Imperialist Competitive Algorithm with Three Empires for Energy-Efficient Parallel Batch Processing Machine Scheduling with Preventive Maintenance

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

My comments:

1. Abstract
   1. The phrase "a new way is applied to construct three initial empires with new structure" is somewhat vague. Briefly alluding to the core characteristic of this "new structure" (e.g., fixed number of imperialists per empire) within the abstract could be more informative.
   2. Could the abstract briefly mention the specific nature of the "adaptive imperialist competition"? For instance, what aspect is adaptive?
   3. While "promising advantages" are stated, could a more concrete, albeit brief, indication of the performance improvement (e.g., "outperforming compared algorithms on X% of instances for key metrics") be included?
   4. The claim of TEICA's effectiveness is made; it might be beneficial to state that this effectiveness is in comparison to other established or adapted algorithms.
2. Introduction
   1. The statement "PM of parallel BPM is hardly studied” is a strong claim. While likely true, providing a more direct, brief synthesis from your literature review that leads to this conclusion, rather than just stating it post-review, would bolster this assertion.
   2. The rationale for choosing ICA as the base algorithm for this problem is provided. However, could you briefly elaborate on why ICA, specifically, is considered more advantageous than other population-based metaheuristics (e.g., GA, PSO, DE which are also mentioned) for this particular BPM scheduling problem's characteristics?
   3. The introduction mentions "new strategies such as new empire structure". It would be beneficial to briefly hint at how this new structure is theorised to overcome known limitations of traditional ICA (e.g., premature convergence, parameter sensitivity of (N_im).
   4. The assertion that "Parallel BPM scheduling problem is the extended version of PMSP” is correct. However, for readers less familiar, briefly outlining the unique combinatorial complexities introduced by the batch formation aspect, beyond standard PMSP, could better frame the problem's difficulty.
   5. The introduction notes several constraints considered in literature but states that "PM is often applied... however, PM of parallel BPM is hardly studied". Are there specific aspects of PM in the context of BPMs (e.g., PM duration affecting batch availability, PM impact on energy consumption patterns) that make this integration particularly challenging or novel?
   6. The objectives are minimizing makespan and total energy consumption. A brief mention in the introduction about the potential trade-offs between these two objectives could better motivate the bi-objective approach from the outset.
   7. The manuscript focuses on unrelated parallel BPMs. Is there a reason this was chosen over identical or uniform, and does this choice significantly influence the TEICA design?
3. Problem description
   1. The constraint "All the machines are available at all times” appears to conflict with the concept of PM, during which a machine would presumably be unavailable. Please clarify if this means machines are available except during scheduled PM activities.
   2. The "index Si " is stated as potentially being "size, weight". Could you offer a slightly broader range of examples for Si to illustrate its versatility in representing job characteristics relevant to batching capacity qk?
   3. PM start time is defined as g x uk, implying a fixed, deterministic PM schedule. While this simplifies modelling, how realistic is this assumption in industrial settings where PM might be condition-based or offer some flexibility? A brief comment on this could be useful.
   4. The batch processing time PT_kv =max{p_ki,i∈B_kv} is standard. Does the energy consumption model (Equation 2 ) assume the machine operates at full processing power ek for this entire PT_kv, even if some jobs within the batch finish earlier?
   5. In Equation (2) for TEC, is PMT_k (total maintenance time) a direct consequence of the fixed PM schedule (u_k,w) and the makespan, or is it a decision variable in some nuanced way?
   6. Regarding the example in Section 2 and Figure 1: The text states "an example with 15 jobs". However, the Gantt chart in Figure 1 clearly shows jobs labelled 16, 18, 20 on M₁ and 17, 19 on M₂. This implies more than 15 jobs are being scheduled. The matrix (p_ki)_2×15 also suggests 15 jobs. This is a significant point of confusion and needs thorough clarification and correction. Which job set does the example actually refer to?
4. Initialization and initial empires
   1. The two-string representation (machine assignment and scheduling string) is conventional. Was any consideration given to a more integrated representation that might handle batch formation more directly within the chromosome, or is the sequential decoding process considered more effective?
   2. In the decoding process, "choose all jobs with the type of ti from the permutation sequentially under capacity construct". Could "capacity construct" be clarified? Does it imply a specific heuristic for filling the batch (e.g., first-fit, best-fit for jobs of the same type)?
   3. The heuristic for generating six initial solutions assigns jobs to Mk with the smallest pki. Does this heuristic tend to create solutions with good makespan but potentially poor TEC, or vice-versa, and how might this initial bias influence the search?
   4. The choice of exactly six initial solutions generated by the heuristic and N−6 random solutions seems specific. Is there a rationale or empirical basis for this particular ratio?
   5. The new empire structure involves each of the three empires having two imperialists. What is the theoretical advantage or expected synergy of having two imperialists per empire guiding the colonies, as opposed to the traditional single imperialist?
   6. Equation (7) for calculating the normalised cost c_IMk for imperialists incorporates both rank and crowding distance. Is this a standard ICA approach when dealing with multi-objective problems, or a novel adaptation specific to TEICA for selecting imperialists?
   7. The number of empires Nim is fixed at 3, reducing a typical ICA parameter. While the justification is provided later, how was 3 determined to be optimal or a robust choice? Was sensitivity to this number explored in preliminary studies?
5. New assimilation and revolution
   1. The new assimilation process involves two-point crossover for both machine assignment and scheduling strings between a colony and a randomly chosen imperialist within its empire. Why was two-point crossover selected over other crossover operators (e.g., uniform crossover, single-point crossover)?
   2. Imperialists within an empire are also updated via two-point crossover with each other. What is the intended effect of this inter-imperialist crossover? Is it to promote diversity among imperialists or to encourage convergence towards a shared promising region?
   3. The revolution step employs seven neighbourhood structures, applied in two different sequences. What was the rationale for selecting this specific set of seven structures and their particular orderings in the sequences? Are some more focused on machine assignment and others on scheduling aspects?
