# Generalized Method of Modeling Minute-in-Trail Strategy for Air Traffic Flow Management

Authors: Jinglei Huang; Qiucheng Xu; Yongjie Yan; Hui Ding; Jing Tian
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

With the rapidly increasing air traffic demand, the demand-capacity imbalance problem of sector is surfaced gradually. And, minute-in-trail/miles-in-trail (MIT) is an effective strategy to balance the traffic demands and capacity. In this work, we consider the MIT strategy generation problem for the situation that a sector withNC corridors is affected by convection weather for Timb time periods. Given the sector capacity Cwt, t=1,…,Timb, under convection weather, we propose a three-phase optimization framework to generate E-MIT strategy to achieve the demand-capacity balance. First, we take the sector capacity of Timb time periods under convection weather as a whole, that is, ∑t=1TimbCwt, and then a dynamical programming-based method is proposed to allocate ∑t=1TimbCwt for NC corridors such that the capacity resources Awi of each corridor CORi, i=1,…,NC, can be determined. Second, a 0-1 combination algorithm is used to allocate the capacity resources Awi into Timb time periods for each corridor CORi such that the candidate strategies set CSi of each corridor can be determined, where a strategy solji∈CSi is an array with Timb numbers and each number represents the maximum allowed number of flights entering into sector from CORi in one time period. Finally, a modified shortest path algorithm based on the backtracking method is taken to select the optimal strategy from CSi for NC corridors such that the total delay cost and air traffic control load are minimized. Additionally, a dynamical programming-based method is proposed to generate E-MIT strategy for the special case that the sector capacities of different time periods under convection weather are the same, that is, Cw1=Cw2=⋯=CwTimb, and the generated strategies of Timb time periods for a corridor are also the same. Experimental results show that compared with the proposed three-phase optimization method, rate-based method and need-based method will spend more 8.1% and 6.3% of delay cost, respectively. When considering the special case, the experimental results show that compared with the proposed dynamical programming-based method, the rate-based method and need-based method will spend more 10.2% and 7.5% of delay cost, respectively.


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## Body

## 1. Introduction

With the rapidly growing air traffic demands, air traffic control load [1] is increased, and the air traffic problems [2–4], such as air traffic congestion, the demand-capacity imbalance problem [5], and flight delays [6], are surfaced gradually. And, the demand-capacity imbalance problem is often occurred, particularly when capacity has been reduced due to the convection weather.Generally, the demand-capacity imbalance problem can be dealt with in two ways [7]. One is the long-term strategy through the construction of infrastructure, such as new airports and runways to enlarge capacity. However, this procedure would take a long time and high cost.The second way is to regulate the traffic flow by using the traffic management initiatives (TMIs) [8], such that the limited capacity can be used efficiently and the impact of unavoidable delays would be reduced [9]. And, the TMIs can be classified into two categories: (1) strategic actions, which are the strategies that taken before the aircraft has been taken off, including ground delay program [10], ground stop [11], airspace flow program [12], minute-in-trail or miles-in-trail [13], and Collaborative Trajectory Options Program [14] and (2) tactical actions, which are the strategies that taken after the aircraft is airborne, consisting of rerouting [15], speed adjustment, airborne holding [16], and fix balancing.Among these strategies, the minute-in-trail/miles-in-trail is the most frequently used TMI because of its simplicity and ease of implementation [17], which is the strategy that imposes the time/distance spacing restrictions between every two adjacent aircraft flying along a routing path from the same corridor. Because the minute-in-trail strategy and miles-in-trail strategy can be interconverted, this paper focuses on the minute-in-trail strategy, hereafter referred to as MIT.However, most of traffic managers rely largely on experience to determine the MIT restrictions, and no tool is available to support the traffic managers to balance the interests of all stakeholders, airlines, and passengers, resulting in the larger delay cost. Therefore, it is necessary to study the MIT strategy generation problem.Many studies have focused on modeling the strategy of MIT. In [17], the authors presented a perspective on the MIT strategy, where the strengths and shortcomings of MIT were discussed in details and proved that MIT is an effective traffic management strategy for high-density sectors. Ostwald et al. [18] proposed an operational concept for arrival MIT restrictions using the MIA capability, where MIA is the tool used to evaluate the impacts of the proposed MIT restrictions on resources and flights before implementing them. Sheth et al. [19, 20] developed a model to compute MIT and pass back restrictions in the NAS for current traffic conditions, and the maximum ground delay and absorbable airborne delay are incorporated in the model.In [21], the authors proposed a method based on the genetic algorithm for generating the MIT strategy to make full use of the storage capacities of the sector. Unfortunately, the stochastic optimization algorithm would lead to the different MIT strategies in different runs for the same scene; thus, the method cannot be used in the actual control practice. To achieve the goal of demand-capacity balance, Yuanhua and Zhang [22] proposed a method to restrict the interval of every two flights from surrounding areas and the departure time of flights of airports in this area through the strategy of GDP so that the total delay is minimized.Machine learning has been applied to solve air traffic management problems. In [23], four machine learning algorithms, including support vector machine, random forest, decision tree algorithm, and softmax regression algorithm, are used to evaluate the miles-in-trail. Wang and Grabbe [24] offered an update analysis of the cause, frequency, and duration of historical MIT restrictions and subsequently using machine learning techniques to predict the occurrence of MIT restrictions to manage arrivals into the ATL airport.However, most of the previous works mainly focused on generating a control strategy to restrict the flights into a sector with minimization of the total delay while the types of aircraft and passengers are not considered, resulting in the larger delay cost. In addition, the generated MIT strategy of the previous work is difficult to operate for air traffic managers because of the frequent change of traffic flow management strategy would lead to higher air traffic control load [25].In [26], the authors proposed an evolutionary algorithm to generate the MIT strategy, where a traffic flow-capacity matching model based on workload restriction is presented and the stability function for evaluating the strategy in each slot time is defined, and an evolutionary algorithm is proposed to generate the MIT strategy. Unfortunately, the stochastic optimization algorithm would lead to the different strategies in different runs, resulting in the larger air traffic control load.Motivated by these arguments, we propose a method for generating theE-MIT strategy to control the aircraft from different directions heading towards a sector whose capacity is decreased from the normal operation capacity due to the convective weather. The main contributions of this paper are outlined as follows:(1)
Given the sector capacityCwt, t=1,…,Timb, under convection weather, we propose a three-phase optimization framework to generate the E-MIT strategy to achieve the demand-capacity balance. A dynamical programming-based method is proposed to allocate the total sector capacity of Timb time periods under convection weather, ∑t=1TimbCwt, for NC corridors, and a 0-1 combination algorithm is used to determine the candidate strategies set CSi for each corridor. Finally, a modified shortest path algorithm based on the backtracking method is taken to select the optimal strategy from CSi for all corridors with minimization of the total delay cost and air traffic control load.(2)
Additionally, a dynamical programming-based method is proposed to solve theE-MIT strategy generation problem for the special case that sector capacities of different time periods under convection weather are the same, that is, Cw1=Cw2=⋯=CwTimb, and the generated strategies of Timb time periods for a corridor are also the same.Experimental results show that compared with the rate-based method and need-based method, the proposed generalized method can reduce the average cost by 9.1% and 5.2%, respectively. When considering the special case, the experimental results show that compared with the rate-based method and need-based method, the proposed dynamical programming-based method can reduce the average cost by 9.2% and 7.0%, respectively.The remainder of the paper is organized as follows: Section2 describes the problem definition. Section 3 gives the details of computing delay cost and air traffic control load. The generalized E-MIT strategy generation method is discussed in Section 4. Section 5 discusses the generation method for a special case. Experimental results and conclusions are shown and discussed in Sections 6 and 7, respectively.


## 2. Problem Description

### 2.1. Extended MIT Strategy

Minute-in-trail/miles-in-trail (MIT) is the strategy that requires the flights in a flow of air traffic crossing a certain corridor of sector must be separated by a certain number of minutes or miles. Through this strategy, we can control the volume of air traffic into sectors and airports at a safe level.In the actual control practice, MIT, along with the maximum allowed number of flights, is the frequently used traffic management initiatives to balance the traffic flows and sector capacity under converse weather. That is, the generated strategy includes two restrictions for flights across one transfer-of-control point (or, a corridor) of sector:(i)
The maximum allowed number of flights in a time period(ii)
The minimal time interval between every two adjacent flights from the same corridorIn this paper, we focus on the MIT strategy, along with the maximum allowed number of flights, hereafter referred to as the extended MIT strategy (E-MIT).


