1 Introduction
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• Consideration of crucial management indices, such as cost and time, in the process of prioritizing risks with SOD factors.
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• Assignation of different weights to risk factors according to the uncertainty of decision-makers’ preferences with the aim of overcoming the deficiencies of traditional RPN score and making results more interpretable.
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• Simultaneous consideration of the concepts of uncertainty (U) and reliability (R) in the processes of weighting risk factors and prioritizing failures by using Z-number theory.
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• Complete prioritization of failures and distinction between failure ranks by using the SWARA–MOORA integrated method based on Z-number theory.
2 Literature Review
2.1 FMEA and Hybrid Approaches
2.2 SWARA and MOORA Applications
2.3 Z-Number Theory
3 Methodology
3.1 Fuzzy Logic Theory
Definition 1.
Definition 2.
Definition 3.
(3)
\[\begin{aligned}{}& \tilde{A}\oplus \tilde{B}=({l_{1}}+{l_{2}},{m_{1}}+{m_{2}},{u_{1}}+{u_{2}}),\end{aligned}\](4)
\[\begin{aligned}{}& \tilde{A}\otimes \tilde{B}=({l_{1}}{l_{2}},{m_{1}}{m_{2}},{u_{1}}{u_{2}}),\end{aligned}\](5)
\[\begin{aligned}{}& \tilde{A}\square \tilde{B}=({l_{1}}-{u_{2}},{m_{1}}-{m_{2}},{u_{1}}-{l_{2}}),\end{aligned}\]3.2 Z-Numbers
3.3 Z-SWARA
Table 1
Linguistics terms | Membership function |
Equally Important (EI) | $(1,1,1)$ |
Moderately less important (MOL) | $(2/3,1,3/2)$ |
Less important (LI) | $(2/5,1/2,2/3)$ |
Very less Important (VLI) | $(2/7,1/3,2/5)$ |
Much less important (MUL) | $(2/9,1/4,2/7)$ |
Table 2
Linguistic variables | Very Low (VL) | Low (L) | Medium (M) | High (H) | Very High (VH) |
TFNs | $(0,0,0.3)$ | $(0.1,0.3,0.5)$ | $(0.3,0.5,0.7)$ | $(0.5,0.7,0.9)$ | $(0.7,1.0,1.0)$ |
Table 3
Linguistics terms | Membership function | Linguistics terms | Membership function |
(EI, VL) | $(1,1,1)$ | (EI, L) | $(1,1,1)$ |
(EI, M) | $(1,1,1)$ | (EI, H) | $(1,1,1)$ |
(EI, VH) | $(1,1,1)$ | (MOL, VL) | $(0.212,0.316,0.474)$ |
(MOL, L) | $(0.367,0.548,0.822)$ | (MOL, M) | $(0.474,0.707,1.061)$ |
