Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript covers power battery supply chains with a Nash bi-level optimization that I believe have a high potential to contribute the current literature. I have noted some points to improve the manuscript quality. 

First of all, the abstract is well written by addressing background, purpose, methodology, findings and implications. It is clear and easy to understand at a first glance. However, Introduction has to be improved. For example, authors use citations like "chains [1–4]" which is not an obvious citation type. Which statement belongs to which publication? You have to state them clearly. You should add individual references to the related statements. And this situation is valid for the all manuscript like " regulation [1,5,9,10]", and so on. It makes audience feels to have a high level of LLM use inyour manuscript. Please be more specific in citing all over the manuscript text. Secondly, again in the Introduction, between the motivation and contribution parts, you should separately give the literature gap, the purpose of this study, and the applied methodology with its advantages written as paragraphs. 

Liteature Review part is clear and understandable; however, Table 1 is really hard to read in this current version. This table is really nice to have to compare your work, and approve that this research is really needed. Thuhs, you have to present it by guaranteeing the readability. 

Similarly, 3. Materials and Methods , 4. Results sections are well-presented, I did enjoy reading these parts. There are just a few typos, please re-read the paper carefully. 

After giving such important findings with high-quality figures, Discussion and Conclusion sections remain weak. In the Discussion, please highlight the comparison of your your work with the current literature results. And please also include the aligning and contradictory theories with your results by writing separate paragraphs. This will strengthen your work's validity. In the Conclusion, after giving the findings at the beginning, give more details about you practical and managerial implications, theoretical contributions, limitations, and further research ideas with separate paragraphs.  

 

 

Author Response

Comments 1: The manuscript uses broad grouped citations in the Introduction and elsewhere, making it unclear which statement is supported by which publication. Citations should be more specific throughout the manuscript.
Response 1: Thank you for pointing this out. We agree with this comment. We revised the Introduction and Literature Review to make citation placement more specific and to avoid broad grouped citation blocks. In particular, we split several broad citation clusters into smaller, claim-specific citation groups and reworded the opening motivation paragraphs so that the role of each cited study is easier to trace. These changes can be found in Sections 1 and 2 of the revised manuscript.
Comments 2: The Introduction should separately present the literature gap, research purpose, applied methodology, and methodological advantages.
Response 2: We agree. We clarified the literature gap, research purpose, applied method, and methodological advantage in the Introduction, and further distinguished methodological novelty from application novelty in the Literature Review. Specifically, we added a gap-purpose-method-advantage paragraph in Section 1 and added a clearer novelty statement at the end of Section 2. These revisions explain that the methodological novelty lies in combining a shared-constraint competitive structure, exact lower-level response evaluation, and finite-candidate approximate-equilibrium screening, while the application novelty lies in applying this framework to competing sustainable closed-loop supply chains in the power battery industry.
Comments 3: Table 1 is useful but difficult to read in the current version.
Response 3: Thank you for this helpful suggestion. We revised the explanatory text around Table 1 and clarified the meaning of the qualitative criteria. We added an explanatory sentence before Table 1 and expanded the table note to define the compact labels used in the comparison. In particular, `P` now denotes partial coverage, and `Weak`, `Medium`, and `Strong` denote increasing degrees of explicit strategic-operational coupling rather than normative rankings of study quality. The revised explanation can be found around Table 1 in Section 2.
Comments 4: Methods and Results are generally good, but the manuscript needs proofreading for typos and grammar.
Response 4: We agree and have completed a final proofreading and condensation pass after the technical revisions. We refined the manuscript language throughout the revised copy, shortened response-oriented explanatory passages in the main text, and kept detailed justifications in the response document. Revised or condensed passages remain marked in red in the revised manuscript.
Comments 5: The Discussion and Conclusion are weak and should include comparison with existing literature, practical and managerial implications, theoretical contributions, limitations, and future research.
Response 5: We agree that these sections should be strengthened. We revised the Discussion and Conclusions to make them more focused and contribution-oriented. In the Discussion, we added a comparison between the present homogeneous-duopoly formulation and prior battery closed-loop studies, clarified when Nash, Stackelberg, and centralized structures are appropriate, and added managerial implications for battery manufacturers and policymakers. In the Conclusions, we clarified the theoretical contribution, practical implications, limitations, and future research directions. These revisions can be found in Sections 5 and 6 of the revised manuscript.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

<Overall Evaluation>

This paper proposes a framework that combines the Nash game and bi-level optimization to simultaneously consider strategic decision-making and operational planning within the competitive Closed-Loop Supply Chain (CLSC) environment of the power battery industry. The attempt to comprehensively consider production, transportation, recycling, inventory, and carbon emissions aligns well with the scope of the journal *Logistics*. Furthermore, the overall structure of the study is relatively systematic, encompassing algorithm validation, comparisons with alternative models, and sensitivity analysis.

However, the current manuscript fails to present the theoretical consistency of the proposed Nash bi-level model, as well as the definition and verification methods of equilibrium, in a sufficiently persuasive manner. Additionally, the core conclusions of the study rely heavily on assumptions and evaluation metrics established by the authors. In particular, the interdependence between the upper-level and lower-level, the definition of Nash equilibrium under shared constraints, the reliability of approximate equilibrium, and the generality of the conclusion that the Nash model is superior to other competitive structures require more rigorous explanation and further verification. Accordingly, my decision is to proceed with a Major Revision.

 

<Major Concerns>

(1) In the current manuscript, the demand allocation and sustainability performance of the upper-level depend on the results of lower-level operations, while the lower-level also uses the strategic variables determined at the upper-level as input. However, the order in which these interdependent relationships are calculated within the algorithm and how they form a consistent equilibrium are not sufficiently explained. It is necessary to describe more clearly whether this is a simple bi-level optimization or an equilibrium problem involving a fixed-point structure.

(2) The two supply chains simultaneously share various shared constraints, such as market demand, scarce raw materials, and subsidies. In such cases, each firm's feasible strategy set itself is influenced by the competitor's strategy, which distinguishes it from a typical Nash game. While the current manuscript describes this as a simple Nash game, it is necessary to discuss the relationship with GNE more clearly through a comparison with relevant literature and to rigorously define the equilibrium concept of this study.

(3) Production quantity q_i is defined as a strategic variable at the upper-level, while at the lower-level, only the production plan to satisfy this is determined. Therefore, there is some confusion as to whether the lower-level actually performs "production decisions" or merely production allocation. It is necessary to explain the roles of strategic decision-making and operational planning more clearly.

(4) The exhaustive search performed during the validation phase found a reference solution in a discretized strategy space and did not prove exact Nash equilibrium in a continuous strategy space. Therefore, the expression "exact equilibrium validation" appears to be an overinterpretation. Accordingly, it is appropriate to modify expressions such as "exact validation" and "exact equilibrium" to more cautious language, and it is desirable to clearly state that the current results represent a benchmark level in a limited strategy space.

(5) Equilibrium screening relies on a limited unilateral deviation set and a pre-set threshold. However, there is a lack of discussion regarding the rationale for selecting the threshold value and whether the deviation set sufficiently represents the actual equilibrium. It is necessary to additionally present robustness analysis regarding threshold changes or the stability of results based on deviation radius changes, and to provide further justification for the definition of approximate equilibrium and the verification method.

(6) The paper compares the Nash, Stackelberg, and Centralized models and concludes that the Nash model is the most appropriate. However, the Stackelberg model is structured in a disadvantageous way from the start, based on the assumption that no leader exists, and the Centralized model is also a benchmark model that is difficult to apply in actual competitive markets. Therefore, it is more appropriate to interpret the current comparison as demonstrating which environment each model is suitable for, and it would be better to soften the expression that the Nash model is generally superior.

(7) The proposed algorithm uses significantly longer computation time than the comparison algorithms. Comparing solely based on the same number of iterations cannot be considered a fair comparison in terms of computational cost. A more balanced comparison is needed that also considers the number of function evaluations, the number of lower-level solver calls, and wall-clock time.

(8) The paper explains that benchmark instances were constructed based on public data, but it does not sufficiently present how the actual parameters were set or how they were calibrated in the public source. To improve reproducibility, it is necessary to describe the benchmark data and parameter generation procedure in more detail or provide them as supplementary material.

