Abstract
The preference values in group decision-making (GDM) process can differ significantly between different experts, which may yield a low level of group consensus. Therefore, different consensus models have been developed for the modification of preference values to assist experts in improving their consensus degrees. However, most consensus models do not consider collective intelligence (CI) that may decrease as the consensus degree increases under certain circumstances. From the perspective of CI, the distrust relationship allows the group to better explore the decision space, rather than prematurely converge on an agreed suboptimal solution. Inspired by this idea, a theoretical framework of solving intuitionistic fuzzy GDM problems with low group consensus is proposed in this paper, which mainly includes two steps: (1) building the trust/distrust relationships and (2) establishing a consensus model. For two experts with a direct relationship, the trust/distrust relationships between them are constructed by fusing their knowledge levels and representativeness levels. For two experts with an indirect relationship, a new operator is designed to construct the trust/distrust relationships between them, which can describe the information attenuation of the decreasing trust along with the increasing distrust. Additionally, a consensus model based on the social network relationships density and trust/distrust relationships is proposed, which improves consensus degree and CI level conducively. Finally, a ranking of alternatives is constructed to select the optimal alternative. An illustrative example is used to demonstrate the effectiveness and applicability of the proposed method.
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References
Tian, J.F., Zhang, Z.M., Ha, M.H.: An additive consistency and consensus-based approach for uncertain group decision making with linguistic preference relations. IEEE Trans. Fuzzy Syst. 27, 873–887 (2018)
Xu, Y.J., Wen, X.W., Sun, H., Wang, H.M.: Consistency and consensus models with local adjustment strategy for hesitant fuzzy linguistic preference relations. Int. J. Fuzzy Syst. 20, 2216–2233 (2018)
Xu, Y.J., Xi, Y.S., Cabrerizo, F.J., Herrera-Viedma, E.: An alternative consensus model of additive preference relations for group decision making based on the ordinal consistency. Int. J. Fuzzy Syst. 21, 1818–1830 (2019)
Zhang, H.J., Dong, Y.C., Chiclana, F., Yu, S.: Consensus efficiency in group decision making: a comprehensive comparative study and its optimal design. Eur. J. Oper. Res. 275, 580–598 (2019)
Gong, Z.W., Guo, W.W., Herrera-Viedma, E., Gong, Z.J., Wei, G.: Consistency and consensus modeling of linear uncertain preference relations. Eur. J. Oper. Res. 283, 290–307 (2020)
Perez, L.G., Mata, F., Chiclana, F.: Social network decision making with linguistic trustworthiness-based induced OWA operators. Int. J. Intell. Syst. 29, 1117–1137 (2014)
Wu, J., Chiclana, F., Fujita, H., Herrera-Viedma, E.: A visual interaction consensus model for social network group decision making with trust propagation. Knowl. Based Syst. 122, 39–50 (2017)
Wu, J., Dai, L.F., Chiclana, F., Fujita, H., Herrera-Viedma, E.: A minimum adjustment cost feedback mechanism based consensus model for group decision making under social network with distributed linguistic trust. Inf. Fusion 41, 232–242 (2018)
Wu, T., Liu, X.W., Gong, Z.W., Zhang, H.H., Herrera, F.: The minimum cost consensus model considering the implicit trust of opinions similarities in social network group decision-making. Int. J. Fuzzy Syst. 35, 470–493 (2020)
Leimeister, J.M.: Collective intelligence. Bus. Inform. Syst. Eng. 2, 245–248 (2010)
Woolley, A.W., Chabris, C.F., Pentland, A., Hashmi, N., Malone, T.W.: Evidence for a collective intelligence factor in the performance of human groups. Science 330, 686–688 (2010)
Massari, G.F., Giannoccaro, I., Carbone, G.: Are distrust relationships beneficial for group performance? The influence of the scope of distrust on the emergence of collective intelligence. Int. J. Prod. Econ. 208, 343–355 (2019)
Liu, Y.J., Liang, C.Y., Chiclana, F., Wu, J.: A trust induced recommendation mechanism for reaching consensus in group decision making. Knowl. Based Syst. 119, 221–231 (2017)
Hoogendoorn, M., Jaffry, S.W., Van Maanen, P.P., Treur, J.: Design and validation of a relative trust model. Knowl. Based Syst. 57, 81–94 (2014)
Victor, P., Cornelis, C., De Cock, M., Da Silva, P.P.: Gradual trust and distrust in recommender systems. Fuzzy Sets Syst. 160, 1367–1382 (2009)
Victor, P., Cornelis, C., De Cock, M., Herrera-Viedma, E.: Practical aggregation operators for gradual trust and distrust. Fuzzy Sets Syst. 184, 126–147 (2011)
