# Reliability Modelling through the Three-Parametric Weibull Model Based on Microsoft Excel Facilities

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## Abstract

**:**

## 1. Introduction

- Estimating the function of distribution of failure times using the BETA.INV function;
- Identification of the bi-parametric Weibull model by the known method of linear regression;
- Identification of the optimal location parameter—for which the resulting three-parametric Weibull model has the maximum coefficient of determination R;
- Argumentation the efficiency of the proposed method, based exclusively on the facilities of the Microsoft Excel program, by the superior likelihood of the three-parametric Weibull model that are obtained compared to the bi-parametric model.

## 2. Methods and Results

#### 2.1. The Three-Parametric Weibull Model Parameters

- -
- γ—location parameter or position parameter, this being a constant value that defines when the variation of the survival function R(t) starts.
- -
- η—scale parameter, which characterizes the extent of the distribution on the time axis. So, in the particular case when (t − γ) is equal to η, this parameter can be highlighted; because R(t) calculated with the relation (1) will be:

- -
- β—shape parameter, is dimensionless and is the parameter that determines the variation of the variation curves for the reliability indicators.

#### 2.2. Choosing the Optimal Estimator for the Distribution Function F(t)

_{i}) can be estimated with various computational relationships.

_{i}is the value of the probability of failure of i products out of the total of n products, F(t

_{i}), for which the time t

_{i}represents the quintile of 50%. That is, MR

_{i}estimate for F(t

_{i}) represents the value for which at that moment t

_{i}the probability that the true value is higher than F(t

_{i}) is equal to the probability that the true value is lower than F(t

_{i})—therefore equal to 0.5.

_{i}) values result, according to the binomial law (Bernoulli), from the relation:

_{i}) for any order number i, from 1 to n).

- -
- the Herd–Johnson estimator, used in 1960–1964, but also proposed by Weibull in 1939 [4]:

- -
- the Hazen estimator (2014):

- -
- the Benard estimator (1953):

- -
- the Blom estimator (1958) [20]:

_{i}) value that ensures 50% confidence level, and this essentially means that this is the best estimate for the unreliability.

_{i}values by calculating the values of the BETA.INV function (0.5, i, n + 1 − i), which is the inverse of the BETA.INV function for the cumulative density of the probability of failure [23]. The way the program works is iterative, calculating with a minimum degree of confidence the value of the variable corresponding to the specified probability.

_{i}[hours] were recorded (Table 1) and the graphs for F(t) that were obtained with the respective estimators (Figure 2).

#### 2.3. Determining the Parameters of the Three-Parametric Weibull Model

_{i}(X

_{i}) using the “trendline” function that is available in current Microsoft Excel spreadsheets [26,27,28].

_{i}, Y

_{i}) is constructed, for which the analytical relation and the value of the square of the degree of correlation are obtained R

^{2}.

_{i}(I = 1…10) that constitute the case study, the F(t

_{i}) estimator was determined using the BETA.INV function, and then, the quantities ln(t

_{i}) and ln [1/(1 − F)], presented in Table 2 with which the graph that is presented in Figure 3 was built.

^{2}= 0.8932.

^{2}= 0.8932. It is obvious that in this case, the bi-parametric Weibull model is inappropriate so it is necessary to adopt a three-parametric model, so a model in which to use the location parameter [9]. It will certainly provide a better likelihood degree. For the estimation of the third parameter (location), the existence of the parameter estimate is an important aspect. This is demonstrated in the paper [20]. There are various methods to identify this parameter, but are more laborious, including the one that is presented in the paper [11]. But very good results are also obtained by adopting for this parameter a value that is close to the first failure time, but strictly lower, because otherwise it will appear as the first value t − γ = 0, for which the logarithmic function is not defined.