(9) Currently, only three factors—market demand, scarce resources, and low-carbon transportation costs—are being analyzed. However, since key variables of this model, such as subsidy budget, carbon penalty, recycling efficiency, and price elasticity, may also have a significant impact on the results, the persuasiveness of the study would be further enhanced if additional sensitivity analysis were performed on a wider range of key parameters.

 

<Minor Concerns>

(1) The necessity of the research and the content of its contribution, which appear repeatedly in the Introduction and Literature Review, overlap significantly. It is necessary to reduce redundant descriptions to condense the paper overall.

(2) The Literature Review needs to provide a more critical comparison of the limitations of existing research. Currently, there is a tendency to simply list existing studies, and the distinctiveness from this study is not sufficiently emphasized.

(3) While the comparison criteria in Table 1 are useful, qualitative evaluations such as "Weak," "Strong," and "Partial" appear somewhat subjective. It would be better to explain the evaluation criteria more clearly.

(4) The definitions of mathematical symbols are relatively faithful, but some variables and functions lack intuitive explanation when they first appear in the text. In particular, it would be good to supplement the explanations to make the relationship between strategic variables and operational variables easier to understand.

(5) Figures 3–7 are helpful for understanding overall trends, but readability would be improved if the axes (labels), units, and legends for some figures were displayed more clearly.

(6) In the Discussion, the comparison with existing research remains at a somewhat declarative level. Please discuss in more detail the practical implications that the proposed model offers for actual battery supply chain management.

(7) The limitations of the study are described relatively briefly in the Conclusion. It would be beneficial to discuss in more detail the constraints that homogeneous duopoly, deterministic assumptions, and static competition face in the actual industry.

(8) In future research, presenting the potential for expansion to dynamic competition, stochastic environments, heterogeneous firms, and incomplete information games in more detail would clearly demonstrate the direction of research development.

(9) Some sentences are quite long, which somewhat reduces readability. The manuscript would be easier to read if long sentences were appropriately separated, particularly in the Methods and Results sections.

(10) While the References generally include a reasonable amount of recent literature, the literature review would be more comprehensive if additional recent research published over the past two to three years regarding competitive battery supply chains and generalized Nash equilibrium were included.

Author Response

Comments 1: The manuscript should more clearly describe how upper-level strategies, lower-level responses, feedback, and equilibrium formation are calculated, and whether the problem involves a fixed-point structure.
Response 1: Thank you for pointing this out. We agree that the original manuscript did not explain the computational order sufficiently. We clarified that upper-level candidate strategies determine shared allocations and lower-level feasible regions; then the lower-level exact response is solved and fed back to evaluate upper-level objectives. We also added response-mapping and feedback explanations in the model interpretation and solution-methodology sections. These changes can be found in Sections 3.3 and 3.4.5 of the revised manuscript.
Comments 2: Shared market demand, scarce raw materials, subsidies, and recyclable batteries make the problem closer to a generalized Nash equilibrium. The manuscript should discuss the relationship with GNE and define the study's equilibrium concept rigorously.
Response 2: We agree. Since market demand, scarce materials, subsidy resources, and recyclable batteries are shared, the feasible set of one chain depends on the rival's strategy. We now explicitly acknowledge this generalized Nash-game feature while avoiding the overclaim that the algorithm analytically solves a continuous-space generalized Nash equilibrium. The revised manuscript interprets the reported results as finite-candidate approximate equilibria obtained from a bounded search space and finite unilateral-deviation screening. These revisions can be found in Sections 3.3, 3.4.5, and the model-comparison discussion.
Comments 3: The relationship between upper-level production quantity `q_i` and lower-level production decisions is unclear.
Response 3: We agree that this distinction should be made clearer. We clarified that `q_i` is an upper-level strategic planned output commitment, while the lower level determines feasible operational allocation. Specifically, the lower-level model allocates the committed quantity across production lines, production schemes, material options, transport modes, and inventories. We revised the model assumptions, notation, and explanatory text around the upper-level strategy definition. These changes can be found in Sections 3.2 to 3.4.
Comments 4: The previous wording about exact equilibrium validation is too strong. Exhaustive search only validates a reference solution in a discretized strategy space and does not prove exact Nash equilibrium in continuous space.
Response 4: We agree. We revised the wording throughout the manuscript to avoid overclaiming. In the Abstract, we replaced exact-validation wording with "benchmark validation on a discretized small instance" and "discretized reference equilibrium." In the Results section, we renamed the validation task as finite-strategy-space benchmark validation, revised the table caption and headers, and added an explicit statement that the experiment validates the method on the finite grid rather than proving a continuous-space equilibrium. These changes appear in the Abstract and Section 4.2.
Comments 5: Equilibrium screening relies on a limited unilateral-deviation set and preset threshold. The rationale and robustness should be explained.
Response 5: We agree. We clarified that the deviation-based safety rule is finite and algorithm-assisted, not a proof of continuous-space Nash equilibrium. We revised the equilibrium-screening statement and added an operational interpretation of the threshold as a bounded local tolerance. The manuscript now states that the deviation test uses a finite neighbourhood and does not exhaust the continuous strategy space. The model-comparison discussion also reports the `+/-5%` and `+/-10%` deviation-radius evidence. These revisions can be found in Section 3.5 and the model-comparison results.
Comments 6: The manuscript should not claim that the Nash model is generally superior to Stackelberg or centralized models; it should explain which model is suitable under which practical environment.
Response 6: We agree. We revised the interpretation to emphasize environmental suitability rather than universal superiority. The centralized model is now presented as a coordinated efficiency upper bound, the Stackelberg model as suitable under explicit power asymmetry or sequential competition, and the Nash bi-level model as suitable for the homogeneous parallel competition considered in this study. We also added that Stackelberg structures remain useful when leadership is externally determined, one chain has durable first-mover advantage, a dominant platform controls the system, or policy design assigns a guiding role to a dominant participant. These revisions can be found in the model-comparison subsection, Discussion, and Conclusions.
Comments 7: The algorithm comparison is not fully fair if it only uses equal iteration counts. Function evaluations, lower-level solver calls, and wall-clock time should also be considered.
Response 7: We agree that computational cost should be reported more carefully. We added a controlled-comparison statement explaining that the three algorithms use the same case data, lower-level exact-response evaluator, random-seed protocol, and post-processing criteria. We also clarified that equal generation numbers do not imply identical candidate-evaluation or lower-level solver-call counts; therefore, runtime is reported together with equilibrium-quality indicators. The revised manuscript presents the comparison as a controlled computational comparison rather than proof of universal algorithmic dominance under an equal-evaluation budget. These revisions can be found in Section 4.2.
Comments 8: Parameter sources and calibration from public sources need more detail or supplementary material.
Response 8: We agree. We added a reproducibility paragraph explaining that the cases are public-data-adapted computational benchmarks rather than confidential firm-level replicas. The revised paragraph describes the use of public regional and industry information, traceable configuration rules, controlled random seeds, algorithm settings, and the planned release of configuration files, representative outputs, and plotting scripts as supplementary material or through a repository after de-anonymization. These changes can be found in Section 4.1 and the Data Availability statement.
Comments 9: The sensitivity analysis dimensions are limited. Additional key factors such as subsidy budget, carbon penalty, recycling efficiency, and price elasticity may affect the results.
Response 9: Thank you for this suggestion. We clarified the interpretation of the selected sensitivity dimensions and tightened the wording so that data-supported findings are separated from cautious explanatory interpretation. In the revised manuscript, we state that expected demand shows the clearest directional effect within the tested benchmark, while scarce-material supply and low-carbon transport cost mainly lead to non-monotonic equilibrium reconfiguration. We also avoided presenting these benchmark-specific patterns as universal monotonic laws. These revisions can be found in Section 4.5, Discussion, and Conclusions.
Comments 10: Several minor issues should be addressed, including redundancy, critical literature review, qualitative criteria in Table 1, symbol explanations, figure readability, practical implications, limitations, future research, long sentences, and recent literature coverage.
Response 10: We agree with these helpful comments and conducted an additional clarity and concision pass. We added a notation clarification before Table 2 to explain the main subscripts and superscripts used in the model. We also added a result-table abbreviation note before the performance indicators to define `Rep.`, `sust.`, `Eq.`, `Uniq.`, and `Saf.` The qualitative criteria in Table 1 were clarified in the Literature Review. Recent 2024-2025 studies are now explicitly reflected in the Introduction, Literature Review, and Table 1. We also strengthened practical implications, limitations, and future research directions in the Discussion and Conclusions.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Overall, I believe the manuscript is suitable for publication after minor revision. I only have a few comments that may further strengthen the paper.

