INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
ISSN 1841-9836, e-ISSN 1841-9844, 14(5), 672-691, October 2019.

Risk Evaluation in Failure Mode and Effects Analysis Based on
D Numbers Theory

B. Liu, Y. Deng

Baoyu Liu
1. Institute of Fundamental and Frontier Science
University of Electronic Science and Technology of China
Chengdu, 610054, China
2. School of Data and Computer Science
Sun Yat-sen University, China
Guangzhou, China
Baoyu.L@hotmail.com

Yong Deng*
Institute of Fundamental and Frontier Science
University of Electronic Science and Technology of China
Chengdu, 610054, China
*Corresponding author: prof.deng@hotmail.com

Abstract:
Failure mode and effects analysis (FMEA) is a useful technology for identifying the
potential faults or errors in system, and simultaneously preventing them from occur-
ring. In FMEA, risk evaluation is a vital procedure. Many methods are proposed to
address this issue but they have some deficiencies, such as the complex calculation
and two adjacent evaluation ratings being considered to be mutually exclusive. Aim-
ing at these problems, in this paper, A novel method to risk evaluation based on D
numbers theory is proposed. In the proposed method, for one thing, the assessments
of each failure mode are aggregated through D numbers theory. For another, the com-
bination usage of risk priority number (RPN) and the risk coefficient newly defined
not only achieve less computation complexity compared with other methods, but also
overcome the shortcomings of classical RPN. Furthermore, a numerical example is
illustrated to demonstrate the effectiveness and superiority of the proposed method.

Keywords: failure mode and effects analysis, Dempster-Shafer evidence theory, D numbers, risk
evaluation, aggregate assessment.

1 Introduction

Failure mode and effects analysis (FMEA) is an efficient technology for identifying potential
faults, problems, risk, and errors from the system, procedure, and service. It improves the
reliability by preventing these faults, problems, risks, and errors from occurring. Risk evaluation
is a pivotal procedure in FMEA. Nowadays, FMEA is developed so fast that it is extensively
applied in many fields, such as medical care [3,29,32,49,60], society [51], environmental protection
[1,8], financial service [65], industry [2,24,69,78], and so on.

A traditional method for risk evaluation in FMEA is the classical risk priority number
(RPN), which is obtained by multiplying the grades of the occurrence assessment, severity as-
sessment, and detection assessment. Therefore, how to aggregate the assessment information of

Copyright ©2019 CC BY-NC



Risk Evaluation in Failure Mode and Effects Analysis Based on D Numbers Theory 673

these three risk factors is a significant issue, especially when it comes to the information with
uncertainty. Focusing on this problem, many math models such as fuzzy sets [14, 16, 17, 79], R
numbers [53], D numbers [20], Z numbers [21,26,27,39,76] and evidence theory [10,55], have been
applied to the real applications [37, 50]. In [28], Kim et al. present a general model to explain
the functional relationship among the three factors, and use the model to discuss the unique role
of each factor for comparing the risk of different failure modes. In [77], a new method to risk
evaluation based on Dempster-Shafer evidence theory is proposed. Some other evidential FEMA
are presented recently [5].

Nevertheless, although classical RPN is easy to use because of its succinct form, it is still
criticized for its weaknesses. For example, the experts usually give the assessment with uncer-
tainty or fuzzy information, but classical RPN is not appropriated to treat the fuzzy assessment.
Furthermore, different combination of risk factors might acquire the same RPN, however, the
potential risk might be totally different so that they might have different priorities. With the
aim of overcoming these weaknesses, many methods, such as Chin’s method [9], are proposed.
However, existing methods almost is not only too complex to calculation, but also do not take
the non-exclusiveness between two adjacent rankings into account. Actually, because of the sub-
jectivity of the experts, two adjacent estimation scales are supposed to be not exclusive mutually.
In order to solve the problems, a novel method based on D number theory [13] is proposed in
this paper. On the one hand, the assessments for each failure mode are aggregated through
constructing and combining D numbers because two propositions are allowed to be non-exclusive
in D numbers theory. On the other hand, RPN is applied in the proposed method so as to reduce
the computation complexity, simultaneously, novel compute mode to RPN and risk coefficient
present in this paper is capable to get rid of the weaknesses of the classical RPN. Last but not
the least, an illustrative example is used to show the effectiveness and superiority of the proposed
method.

