## 2 Materials and Methods
### Analytical Modeling MC-MD System Methodology  
Based on a method of complex product-systems development, we proposed a structured approach to map, one-to-one [49; 57], the instances of congruence between the environment's architecture and the work organization process in an ED.  
The mapping of congruence was achieved using the following equation for determining the potential of interactions between the domains [49; 52]:  
\[eP_{mm}=R_{mm}^{t}.A_{nn}.R_{nm} \tag{1}\]  
\[aP_{nn}=R_{nm}.E_{mm}.R_{mn}^{t} \tag{2}\]  
where the matrices are identified as follows:  
* \(eP-\)Potential environment relationships matrix;
* \(aP-\)Potential activity relationships matrix;
* \(E-\)Environment matrix;
* \(A-\)Activities matrix;
* \(R-\)Relationship matrix between environment and activities.  
Equation (1) indicates the environmental congruence with the activities, and Equation (2) indicates the congruence of the activities with the environment. The \(R\)-transposed matrix (\(R^{t}_{mm}\)) allows all relationships between environments and activities to be considered even when the other has a different size [49].  
The \(A\) and \(E\) matrices are binary DSMs. \(A_{nm}\) indicates if activity \(j\) depends on, or is sequential to, activity \(i\); whereas \(E_{mn}\) indicates if environment \(j\) is connected to environment \(i\). Both matrices accept only 1 or 0 values for yes or no. The \(R\) matrix defines the relationship between the environment and the activities; therefore, it is a multi-domain matrix (MDM). This matrix accepts three values (Table 1) to establish strong and weak relationships between each activity and environment. The intermediary activities are caied out to make the end activity possible; end activities respond to service goals and add value.  
Finally, the comparison defines the organizational alignment between environments and activities in the congruence matrices. There are two matrices. The activity congruence matrix (\(aC\)) indicates the activities that interact because they occupy the same environment. The environment congruence matrix (\(eC\)) indicates environments that have an interface because they accommodate actions that are necessary for an activity (taking into consideration strong interactions).  
Three conditions were determined for each comparison matrix, as shown in Equation (3) for environments (\(eC_{mm}\)) and Equation (4) for activities (\(aC_{mn}\)), as follows:  
\[eC_{mm}=\begin{cases}eC_{ij}=3&\text{if}\quad E_{ij}=eP_{ij}\\ eC_{ij}=2&\text{if}\quad E_{ij}>eP_{ij}\\ eC_{ij}=1&\text{if}\quad E_{ij}<eP_{ij}\end{cases} \tag{3}\]  
\[aC_{mn}=\begin{cases}aC_{ij}=3&\text{if}\quad A_{ij}=aP_{ij}\\ aC_{ij}=2&\text{if}\quad A_{ij}>aP_{ij}\\ aC_{ij}=1&\text{if}\quad A_{ij}<aP_{ij}\end{cases}. \tag{4}\]  
These three conditions were used to fill each \(eC_{ij}\) or \(aC_{ij}\) cell in the congruence matrices. To facilitate understanding and visualization, the values of each cell were represented by a color and its respective meaning (Table 2).  
The input data used to relate the activities and environments linked to the ES were treated according to this analytical model. In addition to identifying data from these two domains, stakeholders, processes, and organizations were considered as variables.