## 7 Conclusions  
A new mathematical model for drawing a hierarchical hypergraph with vertex ports and multi-source orthogonal hyperedges is discussed in this paper. The model is based on the formulation of a mixed-integer program. One of the advantages of the model is that it enables one to simultaneously take into account a set of optimization criteria, including aesthetics ones (for overall hypergraph drawing), as well as the minimization of hyperedge crossings. The numerical results demonstrate that the number of hyperedge crossings can be minimized compared to the mathematical heuristic approach. One of the application areas of the model is in financial flow visualization in information systems to create an initial drawing of a hypergraph. The initial layout is corrected when actual data are updated using the dynamic hierarchical graph-drawing algorithm [9]. Further directions of research include:  
* The generalization of the proposed mathematical model for reverse-directed hyperedges by adding port vertices to the model on the left and right sides of the generating vertices;
* The development of a fast and efficient algorithm (heuristic, metaheuristic, column generation, etc.) for drawing large-scale hypergraphs. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**ID**} & \multirow{2}{*}{\((|\mathbf{V}|_{i}|\mathbf{E}_{\mathbf{H}}|)\)} & \multicolumn{3}{c}{**Time, in sec**} & \multicolumn{3}{c}{**Number of Crossings**} \\ \cline{3-7}  & & **MH** & **MIP** & **MH+MIP** & **MH** & **MIP/MH+MIP** \\ \hline ms\_n8\_he4 & (8, 4) & 0.5 & 1.6 (\(<\)1) & 1.34 (1) & 3 & 1 \\ ms\_n12\_he6 & (12, 6) & 0.52 & 2.78 (2) & 2.52 (2) & 4 & 1 \\ ms\_n14\_he7 & (14, 7) & 0.53 & 3.02 (2) & 2.91 (2) & 3 & 2 \\ ms\_n15\_he7 & (15, 7) & 0.59 & 10.12 (5) & 5.82 (4) & 16 & 3 \\ ms\_n15\_he8 & (15, 8) & 0.5 & 3.05 (2) & 2.4 (2) & 2 & 0 \\ ms\_n16\_he9 & (16, 9) & 0.57 & 21.07 (14) & 4.61 (4) & 3 & 1 \\ ms\_n18\_he10 & (18, 10) & 0.63 & 44.87 (24) & 6.9 (6) & 13 & 4 \\ ms\_n19\_he10 & (19, 10) & 0.58 & 130.2 (91) & 9.51 (8) & 7 & 3 \\ ms\_n20\_he11 & (20, 11) & 0.63 & 497.9 (406) & 15.86 (14) & 17 & 7 \\ ms\_n27\_he11 & (27, 11) & 0.86 & 34.34 (12) & 8.79 (8) & 22 & 8 \\ ms\_n22\_he14 & (22, 14) & 0.64 & 39.56 (33) & 7.02 (6) & 14 & 4 \\ ms\_n41\_he19 & (41, 19) & 2.29 & 690.8 (411) & 26.33 (24) & 70 & 16 \\ ss\_n10\_he7 & (10, 7) & 0.53 & 4.48 (4) & 3.89 (3) & 3 & 1 \\ ss\_n11\_he7 & (11, 7) & 0.52 & 1.71 (\(<\)1) & 1.69 (1) & 4 & 0 \\ ss\_n20\_he13 & (20, 13) & 0.57 & 46.81 (44) & 6.25 (6) & 2 & 0 \\ ss\_n27\_he21 & (27, 21) & 0.75 & 35.81 (5) & 7.84 (6) & 6 & 1 \\ ss\_n40\_he26 & (40, 26) & 1.43 & 136.4 (85) & 11.85 (10) & 25 & 7 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numerical results obtained using the mathematical heuristic (MH) and mixed-integer program (MIP) for a set of generated hypergraphs. **Author Contributions:** Conceptualization, G.F. and Y.V. ; methodology, Y.V. ; software, V.P. ; validation, V.P. ; formal analysis, G.F.; investigation, G.F., Y.V. and V.P. ; resources, V.R. ; data curation, G.F.; writing--original draft preparation, V.P. ; writing--review and editing, G.F. and Y.V. ; visualization, V.P. ; supervision, G.F.; project administration, V.R. ; funding acquisition, V.R. All authors have read and agreed to the published version of the manuscript. **Funding:** The research was partially funded by the Ministry of Science and Higher Education of the Russian Federation as part of the World-Class Research Center program: Advanced Digital Technologies (contract No. 075-15-2020-903 dated 16.11.2020). **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Collection of the hypergraphs generated: [[https://drive.google.com/drive/folders/1](https://drive.google.com/drive/folders/1) GQL3 gKQsvODMrAt-YkHCYhaBabM7 wBN-]([https://drive.google.com/drive/folders/1](https://drive.google.com/drive/folders/1) GQL3 gKQsvODMrAt-YkHCYhaBabM7 wBN-) (Accessed: 14 December 2021).