Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper proposes a novel approach to solve the no wait continuous time flow shop problem to minimize the makespan and tardiness, the optimization algorithms and constraints are well defined, but I believe the manuscript writing can be improved significantly. In addition to clarifying few points:

1. Abstract is quite vague, what metrics are used and how the work can be evaluated with respect to the literature, please rewrite it stating clearly what is measured

2.  

   at the introduction part, the authors focus on the use case not the general specs of the flow shop problem itself, the baking of biscuits is the use case and I believe the methodology could be applied on another use case. the readers would be more interested in a generic description of the flow shop problem in hand.

3. There is no citation in the introduction section and up to the literature review including its first sections, also at the literature review the pros and cons of the methods used in Table 1 were not mentioned in the table, also which ones were continuous or discrete and does it affect the choice of the optimization algorithm

4.  

   the authors need to explain why in specific they used GA and Heuristic method, since all that was mentioned was the inclusion of setup times and other variables in their approach

5.  

   A list of abbreviations at the end of the paper would be beneficial, as it is very confusing to trace them, and they are explained multiple times in the paper

6. sensitivity figure is not clear, so please update it
7. in the results section it would make more sense to show the results of the two metrics separately and then combine them not the other way around, unless there is a reason for this, can the authors explain this?
8. finally, there is no quantative or qualitative comparison at the end of the paper with work in the literature, which would emphasize the results and contributions of the manuscript, can this be added at the end. especially, the authors mentioned future work directions that mostly are implemented in literature already like scheduling using ML.

Author Response

Comments 1: Abstract is quite vague, what metrics are used and how the work can be evaluated with respect to the literature, please rewrite it stating clearly what is measured 

Response 1: Thank you for pointing this out. In the revised abstract, we have explicitly emphasized that our optimization objective is the sum of the makespan and the maximum tardiness, and that the proposed approach is evaluated using the relative deviation between the obtained solutions and the analytical lower bound of the problem. At the same time, we have explicitly stated that the Earliest Due Date (EDD) rule, which is widely used both in practice and in the scheduling literature, and an DE algorithm, are adopted as a baseline.  We have highlighted the significant performance improvements achieved by our two algorithms. Furthermore, in accordance with Comment 7, we have adjusted the order of the sentences in the abstract to make the presentation more coherent. Please refer to the Abstract section.

Comments 2: At the introduction part, the authors focus on the use case not the general specs of the flow shop problem itself, the baking of biscuits is the use case and I believe the methodology could be applied on another use case. the readers would be more interested in a generic description of the flow shop problem in hand.

Response 2: Thank you for pointing this out. In the revised introduction, we now use the biscuit-baking case only as a motivating example, and throughout the manuscript we have replaced it with a more general description. More specifically, we now refer to the oven as the upstream processor and to the packing machines as processing machines. We hope that this more generic terminology will make the problem more appealing to readers and better highlight the general applicability of our study. Please refer to the Introduction section for the statement ‘In many modern production systems, …very broad applicability in practice’.

Comments 3: There is no citation in the introduction section and up to the literature review including its first sections, also at the literature review the pros and cons of the methods used in Table 1 were not mentioned in the table, also which ones were continuous or discrete and does it affect the choice of the optimization algorithm.

Response 3: Thank you for your great comment. The original comparison table in the literature review did not distinguish whether the flow shop problems were continuous or discrete. In the revised manuscript, we have added the corresponding citations to the introduction and the literature review and updated the comparison table accordingly.

Comments 4: The authors need to explain why in specific they used GA and Heuristic method, since all that was mentioned was the inclusion of setup times and other variables in their approach

Response 4: Thank you very much for this helpful comment. We agree that the reason behind our choice of the genetic algorithm and the heuristic method was not clearly explained in the original manuscript. In the revised manuscript, we have clarified that the heuristic algorithm is developed to quickly generate high-quality feasible solutions, while the genetic algorithm is employed to explore a broader range of production possibilities and obtain solutions of higher quality. Please refer to the beginning of the Solution Method section for the statement ‘First, based on the lower bound …In summary’.

