## 3 Solving Method
### Algorithm Improvement  
Considering the discrete nature of the space and the specific structure of gas field pipeline networks, the traditional GA and its associated parameters may not fully align with the mathematical model proposed in this paper. Therefore, it is essential to enhance the basic GA to better suit these characteristics and optimize the pipeline network design effectively. The improvement includes the overall coding of well-station correspondence, the formulation of the fitness function of the network solution, and the redesign of the crossover operator and mutation operator. (1) Coding design. Taking into account the characteristics of the constraints, using the connection relationship between wells and stations as a coding basis, real number coding is used to improve the efficiency of the GA. The chromosome-encoding sequence is structured such that it directly impacts the objective function, which in this case is the total investment. Each chromosome's length corresponds to the total number of wells, and within the sequence, the identifier of each well is systematically represented in ascending order. This encoding method ensures that the GA optimally addresses the design constraints and effectively targets the optimization of the gas pipeline network's configuration. Each well's corresponding station number is stochastically generated at every position within the chromosome to denote the linkage between the well and its respective station. In the proposed chromosome coding scheme, each digit in a chromosome = (\(C_{1}\), \(C_{2}\), \(\ldots\)\(C_{m}\)) represents the number of the station corresponding to each well. (2) Crossover operator design. For the GA, the crossover operation method and cross probability are extremely important parameters which directly affect the optimization quality of the algorithm. Using CAP101 data based on OR-LIBRARY (50 wells, 25 stations) [25], the crossover operation method and crossover probability are tested for their influence on the solution, and the best crossover mode and crossover probability are determined. Each of the crossover techniques was tested 50 times, and their results were averaged for comparison. As the iterative generations increased, the algorithm progressively con  
Figure 7: Flow diagram of the basic GA.  
verged toward the optimal solution. Among the various crossover methods evaluated, the multi-point crossover method demonstrated a superior performance, leading to its selection. Figure 8 illustrates the average test outcomes. The crossover probability varies between 0.5 and 0.9. A comparison of the effects of these different crossover probabilities on the results was conducted by averaging the outcomes from 50 tests. When the crossover probability is between 0.7 and 0.9, the results are similar. Increasing the crossover probability properly can improve the convergence of the solution, so the crossover probability should be kept at a high value in the design. Figure 9 shows the effect of different crossover probabilities on the results in the multi-point crossing method. (3) Mutation operator design. The mutation operator is a measure that prevents the GA from falling into the local optimum by simulating the genetic and evolutionary values of the rupture allele [26]. In the process of debugging the mutation operator, a variation function is tested for the GA to  
Figure 8: Convergence curve of different crossover operation methods. Figure 9: Convergence curve of different crossover probabilities. mitigate the risk of falling into local optima in later stages. The mutation operators employ an adaptive approach, as indicated in Formula (14). \[\left\{ \begin{array}{l}MP=\varepsilon\cdot\left(\frac{Fit_{\max}-Fit}{Fit_{ \max}-Fit_{\min}}\right)\quad Fit\geq\frac{Fit_{\max}+Fit_{\min}}{2}\\ MP=\varepsilon\quad\quad\quad\quad\quad Fit<\frac{Fit_{\max}+Fit_{\min}}{2} \end{array} \right.