Risk Assessment Model for Oil & Gas Industry Safety Risks

Knowledge-Based Systems 156 (2018) 62–73
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Knowledge-Based Systems
journal homepage: www.elsevier.com/locate/knosys
A multi-experts and multi-criteria risk assessment model for safety risks in
oil and gas industry integrating risk attitudes☆
T
⁎
Donghong Tian ,a, Bowen Yangb, Junhua Chenb, Yi Zhaob
a
b
School of Sciences, Southwest Petroleum University, Chengdu, China
School of Economics and Management, Southwest Petroleum University, Chengdu, China
A R T I C LE I N FO
A B S T R A C T
Keywords:
Safety risk
Risk matrix
OWA operator
Risk attitude
Utility function
Multi-criteria
This paper mainly proposes a novel method to establish a risk matrix for assessing safety risks in oil and gas
industry. The frequency and the consequence of risk are two desired criteria in the establishment process of a risk
matrix. In fact, more than two criteria and several experts are often involved, so a multi-criteria and multiexperts information integration (MEMCII) model is constructed in this paper. Firstly, the method of the determination of experts’ weights is introduced to integrate experts’ assessment scores based on the objective
weights and the subjective weights. Secondly, the weighted ordered weighted operator (WOWA) with a utility
interpolation function is proposed to derive the overall consequence integrating people’s risk attitudes. Finally, a
risk matrix is established to show which risks are highly dangerous and which can be ignored. In addition, an
application is demonstrated to illustrate the efficiency and flexibility of the proposed model.
1. Introduction
In recent years, some unwanted accidents such as personal injuries,
explosions, fire, man-made destructions and oil evaporations possibly
occurred in Chinese oil and gas industry. For example, on December 23,
2011, a blowout accident occurred in Sichuan Qionglai No.1 well, four
people were injured and one was missing [1]; on November 22, 2013,
an explosion in Qingdao east yellow oil pipeline killed 62 people and
injured 136 people, and the direct economic loss was 751.72 million
yuan(RMB) [2]; on September 21, 2016, a deflagration accident occurred in the third engineering branch of petrochina pipeline, four
people were injured and two died [3]. In order to prevent these accidents, oil and gas company must carry out effective risk management.
How to manage safety risks effectively in oil and gas industry is an
important issue, and risk assessment is a primary and key process
within risk management. Traditionally, qualitative way, semi-quantitative way and quantitative way are three ways to carry out risk assessment. The qualitative methods are used to assess risks when credible and accurate data are missing [4,5]. Quantitative risk assessment
methods are frequently used in situations where there are sufficient
data or data can be derived based on simulation [6–10]. The semiquantitative way that is suitable for data with quantitative and qualitative characteristics is often used in risk assessment [11–15]. The risk
matrix approach (RMA), which is a semi-quantitative way, is a classical
tool for risk assessment. Many scholars have devoted major efforts to
construct a reasonable risk matrix in recent years [14,16,17]. Though
the RMA is a practical tool for risk assessment, it still needs to be improved to deal with safety risks in oil and gas industry, for the following
reasons:
(1) The consequence of safety risk in oil and gas industry mainly
performs in four aspects: accident level, economic loss, reputation loss
and environmental pollution [18]. All the four indicators are consequences of safety risk and not all of them are easily measured with
money, so the consequence of safety risk should be a combination of the
four indicators. But the existing methods, which are used to construct a
risk matrix, consider the consequence from economic loss only or the
overall severity which is derived by using some linguistic quantifiers.
This cannot respond well to the consequences of safety risks, therefore
an innovative approach is needed to construct a reasonable safety risk
assessment matrix based on multiple criteria.
(2) On many occasions, the value records of safety accidents are
missing especially for the risk which performs as a low probability but
high cost event. Experts scoring method is usually adopted to get the
values. Experts’ risk attitudes are inevitably involved when they score
safety risks, so the construction of safety risk matrix should integrate
experts’ risk attitudes [14]. However, the data for risk assessment are
☆
This work was jointly supported by the entrusted project of Sichuan Development Research Center of Oil and Gas under Grant No.SKW14-08, the National Social Science Foundation
of China under Grant No. 16CSH070.
⁎
Corresponding author.
E-mail address: [email protected] (D. Tian).
https://doi.org/10.1016/j.knosys.2018.05.018
Received 15 July 2017; Received in revised form 9 May 2018; Accepted 12 May 2018
Available online 23 May 2018
0950-7051/ © 2018 Elsevier B.V. All rights reserved.
Knowledge-Based Systems 156 (2018) 62–73
D. Tian et al.
paper is organized as follows: Section 2 introduces the background to
the paper, a multi-experts integration model in which the integrated
weights are determined based on the objective and subjective weights is
introduced in Section 3; Section 4 proposes a multi-criteria aggregation
model integrating risk attitudes; an application in certain oil and gas
company and a discussion of model flexibility are demonstrated in
Section 5; Section 6 provides some conclusions.
rarely processed to reduce the effect of risk attitudes according to the
existing literature. For safety risks in oil and gas industry, people may
well present loss aversion that will affect their assessment scores. Hence
the scores should be modified to some extent before risk matrix is
constructed.
Therefore the target of this paper is to propose a novel model to
construct a reasonable risk matrix for safety risks in oil and gas industry. According to the existing literature [12,17], most studies agree
to that the risk is a combination of the consequence and the frequency.
Consistent with this, the risk matrix is established considering two input
variables (consequence and frequency) as its 2 axes. The values of the
consequence and frequency of risks have important effect on the risk
assessment results. Marhavilas and Koulouriotis depicted the gradations
of the harm factor as the consequence based on the severity of human
injuries [10]; the consequence was divided into 5 ranks on account of
economic loss from 0 to 4000 (thousand RMB) [14]. Obviously, only
considering one aspect of safety risk in oil and gas industry as the risk
consequence is not comprehensive. More and more scholars began to
assess the risk consequence from multiple indicators [8,18], but this
idea has not yet been used to build a risk matrix.
In order to construct a reasonable risk matrix for assessing safety
risks in this paper, it is comprehensive that the consequence of safety
risk in oil and gas industry is depicted from several aspects and the
consequence axis is established based on the overall consequence. So
the consequence in a safety risk assessment matrix should be a combination of several criteria, then a multiple criteria aggregation model
is needed in such a problem. There are many methods to construct a
aggregation model [19–22]. The selection of the aggregation model
should consider the nature of safety risk in oil and gas industry. Once a
safety risk occurs, the loss of lives and properties would be great. Based
on this, most people are inevitably loss-averse when they assess safety
risk. In recent years, some scholars introduced the utility theory to
describe the risk attitudes of decision makers in risk assessment [9,14].
Ruan et al. pointed out that integrating risk attitudes in the risk assessment was imperative[14]. Then a novel multi-criteria aggregation
model integrating people’s risk attitudes is constructed in this paper. To
the authors’ knowledge, there is no a such method in the existing literature.
Usually the rules about the severities of safety accidents are made in
oil and gas industry from several aspects (such as economic loss, reputation loss, etc.), but their dimensions are not uniform. According to
the rules, the risk criteria can be divided into several ranks (such as 5)
and experts can score risk criteria basing on these ranks and their experiences and the dimensions are uniformed. Since there are multiple
experts in the risk assessment, this scoring process is essentially a group
decision making problem. Some scholars constructed the integration
models in which several preference relations were considered and the
information of experts were integrated [23–25]. Experts may represent
their opinions about alternatives using different preference representation formats, such as orderings of alternatives, utility functions,
multiplicative preference relations, additive preference relations, ordinal 2-tuple linguistic preference relations and so on. The purpose of
the safety risk assessment in this paper is to give the risk level for each
risk. So the preference relations which indicate the preference on pair of
alternatives are not used. On the other hand, it is not enough to just give
the orderings of risks. In this paper, the experts’ preference relations are
expressed by utility function. One of the main tasks of this paper is to
combine these individual utilities into a collective utility reasonably for
safety risk, and the safety risk assessment matrix should be constructed
on the basis of the collective utility.
Therefore, the safety risk assessment in oil and gas industry is a
multi-experts and multi-criteria information integration (MEMCII)
problem essentially. In the context of current literature, there is no
published paper regarding the MEMCII problem in the establishment
process of a risk matrix. A two-stage safety risk assessment model solving this problem is constructed in this paper. The remainder of this
2. Background
This section introduces the background to the paper, including
safety risk assessment, traditional risk assessment matrix, two aggregating operator and the utility function.
