## 2 Literature Review  
### General Traffic Signal Timing Optimization Problem Framework  
In typical traffic signal optimization problems, the goal is to optimize the traffic flow at intersections by adjusting signal timing. The signal control parameters include the green light time, saturation rate, and signal period for each phase, which affect the efficiency of traffic flow. The optimization problem of traffic signals generally selects the minimum total delay time within the region as the objective function of the optimization model. Currently, the models used to calculate delay at signalized intersections mainly include the Webster delay model [3], the ARRB model [4], and the HCM2000 model [5]. The following is the general framework of this optimization problem.  
1. Objective function: The average delay time of vehicles at intersections can be expressed as Equation (1): \[d=\frac{C(1-\lambda)2}{2(1-\lambda\cdot x)}+\frac{x^{2}}{2q(1-x)}\] (1) Among them: \[\lambda=\frac{g}{C},x=\frac{q}{S}\] (2)  
\(C\) is the signal period, \(\lambda\) is the green signal ratio, x is the saturation rate, \(q\) is the actual traffic flow at the intersection, \(g\) is the green light time, and \(S\) is the intersection saturation flow rate.  
Therefore, the objective function for minimizing the total delay time in the region is expressed as Equation (3): [6]  
\[Delay=\min\sum_{k=1}^{n}\sum_{i=1}^{m}\left[\frac{C(1-\lambda_{i}^{k})^{2}}{2 (1-\lambda_{i}^{k}\cdot x_{i}^{k})}+\frac{{(x_{i}^{k})}^{2}}{2q_{i}^{n}(1-x_{ i}^{k})}\right]\cdot q_{i}^{k} \tag{3}\]  
In the formula, \(n\) is the number of intersections, and \(m\) is the number of phases.
2. Constraint condition According to the general situation, it is stipulated that there are shortest and longest green light durations for each phase in the optimization process, ensuring that each phase can obtain the right-of-way within one cycle [7]. The constraints for solving are: 1. Minimum Green Light Time Constraints: The green light time for each phase cannot be less than a minimum time to ensure the safe passage of vehicles: \[g_{i}\geq g_{\min},\forall i\] (4) Among them, \(g_{\min}\) is the minimum green light time for each signal stage. 2. Cycle Length Constraints: \[C_{\min}\leq C\leq C_{\max}\] (5) Among them, \(C_{\max}\) and \(C_{\min}\) are the maximum and minimum periods of the intersection.
3. Mathematical optimization model The mathematical model framework for optimizing the timing of general traffic signals can be represented by the following Equation (6) and Constraint (7): \[Delay=\min\sum_{k=1}^{n}\sum_{i=1}^{m}\left[\frac{C(1-\lambda_{i}^{k})^{2}}{2 (1-\lambda_{i}^{k}\cdot x_{i}^{k})}+\frac{{(x_{i}^{k})}^{2}}{2q_{i}^{n}(1-x_{ i}^{k})}\right]\cdot q_{i}^{k}\] (6)\[s.t.\begin{cases}\lambda=\frac{g}{\lambda}\\ x=\frac{g}{\lambda}\\ C_{\min}\leq C\leq C_{\max}\\ g_{i}\geq g_{\min},\forall i\end{cases} \tag{7}\]  
At the same time, research on signal timing optimization at intersections has primarily focused on minimizing delay, with the most classic being the Webster timing optimization scheme [8]. In recent years, on the one hand, scholars have innovated signal timing optimization schemes based on considering delay. On the other hand, foreign scholars have also been continuously exploring the consideration of fairness in the signal optimization process.