# Tethered Balloon Cluster Deployments and Optimization for Emergency Communication Networks
## 4 Methodology  
The multi-objective particle swarm optimization (MOPSO) algorithm, inspired by the flocking behavior of birds, is designed to optimize two or more conflicting objectives simultaneously, which means that an increase in the performance of one objective function will result in a decrease in the performance of the remaining objective functions. While traditional Particle Swarm Optimization (PSO) algorithms are designed to solve a single objective function, MOPSO extends PSO to deal with multi-objective optimization problems. Since multi-objective problems usually do not have a unique optimal solution, the method aims to find a set of solutions that consider all the objectives and provide good trade-offs, i.e., the Pareto optimal set (POS). The fitness evaluation of all solutions is based on the multi-objective function of the problem to be optimized. The fitness function is used to measure the performance of the particles under the current deployment scenario. To measure the superiority of the multi-dimensional solutions, MOPSO introduces a selection mechanism to evaluate the fitness based on Pareto optimality, i.e., Pareto dominance. This dominance provides an objective criterion for comparing the advantages and disadvantages of different solutions, enabling the algorithm to make effective trade-offs between multiple objectives. Specifically, solution \(X_{1}\) is said to dominate solution \(X_{2}\) when and only when solution \(X_{1}\) performs non-inferiorly to \(X_{2}\) on all objective functions and strictly outperforms \(X_{2}\) on at least one objective function. The relationship between multi-dimensional fitness and Pareto dominance for \(X_{1}\) and \(X_{2}\) is shown below:  
\[\forall\;X_{1},X_{2}\in X,\;X_{1}\;\;dominates\;X_{2}\iff\left\{ \begin{matrix} \forall\;\;i\in(1,2),\,f_{i}(X_{1})\geq f_{i}(X_{2}),\\ \exists\;\;j\in(1,2),\,f_{j}(X_{1})>f_{j}(X_{2}).\end{matrix} \right. \tag{19}\]  
Assuming that there are \(S\) particles in search space \(X\), MOPSO first initializes and randomly assigns positions to all the particles, which represent potential solutions to the multi-dimensional optimization problem, which are dispersed in the search space expected to maximize the multi-objective function. Each particle possesses four attributes for exploring the search space of the problem: the current velocity of the particle, its current position, the best position experienced by the individual (the optimal solution found by the individual during the iteration process, denoted as \(P_{Best}\)), and the best position experienced by the group (the best-fitness position among all particles, denoted as \(G_{Best}\)). Set the particle's current position to \(P_{Best}\). The particle that performs best in both objective functions is selected as the initial \(G_{Best}\)During each iteration, MOPSO updates the velocities and positions of all particles based on the guidance of \(P_{Best}\) and \(G_{Best}\) (see Equations (20) and (21)) and calculates the fitness to evaluate their current positions to maximize the probability of moving to more optimal solution space and thus obtaining a better fitness. Judge the superiority of the current position \(X_{i}\) and \(P_{Best[i]}\) based on Pareto dominance: if \(X_{i}\) dominates \(P_{Best[i]}\) (i.e., \(X_{i}\) performs better on at least one objective and is not worse than \(P_{Best[i]}\) on all other objectives), update \(P_{Best[i]}\) to \(X_{i}\); if \(P_{Best[i]}\) dominates \(X_{i}\), keep \(P_{Best[i]}\) unchanged. In the same way, the dominant relationship between \(P_{Best}\) and \(G_{Best}\) is judged and \(G_{Best}\) is updated (see Equations (23) and (24)). This process continues until the maximum number of iterations is reached or enough \(G_{Best}\) have been collected to obtain a set of optimal deployment locations finally [25]. The steps of MOPSO can be expressed as Algorithm 1.