# A Failure Mode Assessment Model Based on Neutrosophic Logic for Switched-Mode Power Supply Risk Analysis

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## Abstract

**:**

## 1. Introduction

- (i)
- The proposed FMEA model takes into account the economic perspective, which adds the expected cost (E) as a risk factor.
- (ii)
- The introduction of the Single-Valued Trapezoidal Neutrosophic Numbers (SVTNNs), which integrates neutrosophic set theory into the analysis methods (BWM and WASPAS), can more clearly show the uncertainty of the experts in evaluating risks.
- (iii)
- NBWM significantly reduces the number of pairwise comparisons and achieves a better consistency ratio.
- (iv)
- NWASPAS optimizes traditional FMEA’s RPN calculation method to obtain more reliable ranking results.
- (v)
- The root causes of the failures of the switching power supply are analyzed and provided to the decision-makers and R&D department to formulate improvement measures.

## 2. A Brief Literature Review of MCDM Combined with FMEA

- (i)
- (ii)
- (iii)
- (iv)
- (v)
- There are other important risk factors that are ignored, for example, the expected cost from a financial perspective is not considered [23].
- (vi)
- Uncertainty and ambiguity of information are not considered in traditional FMEA.

## 3. The Proposed FMEA Model

#### 3.1. Neutrosophic Set: SVTNNs

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

- If $sc\left(\tilde{p}\right)>sc\left(\tilde{q}\right)$, then $\tilde{p}>\tilde{q}$, and it indicates that $\tilde{p}$ is absolutely better than $\tilde{q}$.
- If $sc\left(\tilde{p}\right)=sc\left(\tilde{q}\right)$ and $l\left(\tilde{p}\right)>l\left(\tilde{q}\right)$, then $\tilde{p}>\tilde{q}$, and it means that $\tilde{p}$ is better than $\tilde{q}$.
- If $sc\left(\tilde{p}\right)=sc\left(\tilde{q}\right)$, then $\tilde{q}>\tilde{p}$, and it means that $\tilde{p}$ is worse than $\tilde{q}$.

#### 3.2. NBWM

- ▪
- Step 1. Identify the set of risk factors for the decision-making system

- ▪
- Step 2. Decide the most and least important risk factors

- ▪
- Step 3. Take the most important risk factor as a basis and doing pairwise comparisons with other risk factors to produce the neutrosophic Best-to-Others (BO) vector

- ▪
- Step 4. Take the rest of the risk factors as a benchmark, and make pairwise comparisons with the least important risk factor to generate the neutrosophic Others-to-Worst (OW) vector

- ▪
- Step 5. Use Equation (4) to convert SVTNNs into crisp values

- ▪
- Step 6. Calculate the optimal weight value of each risk factor $\left({w}_{1}^{*},\text{}{w}_{2}^{*},\dots ,\text{}{w}_{n}^{*}\right)$

#### 3.3. NWASPAS

- ▪
- Step 1. Obtain the initial neutrosophic assessment matrix

- ▪
- Step 2. Obtain the assessment matrix

- ▪
- Step 3. Use Equation (14) to calculate the average assessment matrix

- ▪
- Step 4. Calculate the normalized matrix

- ▪
- Step 5. Calculate the performance indexes WSM and WSP

## 4. Case Illustration

#### 4.1. Problem Description and Data Collection

#### 4.2. Using NBWM to Obtain Risk Factor Weights

#### 4.3. Using NWASPAS to Rank Failure Modes

## 5. Discussion

#### 5.1. Sensitivity Analysis

#### 5.2. Model Comparisons

#### 5.3. Management Implications

## 6. Conclusions and Future Work

- (i)
- Considering the expected cost (E) as a risk factor.
- (ii)
- Introducing the neutrosophic logic into the analysis methods and clearly showing the uncertainty of experts during the evaluation process.
- (iii)
- Significantly reducing the number of problem items and achieving a better consistency ratio.
- (iv)
- Optimizing the traditional FMEA’s RPN calculation method to obtain more reliable ranking results.
- (v)
- Analyzing root causes of SMPS failures and providing improvement measures to decision-makers and engineers of R&D and quality.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Acronym | Nomenclature |

