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Showing papers on "Membership function published in 2009"


Proceedings ArticleDOI
02 Oct 2009
TL;DR: The hesitant fuzzy sets as mentioned in this paper are a generalization of fuzzy sets where the membership is an interval, instead of being a single value, and they have been used in decision making.
Abstract: Intuitionistic Fuzzy Sets (IFS) are a generalization of fuzzy sets where the membership is an interval. That is, membership, instead of being a single value, is an interval. A large number of operations have been defined for this type of fuzzy sets, and several applications have been developed in the last years. In this paper we describe hesitant fuzzy sets. They are another generalization of fuzzy sets. Although similar in intention to IFS, some basic differences on their interpretation and on their operators exist. In this paper we review their definition, the main results and we present an extension principle, which permits to generalize existing operations on fuzzy sets to this new type of fuzzy sets. We also discuss their use in decision making.

1,009 citations


Journal IssueDOI
TL;DR: This paper proposes a new cognitive model—cloud model, which can synthetically describe the randomness and fuzziness of concepts and implement the uncertain transformation between a qualitative concept and its quantitative instantiations and may be more adaptive for the uncertainty description of linguistic concepts.
Abstract: Randomness and fuzziness are the two most important uncertainties inherent in human cognition, which have attracted great attention in artificial intelligence research. In this paper, regarding linguistic terms or concepts as the basic units of human cognition, we propose a new cognitive model—cloud model, which can synthetically describe the randomness and fuzziness of concepts and implement the uncertain transformation between a qualitative concept and its quantitative instantiations. Furthermore, by analyzing in detail the statistical properties of normal cloud model, that is, an important kind of cloud models based on normal distribution and Gauss membership function, we show that normal cloud model can not only be viewed as a generalized normal distribution with weak constraints but also avoid the flaw of fuzzy sets to quantify the membership degree of an element as an accurate value between 0 and 1 and, therefore, may be more adaptive for the uncertainty description of linguistic concepts. Finally, two demonstration examples about the fractal evolution of plants and network topologies based on cloud models are given to illustrate the promising applications of cloud models in some more complex knowledge representation tasks. © 2009 Wiley Periodicals, Inc.

410 citations


Journal ArticleDOI
TL;DR: A novel accuracy function for interval-valued intuitionistic fuzzy sets (IVIFS) is proposed by taking into account the unknown degree (hesitancy degree) of IVIFSs to overcome the situation of difficult decision of existing accuracy functions to the alternatives in some cases.
Abstract: The interval-valued intuitionistic fuzzy weighted arithmetic average operator, the interval-valued intuitionistic fuzzy weighted geometric average operator, and an accuracy function of interval-valued intuitionistic fuzzy value are introduced in this paper. A novel accuracy function for interval-valued intuitionistic fuzzy sets (IVIFSs) is proposed by taking into account the unknown degree (hesitancy degree) of IVIFSs to overcome the situation of difficult decision of existing accuracy functions to the alternatives in some cases. To identify the best alternative in multicriteria decision-making problems, a multicriteria fuzzy decision-making method is established in which criterion values for alternatives are IVIFSs. We utilize the interval-valued intuitionistic fuzzy weighted aggregation operators to aggregate the interval-valued intuitionistic fuzzy information corresponding to each alternative, and then rank the alternatives and select the most desirable one(s) according to the accuracy degree of the aggregated the interval-valued intuitionistic fuzzy information corresponding to the new accuracy function. Finally, an illustrative example is given to verify the developed approach.

361 citations


Journal ArticleDOI
01 Mar 2009
TL;DR: The interval-valued fuzzy TOPSIS method is presented aiming at solving MCDM problems in which the weights of criteria are unequal, using interval- valued fuzzy sets concepts.
Abstract: Decision making is one of the most complex administrative processes in management. In circumstances where the members of the decision making team are uncertain in determining and defining the decision making criteria, fuzzy theory provides a proper tool to encounter with such uncertainties. However, if decision makers cannot reach an agreement on the method of defining linguistic variables based on the fuzzy sets, the interval-valued fuzzy set theory can provide a more accurate modeling. In this paper the interval-valued fuzzy TOPSIS method is presented aiming at solving MCDM problems in which the weights of criteria are unequal, using interval-valued fuzzy sets concepts.

