An Improved FMEA Method Based on ANP with Probabilistic Linguistic Term Sets

Abstract

Failure modes and effects analysis (FMEA), as a practical and easy-to-use reliability assessment tool, has been widely applied across various fields of researches. At the same time, it also receives criticisms for its limited selection of risk factors as well as its discrete numerical ordinal scales. In an attempt to fill in this gap, this paper proposes an improved FMEA method based on analytic network process (ANP) with probabilistic linguistic term sets (PLTSs). Firstly, the three risk factors of FMEA, namely occurrence, severity and detection, are broken down to more elaborate and specific sub-factors, and a network representing the influential relationship between these sub-factors is constructed. ANP is then utilized to derive the relative weights of factors, sub-factors and failure modes by making pairwise comparisons with the help of PLTS. To verify the rationality and applicability of our proposed method, a case study of hospital information system reliability assessment is carried out. Comparative analyses with other existing FMEA methods are also undertaken to highlight the differences and advantages of our proposed method.

Appendices

Appendix 1: Abbreviation and Notation List

Abbreviation list

AHP	Analytic hierarchy process
ANP	Analytic network process
BWM	Best–worst method
COPRAS	COmplex PRoportional ASsessment of alternatives
FMEA	Failure modes and effects analysis
GRA	Grey relation analysis
HLTS	Hesitant linguistic term set
HIS	Hospital information system
LTS	Linguistic term set
MCDM	Multi-criteria decision making
MOORA	Multi-objective optimization by ratio analysis
MULTIMOORA	Multiple multi-objective optimization by ratio analysis
NASA	National Aeronautics and Space Administration
PLTS	Probabilistic linguistic term set
RPN	Risk prioritization number
SAW	Simple additive weighting
TFN	Triangular fuzzy numbers
TODIM	An acronym in Portuguese for interactive multi-criteria decision making
TOPSIS	Technique for order preference by similarity to an ideal solution
VIKOR	Vlse kriterijumska optimizacija kompromisno resenje

Notation list

$O$	Probability of occurrence
$S$	Severity
$D$	Likelihood of detection
${\text{RPN}}$	Risk prioritization number
$t_{\alpha }$	Linguistic terms
$T$	Additive linguistic term set
$\overline{T}$	Continuous linguistic term set
${\text{neg}}(t_{\alpha } )$	Negation operator of linguistic term set
$\max (t_{\alpha } ,\;t_{\beta } )$	Maximum operator of linguistic term set
$\min (t_{\alpha } ,\;t_{\beta } )$	Minimum operator of linguistic term set
$b_{T}$	Hesitant linguistic term set
$L(p)$	Probabilistic linguistic term set
$L^{(k)} (p^{(k)} )$	Linguistic term $L^{(k)}$ associated with probability $p^{(k)}$
$\# L(p)$	Number of all different linguistic terms in $L(p)$
$g\left( {L(p)} \right)$	Numerical score of $L(p)$
$e_{z} \;(z = 1:Z)$	The $z$th expert
${\text{FM}}_{m} \;(m = 1:M)$	The $m$th failure mode
$O_{i} \;(i = 1:h_{O} )$	The $i$th sub-factors under the category Occurrence $O$
$S_{i} \;(i = 1:h_{S} )$	The $i$th sub-factors under the category Severity $S$
$D_{i} \;(i = 1:h_{D} )$	The $i$th sub-factors under the category Detection $D$
$A$	Comparison matrix of risk factors $O$, $S$ and $D$
$L_{ij} (p)$	Comparing result of the $i$th risk factor to the $j$th risk factor
$\overline{{L_{ij} (p)}}$	Negation of $L_{ij} (p)$
$G(A)$	Crisp-valued matrix obtained by calculating the numerical score
${\text{CL}}_{ij}^{r}$	Consistent preference of the $i$th element over the $j$th element through the $r$th element
${\text{Consistency}}\;(G(A))$	Consistency level of $A$
$\gamma$	Required consistency level threshold
$w$	Weighting vector of three main risk factors
$A^{O} ,\;A^{S} ,\;A^{D}$	Comparison matrix of sub-factors
$\omega^{O} ,\;\omega^{S} ,\;\omega^{D}$	Weighting vector of sub-factors
$Q$	Super-matrix representing the influence of sub-factors
$A^{{O_{i} }}$	Comparison matrix of sub-factors regarding their influences on $O_{i}$
$(\dot{\omega }_{O1}^{Oi} ,\;\dot{\omega }_{O2}^{Oi} , \ldots ,\dot{\omega }_{{Oh_{O} }}^{Oi} )^{{\text{T}}}$	Normalized principal eigenvector of $G(A^{{O_{i} }} )$
$||Q^{c} - Q^{c - 1} ||_{2}$	Euclidean norm of the deviation between $Q^{c}$ and $Q^{c - 1}$
$\ddot{\omega }$	Long-term stable weight vector of sub-factors
$A_{FM}^{{O_{i} }}$	Comparison matrix of failure modes regarding the $i$th sub-factor $O_{i}$
$\varpi^{{O_{i} }}$	Weight of failure modes with respect to the $i$th sub-factor $O_{i}$
$RPN_{m}$	RPN of the $m$th failure mode
${\text{RPN}}_{m}^{C}$	group assessment of the RPN of the $m$th failure mode
$\varphi (L^{(k)} )$	Transformation function from linguistic terms to AHP scale
$\hat{\varphi }(\rho )$	Transformation function from PLTS numerical score to TFN scale

