Document Type : Research Paper

**Authors**

Department of Mathematics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran

**Abstract**

In this paper, we consider the width invariant trapezoidal and triangular

approximations of fuzzy numbers. The presented methods avoid the effortful computation of Karush-Kuhn-Tucker Theorem. Some properties of the new approximation methods are presented and the applicability of the methods is illustrated by examples. In addition, we show that the proposed approximations of fuzzy numbers preserve the expected value too.

approximations of fuzzy numbers. The presented methods avoid the effortful computation of Karush-Kuhn-Tucker Theorem. Some properties of the new approximation methods are presented and the applicability of the methods is illustrated by examples. In addition, we show that the proposed approximations of fuzzy numbers preserve the expected value too.

**Keywords**

[1] S. Abbasbandy and M. Amirfakhrian, The nearest approximation of a fuzzy quantity in

parametric form, Applied Mathematics and Computation, 172 (2006), 624–632.

[2] S. Abbasbandy and M. Amirfakhrian, The nearest trapezoidal form of a generalized left right

fuzzy number, International Journal of Approximate Reasoning, 43 (2006), 166–178.

[3] S. Abbasbandy and B. Asady, The nearest trapezoidal fuzzy number to a fuzzy quantity,

Applied Mathematics and Computation, 156 (2004), 381–386.

[4] S. Abbasbandy and T. Hajjari, Weighted trapezoidal approximation-preserving core of a fuzzy

number, Computers and Mathematics with Applications, 59 (2010),3066–3077.

[5] T. Allahviranloo and M. Adabitabar Firozja, Note on "Trapezoidal approximation of fuzzy

numbers", Fuzzy Sets and Systems, 158 (2007), 755–756.

[6] A. I. Ban, Approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the

expected interval, Fuzzy Sets and Systems, 159 (2008), 1327-1344.

[7] A. I. Ban, Trapezoidal and triangular approximations of fuzzy numbers-inadvertences and

corrections, Fuzzy Sets and Systems, 160 (2009), 3048-3058.

[8] A. I. Ban, A. Brandas, L. Coroianu, C. Negrutiu and O. Nica, Approximations of fuzzy

numbers by trapezoidal fuzzy numbers preserving the ambiguity and value, Computers and

Mathematics with Applications, 61 (2011), 1379-1401.

[9] A. I. Ban and 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.

[10] A. I. Ban and L. Coroianu, Nearest interval, triangular and trapezoidal approximation of

a fuzzy number preserving ambiguity, International Journal of Approximate Reasoning, 53

(2012), 805–836.

[11] A.I. Ban, L. Coroianu, Existence, uniqueness and continuity of trapezoidal approximations

of fuzzy numbers under a general condition, Fuzzy Sets and Systems, 257(2014), 3-22.

[12] A. Brandas, Approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the

core, the ambiguity and the value, Advanced Studies in Contemporary Mathematics, 21

(2011), 247259.

[13] S. Bodjanova, Median value and median interval of a fuzzy number, Information Sciences,

172 (2005), 73-89.

[14] S. Chanas, On the interval approximation of a fuzzy number, Fuzzy Sets and Systems, 122

(2001), 353-356.

[15] L. Coroianu, M. Gagolewski and P. Grzegorzewski, Nearset piecewise linear approximation

of fuzzy numbers, Fuzzy Sets and Systems, 233 (2013), 26-51.

[16] P. Diamond and P. Kloeden, Metric spaces of fuzzy sets, theory and applications, World

Scientific, Singapore, 1994.

[17] D. Dubois and H. Prade, Operations on fuzzy numbers, Int. J. Syst. Sci., 30 (1978), 613-626.

[18] D. Dubois, H. Prade, The mean value of a fuzzy number, Fuzzy Sets and Systems, 24 (1987),

279-300.

[19] P. Grzegorzewski, Metrics and orders in space of fuzzy numbers, Fuzzy Sets and Systems, 97

(1998), 83-94.

[20] P. Grzegorzewski, Nearest interval approximation of a fuzzy number, Fuzzy Sets and Systems,

130 (2002), 321-330.

[21] P. Grzegorzewski, E. Mr´owka, Trapezoidal approximations of fuzzy numbers, Fuzzy Sets and

Systems, 153 (2005), 115-135.

[22] P. Grzegorzewski, E. Mr´owka, Trapezoidal approximations of fuzzy numbers-revisited, Fuzzy

Sets and Systems, 158 (2007), 757-768.

[23] S. Heilpern, The expected value of a fuzzy number, Fuzzy Sets and Systems, 47 (1992) 81-86.

[24] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1986.

[25] C. T. Yeh, A note on trapezoidal approximation of fuzzy numbers, Fuzzy Sets and Systems,

158 (2007), 747-754.

parametric form, Applied Mathematics and Computation, 172 (2006), 624–632.

[2] S. Abbasbandy and M. Amirfakhrian, The nearest trapezoidal form of a generalized left right

fuzzy number, International Journal of Approximate Reasoning, 43 (2006), 166–178.

[3] S. Abbasbandy and B. Asady, The nearest trapezoidal fuzzy number to a fuzzy quantity,

Applied Mathematics and Computation, 156 (2004), 381–386.

