International Journal of Mathematical Analysis

Vol. 8, 2014, no. 50, 2481 - 2492

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ijma.2014.49286

A Hybrid GA - PSO Approach for

Reliability under Type II Censored Data Using

Exponential Distribution

K. Kalaivani 1 and S. Somsundaram 2

1 Dept of Mathematics, RVS College of Engg & Technology

Coimbatore Tamilnadu, India

2 Dept of Mathematics, Coimbatore Institute of Technology

Coimbatore, Tamilnadu, India

Copyright © 2014 K. Kalaivani and S. Somsundaram. This is an open access article distributed

under the Creative Commons Attribution License, which permits unrestricted use, distribution, and

reproduction in any medium, provided the original work is properly cited.

Abstract

Today software plays an important role and its application is used in each and

every domain. In software testing phase, quality risk, reliability and fault detection

are used to analysis and remove the failure of the item. Owing to time constraints

and limited number of testing unit, we cannot fix the experiment until the failure.

In order to observe the failure during testing phase, censoring becomes significant

methodology to estimate model parameters of exponential distributions. The most

common censoring schemes do not have the flexibility to identify the failure in the

terminal point. The most commonly used censoring schemes are Type I and Type

II censoring schemes. To identify the optimum censoring scheme and overcome

these problems optimal technique is used in this paper. Thus, optimal scheme will

improve the output of testing phase with the aid of specific optimal constraints.

Entropy and variance are used as optimal criterion. Determination of Optimal

schemes will be done by Hybrid Genetic Algorithm (GA) and Particle Swarm

Optimization (PSO). The risk values are estimated of the selected optimal

censoring scheme.

2482 K. Kalaivani and S. Somsundaram

Keywords: Reliability, Risk Analysis, Censoring Scheme, GA, PSO and Fast

Kernel entropy

I. Introduction

Software reliability can be improved by introducing various methods and that

techniques are all depend on mathematical theories. This reliability approach is

developed for both the testing and verification. Software testing is one of the

important tasks to check the coding line and debugging the errors and this

technique is useful for software industry [1]. For critical and non critical

applications software can be used. Because of the uses and importance of software

is increased, software quality was most important. So software quality can be

expressed in the ways of reliability, availability, safety and security. Among these

qualities reliability is most important [8]

The computational efficiency is improved and the global optimal solution is

obtained for the censoring data in [2] using genetic algorithm. Type I, Type II and

random censoring are frequently used in reliability analysis to save the

experimental time. Type I and Type II censoring cases are commonly used but

random censoring used in medical studies or clinical trials [3]

Exponential distribution has significant the bias and variance of this estimator

are seen, and it is shown that this estimator is as efficient as the best linear

unbiased estimator. [7] The hybrid censoring scheme is a mixture of Type-I and

Type-II censoring schemes [6].

The exponential distribution allows increasing with decreasing hazard rate

depending on the shape parameter. [4] Reliability is a key driver of safety-critical

systems such as health-care systems and traffic controllers. It is also one of the

most important quality attributes of the systems embedded into our surroundings

[5].

For a discrete probability distribution p on the countable set {x1, x2, …….} with

pi = p(x) the entropy of p is defined as

i

i

i ppph log)(1

For a continuous probability density function p(x) on an interval I, its entropy is

defined as

i

xpxpph )(log)()(

Recently GA has been proved to be a more effective approach for solving the

reliability optimization problem. A new type of hybrid algorithm particle swarm

optimization (PSO) was first proposed by Kennedy and Eberhart The theory of

PSO describes a solution process in which each particle flies through a

multidimensional search space while the particle velocity and position constantly

A hybrid GA - PSO approach for reliability 2483

updated with respect to best previous performance of the particle or particle’s

neighbor, the best performance of the particle in the entire population. [14]

II. Related Work

Reza Azimi, Farhad Yaghmaei [9] a novel reliability approach was handled.

Here the author handled in two ways. First, incorporate the classification of

failures through reliability metrics it is based on the users. Second, to improve or

increase the reliability quality or testing proposed modified adaptive testing

strategy, the idea behind the theory is followed from adaptive Markov control.

From the result observes that modified adaptive testing approach outperforms the

random testing approach and the operational profile-based testing approach.

Debasis Kundu and Avit Joarder [3] introduced Type II progressively hybrid

sensors are used to obtain the maximum likelihood estimators for the unknown

parameters. This paper also introduces the Bayes estimate and credible interval for

this unknown parameter to obtain the reliable data.

Aveek Banerjee and Debasis Kundu [10] implemented the combination of

classical and Bayesian algorithms are used for Type-II hybrid censored and these

algorithms are proposed for Weibull parameters. This paper provides the

maximum likelihood for unknown parameters are provides the system as an

explicitly.

Jung-Hua Lo, Chin-Yu Huang [11] introduced a number of reliability models

have been developed to evaluate the reliability of a software system. The

parameters in these software reliability models are usually directly obtained from

the field failure data. J. Kennedy, R. Eberhart [12] a concept for the optimization

of nonlinear functions using particle swarm methodology is introduced.

