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Nearest interval, triangular and trapezoidal approximation of a fuzzy number preserving ambiguity. (English) Zbl 1258.03079

The paper shows how to approximate a fuzzy number in such a way that an average Euclidean distance between the original fuzzy number and its approximation is minimized under the condition of preservation of ambiguity. Here the ambiguity of a fuzzy number \(A\) is expressed in the form \[ \int^1_0 \alpha(A_U(\alpha)- A_L(\alpha))\,d\alpha \] with \(A_U\) and \(A_L\) computed as \(A_U(\alpha)= \sup\{x\in \mathbb R\mid A(x)\geq \alpha\}\) and \(A_L(\alpha)= \text{inf}\{x\in \mathbb R\mid A(x)\geq \alpha\}\).
The study provides detailed formulas in case of approximation realized by (a) intervals, (b) triangular fuzzy numbers, and (c) trapezoidal fuzzy numbers.

MSC:

03E72 Theory of fuzzy sets, etc.
26E50 Fuzzy real analysis
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[1] Abbasbandy, S.; Amirfakhrian, M., The nearest approximation of a fuzzy quantity in parametric form, Applied mathematics and computation (New York), 172, 624-632, (2006) · Zbl 1097.41001
[2] Abbasbandy, S.; Amirfakhrian, M., The nearest trapezoidal form of a generalized left right fuzzy number, International journal of approximate reasoning, 43, 166-178, (2006) · Zbl 1112.03316
[3] Abbasbandy, S.; Asady, B., The nearest trapezoidal fuzzy number to a fuzzy quantity, Applied mathematics and computation (New York), 156, 381-386, (2004) · Zbl 1058.03516
[4] Abbasbandy, S.; Hajjiri, T., Weighted trapezoidal approximation-preserving core of a fuzzy number, Computers and mathematics with applications, 59, 3066-3077, (2010) · Zbl 1193.26026
[5] Allahviranloo, T.; Adabitabar Firozja, M., Note on “ trapezoidal approximation of fuzzy numbers”, Fuzzy sets and systems, 158, 755-756, (2007) · Zbl 1119.03340
[6] Ban, A.I., Approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval, Fuzzy sets and systems, 159, 1327-1344, (2008) · Zbl 1176.03024
[7] Ban, A.I., On the nearest parametric approximation of a fuzzy number-revisited, Fuzzy sets and systems, 160, 3027-3047, (2009) · Zbl 1183.03047
[8] Ban, A.I., Trapezoidal and triangular approximations of fuzzy numbers-inadvertences and corrections, Fuzzy sets and systems, 160, 3048-3058, (2009) · Zbl 1183.03048
[9] Ban, A.I., Remarks and corrections to the triangular approximations of fuzzy numbers using α-weighted valuations, Soft computing, 15, 351-361, (2011) · Zbl 1237.03033
[10] Ban, A.I.; Brândaş, A.; Coroianu, L.; Negruţiu, C.; Nica, O., Approximations of fuzzy numbers by trapezoidal fuzzy numbers preserving the ambiguity and value, Computers and mathematics with applications, 61, 1379-1401, (2011) · Zbl 1217.26064
[11] Ban, A.I.; Coroianu, L.; Grzegorzewski, P., Trapezoidal approximation and aggregation, Fuzzy sets and systems, 177, 45-59, (2011) · Zbl 1241.03059
[12] Ban, A.I.; Coroianu, L., Metric properties of the nearest extended parametric fuzzy number and applications, International journal of approximate reasoning, 52, 488-500, (2011) · Zbl 1229.03040
[13] A.I. Ban, L. Coroianu, Translation invariance and scale invariance of approximations of fuzzy numbers, in: 7th Conference of the European Society for Fuzzy Logic and Technology, Aix-Les-Bains, 18-22 July 2011.
[14] Ban, A.I.; Coroianu, L.C., Discontinuity of the trapezoidal fuzzy number-valued operators preserving core, Computers and mathematics with applications, 62, 3103-3110, (2011) · Zbl 1232.26051
[15] Bobillo, F.; Straccia, U., Fuzzy ontology representation using OWL 2, International journal of approximate reasoning, 52, 1073-1094, (2011)
[16] Bodjanova, S., Median value and Median interval of a fuzzy number, Information sciences, 172, 73-89, (2005) · Zbl 1074.03018
[17] Carlsson, C.; Fullér, R.; Heikkilä, M.; Majlender, P., A fuzzy approach to R& D project portfolio selection, International journal of approximate reasoning, 44, 93-105, (2007) · Zbl 1286.91146
[18] Chanas, S., On the interval approximation of a fuzzy number, Fuzzy sets and systems, 122, 353-356, (2001) · Zbl 1010.03523
[19] Chon, T.-Y.; Hsu, C.-L.; Chen, M.-C., A fuzzy multicriteria decision model for international tourist hotels location selection, International journal of hospitality management, 27, 293-301, (2008)
[20] Chu, T.-C.; Liu, Y., An extension to fuzzy MCDM, Computers and mathematics with applications, 57, 445-454, (2009) · Zbl 1165.90697
