Qualitative hesitant fuzzy group decision making: An additively consistent probability and consensus-based perspective

Hesitant fuzzy linguistic preference relations (HFLPRs) can efficiently denote the hesitant qualitative judgments of decision makers. Consistency and consensus are two critical topics in group decision making (GDM) with preference relations. This paper uses the additively consistent concept for linguistic fuzzy preference relations (LFPRs) to give an additive consistency definition for HFLPRs. To judge the additive consistency of HFLPRs, 0-1 mixed programming models (0-1-MPMs) are constructed. Meanwhile, additive-consistency-based 0-1-MPMs to ascertain missing values in incomplete HFLPRs are established. Following the consistent probability of LFPRs, an algorithm to calculate the linguistic priority weighting vector is presented. In consideration of the consensus of GDM, a consistency-probability-distance-measure-based consensus index is defined, and an interactive improving consensus method is provided. Finally, a method for GDM with HFLPRs is offered that can address incomplete and inconsistent cases. Meanwhile, numerical examples are offered, and comparative analysis is made.     

基于概率不确定犹豫模糊偏好关系的决策模型

针对犹豫模糊信息在现实决策中难以准确和充分的提供决策者评价信息的问题,引入了概率不确定犹豫模糊偏好关系（PUHFPR）的概念,其能够有效处理概率不确定犹豫模糊元（PUHFE）中元素发生概率信息部分已知和完全未知的决策问题;给出了PUHFPR的期望加行一致性、满意加性期望一致性定义,并以偏差最小化为目标函数构建最优化模型确定PUHFPR元素的发生概率;建立基于一致性调整算法的概率不确定犹豫模糊偏决策模型,得到方案的排序权重向量,从而选择最佳的方案;通过遴选上市公司进行投资的实例说明决策模型的有效性。     

Deriving the priority weights from hesitant multiplicative preference relations in group decision making

In this paper, we investigate the deviation of the priority weights from hesitant multiplicative preference relations (HMPRs) in group decision-making environments. As basic elements of HMPRs, hesitant multiplicative elements (HMEs) usually have different numbers of possible values. To correctly compute or compare HMEs, there are two principles to normalize them, i.e., the α-normalization and the β-normalization. Based on the α-normalization, we develop a new goal programming model to derive the priority weights from HMPRs in group decision-making environments. Based on the β-normalization, a consistent HMPR and an acceptably consistent HMPR are defined, and their desired properties are studied. A convex combination method is then developed to obtain interval weights from an acceptably consistent HMPR. This approach is further extended to group decision-making situations in which the experts evaluate their preferences as several HMPRs. Finally, some numerical examples are provided to illustrate the validity and applicability of the proposed models.     

Prioritization of hesitant multiplicative preference relations based on data envelopment analysis for group decision making

Hesitant multiplicative preference relations (HMPRs) are utilized to describe situations where a decision maker gives several possible values by Saaty’s 1-9 scale in pairwise comparison. For further applications of HMPRs, this paper develops two priority methods based on data envelopment analysis (DEA) for group decision making. These methods include self-weight prioritization and the cross-weight prioritization, which are similar to the self-evaluation efficiency and the cross-evaluation efficiency in DEA theory, respectively. We prove that both of them can generate true priority weights for consistent HMPRs. The mechanisms of these proposed methods are illustrated with numerical examples. Also, comparisons with other methods are performed to show the advantages of the proposed methods.

     

Deriving priority weights from intuitionistic fuzzy multiplicative preference relations

Intuitionistic fuzzy multiplicative preference relations (IFMPRs), as an extension of multiplicative preference relations, can denote the decision-makers’ (DMs’) preferred and nonpreferred degrees simultaneously. Just as any other type of preference relations, consistency is crucial to guarantee the rational ranking orders. Thus, this paper introduces a new consistent concept for IFMPRs that is a natural extension of crisp case and overcomes the issues in the previous concepts of consistency. To judge the consistency of IFMPRs, several programming models are constructed, and an approach to deriving completely consistent IFMPRs is presented. Considering incomplete case, consistency-based models are built to determine missing values that can address incomplete IFMPRs with the ignored objects, namely, all information for them is unknown. After that, group decision-making with IFMPRs is studied. To measure the agreement degree between the DMs’ individual IFMPRs, a new consensus index is defined, and an interactive algorithm to improve the consensus is offered. Based on the consistency and consensus analysis, a new method to group decision-making with IFMPRs is developed. Finally, case studies are offered to show the application of the new procedure and to compare it with previous methods.     