BO | Best-to-Others |

FMEA | Failure Mode and Effects Analysis |

GRA | Grey Relational Analysis |

NBWM | Neutrosophic Best Worst method |

NWASPAS | Neutrosophic Weight Aggregated Sum Product Assessments |

OW | Others-to-Worst |

R&D | Research and Design |

RPN | Risk Priority Number |

SAW | Simple Additive Weighting |

SMPS | Switched-Mode Power Supply |

SVTNNs | Single-Valued Trapezoidal Neutrosophic Numbers |

TERPN | Total efficient risk priority number |

TOPSIS | Technique for Order Preference by Similarity to an Ideal Solution |

VIKOR | VlseKriterijuska Optimizacija I Komoromisno Resenje |

WASPAS | Weight Aggregated Sum Product Assessments |

WSM | Weighted Sum Model |

WPM | Weighted Product Model |

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**Table 1.**Summary of literature review of multiple criteria decision-making (MCDM) combined with failure mode and effects analysis (FMEA) models.

Author | Research Methodology | Application Field |
---|---|---|

Lo et al. [24] | Use Decision Making Trial and Evaluation Laboratory (DEMATEL) to determine influence weight of risk factors, rank and compare the results obtained using different MCDM methods—Simple Additive Weighting (SAW) VIKOR, Grey Relational Analysis (GRA), Complex proportional assessment of alternatives (COPRAS) to generate a final ranking result. | Computer Numerical Control (CNC) rotary machine |

Bian et al. [25] | A risk priority model based on D numbers, and propose technique for order of preference by similarity to ideal solution (TOPSIS) to evaluate the risk in FMEA. | Applying to rotor blades of aircraft’s turbine |

Hu et al. [26] | Applying grey relation analysis (GRA) –TOPSIS to determining risk ranking of identified failure modes. Considering vagueness and uncertainty in FMEA team’s evaluations on failure modes, proposed two-dimensional uncertain linguistic variables to describe the risk evaluation result and reliability of a failure mode. | A healthcare risk analysis case study about suctioning by endotracheal tube (ETT) |

Liu et. al. [27] | Use Analytic Hierarchy Process (AHP) to determine weights of risk factors, fuzzy Graph-Theoretical Matrix calculate risk effect indexes, and identify interrelationships between failure modes by DEMATEL. | Empirical study of rotary switch |

Boral et al. [28] | MCGDM integrating by Interval Type-2 Fuzzy DEMATEL (IT2F-DEMATEL) and Modified Fuzzy Multi-Attribute Ideal Real Comparative Analysis (Modified FMAIRCA). | To extend FMEA approach to reflect benefits on sustainable manufacturing |

Zhu et al. [29] | Use linguistic neutrosophic numbers to capture decision-makers’ evaluation of failure modes on each risk criterion. Combined regret theory and PROMETHEE (Preference ranking organization method for enrichment evaluation) methods to establish a hybrid risk ranking model of FMEA. | Using a supercritical water gasification system as an empirical case study |

Liu et. al. [30] | Integrating cloud model theory with extended GRA for resolving disadvantages of traditional RPN method. | Screening unit in a paper mill |

Srivastava et al. [31] | Combined fuzzy decision support system and fuzzy GRA for estimating RPN scores, then compared with classical RPN scores for realistic prioritization and decision making. | Sugar plants milling system |

Rezaee et al. [32] | A hybrid approach based on the Linguistic FMEA, Fuzzy Inference System (FIS) and Fuzzy Data Envelopment Analysis (DEA) model to calculate scores for covering shortcomings of RPN and prioritizing HSE risks. | Chemical industry. |

Indices | Description | ||

i | Failure mode i, i = 1, 2, …, m | ||

j | Risk factor j, j = 1, 2, …, n | ||

k | Expert k, k = 1, 2, …, p | ||

Variables | Description | Variables | Description |

$\tilde{\theta}$ | SVTNN | ${w}_{j}^{*}$ | Weight of risk factor |

${T}_{\tilde{\theta}}$ | Truth variable | ${\xi}^{L}$ | Consistency index |

${F}_{\tilde{\theta}}$ | Falsity variable | ${\tilde{\mathit{D}}}^{(k)}$ | Initial neutrosophic assessment matrix of expert k |

