Traditional Fuzzy Set Research Articles

Complex Fuzzy Geometric Aggregation Operators

A complex fuzzy set is an extension of the traditional fuzzy set, where traditional [0,1]-valued membership grade is extended to the complex unit disk. The aggregation operator plays an important role in many fields, and this paper presents several complex fuzzy geometric aggregation operators. We show that these operators possess the properties of rotational invariance and reflectional invariance. These operators are also closed on the upper-right quadrant of the complex unit disk. Based on the relationship between Pythagorean membership grades and complex numbers, these operators can be applied to the Pythagorean fuzzy environment.

The Basis and Coordinates in Intuitionistic Fuzzy Environment

In 1983, Atanassov extended the fuzzy set and originally introduced the concept of intuitionistic fuzzy set (IFS). The basic elements in every IFS are called intuitionistic fuzzy numbers, each of which consists of a membership degree, a non-membership degree and a hesitancy degree. Compared with the traditional fuzzy set, the IFS is more flexible and practical to deal with the ambiguity and uncertainty. In this paper, we first try to define the basis in intuitionistic fuzzy environment and then study some of its relevant properties. After that, we propose the coordinates based on the basis and present some theorems of the coordinates. Finally, we show how to find the smallest basis of any region.

A New Feature Selection Method based on Intuitionistic Fuzzy Entropy to Categorize Text Documents

Selection of highly discriminative feature in text document plays a major challenging role in categorization. Feature selection is an important task that involves dimensionality reduction of feature matrix, which in turn enhances the performance of categorization. This article presents a new feature selection method based on Intuitionistic Fuzzy Entropy (IFE) for Text Categorization. Firstly, Intuitionistic Fuzzy C-Means (IFCM) clustering method is employed to compute the intuitionistic membership values. The computed intuitionistic membership values are used to estimate intuitionistic fuzzy entropy via Match degree. Further, features with lower entropy values are selected to categorize the text documents. To find the efficacy of the proposed method, experiments are conducted on three standard benchmark datasets using three classifiers. F-measure is used to assess the performance of the classifiers. The proposed method shows impressive results as compared to other well known feature selection methods. Moreover, Intuitionistic Fuzzy Set (IFS) property addresses the uncertainty limitations of traditional fuzzy set.

An analysis approach to handle uncertain multi-criteria group decision problems in the framework of interval type-2 fuzzy sets theory

Uncertainty is one of the most important aspects of any decision-making process. One of the tools lately used in uncertain decision making is interval type-2 fuzzy sets (IT2FSs). These sets unlike the traditional fuzzy sets have more flexibility in expressing uncertainty and can be applied in situations in which the level of vagueness is very high. Unfortunately, most of the recently published papers on this issue are concerned with extending the existing fuzzy decision-making methods to type-2 fuzzy environment and the improvements were mainly focused on improving the applied fuzzy tool from classic sets to type-2 fuzzy sets. This paper offers a new analysis approach in an uncertain decision-making process that has enhancements in areas of the soft computing. In other words, this paper introduces an IT2F-based approach that has several novelties. Firstly, the decision-making method presents a new approach in computing the decision makers’ weights. This approach gives ideas of each decision maker a weight based on the gathered judgments. Secondly, the approach employs the concept of relative preference relation to evaluate the importance of each criterion. Thirdly, a novel decision-making index is introduced to rank the alternatives. The application of this novel last aggregation method is illustrated through solving an existing example in the literature on e-waste recycling programs assessment in Sri Lanka.

Typhoon disaster assessment based on Dombi hesitant fuzzy information aggregation operators

Hesitant fuzzy set is an extension of the traditional fuzzy set, and it has the membership function which was expressed by several possible numbers. Since it was introduced, it has been received wide attention from scholars due to its powerful ability in describing the uncertainty. In this paper, we first introduce some new operations on hesitant fuzzy elements (HFEs) based on Dombi t-conorm and t-norm and then proposed some new aggregation operators for HFEs, such as hesitant fuzzy Dombi weighted averaging, hesitant fuzzy Dombi ordered weighted averaging, hesitant fuzzy Dombi weighted geometric, hesitant fuzzy Dombi ordered weighted geometric, hesitant fuzzy Dombi hybrid averaging and hesitant fuzzy Dombi hybrid geometric operator. Finally, a multiple attribute group decision-making approach under hesitant fuzzy environment is presented based on these proposed operators. A real example about typhoon disaster assessment is presented to show the advantages of the proposed method.

