A novel approach to emergency risk assessment using FMEA with extended MULTIMOORA method under interval-valued Pythagorean fuzzy environment

Abstract

Purpose
The application of the traditional failure mode and effects analysis (FMEA) technique has been widely questioned in evaluation information, risk factor weights and robustness of results. This paper develops a novel FMEA framework with extended MULTIMOORA method under interval-valued Pythagorean fuzzy environment to solve these problems.

Design/methodology/approach
This paper introduces innovatively interval-value Pythagorean fuzzy weighted averaging (IVPFWA) operator, Tchebycheff metric distance and interval-value Pythagorean fuzzy weighted geometric (IVPFWG) operator into the MULTIMOORA submethods to obtain the risk ranking order for emergencies. Finally, an illustrative case is provided to demonstrate the practicality and feasibility of the novel fuzzy FMEA framework.

Findings
The feasibility and validity of the proposed method are verified by comparing with the existing methods. The calculation results indicate that the proposed method is more consistent with the actual situation of project and has more reference value.

Practical implications
The research results can provide supporting information for risk management decisions and offer decision-making basis for formulation of the follow-up emergency control and disposal scheme, which has certain guiding significance for the practical popularization and application of risk management strategies in the infrastructure projects.

Originality/value
A novel approach using FMEA with extended MULTIMOORA method is developed under IVPF environment, which considers weights of risk factors and experts. The method proposed has significantly improved the integrity of information in expert evaluation and the robustness of results.

Full text

A novel approach to emergency
risk assessment using FMEA with
extended MULTIMOORA method
under interval-valued Pythagorean
fuzzy environment
Huimin Li
Department of Construction Engineering and Management,
North China University of Water Resources and Electric Power,
Zhengzhou, China and
School of Architecture and Built Environment,
Centre for Asian and Middle Eastern Architecture, University of Adelaide, Adelaide,
South Australia, and
Lelin Lv, Feng Li, Lunyan Wang and Qing Xia
Department of Construction Engineering and Management,
North China University of Water Resources and Electric Power,
Zhengzhou, China
Abstract
Purpose –The application of the traditional failure mode and effects analysis (FMEA) technique has been
widely questioned in evaluation information, risk factor weights and robustness of results. This paper develops
a novel FMEA framework with extended MULTIMOORA method under interval-valued Pythagorean fuzzy
environment to solve these problems.
Design/methodology/approach –This paper introduces innovatively interval-value Pythagorean fuzzy
weighted averaging (IVPFWA) operator, Tchebycheff metric distance and interval-value Pythagorean fuzzy
weighted geometric (IVPFWG) operator into the MULTIMOORA submethods to obtain the risk ranking order
for emergencies. Finally, an illustrative case is provided to demonstrate the practicality and feasibility of the
novel fuzzy FMEA framework.
Findings –The feasibility and validity of the proposed method are verified by comparing with the existing
methods. The calculation results indicate that the proposed method is more consistent with the actual situation
of project and has more reference value.
Practical implications –The research results can provide supporting information for risk management
decisions and offer decision-making basis for formulation of the follow-up emergency control and disposal
scheme, which has certain guiding significance for the practical popularization and application of risk
management strategies in the infrastructure projects.
Novel
approach to
emergency risk
assessment
The authors acknowledge with gratitude National Key R&D Program of China (No. 2018YFC0406905),
the MOE (Ministry of Education in China) Project of Humanities and Social Sciences (No. 19YJC630078),
Youth Talents Teachers Scheme of Henan Province Universities (No. 2018GGJS080), the National
Natural Science Foundation of China (No. 71974056, No. 71302191), the Foundation for Distinguished
Young Talents in Higher Education of Henan (Humanities & Social Sciences), China (No. 2017-cxrc-023),
China Scholarship Council (No. 201908410388), 2018 Henan Province Water Conservancy Science and
Technology Project (GG201828). This study would not have been possible without their financial
support.
Conflicts of Interest: The authors declare no conflict of interest.
The current issue and full text archive of this journal is available on Emerald Insight at:
https://www.emerald.com/insight/1756-378X.htm
Received 13 August 2019
Revised 16 October 2019
8 November 2019
Accepted 11 November 2019
International Journal of Intelligent
Computing and Cybernetics
© Emerald Publishing Limited
1756-378X
DOI 10.1108/IJICC-08-2019-0091

