## 2 Dynamic Optimization Strategies for Public Transportation Operations
### Path Optimization Strategy  
In order to ensure driving safety and improve operational efficiency, conventional public transportation vehicles have a fixed path and station. However, in the case of the local stop-skipping control of bus routes, a reasonable limited path adjustment and optimization strategy can be taken in order to avoid road congestion or other reasons for operational delays. When it is feasible to adopt the stop-skipping strategy at stop \(j\) and real-time traffic information shows that short-term congestion will occur nearby stop \(j\), the path optimization strategy can be judged and implemented. The path optimization strategy can be carried out in two steps:  
1. Vehicle operation topology network construction with branching. The basic domain for feasible path selection is constructed first. In order to reduce the number of searches for invalid nodes during path generation, the basic feasible domain of nodes is delineated, as shown in Figure 3. The search domain has \(j\) as its center and \(\varepsilon_{max}(d|j-1,j|,d|j,j+1|)\) as its radius. As the bus does not experience the deceleration, stopping, and acceleration process when traveling over the stop, the average speed of the interval will be higher than that of regular driving, so the range diffusion control coefficient \(\varepsilon\) is set to appropriately expand the search range and avoid the truncation of available paths due to the loss of some nodes. In this paper, we take \(\varepsilon=1.2\). Figure 2: Bus travel speed change process for continuous travel. Next, network weight filling and path generation are performed. The scenario targeted in this study is variable path planning under a future travel state, which faces many uncertainties and large dynamic changes in traffic flow. The Co-evolutionary Path Optimization (CEPO) method can deal with multi-source traffic information more efficiently and reflect the evolution trend of the traffic environment of the road section in the future period better, so this method is adopted for the planning of dynamic paths. Considering the main influencing factors in the operation scenario of public transportation, this study selects the average roadway speed and average node delay as the key indicators of roadway network traffic parameters \(G(t)\). There are:  
\[G(D_{t+k},V_{t+k})=G(D_{t},V_{t})+\partial_{t+k},k\geq 0 \tag{4}\]  
\[[D_{t+2k},V_{t+2k}]=f_{\varphi}(D_{t+k},V_{t+k})+\partial(D_{t+k},V_{t+k}),k\geq 0 \tag{5}\]  
In the above equation, \(D_{t+k}\) denotes the average delay incurred by the bus while passing through the node from \(t\) to \(t+k\); \(V_{t+k}\) denotes the average speed of the bus; \(\partial_{t+k}\) is the error between the predicted and actual values of the roadway network traffic parameters at the \(t+k\) moment; and \(f_{\varphi}\) is a time-varying function that describes the relationship between the roadway network traffic parameters over time. (2) Bus operation strategy construction based on relaxation time window. The construction of the path optimization strategy is carried out with reference to the implementation of the vehicle crossing station combination control strategy. Supposing that, at a certain point in time, bus \(i\) is about to arrive at stop \(j\) and will execute the stop-skipping control strategy, the intended traveling path of bus \(i\) along the stop \(j-1\)\(\rightarrow\)stop \(j\)\(\rightarrow\)stop \(j+1\) is \(j^{j}_{0}\), and the feasible optimization paths are \(l^{j}_{1},l^{j}_{2},\ldots,l^{j}_{n}\). The movement process of bus \(i\) can be approximated to divide it into two components: traveling on the roadway and passing through the stop. Taking path \(l^{j}_{1}\) as an example, the total length of the path is specified as \(s^{j}_{1}\), the average traveling speed is \(\overline{v}^{j}_{1}\), the total number of buses passing through the stops is \(n^{j}_{1}\), and the average waiting time at the stops is \(\overline{t}^{j}_{1}\).