## 5 Experimental Testing and Analysis
### Benchmark Test Functions
#### 5.1.2 Time Complexity Analysis  
Time complexity reflects the efficiency of an algorithm, and as the problem size increases, it directly influences both time and resource costs. The time complexity of the proposed algorithm is primarily determined by the population size, the number of iterations, the dimensionality of the problem, and the time required to update the operators. Let N represent the population size, T denote the number of iterations, and D signify the problem dimensionality. Ignoring the time complexity of the operators, the overall time complexity of the algorithm is as follows:  
\[O(DRSTASA)=O(N\cdot D\cdot T) \tag{22}\]  
In a single iteration, let the time complexity of evaluating the objective function for a single solution be \(t_{1}\), and the time complexity of updating a solution using the four operators be \(t_{2}\). Without incorporating any improvement strategies, the time complexity of the algorithm is as follows:  
\[O(DRSTASA)=O(N\cdot D\cdot T\cdot(t_{1}+t_{2})) \tag{23}\]  
When introducing destruction operators and opposition-based learning strategies, let the time complexities of the destruction operator and opposition-based learning be denoted as \(t_{3}\) and \(t_{4}\), respectively. Consequently, the time complexity of the algorithm incorporating these improvements becomes the following:  
\[O(DRSTASA)=O(N\cdot D\cdot T\cdot(t_{1}+t_{2}+t_{3}+t_{4})) \tag{24}\]  
From the analysis, it can be concluded that the time complexity of DRSTASA does not increase by an order of magnitude compared to the original STASA. Additionally, the destruction operator and opposition-based learning strategies enhance the efficiency of the algorithm, resulting in faster convergence.  
Figure 5 shows the average runtime of nine algorithms across 23 test functions. It is observed that the SCA, AOA, and AHA algorithms have the shortest runtimes, but they exhibit slightly lower accuracy compared to the others. AVOA and GWO fall into the second tier, characterized by higher time costs but with improved accuracy. STA, STASA, and DRSTASA are in the third tier, where DRSTASA has a slightly longer runtime than STASA but achieves significantly higher accuracy, making the additional time acceptable. The FDA requires the longest runtime but offers greater stability, albeit at a considerable time.  
To evaluate the effectiveness and stability of DRSTASA, eight engineering design optimization problems were selected for testing. The DRSTASA, STASA, and social network  
Figure 5: Algorithmic average time.  
From Table 6, it is clear that the performance of DRSTASA in the string design problem is not as strong as that of the other two algorithms; however, the algorithm excels in the remaining engineering challenges. In the cantilever beam, tubular column, and welded beam design problems, although DRSTASA achieves optimal values, the average performance and standard deviation still show a gap compared to the other algorithms, suggesting that further improvements in stability are needed. The comparison results indicate that DRSTASA demonstrates significant improvements over STASA in various engineering problems and surpasses the SNS algorithm, highlighting the potential of this approach for solving real-world engineering challenges.