Hesitant interval-valued Pythagorean fuzzy VIKOR method

In this article, we investigate multiple attribute decision-making problems with hesitant interval-valued Pythagorean fuzzy information. First, the concepts of hesitant interval-valued Pythagorean fuzzy set are defined, and the operation laws, the score function, and accuracy function have been developed. Then several distance measures for hesitant interval-valued Pythagorean fuzzy values have been presented including the Hamming distance, Euclidean distance, and generalized distance, and so on. Based on the operational laws, a series of aggregation operators have been developed including the hesitant interval-valued Pythagorean fuzzy weighted averaging (HIVPFWA) operator, the hesitant interval-valued Pythagorean fuzzy geometric weighted averaging (HIVPFGWA) operator, the hesitant interval-valued Pythagorean fuzzy ordered weighed averaging (HIVPFOWA) operator, and hesitant interval-valued Pythagorean fuzzy ordered weighed geometric averaging (HIVPFOWGA) operator. By using the generalized mean operator, we also develop the generalized hesitant interval-valued Pythagorean fuzzy weighed averaging (GHIVPFWA) operator, the generalized hesitant interval-valued Pythagorean fuzzy weighed geometric averaging (GHIVPFWGA) operator, the generalized hesitant interval-valued Pythagorean fuzzy ordered weighted averaging (GHIVPFOWA) operator, and generalized hesitant interval-valued Pythagorean fuzzy ordered weighted geometric averaging (GHIVPFOWGA) operator operator. We further develop several hybrid aggregation operators including the hesitant interval-valued Pythagorean fuzzy hybrid averaging (HIVPFHA) operator and the generalized hesitant interval-valued Pythagorean fuzzy hybrid averaging (GHIVPFHA) operator. Based on the distance measures and the aggregation operators, we propose a hesitant interval-valued Pythagorean fuzzy VIKOR method to solve multiple attribute decision problems with multiple periods. Finally, an illustrative example for evaluating the metro project risk is given to demonstrate the feasibility and effectiveness of the proposed method.     

Quantile-induced uncertain heavy ordered weighted averaging operator and the application in incentive evaluation problems

To uncertain evaluation problems, we integrate incentive management into the aggregation process and propose an aggregation operator called quantile-induced uncertain heavy ordered weighted averaging (QI-UHOWA) operator, which is an extension of the quantile-induced heavy ordered weighted averaging (QI-HOWA) operator. We provide an approach for determining the quantile order-inducing variables by using the technique for order preference by similarity to ideal solution method and the Hamming distance. In this case, the quantile values are measurements of relative developments of alternatives. Furthermore, we analyze the main properties of the operator including commutativity, boundedness, and monotonicity with uniform development space. The QI-UHOWA weighting vector is calculated using the maximum entropy measure with a given level of incentive attitude. We further expand the weighting method to the case of hierarchical stimulation. Moreover, the QI-UHOWA operator is generalized using the quasi-arithmetic mean. Finally, a numerical example regarding the selection of the optimal candidate(s) is given. The aggregation results are compared with those of the UOWA and QI-UOWA operator to illustrate the validity of the QI-UHOWA operator.     

Fuzzy linguistic induced OWA Minkowski distance operator and its application in group decision making

The Minkowski distance is a distance measure that generalizes a wide range of other distances such as the Euclidean and the Hamming distance. In this paper, we develop a new decision making model using induced ordered weighted averaging operators and the Minkowski distance of the fuzzy linguistic variables. Then, the authors introduce a new aggregation operator called the fuzzy linguistic induced ordered weighted averaging Minkowski distance (FLIOWAMD) operator by defining a fuzzy linguistic variable distance. It is an induced generalized aggregation operator that utilizes induced OWA operator, Minkowski distance measures and uncertain information represented as fuzzy linguistic variables. Some of its main properties and particular cases are studied. And a further generalization that uses quasi-arithmetic means also is presented. A method based on the FLIOWAMD operator for decision making is presented. At last, we end the paper with a numerical example of the new method.     

