INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
ISSN 1841-9836, 13(2), 205-220, April 2018.

New Failure Mode and Effects Analysis based on D Numbers
Downscaling Method

B. Liu, Y. Hu, Y. Deng

Baoyu Liu
College of Infomation Science and Technology
Jinan University, Guangzhou, China
Baoyu.L@hotmail.com

Yong Hu
Big Data Decision Institute,
Jinan University, Tianhe, Guangzhou, 510632, China
henryhu200211@163.com

Yong Deng*
1. Big Data Decision Institute,
Jinan University, Tianhe, Guangzhou 510632, China
2. 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 extensively applied to pro-
cess potential faults in systems, designs, and products. Nevertheless, traditional
FMEA, classical risk priority number (RPN), acquired by multiplying the ratings
of occurrence, detection, and severity, risk assessment, is not effective to process
the uncertainty in FMEA. Many methods have been proposed to solve the issue but
deficiencies exist, such as huge computing quality and the mutual exclusivity of propo-
sitions. In fact, because of the subjectivity of experts, the boundary of two adjacent
evaluation ratings is fuzzy so that the propositions are not mutually exclusive. To
address the issues, in this paper, a new method to evaluate risk in FMEA based
on D numbers and evidential downscaling method, named as D numbers downscal-
ing method, is proposed. In the proposed method, D numbers based on the data
are constructed to process uncertain information and aggregate the assessments of
risk factors, for they permit propositions to be not exclusive mutually. Evidential
downscaling method decreases the number of ratings from 10 to 3, and the frame of
discernment from 210 to 23 , which greatly reduce the computational complexity. Be-
sides, a numerical example is illustrated to validate the high efficiency and feasibility
of the proposed method.
Keywords: Failure mode and effects analysis; Dempster-Shafer evidence theory;
basic belief assignment; belief function; risk priority number; D numbers; evidential
downscaling; aggregate assessment

1 Introduction

Failure mode and effects analysis (FMEA) was first developed to assess known and potential
faults and prevent them from happening in the 1960s [5]. It is an efficient and powerful technology
in engineering and management fields, including defining, identifying, and removing known and
potential faults, errors, and risk from the system, process, design as well as service. [1, 51].
Besides, in other fields, such as risk management, healthcare management and engineer design,
FMEA plays an important role [2, 8, 10, 11, 15, 24, 36]. Furthermore, FMEA not only identifies
factors that induce fault but also classifies the likelihood and severity of such fault [11].

Copyright © 2018 by CC BY-NC



206 B. Liu, Y. Hu, Y. Deng

A traditional method in FMEA is risk priority number (RPN), which is acquired by mul-
tiplying the grades of occurrence, severity, and detection. Thus, how to aggregate the assess-
ments of three risk factors is a key issue, especially when the evaluated information given by
experts is uncertain. These years, a large number of approaches have been proposed to im-
prove FMEA methodology [5, 11, 23, 26, 37, 43]. It’s inevitable to deal with uncertainty in real
world [20–22,29,44]. Due to the efficiency to handle linguistic information of fuzzy sets [13,18],
an intuitionistic fuzzy approach for FMEA is proposed in [15], which offers some advantages over
earlier models as it accounts for degrees of uncertainty in relationships among various criteria
or options, specifically when relations cannot be expressed in definite numbers. In addition,
an integrating hesitant 2-tuple linguistic term sets and an extended QUALIFLEX approach is
proposed by Liu H C et al. in [13]. Furthermore, as is known to all, D numbers [9] and grey
theory are two popular methods to process the uncertain information. Therefore, based on the
two theories, plenty of methods are presented [23].