346 citations


Journal ArticleDOI
01 Aug 2009
TL;DR: Performance comparisons of OS-fuzzy-ELM with other existing algorithms are presented using real-world benchmark problems in the areas of nonlinear system identification, regression, and classification.
Abstract: In this correspondence, an online sequential fuzzy extreme learning machine (OS-fuzzy-ELM) has been developed for function approximation and classification problems. The equivalence of a Takagi-Sugeno-Kang (TSK) fuzzy inference system (FIS) to a generalized single hidden-layer feedforward network is shown first, which is then used to develop the OS-fuzzy-ELM algorithm. This results in a FIS that can handle any bounded nonconstant piecewise continuous membership function. Furthermore, the learning in OS-fuzzy-ELM can be done with the input data coming in a one-by-one mode or a chunk-by-chunk (a block of data) mode with fixed or varying chunk size. In OS-fuzzy-ELM, all the antecedent parameters of membership functions are randomly assigned first, and then, the corresponding consequent parameters are determined analytically. Performance comparisons of OS-fuzzy-ELM with other existing algorithms are presented using real-world benchmark problems in the areas of nonlinear system identification, regression, and classification. The results show that the proposed OS-fuzzy-ELM produces similar or better accuracies with at least an order-of-magnitude reduction in the training time.

335 citations


Journal ArticleDOI
TL;DR: It is proved that set theoretic operations for T2 FSs can be computed using very simple alpha-plane computations that are the set theoretics operations for interval T2 (IT2) FSs.
Abstract: This paper 1) reviews the alpha-plane representation of a type-2 fuzzy set (T2 FS), which is a representation that is comparable to the alpha-cut representation of a type-1 FS (T1 FS) and is useful for both theoretical and computational studies of and for T2 FSs; 2) proves that set theoretic operations for T2 FSs can be computed using very simple alpha-plane computations that are the set theoretic operations for interval T2 (IT2) FSs; 3) reviews how the centroid of a T2 FS can be computed using alpha-plane computations that are also very simple because they can be performed using existing Karnik Mendel algorithms that are applied to each alpha-plane; 4) shows how many theoretically based geometrical properties can be obtained about the centroid, even before the centroid is computed; 5) provides examples that show that the mean value (defuzzified value) of the centroid can often be approximated by using the centroids of only 0 and 1 alpha -planes of a T2 FS; 6) examines a triangle quasi-T2 fuzzy logic system (Q-T2 FLS) whose secondary membership functions are triangles and for which all calculations use existing T1 or IT2 FS mathematics, and hence, they may be a good next step in the hierarchy of FLSs, from T1 to IT2 to T2; and 7) compares T1, IT2, and triangle Q-T2 FLSs to forecast noise-corrupted measurements of a chaotic Mackey-Glass time series.

298 citations


Journal ArticleDOI
TL;DR: The proposed method considers the defuzzified values, the heights and the spreads for ranking generalized fuzzy numbers to provide a useful way for handling the fuzzy risk analysis problems.
Abstract: In this paper, we present a new method for fuzzy risk analysis based on ranking generalized fuzzy numbers with different heights and different spreads. First, we present a new method for ranking generalized fuzzy numbers. The proposed method considers the defuzzified values, the heights and the spreads for ranking generalized fuzzy numbers. Based on the proposed method for ranking generalized fuzzy numbers, we propose a fuzzy risk analysis algorithm to deal with fuzzy risk analysis problems. The proposed method provides a useful way for handling the fuzzy risk analysis problems.

295 citations


Journal ArticleDOI
Guiwu Wei1
TL;DR: A procedure based on the DIFWG and IFWG operators is developed to solve the dynamic intuitionistic fuzzy multiple attribute decision making problems where all the decision information about attribute values takes the form of intuitionists fuzzy numbers collected at different periods.
Abstract: The intuitionistic fuzzy set (IFS) characterized by a membership function and a non-membership function, was introduced by [K. Atanassov, "Intuitionistic fuzzy sets", Fuzzy Sets and Systems20 (1986) 87–96] as a generalization of Zadeh' fuzzy set [L. A. Zadeh, "Fuzzy sets", Information and Control8 (1965) 338–356] to deal with fuzziness and uncertainty. In this paper, the dynamic multiple attribute decision making (DMADM) problems with intuitionistic fuzzy information are investigated. The notions of intuitionistic fuzzy variable and uncertain intuitionistic fuzzy variable are defined, and two new aggregation operators called dynamic intuitionistic fuzzy weighted geometric (DIFWG) operator and uncertain dynamic intuitionistic fuzzy weighted geometric (UDIFWG) operator are proposed. Moreover, a procedure based on the DIFWG and IFWG operators is developed to solve the dynamic intuitionistic fuzzy multiple attribute decision making problems where all the decision information about attribute values takes the form of intuitionistic fuzzy numbers collected at different periods, and a procedure based on the UDIFWG and IIWG operators is developed for uncertain dynamic intuitionistic fuzzy multiple attribute decision making problems under interval uncertainty in which all the decision information about attribute values takes the form of interval-valued intuitionistic fuzzy numbers collected at different periods. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.

