Performance improvement of real coded genetic algorithm with Quadratic Approximation based hybridisation

Int. J. Intelligent Defence Support Systems, Vol. 2, No. 4, 2009 319

Copyright © 2009 Inderscience Enterprises Ltd.

Performance improvement of real coded genetic algorithm with Quadratic Approximation based hybridisation

Kusum Deep* and Kedar Nath Das Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand 247667, India E-mail: [email protected] E-mail: [email protected] *Corresponding author

Abstract: Due to their diversity preserving mechanism, real coded genetic algorithms are extremely popular in solving complex non-linear optimisation problems. In recent literature, Deep and Thakur (2007a, 2007b) proved that the new real coded genetic algorithm (called LX-PM that uses Laplace Crossover and Power Mutation) is more efficient than the existing genetic algorithms that use combinations of Heuristic Crossover along with Non-Uniform or Makinen, Periaux and Toivanen Mutation. However, there are some instances where LX-PM needs improvement. Hence, in this paper, an attempt is made to improve the efficiency and reliability of this existing LX-PM by hybridising it with quadratic approximation (called H-LX-PM). To realise the improvement, a set of 22 benchmark test problems and two real world problems, namely a system of linear equations b frequency modulation parameter identification problem, have been

considered. The numerical and graphical results confirm that H-LX-PM really exhibits improvement over LX-PM in terms of efficiency, reliability and stability.

Keywords: hybrid real coded genetic algorithm; Laplace crossover; PM; power mutation; QA; quadratic approximation.

Reference to this paper should be made as follows: Deep, K. and Das, K.N. (2009) ‘Performance improvement of real coded genetic algorithm with Quadratic Approximation based hybridisation’, Int. J. Intelligent Defence Support Systems, Vol. 2, No. 4, pp.319–334.

Biographical notes: Kusum Deep is an Associate Professor, at Department of Mathematics, Indian Institute of Technology Roorkee, India. Six students have completed their PhD under her supervision and five more are currently in progress. She has authored a text book entitled ‘Optimisation Techniques’. She has around 35 research publications in journals and around 50 in conferences. She is on the editorial board of many international and national journals. She is a member of IEEE, ORSI, IMS, CSI. Her area of specialisation is numerical optimisation and applications. Currently she is working on evolutionary computations, particularly, genetic algorithms, memetic algorithms, particle swarm optimisation.

320 K. Deep and K.N. Das

Kedar Nath Das is a Senior Lecturer in the Department of Mathematics, KIIT University, Bhubaneswar, Orissa, India. He received his PhD degree from the Department of Mathematics, Indian Institute of Technology Roorkee (2008). While pursuing his Doctoral degree, he was awarded with the Ministry of Human Resources fellowship by the Government of India. He is a life member of the Operations Research Society of India (ORSI), Indian Society for Technical Education (ISTE) and Orissa Mathematical Society (OMS). He has ten publications in refereed journals and conferences. He has visited Las Vegas, USA, to present one of his research papers in WORLDCOMP (2006). His research focuses on designing efficient hybrid genetic algorithms to solve real life problems.

1 Introduction

The success story of Real Coded Genetic Algorithms is not disputable any longer as far as solving real world applications are concerned. However, many efforts are being made to improve their efficiency, reliability and stability by designing Hybrid Genetic Algorithms or Memetic Algorithms for obtaining the global optimal solution of non-linear optimisation problems. In Deep and Das (2007), a detailed review of the available literature on hybridised Genetic Algorithms is presented.

In Deep and Thakur (2007a, 2007b) a new real coded genetic algorithm called LX-PM is presented. It uses real encoding of parameters and defines Laplace Crossover and Power Mutation. Its performance is evaluated on a number of test problems taken from the literature by comparing with similar crossovers (Heuristic Crossover) and similar mutations (Non-Uniform or Makinen, Periaux and Toivanen Mutation).

