Showing papers on "Fuzzy number published in 2013"

Pythagorean fuzzy subsets

Abstract: We introduce a new class of non-standard fuzzy subsets called Pythagorean fuzzy subsets and the related idea of Pythagorean membership grades. We focus on the negation operation and its relationship to the Pythagorean theorem. We compare Pythagorean fuzzy subsets with intuitionistic fuzzy subsets. We look at the basic set operations for the Pythagorean fuzzy subsets.

Picture fuzzy sets - A new concept for computational intelligence problems

Abstract: Since Zadeh introduced fuzzy sets in 1965, a lot of new theories treating imprecision and uncertainty have been introduced. Some of these theories are extensions of fuzzy set theory, other try to handle imprecision and uncertainty in different way. In this paper, we introduce a new notion of picture fuzzy sets (PFS), which are directly extensions of fuzzy sets and of intuitonistic fuzzy sets (Atanassov). Then some operations on picture fuzzy sets are defined and some properties of these operations are considered. Here the basic preliminaries of PFS theory are presented.

Comprehensive flood risk assessment based on set pair analysis-variable fuzzy sets model and fuzzy AHP

Abstract: On the basis of the disaster system theory and comprehensive analysis of flood risk factors, including the hazard of the disaster-inducing factors and disaster-breeding environment, as well as the vulnerability of the hazards-bearing bodies, the primary risk assessment index system of flood diversion district as well as its assessment standards were established. Then, a new model for comprehensive flood risk assessment was put forward in this paper based on set pair analysis (SPA) and variable fuzzy sets (VFS) theory, named set pair analysis-variable fuzzy sets model (SPA-VFS), which determines the relative membership degree function of VFS by using SPA method and has the advantages of intuitionist course, simple calculation and good generality application. Moreover, the analytic hierarchy process (AHP) was combined with trapezoidal fuzzy numbers to calculate the weights of assessment indices, thus the weights for flood hazard and flood vulnerability were determined by the fuzzy AHP procedure, respectively. Then SPA-VFS were applied to calculate the flood hazard grades and flood vulnerability grades with rank feature value equation and the confidence criterion, respectively. Under the natural disasters risk expression recommended by the Humanitarian Affairs Department of United Nations, flood risk grades were achieved from the flood hazard grades and flood vulnerability grades with risk grade classification matrix, where flood hazard, flood vulnerability and flood risk were all classified into five grades as very low, low, medium, high and very high. Consequently, integrated flood risk maps could be carried out for flood risk management and decision-making. Finally, SPA-VFS and fuzzy AHP were employed for comprehensive flood risk assessment of Jingjiang flood diversion district in China, and the computational results demonstrate that SPA-VFS is reasonable, reliable and applicable, thus has bright prospects of application for comprehensive flood risk assessment, and moreover has potential to be applicable to comprehensive risk assessment of other natural disasters with no much modification.

Generalized hesitant fuzzy sets and their application in decision support system

Abstract: Hesitant fuzzy sets are very useful to deal with group decision making problems when experts have a hesitation among several possible memberships for an element to a set. During the evaluating process in practice, however, these possible memberships may be not only crisp values in [0,1], but also interval values. In this study, we extend hesitant fuzzy sets by intuitionistic fuzzy sets and refer to them as generalized hesitant fuzzy sets. Zadeh's fuzzy sets, intuitionistic fuzzy sets and hesitant fuzzy sets are special cases of the new fuzzy sets. We redefine some basic operations of generalized hesitant fuzzy sets, which are consistent with those of hesitant fuzzy sets. Some arithmetic operations and relationships among them are discussed as well. We further introduce the comparison law to distinguish two generalized hesitant fuzzy sets according to score function and consistency function. Besides, the proposed extension principle enables decision makers to employ aggregation operators of intuitionistic fuzzy sets to aggregate a set of generalized hesitant fuzzy sets for decision making. The rationality of applying the proposed techniques is clarified by a practical example. At last, the proposed techniques are devoted to a decision support system.

