1. Introduction
Significant attention, both in academia and in practice, has been directed towards the development of new techniques and expertise in the construction processes of buildings. However, the ageing of (Western European) building stock has progressively sparked interest in maintenance and future developments in the field [
1]. As a result, standard practices have evolved from corrective maintenance, where works are performed after the occurrence of a failure in order to bring a component back into a state where it can perform its intended functions [
2,
3], to preventive maintenance, where interventions are performed following a specific schedule [
3,
4].
Preventive maintenance (PM) was originally used to pre-emptively reduce or eliminate the deterioration of building components [
5,
6]. Rapidly, though, scholars and practitioners came to realize that certain components were replaced despite being in good condition, thus incurring unnecessary costs. Consequently, condition-based maintenance (CBM) gained momentum. In contrast with predetermined PM, interventions in CBM are planned based on the condition of the assets under management, which is assessed during periodic inspections [
5,
7]. While the scope and methodologies of these inspections vary across regions and sectors, they commonly rely on the sensory assessment of individual building components’ condition.
Because of the poor accessibility and the complexity of mechanical, electrical, and plumbing (MEP) systems, these sensory inspections are rarely sufficient to reliably evaluate the condition of their components [
8]. Asset managers are thus compelled to obtain condition data through estimates, commonly based on a limited number of parameters (e.g., age and theoretical lifespan; see, e.g., [
9]), or through appraisals from third parties, resulting in a poor integration of these data in the overall maintenance strategies. This tendency led MEP to be the building trade where the highest number of defects are reported [
10]. Clearly, then, implementing new methods to estimate these components’ condition is key to improve buildings’ occupants’ comfort [
11,
12,
13] and minimize repair costs, which can be substantial [
10,
14].
To that end, the present study investigates the applicability of Bayesian networks (BNs) for the estimation of MEP systems’ condition. Bayesian networks, which are probabilistic graphical models used to study probabilistic influence between random variables, were selected because their graphical structure facilitates interactions with practitioners and they robustly handle missing data [
15,
16,
17]. These characteristics are essential in the context of MEP systems given the scarcity of historical condition data. To better address this lack of data, so-called Gaussian copula-based Bayesian networks (GCBN) are adopted in this research. Their formulation, detailed in the next section, enables the involvement of field experts for the quantification of the model through structured expert judgments (SEJ), as demonstrated by past implementations of GCBNs (e.g., [
18,
19,
20]).
Whereas the elicitation of univariate distributions has been investigated in academia with great depth, the assessment of dependence remains a topic yet to be consolidated in SEJ literature. Therefore, this paper focuses on the development of a method for the assessment of (conditional) rank correlations by field experts, while less attention is devoted to the elicitation of the one-dimensional marginal distributions. In contrast with existing research, which has delved into the use of statistical [
19,
21] and conditional fractile estimates [
18,
20,
22] approaches, the relevance of a third type of probabilistic assessment is hereby studied: probabilities of concordance. Given the assumptions underlying GCBNs, unconditional rank correlations can be retrieved from concordance probabilities using a set of closed-form relations, which are defined in
Section 2.2.1.
The next section presents theory on GCBNs and related statistical concepts (
Section 2.1), and introduces the methodology implemented in this paper to retrieve rank correlations from expert judgments (
Section 2.2). Then, the case study selected for the implementation of the aforementioned elicitation method is presented (
Section 3). In
Section 4, the results of the consultations are presented and analyzed, resulting in a quantified network for air handling units. Lastly, the research’s findings are discussed and conclusions are drawn with regards to the research objectives formulated above (
Section 5).
5. Discussion
In the literature, one of two approaches are often adopted to validate a Bayesian network: the model’s predictions are compared to empirical data (when available); or experts, who contributed or not to the model creation, are asked to assess the model’s output when subjected to a set of scenarios [
54,
55,
56]. Clearly, the use of data was excluded in this research, simply because they were unavailable. Therefore, the current model was subjected to three hypothetical scenarios to assess the model output’s logic:
Scenario 1: old AHU, frequent maintenance;
Scenarios 2/3: excellent Design & Construction quality, recent/old AHU.
Scenario 1: old AHU, frequent maintenance.
The first scenario consisted in the following configuration:
- •
‘AHU age’: 40 years,
- •
‘Maintenance interval’: 6 months,
- •
‘Design & Construction quality’: 3.63 (mean value),
- ⇒
.
