Introduction

As an important risk analysis tool in reliability engineering, failure mode and effects analysis (FMEA) is used for defining and identifying potential failures or problems in design, processes, or services before they reach the end users [1,2,3,4]. FMEA ranks the potential risk item based on the risk priority number (RPN) model, which is defined by multiplying three risk factors named the Occurrence (O), Severity (S), Detectability (D), where O means the probability of the occurrence of a FMEA item, S means the severity degree if a failure happens with respect to the corresponding FMEA item, and D is the probability of a potential FMEA item being detected. Briefly, the basic process of applying FMEA is as follows: (1) analyze the target system and obtain the failure modes, (2) obtain the evaluation data of the failure modes regarding the RPN model, (3) rank the failure modes according to the evaluation data based on RPN values, and (4) improve the actual system according to the ranking result. FMEA has been widely applied in practical applications such as industry systems [5, 6], equipment maintenance [7], medical domain [8] and new product development [9]. The hazards in maritime autonomous surface ships during development and operation process are identified with FMEA method in [10]. The failure of Taper–Lock bush used in aggregate batcher plant for construction application is analyzed with FEMA model in [11]. An integrated decision-making algorithm is proposed to overcome some shortcomings in FMEA model and applied in clean energy barriers evaluation in [12]. Sustainable medical waste management systems designing process is analyzed with FMEA model in [13].

Uncertain information modeling and processing is commonly studied in practical applications such as fault diagnosis [14, 15], advanced control in complex systems [16, 17], control of repetitive tasks [18] and classification [55, 56].

Definition 1

Suppose that \(\Omega = \{\theta _1,\theta _2,\ldots ,\theta _N\}\) is a nonempty set with N mutually exclusive and exhaustive events, and \(\Omega \) is called the frame of discernment. The power set of \(\Omega \) consists of \(2^N\) elements denoted as follows:

$$\begin{aligned} 2^{\Omega } = \{\emptyset ,\{\theta _1\},\ldots ,\{\theta _N\}, \{\theta _1,\theta _2\},\ldots ,\{\theta _1,\ldots ,\theta _N\}\}. \end{aligned}$$
(1)

Definition 2

A mass function m is a map** from the power set \(2^{\Omega }\) to the interval \(\left[ 0,\ 1\right] \). m satisfies:

$$\begin{aligned} m(\emptyset ) = 0 ,\ \sum _{A\in 2^{\Omega }}{m\left( A \right) =1}. \end{aligned}$$
(2)

A mass function is also known as basic probability assignment (BPA) or basic belief assignment (BBA). If \(m(A)>0\), then A is called focal element. The m(A) represents the degree that an evidence supports A.

Definition 3

In DST, two independent mass functions \(m_1\) and \(m_2\) can be fused with Dempster’s rule of combination defined as follows:

$$\begin{aligned} m\left( A\right) =\left( m_1 \oplus m_2 \right) \left( A\right) =\frac{1}{k-1}\sum \limits _{B \cap C=A}{m_1 \left( B\right) m_2\left( C \right) }. \end{aligned}$$
(3)

where k is a normalization factor defined as:

$$\begin{aligned} k =\sum \limits _{{B \cap C=\emptyset }}{m_1\left( B \right) m_2 \left( C \right) }. \end{aligned}$$
(4)

When two pieces of evidence have high conflict with each other, the classic Dempster’s rule of combination is not efficient. To overcome this limitation, some discounting methods have been proposed based on Dempster’s rule of combination, and this work adopts the one used in [57]:

$$\begin{aligned} \left\{ \begin{array}{l} m^{'}\left( A \right) =\alpha \times m\left( A \right) ,A\ne \emptyset \\ m^{'}\left( \Omega \right) =\alpha \times m\left( \Omega \right) +1-\alpha \\ \end{array} \right. \end{aligned}$$
(5)

where \(\alpha \) is the discounting coefficient.

The pignistic probability transformation function \(BetP_m\)

Definition 4

Let m be a BPA on \(\Omega \). Its associated pignistic probability transformation function \(BetP_m\) is defined as follows [58]:

$$\begin{aligned} BetP_m {\left( B \right) }= \sum \limits _{A\subseteq \Omega ,B\in A}{\frac{m \left( A \right) }{|A|}}. \end{aligned}$$
(6)

where |A| means the cardinality of the set A. The aim of \(BetP_m\) is to transform a BPA into a probability function for final decision-making in DST framework.

