# A New Class of Irregular Packing Problems Reducible to Sphere Packing in Arbitrary Norms

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## Abstract

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## 1. Introduction

- The problem of Packing Objects Composed by Generalized Spheres (PCGS) is formulated for objects and containers represented by spheres in arbitrary norms.
- Non-overlapping and containment conditions considering allowable distances for irregular objects composed by generalized spheres are introduced. By means of a new composition condition, rotations and reflections of the irregular objects are enabled.
- The generalized balance, homothetic and sparse packing problems for objects composed by the generalized spheres are stated for various norms.

## 2. Mathematical Modeling

#### 2.1. The Main Definitions

- (a)
- a containment condition ensures that all composed objects are completely in the container,
- (b)
- a non-overlapping condition states that there is no overlapping between any pair of the composed objects.

#### 2.2. Useful Norms and Transformations

**Comment 1.**

**Comment 2.**

## 3. Some Cases of PCGS Problems

#### 3.1. Generalized Balance Packing Problems (GBPP)

#### 3.2. Generalized Homothetic Packing Problems (GHPP)

**Comment 3.**

#### 3.3. Generalized Sparse Packing Problems (GSPP)

## 4. Computational Results

#### 4.1. Computational Results for Generalized Balance Packing Problem (GBPP)

**Example 1.**

**Example 2.**

**Example 3.**

- (a)
- $K=3,m=4$, ${A}_{1}={S}_{1}\cup {S}_{2},{A}_{2}={S}_{3},{A}_{3}={S}_{4}$, $\{{r}_{i},i=1,\dots ,4\}=\left\{3,\text{}2,\text{}1,\text{}1\right\}$;
- (b)
- $K=2,m=4$, ${A}_{1}={S}_{1}\cup {S}_{2},{A}_{2}={S}_{3}\cup {S}_{4}$, $\{{r}_{i},i=1,\dots ,4\}=\left\{3,\text{}2,\text{}2,\text{}3\right\}$;
- (c)
- $K=6,$ $m=9$, ${A}_{1}={S}_{1}\cup {S}_{2}\cup {S}_{3}\cup {S}_{4},$ ${A}_{2}={S}_{5},$ ${A}_{3}={S}_{6},$ ${A}_{4}={S}_{7},$ ${A}_{5}={S}_{8},$ ${A}_{6}={S}_{9}$, $\{{r}_{i},i=1,\dots ,9\}=\left\{4,\text{}4,\text{}4,\text{}4,\text{}2,\text{}3,\text{}3,\text{}3,\text{}3\right\}$.

**Example 4.**

- (a)
- $K=2,$ $m=5$, ${A}_{1}={S}_{1}\cup {S}_{2}\cup {S}_{3},$ ${A}_{2}={S}_{4},$ ${A}_{3}={S}_{5}$, $\{{r}_{i},i=1,\dots ,5\}=\left\{3,\text{}2,\text{}1,\text{}3,\text{}2)\right\}$;
- (b)
- $K=2,$ $m=6$, ${A}_{1}={S}_{1}\cup {S}_{2}\cup {S}_{3},$ ${A}_{2}={S}_{4}\cup {S}_{5}\cup {S}_{6}$, $\{{r}_{i},i=1,\dots ,6\}=\left\{4,\text{}2,\text{}1,\text{}4,\text{}2,\text{}1)\right\}$;