(MOL, H) | $(0.561,0.837,1.255)$ | (MOL, VH) | $(0.636,0.949,1.423)$ |
(LI, VL) | $(0.126,0.158,0.212)$ | (LI, L) | $(0.219,0.274,0.367)$ |
(LI, M) | $(0.283,0.354,0.474)$ | (LI, H) | $(0.335,0.418,0.561)$ |
(LI, VH) | $(0.379,0.474,0.636)$ | (VLI, VL) | $(0.092,0.104,0.126)$ |
(VLI, L) | $(0.159,0.181,0.219)$ | (VLI, M) | $(0.205,0.233,0.283)$ |
(VLI, H) | $(0.243,0.276,0.335)$ | (VLI, VH) | $(0.275,0.313,0.379)$ |
(MUL, VL) | $(0.069,0.079,0.092)$ | (MUL, L) | $(0.120,0.137,0.159)$ |
(MUL, M) | $(0.155,0.177,0.205)$ | (MUL, H) | $(0.184,0.209,0.243)$ |
(MUL, VH) | $(0.209,0.237,0.275)$ |
(16)
\[ {\tilde{w}_{j}}=\frac{{\tilde{q}_{j}}}{{\textstyle\textstyle\sum _{k=1}^{n}}{\tilde{q}_{k}}}.\]3.4 Z-MOORA
3.4.1 Z-Ratio Method
(17)
\[ Z=\left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}[({x_{11}^{l}},{x_{11}^{m}},{x_{11}^{u}}),({y_{11}^{l}},{y_{11}^{m}}{,_{11}^{u}})]\hspace{1.5pt}& [({x_{12}^{l}},{x_{12}^{m}},{x_{12}^{u}}),({y_{12}^{l}},{y_{12}^{m}}{,_{12}^{u}})]\hspace{1.5pt}& \dots \hspace{1.5pt}& [({x_{1n}^{l}},{x_{1n}^{m}},{x_{1n}^{u}}),({y_{1n}^{l}},{y_{1n}^{m}},{y_{1n}^{u}})]\\ {} \dots \hspace{1.5pt}& \dots \hspace{1.5pt}& \dots \hspace{1.5pt}\\ {} \dots \hspace{1.5pt}& \dots \hspace{1.5pt}& \dots \hspace{1.5pt}\\ {} [({x_{m1}^{l}},{x_{m1}^{m}},{x_{m1}^{u}}),({y_{m1}^{l}},{y_{m1}^{m}}{,_{m1}^{u}})]\hspace{1.5pt}& [({x_{m2}^{l}},{x_{m2}^{m}},{x_{m2}^{u}}),({y_{m2}^{l}},{y_{m2}^{m}}{,_{m2}^{u}})]\hspace{1.5pt}& \dots \hspace{1.5pt}& [({x_{mn}^{l}},{x_{mn}^{m}},{x_{mn}^{u}}),({y_{mn}^{l}},{y_{mn}^{m}},{y_{mn}^{u}})]\end{array}\right].\]Table 4
Linguistic variables | Very Low (VL) | Low (L) | Medium Low (ML) | Medium (M) | Medium High (MH) | High (H) | Very High (VH) |
TFNs | $(0,0,1)$ | $(0,1,3)$ | $(1,3,5)$ | $(3,5,7)$ | $(5,7,9)$ | $(7,9,10)$ | $(9,10,10)$ |
Table 5
Linguistic Variables | Very Low (VL) | Low (L) | Medium (M) | High (H) | Very High (VH) |
TFNs | $(0,0,0.3)$ | $(0.1,0.3,0.5)$ | $(0.3,0.5,0.7)$ | $(0.5,0.7,0.9)$ | $(0.7,1.0,1.0)$ |
(20)