Appendix 2: Computation Process of Comparative Methods

1.1 Classical FMEA Method

As explained before, in classical FMEA method, the expert is instructed to rate the failure modes with a discrete numerical scale of 1–10. To ensure the fairness of comparison, the first step is to convert the entries in failure modes’ comparison matrix from PLTS to the numerical scale.

For example, the comparison of failure mode ${\text{FM}}_{1}$ to failure model ${\text{FM}}_{2}$ under sub-factor $O_{1}$ is $\{ t_{5} (0.8),\;t_{6} (0.2)\}$, then via Eq. (3) this entry is defuzzied to $5 \times 0.8 + 6 \times 0.2 = 5.2$, then rounded to $5$ because the ratings in classical FMEA is required to be integers. With similar calculations, expert $e_{1}$’s comparison matrices of failure modes with respect to sub-factor $O_{1}$ is obtained, as shown in Table 17.

Table 17 Comparison matrix of failure modes w.r.t. $O_{1}$ for classical FMEA method

In Table 17, the entries in the first row represents the criticality of failure mode ${\text{FM}}_{1}$ compared to all other failure modes with respect to sub-factor $O_{1}$. Thus, the arithmetic mean of the entries in the first row of Table 17 can be seen as the criticality rating of failure mode ${\text{FM}}_{1}$ with respect to sub-factor $O_{1}$. The same argument goes for other failure modes, and their criticality ratings are list in the first row of Table 18. Corresponding calculation results for sub-factor $O_{2}$ and $O_{3}$ are also listed in the second and third row of Table 18, with their calculation process omitted for compactness.

Table 18 Criticality rating of failure modes w.r.t. $O_{i} \;(i = 1:3)$ for classical FMEA method

Seeing that in classical FMEA method, only three main risk factors are considered, the criticality ratings of failure modes under sub-factors $O_{i} \;(i = 1:3)$ need to be further aggregated via the average operator, to derive the ratings under the main risk factor $O$, the result of which is listed in the first row of Table 19. Similar computations are also carried out for main risk factors $S$ and $D$, as shown in the second and third rows of Table 19. Finally, the ${\text{RPN}}$ can be calculated, and the ranking of failure modes via classical FMEA method can be obtained:

$$\begin{gathered} {\text{FM}}_{14} \succ {\text{FM}}_{13} \succ {\text{FM}}_{10} \succ {\text{FM}}_{11} \succ {\text{FM}}_{12} \succ {\text{FM}}_{1} \succ {\text{FM}}_{7} \hfill \\ \succ {\text{FM}}_{3} \succ {\text{FM}}_{8} \succ {\text{FM}}_{4} \succ {\text{FM}}_{5} \succ {\text{FM}}_{9} \succ {\text{FM}}_{6} \succ {\text{FM}}_{2} . \hfill \\ \end{gathered}$$

