Incomplete interval fuzzy preference relations and their applications

This paper investigates incomplete interval fuzzy preference relations. A characterization, which is proposed by Herrera-Viedma et al. (2004), of the additive consistency property of the fuzzy preference relations is extended to a more general case. This property is further generalized to interval fuzzy preference relations (IFPRs) based on additive transitivity. Subsequently, we examine how to characterize IFPR. Using these new characterizations, we propose a method to construct an additive consistent IFPR from a set of n − 1 preference data and an estimation algorithm for acceptable incomplete IFPRs with more known elements. Numerical examples are provided to illustrate the effectiveness and practicality of the solution process.     

Incomplete interval-valued intuitionistic fuzzy preference relations

The aim of this paper is to investigate decision making problems with interval-valued intuitionistic fuzzy preference information, in which the preferences provided by the decision maker over alternatives are incomplete or uncertain. We define some new preference relations, including additive consistent incomplete interval-valued intuitionistic fuzzy preference relation, multiplicative consistent incomplete interval-valued intuitionistic fuzzy preference relation and acceptable incomplete interval-valued intuitionistic fuzzy preference relation. Based on the arithmetic average and the geometric mean, respectively, we give two procedures for extending the acceptable incomplete interval-valued intuitionistic fuzzy preference relations to the complete interval-valued intuitionistic fuzzy preference relations. Then, by using the interval-valued intuitionistic fuzzy averaging operator or the interval-valued intuitionistic fuzzy geometric operator, an approach is given to decision making based on the incomplete interval-valued intuitionistic fuzzy preference relation, and the developed approach is applied to a practical problem. It is worth pointing out that if the interval-valued intuitionistic fuzzy preference relation is reduced to the real-valued intuitionistic fuzzy preference relation, then all the above results are also reduced to the counterparts, which can be applied to solve the decision making problems with incomplete intuitionistic fuzzy preference information.     

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.     

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。     

加型Pythagorean模糊偏好关系的多属性决策方法

考虑Pythagorean模糊偏好关系的多属性决策问题,提出了加性Pythagorean模糊偏好关系的多属性决策方法。基于加性一致性Pythagorean模糊偏好关系提出一种新的Pythagorean模糊权重确定模型。给出了可接受加性一致性Pythagorean模糊偏好关系的定义,并针对不满足可接受加性一致性的Pythagorean模糊偏好关系,提出一种加性一致性调整算法。给出基于Pythagorean模糊偏好关系加性一致性的多属性决策方法,并通过实例分析提出的新方法的可行性和合理性。     

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.     

Multiplicative consistency-based decision support system for incomplete linguistic preference relations

The experts may have difficulty in expressing all their preferences over alternatives or criteria, and produce the incomplete linguistic preference relation. Consistency plays an important role in estimating unknown values from an incomplete linguistic preference relation. Many methods have been developed to obtain a complete linguistic preference relation based on additive consistency, but some unreasonable values may be produced in the estimation process. To overcome this issue, we propose a new characterisation about multiplicative consistency of the linguistic preference relation, present an algorithm to estimate missing values from an incomplete linguistic preference relation, and establish a decision support system for aiding the experts to complete their linguistic preference relations in a more consistent way. Some examples are also given to illustrate the proposed methods.     

A new procedure for hesitant multiplicative preference relations

Hesitant information is powerful and flexible to denote decision maker's judgments. Hesitant multiplicative preference relations (HMPRs) own the advantages of preference relations and hesitant fuzzy sets that permit the decision makers (DMs) to compare objects by using several values. Just as other types of preference relations, how to derive the priority weight vector is a crucial step. According to the principle of the consistency concept for multiplicative preference relations, this paper first introduces a new consistency concept for HMPRs, which avoids the disadvantages of the previous ones. Using the new concept, models to judge the consistency of HMPRs are built. Then, a consistency probability-based method to derive the hesitant fuzzy priority weight vector from HMPRs is offered. Considering the incomplete case, consistency-based programming models to determine the missing values are constructed. To address group decision making with HMPRs, a distance measure is defined to determine the weights of the DMs, and a consensus index is proposed. Then, a consistency and consensus-based group decision-making algorithm is performed. Finally, two practical examples, an investment problem and a water conservancy problem are offered to illustrate the feasibility and efficiency of the new algorithm. Comparison analysis from the numerical and theoretical aspects verifies the potential application of the new procedure.     