Some key notations in this work are summarized in Table 1.

Table 1 Key notation in this work
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An improved pignistic probability transformation function

In this section, an improved pignistic probability transformation function based on the classical \(BetP_m\) is proposed. The new method combines the information both in the propositions of singleton and multiple subsets, which can optimize the strategy of assigning the belief value averagely in the pignistic probability transformation function \(BetP_m\).

Limitation in the pignistic probability transformation function

The traditional pignistic probability transformation function assigns the belief of multiple subsets averagely into the singleton. Consequently, as for the singleton A, the \(BetP_m(A)\) can also be expressed as follows:

$$\begin{aligned} BetP_m {\left( A \right) }{} & {} = m\left( A \right) +\sum \limits _{B\subseteq \Omega ,A\subsetneq B} {\frac{m\left( B \right) }{|B|}}=m\left( A \right) \nonumber \\{} & {} \quad +\sum \limits _{i=1}^{n}{{\frac{m\left( A_i \right) }{|A_i|}}}. \end{aligned}$$
(7)

where n means the number of multiple subsets that include the singleton A. \(A_i\) represents the specific multiple subsets.

Figure 1 shows the visualized effect of \(BetP_m\). The abscissa represents the value of m(A), and the ordinate represents the proportion of belief (\(p_1\)) assigned to A from \(A_n\). Obviously, the line stays parallel with the abscissa. The figure shows the essence of pignistic probability transformation function that the proportion of the belief assigned to the singleton from the multiple subsets is independent of the mass value in the singleton proposition. The averagely assigning method makes the belief being equally utilized by each singleton in spite of the mass value of singleton element. Therefore, it has a good compatibility. However, the complete independence between the singleton mass function value and belief assignment of multiple subsets actually neglects the information given by the original singleton BPA. This strategy excessively narrows the differences in each singleton BPA, which causes inaccuracy and information loss in decision-making section to a certain extent.

Fig. 1
figure 1

An illustration of functional relationship between m(A) and \(p_1\) based on the pignistic probability transformation function \(BetP_m\)

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Considering the original BPA in information fusion result during belief assignment of multiple subsets, define a linear pignistic probability transformation function \(\textrm{Lin}_m\) as follows:

$$\begin{aligned} \textrm{Lin}_m {\left( A \right) }{} & {} = m\left( A \right) {+}\sum \limits _{B\subseteq \Omega ,A\subsetneq B} {\frac{m\left( B \right) m\left( A \right) }{\sum _{C\in B}m\left( C \right) }}, \nonumber \\ |A|{} & {} {=}|C|{=}1. \end{aligned}$$
(8)

where C means the singleton included in the multiple subsets proposition B.

Figure 2 shows the visualized effect of \(\textrm{Lin}_m\). The abscissa represents the value of m(A), and the ordinate represents the proportion of belief (\(p_2\)) assigned to A from B.

Fig. 2
figure 2

An illustration of functional relationship between m(A) and \(p_2\) based on the linear pignistic probability transformation function \(\textrm{Lin}_m\)

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Based on the linear pignistic probability transformation function \(\textrm{Lin}_m\), the \(p_2\) is directly proportional to m(A). The figure shows that the proportion of the belief assigned to one singleton from the multiple subsets proposition has complete positive correlation with the singleton mass function value. This assignment method substantially utilizes the information from the original singleton BPA in information fusion result. However, correspondingly, the information from the multiple subsets is neglected. This strategy excessively holds the gap among each singleton BPA, which may also cause inaccuracy in final decision-making section.

The improved pignistic probability transformation function

To address the disadvantage of aforementioned functions \(BetP_m\) and \(\textrm{Lin}_m\), the information in both the original BPA of singleton element of fusion result and the multiple subsets should be utilized. Based on the above analysis, define the improved pignistic probability transformation function \(\textrm{Exp}_m\) as follows:

$$\begin{aligned} \textrm{Exp}_m {\left( A \right) }&= m\left( A \right) +\sum \limits _{B\subseteq \Omega ,A\subsetneq B}\nonumber \\&\quad \frac{m\left( B \right) exp\left( \gamma \times m \left( A \right) \right) }{\sum _{C\in B}exp \left( \gamma \times m\left( C \right) \right) }, \ \ |A|=|C|=1. \end{aligned}$$
(9)

where C means the singleton included by multiple subsets proposition B, \(\gamma \) is the discounting coefficient.

