Showing papers on "Membership function published in 2012"

On Intuitionistic Fuzzy Sets Theory

Abstract: This book aims to be a comprehensive and accurate survey of state-of-art research on intuitionistic fuzzy sets theory and could be considered a continuation and extension of the authors previous book on Intuitionistic Fuzzy Sets, published by Springer in 1999 (Atanassov, Krassimir T., Intuitionistic Fuzzy Sets, Studies in Fuzziness and soft computing, ISBN 978-3-7908-1228-2, 1999). Since the aforementioned book has appeared, the research activity of the author within the area of intuitionistic fuzzy sets has been expanding into many directions. The results of the authors most recent work covering the past 12 years as well as the newest general ideas and open problems in this field have been therefore collected in this new book.

Dual Hesitant Fuzzy Sets

Abstract: 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.

Complex intuitionistic fuzzy sets

Abstract: This paper presents a new concept of complex intuitionistic fuzzy set (CIFS) which is generalized from the innovative concept of a complex fuzzy set (CFS) by adding the non-membership term to the definition of CFS. The novelty of CIFS lies in its ability for membership and non-membership functions to achieve more range of values. The ranges of values are extended to the unit circle in complex plane for both membership and non-membership functions instead of [0, 1] as in the conventional intuitionistic fuzzy functions. We define basic operations namely complement, union, and intersection on CIFSs. Properties of these operations are derived.

On the Fundamental Differences Between Interval Type-2 and Type-1 Fuzzy Logic Controllers

Abstract: Interval type-2 fuzzy logic controllers (IT2 FLCs) have recently been attracting a lot of research attention. Many reported results have shown that IT2 FLCs are better able to handle uncertainties than their type-1 (T1) counterparts. A challenging question is the following: What are the fundamental differences between IT2 and T1 FLCs? Once the fundamental differences are clear, we can better understand the advantages of IT2 FLCs and, hence, make better use of them. This paper explains two fundamental differences between IT2 and T1 FLCs: 1) Adaptiveness, meaning that the embedded T1 fuzzy sets used to compute the bounds of the type-reduced interval change as input changes; and 2) Novelty, meaning that the upper and lower membership functions of the same IT2 fuzzy set may be used simultaneously in computing each bound of the type-reduced interval. T1 FLCs do not have these properties; thus, a T1 FLC cannot implement the complex control surface of an IT2 FLC given the same rulebase. We also present several methods to visualize and analyze the effects of these two fundamental differences, including the control surface, the P-map, the equivalent generalized T1 fuzzy sets, and the equivalent PI gains. Finally, we examine five alternative type reducers for IT2 FLCs and explain why they do not capture the fundamentals of IT2 FLCs.

Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm

Abstract: Archimedean t-conorm and t-norm are generalizations of a lot of other t-conorms and t-norms, such as Algebraic, Einstein, Hamacher and Frank t-conorms and t-norms or others, and some of them have been applied to intuitionistic fuzzy set, which contains three functions: the membership function, the non-membership function and the hesitancy function describing uncertainty and fuzziness more objectively. Recently, Beliakov et al. [3] constructed some operations about intuitionistic fuzzy sets based on Archimedean t-conorm and t-norm, from which an aggregation principle is proposed for intuitionistic fuzzy information. In this paper, we propose some other operations on intuitionistic fuzzy sets, study their properties and relationships, and based on which, we study the properties of the aggregation principle proposed by Beliakov et al. [3], and give some specific intuitionistic fuzzy aggregation operators, which can be considered as the extensions of the known ones. In the end, we develop an approach for multi-criteria decision making under intuitionistic fuzzy environment, and illustrate an example to show the behavior of the proposed operators.

A Method of Converting Z-number to Classical Fuzzy Number

Abstract: The notion Z-number introduced by Zadeh in 2011 has more capability to describe the uncertain information. Now that the theories about Z-number is not mature, how to convert Z-number to classical fuzzy number is rather signicant for application. In this paper, a method of transforming Z-number to classical fuzzy number is proposed according to the Fuzzy Expectation of fuzzy set. At last, a simple example is used to illustrated the procedure of the proposed approach.