The remainder of this paper is organized as follows. Key concepts and previous theories
are reviewed in short in Section 2. In Section 3, A novel risk evaluation in failure mode and
effects analysis based on D numbers theory is proposed. To demonstrate the effectiveness and
superiority of the proposed method, a numerical example is illustrated in Section 4. Last but
not the least, A brief conclusion is drawn in Section 5.

2 Preliminaries

2.1 Dempster-Shafer evidence theory

The real application is inevitable to deal with uncertainty [4, 6, 45–47]. Dempster-Shafer
evidence theory (D-S theory) [10, 55], is a significant theory to handle uncertainty information
[11]. Compared with Bayesian theory, it needs weaker conditions so that it is often deemed as an
extension of the Bayesian theory. D-S theory is widely applicated in many fields, such as decision
making [6,22,30,44,71], pattern recognition [40,41,43,73], evidential reasoning [15,42,74,83–85],
risk and reliability [52,54], information fusion [59,61,75], uncertainty modelling [19,25,58] and
conflict management [36,68,81].

Definition 1. Let Θ = {H1, H2, · · · , HN} be a finite nonempty set, which consist of N mutually
exclusive elements. Let P(Θ) be the power set of Θ, which is composed of 2N elements. The
basic probability assignment (BPA) function is defined as a mapping of the power set P(Θ) to a
number between 0 to 1, that is m : P (Θ) → [0, 1], and which satisfies the following conditions:

m (∅) = 0; (1)



674 B. Liu, Y. Deng

∑
A∈P(Θ)

m (A) = 1. (2)

The mass m(A) represents how strongly the evidence supports to A.

Definition 2. Let m1, m2 be two BPAs defined on the frame of discernment Θ. The Dempster’s
combination rule, denoted by m = m1

⊕
m2, is defined as follows:

m (A) =

{
1

k−1
∑

B∩C=A m1(B)m2(C), A 6= ∅
0, A = ∅

(3)

with
K =

∑
B∩C=∅

m1(B)m2(C) (4)

where K is a normalization constant which reflects the conflict of two bodies of evidence.

Actually, 0 ≤ K ≤ 1. K = 0 shows the absence of conflict between two bodies of evidence.
While K = 1 shows complete conflict between m1 and m2. Besides, when K = 1, the Dempster’s
combination rule is not any longer applicable, a possible explanation is open world assumption
[32, 33]. In order to make decision in terms of the BPA, a method called pignistic probability
transformation is present in [57], which derive a distribution of probabilities from the BPA. The
pignistic probability transformation function is defined as follows:

Definition 3. Let m be a BPA on the frame of discernment Θ, a pignistic probability transfor-
mation function BetPm : Θ −→ [0, 1] associated to m is defined by

BetPm (x) =
∑

x∈A, A∈Θ

1

|A|
m(A)

1 −m(∅)
(5)

where m (∅) 6= 1 and |A| is the cardinality of proposition A.

2.2 Fuzzy set theory

Fuzzy sets were proposed independently by Zadeh [79] in 1965 as an extension of the classical
notion of set. Fuzzy set theory is widely applied in many fields [23, 33, 72]. It reflects the stay
of the object and its fuzzy concept as a fuzzy set. Then, it sets up the appropriate membership
functions through fuzzy set about operation and transform, and analyzes the fuzzy object based
on the fuzzy mathematics. In the objective world, there are many fuzzy phenomena. For example,
when evaluating a person’s appearance, people usually use linguistic variables whose values are
represented by words or sentences in a natural or artificial language, such as "very pretty",
"pretty", "general", "ugly", and "very ugly".

Definition 4. Denote L as the universe of discourse, a fuzzy set A is described by a membership
function µA satisfying

µA : L −→ [0, 1]

where µA(x) is called the membership degree of x ∈ L belonging to fuzzy set A.

