The following table defines the notations used throughout this paper.

Sets and Parameters	Description
𝓣	The set of periods, $t\in \left\{\mathrm{1,2},3,\dots ,T\right\}$
${\mathcal{F}}^{\mathit{a}\mathit{r}\mathit{r}}$	The set of arrival flights scheduled to land at the airport
${\mathcal{F}}^{\mathit{d}\mathit{e}\mathit{p}}$	The set of departure flights scheduled to take off from the airport
$\mathbb{C}\subset {\mathcal{F}}^{\mathit{a}\mathit{r}\mathit{r}}\times {\mathcal{F}}^{\mathit{d}\mathit{e}\mathit{p}}$	A subset of flight pairs $\left(i,j\right)\in {\mathcal{F}}^{arr}\times {\mathcal{F}}^{dep}$such that flights $i$and $j$are flown by the same aircraft
${\mathit{S}}_{\mathit{i}\mathit{t}}^{\mathit{a}\mathit{r}\mathit{r}}$	$1;\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}i\in {\mathcal{F}}^{arr}\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{d}\mathrm{t}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{n}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{d}t$ 0; otherwise
${\mathit{S}}_{\mathit{j}\mathit{t}}^{\mathit{d}\mathit{e}\mathit{p}}$	$1;\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}j\in {\mathcal{F}}^{dep}\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{d}\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{k}\mathrm{e}\mathrm{o}\mathrm{f}\mathrm{f}\mathrm{n}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{d}t$ 0; otherwise
${\mathit{T}}_{\mathit{i}\mathit{j}}^{\mathit{m}\mathit{i}\mathit{n}}$/${\mathit{T}}_{\mathit{i}\mathit{j}}^{\mathit{m}\mathit{a}\mathit{x}}$	The minimum/maximum turnaround time between flights $i$and $j$such that $\left(i,j\right)\in \mathbb{C}$
${\mathsf{\Lambda }}_{\mathit{t}}^{\mathit{a}\mathit{r}\mathit{r}}$	Maximum number of allowable arrival flights in period $t$
${\mathsf{\Lambda }}_{\mathit{t}}^{\mathit{d}\mathit{e}\mathit{p}}$	Maximum number of allowable departure flights in period $t$
Decision Variables	Description
${\mathit{u}}_{\mathit{i}}^{+},{\mathit{u}}_{\mathit{i}}^{-}$	Positive or negative displacement of arrival flight $i\in {\mathcal{F}}^{arr}$
${\mathit{v}}_{\mathit{j}}^{+},{\mathit{v}}_{\mathit{j}}^{-}$	$\mathrm{P}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{n}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{o}\mathrm{f}\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}jϵ{\mathcal{F}}^{dep}$
${\mathit{w}}_{\mathit{i}\mathit{t}}^{\mathit{a}\mathit{r}\mathit{r}}$	A binary array representing the newly scheduled arrival of flight $i\in {\mathcal{F}}^{arr}$
${\mathit{w}}_{\mathit{j}\mathit{t}}^{\mathit{d}\mathit{e}\mathit{p}}$	$\mathrm{A}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{r}\mathrm{y}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{a}\mathrm{y}\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{l}\mathrm{y}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{o}\mathrm{f}\mathrm{f}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}jϵ{\mathcal{F}}^{dep}$
${{\mathit{\delta }}^{+},\mathit{\delta }}^{-}$	Maximum of all positive or negative flight displacements

It is a common practice to discretize the time into 15-minute periods. However, we use 5-minute intervals in the optimization model to avoid unnecessary large schedule displacements. The novelty of this formulation lies in the structure of the scheduling parameters $S_{it}^{arr}$and $S_{it}^{dep}$ as defined in [8]. The array $S_{it}^{arr}$ (similarly $S_{it}^{dep}$) is of the form $\left(1,\dots ,\mathrm{1,0},\dots ,0\right)$ if flight i is originally scheduled to land (take off) no earlier than period t, where the last 1 in the array corresponds to period t.

The parameters $T_{ij}^{min}$ and $T_{ij}^{max}$ are expressed as numbers of 5-minute periods. If flights i and j are connected by an aircraft, we set the corresponding value of $T_{ij}^{min}$ to the minimum turnaround time required to complete the ground operation for the type of aircraft and make it ready for the next flight. The decision variables $w_{it}^{arr}\mathrm{a}\mathrm{n}\mathrm{d}{w}_{jt}^{dep}$take the same form as the input parameters $S_{it}^{arr}$and $S_{it}^{dep}$. By convention, we assume that $w_{i,\mathrm{T}+1}^{arr}=w_{i,\mathrm{T}+1}^{dep}=1$. The displacement variables ${(u}_i^{+},u_i^{-})$ and ${(v}_j^{+},v_j^{-})$are also expressed as numbers of 5-minute periods and allow for positive or negative displacement, i.e., a flight can be rescheduled for a later or earlier time of the day.

