Flight Routing and Scheduling with Departure ...

Flight Routing and Scheduling with Departure Uncertainties in Air Traffic Flow Management Nadeesha Sandamali Gammana Guruge1 , Rong Su1 , Yicheng Zhang1 , Qing Li2 Abstract— Uncertainty takes different forms in air transportation systems. One form is demand uncertainty which is mainly due to the root cause of aircraft departing in deviated time slots. This can lead to severe consequences, including capacity violation in air routes. The current Air Traffic Flow Management (ATFM) literature does not consider demand uncertainty during flight scheduling. However, in reality, most of the flights do not take off at the designated time slots, thus can lead to an unexpected number of aircraft to appear in some links. This creates several difficulties in air transportation systems, including higher workloads for air traffic controllers, and higher delays and travel costs. In this study, we propose a novel flight routing and scheduling scheme while considering the demand uncertainty present in the system. Our objective is to route and schedule flights to minimize the overall network delay while satisfying all link capacity constraints even under demand uncertainty. To this end, when estimating the flight parameters such as location and time, we consider a range of values that the flight can take due to its departure uncertainty rather than a discrete single value. The problem is formulated as a Mixed Integer Quadratic Programming (MIQP) problem. With the help of simulation, we show that the propose approach helps to eliminate the capacity violation completely from the system while ensuring a low amount of overall flight delay.

I. INTRODUCTION With the rapid growth of demand in the air traffic system, the importance of Air Traffic Flow Management (ATFM) can never be depreciated since ATFM is essential in alleviating the demand-capacity imbalance. Due to the disparity between demand in air transportation and available system capacity, passengers may have to experience unnecessary delays in their travels and airlines may have to struggle with the increment in the cost of travel. A rich literature exists on the study of handling the aforementioned problems incurred by the mismatch of capacity and demand in airspace and airports. Among them, we can find two main approaches, which are sector based and trajectory based. In the sector-based approach, the airspace is considered as a set of sectors and flights are routed among them. In several studies by Bertismas ([1], and [2]) under the sector-based approach, the airplanes have routed with the objective of minimizing both ground delay and airborne *This work is financially supported by the ATMRI-CAAS Project on Air Traffic Flow Managment with the project reference number ATMRI:2014R7-SU. 1 Nadeesha Sandamali Gammana Guruge, Rong Su and Yicheng Zhang are affiliated with School of Electrical and Electronic Engineering at Nanyang Technological University, Singapore 639798. Emails [email protected], [email protected],

[email protected] 2 Qing Li is affiliated with Air Traffic Management Research Institute (ATMRI) at Nanyang Technological University, Singapore 639798. E-mail:

[email protected]

delay, moreover allowing rerouting decisions. In contrary, trajectory-based approach consider the waypoints and link details in routing and scheduling the flights. Sun and Bayen [3] propose a multi-commodity Eulerian-Lagrangian model for en-route air traffic, which take origin-destination (OD) information of the flight into consideration. Y. Zhang et al. [4] propose a distributed flight routing and scheduling algorithm based on trajectory-based approach with added features of aircraft types and cruising speed limits. However, none of the above literature considers the stochastic nature of capacity or demand. In ATFM, uncertainty mainly present in two forms; capacity uncertainty and demand uncertainty. Uncertainty in demand and capacity of the system result for the stochastic nature of the system [5]. The uncertainty presence in the air transportation system makes traffic flow management, unreliable if the planning process is completely deterministic which result in false demand-capacity imbalance reduction. Depending on weather fluctuations and ATC workload, the capacity of sectors/links as well as airports can highly fluctuate. This is named as capacity uncertainty. Uncertainty in demand causes due to the time deviation between actual time and the scheduled time, i.e. aircraft do not take off from the airport at the same scheduled times, as well as they do not pass the follow-up waypoints according to the expected times. The actual time may be earlier or later than the scheduled time, which has published in the flight plan. Therefore, the demand uncertainty can result in the capacity violation in some links due to the unexpected arrival of aircraft. Few studies have attempted to estimate and handle the uncertainty present in ATFM. Terrab and Odoni [6] have proposed a stochastic ground-holding problem while considering the airport capacity as a random variable described through a probability distribution. Clare and Richards [7] have applied chance constrained optimization techniques for uncertainty management, including constraints on the chance of sector capacity violations. They formulated and solved for both cases of probabilities within individual sectors and joint probabilities between sectors. Clare et al. [8] describe the way that the Airline Operations Center (AOC) and the Network Manager (NM) solve the demand-capacity imbalance problem by merging uncertainty. Their approach is based on Model Predictive Control (MPC) with disturbance feedback to produce contingency plans in weather scenarios. Gilbo and Smith [9] propose a regression model for improving aggregate traffic demand predictions in Enhanced Traffic Management System (ETMS) while taking into consideration of uncertainty in demand predictions.The above