Comments 5: A list of abbreviations at the end of the paper would be beneficial, as it is very confusing to trace them, and they are explained multiple times in the paper

Response 5: Thank you very much for this precious comment, and we totally agree with the proposal. We have added a list of abbreviations at the end of the paper, and hope that our paper can be read more easily. Please refer to the Appendix A.

Comments 6: Sensitivity figure is not clear, so please update it.

Response 6: Thank you very much for this helpful comment. We have updated the sensitivity analysis figure to improve its clarity. Specifically, we replaced the original bar chart with a clearer and more distinguishable graphical style. In addition, because the objective values in the numerical experiments are relatively large—which made the differences among the algorithms less visible in the previous figure—we have adjusted the value range of the vertical axis to better highlight the performance differences. We hope these revisions make the sensitivity analysis section easier to read and interpret. Please refer to the figure 4.

Comments 7: In the results section it would make more sense to show the results of the two metrics separately and then combine them not the other way around, unless there is a reason for this, can the authors explain this?

Response 7: Thank you for your precious suggestion. Separating the two components of the objective function is part of analyzing the algorithmic characteristics. It is more appropriate to first examine the two components independently in order to identify the distinct behaviors of the algorithms, and then evaluate the combined objective to observe their overall performance. In the revised manuscript, we have supplemented the decomposition results of the two components for the small-scale instances and moved this part to appear before the numerical experiments of the total objective function. We hope that this adjustment improves the logical flow of the paper. Please refer to the Numerical Results and Analysis section.

Comments 8: Finally, there is no quantative or qualitative comparison at the end of the paper with work in the literature, which would emphasize the results and contributions of the manuscript, can this be added at the end. especially, the authors mentioned future work directions that mostly are implemented in literature already like scheduling using ML.

Response 8: Thank you for pointing this out. To the best of our knowledge, existing studies on continuous flow shop systems similar to ours have not employed machine learning–based algorithms for solution purposes. To validate the effectiveness of our proposed algorithms, we have selected the study by Tang et al. (2013) and have compared our methods with their differential evolution algorithm. Their work focuses on large-scale continuous manufacturing processes in the steel industry and proposes an innovative differential strategy that effectively avoids repetitive search behavior and premature convergence. By conducting comparative analyses under both small-scale and large-scale production settings, we have demonstrated that the improved genetic algorithm developed in this study still exhibits a clear advantage in solution quality. Please refer to the Numerical Results and Analysis section for the statement” For further reference to previous studies, …, EDD, HAFG, GAAM, and DE”.

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

1. Major comments
Comment 1 — Modeling clarity (integral/limit to discrete).
The continuous-time model with integrals and limit operators (constraints 2–20) is interesting but dense. When you move to the discrete-time MILP, you say “meanings remain the same,” but several places (the setup integration windows, the jump constraints for s¹ and s²) actually change semantics slightly. Please add 1–2 paragraphs explicitly mapping: continuous setup window → discrete sum, and limit jump → absolute-difference equality. This will help readers implement the model. 
Comment 2 — Assumption 2 is strong.
You assume that under even distribution the packaging capacity is not lower than the oven speed, so each order can be processed “independently.” In practice, setups will temporarily drop capacity below the oven speed (you mention this later). Clarify how often that happens in your instances and whether the MILP enforces line stops correctly during those periods. A short numerical illustration of one such stop–resume episode would help. 
Comment 3 — Objective mixing (Cmax + Tmax).
Minimizing the sum of makespan and maximum tardiness is reasonable for your scenario, but it couples two scales (time horizon vs. lateness). You partly address this by later decomposing the objective (Table 6), which is good. I suggest: (i) move that decomposition earlier, and (ii) briefly motivate why you didn’t use a lexicographic or weighted objective (e.g. tardiness first, makespan second). Otherwise readers may think the heuristic “prefers” makespan just because Cmax is larger in magnitude. 
Comment 4 — GAAM decoding is doing a lot of work.
The genetic chromosome just encodes priorities, but the real feasibility (respect oven rate, remove redundant machine combos, allocate speeds) is done in the decoder (Appendix B). That’s fine, but then the complexity and novelty live in the decoder, not the GA. Please bring 1–2 key steps from Appendix B into the main text: (a) the two-level redundancy extraction, (b) the two principles for speed allocation. That will make the algorithm contribution clearer. 
Comment 5 — Experiments: add variability.
All tests use the same random ranges for order size (800–1500) and due dates (100–300). It would be useful to add tight due dates and more heterogeneous machine speeds to stress the heuristic. Also, you could report std/median gaps over 10 random seeds so we see stability. Right now results are point estimates. 
Comment 6 — Positioning vs. recent distributed / energy-aware no-wait shops.
Your literature review is thorough, but many of the cited works optimize a different compound objective (e.g. energy + tardiness). Make it explicit that your contribution is not the metaheuristic itself but adapting it to continuous-flow, rate-constrained, with due dates and setups — that’s where your novelty is. 

