DOI: 10.3303/CET24112050 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper Received: 11 June 2024; Revised: 08 September 2024; Accepted: 02 October 2024 
Please cite this article as: Dehghanisanij A., Khakzad N., 2024, Natech Risk Assessment with Imprecise Probabilities, Chemical Engineering 
Transactions, 112, 295-300  DOI:10.3303/CET24112050 
  

 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 112, 2024 

A publication of 

 
The Italian Association 

of Chemical Engineering 
Online at www.cetjournal.it 

Guest Editors: Selena Sironi, Laura Capelli 
Copyright © 2024, AIDIC Servizi S.r.l. 
ISBN 979-12-81206-13-7; ISSN 2283-9216 

Natech Risk Assessment with Imprecise Probabilities 
Alireza Dehghanisanij, Nima Khakzad * 
School of Occupational and Public Health, Toronto Metropolitan University, Toronto, Canada 
nima.khakzad@torontomu.ca  

Natechs are technological accidents that are triggered by natural disasters. The increase in the frequency and 
severity of climatic natural disasters and the growth of industrialization demand the development of dedicated 
methodologies for risk assessment and management of Natechs. Due to the lack of accurate and sufficient 
data, the risk assessment of Natechs has largely been based on subjective assumptions and imprecise 
probabilities, making the assessed risks and the subsequent risk management strategies less cost-effective. 
In the present study, evidence theory, as an effective technique for dealing with imprecise probabilities, and 
Bayesian network, as an effective tool for reasoning under uncertainty, are combined to develop a 
methodology for risk analysis of Natechs. Flotation of oil tanks during floods has been considered to exemplify 
the methodology. It is demonstrated that replacing the interval probabilities with average probabilities can 
result in different yet consistent risk values.   

1. Introduction 
Technological accidents which are triggered by natural disasters are known as natural-technological accidents 
or Natechs. Natechs that occur to chemical and process plants can be catastrophic due to the possibility of 
damage to process units and subsequent release of hazardous chemicals which may cause fire and 
explosions or major environmental pollution. Among the process units, atmospheric storage tanks have 
reportedly been the most vulnerable type of vessels (Godoy, 2007). This is because such vessels have thin 
shells and high volume-weight ratios. A thin shell makes the storage tank very susceptible to lateral forces 
exerted by high winds or floods, whereas a high volume-weight ratio makes the tank susceptible to buoyancy 
force in the event of floods or heavy rainfalls (Godoy, 2007; Qin et al., 2020). Flotation of atmospheric storage 
tanks due to the buoyancy force has been identified as the most common failure mode during floods (Cozzani 
et al., 2010; Landucci et al., 2012; Khakzad and van Gelder, 2017, 2018). 
Compared to conventional technological accidents which are caused by random failures or human error, risk 
assessment and management of natechs are prone to more uncertainty and are thus more challenging. The 
foregoing uncertainty consists of aleatory uncertainty that arises from the randomness of natural disasters or 
failures and epistemic uncertainty that represents our lack of knowledge due to insufficient or inaccurate 
objective data for the Natech of interest. 
Probability theory has effectively been used to account for uncertainty embedded in the occurrence and 
severity of natural disasters as well as the extent of damage they may cause to structures and industrial 
plants. However, in the absence of sufficiently large and reliable datasets or accurate field measurements, 
subject matter experts may inevitably come up with subjective imprecise probabilities that may influence the 
accuracy and credibility of risk analysis if not properly handled. Evidence theory (Dempster, 1967; Shafer, 
1976) is an effective tool for handling imprecise probabilities. In Evidence theory, the propagation of 
uncertainty is based on belief masses rather than probability masses. Belief masses are analyst’s degrees of 
belief about a hypothesis and can be derived from imprecise probabilities. Compared with probability theory, 
the application of evidence theory to the domain of risk assessment and management has not been so 
widespread mainly due to a lack of efficient inference algorithms. Simon and Weber (2009) and Khakzad 
(2019) demonstrated that the Bayesian network (BN) and evidence theory can be combined to form a so-
called evidential network (EN), where BN is used to handle belief masses developed through evidence theory 
the same way as it handles the probabilities. 

