## Appendix A Meta-Heuristic Optimization Algorithms
### Particle Swarm Optimization  
Particle swarm optimization is a well-known and familiar type of SI algorithm proposed by [PERSON] and [PERSON] in 1995 [31; 53]. The main inspiration of PSO is to mimic the collective interactive behavior of non-intelligent social swarms, such as birds and fishes, with collaborating agents in nature. It represents a set of solutions (i.e., particles) that move in groups in the search space. The particles of PSO explore possible solutions in the search area and accelerate to the global solutions based on a predetermined criterion. The term \"particles\" was adopted because particles experience positions, velocities and accelerations of the swarms in accordance with specific equations that update the particles' movement. The particles of the PSO algorithm are updated at each iteration based on the optimal solutions found so far by all particles, as the best solution can be explicitly identified by the best updated particle. The position of each particle is controlled based on the fitness score of each particle as well as the fittest particle of the swarms in the search area. Equation (A1) can be used to update the velocity, \(v\), and the position, \(x\), for each particle for a number of \(n\) particles in PSO.  
\[v_{id}(t+1) = \omega v_{id}(t)+\alpha r_{1}[p_{id}(t)-x_{id}(t)]\] \[+ \gamma r_{2}[p_{gd}(t)-x_{id}(t)]\] \[x_{id}(t+1) = x_{id}(t)+v_{id}(t+1)\]  
where \(i\) identifies the index number of each particle, \(d\) indicates the dimension of the problem under consideration, \(t\) identifies the iteration number, \(v_{id}\) is the velocity of the \(i\)th particle in the \(d\)th dimension, \(p_{id}\) represents the best position of the \(i\)th particle visited so far in the \(d\)th dimension, \(p_{gd}\) represents the global best position of a particle in the swarm in the \(d\)th dimension, \(\omega\) is a positive inertia weight, \(r_{1}\) and \(r_{2}\) are two randomly generated numbers between 0 and 1, \(\alpha\) and \(\beta\) are two constants that represent the values of the degree of influence of \(p_{id}\) and \(p_{gd}\) on the particles velocity, respectively, and \(x_{id}\) is the position of the \(i\)th particle in the \(d\)th dimension.  
Equation (A1) can be used to find a new solution in the search space and to update the global best solution. The key steps of PSO can be summarized as shown by the pseudo-code in Algorithm A2.  
The PSO algorithm embraces exploration search properties, making it well suited for optimizing nonlinear functions and complex real-world problems. It is expected that through the use of PSO, there is a high probability of evading minimal local solutions over and over by encouraging more exploration of the search domain. This issue is concerned with the fact that survival is the ultimate goal for fish schooling and bird flocking. Consequently, PSO exhibits a coordinated behavior that distinguishes the social behavior of bird and fish swarms in nature.  
There are many applications presented in the literature based on PSO that have reported convincing results in a wide range of optimization issues [28; 54]. Today, there is rapid development in the use of meta-heuristic algorithms, and especially PSO, to address real contemporary problems, such as the problem under consideration. The particle representation for the problem under study is represented in Table A1.