# Multi-Objective Gate Allocation Problem Based on Multi-Commodity Network Flow Model
## 5 Numerical Study
### Results Analysis
#### 5.2.2 Trade-Offs of Multiple Objectives  
In order to further explore the influence of different weight combinations on the value of each objective function, weight coefficients are set from 0.1 to 0.8 with a step size of 0.1 (e.g., \(\alpha=0.1\), \(\beta=0.1\), \(\gamma=0.8\)), forming a total of 36 weight combinations. With these weight combinations in mind, firstly, the corresponding optimal solution of the integrated objective function is found and the value of each individual objective function is calculated. On this basis, the K-means clustering algorithm is used to perform cluster analysis on 36 combinations, where the silhouette index is used to select the optimal number of clusters. In this way, clusters with similar objective function values can be easily found to help understand the differences and connections between weight combinations. By providing K sets of weight combinations for decision makers, the problem of weight \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Problem**} & \multirow{2}{*}{**Objective Function**} & \multicolumn{4}{c}{**Performance (*)**} \\ \cline{3-6}  & & **Walking Cost (m)** & **Taxi Cost (min)** & **Gate Cost** & **Running Time (s)** \\ \hline Problem 1 & f1 & 17,239,949 (0.00) & 523,769.1 (0.83) & 7.3 (0.94) & 59.17 \\ Problem 2 & f2 & 20,580,812 (0.98) & 489,939.3 (0.00) & 7.0 (0.86) & 51.68 \\ Problem 3 & f3 & 19,676,709 (0.71) & 515,254.5 (0.62) & 4.0 (0.00) & 53.15 \\ Problem 4 & \(1/3\) (f1\_norm + f2\_norm + f3\_norm) & 18,762,266 (0.44) & 500,280.0 (0.25) & 4.2 (0.06) & 47.47 \\ \hline \hline \end{tabular}  
* Note that the values inside the parentheses are the corresponding normalized values.  
\end{table} 
Table 4: Performance comparison of four optimization problems. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**High ID**} & \multirow{2}{*}{**Aircraft Size *} & **Arrival Time (HH/MM)** & **Depature Time** & **Occupancy Time** & **Number of** \\  & & **(HH/MM)** & **(HH/MM)** & **(min)** & **Passengers (Arr/Dep)** \\ \hline F1 & L & 0803 & 0848 & 45 & 316 \\ F2 & M & 0816 & 0911 & 55 & 238 \\ F3 & M & 0827 & 0924 & 57 & 224 \\ F4 & S & 0832 & 0940 & 68 & 177 \\ F5 & L & 0840 & 0935 & 55 & 243 \\ F6 & S & 0852  
setting commonly existing in the linear weighting method is solved to a certain extent. Overall, a total of 7 clusters are formed and the results are presented in Table 5. Specifically, some interesting phenomena and conclusions can be found: (1) The first two clusters contain 55.6% of the combinations, and both have a minimum gate cost. From the detailed allocation results, it can be inferred that adjustments are made to the adjacent flights under the premise of the same gate size to trade off the benefits of walking distance and taxi time. Cluster 4 and cluster 5 are in a similar situation. (2) Cluster 3 and cluster 7 mainly compromise the cost of taxi and gate, while 1 and 7 compromise the cost of walking and gate. (3) Cluster 6 corresponds to the most extreme type of situation, which maximizes the passenger satisfaction represented by the walking distance at the cost of taxiing time and gate cost. (4) Regardless of the cluster, both remote gates (i.e., G19 and G20) are assigned flights between 15:00 and 18:00. This is because during this time period, all the contact gates are occupied due to the excessive flight volume, so the remote gates are forced to be used. Although cluster analysis can provide decision makers with initial insight into the weight setting, the authors found that different combinations of weights within the same cluster may correspond to the same objective function value. Moreover, further analysis of allocation results reveals that even in the same cluster, its related performance still has great uncertainty, especially the idle time of the gates.