Assets Assembled by Multiple Components

The condition of an asset with multiple components can be evaluated in various ways depending on the type of interaction between components, including in parallel, series, bridge structure, and their variants (see Section 2.3.3). Dynamic Fault Trees (DFTs) is one of the most notable frameworks in modelling a flexible system configuration [108]. Here, we represent the DFTs in the form of event-based Bayesian structure to model parallel and series system, and bridge structure is modelled with a similar structure but with a specific function. Each transition distribution of each elementary component of the system is represented by a continuous random variable. Each transition of the system is connected by several components’ corresponding transitions and characterised by an arithmetic function. The state of the asset given a time instance is modelled with a discrete random variable using the models we described in the last subsection. Since the resulting BN is a hybrid BN with continuous variables, we use dynamic discretisation for its inference.

To specify the joint distribution of the BN, we need to assign the marginal (prior) probability density functions of the basic events (transition distributions of components) and functions of intermediate events (transition distributions of asset system). In the case where deterioration distribution between components is statistically independent, we can define them by their marginal probability distributions. By using the models described in Section 4.1 and 4.2, we can learn these transition distributions from deterioration data and knowledge. The function of an intermediate event is characterised by its parents and its fault tree construct. These functions are illustrated with the examples showed in Figure 4.11.

Figure 4.11 Multi-state asset system configuration: (a)asset assembled in parallel; (b)asset assembled in series; (c)asset assembled with bridge structure.

Consider Asset 1, 2 and 3 (A1, A2 and A3) all consisting of two components, Component 1 (C1) and Component 2 (C2), arranged in parallel, series and with a bridge structure respectively. Transition distributions of all components are learned from the deterioration learning models mentioned before. The system configuration models are depicted in Figure 4.11 for asset rated by multiple states (a three states example is showed: Good, Fair and Poor).

In Figure 4.11(a), the components are assembled in parallel. AND gates are used to describe the relationship between input and output event. AND gate is commonly used in the parallel system, representing the event fails if all the components in the system fail. We denote the continuous transition distribution of Component Ci as TCi, we have TAND= max{TCi}. Each transition of A1 is constructed by an AND gate that connects the corresponding transition from C1 and C2. We also include a categorical variable developed from the last subsection that connects the transition distributions of A1 and the inspection time, representing the state of A1 given an inspection time.

Similarly, in Figure 4.11(b), the components are assembled in series and can be described by OR gates. OR gate is used to represent a situation where the event fails when at least one of the components fails. It can be denoted as TOR= min{TCi}. Each transition of A2 is constructed by an OR gate that connects the corresponding transition from C1 and C2 and evaluated by a categorical variable.

Slightly differently, in Figure 4.11(c), the components are assembled with a bridge structure and evaluated by an aggregation function of its components with weights rather than using the fault tree gates. Each component in the bridge structure system is assumed to have a designated weight contributing towards its failure. And further, aggregated, an example of an aggregated weighted mean (wmean) is discussed here. We denote the weight of Component Ci as wCi, where i = 1, . . . , n, we have the weighted mean transition distribution T of this system T =∑ n i=1wCiTCi ∑n_i=1wCi (4.12)

Each transition of A3 is constructed by a weighted mean function that connects the corresponding transition from C1 and C2. In this example, experts believe that contributing to the transition between Good to Fair of A3 as a system, C1 has a weight of 0.3 and C2 has a weight of 0.7; for the transition between Fair to Poor, C1 has a weight of 0.4 and C2 has a weight of 0.6. These system transitions are further evaluated by a categorical variable to predict the asset condition.

Sometimes, decision makers are more interested in evaluating the state of the system by the state of its components, rather than the transition distributions. For example, in the US,

one criterion in determining whether a bridge is structurally deficient is a condition rating of 4 or less for any of its structures, including deck, superstructure, substructure and culvert (if it exists). We can model them by considering each of these structures as a subsystem, and they are assembled in parallel.

In a simple case where assets are rated by binary states (for example, working or fail), we can model them as traditional fault tree with a Boolean node in the BN. The Boolean node is constructed with a comparative statement to express the logic gates and can be used to perform static fault tree analysis. For example, for a binary-state asset that is evaluated by the states of two components C1 and C2 in parallel, we can define: Asset ∼ i f(C1 ==“Failed” && C2 ==“Failed”, “Failed”, “Working”). Here, C1 == “Failed” means C1 is in “Failed” state. This expression means if both events, event C1 is in “Failed” state, and event C2 is in “Failed” state, are true, then the result is in “Failed” state, otherwise in “Working” state. With BN tools like AgenaRisk [2], this expression for the Boolean operator can be automatically transformed into a corresponding CPT for inference in the BN. The joint probability distribution of the BN that has a collection of independent cause variables Component C consists of Component Ci, where i = 1, 2, . . . , n as the parents of the asset’s

state Y: p(C,Y ) = p(Y |C) n

∏

Therefore, together with the prior probability of the state of components (learned from the condition prediction discussed in the last subsection), after the inference of the BN, we can get the probability of a binary-state asset’s states assembled in parallel, in series or with a bridge structure. Thought we could extend the if-else statements from the Boolean nodes in binary-state systems to form a collection of logic expressions (nested if-else statements) to evaluate the state of a multi-state asset with components assembled in parallel or series, the process for defining these logic expressions becomes tedious with the increase in the state number.