research is based on their previous work [10], which analyze several regression models for factors such as adjacent intervals (both preceding and following), airport specific coefficient, and active and proposed flights. In this paper, we introduce a flight routing and scheduling model while considering the demand uncertainty based on trajectory approach. To the best of our knowledge, we provide the first study on flight scheduling under demand uncertainty in ATFM. We further present a modified flight routing scheme where OD pairs can take multiple flight plans. Based on the above-modified routing scheme, we schedule the flights while considering the maximum possible uncertainty that can take place. Our objective is to eliminate the uncertainty that can occur due to flights departing in deviated time slots by 100% . The remainder of the paper is structured as follows. Section II provides insights on the ATFM system model. In section III, we introduce our flight schedule generation scheme. Section IV discusses the results and finally, section V concludes this paper. II. SYSTEM MODEL DESCRIPTION A. General Overview As depicted in Fig. 1, the air traffic network is modeled as a set of airports, routes, and waypoints. Each airport has its departure link which is the first link that flight passes through after it takes off from the departure airport and the arrival link which is the last link that flight passes before landing at the arrival airport. During the flight path, an aircraft pass set of waypoints which are the control points to notify the control center about its location. The air route is the directed airways between two waypoints. After departing from the airport, the aircraft keeps its track through a grid of network passing waypoints and finally, it reaches to the destination airport. Between each OD pair, there can be more than one flight paths which are different from the optimal path. B. Problem Statement In most cases, flights do not depart from the airport at their planned departure times. In addition, flights may not always Holding Pattern a

w’ Air Route / link j w

Arrival Link (I’)

Departure v Link (I)

Waypoint/ Control point

v’

a’ Airport

Fig. 1.

Simplified air traffic network

follow their scheduled flight plans, owing to reasons of bad weather, speed variation, unexpected deviation, etc.. The above mentioned factors cause demand uncertainty resulting in a difference between the actual flight arrival time and the scheduled flight arrival time to routes/airports. However in this research, we only consider the demand uncertainty due to the deviation of the departure time (schedule uncertainty). The departure deviation can lead to unexpected arrivals of aircraft in en-route airspace, which may lead to en-route capacity violation. We made the assumption of constant aircraft speed since we only consider the uncertainty of departure time. So that, the same deviation between actual and scheduled times will occur in subsequent waypoints/links of the flight path. Moreover, we have not considered the uncertainty in capacity, which makes the capacity of the links and airports still deterministic in our model. We expect to carry out further research on capacity uncertainty to build the model more realistic later on. In order to minimize the issues due to demand uncertainty the control actions of ground holding, airborne speed control and use of some alternative path/ rerouting are considered. Although airlines preferred to stick to the original flight trajectory, in some scenarios they tend to use some alternative paths to reach the destination on time. In this approach, we assume constant aircraft speed, so that, two control actions of ground holding and rerouting are considered. III. FLIGHT SCHEDULE GENERATION A. Air Traffic Network Model The air traffic network is a directed graph denoted as G = (V, E) consists of a set of vertices (V) and edges (E). Waypoints, W ⊆ V and airports, A ⊆ V are considered as nodes and links are the directed connections between nodes. The departure link from departure airport, a to waypoint, v is defined as I = (a, v) ∈ E ⊆ A × W while the arrival link from waypoint, v’ to arrival airport, a’ is defined as I 0 = (v 0 , a0 ) ∈ E ⊆ W × A as shown in Fig. 1. Moreover, any en-route link from waypoint w to w’ is j = (w, w0 ) ∈ E ⊆ W × W . Let Lj and Cj be the length and capacity of link j, respectively. Since the air route is a directional link between two waypoints, Lj is the directed route length between two connected waypoints. The route capacity, Cj is an uncertainty variable since it depends on weather fluctuation and controller workload. In this model, we consider the link capacity as deterministic. i Let Lfa,j be the total air route length of flight fi from departure airport a to link j. fi ∈ F are the set of flights concerned and v fi denotes the constant speed of flight, fi . Let Djfi be the set of downstream links connected with link j that flight fi can follow to reach its destination airport. Two decision variables are, ( 1; if flight fi pass route j fi Yj = 0; otherwise τ fi − Desirable departure time of flight fi

