**6. Conclusions**

60 Bio-Inspired Computational Algorithms and Their Applications

*a)* 200 0.6 0.01 0.4530713342399 180.693674 978 *b)* 200 0.6 0.4 0.2088516234 162.480752 991 *c)* 200 0.8 0.8 0.1039548356200 168.681711 996 *d)* 500 0.6 0.01 0.1441113 250.446997 981 *e)* 500 0.6 0.4 0.07266868 282.191 960 *f)* 500 0.8 0.8 0.0558228894 290.697 987 *g)* 1000 0.6 0.01 0.059796068 468.693084 999 *h)* 1000 0.6 0.4 0.0532988776 480.819397 947 *i)* 1000 0.7 0.5 0.119467646451 457.490696 999 *j)* 1000 0.7 0.7 0.03260650948 524.067239 989 *k)* 1000 0.8 0.5 0.099396876739 536.369984 993 *l)* 1000 0.85 0.8 0.0311870033413 550.612374 972 *m)* 1500 0.6 0.1 0.1962062488 619.52535 981 *n)* 1500 0.6 0.4 0.090105144 672.77304 990 *o)* 1500 0.95 0.85 0.163192355 1046.2808 968 *p)* 2000 0.7 0.7 0.08380448 1116.16188 999 *q)* 2000 0.85 0.8 0.0114246856933 1295.874818 987 *r)* 2000 0.95 0.85 0.0277589482798 1306.641231 1000

Table 5. Adjustment of six-bar mechanism parameters for a specific path.

The optimization process is iterative, and it was demonstrated with the tests that were realized varying the parameters of the genetic algorithm to analyze the behavior of the system, which means that they can be modified until finding a system whose behavior satisfies the expectations and requirements of the designer. The parameters of the GA usually interact with each other in a nonlinear relation, that's why they cannot be optimized in an independent way, been demonstrated in the presented study cases. When existing a change in the population size, this fact will be reflected in time of convergence and accuracy

It was demonstrated that the diversity of individuals in the population is obtained and it remains along with the operator of crossing and the genetic mutation, since in all the analysis, they allow to find better solutions and avoid premature convergence to the maximum premises. Although also it must be mentioned that the elitism and the forced inheritance help to limit the number of individuals that will cover the imposed restrictions. On the other hand, it was observed that the GA has few possibilities of making considerable or necessary a number of reproductions for the optimal solution if it has an insufficient or small population.

**5***.* **Discussion**

in the path generation.

*ni Pc Pm error time generations* 

When operating with a population reduced in number of individuals, a sufficient representative quantity of the different regions of the solution space is not achieved, but the necessary computation time to create a new generation of possible solutions diminishes dramatically. When considering a high percentage of the probability of mutation in the algorithm, one assures a heuristic search made in different regions of the solution space, this combined with the forced inheritance mechanism has demonstrated that for the problem treated in this work, it is a strategy that power the heuristic capacities of the GA, for nonlinear multidimensional problems, non-homogenous, becoming the algorithm metaheuristic; it is demonstrated then that an important improvement in the diminution of the error is obtained, around 20% with respect to the reported works previously. Also it was observed that the increase in the percentage of mutation improves the off-line performance, since all the solutions in the population are taken into account to obtain the optimal value. The off-line performance does not penalize the algorithm to explore poor regions of the search space, as long as it contributes to reach the best possible solutions in terms of aptitude.

It was verified that for the crossover the rule is fulfilled of which applying values smaller to 0.6, the performance is not optimal and it does not change the expected result for a specific problem. In the case of mutation, one demonstrated that this one can change no mattering the number of times and increasing its value to obtain optimal results, reaching almost at the unit, but avoiding to muter totally all the chromosomes eliminating the benefits created by the elitism and the forced inheritance mechanism.

By means of the trial and error, also one concludes that the parameters are not independent, and searching systematically to obtain all the possible combinations of these, is almost

Performance of Simple Genetic Algorithm

*Michigan Press*.

202.

Inserting Forced Inheritance Mechanism and Parameters Relaxation 63

Grosso, P. 1985. Computer Simulations of Genetic Adaption: Parallel Subcompnent Interaction in a Multilocus Model. *Ph.D Dissertation, University of Michigan*. Hidalgo, J. I. & Lanchares, H. R. 2000. Partitioning and placement for multi-fpga systems

Holland, J. H. 1975. Adaptation in natural and artificial system. *Ann Arbor, The University of* 

Kalnas, R. & Kota, S. 2001. Incorporating Uncertaintly intoMechanism Synthesis. *Mechanism* 

Kunjur, A. & Krishnamurty, S. 1997. Genetic Algorithms in Mechanism Synthesis. *Journal of* 

Kuri-Morales, A. & Galaviz-Casas, J. 2002. Algoritmos Genéticos. *Instituto Politécnico* 

Laribi, M. A., Mlika, A., Romdhane, L. & Zeghloul, S. 2004. A combined genetic algorithm-

Levitski, N. L. & Shakvazian, K. K. 1960. Synthesis of four element spatial mechanisms with

Lima, C. A. F. 2005. Combining Competent Crossover and Mutation Operators:a

Lima, C. F., Sastry, K., Goldberg, D. E. & Lobo, F. G. 2005. Combining Competent Crossover

Lugo-González, E., Hernández-Gómez, L. H., Ponce-Reynoso, R., Velázquez-Sánchez, A. T.,

lower pairs. *International Journal of Mechanical Sciences 2***,** 76-92.