Empirical Fuzzy Sets

In this paper, we introduce a new form of describing fuzzy sets (FSs) and a new form of fuzzy rule-based (FRB) systems, namely, empirical fuzzy sets (eFSs) and empirical fuzzy rule-based (eFRB) systems. Traditionally, the membership functions (MFs), which are the key mathematical representation of FSs, are designed subjectively or extracted from the data by clustering projections or defined subjectively. eFSs, on the contrary, are described by the empirically derived membership functions (eMFs). The new proposal made in this paper is based on the recently introduced Empirical Data Analytics (EDA) computational framework and is closely linked with the density of the data. This allows to keep and improve the link between the objective data and the subjective labels, linguistic terms, and classes definition. Furthermore, eFSs can deal with heterogeneous data combining categorical with continuous and/or discrete data in a natural way. eFRB systems can be extracted from data including data streams and can have dynamically evolving structure. However, they can also be used as a tool to represent expert knowledge. The main difference from the traditional FSs and FRB systems is that the expert does not need to define the MF per variable; instead, possibly multimodal, densities will be extracted automatically from the data and used as eMFs in a vector form for all numerical variables. This is done in a seamless way whereby the human involvement is only required to label the classes and linguistic terms. Moreover, even this intervention is optional. Thus, the proposed new approach to define and design the FSs and FRB systems puts the human “in the driving seat.” Instead of asking experts to define features and MFs correspondingly, to parameterize them, to define algorithm parameters, to choose types of MFs, or to label each individual item, it only requires (optionally) to select prototypes from data and (again, optionally) to label them. Numerical examples as well as a naive empirical fuzzy (eF) classifier are presented with an illustrative purpose. Due to the very fundamental nature of the proposal, it can have a very wide area of applications resulting in a series of new algorithms such as eF classifiers, eF predictors, eF controllers, and so on. This is left for the future research.

Normal Type-2 Fuzzy Geometric Curve Modeling: A Literature Review

Type-2 Fuzzy Set Theory (T2FST) is widely used for defining uncertainty data points rather than the traditional fuzzy set theory (type-1) since 2001. Recently, T2FST is used in many fields due to its ability to handle complex uncertainty data. In this paper, a review of normal type-2 fuzzy geometric curve modeling methods and techniques is presented. In particular, there have been recent applications of Normal Type-2 Fuzzy Set Theory (NT2FST) in geometric modeling, where it has helped improving results over type-1 fuzzy sets. In this paper, a concise and representative review of the processes in normal type-2 fuzzy geometrical curve modeling such as the fuzzification is presented.

Multi-attribute decision making based on a novel IF point operator

Intuitionistic fuzzy (IF) point operators transform an intuitionistic fuzzy set to a fuzzy set, or an intuitionistic fuzzy set with a smaller degree of hesitance. The purpose of which is to empower the operations of intuitionistic fuzzy sets with traditional fuzzy set methodologies. In the present paper, a new IF point operator is proposed. Different from the existing IF point operators, the new approach includes two parametric functions with respect to the number of iterations. It is proved that the newly proposed method could be degenerated to the traditional IF point operators. Meanwhile, the new IF point operator can extend our horizons in the exploration of the relationship between fuzzy sets and intuitionistic fuzzy sets. Specifically, a special case of the new IF point operator would be entitled as the memoryless multi-stage voting method, with which the decision makers are assumed to be affected only by the outcomes of the latest round of voting. It is proved that the memoryless multi-stage voting method converges to the limit produced by the infinite number of iterations much faster than the traditional voting method. Furthermore, a numerical example is employed to demonstrate the validity and performance of the memoryless multi-stage voting approach.

