# Sequential Memetic Algorithm Optimization for Allocation Planning in Hostelry Establishments
## 2 Problem Description  
### Table Location Problem in Hostelry Applications  
To achieve optimal space utilization in the competitive world of hostelry, it is essential to address the Table Location Problem (TLP) as a foundational step. The proposed methodology for attaining optimal space utilization in hostelry applications requires addressing the TLP in the first place. The TLP looks to find the Cartesian coordinates of the center of gravity of each of the tables deployed in a particular scenario, both for achieving the maximization of the distance among the tables and for introducing the maximum number of tables in safety conditions in the application scenario. This entails the finding of the optimal subset \((\langle T_{i}\rangle=\langle t_{i},\ldots,t_{n}\rangle)\) containing the Cartesian coordinates of the center of gravity of the \(n\) tables deployed \((\langle t_{i}\rangle=\langle x_{i},y_{i}\rangle)\), within the set of available locations for each table within the scenario of application \((T)\). This problem, exponential in complexity [17], looks to maximize the distance among the tables while achieving uniformity with these distances. In this sense, the Cartesian coordinates of the tables are modified during the optimization (i.e., the decision variables of the problem) also fulfilling some restrictions for ensuring the achievement of valid table dispositions. This states the following mathematical model for the TLP:  
\[\text{Maximize }Z=\textit{ff}(\textit{ff}_{d}(T_{i}),\textit{ff}_{\textit{ pen}}(T_{i})) \tag{1}\]  
Subject to  
\[ \begin{array}{l}x_{l_{1}}\leq x_{i}\leq x_{l_{2}}\quad\forall\ x_{i}\in t_{ i};\ t_{i}\in T_{i};\ t_{i}\
ot\in U\\ y_{l_{1}}\leq y_{i}\leq y_{l_{2}}\quad\forall\ y_{i}\in t_{i};\ t_{i}\in T_{i};\ t_{i}\
ot\in U\\ d_{ij_{CLE}}\geq d_{s}\quad\forall\ i,j\in 1,\ldots,n;\ i\
ot=j\\ d_{ki_{CLE}}\geq d_{w}\quad\forall\ 1,\ldots,n;\ \forall k\in 1,\ldots,n_{obs} \end{array}  \tag{2}\]  
where \(\textit{ff}_{d}\) and \(\textit{ff}_{pen}\) are fitness functions for maximizing the distance among the tables and for guiding the optimizations to valid table distributions, respectively, \(x_{l_{1}},x_{l_{2}},y_{l_{1}},\) and \(y_{l_{2}}\) are the scenario limits for the Cartesian coordinates of the center of gravity of the tables; \(U\) is the subset including the forbidden location for the tables due to the obstacles of the scenario; \(d_{ij_{CLE}}\) represents the distance among the Chair Location Environment (CLE) of the tables \(i\) and \(j\), which must exceed the security distance (\(d_{s}\)) of 1.5 m set in the Spanish legislation during the pandemic [21]; \(n_{obs}\) is the number of obstacles defined in the scenario and \(d_{ki_{CLE}}\) is the distance from the obstacle \(k\) to the closest point in the CLE of the table \(i\) which must allow the passage of the waiters for a proper service exceeding a threshold distance (\(d_{w}\)). An important fact to highlight in this model is the calculation of the distance between the CLE of different tables. The CLE is defined as the region around each table where chairs may be placed. By defining a bounded region around each table for the chair allocation, the TLP optimization is carried out independently of the Chair Location Problem CLP, described in the next subsection, while also considering the interpersonal distancing between variable chair locations in the distance calculation. To achieve this objective, the distancing between tables is deduced from the separation between the two CLE regions of those tables, as depicted in Figure 1. From this scheme, we can compute the distance between potentially seated individuals in a stage in which the number and location of the chairs are unknown. This procedure, shown in Figure 1, guarantees the specified minimum distance separation between any pair of chairs from different tables, as described in [16]. Figure 1: Distance calculation between tables for the TLP. The interpersonal distance \(d_{i,j}\) is as the distance between the CLE regions of each table.