Showing papers on "Fuzzy number published in 2012"

Dual Hesitant Fuzzy Sets

Abstract: In recent decades, several types of sets, such as fuzzy sets, interval-valued fuzzy sets, intuitionistic fuzzy sets, interval-valued intuitionistic fuzzy sets, type 2 fuzzy sets, type 𝑛 fuzzy sets, and hesitant fuzzy sets, have been introduced and investigated widely. In this paper, we propose dual hesitant fuzzy sets (DHFSs), which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as special cases. Then we investigate the basic operations and properties of DHFSs. We also discuss the relationships among the sets mentioned above, use a notion of nested interval to reflect their common ground, then propose an extension principle of DHFSs. Additionally, we give an example to illustrate the application of DHFSs in group forecasting.

Mathematics of Fuzzy Sets and Fuzzy Logic

Abstract: This book presents a mathematically-based introduction into the fascinating topic of Fuzzy Sets and Fuzzy Logic and might be used as textbook at both undergraduate and graduate levels and also as reference guide for mathematician, scientists or engineers who would like to get an insight into Fuzzy Logic. Fuzzy Sets have been introduced by Lotfi Zadeh in 1965 and since then, they have been used in many applications. As a consequence, there is a vast literature on the practical applications of fuzzy sets, while theory has a more modest coverage. The main purpose of the present book is to reduce this gap by providing a theoretical introduction into Fuzzy Sets based on Mathematical Analysis and Approximation Theory. Well-known applications, as for example fuzzy control, are also discussed in this book and placed on new ground, a theoretical foundation. Moreover, a few advanced chapters and several new results are included. These comprise, among others, a new systematic and constructive approach for fuzzy inference systems of Mamdani and Takagi-Sugeno types, that investigates their approximation capability by providing new error estimates.

Adaptive Fuzzy Control of a Class of Nonlinear Systems by Fuzzy Approximation Approach

Abstract: Controlling nonstrict-feedback nonlinear systems is a challenging problem in control theory. In this paper, we consider adaptive fuzzy control for a class of nonlinear systems with nonstrict-feedback structure by using fuzzy logic systems. A variable separation approach is developed to overcome the difficulty from the nonstrict-feedback structure. Furthermore, based on fuzzy approximation and backstepping techniques, a state feedback adaptive fuzzy tracking controller is proposed, which guarantees that all of the signals in the closed-loop system are bounded, while the tracking error converges to a small neighborhood of the origin. Simulation studies are included to demonstrate the effectiveness of our results.

An introduction to semi-tensor product of matrices and its applications

Abstract: Multi-Dimensional Data Semi-Tensor Product of Matrices Multilinear Mappings Among Vector Spaces Right and General Semi-Tensor Products Rank, Pseudo-Inverse, and Positivity of STP Matrix Expression of Logic Mix-Valued Logic Logical Matrix, Fuzzy Set and Fuzzy Logic Fuzzy Relational Equation Fuzzy Control with Coupled Fuzzy Relations Boolean Function with Galois Field Structure Decomposition of Logical Functions Boolean Calculus Lattice, Graph, and Universal Algebra Boolean Network Boolean Control System Game Theory Multi-Variable Polynomials Some Applications to Differential Geometry and Algebra Morgan's Problem Linearization of Nonlinear Control Systems Stability Region of Dynamic System.

Intuitionistic Fuzzy Information Aggregation Using Einstein Operations

Abstract: Aggregation of fuzzy information is a new branch of Atanassov's intuitionistic fuzzy set (AIFS) theory, which has attracted significant interest from researchers in recent years. In this paper, we treat the intuitionistic fuzzy aggregation operators with the help of Einstein operations. We first introduce some new operations of AIFSs, such as Einstein sum, Einstein product, and Einstein scalar multiplication. Then, we develop some intuitionistic fuzzy aggregation operators, such as the intuitionistic fuzzy Einstein weighted averaging operator and the intuitionistic fuzzy Einstein ordered weighted averaging operator, which extend the weighted averaging operator and the ordered weighted averaging operator to aggregate Atanassov's intuitionistic fuzzy values, respectively. We further establish various properties of these operators and analyze the relations between these operators and the existing intuitionistic fuzzy aggregation operators. Moreover, we give some numerical examples to illustrate the developed aggregation operators. Finally, we apply the intuitionistic fuzzy Einstein weighted averaging operator to multiple attribute decision making with intuitionistic fuzzy information.