Wu, J., Cao, M.S., Chiclana, F., Dong, Y.C., Herrera-Viedma, E.: An optimal feedback model to prevent manipulation behaviours in consensus under social network group decision making. IEEE Trans. Fuzzy Syst. (2020). https://doi.org/10.1109/tfuzz.2020.2985331
Xu, Z.S., Liao, H.C.: Intuitionistic fuzzy analytic hierarchy process. IEEE Trans. Fuzzy Syst. 22, 749–761 (2014)
Xu, Z.S., Liao, H.C.: A survey of approaches to decision making with intuitionistic fuzzy preference relations. Knowl. Based Syst. 80, 131–142 (2015)
Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986)
Xu, Z.S.: Intuitionistic preference relations and their application in group decision making. Inf. Sci. 177, 2363–2379 (2007)
Tan, J.Y., Zhu, C.X., Zhang, X.Z., Zhu, L.: The method of hesitant fuzzy multiple attribute decision making based on group consistency. Oper. Res. Manage. Sci. 177, 105–109 (2016)
Li, S.L., Wei, C.P.: Modeling the social influence in consensus reaching process with interval fuzzy preference relations. Int. J. Fuzzy Syst. 21, 1755–1770 (2019)
Tang, M., Liao, H.C., Xu, J.P., Streimikiene, D., Zheng, X.S.: Adaptive consensus reaching process with hybrid strategies for large-scale group decision making. Eur. J. Oper. Res. 282, 957–971 (2020)
Tian, Z.P., Nie, R.X., Wang, J.Q.: Social network analysis-based consensus-supporting framework for large-scale group decision-making with incomplete interval type-2 fuzzy information. Inf. Sci. 502, 446–471 (2019)
Mayer, R.C., Davis, J.H., Schoorman, F.D.: An integrative model of organizational trust. Acad. Manage. Rev. 20, 709–734 (1995)
Wang, H., Huang, L., Ren, P.Y., Zhao, R., Luo, Y.Y.: Dynamic incomplete uninorm trust propagation and aggregation methods in social network. J. Intell. Fuzzy Syst. 33, 3027–3039 (2017)
Geffroy, B., Bru, N.: Dossou-Gbete, Simplice, Tentelier, Cedric, Bardonnet, Agnes: the link between social network density and rank-order consistency of aggressiveness in juvenile eels. Behav. Ecol. Sociobiol. 37, 1073–1083 (2014)
Liang, Q., Liao, X.W., Liu, J.P.: A social ties-based approach for group decision-making problems with incomplete additive preference relations. Knowl. Based Syst. 119, 68–86 (2017)
Wang, Z.J., Li, K.W.: A multi-step goal programming approach for group decision making with incomplete interval additive reciprocal comparison matrices. Eur. J. Oper. Res. 242, 890–900 (2015)
Hong, D.H., Choi, C.H.: Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst. 114, 103–113 (2000)
Liao, H.C., Xu, Z.S., Zeng, X.J., Merigo, J.M.: Framework of group decision making with intuitionistic fuzzy preference information. IEEE Trans. Fuzzy Syst. 23, 1211–1227 (2015)
Capuano, N., Chiclana, F., Fujita, H., Herrera-Viedma, E., Loia, V.: Fuzzy group decision making with incomplete information guided by social influence. IEEE Trans. Fuzzy Sys. 99, 1704–1718 (2017)
Kamis, N.H., Chiclana, F., Levesley, J.: Preference similarity network structural equivalence clustering based consensus group decision making model. Appl. Soft Comput. 67, 706–720 (2018)
Dong, Y.C., Zha, Q.B., Zhang, H.J., Kou, G., Fujita, H., Chiclana, F., Herrera-Viedma, E.: Consensus reaching in social network group decision making: research paradigms and challenges. Knowl. Based Syst. 162, 3–13 (2018)
Li, S.L., Wei, C.P., Song, Y.H.: Group decision making method for fuzzy complementary judgment matrices based on trust relationships. Contr. Decis. 35, 1240–1246 (2020)
Acknowledgements
This research was supported by the National Natural Science Foundation of China (No. 72071056), the Project of Key Research Institute of Humanities and Social Science in University of Anhui Province (No. SK2017A0055), and the NSFC-Zhejiang Joint Fund for the Integration of Industrialization and Informatization under the Grant (No. U1709215).
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Appendix A
Appendix A
1.1 Proof of Theorem 1
Suppose \(\sigma\) is a number between the minimum consensus degree and the second minimum consensus degree of all experts. The consensus degree of \(e_{m}\) is below the predefined threshold value, and \(e_{m}\) is the expert with the lowest consensus degree, i.e., \({\text{ACD}}^{m} < \sigma\). For other experts \(e_{k}\) (\(k = 1,2, \ldots ,m - 1\)), there is \({\text{ACD}}^{k} > \sigma\). \({\text{TS}}^{m} = \left\{ {e_{\left( 1 \right)}^{m} ,e_{\left( 2 \right)}^{m} , \ldots ,e_{\left( t \right)}^{m} } \right\}\) and \({\text{DS}}^{m} = \left\{ {e_{{\left( {\left( 1 \right)} \right)}}^{m} ,e_{{\left( {\left( 2 \right)} \right)}}^{m} , \ldots ,e_{{\left( {\left( d \right)} \right)}}^{m} } \right\}\) are sets of experts trusted and distrusted by \(e^{m}\), respectively.
