Showing papers on "Fuzzy number published in 2011"
TL;DR: The relationship between intutionistic fuzzy set and hesitant fuzzy set is discussed, based on which some operations and aggregation operators for hesitant fuzzy elements are developed and their application in solving decision making problems is given.
1,352 citations
TL;DR: This paper treats supplier selection as a group multiple criteria decision making (GMCDM) problem and obtain decision makers' opinions in the form of linguistic terms which are converted to trapezoidal fuzzy numbers and extended the VIKOR method with a mechanism to extract and deploy objective weights based on Shannon entropy concept.
Abstract: Recently, resolving the problem of evaluation and ranking the potential suppliers has become as a key strategic factor for business firms. With the development of intelligent and automated information systems in the information era, the need for more efficient decision making methods is growing. The VIKOR method was developed to solve multiple criteria decision making (MCDM) problems with conflicting and non-commensurable criteria assuming that compromising is acceptable to resolve conflicts. On the other side objective weights based on Shannon entropy concept could be used to regulate subjective weights assigned by decision makers or even taking into account the end-users' opinions. In this paper, we treat supplier selection as a group multiple criteria decision making (GMCDM) problem and obtain decision makers' opinions in the form of linguistic terms. Then, these linguistic terms are converted to trapezoidal fuzzy numbers. We extended the VIKOR method with a mechanism to extract and deploy objective weights based on Shannon entropy concept. The final result is obtained through next steps based on factors R, S and Q. A numerical example is proposed to illustrate an application of the proposed method.
612 citations
01 Dec 2011
TL;DR: It is proved that these two control approaches can guarantee that all the signals of the closed-loop system are semi-globally uniformly ultimately bounded (SGUUB) in mean square, and the observer errors and the output of the system converge to a small neighborhood of the origin.
Abstract: In this paper, two adaptive fuzzy output feedback control approaches are proposed for a class of uncertain stochastic nonlinear strict-feedback systems without the measurements of the states. The fuzzy logic systems are used to approximate the unknown nonlinear functions, and a fuzzy state observer is designed for estimating the unmeasured states. On the basis of the fuzzy state observer, and by combining the adaptive backstepping technique with fuzzy adaptive control design, an adaptive fuzzy output feedback control approach is developed. To overcome the problem of “explosion of complexity” inherent in the proposed control method, the dynamic surface control (DSC) technique is incorporated into the first adaptive fuzzy control scheme, and a simplified adaptive fuzzy output feedback DSC approach is developed. It is proved that these two control approaches can guarantee that all the signals of the closed-loop system are semi-globally uniformly ultimately bounded (SGUUB) in mean square, and the observer errors and the output of the system converge to a small neighborhood of the origin. A simulation example is provided to show the effectiveness of the proposed approaches.
548 citations
TL;DR: This paper defines the distance and correlation measures for hesitant fuzzy information and then discusses their properties in detail, finding that the results are the smallest ones among those when the values in two hesitant fuzzy elements are arranged in any permutations.
Abstract: A hesitant fuzzy set, allowing the membership of an element to be a set of several possible values, is very useful to express people's hesitancy in daily life. In this paper, we define the distance and correlation measures for hesitant fuzzy information and then discuss their properties in detail. These measures are all defined under the assumption that the values in all hesitant fuzzy elements (the fundamental units of hesitant fuzzy sets) are arranged in an increasing order and two hesitant fuzzy elements have the same length when we compare them. We can find that the results, by using the developed distance measures, are the smallest ones among those when the values in two hesitant fuzzy elements are arranged in any permutations. In addition, the derived correlation coefficients are based on different linear relationships and may have different results. © 2011 Wiley Periodicals, Inc. © 2011 Wiley Periodicals, Inc.
461 citations
TL;DR: This article presents a risk assessment methodology based on the Fuzzy Sets Theory, which is an effective tool to deal with subjective judgement, and on the Analytic Hierarchy Process (AHP), which is used to structure a large number of risks.
458 citations
TL;DR: This paper develops a series of operators for aggregating IFNs, establishes various properties of these power aggregation operators, and applies them to develop some approaches to multiple attribute group decision making with Atanassov's intuitionistic fuzzy information.