Comments for author File: Comments.pdf

Author Response

Comments 1: Section 2 should more explicitly distinguish methodological novelty and application novelty.
Response 1: We agree and have revised Section 2 accordingly. We added a paragraph at the end of the Literature Review that explicitly distinguishes methodological novelty from application novelty. The revised text explains that the methodological novelty lies in combining a shared-constraint competitive structure, exact lower-level response evaluation, and finite-candidate approximate-equilibrium screening, while the application novelty lies in formulating these ideas for competing sustainable closed-loop supply chains in the power battery industry. This revision can be found at the end of Section 2.
Comments 2: The manuscript should briefly discuss practical circumstances under which the Stackelberg structure may still be preferable.
Response 2: We agree. We revised the model-comparison interpretation and Discussion to provide a more balanced statement. The revised manuscript now explains that Stackelberg structures remain appropriate when leadership is externally determined, one chain has a durable first-mover advantage, a dominant platform controls the system, or policy design assigns a guiding role to a dominant participant. We also clarified that the present Nash bi-level structure is not claimed to be universally superior, but is more suitable for the homogeneous parallel rivalry considered in this study. These changes can be found in the model-comparison subsection and Discussion.
Comments 3: The manuscript should discuss whether the public-data benchmark instances represent real industrial battery supply chains and whether future industrial validation is needed.
Response 3: Thank you for this important comment. We clarified that the benchmark cases are public-data-adapted computational benchmarks and structurally representative for the modelling task, but they are not claimed to be confidential firm-level industrial replicas. We also noted that validation with firm-level industrial data remains an important direction for future research. These revisions can be found in the case-generation subsection, Discussion, Conclusions, and Data Availability statement.
Comments 4: The manuscript should strengthen the managerial implications for manufacturers and policymakers.
Response 4: We agree. We strengthened the managerial implications in the Discussion. The revised manuscript emphasizes that battery manufacturers should coordinate resource bidding, price setting, recycling effort, and logistics choice rather than optimizing a single strategic variable in isolation. It also suggests that policymakers should assess subsidy and demand-side instruments together with market structure and operational constraints, because their effects may be absorbed or redirected by competitive responses. These changes can be found in the Discussion section.
Comments 5: The manuscript requires further English proofreading.
Response 5: We agree and have completed a final proofreading and concision pass after the technical revisions. We polished the manuscript for clarity and concision, especially in the Introduction, Methods interpretation, Results interpretation, Discussion, and Conclusions. Revised passages are marked in red in the revised manuscript.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

<Overall Evaluation>

This paper is a study that combines Nash games and bi-level optimization to simultaneously consider strategic decision-making and operational planning in a competitive electric vehicle battery Closed-Loop Supply Chain (CLSC) environment. The research topic itself aligns with the scope of Logistics, and the expression and structure of the manuscript have been significantly improved since the first review. However, several key requirements explicitly requested in the first Major Revision have still not been fully addressed. In particular, regarding matters requiring additional computation, experimentation, and reproducibility, it is observed that the response was to supplement explanations or to tone down existing arguments rather than adding actual analyses. The most obvious example is the sensitivity analysis. While the first review explicitly requested additional analysis of key parameters such as subsidy budget, carbon penalty, recycling efficiency, and price elasticity, the revised manuscript still analyzes only three factors: expected market demand, scarce-material supply, and low-carbon transport cost coefficient. Furthermore, criticisms regarding the computational fairness of algorithm comparison, benchmark parameter calibration, and reproducibility were also only partially reflected.

Therefore, it is considered that another ‘Major Revision’ is required. This revision must not replace the substantive concerns of the previous review merely through descriptive supplementation or the mitigation of claims. The issues pointed out again below must be addressed through actual analysis, verification, and the presentation of data.

 

<Major Concerns>

(1) Following the first review, the authors explicitly acknowledged that the feasible set of each supply chain depends on the competitor's strategy due to shared demand, scarce materials, subsidized resources, and recyclable batteries. However, Section 3.4.2 still defines the upper level as a general complete-information static Nash game and uses a strategy space of the form, and Section 3.5 also defines the criterion equilibrium of the continuous game in the form of a Pareto-Nash equilibrium. On the other hand, Section 3.3 specifies a shared-constraint structure in which the feasible set varies depending on the rival strategy. Therefore, the authors need to clarify more rigorously whether the current model is a general Nash game, a model with the characteristics of a Generalized Nash Equilibrium Problem (GNEP), or a GNE-like shared-constraint structure treated as a finite-candidate approximation. It is not sufficient to merely mention the existence of GNE characteristics; the relationship between the strategy/feasible correspondence and the equilibrium concept actually calculated in this study must be explained mathematically and consistently.

(2) In response to the first criticism, the authors compared ±5% and ±10% unilateral-deviation radii, which is an improvement over the previous study. However, for equilibrium screening, a fixed threshold of Saf = 0.80 is still used. The core requirement of the first review was to verify not only the deviation radius but also the rationale for the threshold selection itself and the robustness of the results. Therefore, it is necessary to additionally present the extent to which the size of the equilibrium set, representative solution, profit/sustainability performance, and key conclusions are maintained when different Saf or safety cutoffs are applied. In the current state, relying on a single arbitrary cutoff, it is difficult to consider this as sufficient verification of the stability of the approximate equilibrium.

(3) In the revised manuscript, the authors acknowledged that the same number of generations does not imply the same number of candidate evaluations or lower-level solver calls, and softened their claim regarding algorithmic superiority. However, this did not perform the computational-effort comparison required in the first review. In Table 6, the computation times for GT-DP-NSGA-II were 316.25, 1449.32, and 7999.16 seconds for small, medium, and large instances, respectively, whereas for Joint-NSGA-II they were 2.83, 102.73, and 629.06 seconds, and for NGA they were 44.51, 343.88, and 698.28 seconds. In other words, the difference in computational cost is very large, especially in medium and large instances.

Therefore, at a minimum, candidate/function evaluation counts, lower-level CPLEX solver calls, and wall-clock time should be reported together to transparently compare the actual computational effort. If it is difficult to apply an identical evaluation budget due to the algorithmic structure, the reason must be clearly explained, and a normalized computational-effort comparison must be presented to the extent possible. Simply softening the wording to "this is not a superiority test of the same computational budget" does not resolve the previous reviewer concerns.

(4) Although Section 4.1 explains that the benchmark is based on public data and public case information and that "limited and traceable adjustments" were performed, it is difficult for readers to sufficiently trace which parameters were derived from which public sources and what transformation or calibration processes were undergone. Table 4 also primarily presents only the structural dimensions of the benchmark. Furthermore, while the Data Availability Statement states that configuration files and representative outputs may be provided through future supplementary material or a repository, "can be made available" is not equivalent to guaranteeing the reproducibility of the current research. Authors must specify the source, raw data values, transformation and calibration rules, and final values ​​for each major parameter in a table or supplementary material, and provide the benchmark configuration and major computational settings in a form that can be actually verified.

(5) This issue is one of the least clearly resolved matters in this revision. In the first review, additional sensitivity analysis was requested for key parameters that directly affect the economic and environmental mechanisms of the model, such as subsidy budget, carbon penalty, recycling efficiency, and price elasticity, in addition to the three existing factors. However, the authors' response explains that they merely modified the interpretation of existing sensitivity results more carefully rather than conducting additional analysis, and Section 4.4 actually analyzes only the three existing factors: expected market demand, scarce-material supply, and low-carbon transport cost coefficient. This cannot be considered a sufficient response to the previous reviewer's request. In this revision, appropriate additional factors must be selected from among the model's core behavioral/policy parameters to conduct actual sensitivity experiments. While it is not necessary to analyze all possible parameters, at least key parameters representing subsidy, environmental penalty/recycling mechanism, and demand elasticity must be included to verify that the main conclusions do not rely excessively on a specific calibration.