### 2.2. Flow Control Time

Table1 shows some notations used in this paper.Table 1
Some used notations.
NC
The number of corridors belong to the sector
COR
i
Thei-th corridor of sector, 1≤i≤NC
C
n
The sector capacity under normal condition in one time period
K
n
i
The maximum allowed number of flights for corridorCORiin a time period under normal condition, 1≤i≤NC
C
w
j
The sector capacity of time periodTPjunder convection weather condition, 1≤j≤T
K
w
i
,
j
Thei-th corridor capacity resource of TPj under convection weather condition, 1≤i≤NC, 1≤j≤T
A
w
i
The total corridor capacity ofCORi of all time periods under convection weather condition, Awi=∑t=1TKwi,t
t
e
i
Estimated time of takeoff for flightfi
t
c
i
Calculated time of takeoff for flightfi
T
imb
The duration time periods of convection weather on sector
T
rec
The time periods needed to recover the over traffic flows ofTimb
T
fc
The duration time ofE-MIT strategy, that is, the total time that needed to balance the sector capacity and traffic flowsOnce theE-MIT strategy is issued, the duration time of strategy, i.e., the flow control time Tfc, must be attached to the E-MIT strategy so that the air traffic managers can control the traffic flows more effectively.Flow control timeTfc is the total time needed to balance the sector capacity and traffic flows using the E-MIT strategy, which is defined as follows:(1)Tfc=Timb+Trec,where Timb is the duration time periods of convection weather on sector, that is, the time periods of demand-capacity imbalance. Because the sector capacity is decreased from the normal operation capacity due to the convection weather, some planned flights are delayed to the subsequent time periods. And, Trec is the time period to process the delayed flights such that the balance of demand-capacity can be recovered, which is defined as follows:(2)Trec=maxTreci1≤i≤NC,where Treci is the time period needed to process the delayed flights and recover the demand-capacity imbalance for corridor CORi.Assume that the sector is affected by convection weather duringtb,te, and let F=Fithe flights set across the corridor CORi,i=1,…,NC} be the flights scheduled to arrive at this sector during tb,−. The time axis can be discretized by subdividing tb,− into a set of consecutive time periods, and the time span τ of each time period can be set any value. However, because of the accuracy of weather forecast, τ is set to 15 minutes, 30 minutes, or 1 hour, practically. For convenience, we define Fit to be the number of flights across the corridor CORi during t-th time period.Thus, the imbalance time periodsTimb can be calculated as follows:(3)Timb=te−tbτ,where τ is the time span of a time period, and in this work, we set one hour.If the maximum allowed number of flights across the corridorCORi in a time period TPt 1≤t≤Timb is Kwi,t under convection weather and the corresponding normal level is Kni in one time period, the demand-capacity recovery time periods Treci is the least value that meets the following in-equation:(4)∑t=1TimbFit−Kwi,t≤∑t=TimbTreciKni−Fit.Figure1 shows an example of calculating the recovery time Trec1 for COR1, where the convection weather occurs from 20:00 to 22:00, and we set each time period at one hour; thus, Timb=2. In these two time periods, the maximum allowed number of flights across the corridor COR1 is Kw1,1=10 flights per hour and Kw1,2=11 flights per hour, respectively. And, the maximum allowed number of flights across COR1 under normal condition Kni is 16 flights per hour.Figure 1
An example of calculating the recovery timeTrec1 for COR1.From Figure1, we can get F11=16, F12=17, F13=10, and F14=9. Thus, according to in-equation (4), we can calculate the recovery time periods Trec1=2 to recover the left 12 flights of the demand-capacity imbalance periods.


### 2.3. Problem Definition

Based on the above definitions, the problem definition for generating theE-MIT strategy is as follows.Given the following inputs,(1)
The number of corridorsNC of a sector, that is CORi 1≤i≤NC(2)
The sector capacity under convection weather condition,Cwj, 1≤j≤Timb(3)
The sector capacityCn of a time period under normal condition(4)
The maximum allowed number of flightsKni (or, corridor capacity) of a time period for corridor CORi, 1≤i≤NC, under normal conditionand the following constraints:(1)
The flights are not allowed to take off in advance, that is,tci>tei(2)
The maximum allowed number of flights acrossCORi is recovered to a normal level Kni once the convection weather is over(3)
The maximum allowed number of flights across a corridor in one time period under convection weather conditions cannot exceed the corresponding normal level, that is,Kwi,j<KniWe try to generate an optimalE-MIT strategy to control the air traffic flows from different corridors that enter into a sector whose capacity is decreased from the normal operation capacity due to the convective weather. And, the E-MIT strategy generally includes two restrictions on flights:(1)
The maximum allowed number of flights across a corridorCORi in a time period TPj, i.e., Kwi,j(2)
the minimal time intervaltinti,j between every two adjacent flights across the same corridor CORi in one time period TPjThus, the total flight delay cost and air traffic control load are minimized.Figure2 gives an example of the E-MIT control strategy for a sector in one hour, where Cw1 is 31 flights/h, and the maximum allowed number of flights for each corridor in one hour is 8 flights/h, 8 flights/h, 12 flights/h, and 3 flights/h, respectively.Figure 2
Example of theE-MIT control strategy for a sector.However, once the maximum allowed number of flightsKwi,j of corridor CORi in time period TPj is determined, the minimal time interval tinti,j can be calculated as follows:(5)tinti,j=maxτKwi,j∗2,tsepi,j,i∈1,NC,j∈1,T,where τ is the duration time of a time period TPj and tsepi,j is the minimal separation time between aircraft i and j to ensure safety, which can be calculated according to Table 2. In Table 2, three types of aircraft, small, large, and heavy, are considered [27]. For example, if a heavy aircraft follows a large aircraft, then their minimal separation time must be at least 61 seconds.Table 2
Aircraft types and minimum separation timetsepi,j (in seconds).
Leading aircraft type
Trailing aircraft type
Small
Large
Heavy
Small
59
88
109
Large
59
61
109
Heavy
59
61
90Thus, theE-MIT strategy generation problem is to determine Kwi,j of corridor CORi in time period TPj, i=1,…,NC, j=1,…,T.LetCosti be the total delay cost of corridor CORi when Kwi,j of all time periods are determined simultaneously, and let Li be the air traffic control load of corridor CORi, which is used to approximate the strategy stability. Therefore, the strategy generation problem can be formulated as a resource allocation problem as follows:(6)minimize∑k=1NCα⋅Costk+β⋅Lk,subject to(7)∑i=1NCKwi,j=Cwj,∀j∈1,Timb,Kwi,j<Kni,∀i∈1,NC,j∈1,Timb,where α and β can make trade-off between the contributing factors.The first sets of constraints define a limited capacity resource in each time periodTPj for sector, and the second sets of constraints show the limited value of the maximum allowed number of flights for each corridor CORi in a time period TPj.To explain the strategy generation problem more clearly, an example is given in Figure3, where the sector has four corridors, and the sector capacities of three time periods under convection weather are Cw1, Cw2, and Cw3, respectively. The strategy generation problem is to determine the values of Kwi,t, i=1,…,4, t=1,…,3, under the constraints that ∑i=14Kwi,j=Cwj,∀j∈1,3 such that the total delay cost and air traffic control load are minimized. Unfortunately, the delay cost Costi and air traffic control load Li of corridor CORi can be calculated (discussed in the next section) only if the values of Kwi,1, Kwi,2, and Kwi,3 are given simultaneously.Figure 3
Example of the strategy generation problem.Therefore, it is very difficult to directly solve this resource allocation problem sinceKwi,j, i=1,…,NC, j=1,…,Timb, cannot be determined simultaneously. And, in this work, a three-phase method is proposed to generate the E-MIT strategy.


## 2.1. Extended MIT Strategy

Minute-in-trail/miles-in-trail (MIT) is the strategy that requires the flights in a flow of air traffic crossing a certain corridor of sector must be separated by a certain number of minutes or miles. Through this strategy, we can control the volume of air traffic into sectors and airports at a safe level.In the actual control practice, MIT, along with the maximum allowed number of flights, is the frequently used traffic management initiatives to balance the traffic flows and sector capacity under converse weather. That is, the generated strategy includes two restrictions for flights across one transfer-of-control point (or, a corridor) of sector:(i)
The maximum allowed number of flights in a time period(ii)
The minimal time interval between every two adjacent flights from the same corridorIn this paper, we focus on the MIT strategy, along with the maximum allowed number of flights, hereafter referred to as the extended MIT strategy (E-MIT).


## 2.2. Flow Control Time

Table1 shows some notations used in this paper.Table 1
Some used notations.
NC
The number of corridors belong to the sector
COR
i
Thei-th corridor of sector, 1≤i≤NC
C
n
The sector capacity under normal condition in one time period
K
n
i
The maximum allowed number of flights for corridorCORiin a time period under normal condition, 1≤i≤NC
C
w
j
The sector capacity of time periodTPjunder convection weather condition, 1≤j≤T
K
w
i
,
j
Thei-th corridor capacity resource of TPj under convection weather condition, 1≤i≤NC, 1≤j≤T
A
w
i
The total corridor capacity ofCORi of all time periods under convection weather condition, Awi=∑t=1TKwi,t
t
e
i
Estimated time of takeoff for flightfi
t
c
i
Calculated time of takeoff for flightfi
T
imb
The duration time periods of convection weather on sector
T
rec
The time periods needed to recover the over traffic flows ofTimb
T
fc
The duration time ofE-MIT strategy, that is, the total time that needed to balance the sector capacity and traffic flowsOnce theE-MIT strategy is issued, the duration time of strategy, i.e., the flow control time Tfc, must be attached to the E-MIT strategy so that the air traffic managers can control the traffic flows more effectively.Flow control timeTfc is the total time needed to balance the sector capacity and traffic flows using the E-MIT strategy, which is defined as follows:(1)Tfc=Timb+Trec,where Timb is the duration time periods of convection weather on sector, that is, the time periods of demand-capacity imbalance. Because the sector capacity is decreased from the normal operation capacity due to the convection weather, some planned flights are delayed to the subsequent time periods. And, Trec is the time period to process the delayed flights such that the balance of demand-capacity can be recovered, which is defined as follows:(2)Trec=maxTreci1≤i≤NC,where Treci is the time period needed to process the delayed flights and recover the demand-capacity imbalance for corridor CORi.Assume that the sector is affected by convection weather duringtb,te, and let F=Fithe flights set across the corridor CORi,i=1,…,NC} be the flights scheduled to arrive at this sector during tb,−. The time axis can be discretized by subdividing tb,− into a set of consecutive time periods, and the time span τ of each time period can be set any value. However, because of the accuracy of weather forecast, τ is set to 15 minutes, 30 minutes, or 1 hour, practically. For convenience, we define Fit to be the number of flights across the corridor CORi during t-th time period.Thus, the imbalance time periodsTimb can be calculated as follows:(3)Timb=te−tbτ,where τ is the time span of a time period, and in this work, we set one hour.If the maximum allowed number of flights across the corridorCORi in a time period TPt 1≤t≤Timb is Kwi,t under convection weather and the corresponding normal level is Kni in one time period, the demand-capacity recovery time periods Treci is the least value that meets the following in-equation:(4)∑t=1TimbFit−Kwi,t≤∑t=TimbTreciKni−Fit.Figure1 shows an example of calculating the recovery time Trec1 for COR1, where the convection weather occurs from 20:00 to 22:00, and we set each time period at one hour; thus, Timb=2. In these two time periods, the maximum allowed number of flights across the corridor COR1 is Kw1,1=10 flights per hour and Kw1,2=11 flights per hour, respectively. And, the maximum allowed number of flights across COR1 under normal condition Kni is 16 flights per hour.Figure 1
An example of calculating the recovery timeTrec1 for COR1.From Figure1, we can get F11=16, F12=17, F13=10, and F14=9. Thus, according to in-equation (4), we can calculate the recovery time periods Trec1=2 to recover the left 12 flights of the demand-capacity imbalance periods.


