Particle Swarm Optimization Combined with Inertia-Free Velocity and Direction Search
Round 1
Reviewer 1 Report
In this paper a new hybrid algorithm for improving the performance of particle swarm optimization algorithm on local searches is developed, by combining two strategies into the original particle swarm optimization algorithm. By implementing and testing the new algorithm on unconstrained benchmark problems and constrained engineering design problems, its performance over other optimization algorithms is proved. The paper contains some new and relevant results. Therefore, I consider it can be accepted for publication.
Author Response
Point 1: In this paper a new hybrid algorithm for improving the performance of particle swarm optimization algorithm on local searches is developed, by combining two strategies into the original particle swarm optimization algorithm. By implementing and testing the new algorithm on unconstrained benchmark problems and constrained engineering design problems, its performance over other optimization algorithms is proved. The paper contains some new and relevant results. Therefore, I consider it can be accepted for publication. 

Response 1: Thanks for your good comments.
Point 2: English language and style are fine/minor spell check required. 

Response 2: Thank you for your comments on the English language and style of our manuscript. We have carefully checked the English spelling again, and have made correction.
Reviewer 2 Report
The authors propose a hybridisation of PSO algorithm. They use velocity updating strategy and direction search. The modified method is called SDPSO. The aim is to improve the exploatation of the search process. The authors tested their idea on well known test functions. I have following remarks:
- line 30 MA is not defined.
- after line 187, formula (3), (t) is better to be not an index
- 3. the same for formulas (4) and (5)
- Take in to account upper indexes, in different formulas they have different meaning, which can confuse the reader.
- Dimension 30 for test functions is too small. Most of the algorithms can find good solutions when the dimension is less than 100, but some of them (PSO and Free search for example) can find good solutions even the dimension is 1000. Thus test the algorithm with dimension minimum 100.
- The literature review is not up to date. Less than 25% of the cited titles are from the last 5 years.
Author Response
Dear Reviewers,
Thank you for your comments. Those comments are all valuable and very helpful for revising and improving our paper. We have studied comments carefully and have made correction for meeting with approval.
Point 1: line 30 MA is not defined.
Response 1: We have modified “MA” to “meta-heuristic”.
Point 2: after line 187, formula (3), (t) is better to be not an index.
Response 2: Thank you for reminding us. In formula (3), “t” denotes the velocity gained by the t-th “static exploitation” (SE) of one particle at generation n. We add “T” to represent the maximum times of “static exploitation” (SE).
Point 3: the same for formulas (4) and (5).
Response 3: We modify it as Point 2.
Point 4: Take into account upper indexes, in different formulas they have different meaning, which can confuse the reader.
Response 4: Thank you for reminding us. We have carefully checked and corrected the upper indexes again to avoid the confusion.
Point 5: Dimension 30 for test functions is too small. Most of the algorithms can find good solutions when the dimension is less than 100, but some of them (PSO and Free search for example) can find good solutions even the dimension is 1000. Thus test the algorithm with dimension minimum 100.
Response 5: We have done the 100-dimensional experiment to test our algorithm. The results show that our algorithm maintains advantages in 100 dimensions by comparing other state-of-the-art PSO variants, Switch-PSO, S-PSO, AIW-PSO, DLI-PSO.
Thanks for your suggestions. With the 100-dimensional experiment, the performance of our algorithm is further verified. We have added these results to Section 6.1.2.
Point 6: The literature review is not up to date. Less than 25% of the cited titles are from the last 5 years.
Response 6: Considering the reviewer’s suggestion, we have studied some recent literatures and added them to section 2. Meanwhile, the literature review is further refined.
Point 7: English language and style are fine/minor spell check required. 

Response 7: Thank you for your comments on the English language and style of our manuscript. We have carefully checked the English spelling again, and have made some correction.
Reviewer 3 Report
Incorrect way of providing the authors' affiliation.
Abbreviations such as: PSO and SDPSO should be explained. SDPSO - is not. The note applies to all text (other abbreviations).
Why did you distinguish the words "static exploitation" and "direction search"?
Figure 1, 4, 5, 6, 7 - Poor drawing quality. It is not legible.
The article is very good. My congratulations! It is interesting.
Author Response
Dear Reviewers,
Thank you for your comments. Those comments are all valuable and very helpful for revising and improving our paper. We have studied comments carefully and have made correction for meeting with approval.
Point 1: Incorrect way of providing the authors' affiliation.
Response 1: Thank you for reminding us.We checked the requirements of the journal and revised it. The revised author information is as follows.
Kun Miao 1,*, Qian Feng 1 and Wei Kuang 1
1 School of Civil Engineering, Central South University, Changsha 410075, China
* Correspondence: [email protected]
Point 2: Abbreviations such as: PSO and SDPSO should be explained. SDPSO - is not. The note applies to all text (other abbreviations).
Response 2: PSO is the abbreviation of the particle swarm optimization algorithm. We have explained it in line 10 of our manuscript.
SDPSO is the abbreviation of our proposed algorithm. Our manuscript presents an improved PSO algorithm with an inertia-free velocity and direction search, which combines the static exploitation (SE) and the direction search (DS) with the original PSO. Thus we call it SDPSO. We have explained it in line 57 of our manuscript.
We have checked other abbreviations and confirmed that they have been explained.
Point 3: Why did you distinguish the words "static exploitation" and "direction search"?
Response 3: They are the two different operators of SDPSO. The "static exploitation (SE)" is a search way with an inertia-free velocity, while the direction search (DS) is another way of searching along the coordinate axis.
Point 4: Figure 1, 4, 5, 6, 7 - Poor drawing quality. It is not legible.
Response 4: Thank you for reminding us. We have updated Figure 1, 4, 5, 6, 7 with high pixel image in our manuscript.
Point 5: The article is very good. My congratulations! It is interesting.
Response 5: Thank you for your good comments.
Point 6: English language and style: I don't feel qualified to judge about the English language and style.
Response 6: Thank you.
Round 2
Reviewer 2 Report
All the reviewers suggestions have taken in to account. The paper can be accepted in its present form.
