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.     

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.     

Generalized intuitionistic fuzzy geometric aggregation operator and its application to multi-criteria group decision making

In general, for multi-criteria group decision making problem, there exist inter-dependent or interactive phenomena among criteria or preference of experts, so that it is not suitable for us to aggregate them by conventional aggregation operators based on additive measures. In this paper, based on fuzzy measures a generalized intuitionistic fuzzy geometric aggregation operator is investigated for multiple criteria group decision making. First, some operational laws on intuitionistic fuzzy values are introduced. Then, a generalized intuitionistic fuzzy ordered geometric averaging (GIFOGA) operator is proposed. Moreover, some of its properties are given in detail. It is shown that GIFOGA operator can be represented by special t-norms and t-conorms and is a generalization of intuitionistic fuzzy ordered weighted geometric averaging operator. Further, an approach to multiple criteria group decision making with intuitionistic fuzzy information is developed where what criteria and preference of experts often have inter-dependent or interactive phenomena among criteria or preference of experts is taken into account. Finally, a practical example is provided to illustrate the developed approaches.     

Linguistic labels for expressing fuzzy preference relations infuzzy group decision making

This paper proposes improvements to pairwise group decision making based on fuzzy preference relations in three ways. First, it extends the fuzzy preference relation representation using linguistic labels. Decision makers may express their preference relations in linguistic labels which are more practically implementable for solving group decision making problems. Second, it modifies the computational procedures by using fuzzy sets representation and computation, and by avoiding the use of strict threshold values. This allows natural representation, preserves the preference accuracy, and produces more intuitively meaningful solutions. Finally, it considers fuzzy criteria of the alternatives explicitly. Solutions are first derived based on each criterion, and then by using neat ordered weighted average (OWA) operator the final solutions which accommodate all criteria are determined. The proposed method is verified for solving fuzzy group decision making problems, i.e., advertising media selection cases.     

基于信任关系和互补判断矩阵的群决策方法

随着信息和网络技术的不断发展,基于社会网络的群决策问题受到越来越多研究者的关注.针对社会网络环境下模糊互补判断矩阵的群决策问题,研究群体共识调整过程和方案选择方法.首先,融合决策者之间的社会关系、身份地位、知识能力3个方面信息来构建决策者两两之间的信任关系;其次,提出一种尽可能减少元素间共识补偿的共识度度量方法,在此基础上建立基于信任关系的共识调整模型,并从理论上证明该模型的有效性;最后通过信任关系矩阵的特征向量中心度分别求出专家的重要性权重,用以集结专家的偏好信息和对方案进行排序选择,算例分析表明了所提出方法的有效性.     

A fuzzy group decision making model with trapezoidal fuzzy preference relations based on compatibility measure and COWGA operator

This paper proposes a fuzzy group decision-making model based on a logarithm compatibility measure with multiplicative trapezoidal fuzzy preference relations (MTFPRs) based on a continuous ordered weighted geometric averaging (COWGA) operator. New concepts are presented to measure deviation between MTFPR and its expected fuzzy preference relation. Then, an iterative algorithm is developed to help individual MTFPR reach acceptable compatibility. To determine the weights of decision makers, an optimal model is constructed using group logarithm compatibility index COWGA operator. Finally, we illustrate an example to show how it works and compare it with the existing methods. The main advantages of the proposed approach are the following: (1) The COWGA operator makes decision making more flexible; (2) an iterative and convergent algorithm is proposed to improve the compatibility of MTFPR; (3) decision makers’ weights in group decision making are determined by an optimal model based on a logarithm compatibility measure.     

Group decision making based on multiple types of linguistic preference relations

In this paper, we investigate group decision making problems with multiple types of linguistic preference relations. The paper has two parts with similar structures. In the first part, we transform the uncertain additive linguistic preference relations into the expected additive linguistic preference relations, and present a procedure for group decision making based on multiple types of additive linguistic preference relations. By using the deviation measures between additive linguistic preference relations, we give some straightforward formulas to determine the weights of decision makers, and propose a method to reach consensus among the individual preferences and the group’s opinion. In the second part, we extend the above results to group decision making based on multiple types of multiplicative linguistic preference relations, and finally, a practical example is given to illustrate the application of the results.     

