## 3 Methodology
### IPSO Algorithm  
After finding the best local optimal solutions with the SCEHO algorithm, the best global optimal solutions are determined with the IPSO algorithm. The PSO algorithm is one of the stochastic optimization algorithms which follows swarm movement and intelligence behaviors [37; 38; 39]. The social interaction concept is used in the conventional PSO algorithm for resolving the optimization problems. The PSO algorithm makes use of agents (particles) which constitute the swarms that move in the search space [40; 41]. For every iteration, the agents (particles) update their positions in order to obtain optimal solutions. In the swarm, every particle moves towards its prior global and personal best position. The Equations (12) and (13) are used to update the velocity and position of the agents (particles).  
The architecture of the IPSO algorithm is specified in Figure 3.  
\[v_{id}(t+1)=I_{w}\times v_{id}(t)+ac_{1}\times r_{1}\times[p_{id}(t)-o_{id}(t )]+ac_{2}\times r_{2}\times\left[p_{gd}(t)-o_{id}(t)\right] \tag{12}\]  
\[o_{id}(t+1)=o_{id}(t)+v_{id}(t+1) \tag{13}\]  
where the random numbers are represented as \(r_{1}\) and \(r_{2}\), the acceleration coefficients are denoted as \(ac_{1}\) and \(ac_{2}\), and the inertia weight used to balance local and global search is specified as \(I_{w}\). The particles' global best position and the personal best position are denoted as \(p_{gd}\) and \(p_{id}\). In the IPSO algorithm, a novel Human Group Optimizer (HGO) is used to influence the agents (particles). The HGO uses an adaptive uniform mutation operator for enhancing the convergence speed of the conventional PSO algorithm. Additionally, nonlinear function \(p_{m}\) is used in the IPSO algorithm for controlling the range and decision  
Figure 3: Architecture of IPSO algorithm.  
of the mutation on each particle. The nonlinear function \(p_{m}\) is updated after every iteration and it is mathematically specified in Equation (14).  
\[p_{m}=0.5\times e^{(-10\times\frac{t}{T})}+0.01 \tag{14}\]  
where \(T\) indicates the maximum iteration, and \(t\) denotes the total number of iterations. The assumed parameters of the IPSO algorithm are listed as follows: the iteration number is 100, the population number is 100, the cognitive constant \(ac_{2}\) is two, and the social constant \(ac_{1}\) is three. The numerical analysis of the proposed SCEHO-IPSO algorithm is detailed in Section 4, and the steps involved in the Algorithm 1 are described below:  
```
Step 1: Initialize the objective functions. Step 2: Create initial population. Step 3: Evaluate fitness value. Step 4: For every task, find the best local optimal solutions using SCEHO algorithm. Step 5: For every task, find the best global optimal solutions using IPSO algorithm. Step 6: Find the hybrid solutions. Step 7: If the hybrid new solution value is higher than the current value, then Step 8: replace the current value with the hybrid new solution. Step 9: Select any resources among the population. Step 10: If the execution time is higher for the selected resource, then eliminate the respective resource and select another resource. Step 11: Update personal best and global best solutions. Step 12: Retain it and rank the best solutions. Step 13: End.
```  
**Algorithm 1** SCEHO-IPSO algorithm.