Elsevier

Applied Soft Computing

Volume 77, April 2019, Pages 135-154
Applied Soft Computing

Measuring and reaching consensus in group decision making with the linguistic computing model based on discrete fuzzy numbers

https://doi.org/10.1016/j.asoc.2019.01.008Get rights and content

Highlights

  • The linguistic model based on discrete fuzzy numbers (LM-DFN) is studied.

  • The method for measuring group consensus with the LM-DFN is proposed.

  • The method for improving group consensus with the LM-DFN is proposed.

  • The proposed methods have some advantages compared with the existing ones.

Abstract

The linguistic computing model based on discrete fuzzy numbers has some good properties compared with other existing models and should be further studied, which has been proved by some researchers. However, the research of group consensus with this linguistic model is insufficient, given that group consensus is an important issue in group decision making. Therefore, this paper would concentrate on this subject. It includes two main issues: research on consensus measure and research on the method for improving group consensus in group decision making based on this linguistic computing model. For research on the consensus measure, this paper first studies on the aggregation method for discrete fuzzy numbers. Then, the index of measuring group consensus is determined. For research on improving the group consensus, considering the characteristics of discrete fuzzy numbers, we present an algorithm to improve group consensus. In addition, an illustrative example of a decision-making problem about investment is stated to show the whole solving process. It also illustrates the feasibility, rationality and validity of all the proposed methods. Finally, the comparisons between some proposals and existing studies are made, which helps point out the advantages of the proposed methods.

Introduction

In modern decision-making problems, decision makers tend to use natural language or language sets rather than concrete abstract numbers to evaluate alternatives. They benefit from its high convenience, reliability and practicality. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27]. Nevertheless, natural language brings some problems. For example, the way to calculate the original language information is needed, and measuring such information quantitatively to ensure its effectiveness and accurateness is complicated etc. Based on this kind of problems, experts and scholars have put forward some language computing models to quantify the linguistic information into numerical values, such as the symbolic linguistic model based on ordinal scales [1], [2], the 2-tuple linguistic model [3], linguistic models based on various kinds of type-2 fuzzy sets [4], proportional 2-tuple linguistic model [5], linguistic model based on discrete fuzzy numbers [6] and linguistic computing model setting numerical scale for hesitant fuzzy linguistic preference relations [28] and for hesitant unbalanced linguistic information [29]. These linguistic computing models have their own advantages and some limitations. The symbolic linguistic model based on ordinal scales assumes a sequential structure between linguistic terms. The 2-tuple linguistic model is a symbolic transformation of linguistic information. The linguistic model based on the type-2 fuzzy sets uses the type-2 membership function to express the linguistic term semantics. The proportional 2-tuple model is the extension of 2-tuple linguistic model, using two linguistic terms with their corresponding proportion to model linguistic information. In these models, the evaluations given by the decision makers must correspond with a single linguistic term in the linguistic term set. However, in most cases, the expert’s opinions cannot be expressed as a single linguistic term, they often use some expressions such as “better than Bad” or “between Fair and Good” to express a more complex point of view [6]. Also, in group decision making, when the individual preferences are aggregated into group preference, most of the existing aggregate operators are defined on the [0,1] interval. First, the linguistic inputs are transformed into real numbers on the unit interval, the aggregation operator on the unit interval is applied and a transformation is performed in the output to recover a linguistic term [6]. However, the aggregation result should be consistent with the original linguistic information, like the idea in [30]. The linguistic models presented in [6], [28], [29] avoid the above limits because decision makers can use them to express more complex information by more linguistic terms and the aggregation result is consistent with the original linguistic information. However, comparing to the models presented in [28], [29], the linguistic computing model based on discrete fuzzy numbers presented by Massanet et al. [6] considers not only the linguistic term set but also corresponding membership values based on the characteristics and advantages of discrete fuzzy numbers. Hence, this model is more applicative and flexible with less loss of information in actual decision making. The study of this paper is based on this linguistic computing model.

