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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online), Volume 5, Issue 12, December (2014), pp. 73-86 © IAEME
73
NEURAL NETWORK FOR THE RELIABILITY ANALYSIS
OF A SERIES - PARALLEL SYSTEM SUBJECTED TO
FINITE COMMON - CAUSE AND FINITE HUMAN ERROR
FAILURES
K. Umamaheswari
Research Scholar, Dept. of Mathematics, Sri Krishnadevaraya University,
Ananthapuramu-515001, A.P., India
A. Mallikarjuna Reddy
Professor, Dept. of Mathematics, Sri Krishnadevaraya University,
Ananthapuramu-515001, A.P, India
ABSTRACT
Artificial neural networks can achieve high computation rates by employing a massive
number of simple processing elements with a high degree of connectivity between the elements.
Neural networks with feedback connections provide a computing model capable of exploiting fine-
grained parallelism to solve a rich class of complex problems. In this paper we discuss a complex
series-parallel system subjected to finite common cause and finite human error failures and its
reliability using neural network method.
Keywords: Reliability, availability, Markov Model, Neural networks and Series-parallel system.
1. INTRODUCTION
Artificial neural networks can achieve high computation rates by employing a massive
number of simple processing elements with a high degree of connectivity between the elements.
Neural networks with feedback connections provide a computing model capable of exploiting fine-
grained parallelism to solve a rich class of complex problems. Network parameters are explicitly
computed based upon problem specifications, to cause the network to overage to an equilibrium that
represents a solution. Recently Mahmoud and Suliman [1990] introduced a new approach to the
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© I A E M E
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online), Volume 5, Issue 12, December (2014), pp. 73-86 © IAEME
74
reliability analysis based on Neural Network approach. In general the reliability of hardware under
design in usually arrived at by assuming suitable values for certain parameters such as the failure
rate, coverage factor and the repair rate whichever is applicable to the design. The reliability of the
system is then computed using discrete or continuous time analysis. If the resulting reliability does
not meet the design requirements, then the whole process is repeated to obtain another set of values.
A table is usually set at the end showing the various reliability values with the corresponding
parameters of the design so that the designer may pick the most suitable features for the system. This
technique is lengthy and complicated when dealing with complex fault tolerant system. Help of
neural network philosophy is taken to avoid complication.
The initial conditions and the desired reliability are fed in to the neural network. When neural
network converges their different weight indicates the appropriate parameters and hence features of
the system under investigation. A computation is performed collectively by the whole network with
the actively distributed over all the computing elements. This collective operation results in a high
degree of parallel computations for past solution of complex problem.
Artificial neural networks (ANN’S) are motivated by biological nervous systems. Modern
computers and algorithmic computations are good at well defined tasks. Biological brains, on the
other hand, easily solve speech and vision problems under a wide range of condition tasks that no
digital computer has solved adequately. This inadequacy has prompted researchers to study
biological neural systems in an attempt to design computational systems with brain-like capabilities.
At the same time, modern analog and digital integrated circuit technology is offering the potential for
implementing massively parallel networks of sample processing elements. Nero computing will
enable us to take advantage of these advances in VLSI by providing the computational model
necessary to program and coordinate the behavior of thousands of processing elements.
Neural network models are providing new approaches to problem solving. Neural network
can be simulated on special purpose neural hardware accelerators as well as conventional machines.
For maximum processing speed they may even be realized using optimal implementations or silicon
VLSI. The key to the utility of ANN’s is that they provide a computational model that can be used to
systematize the process of simple processors. This chapter deals with, the definition of artificial
neural networks, types of neural networks, their use and a systematic approach to the availability
analysis of a “series-parallel system” with repair, which illustrates the neural network approach. The
discrete-time Markov model of a series-parallel system is realized using feed- forward recursive
neural network. The obtained results are verified with the continuous time solutions of the Markov
models and digitals simulation.
2 ARTIFICIAL NEURAL NETWORKS
2.1 Network Models
Networks may be distinguished on the basis of the directions in which signal flow. Basically,
there are two types of networks. Feed Forward and Feedback Networks. A network in which signals
propagate in only one direction from an input stage through intermediate neurons to an output stage
is called a Feed forward network. Feedback networks, on the other area networks in which signals
may propagate from the output of any neuron to the input of any neuron.
2.2 Feed Forward Networks
Fig 1 illustrates a Feed forward network. The first layer serves only to distribute a weighted
version of the input vector to the neurons in the inner layer. Neurons in the inner layer, called hidden
neurons, respond to the accumulated effects of their inputs and propagate their response signals to
neurons in the ouput layer. Neurons in the output layer also accumulate the effect of the signals they
receive and collectively produce an output vector of signals which represents the response of the
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online), Volume 5, Issue 12, December (2014), pp. 73-86 © IAEME
75
network to the input vector. There are several powerful algorithms [1991] & [1988] available for
adopting the strengths of the interconnections between neurons in feed forward network so that the
network learns to map input patterns into desired output patterns. Feed forward networks have been
applied successively to a number of problem areas.