For L = {x1, . . . ,xi, . . . , xn}, the fuzzy set (A,µA) is represent by

µA (x1)

x1
, . . . ,

µA (xi)

xi
, . . . ,

µA (xn)

xn
.



Risk Evaluation in Failure Mode and Effects Analysis Based on D Numbers Theory 675

1

 !  "  #0

 !(")

x

Figure 1: Graphical presentation of the triangular fuzzy number

It is obvious that a fuzzy set is characterized entirely by its membership function. When µA(x)
get value from {0, 1}, fuzzy set A degrade into a classical set. A is a fuzzy subset of the real
number R, and its membership function satisfies

µA (x) : R −→ [0, 1]

where x is real number and there exists an element x0 such that µA = 1. Triangular fuzzy
numbers are the most extensively applied fuzzy numbers. A triangular fuzzy number is usu-
ally expressed as A = (a1,a2, a3), as graphically shown in Figure 1, which has the following
membership function

µA (x) =




0, x < a1
x−a1
a2−a1 , a1 ≤ x ≤ a2
a3−x
a3−a2, a2 ≤ x ≤ a3
0, x > a3

(6)

where a1 < a2 < a3.
In practice, fuzzy numbers are bound up with linguistic variables to describe the fuzzy

evolution to objects.

2.3 Risk priority number

The real systems are too complicated to be modelled. Risk priority number (RPN) is a
traditional and typical method to model and evaluate risk in FMEA. RPN is calculated by
multiplying the grades of occurrence assessment (O), severity assessment (S), and detection
assessment (D). That is

RPN = O×S×D (7)

where O stands for the probability of occurrence of failure mode, S refers to the severity of
failure mode and D refers to the probability of failure being detected. The three risk factors are
evaluated by FMEA experts using a 1 to 10 numeric scale. Besides, occurrence assessment is
expressed in Table 1. The larger RPN is, the more important degree it is supposed to be assigned,
referring to the failure mode should be more priority to be corrected. Although this method is
easy to use because of its sententious form, traditional RPN is still criticized for its weaknesses.
For example, traditional RPN does not take the weights of three risk factors into consideration.
Besides, different combination of risk factors might acquire the same RPN, however, the potential
risk might be totally different so that they might have different priorities.



676 B. Liu, Y. Deng

2.4 D numbers

D number, is a useful tool to model uncertain information, which overcomes the shortcomings
of Dempster-Sharfer theory [20]. Nowadays, D number is widely used in many fields such as
decision making [7,19,31], risk assessment [18,29,35,67], reliability analysis [70,80], data fusion
[8,62,66]. It is defined as follows:

Definition 5. Let Ω be a finite nonempty set, D number is a mapping D : Ω −→ [0, 1], such
that ∑

B⊆Ω
D (B) ≤ 1 and D (∅) = 0 (8)

where ∅ is the empty set and B is a subset of Ω.

If D (B) = 1, the information is regarded to be complete while D (B) < 1, it is considered
to be incomplete. Most importantly, different from D-S theory, D number does not request the
elements of set Ω to be mutually exclusive. In order to express the non-exclusiveness in Ω, a
fuzzy membership function is used to measure the exclusive/non-exclusive degree [82].

Definition 6. Let Ai and Aj be two non-empty elements in 2Ω, the non-exclusive degree between
Ai and Aj is characterized by a fuzzy membership function u¬E as follows:

u¬E : 2
Ω × 2Ω −→ [0, 1] (9)

with

u¬E (Ai,Aj) =

{
1, Ai ∩Aj 6= ∅
p, p ∈ [0, 1], Ai ∩Aj = ∅

(10)

besides,
u¬E (Ai,Aj) = max

x∈Ai, y∈Aj
{u¬E(x,y)} (11)

Let uE be the exclusive degree between Ai and Aj, then uE = 1 −u¬E.

An illustrative example is given as follows to express the calculation of the non-exclusive
degree.