[4] S. Abbasbandy and T. Hajjari, Weighted trapezoidal approximation-preserving core of a fuzzy

number, Computers and Mathematics with Applications, 59 (2010),3066–3077.

[5] T. Allahviranloo and M. Adabitabar Firozja, Note on "Trapezoidal approximation of fuzzy

numbers", Fuzzy Sets and Systems, 158 (2007), 755–756.

[6] A. I. Ban, Approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the

expected interval, Fuzzy Sets and Systems, 159 (2008), 1327-1344.

[7] A. I. Ban, Trapezoidal and triangular approximations of fuzzy numbers-inadvertences and

corrections, Fuzzy Sets and Systems, 160 (2009), 3048-3058.

[8] A. I. Ban, A. Brandas, L. Coroianu, C. Negrutiu and O. Nica, Approximations of fuzzy

numbers by trapezoidal fuzzy numbers preserving the ambiguity and value, Computers and

Mathematics with Applications, 61 (2011), 1379-1401.

[9] A. I. Ban and 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.

[10] A. I. Ban and L. Coroianu, Nearest interval, triangular and trapezoidal approximation of

a fuzzy number preserving ambiguity, International Journal of Approximate Reasoning, 53

(2012), 805–836.

[11] A.I. Ban, L. Coroianu, Existence, uniqueness and continuity of trapezoidal approximations

of fuzzy numbers under a general condition, Fuzzy Sets and Systems, 257(2014), 3-22.

[12] A. Brandas, Approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the

core, the ambiguity and the value, Advanced Studies in Contemporary Mathematics, 21

(2011), 247259.

[13] S. Bodjanova, Median value and median interval of a fuzzy number, Information Sciences,

172 (2005), 73-89.

[14] S. Chanas, On the interval approximation of a fuzzy number, Fuzzy Sets and Systems, 122

(2001), 353-356.

[15] L. Coroianu, M. Gagolewski and P. Grzegorzewski, Nearset piecewise linear approximation

of fuzzy numbers, Fuzzy Sets and Systems, 233 (2013), 26-51.

[16] P. Diamond and P. Kloeden, Metric spaces of fuzzy sets, theory and applications, World

Scientific, Singapore, 1994.

[17] D. Dubois and H. Prade, Operations on fuzzy numbers, Int. J. Syst. Sci., 30 (1978), 613-626.

[18] D. Dubois, H. Prade, The mean value of a fuzzy number, Fuzzy Sets and Systems, 24 (1987),

279-300.

[19] P. Grzegorzewski, Metrics and orders in space of fuzzy numbers, Fuzzy Sets and Systems, 97

(1998), 83-94.

[20] P. Grzegorzewski, Nearest interval approximation of a fuzzy number, Fuzzy Sets and Systems,

130 (2002), 321-330.

[21] P. Grzegorzewski, E. Mr´owka, Trapezoidal approximations of fuzzy numbers, Fuzzy Sets and

Systems, 153 (2005), 115-135.

[22] P. Grzegorzewski, E. Mr´owka, Trapezoidal approximations of fuzzy numbers-revisited, Fuzzy

Sets and Systems, 158 (2007), 757-768.

[23] S. Heilpern, The expected value of a fuzzy number, Fuzzy Sets and Systems, 47 (1992) 81-86.

[24] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1986.

[25] C. T. Yeh, A note on trapezoidal approximation of fuzzy numbers, Fuzzy Sets and Systems,

158 (2007), 747-754.

[26] C. T. Yeh, On improving trapezoidal and triangular approximations of fuzzy numbers, International

Journal of Approximate Reasoning, 48 (2008), 297-313.

[27] C. T. Yeh, Trapezoidal and triangular approximations preserving the expected interval, Fuzzy

Sets and Systems, 159 (2008), 1345–1353.

[28] C. T. Yeh, Weighted trapezoidal and triangular approximations of fuzzy numbers, Fuzzy Sets

and Systems, 160 (2009), 3059–3079.

[29] C. T. Yeh, Weighted semi-trapezoidal approximations of fuzzy numbers, Fuzzy Sets and Systems,

165 (2011), 61-80.

[30] C. T. Yeh, H. M. Chu, Approximations by LR-type fuzzy numbers, Fuzzy Sets and Systems,

257 (2014) 23-40.

[31] W. Zeng, H. Li, Weighted triangular approximation of fuzzy numbers, International Journal

of Approximate Reasoning, 46 (2007), 137–150.

Journal of Approximate Reasoning, 48 (2008), 297-313.

[27] C. T. Yeh, Trapezoidal and triangular approximations preserving the expected interval, Fuzzy

Sets and Systems, 159 (2008), 1345–1353.

[28] C. T. Yeh, Weighted trapezoidal and triangular approximations of fuzzy numbers, Fuzzy Sets

and Systems, 160 (2009), 3059–3079.

[29] C. T. Yeh, Weighted semi-trapezoidal approximations of fuzzy numbers, Fuzzy Sets and Systems,

165 (2011), 61-80.

[30] C. T. Yeh, H. M. Chu, Approximations by LR-type fuzzy numbers, Fuzzy Sets and Systems,

257 (2014) 23-40.

[31] W. Zeng, H. Li, Weighted triangular approximation of fuzzy numbers, International Journal

of Approximate Reasoning, 46 (2007), 137–150.

March and April 2016

Pages 111-130