## 1 Introduction
[30] improved the A* algorithm by incorporating time, distance, and altitude to construct a heuristic function, solving the optimal and fastest cycling routes in the bicycle network in the city of Batu, Indonesia. [PERSON] et al. [31] applied the A* algorithm to solve the optimal path analysis problem for the public transportation network in Incheon, South Korea, using the distance to Manhattan as a heuristic function. [PERSON] et al. [32] designed a dynamic planning algorithm based on the urban transit commuting path planning method, which firstly calculates the shortest path between residential and office nodes using the shortest path algorithm, and then calculates the optimal transit travel plan on the shortest path using the dynamic planning algorithm. While the mentioned literature optimizes and improves path analysis algorithms in a single network of urban transportation, most of the above algorithms are evaluated in terms of distance, without comprehensively considering factors like time, transfers, and walking. Limited research has been conducted on hybrid network-based path analysis algorithms. Some have proposed hierarchical hub location models, treating hub locations as transfer stations, and conducting optimal path analysis by searching the paths from ordinary station to transfer station and vice versa [33, 34, 35]. However, these algorithms often use the number of transfers as the primary evaluation criterion, and neglect factors like the distance, time, and cost. Additionally, they struggle to solve the problem of this path analysis of hybrid networks, when travelers need to arrive at the next transfer station by walking, biking, or cab. [PERSON] et al. [36] used the A* algorithm, constructing heuristic functions based on straight-line distances, travel speeds, and fares, to solve the problem of travel between urban and intercity transportation networks. While effective for transferring between various restricted networks, such as subways, high-speed railways, and airplanes, this algorithm does not involve unrestricted networks, such as road networks. In addition to the typical constrained networks, such as urban transportation networks, there are also maritime beths and aviation networks that fall into the category of restricted networks, and there is corresponding research on their path planning [37,38]. To solve the urban multi-modal transportation mixed travel path planning problem, this paper firstly constructs a hybrid network designed for path planning analysis. This is achieved by establishing the topological relationship between restricted and unrestricted networks. Subsequently, an innovative multivariate heuristic function is proposed. By incorporating a multivariate constraint onto the distance heuristic function, this improve ment aims to optimize and refine the traditional A* algorithm. The number of interchanges, travel time, and travel cost, which are the key concerns of travelers, are also added to the heuristic function as constraint functions to realize the different needs of travelers within both restricted and unrestricted networks, achieved through a controlled prioritization of individual evaluation criteria. The contributions in this paper focus on the following three aspects:  
1. Design a multivariate heuristic function to solve the limitation of a single evaluation standard in the traditional A* algorithm. This improvement fully considers the traveler's experience and feelings. The application of the algorithm aims to furnish travelers with an optimal and highly comfortable travel plan. 2. Propose an improved A* algorithm tailored to restricted networks, specially designed to offer travelers the optimal travel plan for the combination of subway, bus, ferry, tourist line, airport express, and other modes of transport interchange in the urban public transportation network. 3.