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.     

Linguistic power aggregation operators and their application to multiple attribute group decision making

The aim of this study is to develop new linguistic aggregation operators based on the power-average (PA) operator, such as the linguistic power average (LPA) operator, the linguistic weighted PA operator, and the LPOWA operator. We studied some desired properties of the developed operators, such as idempotency and commutativity. Moreover, we developed two approaches to deal with group decision-making problems under linguistic environments. If the weighting vector of the decision makers was known, we developed an approach based on the linguistic weighted PA operator. On the other hand, if the weighting vector of the decision maker was unknown, we developed a different approach based on the LPOWA operator. We also developed new uncertain linguistic operators under uncertain linguistic environments, such as the ULPA operator, the uncertain linguistic weighted PA operator, and the ULPOWA operator. We extended these approaches, which are based on ULWPA and ULPOWA operators, respectively, to solve group decision-making problems under an uncertain linguistic environment. Finally, a practical example is provided to illustrate the multiple-attribute group decision-making process.     

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.     

Generalized triangular fuzzy correlated averaging operator and their application to multiple attribute decision making

We investigate the multiple attribute decision making problems with triangular fuzzy information. Motivated by the ideal of Choquet integral [G. Choquet, Theory of capacities, Ann. Instit. Fourier 5 (1953) 131–295] and generalized OWA operator [R.R. Yager, Generalized OWA aggregation operators, Fuzzy Optim. Dec. Making 3 (2004) 93–107], in this paper, we have developed an generalized triangular fuzzy correlated averaging (GTFCA) operator. The prominent characteristic of the operators is that they cannot only consider the importance of the elements or their ordered positions, but also reflect the correlation among the elements or their ordered positions. We have applied the GTFCA operator to multiple attribute decision making problems with triangular fuzzy information. Finally an illustrative example has been given to show the developed method.     

Some geometric aggregation operators based on interval intuitionistic uncertain linguistic variables and their application to group decision making

With respect to multiple attribute group decision making (MAGDM) problems in which both the attribute weights and the expert weights take the form of crisp numbers, and attribute values take the form of interval-valued intuitionistic uncertain linguistic variables, some new group decision making analysis methods are developed. Firstly, some operational laws, expected value and accuracy function of interval-valued intuitionistic uncertain linguistic variables are introduced. Then, an interval-valued intuitionistic uncertain linguistic weighted geometric average (IVIULWGA) operator and an interval-valued intuitionistic uncertain linguistic ordered weighted geometric (IVIULOWG) operator have been developed. Furthermore, some desirable properties of the IVIULWGA operator and the IVIULOWG operator, such as commutativity, idempotency and monotonicity, have been studied, and an interval-valued intuitionistic uncertain linguistic hybrid geometric (IVIULHG) operator which generalizes both the IVIULWGA operator and the IVIULOWG operator, was developed. Based on these operators, an approach to multiple attribute group decision making with interval-valued intuitionistic uncertain linguistic information has been proposed. Finally, an illustrative example is given to verify the developed approaches and to demonstrate their practicality and effectiveness.     

Uncertain linguistic Bonferroni mean operators and their application to multiple attribute decision making

In this paper, we investigate the multiple attribute decision making (MADM) problems with uncertain linguistic information. Motivated by the ideal of Bonferroni mean and geometric Bonferroni mean, we develop two aggregation techniques called the uncertain linguistic Bonferroni mean (ULBM) operator and the uncertain linguistic geometric Bonferroni mean (ULGBM) operator for aggregating the uncertain linguistic information. We study its properties and discuss its special cases. For the situations where the input arguments have different importance, we then define the uncertain linguistic weighted Bonferroni mean (ULWBM) operator and the uncertain linguistic weighted geometric Bonferroni mean (ULWGBM) operator, based on which we develop two procedures for multiple attribute decision making under the uncertain linguistic environments. Finally, a practical example is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

The extended COWG operators and their application to multiple attributive group decision making problems with interval numbers

The aim of this paper is to develop two extended continuous ordered weighted geometric (COWG) operators, such as the weighted geometric averaging COWG (WG-COWG) and ordered weighted geometric averaging COWG (OWG-COWG) operators. We study some desirable properties of the WG-COWG and OWG-COWG operators, and present their application to multiple attributive group decision making (MAGDM) problems with interval numbers. Finally, an illustrative numerical example is used to verify the developed approaches.     