Some intuitionistic fuzzy Einstein hybrid aggregation operators and their application to multiple attribute decision making

Abstract: Intuitionistic fuzzy information aggregation plays an important part in intuitionistic fuzzy set theory, which has emerged to be a new research direction receiving more and more attention in recent years. In this paper, we investigate the multiple attribute decision making (MADM) problems with intuitionistic fuzzy numbers. Then, we first introduce some operations on intuitionistic fuzzy sets, such as Einstein sum, Einstein product, and Einstein exponentiation, and further develop some new Einstein hybrid aggregation operators, such as the intuitionistic fuzzy Einstein hybrid averaging (IFEHA) operator and intuitionistic fuzzy Einstein hybrid geometric (IFEHG) operator, which extend the hybrid averaging (HA) operator and the hybrid geometric (HG) operator to accommodate the environment in which the given arguments are intuitionistic fuzzy values. Then, we apply the intuitionistic fuzzy Einstein hybrid averaging (IFEHA) operator and intuitionistic fuzzy Einstein hybrid geometric (IFEHG) operator to deal with multiple attribute decision making under intuitionistic fuzzy environments. Finally, some illustrative examples are given to verify the developed approach and to demonstrate its practicality and effectiveness.

Approaches for Reducing the Computational Cost of Interval Type-2 Fuzzy Logic Systems: Overview and Comparisons

Abstract: Interval type-2 fuzzy logic systems (IT2 FLSs) have demonstrated better abilities to handle uncertainties than their type-1 (T1) counterparts in many applications; however, the high computational cost of the iterative Karnik-Mendel (KM) algorithms in type-reduction means that it is more expensive to deploy IT2 FLSs, which may hinder them from certain cost-sensitive real-world applications. This paper provides a comprehensive overview and comparison of three categories of methods to reduce their computational cost. The first category consists of five enhancements to the KM algorithms, which are the most popular type-reduction algorithms to date. The second category consists of 11 alternative type-reducers, which have closed-form representations and, hence, are more convenient for analysis. The third category consists of a simplified structure for IT2 FLSs, which can be combined with any algorithms in the first or second category for further computational cost reduction. Experiments demonstrate that almost all methods in these three categories are faster than the KM algorithms. This overview and comparison will help researchers and practitioners on IT2 FLSs choose the most suitable structure and type-reduction algorithms, from a computational cost perspective. A recommendation is given in the conclusion.

Group multi-criteria decision making based upon interval-valued fuzzy numbers: An extension of the MULTIMOORA method

Abstract: Multi criteria decision making (MCDM) often involves uncertainty which can be tackled by employing the fuzzy set theory. Type-2 fuzzy sets offer certain additional means for the latter purpose. This paper therefore extends the MULTIMOORA method with type-2 fuzzy sets viz. generalized interval-valued trapezoidal fuzzy numbers. The proposed method thus provides the means for multi-criteria decision making related to uncertain assessments. Utilization of aggregation operators also enables to facilitate group multi-criteria decision making. A numerical example of personnel selection demonstrates the possibilities of application of the proposed method in the field of human resource management and performance management in general.

Extended TODIM method for hybrid multiple attribute decision making problems

Abstract: TODIM (an acronym in Portuguese of interactive and multiple attribute decision making) is a method for solving the multiple attribute decision making (MADM) problem considering decision maker's (DM's) behavior, in which the attribute values are in the format of crisp numbers. It cannot be used to handle hybrid MADM problems with various formats of attribute values. In this paper, an extended TODIM method is proposed to solve the hybrid MADM problem. First, three formats of attribute values (crisp numbers, interval numbers and fuzzy numbers) are expressed in the format of random variables with cumulative distribution functions. Then, according to the concept of the classical TODIM method, the gain and loss matrices concerning each attribute are constructed by calculating the gain and loss of each alternative relative to the others. Further, by calculating the dominance degree of each alternative over the others, the overall value of each alternative can be obtained to rank the alternatives. Finally, two numerical examples are used to illustrate the use of the proposed method.

Fuzzy LINMAP approach to heterogeneous MADM considering comparisons of alternatives with hesitation degrees

Abstract: Multiattribute decision making (MADM) with multiple formats of information, which is called heterogeneous MADM for short, is very complex and interesting in applications. The purpose of this paper is to extend the Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP) for solving heterogeneous MADM problems which involve intuitionistic fuzzy (IF) sets (IFSs), trapezoidal fuzzy numbers (TrFNs), intervals and real numbers. In this method, DM's preference is given through pair-wise comparisons of alternatives with hesitation degrees which are represented as IFSs. The IF consistency and inconsistency indices are defined on the basis of pair-wise comparisons of alternatives. Each alternative is assessed on the basis of its distance to a fuzzy ideal solution (FIS) unknown a priori . Based on the defined IF consistency and inconsistency indices, we construct a new fuzzy mathematical programming model, which is solved by the developed method of fuzzy mathematical programming with IFSs. Once the FIS and the attribute weights are obtained, we can calculate the distances of all alternatives to the FIS, which are used to determine the ranking order of the alternatives. A supplier selection example is presented to demonstrate the validity and applicability of the proposed method.

