Multi-Size Facility Allocation Under Competition: A Model with Competitive Decay and Reinforcement Learning-Enhanced Genetic Algorithm

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper studied the Facility Location Problem using an enhanced genetic algorithm that it proposed.  the following is my comment:

1. Please state clearly your contribution in the introduction. is it that you proposed this simplified model for optimization, or that you brought this new genetic algorithm ? if the former, please explain in detail how this novel model is compared to the other models in the previous literature, and why you think you should use it. if the latter, please explain why this is novel (i.e., there is no such enhanced genetic algorithm in the literature?). 
2. In Figure 5, is the result obtained from a single or multiple runs (i.e., you start from a different initial point and do the iterations)? If it is the former, please consider doing multiple runs and comparing the average performances of the traditional and new methods. Also, the difference in the resulting objective function value for the traditional and new methods is like -675000 - (-677500). Is this a significant improvement, given your parameter?  I understand that you are using a utility function for the objective, which is a more abstract concept. However, it will be easier for the readers to know how economically significant this improvement is. 

2. A robustness check is required for your analysis. For example, the objective function has many parameters. How does the result change if some of the parameter values change? It will be an insightful analysis. Also, it will be more comprehensive if you add the procedure for estimating the parameters of your model in the appendix.

3. In the result analysis (from line 349), please consider adding more intuition on why the new method works. If possible, look deeper into the iterative process to see how the group evolution looks in the latest and old algorithms.

4.A minor point: for figures 5(c) and (d), ensure the y-axis range is the same.

Author Response

1. Please state clearly your contribution in the introduction. is it that you proposed this simplified model for optimization, or that you brought this new genetic algorithm ? if the former, please explain in detail how this novel model is compared to the other models in the previous literature, and why you think you should use it. if the latter, please explain why this is novel (i.e., there is no such enhanced genetic algorithm in the literature?). 

We sincerely thank the reviewer for this important comment regarding the clarity of our contributions. We have substantially revised the Introduction section to explicitly address both aspects of our contribution and provide detailed comparisons with existing literature.

 

2. In Figure 5, is the result obtained from a single or multiple runs (i.e., you start from a different initial point and do the iterations)? If it is the former, please consider doing multiple runs and comparing the average performances of the traditional and new methods. Also, the difference in the resulting objective function value for the traditional and new methods is like -675000 - (-677500). Is this a significant improvement, given your parameter?  I understand that you are using a utility function for the objective, which is a more abstract concept. However, it will be easier for the readers to know how economically significant this improvement is. 

The reviewer correctly identifies a critical aspect of our experimental methodology. Figure 5 (now Figure 5 in the revised manuscript) presents results from multiple independent runs rather than single executions. The reviewer raises an excellent point about interpreting the practical significance of our utility function improvements. Our manuscript states: "In terms of numerical fitness values, there is no significant distinction in performance between the two approaches." This is a crucial finding that we should clarify. Both algorithms achieve similar final fitness values, indicating that the RL-GA does not sacrifice solution quality for convergence speed. The primary advantage lies in convergence efficiency rather than superior final solutions.

 

3. A robustness check is required for your analysis. For example, the objective function has many parameters. How does the result change if some of the parameter values change? It will be an insightful analysis. Also, it will be more comprehensive if you add the procedure for estimating the parameters of your model in the appendix.

We sincerely thank the reviewer for this valuable suggestion regarding robustness analysis and parameter estimation procedures. Regarding Robustness Analysis and Parameter Sensitivity, we are pleased to inform the reviewer that our manuscript already includes comprehensive robustness analysis in Section 2.3.4 "Validation and Sensitivity Analysis."

 

4. In the result analysis (from line 349), please consider adding more intuition on why the new method works. If possible, look deeper into the iterative process to see how the group evolution looks in the latest and old algorithms.

Regarding Intuitive Explanations for RL-GA Effectiveness, our manuscript already provides substantial theoretical foundation for why the RL-GA method works. Regarding Deep Analysis of Evolutionary Process, our manuscript provides extensive analysis of the iterative process differences between traditional and RL-enhanced algorithms.

 

5. A minor point: for figures 5(c) and (d), ensure the y-axis range is the same.

We thank the reviewer for this observation. In Figure 5(c) and (d), both algorithms' fitness curves are already plotted on the same graphs with identical y-axis ranges, ensuring direct visual comparison between the traditional GA and RL-GA performance.

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript introduces a facility allocation framework enabling multiple scale of facility size. The allocation method is designed based on Huff model, considering the demand and competition within a catchment range along with budget constraints. An improved genetic algorithm is applied to search the best solution of the allocation objective. The implementation of the method is for bank allocation for better service coverage. The methodology of the work is well designed and illustrated. However, the innovation and validation of this work is not well explained. The conditions and constraints considered for allocation are too simple making the methodology not quite applicable. Some other issues should also be considered.  

