MULTI-OBJECTIVES OPTIMIZATION ON THE DEPARTURE

OF AIRPLANES FROM BUSY AIRPORT

Zhang Yue, Wang Shidong and Yu HaiYang

Institute of Civil Aviation Development, China Academy of Civil Aviation Science and Technology, Beijing, China

Keywords: Scheduling the departure of airplanes, One machine scheduling, Multi-objective.

Abstract: The delay of airplanes has been the core negative factor to influence the service quality and development of

the airline business. The optimized scheduling the arrival and departure of the airplanes is one of good

methods to decrease the delay except the uncontrollable weather. The sequence of take-off can be seen as

one machine scheduling problem with two objectives: minimizing the number of tardy jobs and minimizing

the maximal tardiness of all the jobs. The mathematical model is formulated and multi-objective GA

(Genetic Algorithm) is utilized to solve the Pareto optimization. Computational results show that the

proposed algorithm performs well when compared with traditional heuristic methods and also provide some

choices to the dispatcher which can decide according to the real condition. The process will promote the

flexibility and effectiveness of scheduling the departure of airplanes.

1 INTRODUCTION

The delay of airplane at the busy airport has been

regarded as the most negative factor which influences

the quality of airlines’ service. There are a lot of

reasons: air traffic control, weather, flow control, and

air path restriction, etc (Song, 2010).

If there are two runways at the airport, the type of

parallel pattern is usually used: one runway for

landing, another runway for departure and both are

independent relatively. The departure of airplanes

will be discussed in the paper.

When the airplane is about to leave, it will follow

the instruction from the ATC-silde on the taxiway-

ready to takeoff-the entering point to the airspace

sector-the location point of departure. From the

viewpoint of traditonal scheduling theory, the ready

time to enter the runway can been the release time of

job, the departure time on the plane ticket can be

viewed as due time, if the airplane’s take-off time is

later than that, the delay happens which should be

avoided in the real world. The period between the

waiting time on the runway and the time passing by

the location point point of departure (where means

the plan has been out of controlling by the airport) is

the processing time, the precise value can been

calculated by the flying speed and distance, etc.

Sequencing has been an important issue in the

busy aiport, Atkin discusses how one runway

controller attempts to ﬁnd the best order for aircraft

to take off with uncertain taxi times (Atkin, 2008).

Bai researches on the coordination model with

dynamic and open for arrival/departure airport

dispatching system using software coordination

technology (Bai, 2007). Song introduces immune

algorithm to solve the fleet assignment problem

(Song, 2007). The same point is that these paper

always view the objective as solely. But if the quality

of airline’s service is considered, the objective of

departure’s sequnce should be not single, and be a

trade-off decision. When the controller plans the

schedule of airplane’s take-off, that the airplane

should leave before their due time must be

considered, so one objectiv is minimizing the total

weighted number of delayed planes (1|r

∑

). If

it’s done well, the more flights will depart on time,

the passengers will be satisfied and the airline will

get higher service evaluation. Meanwhile, the

controller also thinks about the average delay time if

it cann’t be avoided, that idea means the minimum of

maximal delay time (1|r

max

) and there should not

be some flights with very long lateness. So the

departure of airplanes from busy airport will be

discussed as one machine scheduling problem with

multi-objectives optimization.

1. Minimizing the total weighted number of delayed

airplanes:

2. Minimizing the maximal delay time

Minimizing the total number of delayed jobs and

minimizing the maximal delay time with release

250

Yue Z., Shidong W. and HaiYang Y..

MULTI-OBJECTIVES OPTIMIZATION ON THE DEPARTURE OF AIRPLANES FROM BUSY AIRPORT.

DOI: 10.5220/0003470202500253

In Proceedings of the 13th International Conference on Enterprise Information Systems (ICEIS-2011), pages 250-253

ISBN: 978-989-8425-54-6

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

time are both showed to be strongly NP-hard

(Lenstra, 1977).Here the trade-off is considered

among the so-called non-dominating (efficient)

solutions, or pareto-optima. There will not be such

solutions which has both objectives better than others

in the result.

Many researchers have fouced on the multi-

objective scheduling problem in different industry.

