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.     

INTUITIONISTIC UNCERTAIN LINGUISTIC WEIGHTED BONFERRONI OWA OPERATOR AND ITS APPLICATION TO MULTIPLE ATTRIBUTE DECISION MAKING

With respect to multiple attribute decision making (MADM) problems, in which attribute values take the form of intuitionistic uncertain linguistic information, a new decision-making method based on the intuitionistic uncertain linguistic weighted Bonferroni OWA operator is developed. First, the score function, accuracy function, and comparative method of the intuitionistic uncertain linguistic numbers are introduced. Then, an intuitionistic uncertain linguistic Bonferroni OWA (IULBOWA) operator and an intuitionistic uncertain linguistic weighted Bonferroni OWA (IULWBOWA) operator are developed. Furthermore, some properties of the IULBOWA and IULWBOWA operators, such as commutativity, idempotency, monotonicity, and boundedness, are discussed. At the same time, some special cases of these operators are analyzed. Based on the IULWBOWA operator, the multiple attribute decision-making method with intuitionistic uncertain linguistic information is proposed. Finally, an illustrative example is given to illustrate the decision-making steps and to demonstrate its practicality and effectiveness.     

Some generalized aggregating operators with linguistic information and their application to multiple attribute group decision making

With respect to multiple attribute group decision making problems with linguistic information, some new decision analysis methods are proposed. Firstly, we develop three new aggregation operators: generalized 2-tuple weighted average (G-2TWA) operator, generalized 2-tuple ordered weighted average (G-2TOWA) operator and induced generalized 2-tuple ordered weighted average (IG-2TOWA) operator. Then, a method based on the IG-2TOWA and G-2TWA operators for multiple attribute group decision making is presented. In this approach, alternative appraisal values are calculated by the aggregation of 2-tuple linguistic information. Thus, the ranking of alternative or selection of the most desirable alternative(s) is obtained by the comparison of 2-tuple linguistic information. Finally, a numerical example is used to illustrate the applicability and effectiveness of the proposed method.     

Induced generalized intuitionistic fuzzy OWA operator for multi-attribute group decision making

With respect to multi-attribute group decision making (MAGDM) problems in which both the attribute weights and the decision makers (DMs) weights take the form of real numbers, attribute values provided by the DMs take the form of intuitionistic fuzzy numbers, a new group decision making method is developed. Some operational laws, score function and accuracy function of intuitionistic fuzzy numbers are introduced at first. Then a new aggregation operator called induced generalized intuitionistic fuzzy ordered weighted averaging (IG-IFOWA) operator is proposed, which extend the induced generalized ordered weighted averaging (IGOWA) operator introduced by Merigo and Gil-Lafuente [Merigo, J. M., & Gil-Lafuente, A. M. (2009). The induced generalized OWA operator. Information Sciences, 179, 729-741] to accommodate the environment in which the given arguments are intuitionistic fuzzy sets that are characterized by a membership function and a non-membership function. Some desirable properties of the IG-IFOWA operator are studied, such as commutativity, idempotency, monotonicity and boundary. And then, an approach based on the IG-IFOWA and IFWA (intuitionistic fuzzy weighted averaging) operators is developed to solve MAGDM problems with intuitionistic fuzzy information. Finally, a numerical example is used to illustrate the developed approach.     

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.     

The induced generalized aggregation operators for intuitionistic fuzzy sets and their application in group decision making

In this paper, we present the induced generalized intuitionistic fuzzy ordered weighted averaging (I-GIFOWA) operator. It is a new aggregation operator that generalized the IFOWA operator, including all the characteristics of both the generalized IFOWA and the induced IFOWA operators. It provides a very general formulation that includes as special cases a wide range of aggregation operators for intuitionistic fuzzy information, including all the particular cases of the I-IFOWA operator, GIFOWA operator and the induced intuitionistic fuzzy ordered geometric (I-IFOWG) operator. We also present the induced generalized interval-valued intuitionistic fuzzy ordered weighted averaging (I-GIIFOWA) operator to accommodate the environment in which the given arguments are interval-valued intuitionistic fuzzy sets. Further, we develop procedures to apply them to solve group multiple attribute decision making problems with intuitionistic fuzzy or interval-valued intuitionistic fuzzy information. Finally, we present their application to show the effectiveness of the developed methods.     