Although most of the problems could be solved, there is still some scope for improvement. Hence, an attempt is made in this paper to incorporate the Quadratic Approximation (QA) as an additional operator in the GA cycle. The motivation behind this is that QA is a 3-parent, 1-child operator and the current generation can be exploited further to search for better individuals. Also, it may help in fighting premature convergence. This hybrid version is called H-LX-PM. The performance of LX-PM and H-LX-PM is evaluated on a set of 22 standard benchmark problems of varying difficulty levels.

The paper is organised as follows. In Section 2, the Real Coded GA, namely LX-PM is described. The proposed hybrid version of LX-PM, namely, H-LX-PM is presented in Section 3. In Section 4, the discussion of numerical results on 22 scalable benchmark test problems is given. In Section 5, both LX-PM and H-LX-PM are used to solve two real world problems. Finally, the conclusions are presented in Section 6.

2 LX-PM algorithm

LX-PM is a Real Coded Genetic Algorithm presented in Deep and Thakur (2007a). The working principle of LX-PM in the form of pseudo-code is given as follows.

Performance improvement of real coded genetic algorithm 321

We now state the Laplace Crossover Operator and the Power Mutation Operator used in LX-PM.

2.1 The Laplace crossover operator

The Laplace Crossover Operator (LX) by Deep and Thakur (2007a) is a parent centric operator that uses Laplace Distribution. The Density Function of Laplace distribution is given by

1( ) exp ,2

x af x x

b b −

= − − ∞ < < ∞

(1)

and distribution function of Laplace distribution is given by

1 exp ,2

( )11 exp ,2

x ax a

bF x

x ax a

b

− ≤

= − − − >

(2)

where, a R∈ is called the location parameter and 0b > is termed as the scale parameter. The lower the value of b, the higher is the probability of creating offsprings near the parents. For example, if 0.5b = the probability of creating offsprings near the parents is higher and for 1,b = distant points are likely to be created as offsprings.

Using LX, two offsprings (1) (1) (1) (1)1 2( , , , )ny y y y= … and (2) (2) (2) (2)

1 2( , , , )ny y y y= … are generated from a pair of parents (1) (1) (1) (1)

1 2( , , , )nx x x x= … and (2) (2) (2) (2)1 2( , , , )nx x x x= … in

the following way. First, two uniformly distributed random numbers , [0, 1]i iu u′ ∈ are generated.

Then a random number iβ is generated that follows the Laplace distribution just by inverting the distribution function of Laplace distribution as follows

1log ( ),2 .1log ( ),2

e i i

i

e i i

a b u u

a b u uβ

′− ≤= ′+ >

(3)


The offsprings are given by the equation (1) (1) (1) (2)i i i i iy x x xβ= + − (4)

(2) (2) (1) (2) .i i i i iy x x xβ= + − (5)

2.2 The Power Mutation operator

The Power Mutation (PM) by Deep and Thakur (2007b) is based on Power Distribution. Its distribution function is given by

1( ) 0 1pf x px x−= ≤ ≤ (6)

and the density function is given by

( ) 0 1pF x x x= ≤ ≤ (7)

where 0p > is the index of the distribution. PM is used to create a solution y in the vicinity of a parent solution x in the following manner. First, a uniform random number t between 0 and 1 is created and a random number s is created which follows the above mentioned distribution. Then following formula is used to create the muted solution

( ) if( ) if

l

u

x s x x t ry

x s x x t r − − <

= + − ≥

(8)

where /l u li i it x x x x= − − and lx and ux are lower and upper bounds of the decision

variable and r is a uniformly distributed random number between 0 and 1. The strength of mutation is governed by the index of the mutation (p). For small values of p less perturbance in the solution is expected and for large values of p more diversity is achieved. The probability of producing a mutated solution y on left (right) side of x is proportional to distance of x from ( )l ux x and the muted solution is always feasible.

3 The proposed hybridised real coded GA, H-LX-PM

In this section, we present the proposed hybridised Real Coded GA (H-LX-PM). In order to hybridise LX-PM, an additional operator called QA (Mohan and Nguyen, 2004) is used. The motivation behind this is to investigate each generation further in order to produce few better fit individuals. The QA is a three parent operator which produces one child. QA generates the minima of the quadratic hyper surface generated by three chosen individuals. Thus, even in a narrow search space it helps to reach faster to the nearest local or global optimal solution. The QA operator works in two steps, as follows.

