A modified method for risk evaluation in failure modes and effects analysis of aircraft turbine rotor blades

Abstract

Failure mode and effects analysis is an important methodology, which has been extensively used to evaluate the potential failures, errors, or risks in a system, design, or process. The traditional method utilizes the risk priority number ranking system. This method determines the risk priority number by multiplying failure factor values. Dempster–Shafer evidence theory has been combined with failure mode and effects analysis due to its effectiveness in dealing with uncertain and subjective information. However, since the risk evaluation of different experts may be different and some even conflict with each other, Dempster’s combination rule may become invalid. In this article, for better performance of application of evidence theory in failure mode and effects analysis, a modified method is proposed to reassign the basic believe assignment taking into consideration a reliability coefficient based on evidence distance. We illustrate several numerical examples and use the modified method to obtain the risk priority numbers for risk evaluation in failure modes of aircraft engine rotor blades. The results show that the proposed method is more reasonable and effective for real applications.

Keywords

Dempster–Shafer evidence theory failure mode and effects analysis risk priority number basic believe assignments reliability coefficient aircraft turbine rotor blades

Introduction

Failure mode and effects analysis (FMEA) is an efficient approach used to define, identify, and eliminate known or potential failures and errors from system, design, or process.^1,2 This methodology is widely applied to several industry fields such as aerospace, engineering design, and manufacturing to gather important information.^3–6 FMEA can not only help analysts to identify known and potential failure modes and their causes and effects but also help them to prioritize the identified failure modes. It can help designers to adjust the existing programs, take recommended measures to reduce likelihood of failures, decrease probability of failure rates, and avoid potential accidents.^7,8

In general, the priority of a failure mode is determined by the risk priority number (RPN), which is obtained by multiplying the values of occurrence (O), severity (S), and detection (D) of a failure mode. The three factors O, S, and D are all evaluated using ratings (also called rankings or scores) from 1 to 10. The failures with higher RPNs are assumed to be more important and should be given higher priority. FMEA has been proved to be one of the most important early preventative initiatives during the design stage of a system, product, process, or service. However, the RPN-based approach has been extensively criticized for various reasons:^3,9,10

The relative importance among risk factors is not taken into consideration in determining the priority of the failures. The three factors are assumed to be of equal importance, but this may not be the case in practical applications.

The RPN considers only three factors mainly in terms of safety. Other possible influencing factors such as economical aspects are ignored.

It is usually difficult or even impossible to give exact numerical evaluations of practically intangible quantities associated with the risk factors.

Different sets of O, S, and D ratings may produce exactly the same value of RPN, although their hidden risk implications may be totally different. For example, two different failures with the O, S, and D values of 1, 4, 9 and 1, 6, 6, respectively, have the same RPN value of 36.

The RPN elements have many duplicate numbers. Although 1000 numbers are assumed to be produced from the product of O, S, and D, only 120 of them are unique.

Several techniques were developed in order to improve the FMEA methodology.^7,9,11–14 Chang¹⁵ proposed a more general RPN method considering situation parameters and relationships. Sankar and Prabhu¹⁰ proposed the modified approach for prioritization of failures. This method defines the new ratings from 1 to 1000 to represent possible severity–occurrence–detection states and provides a much more sensitive ranking process to quantitate risk factors. Fuzzy set theory¹⁶ is widely used in many applications due to its efficiency to model fuzzy information:^17–23 for example, environmental impact assessment,¹⁷ decision-making,¹⁹ and marketing mix planning.²¹ It is also applied in FMEA under uncertain environment.^24,25 For example, Bowles and Pelaez¹³ described a fuzzy logic–based approach for prioritizing failure in a system failure mode. Pillay and Wang⁹ proposed a fuzzy rule–based approach with the fuzzy rule base and gray relation theory.

Based on the above review, the limitations of the traditional RPN have been investigated. In real applications, like the analysis of aircraft turbine rotor blades, risk factors occurrence (O), severity (S), and detection (D) are difficult to be determined precisely. Besides, FMEA is a group decision behavior and cannot be performed on an individual basis.^9,26 Considering their different expertise and backgrounds, various uncertainties are present in FMEA expert group’s subjective and qualitative assessments, such as imprecision, fuzziness, incompleteness, and conflict. Therefore, one key issue of FMEA is the representation and handling of various types of uncertainties in evaluating failure modes with respect to the risk factors. Liu et al.²⁶ proposed a new risk priority model, which is based on a more effective representation of uncertain information, called D numbers, and an improved gray relational analysis method, gray relational projection (GRP). In the proposed model, the assessment results of risk factors given by FMEA expert group are expressed and modeled by D numbers. The GRP method is used to determine the risk priority order of the failure modes that have been identified.

The Dempster–Shafer (D–S) evidence theory has been employed to quantify the imprecision and uncertainty in reliability and failure analysis. Yang et al.⁷ employed the D–S evidence theory to analyze different failure modes and applied it to the risk priority evaluation of failure modes of rotor blades of an aircraft engine. However, when experts give different and precise values of the risk evaluation factors, the basic believe assignments (BBAs) constructed by Yang et al.’s⁷ method become highly conflicting evidence which cannot be fused by Dempster’s combination rule directly.²⁷ Su et al.²⁸ proposed a modification of Yang et al.’s⁷ method, using uncertain reasoning method based on Gaussian distribution. However, this method can only be used to deal with conflicting situations. To solve this problem, many approaches have been proposed and can be divided into two fields: one is to improve the combination rule²⁹ and the other is to modify the original evidences.³⁰ In this article, we proposed a method based on the second idea, that is, reassign the BBAs using reliability coefficient with evidence distance³¹ to solve the problem of conflicting. This method is completely driven by data, and the weights of evidences can be obtained dynamically. It has the merits of practicability compared with the methods developed in Yang et al.⁷ and Su et al.²⁸ and simplicity compared with the approach proposed by Liu et al.²⁶

The rest of the article is organized as follows. Section “Theoretical background” recalls the basic theoretical background of failure risk analysis and outlines the most important methods available in the literature. Section “Description of the new method for risk evaluation” describes the new method developed in this study. In section “Test problems and discussion of results,” the validity of the proposed approach is tested in ad hoc designed examples and in a real problem. The study is briefly summarized in section “Conclusion.”

Theoretical background

In this section, we briefly introduce the basic concepts including RPN and D–S evidence theory.^32,33 Due to its efficiency to represent and fuse uncertain information, D–S evidence theory is widely used in many real systems,^34–43 such as credal classification³⁷ and reliability analysis.³⁸

RPN

In FMEA, the risk evaluation is determined using the RPN,^15,44 which is defined as follows

RPN = S \times O \times D

(1)

where S is the severity of a failure effect, O is the probability of occurrence of a failure mode, and D is the probability of a failure being detected. Each risk factor is rated from 1 to 10 as described in Tables 1 –3.

Table 1.

Traditional ratings for occurrence of a failure.¹⁰

Rating	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	1/15,000
2	Remote	1/150,000
1	Nearly impossible	≤1/1,500,000

Table 2.

Traditional ratings for severity of a failure.¹⁰

Rating	Effect	Severity of effect
10	Hazardous without warning	Very high severity ranking when a potential failure mode affects safe vehicle operation and/or involves noncompliance with government regulations without warning
9	Hazardous with warning	Very high severity ranking when a potential failure mode affects safe vehicle operation and/or involves noncompliance with government regulations with warning
8	Very high	Vehicle/item inoperable, with loss of primary function
7	High	Vehicle/item operable, but at reduced level of performance. Customer dissatisfied
6	Moderate	Vehicle/item operable, but comfort/convenience item(s) inoperable. Customer experiences discomfort
5	Low	Vehicle/item operable, but comfort/convenience item(s) operable at reduced level of performance. Customer experiences some dissatisfaction
4	Very low	Cosmetic defect in finish, fit, and finish/squeak or rattle item that does not conform to specifications. Defect noticed by most customers
3	Minor	Cosmetic defect in finish, fit, and finish/squeak or rattle item that does not conform to specifications. Defect noted by average customers
2	Very minor	Cosmetic defect in finish, fit, and finish/squeak or rattle item that does not conform to specifications. Defect noted by discriminating customers
1	None	No effect

Table 3.