Enhanced Interval Approach for Encoding Words Into Interval Type-2 Fuzzy Sets and Its Convergence Analysis

Abstract: Construction of interval type-2 fuzzy set models is the first step in the perceptual computer, which is an implementation of computing with words. The interval approach (IA) has, so far, been the only systematic method to construct such models from data intervals that are collected from a survey. However, as pointed out in this paper, it has some limitations, and its performance can be further improved. This paper proposes an enhanced interval approach (EIA) and demonstrates its performance on data that are collected from a web survey. The data part of the EIA has more strict and reasonable tests than the IA, and the fuzzy set part of the EIA has an improved procedure to compute the lower membership function. We also perform a convergence analysis to answer two important questions: 1) Does the output interval type-2 fuzzy set from the EIA converge to a stable model as increasingly more data intervals are collected, and 2) if it converges, then how many data intervals are needed before the resulting interval type-2 fuzzy set is sufficiently similar to the model obtained from infinitely many data intervals? We show that the EIA converges in a mean-square sense, and generally, 30 data intervals seem to be a good compromise between cost and accuracy.

Fuzzy multiple attributes group decision-making based on ranking interval type-2 fuzzy sets

Abstract: In this paper, we present a new method to deal with fuzzy multiple attributes group decision-making problems based on ranking interval type-2 fuzzy sets. First, we propose a new method for ranking interval type-2 fuzzy sets. Then, we propose a new method for fuzzy multiple attributes group decision-making based on the proposed ranking method of interval type-2 fuzzy sets. We also use some examples to illustrate the fuzzy multiple attributes group decision-making process of the proposed method. The proposed method is simpler than the methods presented in Chen and Lee (2010a, 2010b) for fuzzy multiple attributes group decision-making based on interval type-2 fuzzy sets. It provides us with a useful way for dealing with fuzzy multiple attributes group decision-making problems based on interval type-2 fuzzy sets.

Multi-attribute group decision making models under interval type-2 fuzzy environment

Abstract: Interval type-2 fuzzy sets (IT2 FSs) are a very useful means to depict the decision information in the process of decision making. In this article, we investigate the group decision making problems in which all the information provided by the decision makers (DMs) is expressed as IT2 fuzzy decision matrices, and the information about attribute weights is partially known, which may be constructed by various forms. We first use the IT2 fuzzy weighted arithmetic averaging operator to aggregate all individual IT2 fuzzy decision matrices provided by the DMs into the collective IT2 fuzzy decision matrix, then we utilize the ranking-value measure to calculate the ranking value of each attribute value and construct the ranking-value matrix of the collective IT2 fuzzy decision matrix. Based on the ranking-value matrix and the given attribute weight information, we establish some optimization models to determine the weights of attributes. Furthermore, we utilize the obtained attribute weights and the IT2 fuzzy weighted arithmetic average operator to fuse the IT2 fuzzy information in the collective IT2 fuzzy decision matrix to get the overall IT2 fuzzy values of alternatives by which the ranking of all the given alternatives can be found. Finally, we give an illustrative example.

A hybrid particle swarm optimization based fuzzy expert system for the diagnosis of coronary artery disease

Abstract: This paper presents a particle swarm optimization (PSO)-based fuzzy expert system for the diagnosis of coronary artery disease (CAD). The designed system is based on the Cleveland and Hungarian Heart Disease datasets. Since the datasets consist of many input attributes, decision tree (DT) was used to unravel the attributes that contribute towards the diagnosis. The output of the DT was converted into crisp if-then rules and then transformed into fuzzy rule base. PSO was employed to tune the fuzzy membership functions (MFs). Having applied the optimized MFs, the generated fuzzy expert system has yielded 93.27% classification accuracy. The major advantage of this approach is the ability to interpret the decisions made from the created fuzzy expert system, when compared with other approaches.

