Showing papers on "Membership function published in 2016"

Picture fuzzy cross-entropy for multiple attribute decision making problems

Abstract: In this paper, we investigate the multiple attribute decision making problems with picture fuzzy information. The advantage of picture fuzzy set is easily reflecting the ambiguous nature of subjective judgments because the picture fuzzy sets are suitable for capturing imprecise, uncertain, and inconsistent information in the multiple attribute decision making analysis. Thus, the cross entropy of picture fuzzy sets, called picture fuzzy cross entropy, is proposed as an extension of the cross entropy of fuzzy sets. Then, a multiple attribute decision making method based on the proposed picture fuzzy cross entropy is established in which attribute values for alternatives are picture fuzzy numbers. In decision making process, we utilize the picture fuzzy weighted cross entropy between the ideal alternative and an alternative to rank the alternatives corresponding to the cross entropy values and to select the most desirable one(s). Finally, a practical example for enterprise resource planning system se...

Symmetric Pythagorean Fuzzy Weighted Geometric/Averaging Operators and Their Application in Multicriteria Decision-Making Problems

Abstract: Pythagorean fuzzy sets PFSs, originally proposed by Yager, are a new tool to deal with vagueness with the square sum of the membership degree and the nonmembership degree equal to or less than 1, which have much stronger ability than Atanassov's intuitionistic fuzzy sets to model such uncertainty. In this paper, we modify the existing score function and accuracy function for Pythagorean fuzzy number to make it conform to PFSs. Associated with the given operational laws, we define some novel Pythagorean fuzzy weighted geometric/averaging operators for Pythagorean fuzzy information, which can neutrally treat the membership degree and the nonmembership degree, and investigate the relationships among these operators and those existing ones. At length, a practical example is provided to illustrate the developed operators and to make a comparative analysis.

The Properties of Continuous Pythagorean Fuzzy Information

Abstract: In practical decision-making processes, we can utilize various types of fuzzy sets to express the uncertain and ambiguous information. However, we may encounter such the situations: the sum of the support membership degree and the against nonmembership degree to which an alternative satisfies a criterion provided by the decision maker may be bigger than 1 but their square sum is equal to or less than 1. The Pythagorean fuzzy sets PFS, as the generalization of the fuzzy sets, can be used to effectively deal with this issue. Therefore, to enrich the theory of PFS, it is very necessary to investigate the fundamental properties of Pythagorean fuzzy information. In this paper, we first describe the change values of Pythagorean fuzzy numbers PFNs, which are the basic components of PFSs, when considering them as variables. Then we divide all the change values into the eight regions by using the basic operations of PFNs. Finally, we develop several Pythagorean fuzzy functions and study their fundamental properties such as continuity, derivability, and differentiability in detail.

Feature subset selection based on fuzzy neighborhood rough sets

Abstract: Rough set theory has been extensively discussed in machine learning and pattern recognition. It provides us another important theoretical tool for feature selection. In this paper, we construct a novel rough set model for feature subset selection. First, we define the fuzzy decision of a sample by using the concept of fuzzy neighborhood. A parameterized fuzzy relation is introduced to characterize fuzzy information granules for analysis of real-valued data. Then, we use the relationship between fuzzy neighborhood and fuzzy decision to construct a new rough set model: fuzzy neighborhood rough set model. Based on this model, the definitions of upper and lower approximation, boundary region and positive region are given, and the effects of parameters on these concepts are discussed. To make the new model tolerate noises in data, we introduce a variable-precision fuzzy neighborhood rough set model. This model can decrease the possibility that a sample is classified into a wrong category. Finally, we define the dependency between fuzzy decision and condition attributes and employ the dependency to evaluate the significance of a candidate feature, using which a greedy feature subset selection algorithm is designed. The proposed algorithm is compared with some classical algorithms. The experiments show that the proposed algorithm gets higher classification performance and the numbers of selected features are relatively small.

