An extension of Universal Generating Function in Multi-State Systems Considering Epistemic Uncertainties

A Contribution to the Evaluation of Imprecise Availability of Complex Systems Using Markov Models

“…The exact method is a technique where we find all the possible transition matrices of a system and then for each matrix we solve (6) and we find a vector of stationarity, to form at the end a vector of intervals which contains all of the obtained vectors, this method gives an exact result but its complexity increases drastically with the system's size since it computes all the possible state values. BUGF (Believe Universal Generated Function) [12] and IUGF (Interval Universal Generated Function) [11] are two methods based on the UGF (Universal Generated Function) but they are applied on intervals and therefore are used in the case of interval-modeled imprecision, these two methods are efficient and give good results, the IUGF is also noted as more efficient than the BUGF in [12]. In our approach we propose to determine the availability of the system by using a new technique applied on intervals, that is the technique of contractors [4] which we introduce in section 3.…”

“…All state performances of the components are precise. We want to calculate the availability of the system by using Markov chain and we will compare to the IUGF proposed in [11] and the BUGF proposed in [12]. For every component :…”

Reliability analysis and optimization of equal load-sharing k-out-of-n phased-mission systems

“…It is widely adopted in reliability analyses because of its efficiency. It is particularly useful in the reliability analysis of systems with multi-states and large numbers of components [12,18,19]. In the traditional UGF method, the UGF of a component i over state space G i is written as a z-transformation polynomial:…”

Reliability analysis of multi‐state series systems with performance sharing mechanism under epistemic uncertainty

“…Xiao et al proposed a mixed UGF for engineering problems under both epistemic uncertainty and aleatory uncertainty with mixed random, interval, and p‐box variables. Destercke and Sallak proposed belief UGF (BUGF) based on belief functions theory to evaluate the precise reliability interval of MSSs under aleatory and epistemic uncertainties. Li et al proposed a hybrid UGF to represent random fuzzy variables defined on a finite set of fuzzy variables and evaluated availability p‐boxes of MSSs under aleatory and epistemic uncertainties.…”

“…Belief functions theory can be regarded as a generalization of many uncertainty theories, including probabilities, sets, fuzzy sets, p‐boxes, possibilities, and so on. It has the following advantages: •Belief functions can model the lack of knowledge or insufficiency of information more efficiently than classical probabilities do. •Because they encompass fuzzy sets and probabilities, belief functions can represent linguistic experts' opinions as well as statistics of components. •Belief functions can be associated to probability bounds; thus, they correspond to a robust version of classical probabilities. …”