_{1}= 210 h, the respective graphs were constructed in the same way as in Figure 4 and thus the respective values for the coefficient of determination were identified, identifying the highest value: for γ = 206 h, coefficient of determination is R

^{2}= 0.9740, which is clearly superior to that which was obtained for the bi-parametric Weibull model (when γ = 0). Figure 5 shows the graph R

^{2}(γ) made for values γ tested starting with the one immediately below the first failure time, γ = 209 h, noting that the graph shows only one maximum, for γ = 206 h, after which, as the values γ decreases, follows a downward trend that touches for γ = 0 the value that is obtained for the case of the bi-parametric model (R

^{2}= 0.8932).

_{1}. The optimum value can be obtained by successive tests (by probing), for values that are smaller and smaller in relation to the first failure time, until the values that are obtained for R

^{2}begin to decrease, which confirms that the optimal value has been identified, which is unique.

^{2}= 0.9740) resulting, after processing the data from the case study (Table 3) and performing the linear regression (Figure 6), in the values for the other two parameters:

- -
- shape parameter: β = 1.0382.
- -
- the scale parameter: $\eta ={e}^{-\frac{c}{\beta}}={e}^{-\frac{-4.0975}{1.0382}}=51.766\mathrm{h}$.

_{optimum}), it is possible to operate in Microsoft Excel with the function corresponding to the bi-parametric Weibull model, and the operator is, in fact, even the optimal three-parametric model that is defined by the relation:

_{W}model for the parameter γ = 206.

## 3. Discussion

- In many works MR is approximated with various algebraic estimators (i.e., Benard, Hazen etc.), more or less adequate, but now the MR values can be accurately identified, using the BETA.INV function that is available in the Microsoft Excel calculator.
- The use of the location parameter is mandatory in some cases, such as for the product operability, where the location parameter shows the guaranteed operating time of the object.
- By representing in logarithmic coordinates, we determined three-parametric Weibull models for different values that were initially adopted for the location parameter.
- Using as a criterion the coefficient of determination that was obtained using the trendline function for the linear model, it was possible to identify, by successive tests, the optimal value of the location parameter.
- The accuracy with which the optimal parameter γ
_{optimum}is identified is equal to the value of the adopted step t by the user, which can be more or less fine, depending on the desired level of detail. - The proposed methodology can be easily translated into a calculation program, as can be seen in the flowchart that was made.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Highlighting parameters γ and η of the three-parametric Weibull model on the graphs R(t) and F(t).

**Figure 5.**Graph R

^{2}(γ), with the maximum value for R

^{2}corresponding to the value γ

_{optimum}= 206 h.

**Figure 7.**Distribution function graphs F(t) that were obtained by the BETA.INV function and by the bi-parametric Weibull model (γ = 0).

**Figure 8.**Distribution function graphs F(t-206) that were obtained by the BETA.INV function and by the optimal three-parametric Weibull model (γ

_{optimum}= 206 h).

i | t_{i} [Hours] | i/n | i/(n + 1) | (i − 0.5)/n | (i − 0.3)/(n + 0.4) | (i − 0.375)/(n + 0.25) | BETA.INV (0.5; i; n + 1 − i) |
---|---|---|---|---|---|---|---|

1 | 210 | 0.100 | 0.091 | 0.050 | 0.067 | 0.061 | 0.067 |

2 | 215 | 0.200 | 0.182 | 0.150 | 0.163 | 0.159 | 0.162 |

3 | 224 | 0.300 | 0.273 | 0.250 | 0.260 | 0.256 | 0.259 |

4 | 228 | 0.400 | 0.364 | 0.350 | 0.356 | 0.354 | 0.355 |

5 | 234 | 0.500 | 0.455 | 0.450 | 0.452 | 0.451 | 0.452 |

6 | 260 | 0.600 | 0.545 | 0.550 | 0.548 | 0.549 | 0.548 |

7 | 267 | 0.700 | 0.636 | 0.650 | 0.644 | 0.646 | 0.645 |

8 | 285 | 0.800 | 0.727 | 0.750 | 0.740 | 0.744 | 0.741 |

9 | 297 | 0.900 | 0.818 | 0.850 | 0.837 | 0.841 | 0.838 |

10 | 300 | 1.000 | 0.909 | 0.950 | 0.933 | 0.939 | 0.933 |

**Table 2.**Calculated values for estimating and linear regression of the distribution function F(t) for γ = 0.

i | t_{i} [Hours] | ln(t_{i}) | F = BETA.INV (0.5; i; n + 1 − i) | lnln [1/(1 − F)] |
---|---|---|---|---|