Abstract: Intuitionistic fuzzy numbers (IFNs) are very suitable to be used for depicting uncertain or fuzzy information. Motivated by the idea of power aggregation [R.R. Yager, The power average operator, IEEE Transactions on Systems, Man, and Cybernetics-Part A 31 (2001) 724-731], in this paper, we develop a series of operators for aggregating IFNs, establish various properties of these power aggregation operators, and then apply them to develop some approaches to multiple attribute group decision making with Atanassov's intuitionistic fuzzy information. Moreover, we extend these aggregation operators and decision making approaches to interval-valued Atanassov's intuitionistic fuzzy environments.
411 citations
TL;DR: The fuzzy VIKOR method has been developed to solve fuzzy multicriteria problem with conflicting and noncommensurable (different units) criteria in a fuzzy environment where both criteria and weights could be fuzzy sets.
Abstract: The fuzzy VIKOR method has been developed to solve fuzzy multicriteria problem with conflicting and noncommensurable (different units) criteria. This method solves problem in a fuzzy environment where both criteria and weights could be fuzzy sets. The triangular fuzzy numbers are used to handle imprecise numerical quantities. Fuzzy VIKOR is based on the aggregating fuzzy merit that represents distance of an alternative to the ideal solution. The fuzzy operations and procedures for ranking fuzzy numbers are used in developing the fuzzy VIKOR algorithm. VIKOR (VIsekriterijumska optimizacija i KOmpromisno Resenje) focuses on ranking and selecting from a set of alternatives in the presence of conflicting criteria, and on proposing compromise solution (one or more). It is extended with a trade-offs analysis. A numerical example illustrates an application to water resources planning, utilizing the presented methodology to study the development of a reservoir system for the storage of surface flows of the Mlava River and its tributaries for regional water supply. A comparative analysis of results by fuzzy VIKOR and few different approaches is presented.
410 citations
01 Mar 2011
TL;DR: Experimental results show the effectiveness of the proposed method in contrast to conventional fuzzy C means algorithms and also type II fuzzy algorithm.
Abstract: This paper presents a novel intuitionistic fuzzy C means clustering method using intuitionistic fuzzy set theory. The intuitionistic fuzzy set theory considers another uncertainty parameter which is the hesitation degree that arises while defining the membership function and thus the cluster centers may converge to a desirable location than the cluster centers obtained using fuzzy C means algorithm. Also a new objective function which is the intuitionistic fuzzy entropy is incorporated in the conventional fuzzy C means clustering algorithm. This is done to maximize the good points in the class. This clustering method is used in clustering different regions of the CT scan brain images and these may be used to identify the abnormalities in the brain. Experimental results show the effectiveness of the proposed method in contrast to conventional fuzzy C means algorithms and also type II fuzzy algorithm.
334 citations
TL;DR: This method limits the order of the associations in the association rule extraction and considers the use of subgroup discovery, which is based on an improved weighted relative accuracy measure to preselect the most interesting rules before a genetic postprocessing process for rule selection and parameter tuning.
Abstract: The inductive learning of fuzzy rule-based classification systems suffers from exponential growth of the fuzzy rule search space when the number of patterns and/or variables becomes high. This growth makes the learning process more difficult and, in most cases, it leads to problems of scalability (in terms of the time and memory consumed) and/or complexity (with respect to the number of rules obtained and the number of variables included in each rule). In this paper, we propose a fuzzy association rule-based classification method for high-dimensional problems, which is based on three stages to obtain an accurate and compact fuzzy rule-based classifier with a low computational cost. This method limits the order of the associations in the association rule extraction and considers the use of subgroup discovery, which is based on an improved weighted relative accuracy measure to preselect the most interesting rules before a genetic postprocessing process for rule selection and parameter tuning. The results that are obtained more than 26 real-world datasets of different sizes and with different numbers of variables demonstrate the effectiveness of the proposed approach.
320 citations
TL;DR: An entropy measure for interval-valued intuitionistic fuzzy sets is proposed, which generalizes three entropy measures defined independently by Szmidt, Wang and Huang, for intuitionism fuzzy sets and is applied to solve problems on pattern recognitions, multi-criteria fuzzy decision making and medical diagnosis.
294 citations
TL;DR: This study proposes integrated fuzzy techniques for order preference by similarity to ideal solution (TOPSIS) and multi-choice goal programming (MCGP) approach to solve the supplier selection problem.
