Some induced intuitionistic fuzzy aggregation operators applied to multi-attribute group decision making

In this paper, we define various induced intuitionistic fuzzy aggregation operators, including induced intuitionistic fuzzy ordered weighted averaging (OWA) operator, induced intuitionistic fuzzy hybrid averaging (I-IFHA) operator, induced interval-valued intuitionistic fuzzy OWA operator, and induced interval-valued intuitionistic fuzzy hybrid averaging (I-IIFHA) operator. We also establish various properties of these operators. And then, an approach based on I-IFHA operator and intuitionistic fuzzy weighted averaging (WA) operator is developed to solve multi-attribute group decision-making (MAGDM) problems. In such problems, attribute weights and the decision makers' (DMs') weights are real numbers and attribute values provided by the DMs are intuitionistic fuzzy numbers (IFNs), and an approach based on I-IIFHA operator and interval-valued intuitionistic fuzzy WA operator is developed to solve MAGDM problems where the attribute values provided by the DMs are interval-valued IFNs. Furthermore, induced intuitionistic fuzzy hybrid geometric operator and induced interval-valued intuitionistic fuzzy hybrid geometric operator are proposed. Finally, a numerical example is presented to illustrate the developed approaches.     

Some induced correlated aggregating operators with intuitionistic fuzzy information and their application to multiple attribute group decision making

In this paper, some multiple attribute group decision making (MAGDM) problems in which both the attribute weights and the expert weights are usually correlative, attribute values take the form of intuitionistic fuzzy values or interval-valued intuitionistic fuzzy values, are investigated. Firstly, some operational law, score function and accuracy function of intuitionistic fuzzy values or interval-valued intuitionistic fuzzy values are introduced. Then two new aggregation operators: induced intuitionistic fuzzy correlated averaging (I-IFCA) operator and induced intuitionistic fuzzy correlated geometric (I-IFCG) operator are developed and some desirable properties of the I-IFCA and I-IFCG operators are studied, such as commutativity, idempotency and monotonicity. An I-IFCA and IFCA (intuitionistic fuzzy correlated averaging) operators-based approach is developed to solve the MAGDM problems in which both the attribute weights and the expert weights usually correlative, attribute values take the form of intuitionistic fuzzy values. Then, we extend the developed models and procedures to the interval-valued intuitionistic fuzzy environment. Finally, some illustrative examples are given to verify the developed approach and to demonstrate its practicality and effectiveness.     

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.     

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 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.     

A ranking method for multiple attribute decision-making problems based on the possibility degrees of trapezoidal intuitionistic fuzzy numbers

To solve multiple attribute decision-making problems with attribute values or decision values characterized by trapezoidal intuitionistic fuzzy numbers (TIFNs), we define a trapezoidal intuitionistic fuzzy induced ordered weighted arithmetic averaging (TIFIOWA) operator, which is an extension of the induced ordered weighted arithmetic averaging operator. We derive and prove some related properties and conclusions of the TIFIOWA operator. To compare the TIFNs, we define possibility degrees of the TIFNs. Based on the possibility degrees of the TIFNs and the TIFIOWA operator, we construct a new method to determine the order of alternatives in multiple attribute decision making and to choose the best alternative. Finally, a numerical example shows that the developed method is feasible and effective.     

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.     

An interactive method for dynamic intuitionistic fuzzy multi-attribute group decision making

This paper investigates the dynamic intuitionistic fuzzy multi-attribute group decision making (DIF-MAGDM) problems, in which all the attribute values provided by multiple decision makers (DMs) at different periods take the form of intuitionistic fuzzy numbers (IFNs), and develops an interactive method to solve the DIF-MAGDM problems. The developed method first aggregates the individual intuitionistic fuzzy decision matrices at different periods into an individual collective intuitionistic fuzzy decision matrix for each decision maker by using the dynamic intuitionistic fuzzy weighted averaging (DIFWA) operator, and then employs intuitionistic fuzzy TOPSIS method to calculate the individual relative closeness coefficient of each alternative for each decision maker and obtain the individual ranking of alternatives. After doing so, the method utilizes the hybrid weighted averaging (HWA) operator to aggregate all the individual relative closeness coefficients into the collective relative closeness coefficient of each alternative and obtain the aggregate ranking of alternatives, by which the optimal alternative can be selected. In addition, the spearman correlation coefficient for both the aggregate ranking and individual ranking of alternatives is calculated to measure the consensus level of the group preferences. Finally, a numerical example is used to illustrate the developed method.     

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.     