The result shows that most problems ususally are

complex and cann’t be solved by conventional

optimization techniques (Chen, 1994 and Koksalan,

1998). So there are some proposed heursitc methods

to solve these NP problem, such as a multi-objective

simulated annealing (MOSA) method (Loukil, 2004),

tabu search algorithm (Michael, 2002).

The paper is organized as follows: one machine

problem with the multi-objective optimization

derived from airplane take-off sequence problem is

described in Section 1. In Section 2, GA (Genetic

Algorithm) will be introduced. The test instances and

results are described in Section 3 and conclusions are

given in Section 4.

2 MATHEMATIC MODEL

2.1 Genetic Algorithm

Genetic algorithm (GA) is a powerful and broadly

method for many problems which are very hard to

solve by mathematical OR techniques and has been

successfully applied to solve many scheduling

optimization problems. GA simulates the evolution in

nature by operators (such as crossover and mutation)

and evaluates the middle solution during the

searching process, and then the global solution will

be found in a high probability.

In GA, each solution is encoded to be a

chromosome. By selecting the individuals with best

fitness, the better solutions will survive. The method

is flexible enough to provide sub-optimal solution of

large-scale optimization problems, but will cost a lot

of time (Wang, 2003). This paper uses NSGA-II

(Deb, 2002) to the solve problem and find the

frontier curve E(P) composed of non-dominating

solutions.

2.2 Solution and Chromosome

The chromosome scheme must express the solution

which is made of n genes, where n is the number of

airplanes. The feasibility of the crossover operator

depends greatly on the scheme.

The solution here can be encoded as π (an

permutation of n) which the each digit means the

position where each airplane is scheduled on. The

formation is easy to handle because each

chromosome represents one feasible solution which

will have no conflict. So a solution is formatted as

,…,j

, where gene j

denotes that the airplane

j is operated on the position j

After definition of chromosome, initial populations

can be achieved by generating some individuals

whose chromosome is a randomized permutation of n.

Figure 1 illustrates an example of chromosome

when to schedule 5 airplanes (n = 5) which sequence

is {5, 3, 1, 2,4}.

Figure 1: A chromosome encoded in permutation structure.

2.3 Fitness Function

The fitness function is used to provide a measure of

how individuals have performed in the problem

domain. In this problem, the most fit individuals will

have the minimal value of the associated objective

functions. The aim is to find appropriate f and g to

construct the curve E(P) so that the different

objectives have to search on the non-dominating area

as possible as near, the crowding distance will be

used to describe the distance between the solution on

the frontier curve and other close solution (Deb,

2002). More crowding distance means the well-

distribution of all solutions and is the searching

direction. For the non-dominating solution, the

fitness

()

is calculated as followed.

Step 1: calculate objective f and g of non-dominating

solutions respectively, and rank the current N non-

dominating solutions, set f

,…,f

and g

,…,g

for each solution in the frontier of feasible area .

Step 2: set y

[k,f]

and y

[k,g]

represent the final rank

according to the objective f and g respectively, set

[1,f]

)=∞, cd

[1,g]

)=∞, cd

[R,f]

)=∞ and cd

[R,g]

)

=∞, for other k=2,…,N-1, there is

[ 1,] [ 1,]

[, ]

max min

()()

()

kf kf

fkf

fy fy

cd y

+−

−

(10)

[ 1,] [ 1,]

[,]

max min

()()

()

kg kg

gkg

fy fy

cd y

+−

−

(11)

Where f

max

, g

max

and f

min

, g

min

are the minimums

and maximums of f and g respectively.

Step 3: the crowding distance is defined as

()

kfg

cd cd

(12)

MULTI-OBJECTIVES OPTIMIZATION ON THE DEPARTURE OF AIRPLANES FROM BUSY AIRPORT

251

2.4 Selection

The selection criterion is used to select the two

parents in non-dominating solutions to apply the

crossover operator which will produce successive

generations, and a good method will lead to a fast

convergence. The method in which the best fitness

have more chance to be selected as parents for

creating offspring of subsequent generation and

called Rolette Wheel Sampling (RWS) is one of the

most common strategies. The parameter selection

probability P

(0 < P

< 1) defines the proportion of

the previous generation best chromosomes that are

copied to the next. By this scheme, the excellent

genes in the chromosome can be inherited to next

generation in a higher probability.