2.1. Safety risk assessment
Petrochemical industry has become a high-risk industry.
Inflammability, high temperature, high pressure, toxicant, easy corrosion etc. are all the reasons that cause safety risks in the productive
process of petrochemical industry. They are serious threats to the enterprise staffs’ and the surrounding community residents’ lives and health.
Countless bloody accidents tell us that ensuring the safety in oil and gas
industry is a very important work. Safety risk management is mainly to
control safety risks and make them acceptable. Before controlling the
safety risks, the first work is to identify and assess them.
Most studies agree to that the risk is a combination of the consequence and the frequency. Safety risk assessment is carried out based
on the two indexes. The frequency refers to the time needed for a safety
risk happening once. The consequence of safety risk in oil and gas industry mainly performs in four aspects: accident level, economic loss,
reputation loss and environmental pollution. In general, accident level
is used to show how many injuries in a risk accident, the direct economic loss in a risk accident is expressed by economic loss and measured in RMB, reputation loss means the size of the criticized scope
because of a risk accident, the scope of environmental pollution because
of a risk accident is expressed by environmental pollution. All the four
aspects are consequences of safety risk and not all of them are easily
measured with money. In many oil and gas industries, the specific values of safety accidents are missing especially for the risk which performs as a low probability but high cost event. To derive the values,
expert scoring is usually adopted as the evaluating method. Therefore,
semi-quantitative assessment method is used to implement the safety
risk assessment problem.
2.2. Traditional risk assessment matrix
Risk assessment matrix is a useful tool to implement semi-quantitative risk assessment. The frequency and the consequence of risk are
the desired indexes. There are mainly four steps to establish a traditional risk assessment matrix:
1. The severities of frequency and consequence are categorized and
scaled.
2. The output risk index is divided into different levels.
3. The IF-THEN rule between all input and output is built up, that is
to say, IF frequency is fm and consequence is cn THEN risk is rk, where
fm, cn and rk are ranks of frequency, consequence and risk respectively.
4. The graphical depiction of the risk matrix is created.
Fig. 1 shows an example of traditional risk assessment matrix with
25 cells [26]. The frequency and consequence are all categorized into
five different levels and the output risk index is divided into three
different levels: Low(L), Medium(M), High(H). Within the matrix, the
green zone denotes the low risk level, the yellow and red zones stand
for the medium and high risk levels respectively.
In recent years, many scholars focused on how to establish a reasonable risk assessment matrix. Considering the partial knowledge and
information, Markowski and Sam Mannan constructed fuzzy risk
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Knowledge-Based Systems 156 (2018) 62–73
D. Tian et al.
5
4.5
M
H
H
H
H
M
M
M
H
H
L
M
M
M
H
L
L
M
M
H
L
L
L
M
M
4
Consequence
3.5
3
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
Frequency
Fig. 1. The traditional risk assessment matrix with 25 cells.
The popular operator which is used to modify the effect of bigger or
smaller values is the OWA opertor [19,27,29,30,36–39,42]. When the
OWA operator is applied, the assessment values need to be reordered in
a descending order and the weights associated to the positions need to
be determined. This method can flexibly deal with the preference to the
bigger or smaller values. Then the risk attitudes of assessment experts
can be considered.
matrixes and three relations (HARD, STANDARD and EASY) between
all input and output were shown in their paper [17]. Ni et al. pointed
out that the risk tie was inevitable in risk matrix and put forward some
extensions on risk matrix approach [12]. Ruan et al. pointed out that it
was necessary to consider decision makers’ risk attitudes in risk assessment and utility theory was integrated in the construction process
of risk matrix [14]. Four risk assessment matrices were constructed
form reputation, assets, people and environment respectively and the
ranks in four matrices were integrated to derive the overall ranks of
risks [18]. On the base of potential risk influence, an analysis framework of risk matrix was made for risk assessment and ranking [15].
The bases of risk assessment matrices in these literature are the
concept that the risk is a combination of frequency and consequence.
The consequence of risk was only depicted from economic loss only or
the overall severity which was derived by using some linguistic quantifiers and the scores of frequency and consequence were given only by
one expert [12,15,17,18]. In most existing literature, the specific intervals of index value of safety risk consequence are not made except
for some linguistic quantifiers. Due to the consequence of safety risk in
oil and gas industry mainly performs in four aspects: accident level,
economic loss, reputation loss and environmental pollution and the
scores of risks are often given by several experts, it is necessary to
propose a novel method to construct a safety risk assessment matrix
based on multiple experts and multiple criteria. This paper primarily
concerns the problem of aggregating multiple criteria to form the
overall risk consequence.
Definition 1. Suppose that a1, a2 , …, an are the scores on n criteria of a
risk, the weighted average operator (WA) is defined as
WA (a1, a2, …, an ) = a1 w1 + a2 w2 + ⋯+an wn,
n
where ∑i = 1 wi = 1, 0 ≤ wi ≤ 1.
Definition 2. [19] A mapping FOWA from
I n→I
is called an Ordered Weighted Aggregation (OWA) operator if there is a
weight vector W = (w1, w2, …, wn ), such that (1) wi ∈ (0, 1), (2) ∑i wi = 1
and
FOWA (a1, a2, …, an ) = w1 b1 + w2 b2 + …+wn bn ,
(1)
where bi is the ith largest value of a1, a2, … , an.
The OWA operator fulfills the following properties: Monotonicity,
Boundedness, Commutativity, Symmetry and Idempotency, these
properties were demonstrated in detail in Yager’s paper [19]. The arithmetic mean, minimum, maximum, median, etc. are all examples of
the OWA operator and can be derived though ascribing corresponding
weights in the operator. Two measures were put forward by Yager for
depicting the characters of the OWA operator [19].
The first measure named attitudinal character is defined as
n
n−i
AC (W ) = ∑i = 1 n − 1 wi, AC(W) lies in the interval [0,1] and characters
the extent of the preference to the smaller or bigger values. When the
value of w1 is 1 and other weights take 0, AC(W) takes 1 and the preference to the biggest value reaches maximum. When the value of wn is
1 and other weights take 0, AC(W) takes 0 and the preference to the
smallest value reaches maximum. It is worth noting that when the value
of AC(W) is closer to 0.5, the operator is more neutral to the bigger and
smaller values.
2.3. Aggregating operator
From the existing literature [18,20,22,33,35,40], the usually used
method in a multi-criteria aggregation problem is assigning appropriate
weights to criteria to construct an aggregation operator. The selection
of aggregation operator in risk assessment problem should consider the
nature of safety risk. When one assesses safety risk, the risk attitude is
inevitably involved. Then, the risk attitudes of assessment experts need
to be described and the assessment scores need to be adjusted to some
extent. Therefore, the aggregation operator which can modify the effect
of bigger or smaller assessment scores should be considered.
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Knowledge-Based Systems 156 (2018) 62–73
D. Tian et al.
three properties: (1) f (0) = 0 ; (2) f (1) = 1; (3) f(x) ≥ f(y); if x ≥ y was
also a weight generating function [42].
The second measure called dispersion measure is defined as
n
Disp (W ) = − ∑i = 1 wi ln (wi ) which measures the degree of the use of all
values. We can see that ln(n) is the maximum of Disp(W) when the
1
values of w1, w2, …, wn are all n , and 0 is the minimum of Disp(W) when
some weight takes 1 and the other weights are all 0. The greater the
value of Disp(W), the higher degree of the use of all information.
2.4. Utility function
Definition 3. A real-valued function U: [0, 1] → [0, 1] is called a loss
utility function if the following properties hold: (1) U˙ (x ) > 0 ;
(2) U¨ (x ) > 0, where x denotes the degree of loss, and U(x) denotes the
degree of dissatisfaction.
Example 1. Suppose that a = (2, 3, 1, 4) is the four-dimensional vector
to be aggregated of a risk, and W = (0.1, 0.3, 0.4, 0.2) is the predetermined weight vector. Thus,
⎡4⎤
FOWA (a) = [0.1, 0.3, 0.4, 0.2] ⎢ 3 ⎥
⎢2⎥
⎣1⎦
= 0.1 × 4 + 0.3 × 3 + 0.4 × 2 + 0.2 × 1
= 2.3.