1 From the current generation, select the individual R1, with the best fitness value and two random distinct individuals, R2 and R3.

2 Find the point of minima (child) of the quadratic surface passing through R1, R2 and R3 defined as:


2 2 2 2 2 22 3 1 3 1 2 1 2 3

2 3 1 3 1 2 1 2 3

( ) ( ) ( ) ( ) ( ) ( )Child = 0.5

( ) ( ) ( ) ( ) ( ) ( )R R f R R R f R R R f RR R f R R R f R R R f R

− + − + − − + − + −

(9)

where 1( ),f R 2( )f R and 3( )f R are the fitness function values at R1, R2 and R3, respectively.

The pseudo-code of H-LX-PM becomes:

Note that the QA is introduced at the end of each GA cycle just after completion of the complete elitism. Hence it helps in replacing few individuals in the population obtained after selection, crossover, mutation and elitism by probably few improved individuals; here, ‘improved’ in the sense of having better objective function values closer towards the minima.

4 Numerical results and discussion (for benchmark problems)

In order to demonstrate the effectiveness of the hybridisation of the Real Coded Genetic Algorithm, LX-PM, a set of 22 benchmark scalable test problems is selected from literature. These problems are of varied difficulty levels. The problems include unimodal and multi-modal functions which are scalable (the problem size can be varied as per the user’s choice). Here, the problem size for all problems is kept fixed to 10. However, more or less the parallel results are found to be obtained by varying the size of the problems (showing all the results is not within the scope of this paper). The test bed of the problems is listed in Table 1.

LX-PM and H-LX-PM are implemented in C++ and the experiments are carried out on a P-IV, 2.8 GHz machine with 512 MB RAM under WINXP platform. Extensive experiments have been exhibited to fine tune the parameters present in the crossover and mutation operators. Finally, in crossover we fix a = 0, b = 0.35, Pc = 0.55, Pm = 0.005, and in mutation, p = 0.25. The population size is kept fixed to ten times the number of decision variables. A total of 100 runs is performed for each of the two algorithms, namely, LX-PM and H-LX-PM, each time using a different seed for the generation of random numbers. However, the same seeds are used for each run to start from unique initial population. To stop a run, the criterion is either there is no further improvement in the objective function value (within a radius of 0.01) for 100 consecutive generations


or a maximum of 2000 generations is attained. A run is considered to be a success if the objective function value obtained by the algorithm is within 1% accuracy of the known optimal value.

Table 1 $Functions used as test bed

S. No. Name Function Bounds

1 Ackley 2

1 1

1 120exp 0.02 exp cos(2 ) 20n n

i ii i

x x en n

π= =

− − − + + ∑ ∑ [–30, 30]

2 Cosine mixture

2

1 10.1 0.1 cos(5 )

n n

i ii i

n x xπ= =

+ −∑ ∑ [–1, 1]

3 Exponential 2

1

1 exp 0.5n

ii

x=

− −

∑ [–1, 1]

4 Griewank 2

1 1

11 cos4000

nni

ii i

xxi= =

+ −

∑ ∏ [–600, 600]

5 Levy and Montalvo-1

12 2 2 2

1 11

10sin ( ) ( 1) [1 10sin ( )] ( 1) ,

11 ( 1)4

n

i i ni

i i

y y y yn

y x

π π π−

+=

+ − + + −

= + +

∑ [–10, 10]

6 Levy and Montalvo-2

))

12 2 2

1 11

2 2

0.1 sin (3 ) ( 1) 1 sin (3 )

( 1) 1 sin (2 )

n

i ii

n n

x x x

x x

π π

π

−

+=

+ − + +

− +

∑ [–5, 5]

7 Paviani 0.21010

2 2

1 1

45.778 (ln( 2)) (ln(10 ))i i ii i

x x x= =

+ − + − − ∑ ∏ [2, 10]

8 Rastrigin 2

110 10cos(2 )

n

i ii

n x xπ=

+ − ∑ [–5.12, 5.12]