A new approach for solving fuzzy transportation problems using generalized trapezoidal fuzzy numbers

Abstract: In the literature, several algorithms are proposed for solving the transportation problems in fuzzy environment but in all the proposed algorithms the parameters are represented by normal fuzzy numbers. Chen [Operations on fuzzy numbers with function principal, Tamkang Journal of Management Science 6 (1985) 13-25] pointed out that in many cases it is not to possible to restrict the membership function to the normal form and proposed the concept of generalized fuzzy numbers. There are several papers in the literature in which generalized fuzzy numbers are used for solving real life problems but to the best of our knowledge, till now no one has used generalized fuzzy numbers for solving the transportation problems. In this paper, a new algorithm is proposed for solving a special type of fuzzy transportation problems by assuming that a decision maker is uncertain about the precise values of transportation cost only but there is no uncertainty about the supply and demand of the product. In the proposed algorithm transportation costs are represented by generalized trapezoidal fuzzy numbers. To illustrate the proposed algorithm a numerical example is solved and the obtained results are compared with the results of existing approaches. Since the proposed approach is a direct extension of classical approach so the proposed approach is very easy to understand and to apply on real life transportation problems for the decision makers.

General Type-2 Fuzzy C-Means Algorithm for Uncertain Fuzzy Clustering

Abstract: Pattern recognition in real-world data is subject to various sources of uncertainty that should be appropriately managed. The focus of this paper is the management of uncertainty associated with parameters of fuzzy clustering algorithms. Type-2 fuzzy sets (T2 FSs) have received increased research interest over the past decade, primarily due to their potential to model various uncertainties. However, because of the computational intensity of the processing of general T2 fuzzy sets (GT2 FSs), only their constrained version, i.e., the interval T2 (IT2) FSs, were typically used. Fortunately, the recently introduced concepts of α-planes and zSlices allow for efficient representation and computation with GT2 FSs. Following this recent development, this paper presents a novel approach for uncertain fuzzy clustering using the general type-2 fuzzy C-means (GT2 FCM) algorithm. The proposed method builds on top of the previously published IT2 FCM algorithm, which is extended via the α- planes representation theorem. The fuzzifier parameter of the FCM algorithm can be expressed using linguistic terms such as “small” or “high,” which are modeled as T1 FSs. This linguistic fuzzifier value is then used to construct the GT2 FCM cluster membership functions. The linguistic uncertainty is transformed into uncertain fuzzy positions of the extracted clusters. The GT2 FCM algorithm was found to balance the performance of T1 FCM algorithms in various uncertain pattern recognition tasks and to provide increased robustness in situations where noisy or insufficient training data are present.

On Robust Fuzzy Rough Set Models

Abstract: Rough sets, especially fuzzy rough sets, are supposedly a powerful mathematical tool to deal with uncertainty in data analysis. This theory has been applied to feature selection, dimensionality reduction, and rule learning. However, it is pointed out that the classical model of fuzzy rough sets is sensitive to noisy information, which is considered as a main source of uncertainty in applications. This disadvantage limits the applicability of fuzzy rough sets. In this paper, we reveal why the classical fuzzy rough set model is sensitive to noise and how noisy samples impose influence on fuzzy rough computation. Based on this discussion, we study the properties of some current fuzzy rough models in dealing with noisy data and introduce several new robust models. The properties of the proposed models are also discussed. Finally, a robust classification algorithm is designed based on fuzzy lower approximations. Some numerical experiments are given to illustrate the effectiveness of the models. The classifiers that are developed with the proposed models achieve good generalization performance.