#### 4.2. Computational Results for Generalized Homothetic Packing Problem (GHPP)

**Example 5.**

#### 4.3. Computational Results for Generalized Sparse Packing Problem (GSPP)

**Example 6.**

**Example 7.**

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Bennell, J.A.; Oliveira, J.F. The geometry of nesting problems: A tutorial. Eur. J. Oper. Res.
**2008**, 184, 397–415. [Google Scholar] [CrossRef] - Bennell, J.A.; Oliveira, J.F. A tutorial in irregular shape packing problems. J. Oper. Res. Soc.
**2009**, 60 (Suppl. 1), 93–105. [Google Scholar] [CrossRef] - Wang, J. Packing of unequal spheres and automated radiosurgical treatment planning. J. Comb. Optim.
**1999**, 3, 453–463. [Google Scholar] [CrossRef] - Sutou, A.; Dai, Y. Global optimization approach to unequal sphere packing problems in 3D. J. Optim. Theory Appl.
**2002**, 114, 671–694. [Google Scholar] [CrossRef] - Burtseva, L.; Valdez Salas, B.; Romero, R.; Werner, F. Recent advances on modelling of structures of multi-component mixtures using a sphere packing approach. Int. J. Nanotechnol.
**2016**, 13, 44–59. [Google Scholar] [CrossRef] - Ma, P.; Chan, H.-K. Densest-Packed Columnar Structures of Hard Spheres: An Investigation of the Structural Dependence of Electrical Conductivity. Front. Phys.
**2021**, 9, 778001. [Google Scholar] [CrossRef] - Marín-Aguilar, S.; Camerin, F.; van der Ham, S.; Feasson, A.; Vutukuri, H.R.; Dijkstra, M. A colloidal viewpoint on the sausage catastrophe and the finite sphere packing problem. Nat. Commun.
**2023**, 14, 7896. [Google Scholar] [CrossRef] [PubMed] - Conway, J.H.; Sloane, N.J.A. Sphere Packings, Lattices and Groups; Springer: New York, NY, USA, 1999. [Google Scholar]
- Cullina, D.; Kiyavash, N. Generalized sphere-packing bounds on the size of codes for combinatorial channels. IEEE Trans. Inf. Theory
**2016**, 62, 4454–4465. [Google Scholar] [CrossRef] - Duriagina, Z.; Lemishka, I.; Litvinchev, I.; Marmolejo, J.A.; Pankratov, A.; Romanova, T.; Yaskov, G. Optimized filling of a given cuboid with spherical powders for additive manufacturing. J. Oper. Res. Soc. China
**2021**, 9, 853–868. [Google Scholar] [CrossRef] - Liu, F.; Chen, M.; Wang, L.; Luo, T.; Chen, G. Stress-field driven conformal lattice design using circle packing algorithm. Heliyon
**2023**, 9, e14448. [Google Scholar] [CrossRef] [PubMed] - Castillo, I.; Kampas, F.J.; Pinter, J.D. Solving circle packing problems by global optimization: Numerical results and industrial applications. Eur. J. Oper. Res.
**2008**, 191, 786–802. [Google Scholar] [CrossRef] - Chen, D. Sphere Packing Problem. In Encyclopedia of Algorithms; Kao, M.Y., Ed.; Springer: Boston, MA, USA, 2008. [Google Scholar] [CrossRef]
- Hifi, M.; M’Hallah, R. A literature review on circle and sphere packing problems: Models and methodologies. Adv. Oper. Res.
**2009**, 2009, 150624. [Google Scholar] [CrossRef] - Fischer, A.; Scheithauer, G. Cutting and packing problems with placement constraints. In Optimized Packings with Applications; Fasano, G., Pintér, J.D., Eds.; Springer Optimization and Applications; Springer: Berlin/Heidelberg, Germany, 2015; Volume 105, pp. 119–156. [Google Scholar]
- Kampas, F.J.; Pintér, J.D.; Castillo, I. Packing ovals in optimized regular polygons. J. Glob. Optim.
**2020**, 77, 175–196. [Google Scholar] [CrossRef] - Rao, Y.; Luo, Q. Intelligent Algorithms for Packing and Cutting Problem; Springer: Berlin/Heidelberg, Germany, 2022. [Google Scholar]
- Dechant, P.-P.; Twarock, R. Models of viral capsid symmetry as a driver of discovery in virology and nanotechnology. Biochemist
**2021**, 43, 20–24. [Google Scholar] [CrossRef] - Tetter, S.; Terasaka, N.; Steinauer, A.; Bingham, R.J.; Clark, S.; Scott, A.J.; Patel, N.; Leibundgut, M.; Wroblewski, E.; Ban, N.; et al. Evolution of a virus-like architecture and packaging mechanism in a repurposed bacterial protein. Science
**2021**, 372, 1220–1224. [Google Scholar] [CrossRef] [PubMed] - Phua, A.; Smith, J.; Davies, C.H.J.; Cook, P.S.; Delaney, G.W. Understanding the structure and dynamics of local powder packing density variations in metal additive manufacturing using set Voronoi analysis. Powder Technol.
**2023**, 418, 118272. [Google Scholar] [CrossRef] - Zhao, C.; Gao, Q.; Chen, Y.; Li, C. Application of parametric function in construction of particle shape and discrete element simulation. Powder Technol.
**2021**, 387, 481–493. [Google Scholar] [CrossRef] - Soltanbeigi, B.; Podlozhnyuk, A.; Kloss, C.; Pirker, S.; Ooi, J.Y. Papanicolopulos, S.A. Influence of various DEM shape representation methods on packing and shearing of granular assemblies. Granul. Matter
**2021**, 23, 26. [Google Scholar] [CrossRef] - Ma, H.; Xia, X.; Zhou, L.; Xu, C.; Liu, Z.; Song, T.; Zou, G.; Liu, Y.; Huang, Z.; Liao, X.; et al. A comparative study of the performance of different particle models in simulating particle charging and burden distribution in a blast furnace within the DEM framework. Energies
**2023**, 16, 3890. [Google Scholar] [CrossRef] - Leao, A.A.S.; Toledo, F.M.B.; Oliveira, J.F.; Carravilla, M.A.; Alvarez-Valdes, R. Irregular packing problems: A review of mathematical models. Eur. J. Oper. Res.