An approach to group decision making with hesitant information and its application in credit risk evaluation of enterprises

Hesitant multiplicative preference relation (HMPR) contains much more comprehensive information than the traditional multiplicative preference relations. The HMPR is a useful tool to help the decision makers express their preferences in group decision making under uncertainty. The key of group decision making with the HMPR is to derive the priority weights from the HMPR. Thus, an efficient and practical priority method should be put forward so as to ensure the reasonability of the final decision result. In order to do that, in this paper, we first introduce the expected value and the geometric average value of hesitant multiplicative element (HME) which is the component of the HMPR. Then from different perspectives, we utilize the error-analysis technique to put forward three novel methods for the priorities of the HMPR, i.e., the expectation value method, the geometric average value method, and the multiplicative deviation method. We also investigate the relationships among these methods, and develop an approach to group decision making with the HMPR by using the methods and the possibility degree formula. Finally, by constructing the indicator system for credit risk evaluation of supply chain enterprises, we make a detailed case study concerning Lu-Zhou-Lao-Jiao (the well-known liquor enterprise in China) to demonstrate our approach.     

基于概率信息不完备的群决策模型

针对犹豫模糊元中元素发生的概率信息不完备的群决策问题,提出一种基于最优化模型和一致性调整算法的群决策模型。该模型首先引入了概率不完备犹豫模糊偏好关系（PIHFPR）、概率不完备犹豫模糊偏好关系的期望一致性以及概率不完备犹豫模糊偏好关系的满意加性期望一致性等概念;其次,以PIHFPR和排序权重向量间的偏差最小化作为目标函数,构建线性最优化模型计算得到PIHFPR中不完备的概率信息;随后,通过提出的加权概率不完备犹豫模糊偏好关系集成算子确定综合的PIHFPR,同时设计一种群体一致性调整算法,不仅使得调整后的PIHFPR具有满意加性期望一致性,还可以计算方案的排序权重。最后,将群决策模型应用于区块链的选择实例中。实验结果表明,决策结果合理可靠,且更能反映实际决策情况。     

A new method of obtaining the priority weights from an interval fuzzy preference relation

In analyzing a multiple criteria decision-making problem, the decision maker may express her/his opinions as an interval fuzzy or multiplicative preference relation. Then it is an interesting and important issue to investigate the consistency of the preference relations and obtain the reliable priority weights. In this paper, a new consistent interval fuzzy preference relation is defined, and the corresponding properties are derived. The transformation formulae between interval fuzzy and multiplicative preference relations are further given, which show that two preference relations, consistent interval fuzzy and multiplicative preference relations, can be transformed into each other. Based on the transformation formula, the definition of acceptably consistent interval fuzzy preference relation is given. Furthermore a new algorithm for obtaining the priority weights from consistent or inconsistent interval fuzzy preference relations is presented. Finally, three numerical examples are carried out to compare the results using the proposed method with those using other existing procedures. The numerical results show that the given procedure is feasible, effective and not requisite to solve any mathematical programing.     

Multi-criteria group decision making with incomplete hesitant fuzzy preference relations

In order to simulate the hesitancy and uncertainty associated with impression or vagueness, a decision maker may give her/his judgments by means of hesitant fuzzy preference relations in the process of decision making. The study of their consistency becomes a very important aspect to avoid a misleading solution. This paper defines the concept of additive consistent hesitant fuzzy preference relations. The characterizations of additive consistent hesitant fuzzy preference relations are studied in detail. Owing to the limitations of the experts’ professional knowledge and experience, the provided preferences in a hesitant fuzzy preference relation are usually incomplete. Consequently, this paper introduces the concepts of incomplete hesitant fuzzy preference relation, acceptable incomplete hesitant fuzzy preference relation, and additive consistent incomplete hesitant fuzzy preference relation. Then, two estimation procedures are developed to estimate the missing information in an expert's incomplete hesitant fuzzy preference relation. The first procedure is used to construct an additive consistent hesitant fuzzy preference relation from the lowest possible number, (n − 1), of pairwise comparisons. The second one is designed for the estimation of missing elements of the acceptable incomplete hesitant fuzzy preference relations with more known judgments. Moreover, an algorithm is given to solve the multi-criteria group decision making problem with incomplete hesitant fuzzy preference relations. Finally, a numerical example is provided to illustrate the solution processes of the developed algorithm and to verify its effectiveness and practicality.     