## 2.3. Problem Definition

Based on the above definitions, the problem definition for generating theE-MIT strategy is as follows.Given the following inputs,(1)
The number of corridorsNC of a sector, that is CORi 1≤i≤NC(2)
The sector capacity under convection weather condition,Cwj, 1≤j≤Timb(3)
The sector capacityCn of a time period under normal condition(4)
The maximum allowed number of flightsKni (or, corridor capacity) of a time period for corridor CORi, 1≤i≤NC, under normal conditionand the following constraints:(1)
The flights are not allowed to take off in advance, that is,tci>tei(2)
The maximum allowed number of flights acrossCORi is recovered to a normal level Kni once the convection weather is over(3)
The maximum allowed number of flights across a corridor in one time period under convection weather conditions cannot exceed the corresponding normal level, that is,Kwi,j<KniWe try to generate an optimalE-MIT strategy to control the air traffic flows from different corridors that enter into a sector whose capacity is decreased from the normal operation capacity due to the convective weather. And, the E-MIT strategy generally includes two restrictions on flights:(1)
The maximum allowed number of flights across a corridorCORi in a time period TPj, i.e., Kwi,j(2)
the minimal time intervaltinti,j between every two adjacent flights across the same corridor CORi in one time period TPjThus, the total flight delay cost and air traffic control load are minimized.Figure2 gives an example of the E-MIT control strategy for a sector in one hour, where Cw1 is 31 flights/h, and the maximum allowed number of flights for each corridor in one hour is 8 flights/h, 8 flights/h, 12 flights/h, and 3 flights/h, respectively.Figure 2
Example of theE-MIT control strategy for a sector.However, once the maximum allowed number of flightsKwi,j of corridor CORi in time period TPj is determined, the minimal time interval tinti,j can be calculated as follows:(5)tinti,j=maxτKwi,j∗2,tsepi,j,i∈1,NC,j∈1,T,where τ is the duration time of a time period TPj and tsepi,j is the minimal separation time between aircraft i and j to ensure safety, which can be calculated according to Table 2. In Table 2, three types of aircraft, small, large, and heavy, are considered [27]. For example, if a heavy aircraft follows a large aircraft, then their minimal separation time must be at least 61 seconds.Table 2
Aircraft types and minimum separation timetsepi,j (in seconds).
Leading aircraft type
Trailing aircraft type
Small
Large
Heavy
Small
59
88
109
Large
59
61
109
Heavy
59
61
90Thus, theE-MIT strategy generation problem is to determine Kwi,j of corridor CORi in time period TPj, i=1,…,NC, j=1,…,T.LetCosti be the total delay cost of corridor CORi when Kwi,j of all time periods are determined simultaneously, and let Li be the air traffic control load of corridor CORi, which is used to approximate the strategy stability. Therefore, the strategy generation problem can be formulated as a resource allocation problem as follows:(6)minimize∑k=1NCα⋅Costk+β⋅Lk,subject to(7)∑i=1NCKwi,j=Cwj,∀j∈1,Timb,Kwi,j<Kni,∀i∈1,NC,j∈1,Timb,where α and β can make trade-off between the contributing factors.The first sets of constraints define a limited capacity resource in each time periodTPj for sector, and the second sets of constraints show the limited value of the maximum allowed number of flights for each corridor CORi in a time period TPj.To explain the strategy generation problem more clearly, an example is given in Figure3, where the sector has four corridors, and the sector capacities of three time periods under convection weather are Cw1, Cw2, and Cw3, respectively. The strategy generation problem is to determine the values of Kwi,t, i=1,…,4, t=1,…,3, under the constraints that ∑i=14Kwi,j=Cwj,∀j∈1,3 such that the total delay cost and air traffic control load are minimized. Unfortunately, the delay cost Costi and air traffic control load Li of corridor CORi can be calculated (discussed in the next section) only if the values of Kwi,1, Kwi,2, and Kwi,3 are given simultaneously.Figure 3
Example of the strategy generation problem.Therefore, it is very difficult to directly solve this resource allocation problem sinceKwi,j, i=1,…,NC, j=1,…,Timb, cannot be determined simultaneously. And, in this work, a three-phase method is proposed to generate the E-MIT strategy.


## 3. Computation of Objection Function

### 3.1. Computation of Delay Cost

Once theE-MIT strategy is generated, it will be feed back to the collaborative decision-making system of local airport and regenerate the calculated time of takeoff (CTOT) for flights such that the number of flights across a corridor in a time period and the minimal time interval between every two adjacent flights could meet the constraints of E-MIT strategy.Here, we take a corridor, for example, as shown in Figure3, given the maximum allowed number of flights across COR1 in one time period under convection weather conditions, that is, Kw1,1, Kw1,2, and Kw1,3, and the CTOT of flight fi, tci, can be calculated via the ration-by-schedule algorithm [28], which is based on the first-scheduled-first-served principle. The main steps are listed as follows:(a)
Determine the number of time slotsnslot and the time span of each slot tslot(b)
Sort the flightsFctl according to their estimated time of takeoff tei(c)
Allocate the time slots for the sorted flights in order, and the allocated slot is at or later than the flight’s estimated time of takeofftei(d)
The beginning time of slot is taken astciThe procedure is repeated until each flight is allocated a time slot. At last, the delay timetdi of flight fi can be determined.In Step (a), the number of time slotsnslot is the maximum allowed number of flights across COR1 in one time period, and the time span of each slot tslot can be defined as follows:(8)tslot=τnslot,where τ is the time span of a time period.Figure4 gives an example of calculating tci for flight fi, where Kw1,1=4, Kw1,2=3, Kw1,3=4, and Kn1=8 flights per hour. Because the time span of one time period is one hour, the time span of each slot are 60/4 = 15 minutes, 60/3 = 20 minutes, 60/4 = 15 minutes, and 60/8 = 7 minutes, respectively. Based on the first-scheduled-first-served principle, we allocate these slot times to the flights according to their estimated time of takeoff tei (which is given as input) as shown in Steps (b) and (c).Figure 4
Example of calculatingtci for flight fi.As shown in Figure4, for flight f2, whose estimated takeoff time tei is 20:07 and through the procedure of allocating time slots, its calculated takeoff time tci is 20:15; thus, the delay time tdi is 8 minutes.Once the delay timetdi of flights fi are obtained, the delay cost Costk for corridor CORk can be calculated as follows:(9)Costk=Cf+Cp,where Cf represents the delay cost of flights and Cp is the delay cost of passengers.The delay cost of flightsCf depends on the delay time of flights tdi, which can be calculated as follows:(10)Cf=∑i=1Fctlαi⋅tdi,where Fctl is the set of affected flights, which can be determined according to the flow control time Tfc. αi represents the delay cost per hour for flight fi, which depends on the type of aircraft as shown in Table 3, where three types of aircraft are given.Table 3
Flight delay cost model.
Aircraft type
H
M
L
Unit delay costαi (h−1)
4167
2916
208And, the delay cost of passengersCp can be calculated as follows:(11)Cp=∑i=1Fctlβi⋅npi⋅tdi,where npi represents the number of passengers of flight fi and βi represents the delay cost of one people of flight fi. In this work, we take the cost analysis model [29] to estimate the delay cost of passengers, where the unit delay cost βi of ordinary passengers is set 50 per hour and the important passenger is 100 per hour.


### 3.2. Computation of Air Traffic Control Load

In this work, a method similar to [26] is taken to calculate the air traffic control load Lg for corridor CORg, which is defined as follows:(12)Lg=∑j=1TimbKwg,j+1−Kwg,j2.What is more, the air traffic control loadLg can also represent the strategy stability, where the bigger value of Lg will cause the less strategy stability.


## 3.1. Computation of Delay Cost

Once theE-MIT strategy is generated, it will be feed back to the collaborative decision-making system of local airport and regenerate the calculated time of takeoff (CTOT) for flights such that the number of flights across a corridor in a time period and the minimal time interval between every two adjacent flights could meet the constraints of E-MIT strategy.Here, we take a corridor, for example, as shown in Figure3, given the maximum allowed number of flights across COR1 in one time period under convection weather conditions, that is, Kw1,1, Kw1,2, and Kw1,3, and the CTOT of flight fi, tci, can be calculated via the ration-by-schedule algorithm [28], which is based on the first-scheduled-first-served principle. The main steps are listed as follows:(a)
Determine the number of time slotsnslot and the time span of each slot tslot(b)
Sort the flightsFctl according to their estimated time of takeoff tei(c)
Allocate the time slots for the sorted flights in order, and the allocated slot is at or later than the flight’s estimated time of takeofftei(d)
The beginning time of slot is taken astciThe procedure is repeated until each flight is allocated a time slot. At last, the delay timetdi of flight fi can be determined.In Step (a), the number of time slotsnslot is the maximum allowed number of flights across COR1 in one time period, and the time span of each slot tslot can be defined as follows:(8)tslot=τnslot,where τ is the time span of a time period.Figure4 gives an example of calculating tci for flight fi, where Kw1,1=4, Kw1,2=3, Kw1,3=4, and Kn1=8 flights per hour. Because the time span of one time period is one hour, the time span of each slot are 60/4 = 15 minutes, 60/3 = 20 minutes, 60/4 = 15 minutes, and 60/8 = 7 minutes, respectively. Based on the first-scheduled-first-served principle, we allocate these slot times to the flights according to their estimated time of takeoff tei (which is given as input) as shown in Steps (b) and (c).Figure 4
Example of calculatingtci for flight fi.As shown in Figure4, for flight f2, whose estimated takeoff time tei is 20:07 and through the procedure of allocating time slots, its calculated takeoff time tci is 20:15; thus, the delay time tdi is 8 minutes.Once the delay timetdi of flights fi are obtained, the delay cost Costk for corridor CORk can be calculated as follows:(9)Costk=Cf+Cp,where Cf represents the delay cost of flights and Cp is the delay cost of passengers.The delay cost of flightsCf depends on the delay time of flights tdi, which can be calculated as follows:(10)Cf=∑i=1Fctlαi⋅tdi,where Fctl is the set of affected flights, which can be determined according to the flow control time Tfc. αi represents the delay cost per hour for flight fi, which depends on the type of aircraft as shown in Table 3, where three types of aircraft are given.Table 3
Flight delay cost model.
Aircraft type
H
M
L
Unit delay costαi (h−1)
4167
2916
208And, the delay cost of passengersCp can be calculated as follows:(11)Cp=∑i=1Fctlβi⋅npi⋅tdi,where npi represents the number of passengers of flight fi and βi represents the delay cost of one people of flight fi. In this work, we take the cost analysis model [29] to estimate the delay cost of passengers, where the unit delay cost βi of ordinary passengers is set 50 per hour and the important passenger is 100 per hour.


