Improving the Quantum Multi-Swarm Optimization with Adaptive Differential Evolution for Dynamic Environments

Round 1

Reviewer 1 Report

1. Results: Recommend to be Major revisions

This paper proposes the modification of the quantum multi-swarm optimization algorithm for dynamic optimization problems. The modification implies using the search operators from differential evolution algorithm with certain probability to improve the algorithms search capabilities in dynamically changing environments. For algorithms testing the Generalized Moving Peaks Benchmark was used. The experiments performed in two benchmarks with different changes frequency have shown that the proposed modification with dynamically changing probability of applying differential operators allows better tracking of the changing environment of the dynamic optimization problems.

This paper is with none merits for Algorithms, i.e., poor writing skills and lacking of insight analysis, it requires major revisions.

Firstly, for Section 1, authors should provide the comments of the cited papers after introducing each relevant work. What readers require is, by convinced literature review, to understand the clear thinking/consideration why the proposed approach can reach more convinced results. This is the very contribution from authors. In addition, authors also should provide more sufficient critical literature review to indicate the drawbacks of existed approaches, then, well define the main stream of research direction, how did those previous studies perform? Employ which methodologies? Which problem still requires to be solved? Why is the proposed approach suitable to be used to solve the critical problem? We need more convinced literature reviews to indicate clearly the state-of-the-art development.

For Section 2, authors should introduce their proposed research framework more effective, i.e., some essential brief explanation vis-à-vis the text with a total research flowchart or framework diagram for each proposed algorithm to indicate how these employed models are working to receive the experimental results. It is difficult to understand how the proposed approaches are working.

For Section 3, authors should use more alternative models as the benchmarking models, authors should also conduct some statistical test to ensure the superiority of the proposed approach, i.e., how could authors ensure that their results are superior to others? Meanwhile, authors also have to provide some insight discussion of the results. Authors can refer the following references for conducting statistical test.

Applications of Random forest in multivariable response surface for short-term load forecasting. International Journal of Electrical Power & Energy Systems, 2022, 139, 108073.

Author Response

Open Review

(x) I would not like to sign my review report

( ) I would like to sign my review report

English language and style

( ) Extensive editing of English language and style required

( ) Moderate English changes required

(x) English language and style are fine/minor spell check required

( ) I don't feel qualified to judge about the English language and style

Yes     Can be improved      Must be improved     Not applicable

Does the introduction provide sufficient background and include all relevant references?

( )        (x)       ( )        ( )

Is the research design appropriate?

( )        (x)       ( )        ( )

Are the methods adequately described?

( )        (x)       ( )        ( )

Are the results clearly presented?

( )        (x)       ( )        ( )

Are the conclusions supported by the results?

( )        (x)       ( )        ( )

Comments and Suggestions for Authors

Results: Recommend to be Major revisions

This paper proposes the modification of the quantum multi-swarm optimization algorithm for dynamic optimization problems. The modification implies using the search operators from differential evolution algorithm with certain probability to improve the algorithms search capabilities in dynamically changing environments. For algorithms testing the Generalized Moving Peaks Benchmark was used. The experiments performed in two benchmarks with different changes frequency have shown that the proposed modification with dynamically changing probability of applying differential operators allows better tracking of the changing environment of the dynamic optimization problems.

This paper is with none merits for Algorithms, i.e., poor writing skills and lacking of insight analysis, it requires major revisions.

Answer: thank you for your valuable comments. We have revised every section of the manuscript, from introduction to conclusion in order to improve its quality.

Firstly, for Section 1, authors should provide the comments of the cited papers after introducing each relevant work. What readers require is, by convinced literature review, to understand the clear thinking/consideration why the proposed approach can reach more convinced results. This is the very contribution from authors. In addition, authors also should provide more sufficient critical literature review to indicate the drawbacks of existed approaches, then, well define the main stream of research direction, how did those previous studies perform? Employ which methodologies? Which problem still requires to be solved? Why is the proposed approach suitable to be used to solve the critical problem? We need more convinced literature reviews to indicate clearly the state-of-the-art development.

Answer: In the introduction we have added an explanation of why the developed algorithm is important for the field and added the main contributions of the study.