Table 19 Criticality ratings and RPNs of failure modes

1.2 Kiani Aslani et al.’s Method [32]

To counter the problem of equal weight assumption, Kiani Aslani R. et al. took the advantage of AHP to derive the relative weights of main risk factors, and defined the RPN as the weighted sum of these three factors. In [32], TFN is utilized to accommodate the uncertainty of expert judgements. Therefore, the comparison matrix in Table 5 needs to be converted into TFN to ensure the fairness of comparison. This conversion process is carried out in two steps. Firstly, the PLTS is converted into triangular forms by setting the lower and higher bounds as $L^{{(\min \{ k|L^{(k)} (p^{(k)} ) \in L(p)\} )}}$ and $L^{{(\max \{ k|L^{(k)} (p^{(k)} ) \in L(p)\} )}}$ respectively, and the center as $L^{(g(L(p)))}$. For instance, in Table 5, the comparison result of risk factor $O$ to risk factor $S$ is $\{ t_{3} (0.3),\;t_{4} (0.6)\}$. It is easy to see that $\min \{ 3,\;4\} = 3$, $\max \{ 3,\;4\} = 4$ and $g(L(p)) = 3 \times 0.3$$+ 4 \times 0.6 = 3.3$. Hence, the triangular form of $\{ t_{3} (0.3),\;t_{4} (0.6)\}$ is $(t_{3} ,\;t_{3.3} ,\;t_{4} )$.

Secondly, the triangular form of PLTS is transformed to TFN according to their practical meanings. In AHP method [64], the pairwise comparison of two elements are conducted with instructions in Table 20.

Table 20 Practical meanings of pairwise comparison result for AHP

Thus, according to the practical meanings in Table 4, the linguistic terms can be translated to the scale of AHP with the following equation:

$$\varphi (L^{(k)} ) = \left\{ {\begin{array}{*{20}c} {2(k - 4) + 1,} & {k \ge 4,} \\ {\frac{1}{2(4 - k) + 1},} & {{\text{otherwise}}.} \\ \end{array} } \right.$$

(17)

And the TFN obtained after conversion is:

$$\left( {\varphi (\min \{ k|L^{(k)} (p^{(k)} ) \in L(p)\} ),\;\varphi (g(L(p))),\;\varphi (\max \{ k|L^{(k)} (p^{(k)} ) \in L(p)\} )} \right).$$

(18)

Continuing with the above instance, via Eq. (17) the linguistic terms $t_{3}$, $t_{3.3}$ and $t_{4}$ are translated to:

$$\varphi (t_{3} ) = 1/(2 \times (4 - 3) + 1) = 1/3 \approx 0.33,$$

$$\varphi (t_{3.3} ) = 1/(2 \times (4 - 3.3) + 1) = 1/2.4 \approx 0.42,$$

$$\varphi (t_{4} ) = 1/(2 \times (4 - 4) + 1) = 1/1 = 1.$$

Therefore, the PLTS $\{ t_{3} (0.3),\;t_{4} (0.6)\}$ can be converted to a TFN $(0.33,\;0.42,\;1)$. Identical conversions are also carried out for other entries in Table 5, the results are presented in Table 21.

Table 21 Pairwise comparison matrix of the three main risk factors for method in [32]

For two TFNs $\lambda_{1} = (a_{1} ,\;b_{1} ,\;c_{1} )$ and $\lambda_{2} = (a_{2} ,\;b_{2} ,\;c_{2} )$, their operation laws [65] are defined as:

• (1) $(a_{1} ,\;b_{1} ,\;c_{1} ) \oplus (a_{2} ,\;b_{2} ,\;c_{2} ) = (a_{1} + a_{2} ,\;b_{1} + b_{2} ,\;c_{1} + c_{2} )$;

• (2) $(a_{1} ,\;b_{1} ,\;c_{1} ) \otimes (a_{2} ,\,b_{2} ,\;c_{2} ) = (a_{1} a_{2} ,\;b_{1} b_{2} ,\;c_{1} c_{2} )$;

• (3) $\mu \otimes (a_{1} ,\;b_{1} ,\;c_{1} ) = (\mu a_{1} ,\;\mu b_{1} ,\;\mu c_{1} )$;

• (4) $(a_{1} ,\;b_{1} ,\;c_{1} )^{ - 1} = (1/c_{1} ,\;1/b_{1} ,\;1/a_{1} )$;

• (5) $\tilde{g}(\lambda_{1} ) = (a_{1} + 4b_{1} + c_{1} )/6$.