The increasing complexity of the socio-economic environment makes it difficult for a single decision maker to take into account all the relevant aspects of a decision-making problem. Therefore, most of the practical issues in management science, operational research and industrial engineering fields are solved by group decision [21], [22], [23], [24], [25], [26], [27], [31], [32], [33], [34], [35], [36]. The purpose of group decision making is to balance the decision preferences of different individuals and to find a viable and acceptable alternative for the whole group. An important concept in group decision making is group consensus. Group consensus in the traditional sense refers to the ideal situation that all the decision-makers of the group are highly satisfied. However, in the actual decision-making process, this ideal situation is almost impossible to achieve. Therefore, in modern group decision-making research, group consensus is used to measure the consensus level of group decision making problem [37]. In decision-making process, decision makers expect higher group consensus to maximize the satisfaction of the entire group. The higher the consensus value decision makers achieve, the more satisfied they will be and the more effectiveness the group decision making will obtain. Therefore, it is necessary to study measuring and improving group consensus of a group decision-making problem. There has been a lot of research on the group consensus [21], [22], [23], [24], [25], [26], [27], [31], [32], [33], [34], [35], [36], [38], [39]. However, for the linguistic computing model based on discrete fuzzy numbers proposed by Massanet et al. in [6], the relevant research is scarce, which is also pointed out by Cabrerizo et al. in [37]. Massanet et al. [40] proposed a consensus model based on this linguistic model, their model was able to achieve consensus without any loss of information or imposing any drastic change to the experts on their initial opinions. Nevertheless, their model considers a moderator in the consensus reaching process which has some drawbacks (more details in Section 7.2). Therefore, it is necessary to study the group consensus of the group decision making problems with this linguistic computing model based on discrete fuzzy numbers.

From the above description, the motivation of this paper is obvious and as follows.

(1) The linguistic computing model based on discrete fuzzy numbers proposed by Massanet et al. [6] has some advantages compared with other existing linguistic models and thus is more applicative and flexible in actual decision making.

(2) It is necessary to study group consensus for the group decision making problems with the linguistic computing model based on discrete fuzzy numbers. It is because group consensus is an important issue in actual decision-making and so far the relevant research for the linguistic model using discrete fuzzy numbers is insufficient.

The studies of group consensus generally include two issues: the measurement of group consensus and the improvement of group consensus. Due to the fact that these two issues are sequentially connected in actual decision making, we study these two issues together in this paper to propose a complete scheme relating to group consensus for group decision making with the linguistic model based on discrete fuzzy numbers.

Thus, the main contribution of this paper is proposing a complete scheme relating to group consensus for group decision making problems with the linguistic model based on discrete fuzzy numbers and includes the following 2 issues.

  • (1)

    The method for measuring group consensus with the linguistic computing model based on discrete fuzzy numbers.

    • Aggregation methods for discrete fuzzy numbers (including weights determination)

    • The index for measuring group consensus.

  • (2)

    The method for improving group consensus with the linguistic computing model based on discrete fuzzy numbers.

The structure of this paper is as follows. In Section 2, we provide an overview about the research status of discrete fuzzy numbers and group consensus models. In Section 3, we recall some related concepts and theories of existing research, including concepts on discrete fuzzy numbers, linguistic computing model based on discrete fuzzy numbers and the existing aggregation methods for this linguistic model. In Section 4, we present the method of measuring group consensus, together with the aggregation method and the definition of group consensus index. In Section 5, we propose an iterative algorithm to improve group consensus and illustrate the operation process and effectiveness of this algorithm with examples. In Section 6, we use an illustrative example to further and fully demonstrate the calculation procedure of the methods proposed in Sections 4 The measurement of group consensus, 5 The method of improving group consensus, which verifies the validity, rationality and accuracy of the proposed methods. In Section 7, we develop the comparison between the research of this paper and the existing research, including the comparison of the aggregation methods and the comparison of the methods to improve group consensus, through which we summarize some advantages of the methods proposed in this paper. Finally, the main work of this paper and some possible future work are summarized in the Conclusion.

Section snippets

Related work

In this section, we show the research status of discrete fuzzy numbers and the studies of group consensus with different models based on different types of decision information. From the research status of discrete fuzzy numbers, we can conclude what needs to be done or improved relating to measuring and reaching group consensus. Also, from the existing studies of group consensus with different models, we can draw on some good thoughts from them to design new methods for the model based on

Preliminaries

In this section, we recall some concepts, definitions and results about the linguistic computing model based on discrete fuzzy numbers and some aggregation methods that will be used in the following sections.

The measurement of group consensus

The research of group consensus with the linguistic computing model based on discrete fuzzy numbers includes the research of aggregation method, the method of measuring the deviation and the definition of group consensus index.

The method of improving group consensus

In this section, we present the method of improving group consensus when its value does not reach the required threshold value in decision making problems.

