Optimal Randomness in Swarm-Based Search

Round 1

Reviewer 1 Report

It is a mature, well-written paper. I am not a native English speaker but it same to be very good even from the language point of view. It is a rare paper, where there is not too much to be commented in a negative way or suggested. It is appropriate from every point of view. 

In the abstract, it is used the acronym “CS” but is not introduced, mentioned at the first apparition its long name.

Page 1, Line 29

Substitute the word “famous” with a more appropriate one.

Page 4, Algorithm 1

At the beginning of the algorithm include the input and the output.

Page 7, line 200

Substitute the word “obvious” with a more appropriate one.

Page 9, line 308

Substitute the word “discuss” with a more appropriate one, for instance, the word “study”.

Author Response

Please refer to the attached reply in pdf format. Thank you!

Author Response File: Author Response.pdf

Reviewer 2 Report

The idea of using other heavy-tailed distributions alternatively to Lévy distributions is explored and the results presented for the considered benchmark functions and problem indicate clearly performance improvements.

As stated by authors the tested alternative distributions also depend on additional adjustable parameters by the user.  A study was performed based on two functions supporting the selected parameters. 

The paper is generally well-written, with a good structure. However, it should be reviewed in terms of English.

Also I would recommend to revise the following: Always introduce the terms before using a abbreviation.

Author Response

Please refer to the attached reply in pdf format. Thank you!

Author Response File: Author Response.pdf

Reviewer 3 Report

Multiplication operator should be carefully changed in this paper.
Standard multiplication operator is omitted or /cdot is used. There many places with \times operator that are standard multiplications.

Eq.4,8 larger brackets (scalable) are required

Author Response

Please refer to the attached reply in pdf format. Thank you!

Author Response File: Author Response.pdf

Reviewer 4 Report

-Lines 280-283 “where q1; q2; q3 and a; b; c are fractional orders and systematic parameters. When ..... the system above is chaotic. Figure 4 depicts the distribution figure of system (16) for the objective function values”.

It is not clear which is the objective function.

-Lines 287-288 “Table 5 lists the statistical results of the average identified values, the corresponding relative error values, and the objective function values”.

The authors should present the equation of the relative error that was used as an evaluation metric.

-The acronyms in table 2 should be explained what they mean. E.g. Fsph.

-The mathematical expression of the used functions in table 2 should be given in the text or in the table.

- Line 209 “Error’ represents the average error to the optimal value” The mathematical formula of the used average error metric should be presented. The “Error” should be clarified (E.g. RMSE, MSE, etc.).

-Line 208 “γ varies within interval [0.5, 4.5] in steps of 0.5, β varies from 0.1 to 0.9 in steps of 0.1”

It is not clear why the authors have used the specific intervals in the “4.2. Parameter Tuning”. Some literature should be mentioned.

-Lines 213-214 “we set the values of γ and β to 0.8 and 4.5 for all the experiments being conducted in the next subsections”. This is not consistent with the lines 208-209: “γ varies within interval [0.5, 4,5] in steps of 0.5, β varies from 0.1 to 0.9”.

Author Response

Please refer to the attached reply in pdf format. Thank you!

Author Response File: Author Response.pdf

Reviewer 5 Report

The paper introduces four variations of the Cuckoo Search algorithm, with various heavy-tailed probability distributions. Various experiments are performed in order to comparatively analyse the algorithms. The corresponding sources are uploaded to Matlab Central File Exchange. The authors pass the relevant theoretical study as a follow-up step in this research.

The authors should add a section on analysis of the state-of-the-art in terms of various probability-based extensions to standard search algorithms and argue, both conceptually and numerically, on the superiority of their approach.

What do we learn from the experiments? An analysis of the actual experimental results is necessary, not only an immediate comparison of the numbers themselves.

The use of English language has to be improved. I recommend the authors to carefully reread the text and clean the style and phrasing.

Author Response

Please refer to the attached reply in pdf format. Thank you!

Author Response File: Author Response.pdf

Round 2

Reviewer 4 Report

The manuscript has been improved but there are several issues unsolved:

-The equations of the all the benchmark functions and the values of the parameters of the functions is important to be presented (Preferably in a table).

E.g. the “Generalized Penalized Function 1” has some undefined parameters: a, k, m.

