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Journal of Fuzzy Set Valued Analysis

Operation of Hybrid Evolutionary Algorithm and Fuzzy Sets for Reliability Optimization of Engineering Systems

Article ID: Jfsva-00255, Publishing Date: December 2015, 8 Pages.

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DOI: 10.5899/2015/jfsva-00255

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Operation of Hybrid Evolutionary Algorithm and Fuzzy Sets for Reliability Optimization of Engineering Systems

Mohammad Mohammadi Najafabadi^1*, Esfandiar Eslami², Sara Mansouri³

(1) Department of Computer Science, Payame Noor University (PNU), P.O.Box, 19395-3697 Tehran, Iran.

(2) Department of Mathematics, Shahid Bahonar University of Kerman, Kerman.

(3) Department of Teaching English Language, Humanities faculty, IslamicAzad University-Najafabad Branch, Najafabad, Isfahan.

Copyright 2015 © Mohammad Mohammadi Najafabadi, Esfandiar Eslami and Sara Mansouri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This article uses a specific hybrid evolutionary algorithmto solve the series parallel redundancy optimization problem which is in a fuzzy framework. Reliability optimization provides a means to help the reliability engineer achieve such a goal. Most methods of reliability optimization assume that systems have redundancy components in series and /or parallel systems and that alternative designs are available. Optimization concentrates on optimal allocation of redundancy components and optimalselection of alternative designs to meet system requirements. A fuzzy simulation-based evolutionary algorithm is then employed to solve these kinds of fuzzy programming with fuzzy goal and fuzzy constraints. Finally, numerical examples are also given.

Keywords: Reliability system, Fuzzy optimization, Hybrid evolutionary algorithm, Reliability optimization

1 Introduction

As systems have grown more complex, the consequences of their unreliable behavior have become severe in terms of cost, effort, lives, and so on, and the interest in assessing system reliability and the need for improving the reliability of products and systems have become very important. The reliability of a system can be defined as the probability that the system has operated successfully over a specified interval of time under stated conditions. In the past few decades, the field of reliability has grown sufficiently large to include separate specialized subtopics, such as reliability analysis, failure modeling, reliability optimization, reliability growth and its modeling, reliability testing,reliability data analysis, accelerated testing, and life cycle cost [8].

Reliability optimization provides a means to help the reliability engineer achieve such a goal. Most methods of reliability optimization assume that systems have redundancy components in series and/ or parallel

Journal of Fuzzy Set Valued Analysis 2015 No.3 (2015) 269-276

Available online at www.ispacs.com/jfsva

Volume 2015, Issue 3, Year 2015 Article ID jfsva-00255, 8 Pages doi:10.5899/2015/jfsva-00255

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Journal of Fuzzy Set Valued Analysis 2015 No.3 (2015) 269-276 270 http://www.ispacs.com/journals/jfsva/2015/jfsva-00255/

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systems and thatalternative designs are available. Optimization concentrates on optimal allocation of redundancy components and optimal selection of alternative designs to meet system requirements. In the past two decades, numerous reliability optimization techniques have been proposed. Generally, these techniques can be classified as linear programming, dynamic programming, integer programming, and geometric programming, Heuristic method, Lagrange multiplier, and so on. [4].

2 Reliability optimization models with several failure modes

Using redundant components is well accepted as a technique to improve the reliability of asystem. Usually this problem can be formulated as a nonlinear integer program. [3, 6]

2.1. Notation and Nomenclature

k- Out - of - n: F the system is failed if and only if at least k out of its n elements are failed.

k - Out - of - n: G the systems good if and only if at least k out of its n elements are good.

Class O failure modes: Those of which subsystem i is 1 - out - of - m_i: F Class A failure modes: Those of which subsystem i is 1- out - of - m_i: G m_i: number of elements in subsystem i and

m = [m₁, m₂,…, m_N]

S_i: total number of failure modes in subsystem i

h_i: number of class O failure modes insubsystem i (u = 1,2,…, h_i)

S_i- h_i: number of class A failure modes in subsystem i (u=h_i+1, h_i+2,…si) q_iu: probability of failure mode u for each element in subsystem i

u_i: upper bound of m_i element in subsystem i

b_t: available amount of system resource(t =1, 2, …, T)

g_ti(m_i) : subsystem i requires this amount of resource t when it contains mi elements

) ( , )

( _i _u^A _i

o

u m Q m

Q : failure probability in subsystem i (m_i elements) for failure mode u of the class of O or A failure modes

Q^o( m_i),Q^A(m_i ) : failure probability in subsystem i (m_ielements) subject to class of O or A failure modes.

