Uncertain Rule-Based Fuzzy Systems pp 307-383 | Cite as

# Working with Type-2 Fuzzy Sets

## Abstract

This chapter explains how to work with type-2 fuzzy sets (T2 FSs). Most of its topics are needed in the rest of this book. Coverage includes: set-theoretic operations (union, intersection, and complement) for general type-2 fuzzy sets (GT2 FSs) computed using the Extension Principle, set-theoretic operations for interval type-2 fuzzy sets (IT2 FSs), set-theoretic operations for GT2 FSs computed using horizontal slices, type-2 relations and compositions on the same product space and on different product spaces, compositions of a T2 FS with a type-2 relation, type-2 hedges, Extension Principle for IT2 and GT2 FSs, functions of GT2 FSs computed using \( \alpha \)-planes, Cartesian product of T2 FSs, implications, an appendix about the properties of T2 FSs and an appendix that has detailed proofs of many theorems. 27 examples are used to illustrate the chapter’s important concepts.

## References

- Bilgin, A., H. Hagras, A. Malibari, M. Alhaddad, and D. Alghazzawi. 2012a. Towards a general type-2 fuzzy logic approach for computing with words using linear adjectives. In
*Proceedings of FUZZ-IEEE 2012*, 1130–1137. Brisbane, AU.Google Scholar - Bilgin, A., J. Dooley, L. Whittington, H. Hagras, M. Henson, C. Wagner, A. Malibari, A. Al-Ghamdi, M. Alhaddad, and D. Alghazzawi. 2012b. Dynamic profile-selection for zslices based type-2 fuzzy agents controlling multi-user ambient intelligent environments. In
*Proceedings of FUZZ-IEEE 2012*, 1392–1399. Brisbane, AU.Google Scholar - Bilgin, A., H. Hagras, A. Malibari, M. J. Alhaddad, and D. Alghazzawi. 2012c. A general type-2 fuzzy logic approach for adaptive modeling of perceptions for computing with words. In
*Proceedings of 2012 12th UK Workshop on Computational Intelligence*(*UKCI*), 1–8.Google Scholar - Bilgin, A., H. Hagras, A. Malibari, M. J. Alhaddad, and D. Alghazzawi. 2013a. An experience based linear general type-2 fuzzy logic approach for computing with words. In
*Proceedings of IEEE Int’l. Conference on Fuzzy Systems*, Paper #1139, Hyderabad, India.Google Scholar - Bilgin, A., H. Hagras, A. Malibari, M. J. Alhaddad, and D. Alghazzawi. 2013b. Towards a linear general type-2 fuzzy logic based approach for computing with words.
*Soft Computing*.Google Scholar - Bustince, H., et al. 2016. A historical account of types of fuzzy sets and their relationships.
*IEEE Transactions on Fuzzy Systems*24: 179–194.CrossRefGoogle Scholar - Chen, Q., and S. Kawase. 2000. On fuzzy-valued fuzzy reasoning.
*Fuzzy Sets and Systems*113: 237–251.MathSciNetCrossRefzbMATHGoogle Scholar - Cornelis, C., and E. Kerre. 2004. Inclusion measures in intuitionistic fuzzy set theory.
*Lecture Notes in Computer Science*2711: 345–356.MathSciNetCrossRefzbMATHGoogle Scholar - Coupland, S. and R. I. John. (2004) A new and efficient method for the type-2 meet operation. In
*Proceedings of IEEE FUZZ Conference*, 959–964. Budapest, Hungary.Google Scholar - Coupland, S., and R. I. John. 2005. Towards more efficient type-2 fuzzy logic systems. In
*Proceedings of IEEE FUZZ Conference*, 236–241. Reno, NV, May 2005.Google Scholar - Coupland, S., and R.I. John. 2007. Geometric type-1 and type-2 fuzzy logic systems.
*IEEE Transactions on Fuzzy Systems*15: 3–15.CrossRefzbMATHGoogle Scholar - Coupland, S., and R. I. John. 2013. Geometric type-2 fuzzy sets. In
*Advances in type-2 fuzzy sets and systems: Theory and applications, edited by*A. Sadeghian, J. M. Mendel, and H. Tahayori. Springer, New York.Google Scholar - Dubois, D., and H. Prade. 1978. Operations on fuzzy numbers.
*International Journal of Systems Science*9: 613–626.MathSciNetCrossRefzbMATHGoogle Scholar - Dubois, D., and H. Prade. 1979. Operations in a fuzzy-valued logic.
*Information and Control*43: 224–240.MathSciNetCrossRefzbMATHGoogle Scholar - Dubois, D., and H. Prade. 1980.
*Fuzzy sets and systems: Theory and applications*. NY: Academic Press.zbMATHGoogle Scholar - Greenfield, S., and R. I. John. 2007. Optimized generalized type-2 join and meet operations. In
