Intuitionistic preference relations and their application in group decision making

Intuitionistic fuzzy set, characterized by a membership function and a non-membership function, was introduced by Atanassov [Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87-96]. In this paper, we define the concepts of intuitionistic preference relation, consistent intuitionistic preference relation, incomplete intuitionistic preference relation and acceptable intuitionistic preference relation, and study their various properties. We develop an approach to group decision making based on intuitionistic preference relations and an approach to group decision making based on incomplete intuitionistic preference relations respectively, in which the intuitionistic fuzzy arithmetic averaging operator and intuitionistic fuzzy weighted arithmetic averaging operator are used to aggregate intuitionistic preference information, and the score function and accuracy function are applied to the ranking and selection of alternatives. Finally, a practical example is provided to illustrate the developed approaches.     

A novel approach to multi-attribute decision making based on intuitionistic fuzzy sets

In this paper we present a new approach to solve multi-attribute decision making problems in intuitionistic fuzzy environment. This approach is based on a new ranking method of intuitionistic fuzzy sets, in which the evaluated values (in the form of intervals) of the same alternative with different attributes are considered as one unified entity. According to people’s intuition, the ranking method proposed in this paper is mainly grounded on a revised score function and a revised accuracy function of intuitionistic fuzzy sets. Different from the traditional methods, in this new approach, the degree of membership, the degree of nonmembership and the degree of hesitation are considered with various importance in reflecting the true image of the respective alternative. Furthermore, an optimization model is established to estimate the relative degree of importance of each quantity. Finally, two practical examples are provided to illustrate our approach.     

Triangular norm‐based cuts and possibility characteristics of triangular intuitionistic fuzzy numbers for decision making

Triangular intuitionistic fuzzy numbers (TIFNs) is one of the useful tools to manage the fuzziness and vagueness in expressing decision data and solving decision making problems. In this paper, triangular norm (t‐norm) based cuts of TIFNs are developed to synthesize the membership and nonmembership functions in describing the cut sets, then the possibility characteristics of TIFNs, i.e., the possibility mean, the possibility variance, and the possibility mean‐standard deviation ratio, are given. Thereby, on the ground of the possibility mean‐standard deviation ratio, a ranking method of TIFNs is introduced. With these elements, an approach to multiple attributes decision making (MADM) is proposed and illustrated by a numerical example. It is shown that the approach to MADM comprehensively considers both the membership and nonmembership functions and can lead to objective and reasonable results.     

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.     

属性权重不确定条件下的区间直觉模糊多属性决策

在区间直觉模糊集(Interval-valued intuitionistic fuzzy set, IVIFS)的框架内,重点研究了属性权重在一定约束条件下和属性权重完全未知的 多属性群决策问题.首先利用区间直觉模糊集成算子获得方案在属性上的综合区间直觉模糊决策矩阵,进一步依据逼近理想解排序法(Technique for order preference by similarity to an ideal solution, TOPSIS) 的思想计算候选方案和理想方案的加权距离,最后确定方案排序.其中针对属性权重在一定约束条件下的决策问题,提出了基于 区间直觉模糊集精确度函数的线性规划方法,用以解决属性权重求解问题.针对属性权重完全未知的决策问题,首先定义了区间直觉 模糊熵,其次通过熵衡量每一属性所含的信息量来求解属性权重.实验结果验证了决策方法的有效性和可行性.     

Multiple attribute group decision-making methods with unknown weights in intuitionistic fuzzy setting and interval-valued intuitionistic fuzzy setting

To better solve the corresponding multiple attribute group decision-making problem with unknown weights, multiple attribute group decision-making methods with completely unknown weights of decision-makers and incompletely known weights of attributes are proposed in intuitionistic fuzzy setting and interval-valued intuitionistic fuzzy setting. In the group decision-making method, two weight models are proposed based on the score function to determine the weights of both experts and attributes from the intuitionistic fuzzy decision matrices and the interval-valued intuitionistic fuzzy decision matrices. Then, overall evaluation formulas of weighted scores for each alternative are introduced in the intuitionistic fuzzy setting and the interval-valued intuitionistic fuzzy setting to obtain the ranking order of alternatives and the most desirable one(s). Finally, two numerical examples demonstrate the applicability and benefit of the proposed methods.     

