Real-time optimal delay minimization algorithms for aircraft on a same runway and dual runways

1 Real-time optimal delay minimization algorithms for aircraft on a same runway and dual runw ays Peng Lin, Haopeng Y ang Abstract—In this paper , scheduling problems of air craft minimizing the total delays on a same runway and on dual runways are studied. In contrast to the algorithms based on mixed-integer optimization models in existing works, where the optimality and the r eal-time performance are usually unable to be dealt with at the same time, our work focuses on the interaction mechanism between aircraft coupling with delays and two real-time optimal algorithms are proposed for the four scheduling problems by fully exploiting the combinations of different classes of aircraft based on parallel computing technology . When 100 aircraft on dual runways is considered, by using the algorithm in this paper , the optimal solution can be obtained within less than 10 seconds, while by using the CPLEX software to solve the mix-integer optimization model, the optimal solution cannot be obtained within 1 hour . Keyw ords—Aircraft scheduling, relev ance, landing and takeoff aircraft, dual runways, delays I . I N T RO D U C T I O N W ith the continuous growth of air transportation, aircraft rescheduling has become more and more im- portant to enhance runway utilization and improve flight punctuality rate in particular when conflicts arise during aircraft operations or large scale flight delays occur . Existing studies on aircraft reschedul- ing can be categorized based on their objectiv e functions: minimizing total delay time [1]–[13], minimizing the total de viation of scheduled takeof f and landing times [14]–[22], minimizing the total delay time for arri v al and departure flights [23]– [27], and minimizing overall operational costs [28]– [30]. Among these objecti ve functions, minimizing total delay time is the most widely studied due to its significant practical relev ance. According to the U.S. Department of T ransportation [31], the flight arri v al delay rate has increased to 21 . 90% in 2024, and the number of tarmac delays-defined as domestic flights e xperiencing delays of more than three hours-rose by 51 . 2% from 2023 to 2024, leading to substantial economic losses and sev ere incon venience for airlines, airports, and passengers. The ef fects of flight delays become e ven more pronounced when airports encounter irregular op- erations, such as se vere weather ev ents or system failures, during which a large number of flights are simultaneously disrupted. In such situations, ef fectiv e rescheduling and reco very strate gies are ur - gently needed to swiftly and orderly restore airport operations and minimize cascading delays through- out the air transportation network. Most of the existing works on aircraft reschedul- ing are based on mixed-inte ger programming (MIP) and the corresponding resolving algorithms can generally be categorized into four categories: MIP- based algorithms [5]–[7], [14]–[16], [26]–[28], dy- namic programming algorithms [8]–[10], [20], [29], [30], heuristic algorithms [21]–[24], and meta- heuristic algorithms [1]–[4], [11]–[13], [17]–[19], [25]. The biggest problem with these algorithms is that the optimality and the real-time performance cannot be dealt with at the same time. Dif ferent from [1]–[30], a ne w theoretical frame work was established and real-time optimal algorithms were proposed for scheduling problem on a same runway and dual runways in [33]. But the objectiv e function considered in [33] is the minimization of overall operational time. Follo wing the framew ork of [33], we consider the total delay time for arri val and departure flights as the objectiv e function, and propose two real-time optimal algorithms for scheduling problem on a same runway and dual runways. Compared to the total operation time, the total delay time need to additionally consider the scheduled operation times to calculate out the delays of all aircraft besides the constraints of time windows and the minimum separation times, which are more complicated to an- alyze. Moreov er , in contrast to most of the existing works [1]–[30], where the obtained solutions were usually not the optimal ones and the errors were also unknown, our algorithms are giv en based on the analysis of the minimum separation times and 2 the technology of parallel computing technology and can be applied in real-time to find an optimal solution for scheduling problems on a same runway and dual runways. In addition, our algorithms can be used to the RECA T systems, which might hav e 6 or more classes of aircraft. This is different from the existing algorithms based on the ICA O separations, where aircraft are usually classified into 3 classes. Notations. The operation A − B represents the set that consists of the elements of A which are not elements of B ; the operation ϕ 0 − ϕ 1 represents the sequence which is obtained through modifying the sequence ϕ 0 by remo ving the aircraft belonging to the sequence ϕ 1 and keeping the orders of the rest aircraft unchanged; the symbol cl i represents the class of aircraft T cf i ; the symbol / represents the meaning of “or”. I I . P RO B L E M F O R M U L A T I O N W ithout considering other airport constraints, in order to ensure the safety of the aircraft, the mini- mum separation time between each aircraft and its leading aircraft is only related to their o wn classes. Suppose that aircraft can be partitioned into η classes in descending order of wak e impact, repre- sented as I = { 1 , 2 , · · · , η } , where η is a positi ve integer , and in general the class 1 usually refers to A380 aircraft. Let P i represent the scheduled takeof f/landing time of aircraft T cf i . Let F ( ϕ, S r ( ϕ ) , P r 0 ) = P n i =1 h i ( ϕ ) denote the total delays for the aircraft sequence ϕ = ⟨ T cf 1 , T cf 2 , · · · , T cf n ⟩ to complete the operation (landing and takeof f) tasks, where S r ( ϕ ) = ( S 1 ( ϕ ) , S 2 ( ϕ ) , · · · , S n ( ϕ )) represents the rescheduled takeof f and landing times of aircraft T cf 1 , T cf 2 , · · · , T cf n with S 1 ( ϕ ) ≤ S 2 ( ϕ ) ≤ · · · ≤ S n ( ϕ ) , P r 0 = ( P 1 , P 2 , · · · , P n ) , h i ( ϕ ) = S i ( ϕ ) − P i if S i ( ϕ ) − P i > 0 and h i ( ϕ ) = 0 if S i ( ϕ ) − P i ≤ 0 . For the sak e of e xpression con venience, when no confusion arises, F ( ϕ, S r ( ϕ ) , P r 0 ) can be abbrevi- ated as F ( ϕ ) or F ( ϕ, S r ( ϕ )) . The purpose of this paper is to find appropriate operation sequence of aircraft and the corresponding takeof f and landing times to minimize the objectiv e function F ( ϕ, S r ( ϕ ) , P r 0 ) , i.e., the total delays of all aircraft, so as to