# Fuzzy MCDM Methodology for Analysis of Fibre Laser Cutting Process
## 3 Results and Discussion
### Modelling of the Aggregate Fuzzy Decision-Making Rule  
The application of the proposed fuzzy MCDM methodology has so far enabled the assessment and ranking of existing solutions, i.e., laser cutting conditions that were tested in real manufacturing conditions, in the conducted experiment. In order to obtain insight into how laser cutting parameters (inputs) affect the ranking and provide useful information regarding the selection of favourable input values, with respect to the considered criteria, one needs to find a suitable approximation of the functional relationship between inputs (independent variables) and output (dependent variable), which in this study may refer to the average utility value assigned to each alternative. To this aim, given the type and resolution of the experimental matrix, in Step 6 of the proposed fuzzy MCDM methodology, a full quadratic model was developed (Table 4). Figure 5: The changes of alternative ranks relative to the solution obtained with original weights: (a) Case 1 (\(\pm\)10%), (b) Case 2 (\(\pm\)25%). As can be observed from Table 4, the linear and the quadratic effect of the cutting speed are statistically significant, i.e., one can argue that the cutting speed has dominant influence regarding the considered criteria with associated criteria weights and consequent ranking of a given cutting condition. In addition, the results indicate a statistically significant second-order model (\(p<0.05\)), which is able to explain 96.6% of variation in the response (average utility value). Therefore, the developed model could serve to identify regions in the covered experimental hyper-space which have higher response values. Considering that it was determined that the assist gas pressure (\(p\)) has no statistically significant effect on the response, and as it is well known that cutting costs rise with an increase in assist gas pressure, a single contour diagram was developed (Figure 6). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Term** & **Coefficient** &
\begin{tabular}{c} **SE of** \\ **Coefficients** \\ \end{tabular} & **T** & **P** \\ \hline Constant & 0.5213 & 0.0411 & 12.670 & 0.001 \\ \(f\) & 0.0158 & 0.0145 & 1.087 & 0.357 \\ \(v\) & 0.1067 & 0.0145 & 7.334 & 0.005 \\ \(p\) & \(-\)0.0223 & 0.0145 & \(-\)1.535 & 0.222 \\ \(f\times v\) & 0.0468 & 0.0206 & 2.277 & 0.107 \\ \(f\times p\) & \(-\)0.0104 & 0.0206 & \(-\)0.504 & 0.649 \\ \(v\times p\) & \(-\)0.0102 & 0.0206 & \(-\)0.498 & 0.653 \\ \(f^{2}\) & \(-\)0.0108 & 0.0272 & \(-\)0.398 & 0.717 \\ \(v^{2}\) & -0.1024 & 0.0272 & \(-\)3.762 & 0.033 \\ \(p^{2}\) & 0.0301 & 0.0272 & 1.107 & 0.349 \\ \hline \hline \(S=0.0411\), \(R^{2}=0.966\), \(R^{2}_{adj}=0.866\) & & & & \\ \hline
**Source of** & **Sum of** & **Degree of** & **Mean Square** & **F** & **P** \\
**Variation** & **Squares** & **Freedom** & **Mean Square** & **F** & **P** \\ \hline Second order & 0.1464 & 9 & 0.0163 & 9.60 & 0.044 \\ \(\begin{array}{c}\text{Linear}\\ \text{Interaction}\\ \text{Square}\\ \text{Total}\end{array}\) & 0.0971 & 3 & 0.0324 & 19.11 & 0.019 \\ \(\begin{array}{c}\text{Interaction}\\ \text{Square}\\ \text{Total}\end{array}\) & 0.0096 & 3 & 0.0032 & 1.90 & 0.306 \\ \(\begin{array}{c}\text{Square}\\ \text{Total}\end{array}\) & 0.0397 & 3 & 0.0132 & 7.81 & 0.063 \\ \(\begin{array}{c}\text{Residual error}\\ \text{Total}\end{array}\) & 0.0051 & 3 & 0.0017 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Model summary and analysis of variance (ANOVA). Figure 6: Contour diagram showing the change in response for different values of cutting speed (\(v\)) and focus position (\(f\)) for constant assist gas pressure (\(p\) = 0.75 bar). From Figure 6, one can observe that smaller cutting speeds, irrespective of focus position, result in lower response values, i.e., one obtains cuts which are less favourable. In contrast, the process window with the highest response values is enclosed within middle to high cutting speeds (\(v\) = 2.7-3.6 m/min) and higher focus positions (\(f\) = 1.5-2 mm).