# A Multi-Objective Path-Planning Approach for Multi-Scenario Urban Mobility Needs
## 3 Multi-Objective Path Planning
### Optimization Objectives  
This study adopts the weighted sum method as the core algorithm for multi-criteria decision analysis, primarily due to its low sensitivity to weight variations and low computational complexity, which make it highly applicable to path planning. In contrast, other multi-criteria decision analysis methods have certain limitations. For instance, the multiplicative convolution method may amplify interactions between objectives, reducing the robustness of the algorithm. Similarly, although the TOPSIS method performs well in ranking solutions, its computational cost increases significantly in high-dimensional objective scenarios. Therefore, considering these factors, the weighted sum method is the optimal choice for this study. The goal of multi-objective path planning is to minimize a weighted comprehensive objective function \(F_{\text{total}}(p)\). The path \(p\) is defined as a sequence of consecutive nodes \((n_{1},n_{2},\ldots,n_{k})\), where each pair of nodes \((n_{i},n_{i+1})\) forms an edge in the graph \(G\). This function consists of multiple weighting factors and corresponding objective terms, each representing a different optimization requirement. Each weighting factor is dynamically adjusted during the path-planning process based on the demands of real-world scenarios, enabling multi-objective optimization of the path. To ensure comparability among the optimization objectives, each objective needs to be normalized. The normalization method is detailed in Formula (16). The mathematical model for this multi-objective path optimization problem is as follows:  
\[ \begin{array}{ll}\text{min}_{p}F_{\text{total}}(p)&=\alpha(t)\cdot f_{\text {length}}(p)\\ &+\beta(t)\cdot f_{\text{congestion}}(p)+\gamma(t)\cdot f_{\text{time}}(p)\\ &+\delta(t)\cdot f_{\text{energy}}(p)+\varepsilon(t)\cdot f_{\text{risk}}(p) \end{array}  \tag{5}\]  
\[ \begin{array}{ll}\text{s.t.}&\left\{\begin{array}{ll}f_{\text{time}}(p) \leq T_{\text{max}}\\ f_{\text{congestion}}(p)\leq C_{\text{max}}\\ f_{\text{energy}}(p)\leq E_{\text{max}}\\ f_{\text{risk}}(p)\leq R_{\text{max}}\end{array}\right.\end{array}  \tag{6}\]\[\forall i\in\{1,2,\ldots,k-1\},\ (n_{i},n_{i+1})\in E \tag{7}\]  
where \(\alpha(t)\), \(\beta(t)\), \(\gamma(t)\), \(\delta(t)\), and \(\epsilon(t)\) are the weighting coefficients dynamically adjusted according to the scenario. \(f_{\text{length}}(p)\) represents the total length of path \(p\); \(f_{\text{congestion}}(p)\) is the congestion length of path \(p\); \(f_{\text{time}}(p)\) denotes the total time of path \(p\); \(f_{\text{energy}}(p)\) represents the energy consumption of path \(p\), and \(f_{\text{risk}}(p)\) is the risk index of path \(p\). Equation (6) describes the constraints of this optimization problem. The path must be composed of consecutive nodes from the start to the destination, meaning that there is an edge between each pair of adjacent nodes \(n_{i}\) and \(n_{i+1}\) in graph \(G\). The total travel time \(f_{\text{time}}(p)\) has an upper limit \(T_{\text{max}}\), representing the maximum acceptable travel time. The congestion length \(f_{\text{congestion}}(p)\) has an upper limit \(C_{\text{max}}\) to avoid paths passing through severely congested areas. The energy consumption \(f_{\text{energy}}(p)\) has an upper limit \(E_{\text{max}}\), meeting the demand for energy conservation. The risk index \(f_{\text{risk}}(p)\) has an upper limit \(R_{\text{max}}\) to ensure driving safety. Equation (7) describes the decision variable constraints of the optimization problem. The decision variable is the feasibility constraint of the path: the path must consist of consecutive nodes from the start to the end, with an edge connecting each pair of adjacent nodes \(n_{i}\) and \(n_{i+1}\). \(k\) represents the total number of nodes in the path, and \((n_{i},n_{i+1})\in E\) indicates that there is an edge between node \(n_{i}\) and node \(n_{i+1}\). This study selects five optimization objectives: total distance; congestion distance; time; energy consumption; and risk index.