Department of Computer Science and Technology, University of Science and Technology of China, Hefei, 230026, P.R.China, Anhui Province Key Laboratory of Software in Computing and Communication, Hefei, 230026, P.R.China 86-551-3601545 [email protected], [email protected], [email protected] 2

School of Engineering Science, University of Science and Technology of China Hefei 230026 P.R.China 86-551-3600127 [email protected] 3

School of Electronic and Information Engineering, Beihang University, Beijing, 100083, P.R.China [email protected]

ABSTRACT

1. INTRODUCTION

The Aircraft Landing Scheduling (ALS) problem is a typical hard multi-constraint optimization problem. In real applications, it is not most important to find the best solution but to provide a feasible landing schedule in an acceptable time. We propose a novel approach which can effectively solve the ALS while satisfying the real-time need. It consists of two steps: (i) Use CA to simulate the landing process in the terminal airspace and to find a considerably good landing sequence; (ii) a simple Genetic Algorithm associated with a Relaxation Operator is used to obtain a better result based on the CA result. Experiments have shown that our method is much faster and suitable for real-time ALS problem compared with traditional optimization methods. For all the 13 data sets, the proposed approach can find satisfactory solutions in less than 2 seconds.

Aircraft Landing Scheduling (ALS) is an important part of the Air Traffic Control which determines an efficient landing sequence and landing time for a given set of aircrafts [1]. The ALS problem contains many constraints, e.g. each aircraft must land in a specified time window, and the separation criteria between each pair of landing aircrafts should be guaranteed. It is pointed out that ALS is a typical NP-hard problem [2]. Due to ALS’s largescale and multi-constraint characteristics, it’s hard to find the optimal solution. Thus it draws continuous attentions from different scientific communities. A widely used strategy in applications is the First-Come-First-Served (FCFS) mechanism where the order in which the planes land is the same as the order they enter the radar range. Though FCFS is simple and fast, it cannot result in an effective scheduling. Consequently, the LinearProgramming-based (LP) algorithm is introduced. Beasley et al. presented a method based on mixed integer linear programming formulation and genetic algorithm for static aircraft landing problem [1] and optimal solutions of 8 datasets with up to 50 planes are generated. Balakrishnan and Chandran used a Dynamic-Programming-based approach in a Constrained Position Shifting environment [2]. The LP algorithms can get a nice result, but due to its high complexity, it cannot be applied to large-scale real-time ALS problems. Another powerful framework is the computational intelligence. Randall used an ant colony optimization to solve the problem [3]. Ciesielsk and Scerri used the standard GA with the context of sliding window to solve ALS dynamically [4]. Beasley et al. presented a population heuristic algorithm considering multi-objective functions, where a population replacement scheme was used to deal with infeasible solutions [5]. In Ref.[6], Beasley et al. applied two population heuristic algorithms into ALS. Xiao-Bing Hu and Wen-Hua Chen adopted a fast Genetic Algorithm on ALS with objective of minimizing total delay [7]. Besides, Guo[8], Tang[9] solved the ALS problem using a multiobjective evolutionary algorithm, respectively. However, in real applications, a schedule for aircraft landing

Categories and Subject Descriptors F.1.1 [Computation by abstract devices]: Models of Computation; I.2.8 [Artificial Intelligence]: Problem Solving, Control Methods, and Search—scheduling.

General Terms Algorithms.

Keywords Aircraft Landing Scheduling, Cellular Automaton, Optimization, Genetic Algorithm

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should be generated within a very short time, which is the scanning period for the radar to scan the whole terminal airspace. As most traditional methods, including global search algorithm and LP based algorithm, are aiming at searching for a best result in the solution space, they are destined to make a compromise between speed and accuracy and often cannot find a nice result in a short time. In this article, we introduce an algorithm named CAO (Cellular Automaton-based Optmization) to address this problem from a different aspect. It can give an acceptable good result in very short time, and thus it can efficiently solve the ALS problem in realtime. We use the one-dimension CA (Cellular Automaton) model[10] to simulate the approaching process of planes. The updating rules for planes have the purpose to minimize the total cost, which is linearly related to deviation from its target time. After the landing sequence of planes has been achieved, simple Genetic Algorithm (GA) associated with a relaxation operator is applied to the queue to get a better result. The rest of this paper is organized as follows. Section 2 briefly outlines the aircraft landing scheduling problem and provides the corresponding mathematical formulation. Section 3 describes our CAO in detail, including the special design of CA rules, a simple GA and the Relaxation operator. Section 4 demonstrates the effectiveness of our method through extensive comparative experiments. The conclusion is given in section 5

X i : The landing time for plane i .

