Footprint of uncertainty in the context of Bow-tie risk tool using fuzzy logic

The Bow-tie method is a qualitative risk analysis tool known for its effectiveness in visualizing the relationships between causes, barriers, and consequences. In this paper, we present a novel approach to quantifying Bow-tie diagram outcomes by incorporating uncertainty in both input and output parameters. We utilize fuzzy inference to aggregate frequencies and probabilities of failure on demand (PFD). Two fuzzy logic systems are tested: the Type 1 Fuzzy Logic System (T1FLS), and the Interval Type 2 Fuzzy Logic System (IT2FLS), applied here for the first time. The primary innovation of this work lies in the application of IT2FLS, which introduces the concept of the footprint of uncertainty (FOU) to better account for uncertainty in the membership functions of linguistic variables. To validate these models, both were adjusted using expert knowledge to replicate the behavior of the Bow-tie combined with Layers of Protection Analysis (probabilistic model). Simulations compared the proposed methods to the probabilistic model, with sensitivity analyses examining variations in protection barrier failure probabilities. The T1FLS achieved a normalized root mean square error (NRMSE) of 9.54%, while the IT2FLS reached 12.82%. For the normalized root mean square logarithmic error (NRMSLE), T1FLS yielded 4.65%, and IT2FLS 6.30%. The methods showed 87.32% similarity in ranking protection barrier sensitivity indices. The findings suggest both fuzzy systems exhibit strong potential for accurately representing complex systems with inherent uncertainties, making them valuable tools for risk analysis.