For Section 2, authors should introduce their proposed research framework more effective, i.e., some essential brief explanation vis-à-vis the text with a total research flowchart or framework diagram for each proposed algorithm to indicate how these employed models are working to receive the experimental results. It is difficult to understand how the proposed approaches are working.

Answer: we have expanded the literature review with more recent approaches and analyzed the trends in the field of dynamic optimization. We have also added a more detailed description of the algorithm and a pseudo-code to allow other authors reproduce the proposed approach.

For Section 3, authors should use more alternative models as the benchmarking models, authors should also conduct some statistical test to ensure the superiority of the proposed approach, i.e., how could authors ensure that their results are superior to others? Meanwhile, authors also have to provide some insight discussion of the results. Authors can refer the following references for conducting statistical test.

Applications of Random forest in multivariable response surface for short-term load forecasting. International Journal of Electrical Power & Energy Systems, 2022, 139, 108073.

Answer: we have added the comparison with alternative approaches. To do so, we had to change the benchmark parameters, as there were no known results available for the new version of the benchmark (used before revision). As for statistical tests, we have used the Mann-Whitney tests to compare the proposed algorithm with the baseline, and used the ranking procedure from Friedman statistical test to compare the tested versions.

Author Response File: Author Response.pdf

Reviewer 2 Report

The multi-swarm Quantum Swarm Optimization [18] algorithm is modified with the Differential Evolution (DE) search operator in this work. Two versions of modified algorithms, mQSODE and mQSOaDE, are proposed.
The experiments are performed on the generalized moving peaks benchmark with two scenarios.

However, my recommendation is rejected due to the following reasons. 
1. Except Ref. [19], it has almost no recent reference from the last ten years. Recent and state-of-the-art references should be cited on the topic.
2. There are many journal paper concerning the methods for solving dynamic optimization problem (DOP) from the last two years. 
Authors should survey and discuss these state-of-the-art methods in Section 1.
3. There are many journal paper concerning the Moving Peaks Benchmark (MPB) from the last two years. 
Authors should survey and discuss these state-of-the-art methods in Section 2.
(1) Moving peak drone search problem: An online multi-swarm intelligence approach for UAV search operations, 2021.
(2) A Novel Parametric benchmark generator for dynamic multimodal optimization, 2021.
(3) Adaptive multi-swarm in dynamic environments, 2021.
(4) A Culture-Based Artificial Bee Colony Algorithm for Optimization in Dynamic Environments, 2021.
4. There are many journal paper concerning the variants of Quantum Swarm Optimization algorithm from the last two years. 
Authors should survey and discuss these state-of-the-art methods in Section 3.
(1) A Two-Stage Multi-Swarm Particle Swarm Optimizer for Unconstrained and Constrained Global Optimization, 2020.
(2) Fitness peak clustering based dynamic multi-swarm particle swarm optimization with enhanced learning strategy, 2022.
(3) Dynamic multi-swarm global particle swarm optimization, 2020.
(4) A Dynamic Multi-Swarm Particle Swarm Optimizer for Multi-Objective Optimization of Machining Operations Considering Efficiency and Energy Consumption, 2020.
5. It is hard to link the mQSOaDE with the test examples.
A simple numerical example is needed to provide an illustration of the stochastic model in Section 3.
6. Authors should add a new section to discuss how to find a set of proper initial parameters of the mQSOaDE method? It seems to me that it is not an easy job and is really problem dependent to select the parameters, since there are lots of parameters influencing the performance of the proposed method together. 
7. Authors should compare the performance of the proposed mQSOaDE method with other existing state-of-the-art methods for solving test examples in Section 3.
8. Provide all parameter setting of the mQSO, mQSODE, mQSOaDE in section 3.

Author Response

Open Review

(x) I would not like to sign my review report

( ) I would like to sign my review report

English language and style

( ) Extensive editing of English language and style required

( ) Moderate English changes required

( ) English language and style are fine/minor spell check required

(x) I don't feel qualified to judge about the English language and style

Yes     Can be improved      Must be improved     Not applicable

Does the introduction provide sufficient background and include all relevant references?

