Intelligent Information Management, 2009, 1, 145-149
doi:10.4236/iim.2009.13021 Published Online December 2009 (http://www.scirp.org/journal/iim)
Copyright © 2009 SciRes IIM
145
Reliability Optimization of Entropy Based
Series-Parallel System Using Global Criterion
Method
G. S. MAHAPATRA
Department of Engineering Sciences and Humanities, Siliguri Institute of Technology, Siliguri, India
Email: g_s_mahapatra@yahoo.com
Abstract: In this paper, we have considered a series-parallel system to find out optimum system reliability
with an additional entropy objective function. Maximum system reliability of series-parallel system is de-
pending on proper allocation of redundancy component in different stage. The goal of entropy based reliabil-
ity redundancy allocation problem is to find optimal number of redundancy component in each stage such a
manner that maximize the system reliability subject to available total system cost. Global criterion method is
used to analyze entropy based reliability optimization problem with different weight function of objective
functions. Numerical examples have been provided to illustrate the model.
Keywords: reliability, series-parallel system, redundancy, entropy, global criterion method
1. Introduction
The problem of reliability redundancy allocation is to
find out optimal system reliability by optimal allocation
of redundancy components in series-parallel system. Re-
liability of a multi-stage system can be improved by add-
ing similar or some different components redundancy to
each sub-system as design alternatives. The design of a
reliable system was improved by Hikita et al. [3] by the
addition of redundant components. Several researcher [1,
2,4,5,12,13,17,20–22] presented redundancy optimization
with multiple objective functions of system reliability,
system cost and system weight etc. and solve that objec-
tive redundancy allocation problem by different algori-
thm and nonlinear optimization techniques for multi-
objective system reliability design optimization in fuzzy
and crisp environments. Tillman et al. [19] presented a
comprehensive survey of previous works for system re-
liability with redundancy. Singh and Misra [18], Kuo and
Prasad [7], Kuo et al. [8], and Misra [10] presented reliabil-
ity redundant allocation problem to increase the system
reliability, which is important in reliability engineering.
For series-parallel reliability redundancy allocation
problem, entropy represents the lack of the information
about the state of the each sub-system. Very few authors
discussed reliability analysis with entropy consideration.
Musto and Saridis [11] presented entropy-based reliabil-
ity assessment technique. The technique was demon-
strated in a case study of a robotic system. Rocchi [15]
introduced the entropy function in order to study the re-
liability and repairability of systems. Ridder [14] inves-
tigated the application and usability of the cross-entropy
method for rare event simulation in Markovian reliability
models. Rocchi [16] discussed and calculated the reli-
ability function during system again through the stochas-
tic entropy. Kroese et al. [6] introduced a new approach
based on cross-entropy method for optimization of net-
work reliability.
Here, we have considered a multi-objective entropy
based reliability redundancy problem to finding the op-
timum number of redundant components, which maxi-
mize the system reliability with entropy as an additional
objective function subject to available system cost. The
redundancy reliability optimization problem is consid-
ered with two objective functions such as maximum sys-
tem reliability and maximum entropy amount simulta-
neously with restriction on system cost. Numerical ex-
ample is presented using global criterion method.
2. Reliability Redundancy Allocation Model
2.1. Notations
Series Parallel system, reliability redundancy allocation
model is developed under the following notations.
Ri reliability of each component of reliability model in
the ith stage,
Ci cost of each component of reliability model in the
ith stage,
C available system cost of the reliability model,
xi number of redundancy components in the ith stage
(decision variables),
G. S. MAHAPATRA
146
12
, ,...,
s
n
Rxxx system reliability function of the
reliability model,
12
, ,...,
s
n
Cxxx system cost function of the re-
liability model,
12
,,..., n
En xxx
i
entropy function of the reli-
ability model.
2.2. Reliability Redundancy Allocation Problem
It is to be considered that an n stage series system and at
each stage added (xi-1) redundant components in parallel,
the objectives are to determine the number of redundant
components at each stage such that the system reliability
will be maximize subject to related cost constraints.
Therefore the maximization of Rs subject to the limited
available cost C has to be found.
Therefore the problem becomes


12
, ,...,11i
nx
sn
i
Max RxxxR
(1)
subject to

12
1
, ,...,exp4
n
i
snii
i
x
CxxxCxC






xi >1 for i=1,2,...,n.
2.3. Entropy in Series Parallel System
Entropy has important physical implications as the
amount of “disorder” of a system. A more abstract defi-
nition is used in mathematics.
The Shannon entropy of a variable X is defined as
 
ln
x
En Xp xp x
(2)
where p(x) is the probability that X is the state x, and p(x)
log p(x) is defined as 0 if p(x) = 0.
Reliability redundancy allocation problem is the re-
dundancy distribution of each stage of a series-parallel
system. To determine a suitable measure of allocation, let
us consider a n-stage series-parallel system with xi
(i=1,2,…,n) number of redundant component of each ith
stage of the system. It is known that xi are positive inte-
ger and total number of components is i
i
x
. Redun-
dancy allocation of components share of ith stage is the
share of the total number of redundant component
is i
i
i
x
i
x
p. Normalizes the redundancy numbers xi by
dividing them by the total number of redundant compo-
nents i
i
x
then the probability distribution i
i
i
x
i
x
p
is
found.
The measure of allocation shall be defined as the ex-
pected information of the message which transforms the
system shares into the share of each stage.
Figure 1. A schematic diagram of the n-stage system
So (3)

