A new distance measure for trapezoidal fuzzy numbers. (English) Zbl 1296.03029
Summary: We propose a new distance measure for the space of all trapezoidal fuzzy numbers using centroid point and left/right spread of trapezoidal fuzzy numbers. Moreover, the metric properties of suggested distance measure are investigated. Indeed, we show that for two arbitrary trapezoidal fuzzy numbers if the distance between centroid points and also the distance between left spreads and right spreads go to zero, then two given fuzzy numbers are equal. Consequently, we complete discussion about the relation between fuzzy number and its centroid which is the firstly discussed by A. Hadi-Vencheh and M. Allame [Comput. Math. Appl. 59, No. 11, 3578–3582 (2010; Zbl 1197.26051)]. To the best of our knowledge, this is first time in the literature that such metric is applied by centroid point.
Citations:
Zbl 1197.26051References:
| [1] | L. A. Zadeh, “Fuzzy sets,” Information and Computation, vol. 8, pp. 338-353, 1965. · Zbl 0139.24606 · doi:10.1016/S0019-9958(65)90241-X |
| [2] | A. Kaufmann and M. M. Gupta, Introduction to Fuzzy Arithmetic: Theory and Application, Van Nostrand Reinhold, New York, NY, USA, 1991. · Zbl 0754.26012 |
| [3] | H. J. Zimmermann, Fuzzy sets theory and its applications, Kluwer Academic, Dordrecht, The Netherlands, Second edition, 1992. |
| [4] | A. Ban, L. Coroianu, and P. Grzegorzewski, “Trapezoidal approximation and aggregation,” Fuzzy Sets and Systems, vol. 177, pp. 45-59, 2011. · Zbl 1241.03059 · doi:10.1016/j.fss.2011.02.016 |
| [5] | P. Grzegorzewski, “Metrics and orders in space of fuzzy numbers,” Fuzzy Sets and Systems, vol. 97, no. 1, pp. 83-94, 1998. · Zbl 0930.03073 · doi:10.1016/S0165-0114(96)00322-3 |
| [6] | P. Grzegorzewski, “Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric,” Fuzzy Sets and Systems, vol. 148, no. 2, pp. 319-328, 2004. · Zbl 1056.03031 · doi:10.1016/j.fss.2003.08.005 |
| [7] | A. Hadi-Vencheh and M. Allame, “On the relation between a fuzzy number and its centroid,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3578-3582, 2010. · Zbl 1197.26051 · doi:10.1016/j.camwa.2010.03.051 |
| [8] | M. Friedman, M. Ma, and A. Kandel, “Numerical solutions of fuzzy differential and integral equations,” Fuzzy Sets and Systems, vol. 106, no. 1, pp. 35-48, 1999, Fuzzy modeling and dynamics. · Zbl 0931.65076 · doi:10.1016/S0165-0114(98)00355-8 |
| [9] | Y. M. Wang, J. B. Yang, D.-L. Xu, and K.-S. Chin, “On the centroids of fuzzy numbers,” Fuzzy Sets and Systems, vol. 157, no. 7, pp. 919-926, 2006. · Zbl 1099.91035 · doi:10.1016/j.fss.2005.11.006 |
| [10] | S. Abbasbandy and B. Asady, “Ranking of fuzzy numbers by sign distance,” Information Sciences, vol. 176, no. 16, pp. 2405-2416, 2006. · Zbl 1293.62008 · doi:10.1016/j.ins.2005.03.013 |
| [11] | S. Abbasbandy and T. Hajjari, “A new approach for ranking of trapezoidal fuzzy numbers,” Computers & Mathematics with Applications, vol. 57, no. 3, pp. 413-419, 2009. · Zbl 1165.03337 · doi:10.1016/j.camwa.2008.10.090 |
| [12] | B. Asady, “The revised method of ranking LR fuzzy number based on deviation degree,” Expert Systems with Applications, vol. 37, no. 7, pp. 5056-5060, 2010. · doi:10.1016/j.eswa.2009.12.005 |
| [13] | B. Asady and A. Zendehnam, “Ranking fuzzy numbers by distance minimization,” Applied Mathematical Modelling, vol. 31, no. 11, pp. 2589-2598, 2007. · Zbl 1211.03069 · doi:10.1016/j.apm.2006.10.018 |
| [14] | S. Salahshour, S. Abbasbandy, and T. Allahviranloo, “Ranking fuzzy numbers using fuzzy maximizingminimizing points,” in Proceedings of the 7th Conference of the European Society for Fuzzy Logic and Technology (IFSA-EUSFLAT ’11), vol. 1 of Advances in Intelligent Systems Research, pp. 763-769, 2011. · Zbl 1254.90087 · doi:10.2991/eusflat.2011.115 |
| [15] | Z. X. Wang, Y. J. Liu, Z. P. Fan, and B. Feng, “Ranking L-R fuzzy number based on deviation degree,” Information Sciences, vol. 179, no. 13, pp. 2070-2077, 2009. · Zbl 1166.90349 · doi:10.1016/j.ins.2008.08.017 |
| [16] | Y. M. Wang and Y. Luo, “Area ranking of fuzzy numbers based on positive and negative ideal points,” Computers & Mathematics with Applications, vol. 58, no. 9, pp. 1769-1779, 2009. · Zbl 1189.03061 · doi:10.1016/j.camwa.2009.07.064 |
| [17] | S. M. Chen and J. H. Chen, “Fuzzy risk analysis based on ranking generalized fuzzy numbers with different heights and different spreads,” Expert Systems with Applications, vol. 36, no. 3, pp. 6833-6842, 2009. · doi:10.1016/j.eswa.2008.08.015 |
| [18] | P. Grzegorzewski, “Nearest interval approximation of a fuzzy number,” Fuzzy Sets and Systems, vol. 130, no. 3, pp. 321-330, 2002. · Zbl 1011.03504 · doi:10.1016/S0165-0114(02)00098-2 |
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