Generalized aggregation operators for intuitionistic fuzzy sets

The generalized ordered weighted averaging (GOWA) operators are a new class of operators, which were introduced by Yager (Fuzzy Optim Decision Making 2004;3:93–107). However, it seems that there is no investigation on these aggregation operators to deal with intuitionistic fuzzy or interval‐valued intuitionistic fuzzy information. In this paper, we first develop some new generalized aggregation operators, such as generalized intuitionistic fuzzy weighted averaging operator, generalized intuitionistic fuzzy ordered weighted averaging operator, generalized intuitionistic fuzzy hybrid averaging operator, generalized interval‐valued intuitionistic fuzzy weighted averaging operator, generalized interval‐valued intuitionistic fuzzy ordered weighted averaging operator, generalized interval‐valued intuitionistic fuzzy hybrid average operator, which extend the GOWA operators to accommodate the environment in which the given arguments are both intuitionistic fuzzy sets that are characterized by a membership function and a nonmembership function, and interval‐valued intuitionistic fuzzy sets, whose fundamental characteristic is that the values of its membership function and nonmembership function are intervals rather than exact numbers, and study their properties. Then, we apply them to multiple attribute decision making with intuitionistic fuzzy or interval‐valued intuitionistic fuzzy information. © 2009 Wiley Periodicals, Inc.     

Uncertain Induced Heavy Aggregation Distance Operator and its Application to Decision Making

We introduce a new decision-making approach for dealing with uncertain information and apply it to decision making. We have developed the uncertain induced heavy ordered weighted averaging distance (UIHOWAD) operator. It is a new aggregation operator that generalizes a wide range of uncertain aggregation operators such as the uncertain minimum distance, the uncertain weighted Hamming distance (UWHD), the uncertain OWA distance (UOWAD) operator, and the uncertain heavy OWA distance (UHOWAD) operator. We studied some of its main properties and different particular cases. We also present its applicability in a decision-making problem concerning the selection of investments.     

Generalized power aggregation operators and their applications in group decision making

In this paper, we investigate a generalized power average (GPA) operator and its weighted form, which are on the basis of the power average (PA) operator and the generalized mean, and develop a generalized power ordered weighted average (GPOWA) operator based on the power ordered weighted average (POWA) operator. Then, we extend these operators to uncertain environments and present an uncertain generalized power average (UGPA) operator and its weighted form, and an uncertain generalized power ordered weighted average (UGPOWA) operator to aggregate the input arguments taking the form of interval of numerical values. We also extend the GPA operator and the GPOWA operator to intuitionistic fuzzy environment, and obtain the generalized intuitionistic fuzzy power averaging (GIFPA) operator and the generalized intuitionistic fuzzy power ordered weighted averaging (GIFPOWA) operator. Moreover, some properties of these operators are studied. We also present new approaches on the basis of the proposed operators in an example of strategic decision making.     

An overview of operators for aggregating information

In this work, we first make a survey of the existing main aggregation operators and then propose some new aggregation operators such as the induced ordered weighted geometric averaging (IOWGA) operator, generalized induced ordered weighted averaging (GIOWA) operator, hybrid weighted averaging (HWA) operator, etc., and study their desirable properties. Finally, we briefly classify all of these aggregation operators. © 2003 Wiley Periodicals, Inc.     

Fuzzy Entropic Ordered Weighted Averaging Operator and Its Application in Group Decision Making

In the paper, we develop a new method for multiple attribute group decision making) for fuzzy numbers. The fuzzy entropic weighted averaging (FEOWA) operator is an extension of the entropic ordered weighted averaging operator, which unifies the fuzzy entropy and the ordered weighted averaging operator in the same formulation. Then, some of its main properties by utilizing some operational laws of fuzzy numbers are studied. We also present the generalized entropic ordered weighted averaging operator and the fuzzy generalized entropic ordered weighted averaging operator. Moreover, a method based on the FEOWA operator for decision making is presented. Finally, a numerical example illustrates the applicability and effectiveness of the proposed method.     

A Novel Method for Multiattribute Decision Making with Interval‐Valued Pythagorean Fuzzy Linguistic Information

Pythagorean fuzzy set (PFS), proposed by Yager (2013), is a generalization of the notion of Atanassov's intuitionistic fuzzy set, which has received more and more attention. In this paper, first, we define the weighted Minkowski distance with interval‐valued PFSs. Second, inspired by the idea of the Pythagorean fuzzy linguistic variables, we define a new fuzzy variable called interval‐valued Pythagorean fuzzy linguistic variable set (IVPFLVS), and the operational laws, score function, accuracy function, comparison rules, and distance measures of the IVPFLVS are defined. Third, some aggregation operators are presented for aggregating the interval‐valued Pythagorean fuzzy linguistic information such as the interval‐valued Pythagorean fuzzy linguistic weighted averaging (IVPFLWA), interval‐valued Pythagorean fuzzy linguistic ordered weighted averaging (IVPFLOWA) , interval‐valued Pythagorean fuzzy linguistic hybrid averaging, and generalized interval‐valued Pythagorean fuzzy linguistic ordered weighted average operators. Fourth, some desirable properties of the IVPFLWA and IVPFLOWA operators, such as monotonicity, commutativity, and idempotency, are discussed. Finally, based on the IVPFLWA or interval‐valued Pythagorean fuzzy linguistic geometric weighted operator, a practical example is provided to illustrate the application of the proposed approach and demonstrate its practicality and effectiveness.     