Probabilistic Model Building Approach. *GECCO'05*.

*Nacional, Universidad NAcional Autonoma de México, Fondo de Cultura Económica,***,**

fuzzy logic method (GA-FL) in mechanisms synthesis. *Mechanism and machine* 

and Mutation Operators: a Probabilistic Model Building Approach. *GECCO'05,*

Urriolagoitia-Sosa, G., Merchán-Cruz, E. A. & Ramírez-Gordillo, J. 2010. Performance Optimization of GA Based Planar Mechanism Synthesis. *In Proceedings of the 2010 IEEE Electronics, Robotics and Automotive Mechanics Conference (September 28 - October 01, 2010). IEEE Computer Society, Washington, DC***,** 126-131. Lugo González, E. 2010. *Diseño de mecanismos utilizando algoritmos genéticos con aplicaciòn en prótesis para miembro inferior.*Doctorado, Instituto Politécnico Nacional. Merchán-Cruz, E. A. 2005. *Soft-computing techniques in the trajectory planning of robot manipulators sharing a common workspace.* Doctor of Philosophy, Sheffield. Merchán-Cruz, E. A., Hernández-Gómez, L. H., Velázquez-Sánchez, A. T., Lugo-González,

E. & Urriolagoitia-Sosa, G. 2007. Exploiting monotony on a genetic algorithm based trajectory planner (GABTP) for robot manipulators. ). F. *In the 16th IASTED international Conference on Applied Simulation and Modelling (Palma de Mallorca, Spain,* 

Campos-Padilla, I. Y., Muňoz-César, J. J. & Lugo-González, E. 2008. GA Based Trajectory Planner for Robot Manipulators Sharing a Common Workspace with Adaptive *Population Size. In Proceedings of the 2008 Electronics, Robotics and Automotive Mechanics Conference (September 30 - October 03, 2008). IEEE Computer* 

*August 29 - 31, 2007,* De Felice, Ed. ACTA Press, Anaheim, CA**,** 300-305. Merchán-Cruz, E. A., Urriolagoitia-Sosa, G., Ramírez-Gordillo, J., Rodríguez-Caňizo, R.,

Michalewicz, Z. 1999. Genetic Algorithms + Data Structure = Evolution Programs. *tercera ed.* 

Norton, R. L. 1995. *Diseño de Maquinaria,* Impreso en México, Mc. Graw Hill.

using genetic algorithms. *In Proceedings of the Euromicro DSD 2000*.

*and machine theory (Mech. mach. theory),* 36, No.3**,** 843-851.

*Applied Mechanisms and Robotics,* 4 No. 2**,** 18-24.

*theory (Mech. mach. theory),* 39**,** 717-735.

ACM 1595930108/ 05/0006.

*Society, Washington, DC***,** 520-525.

*Nueva York: Springer*.

impossible; but if the parameters were optimized one at the time, it is then possible to handle its interactions and, for a given problem, the values of the selected parameters are not necessarily the optimal ones, but if they are analyzed uniformly they will generate more significant values.

#### **7. Acknowledgment**

The authors thank to Instituto Politécnico Nacional, Project Number. 20113426, for the facilities and means for the development of this research.

#### **8. References**


62 Bio-Inspired Computational Algorithms and Their Applications

impossible; but if the parameters were optimized one at the time, it is then possible to handle its interactions and, for a given problem, the values of the selected parameters are not necessarily the optimal ones, but if they are analyzed uniformly they will generate more

The authors thank to Instituto Politécnico Nacional, Project Number. 20113426, for the

A. K. Mallik & A. Ghosh 1994. Kinematic Analysis and Synthesis of Mechanisms. *CRC-Press***,**

Beasley, D., Bull, D. R. & Martin, R., And R. 1993. An overview of genetic algorithms: part 1,

Bethke, A. 1976. Comparison of Genetic Algorithms and Gradient-Based Optimizers on

Bulatovic, R. R. & Djordjevic, S. R. 2004. Optimal Synthesis of a Four-Bar Linkage by Method

Cabrera, J. A., Simon, A. & Prado, M. 2002. Optimal synthesis of mechanisms with genetic

Cantu-Paz, E. 2000. Efficient and Accurate Parallel Genetic Algorithms. *Kluwer, Boston, MA*. Coello-Coello, C. A. 2007. Introducción a la computación evolutiva. *In:* CINVESTAV-IPN

De Jong, K. A. 1975. *An analysis of the behaviour of a class of genetic adaptive systems.* Tesis

Endre Eiben, A., Hinterding, R. & Michalewicz, Z. 1999. Parameter Control in Evolutionary Algorithms. *IEEE Transactions on Evolutionary Computation,* 3, No. 2. Fogarty, T. 1989. Varying the probability of mutation in the genetic algorithm. *Proc. 3rd Int. Conf. Genetic Algorithms, J. D. Schaffer, Ed. San Mateo, CA: Morgan Kaufmann*. Freudenstein, F. 1954. An analitical approach to the design of four link mechanism.

