# A Novel Multi-Objective Five-Elements Cycle Optimization Algorithm
## 4 The Proposed Multi-Objective Five-Elements Cycle Optimization Algorithm
### Mutation of Individuals  
To prevent individuals from falling into local optimum, a mutation operation was introduced in MOFECO. In standard genetic algorithms, the mutations operation often flip each binary bit with a small probability \(P_{m}\), which is used to introduce missing genes into the population to achieve global search of the solution space. To this end, researchers designed a number of mutation operators, such as uniform mutation, non-uniform mutation, Gaussian mutation, Cauchy mutation, polynomial mutation and so on [25]. Reference [36] systematically analyzed the global search ability and local search ability of Gauss distribution, Cauchy distribution and Uniform distribution mutation operator. Theory and practice show that the Gauss distribution operator has strong local search ability, the uniform distribution mutation operator has strong global search ability, and the Cauchy distribution operator has a centered ability of local search and global search.  
Hence, we combine these three different mutation operators as follows. In the early stage of evolution, we use the uniform mutation operator to acquire strong global search ability; In the late evolution, we use the Gauss mutation operator to enhance the local search ability of the algorithm; While in the middle of evolution, the Cauchy distribution mutation operator is used to make the algorithm suitable for global and local search. The specific operation is as follows:  
\[XM_{ij}(k+1)=\begin{cases} XC_{ij}(k+1)+U(-\sigma_{1},\sigma_{1})&if\;(\;k+1\; \leq\;\frac{T}{4})\\ XC_{ij}(k+1)+C(0,\sigma_{2})&if\;(\;\frac{T}{4}\;<\;k+1\;\leq\;\frac{3T}{4})\\ XC_{ij}(k+1)+N(0,\sigma_{3})&if\;(\;\frac{3T}{4}\;<\;k+1\;\leq\;T)\end{cases} \tag{15}\]  
where \(XM_{ij}(k+1)\) represents the individual after mutation; \(U(-\sigma_{1},\sigma_{1})\) represents the uniform distribution mutation operator and \(\sigma_{1}\) is called the variation domain and usually has \(\sigma_{1}=0.1\times(u_{p}-l_{w})\), where \(u_{p}\) and \(l_{w}\) represent the upper and lower bounds of the decision variable; \(C(0,\sigma_{2})\) represents a random number of Cauchy distribution centered at zero and a scale parameter of \(\sigma_{2}\); \(N(0,\sigma_{3})\) represents the Gaussian distribution random number.  
After the update and mutation operation, a new population is generated, which contains the retained individuals and the updated individuals, and the population size is still \(N\). Then, the fast non-dominated sorting and the crowded distance method in NSGA-II are adopted to sort the new population. The best solution in each cycle and the optimal solution in the new population are picked out as the neighborhood solutions that produce new solutions when the next generation is updated. At the same time, the parent population and the updated individuals are combined to get a new population \(R\), and \(R\) is sorted by using fast non-dominated sorting and crowded distance calculation  
Figure 4: Nonlinear variation of \(P_{s}\).  
method in NSGA-II. So the solutions with \(Rank\) value of 1 and the crowded distance from large to small in the sorted population \(R\) are selected as the current Pareto optimal solution set. Then, the above process is repeated until the maximum number of iterations is reached.