\[ \tilde{D}=\left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}({d_{11}^{l}},{d_{11}^{m}},{d_{11}^{u}})\hspace{1em}& ({d_{12}^{l}},{d_{12}^{m}},{d_{12}^{u}})\hspace{1em}& \dots \hspace{1em}& ({d_{1n}^{l}},{d_{1n}^{m}},{d_{1n}^{u}})\\ {} \dots \hspace{1em}& \dots \hspace{1em}& \dots \hspace{1em}\\ {} \dots \hspace{1em}& \dots \hspace{1em}& \dots \hspace{1em}\\ {} ({d_{m1}^{l}},{d_{m1}^{m}},{d_{m1}^{u}})\hspace{1em}& ({d_{m2}^{l}},{d_{m2}^{m}},{d_{m2}^{u}})\hspace{1em}& \dots \hspace{1em}& ({d_{mn}^{l}},{d_{mn}^{m}},{d_{mn}^{u}})\end{array}\right].\]Table 6
Linguistics terms | Membership function | Linguistics terms | Membership function |
(VH, VH) | $(8.54,9.49,9.49)$ | (VH, H) | $(7.53,8.37,8.37)$ |
(VH, M) | $(6.36,7.07,7.07)$ | (VH, L) | $(4.93,5.48,5,48)$ |
(VH, VL) | $(2.85,3.16,3.16)$ | (H, VH) | $(6.64,8.54,9.49)$ |
(H, H) | $(5.86,7.53,8.37)$ | (H, M) | $(4.95,6.36,7.07)$ |
(H, L) | $(3.84,4.93,5.48)$ | (H, VL) | $(2.21,2.85,3.16)$ |
(MH, VH) | $(4.74,6.64,8.54)$ | (MH, H) | $(4.18,5.86,7.53)$ |
(MH, M) | $(3.54,4.95,6.36)$ | (MH, L) | $(2.74,3.84,4.93)$ |
(MH, VL) | $(1.58,2.21,2.85)$ | (M, VH) | $(2.85,4.74,6.64)$ |
(M, H) | $(2.51,4.28,5.86)$ | (M, M) | $(2.12,3.54,4.95)$ |
(M, L) | $(1.64,2.74,3.83)$ | (M, VL) | $(0.95,1.58,2.21)$ |
(ML, VH) | $(0.95,2.85,4.74)$ | (ML, H) | $(0.84,2.51,4.18)$ |
(ML, M) | $(0.71,2.12,3.54)$ | (ML, L) | $(0.55,1.64,2.74)$ |
(ML, VL) | $(0.32,0.95,1.58)$ | (L, VH) | $(0,0.95,2.85)$ |
(L, H) | $(0,0.84,2.51)$ | (L, M) | $(0,0.71,2.12)$ |
(L, L) | $(0,0.55,1.64)$ | (L, VL) | $(0,0.32,0.95)$ |
(VL, VH) | $(0,0,0.95)$ | (VL, H) | $(0,0,0.84)$ |
(VL, M) | $(0,0,0.71)$ | (VL, L) | $(0,0,0.55)$ |
(VL, VL) | $(0,0,0.32)$ |
(21)
\[\begin{aligned}{}& {d_{ij}^{m\ast }}=\frac{{d_{ij}^{m}}}{\sqrt{{\textstyle\textstyle\sum _{i=1}^{m}}[{({d_{ij}^{l}})^{2}}+{({d_{ij}^{m}})^{2}}+{({d_{ij}^{u}})^{2}}]}},\\ {} & {d_{ij}^{u\ast }}=\frac{{d_{ij}^{u}}}{\sqrt{{\textstyle\textstyle\sum _{i=1}^{m}}[{({d_{ij}^{l}})^{2}}+{({d_{ij}^{m}})^{2}}+{({d_{ij}^{u}})^{2}}]}}.\end{aligned}\](22)
\[ \tilde{P}=\left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}({p_{11}^{l}},{p_{11}^{m}},{p_{11}^{u}})\hspace{1em}& ({p_{12}^{l}},{p_{12}^{m}},{p_{12}^{u}})\hspace{1em}& \dots \hspace{1em}& ({p_{1n}^{l}},{p_{1n}^{m}},{p_{1n}^{u}})\\ {} \dots \hspace{1em}& \dots \hspace{1em}& \dots \hspace{1em}\\ {} \dots \hspace{1em}& \dots \hspace{1em}& \dots \hspace{1em}\\ {} ({p_{m1}^{l}},{p_{m1}^{m}},{p_{m1}^{u}})\hspace{1em}& ({p_{m2}^{l}},{p_{m2}^{m}},{p_{m2}^{u}})\hspace{1em}& \dots \hspace{1em}& ({p_{mn}^{l}},{p_{mn}^{m}},{p_{mn}^{u}})\end{array}\right].\](23)