A multi-attribute model for construction site layout using intuitionistic fuzzy logic

Abstract Most researchers have concentrated on studying optimization models to produce optimal construction site layout plans using different algorithms, while the overall method for evaluating and selecting the best site layout generated from optimization models has received less attention. In an optimization model, construction cost is generally considered in the objective function. However, several objectives, such as security and tie-in with external transportation, are difficult to quantify in the objective function and were not considered in previous studies. This paper focuses on evaluating and selecting the construction site layout considering qualitative objectives. An intuitionistic fuzzy multi-attribute decision-making model is developed that combines intuitionistic fuzzy set theory and the technique for order preference by similarity to the ideal solution (TOPSIS). This model overcomes the shortcomings of a traditional fuzzy set when describing ambiguous and unclear circumstances by using membership functions. The application of this model for site layout selection is shown to be reasonable and effective based on data from a real construction project.

Generalized picture distance measure and applications to picture fuzzy clustering

Picture fuzzy set (PFS), which is a generalization of traditional fuzzy set and intuitionistic fuzzy set, shows great promises of better adaptation to many practical problems in pattern recognition, artificial life, robotic, expert and knowledge-based systems than existing types of fuzzy sets. An emerging research trend in PFS is development of clustering algorithms which can exploit and investigate hidden knowledge from a mass of datasets. Distance measure is one of the most important tools in clustering that determine the degree of relationship between two objects. In this paper, we propose a generalized picture distance measure and integrate it to a novel hierarchical picture fuzzy clustering method called Hierarchical Picture Clustering (HPC). Experimental results show that the clustering quality of the proposed algorithm is better than those of the relevant ones.

Cooperative grey games and an application on economic order quantity model

Purpose – The purpose of this paper is to extend the results of Meca et al. (2004) depending on the grey information revealed by the individual firms. Design/methodology/approach – The authors introduce cooperative grey games and focus on sharing ordering cost rule (SOC-rule) to distribute the joint cost. Findings – In this study, the authors introduce a model, where inventory costs are assumed as grey numbers instead of crisp or stochastic ones studied in literature. At first, grey numbers and classical cooperative inventory games are recalled. Then, cooperative grey games are introduced and related results are given. Finally, an application is performed for three shotgun companies in Turkey. Originality/value – It is an effective approach for theoretical analysis of systems with imprecise information and incomplete samples. Therefore, grey system theory, rather than the traditional probability theory and fuzzy set theory, is better suited to model the inventory problems by using cooperative game theory. To the best of the knowledge no study exists modeling inventory situations by using cooperative grey games. From this point of view this study is a pioneering work on a promising topic.

An interval neutrosophic linguistic multi-criteria group decision-making method and its application in selecting medical treatment options

Selecting medical treatments is a critical activity in medical decision-making. Usually, medical treatments are selected by doctors, patients, and their families based on various criteria. Due to the subjectivity of decision-making and the large volume of information available, accurately and comprehensively evaluating information with traditional fuzzy sets is impractical. Interval neutrosophic linguistic numbers (INLNs) can be effectively used to evaluate information during the medical treatment selection process. In this study, a medical treatment selection method based on prioritized harmonic mean operators in an interval neutrosophic linguistic environment, in which criteria and decision-makers are assigned different levels of priority, is developed. First, the rectified linguistic scale functions of linguistic variables, new INLN operations, and an INLN comparison method are developed in order to prevent data loss and distortion during the aggregation process. Next, a generalized interval neutrosophic linguistic prioritized weighted harmonic mean operator and a generalized interval neutrosophic linguistic prioritized hybrid harmonic mean operator are developed in order to aggregate the interval neutrosophic linguistic information. Then, these operators are used to develop an interval neutrosophic linguistic multi-criteria group decision-making method. In addition, the proposed method is applied to a practical treatment selection method. Furthermore, the ranking results are compared to those obtained using a traditional approach in order to confirm the practicality and accuracy of the proposed method.