A consensus model for group decision making problems with linguistic interval fuzzy preference relations

Sometimes, we find decision situations in which it is difficult to express some preferences by means of concrete preference degrees. In this paper, we present a consensus model for group decision making problems in which the experts use linguistic interval fuzzy preference relations to represent their preferences. This model is based on two consensus criteria, a consensus measure and a proximity measure, and on the concept of coincidence among preferences. We compute both consensus criteria in the three representation levels of a preference relation and design an automatic feedback mechanism to guide experts in the consensus reaching process.     

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.     

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.     

A peer IF-TOPSIS based decision support system for packaging machine selection

Selecting the appropriate manufacturing machine is a very important and complex problem for firms which usually have to deal with both qualitative and quantitative criteria and involve different decision makers whose knowledge is often vague and imprecise.This paper proposes a peer-based modification to intuitionistic fuzzy multi-criteria group decision making with TOPSIS method (peer IF-TOPSIS) and applies it to a packaging machine selection problem. Intuitionistic fuzzy weighted averaging (IFWA) operator has been selected both to obtain the group opinion on the relevance of the single decision makers and to aggregate individual opinions of decision makers for rating the importance of criteria and alternatives.A case study illustrates the application of the modified IF-TOPSIS method in order to select a Vertical Form Fill and Seal (VFFS) for Double Square Bottom Bag (DSBB) machine in food packaging.     

Consensus-reaching methods for hesitant fuzzy multiple criteria group decision making with hesitant fuzzy decision making matrices

Group decision making plays an important role in various fields of management decision and economics. In this paper, we develop two methods for hesitant fuzzy multiple criteria group decision making with group consensus in which all the experts use hesitant fuzzy decision matrices (HFDMs) to express their preferences. The aim of this paper is to present two novel consensus models applied in different group decision making situations, which are composed of consensus checking processes, consensus-reaching processes, and selection processes. All the experts make their own judgments on each alternative over multiple criteria by hesitant fuzzy sets, and then the aggregation of each hesitant fuzzy set under each criterion is calculated by the aggregation operators. Furthermore, we can calculate the distance between any two aggregations of hesitant fuzzy sets, based on which the deviation between any two experts is yielded. After introducing the consensus measure, we develop two kinds of consensus-reaching procedures and then propose two step-by-step algorithms for hesitant fuzzy multiple criteria group decision making. A numerical example concerning the selection of selling ways about ‘Trade-Ins’ for Apple Inc. is provided to illustrate and verify the developed approaches. In this example, the methods which aim to reach a high consensus of all the experts before the selection process can avoid some experts’ preference values being too high or too low. After modifying the previous preference information by using our consensus measures, the result of the selection process is much more reasonable.     

A multi-criteria interval-valued intuitionistic fuzzy group decision making with Choquet integral-based TOPSIS

An extension of TOPSIS, a multi-criteria interval-valued intuitionistic fuzzy decision making technique, to a group decision environment is investigated, where inter-dependent or interactive characteristics among criteria and preference of decision makers are taken into account. To get a broad view of the techniques used, first, some operational laws on interval-valued intuitionistic fuzzy values are introduced. Based on these operational laws, a generalized interval-valued intuitionistic fuzzy geometric aggregation operator is proposed which is used to aggregate decision makers’ opinions in group decision making process. In addition, some of its properties are discussed. Then Choquet integral-based Hamming distance between interval-valued intuitionistic fuzzy values is defined. Combining the interval-valued intuitionistic fuzzy geometric aggregation operator with Choquet integral-based Hamming distance, an extension of TOPSIS method is developed to deal with a multi-criteria interval-valued intuitionistic fuzzy group decision making problems. Finally, an illustrative example is used to illustrate the developed procedures.     

MAGDM Linear-Programming Models With Distinct Uncertain Preference Structures

Group decision making with preference information on alternatives is an interesting and important research topic which has been receiving more and more attention in recent years. The purpose of this paper is to investigate multiple-attribute group decision-making (MAGDM) problems with distinct uncertain preference structures. We develop some linear-programming models for dealing with the MAGDM problems, where the information about attribute weights is incomplete, and the decision makers have their preferences on alternatives. The provided preference information can be represented in the following three distinct uncertain preference structures: 1) interval utility values; 2) interval fuzzy preference relations; and 3) interval multiplicative preference relations. We first establish some linear-programming models based on decision matrix and each of the distinct uncertain preference structures and, then, develop some linear-programming models to integrate all three structures of subjective uncertain preference information provided by the decision makers and the objective information depicted in the decision matrix. Furthermore, we propose a simple and straightforward approach in ranking and selecting the given alternatives. It is worth pointing out that the developed models can also be used to deal with the situations where the three distinct uncertain preference structures are reduced to the traditional ones, i.e., utility values, fuzzy preference relations, and multiplicative preference relations. Finally, we use a practical example to illustrate in detail the calculation process of the developed approach.      