solve the following optimization problem: min F ( ϕ, S r ( ϕ ) , P r 0 ) Subject to S k ∈ T f k = [ f min k , f max k ] , k = 1 , · · · , n, S j − S i ≥ Y ij , ∀ i < j, i, j = 1 , · · · , n, (1) where T f k = [ f min k , f max k ] represents the set of allo wable takeof f or landing times, i.e., the time windo w constraint, for the aircraft T cf k with two constants f min k ≤ f max k , Y ij represents the minimum separation time between an aircraft T cf i and its trailing aircraft T cf j . If all of the landing and takeof f operations are interrupted for a period of time, denoted by [ I d 1 , I d 2 ] for two constants I d 1 < I d 2 , then the time win- do w constraint T f k can be changed into T f k = [ f min k , f max k + I d 2 − I d 1 ] − [ I d 1 , I d 2 ] . For simplicity , we still use T f k = [ f min k , f max k ] to represent the time windo w constraint for each aircraft T f k . Definition 1: (Rele vance) Consider an aircraft sequence ϕ = ⟨ T cf 1 , T cf 2 , · · · , T cf n ⟩ . Let S ij = S j − S i represent the takeoff or landing separation time between an aircraft T cf i and its trailing aircraft T cf j for i < j . If S ij = Y ij , i < j , it is said that aircraft T cf j is rele vant to the aircraft T cf i . I I I . S O M E N E C E S S A RY A S S U M P T I O N S A N D D E FI N I T I O N S In this section, we give some necessary assump- tions and definitions, which are first introduced in [33], to make preparations for the main theoretical results and algorithms. A. Landing sequences Let T ij represent the minimum separation time between an aircraft of class i and a trailing aircraft of class j on a single runway without considering the influence of other aircraft. Let T 0 denote the minimum value of all possible separation times between aircraft, which is usually taken as 1 minute. Based on the landing separation time standards at Heathro w Airport and the understanding of the physical landing process, we propose the following assumptions. Assumption 1: (1) For i = 1 , 2 , T ii = 1 . 5 T 0 . (2) For i = ρ 1 , ρ 2 , T ii = T 0 + δ , where T 0 / 8 < δ < T 0 / 6 is a positiv e integer , ρ 1 = 3 and ρ 2 = 5 . (3) For i  = 1 , 2 , ρ 1 , ρ 2 , T ii = T 0 . 3 Assumption 2: (1) T 21 = 1 . 5 T 0 . (2) For i > j and i  = 2 , T ij = T 0 . Assumption 3: (1) For all i < k < j , T ik ≤ T ij ≤ 3 T 0 and T kj < T ij ≤ 3 T 0 . (2) For all k ≤ j ≤ i , T ik < T ij + T j k and T ki < T j i + T kj . Definition 2: (Breakpoint aircraft) Consider a landing/takeof f sequence ϕ = ⟨ T cf 1 , T cf 2 , · · · , T cf n ⟩ . If the classes of two consecutiv e aircraft satisfy that cl i < cl i +1 , it is said that the aircraft T cf i is a breakpoint aircraft of ϕ . Definition 3: (Resident-point aircraft) Con- sider a landing or takeof f sequence ϕ = ⟨ T cf 1 , T cf 2 , · · · , T cf n ⟩ . If S 1 > t 0 , T cf 1 is called a resident-point aircraft of ϕ , and S 1 − t 0 is called the resident time of T cf 1 . If the aircraft T cf i is not rele vant to the aircraft T cf i − 1 , i.e., S ( i − 1) i − Y ( i − 1) i > 0 , for i = 2 , 3 , · · · , n , it is said that T cf i is a resident-point of ϕ and S ( i − 1) i − Y ( i − 1) i is the resident time of T cf i . Consider a mixed landing and takeof f sequence ϕ = ⟨ T cf 1 , T cf 2 , · · · , T cf n ⟩ . Let µ 1 and µ 2 be the lar gest inte gers smaller than i such that air - craft T cf µ 1 is a landing aircraft and aircraft T cf µ 2 is a tak eoff aircraft. If the aircraft T cf i is not relev ant to the aircraft T cf µ 1 and T cf µ 2 , i = 3 , 4 , · · · , n , it is said that T cf i is a resident-point aircraft of ϕ and min { S µ 1 i ( ϕ ) − Y µ 1 i , S µ 2 i ( ϕ ) − Y µ 2 i } is the resident time of T cf i . Assumption 4: (1) For k = ρ 2 , T ( k − 1) k = T 0 + δ . For k  = ρ 2 , T ( k − 1) k ≥ 1 . 5 T 0 . (2) For all i ≤ k , when ( i, k )  = ( ρ 2 , ρ 2 ) , T ( i − 1) k − T ik > 2 δ . (3) For k = 1 , T k ( k +1) > 2 T 0 , and for k = 2 , T k ( k +1) > 1 . 5 T 0 + 2 δ . (4) For k = 1 , all k + 2 ≤ h ≤ η , and all h ≤ j ≤ η , T kj − T hj > 0 . 5 T 0 . (5) Let E = {⟨ 3 , 4 ⟩ , ⟨ 3 , η ⟩} be a sequence set such that 0 . 5 T 0 − δ ≤ T 2 j − T kj < 0 . 5 T 0 for all ⟨ k , j ⟩ ∈ E and for all ⟨ k , j ⟩ / ∈ E with 2 < k < j , T 2 j − T kj > 0 . 5 T 0 . Definition 4: (Class-monotonically-decreasing sequence) If a landing/takeof f sequence ϕ = ⟨ T cf 1 , T cf 2 , · · · , T cf n ⟩ satisfies cl 1 ≥ cl 2 ≥ · · · ≥ cl n , then the aircraft sequence ϕ is called a class- monotonically-decreasing sequence. B. T akeoff sequences Let D ij represent the minimum separation time between an aircraft of class i and a trailing aircraft of class j on a single runway without considering the influence of other aircraft. Assumption 5: (1) When i = 1 , 2 , 3 , η , D ii = (1 + 1 / 3) T 0 . (2) When i  = 1 , 2 , 3 , η , D ii = T 0 . Assumption 6: (1) D 21 = (1 + 1 / 3) T 0 . (2) When i > j and i ≥ 3 , D ij = T 0 . Assumption 7: (1) For all k ≤ i ≤ j , D ij ≤ D kj ≤ 3 T 0 and D ki ≤ D kj ≤ 3 T 0 . (2) For all k ≤ j ≤ i , D ik < D ij + D j k , D ki < D j i + D kj . Assumption 8: (1) F or k = 1 , 2 , D k ( k +1) = D kk + T 0 6 . (2) For k = 3 , ρ 2 − 1 , D k ( k +1) = D kk . (3) For k = 1 , D k 3 = D 23 + T 0 / 3 . For k = 1 and all k + 2 ≤ j ≤ η , D kj − D ( k +1) j = 2 T 0 / 3 . (4) For k = 3 , D kη = D ( k +1) η , and for k = ρ 2 − 1 , D kη = D ρ 2 ρ 2 + T 0 . (5) For all 2 ≤ k < j and j ≥ 3 such that ( k , j )  = (3 , η ) , ( k , j )  = ( ρ 2 − 1 , ρ 2 ) and ( k , j )  = ( ρ 2 − 2 , ρ 2 ) , D kj = D ( k +1) j + T 0 / 3 . C. Mixed landing and takeof f sequences on a same runway In this section, the scheduling of mixed take- of fs and landings on a same way for aircraft is discussed. Let D T denote the minimum separation time between a landing aircraft and a leading tak eoff aircraft. Let T D denote the minimum separation time between a takeof f aircraft and a leading landing aircraft. Assumption 9: Suppose that T 0 ≤ T D < 1 . 5 T 0 and T 0 ≤ D T < 1 . 