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See discussions, stats, and author profiles for this publication at: http://www.researchgate.net/publication/257267405
Reliability optimization with high and low levelredundancies in interval environment via
Genetic Algorithm
ARTICLE in INTERNATIONAL JOURNAL OF SYSTEMS ASSURANCE ENGINEERING AND MANAGEMENT OCTOBER2013
DOI: 10.1007/s13198-013-0199-9
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3 AUTHORS, INCLUDING:
Laxminarayan Sahoo
Burdwan Raj College
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Asokekumar Bhunia
University of Burdwan
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Available from: Laxminarayan Sahoo
Retrieved on: 09 September 2015
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O R I G I N A L A R T I C L E
Reliability optimization with high and low level redundanciesin interval environment via genetic algorithm
Laxminarayan Sahoo Asoke Kumar Bhunia
Dilip Roy
Received: 15 May 2013 / Revised: 23 September 2013
The Society for Reliability Engineering, Quality and Operations Management (SREQOM), India and The Division of Operation and
Maintenance, Lulea University of Technology, Sweden 2013
Abstract This paper deals with redundancy allocationproblem in interval environment that maximizes the overall
system reliability subject to the given resource constraints
and also minimizes the overall system cost subject to the
given resources including an additional constraint on sys-
tem reliability. Here, the reliability of each component is
assumed as interval valued and the cost coefficients as well
as the amount of resources are imprecise and interval
valued. These types of problems have been formulated as
an interval valued nonlinear integer programming problem.
In this paper, we have formulated two types of redundancy,
viz. component level redundancy known as low-level
redundancy and the system level redundancy known ashigh-level redundancy. These problems have been trans-
formed as an unconstrained optimization problem using
penalty function technique and solved using genetic algo-
rithm. Finally, two numerical examples (one for low-level
redundancy and another for high-level redundancy) have
been solved and the computational results have been
compared.
Keywords Genetic algorithm Intervalenvironment Low-level redundancy High-levelredundancy Reliability optimization
1 Introduction
Development of modern technological system design
depends on the selection of components and configurations
to meet the functional requirements as well as performance
specifications. For a system with known cost, reliability,
weight, volume and other system parameters, the corre-
sponding design problem becomes a combinatorial opti-mization problem. The best known reliability design
problem of this type is referred as the redundancy alloca-
tion problem. The basic objective of redundancy allocation
problem is to find the number of redundant components
that either maximize the system reliability or minimize the
system cost under several resource constraints. Redun-
dancy allocation problem is basically a nonlinear integer
programming problem. Most of these problems can not be
solved by direct/indirect or mixed search methods due to
discrete search space. According to Chern (1992), redun-
dancy allocation problem with multiple constraints is quite
often hard to find feasible solutions. This redundancyallocation problem is NP-hard and it has been well dis-
cussed in Tillman et al. (1977) and Kuo and Prasad (2000).
Earlier, several deterministic methods like heuristic meth-
ods (Nakagawa and Nakashima1977; Kim and Yum1993;
Kuo et al. 1978; Aggarwal and Gupta 2005; Ha and Kuo
2006), reduced gradient method (Hwang et al. 1979),
branch and bound method (Kuo et al. 1987; Tillman et al.
1977; Sun and Li 2002; Sung and Cho 1999), integer
programming (Misra and Sharma 1991), dynamic
L. Sahoo (&)
Department of Mathematics, Raniganj Girls College, Raniganj713358, India
e-mail: [email protected]
A. K. Bhunia
Department of Mathematics, The University of Burdwan,
Burdwan 713104, India
e-mail: [email protected]
D. Roy
Centre for Management Studies, The University of Burdwan,
Burdwan 713104, India
e-mail: [email protected]
1 3
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programming (Nakagawa and Miyazaki1981; Hikita et al.
1992) and other well-developed mathematical program-
ming techniques were used to solve such redundancy
allocation problem. However, these methods have both
advantages and disadvantages. Dynamic programming is
not useful for reliability optimization of a general system as
it can be used only for few particular structures of the
objective function and constraints that are decomposable.In branch and bound method, the effectiveness depends on
sharpness of the bound and required memory increases
exponentially with the problem size. As a result, with the
development of genetic algorithm (Goldberg 1989) and
other evolutionary algorithms, most of the researchers have
given more attention on redundancy allocation problem as
these methods provide more flexibility and require fewer
assumptions on the objective as well as the constraints.