The following example is proposed to illustrate the calculation procedure of the improved pignistic probability transformation function \(\textrm{Exp}_m\).

Example 1

Define that a frame of discernment is \(\Omega =\{a,b\}\), the following BPAs of information fusion result are constructed:

$$\begin{aligned}{} & {} m\left( a\right) =t,m\left( b\right) =0.1, \\{} & {} m\left( a,b\right) =0.9-t. \end{aligned}$$

where t represents the value of m(a), \(t \in \left[ {0,0.9} \right] \). Based on the improved pignistic probability transformation function, the belief assignment result for this example is calculated as follows:

$$\begin{aligned} m^{'}\left( a\right)= & {} \textrm{Exp}_m{\left( a\right) }=t+\left( 0.9-t\right) \\{} & {} \times {\frac{exp\left( \gamma \times t\right) }{exp\left( \gamma \times t\right) +exp\left( \gamma \times 0.1\right) }} \\ m^{'}\left( b\right)= & {} \textrm{Exp}_m{\left( b\right) }=0.1+\left( 0.9-t\right) \\{} & {} \times {\frac{exp\left( \gamma \times {{0.1}}\right) }{exp\left( \gamma \times t\right) +exp\left( \gamma \times 0.1\right) }} \end{aligned}$$

Define that:

$$\begin{aligned} p_3=\frac{exp\left( \gamma \times t\right) }{exp\left( \gamma \times t\right) +exp\left( \gamma \times 0.1\right) } \end{aligned}$$

Figure 3 shows the visualized effect of the above example (Here, set \(\gamma \)=3 as a numerical example). The abscissa represents the value of t, and the ordinate represents the proportion of belief (\(p_3\)) assigned to \(\{a\}\) from \(\{a,b\}\).

Fig. 3
figure 3

An illustration of functional relationship between t and \(p_3\) based on the improved pignistic probability transformation function \(\textrm{Exp}_m\)

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Based on the improved pignistic probability transformation function, the \(p_3\) is directly proportional to t. The proportion of the belief assigned to one singleton from the multiple subsets proposition is a non-liner and exponential-like function of the singleton mass function value. The ascending rate is controlled by the discounting coefficient \(\gamma \), which can be modified to adapt the requirement of different data set. Consequently, the designed strategy combines the information from both the multiple subsets and the original singleton BPA in information fusion result. It also has a good compatibility with the traditional \(BetP_m\) because the improved pignistic probability transformation function \(\textrm{Exp}_m\) will be degenerated to \(BetP_m\) if \(\gamma \) = 0. The effectiveness and practicality of the improved pignistic probability transformation function will be verified in Sect. “A novel FMEA model utilizing the improved pignistic probability transformation function and GRPM”.

A novel FMEA model utilizing the improved pignistic probability transformation function and GRPM

To improve the efficiency of traditional FMEA, a novel FMEA model utilizing the improved pignistic probability transformation function and grey relational projection method (GRPM) is proposed, as shown in Fig. 4.

Fig. 4
figure 4

The flowchart for the novel FMEA model using the improved pignistic probability function in the evidence theory and grey relational projection method

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Table 2 Criteria of rating for occurrence
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Step 1. Define the failure modes and the level of risk assessment regarding risk factors

Suppose there are k experts involved in the FMEA team (\(E_1,E_2,\ldots ,E_i,\ldots ,E_k\)) for assessing of m failure modes (\(FM_1,FM_2,\ldots ,FM_i,\ldots ,FM_m\)) with respect to three risk factors \(F_R\) denoted as Occurrence (\(F_1\)), Severity (\(F_2\)) and Detectability (\( F_3\)). Considering the flexibility and accuracy of the judgements, each risk factor will be ranked from the lowest level 1 to the highest level 10 [1,2,3, 59], as expressed in Tables 2, 3 and 4 respectively. Each expert will utilize a rating number to assess each risk factor with their belief degree on the assessment. According to the belief structure data of the evaluations, the basic probability assignment can be constructed to represent the belief degree of the assessment coming from the expert.

