Improved Whale Algorithm for Solving the Flexible Job Shop Scheduling Problem
Round 1
Reviewer 1 Report
The paper presents an optimized Whale Algorithm in conjunction with FJSP and falls into a large number of "Honey Bees Inspired Optimization" publications.
The article needs more introduce the underlying methodology.
To evaluate the significance of the results of the simulations, it is necessary to specify the conditions for selecting the parameters.
Based on what were algorithms and simulation conditions chosen?
Among other things, it is necessary to correct:
1) typos in the text,
2) equations formatting, and
3) “Error! Reference source not found "on lines: 141, 143, 145.
Author Response
1. The reviewer’s comment:
The article needs more introduce the underlying methodology.
The author’s response:
I've completed some content of the whale optimization algorithm in the paper. Please see page 4-5.
2. The reviewer’s comment:
To evaluate the significance of the results of the simulations, it is necessary to specify the conditions for selecting the parameters.
The author’s response:
The parameter setting here is mainly based on a large number of experiments, and different combinations of parameters were tested to determine the optimal set of parameters.
3. The reviewer’s comment:
Based on what were algorithms and simulation conditions chosen?
The author’s response:
In the first experiment, we use four algorithms, WOA, IWOA-1, IWOA-2, IWOA to solve the fifteen famous benchmarks, respectively. And then we analysis the results in terms of computational time, the number of optimal values, the number of optimal mean values, to verify the effectiveness of the three strategies in WOA.
In the second experiment, we use seven algorithms, IWOA, KBACO , TSPCB , HGWO , AIA, PSO+TS, TS3,to solve the fifteen famous benchmarks, respectively. And compare the results of IWOA with the next six algorithms, which are famous algorithm for FJSP in recent years and we can easily obtain the results of experiment by consulting corresponding literature.
4. The reviewer’s comment:
1) typos in the text,
The author’s response:
It has been corrected in the paper.
5. The reviewer’s comment:
2) equations formatting, and
The author’s response:
It has been corrected in the paper.
6. The reviewer’s comment:
3) “Error! Reference source not found "on lines: 141, 143, 145.
The author’s response:
It has been corrected in the paper. Please see page 4.
Thanks for the valuable comments of editor and reviewers.
Reviewer 2 Report
The paper addresses the issue of solving the job shop scheduling problem by using an alternative algorithm. The introduction, abstract and literature review are well written and clearly highlight the purpose of the paper. The proposed algorithm is well explained. The numerical example is quite vague and I think that the data in Table 1 should be better explained. Also, the proposed method should be compared in terms of relative performance values with other methods from the literature. What are [35], [36], ... , [40] in Table 3?
Also, please fix the following:
- increase the readability of equations 10 and 11;
- better explain section 3.2.1. Can you please make the replacement within the paper and properly interpret the results?
- fix the errors in rows 140 -145;
- explain equation 12 - I am not sure why the two values are possible;
- why do you have negative values in Figure 2? (I think it should be a table rather than a figure);
-give the full names of the algorithms presented in rows 385-388 and few characteristics;
Thank you!
Author Response
1. The reviewer’s comment:
The paper addresses the issue of solving the job shop scheduling problem by using an alternative algorithm. The introduction, abstract and literature review are well written and clearly highlight the purpose of the paper. The proposed algorithm is well explained. The numerical example is quite vague and I think that the data in Table 1 should be better explained.
The author’s response:
In first experiment, I have added the comparison in terms of ‘Avg’ value between our proposed IWOA with other methods from the literature. Please see page 11.
2. The reviewer’s comment:
Also, the proposed method should be compared in terms of relative performance values with other methods from the literature.
The author’s response:
In third experiment, I have added the comparison in terms of relative percentage deviation to ‘LB’ (RPD) between our proposed IWOA with other methods from the literature. Please see page 11- 12.
3. The reviewer’s comment:
What are [35], [36], ... , [40] in Table 3?
The author’s response:
[35], [36], ... , [40] in Table 3 mean the algorithm proposed in the corresponding papers. Please see page 11-12.
4. The reviewer’s comment:
Increase the readability of equations 10 and 11.
The author’s response:
The equations 10 and 11 have been increased by completing the corresponding content. Please see page 5.
5. The reviewer’s comment:
Better explain section 3.2.1. Can you please make the replacement within the paper and properly interpret the results?