Also these methods are also effective irrespective of
whether the search space is discrete or not.
In almost all the studies referred above, the design
parameters of redundancy allocation problem have usuallybeen taken to be precise values. This means that complete
probabilistic information about the system is known. In this
case, it is usually assumed that for every event probability
involved is perfectly determinable. However, in real life
situations, there are not sufficient statistical data available
in most of the cases. In this case the system is either new
or it exists only as a project in which the data/information
can not be collected precisely due to human errors,
improper storage facilities and other unexpected factors
relating to environment. Therefore, one has to consider
situations where parameters are imprecise. To tackle the
problem with such imprecise numbers, generally stochas-
tic, fuzzy and fuzzy-stochastic approaches are applied and
the corresponding problems are converted to deterministic
problems for solving them. In stochastic approach, the
parameters are assumed as random variables with known
probability distributions. In fuzzy approach, the parame-
ters, constraints and goals are considered as fuzzy sets with
known membership functions or fuzzy numbers. On the
other hand, in fuzzy-stochastic approach, some parameters
are viewed as fuzzy sets and other as random variables.
However, for a decision maker to specify the appropriate
membership function for fuzzy approach and probability
distribution for stochastic approach and both for fuzzy-
stochastic approach is a formidable task.
To overcome these difficulties for handling the impre-
cise numbers by different approaches, one may represent
the same by interval valued number as this representation is
the best representation among others. Due to this new
representation, the objective function as well as constraint
functions of the reduced redundancy allocation problem is
interval valued, which is to be maximized under given
constraints. As all the parameters viz., reliability, cost,
weight and amount of resources are interval valued so we
call this problem as interval valued programming problem
(IVPP).
In this paper, we have formulated two types of redun-
dancy, viz. component level redundancy known as low-
level redundancy and the system level redundancy known
as high-level redundancy, for a five-link bridge system
where the objective function as well as constraint functionsare in interval environment. Studies of the system reli-
ability, with component reliability as interval valued, have
already been initiated by some researchers (Gupta et al.
2009; Bhunia et al.2010; Sahoo et al.2010,2012b; Bhunia
and Sahoo2011; Mahato et al. (2012)). On the other hand,
considering non-interval valued (precise/fuzzy/stochastic)
component reliability, a number of researchers has pre-
sented different situations and solutions methodologies on
redundancy allocation problem. Over the last few years,
several techniques were proposed for solving constrained
optimization problem with precise coefficients with the
help of genetic algorithm. Among these, penalty functiontechniques are very popular in solving the same by genetic
algorithms (Miettinen et al. 2003). In this work, we have
used GA based approaches for solving interval valued
nonlinear integer programming problem (IVNLP) type
redundancy allocation problem. To find the optimal solu-
tion of this type of problem by GA, order relations of
interval numbers take an important role for GA operators.
In these works, we have used the definitions of Sahoo et al.
(2012a) for order relations between interval numbers. For
solving such types of problems, we have developed a real
coded elitist GA with tournament selection, uniform
crossover and 1-neighborhood mutation (Bhunia et al.
2010). Finally, to illustrate the proposed method, for high-
level as well as low-level redundancies, two numerical
examples have been considered and solved.
2 Assumptions and notations
The following assumptions and notations have been used in
the entire paper.
2.1 Assumptions
1. Reliability of each component is imprecise and inter-
val valued.
2. Failures of components are statistically independent.
3. All redundancy is active and there is no provision for
repair.
4. The components as well as the system have two
different states, viz. operating state and failure state.
5. The cost coefficients as well as the amount of
resources are imprecise and interval valued.