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mailto:nima.khakzad@torontomu.ca


The present study aims to demonstrate an application of EN to risk assessment and management of natechs 
when due to a lack of knowledge the analyst may express his uncertainty in the form of interval probabilities. 
Section 2 briefly reviews the evidence theory and how it can be combined with BN; in Section 3 the 
methodology is applied to risk assessment of tank flotation during floods; Section 4 concludes the work. 

2. Evidential theory and evidential network 
In this section, the basics of evidence theory and evidential networks are briefly reviewed.  

2.1 Evidence theory 

Using the evidence theory (Dempster, 1967; Shafer, 1976), all the possible states of a random variable can be 
presented in a set, named the frame of discernment Ω. To each subset of Ω such as Ai, which is a hypothesis 
about the state of the variable, a weight 0.0 ≤ m(Ai) ≤ 1.0 can be assigned to express the degree of belief, 
based on objective data or subjective opinion, in the claim that the variable’ state belongs to Ai (Rakowsky, 
2007). Having m(Ai), which is also known as the belief mass of Ai, the belief bel(Ai) and plausibility pls(Ai) can 
be determined. For the sake of clarity, consider a binary component X with the two states, fail and work, and 
thus a frame of discernment as ΩX = {fail, work}. Therefore, the set of all the subsets of ΩX would be A: {{∅}, 
{fail}, {work}, {fail, work}}, where A1 = {∅}, A2 = {fail}, A3 = {work}, and A4 = {fail, work}. Each member of A for 
which m(Ai) > 0 is called a focal set. If we are certain that all the states of the variable are included in the 
frame of discernment, then m(∅) = 0. It must always hold that: 

∑ m(Ai) = 1Ai    (1) 

Having the belief masses determined, the belief and plausibility measures of each focal set can be defined 
using: 

bel(Ai) = ∑ m(Aj)Aj|Aj⊆Ai    (2) 

pls(Ai) = ∑ m(Aj)Aj|Aj∩Ai≠∅    (3) 

For instance, an expert may assign the weights mX({fail}, {work}, {fail, work}) = (0.15, 0.8, 0.05), in which 
mX({fail, work}) = 0.05 refers to the expert’s uncertainty about the state of X. Using Equations (2) and (3), the 
belief and plausibility of X = {fail} can be calculated, respectively, as bel(X = {fail}) = m({fail}) = 0.15 and pls(X 
= {fail}) = m({fail}) + m({fail, work}) = 0.15 + 0.05 = 0.2. It should be noted that in calculating the plausibility of X 
= {fail} using Equation (3), the mass of {fail, work} should be considered because {fail} ∩ {fail, work} ≠ ∅; 
however, it should not be considered in calculating the belief of X = {fail} via Equation (2) because {fail, work} 
⊈ {fail}.  
Further, the amount of uncertainty Unc(Ai) of a focal set can be expressed as the difference between pls(Ai) 
and bel(Ai) as (Rakowsky, 2007): 

Unc(Ai) = pls(Ai) − bel(Ai)   (4) 

Since mX({fail, work}) represents the uncertainty about the state of X, Equation (4) can be used to calculate it 
as mX({fail, work}) = pls(X = {fail}) – bel(X = {fail}) = 0.05. Subsequently, bel(Ai) and pls(Ai), which are non-
additive, can be taken as lower and upper probability bounds of Ai, respectively (Shafer, 1976): 

bel(Ai) ≤ P(Ai) ≤ pls(Ai)   (5) 

bel(Ai
c) = 1 − pls(Ai)   (6) 

pls(Ai
c) = 1 − bel(Ai)   (7) 

where Ai
C is the complement of Ai in the sense that Ai

c = Ω − Ai.  
According to Equation (5), 0.15 ≤ P(X = fail) ≤ 0.2. Moreover, according to Equations (6) and (7), bel(X = 
{work}) = 1 – pls(X = {fail}) = 0.8 and pls(X = {work}) = 1 – bel(X = {fail}) = 0.85, and thus 0.8 ≤ P(X = work) ≤ 
0.85. Having the bel and pls functions, the belief mass of a focal set can be determined using the möbius 
transformation as (Smets, 2002): 

m(Ai) = ∑ (−1)�Ai−Aj� bel(Aj)Aj|Aj⊆Ai    (8) 

where |Ai – Aj| refers to the difference between the number of elements of Ai and Aj. 