Traditional ratings for detection of a failure.¹⁰

Rating	Detection	Criteria
10	Absolutely impossible	Design control will not and/or cannot detect a potential cause/mechanism and subsequent failure mode; or there is no design control
9	Very remote	Very remote chance the design control will detect a potential cause/mechanism and subsequent failure mode
8	Remote	Remote chance the design control will detect a potential cause/mechanism and subsequent failure mode
7	Very low	Very low chance the design control will detect a potential cause/mechanism and subsequent failure mode
6	Low	Low chance the design control will detect a potential cause/mechanism and subsequent failure mode
5	Moderate	Moderate chance the design control will detect a potential cause/mechanism and subsequent failure mode
4	Moderately high	Moderately high chance the design control will detect a potential cause/mechanism and subsequent failure mode
3	High	High chance the design control will detect a potential cause/mechanism and subsequent failure mode
2	Very high	Very high chance the design control will detect a potential cause/mechanism and subsequent failure mode
1	Almost certain	Design control will almost certainly detect a potential cause/mechanism and subsequent failure mode

D–S evidence theory

The D–S evidence theory, as introduced by Dempster³² and then developed by Shafer,³³ has emerged from their works on statistical inference and uncertain reasoning. This theory is widely applied to decision-making,^45–49 information fusion,⁵⁰ and uncertain information processing.⁵¹

Definition 2.1

Let $Θ$ be a set of mutually exclusive and collectively exhaustive events, indicated by

Θ = {θ_{1}, θ_{2}, \dots, θ_{i}, \dots, θ_{N}}

(2)

where set $Θ$ is called a frame of discernment. The power set of $Θ$ is indicated by $2^{Θ}$ , namely

2^{Θ} = {\emptyset, {θ_{1}}, \dots {θ_{N}}, {θ_{1}, θ_{2}}, \dots, {θ_{1}, θ_{2}, \dots θ_{i}}, \dots, Θ}

(3)

Definition 2.2

A mass function is a mapping m from $2^{Θ}$ to [0, 1], formally defined by

m : 2^{Θ} \to [0, 1]

(4)

which satisfies the following condition

m (\emptyset) = 0

(5)

\sum_{A \in 2^{Θ}} m (A) = 1

(6)

When m(A) > 0, A, which is a member of the power set, is called a focal element of the mass function.

Definition 2.3

In D–S evidence theory, a mass function is also called a BBA. Let us assume there are two BBAs, operating on two sets of propositions B and C, respectively, indicated by m₁ and m₂. The Dempster’s³² combination rule is used to combine them as follows

m (A) = {\begin{matrix} 0, & A \neq \emptyset \\ \frac{1}{1 - K} \sum_{B \cap C = A} m_{1} (B) m_{2} (C), & A = \emptyset \end{matrix}

(7)

K = \sum_{B \cap C = \emptyset} m_{1} (B) m_{2} (C)

(8)

In equations (7) and (8), K reflects the conflict between the two BBAs m₁ and m₂.

Evidence distance

The evidence distance assesses the degree of inconsistency of two evidences: the higher the conflict, the higher the value of evidence distance. If two evidences are completely opposite, the distance will be equal to 1.

Definition 2.4

Let $m_{1}$ and $m_{2}$ be two BBAs on the same frame of discernment $Θ$ , containing N mutually exclusive and exhaustive hypotheses. The distance between $m_{1}$ and m₂ is represented by³¹

d_{BBA} (m_{1}, m_{2}) = \sqrt{\frac{1}{2} {({\vec{m}}_{1} - {\vec{m}}_{2})}^{T} \underset{=}{D} ({\vec{m}}_{1} - {\vec{m}}_{2})}

(9)

where ${\vec{m}}_{1}$ and ${\vec{m}}_{2}$ are the respective BBAs in vector notation, and $\underset{=}{D}$ is an $2^{N} \times 2^{N}$ matrix whose elements are $D (A, B) = | A \cap B | / | A \cup B |$ , where $A, B \in 2^{Θ}$ .

Pignistic probability transformation

Definition 2.5

Let m be a BBA on $Θ$ . The resulting pignistic probability transformation for the singleton $x \in Θ$ is given by⁵²

BetP ({x}) = \sum_{x \in A \subseteq Θ} \frac{1}{| A |} \frac{m (A)}{1 - m (\emptyset)}, m (\emptyset) \neq 1

(10)

where $| A |$ is the number of elements of A.

Available methods

In this section, we briefly review the methods developed by Yang et al.⁷ and Su et al.²⁸ for risk priority evaluation of a failure mode.

In Yang et al.,⁷ the weight of an expert is considered and the modified evidence theory is used to combine the different information from multiple experts. Their approach is shown briefly as follows:

Step 1: simplify the discernment frame as

\begin{matrix} Θ_{ij} = (\min X |_{X \subseteq Θ_{ij}}, \min X |_{X \subseteq Θ_{ij}} + 1, \dots, \max X |_{X \subseteq Θ_{ij}}) \\ i = O, S, D; j = 1, 2, 3, \dots, N \end{matrix}

(11)

where $\min X |_{X \subseteq Θ_{ij}}$ and $\max X |_{X \subseteq Θ_{ij}}$ , respectively, are the minimum and the maximum of the rank of the jth failure mode to the ith risk factor in the evaluations of L experts.

Step 2: construct the belief function

m (\emptyset) = 0

(12)

\sum_{X \in 2^{Θ}} m (X) = 1

(13)

where Θ = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10). The function $m (X)$ is a mapping: $m (X) : 2^{Θ} \to [0, 1]$

Step 3: consider the weight and modify the BBAs, then combine the modified BBAs with evidence theory under the new discernment frame. The modified BBA is represented as ${\bar{m}}_{ij}^{l} (•)$

{\bar{m}}_{ij}^{l} (C) = w_{i}^{l} \times m_{ij}^{l} (C), C \subset Θ_{ij}, C \neq Θ_{ij}

(14)

{\bar{m}}_{ij}^{l} (Θ_{ij}) = 1 - \sum_{B \subset Θ_{ij}} w_{i}^{l} \times m_{ij}^{l} (B), B \neq Θ_{ij}

(15)

where i = O, S, D; j = 1, 2,…, N, N is the number of failure modes; l = 1, 2,…, L, L is the number of experts.

The new combination rule of D–S evidence theory can be written as

m_{i j}^{l k} (C) = ({\bar{m}}_{i j}^{l} \oplus {\bar{m}}_{i j}^{k}) (C) = {\begin{array}{l} 0, & C = \emptyset \\ \frac{\sum_{X \cap Y = C, \forall X, Y \in Θ_{i j}} (w_{i}^{l} \cdot m_{i j}^{l} (X)) \times (w_{i}^{k} \cdot m_{i j}^{k} (Y))}{1 - \sum_{X \cap Y = \emptyset, \forall X, Y \in Θ_{i j}} (w_{i}^{l} \cdot m_{i j}^{l} (X)) \times (w_{i}^{k} \cdot m_{i j}^{k} (Y))}, & C \neq \emptyset \end{array}

(16)

The synthetic effects of all sources of evidence from L different experts can be represented by M_ij

\begin{array}{l} M_{i j} = {\bar{m}}_{i j}^{1} \oplus {\bar{m}}_{i j}^{2} \oplus {\bar{m}}_{i j}^{3} \oplus \dots \oplus {\bar{m}}_{i j}^{L} \\ = (((({\bar{m}}_{i j}^{1} \oplus {\bar{m}}_{i j}^{2}) \oplus {\bar{m}}_{i j}^{3}) \oplus \dots) \oplus {\bar{m}}_{i j}^{L}) \end{array}

(17)

Step 4: calculate the mean value of RPN.

The RPN is a discrete random variable with several different ratings and the corresponding probabilities. Suppose RPN has several ratings $({RPN}_{j}^{1}, \dots, {RPN}_{j}^{q})$ with respective probabilities $(P ({RPN}_{j}^{1}), \dots, P ({RPN}_{j}^{q}))$ for the jth failure mode. In order to compare the overall risk of each failure mode, the mean value of RPN (MVRPN) is defined by calculating the expectation of RPN

MVRP N_{j} = E (RP N_{j}) = \sum_{q} ({RPN}_{j}^{q}) \cdot P ({RPN}_{j}^{q})

(18)

where $E (\cdot)$ is an operator for calculating mathematical expectation. q indicates the total number of rating levels involved in the evaluations. For example, when experts used levels 6 and 9, there were four levels entailed by computations: 6, 7, 8, and 9.