On Pythagorean and Complex Fuzzy Set Operations

Abstract: Complex fuzzy logic is a new multivalued logic system that has emerged in the last decade. At this time, there are a limited number of known instances of complex fuzzy logic, and only a partial exploration of their properties. There has also been relatively little progress in developing interpretations of complex-valued membership grades. In this paper, we address both problems by examining the recently developed Pythagorean fuzzy sets (a generalization of intuitionistic fuzzy sets). We first characterize two lattices that have been suggested for Pythagorean fuzzy sets and then extend these results to the unit disc of the complex plane. We thereby identify two new complete, distributive lattices over the unit disc, and explore interpretations of them based on fuzzy antonyms and negations.

Developing an MCDM method for robot selection with interval type-2 fuzzy sets

Abstract: Industrial robots have been increasingly used by many manufacturing firms in different industries. The selection of robots for a particular application and manufacturing environment is a difficult task for decision makers. It has become more and more complicated due to increase in complexity, advanced features and facilities that are continuously being incorporated into the robots by different manufacturers. The decision maker needs to identify and select the best-suited robot in order to achieve the desired output with respect to many criteria. In the decision-making process, we usually confront with ambiguity and uncertainty for evaluating the criteria weights and alternatives of the problem. Interval type-2 fuzzy sets which are characterized by an interval membership function can provide us with more degrees of freedom to represent the uncertainty of the real-world problems. This paper presents a multi-criteria group decision-making approach for robot selection in the context of type-2 fuzzy sets. We propose a method based on Vlsekriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method with interval type-2 fuzzy numbers to handle this problem. The best alternative (robot) is selected in the developed method according to both the ideal and the nadir solutions without defuzzification. An example with eight alternatives (robots) and seven commonly used criteria is used to illustrate the proposed method. We compare the results with some existing methods to show the validity of the extended method. Seven sets of criteria weights and the Spearman correlation coefficient are also utilized for analyzing the stability of the proposed method. It is observed that the obtained rankings of the proposed method are relatively consistent with the other methods and have good stability in different criteria weights. We focus on fuzzy multi-criteria group decision-making for robot selection.Interval type-2 fuzzy sets are considered in this research.An extended version of VIKOR method is developed for group decision-making.An example is utilized for showing the application of the proposed method in robot selection.We compare and analyze the results of the proposed method to represent the validity of it.

Picture fuzzy clustering: a new computational intelligence method

Abstract: Fuzzy clustering especially fuzzy $$C$$C-means (FCM) is considered as a useful tool in the processes of pattern recognition and knowledge discovery from a database; thus being applied to various crucial, socioeconomic applications. Nevertheless, the clustering quality of FCM is not high since this algorithm is deployed on the basis of the traditional fuzzy sets, which have some limitations in the membership representation, the determination of hesitancy and the vagueness of prototype parameters. Various improvement versions of FCM on some extensions of the traditional fuzzy sets have been proposed to tackle with those limitations. In this paper, we consider another improvement of FCM on the picture fuzzy sets, which is a generalization of the traditional fuzzy sets and the intuitionistic fuzzy sets, and present a novel picture fuzzy clustering algorithm, the so-called FC-PFS. A numerical example on the IRIS dataset is conducted to illustrate the activities of the proposed algorithm. The experimental results on various benchmark datasets of UCI Machine Learning Repository under different scenarios of parameters of the algorithm reveal that FC-PFS has better clustering quality than some relevant clustering algorithms such as FCM, IFCM, KFCM and KIFCM.

A Comprehensive Literature Review of 50 Years of Fuzzy Set Theory

Abstract: Fuzzy sets have a great progress in every scientific research area. It found many application areas in both theoretical and practical studies from engineering area to arts and humanities, from computer science to health sciences, and from life sciences to physical sciences. In this paper, a comprehensive literature review on the fuzzy set theory is realized. In the recent years, ordinary fuzzy sets have been extended to new types and these extensions have been used in many areas such as energy, medicine, material, economics and pharmacology sciences. This literature review also analyzes the chronological development of these extensions. In the last section of the paper, we present our interpretations on the future of fuzzy sets.