Abstract: Supplier selection is an important issue in supply chain management. In recent years, determining the best supplier in the supply chain has become a key strategic consideration. However, these decisions usually involve several objectives or criteria, and it is often necessary to compromise among possibly conflicting factors. Thus, the multiple criteria decision making (MCDM) becomes a useful approach to solve this kind of problem. Considering both tangible and intangible criteria, this study proposes integrated fuzzy techniques for order preference by similarity to ideal solution (TOPSIS) and multi-choice goal programming (MCGP) approach to solve the supplier selection problem. The advantage of this method is that it allows decision makers to set multiple aspiration levels for supplier selection problems. The integrated model is illustrated by an example in a watch firm.
TL;DR: Choquet integral and Dempster-Shafer theory of evidence are applied to aggregate inuitionistic fuzzy information and some new types of aggregation operators are developed, including the induced generalized intuitionistic fuzzy Choquet integral operators and induced generalized intuistic fuzzy Dem pster-shafer operators.
Abstract: We study the induced generalized aggregation operators under intuitionistic fuzzy environments. Choquet integral and Dempster-Shafer theory of evidence are applied to aggregate inuitionistic fuzzy information and some new types of aggregation operators are developed, including the induced generalized intuitionistic fuzzy Choquet integral operators and induced generalized intuitionistic fuzzy Dempster-Shafer operators. Then we investigate their various properties and some of their special cases. Additionally, we apply the developed operators to financial decision making under intuitionistic fuzzy environments. Some extensions in interval-valued intuitionistic fuzzy situations are also pointed out.
TL;DR: A new construction method based on the Lukasiewicz triangular norm is proposed, which is consistent with operations on ordinary fuzzy sets, and therefore is a true generalization of such operations.
TL;DR: A new method is proposed to find the fuzzy optimal solution of same type of fuzzy linear programming problems and it is easy to apply the proposed method compare to the existing method for solving the FFLP problems with equality constraints occurring in real life situations.
TL;DR: An extension of TOPSIS, a multi-criteria interval-valued intuitionistic fuzzy geometric aggregation operator, to a group decision environment is investigated, where inter-dependent or interactive characteristics among criteria and preference of decision makers are taken into account.
Abstract: An extension of TOPSIS, a multi-criteria interval-valued intuitionistic fuzzy decision making technique, to a group decision environment is investigated, where inter-dependent or interactive characteristics among criteria and preference of decision makers are taken into account. To get a broad view of the techniques used, first, some operational laws on interval-valued intuitionistic fuzzy values are introduced. Based on these operational laws, a generalized interval-valued intuitionistic fuzzy geometric aggregation operator is proposed which is used to aggregate decision makers' opinions in group decision making process. In addition, some of its properties are discussed. Then Choquet integral-based Hamming distance between interval-valued intuitionistic fuzzy values is defined. Combining the interval-valued intuitionistic fuzzy geometric aggregation operator with Choquet integral-based Hamming distance, an extension of TOPSIS method is developed to deal with a multi-criteria interval-valued intuitionistic fuzzy group decision making problems. Finally, an illustrative example is used to illustrate the developed procedures.
01 Jun 2011
TL;DR: For any subset X of the universe U, there is a fuzzy subset of U associated with each parameter of the soft set and each soft set over a set U, gives rise to a fuzzysoft set over P(U) and induces a soft equivalence relation over P (U).
Abstract: Concept of an approximation space associated with each parameter in a soft set is discussed and an approximation space associated with the soft set is defined. For any subset X of the universe U, there is a fuzzy subset of U associated with each parameter of the soft set, also there is a fuzzy subset associated with the soft set. Furthermore each soft set over a set U, gives rise to a fuzzy soft set over P(U) and induces a soft equivalence relation over P(U).
TL;DR: A logarithmic fuzzy preference programming (LFPP) based methodology for fuzzy AHP priority derivation is proposed, which formulates the priorities of a fuzzy pairwise comparison matrix as a logarithsmic nonlinear programming and derives crisp priorities from fuzzy Pairwise comparison matrices.
TL;DR: In this paper, a fuzzy linear programming model is proposed to determine how much should be purchased from each supplier, and the capacity of warehouse is considered as a constraint, where the fuzzy logic and triangular fuzzy numbers are integrated with SWOT analysis to deal with vagueness of human thought.