Which values of these parameters were used by the authors? (E.g. k=50, etc.). Also, which is the design space for every benchmark function? E.g. [−3,3]^D . Also, which is the optimum for the selected design space for every benchmark function?

-The error metric should be clearly stated. I assume that the authors use the Mean Absolute Error. The equation should be presented. Generally, only one error metric is not enough to evaluate sufficiently an objective function.

-Figure 2: Axis Y “Average Error”. Probably the authors mean “Mean Absolute Error”.

-Figure 5: the authors refer “Relative error”. The authors should present the equation of the kind of “Relative error” they use. It is not clear why the authors use two kinds of errors (their formulas should be presented): “Absolute Error” and “Relative error” which are not the same.

-The first column of table 5 is confusing for the reader. Which is method a? b, c? Which is the general equation of the relative error? (E.g. the 1.00 used in |a-1.00|/1.00 which parameter represents?).

Author Response

Please see our detailed responses to your kind comments.

Author Response File: Author Response.pdf

Round 3

Reviewer 4 Report

The authors have improved the manuscript according to the suggestions.

Some minor spelling errors should be corrected. E.g. line 364 "Schwfels Problem 2.6" to " Schwefels Problem 2.6".

Author Response

Please see the attached reply document in PDF.

Author Response File: Author Response.pdf

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

The paper investigates how four different types of heavy-tailed distributions (Mittag-Leffler, Pareto, Cauchy, and Weibull) enhanced the searching ability of Cuckoo Search (CS). As such, the paper has some novelty. Unfortunately, the experimental part is weak and must be improved. Comparison must be fair and at least following metaheuristics PSO, ABC, DE, SHADE must be included into comparison.

1) Many claims/statements are questionable/wrong. For example:

a) “Among all the proposed algorithms, Cuckoo search (CS) has been proved to be an efficient approach for global optimum searching due to the combination of Lévy flights, local search capabilities and guaranteed global convergence.” and “Among the newly proposed algorithms, cuckoo search (CS) [2] is considered to be one of the more promising nature-inspired algorithms.” But, CS never won any CEC competition and have never been shown to be superior to other Swarm Intelligence (SI) metaheuristics. Note, that No-Free-Lunch (NFL) Theorem still holds. In many papers CS has been only compared to non-state-of-the-art metaheuristics, such as GA. Hence, we can not generalized those results.

b) “CS searches for new solutions by performing a global explorative random walk together with a local exploitative random walk.” But, instead of random walk the exploration and exploitation are more commonly used terms in the field of SI and Evolutionary Algorithms (EAs).

c) “The salient features of CS lie in its simple concept, limited parameters and easy combination with other intelligent algorithms. Due to these notable benefits and its potential capability … .” The same is true for many other EAs.

d) “Besides, CS with Lévy flights based structured random walk has been demonstrated to perform more effective than many existing metaheuristics such as PSO, ABC, and DE [11].” I have checked [11] and seems that comparison was not fair in [11]. CS termination in Section 2.1 [11] is based on maximum number of cycles/iterations, while experiments are using 2,000,000 fitness evaluations as termination condition. Since CS is using more fitness evaluations per iteration than many other metaheuristics I doubt that fitness evaluations consumed have been correctly counted in [11]. I have also doubt that scout bees in ABC were correctly counted as well. Overall, the authors should include in comparison also PSO, ABC, DE  and its state-of-the-art variants such as SHADE.

2) Justification why CS has been used in this work is weak.

3) The title of the paper: “Optimal Randomness in Swarm-based Search” is too broad and the body of the paper certainly does not justify it. The authors are promising too much. Optimal randomness is not shown, while CS has been only investigated. Not the SI in general. The title must changed.

4) Algorithm 1 (CS) is not described in sufficient details. Seems it is not standard CS at all. Problems:

a) Control parameters are missing.

b) Why iteration number G Is needed if termination is based on maximum number of fitness evaluations consumed (Max_FEs)?

c) It is not clear why two inner loops are used. This is not the standard CS version!

d) It is not clear where nests are abandoned and new random solutions are generated. Note that in this case more fitness evaluations are consumed, which must be counted. The control parameter Pa is not visible in Algorithm 1.

e) What happened if Max_FEs is reached inside the first inner loop? There is no loop termination and more NP solutions will be evaluated. As such a comparison is not fair.

f) Actual code is not provided. Hence reviewer/readers can not check if implementation is indeed correct. I have seen too many examples where actual code doesn’t fit a pseudocode.