An illustrative example

Suppose there is an investment company that needs to make the decision on whether to invest a car company or not. There are 4 experts E=e1,e2,e3,e4 supposed to evaluate the car company. The evaluations are based on discrete fuzzy numbers at the 3rd level in linguistic hierarchy, i.e., l3,9. The original evaluations are represented by A1,A2,A3,A4A1L8 (see Table 4) and the normalized ones are listed in Table 5. The subjective weights of the 4 experts are ωs=ωs1,ωs2,ωs3,ωs4=0.3,0.1,0.5,0.1.

Comparisons

The comparisons in this section include two parts: (1) the numerical comparison between the aggregation method proposed in this paper and the existing aggregation method, (2) the comparison between the principles of the method for improving group consensus proposed in this paper and some existing methods in group decision-making problems.

Conclusion

In this paper, some studies on group consensus for group decision making with the linguistic model based on discrete fuzzy numbers are proposed. The proposed methods help improve the accuracy, effectiveness, acceptability of the final result and the satisfaction degree of all experts. The main work includes two major issues: the method of measuring group consensus and the method of improving group consensus. In the study of the method for measuring group consensus, we research on the

Acknowledgment

The work presented in this paper is supported by the National Natural Science Foundation of China (71701037, 71701038, 71601041).

References (65)

  • DongY. et al.

    The OWA-based consensus operator under linguistic representation models using position indexes

    European J. Oper. Res.

    (2010)
  • DongY. et al.

    On consistency measures of linguistic preference relations

    European J. Oper. Res.

    (2008)
  • SunB. et al.

    An approach to consensus measurement of linguistic preference relations in multi-attribute group decision making and application

    Omega

    (2015)
  • DongY. et al.

    Connecting the linguistic hierarchy and the numerical scale for the 2-tuple linguistic model and its use to deal with hesitant unbalanced linguistic information

    Inform. Sci.

    (2016)
  • LiC.C. et al.

    A consistency-driven approach to set personalized numerical scales for hesitant fuzzy linguistic preference relations

    IEEE Int. Conf. Fuzzy Syst.

    (2017)
  • MengF.Y. et al.

    A new method for group decision making with incomplete fuzzy preference relations

    Knowl.-Based Syst.

    (2015)
  • MengF.Y. et al.

    An approach to incomplete multiplicative preference relations and its application in group decision making

    Inform. Sci.

    (2015)
  • LiC.C. et al.

    Personalized individual semantics in computing with words for supporting linguistic group decision making. An application on consensus reaching

    Inf. Fusion

    (2017)
  • DongY. et al.

    Consensus reaching model in the complex and dynamic MAGDM problem

    Knowl.-Based Syst.

    (2016)
  • WuJ. et al.

    A visual interaction consensus model for social network group decision making with trust propagation

    Knowl.-Based Syst.

    (2017)
  • XuY. et al.

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

    Appl. Soft Comput.

    (2017)
  • WuZ. et al.

    Managing consistency and consensus in group decision making with hesitant fuzzy linguistic preference relations

    Omega

    (2016)
  • ZhangG. et al.

    Consistency and consensus measures for linguistic preference relations based on distribution assessments

    Inf. Fusion

    (2014)
  • MataF. et al.

    Type-1 OWA methodology to consensus reaching processes in multi-granular linguistic contexts

    Knowl.-Based Syst.

    (2014)
  • DongY. et al.

    Minimizing adjusted simple terms in the consensus reaching process with hesitant linguistic assessments in group decision making

    Inform. Sci.

    (2015)
  • ChengL.C. et al.

    Identifying conflict patterns to reach a consensus - A novel group decision approach

    European J. Oper. Res.

    (2016)
  • Herrera-ViedmaE. et al.

    A review of soft consensus models in a fuzzy environment

    Inf. Fusion

    (2014)
  • WuZ. et al.

    A concise consensus support model for group decision making with reciprocal preference relations based on deviation measures

    Fuzzy Sets and Systems

    (2012)
  • LiuX. et al.

    Least square completion and inconsistency repair methods for additively consistent fuzzy preference relations

    Fuzzy Sets and Systems

    (2012)
  • XuZ.

    Induced uncertain linguistic OWA operators applied to group decision making

    Inf. Fusion

    (2006)
  • ZhaoM. et al.

    A method considering and adjusting individual consistency and group consensus for group decision making with incomplete linguistic preference relations

    Appl. Soft Comput.

    (2017)
  • ChenL.H. et al.

    An approximate approach for ranking fuzzy numbers based on left and right dominance

    Comput. Math. Appl.

    (2001)
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