Fig.1: Feed Forward Networks
2.3 Feedback Networks
Fig 2 shows a feedback network with five neurons. Each block dot represents a set of
feedback connections that are analogous to biological synopses. Because the output of a neuron may
be fed back into the networks as an input to other neurons, a neuron may influence its own future
state. Neural models that permit feedback have been employed to develop networks capable of
unsupervised learning, self-organization, retrieving stored memory patterns, and computing solutions
to a variety of optimization problems. The neural solution to each of these problems involves
interpreting the state of the network after it stabilizes. It is therefore necessary to state criteria for the
design of suitable neural networks.
Fig.2: Feedback Networks
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online), Volume 5, Issue 12, December (2014), pp. 73-86 © IAEME
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1
( ) ( ) ( )n
ii i i i ij j j
j
dXa X b X C g X
dt =
= −
∑
2.4 Cohen-Grossberg Stability Results
Grossberg [1963] developed a mathematical model which compasses a variety of neural
network models as well as models from population biology and macromolecular evolution. The
Analysis of this model by Cohen and Grossberg yielded conditions under which the systems of
differential equations used to characterize a number of popular neural network model which
converge to stable states. The model which they analyzed is a dynamical system of mutually
independent differential euqations of the form.
(1)
They showed the existence of Lyapunov function for a system of such equations if the matrix [Cij]
and the functions ai, bi and gi meet three conditions:
1. The matrix [Cij] must be symmetric (i.e., Cij = Cji)
2. The functions ai and bi must be continuous with ai nonnegative.
3. The functions gi must be non-decreasing.
A Lyapunov function for a dynamical system places constraints on the collective behavior of
the equations comprising the system. The central idea is that the system always evolves a manner
that does not increase the value of the Lyapunov function. The existence of Lyapunov function for
system of independent differential equations of the form of equation (1) therefore guarantees that the
system will follow a trajectory leading to a stable state, regardless of the initial state, provided the
above conditions hold. The Cohen-Grossberg proof that establishes the stability of any neural
network model that can be characterized by equation (1). One of the cases of this model was
independently conceived by Hopfield.
2.5 The Hopfield Neural Network
The earliest Neural Network model introduced by Hopfield employed two - state neurons. He
used this model to design neural content - addressable memories. Hopfield later introduced a
modified version of his earlier model which employed a continuous non - linear function to describe
the output behavior of the neurons. It is Hopfields continuous model that corresponds to a special
case of the Grossberg mathematical model for additive neural networks.
In Hopfields continuous model, the behavior of a neuron is characterized by its activation
level ui which is governed by the differential equation.
(2)
Where i
i
n
u− is a passive decay term, Wij is the strength of the interconnection between neuron i,
gj(uj) is the activation function for neuron j, Ij is the external input to neuron I. The activation level
ui is a continuous variable that corresponds to the membrane potential in biological neurons. In the
absence of an external input and inputs from other neurons, the passive decay term cause ui toward 0
at a rate proportional to ni. The output of neuron i can be desired by its mean firing rate vi
corresponding to the activation level ui. The output vi is continuous over its range and is related to ui
1
( )n
i iij j j j
ji
du uW g u I
dt n =
−= + +∑
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online), Volume 5, Issue 12, December (2014), pp. 73-86 © IAEME
77
by the activation function vi = gi(ui). The activation function is typically a smooth sigmoid. A
frequent choice for g(u) is
g(u) = 0.5[1+tanh(gain u)] (3)
As long as g(u) is non-decreasing, it meets the Cohen - Grossberg requirements for stability.
Thus, if the external inputs are maintained at a constant value, a network of neurons modeled by
equation (3) will eventually equilibrate, regardless of the starting state. Hopfield discovered a
Lyapunov function for a network of n neurons characterized by equation (3) which can be expressed
as
(4)
When gain of the activation function is sufficiently high. This expression, which Hopfield
refers to as the networks “computational energy” or just “energy function” can be derived from the
Lyapunov function discovered earlier by Cohen and Grossberg. The term “energy function” stems
from an analogy between the network behavior and that of certain physical systems. Just as physical
systems may evolve toward an equilibrium state, a network of neurons will always evolve toward a
minimum of energy function. The stable states of a network of neurons therefore correspond to the
local minima of the energy function.
Hopfield and Tank [1985] had a key insight when they recognized that it was possible to use
the energy function to perform computations. Because a network of neurons will seek to minimize
the energy function, one may design a neural network for function minimization by associating
variable in an optimization problem with variables in the energy function.
2.6 Series - Parallel System
Consider the Markov model of a series-parallel system consisting n units as shown in the figure 3.
Assumptions The following assumptions are associated with the system under study:
� Failures are statistically independent.
� All system units are active, identical and form a parallel network.
� A unit failure rate is constant.
� A Common - Cause failure or a critical human error leads to system failure.
� A common - cause failure or a critical human error can occur when one (or more) unit is
operating.
� Critical human error and Common - Cause failure rates are constant.
� Failed system repair rates are constant / non-constant.
