On Typical Hesitant Fuzzy Prioritized “or” Operator in Multi‐Attribute Decision Making

Since hesitant fuzzy set was proposed, multi‐attribute decision making (MADM) with hesitant fuzzy information, which is also called hesitant fuzzy MADM, has been a hot research topic in decision theory. This paper investigates a special kind of hesitant fuzzy MADM problems in which the decision data are expressed by several possible values, and the evaluative attributes are in different priority levels. Firstly, we introduce the definitions of hesitant fuzzy t‐norm and t‐conorm by extending the notions of t‐norm and t‐conorm to the hesitant fuzzy environment and explore their constructions by means of t‐norms and t‐conorms. Then motivated by the prioritized “or” operator (R. R. Yager, Prioritized aggregation operators, International Journal of Approximate Reasoning 2008;48:263–274), we develop the typical hesitant fuzzy prioritized “or” operator based on the developed hesitant fuzzy t‐norms and t‐conorms. In this operator, the degree of satisfaction of each alternative in each priority level is derived from a hesitant fuzzy t‐conorm to preserve trade‐offs among the attributes in the same priority level, and the priority weights of attributes are induced by a hesitant fuzzy t‐norm to model the prioritization relationship among attributes. Furthermore, we apply the developed typical hesitant fuzzy prioritized “or” operator to solving the MADM problems in which the decision data are expressed by several possible values and the attributes are in different priority levels. In addition, two numerical examples are given to, respectively, illustrate the applicability and superiority of the developed aggregation operator by comparative analyses with previous research.     

Prioritized aggregation operators of trapezoidal intuitionistic fuzzy sets and their application to multicriteria decision-making

A trapezoidal intuitionistic fuzzy set, some operational laws, score and accuracy functions for trapezoidal intuitionistic fuzzy values are presented in this paper. Then, the trapezoidal intuitionistic fuzzy prioritized weighted averaging (TIFPWA) operator and trapezoidal intuitionistic fuzzy prioritized weighted geometric (TIFPWG) operator are proposed to aggregate the trapezoidal intuitionistic fuzzy information. Furthermore, a multicriteria decision-making method based on the TIFPWA and TIFPWG operators and the score and accuracy functions of trapezoidal intuitionistic fuzzy values is established to deal with the multicriteria decision-making problem in which the criteria are in different priority level. Finally, a practical example about software selection for considering various prioritized relationships between the criteria of decision-making is given to demonstrate its practicality and effectiveness.     

Prioritized intuitionistic fuzzy aggregation operators

In some multi-attribute decision making problems, distorted conclusions will be generated due to the lack of considering various relationships among the attributes of decision making. In this paper, we investigate the prioritization relationship of attributes in multi-attribute decision making with intuitionistic fuzzy information (i.e., partial or all decision information, like attribute values and weights, etc., is represented by intuitionistic fuzzy values (IFVs)). Firstly, we develop a new method for comparing two IFVs, based on which the basic intuitionistic fuzzy operations satisfy monotonicities. In addition, we devise a method to derive the weights with intuitionistic fuzzy forms, which can indicate the importance degrees of the corresponding attributes. Then we develop a prioritized intuitionistic fuzzy aggregation operator, which is motivated by the idea of the prioritized aggregation operators [R.R. Yager, Prioritized aggregation operators, International Journal of Approximate Reasoning 48 (2008) 263–274]. Furthermore, we propose an intuitionistic fuzzy basic unit monotonic (IF-BUM) function to transform the derived intuitionistic fuzzy weights into the normalized weights belonging to the unit interval. Finally, we develop a prioritized intuitionistic fuzzy ordered weighted averaging operator on the basis of the IF-BUM function and the transformed weights.     