Multiple attribute group decision making methods based on intuitionistic linguistic power generalized aggregation operators

With respect to multiple attribute group decision making (MADM) problems in which attribute values take the form of intuitionistic linguistic numbers, some new group decision making methods are developed. Firstly, some operational laws, expected value, score function and accuracy function of intuitionistic linguistic numbers are introduced. Then, an intuitionistic linguistic power generalized weighted average (ILPGWA) operator and an intuitionistic linguistic power generalized ordered weighted average (ILPGOWA) operator are developed. Furthermore, some desirable properties of the ILPGWA and ILPGOWA operators, such as commutativity, idempotency and monotonicity, etc. are studied. At the same time, some special cases of the generalized parameters in these operators are analyzed. Based on the ILPGWA and ILPGOWA operators, two approaches to multiple attribute group decision making with intuitionistic linguistic information are proposed. Finally, an illustrative example is given to verify the developed approaches and to demonstrate their practicality and effectiveness.     

An approach for multiple attribute group decision making problems with interval-valued intuitionistic trapezoidal fuzzy numbers

This article proposes an approach to resolve multiple attribute group decision making (MAGDM) problems with interval-valued intuitionistic trapezoidal fuzzy numbers (IVITFNs). We first introduce the cut set of IVITFNs and investigate the attitudinal score and accuracy expected functions for IVITFNs. Their novelty is that they allow the comparison of IVITFNs by taking into accounting of the experts’ risk attitude. Based on these expected functions, a ranking method for IVITFNs is proposed and a ranking sensitivity analysis method with respect to the risk attitude is developed. To aggregate the information with IVITFNs, we study the desirable properties of the interval-valued intuitionistic trapezoidal fuzzy weighted geometric (IVITFWG) operator, the interval-valued intuitionistic trapezoidal fuzzy ordered weighted geometric (IVITFOWG) operator, and the interval-valued intuitionistic trapezoidal fuzzy hybrid geometric (IVITFHG) operator. It is worth noting that the aggregated value by using these operators is also an interval-valued intuitionistic trapezoidal fuzzy value. Then, based on these expected functions and aggregating operators, an approach is proposed to solve MAGDM problems in which the attribute values take the form of interval-valued intuitionistic fuzzy numbers and the expert weights take the form of real numbers. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

Induced generalized hesitant fuzzy operators and their application to multiple attribute group decision making

In this paper, we develop a series of induced generalized aggregation operators for hesitant fuzzy or interval-valued hesitant fuzzy information, including induced generalized hesitant fuzzy ordered weighted averaging (IGHFOWA) operators, induced generalized hesitant fuzzy ordered weighted geometric (IGHFOWG) operators, induced generalized interval-valued hesitant fuzzy ordered weighted averaging (IGIVHFOWA) operators, and induced generalized interval-valued hesitant fuzzy ordered weighted geometric (IGIVHFOWG) operators. Next, we investigate their various properties and some of their special cases. Furthermore, some approaches based on the proposed operators are developed to solve multiple attribute group decision making (MAGDM) problems with hesitant fuzzy or interval-valued hesitant fuzzy information. Finally, some numerical examples are provided to illustrate the developed approaches.     

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.     

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.     

Multi-attribute decision making based on neutral averaging operators for intuitionistic fuzzy information

In this paper, we construct the probability sum (PS) function and the proportional distribution rules of membership function and non-membership function of intuitionistic fuzzy sets (IFSs), and give their corresponding geometric interpretations. Based on which, we present the neutrality operation and the scalar neutrality operation on intuitionistic fuzzy numbers (IFNs). We propose the intuitionistic fuzzy weighted neutral averaging (IFWNA) operator and the intuitionistic fuzzy ordered weighted neutral averaging (IFOWNA) operator. The properties of the IFWNA operator and the IFOWNA operator are investigated. The principal advantages of the proposed operators are that both the attitude of the decision makers and the interactions between different intuitionistic fuzzy numbers (IFNs) are considered. Furthermore, approaches to multi-criteria decision making based on the proposed IFWNA and IFOWNA operator are given. Finally, an example is illustrated to show the feasibility and validity of the new approaches to the application of decision making.     

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

As a special intuitionistic fuzzy set on a real number set, trapezoidal intuitionistic fuzzy numbers (TrIFNs) have the better capability to model ill-known quantities. The purpose of this paper is to develop some power geometric operators of TrIFNs and apply to multi-attribute group decision making (MAGDM) with TrIFNs. First, the lower and upper weighted possibility means of TrIFNs are introduced as well as weighted possibility means. Hereby, a new lexicographic method is developed to rank TrIFNs. The Minkowski distance between TrIFNs is defined. Then, four kinds of power geometric operators of TrIFNs are investigated including the power geometric operator of TrIFNs, power weighted geometric operator of TrIFNs, power ordered weighted geometric operator of TrIFNs and power hybrid geometric operator of TrIFNs. Their desirable properties are discussed. Four methods for MAGDM with TrIFNs are respectively proposed for the four cases whether the weight vectors of attributes and DMs are known or unknown. In these methods, the individual overall attribute values of alternatives are generated by using the power geometric or power weighted geometric operator of TrIFNs. The collective overall attribute values of alternatives are determined through constructing the multi-objective optimization model, which is transformed into the goal programming model to solve. Thus, the ranking order of alternatives is obtained according to the collective overall attribute values of alternatives. Finally, the green supplier selection problem is illustrated to demonstrate the application and validation of the proposed method.     

