Issue 44

G. G. Bordonaro et alii, Frattura ed Integrità Strutturale, 44 (2018) 1-15; DOI: 10.3221/IGF-ESIS.44.01 12 Figure 12 : Plot of the coefficients of the mathematical model of the response Y 3 (to be minimized). The mathematical model for Y 4 (to be minimized) is: Y 4 = 169 - 57X 1 (***) - 6X 2 + 70X 3 + 71X 4 + 32X 1 X 2 - 14X 1 X 3 - 28X 1 X 4 - 21X 2 X 3 - 22X 2 X 4 + (6) + 15X 3 X 4 + 17X 1 2 - 49X 2 2 - 28X 3 2 + 22X 4 2 Figure 13 shows the plot of the model coefficients. Three linear terms b 1 , b 3 , b 4 are significant. An increase of the terms X 3 , X 4 and a decrease of the term X 1 causes an increase of the response Y 4 , rolls reaction force. Figure 13 : Plot of the coefficients of the mathematical model of the response Y 4 (to be minimized). By taking into account the four individual models, it can be seen that only the linear terms are significant, (the interaction X 3 -X 4 for the response Y 1 , is also statistically significant, but is negligible). The variable X 2 is never significant. As shown by the PCA, the pairs of responses Y 1 -Y 4 and Y 2 -Y 3 are correlated. Each pair requires one response to be maximized and one to be minimized, therefore a compromise has to be reached. This is possible through multicriteria decision-making methods such as the Pareto optimality approach. A definition of acceptability limits and target values for each response allows to narrow down all possible solutions combinations. M ULTICRITERIA D ECISION M AKING areto optimality approach is applied to investigate predicted responses on a set of 729 points, obtained by the following criteria:  temperature (X 1 ): 9 levels in the range 800:50:1200 °C  rolls angular velocity (X 2 ): constant (17.5 RPM)  billet diameter (X 3 ): 9 levels in the range 20:5:60 mm  billet diameter reduction (X 4 ): 9 levels in the range 20:5:60% P