# Risk Evaluation in Failure Mode and Effects Analysis Using Fuzzy Measure and Fuzzy Integral

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Risk Priority Number

- PRN only considers three risk factors including occurrence (O), severity (S), and detection (D) but it ignores some impacts of the other risk factors.
- PRN does not consider the frequency of occurrence, degree of severity, and difficulty of detection to have different importance. It simply assumes that they have the same weight or importance.
- The evaluation of the failure model is only based on the evaluation level from 1–10, without considering that uncertainty will impact the assessment process.
- The final evaluation result (V) is only used to sort the failure mode, without a clear physical quantity that can actually react to the specific risk level of the failure mode.
- Different values of the risk factors may lead to the same V. It hides the potential risk type’s difference about failure modes.

#### 2.2. Fuzzy Set Theory

**Definition**

**1.**

#### 2.3. Fuzzy Measure and Fuzzy Integral

**Definition**

**2.**

- (i)
- μ(∅) = 0
- (ii)
- μ(X) = 1
- (iii)
- μ(A) ≤ μ(B), ∀A,B ∈ P(X), A ⊆ B

**Definition**

**3.**

**Definition**

**4.**

## 3. The Proposed Model

- Evaluate the three risk factors by domain experts including occurrence, severity, and detection and their weights. The results of the assessment include two aspects: one is the importance of each factor, which are defined as g(O), g(S), and g(D) and they are in the range of 0–1; the other is the value of every risk factor in failure modes, which are defined as ${x}_{O}$, ${x}_{S}$, and ${x}_{D}$.
- Generate a fuzzy measure and use the fuzzy measure to generate a $\lambda $-fuzzy measure. The g(O), g(S), and g(D) are regarded as the fuzzy density of the three risk factors. Then generate the ${\mu}_{\lambda}$ of the $\lambda $-fuzzy measure according to Equation (4).
- Fuse evaluation values using the Choquet integral and rank the comprehensive evaluation value. The Choquet integral, by Equation (5), is applied to fuse ${x}_{O}$, ${x}_{S}$, and ${x}_{D}$ based on ${\mu}_{\lambda}$ and the evaluation value by domain experts. Then we can obtain the comprehensive evaluation value ${x}_{c}$, which reflects the comprehensive risk level of the failure mode. Next, rank the failure mode by ${x}_{c}$ to find the high-risk failure mode.
- Prevent and improve failure mode. Domain expert and decision makers can formulate corresponding preventive measures by the ranking. Repeating above steps, the system will be optimized continuously and the reliability will be improved.

## 4. An Illustrative Example

- g(O) = 0.768
- g(S) = 0.787
- g(D) = 0.650

- $\mu $(O) = g(O) = 0.768
- $\mu $(S) = g(S) = 0.787
- $\mu $(O) = g(D) = 0.650

- $\mu $(∅) = 0,
- $\mu $({O}) = 0.768,
- $\mu $({S}) = 0.878,
- $\mu $({D}) = 0.650,
- $\mu $({O,S}) = 0.979,
- $\mu $({O,D}) = 0.924,
- $\mu $({S,D}) = 0.964,
- $\mu $({O,S,D}) = 1.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Table 1.**Traditional FMEA scale for occurrence [41].

Probability of Failure | Possible Failure Rates | Rank |
---|---|---|

Extremely high: Failure almost inevitable | ≥in 2 | 10 |

Very high | 1 in 3 | 9 |

Repeated failures | 1 in 8 | 8 |

High | 1 in 20 | 7 |

Moderately high | 1 in 80 | 6 |

Moderate | 1 in 400 | 5 |

Relatively low | 1 in 2000 | 4 |

Low | 1 in 15,000 | 3 |

Remote | 1 in 150,000 | 2 |

Nearly impossible | 1 in 1,500,000 | 1 |

**Table 2.**Traditional FMEA scale for severity [41].

Effect | Criteria: Severity of Effect | Rank |
---|---|---|

Hazardous | Failure is hazardous and occurs without warning. It suspends operation of the system and/or involves noncompliance with government regulations | 10 |

Serious | Failure involves hazardous outcomes and/or noncompliance with government regulations or standards | 9 |

Extreme | Failure is hazardous and occurs without warning. It system is inoperable | 8 |

Major | Product performance is severely affected but functions. The system may not operate | 7 |

Significant | Product performance is degraded. Comfort or convince functions may not operate | 6 |

Moderate | Moderate effect on product performance. The product requires repair | 5 |

Low | Small effect on product performance. The product does not require repair | 4 |

Minor | Minor effect on product or system performance | 3 |

Very mintor | Very minor effect on product or system performance | 2 |

None | No effect | 1 |

**Table 3.**Traditional FMEA scale for detection [41].

Detection | Criteria: Likelihood of Detection by Design Control | Rank |
---|---|---|

Absolute uncertainty | Design control does not detect a potential cause of failure or subsequent failure mode, or there is no design control | 10 |

Very remote | Very remote chance the design control will detect a potential cause of failure or subsequent failure mode | 9 |

Remote | Remote chance the design control will detect a potential cause of failure or subsequent failure mode | 8 |

Very low | Very low chance the design control will detect a potential cause of failure or subsequent failure mode | 7 |

Low | Low chance the design control will detect a potential cause of failure or subsequent failure mode | 6 |

Moderate | Moderate chance the design control will detect a potential cause of failure or subsequent failure mode | 5 |

Moderately high | Moderately high chance the design control will detect a potential cause of failure or subsequent failure mode | 4 |