Originality/value –A novel approach using FMEA with extended MULTIMOORA method is developed
under IVPF environment, which considers weights of risk factors and experts. The method proposed has
significantly improved the integrity of information in expert evaluation and the robustness of results.
Keywords Failure mode and effects analysis, Emergency, MULTIMOORA, Interval-valued Pythagorean
fuzzy sets, Risk management
Paper type Research paper
1. Introduction
Failure mode and effects analysis (FMEA) method was first proposed in the 1960s (Bowles
and Pel
aez, 1995), which has been widely used in aerospace industry, electric power industry,
nuclear industry, mechanical and medical technologies industry and handicraft industry and
so on to ensure safe and stable production and operation (Song et al., 2014;Chang et al., 2012;
Kutlu and Ekmekcioglu, 2012;Vinodh et al., 2012;Liu et al., 2013,2014a). It is an effective and
scientific tool in evaluating potential failure modes and reducing the frequency of occurrence
(Sankar and Prabhu, 2001;Deng and Jiang, 2017).
Risk evaluation in FMEA is generally conducted by using risk priority number (RPN) to
represent the influence caused by failure mode (Zammori and Gabbrielli, 2012;Xiao et al.,
2011). By analyzing the potential failure modes and their possible effects, the FMEA method
uses an integer scale from 1 to 10 for estimating the actual performance of different failure
modes under the three risk factors of occurrence (O), severity (S) and detection (D). And finally
the risk prioritization is obtained by calculating RPN, that is, RPN ¼O3S3D, where
occurrence (O) denotes the frequency of the failure, severity (S) denotes the seriousness of the
failure and detection (D) denotes the likelihood of the failure not being detected. The higher
the RPN value of a failure mode, the higher the risk level of this failure mode and the greater
the harm to the system. Risk evaluation in traditional FMEA by using RPN method is
considered to be the most effective method of prevention in advance, but the application in
practice of the RPN method has been widely questioned (Liu et al., 2014b,2015;Chen and
Deng, 2018), mainly in the following aspects: (1) the traditional FMEA method uses exact
values to express the risk level for risk factors O,Sand D, but it has certain limitations for it
cannot objectively reflect the complexity and uncertainty of things and the fuzziness of
human thinking in processing information. (2) Weights of risk factors are not considered in
the traditional FMEA risk assessment. These factors are regarded as equally important,
which is inconsistent with the actual situation. (3) It is easy to have the same risk priority
value using RPN method, so that it is difficult to judge the risk ranking order of failure modes,
and the unreasonable information aggregation process will cause information loss. To solve
the aforementioned defects of traditional RPN method and apply FMEA to practical
situations more effectively and scientifically, numerous scholars have put forward many
improvement theories and methods.
Fuzzy set theory was applied to FMEA method in many literatures aiming at the defects of
the traditional RPN method mentioned earlier. The integrated research and application of
FMEA method and fuzzy theory method are widely concerned, because fuzzy method has the
advantage of dealing with risk assessment for failure modes based on expert knowledge and
experience (Zhang et al., 2019). Liu et al. (2016a,2011,2014c,2014d) proposed numerous fuzzy
methods and theories, such as fuzzy digraph and matrix approach, fuzzy evidential reasoning
approach and grey theory, interval two-tuple hybrid weighted distance measure and D
numbers and grey relational projection method, to overcome inherent limitations of the
traditional FMEA. Li et al. (2019) constructed a novel FMEA framework that integrates
interval type-2 fuzzy sets (IT2FSs) and fuzzy Petri nets (FPNs) to overcome the drawbacks
and improve the effectiveness of the traditional FMEA. Vahdani et al. (2015) combined fuzzy
belief structure with TOPSIS to propose a FMEA method based on fuzzy confidence
structure. Mandal and Maiti (2014) proposed a novel method integrating the concepts of
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similarity value measure of fuzzy numbers and possibility theory to solve the shortcoming
for membership functions overlap of FMEA method. Moreover, Peng and Yang (2016)
defined the interval-valued Pythagorean fuzzy sets (IVPFSs) theory according to the fuzzy set
theory, which considered the three kinds of information for membership degree,
nonmembership degree and hesitation degree. Therefore, IVPFSs are more flexible and
practical than other forms in the expression of uncertainty, which when applied to the FMEA
method would be a good effect (Peng, 2019;Peng and Li, 2019).
On the other hand, the essence of the risk ranking order of failure modes in FMEA can also
be treated as a multiple-criteria decision-making (MCDM) problem (Lolli et al., 2015;Liu et al.,
2016b). Together with VIseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR)
(Tian et al., 2018), Technique for Order Preference by Similarity to the Ideal Solution (TOPSIS)
(Bian et al., 2018;Liu et al., 2019;Tekez, 2018), Analytic Hierarchy Process (AHP)
(Abdelgawad and Fayek, 2010;Bao et al., 2017) and Technique for Order Preference by
Similarity to the Ideal Solution (DEMATEL) (Tsai et al., 2017,2018) methods, MCDM is
widely used in FMEA research to improve the traditional ranking order method of risk
priority value. Safari et al. (2016) used the fuzzy VIKOR method instead of the RPN method to
rank enterprise architecture risk factors with respect to the criteria to overcome drawbacks of
the traditional FMEA. Chang et al. (2014) integrated the TOPSIS and the DEMATEL
approach to rank the risk prioritization of failure modes. However, the aforementioned
MCDM methods have single decision-making mode, and the robustness of ranking order still
needs to be improved. The MULTIMOORA method is a robust and flexible MCDM technique,
which comprises the three submethods: the ratio system method, the reference point method
and the full multiplicative form method (Wu et al., 2017). MULTIMOORA method has the
characteristics of simple calculation and strong robustness (Brauers, 2012;Brauers and
Zavadskas, 2012), which has been extended and applied in numerous fields for solving real-
life MCDM problems.
FMEA can help decision-makers adjust the existing programs and employ the
recommended actions to reduce the likelihood of failures, decrease the probability of
failure rates and avoid hazardous accidents (Liu et al., 2016b). FMEA can provide support
information for risk management decisions and improve the performance of the system or
project during construction and operation. According to the analysis results of FMEA, it can
further improve the knowledge structure of emergency risk management and provide
decision-making basis for formulation of the follow-up emergency control and disposal
scheme.
However, the application of traditional FMEA system has been widely questioned in
evaluation information, risk factor weights and robustness of results. As stated previously,
the IVPFSs can better address the expression of linguistic uncertainty, which is more flexible
and practical, and the MULTIMOORA method can further improve the robustness of results.
Hence, the motivation of this paper is to merge IVPFSs into the MULTIMOORA method to
develop a novel FMEA system. Firstly, the linguistic evaluation information is converted into
corresponding interval-valued Pythagorean fuzzy numbers (IVPFNs) to effectively deal with
the uncertainty and vagueness of information in practical application of FMEA. Then,
different priorities are assigned to experts by using the interval-value Pythagorean fuzzy
priority weight average (IVPFPWA) operator, which solve the problem of determining expert
weight and avoid information loss in information aggregation. And the weights of risk
factors are determined by deviation maximization model method. Last but not the least, the
interval-value Pythagorean fuzzy weighted averaging (IVPFWA) operator, Tchebycheff
Metric distance and interval-value Pythagorean fuzzy weighted geometric (IVPFWG)
operator are introduced innovatively into the ratio system, the reference point method and the
full multiplication form model of MULTIMOORA submethods, respectively, to optimize
information aggregation process. This paper applies the proposed FMEA method to the
Novel
approach to
emergency risk
assessment

emergency risk assessment of the East Route of the South-to-North Water Diversion Project
to verify its feasibility and effectiveness.
This paper is organized as follows. The second section introduces the preliminaries about
IVPFSs and traditional MULTIMOORA method. Section 3 proposes a risk evaluation in
FMEA with extended MULTIMOORA method under interval-valued Pythagorean fuzzy
environment. Section 4 provides a case study on East Route of South-to-North Water
Diversion Project’s emergency risk to demonstrate the application and validity of the
proposed method. In Section 5, comparison analysis with the other approaches is given to
show the advantages of the proposed FMEA model. Section 6 provides conclusions and
further research directions.
2. Preliminary
2.1 Interval-valued Pythagorean fuzzy sets
IVPFS is a novel tool to solve uncertainty and vagueness information, which was firstly
proposed by Peng and Yang (2016). The IVPFS is developed on the basis of interval-valued
intuitionistic fuzzy set (IVIFS). Compared with IVIFS, the similarity is that both of them
consider the membership degree and nonmembership degree to describe the fuzzy
characteristics of decision-makers. The difference is that the IVPFS considers the sum of
squares of membership degree and nonmembership degree to be less than or equal to 1, while
the IVIFS only considers the sum of membership degree and nonmembership degree to be
less than or equal to 1. In other words, the IVIFS is a special case of the IVPFS. IVPFS, which
extends IVIFS, gives a wide thinking space for experts with a more general condition. As
shown in Figure 1, the filed I is the thinking space using IVIFS theory, and the filed I þII
when using IVPFS theory. Therefore, the IVPFS fully considers the “true psychological”
behavior of decision experts, and the IVPFS can adapt to more situations and has more
practical application (Wei and Lu, 2018).
Definition 1. (Atanassov, 1986) Let Xdenote a universe of discourse. A single-valued IFS
Aon Xis given as follows:
A¼fhx;uAðxÞ;vAðxÞijx∈Xg;(1)
where uAðxÞ:X→½0;1denotes the degree of membership and vAðxÞ:X→½0;1denotes the
degree of nonmembership of the element x∈Xto P, respectively, and uAðxÞþvAðxÞ≤1.
Definition 2. (Yager and Abbasov, 2013) Let Xdenote a universe of discourse. A single-
valued PFS Pon Xis given as follows:
P¼fhx;uPðxÞ;vPðxÞijx∈Xg¼u−
P;uþ
P;v−
P;vþ
P;(2)
I
II
u
v
1
1
Figure 1.
The fields of IVIFS
and IVPFS
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where uPðxÞ:X→½0;1denotes the degree of membership and vPðxÞ:X→½0;1denotes the
degree of nonmembership of the element x∈Xto P, respectively.
The degree of hesitancy of the element x∈Xto Pis denoted as follows:
π
PðxÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1u2
PðxÞv2
PðxÞ
q;(3)
where
π
Pis called Pythagorean fuzzy index of element, x∈Xto Prepresents the degree of
indeterminacy of xto P. Further, 0 ≤
π
P≤1 for every x∈X.
Moreover, uPðxÞand vPðxÞsatisfy the following condition:
0≤u2
PðxÞþv2
PðxÞ≤1;∀x∈X;(4)
Definition 3. (Peng and Yang, 2016) Let Xdenote a universe of discourse, and IVPFS Pon
Xis given as follows:
P¼fhx;uPðxÞ;vPðxÞijx∈Xg;(5)
where uPðxÞ:X→½0;1denotes the degree of membership and vPðxÞ:X→½0;1denotes the
degree of nonmembership, respectively. Moreover, supðu2
PðxÞÞ þ supðv2
PðxÞÞ ≤1;∀x∈X.
The two tuples ðuPðxÞ;vPðxÞÞ is called the IVPFN, which can be expressed simply as
P¼ð½a;b;½c;dÞ, where ½a;b⊂½0;1,½c;d⊂½0;1, and b2þd2≤1.
Definition 4. (Zhang, 2016) Let P1¼ð½a1;b1;½c1;d1Þ and P2¼ð½a2;b2;½c2;d2Þ are
two IVPFNs, and their natural partial order relations are presented as
follows:
(1) If P1¼P2, then a1¼a2,b1¼b2,c1¼c2, and d1¼d2.
(2) If P1≤P2, then a1≤a2,b1≤b2,c1≥c2, and d1≥d2.
Definition 5. (Peng and Yang, 2016) Let P1¼ð½a1;b1;½c1;d1Þ,P2¼ð½a2;b2;½c2;d2Þ,
and P¼ð½a;b;½c;dÞ are three IVPFNs, then some basic operations with
respect to IVPFNs are provided as follows:
(1) P1⊕P2¼ð½ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ða1Þ2þða2Þ2−ða1a2Þ2
q;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðb1Þ2þðb2Þ2−ðb1b2Þ2
q;½c1c2;d1d2Þ
(2) P1⊗P2¼ð½a1a2;b1b2;½ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðc1Þ2þðc2Þ2−ðc1c2Þ2
q;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðd1Þ2þðd2Þ2−ðd1d2Þ2
qÞ
(3) λP¼ð½ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−ð1−a2Þλ
q;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−ð1−b2Þλ
q;½cλ;dλÞ;λ>0
(4) Pλ¼ð½aλ;bλ;½ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−ð1−c2Þλ
q;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−ð1−d2Þλ
qÞ;λ>0
Definition 6. (Peng and Yang, 2016) Let P1¼ð½a1;b1;½c1;d1Þand P2¼ð½a2;b2;½c2;d2Þ
be two IVPFNs and
π
1¼ð
π
−
1;
π
þ
1Þand
π
2¼ð
π
−
2;
π
þ
2Þare the hesitancy
degrees tuples of the P1and P2, respectively. Then the Euclidean distance
between P1and P2can be defined as follows:
dðP1;P2Þ¼1
2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ða1a2Þ2þðb1b2Þ2þðc1c2Þ2þðd1d2Þ2
q;(6)
Novel
approach to
emergency risk
assessment