2. Minor comments
• Standardize notation: sometimes “secondary packaging specification” is called “secondary processing specification”; stay with one term.
• Figures/tables: enlarge Table 4/5 captions to restate the instance scale; readers may forget what (7,3,8,8) means.
• Time horizon T in the discretized model: tell readers how you chose T (Cmax upper bound?) and granularity (1 time unit? 0.5?). This affects MILP size.
• A small language edit will help: a few long sentences in the Introduction can be split. 

Author Response

1. Major comments

Comment 1 — Modeling clarity (integral/limit to discrete).
The continuous-time model with integrals and limit operators (constraints 2–20) is interesting but dense. When you move to the discrete-time MILP, you say “meanings remain the same,” but several places (the setup integration windows, the jump constraints for s¹ and s²) actually change semantics slightly. Please add 1–2 paragraphs explicitly mapping: continuous setup window → discrete sum, and limit jump → absolute-difference equality. This will help readers implement the model. 

Response 1: Thank you for pointing this out. The explanation in the original manuscript was indeed rather brief at the relevant locations. In the revised version, we have clarified how time is discretized and why time discretization allows the integral and limit formulations to be removed. In addition, for the setup jump constraints and setup integration windows mentioned in your comments, we have provided detailed explanations of their meanings after time discretization. We hope that these revisions improve the readability of the paper and help readers better understand the model. Please refer to the Model Linearization section for the statement” Specifically, …, and the secondary processing machines all become 0.”

Comment 2 — Assumption 2 is strong.
You assume that under even distribution the packaging capacity is not lower than the oven speed, so each order can be processed “independently.” In practice, setups will temporarily drop capacity below the oven speed (you mention this later). Clarify how often that happens in your instances and whether the MILP enforces line stops correctly during those periods. A short numerical illustration of one such stop–resume episode would help. 

Response 2: Thank you for pointing this out. However, we still believe that this assumption is necessary for the following reasons. The processing capacity of a machine combination is determined by the bottleneck capacity of the two stages. Under a static processing mode, the even-distribution connection scheme provides the maximum achievable total processing capacity. If this assumption were not satisfied, it would be impossible to process all orders sequentially according to the EDD rule.

Following your suggestion, in the revised manuscript we have added explanatory text in the nonlinear model section to clarify how the model correctly triggers line interruptions when the downstream processing capacity temporarily falls below the upstream output rate due to setup operations. In addition, as recommended, we have included in the problem description section a numerical example involving a system with only one primary processing machine and one secondary processing machine to illustrate the stop–resume behavior of the production line. Please refer to the Problem Description section for the statement” Figure 1 illustrates … interruption of duration ” and figure 1.

Comment 3 — Objective mixing (Cmax + Tmax).
Minimizing the sum of makespan and maximum tardiness is reasonable for your scenario, but it couples two scales (time horizon vs. lateness). You partly address this by later decomposing the objective (Table 6), which is good. I suggest: (i) move that decomposition earlier, and (ii) briefly motivate why you didn’t use a lexicographic or weighted objective (e.g. tardiness first, makespan second). Otherwise readers may think the heuristic “prefers” makespan just because Cmax is larger in magnitude. 