Process capability indices as fuzzy numbers

Abstract: Most of the traditional methods for assessing the capability of manufacturing processes are dealing with crisp quality. In this paper we discuss the fuzzy quality and introduce fuzzy process capability indices, where instead of precise quality we have two membership functions for specification limits. These indices are necessary when the specification limits are fuzzy and they are helpful for comparing manufacturing processes with fuzzy specification limits. Some interesting relations among the introduced indices are obtained. Numerical examples are given to clarify the method.

Three-way decision spaces based on partially ordered sets and three-way decisions based on hesitant fuzzy sets

Abstract: Three-way decisions on three-way decision spaces are based on fuzzy lattices, i.e. complete distributive lattices with involutive negators. However, now some popular structures, such as hesitant fuzzy sets and type-2 fuzzy sets, do not constitute fuzzy lattices. It limits applications of the theory of three-way decision spaces. So this paper attempts to generalize measurement on decision conclusion in three-way decision spaces from fuzzy lattices to partially ordered sets. First three-way decision spaces and three-way decisions are discussed based on general partially ordered sets. Then this paper points out that the collection of non-empty subset of 0, 1 and the family of hesitant fuzzy sets are both partially ordered sets. Finally this paper systematically discusses three-way decision spaces and three-way decisions based on hesitant fuzzy sets and interval-valued hesitant fuzzy sets and obtains many useful decision evaluation functions.

Extensions of Atanassov's Intuitionistic Fuzzy Interaction Bonferroni Means and Their Application to Multiple-Attribute Decision Making

Abstract: The Bonferroni mean (BM) was originally presented by Bonferroni and had been generalized by many researchers on Atanassov's intuitionistic fuzzy sets (AIFSs) for its capacity to capture the interrelationship between input arguments. Nevertheless, the forms of the combinations of the newly proposed interaction theory on AIFSs with BM are very single, and the existing BMs on AIFSs are not consistent with aggregation operations on the ordinary fuzzy sets. As complements to the existing generalizations of BM under Atanassov's intuitionistic fuzzy environment, this paper develops the extended Atanassov's intuitionistic fuzzy interaction Bonferroni mean (EIFIBM) and the extended weighted Atanassov's intuitionistic fuzzy interaction Bonferroni mean, which can evolve into a series of BMs by taking different generator functions that reflect the different preference attitudes of the decision makers. In addition, some of the EIFIBMs are consistent with aggregation operations on the ordinary fuzzy sets, and some of the EIFIBMs consider the interactions between the membership and nonmembership functions of different Atanassov's intuitionistic fuzzy sets; thus, they can be used in more decision situations. We investigate the properties of these new extensions and apply them to multiple-attribute decision-making problems with admissible orders. Finally, numerical examples show the validity and feasibility of the new approaches.

A new preference scale mcdm method based on interval-valued intuitionistic fuzzy sets and the analytic hierarchy process

Abstract: The intuitionistic fuzzy analytic hierarchy process (IF-AHP), in which intuitionistic fuzzy numbers are utilized in defining decision makers' linguistic judgment, has been used to solve various multi-criteria decision-making problems. Previous theories have suggested that interval-valued intuitionistic fuzzy numbers (IVIFN) with hesitation degree can act as alternative fuzzy numbers that can handle vagueness and uncertainty. This paper proposes a new preference scale in the framework of the interval-valued intuitionistic fuzzy analytic hierarchy process (IVIF-AHP). The comparison matrix judgment is expressed in IVIFN with degree of hesitation. The proposed new preference scale concurrently considers the membership function, the non-membership function and the degree of hesitation of IVIFN. To define the weight entropy of the aggregated matrix of IVIFN, a modified interval-valued intuitionistic fuzzy weighted averaging is proposed, by considering the interval number of the hesitation degree. Three multi-criteria decision-making problems are used to test the proposed method. A comparison of the results is also presented to check the feasibility of the proposed method. It is shown that the ranking order of the proposed method is slightly different from that of the other two methods because of the inclusion of the hesitation degree in defining the preference scale.