## 3.2. Computation of Air Traffic Control Load

In this work, a method similar to [26] is taken to calculate the air traffic control load Lg for corridor CORg, which is defined as follows:(12)Lg=∑j=1TimbKwg,j+1−Kwg,j2.What is more, the air traffic control loadLg can also represent the strategy stability, where the bigger value of Lg will cause the less strategy stability.


## 4.E-MIT

Given the sector capacityCwt of time period TPt (t=1,…,Timb) under the convection weather, a three-phase method is proposed to generate the E-MIT strategy for flights with minimization of delay cost and air traffic control load.


### 4.1. Allocation of Sector Capacity

In this work, we first take the sector capacity ofTimb time periods under convection weather as a whole, that is, SC=∑t=1TimbCwt; thus, the strategy generation problem can be regarded as a classic resource allocation problem that allocates SC for NC corridors with minimization of total delay cost. And, in this work, a dynamical programming-based method is proposed to solve this resource allocation problem such that the capacity resources Awi of each corridor CORi, i=1,…,NC, can be determined.Figure5 shows an example of allocating sector capacity, where the sector has four corridors and SC=Cw1+Cw2+Cw3. Our goal is to determine the capacity resources Aw1, Aw2, Aw3, and Aw4.Figure 5
Example of allocating sector capacity.If corridorCORi is allocated to j resources, we can calculate the delay cost vij for corridor CORi as follows:(i)
Firstly, according to the equation (8), the time span of each slot under convection weather and normal condition can be calculated as Timb∗τ/j and τ/Kni, respectively(ii)
Then, through the procedures of allocating time slots in Section3.1, the delay time tdi of each flight fi can be calculated(iii)
At last, the delay costvij for corridor CORi can be calculated according to equation (9)LetVij be the minimum total delay cost when the first i 1≤i≤NC corridors, COR1,…,CORi, are allocated to the total j resources, which can be calculated according to the following two steps:(1)
The firsti-1 corridors, COR1,…,CORi−1, are allocated to the total m m≤j resources(2)
The remainingj−m resources are allocated to corridor CORiThus, the recursive relation formula of dynamical programming can be formulated as follows:(13)Vij=minVij,Vi−1m+vij−m.Algorithm1 gives a detailed description of the dynamical programming-based method to allocate the total sector capacity SC for NC corridors.Algorithm 1: Sector_capacity_allocation.
(1)
Input the total sector capacitySC and the number of corridors NC;
(2)
Initialize the valuesVi0, ∀i∈1,NC;
(3)
for m⟵1 to minSC,Timb⋅Kni do
(4)
V1m=v1m;
(5)
end for
(6)
total_cap
=
0;
(7)
for i⟵2 to NC do
(8)
total_cap+=Timb⋅Kni−1;
(9)
for j⟵1 to SC do
(10)
MIN_COST = INF;
(11)
m_max = minj,total_cap;
(12)
for m⟵0 to m_max do
(13)
if j−m>Timb⋅Kni then
(14)
Continue;
(15)
end if
(16)
if MIN_COST>Vi−1m+vij−m then
(17)
MIN_COST = Vi−1m+vij−m;
(18)
Roadij = m;
(19)
end if
(20)
end for
(21)
Vij = MIN_COST;
(22)
end for
(23)
end for
(24)
//backtrack output results
(25)
j
=
SC;
(26)
for i⟵NCto 1 do
(27)
Awi=j−Roadij;
(28)
j=Roadij;
(29)
end forIn lines 2–5 in Algorithm1, we initialize the values of Vi0, i=1,…,NC, and V1m. From lines 13 to line 15, we check whether allocating j−m flights of Timb time periods to corridor CORi will violate the constraint (3). If the allocated j−m flights meet the constraints and have less delay cost than the current cost recorded by MIN_COST, we will update the minimum delay cost and record the allocated capacity resource for CORi in lines 17-18.Once Algorithm1 is finished, the total allocated capacity resources Awi of Timb time periods for each corridor CORi, i=1,…,NC, can be determined as shown in line 27.


### 4.2. Candidate Strategies for a Corridor

Once the total capacity resourceAwi of Timb time periods for each corridor CORi(i=1,…,NC) is determined, we will use 0-1 combination algorithm to allocate Awi into Timb time periods for each corridor CORi such that the candidate strategies set CSi of each corridor can be determined, where a strategy solji∈CSi is an array with Timb numbers, and each number represents the maximum allowed number of flights entering into sector from CORi in one time period.To simplify the discussion, the total allocated capacity resourceAwi of corridor CORi is relabeled as N; thus, the corridor capacity resource allocation problem can be formulated as the problem of placing Timb−1 baffles between N numbers, which is the combination optimization problem of selecting Timb−1 from N + 1, and, there has CN+1Timb−1 solutions. In this work, we take the 0-1 combination algorithm to search those solutions, and Algorithm 2 shows the overall flow of the 0-1 combination algorithm.Algorithm 2: 0-1 combination algorithm for CORi.
(1)
Input the capacity resourceN and time periods Timb;
(2)
Initialize the arrayindexwith the size of N + 1;
(3)
//set the firstTimb−1 of array to 1, which corresponds to a solution;
(4)
for i⟵0 to Timb−1 do
(5)
index[i] = 1;
(6)
end for
(7)
Store the solution according to the values of arrayindex;
(8)
while has Done (index, N, Timb) do
(9)
for i⟵0 to N + 1 do
(10)
//find the first 1-0 combination and change it to 0-1 combination;
(11)
if index[i] = = 1 and index[i] = = 0 then
(12)
index[i] = 0;
(13)
index[i] = 1;
(14)
Store the solution fromindex;
(15)
//move 1 of the left of 0-1 combination to the left of array;
(16)
int count = 0;
(17)
for j⟵0 to ido
(18)
if index[j] = = 1 then
(19)
index[j] = 0;
(20)
index[count++] = 1;
(21)
end if
(22)
end for
(23)
end if
(24)
break;
(25)
end for
(26)
end whileIn line 2 in Algorithm2, we initialize the array index with the size of N + 1, and indexi is the flag that represents whether the corresponding baffle is placed or not. If indexi=1, there is a baffle between the i-th number and the (i + 1)-th number. Otherwise, the i-th number and the (i + 1)-th number are allocated to the same time period.The combination optimization problem is solved in lines 8–26, and the main steps of 0-1 combination algorithm are listed as follows:(1)
Set the firstTimb−1 numbers of array index to 1 as shown in lines 4–6, which corresponds to a solution(2)
Find the first 1-0 combination of arrayindex and change it to 0-1 combination as shown in line 11–13(3)
Move all 1 of the left of 0-1 combination to the left of arrayRepeat Steps 2 and 3 until the lastTimb−1 numbers of array index are 1.Once the 0-1 combination algorithm is finished, we can get the candidate strategies setCSi for each corridor CORi (i=1,…,NC) according to the values of array index.Figure6 shows an example of determining candidate strategies CS1 for corridor COR1 using the 0-1 combination algorithm, where Aw1 = 3 and Timb = 3. Firstly, we define an array index with size of 4, and according to Steps 4–25 of Algorithm 2, we can find CS1=6 possible solutions for the combination problem of selecting Timb−1 from Aw1+1, as shown in Figure 6(a). Secondly, for each solution, according to the values of array index, we can get the positions of baffles placed between numbers. Here, we take the third solution (“0110”), for example, because index1 = 1, and we place a baffle between the first number and the second number. Similarly, a baffle is placed between the second number and the third number as shown in Figure 6(b). At last, we calculate the numbers between adjacent baffles, and each number represents the maximum allowed number of flights entering into sector from CORi in one time period.Figure 6
Example of determining candidate strategies for a corridor using the 0-1 combination algorithm.


### 4.3. Strategy Generation Algorithm

For each corridorCORi (i=1,…,NC), there are CSi (=CAwi+1Timb−1) solutions of allocating Awi resources into Timb time periods, and different solutions in CSi will have different delay costs and air traffic control load. In this section, we will introduce a modified shortest path algorithm based on the backtracking method to select the optimal strategy from CSi for each corridor CORi such that the total delay cost and the air traffic control load can be minimized.To represent all the possible candidate strategies of all corridors, a directed search graphGsV,E is constructed, which contains three sets of vertices V=s∪Vi∪t, where s is the start vertex, Vi is the candidate strategies set of corridor CORi i=1,…,NC, and a strategy solji∈Vi is an array with Timb numbers, and t is the end vertex. Besides, edge set E=s⟶V1∪vi,k⟶Vi+1vi,k∈Vi∪VNC⟶t. In addition, the edge costs are defined as follows:(14)cv,u=Costg+Lg,∀v,u∈s⟶V1∪vi,k⟶Vi+1vi,k∈Vi,0,∀u,v∈VNC⟶t,where Costg and Lg are the delay cost and air traffic control load if corridor CORg uses the strategy that node u represents.Figure7 shows an example of a directed search graph GsV,E.Figure 7
Example of a directed search graphGsV,E.Thus, the optimal strategies for each corridor can be determined by finding the shortest path froms to t on GsV,E, and each node represents a strategy.To speed up the search procedure of the strategies for all corridors with minimization of the total delay cost and air traffic control load, we propose a modified shortest path algorithm based on the backtracking method [30] to select the allocation solution for each corridor while meeting the constraint that the corridor capacity of each time period cannot exceed the corresponding normal level, Kwi,j<Kni.Once the modified shortest path algorithm [30] is finished, each node in the returned shortest path represents the best solution, that is, Kwi,j, 1≤i≤NC, 1≤j≤Timb. And, according to formula (5), we can determine the minimal time interval tinti,j between every two adjacent flights from the same corridor CORi and time period TPj.


