LP-Based Row Generation Using Optimization-Based Sorting Method for Solving Budget Allocation with a Combinatorial Number of Constraints

Round 1

Reviewer 1 Report

I have carefully examined the manuscript. Below are my critical comments and suggestions. Please note that these comments reflect a rigorous assessment of the manuscript and aim to ensure the utmost scholarly rigor.

·         The claim that your method solves a problem with over 2 quadrillion constraints in just 1.4 seconds is extraordinary. While impressive, such a claim requires extensive validation. There must be a detailed explanation of the experimental setup, including hardware and software specifications. Without this, the claim lacks the necessary context for replication and validation.

·         While the paper compares the proposed method with traditional LP and ROS methods, there is a lack of comparison with other state-of-the-art methods. The field of optimization is vast, and there may be recent advancements that could challenge the superiority of your approach.

·         The paper focuses on a specific problem size (N=50) and asserts generalizability to larger problems without demonstrating this scalability. Evidence should be provided for a range of problem sizes, and the computational complexity should be formally analyzed.

·         The paper mentions the use of optimization-based sorting without explaining how this integrates with the row generation framework. A detailed algorithmic description is necessary, including pseudocode and a step-by-step breakdown of the process.

·         The paper presents average computation times but does not discuss the statistical significance of the results. It is important to include measures such as standard deviation, confidence intervals, and p-values to support the claims statistically.

·         There is no discussion on the robustness of the proposed method. How does it perform with noisy or incomplete data? How sensitive is it to changes in the parameters or to the initial solution? This analysis is crucial for understanding the practical applicability of the method.

·         The paper fails to address any limitations or potential weaknesses of the proposed approach. No method is without its drawbacks, and these must be discussed transparently to provide a balanced view.

·         There is a lack of discussion about the practical implementation of this method. Theoretical efficiency does not always translate to practical applicability. The paper should include case studies or real-world examples where this method has been applied successfully.

·         The manuscript could benefit from using diverse datasets to validate the method. The use of randomly generated structural problems is a good start, but the inclusion of real-world data sets with inherent complexities would strengthen the validation.

·         The literature review seems to be insufficient for the breadth of the topic. A more comprehensive review could better position the proposed method within the current state-of-the-art.

·         The paper lacks details that would allow for reproducibility of the results. For groundbreaking claims, it is imperative to provide complete transparency, which includes sharing source code, data, and detailed experimental protocols.

·         Finally, there are issues with the presentation. The manuscript is dense and could be better organized to improve readability. Diagrams, flowcharts, and other visual aids would greatly enhance comprehension.

In conclusion, while the paper presents a potentially groundbreaking approach to budget allocation problems, the extraordinary claims made require extraordinary evidence and clarity. Addressing the above points will be necessary to establish the veracity and significance of the proposed method.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

This paper, addressing a budget allocation problem, exhibits several shortcomings that raise concerns about the robustness, reliability, and meaningfulness of the findings. The experimental model details the comparison of three methods—linear programming, row generation, and the proposed model (ROS)—based on processing time. However, the study's robustness is questionable, lacking a systematic exploration of the sensitivity of results to variations in input parameters or experimental conditions. The absence of a comprehensive ablation study, selectively disabling or modifying specific components or methodologies to assess their individual impact, undermines the depth of the analysis. The reliability of the results is compromised by the limited scope of the experiments, focusing solely on processing time without delving into the broader context of solution accuracy, stability, or sensitivity to problem characteristics. The conclusion drawn, emphasizing the efficiency of the proposed methods for larger problem sizes, lacks robust support due to the absence of a rigorous and multifaceted evaluation, diminishing the overall credibility of the computational studies.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

The researchers proposed an innovative approach that combined LP-based row generation with optimization-based sorting to deal with computational challenges due to budget allocation problems with combinatorial constraints. The computational experiments and case studies conducted showed that the presented approach provided better outcomes compared with simplex solutions in terms of solution search time, demonstrating enhanced computational efficiency and solution quality compared to traditional LP methods.

The current paper introduces a relevant topic and fits with the scope of the journal. The manuscript is original, and it has the potential to provide positive feedback to the field of study. The title expresses clearly what the manuscript is about. The abstract is a short and clear summary of the aims, key methods, and important findings. The introduction adequately summarizes the current state of the topic. However, more references are needed to support all the statements used. The introduction clearly explains the objectives, as well as why the study is necessary. The literature review section provides relevant information. As regards the last paragraph in which the structure of the paper is described, it would be better to be placed at the end of the introduction. Maybe it would be useful to add a methodology section to analyze the steps used in the presented study. The Problem Structures and Notations section is adequately described. Tables and figures are relevant, useful, and consistent with the description in the text. The experimental model is adequately described. In the conclusion section, the authors logically explain the findings. It would be useful to mention any limitations of the study. Language is fine. The reference list is in line with the journal’s guidelines.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Accept.

Reviewer 2 Report

Thank you, the experimental part is now much improved and the paper can now be accepted.