Definition 7. (Du, et al., 2017) Give two IVPFNs P1¼ð½a1;b1;½c1;d1Þ and
P2¼ð½a2;b2;½c2;d2Þ, then the score and accuracy function are
calculated as follows:
sðP1Þ¼1
2 a2
1þb2
1
2þ1c2
1þd2
1
2!;(7)
hðP1Þ¼1
4a2
1þb2
1þc2
1þd2
1;(8)
Further, the following facts are true:
(1) If sðP1Þ<sðP2Þ, then P1< P2.
(2) If sðP1Þ¼sðP2Þ, then,
If hðP1Þ¼hðP2Þ, then P1¼P2;
If hðP1Þ<hðP2Þ, then P1< P2;
If hðP1Þ<hðP2Þ, then P1< P2.
Definition 8. (Peng and Yang, 2016) Let Pj¼ð½aj;bj;½cj;djÞ;j¼1;2; :::; nbe a
collection of IVPFNs, then the IVPFPWA operator is defined as follows:
IVPFWAðP1;P2; :::; PnÞ¼w1P1⊕w2P2⊕::: ⊕wnPn
¼0
@2
4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1Y
n
j¼1
ð1ðajÞ2Þwj
v
u
u
t;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1Y
n
j¼11ðbjÞ2Þwj
v
u
u
t3
5;"Y
n
j¼1
cwj
j;Y
n
j¼1
dwj
j#1
A;(9)
Definition 9. (Peng and Yang, 2016) Let Pj¼ð½aj;bj;½cj;djÞ;j¼1;2; :::; nbe a
collection of IVPFNs, then the interval-value Pythagorean fuzzy
weighted geometric (IVPFPWG) operator is defined as follows:
IVPFPWGðP1;P2; :::; PnÞ¼w1P1⊗w2P2⊗::: ⊗wnPn
¼0
@"Y
n
j¼1
awj
j;Y
n
j¼1
bwj
j#;2
4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1Y
n
j¼1
ð1ðcjÞ2Þwj
v
u
u
t;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1Y
n
j¼11ðdjÞ2Þwj
v
u
u
t3
51
A;(10)
2.2 The MULTIMOORA method
Suppose the initial decision-making matrix is constructed as X¼½xij m3n, where
xijði¼1;2; :::; m;j¼1;2; :::; nÞis evaluation value for the alternative Aiunder the cj. The
alternative set is A¼ðA1;A2; :::; AiÞand the attribute set is C¼ðc1;c2; :::; cjÞ. To facilitate
the comparison, normalizing the initial decision matrix X¼½xijm3ninto the standardized
decision matrix X*¼½x*
ijm3n, it can be defined as the following form Brauers and Zavadskas
(2006):
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x*
ij ¼xij
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Pm
i¼1x2
ij
q(11)
where xij is the initial evaluation value, that is, the evaluation of the alternative Aiunder the
attribute cj,i¼1;2; :::; m,j¼1;2; :::; n.mis the number of alternatives and nis the number
of attributes; and x*
ij is the dimensionless evaluation value of the decision matrix.
The traditional MULTIMOORA method comprises the three submethods: the ratio
system method, the reference point method and the full multiplicative form method. The final
ranking of each alternative can be determined based on the results of the three submethods.
The steps of MULTIMOORA method are as follows.
Step 1: The ratio system method
The ration system method is the first part of MULTIMOORA method. After standardization,
the evaluation value of all alternatives under the ration system method can be obtained as
follows:
yi¼X
g
j¼1
x*
ij X
n
j¼gþ1
x*
ij (12)
where gand n−grespectively indicate the number of benefit-type and cost-type attributes. yi
represents the evaluation value of the alternative Ai. The higher the value of yiis, the better
the corresponding alternative is. Therefore, the optimal alternative A*
rs according to the ratio
system method can be obtained as follows (Brauers and Zavadskas, 2006):
A*
RS ¼Aijmax yi
i(13)
Step 2: The reference point method
The reference point method is the second part of MULTIMOORA method. The first step of the
reference point method is to determine the optimal reference point of each attribute. The
optimal reference point of all attributes can be conducted as the following form:
rj¼8
<
:
max
ix*
ij;j≤g
min
ix*
ij;j<g (14)
After determining the optimal reference point, the deviation degree between all attribute
values x*
ij and the corresponding optimal reference point rjcan be obtained, that is, rj−x*
ij.
Therefore, the maximum deviation of each alternative, that is, the evaluation value of each
alternative according to the reference point method, can be expressed as follows:
zi¼max
jrjx*
ij(15)
The smaller the value of ziis, the better the corresponding alternative is. Finally, the optimal
alternative A*
RP according to the reference point method can be obtained as follows (Brauers
and Zavadskas, 2006):
Novel
approach to
emergency risk
assessment

