Multicriteria decision-making method using the Dice similarity measure based on the reduct intuitionistic fuzzy sets of interval-valued intuitionistic fuzzy sets

This paper proposes the concept of the reduct intuitionistic fuzzy sets of interval-valued intuitionistic fuzzy sets (IVIFSs) with respect to adjustable weight vectors and the Dice similarity measure based on the reduct intuitionistic fuzzy sets to explore the effects of optimism, neutralism, and pessimism in decision making. Then a decision-making method with the pessimistic, optimistic, and neutral schemes desired by the decision maker is established by combining adjustable weight vectors and the Dice similarity measure for IVIFSs. The proposed decision-making method is more flexible and adjustable in practical problems and can determine the ranking order of alternatives and the optimal one(s), so that it can overcome the difficulty of the ranking order and decision making when there exist the same measure values of some alternatives in some cases. This adjustable feature can provide the decision maker with more selecting schemes and actionable results for the decision-making analysis. Finally, two illustrative examples are employed to show the feasibility of the proposed method in practical applications.     

Fuzzy decision-making method based on the weighted correlation coefficient under intuitionistic fuzzy environment

A multicriteria fuzzy decision-making method based on weighted correlation coefficients using entropy weights is proposed under intuitionistic fuzzy environment for some situations where the information about criteria weights for alternatives is completely unknown. To determine the entropy weights with respect to a set of criteria represented by intuitionistic fuzzy sets (IFSs), we establish an entropy weight model, which can be used to get the criteria weights, and then propose an evaluation formula of weighted correlation coefficient between an alternative and the ideal alternative. The alternatives can be ranked and the most desirable one(s) can be selected according to the weighted correlation coefficients. Finally, two illustrative examples demonstrate the practicality and effectiveness of the proposed method.     

Correlation coefficient of interval-valued intuitionistic fuzzy sets and its application to multiple attribute group decision making problems

In this paper, we investigate the group decision making problems in which all the information provided by the decision-makers is presented as interval-valued intuitionistic fuzzy decision matrices where each of the elements is characterized by interval-valued intuitionistic fuzzy number (IVIFN), and the information about attribute weights is partially known. First, we use the interval-valued intuitionistic fuzzy hybrid geometric (IIFHG) operator to aggregate all individual interval-valued intuitionistic fuzzy decision matrices provided by the decision-makers into the collective interval-valued intuitionistic fuzzy decision matrix, and then we use the score function to calculate the score of each attribute value and construct the score matrix of the collective interval-valued intuitionistic fuzzy decision matrix. From the score matrix and the given attribute weight information, we establish an optimization model to determine the weights of attributes, and then we use the obtained attribute weights and the interval-valued intuitionistic fuzzy weighted geometric (IIFWG) operator to fuse the interval-valued intuitionistic fuzzy information in the collective interval-valued intuitionistic fuzzy decision matrix to get the overall interval-valued intuitionistic fuzzy values of alternatives, and then rank the alternatives according to the correlation coefficients between IVIFNs and select the most desirable one(s). Finally, a numerical example is used to illustrate the applicability of the proposed approach.     

An improved method on group decision making based on interval-valued intuitionistic fuzzy prioritized operators

Interval-valued intuitionistic fuzzy prioritized operators are widely used in group decision making under uncertain environment due to its flexibility to model uncertain information. However, there is a shortcoming in the existing aggregation operators (interval-valued intuitionistic fuzzy prioritized weighted average (IVIFPWA)) to deal with group decision making in some extreme situations. For example, when an expert gives an absolute negative evaluation, the operators could lead to irrational results, so that they are not effectively enough to handle group decision making. In this paper, several examples are illustrated to show the unreasonable results in some of these situations. Actually, these unreasonable cases are common for operators in dealing with product averaging, not only emerging in IVIFPWA operators. To overcome the shortcoming of these kinds of operators, an improvement of making slight adjustment on initial evaluations is provided. Numerical examples are used to show the efficiency of the improvement.     

