Fuzzy consensus measure on verbal opinions

doi:10.1016/j.eswa.2007.07.040

Expert Systems with Applications

Volume 35, Issue 3, October 2008, Pages 836-842

https://doi.org/10.1016/j.eswa.2007.07.040 Get rights and content

Abstract

In the group decision-making process, consensus is an important indication of group agreement or reliability. Traditional methods to measure consensus refer mostly to gauging variance among the participants’ opinions by transforming them into numbers in the interval scale. In this study we propose a new value-based measure, in which verbal opinions are transformed into values by means of fuzzy membership functions. To understand how the proposed method performs, we conduct two experiments to compare its performance with the variance-based methods and entropy measure. The results show that our method is more appropriate to account for participants’ consensus judgments based on verbal opinions.

Introduction

Group decision making is the process of arriving at a conclusion toward a specific issue based on the opinions of multiple individuals. Under such a circumstance, consensus of the participants is an important indication of group agreement or reliability (Stemler, 2004). It is desired that a good measure on consensus toward an issue can fully reflect the real behavior of the group.

Traditional methods to measure consensus rely heavily on numerical variance among the participants’ opinions. That is, verbal opinions of the participants are transformed into numerical values and variance among these values is then calculated to indicate the disagreement degree. Variance-based measures, however, suffer from major criticism of being imprecise and misleading because of their apparent implication that interval scale is assumed for the ordinal verbal responses (Agresti, 1996).

On the other hand, the concept of entropy is also proposed to measure consensus (Alston, Kearl, & Vaughan, 1992). Entropy is originated from information theory to measure the degree of disorder, which solely depends on the probability of event occurrence. As pointed out by Fuller, Alston, and Vaughan (1995), this measure is nonlinear in nature and, consequently, large changes in the distribution of event occurrence may result in small changes in entropy, i.e. insensitive to the actual distribution of opinions. It seems that entropy is a more robust measure for data in the ordinal scale as verbal opinions.

Nevertheless, it is our idea that variance-based methods are not totally invalid. The major problem lies in that verbal opinions are not equally spaced in the interval scale. On the other hand, several researchers (e.g. Dyer & Sarin, 1979) have shown that fuzzy membership functions can reflect the relative importance of verbal terms in our mind. In this sense, we propose a fuzzy membership function approach to transforming verbal opinions into numbers in the interval scale.

The purpose of this paper is thus to propose a measure of group consensus based on the variability of membership values. In addition, experiments are conducted to compare this measure with other common consensus measures to justify its feasibility in applications.

Section snippets

Literature review

Typical measures on consensus are reviewed in this section, which include variance-based methods and entropy. They are further described as follows.

Membership assessment of verbal opinions

As the major problem for pure variance-based methods comes from the equal-spaced interval scale assumption for ordinal verbal opinions, we propose a fuzzy membership-function approach to transforming verbal opinions into quantitative numbers in the interval scale. Fuzzy membership functions are employed to reflect different degrees of importance of verbal opinions.

To assess the desired fuzzy membership function, we conduct an experiment by asking participating subjects to respond an interval

Comparison of measures

In this section, we desire to compare our proposed membership-based consensus measure with common measures such as VCC, VCC′, and entropy. To do so, two experiments are conducted to obtain subjects’ consensus judgments toward verbal opinions. Various consensus measures are then compared based on their correlations with the experimental results.

Results

We first conducted an MDPref analysis to understand how subjects make consensus judgment on verbal opinions, followed by the utility analyses to understand how the utilities and values elicited from previous studies can be fitted into the current one.

Conclusions

Consensus is essential in the group decision-making process to indicate group agreement or reliability. In this paper, we propose the membership-based measures where verbal opinions are transformed into membership values by means of fuzzy membership functions. To understand how the proposed measures perform, we conduct two experiments to compare the performance with traditional variance-based measures and the entropy measure. The result shows that membership-based measures do improve the

Acknowledgements

This work was supported by the National Science Council, Taiwan, ROC, with grant number NSC92-2416-H-251-003.

References (18)

D. Kahneman et al.
Subjective probability: A judgment of representativeness
Cognitive Psychology
(1972)
A. Agresti
An introduction to categorical data analysis
(1996)
Alston, R. M., Kearl, J. R., & Vaughan, M. B. (1992). “Is there a consensus among economists in the 1990s?” American...
K.D. Bailey
Social entropy theory
(1990)
J.S. Dyer et al.
Measurable multiattribute value functions
Operations Research
(1979)
D.A. Fuller et al.
The split between political parties on economic issues: A survey of Republicans, Democrats, and economists
Eastern Economic Journal
(1995)
F. Herrera et al.
A rational consensus model in group decision making using linguistic assessments
Fuzzy Sets and Systems
(1997)
E. Herrera-Viedma et al.
A consensus model for multiperson decision making with different preference structures
IEEE Transactions on Systems, Man and Cybernetics-Part A: Systems and Humans
(2002)
E. Herrera-Viedma et al.
A consensus support system model for group decision-making problems with multi-granular linguistic preference relations
IEEE Transactions on Fuzzy Systems
(2005)

There are more references available in the full text version of this article.