1) The literature review is not thorough and the references are too outdated. Recent years, large number of advanced methods are applied in optimization strategy, e.g. reinforcement learning. The authors might need to read more recent works before the methodology design.

2) The design of the allocation objective function and constraints is a bit idealistic. It is indeed important to introduce the idea of competition of the service (which most of the advanced allocation methods already considered). Yet, more complicated factors should be taken into consideration, such as the land use type, the availability of the land, influence of different types of banks, etc... especially in metropolises like Beijing.

3) The authors applied designed enhanced genetic algorithm seems to improve the convergence speed of the algorithm according the comparison results. However, there are more advanced algorithms such as NSGA-II and reinforcement learning might achieve better efficiency.

4) The data and spatial unit are not well introduced. The authors claimed that “candidate facility locations are determined based on a grid system” (P7, L267). However, the size of the grid and how they separated or defined the cell is missing.

5) The allocation method is not validated. Only the optimization algorithm is compared with original GA. And, according to the algorithm comparison results, the recommended locations are completely different. Since the authors did not propose a proper validation of allocation result, it is hard to say which results meets the demands better.

The authors might need to redo the literature review and make the methodology more convincing for application. 

Author Response

(1)&(3) The literature review is not thorough and the references are too outdated. Recent years, large number of advanced methods are applied in optimization strategy, e.g. reinforcement learning. The authors might need to read more recent works before the methodology design.

The authors applied designed enhanced genetic algorithm seems to improve the convergence speed of the algorithm according the comparison results. However, there are more advanced algorithms such as NSGA-II and reinforcement learning might achieve better efficiency.

We appreciate the reviewer's emphasis on recent developments in reinforcement learning applications to optimization. Our study directly addresses this gap by proposing a reinforcement learning-enhanced genetic algorithm (RL-GA) that represents a novel contribution to the field.

 

(2) The design of the allocation objective function and constraints is a bit idealistic. It is indeed important to introduce the idea of competition of the service (which most of the advanced allocation methods already considered). Yet, more complicated factors should be taken into consideration, such as the land use type, the availability of the land, influence of different types of banks, etc... especially in metropolises like Beijing.

We fully agree with the reviewer that real-world facility location decisions, particularly in metropolises like Beijing, involve significantly more complex factors than our current model captures. As explicitly stated in our Limitations section: "The current study's scope in characterizing bank facility locations remains constrained by data availability across multiple dimensions. While our RL-GA model demonstrates adaptive capabilities, the underlying mathematical formulation still primarily focuses on competitive dynamics and basic operational costs."

Our study deliberately focuses on establishing the methodological foundation for reinforcement learning-enhanced optimization in facility location problems. The primary contribution lies in the RL-GA algorithmic framework rather than comprehensive real-world modeling. We acknowledge that "The reality of bank branch location decisions involves complex interactions among socioeconomic factors, demographic characteristics, accessibility considerations, market potential indicators, and infrastructure development patterns."

 

(3) The data and spatial unit are not well introduced. The authors claimed that “candidate facility locations are determined based on a grid system” (P7, L267). However, the size of the grid and how they separated or defined the cell is missing.

We thank the reviewer for pointing out the insufficient description of our spatial data structure. We have now added comprehensive details about the grid system implementation in the revised manuscript.

 

(4) The allocation method is not validated. Only the optimization algorithm is compared with original GA. And, according to the algorithm comparison results, the recommended locations are completely different. Since the authors did not propose a proper validation of allocation result, it is hard to say which results meets the demands better.

We thank the reviewer for this important observation. We acknowledge that our study focuses primarily on algorithmic performance comparison rather than validation of the allocation results themselves. While we demonstrate that RL-GA achieves faster convergence than traditional GA, the different spatial distributions produced by both algorithms indeed require additional validation to determine which better meets real-world banking requirements. Our contribution is primarily methodological - demonstrating the RL-GA framework for facility location optimization. The Beijing case study serves as a proof-of-concept for the algorithmic approach rather than validated location recommendations. We will revise the manuscript to clearly position our work as a methodological contribution and acknowledge this validation limitation in the discussion section.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Review of “Multi-Size Facility Allocation under Competition: A Model with Competitive Decay and Enhanced Genetic Algorithms”

 

This paper examines the possible optimization methods of multi-size bank location by proposing a new modeling approach incorporating competitive decay. The new model is well developed based on the close examination of pre-existing models. By comparing it to real-world data and the previous models, simulation results using the new model seem to fit the data better. Also in the discussion section the authors discuss the limitations of their approach and mention issues for future research.

 

I think this kind of numerical approach for decision making processes in general is versatile, and can be applied to other areas.  After some revisions mentioned below, I think the paper can be published.   