Sij : The required time interval between aircraft i and j . The following constraints must be met:

Ei ≤ X i ≤ Li ( i = 1...P )

(1)

X i − X j ≤ Sij ( i = 1...P, j = 1...P, i ≠ j )

(2)

Eq.(1) makes planes land in their time window. Eq.(2) ensures the separation criteria between aircraft i and j . The objective of ALS is to minimize the cost of all aircrafts. In this paper, we adopted the linear cost, which could be formulated as follows. The cost of aircraft i is: ⎪⎧ g i (Ti − X i ) Zi ⎨ ⎪⎩hi ( X i − Ti )

X i ≤ Ti X i > Ti

(3)

The total cost is: Z total = ∑ i =1... P Z i

(4)

3. CAO Algorithm Previous works [1, 11-13] have shown that the aircraft landing optimization process consists of two parts: searching for the best permutation and optimizing the landing times based on a permutation. Thus we firstly propose a CA model to simulate the approaching process of the planes to give an effective landing sequence. Then a Genetic Algorithm associated with a Relaxation Operator is adopted to further optimize the landing schedule generated by the CA simulation. Compared with traditional works that searches in the possible solution space to find an optimal result, our approach can get a nice landing sequence via a self organizing process and can provide an efficient optimization method for ALS problem.

2. Aircraft Landing Scheduling The ALS problem was firstly presented in Ref.[1]. Upon entering the radar range (radar horizon) of air traffic controller (ATC) at an airport, a plane requires ATC to assign it a landing time. The landing time must lie in a given time window, bounded by an earliest time (E) and a latest time (L). Since each plane has its maximum speed, it cannot land before its earliest time. The fuel capacity of the plane must be limited, thus a plane cannot land after its latest time either. Besides, considering the aerodynamic restraints, the separation criteria between any pair of two planes must be satisfied. Every plane has a most economical speed like its cruise speed. The most preferred landing time, or target time (T) is the time when a plane lands with its cruise speed in the whole trajectory. If the plane is to speed up or to slow down, there will be a cost relating to the earliness and tardiness penalties.

3.1 Cellular Automaton Model Cellular Automaton is a simple model of computation which exhibit fascinatingly complex behavior. They have captured the attention of several generations of researchers, leading to an extensive body of work [10].

The goal of ALS is to decide a landing time for each aircraft while satisfying the numerous constraints such as time windows and separation criteria. The ALS problem discussed in this article is formulated as follows:

Cellular Automaton is a dynamical model that has many but discrete degrees of freedom [14]. In fact there is a threefold discreteness in Cellular Automaton model: discrete space, discrete time, and discrete number of states. When applied to traffic research, CA uses cellular state to describe the position and velocity of each vehicle. The updating rules of cellular states are introduced by practical traffic experiences. The whole system’s dynamical evolvement and final steady result are achieved by simulation. CA models can model the complexities of nonlinear characters in traffic problems [14-16], and offer more intuitive physical images. As far as it works well in simulating traffic flow, a variant of CA is introduced here for simulating the landing process.

P : The number of planes.

Ai : The time at which plane i enters radar range. Ei : The earliest landing time of plane i . Li : The latest landing time of plane i . Ti : The target landing time of plane i . g i : The penalty (positive) per unit of time for landing before target time for plane i . hi : The penalty (positive) per unit of time for landing after target time for plane i .

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Here we use CA model in which a plane means an automaton runs following some rules.

3.2 Details of Our Approach We propose a Cellular Automaton based Optimization to address the ALS problem. The flow chart is shown in Fig.1. As can be seen in Fig.1, our method consists of two parts: (i) A Variant of the CA model, (ii) Extra steps consists of Genetic Algorithm and Relaxation Operator.

For simplicity, we divide the distance from the airport to boundary of radar range into M distance unit. The radar range is turned into M sites. Site r stands for a ring r units far from airport and has a width of 1 unit. Site 0 represents airport while site M-1 stands for boundary of the radar range. Planes in the radar range of ATC can be arranged to several sites. Every site can hold more than one plane.

r da (ra e) M ng ra r+1 r

Fig.1 The flow chart of CA based Optimization for ALS problem.