Su et al.²⁸ modified the above described approach. If the situation cannot be figured out because different and precise values of evaluation factors are given by the experts, the BBAs may become highly conflicting evidence which cannot be fused by Dempster’s³² combination rule. In order to solve this problem, the following modified method was developed by Su et al.²⁸

Definition 2.6

Assume that $m (X)$ expresses the probability given by experts in support of proposition X, and R denotes the numeric scale (rating) from 1 to 10. If X is a single possible rating of the ith risk factor for the jth failure mode, then the BBA will be constructed as

m (\emptyset) = 0

(19)

m (R) = f (R; X, {(0.5)}^{2}) = \frac{1}{0.5 \sqrt{2 π}} e^{- \frac{{(R - X)}^{2}}{2 \times {(0.5)}^{2}}}

(20)

The above equation defining a BBA is plotted in Figure 1. This modified method can be illustrated in the following example.

Figure 1.

Definition of BBA from equation (20).

Example 2.1

Two experts give their opinions on the risk factor “severity of a failure effect (S)” of failure mode 1. Suppose the weight of each expert is equal, and $w_{i}^{l}$ is equal to 1. Expert 1 evaluated it as “significant,” which can be represented as $S (6, 100 %)$ ; Expert 2 evaluated it as “major,” which can be represented as $S (7, 100 %)$ . In this case, the discernment frames can be constructed as $Θ_{ij} = {5, 6, 7, 8}$ , and the modified belief function can be obtained using equation (20)

\begin{matrix} Expert 1 : m_{S 1}^{1} (5) = 0.1, m_{S 1}^{1} (6) = 0.8, m_{S 1}^{1} (7) = 0.1 \\ Expert 2 : m_{S 1}^{2} (6) = 0.1, m_{S 1}^{2} (7) = 0.8, m_{S 1}^{2} (8) = 0.1 \end{matrix}

Suppose set $X, Y \subseteq Θ_{S 1}$ , the combined belief function can be obtained using equation (16) between Expert 1 and Expert 2

\begin{matrix} m_{S 1}^{12} (5) & = {\bar{m}}_{S 1}^{1} (X) \oplus {\bar{m}}_{S 1}^{2} (Y) \\ = \frac{\sum_{X \cap Y = 5, \forall X, Y \subseteq Θ_{S 1}} {\bar{m}}_{S 1}^{1} (X) \times {\bar{m}}_{S 1}^{2} (Y)}{1 - \sum_{X \cap Y = \emptyset, \forall X, Y \subseteq Θ_{S 1}} {\bar{m}}_{S 1}^{1} (X) \times {\bar{m}}_{S 1}^{2} (Y)} \\ = \frac{0}{m_{s 1}^{1} (6) \times m_{s 1}^{2} (6) + m_{s 1}^{1} (7) \times m_{s 1}^{2} (7)} \\ = \frac{0}{0.16} \\ = 0 \end{matrix}

\begin{matrix} m_{S 1}^{12} (6) & = {\bar{m}}_{S 1}^{1} (X) \oplus {\bar{m}}_{S 1}^{2} (Y) \\ = \frac{\sum_{X \cap Y = 6, \forall X, Y \subseteq Θ_{S 1}} {\bar{m}}_{S 1}^{1} (X) \times {\bar{m}}_{S 1}^{2} (Y)}{1 - \sum_{X \cap Y = \emptyset, \forall X, Y \subseteq Θ_{S 1}} {\bar{m}}_{S 1}^{1} (X) \times {\bar{m}}_{S 1}^{2} (Y)} \\ = \frac{m_{s 1}^{1} (6) \times m_{s 1}^{2} (6)}{m_{s 1}^{1} (6) \times m_{s 1}^{2} (6) + m_{s 1}^{1} (7) \times m_{s 1}^{2} (7)} \\ = \frac{0.8 \times 0.1}{0.16} \\ = 0.5 \end{matrix}

\begin{matrix} m_{S 1}^{12} (7) & = {\bar{m}}_{S 1}^{1} (X) \oplus {\bar{m}}_{S 1}^{2} (Y) \\ = \frac{\sum_{X \cap Y = 7, \forall X, Y \subseteq Θ_{S 1}} {\bar{m}}_{S 1}^{1} (X) \times {\bar{m}}_{S 1}^{2} (Y)}{1 - \sum_{X \cap Y = \emptyset, \forall X, Y \subseteq Θ_{S 1}} {\bar{m}}_{S 1}^{1} (X) \times {\bar{m}}_{S 1}^{2} (Y)} \\ = \frac{m_{s 1}^{1} (7) \times m_{s 1}^{2} (7)}{m_{s 1}^{1} (6) \times m_{s 1}^{2} (6) + m_{s 1}^{1} (7) \times m_{s 1}^{2} (7)} \\ = \frac{0.8 \times 0.1}{0.16} \\ = 0.5 \end{matrix}

\begin{matrix} m_{S 1}^{12} (8) & = {\bar{m}}_{S 1}^{1} (X) \oplus {\bar{m}}_{S 1}^{2} (Y) \\ = \frac{\sum_{X \cap Y = 8, \forall X, Y \subseteq Θ_{S 1}} {\bar{m}}_{S 1}^{1} (X) \times {\bar{m}}_{S 1}^{2} (Y)}{1 - \sum_{X \cap Y = \emptyset, \forall X, Y \subseteq Θ_{S 1}} {\bar{m}}_{S 1}^{1} (X) \times {\bar{m}}_{S 2}^{1} (Y)} \\ = \frac{0}{m_{s 1}^{1} (6) \times m_{s 1}^{2} (6) + m_{s 1}^{1} (7) \times m_{s 1}^{2} (7)} \\ = \frac{0}{0.16} \\ = 0 \end{matrix}

The mean value of the factor “severity of a failure effect (S)” of RPN can be determined using equation (18) as

\begin{array}{l} M V R P N_{S 1} = E (R P N_{S 1}) \\ = \sum_{q = 1}^{4} (R P N_{q}) \cdot P (R P N_{q}) = 0 + 6 \times 0.5 \\ + 7 \times 0.5 + 0 = 6.5 \end{array}

Description of the new method for risk evaluation

The two methods described in section “Available methods” account for the weight of an expert because different experts may provide imprecise or uncertain risk evaluations for a given failure mode. However, they do not evaluate weights of different experts which are all supposed to be equal to 1. This causes the following problem: when different and precise values of the risk evaluation factors are given by the experts, the BBAs constructed by their methods may become highly conflicting evidence which cannot be fused by the Dempster’s³² combination rule. Although some evidences can apparently be fused using the Dempster’s combination rule, the resulting BBAs are not logical; therefore, using the obtained BBAs to generate RPN lacks credibility. The following example can illustrate this case.

Example 3.1

Suppose two experts give their opinions upon the risk factor “occurrence of a failure (O)” of failure mode 1. Expert 1 evaluated it as O(6,100%) which means the probability of occurrence is “moderately high”; Expert 2 evaluated it as “very high” and can be represented as O(9,100%). The BBAs can be represented as

Expert 1 : m_{O 1}^{1} (6) = 1

Expert 2 : m_{O 1}^{2} (9) = 1

These two BBAs highly conflict with each other and cannot be fused by the Yang et al.’s⁷ method. Although the opinions of the two experts look similar, the method developed by Su et al.²⁸ neither can solve the above-mentioned problem. It can be explained in Figure 2 that the BBAs constructed by Su et al.’s method are as follows

Expert 1 : m_{O 1}^{1} (5) = 0.1, m_{O 1}^{1} (6) = 0.8, m_{O 1}^{1} (7) = 0.1

Expert 2 : m_{O 1}^{2} (8) = 0.1, m_{O 1}^{2} (9) = 0.8, m_{O 1}^{2} (10) = 0.1

Figure 2.

Definition of BBAs in Su et al.’s²⁸ method.

It can be seen that the resulting BBAs are still highly conflicting, and therefore this method cannot efficiently deal with this conflict.

The above example shows the limitations of the methods developed in Yang et al.⁷ and Su et al.²⁸ in solving problems characterized by highly conflicting evidence associated with different experts. Since the Dempster’s combination rule cannot be applied, it is necessary to find an effective way to utilize evidence theory in FMEA.

Our modified method can be described as follows:

Step 1: simplify the discernment frame according to equation (11).