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2.2  Evidential network 

Simon and Weber (2009) and Khakzad (2019) showed that belief masses can be used in BN the same way as 
the probabilities and thus the algorithms developed for BN could be employed to propagate belief masses in a 
system. Since the belief masses allocated to the focal sets of each random variable must add up to 1.0, they 
can be treated as marginal probabilities for the root nodes in a BN.   
Considering a parallel system in Figure 1, comprising two binary components X and Y. Further, assume that 
due to a lack of sufficient knowledge, the analyst cannot assign precise probabilities to the states of X and Y 
and decides to express their uncertainty in the form of interval probabilities as 0.15 ≤ P(X = fail) ≤ 0.35 and 0.2 
≤ P(Y = fail) ≤ 0.5. 
 

 

Figure 1. BN for failure assessment of System using belief masses of X and Y. Nodes X and Y are connected 
to node System by AND agte.  

Having these interval probabilities, the belief masses of the focal sets of X and Y can be identified. For 
instance, consider P(X = fail), where its lower and upper bounds can be taken as the bel and pls functions, 
respectively: 

• bel(X = {fail}) = 0.15 → mX({fail}) = 0.15  
• bel(X = {work}) = 1 – pls(X = {fail}) = 1 – 0.35 = 0.65 → mX({work}) = 0.65 
• mX({fail, work}) = 1 – mX({fail}) – mX({work}) = 1 – 0.15 – 0.65 = 0.2 
• As a result: mX({fail}, {work}, {fail, work}) = (0.15, 0.65, 0.2).  

Following the same procedure, mY({fail}, {work}, {fail, work}) = (0.2, 0.5, 0.3). The calculated mass beliefs can 
now be used in BN to compute the belief masses of the system. In the BN shown in Figure 1, the focal sets of 
X and Y have been considered as the states of nodes X and Y while the respective belief masses have been 
considered as their probabilities. The same focal sets have also been considered for the node System, which 
is connected to nodes X and Y via AND gate. The truth table shown in Table 1 can be used to populate the 
conditional belief table of node System (Simon and Weber, 2009) 

Table 1. Truth table used to populate the conditional belief table of node System in Figure 1 in case of AND 
gate and OR gate. 

Component 
X 

 
Y 

System: AND  
{fail} 

gate 
{work} 

 
{fail, work} 

System: OR gate 
{fail} 

 
{work} 

 
{fail, work} 

{fail} 
{fail} 
{fail} 
{work} 
{work} 
{work} 
{fail, work} 
{fail, work} 
{fail, work} 

{fail} 
{work} 
{fail, work} 
{fail} 
{work} 
{fail, work} 
{fail} 
{work} 
{fail, work} 

1 
0 
0 
0 
0 
0 
0 
0 
0 

0 
1 
0 
1 
1 
1 
0 
1 
0 

0 
0 
1 
0 
0 
0 
1 
0 
1 

1 
1 
1 
1 
0 
0 
1 
0 
0 

0 
0 
0 
0 
1 
0 
0 
0 
0 

0 
0 
0 
0 
0 
1 
0 
1 
1 

 
The developed BN can accordingly be used to calculate the belief masses of node System. Having the belief 
masses of System as mSystem({fail}, {work}, {fail, work}) = (0.03, 0.825, 0.145), the bel and pls functions or the 
lower and upper bound probabilities of (System = fail) can be calculated as bel(System = {fail}) = 0.03 and 
pls(System = {fail}) = 0.03 + 0.145 = 0.175, resulting in 0.03 ≤ P(System = fail) ≤ 0.175.  

297



3. Risk assessment with imprecise probabilities 
In this section, using an illustrative case study, the development of the methodology is demonstrated, and the 
results are presented and discussed regarding the previous studies.  

3.1 Case study 

Flotation of oil storage tanks has reportedly been the most frequent failure mode during floods. Flotation of 
storage tanks occurs if the upthrust force of flood (buoyancy force FB) exceeds the bulk weight of the storage 
tank (weight of the tank WT plus the weight of its containment WL). Considering that such tanks are usually 
unanchored and thus no resisting force is exerted on them from their foundation, FB, WT and WL are the only 
forces considered for the flotation of the tank in Figure 2 (Khakzad and van Gelder, 2017; Dehghanisanij et al., 
2024). Given the tank’s dimension and the flood’s inundation depth, the foregoing forces can be modeled as: 

d

H

L

h

t
WT

FB
WL

 

Figure 2. Schematic of flotation-related loading and resisting forces acting on an oil storage tank.  