The author’s response:
The section 3.2.1 has been explained clearly in the paper. Please see page 4.
6. The reviewer’s comment:
Fix the errors in rows 140 -145.
The author’s response:
The errors have been fixed. Please see page 4.
7. The reviewer’s comment:
Explain equation 12 - I am not sure why the two values are possible.
The author’s response:
During the searching process of WOA, whales move around the prey in a shrink-ing circle and along a spiral path simultaneously, which are chosen according to a pro-bability of 50% . Thus the mathematical model of the bubble-net attacking behavior of humpback whales can be defined by Equation (12), where p is a random number inside [0,1]. Please see page 5.
8. The reviewer’s comment:
Why do you have negative values in Figure 2? (I think it should be a table rather than a figure).
The author’s response:
Because the individual position is still denoted as a multi-dimensional real vector, it belongs to [-, presents the number of the jobs, so sometime, it may be a negative values.
Here we use the figure to clearly describe an concept of the individual position vector not to analysis the data, so we use the tool Visio to draw a figure. Please see page 5.
9. The reviewer’s comment:
Give the full names of the algorithms presented in rows 385-388 and few characteristics.
The author’s response:
The full names of the algorithm and its characteristics as follows:
KBACO is a knowledge-based ant colony optimization, which provides an effective integration between Ant Colony Optimization (ACO) model and knowledge model. In the KBACO algorithm, knowledge model learns some available knowledge from the optimization of ACO, and then applies the existing knowledge to guide the current heuristic searching.
TSPCB is a hybrid tabu search algorithm with an efficient neighborhood structure, First, a mix of four machine assignment rules and four operation scheduling rules is developed to improve the quality of initial solutions to empower the hybrid algorithm with good exploration capability. Second, an effective neighborhood structure to conduct local search in the machine assignment module is proposed, which integrates three adaptive approaches.
Third, a speedup local search method with three kinds of insert and swap neighborhood structures based on public critical block theory is presented. With the fast neighborhood structure, the TSPCB algorithm can enhance its exploitation capability.
AIA is a artificial immune algorithm, this algorithm uses several strategies for generating the initial population(the initial assignments is obtained by AssignmentRule1 and AssignmentRule2 and then the sequencing of the initial assignments is obtained by a mix of three known dispatching rules: random selection of a job (Random), most work remaining (MWR) and most operation remaining (MOR) and selecting the individuals for reproduction. Different mutation operators are also utilized for reproducing new individuals(Assignment mutation , Precedence preserving shift mutation ) to improve the diversification of the population.
PSO+TS is a particle swarm optimization combined with tabu search, To avoid the trap in local optima, tabu search uses a temporary memorization structure in which it memorizes the last moves performed to prohibit the return to solutions recently visited. An adaptation of the parameters of particle swarm optimization is presented, the first step of this adaptation is to create a structure of particles and then propose an algorithm showing the way to follow for better solving of FJSP.
TS3 is an approximation algorithm based on the tabu search metaheuristic which includes a new neighborhood structure called ”golf neighborhood”. The main idea of the proposed neighborhood is to execute a 'long shot' of an operation from the current operation's machine to another machine of the same type, and then to the make a small move by using a local optimization algorithm without changing operations-to-machines assignment. TS3 algorithm provides better solutions and small relative error in the fixed execution time. Please see page10- 11.
Thanks for the valuable comments of editor and reviewers.
Author Response File: 
Author Response.docx
Reviewer 3 Report
This paper applies a whale optimization algorithm for a flexible job shop scheduling problem (FJSP). The authors present an encoding scheme for the application and propose some strategies for local search. They compare the performance of the algorithm with other meta-heuristics using different data sets. However, I would like to recommend you use a proof-reading service before detailed discussion.
For instance, 'job' is misspelled as 'jop' in many places and the table contents and equations are not properly organized.
In equations, I can see unexpected sentences (e.g. see equation 7 and 8). Such things make my reading disturbed.
I highly recommend you improve the explanation on your optimization model described in Equations (1) – (4). It would be better you define more clearly the sets, parameters, and variables with better explanations such as better explanation on your objective function and the type of the variables, etc.
Author Response
1. The reviewer’s comment:
The reviewer recommend to use a proof-reading service before detailed discussion.For instance, 'job' is misspelled as 'jop' in many places and the table contents and equations are not properly organized.