Table 1: Assessment rankings for occurrence in FMEA

Ranking Probability of Occurrence Possible Failure Rate
10 Extremely high: failure almost inevitable ≥ 1/2
9 Very high 1/3
8 Repeated failures 1/8
7 High 1/20
6 Moderately high 1/80
5 Moderate 1/400
4 Relatively low 1/2000
3 Low 1/15000
2 Remote 1/150000
1 Nearly impossible ≤ 1/150000



Risk Evaluation in Failure Mode and Effects Analysis Based on D Numbers Theory 677

1 2 3 4 5

1

0

a b c

Figure 2: Graphically presentation of fuzzy variables in Table 2

Example 7. Suppose there is a non-empty set Ω = {a,b,c}, where a,b,c are three fuzzy variables
represented by triangular fuzzy numbers given in Table 2 and shown in Figure 2. The non-
exclusiveness degree can be calculated as follows [12]:

u¬E (A,B) =
AreaA∩B

AreaA + AreaB −AreaA∩B
(12)

where the areas of fuzzy numbers A and B are characterized by AreaA and AreaB, and the area
of the overlap of A and B is AreaA∩B. Thus, according to Eq.12, we have

u¬E (a,b) =
Areaa∩b

Areaa + Areab −Areaa∩b
=

0.25

1 + 1 − 0.25
≈ 0.1429

u¬E (b,c) =
Areab∩c

Areab + Areac −Areab∩c
=

0.25

1 + 1 − 0.25
≈ 0.1429

u¬E (a,b,c) = max{u¬E (a,b) ,u¬E (b,c)} = max{0.1429, 0.1429} = 0.1429
u¬E (a,{a,b}) = max{u¬E (a,a) ,u¬E (a,b)} = max{1, 0.1429} = 1

Similar with Dempster combination rule, the combination rule is discussed under two cases:
complete information and incomplete information.

Definition 8. (complete information). Let D1 and D2 be two numbers over Ω with∑
A⊆Ω D1 (A) = 1 and

∑
A⊆Ω D2 (A) = 1, the combination of D1 and D2, indicated by

D = D1
⊙
D2, is defined by

D (A) =




0, A = ∅
1

1−KD
(
∑

B∩C=A
u¬E (B,C) D1 (B) D2 (C) +∑

B∪C=A,B∩C=∅
u¬E (B,C) D1 (B) D2 (C)), A 6= ∅

(13)

with
KD =

∑
B∩C=∅

(1 −u¬E (B,C) )D1 (B) D2 (C) (14)

Table 2: The fuzzy number of fuzzy variables

Fuzzy Variables Fuzzy Numbers
a (1, 2, 3)
b (2, 3, 4)
c (3, 4, 5)



678 B. Liu, Y. Deng

It is worth mentioning that this combination rule can be degenerated to the classical Demp-
ster’s rule if u¬E = 0 for any B∩C = ∅. a numerical example is used to illustrate the combination
of two D numbers under the complete information situation as follows:

Example 9. There are two D numbers over Ω = {a,b}:

D1 (a) = 0.4, D1 (b) = 0.6

D2 (a) = 0.9, D2 (b) = 0.1

And assume u¬E (a,b) = 0.1429. Thus, we have D′ = D1
⊙
D2 that

KD′ = (1 − 0.1429) × [0.4 × 0.1 + 0.6 × 0.9] = 0.4971

D′(a) =
1

1 − 0.4971
× 1 × 0.4 × 0.9 = 0.7159

D′(b) =
1

1 − 0.4971
× 1 × 0.6 × 0.1 = 0.1193

D′(a,b) =
1

1 − 0.4971
× 0.1429 × [0.4 × 0.1 + 0.6 × 0.9] = 0.1648

Definition 10. (incomplete information). Let D1 and D2 be two numbers over Ω with∑
A⊆Ω D1 (A) < 1 and∑
A⊆Ω D2 (A) < 1, the combination of D1 and D2, defined by D = D1

⊙
D2 and calculated as

follows:

D(A) =

{
0, A = ∅
f (Q1,Q2)

Dt(A)∑
B⊆Ω Dt(B)

, A 6= ∅ (15)

with

Dt (A) =
∑

B∩C=A
u¬E (B,C) D1 (B) D2 (C) +

∑
B∪C=A,B∩C=∅

u¬E (B,C) D1 (B) D2 (C) ,∀ A ∈ Ω

and
Q1 =

∑
A⊆Ω

D1(A), Q2 =
∑
A⊆Ω

D2(A) (16)

where f (Q1,Q2) is a function satisfying 0 ≤ f (Q1,Q2) ≤ max{Q1,Q2}, f (Q1,Q2) = 1 if Q1 = 1
and Q2 = 1.