(6) It is appropriate for the authors to mitigate the claim of universal superiority of the Nash model and to reinterpret the centralized model as a coordinated efficiency benchmark and the Stackelberg model as a structure suitable for asymmetric/sequential competition. Additionally, the comparison of efficiency, balance, local deviation, role sensitivity, and participation gap in Tables 7–8 is also positive. However, these results are computational evidence derived from a single stylized homogeneous-duopoly benchmark setting. Therefore, care must be taken not to interpret the conclusion that “the Nash bi-level model is most consistent” or “more suitable representation” as a verification of the actual competitive structure of the battery industry itself. This study demonstrates that the Nash formulation is internally more suitable under assumed homogeneous parallel competition conditions, not as an empirical validation that the actual competitive structure of the industry is Nash. I hope that this boundary is maintained more clearly in the Abstract, Discussion, and Conclusion.

(7) Currently, in various comparisons and sensitivity analyses, representative equilibria are selected using ideal-point closeness, and results are reported based on this. While this approach is useful for summarizing results in situations where multiple approximate equilibria exist, there is a possibility that the representative-solution selection rule itself may influence the conclusion. This issue is particularly significant considering that GT-DP-NSGA-II generates a considerable number of approximate equilibria in the medium case.

Therefore, please ensure that the results of key model/algorithm comparisons do not depend on specific ideal-point selection rules by utilizing alternative representative-selection criteria or equilibrium-set level statistics, at the very least, by providing a robustness check for the core conclusions.

(8) In some sections of the current Response Letter, despite the reviewer requesting additional robustness tests or sensitivity analyses, the response presents modifications to interpretations or wording without actually performing the analysis. The response to Comment 9 is a prime example of this. In the next revision, for each reviewer comment, please provide a response that clearly distinguishes between (i) the analysis actually performed, (ii) the changes made in the manuscript, and (iii) the reasons for any requirements not fulfilled. If unperformed analysis is replaced with descriptive revisions, you must explicitly justify why such replacement is academically sufficient. This is not merely a matter of response-letter formatting, but a critical factor in determining the substantive compliance of this revision.

 

<Minor Concerns>

(1) While competitive supply chain and game-theoretic literature have been added to the current bibliography, if shared constraints and strategy-dependent feasibility are the core theoretical issues of this study, it is advisable to systematically review the Generalized Nash Equilibrium and shared-constraint game literature that directly corresponds to these issues.

(2) Although the qualitative classification in Table 1 has become clearer than before, there is a need to further improve the reproducibility of the evaluation. The authors' explanation that Weak, Medium, and Strong are distinguished based on the explicitness of operational execution is an improvement. However, since the judgment classifying some studies into the relevant categories still relies significantly on the author's interpretation, please ensure that the operational definition of each category is applied more clearly and consistently.

(3) Figure 1 is intended to explain the upper-level strategic interaction and lower-level operational feedback, which are the core of this study. However, the expression "Stackelberg game" appears on both sides of the figure, which may cause confusion for readers encountering the Nash bi-level structure of this study for the first time. Please clearly distinguish whether the figure explains the model itself or is a conceptual diagram that includes alternative structures, and simplify the figure if necessary.

(4) For example, the connection of equation numbering between lower-level constraints and alternative model formulations is somewhat complex, and identical or similar performance/safety measures are redefined in Methods and Results. In the final manuscript, please review the notation and equation numbering comprehensively to reduce unnecessary duplication.

(5) The authors appropriately relaxed the exact claim regarding upper-level equilibrium, but continued to use the expression “exact solution” for lower-level equilibrium. This clarifies that CPLEX optimally solves a defined lower-level mathematical program under given conditions, and consistency in terminology should be maintained to avoid confusion with the exact solution of the entire bi-level/game problem.

(6) In the results of Table 6, it is desirable to visually distinguish between computational quality and computational efficiency. Currently, Figure 3 highlights various quality indicators, but the significant runtime differences observed in Table 6 are important for evaluating the practical applicability of the study. In the Discussion section, it is hoped that the trade-off between improving solution quality and computational burden will be discussed more explicitly.

(7) While the addition of implications for managers and policymakers in the Discussion is positive, they remain at a relatively general level. For example, the practical contribution would become clearer if it were explained in conjunction with numerical results under what conditions procurement bidding, recycling effort, transport choice, or subsidy design need to be coordinated.

(8) Currently, the limitations section adequately mentions static complete-information setting, homogeneous duopoly, and finite-candidate approximate equilibrium. It is necessary to clearly include the limitation that computational cases are not empirical cases that directly reproduce actual observational data from individual companies, but rather public-data-adapted benchmark instances.

(9) Although readability has improved compared to the previous version, there are still parts in Methods and Results where definitions, interpretations, and justifications are repeated. In particular, the communicative power of the manuscript will be improved if redundant explanations throughout the manuscript are compressed while maintaining clarity in equilibrium terminology and algorithm descriptions.