Q_i(m_i) : unreliability in subsystem i (m_ielements)

R(m): reliability of the system when the element allocation is m

G_t(m) : the system, requires this amount of resource t when the element allocation is m.

2.2. Mathematical model

The following assumptions are made for the problem:

1. All elements in subsystem i ares–independent with respect to failure mode u, for each i, u combination taken by itself.

2. The system is 1 - out - of - N: F with respect to its subsystems.

3. The failure modes are a partitioning (mutually exclusive and exhaustive) of the failure event. Thus failure probabilities for failure modes combine to obtain total failure probability.

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4. Within a subsystem, the elements are 1 - out - of - m_i: G for some failure modes while at the same time are 1 - out - of - m_i: F for the other failure modes.

Failures occur when the entire subsystem is subjected to the failure condition. All elements in subsystem i can be failed by only one of the si modes at any given time, and these failures are divided into the class O or class A failure modes. The failure probability Q^o_u(m_i) in subsystem i for failure mode u of class O failure modes is Q_u^o(m_i)1(1q_iu)^mⁱ,u = 1,2,…, hi

The failure probability Q^o(m_i) in subsystem i subject to the class O failure modes is





 ^hⁱ

u i o u i

o m Q m

Q

1

) ( )

(

The failure probability Q_u^A(m_i) in subsystem i for failure modes u of class A is

mi iu i A

u m q

Q ( )( ) u=h_i+1, h_i+2, …, s

i

The failure probability Q^A(m_i) in subsystem i subject to the class A failure modes is







 ⁱ

i

S

h u

i A u i

A m Q m

Q

1

) ( )

(

The unreliability Q_i(m_i) of subsystem i is obtained by adding the failure probabilities for class O failure modes to those for class A failure modes ; that is: _Q_i₍_m_i₎__Q^o₍_m_i₎__Q^A₍_m_i₎

Then the reliability optimization of redundant system with several failure modes can be formulated as follows:

Max







 ^N

i

i m

Q m

R

1

)) ( 1 ( )

( (2.1)

St







 ^N

i

t i ti

t m g m b

G

1

) ( )

( t = 1,2,..,T (2.2) ¹^^mⁱ ^^uⁱ i = 1,2,…,N (2.3) The optimization problem is to determine the element allocation which maximizes the nonlinear system reliability (2.1) subject to:

 The T nonlinear constrains

 The nonnegative conditions with upper bound (2.3).

This is a nonlinear integer programming.

3 Reliability optimization with fuzzy Goal and fuzzy constraints 3.1. Problem Formulation

The reliability optimization of redundant system with fuzzy goal and fuzzy constraints can be formulated as follows [1,2]:

Max

0 1

) ( 1 ( )

(m Q m g

R _i _l

N

L

 





(3.4) S.t

t i ti N

i

t m g m t

G( ) ( )

1





 t=1,2,…,T (3.5) ¹^^mⁱ ^^uⁱ , i=1,2,….,N (3.6) The symbols  and  represent fuzzy inequality, g0 represents the desired goal of the reliability R(m) given by decision maker and the other variable have been defined in the previous section.

