Using evolutionary algorithms for approximating solutions in multi-objective optimization
problems has been a popular topic in the field of evolutionary computation. Because EAs
can o↵er simultaneously a set of trade-o↵ solutions. Up to date, there has been a large set
of MOEAs in the literature addressing a widely range of problems with di↵erent properties.
Directions of improvement have been discussed, conceptualized and used to guide MOEAs
during the search process towards POF. The major concern of the thesis is how to use
the directions of improvement in an e↵ective way to guide the evolutionary process of the
MOEAs in both aspects: 1) Automatically guiding the evolutionary process to make MOEAs
balanced between exploitation and exploration. 2) Combining decision maker’s preference
with directions of improvement to guide the MOEAs during optimal process towards the
most preferred region in objective space

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ocess.
This information is used during the MOEA reproduction phase. At every genera-
tion, the archive is exploited to determine directions of improvement. Randomly-
selected parental solutions are then perturbed along those directions in order to
produce o↵spring solutions. That type of selection is similar to the conventional
selection in evolutionary algorithms. However, the niching mechanism requires
to spread solutions along the POF. This is a ’hard’ niching method and it might
make the convergence slow, the selection of the parent is also random, it has an
e↵ect on the convergence. To avoid this e↵ect the thesis suggests to replace orig-
inal selection scheme by a new one in which parent at a loop will be changed by
a solution in the population. Note that, the index for that solution ranges from 1
to the half of population size; With new selection scheme, the work’s reasoning is
126
6.1. CONCLUSIONS
that more selection pleasure on DMEA is placed so that the convergence will be
balanced with diversity.
The results of the proposed DMEA-II indicates a better performance of DMEA-II over
the original DMEA. The thesis also conducts a comparison between DMEA-II’s perfor-
mance with other 5 MOEAs on four metrics: GD, IGD, HYP and SC. DMEA-II and the
other techniques were competitive in comparison with these algorithms with respect to
both convergence and spread. Several analysis on the behaviors of the algorithm were
thoroughly investigated.
The main features of the evolutionary process in DMEA-II are: The evolutionary pro-
cess is automated guided by directions of improvement to keep the balance between
exploitation and exploration. For the keeping, an adaptive ratio of these directions is
used in di↵erent stages of the evolutionary process: At early stages, the evolutionary
process is guided to give priority to generating individuals of convergence directions, at
the later stages, it is guided to give priority to generating individuals of spread direc-
tions. During the generations, a system of rays is used for a ray based niching method
to maintain the diverse of the main population. It also used for a selection scheme to
fill the second half of the population.
The e↵ective techniques are individual analyzed in [77, 75, 76] and the algorithm was
proposed in [78].
2. The thesis proposed an e↵ective interactive method for DMEA-II: To use
guided information in an e↵ective way for guiding the evolutionary process to make the
population towards DM’s preferred region, this thesis proposes a ray based interactive
method for DMEA-II with three ray based approaches:
• Rays Replacement: The furthest rays from DM’s preferred region are replaced by
new rays that generated from a set of reference points.
• Rays Redistribution: Redistribute the system of rays to be in DM’s preferred
region.
• Value Added Niching: Based on the distances from non-dominated solutions in
archive to DM’s preferred region, the niching values for the solutions is increased
127
6.1. CONCLUSIONS
to be priority selected.
With the interactive method, the evolutionary process in DMEA-II can be guided to
converge to DM’s preferred regions through information from reference points which
are given by DM during the optimization process. The interactive method can help
DM to find their optimal solutions more quickly.
The interactive method for DMEA-II was proposed in [80, 79].
3. DMEA-II is applied for a Spam Email Detection System in network security
area: A framework applied DMEA-II to solve the problem of a Spam Email Detection
System (a SpamAssassin based system) as an applied MOEAs for real applications,
which validates the usefulness of distributions on this thesis. In fact, traditional anti-
spam approaches have optimized the spam detection rate (SDR) and the false alarm
rate (FAR) for years and gained specific results. However, the achievement has been
optimized for the single objective only. With the-multi objective optimization with
interactive approach, not only one pair of SDR and FAR for each threshold has been
worked out but a set of solutions with di↵erent tradeo↵ levels are computed. They are
all feasible depending on specific email users’ demands. More importantly, the score
set of selected solutions are always ready to use without any training needed. The
experiments on Vietnamese database and rules are implemented, the results indicated
that, when solving the problem using DMEA-II with the proposed interactive method,
it not only achieved more ecient results but also created a set of ready-to-use rule
scores. These scores support di↵erent levels of the trade-o↵ between SDR and FAR.