A soft computing method of performance evaluation with MCDM based on interval-valued fuzzy numbers

Abstract: This study presented a new performance evaluation method for tackling fuzzy multicriteria decision-making (MCDM) problems based on combining VIKOR and interval-valued fuzzy sets. The performance evaluation problem often exists in complex administrative processes in which multiple evaluation criteria, subjective/objective assessments and fuzzy conditions have to be taken into consideration simultaneously in management. Here, the subjective, imprecise, inexact and uncertain evaluation processes are modeled as fuzzy numbers by means of linguistic terms, as fuzzy theory can provide an appropriate tool to deal with such uncertainties. However, the presentation of linguistic expressions in the form of ordinary fuzzy sets is not clear enough [15,21]. Interval-valued fuzzy sets can provide more flexibility [4,14] to represent the imprecise/vague information that results, and it can also provide a more accurate modeling. This paper presents the interval-valued fuzzy VIKOR, which aims to solve MCDM problems in which the weights and performances of criteria are unequal by using the concepts of interval-valued fuzzy sets. A case study for evaluating the performances of three major intercity bus companies from an intercity public transport system is conducted to illustrate the effectiveness of the method.

A dissimilarity measure for the k-Modes clustering algorithm

Abstract: Clustering is one of the most important data mining techniques that partitions data according to some similarity criterion. The problems of clustering categorical data have attracted much attention from the data mining research community recently. As the extension of the k-Means algorithm, the k-Modes algorithm has been widely applied to categorical data clustering by replacing means with modes. In this paper, the limitations of the simple matching dissimilarity measure and Ng's dissimilarity measure are analyzed using some illustrative examples. Based on the idea of biological and genetic taxonomy and rough membership function, a new dissimilarity measure for the k-Modes algorithm is defined. A distinct characteristic of the new dissimilarity measure is to take account of the distribution of attribute values on the whole universe. A convergence study and time complexity of the k-Modes algorithm based on new dissimilarity measure indicates that it can be effectively used for large data sets. The results of comparative experiments on synthetic data sets and five real data sets from UCI show the effectiveness of the new dissimilarity measure, especially on data sets with biological and genetic taxonomy information.

Multicriteria fuzzy decision making based on interval-valued intuitionistic fuzzy sets

Abstract: In this paper, we present a new method for multicriteria fuzzy decision making based on interval-valued intuitionistic fuzzy sets, where interval-valued intuitionistic fuzzy values are used to represent evaluating values of the decision-maker with respect to alternatives. First, we propose a new method for ranking interval-valued intuitionistic fuzzy values. Based on the proposed fuzzy ranking method of interval-valued intuitionistic fuzzy values, we propose a new method for multicriteria fuzzy decision making. The proposed multicriteria fuzzy decision making method outperforms Ye's method (2009) due to the fact that the proposed method can overcome the drawback of Ye's method (2009), where the drawback of Ye's method is that it can not distinguish the ranking order between alternatives in some situations. The proposed method provides us with a useful way for dealing with multicriteria fuzzy decision making problems based on interval-valued intuitionistic fuzzy sets.

Comparative analysis of SAW and TOPSIS based on interval-valued fuzzy sets: Discussions on score functions and weight constraints

Abstract: Interval-valued fuzzy sets involve more uncertainties than ordinary fuzzy sets and can be used to capture imprecise or uncertain decision information in fields that require multiple-criteria decision analysis (MCDA). This paper takes the simple additive weighting (SAW) method and the technique for order preference by similarity to an ideal solution (TOPSIS) as the main structure to deal with interval-valued fuzzy evaluation information. Using an interval-valued fuzzy framework, this paper presents SAW-based and TOPSIS-based MCDA methods and conducts a comparative study through computational experiments. Comprehensive discussions have been made on the influence of score functions and weight constraints, where the score function represents an aggregated effect of positive and negative evaluations in performance ratings and the weight constraint consists of the unbiased condition, positivity bias, and negativity bias. The correlations and contradiction rates obtained in the experiments suggest that evident similarities exist between the interval-valued fuzzy SAW and TOPSIS rankings.