Author Response

### Major Concern 1: Model type and equilibrium concept
**Reviewer comment.** The manuscript should clarify whether the upper-level interaction is a standard Nash game, a generalized Nash game, or a finite-candidate shared-constraint approximation.
**Response.** We agree. We revised the Abstract, Introduction, model assumptions, upper-level model, and solution methodology to make the distinction explicit.
The model is now consistently described as a Nash-type bi-level game with shared-constraint, GNE-like feasibility. The feasible set of each chain depends on the rival strategy through shared demand, scarce materials, subsidies, and recyclable batteries. The reported equilibria are finite-candidate approximate equilibria screened by objective quality and local unilateral-deviation pressure. The manuscript does not claim an analytical solution of a continuous-space GNEP.
**Revision.** We introduced the strategy-dependent feasible correspondence $\mathcal{F}_i(s_{-i})$, clarified that the implemented game has a shared-constraint structure, and cited direct generalized Nash equilibrium, shared-constraint game, and variational-inequality literature.
### Major Concern 2: Safety threshold robustness
**Reviewer comment.** The manuscript should show how equilibrium-set size, representative solutions, and key indicators respond to different Safety cutoffs.
**Response.** We performed a post hoc robustness check on the archived medium-scale GT-DP-NSGA-II equilibrium sets under ten random seeds. The current working outputs matched the Repository archive file by file.
Supplementary files:
- Supplementary Material S1.
The mean results are shown below.
| Safety cutoff | Non-empty seeds | Mean retained equilibria | Mean representative profit (CNY) | Mean representative sustainability | Mean ideal closeness | Mean safety score | |---:|---:|---:|---:|---:|---:|---:| | 0.70 | 10 | 59.8 | 3,076,461.65 | 2.4585 | 0.9921 | 0.8732 | | 0.75 | 9 | 55.9 | 3,080,437.63 | 2.4587 | 0.9948 | 0.8889 | | 0.80 | 9 | 46.5 | 3,079,768.47 | 2.4588 | 0.9946 | 0.8920 | | 0.85 | 9 | 38.2 | 3,076,474.07 | 2.4579 | 0.9940 | 0.9127 | | 0.90 | 8 | 23.2 | 3,049,485.64 | 2.4586 | 0.9904 | 0.9349 |
As the cutoff increases, the retained equilibrium count decreases. Representative profit, sustainability, and ideal-point closeness remain stable. Therefore, $\underline{\mathrm{Saf}}=0.80$ is retained as a reporting threshold for the present benchmark rather than asserted to be theoretically optimal or uniquely justified.
**Revision.** We added the robustness table and the reporting-threshold statement in Section 4.2 (Algorithm validation).
### Major Concern 3: Computational workload comparison
**Reviewer comment.** The manuscript should report candidate/function evaluations, lower-level CPLEX calls, and wall-clock time, or explain why a common evaluation budget is not possible.
**Response.** We added solver counters without changing the algorithm logic. The medium-scale audit covers GT-DP-NSGA-II, Joint-NSGA-II, and NGA.
The revised code modules are included in the Repository archive: `solver_stats.py`, `lower_milp.py`, `benchmark_public_cases_enhanced.py`, `benchmark_joint_nsga2.py`, and `benchmark_nga.py`.
Audit outputs are provided as Supplementary Material S4.
For the ten-seed medium-scale audit, candidate evaluations are actual unique upper-level strategy-pair evaluations. For GT-DP-NSGA-II and Joint-NSGA-II, lower response/function evaluations are conditional lower-level CPLEX subproblem solves. For NGA, lower response/function evaluations are heuristic lower-level plan evaluations.
| Algorithm | Seeds | Mean wall-clock time (s) | Mean upper candidate / strategy-pair evals | Mean lower response / function evals | Mean CPLEX calls | Mean equilibria | |---:|---:|---:|---:|---:|---:|---:| | GT-DP-NSGA-II | 10 | 626.56 $\pm$ 75.84 | 18,684.30 $\pm$ 255.47 | 38,268.60 $\pm$ 510.95 | 38,268.60 $\pm$ 510.95 | 120.0 | | Joint-NSGA-II | 10 | 41.16 $\pm$ 5.30 | 1,203.70 $\pm$ 59.47 | 2,727.40 $\pm$ 323.56 | 2,727.40 $\pm$ 323.56 | 4.0 | | NGA | 10 | 42.56 $\pm$ 10.40 | 1,183.70 $\pm$ 37.29 | 770,316.00 $\pm$ 194,600.53 | 0 | 7.8 |
NGA uses a heuristic lower-level response search and therefore has zero CPLEX calls in this implementation. Its lower response/function evaluations are obtained from the response-search logs.
The historical manuscript wall-clock times were collected in an earlier VS Code `launch.json` environment, whereas this audit was run in a PowerShell batch environment. The two batches use the same case configuration, algorithms, and seeds, but differ in launch method, runtime environment, and system load. Wall-clock time is therefore not directly comparable across batches. We did not replace the historical manuscript times with the new audit times. Within the audit, wall-clock time is used for a batch-specific relative comparison; CPLEX calls and evaluation counts are the more stable workload indicators.
**Revision.** The manuscript reports wall-clock time, upper candidate evaluations, lower response/function evaluations, CPLEX calls, and normalized ratios in the computational-effort audit. It explicitly states that the comparison is a transparent workload-and-quality comparison rather than an equal-evaluation-budget dominance test.
### Major Concern 4: Benchmark provenance and reproducibility
**Reviewer comment.** The manuscript should clearly identify parameter sources, transformations, calibration rules, and final benchmark values.
**Response.** We added a case-level overview as Table 5 (`Benchmark parameter provenance and calibration overview`). The parameter-level documentation is Supplementary Material S3 (`parameter_provenance_chengdu_medium.md`).
The Repository archive (`submission_data_package_reproduction.zip`) contains the formal benchmark case configuration `chengdu_medium.yaml`, algorithm-specific configurations under `baselines/`, the seed protocol, and representative outputs. The manuscript also distinguishes case YAML, algorithm-specific YAML, validation YAML, and behaviour-validation YAML.
The response and manuscript do not rely on author-local filesystem paths. Final portal locations should be determined by the submission system's supplementary-material and code/reproduction fields.
### Major Concern 5: Sensitivity analysis
**Reviewer comment.** The revision should include substantive sensitivity experiments for key policy, environmental, recycling, and demand-elasticity parameters.
**Response.** We completed an eight-factor single-factor sensitivity analysis on the Chengdu medium instance using five equally spaced levels and seeds 5, 37, and 70. The factors are:
- expected market demand; - scarce-material supply; - government subsidy; - carbon-emission standard; - consumer green preference; - low-carbon transport cost coefficient; - recovery yield factor; - demand elasticity $\beta$.
Demand and scarce-material supply have the clearest directional effects on profit and material structure. Subsidy, carbon standard, green preference, and demand elasticity mainly affect profit allocation, market share, equilibrium multiplicity, and evaluation structure, while realised emission response remains limited. Low-carbon transport cost and recovery yield have weaker, mostly structural effects.
For demand elasticity, $\beta$ is varied from 0.6 to 1.4 and evaluated through total profit, market-share gap, and unique equilibria. The response is non-monotonic for profit and market allocation, while realised emission response remains limited. The `scenario_sample_count=1` setting remains a computational control for scenario noise and does not conflict with including $\beta$ as a formal sensitivity factor.
### Major Concern 6: Scope of model-comparison conclusions
**Reviewer comment.** The results should not be interpreted as empirical evidence of the real battery-industry competition structure.
**Response.** We revised the Abstract, Section 4.3 (Model comparison), Section 5 (Discussion), Section 6 (Conclusions), and Data Availability to state that the conclusions apply to the stylised homogeneous parallel-competition benchmark. The centralized model is interpreted as a coordination-efficiency upper bound, Stackelberg as a reference for asymmetric or sequential competition, and Nash-type bi-level as suitable under the assumed homogeneous parallel-competition setting.
### Major Concern 7: Representative-selection-rule robustness
**Reviewer comment.** The manuscript should show whether conclusions depend on the ideal-point closeness selection rule.
**Response.** We performed a post-processing robustness check on the archived medium-scale GT-DP-NSGA-II equilibrium sets without rerunning the algorithms. In addition to `ideal_closeness`, we compared `safety_score`, `fairness_score`, `total_profit`, and `nash_welfare`.
Supplementary files:
- Supplementary Material S2.
| Selection criterion | Representative profit (10^4 CNY) | Representative sustainability | Mean ideal closeness | Mean safety score | Mean fairness score | |---|---:|---:|---:|---:|---:| | Ideal closeness | 307.58 | 2.4584 | 0.9938 | 0.8389 | 0.8759 | | Safety score | 303.19 | 2.4569 | 0.9739 | 0.9302 | 0.8628 | | Fairness score | 301.71 | 2.4570 | 0.9679 | 0.7951 | 0.9061 | | Total profit | 313.57 | 2.4586 | 0.9758 | 0.7548 | 0.8658 | | Nash welfare | 313.30 | 2.4575 | 0.9728 | 0.7705 | 0.8719 |
The representative sustainability indicator remains in the narrow interval 2.4569--2.4586. Therefore, the main environmental-performance conclusion is stable. Representative profit, ideal-point closeness, Safety, and fairness nevertheless vary with the selection criterion, so these local economic, stability, and fairness indicators remain selection-rule dependent.
**Revision.** We added the representative-selection robustness table and the limited-scope conclusion in Section 4.2 (Algorithm validation).
### Major Concern 8: Distinguishing completed analyses from planned revisions
**Reviewer comment.** Each point should distinguish completed analyses, manuscript revisions, and unfinished items.
**Response.** This response follows that structure. Completed analyses are linked to output files and results; manuscript changes are identified by location; unfinished submission-level items are not described as completed.
## Minor Concerns
### Minor Concern 1: Generalized Nash and shared-constraint literature
We added direct literature on generalized Nash equilibrium problems, shared-constraint games, and variational inequalities. The manuscript clearly states that the computed target is a finite-candidate approximate equilibrium rather than a full continuous-space GNEP solution.
### Minor Concern 2: Operational definitions in Table 1
We clarified the coding rules for `Weak`, `Medium`, and `Strong` before Table 1 and in the table note. The labels apply only to the strategic--operational coupling column.
- `Weak`: operational execution is missing, only qualitatively discussed, or reduced to an aggregate parameter. - `Medium`: operational factors enter the model or experiment, but are not solved as an explicit lower-level response coupled to strategic decisions. - `Strong`: operational decisions are explicitly optimized and feed back into strategic objectives, feasibility, or equilibrium evaluation.
### Minor Concern 3: Figure 1 labels
We replaced the potentially confusing `Stackelberg game` labels. Figure 1 now distinguishes the upper-level `Duopoly Nash Game` between chains from the bidirectional `bi-level strategic--operational coupling` inside each chain. The caption is `Conceptual structure of the Nash-type bi-level strategic--operational coupling`.
### Minor Concern 4: Symbols and equation numbering
We removed duplicate definitions and aligned equation numbering and notation. In particular, $\pi_i$ and $G_i$ denote chain profit and sustainability; $F^{\pi}$ and $F^{G}$ denote balance indicators; $F$ denotes average fairness. Safety is defined once in Methods, and the model-comparison deviation gains use $\Delta_i^{\pi,\mathrm{loc}}$ and $\Delta_i^{G,\mathrm{loc}}$.
### Minor Concern 5: Exact-solution terminology
We removed `exact solution`, `exact lower-level response`, and `exact lower-level optimisation` when describing the full bi-level solution. The manuscript now uses:
- `conditional lower-level optimisation`; - `conditional lower-level response`; - `conditional lower-level post-evaluation`.
CPLEX optimality applies only to a fixed upper-level strategy and fixed $\varepsilon$ lower-level subproblem. Exact enumeration is retained only for the discretized small-scale validation grid.
### Minor Concern 6: Quality versus computational burden
We added the computational-effort audit and the quality--burden explanation in Section 4.2 (Algorithm validation) and Section 5 (Discussion). For the medium-scale case, GT-DP-NSGA-II required 626.56 s and 38,268.6 CPLEX calls per seed on average, compared with 41.16 s and 2,727.4 calls for Joint-NSGA-II. The wall-clock times are from the same PowerShell audit batch; the historical manuscript times were collected in an earlier VS Code environment, so absolute times are not directly compared across batches. CPLEX calls and evaluation counts are reported as more stable workload indicators.
### Minor Concern 7: Managerial implications
We added conditional managerial implications tied to numerical observations. For example, when demand expansion or scarce-material tightness is the main pressure, coordination should prioritise procurement bidding and material allocation. When the policy goal is realised emission reduction, subsidies and preference incentives should be paired with a wider set of low-carbon production, recycling, and transport options.
The statement that centralized coordination is implementable only when the participation loss of the disadvantaged chain can be compensated follows from the participation-gap and unilateral-deviation analysis rather than from firm-level empirical observation.
### Minor Concern 8: Public-data-adapted benchmark status
We revised Section 4.1 (Case generation and parameter setting), Section 6 (Conclusions), and Data Availability to state that the cases are public-information-adapted benchmark instances, not direct replications of firm-level observations or industry-wide empirical data.
### Minor Concern 9: Language and redundancy
We completed a final condensation pass over the Abstract, Section 3 (Materials and Methods), Section 4.2 (Algorithm validation), Section 4.3 (Model comparison), Section 4.4 (Sensitivity analysis), Section 5 (Discussion), and Section 6 (Conclusions). Definitions and explanations are concentrated at their first occurrence, while model boundaries, algorithm flow, core evidence, and applicability conditions are retained.