Step 2: construct the belief function and then calculate the reliability coefficient k.

Definition 3.1

The average of evidence distance among multiple experts denoted as $d^{*}$ is defined as

d^{*} = \frac{1}{n (n - 1) / 2} \sum_{i < j} d_{ij}

(21)

$d_{ij}$ represents the evidence distance between ith expert and jth expert, and n is the number of experts.

Definition 3.2

k is the reliability coefficient also called discounting coefficient and it can be defined as

k = 2^{- d^{*}}

(22)

Since $d^{*}$ reflects the level of evidence inconsistency, it is reasonable to use the average evidence distance as a criterion in evidence fusion. For a certain failure mode, if the experts’ opinions are identical, that is $d^{*} = 0$ and k = 1, then we believe the evidence is quite believable. Thus, there is no need to adjust the BBAs. If two experts’ opinions are completely opposite and the distance of the two evidences is 1, then k is 0.5. We believe the two experts’ opinions are not reliable, then we give both of them a half discount and spare the remaining values to the group set.

From a mathematical point of view, the reliability coefficient k acts as a discounting coefficient, and therefore, the conflict between the evaluations of different experts can be reduced by it. Unlike $w_{i}^{l}$ coefficients defined in Yang et al.,⁷k is obtained according to the degree of inconsistency between the experts’ opinions. In other words, this method is completely driven by data. We just need to dynamically analyze the given data from experts or sensors. Therefore, this method is more practical and flexible.

Step 3: take the reliability coefficient k into consideration to modify the belief function; Dempster’s³² combination rule is then used for fusing multiple information.

The BBAs can be modified as follows

\begin{matrix} {\bar{m}}_{ij}^{l} (C) = k \times m_{ij}^{l} (C), C \subset Θ_{ij}, C \neq Θ_{ij} \\ m_{ij}^{l} (Θ_{ij}) = 1 - \sum_{B \subset Θ_{ij}} k \times m_{ij}^{l} (B), B \neq Θ_{ij} \end{matrix}

(23)

where i = O, S, D; l = 1, 2,…, L, L is the number of experts; j = 1, 2,…, N, N is the number of failure modes, and k is the reliability coefficient. The synthetic effects of all sources of evidence from L different experts can be represented by the $M_{ij}$ terms that are calculated with equation (17).

Step 4: calculate the MVRPN of the risk factors O, S, and D with equation (24).

Suppose the ith risk factor has several ratings $({RPN}_{ij}^{1}, \dots, {RPN}_{ij}^{q})$ for the jth failure mode, then the MVRPN of it is defined as follows

MVRP N_{ij} = \sum_{q} BetP ({RPN}_{ij}^{q}) \cdot ({RPN}_{ij}^{q})

(24)

where q indicates the total number of rating levels involved in the evaluations. $BetP (RPN)$ is the pignistic probability of a rating level and can be determined using equation (10).

Step 5: obtain the RPN using equation (1).

Test problems and discussion of results

Let us solve Problem 3.1 with the present methodology. The problem will be denoted as Example 4.1 in the rest of the article. The aim is to demonstrate that we are able to successfully deal with risk assessment problems characterized by highly conflicting experts’ evaluations.

Example 4.1

Expert 1 : m_{O 1}^{1} (6) = 1

Expert 2 : m_{O 1}^{2} (9) = 1

According to equations (9), (21), and (22), the average evidence $d^{*}$ and reliability coefficient k can be obtained

\begin{array}{l} d^{*} = d_{12} = \sqrt{\frac{1}{2} {({\vec{m}}_{1} - {\vec{m}}_{2})}^{T} \underset{=}{D} ({\vec{m}}_{1} - {\vec{m}}_{2})} = 1; \\ k = 2^{- 1} = 0.5 \end{array}

Then, the new BBAs can be represented using equation (23)

Expert 1 : \begin{matrix} {\bar{m}}_{O 1}^{1} (6) = k \times m_{O 1}^{1} (6) = 0.5 \times 1 = 0.5 \\ m_{O 1}^{1} (Θ) = 1 - {\bar{m}}_{O 1}^{1} (6) = 1 - 0.5 = 0.5 \end{matrix}

Expert 2 : \begin{matrix} {\bar{m}}_{O 1}^{2} (9) = k \times m_{O 1}^{2} (9) = 0.5 \times 1 = 0.5 \\ m_{O 1}^{2} (Θ) = 1 - {\bar{m}}_{O 1}^{2} (9) = 1 - 0.5 = 0.5 \end{matrix}

Suppose set $X, Y \subseteq Θ$ , then combined BBAs are calculated

\begin{matrix} m_{O 1}^{12} (6) & = {\bar{m}}_{O 1}^{1} (X) \oplus {\bar{m}}_{O 1}^{2} (Y) \\ = \frac{\sum_{X \cap Y = 6, \forall X, Y \in Θ} {\bar{m}}_{O 1}^{1} (X) \times {\bar{m}}_{O 1}^{2} (Y)}{1 - \sum_{X \cap Y = \emptyset, \forall X, Y \in Θ} {\bar{m}}_{O 1}^{1} (X) \times {\bar{m}}_{O 1}^{2} Y)} \\ = \frac{{\bar{m}}_{O 1}^{1} (6) \times m_{O 1}^{2} (Θ)}{1 - {\bar{m}}_{O 1}^{1} (6) \times {\bar{m}}_{O 1}^{2} (9)} \\ = \frac{0.5 \times 0.5}{1 - 0.25} \\ = 1 / 3 \end{matrix}

\begin{matrix} m_{O 1}^{12} (9) & = {\bar{m}}_{O 1}^{1} (X) \oplus {\bar{m}}_{O 1}^{2} (Y) \\ = \frac{\sum_{X \cap Y = 9, \forall X, Y \in Θ} {\bar{m}}_{O 1}^{1} (X) \times {\bar{m}}_{O 1}^{2} (Y)}{1 - \sum_{X \cap Y = \emptyset, \forall X, Y \in Θ} {\bar{m}}_{O 1}^{1} (X) \times {\bar{m}}_{O 1}^{2} Y)} \\ = \frac{m_{O 1}^{1} (Θ) \times {\bar{m}}_{O 1}^{2} (9)}{1 - {\bar{m}}_{O 1}^{1} (6) \times {\bar{m}}_{O 1}^{2} (9)} \\ = \frac{0.5 \times 0.5}{1 - 0.25} \\ 1 / 3 \end{matrix}

m_{O 1}^{12} (Θ) = 1 - \frac{2}{3} = \frac{1}{3}

In this case, the minimum and the maximum of the rank of the failure mode 1 to risk factor O are 6 and 9, and therefore, the simplified discernment frame can be constructed as {6, 7, 8, 9} according to equation (11). Then the pignistic probability of each rating can be calculated with equation (10)

BetP (6) = m_{O 1}^{12} (6) + \frac{m_{O 1}^{12} (Θ)}{4} = \frac{5}{12}

BetP (7) = \frac{m_{O 1}^{12} (Θ)}{4} = \frac{1}{12}

BetP (8) = \frac{m_{O 1}^{12} (Θ)}{4} = \frac{1}{12}

BetP (9) = m_{O 1}^{12} (9) + \frac{m_{O 1}^{12} (Θ)}{4} = \frac{5}{12}

The mean value of the factor “O” of RPN can be determined using equation (24)

\begin{matrix} MVRP N_{O 1} & = \sum_{q = 1}^{4} BetP ({RPN}_{O 1}^{q}) \cdot ({RPN}_{O 1}^{q}) \\ = 5 / 12 \times 6 + 1 / 12 \times 7 + 1 / 12 \times 8 + 5 / 12 \times 9 \\ = 7.5 \end{matrix}

If the two experts give their opinions on the risk factor “occurrence of a failure (O)” of failure mode 1 and evaluate it as O(6,100%), then the BBAs can both be represented as $m_{O 1} (6) = 1$ . The reliability coefficient k can be calculated as 1. In this case, the proposed method is consistent with Yang et al.⁷ The mean value of the factor “O” of RPN is 6.

