# Note on “symmetric triangular approximations of fuzzy numbers under a general condition and properties”

Foundations

First Online:

- 86 Downloads

## Abstract

We show by a counterexample that Theorem 2 in Ban, Coroianu [Soft Computing (2016) 20:1249-1261] is not always the symmetric triangular fuzzy number nearest to a given fuzzy number. In addition, a corrected version is provided.

## Keywords

Fuzzy numbers Symmetric triangular approximation## Notes

### Acknowledgements

The author is very grateful to the anonymous referees for their detailed comments and valuable suggestions. This research has been supported by the Ministry of Science and Technology, Taiwan (105-2115-M-024-003).

### Compliance with ethical standards

### Conflict of interest

The author declares that he has no conflict of interest.

### Ethical approval

This article does not contain any studies with human participants or animals performed by the author.

## References

- Abbasbandy S, Ahmady E, Ahmady N (2010) Triangular approximations of fuzzy numbers using \(\alpha \)-weighted valuations. Soft Comput 14:71–79CrossRefMATHGoogle Scholar
- Abbasbandy S, Hajjari T (2010) Weighted trapezoidal approximation preserving cores of a fuzzy numbers. Comput Math Appl 59:3066–3077MathSciNetCrossRefMATHGoogle Scholar
- Allahviranloo T, Adabitabar Firozja M (2007) Note on trapezoidal approximation of fuzzy numbers. Fuzzy Sets Syst 158:755–756MathSciNetCrossRefMATHGoogle Scholar
- Ban AI (2008) Approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval. Fuzzy Sets Syst 159:1327–1344MathSciNetCrossRefMATHGoogle Scholar
- Ban AI (2011) Remarks and corrections to the triangular approximations of fuzzy numbers using \(\alpha \)-weighted valuations. Soft Comput 15:351–361CrossRefMATHGoogle Scholar
- Ban AI, Brândaş A, Coroianu L, Negruţiu C, Nica O (2011) Approximations of fuzzy numbers by trapezoidal fuzzy numbers preserving the ambiguity and value. Comput Math Appl 61:1379–1401MathSciNetCrossRefMATHGoogle Scholar
- Ban AI, Coroianu L (2012) Nearest interval, triangular and trapezoidal approximation of a fuzzy number preserving ambiguity. J Approx Reason 53:805–836MathSciNetCrossRefMATHGoogle Scholar
- Ban AI, Coroianu L (2014) Existence, uniqueness and continuity of trapezoidal approximations of fuzzy numbers under a general condition. Fuzzy Sets Syst 257:3–22MathSciNetCrossRefMATHGoogle Scholar
- Ban AI, Coroianu L (2015) Existence, uniqueness, calculus and properties of triangular approximations of fuzzy numbers under a general condition. Int J Approx Reason 62:1–26MathSciNetCrossRefMATHGoogle Scholar
- Ban AI, Coroianu L (2016) Symmetric triangular approximations of fuzzy numbers under a general condition and properties. Soft Comput 20:1249–1261CrossRefMATHGoogle Scholar
- Chanas S (2001) On the interval approximation of a fuzzy number. Fuzzy Sets Syst 122:353–356MathSciNetCrossRefMATHGoogle Scholar
- Coroianu L (2011) Best Lipschitz constant of the trapezoidal approximation operator preserving the expected interval. Fuzzy Sets Syst 165:81–97MathSciNetCrossRefMATHGoogle Scholar
- Coroianu L (2012) Lipschitz functions and fuzzy number approximations. Fuzzy Sets Syst 200:116–135MathSciNetCrossRefMATHGoogle Scholar
- Delgado M, Vila MA, Voxman W (1998) On a canonical representation of fuzzy number. Fuzzy Sets Syst 93:125–135MathSciNetCrossRefMATHGoogle Scholar
