## 4 Description of IDMMFO-GM Design
### Directional Mutation Strategy  
At the end of the optimization of the MFO algorithm, the moth population is easily affected by the flame set, thus leading a high risk of local convergence. By introducing the mutation operator of the genetic mechanism, the diversity of the moth population can be significantly enhanced, to reduce the risk of the algorithm falling into local convergence. Compared with traditional random mutation based on a certain mutation probability, directional mutation guided by gradient information significantly enhances the convergence speed and optimization precision of the algorithm. The specific process of directional mutation in parking reference trajectory optimization is outlined as follows:  
Figure 5: Schematic diagram of non-linear decreasing function about weight coefficient. Parking reference trajectory optimization is a specific type of optimization problem aimed at minimizing a single objective. Generally, this class's problems can be formally described as follows:  
\[\min y=F(u) \tag{22}\]  
where \(y\) represents the optimization objective, \(u\) denotes the decision variable, and \(F(u)\) signifies the optimization mapping model for solving the optimization objective \(y\) of decision variable \(u\). For the parking reference trajectory optimization problem, the decision variable \(u\) is defined as the center point set \(PS\) of the fixed-point region to be decided, \(PS=\{P_{2},\cdots,P_{ns}\}\), while the objective function \(F(u)\) represents the length \(L(tf(PS))\) of the reference trajectory. The original decision variable to be directed mutation is denoted as \(PS_{k}\), the gene inheritance decision variable for directional mutation is denoted as \(PS_{i}\), and the gene changing decision variable for directional mutation is denoted as \(PS_{j}\). Subsequently, the directed vector \(d\big{(}PS_{j},PS_{i}\big{)}\) can be formally defined as follows:  
\[d\big{(}PS_{j},PS_{i}\big{)}=\text{sgn}\big{(}\big{\{}P_{2,j}-P_{2,i}\cdots,P_ {ns,j}-P_{ns,i}\big{\}}\big{)} \tag{23}\]  
where sgn represents a symbolic function that denotes the sign of the parameters, including \(0\), \(-1\), and \(1\). If \(L\big{(}tf(PS_{j})\big{)}<L(tf(PS_{i})\big{)}\), the vector \(d\big{(}PS_{i},PS_{j}\big{)}\) represents the evolutionary direction of objective function \(L(tf(PS))\) from the gene inheritance decision variable \(PS_{i}\) to the gene changing decision variable \(PS_{j}\). Conversely, the vector \(d\big{(}PS_{i},PS_{j}\big{)}\) denotes its degenerate direction. Gene inheritance decision variable \(PS_{i}\) usually selects the original decision variable \(PS_{k}\) or its direct parent, while selecting the gene changing decision variable, it should be chosen to be as far as possible from the original decision variable \(PS_{k}\). Through the application of direction vector \(d\big{(}PS_{j},PS_{i}\big{)}\), the directional mutation of the original decision variable \(PS_{i}\) can be carried out. The specific formulation is as follows:  
\[{PS_{k}}^{\prime}=PS_{i}+\omega_{m}\cdot r\cdot d\big{(}PS_{j},PS_{i}\big{)} \tag{24}\]  
where \({PS_{i}}^{\prime}\) represents the original decision variable to be directed mutation, \(PS_{k}\) denotes the updated decision variable after directional mutation, and \(\omega_{m}\) signifies the adjustment coefficient for directional mutation, \(\omega_{m}\in(0,1)\). It is assumed that at the \(i\)-th iteration, the \(i\)-th moth \(M_{t,i}\) in the current moth population \(M_{t,i}\) is to be directed mutation, and it is taken as the gene inheritance decision variable, and the \(j\)-th flame \(F_{t,j}\) in flame set \(F_{t}\) is randomly selected as the gene changing decision variable of its directional mutation. The gene inheritance decision variable determined by moth \(M_{t,i}\) is denoted as \(PS_{i}\), while the gene changing decision variable determined by flame \(F_{t,j}\) is represented as \(PS_{j}\).