Intuitionistic uncertain linguistic powered einstein aggregation operators and their application to multi-attribute group decision making

The intuitionistic uncertain fuzzy linguistic variable can easily expressthe fuzzy information, and the power average (PA) operator is a usefultool which provides more versatility in the information aggregation procedure.At the same time, Einstein operations are a kind of various t-normsand t-conorms families which can be used to perform the corresponding intersectionsand unions of intuitionistic fuzzy sets (IFSs). In this paper, wewill combine the PA operator and Einstein operations to intuitionistic uncertainlinguistic environment, and propose some new PA operators. Firstly,the definition and some basic operations of intuitionistic uncertain linguisticnumber (IULN), power aggregation (PA) operator and Einstein operationsare introduced. Then, we propose intuitionistic uncertain linguistic fuzzypowered Einstein averaging (IULFPEA) operator, intuitionistic uncertain linguisticfuzzy powered Einstein weighted (IULFPEWA) operator, intuitionisticuncertain linguistic fuzzy Einstein geometric (IULFPEG) operator and intuitionisticuncertain linguistic fuzzy Einstein weighted geometric (IULFPEWG)operator, and discuss some properties of them in detail. Furthermore, we developthe decision making methods for multi-attribute group decision making(MAGDM) problems with intuitionistic uncertain linguistic information andgive the detail decision steps. At last, an illustrate example is given to showthe process of decision making and the effectiveness of the proposed method.     

Continuous interval-valued intuitionistic fuzzy aggregation operators and their applications to group decision making

In this paper, we introduce a new operator called the continuous interval-valued intuitionistic fuzzy ordered weighted averaging (C-IVIFOWA) operator for aggregating the interval-valued intuitionistic fuzzy values. It combines the intuitionistic fuzzy ordered weighted averaging (IFOWA) operator and the continuous ordered weighted averaging (C-OWA) operator by a controlling parameter, which can be employed to diminish fuzziness and improve the accuracy of decision making. We further apply the C-IVIFOWA operator to the aggregation of multiple interval-valued intuitionistic fuzzy values and obtain a wide range of aggregation operators including the weighted C-IVIFOWA (WC-IVIFOWA) operator, the ordered weighted (OWC-IVIFOWA) operator and the combined C-IVIFOWA (CC-IVIFOWA) operator. Some desirable properties of these operators are investigated. And finally, we give a numerical example to illustrate the applications of these operators to group decision making under interval-valued intuitionistic fuzzy environment.     

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 coefficient of dual hesitant fuzzy sets and its application to multiple attribute decision making

A dual hesitant fuzzy set (DHFS) consists of two parts, that is, the membership hesitancy function and the nonmembership hesitancy function, supporting a more exemplary and flexible access to assign values for each element in the domain, and can handle two kinds of hesitancy in this situation. It can be considered as a powerful tool to express uncertain information in the process of group decision making. Therefore, we propose a correlation coefficient between DHFSs as a new extension of existing correlation coefficients for hesitant fuzzy sets and intuitionistic fuzzy sets and apply it to multiple attribute decision making under dual hesitant fuzzy environments. Through the weighted correlation coefficient between each alternative and the ideal alternative, the ranking order of all alternatives can be determined and the best alternative can be easily identified as well. Finally, a practical example of investment alternatives is given to demonstrate the practicality and effectiveness of the developed approach.     

Power average operators of trapezoidal intuitionistic fuzzy numbers and application to multi-attribute group decision making

Trapezoidal intuitionistic fuzzy numbers (TrIFNs) is a special intuitionistic fuzzy set on a real number set. TrIFNs are useful to deal with ill-known quantities in decision data and decision making problems themselves. The focus of this paper is on multi-attribute group decision making (MAGDM) problems in which the attribute values are expressed with TrIFNs, which are solved by developing a new decision method based on power average operators of TrIFNs. The new operation laws for TrIFNs are given. From a viewpoint of Hausdorff metric, the Hamming and Euclidean distances between TrIFNs are defined. Hereby the power average operator of real numbers is extended to four kinds of power average operators of TrIFNs, involving the power average operator of TrIFNs, the weighted power average operator of TrIFNs, the power ordered weighted average operator of TrIFNs, and the power hybrid average operator of TrIFNs. In the proposed group decision method, the individual overall evaluation values of alternatives are generated by using the power average operator of TrIFNs. Applying the hybrid average operator of TrIFNs, the individual overall evaluation values of alternatives are then integrated into the collective ones, which are used to rank the alternatives. The example analysis shows the practicality and effectiveness of the proposed method.     

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.     

A multi-attribute group decision-making method based on weighted geometric aggregation operators of interval-valued trapezoidal fuzzy numbers

With respect to the multiple attribute group decision making problems in which the attribute values take the form of generalized interval-valued trapezoidal fuzzy numbers (GITFN), this paper proposed a decision making method based on weighted geometric aggregation operators. First, some operational rules, the distance and comparison between two GITFNs are introduced. Second, the generalized interval-valued trapezoidal fuzzy numbers weighted geometric aggregation (GITFNWGA) operator, the generalized interval-valued trapezoidal fuzzy numbers ordered weighted geometric aggregation (GITFNOWGA) operator, and the generalized interval-valued trapezoidal fuzzy numbers hybrid geometric aggregation (GITFNHGA) operator are proposed, and their various properties are investigated. At the same time, the group decision methods based on these operators are also presented. Finally, an illustrate example is given to show the decision-making steps and the effectiveness of this method.     