Classical RPN is criticized due to its several shortcomings. For example, the importance
weights of three risk factors are taken into consideration. Besides, classical RPN can do nothing
about the uncertain information, etc. Aiming at these problems, a method which is effective to
dispose the uncertainty of assessment is proposed by Chin et al. [4]. However, the algorithm is too
complex computationally, the reason is that the numeric ratings of every risk factor are from 1 to
10 so that the number of frame of discernment is 210, which greatly increase the computational
load [11]. In other to reduce the computational complexity, an evidential downscaling method
is proposed in [11]. Nevertheless, the D-S combination theory it uses requests that propositions
are exclusive mutually. As a matter of fact, traditional FMEA ratings are obtained by subjective
judgment of the experts. Therefore, the boundary of two adjacent ratings is fuzzy, hence the
propositions are not actually mutually exclusive. For purpose of solving this problem, in this
paper, a new method to evaluate risk in FMEA based on D numbers and evidential downscaling
method, named as D numbers downscaling method, is proposed. In the proposed method, on
the one hand, the evidential downscaling is utilized to decrease the frame of discernment so that
greatly reduces the computational complexity. On the other hand, according to the data, D
number is constructed to processing uncertain information and aggregate the assessment of risk
factors because D numbers permit propositions that are not exclusive mutually.

The rest of the paper is organized as follows. Key concepts and previous theories are briefly
reviewed in Section 2. A new method to evaluate risk based on D numbers downscaling method
is proposed in Section 3. A numerical example is illustrated to show the feasible of the proposed
method in Section 4. A brief conclusion is drawn in Section 5.

2 Preliminaries

2.1 Risk priority number

The real systems are very complicated to be modelled [14,27,46]. Risk priority number (RPN)
is a traditional and typical method to model and evaluate risk in FMEA of the complicated
systems, which is acquired by multiplying the grades of occurrence assessment (O), severity
assessment (S), and detection assessment (D). That is

RPN = OSD,

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, as expressed in Tables 1 to 3 [11].



New Failure Mode and Effects Analysis based on D Numbers Downscaling Method 207

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.

However, as a traditional method for FMEA, classical RPN has been controversial all the
time for many reasons, and the most important ones are listed as follows:

• The three risk factors have the same importance in RPN, but in practical applications of
FMEA, they might be treated with different weight.

• Other factors are not taken into consideration, such as economy factor.

• Different combination of risk factors might obtain the same RPN, nevertheless, the potential
risk might be totally different.

• The mathematical formula for calculating RPN lacks the scientific basis. There is not any
proof to clarify the reason why O, S and D should be multiplied to obtain RPN.

• In fact, the scores of the three factors are difficult to be determined accurately. Therefore,
FMEA experts usually provide different types of assessment information, some of which
are uncertain and incomplete data.

2.2 Dempster-Shafer evidence theory

Dempster-Shafer evidence theory (D-S evidence theory) constructs a basic probability as-
signment(BPA) in the frame of discernment. Through combining BPAs the imprecise and
uncertain information can be fused [5]. With its rapid development, it is regarded as an im-
portant method that is extensively applied in many fields such as complex networks and sys-
tems [8, 10, 12, 19, 45, 50, 52], multisource information fusion [6, 7, 17, 28, 41, 42, 49], uncertainty
modelling [1,2,16,48,53], pattern recognition [30,31] and Imprecise payoff [34,35].

Definition 1. Let Θ = {H1,H2, · · · ,HN} be a finite nonempty set, which is composed of N
mutually exclusive and exhaustive elements. Denote P(Θ) as the power set composed of 2N

elements of Θ. The BBAs function is defined as a mapping of the power set P(Θ) to a number
between 0 and 1, that is, m : P(Θ)− > [0, 1], and which satisfies the following conditions:

m(∅) = 0; (1)∑
A⊆P(Θ)

m(A) = 1. (2)

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

Definition 2. Two bodies of evidence A and B of Θ can be used to calculate the belief level for
some new hypothesis C shown as follows: The measure of conflict K that is also called as the
conflict coefficient between A and B is given as:

K =
∑

A∩B=∅,∀A,B⊆Θ
mi(A)mi′(B); (3)

and the mass function after combination is:

m(C) =



∑

A∩B=C,∀A,B⊆Θ mi(A)mi′(B)

1 −K
, when A 6= ∅

0, when A = ∅,
(4)



208 B. Liu, Y. Hu, Y. Deng

where K reflects the conflict between two bodies of evidence. Absolutely, 0 ≤ K ≤ 1. K = 0
shows the absence of conflict between two bodies of evidence. K = 1 shows complete conflict
between mi and mi′. When K = 1, the Dempster ’s rule of combination is no longer applicable.