Goldberg, D. 2002. Lessons from and for Competent Genetic Algorithms. *Kluwer, Boston,* 

Goldberg, D. E. 1989. *Genetic algorithms in search, optimization, and machine learning,* USA,

Grefenstette, J. J. 1986. Optimization of control parameters for genetic algorithms. *IEEE* 

Denavit, J. & Hartenberg, R. S. 1964. Kinematic Synthesis of Linkages. *USA: Mc. Graw Hill*. Dewen, J., Ruihong, Z., Ho, D., Rencheng, W. & Jichuan, Z. 2003. Kinematic and dynamic

*Rehabilitation Research and Development,* 40,No. 1**,** 39–48.

Parallel Processors: Efficiency of Use of Processing Capacity. *Tech. Rep. No. 197,* 

of Controlled Deviation. *The first international conference on computational mechanics* 

algorithms. *Mechanism and machine theory (Mech. mach. theory),* 37 No10**,** 1165-

performance of prosthetic knee joint using six-bar mechanism *Journal of* 

significant values.

**8. References** 

688.

1177.

*MA*.

Addison - Wesley.

(ed.). Mèxico.

**7. Acknowledgment** 

facilities and means for the development of this research.

fundamentals. *University Computing,* 15**,** 58-69.

*Logic of Computers Group, University of Michigan*.

*(CM'04),* 31,No.3-4**,** 265-280.

doctoral, University of Michigan.

*Transactions of the ASME 76***,** 483-492.

*Trans. Systems, Man, Cybern,* 16, no. 1**,** 122-128.


**0**

**4**

*Republic of Korea*

**The Roles of Crossover and Mutation in**

<sup>1</sup>*School of Computer Science and Engineering, Seoul National University, Seoul* <sup>2</sup>*Department of Computer Science and Engineering, Kwangwoon University, Seoul*

We recognized that the roles of crossover and mutation in real encoding are quite different from those in binary encoding during performing previous work with real-coded genetic algorithms (Yoon et al., 2012). In this study, we are to argue the distinct roles of genetic

Recently many studies on evolutionary algorithms using real encoding have been done. They include ant colony optimization (Socha & Dorigo, 2008), artificial bee colony algorithm (Akay & Karaboga, 2010; Kang et al., 2011), evolution strategies (ES) (Beyer, 2001), differential evolution (Das & Suganthan, 2011; Dasgupta et al., 2009; Kukkonen & Lampinen, 2004; 2005; Mezura-Montes et al., 2010; Noman & Iba, 2005; Rönkkönen et al., 2005; Storn & Price, 1997; Zhang et al., 2008), particle swarm optimization (Chen et al., 2007; Huang et al., 2010; Juang et al., 2011; Krohling & Coelho, 2006; l. Sun et al., 2011), and so on. In particular, in the field of ES, we can find many studies based on self-adaptive techniques (Beyer & Deb, 2001; Hansen & Ostermeier, 2001; Igel et al., 2007; 2006; Jägersküpper, 2007; Kita, 2001; Kramer, 2008a;b;

Many researchers have also concentrated on using real-valued genes in genetic algorithms (GAs), as in (Ripon et al., 2007). It is reported that, for some problems, real-coded representation and associated techniques outperform conventional binary representation (Eshelman & Schaffer, 1993; Herrera et al., 1998; Janikow & Michalewicz, 1991; Lozano et al., 2004; Ono et al., 1999; Ono & Kobayashi, 1997; Surry & Radcliffe, 1996; Wright, 1991). Several theoretical studies of real-coded GAs have also been performed (Goldberg, 1991; Higuchi et al., 2000; Kita et al., 1998; Qi & Palmieri, 1994a;b). However, the role and behavior of genetic operators in real-coded GAs are fundamentally different from those in binary encodings

In this chapter, we try to verify different properties of crossover and mutation in real encodings from those in binary encodings through various experiments. We especially concentrate on the effect of genetic operators (the bias and functions of crossover and

Kramer et al., 2007; Meyer-Nieberg & Beyer, 2007; Wei et al., 2011).

although motivation of the operators and the framework of GAs are similar.

mutation) when they are used in real-coded GAs.

\*Corresponding author: Yong-Hyuk Kim

**1. Introduction**

operators in real encodings.

**Real-Coded Genetic Algorithms**

Yourim Yoon1 and Yong-Hyuk Kim2\*

