## 4 Analysis
### Sensitivity and Uncertainty Analyses of HOE Variables and ERL
#### 4.2.1 Sensitivity Analysis of HOE Variables and ERL  
Different weights are used for different performance shaping factors (PSFs) in the analysis. The most influencing HOE variables are identified from the analysis. For each test case, ERL values and variations are estimated using the Monte Carlo approach. It is to be noted that the ERL values are necessarily point estimates, where the probabilities of events  
Figure 7: ERL for design scenarios for Case #1 to #4. Note: CS—Concealed Space, CF—Challenging Fire, RC1—Robustness Check Detection Failure, RC2—Robustness Check Sprinkler Failure, RC3—Robustness Check Building Alarm Failure, BE—Blocked Exit, SOU—Sole Occupancy Unit (not CS), BEN—Exit is NOT Blocked. occurring do not take uncertainty into account. The uncertainty inherent in point estimates of HOEs can be considered by estimating a range or distribution in which the probabilities lie (various distributions are described in Appendix B, Table A5). [PERSON] et al. [36] have developed a method for the inclusion of uncertainty by adopting probability distributions in place of point values. This can be performed by transforming the point estimates into probability distributions. The number of parameters in the model is large, thus there is a need to focus on a narrow set of significant HOE variables. Consequently, a sensitivity analysis is conducted to determine the most influential HOE variables on the outcome. To analyze the uncertainty associated with the HOE variables, the beta distribution is assumed because this distribution allows for updating with new HOE data by combining prior with posterior probability; as the number of observations increase, the distribution will become narrower as there is less uncertainty in probability of errors [36]. Beta distributions are useful to express failure probability density functions (PDFs), described by the following equation:  
\[f(x)=\frac{(x-p)^{(\alpha-1)}(q-x)^{(\beta-1)}}{B(\alpha,\beta)(q-p)^{(\alpha +\beta+1)}} \tag{2}\]  
where \(\alpha\) and \(\beta\) indicate the number of successes and failures, respectively. The conversion from point estimation to \(\alpha\) and \(\beta\) values that are required to plot a beta distribution is possible using the three-point estimation method. These three points are the lowest realistic (min), the modal (mod), and the maximum (max); the normal value for each HOE is estimated by expert judgment. PSFs are then used to determine modal, the worst-case and the best-case probability of failure through multipliers that weigh the impact of each factor. The best-case estimation was based on a scenario with realistic HOEs while in the worst-case scenario, the HOEs were assumed to have deteriorated to a point that would still realistically allow an organization to remain functional. For simplicity, the nominal modal level for each HOE-variable is considered, as represented in Table 6: \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**PSF** & **Modal Level** & **Modal Multiplier** & **Best Case Multiplier** &
\begin{tabular}{c} **Worst Case** \\ **Multiplier** \\ \end{tabular} \\ \hline Available time & Nominal & 1 & 1 & 1 \\ Stress and stressors & Nominal & 1 & 1 & 2 \\ Complexity & Nominal & 1 & 1 & 1 \\ Experience and training & Nominal & 1 & 0.1 & 1 \\ Procedures & Nominal & 1 & 0.5 & 1 \\ Ergonomics & Nominal & 1 & 1 & 1 \\ Fitness for duty & Nominal & 1 & 1 & 1 \\ Work processes & Nominal & 1 & 0.8 & 2 \\ Multipliers & & 1 & 0.04 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Performance shaping factors (PSFs) and range for associated multipliers. The results indicate that the most influencing HOE factors are 'not comply with instructions', 'deficient training', and 'inefficient emergency plan'. The mean value of the  
Figure 9: Sensitivity analysis of HOE variables -Tornado plots for case #1 to #4. ERL is highest for Case #4 when considering the HOE 'not comply with the instruction'.