B. Flight Dispatching Constraint Flight dispatching constraints are used to route each flight from its origin to destination airport while passing follow up waypoints along its path. Since the departure route and the arrival route (i.e. first and last route) are independent of the flight path, the condition for flight fi is represented as, YIfi = YIf0i = 1

(1a)

Constraint (1b) defines the next route that flight fi can take after passing link j. Basically, a flight passing through link j, would take one of link j 0 s immediate downstream links. X Yjf0i (1b) Yjfi = j 0 ∈Djf

i

Total flight time of flight fi through the selected routing path is given by, P fi j∈Jfi Lj Yj fi (1c) R = v fi C. Worst Case Link Capacity Constraint Due to the demand uncertainty, the number of flights which may occupy a certain route at a given time can deviate from the original estimated value. To ensure that this will not lead to any capacity violation at all, we have considered the each flight’s maximum possible time period of stay in the route including the uncertainty time. So here, the lower bound of arrival and upper bound of the exit of each of the flight is taken into consideration for the link volume. The distribution of the difference between the actual departure time and the scheduled departure time is shown in Fig. 2. The terms, 4f1i and 4f2i represent the maximum bound of early and late arrival respectively. Due to the assumption of constant speed, the link uncertainty distribution is same with the departure uncertainty distribution. fk Let T˜in,j be the lower bound of entry time of flight fk fk to link j and T˜out,j be the upper bound of exit time of flight fk from link j. So flight fk could be in link j at any Probability

fk fk ]. Moving , T˜out,j moment during the time period of [T˜in,j further, the lower bound of entry time can be represented using desirable departure time, travel time from departure airport to the considered route and the maximum possible uncertainty due to early arrival as in (2a). Moreover, the travel time through the route and the maximum amount of uncertainty due to late arrival are considered for the upper bound of exit time as in (2b). k Lfa,j (2a) − 4f1k v fk k Lfa,j Lj fk T˜out,j = τ fk + f + f + 4f2k (2b) v k v k The above mathematical representation is graphically ilfk lustrated in Fig. 3. Tin,j is the deterministic entry time of flight fk to link j. Flight fk can enter into link j within the time period of fk fk [Tin,j − 4f1k , Tin,j + 4f2k ] due to early or late arrival of the flight. Within this time period, the total number of flights which pass through link j including flight fk , should be less than the air route capacity. If we consider a flight in general, fi which is going through the same route j, flight fi also has two limits of fi fi T˜in,j and T˜out,j based on its uncertainty. Since our goal is to ensure zero(0) capacity violations, we have considered minimum lower bound of entry and maximum upper bound of exit time of flight fi when estimating the flight’s route occupancy time. Flight fi is countered as in route j when fi fk flight fk enters the route, if and only if, T˜in,j ≤ Tin,j + 4f2k fk fk f i and Tin,j − 41 ≤ T˜out,j . We can simplify the condition further as follows.

fk = τ fk + T˜in,j

fi fk fk fi T˜in,j − 4f2k ≤ Tin,j , Tin,j ≤ T˜out,j + 4f1k fi fk fi T˜in,j − 4f2k ≤ Tin,j ≤ T˜out,j + 4f1k

𝑓

𝑓

∆2 𝑖

Departure uncertainty distribution

∆2 𝑘

∆1 𝑘

t 𝑓

Fig. 2.