2.3 The evidential downscaling method

An evidential downscaling method is proposed in [11], which is based on Euclidean distance
from multi-scale to construct the BBAs. The new method , decreasing the frame of discernment
from 210 to 23, is shown as follows:

Step 1. Calculate the group assessment into a crisp number with weighted average method.

Step 2. Construct the BBAs by Definition 2 with the Euclidean distances between group assessment
and multi-scale ratings. When calculating the distance, an evidential downscaling method
is presented to divide the rating into three scales: ‘10’ for ‘bad’, ‘1’ for ‘good’, and ‘5’ for
‘bad or good’ because rating 10 expresses the most serious degree, rating 1 expresses that
failure is almost impossible and rating 5 expresses the moderate degree, which decreases
the number of frame of discernment from 210 to 23.
The BBAs can be constructed as follows:

m({B}) =
d(G)

d(G) + d(B,G) + d(G)
, (5)

m({G}) =
d(B)

(d(G) + d(B,G) + d(G)
, (6)

m({B,G}) = 1 −m({B}) −m({G}). (7)

Step 3. Combine the BBAs of risk factors by Dempster’s combination rule.

Step 4. Acquire the risk priority according to the aggregation of ‘bad’. Let m({B})OSD, m({G})OSD
and m({B,G})OSD be the aggregation of three risk factors for ‘bad’, ‘good’, and ‘bad or
good’. The final assessment of ‘bad’ and ‘good’ can be calculated as follows:

m({B})′OSD = m({B})OSD +
m({B,G})OSD

2
, (8)

m({G})′OSD = m({G})OSD +
m({B,G})OSD

2
. (9)

2.4 D numbers [9]

D numbers is an effective method to deal with uncertain information, which overcomes the
shortcomings of Dempster-Shafer (D-S) theory. Nowadays, D number is extensively used in
many fields such as dependence assessment, emergency management, and aggregating opera-
tor [33,54,55]. It can be defined as follows:

Definition 3. 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, (10)



New Failure Mode and Effects Analysis based on D Numbers Downscaling Method 209

where ∅ is an empty set and B is a subset of Ω. More importantly, different from the concept of
frame of discernment in D-S theory, the elements of set Ω do not require to be mutually exclusive
in D numbers. If

∑
B⊆Ω D(B) = 1, the information is considered to be complete; otherwise, the

information is regarded to be incomplete.

For a set Ω = {b1,b2, · · · ,bi, · · · ,bn}, where bi 6= bj, if i 6= j. Then, a special D number can
be expressed by

D({b1}) = v1,

D({b2}) = v2,

· · · · · · ,

D({bi}) = vi,

· · · · · · ,

D({bn}) = vn

or D = {(b1,v1), (b2,v2), · · · , (bi,vi), · · · , (bn,vn)}, where vi > 1 and
∑n

i=1 vi ≤ 1. For the de-
tailed information about D numbers combination rule, please refer [9]. A combination rule, a
kind of add operation, is proposed to combine two D numbers.

3 The proposed method

As a traditional method for FMEA, classical RPN is acquired by multiplying the rating of
three risk factors. Nevertheless, classical RPN is criticized for several disadvantages mentioned
in 2.1. With the aim of overcoming these shortcomings, a large number of methods to evaluate
risk for FMEA are proposed. However, existing methods either have a huge computing quantity,
such as the Chinś method, or require the propositions to be mutually exclusive, like the method
proposed by Du Y et al.. As a matter of fact, traditional ratings are divided by the subjective
judgments of the experts which are based on the individual experience. Therefore, the boundary
of two adjacent ratings is fuzzy, which means that two propositions in FMEA are not mutually
exclusive. Hence, it is obviously unreasonable that Dempster-Shafer combination theory is used
to aggregate assessment in the evidential downscaling method proposed in [11]. Aiming at
these problems, a new method to evaluate risk in FMEA based on D numbers and evidential
downscaling method is proposed. Based on the three risk factors as well, the proposed method,
for one thing, decreases the number of the frame of discernment from 210 to 23 by making use
of the evidential downscaling method, which greatly reduce the computational complexity. For
another, D numbers are utilized to manage the uncertain information. Because D numbers
allow the propositions to not be exclusive, it is reasonable that D numbers are used to process
uncertain information and aggregate the assessments. Furthermore, in consideration of the fuzzy
information given by the experts on different failure modes, weighted averages are calculated to
substitute them in the proposed method. Suppose there are N failure modes and M experts,
the specific steps are shown as follows: The failures with higher values of ‘bad’ are assumed to
be more important and should be given higher priorities.