*Proceeding of FUZZ-IEEE 2007,*141–146. London, UK.Google Scholar - Hamrawi, H., S. Coupland, and R. John. 2010. A novel alpha-cut representation for type-2 fuzzy sets. In
*Proceeding of FUZZ-IEEE 2010*,*IEEE World Congress on Computational Intelligence*, 351–358. Barcelona, Spain.Google Scholar - Hao, M., and J.M. Mendel. 2014. Similarity measures for general type-2 fuzzy sets based on the α-plane representation.
*Information Sciences*277: 197–215.MathSciNetCrossRefzbMATHGoogle Scholar - Harding, J., C. Walker, and E. Walker. 2010. The variety generated by the truth value algebra of type-2 fuzzy sets.
*Fuzzy Sets and Systems*161: 735–749.MathSciNetCrossRefzbMATHGoogle Scholar - John, R., J. Mendel, and J. Carter. 2006. The extended sup-star composition for type-2 fuzzy sets made simple. In
*Proceeding of 2006 IEEE International Conference on Fuzzy Systems*, 1441–1445. Vancouver, BC, Canada.Google Scholar - Karnik, N. N., and J. M. Mendel. 1988a. Introduction to type-2 fuzzy logic systems. In
*Proceedings of 1998 IEEE FUZZ Conference*, 915–920. Anchorage, AK.Google Scholar - Karnik, N. N., and J. M. Mendel. 1998b.
*Operations on Type-2 Fuzzy Sets*. USC-SIPI Report # 319, University of Southern California, Los Angeles, CA. This can be accessed at: http://sipi.usc.edu/research; then choose “sipi technical reports/319.”. - Karnik, N. N., and J. M. Mendel. 1998c.
*An Introduction to Type-2 Fuzzy Logic Systems*, USC-SIPI Report #418, University of Southern California, Los Angeles, CA. This can be accessed at: http://sipi.usc.edu/research; then choose “sipi technical reports/418.”. - Karnik, N.N., and J.M. Mendel. 2001. Operations on type-2 fuzzy sets.
*Fuzzy Sets and Systems*122: 327–348.MathSciNetCrossRefzbMATHGoogle Scholar - Klir, G.J., and T.A. Folger. 1988.
*Fuggy sets, uncertainity, and information*. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar - Klir, G.J., and B. Yuan. 1995.
*Fuzzy sets and fuzzy logic: Theory and applications*. Upper Saddle River, NJ: Prentice Hall.zbMATHGoogle Scholar - Kreinovich, V., and G. Xiang. 2008. Towards fast algorithms for processing type-2 fuzzy data: Extending Mendel’s algorithms from interval-valued to a more general case, In
*Proceeding of NAFIPS 2008*, Paper #60106, New York City.Google Scholar - Lee, L.-W., and S.-M. Chen. 2008. A new method for fuzzy multiple attributes group decision-making based on the arithmetic of interval type-2 fuzzy sets. In
*Proceeding of the Seventh International Conference on Machine Learning and Cybernetics*, 3084–3089. Kunming, China.Google Scholar - Lin, C.-T., and C.S.G. Lee. 1996.
*Neural fuzzy systems: A neuro-fuzzy synergism to intelligent systems*. Upper Saddle River, NJ: Prentice-Hall PTR.Google Scholar - Livi, L., H. Tahayori, A. Sadeghian, and A. Rizzi. 2014. Distinguishability of interval type-2 fuzzy sets data by analyzing upper and lower membership functions.
*Applied Soft Computing*17: 79–89.CrossRefGoogle Scholar - Mendel, J.M. 2001.
*Introduction to rule-based fuzzy logic systems*. Upper Saddle River, NJ: Prentice-Hall.zbMATHGoogle Scholar - Mendel, J.M. 2009. On answering the question ‘Where do I start in order to solve a new problem involving interval type-2 fuzzy sets?’.
*Information Sciences*179: 3418–3431.MathSciNetCrossRefzbMATHGoogle Scholar - Mendel, J. M. 2011. On the geometry of join and meet calculations for general type-2 fuzzy sets. In
*Proceedings of FUZZ-IEEE 2011*, 2407–2413. Taipei, Taiwan.Google Scholar - Mendel, J.M., R.I. John, and F. Liu. 2006. Interval type-2 fuzzy logic systems made simple.
*IEEE Transactions on Fuzzy Systems*14: 808–821.CrossRefGoogle Scholar - Mendel, J.M., F. Liu, and D. Zhai. 2009. Alpha-plane representation for type-2 fuzzy sets: Theory and applications.
*IEEE Transactions on Fuzzy Systems*17: 1189–1207.CrossRefGoogle Scholar - Mendel, J.M., and D. Wu. 2010.