9 Rosenbrock 1

2 2 21

1100( ) ( 1)

n

i i ii

x x x−

+=

− + − ∑ [–30, 30]

10 Schwefel ( )1

418.9829 sin | |n

i ii

n x x=

−∑ [–500, 500]

11 Sinusoidal 1 1

3.5 2.5 sin sin 56 6

n n

i ii i

x xπ π= =

− − + −

∏ ∏ [0, π]

12 Zakharov’s 2 4

2

1 1 12 2

n n n

i i ii i i

i ix x x= = =

+ +

∑ ∑ ∑ [–5.12, 5.12]

13 Sphere 2

1

n

ii

x=∑ [–5.12, 5.12]


Table 1 $Functions used as test bed (continued)

S. No. Name Function Bounds

14 Axis parallel hyper ellipsoid

2

1

n

ii

ix=∑ [–5.12, 5.12]

15 Schewefel-3 1 1

nn

i ii i

x x= =

+∑ ∏ [–10, 10]

16 Neumaier 3 21

1 2

( 4)( 1) ( 1)6

n n

i i ii i

n n n x x x −= =

+ − + − −∑ ∑ 2 2[ , ]n n−

17 Salomon ( ) 21

1-cos 2 x 0.1 x , nii

x xπ=

+ = ∑ [–100, 100]

18 Ellipsoidal 2

1 ( )

n

ii

x i=

−∑ [ ],n n−

19 Schaffer 1 2 2

2 2

1 10.5 sin 0.5 1 0.001

n n

i ii i

x x= =

+ − + ∑ ∑ [–100, 100]

20 Brown3 2 2

1

1( 1) ( 1)2 2

11

( ) ( )i i

nx x

i ii

x x+

−+ +

+−

+ ∑ [–1, 4]

21 New function 2 2

1(0.2 0.1 sin 2 )

n

i i ii

x x x=

+∑ [–10, 10]

22 Cigar 2 21

2100000

n

ii

x x=

+ ∑ [–10, 10]

$All are minimisation problems having minimum value as 0.

Of the successful runs, the percentage of success and the average number of function evaluations (Table 2), the computational time (Table 3), the mean and the standard deviation of the optimal fitness (objective function) value (Table 4) using LX-PM and H-LX-PM are recorded. A brief view of comparative performance from Tables 2–4 is presented in Table 5. Now, from Table 5 it is clear that only one problem (No. 17) could not be solved by either method. Out of the balance 21 problems, the number of function evaluations, and hence the computational time required to solve by H-LX-PM, is a bit higher than that of LX-PM. However, the success rate is much higher, the average objective function value is much better and the standard deviation is much smaller in H-LX-PM than the corresponding LX-PM. It is worth here to note that with H-LX-PM, 14 problems could be solved with 100% success, whereas no problem with 100% could be solved using LX-PM. Further, with the availability of more and more high-speed computing facilities, the issue of computational time difference in seconds is not of much concern. Still, it is wise to record them. Thus, we can say that H-LX-PM out-performs LX-PM.


Table 2 Comparative success rate and average number of function evaluations using LX-PM and H-LX-PM

Success rate Average function evaluations Prob. LX-PM H-LX-PM LX-PM H-LX-PM 1 0 1 * 35700 2 35 100 18505 22526 3 97 100 14047 21214 4 0 24 * 32833 5 19 100 16847 22242 6 53 92 14286 21593 7 65 100 19433 26936 8 0 7 * 34557 9 8 100 43800 370352 10 0 87 * 61831 11 59 98 18530 23640 12 19 100 22021 23690 13 68 100 18216 22618 14 60 100 18028 22492 15 38 100 20913 26272 16 2 100 39550 162066 17 0 0 * * 18 42 100 18930 24058 19 0 2 * 25600 20 78 100 18343 23286 21 64 100 18756 22470 22 5 100 28920 27404

*It is only problem (No. 17) which could not be solved by both LX-PM and H-LX-PM.