261 citations


Journal ArticleDOI
TL;DR: This paper presents a method to construct interval-valued fuzzy sets from a matrix, in such a way that the length of the interval representing the membership of any element to the new set is obtained from the differences between the values assigned to that element and its neighbors in the starting matrix.

212 citations


Journal ArticleDOI
TL;DR: In this article, the authors proposed a fuzzy multiobjective approach for the placement of switches in distribution networks in the presence of distributed generation sources, where the primary objective is reliability improvement with consideration of economic aspects.
Abstract: This paper proposes a methodology for placement of sectionalizing switches in distribution networks in the presence of distributed generation sources. The multiobjective considerations are handled using a fuzzy approach. The primary objective is reliability improvement with consideration of economic aspects. Thus, the objectives are defined as reliability improvement and minimization of the cost of sectionalizing switches. A fuzzy membership function is defined for each term in the objective function according to relevant conditions. Some considerations incorporated in the proposed model are relocation of existing switches and operating constraints on distribution networks and distributed generation (DG) sources during post-fault service restoration. The ant colony optimization (ACO) algorithm is adopted to solve the fuzzy multiobjective problem efficiently. The performance of the proposed approach is assessed and illustrated by various case studies on a test distribution system and also a real distribution network.

191 citations


Journal ArticleDOI
TL;DR: A new fuzzy data-mining algorithm for extracting both fuzzy association rules and membership functions by means of a genetic learning of the membership functions and a basic method based on the 2-tuples linguistic representation model is presented.

Journal ArticleDOI
TL;DR: This paper explores the application of fuzzy sets theory for basic tools used in process safety analysis such as fault and event tree methods which can be further used in the “bow-tie” approach for accident scenario risk assessment.
Abstract: Fuzzy logic deals with uncertainty and imprecision, and is an efficient tool for solving problems where knowledge uncertainty may occur. Such situations frequently arise in a quantitative fault and event tree analysis in safety and risk assessment of different processes. The lack of detailed data on failure rates, uncertainties in available data, imprecision and vagueness may lead to uncertainty in results, thus producing an underestimated or overestimated process risk level. This paper explores the application of fuzzy sets theory for basic tools used in process safety analysis such as fault and event tree methods which can be further used in the “bow-tie” approach for accident scenario risk assessment. In the traditional fault and event tree analyses, the input variables are treated as exact values and the exact outcome data are received by an appropriate mathematical approach. In the fuzzy method, all variables are replaced by fuzzy numbers in the process of fuzzification and subsequently using fuzzy arithmetic, fuzzy probability of the top event for fault tree, and fuzzy outcome probabilities for event tree are calculated. A single value for each of the outcome event result is obtained with the use of one of the defuzzification methods. A typical case study comprising a fault tree for rupture of the isobutane storage tank and the event tree for its consequences is performed and a comparison between the traditional approach and fuzzy method is made.

Journal ArticleDOI
TL;DR: In this paper, three novel interval type-2 fuzzy membership function (IT2 FMF) generation methods are proposed, based on heuristics, histograms, and interval type -2 fuzzy C-means, which are evaluated by applying them to back-propagation neural networks (BPNNs).

Journal ArticleDOI
TL;DR: This work combines the fuzzy analytic hierarchy process (AHP) with the portfolio selection problem and shows that both of the models provide both ranking and weighting information, via fuzzy AHP, to the investors in this financial scenario.

Journal ArticleDOI
TL;DR: A case-based reasoning (CBR) approach is focused on which is a recently recommended method for solving the vendor selection problem (VSP) by making use of previous similar situations.

Journal ArticleDOI
TL;DR: In this article, the concept of intuitionistic fuzzy set is applied to AHP, called IF-AHP to handle both vagueness and ambiguity related uncertainties in the environmental decision-making process, which is demonstrated with an illustrative example to select best drilling fluid (mud) for drilling operations under multiple environmental criteria.
Abstract: Analytic hierarchy process (AHP) is a utility theory based decision-making technique, which works on a premise that the decision-making of complex problems can be handled by structuring them into simple and comprehensible hierarchical structures. However, AHP involves human subjective evaluation, which introduces vagueness that necessitates the use of decision-making under uncertainty. The vagueness is commonly handled through fuzzy sets theory, by assigning degree of membership. But, the environmental decision-making problem becomes more involved if there is an uncertainty in assigning the membership function (or degree of belief) to fuzzy pairwise comparisons, which is referred to as ambiguity (non-specificity). In this paper, the concept of intuitionistic fuzzy set is applied to AHP, called IF-AHP to handle both vagueness and ambiguity related uncertainties in the environmental decision-making process. The proposed IF-AHP methodology is demonstrated with an illustrative example to select best drilling fluid (mud) for drilling operations under multiple environmental criteria.