Risk evaluation in failure mode and effects analysis with extended VIKOR method under fuzzy environment

Abstract: Failure mode and effects analysis (FMEA) is a widely used risk assessment tool for defining, identifying, and eliminating potential failures or problems in products, process, designs, and services In traditional FMEA, the risk priorities of failure modes are determined by using risk priority numbers (RPNs), which can be obtained by multiplying the scores of risk factors like occurrence (O), severity (S), and detection (D) However, the crisp RPN method has been criticized to have several deficiencies In this paper, linguistic variables, expressed in trapezoidal or triangular fuzzy numbers, are used to assess the ratings and weights for the risk factors O, S, and D For selecting the most serious failure modes, the extended VIKOR method is used to determine risk priorities of the failure modes that have been identified As a result, a fuzzy FMEA based on fuzzy set theory and VIKOR method is proposed for prioritization of failure modes, specifically intended to address some limitations of the traditional FMEA A case study, which assesses the risk of general anesthesia process, is presented to demonstrate the application of the proposed model under fuzzy environment

Complex intuitionistic fuzzy sets

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

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

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

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

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

The Shapley Value

Abstract: In fuzzy Multi-criteria decision-making problems, the ranking of alternatives must take into account their fuzzy scores in all criteria, the weights assigned to each decision criterion, the possible difficulties of comparing two alternatives when one is significantly better than the other on a subset of criteria, but much worse on at least one criterion from the complementary subset of criteria, and the decision maker’s attitude towards the risk associated with evaluation.

A Method of Converting Z-number to Classical Fuzzy Number

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

A utility-based fuzzy TOPSIS method for energy efficient network selection in heterogeneous wireless networks

Abstract: Mobile terminals in 4G heterogeneous wireless networks continuously undergo horizontal and vertical handovers. In order for a mobile terminal to be connected to a network in the best possible way in terms of QoS performance and energy consumption, access network selection as the main decision within the handover process is obviously crucial. This paper presents a novel method that takes into account user preferences, network conditions, QoS and energy consumption requirements in order to select the optimal network which achieves the best balance between performance and energy consumption. The proposed network selection method incorporates the use of parameterized utility functions in order to model diverse QoS elasticities of different applications, and adopts different energy consumption metrics for real-time and non-real-time applications. User preferences are easily configured for different application and situation contexts through the use of linguistic assessments and their representation as triangular fuzzy numbers. The aggregation of multiple criteria for the calculation of the overall rating of the networks is performed through the use of the Fuzzy Set Representation TOPSIS method that resolves the issue of inconsistency related to conflicting decision criteria and is modified through the use of the employed utility functions for the elimination of the ranking abnormality problem. Finally, simulations are conducted in order to demonstrate how the proposed method would work and confirm its suitability and effectiveness.

Strong Intuitionistic Fuzzy Graphs

Abstract: We introduce the notion of strong intuitionistic fuzzy graphs and investigate some of their properties. We discuss some propositions of self complementary and self weak complementary strong intuitionistic fuzzy graphs. We introduce the concept of intuitionistic fuzzy line graphs.

Combining prospect theory and fuzzy numbers to multi-criteria decision making

Abstract: Many multi-criteria decision making (MCDM) methods have been proposed to handle uncertain decision making problems. Most of them are based on fuzzy numbers and they are not able to cope with risk in decision making. In recent years, some MCDM methods based on prospect theory to handle risk MCDM problems have been developed. In this paper, we propose a hybrid approach combining prospect theory and fuzzy numbers to handle risk and uncertainty in MCDM problems. So, it is possible to tackle more challenging MCDM problems. A case study involving oil spill in the sea illustrates the application of the novel method.

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

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

On Z-valuations using Zadeh's Z-numbers

Abstract: We first recall the concept of Z-numbers introduced by Zadeh. These objects consist of an ordered pair (A, B) of fuzzy numbers. We then use these Z-numbers to provide information about an uncertain variable V in the form of a Z-valuation, which expresses the knowledge that the probability that V is A is equal to B. We show that these Z-valuations essentially induce a possibility distribution over probability distributions associated with V. We provide a simple illustration of a Z-valuation. We show how we can use this representation to make decisions and answer questions. We show how to manipulate and combine multiple Z-valuations. We show the relationship between Z-numbers and linguistic summaries. Finally, we provide for a representation of Z-valuations in terms of Dempster–Shafer belief structures, which makes use of type-2 fuzzy sets. © 2012 Wiley Periodicals, Inc. © 2012 Wiley Periodicals, Inc.