Table 3 Criteria of rating for severity of a failure
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Table 4 Criteria of rating for detection of a failure
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Step 2. Construct the frame of discernment

Based on DST, the frame of discernment for the three risk factors can be defined as follows:

$$\begin{aligned} \Omega _{R}^{j}{} & {} =\{1,2,3,4,5,6,7,8,9,10\} \ \nonumber \\ R{} & {} =1,2,3.\ \ j=1,2,\ldots ,m. \end{aligned}$$
(10)

where \(R=1\), \(R=2\) and \(R=3\) correspond to the risk factor O, S and D respectively, j means the jth FMEA item.

Step 3. Combine the judgement results of experts in FMEA team

Define \(m_{iR}^{j}\) (\(i=1,2,\ldots ,m,R=1,2,3,j=1,2,\ldots ,k\)) as the BPA collected from \(E_j\) on the assessment of \(FM_i\) regarding the risk factor \(F_R\). Then the \(m_{iR}^{'j}\) can be calculated using Eq. (5). Combine BPAs from all the experts by using Eq. (3) and Eq. (4), shown as follows:

$$\begin{aligned} m_{iR}=m_{iR}^{'1}\oplus m_{iR}^{'2}\oplus \ldots \oplus m_{iR}^{'k} \end{aligned}$$
(11)

Step 4. Calculate \(x_{ij}\) by utilizing the improved pignistic probability transformation function \(\textrm{Exp}_m\)

After combining the judgement results in Step 3, the integrated representation \(x_{ij}\) (\(i=1,2,\ldots ,m,j=1,2,3\)) of the aggregated assessment \(m_{iR}\) can be constructed through discounting combination rule. The value of \(x_{ij}\) is calculated as follows by utilizing the improved pignistic probability transformation function \(\textrm{Exp}_m\):

$$\begin{aligned} x_{ij}=\sum \limits _{\delta \in \Omega }{\textrm{Exp}_m \left( \delta \right) \times \delta } \end{aligned}$$
(12)

Step 5. Construct comparative sequence X

In comparative sequence X, \(x_{ij}\) represents the jth factor of \(X_i\). The matrix format of X is defined as follows:

$$\begin{aligned} X=\left[ \begin{array}{c} X_1 \\ X_2 \\ \vdots \\ X_m \end{array}\right] =\left[ \begin{array}{ccc} x_{11} &{} x_{12} &{} x_{13} \\ x_{21} &{} x_{22} &{} x_{23} \\ \vdots &{} \vdots &{} \vdots \\ x_{m1} &{} x_{m2} &{} x_{m3} \end{array}\right] \end{aligned}$$

Step 6. Construct reference sequence \(X^+\) and \(X^-\)

The positive sequence means that the evaluation of failure mode on each risk factor is on the lowest risk level.Correspondingly, the negative sequence means that the evaluation of failure mode on each risk factor is on the highest risk level. The double-facet reference sequences improve the reliability of FMEA. \(X^+\) and \(X^-\) are defined as follows:

$$\begin{aligned} {X^+}&=\left( 1\right) _{3 \times m} \end{aligned}$$
(13)
$$\begin{aligned} {X^-}&=\left( 10\right) _{3 \times m} \end{aligned}$$
(14)

Step 7. Construct the grey relational matrices \(Y^+\) and \(Y^-\)

First, calculate the grey relation coefficient \(Y_{ij}^{+}\) and \(Y_{ij}^{-}\):

$$\begin{aligned} Y_{ij}^{+}&=\frac{min_{1\le i\le m}min_{1\le j\le 3}|x_{0j}^{+} -x_{ij}|+\xi max_{1\le i\le m}max_{1\le j\le 3}|x_{0j}^{+} -x_{ij}|}{|x_{0j}^{+}- x_{ij}|+\xi max_{1\le i\le m} max_{1\le j\le 3}|x_{0j}^{+}- x_{ij}|} \end{aligned}$$
(15)
$$\begin{aligned} Y_{ij}^{-}&=\frac{min_{1\le i\le m}min_{1\le j\le 3}|x_{0j}^{-} -x_{ij}|+\xi max_{1\le i\le m}max_{1\le j\le 3}|x_{0j}^{-} -x_{ij}|}{|x_{0j}^{-}- x_{ij}|+\xi max_{1\le i\le m} max_{1\le j\le 3}|x_{0j}^{-}- x_{ij}|} \end{aligned}$$
(16)

