Showing papers on "Membership function published in 2007"

Intuitionistic Fuzzy Aggregation Operators

Abstract: An intuitionistic fuzzy set, characterized by a membership function and a non-membership function, is a generalization of fuzzy set. In this paper, based on score function and accuracy function, we introduce a method for the comparison between two intuitionistic fuzzy values and then develop some aggregation operators, such as the intuitionistic fuzzy weighted averaging operator, intuitionistic fuzzy ordered weighted averaging operator, and intuitionistic fuzzy hybrid aggregation operator, for aggregating intuitionistic fuzzy values and establish various properties of these operators.

Type-2 fuzzy sets and systems: an overview

Abstract: This paper provides an introduction to and an overview of type-2 fuzzy sets (T2 FS) and systems. It does this by answering the following questions: What is a T2 FS and how is it different from a T1 FS? Is there new terminology for a T2 FS? Are there important representations of a T2 FS and, if so, why are they important? How and why are T2 FSs used in a rule-based system? What are the detailed computations for an interval T2 fuzzy logic system (IT2 FLS) and are they easy to understand? Is it possible to have an IT2 FLS without type reduction? How do we wrap this up and where can we go to learn more?

Type-2 Fuzzy Logic: Theory and Applications

Abstract: Type-2 fuzzy sets are used for modeling uncertainty and imprecision in a better way. These type-2 fuzzy sets were originally presented by Zadeh in 1975 and are essentially "fuzzy fuzzy" sets where the fuzzy degree of membership is a type-1 fuzzy set. The new concepts were introduced by Mendel and Liang allowing the characterization of a type-2 fuzzy set with a superior membership function and an inferior membership function; these two functions can be represented each one by a type-1 fuzzy set membership function. The interval between these two functions represents the footprint of uncertainty (FOU), which is used to characterize a type-2 fuzzy set.

Uncertain Fuzzy Clustering: Interval Type-2 Fuzzy Approach to $C$ -Means

Abstract: In many pattern recognition applications, it may be impossible in most cases to obtain perfect knowledge or information for a given pattern set. Uncertain information can create imperfect expressions for pattern sets in various pattern recognition algorithms. Therefore, various types of uncertainty may be taken into account when performing several pattern recognition methods. When one performs clustering with fuzzy sets, fuzzy membership values express assignment availability of patterns for clusters. However, when one assigns fuzzy memberships to a pattern set, imperfect information for a pattern set involves uncertainty which exist in the various parameters that are used in fuzzy membership assignment. When one encounters fuzzy clustering, fuzzy membership design includes various uncertainties (e.g., distance measure, fuzzifier, prototypes, etc.). In this paper, we focus on the uncertainty associated with the fuzzifier parameter m that controls the amount of fuzziness of the final C-partition in the fuzzy C-means (FCM) algorithm. To design and manage uncertainty for fuzzifier m, we extend a pattern set to interval type-2 fuzzy sets using two fuzzifiers m1 and m2 which creates a footprint of uncertainty (FOU) for the fuzzifier m. Then, we incorporate this interval type-2 fuzzy set into FCM to observe the effect of managing uncertainty from the two fuzzifiers. We also provide some solutions to type-reduction and defuzzification (i.e., cluster center updating and hard-partitioning) in FCM. Several experimental results are given to show the validity of our method

Some similarity measures of intuitionistic fuzzy sets and their applications to multiple attribute decision making

Abstract: Atanassov (1986) defined the notion of intuitionistic fuzzy set, which is a generalization of the notion of Zadeh' fuzzy set. In this paper, we first develop some similarity measures of intuitionistic fuzzy sets. Then, we define the notions of positive ideal intuitionistic fuzzy set and negative ideal intuitionistic fuzzy set. Finally, we apply the similarity measures to multiple attribute decision making under intuitionistic fuzzy environment.

Rough Set Based Generalized Fuzzy $C$ -Means Algorithm and Quantitative Indices

Abstract: A generalized hybrid unsupervised learning algorithm, which is termed as rough-fuzzy possibilistic C-means (RFPCM), is proposed in this paper. It comprises a judicious integration of the principles of rough and fuzzy sets. While the concept of lower and upper approximations of rough sets deals with uncertainty, vagueness, and incompleteness in class definition, the membership function of fuzzy sets enables efficient handling of overlapping partitions. It incorporates both probabilistic and possibilistic memberships simultaneously to avoid the problems of noise sensitivity of fuzzy C-means and the coincident clusters of PCM. The concept of crisp lower bound and fuzzy boundary of a class, which is introduced in the RFPCM, enables efficient selection of cluster prototypes. The algorithm is generalized in the sense that all existing variants of C-means algorithms can be derived from the proposed algorithm as a special case. Several quantitative indices are introduced based on rough sets for the evaluation of performance of the proposed C-means algorithm. The effectiveness of the algorithm, along with a comparison with other algorithms, has been demonstrated both qualitatively and quantitatively on a set of real-life data sets.