Utility function which can reflect the risk attitude is widely used in
risk assessment [9,14,31]. Utility is a kind of subjective feeling, and it
can be used to quantify the preference relation. If X denotes the value of
wealth, the holder’s utility function U(x) satisfies the following characters: (1) U˙ (x ) > 0, the utility function is a monotonic increasing
function, utility increases with wealth increasing; (2) U¨ (x ) < 0, the
utility function meets the law of diminishing marginal utility. If X denotes the degree of loss, according to people’s habits, the utility function should meet U˙ (x ) > 0 and U¨ (x ) > 0, that is, the increase of people’s dissatisfaction (the extent of loss effect) speeds up when the degree
of loss increases. The descriptions of the utility function are different
under different conditions, exponential function is a best fit for utility
function in the risk assessment problem [9], and it’s equation is expressed as follows: U (x ) = ae bx − 1, a > 0, b > 0, it obviously satisfies
U˙ (x ) > 0 and U¨ (x ) > 0 . The value of x denotes the degree of the loss of
a risk, there is no loss when the value is 0, conversely, 1 denotes the
maximal loss. The value of U(x) denotes the severity of dissatisfaction,
and the situation of it’s value is similar to x. When U(x) interpolates the
points (0,0) and (1,1), the utility function can be used for a weight
generating function in the OWA operator.
Then, the attitudinal character and dispersion measure can be
calculated.
4
n−i
2
1
AC (W ) = ∑i = 1 n − 1 wi = 0.1 × 1 + 0.3 × 3 + 0.4 × 3 + 0.2 × 0 = 0.4333
. Obviously, the value of AC(W) is close to 0.5, then the aggregation
operator is relatively neutral to the bigger and smaller values.
4
Disp (W ) = − ∑i = 1 wi ln (wi ) = 1.28 is slightly smaller than the
maximum 1.386, then the OWA operator has a high degree of the use
of all the values. These conclusions are consistent to the actual
situation.
Example 1 shows that the OWA operator is a weighted mean
function to aggregate the values based on a weight vector, the weights
measure the importance of the positions of the scores in relation to their
orders. This method can flexibly adjust the preference to the bigger or
smaller assessment scores by changing the weights. This characteristic
can be used to modify the assessment scores that are affected by experts’ risk attitudes.
There are three steps to be implemented when the OWA operator is
used. Firstly, the assessment scores need to be reordered in a descending order. Secondly, the weights associated to the positions need to
be determined. Thirdly, the overall score can be derived though multiplying the weights vector and the reordered score vector. Obviously,
the weight determination method is a key problem.
In 1996, Yager proposed an approach to determine the OWA
weights basing on some linguistic quantifiers, such as “for all”,“most”, “there exists” and so on [27]; O’Hagan developed a constrained
optimization model to derive the OWA weights [28]; A similar model
was presented by Xu and Da based on adding the weight information to
the constraints [29]; Xu determined the weights by using the normal
distribution and validated that the method could relieve the preference
to the extreme scores [30]. According to the existing literature, the
commonly used approach is the constrained optimization model suggested by O’Hagan. This method is finding the values of w1, w2, …, wn to
satisfy the following constrained optimization problem:
3. A multi-experts integration model
In the classical risk assessment problem, several experts are often
needed to score the risk indicators, and experts’ scores are expressed on
the basis of ranking sets of the specific intervals of the risk indicators.
Since experts have different cognitive information, their scores are inevitably different and need to be integrated. Combining the individual
scores into a integrated score is one of the tasks in this paper. A multiexperts integration model is introduced in this section.
Suppose that R = {r1, r2, …, rq} is a finite set of q risks to be assessed,
E = {e1, e2, …, em} is a finite set of m experts, and C = {c1, c2, …, cn} is a
finite set of n criteria of risk. Usually the rules about the severity of
safety risk are made in oil and gas industry from several aspects, but the
dimensions of different aspects are different. The rules about risk criteria such as accident level, economic loss, reputation loss and environmental pollution all can be easily converted to rank scores.
Without loss of generality, suppose that there are 5 ranks of
as
shown
in
Table
1,
where
ci (i = 1, 2, …, n)
[aij , bij )(i = 1, 2, …, n, j = 1, 2, …, 5) denotes the jth value range of ci
corresponding to the rules in oil and gas industry.
Each criterion has a set of value range like that in Table 1, experts
score the criterion basing on their experiences and the value range set.
Experts’ assessment scores constitute assessment matrices. The assessment matrix of the ith risk can be expressed as follows:
n
Maximize:
− ∑i = 1 wi lnwi
Subject to:
n
1
∑i = 1 (n − i ) wi = β ,
n−1
n
∑i = 1 wi = 1,
0≤β≤1
0 ≤ wi ≤ 1,
i = 1, 2, …, n,
(2)
the solutions of (2) can maximize the extent of the use of all values with
a pre-defined value of attitudinal characterwhich is denoted β.
In the MEMCII problem, the assessment scores possibly tend to be
larger than the actual levels and these scores need to be adjusted by
appropriate weights. When the value of β is smaller than 0.5, the
weights which are the solutions of (2) can not only make maximum of
the use of all information, but also reduce the preference to the bigger
scores.
Many weight determination methods were introduced in Yager’s
paper, and Yager pointed out the function which met the following
Table 1
Ranking scores of ci, i = 1, 2, …, n .
65
Ranking score
0-1
1-2
2-3
3-4
4-5
Criterion range
Description
[ai1, bi1)
Negligible
[ai2, bi2)
Minor
[ai3, bi3)
Moderate
[ai4, bi4)
Serious
[ai5, ∞)
Critical
Knowledge-Based Systems 156 (2018) 62–73
D. Tian et al.
x
x i12 … x i1n
⎤
⎡ i11
x i21 x i22 … x i2n ⎥
⎢
Di =
,
⋮ ⋱ ⋮ ⎥
⎢ ⋮
⎥
⎢
x
x
x
…
imn ⎦
⎣ im1 im2
of 4 columns in D are 3,2,2.6667,2.3333 in turn.
Firstly, the proportions of assessment deviation of three experts on
∼ ⎡ 0.5 0.5 0.5 0.5 ⎤
four criteria construct a deviation matrix D = ⎢ 0 0 0.25 0.25⎥.
⎣ 0.5 0.5 0.25 0.25⎦
Secondly, the credibilities of the assessment scores of three experts
are
calculated
and
construct
a
credibility
matrix
⎡ 0.5 0.5 0.5 0.5 ⎤
D̆ = ⎢ 1 1 0.75 0.75⎥.
⎣ 0.5 0.5 0.75 0.75⎦
Thirdly, the proportions of the credibilities of three experts on four
criteria are calculated and construct a credibility proportion matrix
0.25 0.25 0.25 0.25
 = ⎡ 0.5 0.5 0.375 0.375⎤.
D
⎥
⎢
⎣ 0.25 0.25 0.375 0.375⎦
Finally, the objected weights of three experts are respectively obtained by using Definition 4, then the objective weights of three experts
are 0.25,0.4375,0.3125 in turn. Compared to the initial equal weights,
the objective weights are more reasonable with the consideration of the
credibility of data.
where x ijk (i = 1, 2, …, q, j = 1, 2, …, m , k = 1, 2, …, n) denotes the assessment score of the jth expert on the kth criterion of the ith risk, then there
are q assessment matrices: D1, D2 , …, Dq . In the risk assessment, m experts’ scores need to be integrated to derive Xi . k , where Xi . k denotes the
collective score on the kth criterion of the ith risk. The commonly used
integration operator is determining the suitable experts’ weights to
m
construct a linear integration function:
Xi . k = ∑ λj x ijk , where
j=1
λj (j = 1, 2, …, m) denotes the weight of the jth expert.
Due to the experts have different knowledge levels, educational
backgrounds and experiences, the weights of experts should be different. In the existing literature [32–34], there are some subjective
weight determination approaches and objective weight determination
approaches used to calculate the experts’ weights. According to the
before abilities, knowledge, seniorities, etc. to determine the weights is
called the subjective weight determination approach, such as Delphi
method , weighted least square method, eigenvector method, analytic
hierarchy process etc. Only based on the credibilities of assessment
scores to determine the weights is called the objective weight determination approach, such as factor analysis, multiple objective programming model, entropy method etc. Subjective weights determined
by the subjective methods only consider some subjective information
but the credibilities of scores, objective weights are just the opposite.
Obviously, it is more comprehensive to integrate the subjective weight
determination approach and the objective weight determination approach to determine the weights of experts.