- Dubois D, Prade H (1978) Operations on fuzzy numbers. Int J Syst Sci 9:613–626MathSciNetCrossRefMATHGoogle Scholar
- Dubois D, Prade H (1987) The mean value of a fuzzy number. Fuzzy Sets Syst 24:279–300MathSciNetCrossRefMATHGoogle Scholar
- Grzegorzewski P (1998) Metrics and orders in space of fuzzy numbers. Fuzzy Sets Syst 97:83–94MathSciNetCrossRefMATHGoogle Scholar
- Grzegorzewski P, Mrówka E (2005) Trapezoidal approximations of fuzzy numbers. Fuzzy Sets Syst 153:115–135MathSciNetCrossRefMATHGoogle Scholar
- Grzegorzewski P, Mrówka E (2007) Trapezoidal approximations of fuzzy numbers-revisited. Fuzzy Sets Syst 158:757–768CrossRefMATHGoogle Scholar
- Grzegorzewski P (2008) New algorithms for trapezoidal approximation of fuzzy numbers preserving the expected interval. In: Ojeda M, Verdegay JL (eds) Magdalena L. Proceedings on information processing and management of uncertainty in knowledge-based system conference, Malaga, pp 117–123Google Scholar
- Grzegorzewski P (2008) Trapezoidal approximations of fuzzy numbers preserving the expected interval - algorithms and properties. Fuzzy Sets Syst 159:1354–1364MathSciNetCrossRefMATHGoogle Scholar
- Heilpern S (1992) The expected value of a fuzzy number. Fuzzy Sets Syst 47:81–86MathSciNetCrossRefMATHGoogle Scholar
- Li J, Wang ZX, Yue Q (2012) Triangular approximation preserving the centroid of fuzzy numbers. Int J Comput Math 89:810–821MathSciNetCrossRefMATHGoogle Scholar
- Tripathy BC, Das PC (2012) On convergence of series of fuzzy real numbers. Kuwait J Sci Eng 39(1A):57–70MathSciNetGoogle Scholar
- Tripathy BC, Ray GC (2012) On Mixed fuzzy topological spaces and countability. Soft Comput 16(10):1691–1695CrossRefMATHGoogle Scholar
- Tripathy BC, Sarma B (2012) On I-convergent double sequences of fuzzy real numbers. Kyungpook Math J 52(2):189–200MathSciNetCrossRefMATHGoogle Scholar
- Tripathy BC, Sen M, Nath S (2012) I-convergence in probabilistic n-normed space. Soft Comput 16:1021–1027CrossRefMATHGoogle Scholar
- Yeh CT (2007) A note on trapezoidal approximation of fuzzy numbers. Fuzzy Sets Syst 158:747–754MathSciNetCrossRefMATHGoogle Scholar
- Yeh CT (2008a) On improving trapezoidal and triangular approximations of fuzzy numbers. J Approx Reason 48:297–313MathSciNetCrossRefMATHGoogle Scholar
- Yeh CT (2008b) Trapezoidal and triangular approximations preserving the expected interval. Fuzzy Sets Syst 159:1345–1353MathSciNetCrossRefMATHGoogle Scholar
- Yeh CT (2009) Weighted trapezoidal and triangular approximations of fuzzy numbers. Fuzzy Sets Syst 160:3059–3079MathSciNetCrossRefMATHGoogle Scholar
- Yeh CT (2011) Weighted semi-trapezoidal approximations of fuzzy numbers. Fuzzy Sets Syst 165:61–80MathSciNetCrossRefMATHGoogle Scholar
- Yeh CT, Chu HM (2012) Approximations by LR-type fuzzy numbers. Fuzzy Sets Syst 257:23–40MathSciNetCrossRefMATHGoogle Scholar
- Yeh CT (2017) Existence of interval, triangular, and trapezoidal approximations of fuzzy numbers under a general condition. Fuzzy Sets Syst 310:1–13MathSciNetCrossRefMATHGoogle Scholar
- Zeng W, Li H (2007) Weighted triangular approximation of fuzzy numbers. J Approx Reason 46:137–150MathSciNetCrossRefMATHGoogle Scholar

## Copyright information

© Springer-Verlag GmbH Germany 2017