## 4.1. Allocation of Sector Capacity

In this work, we first take the sector capacity ofTimb time periods under convection weather as a whole, that is, SC=∑t=1TimbCwt; thus, the strategy generation problem can be regarded as a classic resource allocation problem that allocates SC for NC corridors with minimization of total delay cost. And, in this work, a dynamical programming-based method is proposed to solve this resource allocation problem such that the capacity resources Awi of each corridor CORi, i=1,…,NC, can be determined.Figure5 shows an example of allocating sector capacity, where the sector has four corridors and SC=Cw1+Cw2+Cw3. Our goal is to determine the capacity resources Aw1, Aw2, Aw3, and Aw4.Figure 5
Example of allocating sector capacity.If corridorCORi is allocated to j resources, we can calculate the delay cost vij for corridor CORi as follows:(i)
Firstly, according to the equation (8), the time span of each slot under convection weather and normal condition can be calculated as Timb∗τ/j and τ/Kni, respectively(ii)
Then, through the procedures of allocating time slots in Section3.1, the delay time tdi of each flight fi can be calculated(iii)
At last, the delay costvij for corridor CORi can be calculated according to equation (9)LetVij be the minimum total delay cost when the first i 1≤i≤NC corridors, COR1,…,CORi, are allocated to the total j resources, which can be calculated according to the following two steps:(1)
The firsti-1 corridors, COR1,…,CORi−1, are allocated to the total m m≤j resources(2)
The remainingj−m resources are allocated to corridor CORiThus, the recursive relation formula of dynamical programming can be formulated as follows:(13)Vij=minVij,Vi−1m+vij−m.Algorithm1 gives a detailed description of the dynamical programming-based method to allocate the total sector capacity SC for NC corridors.Algorithm 1: Sector_capacity_allocation.
(1)
Input the total sector capacitySC and the number of corridors NC;
(2)
Initialize the valuesVi0, ∀i∈1,NC;
(3)
for m⟵1 to minSC,Timb⋅Kni do
(4)
V1m=v1m;
(5)
end for
(6)
total_cap
=
0;
(7)
for i⟵2 to NC do
(8)
total_cap+=Timb⋅Kni−1;
(9)
for j⟵1 to SC do
(10)
MIN_COST = INF;
(11)
m_max = minj,total_cap;
(12)
for m⟵0 to m_max do
(13)
if j−m>Timb⋅Kni then
(14)
Continue;
(15)
end if
(16)
if MIN_COST>Vi−1m+vij−m then
(17)
MIN_COST = Vi−1m+vij−m;
(18)
Roadij = m;
(19)
end if
(20)
end for
(21)
Vij = MIN_COST;
(22)
end for
(23)
end for
(24)
//backtrack output results
(25)
j
=
SC;
(26)
for i⟵NCto 1 do
(27)
Awi=j−Roadij;
(28)
j=Roadij;
(29)
end forIn lines 2–5 in Algorithm1, we initialize the values of Vi0, i=1,…,NC, and V1m. From lines 13 to line 15, we check whether allocating j−m flights of Timb time periods to corridor CORi will violate the constraint (3). If the allocated j−m flights meet the constraints and have less delay cost than the current cost recorded by MIN_COST, we will update the minimum delay cost and record the allocated capacity resource for CORi in lines 17-18.Once Algorithm1 is finished, the total allocated capacity resources Awi of Timb time periods for each corridor CORi, i=1,…,NC, can be determined as shown in line 27.


## 4.2. Candidate Strategies for a Corridor

Once the total capacity resourceAwi of Timb time periods for each corridor CORi(i=1,…,NC) is determined, we will use 0-1 combination algorithm to allocate Awi into Timb time periods for each corridor CORi such that the candidate strategies set CSi of each corridor can be determined, where a strategy solji∈CSi is an array with Timb numbers, and each number represents the maximum allowed number of flights entering into sector from CORi in one time period.To simplify the discussion, the total allocated capacity resourceAwi of corridor CORi is relabeled as N; thus, the corridor capacity resource allocation problem can be formulated as the problem of placing Timb−1 baffles between N numbers, which is the combination optimization problem of selecting Timb−1 from N + 1, and, there has CN+1Timb−1 solutions. In this work, we take the 0-1 combination algorithm to search those solutions, and Algorithm 2 shows the overall flow of the 0-1 combination algorithm.Algorithm 2: 0-1 combination algorithm for CORi.
(1)
Input the capacity resourceN and time periods Timb;
(2)
Initialize the arrayindexwith the size of N + 1;
(3)
//set the firstTimb−1 of array to 1, which corresponds to a solution;
(4)
for i⟵0 to Timb−1 do
(5)
index[i] = 1;
(6)
end for
(7)
Store the solution according to the values of arrayindex;
(8)
while has Done (index, N, Timb) do
(9)
for i⟵0 to N + 1 do
(10)
//find the first 1-0 combination and change it to 0-1 combination;
(11)
if index[i] = = 1 and index[i] = = 0 then
(12)
index[i] = 0;
(13)
index[i] = 1;
(14)
Store the solution fromindex;
(15)
//move 1 of the left of 0-1 combination to the left of array;
(16)
int count = 0;
(17)
for j⟵0 to ido
(18)
if index[j] = = 1 then
(19)
index[j] = 0;
(20)
index[count++] = 1;
(21)
end if
(22)
end for
(23)
end if
(24)
break;
(25)
end for
(26)
end whileIn line 2 in Algorithm2, we initialize the array index with the size of N + 1, and indexi is the flag that represents whether the corresponding baffle is placed or not. If indexi=1, there is a baffle between the i-th number and the (i + 1)-th number. Otherwise, the i-th number and the (i + 1)-th number are allocated to the same time period.The combination optimization problem is solved in lines 8–26, and the main steps of 0-1 combination algorithm are listed as follows:(1)
Set the firstTimb−1 numbers of array index to 1 as shown in lines 4–6, which corresponds to a solution(2)
Find the first 1-0 combination of arrayindex and change it to 0-1 combination as shown in line 11–13(3)
Move all 1 of the left of 0-1 combination to the left of arrayRepeat Steps 2 and 3 until the lastTimb−1 numbers of array index are 1.Once the 0-1 combination algorithm is finished, we can get the candidate strategies setCSi for each corridor CORi (i=1,…,NC) according to the values of array index.Figure6 shows an example of determining candidate strategies CS1 for corridor COR1 using the 0-1 combination algorithm, where Aw1 = 3 and Timb = 3. Firstly, we define an array index with size of 4, and according to Steps 4–25 of Algorithm 2, we can find CS1=6 possible solutions for the combination problem of selecting Timb−1 from Aw1+1, as shown in Figure 6(a). Secondly, for each solution, according to the values of array index, we can get the positions of baffles placed between numbers. Here, we take the third solution (“0110”), for example, because index1 = 1, and we place a baffle between the first number and the second number. Similarly, a baffle is placed between the second number and the third number as shown in Figure 6(b). At last, we calculate the numbers between adjacent baffles, and each number represents the maximum allowed number of flights entering into sector from CORi in one time period.Figure 6
Example of determining candidate strategies for a corridor using the 0-1 combination algorithm.


## 4.3. Strategy Generation Algorithm

For each corridorCORi (i=1,…,NC), there are CSi (=CAwi+1Timb−1) solutions of allocating Awi resources into Timb time periods, and different solutions in CSi will have different delay costs and air traffic control load. In this section, we will introduce a modified shortest path algorithm based on the backtracking method to select the optimal strategy from CSi for each corridor CORi such that the total delay cost and the air traffic control load can be minimized.To represent all the possible candidate strategies of all corridors, a directed search graphGsV,E is constructed, which contains three sets of vertices V=s∪Vi∪t, where s is the start vertex, Vi is the candidate strategies set of corridor CORi i=1,…,NC, and a strategy solji∈Vi is an array with Timb numbers, and t is the end vertex. Besides, edge set E=s⟶V1∪vi,k⟶Vi+1vi,k∈Vi∪VNC⟶t. In addition, the edge costs are defined as follows:(14)cv,u=Costg+Lg,∀v,u∈s⟶V1∪vi,k⟶Vi+1vi,k∈Vi,0,∀u,v∈VNC⟶t,where Costg and Lg are the delay cost and air traffic control load if corridor CORg uses the strategy that node u represents.Figure7 shows an example of a directed search graph GsV,E.Figure 7
Example of a directed search graphGsV,E.Thus, the optimal strategies for each corridor can be determined by finding the shortest path froms to t on GsV,E, and each node represents a strategy.To speed up the search procedure of the strategies for all corridors with minimization of the total delay cost and air traffic control load, we propose a modified shortest path algorithm based on the backtracking method [30] to select the allocation solution for each corridor while meeting the constraint that the corridor capacity of each time period cannot exceed the corresponding normal level, Kwi,j<Kni.Once the modified shortest path algorithm [30] is finished, each node in the returned shortest path represents the best solution, that is, Kwi,j, 1≤i≤NC, 1≤j≤Timb. And, according to formula (5), we can determine the minimal time interval tinti,j between every two adjacent flights from the same corridor CORi and time period TPj.


## 5. A Special Case

In this section, we consider a situation that the input sector capacities of theTimb time periods are the same, that is, Cw1=Cw2=⋯=CwTimb, and the generated strategies of different time periods for a corridor are also same (that is, for a corridor CORi, Kwi,1=Kwi,2=,⋯,=Kwi,Timb).Figure8 shows an example of a special case. For this case, if the strategy of a time period TPt for each corridor, that is, Kwi,t, i=1,…,NC, is given, the delay cost Costi for corridor CORi, i=1,…,NC, can be determined.Figure 8
An example of the special case.Thus, this special case can be solved by allocating the sector capacity resourceCw1 into NC corridors with minimization of total delay cost. And, a dynamical programming-based method similar to Section 4.1 is used to solve this special case.