Group decision making with incomplete fuzzy linguistic preference relations

The aim of this paper is to propose a procedure to estimate missing preference values when dealing with incomplete fuzzy linguistic preference relations assessed using a two‐tuple fuzzy linguistic approach. This procedure attempts to estimate the missing information in an individual incomplete fuzzy linguistic preference relation using only the preference values provided by the respective expert. It is guided by the additive consistency property to maintain experts' consistency levels. Additionally, we present a selection process of alternatives in group decision making with incomplete fuzzy linguistic preference relations and analyze the use of our estimation procedure in the decision process. © 2008 Wiley Periodicals, Inc.     

基于粒计算的犹豫模糊多准则决策方法*

提出基于粒计算的犹豫模糊多准则决策方法.给出各个准则下对应的犹豫模糊集中犹豫模糊元的大于可能度定义,并构造相应准则下的加性一致的模糊偏好矩阵.根据各准则的模糊偏好矩阵对应的预序熵及预序粒结构相似度确定属性的权重,对各个准则下模糊偏好矩阵的排序向量加权平均得到最终的排序向量.文中方法以评价数据序信息量及准则序与整体之间的关系确定准则权重,通过计算加权两两比较下的排序向量得到最终的排序决策结果.最后运用实例验证算法的有效性及可行性.     

A group decision-making model based on incomplete comparative expressions with hesitant linguistic terms

As a result of uncertainty and complexity for environments of decision-making, it is more suitable for decision makers to use hesitant fuzzy linguistic information. In this paper, a novel group decision making (GDM) model based on fuzzy linear programming is proposed for incomplete comparative expressions with hesitant fuzzy linguistic term set (HFLTSs). We establish an equivalence theorem of additive consistency between 2-tuple fuzzy linguistic preference relation (FLPR) and corresponding fuzzy preference relation. Based on this framework, a fuzzy linear programming is established to address incomplete comparative expressions with HFLTSs. It is more important that the proposed fuzzy linear programming has a double action, finding the highest consistent incomplete 2-tuple FLPR and increasing inconsistent 2-tuple FLPR to the additive consistent 2-tuple FLPR based on given incomplete comparative expressions with HFLTSs. By this means, a novel GDM model is constructed based on importance induced ordered weighted averaging operator. Finally, an investment decision-making in real-world is solved by the proposed model, which shows the result of GDM is effectiveness.     

Group decision making based on incomplete multiplicative and fuzzy preference relations

The main aim of this paper is to investigate the group decision making on incomplete multiplicative and fuzzy preference relations without the requirement of satisfying reciprocity property. This paper introduces a new characterization of the multiplicative consistency condition, based on which a method to estimate unknown preference values in an incomplete multiplicative preference relation is proposed. Apart from the multiplicative consistency property among three known preference values, the method proposed also takes the multiplicative consistency property among more than three values into account. In addition, two models for group decision making with incomplete multiplicative preference relations and incomplete fuzzy preference relations are presented, respectively. Some properties of the collective preference relation are further discussed. Numerical examples are provided to make a discussion and comparison with other similar methods.     

Linguistic intuitionistic fuzzy preference relations and their application to multi-criteria decision making

Linguistic intuitionistic fuzzy sets can be regarded as a qualitative form of intuitionistic fuzzy sets. This type of fuzzy sets uses a linguistic membership degree and a linguistic non-membership degree to represent the qualitative preferred and non-preferred judgments of decision makers. Preference relation is a useful and efficient tool for decision making that only requires the decision makers to compare two objects at one time. Taking the advantages of linguistic intuitionistic fuzzy sets and preference relations, this paper introduces linguistic intuitionistic fuzzy preference relations (LIFPRs) and studies their application to decision making. To ensure the ranking of objects reasonably, an additive consistency concept is introduced, and several of its desirable properties are discussed. To cope with inconsistent and incomplete LIFPRs, programming model-based methods to derive additively consistent LIFPRs and determine missing values are constructed, respectively. Subsequently, an approach to multi-criteria decision making with LIFPRs is offered, and the application of the new approach is illustrated by using a decision-making problem about evaluating mobile phones.     

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.     

