## 5 Comparison of the Proposed Method with the Weighted Sum Methods
### Loads Are Borne by the Entire Structure  
#### 5.1.1 Problem 1  
Consider a cantilever beam subjected to two simple loading conditions (concentrated loads), as shown in Figure 9. In this problem (Problem 1), we consider several load combinations (i.e., situations with different load correlations between the magnitudes of LC1 and LC2), in which the magnitude of LC2 remains the same in all load combinations, and the magnitude of LC1 is changed from small to large, representing the change in the importance of the two loading conditions. A mesh with 60 elements in the horizontal direction and 40 elements in the vertical direction was used in this  
As shown in Figure 11, the algorithm underwent 11 outer loops with approximately 500 inner iterations. During this process, compliance under LC2 was always treated as the objective function (the number shown on the graph in Figure 11a is the index of the maximum compliance occurred at every outer loop, namely, \"2\" for the compliance under LC2). Figure 11b also shows that a reduction in the compliance under LC2 (the most important loading condition) can be achieved when increasing the compliance under LC1. This is a good example of the trade-off relationship between different loading conditions. When using the WS methods, by changing the weight (\(w_{1}\)), it is possible to find the weight where the compliances under LC1 and LC2 are the same (at the intersection of _c1_WS_ and _c2_WS_ solid curves in Figure 12a). The compliance at that point is the same as the maximum compliance in the structure obtained via the proposed method (denoted by _c1_BCM_ and _c2_BCM_ dotted lines in Figure 12a). In this case, the two dotted lines (_c1_BCM_ and _c2_BCM_) overlap. It is noted that the advantage of the proposed method is that a solution with minimized maximum compliance can be obtained automatically, i.e., without any weighting of the importance of the load cases or other subjective manners. The stress levels in the structure considering the two loading conditions using the WS methods with different weights are plotted in Figure 12b (_Stress1_WS_ and _Stress2_WS_ solid  
Figure 11: Evolution of compliances in Problem 1 with \(F_{1}=0.2\) and \(F_{2}=1.0\) during Stage 2: (a) the maximum compliance only, with the indices of the corresponding loading conditions; (b) all compliances under each loading condition. Figure 12: Structure performance using the weighted sum (WS) methods and the bisection constraint method (BCM) for Problem 1 with \(F_{1}=0.2\) and \(F_{2}=1.0\): (a) compliance; (b) stress level. curves). In this graph, the stress levels in the structure obtained using the proposed method can also be observed, as represented by the blue (_Stress1_BCM_) and red (_Stress2_BCM_) horizontal dotted lines. A comparison of the stress levels in structures obtained using the WS methods and the structure obtained using the proposed BCM method shows that the maximum stress level is minimized in the BCM structure (the _Stress2_BCM_ line seems to intersect with the _Stress2_WS_ curve near the lowest point of the _Stress2_WS_ curve). This example illustrates that in this loading correlation, the proposed method can also provide a solution with a low stress level in addition to minimizing the maximum compliance. Moreover, Figure 12 illustrates that in order to obtain a unique solution having whether minimized maximum compliance or minimized stress level, conventional WS method necessitates numerous trials with different weight combinations, whereas the proposed method requires only one single run. When \(F_{1}=0.9\) and \(F_{2}=1.0\)  
In this load combination, the magnitudes of loads in LC1 and LC2 are nearly the same; hence, the structure obtained using the TEWS method (Figure 13a) is nearly symmetric. The structure obtained via the BCM method (Figure 13b) has the same topology as the TEWS structure, except for the size of the ties inside the structure; i.e., the tie bearing LC2 became wider, and the tie bearing LC1 became thinner.