## 6. Experiments

### 6.1. Experimental Setup

The proposed method has been implemented in C language on a Linux 64 bit workstation (Intel 2.4 GHz, 256 GB RAM).To explore the effectiveness of the proposed algorithm, we construct a case benchmark by referring to the typical operating day of sector 06 in the terminal area of Beijing as shown in Table4, which has four corridors, COR1, COR2, COR3, and COR4, and the sector capacity Cn is 48 in one time period under normal condition.Table 4
Information of sector and traffic flows.
Corridor
COR
1
COR
2
COR
3
COR
4
NOC (flights/h)
16
12
15
5
Traffic flow (flight number)
20:00-21:00
16
14
20
6
21:00-22:00
17
13
16
6
22:00-23:00
10
5
7
2
23:00-24:00
9
3
5
2
00:00-01:00
5
4
6
2
01:00-02:00
5
4
6
1In Table4, the first row NOC represents the maximum allowed number of flights Kni across the corridor CORi in a time period under normal conditions. The other rows list the traffic flows from each direction across a corridor in different time periods, where the used corridors of flights are predetermined and the sequence of flights are kept unchanged. In addition, the aircraft type, the estimated arrival time of aircraft and the type of passengers are also given.


### 6.2. Results and Analysis

To explore the effectiveness of the proposed generalized method, we perform two other methods, rate-based method and need-based method, which are often used in the actual control practice:(i)
Rate-based method: based on the descending ratio of capacity in time periodTPj(j=1,…,Timb), λj=Cwj/Cn (λj<1), the maximum allowed number of flights Kwi,j of corridor CORi in time period TPj is set λj⋅Kni.(ii)
Need-based method: letβi,j be the rate that the traffic flows of corridor CORi to the total flows of NC corridors in time period TPj. That is, for each time period TPj, ∑i=1NCβi,j=1, the capacity Kwi,j is set βi,j⋅Kni.As a baseline situation, we solve the resource allocation problem using an integer linear programming (“ILP”), and the manual in the study of Gurobi [31] is used as the ILP solver to find the optimal solution.In experiments, we setTimb=2; that is, the sector in 20:00-21:00 and 21:00-22:00 are affected by the convective weather. And, the sector capacities of these two time periods, Cw1 and Cw2, are assumed to be 24 and 28 flights per hour, respectively.Table5 shows the experimental results. AF represents the number of affected flights, whose estimated time of takeoff tei is not equal to the calculated time of takeoff tci. The total delay time TD is defined as follows:(15)TD=∑i=1AFtdi.Table 5
Comparison with therate- and need-based methods under Timb=2 (Cw1=24 and Cw2=28).
—
Rate-based method
Need-based method
Three-phase method
“ILP”
Strat.
COR
1
T
0: 8 (4 min/flight), T1: 9 (3 min/flight)
T
0: 6 (5 min/flight), T1: 9 (3 min/flight)
T
0: 8 (4 min/flight), T1: 5 (6 min/flight)
T
0: 9 (3 min/flight), T1: 16 (2 min/flight)
T
2: 16 (2 min/flight), T3: 16 (2 min/flight)
T
2: 16 (2 min/flight), T3: 16 (2 min/flight)
T
2: 16 (2 min/flight), T3: 16 (2 min/flight)
T
2: 16 (2 min/flight), T3: 16 (2 min/flight)
T
4: 16 (2 min/flight)
T
4: 16 (2 min/flight)
T
4: 16 (2 min/flight)
T
4: 16 (2 min/flight)
COR
2
T
0: 7 (4 min/flight), T1: 8 (4 min/flight)
T
0: 7 (4 min/flight), T1: 8 (4 min/flight)
T
0: 4 (8 min/flight), T1: 9 (3 min/flight)
T
0: 4 (8 min/flight), T1: 6 (5 min/flight)
T
2: 12 (3 min/flight), T3: 12 (3 min/flight)
T
2: 12 (3 min/flight), T3: 12 (3 min/flight)
T
2: 12 (3 min/flight), T3: 12 (3 min/flight)
T
2: 12 (3 min/flight), T3: 12 (3 min/flight)
T
4: 12 (3 min/flight)
T
4: 12 (3 min/flight)
T
4: 12 (3 min/flight)
T
4: 12 (3 min/flight)
COR
3
T
0: 7 (4 min/flight), T1: 9 (2 min/flight)
T
0: 8 (4 min/flight), T1: 8 (4 min/flight)
T
0: 9 (3 min/flight), T1: 11 (3 min/flight)
T
0: 8 (4 min/flight), T1: 4 (8 min/flight)
T
2: 15 (2 min/flight), T3: 15 (2 min/flight)
T
2: 15 (2 min/flight), T3: 15 (2 min/flight)
T
2: 15 (2 min/flight), T3: 15 (2 min/flight)
T
2: 15 (2 min/flight), T3: 15 (2 min/flight)
T
4: 15 (2 min/flight)
T
4: 15 (2 min/flight)
T
4: 15 (2 min/flight)
T
4: 15 (2 min/flight)
COR
4
T
0: 2 (15 min/flight), T1: 2 (15 min/flight)
T
0: 3 (10 min/flight), T1: 3 (10 min/flight)
T
0: 3 (10 min/flight), T1: 3 (10 min/flight)
T
0: 3 (10 min/flight), T1: 2 (15 min/flight)
T
2: 5 (6 min/flight), T3: 5 (6 min/flight)
T
2: 5 (6 min/flight), T3: 5 (6 min/flight)
T
2: 5 (6 min/flight), T3: 5 (6 min/flight)
T
2: 5 (6 min/flight), T3: 5 (6 min/flight)
T
4: 5 (6 min/flight)
T
4: 5 (6 min/flight)
T
4: 5 (6 min/flight)
T
4: 5 (6 min/flight)
Total cost
1156187 (1.081)
1136662 (1.063)
1069118 (1.0)
1053433 (0.985)
AF
121
121
118
123
TD (min.)
4981
4899
4852
4830
AD (min.)
41.2
40.5
41.1
39.3And, the average delay timeAD is defined as AD=TD/AF.As shown in Table5, compared with the proposed three-phase optimization method, the rate-based method and need-based method will spend more 8.1% and 6.3% of delay cost, respectively, which shows the effectiveness of the proposed method. In addition, the proposed three-phase optimization method can reduce the affected flights AF of 3 aircraft and 3 aircraft on average when compared to the rate-based method and need-based method, respectively.In Table5, the delay cost obtained by the proposed method is a litter higher (1.5%) than the baseline situation of “ILP,” which demonstrates the effectiveness of the proposed method.Table5 also lists the strategy generated by the proposed three-phase method for each directions to control the maximum allowed number of flights and the minimal time interval between every two adjacent flights in different time periods, and the minimal interval between every adjacent flights is determined according to formula (5). Here, we take corridor 2, for example, whose flow control time is five hours and the strategies are listed as follows in detail:(i)
20:00 to 21:00: the maximum allowed number of flights is 4 flights/h, and the minimal time interval between adjacent flights is 8 minutes(ii)
21:00 to 22:00: the maximum allowed number of flights is 9 flights/h, and the minimal interval between adjacent flights is 3 minutes(iii)
22:00 to 23:00: the maximum allowed number of flights is 12 flights/h, and the minimal interval between adjacent flights is 3 minutes(iv)
23:00 to 00:00: the maximum allowed number of flights is 12 flights/h, and the minimal interval between adjacent flights is 3 minutes(v)
00:00 to 01:00: the maximum allowed number of flights is 12 flights/h, and the minimal interval between adjacent flights is 3 minutes


### 6.3. Impacts of Sector Capacity onE-MIT

Further to demonstrate the effectiveness of the proposed three-phase optimization method, we perform the comparison experiments on different sector capacities of these two time periods, whereCw1 is set to 30 and Cw2 is ranged from 25 to 35. Figure 9 shows the changing trend of the delay cost, the number of affected flights, air traffic control load, and average delay time with changing Cw2 under the same Cw1. Form Figure 9, it can be seen that the proposed three-phase method can generate the control strategy with the least delay cost and the least air traffic load when compared to the rate-based method and need-based method. And, as shown in Figures 9(b) and 9(d), the proposed three-phase method can achieve the least average delay time with a little more of affected flights.Figure 9
The comparison results of (a) delay cost, (b) the number of affected flights, (c) air traffic control load, and (d) average delay time under the different sector capacity, whereKw1 is set to 30 and Kw2 is ranged from 25 to 35.
(a)
(b)
(c)
(d)


### 6.4. Impacts ofTimb on E-MIT

To evaluate the impact ofTimb on the generated strategy E-MIT, we perform the experiments on the test benchmark as shown in Figure 10, where the duration time periods of convection weather Timb ranged from 2 to 5, and in each time period, the sector capacity Cwt, t=1,…,Timb, is set to 31.Figure 10
Impacts ofTimb on the generated strategy E-MIT, where Timb ranges from 2 to 5. (a) The results of delay cost. (b) The results of delay time.
(a)
(b)As shown in Figure10, with the increasing Timb, the delay cost and average delay time are increased, which is consistent with the actual situation. What is more, compared with the rate-based method and need-based method, the proposed three-phase optimization method can achieve the least delay cost and delay time.