\[ {\tilde{y}_{i}}={\sum \limits_{j=1}^{g}}{\tilde{p}_{ij}}-{\sum \limits_{j=g+1}^{n}}{\tilde{p}_{ij}},\]4 Proposed Approach
Table 7
Rating | Severity (S) | Occurrence (O) | Detection (D) | Expected Cost (C) | Time (T) |
10 | Hazardous without warning | Very high: failure is almost inevitable | Absolute uncertainty | Product/system repair cost is close to the original price | Product/system repair time extremely high |
9 | Hazardous with warning | Product/system repair cost extremely high | |||
8 | Very high | High: repeated failures | High: repeated failures | Product/system repair cost very high | Product/system repair time very high |
7 | High | Product/system repair cost high | Product/system repair time high | ||
6 | Moderate | Moderate: occasional failures | Moderate: occasional failures | Product/system repair cost moderately high | Product/system repair time moderate |
5 | Low | Product/system repair cost moderate | |||
4 | Very low | Product/system repair cost relatively low | |||
Product/system repair time low | |||||
3 | Minor | Product/system repair cost low | |||
2 | Very minor | Low: relatively few failures | Low: relatively few failures | Product/ system repair cost very low | Product/ system repair cost very low |
1 | None | Remote: failure is unlikely | Remote: failure is unlikely | Repair at nearly no cost |
5 Analysing the Results
5.1 Case Study
5.2 Results
Table 8
Failure | Failure name | Severity (S) | Occurrence (O) | Detection (D) | Cost (C) | Time (T) | ||||||||||
TM1 | TM2 | TM3 | TM1 | TM2 | TM3 | TM1 | TM2 | TM3 | TM1 | TM2 | TM3 | TM1 | TM2 | TM3 | ||
F1 | Falling parts when lifting them | 4 | 5 | 6 | 6 | 7 | 8 | 7 | 6 | 5 | 2 | 3 | 4 | 4 | 3 | 5 |
F2 | Pallet falling down when load transportation | 10 | 9 | 8 | 6 | 5 | 4 | 4 | 3 | 2 | 6 | 4 | 5 | 5 | 4 | 6 |
F3 | Collision with personnel in loading | 8 | 7 | 9 | 2 | 3 | 4 | 5 | 4 | 6 | 4 | 3 | 2 | 4 | 5 | 3 |
F4 | Lift truck collision with shelves and falling down of shelves | 9 | 7 | 8 | 5 | 4 | 6 | 2 | 3 | 4 | 5 | 6 | 7 | 3 | 4 | 5 |