Models for multiple attribute decision making with intuitionistic fuzzy information

Abstract: The intuitionistic fuzzy set (IFS) characterized by a membership function and a non-membership function, was introduced by Atanassov [K. Atanassov, "Intuitionistic fuzzy sets", Fuzzy Sets and Systems 20 (1986) 87–96] as a generalization of Zadeh' fuzzy set [L. A. Zadeh, "Fuzzy Sets", Information and Control 8 (1965) 338–353] to deal with fuzziness and uncertainty. In this paper, we investigate the multiple attribute decision making (MADM) problems, in which the information about attribute weights is incomplete, and the attribute values are expressed in intuitionistic fuzzy numbers (IFNs). We first define the concept of intuitionistic fuzzy ideal solution (IFIS), and then, based on the IFIS and the distance measure, we establish some optimization models to derive the attribute weights. Furthermore, based on the developed models, we develop some procedures for the rankings of alternatives under different situations, and extend the developed models and procedures to handle the MADM problems with interval-valued intuitionistic fuzzy information. Finally, we give some illustrative examples to verify the effectiveness and practicability of the developed models and procedures.

On Geometric Aggregation over Interval-Valued Intuitionistic Fuzzy Information

Abstract: The notion of interval-valued intuitionistic fuzzy set (IVIFS) was introduced by Atanassov and Gargov as a generalization of an intuitionistic fuzzy set. The fundamental characteristic of IVIFS is that the values of its membership function and non-membership function are intervals rather than exact numbers. Some operators have been proposed for aggregating intuitionistic fuzzy sets. However, it seems that there is little investigation on aggregation techniques for dealing with interval-valued intuitionistic fuzzy information. In this work, we develop some interval-valued intuitionistic fuzzy geometric operators, such as the interval-valued intuitionistic fuzzy ordered weighted geometric (IIFOWG) operator, and interval-valued intuitionistic fuzzy hybrid geometric (IIFHG) operator, etc., which are the generalizations of the geometric aggregation operators based on intuitionistic fuzzy sets. Then we apply the developed operators to solve a multiple attribute decision-making problem involving the prioritization of a set of information technology improvement projects.

A Proposal for the Genetic Lateral Tuning of Linguistic Fuzzy Systems and Its Interaction With Rule Selection

Abstract: Linguistic fuzzy modeling allows us to deal with the modeling of systems by building a linguistic model which is clearly interpretable by human beings. However, since the accuracy and the interpretability of the obtained model are contradictory properties, the necessity of improving the accuracy of the linguistic model arises when complex systems are modeled. To solve this problem, one of the research lines in recent years has led to the objective of giving more accuracy to linguistic fuzzy modeling without losing the interpretability to a high level. In this paper, a new postprocessing approach is proposed to perform an evolutionary lateral tuning of membership functions, with the main aim of obtaining linguistic models with higher levels of accuracy while maintaining good interpretability. To do so, we consider a new rule representation scheme base on the linguistic 2-tuples representation model which allows the lateral variation of the involved labels. Furthermore, the cooperation of the lateral tuning together with fuzzy rule reduction mechanisms is studied in this paper, presenting results on different real applications. The obtained results show the good performance of the proposed approach in high-dimensional problems and its ability to cooperate with methods to remove unnecessary rules.

Computing with Words: Zadeh, Turing, Popper and Occam

Abstract: In this article, after explaining Zadeh's computing with words (CWW) paradigm, I argue that for this paradigm to be embraced, it must be validated using a Turinglike test, use a scientifically correct fuzzy set model for words, namely interval type-2 fuzzy sets (IT2 FSs), and be simple, meaning that fuzzy set operations should be as simple as possible. These conclusions are drawn using the ideas of Turing, Popper and Occam. Short descriptions are provided for a Perceptual Computer (Per-C), which is an architecture for CWW for making subjective judgments, IT2 FSs, IT2 FS models for words, and why an IT2 FS model captures first-order uncertainties about a word. Short biographies of Zadeh, Turing, Popper and Occam are also provided.