A*
RP ¼Aijmin
izi(16)
Step 3: The full multiplicative method
The full multiplicative method is the third part of MULTIMOORA method. It embodies the
minimization and maximization problems of purely multiplicative utility function (Brauers
and Zavadskas, 2010). Based on this, the evaluation values of all alternatives under the full
multiplicative form method can be expressed as follows:
Ui¼Yg
j¼1x*
ij
Yn
j¼gþ1x*
ij
(17)
where Qg
j¼1x*
ij represents the product of evaluation value of all benefit-type attributes,
similarly, Qn
j¼gþ1x*
ij represents the product of evaluation value of all cost-type attributes.
The higher the value of Uiis, the better the corresponding alternative is. Therefore, the
optimal alternative A*
RP according to the full multiplicative method can be obtained as follows
(Brauers and Zavadskas, 2010):
A*
FM ¼Aijmax
iUi(18)
Step4: The final ranking of alternatives based on dominance theory
Based on the fundamental idea of dominance theory, the ranking results obtained in Step 1,
Step 2 and Step 3 are integrated to get the final ranking result, which is called
MULTIMOORA ranking.
3. The proposed model for risk prioritization with FMEA using extended
MULTIMOORA
In this section, we first provide FMEA method to solve the emergency risk assessment
information problem with interval-valued Pythagorean fuzzy and then propose an extended
MULTIMOORA approach based on interval-valued Pythagorean fuzzy for emergency risk
prioritization. The flowchart in Figure 2 shows the proposed novel ranking method for
emergency risk prioritization in the FMEA framework.
Stage 1 Identify potential emergencies
Step 1.1 Determine risk assessment objective and FMEA scope. The objective of risk
assessment is defined and the factors of risk assessment are confirmed, which play a
crucial role in the following risk evaluating and ranking processes (Zhao et al., 2017). In the
paper, FMEA is used for emergency risk analysis. The emergency risk is generally
defined as the integration of the probability and consequences that the research object
cannot achieve the expected goal or function under a certain environment (Jiang, 2016).
And the potential emergencies are treated as failure modes. FMEA experts will evaluate
emergency risk level from three-type risk factors: occurrence (O), severity (S) and
detection (D), where occurrence denotes the frequency of the emergency, severity denotes
the seriousness of the emergency and detection denotes the likelihood of the emergency
not being detected.
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Step 1.2 List potential emergencies. The FMEA team has experts with different knowledge
structures, knowledge backgrounds and field experience, which can explore
systematically the relationships among occurrence, severity and detection of
emergency. According to the previous researches, news report and actual accident
data, the potential emergencies for an operational infrastructure project can be listed.
Subsequently, experts evaluate the risk degree of emergencies from three aspects:
occurrence, severity and detection. The structure of the emergency risk assessment
problem with FMEA is shown in Figure 3.
Step 1.3 Describe the FMEA for emergencies risk assessment problem. Consider FMEA for
emergencies risk assessment problem as a MCDM problem under the interval-valued
Pythagorean fuzzy environment. We suppose that there are sprofessional experts
EXkðk¼1;2; :::; sÞin an FMEA assessment team. The FMEA experts team is responsible
for the risk assessment of memergencies, which denote as the FMiði¼1;2; :::; mÞin
terms of nrisk factors RFjðk¼1;2; :::; nÞ. Meanwhile, in order to obtain weights of risk
Emergency Risk
Assessment
Occurrence (O)
Severity (S)
Detection (D)
FM
1
FM
2
FM
3
FM
4
FM
m
......
Stage 1
Identify potential emergencies
Stage 2
Aggregate the linguistic assessment
information of experts for each emergency
Step 2.1: Obtain the linguistic evaluation information of experts
Step 2.2: Convert linguistic values into a group IVPF evaluation matrix
Step 2.3: Construct the normalized IVPF emergency risk evaluation matrix
Step 2.4: Calculate the comprehensive IVPF evaluation matrix
Stage 3
Calculate weights of risk factors using
deviation maximization model method
Step 3.1: Construct the deviation maximization model
Step 3.2: Determine the weights of risk factors
Step 1.1: Determine risk assessment objective and FMEA scope
Step 1.2: List the potential emergencies
Step 1.3: Describe the FMEA for emergencies risk evaluation problem
Stage 4
Determine final ranking order for emergencies
using the extend MULTIMOORA method
Step 4.1: The IVPF ratio system
Step 4.2: The IVPF reference point method
Step 4.3: The IVPF full multiplicative form method
Step 4.4: Determine final risk ranking by dominance theory
Figure 3.
The structure of the
emergency risk
assessment problem
with FMEA
Figure 2.
The flowchart of risk
assessment using
FMEA with IVPF-
MULTIMOORA
method
Novel
approach to
emergency risk
assessment

factors, these experts also evaluate the important of risk factors by using linguistic
variable. Then, the assessment results are transformed into relative IVPFNs. Where the

Pk¼ð

Pk
ijÞm3ndenotes an IVPF assessment matrix for the emergencies given by the
experts EXk.
Pk
ij ¼ðuk
ij;vk
ijÞ¼ð½uk−
ij ;ukþ
ij ;½vk−
ij ;vkþ
ij Þ denotes an IVPFN of emergency FMi
with respect to the risk factor RFj.
Wk¼ð

Pk
jÞ13nis an IVPF assessment matrix for the
weight of risk factors given by EXk.
Stage 2 Aggregation of the linguistic assessment information of experts for each emergency
Step 2.1 Obtain the linguistic evaluation of experts. As mentioned earlier, the FMEA team
can evaluate emergency risk through O,Sand Dof actual accident data. However, most of
the available accident data are incomplete or difficult to obtain. And uncertainty and
incomplete information exist in practice, experts find it difficult to accurately assess by
real numbers. Therefore, in the section, the judgment of experts can be described by using
linguistic variables, and then emergency risk assessment information is also obtained.
Step 2.2 Convert linguistic values into a group IVPF evaluation matrix. Experts tend to use
linguistic variables to evaluate the actual performance of emergency aspects of risk
factors. As stated before, IVPFNs are well suitable for describing the uncertainty and
vagueness of risk assessment information (Ding et al., 2019;Chen, 2018). Therefore, the
linguistic variables can be converted to the corresponding IVPFNs according to certain
rules in Table 1 and IVPF evaluation matrix Pk¼ðPk
ijÞm3nof the emergency can be
obtained.
Step 2.3: Construct the normalized IVPF risk emergency evaluation matrix. The IVPF-
MULTIMOORA ranking method needs to subtract and divide the assessment information
according to the types of risk factors, but the corresponding operation rules of IVPFNs
have not been unified. To enhance the universality of the method, the Pk¼ðuk
ij;vk
ijÞis
transformed as follows:
Pk
ij ¼8
<
:
Pk
ij ¼uk
ij;vk
ij;j∈A
NegPk
ij ¼vk
ij;uk
ij;j∈B
;(19)
Then Pk
ij ¼ðuk
ij;vk
ijÞis obtained, where Ais the benefit-type risk factor subset, and Bis the
cost-type risk factor subset. The specific transform rule is described in Table 1.
Linguistic variables IVPFNs for benefit-type IVPFNs for cost-type
Extremely low (EL) ([0.00,0.10],[0.90,0.95]) ([0.90,0.95],[0.00,0.10])
Very low (VL) ([0.10,0.20],[0.80,0.90]) ([0.80,0.90],[0.10,0.20])
Low (L) ([0.30,0.40],[0.70,0.80]) ([0.70,0.80],[0.30,0.40])
Medium low (ML) ([0.40,0.50],[0.50,0.60]) ([0.50,0.60],[0.40,0.50])
Medium (M) ([0.50,0.50],[0.50,0.50]) ([0.50,0.50],[0.50,0.50])
Medium high (MH) ([0.50,0.60],[0.40,0.50]) ([0.40,0.50],[0.50,0.60])
High (H) ([0.70,0.80],[0.30,0.40]) ([0.30,0.40],[0.70,0.80])
Very high (VH) ([0.80,0.90],[0.10,0.20]) ([0.10,0.20],[0.80,0.90])
Extremely high (EH) ([0.90,0.95],[0.00,0.10]) ([0.00,0.10],[0.90,0.95])
Table 1.
Linguistic variables for
rating emergency
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Step 2.4: Calculate the comprehensive interval-valued evaluation matrix. Most existing
FMEA-related researches directly give weights to experts, which have certain impact on
the accuracy of research results. It is difficult to accurately determine the weight of
experts by subjective weighting method, but it is simple and feasible to determine the
priority levels of experts according to the difference in knowledge structure and
experience. Therefore, this section considers the priority of experts during the
aggregation process for evaluation information of each emergency, which improves
the accuracy of results.
Suppose the FMEA team consists of sexperts. According to different knowledge structure
and domain experience, the experts are divided into spriority levels. The knowledge structure
of EX1is closer to the evaluation objects of FMEA and the domain experience is more
extensive, with the highest priority level. That is, its evaluation information is given priority.
So, the priority level of EXsis the lowest. Then, calculate the comprehensive interval-valued
evaluation matrix using the interval-value Pythagorean fuzzy priority power weight average
(IVPFPPWA) operator, which is defined based on the priority average operator constructed
by Yager (2008) as follows:
IVPFPWAP1
j;P2
j; :::; Ps
j¼Pj
¼0
B
B
@2
6
6
4
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1Y
n
j¼1
ð1a2
j
Tk=X
s
k¼1
Tk
v
u
u
u
u
t;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1Y
n
j¼11b2
j
Tk=X
s
k¼1
Tk
v
u
u
u
u
t3
7
7
5
;2
6
6
4Y
n
j¼1
c
Tk=X
s
k¼1
Tk
j;Y
n
j¼1
d
Tk=X
s
k¼1
Tk
j3
7
7
51
C
C
A
;
(20)
where Tk¼Qk−1
t¼1sPt
ij;T1¼1, in the process of information aggregation, the expert
weight information is determined according to the score function value of information itself
and then IVPF comprehensive evaluation matrix P¼ðPij Þm3nfor emergency is obtained as
follows:
P¼ðPijÞm3n
¼0
B
B
@
ð½a11;b11 ;½c11;d11 Þ ð½a12;b12;½c12 ;d12Þ ::: ð½a1n;b1n;½c1n;d1nÞ
ð½a21;b21 ;½c21;d21 Þ ð½a22;b22;½c22 ;d22Þ ::: ð½a2n;b2n;½c2n;d2nÞ
::: ::: ::: :::
ð½am1;bm1;½cm1;dm1Þ ð½am2;bm2;½cm2;dm2Þ ::: ð½amn;bmn ;½cmn;dmnÞ
1
C
C
A
;(21)
where aij ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−Qs
k¼1ð1−ðak
ijÞ2Þ
TkP
s
k¼1
Tk
s,bij ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−Qs
k¼1ð1−ðbk
ijÞ2Þ
TkP
s
k¼1
Tk
s,
cij ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−Qs
k¼1ck
TkP
s
k¼1
Tk
ij
s, and dij ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−Qs
k¼1dk
TkP
s
k¼1
Tk
ij
sfor i¼1;2; :::; m;j¼1;2; :::; n
Stage 3 Calculate weights of risk factors using deviation maximization model method
Step 3.1: Construct the deviation maximization model. This subsection will determine the
weight of risk factors by deviation maximization model method, so as to overcome the gap
that traditional FMEA not considering the weight of risk factors. Under the circumstance
that the attribute weight is completely unknown in MCDM problems, according to the
information theory, if all alternatives have similar attribute values with respect to an
Novel
approach to
emergency risk
assessment

