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.     

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.     

Two flexibility degrees-driven consensus model in group decision making with intuitionistic fuzzy preference relations

A group of experts are commonly invited to find an optimal solution to a complex decision making problem. When the bipolarity of decision information should be considered in group decision making (GDM), intuitionistic fuzzy values (IFVs) have the capability to model such opinions of decision makers (DMs). This paper develops a consensus model in GDM under intuitionistic fuzzy environments with flexibility. First, it is assumed that the initial opinions of DMs are expressed as intuitionistic fuzzy preference relations (IFPRs). A novel additive consistency index is constructed to measure the deviation degree of IFPRs from fuzzy preference relations (FPRs) with additive consistency, where the non-determinacy degree of IFPRs is incorporated. The thresholds of the proposed index corresponding to IFPRs with acceptable additive consistency are discussed and computed. Second, the consensus level of DMs is defined using the similarity degree between two IFVs. An optimization problem is established by maximizing the fitness function, which is constructed by linearly combining the proposed additive consistency index and consensus level. Two flexibility degrees are offered to each DM such that the initial opinions with the bipolarity can be adjusted correspondingly. Third, individual IFPRs in GDM are optimized using the particle swarm optimization (PSO) algorithm. Numerical examples are carried out to illustrate the proposed consensus model by comparing with the existing ones. The obtained results reveal that the proposed additive consistency index can reflect the inherent property of IFPRs. Different with the previous studies, two original flexibility degrees are proposed to characterize the multi-granularity of decision information in GDM.     

Consensus reaching models of linguistic preference relations based on distance functions

In group decision making (GDM) using linguistic preference relations to obtain the maximum degree of agreement, it is desirable to develop a consensus process prior to the selection process. This paper proposes two consensus models with linguistic information to support the GDM consensus reaching process. Two different distance functions between linguistic preference relations are introduced to measure both individual consistency and group consensus. Based on these measures, the consensus reaching models are developed. The two models presented have the same concept that the expert whose preference is farthest from the group preference needs to update their opinion according to the group preference relation. In addition, the convergence of the models is proved. After achieving the predefined consensus level, each expert’s consistency indexes are still acceptable under the condition that the initial preference relations are of satisfactory consistency. Finally, an example is given to show the effectiveness of the models and to verify the theoretical results.     

Deriving the personalized individual semantics of linguistic information from flexible linguistic preference relations

In group decision making, flexible linguistic preference relations (FLPRs) are very useful with the pairwise comparisons taking the form of flexible linguistic expressions (FLEs). Due to the fact that different decision makers have different understandings of words, this paper investigates the personalized individual semantics (PISs) of the linguistic information in FLPRs. Two optimization models are constructed to compute a linguistic distribution which is closest to an incomplete FLE. The FLPRs are transformed into fuzzy preference relations by using optimization models which maximize consistency and consensus of the fuzzy preference relations. The PISs of linguistic terms and subsets of the linguistic term set are obtained in this process. A group decision making model based on FLPRs is presented and a green supplier selection problem in automotive industry is solved by using the proposed model. The comparative analysis is presented to show the feasibility of the group decision making model.     

Adaptive consensus model with multiplicative linguistic preferences based on fuzzy information granulation

An adaptive consensus model based on fuzzy information granulation (fuzzy IG) is presented for group consensus decision-making problems with multiplicative linguistic preference relations (MLPRs). Firstly, a granular representation of linguistic terms is concerned with the triangular fuzzy formation of a family of information granules over given Analytical Hierarchy Process (AHP) numerical scales. On this basis, the individual consistency and group consensus measure indices using fuzzy granulation technique are constructed, respectively. Then, the optimal cut-off points of fuzzy information granules are obtained by establishing a multi-objective optimization model together with a multi-objective particle swarm optimization (MOPSO) algorithm. A novel group consensus decision-making approach where consensus reaching process (CRP) is achieved by adaptively adjusting individual preferences through the optimization of the cut-off points is proposed. After conflict elimination, the obtained group preference gives the ranking of the alternatives. Finally, a real emergency decision-making case for liquid ammonia leak is given to illustrate the application steps of the proposed method and comparative analysis with the existing GDM methods. Comparative results demonstrate that the proposed method has some advantages in aspects of avoiding information loss or distortion and improving consensus performance.     

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.     