**2020**, 282, 803–822. [Google Scholar] [CrossRef] - Labrada-Nueva, Y.; Cruz-Rosales, M.H.; Rendón-Mancha, J.M.; Rivera-López, R.; Eraña-Díaz, M.L.; Cruz-Chávez, M.A. Overlap Detection in 2D Amorphous Shapes for Paper Optimization in Digital Printing Presses. Mathematics
**2021**, 9, 1033. [Google Scholar] [CrossRef] - Guo, B.; Zhang, Y.; Hu, J.; Li, J.; Wu, F.; Peng, Q.; Zhang, Q. Two-dimensional irregular packing problems: A review. Front. Mech. Eng.
**2022**, 8, 966691. [Google Scholar] [CrossRef] - Luo, Q.; Rao, Y. Improved Sliding Algorithm for Generating No-Fit Polygon in the 2D Irregular Packing Problem. Mathematics
**2022**, 10, 2941. [Google Scholar] [CrossRef] - Fang, J.; Rao, Y.; Zhao, X.; Dum, B. A Hybrid Reinforcement Learning Algorithm for 2D Irregular Packing Problems. Mathematics
**2023**, 11, 327. [Google Scholar] [CrossRef] - Fasano, G. Solving Non-Standard Packing Problems by Global Optimization and Heuristics; Springer: Cham, Switzerland, 2014. [Google Scholar]
- Adler, J.R.; Schweikard, A.; Achkire, Y.; Blanck, O.; Bodduluri, M.; Ma, L.; Zhang, H. Treatment Planning for Self-Shielded Radiosurgery. Cureus
**2017**, 9, e1663. [Google Scholar] [CrossRef] [PubMed] - Boles, M.A.; Talapin, D.V. Many-Body Effects in Nanocrystal Superlattices: Departure from Sphere Packing Explains Stability of Binary Phases. J. Am. Chem. Soc.
**2015**, 137, 4494–4502. [Google Scholar] [CrossRef] - Banhelyi, B.; Palatinus, E.; Levai, B.L. Optimal circle covering problems and their applications. Cent. Eur. J. Oper. Res.
**2015**, 23, 815–832. [Google Scholar] [CrossRef] - Romanova, T.; Litvinchev, I.; Pankratov, A. Packing ellipsoids in an optimized cylinder. Eur. J. Oper. Res.
**2020**, 285, 429–443. [Google Scholar] [CrossRef] - Prvan, M.; Ozegovic, J.; Misura, A.B. On calculating the packing efficiency for embedding hexagonal and dodecagonal sensors in a circular container. Math. Probl. Eng.
**2019**, 2019, 9624751. [Google Scholar] [CrossRef] - Arruda, V.P.R.; Mirisola, L.G.B.; Soma, N.Y. Almost squaring the square: Optimal packings for non-decomposable squares. Pesqui. Oper.
**2022**, 42, e262876. [Google Scholar] [CrossRef] - Litvinchev, I.; Infante, L.; Ozuna, L. Approximate Packing: Integer Programming Models, Valid Inequalities and Nesting; Springer Optimization and Its Applications; Springer: Cham, Switzerland, 2015; Volume 105, pp. 117–135. [Google Scholar]
- Litvinchev, I.; Infante, L.; Ozuna, L. Packing circular-like objects in a rectangular container. J. Comput. Syst. Sci. Int.
**2015**, 54, 259–267. [Google Scholar] [CrossRef] - Litvinchev, I.; Infante, L.; Ozuna, L. Using different norms in packing circular objects. In Intelligent Information and Database Systems; Nguyen, N.T., Trawineski, B., Kosala, R., Eds.; Springer: Cham, Switzerland, 2015; pp. 540–548. [Google Scholar]
- Narici, L.; Beckenstein, E. Topological Vector Spaces. In Pure and applied mathematics, 2nd ed.; CRC Press: Boca Raton, FL, USA, 2011. [Google Scholar]
- Khajavirad, A.; Sahinidis, N.V. A hybrid LP/NLP paradigm for global optimization relaxations. Math. Program. Comput.
**2018**, 10, 383–421. [Google Scholar] [CrossRef] - Sahinidis, N. BARON User Manual v. 2024.3.10. Available online: https://minlp.com/downloads/docs/baron%20manual.pdf (accessed on 11 March 2024).
- Tawarmalani, M.; Sahinidis, N.V. A polyhedral branch-and-cut approach to global optimization. Math. Program.
**2005**, 103, 225–249. [Google Scholar] [CrossRef] - Fourer, R.; Gay, D.M.; Kernighan, B. AMPL: A Modeling Language for Mathematical Programming, 2nd ed.; Duxbury: Thomson, Georgia, 2003. [Google Scholar]
- Stetsyuk, P.; Romanova, T.; Scheithauer, G. On the global minimum in a balanced circular packing problem. Optim. Lett.
**2016**, 10, 1347–1360. [Google Scholar] [CrossRef] - Kallrath, J. Cutting and Packing Beyond and within Mathematical Programming. In Business Optimization Using Mathematical Programming; International Series in Operations Research & Management Science; Springer: Cham, Switzerland, 2021; Volume 307. [Google Scholar] [CrossRef]
- Yagiura, M.; Umetani, S.; Imahori, S. Cutting and Packing Problems: From the Perspective of Combinatorial Optimization; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar]
- Scheithauer, G. Introduction to Cutting and Packing Optimization: Problems, Modeling Approaches, Solution Methods; Springer: Berlin/Heidelberg, Germany, 2018. [Google Scholar]
- Lai, X.; Hao, J.K.; Xiao, R.; Glover, F. Perturbation-based thresholding search for packing equal circles and spheres. INFORMS J. Comput.
**2023**, 35, 711–908. [Google Scholar] [CrossRef]