Abstract: Supplier selection is a multi criteria decision-making problem that comprises tangible and intangible factors. The majority of previous supplier selection techniques do not consider strategic perspective. Besides, uncertainty is one of the most important obstacles in supplier selection. In this paper, quantified SWOT is applied in the context of supplier selection for the first time. SWOT (Strengths, Weaknesses, Opportunities and Threats) is one of the most well-known techniques for conducting a strategic study. In addition, the fuzzy logic and triangular fuzzy numbers are integrated with SWOT analysis - as a novel innovation - to deal with vagueness of human thought. SWOT analysis can consider both qualitative and quantitative criteria. The managers can understand the position of suppliers in a competitive environment with a glance at SWOT matrix. Moreover, a fuzzy linear programming model is proposed to determine how much should be purchased from each supplier. It is supposed that the demand is a fuzzy number. Besides, the capacity of warehouse is considered as a constraint. A case study is utilized concurrently to show the efficiency of the proposed model.
TL;DR: A new method for handling multi-criteria fuzzy decision-making problems based on interval-valued intuitionistic fuzzy sets is presented, which can provide a useful way to efficiently help the decision-maker to make his decision.
Abstract: Out of several generalizations of fuzzy set theory for various objectives, the notions introduced by Atanassov (1983) and Atanassov and Gargov (1989) in defining intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets are interesting and very useful in modeling real life problems. Ranking of interval-valued intuitionistic fuzzy sets plays a vital role in decision-making, data analysis, artificial intelligence and socioeconomic system and it was studied in Xu (2007c), Xu and Chen (2007a) and Ye (2009). In this paper a new method for ranking interval-valued intuitionistic fuzzy sets has been introduced and studied. The method is illustrated by numerical examples and compared with other methods. And then a new method for handling multi-criteria fuzzy decision-making problems based on interval-valued intuitionistic fuzzy sets is presented in which criterion values for alternatives are interval-valued intuitionistic fuzzy sets. The method proposed here can provide a useful way to efficiently help the decision-maker to make his decision. An illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.
TL;DR: An optimization model based on the negative ideal solution and max-min operator, by which the attribute weights can be determined, is established and utilization of the interval-valued intuitionistic fuzzy weighted averaging operator proposed by Xu is utilized.
Abstract: With respect to multiple attribute decision-making problems with interval-valued intuitionistic fuzzy information, some operational laws of interval-valued intuitionistic fuzzy numbers, correlation and correlation coefficient of interval-valued intuitionistic fuzzy sets are introduced. An optimization model based on the negative ideal solution and max-min operator, by which the attribute weights can be determined, is established. We utilize the interval-valued intuitionistic fuzzy weighted averaging operator proposed by Xu (Control Decis 22(2):215–219, 2007) to aggregate the interval-valued intuitionistic fuzzy information corresponding to each alternative, and then rank the alternatives and select the most desirable one(s) according to the correlation coefficient. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.
TL;DR: A new MCDM methodology, using FST and DST, based on the main idea of the technique for order preference by similarity to an ideal solution (TOPSIS), is developed to deal with supplier selection problem.
Abstract: Supplier selection is a multi-criterion decision making problem under uncertain environments. Hence, it is reasonable to hand the problem in fuzzy sets theory (FST) and Dempster Shafer theory of evidence (DST). In this paper, a new MCDM methodology, using FST and DST, based on the main idea of the technique for order preference by similarity to an ideal solution (TOPSIS), is developed to deal with supplier selection problem. The basic probability assignments (BPA) can be determined by the distance to the ideal solution and the distance to the negative ideal solution. Dempster combination rule is used to combine all the criterion data to get the final scores of the alternatives in the systems. The final decision results can be drawn through the pignistic probability transformation. In traditional fuzzy TOPSIS method, the quantitative performance of criterion, such as crisp numbers, should be transformed into fuzzy numbers. The proposed method is more flexible due to the reason that the BPA can be determined without the transformation step in traditional fuzzy TOPSIS method. The performance of criterion can be represented as crisp number or fuzzy number according to the real situation in our proposed method. The numerical example about supplier selection is used to illustrate the efficiency of the proposed method.
TL;DR: In this paper, the concept of fuzzy soft topology is introduced and some of its structural properties such as neighborhood of a fuzzysoft set, interior fuzzy soft set, fuzzy soft basis, and fuzzy soft subspace topology are studied.
Abstract: In this paper, the concept of fuzzy soft topology is introduced and some of its structural properties such as neighborhood of a fuzzy soft set, interior fuzzy soft set, fuzzy soft basis, fuzzy soft subspace topology are studied.
TL;DR: This paper considers the situation with intuitionistic fuzzy information and develops an intuistic fuzzy ordered weighted distance (IFOWD) operator, which is very suitable to deal with the situations where the input data are represented in intuitionism fuzzy information.
