## 6 Example: Results and Discussion
\begin{array}{cccccccccccc}0&11.5&\infty&\infty&\infty&\infty&\infty& \infty&17.6&\infty\\ \infty&0&11.0&\infty&\infty&\infty&17.6&\infty&\infty\\ \infty&\infty&0&10.9&\infty&17.6&\infty&\infty&8.4\\ \infty&\infty&\infty&0&13.6&\infty&\infty&\infty&\infty\\ \infty&\infty&\infty&13.6&0&\infty&\infty&\infty&\infty\\ \infty&\infty&\infty&\infty&11.6&0&\infty&\infty&9.2\\ \infty&\infty&\infty&\infty&\infty&11.0&0&\infty&\infty\\ \infty&\infty&\infty&\infty&\infty&\infty&11.5&0&\infty\\ \infty&\infty&\infty&\infty&\infty&\infty&0&0\end{array} \right)\]  
As shown in matrix \(D\), there is only one passable path from T1 to parking space 18: starting from T1, go through T2, subsequently arrive at T3, then turn right to parking space 18, which is also the shortest path, with a total distance of 30.9 m. Through this method, under the scenario of the presence of one HV and one AV in this example, we simultaneously predict the parking space preference of the driver and allocate a parking space to the AV through V2X communication based on the driver preferences. Moreover, a parking lot may be constrained with traffic rules (e.g., one way road) and limited space, the adjacent matrix \(D\) can, in addition to providing the shortest route navigation, also reflect the constraints of the traffic rules and other conditions. Using the same approach as used in scenario one, we can also solve the similar issues, such as in the following two scenarios:  
Scenario 2:  
Step 1: Set the properties table of all available parking spaces currently in the example parking scenario (Figure 7). As shown in Table 3.  
Figure 7: Example of a parking scenario 2. Based on the values from Table 3 and assignment rule in Section 3, the evaluation matrix \(A\) is:  
\[A=\left( \begin{array}{cccc}43.5&29.2&9.0&5.0\\ 44.3&30.0&9.0&5.0\\ 27.8&36.5&9.0&5.0\\ 21.8&42.5&9.0&6.0\\ 3.2&55.7&9.0&6.0\\ 11.1&63.6&9.0&5.0\\ \end{array} \right)\]  
Step 2: Normalize the evaluation matrix \(A\) to form the fuzzy comprehensive evaluation matrix \(R=\left(r_{ij}\right)_{n\times m}\) according to Equations (2) and (3):  
\[R=\left( \begin{array}{cccc}0.074&1.000&1.000&0.833\\ 0.072&0.973&1.000&0.833\\ 0.115&0.799&1.000&0.833\\ 0.147&0.687&1.000&1.000\\ 1.000&0.524&1.000&1.000\\ 0.288&0.459&1.000&0.833\\ \end{array} \right)\]  
Step 3: With the average weight vector calculated in scenario one and Equation (5), the priority vector that contains the weights of every available parking space in scenario two can be obtained:  
\[z_{i}(w)=\left(0.761,\ 0.756,\ 0.737,\ 0.773,\ 0.944,\ 0.719\right)\]  
Step 4: Ranking the priorities of the available parking spaces in this example parking lot according to \(z_{i}(w)\) gives: \(z_{20}(w)>z_{19}(w)>z_{2}(w)>z_{29}(w)>z_{12}(w)>z_{23}(w)\). Based on Step 4, parking space 20 has the highest priority for the driver of the orange vehicle based on scenario two. Thus, parking space 19 will be allocated to the blue autonomous vehicle by the central command station. Scenario 3:  
Step 1: Set the properties table of all available parking spaces currently in the example parking scenario (Figure 8). As shown in Table 4.  
Based on the values from Table 4 and assignment rule in Section 3, the evaluation matrix \(A\) is:  
\[A=\left( \begin{array}{cccc}51.0&36.7&9.0&6.0\\ 41.8&27.5&9.0&5.0\\ 25.3&34.0&9.0&6.0\\ 27.8&36.5&9.0&7.0\\ 21.8&42.5&9.0&6.0\\ 3.2&55.7&9.0&6.0\\ 11.1&63.6&9.0&5.0\\ \end{array} \right)\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parking** & **Walking** & **Distance to** & **Status of** & **Status of Available** \\
**Space** & **Distance (m)** & **Parking (m)** & **Parking** & **Parking Spaces** \\ \hline
2 & 43.5 & 29.2 & clear & both sides are occupied \\ \hline
9 & 44.3 & 30.0 & clear & both sides are occupied \\ \hline
12 & 27.8 & 36.5 & clear & both sides are occupied \\ \hline
19 & 21.8 & 42.5 & clear & adjacent to the road \\ \hline
20 & 3.2 & 55.7 & clear & adjacent to the road \\ \hline
23 & 11.1 & 63.6 & clear & both sides are occupied \\ \hline \hline \end{tabular}
\end{table}