Induced and uncertain heavy OWA operators

In this paper, we analyse in detail the ordered weighted averaging (OWA) operator and some of the extensions developed about it. We specially focus on the heavy aggregation operators. We suggest some new extensions about the OWA operator such as the induced heavy OWA (IHOWA) operator, the uncertain heavy OWA (UHOWA) operator and the uncertain induced heavy OWA (UIHOWA) operator. For these three new extensions, we consider some of their main properties and a wide range of special cases found in the weighting vector such as the heavy weighted average (HWA) and the uncertain heavy weighted average (UHWA). We further generalize these models by using generalized and quasi-arithmetic means obtaining the generalized heavy weighted average (GHWA), the induced generalized HOWA (IGHOWA) and the uncertain IGHOWA (UIGHOWA) operator. Finally, we develop an application of the new approach in a decision-making problem.     

广义犹豫模糊信息集成及其多属性群决策

在犹豫模糊环境下,主要研究了基于阿基米德范数的广义信息集成算法,并提出了一种新的多属性群决策方法。基于阿基米德T-范数和S-范数,定义了广义犹豫模糊运算法则;运用新定义的广义犹豫模糊运算法则,提出了广义犹豫模糊有序加权平均(G-HFOWA)算子,研究了其优良性质;探讨了在某些特殊情况下,广义犹豫模糊有序加权平均算子将转化为一些常见的犹豫模糊信息集成算子,包括犹豫模糊有序加权平均算子、犹豫模糊Einstein有序加权平均算子、犹豫模糊Hamacher有序加权平均算子以及犹豫模糊Frank有序加权平均算子;基于广义信息集成算子,构建了一种新的犹豫模糊多属性群决策方法,并将其应用于区域经济协调发展研究过程中,以验证提出的决策方法是可行的与有效的。     

A method for fuzzy group decision making based on induced aggregation operators and Euclidean distance

In this paper, we present the fuzzy‐induced Euclidean ordered weighted averaging distance (FIEOWAD) operator. It is an extension of the ordered weighted averaging (OWA) operator that uses the main characteristics of the induced OWA (IOWA), the Euclidean distance and uncertain information represented by fuzzy numbers. The main advantage of this operator is that it is able to consider complex attitudinal characters of the decision maker by using order‐inducing variables in the aggregation of the Euclidean distance. Moreover, it is able to deal with uncertain environments where the information is very imprecise and can be assessed with fuzzy numbers. We study some of its main properties and particular cases such as the fuzzy maximum distance, fuzzy minimum distance, fuzzy‐normalized Euclidean distance (FNED), fuzzy‐weighted Euclidean distance (FWED), and fuzzy Euclidean ordered weighted averaging distance (FEOWAD) operator. Finally, we present an application of the operator to a group decision‐making problem concerning the selection of strategies.     

Generalized prioritized aggregation operators

This paper deals with multicriteria decision‐making problems in which the criteria are partitioned into q categories, and a prioritization relationship exists over categories. We aggregate the criteria in the same priority category by a weighted OWA (ordered weighted averaging) operator and introduce two averaging operators, a generalized prioritized averaging operator and a generalized prioritized OWA operator. In the case with one criterion in each priority category, the two operators reduce to the prioritized averaging operator and the prioritized OWA operator as proposed by Yager. © 2012 Wiley Periodicals, Inc.     

Heavy Moving Averages and Their Application in Econometric Forecasting

This paper presents the heavy ordered weighted moving average (HOWMA) operator. It is an aggregation operator that uses the main characteristics of two well-known techniques: the heavy ordered weighted averaging (OWA) and the moving averages. Therefore, this operator provides a parameterized family of aggregation operators from the minimum to the total operator and includes the OWA operator as a special case. It uses a heavy weighting vector in the moving average formulation and it represents the information available and the knowledge of the decision maker about the future scenarios of the phenomenon, according to his attitudinal character. Some of the main properties of this operator are studied, including a wide range of families of HOWMA operators such as the heavy moving average and heavy weighted moving average operators. The HOWMA operator is also extended using generalized and quasi-arithmetic means. An example concerning the foreign exchange rate between US dollars and Mexican pesos is also presented.     