Design and Implementation of Type-2 Fuzzy Logic Controller for DFIG-Based Wind Energy Systems in Distribution Networks

Abstract: Handling the uncertainties in the wind speed and the grid disturbances is a major challenge to the DFIGs to fulfill the modern grid code requirements. This paper proposes the design and implementation of a novel control strategy using interval type-2 fuzzy sets for grid integration of doubly fed induction generator (DFIG) based wind turbines. The presence of third dimension in the type-2 membership function offers an additional degree of freedom in the design of the proposed controller to contribute to power oscillations damping and voltage recovery following parameter uncertainties in the network. The vector control with proposed strategy for DFIG is able to handle uncertainties in the operating conditions of the distributed network like faults, load changes, and wind speed. The performance of the controller is evaluated by connecting the wind turbine to IEEE 34-bus test system considering the various uncertainties. The real time simulations are carried out using real time digital simulator (RTDS) with hardware in loop (HIL) configuration to support the feasibility of the controller for real time applications.

An Extended PROMETHE II Multi-Criteria Group Decision Making Technique Based on Intuitionistic Fuzzy Logic for Sustainable Energy Planning

Abstract: The implementations of Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE) category to complex multi-criteria group decision making (MCGDM) scenarios have been included in thousands areas. Outranking methods such as PROMETHEE II are also greatly employed in energy planning application. In MCGDM methods if decision makers (DMs) are not able to treat precise data in order to define their preferences, the intuitionistic fuzzy set (IFS) theory enables them. IFS attributes are connected with the degree of membership and non-membership, and can be used to draw uncertainty in group decision-making situations. In this paper, a new version of the PROMETHEE II method is proposed, aiming at solving MCGDM problems. Linguistic variables are expressed in the membership function and non-membership function of IFS which are used to assess the weights of all criteria and the ratings of each alternative with respect to each criteria. Conditional normalized Euclidean distance measure is adopted to measure deviations between alternatives on intuitionistic fuzzy set. Then, a ranking algorithm is applied to indicate the order of superiority of alternatives. Finally, a practical example is given to an application of sustainable energy planning to verify our proposed method. Additionally, a comparative analysis is done among the proposed PROMETHEE II method and the intuitionistic fuzzy technique for order preference by similarity to ideal solution (IF-TOPSIS) method and elimination and choice translating reality method (IF-ELECTRE).

Designing Fuzzy Sets With the Use of the Parametric Principle of Justifiable Granularity

Abstract: This study is concerned with a design of membership functions of fuzzy sets. The membership functions are formed in such a way that they are experimentally justifiable and exhibit a sound semantics. These two requirements are articulated through the principle of justifiable granularity. The parametric version of the principle is discussed in detail. We show linkages with type-2 fuzzy sets, which are constructed on a basis of type-1 fuzzy sets. Several experimental studies are reported, which illustrate a behavior of the introduced method.

Fuzzy probabilistic rough sets and their corresponding three-way decisions

Abstract: The concept of probabilistic rough sets, as a main subject of this paper, is intimately connected with the concept of decision-theoretic rough sets. This paper investigates fuzzy and interval-valued fuzzy probabilistic rough sets within frameworks of fuzzy and interval-valued fuzzy probabilistic approximation spaces, respectively. Four types of fuzzy probabilistic rough sets as well as interval-valued fuzzy probabilistic rough sets are established in terms of different constraints on parameters. To find a suitable way of explaining and determining these parameters in each model, three-way decisions are studied based on Bayesian minimum risk decision procedure, i.e., the decision-theoretic rough set approach. The proposed models in this paper broaden applications of probabilistic rough sets due to their abilities of directly dealing with real-valued and interval-valued data.