F5 | Ergonomic problems when using the computer | 7 | 6 | 5 | 7 | 6 | 5 | 6 | 5 | 4 | 5 | 4 | 3 | 3 | 2 | 1 |
F6 | Fire in warehouse | 7 | 6 | 8 | 6 | 5 | 4 | 5 | 3 | 4 | 4 | 5 | 6 | 3 | 5 | 4 |
F7 | Falling parts when setting them | 8 | 10 | 9 | 4 | 2 | 3 | 6 | 5 | 4 | 6 | 5 | 4 | 5 | 3 | 4 |
F8 | Dealing with people and equipment when driving lift trucks | 5 | 6 | 4 | 2 | 3 | 4 | 4 | 6 | 5 | 6 | 4 | 5 | 9 | 8 | 10 |
Table 9
Failure mode | S | O | D | C | T | ||||||||||
TM1 | TM2 | TM3 | TM1 | TM2 | TM3 | TM1 | TM2 | TM3 | TM1 | TM2 | TM3 | TM1 | TM2 | TM3 | |
F1 | ML, VH | M, H | M, H | MH, H | MH, H | H, VH | MH, M | M, H | M, M | L, VH | ML, M | M, VH | ML, M | ML, H | MH, VH |
F2 | VH, VH | H, VH | H, VH | MH, H | MH, VH | M, VH | M, VH | ML, VH | L, M | M, H | ML, VH | M, M | M, VH | ML, VH | MH, H |
F3 | H, H | MH, VH | VH, H | L, H | ML, VH | M, H | M, M | ML, M | MH, H | ML, M | ML, VH | L, VH | ML, H | MH, VH | ML, M |
F4 | H, M | M, VH | MH, VH | M, VH | ML, VH | MH, VH | L, M | ML, VH | M, H | M, H | MH, VH | MH, H | M, H | ML, VH | MH, VH |
F5 | MH, M | MH, VH | M, H | H, VH | MH, H | M, M | M, H | ML, H | ML, H | MH, VH | ML, VH | ML, H | ML, M | L, H | L, VH |
F6 | MH, VH | H, M | H, VH | ML, H | M, M | ML, VH | M, H | ML, VH | MH, M | ML, M | ML, VH | M, H | M, VH | MH, M | ML, H |
F7 | MH, M | H, VH | MH, M | ML, VH | L, VH | ML, M | MH, VH | MH, M | ML, H | M, H | M, M | M, VH | M, M | L, H | ML, M |
F8 | M, VH | MH, VH | MH, M | L, VH | ML, M | M, VH | MH, H | MH, VH | M, VH | ML, H | M, H | MH, VH | H, M | H, VH | MH, H |
Table 10
TM1 | TM2 | TM3 | |||
Risk factor | Relative importance | Risk factor | Relative importance | Risk factor | Relative importance |
S | C | S | |||
C | (MOL, H) | S | (LI, VH) | C | (MOL, H) |
D | (LI, VH) | D | (MOL, H) | D | (VLI, M) |
O | (MUL, VH) | O | (VLI, M) | T | (VLI, H) |
T | (VLI, M) | T | (MUL, VH) | O | (MUL, VH) |
Table 11
Team number | Risk factor | Comparative importance of average value ${\tilde{S}_{j}}$ | Coefficient ${\tilde{k}_{j}}={\tilde{s}_{j}}+\tilde{1}$ | Recalculated weight ${\tilde{q}_{j}}=\frac{{\tilde{x}_{j-1}}}{{\tilde{k}_{j}}}$ | Weight ${\tilde{w}_{j}}=\frac{{\tilde{q}_{j}}}{{\textstyle\sum _{k=1}^{n}}{\tilde{q}_{k}}}$ | ||||||||
