Abstract

When a quantity is measured, the outcome depends on several factors like the measuring system, the measurement procedure, the skill of the operator, the environment, and other effects. In case of inexact quantity the outcome may also depends on the uncertainty representation. Recently inexact numbers got a lot of attention from many researchers in different fields. In this paper, we have tried to give an overview for the different representations of the inexact granular numbers. The objective of this overview is providing a certain insight into the essence of granular data representation being regarded as a framework of representing and manipulation of inexact information, and introduce a new representation of granular uncertain number to be as step in formulate a general form for all uncertain numbers.

References

• Young, R. C. , 1931" The algebra of many-valued quantities", Annals of Mathematics 31, pp. 260–290.
• Moore, R. E. , 1979" Method and Application of Interval Analysis", Prentice Hall, London.
• Zadeh, L. A. , 1965" Fuzzy sets", Information and Control 8, pp. 338–353.
• Zadeh, L. A. , 1975" The concept of a linguistic variable and its application to approximate reasoning", Information Sciences, 8 pp 199–249.
• Pawlak, Z. , 1982" Rough sets", International Journal of Information & Computer Sciences 11, pp. 341–356.
• Deng, J. L. , 1982" The Control problem of grey systems", System & Control Letters 1(5), pp. 288–294.
• Atanassov, K. T. , 1983" Intuitionistic fuzzy sets", VII ITKR's Session, pp. 391-396.
• Buehrer, D. J. 1992 "Vague logic: A first-order extension of interval probability theory" in Proc. IEEE Workshop Imprecise and Approximate Computation, Phoenix, AZ, pp. 83-87.
• Pedrycz, W. , 2005 "Knowledge-based clustering: from data to information granules", J. Wiley, Hoboken, NJ.
• Thomas Jech, , 2003" Set Theory" Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, pp. 642-646.
• Tiles, Mary, 2004" The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise". Dover Publications.
• Tobias Dantzig, 1930 "Number, the language of science";, New York, The Macmillan company.
• Sunaga, T. , 1958 "Theory of interval algebra and its application to numerical analysis", Tokyo, Japan, 2, pp. 29-46.
• Petkovi, M. and Petkovi, L. , 1998 "Complex interval arithmetic and its applications", Wiley.
• Zadeh, L. , 1996 "Fuzzy Sets, Fuzzy Logic, Fuzzy Systems", World Scientific Press, Singapore.
• Gerla, G. , 2006"Effectiveness and Multivalued Logics". Journal of Symbolic Logic 71 (1): pp. 137–162.
• Zimmermann, H. , 2001" Fuzzy set theory and its applications". Boston: Kluwer Academic Publishers.
• Novák, V. ; Perfilieva, I. and Mo?ko?, J. , 1999 "Mathematical principles of fuzzy logic", Dordrecht: Kluwer Academic.
• Buckley, J. J. 1991 "On the Hamacher sum of triangular fuzzy numbers", Fuzzy Sets and Systems 42 pp. 205-212.
• Wang, C. and Chen, S. , 2006 "A new method for appraising the performance of high school teachers based on simplified fuzzy number arithmetic operations". In: Proceedings of the 19th international conference on industrial, engineering& other applications of applied intelligent systems. Annecy, France, pp 432–441.
• Chen, L. , Cheng, C. , 2005 "Selecting IS personnel using ranking fuzzy number by metric distance method". Eur J Oper Res 160(3), pp. 803–820.
• Jerry M. Mendel, Robert I. and Bob John, 2002 "Type-2 Fuzzy Sets Made Simple", IEEE, April 2002.
• Juan R. Castro, Oscar and Castillo, 2007 "Interval Type-2 Fuzzy Logic for Intelligent Control Applications", IEEE.
• Pawlak, Z. 1992 "Rough sets: a new approach to vagueness," in: Fuzzy Logic for the Management of Uncertainty, John Wiley & Sons, New York, pp. 105-118.
• Pawlak, Z. 1994 "Hard and soft sets," in: Rough Sets, Fuzzy Sets and Knowledge Discovery, Springer-Verlag, London, pp. 130-135.
• Pawlak, Z. , 1996" Rough sets, rough relations and rough functions", Fundamental Informatics, 27(2-3), 103–108.
• Marcin M. , 2011 "Rough Numbers and Rough Regression", Lecture Notes in Computer Science Springer Moscow, Russia.
• Deng, J. L. , 2002"The Foundation of Grey Systems Theory". Wuhan: The Press of the Central China University of Science and Technology.
• Lin, Y. , Chen, M. Y. and Liu, S. , 2004" Theory of grey systems: capturing uncertainties of grey information", The International Journal of Systems and Cybernetics 33(2), pp. 196–218.
• Yang, Y. and Hinde,C. 2010 "A new extension of fuzzy sets using rough sets: R-fuzzy sets", Information Sciences 180 (3) pp. 354–365.
• Watanabe, A. , 2010 "Vague Quantity, Numerals, and Natural Numbers", the Dynamics of Complex Systems, 120(7), pp. 37–77.
• G. Beliakova, H. Bustinceb, D. Goswamic, U. Mukherjeed, N. Pale, 2011 "On averaging operators for atanassov's intuitionistic fuzzy sets", Information Sciences 181 (6) pp. 1116–1124.
• Pedrycz, W. and Gomide, F. , 2007" Fuzzy systems engineering: toward human-centric computing", John Wiley, Hoboken, NJ.
• Pedrycz, W. and Valente de Oliveira, J. , 2008 "A development of fuzzy encoding and decoding through fuzzy clustering", IEEE Transactions on Instrumentation and Measurement, 57, N. 4, pp. 829-837.
• Pedrycz, W. and Liu, X. , 2009" Axiomatic fuzzy set theory and its applications", Springer Verlag, Berlin 244, pp. 2475–2486.
• Pedrycz, W. , Loia, V. and Senatore, S. 2010 " Fuzzy clustering with viewpoints", IEEE Trans. on Fuzzy Systems, 161(1) pp. 56-74.