Control Synthesis of Discrete-Time T–S Fuzzy Systems Based on a Novel Non-PDC Control Scheme

Abstract: This paper proposes relaxed stabilization conditions of discrete-time nonlinear systems in the Takagi-Sugeno (T-S) fuzzy form. By using the algebraic property of fuzzy membership functions, a novel nonparallel distributed compensation (non-PDC) control scheme is proposed based on a new class of fuzzy Lyapunov functions. Thus, relaxed stabilization conditions for the underlying closed-loop fuzzy system are developed by applying a new slack variable technique. In particular, some existing fuzzy Lyapunov functions and non-PDC control schemes are special cases of the new Lyapunov function and fuzzy control scheme, respectively. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed method.

Multi-criteria decision making method based on possibility degree of interval type-2 fuzzy number

Abstract: This paper proposes a new approach based on possibility degree to solve multi-criteria decision making (MCDM) problems in which the criteria value takes the form of interval type-2 fuzzy number. First, a new expected value function is defined and an optimal model based on maximizing deviation method is constructed to obtain weight coefficients when criteria weight information is partially known. Then, the overall value of each alternative is calculated by the defined aggregation operators. Furthermore, a new possibility degree, which is proposed to overcome some drawbacks of the existing methods, is introduced for comparisons between the overall values of alternatives to construct a possibility degree matrix. Based on the constructed matrix, all of the alternatives are ranked according to the ranking vector derived from the matrix, and the best one is selected. Finally, the proposed method is applied to a case study on the overseas minerals investment for one of the largest multi-species nonferrous metals companies in China and the results demonstrate the feasibility of the method.

Simplified Interval Type-2 Fuzzy Logic Systems

Abstract: Type reduction (TR) followed by defuzzification is commonly used in interval type-2 fuzzy logic systems (IT2 FLSs). Because of the iterative nature of TR, it may be a computational bottleneck for the real-time applications of an IT2 FLS. This has led to many direct approaches to defuzzification that bypass TR, the simplest of which is the Nie-Tan direct defuzzification method (NT method). This paper provides some theoretical analyses of the NT method that answer the question “Why is the NT method good to use?” This paper also provides a direct relationship between TR followed by defuzzification (using KM algorithms) and the NT method. It also provides an improved NT method. Numerical examples illustrate our theoretical results and suggest that the NT method is a very good way to simplify an interval type-2 fuzzy set.

Identification and Prediction of Dynamic Systems Using an Interactively Recurrent Self-Evolving Fuzzy Neural Network

Abstract: This paper presents a novel recurrent fuzzy neural network, called an interactively recurrent self-evolving fuzzy neural network (IRSFNN), for prediction and identification of dynamic systems. The recurrent structure in an IRSFNN is formed as an external loops and internal feedback by feeding the rule firing strength of each rule to others rules and itself. The consequent part in the IRSFNN is composed of a Takagi-Sugeno-Kang (TSK) or functional-link-based type. The proposed IRSFNN employs a functional link neural network (FLNN) to the consequent part of fuzzy rules for promoting the mapping ability. Unlike a TSK-type fuzzy neural network, the FLNN in the consequent part is a nonlinear function of input variables. An IRSFNNs learning starts with an empty rule base and all of the rules are generated and learned online through a simultaneous structure and parameter learning. An on-line clustering algorithm is effective in generating fuzzy rules. The consequent update parameters are derived by a variable-dimensional Kalman filter algorithm. The premise and recurrent parameters are learned through a gradient descent algorithm. We test the IRSFNN for the prediction and identification of dynamic plants and compare it to other well-known recurrent FNNs. The proposed model obtains enhanced performance results.