attribute, then a small weight should be assigned to the attribute. This is due to that such
attribute does not help in differentiating alternatives (Zeleny and Cochrane, 1982). An
IVPF evaluation matrix P¼Pijm3ncan be obtained based on the aforementioned
principles and emergencies. Let the deviation between FMiand other emergencies with
respect to the risk factors RFjas dijðwÞ¼Pm
h¼1dPij;Phj wj, where dPij;Phjis the
Euclidean distance between Pij and Phj. Then, the total deviation for evaluation
information of emergencies denotes dðwÞ¼Pn
j¼1Pm
i¼1Ph≠idPij;Phj wj. The deviation
maximization model is constructed as follows (Xu, 2010):
max X
n
j¼1X
m
i¼1X
h≠i
dðPij;Phj Þwj
s:t:X
n
j¼1
ðwjÞ2¼1;wj≥0;j¼1;2; :::; n;
(22)
Step 3.2: Determine the weights of risk factors. To solve this optimization model, we
construct the Lagrange function as follows:
Lðw;λÞ¼dðwÞþλ
2 X
n
j¼1
ðwjÞ21!;(23)
where λis the Lagrange multiplier.
The partial derivatives of Equation (23) are calculated with respect to wjand λ,
respectively, and the partial derivatives are set equal to zero as follows:
8
>
>
>
>
>
<
>
>
>
>
>
:
vLðw;λÞ
vwj
¼X
m
i¼1X
h≠i
dðPij;Phj Þwjþλwj¼0
vLðw;λÞ
vλ¼X
n
j¼1
wj1¼0
;(24)
The optimal solution of objective weight of risk factor by solving Equation (24) is then
normalized as follows:
wj¼Pm
i¼1Ph≠idðPij;Phj Þ
Pn
j¼1Pm
i¼1Ph≠idðPij;Phj Þ;j¼1;2; :::; n;(25)
Stage 4 Determine risk ranking order for emergencies using the extended MULTIMOORA
method
The IVPFPWA operator and IVPFPGA operator are introduced into the ratio system and the
full multiplicative model to avoid the information loss, respectively. And the improved
Euclidean distance is calculated between the evaluation information and the reference point
in the reference point method. The emergencies risk ranking order based on extended
MULTIMOORA method can be constructed as follows:
Step 4.1: Construct the IVPF–ratio system. For each IVPFN P¼ð½a;b;½c;dÞ, satisfying
½a;b⊂½0;1,½c;d⊂½0;1and bþd≤1, so there is no need to standardize the evaluation
information. According to the definition 8, the comprehensive utility value of different
emergencies risk is obtained as follows:
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yi¼IVPFWAðPi1;Pi2; :::; PinÞ
¼0
@2
4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1Y
n
j¼1
ð1ðaij Þ2wj
v
u
u
t;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1Y
n
j¼11ðbij Þ2Þwj
v
u
u
t3
5;"Y
n
j¼1
ðcijÞwj;Y
n
j¼1
ðdijÞwj#1
A;
(26)
where yidenotes the comprehensive utility value under all risk factors of FMi. According
to definition 7 and Equations (7) and (8), the score function value sðPÞand accuracy
function value hðPÞof comprehensive utility value of different emergencies risk are
obtained. Then, the risk of emergencies is ranked based on the comprehensive utility value
yi. The smaller the yivalue is, the higher the risk rank.
Step 4.2: Construct the IVPF–reference point method. Two kinds of reference points are
present: (1) the maximum or minimum value of emergency evaluation information under
different risk factors; (2) positive and negative ideal reference points. This study adopted
positive ideal reference points, that is, θj¼ð½1;1;½0;0Þ. Then, the distances between
emergencies FMiand the reference point are calculated under different risk factors, which
can be obtained by the Minkowski metric method (Zhang et al., 2013):
dðθj;FMiÞ¼(X
n
j¼1
½dðθj;PijÞγ)1=γ
;γ∈Nþ;(27)
The robustness of the optimal ranking problem based on Minkowski metric increases with the
increase of γvalue (Brauers, 2012), so that γ→∞. The distance was called as Tchebycheff
metric, and it was calculated as follows:
dðθj;FMiÞ¼ max
1≤j≤ndðθj;PijÞ;(28)
Combining with Equation (6), the risk factor weight vector wj¼ðw1;w2; :::; wnÞis introduced
as the significance coefficient. And the Minkowski metric between emergencies FMiand the
reference point is calculated under different risk factors, which can be obtained as follows:
dðθj;FMiÞ¼ max
1≤j≤ndðθj;PijÞ¼ max
1≤j≤n
wj
2hðaij 1Þ2þðbij 1Þ2þc2
ij þd2
iji;(29)
Where dðθj;FMiÞdenotes the Minkowski metric distance.The higher the dðθj;FMiÞvalue is,
the higher the risk ranking order is.
Step 4.3: Construct the IVPF–full multiplicative form method. According to the definition 9,
the multiplicative utility value of FMiis obtained by using IVPF evaluation matrix
P¼Pijm3nand weight vector wj¼ðw1;w2; :::; wnÞof risk factor as follows:
Ui¼IVPFWGðPi1;Pi2; :::; PinÞ
¼0
@"Y
n
j¼1
ðajÞwj;Y
n
j¼1
ðbjÞwj#;2
4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1Y
n
j¼11ðcjÞ2wj
v
u
u
t;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1Y
n
j¼11ðdjÞ2wj
v
u
u
t3
51
A;
(30)
where Uidenotes the multiplicative utility value under all risk factors of emergencies.
Novel
approach to
emergency risk
assessment

