## 3 Methodology
### Analysis of Experiments  
The analysis of variance (ANOVA) based on a general linear model (GLM) for the collected experimental data was performed by MINITAB 18 [53]. For each performance measure, three ANOVA tests for the individual design cases were separately performed to compare the effects of the process parameters. Equation (2) describes the simplified expression of the GLM considered in this study. Statistically significant terms (\(p\)-value \(\leq\) 0.05) in the ANOVA results were identified to interpret the effects of the process parameters on the manufacturing performance. Then, each model was fitted again only with significant terms to identify optimal parameter levels:  
\[y_{ijk}=\begin{array}{l}\beta_{0}+\sum_{i}\beta_{i}x_{i}+\sum_{j}\beta_{j}x_{ j}+\sum_{k}\beta_{k}x_{k}+\sum_{i}\sum_{j}\beta_{ji}x_{i}x_{j}+\sum_{i}\sum_{k} \beta_{ik}x_{i}x_{k}\\ \qquad\qquad\qquad+\sum_{j}\sum_{k}\beta_{jk}x_{j}x_{k}+\sum_{i}\sum_{j}\sum_{ k}\beta_{ijk}x_{i}x_{j}x_{k}+\varepsilon_{ijk},\end{array} \tag{2}\]  
where \(i\) is the factor level of layer thickness (\(i=1,2\)), \(j\) is the factor level of build orientation (\(j=1,2\)), \(k\) is the factor level of printing speed (\(k=1,...,3\)), \(y_{ijk}\) is the value of a response variable (i.e., printing time, dimensional accuracy, and material cost), \(\beta_{0}\) is an intercept, \(\beta_{i}\), \(\beta_{j}\), and \(\beta_{k}\) are the main effect coefficients, \(\beta_{ij}\), \(\beta_{ik}\), and \(\beta_{jk}\) are two-way interaction coefficients, \(\beta_{ijk}\) is a three-way interaction coefficient, \(x\) is the coded value (\(-1,0,+1\)) of each factor level, and \(\varepsilon_{ijk}\) is an error term.  
To derive parameter settings to simultaneously optimize all the manufacturing performance measures for each design case, the Derringer-Sueich method for multi-response optimization [54] was used. Since all the manufacturing performance variables have the same optimality direction (i.e., minimization), the one-sided desirability function (\(d_{i}\)) expressed in Equation (3) was employed for each manufacturing performance measure:  
\[d_{i}=\left(\frac{\hat{y}_{i}-U_{i}}{T_{i}-U_{i}}\right)^{t}, \tag{3}\]  
where \(T_{i}\leq\hat{y}_{i}\leq U_{i}\), \(\hat{y}_{i}\) is a predicted value of performance measure \(i\), \(T_{i}\) = a minimum value of \(i\), \(U_{i}\) = an upper limit of \(i\), and \(t\) = a weight to express the shape of the desirability function.  
For all the predicted values of \(i\), \(T_{i}\) and \(U_{i}\) that satisfy \(\hat{y}_{i}\leq T_{i}\) and \(U_{i}\leq\hat{y}_{i}\), respectively, were chosen to derive \(d_{i}\) (\(0\leq d_{i}\leq 1\)). If \(\hat{y}_{i}\) is at its goal \(T_{i}\), then \(d_{i}\) becomes 1. Then, the desired parameter settings to satisfy the overall manufacturing performance were obtained to maximize the composite desirability (\(D\)) in Equation (4):  
\[D=\left(\prod\
olimits_{i=1}^{n}d_{i}\right)^{1/n}, \tag{4}\]  
For the derivation of D, the response optimizer tool provided in MINITAB 18 was employed in this study; MINITAB uses a reduced gradient algorithm to identify the optimal solution to maximize the composite desirability [55, 56]. Based on the general linear regression model only including the significant factors in each design case, the optimal parameter settings to maximize composite desirability were derived by assuming the linearity of the individual desirability functions (\(t=1\)).  
Figure 4: Measured dimensional specifications of each design type. (**a**) ASTM D638; (**b**) ASTM D695; (**c**) ASTM D3039.