficiency, cogeneration, life-cycle and environmental impacts. The fuzzy-based expert system considers three different settings with two different constraints: costs and environmental impacts. One of the most relevant aspects presented by this work is about the previous criteria rank. It was created according to the different relevance observed among the attributes. The purpose of the proposed arrangement is to facilitate the understanding of the methodology and to increase the possibility of incorporating the decision makers' preferences on the decision-aid process. These aspects are essential to strengthen the final decision.

An Outline of the Development of the Concept of Fuzzy Multisets

 Abstract—The paper argues that the classical set-theoretic foundation for mathematics is too restrictive to be able to model a large class of real-life problems which intrinsically involve ambiguities. Further, it describes how by relaxing the restrictions of definiteness and distinctness imposed on the nature of objects to form a cantorian set, the notions of fuzzy sets and multisets respectively get introduced. Finally, it explicates the relevance of generalizing fuzzy sets to fuzzy multisets.

Correlation Measures of Dual Hesitant Fuzzy Sets

The dual hesitant fuzzy sets (DHFSs) were proposed by Zhu et al. (2012), which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as special cases. Correlation measures analysis is an important research topic. In this paper, we define the correlation measures for dual hesitant fuzzy information and then discuss their properties in detail. One numerical example is provided to illustrate these correlation measures. Then we present a direct transfer algorithm with respect to the problem of complex operation of matrix synthesis when reconstructing an equivalent correlation matrix for clustering DHFSs. Furthermore, we prove that the direct transfer algorithm is equivalent to transfer closure algorithm, but its asymptotic time complexity and space complexity are superior to the latter. Another real world example, that is, diamond evaluation and classification, is employed to show the effectiveness of the association coefficient and the algorithm for clustering DHFSs.

On Distance and Similarity Measures of Intuitionistic Fuzzy Multi Set

On Distance and Similarity Measures of Intuitionistic Fuzzy Multi Set

The multi-fuzzy soft set and its application in decision making

Molodtsov’s soft set theory was originally proposed as a general mathematical tool for dealing with uncertainty. By combining the multi-fuzzy set and soft set models, the purpose of this paper is to introduce the concept of multi-fuzzy soft sets. Some operations on a multi-fuzzy soft set are defined, such as complement operation, “AND” and “OR” operations, Union and Intersection operations. Then, the DeMorgan’s laws are proved. Finally, by means of level soft set, an algorithm is presented, and a decision problem is analyzed using multi-fuzzy soft set.

Dual Hesitant Fuzzy Sets

In recent decades, several types of sets, such as fuzzy sets, interval‐valued fuzzy sets, intuitionistic fuzzy sets, interval‐valued intuitionistic fuzzy sets, type 2 fuzzy sets, type n fuzzy sets, and hesitant fuzzy sets, have been introduced and investigated widely. In this paper, we propose dual hesitant fuzzy sets (DHFSs), which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as special cases. Then we investigate the basic operations and properties of DHFSs. We also discuss the relationships among the sets mentioned above, use a notion of nested interval to reflect their common ground, then propose an extension principle of DHFSs. Additionally, we give an example to illustrate the application of DHFSs in group forecasting.

Fuzzy two-term controllers with multi-fuzzy sets: mathematical models and analysis

This paper first presents mathematical models for fuzzy two-term (PI/PD) controllers which employ N1 (³3) number of symmetric fuzzy sets for the input variable displacement, N2 (³3) number of symmetric fuzzy sets for the input variable velocity and N1 + N2 – 1 number of symmetric fuzzy sets for the output variable. These models are derived via triangular membership functions for fuzzification of the inputs and output variables, linear control rules, minimum/algebraic product triangular norm, different triangular conorms, different inference methods and centre of sums (COS) defuzzification method. Properties of such models are investigated. Using the well-known small-gain theorem bounded-input bounded-output (BIBO) stability analysis of feedback systems involving fuzzy PD controller as a subsystem is presented. Next, mathematical models with N1 and N2 number of asymmetric input fuzzy sets and N1 + N2 – 1 number of asymmetric output fuzzy sets (both triangular and trapezoidal) are also presented. Finally, s...

A class of fuzzy multisets with a fixed number of memberships

The main aim of this work is to present a generalization of Atanassov’s operators to higher dimensions. To do so, we use the concept of fuzzy set, which can be seen as a special kind of fuzzy multiset, to define a generalization of Atanassov’s operators for n-dimensional fuzzy values (called n-dimensional intervals). We prove that our generalized Atanassov’s operators also generalize OWA operators of any dimension by allowing negative weights. We apply our results to a decision making problem. We also extend the notions of aggregating functions, in particular t-norms, fuzzy negations and automorphism and related notions for n-dimensional framework.

