# An Improved Multi-Objective Grey Wolf Optimizer for Aerodynamic Optimization of Axial Cooling Fans
## 2 Improved Multi-Objective Grey Wolf Optimizer
### Nonlinear Convergence Factor  
In the GWO algorithm, \(A\) is an important parameter that controls the hunting behavior of the grey wolf population. The following equations were proposed in order to simulate the encircling behavior of grey wolves during hunting [14].  
\[D=CX_{p}(t)-X(t), \tag{6}\]  
\[X(t+1)=X_{p}(t)-AD, \tag{7}\]  
where \(t\) indicates the current iteration, \(X_{p}\) is the position of the prey, and \(X\) is the position vector of a grey wolf. \(A\) and \(C\) are coefficients, which can be calculated as follows:  
\[A=2 ar_{1}-a \tag{8}\]  
\begin{table}
\begin{tabular}{l} \hline \hline For each individual \(i\) in the population size (Popsize): \\ Set Grey Wolves[\(i\)].Velocity to 0 \\ Initialize Grey Wolves[\(i\)].Position as a zero vector of length \(n\)Var \\ \% The three coordinates of qubits \\ For each variable \(j\) in \(n\)Var: \\ Generate random angle \(\varphi\) in the range [0, 2\(\pi\)] \\ Generate random angle \(\theta\) in the range [0, 2\(\pi\)] \\ Calculate chrom1[\(j\)] as \(\cos(\varphi)\) sin(\(\theta\)) \\ Calculate chrom2[\(j\)] as \(\sin(\varphi)\) sin(\(\theta\)) \\ Calculate chrom3[\(j\)] as \(\cos(\theta)\) \\ End For \\ \% Transform to the solution space of this problem \\ Calculate quantum[1].Position using the transformation formula with chrom1 \\ Calculate quantum[2].Position using the transformation formula with chrom2 \\ Calculate quantum[3].Position using the transformation formula with chrom3 \\ \% Evaluate the cost of each quantum position \\ Set quantum[1].Cost to the result of the objective function fobj evaluated at  
quantum[1].Position \\ Set quantum[2].Cost to the result of the objective function fobj evaluated at  
quantum[2].Position \\ Set quantum[3].Cost to the result of the objective function fobj evaluated at  
quantum[3].Position \\ \% Determine domination status among quantum solutions \\ Determine domination status for quantum solutions \\ Create a list domi containing the domination status of quantum[1], quantum[2], and \\ \%end{tabular} \\
\begin{tabular}{l} Find indices of non-dominated solutions and store in num \\ If num is empty: \\ Set Grey Wolves[\(i\)].Position to quantum[1].Position \\ Set Grey Wolves[\(i\)].Cost to quantum[1].Cost \\ Else: \\ Set Grey Wolves[\(i\)].Position to the position of the first non-dominated solution in  
quantum[num[1]] \\ Set Grey Wolves[\(i\)].Cost to the cost of the first non-dominated solution in  
quantum[num[1]] \\ End If \\ Set Grey Wolves[\(i\)].Best.Position to Grey Wolves[\(i\)].Position \\ Set Grey Wolves[\(i\)].Best.Cost to Grey Wolves[\(i\)].Cost \\ End For \\ Clear variables quantum, domi, num \\ \end{tabular}
\end{table}
Table 1: The pseudo-code of the Bloch qubit spherical initialization population.  
\[C=2r_{2} \tag{9}\]  
where \(r_{1}\) and \(r_{2}\) are random numbers in \([0,1]\). When \(A>1\), the grey wolves tend to perform a wide-range global search; when \(A<1\), the grey wolves attack their prey, i.e., perform a local search near the optimal solution. The convergence factor \(a\) directly affects the value of parameter \(A\). In the original GWO, \(a\) decreases linearly from 2 to 0. However, when dealing with complex problems, the linear decrement strategy often leads to insufficient exploration of the search space. It is more efficient for \(a\) to maintain a larger value in the early stages for thorough global exploratio, and a smaller value in the later stages to focus on local exploitation and accelerate convergence. Therefore, it theoretically supports a better exploration-exploitation balance. A nonlinear convergence factor \(a\) is proposed, expressed as follows:  
\[a=1+\cos(\pi t/T_{max}) \tag{10}\]  
where \(T_{max}\) is the maximum number of iterations. An example of the change in the convergence factor before and after the improvement for \(T_{max}=100\) is shown in Figure 3.