## 5 Comparison of the Proposed Method with the Weighted Sum Methods
### Loads Are Borne by the Entire Structure
#### 5.1.1 Problem 1
Because of the size change, the compliance under LC2 was reduced from 46.96 (TEWS) to 44.43 (BCM) (see Table 3), slightly improving the performance of the structure. The graph in Figure 14a shows that when changing the weights using the WS methods, it is also possible to find a design, where the compliance under LC1 is the same as the compliance under LC2 (at the intersection of _c1_WS_ and _c2_WS_ curves). The compliance value at that point is greater than the maximum compliance obtained via the BCM method (approximately 45.5 compared with 44.43). This indicates that the minimized maximum compliance may not be found by changing the weights in the WS methods. The graph in Figure 14b illustrates that the structure with the lowest stress level can be obtained using the WS methods at around \(w_{1}\approx 0.34\) (the intersection of the _Stress1_WS_ curve and the _Stress2_WS_ curve, showing the same stress level when considering LC1 and LC2 individually), while the proposed BCM method resulted in a structure with not-so-different stress levels (_Stress1_BCM_ and _Stress2_BCM_), which quite close to that intersection point. Note that in the proposed method, the structure can be obtained automatically and uniquely. This observation demonstrates the capability of the proposed method in terms of reducing the stress level in this load combination. Figure 13: Structures in Problem 1 with \(F_{1}=0.9\) and \(F_{2}=1.0\): (**a**) using the traditional equally weighted sum method (Stage 1); (**b**) using the proposed method. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & \(\mathbf{c_{1}}\) & \(\mathbf{c_{2}}\) & **Stress Level 1** & **Stress Level 2** \\ \hline TEWS & 40.36 & 46.96 & 0.1435 & 0.1642 \\ BCM & 44.42 & 44.43 & 0.1623 & 0.1514 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Compliances and stress levels using the traditional equally weighted sum (TEWS) method and the bisection constraint method (BCM) for Problem 1 with \(F_{1}=0.9\) and \(F_{2}=1.0\). The same conclusion can be drawn from Figure 14 regarding the single run required when using the proposed method compared to the numerous trials with different combinations of weights using the WS method to obtain a unique solution for the considered TO problem under multiple-loading conditions. This emphasizes the distinct ability of the proposed method to automatically generate solutions for designated problems. When \(F_{1}=1.0\) and \(F_{2}=1.0\)  
This is a special load combination, referred to as symmetric loading conditions, for which it is expected that the final topology should be symmetric. In fact, in the proposed framework, the optimization algorithm stops after Stage 1, which consists of TO using the TEWS method. This is because after Stage 1, a symmetric structure is obtained (Figure 15) with the same compliance when applying either LC1 or LC2. Thus, there is no greater compliance to minimize. The graph in Figure 15(a) shows that a structure with the same compliance values under LC1 and LC2 obtained via WS methods can be found at the intersection of the two compliance curves (\(c1\) and \(c2\) curves) at \(w_{1}=0.5\), which is the TEWS method. As previously mentioned, this structure is the same as the one obtained using the proposed framework. By adopting the TEWS method in Stage 1, the proposed method is applicable for this symmetric load scenario. The stress levels in the structures using the WS methods with different weights (\(w_{1}\)) and using the BCM method can be observed in Figure 15(b). This graph illustrates that a structure with minimized stress levels can be found uniquely using the WS methods at \(w_{1}=0.5\), which is the same point as in the TEWS method (or Stage 1 of the proposed framework). Moreover, this point is the same as that of the design with minimized compliance as regarding Figure 15(a). Hence, in this case, both the TEWS and BCM methods yielded a true and unique solution.