Suppose the pairwise comparison matrix is $\tilde{A} = \left[ {\lambda_{i}^{j} } \right]_{n \times n}$, then the TFN weight of each element is calculated with the following formula [65]:

$$\tilde{w}_{i} = \sum\limits_{j = 1}^{n} {\lambda_{i}^{j} \otimes \left[ {\sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{n} {\lambda_{i}^{j} } } } \right]}^{ - 1} .$$

(19)

Thus, according to the comparison matrix in Table 21, the TFN weights of the three risk factors can be calculated via Eq. (19):

$$\tilde{w}_{O} = \left( {4.33,\;5.02,\;9} \right) \otimes \left( {\frac{1}{19.66},\;\frac{1}{12.1},\;\frac{1}{10.49}} \right) = \left( {0.22,\;0.41,\;0.86} \right),$$

$$\tilde{w}_{S} = \left( {4.83,\;5.60,\;9} \right) \otimes \left( {\frac{1}{19.66},\;\frac{1}{12.1},\;\frac{1}{10.49}} \right) = \left( {0.25,\;0.46,\;0.86} \right),$$

$$\tilde{w}_{D} = \left( {1.33,\;1.48,\;1.66} \right) \otimes \left( {\frac{1}{19.66},\;\frac{1}{12.1},\;\frac{1}{10.49}} \right) = \left( {0.07,\;0.12,\;0.16} \right).$$

Similar aggregation process as described in “Appendix 2.1” is also undertaken to derive the ratings of failure modes with respect to three main risk factors. The only difference is that in “Appendix 2.1” the ratings are given in integers, while here the ratings are transformed to TFNs, as shown in the second to fourth columns of Table 22.

Table 22 Ratings and RPNs of failure modes for Kiani Aslani et al.’s method [32]

Suppose the ratings of failure modes under risk factors $O$, $S$ and $D$ are $\lambda^{O}$, $\lambda^{S}$ and $\lambda^{D}$, respectively, then the RPN of failure modes are defined as:

$${\text{RPN}}_{i} = \tilde{g}\left( {(\tilde{w}_{O} \otimes \lambda^{O} ) \oplus (\tilde{w}_{S} \otimes \lambda^{S} ) \oplus (\tilde{w}_{D} \otimes \lambda^{D} )} \right).$$

The calculation results are shown in the last two columns of Table 22, and finally the ranking of failure modes can be obtained for Kiani Aslani R. et al.’s method [32]:

$\begin{gathered} {\text{FM}}_{14} \succ {\text{FM}}_{13} \succ {\text{FM}}_{10} \succ {\text{FM}}_{11} \succ {\text{FM}}_{12} \succ {\text{FM}}_{9} \succ {\text{FM}}_{7} \hfill \\ \succ {\text{FM}}_{1} \succ {\text{FM}}_{5} \succ {\text{FM}}_{4} \succ {\text{FM}}_{3} \succ {\text{FM}}_{6} \succ {\text{FM}}_{8} \succ {\text{FM}}_{2} . \hfill \\ \end{gathered}$.

1.3 Abdelgawad and Fayek’s Method [43]

In [43], the authors broke down risk factor $S$ to sub-factors and utilized classical AHP to derive their relative weights. Similar to the conversion process in “Appendix 2.2”, the entries of comparison matrix in Table 7 are first defuzzied via Eq. (3), then translated to the scale of AHP method via Eq. (17). Take the comparison result of sub-factor $S_{1}$ to sub-factor $S_{2}$ as an example, $\{ t_{5} (0.5),\;t_{6} (0.2)\}$ is first converted to $5 \times 0.5 + 6 \times 0.2 = 3.7$. Then according to the practical meanings listed in Table 20, linguistic term $t_{3.7}$ corresponds to $1/(2 \times (4 - 3.7) + 1) = 1/1.6 = 0.625$ in classical AHP method. The converted comparison matrix is as shown in Table 23. Moreover, the entries of the same column in Table 23 are normalized, then entries of the same row are summed together, which are then normalized to obtain the weights of corresponding sub-factors, as listed in the last column in Table 23.

Table 23 Comparison matrix and the corresponding weights of sub-factors for AHP method

Similar as in “Appendix 2.1”, the ratings of failure modes under different sub-factors are derived by calculating the arithmetic mean of the entries in the same row of the pairwise comparison matrices like Table 17. Because in [43] only the risk factor $S$ is broken down to sub-factors while $O$ and $D$ remains the same as in classical FMEA method, here the ratings of failure modes under risk factor $S$ is calculated as the weighted sum of ratings under $S_{i} \;(i = 1:5)$ with the weighting vector from Table 23. The ratings under risk factors $O$ and $D$ are the same as in “Appendix 2.1”, where sub-factors $O_{i} \;(i = 3)$ and $D_{i} \;(i = 1:3)$ are assigned equal importance. The calculation results are as listed in Table 24.

