## 2 Background Work
### Multi Criteria Decision Methods as a Formal Support to an Objective Design Process  
In the context of fostering the spread of AM applications, it is beneficial to help designers understand when and how to adopt AM technologies to improve designs with better performing and cost-saving solutions. There is a need for a broader and holistic assessment that includes diverse aspects, as well as the possibility of adopting different and new design principles. The discipline of DFAM targets the geometric shape definition phase, while additional efforts are needed to fully understand the potential of AM in an earlier stage, when multiple, conflicting, and often fuzzy requirements must be evaluated. Therefore, to define a structured evaluation workflow to assess the design opportunities, decision-making approaches must be considered.  
Multi Criteria Decision Analysis (MCDA) [28] is the discipline aimed at supporting the decision-maker while operating with numerous and conflicting evaluations, allowing them to obtain a compromise solution in a structured way [29; 30]. Contrary to numeric simulation tools, which rely on precise quantities, MCDA is better suited to weight subjected, discrete or fuzzy evaluations.  
Scarce applications of such methods in early AM design phases are found in the literature. Among them, in [31] the authors have used the Analytical Hierarchical Process (AHP) procedure to rank the most appropriate technologies and machines. Another study in this sense was developed by [PERSON] et al. in [32], where the authors have introduced a generic decision methodology to provide a set of compromised AM materials, processes, and machines and to guide designers to achieve feasible material-machine combinations from a current database of AM vendors in the world.  
Weighted Sum Model (WSM) and AHP appear to be able to support the qualitative and quantitative analyses required by the scope of this paper. The two approaches are briefly recalled here. The AHP method is based on a series of pairwise comparisons between a set of criteria, giving them a score of relative importance and assigning percentage weights [33]. An example of the values for the pair evaluation is reported in Table 2 below.  
Subsequently, a matrix \(nxxn\) is constructed, where \(n\) is equal to the number of criteria to be compared. Supposing that \(n\) = 3 (D1, D2 and D3), the matrix has the following construction (Table 3):  
In Table 3 X is the relative importance value of D1 compared to D2, Y is the relative importance value of D1 compared to D3 and Z is the relative importance value of D2 compared to D3. To calculate the vector of the percentage weights to assign to each driver, it is enough to determine the maximum eigenvalue \(\lambda\) and the relative eigenvector \(V_{\lambda}\) of the matrix [32]. Normalizing the eigenvector \(V_{\lambda}\) so that the sum of its elements is equal to 1, the vector of the percentage weights or priorities relating to the driver Di is obtained. Once the priority vector has been determined, it is fundamental to understand the quality of the pairwise matrix, measuring the consistency of the decision-maker's subjective judgments. To this aim, the Consistency Index (\(CI\)) is defined as (Equation (1)):  
\[CI=\frac{\lambda-n}{n-1}, \tag{1}\]  
Finally, the Consistency Ratio (\(CR\)) is defined as follows (Equation (2)):  
\[CR=\frac{CI}{RI}, \tag{2}\]  
where \(RI\) is the Random Index, which is a value that depends on the number of criteria as explained in [33]. For \(n>4\), the pairwise matrix is consistent if \(CR\) is lower than 0.1.  
On the other hand, the WSM is used to find the best solution among those available, thus solving a decision-making problem. Here, the criteria values (\(aij\)) are multiplied by the weights calculated with AHP (\(w\)), and finally the sum is performed for each alternative (\(Ai\)), as in Equation (3). The criteria values should be normalized to have the same unit [34].  
\[Ai=\sum_{j=1}^{n}wj\times aij, \tag{3}\]