Example 4.1 shows how the methods developed in Yang et al.⁷ and Su et al.²⁸ cannot solve risk assessment problems where evaluations given by different experts are highly conflicting. Actually, this situation can happen in many complicated FMEA, since it is difficult or impossible for experts to give exact numerical evaluations of practically intangible quantities associated with the risk factors. Conversely, Example 4.1 proves that the proposed method can overcome the problem highlighted by Example 3.1. Through Step 3, a nonzero value is assigned to the simplified group set $Θ$ , thus reducing conflicts between BBAs. Similarly, the problem of high conflicting evidence’s combination can be solved by the proposed method.

It should be noted that when conflicts between experts’ evaluations are not very evident, the Dempster’s rule can still be used in Yang et al.⁷ and Su et al.²⁸ However, solutions may be unreasonable in some cases as it will be illustrated by Example 4.2.

Example 4.2

Suppose three experts give their opinions upon the risk factor “detection of a failure ” (D) of failure mode 1. Expert 1 evaluated it as $D (5, 10 %; 6, 90 %)$ which means he or she considers the percentage of “moderate chance the design control will detect a potential cause/mechanism and subsequent failure mode” is 10% and the percentage of “low chance the design control will detect a potential cause/mechanism and subsequent failure mode” is 90%; Expert 2 evaluated it as $D (5, 20 %; 6, 80 %)$ which he or she believes the probability of situation “moderate” is 20% and “low” is 80%; Expert 3 evaluated it as $D (4, 80 %; 5, 20 %)$ which means the percentage of “moderately high” is 80% and “moderate” is 20%. In this case, the discernment frame can be simplified as Θ = {4, 5, 6}. The BBAs are represented as

Expert 1 : m_{D 1}^{1} (5) = 0.1, m_{D 1}^{1} (6) = 0.9

Expert 2 : m_{D 1}^{2} (5) = 0.2, m_{D 1}^{2} (6) = 0.8

Expert 3 : m_{D 1}^{3} (4) = 0.8, m_{D 1}^{3} (5) = 0.2

The reliability coefficient k can be calculated according to equations (9), (21), and (22)

d_{12} = \sqrt{\frac{1}{2} {({\vec{m}}_{1} - {\vec{m}}_{2})}^{T} \underset{=}{D} ({\vec{m}}_{1} - {\vec{m}}_{2})} = 0.1

d_{13} = \sqrt{\frac{1}{2} {({\vec{m}}_{1} - {\vec{m}}_{3})}^{T} \underset{=}{D} ({\vec{m}}_{1} - {\vec{m}}_{3})} = 0.85

d_{23} = \sqrt{\frac{1}{2} {({\vec{m}}_{2} - {\vec{m}}_{3})}^{T} \underset{=}{D} ({\vec{m}}_{2} - {\vec{m}}_{3})} = 0.8

d^{*} = \frac{(0.1 + 0.85 + 0.8)}{3} = 0.58

k = 2^{- d^{*}} = 0.67

Then, the modified BBAs using equation (23) is shown as follows:

Expert 1

{\bar{m}}_{D 1}^{1} (5) = k \times m_{D 1}^{1} (5) = 0.67 \times 0.1 = 0.07

{\bar{m}}_{D 1}^{1} (6) = k \times m_{D 1}^{1} (6) = 0.67 \times 0.9 = 0.6

m_{D 1}^{1} (Θ) = 1 - {\bar{m}}_{D 1}^{1} (5) - {\bar{m}}_{D 1}^{1} (6) = 1 - 0.07 - 0.6 = 0.33

Expert 2

{\bar{m}}_{D 1}^{2} (5) = k \times m_{D 1}^{2} (5) = 0.67 \times 0.2 = 0.13

{\bar{m}}_{D 1}^{2} (6) = k \times m_{D 1}^{2} (6) = 0.67 \times 0.8 = 0.54

\begin{array}{l} m_{D 1}^{2} (Θ) = 1 - {\bar{m}}_{D 1}^{2} (5) - {\bar{m}}_{D 1}^{2} (6) \\ = 1 - 0.13 - 0.54 = 0.33 \end{array}

Expert 3

{\bar{m}}_{D 1}^{3} (4) = k \times m_{D 1}^{3} (4) = 0.67 \times 0.8 = 0.54

{\bar{m}}_{D 1}^{3} (5) = k \times m_{D 1}^{3} (5) = 0.67 \times 0.2 = 0.13

m_{D 1}^{3} (Θ) = 1 - {\bar{m}}_{D 1}^{3} (4) - {\bar{m}}_{D 1}^{3} (5) = 1 - 0.54 - 0.13 = 0.33

Suppose set $X, Y \subseteq Θ$ , the combined belief function can be obtained

\begin{matrix} m_{D 1}^{12} (4) & = {\bar{m}}_{D 1}^{1} (X) \oplus {\bar{m}}_{D 1}^{2} (Y) \\ = \frac{\sum_{X \cap Y = 4, \forall X, Y \in Θ} {\bar{m}}_{D 1}^{1} (X) \times {\bar{m}}_{D 1}^{2} (Y)}{1 - \sum_{X \cap Y = \emptyset, \forall X, Y \in Θ} {\bar{m}}_{D 1}^{1} (X) \times {\bar{m}}_{D 1}^{2} (Y)} \\ = \frac{0}{1 - {\bar{m}}_{D 1}^{1} (5) \times {\bar{m}}_{D 1}^{2} (6) - {\bar{m}}_{D 1}^{1} (6) \times {\bar{m}}_{D 1}^{2} (5)} \\ = 0 \end{matrix}

\begin{matrix} m_{D 1}^{12} (5) & = {\bar{m}}_{D 1}^{1} (X) \oplus {\bar{m}}_{D 1}^{2} (Y) \\ = \frac{\sum_{X \cap Y = 5, \forall X, Y \in Θ} {\bar{m}}_{D 1}^{1} (X) \times {\bar{m}}_{D 1}^{2} (Y)}{1 - \sum_{X \cap Y = \emptyset, \forall X, Y \in Θ} {\bar{m}}_{D 1}^{1} (X) \times {\bar{m}}_{D 1}^{2} (Y)} \\ = \frac{{\bar{m}}_{D 1}^{1} (5) \times {\bar{m}}_{D 1}^{2} (5) + {\bar{m}}_{D 1}^{1} (5) \times m_{D 1}^{2} (Θ) + {\bar{m}}_{D 1}^{2} (5) \times m_{D 1}^{1} (Θ)}{1 - {\bar{m}}_{D 1}^{1} (5) \times {\bar{m}}_{D 1}^{2} (6) - {\bar{m}}_{D 1}^{1} (6) \times {\bar{m}}_{D 1}^{2} (5)} \\ = \frac{0.07 \times 0.13 + 0.07 \times 0.33 + 0.13 \times 0.33}{1 - 0.12} \\ = 0.09 \end{matrix}

\begin{matrix} m_{D 1}^{12} (6) & = {\bar{m}}_{D 1}^{1} (X) \oplus {\bar{m}}_{D 1}^{2} (Y) \\ = \frac{\sum_{X \cap Y = 6, \forall X, Y \in Θ} {\bar{m}}_{D 1}^{1} (X) \times {\bar{m}}_{D 1}^{2} (Y)}{1 - \sum_{X \cap Y = \emptyset, \forall X, Y \in Θ} {\bar{m}}_{D 1}^{1} (X) \times {\bar{m}}_{D 1}^{2} (Y)} \\ = \frac{{\bar{m}}_{D 1}^{1} (6) \times {\bar{m}}_{D 1}^{2} (6) + {\bar{m}}_{D 1}^{1} (6) \times m_{D 1}^{2} (Θ) + {\bar{m}}_{D 1}^{2} (6) \times m_{D 1}^{1} (Θ)}{1 - {\bar{m}}_{D 1}^{1} (5) \times {\bar{m}}_{D 1}^{2} (6) - {\bar{m}}_{D 1}^{1} (6) \times {\bar{m}}_{D 1}^{2} (5)} \\ = \frac{0.6 \times 0.54 + 0.6 \times 0.33 + 0.54 \times 0.33}{1 - 0.12} \\ = 0.79 \end{matrix}

m_{D 1}^{12} (Θ) = 1 - m_{D 1}^{12} (4) - m_{D 1}^{12} (5) - m_{D 1}^{12} (6) = 0.12