Table 3 Comparative computational time (in seconds) using LX-PM and H-LX-PM

Prob. LX-PM H-LX-PM

1 * 0.531 2 0.281 0.343 3 0.189 0.270 4 * 0.517 5 0.257 0.460 6 0.285 0.546 7 0.322 0.490 8 * 0.462 9 0.770 7.301


Table 3 Comparative computational time (in seconds) using LX-PM and H-LX-PM (continued)

Prob. LX-PM H-LX-PM

10 * 0.875 11 0.246 0.401 12 0.372 0.492 13 0.224 0.347 14 0.222 0.346 15 0.230 0.375 16 0.477 2.356 17 * * 18 0.235 0.362 19 * 0.399 20 0.404 0.575 21 0.286 0.481 22 0.363 0.395


Table 4 Comparative mean and standard deviation of the optimal objective function value using LX-PM and H-LX-PM

Mean Standard deviation

Prob. LX-PM H-LX-PM LX-PM H-LX-PM 1 * 6.10E-14 * 0.00E+00 2 9.72E-04 1.57E-15 1.40E-03 1.50E-15 3 9.54E-04 1.32E-15 1.97E-03 2.68E-15 4 * 4.01E-03 * 4.43E-03 5 1.33E-03 1.18E-12 2.41E-03 3.07E-18 6 4.78E-04 1.89E-13 1.25E-03 2.64E-18 7 1.87E-03 4.70E-04 2.54E-03 7.23E-14 8 * 1.65E-14 * 1.67E-14 9 9.55E+00 1.29E+00 3.37E+00 1.81E+00 10 * 1.27E-04 * 6.82E-13 11 1.46E-03 3.46E-15 2.34E-03 1.39E-15 12 1.65E-03 7.41E-21 1.91E-03 6.04E-20 13 1.75E-03 1.94E-22 2.30E-03 5.32E-22 14 1.54E-03 1.73E-21 2.48E-03 1.17E-20


Table 4 Comparative mean and standard deviation of the optimal objective function value using LX-PM and H-LX-PM (continued)

Mean Standard deviation Prob. LX-PM H-LX-PM LX-PM H-LX-PM 15 2.08E-03 1.43E-13 2.58E-03 2.79E-13 16 3.40E-01 2.98E-03 2.14E-01 1.05E-03 17 * * * * 18 1.50E-03 1.78E-15 2.39E-03 1.19E-15 19 * 9.72E-03 * 2.78E-17 20 1.63E-03 3.81E-21 2.29E-03 2.25E-20 21 1.54E-03 1.76E-21 2.29E-03 1.01E-20 22 3.08E-03 3.93E-21 3.19E-03 2.76E-20


Table 5 Comparison of LX-PM with H-LX-PM in analysing Tables 2–4

Quality of H-LX-PM over LX-PM

Success rate (ref. Table 2)

Average no. of function evaluations

(ref. Table 2)

Time (in seconds) (ref. Table 3)

Mean obj. function value (of successful runs) (ref. Table 4)

Mean SD (of successful runs)

(ref. Table 4) Better 21 5 5 21 21 Equal 1* 1* 1* 1* 1* Worse 0 16 16 0 0


In view of these conflicting conclusions based on different criteria, it is difficult to judge the better algorithm between LX-PM and H-LX-PM. Hence, to measure the winner, a Performance Index (PI) (Mohan and Nguyen, 2004) in its extended form is used and is discussed below.

Consider three criteria, namely:

1 success rate to measure the reliability

2 average number of function evaluations to measure the efficiency

3 the mean of the optimal objective function value attained to measure the accuracy of the algorithms.

A PI, is designed as follows:

1 1 2 2 3 31

1PI ( )pN

i i i

ip

k k kN

α α α=

= + +∑ (10)

where

1 ,i

ii

SrTr

α =


2, if 0 and

0, if 0

ii

i i

i

Mo SrAo

Srα

>=

=

3

, if 0, for 1, 2, ,

0, if 0

ii

i ip

i

Mf Sri NAf

Srα

>= =

=

…

Sri: Number of successful runs of ith problem Tri: Total number of runs of ith problem Aoi: Mean optimal objective function value obtained by an algorithm of ith problem Moi: Minimum of mean optimal objective function value obtained by all the algorithms of ith problem Af i: Average number of function evaluations of successful runs required by an algorithm in obtaining the solution of ith problem Mf

i: Minimum of average number of function evaluations of successful runs required by all algorithms in obtaining the solution of ith problem Np: Total number of problems analysed.