[21] Coroianu, L., Best Lipschitz constant of the trapezoidal approximation operator preserving the expected interval, Fuzzy sets and systems, 165, 81-97, (2011) · Zbl 1226.03057
[22] Coroianu, L., Lipschitz functions and fuzzy number approximations, Fuzzy sets and systems, (2012) · Zbl 1258.26016
[23] Diamond, P.; Kloeden, P., Metric spaces of fuzzy sets. theory and applications, (1994), World Scientific Singapore · Zbl 0873.54019
[24] Delgado, M.; Vila, M.A.; Voxman, W., On a canonical representation of a fuzzy number, Fuzzy sets and systems, 93, 125-135, (1998) · Zbl 0916.04004
[25] Dubois, D.; Prade, H., The Mean value of a fuzzy number, Fuzzy sets and systems, 24, 279-300, (1987) · Zbl 0634.94026
[26] Grzegorzewski, P., Metrics and orders in space of fuzzy numbers, Fuzzy sets and systems, 97, 83-94, (1998) · Zbl 0930.03073
[27] Grzegorzewski, P., Nearest interval approximation of a fuzzy number, Fuzzy sets and systems, 130, 321-330, (2002) · Zbl 1011.03504
[28] Grzegorzewski, P.; Mrówka, E., Trapezoidal approximations of fuzzy numbers, Fuzzy sets and systems, 153, 115-135, (2005) · Zbl 1067.03508
[29] Grzegorzewski, P.; Mrówka, E., Trapezoidal approximations of fuzzy numbers-revisited, Fuzzy sets and systems, 158, 757-768, (2007) · Zbl 1119.03052
[30] Grzegorzewski, P., Trapezoidal approximations of fuzzy numbers preserving the expected interval-alghoritms and properties, Fuzzy sets and systems, 159, 1354-1364, (2008) · Zbl 1176.03031
[31] P. Grzegorzewski, New algorithms for trapezoidal approximation of fuzzy numbers preserving the expected interval, in: L. Magdalena, M. Ojeda, J.L. Verdegay (Eds.), Proceedings of 12th Conference Information Processing and Management of Uncertainty in Knowledge-Based Systems, Malaga, 2008, pp. 117-123.
[32] P. Grzegorzewski, Intuitionistic fuzzy numbers-Principles, metrics and ranking, in: K.T. Atanassov, O. Hryniewicz, J. Kacprzyk (Eds.), Soft Computing. Foundations and Theoretical Aspects, Exit, Warszawa, 2004, pp. 235-249.
[33] Heilpern, S., The expected value of a fuzzy number, Fuzzy sets and systems, 47, 81-86, (1992) · Zbl 0755.60004
[34] W. Karush, Minima of functions of several variables with inequalities as side constraints, M.Sc. Dissertation, University of Chicago, 1939.
[35] Kasemi, N.; Ehsani, E.; Jaber, M.Y., An inventory model with backorders with fuzzy parameters and decision variables, International journal of approximate reasoning, 51, 964-972, (2010) · Zbl 1230.90018
[36] Kuhn, H.W.; Tucker, A.W., Nonlinear programming, (), 481-492 · Zbl 0044.05903
[37] Nasibov, E.N.; Peker, S., On the nearest parametric approximation of a fuzzy number, Fuzzy sets and systems, 159, 1365-1375, (2008) · Zbl 1176.03038
[38] Palacios, A.M.; Sánchez, L.; Couso, I., Linguistic cost-sensitive learning of genetic fuzzy classifiers for imprecise data, International journal of approximate reasoning, 52, 841-862, (2011)
[39] Rockafellar, R.T., Convex analysis, (1970), Princeton University Press Princeton, NJ · Zbl 0229.90020
[40] Rudin, W., Real and complex analysis, (1986), McGraw-Hill New York
[41] Starcezewski, J.T., Efficient triangular type-2 fuzzy logic systems, International journal of approximate reasoning, 50, 799-811, (2009) · Zbl 1191.68698
[42] Vijayan, T.; Kumaran, M., Fuzzy economic order time models with random demand, International journal of approximate reasoning, 50, 529-540, (2009) · Zbl 1183.90036
[43] Yeh, C.-T., A note on trapezoidal approximation of fuzzy numbers, Fuzzy sets and systems, 158, 747-754, (2007) · Zbl 1119.03057
[44] Yeh, C.-T., On improving trapezoidal and triangular approximations of fuzzy numbers, International journal of approximate reasoning, 48, 297-313, (2008) · Zbl 1189.03066
[45] Yeh, C.-T., Trapezoidal and triangular approximations preserving the expected interval, Fuzzy sets and systems, 159, 1345-1353, (2008) · Zbl 1176.03041
[46] Yeh, C.-T., Weighted trapezoidal and triangular approximations of fuzzy numbers, Fuzzy sets and systems, 160, 3059-3079, (2009) · Zbl 1183.03058
[47] Yeh, C.-T., Weighted semi-trapezoidal approximations of fuzzy numbers, Fuzzy sets and systems, 165, 61-80, (2011) · Zbl 1226.03058
[48] Zeng, W.; Li, H., Weighted triangular approximation of fuzzy numbers, International journal of approximate reasoning, 46, 137-150, (2007) · Zbl 1136.03332
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