Table 24 Ratings and RPNs of failure modes for method in [43]

With the ratings under $O$, $S$ and $D$ obtained, the RPNs of failure modes can be calculated, as shown in the last column of Table 24. Thus, the ranking of failure modes via Abdelgawad and Fayek’s method [43] is:

$$\begin{gathered} {\text{FM}}_{14} \succ {\text{FM}}_{13} \succ {\text{FM}}_{10} \succ {\text{FM}}_{11} \succ {\text{FM}}_{12} \succ {\text{FM}}_{8} \succ {\text{FM}}_{1} \hfill \\ \succ {\text{FM}}_{4} \succ {\text{FM}}_{7} \succ {\text{FM}}_{6} \succ {\text{FM}}_{3} \succ {\text{FM}}_{5} \succ {\text{FM}}_{2} \succ {\text{FM}}_{9} . \hfill \\ \end{gathered}$$

1.4 Zandi et al.’s Method [29]

In [29], Zandi P. et al. broke down the risk factor $S$ to sub-factors and utilized AHP to derive both the weights of three main risk factors and the weights of sub-factors under the category of $S$. Seeing that in [29] the experts express their assessments in the form of TFNs, to ensure the fairness of comparison, first the comparison matrices in Tables 5 and 7 need to be converted to relevant forms.

As described in Table 25, a special kind of TFN scale is used in [29] where the deviation between lower bound and center of the TFN, as well as the deviation between the center and higher bound of the TFN is 2. In order to conform to this norm, first the PLTS entries in Tables 5 and 7 is defuzzied via the numerical score function in Eq. (3), then rounded to a positive integer. Suppose $\rho = \left\lfloor {g(L(p))} \right\rfloor$, then the conversion is completed by the following function:

$$\hat{\varphi }\left( \rho \right) = \left( {\begin{array}{*{20}c} {\left( {2(\rho - 4) - 1,\;2(\rho - 4) + 1,\;\min \left( {2(\rho - 4) + 3,\;9} \right)} \right),} & {\rho > 4,} \\ {\left( {1,\;1,\;1} \right),} & {\rho = 4,} \\ {(1/\min \left( {2(4 - \rho ) + 3,\;9} \right),\;1/(2(4 - \rho ) + 1),\;1/(2(4 - \rho ) - 1)),} & {{\text{otherwise}}{.}} \\ \end{array} } \right.$$

(20)

Table 25 Practical meaning of TFN scales used in [29]

The comparison result of risk factor $O$ to risk factor $S$ in Table 5, $\{ t_{3} (0.3),\;t_{4} (0.6)\}$, is taken once again as an example. It is easy to see that $g\left( {L(p)} \right) = 3 \times 0.3 + 4 \times 0.6 = 3.3$, then $\rho = \left\lfloor {g(L(p))} \right\rfloor$$= 3$. Via Eq. (20), the TFN scale after conversion is:

$$\hat{\varphi }\left( \rho \right) = (1/\min \left( {2(4 - 3) + 3,\;9} \right),\;1/(2(4 - 3) + 1),\;1/(2(4 - 3) - 1)) = (1/5,\;1/3,\;1).$$

Likewise, other entries in Tables 5 and 7 are converted following the above procedures, the results are presented in Tables 26 and 27.

Table 26 Pairwise comparison matrix of the three main risk factors for method in [29]

Table 27 Pairwise comparison matrix of sub-factors for method in [29]

Here, we introduce the principle of comparing TFNs. For any two TFNs $\lambda_{1} = (a_{1} ,\;b_{1} ,\;c_{1} )$ and $\lambda_{2} = (a_{2} ,\;b_{2} ,\;c_{2} )$, the possibility degree of $\lambda_{1} \ge \lambda_{2}$ is defined as:

$$\sigma \left( {\lambda_{1} \ge \lambda_{2} } \right) = \left\{ {\begin{array}{*{20}c} {1,} & {b_{1} \ge b_{2} ,} \\ {\frac{{a_{2} - c_{1} }}{{(b_{1} - c_{1} ) - (b_{2} - a_{2} )}},} & {{\text{otherwise}}.} \\ \end{array} } \right.$$

(21)