考虑优先级的广义犹豫模糊信息集成方法

研究了在属性之间存在优先级的情况下的广义犹豫模糊信息集成问题。考虑到属性优先级以及属性元素的统一程度的双重影响,首先给出了犹豫模糊信息下的熵值求法,并在其基础上提出了优先级混合赋权方法。之后,在该优先级混合赋权方法的基础上提出了广义犹豫模糊优先级混合几何(GHFPHG)算子,并给出了该类算子的优良特性。最后,利用案例验证了本文所提方法的实用性和有效性。     

Multicriteria decision making based on intuitionistic fuzzy prioritized arithmetic mean

Atanassov’s intuitionistic fuzzy sets (AIFSs), characterized by a membership function, a nonmembership function, and a hesitancy function, is a generalization of a fuzzy set. Various aggregation operators are defined for AIFSs to deal with multicriteria decision‐making problems in which there exists a prioritization of criteria. However, these existing intuitionistic fuzzy prioritized aggregation operators are not monotone with respect to the total order on Atanassov’s intuitionistic fuzzy values (AIFVs), which is undesirable. We propose an intuitionistic fuzzy prioritized arithmetic mean based on the ?ukasiewicz triangular norm, which is monotone with respect to the total order on AIFVs, and therefore is a true generalization of such operations. We give an example that a consumer selects a car to illustrate the validity and applicability of the proposed method aggregation operator.     

直觉模糊POWA算子及其在多准则决策中的应用

为了解决具有优先级的直觉模糊多准则决策问题,定义了直觉模糊优先有序加权平均(IFPOWA)算子.基于优先关系.利用直觉模糊值修正得分函数给出其关联权重向量的计算方法,分析并证明了IFPOWA算子的性质;提出了基于IFPOWA算了的具有优先级的直觉模糊多准则决策方法.最后,利用实例对方法的有效性进行了分析.     

毕达哥拉斯模糊优先集成算子及其决策应用

作为直觉模糊集的推广形式,毕达哥拉斯模糊数能更好地刻画现实中的不确定性,此外在某些问题上,方案的属性之间往往具有优先关系,针对此类信息的集成问题,将毕达哥拉斯模糊数与优先集成算子相结合,提出了毕达哥拉斯模糊优先集成算子,包括毕达哥拉斯模糊优先加权平均算子和毕达哥拉斯模糊优先加权几何算子,并讨论了这些算子的性质。在此基础上,提出了毕达哥拉斯模糊优先集成算子的多属性决策方法,最后将其应用于国内四家航空公司服务质量评价中,说明了该算子的有效性和可行性。     

Multiattribute group decision-making based on Pythagorean fuzzy Einstein prioritized aggregation operators

Pythagorean fuzzy set (PFS) is a powerful tool to deal with the imprecision and vagueness. Many aggregation operators have been proposed by many researchers based on PFSs. But the existing methods are under the hypothesis that the decision-makers (DMs) and the attributes are at the same priority level. However, in real group decision-making problems, the attribute and DMs may have different priority level. Therefore, in this paper, we introduce multiattribute group decision-making (MAGDM) based on PFSs where there exists a prioritization relationship over the attributes and DMs. First we develop Pythagorean fuzzy Einstein prioritized weighted average operator and Pythagorean fuzzy Einstein prioritized weighted geometric operator. We study some of its desirable properties such as idempotency, boundary, and monotonicity in detail. Moreover we propose a MAGDM approach based on the developed operators under Pythagorean fuzzy environment. Finally, an illustrative example is provided to illustrate the practicality of the proposed approach.     

Prioritized Information Fusion Method for Triangular Intuitionistic Fuzzy Set and its Application to Teaching Quality Evaluation

In this article, we examine the issue of triangular intuitionistic fuzzy information fusion. We first propose some new triangular intuitionistic fuzzy aggregation operators based on the prioritized average operator, such as the triangular intuitionistic fuzzy prioritized weighted average and the triangular intuitionistic fuzzy prioritized weighted geometric operators. We study some desired properties of the proposed operators, such as idempotency, noncompensatory, and boundary. We then develop an approach to deal with group decision‐making problems under triangular intuitionistic fuzzy environments. Finally, a practical example about teaching quality evalution is provided to illustrate the group decision‐making process.     