A complex multi-attribute large-group PLS decision-making method in the interval-valued intuitionistic fuzzy environment

In the complex multi-attribute large-group decision-making (CMALGDM) problems in interval-valued intuitionistic fuzzy (IVIF) environment, attributes of the alternatives are often stratified and correlated. This paper proposes a decision-making method for these problems based on partial least squares (PLS) path modelling, which not only fully exploits the decision information of decision makers (DMs), but also effectively addresses the relativity problem in the decision attributes and objectively assigned weights to the primary decision attributes (i.e., “latent variables for decision making”). The method can be outlined in three steps. First, a two-stage method is proposed to transform the interval-valued intuitionistic fuzzy number (IVIFN) samples into single-valued samples. In this step, an improved C-OWA operator is first given to transform the IVIFN samples into intuitionistic fuzzy number (IFN) samples, which makes the preference information of the DMs more objectively aggregated. Then a proposed membership-based method is applied to reduce the information loss and transform the IFN samples into single-valued samples. Second, the estimated values and weights of the “latent variables for decision-making” are obtained by means of the PLS path modelling algorithm. Finally, a multi-alternative sorting method is devised in accordance with the estimated values and weights. An example is provided to illustrate the proposed technique and evaluate its feasibility and validity.     

The extended linear assignment method for multiple criteria decision analysis based on interval-valued intuitionistic fuzzy sets

The theory of interval-valued intuitionistic fuzzy sets is useful and beneficial for handling uncertainty and imprecision in multiple criteria decision analysis. In addition, the theory allows for convenient quantification of the equivocal nature of human subjective assessments. In this paper, by extending the traditional linear assignment method, we propose a useful method for solving multiple criteria evaluation problems in the interval-valued intuitionistic fuzzy context. A ranking procedure consisting of score functions, accuracy functions, membership uncertainty indices, and hesitation uncertainty indices is presented to determine a criterion-wise preference of the alternatives. An extended linear assignment model is then constructed using a modified weighted-rank frequency matrix to determine the priority order of various alternatives. The feasibility and applicability of the proposed method are illustrated with a multiple criteria decision-making problem involving the selection of a bridge construction method. Additionally, a comparative analysis with other methods, including the approach with weighted aggregation operators, the closeness coefficient-based method, and the auxiliary nonlinear programming models, has been conducted for solving the investment company selection problem to validate the effectiveness of the extended linear assignment method.     

Extension of the TOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environment

TOPSIS is one of the well-known methods for multiple attribute decision making (MADM). In this paper, we extend the TOPSIS method to solve multiple attribute group decision making (MAGDM) problems in interval-valued intuitionistic fuzzy environment in which all the preference information provided by the decision-makers is presented as interval-valued intuitionistic fuzzy decision matrices where each of the elements is characterized by interval-valued intuitionistic fuzzy number (IVIFNs), and the information about attribute weights is partially known. First, we use the interval-valued intuitionistic fuzzy hybrid geometric (IIFHG) operator to aggregate all individual interval-valued intuitionistic fuzzy decision matrices provided by the decision-makers into the collective interval-valued intuitionistic fuzzy decision matrix, and then we use the score function to calculate the score of each attribute value and construct the score matrix of the collective interval-valued intuitionistic fuzzy decision matrix. From the score matrix and the given attribute weight information, we establish an optimization model to determine the weights of attributes, and construct the weighted collective interval-valued intuitionistic fuzzy decision matrix, and then determine the interval-valued intuitionistic positive-ideal solution and interval-valued intuitionistic negative-ideal solution. Based on different distance definitions, we calculate the relative closeness of each alternative to the interval-valued intuitionistic positive-ideal solution and rank the alternatives according to the relative closeness to the interval-valued intuitionistic positive-ideal solution and select the most desirable one(s). Finally, an example is used to illustrate the applicability of the proposed approach.     

Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis

Hesitant fuzzy sets (HFSs), which allow the membership degree of an element to a set represented by several possible values, can be considered as a powerful tool to express uncertain information in the process of group decision making. We derive some correlation coefficient formulas for HFSs and apply them to clustering analysis under hesitant fuzzy environments. Two real world examples, i.e. software evaluation and classification as well as the assessment of business failure risk, are employed to illustrate the actual need of the clustering algorithm based on HFSs, which can incorporate the difference of evaluation information provided by different experts in clustering processes. In order to extend the application domain of the clustering algorithm in the framework of HFSs, we develop the interval-valued HFSs and the corresponding correlation coefficient formulas, and then demonstrate their application in clustering with interval-valued hesitant fuzzy information through a specific numerical example.     