   4. In the multiple neighbourhood search, the first improving or non-dominating solution found is accepted, and the search moves to the next structure or terminates. Is there a mechanism for escaping deeper local optima if an entire sequence of neighbourhood moves fails to yield an improvement?
   5. The use of an external archive A to store non-dominated solutions is good practice. Is the size of this archive bounded, and if so, how is it maintained (e.g., pruning by crowding distance if full)?
   6. Updating imperialists after assimilation and revolution by choosing the two solutions with the smallest rank and biggest crowding distance from all solutions within the empire is a strong elitist strategy. Does this apply to the colonies as well, or only to select the two leading imperialists?
6. Imperialist competition
   1. The new imperialist competition provides additional searches for the winning empire, rather than reallocating colonies from weaker empires. What is the perceived advantage of this 'search intensification' approach over the traditional 'colony reallocation' in terms of search dynamics (e.g., exploration vs. exploitation balance)?
   2. The mechanism to support empires that have not won for A consecutive generations (rk ≥A) by intensifying them using best solutions from other empires is novel. How sensitive is the algorithm's performance to the choice of parameter A? Is there a risk of this mechanism prematurely focusing the search or disrupting the natural competitive evolution?
   3. The justification for fixing Nim at 3 mentions a "good balance on diversity of population and effect of imperialist competition". Could this be elaborated with more specific theoretical arguments or references to preliminary empirical findings that support three empires as being more effective than two or, say, four or five, especially with the new competition rules?
   4. Equation (9) defines $TC_k = C_k + \xi \cdot \text{mean_Cost(colonies of empire k)}$. Given that each empire in TEICA has two imperialists, how is the single Ck (cost of the imperialist) determined for this calculation? Is it an average of the two imperialists' costs, or the cost of the 'stronger' one?
7. Algorithm description
   1. The list of TEICA's distinguishing features (fixed three empires, no empire elimination, new structure with two imperialists, novel assimilation, and support for weaker empires during competition) clearly summarises its innovations.
   2. The flowchart in Figure 2 accurately represents the algorithm's iterative process.
8. Instances, metrics and comparative algorithmsomputational experiments
   1. For the generated instances, job types ti ∈{2,4,6} and sizes Sj ∈[1,10] are specified. Are the processing times pki correlated in any way with job types or sizes, or are they generated independently? This can influence scheduling difficulty.
   2. The choice of C-NSGA-A and BOACO as comparative algorithms is reasonable, given their application to similar problems. Was consideration given to comparing TEICA against other ICA variants to more directly isolate the performance contributions of the novel "three empires" structure and associated mechanisms?
   3. The "general ICA" constructed for comparison: it would be beneficial to specify which standard ICA implementation or well-known variant this is based on, including its key operator details, to better contextualise the performance differences observed.
   4. The performance metrics C, ρ, and IGD are well-established and appropriate for evaluating multi-objective algorithms.
9. Parameter settings
   1. The stopping condition is 0.3×n seconds of CPU time. While pragmatic, does this provide a fair basis for comparison if algorithms exhibit different convergence speeds? The statement "TEICA, ICA, C-NSGA-A and BOACO can converge fully on all instances" within this time is a very strong claim; perhaps "reach a stable set of solutions" would be more cautious.
   2. The Taguchi method was applied to tune TEICA's parameters (N, UR, R, A) using instance 37 (n=50,m=5). Is there a concern that optimal parameter settings derived from a single medium-sized instance might not be optimal for the full range of instance sizes (n=10 to n=500)?
   3. The parameter Nim for the comparative general ICA was set to 5. How was this value chosen? Was it also tuned, or selected based on common practice or the original ICA paper? This is relevant as Nim can significantly affect ICA performance.
   4. It is good practice that parameters for C-NSGA-A and BOACO were taken from their original publications and their continued effectiveness under the new stopping condition was verified.
10. Results and discussions
    1. In Table 2, TEICA achieves an IGD of 0.000 on many instances. This implies that the non-dominated set found by TEICA perfectly matches the reference set Ω∗ (comprising all non-dominated solutions from all algorithms combined). This is a remarkable result if Ω∗ often contains solutions from other algorithms. Could this imply that on these instances, other algorithms did not contribute any solutions to Ω∗ that TEICA did not also find?
    2. Figure 5 presents mean plots with 95% confidence intervals for the metrics. While visual differences are apparent, have any statistical significance tests (e.g., ANOVA followed by post-hoc tests, or Wilcoxon rank-sum tests for pairwise comparisons) been performed on the metric values over the 10 runs to formally establish the superiority of TEICA?
    3. The paper states, "TEICA provides all members for reference set Ω∗" for metric ρ on 89 instances (Table 3). This is a very strong performance. It would be interesting to briefly discuss if there are any discernible characteristics of the few instances where TEICA did not achieve ρ=1.
    4. The discussion attributes TEICA's good performance to its new strategies, such as the three-empire structure maintaining diversity and new assimilation intensifying exploration. Is it possible to further dissect which specific new strategy (e.g., the two imperialists per empire, the adaptive competition, the support for weaker empires) contributes most significantly to these observed improvements, perhaps through an ablation study (even if limited)?
    5. The visualisations in Figure 4 are helpful. For instances like #104 where solutions are densely clustered in the TEC dimension, a log scale or a zoomed-in inset for that region might offer better visual discrimination if space permits.
11. Conclusions
    1. The conclusion that "TEICA has promising advantages in solving the considered BPM scheduling problem" is well supported by the presented experimental results.
    2. The statement that "BPM scheduling problem has higher complexity than the classical scheduling problem" is true. A brief mention of why (e.g., the added combinatorial layer of batch formation and assignment) would reinforce this for a broader audience.
    3. The future work outlined, focusing on more complex shop environments (hybrid flow shop, flexible job shop) and real-life manufacturing processes, is ambitious and logical.
    4. The paper mentions trying "meta-heuristics with new optimization mechanisms including learning, cooperation, feedback and competition". Could this be slightly more specific about how these mechanisms might address particular challenges in the proposed future research areas?
12. General
    1. Please include a nomenclature table listing all symbols and parameters used, especially those introduced in Section 2 (Problem Description) and Section 3 (TEICA algorithm). This would significantly aid readability for an expert audience navigating numerous variables.

