Multi-Objective, Reliability-Based Design Optimization of a Steering Linkage
Round 1
Reviewer 1 Report
The paper is interesting but there are some problems that make some parts hard to follow:
- I think the angle ΘI should be indicated in fig. 2
- the caption of figure 3 is on page 4 while the figure itself is on page 3
- the symbols (such as R or XO) should not overlap other elements of the figure
- text formatting needs to be improved (missing or unecessary spaces, equation font sizes etc. - on page 5 for example)
- grammar needs to be checked (I'm not an expert but in my opinion "the extension may causes the feasible design..." (line 198) or "the optimizer uses in this study is an adaptation..." (line 202) are examples of grammar mistakes that make the paper harder to read)
- empty parentheses in line 317
In general the paper requires editing to improve it's readability.
Author Response
I think the angle Θ should be indicated in fig. 2
Answer: We have corrected the figure as suggested.
the caption of figure 3 is on page 4 while the figure itself is on page 3
Answer: We have improved the figure in the revision.
the symbols (such as R or X ) should not overlap other elements of the figure
Answer: We have rechecked and adjusted the symbols. Thank you for your suggestion.
text formatting needs to be improved (missing or unecessary spaces, equation font sizes etc. - on page 5 for example)
Answer: We have rechecked it according to your suggestion.
grammar needs to be checked (I'm not an expert but in my opinion "the extension may causes the feasible design..." (line 198) or "the optimizer uses in this study is an adaptation..." (line 202) are examples of grammar mistakes that make the paper harder to read) empty parentheses in line 317
In general the paper requires editing to improve it's readability.
Answer: The revision has been proofread by native English speaker. For the empty parentheses below the table is caption for the data from previous study. Thank you for your suggestions, which leads to improvement of the manuscript.
Reviewer 2 Report
The authors optimize a steering linkage solving a multiobjective non-linear programing problem under uncertainty. Particularly, the uncertainty is limited in the constraints and arises in two parameters (tolerances of two lengths), which are described as fuzzy variables with specified membership functions. The alpha-cut approach generates an additional objective to the original objective (a distance deviation), and the resulting multiobjective problem is solved using a derivative-free algorithm. The proposed optimization technique reduces the complexity of the design process and provides tradeoff solutions between accuracy and reliability.
In my view, this is an interesting contribution that highlights the importance of optimization in engineering systems. The paper is clearly written, the case study is described in detail with enough information to be validate. In addition, the problem statement is well-defined. Before final acceptance the authors could address the following points:
- Do the authors know if this problem has been previously solved as a stochastic model using mathematical programming techniques? If it is not the case, as the initial deterministic problem is not a large-size problem (only 3 decision variables), then I guess that a state-of-the-art gradient based solver could exhibit a high performance solving the resulting stochastic problem, which is formulated once the uncertainties are incorporated by Montecarlo Sampling.
- If the solver that the authors apply, RPBIL-DE, it is not commonly used by the readers of this journal, in my opinion, the authors should provide more details about it.
- The authors claim that they reduce the complexity of the optimization problem by their proposed technique, in this regard it would be useful to know the degree of such complexity by adding the solver statistics.
Minor tips:
Figure 2: the notation for the wheelbase “W_b” is mising in the figure, and the notation for inner wheel angle is different from the one used in the text, “\theta_t”.
Page 8, line 272: spelling error in word “variables”.
Page 9, line 297: “are shows” must be replaced with “are shown”.
Page 14, Figure 8: Perhaps, fewer drawing objects could improve its readability.
Author Response
The authors optimize a steering linkage solving a multiobjective non-linear programing problem under uncertainty. Particularly, the uncertainty is limited in the constraints and arises in two parameters (tolerances of two lengths), which are described as fuzzy variables with specified membership functions. The alpha-cut approach generates an additional objective to the original objective (a distance deviation), and the resulting multiobjective problem is solved using a derivative-free algorithm. The proposed optimization technique reduces the complexity of the design process and provides tradeoff solutions between accuracy and reliability.
In my view, this is an interesting contribution that highlights the importance of optimization in engineering systems. The paper is clearly written, the case study is described in detail with enough information to be validate. In addition, the problem statement is well-defined. Before final acceptance the authors could address the following points:
- Do the authors know if this problem has been previously solved as a stochastic model using mathematical programming techniques? If it is not the case, as the initial deterministic problem is not a large-size problem (only 3 decision variables), then I guess that a state-of-the-art gradient based solver could exhibit a high performance solving the resulting stochastic problem, which is formulated once the uncertainties are incorporated by Montecarlo Sampling.