3.2.1 A Variant of the CA model

Fig.2 Illustration of the Cellular Automaton model for the ALS problem. (a) The radar range of the air traffic controller is divided into M cells, where cell zero represents the airport and cell M-1 represents the boundary of radar range. (b) The planes’ position and speed are recorded as the state of the cells. Updating rules are then adopted to mimic the approaching processes of the planes

After entering the radar range, aircraft should fly according to the instructions from airport control center until landing at the airport. As shown in Fig.2, the distance from the airport to boundary of radar range is divided into M cells. Each cell stands for a ring with the width of one unit. Cell zero represents airport while cell M-1 stands for the boundary of radar range. If a plane arrives at cell zero, it is interpreted as arriving at the airport. It will be removed from the system and the successive plane becomes the leading plane. Different from most CA models for road traffic, here each site can hold more than one plane. Moreover, in order to satisfy the separation criterion, there should be a safe distance for the planes within which the planes cannot overtake each other.

In order to achieve an optimal landing sequence of plane set, the cost caused by landing before (or after) the target time is calculated for each plane supposing that the plane keeps its velocity until it reaches the airport

After entering radar range, aircraft should fly following instructions from airport control center until landing at the airport. ⎧⎪ ⎛ ⎫⎪ ⎞ D f l ( i ) = max ⎨0, ⎜ t + i − Ti ⎟ × g i ⎬ (5) ⎠ ⎩⎪ ⎝ Vi ⎭⎪

⎫⎪ ⎞ D ⎪⎧ ⎛ f e ( i ) = max ⎨0, ⎜ Ti − i − t ⎟ × hi ⎬ Vi ⎠ ⎩⎪ ⎝ ⎭⎪

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(6)

In this paper, we set the minimal speed for planes to zero. In the following experiments, we find that the requirement of latest landing time can be always satisfied in the CA model.

Where Di stands for the position of plane i in the cell space and t is the current time. So f i is the current earliness penalty and

f e is the current tardiness penalty. An optimal method should try to minimize the above penalty for the whole system and/or the current plane while satisfying the constraints. In order to achieve these goals, we propose that the state of each plane is updated in parallel with the following updating rules.

In the velocity update, planes can overtake each other for a better landing sequence if their positions are beyond the safe distance： Di > Dsafe . Eq.(8),(9),(10) are the keys for optimizing landing sequence. If the tardiness penalty of the current plane is higher than that of the previous plane, rule (8) is applied with an acceleration alpha to allow the plane to overtake the previous plane. Simultaneously, if the earliness penalty of the current plane is higher than that of the successive plane, rule (9) is applied with a deceleration beta to allow the successive plane to overtake. Thus the optimization of minimizing total penalty for the whole system is naturally incorporated in the CA rules.

Considered the acceleration and deceleration of a plane should be limited, we propose α × Vi ,max and β × Vi ,max as the accelerations for accelerating and decelerating of plane i . Where α and β stands for acceleration rate based on Vmax of plane i , and we let them between 0.1 and 1.0. In every step, the velocities and position of the planes are updated according to the following rules:

However, when Di ≤ Dsafe , the updating of planes is driven by the

1. Arrival of planes: if the arrival time of a plane is equal to current time t, it is inserted into the system with the best velocity: Vbest =

M s Ti − t

requirement of separation criterion. They cannot overtake each other. If the current plane is not the leading plane, it may need to decelerate to meet the separation criterion with the previous plane, as shown in Eq.(11). Otherwise it can simply accelerate to reach the airport, as shown in Eq.(12).

(7)

2. Velocity update:

3.2.2 Genetic Algorithm and Relaxation Operator

If Di > Dsafe

Vi →min{Vi +α ×Vi,max ,Vi,max} if fl ( i) > fl ( i −1)

(8)

Vi → max {Vi − β × Vi ,max ,0} else if f e ( i ) > f e ( i − 1)

(9)

Vi → Vi

(10)

otherwise

The result obtained by CA model is a landing queue, with landing times and sequence of the planes under scheduling. Obviously, CA is a self-organizing model without the capability of global optimization. It can generate a feasible landing sequence but the sequence can hardly be the best, as well as the result consists of the landing results has not provide the optimal landing times for every plane. Thus we introduce a simple Genetic Algorithm (GA) associated with a Relaxation Operator (RO) to the landing sequence and obtain a better result. The RO can adjust each plane in scheduling to an optimal landing time based on a feasible landing sequence. And GA can obtain better landing sequence via simply adjusting the CA resulting landing sequence.

