This paper presents an account of fuzzy similarity metrics that have been proposed to quantify consensus in Multi Criteria Group Decision Making. Fuzzy similarity metrics are indispensable to determine consensus when experts evaluate alternatives in fuzzy terms, which capture experts’ uncertainty and hesitancy. Furthermore, factors such as the level of expertise or cognitive bias lead to disagreements within the group. The fuzzy similarity metrics described in this article are used to measure the similarity between type 1 and type 2 fuzzy sets, and fuzzy numbers. Consensus can be quantified at three different levels: criteria judgement, alternative judgement, or expert preferences. Promising future work includes the incorporation of social fuzzy measures under the umbrella of multi-agent systems as well as the analysis of fuzzy intuitionistic sets.
T.L.Saaty, A scaling method for priorities in hierarchical structures, Journal of Mathematical Psychology15 (1977), 57–68.
2.
I.-J.Perez, F.-J.Cabrerizo, A.Sergio, F.Chiclana and E.Herrera-Viedma, Soft Consensus Models in Group Decision Making, Springer International Publishing, Cham, 2016, pp. 135–153.
3.
J.M-Merigo, D.Palacios-Marques and S.Zeng, Subjective and objective information in linguistic multi-criteria group decision making, European Journal of Operational Research248 (2016), 522–531.
4.
E.Forman and K.Peniwati, Aggregating individual judgements and priorities with the analytic hierarchy process, European Journal of Operational Research108 (1998), 165–169.
5.
L.A.Zadeh, Fuzzy sets, Informatics and Control8(3) (1965), 338–352.
6.
Q.Dong and T.L.Saaty, An analytic hierarchy process model of group consensus, Journal of Systems Science and Systems Engineering23(3) (2014), 362–374.
7.
J.Kacprzyk, Group decision making with a fuzzy linguistic majority, Fuzzy Sets and Systems18(2) (1986), 105–118.
8.
S.M.Chen and L.W.Lee, Fuzzy multiple attributes group decision-making based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets, Expert Systems with Applications37(1) (2010), 824–833.
9.
G.Preuss, Foundations of Topology. An approach to convenient topology. Kluwer Academic Publishers, The Netherlands, first edition, 2002.
10.
J.N.Mordeson and P.S.Nair, Fuzzy Mathematics. An introduction for engineers and scientists, Springer - Verlag, Heidelberg, second edition, 2001.
11.
K.Chakrabarty, R.T.Biswas and S.Nanda, On fuzzy metric spaces, Fuzzy Sets and Systems99 (1998), 114–114.
12.
I.Sánchez and M.Sanchis, Complete invariant fuzzy metrics on groups, Fuzzy Sets and Systems330 (2018), 41–51.
13.
J.Rodríguez-López and S.Romaguera, The hausdorff fuzzy metric on compact sets, Fuzzy Sets and Systems147 (2004), 273–283.
14.
J.Kacprzyk, Multi-stage fuzzy control, Wiley, Chichester, first edition, 1997.
15.
P.Grzegorzewski, Metrics and orders in space of fuzzy numbers, Fuzzy Sets and Systems97 (1998), 83–94.
16.
J.Yen and R.Langari, Fuzzy Logic. Intelligence, Control and Information. Prentice - Hall, USA, first edition, 1999.
17.
N.Gueorguieva, I.Valova and G.Georgiev, M&mfcm: Fuzzy c-means clustering with mahalanobis and minkowski distance metrics, Procedia Computer Science114 (2017), 224–233.
18.
O.Lóipez-Ortega and M.-A.Rosales, An agent-oriented decision support system combining fuzzy clustering and the ahp, Expert Systems with Applications38(7) (2011), 8275–8284.
19.
C.Lu, J.Lan and Z.Wang, Aggregation of fuzzy opinions under group decision-making based on similarity and distance, Journal of Systems Science and Complexity19(1) (2006), 63–71.
20.
R.O.Parreiras, P.Ya.Ekel and D.C.Morais, Fuzzy set based consensus schemes for multicriteria group decision making applied to strategic planning, Group Decision and Negotiation21 (2012), 153–183.
21.
Y.C.Dong, H.Zhang and and E.Herrera-Viedma, Consensus reaching model in the complex and dynamic magdm problem, Knowledge-Based Systems106 (2016), 206–219.
22.