 

• The authors use two variables for the bank facilities: travel time and size. As the authors themselves state in the discussion, these are oversimplified variables. In addition, there is one more fundamental consideration which should be discussed. Nowadays, internet banking has been rapidly expanding and cashless transactions have prevailed. The real importance of in-person bank service, including ATM machines, has been dramatically diminished. The number of bank branches has sharply declined, and ATMs are often shared by partnerships among various banks. In view of this trend, the authors should clarify the importance of a study based on the physical location of bank facilities.
• This paper aims to inform the real demands of decision makers. The heavy computational load may be a serious bottle neck for actual implementation. If the authors could make any suggestions on how to reduce the computational burden and/or improve efficiency of the calculations this would contribute to further applications of the methods developed in this study.

 

Author Response

The authors use two variables for the bank facilities: travel time and size. As the authors themselves state in the discussion, these are oversimplified variables. In addition, there is one more fundamental consideration which should be discussed. Nowadays, internet banking has been rapidly expanding and cashless transactions have prevailed. The real importance of in-person bank service, including ATM machines, has been dramatically diminished. The number of bank branches has sharply declined, and ATMs are often shared by partnerships among various banks. In view of this trend, the authors should clarify the importance of a study based on the physical location of bank facilities.

The reviewer correctly identifies that internet banking and cashless transactions have significantly reduced reliance on traditional physical banking services, particularly in China where digital payment adoption is widespread. We acknowledge our model's simplified approach using travel time and facility size. While traditional bank branch optimization may have diminished practical relevance, our RL-GA framework represents a methodological contribution applicable to various facility location problems where physical presence remains critical (healthcare, emergency services, retail). We have clarified in the revised manuscript that modern bank branches are evolving into strategic service hubs rather than simple transaction centers, making location optimization increasingly important for maximizing service efficiency and customer accessibility in the hybrid digital-physical banking ecosystem.

 

This paper aims to inform the real demands of decision makers. The heavy computational load may be a serious bottle neck for actual implementation. If the authors could make any suggestions on how to reduce the computational burden and/or improve efficiency of the calculations this would contribute to further applications of the methods developed in this study.

We have addressedthis concern by incorporating comprehensive computational optimization strategies in our revised manuscript. As detailed in Section 2.2.6 " Algorithm Implementation and Computational Complexity Analysis," we have implemented multiple optimization strategies to reduce computational burden.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have worked hard and made great improvement on the manuscript. I am impressed by the efficient redesign of the whole methodology. The innovation of the study is well illustrated and the validation and comparison between two different methods are demonstrated. Yet, there are still minor issues should be noticed.

(1) The list of references is still not very up-to-date. I suggest the authors add more recent relative work.

(2) The explanation on the allocation result needs to be more specific. For example, in Line 668, the authors claimed that “The spatial arrangement aligns well with strategic considerations, 668 such as maximizing market penetration and resource utilization efficiency.” But how?

Author Response

Comment(1) The list of references is still not very up-to-date. I suggest the authors add more recent related work.

Respond(1) We appreciate the reviewer's feedback on updating our reference list. In response, we have added three recent publications on genetic algorithms in facility location at line 326 (Sachdeva et al., 2022; Lazari and Chassiakos, 2023; Salami et al., 2023) and three recent works on reinforcement learning applications in spatial optimisation at line 905 (Su et al., 2024; Bagga and Delarue, 2023; Wu et al., 2025). These additions ensure our literature review reflects the most current developments in both methodological areas relevant to our research.

Comment(2) The explanation of the allocation result needs to be more specific. For example, in Line 668, the authors claimed that “The spatial arrangement aligns well with strategic considerations, 668 such as maximising market penetration and resource utilisation efficiency.” But how?

Respond(2) We sincerely appreciate the reviewer's valuable feedback. We have significantly improved the specific explanation of allocation results in line 685, now providing detailed elaboration on the specific mechanisms by which spatial arrangements achieve strategic objectives. The revised content explicitly explains that market penetration maximization is achieved through strategic positioning that avoids oversaturated areas while covering underserved regions, as evidenced by the deliberate placement away from existing competitor clusters (blue dots) and toward areas with higher demand density; resource utilization efficiency is optimized through the balanced distribution of different facility scales, where larger facilities are strategically positioned in areas with optimal demand-to-competition ratios, reducing redundant service overlap and maximizing coverage per invested resource; spatial coverage optimization ensures that the minimum distance from any demand point to the nearest facility is minimized while maintaining cost-effectiveness, resulting in the observed even distribution pattern that balances accessibility with economic constraints. While we understand the reviewer's expectation for more quantitative indicators, considering that the core contribution of this research lies in algorithmic innovation rather than specific urban planning analysis, and given the difficulties in obtaining real market data, we believe that the current qualitative analysis adequately addresses the "how" question while maintaining the generalizability and universal applicability of our research methodology.