Reliability Modelling through the Three-Parametric Weibull Model Based on Microsoft Excel Facilities
, Andrei Alexandru Boroiu 2, Alexandru Boroiu 2, Mihai Dragomir 3, Alina Bianca Pop 4,* and Stefan Titu 5,6,*
Round 1
Reviewer 1 Report
The paper does not provide a comparison between the proposed method and other methods. It is not enough to demonstrate the validity of the model.
Author Response
Dear Reviewer,
We appreciate your precious time and invaluable comments on our manuscript. We have carefully taken the comments into consideration for improving the quality of the paper in our revision. The corresponding changes and refinements made in the revised paper are summarized in our response below. The modified text is highlighted in revised manuscript.
With Regards,
Authors.
Reviewer 1
The paper does not provide a comparison between the proposed method and other methods. It is not enough to demonstrate the validity of the model.
Response:
The paper aims to develop a method based exclusively on the facilities of the Microsoft Excel spreadsheet (found in the MS Office package) to obtain a three-parametric Weibull model as appropriate as possible. We have removed from the text all that induced the idea that this method leads to a three-parametric Weibull model more likely than the models obtained by other methods.
- the method is elaborated, and its feasibility is demonstrated.
- the estimated Median Rank is calculated exactly with the BETA.INV function, not using algebraic estimators (Hazen, Benard, Blom…), as is done in many works where the confusion persists that these algebraic estimators would be Median Rank itself.
- a way is proposed for identifying the most suitable three-parametric Weibull model by searching, by probing with a certain step, the value of the localization parameter that leads to a correlation coefficient as high as possible.
- the search for the optimal value for the location parameter can even be improved by using a finer step.
The proposed method is original, and the gain consists in:
- total feasibility only based on the extremely accessible and common Microsoft Excel program
- the exact calculation of the estimated Median Rank with the BETA.INV function.
- the possibility to identify a three-parametric Weibull model as appropriate as possible depending on the value adopted for the location parameter.
- the possibility to improve the search for the optimal value by using a step as fine as possible.
Author Response File: 
Author Response.docx
Reviewer 2 Report
Word file attached
Comments for author File: Comments.docx
Author Response
Dear Reviewer,
We appreciate your precious time and invaluable comments on our manuscript. We have carefully taken the comments into consideration for improving the quality of the paper in our revision. The corresponding changes and refinements made in the revised paper are summarized in our response below. The modified text is highlighted in revised manuscript.
With Regards,
Authors.
Reviewer 2
Mathematically a very strong paper with a new paradigm. But before the publication the following should be incorporated/corrected.
To ensure consistency throughout the manuscript use the phrase three-parametric Weibull model
Response: It has been replaced everywhere with the three-parametric Weibull model.
Line 44: English needs to be improved e.g.:
“The three parametric Weibull model is considered as most suitable for modeling reliability”.
Response: The expression has been replaced with: The three parametric Weibull model is considered as most suitable for reliability modeling.
Line 50: “MS Office” instead of Office
Response: “MS Office” was used instead of Office
Line52: “which allows for the development …”
Response: “allows” was used instead of “allow”
Line 54,55,56 to be rewritten
Response: Lines 54,55,56 was rewritten.
“…including in the case of online education, widespread in recent years due to the pandemic situation…”
Line 67 to 72: Sentence is too long break into two sentences.
Response: Instead of:
“The most appropriate way to develop a method for identifying the three-parametric Weibull model based on the Microsoft Excel utility is the graphical path, since this utility contains the regression function through which the Weibull function can be linearized in order to identify the bi-parametric model (for which the parameter of location is equal to zero) and allows the resumption of this regression for various values proposed for the location parameter (non-zero), creating, in fact, a three-parametric model. ”
the two sentences are:
“The most appropriate way to develop a method for identifying the three-parametric Weibull model based on the Microsoft Excel utility is the graphical path, because this utility contains the linear regression function through which the Weibull function can be linearized. Thus, using as a variable the failure times we can identify the bi-parametric Weibull model (for which the location parameter is equal to zero), but we can do more than that: using as a variable the failure times adjusted with the location parameter (a non-zero value), a three-parametric model is actually obtained.”