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2.2 Notations
n The number of subsystems
xj The number of components in
j-th subsystem, arranged in
parallel in case of low-level
redundancy
x (x1, x2,,xn)
h The number of redundant
subsystems, arranged in
parallel in case of high-level
redundancy
~ri rjL; rjR Interval valued reliability of j-th component
~qi qiL; qiR 1 - [riL, riR]~Rjxj RjLxj;RjRxj 1 1 rjL; rjRxj , the reli-
ability of j-th parallel
subsystem~Qj QjL; QjR 1 - [RjL, RjR]~RSx RSLx;RSRx Interval valued system
reliability
~CS CSL; CSR Interval valued system cost~cj cjL; cjR Interval valued cost
coefficients for the j-th
component
~wj wjL; wjR Interval valued weightcoefficients for the j-th
component
~RT
RTL;RTR
Interval valued target system
reliability~gix giLx; giRx i-th constraint function,
i = 1, 2, , m
~bi biL; biR Availability of i-th resource(i1; 2; . . .; m
lj, uj Lower and upper bounds ofxjpc Probability of crossover/
crossover rate
pm Probability of mutation/
mutation rate
max_gen Maximum number of
generation
yb c Integral value ofy
3 Low-level and high-level redundancy
Let us consider an component system. Now, we can either
provide redundant components, which give a system design
diagram as shown in Fig. 1, or provide a total redundant
system as shown in Fig. 2. The component level redun-
dancy is known as low-level redundancy whereas the
system level redundancy known as high-level redundancy.
4 Formulation of reliability-redundancy optimization
problems
Let us consider a network with n subsystems. The goal of
the redundancy allocation problem is to determine the
number of redundant components in each of n parallel
subsystems so as to maximize the overall system reliability
subject to the given resource constraints and also to mini-
mize the overall system cost subject to the given constraint
on system reliability.
2
1x 2x nx
22
1 1 1Fig. 1 Low-level redundancy
1 1 1
2 2 2
h h h
Fig. 2 High-level redundancy
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The general form of the redundancy allocation problem
is as follows:
Maximize ~RSxsubject to ~gix ~bi; i1; 2; ; m1 ljxj uj; xj integer, j1; 2; . . .; n
1
The goal of the formulation (1) is to determine thenumber of redundant components so as to maximize the
overall system reliability. This problem belongs to the
category of constrained nonlinear integer programming
problems (NIPP).
The general form of the cost minimization problem is as
follows:
Minimize ~CSxsubject to ~RSx ~RT
2
This formulation is designed to achieve a minimum total
system cost, subject to ~
RT, a target limit on the systemreliability.
For low-level redundant system (see. Fig. 1), the cor-
responding optimization problems are as follows:
Maximize ~RSx f~R1x1; ~R2x2; . . .; ~Rqxq; . . .; ~Rnxnsubject to ~gix ~bi; i1; 2; . . .; m1 ljxj uj; xj integer, j1; 2; . . .; n
3where ~Rjxj 1 1 ~rjxj ;j1; 2; . . .; n and~rj2 0; 1Minimize ~C
Sx
subject to ~RSx f~R1x1; ~R2x2; . . .; ~Rqxq; . . .; ~Rnxn ~RT4
where ~Rjxj 1 1 ~rjxj ;j1; 2; ; nFor high-level redundant system (see Fig.2), the cor-
responding optimization problems are as follows:
Maximize ~RSh 1 1f~r1;~r2;~r3; . . .;~rq; . . .;~rnhsubject to ~gih ~bi; i1; 2; . . .; m
5l
h
u;
h integer and ~ri2
0;
1
, i
1;
2;. . .n
Minimize ~CShsubject to ~RSh ~RTwhere ~RSh 1 1f~r1;~r2;~r3; . . .;~rq; . . .;~rnh
6
5 Prerequisites mathematics
5.1 Interval
An interval number ~A is a closed interval denoted by ~AaL; aR and is defined by ~A aL; aR fx : aL
x aR;x2
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5.4 Order relation of interval numbers
According to the assumption (1), the objective function of
redundancy allocation problem would be interval valued.
So, to find the optimal solution of the said problem, the
order relations of interval numbers take important role in
decision-making.
Let ~A aL; aR and ~B bL; bR be two closed inter-vals. Then these two intervals may be of the following
three types:
Type I: Two intervals are disjoint
Type II: Two intervals are partially overlapping
Type III: One of the intervals contained in the other one
It is to be noted that both the intervals A =[aL,aR] and
B = [bL, bR] will be equal in case of fully overlapping
intervals. That is A = B ifaL =bLand aR = bR.
