## 4 Analysis of Single Objective Algorithms Overview
### Practical Swarm Optimization  
PSO can be defined as a heuristic search approach that simulates movements of a swarm of birds seeking food. Its developers, refer to PSO as an evolutionary algorithm. Due to its population-based methodology and simple nature, PSO has been positioned as a good option for diversifying into MOO [6; 40]. [PERSON] and [PERSON] first presented the PSO method for optimization. PSO is a population-based search algorithm that mimics the social behavior of a bird flock. PSO was originally developed as a global optimizer for neural network (NN) weight balancing [53]. It immediately became well liked, especially for problems requiring real number choice variables. There are two main distinctions between an evolutionary algorithm and PSO, according to [PERSON]. Parent representation, individual selection, and parameter fine-tuning are the three mechanisms that evolutionary algorithms rely on. In PSO, there are only two approaches that are used; an explicit selection function is not used. PSO uses leaders to guide the search to compensate for the absence of a selection mechanism. PSO is not related to the idea of offspring generation as an evolutionary algorithm. The management of the individuals is another way that evolutionary algorithms and PSO differ from one another. Using an operator, PSO can determine a particle's velocity and give it a specific direction. This is a directional mutation operator that depends on both the swarm's global best as well as the particle's personal best. There will be less investigation when the prospective directions have a modest angle in the case when the personal best direction coincides with the global best direction. On the other hand, a larger angle will enable more comprehensive research. Evolutionary algorithms use a mutation operator that can orient an individual in multiple ways, each with a different probability. Mutation operators similar to evolutionary algorithms have been adopted because of the directed mutation limitations of PSO [6]. There are two main characteristics that we believe have contributed to the popularity of PSO. First, the primary algorithm of PSO is simple due to its use of a single operator for generating new solutions, which sets it apart from most evolutionary algorithms, making its implementation uncomplicated. Moreover, a significant amount of PSO source code is accessible in the public domain. PSO has demonstrated high effectiveness across a diverse range of applications, yielding excellent outcomes with minimal computing expenses [53; 54; 55; 12]. To learn more, see [6]. In PSO, particles move through a hyper-dimensional search space. Particle movements in the search space are driven by people's tendency, depending on social-psychological reasons, to imitate other people's success. Depending on its own experiences and those of its nearest neighbor, each particle adjusts its position using the equations in [6]. The position of particle pi at time step t is represented by \(Xi(t)\). Adding a velocity \(Vi(t)\) to pi's current position modifies its position. \[X_{i}(t)=X_{i}(t-1)+V_{i}(t) \tag{3}\]  
The velocity vector, which is commonly defined as follows, depicts the information conveyed within society. \[V_{i}(t)=WV_{i}(t-1)+c1r1\Big{(}X_{pbest,i}-X_{i}(t)\Big{)}+c2r2(X_{leader}-X_ {i}(t)) \tag{4}\]  
Random numbers between 0 and 1 make up \(c2\), \(c1\), \(r2\), and \(r1\). Particles are frequently affected by the accomplishments of those with whom they are associated. Instead of being close in parameter space, such neighbors are particles that are close to one another depending on a neighborhood topology that characterizes the social structure of the swarm. For single optimization as seen in below Algorithm 3, the fundamental PSO algorithm functions. First, the swarm would be initialized. Velocities and locations are both initialized in this way. The leader is identified by choosing the gbest solution, and each particle's pbest is set.