5) Jump lengths in Figure 1 are based on iterations. This is another indication that authors used termination based on iterations and not termination based on maximum fitness evaluations consumed as indicated in Algorithm 1. Algorithm and graphs must be consistent.

6) Experimental part is weak and must be improved.

a) It is not clear how basic CS control parameters have been set. Have any tuners being used (e.g., CRS-Tuning, F-Race, REVAC). At least some discussion on tuning must be included. For CSML, CSP, CSC and CSW the additional parameters have been tuned only on two optimization problems, which might be considered insufficient.

b) Comparison with other meta-heuristics is completely omitted. The authors must include PSO, ABC, DE, and SHADE. If basic CS is performed badly the proposed CS enhancements might have little scientific value if there are still worse than other meta-heuristics.

c) The authors must pay full attention that the comparison is fair (see my comments in this report and examples of violations).

d) In the Section 4.4 termination is based on maximum number of iterations (“maximum iteration number is set to 200 and the population size is set to 40”). When comparing with PSO, ABC, DE and SHADE this termination conditions would be unfair. Anyway, the termination condition is not the same as explained in Algorithm 1.

e) Null Hypothesis Significance Testing has not been applied in Section 4.4.

7) The authors wrote: “In the experimental studies, Max_FEs is taken as the termination criterion and set to 10, 000 x D.” This would be fair comparison if all fitness evaluations are counted and the inner loops terminated immediately when termination condition holds. Currently, this is not the case – see Algorithm 1.

8) Figure 2 legends are too small and unreadable.

9) Too many typos:

Cuckoo search (CS)

->

Cuckoo Search (CS)

Pareto distribution Cauchy distribution

->

Pareto distribution, Cauchy distribution

CS with Mittag-Leffler distribution, denoted as CSP

->

CS with Pareto distribution, denoted as CSP

Pareto(b; a) in Equation (12) represents a random number drawn from Cauchy distribution

->

Pareto(b; a) in Equation (12) represents a random number drawn from Pareto distribution

References used in this review report:

========================================

Črepinšek et al. 2013: Exploration and exploitation in evolutionary algorithms: A survey. ACM Comput. Surv. 45(3): 35, 2013.

Mernik et al. 2015: On clarifying misconceptions when comparing variants of the Artificial Bee Colony Algorithm by offering a new implementation. Information Sciences, Vol 291, pages 115 – 127, 2015.

Draa 2015: On the performances of the flower pollination algorithm – Qualitative and quantitative analyses. Applied Soft Computing, 34(2015), Pages 349–371, 2015. 

Črepinšek et al. 2014: Replication and comparison of computational experiments in applied evolutionary computing: Common pitfalls and guidelines to avoid them. Applied Soft Computing, 19 (2014) 161–170. 

Birattari et al. 2002: A racing algorithm for configuring metaheuristics, Genetic Evol. Comput. Conf. 2 (2002) 11–18.

Nannen et al. 2007; Relevance estimation and value calibration of evolutionary algorithm parameters, Int. Joint Conf. Artif. Intell. 7, 975–980.

Veček et al. 2016: Parameter tuning with Chess Rating System (CRS-Tuning) for metaheuristic algorithms, Information Sciences 372 (2016) 446–469 

Garcia-Martinez et al. 2017. Since CEC 2005 competition on real-parameter optimization: a decade of research, progress and comparative analysis’s weakness. Soft Computing, Issue 19.

Tanabe and Fukunaga 2013. Success-history based parameter adaptation for differential evolution. In: IEEE Congress on Evolutionary Computation (CEC’13), pp 71–78.

Author Response

please see the attached pdf for detailed replies - thanks!

Author Response File: Author Response.pdf

Reviewer 2 Report

This study focuses on swarm intelligence algorithms. In particular, four different types of commonly used heavy-tailed distributions are evaluated. The numerical experiments are discussed in the paper. The key findings are highlighted. In general, I think that the paper fits well into the scope of the journal. However, some revisions are required before the paper can be considered for publication. Certain segments of the paper must be strengthened. Below please find more specific comments:

*Page 1 line 1: “Among all the proposed algorithms” – sounds a bit vague. I assume the authors refer to “Among all the proposed swarm intelligence algorithms”.