( )        ( )        (x)       ( )

Is the research design appropriate?

( )        ( )        (x)       ( )

Are the methods adequately described?

( )        ( )        (x)       ( )

Are the results clearly presented?

( )        (x)       ( )        ( )

Are the conclusions supported by the results?

( )        (x)       ( )        ( )

Comments and Suggestions for Authors

The multi-swarm Quantum Swarm Optimization [18] algorithm is modified with the Differential Evolution (DE) search operator in this work. Two versions of modified algorithms, mQSODE and mQSOaDE, are proposed.

The experiments are performed on the generalized moving peaks benchmark with two scenarios.

Answer: thank you for your valuable comments. We have revised every section of the manuscript, from introduction to conclusion in order to improve its quality.

However, my recommendation is rejected due to the following reasons.

1. Except Ref. [19], it has almost no recent reference from the last ten years. Recent and state-of-the-art references should be cited on the topic.
2. There are many journal paper concerning the methods for solving dynamic optimization problem (DOP) from the last two years.

Authors should survey and discuss these state-of-the-art methods in Section 1.

3. There are many journal paper concerning the Moving Peaks Benchmark (MPB) from the last two years.

Authors should survey and discuss these state-of-the-art methods in Section 2.

(1) Moving peak drone search problem: An online multi-swarm intelligence approach for UAV search operations, 2021.

(2) A Novel Parametric benchmark generator for dynamic multimodal optimization, 2021.

(3) Adaptive multi-swarm in dynamic environments, 2021.

(4) A Culture-Based Artificial Bee Colony Algorithm for Optimization in Dynamic Environments, 2021.

4. There are many journal paper concerning the variants of Quantum Swarm Optimization algorithm from the last two years.

Authors should survey and discuss these state-of-the-art methods in Section 3.

(1) A Two-Stage Multi-Swarm Particle Swarm Optimizer for Unconstrained and Constrained Global Optimization, 2020.

(2) Fitness peak clustering based dynamic multi-swarm particle swarm optimization with enhanced learning strategy, 2022.

(3) Dynamic multi-swarm global particle swarm optimization, 2020.

(4) A Dynamic Multi-Swarm Particle Swarm Optimizer for Multi-Objective Optimization of Machining Operations Considering Efficiency and Energy Consumption, 2020.

Answer: we have expanded the literature review with more recent studies in Sections 1 and 2, and added the recommended references, as well as some additional ones.

5. It is hard to link the mQSOaDE with the test examples.

A simple numerical example is needed to provide an illustration of the stochastic model in Section 3.

Answer: we have added a more detailed explanation of the algorithm in Section 2 and added the pseudo-code. The benchmark parameters were changed to an older version, so that it would be possible to compare the results with the ones presented in the literature.

6. Authors should add a new section to discuss how to find a set of proper initial parameters of the mQSOaDE method? It seems to me that it is not an easy job and is really problem dependent to select the parameters, since there are lots of parameters influencing the performance of the proposed method together.

Answer: we have performed the sensitivity analysis for the most important parameters of the mQSODE algorithm, and added a discussion section, highlighting the main contributions of the study, and also added them to the introduction section.

7. Authors should compare the performance of the proposed mQSOaDE method with other existing state-of-the-art methods for solving test examples in Section 3.

Answer: thanks to the changed benchmark parameters we were able to perform the comparison with alternative methods tested on the GMPB.

8. Provide all parameter setting of the mQSO, mQSODE, mQSOaDE in section 3.

Answer: we have added the description of all the parameters used in the algorithm, including those fixed and altered in the experiments.

Author Response File: Author Response.pdf

Reviewer 3 Report

The work deals with metaheuristics, biologically inspired approach designed for heuristic optimization in the time-varying mid-dimensional environments. These types of studies have gained much popularity because of considerable potential in the engineering areas.  

A combined (hybrid) approach is presented that exploits local intergenerational fitness changes to obtain an instantaneous stochastic preference for the use of a DE-based or QSO-based technique. The paper introduces a new aspect of the methodology that takes into account probabilistic optimization features. Although this idea may have merit, one of the problematic aspects of the study is that its formulation and the relationships between variables are not fully and flawlessly explained in some parts, making it difficult to read. 

