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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