Fuzzy extension of TOPSIS model for group decision making under multiple criteria

Multi criteria group decision making methods are broadly used in the real-world decision circumstances for homogeneous groups. Significant and vital decisions are usually made by the heterogeneous groups of managers or experts in organizations. In many situations, experts may decide on the basis of imprecise information coming from a variety of sources about alternatives. In fact, some criteria are completely quantifiable, some partially quantifiable, and others completely subjective. This paper proposes a fuzzy extension of TOPSIS method for heterogeneous group decision making models under fuzzy environment. It converts the decision makers’ fuzzy decision matrices into an aggregated decision matrix to determine the most preferable choice among all possible alternatives. The results of a numerical example of proposed method have been highly consistent based on the Spearman’s rank correlation coefficient and showed good agreement with other methods.     

Consistency improvement under a personalized individual semantics context in distributed linguistic group decision making

Consistency measurement is a significant issue in linguistic decision making when preferences are expressed via linguistic preference relations. However, the extant literatures on linguistic consistency generally overlook the fact that even the same word can have diverse meanings for different people, which indicates that people usually possess personalized individual semantics (PISs) over words. Furthermore, with the complexity of the practical decision-making problem increases, decision makers become more likely to be uncertain and hesitant to make their preferences due to the lack of knowledge, therefore, their linguistic preferences may be represented through distributed linguistic representations. However, there are few consistency improving studies on distributed linguistic representations. Therefore, in this study we devise a novel consistency improving approach for distribution linguistic preference relations under a PISs context. Furthermore, the usability and effectiveness of the PISs based consistency improvement method are verified through the detail numerical analysis and comparative study.     

Possibilistic programming approach for fuzzy multidimensional analysis of preference in group decision making

Multiple attribute decision making (MADM) problems are the most encountered problems in decision making. Fuzziness is inherent in decision making process and linguistic variables are well suited to assessing an alternative on qualitative attributes using fuzzy rating. A few techniques in MADM assess the weights of attributes based on preference information on alternatives. But they are not practical any more when the set of all paired comparison judgments from decision makers (DMs) on attributes are not crisp and also we have to deal with fuzzy decision matrix. This paper investigates the generation of a possibilistic model for multidimensional analysis of preference (LINMAP). The model assesses the fuzzy weights as well as locating the ideal solution with fuzzy decision making preference on attributes and fuzzy decision matrix. All of the information is assumed as triangular fuzzy numbers (TFNs). This method is developed in group decision making environments and formulates the problem as a possibilistic programming with multiple objectives.     

基于直觉梯形模糊TOPSIS的多属性群决策方法

提出一种改进的逼近理想解排序(TOPSIS)方法,即直觉梯形模糊TOPSIS多属性群决策方法。首先,应用直觉梯形模糊数形式表示方案属性偏好和属性权重信息且专家权重完全未知;然后,利用直觉梯形模糊数间距离测度和期望值及直觉梯形模糊加权平均算子来确定决策者权重信息和属性权重信息;进而给出直觉梯形模糊环境下方案优选的算法;最后,通过算例进一步说明了该直觉梯形模糊TOPSIS方法的有效性。     

Compatibility of interval fuzzy preference relations with the COWA operator and its application to group decision making

We develop a new compatibility for the interval fuzzy preference relations based on the continuous ordered weighted averaging (COWA) operator and use it to determine the weights of experts in group decision making (GDM). We define some concepts of the compatibility degree and the compatibility index for the two interval fuzzy preference relations based on the COWA operator. We study some desirable properties of the compatibility index and investigate the relationship between the each expert’s interval fuzzy preference relation and the synthetic interval fuzzy preference relation. The prominent characteristic of the compatibility index based on the COWA operator is that it can deal with the compatibility of all the arguments by using a controlled parameter considering the attitude of decision maker rather than the compatibility of the simply two points in intervals. To determine the experts’ weights in the GDM with the interval fuzzy preference relations, we propose an optimal model based on the criterion of minimizing the compatibility index. In the end, we give a numerical example to develop the new approach to GDM with interval fuzzy preference relations.