Fuzzy evidence theory and Bayesian networks for process systems risk analysis

Abstract

Quantitative risk assessment (QRA) approaches systematically evaluate the likelihood, impacts, and risk of adverse events. QRA using fault tree analysis (FTA) is based on the assumptions that failure events have crisp probabilities and they are statistically independent. The crisp probabilities of the events are often absent, which leads to data uncertainty. However, the independence assumption leads to model uncertainty. Experts’ knowledge can be utilized to obtain unknown failure data; however, this process itself is subject to different issues such as imprecision, incompleteness, and lack of consensus. For this reason, to minimize the overall uncertainty in QRA, in addition to addressing the uncertainties in the knowledge, it is equally important to combine the opinions of multiple experts and update prior beliefs based on new evidence. In this article, a novel methodology is proposed for QRA by combining fuzzy set theory and evidence theory with Bayesian networks to describe the uncertainties, aggregate experts’ opinions, and update prior probabilities when new evidences become available. Additionally, sensitivity analysis is performed to identify the most critical events in the FTA. The effectiveness of the proposed approach has been demonstrated via application to a practical system.

Keywords:

• risk analysis
• fault tree analysis
• process safety
• evidence theory
• fuzzy set theory
• Bayesian networks
• uncertainty analysis

Acknowledgment

The research of Sohag Kabir was partly funded by the DEIS project (Grant Agreement 732242).

Nomenclature

BBA	=	basic belief assignment
BIM	=	birnbaum importance measure
BE	=	basic event
Bel	=	belief measure
Bet	=	the pignistic probability function in the belief structure
BN	=	Bayesian network
BT	=	bow-tie model
CSB	=	chemical Safety Board
CP	=	crisp possibility
CPT	=	conditional probability tables
D-S	=	Dempster-Shafer theory
E	=	evidence
ETA	=	event tree analysis
F	=	failure
FMEA	=	failure mode and effect analysis
FP	=	failure probability
FT	=	fault tree
FTA	=	fault tree analysis
Hazmat	=	hazardous material
HAZOP	=	hazard and operability study
HTHA	=	high temperature hydrogen attack
I	=	importance measures
IE	=	intermediate event
LOPA	=	layer of protection analysis
P	=	the probability of an input event
PDF	=	probability density function
pl	=	plausibility measure
QRA	=	quantitative risk assessment
RPB	=	release prevention barrier
RoV	=	ratio of variation
S	=	success
SA	=	sensitivity analysis
TE	=	top event

Symbols

$\mathbf{X}$	=	decision matrix
${\mathbf{x}}_{\mathbf{j}}$	=	the column vector of decision matrix regarding to jth attributes
${\mathit{A}}_{\mathbf{m}}$	=	the row of decision matrix representing alternative
${\mathit{P}}_{\mathit{i}\mathit{j}}$	=	the projected outcomes of attributes j
m	=	the number of evaluated alternative
n	=	the number of criteria
${\mathit{E}}_{\mathit{j}}$	=	the set of projected outcomes of criterion j
${\mathit{d}\mathit{i}\mathit{v}}_{\mathit{j}}$	=	the degree of diversification
${\mathit{w}}_{\mathit{j}}$	=	the objective weight of criteria j
${\mathit{n}}_{\mathit{m}\mathit{n}}$	=	the normalized value regarding to criteria n from alternative m
${\mathit{W}}_{{\mathit{A}}_{\mathit{m}}}$	=	the weight of each alternative
${\mathit{P}}_{\mathit{L}}$	=	the lower boundary of triangular fuzzy number
${\mathit{P}}_{\mathit{m}}$	=	the most likely boundary of triangular fuzzy number
${\mathit{P}}_{\mathit{U}}$	=	the upper boundary of triangular fuzzy number
${\mathit{\mu }}_{\mathit{g}}$	=	fuzzy membership function having g vectors
$\mathit{x}$	=	the output variable in fuzzy membership function
X*	=	the defuzzified output of fuzzy membership function
N	=	the number of elements using in Dempster-Shafer theory
$\mathit{\Omega }$	=	the set of N elements in Dempster-Shafer theory
PS	=	the power set
$\mathit{m}({\mathit{p}}_{\mathit{i}})$	=	the belief mass
$\overline{{\mathbf{p}\mathbf{s}}_{\mathbf{i}}}$	=	the negation of a hypothesis${\mathrm{ }\mathit{p}\mathit{s}}_{\mathit{i}}$
U	=	the set of event used in BN
$\mathit{\pi }$	=	posterior probability of BE
$\mathit{\theta }$	=	prior probability of BE