# Analysis of Mathematical Methods of Integral Expert Evaluation for Predictive Diagnostics of Technical Systems Based on the Kemeny Median

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## Abstract

**:**

## 1. Introduction

- -
- Arithmetic mean;
- -
- The Kemeny median (as the median from the point of view of probability theory and mathematical statistics more correctly reflects the property of the statistical population);
- -
- The pairwise comparison of expert assessments based on the Saaty method, including the results of expert judgments from several experts to diagnose the current state of electrical equipment, as well as a discussion of the results, conclusion, and a list of references.

## 2. Materials and Methods

#### 2.1. Evaluation of the Consistency of Expert Opinions Based on the Standard Deviation Method

_{ij}—evaluation of the i-th expert for the j-th factor and n—number of experts.

_{j}= 0.7–0.9, it demonstrates the high consistency of experts with no conspiracy between them [46].

_{i}is a symptom and Y

_{i}is a controlled parameter.

_{1}—continuous uninterruption of a through short circuit current at the low voltage side of a transformer;

_{2}—insufficient electrodynamic strength of windings to short circuit currents;

_{3}—cooling system failure;

_{4}—reduction in mechanical strength of insulation;

_{1}—winding overheating;

_{2}—winding deformation;

_{3}—moistening and contamination of winding insulation;

_{4}—winding insulation deterioration.

_{1}, X

_{2}, X

_{3}, and X

_{4}and their consistency for the second parameter Y

_{2}.

_{2}correspond to the recommended level of consistency, being in the range of 0.7–0.9 [47,48,49,50,51,52,53,54]. The consistency of expert opinions for other parameters Y

_{1}, Y

_{3}, and Y

_{4}was determined in a similar way.

#### 2.2. Evaluation of the Consistency of Expert Opinions Based on the Kemeny Median Method

_{ik}—evaluation of the i-th expert for the k-th symptom; and c

_{kk}—evaluation of an expert using the Kemeny median method for the k-th symptom.

_{2}are presented in Table 3 and Table 4.

_{2}correspond to the recommended level of consistency, being in the range of 0.9 ≤ CR < 1.

_{2}with considerable deviations of expert No.1’s opinions (Table 6 and Table 7).

_{j}= 0.6, which does not fall within the generally accepted interval [55].

#### 2.3. Evaluation of the Current Technical Condition Based on the Analytic Hierarchy Process

_{max}(called the maximum or principal eigenvalue) used for evaluation of consistency representing proportionality of importance. The closer λ

_{max}is to n (the number of objects or actions in the matrix), the more consistent the result is. Departure from the consistency can be calculated in the following way:

_{max}− n)/(n − 1)

_{1}—winding overheating; Y

_{2}—winding deformation; Y

_{3}—moistening and contamination of winding insulation; and Y

_{4}—winding insulation deterioration.

_{1}—continuous uninterruption of a through short circuit current at the low voltage side of a transformer; X

_{2}—insufficient electrodynamic strength of windings to short circuit currents; X

_{3}—cooling system failure; and X

_{4}—reduction in mechanical strength of windings.

_{1}—malfunction of relay protection; P

_{2}—seal failure of a casing; P

_{3}—lack of a surge arrester or its malfunction; and P

_{4}—violation of operating regulations.

_{max}), consistency index (CI), and consistency ratio (CR) were calculated (Table 8).

_{max}= 4.05.

_{max}= 4.093.

## 3. Discussion of Results

_{j}= 0.6, which is out of the conventional consistency interval of μ

_{j}= 0.7–0.9. In addition, the same deviation of evaluation for the second symptom using the Kemeny median method leads to the substitution of expert No. 5 by expert No. 6, who represents the expert group opinion in the best way. However, the consistency of opinions, in general, was decreased, but it did not exceed the conventional consistency interval of μ

_{j}= 0.7–0.9. In turn, the expert evaluations of the possible reasons and consequences of failures organized into four levels of the hierarchy, in spite of several deviations at the medium level, have shown the same result; the winding overheating is the main reason for the failures of an oil-filled power transformer.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Reasons | |||||
---|---|---|---|---|---|

X_{1} | X_{2} | X_{3} | X_{4} | ||

Failures | Y_{1} | 0.9 | 0.5 | 0.8 | 0.3 |

Y_{2} | 0.6 | 0.5 | 0.3 | 0.7 | |

Y_{3} | 0.4 | 0.3 | 0.8 | 0.7 | |

Y_{4} | 0.7 | 0.4 | 0.6 | 0.6 |

Reasons | ||||
---|---|---|---|---|

Y_{2} | X_{1} | X_{2} | X_{3} | X_{4} |

Experts | ||||

1 | 1.0 | 0.8 | 0.3 | 0.7 |

2 | 0.9 | 0.7 | 0.4 | 0.5 |

3 | 0.6 | 0.5 | 0.3 | 0.7 |

4 | 1.0 | 0.6 | 0.5 | 0.8 |

5 | 0.9 | 0.5 | 0.4 | 0.7 |

6 | 0.7 | 0.8 | 0.5 | 0.5 |

7 | 0.9 | 0.8 | 0.3 | 0.7 |

8 | 0.7 | 0.6 | 0.5 | 0.5 |

9 | 0.8 | 0.7 | 0.4 | 0.8 |

Consistency for Y_{2} | ||||

6 experts | ||||

${\mu}_{j}$ | 0.807 | 0.788 | 0.776 | 0.811 |

9 experts | ||||

${\mu}_{j}$ | 0.830 | 0.839 | 0.783 | 0.812 |

**Table 3.**Kemeny median between the opinion of one expert in relation to the others for the second parameter for 6 experts.