Sometimes experts do not always know the same about each risk,
this means that the credibilities of assessment scores are different about
each risk. So it is more reasonable to determine different objective
weights for different risks. Let λij (i = 1, 2, …, q, j = 1, 2, …, m) be the integrated weight of the jth expert about the ith risk, θj be the subjective
weight of the jth expert and it is same for each risk, and μji be the ob-
Step3: Determine the integrated weight λij .
Definition 5. [34] Weight coefficients λ1i , λ 2i , …, λmi are called the
integrated weights of m experts about the ith risk when these
coefficients integrate the subjective weights and the objective weights
m
simultaneously, and ∑ λij = 1, 0 ≤ λij ≤ 1, with
j=1
λij =
Example 3. In Example 2, the subjective weights of three experts are
0.3,0.3,0.4 given by the decision makers, and the objective weights are
0.25,0.4375,0.3125 calculated by using (3). The integrated weights of
experts can be derived by using (4) and are different with different
values of α. Their values are shown in Table 2, Table 2 shows that when
the value of α is closer to 0, the integrated weights are closer to the
objected weights, while the situation is opposite when α is closer to 1.
j−1
Then the assessment scores of four criteria in Example 2 can be
integrated into the collective scores based on the integrated weights.
The collective scores will be derived after α is determined, 5 kinds of
collective scores with 5 values of α are displayed in Table 2. Table 2
shows that the collective scores are more sensitive to α and the differences are bigger when the experts’ scores on criteria change smaller.
There are q assessment matrices: D1, D2 , …, Dq in a MEMCII problem, m rows of each matrix can be integrated to one row by using the
Step2: Determine the objective weight μji .
m
1
x ijk , then x i . k denotes the simple average level of m
m
j=1
experts on the jth criterion of the ith risk regardless of the subjective
weights. The assessment deviation dik
j between the j − th expert and the
group on the kth criterion of the ith risk can be derived, and
dijk = x ijk − x i . k .
Let x i . k = ∑
m
Definition 4. Weight coefficients μ1i , μ2i , …, μmi are called the objective
weights of m experts when these coefficients reflect the credibilities of
linear aggregation model: Xi . k = ∑ λij x ijk , then q line matrices of q risks
j=1
m
are derived, that is to say, the collective assessment scores on criteria
the assessment scores about the ith risk, and ∑ μji = 1, 0 ≤ μji ≤ 1, with
j=1
i
j = 1 djk
(4)
1−α
m
∼i = 1 −
where μ
jk
∑m
,
i
problem etc. and ∑ θj = 1, 0 < θj < 1.
dijk
i
m
∑ j = 1 λ͠ j
where λ͠ j = θjα ⎛⎜μji ⎞⎟ , 0 ≤ α ≤ 1 is calculated based on the subjective
⎝ ⎠
weight θj and the objective weight μji with the value of α.
The value of α in Definition 5 reflects the attitudes of decision
makers to the subjective weights and the objective weights, the higher
the value of α, the more confidence to the subjective weights.
jective weight. In this paper, λij is calculated by integrating θj and μji
simultaneously.
Step1: Determine the subjective weight θj.
θj which has strong subjectivity is given by the decision makers
based on the expert’s before abilities, familiarity with the decision
n
1
∼ik
∑k = 1 μ
j
μji = mn 1 n ∼ik ,
∑ j = 1 n ∑k = 1 μ j
i
λ͠ j
Table 2
The values of λj and ck with different α.
(3)
reflects the proportion of the assessment
credibility of the j − th expert about the ith risk.
⎡ 4 1 2 3⎤
Example 2. Let D = ⎢ 3 2 3 2 ⎥ be the assessment matrix of a risk,
⎣2 3 3 2⎦
obviously, there are three experts and four criteria. The simple averages
66
α
α=0
α = 0.25
α = 0.5
α = 0.75
α=1
λ1
λ2
λ3
c1
c2
c3
c4
0.2500
0.4375
0.3125
2.9325
2.0625
2.7500
2.2500
0.2637
0.4013
0.3350
2.9287
2.0713
2.7363
2.2637
0.2767
0.3661
0.3572
2.9195
2.0805
2.7233
2.2767
0.2888
0.3322
0.3790
2.9099
2.0901
2.7112
2.2888
0.3
0.3
0.4
2.9
2.1
2.7
2.3
Knowledge-Based Systems 156 (2018) 62–73
D. Tian et al.
being a monotone increasing function that interpolates these points:
n
1
2
(0, 0), n , w1 ,( n , Σj ≤ 2 wj,…, ( n , Σj ≤ n wj ) by means of straight lines, wi (i
…,
= 1,2,
n) and pi (i = 1,2, …, n) being the weight vectors that reflect
the importance of the orders of the values and the importance of the
information sources respectively, and
are derived. Nextly, the elements in these line matrices need to be aggregated to form the overall scores to build a risk matrix.
(
4. A multi-criteria aggregation model
There are q risks, m experts and n criteria in the MEMCII problem.
The assessment scores constitute q matrices with m rows and n columns,
and the m rows in each matrix are integrated to one row by using the
multi-experts integration model in Section 3. For example, the ith assessment matrix is transformed to
FWOWA (a1, a2, …, an ) = ω1 b1 + ω2 b2 + …+ωn bn ,
The WOWA operator is still a weighted average function, the value
of ωi depends on wi and pi at the same time. In recent years, this operator is more and more widely used in many areas. For example, six
people error events were assessed based on the WOWA operator considering four criteria [18].
′
where Xi . j (j = 1, 2, …, n) denotes the collective score on the jth criterion
of the ith risk. Suppose that the n criteria are all consequences of safety
risk from n aspects, and they need to be aggregated into the overall
consequence which will be used in the construction process of a risk
matrix. Then, this is a problem of aggregating multi-criteria into an
overall index.
From the existing literature [35,36], the usually used method in a
multi-criteria aggregation problem is assigning appropriate weight values to criteria to construct a linear aggregation operator. But the selection of aggregation operator in this paper should consider the nature
of safety risk. When one assesses safety risk, the risk attitude is inevitably involved in the assessment process. Because the occurrence of
safety risk is often accompanied by losses, loss aversion is the probable
risk attitude and the assessment scores are always larger than the actual
levels. Then, the risk attitudes of assessment experts need to be described and the assessment scores need to be adjusted to some extent.
In addition to the above reason, there is another situation that
should be considered. The criteria describing the consequence of safety
risk from different aspects should be assigned different weights. If the
direct economic loss is more important to decision makers, the criterion
that describes economic loss should be assigned more weight. Which
criterion is more important, different decision makers have different
views and the views would change with the time and circumstance.
Then, the weights in the aggregation operator should be adjusted
flexibly based on the changes of the importance of risk criteria.
According to the above reasons, the Weighted Ordered Weighted
Aggregation (WOWA) operator with a new weight generation function
is constructed in this section. This operator can adjust both the preference to the larger or smaller values and the importance of risk criteria flexibly.
In this section, a multi-criteria aggregation model is proposed.
Firstly, the methods determining the weights in the WOWA operator are
introduced. Secondly, a utility interpolation function is presented as a
new weight generating function in the MEMCII problem.
Example 4. Let a = (2, 3, 1, 4) be the four-dimensional vector,
w = (0.1, 0.3, 0.4, 0.2) be the position weight vector considering the
orders of the four values, and p = (0.25, 0.25, 0.3, 0.2) be the criterion
weight vector considering the preference to the four criteria. The
monotone increasing function w* is defined firstly, because w* is a
linear interpolation function with the five points (0, 0), (0.25, 0.1),
(0.5, 0.4), (0.75, 0.8) and (1, 1), thus,
w * (x ) =
⎧ 0.4x ,
⎪1.2x − 0.2,
⎨1.6x − 0.4,
⎪ 0.8x + 0.2,
⎩
if 0 ≤ x ≤ 0.25,
if 0.25 < x ≤ 0.5,
if 0.5 < x ≤ 0.75,
if 0.75 < x ≤ 1.
Then,
the
weight
can
be
derived,
ωi (i = 1, 2, 3, 4)
ω1 = w * (0.2) − w * (0) = 0.08, ω2 = w * (0.45) − w * (0.2) = 0.26, ω3 =
w * (0.7) − w * (0.45) = 0.38, ω4 = w * (1) − w * (0.7) = 0.28. Finally, the
aggregation value of the four values is computed, and
FWOWA (a) = 0.08 × 4 + 0.26 × 3 + 0.38 × 2 + 0.28 × 1 = 2.14 .
