# Hybrid Particle Swarm Optimization-Jaya Algorithm for Team Formation
## 3 Preliminaries
### Notation  
This section provides the notations used in the TFP, shown in Table 2, and gives team formation examples in Table 3 and Figure 1. \begin{table}
\begin{tabular}{l l} \hline \hline
**Abbreviation** & **Meaning** \\ \hline PSO & Particle Swarm Optimization \\ \hline I-PSONSO & Improved Particle Swarm Optimization with New Swap Operator \\ \hline I-JMSO & Improved Java algorithm with Modified Swap Operator \\ \hline GWO & Grey Wolf Optimization \\ \hline I-GWO & Improved Grey Wolf Optimization \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Abbreviations. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Symbol** & **Description** & **Symbol** & **Description** \\ \hline \(G=(V,E)\) & Communication Graph & \(TC\) & Total Communication Cost \\ \hline \(TF\) & Team Formation & \(T\) & Task \\ \hline \(w\) & Weight & \(S\) & Skills \\ \hline \(CC\) & Communication Cost & \(MSO\) & Modified Swap Operator \\ \hline \(u,v\) & Agents and an Edge & \(Min\) & Minimum Communication Cost \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Notation of Team Formation Problem. The team formation problem is a combinatorial optimization problem that involves grouping individuals into teams based on their skills, preferences, and other characteristics. The goal is to form efficient teams that meet specific objectives. Definition 1: [19]: [Team Formation] Given a set of agents, skills of agents, and tasks--an agent has a set of skills (abilities) and a task is defined by a set of skills--the team formation problem requires finding a subset of agents whose combined skills (union of the skills of the individual agents) are a superset of the skills required for the task. Definition 2: [19]: [Communication Graph] A communication graph \(G=(V,E)\) represents an undirected weighted graph. The graph's vertices correspond to the agents, and an edge \((u,v)\) is assigned a weight \(w\), where \(0\leq w\leq 1\). A lower weight indicates that the agents can communicate with each other more easily. Definition 3: [19]: [Communication Cost (CC) of Two Agents] Let the agents \(A_{i},A_{j}\) have skills \(S_{i},S_{j}\), respectively. The communication cost of the agents is given as:  
\[CC_{i,j}=1-|(S_{i}\cap S_{j})|/|(S_{i}\cup S_{j})|. \tag{1}\]  
Figure 1 shows the communication cost among the agents. The skills of the agents are given in Table 3.  
Consider two _Agents_, \(A_{1}\) and \(A_{2}\), with skills:  
\(A_{1}=\) {distributed data mining, agent computing, intrusion detection}, and  
\(A_{2}=\) speech acts, agent computing}, respectively. The intersection of skills between \(A_{1}\) and \(A_{2}\) is the skill 'agent computing', i.e., 1, while the union of their skills consists of 4 distinct skills. To calculate the communication cost between \(A_{1}\) and \(A_{2}\), you divide the number of skills in the intersection by the total number of skills in the union, which equals 0.25. This division helps quantify the communication efficiency between the two agents. Subtracting this value from 1 gives us the communication cost between \(A_{1}\) and \(A_{2}\), which is 0.75. You can apply this method consistently to calculate the communication cost between various pairs of agents within the network, ensuring a systematic approach to evaluating communication efficiency. Definition 4: [19]: [Swap Operator (SO)] The swap operator works as follows. Consider the j-th column of any two populations numbered, say, P1 and P2; swap(P1, P2, j) means the agents at position j corresponding to these populations are interchanged. Accordingly, the total communication cost of the teams would change. For example, consider two teams (particles):  
\[Team_{1}=\{A_{1},A_{2},A_{3},A_{4}\},\]  
\[Team_{2}=\{B_{1},B_{2},B_{3},B_{4}\},\]