Here we shall consider the definitions of order relations
developed by Sahoo et al. (2012a).
Definition 5.4.1: The order relation [max between
the intervals ~A aL; aR ac; awh i and ~B bL; bR bc; bwh i, then for maximization problems
1. ~A[max ~B,ac[bc for Type I and Type II intervals,2. ~A[max ~B, either ac bc^ aw\bw or
ac bc^ aR [ bR for Type III intervalsAccording to this definition, the interval ~A is accepted
for maximization case. Clearly the order relation ~A[max ~B
is reflexive and transitive but not symmetric.
Definition 5.4.2: The order relation \min between theintervals ~A aL; aR ac; awh i and ~B bL; bR
bc; bwh i, then for minimization problems1. ~A\min ~B,ac\bc for Type I and Type II intervals,2. ~A\min ~B, either ac bc^ aw\bw or
ac bc^ aL\bLfor Type III intervals,According to this definition, the interval A is accepted
for minimization case. Clearly the order relation ~A\min ~Bis
reflexive and transitive but not symmetric.
5.5 Mean, variance and standard deviation of interval
numbers
According to Bhunia et al. (2010), mean, variance and
standard deviation of n interval numbers are defined as
follows:
Let ~xi xiL;xiR, i1; 2; . . .; n, be the ith observationwhich is an interval number. Then mean, variance and
standard deviation of ~x1; ~x2; ; ~xn are given by
~x xL;xR 1n
Xni1
xiL;1
n
Xni1
xiR
" #;
Var(~x r2L;r2R
1nX
n
i1xiL1
nXn
i1xiR;xiR1
nXn
i1xiL
!2
and r~x rL;rR ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Var ~xp 1n
Pni1
xiL1nPni1
xiR;
xiR1nPni1
xiL21=2
6 Constraint handling technique
In the application of genetic algorithm for solving the said
reliability optimization problems with interval objectives,
there arises a major huddle question for handling the
constraints. Over the last few years, several techniques
have been proposed to handle the constraints in geneticalgorithms for solving problems with non-interval/fixed
objectives. In our work, we have used penalty function
technique to solve the constrained optimization problem
with interval objective. In this method, the constrained
optimization problem is converted into an unconstrained
optimization problem in which the reduced objective
function involves objective function and a penalty for
violating the constraints. Here, we have used Big-M pen-
alty technique (Gupta et al. 2009).
For the constrained optimization problem
Maximize ~RSxsubject to ~gix ~bi; i1; 2; ; m1 ljxj uj; xj integer, j1; 2; . . .; nthe general form of reduced problem by Big-M penalty
technique is as follows:
Maximize^RSLx; ^RSRx RSLx;RSRx ifx2 SM; M; ifx62S
7andS fx : ~gix ~bi;i1; 2; ; m and1 ljxj uj;
xj integer, j1; 2; . . .; ngFor the constrained optimization problem
Minimize ~CS
subject to ~RSx ~RTthe general form of reduced problem by Big-M penalty
technique is as follows:
Maximize CsLx; CsRx CSLx; CSRx ifx2SM; M ifx62 S
8
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and S x : ~RSx ~RT;
and 1 ljxj uj; xj integer,j1; 2; . . .; n g
For the constrained optimization problem
Maximize ~RShsubject to ~gih ~bi; i1; 2; . . .; ml
h
u; h integer
the general form is as follows:
Maximize^RSLh; ^RSRh RSLh;RSRh ifh2 SM; M ifh62 S
9and S fh: ~gih~bi; i1; 2; . . .; mand1 lh u; hintegerg
For the constrained optimization problem
Minimize ~CShsubject to ~RSh ~RTthe general form is as follows:
MaximizeCsLh; CsRh ~CSLh; ~CSRh ifh2S
M; M ifh62S
10and S h : ~RSh ~RT; and 1 l h u; h integer
Here ^RSLx; ^RSRx,CsLx; CsRx, ^RSLh; ^RSRh
and CsLh; CsRh are the interval valued auxiliaryobjective function. Problem (7) and (8) are integer non-
linear unconstrained optimization problem with interval
objective of n integer variables x1;x2; . . .;xn whereas
problem (9) and (10) are integer non-linear unconstrainedoptimization problem with interval objective of integer
variableh. For solving these problems, we have developed
a real coded genetic algorithm (GA) with advanced oper-
ators for integer variable(s).