The dependence structure elicited from experts indicates that the age of the unit and the frequency at which it is maintained overwhelmingly affect its condition. The outcome of Scenario 1 is shown in
Figure 12. Unsurprisingly, the filters’ condition has significantly improved due to its connection with maintenance. However, this outcome, while consistent with our earlier assumptions, appears to be somewhat unrealistic from a physical standpoint. Expert D illustrated the relationship between ‘Filters’ and ‘Maintenance’ with the example of Schiphol airport, the Netherlands’ main international airport, where filters are replaced three to four times a year due to air pollution. In fact, experts almost unanimously (4/5) indicated that variables describing
environmental conditions should be included in the model because of their impact on the filters’ deterioration. Clearly then, this scenario showcases the model’s disproportionate response as the probabilities associated to states 3 and above (for ’Filters’) should not be null, as demonstrated by the example of Schiphol.
Similarly, the distribution of the variable ‘Coils’ shifted to the left, reflecting an improvement from the unconditional case. This finds explanation in the dependence structure of the BN, where the correlation between ‘Coils’ and ‘Maintenance interval’ is substantially higher than between ‘Coils’ and ‘Age’ (0.686 and 0.275, respectively). Because the main mode of deterioration of the coils is by corrosion, accelerated by frost and the accumulation of particles, consistently cleaning them allows one to temper the phenomenon. Moreover, the shift in the distribution of ‘Filters’ also influences the one of ‘Coils’ since these variables are positively correlated. Interestingly, the probabilities of states 3, 4 and 5 are relatively low (0.15, 0 and 0, respectively) given the advanced age of the unit and the theoretical lifespan of the coils (∼20–25 years). For the same reason, the probability that the coils are in condition 1 (0.26) is abnormally high, indicating that the model’s capacity to handle extreme cases is limited.
Finally, the fans’ condition was inversely impacted by the input values: the probability that the component is in condition 4 has dramatically increased (from 0.39 to 0.65), again in accordance with the correlation of ‘Fans’ with ‘Age’ being higher than with ‘Maintenance interval’ (0.52 and 0.219, respectively). Failure in the fans mainly involves mechanical malfunctions such as exhaustion of the motor or failure of the bearings, whose maintenance has limited impact on their lifespan. The conditional probability that the component is in reasonable condition or better seems high (0.31) but most of this density is in state 3, which is conform with the previous comments.
Scenarios 2/3: old AHU, very poor/excellent Design & Construction quality.
The second and third scenarios consisted in the following configurations:
- •
‘AHU age’: 40 years,
- •
‘Maintenance interval’: 1.20 (mean value),
- •
‘Design & Construction quality’: 1 (very poor, Scen. 2)/5 (excellent, Scen. 3),
- ⇒
; .
Scenarios 2 and 3 aim to determine whether an investment in an excellent quality installation significantly affects the long-term condition of the air handling unit, and whether that is reflected by the model’s outputs. The latter are displayed in
Figure 13.
For old systems, an increase in ‘D&C quality’ evidently results in a slower deterioration for both ‘Coils’ and ‘Fans’, with a substantial share of the distributions being below the threshold values: and , against and in Scenario 2. In alignment with the observations for Scenario 5, the impact of an excellent quality on the components is too high. While it is logical to witness an improvement from Scenario 2, the unit’s age (40 years) must translate in medium-to-high likelihoods for states 4 and 5. Conversely, the probabilities of states 1 and 2 are too high as both components (almost) have reached their theoretical lifespan. Clearly then, Scenario 2 indicates that the rank correlations associated to the edges ‘D&C quality’ → ‘Coils’ (−0.324) and ‘D&C quality’ → ‘Fans’ (−0.443) are possibly too high (in absolute values), which may partly stem from expert C’s strong assessment for D&C quality’ → ‘Fans’ (−0.795). His assessment, which is substantially higher than the rest of the experts’, strongly contributes to the final decision maker because of his excellent dependence calibration score.
All in all, the influence of the basic quality of the air handling unit’s components is correctly translated, even though some adjustments to the model’s parameters are still needed to obtain more realistic outputs. We notably observe that high-quality materials, design and construction can significantly extend the components’ lifespan.