FB = ρw g π d2

4
h  (9) 

WT = ρs g �πdH + 2 π d2

4
� t  (10) 

WL = ρl g
π d2

4
L  (11) 

Consider a case in which the characteristics of the storage tank and an imminent flood are measured or 
predicted as listed in Table 2.  

Table 2. Parameters used to develop the limit state equation for flotation of the storage tank. 

Parameters  Symbols Values 
Tank’s height 
Tank’s diameter  
Tank’s shell thickness 
Chemical inventory height 
Tank material density (steel) 
Flood water density 
Chemical inventory density (gasoline) 

H (m) 
d (m) 
t (m) 
L (m) 
ρs (kg/m3) 
ρw (kg/m3) 
ρl (kg/m3) 

6 
10 
0.01 
(0.5, 1.0, 1.5) 
7900 
1024 
850 

Flood inundation height h (m) N (µ = 1, σ = 0.2) 

 
Given the foregoing forces, the limit-state equation (LSE) for the flotation of the tank can be developed 
(Landucci et al., 2012; Khakzad and van Gelder, 2017): 

LSE = FB − WT − WL  (12) 

As such, the flotation probability of the tank can be presented as the probability that LSE > 0.  
Based on historical data, the flood’s inundation height is expected to follow a normal distribution (Table 2). 
Further, assume that although the specific values of the tank’s dimension are known, the initial amount of 
crude oil inside the tank is unknown as the mechanical and automatic level indicators each show a different 
number: the mechanical level indicator, which is likely to malfunction and is thus not so reliable, shows the 
crude level as L = 1.0 m but the automatic gauge shows the level as L = 0.5 m. Therefore, the operator 
decides to take a glance at the crude level via the top manhole, estimating the crude level as L = 1.5 m. As 
such, the operator’s uncertainty about the level of crude oil can be modeled as a tertiary random variable L 
with three states as L1 = 0.5 m, L2 = 1 m, and L3 = 1.5 m, with the following interval probabilities based on his 
confidence in the mechanical and automatic gauges and his own estimate:  

• 0.2 ≤ P(L1) ≤ 0.5  

298



• 0.3 ≤ P(L2) ≤ 0.5 
• 0.2 ≤ P(L3) ≤ 0.3. 

3.2 Risk assessment 

According to the parameters in Table 2, the magnitudes of the three forces can be calculated as WT = 268 
(KN), WL = 655×L (KN), and FB = 789×h (KN). The probability that the storage tank floats due to the buoyancy 
force can thus be calculated as: 

P(Flotation = yes) = P(FB > WT + WL) = P(789 h > 268 + 655 L) = P �h > 268 + 655 L
789

�  (13) 

Considering L as an uncertain variable with three states as L1, L2 and L3, its frame of discernment can be 
developed as ΩL = {L1, L2, L3}. Consequently, the set of its focal sets would be AL: {{L1}, {L2}, {L3}, {L1, L2}, {L1, 
L3}, {L2, L3}, {L1, L2, L3}}. The belief mass of each focal set can subsequently be determined. For example, for 
the first focal set {L1} with the lower and upper bound probabilities as 0.2 < P(L1) < 0.5, the belief and 
plausibility functions can be determined as bel({L1}) = 0.2 and pls({L1}) = 0.5. Since {L1} is a singleton, m({L1}) 
= bel({L1}) = 0.2. Similarly, m({L2}) = 0.3 and m({L3}) = 0.2. 
Further, consider the focal set {L1, L2}. Since {L1}, {L2}, and {L1, L2} are all the subsets of {L1, L2}, we will have 
m({L1, L2}) = bel({L1, L2}) – bel({L1}) – bel({L2}). Furthermore, bel({L1, L2}) = 1 – pls({L3}) = 1 – 0.3 = 0.7. As a 
result, m({L1, L2}) = 0.7 – 0.2 – 0.3 = 0.2. Following the same procedure, m{{L1}, {L2}, {L3}, {L1, L2}, {L1, L3}, {L2, 
L3}, {L1, L2, L3}} = (0.2, 0.3, 0.2, 0.2, 0.1, 0.0, 0.0). Since m({L1, L3}) = m({L1, L2, L3}) = 0.0, they would not be 
considered focal sets anymore.  
As can be seen from Equation (13), the only influential parameters in estimating the probability of tank flotation 
are the flood inundation height (h) and the height of the chemical inside the tank (L). To facilitate the 
propagation of uncertainty – aleatory uncertainty in h and epistemic uncertainty in L – the BN in Figure 3 can 
be developed.  
 