� At least one unit must operate normally for the system’s success.
Symbols The following symbols are associated with this model;
n - Number of units in the parallel system
λ - constant failure rate of a unit
i - system up state as shown in boxes fig 3
i = 0 (all units operating normally),
i = 1 (one unit failed, (n-1) operating),
i = 2 (two units failed, (n-2) operating),
1 1 1
1( ) ( ) ( )
2
n n n
ij i j j j i i i
i j i
E W g u g u I g u= = =
= −∑∑ ∑
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online), Volume 5, Issue 12, December (2014), pp. 73-86 © IAEME
78
i = 3 (three units failed, (n-3) operating),
i = 4 (four units failed, (n-4) operating),
i = k (k units failed, (n-k) operating).
K - number of failed units in the system and corresponding up state of the system, for k
= 1, 2.........(n-1)
λcliλc
2i- Constant common cause failure rate from system up state i, for i=0,1,2......k
λh1i,
λh2i-Constant critical human error rate from system up state i, for i=0,1,2, .........k
i = System down state as shown in boxes of fig.3 j=n (all units failed other than due to a
common-cause failure or a critical human error)
j = c1, c
2, c
3..............c
k (system failed due to common cause failure)
j = h1, h
2, h
3..............h
k (system failed due to critical human error)
Pi(t) - Probability that the system is in up state i at time t for i = 0, 1, 2, 3,........k
Pj(t) - Probability that the system is in down state j at time t for j=n,c
1,c
2,..........c
k,h
1,h
2.............h
k
Pi - Steady state probability that the system is in up state i, for i = 0, 1, 2,.........k
Pj - Steady state probability that the system is in down state j, for
j=n,c1,c
2,..........c
k,h
1,h
2.............h
k
GENERAL MODEL
The system transition diagram is shown in Fig.3 The discrete time equations for the Markov
model are given by
The system transition diagram is shown in Fig.3
The discrete time equations for the Markov model are given by
P0(t+∆t) = P
0(t)[1-nλ∆t - λ
c10∆t] - λ
c20∆t -λ
c30∆t............... - λ
ck0∆t -λ
h10∆t -λ
h20∆t -λ
h30∆t ............. - λ
hk0∆t
]+P1(t)nλ∆t +P
c1(t)λ
c10∆t +P
c2(t)λ
c20∆t +P
c3(t)λ
c30∆t +.............+P
ckλ
ck0∆t +P
h1(t)λ
h10∆t +P
h2(t)λ
h20∆t
+Ph3
(t) λh30
∆t +...............+Phk0
λhk0
∆t.
Pc1
(t+∆t) = P0(t)r
1∆t+P
c1(t)[1-r
1∆t]
Pc2
(t+∆t) = P0(t)r
2∆t+P
c2(t)[1-r
2∆t]
Pc3
(t+∆t) = P0(t)r
3∆t+P
c3(t)[1-r
3∆t]
.
.
.
Pck(t+∆t) = P
0(t)r
k∆t+P
ck(t)[1-r
k∆t]
Ph1
(t+∆t) = P0(t)z
1∆t+P
h1(t)[1-z
1∆t]
Ph2
(t+∆t) = P0(t)z
2∆t+P
h2(t)[1-z
2∆t]
Ph3
(t+∆t) = P0(t)z
3∆t+P
h3(t)[1-z
3∆t]
.
.
.
Phk
(t+∆t) = P0(t)z
k∆t+P
hk(t)[1-z
k∆t]
Fig.3 System Transition
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
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P1(t+∆t) = P
c1(t)λ
c11∆t +P
c2(t)λ
c21∆t +P
c3(t)λ
c31∆t +...................+P
ck(t)λ
ck1∆t +P
h1(t)λ
h11∆t +P
h2(t)λ
h21∆t
+Ph3
(t)λh31
∆t +..................+Phk
(t)λhk1
∆t +P2(t)(n-1) λ∆t +P
1(t)[1-λ
c11∆t -λ
c21∆t -λ
c31∆t - .............. -
λck1
∆t -λh11
∆t -λh21
∆t -λh31
∆t - ................. - λhk1
∆t - (n-1)λ ∆t ]
P2(t+∆t) = P
c1(t)λ
c12∆t +P
c2(t)λ
c22∆t +P
c3(t)λ
c32∆t +...................+P
ck(t)λ
ck2∆t +P
h1(t)λ
h12∆t +P
h2(t)λ
h22∆t
+Ph3
(t)λh32
∆t +..................+Phk
(t)λhk2
∆t +P3(t)(n-2) λ∆t +P
2(t)[1-λ
c12∆t -λ
c22∆t -λ
c32∆t - .............. -
λck2
∆t -λh12
∆t -λh22
∆t -λh32
∆t - ................. - λhk2
∆t]
P3(t+∆t) = P
c1(t)λ
c13∆t +P
c2(t)λ
c23∆t +P
c3(t)λ
c33∆t +...............+P
ck(t)λ
ck∆
3t +P
h1(t)λ
h13∆t +P
h2(t)λ
h23∆t
+Ph3
(t)λh33
∆t +................+Phk
(t)λhk3
∆t +P3(t)[1-λ
c13∆t -λ
c23∆t -λ
c33∆t - ........... - λ
ck3∆t -λ
h13∆t -λ
h23∆t -
λh33
∆t - .......... - λhk3
∆t - (n-3)λ ∆t ]
Pk(t+∆t) = P
c1(t)λ
c1k∆t +P
c2(t)λ
c2k∆t +P
c3(t)λ
c3k∆t +...................+P
ck(t)λ
ckk∆t +P
h1(t)λ
h1k∆t +P
h2(t)λ
h2k∆t
+Ph3
(t)λh3k
∆t +..................+Phk
(t)λhkk
∆t +Pk+1
(t)(n-k)λ ∆t -Pk(t)[1−λ
c1k∆t-λ
c2k∆t -λ
c3k∆t - .............. -
λckk
∆t -λh1k
∆t -λh2k
∆t -λh3k
∆t - ................. - λhkk
∆t - (n-k)λ ∆t ]
Pn(t+∆t) = P
0(t)µ ∆t+P
n(t)[1-µ∆t].