Abstract: The ordered weighted distance and is a new decision-making technique, having been proved useful for the treatment of input data in the form of exact numbers. In this paper, we consider the situation with intuitionistic fuzzy information and develop an intuitionistic fuzzy ordered weighted distance (IFOWD) operator. The IFOWD operator is very suitable to deal with the situations where the input data are represented in intuitionistic fuzzy information and includes a wide range of distance measures and aggregation operators. We study some of its main properties and different families of IFOWD operators. Finally, we develop an application of the new approach in a group decision-making under intuitionistic fuzzy environment and illustrate it with a numerical example.
TL;DR: This work defines a fuzzy soft set theory and its related properties and then defines fuzzy soft aggregation operator that allows constructing more efficient decision making method.
Abstract: In this work, we define a fuzzy soft set theory and its related properties. We then define fuzzy soft aggregation operator that allows constructing more efficient decision making method. Finally, we give an example which shows that the method can be successfully applied to many problems that contain uncertainties.
TL;DR: Two models for prioritizing failures modes through a crisp risk priority number (RPN) are proposed, specifically intended to overcome limitations of traditional FMEA.
Abstract: Traditional Failure Mode and Effects Analysis (FMEA) has shown its effectiveness in defining, identifying, and eliminating known and/or potential failures or problems in products, process, designs, and services to help ensure the safety and reliability of systems applied in a wide range of industries. However, its approach to prioritize failure modes through a crisp risk priority number (RPN) has been highly controversial. This paper proposes two models for prioritizing failures modes, specifically intended to overcome such limitations of traditional FMEA. The first proposed model treats the three risk factors as fuzzy linguistic variables, and employs alpha level sets to provide a fuzzy RPN. The second model employs an approach based on the degree of match and fuzzy rule-base. This second model considers the diversity and uncertainty in the opinions of FMEA team members, and converts the assessed information into a convex normalized fuzzy number. The degree of match (DM) is used thereafter to estimate the matching between the assessed information and the fuzzy number characterizing the linguistic terms. The proposed models are suitably supplemented by illustrative examples.
TL;DR: With respect to risk decision making problems with interval probability in which the attribute values take the form of the uncertain linguistic variables, a multi-attribute decision making method based on prospect theory is proposed.
Abstract: With respect to risk decision making problems with interval probability in which the attribute values take the form of the uncertain linguistic variables, a multi-attribute decision making method based on prospect theory is proposed To begin with, the uncertain linguistic variables can be transformed into the trapezoidal fuzzy number, and the prospect value function of the trapezoidal fuzzy number based on the decision-making reference point of each attribute and the weight function of interval probability can be constructed; then the prospect value of attribute for every alternative is calculated through prospect value function of the trapezoidal fuzzy number and the weight function of interval probability, and the weighted prospect value of alternative is acquired by using weighted average method according to attribute weights, and all the alternatives are sorted according to the expected values of the weighted prospect values; Finally, an illustrate example is given to show the decision-making steps, the influence on decision making for different parameters of value function and different decision-making reference point, and the feasibility of the method
TL;DR: The Cartesian product, composition, union and join is defined on interval-valued fuzzy graphs and some properties of self-complementary and self-weak complementary interval- valued fuzzy complete graphs are presented.
Abstract: We define the Cartesian product, composition, union and join on interval-valued fuzzy graphs and investigate some of their properties. We also introduce the notion of interval-valued fuzzy complete graphs and present some properties of self-complementary and self-weak complementary interval-valued fuzzy complete graphs.
Book•
14 Feb 2011
TL;DR: The purpose of this monograph is to explore the relationship between Fuzzy Data, Bayes' Theorem, and Problems, and the role that these models play in the development of knowledge and understanding of fuzzy data.
