Some methods for strategic decision‐making problems with immediate probabilities in Pythagorean fuzzy environment

In this article, a new decision‐making model with probabilistic information and using the concept of immediate probabilities has been developed to aggregate the information under the Pythagorean fuzzy set environment. In it, the existing probabilities have been modified by introducing the attitudinal character of the decision maker by using an ordered weighted average operator. Based on it, we have developed some new probabilistic aggregation operator with Pythagorean fuzzy information, namely probabilistic Pythagorean fuzzy weighted average operator, immediate probability Pythagorean fuzzy ordered weighted average operator, probabilistic Pythagorean fuzzy ordered weighted average, probabilistic Pythagorean fuzzy weighted geometric operator, immediate probability Pythagorean fuzzy ordered weighted geometric operator, probabilistic Pythagorean fuzzy ordered weighted geometric, etc. Furthermore, we extended these operators by taking interval‐valued Pythagorean fuzzy information and developed their corresponding aggregation operators. Few properties of these operators have also been investigated. Finally, an illustrative example about the selection of the optimal production strategy has been given to show the utility of the developed method.     

Generalized intuitionistic fuzzy geometric interaction operators and their application to decision making

Considering that there may exist some interactions between membership function and non-membership function of different intuitionistic fuzzy sets, we present some new operational laws from the probability point of view and give a geometric interpretation of the new operations. Based on which, a new class of generalized intuitionistic fuzzy aggregation operators are developed, including the generalized intuitionistic fuzzy weighted geometric interaction averaging (GIFWGIA) operator, the generalized intuitionistic fuzzy ordered weighted geometric interaction averaging (GIFOWGIA) operator and the generalized intuitionistic fuzzy hybrid geometric interaction averaging (GIFHGIA) operator. The properties of these new generalized aggregation operators are investigated. Moreover, approaches to multiple attributes decision making are given based on the generalized aggregation operators under intuitionistic fuzzy environment, and an example is illustrated to show the validity and feasibility of new approach. Finally, we give a systematic comparison between the work of this paper and that of other papers.     

Novel neutrality operation–based Pythagorean fuzzy geometric aggregation operators for multiple attribute group decision analysis

Pythagorean fuzzy sets (PFSs) accommodate more uncertainties than L_x the intuitionistic fuzzy sets and hence its applications are more extensive. Under the PFS, the objective of this paper is to develop some new operational laws and their corresponding weighted geometric aggregation operators. For it, we define some new neutral multiplication and power operational laws by including the feature of the probability sum and the interaction coefficient into the analysis to get a neutral or a fair treatment to the membership and nonmembership functions of PFSs. Associated with these operational laws, we define some novel Pythagorean fuzzy weighted, ordered weighted, and hybrid neutral geometric operators for Pythagorean fuzzy information, which can neutrally treat the membership and nonmembership degrees. The desirable relations and the characteristics of the proposed operators are studied in details. Furthermore, a multiple attribute group decision-making approach based on the proposed operators under the Pythagorean fuzzy environment is developed. Finally, an illustrative example is provided to show the practicality and the feasibility of the developed approach.     

Symmetric Pythagorean Fuzzy Weighted Geometric/Averaging Operators and Their Application in Multicriteria Decision‐Making Problems

Pythagorean fuzzy sets (PFSs), originally proposed by Yager, are a new tool to deal with vagueness with the square sum of the membership degree and the nonmembership degree equal to or less than 1, which have much stronger ability than Atanassov's intuitionistic fuzzy sets to model such uncertainty. In this paper, we modify the existing score function and accuracy function for Pythagorean fuzzy number to make it conform to PFSs. Associated with the given operational laws, we define some novel Pythagorean fuzzy weighted geometric/averaging operators for Pythagorean fuzzy information, which can neutrally treat the membership degree and the nonmembership degree, and investigate the relationships among these operators and those existing ones. At length, a practical example is provided to illustrate the developed operators and to make a comparative analysis.     

