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Analytical expressions for the addition of fuzzy intervals. (English) Zbl 0919.04005
Summary: The addition of fuzzy quantities is undeniably the most important operation in fuzzy arithmetic. In this paper, we consider the general case of the addition of fuzzy intervals based on a continuous triangular norm. We start our discussion with an extensive literature review in which we recall explicit formulae for the addition based on the strongest and the weakest triangular norm, for the addition based on the algebraic product, and several recent results for the addition based on a continuous Archimedean triangular norm, including specific theorems for strict and nilpotent triangular norms. Subsequently, the addition based on an ordinal sum is studied, and it is shown how this addition can be transformed into a series of additions based on the summands of this ordinal sum. This important observation implies that the addition based on an arbitrary continuous triangular norm can be practically performed, provided the summands of the corresponding ordinal sum representation and the fuzzy intervals involved fulfil the appropriate conditions mentioned in the overview. This is illustrated by means of several examples.

##### MSC:
 3e+72 Theory of fuzzy sets, etc.
Full Text:
##### References:
  De Baets, B.; Marková, A., Addition of LR-fuzzy intervals based on a continuous t-norm, (), 353-358, Granada, Spain  Dubois, D.; Prade, H., Additions of interactive fuzzy numbers, IEEE trans. automat. control, 26, 926-936, (1981)  Dubois, D.; Prade, H., Fuzzy numbers: an overview, (), 3-39  Fullér, R.; Keresztfalvi, T., T-norm-based addition of fuzzy intervals, Fuzzy sets and systems, 51, 155-159, (1992)  Hong, D., A note on the product-sum of L-R fuzzy numbers, Fuzzy sets and systems, 66, 381-382, (1994) · Zbl 0844.04005  Hong, D.; Hwang, S., On the compositional rule of inference under triangular norms, Fuzzy sets and systems, 66, 25-38, (1994) · Zbl 1018.03511  E.-P. Klement, R. Mesiar, E. Pap, Triangular norms, in preparation. · Zbl 0972.03002  Kolesárová, A., Triangular norm-based additions of fuzzy numbers and preserving of similarity, Busefal, 69, 43-54, (1997)  Ling, C., Representation of associative functions, Publ. math. debrecen, 12, 189-212, (1965) · Zbl 0137.26401  Mareš, M., Computation over fuzzy quantities, (1994), CRC Press Boca Raton · Zbl 0859.94035  Marková, A., Additions of L-R fuzzy numbers, Busefal, 63, 25-29, (1995)  Marková, A., T-sum of L-R fuzzy numbers, Fuzzy sets and systems, 85, 379-384, (1997) · Zbl 0904.04007  Marková-Stupňanová, A., A note to the addition of fuzzy numbers based on a continuous Archimedean t-norm, Fuzzy sets and systems, 91, 251-256, (1997) · Zbl 0919.04010  Mesiar, R., Computation over LR-fuzzy numbers, (), 165-176  Mesiar, R., A note to the T-sum of L-R fuzzy numbers, Fuzzy sets and systems, 79, 259-261, (1996) · Zbl 0871.04010  Mesiar, R., Shape preserving additions of fuzzy intervals, Fuzzy sets and systems, 86, 73-78, (1997) · Zbl 0921.04002  Schweizer, B.; Sklar, A., Probabilistic metric spaces, (1983), North-Holland New York · Zbl 0546.60010  Triesch, E., On the convergence of product-sum series of L-R fuzzy numbers, Fuzzy sets and systems, 53, 189-192, (1993) · Zbl 0874.26019
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