On prioritized weighted aggregation in multi-criteria decision making

This paper deals with multi-criteria decision making (MCDM) problems with multiple priorities, in which priority weights associated with the lower priority criteria are related to the satisfactions of the higher priority criteria. Firstly, we propose a prioritized weighted aggregation operator based on ordered weighted averaging (OWA) operator and triangular norms (t-norms). To preserve the tradeoffs among the criteria in the same priority level, we suggest that the degree of satisfaction regarding each priority level is viewed as a pseudo criterion. On the other hand, t-norms are used to model the priority relationships between the criteria in different priority levels. In particular, we show that strict Archimedean t-norms perform better in inducing priority weights. As Hamacher family of t-norms provide a wide class of strict Archimedean t-norms ranging from the product to weakest t-norm, Hamacher parameterized t-norms are used to induce the priority weight for each priority level. Secondly, considering decision maker (DM)’s requirement toward higher priority levels, a benchmark based approach is proposed to induce priority weight for each priority level. In particular, ?ukasiewicz implication is used to compute benchmark achievement for crisp requirements; target-oriented decision analysis is utilized to obtain the benchmark achievement for fuzzy requirements. Finally, some numerical examples are used to illustrate the proposed prioritized aggregation technique as well as to compare with previous research.     

基于Zhenyuan积分的直觉模糊多属性决策方法

针对属性之间具有相互关联关系的直觉模糊多属性决策问题,提出一种基于Zhenyuan积分的决策方法.首先提出直觉模糊Zhenyuan积分平均(IFZA)算子;然后探讨IFZA算子的优良性质以及与现有直觉模糊集成算子的关系,研究表明,IFZA算子可以改进现有直觉模糊奇异积分算子的缺陷,能够全面度量属性之间的相互关联关系;最后提出一种基于IFZA算子的属性间具有相互关联关系的直觉模糊多属性决策方法,并通过实例验证所提出方法的有效性和可行性.     

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.     

Weakly Prioritized Measure Aggregation in Prioritized Multicriteria Decision Making

This paper mainly investigates a special kind of multicriteria decision–making problem, in which all the criteria can be divided into several hierarchies and the criteria in the higher hierarchy have priorities over those in the lower hierarchy. It implies that the loss of the higher priority criterion can't be compensated by the gain of the lower prioritized criteria. As we know, fuzzy measures can well represent the interactions between criteria. In this situation, we develop a new fuzzy measure called weakly ordered prioritized measure (WOPM) to express the priority rule among the weakly ordered prioritized criteria. On the basis of the WOPM, we use discrete Choquet integral to construct a new WOPM‐guided aggregation (WOPMGA) operator. To understand the priority property of this aggregation operator deeply, we get all the criteria's Shapley values and make an analysis of all criteria's Shapley values with different parameter values. Through analysis, we can find that the WOPMGA operator has the properties of boundedness, idempotency and monotonicity. Finally, we give several practical examples to illustrate the effectiveness of this aggregation operator.     

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.     

Generalized Atanassov’s intuitionistic fuzzy power geometric operators and their application to multiple attribute group decision making

In this paper, we extend the power geometric (PG) operator and the power ordered weighted geometric (POWG) operator [Z.S. Xu, R.R. Yager, Power-geometric operators and their use in group decision making, IEEE Transactions on Fuzzy Systems 18 (2010) 94–105] to Atanassov’s intuitionistic fuzzy environments, i.e., we develop a series of generalized Atanassov’s intuitionistic fuzzy power geometric operators to aggregate input arguments that are Atanassov’s intuitionistic fuzzy numbers (IFNs). Then, we study some desired properties of these aggregation operators and investigate the relationships among these operators. Furthermore, we apply these aggregation operators to develop some methods for multiple attribute group decision making with Atanassov’s intuitionistic fuzzy information. Finally, two practical examples are provided to illustrate the proposed methods.     

