52 A. J. Hughes et al. Fig. 3 An influence diagram representing the partially observable Markov decision process over one time-slice for determining the utility-optimal maintenance strategy for a simple structure comprised of four components. The fault tree failure-mode model for time t +1 has been represented as the node F t+1 for compactness UFt FS hs2 hs1 hc3 hc4 hc1 hc2 UFt+1 F t+1 Ht Ht+1 νt dt Udt Fig. 4 An influence diagram representing the transition sub-model of the overall SHM decision process Ht Ht+1 dt and that the future health-state Ht+1 is dependent only on the current health-state and the action decided in the current time-slice. An underlying assumption of the decision framework presented in [2], that facilitates the modelling process, is that structures can be represented as a hierarchical combination of discrete substructures/regions. A consequence of this assumption is that the health-states of interest are all represented as discrete random variables, hence, the transition models required are matrices. For a given decided action a, and assuming a finite number N of possible discrete global healthstates, the conditional probability table P(Ht+1|Ht ,dt =a) is given by an N×N square matrix whose i,jth entry is the probability of transitioning from the ith to the jth health-state and i,j ∈Z: 1 ≤i,j ≤N. Additionally, it is assumed that the Markov decision process is stationary, i.e. P(Ht+1|Ht ,dt =a) is invariant with respect tot. Because of this stationarity, assuming no intervention is made (dt =‘donothing ∀t), the future global structural health-state is forecast as, P(Ht+n) =P(Ht ) · P(Ht+1|Ht ,dt =‘do nothing’) n (2)
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