High | High chance the design control will detect a potential cause of failure or subsequent failure mode | 3 |

Very high | Very high chance the design control will detect a potential cause of failure or subsequent failure mode | 2 |

Almost certain | Design control will almost certainty detect a potential cause of failure or subsequent failure mode | 1 |

Risk Factors | Team Members | ||||
---|---|---|---|---|---|

DM 1 | DM 2 | DM 3 | DM 4 | DM 5 | |

O | H | H | VH | H | MH |

S | VH | VH | H | VH | VH |

D | MH | MH | M | H | MH |

Team Members | FM 1 | FM 2 | FM 3 | FM 4 | FM 5 | FM 6 |
---|---|---|---|---|---|---|

O | ||||||

DM 1 | M | H | VH | M | M | MH |

DM 2 | M | MH | MH | M | ML | H |

DM 3 | M | H | VH | L | M | M |

DM 4 | MH | MH | VH | M | M | MH |

DM 5 | M | MH | VH | M | M | M |

S | ||||||

DM 1 | ML | H | MH | M | M | H |

DM 2 | ML | MH | MH | M | MH | H |

DM 3 | ML | H | MH | ML | MH | H |

DM 4 | M | H | MH | M | M | H |

DM 5 | M | H | MH | M | M | H |

D | ||||||

DM 1 | M | M | MH | VL | L | L |

DM 2 | ML | M | M | ML | ML | M |

DM 3 | ML | ML | MH | VL | L | L |

DM 4 | ML | M | MH | ML | L | L |

DM 5 | ML | M | M | VL | L | VL |

Failure Modes | O | S | D |
---|---|---|---|

FM 1 | (4, 5.2, 5.4, 8) | (2, 3.8, 4.4, 6) | (2, 3.4, 4.2, 6) |

FM 2 | (5, 6.8, 7.4, 9) | (5, 7.6, 7.8, 9) | (2, 4.6, 4.8, 6)) |

FM 3 | (5, 8.4, 9.4, 10) | (5, 6, 7, 8) | (4, 5.6, 6.2, 8) |

FM 4 | (1, 4.4, 4.4, 6) | (2, 4.6, 4.8, 6) | (0, 1.2, 2.2, 5) |

FM 5 | (2, 4.6, 4.8, 6) | (4, 5.4, 5.8, 8) | (1, 2.2, 2.4, 5) |

FM 6 | (4, 6, 6.4, 9) | (7, 8, 8, 9) | (0, 2.2, 2.4, 6) |

Weight | (0.5, 0.78, 0.82, 1) | (0.7, 0.88, 0.96, 1) | (0.4, 0.62, 0.68, 0.9) |

Failure Modes | O | S | D |
---|---|---|---|

FM 1 | 5.756 | 4.038 | 3.922 |

FM 2 | 7.038 | 7.244 | 4.244 |

FM 3 | 8.044 | 6.500 | 5.962 |

FM 4 | 3.800 | 4.244 | 2.189 |

FM 5 | 4.244 | 5.855 | 2.756 |

FM 6 | 6.393 | 8.000 | 2.759 |

Weight | 0.768 | 0.878 | 0.650 |

Failure Modes | Values of the Fuzzy Integral ${\mathit{x}}_{\mathit{c}}$ | Ranking |
---|---|---|

FM 1 | 5.355 | 5 |

FM 2 | 7.161 | 3 |

FM 3 | 7.675 | 2 |

FM 4 | 4.157 | 6 |

FM 5 | 5.628 | 4 |

FM 6 | 7.729 | 1 |

**Table 9.**The values of S, R, and Q for all failure modes based on extended VIKOR method under a fuzzy environment [59].

Failure Modes | S | R | Q |
---|---|---|---|

FM 1 | 0.653 | 0.354 | 0.343 |

FM 2 | 1.650 | 0.710 | 0.817 |

FM 3 | 1.964 | 0.768 | 0.943 |

FM 4 | 0.046 | 0.046 | 0 |

FM 5 | 0.581 | 0.403 | 0.354 |

FM 6 | 1.445 | 0.878 | 0.865 |

**Table 10.**The ranking of the failure modes by S, R, and Q in decreasing order based on extended the VIKOR method under a fuzzy environment [59].

Failure Modes | S | R | Q |
---|---|---|---|

FM 1 | 4 | 5 | 5 |

FM 2 | 2 | 3 | 3 |

FM 3 | 1 | 2 | 1 |

FM 4 | 6 | 6 | 6 |

FM 5 | 5 | 4 | 4 |

FM 6 | 3 | 1 | 2 |

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**MDPI and ACS Style**

Liu, H.; Deng, X.; Jiang, W.
Risk Evaluation in Failure Mode and Effects Analysis Using Fuzzy Measure and Fuzzy Integral. *Symmetry* **2017**, *9*, 162.
https://doi.org/10.3390/sym9080162

**AMA Style**

Liu H, Deng X, Jiang W.
Risk Evaluation in Failure Mode and Effects Analysis Using Fuzzy Measure and Fuzzy Integral. *Symmetry*. 2017; 9(8):162.
https://doi.org/10.3390/sym9080162

**Chicago/Turabian Style**

Liu, Haibin, Xinyang Deng, and Wen Jiang.
2017. "Risk Evaluation in Failure Mode and Effects Analysis Using Fuzzy Measure and Fuzzy Integral" *Symmetry* 9, no. 8: 162.
https://doi.org/10.3390/sym9080162