The induced generalized OWA operator

We present the induced generalized ordered weighted averaging (IGOWA) operator. It is a new aggregation operator that generalizes the OWA operator, including the main characteristics of both the generalized OWA and the induced OWA operator. This operator uses generalized means and order-inducing variables in the reordering process. It provides a very general formulation that includes as special cases a wide range of aggregation operators, including all the particular cases of the IOWA and the GOWA operator, the induced ordered weighted geometric (IOWG) operator and the induced ordered weighted quadratic averaging (IOWQA) operator. We further generalize the IGOWA operator via quasi-arithmetic means. The result is the Quasi-IOWA operator. Finally, we present a numerical example to illustrate the new approach in a financial decision-making problem.     

Construction of Choquet integrals through unimodal weighting vectors

Semiuninorm‐based ordered weighted averaging (SUOWA) operators are a specific case of Choquet integrals that allow us to generalize simultaneously weighted means and ordered weighting averaging (OWA) operators. Although SUOWA operators possess some very interesting properties, their main weakness is that, sometimes, the game used in their construction is not monotonic and it is necessary to calculate its monotonic cover. In this paper, we introduce a new family of weighting vectors, called unimodal weighting vectors, which embrace some of the most outstanding weighting vectors used in the framework of OWA operators, and we show that when using these weighting vectors and a specific semiuninorm we directly get normalized capacities. Moreover, we also show that these operators satisfy some properties which are very useful in practice.     

The ordered weighted geometric averaging operators

The ordered weighted averaging (OWA) operator was introduced by Yager. 1 The fundamental aspect of the OWA operator is a reordering step in which the input arguments are rearranged in descending order. In this article, we propose two new classes of aggregation operators called ordered weighted geometric averaging (OWGA) operators and study some desired properties of these operators. Some methods for obtaining the associated weighting parameters are discussed, and the relationship between the OWA and DOWGA operators is also investigated. © 2002 Wiley Periodicals, Inc.     

Decision-making with distance measures and induced aggregation operators

In this paper, we present a new decision-making approach that uses distance measures and induced aggregation operators. We introduce the induced ordered weighted averaging distance (IOWAD) operator. IOWAD is a new aggregation operator that extends the OWA operator by using distance measures and a reordering of arguments that depends on order-inducing variables. The main advantage of IOWAD is that it provides a parameterized family of distance aggregation operators between the maximum and the minimum distance based on a complex reordering process that reflects the complex attitudinal character of the decision-maker. We studied some of IOWAD’s main properties and different particular cases and further generalized IOWAD by using Choquet integrals. We developed an application in a multi-person decision-making problem regarding the selection of investments. We found that the main advantage of this approach is that it is able to provide a more complete picture of the decision-making process, enabling the decision-maker to select the alternative that it is more in accordance with his interests.     

The uncertain induced quasi‐arithmetic OWA operator

We present the uncertain induced quasi‐arithmetic OWA (Quasi‐UIOWA) operator. It is an extension of the OWA operator that uses the main characteristics of the induced OWA (IOWA), the quasi‐arithmetic OWA (Quasi‐OWA) and the uncertain OWA (UOWA) operator. Thus, this generalization uses quasi‐arithmetic means, order inducing variables in the reordering process and uncertain information represented by interval numbers. A key feature of the Quasi‐UIOWA operator is that it generalizes a wide range of aggregation operators such as the uncertain quasi‐arithmetic mean, the uncertain weighted quasi‐arithmetic mean, the UOWA, the uncertain weighted generalized mean, the uncertain induced generalized OWA (UIGOWA), the Quasi‐UOWA, the uncertain IOWA, the uncertain induced ordered weighted geometric (UIOWG), and the uncertain induced ordered weighted quadratic averaging (UIOWQA) operator. We study some of the main properties of this approach including how to obtain a wide range of particular cases. We further generalize the Quasi‐UIOWA operator by using discrete Choquet integrals. We end the article with an application of the new approach in a decision making problem about investment selection. © 2010 Wiley Periodicals, Inc.     

Cubic fuzzy Einstein aggregation operators and its application to decision-making

In this paper, we define some Einstein operations on cubic fuzzy set (CFS) and develop three arithmetic averaging operators, which are cubic fuzzy Einstein weighted averaging (CFEWA) operator, cubic fuzzy Einstein ordered weighted averaging (CFEOWA) operator and cubic fuzzy Einstein hybrid weighted averaging (CFEHWA) operator, for aggregating cubic fuzzy data. The CFEHWA operator generalises both the CFEWA and CFEOWA operators. Furthermore, we develop various properties of these operators and derive the relationship between the proposed operators and the exiting aggregation operators. We apply CFEHWA operator to multiple attribute decision-making with cubic fuzzy data. Finally, a numerical example is constructed to demonstrate the established approach.