Two-Dimensional Age Replacement Decision for Structural Dependence Parallel Systems via Intelligent Optimization Algorithm
Round 1
Reviewer 1 Report
The comments and questions are in the file. My opinion is that the material presented by the authors needs significant improvement.
Comments for author File: 
Comments.pdf
Author Response
Dear Editor,
Thank you for allowing a resubmission of our manuscript, with an opportunity to address the reviewers’ comments. We have carefully revised the paper according to the opinions of the reviewers. Thank the reviewers for their valuable comments very much, which will greatly improve the quality of the paper.
We are uploading our point-by-point response to the comments (response to reviewers), an updated manuscript with yellow highlighting indicating changes.
Best regards,
<Enzhi Dong> et al.
Reviewer #1 Concern # 1: You give a section as notation, and then duplicate part of it. Usually the notation is given at the end of the article.
Author response: We agree with the opinions of the reviewer.
Author action: We put the Notation section at the end of the paper.
Reviewer #1 Concern # 2: You declare the theorem, but there is no proof.
Reviewer #1 Concern # 3: Theorems 1-3 are a kind of definition of the Copula function. Do you present these theorems as your personal contribution?
Author response: Full consideration to the opinions of the reviewer and make serious changes.
Author action: Theorems 1-3 are the basic theorems of copula function. At present, the proof of these three theorems has been very mature. This paper mainly uses copula function to solve engineering problems, and does not focus on the derivation of theorems. Therefore, these theorems are deleted from the paper, and a reference is added near formula (1), through which readers can learn more about copula function.
Reviewer #1 Concern # 4: Step 4 of the Model analysis is incorrectly stated.
Author response: We agree with the opinions of the reviewer.
Author action: The original meaning of this sentence is that an interval is regarded as a specific value for operation. We rewrite this sentence as follows: the average utilization rate of this interval is used to replace this interval for calculation.
Reviewer #1 Concern # 5: All these algorithms are well known. You have not made any modification, is it worth giving a description of them?
Author response: Full consideration to the opinions of the reviewer and make serious changes.
Author action: As the reviewer said, we have no algorithm to modify and innovate, but only compare three algorithms commonly used in the warranty decision-making field to determine the optimal algorithm for solving the model proposed in the manuscript, so as to provide guidance for solving the same type of problems. We all agree that there will be some redundancy in introducing the algorithm again when the algorithm is very mature. Therefore, we delete Section 5 in the manuscript and add references to the upper paragraph of Figure 7. Readers can obtain specific knowledge about these algorithms by consulting these references.
Reviewer #1 Concern # 6: How do you explain the weak dependance on T0 of System availability?
Author response: It can be seen from Figure 5 and Figure 6 that T0 has little impact on the system warranty cost and availability, that is, when U0 is fixed, the change in the time interval for replacing the system will not cause a significant change in the system warranty cost and availability. On the one hand, the main failure mode of the fuel fine filter is that the fuel filter is blocked and dirty, resulting in efficient fuel supply. Therefore, the mileage has a great impact on the failure of the fuel fine filter, and the two have a strong correlation. On the other hand, as time goes by, the components that make up the fuel fine filter may corrode, but this process is very slow. Therefore, calendar time is not the main factor that causes the failure of the fuel fine filter. So there is weak dependency between T0 and system availability/warranty cost. This is also the reason why this paper chooses to use AFT model to construct failure rate function.
The above explanatory content has been added in the manuscript.
Reviewer #1 Concern # 7: What about solution stability? The listed algorithms may give different solutions on multiple runs. Were the results given in the table stable at different runs of the algorithms?
Author response: Full consideration to the opinions of the reviewer and make serious changes.
Author action: The above three algorithms are global optimization algorithms, which can obtain the best possible global optimal solution region in the sense of probability through probability search. Therefore, the results of each run may be different. In order to test the stability of the results, the three algorithms are run 1000 times respectively with the same parameters, and the average value of the results of 1000 runs is calculated for comparison. The optimal value obtained after each run of the program is shown in Figure 9. The average value of 1000 operations is shown in Table 3.
Reviewer #1 Concern # 8: The variable r is absent in Equation 40.
