## 3 Multi-Objective Optimization
### Multi-Objective Optimization Based RSM  
Multi-objective optimization (MOO) is a method of multiple criteria decision making, concerned with mathematical optimization problems where more than one objective function needs to be optimized  
Figure 8: Effects of the three factors on the machine’s performance. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\mathrm{w_{t}}\) & \(\mathrm{w_{am}}\) & \(\mathrm{t_{tm}}\) & \(\mathrm{P_{m}}\) & \(\mathrm{P_{av}}\) & \(\mathrm{P_{d\_m}}\) & \(\mathrm{P_{d\_av}}\) & \(\mathrm{F_{m}}\) \\ (mm) & (mm) & (mm) & (W) & (W) & (W/cm\({}^{3}\)) & (W/cm\({}^{3}\)) & (N) \\ \hline
6 & 8 & 8 & 335.67 & 111.71 & 0.328 & 0.109 & 2729.24 \\
6 & 8 & 3 & 284.10 & 104.73 & 0.277 & 0.102 & 2444.29 \\
6 & 12 & 8 & 262.85 & 105.12 & 0.258 & 0.103 & 3826.74 \\
6 & 12 & 3 & 229.62 & 100.34 & 0.225 & 0.098 & 3481.17 \\
10 & 8 & 8 & 302.74 & 100.40 & 0.277 & 0.092 & 1789.28 \\
10 & 8 & 3 & 275.41 & 99.80 & 0.252 & 0.091 & 1635.07 \\
10 & 12 & 8 & 308.61 & 106.34 & 0.283 & 0.098 & 2876.06 \\
10 & 12 & 3 & 288.78 & 108.33 & 0.265 & 0.100 & 2835.73 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Factors and machine’s performance. simultaneously. The output of the MOO is a set of solutions that define the best trade-off between various physical responses [18; 19; 20]. In electrical machine design, MOO using the RSM is widely employed for modeling performance. The main idea of the RSM is to use a sequence of designed experiments to obtain an optimal response [21; 22; 23; 24; 25; 26]. In RSM, a polynomial model, called a fitted model, is generally constructed to represent the relationship between the performance and the design parameters [21]. Figure 9 shows the effect plots for the major factors. To evaluate the effectiveness of the MOO, several cases were studied as shown in Table 4.  
The relationship between the machine's performance and the two selected factors can be presented by fitted second-order polynomial functions as follows:  
\[P_{m}=195.9+35.03w_{t}+7.49w_{am}-4.74w_{t}^{2}-2.743w_{am}^{2}+4.541w_{t}w_{am} \tag{2}\]  
\[P_{av}=44.6+14.17w_{t}+6.93w_{am}-1.4745w_{t}^{2}-0.7897w_{am}^{2}+0.894w_{t}w_ {am} \tag{3}\]  
\[P_{d\_m}=0.2263+0.0284w_{t}+0.00584w_{am}-0.004552w_{t}^{2}-0.002584w_{am}^{2} +0.004456w_{t}w_{am} \tag{4}\]  
\[P_{d\_av}=0.0566+0.01149w_{t}+0.00637w_{am}-0.00139w_{t}^{2}-0.00074w_{am}^{2} +0.000867w_{t}w_{am} \tag{5}\]  
\[F_{m}=229+18w_{t}+424w_{am}-27.26w_{t}^{2}-12.09w_{am}^{2}+15.7w_{t}w_{am} \tag{6}\]  
Firstly, if power and power density are optimized while electromagnetic force is freely varied without any constraints, the responses for tooth width \(\rm{w_{t}}\) and axial PM width \(\rm{w_{am}}\) are 6.8 and 8.2 mm, respectively. In this case, maximum \(\rm{P_{m}}\) and average power \(\rm{P_{av}}\) are 345.7 and 126.4 W, maximum \(\rm{P_{d\_m}}\) \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Item**} & \multicolumn{1}{c}{**Case 1**} & \multicolumn{1}{c}{**Case 2**} & \multicolumn{1}{c}{**Case 3**} & \multicolumn{1}{c}{**Target**} \\ \hline \multirow{5}{*}{Setup} & \(\rm{P_{m}}\) & Maximize & Maximize & Maximize & 250 \\  & \(\rm{P_{av}}\) & Maximize & Maximize & Maximize & 100 \\  & \(\rm{P_{d\_m}}\) & Maximize & Maximize & Maximize & 0.250 \\  & \(\rm{P_{d\_av}}\) & Maximize & Maximize & Maximize & 0.100 \\  & \(\rm{F_{m}}\) & Free & Minimize & \(\leq\)2000 & \(\leq\)2000 \\ \hline \multirow{5}{*}{Response} & \(\rm{w_{t}}\) & 6.8 & 7.4 & 8.6 & - \\  & \(\rm{w_{am}}\) & 8.2 & 7.2 & 7.2 & - \\  & \(\rm{P_{m}}\) & 345.71 & 349.20 & 339.32 & 250 \\  & \(\rm{P_{av}}\) & 126.43 & 125.22 & 121.60 & 100 \\  & \(\rm{P_{d\_m}}\) & 0.332 & 0.333 & 0.318 & 0.250 \\  & \(\rm{P_{d\_av}}\) & 0.121 & 0.119 & 0.114 & 0.100 \\  & \(\rm{F_{m}}\) & 2683.30 & 2121.39 & 1755.07 & \(\leq\)2000 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Factors and machine’s performance. Figure 9: Main effect plot. and average power density \(\mathrm{P_{d\_av}}\) are 0.332 and 0.121 W/cm\({}^{3}\), respectively.