where \(i=1,2,\ldots ,m,j=1,2,3\). \(Y_{ij}^{+}\) is the grey relation coefficient of \(x_{ij}\) with respect to the positive ideal risk factor \(x_{0j}^{+}\). \(Y_{ij}^{-}\) is the grey relation coefficient of \(x_{ij}\) with respect to the negative ideal risk factor \(x_{0j}^{-}\). \(\xi \) is the distinguishing coefficient and \(\xi \in \left[ 0,1\right] \). In this method, the medium value \(\xi \) = 0.5 is applied.

Then, based on the coefficients \(Y_{ij}^{+}\) and \(Y_{ij}^{-}\), the grey relational matrices \(Y^+\) and \(Y^-\) can be denoted as follows:

$$\begin{aligned} Y^+=\left[ \begin{array}{ccc} Y_{11}^{+} &{} Y_{12}^{+} &{} Y_{13}^{+} \\ Y_{21}^{+} &{} Y_{22}^{+} &{} Y_{23}^{+} \\ \vdots &{} \vdots &{} \vdots \\ Y_{1m}^{+} &{} Y_{2m}^{+} &{} Y_{3m}^{+} \end{array}\right] , \ \ Y^-=\left[ \begin{array}{ccc} Y_{11}^{-} &{} Y_{12}^{-} &{} Y_{13}^{-} \\ Y_{21}^{-} &{} Y_{22}^{-} &{} Y_{23}^{-} \\ \vdots &{} \vdots &{} \vdots \\ Y_{1m}^{-} &{} Y_{2m}^{-} &{} Y_{3m}^{-} \end{array}\right] \end{aligned}$$

Step 8. Calculate the grey relation projection \(P^+\) and \(P^-\)

The grey relation projections of \(FM_i\) on the reference sequence \(X^+\) and \(X^-\) respectively are calculated as follows:

$$\begin{aligned} P_{i}^{+}&=\frac{1}{3}\sum \limits _{j=1}^{3}Y_{ij}^{+} \end{aligned}$$
(17)
$$\begin{aligned} P_{i}^{-}&=\frac{1}{3}\sum \limits _{j=1}^{3}Y_{ij}^{-} \end{aligned}$$
(18)
Table 5 Evaluation information of failure modes provided by the FMEA team members
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Step 9. Calculate the relative projection \(RP_i\) and rank failure modes

The relative projection of each failure mode to the positive reference sequence is defined as follows:

$$\begin{aligned} RP_i=\frac{P_{i}^{+}}{P_{i}^{+}+P_{i}^{-}} \end{aligned}$$
(19)

In FMEA, the relative projection value denotes the relative size of each failure mode. The higher the relative projection value, the smaller the impact of the failure mode and correspondingly the lower the ranking priority.

Application

Experiment 1: Application in aircraft turbine rotor blades

In this section, the proposed method is applied to a FMEA case study of rotor blades for an aircraft turbine adopted from [60,61,62,63,64]. Rotor blades are the most damageable component in aircraft turbines due to the complex working environments. The FMEA method is introduced to improve the reliability and safety of the rotor blades’ design.

Experiment process

Step 1. Define the failure modes and the level of risk assessment regarding risk factors. According to previous studies [60,61,62,63,64], eight failure modes (\(FM_i,i=1,2,\ldots ,8\)) are considered with the improved FMEA model and will be ranked for efficient management of risk level. Three experts (\(E_1,E_2,E_3\)) provide the assessment of \(FM_i\) by utilizing 10 levels in terms of the three risk factors denoted as \(F_R\) Occurrence (\(F_1\)), Severity (\(F_2\)) and Detectivity (\(F_3\)).