Author Response File: Author Response.pdf

Round 3

Reviewer 2 Report

Comments and Suggestions for Authors

<Overall Evaluation>

This manuscript has undergone substantial improvements following two previous major revisions. In particular, regarding the key methodological concerns raised in the previous rounds, the authors did not stop at simple descriptive clarifications or wording corrections; instead, they conducted additional computational experiments and robustness verifications. As a result, we assess that the major concerns raised in the previous rounds have been largely resolved. Therefore, we do not believe that the current manuscript still contains issues severe enough to require a new large-scale model redesign or a complete re-run of key experiments. However, there are still several important matters that must be clearly resolved before final publication. Accordingly, I recommend a (Strong) Minor Revision in this round.

 

<Major Concerns>

(1) The significant improvement in reproducibility in this revision is positive. However, the wording in the text is still inconsistent. Section 4.1 refers to the parameter-level provenance table as a “supplementary-material draft,” and the Data Availability Statement states that the replication package may be provided as supplementary material or through a public repository after de-anonymization. In the final manuscript, the exact status and availability of S1–S4 and the reproduction archive must be clearly described, and linked to actual supplementary files or a permanent repository to the extent permitted by the editorial department. In particular, since many of the robustness and reproducibility claims of this study depend on these materials, it should not remain at the level of simply being “available upon request” or scheduled for future provision.

(2) The computational-effort audit in Table 8 is a significant improvement over the previous version. However, while GT-DP-NSGA-II requires an average of 38,268.6 CPLEX calls in the medium-scale case, Joint-NSGA-II requires only 2,727.4 calls, and the number of candidate evaluations also differs significantly. Therefore, the solution-quality differences observed in Table 7 and Figures 3–4 should not be interpreted as algorithmic superiority achieved within the same computational budget. The authors must consistently interpret the algorithm comparison from the perspective of a trade-off between solution quality and computational burden.

(3) The newly added eight-factor sensitivity analysis addresses a significant shortcoming of the previous version. However, while the main algorithm comparison uses 10 random seeds, the sensitivity analysis uses only three seeds: 5, 37, and 70. At this stage, it is not required that the entire sensitivity experiment be re-run with 10 seeds. However, it should be explained what criteria were used to select the three seeds, and it should be specified that due to this limited replication, the sensitivity results should be interpreted primarily as directional and benchmark-specific evidence. In particular, it is necessary to acknowledge in the Discussion or Limitations that it is difficult to completely separate stochastic search variation from genuine parameter sensitivity for some factors where weak or non-monotonic effects have been reported.

(4) This revision has largely resolved previous concerns by specifying that the Nash-type structure does not empirically identify the competitive structure of the actual power-battery industry. In particular, the distinction made in Section 4.3 regarding the roles of centralized coordination, Stackelberg structure, and Nash-type competition is appropriate. However, it must be more clearly recognized that the conclusion “the Nash-type bi-level model is most consistent with the present homogeneous-duopoly setting” is largely connected to the homogeneous and simultaneous competition assumptions pre-set in the model. In other words, the authors should strictly limit the scope of the model comparison results to within the stylized benchmark.

(5) The manuscript is now much more accurate than before as it clearly distinguishes between the continuous-space GNE and the actual computed finite-candidate approximate equilibrium. However, expressions such as Nash-type, GNE-like, Pareto-Nash-type, approximate equilibrium, and equilibrium-screened solution are used in parallel across several sections, potentially confusing the reader's conceptual status. It is recommended that (a) the underlying conceptual game, (b) the continuous benchmark equilibrium concept, and (c) the operational computational target produced by the actual algorithm be explicitly defined once at the beginning of the Methods section, and that the same terminology be used consistently thereafter.

 

<Minor Concerns>

(1) Table 9 shows that the representative profit and sustainability are quite stable, but the number of non-empty seeds decreases from 10 to 8 as the cutoff increases. Therefore, the authors request that the interpretation of the Safety cutoff robustness results be expressed a little more precisely.

(2) Table 10 shows the significant result that while sustainability results are very stable, profit, safety, and fairness vary depending on the selection rule. Since this has significance beyond a simple technical robustness check, it is advisable to clearly mention, in the Conclusions or Limitations, that some economic and distributive interpretations based on the representative solution are selection-rule dependent.

(3) NGA's lower response/function evaluations are heuristic evaluations, whereas the other two methods refer to CPLEX subproblem solves. Therefore, the number 770,316 should not be directly compared to the lower-level evaluation counts of other algorithms as if they were the same computational unit. The authors should provide a clearer explanation regarding the comparability between the workload measures in Table 8.

(4) Brief explanations should be added to some of the counter-intuitive sensitivity results in Table 13. For example, the result that profit decreases as scarce-material supply is alleviated can be interpreted contrary to general intuition. The interpretability of the sensitivity analysis would be significantly improved if a brief explanation were provided regarding which mechanism—such as endogenous allocation, bidding behavior, native/recycled material substitution, or representative-equilibrium switching—primarily generates these results.

(5) While the manuscript defines (G_i) as the sustainability-performance level, the term “greenness” is used in some results and figures. If both expressions refer to the exact same quantity, it is advisable to unify them under a single key term or to specify the relationship when first using them.

(6) Although the managerial implications in Section 5 are much more specific than before, this study is not an empirical policy evaluation using actual corporate data. Therefore, it is appropriate to consistently use conditional expressions such as “the benchmark results suggest,” “under the modeled conditions,” and “may be appropriate” rather than strong prescriptive expressions such as “should.”

(7) The authors are requested to improve the readability of Figures 7–8 at the final publication stage.