Journal ArticleDOI
TL;DR: This paper investigates various operation properties and proposes a distance measure for complex fuzzy sets and defines @d-equalities ofcomplex fuzzy sets which coincide with those of fuzzy sets already defined in the literature if complex fuzzy set reduce to real-valued fuzzy sets.

Journal ArticleDOI
TL;DR: This paper gets a fuzzy solution for this class of fuzzy differential equations where the dynamics is given by a continuous fuzzy mapping which is obtained via Zadeh's extension principle.

Journal ArticleDOI
TL;DR: This brief presents an approach to detect premature ventricular contractions (PVCs) using the neural network with weighted fuzzy membership functions (NEWFMs), and it is shown that the locations of the eight features are not only around the QRS complex that represents ventricular depolarization in the electrocardiogram (ECG) containing a Q wave, an R wave, and an S wave.
Abstract: Fuzzy neural networks (FNNs) have been successfully applied to generate predictive rules for medical or diagnostic data. This brief presents an approach to detect premature ventricular contractions (PVCs) using the neural network with weighted fuzzy membership functions (NEWFMs). The NEWFM classifies normal and PVC beats by the trained bounded sum of weighted fuzzy membership functions (BSWFMs) using wavelet transformed coefficients from the MIT-BIH PVC database. The eight generalized coefficients, locally related to the time signal, are extracted by the nonoverlap area distribution measurement method. The eight generalized coefficients are used for the three PVC data sets with reliable accuracy rates of 99.80%, 99.21%, and 98.78%, respectively, which means that the selected features are less dependent on the data sets. It is shown that the locations of the eight features are not only around the QRS complex that represents ventricular depolarization in the electrocardiogram (ECG) containing a Q wave, an R wave, and an S wave, but also the QR segment from the Q wave to the R wave has more discriminate information than the RS segment from the R wave to the S wave. The BSWFMs of the eight features trained by NEWFM are shown visually, which makes the features explicitly interpretable. Since each BSWFM combines multiple weighted fuzzy membership functions into one using the bounded sum, the eight small-sized BSWFMs can realize real-time PVC detection in a mobile environment.

Book ChapterDOI
07 Jul 2009
TL;DR: This paper revisits the hybridization of rough sets and fuzzy sets by introducing vague quantifiers like "some" or "most" into the definition of upper and lower approximation, and develops a vaguely quantified rough set model that is closely related to Ziarko's variable precision rough set (VPRS) model.
Abstract: The hybridization of rough sets and fuzzy sets has focused on creating an end product that extends both contributing computing paradigms in a conservative way As a result, the hybrid theory inherits their respective strengths, but also exhibits some weaknesses In particular, although they allow for gradual membership, fuzzy rough sets are still abrupt in a sense that adding or omitting a single element may drastically alter the outcome of the approximations In this paper, we revisit the hybridization process by introducing vague quantifiers like "some" or "most" into the definition of upper and lower approximation The resulting vaguely quantified rough set (VQRS) model is closely related to Ziarko's variable precision rough set (VPRS) model

Journal ArticleDOI
TL;DR: The concept of a mapping on classes of fuzzySoft sets is defined and the properties of fuzzy soft images and fuzzy soft inverse images of fuzzysoft sets are studied.
Abstract: We define the concept of a mapping on classes of fuzzy soft sets and study the properties of fuzzy soft images and fuzzy soft inverse images of fuzzy soft sets, and support them with examples and counterexamples.

Journal ArticleDOI
TL;DR: A fuzzy proportional-integral-derivative (fuzzy-PID) control strategy, and applies it to the nuclear reactor power control system, is introduced and simulation results show the favorable performance of the fuzzy-PIDs controller.

Journal ArticleDOI
Ying-Ming Wang1
TL;DR: The two methods of centroid defuzzification and the maximizing set and minimizing set methods are reinvestigated when explicit membership functions are not known but alpha level sets are available.

