Many-Objectives Optimization: A Machine Learning Approach for Reducing the Number of Objectives

Round 1

Reviewer 1 Report

Some suggestions:

1. abs:  not all the EMO algorithms use non-dominance such as hypervolume based and decomposition based algorithms. 

2.  Some recent methods for dimension reduction has been reported. You  can find some in TEVC.  You should mention them.

3. Methodology part:  you should explain why DAMICORE and FS-OPA are good tools for objective reduction, particularly, in comparison with other techniques such as PCA. 

 

Author Response

We thank the reviewer for the valuable comments that helped to improve the final manuscript.

 

QUESTION 1

1. abs: not all the EMO algorithms use non-dominance such as hypervolume based and decomposition based algorithms.

ANSWER

Our idea is to refer that in a Multi-Objective environment the concept of dominance is essential. Anyway, we changed the following sentence at the beginning of the second paragraph of section 1:

“There are at least three basic types of population-based algorithms commonly employed to solve Multiobjective Optimization Problems MOPs, namely, evolutionary algorithms, swarm-based methods and colony-based algorithms, which can use the dominance concept, the metric indicators or the decomposition strategy [3]”.

 

QUESTION 2

2. Some recent methods for dimension reduction has been reported. You can find some in TEVC. You should mention them.

ANSWER

A recent work based on the use of MOEAs was added to the text. Please, see the following sentence:

“Yuan et al. [13] proposed a methodology based on the use of multi-objective evolutionary algorithms to solve a MOOP formulation. The authors applied this approach to some benchmark problems and two real optimization problems. In both cases, the calculation of the objective functions is based on simple analytical equations where the computational cost is not relevant when compared with the problems that we intend to solve here, which are based on numerical calculation. Therefore, besides is performance, this type of methodology will not be explored in the present work.”

 

QUESTION 3

3. Methodology part: you should explain why DAMICORE and FS-OPA are good tools for objective reduction, particularly, in comparison with other techniques such as PCA.

ANSWER

According to references [5] and [12], NL-MVU-PCA was chosen over PCA since the former can deal with datasets with different probability distributions. Moreover, the current paper focuses on real-world problems with datasets with few samples and, thus, no prior knowledge of variable distributions is assumed. As a consequence, FS-OPA is only compared to its direct rival.

Reviewer 2 Report

This paper proposes a machine learning methodology, namely, FS-OPA, to reduce the number of objectives to use in a given optimization problem. The proposed method has been accessed on benchmark DTLZ problems, and then, has been applied to solve a real-world polymer extrusion problem. The methodology has been described well and the results presented clearly.

- The authors may note that there is some discrepancy in the comparison of NL-MVU-PCA and FS-OPA presented in Table 1 on Page 7, as below. 

1. For the row named "kernel function choice", the original study of NL-MVU-PCA (citation [5] in the paper) clearly states that the kernel matrix is developed by solving an optimization problem. Hence, there is no kernel function choice to be made apriori. 

2. For the row named "parameter optimization", the study in [5] has recommended an appropriate value for the parameter, which need not be refined (or optimized) while solving any new problem. 

3. For the row named "mixed", a real-world gearbox problem has been solved in [5], which is a mixed-variable problem, depicting that NL-MVU-PCA can be used on mixed variable problems as well. 

4. For the row named "explainability and the why", the authors may refer to reference [12] in their paper, where NL-MVU-PCA has been used to determined a ranking order between the objectives. Such a ranking can be considered as an explicit knowledge representation to benefit the decision-maker interaction.

I'd suggest the authors to rectify these claims in the Table, and appropriately in the text, say on Page 3 and so on. 

- In Section 3 (Page 8), the authors have demonstrated the proposed approach on the benchmark DTLZ problems. However, the authors should also present the results in terms of objective reduction (the final essential objective set), as it is the main focus of this paper. 

Author Response

We thank the reviewer for the valuable comments that helped to improve the final manuscript.

 

This paper proposes a machine learning methodology, namely, FS-OPA, to reduce the number of objectives to use in a given optimization problem. The proposed method has been accessed on benchmark DTLZ problems, and then, has been applied to solve a real-world polymer extrusion problem. The methodology has been described well and the results presented clearly.

 

QUESTION 1

- The authors may note that there is some discrepancy in the comparison of NL-MVU-PCA and FS-OPA presented in Table 1 on Page 7, as below.

1. For the row named "kernel function choice", the original study of NL-MVU-PCA (citation [5] in the paper) clearly states that the kernel matrix is developed by solving an optimization problem. Hence, there is no kernel function choice to be made apriori.
2. For the row named "parameter optimization", the study in [5] has recommended an appropriate value for the parameter, which need not be refined (or optimized) while solving any new problem.
3. For the row named "mixed", a real-world gearbox problem has been solved in [5], which is a mixed-variable problem, depicting that NL-MVU-PCA can be used on mixed variable problems as well.
4. For the row named "explainability and the why", the authors may refer to reference [12] in their paper, where NL-MVU-PCA has been used to determined a ranking order between the objectives. Such a ranking can be considered as an explicit knowledge representation to benefit the decision-maker interaction.

