# A Multi-Objective Path-Planning Approach for Multi-Scenario Urban Mobility Needs
## 3 Multi-Objective Path Planning
### Optimization Objectives
These objectives comprehensively assess the feasibility of each path based on the actual needs of users. Each optimization objective is described in detail below. 1. Total Distance  
The total distance of the path is defined as the cumulative length from the starting point to the endpoint. Given a path node sequence \(n_{1},n_{2},\ldots,n_{k}\), where \(k\) is the number of nodes in the path, the total distance \(f_{\text{length}}(p)\) of the path is expressed as  
\[f_{\text{length}}(p)=\sum_{i=1}^{k-1}d(n_{i},n_{i+1}) \tag{8}\]  
where \(d(n_{i},n_{i+1})\) represents the distance between nodes \(n_{i}\) and \(n_{i+1}\);  
2. Congestion Distance  
The congestion distance refers to the cumulative length of all congested segments in the path and is used to assess the traffic condition of the path. Congestion status is determined by the \"congestion tag\" in the path. The congestion distance \(f_{\text{congestion}}(p)\) of the path is defined as follows:  
\[f_{\text{congestion}}(p)=\sum_{i=1}^{k-1}\left\{ \begin{array}{l}d(n_{i},n_ {i+1}),\text{if congestion}(n_{i},n_{i+1})>0\\ 0,\text{otherwise}\end{array} \right. \tag{9}\]  
where congestion\((n_{i},n_{i+1})\) is the congestion value of edge \((n_{i},n_{i+1})\). If the congestion value is greater than zero, the segment distance is included in the cumulative congestion distance;  
3. Total Time  
The total time \(f_{\text{time}}(p)\) of the path comprises the driving time and the waiting time at red lights. In this study, the average vehicle speed is set to \(v\) = 60 km/h, and the red light waiting time is assumed to be 1 min per stop. The total time can be expressed as  
\[f_{\text{time}}(p)=\frac{f_{\text{length}}(p)}{v}+r\cdot t_{r} \tag{10}\]where \(f_{\text{length}}(p)\) is the total length of the path; \(r\) is the number of red lights along the path, and \(t_{r}\) is the waiting time per red light;
4. Energy Consumption  
Based on the phenomenon of increased energy consumption under congested conditions, this study evaluates energy consumption according to the congestion status of the path. Due to frequent acceleration, deceleration, and idling, vehicles experience additional energy consumption. The value of 20% is used as an average to represent the extra impact of congestion on energy consumption. The total energy consumption \(f_{\text{energy}}(p)\) of the path can be represented as  
\[f_{\text{energy}}(p)=\sum_{i=1}^{k-1}e(n_{i},n_{i+1}) \tag{11}\]  
where \(e(n_{i},n_{i+1})\) denotes the unit energy consumption from node \(n_{i}\) to node \(n_{i+1}\), defined as  
\[e(n_{i},n_{i+1})=d(n_{i},n_{i+1})\times \begin{cases}1.2,&\text{ifcongestion}( n_{i},n_{i+1})>0\\ 1,&\text{otherwise}\end{cases}  \tag{12}\]  
When a segment of the path is congested, the energy consumption increases by 20% due to congestion; otherwise, it is calculated at the normal rate;
5. Risk Index;
6. In this study, the risk index is measured by the sum of the \"road type score\" and \"traffic flow score\". The risk index \(f_{\text{risk}}(p)\) of the path is calculated as  
\[f_{\text{risk}}(p)=\frac{d(n_{i},n_{i+1})}{f_{\text{length}}}\sum_{i=1}^{k-1} \left(\text{RL}(n_{i},n_{i+1})+\text{TL}(n_{i},n_{i+1})\right) \tag{13}\]  
where \(\text{RL}(n_{i},n_{i+1})\) represents the road type score from node \(n_{i}\) to node \(n_{i+1}\); \(\text{TL}(n_{i},n_{i+1})\) represents the traffic flow score from node \(n_{i}\) to node \(n_{i+1}\); \(d(n_{i},n_{i+1})\) is the length of the edge from node \(n_{i}\) to node \(n_{i+1}\), and \(f_{\text{length}}(p)\) is the total length of the path. The road type score is as follows:  
\[\text{RL}(n_{i},n_{i+1})= \begin{cases}5,&\text{Highway}\\ 4,&\text{Urban Main Road}\\ 3,&\text{Urban Secondary Road}\\ 2,&\text{Rural Road, Alley, Side Road}\\ 1,&\text{Other}\end{cases}  \tag{14}\]  
The traffic flow score is as follows:  
\[\text{TL}(n_{i},n_{i+1})=\left\{ \begin{array}{l}1,\geq 2000\text{vehiches}/ \text{day}\\ 3,500\,\sim 2000\text{ vehicles}/\text{day}\\ 5,\leq 500\text{ vehicles}/\text{day}\end{array} \right.