Error Analysis Methods for Group Decision Making Based on Hesitant Fuzzy Preference Relation

Hesitant fuzzy preference relation (HFPR) is an effective way to depict the decision makers’ preferences over the objects (alternatives or attributes) in the process of group decision making. Each component of the HFPR is characterized by several possible values and can express the decision makers’ hesitant information comprehensively. To make a decision with the HFPR, it is very necessary to find a proper technique for deriving the priority weights from the HFPR. In this paper, we use the error analysis as a tool to develop several straightforward methods for the priorities of the HFPR. We first define the expected value and the average value of each hesitant fuzzy element in the HFPR. Then based on the error analysis, we come up with the interval midpoint method, the average value method, and the difference method to derive the priority weights from the HFPR. After that, we discuss the relations among these methods, and utilize them and the possibility degree formula to develop an approach to decision making with the HFPR. Finally, we demonstrate the effectiveness and practicality of our approach through a case study concerning the investment problem in liquor enterprise.     

基于三角犹豫Einstein算法的多属性决策模型

针对属性信息为三角犹豫模糊信息的多属性决策问题,结合Einstein运算,构建了一种基于三角犹豫模糊Einstein集成算法的多属性决策方法。首先,考虑到决策信息为三角犹豫模糊数且属性间存在一定的内在联系,基于三角犹豫模糊数的运算法则,提出了三角犹豫模糊Einstein加权平均（THFEWA）算子和三角犹豫模糊Einstein加权几何（THFEWG）算子;其次,针对三角犹豫模糊元的有序位置存在具有不同权重的情况,构建了三角犹豫模糊Einstein有序加权平均（THFEOWA）算子和三角犹豫模糊Einstein有序加权几何（THFEOWG）算子,并讨论了它们相应的基本性质;最后建立了基于THFEOWA算子和THFEOWG算子的多属性决策模型,并通过实例说明提出的决策模型是合理和有效的。     

Interval multiplicative transitivity for consistency, missing values and priority weights of interval fuzzy preference relations

In this paper, the concept of multiplicative transitivity of a fuzzy preference relation, as defined by Tanino [T. Tanino, Fuzzy preference orderings in group decision-making, Fuzzy Sets and Systems 12 (1984) 117-131], is extended to discover whether an interval fuzzy preference relation is consistent or not, and to derive the priority vector of a consistent interval fuzzy preference relation. We achieve this by introducing the concept of interval multiplicative transitivity of an interval fuzzy preference relation and show that, by solving numerical examples, the test of consistency and the weights derived by the simple formulas based on the interval multiplicative transitivity produce the same results as those of linear programming models proposed by Xu and Chen [Z.S. Xu, J. Chen, Some models for deriving the priority weights from interval fuzzy preference relations, European Journal of Operational Research 184 (2008) 266-280]. In addition, by taking advantage of interval multiplicative transitivity of an interval fuzzy preference relation, we put forward two approaches to estimate missing value(s) of an incomplete interval fuzzy preference relation, and present numerical examples to illustrate these two approaches.     

犹豫模糊Einstein几何算子及应用

研究了属性权重信息已知条件下的犹豫模糊信息集结算子及其在多属性群决策问题中的应用。基于Einstein运算定义了犹豫模糊Einstein和、犹豫模糊Einstein积以及犹豫模糊Einstein幂运算,并且研究了犹豫模糊Einstein运算法则间的关系。提出了四种犹豫模糊信息集结算子,即犹豫模糊Einstein加权几何（HFEWG）算子、犹豫模糊Einstein有序加权几何（HFEOWG）算子、犹豫模糊Einstein混合几何（HFEHG）算子和犹豫模糊Einstein诱导有序加权几何（HFEIOWG）算子,并分析了这些算子的性质。给出了基于HFEIOWG算子的犹豫模糊多属性决策方法,并结合投资公司对金融产品的选择来验证提出的决策方法是可行有效的。     

Group Decision Making With Consistency of Intuitionistic Fuzzy Preference Relations Under Uncertainty

Intuitionistic fuzzy preference relation (IFPR) is a suitable technique to express fuzzy preference information by decision makers (DMs). This paper aims to provide a group decision making method where DMs use the IFPRs to indicate their preferences with uncertain weights. To begin with, a model to derive weight vectors of alternatives from IFPRs based on multiplicative consistency is presented. Specifically, for any IFPR, by minimizing its absolute deviation from the corresponding consistent IFPR, the weight vectors are generated. Secondly, a method to determine relative weights of DMs depending on preference information is developed. After that we prioritize alternatives based on the obtained weights considering the risk preference of DMs. Finally, this approach is applied to the problem of technical risks assessment of armored equipment to illustrate the applicability and superiority of the proposed method.      