A Novel Method of Ranking Hesitant Fuzzy Values for Multiple Attribute Decision-Making Problems

Abstract: Despite of several generalizations of fuzzy set theory, the notion of hesitant fuzzy set (HFS), which permits the membership having a set of possible values, is interesting and very useful in modeling real-life problems with anonymity. In this article, we introduce a new score function for ranking hesitant fuzzy elements (HFEs), which are the fundamental units of HFSs. Comparison with the existing score function shows that the proposed method meets all the well-known properties of a ranking measure and has no counterintuitive examples. On the basis of the relationships between the aggregation operators for HFEs, we derive a series of interesting properties of the new score function. Finally, we apply the proposed score function to solve the hesitant fuzzy multiattribute decision-making problems.

Short Communication: IF-TODIM: An intuitionistic fuzzy TODIM to multi-criteria decision making

Abstract: The recently developed fuzzy TODIM (an acronym in Portuguese for iterative multi-criteria decision making) method using fuzzy numbers has been applied to uncertain MCDM problems with promising results. In this paper, a more general approach to the fuzzy TODIM, which takes into account the membership and the non-membership of the fuzzy information is considered. So, the fuzzy TODIM method has been extended to handle intuitionistic fuzzy information. This way, it is possible to tackle more challenging MCDM problems. Two case studies are used to illustrate and show the suitability of the developed method.

γ-Open sets and γ-continuous mappings in fuzzy bitopological spaces

Abstract: In this paper we introduce the notion of γ-open sets and γ-continuous functions in fuzzy bitopological spaces. We examine some basic properties and prove some characterization theorems for the said functions. It is observed that every pairwise fuzzy γ-continuous functions is pairwise fuzzy precontinuous but the converse not true.

Weighted Fuzzy Spiking Neural P Systems

Abstract: Spiking neural P systems (SN P systems) are a new class of computing models inspired by the neurophysiological behavior of biological spiking neurons. In order to make SN P systems capable of representing and processing fuzzy and uncertain knowledge, we propose a new class of spiking neural P systems in this paper called weighted fuzzy spiking neural P systems (WFSN P systems). New elements, including fuzzy truth value, certain factor, weighted fuzzy logic, output weight, threshold, new firing rule, and two types of neurons, are added to the original definition of SN P systems. This allows WFSN P systems to adequately characterize the features of weighted fuzzy production rules in a fuzzy rule-based system. Furthermore, a weighted fuzzy backward reasoning algorithm, based on WFSN P systems, is developed, which can accomplish dynamic fuzzy reasoning of a rule-based system more flexibly and intelligently. In addition, we compare the proposed WFSN P systems with other knowledge representation methods, such as fuzzy production rule, conceptual graph, and Petri nets, to demonstrate the features and advantages of the proposed techniques.

IVTURS: A Linguistic Fuzzy Rule-Based Classification System Based On a New Interval-Valued Fuzzy Reasoning Method With Tuning and Rule Selection

Abstract: Interval-valued fuzzy sets have been shown to be a useful tool to deal with the ignorance related to the definition of the linguistic labels. Specifically, they have been successfully applied to solve classification problems, performing simple modifications on the fuzzy reasoning method to work with this representation and making the classification based on a single number. In this paper, we present IVTURS, which is a new linguistic fuzzy rule-based classification method based on a new completely interval-valued fuzzy reasoning method. This inference process uses interval-valued restricted equivalence functions to increase the relevance of the rules in which the equivalence of the interval membership degrees of the patterns and the ideal membership degrees is greater, which is a desirable behavior. Furthermore, their parametrized construction allows the computation of the optimal function for each variable to be performed, which could involve a potential improvement in the system's behavior. Additionally, we combine this tuning of the equivalence with rule selection in order to decrease the complexity of the system. In this paper, we name our method IVTURS-FARC, since we use the FARC-HD method to accomplish the fuzzy rule learning process. The experimental study is developed in three steps in order to ascertain the quality of our new proposal. First, we determine both the essential role that interval-valued fuzzy sets play in the method and the need for the rule selection process. Next, we show the improvements achieved by IVTURS-FARC with respect to the tuning of the degree of ignorance when it is applied in both an isolated way and when combined with the tuning of the equivalence. Finally, the significance of IVTURS-FARC is further depicted by means of a comparison by which it is proved to outperform the results of FARC-HD and FURIA, which are two high performing fuzzy classification algorithms.