Benchmark testing of the paradigm is described, and applications, including

nonlinear function optimization and neural network training, are proposed. The

relationships between particle swarm optimization and both artificial life and

genetic algorithms are described. Luo, J. X. Defeng Wu; Zi Ma [13] introduced

Particle swarm optimization (PSO) algorithm and genetic algorithm (GA) are

employed to optimize the berth allocation planning to improve the SR in CT.

Young Su Yun [14] developed an adaptive hybrid GA – PSO approach to

maximize the overall system reliability and minimizing system cost to identify the

optimal solution and improve computation efficiency.

M. Sheikhalishahi, V. Ebrahimipour [15] proposed optimization hybrid

approach using adaptive GA and improved PSO is proposed. Here they propose

an aadaptive scheme is incorporated into GA loop & it adaptively regulates

crossover & mutation rates during the genetic process. For proving the

performance of proposed hybrid approach conventional type algorithms using GA

& PSO are presented and their performances are compared with that of the

proposed hybrid algorithm using several reliability optimization problems which

have been often used in many conventional studies.

2484 K. Kalaivani and S. Somsundaram

III. Methodology

Software reliability is most important one for most system. A number of

reliability models are proposed for prediction and detection. In this paper

reliability is measure or evaluated using hybrid approach of Genetic Algorithm

(GA) with Particle Swap Optimation (PSO), rescaled bootstrap method for

variance estimation, fast kernel entropy estimation and kernel failure rate function

estimator. Some existing method use PSO for software reliability, where swarm

may prematurely converge in it. To overcome this in this paper introduces the

hybrid algorithm of GA with PSO. The advantage of the proposed algorithm is

simplicity one and it is able to control convergence.

A. Hybrid approach of Genetic algorithm with PSO:

Hybrid approach can be used when the information of the problem is little. This

approach can be used suitable for multimedia and multivariate nonlinear

optimization problem and it is used for noisy data also.

Genetic algorithm is one of the recognized powerful techniques for optimization

and global search problems. It is originated from in 1970 by John Holland; here

they process the genetic evolution such as natural selection, mutation and

crossover are developed for their target problems. For optimization problem GA

populations are formulated a chromosomes of candidate solution. This technique

maintains a population of M individuals for each iteration

g, potential solution of the population is denoted by each individual. Using

selection process, new populations are obtained and these are all based on

individual adaptation and some genetic operators. In each generation, the fitness

of every individual in the population is evaluated. Depend on muting and

crossover, generating and reproducing new individuals. Further genetic operation

process, new tentative populations are formed by selected individuals and this

selection process repeated several times. Then from this process some of the new

tentative population are undergoes into transformation. After the selection

process, crossover creates two new individual by combining parts from two

randomly selected individuals of the population. The system needs high crossover

probability and low mutation for good GA performances as an output. Mutation is

a unitary transformation which creates, with certain probability, ( mp ), a new

individual by a small change in a single individual.

Below algorithm gives the methodology of the proposed hybrid algorithm; in

selection process of the genetic algorithm Particle Swarm Optimization (PSO)

method is introduced. This algorithm is implemented in 1995 by Kennedy and

Eberhart and this is one of the well optimization techniques [20]. In PSO, each

particle represents a candidate solution within a n-dimensional search space.

A hybrid GA - PSO approach for reliability 2485

Figure 1: Algorithm for proposed hybrid approach

The position of a particle i at iteration t is denoted by ,.....}{)( 2,1 iii xxtx

For each iteration, every particle moves through the search space with a velocity

,........}2,1{)( iii vvtv calculated as follows

In the above equation dimension of the search space is denoted by j and

,......2,1j , inertia weight is given by w, iy is the personal best position of the

Pseudocode of the proposed hybrid algorithm

Ste p 1: Initialization

Initialize individuals to random values

end for

{Main Loop}

While (not terminate condition)

{Genetic Operators}

Pop: Selection process

Step 1: Initialize the swarm s in the search space

Step 2: While maximum number of iterations is not reached do

Step 3: for all particle i of the swarm do

Step 4: calculate the fitness of the particle i

Step 5: set the personal best position

Step 6: end for

Step 7: set the global best position of the swarm

Step 8: for all particle i of the swarm do

Step 9: Update the velocity

Step 10: Update the position

Step 11: end for

Step 12: end while

{Evaluation Loop}

for i=1 to M do

end for

end while

2486 K. Kalaivani and S. Somsundaram

particle i at iteration t and is the global best position of the swarm iteration t.

is the personal best solution and it denotes the best position found by the

particle search process until the iteration t. is the global best position and it

identify the best position found by the entire swarm until the iteration t.

Acceleration coefficient and random numbers are given by

parameter . },{ maxmin vv is the limited range for the velocity. . After

updating velocity, the new position of the particle i at iteration t+1 is calculated

using the following equation:

From the above equation the parameter w reduces gradually as the iteration

increases and the parameter denotes the inertia and final weight and

maximum number of iteration are denoted by

B. Bootstrap variance estimation

is the given estimation and the bootstrap estimate of the variance

this is given by the empirical variance of the bootstrapped

estimates, that is in the above equation the

average of the bootstrapped estimates.