2.7 The Neural Network for A Series - Parallel System
A feed forward cascade recursive network is set to represent the parallel system. As shown in
Fig.4 the network consists of two layers of neurons: one form by input and other forms the output,
the number of neurons in each layer equals to the number of states, in Markov model. The weights
connecting the input and output neurons represent the entries of the transition matrix of the
differential equations. In other words, the weights of the neural network are related as follows to the
Markov model.
W12 = λc10∆t W21 = r1∆t W2k+2,2= λc11∆t W3k+1,2= λc1k∆t
W13 = λc20∆t W31 = r2∆t W2k+2,3= λc21∆t W3k+1,3= λc2k∆t
W14 = λc30∆t W41 = r3∆t W2k+2,4= λc31∆t W3k+1,4= λc3k∆t
W1,k+1= λcko∆t Wk+1,1= rk∆t W2k+2,k+1=λck1∆t W3k+1,k+1= λckk∆t
W1,k+2= λh10∆t Wk+2,1= z1∆t W2k+3,2=λc12∆t W3k+1,k+2= λh1k∆t
W1,k+3= λh20∆t Wk+3,1= z2∆t W2k+3,3=λc22∆t W3k+1,k+3= λh2k∆t
W1,k+4= λh30∆t Wk+4,1= z3∆t W2k+3,4=λc32∆t W3k+1,k+4= λh3k∆t
W1,2k+1= λhko∆t W2k+1,1= zk∆t W2k+3,k+1=λck2∆t W3k+1,2k+1= λhkk∆t
W1,2k+2= nλ∆t
W2k+4,2 = λc13∆t W3k+1,2k+2= (n-k)λ ∆t
W2k+4,3 = λc23∆t W2k+n+1,1 = µ∆t
W2k+2,2k+3= (n-1)λ ∆t W2k+4,4 = λc33∆t
W2k+4,k+1 = λck3∆t
W2k+3,2K+4= (n-2)λ ∆t
W2k+4,2K+5= (n-3)λ ∆∆t
W11=1-[W12+W13+W14+W1,k+1+W1,k+2+W1,k+3+W1,k+4+W1,2k+1+W1,2k+2]
W22=1-W21
W33=1-W31
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
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W44=1-W41
.
.
Wk+1,k+1=1-Wk+1,1
.
.
.
Wk+2,k+2=1-Wk+2,1
Wk+3,k+3=1-Wk+3,1
Wk+4,k+4=1-Wk+4,1
.
.
.
.
W2k+1,2k+1=1-W2k+1,1
W2k+2,2k+2=1-W2k+2,2
W2k+3,2k+3=1-W2k+3,2
W2k+4,2k+4=1-W2k+4,2
W3k+1,3k+1=1-W3k+1,2
W2k+n+1,2k+n+1=1-W2k+n+1,1
W2k+1,2k+1=1-W2k+1,1
At any time ‘t’ during operation of
the system
X1 = p0(t)
X2 = pc1(t)
X3 = pc2(t)
X4 = pc3(t)
Xk+1 = pck(t)
Xk+2 = ph1(t)
Xk+3 = ph2(t) Fig.4 Series-Parallel System
Xk+4 = ph3(t)
:
X2k+1 = phk(t)
X2k+2 = p1(t)
X2k+3 = p2(t)
X2k+4 = p3(t)
X3k+1 = pk(t)
X2k+n+1= pn(t)
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
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Y1 = p0(t+ ∆ t)
Y2 = pc1(t+ ∆ t)
Y3 = pc2(t+ ∆ t)
Y4 = pc3(t+ ∆ t)
:
Yk+1 = pck(t+ ∆ t)
Yk+2 = ph1(t+ ∆ t)
Yk+3 = ph2(t+ ∆ t)
Yk+4 = ph3(t+ ∆ t)
:
Y2k+1 = phk(t+ ∆ t)
Y2k+2 = p1(t+ ∆ t)
Y2k+3 = p2(t+ ∆ t)
Y2k+4 = p3(t+ ∆ t)
Y3k+1 = pk(t+ ∆ t)
Y2k+n+1 = pn(t+ ∆ t)
The initial conditions are given by
X1 = 1 X2 = X3 = X4 = X5 = ................. = X2k+n+1 = 0
The basic equations of the Neural network are
Y1 = W11X1 + W21X2 + W31X3 + W41X4+ Wk+1,1Xk+1+ Wk+2,1Xk+2+ Wk+3,1Xk+3+
Wk+4,1 Xk+4+ W2k+1,1 X2k+1+ W2k+n+1,1X2k+n+1
Y2 = W12X1 + W22X2 + W2k+2,2X2K+2 + W2k+3,2X2K+3 + W2k+4,2X2K+4 +.......................