Consistency and consensus improvement models driven by a personalized normalization method with probabilistic linguistic preference relations

Probabilistic linguistic preference relation (PLPR) provides an effective and flexible tool with which preference degrees of decision-makers can be captured when they vacillatingly express linguistic preference values among several linguistic terms. Individual consistency and group consensus are two important research topics of PLPRs in group decision making (GDM). Considering the problems associated with these two topics, this study proposes a novel GDM framework with consistency-driven and consensus-driven optimization models based on a personalized normalization method for managing complete and incomplete PLPRs. First, existing limitations of the traditional normalization method for probabilistic linguistic term sets (PLTSs) managing ignorance information are specifically discussed. Given the potential valuable information hidden in PLTSs, a personalized normalization method is newly proposed through a two-stage decision-making process with a comprehensive fusion mechanism. Then, based on the proposed normalization method for PLTSs, consistency-driven optimization models that aim to minimize the overall adjustment amount of a PLPR are constructed to improve consistency. Moreover, the developed models are extended to improve consistency and estimate the missing values of an incomplete PLPR. Subsequently, a consensus-driven optimization model that aims to maximize group consensus by adjusting experts’ weights is constructed to support the consensus-reaching process. Finally, an illustrative example, followed by some comparative analyses is presented to demonstrate the application and advantages of the proposed approach.     

A Consensus Model for Group Decision Making With Incomplete Fuzzy Preference Relations

Two processes are necessary to solve group decision making problems: A consensus process and a selection process. The consensus reaching process is necessary to obtain a final solution with a certain level of agreement between the experts; and the selection process is necessary to obtain such a final solution. In a previous paper, we present a selection process to deal with group decision making problems with incomplete fuzzy preference relations, which uses consistency measures to estimate the incomplete fuzzy preference relations. In this paper we present a consensus model. The main novelty of this consensus model is that of being guided by both consensus and consistency measures. Also, the consensus reaching process is guided automatically, without moderator, through both consensus and consistency criteria. To do that, a feedback mechanism is developed to generate advice on how experts should change or complete their preferences in order to reach a solution with high consensus and consistency degrees. In each consensus round, experts are given information on how to change their preferences, and to estimate missing values if their corresponding preference relation is incomplete. Additionally, a consensus and consistency based induced ordered weighted averaging operator to aggregate the experts' preferences is introduced, which can be used in consensus models as well as in selection processes. The main improvement of this consensus model is that it supports the management of incomplete information and it allows to achieve consistent solutions with a great level of agreement.     

Compatibility measures and consensus models for group decision making with intuitionistic multiplicative preference relations

Compatibility is a very efficient tool for measuring the consensus level in group decision making (GDM) problems. The lack of acceptable compatibility can lead to unsatisfied or even incorrect results in GDM problems. Preference relations can be given in various forms, one of which called intuitionistic multiplicative preference relation is a new developed preference structure that uses an unsymmetrical scale (Saaty's 1–9 scale) to express the decision maker's preferences instead of the symmetrical scale in an intuitionistic fuzzy preference relation. This new preference relation can reflect our intuition more objectively. In this paper, we first develop some compatibility measures for intuitionistic multiplicative values and intuitionistic multiplicative preference relations in GDM. Their desirable properties are also studied in detail. Furthermore, based on compatibility measures, we further develop two different consensus models with respect to intuitionistic multiplicative preference relations for checking, reaching and improving the group consensus level. Finally, a numerical example is given to illustrate the effectiveness of our measures and models.     

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.     

Analysis of self-confidence indices-based additive consistency for fuzzy preference relations with self-confidence and its application in group decision making

Preference relations have been widely used in group decision-making (GDM) problems. Recently, a new kind of preference relations called fuzzy preference relations with self-confidence (FPRs-SC) has been introduced, which allow experts to express multiple self-confidence levels when providing their preferences. This paper focuses on the analysis of additive consistency for FPRs-SC and its application in GDM problems. To do that, some operational laws for FPRs-SC are proposed. Subsequently, an additive consistency index that considers both the fuzzy preference values and self-confidence is presented to measure the consistency level of an FPR-SC. Moreover, an iterative algorithm that adjusts both the fuzzy preference values and self-confidence levels is proposed to repair the inconsistency of FPRs-SC. When an acceptable additive consistency level for FPRs-SC is achieved, the collective FPR-SC can be computed. We aggregate the individual FPRs-SC using a self-confidence indices-based induced ordered weighted averaging operator. The inherent rule for aggregation is to give more importance to the most self-confident experts. In addition, a self-confidence score function for FPRs-SC is designed to obtain the best alternative in GDM with FPRs-SC. Finally, the feasibility and validity of the research are demonstrated with an illustrative example and some comparative analyses.     