Research on AHP with interval-valued intuitionistic fuzzy sets and its application in multi-criteria decision making problems

This paper investigates an approach for multi-criterion decision making (MCDM) problems with interval-valued intuitionistic fuzzy preference relations (IVIFPRs). Based on the novel interval score function, some extended concepts associated with IVIFPRs are defined, including the score matrix, the approximate optimal transfer matrix and the possibility degree matrix. By using these new matrixes, a prioritization method for IVIFPRs is proposed. Then, we investigate an interval-valued intuitionistic fuzzy AHP method for multi-criteria decision making (MCDM) problems. In the end, a numerical example is provided to illustrate the application of the proposed approach.     

A novel approach to interval-valued intuitionistic fuzzy soft set based decision making

The soft set theory, originally proposed by Molodtsov, can be used as a general mathematical tool for dealing with uncertainty. The interval-valued intuitionistic fuzzy soft set is a combination of an interval-valued intuitionistic fuzzy set and a soft set. The aim of this paper is to investigate the decision making based on interval-valued intuitionistic fuzzy soft sets. By means of level soft sets, we develop an adjustable approach to interval-valued intuitionistic fuzzy soft sets based decision making and some numerical examples are provided to illustrate the developed approach. Furthermore, we also define the concept of the weighted interval-valued intuitionistic fuzzy soft set and apply it to decision making.     

Multiple criteria group decision-making with generalized interval-valued fuzzy numbers based on signed distances and incomplete weights

Decision-making information provided by decision makers is often imprecise or uncertain, due to lack of data, time pressure, or the decision makers’ limited attention and information-processing capabilities. Interval-valued fuzzy sets are associated with greater imprecision and more ambiguity than are ordinary fuzzy sets. For these reasons, this paper presents a signed distance-based method for handling fuzzy multiple-criteria group decision-making problems in which individual assessments are provided as generalized interval-valued trapezoidal fuzzy numbers, and the information about criterion weights are not precisely but partially known. First, concerning the relative importance of decision makers and the group consensus of fuzzy opinions, all individual decision opinions were aggregated into group opinions using a hybrid average with weighted averaging and signed distance-based ordered weighted averaging operations. Next, considering a decision situation with incomplete weight information of criteria, an integrated programming model was developed to estimate criterion weights and to order the priorities of various alternatives based on signed distances. In addition, several deviation variables were introduced to mitigate the effect of inconsistent evaluations on the importance of criteria. Finally, the feasibility of the proposed method is illustrated by a numerical example of a multi-criteria supplier selection problem. Furthermore, a comparative analysis with other methods was conducted to validate the effectiveness and applicability of the proposed methodology.     

The interval-valued fuzzy TOPSIS method and experimental analysis

The purpose of this paper is to extend the TOPSIS method based on interval-valued fuzzy sets in decision analysis. Hwang and Yoon developed the technique for order preference by similarity to ideal solution (TOPSIS) in 1981. TOPSIS has been widely used to rank the preference order of alternatives and determine the optimal choice. Considering the fact that it is difficult to precisely attach the numerical measures to the relative importance of the attributes and to the impacts of the alternatives on these attributes in some cases, therefore, the TOPSIS method has been extended for interval-valued fuzzy data in this paper. In addition, a comprehensive experimental analysis to observe the interval-valued fuzzy TOPSIS results yielded by different distance measures is presented. A comparative analysis of interval-valued fuzzy TOPSIS rankings from each distance measure is illustrated with discussions on consistency rates, contradiction rates, and average Spearman correlation coefficients. Finally, a second-order regression model is provided to highlight the effects of the number of alternatives, the number of attributes, and distance measures on average Spearmen correlation coefficients.     