Let Z₀ be the worst tolerable value of system reliability. The membership function _₀₍_m₎ for objective (3.4) is defined as follows [9]:

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













 







0 0 0

0 0

0

) ( 0

) ) (

(

) ( 1

) (

Z m R

g m R Z Z

g Z m R

g m R

 m

(3.7)

Let: _tbe the tolerable overuse of the t – The member ship function __t for constraint (3.5) is defined as follows:



















 







t t t

t t t t t

t t

t

b m G

b m G b b

m G

b m G m



 



) ( 0

) ) (

1 (

) ( 1

) (

(3.8)

Fuzzy reliability optimization problem can transform from (3.4) to (3.6) in to equivalent crisp problem as follows:

Max ₍ ₎ ₍ ₎

0

m w m

f _t _t

T

t







(3.9) S.t. R(m)(g₀z₀)₀(m)z₀ (3.10) ⁽^^t⁽^m⁾^¹⁾^^t^^G^t⁽^m⁾^^b^t^,^t^¹^,²^,...,^T (3.11) Where wt, t = 0, 1,…, T are the weights of membership function __t given by the decision maker with∑^𝑇_𝑡=0𝑤_𝑡 = 1.The fuzzy reliability optimization problem is to determine the best number of redundancy units for each subsystem to maximize the grades of system designers' satisfaction for both the achievement of objective of reliability and the best utilization of resources simultaneously [7].

4 Hybrid Evolutionary Algorithm

Hybrid evolutionary algorithm is stochastic search techniques based on the mechanism of natural selection and natural genetics. Hybrid Evolutionary algorithms, deferring from conventional search techniques, start with an initial set of random solution called population [3]. The strength of each chromosome is the objective function (fitness) that must be optimized. The new generation is formed by three basic operations:

reproduction, crossover and mutation from last generation according to their fitness value which provide them with higher probabilities of being selected finally, the population stabilizes, because no better individual can be found. When algorithm converges, and most of individuals in the population are almost identical, it represents a suboptimal solution. A hybrid evolutionary algorithm has three parameters: the population size, crossover rate and mutation rate, these parameter are important to determine the performance of the algorithm. The detailed description can be found in [10, 11].

4.1. Presentation of control variable

To apply hybrid evolutionary algorithm to solve a specific problem, one has to define the solution

representation and the coding of the control variables. Here chromosome is defined as an ordered list of the number of redundant units’ mxi shown as follows:

Vx= [mx1 mx2….mx6]

Where Vk is the k – th chromosome.

4.2. Initialization

The initialized procedure will select the initial population within the range of the control variable with a random number generator. The user can specify the population number in this procedure [11].

4.3. Fitness Evaluation

When evaluating chromosome, each legal one is assigned the value of objective function f(m) and each illegal chromosome is penalized as 0.

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





















Otherwise 0

t i, u m 1

δ , b (m) G , Z R(m) f(m)

)

fitness(V ⁱ ⁱ

t t t 0

k

(4.12)

4.4. Crossover

Crossover is one of the main distinguishing features of hybrid evolutionary algorithm that make them different from other algorithms. Its main aim is to recombine blocks on different individual to make a new one. Discrete recombination form is used here as crossover operator with Pc=0.74 as crossover probability [10]

{0,1}

λ λ , P

λ ) (1 λ P

O

P λ ) (1 λ P O

2 1 1

2 2

2 1 1

1 1















4.5. Mutation

Mutation is performed as random perturbation in the population to avoid premature convergence to local optimum. For a selected chromosome's gene nki, it will be replaced by a random integer with in [1-3] Let the probability of mutation, Pm be 0.03. [10]

4.6 Selection

4.6.1. Roulette wheel selection

The general selection operator is roulette-wheel, also called stochastic sampling with replacement. This is a stochastic algorithm and involves the following technique. The individuals are mapped to contiguous segments of a line, such that each individual's segment is equal in size to its fitness. A random number is generated and the individual whose segment spans the random number is selected. The process is repeated until the desired number of individuals is obtained (called mating population). This technique is analogous to a roulette wheel with each slice proportional in size to the fitness. But here rank of each individuals use instead of fitness. In ranking selection, the population is sorted according to the objective values. The fitness assigned to each individual depends only on its position in the individuals rank end not on the actual objective value [5].