Answer: According to our review, there have been a few researches studying the reliability-based design optimization of a steering linkage using mathematical programming techniques. The difficulty of the gradient-based optimizers in solving this problem is that the gradient calculation is somewhat noisy and unreliable as the reliability index depends on the Monte Carlo randomization. Furthermore, the combination of gradient based and MCS requires significantly greater computing time so that it is adequately accurate. The present idea can avoid the difficulties and can find a solution set within one optimization run while gradient-based optimizers cannot deal with exploring a Pareto front within one run like a metaheuristic.
- If the solver that the authors apply, RPBIL-DE, it is not commonly used by the readers of this journal, in my opinion, the authors should provide more details about it.
Answer: We have added detail of the algorithm in section 2.2. Thank you for your suggestion.
- The authors claim that they reduce the complexity of the optimization problem by their proposed technique, in this regard it would be useful to know the degree of such complexity by adding the solver statistics.
Answer: For the term complexity here, we mean the reliability optimization using our proposed idea doesn’t need a double loop or triple loop optimization runs. The single-objective optimization problem with uncertainties is treated to be equivalent to a deterministic multi-objective optimization problem in which any good multi-objective metaheuristic can solve the problem. We have added the average computing time of the RPBILDE algorithm in Table 2.
Minor tips: Figure 2: the notation for the wheelbase “W_b” is mising in the figure, and the notation for inner wheel angle is different from the one used in the text, “\theta_t”.
Answer: we have corrected as suggested.
Page 8, line 272: spelling error in word “variables”.
Answer: We have corrected the word.
Page 9, line 297: “are shows” must be replaced with “are shown”.
Answer: We have corrected as suggested.
Page 14, Figure 8: Perhaps, fewer drawing objects could improve its readability.
Answer: We have adjusted the figure. Thank you for your suggestions, which leads to improvement of the manuscript.
Reviewer 3 Report
The authors presented a multi-objective reliability-based design optimization formulation and its applications to different problems. Although the applications are interesting and well-presented, major revisions are needed for this article to be recommended as a publication. My suggestions are as follows:
- The manuscript should be edited for grammar.
- A more generic definition for uncertainty and RBDO must be given in the Introduction. What type of uncertainties can the presented RBDO method model (i.e., aleatoric or epistemic)? Please evaluate in more detail in the Introduction.
- There is a problem with the visualization of Equation 2.
- There are some formatting issues throughout the paper (regarding the spacing of the lines, I believe it is not fixed at some parts).
- The formatting of equations is not fixed as well, they appear in different sizes throughout the manuscript.
- How is the design constraint defined in terms of the worst-case scenario? More details should be given on the definition of the design constraint.
- The optimization problem definition still contains a single objective function, while the worst-case scenario is defined through a constraint. How is the justification of treating this problem as a multi-objective problem when there is a single objective function?
Author Response
The authors presented a multi-objective reliability-based design optimization formulation and its applications to different problems. Although the applications are interesting and well-presented, major revisions are needed for this article to be recommended as a publication. My suggestions are as follows:
The manuscript should be edited for grammar.
Answer: The revision has been proofread by a native English speaker as suggested.
A more generic definition for uncertainty and RBDO must be given in the Introduction. What type of uncertainties can the presented RBDO method model (i.e., aleatoric or epistemic)? Please evaluate in more detail in the Introduction.
Answer: We have added the generic definition of uncertainty in the introduction part.
There is a problem with the visualization of Equation 2. There are some formatting issues throughout the paper (regarding the spacing of the lines, I believe it is not fixed at some parts).
Answer: We have rechecked following the reviewer’s comment in all parts.
The formatting of equations is not fixed as well, they appear in different sizes throughout the manuscript.
Answer: We have corrected the format of all equation as the reviewer mentioned.
How is the design constraint defined in terms of the worst-case scenario? More details should be given on the definition of the design constraint.
Answer: We have added the explanation in the part of MORBDO. Thank you for your suggestion.
The optimization problem definition still contains a single objective function, while the worst-case scenario is defined through a constraint. How is the justification of treating this problem as a multi-objective problem when there is a single objective function?
Answer: We have added some explanation of the technique in general to make it clearer. Our aim of this research is to propose a reliability-based design of a steering linkage, which is usually performed by means of single-objective optimization. It causes complexity in solving a double-loop nested problem. The present idea alters the problem with a double loop optimization run to an equivalent deterministic multi-objective optimization that can be solved within one optimization run if the optimizer is a multi-objective evolutionary algorithm. We included the worst-case scenario value (WSCV) as one of the objective functions. The design objective function in design of the steering linkage still as one, but in the reliability-based design problem, one additional objective function is WSCV. Thank you for your suggestions, which leads to improvement of the manuscript.
Round 2
Reviewer 3 Report
The authors have addresses the points that I have raised in my review. The article can be published with its current form.