Via obtained solutions, it gives users more flexibility and eciency for system configu-
ration.
The application is introduced in [79, 117].
Through the survey and analysis of methods of using directions of improvement to guide
MOEAs, the thesis proposes a new direction based multi-objective evolutionary algorithm -
DMEA-II. With the techniques used in choosing the direction, niching and selection approach,
and through using diverse of standard problems, well-known measurements in comparison
128
6.2. FUTURE DIRECTIONS
with other popular MOEAs, DMEA-II is considered as a relatively good algorithms. Along
with an interactive method, DMEA-II is added with interactive feature between human
and machine which makes it be able to fit the actual problem. DMEA-II algorithm with
interactive feature, which was applied in spam email detection problem in the field of network
security, is a testament to the usability of the proposed algorithm.
6.2 Future directions
There are several possibilities to extend the research on the using directions of improvement
in the thesis:
• Design of more mechanisms that allow an automated control of the pa-
rameters of DMEA-II : the concepts of using directions of improvement and related
techniques which are used in DMEA-II should be analysed. The properties of the used
techniques might suggest to determine e↵ective parameters to make the DMEA-II qual-
ity to be better in convergence and diversity. In other words, it might make DMEA-II
to be good keeping balanced in exploitation and exploration automatically.
• Parallelization the optimization process of DMEA-II : this is an interesting
idea to improve the algorithm performance when calculating during the evolutionary
process. The Parallel computation might make DMEA-II to be faster than existed
sequential computation during the evolutionary process on the fitness functions assess-
ment. A parallel computation is capable of multi-threading on a large scale, it can
simultaneously process several calculations of fitness functions.
• Extending the capacity of DMEA-II on multi-objective discrete optimiza-
tion problem : DMEA-II was experimented and validated in real space, however,
there are many multi-objective discrete optimization problems in the real world, so the
idea of extending DMEA-II to work with discrete multi-objective discrete optimization
problems is a meaningful research to make DMEA-II widely applied in many areas of
real applications.
129
6.2. FUTURE DIRECTIONS
Publications
Journal articles
[1] Long Nguyen, Lam Thu Bui, Hussein Abbass. DMEA-II: A Direction based Multi-
objective Evolutionary Algorithm-II. Journal of Soft Computing, Volume 18, Issue 11, pp.
2119–2134, 2014, Springer-Verlag.[ISI Journal].
[2] Long Nguyen, Lam Thu Bui. A Decomposition-Based Interactive Method for Multi-
Objective Evolutionary Algorithm. Journal of Development and Application on Information
and Telecommunication Technology, pp. 17–24, Volume E-2, No.5 (9), 2012.
[3] Long Nguyen, Lam Thu Bui. A New Selection Strategy for the Direction-based Multi-
objective Evolutionary Algorithm. Journal of Development and Application on Information
and Telecommunication Technology, pp. 35–48, Volume E-2, No.6 (10), 2013.
[4] Long Nguyen, Lam Thu Bui, Anh Quang Tran. Toward an Interactive Method for DMEA-
II and Application to the Spam-Email Detection System. VNU Journal of Computer Science
and Communication Engineering, Volume 1, No.4, pp. 29–44, 2014.
Conference papers
[1] Long Nguyen, Lam Thu Bui, Hussein Abbass. A new niching method for the direction-
based multi-objective evolutionary algorithm. In proceeding of Computational Intelligence in
Multi-Criteria Decision-Making (MCDM), IEEE Symposium Series on Computational Intel-
ligence, Volume 1, No.8, pp. 16–19, Singapore, 2013, IEEE.
[2] Long Nguyen, Lam Thu Bui. A Ray Based Interactive Method for Direction Based Multi-
objective Evolutionary Algorithm. In proceeding of The fifth International Conference on
Knowledge and Systems Engineering (KSE 2013), Volume 2, pp. 173–184, Hanoi, Vietnam,
2013, Springer-Verlag.
[3] Long Nguyen, Lam Thu Bui. A Multi-Point Interactive Method For Multi-objective Evo-
lutionary Algorithms. In proceeding of The fourth International Conference on Knowledge
and Systems Engineering (KSE 2012), Volume 1, pp. 107–112, Da nang, Vietnam, 2013,
IEEE.
