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.     

基于加权直觉模糊集合的聚类模型

针对已有基于直觉模糊集的聚类方法的局限性,提出了一种基于加权直觉模糊集合的聚类模型——WIFSCM。在该模型中,提出了特定特征空间下的等价样本和加权直觉模糊集合的概念;并推导出基于等价样本和加权直觉模糊集合的直觉模糊聚类算法的目标函数,利用该目标函数推导出直觉模糊聚类中心迭代算法和隶属度矩阵迭代算法;定义了基于加权直觉模糊集合的密度函数,确定了初始聚类中心,减少了迭代次数。通过灰度图像分割实验,证明了该模型的有效性,同时与普通直觉模糊集FCM聚类算法(IFCM)相比,聚类速度提高近百倍。     

An interval-valued intuitionistic fuzzy permutation method with likelihood-based preference functions and its application to multiple criteria decision analysis

This paper presents an interval-valued intuitionistic fuzzy permutation method with likelihood-based preference functions for managing multiple criteria decision analysis based on interval-valued intuitionistic fuzzy sets. First, certain likelihood-based preference functions are proposed using the likelihoods of interval-valued intuitionistic fuzzy preference relationships. Next, selected practical indices of concordance/discordance are established to evaluate all possible permutations of the alternatives. The optimal priority order of the alternatives is determined by comparing all comprehensive concordance/discordance values based on score functions. Furthermore, this paper considers various preference types and develops another interval-valued intuitionistic fuzzy permutation method using programming models to address multiple criteria decision-making problems with incomplete preference information. The feasibility and applicability of the proposed methods are illustrated in the problem of selecting a suitable bridge construction method. Moreover, certain comparative analyses are conducted to verify the advantages of the proposed methods compared with those of other decision-making methods. Finally, the practical effectiveness of the proposed methods is validated with a risk assessment problem in new product development.     

基于改进直觉模糊熵和信息集成算子的多准则群决策方法

针对直觉模糊环境下准则权重和决策者权重完全未知的群决策问题,提出了一种基于改进直觉模糊熵和信息集成算子的多准则群决策方法.首先,借助直觉模糊数、语言变量法和定量指标值转化公式,将决策者的初始评价准则值规范化;其次,引入改进的直觉模糊熵度量方法确定准则权重和决策者权重;进一步对直觉模糊决策矩阵用直觉模糊加权平均算子和直觉模糊有序加权平均算子进行信息集成,利用新的得分函数对备选方案排序择优.从理论上论证了该方法的可行性,结合算例验证了该方法的有效性.     

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.     

一种加权直觉模糊熵分割法

针对现有的直觉模糊熵阈值法对于光照不均匀图像的分割很不理想,提出了一种新的加权直觉模糊熵,将其应用于图像分割最佳阈值的选取,最后给出了加权直觉模糊熵阈值化准则的权重选取方法。实验结果表明,该加权直觉模糊熵阈值法是可行的,可以弥补现有直觉模糊熵阈值法的不足。     

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.     

基于改进熵和新得分函数的区间直觉模糊多属性决策

针对决策信息为区间直觉模糊数且属性权重完全未知的多属性决策问题, 提出基于改进的区间直觉模糊熵和新得分函数的决策方法. 首先, 利用改进的区间直觉模糊熵确定属性权重; 然后, 利用区间直觉模糊加权算术平均算子集成信息, 得到各备选方案的综合属性值, 进而指出现有得分函数存在排序失效或排序不符合实际的不足, 同时给出一个新的得分函数, 并以此对方案进行排序; 最后, 通过实例表明了所提出方法的有效性.

     

A comparative analysis of score functions for multiple criteria decision making in intuitionistic fuzzy settings

The purpose of this paper was to conduct a comparative study of score functions in multiple criteria decision analysis based on intuitionistic fuzzy sets. The concept of score functions has been conceptualized and widely applied to multi-criteria decision-making problems. There are several types of score functions that can identify the mixed results of positive and negative parts in a bi-dimensional framework of intuitionistic fuzzy sets. Considering various perspectives on score functions, the present study adopts an order of preference based on similarity to the ideal solution as the main structure to estimate the importance of different criteria and compute optimal multi-criteria decisions in intuitionistic fuzzy evaluation settings. An experimental analysis is conducted to examine the relationship between the results yielded by different score functions, considering the average Spearman correlation coefficients and contradiction rates. Furthermore, additional discussions clarify the relative differences in the ranking orders obtained from different combinations of numbers of alternatives and criteria as well as different importance conditions.     