Interval-valued intuitionistic fuzzy cosine similarity measures for multiple attribute decision-making

In this work, considering all information carried by the membership degrees, nonmembership degree, and hesitancy degree in interval-valued intuitionistic fuzzy sets (IVIFSs) as 3-D vector representations, we propose a cosine similarity measure and a weighted cosine similarity measure for IVIFSs based on the extension of the cosine similarity measure (angular coefficient) between intuitionistic fuzzy sets in 2-D vector space. Then, the weighted cosine similarity measure for IVIFSs is applied to multiple attribute decision-making problems under the interval-valued intuitionistic fuzzy environment. Through the similarity measure between the ideal alternative and each alternative, the ranking order of all alternatives can be determined and the best alternative can be easily identified as well. Finally, two illustrative examples are given to demonstrate the applications and efficiency of the proposed decision-making method.     

Nonlinear optimization models for multiple attribute group decision making with intuitionistic fuzzy information

Intuitionistic fuzzy numbers are very useful for experts to depict in depth their fuzzy preference information over objects. In this work, we investigate multiple attribute group decision‐making problems in which the attribute values provided by experts are expressed in intuitionistic fuzzy numbers, each of which is composed of a membership degree, a nonmembership degree and a hesitancy degree, and the weight information about both the experts and the attributes is to be determined. We first make different types of attribute values uniform so as to facilitate interattribute comparisons and employ the simple additive weighting method to fuse all the individual opinions into the group one. We then develop two nonlinear optimization models, one minimizing the divergence between each individual opinion and the group one, and the other minimizing the divergence among the individual opinions, from which two exact formulae can be obtained to derive the weights of experts. Similarly, from the viewpoint of maximizing group consensus, we establish a nonlinear optimization model based on all the individual intuitionistic fuzzy decision matrices to determine the weights of attributes. The simple additive weighting method is used to aggregate all the intuitionistic fuzzy attribute values corresponding to each alternative, and then the score function and the accuracy function are employed to rank and select the given alternatives. Moreover, we extend all the above results to interval intuitionistic fuzzy situations, and finally apply the developed models to an air‐condition system selection problem. © 2010 Wiley Periodicals, Inc.     

A possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision making

The ranking of interval-valued intuitionistic fuzzy sets (IVIFSs) is very important for the interval-valued intuitionistic fuzzy decision making. From the probability viewpoint, the possibility degree of comparison between two interval-valued intuitionistic fuzzy numbers (IVIFNs) is defined by using the notion of 2-dimensional random vector, and a new method is then developed to rank IVIFNs. Hereby the ordered weighted average operator and hybrid weighted average operator for IVIFNs are defined based on the Karnik–Mendel algorithms and employed to solve multi-attribute group decision making problems with IVIFNs. The individual overall attribute values of alternatives are obtained by using the weighted average operator for IVIFNs. By using the hybrid weighted average operator for IVIFNs, we can obtain the collective overall attribute values of alternatives, which are used to rank the alternatives. A numerical example is examined to illustrate the effectiveness and flexibility of the proposed method in this paper.     

A PROMETHEE-based outranking method for multiple criteria decision analysis with interval type-2 fuzzy sets

This paper develops new methods based on the preference ranking organization method for enrichment evaluations (PROMETHEE) that use a signed distance-based approach within the environment of interval type-2 fuzzy sets for multiple criteria decision analysis. The theory of interval type-2 fuzzy sets provides an intuitive and computationally feasible way of addressing uncertain and ambiguous information in decision-making fields. Many studies have developed multiple criteria decision analysis methods in the context of interval type-2 fuzzy sets; most of these methods can be characterized as scoring or compromising models. Nevertheless, the extended versions of outranking methods have not been thoroughly investigated. This paper establishes interval type-2 fuzzy PROMETHEE methods for ranking alternative actions among multiple criteria based on the concepts of signed distance-based generalized criteria and comprehensive preference indices. We develop interval type-2 fuzzy PROMETHEE I and interval type-2 fuzzy PROMETHEE II procedures for partial and complete ranking, respectively, of the alternatives. Finally, the feasibility and applicability of the proposed methods are illustrated by a practical problem of landfill site selection. A comparative analysis is also performed with ordinary fuzzy PROMETHEE methods to validate the effectiveness of the proposed methodology.     