A narrow band interval type-2 fuzzy approach for image segmentation

With the extension of traditional fuzzy sets into interval type-2 fuzzy sets, uncertain knowledge or information of an image will be well described by uncertain membership values.Pixels within a small selected region are updated at each iteration, which enables the contour to evolve gradually and reduces the computational cost on the employment of the interval type-2 fuzzy set.The spatial constraint providing more image details has been intensively used in image segmentation algorithms. In the proposed model, the weighted fuzzy factor which takes both spatial and gray constraints into consideration is designed for computing the type-2 fuzzy membership value. Traditional fuzzy sets capture vagueness through precise numeric membership degrees. This poses a dilemma of excessive precision in describing uncertain phenomenon. Interval type-2 fuzzy sets have shown its effectiveness in handling uncertainties in comparison to the traditional fuzzy sets. In this paper, the interval type-2 fuzzy approach is introduced into the framework of active contour model, which effectively segment images with large uncertainties. However, the computational cost is largely increased by employing the interval type-2 fuzzy set. Therefore, we try to update the pixels within a narrow band region near the contour boundary for reducing the computational cost caused by employing the interval type-2 fuzzy set. Moreover, both spatial and gray constraints are taken into consideration when calculating the fuzzy membership value to retain more image details. Experimental results on synthetic and real images show that the proposed method is effective and efficient, and is relatively independent of initial conditions.

Fuzzy-valued linguistic soft set theory and multi-attribute decision-making application

In this work, we propose the theory of fuzzy linguistic soft set (FLSS) to represent the uncertainty and multi-angle of view when decision makers evaluate an object during decision-making. FLSS integrates fuzzy set theory, linguistic variable and soft set theory. It allows decision makers to utilize linguistic variables to evaluate an object and utilize fuzzy values to describe the corresponding grade of their support of their decisions. Meanwhile, because of the flexibility of soft set, decision makers can use more than one pair of fuzzy-linguistic evaluations to express their opinions from multiple perspectives directly, if necessary. Therefore, it is more flexible and practical than traditional fuzzy set or 2-dimension uncertainty linguistic variable. We also develop a generalized weighted aggregation operator for FLSSs to solve corresponding decision-making issues. Finally, we give a numerical example to verify the practicality and effectiveness of the proposed method.

A comprehensive review of multi criteria decision making approaches based on interval type-2 fuzzy sets

Multi criteria decision making (MCDM) is a discipline of operations research which has widely studied by researchers and practitioners. It deals with evaluating and ranking alternatives from the best to the worst under conflicting criteria with respect to decision maker(s) preferences. Since, many real-world systems include uncertainty and vagueness in information, MCDM uses fuzzy sets. In recent years, as an extension of the traditional fuzzy sets concept, type-2 fuzzy sets are preferred to have the capability of handling more uncertainty, and hence, to produce more accurate and robust results, MCDM approaches based on interval type-2 fuzzy sets (IT2FSs) have been published in various subjects. This paper reviews 82 different papers using various MCDM approaches based on IT2FSs which are classified into 35 categories. All papers with respect to single and hybrid approaches are discussed, pointing out their real applications or empirical results and limitations. Furthermore, the papers are statistically analyzed to show new trends within the context of IT2FSs. This systematic and comprehensive review study provides an insight for researchers on interval type-2 fuzzy MCDM in terms of showing current state and potential areas to be focused in the future.

Picture fuzzy clustering: a new computational intelligence method

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.

HIFCF: An effective hybrid model between picture fuzzy clustering and intuitionistic fuzzy recommender systems for medical diagnosis