130
6.2. FUTURE DIRECTIONS
[4] Long Nguyen, Lam Thu Bui. The e↵ects of di↵erent selection schemes on the direction
based multi-objective evolutionary algorithm. In proceeding of The first NAFOSTED Con-
ference on Information and Computer Science 2014 (NICS14), pp. 428–437, Hanoi, Vietnam,
2014.
[5] Long Nguyen, Quang Anh Tran, Lam Thu Bui. DMEA-II and its application on spam
email detection problems. In proceeding of Seventh IEEE Symposium on Computational
Intelligence for Security and Defense Applications (IEEE CISDA 2014), Hanoi, Vietnam,
2014.
131
Appendix A
Benchmark sets
132
Tab. A.1: ZDT Problems
MOP POF
ZDT1 :
f1(
!x ) = x1,
f2(
!x , g) = g(!x ).(1
s
f1(
!x )
g(!x ) ),
g(!x ) = 1 + 9
n 1
nX
i=2
xi.
where n = 30, and xi 2 [0, 1]; g(!x ) = 1. The POF
is convex.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
F1
F2
ZDT2 :
f1(
!x ) = x1,
f2(
!x , g) = g(!x ).(1 (f1(
!x )
g(!x ) )
2),
g(!x ) = 1 + 9
n 1
nX
i=2
xi.
where n = 30, and xi 2 [0, 1]; g(!x ) = 1. The POF
is concave.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
F1
F2
ZDT3 :
f1(
!x ) = x1,
f2(
!x , g) = g(!x ).(1
s
f1(
!x )
g(!x )
f1(
!x )
g(!x ) . sin(10⇡f1(
!x ))),
g(!x ) = 1 + 9
n 1
nX
i=2
xi.
where n = 30, and xi 2 [0, 1]; g(!x ) = 1. The POF
is disconnected and convex.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
F1
F2
133
MOP POF
ZDT4 :
f1(
!x ) = x1,
f2(
!x , g) = g(!x ).(1
s
f1(
!x )
g(!x ) ),
g(!x ) = 1 + 10.(n 1) +
nX
i=2
(x2i 10 cos(4⇡xi)).
where n = 10, x1 2 [0, 1] and x2, ..., xn 2 [5, 5];
g(!x ) = 1. The POF is multi-modal.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
F1
F2
ZDT6 :
f1(
!x ) = 1 exp(4x1). sin6(6⇡x1),
f2(
!x , g) = g(!x ).(1 (f1(
!x )
g(!x ) )
2),
g(!x ) = 1 + 9(1
9
.
nX
i=2
(xi)).
where n = 10, xi 2 [0, 1]; g(!x ) = 1. The POF is
non-convex.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
F1
F2
134
Tab. A.2: DTLZ Problems
MOP POF
DTLZ1 :
f1(
!x ) = 1
2
x1x2...xM1(1 + g(!xM )),
f2(!x ) = 1
2
x1x2...(1 xM 1)(1 + g(!xM )),
...,
fM1(!x ) = 1
2
x1(1 x2)(1 + g(!xM )),
fM (
!x ) = 1
2
(1 x1)(1 + g(!xM )).
subject to 0 xi 1, 8i = 1, 2, ..., n where:
!xM = M,M +1, ..., xn and g(!xM ) = 100[|!xM |+P
xi2!xM (xi 0.5)2 cos(20⇡(xi 0.5))].
The Pareto-optimal solution corresponds to
!x ⇤M = 0.5 and the objective function values on
the linear hyper-plane:
PM
m=1 fi = 0.5. The POF
is linear, separable and multi-modal.
00.10.20.30.40.5
0 0.1 0.2 0.3 0.4 0.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
f1
f2
f3
DTLZ2 :
f1(
!x ) = (1 + g(!xM )) cos(x1 ⇡
2
) cos(x2
⇡
2
)...
cos(xM2
⇡
2
) cos(xM1
⇡
2
),
f2(
!x ) = (1 + g(!xM )) cos(x1 ⇡
2
) cos(x2
⇡
2
)...
cos(xM2
⇡
2
) sin(xM1
⇡
2
),
f3(
!x ) = (1 + g(!xM )) cos(x1 ⇡
2
)cos(x2
⇡
2
)...sin(xM2
⇡
2
),
...,
fM1(!x ) = (1 + g(!xM )) cos(x1 ⇡
2
)sin(x2
⇡
2
),
fM (
!x ) = (1 + g(!xM ))sin(x1 ⇡
2
).
subject to 0 xi 1, 8i = 1, 2, ..., n
where: !xM = M,M + 1, ..., xn and
g(!xM ) =
P
xi2!xM (xi 0.5)2
The Pareto-optimal solutions corresponds to
xi = 0.5 for all xi 2 !xM , 8i = M,M + 1, ..., n
and all objective function values must satisfy:PM
i=1(fi)
2 = 1. The POF is concave, scalable and
multi-modal.
