# Probabilistic Fuzzy System for Evaluation and Classification in Failure Mode and Effect Analysis

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Linguistic and stochastic uncertainty is integrated into a robust mathematical model.
- Poisson distribution is used to determine the probabilistic values of the occurrence criterion.
- Binomial distribution determines the probabilistic values of the detection and severity criteria.
- The outputs of the probabilistic fuzzy system are obtained in a simple and easy range of [1, 10], which allows straightforward interpretation when classifying each failure mode.

## 2. Materials and Methods

#### 2.1. FMEA

- Establish the objective and integrate a multidisciplinary team of experts.
- Identify and analyze failure modes, their effects, causes, and detection controls.
- Establish the level of occurrence, severity, and detection.
- Obtain the risk priority number.
- Formulate a final report with the recommended actions and modifications to reduce or eliminate risks in the system/process.

#### 2.2. Evaluation System

## 3. Case Study

#### 3.1. Expert Team Integration

#### 3.2. Failure Analysis

#### 3.3. Opinions

#### 3.4. Frequency

#### Occurrence

- $\lambda =$ Average rate or average number of times a failure mode occurs.
- $e=$ 2.71828
- $x=$ the number of occurrences of the failure mode
- $f\left(x\right)$ = probability of x occurrences in the interval

#### 3.5. Assignment

#### Severity and Detection

- $n=$ the number of trials
- $\rho =$ the probability of success in a trial
- $x=$ number of successes in n trials
- $f\left(x\right)=$ probability of x successes in n trials

#### 3.6. Knowledge Base

#### 3.7. Implication and Aggregation

#### 3.8. Defuzzification

- $\mu =$ mean of the random variable $x$
- ${\sigma}^{2}=$ variance of the random variable $x$
- $\sigma =$ standard deviation of the random variable $x$
- $f\left(x\right)=$ probability of the random variable $x$