7 Genetic algorithm
Genetic algorithm is a well-known stochastic method of
global optimization based on the evolutionary theory of
Darwin: The survival of the fittest and natural genetics
(Goldberg1989). It has successfully been applied in solv-
ing optimization problems of different real world applica-
tion problems. This algorithm is based on the evaluation of
a set of solutions, called population. Basically, the popu-
lation is initialized by randomly generated individuals.
These populations will be improved from generation to
generation by an artificial evolution process. During each
generation, each chromosome in the entire population is
evaluated using the measure of fitness and the population
of the next generation is created through different genetic
operators. This algorithm can be implemented easily with
the help of computer programming. In particular, it is very
useful for solving complicated optimization problems
which cannot be solved easily by analytical/direct/gradient
based mathematical techniques.
For implementing the GA in solving the optimization
problems, the following basic components are to be
considered.
GA Parameters.
Chromosome representation.
Initialization of population.
Evaluation of fitness function.
Selection process.
Genetic operators (crossover, mutation and elitism).
Termination criteria.
There are basically four parameters used in the genetic
algorithm, viz. p_size (population size i.e., the number of
individuals in the population), m_gen (maximum number
of generations/iterations), pc(probability of crossover) andpm(probability of mutation/mutation rate). There is no hard
and fast rule for the choice of the values of first two
parameters i.e.,p_size and m_gen. These vary from prob-
lem to problem according to their dimension. Again, from
the natural genetics, it is obvious that the crossover rate
should be greater than the mutation rate. Generally, the
crossover rate varies from 0.8 to 0.95 whereas the mutation
rate varies from 0.05 to 0.30.
For successful applications of GA, the appropriate
chromosome/individual representation of solutions for the
given problem is an important task. There are different
types of representations available in the existing literature,
viz. binary, octal, hexadecimal and real. Among these
representations, real coding representation is very popular.
In this representation, for a given problem withn decision
variables, an-component vectorx x1;x2; . . .;xnis usedas a chromosome to represent a solution to the problem. A
chromosome, denoted by vkk1; 2; . . .;p size is anordered list ofngenes sayvk xk1;xk2; . . .;xkn. An initialpopulation of sizep_size is randomly generated within the
bounds of the corresponding decision variables according
to the uniform distribution.
In GA, fitness function plays an important role. It rep-
resents the fitness value of a chromosome. In this work, the
transformed unconstrained objective function due to pen-
alty technique is considered as the fitness function.
In artificial genetics, the selection operator is the first
operator which is applied to the population to form the
population for the next generation by selecting the above
average chromosome and eliminating the below average
chromosome. Here, we have used the well-known tourna-
ment selection scheme of size two with replacement as
the selection strategy. This process selects the better
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chromosome/individual from randomly selected two chro-
mosomes/individuals. This selection procedure is based on
the following assumptions:
1. For feasible chromosomes/individuals, the better one
with respect to fitness value is selected.
2. For one feasible and another infeasible chromosomes/
individuals, the feasible one is selected.3. For both infeasible chromosomes/individuals with
unequal constraint violations, the chromosome with
less constraint violation is selected.
4. For both infeasible chromosomes/individuals with
equal constraint violations, then any one chromo-
some/individual is selected.