Consider N the number of individual in this population (least fit individual has p = 1, the fittest individual P

= N) and SP is selected random number in [1, 2].

rank (P) = 2–sp+2+(sp -1)(p-1)/(n-1) (4.13) 4.6.2. Elitism

In elitism one or more of the best individuals is selected and transfer directly to next generation here the best selected chromosome is transfer. [10]

5 Case study and results analysis

The following example, modified by substitution last parameter with defined numbers to maximize the nonlinear reliability objective subject to fuzzy nonlinear constraints:

Max

 

 



 



 ⁶

1

4

1 1 1

1) ) ( ) ] 0.93

1 ( 1 ( 1 [ ) (

i u

m iu m

i

i q

q m

R 

(4.14)

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1,2,...,6 i : integer positive : 3 m 1

140 /4)}

m exp(

{m 20 (m) G

260 )}

m exp(

{m 20 (m) G

51 ) (m ) (m 2) (m ) (m ) (m ) (m (m) G

i 6

1 i

i i 3

6

1 i

i i

2

2 6 2 5 2 4 2 3 2 2 2 1 1



























(4.15)

Where m = [m1…m6]. The subsystems are subject to four failure modes (si=4)

With one O failure (hi=1) and three A failure, for subsystems i = 1,…, 6. For each subsystem the failure probabilities are shown in Table (1).

Let: g₀0.93,z₀0.86,δ₁14.0,δ₂8.0,δ₃8.0,w₀0.85,w₁0.05,andw₃0.05respectively. The above problem can be transformed into the following equivalent crisp problem:

Max f(m)0.85₀(m)0.05₁(m)0.05₂(m)0.05₃(m)

1,2,....6 i integer, 0 m

140 /4)}

m exp(

{m 20 (m))δ (1 μ

260 } m exp(

{m 20 (m))δ (1 μ

51 ) (m ) (m 2) (m ) (m ) (m ) (m 1)δ (m) (μ

z (m) )μ z (g R(m)

i

6 1 i

i i

3 3

6 1 i

i i

2 2

6 2 5 2 4 2

3 2 2 2 1 2 1 1

0 0 0 0

































Reliability membership is shown as follows which depicted in figure land in figure 2 membership function for first constraint is illustrated which shows legal tolerance for the first constraint.

Figure 2: First constraint membership function

Second and third membership function is depicted simultaneously in figure 3 & 4 which shown as follows.

Figure 3: Second constraintmembershipfunction Figure 4: Third constraint membership function

Search process of hybrid evolutionary algorithm started by sampled chromosomes as below by 100 populations: V= [v1 v2 v3 v4 v5 v6]. By using mentioned algorithm and operators search process performed

0 0.2 0.4 0.6 0.8 1

45 55 65

Degree of membership

first constraint

0 0.2 0.4 0.6 0.8 1

250 255 260 265

Degree of membership

Scond constraint

0 0.2 0.4 0.6 0.8 1

130 135 140 145

Degree of memebership

Third constraint

Figure 1: reliability membership function

0 0.2 0.4 0.6 0.8 1

0.8 0.9 1

Degree of Membership

relibility

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to reach solution the best solution from a random run of hybrid evolutionaryalgorithm with 100 generation is V^*= [3 2 3 3 2 2].

The values of corresponding membership function are: µ0=0.9445 µ1=0.7143 µ2=1 µ3=1.

6 Conclusion and implication

Nowadays, the development of sciences and technology and the complexity of engineering systems and their results have caused the effects of their unreliable behavior in different areas. These consequences are one of the real sources of financial, life-threatening and environmental destructions. The harmful effects of engineering systems’ unreliable behavior are evident in aerospace, army and nuclear industries. Therefore, the optimization of reliability coefficient of engineering systems has got increasingly great importance. In this article, we used special evolutionary algorithms to optimize the reliability problems with fuzzy goals and constrains. This enabled us to present a better and simpler answer in comparison with other optimization methods such as linear programming and lagrangean multiplier.

Table 1: failureprobabilities Subsystem i Failure modes

S_i=4 h_i =1 Failure Probabilities q_iu

1

O A A A

0.002 0.05 0.10 0.18

2

O A A A

0.004 0.02 0.15 0.12

3

O A A A

0.005 0.05 0.20 0.10

4

O A A A

0.003 0.01 0.18 0.10

5

O A A A

0.002 0.02 0.10 0.07

6

O A A A

0.002 0.02 0.10 0.10

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