• Victor Bocharnikov National Aviation University



Fuzzy measure, index of clearness, quality of control, sets


Objective: To justify the need and propose a new clarity index for the distribution of a fuzzy measure with an arbitrary modality. Conduct a study of the new index and show its effectiveness, sensitivity and ease of use for analyzing fuzzy data. Methods: for solving the problem using methods of set theory, fuzzy measures theory and functional analysis, and formal logic. Results: A new index is justified and proposed, which provides an estimate of the clarity for the distribution of a fuzzy measure with an arbitrary modality. Formulas are proposed for calculating the clarity index for a fuzzy measure on a discrete and continuous space. It is proved that the proposed index satisfies the properties that are advanced to the clarity indices. Additional dependencies are obtained to calculate the clarity index based on the use of level sets for the fuzzy measure density function. Discussion: the results of the calculation of the clarity index for a family of fuzzy measures with various modalities are presented. It is shown that the proposed index completely satisfies the advanced requirements to the logic of the performance of the clarity index and takes into account the modality of the fuzzy measure.

Author Biography

Victor Bocharnikov, National Aviation University

Doctor of Science. Professor.

Honored Worker of Science and Technology of Ukraine

Aerospace Control Systems Department, National Aviation University, Kyiv, Ukraine

Education: Kiev Higher Military Aviation Engineering School (1987)

Research area: Artificial intelligence, modeling and solving analytical problems under uncertainty, the theory of fuzzy sets, measures, and integrals.


Pospelov D.A., eds. (1986), Nechetkie mnozhestva v modelyakh upravleniya i iskusstvennogo intellekta. [Fuzzy sets in control models and artificial intelligence]. Moscow, Nauka Publ., 396 p.

Klir G. (2006) Uncertainty and information: foundations of generalized information theory. New Jersey, Published by John Wiley & Sons, Inc., Hoboken, 539 p.

Dubois D., Prade H. (1988) Théorie des possibilités: application á la représentation des connaissances en informatique. Paris,. Masson, 287 p.

Klir G. Elias D. (1985) Architecture of Systems Problem Solving. New York, Plenum Press, 354 p.

Yager R. (1982) Measuring Tranquility and Anxiety in Decision Making: an Application of Fuzzy Sets, Int. J. General Systems, Vol.8, pp. 139-146.

Deza M-M, Deza E. (2008) Encyclopedia of distances. Berlin, Springer, 2008. 412 p. (Russ. ed.: Deza M-M, Deza E. Entsiklopedicheskii slovar' rasstoyanii. Moscow, Nauka Publ., 444 p)

Bocharnikov V., Bocharnikov I. (2010) Discrete Fuzzy Filter Of Uav’s Flight Parameters. Proceedings of the NAU, no 3, pp.30-38.

Bocharnikov V., Bocharnikov I. (2012) Optimal discrete fuzzy filter jf UAV’s flight parameters. Proceedings of the NAU, no 3, pp. 22-29.

Bocharnikov V. (2000) Fuzzy-tekhnologiya: Matematicheskie osnovy. Praktika modelirovaniya v ekonomike. [Fuzzy-technology: Mathematical basics. The practice of modeling in economics]. St.Peterburg, Nauka Publ., 328p.

Asai K., Vatada D., Ivai S. eds. (1994) Applied fuzzy systems. Translation from Japanese, Under ed. T. Terano, K. Asai, M. Sugeno. Tokyo, AP Professional, 1994 (Russ. ed.: Asai K., Vatada D., Ivai S. Prikladnye nechetkie sistemy. Moscow, Mir Publ., 386 p.)

Tsukamoto Y. (1972) Identification of preference measure by means of fuzzy integral. Ann. Conf. of JORS, pp. 131-135.

Sugeno M. (1972) Fuzzy Measure and Fuzzy Integral. Transaction of the Sosiety of Instrument and Control Engineers, vol. 10, no 2, pp. 218-226.

Korn G., Korn T. (1984) Spravochnik po matematike dlya nauchnykh rabotnikov i inzhenerov. [Mathematical Handbook for Scientists and Engineers]. Moscow, Nauka Publ., 831 p.



How to Cite

Bocharnikov, V. (2018). DETERMINATION THE INDEX OF CLEARNESS FOR THE DISTRIBUTION OF FUZZY MEASURES. Proceedings of National Aviation University, 76(3), 44–55.