Derivation of an equation for quantitative SIL assignment

Quantitative risk analysis for selection of safety integrity levels (SIL's) is becoming more prevalent due to the high equipment costs associated with qualitative methods. Procedures for applying quantitative methods to the SIL selection problem are in the early stages of development. Literature explaining quantitative methods is scarce and many engineers are forced to develop ad hoc procedures, some of which are mathematically flawed. One of the equations popularly used to assign a probability of failure on demand (PFD) to a safety instrumented function (SIF) based on tolerable risk and unmitigated event frequency is shown in the equation below. Variations of this equation are in use by various SIS practitioners: PFD(SIF) = f(Tolerable)_Risk/f(Unmitigated)_Event. This paper presents a rigorous derivation of this equation based on scientific laws of probability math and standardized definitions used in the field of SIS engineering. By publishing this derivation the author hopes to lend more credibility to this type of SIL selection process and increase its usage in the process industries.