### 6.5. Results of the Special Case

In this section, to demonstrate the effectiveness of the proposed dynamical programming-based method for the special case, we perform experiments with the rate-based method and need-based method as shown in Table6, where the sectors in 20:00-21:00 and 21:00-22:00 are affected by the convective weather, and the corresponding sector capacity of these two time periods are both set to be 31 flights per hour.Table 6
Comparison with therate- and need-based methods under T=2 for the special case (Cw1=31 and Cw2=31).
—
Rate-based method
Need-based method
DP-based method
Strategy
COR
1
T
0: 10 (3 min/flight), T1:10 (3 min/flight)
T
0: 10 (3 min/flight), T1: 10 (3 min/flight)
T
0: 8 (3 min/flight), T1: 8 (3 min/flight)
T
2: 16 (2 min/flight), T3: 16 (2 min/flight)
T
2: 16 (2 min/flight), T3: 16 (2 min/flight)
T
2: 16 (2 min/flight), T3: 16 (2 min/flight)
T
4: 16 (2 min/flight)
COR
2
T
0: 9 (4 min/flight), T1: 9 (4 min/flight)
T
0: 8 (4 min/flight), T1: 8 (4 min/flight)
T
0: 8 (4 min/flight), T1: 8 (4 min/flight)
T
2: 12 (3 min/flight), T3: 12 (3 min/flight)
T
2: 12 (3 min/flight), T3: 12 (3 min/flight)
T
2: 12 (3 min/flight), T3: 12 (3 min/flight)
T
4: 12 (3 min/flight)
COR
3
T
0: 9 (4 min/flight), T1: 9 (4 min/flight)
T
0: 10 (3 min/flight), T1: 10 (3 min/flight)
T
0: 12 (3 min/flight), T1: 12 (3 min/flight)
T
2: 15 (2 min/flight), T3: 15 (2 min/flight)
T
2: 15 (2 min/flight), T3: 15 (2 min/flight)
T
2: 15 (2 min/flight), T3: 15 (2 min/flight)
T
4: 15 (2 min/flight)
COR
4
T
0: 3 (10 min/flight), T1: 3 (10 min/flight)
T
0: 3 (10 min/flight), T1: 3 (10 min/flight)
T
0: 3 (10 min/flight), T1: 3 (10 min/flight)
T
2: 5 (6 min/flight), T3: 5 (6 min/flight)
T
2: 5 (6 min/flight), T3: 5 (6 min/flight)
T
2: 5 (6 min/flight), T3: 5 (6 min/flight)
T
4: 5 (6 min/flight)
Total cost
787041 (1.102)
768141 (1.075)
714464 (1.0)
AF
101
101
104
TD (min.)
3341
3307
3236
AD (min.)
33.1
32.7
31.1From Table6, it can be seen that, compared with the proposed DP-based method, the rate-based method and need-based method will spend more 10.2% and 7.5% of delay cost, respectively, which shows the effectiveness of the proposed method for the special case. In addition, the proposed DP-based method can save the average delay AD of 2.0 minutes and 1.6 minutes on average when compared to the rate-based method and need-based method, respectively.In addition, we also compare the DP-based method with the proposed three-phase method as shown in Table7. In this experiment, the sector capacities Cw1 and Cw2 are set to the same values, which are ranging from 15 to 35 flights per hour. As shown in Table 7, compared with the DP-based method, the proposed three-phase method can save 2% delay cost because it could explore larger solution space, while the DP-based method assumes that the generated strategies of different time periods for a corridor to be same; thus, the air traffic control load is zero as shown in Table 7.Table 7
Comparison with the proposed three-phase method.
C
w
1
C
w
2
DP-based method
Three-phase method
T.Cost
A.Flight
T.Delay
A.Delay
Load
T.Cost
A.Flight
T.Delay
A.Delay
Load
15
15
2028187
141
8907
63.2
0
1986784
141
8912
63.2
615
16
16
1931099
141
8491
60.2
0
1899271
139
8512
61.2
500
17
17
1834914
140
8074
57.7
0
1810957
139
8099
58.3
467
18
18
1739630
140
7660
54.7
0
1721609
133
7657
57.6
444
19
19
1650660
136
7318
53.8
0
1633627
130
7317
56.3
525
20
20
1561976
138
6869
49.8
0
1535481
130
6857
52.7
308
21
21
1473005
134
6527
48.7
0
1447824
126
6496
51.6
279
22
22
1385355
131
6140
46.9
0
1368293
124
6129
49.4
238
23
23
1298882
130
5759
44.3
0
1275632
122
5801
47.6
223
24
24
1216527
129
5471
42.4
0
1193387
120
5376
44.8
238
25
25
1137924
127
5143
40.5
0
1107526
118
5046
42.8
227
26
26
1061984
117
4748
40.6
0
1043709
117
4731
40.4
218
27
27
983381
115
4420
38.4
0
970058
109
4369
40.1
159
28
28
910680
113
4102
36.3
0
914595
108
4076
37.7
122
29
29
843689
112
3835
34.2
0
848398
106
3785
35.7
111
30
30
778042
105
3526
33.6
0
775967
104
3511
33.8
108
31
31
714465
104
3236
31.1
0
708205
103
3251
31.6
101
32
32
651009
103
2982
29.0
0
643665
102
2974
29.2
104
33
33
587866
101
2695
26.7
0
578850
101
2691
26.6
87
34
34
526066
98
2454
25.0
0
518742
98
2390
24.4
108
35
35
476363
94
2180
23.2
0
468282
96
2139
22.3
69
Average
1.0
121
5263.7
41.9
0
0.98
117
5243.7
43.2
249.5


## 6.1. Experimental Setup

The proposed method has been implemented in C language on a Linux 64 bit workstation (Intel 2.4 GHz, 256 GB RAM).To explore the effectiveness of the proposed algorithm, we construct a case benchmark by referring to the typical operating day of sector 06 in the terminal area of Beijing as shown in Table4, which has four corridors, COR1, COR2, COR3, and COR4, and the sector capacity Cn is 48 in one time period under normal condition.Table 4
Information of sector and traffic flows.
Corridor
COR
1
COR
2
COR
3
COR
4
NOC (flights/h)
16
12
15
5
Traffic flow (flight number)
20:00-21:00
16
14
20
6
21:00-22:00
17
13
16
6
22:00-23:00
10
5
7
2
23:00-24:00
9
3
5
2
00:00-01:00
5
4
6
2
01:00-02:00
5
4
6
1In Table4, the first row NOC represents the maximum allowed number of flights Kni across the corridor CORi in a time period under normal conditions. The other rows list the traffic flows from each direction across a corridor in different time periods, where the used corridors of flights are predetermined and the sequence of flights are kept unchanged. In addition, the aircraft type, the estimated arrival time of aircraft and the type of passengers are also given.


## 6.2. Results and Analysis

To explore the effectiveness of the proposed generalized method, we perform two other methods, rate-based method and need-based method, which are often used in the actual control practice:(i)
Rate-based method: based on the descending ratio of capacity in time periodTPj(j=1,…,Timb), λj=Cwj/Cn (λj<1), the maximum allowed number of flights Kwi,j of corridor CORi in time period TPj is set λj⋅Kni.(ii)
Need-based method: letβi,j be the rate that the traffic flows of corridor CORi to the total flows of NC corridors in time period TPj. That is, for each time period TPj, ∑i=1NCβi,j=1, the capacity Kwi,j is set βi,j⋅Kni.As a baseline situation, we solve the resource allocation problem using an integer linear programming (“ILP”), and the manual in the study of Gurobi [31] is used as the ILP solver to find the optimal solution.In experiments, we setTimb=2; that is, the sector in 20:00-21:00 and 21:00-22:00 are affected by the convective weather. And, the sector capacities of these two time periods, Cw1 and Cw2, are assumed to be 24 and 28 flights per hour, respectively.Table5 shows the experimental results. AF represents the number of affected flights, whose estimated time of takeoff tei is not equal to the calculated time of takeoff tci. The total delay time TD is defined as follows:(15)TD=∑i=1AFtdi.Table 5
Comparison with therate- and need-based methods under Timb=2 (Cw1=24 and Cw2=28).
—
Rate-based method
Need-based method
Three-phase method
“ILP”
Strat.
COR
1
T
0: 8 (4 min/flight), T1: 9 (3 min/flight)
T
0: 6 (5 min/flight), T1: 9 (3 min/flight)
T
0: 8 (4 min/flight), T1: 5 (6 min/flight)
T
0: 9 (3 min/flight), T1: 16 (2 min/flight)
T
2: 16 (2 min/flight), T3: 16 (2 min/flight)
T
2: 16 (2 min/flight), T3: 16 (2 min/flight)
T
2: 16 (2 min/flight), T3: 16 (2 min/flight)
T
2: 16 (2 min/flight), T3: 16 (2 min/flight)
T
4: 16 (2 min/flight)
T
4: 16 (2 min/flight)
T
4: 16 (2 min/flight)
T
4: 16 (2 min/flight)
COR
2
T
0: 7 (4 min/flight), T1: 8 (4 min/flight)
T
0: 7 (4 min/flight), T1: 8 (4 min/flight)
T
0: 4 (8 min/flight), T1: 9 (3 min/flight)
T
0: 4 (8 min/flight), T1: 6 (5 min/flight)
T
2: 12 (3 min/flight), T3: 12 (3 min/flight)
T
2: 12 (3 min/flight), T3: 12 (3 min/flight)
T
2: 12 (3 min/flight), T3: 12 (3 min/flight)
T
2: 12 (3 min/flight), T3: 12 (3 min/flight)
T
4: 12 (3 min/flight)
T
4: 12 (3 min/flight)
T
4: 12 (3 min/flight)
T
4: 12 (3 min/flight)
COR
3
T
0: 7 (4 min/flight), T1: 9 (2 min/flight)
T
0: 8 (4 min/flight), T1: 8 (4 min/flight)
T
0: 9 (3 min/flight), T1: 11 (3 min/flight)
T
0: 8 (4 min/flight), T1: 4 (8 min/flight)
T
2: 15 (2 min/flight), T3: 15 (2 min/flight)
T
2: 15 (2 min/flight), T3: 15 (2 min/flight)
T
2: 15 (2 min/flight), T3: 15 (2 min/flight)
T
2: 15 (2 min/flight), T3: 15 (2 min/flight)
T
4: 15 (2 min/flight)
T
4: 15 (2 min/flight)
T
4: 15 (2 min/flight)
T
4: 15 (2 min/flight)
COR
4
T
0: 2 (15 min/flight), T1: 2 (15 min/flight)
T
0: 3 (10 min/flight), T1: 3 (10 min/flight)
T
0: 3 (10 min/flight), T1: 3 (10 min/flight)
T
0: 3 (10 min/flight), T1: 2 (15 min/flight)
T
2: 5 (6 min/flight), T3: 5 (6 min/flight)
T
2: 5 (6 min/flight), T3: 5 (6 min/flight)
T
2: 5 (6 min/flight), T3: 5 (6 min/flight)
T
2: 5 (6 min/flight), T3: 5 (6 min/flight)
T
4: 5 (6 min/flight)
T
4: 5 (6 min/flight)
T
4: 5 (6 min/flight)
T
4: 5 (6 min/flight)
Total cost
1156187 (1.081)
1136662 (1.063)
1069118 (1.0)
1053433 (0.985)
AF
121
121
118
123
TD (min.)
4981
4899
4852
4830
AD (min.)
41.2
40.5
41.1
39.3And, the average delay timeAD is defined as AD=TD/AF.As shown in Table5, compared with the proposed three-phase optimization method, the rate-based method and need-based method will spend more 8.1% and 6.3% of delay cost, respectively, which shows the effectiveness of the proposed method. In addition, the proposed three-phase optimization method can reduce the affected flights AF of 3 aircraft and 3 aircraft on average when compared to the rate-based method and need-based method, respectively.In Table5, the delay cost obtained by the proposed method is a litter higher (1.5%) than the baseline situation of “ILP,” which demonstrates the effectiveness of the proposed method.Table5 also lists the strategy generated by the proposed three-phase method for each directions to control the maximum allowed number of flights and the minimal time interval between every two adjacent flights in different time periods, and the minimal interval between every adjacent flights is determined according to formula (5). Here, we take corridor 2, for example, whose flow control time is five hours and the strategies are listed as follows in detail:(i)
20:00 to 21:00: the maximum allowed number of flights is 4 flights/h, and the minimal time interval between adjacent flights is 8 minutes(ii)
21:00 to 22:00: the maximum allowed number of flights is 9 flights/h, and the minimal interval between adjacent flights is 3 minutes(iii)
22:00 to 23:00: the maximum allowed number of flights is 12 flights/h, and the minimal interval between adjacent flights is 3 minutes(iv)
23:00 to 00:00: the maximum allowed number of flights is 12 flights/h, and the minimal interval between adjacent flights is 3 minutes(v)
00:00 to 01:00: the maximum allowed number of flights is 12 flights/h, and the minimal interval between adjacent flights is 3 minutes