TM1 | S | 1 | 1 | 1 | 1.000 | 1.000 | 1.000 | 0.377 | 0.434 | 0.510 | |||
C | 0.561 | 0.837 | 1.255 | 1.561 | 1.837 | 2.255 | 0.443 | 0.544 | 0.641 | 0.167 | 0.237 | 0.327 | |
D | 0.379 | 0.474 | 0.636 | 1.379 | 1.474 | 1.636 | 0.271 | 0.369 | 0.465 | 0.102 | 0.160 | 0.237 | |
O | 0.209 | 0.237 | 0.275 | 1.209 | 1.237 | 1.275 | 0.213 | 0.299 | 0.384 | 0.080 | 0.130 | 0.196 | |
T | 0.205 | 0.233 | 0.283 | 1.205 | 1.233 | 1.283 | 0.166 | 0.242 | 0.319 | 0.062 | 0.105 | 0.163 | |
TM2 | C | 1 | 1 | 1 | 1.000 | 1.000 | 1.000 | 0.364 | 0.408 | 0.466 | |||
S | 0.379 | 0.474 | 0.636 | 1.379 | 1.474 | 1.636 | 0.611 | 0.678 | 0.725 | 0.222 | 0.277 | 0.338 | |
D | 0.561 | 0.837 | 1.255 | 1.561 | 1.837 | 2.255 | 0.271 | 0.369 | 0.465 | 0.099 | 0.150 | 0.217 | |
O | 0.205 | 0.233 | 0.283 | 1.205 | 1.233 | 1.283 | 0.211 | 0.300 | 0.386 | 0.077 | 0.122 | 0.180 | |
T | 0.209 | 0.237 | 0.275 | 1.209 | 1.237 | 1.275 | 0.166 | 0.242 | 0.319 | 0.060 | 0.099 | 0.149 | |
TM3 | S | 1 | 1 | 1 | 1.000 | 1.000 | 1.000 | 0.367 | 0.420 | 0.489 | |||
C | 0.561 | 0.837 | 1.255 | 1.561 | 1.837 | 2.255 | 0.443 | 0.544 | 0.641 | 0.163 | 0.230 | 0.313 | |
D | 0.205 | 0.233 | 0.283 | 1.205 | 1.233 | 1.283 | 0.346 | 0.441 | 0.532 | 0.127 | 0.185 | 0.259 | |
T | 0.243 | 0.276 | 0.335 | 1.243 | 1.276 | 1.335 | 0.259 | 0.346 | 0.428 | 0.095 | 0.145 | 0.209 | |
O | 0.209 | 0.237 | 0.275 | 1.209 | 1.237 | 1.275 | 0.203 | 0.280 | 0.354 | 0.074 | 0.117 | 0.172 |
Table 12
Risk factors | Final weight | ||
S | 0.322 | 0.377 | 0.445 |
C | 0.231 | 0.291 | 0.369 |
D | 0.109 | 0.166 | 0.238 |
O | 0.077 | 0.123 | 0.183 |
T | 0.073 | 0.116 | 0.173 |
Table 13
Factor | Failure | TM1 | TM2 | TM3 | ||||||
l | m | u | l | m | u | l | M | u | ||
S | F1 | 0.95 | 2.85 | 4.74 | 2.51 | 4.28 | 5.86 | 2.51 | 4.28 | 5.86 |
F2 | 8.54 | 9.49 | 9.49 | 6.64 | 8.54 | 9.49 | 6.64 | 8.54 | 9.49 | |
F3 | 5.86 | 7.53 | 8.37 | 4.74 | 6.64 | 8.54 | 7.53 | 8.37 | 8.37 | |
F4 | 4.95 | 6.36 | 7.07 | 2.85 | 4.74 | 6.64 | 4.74 | 6.64 | 8.54 | |
F5 | 3.54 | 4.95 | 6.36 | 4.74 | 6.64 | 8.54 | 2.51 | 4.28 | 5.86 | |
F6 | 4.74 | 6.64 | 8.54 | 4.95 | 6.36 | 7.07 | 6.64 | 8.54 | 9.49 | |
F7 | 3.54 | 4.95 | 6.36 | 6.64 | 8.54 | 9.49 | 3.54 | 4.95 | 6.36 | |