Rescaled bootstrap was selected because it translates well to replicate weights

and it is very easy for selecting the samples and setup for random samples and this

rescaled bootstrap is used by some statistical institutes.

This Rescaled bootstrap was introduced by Rao and Wu in 1988. In case of

strata indexed by with units in each of them. Out of which are

sampled without replacement. Sampling friction is given by , variable of

interest is denoted by and weight is indicated by . Following rules are

done in rescaled bootstrap variance estimation.

Step 1: A sample of size is taken with replacement from each stratum.

Step 2: Writing the number of times unit is resample, the weights are

adjusted as follows with

Step 3: Total bootstrap is evaluated by

Step 4: Bootstrap Variance is calculated using where

mean of the bootstrap total over all B iterations.

A hybrid GA - PSO approach for reliability 2487

C. Fast Kernel entropy estimation

In most of the signal processing problem, entropy is used. For efficient

optimation sensitive and entropy analysis are implemented. Calculate the entropy

value using forward propagation. It consists of following steps. Figure 2 provides

the block diagram for forward approach for estimating the entropy value.

Step 1:

Step 2:

Figure 2: Forward Propagation for estimating the entropy

∑

f (y,W)

y

W

:

φ

∑

log

∑

:

2488 K. Kalaivani and S. Somsundaram

Figure 3: Pseudocode for calculating the entropy

Step 3:

Save

In the above equation is the derivative of the L.

Then overall complexity of the forward propagation is

D. Exponential distribution for the hazard function

The formula for the hazard function of the exponential distribution is

The hazard function is defined as the ration of the probability density function to

the survival function, s(x)

IV. Experimental Results

The proposed framework provides the reliability using different values of alpha

and beta. Software quality is given by reliability value, entropy, and hazard and

Pseudocode for calculating the entropy

Input: w,s

Output:

Algorithm:

for

end

for

for

end

end

A hybrid GA - PSO approach for reliability 2489

risk value. In this proposed method, the optimal censoring scheme is selected

based on Hybrid GA-PSO approach. Table I represents the sample data generated.

Based on these values the hazard rate and reliability values are calculated

Table 1: Generated Data set

i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

xi:35:50 0.4 0.6 1 1 1 2 2 2 3 6 7 11 18 18 19

Ri 0 0 3 3 0 0 3 0 0 0 3 0 0 0 0

i 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

xi:35:50 36 45 49 50 54 54 60 63 67 75 75 75 83 84 84

Ri 0 0 3 0 3 0 3 3 0 0 0 0 3 3 3

The censor data samples are generated with m=35 and n=50 and censoring

scheme R3=R4=R11=R21=R23=R26=R31=R32=R33=3, Ri= 0, and obtained the ML

parameters such as [α, β]. From those values, the hazard rate and reliability are

calculated which is represented in Table 1

Table 2: Reliability for alpha and beta value

α β Reliability

(%)

0.4 0.01 0.8193

0.2 0.05 0.8928

0.1 0.07 0.9557

Table 3: Estimation of entropy for different value

α β Entropy

0.4 0.01 0.9183

0.2 0.05 0.9183

0.1 0.07 0.9098

Table 4: Hazard and risk value

α β Hazard

(%)

Risk

(%)

0.4 0.01 4.0729 0.3848

0.2 0.05 3.7424 1.1638

0.1 0.07 3.4978 1.4044

Table 2, 3 and 4 provides the reliability, entropy, hazard and risk value for

different alpha and beta value. Reliability for beta value and alpha value are given

in table I. Here 81.93% reliability is obtained from the 0.4 and 0.01 alpha and beta

value.89.28% of reliability is obtained from the 0.2 and 0.05 alpha and beta value.

95.57% of reliability is obtained from the 0.1 and 0.07 alpha and beta value.

0.9183 and 0.0908 entropy is obtained for these values of α & β

2490 K. Kalaivani and S. Somsundaram

Figure 4: Reliability and hazard rate for the proposed system

Figure 5: Risk rate for the proposed system

Figure 4 and 5 illustrates the reliability, hazard and risk rate for the proposed

system.

V. Conclusion

In present software environment, the problems such as quality risk, reliability

and fault detection arises. This framework for software testing reliability is

presented in this research paper. Software testing problem is solved by using

quality tools such as reliability, verification and fault detection and debugging. In

this reliability is obtained using hybrid PSO - genetic algorithm approach and the

corresponding hazard rate is also obtained. Risk analysis is performed under type

A hybrid GA - PSO approach for reliability 2491

II censored data. Then this technique is evaluated and reliability, entropy, hazard

and risk value’s are measured for the type II censored data. For different alpha and

beta value they are taken and by the obtained result, the proposed technique

proves by better results.

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Received: September 17, 2014; Published: November 13, 2014