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.     

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.     

A web based consensus support system for group decision making problems and incomplete preferences

Reaching a high level of consensus among experts is critical in group decision making problems. Usually, it is the moderator task to assure that the consensus process is carried out properly and, if possible, to offer recommendations to the expert in order to change their opinions and narrow their differences.In this paper we present an implemented web based consensus support system that is able to help, or even replace, the moderator in a consensus process where experts are allowed to provide their preferences using one of many types (fuzzy, linguistic and multi-granular linguistic) of incomplete preference relations.This system is based on both consistency and consensus measures and it has been designed to provide advice to the experts to increase group consensus level while maintaining the individual consistency of each expert. The consistency measures are characterized by and computed using uninorm operators. When appropriate, the system also helps experts to reduce the incompleteness of their preference relations. The web interface allows to carry out distributed consensus processes and thus, experts do not necessarily need to physically meet together.     

A decision support model for group decision making with intuitionistic fuzzy linguistic preferences relations

As a new preference structure, the intuitionistic fuzzy linguistic preference relation (IFLPR) was introduced to efficiently cope with situations in which the membership degree and non-membership degree are represented as linguistic terms. For group decision making (GDM) problems with IFLPRs, two significant and challenging issues are individual consistency and group consensus before deriving the reliable priority weights of alternatives. In this paper, a novel decision support model is investigated to simultaneously deal with the individual consistency and group consensus for GDM with IFLPRs. First, the concepts of multiplicative consistency and weak transitivity for IFLPRs are introduced and followed by a discussion of their desirable properties. Then, a transformation approach is developed to convert the normalized intuitionistic fuzzy priority weights into multiplicative consistent IFLPR. Based on the distance of IFLPRs, the consistency index, individual consensus degree and group consensus degree for IFLPRs are further defined. In addition, two convergent automatic iterative algorithms are proposed in the investigated decision support model. The first algorithm is utilized to convert an unacceptable multiplicative consistent IFLPR to an acceptable one. The second algorithm can assist the group decision makers to achieve a predefined consensus level. The main characteristic of the investigated decision support model is that it guarantees each IFLPR is still acceptable multiplicative consistent when the predefined consensus level is achieved. Finally, several numerical examples are provided, and comparative analyses with existing approaches are performed to demonstrate the effectiveness and practicality of the investigated model.

     

基于群体共识算法的深度学习推荐系统评价

针对群决策过程中专家群体共识水平不高的问题,构建一种基于调整成本最小化的群决策算法。该算法在三个层面定义一致性-共识性测度,用于衡量模糊判断矩阵的一致性和共识性水平;建立基于最小调整成本的共识调整反馈机制来识别需要调整的决策者和偏好值;建立以共识调整总成本最小为目标函数的最优化模型计算每个决策者的最优调整参数和共识阈值上界;设计基于调整成本最小化的群决策算法,并验证其收敛性。通过深度学习推荐系统的优选实验表明,构建的群决策算法在成本和效率方面更有效。     

A consistency and consensus based decision support model for group decision making with multiplicative preference relations

In group decision making (GDM) with multiplicative preference relations (also known as pairwise comparison matrices in the Analytical Hierarchy Process), to come to a meaningful and reliable solution, it is preferable to consider individual consistency and group consensus in the decision process. This paper provides a decision support model to aid the group consensus process while keeping an acceptable individual consistency for each decision maker. The concept of an individual consistency index and a group consensus index is introduced based on the Hadamard product of two matrices. Two algorithms are presented in the designed support model. The first algorithm is utilized to convert an unacceptable preference relation to an acceptable one. The second algorithm is designed to assist the group in achieving a predefined consensus level. The main characteristics of our model are that: (1) it is independent of the prioritization method used in the consensus process; (2) it ensures that each individual multiplicative preference relation is of acceptable consistency when the predefined consensus level is achieved. Finally, some numerical examples are given to verify the effectiveness of our model.     

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.     

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.