F8 | 2.85 | 4.74 | 6.64 | 4.74 | 6.64 | 8.54 | 3.54 | 4.95 | 6.36 | |
O | F1 | 4.18 | 5.86 | 7.53 | 4.18 | 5.86 | 7.53 | 6.64 | 8.54 | 9.49 |
F2 | 4.18 | 5.86 | 7.53 | 4.74 | 6.64 | 8.54 | 2.85 | 4.74 | 6.64 | |
F3 | 0 | 0.84 | 2.51 | 0.95 | 2.85 | 4.74 | 2.51 | 4.28 | 5.86 | |
F4 | 2.85 | 4.74 | 6.64 | 0.95 | 2.85 | 4.74 | 4.74 | 6.64 | 8.54 | |
F5 | 6.64 | 8.54 | 9.49 | 4.18 | 5.86 | 7.53 | 2.12 | 3.54 | 4.95 | |
F6 | 0.84 | 2.51 | 4.18 | 2.12 | 3.54 | 4.95 | 0.95 | 2.85 | 4.74 | |
F7 | 0.95 | 2.85 | 4.74 | 0 | 0.95 | 2.85 | 0.71 | 2.12 | 3.54 | |
F8 | 0 | 0.95 | 2.85 | 0.71 | 2.12 | 3.54 | 2.85 | 4.74 | 6.64 | |
D | F1 | 3.54 | 4.95 | 6.36 | 2.51 | 4.28 | 5.86 | 2.12 | 3.54 | 4.95 |
F2 | 2.85 | 4.74 | 6.64 | 0.95 | 2.85 | 4.74 | 0 | 0.71 | 2.12 | |
F3 | 2.12 | 3.54 | 4.95 | 0.71 | 2.12 | 3.54 | 4.18 | 5.86 | 7.53 | |
F4 | 0 | 0.71 | 2.12 | 0.95 | 2.85 | 4.74 | 2.51 | 4.28 | 5.86 | |
F5 | 2.51 | 4.28 | 5.86 | 0.84 | 2.51 | 4.18 | 0.84 | 2.51 | 4.18 | |
F6 | 2.51 | 4.28 | 5.86 | 0.95 | 2.85 | 4.74 | 3.54 | 4.95 | 6.36 | |
F7 | 4.74 | 6.64 | 8.54 | 3.54 | 4.95 | 6.36 | 0.84 | 2.51 | 4.18 | |
F8 | 4.18 | 5.86 | 7.53 | 4.74 | 6.64 | 8.54 | 2.85 | 4.74 | 6.64 | |
C | F1 | 0 | 0.95 | 2.85 | 0.71 | 2.12 | 3.54 | 2.85 | 4.74 | 6.64 |
F2 | 2.51 | 4.28 | 5.86 | 0.95 | 2.85 | 4.74 | 2.12 | 3.54 | 4.95 | |
F3 | 0.71 | 2.12 | 3.54 | 0.95 | 2.85 | 4.74 | 0 | 0.95 | 2.85 | |
F4 | 2.51 | 4.28 | 5.86 | 4.74 | 6.64 | 8.54 | 4.18 | 5.86 | 7.53 | |
F5 | 4.74 | 6.64 | 8.54 | 0.95 | 2.85 | 4.74 | 0.84 | 2.51 | 4.18 | |
F6 | 0.71 | 2.12 | 3.54 | 0.95 | 2.85 | 4.74 | 2.51 | 4.28 | 5.86 | |
F7 | 2.51 | 4.28 | 5.86 | 2.12 | 3.54 | 4.95 | 2.85 | 4.74 | 6.64 | |
F8 | 0.84 | 2.51 | 4.18 | 2.51 | 4.28 | 5.86 | 4.74 | 6.64 | 8.54 | |
T | F1 | 0.71 | 2.12 | 3.54 | 0.84 | 2.51 | 4.18 | 4.74 | 6.64 | 8.54 |
F2 | 2.85 | 4.74 | 6.64 | 0.95 | 2.85 | 4.74 | 4.18 | 5.86 | 7.53 | |
F3 | 0.84 | 2.51 | 4.18 | 4.74 | 6.64 | 8.54 | 0.71 | 2.12 | 3.54 | |
F4 | 2.51 | 4.28 | 5.86 | 0.95 | 2.85 | 4.74 | 4.74 | 6.64 | 8.54 | |
F5 | 0.71 | 2.12 | 3.54 | 0 | 0.84 | 2.51 | 0 | 0.95 | 2.85 | |
F6 | 2.85 | 4.74 | 6.64 | 3.54 | 4.95 | 6.36 | 0.84 | 2.51 | 4.18 | |
F7 | 2.12 | 3.54 | 4.95 | 0 | 0.84 | 2.51 | 0.71 | 2.12 | 3.54 | |