W3k+1,2X3K+1 + .................... + W2k+n,2X2K+n
Y3 = W13X1 + W33X3 + W2k+2,3X2K+2 + W2k+3,3X2K+3 + W2k+4,3X2K+4 +.......................
W3k+1,3X3K+1 + .................... + W2k+n,3X2K+n
Y4 = W14X1 + W44X4 + W2k+2,4X2K+2 + W2k+3,4X2K+3 + W2k+4,4X2K+4 +.......................
W3k+1,4X3K+1 + .................... + W2k+n,4X2K+n
Yk+1 = W1,k+1X1+Wk+1,k+1Xk+1+W2k+2,k+1X2k+2+W2k+3,k+1X2k+3+W2k+4,k+1X2k+4+............+
W3k+1,k+1X3k+1+ .............+W2k+n,k+1X2k+n
Yk+2 = W1,k+2X1+Wk+2,k+2Xk+2+ W2k+2,k+2X2k+2 + W2k+3,k+2X2k+3+ W2k+4,k+2X2k+4+............+
W3k+1,k+2X3k+1+ .............+W2k+n,k+2X2k+n
Yk+3 = W1,k+3X1+Wk+3,k+3Xk+3+ W2k+2,k+3X2k+2 + W2k+3,k+3X2k+3+ W2k+4,k+3X2k+4+............+
W3k+1,k+3X3k+1+ .............+W2k+n,k+3X2k+n
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1
n
m=
∑
Yk+4 = W1,k+4X1+Wk+4,k+4Xk+4+ W2k+2,k+4X2k+2 + W2k+3,k+4X2k+3+ W2k+4,k+4X2k+4+............+
W3k+1,k+4X3k+1+ .............+W2k+n,k+4X2k+n
Y2k+1 = W1,2k+1X1+W2k+1,2k+1X2k+1+W2k+2,2k+1X2k+2+W2k+3,2k+1X2k+3+ W2k+4,2k+1X2k+4 +
...........+ W3k+1,2k+1X3k+1+ .............+W2k+n,2k+1X2k+n
Y2k+2 = W12,k+2X1+W2k+2,2k+2X2k+2
Y2k+3 = W2k+2,2k+3X2k+2+W2k+3,2k+3X2k+3
Y2k+4 = W2k+3,2k+4X2k+3+W2k+4,2k+4X2k+4
Y3k+1 = W3k,3kX3k+W3k+1,3k+1X3k+1
K = 2, 3, 4,..............n-1
Y2k+n+1 = W2k,2kX2k+W2k+n+1,2k+n+1X2k+n+1
The energy function E for the neural network and update equations are obtained using the
least mean square, gradient - descent learning procedure as follows:
Where Yi is the output of neuron i in the output layer corresponding to the probability of
system being in state i. Di is the desired output of neuron I, equivalent to the design requirement for
the probability of the system being in the state I after a specified time of operation, and it is to be
determined from the target reliability of the design.
By the least mean square, gradient descent procedure, the update equation for the neural
network is derived as follows.
The change in the weight Wj denoted by WIJ is the related to the energy function by the
following update relation.