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f1
f2
f3
135
MOP POF
DTLZ3 :
f1(
!x ) = (1 + g(!xM )) cos(x1 ⇡
2
) cos(x2
⇡
2
)...
cos(xM2
⇡
2
) cos(xM1
⇡
2
),
f2(
!x ) = (1 + g(!xM )) cos(x1 ⇡
2
) cos(x2
⇡
2
)...
cos(xM2
⇡
2
) sin(xM1
⇡
2
),
f3(
!x ) = (1 + g(!xM )) cos(x1 ⇡
2
)cos(x2
⇡
2
)...
sin(xM2
⇡
2
),
...,
fM1(!x ) = (1 + g(!xM )) cos(x1 ⇡
2
)sin(x2
⇡
2
),
fM (
!x ) = (1 + g(!xM ))sin(x1 ⇡
2
).
subject to 0 xi 1, 8i = 1, 2, ..., n where:
!xM = M,M + 1, ..., xn and g(!xM ) =
100[|!xM |+
P
xi2!xM (xi0.5)2cos(20⇡(xi0.5))]
It is suggested that k = |xM | = 10 ; g⇤ = 1.
00.10.20.30.40.5
0 0.1 0.2 0.3 0.4 0.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
f1
f2
f3
DTLZ4 :
f1(
!x ) = (1 + g(!xM )) cos(x⇡1
⇡
2
) cos(x⇡2
⇡
2
)...
cos(x⇡M2
⇡
2
) cos(x⇡M2
⇡
2
),
f2(
!x ) = (1 + g(!xM )) cos(x⇡1
⇡
2
) cos(x⇡2
⇡
2
)...
cos(x⇡M2
⇡
2
) sin(x⇡M1
⇡
2
),
f3(x) = (1 + g(
!xM )) cos(x⇡1
⇡
2
) cos(x⇡2
⇡
2
)...
sin(x⇡M2
⇡
2
),
...,
fM1(x) = (1 + g(!xM )) cos(x⇡1
⇡
2
) sin(x⇡2
⇡
2
),
fM(x) = (1 + g(!xM )) sin(x⇡1
⇡
2
).
subject to 0 xi 1, 8i = 1, 2, ..., n
where: !xM = M,M + 1, ..., xn and
g(!xM ) =
P
xi2!xM (xi 0.5)2.
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f1
f2
f3
136
MOP POF
DTLZ5 :
f1(
!x ) = (1 + g(!xM )) cos(1 ⇡
2
) cos(2
⇡
2
)...
cos(M2
⇡
2
) cos(M1
⇡
2
),
f2(
!x ) = (1 + g(!xM )) cos(1 ⇡
2
) cos(2
⇡
2
)...
cos(M2
⇡
2
) sin(M1
⇡
2
),
f3(
!x ) = (1 + g(!xM )) cos(1 ⇡
2
)cos(2
⇡
2
)...
sin(M2
⇡
2
),
...,
fM1(!x ) = (1 + g(!xM )) cos(1 ⇡
2
)sin(2
⇡
2
),
fM (
!x ) = (1 + g(!xM ))sin(1 ⇡
2
).
subject to 0 xi 1, 8i = 1, 2, ..., n where: i =
⇡
4(1+g(!xM )) (1+2g(
!xM )xi), for i = 2, 3, ..., (M1)
and g(!xM ) =
P
xi2!xM (xi 0.5)2.
0 0.2 0.4 0.6 0.8 0
0.2 0.4
0.6 0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f2
f1
f3
DTLZ6 :
f1(
!x ) = (1 + g(!xM )) cos(1 ⇡
2
) cos(2
⇡
2
)...
cos(M2
⇡
2
) cos(M1
⇡
2
),
f2(
!x ) = (1 + g(!xM )) cos(1 ⇡
2
) cos(2
⇡
2
)...
cos(M2
⇡
2
) sin(M1
⇡
2
),
f3(
!x ) = (1 + g(!xM )) cos(1 ⇡
2
)cos(2
⇡
2
)...
sin(M2
⇡
2
),
...,
fM1(!x ) = (1 + g(!xM )) cos(1 ⇡
2
)sin(2
⇡
2
),
fM (
!x ) = (1 + g(!xM ))sin(1 ⇡
2
).
subject to 0 xi 1, 8i = 1, 2, ..., n where:
i =
⇡
4(1+g(!xM )) (1 + 2g(
!xM )xi), for
i = 2, 3, ..., (M1) and g(!xM ) =
P
xi2!xM (xi)
0.1.