Notes:

(1) References to works on optimization in time-varying environments have been rather poorly mentioned in the first part of the Introduction. Many important references - "milestones" for the domain of variable environments - objective functions are missing. This does not apply to the lines > 40, where the population optimization methods are discussed and where the style improves.

2. The text around line 53 is misleading. In a relatively small space, the authors wish to say a lot and the resulting effect is confusion, despite the fact that they are really only describing a well-known item.

3. The role of the subset F(t) in formula 1 needs at least some discussion.

4. In the part of the text after line 123, Eq. 3, I did not find the specification of the parameter c ( governing the shifts in the peak positions...). 

5. In the section - Proposed Algorithm, I found the mQSO algorithm written in the incorrect form. There are some problems that go beyond what I can be listed, so I request a more detailed analysis/correction and ask to avoid further errors. I provide some explanation:

5.a: The optimization relations are of iterative type, so the if variables in the corresponding equations should contain discrete time. If not written that way, then relations should be written in the algorithmic form, which implies  updating of the respective (memorized) variable (usually denoted by an arrow).

5.b: In the formula, the bracketed part is written twice as if it were the same thing. Clearly this is an error, it does not distinguish between local and global.

(5.c) If the random vectors in the equation are used epsilon_{1.2}, then the scalar multiplication would give only a scalar, contradicting the vector nature of the optimizing formula.  Surely the vectors cannot be from [0,1]. The only correct way is that only the components have a value from [0,1]. 

6. Unfortunately, equation 10 on line 231 is not sufficiently explained in terms of the newly used variables. It is not clearly stated what represents theta and what represents phi. 

7. Formal: The standard deviations in Table 2 should be written with a +/- rather than just with pluses. 

8. Friedman's ranking is an incredibly valuable tool for assessing multi-criterion results.

9. The main work's findings presented in the conclusions are relatively shallow and do not adequately reflect the knowledge gleaned from the study with the algorithm under consideration.

Author Response

Open Review

(x) I would not like to sign my review report

( ) I would like to sign my review report

English language and style

( ) Extensive editing of English language and style required

( ) Moderate English changes required

(x) English language and style are fine/minor spell check required

( ) I don't feel qualified to judge about the English language and style

Yes     Can be improved      Must be improved     Not applicable

Does the introduction provide sufficient background and include all relevant references?

( )        ( )        (x)       ( )

Is the research design appropriate?

( )        (x)       ( )        ( )

Are the methods adequately described?

( )        ( )        (x)       ( )

Are the results clearly presented?

( )        ( )        (x)       ( )

Are the conclusions supported by the results?

( )        ( )        (x)       ( )

Comments and Suggestions for Authors

The work deals with metaheuristics, biologically inspired approach designed for heuristic optimization in the time-varying mid-dimensional environments. These types of studies have gained much popularity because of considerable potential in the engineering areas. 

A combined (hybrid) approach is presented that exploits local intergenerational fitness changes to obtain an instantaneous stochastic preference for the use of a DE-based or QSO-based technique. The paper introduces a new aspect of the methodology that takes into account probabilistic optimization features. Although this idea may have merit, one of the problematic aspects of the study is that its formulation and the relationships between variables are not fully and flawlessly explained in some parts, making it difficult to read.

Answer: thank you for your valuable comments. We have revised every section of the manuscript, from introduction to conclusion in order to improve its quality.

Notes:

1. References to works on optimization in time-varying environments have been rather poorly mentioned in the first part of the Introduction. Many important references - "milestones" for the domain of variable environments - objective functions are missing. This does not apply to the lines > 40, where the population optimization methods are discussed and where the style improves.

Answer: we have expanded the literature review in both introduction and related work subsection.

2. The text around line 53 is misleading. In a relatively small space, the authors wish to say a lot and the resulting effect is confusion, despite the fact that they are really only describing a well-known item.

Answer: the description of the problem considered in this study, as well as possible ways to solve it have been described in more details.

3. The role of the subset F(t) in formula 1 needs at least some discussion.

Answer: the description of DOP is improved, and F(t) role is explained.