12
1
, ,...,ln
n
ni
i
En xxxpp

i
where i
i
i
i
x
p
x
The each stage share i
i
i
i
x
p
satisfying the con-
dition and defines a
probability distribution and the Shannon-entropy meas-
ure the diversity of the probability distribu-
tion
0 (i=1,2,...,n)
i
p 1
i
i
p
12
, ,...,.
n
pp p
12
... 1/
n
p n
Maximum is reached when
pp
  i.e. when allocation of all
stage have the same no of redundant components. Since
increasing of xi, maximizing Inpi is equivalent to
maximizing entropy as defined above. This is one of the
reasons why the entropy optimization model is particu-
larly suitable for the redundancy allocation problem. In
redundancy allocation problem, entropy acts as a meas-
ure of dispersal of allocation between stages. So it will
be more potential if we would like to have maximum
system reliability as well as maximize entropy measure.
2.4. Multi-Objective Entropy Redundancy
Allocation Problem
Taking entropy function as additional objective func-
tion the problem (1) becomes



12
, ,...,11i
nx
sn
i
Maximize RxxxR
i
(4)

12
1
, ,...,log
ii
ii
n
xx
n
x
x
i
Maximize Enxxx
 
Subject to same constraint and restriction as in (1)
3. Method of Global Criterion
A multi-objective non-linear problem may be taken in
the following form
Maximize/Minimize f(x) = [f1(x), f2(x),…, fk(x) ]T (5)
subject to x X={x Rn : gj(x) b
j for j=1,…,m;
lixiui for i=1,2,...,n}.
Solve the multi-objective non-linear problem (5) as a
Copyright © 2009 SciRes IIM
G. S. MAHAPATRA 147
single objective non-linear problem k times for each
problem by taking one of the objective at a time and ig-
noring the others. From the result, determine the corre-
sponding values for every objective for each derived
solution. For each objective
r
f
x, find lower bound
(minimum) l
r
f
and the upper bound (maximum)u
r
f
.
In the global criterion method [9], the distance be-
tween some reference point and the feasible objective
region is minimized. The analyst has to select the refer-
ence point and the metric for measuring the distances.
Suppose that the weighting coefficients wr are real num-
bers such that and
0,1, 2,...,
r
wr k
1
1
k
r
r
w
.
Here we examine the method where the ideal objective
vector is used as a reference point and Lp-metrics are
used for measuring. In this case, the weighted Lp
-problem for minimizing distances is stated as
Minimize



1
1
p
p
l
m
rr
pr
ul
rir
fx f
Lfxwff
(6)
Subject to
x
X for 1 p
The exponent 1/p may be dropped. Problems with or
without the exponent 1/p are equivalent for 1p
.
The solution obtained depends greatly on the value cho-
sen for p, commonly used choices are p=1,2 or
.
For p=1,



1
1
l
m
r
rul
rrr
r
f
xf
Lfxw
f
f
(7)
The objective function

1
L
fx is the sum of the
normalized weighted deviations, which is to be mini-
mized.
For p=2,



1
22
2
1
l
m
rr
rul
rrr
fx f
Lfxw ff




(8)
When p becomes larger, the minimization of the de-
viation becomes more and more important.
If p= , GCM (6) is of the form
Minimize

1,2,...,
max
l
r
rul
rk rr
r
f
xf
w
f
f
(9)
Subject to
x
X.
The problem (9) can be transformed into the following
form
Minimize
(10)
Subject to

l
rr
rul
rr
fx f
wff
for all r=1,2,…,k.
x
X
where both n
x
and
 are variables.
4. Global Criterion Method on Entropy
Based Reliability Redundancy Problem
In entropy based reliability optimization of series-parall-
el system, maximum system reliability
s
Rx and
maximum entropy
En x has to be found, having sub-
ject to the system cost constraint with goal of
system cost is C. So the problem is a multi-objective
entropy reliability redundancy allocation problem as fol-
lows

s
Cx

s
M
aximize Rx (11)

M
aximize En x
Subject to
s
CxC
Where
12
,,..., n
x
xx xand xi >1 for i=1,2,...,n.
To solve the above multi-objective reliability optimi-
zation problem (11), according to section 3 pay-off ma-
trix is formulated as follows:
1
2
x
x



**
**
()
()
()
s
s
s
Rx Enx
Rx Enx
RxEn x
Now lower and upper bounds of

s
RxandEnx
,,
are identified and denoted as
L
ULU
and EnEn
ss
R R re-
spectively.
Using Global criterion method for the problem (11),
the weighted Lp-problem for minimizing distances is
stated as
Minimize
 
