A stochastic consideration to compare T-transitive proximity relations

Garmendia and Recasens proposed three methods to approximate a proximity relation R by a T- transitive one where T is a continuous Archimedean t-norm are given. However, to obtain the closest approximation E to R with respect to the Euclidean distance could be very expensive. Indeed, the calculation of E becomes then a nonlinear programming problem. In this paper, we show that weighted quasi-arithmetic means of T-indistinguish abilities can be an relatively efficient way to find the closest E for Tthe ?ukasiewiczt-norm without solving complicate nonlinear programming problems. We also give formal proofs of some useful propositions and theorems.