A new attitude coupled with the basic fuzzy thinking to distance between two fuzzy numbers

Document Type: Research Paper

Authors

1 Department of Mathematics Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran

2 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

Abstract

Fuzzy measures are suitable in analyzing human subjective evaluation processes. Several different strategies have been proposed for distance of fuzzy numbers. The distances introduced for fuzzy numbers can be categorized in two groups:\\
1. The crisp distances which explain crisp values for the distance between two fuzzy numbers.\\
2. The fuzzy distance which introduce a fuzzy distance for normal fuzzy numbers. It was introduced by Voxman \cite{33} for the first time through using  $\alpha$-cut.\\
However, both mentioned concepts can lead to unsatisfactory results from the applications point of view, but there is no method, which gives a satisfactory result to all situations. In this paper,  a new attitude coupled with fuzzy thinking to the fuzzy distance function on the set of fuzzy numbers is proposed. In this new fuzzy distance, we considered both mentioned attitudes,  then we introduced new fuzzy distance based on a combination (hybrid) of those two. Some properties of the proposed fuzzy distance have been discussed. Finally, several examples have been provided to explain the application of the proposed method and compare this methods with others.

Keywords


[1] S. Abbasbandy and T. Hajjari, A new approach for ranking of trapezoidal fuzzy numbers,
Computers and Mathematics with Applications, 57 (2009), 413-419.
[2] F. Abbasi, T. Allahviranloo and S. Abbasbandy,A new attitude coupled with fuzzy thinking
to fuzzy rings and elds, Journal of Intelligent and Fuzzy Systems, 29 (2015), 851-861.
[3] F. Abbasi, T. Allahviranloo and S. Abbasbandy,A new attitude coupled with fuzzy thinking
to fuzzy group and subgroup, Journal of Fuzzy Set Valued Analysis, 4 (2015), 1-18.
[4] F. Abbasi, T. Allahviranloo and S. Abbasbandy, A new attitude coupled with the basic think-
ing to ordering for ranking fuzzy numbers, International Journal of Industrial Mathematics,
8(4) (2016), 365-375.
[5] D. Altman, Fuzzy set theoretic approaches for handling imprecision in spatial analysis, International
Journal of Geographical Information Systems, 8 (1994), 271-289.
[6] M. Ali Beigi, T. Hajjari and E. Ghasem Khani,An Algorithm to Determine Fuzzy Distance
Measure, 13th Iranian Conference on Fuzzy Systems (IFSC), 2013.
[7] I. Bloch,On fuzzy distances and their use in image processing under imprecisionm, Pattern
Recognition, 32(11) (1999), 1873-1895.
[8] C. Chakraborty and D. Chakraborty, Atheoretical development on a fuzzy distance measure
for fuzzy numbers, Mathematical and Computer Modelling, 43(3-4) (2006), 254-261.
[9] S. H. Chen and C. C. Wang,Fuzzy distance of trapezoidal fuzzy numbers, In: Proceedings of
the 9th Joint Conference on Information Sciences, JCIS 2006.

[10] C.H. Cheng,A new approach for ranking fuzzy numbers by distance method, Fuzzy Sets and
Systems, 95(3) (1998), 307-317.
[11] S. H. Chen and C. H. Hsieh,Graded mean integration representation of generalized fuzzy
number, Proceeding of TFSA, 1998.
[12] C. Chakraborty and D. Chakraborty, A theoretical development on a fuzzy distance measure
for fuzzy numbers, Mathematical and Computer Modeling, 43( 2006), 254-261, .
[13] T. C. Chu and C. T. Tsao,Ranking fuzzy numbers with an area between the centroid point
and original point, Computers Math. Applications, 43 (2002), 111-117.
[14] M. M. Deza and E. Deza, Encyclopedia of Distances, 2009.
[15] P. D'Urso and P. Giordani, A weighted fuzzy c-means clustering model for fuzzy data, Computational
Statistics and Data Analysis, 50(6) (2006), 1496-1523.
[16] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, 1980.
[17] R. Fullor,Fuzzy reasoning and fuzzy optimization, On leave from Department of Operations
Research, Eotvos Lorand University, Budapest, 1998.
[18] D. Guha and D. Chakraborty,A new approach to fuzzy distance measure and similarity mea-
sure between two generalized fuzzy numbers, Applied Soft Computing, 10(1) (2010), 90-99.
[19] J. Kacprzyk,Multistage fuzzy control: a prescriptive approach, John Wiley and Sons, Inc,
1997.
[20] G. J. Klir and B. Yuan,Fuzzy sets and fuzzy logic: theory and applications, Prentice-Hall
PTR, Upper Saddlie River, 1995.
[21] S. Nezhad, A. Noroozi and A. Makui,Fuzzy distance of triangular fuzzy numbers, Journal of
Intelligent and Fuzzy Systems, 2012.
[22] A. Rosenfeld,Distance between Fuzzy Sets. Pattern Recognition Letters, 3 (1985), 229-233.
[23] H. Rouhparvar, A. Panahi and A. Noorafkan Zanjani,Fuzzy distance measure for fuzzy num-
bers, Australian Journal of Basic and Applied Sciences, 5(6) (2011), 258-265.
[24] C. Shan-Huo and W. Chien-Chung, Fuzzy distance using fuzzy absolute value, In: Machine
Learning and Cybernetics, International Conference, 2009.
[25] K. Sridharan and H. E. Stephanou,Fuzzy distance functions for motion planning, In: Tools
with Arti cial Intelligence. TAI '92, Proceedings., Fourth International Conference, 1992.
[26] S. R. Sudharsanan, Fuzzy distance approach to routing algorithms for optimal web path
estimation, In: Fuzzy Systems, The 10th IEEE International Conference, 2001.
[27] L. Stefanini, A generalization of hukuhara di erence and division for interval and fuzzy
arithmetic, Fuzzy Sets and Systems, 161 (2010), 1564-1584.
[28] L. Tran and L. Duckstein, Comparison of fuzzy numbers using a fuzzy distance measure,
Fuzzy Sets and Systems, 130(3) (2002), 331-341.
[29] W. Voxman, Some remarks on distances between fuzzy numbers, Fuzzy Sets and Systems,
100(1-3) (1998), 353365.