Example 4 shows that the aggregation value depends on both the
weight vector and the score vector. If the score vector to be aggregated
is (1, 2, 3, 4) and the position weight vector w is the same, then the
weight ωi needs to be recalculated, ω1 = w * (0.2) − w * (0) = 0.08,
ω2 = w * (0.5) − w * (0.2) = 0.32, ω3 = w * (0.75) − w * (0.5) = 0.4, ω4 =
and
FWOWA (a) = 0.08 × 4 + 0.32 ×
w * (1) − w * (0.75) = 0.2,
3 + 0.4 × 2 + 0.2 × 1 = 2.28. That is to say, ωi needs to be calculated
again when the assessment vector is changed.
4.2. A new interpolation function: Utility function
A weight generating function in a MEMCII problem is needed to
produce weights. The selection of the weight generating function in a
risk assessment problem should consider the nature of risk and experts’
risk attitudes simultaneously. According to the existing literature, most
scholars agree to that the risk is a combination of the frequency and the
loss consequence, and the assessment scores of risks are inevitably influenced by the evaluator’s risk attitude.
Risk attitude is a person’s attitude toward the situation of uncertainty, and it is generally divided into three types: risk-averse, riskneutral and risk-appetite. People tend to be loss-averse when they are in
the face of loss situation [43], and the common opinion in economics is
that people tend to be more loss-averse when the loss increases. The
performances of this situation are that people often provide the bigger
values than the actual levels and the scores increase faster when the loss
increases. In order to carry out a fair and real risk assessment, these
performances need to be depicted and adjusted.
The influence of the bigger scores can be eliminated by using
smaller weights to some extent. These smaller weights can be derived
by using a appropriated β in (2). The more loss aversion when the loss
increases can be reflected by the weight generating function which has
a positive second derivative. But the straight line in Example 4 cannot
depict this situation, for example, the slope of the straight line in interval (0.5,0.75] is a constant, and this constant cannot depict the speed
4.1. The WOWA operator
The OWA operator was widely used after it’s appearance [37–40].
But some scholars proposed that the weights in the OWA operator were
not enough comprehensive only based on the orders of the values. In
1997, Torra proposed the Weighted Ordered Weighted Aggregation
(WOWA) operator considering the information sources and the orders
of the assessment scores at the same time [41].
Definition 6. [41] A mapping FWOWA from
I n→I
is called a Weighted Ordered Weighted Aggregation (WOWA) operator
if there is a weight vector ω = (ω1, ω2 , …, ωn ), where ωi is defined as
⎛
⎞
⎛
⎞
− w * ∑ pσ (j) ,
p
⎜
⎟
⎜ ∑ σ (j) ⎟
⎝ j<i
⎠
⎝ j≤i
⎠
(6)
where bi is the ith largest value of a1, a2 , …, an .
Di = [ Xi .1 Xi .2 … Xi . n ],
ωi = w *
)
(5)
with {σ(1), σ(2), …, σ(n)} being a permutation of {1, 2, …, n} and w*
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Knowledge-Based Systems 156 (2018) 62–73
D. Tian et al.
of the increase of risk aversion when x increases in interval (0.5,0.75],
where x denotes the extent of the loss in a MEMCII problem. But the
slope of a utility function curve which is used to depict the experts’ risk
aversions is not a constant, the speed of increase could be depicted by
the increasing slope of the utility function curve if the utility function
satisfies some properties [42] and is used for a weight generating
function.
Table 3
The results of different weights.
OWA
WOWAL
WOWAU
Example 5. The weight generating function in Example 4 is a linear
function, now, U (x ) = ae bx − 1, a > 0, b > 0 replaces the linear
function as a new weight generating function. Because U(x) is a
monotonic increasing function with interpolating the five points (0,
0), (0.25, 0.1), (0.5, 0.4), (0.75, 0.8) and (1, 1), thus,
ω2
ω3
ω4
AC(ω)
disp(ω)
Overall score
0.1
0.08
0.0792
0.3
0.26
0.2549
0.4
0.38
0.3777
0.2
0.28
0.2882
0.4333
0.3800
0.3750
1.2799
1.2764
1.2756
2.3
2.14
2.138
A same vector a = (2, 3, 1, 4) is aggregated respectively in
Example 1, 4 and 5. The weights in Example 1 merely consider the
orders of values. The WOWA operator is used to generate weights
considering the orders of values and the importance of criteria at the
same time based on a linear interpolation function in Example 4. The
situation in Example 5 is similar to Example 4 but a loss utility interpolation function. The results are different as shown in Table 3. OWA
weights mean the weights in the OWA operator in Example 1, and the
overall score is 2.3. WOWAL weights and WOWAU weights mean the
weights in the WOWA operator based on the linear interpolation
function and the loss utility interpolation function respectively. Table 3
shows that the effect of WOWAU operator in reducing the impact of the
bigger values is most significant and the value of AC(ω) is the smallest.
ω1 is the weight attached to the biggest value and it’s value is reduced
to 0.0792 by using the WOWAU operator. More accurate values are
derived based on the WOWAU operator with small loss in disp(ω), so the
WOWAU operator plays a practical and effective role in distinguishing
and sorting risks.
There are q decision matrices: D1, D2 , …, Dq in the MEMCII problem,
m rows of each matrix are integrated into one row by using the model in
Section 3, then q row matrices are derived. The columns in each row
matrix depict the consequence of a risk from several aspects, and they
can be aggregated into the overall consequence score based on the
model in Section 4. Finally, the overall consequence of a risk can be
used to established a risk matrix.
1
U (x ) =
ω1
if 0 ≤ x ≤ 4 ,
⎧ e 0.3812x − 1,
⎪
⎪ 0.8643e 0.9646x − 1, if 1 < x ≤ 1 ,
⎪
4
2
⎨ 0.8469e1.0053x − 1, if 1 < x ≤ 3 ,
2
4
⎪
⎪
3
x
0.4214
⎪1.3122e
− 1, if 4 < x ≤ 1.
⎩
Then, the weight ωi (i = 1, 2, 3, 4) is derived by using U(x),
ω1 = U (0.2) − U (0) = 0.0792, ω2 = U (0.45) − U (0.2) = 0.2549, ω3 =
U (0.7) − U (0.45) = 0.3777, ω4 = U (1) − U (0.7) = 0.2882. Finally, the
aggregation
of
the
risk
is
computed,
and
FWOWA (x ) = 0.0792 × 4 + 0.2549 × 3 + 0.3777 × 2 + 0.2882 × 1 = 2.138
.
The two weight generating functions are shown in Example 4 and
Example 5, their values are different especially in interval [0.5, 0.75] as
shown in Fig. 2. What causes the difference is mainly the different
expressions of the two functions. The linear function line has the same
slope in every piecewise interval, this situation can’t response to people’s habits when they are in the face of loss situation. Conversely, the
slope of the utility function curve can response to the increase of dissatisfaction when the degree of loss increases in every piecewise interval. The position weight vector is (0.1, 0.3, 0.4, 0.2) in Example 4,
obviously, the weight of the ith largest value are bigger than the weights
of other values, so the difference of the two functions in interval
[0.5,0.75] is more obvious as shown in Fig. 2.
5. Application
Suppose there are six safety risks in some petrochemical enter-prise
without consideration the interaction among them and these risks are
0.85
0.8
the utility weight generating function
the linear weight generating function
the degree of consequence
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.5
0.55
0.6
0.65
the degree of loss
Fig. 2. Two situations in interval [0.5,0.75].
68
0.7
0.75
Knowledge-Based Systems 156 (2018) 62–73
D. Tian et al.
model in Section 3 based on the subjective weights and the objected
weights at the same time. The subjective weights are determined by the
decision makers, and the objective weights are determined based on the
credibilities of the assessment scores. The objected weights of a same
expert about different risks are not always same and need to be calculated respectively about 6 risks.
The assessment scores in Table 5 can be arranged into six 3 × 5
decision matrices. Firstly, the simple mean of every column in each
decision matrix is calculated. Secondly, the proportion of the deviation
between each expert and the simple mean is calculated. Thirdly, the
objected weights of three experts are derived by using (3). Finally, the
integrated weights of three experts are calculated by using the
Definition 5 with α taking 0.5. The main results of the calculation
process are shown in Table 6, and Table 6 shows that the values of the
integrated weight of 3 experts about six risks are different.