A new time invariant fuzzy time series forecasting method based on particle swarm optimization

Abstract: In the analysis of time invariant fuzzy time series, fuzzy logic group relationships tables have been generally preferred for determination of fuzzy logic relationships. The reason of this is that it is not need to perform complex matrix operations when these tables are used. On the other hand, when fuzzy logic group relationships tables are exploited, membership values of fuzzy sets are ignored. Thus, in defiance of fuzzy set theory, fuzzy sets' elements with the highest membership value are only considered. This situation causes information loss and decrease in the explanation power of the model. To deal with these problems, a novel time invariant fuzzy time series forecasting approach is proposed in this study. In the proposed method, membership values in the fuzzy relationship matrix are computed by using particle swarm optimization technique. The method suggested in this study is the first method proposed in the literature in which particle swarm optimization algorithm is used to determine fuzzy relations. In addition, in order to increase forecasting accuracy and make the proposed approach more systematic, the fuzzy c-means clustering method is used for fuzzification of time series in the proposed method. The proposed method is applied to well-known time series to show the forecasting performance of the method. These time series are also analyzed by using some other forecasting methods available in the literature. Then, the results obtained from the proposed method are compared to those produced by the other methods. It is observed that the proposed method gives the most accurate forecasts.

Induced generalized intuitionistic fuzzy OWA operator for multi-attribute group decision making

Abstract: With respect to multi-attribute group decision making (MAGDM) problems in which both the attribute weights and the decision makers (DMs) weights take the form of real numbers, attribute values provided by the DMs take the form of intuitionistic fuzzy numbers, a new group decision making method is developed. Some operational laws, score function and accuracy function of intuitionistic fuzzy numbers are introduced at first. Then a new aggregation operator called induced generalized intuitionistic fuzzy ordered weighted averaging (IG-IFOWA) operator is proposed, which extend the induced generalized ordered weighted averaging (IGOWA) operator introduced by Merigo and Gil-Lafuente [Merigo, J. M., & Gil-Lafuente, A. M. (2009). The induced generalized OWA operator. Information Sciences, 179, 729-741] to accommodate the environment in which the given arguments are intuitionistic fuzzy sets that are characterized by a membership function and a non-membership function. Some desirable properties of the IG-IFOWA operator are studied, such as commutativity, idempotency, monotonicity and boundary. And then, an approach based on the IG-IFOWA and IFWA (intuitionistic fuzzy weighted averaging) operators is developed to solve MAGDM problems with intuitionistic fuzzy information. Finally, a numerical example is used to illustrate the developed approach.

A novel approach to multi-attribute decision making based on intuitionistic fuzzy sets

Abstract: Highlights? A new approach to solve multi-attribute decision making problems in intuitionistic fuzzy environment. ? The same alternative with different attributes are considered as one unified entity. ? A revised score function and a revised accuracy function of intuitionistic fuzzy sets based on human intuition. ? The degree of membership, the degree of nonmembership and the degree of hesitation are with various importance. ? An optimization model is established to estimate the relative degree of importance. In this paper we present a new approach to solve multi-attribute decision making problems in intuitionistic fuzzy environment. This approach is based on a new ranking method of intuitionistic fuzzy sets, in which the evaluated values (in the form of intervals) of the same alternative with different attributes are considered as one unified entity. According to people's intuition, the ranking method proposed in this paper is mainly grounded on a revised score function and a revised accuracy function of intuitionistic fuzzy sets. Different from the traditional methods, in this new approach, the degree of membership, the degree of nonmembership and the degree of hesitation are considered with various importance in reflecting the true image of the respective alternative. Furthermore, an optimization model is established to estimate the relative degree of importance of each quantity. Finally, two practical examples are provided to illustrate our approach.

