# Optimizing Ambulance Allocation in Dynamic Urban Environments: A Historic Data-Driven Approach
## 4 Numerical Experiments
### Result  
Table 2 is a table that compares the results of the experiments for each term of the objective function (1).  
For each column, the number in parentheses indicates the index assigned to each term in the objective function (1). The \"Before\" column represents the current operational strategy without applying the proposed mathematical model.  
The sum of the objective values improved from 222.78 before to 445.43 after applying the mathematical model, an improvement of about 99.94%. Looking at the values of the main terms, Term (a) for the demand cover reward improved by 1.33%. Term (d), which rewards cells where an ambulance can be dispatched within the golden time, improved by 138.46%, from 156 before to 372 after applying the mathematical model.  
Figure 2 compares the current practice and the results from the mathematical model regarding the number of ambulances deployed in each region. For Seongbuk-gu, the number of ambulances remained the same at 6 during spring, while it was reduced to 5 for the rest of the seasons. In Dongdaemun-gu, 6 ambulances were deployed in winter and 7 in the other seasons. In Seongdong-gu, 4 ambulances were deployed in spring as before, but the number increased to 5 in summer and fall and to 6 in winter.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Before**} & \multicolumn{4}{c}{**Mathematical Model**} \\ \hline
**Demand** & **Open Cost** & **Relocation** & **Non-Zero** & **Demand** & **Open Cost** & **Relocation** & **Non-Zero** \\
**Cover (a)** & **(b)** & **Cost (c)** & **Roward (d)** & **Cover (a)** & **(b)** & **Cost (c)** & **Roward (d)** \\ \hline
**Value** & 66.78 & 0 & 0 & 156 & 67.67 & 6 & \(-2.4\) & 372 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Experiment results.  
Figure 1: (a) shows the regions to be analyzed. (b) is a map of the demand information after dividing the region into hexagonal cells from the historical data for the three regions we want to analyze. The darker the color, the higher the demand is.  
This result is likely due to Dongdaemun-gu being located at the center of the three regions, with the main demand stemming from the center, as illustrated in Figure 1b. For Seongdong-gu, even though the demand was not lower than in the other two regions, the current number of ambulances was insufficient. Therefore, more ambulances were deployed to cover more demand cells within the golden time.  
Figure 3 is a chart comparing the number of cells that cannot be covered within the golden time, by region and season, before and after applying the mathematical model.  
We defined the zero zone as the cells with \(y_{i1t}=0\) that are not covered by at least one ambulance. Prior to the mathematical model, there were 171 zero zones. In the mathematical model solution, there were 90 on average for the four planning horizons. The number of zero zones decreased by 47.37%, especially for Seongbuk-gu, which decreased by about 61.22% from 98 to an average of 38.  
Figure 4 shows the change in ambulance deployment over the seasons on a map.  
When analyzing Seongbuk-gu, we see that even though one fewer ambulance is deployed compared to before, as shown in Figure 2, the number of zero zones was reduced by more than half, as illustrated in Figure 3. This reduction can likely be attributed to ambulances being deployed across a broader area than in the previous setup. For fall and winter, while the center locations remained consistent from a regional perspective, the distribution of ambulances across the three regions varied seasonally, as depicted in Figure 2.  
Figure 3: Number of zero zones where it is difficult for an ambulance to cover within the golden time.  
Figure 2: Number of ambulance deployments for each time period.