#### 3.9. Simulation Process

#### 3.10. Classification

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Jahangoshai, M.; Yousefi, S.; Valipour, M.; Dehdar, M.M. Risk Analysis of Sequential Processes in Food Industry Integrating Multi-Stage Fuzzy Cognitive Map and Process Failure Mode and Effects Analysis. Comput. Ind. Eng.
**2018**, 123, 325–337. [Google Scholar] [CrossRef] - Jin, C.; Ran, Y.; Zhang, G. Interval-Valued q-Rung Orthopair Fuzzy FMEA Application to Improve Risk Evaluation Process of Tool Changing Manipulator. Appl. Soft Comput.
**2021**, 104, 107192. [Google Scholar] [CrossRef] - Kalathil, M.J.; Renjith, V.R.; Augustine, N.R. Failure Mode Effect and Criticality Analysis Using Dempster Shafer Theory and Its Comparison with Fuzzy Failure Mode Effect and Criticality Analysis: A Case Study Applied to LNG Storage Facility. Process Saf. Environ. Prot.
**2020**, 138, 337–348. [Google Scholar] [CrossRef] - Godina, R.; Silva, B.G.R.; Espadinha-Cruz, P. A DMAIC Integrated Fuzzy FMEA Model: A Case Study in the Automotive Industry. Appl. Sci.
**2021**, 11, 3726. [Google Scholar] [CrossRef] - Liu, H.-C.; Chen, X.-Q.; Duan, C.-Y.; Wang, Y.-M. Failure Mode and Effect Analysis Using Multi-Criteria Decision Making Methods: A Systematic Literature Review. Comput. Ind. Eng.
**2019**, 135, 881–897. [Google Scholar] [CrossRef] - Huang, J.; You, J.-X.; Liu, H.-C.; Song, M.-S. Failure Mode and Effect Analysis Improvement: A Systematic Literature Review and Future Research Agenda. Reliab. Eng. Syst. Saf.
**2020**, 199, 106885. [Google Scholar] [CrossRef] - Wan, C.; Yan, X.; Zhang, D.; Qu, Z.; Yang, Z. An Advanced Fuzzy Bayesian-Based FMEA Approach for Assessing Maritime Supply Chain Risks. Transp. Res. Part E Logist. Transp. Rev.
**2019**, 125, 222–240. [Google Scholar] [CrossRef] - Zhang, H.; Dong, Y.; Palomares-Carrascosa, I.; Zhou, H. Failure Mode and Effect Analysis in a Linguistic Context: A Consensus-Based Multiattribute Group Decision-Making Approach. IEEE Trans. Rel.
**2019**, 68, 566–582. [Google Scholar] [CrossRef] - Di Nardo, M.; Murino, T.; Osteria, G.; Santillo, L.C. A New Hybrid Dynamic FMECA with Decision-Making Methodology: A Case Study in an Agri-Food Company. ASI
**2022**, 5, 45. [Google Scholar] [CrossRef] - Kahneman, D. Thinking, Fast and Slow, 1st ed.; Farrar, Straus and Giroux: New York, NY, USA, 2013. [Google Scholar]
- Akbarzadeh-T, M.-R.; Bemani-N, A. Probabilistic Fuzzy Systems, Expressions and Approaches. In Proceedings of the 2015 4th Iranian Joint Congress on Fuzzy and Intelligent Systems (CFIS), Zahedan, Iran, 9–11 September 2015; IEEE: Zahedan, Iran, 2015; pp. 1–6. [Google Scholar] [CrossRef]
- Yigin, B.; Celik, M. A Prescriptive Model for Failure Analysis in Ship Machinery Monitoring Using Generative Adversarial Networks. J. Mar. Sci. Eng.
**2024**, 12, 493. [Google Scholar] [CrossRef] - Testik, O.M.; Unlu, E.T. Fuzzy FMEA in Risk Assessment for Test and Calibration Laboratories. Qual. Reliab. Eng.
**2023**, 39, 575–589. [Google Scholar] [CrossRef] - Ilczuk, P.; Kycko, M. Risk Assessment in the Design of Railroad Control Command and Signaling Devices Using Fuzzy Sets. Appl. Sci.
**2023**, 13, 12460. [Google Scholar] [CrossRef] - Goksu, S.; Arslan, O. A Quantitative Dynamic Risk Assessment for Ship Operation Using the Fuzzy FMEA: The Case of Ship Berthing/Unberthing Operation. Ocean. Eng.
**2023**, 287, 115548. [Google Scholar] [CrossRef] - Cruz-Rivero, L.; Méndez-Hernández, M.L.; Mar-Orozco, C.E.; Aguilar-Lasserre, A.A.; Barbosa-Moreno, A.; Sánchez-Escobar, J. Functional Evaluation Using Fuzzy FMEA for a Non-Invasive Measurer for Methane and Carbone Dioxide. Symmetry
**2022**, 14, 421. [Google Scholar] [CrossRef] - Ribas, J.R.; Severo, J.C.R.; Guimarães, L.F.; Perpetuo, K.P.C. A Fuzzy FMEA Assessment of Hydroelectric Earth Dam Failure Modes: A Case Study in Central Brazil. Energy Rep.