Relaxed Stability and Performance Conditions for Takagi–Sugeno Fuzzy Systems With Knowledge on Membership Function Overlap

Abstract: This correspondence presents a relaxation of some earlier linear matrix inequality (LMI) conditions, which allow setting up less conservative stability or performance conditions for Takagi-Sugeno fuzzy models Unlike the previous literature, this correspondence takes into account the knowledge of the membership functions' shape by considering bounds on them and their cross products (interpreted as an overlap measure), introducing auxiliary LMI variables Numerical examples illustrate the achieved improvements

Fuzzy inference to risk assessment on nuclear engineering systems

Abstract: This paper presents a nuclear case study, in which a fuzzy inference system (FIS) is used as alternative approach in risk analysis. The main objective of this study is to obtain an understanding of the aging process of an important nuclear power system and how it affects the overall plant safety. This approach uses the concept of a pure fuzzy logic system where the fuzzy rule base consists of a collection of fuzzy IF-THEN rules. The fuzzy inference engine uses these fuzzy IF-THEN rules to determine a mapping from fuzzy sets in the input universe of discourse to fuzzy sets in the output universe of discourse based on fuzzy logic principles. The risk priority number (RPN), a traditional analysis parameter, was calculated and compared to fuzzy risk priority number (FRPN) using scores from expert opinion to probabilities of occurrence, severity and not detection. A standard four-loop pressurized water reactor (PWR) containment cooling system (CCS) was used as example case. The results demonstrated the potential of the inference system for subsiding the failure modes and effects analysis (FMEA) in aging studies.

RFCM: A Hybrid Clustering Algorithm Using Rough and Fuzzy Sets

Abstract: A hybrid unsupervised learning algorithm, termed as rough-fuzzy c-means, is proposed in this paper. It comprises a judicious integration of the principles of rough sets and fuzzy sets. While the concept of lower and upper approximations of rough sets deals with uncertainty, vagueness, and incompleteness in class definition, the membership function of fuzzy sets enables efficient handling of overlapping partitions. The concept of crisp lower bound and fuzzy boundary of a class, introduced in rough-fuzzy c-means, enables efficient selection of cluster prototypes. Several quantitative indices are introduced based on rough sets for evaluating the performance of the proposed c-means algorithm. The effectiveness of the algorithm, along with a comparison with other algorithms, has been demonstrated on a set of real life data sets.

Uncertain Fuzzy Reasoning: A Case Study in Modelling Expert Decision Making

Abstract: This paper presents a case study in which the introduction of vagueness or uncertainty into the membership functions of a fuzzy system was investigated in order to model the variation exhibited by experts in a medical decision-making context. A conventional (type-1) fuzzy expert system had previously been developed to assess the health of infants immediately after birth by analysis of the biochemical status of blood taken from infants' umbilical cords. Variation in decision making was introduced into the fuzzy expert system by means of membership functions which altered in small, predetermined manners over time. Three types of variation in membership functions were investigated: i) variation in the centre points, ii) variation in the widths, and iii) the addition of "white noise". Different levels (amounts) of uniformly distributed random variation were investigated for each of these types. Monte Carlo simulations were carried out to propagate the variation through the inferencing process in order to determine distributions of the conclusions reached. Interval valued type-2 fuzzy systems were also implemented to investigate the boundaries of variability in decisions. The results obtained were compared to the experts' decisions in order to determine which type and size of membership function variability best matched the experts' variability. The novel reasoning technique introduced in this study is termed nonstationary fuzzy reasoning

Neurofuzzy-Based Approach to Mobile Robot Navigation in Unknown Environments

Abstract: In this paper, a neurofuzzy-based approach is proposed, which coordinates the sensor information and robot motion together. A fuzzy logic system is designed with two basic behaviors, target seeking and obstacle avoidance. A learning algorithm based on neural network techniques is developed to tune the parameters of membership functions, which smooths the trajectory generated by the fuzzy logic system. Another learning algorithm is developed to suppress redundant rules in the designed rule base. A state memory strategy is proposed for resolving the "dead cycle" problem. Under the control of the proposed model, a mobile robot can adequately sense the environment around, autonomously avoid static and moving obstacles, and generate reasonable trajectories toward the target in various situations without suffering from the "dead cycle" problems. The effectiveness and efficiency of the proposed approach are demonstrated by simulation studies.