A theoretical extension on the operational law for monotone functions of uncertain variables

The classical operational law of uncertain variables proposed by Liu makes an important contribution to the development of the uncertainty theory in both theories and applications. It provides a powerful and practical approach for calculating the uncertainty distribution of strictly monotone function of uncertain variables. However, the restriction on strictly monotone functions of the operational law limits its applications since many practical problems cannot be modeled by strictly monotone functions but general monotone functions. Therefore, an extension of the original operational law is needed. For this purpose, some properties concerning the uncertainty distributions of monotone functions of uncertain variables as well as the generalized inverse uncertainty distributions are presented first in this paper. On the basis of these discussions, a generalized operational law is proposed as a natural extension of the original operational law. Then the uncertainty distribution of a general monotone function of independent regular uncertain variables can be derived, which is analogous to the way that suggested by the original operational law for dealing with strictly monotone functions. Furthermore, as an application of the generalized operational law, a theorem for calculating the expected values of general monotone functions of uncertain variables is presented as well.