1 Introduction
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i. Considering that a heavy-duty machine tool is composed of many subsystems with different functions and its reliability depends primarily on the reliability of the weakest subsystem, a weakness ranking method based on the generalized FMECA information is proposed to determine the weakest subsystem of the heavy-duty machine tool.
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ii. Considering the complex maintenance of the heavy-duty machine tool, the maintainability and maintenance cost are considered in the generalized FMECA information. Subsequently, eight reliability indexes are considered as the FMECA information to comprehensively analyze the failure of the subsystems.
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iii. To reduce the effects of subjective cognitive differences on the analysis result, the CBWM is applied to calculate the weight of each screened index, and the TOPSIS is used to rank the weaknesses of all subsystems accurately and rapidly.
2 Generalized FMECA Information and Data Preprocessing
2.1 Generalized FMECA Information
2.1.1 Failure Rate
2.1.2 Failure Impact
Category | Description | ESR Score | Trapezoidal fuzzy number ESR Score |
---|---|---|---|
Class I (Disastrous) | Major failure occurs; loses specified function; causes major safety accidents, casualties, and significant damage | 10, 9 | \(\widetilde{R}_{1}\)= (8, 9, 10, 10) |
Class II (Deadly) | Severe damage; loses specified function; no casualties occurs | 8, 7 | \(\widetilde{R}_{2}\) = (6, 7, 8, 9) |
Class III (Crisis) | Specified function degrades; loses partial performance | 6, 5, 4 | \(\widetilde{R}_{3}\) = (3, 4, 6, 7) |
Class IV (Mild) | Minor failure occurs; specified function degrades; acceptable performance | 3, 2, 1 | \(\widetilde{R}_{4}\) = (1, 1, 3, 4) |
2.1.3 Detection Difficulty
Category | Description | DDR Score | Trapezoidal fuzzy number DDR Score |
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Class I (Cannot detect) | Almost impossible to be detected | 10 | \(\widetilde{r}_{1}\) = (9, 10, 10, 10) |
Class II (Very difficult to detect) | Slight possibility of being detected | 9, 8, 7 | \(\widetilde{r}_{2}\) = (6, 7, 9, 10) |
Class III (Difficult to detect) | Can be detected on the spot or during disassembly | 6, 5, 4 | \(\widetilde{r}_{3}\) = (3, 4, 6, 7) |
Class IV (Able to detect) | Self-warning | 3, 2 | \(\widetilde{r}_{4}\) = (1, 2, 3, 4) |
Class V (Easy to detect) | Visual detection | 1 | \(\widetilde{r}_{5}\) = (1, 1, 1, 2) |
2.1.4 Criticality Degree
Category | Description | Reference value of failure frequency | OPR score |
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Class I | Occur frequently | > 20% | 10 |
Class II | Occur sometimes | 10%–20% | 9, 8, 7 |
Class III | Occur occasionally | 1%–10% | 6, 5, 4 |
Class V | Occur less | 0.1%–1% | 3, 2 |
Class V | Occur rarely | < 0.1% | 1 |
Classes of occurrence degree | Description |
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Destroys the product or nullifies functions | 1.00 |
Renders the product inoperable or degrades the functions | 0.1–1.00 |
Reduces the product function or causes defects | 0–0.1 |
No appreciable impact | 0 |
Destroys the product or nullifies the functions | 1.00 |
2.1.5 Maintainability
2.1.6 Maintenance Cost
2.2 Ranking Indexes
2.3 Preprocessing
3 Screening and Determination of Indexes
3.1 Index Screening
3.2 Index Determination
Maximum-criticality index | \(\overline{\lambda }_{i}\) | Ei | Di | Ti | Fi |
---|---|---|---|---|---|
RPNi | √ | √ | √ | ||
Ci | √ | √ | √ | √ | |
CRi | √ | √ | √ | √ |
4 Determination of Information Weight Based on CBWM
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Step 1: Determine the set of criteria. The set of criteria in this study is the effects of weak links after screening, defined as \(G = \left\{ {g^{1} ,g^{2} , \ldots ,g^{{v_{*} }} } \right\}\).