Step 1. Calculate the mathematical expectation of the score given by experts for evaluating each
risk factor by Eq. (11).



210 B. Liu, Y. Hu, Y. Deng

Definition 4. Let Si jl be the mathematical expectation of the score S of failure mode i
given by expert j in the assessment of risk factor l, it is calculated as follows:

S
i j
l =




S, if S is integer
a× c% + b×d%, if S is (a : c% , b : d%) with c% + d% = 100%

a× c% +
1

9
× (55 −a) × (1 − c%), if S is (a : c%) but 0 < c% < 100%

5.5, if S is missing
(11)

where i = 1, · · · ,N, j = 1, · · · ,M, l = O,S,D.

Step 2. Calculate the Euclidean distances between group assessment and multi-scale ratings. Con-
sistent with the evidential downscaling method mention in 2.3, then the new score SGi jl
of each factor of each failure mode after downscaling can be obtained as follows:

d(B)
i j
l = |S

i j
l − 10|, (12)

d(G)
i j
l = |S

i j
l − 1|, (13)

d(B,G)
i j
l = |S

i j
l − 5|, (14)

SG
i j
l =

d(B)
i j
l

(d(G)
i j
l + d(B,G)

i j
l + d(G)

i j
l )

, (15)

where i = 1, · · · ,N, j = 1, · · · ,M, l = O,S,D.

Step 3. Construct D numbers on the basis of SGi jl in the light of Definition 5.

Definition 5. Let DEjFi be the D number of failure mode i of experts j, which stands for
the three assessments of experts j towards failure mode i, is modeled as follows:

D
Ej
Fi

= {(bijO,vO), (b
ij
S ,vS), (b

ij
D,vD)} (16)

with

b
i j
l = λj ×SG

i j
l (17)

where λj is the weight for expert j, and vl (l = O,S,D) is the weight of the risk factor
standing for the importance of it in FMEA. In addition, i = 1, · · · ,N, j = 1, · · · ,M,
l = O,S,D.

Step 4. According to the processes of DFi numbers mentioned in [9] and Definition 6, for each
failure mode, the combination of D numbers can be calculated and the I(D) of each failure
mode can be obtained as well. Most importantly, the failure modes with the lower values
of I(D) are assumed to be more important and should be given higher priorities.

Definition 6. Let DFi be the D number of failure mode i (i = 1, · · · ,N) aggregate the
assessment of M experts, thus, it can be calculate as follows:

DFi = D
E1
Fi
⊕DE2Fi ⊕···⊕D

Ej
Fi
⊕···⊕DEMFi (18)

Figure 1 is the sub-flowsheet of the proposed method, that is, the procedure of downscaling and
constructing D number. Figure 2 is the general flow-chart.



New Failure Mode and Effects Analysis based on D Numbers Downscaling Method 211

Figure 1: The procedure of downscaling and constructing D number

Figure 2: The general flow-chart of the proposed method

Table 1: Occurrence assessment by FMEA team members

Failure
mode Expert1 Expert2 Expert3 Expert4 Expert5
1 1 1 1 1
2 1 : 50%, 2 : 50% 1 1 1 1
3 2 2 : 90% 2 2 2
4 8 8 8 : 80%, 9 : 20% 8 8
5 6 6 6 6 6
6 2 13 2 2 2-3
7 2 2 2 9 2
8 1 1 : 75%, 2 : 25% 1 1 1
9 3 3 3 3 3
10 1 : 80%, 2 : 20% 1 1 1 1−2 : 85%, 3 : 15%
11 4 4 4 3−4 : 75%, 5 : 25% 4
12 9 9 9 9 9
13 8 8 : 80% 8 8 8
14 3 3 4 3 3
15 3 3 3 3 3 : 70%, 4 : 30%
16 1 1 1 1 1
17 3−5 : 90%, 6 : 10% 4 4 4 4
18 2 2 2 : 90% 2 2
19 7 7 7 7 7:80%
20 9 9 9 7 : 30%, 8−9 : 70% 9
21 9 8 − 9 9 9 9