If Di ≤ Dsafe Vi → max {Vi − β × Vi ,max ,0} if

Di − D( i −1) Vi

< Si −1,i (11)

and i is not the leading plane Vi → min {Vi + α × Vi ,max ,Vi ,max } otherwise

a). Relaxation Operator

(12)

As the deterministic optimization algorithm can give optimal landing times of a given landing sequence [1,6,12,13], we propose a Relaxation Operator that can get optimal landing time based on the landing sequence given by CA. The Relaxation Operator is partially inspired by the algorithm reported in Ref.[13]. To get an optimal landing time series, the original algorithm of Ref.[13] separates the plane set into several chains, and then shifts the chains simultaneously by testing the benefit of shifting earlier or later. In our paper, we calculate the benefits for shifting each chain, and then shift the chains by the order of magnitude of the benefits.

3. Position update: Di → max {0, Di − Vi }

(13)

4. Landing: If Di = 0 , the plane is removed from the system and the successive plane will become the leading plane. Here, the planes are ordered by their distance from the airport. For the planes in the same cell, they are ordered by the entering sequence. So plane i − 1 is the previous neighbor of plane i . Vi is the current speed of plane i .

Vi ,max

is the maximum speed of

b). GA

plane i which ensures the landing time is after the earliest landing time: Vi ,max =

M Ei − Ai

The simple GA is used to search for several better landing sequences based on sequence generated by CA. Details of the GA are depicted as follows:

(14) 1) Structure of the Chromosomes

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Each chromosome standing for a landing sequence is formulated as ( a1 ,..., a p ) as shown in Fig.3. In the chromosome, each gene represents a plane of the landing sequence.

Fig.4 Mutation Operator Fig.3 Structure of a chromosome.

4. Experiment

2) Fitness Evaluating

The benchmarks used here have been described in Ref.[1,11] and they are widely used in previous works. In Ref.[6], the current best results of 13 datasets have been given. So we test our optimization method with these datasets. Without losing generality, we set the length of the Automaton as L = 2600 , and the acceleration rate as α = 0.25 and β = 0.25 . Dsafe is set to 500.

As each chromosome stands for a landing sequence, we apply Relaxation Operator on each sequence to generate a new landing schedule. If a feasible schedule can be generated, the fitness can be calculated according to the penalty of the final landing schedule. Otherwise, this chromosome is abandoned. As formulated in Eq.(4),

Z total

The results are given in table 1 and 2. The problem introduced in this article was programmed in C++ and run on a PC (with CPU of 1.60GHz, 1.5GB main memory).

is the total penalty of a landing

schedule. Let Z total ( k ) be the penalty of chromosome k, the

Table 1 shows the comparison results on 8 datasets, each dataset contains no more than 50 planes. We compared our algorithm in Ref.[1] with a deterministic optimal algorithm which gives the optimal results. P represents the number of planes of the corresponding datasets, Z represents total earliness and tardiness penalty of results given by certain algorithms. For optimal results for rest of the datasets are not given in [1], we compared our results to a Scatter Search algorithm proposed in Ref.[6], in which the best results has been given. The comparison results are given

fitness of chromosome k can be simply defined as f ( k ) = − Z total ( k )

(15)

3) Genetic Operators Basically, there are two kinds of genetic operators in GA: crossover and mutation.

in table 2.

Colum titled by SS is the best results generated by Scatter Search in 10 replications and the time it cost. The results presented in Tables 1 and 2 are the best results from a set of independent runs.

Here we only adopt the mutation operator, which is illustrated in Fig.4. The order of a pair of planes ai and a j is exchanged by a mutation. For the case of the ALS problem, if

Z best is the total penalty of the best results of Ref.[6].

( j − i ) is too big,

One can see that the final solution provided by CAO is always feasible and dominates first-come-first-served (FCFS) solution. For the eight problems in table 1, there are less than 50 planes. The optimal results can be achieved by using LP-based deterministic algorithm. One can see that our method can provide optimal results for most the 8 datasets (except for 5 & 8) within a very short time. For data 5 and 8, separation criteria Sij for most

the landing sequence will be infeasible because it may violate some constraints. So only the pairs of planes that are not far away from each other in a sequence can be exchanged. Here we adopt a maximum distance as 4, which means ( j − i ) cannot surpass 4. 4) Process of the GA

pairs of planes are quite small, so the space for planes to adjust the position in the landing sequence is large. CA rules for generating better results on these data sets maybe extraordinary. But to be realistic, our rules for CA are moderate, e.g. the acceleration rate α & β are set as 0.25. So we didn’t get optimal schedules on the data sets (e.g. 5 and 8). As methods for applications, our approach may be more realistic because our CA model simulates the approaching process very like the reality. So we didn’t set α & β to a value that can generate a nice but far from reality.