Step 2. Construct the frame of discernment. According to Eq. (10), the frame of discernment can be defined as follows:

$$\begin{aligned} \Omega _{R}^{j}{=}\{1,2,3,4,5,6,7,8,9,10\} \ \ R{=}1,2,3.\ \ j{=}1,2,\ldots , 8. \end{aligned}$$

Step 3. Combine the judgement results of experts in FMEA team. The original BPA of the risk factors derived from experts’ assessment is shown in Table 5. The combination result of original BPA using Eqs. (3), (4) and (5) is shown in Table 6.

Table 6 Group assessments of the FMEA team members
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Step 4. Calculate \(x_{ij}\) by utilizing the improved pignistic probability transformation function \(\textrm{Exp}_m\). First, transform the combining result of BPA into probability function by utilizing the improved pignistic probability transformation function \(\textrm{Exp}_m\) which is denoted in Eq. (9). Then, the value of \(x_{ij}\) can be calculated by using Eq. (12).

To select the ideal discounting coefficient \(\gamma \), we set \(\gamma \) as different values (0, 3, 6, and 9) and calculate the results respectively. In the following steps, we only show the detail of calculating process under the parameter of \(\gamma =3\).

Step 5. Construct comparative sequence X. The comparative sequence X is shown as follows based on the calculating result of \(x_{ij}\):

$$\begin{aligned} X=\left[ \begin{array}{ccc} 3.8259 &{} 6.7137 &{} 2.5351 \\ 2.6691 &{} 7.1548 &{} 4.2293 \\ 1.9099 &{} 9.2359 &{} 3.4062 \\ 1.8587 &{} 5.9198 &{} 3.3928 \\ 1.8587 &{} 3.4778 &{} 2.1774 \\ 2.6691 &{} 5.9151 &{} 5.0764 \\ 2.0198 &{} 6.7321 &{} 3.3822 \\ 3.4794 &{} 5.1496 &{} 1.6880 \end{array}\right] \end{aligned}$$

Step 6. Construct reference sequence \(X^+\) and \(X^-\). The reference sequences \(X^+\) and \(X^-\) can be constructed by using Eqs. (13) and (14).

Step 7. Construct the grey relational matrices \(Y^+\) and \(Y^-\). According to Eqs. (15) and (16), the grey relation matrices \(Y^+\) and \(Y^-\) are calculated, shown as follows:

$$\begin{aligned} Y^+=\left[ \begin{array}{ccc} 0.6921 &{} 0.4888 &{} 0.8502 \\ 0.8304 &{} 0.4678 &{} 0.6541 \\ 0.9559 &{} 0.3890 &{} 0.7366 \\ 0.9657 &{} 0.5318 &{} 0.7382 \\ 0.9657 &{} 0.7287 &{} 0.9076 \\ 0.8305 &{} 0.5320 &{} 0.5865 \\ 0.9354 &{} 0.4879 &{} 0.7394 \\ 0.7285 &{} 0.5813 &{} 1.0000 \end{array}\right] ,\\ Y^-=\left[ \begin{array}{ccc} 0.4763 &{} 0.6611 &{} 0.4234 \\ 0.4283 &{} 0.7028 &{} 0.4956 \\ 0.4018 &{} 1.0000 &{} 0.4577 \\ 0.4001 &{} 0.5974 &{} 0.4571 \\ 0.4001 &{} 0.4608 &{} 0.4107 \\ 0.4283 &{} 0.5970 &{} 0.5419 \\ 0.4054 &{} 0.6628 &{} 0.4567 \\ 0.4608 &{} 0.5463 &{} 0.3946 \end{array}\right] \end{aligned}$$

Step 8. Calculate the grey relation projection \(P^+\) and \(P^-\). \(P^+\) is the grey relation projection of \(FM_i\) on the positive reference sequence and \(P^-\) is the grey relation projection of \(FM_i\) on the negative reference sequence. \(P^+\) and \(P^-\) are calculated by using Eqs. (17) and (18). The calculation result is shown in Table 7.

Step 9. Calculate the relative projection \(RP_i\) and rank failure modes. The relative projection of each failure mode to positive reference sequence \(RP_i\) can be calculated by using Eq. (19). The \(RP_i\) values and the risk priority ranking of all the 8 failure modes in aircraft turbine rotor blades are also shown in Table 7.