Crossover is a key operator of genetic algorithm. The
purpose of this operator is to generate the rearrangement of
co-adapted groups of information from high performance
structures. Here, we have used the uniform crossover as the
crossover operator. The different steps of this operator are
as follows:
Step-1 Find the integral value of pcp sizeb c and storeit in Nc
Step-2 Select two chromosomesvkand vi randomly from
the population
Step-3 Compute the components xkj and xijj1; 2; . . .; n of two offspring by either xkjxkjgand xijxijgifxkj [xij or, xkjxkjgandxijxijg, whereg is a random integer numberbetween 0 and xkjxij
, j1; 2; . . .; n
Step-4 Repeat Step-2 and Step-3 for Nc2
times
Mutation is a background operator that produces random
changes in various chromosomes. Sometimes, it helps to
regain the information lost in earlier generations. Mainly,
this operator is responsible for fine tuning capabilities of the
system. This operator is applied to a single chromosome
only. Mutation attempts to bump the population gently into a
slightly better way, i.e., the mutation changes single or all
the genes of a randomly selected chromosome slightly. The
mutation operator used herein is the one-neighborhood
mutation. The different steps of this operator are as follows:
Step-1 Find the integral value of pm
p size
b cand store
it in Nm.
Step-2 Select a chromosome vi randomly from the
population.
Step-3 Select a particular gene xikk1; 2; . . .; n onchromosomev i for mutation and domain ofxik is
lik; uik.Step-4 Create new genex0ikcorresponding to the selected
gene x ikby mutation process as follows:
For k1; 2; . . .; n
x0ikxik1 ifxiklikxik1 ifxikuikxik1 if r[ 0:5xik1 ifr 0:5
8>:
where r is a random number between 0 and 1.
Step-5 Repeat Step-2 to Step-4 for Nm times
Sometimes, in any generation, there is a chance that the
best chromosome may be lost when a new population is
created by crossover and mutation operations. To remove
this situation the worst individual/chromosome may be
replaced by that best individual/chromosome in the current
generation. This process is called elitism. In this operation
instead of single chromosome one or more chromosomes
may take part. Elitism guarantees that the objective func-
tion values do not improve from one generation to another.
7.1 Termination criteria
The termination condition is a condition for which the
algorithm is going to stop. For this purpose any one of the
following three conditions is considered as the termination
condition.
1. the best individual does not improve over specified
generations.
2. the total improvement of the last certain number of
best solutions is less than a pre-assigned small positive
number or
3. The number of generations reaches a maximum
number of generation i.e., max_gen.
7.2 Algorithm
Step-1 Set population size (p_size), crossover
probability (pc), mutation probability (pm),
maximum generation (m-gen) and bounds of
the variables li; ui i1; . . .; nStep-2 t =0 [t represents the number of current
generation]
Step-3 Initialize the chromosome of the population
P
t
[P
t
represents the population at
tthgeneration]Step-4 Evaluate the fitness function of each
chromosome of Ptconsidering the objectivefunction as the fitness function
Step-5 Find the best chromosome from the population
PtStep-6 tis increased by unity
Step-7 If the termination criterion is satisfied go to step-
14, otherwise, go to next step
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Step-8 Select the population Pt from the populationPt1 of earlier generation by tournamentselection process
Step-9 Alter the populationPtby crossover, mutationand elitism operators
Step-10 Evaluate the fitness function value of each
chromosome ofP
t
Step-11 Find the best chromosome fromPtStep-12 Compare the best chromosome of Pt and
Pt1 and store better oneStep-13 Go to step-6
Step-14 Print the best chromosome (which is the solution
of the optimization problem)
Step-15 End
8 Numerical illustrations
In thissection, we have considered theredundancy allocation
problem for low-level redundancy (see Fig. 3) and for high-level redundancy of five-link bridge system (see Fig. 4) for
numerical experiments. In bridge system network, subsys-
tem-5 represents a hub whereas other subsystems represent
servers/client with processors arranged in parallel.
This five-link bridge network system (Kuo et al. 2001)
works successfully as long as one of the paths, (subsystem-
1-subsystem-2) or (subsystem-3- subsystem-4), is active
independently of subsystem-5. However, if the pair of
subsystems (1, 4) or (2, 3) fails, then subsystem-5 plays an
important role in the system operation. In each subsystem-i
i1; 2; 3; 4; 5, there is a parallel configuration consistingofxi identical components having reliability ~ri. If ~Ri be thesystem reliability of subsystem-i, i1; 2; 3; 4; 5 then~Ri1 1 ~rixi ; i1; 2; 3; 4; 5:
The system reliability of the low-level five-link bridge
network system is given by the expression as follows:
~RSx ~R1 ~R2 ~Q2 ~R3 ~R4 ~Q1 ~R2 ~R3 ~R4 ~R1 ~Q2 ~Q3 ~R4 ~R5 ~Q1 ~R2 ~R3 ~Q4 ~R5; where ~Ri
1 ~Qi; i1; 2; 3; 4; 5The system reliability of the high-level five-link bridge
network system is given by the expression as follows:
~RS
h
1
1
~r1~r2
~q2~r3~r4
~q1~r2~r3~r4
~r1 ~q2 ~q3~r4~r5
~q1~r2~r3 ~q4~r5h, where ~ri 1~qi,i 1;2;3;4;5 andh be thenumber of redundant subsystems, arranged in parallel.