212 B. Liu, Y. Hu, Y. Deng

Table 2: Severity assessment by FMEA team members

Failure
mode Expert1 Expert2 Expert3 Expert4 Expert5
1 7 : 20%, 8 : 80% 8 8 6−7 : 50%, 8−9 : 50% 8
2 8 8 8 8
3 7 − 9 : 90% 8 6-8 8 8
4 8 8 8 8 7 − 9 : 80%
5 8 7 − 9 : 90% 8 8 8
6 8 8 8 6-8 8
7 9 : 75%, 8 : 25% 9 9 9 9
8 4 4 4 4 : 50%, 5 : 50% 3−5 : 75%, 6−7 : 25%
9 2 2 2 2 2
10 2 2 1−2 : 60%, 3−4 : 60% 2 2
11 2 2-3 2 2 2
12 3 3 3 : 60%, 4 : 40% 3 3
13 2−3 : 80%, 3−4 : 20% 3 3 3 3
14 7 8 7 7 7
15 3 3 3 3 3
16 8 8 8 8
17 8 4 8 8 8
18 7 7 7 7 7
19 2 1−2 : 75%, 2−3 : 25% 2 2 2
20 8 8 8 8
21 3 3 3 3 3

4 Numerical example

Suppose that there are 5 experts evaluating risk in FMEA, who give their assessments on the
three risk factors of 21 failure modes as shown in Table 1 to 3. The weights for the five experts
are assumed to be 0.3, 0.3, 0.2, 0.1 and 0.1.

Take failure mode 1 of expert 1 for example. As shown in Table 1, three ratings the expert 1
gives are 1, (7:20%, 8:80%), and 3. Thus, according to Eq. (11), the mathematical expectation
of score are 1, 7.8, and 3.

Then, the Euclidean distances between it and ‘bad’, ‘good’ and ‘bad or good’ are calculating
as in Table 4.

Therefore, the new scores can be obtained as follows:

SG1 1O =
d(B)1 1O

d(G)1 1O + d(B,G)
1 1
O + d(G)

1 1
O

=
9

9 + 0 + 4
≈ 0.6923.

Table 3: Detection assessment by FMEA team members

Failure
mode Expert1 Expert2 Expert3 Expert4 Expert5
1 3 3 3 3 3 : 90%
2 3 3 3 3 3
3 4 4 4 4 3−4 : 80%, 5−6 : 20%
4 5 5 5 5 5
5 6 6 6 : 85%, 7 : 15% 6 6
6 1 1 : 85%, 2 : 15% 2 1
7 3 2 2 1−2 : 75%, 3−4 : 25% 2
8 3 3 3 : 80%, 4 : 20% 3 3
9 3 3 3 − 4 : 60%, 5 : 40% 3 3
10 4 4 4 4 4
11 3 : 70%, 5 : 30% 3 3 3
12 7 7 7 7 7
13 6 6 6 5-7 6
14 4 4 4 4 4
15 4 4 : 95% 4 4 4
16 3 3 3 3 3
17 5 5 5 5 5
18 7 6-8 7 7 7
19 4 4 4 8 − 9 : 90% 4
20 4 : 60% 9 9 9 9
21 6 6 4-6 6 4 : 25%, 5 − 7 : 75%



New Failure Mode and Effects Analysis based on D Numbers Downscaling Method 213

Table 4: Euclidean distances between it and "bad", "good" and "bad or good"

Bad Good Bad or Good
d(B)1 1O = |1−10| = 9, d(G)

1 1
O = |1−1| = 0, d(B, G)

1 1
O = |1−5| = 4;

d(B)1 1S = |7.8−10| = 2.2, d(G)
1 1
S = |7.8−1| = 6.8, d(B, G)