Then, the final consequence is

\begin{matrix} m_{D 1}^{123} (4) & = m_{D 1}^{12} (X) \oplus m_{D 1}^{3} (Y) \\ = \frac{\sum_{X \cap Y = 4, \forall X, Y \in Θ} m_{D 1}^{12} (X) \times {\bar{m}}_{D 1}^{3} (Y)}{1 - \sum_{X \cap Y = \emptyset, \forall X, Y \in Θ} m_{D 1}^{12} (X) \times m_{D 1}^{3} (Y)} \\ = \frac{m_{D 1}^{12} (Θ) \times {\bar{m}}_{D 1}^{3} (4)}{1 - m_{D 1}^{12} (4) \times {\bar{m}}_{D 1}^{3} (5) - m_{D 1}^{12} (5) \times {\bar{m}}_{D 1}^{3} (4) - m_{D 1}^{12} (6) \times {\bar{m}}_{D 1}^{3} (4) - m_{D 1}^{12} (6) \times {\bar{m}}_{D 1}^{3} (5)} \\ = \frac{0.12 \times 0.54}{1 - 0.56} \\ = 0.15 \end{matrix}

\begin{matrix} m_{D 1}^{123} (5) & = m_{D 1}^{12} (X) \oplus m_{D 1}^{3} (Y) \\ = \frac{\sum_{X \cap Y = 5, \forall X, Y \in Θ} m_{D 1}^{12} (X) \times {\bar{m}}_{D 1}^{3} (Y)}{1 - \sum_{X \cap Y = \emptyset, \forall X, Y \in Θ} m_{D 1}^{12} (X) \times m_{D 1}^{3} (Y)} \\ = \frac{m_{D 1}^{12} (5) \times {\bar{m}}_{D 1}^{3} (5) + m_{D 1}^{12} (Θ) \times {\bar{m}}_{D 1}^{3} (5) + m_{D 1}^{12} (5) \times {\bar{m}}_{D 1}^{3} (Θ)}{1 - m_{D 1}^{12} (4) \times {\bar{m}}_{D 1}^{3} (5) - m_{D 1}^{12} (5) \times {\bar{m}}_{D 1}^{3} (4) - m_{D 1}^{12} (6) \times {\bar{m}}_{D 1}^{3} (4) - m_{D 1}^{12} (6) \times {\bar{m}}_{D 1}^{3} (5)} \\ = \frac{0.09 \times 0.13 + 0.12 \times 0.13 + 0.09 \times 0.33}{1 - 0.56} \\ = 0.13 \end{matrix}

\begin{matrix} m_{D 1}^{123} (6) & = m_{D 1}^{12} (X) \oplus m_{D 1}^{3} (Y) \\ = \frac{\sum_{X \cap Y = 6, \forall X, Y \in Θ} m_{D 1}^{12} (X) \times {\bar{m}}_{D 1}^{3} (Y)}{1 - \sum_{X \cap Y = \emptyset, \forall X, Y \in Θ} m_{D 1}^{12} (X) \times {\bar{m}}_{D 1}^{3} (Y)} \\ = \frac{m_{D 1}^{12} (6) \times {\bar{m}}_{D 1}^{3} (Θ)}{1 - m_{D 1}^{12} (4) \times {\bar{m}}_{D 1}^{3} (5) - m_{D 1}^{12} (5) \times {\bar{m}}_{D 1}^{3} (4) - m_{D 1}^{12} (6) \times {\bar{m}}_{D 1}^{3} (4) - m_{D 1}^{12} (6) \times {\bar{m}}_{D 1}^{3} (5)} \\ = \frac{0.79 \times 0.33}{1 - 0.56} \\ = 0.60 \end{matrix}

\begin{matrix} m_{D 1}^{123} (Θ) = 1 - m_{D 1}^{123} (4) - m_{D 1}^{123} (5) - m_{D 1}^{123} (6) = 0.12 \end{matrix}

By equation (10), the pignistic probabilities can be calculated as follows

BetP (4) = m_{D 1}^{123} (4) + \frac{m_{D 1}^{123} (Θ)}{3} = 0.19

BetP (5) = m_{D 1}^{123} (5) + \frac{m_{D 1}^{123} (Θ)}{3} = 0.17

BetP (6) = m_{D 1}^{123} (6) + \frac{m_{D 1}^{123} (Θ)}{3} = 0.64

The mean value of the factor “D” of RPN can be determined using equation (24)

\begin{matrix} MVRP N_{D 1} = \sum_{q = 1}^{3} BetP ({RPN}_{D 1}^{q}) \cdot ({RPN}_{D 1}^{q}) \\ = 0.19 \times 4 + 0.17 \times 5 + 0.64 \times 6 = 5.45 \end{matrix}

The combined belief function obtained with the direct application of the Dempster’s combination rule is

m_{D 1}^{123} (4) = 0, m_{D 1}^{123} (5) = 1, m_{D 1}^{123} (6) = 0

Therefore, the final BBA obtained from the methods described in Yang et al.⁷ and Su et al.²⁸ is $m_{D 1}^{123} (5) = 1$ . However, this is not logical because experts thought moderate (i.e. rating 5) is possible but at a much lower level of probability.

The results obtained for Examples 4.1 and 4.2 demonstrate that the present approach improves the Yang et al.’s⁷ method, thus overcoming the limitation that conflicting evidence cannot be combined by Dempster’s combination rule. The following example will prove the efficiency of the proposed method in the case of non-conflicting evidence.

Example 4.3

Suppose three experts give their opinions upon the risk factor “detection of a failure” (D) of mode 1. Expert 1 evaluated it as $D (5, 10 %; 6, 90 %)$ which means he or she considers the percentage of “moderate chance the design control will detect a potential cause/mechanism and subsequent failure mode” is 10% and the percentage of “low chance the design control will detect a potential cause/mechanism and subsequent failure mode” is 90%; Expert 2 evaluated it as $D (5, 20 %; 6, 80 %)$ which he or she believes the probability of situation “moderate” is 20% and “low” is 80%; Expert 3 evaluated it as $D (5, 30 %; 6, 70 %)$ , similarly. In this case, the discernment frame can be simplified as Θ = {5, 6}. The BBAs can be represented as follows

Expert 1 : m_{D 1}^{1} (5) = 0.1, m_{D 1}^{1} (6) = 0.9

Expert 2 : m_{D 1}^{2} (5) = 0.2, m_{D 1}^{2} (6) = 0.8

Expert 3 : m_{D 1}^{3} (5) = 0.3, m_{D 1}^{3} (6) = 0.7

The reliability coefficient k can be calculated according to equations (9), (21), and (22)

d_{12} = \sqrt{\frac{1}{2} {({\vec{m}}_{1} - {\vec{m}}_{2})}^{T} \underset{=}{D} ({\vec{m}}_{1} - {\vec{m}}_{2})} = 0.1

d_{13} = \sqrt{\frac{1}{2} {({\vec{m}}_{1} - {\vec{m}}_{3})}^{T} \underset{=}{D} ({\vec{m}}_{1} - {\vec{m}}_{3})} = 0.2

d_{23} = \sqrt{\frac{1}{2} {({\vec{m}}_{2} - {\vec{m}}_{3})}^{T} \underset{=}{D} ({\vec{m}}_{2} - {\vec{m}}_{3})} = 0.1

d^{*} = \frac{(0.1 + 0.2 + 0.1)}{3} = 0.13

k = 2^{- d^{*}} = 0.9

Then, the modified BBAs using equation (23) is shown as follows:

Expert 1

{\bar{m}}_{D 1}^{1} (5) = k \times m_{D 1}^{1} (5) = 0.9 \times 0.1 = 0.09

{\bar{m}}_{D 1}^{1} (6) = k \times m_{D 1}^{1} (6) = 0.9 \times 0.9 = 0.81

m_{D 1}^{1} (Θ) = 1 - {\bar{m}}_{D 1}^{1} (5) - {\bar{m}}_{D 1}^{1} (6) = 1 - 0.09 - 0.81 = 0.1

Expert 2

{\bar{m}}_{D 1}^{2} (5) = k \times m_{D 1}^{2} (5) = 0.9 \times 0.2 = 0.18

{\bar{m}}_{D 1}^{2} (6) = k \times m_{D 1}^{2} (6) = 0.9 \times 0.8 = 0.72

m_{D 1}^{2} (Θ) = 1 - m_{D 1}^{2} (5) - m_{D 1}^{2} (6) = 1 - 0.18 - 0.72 = 0.1

Expert 3

{\bar{m}}_{D 1}^{3} (5) = k \times m_{D 1}^{3} (5) = 0.9 \times 0.3 = 0.27

{\bar{m}}_{D 1}^{3} (6) = k \times m_{D 1}^{3} (6) = 0.9 \times 0.7 = 0.63

m_{D 1}^{3} (Θ) = 1 - {\bar{m}}_{D 1}^{3} (5) - {\bar{m}}_{D 1}^{3} (6) = 1 - 0.27 - 0.63 = 0.1

