## 4 Algorithm Design
### Conflict Resolution Mechanisms for Limited Charging Resources
#### 4.1.1 Charging Sequence Feasibility Theorem  
This section proposes the charging sequence feasibility theorem by analyzing the time constraints of replenishment for different paths at CI. The theorem establishes the necessary conditions for conflict elimination through charging sequence adjustments between two vehicle paths sharing the same CI, providing a theoretical foundation for the conflict resolution mechanism at energy replenishment. Assuming that the available internal capacity of the CI is 1, which means only one vehicle is allowed to charge during a specific period. Let \(ES_{i}\) denote the earliest start time of charging for vehicle \(i\), \(T_{i}\) represent the required charging duration, and \(EF_{i}\) be the earliest finish time, where \(EF_{i}=ES_{i}+T_{i}\). In an ideal scenario without capacity constraints, all vehicles can start charging immediately upon arrival, where \(ES_{i}\) and \(EF_{i}\) become deterministic values solely dependent on the current route's structure. \(T_{max}\) represents the longest path duration among all routes, analogous to the total project completion time. \(T_{back(i)}\) represents the time at which vehicle \(i\) returns to the depot under the current routing plan, allowing for a charging shift time of \(T_{shift(i)}=T_{max}-T_{back(i)}\). Consequently, the latest start time for charging is \(LS_{i}=ES_{i}+T_{shift(i)}\), and the latest finish time is \(LF_{i}=LS_{i}+T_{i}\). Let \(S_{i}\) and \(F_{i}\) denote the actual start and finish times of charging for vehicle \(i\), respectively,with \(F_{i}=S_{i}+T_{i}\). The definitions of the aforementioned time parameters are illustrated in Figure 3.  
If vehicles \(a\) and \(b\) arrive at the CI sequentially and satisfy \(ES_{a}\leq ES_{b}\leq EF_{a}\), the FIFO principle prioritizes vehicle \(a\), requiring vehicle \(b\) to start charging only after \(a\) completes its task. This results in a charging delay of \(T_{delay}=EF_{a}-ES_{b}\). Notably, such a delay occurs if and only if \(LS_{b}\geq EF_{a}\). The mechanism of charging delay for vehicle \(b\) is shown in Figure 4.  
However, if \(T_{shift(b)}\) is too small, indicating a narrow charging shift time window for vehicle \(b\), there is a probability that \(LS_{b}<EF_{a}\). In such cases, adhering to the FIFO principle for vehicle \(a\) may prevent vehicle \(b\) from being charged. By reversing the charging order of \(a\) and \(b\), it becomes possible to satisfy both charging demands simultaneously through deferred charging, thereby improving the utilization of capacity-limited CI. Figure 5 depicts charging conflict resolution via this scheduling strategy. Based on this insight, the following theorem is proposed. Theorem 1: _If the internal capacity of the CI is 1 at a given time, and electric logistics vehicle a arrives at the CI before vehicle \(b\) (i.e., \(ES_{a}\leq ES_{b}\)), the following condition on their charging time parameters must be satisfied:_  
\[ \begin{cases}LS_{b}<EF_{a}\\ LS_{a}>EF_{b}\end{cases}  \tag{27}\]  
_Then, the following is true:_  
* _If the charging task of a is an immediate predecessor, the charging demand of_ \(b\) _cannot be met._
* _If the charging task of vehicle_ \(b\) _is an immediate predecessor, a feasible deferred charging schedule exists to simultaneously satisfy the charging demands of both_ \(a\) _and_ \(b\)  
Figure 4: Deferrable initiation of charging task \(b\) after the completion of task \(a\). Figure 3: Schematic diagram of time parameters under CPM. Proof of Theorem 1.: The charging service interval of vehicle \(i\) is defined as the closed interval \([S_{i},F_{i}]\), where \(ES_{i}\leq S_{i}\leq LS_{i}\). (1) Scenario 1. Charging task \(a\) is the immediate predecessor. Assume that a feasible schedule exists, where the charging interval for vehicle \(a\) is \([S_{a},F_{a}]\), thus satisfying  
\[ES_{a}\leq S_{a}\leq LS_{a} \tag{28}\]  
\[F_{a}=S_{a}+(EF_{a}-ES_{a}) \tag{29}\]  
Vehicle \(b\) starts charging only after the completion of vehicle \(a\)'s charging task, implying \(S_{b}\geq F_{a}\).