## 2 Background of Soft Computing Methods
### Equilibrium Optimizer (EO)  
The underlying nature of EO is based on the law of conservation of mass, where mass changing in time is equivalent to the amount of mass entering into a system plus the generated amount of mass inside the system minus the amount of mass leaving that system. More details about the inspiration of EO are given in [39]. Like other population-based optimization techniques, EO has a number of particles (population) as candidate solution and particle's position is termed as concentration. Each particle's fitness function is evaluated to determine the equilibrium states of populations for the selection of best solution. The following are the steps of EO:  
Step 1: Initialization of particle's concentrations  
The concentrations are initialized randomly for each particle using the upper bound and lower bound of the decision variable which is formulated as:  
\[C_{i}^{init}=C_{lower\_bound}+rand_{i}\times\Big{(}C_{upper\_bound}-C_{lower\_ bound}\Big{)},i=1,2,\dots,N \tag{3}\]  
where \(C_{i}^{init}\) is \(i^{\text{th}}\) particle's initial concentrations, \(C_{upper\_bound}\) and \(C_{lower\_bound}\) are the of concentrations upper bound and lower bound respectively, \(rand_{i}\) is the generated random number between 0 and 1, and N is the population size. Step 2: Candidate solutions in equilibrium pool \(\left(\overset{\rightarrow}{P}_{eq,\ pool}\right)\)  
The equilibrium pool is a collection of best candidate solutions which consists of five types of global optimal candidates. In the initial optimization period, the positions of the equilibrium candidates are unknown, which are determined in each run of the main loop to provide the knowledge about the search pattern. Among the five global optimum solutions, the four best equilibrium candidates are selected in each run with another one which is an arithmetic mean of four best solutions. The four best solutions are useful for the exploration task and the average one helps in the exploitation phase of the EO algorithm. The number of candidate solutions in equilibrium pool may depend on the nature of the evaluated function. In this work, the authors choose five equilibrium candidates for the problem space. The following is the equilibrium pool:  
\[\overset{\rightarrow}{P}_{eq,\ pool}=\left\{\overset{\rightarrow}{P}_{eq[1] }\overset{\rightarrow}{P}_{eq[2]}\overset{\rightarrow}{P}_{eq[3]}\overset{ \rightarrow}{P}_{eq[4]}\overset{\rightarrow}{P}_{eq[mean]}\right\} \tag{4}\]  
Particles are randomly selected to update their concentrations with the same probability. During this optimization process all particles receive approximately the same number of updating processes. Step 3: particle's concentration updating process  
After random selection of particle, the first term is the main concentration updating formula (Equation (5)) called exponential term (\(F\)). \[\overset{\rightarrow}{F}=e^{-\overset{\rightarrow}{\lambda}(t-t_{0})} \tag{5}\]  
where \(\lambda\) is the turnover rate, i.e., rate of change in concentration within a control volume, which is varying with time and defined as a random vector in between 0 and 1. Variable \(t\) (time) is a function of iteration and which has decreasing pattern in each iteration (Equation (6)). \[t=\left(1-\frac{Iteration}{Max\_iteration}\right)^{\left(a_{2}\times\frac{Iteration }{Max\_iteration}\right)} \tag{6}\]  
where _Iteration_ and _Max_iteration_ are the present value and maximum value of iteration, respectively and \(a_{2}\) is used to control the exploitation task which is a manually specified constant value. To guarantee convergence and balance in between intensification and diversification, \(t_{0}\) is considered as:  
\[t_{0}=\frac{1}{\underset{\lambda}{\rightarrow}}\times\ln\left(-a_{1}*sign \left(\overset{\rightarrow}{r}-0.5\right)\left[1-e^{-\overset{\rightarrow}{ \lambda}t}\right]\right)+t \tag{7}\]  
where \(a_{1}\) is used to control the exploration task which is a manually specified constant value. The higher value of \(a_{1}\) means good exploration ability and lower exploitation efficiency.