Fuzzy multicriteria decision making method based on the improved accuracy function for interval-valued intuitionistic fuzzy sets

Abstract: A new accuracy function for the theory of interval-valued intuitionistic fuzzy set, which overcomes some difficulties arising in the existing methods for determining rank of interval-valued intuitionistic fuzzy numbers, is proposed by taking into account the hesitancy degree of interval-valued intuitionistic fuzzy sets. By comparing it with several proposed accuracy functions, the necessity and efficiency of our accuracy function are provided by giving related examples. A fuzzy multicriteria decision making method is established to select the best alternative in multicriteria decision making process which is taken as interval-valued intuitionistic fuzzy set of criterion values for alternatives. While aggregating the interval-valued intuitionistic fuzzy information corresponding to each alternative, we utilize the interval-valued intuitionistic fuzzy weighted aggregation operators. Then the accuracy degree of the aggregated interval-valued intuitionistic fuzzy information is computed via the new proposed accuracy function. Thus, we can rank all the alternatives according to the accuracy function and choose the optimal one(s). Finally, an illustrative example is given to demonstrate the practicality and effectiveness of the proposed approach.

Fuzzy inference-based fall detection using kinect and body-worn accelerometer

Abstract: Graphical abstractDisplay Omitted HighlightsA new approach for reliable fall detection.In case of potential fall a threshold-based algorithm launches the fuzzy system to authenticate the fall event. The fuzzy system consists of two input Mamdani engines and a triggering alert Sugeno engine.The output of the first engine is a fuzzy set, which assigns grades of membership to the possible values of dynamic transitions, whereas the output of the second one is another fuzzy set assigning membership grades to possible body poses.Since the Mamdani engines perform fuzzy reasoning on disjoint subsets of the linguistic variables, the total number of the fuzzy rules needed for input-output mapping is far smaller. In this paper, we present a new approach for reliable fall detection. The fuzzy system consists of two input Mamdani engines and a triggering alert Sugeno engine. The output of the first Mamdani engine is a fuzzy set, which assigns grades of membership to the possible values of dynamic transitions, whereas the output of the second one is another fuzzy set assigning membership grades to possible body poses. Since Mamdani engines perform fuzzy reasoning on disjoint subsets of the linguistic variables, the total number of the fuzzy rules needed for input-output mapping is far smaller. The person pose is determined on the basis of depth maps, whereas the pose transitions are inferred using both depth maps and the accelerations acquired by a body worn inertial sensor. In case of potential fall a threshold-based algorithm launches the fuzzy system to authenticate the fall event. Using the accelerometric data we determine the moment of the impact, which in turn helps us to calculate the pose transitions. To the best of our knowledge, this is a new application of fuzzy logic in a novel approach to modeling and reliable low cost detecting of falls.

Medical diagnosis with the aid of using fuzzy logic and intuitionistic fuzzy logic

Abstract: The objective of the present study is to develop/establish a web-based medical diagnostic support system (MDSS) by which health care support can be provided for people living in rural areas of a country. In this respect, this research provides a novel approach for medical diagnosis driven by integrating fuzzy and intuitionistic fuzzy (IF) frameworks. Subsequently, based on the proposed approach a web-based MDSS is developed. The proposed MDSS comprises of a knowledge base (KB) and intuitionistic fuzzy inference system (IFIS). Based on the observation that medical data cannot be described with both precision and certainty, a medical KB is constructed in the form of a set of if-then decision rules by employing both fuzzy and IF logics. After constructing the medical KB, a new set of patients is considered for diagnosing the diseases. For each patient, linguistic values of the patients' symptoms are considered as inputs of the proposed IFIS and modeled by using the generalized triangular membership functions. Subsequently, integrated fuzzy and IF rule-based inference system is used to find a valid conclusion for the new set of patients. In a nutshell, in this paper fuzzy rule-based and IFS based inference systems are combined for better and more realistic representation of uncertainty of the medical diagnosis problem and for more accurate diagnostic result. The method is composed of following four steps: (1) the modeling of antecedent part of the rules, which consist of linguistic assessments of the patients' symptoms provided by the doctors/medical experts with their corresponding confidence levels, by using generalized fuzzy numbers; (2) the modeling of consequent part, which reveals the degree of association and the degree of non-association of diseases into the patient, by using IFSs; (3) the use of IF aggregation operator in inference process; (4) the application of relative closeness function to find the final crisp output for a given diagnosis. Finally, the applicability of the proposed approach is illustrated with a suitable case study. This article has also justified the proposed approach by using similarity measurement.