Note that 1 2,k k and 3 1 2 3 1 2 3( 1 and 0 , , 1)k k k k k k k+ + = ≤ ≤ are the weights assigned to percentage of success, mean optimal objective function value and average number of function evaluations of successful runs, respectively. From this definition it is seen that PI is a function of 1 2,k k and 3.k However, since 1 2 3 1,k k k+ + = one of , 1, 2, 3ik i = can be eliminated to reduce the number of dependent variables from the expression of PI. But it is still difficult to analyse the behaviour of PI, because the surface plots of PI for all three algorithms are overlapping and it is difficult to visualise them. Therefore, if equal weights are assigned to two terms at a time in the PI expression, then PI becomes a function of one variable only. The resultant cases are as follows.

i 1 2 31, , where 0 1

2wk w k k w−= = = ≤ ≤

ii 2 1 31, , where 0 1

2wk w k k w−= = = ≤ ≤

iii 3 1 21, , where 0 1.

2wk w k k w−= = = ≤ ≤

In each of the Figures 1–3, the horizontal axis represents the weights corresponding to the first, second and third term of the expression of PI, namely, percentage of success, mean optimal objective function value and average number of function evaluations of successful runs, respectively. This weight is varied from 0 to 1. The vertical axis represents the value of PI, also lying between 0 and 1. The graphs corresponding to LX-PM and H-LX-PM are superimposed and the one that has higher PI is taken to be the better than the other.


Figure 1 Performance Index of LX-PM and H-LX-PM, when percentage of success is varied between 0 and 1

Figure 2 Performance Index of LX-PM and H-LX-PM, when mean optimal objective function value is varied between 0 and 1

Figure 3 Performance Index of LX-PM and H-LX-PM, when average number of function evaluations of successful runs is varied between 0 and 1

From Figure 1 it is clear that with respect to percentage of success, H-LX-PM is always better than LX-PM. From Figure 2 it is observed that with respect to mean optimal objective function value, although H-LX-PM is always better than LX-PM the difference between PI is very large if this weight is given a value 0, and is very marginal if this weight is given a value 1. From Figure 3 it is seen that with respect to average number of function evaluations of successful runs, although H-LX-PM is always better than LX-PM the difference between their PI is small if this weight is given 0 value, and is very large if


this weight is given 1 value. Therefore, H-LX-PM is more reliable, more efficient and more accurate than LX-PM. However, depending upon the priority that the user might like to give to the three factors a suitable choice between the two algorithms can be made. However, in totality, H-LX-PM outperforms LX-PM.

5 Real world applications

In this section we present two popular real life non-linear optimisation problems and use the LX-PM and H-LX-PM to solve these problems.

Problem 1: System of linear equations

This problem is taken from Eshelman et al. (1997). The problem may be stated as solving for the elements of a vector, X, given the matrix A and vector B in the expression: A⋅X = B. The evaluation function used for these experiments is as follows:

sle 1 21 1

( , , , ) ( ) .n n

n ij j ii j

P x x x a x b= =

= − −∑∑…

Clearly, the best value for this objective function is sle ( *) 0.P x = Inter-parameter linkage (i.e., non-linearity) is easily controlled in systems of linear equations, their non-linearity does not deteriorate as increasing numbers of parameters are used, and they have proven to be quite difficult.