Enhancing PROMETHEE method with intuitionistic fuzzy soft sets

The notion of intuitionistic fuzzy soft sets (IFSSs) provides an effective tool for solving multiple attribute decision making with intuitionistic fuzzy information. The most crucial issue in decision making based on IFSSs is how to derive the ranking of alternatives from the information quantified in terms of intuitionistic fuzzy values. In this study, we propose a new extension of the preference ranking organization method for enrichment evaluation (PROMETHEE), by taking advantage of IFSSs. In addition to presenting a myriad of new notions, such as intuitionistic fuzzy membership (or nonmembership) deviation matrices, intuitionistic fuzzy membership (or nonmembership) preference matrices, and aggregated intuitionistic fuzzy preference matrices, we put more emphasis on the construction of three distinct preference structures and related utility functions on the corresponding weakly ordered sets by considering the positive, negative, and net flows of the alternatives based on the aggregated intuitionistic fuzzy preference matrix. We present a new algorithm for solving multiple attribute decision-making problems with the extended PROMETHEE method based on IFSSs. Moreover, a benchmark problem concerning risk investment is investigated to give a comparative analysis and show the feasibility of our approach.     

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。     

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.     

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.     

A risk attitudinal ranking method for interval-valued intuitionistic fuzzy numbers based on novel attitudinal expected score and accuracy functions

This article investigates new score and accuracy functions for ranking interval-valued intuitionistic fuzzy numbers (IVIFNs). The novelty of these functions is that they allow the comparison of IVIFNs by taking into account of the decision makers’ attitudinal character. The new attitudinal expected score and accuracy functions extend Xu and Chen's score and accuracy degree functions, and verify the following set of properties: (1) boundedness; (2) monotonicity; (3) commutativity; and (4) symmetry. These novel functions are used to propose a total order on the set of IVIFNs, and to develop an interval-valued intuitionistic fuzzy multi-attribute decision making selection process in which the final result depends on the decision maker's risk attitude. In addition, a ranking sensitivity analysis with respect to the risk attitude is provided.     

带偏好度量的直觉模糊序决策信息系统的部分一致约简

现实生活中,不同的需求导致许多信息系统的属性值是基于直觉模糊数的。针对这一现象,在加权得分函数的基础上建立了一种直觉模糊序关系,并给出了不协调带偏好度量的直觉模糊序决策信息系统。进一步,在该复杂系统中引入了部分一致函数,并通过部分一致可辨识矩阵研究求解部分一致约简的方法。最后,通过案例分析验证了该方法的可行性与有效性。     

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.     

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.     

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.     

On intuitionistic fuzzy decision-making using soft likelihood functions

Inspired by Yager, in this paper, we present the concept of likelihood for intuitionistic fuzzy sets (IFSs), and propose an approach for flexible computation of likelihood functions of IFSs for multicriteria decision-making (MCDM). We employ ordered weighted average (OWA) aggregation method to soften the strong likelihood constraint condition. The OWA measure can be considered as the attitudinal character, which determines OWA weights, including optimistic or pessimistic likelihood values. Then the reliability-based soft likelihood function is developed by considering the reliability of intuitionistic fuzzy information. Some examples are conducted by using the proposed (reliability-based) soft likelihood functions in intuitionistic fuzzy environment for MCDM problem, and the results are analyzed in detail.     

直觉模糊集的结构化分析

直觉模糊集理论采用隶属度函数和非隶属度函数刻画不确定性信息,具有一定程度的主观性。为了研究直觉模糊集的本质特征,提出一种直觉模糊集的结构化分析方法,定义了直觉模糊相容关系,给出了直觉模糊集的同构原理,讨论了直觉模糊集的结构化特征。所得结果表明,直觉模糊集也具有客观性的一面。