The initial consensus degree of \(e_{m}\) is \({\text{ACD}}^{m} = \frac{1}{m - 1}\left( {m - 1 - \frac{1}{2q}\mathop \sum \limits_{k = 1}^{m - 1} \mathop \sum \limits_{{\left( {i,j} \right) \in {\text{APS}}^{m} }} \left( {\left| {\mu_{ij}^{m} - \mu_{ij}^{k} } \right| + \left| {\gamma_{ij}^{m} - \gamma_{ij}^{k} } \right|} \right)} \right)\).
To simplify the proof, let \(\left| {\mu^{m} - \mu^{k} } \right| + \left| {\gamma^{m} - \gamma^{k} } \right| = \frac{1}{2q}\mathop \sum \limits_{{\left( {i,j} \right) \in {\text{APS}}^{m} }} \left( {\left| {\mu_{ij}^{m} - \mu_{ij}^{k} } \right| + \left| {\gamma_{ij}^{m} - \gamma_{ij}^{k} } \right|} \right)\).
Then, we have \({\text{ACD}}^{m} = \frac{1}{m - 1}\left( {m - 1 - \mathop \sum \limits_{k = 1}^{m - 1} \left( {\left| {\mu^{m} - \mu^{k} } \right| + \left| {\gamma^{m} - \gamma^{k} } \right|} \right)} \right) < \sigma\),
which can be expressed as
Then for any \(h \in \left\{ {1,2, \ldots ,m - 1} \right\}\), there is
Now we need to prove that
According to Eq. (16), when \(\rho \le 0.3\), for any \(k \in \left\{ {1,2, \ldots ,m - 1} \right\},\) there is \(\begin{aligned} \left| {\overline{\mu }^{m} - \mu^{k} } \right| + \left| {\overline{\gamma }^{m} - \gamma^{k} } \right| = & \left| {\left( {1 - \theta_{1} } \right)\mu^{m} + \frac{{\theta_{1} }}{t}\mathop \sum \limits_{i = 1}^{t} \mu_{\left( i \right)}^{m} - \mu^{k} } \right| + \left| {\left( {1 - \theta_{1} } \right)\gamma^{m} + \frac{{\theta_{1} }}{t}\mathop \sum \limits_{i = 1}^{t} \gamma_{\left( i \right)}^{m} - \gamma^{k} } \right| \\ & \le \left( {1 - \theta_{1} } \right)\left( {\left| {\mu^{m} - \mu^{k} } \right| + \left| {\gamma^{m} - \gamma^{k} } \right|} \right) + \frac{{\theta_{1} }}{t}\mathop \sum \limits_{i = 1}^{t} \left( {\left| {\mu_{\left( i \right)}^{m} - \mu^{k} } \right| + \left| {\gamma_{\left( i \right)}^{m} - \gamma^{k} } \right|} \right). \\ \end{aligned}\)
Then, there is
Because \(\left\{ {\left( 1 \right),\left( 2 \right), \ldots ,\left( t \right)} \right\} \subseteq \left\{ {1,2, \ldots ,m - 1} \right\}\). According to Eq. (23), for any \(i \in \left\{ {1,2, \ldots ,t} \right\}\), there is \(\mathop \sum \nolimits_{k = 1}^{m - 1} \left( {\left| {\mu_{\left( i \right)}^{m} - \mu^{k} } \right| + \left| {\gamma_{\left( i \right)}^{m} - \gamma^{k} } \right|} \right) < \left( {m - 1} \right)\left( {1 - \sigma } \right)\).
As a result, there is \(\frac{{\theta_{1} }}{t}\mathop \sum \nolimits_{k = 1}^{m - 1} \mathop \sum \nolimits_{i = 1}^{t} \left( {\left| {\mu_{\left( i \right)}^{m} - \mu^{k} } \right| + \left| {\gamma_{\left( i \right)}^{m} - \gamma^{k} } \right|} \right) < \theta_{1} \left( {m - 1} \right)\left( {1 - \sigma } \right)\).
Thus,
According to Eq. (22), there is
Then Eq. (24) is proved, which means \(\overline{\text{ACD}}^{m} > {\text{ACD}}^{m}\).
Similarly, we can prove that \(\overline{\text{ACD}}^{m} > {\text{ACD}}^{m}\) when \(\rho > 0.3\).
According to the above two situations, the consensus model proposed in this paper satisfies Theorem 1.
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Pei, F., He, YW., Yan, A. et al. A Consensus Model for Intuitionistic Fuzzy Group Decision-Making Problems Based on the Construction and Propagation of Trust/Distrust Relationships in Social Networks. Int. J. Fuzzy Syst. 22, 2664–2679 (2020). https://doi.org/10.1007/s40815-020-00980-0
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DOI: https://doi.org/10.1007/s40815-020-00980-0