Next, a simple example is used to illustrate the combination process of D numbers according
to Definition 10.

Example 11. There are two D numbers over Ω = {a,b}:

D1 (a) = 0.7, D1 (b) = 0.2

D2 (a) = 0.5, D2 (b) = 0.3



Risk Evaluation in Failure Mode and Effects Analysis Based on D Numbers Theory 679

Assume u¬E (a,b) = 0.1429, and let f (Q1,Q2) = Q1 ×Q2. Thus, we have D = D1
⊙
D2 that

f (Q1,Q2) = (0.7 + 0.2) × (0.5 + 0.3) = 0.72
Dt (a) = 0.7 × 0.5 = 0.35
Dt (b) = 0.2 × 0.3 = 0.06

Dt (a,b) = 0.1429 × (0.7 × 0.3 + 0.2 × 0.5) ≈ 0.044

Thus, ∑
B⊆Ω

D1(B) = 0.35 + 0.06 + 0.044 = 0.454

D (a) = f (Q1,Q2)
Dt (a)∑

B⊆Ω
D1(B)

≈ 0.555

D (b) = f (Q1,Q2)
Dt (b)∑

B⊆Ω
D1(B)

≈ 0.095

D (a,b) = f (Q1,Q2)
Dt (a,b)∑

B⊆Ω
D1(B)

≈ 0.070

If there are n D numbers expressed as D1, D2, · · · , Dn, whose weights are w1,w2, · · · , wn,
satisfying

∑n
i=1 wi = 1. At first, the average D number among D1,D2, · · · ,Dn is defined as

D̄ (A) =
n∑
i=1

wiDi(A),∀A ⊆ Ω. (17)

Then, the result of combination D1,D2, · · · ,Dn is acquired by combining average D number D̄
with itself n− 1 times.

D = D̄
⊙

D̄
⊙

· · ·
⊙

D̄ (18)

where
⊙

is the combination rule given in Definition 8 and Definition 10.

3 Proposed method

Failure mode and analysis (FMEA) is an efficient technology to identify and remove potential
faults, errors and risk from systems. In FMEA, risk evaluation is a significant procession. Tradi-
tionally, risk priority number (RPR) is used to evaluate risk, which is calculated by multiplying
the grades of occurrence assessment (O), severity assessment (S), and detection assessment (D).
Although classical RPN is easy to use for its concise form, it is still criticized for its weaknesses.
For example, traditional RPN does not take the weights of three risk factors into consideration.
Besides, different combination of risk factors might acquire the same RPN, however, the potential
risk might be totally different so that they might have different priorities. So far, many methods
are presented. Nevertheless, most of them not only have rather complex algorithm, but also do
not consider the non-exclusiveness between two rankings in assessment.

With the aim of solving these problems, in this paper, a novel method to risk evaluation
based on D numbers theory is proposed. First of all, each assessment rating is treated as a fuzzy



680 B. Liu, Y. Deng

variable which is represented as a fuzzy number. Then, non-exclusive degree between two ratings
can be calculated. Next, for each failure mode, the assessments of experts are aggregated by D
numbers theory. The assessments are treated as D numbers and then are combined by using D
numbers combination rule. Furthermore, RPNs are calculated for ranking the failure mode. Last
but not the least, when coming to the same RPN of some failure modes, a variable, named risk
coefficient, is defined to rank the failure modes with the same RPN. The large risk coefficient is,
the more important degree it is supposed to be assigned, referring to the failure mode should be
more priority to be corrected.

Step 1. Make sure the triangular fuzzy numbers for each rankings in assessment and
calculate the non-exclusive degrees by 12.