Table 7 Risk priority ranking in aircraft turbine rotor blades
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Experiment result

Figure 5 shows the ranking results of the 8 failure modes under the condition that \(\gamma \) is assigned with different coefficient values: 0, 3, 6, and 9. If \(\gamma =0\), the experimental result is mathematically equivalent to the result with the pignistic probability transformation function \(BetP_m\). Among the coefficients of \(\gamma =3\), \(\gamma =6\) and \(\gamma =9\), the result under the condition of \(\gamma =3\) has a proximal ranking trend with \(\gamma =0\) according to Fig. 5. In this work, we choose the result under the condition of \(\gamma =3\) for the following experiment and analysis.

Fig. 5
figure 5

The ranking of failure modes under the conditions of \(\gamma =0,3,6,9\)

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Fig. 6
figure 6

The ranking of failure modes based on the proposed method in comparison with other methods in Experiment 1

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The ranking result for \(FM_i\) under the condition of \(\gamma =3\) is \(FM_3 \succ FM_2 \succ FM_6 \succ FM_1 \succ FM_7 \succ FM_6 \succ FM_8 \succ FM_5\), where \( \succ \) means prior to. The ranking result of the 8 failure modes is compared with the result of some previous studies in [60,61,62,63,64], as shown in Fig. 6. Most of the methods rank \(FM_3\) as the most important failure mode that is supposed to be addressed with the highest priority and \(FM_5\) as the least important failure mode that is supposed to be addressed with the lowest priority. From this perspective, the result of the proposed method is consistent with the previous works. Meanwhile, in general, the proposed method has a similar ranking trend on all the failure modes in comparison with the results of previous studies, which shows the rationality and effectiveness of the proposed method. There is no duplicated ranking level for all the 8 FMEA items with the proposed method.

Fig. 7
figure 7

The ranking of failure modes based on the proposed method in comparison with other methods in Experiment 2

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Experiment 2: application in steel production process

To better verify the generalization ability of the proposed FMEA method, we adopt another risk analysis example in [57] as the Experiment 2. In this case study [57], 10 failure modes of sheet steel production process in a steel factory are evaluated by the proposed method with respect to the three risk factors.

The original data of risk analysis in the sheet steel production process is adopted from [57]. The calculation steps based on the proposed method are similar to the application of aircraft turbine rotor blades in Experiment 1. For this reason, the detailed calculation process is omitted here. Table 8 shows the calculation and ranking results with the proposed method. The ranking result of all the 10 failure modes is \(FM_4 \succ FM_7 \succ FM_3 \succ FM_8 \succ FM_5 \succ FM_1 \succ FM_6 \succ FM_{10} \succ FM_9 \succ FM_2\), where \( \succ \) represents prior to.

The comparison of the ranking result with other methods in previous studies of [22, 57, 64, 65] is shown in Fig. 7. The figure indicates that the ranking result obtained by the proposed method has the same trend obtained by the other methods, especially for Wu et al. method [22] and Liu et al. method [64]. The failure mode \(FM_4\) is ranked as the first FMEA item in the proposed method, which is consistent with the results of other three methods. In practical application, a higher priority should be given to the failure mode \(FM_4\) because it has the highest risk level according to the result in the proposed method as well as some other different methods in comparison. It should be noted that there is no duplicated ranking level for all the 10 FMEA items with the proposed method, which shows a good distinguish ability of the proposed method on different FMEA items.

Table 8 Risk priority ranking in steel factory
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Discussion

Sensitivity and robustness analysis

In general, the proposed novel FMEA model makes good use of the information given by the original singleton BPA of fusion result in principle through the improved pignistic probability transformation function and further ensures the accuracy of the results by GRPM. The reliability and rationality of the proposed method are verified by two experiments in different applications. There are two variable parameters included in the proposed method including the \(\gamma \) in the improved pignistic probability transformation function and \(\xi \) in GRPM process which will be adopted for sensitivity and robustness analysis.

For testing the sensitivity and robustness of the parameter \(\xi \) in Eqs. (15) and (16) of GRPM method, the application in Experiment 1 is taken as the example where another experiment is designed and implemented to observe the ranking results of the distinguishing coefficient \(\xi \) varying from 0.1 to 1 with a step length of 0.1. With the change of \(\xi \) from 0.1 to 1 with the step length of 0.1, the corresponding values of \(RP_i\) and the ranking results of 8 failure modes in Experiment 1 are shown in Table 9. It shows that the \(RP_i\) values are different in each experiment but the ranking results of all the FMEA items stay the same, which shows the robustness of the proposed method.