I'd suggest the authors to rectify these claims in the Table, and appropriately in the text, say on Page 3 and so on.

ANSWER

Item 1

Table 1 was modified to indicate that NL-MVU-PCA may require an optimization process to determine a kernel matrix, avoiding the misunderstanding that a practitioner must choose a kernel function. Moreover, a remark in Table 1 explains that using previously derived "data-dependent" kernels can bypass the limitation of an optimization process [5]. However, such a strategy may fail for real-world problems with poor previous knowledge and few samples, which are the problem that the proposed paper focuses on.

Item 2

A remark in Table 1 clarifies that NL-MVU-PCA may not require a parameter optimization phase (to choose a kernel) when solving a new problem if the most constrained case (that corresponds to parameter q equals to M − 1) is recommended and used.

Item 3

The new version of Table 1 annotates that NL-MVU-PCA has solved a mixed-variable problem, the gearbox problem [5], with the variables gear thickness, the number of teeth, the power, and the module, and the following objectives overall volume of gear material used, the power delivered by the gearbox, and the centre distance between input and output shafts. NL-MVU-PCA is applied to the non-dominated points in the objective space, i.e., to continuous vectors only. Moreover, the explaining level of "The Why" requires their relations to variables (see the answer to item 4 below. FS-OPA performs such analysis involving vectors of mixed types for any combination of continuous, discrete, ordinal, and nominal vectors in the decision and objective space. The new version of Table 1 and its remarks highlights the importance of dealing with mixed vectors from decision and objective space and their relevance for the explicit explainability of a final essential objective set.

Item 4

The explicability issue in Table 1 was rearranged into two levels of explainability: implicit and explicit (also referred to as "The Why"). NL-MVU-PCA can rank objectives according to their priority for optimization. Moreover, FS-OPA indicates the potential influence of variables on objectives and the groups of strongly related objectives. At a certain level, such influences are clues for causality ("The Why"), another class of explainability.

 

 

QUESTION 2

- In Section 3 (Page 8), the authors have demonstrated the proposed approach on the benchmark DTLZ problems. However, the authors should also present the results in terms of objective reduction (the final essential objective set), as it is the main focus of this paper.

ANSWER

The final essential objective sets found for DTLZ's by FS-OPA were now highlighted in the paper.

Round 2

Reviewer 1 Report

The quality has been improved. it is fine with me. 

Author Response

We thank the reviewer for the valuable comments that helped to improve the final manuscript.

Reviewer 2 Report

I appreciate the authors revising the manuscript based on my previous comments. However, I would highlight some critical concerns pertaining to the modifications made. 

1. For theoretical validation, the authors have used the original unconstrained version of DTLZ5. However, the paper in reference [5] demonstrated that DTLZ5 is only a redundant problem with application of constraints. I would suggest the authors to use the constrained formulation of DTLZ5. 

2. On page-10, the authors have reported reduction of objectives in problems DTLZ1 - DTLZ4. This seems to be incorrect since these problems do not have any redundant objective. The authors can create the parallel coordinate plots for these problems and see that the reported reduction in objectives is not genuine. 

3. The above two points raise a concern since the results of the proposed framework on the benchmark problems cannot be validated. I would suggest the authors to explicitly mention that the proposed method did not work on the benchmark test problems. The pertaining question is, how to validate the results? 

Overall, I would suggest the authors to revise the results on benchmark problems, or mention that their proposed framework does not work on these problems. This is rather imperative since it raises a question on the applicability of the proposed method on the real-world problem solved in this paper. 

Author Response

We thank the reviewer for the valuable comments that helped to improve the final manuscript.

 

GENERAL ANSWER

The reviewer is correct since we didn´t explain our work’s aim well. The main aim of the (previous) works cited in our paper on reducing objectives was to find a reduced set of objectives that produce exactly the same results as the original set. This implies that only the redundant objectives can be discarded after the reduction process. This is not the case in the present work since our aim is to apply the methodology proposed to real-world and complex problems in which the relations between the decision variables and the objectives are complex and do not allow for eliminating objectives as they are not completely redundant.

Therefore, we develop a methodology able to capture those complex relations and define the level of importance of the objectives based on the determination of the relations objectives-objectives. By doing this was possible to determine that some objectives can be discarded, but with a certain error, i.e., the approximation of the Pareto optimal found with the reduced number of objectives have some error when compared with the approximation to the optimal Pareto front when using all the objectives. This has at least two important advantages. First, it will be easier for the optimization algorithm to find the approximation to the optimal Pareto front; and second, it will be easy to explain to the DM the results found.

The results obtained for the problem under study (Table 7 in the paper) clarify that this methodology is valid as the results obtained with the reduced number of objectives have an error that is within the limits of the error of the MOEA used, i.e., very small. The optimal Pareto front (POF) was not known for such a problem, while a POF is very well known for “toy” problems, generally used to test these types by methodologies.