5 T 0 . Assumption 10: Consider an aircraft sequence ϕ = ⟨ T cf 1 , T cf 2 , · · · , T cf n ⟩ . Suppose that the air- craft T cf j 1 is relev ant to T cf j 0 , the aircraft T cf j 2 is rele vant to T cf j 1 , Y j 0 j 1 ≥ T D + D T and Y j 1 j 2 ≥ T D + D T . If the aircraft T cf j 3 is relev ant to T cf j 2 , then Y j 2 j 3 < T D + D T . Definition 5: (P ath) Consider a sequence ϕ = ⟨ T cf 1 , T cf 2 , · · · , T cf n ⟩ . For any gi ven aircraft T cf i and T cf j , if there exists an aircraft sub- sequence ⟨ T cf 0 i , T cf 1 i , · · · , T cf ρ i ⟩ for some positi ve integer ρ > 0 such that T cf 0 i = T cf i , T cf ρ i = T cf j and each aircraft T cf h i is relev ant to air- craft T cf h − 1 i , i = 1 , 2 , · · · , ρ , then the sequence ⟨ T cf 0 i , T cf 1 i , · · · , T cf ρ i ⟩ is said to be a path from the aircraft T cf j to the aircraft T cf i . It is assumed by default that each aircraft has a path to itself. Assumption 11: Suppose that the aircraft se- quence ϕ = ⟨ T cf 1 , T cf 2 , · · · , T cf n ⟩ is fixed. The 4 aircraft T cf 2 is rele vant to aircraft T cf 1 , and aircraft T cf i +2 is rele vant to aircraft T cf i +1 or aircraft T cf i , i = 1 , 2 , · · · , n − 2 . Definition 6: Consider an aircraft sequence ⟨ T cf i ,T cf j ⟩ . If the aircraft T cf i is a takeof f aircraft, and the aircraft T cf j is a landing aircraft, it is said that the aircraft sequence forms a takeof f-landing transition at the aircraft T cf i . If the aircraft T cf i is a landing aircraft, and the aircraft T cf j is a takeof f aircraft, it is said that the aircraft sequence forms a landing-takeof f transition at the aircraft T cf i . D. Mixed landing and takeoff air cr aft on dual run- ways In this section, scheduling of mixed landing and takeof f on dual runways whose spacing is no lar ger than 760 m , where all of the landing aircraft lands on one runway and all of the takeof f aircraft take of f from the other runway . Let P D denote the minimum separation time between a takeof f aircraft and a leading landing aircraft. Let D P denote the minimum separation time between a landing aircraft and a leading takeoff aircraft. Assumption 12: Suppose that D P = T 0 , P D = 0 , and for any two aircraft T cf i and T cf j with the same operation tasks, Y ij = T cl i cl j or Y ij = D cl i cl j . Consider a group of landing aircraft and take- of f aircraft operating on dual runways. Let Φ = ⟨ T cf 1 , T cf 2 , · · · , T cf n + m ⟩ denote the whole mixed landing and takeof f sequence on dual runway , Φ 0 = ⟨ T cf 0 1 , T cf 0 2 , · · · , T cf 0 n ⟩ denote the landing aircraft sequence in Φ , Φ 1 = ⟨ T cf 1 1 , T cf 1 2 , · · · , T cf 1 m ⟩ denote the takeof f aircraft sequence in Φ . From the definition of aircraft sequence, it follo ws that S 1 (Φ) ≤ S 2 (Φ) ≤ · · · ≤ S n + m (Φ) , S 1 (Φ 0 ) ≤ S 2 (Φ 0 ) ≤ · · · ≤ S n (Φ 0 ) and S 1 (Φ 1 ) ≤ S 2 (Φ 1 ) ≤ · · · ≤ S m (Φ 1 ) . Assumption 13: Suppose that for each aircraft T cf i ∈ { 2 , 3 , · · · , n + m } in Φ , the aircraft T cf i is rele vant to the aircraft T cf k for k ∈ { T cf i − 4 , T cf i − 3 , T cf i − 2 , T cf i − 1 } . If aircraft T cf i and T cf i − 1 hav e different (takeof f/landing) opera- tion tasks and aircraft T cf i is relev ant to T cf i − 1 for i ∈ { 2 , 3 , · · · , n + m } , there is at least an aircraft T cf j ∈ { T cf 1 , T cf 2 , · · · , T cf k − 1 } such that S j i (Φ) − T 0 < Y j i . Definition 7: (T -block, D-block subsequences) Consider a subsequence of the sequence Φ , denoted by ϕ = ⟨ T cf i 0 , T cf i 0 +1 , · · · , T cf i 1 ⟩ for two integers 0 < i 0 < i 1 ≤ n + m . Let ϕ 0 denote the landing aircraft subsequence of ϕ , and ϕ 1 denote the takeof f aircraft subsequence of ϕ . Suppose that there is a path from aircraft T cf i 1 to aircraft T cf i 0 , the operation (landing/takeoff) tasks of T cf i 0 , T cf i 0 +1 are dif ferent, and T cf i 0 +1 is rele vant to T cf i 0 . (1) Suppose that T cf i 1 − 1 ∈ ϕ 0 , T cf i 1 ∈ ϕ 1 , for the sequence ϕ 0 , there is a path from its last aircraft T cf i 1 − 1 to its first aircraft, and for the sequence ϕ 1 , there is a path from its second last aircraft to its first aircraft. If T cf i 1 is only relev ant to T cf i 1 − 1 , it is said that the sequence ϕ is a T -block subsequence of Φ , and ( | ϕ 1 | − 1) / ( | ϕ 0 | − 1) and S i 1 ( ϕ ) − S i 2 ( ϕ ) − Y i 2 i 1 are the length and the takeof f time increment of T - block subsequence, where T cf i 2 is the second last takeof f aircraft in ϕ . (2) Suppose that T cf i 1 − 1 ∈ ϕ 1 , T cf i 1 ∈ ϕ 0 , for the sequence ϕ 1 , there is a path from its last aircraft T cf i 1 − 1 to its first aircraft, and for the sequence ϕ 0 , there is a path from the second last aircraft to the first aircraft in ϕ 0 . If T cf i 1 is only relev ant to T cf i 1 − 1 ,it is said that the sequence ϕ is a D- block subsequence of Φ , and ( | ϕ 0 | − 1) / ( | ϕ 1 | − 1) and S i 1 ( ϕ ) − S i 2 ( ϕ ) − Y i 2 i 1 are the length and the landing time increment of D-block subsequence, where T cf i 2 is the second last landing aircraft in ϕ . In fact, under Assumption 13, the sequence Φ can be expressed as a group of block subsequences in the form of, e.g., ⟨ T B 1 , T B 2 , D B 1 , T B 3 , D B 2 , D B 3 , T B 4 , T B 5 , · · · ⟩ , where each T B i denotes a T -block subsequence and each D B i denotes a D-block subsequence. Based on the definitions of block subsequences, we can focus on the switching between T -block subsequences and D-block subsequences to study the time increments of the landing sequence and the takeof f sequence of the whole sequence Φ . I V . S O M E T H E O R E T I C A L R E S U LT S In this section, we present some typical theoreti- cal results which might be frequently used when the algorithms proposed in this paper are implemented. Theorem 1: [33] Consider a landing/takeoff se- quence ϕ = ⟨ T cf 1 , T cf 2 , · · · , T cf n ⟩ . Suppose that T f k = [ t 0 , + ∞ ] for all k . Under Assumptions 1-8, if the aircraft T cf j is relev ant to T cf i , then j = i + 1 . Theorem 2: [33] Consider a mixed landing and takeof f sequence ϕ = ⟨ T cf 1 , T cf 2 , · · · , T cf n ⟩ on a 5 same runw ay or dual runways with spacing no lar ger than 760 m. Suppose that T f j = [ t 0 , + ∞ ] for all j . Under Assumptions 1-13, the following statements hold. (1) If aircraft T cf j is rele vant to aircraft T cf i and aircraft T cf i and T cf j hav e different operation tasks, then j = i + 1 . (2) If aircraft T cf j is rele vant to aircraft T cf i and aircraft T cf i and T cf j hav e the same operation tasks, then aircraft T cf i +1 , T cf i +2 , · · · , T cf j − 1 hav e dif ferent operation tasks from aircraft T cf i and T cf j . Remark 1: If an aircraft T cf j is relev ant to its leading aircraft T cf i , then the separation time between aircraft T cf i and T cf j is equal to Y ij . Theorem 3: Consider two sequences Φ a = ⟨ ϕ 1 , T cf i 0 , T cf i 1 , T cf i 2 , ϕ 2 , T cf i 3 , T cf i 4 , ϕ 3 ⟩ , and Φ b = ⟨ ϕ 1 , T cf i 0 , T cf i 2 , ϕ 2 , T cf i 3 , T cf i 1 , T cf i 4 , ϕ 3 ⟩ , where ϕ 1 , ϕ 2 , ϕ 3 are three aircraft subsequences containing m 1 , m 2 and m 3 aircraft. Suppose that each aircraft T cf i is relev ant to its leading aircraft for all i  = 1 , P i > t 0 for all i , and S i (Φ a ) > P i and S i (Φ b ) > P i for all i . Under Assumptions 1-13, the follo wing statements hold. (1) F (Φ a ) − F (Φ b ) = S i 1 (Φ a ) − S i 1 (Φ b ) + D 1 ( m 2 + 2) + D 2 ( m 3 + 1) , where D 1 = Y i 0 i 1 + Y i 1 i 2 − Y i 0 i 2 and D 2 = Y i 0 i 1 + Y i 1 i 2 + Y i 3 i 4 − Y i 0 i 2 − Y i 3 i 1 − Y i 1 