## 5 Comparison of the Proposed Method with the Weighted Sum Methods  
From Section 4, it is found that the proposed method yields good results by minimizing the maximum compliance when solving TO problems under multiple-loading conditions compared to the TEWS method. On the other hand, past research has considered that the proper optimization solution can be obtained for multiple objectives with different degrees of importance by changing the weight factors (WS methods, [29; 34; 35; 36; 37; 38; 39; 40; 41; 42]). In this section, the features of the proposed method and its advantages are discussed by comparing the TO results between this method and the WS methods (including the TEWS method in Stage 1 of the proposed framework).  
In this section, the aforementioned methods were used to solve the same TO problems. For better visualization using graphs, simple 2D problems with only two loading conditions were considered. During the optimization processes, the Von Mises (VM) stress in the structures was observed. Calculations were performed to determine the VM stress for each load case. The VM stress at the center of each element was computed by averaging the stress tensors at the Gauss points (integration points). For the FE analysis, we employed 2D square elements with four Gauss points because the design domains used in the examples were rectangular.  
All the methods were formulated based on the SIMP approach and implemented in MATLAB. Similar to examples used in Section 4, the material was assumed to have [PERSON]'s modulus \(E=1\) and [PERSON]'s ratio \(\
u=0.3\). A volume fraction \(f=0.5\) was considered as a constraint. The derivative of the cost functions (compliance functions) is computed following [PERSON] and [PERSON] [1].  
To assess the performance of the obtained structures, a representative stress level was determined by calculating the average stress level of the 10 elements with the highest VM stress values observed throughout the structures. Subsequently, in this paper, the representative stress level is simply referred to as the stress level, indicating the performance of the structures. This approach effectively captures the stress level in a localized region where high stress occurs while avoiding reliance on outlier values.  
When using the WS methods, all compliance functions can be aggregated into one objective function using Equation (6). For problems with two loading conditions, only two compliance functions associated with the two loading conditions are considered. Hence, the trade-off relationship can be represented using only one parameter: \(w_{1}\) (the weight factor of the first load case). The weight factor of the remaining loading condition can be determined automatically as follows:  
\[w_{2}=1-w_{1}. \tag{18}\]Additionally, in practice, it is common for loads to spread across the entire structure owing to interactions between adjacent elements. This implies that the loads are borne by the entire structure, and as a result, the majority of the elements within the structure contribute to its global stiffness to withstand the applied loads. In such cases, when the compliance sum of all elements is reduced, there is a possibility that the stress level will also reduce. However, in extreme scenarios in which only a limited region of the structure is affected by the applied loads, maintaining a small compliance may not effectively reduce the stress level throughout the structure (see also the discussion in Section 5.2). To evaluate the effectiveness of the proposed method in various cases, two scenarios for testing were considered: (1) the most common scenario, when loads are borne by the entire structure; and (2) the extreme scenario, when loads are borne by local regions of the structure.