**3. The Proposed Work**

As explained previously, the traditional PSO algorithm is facing challenges. The main challenge in the PSO process is premature convergence and lack of diversity problems due to unbalance between exploration and exploitation searches of the particles. The PSO technique demands significant testing in order to establish the right parameters required to address the aforementioned difficulties. Therefore, we developed a novel strategy for the control parameter and presented a modified mutation mechanism for the personal best and global best particles.

In the traditional PSOs, the inertia weight value is constant in the search process, so the particles are unable to find the best solution. On the other hand, many researchers are practices the maximum and minimum inertia weight values for exploration and exploitation searches respectively. As the values of inertia weight have an imperative role in a dynamic environment, to solve real world problems in a dynamic environment, we developed a novel strategy for the inertia weight which will try to maintain the best balance between exploration and exploitation search of the candidates in the PSO process. Based on global best function particle values, the inertia weight value is frequently changed during the development process. In the search procedure, the proposed inertia weight strategy is important and works with the current mutation mechanism, and this process mathematically stated as:

$$w\_i = \frac{G\_{best\_{valm}}}{M\_\mathcal{S}} \tag{8}$$

where *w* is inertia weight, *i* denotes the *i*th particle, *Gbest* value is the best objective function value of global best particle and *Mg* represents the maximum number of generation.

Furthermore, because of the presence of static fitness, the traditional PSO technique experiences a lack of diversity problem in the early phases of the evolution process for global best particle *gbest* and personal best particles *pbest*. During the search process, all the particles follow the *gbest* particle, it may be possible that if the *gbest* does not know the best solution, then all the particles are trapped in a local optimal region. During the optimization process, the difference between the global best particle and the current particle is so small due to the increasing number of generations that it causes the particles to become static or stagnant, and as a consequence, the particle velocity is approaching zero, which causes the algorithm to prematurely convergence.

To tackle the aforementioned issues and difficulties in the conventional PSO algorithm, we introduced a new mechanism and strategy that chooses a different mutation operator

based on the selection ratio. The mutation operators are accompanied by personal best particles and global best particle for the purpose of enhancing the performance of the PSO process as well as preserving the diversity of the swarm. The proposed adaptive mutation operators are mathematically expressed by:

$$Q\_1 = pbest\_{i\bar{j}}^1 = pbest\_{i\bar{j}} + Rly\_{i\bar{j}} \tag{9}$$

$$Q\_2 = gbest\_j^1 = g\_{best\_j} + Rly\_j \tag{10}$$

$$Q\_3 = pbest\_{i\bar{j}}^1 = pbest\_{i\bar{j}} + std\_{i\bar{j}} \tag{11}$$

$$Q\_4 = gbest\_j^1 = gbest\_j + std\_j \tag{12}$$

$$Q\_5 = pbest\_{i\bar{j}}^1 = pbest\_{i\bar{j}} + \gamma ama\_{i\bar{j}} \tag{13}$$

$$Q\_6 = \emptyset \textit{best}\_j^1 = \emptyset \textit{best}\_j + \textit{gamma}\_j \tag{14}$$

The inspiration of the mutation operators is described in the following paragraph.

The basic PSO is inspired by the flocking of birds or school fishes, such as the birds flying in the air randomly, and the learning rate of each particle in the PSO process is randomized as well. Also, during the motion of birds, the wings of birds play an imperative role in order to continue flight. At the same time, the wings of the birds need randomized energy for their flight to spend more time in the air. Consequently, in the flying mode, the wings of birds are tired due to the presence of less energy during a long journey, and as a consequence, the birds are unable to explore more search space. Viewing the same procedure in the PSO process, where the two particles play a primary role during the search procedure, if the values of personal best and global best particles (energy of the given particles) are less or reduced during the passing of computational time, the velocity of the particles approaches zero, and as a result, the algorithm converges prematurely. In order to avoid this kind of issue, we conducted the mutation operators on particles with the purpose of improving the searching process of the PSO process and enabling the personal and global best particle to explore more optima space. Thus, the novel mutation operators generate random numbers that will provide more energy to the particles and explore more space regions in the evolution process.

In the PSO optimization process, each mutation operator plays a key role in the proposed strategy and has a self-determining selection ratio. The optimum proposed ratios of *Q*<sup>1</sup> and *Q*<sup>2</sup> denoted by X, *Q*<sup>3</sup> and *Q*<sup>4</sup> by *Y* and *Q*<sup>5</sup> and *Q*<sup>6</sup> by *Z* respectively. Where *X*,*Y* and *Z* are all set to 0.3 during the initial phases of the optimization process, which ensures that each mutation is chosen an equal number of times. The mutation ratio is updated during the search process depending on the previous mutation operator success rate to summarize the information gained from the history of the objective function. Explicitly, the following updated equations for the novel mutation of mechanism as:

$$X = l + (l - 3l)\frac{out\_{Rly}}{out\_n} \tag{15}$$

$$Y = l + (l - \mathfrak{J}l)\frac{out\_{std}}{out\_n} \tag{16}$$

$$Z = l + (l - 3l)\frac{out\_{\text{\ $}}}{out\_{\text{\$ }}}\tag{17}$$

The number of successful mutations of unique mutation operators in the primary mutation operations is represented by probability (*out*) in the above equation. The minimum ratio of each mutation operator is predefined by a constant *l*, and its value is set to 0.04. Furthermore, during the evolution process the values of *X*,*Y* and *Z* are updated after every generation. The selection process of the best mutation is adapted to the roulette wheel selection method on the basis of the selection ratio of mutation operators, as the roulette

wheel selection mechanism is such that the ratio of mutation operators having a longer stay (high selection ratio) will be chosen with a high probability.