Table 9 The \(RP_i\) values and ranking results with the variation of \(\xi \) in Experiment 1
Full size table

For the discounting coefficient \(\gamma \) in the improved pignistic probability transformation function, in Experiment 1, we test different values of \(\gamma \) as 0, 3, 6, and 9, the corresponding ranking results with different values of \(\gamma \) in the improved FMEA model are shown in Fig. 5. The discounting coefficient \(\gamma \) has a significant effect on ranking results. Thus, we can improve the accuracy of ranking results through different strategies of the selection of \(\gamma \), which can be an open issue in the improved pignistic probability transformation function.

According to all the ranking results listed in Tables 7, 8, 9, Figs. 6, and  7, there is no duplicated ranking level for both Experiment 1 and Experiment 2 with the proposed method, which shows a good distinguish ability of the proposed method on different FMEA items. The high distinguish ability on similar failure modes can be a significant feature and superiority of the proposed method.

Complexity of the proposed method

The time complexity in the proposed method is mainly based on the scale of the risk analysis problem in practical application. Suppose that there are m failure modes, n risk factors and k team members in a risk analysis problem with FMEA model. In the process with DST theory, there are mnk pieces of BPAs, so the combination of BPAs can be finished in time O(mnk). In the process with GRPM theory, establishing comparative sequence needs O(m), determining the grey relation matrices can be done with time of O(2mn), the grey relation projections can be finished with time of O(2mn), computing the relative projection and ranking failure modes need the time of \(O(m\log _{2}m)\). In conclusion, the time complexity in the proposed FMEA model is \(O(mnk+4mn+m+m\log _{2}m)\) which is more complex than traditional FMEA theory with RPN model. Theoretically, the time complexity is cubic (mnk). It can be understand that more accurate on modeling and computing of uncertain information means more cost. It should be noted that, in FMEA, there are three risk factors and n equals to 3.

Limitation and open issue

There are limitations and open issues in the work. First, the strategy of choosing the coefficient \(\gamma \) in the improved pignistic probability transformation function is mainly based on intuition or experience. Some intelligent methods may be adopted for addressing this issue in some complex problems. Second, the time complexity of the proposed method is cubic which is higher than the classical method, which may be a potential issue while it is used in some very complex system. It may be solved with more computing resources. Third, there are many modified evidence combination rules in DST. It can be a problem while choosing a proper combination rule in different practical applications.

Conclusion

In this work, an improved pignistic probability transformation function in Dempster–Shafer evidence theory is proposed to improve the probability transformation accuracy of basic probability assignment in decision-making in the evidence theory. Based on the improved pignistic probability transformation function, an improved FMEA model using the grey relational projection method is proposed. The experts in FMEA team firstly use basic probability assignment to evaluate failure modes regarding different risk factors, then Dempster’s rule of combination is used to fuse the evaluation coming from each expert. Meanwhile, the improved pignistic probability transformation function is used to transform the fusion results of BPA into probability function for more accurate decision-making result. Finally, the grey relational projection method is adopted to calculate the risk priorities of all the failure modes. The rationality and generality of the proposed method is verified through comparative studies of FMEA in applications of aircraft turbine rotor blades and steel factory process.

Some possible further research directions can be as follows. First, a widely accepted BPA transformation strategy other than the pignistic probability transformation function is still an open issue [45, 66]. The improved pignistic probability transformation function can complete the principles of setting the discounting coefficient. The assignment of different mass function value in the frame of discernment is expected to be concretely combined with choosing the discounting coefficient to adapt complex environment efficiently. Second, uncertainty measures such as belief entropy [67,68,69] and correlation coefficient [70] can be adopted to quantify and manage the uncertainty in the process of risk analysis such as the relative importance among difference experts and risk factors. Third, the interdependencies among the FMEA items as well as the FMEA experts should be taken into consideration and can be properly modeled with the dependent evidence fusion method [71]. Fourth, approximate calculation method may be adopted to address the computation complexity of the proposed method.