## 6.3. Impacts of Sector Capacity onE-MIT

Further to demonstrate the effectiveness of the proposed three-phase optimization method, we perform the comparison experiments on different sector capacities of these two time periods, whereCw1 is set to 30 and Cw2 is ranged from 25 to 35. Figure 9 shows the changing trend of the delay cost, the number of affected flights, air traffic control load, and average delay time with changing Cw2 under the same Cw1. Form Figure 9, it can be seen that the proposed three-phase method can generate the control strategy with the least delay cost and the least air traffic load when compared to the rate-based method and need-based method. And, as shown in Figures 9(b) and 9(d), the proposed three-phase method can achieve the least average delay time with a little more of affected flights.Figure 9
The comparison results of (a) delay cost, (b) the number of affected flights, (c) air traffic control load, and (d) average delay time under the different sector capacity, whereKw1 is set to 30 and Kw2 is ranged from 25 to 35.
(a)
(b)
(c)
(d)


## 6.4. Impacts ofTimb on E-MIT

To evaluate the impact ofTimb on the generated strategy E-MIT, we perform the experiments on the test benchmark as shown in Figure 10, where the duration time periods of convection weather Timb ranged from 2 to 5, and in each time period, the sector capacity Cwt, t=1,…,Timb, is set to 31.Figure 10
Impacts ofTimb on the generated strategy E-MIT, where Timb ranges from 2 to 5. (a) The results of delay cost. (b) The results of delay time.
(a)
(b)As shown in Figure10, with the increasing Timb, the delay cost and average delay time are increased, which is consistent with the actual situation. What is more, compared with the rate-based method and need-based method, the proposed three-phase optimization method can achieve the least delay cost and delay time.


## 6.5. Results of the Special Case

In this section, to demonstrate the effectiveness of the proposed dynamical programming-based method for the special case, we perform experiments with the rate-based method and need-based method as shown in Table6, where the sectors in 20:00-21:00 and 21:00-22:00 are affected by the convective weather, and the corresponding sector capacity of these two time periods are both set to be 31 flights per hour.Table 6
Comparison with therate- and need-based methods under T=2 for the special case (Cw1=31 and Cw2=31).
—
Rate-based method
Need-based method
DP-based method
Strategy
COR
1
T
0: 10 (3 min/flight), T1:10 (3 min/flight)
T
0: 10 (3 min/flight), T1: 10 (3 min/flight)
T
0: 8 (3 min/flight), T1: 8 (3 min/flight)
T
2: 16 (2 min/flight), T3: 16 (2 min/flight)
T
2: 16 (2 min/flight), T3: 16 (2 min/flight)
T
2: 16 (2 min/flight), T3: 16 (2 min/flight)
T
4: 16 (2 min/flight)
COR
2
T
0: 9 (4 min/flight), T1: 9 (4 min/flight)
T
0: 8 (4 min/flight), T1: 8 (4 min/flight)
T
0: 8 (4 min/flight), T1: 8 (4 min/flight)
T
2: 12 (3 min/flight), T3: 12 (3 min/flight)
T
2: 12 (3 min/flight), T3: 12 (3 min/flight)
T
2: 12 (3 min/flight), T3: 12 (3 min/flight)
T
4: 12 (3 min/flight)
COR
3
T
0: 9 (4 min/flight), T1: 9 (4 min/flight)
T
0: 10 (3 min/flight), T1: 10 (3 min/flight)
T
0: 12 (3 min/flight), T1: 12 (3 min/flight)
T
2: 15 (2 min/flight), T3: 15 (2 min/flight)
T
2: 15 (2 min/flight), T3: 15 (2 min/flight)
T
2: 15 (2 min/flight), T3: 15 (2 min/flight)
T
4: 15 (2 min/flight)
COR
4
T
0: 3 (10 min/flight), T1: 3 (10 min/flight)
T
0: 3 (10 min/flight), T1: 3 (10 min/flight)
T
0: 3 (10 min/flight), T1: 3 (10 min/flight)
T
2: 5 (6 min/flight), T3: 5 (6 min/flight)
T
2: 5 (6 min/flight), T3: 5 (6 min/flight)
T
2: 5 (6 min/flight), T3: 5 (6 min/flight)
T
4: 5 (6 min/flight)
Total cost
787041 (1.102)
768141 (1.075)
714464 (1.0)
AF
101
101
104
TD (min.)
3341
3307
3236
AD (min.)
33.1
32.7
31.1From Table6, it can be seen that, compared with the proposed DP-based method, the rate-based method and need-based method will spend more 10.2% and 7.5% of delay cost, respectively, which shows the effectiveness of the proposed method for the special case. In addition, the proposed DP-based method can save the average delay AD of 2.0 minutes and 1.6 minutes on average when compared to the rate-based method and need-based method, respectively.In addition, we also compare the DP-based method with the proposed three-phase method as shown in Table7. In this experiment, the sector capacities Cw1 and Cw2 are set to the same values, which are ranging from 15 to 35 flights per hour. As shown in Table 7, compared with the DP-based method, the proposed three-phase method can save 2% delay cost because it could explore larger solution space, while the DP-based method assumes that the generated strategies of different time periods for a corridor to be same; thus, the air traffic control load is zero as shown in Table 7.Table 7
Comparison with the proposed three-phase method.
C
w
1
C
w
2
DP-based method
Three-phase method
T.Cost
A.Flight
T.Delay
A.Delay
Load
T.Cost
A.Flight
T.Delay
A.Delay
Load
15
15
2028187
141
8907
63.2
0
1986784
141
8912
63.2
615
16
16
1931099
141
8491
60.2
0
1899271
139
8512
61.2
500
17
17
1834914
140
8074
57.7
0
1810957
139
8099
58.3
467
18
18
1739630
140
7660
54.7
0
1721609
133
7657
57.6
444
19
19
1650660
136
7318
53.8
0
1633627
130
7317
56.3
525
20
20
1561976
138
6869
49.8
0
1535481
130
6857
52.7
308
21
21
1473005
134
6527
48.7
0
1447824
126
6496
51.6
279
22
22
1385355
131
6140
46.9
0
1368293
124
6129
49.4
238
23
23
1298882
130
5759
44.3
0
1275632
122
5801
47.6
223
24
24
1216527
129
5471
42.4
0
1193387
120
5376
44.8
238
25
25
1137924
127
5143
40.5
0
1107526
118
5046
42.8
227
26
26
1061984
117
4748
40.6
0
1043709
117
4731
40.4
218
27
27
983381
115
4420
38.4
0
970058
109
4369
40.1
159
28
28
910680
113
4102
36.3
0
914595
108
4076
37.7
122
29
29
843689
112
3835
34.2
0
848398
106
3785
35.7
111
30
30
778042
105
3526
33.6
0
775967
104
3511
33.8
108
31
31
714465
104
3236
31.1
0
708205
103
3251
31.6
101
32
32
651009
103
2982
29.0
0
643665
102
2974
29.2
104
33
33
587866
101
2695
26.7
0
578850
101
2691
26.6
87
34
34
526066
98
2454
25.0
0
518742
98
2390
24.4
108
35
35
476363
94
2180
23.2
0
468282
96
2139
22.3
69
Average
1.0
121
5263.7
41.9
0
0.98
117
5243.7
43.2
249.5


## 7. Conclusion

In this work, we consider the MIT strategy generation problem for the situation that a sector withNC corridors is affected by convection weather for Timb time periods. Given the sector capacity Cwt, t=1,…,Timb, under convection weather, we propose a three-phase optimization framework to generate E-MIT strategy to achieve the demand-capacity balance. Firstly, we take the sector capacity of Timb time periods under convection weather as a whole, that is, ∑t=1TimbCwt, and then a dynamical programming-based method is proposed to allocate ∑t=1TimbCwt for NC corridors such that the capacity resources Awi of each corridor CORi, i=1,…,NC, can be determined. Secondly, a 0-1 combination algorithm is used to allocate the capacity resources Awi into Timb time periods for each corridor CORi such that the candidate strategies set CSi of each corridor can be determined, where a strategy solji∈CSi is an array with Timb numbers, and each number represents the maximum allowed number of flights entering into sector from CORi in one time period. Finally, a modified shortest path algorithm based on the backtracking method is taken to select the optimal strategy from CSi for NC corridors such that the total delay cost and air traffic control load are minimized. Additionally, a dynamical programming-based method is proposed to generate the E-MIT strategy for the special case that the sector capacities of different time periods under convection weather are the same, that is, Cw1=Cw2=⋯=CwTimb, and the generated strategies of Timb time periods for a corridor are also the same. Experimental results show that compared with the proposed three-phase optimization method, rate-based method and need-based method will spend more 8.1% and 6.3% of delay cost, respectively. When considering the special case, the experimental results show that compared with the proposed dynamical programming-based method, the rate-based method and need-based method will spend more 10.2% and 7.5% of delay cost, respectively.



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*Source: 6059608-2019-12-26.xml*