Research Journal of Applied Sciences, Engineering and Technology 4(2): 90-92 , 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: September 23, 2011
Accepted: October 24 , 2011
Published: January 15, 2012
Dynamic Multi-objective Optimization Algorithm Based
On GEP and Virus Evolution
Weihong Wang, Yanye Du, Qu Li and Zhaolin Fang
College of Computer Science, Zhejiang University of Technology, China
Abstract: Dynamic Multi-objective Optimization (DMO) is very popular nowadays. A new algorithm for DMO
called Virus-GEP Dynamic based on Gene Expression Programming (GEP) and virus evolution is proposed.
Experiments on two test problems have shown that the algorithm has better performance on convergence,
diversity and the breadth of the distribution.
Key words: Dynamic Multi-objective Optimization (DMO), Gene Expression Programming (GEP), virus
evolution
INTRODUCTION
optimization algorithm NSGA-II (Deb et al., 2002) to
make it suitable for DMO, while using GEP for encoding
and virus evolution (Xian-bin et al., 2001; Shicheng et al.,
2003) for evolution mechanism.
Dynamic multi-objective optimization: Many systems
need to consider dynamic scheduling problems, and these
constraints are called dynamic constraints. Mathematical
models abstracted from problems with multiple objectives
and related with time factors are Dynamic Multi-objective
Optimization (referred to as DMO) (Farina et al., 2002;
Shang et al., 2007).
Initialization: Initialization of the host population
hostPop(t) and the virus population hostPop(t).
Generate the subpopulation S by the host population
P. Take genetic operations and selection operation to gain
the new next host population hostPop(t+1). Then take
virus-infection operation on each host individual, and the
host individuals infected by the virus individual
composesthe set U. Set the subpopulation S = hostPop
(t+1). Then update the virus population.
Generate the next host population by Pt and S:
GEP: Gene Expression Programming (GEP) (Ferreira,
2001) was invented by Candida Ferreira. A GEP gene is
the basic unit of a GEP genome and consists of head and
tail parts. The gene is then mapped into an Expression
Tree (ET) by following a width-first fashion and the ET
is easy to be converted into a mathematical expression.
•
Virus evolutionary mechanism: As a major component
of the biological immune system, virus system has many
information processing mechanisms and functional
features, and it has a great significance for further
improvements in genetic algorithm, gene expression
programming, and so on (Xian-bin et al., 2001).
•
New dynamic multi-objective optimization algorithm
Virus-GEP dynamic coding: Each individual
(chromosome) are composed of n genes, where n is equal
to the number of decision variables of the multiobjective optimization problem. Each gene is responsible
for one decision variable assignment. Suppose the genes
{G1, G2, ..., Gn}, the corresponding terminal set for each
gene Gi(i = 1, 2, ..., n) is Ti = {0, 1, 2, ..., 9, g1, g2, ..., gi-1}
(i = 1, 2, ...., n), where figures 0, 1, ..., 9 show its location
in the constants filed C.
Generate the new population R = PcS, and do nondominated sorting of R to get the non-dominated
fronts F!, F2, ....
For Fi all, do sorting according to the CrowdedComparison Operator, and choose the N best
individuals to compose the population P!.
EXPERIMENTS AND ANALYSIS
Convergence and diversity measures: For convergence
measure, we use the measure of the objective space used
in (Farina et al., 2002), which is shown in Eq (1).
ef t  
1
np
np

j 1
min S p, i (t )  X sol
j
i 1: nh
(1)
For diversity measure, we use the distribution of the
optimal solutions obtained in the objective space to
describe the diversity of the solutions.