*Page 1 line 9: “Pareto distribution Cauchy distribution” – I believe comma is missing between “Pareto distribution” and “Cauchy distribution”.

*Page 1: At the beginning on the introduction section, the authors start with discussion regarding the importance of the swarm intelligence algorithms. I recommend adding another discussion, highlighting the effectiveness of the swarm intelligence and evolutionary-based algorithms for solving complex combinatorial decision problems. Also, please provide reference to the recent and relevant studies that can support this discussion. In particular, references to the following recent studies are recommended:

Slowik, A., & Kwasnicka, H. (2018). Nature inspired methods and their industry applications—Swarm intelligence algorithms. IEEE Transactions on Industrial Informatics, 14(3), 1004-1015.

Dulebenets, M. A. (2018). A Comprehensive Evaluation of Weak and Strong Mutation Mechanisms in Evolutionary Algorithms for Truck Scheduling at Cross-Docking Terminals. IEEE Access, 6, 65635-65650.

Govindan, K., Jafarian, A., & Nourbakhsh, V. (2018). Designing a sustainable supply chain network integrated with vehicle routing: A comparison of hybrid swarm intelligence metaheuristics. Computers & Operations Research.

Zhao, X., Wang, C., Su, J., & Wang, J. (2019). Research and application based on the swarm intelligence algorithm and artificial intelligence for wind farm decision system. Renewable Energy, 134, 681-697.

Brezočnik, L., Fister, I., & Podgorelec, V. (2018). Swarm intelligence algorithms for feature selection: a review. Applied Sciences, 8(9), 1521.

Dulebenets, M. A. (2017). A novel memetic algorithm with a deterministic parameter control for efficient berth scheduling at marine container terminals. Maritime Business Review, 2(4), 302-330.

Anandakumar, H., & Umamaheswari, K. (2018). A bio-inspired swarm intelligence technique for social aware cognitive radio handovers. Computers & Electrical Engineering, 71, 925-937.

*Page 2 line 59: “Cuckoo search (CS), recently developed by Yang and Deb” – does not sound accurate. Cuckoo search was developed a while ago.

*Page 3: Please check spacing between consecutive lines in Algorithm 1. It seems that consecutive lines are almost overlapping.

*Page 6 line 163: “All the algorithm are evaluated for 50 times” should be replaced with ““All the algorithms are evaluated for 50 times”.

*Page 6 line 163: “All the algorithm are evaluated for 50 times” – please provide a justification why 50 times (not 100).

*Page 6 line 170: I am glad to see that the authors conducted a detailed parameter tuning analysis; as, otherwise, the comparative analysis might be misleading.

*Page 11: The conclusions section should be strengthened. The authors should clearly highlight limitations of this study and how they will be addressed in future research.

Author Response

please see the attached pdf for detailed replies - thanks!

Author Response File: Author Response.pdf

Reviewer 3 Report

The contribution ot this paper is very high because and goes beyong Cuckoo search since it is transversal with respect to the many Swarm based approaches based on metaheuristics

inspired by the behaviour of foreaging animals/

The paper proposes and discusses the advantages of using  Lévy flight, instead of Gaussian distribution since it is  heavy-tailed distribution which is simple to compute and easy to analyze.

The proposal is strongly supported by experiments and it should be definitely accepted for publication.

Please improve the English avoiding colloquial expressions and throughout spell check the entire text.

Author Response

please see the attached pdf for detailed replies - thanks!

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Unfortunately, the authors didn’t address the most important comments. Hence, I can not support acceptance of this paper.

1) What happened if Max_FEs is reached inside the first/second inner loop of Algorithm 1? There is no loop termination and more solutions will be evaluated. As such a comparison is not fair.

2) Actual code is not provided. Hence reviewer/readers can not check if implementation is indeed correct. I have seen too many examples where actual code doesn’t fit a pseudocode.

3) My previous comment was that the authors must compare their algorithms with PSO, ABC, DE, and SHADE. While PSO, ABC, DE are original variant and SHADE is state-of-the-art DE. However, the authors instead of SHADE again used non-state-of-the-art metaheuristics such as FA and FPA both known that are consuming more fitness evaluations per iterations than DE or PSO. For ABC they are using version [19] where scout bee phase have not been counted towards cumulative fitness evaluations consumed. Hence, comparison has not been fair. Why authors refuse to use SHADE? I can only speculate, but seems that results were much better than the proposed algorithms.