Similar to Step 1, the scoring function value and accuracy function value of multiplication
utility value can be obtained and the emergency risk ranking order can be determined. The
smaller the U
i
value is, the higher the risk ranking order is.
Step 4.4: Determine final rank based on dominance theory. IVPF-MULTIMOORA method
comprises the IVPF–ratio system, the IVPF–reference point and the IVPF–full
multiplicative method, that is, three kinds risk ranking for emergencies exist, and equal
importance. According to dominance theory, the final risk rank of the emergencies is
determined based on the three risk orders under each emergency.
4. Application
The emergency risk of the East Route of the South-to-North Water Diversion Project mainly
refers to all kinds of emergencies that affect the water supply, water quality and engineering
safety accidents in the water trunk canal. In view of the characteristics of the East Route
engineering system and the causes of accidents, according to the types of risk sources, Zhao
et al. (2017) determined six categories for potential emergencies in the East Route of the South-
to-North Water Diversion Project. It mainly includes the following six categories. Category
I(FM
1
): Water pollution caused by leakage of highly toxic substances into canal due to land
transport incidents. Category II(FM
2
): Water pollution caused by leakage of oil and sewage
due to shipping traffic incidents. Category III(FM
3
): Water pollution caused by maliciously
poisoning by human. Category IV(FM
4
) : Engineering safety accidents caused by geological
landslides, earthquakes, ice jams and other natural disasters. Category V(FM
5
): Engineering
safety accidents caused by machine fault. Category VI(FM
6
): Engineering safety accidents
caused by inundation of pumping stations and dam break. Category VII(FM
7
): Engineering
safety accidents caused by explosions or terrorist attacks. In this section, a novel approach
using extended MULTIMOORA method based on IVPF is proposed to evaluate the
emergencies risk prioritization. The specific steps are described as follows:
Stage 1: Determine the comprehensive evaluation matrix
Suppose the experts team for FMEA comprises three members, according to different
knowledge structure and domain experience, the experts are divided into three priority levels,
that is, EX1,EX2and EX3. Firstly, the actual performance of these failure modes under the
three risk factors (O, S, D) and the importance of risk factors are evaluated by the three
experts, as shown in Table 2. Secondly, the evaluation information of linguistic variables is
converted into the corresponding IVPFNs, and then IVPF evaluation matrix Pk¼ðPk
ijÞm3nis
constructed. Thirdly, O,Sand Dare cost-type risk factors, so the evaluation matrix needs to
be transformed according to Equation (19), as shown in Table 3. Finally, according to
Risk factors OSD
Experts
DM
1
DM
2
DM
3
DM
1
DM
2
DM
3
DM
1
DM
2
DM
3
Emergencies
FM
1
M ML ML H VH VH ML M M
FM
2
H MHVHVHEHVHVHH H
FM
3
MLMMLHHHMHMH
FM
4
HHMHHHVHMMLL
FM
5
HMHHHMMHHHH
FM
6
VL VL EL VH VH VH ML L L
FM
7
EL EL VL EH EH VH M ML M
Table 2.
Evaluation
information of
linguistic variables for
experts
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Risk factors OSD
Experts DM
1
DM
2
DM
3
DM
1
DM
2
DM
3
DM
1
DM
2
DM
3
Emergencies
FM
1
([0.50,0.50],
[0.50,0.50])
([0.50,0.60],
[0.40,0.50])
([0.50,0.60],
[0.40,0.50])
([0.30,0.40],
[0.70,0.80])
([0.10,0.20],
[0.80,0.90])
([0.10,0.20],
[0.80,0.90])
([0.50,0.60],
[0.40,0.50])
([0.50,0.50],
[0.50,0.50])
([0.50,0.50],
[0.50,0.50])
FM
2
([0.30,0.40],
[0.70,0.80])
([0.40,0.50],
[0.50,0.60])
([0.10,0.20],
[0.80,0.90])
([0.10,0.20],
[0.80,0.90])
([0.00,0.10],
[0.90,0.95])
([0.10,0.20],
[0.80,0.90])
([0.10,0.20],
[0.80,0.90])
([0.30,0.40],
[0.70,0.80])
([0.30,0.40],
[0.70,0.80])
FM
3
([0.50,0.60],
[0.40,0.50])
([0.50,0.50],
[0.50,0.50])
([0.50,0.60],
[0.40,0.50])
([0.30,0.40],
[0.70,0.80])
([0.30,0.40],
[0.70,0.80])
([0.30,0.40],
[0.70,0.80])
([0.40,0.50],
[0.50,0.60])
([0.50,0.50],
[0.50,0.50])
([0.30,0.40],
[0.70,0.80])
FM
4
([0.30,0.40],
[0.70,0.80])
([0.30,0.40],
[0.70,0.80])
([0.40,0.50],
[0.50,0.60])
([0.30,0.40],
[0.70,0.80])
([0.30,0.40],
[0.70,0.80])
([0.10,0.20],
[0.80,0.90])
([0.50,0.50],
[0.50,0.50])
([0.50,0.60],
[0.40,0.50])
([0.70,0.80],
[0.30,0.40])
FM
5
([0.30,0.40],
[0.70,0.80])
([0.40,0.50],
[0.50,0.60])
([0.30,0.40],
[0.70,0.80])
([0.30,0.40],
[0.70,0.80])
([0.50,0.50],
[0.50,0.50])
([0.40,0.50],
[0.50,0.60])
([0.30,0.40],
[0.70,0.80])
([0.30,0.40],
[0.70,0.80])
([0.30,0.40],
[0.70,0.80])
FM
6
([0.80,0.90],
[0.10,0.20])
([0.80,0.90],
[0.10,0.20])
([0.90,0.95],
[0.00,0.10])
([0.10,0.20],
[0.80,0.90])
([0.10,0.20],
[0.80,0.90])
([0.10,0.20],
[0.80,0.90])
([0.50,0.60],
[0.40,0.50])
([0.70,0.80],
[0.30,0.40])
([0.70,0.80],
[0.30,0.40])
FM
7
([0.90,0.95],
[0.00,0.10])
([0.90,0.95],
[0.00,0.10])
([0.80,0.90],
[0.10,0.20])
([0.00,0.10],
[0.90,0.95])
([0.00,0.10],
[0.90,0.95])
([0.10,0.20],
[0.80,0.90])
([0.50,0.50],
[0.50,0.50])
([0.50,0.60],
[0.40,0.50])
([0.50,0.50],
[0.50,0.50])
Table 3.
Interval-valued
Pythagorean fuzzy
evaluation matrix
Novel
approach to
emergency risk
assessment

Equation (20), three experts’evaluation information by using IVPFPWA operator is
aggregated, and then the IVPF comprehensive evaluation matrix P¼Pijm3nfor
emergencies is obtained in Table 4.
Stage 2: Determine the weights of risk factors
According to the obtained IVPF comprehensive evaluation matrix P¼Pij m3nof risk
factors in stage 1, the weights of risk factors are calculated using deviation maximization
model method. Firstly, the dPij;Phj between Pij and Phj is obtained by calculating
Equation (6), and then the deviations, that is, dðPiO;PhO Þ,dðPiS ;PhS Þ,dðPiD;PhD Þbetween
evaluation information of different emergencies can be obtained as shown in Table 5. Finally,
the weight vectors wj¼ð0:390;0:319;0:291Þof risk factors are obtained by solving the
deviation maximization optimization model (22). The computational procedure in detail is as
follows based on Equation (25).
First, the deviations among emergencies with respect to the risk factors can be obtained as
follows:
X
m
i¼1X
h≠i
dðPiO;PhO Þ¼dðP1O;PhOÞþdðP2O;PhOÞþdðP3O;PhO Þ
þdðP4O;PhOÞþdðP5O;PhO ÞþdðP6O;PhOÞþdðP7O;PhOÞ
¼1:531 þ1:758 þ2:084 þ2:483 þ2:287 þ1:257 þ2:241 ¼13:642
Emergencies OSD
FM
1
([0.500,0.541],
[0.461,0.500])
([0.267,0.364],
[0.723,0.823)
([0.500,0.559],
[0.442,0.500])
FM
2
([0.309,0.408],
[0.672,0.773])
([0.093,0.190],
[0.812,0.906])
([0.153,0.246],
[0.783,0.883])
FM
3
([0.500,0.577],
[0.424,0.500])
([0.300,0.400],
[0.700,0.800])
([0.420,0.488],
[0.523,0.594])
FM
4
([0.305,0.405],
[0.690,0.790])
([0.293,0.392],
[0.705,0.805])
([0.542,0.600],
[0.434,0.483])
FM
5
([0.317,0.417],
[0.666,0.767])
([0.362,0.433],
[0.634,0.709])
([0.300,0.400],
[0.700,0.800])
FM
6
([0.835,0.917],
[0.000,0.166])
([0.100,0.200],
[0.800,0.900])
([0.615,0.718],
[0.348,0.449])
FM
7
([0.877,0.938],
[0.000,0.124])
([0.007,0.101],
[0.899,0.950])
([0.500,0.532],
[0.470,0.500])
Emergencies dðP
iO;P
hOÞdðP
iS;P