 

 

Comments on the Quality of English Language

Several sections would benefit from further attention to enhance clarity, conciseness, and idiomatic expression, particularly for an international audience familiar with UK English conventions. Some sentences are overly long and complex, which can occasionally obscure the intended meaning or reduce readability. A thorough proofread by a native English speaker, focusing on refining sentence structure, improving flow between ideas, and ensuring precise use of terminology, would contribute to the manuscript's overall polish and impact. Addressing these minor linguistic points will help ensure the research is communicated as effectively as possible.

Author Response

We response to comments in PDF file Reviewer 1

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Suggestions for improvement:
Clarify Contribution: Specify the unique contribution of the paper. What novel insights or solutions does the proposed system offer compared to existing approaches? Highlighting this will strengthen the paper's significance.
Consider Future Work: Conclude the abstract by suggesting potential avenues for future research or improvements to the proposed system. This demonstrates a forward-looking perspective and can encourage further exploration in the field.
Ambiguity in control strategy and algorithm: The abstract mentions the use of a three-empire imperial competitive algorithm (TEICA) without providing details about the structure of the TEICA algorithm or the methodology of the TEICA algorithm. This ambiguity obscures the novel aspects and effectiveness of the strategies used.
Oversimplification of components: The description of the TEICA algorithm lacks depth. There is a need to clarify the technical aspects, specific configurations, and their role in achieving optimal utilization and reliability.
Explanation of limited scope: There is no explanation of setting up an environment to minimize production time and total energy consumption, or validation against real-world data.
Lack of comparative analysis with existing systems: The absence of references to compare performance with existing systems or algorithms for minimizing production time and total energy consumption limits understanding of how this proposed system improves upon or differs from existing solutions. Missing performance metrics: The summary lacks explicit mention of performance metrics for reducing production time and overall energy consumption, such as efficiency improvement percentages, accuracy, or energy usage improvements, which could quantify the benefits of the system and measure its success.