Line 74 to 81: Sentence is too long break into two sentences.
Response: Instead of:
“Thus, the main steps that will be taken are: 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 obtained compared to the bi-parametric model.”
The phrase was arranged for a better readability, highlighting successively the 4 steps to follow:
“Thus, the main steps that will be taken are:
1 - estimating the function of distribution of failure times using the BETA.INV function;
2 - identification of the bi-parametric Weibull model by the known method of linear regression;
3 - identification of the optimal location parameter - for which the resulting three-parametric Weibull model has the maximum coefficient of determination R;
4 - 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 obtained compared to the bi-parametric model.
Line 83: Rephrase the subheading 2.1 “The three parametric Weibull Model”
Response: Instead of: “2.1. Weibull three-parametric model” it was written “2.1. The three-parametric Weibull model”
Line 98,99: What do you mean by “variation in the variation curve”
Response: Instead of: “the parameter that determines the variation of the variation curve for the reliability indicators“, it was written “the parameter that determines the allure of the variation curves for the reliability indicators”
Line 104: Instead of “The most used indicator” use “Most frequently used indicator”
Response: Instead of “The most used indicator use” it was written “Most frequently used indicator”
Line 172 to 176: Sentence is too long break into two sentences.
Response: Instead of “The method to be proposed has as a starting point the observation that, under the conditions in which the parameter γ is initially determined, the analytical expression for the Weibull line can be used in relation (12), which contains the form parameter β and the scale parameter η, to determine parameter values β and η by linear regression of point distribution Yi(Xi) using the “Trendline” function available in current Microsoft Excel spreadsheets [27-29].”
it was written:
“The method to be proposed has as a starting point the observation that, under the conditions in which the parameter γ is initially determined, the analytical expression for the Weibull line is reduced to linear relation (13), which contains the shape parameter β and the scale parameter η. This means that the values of the parameters β and η can be determined by linear regression of point distribution Yi(Xi) using the “Trendline” function available in current Microsoft Excel spreadsheets [27-29].”
Line 194 to 218: Too many sentences group these in two paragraphs.
Response: Instead of:
“It is found that the parameter β has a value above 6 - too high, as specified in the paper [2], so that the model does not have a proper probability (this is due to the fact that the first failure time is very long, relative to the amplitude distribution, which corresponds to a coefficient of variation ν= σ/m very small), which is confirmed both by the graphic image and by the low value of coefficient of determination: R2 = 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 more laborious, including the one presented in the paper [11].
But very good results are also obtained by adopting for this parameter a value 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.
Thus, successively using the values for γ smaller than t1 = 210 hours, the respective graphs were constructed in the same way as in Figure 3 and thus the respective values for coefficient of determination were identified, identifying the highest value: for γ = 206 hours, coefficient of determination is R2 = 0.9740, which is clearly superior to that obtained for the bi-parametric Weibull model (when γ = 0).
Figure 4 shows the graph R2(γ) made for values γ tested starting with the one immediately below the first failure time, γ = 209 hours, noting that the graph shows only one maximum, for γ = 206 hours, after which, as the values γ decreases, follows a downward trend that touches for γ = 0 the value obtained for the case of the bi-parametric model (R2 = 0.8932).”
it was written:
“It is found that the parameter β has a value above 6 - too high, as specified in the paper [2], so that the model does not have a proper probability (this is due to the fact that the first failure time is very long, relative to the amplitude distribution, which corresponds to a coefficient of variation ν= σ/m very small), which is confirmed both by the graphic image and by the low value of coefficient of determination: R2 = 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 more laborious, including the one presented in the paper [11]. But very good results are also obtained by adopting for this parameter a value 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.
Thus, successively using the values for γ smaller than t1 = 210 hours, the respective graphs were constructed in the same way as in Figure 3 and thus the respective values for coefficient of determination were identified, identifying the highest value: for γ = 206 hours, coefficient of determination is R2 = 0.9740, which is clearly superior to that obtained for the bi-parametric Weibull model (when γ = 0). Figure 4 shows the graph R2(γ) made for values γ tested starting with the one immediately below the first failure time, γ = 209 hours, noting that the graph shows only one maximum, for γ = 206 hours, after which, as the values γ decreases, follows a downward trend that touches for γ = 0 the value obtained for the case of the bi-parametric model (R2 = 0.8932).”
