In order to aggregate all uncertain information of the footprint of uncertainty (FOU) and improve the speed of defuzzification for interval type-2 fuzzy sets (IT2 FS), the normal ordered weighted averaging (NOWA) weighted sampling method is proposed to transform an IT2 FS into a type-1 fuzzy set (T1 FS), whose defuzzified values closely approximate those of the expected values of all embedded T1 FSs. Compared with the existing methods, the proposed method is more simple and practical, and the result is mainly determined by the numbers and weights of random samples in each vertical slice of FOU. Finally, based on the NOWA weighted sampling method, a new possibility degree is introduced to solve multi-attribute decision-making problems in which the attribute values take the form of interval type-2 fuzzy numbers.
UluC., GuzelkayaM. and EksinI., A closed form type reduction method for piecewise linear interval type-2 fuzzy sets, International Journal of Approximate Reasoning54 (2013), 1421–1433.
2.
WuD.R. and MendelJ.M., Aggregation using the linguistic weighted average and interval type-2 fuzzy sets, IEEE Transactions on Fuzzy Systems15 (2007), 1145–1161.
3.
WuD.R. and MendelJ.M., A vector similarity measure for linguistic approximation: Interval type-2 and type-1 fuzzy sets, Information Sciences178 (2008), 381–402.
4.
WuD.R. and MendelJ.M., A comparative study of ranking methods, similarity measures and uncertainty measures for interval type-2 fuzzy sets, Information Sciences179 (2009), 1169–1192.
5.
JanaD.K., PramanikS. and MaitiM., Mean and CV reduction methods on Gaussian type-2 fuzzy set and its application to a multilevel profit transportation problem in a two-stage supply chain network, Neural Computing & Applications1 (2016), 1–24.
6.
WuD.R. and MendelJ.M., Uncertainty measures for interval type-2 fuzzy sets, Information Science177 (2007), 5378–5393.
7.
MendelJ.M., Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions, Upper Saddle River, Prentice-Hall, 2001.
8.
MendelJ.M. and WuH.W., Type-2 fuzzistics for symmetric interval type-2 fuzzy sets: Part 1, forward problems, IEEE Transactions on Fuzzy Systems14 (2006), 781–792.
9.
MendelJ.M. and WuH.W., Type-2 fuzzistics for symmetric interval type-2 fuzzy sets: Part 2, inverse problems, IEEE Transactions on Fuzzy Systems15 (2007), 301–308.
10.
Figueroa GarcíaJ.C., An approximation method for Type Reduction of an Interval Type-2 fuzzy set based on α-cuts, Proceedings of the FedCSIS 2012, Wrocław, (2012), pp. 49–54.
11.
HuJ.H., ZhangY., ChenX.H. and LiuY.M., Multi-criteria decision making method based on possibility degree of interval type-2 fuzzy number, Knowledge-Based Systems43 (2013), 21–29.
12.
ZadehL., The concept of a linguistic variable and its application to approximate reasoning, Part 1, Information Sciences8 (1975), 199–249.
13.
AbbasiM. and KavianiM.A., Operational efficiency-based ranking framework using uncertain DEA methods: An application to the cement industry in Iran, Management Decision54 (2016), 902–928.
14.
NieM. and TanW.W., Towards an efficient type-reduction method for interval type-2 fuzzy logic systems, Proceedings of FUZZ-IEEE 2008, Hong Kong, (2008), pp. 1425–1432.
15.
KarnikN.N. and MendelJ.M., Centroid of a type-2 fuzzy set, Information Sciences132 (2001), 195–220.
16.
SalazarO. and SorianoJ., Generating embedded type-1 fuzzy sets by means of convex combination, Proceedings of IFSA-NAFIPS 2013, Edmonton, (2013), pp. 51–56.
17.
QinR., LiuY. and LiuZ., Methods of critical value reduction for type-2 fuzzy variables and their applications, Journal of Computational and Applied Mathematics235 (2011), 1454–1481.
18.
YagerR.R., On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Transactions on Systems Man & Cybernetic18 (1988), 183–190.
19.
GreenfieldS., ChiclanaF., CouplandS. and JohnR.I., The collapsing method of defuzzification for discretised interval type-2 fuzzy sets, Information Sciences179 (2008), 2055–2069.
20.
GreenfieldS., ChiclanaF., JohnR. and CouplandS., The sampling method of defuzzification for type-2 fuzzy sets: Experimental evaluation, Information Sciences189 (2012), 77–92.
21.
ChenS.M. and LeeL.W., 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 (2010), 824–833.
22.
XuZ.S., An overview of methods for determining OWA weights, International Journal of Intelligent Systems20 (2005), 843–865.
23.
XuZ.S., A ranking arithmetic for fuzzy mutual complementary judgment matrices, Journal of Systems Engineering16 (2001), 311–314.
