## 2 Dynamic Optimization Strategies for Public Transportation Operations
### Stop-Skipping Strategy  
Skipping a stop means that passengers disembarking at the stop must spend more time waiting for the next bus or choose to disembark at a nearby stop, making the passengers' traveling experience decline. Therefore, the use of the stop-skipping strategy must fully consider the actual level of passenger demand and the distribution characteristics of the line. To improve the overall operational efficiency of the line, a reasonable start threshold should be set as far as possible to reduce the negative impact of bus skipping on the passenger experience. Suppose that, at some point in time, a bus on the route is about to arrive at bus stop \(j\), where the adjacent previous bus stop is \(j-1\) and the next stop is \(j+1\). Taking the interval between stop \(j\) and \(j+1\) as an example, as shown in Figure 1, the bus operation process can be described as follows: at the very first moment, bus \(i\) starts traveling with uniform acceleration from stop \(j\), and the speed reaches the highest speed \(v_{max}\) of the interval operation and is then maintained; when near stop \(j+1\), bus \(i\) enters into the process of uniform deceleration until its speed is reduced to 0, and stays at stop \(j+1\). Figure 2 shows the process of a bus traveling continuously between two or more bus stops. Point \(p\) is the location where bus \(i\) has the greatest speed \(v_{max}\) and is about to enter the deceleration process as it travels from stop \(j\) to \(j+1\). Similarly, point \(q\) is the location where bus \(i\) has just completed the acceleration process, and speed \(v_{max}\) is reached. When the stop-skipping strategy is not adopted, the operation process between \(p\) and \(q\) can be divided into three phases: deceleration to stop, passengers embarking or disembarking, and accelerating to normal speed. The total running time \(T_{s1}^{i,j}\) consists of deceleration time \(T_{a}^{i,j}\), embark/disembark time \(T_{k}^{i,j}\), and acceleration time \(T_{b}^{i,j}\): \(T_{s1}^{i,j}=T_{a}^{i,j}+T_{k}^{i,j}+T_{k}^{i,j}+T_{b}^{i,j}\). The additional cost of bus operations can be described as \(C_{s1}^{i,j}=\alpha\left(T_{a}^{i,j}+T_{b}^{i,j}\right)+\beta T_{k}^{i,j}\), where \(\alpha\) is the control variable of the bus deceleration and acceleration costs and \(\beta\) is the control variable for the cost of the bus-stopping operation time at stop \(j\). Figure 1: Bus travel speed change process between stops. When adopting the stop-skipping strategy, i.e., bus \(i\) travels from point \(p\) at a constant speed \(v_{max}\) until it passes point \(q\), the total running time \(T_{s2}^{i,j}=T_{p}^{i,j}+T_{q}^{i,j}=\frac{s_{p}+s_{q}}{v}\). The additional cost of bus operations and the cost of the stopping time at stop \(j\) is 0. However, stop-skipping will lead to passengers waiting at stop \(j\) not being able to get on the bus and passengers on bus \(i\) not being able to disembark at stop \(j\), so it is considered that there needs to be stop-skipping penalty cost, which can be described as \(C_{s2}^{i,j}=\gamma N_{d}^{j}+\delta N_{u}^{j}\), where \(\gamma\) and \(\delta\) are the control variables of passenger loss costs and \(N_{d}^{j}\) and \(N_{u}^{j}\) are the numbers of passengers needing to embark/disembark. In summary, the combined bus stop-skipping strategy in this study can be described as follows:  
\[C_{sd}^{i,j}= \begin{cases}C_{s1}^{i,j},N_{d}^{j}\geq N_{d_{max}}^{j}\\ min\big{[}C_{s1}^{i,j},C_{s2}^{i,j}\big{]},0\leq N_{d}^{j}<N_{dmax}^{j}\end{cases}  \tag{1}\]  
\[C_{su}^{i,j}= \begin{cases}C_{s1}^{i,j},N_{u}^{i}\geq N_{umax}^{j}\\ min\big{[}C_{s1}^{i,j},C_{s2}^{i,j}\big{]},0\leq N_{u}^{j}<N_{umax}^{j}\end{cases}  \tag{2}\]  
\[C_{s}^{i,j}=max\big{[}C_{sd}^{i,j},C_{su}^{i,j}\big{]} \tag{3}\]  
In the above equation, \(N_{dmax}^{j}\) and \(N_{umax}^{j}\) are the thresholds for the activation of embarking and disembarking for stop-skipping control.