# VoI-Based Optimization of Structural Assessment for Spatially Degrading RC Structures
## 6 Scatter or Uncertainty on the Posterior Costs  
The calculation of the VoI is based on estimated values of the costs, averaged over different possible monitoring outcomes. This results in a distribution for the costs. The expected value of this distribution is then calculated and used in the determination of the VoI, as also illustrated in the flowchart in Figure 1. Nevertheless, besides the expected value, the scatter on the calculated costs could also be accounted for. For the prior costs, a coefficient of variation of about 0.25 is found for all seven different repair strategies. For the posterior costs, some differences on this COV are found, as given in Table 7.  
Furthermore, when increasing the number of locations where the strain mode shapes are measured (strategy 2), the uncertainty on the posterior cost decreases from a COV of 0.012 to a COV of 0.006. For the static strains (strategy 3), the uncertainty even reduces from a COV of 0.09 to 0.02 when increasing the number of sensors from 20 to 50. Under these specific assumptions, the uncertainty on the costs is also much larger for the mode shapes (strategy 1) than for the other monitoring data. Hence, data with a larger (measurement and/or model) error leads to a larger posterior uncertainty. This is also illustrated in Figure 12, where the mean value and uncertainty of the posterior cost is given for the different measurement errors given in Table 8. Besides the results visualized in Figure 12, it is also found that, when the prior COV of the corrosion parameters is increased, the COV of the costs also increases. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Exposure Class** & \(\mu\) **[mm/year]** & \(\sigma\) **[mm/year]** \\ \hline
1 & Wet—rarely dry & 0.004 & 0.006 \\
2 & Cyclic wet–dry & 0.030 & 0.020 \\
3 & Airborne seawater & 0.030 & 0.040 \\
4 & Tidal zone & 0.070 & 0.070 \\ \hline \end{tabular}
\end{table} 
Table 6: Different parameters for the distribution of the corrosion rate based on [53]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Repair Model** & **COV Posterior Cost** \\ \hline
1 & 0.013 \\
2 & 0.012 \\
3 & 0.009 \\
4 & 0.017 \\
5 & 0.006 \\
6 & 0.006 \\
7 & 0.012 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: COV of the posterior costs for the different repair models. Since the posterior uncertainty of the costs also depends on the monitoring strategy and the accompanying (measurement and/or model) errors, it might be necessary to not only take into account the expected values of the costs in the calculation of the Vol but also to include the accompanying uncertainties. A method to account for the uncertainties on the costs can be based on robust design optimization, where uncertainties on the objective function \(f(x)\) are accounted for by changing the minimization according to Equation (12). \[\textit{minimize}\ W\frac{\mu_{f(x)}}{\mu_{f(x)}^{0}}+(1-W)\frac{\sigma_{f(x)} }{\sigma_{f(x)}^{0}} \tag{12}\]  
where \(W\) is a weighting factor within the range [0, 1], and (\(\mu_{f(x)}^{0}\), \(\sigma_{f(x)}^{0}\)) is the utopia point corresponding to \(W\) = 0 or \(W\) = 1. By inserting this weighting factor \(W\) in the equation, a trade-off is found between performance and robustness. In the case of the optimization of monitoring strategies, the objective function equals the Vol. This Vol should be maximized, which corresponds to a minimization of the posterior cost, since the prior cost is the same for all monitoring strategies to be compared. Instead of minimizing the expected value of the posterior cost, one could also consider \(W\mu_{post}+(1-W)\sigma_{post}\) to account for a balance between expected value and the uncertainty on it. As an example, when applying the latter formula to estimate the posterior costs following the strain mode shapes at different monitoring times (strategy 2), the results are given in Figure 13. Here, it can be seen that the general pattern of the posterior cost as a  
Figure 12: Influence of the measurement error on the uncertainty of the posterior cost.