Simplified neutrosophic sets and their applications in multi-criteria group decision-making problems

As a variation of fuzzy sets and intuitionistic fuzzy sets, neutrosophic sets have been developed to represent uncertain, imprecise, incomplete and inconsistent information that exists in the real world. Simplified neutrosophic sets (SNSs) have been proposed for the main purpose of addressing issues with a set of specific numbers. However, there are certain problems regarding the existing operations of SNSs, as well as their aggregation operators and the comparison methods. Therefore, this paper defines the novel operations of simplified neutrosophic numbers (SNNs) and develops a comparison method based on the related research of intuitionistic fuzzy numbers. On the basis of these operations and the comparison method, some SNN aggregation operators are proposed. Additionally, an approach for multi-criteria group decision-making (MCGDM) problems is explored by applying these aggregation operators. Finally, an example to illustrate the applicability of the proposed method is provided and a comparison with some other methods is made.     

Interval-valued hesitant fuzzy Einstein prioritized aggregation operators and their applications to multi-attribute group decision making

Interval-valued hesitant fuzzy information aggregation plays an important role in interval-valued hesitant fuzzy set theory, which has received more and more attention in recent years. In this paper, we investigate interval-valued hesitant fuzzy multi-attribute group decision-making problems in which there exists a prioritization relationship among the attributes. Firstly, we introduce some Einstein operational laws on interval-valued hesitant fuzzy sets, and discuss some relations of these operations. Then, we develop two interval-valued hesitant fuzzy prioritized aggregation operators with the help of Einstein operations, such as the interval-valued hesitant fuzzy Einstein prioritized weighted average (IVHFEPWA) operator and the interval-valued hesitant fuzzy Einstein prioritized weighted geometric (IVHFEPWG) operator, whose desirable properties are investigated in detail. We further analyze the relationship between these proposed operators and the existing interval-valued hesitant fuzzy prioritized aggregation operators. Moreover, an approach to interval-valued hesitant fuzzy multi-attribute group decision making is given on the basis of the proposed operators. Finally, a numerical example is provided to demonstrate their effectiveness.     

Linguistic Pythagorean fuzzy sets and its applications in multiattribute decision‐making process

In this article, a new linguistic Pythagorean fuzzy set (LPFS) is presented by combining the concepts of a Pythagorean fuzzy set and linguistic fuzzy set. LPFS is a better way to deal with the uncertain and imprecise information in decision making, which is characterized by linguistic membership and nonmembership degrees. Some of the basic operational laws, score, and accuracy functions are defined to compare the two or more linguistic Pythagorean fuzzy numbers and their properties are investigated in detail. Based on the norm operations, some series of the linguistic Pythagorean weighted averaging and geometric aggregation operators, named as linguistic Pythagorean fuzzy weighted average and geometric, ordered weighted average and geometric with linguistic Pythagorean fuzzy information are proposed. Furthermore, a multiattribute decision‐making method is established based on these operators. Finally, an illustrative example is used to illustrate the applicability and validity of the proposed approach and compare the results with the existing methods to show the effectiveness of it.     

Some geometric aggregation operators based on intuitionistic fuzzy sets

The weighted geometric (WG) operator and the ordered weighted geometric (OWG) operator are two common aggregation operators in the field of information fusion. But these two aggregation operators are usually used in situations where the given arguments are expressed as crisp numbers or linguistic values. In this paper, we develop some new geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator, which extend the WG and OWG operators to accommodate the environment in which the given arguments are intuitionistic fuzzy sets which are characterized by a membership function and a non-membership function. Some numerical examples are given to illustrate the developed operators. Finally, we give an application of the IFHG operator to multiple attribute decision making based on intuitionistic fuzzy sets.     