0
0.2
0.4
0.6
0.8 0 0.1
0.2 0.3
0.4 0.5
0.6 0.7
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f2f1
f3
137
MOP POF
DTLZ7 :
f1(
!x ) = (1 + g(!xM )) cos(x1 ⇡
2
) cos(x2
⇡
2
)...
cos(xM2
⇡
2
) cos(xM1
⇡
2
),
f2(
!x ) = (1 + g(!xM )) cos(x1 ⇡
2
) cos(x2
⇡
2
)...
cos(xM2
⇡
2
) sin(xM1
⇡
2
),
f3(
!x ) = (1 + g(!xM )) cos(x1 ⇡
2
)cos(x2
⇡
2
)...sin(xM2
⇡
2
),
...,
fM1(!x ) = (1 + g(!xM )) cos(x1 ⇡
2
)sin(x2
⇡
2
),
fM (
!x ) = (1 + g(!xM ))sin(x1 ⇡
2
).
subject to 0 xi 1, 8i = 1, 2, ..., n
where: !xM = M,M + 1, ..., xn and
g(!xM ) =
P
xi2!xM (xi 0.5)2
The Pareto-optimal solutions corresponds to
xi = 0.5 for all xi 2 !xM , 8i = M,M + 1, ..., n
and all objective function values must satisfy:PM
i=1(fi)
2 = 1. The POF is concave, scalable and
multi-modal.
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
2.5
3
3.5
4
4.5
5
5.5
6
f1
f2
f3
138
Tab. A.3: UF Problems
MOP POF
UF1 :
f1(
!x ) = x1 + 2| J1|
X
j2J1
[xj sin(6⇡x1 + j⇡
n
)]2,
f2(
!x ) = 1px1 + 2|J2|
X
j2J2
[xj sin(6⇡x1 + j⇡
n
]2
where J1 = {j|j is odd and (2 j n} and J2 = {j|j is even
and 2 j n}. The search space is [0, 1]⇥ [1, 1]n1. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
F1
F2
UF2 :
f1(
!x ) = x1 + 2|J1|
X
j2J1
y2j ,
f2(
!x ) = 1px1 + 2|J2|
X
j2J2
y2j
where J1 = {j|j is odd and (2 j n} and J2 = {j|j is even
and 2 j n} and
yj =
8>:
xj [0.3x21 cos(24⇡x1 + 4j⇡n ) + 0.6x1] cos(6⇡x1 + j⇡n )j 2 J1
xj [0.3x21 cos(24⇡x1 + 4j⇡n ) + 0.6x1] sin(6⇡x1 + j⇡n )j 2 J2
The search space is [0, 1]⇥ [1, 1]n1.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
F1
F2
UF3 :
f1(
!x ) = x1 + 2|J1| (4
X
j2J1
y2j 2
Y
j2J1
cos(
20yj⇡p
j
) + 2),
f2(
!x ) = 1px1 + 2|J2| (4
X
j2J2
y2j 2
Y
j2J2
cos(
20yj⇡p
j
) + 2)
where J1 and J2 are the same as those of UF1, and
yj = xj x
0.5(1.0+
3(j2)
n2 )
1 , j = 2, ..., n,. The search space is
[0, 1]n.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
F1
F2
139
MOP POF
UF4 :
f1(
!x ) = x1 + 2|J1|
X
j2J1
h(yj),
f2(
!x ) = 1 x21 +
2
|J2|
X
j2J2
h(yj)
where J1 = {j|j is odd and (2 j n} and J2 = {j|j is even
and 2 j n}
yi = xj sin(6⇡x1 + j⇡n ), j = 2, ..., n and h(t) = |t|1+e2|t| . The
search space is [0, 1]⇥ [2, 2]n1.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
F1
F2
UF5 :
f1(
!x ) = x1 + ( 1
2N
+ ✏)| sin(2N⇡x1)|+ 2|J1|
X
j2J1
h(yj),
f2(
!x ) = 1 x1 + ( 1
2N
+ ✏)| sin(2N⇡x1)|+ 2|J2|
X
j2J2
h(yj)
where J1 = {j|j is odd and (2 j n} and J2 = {j|j is even
and 2 j n}
yi = xjsin(6⇡x1+ j⇡n ), j = 2, ..., n and h(t) = 2t2cos(4⇡t)+1.