(8) A significant condensation has been achieved in this revision. Nevertheless, the safety definition, the range of finite-candidate equilibrium, the meaning of conditional lower-level optimization, and benchmark limitations are repeated in several places. While maintaining the core methodological caveat, the flow and readability of the manuscript will be improved if the parts where the same explanations are repeated are compressed slightly in the Methods and Results sections.

(9) Authors are requested to finally verify and check the notation, equation numbering, and typographical consistency.

Author Response

Comments 1: The significant improvement in reproducibility in this revision is positive. However,
the wording in the text is still inconsistent. Section 4.1 refers to the parameter-level provenance
table as a “supplementary-material draft,” and the Data Availability Statement states that the
replication package may be provided as supplementary material or through a public repository
after de-anonymization. In the final manuscript, the exact status and availability of S1–S4 and
the reproduction archive must be clearly described, and linked to actual supplementary files or
a permanent repository to the extent permitted by the editorial department. In particular, since
many of the robustness and reproducibility claims of this study depend on these materials, it should
not remain at the level of simply being “available upon request” or scheduled for future provision.
Response 1: Thank you for pointing this out. We agree with this comment. We have revised
Section 4.1 and the Data Availability Statement to remove the ambiguous wording and to distin-
guish clearly between the submission-package supplementary materials and the public reproduction
archive. The revision can be found in the revised manuscript on page 14, paragraph beginning “A
parameter-level provenance...”, lines 487–493, and on page 25, Data Availability Statement, lines
760–765. We also clarified the file-level location of S1–S4 in the response materials: S1–S4 are lo-
cated in the final submission package under 04 supplementary materials/, not inside the Figshare
reproduction zip. The file-level mapping is as follows:
ˆ S1: parameter_provenance_chengdu_medium.md.
ˆ S2: existing_quality_metric_seed_audit.csv; existing_quality_metric_summary.csv.
ˆ S3: safety_cutoff_seed_level.csv; safety_cutoff_summary.csv; representative_selec
tion_seed_level.csv; representative_selection_summary.csv.
ˆ S4: computational_effort_summary.csv; medium_solver_counter_seed_level.csv;
medium_solver_counter_summary.csv; medium_solver_counter_summary.md.
The public reproduction archive is available through the permanent Figshare DOI: https://doi.
org/10.6084/m9.figshare.32054619.
Revised manuscript text: “A parameter-level provenance and calibration table is provided as Sup-
plementary Material S1... Supplementary Materials S2–S4 provide quality-metric audits, robust-
ness checks, and the computational-effort audit, respectively. These supplementary materials are
provided separately from the reproduction archive; the archived reproduction package, which con-
tains the code, configurations, data, figures, and archived outputs used for replication, is publicly
available at https://doi.org/10.6084/m9.figshare.32054619.”

Comments 2: The computational-effort audit in Table 8 is a significant improvement over the
previous version. However, while GT-DP-NSGA-II requires an average of 38,268.6 CPLEX calls
in the medium-scale case, Joint-NSGA-II requires only 2,727.4 calls, and the number of candidate
1
evaluations also differs significantly. Therefore, the solution-quality differences observed in Table
7 and Figures 3–4 should not be interpreted as algorithmic superiority achieved within the same
computational budget. The authors must consistently interpret the algorithm comparison from the
perspective of a trade-off between solution quality and computational burden.
Response 2: Thank you for this important clarification. We agree that the algorithm comparison
should not be interpreted as an equal-budget dominance test. We have revised the algorithm-
comparison discussion to emphasize the trade-off between solution quality and computational
burden. The changes appear on page 17, paragraph following Table 7, lines 565–568; page 18,
computational-effort audit discussion, lines 580–585; and page 19, validation summary, lines 603–
607.
Revised manuscript text: “These quality differences indicate a quality–workload trade-off, not
algorithmic superiority under an equal computational budget.”
Revised manuscript text: “The results should therefore be read as evidence on the trade-off between
solution quality and computational burden, not as an equal-budget dominance test.”

Comments 3: The newly added eight-factor sensitivity analysis addresses a significant shortcom-
ing of the previous version. However, while the main algorithm comparison uses 10 random seeds,
the sensitivity analysis uses only three seeds: 5, 37, and 70. At this stage, it is not required that
the entire sensitivity experiment be re-run with 10 seeds. However, it should be explained what
criteria were used to select the three seeds, and it should be specified that due to this limited repli-
cation, the sensitivity results should be interpreted primarily as directional and benchmark-specific
evidence. In particular, it is necessary to acknowledge in the Discussion or Limitations that it is
difficult to completely separate stochastic search variation from genuine parameter sensitivity for
some factors where weak or non-monotonic effects have been reported.
Response 3: Thank you for this helpful suggestion. We have added the seed-selection rationale at
the beginning of the sensitivity analysis and limited the interpretation of the results to directional,
benchmark-specific evidence. We also added a limitation that stochastic search variation cannot
be fully separated from genuine parameter sensitivity for weak or non-monotonic responses. These
revisions can be found on page 21, Section 4.4, paragraph beginning “We conduct an eight-factor...”,
lines 673–678, and in the Conclusions on page 25, lines 743–746.
Revised manuscript text: “The three seeds were selected from the formal ten-seed protocol to
cover representative low-, middle-, and high-index search trajectories while keeping the additional
computational burden manageable. Because this analysis uses three replications rather than the
ten seeds used in the main algorithm comparison, the results are interpreted as directional and
benchmark-specific evidence. For weak or non-monotonic responses, stochastic search variation
cannot be fully separated from genuine parameter sensitivity.”

Comments 4: This revision has largely resolved previous concerns by specifying that the Nash-
type structure does not empirically identify the competitive structure of the actual power-battery
industry. In particular, the distinction made in Section 4.3 regarding the roles of centralized coordi-
nation, Stackelberg structure, and Nash-type competition is appropriate. However, it must be more
clearly recognized that the conclusion “the Nash-type bi-level model is most consistent with the
present homogeneous-duopoly setting” is largely connected to the homogeneous and simultaneous
competition assumptions pre-set in the model. In other words, the authors should strictly limit the
scope of the model comparison results to within the stylized benchmark.

Response 4: Thank you for this precise comment. We agree and have tightened the scope of
the model-comparison conclusion. The revised manuscript now states that the model-comparison
result is limited to the stylised homogeneous-duopoly benchmark and should not be interpreted
as empirical identification of the actual competitive structure of the power-battery industry. The
changes appear in the Abstract on page 1, lines 2–13; in the model-comparison discussion on page
21, lines 666–669; and in the Discussion and Conclusions on pages 24–25, lines 722–746.
Revised manuscript text: “In the stylised homogeneous-duopoly benchmark, the Nash-type struc-
ture is more suitable than the selected Stackelberg and centralized-cooperative alternatives when
competitive stability and implementation conditions are considered jointly, although this result
does not identify the competitive structure of the real battery industry.”
Revised manuscript text: “Central coordination is implementable only when the participation gap
of the disadvantaged chain can be compensated, whereas a Nash-type structure is more appropriate
for structurally comparable competitors without a designated leader in the stylised benchmark.”

Comments 5: The manuscript is now much more accurate than before as it clearly distinguishes
between the continuous-space GNE and the actual computed finite-candidate approximate equi-
librium. However, expressions such as Nash-type, GNE-like, Pareto-Nash-type, approximate equi-
librium, and equilibrium-screened solution are used in parallel across several sections, potentially
confusing the reader’s conceptual status. It is recommended that (a) the underlying conceptual
game, (b) the continuous benchmark equilibrium concept, and (c) the operational computational
target produced by the actual algorithm be explicitly defined once at the beginning of the Methods
section, and that the same terminology be used consistently thereafter.
Response 5: Thank you for this suggestion. We have added a terminology paragraph at the
beginning of the Methods section and then revised the later algorithm, pseudocode, results, and
conclusion sections to use the same terminology. The revised manuscript distinguishes three levels:
the modelling framework, the continuous-space conceptual benchmark, and the operational com-
putational target. This change can be found on page 6, Section 3.1, paragraph beginning “Three
terminology levels...”, lines 217–225, and is reflected later in the solution methodology on pages
13–14, lines 447–493.
Revised manuscript text: “Three terminology levels are used in this study. First, the modelling
framework is a Nash-type bi-level model... Second, the continuous-space conceptual reference is a
shared-constraint generalized Nash equilibrium benchmark... Third, the actual computational tar-
get is a finite-candidate approximate equilibrium... Unless explicitly stated otherwise, all reported
equilibria in the computational sections refer to this finite-candidate approximate-equilibrium con-
cept rather than to an analytically solved continuous-space equilibrium.”