Since the integrated weights of three experts have been derived, the
assessment scores can be integrated into collective scores with these
weights. The final results are shown in Table 7, and Table 7 shows that
three rows are integrated into one in each decision matrix, that is, the
collective scores on c1 to c5 of 6 risks are derived. There are five criteria
depicting safety risks in Table 7, c1 which measures the frequency is one
of the input variables of the risk matrix, c2, c3, c4, c5 are four criteria
that account for the consequence from four aspects. To construct a risk
matrix, the overall consequence of each risk which is another input
variable of the risk matrix needs to be calculated, therefore, the multicriteria aggregation model in Section 4 is needed to implement this
work.
denoted by r1, r2, r3, r4, r5, r6 in turn.
Because of the lack of historical data, the collection of assessment
data mainly depends on the Expert Scoring Method, and the Stratified
Sampling Method is used to select experts to participate the questionnaire survey. A department head who is responsible for the safety in
the process of production, a team leader who has rich work experience,
and an operating employee who works at the first line for a long time
are selected as the evaluation experts, these experts are denoted by e1,
e2, e3 in turn. Owing to the rich experience of the three evaluators, the
results given by them have strong accuracy and credibility. Three experts’ risk attitudes are all loss-averse and their subjective weights are
respectively 0.4, 0.3, 0.3 based on the opinions of decision makers.
Five criteria which are denoted by c1, c2, c3, c4, c5 are used to
measure and prioritize safety risks, they are frequency, accident level,
economic loss, reputation loss and environmental pollution in turn. c2,
c3, c4, c5 describe the consequence of a safety risk from four aspects and
need to be aggregated to derive the overall consequence. These five
criteria are all divided into five grades respectively according to the
rules in the company who is involved in the application, and the specific classifications are shown in Table 4. In combination with the grade
levels of the Table 4 the experts can score the six risks, which, to a
certain extent, can help the experts to give more reasonable and correct
results.
Three experts are selected to take part in the questionnaire survey
and score the safety risks. They complete the assessment scores on c1 to
c5 of six risks as shown in Table 5. c1 is used to depict the frequency of a
risk, c2 to c5 are used to describe the consequence of a risk from four
aspects. There are three ways (quantitative way, semi-quantitative way
and qualitative way) to complete the risk assessment. The main purpose
of the risk assessment in this paper is to give which risks are unacceptable and need to be carried out some measures immediately. The
risk matrix approach, which is a practical tool, is used to achieve the
purpose . Only two input variables(the frequency and the overall consequence) are needed in the process of the matrix construction. Obviously, c2 to c5 are needed to be congregated to derive the overall
consequence. To derive the scores of the frequency and the overall
consequence, the method that integrates the scores of 3 experts to
collective scores is needed firstly.
5.2. Calculation the overall consequence
There are four criteria depicting the consequence of safety risk from
four aspects, c2 shows the degree of the injuries, c3 refers to the direct
economic loss, c4 measures the loss of the reputation of the company, c5
accounts for the extent of the environmental pollution. A aggregation
model is needed to aggregate c2, c3, c4, c5 to derive the overall consequence with appropriate weights ω1, ω2, ω3, ω4. According to the
nature of safety risk and experts’ risk attitudes, the WOWAU operator is
used in this subsection. There are mainly four steps as follows:
Step 1: Determine the position weight vector w = (w1, w2, w3, w4 ) .
w1, w2, w3, w4 are the position weights of c2, c3, c4, c5 about their
orders, and their values can be given by the decision makers, also can
be determined with some kinds of mathematical models. If cautious
attitudes and ideas preventing risk accidents are taken in the safety risk
assessment, the more preference should be taken to the larger scores
and the overall consequence would take a larger value. On the other
hand, optimistic attitudes might produce a smaller overall consequence
and the more preference should be taken to the smaller scores.
Therefore, a model that can flexibly handle these above situations is
needed. In this paper, three experts are loss-averse basing on their experiences and the decision makers want to get the fair assessment results. So a constrained optimization model is developed with a smaller
β, and the expression of the model is shown as follows:
5.1. Experts scores integration
The integrated weights of three experts can be determined by the
Table 4
Levels of c1–c5.
Score
Severity
c2(Injuries, people)
c3(Economic loss,
RMB)
0-1
1-2
2-3
Negligible
Minor
Moderate
Less than 5000
5000-50000
50000-500000
3-4
Serious
4-5
Critical
Score
0-1
1-2
Severity
Negligible
Minor
No
1-3 slightly injured
4-10 slightly injured or 1
seriously injured
More than 10 slightly injured or
2–4 seriously injured
Someone is dead or more than 4
seriously injured
c4(Criticized scope)
In the team
In the production stations
2-3
3-4
4-5
Score
0-1
1-2
2-3
3-4
4-5
Moderate
Serious
Critical
Description
Remote
Unlikely
Likely
Highly likely
Near certainty
In the department
In the branch
In the parent company
c1(Time for happening once)
More than 5 years
3 to 5 years
1 to 3 years
6 mouths to 1 year
Less than 6 mouths
500000-5000000
4
More than 5000000
Maximize:
− ∑i = 1 wi lnwi
c5(Pollution scope)
In the production spot
In the surrounding
villages
In the local state
In autonomous region
Above the country
Subject to:
4
1
∑i = 1 (4 − i) wi = 0.4,
4−1
4
∑i = 1 wi = 1,
0 ≤ wi ≤ 1,
0≤β≤1
i = 1, 2, …, 4.
(7)
The target of this model is to maximize dispersion measure with a
pre-defined attitudinal character. The full use of all information and the
neutral risk attitude are all important in the safety risk assessment.
Therefore the value of β should be controlled. Three experts involved in
the risk assessment problem are loss-averse when they are in the face of
the loss situation. That is to say, the values of c2, c3, c4, c5 are possibly
greater than their actual levels especially when the loss increases. In
this paper, the value of β is set to 0.4 so that the risk attitudes can be
69
Knowledge-Based Systems 156 (2018) 62–73
D. Tian et al.
Table 5
The investigated scores of c1 to c5.
Expert
r1
c1
c2
c3
c4
c5
r2
r3
r4
r5
r6
e1
e2
e3
e1
e2
e3
e1
e2
e3
e1
e2
e3
e1
e2
e3
e1
e2
e3
2
3
4
3
4
1
2
3
3.7
2
1.5
3
3
4
3
1
4
5
3.2
2
2
4.6
4
5
4
2
4
3
4
3
3
2
3
2
3
4
2
2
3
4
3
1
3
2
2
1.5
4
3.8
4
3
2
5
4
3.5
5
1
4
5
4
4
2
2
1.8
3
2
1
2
2
3
4
2
3
2
1
1
2
2
3
1
2
1.3
1
2
2
1.4
1
2
3
1
2
Table 6
The weights of three experts about six risks (α = 0.5 ).
Expert
e1
e2
e3
θj
0.4
0.3
0.3
r1
r2
r3
r4
λ1j
μj2
λ2j
μj3
λ3j
μj4
λj4
μj5
λ5j
μj6
λ6j
0.2750
0.3132
0.4118
0.3351
0.3098
0.3551
0.2786
0.3250
0.3964
0.3369
0.3151
0.3480
0.4000
0.2417
0.3583
0.4012
0.2700
0.3288
0.3682
0.2818
0.3500
0.3843
0.2912
0.3245
0.3650
0.3250
0.3100
0.3824
0.3125
0.3052
0.3437
0.2978
0.3585
0.3717
0.2996
0.3287
Table 8
The results of the safety risk assessment.
c1
c2
c3
c4
c5
1.5127
1.6631
3.2700
1.4834
1.6877
1.4616
2.6902
4.1891
1.6712
4.2912
2.3054
1.7004
3.3351
3.9889
2.7300
4.2476
1.9237
2.7004
3.5720
4.0456
2.2700
3.8544
2.3899
1.2996
3.0253
2.9782
2.9412
3.9069
2.3200
1.8202
Risk
r1
r2
r3
r4
r5
r6
w1 =
corrected
to
some
extent,
then
0.1671, w2 = 0.2133, w3 = 0.2722, w4 = 0.3474 . The results show that the
larger scores are assigned smaller weights.
Step 2: Determine the criterion weights p1, p2, p3, p4.
p1, p2, p3, p4 are the criterion weights of c2, c3, c4, c5 about their
importance, the value of pi is more larger, the more importance is set to
the corresponding criterion. c2, c3, c4, c5 are accident level, economic
loss, reputation loss, and environmental pollution respectively, and
their weights are set to 0.30, 0.30, 0.20, 0.20 in turn based on the
opinions of the decision makers.