1 | 210 | 5.3471 | 0.0670 | −2.6691 |

2 | 215 | 5.3706 | 0.1623 | −1.7313 |

3 | 224 | 5.4116 | 0.2586 | −1.2067 |

4 | 228 | 5.4293 | 0.3551 | −0.8240 |

5 | 234 | 5.4553 | 0.4517 | −0.5093 |

6 | 260 | 5.5607 | 0.5483 | −0.2297 |

7 | 267 | 5.5872 | 0.6449 | 0.0347 |

8 | 285 | 5.6525 | 0.7414 | 0.3020 |

9 | 297 | 5.6937 | 0.8377 | 0.5980 |

10 | 300 | 5.7038 | 0.9330 | 0.9946 |

**Table 3.**Calculated values for estimating and linear regression of the distribution function F(t) for γ

_{optimum}= 206 h.

i | t [Hours] | Int | F = BETA.INV (0.5; i; n + 1 − i) | lnln [1/(1 − F)] |
---|---|---|---|---|

1 | 4 | 1.3863 | 0.0670 | −2.6691 |

2 | 9 | 2.1972 | 0.1623 | −1.7313 |

3 | 18 | 2.8904 | 0.2586 | −1.2067 |

4 | 22 | 3.0910 | 0.3551 | −0.8240 |

5 | 28 | 3.3322 | 0.4517 | −0.5093 |

6 | 54 | 3.9890 | 0.5483 | −0.2297 |

7 | 61 | 4.1109 | 0.6449 | 0.0347 |

8 | 79 | 4.3694 | 0.7414 | 0.3020 |

9 | 91 | 4.5109 | 0.8377 | 0.5980 |

10 | 94 | 4.5433 | 0.9330 | 0.9946 |

**Table 4.**Values that were calculated for the distribution function F(t), for t, and (t – γ

_{optimum}).

T [Hours] | BETA.INV | F(t) | |
---|---|---|---|

WEIBULL.DIST (t; beta; eta; TRUE) | WEIBULL.DIST (t-206; beta; eta; TRUE) | ||

210 | 0.0670 | 0.1404 | 0.0564 |

215 | 0.1623 | 0.1663 | 0.1322 |

224 | 0.2586 | 0.2218 | 0.2623 |

228 | 0.3551 | 0.2503 | 0.3157 |

234 | 0.4517 | 0.2975 | 0.3903 |

260 | 0.5483 | 0.5532 | 0.6392 |

267 | 0.6449 | 0.6292 | 0.6884 |

285 | 0.7414 | 0.8086 | 0.7877 |

297 | 0.8377 | 0.8981 | 0.8365 |

300 | 0.9330 | 0.9155 | 0.8469 |

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**MDPI and ACS Style**

Titu, A.M.; Boroiu, A.A.; Boroiu, A.; Dragomir, M.; Pop, A.B.; Titu, S.
Reliability Modelling through the Three-Parametric Weibull Model Based on Microsoft Excel Facilities. *Processes* **2022**, *10*, 1585.
https://doi.org/10.3390/pr10081585

**AMA Style**

Titu AM, Boroiu AA, Boroiu A, Dragomir M, Pop AB, Titu S.
Reliability Modelling through the Three-Parametric Weibull Model Based on Microsoft Excel Facilities. *Processes*. 2022; 10(8):1585.
https://doi.org/10.3390/pr10081585

**Chicago/Turabian Style**

Titu, Aurel Mihail, Andrei Alexandru Boroiu, Alexandru Boroiu, Mihai Dragomir, Alina Bianca Pop, and Stefan Titu.
2022. "Reliability Modelling through the Three-Parametric Weibull Model Based on Microsoft Excel Facilities" *Processes* 10, no. 8: 1585.
https://doi.org/10.3390/pr10081585