Suppose set $X, Y \subseteq Θ$ , the combined belief function can be obtained

\begin{matrix} m_{D 1}^{12} (5) & = {\bar{m}}_{D 1}^{1} (X) \oplus {\bar{m}}_{D 1}^{2} (Y) \\ = \frac{\sum_{X \cap Y = 5, \forall X, Y \in Θ} {\bar{m}}_{D 1}^{1} (X) \times {\bar{m}}_{D 1}^{2} (Y)}{1 - \sum_{X \cap Y = \emptyset, \forall X, Y \in Θ} {\bar{m}}_{D 1}^{1} (X) \times {\bar{m}}_{D 1}^{2} (Y)} \\ = \frac{{\bar{m}}_{D 1}^{1} (5) \times {\bar{m}}_{D 1}^{2} (5) + {\bar{m}}_{D 1}^{1} (5) \times m_{D 1}^{2} (Θ) + {\bar{m}}_{D 1}^{2} (5) \times m_{D 1}^{1} (Θ)}{1 - {\bar{m}}_{D 1}^{1} (5) \times {\bar{m}}_{D 1}^{2} (6) - {\bar{m}}_{D 1}^{1} (6) \times {\bar{m}}_{D 1}^{2} (5)} \\ = \frac{0.09 \times 0.18 + 0.09 \times 0.1 + 0.18 \times 0.1}{1 - 0.21} \\ = 0.05 \end{matrix}

\begin{matrix} m_{D 1}^{12} (6) & = {\bar{m}}_{D 1}^{1} (X) \oplus {\bar{m}}_{D 1}^{2} (Y) \\ = \frac{\sum_{X \cap Y = 6, \forall X, Y \in Θ} {\bar{m}}_{D 1}^{1} (X) \times {\bar{m}}_{D 1}^{2} (Y)}{1 - \sum_{X \cap Y = \emptyset, \forall X, Y \in Θ} {\bar{m}}_{D 1}^{1} (X) \times {\bar{m}}_{D 1}^{2} (Y)} \\ = \frac{{\bar{m}}_{D 1}^{1} (6) \times {\bar{m}}_{D 1}^{2} (6) + {\bar{m}}_{D 1}^{1} (6) \times m_{D 1}^{2} (Θ) + {\bar{m}}_{D 1}^{2} (6) \times m_{D 1}^{1} (Θ)}{1 - {\bar{m}}_{D 1}^{1} (5) \times {\bar{m}}_{D 1}^{2} (6) - {\bar{m}}_{D 1}^{1} (6) \times {\bar{m}}_{D 1}^{2} (5)} \\ = \frac{0.81 \times 0.72 + 0.81 \times 0.1 + 0.72 \times 0.1}{1 - 0.21} \\ = 0.93 \end{matrix}

m_{D 1}^{12} (Θ) = 1 - m_{D 1}^{12} (5) - m_{D 1}^{12} (6) = 0.02

Then, the final consequence is

\begin{matrix} m_{D 1}^{123} (5) & = m_{D 1}^{12} (X) \oplus {\bar{m}}_{D 1}^{3} (Y) \\ = \frac{\sum_{X \cap Y = 5, \forall X, Y \in Θ} m_{D 1}^{12} (X) \times {\bar{m}}_{D 1}^{3} (Y)}{1 - \sum_{X \cap Y = \emptyset, \forall X, Y \in Θ} m_{D 1}^{12} (X) \times {\bar{m}}_{D 1}^{3} (Y)} \\ = \frac{m_{D 1}^{12} (5) \times {\bar{m}}_{D 1}^{3} (5) + m_{D 1}^{12} (Θ) \times {\bar{m}}_{D 1}^{3} (5) + m_{D 1}^{12} (5) \times m_{D 1}^{3} (Θ)}{1 - m_{D 1}^{12} (5) \times {\bar{m}}_{D 1}^{3} (6) - m_{D 1}^{12} (6) \times {\bar{m}}_{D 1}^{3} (5)} \\ = \frac{0.05 \times 0.27 + 0.02 \times 0.27 + 0.05 \times 0.1}{1 - 0.28} \\ = 0.03 \end{matrix}

\begin{matrix} m_{D 1}^{123} (6) & = m_{D 1}^{12} (X) \oplus {\bar{m}}_{D 1}^{3} (Y) \\ = \frac{\sum_{X \cap Y = 6, \forall X, Y \in Θ} m_{D 1}^{12} (X) \times {\bar{m}}_{D 1}^{3} (Y)}{1 - \sum_{X \cap Y = \emptyset, \forall X, Y \in Θ} m_{D 1}^{12} (X) \times {\bar{m}}_{D 1}^{3} (Y)} \\ = \frac{m_{D 1}^{12} (6) \times {\bar{m}}_{D 1}^{3} (6) + m_{D 1}^{12} (Θ) \times {\bar{m}}_{D 1}^{3} (6) + m_{D 1}^{12} (6) \times m_{D 1}^{3} (Θ)}{1 - m_{D 1}^{12} (5) \times {\bar{m}}_{D 1}^{3} (6) - m_{D 1}^{12} (6) \times {\bar{m}}_{D 1}^{3} (5)} \\ = \frac{0.93 \times 0.63 + 0.63 \times 0.02 + 0.93 \times 0.1}{1 - 0.28} \\ = 0.96 \end{matrix}

m_{D 1}^{123} (Θ) = 1 - m_{D 1}^{123} (5) - m_{D 1}^{123} (6) = 0.01

The pignistic probabilities can be obtained as follows

BetP (5) = m_{D 1}^{123} (5) + \frac{m_{D 1}^{123} (Θ)}{2} = 0.035

BetP (6) = m_{D 1}^{123} (6) + \frac{m_{D 1}^{123} (Θ)}{2} = 0.965

The mean value of the factor “D” of RPN can be determined using equation (24)

\begin{matrix} MVRP N_{D 1} & = \sum_{q = 1}^{2} BetP ({RPN}_{D 1}^{q}) \cdot ({RPN}_{D 1}^{q}) \\ = 0.035 \times 5 + 0.96 \times 6 \\ = 6 \end{matrix}

The traditional combination rule leads to the following result

m_{D 1}^{123} (5) = 0.01, m_{D 1}^{123} (6) = 0.99

\begin{array}{l} M V R P N_{D 1} = E (R P N_{D 1}) = \sum_{q = 1}^{2} (R P N_{q}) \cdot P (R P N_{q}) \\ = 5 \times 0.01 + 6 \times 0.99 = 6 \end{array}

Example 4.3 indicates that when experts give similar opinions on a failure mode, the BBAs and mean value of factor D are consistent with Yang et al.⁷ Examples 4.1–4.3 prove that the proposed method can efficiently deal with multiple information in FMEA.

In order to demonstrate the effectiveness of the proposed method in FMEA, we use this method to solve the same problem considered in Yang et al.⁷: the risk priority evaluation of rotor blades of an aircraft engine. The BBAs on 17 failure modes evaluated by three experts are shown in Table 4. The results of the present approach are compared with the literature in Figure 3.

Table 4.

Aero-engine turbo group failure analysis: BBAs on 17 failure modes evaluated by three experts.