The search space is [0, 1]⇥ [1, 1]n1.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
F1
F2
140
MOP POF
UF6 :
f1(
!x ) = x1 +max{0, 2( 1
2N
+ ✏) sin(2N⇡x1)}
+
2
|J1| (4
X
j2J1
y2j 2
Y
j2J1
cos(
20yj⇡p
j
) + 2),
f2(
!x ) = 1 x1 +max{0, 2( 1
2N
+ ✏) sin(2N⇡x1)}
+
2
|J2| (4
X
j2J2
y2j 2
Y
j2J2
cos(
20yj⇡p
j
) + 2)
where J1 = {j|j is odd and (2 j n} and J2 = {j|j is even
and 2 j n}
yi = xj sin(6⇡x1 + j⇡n ), j = 2, ..., n. The search space is
[0, 1]⇥ [1, 1]n1.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
F1
F2
UF7 :
f1(
!x ) = 5px1 + 2
J1
X
j2J1
y2j ,
f2(
!x ) = 1 5px1 + 2
J2
X
j2J2
y2j
where J1 = {j|j is odd and (2 j n} and J2 = {j|j is even
and 2 j n}
yi = xj sin(6⇡x1 + j⇡n ), j = 2, ..., n. The search space is
[0, 1]⇥ [1, 1]n1.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
F1
F2
UF8 :
f1(
!x ) = cos(0.5x1⇡) cos(0.5x2⇡)+
2
|J1|
X
j2J1
(xj 2x2 sin(2⇡x1 + j⇡
n
))2,
f2(
!x ) = cos(0.5x1⇡) sin(0.5x2⇡)+
2
|J2|
X
j2J2
(xj 2x2 sin(2⇡x1 + j⇡
n
))2,
f3(
!x ) = sin(0.5x1⇡) + 2|J3|
X
j2J3
(xj 2x2 sin(2⇡x1 + j⇡
n
))2,
where J1 = {j|3 j n, and j 1 is a multiplication of
3}, J1 = {j|3 j n, and j 2 is a multiplication of 3}, J1 =
{j|3 j n, and j is a multiplication of 3}. The search space
is [0, 1]2 ⇥ [2, 2]n2.
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
f1f2
f3
141
MOP POF
UF9 :
f1(
!x ) = 0.5[max{0, (1 + ✏)(1 4(2x1 1)2)}+ 2x1]]x2
+
2
|J1|
X
j2J1
(xj 2x2 sin(2⇡x1 + j⇡
n
))2,
f2(
!x ) = 0.5[max{0, (1 + ✏)(1 4(2x1 1)2)} 2x1 + 2]x2
+
2
|J2|
X
j2J2
(xj 2x2 sin(2⇡x1 + j⇡
n
))2,
f3(
!x ) = 1 x2 + 2|J3|
X
j2J3
(xj 2x2 sin(2⇡x1 + j⇡
n
))2
where J1 = {j|3 j n, and j 1 is a multiplication of
3}, J1 = {j|3 j n, and j 2 is a multiplication of 3}, J1 =
{j|3 j n, and j is a multiplication of 3} and ✏ = 0.1. The
search space is [0, 1]2 ⇥ [2, 2]n2.
00.010.020.030.040.05
0
0.5
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f1f2
f3
UF10 :
f1(
!x ) = cos(0.5x1⇡) cos(0.5x2⇡) + 2
J1
X
j2J1
[4y2j cos(8⇡yj) + 1],
f2(
!x ) = 0 cos(0.5x1⇡) sin(0.5x2⇡) + 2
J2
X
j2J1
[4y2j cos(8⇡yj) + 1],
f3(
!x ) = sin(0.5x2⇡) + 2
J3
X
j2J1
[4y2j cos(8⇡yj) + 1]
where J1 = {j|3 j n, and j1 is a multiplication of 3}, J1 =
{j|3 j n, and j2 is a multiplication of 3}, J1 = {j|3 j
n, and j is a multiplication of 3} and yj = xj 2x2 sin(2⇡x1 +
j⇡
n ), j = 3, ..., n. The search space is [0, 1]
2 ⇥ [2, 2]n2.
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
f1f2
f3
142
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