i 4 . (2) If D 2 = 0 and D 1 ≤ [ S i 1 (Φ b ) − S i 1 (Φ a )] / ( m 2 + 1) , F (Φ a ) − F (Φ b ) ≤ 0 . Proof: Theorem 5, 7 and this theorem can be proved by calculations and the detailed proofs are omitted. Remark 2: Theorem 3(1) gives a calculation method when the order of an aircraft is adjusted. More detailed discussion can be made according to the specific v alues of m 2 , m 3 , D 1 , D 2 and S i 1 (Φ a ) − S i 1 (Φ b ) which can reduce the computational amount of the algorithms proposed in this paper . Theorem 4: Consider a class-monotonically- decreasing landing/takeof f sequence Φ a = ⟨ T cf 1 , T cf 2 , · · · , T cf n ⟩ . Suppose that each aircraft T cf i is relev ant to its leading aircraft for all i  = 1 , S i ( ϕ ) > P i > 0 for all i ∈ { 1 , 2 , · · · , n } and all possible sequences ϕ . Let E 1 = Y 12 and E i = Y ( i − 1) i + Y i ( i +1) − Y ( i − 1)( i +1) for i ∈ { 2 , · · · , n − 1 } . Under Assumptions 1-13, the follo wing statements hold. (1) Generate a new sequence Φ b by moving T cf j 0 to be between T cf k 0 and T cf k 0 +1 for j 0 < k 0 . If E j 0 = T 0 , then F (Φ b ) ≥ F (Φ a ) . (2) Suppose that Y k ( k +1) ≤ Y ( k +1)( k +2) for all k ∈ { 1 , 2 , · · · , n − 2 } . Then (Φ a , S r (Φ a )) is an optimal solution of the optimization problem (1). Proof: (1) Since Y ij ≥ T 0 for all i, j when land- ing/takeof f aircraft are considered, it follo ws from Theorem 3 that F (Φ b ) ≥ F (Φ a ) . (2) In the following, we only discuss the case where the number of the aircraft of each class in the sequence Φ a is no larger than 1 and the general case can be discussed in a similar way . Suppose that this statement does not hold. That is, (Φ a , S r (Φ a )) is not an optimal solution of the optimization problem (1). Let (Φ c , S r (Φ c )) be an optimal solution of the optimization problem (1), where Φ c = ⟨ T cf e 1 , T cf e 2 , · · · , T cf e n ⟩ . It follows that F (Φ c ) < F (Φ a ) . Suppose that e 1 = 1 , e 2 = 2 , · · · , e b = b, e b +1  = b + 1 < n . Note that Φ a is a class-monotonically-decreasing landing/takeof f sequence and the number of the aircraft of each class in the sequence Φ a is no larger than 1 . Hence, cl e 1 > cl e 2 > · · · > cl e b > cl b +1 > cl e b +1 . It follo ws that d > b + 1 , T cf e d − 1 is a breakpoint aircraft in Φ c , and cl e d > cl e d +1 , where e d = b + 1 . Generate a sequence Φ q = ⟨ T cf q 1 , T cf q 2 , · · · , T cf q n ⟩ by moving T cf b +1 to be between T cf e b and T cf e b +1 in Φ c , where T cf e k = T cf q k for k = 1 , 2 , · · · , b . Note that Y k ( k +1) ≤ Y ( k +1)( k +2) . It is clear that S e i (Φ c ) − S q i (Φ q ) = Y be b +1 − ( Y b ( b +1) + Y ( b +1) e b +1 ) + Y e d − 1 e d + Y e d e d +1 − Y e d − 1 e d − 1 ≥ 0 for i = d + 1 , d + 2 , · · · , n and S e i (Φ c ) − S q i (Φ q ) ≥ 0 for i = b + 1 , b + 2 , · · · , d . From Theorem 3, it can be obtained that F (Φ d ) ≤ F (Φ c ) . By using a similar approach, it can be prov ed that F (Φ a ) ≤ F (Φ c ) , which yields a contradiction. Therefore, (Φ a , S r (Φ a )) is an optimal solution of the optimization problem (1). Theorem 4 gi ves two rules to judge whether the aircraft orders are optimal. Using similar analysis approaches, we can obtain the follo wing two lem- mas. Lemma 1: Consider a landing sequence Φ a = ⟨ ϕ 1 , ϕ 2 ⟩ , where ϕ 1 = ⟨ T cf 1 , T cf 2 , · · · , T cf n 1 ⟩ , ϕ 2 = ⟨ T cf n 1 +1 , T cf n 1 +2 , · · · , T cf n 2 ⟩ , ϕ 1 and ϕ 2 are both class-monotoni- cally-decreasing sequences, cl n 1 < cl n 1 +1 , and n 1 , n 2 are two positi ve in- tegers. Merge the aircraft sequences ϕ 1 and ϕ 2 to form a new class-monotonically-decreasing se- quence Φ b = ⟨ T cf s 1 , T cf s 2 , · · · , T cf s n 1 + n 2 ⟩ . Sup- pose that Assumptions 1-4 hold for each landing aircraft sequence, each aircraft T cf i is relev ant to its leading aircraft for all i ∈ { 2 , 3 , · · · , n 2 } and S i (Φ a ) > P i > 0 and S i (Φ b ) > P i > 0 for all 6 i ∈ { 1 , 2 , · · · , n 2 } , the following statements hold. (1) If ( cl i , cl n 1 , cl n 1 +1 ) = ( ρ 2 , ρ 2 − 1 , ρ 2 ) for some i ∈ { 1 , 2 , · · · , n 1 − 1 } , F (Φ a ) < F (Φ b ) . (2) If cl n 1 / ∈ { 1 , 2 } and ( cl i , cl n 1 , cl n 1 +1 )  = ( ρ 2 , ρ 2 − 1 , ρ 2 ) for all i ∈ { 1 , 2 , · · · , n 1 − 1 } , F (Φ a ) − F (Φ b ) > 0 . (3) If cl n 1 ∈ { 1 , 2 } , F (Φ a ) − F (Φ b ) > 0 . Lemma 2: Consider a takeof f sequence Φ a = ⟨ ϕ 1 , ϕ 2 ⟩ , where ϕ 1 = ⟨ T cf 1 , T cf 2 , · · · , T cf n 1 ⟩ , ϕ 2 = ⟨ T cf n 1 +1 , T cf n 1 +2 , · · · , T cf n 2 ⟩ , ϕ 1 and ϕ 2 are both class-monotoni- cally-decreasing sequences, cl n 1 < cl n 1 +1 , and n 1 > 1 and n 2 > 1 are two positiv e integers. Merge the aircraft sequences ϕ 1 and ϕ 2 to form a new class-monotonically- decreasing sequence Φ b . Suppose that Assumptions 5-8 hold for each takeoff aircraft sequence, each aircraft T cf i is rele v ant to its leading aircraft for all i ∈ { 2 , 3 , · · · , n 2 } and S i (Φ a ) > P i > 0 and S i (Φ b ) > P i > 0 for all i ∈ { 1 , 2 , · · · , n 2 } , the follo wing statements hold. (1) Suppose that ( cl n 1 , cl n 1 +1 ) = ( ρ 2 − 1 , ρ 2 ) . Then F (Φ a ) − F (Φ b ) = 0 . (2) Suppose that ( cl n 1 , cl n 1 +1 , j ) = (3 , 4 , 3) for some j ∈ { n 1 + 2 , n 1 + 3 , · · · , n 1 + n 2 } . Then F (Φ a ) > F (Φ b ) . (3) Suppose that ( cl n 1 , cl n 1 +1 , j ) = ( η − 1 , η , η ) for some j ∈ { 1 , 2 , · · · , n 1 − 1 } . Then F (Φ a ) < F (Φ b ) . (4) Suppose that cl n 2 ≤ 2 and ( cl n 1 , cl n 1 +1 , j ) = (2 , 3 , 3) for some j ∈ { 1 , 2 , · · · , n 1 − 1 } . Then F (Φ a ) = F (Φ b ) . Theorem 5: Consider a landing/takeof f sequence Φ a = ⟨ T cf 1 , T cf 2 , · · · , T cf n ⟩ on a same runway . Generate a new sequence Φ b by moving T cf j 0 to be between T cf k 0 and T cf k 0 +1 for j 0 > k 0 + 1 . Suppose that S 1 (Φ a ) = S 1 (Φ b ) = t 0 and each aircraft T cf i is rele v ant to its leading aircraft for all i  = 1 in Φ a and Φ b . Under Assumptions 1-13, the follo wing statements hold. (1) Suppose that S j 0 (Φ a ) ≤ P j 0 . It follo ws that F (Φ b ) ≥ F (Φ a ) . If P j < S j (Φ b ) for some j ∈ { k 0 + 1 , k 0 + 2 , · · · , j 0 − 1 } , then F (Φ b ) > F (Φ a ) . (2) Suppose that S j 0 (Φ b ) > P j 0 , Y j 0 ( k 0 +1) + Y k 0 j 0 − Y k 0 ( k 0 +1) = T 0 and Y j ( j +1) ≥ T 0 for all j ∈ { 1 , 2 , · · · , n − 1 } . It follows that F (Φ b ) ≤ F (Φ a ) . Further , if Y j 0 ( j 0 +1) + Y ( j 0 − 1) j 0 − Y ( j 0 − 1)( j 0 +1) > T 0 or Y j ( j +1) > T 0 for some j ∈ { k 0 + 1 , k 0 + 2 , · · · , j 0 − 2 } , then F (Φ b ) < F (Φ a ) . Remark 3: Theorem 5(1) shows that if the forward mov ement of an aircraft without delay in sequences would result in nondecreasement of the objecti ve function F ( · ) . Theorem 6: Consider a sequence Φ a = ⟨ T cf 1 , T cf 2 , · · · , T cf n ⟩ on a same runway or dual runways with spacing no larger than 760 m. Suppose that S k ( ϕ ) > P k > 0 for all k ∈ { 1 , 2 , · · · , i } and all possible sequences ϕ . and ( ϕ i , S r ( ϕ i )) is one solution of the optimization problem (1), where ϕ i = ⟨ T cf 1 , T cf 2 , · · · , T cf i ⟩ . Suppose that each aircraft T cf i is relev ant to its leading aircraft for all i  = 1 and i  = j 0 + 1 . If S j ( ϕ i ) + T σ ∈ [ f min j , f max j ] for all j ∈ { j 0 , j 0 + 1 , · · · , i } and the operation times of T cf j 0 − 1 and T cf j 0 are fix ed as S j 0 − 1 ( ϕ i ) and S j 0 ( ϕ i ) + T σ for a constant T σ > 0 , then the function of F ( ϕ, S r ( ϕ ) , P r 0 ) − S ( j 0 − 1) j 0 ( ϕ ) can be minimized by ( ϕ i , S r ( ϕ i )) , where S r ( ϕ i ) = [ S 1 ( ϕ i ) , · · · , S j 0 − 2 ( ϕ i ) , ¯ S j 0 − 1 ( ϕ i ) , ¯ S j 0 ( ϕ i ) , S j 0 +1 ( ϕ i )+ T σ , · · · , S i ( ϕ i ) + T σ ] . Proof: For landing/takeof f sequences, each air- craft is relev ant to only its leading aircraft. Since ¯ S j ( ϕ i ) ∈ [ f min j , f max j ] for all j ∈ { j 0 , j 0 + 1 , · · · , i } and ( ϕ i , S r ( ϕ i )) is one solution of the optimization problem (1) for aircrafts T cf 11 , T cf 12 , · · · , T cf 1 i , the function of F ( ϕ, S r ( ϕ )) − S ( j 0 − 1) j 0 ( ϕ ) can be minimized by ( ϕ i , S r ( ϕ i )) . This theorem actually studies a preserv ation prob- lem of the sequence optimality when some time in- terv al is occupied by some operations, e.g., new air- craft are inserted. Specifically , Theorem 6 discusses the case when the optimality can be maintained. This theorem is v ery useful to analyze the optimality of the sequences to reduce the computation of algorithms when ne w aircrafts are inserted into the original sequences while keeping the orders of its adjacent aircraft unchanged. Theorem 7: Consider a mixed landing and takeof f sequence Φ a = ⟨ ϕ a 1 , ϕ a 2 ⟩ on a same runway or dual runways, where ϕ a 1 = ⟨ T cf 1 , T cf 2 , · · · , T cf m 0 ⟩ for some integer m 0 > 0 , ϕ a 2 = ⟨ T cf m 0 +1 , T cf m 0 +2 , · · · , T cf m 0 + m 1 ⟩ for some integer m 1 > 0 and ϕ a 2 contains m 2 delayed aircraft. Let Φ b = ⟨ ϕ b 1 , ϕ a 2 ⟩ be a sequence with the same group of aircraft as Φ a , where ϕ a 2 contains at least m 2 delayed aircraft. Suppose that each aircraft T cf i is rele v ant to its leading aircraft for all i  = 1 in Φ a and Φ b . Under Assumptions 1-13, if m 2 [ S m 0 +1 (Φ b ) − S m 0 +1 (Φ a )] + F ( ϕ b 1 , S r ( ϕ b 1 )) − F ( ϕ a 1 , S r ( ϕ a 1 )) > 0 , then F (Φ a ) − F (Φ b ) < 0 . Under this theorem, when the number of delayed aircraft in ϕ a 2 is lar ge, by simple calculations similar 7 to Theorem 3, it can be obtained that the operation time of the first aircraft of ϕ a 2 should be minimized when F ( ϕ b 1 , S r ( ϕ b 1 )) − F ( ϕ a 1 , S r ( ϕ a 1 )) is small, which might be equiv alent to be a makespan prob- lem and might be solved by using the algorithm in [33]. Theorem 8: [33] Consider a mixed landing and takeof f sequence Φ a = ⟨ T cf 1 , T cf 2 , T cf 3 , T cf 4 ⟩ on a same runway , where S 12 (Φ a ) ≥ Y 12 , aircraft T cf 3 is relev ant to aircraft T cf 1 or T cf 2 and aircraft T cf 4 is relev ant to aircraft T cf 2 or T cf 3 . Generate a new sequence Φ b by inserting an aircraft T cf 5 to be between aircraft T cf 2 and T cf 3 , where S 1 (Φ a ) = S 1 (Φ b ) , S 2 (Φ a ) = S 2 (Φ b ) , aircraft T cf 5 is relev ant to aircraft T cf 1 or T cf 2 , aircraft T cf 3 is relev ant to aircraft T cf 2 or T cf 5 and aircraft T cf 4 is rele vant to aircraft T cf 5 or T cf 3 . Let ω min (Φ a , T cf 5 ) = min { S 3 (Φ b ) − S 3 (Φ a ) , S 4 (Φ b ) − S 4 (Φ a ) } . The fol- lo wing statements hold. (1) Suppose that aircraft T cf 2 , T cf 3 and T cf 5 are all landing (takeof f) aircraft. It follo ws that ω min (Φ a , T cf 5 ) = Y 25 + Y 53 − Y 23 . (2) Suppose that aircraft T cf 2 and T cf 5 are both landing (takeof f) aircraft and T cf 3 is a takeof f (landing) aircraft. It follo ws that ω min (Φ a , T cf 5 ) ≥ min { T D + D T − Y 13 , T D + D T − Y 24 , 0 } + Y 25 . (3) Suppose that aircraft T cf 3 and T cf 5 are both landing (takeof f) aircraft and T cf 2 is a takeof f (landing) aircraft. It follo ws that ω min (Φ a , T cf 5 ) ≥ min { T D + D T − Y 13 , T D + D T − Y 24 , 0 } + Y 53 . (4) Suppose that aircraft T cf 2 and T cf 3 are both landing (takeof f) aircraft and T cf 5 is a takeof f (landing) aircraft. It follo ws that ω min (Φ a , T cf 5 ) ≥ max { D T + T D − Y 23 , 0 } . Theorem 9: Consider a mixed landing and take- of f sequence Φ = ⟨ T cf 1 , T cf 2 , · · · , T cf n + m ⟩ on dual runways with spacing no larger than 760 m . Suppose that T cf i , T cf i +1 , T cf i +2 , · · · , T cf n + m are the last n + m − i + 1 aircraft of Φ , aircraft T cf i +1 and T cf i +2 are landing aircraft, T cf i , T cf i +3 , T cf i +4 , · · · , T cf n + m are tak eoff air- crafts, and T cf i , T cf i +1 , T cf i +2 , T cf i +3 are part of a T -block subsequence where T cf i +3 is not relev ant to T cf i . Generate a new sequence Φ a by exchang- ing the orders of the aircraft T cf i +2 and T cf i +3 . Suppose that aircraft T cf i +3 is relev ant to aircraft T cf i +2 in Φ , aircraft T cf i +3 is rele vant to aircraft T cf i +1 in Φ a and each T cf k is rele vant to aircraft T cf k − 1 for all k = i + 4 , · · · , n + m in Φ and Φ a . If ( n + m − i − 2)( S i +3 (Φ) − S i +3 (Φ a )) + S i +2 (Φ) − S i +2 (Φ a ) > 0 , then F (Φ) − F (Φ a ) > 0 . This theorem is important and can be used to determine whether the sequence is optimal for the optimization problem (1) by analyzing the aircraft at the tail of the sequence. V . A L G O R I T H M S In this section, we propose algorithms to find the optimal solution for the optimization problem (1) based on the theoretical results gi ven in [33] and this paper . A. Algorithm 1 Algorithm 1. Suppose that there are totally n aircraft, denoted by T cf 01 , T cf 02 , · · · , T cf 0 n , and there is at least a feasible sequence for them to land/takeof f without conflicts. Step 1. Arrange the aircraft in ascending order of their scheduled landing/takeof f times, denoted by ⟨ T cf 11 , T cf 12 , · · · , T cf 1 n ⟩ . Step 2. Search for the optimal sequence ϕ 2 for the optimization problem (1). Step 3. Suppose that ϕ 2 = ⟨ T cf 1 , T cf 2 ⟩ is an optimal sequence of the optimization problem (1) for aircraft T cf 1 and T cf 2 . Search for the optimal sequence of the optimization problem (1) for aircraft T cf 1 , T cf 2 and T cf 13 . Step i , i = 4 , 5 , · · · , n . Suppose that ϕ i − 1 = ⟨ T cf 1 , T cf 2 , · · · , T cf i − 1 ⟩ is an optimal sequence of the optimization problem (1) for aircraft T cf 1 , T cf 2 , · · · , T cf i − 1 . Search for an optimal se- quence of the optimization problem (1) for aircraft T cf 1 , T cf 2 , · · · , T cf i . as follows. ( i − 1) . Insert aircraft T cf 1 i between any two adjacent aircraft in ϕ i − 1 under the time windo w constraints without changing the orders of the air- craft in ϕ i − 1 . Let f i inc be the smallest value of the objecti ve functions F ( · ) among all the generated ne w sequences and ϕ i inc denote the corresponding sequence. ( i − 2) . Construct a sequence set F inc = {⟨ h 1 , h 2 , 1 i, h 3 , h 4 ⟩ | h 1 , h 2 , h 3 , h 4 ∈ { 1 , 2 , · · · , i − 1 }} . Note here that since the minimum separation times are related to only the classes and the opera- tion tasks of the aircraft, we can classify the aircraft to form the set F inc according to the classes and the operation tasks of the aircraft. 