𝑓

𝑓

(Actual Departure Time – Scheduled Departure Time)

(3b)

This means, if the deterministic entry time of flight fk to fi fi link j is within the range of [T˜in,j −4f2k , T˜out,j +4f1k ], then flight fi is countered for link volume since it is within the link, when fk enters the link. In this way, we can count the number of flights which may feasibly occupy link j when flight fk enters the link. To check the route capacity utilization, we select the each flight’s entry time to the route and check the route capacity constraint at each time, a flight is scheduled to enter a route. This will make sure the link will be free from capacity violations since we consider the uncertainty of flights entry time as well. The matrix shown in Fig. 4 indicates this concept more clearly. Each column in the matrix represents all flights fi in link j when flight fk reaches the link and so on. By multiplying this matrix with n × 1 column matrix

0 ∆1 𝑖

(3a)

𝑖 𝑇𝑖𝑛,𝑗

Fig. 3.

𝑓 𝑇𝑖𝑛𝑘,𝑗

−

𝑓 ∆1 𝑘

𝑓 𝑇𝑖𝑛𝑘,𝑗

𝑓 𝑇𝑖𝑛𝑘,𝑗

+

𝑓 ∆2 𝑘

𝑓

𝑖 𝑇𝑜𝑢𝑡,𝑗

Time axis representation of link arrival and departure time

𝑓𝑖

Ofi )2 is to minimize the rerouting path cost. These two deviation terms are not equally weighted in the objective function since airlines prefer to use ground delay rather than rerouting due to the objection for deviating from the original plan as well as the cost of fuel (CR > CD ).

𝑓1 𝑓2 𝑓3 . . . . . . . 𝑓𝑛 𝑓1 1 0 1 . . . . . . . .

𝑓𝑘

𝑓2

IV. CONVERSION OF MIXED LOGIC CONSTRAINTS

𝑓3 . . . .

The logical condition (4) should convert into linear forms using mixed logic constraints. Mixed logic constraints are used to transform logical facts into linear inequalities as well as for the translation of the nonlinear form into linear form [11]. By introducing two auxiliary logical variables δ1 (j, fk , fi ) and δ2 (j, fk , fi ), as shown in Fig. 5, condition (4) can approximate by mixed logic constraints as follows.

𝑓𝑛 Fig. 4.

𝑛×𝑛 Matrix representation of link volume

with all element as 1, will give the link volume of route j when flight fk enters the link. To represent the above matrix in mathematical notation, fi fi we define a new variable as X j fk (T˜in,j , T˜out,j ). The variTin,j

able will be 1 if flight fi is in the route, when flight fk enters the route. Otherwise, it will be zero, resulting that flight fi is not countered for link volume since it is not within link j when flight fk reaches the link. For all fi ∈ F  fk fk ˜fi  1; Tin,j − 42 ≤ Tin,j ≤ fi fi fi X j fk (T˜in,j , T˜out,j )= T˜out,j + 4f1k (4) Tin,j   0; otherwise

fi fk + 4f1k ] ←→ [δ2 (j, fk , fi ) = 1] ≤ T˜out,j [Tin,j fk fk fi [Tin,j ≥ T˜in ,j − 42 ] ←→ [δ1 (j, fk , fi ) = 1] [T fk ≥ T˜fi + 4fk ] ←→ [δ2 (j, fk , fi ) = 0] in,j fk [Tin,j

≤

out,j fi ˜ Tin ,j

−

1 4f2k ]

←→ [δ1 (j, fk , fi ) = 0]

(7a) (7b) (7c) (7d)

The logical conditions (7a), (7b), (7c) and (7d) can be rewritten as, fi fk  + (m − )δ2 (j, fk , fi ) + (T˜out,j + 4f1k ) ≤ Tin,j ≤ M (1 − δ2 (j, fk , fi )) + (T˜fi + 4fk )

(8a)

fk fi  + (m − )δ1 (j, fk , fi ) + Tin,j ≤ (T˜in,j − 4f2k ) ≤ M (1 − δ1 (j, fk , fi )) + T˜fk

(8b)

out,j

1

in,j

Eventually, the number of flights in link j should be less than the link capacity Cj at the time flight fk enters. So, the capacity constraint is given as, for all fk ∈ F ,∀j X j fi fi fk X fk (T˜in,j , T˜out,j ) ∗ Yjfi ≤ Cj (Tin,j + 4l ) (5) fi ∈F