**Figure 1.**Composed object $A={\displaystyle \underset{i=1}{\overset{5}{\cup}}{S}_{i}}({\xi}_{i})$: (

**a**) $p=1$, (

**b**) $p=2$, (

**c**) $p=\infty $.

**Figure 4.**Layouts corresponding to the global solutions: (

**a**–

**e**) in Example 1 for $p=1,2,3,6,\infty $ and (

**f**) in Example 2 for the composite norm.

**Figure 6.**Layouts corresponding to the global solutions in Example 3(b) for $p=1,3$ and in Example 3(c) for $p=1$.

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**MDPI and ACS Style**

Litvinchev, I.; Fischer, A.; Romanova, T.; Stetsyuk, P.
A New Class of Irregular Packing Problems Reducible to Sphere Packing in Arbitrary Norms. *Mathematics* **2024**, *12*, 935.
https://doi.org/10.3390/math12070935

**AMA Style**

Litvinchev I, Fischer A, Romanova T, Stetsyuk P.
A New Class of Irregular Packing Problems Reducible to Sphere Packing in Arbitrary Norms. *Mathematics*. 2024; 12(7):935.
https://doi.org/10.3390/math12070935

**Chicago/Turabian Style**

Litvinchev, Igor, Andreas Fischer, Tetyana Romanova, and Petro Stetsyuk.
2024. "A New Class of Irregular Packing Problems Reducible to Sphere Packing in Arbitrary Norms" *Mathematics* 12, no. 7: 935.
https://doi.org/10.3390/math12070935