On characterization of generalized interval-valued fuzzy rough sets on two universes of discourse

This paper proposes a general study of (I,T)-interval-valued fuzzy rough sets on two universes of discourse integrating the rough set theory with the interval-valued fuzzy set theory by constructive and axiomatic approaches. Some primary properties of interval-valued fuzzy logical operators and the construction approaches of interval-valued fuzzy T-similarity relations are first introduced. Determined by an interval-valued fuzzy triangular norm and an interval-valued fuzzy implicator, a pair of lower and upper generalized interval-valued fuzzy rough approximation operators with respect to an arbitrary interval-valued fuzzy relation on two universes of discourse is then defined. Properties of I-lower and T-upper interval-valued fuzzy rough approximation operators are examined based on the properties of interval-valued fuzzy logical operators discussed above. Connections between interval-valued fuzzy relations and interval-valued fuzzy rough approximation operators are also established. Finally, an operator-oriented characterization of interval-valued fuzzy rough sets is proposed, that is, interval-valued fuzzy rough approximation operators are characterized by axioms. Different axiom sets of I-lower and T-upper interval-valued fuzzy set-theoretic operators guarantee the existence of different types of interval-valued fuzzy relations which produce the same operators.     

Multi-criteria decision-making methods based on intuitionistic fuzzy sets

In this paper we present new methods for solving multi-criteria decision-making problem in an intuitionistic fuzzy environment. First, we define an evaluation function for the decision-making problem to measure the degrees to which alternatives satisfy and do not satisfy the decision-maker’s requirement. Then, we introduce and discuss the concept of intuitionistic fuzzy point operators. By using the intuitionistic fuzzy point operators, we can reduce the degree of uncertainty of the elements in a universe corresponding to an intuitionistic fuzzy set. Furthermore, a series of new score functions are defined for multi-criteria decision-making problem based on the intuitionistic fuzzy point operators and the evaluation function and their effectiveness and advantage are illustrated by examples.     

Some similarity measures of intuitionistic fuzzy sets and their applications to multiple attribute decision making

Atanassov (1986) defined the notion of intuitionistic fuzzy set, which is a generalization of the notion of Zadeh’ fuzzy set. In this paper, we first develop some similarity measures of intuitionistic fuzzy sets. Then, we define the notions of positive ideal intuitionistic fuzzy set and negative ideal intuitionistic fuzzy set. Finally, we apply the similarity measures to multiple attribute decision making under intuitionistic fuzzy environment.     

Intuitionistic and interval-valued intutionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group

Szmidt and Kacprzyk (Lecture Notes in Artificial Intelligence 3070:388–393, 2004a) introduced a similarity measure, which takes into account not only a pure distance between intuitionistic fuzzy sets but also examines if the compared values are more similar or more dissimilar to each other. By analyzing this similarity measure, we find it somewhat inconvenient in some cases, and thus we develop a new similarity measure between intuitionistic fuzzy sets. Then we apply the developed similarity measure for consensus analysis in group decision making based on intuitionistic fuzzy preference relations, and finally further extend it to the interval-valued intuitionistic fuzzy set theory.     

A group decision making approach based on aggregating interval data into interval-valued intuitionistic fuzzy information

Group decision making is one of the most important problems in decision making sciences. The aim of this article is to aggregate the interval data into the interval-valued intuitionistic fuzzy information for multiple attribute group decision making. In this model, the decision information is provided by decision maker, which is characterized by interval data. Based on the idea of mean and variance in statistics, we first define the concepts of satisfactory and dissatisfactory intervals of attribute vector against each alternative. Using these concepts, we develop an approach to aggregate the attribute vector into interval-valued intuitionistic fuzzy number under group decision making environment. A practical example is provided to illustrate the proposed method. To show the validity of the reported method, comparisons with other methods are also made.     

Lattice valued intuitionistic fuzzy sets

In this paper a new definition of a lattice valued intuitionistic fuzzy set (LIFS) is introduced, in an attempt to overcome the disadvantages of earlier definitions. Some properties of this kind of fuzzy sets and their basic operations are given. The theorem of synthesis is proved: For every two families of subsets of a set satisfying certain conditions, there is an lattice valued intuitionistic fuzzy set for which these are families of level sets. The research supported by Serbian Ministry of Science and Technology, Grant No. 1227.