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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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University of Burdwan

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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


D. Roy

Centre for Management Studies, The University of Burdwan,

Burdwan 713104, India


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Int J Syst Assur Eng Manag

DOI 10.1007/s13198-013-0199-9


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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


1 3


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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


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


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


1 3


10/12

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]


1 3


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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


1 3


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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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high and low level redundancy

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