## 4 Results
### F-N Curve Assessment of Case Studies  
Societal risks for the case studies are presented as F-N curves and constructed using the following equation:  
\[F=k\times N^{-a} \tag{10}\]  
where \(F\) is the cumulative frequency of \(N\) or more fatalities, \(N\) is the number of fatalities, \(a\) is the aversion factor and \(k\) is a constant (please refer to Appendix B for inputs). The individual tolerability limit based on PD 7974-7:2019 is given as \(1\times 10^{-4}\)/year while the de minimis limit is set at \(1\times 10^{-6}\)[6]. From the draft ABCB Tolerable Risk Handbook, the Upper and Lower Individual Tolerance Limits are set as \(5\times 10^{-4}\)/year and \(5\times 10^{-6}\)/year respectively for residential classes. [PERSON] et al. [49] suggest a method for building the tolerability limit curve mathematically from raw data of previous events. An \(F\) function is minimized to obtain some parameters that define the tolerability limit curve. The method also aims at extrapolating the curve values for low probabilities/high consequences events when data for such events are not available. To create an approximating function that fits with the curve, a minimizing function \(F\) is introduced. The equations for generating the tolerability limit curve are provided in Appendix C. Figure 5 shows the F-N curves for Case #1 to Case #7: Trial Design TD01 to Trial Design TD04: [noHOE, HOE, fullHOE]. The F-N curves for the other Trial Designs TD05-TD16 are provided in Supplementary Material--Figures S1-S3. F-N curves for each individual Trial Designs for each Case #1 to #7 (total 112 individual F-N graphs) are provided in Supplementary Material--Figures S4-S10. The tolerability and de minimis limit curves are indicated in Figure 5 as follows:  
* The de minimis limit (denoted by dotted lines with black circle as markers) is the lower threshold below which designs are denoted as 'broadly acceptable' indicating no further requirement to investigate further risk reduction measures below this threshold [2]. * The gradient of tolerability limit curve (denoted by circle markers) is shallow for low consequences and steep for high consequences to explicitly acknowledge risk aversion. * The shape of the tolerability and de-minimis limits results in a wider ALARP region, increasing the range of design solutions requiring explicit cost-benefit assessment to maximize social welfare [2; 3]. In the ALARP region, safety measures should be implemented to reduce the risk to ALARP so the design can be considered acceptable. Figure 5: F-N curves for Case #1 to Case #7: TD01 to TD04: noHOE, HOE, fullHOE. * The F-N curve moves upward when HOEs are considered indicating increasing risks and greater severity in terms of frequency and number of fatalities. ABCB tolerability curves are represented graphically in Figure 5 and the bounds set by BSI 7974 for their tolerance are superimposed using black solid and dashed lines. Overall, the limits set by ABCB are similar to BSI PD-7974-7:2019. However, the rate of change in allowable frequency is also much faster (steeper slope) than BSI. The ABCB slope of \(-1.5\) indicates a higher risk aversion that BSI's neutral risk aversion slope of \(-1\). Following are observations from Figure 5:  
* Generally, HOEs result in an increase in risk levels for all seven case studies. In TD01 (with all active systems present), only Case #7 exceeded the upper tolerability limit of \(1\times 10^{-4}\) based on BSI PD7974-7 tolerability. Against ABCB tolerability bounds, Case #7 did not exceed the upper tolerance while it is at the borderline of the upper Kaneko curve. For Case #7, the difference in risk with and without HOEs is \(\sim\)33%. On the other hand, the risk levels in Cases #1, #2, #3, #5, and #6 (with sprinkler systems, BOWS, smoke detection, and natural ventilation in corridors), stayed within the ALARP region even when HOEs were considered. * Societal risks remain high if no active safety systems are present. For TD04, the F-N curve exceeds the ABCB tolerability curve for cases #1, 3, 6, and 7 (noHOE).