Moreover, the possibility degree that one TFN $\lambda = (a,\;b,\;c)$ is greater than $k$ other TFNs $\lambda_{i} \;(i = 1:k)$ is defined as:

$$\sigma \left( {\lambda \ge \lambda_{1} , \ldots ,\lambda_{k} } \right) = \min \left( {\sigma (\lambda \ge \lambda_{1} ), \ldots ,\sigma (\lambda \ge \lambda_{k} )} \right).$$

(22)

Next, let us suppose the pairwise comparison matrix is $\hat{A} = \left[ {\hat{\lambda }_{i}^{j} } \right]_{n \times n}$, then the synthetic extent with respect to the $i$th element is defined as:

$$V_{i} = \sum\limits_{j = 1}^{n} {\hat{\lambda }_{i}^{j} \otimes \left[ {\sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{n} {\hat{\lambda }_{i}^{j} } } } \right]}^{ - 1} .$$

(23)

Then, the normalized possibility degree of $V_{i} \ge V_{j} \;(j \ne i)$ is regarded as the weight of the $i$th element derived from comparison matrix.

For instance, according to the comparison matrix in Table 26, via Eq. (23) we have:

$$V_{1} = \left( {2.2,\;4.33,\;7} \right) \otimes \left( {\frac{1}{6.54},\;\frac{1}{10.86},\;\frac{1}{16.33}} \right) = \left( {0.13,\;0.40,\;1.07} \right),$$

$$V_{2} = \left( {3,\;5,\;7} \right) \otimes \left( {\frac{1}{6.54},\;\frac{1}{10.86},\;\frac{1}{16.33}} \right) = \left( {0.18,\;0.46,\;1.07} \right),$$

$$V_{3} = \left( {1.34,\;1.53,\;2.33} \right) \otimes \left( {\frac{1}{6.54},\;\frac{1}{10.86},\;\frac{1}{16.33}} \right) = \left( {0.08,\;0.14,\;0.36} \right).$$

Using Eqs. (21) and (22),

$$\sigma (V_{1} \ge V_{2} ) = \frac{0.18 - 1.07}{{(0.40 - 1.07) - (0.46 - 0.18)}} \approx 0.94,$$

$$\sigma (V_{1} \ge V_{3} ) = 1,\quad \sigma (V_{2} \ge V_{1} ) = 1,\quad \sigma (V_{2} \ge V_{3} ) = 1,$$

$$\sigma (V_{3} \ge V_{1} ) = \frac{0.13 - 0.36}{{(0.14 - 0.36) - (0.40 - 0.13)}} \approx 0.46,$$

$$\sigma (V_{3} \ge V_{2} ) = \frac{0.18 - 0.36}{{(0.14 - 0.36) - (0.46 - 0.18)}} \approx 0.35,$$

$$\sigma (V_{1} \ge V_{2} ,\;V_{3} ) = \min (0.94,\;1) = 0.94,$$

$$\sigma (V_{2} \ge V_{1} ,\;V_{3} ) = \min (1,\;1) = 1,$$

$$\sigma (V_{3} \ge V_{1} ,\;V_{2} ) = \min (0.46,\;0.35) = 0.35.$$

Furthermore, the vector $(0.94,\;1,\;0.35)^{{\text{T}}}$ is normalized to $(0.41,\;0.44,\;0.15)^{{\text{T}}}$, which serves as the weighting vector of the three main risk factors. The same computing process is also performed for the comparison matrix in Table 27, the result of which is shown in the first row of Table 28. Same as in “Appendix 2.3”, the ratings of failure modes are transformed to TFN then defuzzied, as shown in Table 28. Then, the RPN of failure modes can be defined as their weighted sum, as presented in the last column of Table 28. Finally, the ranking of failure modes via Zandi et al.’s method [29] is obtained:

Table 28 Ratings and RPNs of failure modes for method in [29]

$\begin{gathered} {\text{FM}}_{13} \succ {\text{FM}}_{14} \succ {\text{FM}}_{10} \succ {\text{FM}}_{11} \succ {\text{FM}}_{12} \succ {\text{FM}}_{1} \succ {\text{FM}}_{9} \hfill \\ \succ {\text{FM}}_{7} \succ {\text{FM}}_{4} \succ {\text{FM}}_{5} \succ {\text{FM}}_{6} \succ {\text{FM}}_{3} \succ {\text{FM}}_{8} \succ {\text{FM}}_{2} . \hfill \\ \end{gathered}$.