# Multi-Objective Gate Allocation Problem Based on Multi-Commodity Network Flow Model
## 5 Numerical Study  
### Experimental Data and Settings  
In order to verify the proposed model in the problem of gate allocation, one of the busiest hub airports in the world, Shanghai Pudong International Airport, is used as the research object, where 20 representative gates in Terminal 1 are selected, as shown in Figure 3. In addition to 18 contact gates, 2 remote gates are also considered to prevent the absence of feasible solutions due to the accumulation of flights exceeding the inherent capacity of contact gates at a specific time.  
Specifically, according to the size of aircraft wingspan that the gate can accommodate, all these gates are divided into three categories, which are small gates (G1-G5), medium gates (G6-G13), and large gates (G14-G20). Furthermore, based on historical operations, we filtered flights for each gate and calculated the average taxi time as the taxi cost. The walking distance of passengers related to each gate was also preliminarily estimated, and the detailed information is presented in Table 2.  
Meanwhile, this paper considers the actual operational data of the flight, which involves a total of 88 pairs of flights in the time range of 08:00-20:00. All flights are also divided into three categories based on the aircraft wingspan length, including 39 small flights, 28 medium flights, and 21 large flights. As for the number of passengers, it is set to the full capacity of each flight for simplicity. More detailed flight information is shown in Table 3.  
In the following experiments, the buffer time between two adjacent flights for each gate is set to 30 min. All the models related to this paper are created on a Dell G15 laptop consisting of an [EMAIL_ADDRESS] GHz Intel Core and a 16 GB DDR3 RAM. With PyCharm as the implementation platform, Gurobi Commercial Optimizer 9.5.2 and Python 3.9.9 are used as core tools to solve the Binary Integer Linear Programming problem and obtain the optimal results of gate allocation. Specifically, the branch-and-cut (B&C) [42] algorithm is used to solve the integer programming problem, which is an exact algorithm composed of a cutting plane method and a branch-and-bound algorithm. The parameter settings  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Gate ID**} & \multirow{2}{*}{**Size \({}^{*}\)**} & \multicolumn{3}{c}{**Taxi Time (min)**} & \multicolumn{3}{c}{**Walking Distance (m)**} & \multirow{2}{*}{**Gate ID**} & \multirow{2}{*}{**Size \({}^{*}\)**} & \multicolumn{3}{c}{**Taxi Time (min)**} & \multirow{2}{*}{**Walking Distance (m)**} \\ \cline{3-3} \cline{8-11}  & & & **Taxi-In** & & & & & & **Taxi-In** & & **Taxi-Out** & **Arrival** & **Departure** \\ \hline
1 & S & 9.8 & 11.5 & 402 & 453 & 11 & M & 15.5 & 17.2 & 144 & 239 \\
2 & S & 11.4 & 12.3 & 581 & 643 & 12 & M & 8.5 & 10.4 & 267 & 360 \\
3 & S & 16.0 & 16.5 & 724 & 769 & 13 & M & 13.4 & 15.8 & 423 & 558 \\
4 & S & 15.4 & 16.7 & 757 & 814 & 14 & L & 9.3 & 11.4 & 553 & 678 \\
5 & S & 14.5 & 15.9 & 639 & 696 & 15 & L & 12.9 & 14.7 & 688 & 814 \\
6 & M & 10.5 & 11.1 & 451 & 522 & 16 & L & 13.6 & 15.5 & 724 & 830 \\
7 & M & 14.8 & 15.0 & 325 & 415 & 17 & L & 12.7 & 13.3 & 641 & 758 \\
8 & M & 16.3 & 17.8 & 146 & 287 & 18 & L & 14.3 & 15.8 & 497 & 616 \\
9 & M & 17.3 & 18.3 & 129 & 212 & 19 & L & 23.2 & 23.7 & 964 & 1080 \\
10 & M & 12.2 & 10.9 & 124 & 217 & 20 & L & 21.7 & 22.2 & 1088 & 1173 \\ \hline \hline \end{tabular}  
* The size of the gate depends on the aircraft wingspan it can accommodate. S: <52 m; M: 52-65 m; L: >65 m.  
\end{table}
Table 2: Basic information of gate resources.  
Figure 3: Layout of Terminal 1 in Shanghai Pudong International Airport.  
and branching strategies related to the algorithm are based on the default configuration of the Gurobi solver. According to the operation results of the solver, the optimal values for decision variables and objective functions are obtained, as well as the corresponding running time.