hSÞdðP
iD;P

hDÞ
FM
1
1.531 0.626 0.826
FM
2
1.758 1.555 1.595
FM
3
2.084 1.797 1.548
FM
4
2.483 2.157 1.868
FM
5
2.287 1.982 1.750
FM
6
1.257 1.089 0.951
FM
7
2.241 1.980 1.658
Table 4.
Interval-valued
Pythagorean fuzzy
comprehensive
evaluation matrix
Table 5.
Relevant parameters of
the weight vector for
risk factors
IJICC

X
m
i¼1X
h≠i
dðPiS ;PhS Þ¼dðP1S;PhS ÞþdðP2S;PhSÞþdðP3S;PhS ÞþdðP4S;PhS Þ
þdðP5S;PhSÞþdðP6S;PhSÞþdðP7S;PhS Þ
¼0:626 þ1:555 þ1:797 þ2:157 þ1:982 þ1:089 þ1:980 ¼11:185
X
m
i¼1X
h≠i
dðPiD;PhD Þ¼dðP1D;PhDÞþdðP2D;PhD ÞþdðP3D;PhDÞþdðP4D;PhD Þ
þdðP5D;PhDÞþdðP6D;PhD ÞþdðP7D;PhDÞ
¼0:826 þ1:595 þ1:548 þ1:868 þ1:750 þ0:951 þ1:658 ¼10:196
Then, the total deviation for evaluation information of emergencies is obtained as follows:
X
n
j¼1X
m
i¼1X
h≠i
dðPij;Phj Þ¼X
m
i¼1X
h≠i
dðPiO;PhO ÞþX
m
i¼1X
h≠i
dðPiS ;PhS ÞþX
m
i¼1X
h≠i
dðPiD;PhD Þ
¼13:642 þ11:185 þ10:196 ¼35:023
Finally, the weight vectors wj¼ðw1
j;w2
j;w3
jÞof risk factors are calculated as follows:
w1
j¼P
m
i¼1P
h≠i
dðPiO;PhO Þ
P
n
j¼1P
m
i¼1P
h≠i
dðPij;Phj Þ
¼13:642
35:023 ¼0:390;w2
j¼P
m
i¼1P
h≠i
dðPiS ;PhS Þ
P
n
j¼1P
m
i¼1P
h≠i
dðPij;Phj Þ
¼11:185
35:023 ¼0:319;w3
j¼P
m
i¼1P
h≠i
dðPP iD;PhD Þ
P
n
j¼1P
m
i¼1P
h≠i
dðPij;Phj Þ
¼10:196
35:023 ¼0:291;
Stage 3: Determine the final risk ranking order of potential emergencies
The comprehensive utility value y
i
, the Tchebycheff Metric distance d
max
and the
multiplicative utility value U
i
of potential emergencies are obtained using Equations (26),
(29) and (30) in Table 6. Then, the risk ranking of potential emergencies is obtained as
expressed in Table 7 by IVPF–ratio system, IVPF–reference point method and IVPF–full
multiplicative form method. Finally, the final risk ranking of potential emergencies is
determined based on dominance theory as seen in the last column of Table 7. Therefore,
Emergencies y
i
d
max
U
i
FM
1
([0.444,0.501],[0.526,0.586]) 0.234 ([0.409,0.481],[0.570,0.654])
FM
2
([0.218,0.310],[0.746,0.845]) 0.274 ([0.172,0.276],[0.805,0.860])
FM
3
([0.425,0.504],[0.529,0.611]) 0.224 ([0.404,0.489],[0.652,0.657])
FM
4
([0.393,0.473],[0.607,0.689]) 0.271 ([0.356,0.449],[0.710,0.740])
FM
5
([0.328,0.418],[0.665,0.757]) 0.299 ([0.326,0.417],[0.725,0.761])
FM
6
([0.675,0.781],[0.000,0.380]) 0.272 ([0.388,0.526],[0.651,0.674])
FM
7
([0.693,0.778],[0.000,0.356]) 0.264 ([0.161,0.390],[0.744,0.751])
Table 6.
Relevant parameters of
the IVPF-
MULTIMOORA
method
Novel
approach to
emergency risk
assessment

Category I(FM
2
): Water pollution caused by leakage of oil and sewage due to shipping traffic
incidents is the highest risk degree among the emergencies. Category II(FM
1
): Water
pollution caused by leakage of highly toxic substances into canal due to land transport
incidents is the lowest risk degree among the emergencies. The risk ranking order for the
remaining emergencies is as follows: Category III(FM
3
): Water pollution caused by
maliciously poisoning by human. Category IV(FM
4
): Engineering safety accidents caused
by geological landslides, earthquakes, ice jams and other natural disasters. Category V(FM
5
):
Engineering safety accidents caused by machine fault. Category VI(FM
6
): Engineering safety
accidents caused by inundation of pumping stations and dam break. Category VII(FM
7
):
Engineering safety accidents caused by explosions or terrorist attacks. From the results, we
can determine that the risk ranking order of emergencies is acceptable for practical
applications.
5. Comparison analysis and discussion
To verify the feasibility and validity of the proposed method in this paper, the risk ranking
results for emergencies using extended MULTIMOORA method under interval-valued
Pythagorean fuzzy environment are compared with the RPN method, the MULTIMOORA
method and the IVPF-TOPSIS method, as shown in Figure 3. According to the results in
Figure 4, it can be seen that the FMEA method proposed in this paper is superior to other
methods, which is stated in detail as follows.
First of all, the risk ranking order of emergencies FM
2
,FM
4
and FM
5
using the RPN
method remains the same as the proposed method, and the risk ranking order of other
Emergencies
IVPF–ratio
system
IVPF–reference
point
IVPF–full multiplicative
form
IVPF-
MULTIMOORA
FM
1
56 7 7
FM
2
12 1 1
FM
3
47 5 6
FM
4
34 4 3
FM
5
21 3 2
FM
6
73 6 5
FM
7
65 2 4
1
0
1
2
3
4
5
6
7
8Rank
The proposed method The RPN method
The IVPF-TOPSIS methodThe MULTIMOORA method
234 567
FM
Table 7.
The final risk ranking
order of failure modes
with IVPF-
MULTIMOORA
method
Figure 4.
Results of comparison
analysis
IJICC