Author response: We agree with the opinions of the reviewer.
Author action: In formula 40, x should be r, which has been modified.
Reviewer #1 Concern # 9: You define rather narrow range of r, so the condition that the integral of the probability density function is equal to 1 on this interval is not met.
Author response: Full consideration to the opinions of the reviewer and make serious changes.
Author action: In the field of two-dimensional warranty research, uniform distribution, normal distribution and Weibull distribution are the most common distribution forms of utilization rate r. To facilitate comparison, it is assumed that the three distribution forms of utilization have the same upper and lower limits, and the maximum utilization rate is 3.6×104 km/year, the minimum utilization rate is 0.36× 104 km/year. Moreover, it is assumed that the mean value of normal distribution is 2× 104km/year, the standard deviation is 0.8. It should be emphasized here that since the value range of the normal distribution variable is (-∞,+∞), but the utilization rate cannot be negative, this paper makes an approximate estimation of the parameters of the normal distribution to ensure that the integral value within the utilization rate interval (0.36, 3.6) is greater than 0.95.
Reviewer #1 Concern # 10: There is an English grammar error above Formula 40.
Author response: We agree with the opinions of the reviewer.
Author action: We have revised the English expression here.
Reviewer #1 Concern # 11: “the availability is 0.687.” You got rather low availability rate. What is the reason for this?
Author response: Full consideration to the opinions of the reviewer and make serious changes.
Author action: This is mainly due to the high failure rate of the fuel fine filter in this case. Because of the high failure rate of the fuel fine filter, within a preventive replacement interval, the fuel fine filter needs to be replaced after multiple failures, which not only increases the warranty cost, but also reduces the availability. In addition, this model also considers the failure interaction between components, and introduces parameter α to describe the correlation of failure time between components, which further increases the failure rate of the fuel fine filter and reduces the availability. 0.687 is the availability corresponding to the optimal two-dimensional replacement scheme found when the utilization rate follows the two-parameter Weibull distribution. This value is less than the availability corresponding to the optimal two-dimensional age replacement scheme found when the utilization rate follows the uniform distribution and normal distribution. This is mainly because when the utilization rate follows the two-parameter Weibull distribution, the higher utilization rate appears with a higher probability, so the failure of the fuel fine filter is more frequent. This factor can also lead to reduced availability.
Reviewer #1 Concern # 12: What is the physical meaning of a negative alpha value? Can the failure of one component increase the reliability of the other?
Author response: Full consideration to the opinions of the reviewer and make serious changes.
Author action: α is the parameter of the Copula function, which measures the correlation strength between variables. In the case study, α indicates the correlation between the failure times of the two filter bodies of the fuel fine filter, so it represents the physical quantity of failure interaction between components. The value of α cannot be negative. We have corrected Table 7 and Figure 16.
Reviewer #1 Concern # 13: U0=100000KM??? Why do you set such value – in the tables you present U0 in the range 2000-13000KM??
Author response: Full consideration to the opinions of the reviewer and make serious changes.
Author action: In this case, the value range of U0 is (0KM, 105KM), which has been defined in 5.1. Here, the maximum value 105KM of U0 is calculated and fixed, that is, preventive replacement is not considered in the mileage dimension. In this way, two-dimensional age replacement becomes one-dimensional age replacement. Through comparison, highlighting the effectiveness of two-dimensional service age replacement. The effectiveness of two-dimensional age replacement is reflected.
Reviewer #1 Concern # 14: Why do you limit your research U0=5000KM? In the page 19 you mention about U0=11000KM and U0=13000KM. Will the conclusion be the same?
Author response: We agree with the opinions of the reviewer. Only one special value is fixed for research, and the conclusion may be one-sided and unrepresentative, so we delete this part of the research from the manuscript. And we rewrite the conclusion.
Reviewer #1 Concern # 15: What is the advantage of your approach compared to the Pareto optimal solution to the problem?