∆wij = -k∂ E / ∂ wij
Where K is the constant of proportionality. Now by using the chain rule
∂ E / ∂ wij (∂ E / ∂ Ym) (∂ Ym / ∂wij)
and since the energy function is quadratic, then
∂ E / ∂ Ym = 2(Ym - Dm)
And so from equation
∆wij = - 2k ∑ (Ym - Dm) ∂ Ym / ∂ wij
∆wij = - 2k ∑ Em∂ Ym / ∂ wij
∆w12 = - k∂ E / ∂ w12
= 2kX1 [E2 - E1]
∆w13 = - k∂ E / ∂ w13
[ ]22 1
1
k n
i i
i
E Y D+ +
=
= −∑
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online), Volume 5, Issue 12, December (2014), pp. 73-86 © IAEME
83
= 2kX1 [E3 - E1]
∆w14 = 2kX1 [E4 - E1]
∆W1,k+1= 2KX1 [Ek+1 - E1]
∆W1,k+2= 2KX1 [Ek+2 - E1]
∆W1,k+3= 2KX1 [Ek+3 - E1]
∆W1,k+4= 2KX1 [Ek+4 - E1]
∆W1,2k+1= 2KX1 [E2k+1 - E1]
∆W1,2k+2= 2KX1 [E2k+2 - E1]
∆W21 = 2KX2 [E1 - E2]
∆W31 = 2KX3 [E1 - E3]
∆W41 = 2KX4 [E1 - E4]
∆Wk+1,1= 2KXk+1 [E1 - Ek+1]
∆Wk+2,1= 2KXk+2 [E1 - Ek+2]
∆Wk+3,1= 2KXk+3 [E1 - Ek+3]
∆Wk+4,1= 2KXk+4 [E1 - Ek+4]
∆W2K+1,1= 2KX2K+1 [E1 - E2K+1]
∆W2K+2,2= 2KX2K+2 [E2 - E2K+2]
∆W2K+2,3= 2KX2K+2 [E3 - E2K+2]
∆W2K+2,4= 2KX2K+2 [E4 - E2K+2]
∆W2K+2,K+1= 2KX2K+2 [EK+1 - E2K+2]
∆W2K+2,K+2= 2KX2K+2 [EK+2 - E2K+2]
∆W2K+2,K+3= 2KX2K+2 [EK+3 - E2K+2]
∆W2K+2,K+4= 2KX2K+2 [EK+4 - E2K+2]
∆W2K+2,2K+1= 2KX2K+2 [E2K+1 - E2K+2]
∆W2K+2,2K+3= 2KX2K+2 [E2K+3 - E2K+2]
∆W2K+3,2= 2KX2K+3 [E2 - E2K+3]
∆W2K+3,3= 2KX2K+3 [E3 - E2K+3]
∆W2K+3,4= 2KX2K+3 [E4 - E2K+3]
∆W2K+3,K+1= 2KX2K+3 [EK+1 - E2K+3]
∆W2K+3,K+2= 2KX2K+3 [EK+2 - E2K+3]
∆W2K+3,K+3= 2KX2K+3 [EK+3 - E2K+3]
∆W2K+3,K+4= 2KX2K+3 [EK+4 - E2K+3]
∆W2K+3,2K+1= 2KX2K+3 [E2K+1 - E2K+3]
∆W2K+3,2K+2= 2KX2K+3 [E2K+2 - E2K+3]
∆W2K+3,2K+4= 2KX2K+3 [E2K+4 - E2K+3]
∆W2K+4,2 = 2KX2K+4 [E2 - E2K+4]
∆W2K+4,3 = 2KX2K+4 [E3 - E2K+4]
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online), Volume 5, Issue 12, December (2014), pp. 73-86 © IAEME
84
∆W2K+4,4 = 2KX2K+4 [E4 - E2K+4]
∆W2K+4,k+1 = 2KX2K+4 [EK+1 - E2K+4]
∆W2K+4,k+2 = 2KX2K+4 [EK+2 - E2K+4]
∆W2K+4,k+3 = 2KX2K+4 [EK+3 - E2K+4]
∆W2K+4,k+4 = 2KX2K+4 [EK+4 - E2K+4]
∆W2K+4,2k+1 = 2KX2K+4 [E2K+1 - E2K+4]
∆W2K+4,2k+2 = 2KX2K+4 [E2K+2 - E2K+4]
∆W2K+4,2k+3 = 2KX2K+4 [E2K+3 - E2K+4]
∆W2K+4,2k+5 = 2KX2K+4 [E2K+5 - E2K+4]
∆W3K+1,2 = 2KX3K+1 [E2 - E3K+1]
∆W3K+1,3 = 2KX3K+1 [E3 - E3K+1]
∆W3K+1,4 = 2KX3K+1 [E4 - E3K+1]
∆W3K+1,K+1 = 2KX3K+1 [EK+1 - E3K+1]
∆W3K+1,K+2 = 2KX3K+1 [EK+2 - E3K+1]
∆W3K+1,K+3 = 2KX3K+1 [EK+3 - E3K+1]
∆W3K+1,K+4 = 2KX3K+1 [EK+4 - E3K+1]
∆W3K+1,2K+1 = 2KX3K+1 [E2K+1 - E3K+1]
∆W3K+1,3K+2 = 2KX3K+1 [E2K+2 - E3K+1]
∆W2K+n+1,1 = 2KX2K+n+1 [E1 - E2K+n+1]
Where errorm=Em=(Ym-Dm) or the difference between the actual and the desired output of neuron
m in the output layer.
3 SIMULATION RESULTS AND DISCUSSION
Computer software is developed for simulation of neural network representing the series -
parallel system and is tested for 4-unit series - parallel system. The time of operation of the system is
taken as t=10hrs and t = 0.1 sec. The initial failure and repair rates chosen with in an attainable
practical range. Samples of the results obtained from the simulation are shown in Table 1.
Discrete time Markov model of a series-parallel system can be realized by a neural network,
after feeding in the desired reliability as shown in table. The main interesting feature of this method
is the utilization of the collective computational abilities of neural network in the analyzed.
