In this paper, we introduce the concept of complex multi-fuzzy sets (CMkFSs) as a generalization of the concept of multi-fuzzy sets by adding the phase term to the definition of multi-fuzzy sets. In other words, we extend the range of multi-membership function from the interval [0,1] to unit circle in the complex plane. The novelty of CMkFSs lies in the ability of complex multi-membership functions to achieve more range of values while handling uncertainty of data that is periodic in nature. The basic operations on CMkFSs, namely complement, union, intersection, product and Cartesian product are studied along with accompanying examples. Properties of these operations are derived. Finally, we introduce the intuitive definition of the distance measure between two complex multi-fuzzy sets which are used to define δ-equalities of complex multi-fuzzy sets.
ZadehL.A., Fuzzy set, Information and Control8(3) (1965), 338–353.
2.
CabrerizoF.J., ChiclanaF., Al-HmouzR., MorfeqA., BalamashA.S. and Herrera-ViedmaE., Fuzzy decision making and consensus: Challenges, Journal of Intelligent and Fuzzy Systems29(3) (2015), 1109–1118.
3.
Sarkodie-GyanT., YuH., AlaqtashM., BogaleM.A., MoodyJ. and BrowerR., Application of fuzzy sets for assisting the physician’s model of functional impairments in human locomotion, Journal of Intelligent and Fuzzy Systems25(4) (2013), 1001–1011.
4.
LeeL.W. and ChenS.M., Fuzzy decision making and fuzzy group decision making based on likelihood-based comparison relations of hesitant fuzzy linguistic term sets, Journal of Intelligent and Fuzzy Systems29(3) (2015), 1119–1137.
5.
AtanassovK.T., Intuitionistic fuzzy sets, Fuzzy Sets and Systems20(1) (1986), 87–96.
6.
SmarandacheF., Neutrosophic set- A generalisation of the intuitionistic fuzzy sets, International Journal of Pure and Applied Mathematics24(3) (2005), 287–297.
7.
SebastianS. and RamakrishnanT.V., Multi-fuzzy sets, International Mathematical Forum5(50) (2010), 2471–2476.
8.
SebastianS. and RamakrishnanT.V., Multi-fuzzy sets: An extension of fuzzy sets, Fuzzy Information and Engineering3(1) (2011), 35–43.
9.
SmarandacheF., n-Valued refined neutrosophic logic and its applications in Physics, Progress in Physics4 (2013), 143–146.
10.
RamotD., MiloR., FriedmanM. and KandelA., Complex fuzzy sets, IEEE Transactions on Fuzzy Systems10(2) (2002), 171–186.
11.
RamotD., FriedmanM., LangholzG. and KandelA., Complex fuzzy logic, IEEE Transactions on Fuzzy Systems11(4) (2003), 450–461.
AliM. and SmarandacheF., Complex neutrosophic set, Neural Computing and Applications (2016). doi: 10.1007/s00521-015-2154-y.
14.
PappisC.P., Value approximation of fuzzy systems variables, Fuzzy Sets and Systems39(1) (1991), 111–115.
15.
HongD.H. and HwangS.Y., A note on the value similarity of fuzzy systems variables, Fuzzy Sets and Systems66(3) (1994), 383–386.
16.
CaiK.Y., δ-Equalities of fuzzy sets, Fuzzy Sets and Systems76(1) (1995), 97–112.
17.
CaiK.Y., Robustness of fuzzy reasoning and δ-equalities of fuzzy sets, IEEE Transactions on Fuzzy Systems9(5) (2001), 738–750.
18.
ZhangG., DillonT.S., CaiK.Y., MaJ. and LuJ., Operation properties and δ-equalities of complex fuzzy sets, International Journal of Approximate Reasoning50(8) (2009), 1227–1249.
19.
BuckleyJ.J., Fuzzy complex numbers, Fuzzy Sets and Systems33(1) (1989), 333–345.
20.
DeyA. and PalM., Multi-fuzzy complex numbers and multifuzzy complex sets, International Journal of Fuzzy Systems Applications4(2) (2014), 15–27.
21.
JunM., ZhangG. and LuJ., A method for multiple periodic factor prediction problems using complex fuzzy sets, IEEE Transactions on Fuzzy Systems20(1) (2012), 32–45.
22.
Al-QudahY., Complex multi-fuzzy set, MSc Research Project, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 2016.