${I}_{\tilde{\theta}}$ | Indeterminacy variable | $\mathit{A}$ | Average assessment matrix |

${\mu}_{\tilde{\theta}}$ | Truth-membership function | $\mathit{P}$ | Normalized assessment matrix |

${\lambda}_{\tilde{\theta}}$ | Falsity-membership function | $S{Q}_{i}$ | Performance indexes WSM of failure mode i |

${\nu}_{\tilde{\theta}}$ | Indeterminacy-membership function | $P{Q}_{i}$ | Performance indexes WSP of failure mode i |

${\tilde{A}}_{Bj}$ | BO vector | ${Z}_{i}$ | Integrated performance index of failure mode i |

${\tilde{A}}_{jW}$ | OW vector |

**Table 3.**Linguistic variables and corresponding fuzzy numbers in the Best Worst Method (BWM) survey.

Linguistic Variables | Fuzzy Numbers |
---|---|

Equally important | (1, 1, 1, 1) |

Weakly important | (1, 1.5, 1.5, 1.5) |

Fairly important | (1.5, 2, 2, 2.5) |

Important | (2.5, 3, 3, 3.5) |

Very important | (3.5, 4, 4, 4.5) |

Absolutely important | (4.5, 4.5, 4.5, 5) |

Linguistic Variables | Fuzzy Numbers |
---|---|

No confidence | (0, 0, 0) |

Low confidence | (0.6, 0.2, 0.2) |

Fairly low confidence | (0.7, 0.1, 0.1) |

Medium confidence | (0.8, 0.1, 0) |

Fairly high confidence | (0.8, 0.2, 0.2) |

High confidence | (0.9, 0.1, 0.1) |

Absolutely high confidence | (1, 0, 0) |

S | O | D | E | Fuzzy Numbers |
---|---|---|---|---|

Very hazardous | Failure almost inevitable | Absolute uncertainty | Almost close to the original price | (1, 1, 1, 2) |

Hazardous | Very high | Very remote | Extremely high | (1, 2, 2, 3) |

Extreme | Repeated failures | Remote | Very high | (2, 3, 3, 4) |

Major | High | Very low | High | (3, 4, 4, 5) |

Significant | Moderately high | Low | Moderately high | (4, 5, 5, 6) |

Moderate | Moderate | Moderate | Moderate | (5, 6, 6, 7) |

Low | Relatively low | Moderately high | Relatively low | (6, 7, 7, 8) |

Minor | Low | High | Low | (7, 8, 8, 9) |

Very minor | Remote | Very high | Remote | (8, 9, 9, 10) |

None | Nearly impossible | Almost certain | Nearly no cost | (9, 10, 10, 10) |

Category | Description | Failure Mode | |
---|---|---|---|

No Output | No power output | FM11 | Capacitor burst |

FM12 | Transistor failure | ||

FM13 | Components burned out | ||

FM14 | Blown fuses | ||

Electrical | Electrical | FM21 | Insufficient output voltage |

Defect | characteristics | FM22 | No power-good signal |

failed to meet | FM23 | Cooling fan no operation | |

specifications | FM24 | Insufficient output wattage | |

FM25 | Efficiency recession | ||

FM26 | High frequency noise | ||

FM27 | Radiation interference | ||

Appearance | Appearance out of | FM31 | Tooling abrasion |

Issue | product specification | FM32 | Case Scratch |

FM33 | Connector deformation | ||

FM34 | Label broken | ||

FM35 | Printing blurred | ||

Dimension | Mechanical dimension | FM41 | Direct Current (DC) cable dimension |

Defect | Out of specifications | FM42 | Screw hole offset |

FM43 | Improper heat dissipation hole | ||

FM44 | Assembly problem |

Expert No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Most important | S | S | E | S | S | E | S | S | E | S | E | S | S | S | E |