Abstract: Preface. Part I FUZZY INFORMATION. 1. Fuzzy Data. 1.1 One-dimensional Fuzzy Data. 1.2 Vector-valued Fuzzy Data. 1.3 Fuzziness and Variability. 1.4 Fuzziness and Errors. 1.5 Problems. 2. Fuzzy Numbers and Fuzzy Vectors. 2.1 Fuzzy Numbers and Characterizing Functions. 2.2 Vectors of Fuzzy Numbers and Fuzzy Vectors. 2.3 Triangular Norms. 2.4 Problems. 3. Mathematical Operations for Fuzzy Quantities. 3.1 Functions of Fuzzy Variables. 3.2 Addition of Fuzzy Numbers. 3.3 Multiplication of Fuzzy Numbers. 3.4 Mean Value of Fuzzy Numbers. 3.5 Differences and Quotients. 3.6 Fuzzy Valued Functions. 3.7 Problems. Part II DESCRIPTIVE STATISTICS FOR FUZZY DATA. 4. Fuzzy Samples. 4.1 Minimum of Fuzzy Data. 4.2 Maximum of Fuzzy Data. 4.3 Cumulative Sum for Fuzzy Data. 4.4 Problems. 5. Histograms for Fuzzy Data. 5.1 Fuzzy Frequency of a Fixed Class. 5.2 Fuzzy Frequency Distributions. 5.3 Axonometric Diagram of the Fuzzy Histogram. 5.4 Problems. 6. Empirical Distribution Functions. 6.1 Fuzzy Valued Empirical Distribution Function. 6.2 Fuzzy Empirical Fractiles. 6.3 Smoothed Empirical Distribution Function. 6.4 Problems. 7. Empirical Correlation for Fuzzy Data. 7.1 Fuzzy Empirical Correlation Coefficient. 7.2 Problems. Part III FOUNDATIONS OF STATISTICAL INFERENCE WITH FUZZY DATA. 8. Fuzzy Probability Distributions. 8.1 Fuzzy Probability Densities. 8.2 Probabilities based on Fuzzy Probability Densities. 8.3 General Fuzzy Probability Distributions. 8.4 Problems. 9. A Law of Large Numbers. 9.1 Fuzzy Random Variables. 9.2 Fuzzy Probability Distributions induced by Fuzzy Random Variables. 9.3 Sequences of Fuzzy Random Variables. 9.4 Law of Large Numbers for Fuzzy Random Variables. 9.5 Problems. 10. Combined Fuzzy Samples. 10.1 Observation Space and Sample Space. 10.2 Combination of Fuzzy Samples. 10.3 Statistics of Fuzzy Data. 10.4 Problems. Part IV CLASSICAL STATISTICAL INFERENCE FOR FUZZY DATA. 11. Generalized Point Estimations. 11.1 Estimations based on Fuzzy Samples. 11.2 Sample Moments. 11.3 Problems. 12. Generalized Confidence Regions. 12.1 Confidence Functions. 12.2 Fuzzy Confidence Regions. 12.3 Problems. 13. Statistical Tests for Fuzzy Data. 13.1 Test Statistics and Fuzzy Data. 13.2 Fuzzy p-Values. 13.3 Problems. Part V BAYESIAN INFERENCE AND FUZZY INFORMATION. 14. Bayes' Theorem and Fuzzy Information. 14.1 Fuzzy a-priori Distributions. 14.2 Updating Fuzzy a-priori Distributions. 14.3 Problems. 15. Generalized Bayes' Theorem. 15.1 Likelihood Function for Fuzzy Data. 15.2 Bayes' Theorem for Fuzzy a-priori Distribution and Fuzzy Data. 15.3 Problems. 16. Bayesian Confidence Regions. 16.1 Confidence Regions based on Fuzzy Data. 16.2 Fuzzy HPD-Regions. 16.3 Problems. 17. Fuzzy Predictive Distributions. 17.1 Discrete Case. 17.2 Discrete Models with Continuous Parameter Space. 17.3 Continuous Case. 17.4 Problems. 18. Bayesian Decisions and Fuzzy Information. 18.1 Bayesian Decisions. 18.2 Fuzzy Utility. 18.3 Discrete State Space. 18.4 Continuous State Space. 18.5 Problems. References. Part VI REGRESSION ANALYSIS AND FUZZYINFORMATION. 19 Classical regression analysis. 19.1 Regression models. 19.2 Linear regression models with Gaussian dependent variables. 19.3 General linear models. 19.4 Nonidentical variances. 19.5 Problems. 20 Regression models and fuzzy data. 20.1 Generalized estimators for linear regression models based on the extension principle. 20.2 Generalized confidence regions for parameters. 20.3 Prediction in fuzzy regression models. 20.4 Problems. 21 Bayesian regression analysis. 21.1 Calculation of a posteriori distributions. 21.2 Bayesian confidence regions. 21.3 Probabilities of hypotheses. 21.4 Predictive distributions. 21.5 A posteriori Bayes estimators for regression parameters. 21.6 Bayesian regression with Gaussian distributions. 21.7 Problems. 22 Bayesian regression analysis and fuzzy information. 22.1 Fuzzy estimators of regression parameters. 22.2 Generalized Bayesian confidence regions. 22.3 Fuzzy predictive distributions. 22.4 Problems. Part VII FUZZY TIME SERIES. 23 Mathematical concepts. 23.1 Support functions of fuzzy quantities. 23.2 Distances of fuzzy quantities. 23.3 Generalized Hukuhara difference. 24 Descriptive methods for fuzzy time series. 24.1 Moving averages. 24.2 Filtering. 24.2.1 Linear filtering. 24.2.2 Nonlinear filters. 24.3 Exponential smoothing. 