The author’s response:
It has been corrected in the paper.
2. The reviewer’s comment:
In equations, I can see unexpected sentences (e.g. see equation 7 and 8). Such things make my reading disturbed.
The author’s response:
Some content has been added in the corresponding part of the paper to explain the problem. In addition, because the space limit in our paper , the traditional whale algorithm cannot be discussed particularly, and the detailed content can be referred to the literature(Mirjalili, S.; Lewis, A.The Whale Optimization Algorithm. Adv. Eng. Soft. 2016, 95,51-67). Please see page 4.
3. The reviewer’s comment:
I highly recommend you improve the explanation on your optimization model described in Equations (1) – (4). It would be better you define more clearly the sets, parameters, and variables with better explanations such as better explanation on your objective function and the type of the variables, etc.
The author’s response:
It has been improved in the paper. Please see page 3.
Thanks for the valuable comments of editor and reviewers.
Round 2
Reviewer 1 Report
The author has more or less fulfilled the comments sent, please check the formatting and “Error! Reference source not found" still occurs in the text.
Author Response
1. The reviewer’s comment:
The author has more or less fulfilled the comments sent, please check the formatting and “Error! Reference source not found" still occurs in the text.
The author’s response:
I have proofread the whole paper and addressed the error you mentioned.
Thanks for the valuable comments of editor and reviewers.
Reviewer 3 Report
1. I think that the mathematical model defined in Equation (1) - (4) should be improved further. The model still does not define which variables are decision variables or given parameters clearly. Moreover, I guess the presented model would not work properly. For instance, if all 'y' variables have 0 value, all the constraints do not work anymore. I would like to recommend the authors describe the relationship between the model and the solution representation scheme of the meta-heuristic. If doing so, the authors can address the implicit constraints that are naturally included in the solution representation scheme.
2. I would like to ask about the solution representation scheme in Figure 1. The authors state that the first segment indicates the assigned machines to the operations and the second segment indicates the processing sequence on each machine. Can you explain the meaning of the given solution in detail here? For instance, what does '3' of OP_21 in first row and third column mean? I would like to recommend you to use the numbers in Figure 1 when the figure is explained.
3. There are obvious grammar errors in the paper. I cannot address all the errors but for example:
- line 15: 'a' chaotic reverse learning strategies -> remove 'a'.
- line 27: in real manufacturing? In a real manufacturing environment?
- line 39: the FJSP has 'became' -> 'become'.
- line 59: two subproblem -> subproblems.
- line 63-64: MOFJSP with three objectives, i.e., makespan, total workload and critical workload. (you address all three objectives so may not need 'i.e.?'.
- line 124-126: Constraint (2), (3), and (4) -> Equation (2), (3), and (4).
- and so on…
Author Response
Thanks for your suggestions. I attach the responses.
Author Response File: 
Author Response.pdf
Round 3
Reviewer 3 Report
1. The mathematical model can stand alone and it can be solved by other approaches. In your mathematical model, there is no constraint that makes sure that each operation is processed only once. Of course, when the whale algorithm determines the optimal solution, solution candidates may satisfy the constraint because the solution representation and the way of creating the solutions make sure that the constraint, 'each operation is processed only once', becomes satisfied. In the field of operations research, the type and range of variables should be defined clearly. You introduced the starting and completion time variables in text. However, there is no mathematical definition that defines the range and type of such dependent variables. I think in the model all variables are independent or dependent variables except of the processing time which means their range and type should be defined in equations.
2. In your model the number of operations in a job is a fixed value 'm'. Does it correct? I guess each job can have a different number of operations. If it is, the number of operations should be defined using subsets?
3. In the solution representation, I will don't understand the conversion. In the example, there are two machines. In the example solution, each element defined by the job index and operation index has a number. I thought that the number in the assignment segment indicates the machine index and the number in the operation sequence indicates the sequence in the assigned machine (which may be defined by the first segment). I think my understanding is wrong because it does not fit your explanation in the second corresponding letter. I am kindly asking the explanation on this again. Also, the o_ij sequences in the machine assignment segment and the operation sequence segment are different. Why is the sequence is different?
4. There are still many obvious grammar errors. And the authors need to be careful about font size of equation.
Author Response
please see the attached file. Thank you.
Author Response File: Author Response.pdf