Tin,j

where 4l is the uncertainty (time) attached in flight fk entering route j. D. Objective Function Our objective is to minimize the expected total delay time of each aircraft by minimizing departure deviation and rerouting delay. Due to the assumption of constant speed, the arrival deviation is the sum of departure deviation and rerouting delay. Hence, the objective function reflects the minimization of total arrival deviation. X min CD ∗ (τ fi − Dfi )2 + CR ∗ (Rfi − Ofi )2 (6) fi ∈F

fk fi ( − m) + δ2 (j, fk , fi ) + m + Tin,j ≤ (T˜out,j + fk 4f1k ) ≤ M δ2 (j, fk , fi ) + Tin,j fk fk ( − m)δ1 (j, fk , fi ) + m + (T˜in,j − 4f2k ) ≤ Tin,j ≤ M δ1 (j, fk , fi ) + (T˜fi − 4fk ) in,j

(8d)

2

where M is a sufficiently big integer and m is a small integer.  is the machine precision, beyond which the constraint is violated. A new auxiliary logical variable δ3 (j, fk , fi ) is defined as [δ3 (j, fk , fi )] ←→ [δ1 (j, fk , fi )] ∧ [δ2 (j, fk , fi )] to represent Z 𝛅𝟏 𝐣, 𝐟𝐤 , 𝐟𝐢 = 𝟏

1

τ fi −Desirable departure time of flight fi Dfi −Scheduled departure time of flight fi Rfi −Total travel time of flight fi through the rerouting path Ofi −Total travel time of flight fi through the optimal path

𝑓

𝑓𝑖 𝑇𝑖𝑛,𝑗

−

𝑓 ∆2 𝑘

𝑓𝑖 𝑇𝑜𝑢𝑡 ,𝑗

+

𝑓 ∆1 𝑘

𝛅𝟐 𝐣, 𝐟𝐤 , 𝐟𝐢 = 𝟏

The term, (τ fi − Dfi )2 of the objective function is to minimize the total departure deviation and the term, (Rfi −

(8c)

Fig. 5.

Time range for δ1 (j, fk , fi ) and δ2 (j, fk , fi )

𝑇𝑖𝑛𝑘,𝑗

fi fi ) in condition (4). The re, T˜out,j the variable X j fk (T˜in,j Tin,j

lationship of δ3 (j, fk , fi ) = δ1 (j, fk , fi ) ∗ δ2 (j, fk , fi ) can express using mixed logic constraints as, δ3 (j, fk , fi ) − δ1 (j, fk , fi ) ≤ 0

(8e)

δ3 (j, fk , fi ) − δ2 (j, fk , fi ) ≤ 0

(8f)

δ1 (j, fk , fi ) + δ2 (j, fk , fi ) − δ3 (j, fk , fi ) ≤ 1

(8g)

So that, the constraint (5) can replace as, X fk + 4l ); ∀fk ∈ F , ∀j δ3 (j, fk , fi ) ∗ Yjfi ≤ Cj (Tin,j fi ∈F

(8h) Introducing new auxiliary logical variable to convert (8h) into linear form as δ4 (j, fk , fi ) = δ3 (j, fk , fi ) ∗ Yjfi which is equivalent to, −δ3 (j, fk , fi ) + δ4 (j, fk , fi ) ≤ 0 −Yjfi δ3 (j, fk , fi ) + Yjfi

(8i)

+ δ4 (j, fk , fi ) ≤ 0

(8j)

− δ4 (j, fk , fi ) ≤ 1

(8k)

Finally, the link capacity constraint in (5) can be represented as, X fk δ4 (j, fk , fi ) ≤ Cj (Tin,j + 4l ); ∀fk ∈ F , ∀j (9) fi ∈F

The optimization problem is a mixed integer quadratic programming (MIQP) with continuous and binary decision variables, quadratic cost function (6), and linear constraints for flight dispatching (1) and link capacity (8), (9). V. EXPERIMENTAL RESULTS In this section, we present our simulation results for a simplified network with 6 airports, 19 waypoints and 47 air routes for a sub-region of ASEAN. The air transportation network map is shown in Fig.6. We considered Changi and Penang as two departure airports with notation of 1 and 2 respectively, while all other airports are arrival airports. Flights from Changi and Penang are scheduled to fly for 4 other airports which are Phuket, Bangkok, Ho Chi Minh City and Kota Kinabalu. 4

Legend

- Airport - Waypoint - Air route / link

1 4 2

8

5

3 9 6

16

10

Departure Uncertainty Distribution of Changi (WSSS) Airport

7

5

25 25

18 12 13

15

Percentage

11

20

15

10

0

19

The simulated network

15

5

0

12

24

36

(actual-scheduled)departure time difference

(a) Changi(WSSS) airport Fig. 6.

