## 5 Simulation and Results  
Firstly, the methods will be evaluated based on standard parameters defined as follows:  
The average travel time represents the total time spent by all vehicles within the transport network, divided by the number of vehicles that have entered the simulation. This parameter is mathematically expressed in Equation (8) as:  
\[\bar{t}=\frac{1}{N}\sum_{i=0}^{N}\left(t_{i,\text{ end }}-t_{i,\text{ start }}\right) \tag{8}\]  
where the variable \(\bar{t}\) represents the average time calculated over a total of N vehicles. For each vehicle, \(i\), \(t_{i,start}\) denotes the start time, and this value is used alongside other time data to compute the overall average time for the vehicles within the traffic system, \(t_{i,\,\text{end}}\)  is the end time for vehicle i. The relationship between these variables helps assess traffic flow and efficiency in the model. The average waiting time Equation (9) is determined by dividing the total accumulated waiting time of all vehicles by the total number of vehicles in the system. \[\overline{t_{w}}=\frac{1}{N}\sum_{i=0}^{N}t_{i,w} \tag{9}\]  
where the variable \(\overline{t_{w}}\) represents the average waiting time across N vehicles, where \(t_{i,w}\) is the specific waiting time for vehicle i. This metric is used to evaluate traffic efficiency and delays within the system. Each lane's queue length is determined by counting the number of halted vehicles within it. The statistical queue length can be calculated by adding the queue lengths across all lanes and dividing by the total number of lanes, resulting in a single value indicating the length of the network. Equation (10) describes how to compute this value:  
\[q_{l}=\frac{\sum_{i=0}^{L}q_{l}}{L} \tag{10}\]  
where the variable \(q_{l}\) represents the average queue length across L lanes, where \(\text{ql}_{\text{i}}\) is the specific queue length for lane i.  
Multi-Objective Optimization of Traffic Signal Prioritization: Balancing Emergency Response, System Efficiency, and Fairness  
Figure 3 includes two subfigures to compare traffic signal optimization and traffic signal delay for EV travel time. Figure 3a Trade-off between EV Time and Fairness shows a distribution diagram where each point represents a simulation example, the x-axis indicates the average EV travel time in seconds, and the \(y\)-axis shows the Gini coefficient corresponding to the metric to quantify the fairness of the travel time distribution of all vehicles. Color gradient encodes EV priority weight (\(\beta\)). This graph shows that the lower travel time of EVs is generally associated with a higher value \(\beta\), which means a better priority for emergency vehicles. However, with increasing and EV travel time declining, the Gini coefficient also tends to increase, indicating increased system unfairness. This demonstrates a trade-off where improving EV response times may lead to equity losses among regular vehicle users. Figure 3b accentuates this by presenting the values of the fairness penalty (\(\gamma\)) as contour levels on the same two variables: the EV travel time and the Gini coefficient. The contour map shows how varying \(\gamma\) affects the system's ability to balance performance and fairness. Lower \(\gamma\) values dominate regions with longer EV travel times and an equal traffic flow (lower Gini), while higher \(\gamma\) values appear in areas where fairness is compromised, indicating shorter EV periods. This supports the concept that \(\gamma\) is a regulatory parameter and penalizes solutions that undermine fairness, even if they are beneficial to EV travel time. Collectively, these two subfigures illustrate the dynamic tension between efficiency and equity in traffic management and indicate that adjustment of weights (\(\beta\) and \(\gamma\)) is required to achieve an appropriate balance based on operational objectives. Figure 4 shows the evolution of optimization weights over 50 epochs.