# Multi-Objective Gate Allocation Problem Based on Multi-Commodity Network Flow Model
## 4 Modeling
The second optimization objective is provided as follows:  
\[\text{min}f_{2}=\text{min}\sum_{i\in F_{a}}\sum_{g\in G}n_{i}^{a}l_{g}^{a}(x_{Oi }^{g}+\sum_{j\in F_{d}}x_{ji}^{g})+\sum_{j\in F_{d}}\sum_{g\in G}n_{j}^{d}l_{g}^ {out}(x_{jD}^{g}+\sum_{i\in F_{a}}x_{ji}^{g}) \tag{4}\]  
where \(t_{g}^{in}\) and \(t_{g}^{out}\) are the average taxi-in and taxi-out time of gate \(g\). Since it is calculated from historical operating data, it has potentially considered different taxiing paths, taxiing conflicts, etc. Similar to objective 1, the number of passengers is also included as a weight for the purpose of measuring the importance of taxi cost. (3) Minimize gate resource allocation cost. To fully use the gate resource, the size of the allocated gate needs to match the aircraft size. Therefore, the waste of surface resources can be minimized only by allocating the most appropriate gate for the corresponding aircraft and not allocating remote gates as far as possible. The third optimization objective is provided as follows:  
\[\text{min}f_{3}=\text{min}\sum_{i\in F_{a}}\sum_{g\in G}C_{ig}(x_{Oi}^{g}+\sum_ {j\in F_{d}}x_{ji}^{g})+\sum_{j\in F_{d}}\sum_{g\in G}C_{ig}(x_{jD}^{g}+\sum_{ i\in F_{a}}x_{ji}^{g}) \tag{5}\]  
where \(C_{ig}\) is the cost of size mismatch between arrival flight \(i\) and gate \(g\). As a rule of thumb, different levels of penalty values are defined as shown in Equation (6). \(S_{g}^{\text{gate}}\) is the size of gate \(g\), and \(S_{i}^{\text{flight}}\) is the size of arrival flight \(i\). Especially, we assign the largest penalty to the remote gate, which is to minimize its frequency of use unless all contact gates are occupied. \(C_{ig}\) is defined, likewise:  
\[C_{ig}= \begin{cases}0,&S_{g}^{\text{gate}}=S_{i}^{\text{flight}}\\ 0.2,&S_{g}^{\text{gate}}=S_{i}^{\text{flight}}+1\\ 0.5,&S_{g}^{\text{gate}}=S_{i}^{\text{flight}}+2\\ 1,&\text{gate }g\text{ is a remote gate}\end{cases}  \tag{6}\]  
In order to balance the interests of multiple parties, multiple optimization objectives are integrated together through a linear weighting method, as follows:  
\[\text{min}f=\text{min}\alpha f_{1}+\beta f_{2}+\gamma f_{3} \tag{7}\]  
where \(\alpha\), \(\beta\), and \(\gamma\) are weight coefficients for each objective, and \(\alpha+\beta+\gamma=1\  
Furthermore, to obtain a reliable gate allocation scheme for each flight, three types of constraints are considered, namely multi-commodity network flow constraints, flight exclusive constraint, and gate compatible constraint. (1) Multi-commodity network flow constraints. Considering the complex interactions between commodities (i.e., gates) and obtaining valid commodity flows, the following basic constraints must be observed:  
\[x_{OD}^{g}+\sum\limits_{i\in F_{a}}x_{Oi}^{g}=1,\forall g\in G \tag{9}\]  
\[x_{OD}^{g}+\sum\limits_{j\in F_{d}}x_{jD}^{g}=1,\forall g\in G \tag{10}\]  
\[x_{Oi}^{g}+\sum\limits_{j\in F_{d}}I_{j^{\prime}}x_{j^{\prime}}^{g}=x_{ij^{ \prime}}^{g},\forall g\in G,\forall i\in F_{a},\forall j\in F_{d},|i|=|j| \tag{11}\]  
\[x_{jD}^{g}+\sum\limits_{i^{\prime}\in F_{a}}I_{j^{\prime}}x_{j^{\prime}}^{g}=x_ {ij^{\prime}}^{g},\forall g\in G,\forall i\in F_{a},\forall j\in F_{d},|i|=|j| \tag{12}\]  
\[\sum\limits_{g\in G}x_{ij}^{g}=1,\forall i\in F_{a},\forall j\in F_{d},|i|=|j| \tag{13}\]  
\[x_{Oi^{\prime}}^{g},x_{jD}^{g},x_{j^{\prime}}^{g},x_{ji^{\prime}}^{g},x_{ji^{ \prime}}^{g},x_{ij^{\prime}}^{g}\leq\{0,1\} \tag{14}\]  
Equation (9) describes the constraint of flow-in, indicating that each gate needs to choose the entering arc or the directing arc (i.e., whether there is a flight using the gate). Correspondingly, Equation (10) provides the constraint of flow-out. As for Equation (11), it represents the flow balance constraint of the flight arrival nodes. Here, \(|i|\)=\(|j|\) means arrival flight \(i\) and departure flight \(j\) form a flight pair. Similarly, Equation (12) presents the flow balance constraint of the flight departure nodes.