## 4 Analysis on Air Distance, Road Distance and Road Time
### Closeness Decived by Air Distance  
There is no doubt that air distance can be deceiving when measuring closeness in transportation networks. However, the question of how often and how much has rarely been studied.  
Figures 8 and 9 highlight the frequency and magnitude, respectively. More specifically, Figure 8 presents the rate at which the use of air distances and road distances produce different results. Suppose there are three nodes, A, B, and C. The air distances from A to B and from A to C are measured and compared. This same comparison is subsequently carried out using road distance or road time. If the comparison using air distance versus road distance or road time creates different winners, it is assumed that the air distances are inaccurate. Ten distance range boundaries are set for the analyses. The boundaries are based on air distances. For a boundary of [0-1], the air distances from A to B and from A to C are both within 1 km. Two different average values are displayed in the figure. In a nutshell, average (1) is calculated with nodes with similar air distances, while average (2) also includes cases with dissimilar air distances. The average (1) refers to the average value from all boundaries [0-1, 1-2,..., 9-10]. On the other hand, average (2) is the average value from a different boundary, which is [0-10]. It is natural that average (2) has a lower rate than average (1), since average (2) includes cases where differences in the air distance  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Cost** & **Vehicle** & **Capacity** & **Multiple** & **Name** \\ \hline \multirow{4}{*}{Distance (D)} & 1-ton trucks & & 5 times (M5) & DV1M5 \\  & (V1) & 5 cubic meters & 10 times (M10) & DV1M10 \\  & & & 20 times (M20) & DV1M20 \\ \cline{2-5}  & 2.5-ton trucks (V2) & & 5 times (M5) & DV2M5 \\  & 2.5-ton trucks & 12.5 cubic meters & 10 times (M10) & DV2M10 \\  & & & 20 times (M20) & DV2M20 \\ \hline \multirow{4}{*}{Time (T)} & 1-ton trucks (V1) & & 5 times (M5) & TV1M5 \\  & (V1) & 5 cubic meters & 10 times (M10) & TV1M10 \\  & & & 20 times (M20) & TV1M20 \\ \cline{1-1} \cline{2-5}  & 2.5-ton trucks (V2) & & 5 times (M5) & TV2M5 \\  & (V2) & 12.5 cubic meters & 10 times (M10) & TV2M10 \\  & & & 20 times (M20) & TV2M20 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Options for varying proposed benchmark instances.  
from A to B and from A to C are within 10 km, while average (1) only includes cases where the differences are within 1 km.  
As shown in Figure 8, within the boundaries of 1 km, there is a 34.9% chance that the air distance is wrong for the distance measurement, and a 40.6% chance that it is wrong for the time measurement. Even if the boundary is set to be within 10 km, there still exists a high chance that the air distance will be wrong: 11.6% for distance and 16.4% for time.  
In Figure 8, it is noted that using air distance in a time measurement is more likely to be inaccurate than for a distance measurement. This is true because the criterion used is the air distance rather than the air time. It is also noted that as the distance range increases, the rate at which air distance produces inaccurate results also increases.  
However, as illustrated in Figure 9, the magnitude of inaccuracy decreases when the distance range increases. For example, in a short distance rage of {0-1} km, the average difference is as high as 2.5 times. That is, if the air distance measurement from A to B is smaller than from A to C, there is about a 25% chance the measurement will be wrong, as  
Figure 8: The ratio of closeness deceived by air distance: (**a**) Road distance; (**b**) road time.  
Figure 9: The magnitude of closeness deceived by air distance: (**a**) Road distance; (**b**) road time.  
shown in Figure 8a. In addition, the road distance measurement (with respect to cost) from A to B will, in fact, be larger than from A to C by 2.5 times. As provided in Figure 9, within the range {0-10}, the average differences are calculated as 27.9% for road distance and 29% for road time.