Flow of virus-GEP dynamic algorithm: The new
algorithm is designed on the classic multi-objective
Corresponding Author: Weihong Wang, College of Computer Science, Zhejiang University of Technology, China
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Res. J. Appl. Sci. Eng. Technol., 4(2): 90-92, 2012
(a) Results of DBM[1]
(b) Results of D-GEP Chaotic NSGA-II
1.5
1.0
-4.0
-4.5
-5.0
-0.5
lgeƒ
lgeƒ
0.5
0
-1.0
-5.5
1.5
-6.0
-2.0
-6.5
-2.5
-3.0
-7.0
0
5
10
t
15
0
20
(c) Results of DBM[1]
Fig. 1: Result of FDA
5
10
t
15
20
(d) Results of D-GEP Chaotic NSGA-II
2.0
2.0
1.5
f2
ƒ2
1.5
1.0
1.0
0.5
0.5
0
0
0
0.5
1.0
f1
1.5
0
2.0
1.0
f-1
1.5
2.0
(b) Results of D-GEP Chaotic NSGA-II
0
1.5
1.0
0.5
0
-0.5
-1.0
1.5
-2.0
-2.5
-3.0
-3.5
-1
lgeƒ
lgeƒ
(a) Results of DBM[1]
0.5
-2
-3
-4
0
2
4
6
t
8
10
-5
12
0
(c) Results of DBM[1]
2
4
6
t
8
10
(d) Results of D-GEP Chaotic NSGA-II
Fig. 2: Result of FDA2
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Res. J. Appl. Sci. Eng. Technol., 4(2): 90-92, 2012
Analysis of the experimental results: We use FDA1,
FDA2 as test problems from (Farina et al., 2002) to study
the dynamic multi-objective optimization problems.
ACKNOWLEDGMENT
The work is supported by the National Natural
Science Foundation under grant 60873033, Natural
Science Foundation of Zhejiang Province under grant
R1090569 and Project of Department of Science and
Technology of Zhejiang Province under grant
2009C31108.
FDA1 test problem: Compare Fig. 1a with b, we find
that, the new algorithm can find the optimal solutions in
a wide range at each time step, that is, distribution of the
obtained solutions is broader and uniform in the objective
space. On the other hand, compare Fig. 1c with d, we can
see that, convergence of the new algorithm is much better
than DBM, the value of lg(ef) can keep change in the
range of [-6.4, -4.3].
REFERENCES
Deb, K., A. Pratap, S. Agarwal and T. Meyarivan, 2002.
A fast and elitist multi-objective genetic algorithm:
NSGA-II. IEEE T. Evol Comput., 6(2): 182-197.
Farina, M., K. Deb and P. Amato, 2002. Dynamic
multiobjective optimization problems: test cases,
approximations and applications. IEEE T. Evol.
Comput., 8(5): 425-442.
Ferreira, C., 2001. Gene expression programming: A new
adaptive algorithm for solving problems. Complex
Syst., 13(2): 87-129.
Shang, R., L. Jiao, M. Gong and M.A. wp, 2007. An
immune clonal algorithm for dynamic multi-objective
optimization. J. Software, 18(11): 2700-2711, (In
Chinese).
Shicheng, H., X. Xiaofei and D. Zhan, 2003. A Virus
Evolutionary Genetic Algorithm for Large Product
Structure Optimization Problem. J. Comp. Integrated
Manufacturing Syst., 9(3): 202-205.
Xian-bin, C., B. Wang, and X. Wang, 2001. A virus
evolutional genetic algorithm [J]. Mini-micro Syst
doi:cnki:sun:xxwx.0.2001-0.1-014.
FDA2 test problem: Compare Fig. 2a with b, we can see
that, uniformity of the new algorithm is poor, but to
maintain a good diversity. On the other hand, compare
Fig. 2c with d, convergence of the new algorithm is much
better than DBM, the value of lg(ef) can keep change in
the range of [-4.8, -0.7].
CONCLUSION
We proposed a new algorithm Virus-GEP Dynamic
based on GEP and virus evolution to solve DMO
problems. Through the horizontal infection operation, this
algorithm increases the diversity of the host population
during the evolutional process, and make it easier to jump
out of the local optimum, resulting in faster search for
better solution. We took experiments on two test
problems, and the results shown that the algorithm has
better performance on convergence, diversity and the
breadth of the distribution.
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