Step 3: Determine the integrated weights ω1, ω2, ω3, ω4.
The integrated weight ωi is calculated by integrating the position
weight and the criterion weight, and the weight generating function is
the utility function U (x ) = ae bx − 1(a > 0, b > 0) . Because U(x) is a
monotonic function with interpolating the five points (0, 0), (0.25,
0.1671), (0.5, 0.3804), (0.75, 0.6526) and (1, 1), the expression of U(x)
can be derived, as follows:
Integrated weights
ω1
ω2
ω3
ω4
0.1316
0.2070
0.1316
0.2070
0.1316
0.2070
0.2489
0.1734
0.2489
0.2763
0.1591
0.1734
0.2136
0.3365
0.2136
0.2336
0.3033
0.3365
0.4059
0.2831
0.4059
0.2831
0.4060
0.2831
Consequence
Frequency
Rank
3.0383
3.7540
2.3117
4.0657
2.1660
1.8147
1.5127
1.6631
3.2700
1.4834
1.6877
1.4616
M
M
M
H
M
L
the orders of scores of c2, c3, c4, c5 are same about r1, r3, so the final
integrated weights are identical. The situation is same about r2, r6.
Step 4: Calculate the scores of the overall consequence.
The score of the overall consequence of a risk can be derived by
c
using
the
linear
aggregation
function:
(x ) = ω1 cid (1) (x ) + ω2 cid (2) (x ) + ω3 cid (3) (x ) + ω4 cid (4) (x ), where cid(i)(x)
denotes the ith largest score of c2, c3, c4, c5 and ωi(x) denotes the integrated weight corresponding the ith largest score. Firstly, the values of
c2, c3, c4, c5 are needed to be ordered from largest to smallest, then
cid(1), cid(2), cid(3), cid(4) multiply ω1, ω2, ω3, ω4 in turn, finally, the
overall consequence of each risk is derived and shown in Table 8. The
overall consequence and the frequency are the final indicators to construct a risk matrix.
In this paper, the traditional risk matrix approach is taken into account to complete the risk assessment. The risk matrix is shown in
Fig. 3. In the matrix, Frequency and Consequence are two input indicators to generate the assessment results. The output results are divided into three different levels: High(H), Medium(M) and Low(L),
these levels are colored by red, yellow and green respectively. Risks are
categorized according to the zone in which they locate. Risk locates in
the green zone means that the risk is acceptable, no further action is
required. Risk locates in the yellow zone means that the rank of the risk
is medium, additional actions and measures are needed. Risk locates in
the red zone means that the risk is unacceptable, safety actions and
measures need to be carried out immediately.
As shown in Fig. 3, r4 locates in the red zone, its output is high and
safety actions should be taken immediately. The outputs of r1, r2, r3, r5
are medium, the consequences and the frequencies of these risks are not
very high, but additional measures should be considered to prevent
these risks. r6 locates in the green zone and its output is low and acceptable. On the other hand, the risk matrix can present some risks
1
U (x ) =
r6
μ1j
Table 7
Collective scores on five criteria of six risks.
r1
r2
r3
r4
r5
r6
r5
if 0 ≤ x ≤ 4 ,
⎧ e 0.6181x − 1,
⎪
⎪ 0.9868e 0.6714x − 1, if 1 < x ≤ 1 ,
⎪
4
2
⎨ 0.9631e 0.7199x − 1, if 1 < x ≤ 3 ,
2
4
⎪
⎪
3
x
0.7632
⎪ 0.9324e
− 1, if 4 < x ≤ 1.
⎩
Then, the integrated weight ωi (i = 1, 2, 3, 4) can be calculated, and
ωi = U ⎜⎛ ∑j ≤ i pσ (j) ⎟⎞ − U ⎜⎛ ∑j < i pσ (j) ⎞⎟, where id(j) is a index function so
⎝
⎠
⎝
⎠
that pid(j) is the weight corresponding to the jth largest value. The orders
of the values are not always the same about different risks, therefore the
integrated weights are different for different risks and they are calculated by using the method in Section 4respectively. The values of
ωi (i = 1, 2, 3, 4) of six risks are shown in Table 8. Table 8 shows that
70
Knowledge-Based Systems 156 (2018) 62–73
D. Tian et al.
5
H
M
4.5
H
H
H
M
H
H
M
H
*r4
4
*r2
Consequence
M
M
3.5
*r1
3
M
M
L
2.5
*r3
*r5
2
*r6
1.5
L
L
M
M
H
L
L
L
M
M
1
0.5
0
0
1
2
3
4
5
Frequency
Fig. 3. The results of safety risk assessment.
aspects based on the purpose. The multi-criteria aggregation model in
this paper can deal with the criterion selection problem flexibly by
adjusting the weight vector in the model.
c2, c3, c4, c5 are four criteria depicting the consequence of safety
risk, w = (w1, w2, w3, w4 ) and p = (p1 , p2 , p3 , p4 ) are the position weight
vector and the criterion weight vector respectively. If only c3 needs to
be considered as the consequence based on the purpose and the preference to the larger or smaller scores is unchanged, just let
p = (0, 1, 0, 0) . Because the weight generating function is same to U(x)
in Section 5.2, the integrated weights can be derived in the same way,
which are of low-frequency and high-consequence and help decision
makers to recognize these risks. The outputs of such risks would not
always be high and could be ignored, but this kind of risks should be
attached great attention in practical.
5.3. Discussion of model flexibility
The risk assessment in this section is implemented mainly by two
processes: integrating the scores of three experts and aggregating four
criteria to derive the overall consequence. Two models are used: a
multi-experts integration model and a multi-criteria aggregation model.
The first model is flexible to handle the integrated weights of the experts, the second one can adjust flexibly the weights of criteria and
people’s risk attitudes according to the need.
(1) The sensitivity analysis of α.
The integrated weights of three experts are calculated by considering the subjective weights and the objective weights simultaλ͠ j
(j = 1, 2, 3), where λ͠ j = θjα μ1 − α , 0 ≤ α ≤ 1, θj is
neously and λj =
∑3i = 1 λ͠ j
and ωi = U ⎜⎛ ∑j ≤ i pσ (j) ⎟⎞ − U ⎜⎛ ∑j < i pσ (j) ⎞⎟. For r1, its consequences on
⎠
⎝
⎠
⎝
four criteria are 2.6902, 3.3351, 3.5720, 3.0253 in turn, therefore
ω1 = U (0) − U (0) = 0, ω2 = U (1) − U (0) = 1, ω3 = U (1) − U (1) = 0,
ω4 = U (1) − U (1) = 0 . Then only c3 is considered in the calculation
process of the overall consequence, and the overall consequence of r1
takes 3.3351.
If only c4 and c5 are considered as the consequence with equal criterion weights, and the preference to the larger or smaller scores is
unchanged also, it just needs to let p = (0, 0, 0.5, 0.5) . The integrated
weights in Section 5.2 can be derived also in this situation. For r1,
ω3 =
ω2 = U (0.5) − U (0.5) = 0,
ω1 = U (0.5) − U (0) = 0.3805,
U (1) − U (0.5) = 0.6195, ω4 = U (1) − U (1) = 0 . Then the overall conc
sequence
of
r1
can
be
calculated:
(x ) = 3.5720 × 0.3805 + 3.3351 × 0 + 3.0253 × 0.6195 + 2.6902 × 0 .
j
the subjective weight, μj is the objective weight and λj is the integrated
weight.
If only the subjective weights are considered, just make α equals to
1. When α = 0, then the subjective weights are not considered. These
two kinds of weights have some advantages and disadvantages, and the
best is to combine them. The preference to the two kinds of weights can
be adjusted by α. Five conditions are shown in Table 9. Where ‘F’ and
‘C’ denote the frequency and the overall consequence respectively.
Table 9 shows that the change of frequency is larger than the overall
consequence under the same condition. Each frequency value is derived
through integrating 3 original data, but the value of each overall consequence is calculated based on 12 original data. So the weights of
experts need to be carefully determined, especially with a small amount
of data.
(2) The flexibility of criterion selection.
