## 5 Comparison of the Proposed Method with the Weighted Sum Methods
### Loads Are Borne by a Local Region of the Structure
#### 5.2.1 Problem 3  
In this problem, two loading conditions with concentrated loads (LC1 and LC2) are applied to the upper-left and upper-right corners, as shown in Figure 23. Similar to the previous problems, the correlation between the two loading conditions is gradually changed by changing the magnitudes of the forces \(P_{1}\) and \(P_{2}\). Computational results for Problem 3 are provided in Table 8, where \(c_{1}\) and \(c_{2}\) are the compliances under LC1 and LC2, respectively, and Stress level 1 and Stress level 2 represent the stress levels under LC1 and LC2, respectively. The results given in Table 8 show that for all load combinations of this problem, the maximum compliance is always reduced after applying the BCM method (Stage 2) to the TEWS structures (Stage 1). This demonstrates the efficiency of the proposed method in terms of minimizing the maximum compliance. In addition, the stress levels of the BCM structures (Stage 2) are always lower than those of the TEWS structures (Stage 1). More importantly, the maximum stress levels arising under the worst load case in the BCM structures are lower than those in the TEWS structures, implying the effectiveness of the proposed method. Figure 23: Loading conditions for Problem 3: (**a**) Load case 1; (**b**) Load case 2. One important observation in Problem 3 is that whenever the load magnitudes are different, the stress level under the more severe loading condition is consistently much higher than that under the less severe loading condition, no matter which method is used (e.g., Figure 24b). Consequently, it is challenging to find a structure with the same stress level under the two loading conditions. For instance, in the case of \(P_{1}=3.0\) and \(P_{2}=7.0\), despite the load magnitudes in the two loading conditions not differing significantly, the stress level under the more severe load case (i.e., LC2) is significantly higher than the stress level under the less severe load case (i.e., LC1). In addition, Figure 24b shows that the stress level under LC2 is nearly the same for most weight factors (\(w_{1}\)) in the WS methods (from \(w_{1}=0\) to \(w_{1}=0.7\)). Most of the time, the compliance under LC2 is also significantly higher than that under LC1, and the two compliances can only be the same at a very small \(w_{1}\) in the WS methods (see Figure 24a). On the other hand, the BCM method generated a structure in which not only was the compliance the same in both loading conditions (503.04 (see Figure 24a and Table 8)); the stress level under LC2 was also minimized while keeping the stress level under LC1 small. In Problem 3, it is also noticed that the stress level is reduced by minimizing compliance, even though the loads are borne by local regions of the structure. In other words, when concentrated loads are applied to the structure, it is possible to control the stress level by controlling the compliance of the structures. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Load** & **Method** & **Topology** & \(c_{1}\) & \(c_{2}\) & \
\begin{tabular}{c} **Stress** \\ **Level 1** \\ \end{tabular} &
\begin{tabular}{c} **Stress** \\ **Level 2** \\ \end{tabular} \\ \hline \multirow{3}{*}{\(P_{1}=2.0\), \(P_{2}=8.0\)} & TEWS & & 49.56 & **662.47** & 0.285 & **1.119** \\  & BCM & & 635.91 & **654.71** & 0.614 & **1.118** \\ \hline \multirow{3}{*}{\(P_{1}=3.0\), \(P_{2}=7.0\)} & TEWS & & 104.04 & **511.08** & 0.423 & **0.9789** \\  & BCM & & 503.04 & 503.04 & 0.853 & **0.9787** \\ \hline \multirow{3}{*}{\(P_{1}=5.0\), \(P_{2}=5.0\)} & TEWS & & 271.911 & 271.911 & 0.7007 & 0.7007 \\  & BCM & & 271.911 & 271.911 & 0.7007 & 0.7007 \\ \hline \multirow{3}{*}{\(P_{1}=8.0\), \(P_{2}=2.0\)} & TEWS & & **662.47** & 49.56 & **1.119** & 0.285 \\  & BCM & & **654.71** & 635.91 & **1.118** & 0.614 \\ \hline \multirow{3}{*}{\(P_{1}=9.0\), \(P_{2}=1.0\)} & TEWS & & 833.06 & 14.75 & **1.2584** & 0.1626 \\  & BCM & & **827.99** & 806.55 & **1.2583** & 0.3133 \\ \hline \end{tabular} Bold font represents the maximum value(s).  
\end{table}