8 ( i − 3) . According to the elements of the set F inc , adjust the aircraft orders in the se- quence ϕ i − 1 and insert the aircraft T cf 1 i to gen- erate a ne w sequence ¯ ϕ i under the condition F ( ¯ ϕ i ) < f i inc such that ¯ ϕ i contains the sub- sequence ⟨ T cf h 1 , T cf h 2 , T cf 1 i , T cf h 3 , T cf h 4 ⟩ and ⟨ h 1 , h 2 , h 3 , h 4 ⟩ ∈ F inc . Minimize F ( ¯ ϕ i ) based on the obtained theoretical results, and compare the v alues of all possible F ( ¯ ϕ i ) so as to find the optimal sequence of aircraft T cf 11 , T cf 12 , · · · , T cf 1 i . When landing/takeof f sequences are considered and all aircraft are delayed in all possible sequences, if the aircraft T cf 1 i is the k th aircraft of ¯ ϕ i and the inequality is not satisfied ω min ( ⟨ T cf h 1 , T cf h 2 , T cf h 3 , T cf h 4 ⟩ , T cf 1 i ) × ( i − k ) + h 1 i ( ¯ ϕ i ) < f i inc − F ( ϕ i − 1 ) (2) for any ⟨ h 1 , h 2 , h 3 , h 4 ⟩ ∈ F inc , then F ( ¯ ϕ i ) > f i inc . This fact can be used to reduce the computation amount of the algorithm. In step i. ( i − 3) of Algorithm 1, we can discuss the sequence optimality in the following cases and search for the optimal sequence by comparing the v alues of the objective function F ( · ) of all possible sequences based on parallel computing technology . Case 1.1 The number of breakpoint aircraft in ϕ i is no larger than that in ϕ i − 1 for landing/takeof f sequences. Case 1.2. The number of breakpoint aircraft in ϕ i is larger than that in ϕ i − 1 for landing/takeof f sequences. Case 2.1. The number of landing-takeof f and takeof f-landing transitions in ϕ i is no lar ger than that in ϕ i − 1 for mixed landing and tak eoff sequences on a same runway . Subcase 2.1.1. The number of breakpoint aircraft in ϕ i is no larger than that in ϕ i − 1 . Subcase 2.1.2. The number of breakpoint aircraft in ϕ i is lar ger than that in ϕ i − 1 . Case 2.2. The number of breakpoint aircraft in ϕ 0 i is lar ger than that in ϕ i − 1 for mix ed landing and takeof f sequences on a same runway . Subcase 2.2.1. The number of breakpoint aircraft in ϕ i is no larger than that in ϕ i − 1 . Subcase 2.2.2. The number of breakpoint aircraft in ϕ i is lar ger than that in ϕ i − 1 . When the number of breakpoint aircraft in ϕ i is 2 larger than that in ϕ i − 1 , ϕ i is usually not the optimal sequence when the aircraft subsequence ⟨ T cf i , T cf j ⟩ with cl i = ρ 2 − 1 and cl j = ρ 2 is not in volv ed. Note that ϕ i − 1 might be composed of mul- tiple class-monotonically-decreasing subsequences ϕ 1 , ϕ 2 , · · · , ϕ s . W e can consider the insertion of T cf 1 i on each subsequence ϕ k with some corre- sponding aircraft order adjustments so as to reduce the comple xity of the analysis of the optimality of the sequence. Note that each aircraft might be rele vant to its leading aircraft and nearest aircraft ahead with the same operation task for mix ed landing and take- of f sequences on a same runway . The insertion of the aircraft T cf 1 i might have direct impact on the follo wing two aircraft and be af fected by the preceding two aircraft. So, when the aircraft T cf 1 i is inserted, we might need to consider all possible combinations of 5 consecutiv e aircraft of the form ⟨ T cf h 1 , T cf h 2 , T cf 1 i , T cf h 3 , T cf h 4 ⟩ in the new gen- erated sequence. From Theorem 5, when the number of delayed aircraft in the considered sequence is large, the operation time of the first half of the sequence should be minimized for the optimization problem (1). Based on the obtained theoretical results, es- pecially Theorem 5, we can simplify the genera- tion of the combinations of 5 consecutiv e aircraft of the form ⟨ T cf h 1 , T cf h 2 , T cf 1 i , T cf h 3 , T cf h 4 ⟩ to reduce the computation complexity . More- ov er , since aircraft might be rele vant to only its leading aircraft, then the combination of ⟨ T cf h 1 , T cf h 2 , T cf 1 i , T cf h 3 , T cf h 4 ⟩ can be simpli- fied as ⟨ T cf h 2 , T cf 1 i , T cf h 3 ⟩ in particular for land- ing/takeof f sequences. Due to the constraints of time windows of air - craft, breakpoint aircraft and resident-point aircraft might be generated which might increase the v alue of the objecti ve function F ( · ) . T o analyze the opti- mal sequence of the aircraft, the emphasis should be imposed on the breakpoint aircraft and the resident- point aircraft as well as the separation times be- tween the consecutiv e aircraft of the same class that are lar ger than T 0 by trying not to increase their numbers. Moreov er , when the orders of aircraft need to be adjusted, it is better to fully consider the classes of the aircraft rather than the aircraft themselves. 9 B. Algorithm 2 In the follo wing, we propose an algorithm to deal with the scheduling problem of landing and takeof f aircraft on dual runways with spacing no larger than 760 m . Algorithm 2. Suppose that there are to- tally n + m aircraft composed of n land- ing aircraft and m takeof f aircraft, denoted by T cf 1 , T cf 2 , · · · , T cf n + m , and there is at least a feasible sequence for them to land/takeof f without conflicts. Let T cf 0 1 , T cf 0 2 , · · · , T cf 0 n denote all the landing aircraft, and T cf 1 1 , T cf 1 2 , · · · , T cf 1 m denote all the takeof f aircraft. The main steps are as fol- lo ws. Step 1. Use Algorithm 1 to find the optimal sequence, denoted by Φ a , for the landing aircraft to minimize F (Φ a ) , and the optimal sequence for the takeof f aircraft, denoted by Φ b to minimize F (Φ b ) . Suppose that (Φ , S r (Φ)) is an optimal solution of the optimization problem (1). It is clear that F (Φ) ≥ F (Φ a ) + F (Φ b ) . Step 2. (2.1) According to the landing/takeof f time increments of block subsequences, classify all possible block subsequences into sev eral sets, which can be processed offline. It should be noted that all possible block subse- quences can be obtained by an enumeration method. It should also be noted that when the number of breakpoint aircraft in