Study on the ranking problems in multiple attribute decision making based on interval‐valued intuitionistic fuzzy numbers

This paper puts forward a new ranking method for multiple attribute decision‐making problems based on interval‐valued intuitionistic fuzzy set (IIFS) theory. First, the composed ordered weighted arithmetic averaging operator and composed ordered weighted geometric averaging operator are extended to the IIFSs in which they are, respectively, named interval‐valued intuitionistic fuzzy composed ordered weighted arithmetic averaging (IIFCOWA) operator and interval‐valued intuitionistic composed ordered weighted geometric averaging (IIFCOWG) operator. Afterwards, to compare interval‐valued intuitionistic fuzzy numbers, we define the concepts of the maximum, the minimum, and ranking function. Some properties associated with the concepts are investigated. Using the IIFCOWA or IIFCOWG operator, we establish the detailed steps of ranking alternatives (or attributes) in multiple attribute decision making. Finally, an illustrative example is provided to show that the proposed ranking method is feasible in multiple attribute decision making.     

A new method for probabilistic linguistic multi-attribute group decision making: Application to the selection of financial technologies

“No technology, no finance” has been the consensus in banking industry. Under the background of financial technology (Fintech), how to select an appropriate technology company to cooperate for the banks has become a key. The technology company selection can be regarded as a kind of multi-attribute group decision making (MAGDM) problems. The probabilistic linguistic term set (PLTS) is a useful tool to express decision makers’ (DMs’) evaluations in the technology company selection. This paper develops a new method for MAGDM with PLTSs. Firstly, the possibility degree and range value of PLTSs are defined. Then a possibility degree algorithm is designed for ranking PLTSs. An Euclidean distance measure between PLTSs is presented and extended to probabilistic linguistic matrices. Based on Archimedean t-norm and s-norm, some new operational laws for PLTSs are defined and some desirable properties are discussed. Then, a generalized probabilistic linguistic Hamacher weighted averaging (GPLHWA) operator and a generalized probabilistic linguistic Hamacher ordered weighted averaging (GPLHOWA) operator are developed. Some useful properties for these operators are investigated in detail. Combined with the subjective weights of DMs, the DMs’ weights are determined by the adjusted coefficients. Using the GPLHWA operator, the collective decision matrix is obtained by aggregating all the individual decision matrices. By maximizing the total weighted square possibility degree, a multi-objective program is constructed to derive the attribute weights. The ranking order of alternatives is generated by integrating ELECTRE and TOPSIS. Thereby, a new method is put forward for MAGDM with PLTSs. A Fintech example is analyzed to show the effectiveness of the proposed method. The sensitivity analysis and comparative analyses are conducted to illustrate its advantages.     

A Novel Intuitionistic Fuzzy Induced Ordered Weighted Euclidean Distance Operator and Its Application for Group Decision Making

In the paper, we develop the novel intuitionistic fuzzy induced ordered weighted Euclidean distance (NIFIOWED) operator based on the generalized intuitionistic fuzzy distance for multiple attribute group decision making problems. The NIFIOWED operator's attribute values take the form of Atanassov's intuitionistic fuzzy numbers(A‐IFNs), and the principal component x of A‐IFN is taken into account first. The prime properties of the NIFIOWED operator are investigated. Finally, a new method and a numerical example are provided to reveal the availability and practicability of the NIFIOWED operator.     

Some new Shapley 2-tuple linguistic Choquet aggregation operators and their applications to multiple attribute group decision making

In this paper, we investigate the multiple attribute group decision making (MAGDM) problems with 2-tuple linguistic information. Firstly, motivated by the ideas of Choquet integral and Shapley index, we propose three 2-tuple linguistic aggregation operators called Shapley 2-tuple linguistic Choquet averaging operator, Shapley 2-tuple linguistic Choquet geometric operator and generalized Shapley 2-tuple linguistic Choquet averaging operator. Then we discuss some properties of these operators, such as idempotency, monotonicity, boundary and commutativity. Secondly, if the information about the weights of decision makers (DMs) and attributes is incompletely known, we build two models to determine the optimal fuzzy measures on DM set and attribute set, respectively. Furthermore, we develop a new method for multiple attribute group decision making under 2-tuple linguistic environment based on the proposed operators. Finally, we apply the developed MAGDM method to select the most desirable emergency alternative and the validity of the developed method is verified by comparing the evaluation results with those obtained from the existing 2-tuple correlated aggregation operators.