Flow chart shown in Figure 8 should be given at the beginning of subheading 2.3
Response: figure 8 has been moved to the beginning of subheading 2.3 and the figures have been renumbered.
The text “Figure 8 shows the flow chart for the proposed methodology, which can be easily translated into a calculation program.”
became
“Figure 8 shows the flowchart for the proposed methodology, which could be easily translated into a calculation program.”
Discussion and Conclusion paragraphs need more English structuring by a professional.
Response: We have verified the English language with the help of a specialist.
Line 300 to 307: Sentence is too long break into two sentences.
Response: Instead of:
“By the well-known grapho-analytical method for estimating the biparametric Weibull model, the two parameters can be identified - the shape parameter and the scale parameter - but it is possible to identify the most suitable three-parametric model (which also contains the location parameter), by the correlation method, appreciated as the one that leads to the best results: the value of the location parameter corresponding to the three-parametric Weibull model resulting from the linear regression of the distribution function obtained based on the Median Ranks values for which the coefficient of determination value is identified is maximum.”
it was written:
“By the well-known graphic-analytical method for estimating the bi-parametric Weibull model, the two parameters can be identified - the shape parameter and the scale parameter - but it is possible to identify an appropriate three-parametric model (which also contains the location parameter) by the method presented. The optimum value of the location parameter corresponds to the three-parametric Weibull model resulting from the linear regression of the distribution function for which the coefficient of determination value is maximum.”
Author Response File: Author Response.docx
Reviewer 3 Report
General remarks:
My main point would be to justify the choice of estimators of F(t) such as those introduced by equations (5) to (8). Indeed, the illustrative example (figure 2) seems to me counterproductive. At t=300, all the 10 systems fail. The reliability of the set of system is therfore necessary R(300)=0. Then, the value of F(300) should clearly be 1. Could you more precisely introduce the interest of these "biased" estimators in comparison with the simple i/n...
Detailed remarks:
Page 2, equation (2): F(t) is also introduced as the cumulative function...
Page 8, table 2: Is this table cited and commented ?
Page 12, line 279: "Which using onlu the support of the usual..." wich uses... or Which is using... ?
Author Response
Response to Reviewer 3 Comments
Manuscript ID: processes-1757581
Title of manuscript: Reliability modelling through the three-parametric Weibull model based on Microsoft Excel facilities
Dear Reviewer,
We appreciate your precious time and invaluable comments on our manuscript. We have carefully taken the comments into consideration for improving the quality of the paper in our revision. The corresponding changes and refinements made in the revised paper are summarized in our response below. The modified text is highlighted in revised manuscript.
With Regards,
Authors.
Reviewer 3
General remarks:
My main point would be to justify the choice of estimators of F(t) such as those introduced by equations (5) to (8). Indeed, the illustrative example (figure 2) seems to me counterproductive. At t=300, all the 10 systems fail. The reliability of the set of system is therefore necessary R(300)=0. Then, the value of F(300) should clearly be 1. Could you more precisely introduce the interest of these "biased" estimators in comparison with the simple i/n...
Response:
Thanks for the remark. Indeed, it must be specified that although the “naive estimator” expresses exactly the cumulative frequency of failures for the monitored batch of products, to obtain a mathematical model that describes as adequately as possible the reliability of the entire population (reliability modeling), the Median Ranks indicator is the most suitable. It represents the F(ti) value that ensures 50% confidence level, and this essentially means that this is the best estimate for the unreliability.
Thus, the more the number of ti values increases, the more the last value of the cumulative function, F(tn) will be closer to 100%, but it will never be 100%. Although for the sample at time tn 100% of the products have failed, the reliability model will always lead to an F(tn) value less than 100%.
In a simple way, Median Ranks can be approximated quite well by algebraic estimators calculated with one of the relations (5) … (8), which is highlighted in figure 2 (only the i/n indicator is not a good approximation), but it can be determined exactly with the inverse of the beta function, BETA.INV, now integrated in Microsoft Excel software.
