×
Hong, Dug Hun

On shape-preserving additions of fuzzy intervals. (English) Zbl 0993.03070

J. Math. Anal. Appl. 267, No. 1, 369-376 (2002).
For given left and right shapes \(L\) and \(R\), the corresponding \(LR\)-fuzzy intervals are characterized by 4 parameters describing their kernel and their support. Triangular norm-based addition of fuzzy quantities is, in general, a process with high computational complexity. However, in the case of processing \(LR\)-fuzzy numbers (intervals) by means of some specific t-norm preserving the shapes \(L\) and \(R\), the processing is reduced to processing the parameters only, thus efficiently reducing the computational complexity.
Specific cases of shape-preservation were given by A. Kolesárová [Tatra Mt. Math. Publ. 6, 75-81 (1995; Zbl 0851.04005)] (for linear shapes) and by R. Mesiar [Fuzzy Sets Syst. 86, 73-78 (1997; Zbl 0921.04002)], requiring the convexity of composite functions \(fL\) and \(fR\), where \(f\) is an additive generator of the t-norm \(T\) [see E. P. Klement, R. Mesiar and E. Pap, Triangular norms, Dordrecht: Kluwer (2000; Zbl 0972.03002)].
The present paper completely solves the characterization of all shape preserving additions of fuzzy numbers based on continuous t-norms, confirming the hypothesis of Mesiar from 1997. Note that the characterizations of non-continuous t-norms leading to the shape-preserving additions of \(LR\)-fuzzy numbers (intervals) is still an open problems.
Reviewer: Radko Mesiar (Bratislava)

Cited in 12 Documents

MSC:

03E72 Theory of fuzzy sets, etc.

Keywords:

triangular norm; LR-fuzzy intervals; LR-fuzzy numbers; shape preserving additions

Citations:

Zbl 0851.04005; Zbl 0921.04002; Zbl 0972.03002
PDFBibTeX XMLCite
\textit{D. H. Hong}, J. Math. Anal. Appl. 267, No. 1, 369--376 (2002; Zbl 0993.03070)
Full Text: DOI Link

References:

[1] Dubois, D.; Prade, H., Fuzzy Sets and Systems: Theory and Applications (1980), Academic Press: Academic Press New York · Zbl 0444.94049
[2] Dubois, D.; Prade, H., Additions of interactive fuzzy intervals, IEEE Trans. Automat. Control, 26, 926-936 (1981) · Zbl 1457.68262
[3] Fullér, R.; Keresztfalvi, T., \(t\)-Norm-based addition of fuzzy intervals, Fuzzy Sets and Systems, 51, 155-159 (1992)
[4] Fullér, R.; Zimmermann, H.-J., On computation of the compositional rule of inference under triangular norms, Fuzzy Sets and Systems, 51, 267-275 (1992) · Zbl 0782.68110
[5] Hong, D. H.; Hwang, S. Y., The convergence of \(T\)-product of fuzzy numbers, Fuzzy Sets and Systems, 85, 373-378 (1997) · Zbl 0904.04004
[6] Hong, D. H., A note on \(t\)-norm-based addition of fuzzy intervals, Fuzzy Sets and Systems, 75, 73-76 (1995) · Zbl 0947.26024
[7] Hong, D. H.; Do, H. Y., Fuzzy system reliability analysis by the use of \(T_W\) (the weakest \(t\)-norm) on fuzzy number arithmetic operations, Fuzzy Sets and Systems, 90, 307-316 (1997)
[8] Hong, D. H.; Hwang, C., A \(T\)-sum bound of LR-fuzzy numbers, Fuzzy Sets and Systems, 91, 239-252 (1997) · Zbl 0920.04010
[9] Hong, D. H.; Kim, H., A note to the sum of fuzzy variables, Fuzzy Sets and Systems, 93, 121-124 (1998) · Zbl 0919.04007
[10] Hong, D. H.; Hwang, S. Y., On the compositional rule of inference under triangular norms, Fuzzy Sets and Systems, 66, 25-38 (1994) · Zbl 1018.03511
[11] Hong, D. H., Shape preserving multiplications of fuzzy intervals, Fuzzy Sets and Systems, 123, 93-96 (2001)
[12] Hong, D. H., Some results on the addition of fuzzy intervals, Fuzzy Sets and Systems, 122, 349-352 (2001) · Zbl 1010.03524
[13] Keresztfalvi, T.; Kovócs, M., \(g, p\)-fuzzification of arithmetic operations, Tatra Mt. Math. Publ., 1, 65-71 (1992) · Zbl 0786.04004
[14] Klement, E. P.; Mesiar, R.; Pap, E., A characterization of the ordering of continuous \(t\)-norms, Fuzzy Sets and Systems, 86, 189-195 (1997) · Zbl 0914.04006
[15] Kolesárová, A., Triangular norm-based addition preserving linearity of \(t\)-sums of fuzzy intervals, Mathware Soft Comput., 5, 97-98 (1998) · Zbl 0934.03064
[16] Kolesárová, A., Triangular norm-based addition of linear fuzzy numbers, Tatra Mt. Math. Publ., 6, 75-82 (1995) · Zbl 0851.04005
[17] Lee, J. J.; Hong, D. H.; Hwang, S. Y., A learning algorithm of fuzzy neural networks using a shape preserving operation, J. Electr. Engrg. Inform. Sci., 3, 131-138 (1998)
[18] Marková, A., \(T\)-sum of \(L-R\)-fuzzy numbers, Fuzzy Sets and Systems, 85, 379-384 (1996) · Zbl 0904.04007
[19] Marková, A., A note to the addition of fuzzy numbers based on a continuous Archimedean \(T\)-norm, Fuzzy Sets and Systems, 91, 253-258 (1997) · Zbl 0919.04010
[20] R. Mesiar, LRin; R. Mesiar, LRin
[21] Mesiar, R., A note on the \(T\)-sum of LR fuzzy numbers, Fuzzy Sets and Systems, 79, 259-261 (1996) · Zbl 0871.04010
[22] Mesiar, R., Shape preserving additions of fuzzy intervals, Fuzzy Sets and Systems, 86, 73-78 (1997) · Zbl 0921.04002
[23] Robert, A. W.; Varberg, D. E., Convex Functions (1980), Academic Press: Academic Press New York
[24] Yager, R. R., On a general class of fuzzy connectives, Fuzzy Sets and Systems, 4, 235-242 (1980) · Zbl 0443.04008
[25] Zadeh, L. A., The concept of a linguistic variable and its applications to approximate reasoning, Parts I-III, Inform. Sci., 8, 199-251 (1975) · Zbl 0397.68071
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.