## 5 Results
### A Small Example  
In this subsection, a small example is introduced to illustrate the method proposed in this paper. The start locations and end locations of the driver and passengers in this example are listed in Tables 4 and 5, respectively. For brevity, data regarding the time requirements of the itineraries are not shown in the tables. The original data files associated with this example can be downloaded from the link in [61]. The number of seats and the price information in the bids submitted by the drivers are shown in Tables 6 and 7, respectively. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Driver** & \
\begin{tabular}{c} **Start Location** \\ **(Latitude/Longitude)** \\ \end{tabular} &
\begin{tabular}{c} **Destination Location** \\ **(Latitude/Longitude)** \\ \end{tabular} \\ \hline Driver 1 & 24.23785/120.66993 & 24.11308/120.65914 \\ Driver 2 & 24.1692536/120.6848233 & 24.20195/120.56815 \\ Driver 3 & 24.13425/120.5539 & 24.14289/120.70549 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Start locations and end locations of drivers. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Driver ID (_d_)** & \(j\) & _q\({}_{d\text{j1}}\)_ & _q\({}_{d\text{j2}}\)_ & _q\({}_{d\text{j3}}\)_ & _q\({}_{d\text{j4}}\)_ & _q\({}_{d\text{j5}}\)_ & _q\({}_{d\text{j6}}\)_ & _q\({}_{d\text{j7}}\)_ & _q\({}_{d\text{j8}}\)_ & _q\({}_{d\text{j9}}\)_ & _q\({}_{d\text{j10}}\)_ & _\(\boldsymbol{\Gamma_{dj}}\)_ \\ \hline
1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \{5 \} \\
2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \{10 \} \\
3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \{9 \} \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The number of seats in the bids submitted by drivers. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Driver ID (_d_)** & \(j\) & _o\({}_{d\text{j}}\)_ & _c\({}_{d\text{j}}\)_ \\ \hline
1 & 1 & 50.4025 & 51.4975 \\
2 & 1 & 36.745 & 41.1575 \\
3 & 1 & 57.485 & 57.485 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Price information in the bids submitted by drivers. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Driver Decision Variable** & _x\({}_{\text{11}}\)_ & _x\({}_{\text{21}}\)_ & _x\({}_{\text{31}}\)_ \\ \hline Value & 1 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Bids submitted by passengers. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Driver** & **Start Location** & **Destination Location** \\  & **(Latitude/Longitude)** & **(Latitude/Longitude)** \\ \hline Passenger 1 & 24.21872/120.6469 & 24.12877/120.66223 \\ Passenger 2 & 24.1790507/120.6657476 & 24.17369/120.61354 \\ Passenger 3 & 24.0611/120.64342 & 24.1465287/120.6532456 \\ Passenger 4 & 24.07962/120.69454 & 24.12438/120.66244 \\ Passenger 5 & 24.19422/120.69538 & 24.15288/120.69704 \\ Passenger 6 & 24.16048/120.69173 & 24.16359/120.65138 \\ Passenger 7 & 24.0611/120.64342 & 24.11009/120.64146 \\ Passenger 8 & 24.19422/120.69538 & 24.13046/120.7047 \\ Passenger 9 & 24.13623/120.60693 & 24.13527/120.6571 \\ Passenger 10 & 24.16429/120.68522 & 24.15345/120.65495 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Start locations and end locations of passengers. The above solution involved three shared rides. The cost savings discount for each of the drivers and passengers in the shared rides could be calculated as follows. For driver 1 and passenger 5 in the shared ride of bid (\(d_{j}\)) = (1,1) of driver 1,  
\[F_{11}(x,y)=o_{11}+f_{5}-c_{11}=50.4025+14.1675-51.4975=13.0625\]  
\[cf_{511}=14.1675\]  
For driver 1 and passenger 5 in their shared ride, the cost savings discount is \(\frac{F_{11}(x,y)}{c_{511}+c_{11}}=\frac{13.0725}{14.1675+51.4975}=\frac{13.0725 }{65.665}=0.199\). Hence, the cost savings discount for driver 1 and passenger 5 is greater than 0.1(=\(r_{D}=r_{P}\)). The cost savings discount for driver 2 and passenger 10 in their shared ride can be calculated similarly. For driver 2 and passenger 10, the cost savings discount is 0.103, which is greater than 0.1(=\(r_{D}=r_{P}\)).