The consequence of safety risk in oil and gas industry mainly performs in four aspects: accident level, economic loss, reputation loss and
environmental pollution and it is more comprehensive that the consequence should be a combination of the four indicators. But sometimes, risk need to be assessed just from one aspect or two or three
= 3.2333
In the safety risk assessment problem, the results of only considering
c3 and considering c4, c5 are shown in Fig. 4. Fig. 4 shows that the
assessment results are different under the two conditions. The outputs
of r2 and r4 are high when only economic loss is used to depict the
consequence, but there are no high risks when the consequence is derived on the basis of c4 and c5. r5 locates in the green zone with the only
consideration of economics loss, but locates in the yellow zone under
the second condition. Similarly, the risk matrices of other situations can
also be obtained easily.
(3) The influence of risk attitude.
In the process of risk assessment, people’s risk attitudes are inevitably involved. The results of risk assessment should vary with the
71
Knowledge-Based Systems 156 (2018) 62–73
D. Tian et al.
Table 9
The results of risk assessment with different α.
Risk
α = 0.25
α=0
r1
r2
r3
r4
r5
r6
α = 0.5
F
C
F
C
F
C
F
C
F
C
1.55
1.6
3.3
1.5
1.7
1.49
3.0663
3.7560
2.3291
4.0571
2.1589
1.8146
1.4947
1.6807
3.3089
1.4909
1.6686
1.4334
3.0201
3.7650
2.3078
4.0898
2.1715
1.8057
1.5127
1.6631
3.2700
1.4834
1.6877
1.4616
3.0383
3.7540
2.3117
4.0657
2.1660
1.8147
1.485
1.7081
3.6630
1.4742
1.6728
1.4332
3.0180
3.7543
2.5517
4.0802
2.1724
1.8122
1.4809
1.6079
3.2417
1.4659
1.675
1.4106
3.0162
3.5566
2.7939
4.0746
2.1728
1.7720
involved risk attitude. So a good model should adapt to this need.
In order to prevent safety risk, some decision makers who are lossaverse believe that the maximum value of safety risk in four aspects
should be the risk consequence regardless of the information source.
That is to say, the importance of four criteria has no difference. This
situation can be achieved by adjusting the weights in the WOWAU operator. Just let w = (1, 0, 0, 0) and p = (0.25, 0.25, 0.25, 0.25), then the
weight generating function is derived, and
U (x ) =
e 2.7726x − 1,
if
⎨1,
⎩
if
⎧
1
U (x ) =
6. Conclusions
Safety risk assessment is often a multi-experts and multi-criteria
information integration (MEMCII) problem, there are several experts
and several criteria. To construct a risk matrix, only two input variables
(the probability and the consequence of a risk) are needed. Then a twostage multi-experts and multi-criteria aggregation model is constructed
in this paper. Firstly, the evaluation scores of several experts are integrated. Secondly, the scores that account for the consequence of a risk
from several aspects are aggregated to derive the overall consequence.
Finally, a risk matrix is established based on the results of this model.
Compared with the existing methods which are used to build a risk
matrix, the proposed two-stage multi-experts and multi-criteria aggregation model incorporates three new practical advantages:
if 0 ≤ x ≤ 4 ,
⎨ 0.125e 2.7726x − 1, if 3 < x ≤ 1.
4
⎩
For
r4,
ω1 = U (0.25) − U (0) = 0,
ω2 = U (0.5) − U (0.25) = 0,
ω3 = U (0.75) − U (0.5) = 0, and ω4 = U (1) − U (0.75) = 1. The consequence of r4 is 3.8544.
If the decision makers are risk neutral and do not prefer to the larger
values and the smaller values, and the weights of four criteria are still
0.3, 0.3, 0.2, 0.2 in turn, just let w = (0.25, 0.25, 0.25, 0.25), then,
5
5
4
M
Economic loss
3.5
H
*r4
*r2
H
M
*r1
M
H
H
H
H
*r3
*r6
M
L
2
H
M
M
*r5
L
L
1.5
H
4
3
2.5
M
4.5
Reputation and Pollution
M
4.5
H
M
M
1
H
H
H
*r4
3.5
M
M
*r2
*r1
M
H
H
L
M
M
*r3 M
H
L
L
*r6
M
M
H
L
L
L
M
M
3
2.5
*r5
2
1.5
1
L
L
0.5
0
⎨1.102e 0.6166x − 1, if 1 < x ≤ 3 ,
2
4
⎪
⎪
3
x
0.5341
⎪1.1724e
− 1, if 4 < x ≤ 1.
⎩
+ 0.2026 × 3.8544 = 4.1120
Other values of the preference to the larger or smaller scores can
also be easily realized. From what have been discussed above, the
model in this paper is highly applicable and flexible.
3
⎧ 0,
if 0 ≤ x ≤ 4 ,
⎧ e 0.8926x − 1,
⎪
⎪1.0417e 0.7293x − 1, if 1 < x ≤ 1 ,
⎪
4
2
For r4, ω1 = U (0.3) − U (0) = 0.2965, ω2 = U (0.6) − U (0.3) = 2988,
ω3 = U (0.8) − U (0.6) = 0.2021, and ω4 = U (1) − U (0.8) = 0.2026. The
consequence
of
r4
can
be
calculated:
c (x ) = 0.2965 × 4.2912 + 0.2988 × 4.2476 + 0.2021 × 3.9069.
1
0 ≤ x ≤ 4,
1
< x ≤ 1.
4
For r4, the four criteria take 4.2912, 4.2476, 3.8544, 3.9069 respectively. Then, ω1 = U (0.25) − U (0) = 1, ω2 = U (0.5) − U (0.25) = 0,
ω3 = U (0.75) − U (0.5) = 0, and ω4 = U (1) − U (0.75) = 0 . The consequence of r4 is 4.2912.
If the risk attitudes of decision makers are risk-appetite and they
think that the minimum value of four criteria should be the consequence of a risk. This situation can also be obtained. let
w = (0, 0, 0, 1) and p = (0.25, 0.25, 0.25, 0.25), then
U (x ) =
α=1
α = 0.75
0
1
2
M
M
L
3
4
0.5
0
5
0
1
Frequency
2
3
4
5
Frequency
(a) Economic loss being considered
(b) Reputation and Pollution being equally considered
Fig. 4. Examples of criterion selection with assessment results.
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Knowledge-Based Systems 156 (2018) 62–73
D. Tian et al.
(1) In this paper, the determination of the integrated weights of
evaluation experts considers not only the subjective weights given by
decision makers but also the credibilities of the assessment scores.
Moreover, the experts do not always know the same about different
risks, the credibilities of the assessment scores about each risk should be
calculated separately, in this paper, the integrated weights of the same
expert about different risks are different. Besides the above, the preferences to the two kinds of weights can be adjusted by the parameter α
flexibly.
(2) The consequence of a safety risk in oil and gas industry mainly
performs in four aspects: accident level, economic loss, reputation loss
and environmental pollution. All the four indicators are the consequences of a safety risk, and the overall consequence of a risk should
be a combination of the four indicators. The overall consequence is
considered as one of the input variables of a risk matrix and this proposal is a new idea for the construction of a risk matrix. So the results of
the safety risk assessment are more reasonable. Moreover, the model
that is constructed in this paper can select the needed criteria flexibly.
(3) In order to carry out fair and real risk assessment, the utility
function is proposed as a weight generating function to reduce the influence of experts’ risk attitudes in this paper. Through comparative
analysis, more accurate values are derived based on the WOWAU operator with small losses in AC(w) and disp(w) and the effect of the
WOWAU operator in reducing the impact of the biggest values is most
significant. In addition, the model that is constructed can deal with
different risk attitudes flexibly. In conclusion, the WOWAU operator
plays a practical role in distinguishing and sorting safety risks.
The multi-experts and multi-criteria risk assessment model is very
flexible. Meanwhile, there are still some places to be improved in the
future.
(1) The preference relations of experts are the same: utility function
(expressing a utility score for each criterion) in this paper. Sometimes,
the involved evaluators may represent their opinions using homogeneous preference relations [44] and even give their self-confidences
[24]. In the future, we will try to incorporate the homogeneous preference relations and the self-confidences into the MEMCII problem.
(2) The three evaluators are all familiar with the safety risks in the
petrochemical enter-prise and they are named experts. Sometimes, the
involved evaluator(such as the people outside the industry) could have
a vague knowledge about the safety risk. But this is likely to happen
when the number of evaluators is larger or the sources of evaluators are
more extensive. Then a more realistic approach such as linguistic assessment [25,45] should be incorporated into the MEMCII problem.
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Acknowledgments
The authors would like to thank the editor and reviewers for their
kind help and constructive comments.
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