**2021**, 7, 4412–4424. [Google Scholar] [CrossRef] - Alizadeh, S.S.; Solimanzadeh, Y.; Mousavi, S.; Safari, G.H. Risk Assessment of Physical Unit Operations of Wastewater Treatment Plant Using Fuzzy FMEA Method: A Case Study in the Northwest of Iran. Environ. Monit. Assess.
**2022**, 194, 609. [Google Scholar] [CrossRef] - Łapczyńska, D.; Burduk, A. Application of Fuzzy Logic to the Risk Assessment of Production Machines Failures. In Proceedings of the 18th International Conference on Soft Computing Models in Industrial and Environmental Applications (SOCO 2023), Salamanca, Spain, 5–7 September 2023; Springer Nature: Cham, Switzerland, 2023; Volume 749, pp. 34–45. [Google Scholar] [CrossRef]
- Pacana, A.; Siwiec, D. Method of Fuzzy Analysis of Qualitative-Environmental Threat in Improving Products and Processes (Fuzzy QE-FMEA). Materials
**2023**, 16, 1651. [Google Scholar] [CrossRef] - Alshehhi, K.; Cheaitou, A.; Rashid, H. Fuzzy Failure Modes Effect and Criticality Analysis of the Procurement Process of Artificial Intelligent Systems/Services. IJACSA
**2023**, 14, 10. [Google Scholar] [CrossRef] - Wu, X.; Wu, J. The Risk Priority Number Evaluation of FMEA Analysis Based on Random Uncertainty and Fuzzy Uncertainty. Complexity
**2021**, 2021, 8817667. [Google Scholar] [CrossRef] - Jang, H.-a.; Min, S. Time-Dependent Probabilistic Approach of Failure Mode and Effect Analysis. Appl. Sci.
**2019**, 9, 4939. [Google Scholar] [CrossRef] - Gul, M.; Yucesan, M.; Celik, E. A Manufacturing Failure Mode and Effect Analysis Based on Fuzzy and Probabilistic Risk Analysis. Appl. Soft Comput.
**2020**, 96, 106689. [Google Scholar] [CrossRef] - De Aguiar, J.; Scalice, R.K.; Bond, D. Using Fuzzy Logic to Reduce Risk Uncertainty in Failure Modes and Effects Analysis. J. Braz. Soc. Mech. Sci. Eng.
**2018**, 40, 516. [Google Scholar] [CrossRef] - Ceylan, B.O. Shipboard Compressor System Risk Analysis by Using Rule-Based Fuzzy FMEA for Preventing Major Marine Accidents. Ocean. Eng.
**2023**, 272, 113888. [Google Scholar] [CrossRef] - Certa, A.; Hopps, F.; Inghilleri, R.; La Fata, C.M. A Dempster-Shafer Theory-Based Approach to the Failure Mode, Effects and Criticality Analysis (FMECA) under Epistemic Uncertainty: Application to the Propulsion System of a Fishing Vessel. Reliab. Eng. Syst. Saf.
**2017**, 159, 69–79. [Google Scholar] [CrossRef] - Ghasemi, F.; Rahimi, J. Failure Mode and Effect Analysis of Personal Fall Arrest System under the Intuitionistic Fuzzy Environment. Heliyon
**2023**, 9, e16606. [Google Scholar] [CrossRef] - Awodi, N.J.; Liu, Y.; Ayo-Imoru, R.M.; Ayodeji, A. Fuzzy TOPSIS-Based Risk Assessment Model for Effective Nuclear Decommissioning Risk Management. Prog. Nucl. Energy
**2023**, 155, 104524. [Google Scholar] [CrossRef] - Ghoushchi, S.J.; Yousefi, S.; Khazaeili, M. An Extended FMEA Approach Based on the Z-MOORA and Fuzzy BWM for Prioritization of Failures. Appl. Soft Comput.
**2019**, 81, 105505. [Google Scholar] [CrossRef] - Buffa, P.; Giardina, M.; Prete, G.; De Ruvo, L. Fuzzy FMECA Analysis of Radioactive Gas Recovery System in the SPES Experimental Facility. Nucl. Eng. Technol.
**2021**, 53, 1464–1478. [Google Scholar] [CrossRef] - Qiu, Y.; Zhang, H. A Modified FMEA Approach to Predict Job Shop Disturbance. Processes
**2022**, 10, 2223. [Google Scholar] [CrossRef] - Cardiel-Ortega, J.J.; Baeza-Serrato, R. Failure Mode and Effect Analysis with a Fuzzy Logic Approach. Systems
**2023**, 11, 348. [Google Scholar] [CrossRef] - Boral, S.; Howard, I.; Chaturvedi, S.K.; McKee, K.; Naikan, V.N.A. An Integrated Approach for Fuzzy Failure Modes and Effects Analysis Using Fuzzy AHP and Fuzzy MAIRCA. Eng. Fail. Anal.
**2020**, 108, 104195. [Google Scholar] [CrossRef] - Ivančan, J.; Lisjak, D. New FMEA Risks Ranking Approach Utilizing Four Fuzzy Logic Systems. Machines
**2021**, 9, 292. [Google Scholar] [CrossRef]