*Perceptual computing: Aiding people in making subjective judgments*. Hoboken, NJ: Wiley and IEEE Press.CrossRefGoogle Scholar - Miyakoshi, M., Y. Sato, and M. Kawaguchi. 1980. A fuzzy-fuzzy relation and its application to the clustering technique.
*Behaviormetrika*8: 15–22.CrossRefGoogle Scholar - Mizumoto, M., and K. Tanaka. 1976. Some properties of fuzzy sets of type-2.
*Information and Control*31: 312–340.MathSciNetCrossRefzbMATHGoogle Scholar - Mizumoto, M., and K. Tanaka. 1981. Fuzzy sets of type-2 under algebraic product and algebraic sum.
*Fuzzy Sets and Systems*5: 277–290.MathSciNetCrossRefzbMATHGoogle Scholar - Nieminen, J. 1977. On the algebraic structure of fuzzy sets of type-2.
*Kybernetica*, 13 (4): 261–273.Google Scholar - Nguyen, H. T., and V. Kreinovich. 2008.Computing degrees of subsethood and similarity for interval-valued fuzzy sets: Fast algorithms. In
*Proceedings of of 9th International Conference on Intelligent Technologies in Tech’08*, 47–55. Samui, Thailand.Google Scholar - Rajati, M.R., and J.M. Mendel. 2013. Novel weighted averages versus normalized sums in Computing With Words.
*Information Sciences*235: 130–149.MathSciNetCrossRefzbMATHGoogle Scholar - Rickard, J. T., J. Aisbett, G. Gibbon, and D. Morgenthaler. 2008. Fuzzy subsethood for type-n fuzzy sets. In
*Proceeding of NAFIPS*, New York.Google Scholar - Ruiz-Garcia, G., H. Hagras, H. Pomares, I. Rojas, and H. Bustince. 2016. Join and meet operations for type-2 fuzzy sets with non-convex secondary memberships.
*IEEE Transactions on Fuzzy Systems*.Google Scholar - Tahayori, H., A. G. B. Tettamanzi, and G. D. Antoni. 2006. Approximated type-2 fuzzy set operations. In
*Proceedings of FUZZ-IEEE 2006*, 9042–9049. Vancouver, B.C. Canada.Google Scholar - Wagner, C., and H. Hagras. 2008. z Slices—towards bridging the gap between interval and general type-2 fuzzy logic. In
*Proceeding of IEEE FUZZ Conference*, Paper # FS0126, Hong Kong, China.Google Scholar - Wagner, C., and H. Hagras. 2010. Towards general type-2 fuzzy logic systems based on zSlices.
*IEEE Transactions on Fuzzy Systems*18: 637–660.CrossRefGoogle Scholar - Wagner, C., and H. Hagras. 2013. zSlices based general type-2 fuzzy sets and systems. In
*Advances in Type-2 Fuzzy Sets and Systems: Theory and Applications,*edited by A. Sadeghian, J. M. Mendel and H. Tahayori. Springer, New York.Google Scholar - Walker, C.L., and E. Walker. 2005. The algebra of fuzzy truth values.
*Fuzzy Sets Systems*149: 309–347.MathSciNetCrossRefzbMATHGoogle Scholar - Walker, C. L., and E. Walker. 2006. Automorphisms of the algebra of fuzzy truth values.
*International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems*, 14: 711–732.Google Scholar - Walker, C.L., and E.A. Walker. 2009. Sets with type-2 operations.
*International Journal of Approximate Reasoning*50: 63–71.MathSciNetCrossRefzbMATHGoogle Scholar - Walker, C.L., and E.A. Walker. 2014. Type-2 operations on finite chains.
*Fuzzy Sets and Systems*236: 33–49.MathSciNetCrossRefzbMATHGoogle Scholar - Wei, S.-H., and S.-M. Chen. 2009. Fuzzy risk analysis based on interval-valued fuzzy numbers.
*Expert Systems with Applications*36: 2285–2299.Google Scholar - Wu, D., and J.M. Mendel. 2007. Uncertainty measures for interval type-2 fuzzy sets.
*Information Sciences*177: 5378–5393.MathSciNetCrossRefzbMATHGoogle Scholar - Wu, D., and J.M. Mendel. 2009. A comparative study of ranking methods, similarity measures and uncertainty measures for interval type-2 fuzzy sets.
*Information Sciences*179: 1169–1192.MathSciNetCrossRefGoogle Scholar - Zadeh, L.A. 1975. The concept of a linguistic variable and its application to approximate reasoning–1.
*Information Sciences*8: 199–249.MathSciNetCrossRefzbMATHGoogle Scholar - Zeng, W. Y. Zhou, and H. Li. 2007. Extension principle of interval-valued fuzzy set. In
*Fuzzy Information and Engineering: Proceedings of the Second International Conference on Fuzzy Information and Engineering*(*ICFIE*), 125–137. Springer.Google Scholar - Zhai, D., and J.M. Mendel. 2011. Uncertainty measures for general type-2 fuzzy sets.
*Information Sciences*181: 503–518.MathSciNetCrossRefzbMATHGoogle Scholar