API RP 581 - 3rd Ed.2016 - Add.2-2020 - Risk-Based Inspection Methodology

RISK-BASED INSPECTION METHODOLOGY, PART 5—SPECIAL EQUIPMENT 5-55
Applying a Bayesian updating approach to problems of this type is common to adjust probabilities
as new information is collected. This approach assumes that the Weibull shape parameter (β
parameter) remains constant based on the historical data, and adjusts the characteristic life (η
parameter), as inspection data are collected. This is analogous to evaluating a one-parameter
Weibull to update the PRD performance. Bayes’ Theorem works best when the error rates for a
test are very small. This is not the case for PRDs. Test effectiveness, shown in Table 6.8, range
from 50 % to 90 %. This uncertainty using Bayes’ Theorem results in an unrealistically high
adjusted POF, particularly for a passed bench test. Therefore, a modified inspection updating
method was devised to provide reasonable adjustments of characteristic life.
Since the default Weibull parameters for a given PRD provide the probability of a failure on
demand vs time, a default POFOD (modified as per Section 6) may be obtained for the device
based on its in-service duration at the time of inspection. This inspection method begins with the
prior POFOD and is calculated using Equation (5.95) as follows:
P
prd
f ,prior
⎡
β
⎛ t ⎞
⎤
= 1−exp
⎢− ⎥
⎢ ⎜
⎝η
⎟
mod ⎠ ⎥
⎣ ⎦
(5.95)
The prior probability that the device will pass on demand is:
prd
Pp,prior
prd
= 1− Pf ,prior
(5.96)
After the inspection, a second POFOD is calculated based upon the conditional probability factor,
or confidence factor (CF) for the effectiveness of the inspection performed (see Table 6.9). This
second, calculated probability is called the conditional POFOD and is calculated using Equation
(5.97) or Equation (5.98) depending on the result of the inspection:
When the PRD passed the inspection, the conditional POFOD is calculated as follows:
( 1 pass )
prd
prd
Pf ,cond CF Pp,prior
= − ⋅ (5.97)
With a failed inspection, the conditional POFOD is calculated as follows:
( 1 )
P prd prd prd
f ,cond CF fail P f ,prior CF pass P p,prior
A weighted POF,
= ⋅ + − ⋅ (5.98)
prd
P f ,wgt
, is then calculated, where the weighting factors have been formulated to
give more credit to tests conducted later in the characteristic life. Using the prior and conditional
probabilities and the weighting factors, an updated or posterior POFOD is calculated using the
equations provided in Table 6.10.
A revised characteristic life may be obtained using Equation (5.99) based on the in-service
duration of the PRD, the known β parameter, and the posterior probability.