2.5 Crossover and Mutation

Two basic types of operators: crossover and mutation

are used to create new solutions based on existing

solutions in the population. Crossover operates will

produce two new individuals by exchanging parent

chromosomes, while mutation change two position to

produce a single new solution. The application of

these two basic types of operators and their

derivatives depends on the chromosome

representation.

The crossover operator is one-point crossover in

the paper and controlled by a pre-specified parameter

(0 < P

< 1) called the crossover probability. After

two chromosomes are chosen from the old population

and the crossover point is decided by crossover

probability, Genes after the position of crossover site

in chromosome will be exchanged between two

chromosomes. Figure 2 shows the sequence of genes

{1, 2, 4} in the first chromosome is {4, 2, 1}, so in

the new chromosome {1, 2, 4} is placed by {4, 2, 1}.

{5, 1, 3} is also changed to {5, 3, 1} at the same

principle.

Figure 2: Crossover operation.

Mutation operator is applied to modify a

chromosome in order to prevent premature

convergence from a local optimization. The genes in

the chromosome will be changed as: j

→j

, j

→j

the quantity and position of mutation is decided by

mutation probability P

(0 < P

< 1).

Figure 3: Mutation operation.

By the definition of these operators, the feasibility of

each solution is kept.

2.6 Termination Criteria

A pre-determined number of generations are satisfied

or the fitness of a population may remain static after

a number of generations.

3 CASE STUDY

Experiments and parameters used in GA are

proposed to validate the algorithm which is coded

with Matlab 6.5 and all experiments are run using an

Intel

(R)

Pentium

(R)

-M 1.70 GHz PC with 512 MB

RAM. The computational instances are randomly

generated as follows: for each airplane j ( j = 1,2,…n,

n=40, 80, 100), an integer processing time p

from the

uniform distribution U[1,100]; an integer release time

from the uniform distribution U[1, P], where P is

the total processing time (P=∑p

). For a given

relative range of due time R (R=0.4, 0.6, 0.8) and a

given average delayed factor T (T=0.2, 0.6, 0.8), the

integer due time d

for each job j is generated from

the uniform distribution U[(1

－T－R/2)P, (1－T＋

R/2)P]. The value R determines the length of the

interval from which the due date is taken. T

determines the relative position of the centre of this

interval between 0 and the sum of the processing

time P. there are 5 instances for each example.

The parameters are set at the following values for GA:

Population size, 80; number of generation, 100;

crossover probability, P

=0.6; selection probability,

=0.1; mutation probability, P

=0.1.

Figure 4: Efficient frontier when N=40.

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252

Fig. 4, 6, 7 represents the corresponding image of

the frontier of non-dominating solutions for one

instance with 40, 80 and 100 airplanes. Fig. 5 shows

the relationship between all solutions and frontier

curve of non-dominating solutions, which is the

detailed information in Fig. 4 when T=R=0.4. the

non-linked points means dominating solutions and

will be discarded.

Figure 5: the Detail of efficient frontier when m=40.

Figure 6: Efficient frontier when N=80.

Figure 7: Efficient frontier when N=100.

4 CONCLUSIONS

The paper discusses a special one machine

scheduling problem with multi-objectives:

minimizing the total weighted number of delayed

airplanes; minimizing the maximal delay time of

airplanes, which derived from the departure of

airplanes from busy airport.

To carry out the playoff between multi-objectives,

pareto-optima is considered to provide various

choices to the controller. To solve the difficult

problem and find the frontier curve containing the

non-dominating solutions as more as possible,

genetic algorithm is proposed which has been applied

to various applications, and proved to be powerful by

very good results. The computational result shows

the GA can provide more practical solutions than

traditional heuristic methods (FCFS and urgency

assessment method) and decrease the delay problem

because of irrational and wrong sequence or with

considering single objective.

ACKNOWLEDGEMENTS

The paper is supported by the National Soft Science

Research Program (2010GXS1B105): Assessment

and allocation models of the capacity of airport and

airspace.

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