**Figure 6.**Failure mode frequencies: (

**a**) large knot, (

**b**) small knot, (

**c**) tensioner out of adjustment and (

**d**) up tension wire.

Linguistic Ranking | |||
---|---|---|---|

Ranking | S | O | D |

1 | None | Very remote | Almost certain |

2 | Very weak | Remote | Very High |

3 | Weak | Rarely | Major |

4 | Extremely low | Not frequently | Important |

5 | Low | Low | Moderate |

6 | Moderately | Moderately | Medium |

7 | High | Moderate high | Scarce |

8 | Extremely high | High | Remote |

9 | Hazardous | Extreme | Very remote |

10 | Extremely | Extremely | None |

Expert | Area | Experience | Education |
---|---|---|---|

E1 | Mechanical | 31 years | Technical |

E2 | Operational | 23 years | Technical |

E3 | Electronic | 29 years | Engineer |

E4 | Manufacture | 13 years | Engineer |

No. | Failure | Component | Effect | Secondary Effect |
---|---|---|---|---|

FM1 | Large knot | Electronic/Mechanical | Breakup of the fiber and detachment of the knit fabric. | Machine stop |

Deformation of the needle hook | Machine stop | |||

FM2 | Small knot | Electronic/Mechanical | Deformation of the needle hook | Machine stop |

Defective knit fabric | Canvas with low quality | |||

FM3 | Tensioner out of adjustment | Mechanical | Alteration in fiber tension | Loop length alteration |

Mark and relief on the canvas | Canvas with low quality | |||

FM4 | Damaged lamp | Electronic | Delay in identifying the faulty device | Machine stop |

FM5 | Up tension wire | Mechanical | Change in yarn tension | Canvas with low quality |

Stages | Characteristics/Parameters |
---|---|

Input variables: | Opinion E1 Opinion E2 Opinion E3 Opinion E4 |

Output variable—Expert System 1: Output variable—Expert System 2: Output variable—Expert System 3: | Group assignment—Occurrence Group assignment—Detection Group assignment—Severity |

Label parameters: | Low (1, 3, 7) Medium (1, 5, 8) High (2, 7, 10) |

Membership function: | Sigmoid function |

Universe: | [1,2,3,4,5,6,7,8,9,10] |

Implication method: | Min |

Aggregation method: | Max |

Defuzzification method | Centroid [33] |

Universe | ${\mathsf{\lambda}}_{\mathbf{H}}=6$ | ${\mathsf{\lambda}}_{\mathbf{M}}=3.68$ | ${\mathsf{\lambda}}_{\mathbf{L}}=2$ |
---|---|---|---|

1 | 0.015 | 0.085 | 0.271 |

2 | 0.045 | 0.161 | 0.271 |

3 | 0.089 | 0.204 | 0.180 |

4 | 0.134 | 0.194 | 0.090 |

5 | 0.161 | 0.148 | 0.036 |

6 | 0.161 | 0.094 | 0.012 |

7 | 0.138 | 0.051 | 0.003 |

8 | 0.103 | 0.024 | 0.001 |

9 | 0.069 | 0.010 | 0.000 |

10 | 0.041 | 0.004 | 0.000 |

$\mathbf{n}=10$ | $\mathbf{n}=10$ | $\mathbf{n}=10$ | |
---|---|---|---|

Universe | ${\mathsf{\rho}}_{\mathbf{H}}=0.9$ | ${\mathsf{\rho}}_{\mathbf{M}}=0.6$ | ${\mathsf{\rho}}_{\mathbf{L}}=0.3$ |

1 | 0.000 | 0.002 | 0.121 |

2 | 0.000 | 0.011 | 0.233 |

3 | 0.000 | 0.042 | 0.267 |

4 | 0.000 | 0.111 | 0.200 |

5 | 0.001 | 0.201 | 0.103 |

6 | 0.011 | 0.251 | 0.037 |

7 | 0.057 | 0.215 | 0.009 |

8 | 0.194 | 0.121 | 0.001 |

9 | 0.387 | 0.040 | 0.000 |

10 | 0.349 | 0.006 | 0.000 |

$\mathbf{n}=10$ | $\mathbf{n}=10$ | $\mathbf{n}=10$ | |
---|---|---|---|

Universe | ${\mathsf{\rho}}_{\mathbf{H}}=0.95$ | ${\mathsf{\rho}}_{\mathbf{M}}=0.65$ | ${\mathsf{\rho}}_{\mathbf{L}}=0.35$ |

1 | 0.000 | 0.001 | 0.072 |

2 | 0.000 | 0.004 | 0.176 |

3 | 0.000 | 0.021 | 0.252 |

4 | 0.000 | 0.069 | 0.238 |

5 | 0.000 | 0.154 | 0.154 |

6 | 0.001 | 0.238 | 0.069 |

7 | 0.010 | 0.252 | 0.021 |

8 | 0.075 | 0.176 | 0.004 |

9 | 0.315 | 0.072 | 0.001 |

10 | 0.599 | 0.013 | 0.000 |

Rule | Antecedent (Criteria) | Consequent (Classification) | ||
---|---|---|---|---|