For low-level redundancy the corresponding system
reliability maximization and cost minimization problems
are of the form as follows:
Example-8.1
Maximize ~RSx ~R1 ~R2 ~Q2 ~R3 ~R4 ~Q1 ~R2 ~R3 ~R4 ~R1 ~Q2 ~Q3 ~R4 ~R5 ~Q1 ~R2 ~R3 ~Q4 ~R5Fig. 3 Low-level redundancy of five-link bridge system
Fig. 4 High-level redundancy of five-link bridge system
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subject to,
~g1x X5j1
~cjxjexp xj4
~b1 0;
~g2x X
5
j
1
~wjxj exp xj
4 ~b2 0;
and
Example-8.2
Minimize ~Csx X5j1
~cj xjexp xj4
h i
subject to; ~RSx ~RTWhere ~RSx ~R1 ~R2 ~Q2 ~R3 ~R4 ~Q1 ~R2 ~R3 ~R4 ~R1 ~Q2 ~Q3~R4 ~R5 ~Q1 ~R2 ~R3 ~Q4 ~R5
For high-level redundancy the corresponding system
reliability maximization and cost minimization problems
are of the form as follows:
Example-8.3
Maximize ~RSh 1 1 ~r1~r2 ~q2~r3~r4 ~q1~r2~r3~r4 ~r1 ~q2 ~q3~r4~r5 ~q1~r2~r3 ~q4~r5h
subject to,
~g1x hexp h
4
X5j1
~cj~b1 0;
~g2x
h exp
h
4 X
5
j1~wj
~b2
0;
and
Example-8.4
Minimize ~Csh hexp h4
X5j1
~cj
subject to; ~RSh ~RTwhere ~RSh 1 1 ~r1~r2~q2~r3~r4 ~q1~r2~r3~r4~r1 ~q2~q3~r4~r5~q1~r2~r3 ~q4~r5h
All the values of the parameters related to problems8.18.4 are given in Table 1:
The proposed method has been coded in C programming
language. The computational work has been done on a PC
with Intel core-i3 processor in Linux environment. For
each example 20 independent runs have been performed to
calculate the best found system reliability and best found
system cost which are nothing but the optimal values of
system reliability and system cost. Also we have been
computed the statistical measure like mean and variance of
system reliability as well as system cost. In this computa-
tion, the values of genetic parameters like p_size,max_gen,
pm and pc have been taken as 100, 100, 0.15 and 0.85
respectively. The computational results have been shown in
Table2.It has been observed from the computational results that
the mean system reliability/mean system cost coincides
with the best found system reliability/system cost. This
strict coincidence is due to the fact that each trial run
provides us optimum solution. Also, the lower ends of the
standard deviations, measured in interval form, assume
zero value. It may also be noted that the average CPU time
required for implementing the genetic algorithm, is also
very less.
