By Sadaaki Miyamoto (auth.), Vicenç Torra, Yasuo Narukawa, Beatriz López, Mateu Villaret (eds.)

This ebook constitutes the refereed complaints of the ninth foreign convention on Modeling judgements for man made Intelligence, MDAI 2012, held in Girona, Catalonia, Spain, in November 2012. The 32 revised complete papers have been conscientiously reviewed and chosen from forty nine submissions and are provided with four plenary talks. The papers are prepared in topical sections on aggregation operators, integrals, facts privateness and safety, reasoning, functions, and clustering and similarity.

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Ordered weighted averages have a lot of applications and Beliakov et al. [3] has discussed how to choose the weights in practice. Ordered weighted averages are described as follows: Let n be a positive integer, and let {w1 , w2 , · · · , wn } be a weighting sequence such that wi ∈ [0, 1](i = n 1, 2, · · · , n) and i=1 wi = 1. e. {(1), (2), · · · , (n)} is a permutation of the index set {1, 2, · · · , n} and x(i−1) ≥ x(i) for i = 2, 3, · · · , n. Extending the ordered weighted averages, Yager [15] and Yager and Xu [17] have studied continuous weighted ordered weighted averages with weights defined by a fuzzy quantifier, and Torra [10], Torra and Narukawa [12], Torra and Godo [11], Narukawa and Torra [7] and Narukawa et al.

So, μ1,2 is normalized in the sense that each of its focal elements can be regarded as a normalized vector of [0, 1]4 . On the other hand, b1,2 is not normalized because (0, 0) = (1, 0) ∧ (0, 1), ρ(0,0) (0, 0) = 1, and hence ρ(0,0) ((0, 0)) · μ1 (b) · μ2 (c) = ρ(0,0) (0, 0) · μ1 (1, 0) · μ2 (0, 1) = 1/8 > 0. b(0, 0) = b∧c=(0,0) The above min-conjunctive combination can easily be extended to well-known MV-operations on fuzzy sets, such as max-disjunction ∨, strong conjunction being one and strong disjunction ⊕, by defining (b1 b2 )(a) = s1,2 (ρa ), for of these operations, and defining ρa (b, c) = ρa (b c).

K, wi ≥ 0, ∀ i, n wi = 1. quasi where the generators of both M1 , M2 can be specified, as well as the weighting vector u associated with M2 and the indices p, q. 3 Enforcing Mandatory Requirements In some applications, it may be desirable to define a model with one or two mandatory requirements, but which still fits the data as well as it can. In this case, we can use projections on w. Denoting the generator of M2 (a weighted quasi-arithmetic mean) by m2 , this will result in the following expression, 1 ⎛ ⎛ ⎞q ⎞ p+q uj ⎝ ⎝xpi m−1 m2 (xj )⎠ ⎠ .

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