Least important | D | D | D | D | E | O | D | D | O | D | D | D | O | O | D |

S | O | D | E | |
---|---|---|---|---|

S (most important) | {(1.0, 1.0, 1.0, 1.0), (1.0, 0.0, 0.0)} | {(3.5, 4.0, 4.0, 4.5), (0.8, 0.1, 0.0)} | {(4.5, 4.5, 4.5, 5.0), (0.8, 0.1, 0.0)} | {(1.5, 2.0, 2.0, 2.5), (0.8, 0.1, 0.0)} |

D (Least Important) | |
---|---|

S | {(4.5, 4.5, 4.5, 5.0), (0.8, 0.1, 0.0)} |

O | {(1.5, 2.0, 2.0, 2.5), (0.8, 0.1, 0.0)} |

D | {(1.0, 1.0, 1.0, 1.0), (1.0, 0.0, 0.0)} |

C | {(3.5, 4.0, 4.0, 4.5), (0.8, 0.1, 0.0)} |

S | O | D | E | |
---|---|---|---|---|

S (most important) | 1.0 | 3.6 | 4.1 | 1.8 |

D (Least Important) | |
---|---|

S | 4.1 |

O | 1.8 |

D | 1.0 |

E | 3.6 |

Risk Factor | Weights | Rank |
---|---|---|

S | 0.4080 | 1 |

O | 0.1879 | 3 |

D | 0.1169 | 4 |

E | 0.2872 | 2 |

Failure Mode | S | O | D | E |
---|---|---|---|---|

FM11 | {(8.0, 9.0, 9.0, 10.0), (0.8, 0.1, 0.0)} | {(6.0, 7.0, 7.0, 8.0), (0.9, 0.1, 0.1)} | {(7.0, 8.0, 8.0, 9.0), (0.9, 0.1, 0.1)} | {(8.0, 9.0, 9.0, 10.0), (0.8, 0.1, 0.0)} |

FM12 | {(8.0, 9.0, 9.0, 10.0), (0.8, 0.2, 0.2)} | {(4.0, 5.0, 5.0, 6.0), (0.8, 0.1, 0.0)} | {(7.0, 8.0, 8.0, 9.0), (0.9, 0.1, 0.1)} | {(7.0, 8.0, 8.0, 9.0), (0.8, 0.2, 0.2)} |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

FM44 | {(1.0, 2.0, 2.0, 3.0), (0.9, 0.1, 0.1)} | {(4.0, 5.0, 5.0, 6.0), (0.8, 0.2, 0.2)} | {(2.0, 3.0, 3.0, 4.0), (0.8, 0.2, 0.2)} | {(2.0, 3.0, 3.0, 4.0), (0.9, 0.1, 0.1)} |

Failure Mode | WSM | WSP | WASPAS | Rank |
---|---|---|---|---|

FM11 | 0.6945 | 0.6905 | 0.6925 | 1 |

FM12 | 0.6100 | 0.5952 | 0.6026 | 3 |

FM13 | 0.6397 | 0.5889 | 0.6143 | 2 |

FM14 | 0.6072 | 0.5504 | 0.5788 | 4 |

FM21 | 0.5235 | 0.4942 | 0.5088 | 5 |

FM22 | 0.4092 | 0.3872 | 0.3982 | 7 |

FM23 | 0.4397 | 0.4012 | 0.4204 | 6 |

FM24 | 0.3865 | 0.3735 | 0.3800 | 8 |

FM25 | 0.3274 | 0.3219 | 0.3247 | 10 |

FM26 | 0.3262 | 0.3130 | 0.3196 | 11 |

FM27 | 0.3527 | 0.3451 | 0.3489 | 9 |

FM31 | 0.2531 | 0.2338 | 0.2434 | 14 |

FM32 | 0.2482 | 0.2036 | 0.2259 | 16 |

FM33 | 0.3307 | 0.3081 | 0.3194 | 12 |

FM34 | 0.1146 | 0.0829 | 0.0987 | 19 |

FM35 | 0.1093 | 0.0708 | 0.0900 | 20 |

FM41 | 0.2465 | 0.2393 | 0.2429 | 15 |

FM42 | 0.1652 | 0.1614 | 0.1633 | 18 |

FM43 | 0.2817 | 0.2757 | 0.2787 | 13 |

FM44 | 0.1962 | 0.1898 | 0.1930 | 17 |

RUN1 | RUN2 | RUN3 | RUN4 | RUN5 | RUN6 | RUN7 | RUN8 | RUN9 | |||
---|---|---|---|---|---|---|---|---|---|---|---|

WSM | WSP | λ = 0.1 | λ = 0.2 | λ = 0.3 | λ = 0.4 | λ = 0.5 | λ = 0.6 | λ = 0.7 | λ = 0.8 | λ = 0.9 | |