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Step 2: Determine the best criteria gB and the worst criteria \(g_{w}\).
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Step 3: Compare gB with other criteria to establish a comparison vector \(A_{B} = \left( {a_{1B} ,a_{2B} , \ldots ,a_{{v_{*} B}} } \right)\) based on the scale shown in Table 6.Table 6Difference scaleScaleDifferenceEqually important0Slightly important1More important2Moderately important3Obviously important4Very important5Especially important6Greatly important7Extremely important8
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Step 4: Compare \(g_{W}\) with other criteria to establish the comparison vector \(A_{W} = \left( {a_{1W} ,a_{2W} , \ldots ,a_{{v_{*} W}} } \right)\), according to the same scale shown in Table 6.
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Step 5: Judge the consistency of AB and AW based on the consistency index shown in Eq. (36):where aBW is the difference between the most different criteria, and κ is the maximum scale.$$C_{CBWM} = \sqrt {\frac{1}{{v_{*} }}\sum\limits_{v = 1}^{{v_{*} }} {\left( {\frac{{a_{Bv} + a_{vW} - a_{BW} }}{\kappa }} \right)^{2} } } ,$$(36)
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Step 6: Calculate the weight. When AB and AW are exactly the same, the weight is calculated using Eqs. (37)–(38). Otherwise, calculate the weight by minimizing the maximum deviation method based on Eqs. (39)–(40).$$\tau_{v} = \frac{1}{{v_{*} }}\sum\limits_{v = 1}^{{v_{*} }} {a_{Bv} } + \kappa - a_{Bv} ,$$(37)$$\tau_{v} = \kappa - \frac{1}{{v_{*} }}\sum\limits_{v = 1}^{{v_{*} }} {a_{vW} } + a_{vW} ,$$(38)$$W^{v} = \frac{{\tau_{v} }}{{v_{*} \kappa }}\left( {v = 1,2, \cdots ,v_{*} } \right),$$(39)where τv is the intermediate variable; τB and τW are the corresponding values of the optimal and worst criteria, respectively.$$\begin{aligned} & \min \, \mathop {{\text{max}}}\limits_{v} \left\{ {\left| {\tau_{B} - \tau_{v} - a_{Bv} } \right|,\left| {\tau_{v} - \tau_{W} - a_{vW} } \right|} \right\}, \\ & {\text{s}}{.}{\text{t}}{.,}\sum\limits_{v = 1}^{{v_{*} }} {\tau_{v} } = v_{*} \kappa ;\tau_{v} \ge 0, {\text{for}\;{\text{all}}}\;v. \\ \end{aligned}$$(40)
5 Weakness Ranking of Subsystems Based on TOPSIS
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Step 1: Based on the impact factor vector of each subsystem in Eq. (34), the optimal and worst vectors are constructed using Eqs. (41), (42), which are conditions of the weakest and least weak subsystems, respectively.$$L^{ + } = \left( {\max \left\{ {L_{i}^{1} } \right\},\max \left\{ {L_{i}^{2} } \right\}, \ldots ,\max \left\{ {L_{i}^{{v_{*} }} } \right\}} \right),$$(41)where \(i \in \left\{ {1,2, \cdots ,n} \right\}\).$$L^{ - } = \left( {\min \left\{ {L_{i}^{1} } \right\},\min \left\{ {L_{i}^{2} } \right\}, \ldots ,\min \left\{ {L_{i}^{{v_{*} }} } \right\}} \right),$$(42)
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Step 2: Substitute the weights and influence factors into Eqs. (43)–(44) and calculate the distance between all indexes and the distance between L+ and L−.$$d_{i}^{ + } = \sqrt {\sum\limits_{v = 1}^{{v_{*} }} {W^{v} \left( {L_{{}}^{ + v} - L_{i}^{v} } \right)^{2} } } ,$$(43)where L+v and L−v are the vth screening index components of L+ and L−, respectively.$$d_{i}^{ - } = \sqrt {\sum\limits_{v = 1}^{{v_{*} }} {W^{v} \left( {L_{{}}^{ - v} - L_{i}^{v} } \right)^{2} } } ,$$(44)
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Step 3: Calculate the closeness between the weakest conditions of each subsystem using Eq. (45):$$G_{i} = \frac{{d_{i}^{ - } }}{{d_{i}^{ + } + d_{i}^{ - } }}.$$(45)
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Step 4: The subsystems are ranked based on the value of Gi. The closer the value is to 1, the closer is the subsystem to the weakest condition. The closer the value is to 0, the closer is the subsystem to the non-weak condition. In other words, the subsystem with the largest value is the weakest subsystem.