Comments 6: Table 9 shows that the representative profit and sustainability are quite stable,
but the number of non-empty seeds decreases from 10 to 8 as the cutoff increases. Therefore, the
authors request that the interpretation of the Safety cutoff robustness results be expressed a little
more precisely.
Response 6: Thank you for pointing this out. We have revised the explanation of Table 9 to state
explicitly that the stability conclusion is conditional on the retained non-empty seed set. This
revision appears on page 18, paragraph following Table 9, lines 587–594.
Revised manuscript text: “Among the seeds that still retain at least one feasible representative
equilibrium... the robustness statement is conditional on the retained non-empty seed set. There-
fore, Saf = 0.80 is retained as a reporting threshold for the present benchmark rather than asserted
to be theoretically optimal or uniquely justified.”

Comments 7: Table 10 shows the significant result that while sustainability results are very
stable, profit, safety, and fairness vary depending on the selection rule. Since this has significance
beyond a simple technical robustness check, it is advisable to clearly mention, in the Conclusions
or Limitations, that some economic and distributive interpretations based on the representative
solution are selection-rule dependent.
Response 7: Thank you for this useful suggestion. We have added this caveat both after Table 10
and in the Conclusions. The revised text appears on page 19, paragraph following Table 10, lines
599–603, and on page 25, Conclusions, lines 740–746.
Revised manuscript text: “The robustness checks also show that sustainability-performance con-
clusions are relatively stable across representative-selection rules, while some economic, stability,
and distributive interpretations based on a single representative solution remain selection-rule de-
pendent.”

Comments 8: NGA’s lower response/function evaluations are heuristic evaluations, whereas the
other two methods refer to CPLEX subproblem solves. Therefore, the number 770,316 should
not be directly compared to the lower-level evaluation counts of other algorithms as if they were
the same computational unit. The authors should provide a clearer explanation regarding the
comparability between the workload measures in Table 8.
Response 8: Thank you for this correction. We have clarified in the note to Table 8 and in
the paragraph following the table that NGA’s lower response/function evaluations are heuristic
evaluations, whereas GT-DP-NSGA-II and Joint-NSGA-II use CPLEX subproblem solves. There-
fore, Table 8 is used to describe workload structure rather than to make one-to-one comparisons of
identical computational units. This revision can be found on page 18, Table 8 note and following
paragraph, lines 575–585.
Revised manuscript text: “For GT-DP-NSGA-II and Joint-NSGA-II, lower response/function eval-
uations correspond to conditional lower-level CPLEX subproblem solves. For NGA, lower re-
sponse/function evaluations denote heuristic lower-level plan evaluations; therefore CPLEX calls
are zero and the CPLEX-call ratio is not applicable.”

Comments 9: Brief explanations should be added to some of the counter-intuitive sensitivity
results in Table 13. For example, the result that profit decreases as scarce-material supply is
alleviated can be interpreted contrary to general intuition. The interpretability of the sensitiv-
ity analysis would be significantly improved if a brief explanation were provided regarding which
mechanism–such as endogenous allocation, bidding behavior, native/recycled material substitution,
or representative-equilibrium switching–primarily generates these results.
Response 9: Thank you for this helpful recommendation. We have added a mechanism-based
explanation after Table 13. We now state that the counter-intuitive result should not be read as
a general economic law and may arise from endogenous resource allocation, bidding behaviour,
native/recycled material substitution, and representative-equilibrium switching in the benchmark.
The revision appears on page 22, paragraph following Table 13, lines 688–693.
Revised manuscript text: “The counter-intuitive profit decrease observed when scarce-material
supply is relaxed is not interpreted as a general economic law. In this benchmark, scarce-material
allocation is endogenous to bidding behaviour, recycled-material substitution, and representative-
equilibrium selection; relaxing one shared resource can therefore move the reported solution to a
different finite-candidate equilibrium with a different cost and allocation structure.”

Comments 10: While the manuscript defines (G i) as the sustainability-performance level, the
term “greenness” is used in some results and figures. If both expressions refer to the exact same
quantity, it is advisable to unify them under a single key term or to specify the relationship when
first using them.
Response 10: Thank you for pointing out this terminology issue. We have checked the manuscript
and unified the main result term as “sustainability performance” or “sustainability-performance
level”. The symbol Gi is retained as the realized sustainability-performance level. The term
“consumer green preference” is retained only where it denotes the consumer-preference parameter,
not the outcome indicator Gi. The relevant clarification appears in the symbol table on page 7,
lines 250–254, and in the model formulation on page 8, lines 272–274.
Revised manuscript text: “Gi: Realized sustainability-performance level of supply chain i.”

Comments 11: Although the managerial implications in Section 5 are much more specific than
before, this study is not an empirical policy evaluation using actual corporate data. Therefore,
it is appropriate to consistently use conditional expressions such as “the benchmark results sug-
gest,” “under the modeled conditions,” and “may be appropriate” rather than strong prescriptive
expressions such as “should.”
Response 11: Thank you for this comment. We agree that the managerial implications should be
framed conditionally because this study is based on public-information-adapted benchmarks rather
than firm-level empirical policy evaluation. We have revised the Discussion accordingly. The
changes appear on page 24, Discussion, paragraph beginning “The managerial implications...”,
lines 722–729.
Revised manuscript text: “Under the modeled conditions... it may be appropriate... the benchmark
results suggest... may need to be paired with...”

Comments 12: The authors are requested to improve the readability of Figures 7–8 at the final
publication stage.
Response 12: Thank you for the reminder. We checked the original figure resolutions and adjusted
the LaTeX figure width to improve readability in the manuscript. The figure captions also indicate
that the sensitivity plots are based on the three-seed directional check. The revised figures appear
as Figure 7 on page 23 and Figure 8 on page 24.
Revised manuscript text: “Figure 7. Sensitivity responses to market, resource, subsidy, and
emission-standard parameters under the three-seed directional check...”
Revised manuscript text: “Figure 8. Sensitivity responses to consumer preference, operational
parameters, and demand elasticity under the three-seed directional check...”

Comments 13: A significant condensation has been achieved in this revision. Nevertheless, the
safety definition, the range of finite-candidate equilibrium, the meaning of conditional lower-level
optimization, and benchmark limitations are repeated in several places. While maintaining the
core methodological caveat, the flow and readability of the manuscript will be improved if the parts
where the same explanations are repeated are compressed slightly in the Methods and Results
sections.
Response 13: Thank you for this suggestion. We compressed repeated explanations in the Meth-
ods and Results sections. The revised manuscript now defines the conceptual and computational
equilibrium terminology once at the beginning of Methods and then refers to it consistently in the
5
algorithm and results sections. The main changes appear on page 6, lines 217–225; page 13, lines
447–460; and pages 18–19, lines 587–607.
Revised manuscript text: “The same conditional lower-level optimisation workflow is used through-
out... Finite-candidate approximate-equilibrium extraction is performed on the final finite can-
didate set, while Safety and local unilateral-deviation measures are reported as post-processing
indicators.”
Comments 14: Authors are requested to finally verify and check the notation, equation number-
ing, and typographical consistency.

Response 14: Thank you for the reminder. We have checked notation, equation numbering, figure
and table references, capitalization, terminology, and typographical consistency. We recompiled
both the clean and marked PDF versions. The final clean manuscript and the marked manuscript
compile successfully without undefined references or missing citations. The relevant submitted
files are manuscript clean anonymous.pdf and manuscript changes marked red.pdf in the final
submission package.
Thank you again for the constructive comments. They helped us clarify the manuscript’s method-
ological scope, data availability, computational interpretation, and benchmark-specific conclusions.

Author Response File: Author Response.pdf