Multi-person multi-attribute decision making models under intuitionistic fuzzy environment

Intuitionistic fuzzy numbers, each of which is characterized by the degree of membership and the degree of non-membership of an element, 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 is expressed as intuitionistic fuzzy decision matrices where each of the elements is characterized by intuitionistic fuzzy number, and the information about attribute weights is partially known, which may be constructed by various forms. We first use the intuitionistic fuzzy hybrid geometric (IFHG) operator to aggregate all individual intuitionistic fuzzy decision matrices provided by the decision makers into the collective intuitionistic fuzzy decision matrix, then we utilize the score function to calculate the score of each attribute value and construct the score matrix of the collective intuitionistic fuzzy decision matrix. Based on the score 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 intuitionistic fuzzy weighted geometric (IFWG) operator to fuse the intuitionistic fuzzy information in the collective intuitionistic fuzzy decision matrix to get the overall intuitionistic fuzzy values of alternatives by which the ranking of all the given alternatives can be found. Finally, we give an illustrative example.     

Some interval-valued intuitionistic uncertain linguistic Choquet operators and their application to multi-attribute group decision making

In this study a generated admissible order between interval-valued intuitionistic uncertain linguistic numbers using two continuous functions is introduced. Then, two interval-valued intuitionistic uncertain linguistic operators called the interval-valued intuitionistic uncertain linguistic Choquet averaging (IVIULCA) operator and the interval-valued intuitionistic uncertain linguistic Choquet geometric mean (IVIULCGM) operator are defined, which consider the interactive characteristics among elements in a set. In order to overall reflect the correlations between them, we further define the generalized Shapley interval-valued intuitionistic uncertain linguistic Choquet averaging (GS-IVIULCA) operator and the generalized Shapley interval-valued intuitionistic uncertain linguistic Choquet geometric mean (GS-IVIULCGM) operator. Moreover, if the information about the weights of experts and attributes is incompletely known, the models for the optimal fuzzy measures on expert set and attribute set are established, respectively. Finally, a method to multi-attribute group decision making under interval-valued intuitionistic uncertain linguistic environment is developed, and an example is provided to show the specific application of the developed procedure.     

Generalized weighted exponential proportional aggregation operators and their applications to group decision making

We present a new aggregation operator called the generalized ordered weighted exponential proportional averaging (GOWEPA) operator, which is based on an optimal model. We study some properties and different families of the GOWEPA operator. We also generalize the GOWEPA operator. The key advantage of the GOWEPA operator is that it is an aggregation operator with theoretic basis on aggregation. Moreover, we propose an orness measure of the GOWEPA operator and indicate some properties of this orness measure. Furthermore, we introduce the least exponential squares method (LESM) to determine the GOWEPA operator weights based on its orness measure. In the end, we develop an application of the new approach in a case of group decision making in investment selection.     

基于FOWGA算子的动态三角模糊数多属性决策法

针对各决策时段权重未知、属性值以三角模糊数形式给出的动态多属性决策问题,给出模糊有序加权几何平均(FOWGA)算子.算子可以对各时段属性信息进行横向和纵向的集结,从而得到方案的排序.最后通过实例分析说明方法的有效性和可行性.     

Group decision making with expertons and uncertain generalized probabilistic weighted aggregation operators

Expertons and uncertain aggregation operators are tools for dealing with imprecise information that can be assessed with interval numbers. This paper introduces the uncertain generalized probabilistic weighted averaging (UGPWA) operator. It is an aggregation operator that unifies the probability and the weighted average in the same formulation considering the degree of importance that each concept has in the aggregation. Moreover, it is able to assess uncertain environments that cannot be assessed with exact numbers but it is possible to use interval numbers. Thus, we can analyze imprecise information considering the minimum and the maximum result that may occur. Further extensions to this approach are presented including the quasi-arithmetic uncertain probabilistic weighted averaging operator and the uncertain generalized probabilistic weighted moving average. We analyze the applicability of this new approach in a group decision making problem by using the theory of expertons in strategic management.     

A new method for ranking fuzzy numbers and its application to group decision making

In this paper, a new method for comparing fuzzy numbers based on a fuzzy probabilistic preference relation is introduced. The ranking order of fuzzy numbers with the weighted confidence level is derived from the pairwise comparison matrix based on 0.5-transitivity of the fuzzy probabilistic preference relation. The main difference between the proposed method and existing ones is that the comparison result between two fuzzy numbers is expressed as a fuzzy set instead of a crisp one. As such, the ranking order of n fuzzy numbers provides more information on the uncertainty level of the comparison. Illustrated by comparative examples, the proposed method overcomes certain unreasonable (due to the violation of the inequality properties) and indiscriminative problems exhibited by some existing methods. More importantly, the proposed method is able to provide decision makers with the probability of making errors when a crisp ranking order is obtained. The proposed method is also able to provide a probability-based explanation for conflicts among the comparison results provided by some existing methods using a proper ranking order, which ensures that ties of alternatives can be broken.