New Semi-Supervised Classification Method Based on Modified Cluster Assumption

Abstract: The cluster assumption, which assumes that “similar instances should share the same label,” is a basic assumption in semi-supervised classification learning, and has been found very useful in many successful semi-supervised classification methods It is rarely noticed that when the cluster assumption is adopted, there is an implicit assumption that every instance should have a crisp class label assignment In real applications, however, there are cases where it is difficult to tell that an instance definitely belongs to one class and does not belong to other neighboring classes In such cases, it is more adequate to assume that “similar instances should share similar label memberships” rather than sharing a crisp label assignment Here “label memberships” can be represented as a vector, where each element corresponds to a class, and the value at the element expresses the likelihood of the concerned instance belonging to the class By adopting this modified cluster assumption, in this paper we propose a new semi-supervised classification method, that is, semi-supervised classification based on class membership (SSCCM) Specifically, we try to solve the decision function and adequate label memberships for instances simultaneously, and constrain that an instance and its “local weighted mean” (LWM) share the same label membership vector, where the LWM is a robust image of the instance, constructed by calculating the weighted mean of its neighboring instances We formulate the problem in a unified objective function for the labeled, unlabeled data and their LWMs based on the square loss function, and take an alternating iterative strategy to solve it, in which each step generates a closed-form solution, and the convergence is guaranteed The solution will provide both the decision function and the label membership function for classification, their classification results can verify each other, and the reliability of semi-supervised classification learning might be enhanced by checking the consistency between those two predictions Experiments show that SSCCM obtains encouraging results compared to state-of-the-art semi-supervised classification methods

Design of fuzzy expert system for microarray data classification using a novel Genetic Swarm Algorithm

Abstract: Knowledge gained through classification of microarray gene expression data is increasingly important as they are useful for phenotype classification of diseases. Different from black box methods, fuzzy expert system can produce interpretable classifier with knowledge expressed in terms of if-then rules and membership function. This paper proposes a novel Genetic Swarm Algorithm (GSA) for obtaining near optimal rule set and membership function tuning. Advanced and problem specific genetic operators are proposed to improve the convergence of GSA and classification accuracy. The performance of the proposed approach is evaluated using six gene expression data sets. From the simulation study it is found that the proposed approach generated a compact fuzzy system with high classification accuracy for all the data sets when compared with other approaches.

Twelve considerations in choosing between Gaussian and trapezoidal membership functions in interval type-2 fuzzy logic controllers

Abstract: Interval type-2 fuzzy logic controllers (IT2 FLCs) have been attracting great research interests recently. There are many decisions to be made in designing an IT2 FLC. One of them is to determine which membership function type to use, e.g., Gaussian or trapezoidal. There have not been comprehensive studies on this problem so far. In this paper we present 12 considerations in choosing between Gaussian and trapezoidal membership functions for an IT2 FLC, including representation, construction, optimization, adaptiveness, novelty, analytical structure, continuity, monotonicity, stability, robustness, computational cost, and control performance. It can help practitioners select the appropriate membership function type in IT2 FLC design, and researchers identify new research opportunities on IT2 FLCs. Our study shows that each MF type has its own advantages: Gaussian IT2 FLCs are simpler in design because they are easier to represent and optimize, always continuous, and faster for small rulebases, whereas trapezoidal IT2 FLCs are simpler in analysis.

Fuzzy energy management strategy for a hybrid electric vehicle based on driving cycle recognition

Abstract: By considering the effect of the driving cycle on the energy management strategy (EMS), a fuzzy EMS based on driving cycle recognition is proposed to improve the fuel economy of a parallel hybrid electric vehicle The EMS is composed of driving cycle recognition and a fuzzy torque distribution controller The current driving cycle is recognized by learning vector quantization in driving cycle recognition The torque of the engine and the motor is controlled by a fuzzy torque distribution controller based on the required torque of the hybrid powertrain and the battery state of charge The membership functions and rules of the fuzzy torque distribution controller are optimized simultaneously by using particle swarm optimization Based on the identification results of driving cycle recognition, the fuzzy torque distribution controller selects the corresponding membership function and rule to control the hybrid powertrain The simulation research based on ADVISOR demonstrates that this EMS improves fuel economy more effectively than fuzzy EMS without driving cycle recognition