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

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

Fuzzy AHP-based multicriteria decision making systems using particle swarm optimization

Abstract: This paper presents a fuzzy optimization model to solve multicriteria decision making (MCDM) systems based on a fuzzy analytic hierarchy process (fuzzy AHP). To deal with the imprecise judgments of decision makers, a fuzzy AHP decision making model is proposed as an evaluation tool, where the expert's comparison judgments are translated into fuzzy numbers. Unlike the conventional fuzzy AHP methods, the proposed method drives exact weights from consistent and inconsistent fuzzy comparison matrices, which eliminate the need of additional aggregation and ranking procedures. The proposed method transforms a fuzzy prioritization problem into a constrained nonlinear optimization model. An improved particle swarm optimization (PSO) is applied to solve the optimization model as a nonlinear system of equations. Several illustrative examples using existing fuzzy AHP methods are given to demonstrate the effectiveness of the proposed method.

Fuzzy Sets in Approximate Reasoning and Information Systems

Abstract: Series Foreword. Contributing Authors. Introduction. Part I: Reasoning. 1. Fuzzy Sets and Possibility Theory in Approximate and Plausible Reasoning B. Bouchon-Meunier, et al. 2. Weighted Inference Systems V. Novak. 3. Closure Operators in Fuzzy set Theory L. Biacino, G. Gerla. Part II: Learning and Fusion. 4. Learning Fuzzy Decision Rules B. Bouchon-Meunier, C. Marsala. 5. Neuro-Fuzzy Methods in Fuzzy Rule Generation D. Nauck, R. Kruse. 6. Merging Fuzzy Information D. Dubois, et al. Part III: Fuzzy Information Systems. 7. Fuzzy Databases P. Bosc, et al. 8. Fuzzy Set Techniques in Information Retrieval D.H. Kraft, et al. Summary. References

The induced generalized aggregation operators for intuitionistic fuzzy sets and their application in group decision making

Abstract: In this paper, we present the induced generalized intuitionistic fuzzy ordered weighted averaging (I-GIFOWA) operator. It is a new aggregation operator that generalized the IFOWA operator, including all the characteristics of both the generalized IFOWA and the induced IFOWA operators. It provides a very general formulation that includes as special cases a wide range of aggregation operators for intuitionistic fuzzy information, including all the particular cases of the I-IFOWA operator, GIFOWA operator and the induced intuitionistic fuzzy ordered geometric (I-IFOWG) operator. We also present the induced generalized interval-valued intuitionistic fuzzy ordered weighted averaging (I-GIIFOWA) operator to accommodate the environment in which the given arguments are interval-valued intuitionistic fuzzy sets. Further, we develop procedures to apply them to solve group multiple attribute decision making problems with intuitionistic fuzzy or interval-valued intuitionistic fuzzy information. Finally, we present their application to show the effectiveness of the developed methods.

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

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

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

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

On Robust Fuzzy Rough Set Models

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

Towards the Wide Spread Use of Type-2 Fuzzy Logic Systems in Real World Applications

Abstract: Real world applications are characterized by high levels of linguistic and numerical uncertainties. Since the inception of Fuzzy Logic Systems (FLSs), they have been applied with great success to numerous real world applications. The vast majority of FLSs so far have been traditional type-1 FLSs. However, type-1 FLSs cannot fully handle the high levels of uncertainties available in the vast majority of real world applications. This is because type-1 FLSs employ crisp and precise type-1 fuzzy sets. A type-2 FLS can handle higher uncertainty levels to produce improved performance. This paper follows on [1] to show how type-2 FLSs are starting to find their way into a variety of real world applications, promising a continuous growth both in number and variety of type-2 FLS applications in the next decade.

Decision Making Using Z-numbers under Uncertain Environment

Abstract: Multi-criteria decision making (MCDM) under uncertain environment is still an open issue. Recently, Znumber has been developed by Zadeh to model fuzzy numbers with the confidence degree. In this paper, a new MCDM method based on Z-number is proposed to deal with linguistic decision making problems. The decision making process can be easily realized step by step with the arithmetic operations on Znumbers. A numerical example on MCDM is used to illustrate the efficiency of the proposed method.