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This is the approximation problem. There are of course other issues to take into account when choosing an aggregation function, such as simplicity, numerical efficiency, easiness of interpretation, and so on [286]. There are no general rules here, and it is up to the system developer to make an educated choice. In what follows, we concentrate on the first two criteria: to be consistent with semantically important properties of the aggregation procedure, and to fit the desired data. We now formalize the selection problem.

Prade. Fundamentals of Fuzzy Sets. Kluwer, Boston, 2000. D. Dubois and H. Prade. On the use of aggregation operations in information fusion processes. Fuzzy Sets and Systems, 142:143–161, 2004. J. Fodor and M. Roubens. Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer, Dordrecht, 1994. M. T. A. Walker. Fundamentals of Uncertainty Calculi, with Applications to Fuzzy Inference. Kluwer, Dordrecht, 1995. P. Klement, R. Mesiar, and E. Pap. Triangular Norms. Kluwer, Dordrecht, 2000.

The use of the mentioned fitting criteria does not preserve the ranking of outputs, unless they are interpolated. Preservation of ranking of outputs can be done by imposing the constraints f (xk ) ≤ f (xl ) if yk ≤ yl for all pairs k, l. We will consider this in detail in Chapter 5. 16). In this section we briefly outline a number of useful numerical tools that will allow us to solve such problems. 305. An approximation problem involves fitting a function from a certain class to the data (xk , yk ), k = 1, .

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