A hybrid model based on differential fuzzy logic relationships and imperialist competitive algorithm for stock market forecasting

Abstract: Graphical abstractDisplay Omitted HighlightsA differential fuzzy time series model is defined for forecast inside trend data.The differential fuzzy logical relationships and groups are established.The actual value at former state is added to the average of defuzzified values.The proposed method is combined with the ICA to enhance forecast accuracy.Frothy case studies are employed for validating the proposed method. In this study, a new kind of fuzzy set in fuzzy time series' field is introduced. It works as a trend estimator to be appropriate for fuzzy time series forecasting by reconnoitering trend of data appropriately. First, the historical data are fuzzified into differential fuzzy sets, and then differential fuzzy relationships are calculated. Second, differential fuzzy logic groups are established by grouping differential fuzzy relationships. Finally, in the defuzzification step, the forecasts are calculated. However, for increasing the accuracy of the models, an evolutionary algorithm, namely imperialist competitive algorithm is injected, to train the model. A massive stock data from four main stock databases have been selected for model validation. The final project, has shown that outperformed its counterparts in term of accuracy.

Modeling Fuzzy and Interval Fuzzy Preferences Within a Graph Model Framework

Abstract: A methodology is developed to model a decision maker's (DM's) fuzzy and/or interval fuzzy preference over feasible scenarios or states within the framework of the graph model for conflict resolution. This technique uses the DM's fuzzy relative importance of its preference statements and their fuzzy truth values for the feasible states in a conflict under uncertain conditions. A preference statement of a DM is a preferable combination of DM's options or courses of action. The fuzzy importance for one preference statement over another, a value in the interval $[0, 1]$ , is interpreted as the degree to which the first preference statement is more important than the second to the DM. A fuzzy truth value of a preference statement at a feasible state is a number in the interval $[0, 1]$ that represents the degree to which the statement is true at the state. When the DM is confident in its pairwise fuzzy importance degrees over the preference statements and their fuzzy truth values at the feasible states, the methodology provides a fuzzy preference over the states. When there is an ordinary or crisp importance ordering of preference statements and when the truth values of preference statements are classical or crisp at the feasible states, the technique generates a crisp preference over the states. The methodology is illustrated using a case study.

A novel approach for analyzing the reliability of series-parallel system using credibility theory and different types of intuitionistic fuzzy numbers

Abstract: The main objective of this paper is to present an alternative method for computing the membership function of the system under intuitionistic fuzzy set environment. Conventionally, it is not always easy to obtain the system reliability for components with different individual failure probability density function due to necessary but complicated combination and integration of various probability density functions. Also, in the literature, it is assumed that failure rates of all the components of a system follow the same type of fuzzy set which is rarely occurring in the practical situations. Thus, this paper addresses the fuzzy system reliability analysis to construct the membership and non-membership functions by considering the different types of intuitionistic fuzzy failure rates. Functions of intuitionistic fuzzy numbers are calculated using credibility theory. The effectiveness of the proposed approach is illustrated with analyze of the fuzzy reliability of series, parallel and series-parallel systems using different types of intuitionistic fuzzy failure rates. The computed results from the analysis have a less range of uncertainty as the comparability of existing results.