4) Control parameters for ABC, DE, PSO has not been presented. By badly setting control parameters it is hard to achieve good results.

5) Convergence graphs (Fig. 5) are still based on the number of iterations, which are now removed from Algorithm 1. Seems that authors just manually correct the pseudocode and keep the actual code the same. This might be a reason why they didn’t provide actual code as requested.

6) Critical Difference in Friedman test is not computed. Hence, it is not known if outperformance is indeed statistically significant.

7) In Section 4.6 the termination condition is based on maximum number of iterations (“maximum iteration number is set to 200 and the population size is set to 40”). Note, that the termination condition is not the same as explained in Algorithm 1. Yet, another indication that actual code has not been corrected.

8) Even not all typos have been corrected.

Pareto(b; a) in Equation (12) represents a random number drawn from Cauchy distribution

->
Pareto(b; a) in Equation (12) represents a random number drawn from Pareto distribution

References used in this review report:

=======================================

Wang et al. 2017: Firefly algorithm with neighborhood attraction. Information Sciences382–383(2017) 374–387.

Draa 2015: On the performances of the flower pollination algorithm – Qualitative and quantitative analyses. Applied Soft Computing, 34(2015), Pages 349–371, 2015.

Mernik et al. 2015: On clarifying misconceptions when comparing variants of the Artificial Bee Colony Algorithm by offering a new implementation. Information Sciences, Vol 291, pages 115 – 127, 2015.

Tanabe and Fukunaga 2013. Success-history based parameter adaptation for differential evolution. In: IEEE Congress on Evolutionary Computation (CEC’13), pp 71–78.

Author Response

Dear Reviewer#1:

Thank you for your review. We found your comments very useful and constructive for us to improve our presentation. Please view the attached detailed response to our replies. We are open to further revision to our paper for improved future impacts.

Many thanks again!

YangQuan Chen

--

YangQuan Chen, Director of MESA (Mechatronics, Embedded Systems and Automation) Lab

School of Engineering, University of California, 5200 N. Lake Road, Merced, CA 95343, USA

GOT FEW G: scholar.google.com/citations?user=RDEIRbcAAAAJ ; O: SE2-273; T: (209)2284672

F:(209)2284047; E:[email protected]; [email protected] W:mechatronics.ucmerced.edu

UC Merced Castle Research Facility @ 4225 Hospital Rd, Atwater, CA 95301. T: 209-2284398

Author Response File: Author Response.pdf

Reviewer 2 Report

The authors have adequately addressed my comments and I recommend this article for publication.

Author Response

Thank you for your review and approval!

Round 3

Reviewer 1 Report

Unfortunately, the authors still didn’t remove unfair comparison and I can not support acceptance of this paper. In the response letter they claimed: “We have to confess that the comparison is unfair to some extent, but we have to say the comparison cannot be fair and even unnecessary for this paper.” I can not agree with these claims:

- “comparison can not be fair” - Yes, it can be fair.

- “fair comparison is not necessary” – No, it is necessary. It was shown in the literature that when allowing one particular metaheuristics to consume more fitness evaluations by 1%, 5%, 10%, 20%, and 50% the error, or improvement in fitness, can be as great as 18%, 40%, 66%, 92%, and 100%. Hence, comparison based on equal number of fitness evaluations must be incorporated.

Yet, another doubtful responses by authors:

- “Hence, it is not fair to compare CSML, CSP, CSC and CSW with the sophisticated algorithm like SHADE. And that's why we only did comparisons with metaheuristics like ABC, DE, FA, FPA and PSO.” Why CS and its variant can be compared to ABC, DE, FA, FPA and PSO, but not with SHADE? There is no real reason. CS is only competitive if unfair comparison based on equal number of iterations is used where CS is allowed to consume fitness evaluations.

- The authors wrote: “When Max_FEs is reached inside either loop, Algorithm will continue until the condition`FEs < Max_FEs' is justified again and then terminated.” Yes, algorithm will continue and consume more fitness evaluations. Hence, again exact stopping condition is not achieved.