## 2 Background of Soft Computing Methods
### Equilibrium Optimizer (EO)
Equally, the higher value of \(a_{2}\) means good exploitation ability and lower exploration efficiency. \(sign\left(\overset{\rightarrow}{r}-0.5\right)\) component specifies the direction of intensification and diversification of particles and \(r\) is defined as a random vector in between 0 and 1. The final equation of \(F\) is defined by substituting Equation (5) with Equation (7) as follows:  
\[\overset{\rightarrow}{F}=a_{1}\times sign\left(\overset{\rightarrow}{r}-0.5\right)\left[e^{-\overset{\rightarrow}{\lambda}t}-1\right] \tag{8}\]  
The next important term of EO is the generation rate (G) and is used to improve the exploitation task. That controls the search patterns towards the accurate solution. The generation rate is considered as a function of time. Equation (9) shows the generation rate:  
\[\overset{\rightarrow}{G}=\overset{\rightarrow}{G}_{0}\times\overset{ \rightarrow}{F} \tag{9}\]  
\[\overset{\rightarrow}{G}_{0}=\overset{\rightarrow}{G}\overset{\rightarrow} {P}\left(\overset{\rightarrow}{P}_{\alpha q}-\overset{\rightarrow}{\lambda} \times\overset{\rightarrow}{P}\right) \tag{10}\]  
\[G\overset{\rightarrow}{C}P=\left\{ \begin{array}{cc}0.5*r_{1}&r_{2}\geq GP\\ 0&r_{2}<GP\end{array} \right. \tag{11}\]  
where \(r_{1}\) and \(r_{2}\) are random values between 0 and 1. \(G\)\(C\)\(P\) is called generation rate control parameter and generated by replicating same values. The GCP controls the probable contribution of generation term in updating process. This probability specifies how many particles update their positions with the generation term. The generation probability (\(GP\)) is responsible for that act. Equations (10) and (11) are used to determine this process. If \(GP\) = 0 then all concentrations of a particle are updated without \(\overset{\rightarrow}{G}\). \(GP\) = 0.5 provides a good balance between the exploitation and exploration phase. The final concentrations updating rule is as follows:  
\[\overset{\rightarrow}{P}=\overset{\rightarrow}{P}_{\alpha q}+\left( \overset{\rightarrow}{P}-\overset{\rightarrow}{P}_{\alpha q}\right)\cdot \overset{\rightarrow}{F}+\frac{\overset{\rightarrow}{G}}{\overset{ \rightarrow}{\lambda}V}*\left(1-\overset{\rightarrow}{F}\right) \tag{12}\]  
where \(\overset{\rightarrow}{F}\) is define in Equation (8) and \(V\) is assigned as unit. Equation (12) consists of three terms. The first is the concentration of an equilibrium candidate. The second and third define the variations in particle's concentrations. The variations in the second term specifies large variations in concentration (difference between equilibrium candidate and candidate particle) that controls exploration in EO. The third term manages exploitation by fine tuning concentration based on the generation rate (Equation (9)). Based on parameters which are the candidate particle's concentration, equilibrium candidate particles, turnover rate \(\lambda\)), the final two terms may have the opposite and same sign. The same sign indicates global search and opposite sign indicates local search. EO has several parameters for exploration and exploitation tasks which are summarized in Table 1. Algorithm 1 shows the algorithm of EO. ```
1. Set number of particles
2. Assign of EO parameters value (\(a_{1}\), \(a_{2}\), \(GP\))
3. Initialize the fitness of four equilibrium candidates [fitness(\(\overset{\rightarrow}{P}_{eq[1]}\)), fitness(\(\overset{\rightarrow}{P}_{eq[2]}\)), fitness(\(\overset{\rightarrow}{P}_{eq[3]}\)), fitness(\(\overset{\rightarrow}{P}_{eq[4]}\))]
4. for it = 1 to maximum iteration number do
5. for\(i\) = 1 to P do
6.