As mentioned in
Section 2, the addition of new variables is facilitated by the modular nature of GCBNs. Because Scenario 1 underpinned the necessity to include the node ’Environmental conditions’ as an input, and to illustrate the effect of the addition of that variable, let us consider a hypothetical network, which includes the variable ’Environmental conditions’ defined on the following scale (
Table 6):
For illustration purposes, we assigned equal probabilities to each state (i.e., ‘Environmental conditions’
). Then, consultations with the experts indicated that this factor mainly influences the deterioration of the filters, hence the creation of the edge ‘Environmental conditions’ → ‘Filters’. Although this relationship is weaker than that between ‘Maintenance interval’ and ‘Filters’, the conditional correlation associated to the new edge will be high since for two units maintained at the same frequency, the environmental conditions are a strong predictor for the conditions of the filters. Resultingly, we considered
. The resulting model, used in the next scenario, is illustrated in
Figure 14.
Scenario 4: old AHU, frequent maintenance and very unfavorable environmental conditions.
The fourth scenario consisted in the following configuration:
- •
‘AHU age’: 40 years,
- •
‘Maintenance interval’: 6 months,
- •
‘Design & Construction quality’: 3.63 (mean value),
- •
‘Environmental conditions’: 1 (very unfavorable),
- ⇒
.
Figure 15 illustrates the conditional distributions of ‘Filters’ obtained in Scenarios 1 and 4. First, there is an evident change in the distribution. The discussion on Scenario 1 underlined that the probabilities of states 3, 4, and 5 could not realistically be null without information on the environmental conditions. Here, evidence of very unfavorable climatic conditions clearly resulted in a concentration of the distribution around states 3 and 4, with probabilities of 0.646 and 0.268, respectively, aligning with the example of Schiphol airport presented previously. This brief discussion demonstrates that the addition of ‘Environmental conditions’, although not rigorous, was a fairly straightforward endeavor that yielded encouraging results. Still, the previous paragraphs underlined that the developed model is not ready for practical applications.
6. Conclusions
To the authors knowledge, this article presents to date the first application of probabilities of concordance for the assessment of dependence in expert-based GCBNs. While the experts’ feedback indicates that this method is relevant and accessible, two elements may have influenced the validity of the elicited values. First, the closed-form equations used to compute the rank correlations from the concordance probabilities require the normal copula assumption, which often fails to reflect the behaviour of real-life systems. Second, some experts expressed difficulty assessing unconditional concordance probabilities, i.e., accounting for the uncertainty given that evidence on only one (parent) variable is available. Instead, the use of conditional concordance probabilities may help reduce the ‘vagueness’ perceived by some of the respondents.
Let
be a vector of covariates; then, for each
, the concordance probability between two random variables X and Y given
is:
with (
) and (
) two random draws of variables X and Y. To illustrate the practical impact of this modification, let us consider the edge between ‘Maintenance interval’ and ‘Coils’ (
Figure 11). To assess
, an expert would be presented the following question:
“Two buildings A and B are randomly selected among all non-residential buildings in the Netherlands. Given that the AHUs in buildings A and B are both z years old, and that the AHU in building A is maintained more regularly than in building B (), what is the probability that the coils are in better condition in building A than building B ()?”
However, for conditional concordance probabilities to be relevant, additional research should investigate the extent to which their use facilitates the elicitation and whether the protocol used to retrieve rank correlations from unconditional concordance probabilities still applies. The latter is crucial as the validity of the closed-form formulas used to retrieve rank correlations (
Section 2.2.1) is not trivial in the conditional case, and is demonstrated in
Appendix D. Moreover, the dependence in
can be eliminated by assuming that
is constant in
, similarly to assumptions formulated for conditional exceedance probabilities. All in all, the use of conditional probabilities of concordance could enhance the interpretability of the questionnaire presented to the experts, and therefore the quality of the collected assessments.
Furthermore, the application of dependence calibration in this research highlighted the challenge of selecting appropriate seed variables when few to no empirical data are available. Past studies relied on the wide availability of data in their field (e.g., [
20], with traffic data) or knowledge of the ‘true’ dependence structure (e.g., [
46]). However, due to the emerging nature of the method, there is no guideline for its application in data-sparse environments, sometimes constraining assessors to use equal weights decision-makers [
57] or unrelated seed variables as was the case in this study. Therefore, future research should focus on assessing the potential loss of accuracy between a BN quantified with field data and another with ‘common knowledge’ information, such as precipitation or physiological data.