The objective function is to minimize the total displacements of arrival and departure flights on a particular day. The maximal displacement of any flight, denoted by $\delta$, could be considered as the second objective function. However, we restrict it by assigning a fixed value to $\delta$ so that it would not hurt the first objective, based on various initial runs of the model. The optimization model for scheduling interventions at a busy airport can be formulated as follows.

This is a mixed-integer linear programming model. Constraints (2) and (3) ensure that all arriving and departing flights are scheduled. Constraints (4) and (5) define the rescheduled arrival and departure times, respectively, based on the decisions on displacements of the original flight schedules. The combination of constraints (4) and (5) ensures that flight block times are kept unchanged. Constraints (6) and (7) enforce the minimum and maximum connection times. Constraints (8) and (9) ensure that $w_{it}^{arr}\mathrm{a}\mathrm{n}\mathrm{d}{w}_{jt}^{dep}$ are nonincreasing for each flight, consistently with their definition. Constraints (10) and (11) ensure that the aggregate number of scheduled arrivals and departures in each period t do not exceed the runway capacity limits. Constraints (12) define the maximum of all positive and negative displacements for all incoming and outgoing flights and finally (13) define variables’ type and sign constraints.

We have developed an algorithm to simulate the arrival and departure processes using the first come first served (FCFS) discipline while respecting runway capacity limitations in a schedule-coordinated airport. The capacity is defined as the number of arrival or departure slots that can be safely assigned to the incoming or outgoing flights from the airport in each 15-minute time interval during a day. The number of arrival or departure slots for each 15-minute time interval depends on the minimum separation time between two consecutive landings or take-offs for safe operations. The minimum safe separation interval between successive flights is a critical safety measure that mitigates the risk of in-flight incidents by ensuring a sufficient buffer to prevent the aerodynamic effects of one aircraft from adversely affecting another. We assume 3 minutes minimum safe separation time for arrivals and 2 minutes for departures. After assigning a time for a flight in a certain interval, the value of the delay is calculated, which is the difference between the flight time that we can reschedule after slot allocation and the original scheduled time. The arrival delay, caused by shifting some flights to subsequent time intervals, may get propagated to those departure flights that are connected to the arrival flight by some airline resources (aircraft, crew members, and/or passenger connections). In this study, we consider flight connections only by aircraft since other connection data were not available. We assumed fixed taxi and turnaround times under the assumption that there are no capacity limitations for ground operation resources. The simulation of the departure process is done similarly. However, before starting the allocation of flights to runways, we check the minimum ready time after the execution of ground operations following the aircraft's actual arrival. An arrival delay will result in a departure delay if there is not enough slack between the two connected flights. We ensure that a minimum turnaround time of 35 minutes between each pair of connected arrival and departure flights that are performed by a certain aircraft is maintained. After taking into account the ground operation time, the delay is calculated, which is the difference between scheduled and actual times. The details and flowchart of the simulation algorithm can be found in [9].

The data used in our experiments were collected from the FAA website for Hartsfield-Jackson Atlanta International Airport (ATL) on March 24, 2023, which happened to be the busiest day in 2023 [11]. It includes all departure and arrival statistics (scheduled time, actual time, flight delay, etc.) for all the flights going to or departing from the USA airports. The connections between flight pairs were found using the tail numbers of the aircraft performing those flights. ATL is the number two weather-affected airport, largely because of system delays and thunderstorms at ATL [11]. We collected statistics for 881 arrivals and 873 departures as well as for 739 connection flights at ATL on that day.

According to the FAA, the capacity benchmark is defined as the maximum number of flights an airport can routinely handle in an hour, for the most commonly used runway configuration in each specified weather condition. In our research, we analyzed the capacity of the 5 parallel runways in this airport. ATL normally assigns three runways for arrivals and two for departures. We assumed that each runway was designated for either incoming or outgoing flights, but not for both at the same time. To validate the simulation code, we ran the algorithm considering the capacity of 20 arrivals and 20 departures in each 15-minute interval and observed that the simulated data closely matched the actual data [9]. It should be noted that the simulation algorithm enforced runway capacity limitations for safer operation whereas the actual data were the result of ad hoc operation based on FCFS discipline. After validating the simulation code, the algorithm was run with an operational capacity of 15 – 15 flights in each 15-minute interval. This is equivalent to enforcing a 3-minute separation time for arrivals and 2 minutes for departures.