Step 2. The assessments of experts are regarded as D numbers, that is, for each failure
mode, the assessments of each expert is constructed as a D number. Therefore, the aggregation
of experts’ assessment of each failure mode by combining the corresponding D numbers. If
information is complete, the equations in Definition 8 are used to the combination. If the
information is incomplete, the equations in Definition 10 are applied to the combination. What
is more, if different experts have different weights, it is supposed to put the 17 into use.

Step 3. For the results of Step 2, use 5 to calculate pignistic probability transformation
(PPT).

Step 4. Calculate RPN. Firstly, calculate the mathematical expectation of each assessment.
Then, use Eq. (7) to calculate the RPN of each failure mode.

Step 5. Calculate risk coefficient.

Definition 12. Let s be the standard deviation of the ratings of three assessments, defined as
risk coefficient.

As mentioned above, different combination of risk factors might acquire the same RPN,
however, the potential risk might be totally different. In this paper, risk coefficient, the standard
deviation of the ratings of three assessments, is used to evaluate such kinds of failure modes.
As a matter of fact, this method is reasonable. For example, there are two failure modes, the
assessments of which are shown in Table 3.

Table 3: The assessments of three risk factors

occurrence assessment severity assessment detection assessment
FM1 1 10 6
FM2 2 6 5

Apparently, two failure modes have the same RPN. However, as is shown in Table 3, com-
pared with FM2, FM1 has the better grade in occurrence assessment but performs worse in
severity and detection assessment. Therefore, FM1 should be more priority to be corrected.
When observing and analyzing the data, it is not difficult to find that the distribution of FM2’s
data is more concentrated, which has smaller standard deviation. Besides, it is worth mention-
ing that standard deviation is usually used to measure the risk in financial field. Thus, it is
reasonable that the standard deviation of the ratings of three assessments is applied to measure
the risk in risk evaluation.

Step 6. Rank the failure modes though RPN. The larger RPN is, the more priority failure
mode is supposed to be corrected. If the failure modes have the same RPN, their rankings depend
on risk coefficient defined in Definition 12. The failure modes with the lager risk coefficient are
assumed to be more significant and should be given higher priorities.



Risk Evaluation in Failure Mode and Effects Analysis Based on D Numbers Theory 681

Table 4: The results of the risk evaluation

Item Rating of risk factor
Expert 1 Expert 2 Expert 3
O S D O S D O S D

1 3:40% 7 2 3:90% 7 2 3:80% 7 2
4:60% 4:10% 4:20%

2 2 8 4 2 8:70% 4 2 8 4
9:30%

3 1 10 3 1 10 3 1 10 3
4 1 6:80% 3 1 6 3:70% 1 6 3

7:20% 2:30%
5 1 3 2:50% 1 3 1:70% 1 3:60% 1

1:50% 2:30% 2:40%
6 2 6 5 2 6 5 2 6 5
7 1 7 3 1 7 3 1 7 3
8 3 5:60% 1 3 5:80% 1 3 5:80% 1

6:40% 6:20% 7:20%
9 2:90% 10:60% 4 2:75% 10:90% 4 2:80% 10:90% 4

1:10% 9:40% 1:25% 9:10% 1:20% 9:10%
10 1 10 6 1 10 6 1 10 6
11 1 10 5 1 10 5 1 10 5
12 1 10 6:60% 1 10 5:80% 1 10 6:70%

5:40% 4:20% 5:30%
13 1 10 5:80% 1 10 5 1 10 5

4:20%
14 1 10 6 1 10 6:80% 1 10 6

7:20%
15 2 7:95% 3 2 7 3 2 7 3:70%

6:5% 4:30%
16 2:90% 4 3 2:75% 4 3 2:80% 4 3:80%

1:10% 1:25% 1:20% 2:20%
17 2 5:90% 3 2 5:90% 3 2 5:60% 3

6:10% 6:10% 6:40%

4 Numerical example

In order to demonstrate the effectiveness and superiority of the proposed method, a numer-
ical example in [77] is solved in this section. Supposing there are three experts who evaluate
17 failure modes and identify the ratings of the three risk factors. The assessment results are
expressed in Table 4. In this illustrative example, the weights of experts and risk factors are
supposed to be equal to 1. Taking the failure mode 1 for example, the detailed computing is
expressed as follows:

Step 1. The ratings are represented by triangular fuzzy numbers listed in Table 5 and shown
in Figure 3. From Table 4, the occurrence assessments of three experts are (3 : 40%, 4 : 60%),
(3 : 90%, 4 : 10%) and (3 : 80%, 4 : 20%). Therefore, according to the Eq. (12), the non-



682 B. Liu, Y. Deng

0 1 2 3 4 5 6 7 8 9 10

 Assessment Ranking

0

1

 F
u

z
z
y

 M
e
m

b
e
r
sh

ip
 D

e
g

r
e
e

Figure 3: Graphically presentation of fuzzy variables in Table 5

Table 5: The fuzzy numbers of the assessment ratings

Assessment Ratings Fuzzy Numbers
1 (0, 1, 2)
2 (1, 2, 3)
3 (2, 3, 4)
4 (3, 4, 5)
5 (4, 5, 6)
6 (5, 6, 7)
7 (6, 7, 8)
8 (7, 8, 9)
9 (8, 9, 10)
10 (9, 10, 10)

exclusiveness degrees are calculated as follows:

u¬E (3, 3) = 1, u¬E (4, 4) = 1, u¬E (3,{3, 4}) = 1

u¬E (4,{3, 4}) = 1, u¬E (3, 4) =
0.25

1 + 1 − 0.25
≈ 0.1429

Step 2. Construct D numbers and combine D numbers. According to Section 3, corre-
sponding to the evaluations of three experts, three D numbers are constructed as follows:

D1 (3) = 0.4, D1 (4) = 0.6

D2 (3) = 0.9, D2 (4) = 0.1

D3 (3) = 0.8, D3 (4) = 0.2

Then, because these D numbers have complete information, the combination among D1, D2 and
D3 is calculated by 10-14, that is D1

⊙
D2
⊙
D3.

At first, calculate D′ = D1
⊙
D2:

KD′ = (1 − 0.1429) × [0.4 × 0.1 + 0.6 × 0.9] = 0.4971

D′ (3) =
1

1 − 0.4971
× 1 × 0.4 × 0.9 = 0.7159

D′ (4) =
1

1 − 0.4971
× 1 × 0.6 × 0.1 = 0.1193

D′ (3, 4) =
1

1 − 0.4971
× 0.1429 × [0.4 × 0.1 + 0.6 × 0.9] = 0.1648



Risk Evaluation in Failure Mode and Effects Analysis Based on D Numbers Theory 683

0 2 4 6 8 10 12 14 16 18
0

2

4

6

8

10

12

14

16

18

The proposed method

The method of J. Yang et al.

Figure 4: The risk priority of the failure modes using two methods

Next, calculate D = D′
⊙
D3:

KD = (1 − 0.1429) × [0.7159 × 0.2 + 0.1193 × 0.8] = 0.2045

D (3) =
1

1 − 0.2045
× [1 × 0.7159 × 0.8 + 1 × 0.1648 × 0.8] = 0.8857

D (4) =
1

1 − 0.2045
× [1 × 0.1193 × 0.2 + 1 × 0.1648 × 0.2] = 0.0714

D (3, 4) =
1

1 − 0.2045
× 0.1429 × [0.7159 × 0.2 + 0.1193 × 0.8] = 0.0429

Step 3. Calculate pignistic probability transformation (PPT) of the result of Step 2 by 5

BetP (3) = 0.8857 +
0.0429

2
≈ 0.9071BetP (4) = 0.8857 +

0.0429

2
≈ 0.0929

Step 4. The mathematical expectation of the occurrence assessment is 3 × 0.9071 + 4 ×
0.0929 = 3.0929. Besides, three experts have the same evaluation to the severity assessment (S)
and detection assessment (D) of failure mode 1, what is more, the evaluation are real numbers,
7 and 2, without any uncertainty. Thus, in these two assessments, the results through Step 1 to
4 are still 7 and 2.

Step 5. Calculate the RPN of failure mode 1 (RPN1) by 7.

