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Grzegorzewski et al. 1007/978-3-319-10765-3_3 22 I. Montes, E. Miranda, and S. Montes some additional remarks and a discussion of other approaches to this problem. Proofs are omitted because of space limitations. 1 Preliminary Notions Fuzzy Random Variables Fuzzy random variables were introduced simultaneously by Kruse and Meyer [6] and Puri and Ralescu [12]. In this paper, we follow the epistemic approach considered in [6]. Let F (R) denote the set of all fuzzy sets on R. Definition 1 ([6]). A fuzzy random variable is a map X : Ω → F (R) such that the α-cuts Xα are strongly measurable multi-valued mappings.

These works are in preparation. References 1. : Ambiguity of Fuzzy Quantities and a New Proposal for their Ranking. Przeglad Elektrotechniczny-Electrical Review 10, 280–283 (2012) 2. : The total variation of bounded variation functions to evaluate and rank fuzzy quantities. International Journal of Intelligent Systems 28, 927–956 (2013) 3. : Evaluation and interval approximation of fuzzy quantities. In: Proceedings of 8th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2013), pp.

Applying the results concerning normalization of interval weights that were proved in [7], we find out that the following general relations hold: 34 O. Pavlaˇcka and K. Hron Table 3. 253 j, k = 1, . . , D, j = k. Example 1. 932] (see Table 2). 125] . 125]. The interactions among the proportions [cik , cik ], k = 1, . . , D, mean that the proper proportional representation of interval compositional data Xi , i = 1, . . , n, has to be given in the following way: C(Xi ) := C(xi ) ∈ [0, 1]D | xi ∈ [xi1 , xi1 ] × .

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