We appreciate the reviewer’s comments as they allow us to explain this point more rigorously. For that, a sentence was added to the introduction to clarify the aim of this paper. Also, some changes in the paper were made to answer the specific questions below.

 

QUESTION

I appreciate the authors revising the manuscript based on my previous comments. However, I would highlight some critical concerns pertaining to the modifications made.

1. For theoretical validation, the authors have used the original unconstrained version of DTLZ5. However, the paper in reference [5] demonstrated that DTLZ5 is only a redundant problem with application of constraints. I would suggest the authors to use the constrained formulation of DTLZ5.

ANSWER

The results concerning the constrained version of DTLZ5 are presented and discussed as suggested.

 

QUESTION

2. On page-10, the authors have reported reduction of objectives in problems DTLZ1 - DTLZ4. This seems to be incorrect since these problems do not have any redundant objective. The authors can create the parallel coordinate plots for these problems and see that the reported reduction in objectives is not genuine.

ANSWER

The reduction of objectives performed for those problems was expected to account for the work's global aim, as described above in the general answer to the reviewer. We agree with the reviewer that they do not fit the paper's primary purpose; thus, they were removed in the paper's new version.

 

QUESTION

3. The above two points raise a concern since the results of the proposed framework on the benchmark problems cannot be validated. I would suggest the authors to explicitly mention that the proposed method did not work on the benchmark test problems. The pertaining question is, how to validate the results?

ANSWER

The new results presented and discussed show that the methodology proposed was validated using, also, the DTLZ5 problem. The results for the real problem used are validated in Table 7, taking into account the aim of this paper as described above. Please, see the answer to points 1 and 2 above.

 

QUESTION

Overall, I would suggest the authors to revise the results on benchmark problems, or mention that their proposed framework does not work on these problems. This is rather imperative since it raises a question on the applicability of the proposed method on the real-world problem solved in this paper.

ANSWER

Please, see the general answer and the answers for points 1 to 3 above.

Round 3

Reviewer 2 Report

I appreciate the authors revising the paper. The paper has improved significantly, however there are still some issues that should be addressed. My comments are below.

- The changes made in the introduction section (regarding the revised aim of the paper) are not convincing. Objective reduction (with or without inducing error) should not be seen in isolation with the identification of redundant objectives. For any meaningful reduction in a given objective set, first the redundant objectives need to be identified and eliminated, followed by elimination of any of the essential objectives (if desired). The elimination of an essential objective while retaining a redundant one, would be misleading the researchers and practitioners alike. 

- In terms of the results presented, the inclusion of DTLZ5 (constrained) is encouraging since the results are meaningful. However, to convey the complete message to the reader, the strengths and limitations of the proposed approach should also be included. For instance, the authors should include the DTLZ1-4 results (which were there in the earlier submission), and highlight the issues pertaining to accuracy in those problems with no redundant objectives. This will allow the reader to have a balanced view of the paper. 

Author Response

We thank you again for the reviewer comments.

 

QUESTION:

I appreciate the authors revising the paper. The paper has improved significantly, however there are still some issues that should be addressed. My comments are below.

- The changes made in the introduction section (regarding the revised aim of the paper) are not convincing. Objective reduction (with or without inducing error) should not be seen in isolation with the identification of redundant objectives. For any meaningful reduction in a given objective set, first the redundant objectives need to be identified and eliminated, followed by elimination of any of the essential objectives (if desired). The elimination of an essential objective while retaining a redundant one, would be misleading the researchers and practitioners alike.

ANSWER:

We must point out again that this paper aims to develop a methodology to deal with real-world problems. Therefore, we do not totally agree with the reviewer since as we said the methodology needs to work in complex real problems where it is not possible (or at least with the actual knowledge) to identify redundant objectives because of a high probability they do not exist. As was demonstrated in the practical example studied none of the objectives is “completely” redundant, there is always a relation between the objectives and any decision variable (and also with other objectives).

In any part of our text, we refer that the redundant objectives were not eliminated. In fact, the methodology proposed does not allow for maintaining any redundant objective. For example, objectives T and TTb (table 6 and figure 9) are strongly correlated (but not totally), however, the methodology proposed does not allow to keep both objectives (or at least suggest eliminating one).

Anyway, this was clarified in the text by adding the following sentence: “Simultaneously, the redundant objectives are also eliminated.”

 

QUESTION:

- In terms of the results presented, the inclusion of DTLZ5 (constrained) is encouraging since the results are meaningful. However, to convey the complete message to the reader, the strengths and limitations of the proposed approach should also be included. For instance, the authors should include the DTLZ1-4 results (which were there in the earlier submission), and highlight the issues pertaining to accuracy in those problems with no redundant objectives. This will allow the reader to have a balanced view of the paper.

ANSWER:

The strengths and the limitations of the methodology proposed were from the beginning highlighted in Table 1.

As requested, an analysis of DTLZ1-DTLZ4 problems is, again, made in the manuscript.