Response 3: Thank you for your precious suggestion. In the revised manuscript, we have made the following modifications: (1) We have added the decomposition experiment for the small-scale instances and moved this part to appear before the numerical experiments for the total objective function; (2) We have explained in the problem description section why a weighted objective was not adopted; (3) We have added an explanation in the objective-function decomposition section to clarify why the heuristic algorithm tends to optimize the maximum completion time. Specifically, the heuristic algorithm has less flexibility than the genetic algorithm in adjusting the order sequence, which limits its ability to reduce tardiness. However, it significantly reduces the number of interruptions, and thus naturally places more emphasis on improving the maximum completion time. We hope that these revisions enhance the readability of the paper and clearly highlight our contributions. Please refer to the Problem Description section and the Numerical Results and Analysis section.

Comment 4 — GAAM decoding is doing a lot of work.
The genetic chromosome just encodes priorities, but the real feasibility (respect oven rate, remove redundant machine combos, allocate speeds) is done in the decoder (Appendix B). That’s fine, but then the complexity and novelty live in the decoder, not the GA. Please bring 1–2 key steps from Appendix B into the main text: (a) the two-level redundancy extraction, (b) the two principles for speed allocation. That will make the algorithm contribution clearer. 

Response 4: Thank you for pointing this out. We truly appreciate your careful reading and constructive feedback. In the revised manuscript, we have supplemented the fitness-function section with explicit descriptions of (1) the two-step procedure for extracting redundant machine combinations, and (2) the principles used for allocating processing speeds to machines. We hope that these additions help clarify the contributions of our algorithm. Please refer to the Solution Method section.

Comment 5 — Experiments: add variability.
All tests use the same random ranges for order size (800–1500) and due dates (100–300). It would be useful to add tight due dates and more heterogeneous machine speeds to stress the heuristic. Also, you could report std/median gaps over 10 random seeds so we see stability. Right now results are point estimates. 

Response 5: Thank you very much for pointing this out. The intention in our original design was that the current due dates should already be quite tight, which can also be seen from the relatively large values of the maximum tardiness observed in the numerical results. In addition, the main difficulty of the problem in our setting is intended to come from the configuration of machine processing speeds rather than from the nominal values of the due dates themselves. The current machine speeds were chosen to be relatively low so as to impose sufficient pressure on all algorithms. If the machine speeds were set much higher or allowed to vary too widely, the results for all algorithms would tend to become uniformly good, making it harder to clearly distinguish the advantages of the proposed methods. We hope this design choice is understandable and acceptable.

Following your helpful suggestion, the variance of the objective values has now been computed for each of the five small-scale and five large-scale instances, and the stability of the different algorithms has been analyzed. This additional stability analysis provides a more detailed comparison among the algorithms and helps to further clarify their behavioral differences. Please refer to the Numerical Results and Analysis section.

Comment 6 — Positioning vs. recent distributed / energy-aware no-wait shops.
Your literature review is thorough, but many of the cited works optimize a different compound objective (e.g. energy + tardiness). Make it explicit that your contribution is not the metaheuristic itself but adapting it to continuous-flow, rate-constrained, with due dates and setups — that’s where your novelty is. 

Response 6: Thank you very much for pointing this out. Your suggestion is highly valuable. Following your advice, we have revised the wording of the research-gap section to make our positioning clearer. Specifically, we have added two additional columns to the literature comparison table: one distinguishing whether each referenced study considers a continuous-flow or discrete-flow setting, and the other indicating whether machine rated-processing-speed constraints are incorporated. In addition, we have refined the narrative in this section to explicitly highlight that our contribution does not lie in the metaheuristic itself, but rather in adapting it to a continuous, rate-constrained flow shop with due dates and sequence-dependent setups. We hope these revisions clarify the novelty of our work. Please refer to the Research Gaps section for the statement “A comparison of the existing literature … rated processing-speed constraints of machines”.

2. Minor comments

• Standardize notation: sometimes “secondary packaging specification” is called “secondary processing specification”; stay with one term.

Response 7: Thank you very much for pointing this out. In the revised manuscript, we have uniformly used the term “processing specification” instead of “packaging specification” to maintain consistency in terminology.

• Figures/tables: enlarge Table 4/5 captions to restate the instance scale; readers may forget what (7,3,8,8) means.