Suggestions for improvement:
Enhance Flow and Clarity: Consider restructuring sentences for improved flow and clarity. Break down complex ideas into smaller, more digestible sentences to enhance readability.
Emphasize Novelty: Explicitly state the unique contributions or innovations of the research. This can be placed toward the end of the introduction to engage the reader's interest.
Include Objectives: Explicitly state the objectives of the research in the introduction. What specific problems or questions is the study aiming to address? This provides a clear roadmap for readers.
Highlight Practical Applications: In the section discussing applications of standalone or isolated , provide a few practical examples to help readers grasp the real-world relevance of the research.

Suggestions for improvement:
Absence of References in Equations: Equations are presented without references, hindering the opportunity for readers to cross-reference or delve deeper into the theoretical foundations or sources of these mathematical formulations.
Lack of Validation Mention: The section lacks , real-world validations, or case studies that might have demonstrated the efficacy or the performance  , limiting the practical demonstration of the proposed designs.
 No Discussion on Limitations or Challenges: The section does not address the limitations or challenges associated with the proposed models or control strategies, missing an opportunity to highlight potential areas for improvement or future research directions.

Organize Content Logically: Consider grouping related information together to enhance the logical flow of the method section. For instance, discussing the components and models  aspects might make the content more cohesive.
Clarify Equations: Ensure that equations are clearly explained and defined for readers who may not have an in-depth background. Provide brief explanations or interpretations for key variables and parameters.
Include Justifications: Provide brief justifications for the choice of specific models, components, or parameters. This will enhance the reader's understanding of the rationale behind certain decisions.
Visual Clarity: Ensure that figures and diagrams are clear and labeled appropriately. Consider including a brief caption or description for each figure to aid in understanding.
Discuss Limitations: Briefly discuss any limitations or assumptions made during the  process. This provides transparency and helps readers understand the scope and potential constraints of the proposed methodology.

The conclusion section can highlight potential broader implications of your work and areas that need further study. Be careful not to inflate your findings.
Your conclusion is not simply a summary of what you've already written. It should take your paper a step further by answering any outstanding questions. Summarize what you've shown in your study and suggest potential applications and extensions. The main question in your conclusion should be, "What do my findings mean for the research field and my community?"

- Update the conclusion to include the newly formulated theoretical contributions;
- Mention the limitations of the study and prospects for future research;
- Summarize the key results in a compact form and re-emphasize their significance;
- Summarize how the article contributes to new knowledge in the domain.
- Add future work to motivate other researchers to continue the research.

Author Response

We response to comments of reviewer 2 in PDF file Revierer 1

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

• In the abstract, clearly highlight the existing problem first, followed by a concise statement of your research objective, explaining how your work addresses the problem.

• Ensure the objective describes what you aim to achieve, not just what has been done technically.

• Include key numerical results in the abstract to make it more impactful and immediately informative to readers.

• In all the equations, it is unclear whether they are presented in vector or scalar form. If an equation is in vector form, it should be indicated using boldface notation—this has not been applied consistently, making interpretation difficult.

• Specifically, in Equation (9), it appears to be in scalar form; thus, a cross product would not be valid. Please double-check this equation and verify the use of dot and cross products throughout the manuscript to ensure mathematical correctness.

• The flowchart in Figure 2 should be moved to the beginning of the methodology section to provide a clear overview of the algorithmic process.

• Accompany the flowchart with a clear explanation of how the algorithm operates, particularly on the criteria used to determine the stopping condition for iterations.

• In Section 4, you mention using 108 instances for computational testing. However, it is not clear where these instances come from or how they are selected. Justify how this sample size supports the robustness and generality of your algorithm.

• The results section should demonstrate how the total energy consumption is minimized, as this is expected based on your objective. However, the link between the results and your stated objective is currently unclear—please clarify this connection.

• In Figure 4, you present TEC and Cmax, but the units are missing. Please add appropriate units to enhance clarity and interpretation.

• While you have conducted comparative studies among algorithms, the computational accuracy of your method is not well established. Provide a detailed analysis or metrics to demonstrate the reliability and precision of your computational results.

 

Author Response

We response to comments of reviewer 3 in PDF file Reviewer 1

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

It is accepted and hoped that this paper can have a good impact.