20

10

5

14

1

35 30

6

2

Departure Uncertainty Distribution of Penang(PEN/WMKP) Airport

30

17

Percentage

3

We generated 80 flight plans based on our scheduling algorithm as there are 40 flights take off from each airport with 10 flights per od pair. The compressed optimization problem consists of 47720 decision variables, 162981 inequality constraints, 1241 equality constraints. The optimization problem is solved by using Gurobi solver integrated with MATLAB. To verify that the model work as intended, we went through the following procedure. First, we generated flight plans using our model described in section III. Then, we used actual 24 hour flight information from flightradar24 tool [12] and analyzed the departure uncertainty distribution for Changi and Penang airport as shown in Fig.7a and Fig.7b, respectively. We found that the departure uncertainty of Changi airport follows a normal distribution with mean and standard deviation of 18.69min and 8.577min, respectively, which reflect that there is a small tail for early departures (4min) and a long tail for late departures (45min). Moreover, Penang airport follows a Weibull distribution with shape, scale and location parameter values as 2.73, 33.72, -18.02min. It means that 15min of early departure deviation and 44min late departure deviation from the scheduled departure time. Then, we generated random numbers from these two departure distributions and considered them as the actual departure time and embedded them in the departure times of the generated flight plans. Finally, we evaluated the total capacity violations, average flight delay and average capacity utilization with injected uncertainties of the departure time for three different scenarios as described below. For all three scenarios, we used the same set of scheduled departure times. To verify that the proposed model can achieve the outlined objectives even at the most challenging situations, we used an extreme flight scheduling where all the flights destined for the same arrival airport (sets of 20 flights) are scheduled to depart at the same time. We used such extreme case to verify that our model work as intended without any capacity violation even at the intense situations. Further, the above mentioned extreme flight scheduling helps to create highly congested situations in the network, which can lead to possible capacity violations under deterministic scheduling. So it helps to clearly emphasize the importance of our work. It should be noted that this may lead to very high delays in some of the flights. For the analysis, four congested links were selected as route from waypoint 3 to 1, 6 to 2, 11 to 10 and 15 to 18. We used the performance metrics of total capacity violations, average flight delay and average link utilization to evaluate our model.

Fig. 7.

0

-20

-10

0

10

20

30

40

50

(actual-scheduled)departure time difference

(b) Penang(PEN/WMKP) airport

Departure uncertainty distribution

In this study, we didn’t apply any computational efficient methods to boost the run time as our solely concern is to eliminate the en-route capacity violation. We expect to carry out further work on this later. A. Scenario 1: Fixed routing with deterministic scheduling In this case, we used high weight for rerouting path cost in the objective function (CR >> CD ) and very small value for 41 and 42 to convert the model into a fixed routing deterministic scheduling scheme. Since the coefficient of the rerouting cost is high, almost all flight plans attempt to use the optimal path to minimize the rerouting path cost. So, the flight scheduling is focused on minimizing the deviation of the flight path while applying more ground delay. B. Scenario 2: Fixed routing with stochastic scheduling In scenario 2, we simulated our model with the same objective function coefficients (CR >> CD ), but with the actual values for 41 and 42 by using statistics of Changi and Penang airport departure distribution. We used maximum lower and upper bounds for both departure airports to test the model for the stochastic case. C. Scenario 3: Free routing with stochastic scheduling In this case, we used both free routing as well as actual departure uncertainties as proposed in our model. We reduce the weight of rerouting path cost in the objective function to allow rerouting and used actual 41 , 42 values as in the scenario 2 to counter the departure uncertainty effect. The results are summarized in Table I. As it can be seen from the results, there exist a large number of capacity violations (27.5% of total flights) in the scenario 1 as its deterministic nature in flight scheduling leads to unexpected arrivals of flights into the routes. However, when we consider scenario 2 and 3, results reveal that there is no any link capacity violation since we considered the maximum bounds of departure uncertainty for flight scheduling in both of the cases, thus stochastic scheduling. Eventually, this forces the average flight delay to increase significantly and average link utilization to decrease moderately, since a flight’s positioning is no longer considered as a point of time rather as a fixed range of time. Yet, we need to note that these average flight delays are high since we used an extreme case (same scheduled departure time) in our simulation which is hypothetical in the real environment. From all three cases, scenarios 2 display the highest average flight delay because of the fixed path as well as the maximum boundary TABLE I R ESULTS FOR THE THREE SCENARIOS Parameters\ Cases