Author response: Thanks for your question. In this paper, the lowest warranty cost and the highest availability are considered as two independent decision-making objectives. At present, there have been many studies that have taken these two objectives into consideration for joint optimization. Some have taken the lowest warranty cost as the decision objective and the availability as the constraint condition. Some have taken the warranty cost and availability into consideration for multi-objective optimization to obtain the Pareto optimal solution. The purpose of this study is to provide manufacturers with an effective tool for estimating warranty costs and system availability, so as to assist manufacturers in making decisions, and to provide suggestions for system design and warranty scheme optimization. Therefore, the two objectives are not considered together. In the next step of research, we can consider taking the lowest warranty cost and the highest availability as the decision objectives, using NSGA algorithm to solve the optimal warranty scheme, taking into account the interests of both manufacturers and users, and expanding the application scope of the model.
Author Response File: 
Author Response.docx
Reviewer 2 Report
In this paper the expected cost rate is minimized or the availability is maximized, by giving the optimal age replacement interval for large scale aerospace systems or household appliances.
Three optimization algorithms are applied and results compared.
Author Response
Thank you for your review of the paper.
Reviewer 3 Report
1. In the optimization, the expected cost rate is minimized, or the availability is maximized. What’s the relationship between these two objectives? Why are these two objectives not optimized simultaneously, like multiple-objective optimization? Because it is possible that the optimizer which makes the lowest cost rate may lead to the low availability.
2. In the performance comparison of three optimization methods, PSO, GA and SSA, because there are so many hyper-parameters for each method, for instance, GA has the parameters like population size which is actually not trivial, is it fair for side-by-side comparison? Did the authors try hyperparameter optimization for each method, or just try different settings for each optimization method?
3. What’s the purpose of “dimension reduction” analysis? The peak points shown in Figures 7 and 8 correspond to the optimal points when only single variable varies while the other variable is fixed. Do the peaks shown in the figures have relationship with the following optimal points? In addition, when we say dimension reduction analysis, if typically refers the transformation of data from a high-dimensional space into a low-dimensional space like PCA.
4. In the impact analysis of alpha in section 6.4, Tables 6 shows that the value of alphas has a huge impact on the optimal values, in real application, how is the value of alphas determined?
5. For one-dimensional age replacement strategy, why the U0 is taken as its maximum value to get the change of warranty cost rate and availability in Figures 15 and 16?
6. In the paper, the method is only applied to one single example, can this single example represent more general a[plication scenarios?
Author Response
Dear Editor,
Thank you for allowing a resubmission of our manuscript, with an opportunity to address the reviewers’ comments. We have carefully revised the paper according to the opinions of the reviewers. Thank the reviewers for their valuable comments very much, which will greatly improve the quality of the paper.
We are uploading our point-by-point response to the comments (response to reviewers), an updated manuscript with yellow highlighting indicating changes.
Best regards,
<Enzhi Dong> et al.
Reviewer #2 Concern # 1: In the optimization, the expected cost rate is minimized, or the availability is maximized. What’s the relationship between these two objectives? Why are these two objectives not optimized simultaneously, like multiple-objective optimization? Because it is possible that the optimizer which makes the lowest cost rate may lead to the low availability.
Author response: We agree with the opinions of the reviewer.
Author action: In this paper, the lowest warranty cost and the highest availability are considered as two independent decision-making objectives. Increased availability sometimes leads to increased warranty costs. At present, there have been many studies that have taken these two objectives into consideration for joint optimization. Some have taken the lowest warranty cost as the decision objective and the availability as the constraint condition. Some have taken the warranty cost and availability into consideration for multi-objective optimization to obtain the Pareto optimal solution. The purpose of this study is to provide manufacturers with an effective tool for estimating warranty costs and system availability, so as to assist manufacturers in making decisions, and to provide suggestions for system design and warranty scheme optimization. Therefore, the two objectives are not considered together. In the next step of research, we can consider taking the lowest warranty cost and the highest availability as the decision objectives, using NSGA algorithm to solve the optimal warranty scheme, taking into account the interests of both manufacturers and users, and expanding the application scope of the model. In this regard, we put forward this point in the conclusion.
Reviewer #2 Concern # 2: In the performance comparison of three optimization methods, PSO, GA and SSA, because there are so many hyper-parameters for each method, for instance, GA has the parameters like population size which is actually not trivial, is it fair for side-by-side comparison? Did the authors try hyperparameter optimization for each method, or just try different settings for each optimization method?