As a result, the following paragraph was inserted after relation (9):
Although the “naive estimator” expresses exactly the cumulative frequency of failures for the monitored batch of products, in order to obtain a mathematical model that describes as adequately as possible the reliability of the entire population (reliability modeling), the Median Ranks indicator is the most suitable. It represents the F(ti) value that ensures 50% confidence level, and this essentially means that this is the best estimate for the unreliability.
Also, the paragraphs before relation (4), which determines F(ti) values as Median Ranks, has been reformulated, as follows:
That is, MRi estimate for F(ti) represents the value for which at that moment ti the probability that the true value is higher than F(ti) is equal to the probability that the true value is lower than F(ti) - therefore equal to 0.5.
The F(ti) values result, according to the binomial law (Bernoulli), from the relation: …
Detailed remarks:
Page 2, equation (2): F(t) is also introduced as the cumulative function...
Response:
To explain the fact that the indicator F(t) can be calculated analytically with relation (2) or can be estimated with a statistical relation, for mathematical modeling, it was introduced within the chapter 2. Methods and results, at the beginning of the subchapter 2.2. Choosing the optimal estimator for the distribution function F(t) parameters, the next paragraph:
Obviously, when the Weibull mathematical model for product reliability is known, the previous analytical relationships are used. But when the model is not known, obtaining it requires the calculation of one of the reliability indicators through the statistical processing of the values for failure times and then, the modeling of the distribution of these singular values through a continuous mathematical function as appropriate as possible.
Further, at the beginning of the next paragraph, the wording was added:
Due to its cumulative character, most frequently used indicator for modeling reliability is the unreliability function or the distribution function F(t), which expresses the probability that the product will fail at the time t.
Page 8, table 2: Is this table cited and commented?
Response 2
Indeed, this table is not quoted and commented on.
As a result, the following specification was inserted in the content of the work, before the paragraph above table 2:
According to the proposed methodology, for the data ti (i = 1...10) that constitute the case study, the F(ti) estimator was determined using the BETA.INV function and, then, the quantities ln(ti) and ln[1/(1 -F)], presented in table 2, with which the graph presented in figure 3 was built.
Page 12, line 279: "Which using only the support of the usual..." wich uses... or Which is using...?
Response 3
It was wanted to claim that the usual Microsoft Excel program is sufficient for the application of the proposed method, but the expression was not successful, so the respective phrase was reformulated in the text:
This supports the efficiency of the proposed method, for which the usual Microsoft Excel program is sufficient to arrive at a three-parameter Weibull model more likely than the two-parameter Weibull model.
Author Response File: Author Response.docx
Reviewer 4 Report
While the information in your work is no doubt very important, the material is sometimes dry, so please improve your paper giving it more life. Deliver only one of your main results with passion, and it can often times translate into your readers forming their own interest in the subject of your paper.
Author Response
Response to Reviewer 4 Comments
Manuscript ID: processes-1757581
Title of manuscript: Reliability modelling through the three-parametric Weibull model based on Microsoft Excel facilities
Dear Reviewer,
We appreciate your precious time and invaluable comments on our manuscript. We have carefully taken the comments into consideration for improving the quality of the paper in our revision. The corresponding changes and refinements made in the revised paper are summarized in our response below. The modified text is highlighted in revised manuscript.
With Regards,
Authors.
Reviewer 4
While the information in your work is no doubt very important, the material is sometimes dry, so please improve your paper giving it more life. Deliver only one of your main results with passion, and it can often times translate into your readers forming their own interest in the subject of your paper.
Response:
Thanks for the remark. Indeed, support for the proposed method can be improved by insisting on its novelty and on the advantages it brings.
For this, the content of the work was completed with the following paragraphs, included in chapter 4. Conclusions:
As a result, using the exact values for Median Ranks, and not estimated by various algebraic estimators (Benard, Hazen, etc.), a more appropriate reliability model will be obtained.
The applicability of the method can be further improved by transposing it into a calculation program, according to the presented logic scheme.
Author Response File: Author Response.docx