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.     

Multi-criteria group decision making based on trapezoidal intuitionistic fuzzy information

With respect to multi-criteria group decision making (MCGDM) problems under trapezoidal intuitionistic fuzzy environment, a new MCGDM method is investigated. The proposed method can effectively avoid the failure caused by the use of inconsistent decision information and provides a decision-making idea for the case of “the truth be held in minority”. It consists of three interrelated modules: weight determining mechanism, group consistency analysis, and ranking and selection procedure. For the first module, distance measures, expected values and arithmetic averaging operator for trapezoidal intuitionistic fuzzy numbers are used to determine the weight values of criteria and decision makers. For the second module, a consistency analysis and correction procedure based on trapezoidal intuitionistic fuzzy weighted averaging operator and OWA operator is developed to reduce the influence of conflicting opinions prior to the ranking process. For the third module, a trapezoidal intuitionistic fuzzy TOPSIS is used for ranking and selection. Then a procedure for the proposed MCGDM method is developed. Finally, a numerical example further illustrates the practicality and efficiency of the proposed method.     

Extension of the VIKOR method to dynamic intuitionistic fuzzy multiple attribute decision making

In this paper, we extend the VIKOR method for dynamic intuitionistic fuzzy multiple attribute decision making (DIF-MADM). Two new aggregation operators called dynamic intuitionistic fuzzy weighted geometric (DIFWG) operator and uncertain dynamic intuitionistic fuzzy weighted geometric (UDIFWG) operator are presented. Based on the DIFWA and UDIFWA operators respectively, we develop two procedures to solve the DIF-MADM problems where all attribute values are expressed in intuitionistic fuzzy numbers or interval-valued intuitionistic fuzzy numbers, which are collected at different periods. Finally, a numerical example is used to illustrate the applicability of the proposed approach.     

Group decision making based on generalized intuitionistic fuzzy prioritized geometric operator

Intuitionistic fuzzy set was studied by many authors because it is a powerful technique to depict uncertainty, which is a set containing three functions: the membership function; the nonmembership function; and the hesitancy function. The aggregation of intuitionistic fuzzy values (IFVs) is of paramount importance in decision making. In this paper, we research IFVs aggregation problems, where there exist a prioritization relationship over the aggregated arguments. First, we propose the generalized intuitionistic fuzzy prioritized weighted geometric operator based on Archimedean t‐conorm and t‐norm. Then, some of its desirable properties and special cases are investigated in detail. Furthermore, a multicriteria group decision‐making problems is formulated with IFVs using the proposed operator. Finally, the validity and applicability of the proposed method, as well as analysis of the comparison with different generator functions, are illustrated with a real example about talent introduction. © 2012 Wiley Periodicals, Inc.     

A multi-criteria interval-valued intuitionistic fuzzy group decision making with Choquet integral-based TOPSIS

An extension of TOPSIS, a multi-criteria interval-valued intuitionistic fuzzy decision making technique, to a group decision environment is investigated, where inter-dependent or interactive characteristics among criteria and preference of decision makers are taken into account. To get a broad view of the techniques used, first, some operational laws on interval-valued intuitionistic fuzzy values are introduced. Based on these operational laws, a generalized interval-valued intuitionistic fuzzy geometric aggregation operator is proposed which is used to aggregate decision makers’ opinions in group decision making process. In addition, some of its properties are discussed. Then Choquet integral-based Hamming distance between interval-valued intuitionistic fuzzy values is defined. Combining the interval-valued intuitionistic fuzzy geometric aggregation operator with Choquet integral-based Hamming distance, an extension of TOPSIS method is developed to deal with a multi-criteria interval-valued intuitionistic fuzzy group decision making problems. Finally, an illustrative example is used to illustrate the developed procedures.