Failure mode	Rating of risk factor
	Expert 1			Expert 2			Expert 3
	O	S	D	O	S	D	O	S	D
1	$m (3) = 0.4$	$m (7) = 1$	m(2) = 1	m(3) = 0.9	m(7) = 1	m(2) = 1	m(3) = 0.8	m(7) = 1	m(2) = 1
1	m(4) = 0.6			m(4) = 0.1			m(4) = 0.2
2	m(2) = 1	m(8) = 1	m(4) = 1	m(2) = 1	m(8) = 0.7	m(4) = 1	m(2) = 1	m(8) = 1	m(4) = 1
2					m(9) = 0.3
3	m(1) = 1	m(10) = 1	m(3) = 1	m(1) = 1	m(10) = 1	m(3) = 1	m(1) = 1	m(10) = 1	m(3) = 1
4	m(1) = 1	m(6) = 0.8	m(3) = 1	m(1) = 1	m(6) = 1	m(2) = 0.3	m(1) = 1	m(6) = 1	m(3) = 1
4		m(7) = 0.2				m(3) = 0.7
5	m(1) = 1	m(3) = 1	m(1) = 0.5	m(1) = 1	m(3) = 1	m(1) = 0.7	m(1) = 1	m(2) = 0.4	m(1) = 1
5			m(2) = 0.5			m(2) = 0.3		m(3) = 0.6
6	m(2) = 1	m(6) = 1	m(5) = 1	m(2) = 1	m(6) = 1	m(5) = 1	m(2) = 1	m(6) = 1	m(5) = 1
7	m(1) = 1	m(7) = 1	m(3) = 1	m(1) = 1	m(7) = 1	m(3) = 1	m(1) = 1	m(7) = 1	m(3) = 1
8	m(3) = 1	m(5) = 0.6	m(1) = 1	m(3) = 1	m(5) = 0.8	m(1) = 1	m(3) = 1	m(5) = 0.8	m(1) = 1
8		m(6) = 0.4			m(6) = 0.2			m(7) = 0.2
9	m(1) = 0.1	m(9) = 0.4	m(4) = 1	m(1) = 0.25	m(9) = 0.1	m(4) = 1	m(1) = 0.2	m(9) = 0.1	m(4) = 1
9	m(2) = 0.9	m(10) = 0.6		m(2) = 0.75	m(10) = 0.9		m(2) = 0.8	m(10) = 0.9
10	m(1) = 1	m(10) = 1	m(6) = 1	m(1) = 1	m(10) = 1	m(6) = 1	m(1) = 1	m(10) = 1	m(6) = 1
11	m(1) = 1	m(10) = 1	m(5) = 1	m(1) = 1	m(10) = 1	m(5) = 1	m(1) = 1	m(10) = 1	m(5) = 1
12	m(1) = 1	m(10) = 1	m(5) = 0.4	m(1) = 1	m(10) = 1	m(4) = 0.2	m(1) = 1	m(10) = 1	m(5) = 0.3
12			m(5) = 0.6			m(5) = 0.8			m(6) = 0.7
13	m(1) = 1	m(10) = 1	m(4) = 0.2	m(1) = 1	m(10) = 1	m(5) = 1	m(1) = 1	m(10) = 1	m(5) = 1
13			m(5) = 0.8
14	m(1) = 1	m(10) = 1	m(6) = 1	m(1) = 1	m(10) = 1	m(6) = 0.8	m(1) = 1	m(10) = 1	m(6) = 1
14						m(7) = 0.2
15	m(2) = 1	m(6) = 0.05	m(3) = 1	m(2) = 1	m(7) = 1	m(3) = 1	m(2) = 1	m(7) = 1	m(3) = 0.7
15		m(7) = 0.95							m(4) = 0.3
16	m(1) = 0.1	m(4) = 1	m(3) = 1	m(1) = 0.25	m(4) = 1	m(3) = 1	m(1) = 0.2	m(4) = 1	m(2) = 0.2
16	m(2) = 0.9			m(2) = 0.75			m(2) = 0.8		m(3) = 0.8
17	m(2) = 1	m(5) = 0.9	m(3) = 1	m(2) = 1	m(5) = 0.9	m(3) = 1	m(2) = 1	m(5) = 0.6	m(3) = 1
17		m(6) = 0.1			m(6) = 0.1			m(6) = 0.4

Figure 3.

Comparison of the results by the three methods.

It can be seen that all methods obtained practically the same result. Failure mode 2 has the largest RPN in the failure modes of compressor rotor blades, followed by failure modes 6, 1, 3, 7, 4, 8, and 5. For turbine rotor blades, failure mode 9 has largest RPN in the failure modes of compressor rotor blades, followed by failure modes 10, 13, 14, 11, 12, 15, 17, and 16. The larger the RPN, the more attention should be paid to the corresponding failure mode. If the different failure modes have the same RPN, like modes 6, 10, and 14 (i.e. RPN = 60), the same risk attention should be given to them. In conclusion, the proposed approach keeps the advantage of the Yang et al.’s method and has good performance in dealing with multiple information in FMEA.

Another test problem was solved in order to prove the efficiency of the proposed method in the case of highly conflicting evidences. Table 5 shows the BBAs for four failure modes characterized by significant variations in experts’ evaluations.

Table 5.

Aero-engine turbo group failure analysis: highly conflicting BBAs on four failure modes evaluated by three experts.

Item	Rating of risk factor
	Expert 1			Expert 2			Expert 3
	O	S	D	O	S	D	O	S	D
1	m(3) = 0.7	m(6) = 0.1	m(2) = 0.8	m(4) = 0.7	m(7) = 0.8	m(2) = 1	m(5) = 0.8	m(7) = 1	m(2) = 1
1	m(4) = 0.3	m(7) = 0.9	m(3) = 0.2	m(3) = 0.3	m(8) = 0.2		m(6) = 0.2
2	m(7) = 0.2	m(8) = 1	m(7) = 0.8	m(6) = 0.4	m(8) = 1	m(7) = 0.6	m(6) = 1	m(8) = 1	m(7) = 1
2	m(8) = 0.8		m(8) = 0.2	m(7) = 0.6		m(8) = 0.4
3	m(7) = 0.2	m(5) = 1	m(3) = 1	m(6) = 0.5	m(5) = 1	m(3) = 1	m(5) = 1	m(6) = 0.4	m(3) = 1
3	m(8) = 0.2			m(7) = 0.5				m(5) = 0.6
4	m(3) = 0.7	m(2) = 0.35	m(6) = 1	m(5) = 1	m(2) = 0.4	m(6) = 0.5	m(4) = 0.7	m(5) = 1	m(7) = 0.25
4	m(4) = 0.3	m(3) = 0.65			m(3) = 0.6	m(7) = 0.5	m(5) = 0.3		m(8) = 0.75

The results shown in Table 6 clearly demonstrate the superiority of the present approach. In fact, the method developed by Yang et al.⁷ cannot perform the risk-level assessment for any failure mode while the method developed by Su et al.²⁸ is successful for only one failure mode. Since the RPN parameter is determined as the product of the three risk factors O, S, and D, methods described in Yang et al.⁷ and Su et al.²⁸ become invalid if experts’ evaluations on a risk factor diverge considerably. Let us consider in particular failure mode 4. Yang et al.’s method clearly shows its limits in dealing with highly conflicting evidence between O, S, and D. By turning m(X) = 1 into m(X − 1) = 0.1, m(X) = 0.8, and m(X + 1) = 0.1 (where X = 5, 6), the Su et al.’s method can determine the MVRPN of O and D but still fails to reduce the conflict in the evaluation of S. Therefore, neither this method can determine the correct value of RPN for failure mode 4. The new methodology developed in this research found instead logical values for the RPN associated with the four failure modes: 57.96, 379.03, 96.98, and 71.73, respectively, thus leading to rank the level of risk associated with the four failure modes as $FM 2 ≻ FM 3 ≻ FM 4 ≻ FM 1$ .

Table 6.

Results of risk-level assessment in case of highly conflicting experts’ evaluations.

Failure mode	1	2	3	4
Yang et al.
Su et al.		393.12
This study	57.96	389.03	96.96	71.73

RPN: risk priority number.

The dark area indicates that RPN could not be calculated.

Conclusion

This study presented a novel method for properly evaluating the level of risk when basic belief assignments cannot be combined with the Dempster’s combination rule. The main novelty introduced in the article is the use of a reliability coefficient based on evidence distance. The results obtained in some specifically designed numerical examples and the risk analysis of real mechanical components (i.e. the turbo group of an aero-engine) demonstrate the validity of the proposed approach.

The proposed approach has been proved to be useful and practical, but can still be improved in some aspects. For example, we put emphasis on producing the weight of evidence according to the evaluations associated with different experts, while the weights of experts themselves were not accounted for in this study.

Footnotes

Acknowledgements

The authors greatly appreciate the encouragement of the editor and the anonymous reviewers’ valuable comments and suggestions to improve this article.

Academic Editor: Jia-Jang Wu

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported, in part, by a grant from National Natural Science Foundation of China (No. 60904099) and Foundation for Fundament Research of Northwestern Polytechnical University, Grant No. JC20120235.

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