The Normalized Direct Trigonometry Model for the Two-Dimensional Irregular Strip Packing Problem
Tony Wauters
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe presented paper discusses a new MIP model for solving the strip packing problem with irregular shapes. A relevant real-world problem. The authors propose an adaptation to the already published direct trigonometry model. The novelty can be found in the usage of the Euclidean Distance instead of the D-function. This also allows the authors to include an interesting real-world constraint to separate pieces with some minimum distance in between (clearance constraint).
It is great to see the authors make their implementation and data available on GitHub!
The results are good enough to be publishable, both not groundbreaking.
In general, the paper is well written and interesting to read. However, I have the following remarks:
- The authors state that the distance to the container is not considered. But maybe it can be mentioned that the container can be replaced with some fixed pieces to model this particular aspect?
- A figure illustrating the model conventions and notation could be useful
- I'm not a fan of using ni, np or nl to index the last element in the set. Maybe index by the size of the sets?
- The result section could be improved. Foremost, it is not clear to me what is meant with "Individual Results" and "Consensus Results", please clarify. Moreover, I am missing some analysis of the properties of the mathematical models, for example the quality of the linear relaxation of both models. What happens when more than four angles are considered? A scaling analysis could be interesting. More pieces? More rotations?
- Did the authors find any new optimal solutions? Maybe they can provide some Figure of some interesting optimal solutions?
Comments on the Quality of English Language
Typo in Section 4.1: "convention" instead of "convection"
Author Response
Dear Reviewer,
Thank you for your valuable feedback and insightful comments on our manuscript. We have carefully considered your suggestions and made several revisions to address the points raised. Below, we outline our responses to each of your comments and the corresponding changes made to the manuscript.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsThe authors study strip packing problem for irregular objects presented as unions of convex polygons. Discrete rotation is allowed. Corresponding mathematical models are formulated in the form of mixed integer programming problems which are solved by CPLEX. Numerical results are provided to demonstrate efficiency of the proposed approach.
The paper provides interesting results in optimized packing of irregular objects. However, the presentation is poor and must be improved considering the following comments.
1. Please list at the end of Introduction the main findings of the paper.
2. Lines 67-68. “This study introduces the clearance constraint, which mandates that at least a specified distance must separate the packed items within the container.” This constraint, also known as distance constraint, is not a new one. For example, in the context of irregular packing it was used in the work “A New Class of Irregular Packing Problems Reducible to Sphere Packing in Arbitrary Norms” Mathematics 2024, 12, 935. https://doi.org/ 10.3390/math12070935
3. Section 2 is too short, worth merging with another section.
4. Lines 140, 141. “The lines of the convex polygons are oriented such that the center of the convex polygon always lies on one side of the lines.” What do you mean “the lines of the convex polygons”? What is “the center of the convex polygon”? Please be more specific.
5. Lines 156, 157. “In this study, the considered separating lines are colinear to the polygons' edges.” Lines 181, 182 “Moreover, the non-overlapping constraint allows touching; therefore, the vertices of both polygons can be colinear to the separating line.” What do you mean? So, is it true that both vertices and edges are colinear to the separating line? Please be more specific.
6. All considerations, propositions and conclusions presented in Section 4.1 must be proved or at least graphically illustrated.
7. Line 206, 207. “each piece 𝑖𝑖 has a subset of convex parts (sub-polygons) 𝑃𝑃𝑖𝑖:{𝑃𝑃𝑃𝑃𝑃𝑃𝑡𝑡1𝑖𝑖, … , 𝑃𝑃𝑃𝑃𝑃𝑃𝑡𝑡𝑛𝑛𝑝𝑝𝑖𝑖}” Would you like to say that each piece is a union of convex parts?
8. Lines 388, 389. “The implementation of the models, instance data, solution files, and visuals can be found at the following repository: https://github.com/gfpantoja/NDTM.” I can’t find at this link visual representations of the solutions obtained. Only txt files are provided.
9. The paper lacks graphical illustrations. It would be helpful for the reader to visualize the main constructions used to models’ formulations and present graphically some solutions obtained.
Author Response
Dear Reviewer,
Thank you for your valuable feedback and insightful comments on our manuscript. We have carefully considered your suggestions and made several revisions to address the points raised. Below, we outline our responses to each of your comments and the corresponding changes made to the manuscript.
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
Comments and Suggestions for AuthorsThe authors revised the paper significantly considering referees' comments. I think the current version of the paper can be published as it is.
