Proportional and Tournament Selections for
Constrained Optimization Problems Using Genetic
Algorithm {[Tytuł] 16 point bold}
Research paper {[Paper] 11 point bold italic}
{[Name] 10 point}
osyczka@mech.pk.edu.pl
Dept. of Mechanical Engineering, Cracow University of Technology, Al. Jana Pawla II 37, 31-864
Krakow, Poland {[Address] 9 point italic}
Andrzej OSYCZKA
Sourav KUNDU
sourav@kenroku.ipc.kanazawa-u.ac.jp
Dept. of Human and Mechanical Systems Engineering, Faculty of Engineering, Kanazawa University,
2-40-20 Kodatsumo, Kanazawa 920, Japan
Abstract. In this paper the use of proportional and tournament selections in solving constrained
optimization problems is discussed. ... {[Abstract] 9 point, between 150 and 200 words}
Keywords: Genetic algorithm, constrained optimization, {[Keywords] 9 point, between 5 and 10 phrases}
1. Introduction {[Mainheadings] 11 point bold}
Genetic Algorithms (GAs) are wildly used to solve nonlinear programming problems and
many different GA based strategies have been developed recently to solve these problems.
Liepins & Potter (1990) proposed the repairing strategy in which a non-feasible
chromosome is taken and repaired to a feasible chromosome through a repairing procedure.
Michalewicz et. al. (1996) developed the modifying genetic operations strategy in which
specialized genetic operations are used to maintain the feasibility of chromosomes.
{[MainText] 10 point through all the text}
In this paper... {[Indent] indent 0.5 cm}
2. Problem formulation
The general nonlinear programming problem for single criterion optimization is formulated
as follows: Find x* = [x1*, x2*, ..., xN*] which will satisfy the K inequality constraints
g k (x)  0
k = 1, 2, …, K {[Math]
and the M equality constraints
h m (x) = 0
m = 1, 2, …, M
and minimize the objective function f(x)
f(x*) = min f(x)
indent 1.5 cm}
(1)
(2)
(3)
2.1 Subtitles {[Subheadings] 10 point bold}
Table 1 Results of the numerical example. {9 point, centered}
Figure 1. Graphical illustration of the results. {9 point, centered}
References
Goldberg, D.E. (1989) Genetic Algorithms in Search, Optimization and Machine Learning, AddisonWesley, Reading. {[References] 9 point, indent 0,5 cm}
Michalewicz, Z. & Schoenauer, M. (1996) Evolutionary Algorithms for Constrained Parameter
Optimization Problems, Evolutionary Computation, 4(1) 1-32.
Osyczka, A. & Kundu, S. (1996) A Genetic Algorithm Approach to Multicriteria Network
Optimization Problems”. In Proceedings of 20-th International Conference on Computers &
Industrial Engineering, Kyongju, Korea, 329-332.
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