a block subsequence is fixed, the total number of all possible block subsequences is not large under the minimum separation standards of Heathrow Airport and the RECA T -EU system. This observation can be used to reduce the compu- tational b urden in Step 5 of the algorithm. (2.2) F ocus on the combinations of dif ferent block subsequences and analyze the landing/takeof f time increments between any two consecuti ve blocks, where the last two aircraft of the first block are identical to the first two aircraft of the second block. According to the landing/takeof f time increments of two consecutiv e block subsequences, classify all possible combinations of two consecutiv e subse- quences into several sets. Step 3. (3.1) Match the takeof f aircraft with the landing sequence Φ a to generate a sequence, denoted by Φ c = ⟨ T cf c 1 , T cf c 2 , · · · , T cf c ( n + m ) ⟩ , minimizing F (Φ c ) under the condition that F (Φ c LA , S r LA (Φ c )) = F (Φ a ) , where Φ c LA and S r LA (Φ c ) denote the landing sequence and the corresponding landing time vector in Φ c . (3.2) Let c z denote the number of the aircraft of Φ c which take of f in ( S n (Φ a ) , + ∞ ) . Insert the last c z aircraft of Φ c into the subsequence of Φ c , Φ sub c = ⟨ T cf c 1 , T cf c 2 , · · · , T cf c ( n + m − c z ) ⟩ , to gen- erate a new sequence Φ c 3 and minimize F (Φ c 3 ) without changing the orders of the landing aircraft based on the sets defined Step 2. Note here that the sequence Φ c 3 can contain D-block subsequences. It should be noted that the optimal v alue of the objecti ve function F ( · ) for the optimization problem (1) lies in [ F (Φ a ) + F (Φ b ) , F (Φ c 3 )] , and the final optimal sequence usually contains at least c z D- block subsequences, which can be used as an initial condition for the search in step 4. Step 4. (4.1) Match the landing aircraft with the takeof f sequence Φ b to generate a new sequence, denoted by Φ d = ⟨ T cf d 1 , T cf d 2 , · · · , T cf d ( n + m ) ⟩ , and optimize Φ d such that F (Φ d T A , S r T A (Φ d )) = F (Φ b ) , where Φ d T A and S r T A (Φ d ) denote the takeof f sequence and the corresponding takeoff time v ector in Φ d . (4.2) Let d z denote the number of the aircraft of Φ d which land in ( S m (Φ b ) , + ∞ ) . Insert the last d z aircraft of Φ d into the subsequence of Φ d , Φ sub d = ⟨ T cf d 1 , T cf d 2 , · · · , T cf d ( n + m − d z ) ⟩ , to gen- erate a new sequence Φ d 3 and minimize F (Φ d 3 ) without changing the orders of the takeoff aircraft based on the sets defined Step 2. Note here that the sequence Φ d 3 can contain T -block subsequences. It should be noted that the optimal v alue of the objecti ve function F ( · ) for the optimization problem (1) lies in [ F (Φ a ) + F (Φ b ) , min { F (Φ c 3 ) , F (Φ d 3 ) } ] . Step 5. The main idea of this step is to im- plement a full space search for an optimal solu- tion (Φ c 4 , S r (Φ c 4 )) for the optimization problem (1) under the condition that F (Φ a ) + F (Φ b ) ≤ F (Φ c 4 ) ≤ min { F (Φ c 3 ) , F (Φ d 3 ) } starting from the initial condition that Φ c 4 LA = Φ a or Φ c 4 T A = Φ b . Since different D-block/T -block subsequences might ha ve dif ferent landing/takeof f time incre- ments, we can search for the optimal solutions for the optimization problem (1) by increasing or decreasing the number , the landing/takeof f time increments and the time lengths of D-block/T -block subsequences of the sequence Φ c 4 . Note that the quantity F (Φ c 4 ) − F (Φ a ) − F (Φ b ) might correspond to se veral combinations of the landing/takeof f time increments and the increments of the time lengths of dif ferent groups of D-block/T -block subsequences compared to benchmark landing/takeof f sequences 10 T ABLE I M I N I M U M L A N D I N G S E PA R ATI O N T I M E S I N H E AT H RO W A I R P O RT ( S E C ) T railing Aircraft A B C D E F Leading Aircraft A 90 135 158 158 158 180 B 90 90 113 113 135 158 C 60 60 68 90 90 135 D 60 60 60 60 68 113 E 60 60 60 60 68 90 F 60 60 60 60 60 60 Φ a and Φ b . Then we can obtain all the possible combinations of the landing/takeof f time increments and the increments of the time lengths of D-block/T - block subsequences such that the increments of the total delays are no lar ger than the quantity F (Φ c 4 ) − F (Φ a ) − F (Φ b ) . Then based on the ob- tained combinations, we can search for the optimal solutions for the optimization problem (1) within [ F (Φ a ) + F (Φ b ) , min { F (Φ c 3 ) , F (Φ d 3 ) } ] . T o implement the full space search in this step, we can fix the orders of the landing/takeof f aircraft and make discussions based on the number of breakpoints. Moreov er , in this step, we can use the paral- lel computing technology to reduce the computing time. Remark 4: The main idea of Algorithm 2 is to decompose the aircraft sequence into sev eral block subsequences and fully explore combina- tions of the block subsequences along an optimal landing/takeof f subsequence to consider the opti- mization problem (1). Note that the number of all possible block subsequences of dif ferent class combinations is not large according to the minimum separation time standards of Heathrow Airport and the RECA T system, when resident-point aircraft are not taken into account. The use of block subse- quences might significantly reduce the computation amount of the algorithm. V I . S I M U L AT I O N In this section, we e valuate the efficienc y of the proposed algorithm by comparing its computation time and objectiv e function performance against a standard MIP solver . T wo typical aircraft scheduling problems are considered: operations on a same runway and on dual runways with spacing no larger T ABLE II M I N I M U M TAK E O FF S E PA R A T I O N T I M E S B A S E D O N R E C A T - E U ( S E C ) T railing Aircraft A B C D E F Leading Aircraft A 80 100 120 140 160 180 B 80 80 100 100 120 140 C 60 60 80 80 100 120 D 60 60 60 60 60 120 E 60 60 60 60 60 100 F 60 60 60 60 60 80 than 760 m. For the same-runway case, we e xamine four scenarios with | A | = 30 , 40 , 50 , 60 under three dif ferent operations: takeoff only , landing only , and mixed operations (both takeoff and landing). For the dual-runway case, we consider scenarios with | A | = 70 , 80 , 90 , 100 aircraft, each comprising an equal number of tak eoff aircraft and landing aircraft. The aircraft are classified into six categories based on the RECA T -EU frame work ( A, B , C , D , E , F ) , with proportions of 10% , 20% , 25% , 15% , 20% , and 10% , respecti vely . The minimum separation times for aircraft pairs with the same type of operation are provided in T ables I and II. For both problems, the separation time between a takeof f aircraft and a