## 1 Introduction  
Multiple transportation lines mix and coexist in a modern urban transportation system [1]. Urban roads, overpasses, and underpasses constitute the city's road traffic network, while urban rail transit, bus lines, tourist lines, and airport express lines constitute the urban public transport network. The urban transport service network is made up of these overlapping and crisscrossing traffic service lines. For instance, the public transportation network in Shanghai, China, already included more than 1000 public transportation lines by 2022, encompassing subway, light rail, bus, and airport express lines, among others [2]. Travelers are typically bound to the specified line and can't request route adjustments from bus drivers to accommodate their requirements. Meanwhile, other means of transportation, such as walking, bicycles, taxis, and private cars, provide additional travel options. Based on whether travelers' route choice is restricted, the urban traffic service network is split into unrestricted and restricted networks, with both networks amalgamating into a hybrid network in most instances. Such a hybrid network represents the essence of the modern urban traffic service system. Moreover, many global metropolises have gradually developed and evolved into satellite town mode due to ongoing urban expansion [3, 4]. The park-and-ride (P&R) traffic traffic, in which travelers drive to a transfer parking lot in urban suburbs and then use public transportation to reach their destination in the city center, has provided people with a new way of life and travel to solve the traffic challenges between satellite towns and central urban areas [5, 6, 7, 8, 9]. The P&R traffic mode supports integrating the road transportation system into the urban public transportation system and promoting higher use of urban public transportation systems. Taking Shanghai as an example, by the end of 2022, there were 20 P&R transfer parking lots close to the satellite urban rail transit stations. The P&R traffic mode formally complies with the idea of low-carbon transportation, which may significantly lessen traffic congestion and reduce carbon emissions [9]. It is often challenging for travelers to develop a practical travel plan given the intricate transportation service network and wide range of travel alternatives [10,11]. The growing travel demand for P&R parking and transfer mode in particular compels travelers to develop a reasonable travel plan in the hybrid network comprising both road traffic and the public transportation network, introducing considerable difficulty for travelers. The P&R traffic mode has been an important part of the hybrid urban traffic service network. Therefore, based on various forms of traffic services, this paper solves the problem of path analysis in the modern urban traffic service system and provides suitable transportation plans for travelers. A large number of specialists and researchers have carried out extensive studies on path analysis algorithms, with the Dijkstra algorithm [12, 13, 14, 15, 16, 17] and the A* algorithm [18, 19, 20, 21, 22, 23, 24] standing out as the most representative. The majority of these studies have primarily focused on a single transportation network. For example, [PERSON] et al. [25] applied [PERSON]'s algorithm to analyze the shortest bus paths in Yangon City based on bus transit network data. [PERSON] and [PERSON] [26] solved the problem of shortest path analysis for Tirana's urban tram network using the [PERSON]'s algorithm, while [PERSON] et al. [27] focused on cab shortest path analysis. [PERSON] et al. [28] implemented the shortest path travel scheme for tourists in the urban Bekasi road network using the A* algorithm within the Android system. [PERSON] et al. [29] tackled the path analysis of tourist attractions in Java using the A* algorithm, considering factors such as the shortest distance and traffic flow as the primary evaluation criteria. [PERSON] et al.