Scenario 1

Scenario 2

Scenario 3

Average delay per flight

1.67hr

4.7hr

2.28hr

No. of capacity violation

22

0

0

Link utilization

84.4%

38%

54.2%

consideration of departure uncertainty in flight scheduling. Our proposed model is represented by scenario 3, which has both free routing as well as the departure uncertainty consideration in flight scheduling to avoid capacity violation and to minimize the average flight delay. Scenario 3 results clearly indicate that there is no any capacity violation, and also the average delay is notably lower than in scenario 2. However, the proposed system still possesses limitations on the link utilizations and further research is required to handle this issue. VI. CONCLUSION In this paper, we have proposed a novel flight routing and scheduling algorithm, which considers the departure uncertainty of flights. Free routing feature of our model helps to minimize the unnecessary departure delays of flights while the consideration of the stochastic nature of flight’s departure clears out the possible capacity violations. To assess the performance, we have used an extreme flight scheduling pattern and the results show that the proposed system eliminates the capacity violation due to departure uncertainty completely while maintaining a competitive flight delay. In future, we expect to expand our model for the actual ASEAN air traffic network and also to use relaxation mechanisms to boost the computational efficiency. Further research will be carried out to take both speed uncertainty and en-route capacity uncertainty into account to make the model and approach more realistic. R EFERENCES [1] D. Bertsimas and S. S. Patterson, ”The Air Traffic Flow Management Problem with Enroute Capacities,” Oper. Res., vol. 46, pp. 406-422, 1998. [2] D. Bertsimas and S. S. Patterson, ”The Traffic Flow Management Rerouting Problem in Air Traffic Control: A Dynamic Network Flow Approach,” Transportation Science, vol. 34, pp. 239-255, 2000. [3] D. Sun and A. M. Bayen, ”Multicommodity Eulerian-Lagrangian Large-Capacity Cell Transmission Model for En Route Traffic,” Journal of Guidance, Control, and Dynamics, vol. 31, pp. 616-628,2008. [4] Y. Zhang, R. Su, Q. Li, C. Cassandras, and L. Xie, ”Distributed flight routing and scheduling in air traffic flow management,” in 2016 IEEE 55th Conference on Decision and Control (CDC), pp. 1080-1085. [5] S. Smith and E. Gilbo, ”Analysis of Uncertainty in ETMS Aggregate Demand Predictions,” Volpe National Transportation Systems Center, Report no. VNTSC-ATMS-05-05, 2005. [6] M. Terrab and A. R. Odoni, ”Strategic Flow Management for Air Traffic Control,” Operations Research, vol. 41, pp. 138-152, 1993. [7] G. Clare and A. Richards, ”Air traffic flow management under uncertainty: application of chance constraints,” Proceedings of the 2nd International Conference on Application and Theory of Automation in Command and Control Systems, London, United Kingdom, 2012. [8] G. Clare, A. Richards, J. Escartin, D. Martinez, J. Cegarra, and L. J. Alvarez, ”Air Traffic Flow Management under Uncertainty: Interactions Between Network Manager and Airline Operations Centre,” presented at the Second SESAR Innovation Day, 2012. [9] E. Gilbo and S. Smith, ”A New Model to Improve Aggregate Air Traffic Demand Predictions,” in AIAA Guidance, Navigation and Control Conference and Exhibit, ed: American Institute of Aeronautics and Astronautics, 2007. [10] S. Smith and E. Gilbo, ”Analysis of Uncertainty in ETMS Aggregate Demand Predictions,” Volpe National Transportation Systems Center, Report no. VNTSC-ATMS-05-05, 2005. [11] A. Bemporad and M. Morari, ”Control of systems integrating logic, dynamics, and constraints,” Automatica, vol. 35, pp. 407-427, 1999. [12] ”Flightradar24 LIVE AIR TRAFFIC tool”,URL: https://www.flightradar24.com/