# A Two-Stage Optimization Model for Airport Stand Allocation and Ground Support Vehicle Scheduling
## 5 The NSGA-II Algorithm Combining Local Search Strategies
To prevent the algorithm from overly relying on this correlation, a random operator is introduced to enhance the algorithm's diversity in Equation (32). By incorporating randomness, the algorithm avoids becoming too deterministic in its decisions, allowing for a broader exploration of the solution space and potentially avoiding local optima. This balance between correlation-based decisions and randomness helps the algorithm maintain flexibility and adaptability throughout the optimization process. \[\lambda=random([0,1])^{D}\times n^{\prime} \tag{32}\]  
where \(n^{\prime}\) represents the number of remaining flights in the set \(X^{{}^{\prime}}\). The \(\lambda\)-th flight from the set \(X^{{}^{\prime}}\) is chosen to be removed and added to set \(A\). It can be observed that as \(D\rightarrow\infty\), the selected flights have a stronger correlation. Repair method: flights from set \(A\) are randomly assigned to compatible stands. The fitness value of all feasible solutions is calculated, and the solution with the best fitness value is selected as the new feasible solution. This ensures that after the destroy phase, the repaired solution is optimized for fitness, maintaining the integrity of the overall search process. (6) Fitness value evaluation: To assess the feasibility of the solution after the local search, the fitness value is used. \[fitness(x)=Z1+Z2+Z3+\varpi W \tag{33}\]  
Equation (33) calculate the fitness, \(Z1,\ Z2,\ Z3\) represent the values of the objective functions. \(W\) is the total delay time of all flights, and \(\varpi\) is the penalty value, empirically set to 100. The fitness function, therefore, incorporates both the objective function values and the total delay, with the penalty value ensuring that excessive delays are strongly discouraged. (7) Merge offspring and parent populations and select individuals based on fast non-dominated sorting and crowding distance calculation. Identify the non-dominated solution set \(P_{m}\) from the population, assign the current rank as \(\text{rank}(i)=1\), and add these solutions to the non-dominated layer \(F\). Repeat this process iteratively, assigning a non-dominated rank value to all individuals until the entire population is sorted. For individuals with the same \(\text{rank}(i)\), the crowding distance is calculated based on Equations (34) and (35). \[d_{i}=\frac{d_{i}+\left(v_{i+1}^{m}-v_{i-1}^{m}\right)}{v_{max}^{m}-v_{min}^{ m}} \tag{34}\]  
\[d_{0}=d_{n}=M \tag{35}\]  
where \(d_{i}(i\in[1,n-1])\) represents the crowding distance of an individual, with an initial value of \(0\), \(v_{i+1}^{m}\) denotes the function value of the \(m\)-th objective for the individual, \(v_{max}^{m}\) and \(v_{min}^{m}\) are the maximum and minimum values of the \(m\)-th objective, respectively, \(M\) is a very large number. (8) Termination condition: To improve the computational efficiency of the algorithm, the number of iterations is set as the termination condition. LS-NSGA-II algorithm pseudocode see in Algorithm 1. ```
Input & Initial population \(P_{0}\), population size \(N\), maximum iterations \(G_{max}\), crossover probability \(Pc\), mutation probability \(Pm\), flight set \(F\), stand set \(G\) & Output & The population \(P\) after iteration
1 Perform fast non-dominated sorting and crowding distance assignment for \(P\)
2\(g\gets 1\)
3\(P\gets P_{0}\)
4 while\(g<G_{max}\)
5\(F\leftarrow\emptyset\) //Generate the parent population \(F\)
6 For\(i\gets 1\) to \(length[P_{0}]\)
7 Select individual \(x\) from \(P\) using binary tournament selection
8 Put \(x\) into \(F\)
9 endfor
10 Perform crossover and mutation operations on \(F\) to generate the offspring population \(C\). 11 For\(i=1\) to \(length[C]\)
12\(x=C(i)\)
13\(x^{{}^{\prime}}\gets r(d(x))\) //Perform repair and destruction operations on individual \(x\) to generate new individual \(x^{{}^{\prime}}\).