Experts | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

Sum of distances | 2.9 | 2.7 | 3.5 | 3.1 | 3.0 | 3.5 |

**Table 4.**Kemeny median between the opinion of one expert in relation to the others for the second parameter for 9 experts.

Experts | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

Sum of distances | 2.1 | 0.6 | 4.2 | 3.0 | 0.3 | 0.6 | 1.2 | 2.4 | 1.2 |

Symptoms | ||||
---|---|---|---|---|

Number of Experts | X_{1} | X_{2} | X_{3} | X_{4} |

6 experts | 0.945 | 0.955 | 0.970 | 0.955 |

9 experts | 0.966 | 0.975 | 0.981 | 0.959 |

**Table 6.**Kemeny median between one expert’s opinions in relation to the others in the case of the deviations of one expert’s opinions.

Experts | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

Sum of distances | 2.4 | 0.5 | 6.6 | 3.3 | 0.6 | 0.3 | 1.5 | 2.1 | 1.5 |

Reasons | ||||
---|---|---|---|---|

Y_{2} | X_{1} | X_{2} | X_{3} | X_{4} |

Standard deviation method | ||||

μ_{j} | 0.766 | 0.724 | 0.635 | 0.677 |

Kemeny median method | ||||

CR | 0.963 | 0.966 | 0.972 | 0.963 |

**Table 8.**Pairwise comparisons of symptoms under continuous interruption of a through short circuit current.

Expert No. 5 | |||||
---|---|---|---|---|---|

Y_{1} | Y_{2} | Y_{3} | Y_{4} | CVP | |

Y_{1} | 1 | 5 | 2 | 3 | 0.464 |

Y_{2} | 1/5 | 1 | 1/3 | 1 | 0.106 |

Y_{3} | 1/2 | 3 | 1 | 3 | 0.316 |

Y_{4} | 1/3 | 1 | 1/3 | 1 | 0.112 |

CR | 0.021 | ||||

CI | 0.019 | ||||

λ_{max} | 4.057 | ||||

Expert No. 6 | |||||

Y_{1} | 1 | 2 | 5 | 3 | 0.522 |

Y_{2} | 1/2 | 1 | 1 | 1 | 0.166 |

Y_{3} | 1/5 | 1 | 1 | 1 | 0.152 |

Y_{4} | 1/3 | 1 | 1 | 1 | 0.158 |

CR | 0.034 | ||||

CI | 0.031 | ||||

λ_{max} | 4.093 |

Expert No. 5 | ||||
---|---|---|---|---|

X_{1} | X_{2} | X_{3} | X_{4} | |

Y_{1} | 0.464 | 0.437 | 0.457 | 0.457 |

Y_{2} | 0.107 | 0.328 | 0.122 | 0.274 |

Y_{3} | 0.316 | 0.171 | 0.146 | 0.152 |

Y_{4} | 0.113 | 0.063 | 0.274 | 0.116 |

Expert No. 6 | ||||

Y_{1} | 0.523 | 0.393 | 0.349 | 0.459 |

Y_{2} | 0.166 | 0.196 | 0.218 | 0.161 |

Y_{3} | 0.152 | 0.224 | 0.349 | 0.307 |

Y_{4} | 0.158 | 0.187 | 0.083 | 0.073 |

Expert No. 5 | Expert No. 6 | ||
---|---|---|---|

Y_{1} | 0.451 | Y_{1} | 0.435 |

Y_{2} | 0.219 | Y_{2} | 0.186 |

Y_{3} | 0.105 | Y_{3} | 0.164 |

Y_{4} | 0.019 | Y_{4} | 0.105 |

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**MDPI and ACS Style**

Manusov, V.; Kalanakova, A.; Ahyoev, J.; Zicmane, I.; Praveenkumar, S.; Safaraliev, M.
Analysis of Mathematical Methods of Integral Expert Evaluation for Predictive Diagnostics of Technical Systems Based on the Kemeny Median. *Inventions* **2023**, *8*, 28.
https://doi.org/10.3390/inventions8010028

**AMA Style**

Manusov V, Kalanakova A, Ahyoev J, Zicmane I, Praveenkumar S, Safaraliev M.
Analysis of Mathematical Methods of Integral Expert Evaluation for Predictive Diagnostics of Technical Systems Based on the Kemeny Median. *Inventions*. 2023; 8(1):28.
https://doi.org/10.3390/inventions8010028

**Chicago/Turabian Style**

Manusov, Vadim, Aysulu Kalanakova, Javod Ahyoev, Inga Zicmane, Seepana Praveenkumar, and Murodbek Safaraliev.
2023. "Analysis of Mathematical Methods of Integral Expert Evaluation for Predictive Diagnostics of Technical Systems Based on the Kemeny Median" *Inventions* 8, no. 1: 28.
https://doi.org/10.3390/inventions8010028