If P(O) is | and If P(D) is | and If P(S) is | Then, RPN is | |

R_{1} | L | L | L | L |

R_{2} | L | L | M | L |

R_{3} | L | L | H | L |

R_{4} | L | M | L | L |

R_{5} | L | M | H | L |

R_{6} | L | H | L | L |

R_{7} | L | H | M | L |

R_{8} | M | L | L | L |

R_{9} | H | L | L | L |

R_{10} | L | M | M | M |

R_{11} | M | L | M | M |

R_{12} | M | L | H | M |

R_{13} | M | M | L | M |

R_{14} | M | M | M | M |

R_{15} | M | M | H | M |

R_{16} | M | H | L | M |

R_{17} | M | H | M | M |

R_{18} | H | M | M | M |

R_{19} | B | H | H | H |

R_{20} | M | H | H | H |

R_{21} | H | L | M | H |

R_{22} | H | L | H | H |

R_{23} | H | M | L | H |

R_{24} | H | M | H | H |

R_{25} | H | H | L | H |

R_{26} | H | H | M | H |

R_{27} | H | H | H | H |

Rule | Implication | Aggregation L | Rule | Implication | Aggregation M | Rule | Implication | Aggregation H |
---|---|---|---|---|---|---|---|---|

R_{1} | 0.00003 | 0.00317 | R_{10} | 0.00001 | 0.00177 | R_{19} | 0.00000 | 0.00317 |

R_{2} | 0.00003 | R_{11} | 0.00065 | R_{20} | 0.00000 | |||

R_{3} | 0.00000 | R_{12} | 0.00000 | R_{21} | 0.00317 | |||

R_{4} | 0.00001 | R_{13} | 0.00036 | R_{22} | 0.00000 | |||

R_{5} | 0.00000 | R_{14} | 0.00036 | R_{23} | 0.00177 | |||

R_{6} | 0.00000 | R_{15} | 0.00000 | R_{24} | 0.00000 | |||

R_{7} | 0.00000 | R_{16} | 0.00000 | R_{25} | 0.00000 | |||

R_{8} | 0.00065 | R_{17} | 0.00000 | R_{26} | 0.00000 | |||

R_{9} | 0.00317 | R_{18} | 0.00177 | R_{27} | 0.00000 |

${\mathsf{\mu}}_{\mathbf{H}}=8$ | ${\mathsf{\mu}}_{\mathbf{M}}=5$ | ${\mathsf{\mu}}_{\mathbf{L}}=3$ | |
---|---|---|---|

Universe | ${\mathsf{\sigma}}_{\mathbf{H}}=1$ | ${\mathsf{\sigma}}_{\mathbf{M}}=1$ | ${\mathsf{\sigma}}_{\mathbf{L}}=1$ |

1 | 0.0000 | 0.0001 | 0.0540 |

2 | 0.0000 | 0.0044 | 0.2420 |

3 | 0.0000 | 0.0540 | 0.3989 |

4 | 0.0001 | 0.2420 | 0.2420 |

5 | 0.0044 | 0.3989 | 0.0540 |

6 | 0.0540 | 0.2420 | 0.0044 |

7 | 0.2420 | 0.0540 | 0.0001 |

8 | 0.3989 | 0.0044 | 0.0000 |

9 | 0.2420 | 0.0001 | 0.0000 |

10 | 0.0540 | 0.0000 | 0.0000 |

Expert Team Evaluation | |||||
---|---|---|---|---|---|

Expert | O | D | S | Conventional RPN | Classification Probabilistic Fuzzy System |

Opinion E1 | 6 | 7 | 6 | 252 | 5.5902 |

Opinion E2 | 10 | 9 | 5 | 450 | 7.7133 |

Opinion E3 | 7 | 5 | 5 | 175 | 5.7856 |

Opinion E4 | 8 | 4 | 5 | 160 | 5.3911 |

Expert System 1 | Expert System 2 | Expert System 3 | Conventional RPN | Classification Probabilistic fuzzy system | |

Group assessment | 5 | 4 | 5 | 100 | 5.3908 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cardiel-Ortega, J.J.; Baeza-Serrato, R.
Probabilistic Fuzzy System for Evaluation and Classification in Failure Mode and Effect Analysis. *Processes* **2024**, *12*, 1197.
https://doi.org/10.3390/pr12061197

**AMA Style**

Cardiel-Ortega JJ, Baeza-Serrato R.
Probabilistic Fuzzy System for Evaluation and Classification in Failure Mode and Effect Analysis. *Processes*. 2024; 12(6):1197.
https://doi.org/10.3390/pr12061197

**Chicago/Turabian Style**

Cardiel-Ortega, José Jovani, and Roberto Baeza-Serrato.
2024. "Probabilistic Fuzzy System for Evaluation and Classification in Failure Mode and Effect Analysis" *Processes* 12, no. 6: 1197.
https://doi.org/10.3390/pr12061197