# A Particle Swarm and Smell Agent-Based Hybrid Algorithm for Enhanced Optimization
## 3 Results and Discussion  
This section presents the results of evaluating the PSO-SAO algorithm on benchmark test functions, as well as its comparison with the original PSO [1] and SAO algorithms [27]. Additionally, we report the fitness results of PSO-SAO on the benchmark functions for nine specified hopping frequencies. These results are essential to demonstrate the efficacy of the PSO-SAO algorithm's search process. Figures 3 and 4 provide the outcomes of running each function 10 times with hopping frequencies ranging from 0.1 to 0.9, which were transformed to values from 1 to 9 in the charts. In particular, Figure 3 illustrates the minimization process for the PSO-SAO algorithm on the first benchmark function (Adjiman function) using varying hopping frequencies. Notably, PSO-SAO exhibits the fastest convergence at a hopping frequency of 8. Figure 4 presents the average fitness plot for the nine hopping frequencies applied to the Adjiman function. It is apparent from Figure 4 that hopping frequencies between 0.1 and 0.5 yield identical average fitness values, while a hopping frequency of 0.9 produces the highest average fitness. This observation explains why the hopping frequency of 9 (on a scale of 1-10) achieves the best convergence rate in Figure 3. In this figure, the differences in the fitness values are indeed minimal, primarily because the Adjiman function considered here served as an easily solvable problem for the PSO-SAO algorithm. Nevertheless, we included it here because it broadly mirrors how the hopping frequency responds to numerous other functions. Furthermore, this difference in the fitness values only becomes more prominent when dealing with difficult problems with higher-dimensional and multimodal characteristics. Extensive analyses were conducted for all of the test functions in Table 1, with similar outcomes obtained as explained here, but due to space constraints, only the results for the Adjiman test function were reported. Figure 3: Fitness values for the PSO-SAO on Adjiman test function showing convergence of all Hf. Table 3 presents a comprehensive summary of the outcomes pertaining to both the average and standard deviation values for each algorithm when applied to the set of 37 functions. Notably, the PSO-SAO algorithm emerged as particularly successful in its ability to converge to global solutions, as highlighted by the use of the bold red font in Table 3. The ground truth global values, which served as benchmarks for assessing the performance of the algorithms, can be cross-referenced in Table 1. For instance, functions F1, F2, and F3, corresponding to the Adjiman, Beale, and Bird functions, respectively, exhibited true global solutions of \(-2.0218\), \(0\), and \(-106.765\), respectively, as documented in Table 1. Strikingly, a comparison of these global values with those attained by the PSO-SAO algorithm, as indicated in Table 3, revealed a good degree of approximation. The PSO-SAO algorithm achieved an average global solution of \(-2.0218\) for the Adjiman function F1 (see Table 3). A similar result was obtained for SAO, with values of \(-2.0218\) for F1. PSO, on the other hand, obtained an average of \(-2.0134\), thus resulting in a larger error. These results highlight that both the PSO-SAO and SAO algorithms converged to the exact global solution with high precision (\(-2.0218\)) compared with the PSO algorithm. Other results can be observed for the other functions in Table 3.  
We present a concise summary of algorithm rankings based on the number of functions for which they achieved optimal values in Table 4. It is important to note that the determination of optimal values was established through the calculation of the absolute errors between the average fitness values attained by each algorithm and the global solutions for each function, as presented in Table 1. The results indicate that the PSO-SAO algorithm achieved the highest ranking, demonstrating the best performance in 28 out of the 37 functions, resulting in a 76% success rate.