The initial population of GA is composed by 30 copies of landing sequence of CA. In each generation, the mutation operator is applied to all chromosomes to generate new chromosomes. Relaxation Operator is applied to each new chromosome for evaluating the fitness. If the new chromosome has a better fitness value than the parent chromosome, the parent chromosome is replaced with the new chromosome. After 30 replications, the final population is generated. The best landing sequence is chosen as the final landing sequence. As this GA is quite simple, extra computation time caused by it can be neglected.

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competitive short time. Different from previous works which focused on searching good schedules in a possible solution space, we generate the landing sequence by a self organized process. As CA has never been used in this kind of problem as an optimization method, our approach is a significant attempt.

Table 1. Comparative Experiment Results on small-scale datasets. Comparing to optimal results got in Ref.[2] by LPbased deterministic algorithm. Z is the total cost. T is the computation time. Data Num.

P

CAO

Optimal Result[6]

FCFS

The current implementation can be easily adapted to dynamic cases, in which the whole process of the algorithm will be done on the arrival of every plane. In the future work, the multi-runway ALS problem and non-linear objective functions will be considered.

Z

Z

T(s)

Z

T(s)

1

10

700

0.4

700

0.0025

1790

2

15

1480

5.2

1480

0.0034

2610

3

20

820

2.7

820

0.0042

2930

4

20

2520

220.4

2520

0.0043

6290

5

20

3100

922.0

3680

0.0045

8370

6

30

24442

33.1

24442

0.0521

24442

7

44

1550

10.6

1550

0.0857

1550

8

50

1950

111.9

2635

0.0110

26835

It’s shown here Cellular Automaton is efficient and robust in solving large-scale and multi-constraint optimization problems such as ALS. It can be further extended to other optimization problems described by many time constraints.

6. Acknowledgment This work is funded by the National High Technology Research and Development Program of China (2007AA11Z240), Program for New Century Excellent Talents in University (NCET-070787) , the project of CNS/ATM Labs, CAAC, National Basic Research Program of China(No.2006CB705500) and the NNSFC under Key Project No.10532060, Project No. 10872194.

Table 2. Comparative Experiment Results on Large-scale datasets. Experiment Results comparing to the results presented in Ref.[6]. Here the value of Z in SS is average result found by 10 replications of SS, while Z best is the best result found by the Scatter Search algorithm. P

Zbest[6]

SS[6]

CAO

FCFS

Z

T(s)

Z

T(s)

Z

9

100

5611

7303

119

6988

0.1586

36839

10

150

12329

17872

227

19785

0.2579

54113

11

200

12418

14647

256

14966

0.3017

66427

12

250

16209

19799

381

19322

0.4124

81916

13

500

44832

46284

1237

45041

1.0743

178725

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For the five large-scale data sets in table 2, the optimal results have not been provided so far. It is currently impossible to get the optimal results by deterministic algorithms like the LP based tree search. So we compare the results of CAO with the best results reported in Ref.[6]. One can see that three of our results (9,12,13) are better than the average result found by 10 replications of scatter search algorithm. The other two results are quite close to the previous results. All these results are obtained within a very short computation time. Moreover, our method can provide a feasible and better result for the data sets of 12 and 13 that contain numerous planes. For data set 9, 12, and 13, our results are closer to the best results sought out by the Scatter Search algorithm. Our method can make the real-time solution of ALS problem possible.

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5. Conclusion In this paper we considered the single runway Aircraft Landing Scheduling Problem of linear objective function. The proposed method consists of CA and GA as well as a deterministic algorithm which can solve ALS problem effectively in a very short time. A preliminary schedule is generated by Cellular Automaton, in which every plane updates its state with some rules while approaching the destination. Then we obtain better results by using Genetic Algorithm and Relaxation Operator based on the landing sequence generated by Cellular Automaton.

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Compared with previous works, we can get a good landing schedule aiming to minimize the linear objective function in

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