 
Figure 3. BN to estimate the probability of tank floatation. 

Quantifying the BN, the belief masses of the focal sets of “Flotation” can be calculated as mFlotation ({yes}) = 
0.26, mFlotation ({no}) = 0.52, and mFlotation ({yes, no}) = 0.22. Consequently, the belief and plausibility functions 
or the lower and upper bound probabilities of (Flotation = yes) can be calculated as: 0.26 ≤ P(Flotation = yes) 
≤ 0.48. Likewise, 0.52 ≤ P(Flotation = no) ≤ 0.74. 

3.3 Discussion 

Regarding the probability intervals for L, each probability interval may be replaced with an average probability. 
Yager and Kreinovich (1999) proposed that the average probability pj~ for an interval 𝑝𝑝𝑗𝑗 = [𝑝𝑝𝑗𝑗−, 𝑝𝑝𝑗𝑗+] can be 
calculated as: 

pj~ = Σ+−1
Σ+−Σ−

. pj− + 1−Σ−

Σ+−Σ−
. pj+   (14) 

where: 

Σ+ = ∑ pi+n
i=1    (15) 

Σ− = ∑ pi−n
i=1    (16) 

𝑝𝑝𝑗𝑗− and 𝑝𝑝𝑗𝑗+ are the lower and upper probability bounds of the interval, respectively. Therefore, given the 
probability intervals for L as 0.2 ≤ P(L1) ≤ 0.5, 0.3 ≤ P(L2) ≤ 0.5, and 0.2 ≤ P(L3) ≤ 0.3, the values of 𝛴𝛴+ and 𝛴𝛴− 
can be calculated as: 

299



• 𝛴𝛴+ = ∑ 𝑝𝑝𝑖𝑖+3
𝑖𝑖=1 = 𝑝𝑝1+ + 𝑝𝑝2+ + 𝑝𝑝3+ = 0.5 + 0.5 + 0.3 = 1.3  

• 𝛴𝛴− = ∑ 𝑝𝑝𝑖𝑖−3
𝑖𝑖=1 = 𝑝𝑝1− + 𝑝𝑝2− + 𝑝𝑝3− = 0.2 + 0.3 + 0.2 = 0.7  

Subsequently, the average probabilities for L1, L2, and L3 can be calculated as 𝑝𝑝1~ = 0.35, 𝑝𝑝2~ = 0.4, and 𝑝𝑝3~ =
0.25. Now, L can be considered as a discrete variable with a discrete distribution as P(L1, L2, L3) = (0.35, 0.4, 
0.25). Having h as a normal variable (Table 2) with a normal distribution as h ~ N(1, 0.2), Monte Carlo 
simulation can be employed to estimate the probability of flotation via Equation (13). Conducting the 
simulation for 10,000 iterations, the probability of flotation can be estimated as P(Flotation = yes) = 0.38, and 
subsequently P(Flotation = no) = 1 – 0.38 = 0.62. The results derived from evidence theory and the ones from 
the average probabilities are summarized in Table 3.  

Table 3. Comparison between the results obtained from interval probabilities and average probabilities. 

Approach P(Flotation = yes) P(Flotation = no) 
Interval probabilities [0.26, 0.48] [0.52, 0.74] 
Average probabilities 0.38 0.62 

4. Conclusions 
In the present study, we presented an application of evidence theory to risk assessment of oil storage tanks 
during floods. It was demonstrated that replacing the interval probabilities with average probabilities may result 
in different yet consistent risk outcomes. Therefore, in the absence of required resources (time, expertise, etc.) 
for dealing with interval probabilities, the analyst may decide to replace them with average probabilities as the 
results are still consistent with the results obtained from the interval probabilities. However, more applications 
and comparisons are required to determine if the average probabilities could efficiently substitute the interval 
probabilities for risk assessment and if both approached would lead to similar risk management strategies.  

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