Toward Optimal Multi-Agent Robot and Lift Schedules via Boolean Satisfiability

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This research focuses on optimizing how multiple robots use elevators (lifts) in buildings by creating smart schedules using Boolean Satisfiability (SAT). The authors propose and compare two approaches: Time-Expansion Graphs (TEG), which aim for the fastest (optimal) schedules, and Time-Ordered Encoding, which finds workable (feasible) schedules more efficiently when many robots are involved. Their experiments show that good scheduling can make robot systems 4 to 10 times faster than random use of lifts. The work is relevant for real-world settings like hospitals or warehouses and provides open-source tools to help implement these solutions.

 

While the paper introduces a timely and relevant scheduling problem and presents technically sound SAT-based methods, several key issues need to be addressed.

• The evaluation is based mostly on synthetic benchmarks. While the models are reasonable, testing in real-world or simulation-based environments (e.g., Gazebo or a real hospital layout) would strengthen the claims. Include at least one real-world case study or close-to-reality simulation.
• The current model assumes that each lift only carries one robot and ignores priorities, shared rides, or capacity limits. These are common in real-world environments and should be addressed or at least modeled in an extended simulation.
• The encoding logic, especially around the constraint formulations, is hard to follow. The paper would benefit from simplified diagrams or pseudocode for each method. Without this, reproducibility suffers.
• Only two approaches are evaluated. How do they compare with other known scheduling methods like Mixed Integer Programming or Reinforcement Learning?
• The figures are unclear and low quality—many lack proper labels, are hard to read, and do not effectively support the content. They need to be redesigned with higher resolution, clearer annotations, and consistent formatting.

Comments on the Quality of English Language

The paper requires significant rewriting for clarity. As it stands, the writing quality detracts from the technical contributions. The authors should revise for conciseness, consistency in terminology, and accessibility to readers unfamiliar with SAT-based scheduling. Consider professional editing if necessary

Author Response

Thank you once again for your detailed and valuable feedback. Before addressing the specific points below, we would like to note that we have performed a comprehensive revision of the entire manuscript to improve language, clarity, and flow. Given the extensive nature of these edits, providing highlights for every language change would unfortunately clutter the document and severely hinder readability. Therefore, we have opted to present a clean, revised version of the manuscript. We have, however, ensured that all major structural and content-based changes—such as the addition of new sections, algorithms, figures, and tables—are explicitly detailed in our point-by-point responses. We are confident that you will find the revised manuscript much clearer and easier to read.

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Thank you for your comprehensive and constructive review. We have undertaken a significant revision of the manuscript to address the points you raised, which we believe has substantially improved its clarity, rigor, and contribution.

Comment 1: The evaluation is based mostly on synthetic benchmarks. While the models are reasonable, testing in real-world or simulation-based environments (e.g., Gazebo or a real hospital layout) would strengthen the claims. Include at least one real-world case study or close-to-reality simulation.

Response 1: We agree that grounding our evaluation in more realistic scenarios is crucial. To address this, we have added a new "Real-World Inspired Benchmark" (Section 5.3, Page 9, line 261). These new benchmarks are derived from the Open-RMF robotics framework and are based on realistic layouts of a hotel and a hospital. We analyze the performance of our approach in these scenarios, demonstrating significant speedups (up to 4.34x) over a greedy algorithm. The results are presented in two new tables (Table 1 and Table 2, page 11) and discussed in detail on page 14, line 332.

Comment 2: The current model assumes that each lift only carries one robot and ignores priorities, shared rides, or capacity limits. These are common in real-world environments and should be addressed or at least modeled in an extended simulation.

Response 2: We agree with the reviewer that this is an important limitation of the current model. We have explicitly acknowledged this in the Discussion and Future Work sections (page 15, line 358 & page 15, line 351). While a full implementation of these features is beyond the scope of this revision, we have added a new Section 6.1, 'Handling Priorities and Dynamic Events,'(Page 15, line 355) to demonstrate how our framework could be conceptually extended to address such real-world complexities. As for shared lifts, you arer right, we do not address these in this model. We leave it as "Future Work" as we believe there are a number of real-world issues that need to be overcome for this to be relevant (see Page 16 line 381).

Comment 3: The encoding logic, especially around the constraint formulations, is hard to follow. The paper would benefit from simplified diagrams or pseudocode for each method. Without this, reproducibility suffers.

Response 3: We thank the reviewer for this valuable suggestion. To improve the clarity and reproducibility of our methods, we have made two key additions. First, we introduced Algorithm 1 to provide a clear, step-by-step procedure for generating the TEG constraints (page 5). In addition we make the distinction between the two methods clearer using Figure 2 on page 3.

Comment 4: Only two approaches are evaluated. How do they compare with other known scheduling methods like Mixed Integer Programming or Reinforcement Learning?

Response 4: This is an excellent point. We have expanded the "Related Work" section to include a discussion of alternative methods like Mixed Integer Programming (MIP) and Reinforcement Learning (RL). We justify our focus on SAT-based methods by highlighting their suitability as "Anytime Algorithms," which is a critical feature for real-world robotic systems where a timely, good-enough solution is often preferable to a perfect one that takes too long to compute (page 2, line 58). In addition we have use state of the art methods to perform benchmarking in real world inspired scenarios.