Instead, for a multi-state asset that is evaluated by multi-state components, functions such as maximum, minimum or weighted mean can be used to describe the relationships between states of components and asset. To do that, we need to consider representing the states with numerical scales rather than merely using categorical states. Here, their states are modelled with a set of ordinal continuous intervals. The discrete variable (state of components and assets) is mapped onto a continuous scale that is bounded (from 0 to 1) and monotonically ordered. The reason for expressing them on an ordinal scale is that in most asset rating scales, the states are ordered. For example, in the NBI, rating 9 represents a better condition

compares to rating 8. Given the number of asset states, we can discretise the numerical scale accordingly. Hence, the binary-state system becomes one of the special cases with an interval of 0.5. Figure 4.12 presents a three-state scale example. Therefore, the interval for each state is 0.333. Thus, Good state belongs to the interval of [1, 0.667) and Fair state belongs to the interval of [0.667, 0.333) and so forth.

Figure 4.12 Multi-state asset system assembled by parallel components.

Table 4.2 CPT for an asset with two components that are assembled in parallel.

Component 1 Poor Fair Good

Component 2 Poor Fair Good Poor Fair Good Poor Fair Good Asset 1

Poor (0 - 0.333) 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Fair (0.333 - 0.667) 0.0 1.0 0.0 1.0 1.0 0.0 0.0 0.0 0.0 Good (0.667 - 1) 0.0 0.0 1.0 0.0 0.0 1.0 1.0 1.0 1.0

Therefore, with these numerically scaled states, we can use maximum and minimum or weighted mean functions. Since the states, for example, Good to Poor, are mapped to ordered numerical bounds from high to low, and their states are mutually exclusive, the probability of asset under maximum, minimum or weighted mean function becomes the maximum, minimum or weighted mean state scale of their components and later re-marginalised. Figure 4.12 presents an example of two components assembled in parallel. The CPT generated from the maximum function for Asset 4 is given in Table 4.2. The probability of the asset follows the sequence of logic: if either component is in Good state, the asset is in Good state; or if either component is in Fair state, the asset is in Fair state; otherwise, the asset is in Poor state.

Additionally, for a bridge structure system configuration, the quantification of the con- tribution of each cause to the effect (credibility of state of the component to the state of an asset) often involves uncertainty. To consider the uncertainty, we can extend the state node that is discretized into several ordinal continuous intervals to a state node that is characterised by a TNormal distribution using these ordinal continuous intervals. This function is also im- plemented in the ranked nodes developed from Fenton et al. [46] and has been implemented in software like AgenaRisk [2] as discussed in Section 3.2.2. Therefore, we have:

Asset State ∼ TNormal(µ, σ2, 0, 1)

where µ is the weighted mean function of the state of components. σ2is the variance representing the certainty from experts about assigning the weights of components: ranging from 0 to 1, where 0 represents entirely certain about the weights, and 1 represents utterly uncertain about the weights. The lower bound of 0 and upper bound of 1 are applied to avoid probability from expanding into infinity ranges but instead regularise the resulting distribution within the finite range of 0 to 1. Hence, in order to include the uncertainty about assigning the weights, for bridge structure, instead of using a function like wmean (0.4, Component 1, 0.6, Component 2), we can use function TNormal (wmean (0.4, Component 1, 0.6, Component 2), 0.2, 0, 1). It not only includes the wmean function as the µ, but also consists of the uncertainty about the weights of components as σ2, which is 0.2, a relatively high level of uncertainty in this example.

In summary, this subsection presents two types of modelling approaches to evaluate the state of an asset that is assembled by multiple components. Depending on the requirement of the applications, we can either estimate the state of an asset from its components’ deterioration distributions, or its components’ states. Especially for a bridge structure configuration, this subsection also shows how to extend our model to include expert knowledge. This knowledge reflects the expert’s certainty level about assigning the weights of components, which represents the contribution of the state of each component to the state of the asset as a system. Though only three types of system configurations are developed: parallel, series and bridge structure, they can be combined or extended to model more complex configurations, for example, using the AND and OR gates together to model a k-out-of-n system or extend AND gate with a logic expression to form a Priority AND gate to model system redundancy as suggested in Marquez et al. [108].