## 3 Case Study  
The case study was the No. 115 bus in Jinan, China, operating along Jingshi Road, north of Jinan. The Yellow River is nearby, and the south is surrounded by mountains. Jinan is an east-west city. Jingshi Road is one of the east-west trunk roads of Jinan. There are 34 stops for the No. 115 bus, as shown in Figure 2. The service area of the No. 115 bus includes a college town, central business district (CBD), high-tech zone, and tourist attractions, while the travel demand distribution along the No. 115 bus route is very uneven. The total number of passengers is approximately 2800 p/h during peak hours. The headway of the No. 115 is 5 to 8 min during peak hours. The origin-destination (OD) data were mainly investigated through IC cards and supplemented by manual investigation. The stop distance matrix, \(\mathbf{D}\), was obtained by Geographic Information System (GIS). Other input data are shown in Table 1.  
Genetic algorithms (GAs) have been comprehensively applied in the stop design of limited-stop services, such as in the studies by [PERSON] [25], [PERSON] [26], [PERSON] [27], and [PERSON] [28]. The optimal solution obtained by the Lingo software is the global optimal solution. Therefore, this study developed heuristic methods based on the GA and the Lingo software program to solve this problem. The two methods were compared. The application procedures for the genetic algorithm are shown in Figure 3. The problem was modeled in the GA using \"Matlab optimtool\". The decision variables were coded as the double vector. The fitness function was the objective function of the proposed model. In this study, the population size was 50, the crossover fraction was 0.8, and the migration fraction was \begin{table}
\begin{tabular}{c c c} \hline \hline
**Label** & **Value** & **Parameter** \\ \hline \(M\) & 34 & The number of vehicles operating on the two-way line during peak hours (veh); \\ \(L\) & 20.4 & The length of the bus corridor (km); \\ \(\mu\) & 20 & The average time value for the passengers (\$/h); \\ \(\tau\) & 35 & The bus dwelling time at each stop (s/stop); \\ \(\tau^{\prime}\) & 1 & The average passengers’ boarding and outboarding time (s/p); \\ \(\$_{M}\) & 40 & The unit cost for the vehicular purchase per hour (\$/veh-h); \\ \(\$_{V}\) & 2 & The unit cost for the vehicular traveled per kilometer (\$/veh-km); \\ \(\Lambda\) & 2800 & The total number of passengers during peak hours (p/h); \\ \(\lambda\) & 1120 & The average number of passengers (p/h); \\ \(v_{w}\) & 2.5 & The speed of passengers entering or leaving the bus system (km/h); \\ \(v\) & 25 & The cruising speed in the city (km/h), which was constant; \\ \(w_{w}\) & 2.2 & The weight of the waiting time; \\ \(w_{A}\) & 2.1 & The weight of the access and egress times; \\ \(g\) & 1.5 & The ratio of the fare for the limited-stop service to the fare for the local service. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The input values of the parameters. Figure 2: No. 115 bus stops. 0.05 [25]. Crossover combined two parents. To ensure that the children met the constraints, child1 = parent1 + 0.5\(a\) (parent2-parent1). \(a\) was a random number between 0 and 1. Based on \"Matlab optimtool\", migration was set as \"constraint dependent\" to ensure the new chromosomes met the constraints. When the number of iterations reached 150, the solver stopped [25]. The application procedures for the Lingo software included input programming, software solutions, and output results. The results obtained from the Lingo software and the genetic algorithm are given in Tables 2 and 3. In Tables 2 and 3, if stop \(i\) is serviced by a limited-stop bus, \(x_{i}\) = 1, else \(x_{i}\) = 0; \(H_{L}\) and \(H_{e}\) respectively mean the headways of the local service and the limited-stop service during peak hours; \(C\), \(C_{A}\) and \(C_{U}\) respectively mean the total cost, the agency cost, and the passenger cost of the bus service.