## 2 Problem Definition and Scenario of Application
### Real Scenario for the TLP Optimisation  
The TLP represents a considerable problem for a wide variety of establishments. In previous studies [37] a GA optimisation was presented for optimising the table distribution in multiple classrooms of the Marist Brothers School San Jose in the city of Leon, Spain.  
However, the application of this optimisation is not limited to educational scenarios. Through limited variations regarding the table characterisation, it is possible to perform the table distribution optimisation in different environments.  
Therefore, in this paper, the TLP is addressed for the Esla Salon at the Conde Luna Hotel in the city of Leon, Spain. The Esla Salon is a flexible space whose table disposition varies depending on the customer requirements. Hence, the managers of the establishment rearrange the table dimensions and locations depending on the celebration and the number of customers. As told by the managers, this task proved a significant challenge, especially for grand celebrations, where limited space was available for waiters to assist the diners.  
Figure 3: Table Codification implemented for the TLP. Each individual of the population carries a set of tables. Each table contains all the required information, including the gene sequence, the escalated coordinates with respect to the scenario limits, a unique identification number and a particularisation of the geometry and distances of the CLE. The proposed codification is compatible with multiple table geometries, and is independent of the selected scenario of application.  
Moreover, the complexity of this task escalated substantially with the introduction of the COVID-19 measures, restricting the maximum number of people allocated and enforcing a minimum interpersonal distance separation. Thus, the implementation of this scenario for the TLP optimisation proved ideal due to the required flexibility and limitations.  
The modelling of the Esla Salon is based on the definition of three different regions. The first region represents the scenario limits, thus containing the rest of the regions. The second area is defined as the Table Location Environment (TLE), it represents the real limits where a table centre can be located. Therefore, the difference between these two regions depends on the table measurements, the CLE and other considerations such as the introduction of additional space for waiters.  
Finally, the third and last region consist of the obstacles or prohibited regions for table placement. As shown in Figure 4, certain areas of the Esla Salon, such as exits and columns represent a direct restriction for the table placement.  
These obstacles are represented in consideration to their true dimension, however, in the MA optimisation, the prohibited areas are referred to the table centre, taking into consideration the table dimensions and other imposed limits for the CLE.  
Therefore, Table 1 shows the implemented measurements and properties obtained from the studied scenario.  
One of the main advantages of the proposed optimisation is its capability for obtaining the optimal table distribution for the specified table requirements. Therefore, two optimisation strategies are available when planning a table arrangement.  
Figure 4: Model of the Esla Salon without the Table allocation.  
The first strategy is to allocate the maximum number of tables (i.e., maximum capacity) while satisfying the imposed restrictions. On the other hand, the second strategy aims to obtain the most disperse distribution for a sub-optimal number of tables (i.e., maximum dispersion).  
Furthermore, the table dimensions and other parameters shown in Table 1 correspond to a singular configuration of banquet in the Esla Salon. Therefore, it could be possible to modify these parameters depending on the manager requirements, achieving a greater degree of flexibility, thus improving the table distribution for conventional usage, regardless of COVID-19 restrictions.