1 1
S = |7.8−5| = 2.8;

d(B)1 1D = |3−10| = 7, d(G)
1 1
D = |3−1| = 2, d(B, G)

1 1
D = |3−5| = 2;

0 5 10 15 20 25

failure mode

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

th
e
 S

G
 o

f 
th

e
 o

cc
u
rr

e
n
ce

 a
ss

e
ss

m
e
n
t

expert1
expert2
expert3
expert4
expert5

Figure 3: The new score of Occurrence assessment

SG1 1S =
d(B)1 1S

d(G)1 1S + d(B,G)
1 1
S + d(G)

1 1
S

=
2.2

2.2 + 6.8 + 2.8
≈ 0.1864.

SG1 1D =
d(B)1 1D

d(G)1 1D + d(B,G)
1 1
D + d(G)

1 1
D

=
7

7 + 2 + 2
≈ 0.6364.

The data of the other three assessments is treated in the same way, among which the result
of occurrence assessment with the new scores are shown in Figure 3. As shown in Figure 3, the
higher initial rating tends to the lower new score while the lower rating tends to the higher one.

Then, the first element b can be calculated by multiply the weight of the expert 1 and the
new score:

b1 1O = λ1 ×SG
1 1
O = 0.3 × 0.6923 = 0.2077.



214 B. Liu, Y. Hu, Y. Deng

Table 5: The representation of D numbers for failure mode 1 (F1)

F1 D numbers

Expert 1 DE1F1 = {(0.2077,
1

3
), (0.0559,

1

3
), (0.1909,

1

3
)}

Expert 2 DE2F1 = {(0.2077,
1

3
), (0.0500,

1

3
), (0.1909,

1

3
)}

Expert 3 DE3F1 = {(0.0947,
1

3
), (0.0333,

1

3
), (0.1273,

1

3
)}

Expert 4 DE4F1 = {(0.0692,
1

3
), (0.0217,

1

3
), (0.0636,

1

3
)}

Expert 5 DE5F1 = {(0.0692,
1

3
), (0.0167,

1

3
), (0.0627,

1

3
)}

b1 1S = λ1 ×SG
1 1
S = 0.3 × 0.6923 = 0.0559.

b1 1D = λ1 ×SG
1 1
D = 0.3 × 0.6923 = 0.1909.

With the same process, the b of other D numbers can be obtained. Next, the data is treated
using the D numbers. It needs to be emphasized that the weight of each rich factor is treated as
the second parameter v. It is worth mentioning that the judgments for the relative importance
of each criterion are various from different expert.

In order to compare with other methods which do not take it into consideration and prove
the feasibility of the proposed method as well, the importance of three factors are assumed to
be same. That is, v is supposed to be identically equal to 1

3
.

Therefore, the assessment result being disposed through above process are expressed in the
forms of D numbers. According to the data of expert 1, a D number DE1F1 is constructed, where
DE1F1 = {(0.2077,

1
3
), (0.0559, 1

3
), (0.1909, 1

3
)}. Similarity to DE1F1 , According to the data of expert

2, expert 3, expert 4 and expert 5, four D numbers DE2F1 , D
E3
F1

, DE4F1 and D
E5
F1

are generated.
Table 4 shows these D numbers. In the same method, for failure mode Fi, and expert Ej, each
D number DEjFi can be constructed, too.

For each failure mode, all experts’ data represented by D numbers are combined according to
the combination rule of D numbers mentioned in [9]. Take the failure mode 1 (F1) for example,
the integrated assessment of three experts is the aggregation of DE1F1 , D

E2
F1

, DE3F1 , D
E4
F1

and DE5F1 ,

DF1 = D
E1
F1
⊕DE2F1 ⊕D

E3
F1
⊕DE4F1 ⊕D

E5
F1
.

In accordance with the processes of DFi numbers mentioned in [9], DF1 is calculated. Therefore,
the I(DF1 ) of DF1 can be obtained. Using the same method, DFi and I(DFi) are acquired, which
are shown in Table 5. In the meantime, according to I(DFi)(i = 1, 2, · · · , 21), the ranking of
21 failure modes are obtained, where the failure modes with the lower values of I(D) are given
higher priorities.

