## 3 Problem Definition and Scenario of Application
### Scenario of Application  
The TLP is a real-application problem that has been addressed in collaboration with the Marist Brothers School San Jose in the city of Leon, Spain. The government of the autonomous country of Castilla and Leon has promoted the disposition of the students in two kinds of grids for addressing the TLP (i.e., a rectangular and a zig-zag grid) in order to satisfy the 1.5 m separation defined by the Spanish legislation. However, this kind of table disposition is suboptimal and the regular patterns do not necessarily obtain the best results in this type of combinatorial problem [39] since better results can be achieved by using metaheuristics for solving this complex problem.  
Therefore, we have analyzed two different classrooms of this school (i.e., Class A and Class B) with different characteristics, which forces different optimization goals. The first class consists of 16 tables while the second scenario needs to allocate 21 students, although the first scenario is smaller, its student density is considerably lower than the second class. Therefore, the Class A scenario is less restrictive than the second classroom in respect to the 1.5 minimum separation protocol.  
In order to model these classes and the table allocation area, we have defined three different regions for each class. The first area, the bigger one, represents the class limits, both classes were approximately rectangular; thus, both scenarios are considered to be of such shape.  
The second area is defined as the Table Location Environment (TLE) and it is the region within the class limits where the tables can be positioned. For this optimization, the positions of the tables are referenced from the students coordinates; thus, each point of the TLE is a possible location for a student. Therefore, a series of table measurements, shown in Table 1, have been considered in order to properly define the TLE region.  
Moreover, although major obstacles of the classrooms were considered into the TLE limit definition, some prohibited areas remained inside the TLE region, thus the codification of these areas is required. The third region in the scenario is the prohibited areas or obstacles, which are defined as areas where a student cannot be allocated. In the simulations presented in this paper, we have displayed the limits of these obstacles with respect to the student's table for visual clarification. Thus, in the following figures, neither the student nor the table can be positioned inside these obstacle regions.  
As for the original student distribution, the first class, shown in Figure 1a, displays an orthogonal mesh distribution of tables, with this scenario being less restrictive in terms of student allocation. On the other hand, Class B, shown in Figure 1b, requires a greater density of students, thus utilizing a triangular mesh distribution, which generally produces more efficient results in the table optimization.  
Figure 2a,b shows the classroom models used in this study.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Values** \\ \hline Table Length & 69.5 cm \\ Table Width & 49.5 cm \\ Student Space Radius & 10 cm \\ Distance from Student to Table & 20 cm \\ \hline \hline \end{tabular}
\end{table}
Table 1: List of parameters measured from the studied classrooms, implemented into the scenario modeling and table distribution optimization.