Some power Maclaurin symmetric mean aggregation operators based on Pythagorean fuzzy linguistic numbers and their application to group decision making

The power average (PA) operator and Maclaurin symmetric mean (MSM) operator are two important tools to handle the multiple attribute group decision‐making (MAGDM) problems, and the combination of two operators can eliminate the influence of unreasonable information from biased decision makers (DMs) and can capture the interrelationship among any number of arguments. The Pythagorean fuzzy linguistic set (PFLS) is parallel to the intuitionistic linguistic set (ILS), which is more powerful to convey the uncertainty and ambiguity of the DMs than ILS. In this paper, we propose some power MSM aggregation operators for Pythagorean fuzzy linguistic information, such as Pythagorean fuzzy linguistic power MSM operator and Pythagorean fuzzy linguistic power weighted MSM (PFLPWMSM) operator. At the same time, we further discuss the properties and special cases of these operators. Then, we propose a new method to solve the MAGDM problems with Pythagorean fuzzy linguistic information based on the PFLPWMSM operator. Finally, some illustrative examples are utilized to show the effectiveness of the proposed method.     

基于动态P-模糊语言算子的VIKOR决策方法

针对直觉模糊信息解决动态多属性决策问题时存在的不足,将Pythagorean模糊语言信息引入到动态多属性决策问题,提出一种基于Pythagorean模糊语言信息集成算子的多准则妥协排序（VIKOR）决策方法。引入Pythagorean模糊语言得分函数、精确函数、距离计算公式等概念,提出动态 Pythagorean模糊语言加权平均（DPFLWA）算子,并研究DPFLWA算子的基本性质。最后,基于DPFLWA算子和VIKOR方法,构建一种动态 Pythagorean模糊语言多属性决策方法。通过第三方逆向物流服务商的选择实例,表明该方法的可行性和有效性。     

基于广义直觉语言算子的多属性群决策方法

针对属性权重未知,属性值为直觉语言数的多属性决策问题,提出了一种基于直觉语言熵和广义直觉语言算子的群决策方法.定义了直觉语言熵,并利用直觉语言熵确定属性权重,提出了三种直觉语言算子:广义直觉语言加权几何平均(GILWGA)算子、广义直觉语言有序加权几何(GILOWG)算子及广义直觉语言混合几何(GILHG)算子.利用GILWGA和GILHG算子集结信息,采用基于直觉语言数的得分函数及精确函数进行方案排序与择优,最后通过一个算例说明了该方法的有效性和合理性.     

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.     

A novel induced aggregation method for intuitionistic fuzzy set and its application in multiple attribute group decision making

In this paper, by unifying the induced ordered weighted averaging (IOWA) and the weighted average, a novel induced aggregation method for intuitionistic fuzzy set is investigated. More specifically, a new intuitionistic fuzzy (IF) induced aggregation operator called weighted intuitionistic fuzzy IOWA weighted average (WIFIOWAWA) operator is introduced. A significant advantage of the WIFIOWAWA operator is that it can eliminate the drawback of the existing operators by the dual roles of its order‐inducing variables. In addition, some of its desired properties and families are explored. Furthermore, using the proposed operator, a procedure is developed to solve multiple attribute group decision making problems in the case of IF situation. Finally, an illustrative example is provided to demonstrate the effectiveness and practicality of the developed method.     

区间直觉二元语言集成算子及其群决策方法

基于下标以零为中心对称的语言评估标度,将区间不确定二元语言集与区间直觉模糊集结合,提出区间直觉二元语言集及变量的概念;讨论区间直觉二元语言变量的运算及可能度;提出区间直觉二元语言加权算术平均算子、区间直觉二元语言有序加权平均算子,并在此基础上,通过可能度矩阵对区间直觉二元语言变量进行排序提出区间直觉二元语言混合加权平均算子;最后基于这些算子构建了一种新的直觉模糊多属性群决策方法,并将其运用于供应商选择过程中。     

零模的直觉模糊不确定语言集成算子研究

基于零模与共轭零模算子,探讨了直觉模糊不确定语言变量运算法则,得到了基于零模与共轭零模的直觉模糊不确定语言加权几何算子,并给出了一种使用直觉不确定语言变量的集成算子的多属性群决策方法,最后通过Matlab软件分析了直觉模糊不确定语言加权几何算子的K值与语言术语下标间关系。为多属性群决策提供了有价值的参考,有效地解决了一类具有直觉模糊不确定语言评估信息的多属性群决策问题。     

基于IFN 输入的复杂系统关联MCGDM方法

考虑现有直觉模糊熵公理化定义存在的不足,提出改进直觉模糊熵的公理化定义及其计算公式;同时,定义广义幂均算子,验证其相关性质,给出确定幂方参数的方法,并将其推广至广义直觉模糊幂均算子;在以直觉模糊数(IFN)为信息输入的复杂系统框架内,针对决策者及准则之间均存在交互关联关系且权重信息完全未知的多准则群决策(MCGDM)问题,提出基于直觉模糊熵与广义直觉模糊幂均算子的关联MCGDM方法。案例分析表明,所提出的方法是可行且有效的。     

