A sufficient and necessary condition of uncertainty distribution

Abstract

Uncertainty theory is a branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms. A key concept to describe uncertain quantity is uncertain variable. Uncertainty distribution is an important tool to specify an uncertain variable. In this paper, a sufficient and necessary condition of uncertainty distribution is proved to show what function is an uncertainty distribution. and an example is given on how to construct an uncertain measure with respect to a given function.

Chance distribution of the maximum flow of uncertain random network

“…Iwamura [24] proved a sufficient and necessary condition for uncertainty distribution. In addition, the concept of independence was proposed by Liu [25].…”

Method of moments for estimating uncertainty distributions

“…For any x ∈ , the function (x) = {ξ ≤ x} is called the uncertainty distribution of uncertain variable ξ , Peng and Iwamura [13] presented a sufficient and necessary condition of uncertainty distribution that a function : …”

“…Meanwhile, as an application of uncertainty theory, Liu [10] proposed a spectrum of uncertain programming and applied uncertain programming to system reliability design, facility location problem, vehicle routing problem, project scheduling problem, and so on. Other references related to uncertainty theory are Gao [11], Gao et al [12], Peng and Iwamura [13], and Liu and Ha [14]. Nowadays, uncertainty theory was well developed on both theory section and practice section.…”

Subadditivity of chance measure

“…Liu [1] presented the concept of uncertain variable and uncertainty distribution. Then, a sufficient and necessary condition of uncertainty distribution was proved by Peng and Iwamura [2] in 2010. In addition, a measure inversion theorem was proposed by Liu [3] from which the uncertain measures of some events can be calculated via the uncertainty distribution.…”