Multi-fuzzy Sets: An Extension of Fuzzy Sets

In this paper we propose a method to construct more general fuzzy sets using ordinary fuzzy sets as building blocks. We introduce the concept of multi-fuzzy sets in terms of ordered sequences of membership functions. The family of operations T, S, M of multi-fuzzy sets are introduced by coordinate wise t-norms, s-norms and aggregation operations. We define the notion of coordinate wise conjugation of multi-fuzzy sets, a method for obtaining Atanassov's intuitionistic fuzzy operations from multi-fuzzy sets. We show that various binary operations in Atanassov's intuitionistic fuzzy sets are equivalent to some operations in multi-fuzzy sets like M operations, 2-conjugates of the T and S operations. It is concluded that multi-fuzzy set theory is an extension of Zadeh's fuzzy set theory, Atanassov's intuitionsitic fuzzy set theory and L-fuzzy set theory.

Hesitant fuzzy sets

Several extensions and generalizations of fuzzy sets have been introduced in the literature, for example, Atanassov's intuitionistic fuzzy sets, type 2 fuzzy sets, and fuzzy multisets. In this pape...

Hesitant fuzzy sets

Several extensions and generalizations of fuzzy sets have been introduced in the literature, for example, Atanassov's intuitionistic fuzzy sets, type 2 fuzzy sets, and fuzzy multisets. In this paper, we propose hesitant fuzzy sets. Although from a formal point of view, they can be seen as fuzzy multisets, we will show that their interpretation differs from the two existing approaches for fuzzy multisets. Because of this, together with their definition, we also introduce some basic operations. In addition, we also study their relationship with intuitionistic fuzzy sets. We prove that the envelope of the hesitant fuzzy sets is an intuitionistic fuzzy set. We prove also that the operations we propose are consistent with the ones of intuitionistic fuzzy sets when applied to the envelope of the hesitant fuzzy sets. © 2010 Wiley Periodicals, Inc.

Scalar and fuzzy cardinalities of crisp and fuzzy multisets

We define in an axiomatic way scalar and fuzzy cardinalities of finite multisets over ]0, 1], and we obtain explicit descriptions for them. We show that, for multisets over ]0, 1] associated to finite fuzzy sets, the cardinalities defined in this study are equivalent to the cardinalities of the corresponding fuzzy sets previously introduced in the literature. Finally, we also define in an axiomatic way scalar and fuzzy cardinalities of finite fuzzy multisets over any set X, and we use the descriptions of the cardinalities of finite multisets over ]0, 1] to obtain explicit characterizations of the former. © 2009 Wiley Periodicals, Inc.

Generalized and Customizable Sets inR

We present data structures and algorithms for sets and some generalizations thereof (fuzzy sets, multisets, and fuzzy multisets) available for <b>R</b> through the <b>sets</b> package. Fuzzy (multi-)sets are based on dynamically bound fuzzy logic families. Further extensions include user-definable iterators and matching functions.

Algorithms of nonlinear document clustering based on fuzzy multiset model

Fuzzy multiset is applicable as a model of information retrieval because it has the mathematical structure that expresses the number and the degree of attribution of an element simultaneously. Ther...

A hybrid approach to semantic web services matchmaking

Deploying the semantics embedded in web services is a mandatory step in the automation of discovery, invocation and composition activities. The semantic annotation is the “add-on” to cope with the actual interoperability limitations and to assure a valid support to the interpretation of services capabilities. Nevertheless many issues have to be reached to support semantics in the web services and to guarantee accurate functionality descriptions. Early efforts address automatic matchmaking tasks, in order to find eligible advertised services which appropriately meet the consumer’s demand. In the most of approaches, this activity is often entrusted to software agents, able to drive reasoning/planning activities, to discover the required service which can be single or composed of more atomic services. This paper presents a hybrid framework which achieves a fuzzy matchmaking of semantic web services. Central role is entrusted to task-oriented agents that, given a service request, interact to discover approximate reply, when no exact match occurs among the available web services. The matchmaking activity exploits a mathematical model, the fuzzy multiset to suitably represent the multi-granular information, enclosed into an OWLS-based description of a semantic web service.

Rule reduction in fuzzy logic for better interpretability in reservoir operation

AbstractDecision‐making in reservoir operation has become easy and understandable with the use of fuzzy logic models, which represent the knowledge in terms of interpretable linguistic rules. However, the improvement in interpretability with increase in number of fuzzy sets (‘low’, ‘high’, etc) comes with the disadvantage of increase in number of rules that are difficult to comprehend by decision makers. In this study, a clustering‐based novel approach is suggested to provide the operators with a limited number of most meaningful operating rules. A single triangular fuzzy set is adopted for different variables in each cluster, which are fine‐tuned with genetic algorithm (GA) to meet the desired objective. The results are compared with the multi fuzzy set fuzzy logic model through a case study in the Pilavakkal reservoir system in Tamilnadu State, India. The results obtained are highly encouraging with a smaller set of rules representing the actual fuzzy logic system. Copyright © 2007 John Wiley &amp; Sons, Ltd.