Hesitant fuzzy Hamacher aggregation operators for multicriteria decision making

As a fuzzy set extension, the hesitant set is effectively used to model situations where it is allowable to determine several possible membership degrees of an element to a set due to the ambiguity between different values. We first introduce some new operational rules of hesitant fuzzy sets based on the Hamacher t-norm and t-conorm, in which a family of hesitant fuzzy Hamacher operators is proposed for aggregating hesitant fuzzy information. Some basic properties of these proposed operators are given, and the relationships between them are shown in detail. We further discuss the interrelations between the proposed aggregation operators and the existing hesitant fuzzy aggregation operators. Applying the proposed hesitant fuzzy operators, we develop a new technique for hesitant fuzzy multicriteria decision making problems. Finally, the effectiveness of the proposed technique is illustrated by mean of a practical example.     

Interval-valued intuitionistic fuzzy Frank aggregation operators and their applications to multiple attribute group decision making

Interval-valued intuitionistic fuzzy numbers (IVIFNs), which contain three ranges: the membership degree range, the non-membership degree range, and the hesitancy degree range, are very suitable to be used for depicting uncertain or fuzzy information. In this paper, we study the aggregation techniques of IVIFNs with the help of Frank operations. We first extend the Frank t-conorm and t-norm to interval-valued intuitionistic fuzzy environments and introduce several new operations of IVIFNs, such as Frank sum, Frank product, Frank scalar multiplication, and Frank exponentiation, based on which we develop several new interval-valued intuitionistic fuzzy aggregation operators, including the interval-valued intuitionistic fuzzy Frank weighted averaging operator and the interval-valued intuitionistic fuzzy Frank weighted geometric operator. We further establish various properties of these operators, give some special cases of them, and analyze the relationships between these operators. Moreover, we apply these operators to develop an approach for dealing with multiple attribute group decision making with interval-valued intuitionistic fuzzy information. Finally, a numerical example is provided to illustrate the practicality and effectiveness of the developed operators and approach.

     

Pythagorean fuzzy multicriteria group decision making through similarity measure based on point operators

In this paper, a series of similarity measures based on point operators for Pythagorean fuzzy sets are proposed. Using the proposed similarity measures, two new aggregation operators, viz., Pythagorean fuzzy‐dependent averaging operator and Pythagorean fuzzy‐dependent geometric operator, are developed. The advantage of using these operators is that the influence of unfair arguments of aggregated results could be eliminated, since the associated weights are taken from the aggregated Pythagorean fuzzy arguments. Also, the proposed operators have the capability to adjust the degree of aggregated arguments with the controlling parameters. To establish the application potentiality of those operators, a methodology for solving multicriteria group decision‐making problems having Pythagorean fuzzy arguments is developed. A numerical example is provided to demonstrate the proficiency of the proposed method. The achieved results are compared with the results of other existing technique.     

加权犹豫模糊集的群决策方法

对于犹豫模糊元中的不同隶属度值赋予不同的权重,由此构造出一种应用范围更广、更符合实际需要的犹豫模糊集合 ----- 加权犹豫模糊集合.针对加权犹豫模糊集中的加权犹豫模糊元,定义了加权犹豫模糊集合和加权犹豫模糊元的并、交、余、数乘和幂等运算及其运算法则,并讨论它们的运算性质;同时,给出加权犹豫模糊元的得分函数和离散函数,进而给出一种比较加权犹豫模糊元的排序法则.在此基础上,提出两类集成算子:加权犹豫模糊元的加权算术平均算子和加权犹豫模糊元的加权几何平均算子,并针对专家权重(已知和未知)的两种情形,将加权犹豫模糊集合应用于群决策,给出两种基于加权犹豫模糊集合的群决策方法.最后,通过一个应用实例表明所提出的群决策方法的有效性和实用性.     