Table 2 Computational results for examples 8.118.4
Example 1 3 2 4
xs/h (2, 2, 1, 2,
2)
(1) (1, 1, 2,
1, 1)
(3)
Best found
systemreliability
[0.939545,
0.999027]
[0.819842,
0.928286]
Mean value of
system
reliability
[0.939545,
0.999027]
[0.819842,
0.928286]
Best found
system cost
[53.186,
71.904]
[102.34,
138.159]
Mean value of
system cost
[53.186,
71.904]
[102.34,
138.159]
Standard
deviation of
system
reliability
[0,
0.059482]
[0,
0.108444]
Standarddeviation of
system cost
[0,18.718] [0,35.819]
CPU time in
seconds
0.04000 0.07000 0.03000 0.18000
Table 1 Values of the parameters related to Examples 8.18.4
j rj cj b1 wj b2 RT
1 [0.64,
0.66]
[3, 5] [105,
115]
[1.5, 1.6] [30,
35]
[0.99,
0.999]
2 [0.73,
0.76]
[4.5, 5] [2, 2.5]
3 [0.75,0.77] [5.5,7.5] [2, 2.25]
4 [0.83,
0.86]
[5, 7] [1.5,
1.75]
5 [0.88,
0.90]
[2, 2.5] [1.75, 2]
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9 Sensitivity analysis
To investigate the overall performance of the proposed GA
based penalty technique for solving low-level redundancy
as well as high level redundancy, sensitivity analyses have
been carried out graphically on the interval valued system
reliability with respect to different GA parameters sepa-
rately taking other parameters at their original values.
These have been shown in Figs.5,6,7,8. From Fig. 5 it is
observed that both the bounds of the interval valued system
reliability be the same for all the values of population size
greater than or equal to 30. This implies that our proposed
GA is stable when population size exceeds 30. From Fig. 6
it is clear that our proposed GA is stable when maximum
number of generation is greater than or equal to 10. In
Figs.7, 8, the values of interval valued system reliability
have been computed with respect to the probability of
crossover within the range 0.450.95 and the probability of
mutation within the range 0.050.30 respectively. From
these figures, it is clear that the proposed GA is stable with
respect to probability of crossover as well as the probability
of mutation.
0.9
0.92
0.94
0.96
0.98
1
10 20 30 40 50 60
Population size
Intervalvaluedsys
tem
reliability
Lower bound of system
reliability
Upper bound of system
reliability
Fig. 5 P_sizeversus interval valued system reliability for Example 1
0.9
0.92
0.94
0.96
0.98
1
10 20 30 40 50 60
Max_gen
Intervalvalu
edsystem
relia
bility
Lower bound of interval
valued system reliability
Upper bound of interval
valued system reliability
Fig. 6 Max_gen versus interval valued system reliability for Exam-
ple 1
0.92
0.935
0.95
0.965
0.98
0.995
0.45 0.55 0.65 0.75 0.85 0.95
Probability of crossover
intervalvaluedsystem
reliability
Lower bound of interval valued
system reliability
Upper bound of interval valued
system reliability
Fig. 7 P_cross versus interval valued system reliabilityfor Example 1
0.92
0.935
0.95
0.965
0.98
0.995
0 0.05 0.1 0.15 0.2 0.25 0.3
Probability of mutation
Intervalvaluedsystem
reliability
Lower bound of interval valuedsystem reliability
Upper bound of interval valuedsystem reliability
Fig. 8 P_muteversus interval valued system reliability for Example 1
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10 Concluding remarks
In this paper, for the first time, we have formulated two
different redundancies known as low-level redundancy and
high-level redundancy and proposed four problems where
each problem belongs to the category of interval valued
nonlinear integer programming problems. Then we have
solved these problems corresponding to constrained singleobjective interval valued reliability optimization problem.
The reduced problem has been converted to unconstrained
interval valued integer programming problem using Big-M
penalty technique and solved by genetic algorithm. To
solve the problem, we have developed a real coded GA for
integer variables with interval valued fitness function,
tournament selection, uniform crossover and one neigh-
borhood mutation and elitism of size one. It is well known
that the penalty coefficient plays a crucial role in solving
constrained optimization problem by penalty function
technique. However, the selection of the value of this
parameter is a formidable task. To avoid this difficulty, wehave used Big-M penalty technique which does not require
any penalty coefficient. This entire approach opens up the
scope for reliability optimization when reliability values
and other design parameters are interval/imprecise valued.
Thus, it can be claimed that the generalization attempted in
this paper can be handled the real life problem with
imprecise parameters. For further research, one may use
the proposed GA and interval approach in solving interval
valued-integer programming problems as well as interval
valued mixed-integer programming problems relating to
several real life application problems.
Acknowledgments The first author would like to acknowledge the
support of the University Grants Commission (UGC), India, for
conducting this research work.
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