Multicriteria fuzzy decision making based on interval-valued intuitionistic fuzzy sets

In this paper, we present a new method for multicriteria fuzzy decision making based on interval-valued intuitionistic fuzzy sets, where interval-valued intuitionistic fuzzy values are used to represent evaluating values of the decision-maker with respect to alternatives. First, we propose a new method for ranking interval-valued intuitionistic fuzzy values. Based on the proposed fuzzy ranking method of interval-valued intuitionistic fuzzy values, we propose a new method for multicriteria fuzzy decision making. The proposed multicriteria fuzzy decision making method outperforms Ye’s method (2009) due to the fact that the proposed method can overcome the drawback of Ye’s method (2009), where the drawback of Ye’s method is that it can not distinguish the ranking order between alternatives in some situations. The proposed method provides us with a useful way for dealing with multicriteria fuzzy decision making problems based on interval-valued intuitionistic fuzzy sets.     

用区间直觉模糊集方法对属性权重未知的群求解其多属性决策

本文首先提出群区间直觉模糊有序加权几何(groupinterval-valuedintuitionistic fuzzy orderedweighted geometric,GIVIFOWG)算子和群区间直觉模糊有序加权平均(group interval-valued intuitionistic fuzzy ordered weighted averaging,GIVIFOWA)算子.利用GIVIFOWG算子或GIVIFOWA算子聚集群的决策矩阵以获得方案在属性上的综合区间直觉模糊决策矩阵(collectiveinterval-valuedintuitionistic fuzzy decision-matrix,CIVIFDM).然后定义了一个考虑犹豫度的区间直觉模糊熵(interval-valuedintuitionistic fuzzyentropy,IVIFE);通过熵衡量每个属性所含的信息来求解属性权重.最后,提出基于可能度的接近理想解的区间排序法(interval technique for order preference by similarity to an ideal solution,ITOPSIS)和区间得分函数法.在ITOPSIS法中,依据区间距离公式计算候选方案和理想方案的属性加权区间距离,进而采用ITOPSIS准则对各方案进行排序;在区间得分函数法中,算出CIVIFDM中各方案的得分值以及精确值,然后利用区间得分准则对各方案进行排序.实验结果验证了决策方法的有效性和可行性.     

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 novel risk attitudinal ranking method for intuitionistic fuzzy values and application to MADM

The ranking of intuitionistic fuzzy sets (IFSs) is very important for the intuitionistic fuzzy decision making. The aim of this paper is to propose a new risk attitudinal ranking method of IFSs and apply to multi-attribute decision making (MADM) with incomplete weight information. Motivated by technique for order preference by similarity to ideal solution (TOPSIS), we utilize the closeness degree to characterize the amount of information according to the geometrical representation of an IFS. The area of triangle is calculated to measure the reliability of information. It is proved that the closeness degree and the triangle area just form an interval. Thereby, a new lexicographical method is proposed based on the intervals for ranking the intuitionistic fuzzy values (IFVs). Furthermore, considered the risk attitude of decision maker sufficiently, a novel risk attitudinal ranking measure is developed to rank the IFVs on the basis of the continuous ordered weighted average (C-OWA) operator and this interval. Through maximizing the closeness degrees of alternatives, we construct a multi-objective fractional programming model which is transformed into a linear program. Thus, the attribute weights are derived objectively by solving this linear program. Then, a new method is put forward for MADM with IFVs and incomplete weight information. Finally, an example analysis of a teacher selection is given to verify the effectiveness and practicability of the proposed method.     