F8 | 4.95 | 6.36 | 7.07 | 6.64 | 8.54 | 9.49 | 4.18 | 5.86 | 7.53 |
Table 14
S | O | D | C | T | ||||||||||
0.093 | 0.300 | 0.618 | 0.228 | 0.439 | 0.713 | 0.117 | 0.265 | 0.491 | 0.052 | 0.182 | 0.448 | 0.042 | 0.139 | 0.317 |
1.242 | 1.626 | 1.850 | 0.140 | 0.318 | 0.610 | 0.025 | 0.112 | 0.303 | 0.127 | 0.340 | 0.638 | 0.068 | 0.198 | 0.428 |
0.858 | 1.170 | 1.458 | 0.012 | 0.068 | 0.203 | 0.086 | 0.216 | 0.427 | 0.011 | 0.105 | 0.327 | 0.042 | 0.139 | 0.317 |
0.410 | 0.725 | 1.130 | 0.074 | 0.217 | 0.469 | 0.021 | 0.100 | 0.269 | 0.532 | 0.840 | 1.268 | 0.071 | 0.207 | 0.439 |
0.304 | 0.580 | 0.983 | 0.169 | 0.345 | 0.571 | 0.031 | 0.140 | 0.337 | 0.174 | 0.429 | 0.804 | 0.001 | 0.017 | 0.095 |
0.696 | 1.068 | 1.438 | 0.015 | 0.085 | 0.227 | 0.086 | 0.237 | 0.479 | 0.071 | 0.255 | 0.527 | 0.055 | 0.163 | 0.353 |
0.491 | 0.783 | 1.126 | 0.003 | 0.038 | 0.146 | 0.146 | 0.323 | 0.606 | 0.228 | 0.470 | 0.803 | 0.008 | 0.046 | 0.145 |
0.323 | 0.614 | 1.059 | 0.013 | 0.065 | 0.201 | 0.243 | 0.483 | 0.859 | 0.266 | 0.538 | 0.910 | 0.264 | 0.471 | 0.695 |
Table 15
Failure | Z-MOORA | ||||
${\tilde{y}_{i}}$ | ${\bar{y}_{i}}$ | Rank | |||
l | m | u | |||
F1 | 0.531 | 1.325 | 2.586 | 1.481 | 8 |
F2 | 1.602 | 2.593 | 3.829 | 2.675 | 1 |
F3 | 1.009 | 1.697 | 2.732 | 1.813 | 5 |
F4 | 1.108 | 2.088 | 3.575 | 2.257 | 3 |
F5 | 0.678 | 1.511 | 2.789 | 1.660 | 7 |
F6 | 0.923 | 1.808 | 3.025 | 1.919 | 4 |
F7 | 0.876 | 1.660 | 2.826 | 1.787 | 6 |
F8 | 1.109 | 2.171 | 3.723 | 2.335 | 2 |
Table 16
Failure | Conventional FMEA | Fuzzy MOORA | Z-MOORA | |||
RPN | Rank | ${\bar{y}_{i}}$ | Rank | ${\bar{y}_{i}}$ | Rank | |
F1 | 2520 | 5 | 1.596 | 8 | 1.481 | 8 |
F2 | 3375 | 1 | 2.635 | 1 | 2.675 | 1 |
F3 | 1440 | 6 | 2.062 | 6 | 1.813 | 5 |
F4 | 2880 | 2 | 2.403 | 3 | 2.257 | 3 |
F5 | 1440 | 6 | 1.837 | 7 | 1.660 | 7 |
F6 | 2800 | 3 | 2.216 | 5 | 1.919 | 4 |
F7 | 2700 | 4 | 2.224 | 4 | 1.787 | 6 |
F8 | 3375 | 1 | 2.406 | 2 | 2.335 | 2 |
5.3 Sensitivity Analysis
Table 17
S | O | D | C | T | |
Case 0 | 0.379 | 0.125 | 0.168 | 0.294 | 0.119 |
Case 1 | 0.354 | 0.100 | 0.143 | 0.394 | 0.094 |
Case 2 | 0.329 | 0.075 | 0.368 | 0.244 | 0.069 |
Case 3 | 0.319 | 0.365 | 0.108 | 0.234 | 0.059 |
Case 4 | 0.309 | 0.055 | 0.098 | 0.224 | 0.399 |