24.4 Components model. 24.4.1 Model without seasonal component. 24.4.2 Model with seasonal component. 24.5 Difference filters. 24.6 Generalized Holt-Winter method. 24.7 Presentation in the frequency domain. 25 More on fuzzy random variables and fuzzy random vectors. 25.1 Basics. 25.2 Expectation and variance of fuzzy random variables. 25.3 Covariance and correlation. 25.4 Further results. 26 Stochastic methods in fuzzy time series analysis. 26.1 Linear approximation and prediction. 26.2 Remarks concerning Kalman filtering. Part VIII APPENDICES. A1 List of symbols and abbreviations. A2 Solutions to the problems. A3 Glossary. A4 Related literature. References. Index.
TL;DR: An evaluation model based on analytic hierarchy process, fuzzy sets and technique for order performance by similarity to ideal solution (TOPSIS), to tackle the issue in fuzzy environment is proposed.
Abstract: E-alliance is the union of e-commerce and its success and efficiency is related to comprehensive quality of e-commerce. Thus, ranking e-commerce websites in e-alliance is of importance, which is a multi-criteria decision-making (MCDM) problem. This paper proposes an evaluation model based on analytic hierarchy process (AHP), fuzzy sets and technique for order performance by similarity to ideal solution (TOPSIS), to tackle the issue in fuzzy environment. The AHP is applied to analyze the structure of ranking problem and to determine weights of the criteria, fuzzy sets is utilized to present ambiguity and subjectivity with linguistic values parameterized by triangular fuzzy numbers, and TOPSIS method is used to obtain final ranking. Case analysis is conducted to illustrate the utilization of the model for the problem. It demonstrates the effectiveness and feasibility of the proposed model.
TL;DR: An approach to tackle multiple criteria group decision making problems in the context of interval-valued intuitionistic fuzzy sets using an optimization model to obtain criterion weights in exact numbers rather than intervals and subsequently calculates an aggregated IVIFN for each alternative.
Abstract: Research highlights? We develop an approach to tackle multiple criteria group decision making problems in the context of interval-valued intuitionistic fuzzy sets. ? An interval-valued intuitionistic fuzzy preference relation matrix is employed to determine the criterion importance with pairwise comparisons. ? Three families of parametric fuzzy unions and intersections are applied in the aggregation operation with comparisons of alternative rankings. ? The parameters of the aggregation operators have an impact on the ranks of alternatives. ? The non-parametric fuzzy operations in the aggregation operators result in a consistent ranking of alternatives. This study develops an approach to tackle multiple criteria group decision-making problems in the context of interval-valued intuitionistic fuzzy sets. Due to conflicting evaluations and insufficient information about the criteria, an interval-valued intuitionistic fuzzy preference relation matrix is employed to determine the relative importance of criteria in terms of pairwise comparisons. The decision matrix, which indicates the degree of alternatives with respect to each criterion, is expressed by interval-valued intuitionistic fuzzy numbers (IVIFNs). In order to integrate interval-valued intuitionistic fuzzy information, some special aggregation operators are created by altering the aggregation operation of IVIFNs. The three families of parametric fuzzy unions and fuzzy intersections are applied in the aggregation operation with comparisons of the ranking results of alternatives. With a linear programming method, the proposed approach uses an optimization model to obtain criterion weights in exact numbers rather than intervals, and subsequently calculates an aggregated IVIFN for each alternative. The score function and accuracy function assist in discriminating between the aggregated IVIFNs, and in generating a final rank of alternatives. Finally, an illustrative supplier selection problem is used to demonstrate how to apply the proposed approach and to observe the computational consequences resulting from various aggregation operators. The results reveal that the parameters of the aggregation operators indeed have an impact on the ranks of alternatives.