## 3 Adaptive Surrogate-Assisted Particle Swarm Optimization (ASAPSO)
We postulate that amidst a plethora of uncertain solutions, the probability of pinpointing the optimal fitness value point intensifies within this segment of the region. As a result, the ensuing equation is employed to chronicle the uncertainty information over intervals and to influence the amplitude of the k-value grounded on these archived data. \[\mathrm{un}=\mathrm{size}(\mathrm{u})/1000 \tag{10}\]  
where un symbolizes the ensemble of uncertain solutions archived by the PSO algorithm for every iteration. The tally of uncertain solutions is monitored per iteration via Goodstep. The initial step size is set at 0.001, in accordance with the recommendations set forth in  
Figure 1: Diagram of uncertain solution- and optimum fitness value-guided optimization. reference [33], and is subsequently recalibrated following the determination of the number of uncertain solutions within the global surrogate model. \[\text{ustep}=(1-\text{c})*\text{ustep}+\text{c}*\text{mean}_{\text{A}}(\text{Sstep}) \tag{11}\]  
where using the parameter archive storing uncertainty values and their mean value for each iteration. c is a value between 0 and 1. In a typical setting, the threshold is initialized at 0.5 to begin with balanced probabilities. The primary objective of this adaptive adjustment is to identify whether a transition to either a local or a global surrogate model is warranted. Such adaptability broadens the exploration spectrum across the entirety of the population, thereby reducing the potential of the algorithm becoming ensnared in a local optimum. Additionally, this methodology adeptly mitigates the challenges of suboptimal convergence observed when solely depending on a static value for decision-making criteria. In the final analysis, the user is designated its expected value through the application of a normal distribution function and maintains a fixed standard deviation of 0.1. This deliberate decision ensures the procurement of values from the primary half of the normal distribution curve. In tandem, this study harnesses three unique surrogate models which are RBE, PR, and KRG. These models underpin the dual evaluation benchmarks of the amalgamated surrogate model and the uncertainty resolution via the OWS methodology. The selection of individuals not only possesses optimal adaptation metrics, but also exhibits pronounced uncertainty resolutions. Such a discerning selection mechanism steers the PSO algorithm towards pinpointing the apex value. The local surrogate model integrates the foremost 75% of the ordered fitness values extracted from the global surrogate model and the latter 25% from the sorted uncertain solutions. Subsequently, the population is subjected to a de-emphasis procedure, drawing from the global surrogate model, thus shaping the population intended for the local surrogate model. Within this localized model, the amalgamated model becomes pivotal for predicting response values, thereby forecasting the fitness values that the surrogate model anticipates for points situated within this domain. If the fitness value derived from the local surrogate model, less the least initial population value, falls below 0.001, or if said fitness value is inferior to the initial population's minimum, the algorithm persists within the global surrogate model's ambit. On the contrary, if the value meets or surpasses this threshold, the course of action redirects to the global surrogate model. A visual representation of the algorithm's architectural blueprint can be viewed in Figure 2.  
Figure 2: Structure diagram of ASAPSO algorithm. The whole process of ASAPSO can be represented as Algorithm 1.