RPN1 = O1 ×S1 ×D1 = 3.0929 × 7 × 2 = 43.3006

Step 6. Calculate the risk coefficient of the failure mode 1.

x̄1 =
3.0929 + 7 + 2

3
≈ 4.0310

s1 =

√
1

3
· [(3.0929 − 4.0310)2 + (7 − 4.0310)2 + (2 − 4.0310)2] ≈ 2.6287

The data of other failure modes are treated through Step 1 to 6 as mentioned above. Then,
according to the RPNs and risk coefficients, the risk priorities of the failure modes are obtained,
which are shown in Table 6. Meanwhile, in order to demonstrate the effectiveness of the proposed
method, the results are compared with that of the J. Yang et al.’s method [77].

As shown in Table 6, the results of two methods are similar. Apart from failure mode 6,
11 and 13, other failure modes have the same risk priority rankings in both two methods. In
addition, it is indicated that the five of highest risk priority rankings are failure mode 9, 2, 10,



684 B. Liu, Y. Deng

Table 6: The results of the risk evaluation

Failure mode RPN Risk coefficient
The rankings of
the proposed

method

The rankings of
the J. Yang et al.’s

method
1 43.3006 2.6287 9 9
2 64 3.0551 2 2
3 30 4.7258 12 12
4 18 2.5166 15 15
5 3.0726 1.1478 17 17
6 60 2.0817 5 3
7 21 3.0551 14 14
8 15.0657 2.0110 16 16
9 78.10103376 4.1593 1 1
10 60 4.5092 3 3
11 50 4.5092 7 6
12 51.452 4.5047 6 6
13 50 4.5092 7 6
14 60 4.5092 3 3
15 42 2.6458 10 10
16 23.5152 1.0203 13 13
17 30.4038 1.5643 11 11

14, and 6, which refers that these 5 faults are most likely to occur. Furthermore, in both two
methods, failure mode 16, 7, 4, 8, and 5 have the five of lowest priorities, indicating that these
5 failures are almost impossible to happen.

Figure 4 shows the comparison of risk priorities of two methods, in which the ranking is on
the abscissa axis while the failure mode is on the vertical axis. As shown in Figure 4, two curves
have the similar trend, which indicates that the proposed method is as effective as J. Yang et
al.’s method.

Nevertheless, the results also reflect the different evaluations using two methods, which
precisely demonstrates the superiority of the proposed method. As seen in Figure 4, using J.
Yang et al.’s method, failure mode 11, 12, 13 have the same risk priority, while failure mode 12
has the larger risk priority compared with failure mode 11 and failure mode 13 in the proposed
method. Because the highest rating of failure mode 12 in detection assessment is 6, which is
higher than that of failure mode 11, 13 in detection assessment. Thus, failure mode 12 is obviously
supposed to have higher risk priority compare with failure mode 11 and failure mode 13. What
is more, in J. Yang et al.’s method, failure mode 6, 10, 14 have the same risk priority, but in
the proposed method, failure mode 6 has the lower risk priority compared with failure mode 10
and failure mode 14. Although these failure modes have the same RPN, compared with failure
mode 10 and failure mode 14, failure mode 6 has two lower risk ratings in severity assessment
and detection assessment and merely a lager rating in occurrence assessment. Therefore, failure
mode 6 is supposed to have the lower risk priority compared with failure mode 10 and failure
mode 14.



Risk Evaluation in Failure Mode and Effects Analysis Based on D Numbers Theory 685

5 Conclusion

In this paper, a novel method to risk evaluation in failure mode and effects analysis based on
D numbers theory is proposed. In the proposed method, the application of the D numbers not
only aggregates the fuzzy assessment in risk evaluation, but also takes the non-exclusiveness into
account. Besides, the shortcomings of RPN are overcome successfully. Furthermore, the numer-
ical example has demonstrated that the proposed method achieves less computation complexity
compared with most existing method to risk evaluation in FMEA. In conclusion, the proposed
method is an advanced and efficient method to risk evaluation in FMEA.

Acknowledgement

The work is partially supported by National Natural Science Foundation of China (Grant
No. 61573290,61973332).

Conflict of Interest

The authors declare no conflict of interest.

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