6 Numerical Example
Name | Code | Frequence | Frequency |
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Basic component | BC | 0 | 0 |
Headstock | BB | 17 | 0.030 |
Feed system | FS | 47 | 0.083 |
Tool holder | TU | 20 | 0.035 |
CNC system | NC | 28 | 0.050 |
Electrical system | ES | 60 | 0.106 |
Chip removal system | CC | 33 | 0.058 |
Hydraulic system | HS | 143 | 0.253 |
Center rest | CF | 48 | 0.085 |
Spider device | CD | 13 | 0.023 |
Tailstock | TS | 14 | 0.025 |
Grinding device | GD | 7 | 0.012 |
Drilling device | DD | 2 | 0.004 |
Protective device | PD | 41 | 0.073 |
Cooling system | CS | 62 | 0.110 |
Item | No. 1 | No. 2 |
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Failure phenomenon | Scorpion cannot move | Squat motor was not functioning |
Failure type | Technology type | Loosening type |
Failure mode | Positioning accuracy exceeds the standard | Poor contact |
Failure reason | Squat center is not on the established axis | Poor line contact |
Causality classification | Poor assembly | Loose |
Failure treatment | Adjust the center of the sley to the specified axis | Rewiring |
Index | Unknown assignment | Result |
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λ13 | Failure frequency of subsystems | 0.004 |
E13 | \(\tilde{e}_{13,1} = (3,3,4,5)\), \(\tilde{e}_{13,2} = (3,4,4,5)\) | 9.393 |
D13 | \(\tilde{d}_{13,1} = (2,3,4,4)\), \(\tilde{d}_{13,2} = (1,2,3,4)\) | 5.556 |
RPN13 | OPR13,1 = 2, OPR13,2 = 1 | 38.423 |
C13 | ‒ | 9.891 |
CR13 | n13,1 = 1, n13,2 = 1, n13 = 2, β13,1 = 0.2, β13,2 = 0.4 | 0.6 |
T13 | T13,1 = 2, T13,2 = 1 | 3 |
F13 | \(F_{13,1}^{11} = 300\), \(F_{13,1}^{21} = 30\), \(F_{13,1}^{31} = 500\)\(t_{13,1}^{11} = 2\), \(F_{13,1}^{41} = 200\), \(F_{13,1}^{51} = 100\), \(t_{13,1}^{21} = 2\) \(F_{13,2}^{11} = 300\), \(F_{13,2}^{21} = 30\), \(F_{13,2}^{31} = 500\)\(t_{13,2}^{11} = 1\), \(F_{13,2}^{41} = 0\), \(F_{13,2}^{51} = 80\), \(t_{13,2}^{21} = 1\) | 2970 |
Index | \(\overline{\lambda }_{i}\) | ★Ei | ★Di | RPNi | Ci | ★CRi | ★Ti | ★Fi |
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Basic component | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Headstock | 0.030 | 0.056 | 0.037 | 0.017 | 0.041 | 0.047 | 0.135 | 0.128 |
Feed system | 0.083 | 0.092 | 0.047 | 0.052 | 0.080 | 0.127 | 0.191 | 0.187 |
Tool holder | 0.035 | 0.072 | 0.051 | 0.032 | 0.052 | 0.058 | 0.215 | 0.207 |
CNC system | 0.050 | 0.069 | 0.080 | 0.173 | 0.055 | 0.087 | 0.022 | 0.021 |