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online), Volume 5, Issue 12, December (2014), pp. 73-86 © IAEME
85
Table.1 Shows the Convergence results
P0
Desi
red
P1
Valu
es
µ µµµ
r1
r2
r3
r4
r5
r6
z1
z2
z3
z4
z5
z6
λ λλλ
λ λλλc1
0
λ λλλc1
1
λ λλλc12
0.1
4
0.1
5
2.4
154
1.2
63
0.3
553 0.2
211 2
.823
1.2
34
0.1
43
9 2.4
08
0.7
31
1
2.3
23
1.1
97
2.8
50
2.9
54
0.1
71
1
0.0
02769
0.0
00
46
88 0
.005
54
6
0.4
5
0.0
28
1.5
034
2.4
02
1.6
33
2.2
21
1.3
00
0.7
559
2.1
25
1.5
56
2.8
25
1.6
68
0.7
148 2.0
99
1.4
04
0.5
27
8
0.0
00112
9 0
.002
44
4
0.0
01
53
6
0.0
89
0.2
2
1.1
152
0.8
042 0.4
251 2.5
08
0.0
29
67 0.0
2526 1.5
12
2.9
49
1.6
07
2.4
77
0.9
990 0.6
61
7 2.6
43
0.2
70
7
0.0
00770
9 0
.003
39
4
0.0
06
18
9
0.3
6
0.2
0
3.2
201
0.9
059 2.8
96
0.7
288 0
.854
9
0.7
342
1.8
69
1.3
36
2.8
49
2.6
72
1.3
80
1.7
56
2.3
85
0.2
98
3
0.0
03241
0.0
04
89
4
0.0
04
62
3
0.3
3
0.2
5
3.1
443
2.8
47
1.7
37
2.7
60
1.4
58
1.9
79
0.5
86
9 1.2
11
2.3
00
2.8
50
0.1
606 2.3
46
0.7
098
0.3
53
3
0.0
05836
0.0
04
62
0
0.0
05
62
4
0.3
0
0.1
7
3.7
493
0.9
366 2.9
84
2.1
04
1.6
06
2.2
29
1.0
68
0.5
388 2
.243
2.6
05
0.3
248 1.7
68
0.3
395
0.1
76
2
0.0
05173
0.0
03
88
2
0.0
05
40
5
0.1
7
0.2
9
2.5
308
1.2
66
1.1
49
1.2
21
0.2
34
0
2.8
99
0.1
03
6 1.3
14
1.0
49
1.6
29
0.8
807 2.0
59
0.0
4188 0
.392
8
0.0
02961
0.0
04
20
8
0.0
02
78
4
0.2
1
0.0
76
2.5
193
2.9
45
0.3
157 2.4
35
0.6
05
1
1.1
31
0.6
31
6 2.1
28
0.1
20
1
0.7
900 2.5
85
0.7
91
2 2.8
03
0.5
64
9
0.0
04946
0.0
05
62
7
0.0
04
74
1
0.5
0
0.3
8
3.3
016
1.6
66
2.8
09
0.3
457 2
.352
1.1
98
1.5
50
2.7
49
2.6
01
0.2
652 1.7
02
0.3
78
5 0.7
112
0.1
10
1
0.0
01197
0.0
01
38
6
0.0
04
89
0
0.3
4
0.1
7
3.5
862
2.0
29
1.1
82
1.2
65
2.8
58
1.3
27
1.5
07
2.4
97
1.0
66
0.9
200 0.7
750 1.2
75
2.0
06
0.2
72
2
0.0
04436
0.0
03
88
0
0.0
01
58
9
0.2
0
0.0
82
3.0
756
0.2
582 1.8
33
1.3
69
2.6
89
1.4
81
2.7
36
2.7
80
0.5
31
4
0.2
456 2.3
93
2.2
25
0.6
047
0.1
15
8
0.0
02214
0.0
00
13
35 0
.007
04
9
0.0
53
0.0
87
3.6
493
1.5
37
1.3
50
1.0
66
1.4
04
2.8
05
2.4
80
0.7
619 0
.917
1
2.4
20
2.2
10
1.2
86
0.3
532
0.0
53
04 0.0
02598
0.0
03
10
1
0.0
06
55
9
0.1
6
0.0
86
0.7
9659 0.7
729 0.8
554 2.3
69
1.3
28
1.0
81
1.2
81
0.8
590 0
.068
80 1.2
91
1.1
02
1.1
62
1.2
12
0.0
71
33 0.0
04356
0.0
05
43
0
0.0
00
31
72
0.4
9
0.2
2
1.4
996
1.9
94
1.1
67
0.4
753 0
.815
3
2.4
93
0.2
82
8 0.8
916 0
.610
0
0.5
014 1.5
06
1.9
26
1.0
62
0.0
77
84 0.0
05081
0.0
05
26
4
0.0
01
35
2
0.4
0
0.2
3
1.3
423
2.3
69
1.9
97
1.0
37
2.8
57
2.4
93
2.7
60
2.8
00
1.0
36
2.4
94
1.3
34
2.8
83
1.7
11
0.2
81
2
0.0
02213
0.0
03
37
8
0.0
06
71
8
λ λλλc1
3
λ λλλc2
0
λ λλλc2
1
λ λλλc2
2
λ λλλc2
3
λ λλλh
10
λ λλλh
11
λ λλλh1
2
λ λλλh
13
λ λλλh
20
λ λλλh
21
λ λλλh2
2
λ λλλh
23
Tim
e
K
No.