following landing aircraft is set to D T = D P = 60 seconds, while the separation time between a landing aircraft and a follo wing takeoff aircraft is set to T D = 75 seconds for the same-runway case and P D = 0 seconds for the dual-runway case. All simulations are conducted on a computer equipped with an AMD Ryzen 7 7840H processor (3.8 GHz, 16 GB RAM). The MIP formulations are solv ed using CPLEX 12.10 with a time limit of 600 seconds per instance. The proposed algorithm is implemented in MA TLAB R2021b . T o enhance computational efficienc y , we employ MA TLAB’ s parallel computing technology using 8 work ers to deal with the mixed operation scenarios on a same runway , and the dual-runway scheduling scenarios. For the same-runway case, the earliest landing or takeof f times for each aircraft are randomly gen- erated, following a uniform distrib ution within the interv al [0 , T E ] minutes. The time windo w lengths are set to T W = 60 , 90 , and 120 minutes to e valuate the algorithms’ performance under different lev els of scheduling fle xibility . The scheduled time for each aircraft is set as P i = f min i for takeof f aircraft 11 T ABLE III C O M PA R I S O N O F P E R F O R M A N C E A N D C O M P U TA T I O N T I M E S F O R S A M E - R U N W AY A I R C R A F T S C H E D U L I N G P RO B L E M T W (min) Aircraft Number T E (min) Operation Objectiv e Function(s) Computation T imes(s) Gap | A | T ask MIP Our Algorithm MIP Our Algorithm % 60 30 20 T akeoff 14362 14362 600 0.20 0 20 Landing 8763 8279 600 0.15 5.85 20 Mixed 8671 8569 600 1.28 1.19 40 20 T akeoff 36695 35095 600 0.55 4.56 20 Landing 28297 25234 600 0.37 12.14 20 Mixed 24520 24400 600 1.93 0.49 50 30 T akeoff 47607 45813 600 1.27 3.92 30 Landing 37264 33141 600 1.11 12.44 20 Mixed 41914 41490 600 2.86 1.02 60 30 T akeoff 77118 73478 600 3.24 4.95 30 Landing 68173 58214 600 2.69 17.11 20 Mixed 75975 72852 600 4.56 4.29 90 30 20 T akeoff 12164 12144 600 0.18 0.16 20 Landing 9073 8785 600 0.16 3.29 20 Mixed 8453 8439 600 1.21 0.17 40 20 T akeoff 35835 35095 600 0.55 2.11 20 Landing 22738 22566 600 0.51 0.76 20 Mixed 24700 24400 600 1.95 1.23 50 20 T akeoff 61584 60164 600 1.35 2.36 20 Landing 47067 42097 600 1.42 11.81 20 Mixed 43788 43424 600 2.87 0.84 60 20 T akeoff 89136 84736 600 2.65 5.19 20 Landing 79896 71478 600 3.19 11.78 20 Mixed 67749 66484 600 4.32 1.90 120 30 20 T akeoff 15637 15637 600 0.15 0 20 Landing 9120 8928 600 0.14 2.15 20 Mixed 6128 6109 600 1.21 0.31 40 20 T akeoff 28409 28069 600 0.57 1.21 20 Landing 24201 21990 600 0.48 10.05 20 Mixed 20906 20780 600 2.09 0.61 50 20 T akeoff 56504 53864 600 1.37 4.90 20 Landing 53176 46422 600 1.29 14.55 20 Mixed 37394 36829 600 3.19 1.53 60 20 T akeoff 86716 81845 600 2.89 5.95 20 Landing 75769 68811 600 2.62 10.11 20 Mixed 72386 71109 600 4.61 1.80 and P i = f min i + 5 minutes for landing aircraft. T able III presents a comparison of the objectiv e function v alues and computation times between the proposed algorithm and the MIP solver , along with the percentage gap in objecti ve v alues between the two methods. For the dual-runway case, aircraft earliest opera- tion times are also uniformly distributed ov er [0 , T E ] minutes. The time window lengths remain at T W = 60 , 90 , and 120 minutes. T able IV summarizes the performance comparison between our algorithm and the MIP solver in terms of objectiv e function values and computation time, as well as the corresponding objecti ve v alue gaps. From T ables III and IV, it is e vident that the proposed algorithm consistently obtains solutions within 10 seconds, whereas the MIP solv er requires the full 10 -minute time limit. Moreov er , our algo- rithm consistently yields better objecti ve function v alues than those produced by the MIP solver , with the performance gap widening as the number of aircraft increases. In addition, the runtime of the 12 T ABLE IV C O M PA R I S O N O F P E R F O R M A N C E A N D C O M P U TA T I O N T I M E S F O R D UA L - RU N W AY A I R C R A F T S C H E D U L I N G P R O B L E M T W T E Aircraft Number Objecti ve Function(s) Computation T imes(s) Gap (min) (min) | A | MIP Our Algorithm MIP Our Algorithm % 60 20 70 47803 43066 600 4.15 11.00 20 80 71524 60823 600 4.57 17.59 20 90 88989 76128 600 6.77 16.89 20 100 131632 115304 600 8.79 14.16 90 20 70 53389 46611 600 4.04 14.54 30 80 63372 59127 600 5.40 7.18 40 90 94272 82905 600 6.83 13.71 50 100 134749 112959 600 9.40 19.29 120 20 70 47169 43171 600 3.66 9.26 20 80 74232 63237 600 5.57 17.39 20 90 96782 82216 600 8.16 17.71 20 100 121368 101823 600 9.57 19.20 MIP solver has been e xtended to ov er one hour but no significant improv ements were found in the objecti ve function values compared to the results obtained within the 10 -minute time limit. In the same-runway scenario, our algorithm sho ws particularly strong performance in the takeof f-only and landing-only cases. The perfor- mance advantage is less pronounced in the mixed takeof f-and-landing case. This is primarily due to the relativ ely short separation times required be- tween different operation tasks: 60 seconds between a takeoff aircraft and a trailing landing aircraft, and 75 seconds between a landing aircraft and a trailing takeof f aircraft. These moderate separation values enable the MIP solv er to generate relati ve high- quality solutions by frequently using tak eoff-landing and landing-takeof f transitions. Nonetheless, as the number of aircraft increases, the superiority of our algorithm remains increasingly evident. For the dual-runway scenario, we ev aluate 12 instances with progressiv ely larger numbers of air - craft, reflecting the higher operational capacity of dual-runway airports. In particular , when the prob- lem size reaches 90 and 100 aircraft, the perfor - mance gap in objectiv e function v alues between our algorithm and the MIP solv er exceeds 17% , exhibit- ing the scalability and ef ficiency of our proposed algorithms. V I I . C O N C L U S I O N S In this paper , scheduling problems of aircraft minimizing the total delays on a same runway and on dual runw ays are studied. T wo real-time optimal algorithms were proposed for the four schedul- ing problems by fully exploiting the combinations of different classes of aircraft based on parallel computing technology . When 100 aircraft on dual runways were considered, by using the algorithm in this paper , the optimal solution can be obtained within less than 10 seconds, while by using the CPLEX software to solve the mix-inte ger optimiza- tion model, the optimal solution cannot be obtained within 1 hour . R E F E R E N C E S [1] Xiao-Bing Hu and Ezequiel Di Paolo. 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