1
12
,
ps
pp
p
LL
ss
ULU L
ss
LRxEnx
Rx REnxEn
ww
RR EnEn






(12)
Subject to
s
CxC
, for 1p
Putting different value of p (1,2 or ) in (12), we
get as follows
For p=1,

 
1
12
,
s
L
L
ss
ULU L
ss
LRxEnx
R
xR EnxEn
ww
R
REnE


n
(13)
For p=2,
 
2
1
22
2
12
,
s
LL
ss
ULU L
ss
LRxEnx
Rx REnxEn
ww
RR EnEn






(14)
Copyright © 2009 SciRes IIM
G. S. MAHAPATRA
Copyright © 2009 SciRes IIM
148
For p= , (12) is of the form
Minimize
(15)
Subject to

1
L
ss
UL
ss
Rx R
wRR

2
L
UL
En xEn
wEn En
.
To solve the entropy based reliability redundancy al-
location problem (11) using GCM, we have to solve (13),
(14), (15) with same constraints as in Equation (10) for
different weight.
5. Numerical Example
A four stage reliability redundancy allocation problem
with entropy objective function with cost constraints is
considered for numerical exposure. The problem be-
comes as follows:
Maximize


4
1234
,,,11 i
x
s
i
Rxxxx R
i
Maximize

4
1234
1
,,, log
ii
ii
xx
x
x
i
En xxxx
 
subject to
(16)

12
1
, ,...,exp4
n
i
snii
i
x
Cxx xCxC




,
x >1 for i=1,2,3,4.
of the problem (5.1) are given in ta-
bl
:
T 1. Input d food4)
1 2341 2 3 4
ableatar mel (
RR R R C CC C C
0.85 0.9 0.8 0.95 8 9 7 10 200
The pay-off matrix is formulated as follows:
() ()
0.9976021 1.366073
0.8627854 1.386294
s
Rx Enx
1
x
i
Input parameters
e 1.
Solution
2
x
Here 0.9976021, 0.8627854,
ss
RR
1.386294, 1.366073
UL
En En are identified and us-
ing these bounds construct the objective functions. The
optimal solutions of the multi-objective reliability opti-
mization model (16) using global criterion method
UL
(fol-
lo
when thrence to
entropy function tHere
wing (13), (14) and (15)) are given in Table 2 for dif-
ferent preference values of the objective functions.
In case-I, Table 2 shows different optimal solutions
e decision maker supplies more prefe the
han the reliability function.
**
R
sx is maximum when p= 2, whereas
En x is
maximum when p =2 or .
In Table 2, case-II gives different optimal solutions
e decision maker supplies equal pre to
entropy function. Here
**
when thferences
the reliability function and
**
R
sx is maximum when p=1, whereas
En x is
remains unaltered for p.
In case-III, Table 2 shows different optimal solutions
e deciser supplies more preferenc
he entropy function. Here
**
when thion make to the
reliability function than t
**
s
Rx and
**
En x remains unaltered for p.
Table igtages of system reliability (w1) and entropy functions (w2) by GCM 2. Optimal solution for different we
Case w1 w
2 p *
1
*
2
*
3
*
4
**
s
R
x
**
En x
1 2 3 2 3 0.9373444 1.366159
2 2 2 3 2 0.9575832 1.368922
I 0.2 0.8
2 3 2 2 0.9351179 1.368922
1 2 2 3 2 0.9575832 1.368922
2 2 2 2 3 0.9288999 1.368922
II
0.5
0.5
2 2 2 3 0.9288999 1.368922
1 2 2 2 2 0.9266935 1.386294
2 2 2 2 2 0.9266935 1.386294
III
0.8
0.2
2 2 2 2 0.9266935 1.386294
G. S. MAHAPATRA 149
6. Consion
Here redundancy allocation probleseries-parallel
ility and entropy objectives is pre-
erion method is used to solve the
[1] A. O. Charles Elegbede, C. Chu, K. H. Adjallah, and F.
y allocation through cost minimiza-
actions on Reliability, Vol. 52, No. 1, pp.
.
Camge Univs, The Pit Trump-
ington Street, CambridgeKingdom, 2001.
[9] K. Miettinen, “Nonlinear multiobjective optimization,”
4, 1997.
y, Vol. 37, pp. 253–261, 1992.
h the
ization by the surro-
clus
m of
system with reliab
sented. Global crit
Klu
problem of multi-objective reliability redundancy alloca-
tion problem with entropy function. Two objective func-
tions are combined into a single objective function by the
global criterion method. The optimal solutions for dif-
ferent preferences on objective functions are presented.
Decision-maker may obtain the Pareto optimal results
according to his expectation of system cost.
Here it is considered that the problem as to finding
the optimum number of redundancies, which maximize
the system reliability and entropy subject to the limited
system cost. The system reliability increases with in-
creases of redundancies and entropy of the system de-
creases which is expected.
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