Author response: Full consideration to the opinions of the reviewer and make serious changes.
Author action: The three algorithms are global optimization algorithms, which can obtain the best possible global optimal solution region in the sense of probability through probability search. Therefore, the results of each run may be different. In order to test the stability of the results, the three algorithms are run 1000 times respectively with the same parameters, and the average value of the results of 1000 runs is calculated for comparison. The optimal value obtained after each run of the program is shown in Figure 9. The average value of 1000 operations is shown in Table 3. The results show that SAA algorithm is more effective for solving this model. The result of multiple operations is relatively fair.
Reviewer #2 Concern # 3: What’s the purpose of “dimension reduction” analysis? The peak points shown in Figures 7 and 8 correspond to the optimal points when only single variable varies while the other variable is fixed. Do the peaks shown in the figures have relationship with the following optimal points? In addition, when we say dimension reduction analysis, if typically refers the transformation of data from a high-dimensional space into a low-dimensional space like PCA.
Author response: Full consideration to the opinions of the reviewer and make serious changes.
Author action: In this part, we originally wanted to fix one dimension and observe the change of the results with another dimension, but in fact, this trend can be seen from Figure 5 and Figure 6. Therefore, this part is redundant. What’s more, according to the reviewer's opinion, it is inappropriate to apply the statement of " dimension reduction " here, and we have deleted this part from the manuscript.
Reviewer #2 Concern # 4: In the impact analysis of alpha in section 6.4, Tables 6 shows that the value of alphas has a huge impact on the optimal values, in real application, how is the value of alphas determined?
Author response: α is the parameter of the Copula function, which measures the correlation strength between variables. In the case study, α indicates the correlation between the failure times of the two filter bodies of the fuel fine filter, so it represents the physical quantity of failure interaction between components. The correlation coefficient can be determined by the following methods:
- Obtained by probability theory
- Based on experience of designers, manufacturers and maintenance personnel
- Based on mechanical or dynamic estimation
- Based on laboratory experiments
Reviewer #2 Concern # 5: For one-dimensional age replacement strategy, why the U0 is taken as its maximum value to get the change of warranty cost rate and availability in Figures 15 and 16?
Author response: Full consideration to the opinions of the reviewer and make serious changes.
Author action: In this case, the value range of U0 is (0KM, 105KM), which has been defined in 5.1. Here, the maximum value 105KM of U0 is calculated and fixed, that is, preventive replacement is not considered in the mileage dimension. In this way, two-dimensional age replacement becomes one-dimensional age replacement. Through comparison, the effectiveness of two-dimensional service age replacement is highlighted. The effectiveness of two-dimensional age replacement is reflected.
Reviewer #2 Concern # 6: In the paper, the method is only applied to one single example, can this single example represent more general application scenarios?
Author response: This single example can represent more general application scenarios. The application of this model requires two conditions: one is that the warranty object is a multi component parallel system, and the other is that there is a correlation between the failure times of components, such as the Warm Standby Redundant. On the basis of meeting these two conditions, by fitting the failure rate function and utilization rate function as well as estimating the value of α. This model can be used to obtain the two-dimensional age replacement plan under different decision-making objectives.
Author action: We explain the scope of application of the model in the conclusion.
Author Response File: Author Response.docx
Round 2
Reviewer 1 Report
The authors have worked out all the comments made, all questions have been answered, all points that caused objections have been removed from the manuscript. I believe that the ongoing research is far from complete, but the results presented are quite interesting and deserve publication.
New comments -
1) page 15 - please correct "it is assumed that the mean value of normal distribution is 2×10^4km/year, the standard deviation is 0.8"×10^4km/year
2) Table 7, Figure 16 - the relationship between alpha and the lowest expected cost rate is almost linear, and between alpha and the highest availability is non linear - it is interesting to see more points corresponding to alpha=[0.1, 0.3, 0.5, 0.7, 0.9]
Author Response
Thanks for the comments of reviewers, we have carefully revised the manuscript.
