# Multi-Objective Gate Allocation Problem Based on Multi-Commodity Network Flow Model
## 4 Modeling  
The specific components of the multi-objective optimization problem proposed in this paper are shown in Figure 2. Due to the correlation and contradiction of multiple interests, the gate allocation must face the fact that resources are scarce and expensive and deal with and balance the conflicting factors. Therefore, the problem of gate allocation is inherently multi-objective. On the premise of avoiding deadlock at the planning level, optimization strategies should also be designed at the operation level to avoid running conflicts. The importance of the factors needs to be adjusted to allocate gate resources reasonably. The related decision variables are provided in Table 1.  
In general, based on the multi-commodity network flow, the proposed multi-objective model considers three perspectives, namely, passengers, airlines, and airports. In order to characterize the respective interests, passenger walking distance, aircraft taxi-in and taxi-out time, and gate resource allocation cost are defined as three objective functions. (1) Minimize passenger walking distance. From the perspective of passengers, the primary purpose of gate allocation is to improve passenger satisfaction. The number of passengers departing and arriving on the flight is used as the weight to calculate the \begin{table}
\begin{tabular}{c c} \hline \hline
**Decision Variable** & **Explanation** \\ \hline \multirow{3}{*}{\(x_{OI}^{g}\)} & Binary decision variable, equal to 1 if flight \(i\) begins at gate \(g\), that is, the \\  & entering arc \(\widehat{OA}\) from the origin node \(O\) to the flight arrival node \(a_{i}\) exists; \\  & otherwise, equal to 0. \\ \hline \multirow{3}{*}{\(x_{JD}^{g}\)} & Binary decision variable, equal to 1 if the last flight \(j\) flows through the gate \\  & \(g\), that is, the leaving arc \(\widehat{BD}\) from the flight departure node \(d_{j}\) to the \\  & destination node \(D\) exists; otherwise, equal to 0. \\ \hline \multirow{3}{*}{\(x_{OD}^{g}\)} & Binary decision variable, equal to 1 if no flight flow through the gate \(g\), that \\  & is, the directing arc \(\widehat{OD}\) from the origin node \(O\) to the destination node \(D\) \\  & exists; otherwise, equal to 0. \\ \hline \multirow{3}{*}{\(x_{ji}^{g}\)} & Binary decision variable, equal to 1 if flight \(i\) uses the gate \(g\) after flight \(j\), \\  & that is, the transferring arc \(\widehat{BA}\) from the flight departure node \(d_{j}\) to the \\  & flight arrival nodes \(a_{i}\) exists; otherwise, equal to 0. \\ \hline \multirow{3}{*}{\(x_{ij}^{g}\)} & Binary decision variable, equal to 1 if flight \(i\) is serviced at the gate \(g\), that \\  & is, the holding arc \(\widehat{AB}\) from the flight arrival nodes \(a_{i}\) to the flight \\  & departure node \(d_{j}\) exists; otherwise, equal to 0. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Description of decision variables. Figure 2: Basic diagram of the proposed multi-objective optimization problem. corresponding walking distance. The overall walking distance of passengers is taken as the first optimization objective, as follows:  
\[\text{min}f_{1}=\text{min}\sum_{i\in F_{a}}\sum_{g\in G}n_{i}^{a}l_{g}^{a}(x_{Oi} ^{g}+\sum_{j\in F_{d}}x_{ji}^{g})+\sum_{j\in F_{d}}\sum_{g\in G}n_{j}^{d}l_{g}^{ d}(x_{jD}^{g}+\sum_{i\in F_{a}}x_{ji}^{g}) \tag{3}\]  
where \(F_{a}\) and \(F_{d}\) denote the set of arrival and departure flights, \(G\) is the set of gate resource, \(n_{i}^{a}\) and \(n_{j}^{d}\) are the number of passengers for arrival flight \(i\) and departure flight \(j\), and \(l_{g}^{a}\) and \(l_{g}^{d}\) represent the corresponding walking distance between gate \(g\) and the terminal (i.e., baggage waiting for arrival and security check for departure). (2) Minimize aircraft taxi-in and taxi-out time. In the taxing phase, aircraft operating costs such as fuel consumption are directly related to taxi time. Therefore, the taxi-in and taxi-out time of the aircraft is also an important factor in choosing a suitable gate.