FM11 | 0.6945 | 0.6905 | 0.6909 | 0.6913 | 0.6917 | 0.6921 | 0.6925 | 0.6929 | 0.6933 | 0.6937 | 0.6941 |

FM12 | 0.6100 | 0.5952 | 0.5967 | 0.5981 | 0.5996 | 0.6011 | 0.6026 | 0.6040 | 0.6055 | 0.6070 | 0.6085 |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

FM44 | 0.1962 | 0.1898 | 0.1904 | 0.1911 | 0.1917 | 0.1924 | 0.1930 | 0.1937 | 0.1943 | 0.1949 | 0.1956 |

Failure Mode | RPN | Rank | BWM and SAW | Rank | NBWM and NWASPAS | Rank |
---|---|---|---|---|---|---|

FM11 | 4132.95 | 1 | 0.8298 | 1 | 0.6925 | 1 |

FM12 | 2426.67 | 2 | 0.7350 | 4 | 0.6026 | 3 |

FM13 | 1632.15 | 3 | 0.7854 | 2 | 0.6143 | 2 |

FM14 | 1350.85 | 4 | 0.7452 | 3 | 0.5788 | 4 |

FM21 | 1096.68 | 5 | 0.6371 | 5 | 0.5088 | 5 |

FM22 | 675.58 | 6 | 0.4988 | 7 | 0.3982 | 7 |

FM23 | 470.13 | 9 | 0.5402 | 6 | 0.4204 | 6 |

FM24 | 525.50 | 7 | 0.4736 | 8 | 0.3800 | 8 |

FM25 | 401.69 | 10 | 0.3992 | 12 | 0.3247 | 10 |

FM26 | 290.28 | 11 | 0.4035 | 11 | 0.3196 | 11 |

FM27 | 506.15 | 8 | 0.4294 | 9 | 0.3489 | 9 |

FM31 | 154.35 | 14 | 0.3056 | 15 | 0.2434 | 14 |

FM32 | 105.73 | 17 | 0.2996 | 16 | 0.2259 | 16 |

FM33 | 243.30 | 12 | 0.4049 | 10 | 0.3194 | 12 |

FM34 | 18.66 | 19 | 0.1473 | 19 | 0.0987 | 19 |

FM35 | 16.63 | 20 | 0.1367 | 20 | 0.0900 | 20 |

FM41 | 153.21 | 15 | 0.3057 | 14 | 0.2429 | 15 |

FM42 | 71.83 | 18 | 0.2096 | 18 | 0.1633 | 18 |

FM43 | 238.83 | 13 | 0.3458 | 13 | 0.2787 | 13 |

FM44 | 115.93 | 16 | 0.2432 | 17 | 0.1930 | 17 |

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## Share and Cite

**MDPI and ACS Style**

Liou, J.J.H.; Liu, P.C.Y.; Lo, H.-W.
A Failure Mode Assessment Model Based on Neutrosophic Logic for Switched-Mode Power Supply Risk Analysis. *Mathematics* **2020**, *8*, 2145.
https://doi.org/10.3390/math8122145

**AMA Style**

Liou JJH, Liu PCY, Lo H-W.
A Failure Mode Assessment Model Based on Neutrosophic Logic for Switched-Mode Power Supply Risk Analysis. *Mathematics*. 2020; 8(12):2145.
https://doi.org/10.3390/math8122145

**Chicago/Turabian Style**

Liou, James J. H., Perry C. Y. Liu, and Huai-Wei Lo.
2020. "A Failure Mode Assessment Model Based on Neutrosophic Logic for Switched-Mode Power Supply Risk Analysis" *Mathematics* 8, no. 12: 2145.
https://doi.org/10.3390/math8122145