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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
1
,j
2
,…,j
i
,…,j
n
, where gene j
i
denotes that the airplane
j is operated on the position j
i
.
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
()
k
F
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
1
,f
2
,…,f
N
and g
1
,g
2
,…,g
N
,
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
cd
1
(y
[1,f]
)=∞, cd
1
(y
[1,g]
)=∞, cd
1
(y
[R,f]
)=∞ and cd
1
(y
[R,g]
)
=∞, for other k=2,…,N-1, there is
[ 1,] [ 1,]
[, ]
max min
()()
()
kf kf
fkf
fy fy
cd y
ff
+−
−
=
−
(10)
[ 1,] [ 1,]
[,]
max min
()()
()
kg kg
gkg
fy fy
cd y
gg
+−
−
=
−
(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
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