基于折衷型变权向量的直觉语言决策方法

研究一种基于折衷型变权向量的直觉语言决策方法. 首先, 定义折衷型变权向量, 提出与之对应的状态变权向量; 其次, 研究利用马氏效用函数诱导出折衷型变权向量; 再次, 定义直觉语言变量运算法则和大小比较方法, 提出直觉语言信息变权加权平均算子和直觉语言信息变权加权几何平均算子, 进而提出一种初始属性权重确定且属性值以直觉语言形式给出的多属性决策方法; 最后, 通过实例表明了所提出方法的有效性和合理性.

     

Some induced geometric aggregation operators with intuitionistic fuzzy information 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 real numbers, attribute values take the form of intuitionistic fuzzy numbers or interval-valued intuitionistic fuzzy numbers, some new group decision making analysis methods are developed. Firstly, some operational laws, score function and accuracy function of intuitionistic fuzzy numbers or interval-valued intuitionistic fuzzy numbers are introduced. Then two new aggregation operators: induced intuitionistic fuzzy ordered weighted geometric (I-IFOWG) operator and induced interval-valued intuitionistic fuzzy ordered weighted geometric (I-IIFOWG) operator are proposed, and some desirable properties of the I-IFOWG and I-IIFOWG operators are studied, such as commutativity, idempotency and monotonicity. An I-IFOWG and IFWG (intuitionistic fuzzy weighted geometric) operators-based approach is developed to solve the MAGDM problems in which both the attribute weights and the expert weights take the form of real numbers, attribute values take the form of intuitionistic fuzzy numbers. Further, we extend the developed models and procedures based on I-IIFOWG and IIFWG (interval-valued intuitionistic fuzzy weighted geometric) operators to solve the MAGDM problems in which both the attribute weights and the expert weights take the form of real numbers, attribute values take the form of interval-valued intuitionistic fuzzy numbers. Finally, some illustrative examples are given to verify the developed approach and to demonstrate its practicality and effectiveness.     

基于勾股模糊语言幂加权平均算子的多属性群体决策方法

针对多属性群决策问题,采用能够方便专家参考语言集信息进行评价并且取值灵活的勾股模糊语言集进行了处理。首先,基于语言集和勾股模糊集的距离测度给出了勾股模糊语言数距离测度的定义与相关性质;然后,以勾股模糊语言数的距离测度作为幂均（PA）算子的距离度量,提出了勾股模糊语言幂加权平均（PFLPWA）算子用以对群决策过程中不同专家评价矩阵进行融合,并同时在融合过程中考虑专家评价的差异性;最后,基于PFLPWA算子构建了勾股模糊语言环境下的群体决策新方法,并通过案例分析检验了PFLPWA算子应用于群决策中的有效性和适用性。     

一种基于区间灰色语言变量几何加权集成算子的 多属性群决策方法

针对属性值和属性权重均为区间灰色语言变量的多属性群决策问题,提出一种基于区间灰色语言变量的加权几何集成算子的多属性群决策方法.首先,给出区间灰色语言变量的定义和运算规则;然后详细介绍了区间灰色语言变量加权几何集成算子、区间灰色语言变量有序加权几何集成算子、区间灰色语言变量混合加权几何集成算子,以及利用这些算子进行群决策的方法;最后,通过实例说明了所提出方法的决策步骤,并验证了方法的有效性.     

A novel aggregation method for Pythagorean fuzzy multiple attribute group decision making

As an extension of fuzzy set, a Pythagorean fuzzy set has recently been developed to model imprecise and ambiguous information in practical group decision‐making problems. The aim of this paper is to introduce a novel aggregation method for the Pythagorean fuzzy set and analyze possibilities for its application in solving multiple attribute decision‐making problems. More specifically, a new Pythagorean fuzzy aggregation operator called the Pythagorean fuzzy induced ordered weighted averaging‐weighted average (PFIOWAWA) operator is developed. This operator inherits main characteristics of both ordered weighted average operator and induced ordered weighted average to aggregate the Pythagorean fuzzy information. Some of main properties and particular cases of the PFIOWAWA operator are studied. A method based on the proposed operator for multiple attribute group decision making is developed. Finally, we present a numerical example of selection of research and development projects to illustrate applicability of the new approach in a multiple attribute group decision‐making problem.     

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.