## 2 Problem Formulation  
### Fitness Function  
The UA-FLP deals with the arrangement of \(n\) rectangular facilities, each one of a given area \(A_{i}\) within an available space of dimensions \(W\times H\) (Equation (1)) and without overlapping between them. \[\sum_{i}^{n}A_{i}\leq W\times H \tag{1}\]  
Furthermore, there is one constraint related to the geometrical proportions of each of the facilities. In this sense, the parameter _aspect ratio_ is defined as the quotient of the largest side of a facility by the shortest one, and it can be a general restriction that is equal for all the plant's facilities or a particular limit for each one of them. With these restrictions, the parameter to be optimized (minimized) is the material handling cost between facilities, given the distance between the centroids of every two facilities \(d_{ij}\), and the material flow between them (units of product per unit of time, kg per unit of time, etc. ), \(f_{ij}\). The problem consists of finding the disposition of the \(n\) facilities within the plant while respecting the restrictions and minimizing the total material handling cost (Equation (2)). \[minimize\sum_{i=1}^{n}\sum_{j\
eq i}^{n}f_{ij}\times d_{ij} \tag{2}\]  
The general fitness function must consider the minimization of the material handling cost, as shown in Equation (2), and penalizing candidate solutions that do not respect the aspect ratio constraints. On the contrary, if the fitness function only considers the minimization of the material handling costs, the search could converge quickly in unfeasible solutions from the point of view of the aspect ratio of one or more facilities. To avoid this problem, [PERSON] and [PERSON] [51] proposed the use of a penalization for the solutions that contain these unfeasible facilities, adding a penalty value proportional to the number of unfeasible facilities (if any). In this way, the fitness function finally considered is shown in Equation (3). \[V_{t}=\sum_{i=1}^{n}\sum_{j\
eq i}^{n}f_{ij}d_{ij}+(D_{inf})^{k}(V_{feas}-V_{ all}) \tag{3}\]  
where:  
\(V_{t}\) is the value of the fitness function with penalty for a specific plant. \(f_{ij}\) and \(d_{ij}\) are the material flows between departments and the distances between centroids, respectively. \(D_{inf}\) is the number of unfeasible facilities. \(k\) is a penalty parameter that adjusts the grade of penalization (set to 3, according to [PERSON] and [PERSON] [51]). \(V_{feas}\) is the best fitness value found in the set of feasible solutions. \(V_{all}\) is the overall best fitness value found. Note that in the case of \(D_{inf}=0\), the value of the fitness function is simply the material handling cost. Nevertheless, since the collaborative intervention of several expert designers (or decision-makers, DMs) has to be incorporated into the evaluation, the fitness function must be modified [10]. The expert designers' evaluation is a discrete value on a Likert scale from 1 (not satisfactory) to 5 (very satisfactory). To fulfill this new requirement, an extra penalty function has been added (Equation (4)):  
\[U_{t}=(5-x)\times\frac{n}{4} \tag{4}\]  
where:  
\(U_{t}\) is the factor for the extra penalty. \(x\) is the evaluation assigned by the expert. \(n\) is the plant's number of facilities. Therefore, if the DM assigns a score of 5 (very satisfactory), the extra penalty factor is 0, and in the other extreme, if the DM assigns a valuation of 1 (not satisfactory), the extra penalty factor is \(n\). The final fitness function to be applied is shown in Equation (5):  
\[V_{t}=(1+(U_{t})^{k})\sum_{i}^{n}\sum_{j}^{n}f_{ij}d_{ij}+(D_{inf})^{k}(V_{ feas}-V_{all}) \tag{5}\]  
In this way, if the evaluation assigned by the DM is 5 and the number of unfeasible facilities is 0, the value obtained by the plant is equal to the material handling cost; moreover, if the evaluation assigned by the DM is 1, the material handling cost is multiplied by \(1+n\).