Intuitionistic fuzzy geometric aggregation operators based on einstein operations

Intuitionistic fuzzy information aggregation plays an important part in Atanassov's intuitionistic fuzzy set theory, which has emerged to be a new research direction receiving more and more attention in recent years. In this paper, we first introduce some operations on intuitionistic fuzzy sets, such as Einstein sum, Einstein product, Einstein exponentiation, etc., and further develop some new geometric aggregation operators, such as the intuitionistic fuzzy Einstein weighted geometric operator and the intuitionistic fuzzy Einstein ordered weighted geometric operator, which extend the weighted geometric (WG) operator and the ordered weighted geometric (OWG) operator to accommodate the environment in which the given arguments are intuitionistic fuzzy values. We also establish some desirable properties of these operators, such as commutativity, idempotency and monotonicity, and give some numerical examples to illustrate the developed aggregation operators. In addition, we compare the proposed operators with the existing intuitionistic fuzzy geometric operators and get the corresponding relations. Finally, we apply the intuitionistic fuzzy Einstein weighted geometric operator to deal with multiple attribute decision making under intuitionistic fuzzy environments. © 2011 Wiley Periodicals, Inc.     

Intuitionistic Fuzzy Hybrid Weighted Aggregation Operators

Intuitionistic fuzzy sets (IFSs) have attracted more and more scholars’ attention due to their powerfulness in expressing vagueness and uncertainty. In the course of decision making with IFSs, aggregation operators play a very important role since they can be used to synthesize multidimensional evaluation values represented as intuitionistic fuzzy values into collective values. This paper proposes a family of intuitionistic fuzzy hybrid weighted aggregation operators, such as the intuitionistic fuzzy hybrid weighted averaging operator, the intuitionistic fuzzy hybrid weighted geometric operator, the generalized intuitionistic fuzzy hybrid weighted averaging operator, and the generalized intuitionistic fuzzy hybrid weighted geometric operator. All these newly developed operators not only can weight both the arguments and their ordered positions simultaneously but also have some desirable properties, such as idempotency, boundedness, and monotonicity. To show the applications of our proposed intuitionistic fuzzy hybrid weighted aggregation operators, a simple schema for decision making with intuitionistic fuzzy information is developed. An example concerning the human resource management is given to illustrate the validity and applicability of the proposed method and also the hybrid weighted aggregation operators.     

Pythagorean hesitant fuzzy Hamacher aggregation operators and their application to multiple attribute decision making

Hamacher product is a t‐norm and Hamacher sum is a t‐conorm. They are good alternatives to algebraic product and algebraic sum, respectively. Nevertheless, it seems that most of the existing hesitant fuzzy aggregation operators are based on the algebraic operations. In this paper, we utilize Hamacher operations to develop some Pythagorean hesitant fuzzy aggregation operators: Pythagorean hesitant fuzzy Hamacher weighted average (PHFHWA) operator, Pythagorean hesitant fuzzy Hamacher weighted geometric (PHFHWG) operator, Pythagorean hesitant fuzzy Hamacher ordered weighted average (PHFHOWA) operator, Pythagorean hesitant fuzzy Hamacher ordered weighted geometric (PHFHOWG) operator, Pythagorean hesitant fuzzy Hamacher induced ordered weighted average (PHFHIOWA) operator, Pythagorean hesitant fuzzy Hamacher induced ordered weighted geometric (PHFHIOWG) operator, Pythagorean hesitant fuzzy Hamacher induced correlated aggregation operators, Pythagorean hesitant fuzzy Hamacher prioritized aggregation operators, and Pythagorean hesitant fuzzy Hamacher power aggregation operators. The special cases of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the Pythagorean hesitant fuzzy multiple attribute decision making problems. Finally, a practical example for green supplier selections in green supply chain management is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

区间直觉模糊信息的集成方法及其在决策中的应用

对区间直觉模糊信息的集成方法进行了研究.定义了区间直觉模糊数的一些运算法则,并基于这些运算法则,给出区间直觉模糊数的加权算术和加权几何集成算子.定义了区间直觉模糊数的得分函数和精确函数,进而给出了区间直觉模糊数的一种简单的排序方法.最后提供了一种基于区间直觉模糊信息的决策途径,并进行了实例分析.     