A consensus model for hesitant fuzzy preference relations and its application in water allocation management

This paper investigates a consensus model for hesitant fuzzy preference relations (HFPRs). First, we present a revised definition of HFPRs, in which the values are not ordered for the hesitant fuzzy element. Second, we propose an additive consistency based estimation measure to normalize the HFPRs, based on which, a consensus model is developed. Here, two feedback mechanisms are proposed, namely, interactive mechanism and automatic mechanism, to obtain a solution with desired consistency and consensus levels. In the interactive mechanism, the experts are suggested to give their new preference values in a specific range. If the experts are unwilling to offer their updated preferences, the automatic mechanism could be adopted to carry out the consensus process. Induced ordered weighted averaging (IOWA) operator is used to aggregate the individual HFPRs into a collective one. A score HFPR is proposed for collective HFPR, and then the quantifier-guided dominance degrees of alternatives by using an OWA operator are obtained to rank the alternatives. Finally, both a case of study for water allocation management in Jiangxi Province of China and a comparison with the existing approaches are carried out to show the advantages of the proposed method.     

Interval programming method for hesitant fuzzy multi-attribute group decision making with incomplete preference over alternatives

The aim of this study is to employ the main structure of LINMAP (LINear programming technique for Multidimensional Analysis of Preference) to propose an interval programming method for solving multi-attribute group decision making (MAGDM) problems in which the ratings of alternatives are taken as hesitant fuzzy elements (HFEs) and all pair-wise comparison judgments over alternatives are represented by interval numbers. The contribution of this study is fivefold: (1) we define the new consistency and inconsistency indices; (2) we construct an interval programming model to determine the hesitant fuzzy positive ideal solution and the optimal weights of attributes, and at the same time present a decision algorithm; (3) we discuss several special cases of the proposed model in detail; (4) we show that compared with intuitionistic fuzzy LINMAP method (Li et al., 2010), the proposed approach reveals more useful information including the interval preference information, and does not need to transform HFEs into intuitionistic fuzzy numbers but directly deals with MAGDM problems and thus obtains better final decision results; and (5) we demonstrate the applicability and implementation process of the proposed approach by using an energy project selection example.     

Group consensus algorithms based on preference relations

In many group decision-making situations, decision makers’ preferences for alternatives are expressed in preference relations (including fuzzy preference relations and multiplicative preference relations). An important step in the process of aggregating preference relations, is to determine the importance weight of each preference relation. In this paper, we develop a number of goal programming models and quadratic programming models based on the idea of maximizing group consensus. Our models can be used to derive the importance weights of fuzzy preference relations and multiplicative preference relations. We further develop iterative algorithms for reaching acceptable levels of consensus in group decision making based on fuzzy preference relations or multiplicative preference relations. Finally, we include an illustrative example.     

On group decision making with four formats of incomplete preference relations

In this paper, a new approach is proposed to solve group decision making (GDM) problems where the preference information on alternatives provided by decision makers (DMs) is represented in four formats of incomplete preference relations, i.e., incomplete multiplicative preference relations, incomplete fuzzy preference relations, incomplete additive linguistic preference relations, incomplete multiplicative linguistic preference relations. In order to make the collective opinion close each decision maker’s opinion as near as possible, an optimization model is constructed to integrate the four different formats of incomplete preference relations and to compute the collective ranking values of the alternatives. The ranking of alternatives or selection of the most desirable alternative(s) is directly obtained from the derived collective ranking values. A numerical example is also used to illustrate the applicability of the proposed approach.     

Estimating incomplete information in group decision making: A framework of granular computing

A general assumption in group decision making scenarios is that of all individuals possess accurate knowledge of the entire problem under study, including the abilities to make a distinction of the degree up to which an alternative is better than other one. However, in many real world scenarios, this may be unrealistic, particularly those involving numerous individuals and options to choose from conflicting and dynamics information sources. To manage such a situation, estimation methods of incomplete information, which use own assessments provided by the individuals and consistency criteria to avoid discrepancy, have been widely employed under fuzzy preference relations. In this study, we introduce the information granularity concept to estimate missing values supporting the objective of obtaining complete fuzzy preference relations with higher consistency levels. We use the concept of granular preference relations to form each missing value as a granule of information in place of a crisp number. This offers the flexibility that is required to estimate the missing information so that the consistency levels related to the complete fuzzy preference relations are as higher as possible.