## 2 Overview of PSO Schemes and the IPTBK Problem
### MSPSO Algorithms  
Route construction is performed using a velocity-state change rule for position updating based on two factors, namely \(P_{\text{best}}\) and \(P_{\text{gbest}}\), as illustrated in Figure 2. When a PSO system solves the IPTBK problem, each particle swarm constructs a route between the victim and the attacker by repeatedly applying velocity-state updating rules, which are expressed using Equations (1) and (2). As presented in Equation (1), the speed of each particle is updated by updating the velocity of each particle toward its optimal locations \(P_{\text{best}}\) and \(P_{\text{gbest}}\) at each time step in the PSO process. [14]  
\[v_{i}(t\ +\ 1)\ =\ w_{i}v_{i}\ (t)\ +\ c_{1}\text{-}rand()\cdot(P_{\text{ best}}\ -\ x_{i}\ (t))\ +\ c_{2}\text{-}rand()\cdot(P_{\text{gbest}}\ -\ x_{i}t)) \tag{1}\]  
\[\text{x}_{i}\ (t\ +\ 1)\ =\text{x}_{i}\ (t)\ +\ \text{v}_{i}\ (t\ +\ 1) \tag{2}\]  
\[w_{i}=\frac{2}{\left|\left(2-c-\sqrt{c^{2}-4c}\right)\right|} \tag{3}\]  
\[c=c_{1}+c_{2},c\geq 4, \tag{4}\]  
where \(v\) and \(x\) represent the speed and position of particle \(i\) (\(i=1,\ \ldots,\ m\)) at time \(t\), respectively, and \(c_{1}\) and \(c_{2}\) represent the acceleration constants. Suitable acceleration constants (\(c_{1}\) and \(c_{2}\)) can control the speed of the particle's movement. Typically, \(c_{1}\) is equal to \(c_{2}\), and both these parameters are equal to 2. The parameter \(rand()\) represents a random number in the range (0,1).  
Notably, each particle utilises its memory to record its previous best position and the best position discovered by any particle in the swarm. Thus, when the particle moves to a new position, the fitness value of the new position is calculated. If the fitness value of the new position is better than that of the previous best position (i.e., \(P_{\text{best}}\)), then the value of \(P_{\text{best}}\) must be replaced by that of the new position updated according to the particle's optimal experience. Similarly, \(P_{\text{gbest}}\) must be replaced by \(P_{\text{best}}\) if the fitness value of the new position is better than that of \(P_{\text{gbest}}\).  
In searching for the global optimal solution, the movement of each particle in the subswarm is affected by three factors: (1) Inertia, (2) individual experience and (3) swarm experience. [11; 12].  
To prevent the particle swarm from rapidly converging on a single path, the proposed MSPSO algorithm divides a particle swarm into several subswarms. In the proposed MSPSO algorithm, the local update rules in each subswarm are required to search on multimodal optimisation problems. The multimodal optimisation methods generally arrange the exploration and exploitation into different search stages as shown in Figure 3[20]. In the local search stage, each subswarm is used to explore possible solutions. In this stage, the focus is on discovering the local optimal solutions in the region. In the global search stage, the global optimal position is exploited through a regrouping strategy.  
Figure 3: Dynamic behaviour of the particle movement for three multiswarms in multiswarm PSO (MSPSO) [20].  
Figure 2: Attack path reconstruction using a particle swarm optimisation (PSO) scheme [13].  
Two known multimodal optimisation methods use revised PSO schemes: The waves of swarm particles (WOSP) [21] and the dynamic multiswarm PSO (DMS-PSO) [22; 23] algorithms. The two crucial algorithms are introduced as follows.