M-Fedrizzi and G.G.Pasi, Fuzzy Logic Approaches to Consensus Modelling in Group Decision Making, SpringerBerlin Heidelberg, Berlin, Heidelberg, 2008, pp. 19–37.
23.
M.Fedrizzi, M.Fedrizzi, R.A.Marques Pereira and M.Brunelli, The Dynamics of Consensus in Group Decision Making: Investigating the Pairwise Interactions between Fuzzy Preferences, SpringerBerlin Heidelberg, Berlin, Heidelberg, 2010, pp. 159–182.
24.
P.Wu, Q.Wu, L.Zhou, H.Chen, and H.Zhou, A consensus model for group decision making under trapezoidal fuzzy numbers environment, Neural Computing and Applications (2017).
25.
A.Maturo and A.-G.S.Ventre, Fuzzy Numbers and Consensus, Springer International Publishing, Cham, 2018, pp. 441–450.
26.
F.Maturo and V.Ventre, Consensus in Multiperson Decision Making Using Fuzzy Coalitions, Springer International Publishing, Cham, 2018, pp. 451–464.
27.
R.Chutia and M.Krishna Gogoi, Fuzzy risk analysis in poultry farming based on a novel similarity measure of fuzzy numbers, Applied Soft Computing X (2018), 1–39.
28.
D.Guha and D.Chakraborty, Fuzzy multi attribute group decision making method to achieve consensus under the consideration of degrees of confidence of experts' opinions, Computers and Industrial Engineering60 (2011), 493–504.
29.
S.H.Chen, Operations on fuzzy numbers with function principal, Tamkang Journal of Management Science6 (1985), 13–21.
30.
O.Castillo and P.Melin, Type-2 Fuzzy Logic: Theory and Applications.Springer, Heidelberg, first edition, 2008.
31.
X.X.Zhang, B.F.Ge and Y.J.Tan, A consensus model for group decision making under interval type-2 fuzzy environment, Frontiers of Information Technology & Electronic Engineering17(3) (2016), 237–249.
32.
X.X.Zhang and B.F.Ge, A new consensus model for group decision making using fuzzy linguistic preference relations with heterogeneous experts, Journal of Intelligent and Fuzzy Systems30(1) (2015), 171–182.
33.
I.Batyrshin, F.Monroy-Tenorio, A.Gelbukh, L.A.Villa-Vargas, V.Solovyev and N.Kubysheva, Bipolar rating scales: A survey and novel correlation measures based on nonlinear bipolar scoring functions, Acta Polytechnica Hun-garica14(3) (2017), 33–57.
34.
Y.C.Dong, C.-C.Li, Y.Xu and X.Gu, Consensus-based group decision making under multi-granular unbalanced 2-tuple linguistic preference relations, Group Decision and Negotiation24(2) (2015), 217–242.
35.
L.P.Dinu and F.Manea, An efficient approach for the rank aggregation problem, Theoretical Computer Science359 (2006), 455–461.
36.
E.Herrera-Viedma, F.J.Cabrerizzo, J.Kacprzyk and W.Pedrycz, A review of soft consensus models in a fuzzy environment, Information Fusion17 (2014), 4–13.
37.
J.Kacprzyk and S.Zadrozny, Soft computing and web intelligence for supporting consensus reaching, Soft Computing14(8) (2010), 833–846.
38.
A.Cervantes-Herrera, J.Ruiz-León, C.López-Limón, and A.Ramírez-Trevino, Output regulation and consensus of a class of multi-agent systems under switching communication topologies, Computación y Sistemas19(4) (2015), 757–767.
39.
I.Z.Batyrshin, N.Kubysheva, V.Solovyev and L.A.Villa-Vargas, Visualization of similarity measures for binary data and 2 x 2 tables, Computacion y Sistemas20(3) (2016), 345–353.
40.
Y.C.Dong, Z.-G.Ding and F.L.Martínez-Herrera, Managing consensus based on leadership in opinion dynamics, Information Sciences397-398 (2017), 187–205.
41.
Y.C.Dong, M.Zhang, G.Kou, Z.G.Ding and H.M.Liang, A survey on the fusion process in opinion dynamics, Information Fusion43 (2018), 57–65.
42.
K.T.Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems20 (1986), 89–97.
43.
J.Chu, X.Liu, Y.Wuang and K.I.-S.Chin, A group decision making model considering both the additive consistency and group consensus of intuitionistic fuzzy preference relations, Computers and Industrial Engineering101 (2016), 227–242.