emergencies (FM
1
,FM
3
,FM
6
and FM
7
) all fluctuates. The reasons for this result are as
follows. (1) The real number form of 1–10 is adopted to represent the evaluation information
for emergencies and fails to effectively deal with the vagueness of the evaluation information.
(2) The weights of risk factors and experts are equally assigned, which may lead to the RPNs
of emergencies being different from the actual situation. (3) The aggregation form of
evaluation information is simple multiplication for risk factors, which makes the same RPNs
easy to appear so that the risk ranking order is difficult to judge, and there is information loss
phenomenon. In the novel FMEA framework proposed in the paper, the expert team
combined their own experience with knowledge management to evaluate the risk degree of
the failure modes. The linguistic variables are used to represent the evaluation information,
which is more in line with the practical thinking habits of human than traditional methods
using real numbers. It can effectively deal with the uncertainty of experts’evaluation
information and retain the integrality of information. In addition, the proposed method also
solves the problem that risk factors and expert weights are not considered. Therefore, the
proposed method can more accurately reflect the evaluation information of experts.
In addition, the traditional MULTIMOORA method and the RPN method have almost the
same results in risk ranking order for emergencies, but they are also slightly different from
the results in this paper. Such as emergencies FM
3
and FM
6
, they are the fourth and seventh
risk ranking order using the traditional MULTIMOORA method. However, in the actual
situation, the emergency for water pollution caused by maliciously poisoning by human
(FM
3
) has a relatively low frequency, which is detected more easily. And the result is usually
the death of aquatic organisms. It will not lead to a large level of safety accidents. So it is
reasonable to have a sixth risk ranking order for emergency FM
3
. Although emergency for
engineering safety accidents caused by inundation of pumping stations and dam break (FM
6
)
has a low probability of occurrence, it has great harmfulness. Once emergency FM
6
occurs, its
emergency measures are difficult to handle. Therefore, the fifth ranking order of risk in
emergency FM6 is in line with the actual situation. The causes of these differences are as
follows. (1) The traditional MULTIMOORA method has certain limitations using real
numbers for they cannot objectively reflect the complexity and uncertainty of things. (2) The
risk factors are regarded as equally important in the traditional MULTIMOORA method,
which has certain impact on the accuracy of research results. However, the proposed method
in the paper introduces IVPFWA operator, Tchebycheff metric distance and IVPFWG
operator into the ratio system, the reference point method and the full multiplication form
model of MULTIMOORA submethods, respectively, to optimize information aggregation of
FMEA process, which highlights the effect of risk factor weight on risk ranking order.
Moreover, the proposed method in this paper also considers the priority of evaluation experts.
Therefore, compared with traditional MULTIMOORA method, the results using proposed
method in the paper are more accurate and are more consistent with the actual situation.
Finally, the risk ranking order obtained by IVPF-TOPSIS method is also different from the
proposed method in this paper, this is due to the risk ranking order for emergencies being
obtained based on the results of three decision-making methods, that is, the IVPF–the ratio
system, IVPF–the reference point method and IVPF–the full multiplication form model.
Among them, IVPF–the reference point method adopts the Minkowski metric method, not the
TOPSIS method. The reason is that the risk degree of an emergency that is close to the ideal
solution may be closer simultaneously to the negative ideal solution, which cannot fully
reflect the risk degree of emergency, so as to affect the accuracy of decision result (Yang and
Huang, 2017;Hua and Tan, 2004). Moreover, Brauers and Zavadskas (2012) proposed
conditions to measure the robustness of MCDM methods and concluded that the robustness
of MCDM methods combined with multiple decision-making methods is better than that of
MCDM method with a single decision-making method. The IVPF-MULTIMOORA method
comprises three decision-making methods, which fully improves the robustness of the results
Novel
approach to
emergency risk
assessment

to some extent. Therefore, compared with IVPF-TOPSIS method, the proposed method in the
paper is more scientific and more robust.
In a word, in the risk ranking results for emergency obtained by the aforementioned four
methods, FM
2
,FM
4
and FM
5
are the three kinds of the highest risk ranking, and the other
four kinds of emergencies have different risk ranking. As previously discussed, the proposed
method in this paper has been optimized in every step of the FMEA process. Compared with
the other three methods, the calculation results of the proposed method in this paper are more
consistent with the actual situation of the project, thus verifying that the results can provide
decision-makers with more accurate reference basis for risk management. The results could
help operators identify the high-risk emergencies in infrastructure project, take appropriate
measures in advance to decrease the occurrence of emergencies and provide decision-making
basis for formulation of the follow-up emergency control and disposal scheme. In addition, the
results can also provide support information for risk management decisions and improve
system or project performance during their construction and operation stage. It has a certain
guiding significance for the practical popularization and application of risk management
strategies in the operation of infrastructure projects and provides reference for risk
assessment in other areas where data information is scarce.
6. Conclusions
In this paper, to overcome the shortcomings of the RPN method and the traditional
MULTIMOORA method, an innovative method has been proposed to evaluate the risk degree
of emergencies for remedial actions with FMEA using extended MULTIMOORA under the
interval-valued Pythagorean fuzzy environment. The feasibility and effectiveness of the
proposed method are examined by the case study for the emergency risk assessment of East
Route of the South-to-North Water Diversion Project. The application has indicated that the
proposed method is a comprehensive and valid tool to assess the risk of potential emergencies
in fuzzy FMEA. Moreover, this study also compared the results with the RPN method, the
MULTIMOORA method and the IVPF-TOPSIS method. The comparison analysis results
indicated that the novel emergency risk assessment method can effectively deal with the
shortcomings of the traditional FMEA and also superior to these methods, which confirm
that the final results of proposed method are rational and robust. Some superiorities of the
proposed approach are as follows:
(1) The linguistic variables are used to represent the evaluation information, which is
more in line with the human thinking habits than traditional methods using real
numbers. It can effectively deal with the uncertainty of experts’evaluation
information and retain the integrality of information.
(2) Experts’evaluation information is aggregated by the IVPFPWA operator, which
solves the problem of determining expert weight and improves the accuracy of
results.
(3) Comprehensive weighting method gives full consideration to the experts’opinions
and assessment information itself in the risk factors’weight determination, which
makes the risk ranking order closer to the accurate and practice.
(4) The IVPFWA operator, Tchebycheff metric distance and IVPFWG operator are
introduced into MULTIMOORA submethods, which optimize information
aggregation of FMEA process. The proposed method can effectively obtain a more
feasible and practical result and improve the robustness of result.
In the further study, the following research works can be focused. Firstly, the proposed
method can be applied for emergency risk management in the other fields of infrastructure,
IJICC

such as highways, water treatment plant and sewage treatment plants, to further testify its
validity. In addition, in the proposed FMEA framework, the three risk factors (occurrence,
severity and detection) are only considered; future study can explore other risk factors that
were not considered in the paper and determine the relation among risk factors. Finally, the
proposed novel method can be used flexibly, but the computational process for practitioner is
relatively complex and difficult, so further work can develop a programming software to
facilitate the application of the proposed FMEA model.
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About the authors
Huimin Li. He is an associate professor in the Department of Construction Engineering
and Management at North China University of Water Resources and Electric Power,
China. He received his PhD degree in Management Science and Engineering, Hohai
University, Nanjing, China, in 2011. His research interests include decision-making,
trust, risk management and schedule management.
Lelin Lv. She is a master’s student in the Department of Construction Engineering and
Management at the North China University of Water Resources and Electric Power,
China. She received her bachelor’s degree from the Engineering Management, North
China University of Water Resources and Electric Power, China, in 2017. Her research
interests include decision-making, game theory and so on.
Feng Li. He is a lecturer in the Department of Construction Engineering and
Management at North China University of Water Resources and environmentElectric
Power, China. He received his PhD degree in College of Water Conservancy and
Environmental Engineering, Zhengzhou University, Zhengzhou, China, in 2011. His
research interests include decision-making, evaluation, project management and so on.
Feng Li is the corresponding author and can be contacted at: lifeng9406@126.com
IJICC

Lunyan Wang. He is a professor in the Department of Construction Engineering and
Management at the North China University of Water Resources and Electric Power,
China. He received his PhD degree from the Department of Water Conservancy and
Hydropower Engineering, Hohai University, Nanjing, China, in 2014. His research
interests include decision-making, evaluation, project management and so on.
Qing Xia. She is a master’s student in the Department of Construction Engineering and
Management at the North China University of Water Resources and Electric Power,
China. She received her bachelor’s degree from the Engineering Management, North
China University of Water Resources and Electric Power, China, in 2017. Her research
interests include decision-making, sustainability assessment and so on.
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Novel
approach to
emergency risk
assessment