Electrical system | 0.106 | 0.117 | 0.086 | 0.336 | 0.164 | 0.069 | 0.018 | 0.018 |
Chip removal system | 0.058 | 0.038 | 0.027 | 0.027 | 0.030 | 0.017 | 0.014 | 0.012 |
Hydraulic system | 0.253 | 0.140 | 0.189 | 0.082 | 0.229 | 0.344 | 0.111 | 0.116 |
Center rest | 0.085 | 0.073 | 0.117 | 0.060 | 0.073 | 0.039 | 0.028 | 0.036 |
Spider device | 0.023 | 0.015 | 0.024 | 0.006 | 0.011 | 0.027 | 0.004 | 0.004 |
Tailstock | 0.025 | 0.034 | 0.068 | 0.019 | 0.026 | 0.027 | 0.076 | 0.079 |
Grinding device | 0.012 | 0.038 | 0.040 | 0.011 | 0.024 | 0.011 | 0.021 | 0.021 |
Drilling device | 0.004 | 0.011 | 0.013 | 0.003 | 0.008 | 0.004 | 0.001 | 0.001 |
Protective device | 0.073 | 0.127 | 0.038 | 0.051 | 0.089 | 0.099 | 0.090 | 0.099 |
Cooling system | 0.110 | 0.034 | 0.079 | 0.084 | 0.056 | 0.028 | 0.033 | 0.032 |
Lubrication system | 0.030 | 0.045 | 0.067 | 0.042 | 0.037 | 0.012 | 0.035 | 0.032 |
Other | 0.023 | 0.039 | 0.037 | 0.005 | 0.025 | 0.006 | 0.006 | 0.007 |
RPNi | Ci | CRi | |
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RPNi | 1 | 2 | 1/2 |
Ci | ½ | 1 | 1/4 |
CRi | 2 | 4 | 1 |
RPNi | Ci | CRi | |
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Subjective weight | 0.286 | 0.143 | 0.571 |
Relevance weight | 0.337 | 0.335 | 0.328 |
Information weight | 0.405 | 0.202 | 0.393 |
Combined weight | 1.028 | 0.679 | 1.293 |
Index | Ei | Di | CRi | Ti | Fi |
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Basic component | 0.149 | 0.091 | 0.291 | 0.206 | 0.263 |
Name | di+ | di− | Gi | Order |
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BC | 0.248 | 0.000 | 0.000 | 17 |
BB | 0.179 | 0.096 | 0.351 | 5 |
FS | 0.127 | 0.151 | 0.543 | 2 |
TU | 0.162 | 0.151 | 0.483 | 3 |
NC | 0.195 | 0.061 | 0.238 | 7 |
ES | 0.201 | 0.065 | 0.245 | 6 |
CC | 0.231 | 0.021 | 0.084 | 13 |
HS | 0.067 | 0.216 | 0.764 | 1 |
CF | 0.208 | 0.054 | 0.207 | 9 |
CD | 0.233 | 0.017 | 0.070 | 15 |
TS | 0.202 | 0.060 | 0.230 | 8 |
GD | 0.230 | 0.024 | 0.096 | 12 |
DD | 0.244 | 0.006 | 0.025 | 16 |
PD | 0.161 | 0.098 | 0.378 | 4 |
CS | 0.216 | 0.038 | 0.151 | 10 |
LS | 0.223 | 0.036 | 0.138 | 11 |
OT | 0.238 | 0.020 | 0.076 | 14 |
Name | RPNi | Order |
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BC | 0 | 17 |
BB | 0.017 | 12 |
FS | 0.052 | 6 |
TU | 0.032 | 9 |
NC | 0.173 | 2 |
ES | 0.336 | 1 |
CC | 0.027 | 10 |
HS | 0.082 | 4 |
CF | 0.060 | 5 |
CD | 0.006 | 14 |
TS | 0.019 | 11 |
GD | 0.011 | 13 |
DD | 0.003 | 16 |
PD | 0.051 | 7 |
CS | 0.084 | 3 |
LS | 0.042 | 8 |
OT | 0.005 | 15 |