In Table 6, the risk priority rankings of the evidential downscaling method are shown. Com-
pared with the ranking of the evidential downscaling method, the proposed method’s is similar.
As seen from Table 6, failure mode 3, failure mode 9, failure mode 12, failure mode 13, failure
mode 15, failure mode 18, failure mode 20 and failure mode 21 have the same risk priority rank-
ing in both methods. Besides, it is indicated that the five of highest risk priority rankings are



New Failure Mode and Effects Analysis based on D Numbers Downscaling Method 215

0 5 10 15 20 25

failure mode

0

5

10

15

20

25
th

e
 r

a
n

k 
o

f 
tw

o
 m

e
th

o
d

s
the proposed method
the evidential downscaling method

Figure 4: The comparison of risk priority ranking by two method

Figure 5: The occupation of ratings between two method

failure mode 20, 4, 5, 12 and 21, which means that these 5 faults are most likely to occur in both
two methods. Moreover, in both two methods, the five of the lowest priorities are failure mode
8, 9, 10, 11 and 15, indicating that these 5 failures are supposed to be.

Figure 4 shows the comparison of risk priority rankings by the proposed method and the



216 B. Liu, Y. Hu, Y. Deng

Table 6: The comparison of risk priority ranking by two methods

Failure
mode Result The rank of proposed

method
The evidential downscaling
method

1 0.06634 12 10
2 0.07464 16 15
3 0.06597 11 11
4 0.04524 3 2
5 0.04210 2 3
6 0.06309 9 14
7 0.05789 7 9
8 0.08779 20 21
9 0.08743 19 19
10 0.08616 18 20
11 0.08824 21 18
12 0.04596 4 4
13 0.05568 6 6
14 0.06670 13 12
15 0.08518 17 17
16 0.06896 15 16
17 0.0633 10 7
18 0.06064 8 8
19 0.06781 14 13
20 0.02350 1 1
21 0.05345 5 5

evidential downscaling method, in which the X-axis shows the rankings and Y-axis shows the
failure modes. As is shown in Figure 4, it indicates that the rank result generated by proposed
method has similar trend with the evidential downscaling method, which proves the proposed
method is valid. However, in consideration that it is out of reality to request the propositions
to be exclusive mutually, the proposed method is more reasonable. In 5, the occupation of
ratings between two methods for each failure mode is shown. From Figure 5, the edge of the
area around with different color is almost near the midline of the graph, which proves that the
proposed method similarly has a good effect for risk assessment.

5 Conclusion

The conventional FMEA method has been criticized for its deficiencies especially in the
evaluation of risks of failure mods and computation of classical RPN. A large number of new
methods for FMEA have been proposed. Nevertheless, some deficiencies still exist in these
methods, such as huge computing quality. Although the evidential downscaling method presented
by Du Y et al. is capable to greatly reduce the amount of calculation, it is not reasonable that
the propositions are requested to be exclusive mutually. Therefore, a new method to evaluate
risk in FMEA based on D numbers downscaling method is proposed with the purpose of solving
this problem.

Obviously, the proposed method can handle the uncertain information well in FMEA. Com-
pared with the traditional RPN, the proposed method not only can dispose the scores of three
risk factors even if they are uncertain, but also takes into consideration the relative importance
weigths of risk factors. Therefore, the result of the proposed method is more reliable. In com-
parison to the existing method, the proposed has more succinct calculation formulas, which has
a smaller amount of calculation so that it is more worthy of promotion. Furthermore, because of



New Failure Mode and Effects Analysis based on D Numbers Downscaling Method 217

the use of D numbers, the propositions are not required to be exclusive mutually, which proves
the proposed method is more reasonable than the evidential downscaling method. In conclusion,
the proposed method is a feasible and efficient method to risk assessment in failure mode and
effects analysis. In addition, although the proposed method takes the weights of three factors
into consideration, it does not make sure the method to obtain the weights, which would be the
further exploration.

Acknowledgment

The authors greatly appreciate the reviewers’ suggestions and the editor’s encouragement.
The work is partially supported by National Natural Science Foundation of China (Grant Nos.
61573290, 61503237).

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