4. In the part of the text after line 123, Eq. 3, I did not find the specification of the parameter c ( governing the shifts in the peak positions...).

Answer: the description of c parameter is added.

5. In the section - Proposed Algorithm, I found the mQSO algorithm written in the incorrect form. There are some problems that go beyond what I can be listed, so I request a more detailed analysis/correction and ask to avoid further errors. I provide some explanation:

5.a: The optimization relations are of iterative type, so the if variables in the corresponding equations should contain discrete time. If not written that way, then relations should be written in the algorithmic form, which implies  updating of the respective (memorized) variable (usually denoted by an arrow).

5.b: In the formula, the bracketed part is written twice as if it were the same thing. Clearly this is an error, it does not distinguish between local and global.

(5.c) If the random vectors in the equation are used epsilon_{1.2}, then the scalar multiplication would give only a scalar, contradicting the vector nature of the optimizing formula.  Surely the vectors cannot be from [0,1]. The only correct way is that only the components have a value from [0,1].

6. Unfortunately, equation 10 on line 231 is not sufficiently explained in terms of the newly used variables. It is not clearly stated what represents theta and what represents phi.

Answer: we have revised and updated the algorithm description, and to avoid any ambiguities, we have also added the pseudo-code of the proposed approach.

7. Formal: The standard deviations in Table 2 should be written with a +/- rather than just with pluses.

Answer: we have used +/- in all tables.

8. Friedman's ranking is an incredibly valuable tool for assessing multi-criterion results.

Answer: we have applied the Friedman ranking procedure to compare all the tested algorithms, combining several settings and performance metrics.

9. The main work's findings presented in the conclusions are relatively shallow and do not adequately reflect the knowledge gleaned from the study with the algorithm under consideration.

Answer: we have added the discussion section, where some insights about the meaning of the results and directions of further studies are provided.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Authors have completely addressed all my concerns.

Author Response

Authors have completely addressed all my concerns.

Answer: thank you for good evaluation of our work.

Reviewer 2 Report

The authors have carefully addressed the previous comments of the reviewer and significantly improved the manuscript. 

Author Response

The authors have carefully addressed the previous comments of the reviewer and significantly improved the manuscript. 

Answer: thank you for good evaluation of our work.

Reviewer 3 Report

The authors checked their work for typos, included references and appropriate vocabulary, and considered most of my suggestions.

However, I have still some objection regarding the notation used in relation 4. In the formula and after the formula there should be an operation between epsilon_1 and (p_ {ng} -x_ {nl}), as well as between epsilon_2 and also the difference (p_ {nl} -x_ {nl}) better clarified. It is certainly not a common product as between real numbers, nor is it a scalar product (dot product, inner product) because the result is a vector assembly, such as u_ {nl}. There is no obvious reason for vector - cross product, and moreover there is a higher dimension where the interpretation is even less trivial. My candidate for this purpose is Hadamard's product (also known as the element-wise product, entrywise product, also known as Schur product) is a binary operation that takes two matrices / vectors of the same dimensions and produces another matrix of the same dimension. However, this one usually marks as a circle with a dot inside, which would be good to incorporate into the text.

Author Response

The authors checked their work for typos, included references and appropriate vocabulary, and considered most of my suggestions.

However, I have still some objection regarding the notation used in relation 4. In the formula and after the formula there should be an operation between epsilon_1 and (p_ {ng} -x_ {nl}), as well as between epsilon_2 and also the difference (p_ {nl} -x_ {nl}) better clarified. It is certainly not a common product as between real numbers, nor is it a scalar product (dot product, inner product) because the result is a vector assembly, such as u_ {nl}. There is no obvious reason for vector - cross product, and moreover there is a higher dimension where the interpretation is even less trivial. My candidate for this purpose is Hadamard's product (also known as the element-wise product, entrywise product, also known as Schur product) is a binary operation that takes two matrices / vectors of the same dimensions and produces another matrix of the same dimension. However, this one usually marks as a circle with a dot inside, which would be good to incorporate into the text.

Answer: we have corrected eq. 4 and added element-wise product.