Cited by (13)

Granular fuzzy models: A study in knowledge management in fuzzy modeling
2012, International Journal of Approximate Reasoning
In system modeling, knowledge management comes vividly into the picture when dealing with a collection of individual models. These models being considered as sources of knowledge, are engaged in some collective pursuits of a collaborative development to establish modeling outcomes of global character. The result comes in the form of a so-called granular fuzzy model, which directly reflects upon and quantifies the diversity of the available sources of knowledge (local models) involved in knowledge management. In this study, several detailed algorithmic schemes are presented along with related computational aspects associated with Granular Computing. It is also shown how the construction of information granules completed through the use of the principle of justifiable granularity becomes advantageous in the realization of granular fuzzy models and a quantification of the quality (specificity) of the results of modeling. We focus on the design of granular fuzzy models considering that the locally available models are those fuzzy rule-based. It is shown that the model quantified in terms of two conflicting criteria, that is (a) a coverage criterion expressing to which extent the resulting information granules “cover” include data and (b) specificity criterion articulating how detailed (specific) the obtained information granules are. The overall quality of the granular model is also assessed by determining an area under curve (AUC) where the curve is formed in the coverage-specificity coordinates. Numeric results are discussed with intent of displaying the most essential features of the proposed methodology and algorithmic developments.
Linear optimization modeling of consistency issues in group decision making based on fuzzy preference relations
2012, Expert Systems with Applications
Citation Excerpt :
As applications of individual consistency of fuzzy preference relations, Chang and Chen (2009) study the cooperative learning in E-learning, Wang and Chang (2007) propose a method to predict the chance of the successful knowledge management implementation, and Wang and Chen (2007) discuss the partnership selection problem. The consensus model is an important aspect in GDM (Ben-Arieh & Easton, 2007; Ben-Arieh, Easton, & Evans, 2009; Chen & Cheng, 2009; Dong, Xu, Li, & Feng, 2010; Herrera, Herrera-Viedma, & Verdegay, 1996; Hsiao, Lin, & Chang, 2008; Kacprzyk, 1986; Kacprzyk, Fedrizzi, & Nurmi, 1992; Khorshid, 2010; Zhang, Dong, Xu, & Li, in press). Classically, consensus is defined as the full and unanimous agreement of all the decision makers regarding all the possible alternatives.
The consistency measure is a vital basis for group decision making (GDM) based on fuzzy preference relations, and includes two subproblems: individual consistency and consensus consistency. This paper proposes linear optimization models for solving some issues on consistency of fuzzy preference relations, such as individual consistency construction, consensus model and management of incomplete fuzzy preference relations. Our proposal optimally preserves original preference information in constructing individual consistency and reaching consensus (in Manhattan distance sense), and maximizes the consistency level of fuzzy preference relations in calculating the missing values of incomplete fuzzy preference relations. Linear optimization models can be solved in very little computational time using readily available softwares. Therefore, the results in this paper are also of simplicity and convenience for the application of consistent fuzzy preference relations in GDM problems.
Information granules and their use in schemes of knowledge management
2011, Scientia Iranica
Citation Excerpt :
It also invokes some mechanisms of consensus building. In this regard, fuzzy sets play a pivotal role as convincingly documented in the existing literature (cf. [10–17]). In this context, various algorithms of collaboration [18,19] are of significant interest as well.
In system modeling, knowledge management comes vividly into the picture when dealing with a collection of models. These models, being considered as sources of knowledge, are engaged in some collective pursuits of collaboration and consensus formation. We show the use of information granules in these processes by elaborating on their conceptual role. It is revealed that information granules are used to facilitate processes of collaboration and consensus building. Such granular constructs, referred to as granular models, can also emerge as a part of higher order models to reflect and quantify the diversity of the sources of knowledge involved in knowledge management. Several detailed algorithmic schemes are presented, along with related computational aspects associated with Granular Computing. It is also shown how the construction of information granules, through the use of the principle of justifiable granularity, becomes advantageous in the realization of granular models. This study builds upon seminal concepts established in L.A. Zadeh’s Rosetta Stone paper devoted to information granulation.
Risk assessment of renewable energy projects using uncertain information
2022, International Journal of Energy Research
A hybrid risk prioritization approach in construction projects using failure mode and effective analysis
2020, Engineering, Construction and Architectural Management
Human linguistic perception of distances for location-aware systems
2019, International Journal of Mobile Computing and Multimedia Communications

View all citing articles on Scopus

View full text

Fuzzy consensus measure on verbal opinions

Abstract

Introduction

Section snippets

Literature review

Membership assessment of verbal opinions

Comparison of measures

Results

Conclusions

Acknowledgements

Cognitive Psychology

An introduction to categorical data analysis

Social entropy theory

Measurable multiattribute value functions

Operations Research

The split between political parties on economic issues: A survey of Republicans, Democrats, and economists

Eastern Economic Journal

A rational consensus model in group decision making using linguistic assessments

Fuzzy Sets and Systems

A consensus model for multiperson decision making with different preference structures

IEEE Transactions on Systems, Man and Cybernetics-Part A: Systems and Humans

A consensus support system model for group decision-making problems with multi-granular linguistic preference relations

IEEE Transactions on Fuzzy Systems