Response 8: Thank you very much for pointing this out. In the revised manuscript, we have added explicit descriptions of the instance scale to the captions of the two tables you mentioned, in order to remind readers of the meaning of the notation.

• Time horizon T in the discretized model: tell readers how you chose T (Cmax upper bound?) and granularity (1 time unit? 0.5?). This affects MILP size.

Response 9: Thank you very much for pointing this out. In the revised manuscript, we have added an explanation in the constraint interpretation section of the linearized model, stating that a one-minute time slot is appropriate for the time discretization, and that setting  to twice the maximum completion time is sufficient. Please refer to the Model Linearization section

• A small language edit will help: a few long sentences in the Introduction can be split. 

Response 10: Thank you very much for pointing this out. In the revised manuscript, we have carefully reviewed the entire text and split several long and complex sentences into shorter ones, with the aim of improving clarity and making the manuscript easier for readers to follow.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I would like to thank the author for objectively considering my comments, a few editing ones in the updated version:

1. table 1: the rated processing speed is only considered in this research, please remove it from the table and highlight it in the text for better readability of the paper
2. also in Table 1, remove the footer with abbreviations, since you already added the abbreviation list at the end of the paper. 
3. list of abbreviations is not complete and it should be ordered alphabetically, also its not an appendix, as I believe mdpi has a format that includes it 
4. Figure 4 has improved, but it should include different algorithms, not one with different problem complexity scale
5. The comparison is still missing, and the added reference is outdated, either the authors can validate the comparison in a table and add a recent one, or better be removed altogether.

 

Author Response

Comments1: table 1: the rated processing speed is only considered in this research, please remove it from the table and highlight it in the text for better readability of the paper

Response 1: Thank you for pointing this out. In the revised manuscript, we removed the rated processing speed in table 1, and highlight it in the text. Please refer to the Research Gaps section for the statement ‘It is worth noting that … for the literature to incorporate.’.

Comments2: also in Table 1, remove the footer with abbreviations, since you already added the abbreviation list at the end of the paper.

Response 2: Thank you for pointing this out. In the revised manuscript, we have removed the footer and added the abbreviations to the abbreviation list at the end of the paper.

Comments3: list of abbreviations is not complete and it should be ordered alphabetically, also its not an appendix, as I believe mdpi has a format that includes it.

Response 3: Thank you for pointing this out. In the revised manuscript, we have reordered the list of abbreviations alphabetically, using the mdpi format.

Comments4: Figure 4 has improved, but it should include different algorithms, not one with different problem complexity scale

Response 4: Thank you for pointing this out. Figure 4 illustrates the sensitivity analysis with respect to different machine-number combinations. Our intention is to show how the objective value changes under various machine configurations in order to derive managerial insights, rather than to compare the performance of different algorithms. Regarding the choice of algorithm, we use GAAM to compute the objective values because it performs the best among the tested algorithms, ensuring that the observed variations in the objective function are reliable.

Comments5: The comparison is still missing, and the added reference is outdated, either the authors can validate the comparison in a table and add a recent one, or better be removed altogether.

Response 5: Thank you for pointing this out. The production system studied in our work is a three-stage flow shop with continuous-flow characteristics and rated power constraints. Such a setting is relatively uncommon in the existing literature, which makes it difficult to identify recent studies that are highly comparable to ours. We agree with your comment that the previously cited reference used for comparison is outdated. Moreover, in our algorithmic comparison, the core solution procedure still relies on the GAAM decoding method proposed in this paper, which is a major contribution of our work. As a result, even if a comparison were conducted, it would essentially be a comparison between two algorithms that both depend on the same GAAM decoding framework, and thus would not effectively highlight the originality of our approach. Considering these factors and your helpful suggestion, we have removed the DE-related comparison from the revised manuscript. We hope this adjustment meets your expectations.

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have satisfactorily addressed all prior comments and improved the manuscript’s clarity and methodological transparency. I have no remaining concerns and recommend acceptance in its current form

Author Response

We sincerely appreciate your careful comments and valuable suggestions for improvement. Thank you very much.