Journal ArticleDOI
TL;DR: This work suggests a fuzzy TOPSIS model, where ratings of alternatives under criteria and importance weights of criteria are assessed in linguistic values represented by fuzzy numbers and a ranking method can be applied easily to develop positive and negative idea solutions.
Abstract: This work suggests a fuzzy TOPSIS model, where ratings of alternatives under criteria and importance weights of criteria are assessed in linguistic values represented by fuzzy numbers. Criteria can be categorized into benefit and cost. Ratings of alternatives versus criteria and the importance weights of criteria are normalized before multiplication. The membership function of each fuzzy weighted rating can be developed by interval arithmetic of fuzzy numbers. A ranking method can then be applied easily to develop positive and negative idea solutions in order to complete the fuzzy TOPSIS model. Finally, a numerical example demonstrates the feasibility of the proposed method.

Journal ArticleDOI
Ronald R. Yager1
TL;DR: The significant role that duality plays in many aggregation operations involving intuitionistic fuzzy subsets, and a decision paradigm called the method of least commitment is introduced.
Abstract: We first discuss the significant role that duality plays in many aggregation operations involving intuitionistic fuzzy subsets We then consider the extension to intuitionistic fuzzy subsets of a number of ideas from standard fuzzy subsets In particular we look at the measure of specificity We also look at the problem of alternative selection when decision criteria satisfaction is expressed using intuitionistic fuzzy subsets We introduce a decision paradigm called the method of least commitment We briefly look at the problem of defuzzification of intuitionistic fuzzy subsets

Journal ArticleDOI
TL;DR: A new approach based on vague sets group decision is proposed to deal with the supplier selection problem in supply chain systems and a numerical example of the selection problem of suppliers is shown.
Abstract: In practice, in purchasing decision-making, many quantitative and qualitative factors, with vagueness and imprecision, have to be considered. This makes the decision process very complicated and unstructured. Besides the fuzzy sets theory, vague sets theory is one of the methods used to deal with uncertain information. Since vague sets can provide more information than fuzzy sets, it is considered superior in mathematical analysis of uncertain information. In this paper, a new approach based on vague sets group decision is proposed to deal with the supplier selection problem in supply chain systems. The work procedure is shown briefly, as follows: First, linguistic values are used to assess the ratings and weights for quantitative or qualitative factors. Second, degree of similarity and probability of vague sets are used to determine the ranking order of all alternatives. Finally, a numerical example of the selection problem of suppliers is shown, to highlight the procedure of the proposed approach, at the end of this paper.

Journal IssueDOI
TL;DR: It is demonstrated how shadowed sets help in problems in data interpretation in fuzzy clustering by leading to the three-valued quantification of data structure that consists of core, shadowed, and uncertain structure.
Abstract: In this study, we discuss a concept of shadowed sets and present their applications. To establish some sound compromise between the qualitative Boolean (two-valued) description of data and quantitative membership grades, we introduce an interpretation framework of shadowed sets. Shadowed sets are discussed as three-valued constructs induced by fuzzy sets assuming three values (that could be interpreted as full membership, full exclusion, and uncertain membership). The algorithm of converting membership functions into this quantification is a result of a certain optimization problem guided by the principle of uncertainty localization. We revisit fundamental ideas of relational calculus in the setting of shadowed sets. We demonstrate how shadowed sets help in problems in data interpretation in fuzzy clustering by leading to the three-valued quantification of data structure that consists of core, shadowed, and uncertain structure. © 2008 Wiley Periodicals, Inc.

Journal ArticleDOI
TL;DR: This paper presents a general framework for the study of relation-based (I,T)-intuitionistic fuzzy rough sets by using constructive and axiomatic approaches and different axiom sets characterizing the essential properties of intuitionism fuzzy approximation operators associated with various intuitionistic fuzzy relations.

Journal ArticleDOI
TL;DR: This paper presents a new approach for stability analysis and controller design of Takagi-Sugeno (TS) models that considers information derived from existing or induced order relations among the membership functions.

Journal ArticleDOI
TL;DR: This introduction to the R package sets is a (slightly) modied version of Meyer and Hornik (2009a), published in the Journal of Statistical Software.
Abstract: This introduction to the R package sets is a (slightly) modied version of Meyer and Hornik (2009a), published in the Journal of Statistical Software. We present data structures and algorithms for sets and some generalizations thereof (fuzzy sets, multisets, and fuzzy multisets) available for R through the sets package. Fuzzy (multi-)sets are based on dynamically bound fuzzy logic families. Further extensions include user-denable iterators and matching functions.