Approaches to decision making with linguistic preference relations based on additive consistency

The linguistic preference relation (LPR) is introduced to efficiently deal with situations in which the decision makers (DMs) provide their preference information by using linguistic labels over paired comparisons of alternatives. However, the lack of consistency in decision making with LPRs can lead to inconsistent conclusions. In this paper, two new decision making methods are developed to improve the additive consistency of LPRs until they are acceptable, and eventually obtain the reliable decision making results. First, the new concepts of order consistency and additive consistency of LPRs are introduced, and followed by a discussion of the characterization about additive consistent LPRs. Then, a consistency index is defined to measure whether an LPR is of acceptable additive consistency. For an unacceptable additive consistent LPR, two automatic iterative algorithms are further proposed to help DMs improve additive consistency level until it is acceptable. In addition, the proposed algorithms can derive the priority weight vector from LPRs and obtain the ranking of the alternatives. Finally, the proposed methods are applied to an emergency operating center (EOC) selection problem. The comparative analysis demonstrates the applicability and effectiveness of the proposed methods.     

基于加权犹豫Frank几何算法的决策模型

加权犹豫模糊集是一种广义的犹豫模糊集,其可以更准确和全面地刻画决策信息。而Frank三角模运算能够挖掘多个输入参数值间的相互关系。基于Frank三角模思想,在加权犹豫模糊环境下,提出了一种加权犹豫Frank几何平均算法的群决策模型。首先,运用Frank三角模定义了加权犹豫模糊基本运算法则,并构建了新的得分函数;接着,提出了加权犹豫Frank几何平均（WHFGA）算子,分析了WHFGA算子关于参数[r]的相关性质;最后,基于提出的WHFGA算子,建立了加权犹豫模糊多属性决策模型,并通过算例进行分析。实验结果表明,WHFGA算子具有良好的内在一致性。     

基于犹豫模糊群决策模型的云计算服务商选择

针对输入信息有内在联系的犹豫模糊群决策问题,构建了一种基于犹豫模糊Maclaurin对称平均（HFMSM）算法的多属性群决策模型,该模型可以依据决策者偏好而选择合适参数值进行决策。利用Archimedean范数定义新的犹豫模糊运算;结合Maclaurin对称平均提出了HFMSM算子,并详细探讨了HFMSM算子具有的四种优良性质,分析了HFMSM算子的几种特殊情况;建立了一种基于HFMSM算子的犹豫模糊多属性群决策模型,并将模型应用于云计算服务商的实际选择决策过程。实验结果表明,模型的可靠性更优,具有更广的应用前景。     

基于犹豫语言H-几何算子的信息安全系统选择

针对属性值为犹豫模糊语言信息且属性输入变量之间存在相互联系的多属性群决策问题,提出了一种基于犹豫语言Heronian几何算子的多属性群决策模型。基于Archimedean范数定义了新的犹豫模糊语言运算法则,并提出了犹豫语言Heronian几何（HLHG）算子;探讨了HLHG算子的一些优良性质,研究了HLHG算子的几种常见形式,引入了犹豫语言Heronian加权几何（HLHWG）算子;基于HLHWG算子构建了一种新的犹豫模糊语言多属性群决策模型,该群决策模型不仅考虑到了输入决策信息之间的相互联系,而且使得决策者能够依据自身偏好选择不同参数进行决策。结合信息安全系统选择实例验证了提出的群决策模型是可行的和有效的。     

基于犹豫语言判断矩阵的数据产品选择研究

犹豫语言判断矩阵作为一种新的判断矩阵,能够有效地处理决策信息为语言变量且决策者态度犹豫不行的决策问题。针对犹豫模糊语言信息环境下的数据产品选择问题,构建了一种基于犹豫语言判断矩阵的数据产品选择方法。该方法引入了犹豫语言判断矩阵的一些相关概念,包括加性一致性、一致性指数、可接受一致性;研究了犹豫语言判断矩阵一致性判定方法和特征矩阵的构造方法,并设计了一种收敛性算法用以改进犹豫语言判断矩阵的一致性;建立了基于犹豫语言判断矩阵的决策模型,并通过数据产品的选择实例说明提出的决策方法是合理和有效的。     

An approach to group decision making with heterogeneous incomplete uncertain preference relations

For practical group decision making problems, decision makers tend to provide heterogeneous uncertain preference relations due to the uncertainty of the decision environment and the difference of cultures and education backgrounds. Sometimes, decision makers may not have an in-depth knowledge of the problem to be solved and provide incomplete preference relations. In this paper, we focus on group decision making (GDM) problems with heterogeneous incomplete uncertain preference relations, including uncertain multiplicative preference relations, uncertain fuzzy preference relations, uncertain linguistic preference relations and intuitionistic fuzzy preference relations. To deal with such GDM problems, a decision analysis method is proposed. Based on the multiplicative consistency of uncertain preference relations, a bi-objective optimization model which aims to maximize both the group consensus and the individual consistency of each decision maker is established. By solving the optimization model, the priority weights of alternatives can be obtained. Finally, some illustrative examples are used to show the feasibility and effectiveness of the proposed method.