## 2 Theoretical Background and Problem Statement
### Improved Sparrow Search Algorithm (ISSA)  
The SSA is a nature-inspired optimization algorithm inspired by the foraging behavior and predator evasion tactics of sparrows [7]. In order to solve various optimization problems, this algorithm simulates the patterns of behavior exhibited by sparrow populations when searching for food, being vigilant, and escaping from predators. The sparrow population is typically divided into two main roles: leader sparrows (explorers) and followers. Leader sparrows primarily explore unknown areas to find resources, while followers change their positions based on the information provided by the leader sparrows [25]. Firstly, it is necessary to initialize the positions of the sparrow population. Each sparrow represents a potential solution. Sparrows update their positions either randomly or in a way that is partially guided by relevant information in order to simulate foraging. The formula is often expressed as follows:  
\[x_{i}(t+1)=x_{i}(t)+r_{1}\times(x_{best}(t)-x_{i}(t))+r_{2}\times(x_{i}(t)-x_ {worst}(t)) \tag{14}\]  
where \(x_{i}(t)\) is the position of sparrow i at time t, \(x_{best}(t)\) and \(x_{worst}(t)\) are the best and worst positions in the current population, respectively, and \(r_{1}\) and \(r_{2}\) are random numbers. While searching for food, sparrows must also remain vigilant to avoid becoming prey. This is typically achieved through another position updating strategy, such as the following:  
\[x_{i}(t+1)=x_{i}(t)+r_{3}\times sgn(r_{4}-0.5)\times(LB+r_{5}\times(UB-LB)) \tag{15}\]where \(r_{3}\), \(r_{4}\), and \(r_{5}\) are random numbers, \(sgn\) is the sign function, and \(LB\) and \(UB\) are the lower and upper bounds of the solution space, respectively. When sparrows perceive the threat of a predator, they rapidly change location to evade the threat. This process can be modeled with the following formula:  
\[x_{i}(t+1)=x_{rand}(t)-r_{6}+|x_{rand}(t)-x_{i}(t)| \tag{16}\]  
where \(x_{rand}(t)\) is a randomly chosen sparrow's position from the population, and \(r_{6}\) is a random number. As shown in Figure 2, the improved sparrow search algorithm (ISSA) begins by accepting inputs, such as the population size, maximum iterations, and the lower and upper bounds of the search space. Initially, the population is generated through good point set (GPS) mapping, which enhances the diversity of the population and ensures better coverage of the search space. Following the initialization, the fitness of each individual in the population is calculated using the objective function, and the individuals are sorted based on their fitness values in order to identify the best and worst solutions. As the algorithm progresses, a proportion coefficient is calculated based on a warning threshold (ST) and a disturbance factor (k). This is used to dynamically adjust the balance between exploration and exploitation throughout the iterations. The next step involves calculating the number of explorers (discoverers) and joiners within the population. Explorers, responsible for finding new directions in the search space, update their positions using an improved formula or random perturbations, depending on the current fitness landscape and the global best solution. Joiners, on the other hand, update their positions relative to other individuals in the population, incorporating noise or differential vectors to refine their search. After updating positions, the average fitness of the population is calculated, serving as a criterion for determining which individuals undergo mutation-based operations. Individuals with fitness values below the population average are subjected to Cauchy mutation, introducing larger perturbations in order to facilitate exploration. Meanwhile, individuals with higher fitness values experience tent mapping, which finetunes their positions for exploitation. The algorithm incorporates boundary control to ensure that all individuals remain within the defined search space. This is followed by updates to the fitness values based on the new positions.