of
Iterati
on
s
0.0
02470
0.0
07490
0.0
03
47
6
0.0
00851
7 0
.001
72
5
0.0
03723
0.0
05
91
2
0.0
06878
0.0
03
43
2
0.0
02233
0.0
02469
0.0
009689 0.0
03107
0.0
05
0.1
9
24
0.0
02002
0.0
01511
0.0
01
66
2
0.0
04714
0.0
01
62
7
0.0
06731
0.0
02
51
8
0.0
06452
0.0
02
62
9
0.0
05627
0.0
01737
0.0
01380
2.4
14e-0
5
0.0
01
0.3
0
20
0.0
002665 0.0
04247
0.0
00
65
97 0.0
00512
5 0
.005
88
3
0.0
04232
0.0
00
22
64 0.0
01067
0.0
00
49
86 0.0
04919
0.0
01673
0.0
07155
0.0
01954
0.0
03
0.0
56 4
0.0
01865
0.0
05377
0.0
01
87
6
0.0
02330
0.0
03
79
4
0.0
04181
0.0
01
61
9
0.0
02871
0.0
05
93
9
0.0
06751
0.0
04524
0.0
06259
0.0
03057
0.0
05
0.0
81 30
0.0
04561
0.0
01443
0.0
03
69
8
0.0
01913
0.0
01
41
5
0.0
00422
9 0
.004
16
6
0.0
01787
0.0
01
61
7
0.0
03325
0.0
04481
0.0
01923
9.8
48e-0
5
0.0
05
0.2
7
28
0.0
04250
0.0
03359
0.0
05
58
8
0.0
01435
0.0
02
93
0
0.0
02636
0.0
04
63
4
0.0
03023
0.0
03
65
7
0.0
02617
0.0
03558
0.0
02330
0.0
01539
0.0
04
0.2
4
30
0.0
01497
0.0
05375
0.0
05
07
1
0.0
05190
3.2
55
e-05
0.0
07488
0.0
00
94
58 0.0
04125
0.0
03
46
2
0.0
008592 0.0
04278
0.0
008390 0.0
02058
0.0
04
0.2
5
2
0.0
02438
0.0
01133
0.0
01
68
5
0.0
02335
0.0
03
88
8
0.0
05396
0.0
00
46
69 0.0
06016
0.0
04
15
3
0.0
06328
0.0
05654
0.0
005117 0.0
00918
9 0.0
03
0.1
3
4
0.0
05301
0.0
03706
0.0
05
17
7
0.0
01634
0.0
01
32
7
0.0
04494
0.0
00
62
64 0.0
02540
0.0
02
62
2
0.0
06426
0.0
05617
0.0
03751
0.0
04202
0.0
00
9 0
.26
12
0.0
01515
0.0
06930
0.0
02
45
4
0.0
07175
0.0
00
93
21 0
.001982
0.0
00
96
86 0.0
005553 0
.002
86
9
0.0
03266
0.0
04522
0.0
03037
0.0
02230
0.0
03
0.2
5
30
0.0
04233
0.0
07042
0.0
01
19
9
0.0
02796
0.0
01
86
3
0.0
01719
0.0
05
82
1
0.0
07409
0.0
00
37
38 0.0
03064
0.0
03316
0.0
04348
0.0
04338
0.0
06
0.1
3
16
0.0
004429 0.0
00882
7 0
.001
15
0
0.0
03139
0.0
02
82
4
0.0
06001
0.0
05
93
8
0.0
03851
0.0
0107
9
0.0
004140 0.0
05986
0.0
02348
0.0
03226
0.0
03
0.1
2
22
0.0
02856
0.0
07376
0.0
05
30
4
0.0
07953
0.0
01
28
8
0.0
07816
0.0
01
75
4
0.0
005891 0
.005
28
4
0.0
04791
0.0
02514
0.0
07416
0.0
02580
0.0
02
0.1
4
10
0.0
04004
0.0
06747
0.0
01
13
9
0.0
01383
0.0
05
97
3
0.0
07246
0.0
03
01
2
0.0
04768
0.0
04
98
9
0.0
04395
0.0
00106
3 0.0
03959
0.0
00116
6 0.0
00
2 0
.22
26
0.0
008449 0.0
06459
0.0
03
14
6
0.0
01524
0.0
03
59
0
0.0
01575
0.0
04
32
6
0.0
06201
0.0
01
15
0
0.0
05874
0.0
01719
0.0
05978
0.0
03975
0.0
03
0.1
4
28
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online), Volume 5, Issue 12, December (2014), pp. 73-86 © IAEME
86
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