Multiple attribute decision‐making method for dealing with heterogeneous relationship among attributes and unknown attribute weight information under q‐rung orthopair fuzzy environment

A Q‐rung orthopair fuzzy set (q‐ROFS) originally proposed by Yager (2017) is a new generalization of orthopair fuzzy sets, which has a larger representation space of acceptable membership grades and gives decision makers more flexibility to express their real preferences. In this paper, for multiple attribute decision‐making problems with q‐rung orthopair fuzzy information, we propose a new method for dealing with heterogeneous relationship among attributes and unknown attribute weight information. First, we present two novel q‐rung orthopair fuzzy extended Bonferroni mean (q‐ROFEBM) operator and its weighted form (q‐ROFEWEBM). A comparative example is provided to illustrate the advantages of the new operators, that is, they can effectively model the heterogeneous relationship among attributes. We prove that some existing known intuitionistic fuzzy aggregation operators and Pythagorean fuzzy aggregation operators are special cases of the proposed q‐ROFEBM and q‐ROFEWEBM operators. Meanwhile, several desirable properties are also investigated. Then, a new knowledge‐based entropy measure for q‐ROFSs is also proposed to obtain the attribute weights. Based on the proposed q‐ROFWEBM and the new entropy measure, a new method is developed to solve multiple attribute decision making problems with q‐ROFSs. Finally, an illustrative example is given to demonstrate the application process of the proposed method, and a comparison analysis with other existing representative methods is also conducted to show its validity and superiority.     

改进的区间犹豫算子应用于物流企业选择决策

针对现有区间犹豫模糊Hamacher算子存在的缺陷，构建了一种基于改进的区间犹豫模糊Hamacher加权算子的群决策方法。在分析现有区间犹豫模糊Hamacher算子不能满足幂等性的基础上，定义新的区间犹豫模糊Hamacher四则运算；提出两种改进的区间犹豫模糊Hamacher加权算子，包括改进的区间犹豫模糊Hamacher有序加权平均（I-IVHFHOWA）算子和改进的区间犹豫模糊Hamacher有序加权几何（I-IVHFHOWG）算子，并详细探究它们的常用算子形式以及算子之间的内在联系；建立基于I-IVHFHOWA算子和I-IVHFHOWG算子的物流企业选择决策模型，并通过实例说明模型的有效性。     

Pythagorean fuzzy power aggregation operators in multiple attribute decision making

In this paper, we utilize power aggregation operators to develop some Pythagorean fuzzy power aggregation operators: Pythagorean fuzzy power average operator, Pythagorean fuzzy power geometric operator, Pythagorean fuzzy power weighted average operator, Pythagorean fuzzy power weighted geometric operator, Pythagorean fuzzy power ordered weighted average operator, Pythagorean fuzzy power ordered weighted geometric operator, Pythagorean fuzzy power hybrid average operator, and Pythagorean fuzzy power hybrid geometric operator. The prominent characteristic of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the Pythagorean fuzzy multiple attribute decision‐making problems. Finally, a practical example is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

Generalized intuitionistic fuzzy Bonferroni means

Intuitionistic fuzzy set is a widely used tool to express the membership, nonmembership, and hesitancy information of an element to a set. To aggregate the intuitionistic fuzzy information, a lot of aggregation techniques have been developed, especially, the ones which reflect the correlations of the aggregated arguments are the hot research topics, among which Bonferroni mean (BM) is an important aggregation technique. However, the classical BM ignores some aggregation information and the weight vector of the aggregated arguments. In this paper, we introduce the generalized weighted BM and the generalized intuitionistic fuzzy weighted BM, both of which focus on the group opinion. Paying more attention to the individual opinions, we further define the generalized weighted Bonferroni geometric mean and the generalized intuitionistic fuzzy weighted Bonferroni geometric mean. Various families of the existing operators can be obtained when the parameters of the developed aggregation techniques are assigned different values. Finally, we propose an approach to multicriteria decision making on the basis of the proposed aggregation techniques and an example is also given to illustrate our results. © 2011 Wiley Periodicals, Inc.