Considering a 10-parameter problem (Lozano et al., 2004) instance, the matrices are as follows:

5 4 5 2 9 5 4 2 3 1 1 409 7 1 1 7 2 2 6 6 9 1 503 1 8 6 9 7 4 2 1 6 1 478 3 7 3 7 5 3 9 9 5 1 599 5 1 6 3 4 2 3 3 9 1 45

.1 2 3 1 7 6 6 3 3 3 1 351 5 7 8 1 4 7 8 4 8 1 539 3 8 6 3 4 7 1 8 1 1 508 2 8 5 3 8 7 2 7 5 1 552 1 2 2 9 8 7 4 4 1 1 40

=

Problem 2: Frequency modulation sounds parameter identification problem

This problem is taken from Tsutsui and Fujimoto (1993). The frequency modulation sound model is represented by

1 1 2 2 3 32( ) sin( sin( sin( ))), where .100

y t a w t a w t a w t πθ θ θ θ= + + =


The problem is to specify the six parameters 1 1 2 2 3 3, , , , , .a w a w a w The fitness function is defined as the summation of square errors between the evolved data and the model data as follows:

1002

1 1 2 2 3 3 00

( , , , , , ) ( ( ) ( ))fmst

P a w a w a w y t y t=

= −∑

where the model data are given by the following equation:

0 ( ) 1.0 sin(5.0 1.5 sin(4.5 2.0 sin(4.9 ))).y t t t tθ θ θ= × × − × × + × ×

Each parameter is in the range –6.4 to 6.35. This problem is a highly complex multi-modal one having strong epitasis, with minimum value ( *) 0.fmsP x =

6 Results and discussions

Each of the two real world problems, 1 and 2, is run 100 times. The parameter setting and the stopping criteria are as discussed in Section 4. A run is considered to be a success if the obtained objective value is with in 0.5 and 20 for the real life problems 1 and 2, respectively. The mean of the optimal objective function values and the standard deviations are reported for the successful runs only. Both the real world problems are solved by both LX-PM and H-LX-PM. The comparative results for problem 1 and 2 are reported in Tables 6 and 7, respectively.

Table 6 Performance comparison of LX-PM and H-LX-PM to solve the system of linear equations

Methods Success rate (%)


Time (in seconds)

Mean obj. function value (of successful runs)


LX-PM 39 19294 0.201 0.336352 0.105848

H-LX-PM 100 31500 0.477 0.115572 0.061183

Table 7 Performance comparison of LX-PM and H-LX-PM to solve Frequency Modulation Sounds Parameter Identification Problem

Methods Success rate (%)


Time (in seconds)

Mean Obj. function value (of successful runs)


LX-PM 16 23250 2.26 15.3688 5.17363 H-LX-PM 24 45036 4.52 14.5191 4.37356

From Tables 6 and 7, it is observed that although H-LX-PM takes more time to solve the problems with a higher number of function evaluations the H-LX-PM yields a better solution with a higher success rate. Further the standard deviations are also lower in case of H-LX-PM and, hence, are more stable than LX-PM.

In order to visualise the effect of the hybridisation in finding the objective function value (fitness function value), the convergence graphs for LX-PM and H-LX-PM are plotted in Figures 4 and 5 for problems 1 and 2, respectively. The horizontal axis shows


the generation number and the vertical axis shows the optimal objective function for a typical run. It is observed that although both LX-PM and H-LX-PM started from the same starting point the fitness function value improved much more rapidly while using H-LX-PM, particularly during the initial generations of a run.

Figure 4 Generation wise convergence for System of linear equations

Figure 5 Generation wise convergence for frequency modulation sounds parameter identification problem

Hence, it is observed that although H-LX-PM is less efficient it is more reliable, more accurate and more stable than LX-PM. Thus, in totality H-LX-PM is recommended to solve these two typical real life problems to get improved success rates with better optimum values.

7 Conclusions

In an earlier work, the authors designed a Real Coded Genetic Algorithm, called LX-PM, which attempts to determine the global optimal solution of non-linear optimisation problems. With a view to improve its performance with respect to efficiency, reliability and stability, a hybrid Real Coded Genetic Algorithm, called H-LX-PM, is proposed herein. The performance of both the algorithms is evaluated on a wide variety of 22 benchmark test problems taken from literature. Also, two popular real world application problems are solved using LX-PM and H-LX-PM. From the results and discussions it is concluded that H-LX-PM is less efficient as it takes more time and function evaluations to solve a problem in general but, simultaneously, it is more reliable, more accurate and more stable than LX-PM, not only for test problems but also for real life application problems too.


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Documents

Performance improvement of real coded genetic algorithm with Quadratic Approximation based hybridisation