## 2 Problem Definition and Scenario of Application
### Table Characterisation  
The TLP aims to obtain the table distribution that maximises the interpersonal distance separation. However, certain aspects regarding the table characterisation of the problem must be specified for each scenario of application.  
In previous studies [37], a GA was proposed for solving the TLP in a conventional classroom environment. In these scenarios, each table contains a single student, thus the desired separation can be defined as the Euclidean distance from student to student.  
However, the hostelry sector requires a different approach. In common restaurant or pub tables, multiple individuals are expected to be allocated around each table. Moreover, the number and position of the chairs around the table may differ depending on the client's requirements.  
Therefore, in search of a more flexible approach, we have defined a Chair Location Environment (CLE) around the table where the diners are expected to be allocated. The definition of this environment isolate the chair placement from the TLP, resulting in a greater flexibility for the restaurant manager.  
However, the evolution from a singular chair point to the CLE also reshapes the Non-Habitable Environment (NHE) from the resulting circle from the student perspective analysed in previous studies to a rectangular that contains the CLE, as shown in Figure 1.  
However, the implementation of the CLE as a limiting region must be taken into consideration when obtaining the interpersonal distance separation. Nevertheless, knowing the dimensions of each region, it is possible to deduce the equivalent distance of intersection between the CLE of pairs of tables.  
Figure 2 shows the table interaction chart. The geometry of the selected rectangular tables results in a variation of the security distance with the angle of incidence between tables as stated in Equation (3).  
\[d_{\text{sec}_{ij}}=d_{\text{Total}_{ij}}-d_{\text{CLE}_{i}}(\alpha_{i})-d_{ \text{CLE}_{j}}(\alpha_{j}), \tag{3}\]  
where \(d_{\text{Total}_{ij}}\) is the Euclidean distance from table-to-table centres.  
Therefore, the CLE dimensions being known, it is possible to deduce these distances from the angle of intersection (\(\alpha\)) between tables. However, the MA optimisation proposed requires the evaluation of this table interaction during a considerable number of iterations. The resulting CLE distances could be obtained through some trigonometric  
Figure 1: Table Definition for the hostelry perspective. The CLE and NHE dimensions are known and can be referenced from the Table centre, the reference point for the MA optimisation proposed.  
Figure 2: Table interaction for the TLP in hostelry tables. The security distance \(d_{\text{sec}_{ij}}\) is deduced from the tables centre of reference. \(d_{\text{CLE}_{i}}\) is defined as the distance between the table \(i\) centre and the intersection of the CLE with the table-to-table line.  
relations. However, these operations introduce an increment in complexity from the initial Euclidean distance, resulting in a significant increase in the computational cost required for the optimisation.  
Nevertheless, it is possible to improve the efficiency of this evaluation by predetermining a polar distribution of the \(d_{\text{CLE}}\) distance for a discrete distribution of intersection angles. This distribution can be generated before the MA takes place, kept stored inside the table codification for future evaluations, as shown in Figure 3.  
The proposed polar definition of the CLE distances achieved an increase of up to 80% in time efficiency, reducing the complexity of the distance algorithm from trigonometric evaluations to finding values in a stored list, thus improving the overall efficiency of the MA optimisation.