# Sequential Memetic Algorithm Optimization for Allocation Planning in Hostelry Establishments
## 1 Introduction
Such is the case of the chair or table distribution optimization problem. Due to the high complexity of these problems, a direct approach that analyzes each possible solution is unfeasible, so multiple metaheuristic techniques have been implemented for these optimization problems [14]. In our previous works, a Genetic Algorithm (GA) was proposed for optimizing the student distribution in schools during the COVID-19 pandemic [15]. This optimization problem, described as the Table Location Problem (TLP), seeks to obtain the optimal distribution of a given number of tables over a specific scenario that guarantees the minimum distance separation and maximizes the mean interpersonal distance while also taking into consideration the scenario properties. The algorithm located a given number of tables within the scenario limits in an optimized distribution for the scenario geometry, while also respecting the scenario obstacles. Later on, we designed a Memetic Algorithm (MA) for escalating the TLP to hostlery applications [16]. In these scenarios, the space of solutions increases significantly due to the consideration that a variable number of people may be allocated in each table. The proposed MA was further enhanced with the introduction of memory chains for recalling each individual potential for improvement in [17], improving the overall exploration and exploitation of the available space, resulting in an overall better performance. However, our previous approach dissociated the table and chair allocation problem. This decision was selected based on the regular operation of a common hostlery establishment, where each table may accommodate a variable and unknown number of customers [18]. Nevertheless, certain establishments where celebrations or events are held and the number of assistants is known may prefer optimizing the chair and table distribution for allocating a higher number of customers or obtaining a more distanced disposition [19]. The table and chair allocation problem has never been attained simultaneously in an unbounded optimization. [PERSON] et al. [7] analyze different table and chair distribution patterns for different scenario geometries. This provides an efficient approach to the problem; however, the performance of these distributions is compromised for scenarios with a high obstacle density, thus representing a suboptimal approach that does not completely exploit the whole space of solution, resulting in potential empty space being unexploited. [PERSON] et al. [20] apply metaheuristic techniques for optimally allocating tables in hostelry scenarios; nevertheless, this study only focused on locating the tables without analyzing varying seating positions or the presence of obstacle regions. Therefore, this work provides an incomplete solution to the CLP, as the chair arrangement is not taken into consideration when deploying the tables. On the contrary, [PERSON] et al. [6] implemented a heuristic-based algorithm for optimally distributing a certain number of chairs over a given scenario, considering different constraints and obstacles. However, while tables of varying geometry are taken into consideration for the optimization, these table positions are defined as fixed constraints, prior to the optimization, which limits the number of possible solutions as well as the distance distribution optimization potential. Therefore, in this paper, a combined optimization of the table and chair distribution is proposed for hostelry applications. The exploration of the space of solutions growth introduced by establishing variable table positions may disclose solutions that further optimize the distance distribution, whereas the previously defined constrained optimizations are unable to achieve such solutions. However, the combination of two NP-Hard problems, such as the table and chair distribution, may result in a computational cost unacceptable for daily operations in certain establishments.