Comment 5: The figures are unclear and low quality—many lack proper labels, are hard to read, and do not effectively support the content. They need to be redesigned with higher resolution, clearer annotations, and consistent formatting.

Response 5: We sincerely apologize for the poor quality of the figures in the original submission. We have taken this feedback seriously and have revised all figures to improve their clarity, resolution, and labeling. Specifically, we have:
*   Redesigned some of the core diagrams (Figure 2, page 3) with clearer annotations.'
*   Called out the distinction between methods in Figure 2 with acompanying text on Page 3 line 76.
*   Introduced screenshots of real world scenarios (Figure 5, 6).
*   Expanded the captions to ensure the content of each figure is well-explained.
We are confident the revised figures now effectively support the manuscript's content.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The proposed article looks good. It has as mathematical description of problem as well as practical application.

Introduction, analyses, main part, conclusions, references correspond to requirements.

As problem questions I can notice:

1. It is difficult to imagine lift with speed of only 0.15 m/s, especially for hospital, for example in my home it's 1.0 m/s., while it is 40 years old.
2. There is no declaration for patiient's priorities, for emergency cases. 
3. There is no description for case of lifts malfunction, when some number of lifts is out of work (or only one is working).

Author Response

Thank you once again for your detailed and valuable feedback. Before addressing the specific points below, we would like to note that we have performed a comprehensive revision of the entire manuscript to improve language, clarity, and flow. Given the extensive nature of these edits, providing highlights for every language change would unfortunately clutter the document and severely hinder readability. Therefore, we have opted to present a clean, revised version of the manuscript. We have, however, ensured that all major structural and content-based changes—such as the addition of new sections, algorithms, figures, and tables—are explicitly detailed in our point-by-point responses. We are confident that you will find the revised manuscript much clearer and easier to read.
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Comment 1: It is difficult to imagine lift with speed of only 0.15 m/s, especially for hospital, for example in my home it's 1.0 m/s., while it is 40 years old.

Response 1: While we don't include such lifts in our calculations, such lifts are not uncommon where I live. Vertical platform lifts (https://dazenelevator.com/platform-lift-vs-passenger-lift/#:~:text=The%20common%20speed%20for%20platform,installation%20needs%20of%20the%20environment.) are often used in public spaces where they were retrofitted after the space was built, in order to improve accessibility for handicaps. They have speeds of about 0.15m/s for safety reasons. For our study we use 15seconds/level which while on the slow side, this accounts for getting in and out of the lift. Our examples generally are evaluated on shorter buildings. We have also added a few real world inspired benchmarks which directly take the parameters from open-rmf's demo worlds which have faster lifts (see page 9, line 261). 

Comment 2: There is no declaration for patient's priorities, for emergency cases. 

Response 2: A comprehensive study this can be seen as future work. The concern here was to assign all robots to elevators based on deadlines and propose an optimization algorithm. It is possible to adjust priorities by running the algorithm in multiple phases. While not part of this work it can be seen as a future extension. We add a subsection 6.1.1 in the "Discussion" section which outlines how this can be done (see page 15 line 351).

Comment 3: There is no description for case of lifts malfunction, when some number of lifts is out of work.

Response 3: As with my response to comment 2, we have not undertaken a comprehensive study here but, we propose a method to handle scenarios like this in section 6.1.2 on page 15 (see page 15 line 351). 

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The revisions substantially improve the manuscript, particularly the inclusion of real-world inspired benchmarks and clearer methods.

It is good that the limitations are now explicitly acknowledged and that you included a new section on priorities and dynamic events. However, leaving shared-lift capacity entirely as “future work” still feels like a significant gap. Since this is a central real-world issue, even a simplified simulation example (e.g., two robots in one lift) would have been valuable. I encourage the authors to consider at least a basic experiment here, even if full implementation is deferred.

Author Response

Comment 1: It is good that the limitations are now explicitly acknowledged and that you included a new section on priorities and dynamic events. However, leaving shared-lift capacity entirely as “future work” still feels like a significant gap. Since this is a central real-world issue, even a simplified simulation example (e.g., two robots in one lift) would have been valuable. I encourage the authors to consider at least a basic experiment here, even if full implementation is deferred.

Response 1: Thanks for the excellent feedback. We have updated the manuscript to explicitly address the reviewer’s feedback. As you correctly pointed out, we have now added text to the problem definition itself to more clearly state our assumptions.

The formulations in our paper, including both the Time-Ordered and Time-Expansion Graph approaches, are not designed for shared-lift use. They are based on the assumption that a lift can only service one robot at a time. We realize we had not explicitly called this out in our previous Problem Statement. We have hence further refined our problem statement to be clear on this point (see Page 3, line 84). In addition, we have added a citation justifying why our position on this is unchanged.

As we now state in the manuscript, coordinating the movement of multiple robots entering and exiting a confined space is a known practical challenge. Even the single-robot case is quite challenging from a motion planning and perception perspective in a real-world setting. Therefore, while we agree on the importance of this topic, we have identified it as a separate and complex problem that is beyond the scope of this paper.

 

Author Response File: Author Response.pdf