The Operational Law of Uncertain Variables With Continuous Uncertainty Distributions

As we know, the classical operational law of uncertain variables initiated by Liu, which gives a major push to the development of uncertainty theory, restricts the uncertain variables being dealt with to those with regular uncertainty distributions. This restriction makes the operational law no longer applicable when some uncertain variables whose distributions are not regular are involved. Therefore, a generalized operational law is proposed in this paper, for uncertain variables with continuous distributions, of which regular distributions can be treated as special cases. By utilizing this extended operational law, the uncertainty distributions of strictly monotonic functions of uncertain variables with continuous (not necessarily regular) distributions can be analytically deduced, similarly to that suggested by the classical one. Furthermore, some new conclusions on the expected values of uncertain variables with continuous distributions are presented as well. Finally, as an important application of the generalized operational law, it is proved that some types of uncertain programming involving uncertain parameters with continuous distributions can be translated into deterministic counterparts, and then be handled by classic optimization techniques without any other additional efforts.