## 5 Comparison of the Proposed Method with the Weighted Sum Methods
### Loads Are Borne by the Entire Structure
#### 5.1.2 Problem 2
Figure 19a shows that by using the WS methods, it is possible to find a structure with the lowest maximum compliance at weight \(w_{1}\), which is very close to 0, at the intersection of the two compliance curves _c1_WS_ and _c2_WS_. The compliance in the structure obtained using the proposed method is slightly lower than the compliance at the intersection point, emphasizing the effectiveness of the proposed method in determining the minimized maximum compliance. Figure 18: Obtained structures for Problem 2 when \(P_{1}=0.5\) and \(P_{2}=9.5\): **(a)** using the traditional equally weighted sum method; **(b)** using the bisection constraint method. As shown in Figure 19, in this case, the two stress curves of the WS methods (_Stress1_WS_ and _Stress2_WS_ curves) do not intersect with each other. Therefore, the best structure can be roughly estimated at the bottom of the _Stress1_WS_ curve, which is close to the point where \(w_{1}=0\). It is not difficult to realize that the BCM structure possesses satisfactory properties as an optimal design, where the maximum stress level when considering both loading conditions is approximately at the minimum value (see Figure 19). It should again be noted that the structure can be automatically obtained using the BCM method. (b) When \(P_{1}=1.0\) and \(P_{2}=9.0\)  
In this load combination, we encounter the same issue regarding the realism of the obtained structures when using both the TEWS and BCM methods, requiring post-optimization fine-tuning (Figure 20). The material was rearranged from the TEWS structure to the BCM structure; hence, its compliance was reduced from 316.71 to 300.23, as shown in Table 6.  
The graph in Figure 21a shows that the structure with minimized maximum compliance using the WS methods is at some weight \(w_{1}\) between 0.01 and 0.05. That compliance value is almost the same as the compliance of the structure obtained using the BCM method (_c1_BCM_ and _c2_BCM_ lines seem to pass through the intersection between _c1_WS_ and _c2_WS_ curves). This emphasizes the effectiveness of the proposed method in determining the minimized maximum compliance. Figure 21b illustrates the stress levels in the structures obtained via the WS methods with different weight combinations and the structure obtained via the proposed BCM method. The graph shows that the structure with the lowest stress level can be obtained using the WS methods at the weight \(w_{1}\approx 0.27\), where the stress levels are expected to be around 0.16 (the intersection of the _Stress1_WS_ and _Stress2_WS_ curves). Note that when using the WS methods, the weight combination must be assigned such that the stress level is the same in all loading cases, which is generally challenging, especially when there are more than two loading conditions. Additionally, the stress level in the structure obtained via the BCM method when applying LC2 is lower than the stress level at the intersection point mentioned above (the red dotted line is lower than the intersection between _Stress1_WS_ and _Stress2_WS_ curves in Figure 21b). However, the stress level under LC1 in this structure is significantly higher (0.21627 compared with 0.15325, Table 6). In this load combination, LC1 is generally less severe than LC2 due to the significantly smaller load magnitude. As explained in Section 3, the BCM method focuses on improving the performance of the structure considering both loading conditions in terms of minimizing compliance. Therefore, after Stage 1 (the TEWS method), the compliance under LC2 reduced, which was the more severe loading condition. Hence, the stress level under LC2 also reduced (from 0.17372 to 0.15325, (Table 6)). Simultaneously, the stress level under LC1 increased (from 0.13203 to 0.21627, (Table 6)), even when the compliance under LC1 did not exceed that under LC2. This situation necessitates examining the distribution of the Von Mises stress in the structures obtained using the TEWS and BCM methods.