The health care support system is a special type of recommender systems that play an important role in medical sciences nowadays. This kind of systems often provides the medical diagnosis function based on the historic clinical symptoms of patients to give a list of possible diseases accompanied with the membership values. The most acquiring disease from that list is then determined by clinicians’ experience expressed through a specific defuzzification method. An important issue in the health care support system is increasing the accuracy of the medical diagnosis function that involves the cooperation of fuzzy systems and recommender systems in the sense that uncertain behaviors of symptoms and the clinicians’ experience are represented by fuzzy memberships whilst the determination of the possible diseases is conducted by the prediction capability of recommender systems. Intuitionistic fuzzy recommender systems (IFRS) are such the combination, which results in better accuracy of prediction than the relevant methods constructed on either the traditional fuzzy sets or recommender system only. Based upon the observation that the calculation of similarity in IFRS could be enhanced by the integration with the information of possibility of patients belonging to clusters specified by a fuzzy clustering method, in this paper we propose a novel hybrid model between picture fuzzy clustering and intuitionistic fuzzy recommender systems for medical diagnosis so-called HIFCF (Hybrid Intuitionistic Fuzzy Collaborative Filtering). Experimental results reveal that HIFCF obtains better accuracy than IFCF and the standalone methods of intuitionistic fuzzy sets such as De, Biswas & Roy, Szmidt & Kacprzyk, Samuel & Balamurugan and recommender systems, e.g. Davis et al. and Hassan & Syed. The significance and impact of the new method contribute not only the theoretical aspects of recommender systems but also the applicable roles to the health care support systems.

Decomposed fuzzy systems and their application in direct adaptive fuzzy control.

In this paper, a novel fuzzy structure termed as the decomposed fuzzy system (DFS) is proposed to act as the fuzzy approximator for adaptive fuzzy control systems. The proposed structure is to decompose each fuzzy variable into layers of fuzzy systems, and each layer is to characterize one traditional fuzzy set. Similar to forming fuzzy rules in traditional fuzzy systems, layers from different variables form the so-called component fuzzy systems. DFS is proposed to provide more adjustable parameters to facilitate possible adaptation in fuzzy rules, but without introducing a learning burden. It is because those component fuzzy systems are independent so that it can facilitate minimum distribution learning effects among component fuzzy systems. It can be seen from our experiments that even when the rule number increases, the learning time in terms of cycles is still almost constant. It can also be found that the function approximation capability and learning efficiency of the DFS are much better than that of the traditional fuzzy systems when employed in adaptive fuzzy control systems. Besides, in order to further reduce the computational burden, a simplified DFS is proposed in this paper to satisfy possible real time constraints required in many applications. From our simulation results, it can be seen that the simplified DFS can perform fairly with a more concise decomposition structure.

An outranking approach for multi-criteria decision-making with hesitant fuzzy linguistic term sets

The main purpose of this paper is to suggest an approach of multi-criteria decision-making that combines a hesitant fuzzy linguistic term set (HFLTS) with an outranking method involving systematic comparisons of the assessment values of alternatives for each criterion. A HFLTS is more suitable than the traditional fuzzy linguistic set for describing decision-makers’ preferences when they prefer to use several linguistic terms in expressions. After reviewing existing studies of HFLTSs, a novel outranking relational system of HFLTSs is developed and dominance relations for HFLTSs are introduced using outranking degrees of HFLTSs. In order to verify the efficacy of the developed method, an illustrative example is given and comparisons are made between the proposed method and other existing ones.

Enhancing clustering quality of geo-demographic analysis using context fuzzy clustering type-2 and particle swarm optimization

Geo-Demographic Analysis, which is one of the most interesting inter-disciplinary research topics between Geographic Information Systems and Data Mining, plays a very important role in policies decision, population migration and services distribution. Among some soft computing methods used for this problem, clustering is the most popular one because it has many advantages in comparison with the rests such as the fast processing time, the quality of results and the used memory space. Nonetheless, the state-of-the-art clustering algorithm namely FGWC has low clustering quality since it was constructed on the basis of traditional fuzzy sets. In this paper, we will present a novel interval type-2 fuzzy clustering algorithm deployed in an extension of the traditional fuzzy sets namely Interval Type-2 Fuzzy Sets to enhance the clustering quality of FGWC. Some additional techniques such as the interval context variable, Particle Swarm Optimization and the parallel computing are attached to speed up the algorithm. The experimental evaluation through various case studies shows that the proposed method obtains better clustering quality than some best-known ones.