三角直觉模糊决策的权重函数方法

研究了属性值为三角直觉模糊数的多属性决策问题,提出了一种基于权重函数的决策方法。给出了三角直觉模糊数的定义、运算法则和截集;定义了三角直觉模糊数关于隶属函数和非隶属函数的精确值和模糊度,以及精确值的指标和模糊度的指标,给出了三角直觉模糊数的排序方法,并将其应用到属性值为三角直觉模糊数的多属性决策问题中;给出了属性值为三角直觉模糊数的多属性决策的步骤;通过数值算例分析和验证了该方法的有效性。     

Fuzzy multiattribute group decision making based on intuitionistic fuzzy sets and evidential reasoning methodology

In this paper, we propose a new fuzzy multiattribute group decision making method based on intuitionistic fuzzy sets and the evidential reasoning methodology. First, the proposed method uses the evidential reasoning methodology to aggregate each decision maker’s decision matrix and the weights of the attributes to get the aggregated decision matrix of each decision maker. Then, it uses the obtained aggregated decision matrices of the experts, the weights of the experts and the evidential reasoning methodology to get the aggregated intuitionistic fuzzy value of each alternative. Finally, it calculates the transformed value of the obtained intuitionistic fuzzy value of each alternative. The smaller the transformed value, the better the preference order of the alternative. The proposed method can overcome the drawbacks of the existing methods for fuzzy multiattribute group decision making in intuitionistic fuzzy environments.     

A novel group decision making method with intuitionistic fuzzy preference relations for RFID technology selection

A more scientific decision making process for radio frequency identification (RFID) technology selection is important to increase success rate of RFID technology application. RFID technology selection can be formulated as a kind of group decision making (GDM) problem with intuitionistic fuzzy preference relations (IFPRs). This paper develops a novel method for solving such problems. First, A technique for order preference by similarity to ideal solution (TOPSIS) based method is presented to rank intuitionistic fuzzy values (IFVs). To achieve higher group consensus as well as possible, we construct an intuitionistic fuzzy linear programming model to derive experts’ weights. Depending on the construction of membership and non-membership functions, the constructed intuitionistic fuzzy linear programming model is solved by three kinds of approaches: optimistic approach, pessimistic approach and mixed approach. Then to derive the ranking order of alternatives from the collective IFPR, we extend quantifier guided non-dominance degree (QGNDD) and quantifier guided dominance degree (QGDD) to intuitionistic fuzzy environment. A new two-phase ranking approach is designed to generate the ordering of alternatives based on QGNDD and QGDD. Thereby, the corresponding method is proposed for the GDM problems with IFPRs. Some generalizations on the constructed intuitionistic fuzzy linear programming model are further discussed. At length, the validity of the proposed method is illustrated with a real-world RFID technology selection example.     

Linear programming method for multiattribute group decision making using IF sets

The purpose of this paper is to develop a linear programming methodology for solving multiattribute group decision making problems using intuitionistic fuzzy (IF) sets. In this methodology, IF sets are constructed to capture fuzziness in decision information and decision making process. The group consistency and inconsistency indices are defined on the basis of pairwise comparison preference relations on alternatives given by the decision makers. An IF positive ideal solution (IFPIS) and weights which are unknown a priori are estimated using a new auxiliary linear programming model, which minimizes the group inconsistency index under some constraints. The distances of alternatives from the IFPIS are calculated to determine their ranking order. Moreover, some properties of the auxiliary linear programming model and other generalizations or specializations are discussed in detail. Validity and applicability of the proposed methodology are illustrated with the extended air-fighter selection problem and the doctoral student selection problem.     

基于熵和相关系数的直觉模糊多属性决策方法

针对决策信息为直觉模糊信息且属性权重完全未知的多属性决策问题,提出了一种基于直觉模糊熵和得分函数的决策方法。为了准确度量直觉模糊集的直觉性和模糊性,首先提出了一种新的直觉模糊熵,并讨论其相关性质。其次为了减少不确定信息对决策的影响,结合直觉模糊熵建立规划模型,从而确定属性权重。 同时从隶属度,非隶属度和犹豫度三方面构建论域对象与理想对象之间相关系数,并在此基础上根据决策者的决策态度定义得分函数进而得到最优决策。 最后给出一种基于直觉模糊信息的多属性决策方法,并通过候选人评估实例验证了该方法的可行性和有效性。     

对方案有偏好的直觉模糊多属性决策方法

对属性权重信息不完全、属性值和决策者对方案的偏好信息均以直觉模糊数表示的多属性决策问题提出一种决策方法。首先根据决策者对方案的偏好信息建立多目标规划模型,求出属性权重,接着利用觉模糊加权算术平均算子求出方案的综合属性值,由直觉模糊数的得分函数和精确函数确定方案的排序,最后通过实例证明了该方法的实用性和有效性。