Solving reliability redundancy allocation problems with ... - IEEE Xplore

Solving reliability redundancy allocation problems with orthogonal simplified swarm optimization Wei-Chang Yeh Department ofIndustrial Engineering and Engineering Managemen National Tsing Hua University Hsinchu 300, Taiwan [email protected]

Vera Yuk Ying Chung School ofInformation Technology University of Sydney NSW 2006, Australia [email protected]

Abstract-This study applies a penalty guided strategy and Optimization algorithm (SSO) to solve the reliability redundancy allocation problems (RRAP) in the series system, the series­ parallel system, the complex (bridge) system, and the overspeed protection of gas turbine system. For several decades, the RRAP been

one

of

the

most

well

known

techniques.

The

maximization of system reliability, the number of redundant components, and the reliability of corresponding components in each subsystem have to be decided simultaneously with nonlinear constraints, acting as one difficulty for the use of the RRAP. In other words, the objective function of the RRAP is the mixed­ integer programming problem with the nonlinear constraints. The RRAP is of the class of NP-hard. Hence, in this paper, the SSO algorithm is proposed to solve the RRAP and improve computation efficiency for these NP-hard problems. There are four RRAP problems used to illustrate the applicability and the effectiveness of the SSO. The experimental results are compared with previously developed algorithms in literature. Moreover, the maximum-possible-improvement (MPI) is used to measure the amount of improvement of the solution found by the SSO to the previous

solutions.

According

to

the

results,

the

system

reliabilities obtained by the proposed SSO for the four RRAP problems are as well as or better than the previously best-known solutions. Keywords-Reliability;

Redundancy

allocation

problem;

RRAP; Simplified Swarm Optimization algorithm; SSO; Mixed­ integer nonlinear programming

I.

INTRODUCTION

The system reliability optimization has received significant attention since the past several decades [1-8]. To optimize the •

Corresponding author

978-1-4799-1959-8/15/$31.00 @2015

IEEE

School of Software Jiangxi Agricultural University Nanchang, China 330045 School of Software Engineering South China University of Technology Guangzhou, China 510006 *[email protected] Xiangjian He Computer Vision and Recognition Laboratory Research Centre forInnovation inIT Services and Applications (iNEXT) University of Technology, Sydney (UTS) PO Box 123, Broadway NSW 2007, Australia [email protected]

the orthogonal array test (OA) based on the Simplified Swarm

has

' Yun-Zhi Jiang

system reliability is one of the major goals for the reliability engineering industry. Generally, there are two techniques to optimize system reliability. First is to increase the reliability of components, and second is to use the redundant components in the subsystems. The second technique is the more popular one. By the redundancy allocation problems (RAP), the reliability­ redundancy allocation problem (RRAP) is an optimization technique to maximize the system reliability [6]. The reliability which is of continuous value and the redundancy allocation which is of integer value should be considered simultaneously to optimize the objective function of the RRAP. The RRAP is NP-hard problem because the RRAP is the category of mixed­ integer programming with the goal of maximizing system reliability under constraints such as the system cost, weight, and volume, etc.. To improve the computation efficiency and the system reliability, there is numerous researches focusing on developing heuristic optimization algorithms to optimize the reliability for RRAP [1-3, 5, 7-15]. Among the numerous algorithms, the simplified swarm optimization (SSO) was developed by Yeh in 2009 [16] and is a population-based stochastic optimization method [16-38]. To improve the evaluation efficiency, this paper applies the SSO algorithm to optimize RRAP with considering the mixed-inter nonlinear programming models for four problems, including series system, series-parallel system, complex (bridge) system, and an overspeed protection of gas turbine system [1, 12]. The computational results are then compared with the previously developed algorithms in literature. The remainder of this paper is arranged as follows: Section II provides the definition of the RRAP and its four problems.

Section III introduces the SSO. The description of the orthogonal array test is presented in Section IV. The four problems and the results of the experiments are demonstrated in Section V. Finally, the conclusion is presented in Section VI. II.

Fig.3. The complex (bridge) system 2

THE DEFINITION OF RRAP AND FOUR PROBLEMS

The goal of the RRAP is to maximize the overall system reliability with nonlinear constrains that is the mixed-integer nonlinear programming with the highly computational complexity [1-2, 9, 11]. In this study, the RRAP model can be defined as follows. Maximize Rs = fiR, N)

(1)

s.t

(2)

The notations for the models below, where R/nJ = 1 _q;" is the reliability of subsystem i and q=1 - r is the failure I , probability of each component in subsystem i; a" /3, are the physical feature of each component in subsystem i; v., c., W. are the volume, cost, and weight of each component in subsystem i; and V, C, Ware the upper limits on the volume, cost, and weight of the system, respectively. The objective functions of maximizing the systems reliability of the four problems are different due to the system's structure. But they are subject to the similar multiple nonlinear constraints. And The RRAPs are formulated as follows, respectively.

, , ,

l:Sj:S the number of constraint where Ny is the number of subsystems in the system for the subsystem i= I ,2, ... ,Ny, Rs is the system reliability, fiR, N) is the fitness function with respect to R and N, R =( r l, r , . . . , rNY) 2 is the component reliability vector of the system where r. is the reliability of each component in subsystem i, N=(nl, n2, ..., nNY) is the redundancy allocation vector of the system where n. is the number of components in subsystem i. g(R, N) is thejth J constraint function with respect to R and N, and Ii is the resource limitation for the l constraint.

,

Problem 1:

,

This paper applies the SSO to four problems of optimizing RRAP as shown in Figs. 1-3 including series system, series­ parallel system, complex (bridge) system, and an overspeed protection of gas turbine system [1, 12].

The series system as Fig. 1 [12] N,

(3)

Max f(R,N)= I1R,(nj) i =l

S.t

(

gl R, N)=

Nv

LWjv:n,2 :SV, j =1

(4)

g2(R,N) = Ia,(-lOOO/ln 'i l' (n, + exp(n, 14»:S; C, N.

Fig. l. The series system

(5)

i=l

2

3

4

5

g3(R,N)=

2

n, E positive integer,

-

Problem 2:

-

3

5

-

4

i = 1, ..., Nv

The series-parallel system as Fig. 2 [12]

I--

-

�

(6)

, =1

o :S lj :S1, lj E real number,

Fig.2. The series-parallel system 1

Nv

L w,n, exp(n, /4):S W

-

s.t

(8)

gJR,N):S V,

t---

(9)

g3(R,N):S W

(10)

O:S lj :S1, lj E real number,

n, E positive integer,

i = 1, ... , Nv

Problem 3:

The complex (bridge) system as Fig. 3

[12]

Max f(R,N) RJ� + R)R4 + RJR4R5 + �R)R5=

RJR2RJR4 - RJR2R3R, - R1R2R4R, RJR)R4R5- R2R)R4R5 + 2RJ�R)R4R5

0::; r, ::;

1,

The steps of SSO are as following:

r, E real number,

ni E positive integer, Problem 4:

(11)

i=

1, ..., Nv

The overspeed protection system [ 1,

15].

10, 12,

N,

Max f(R,N)= I1[1-(1-ri)"i]

(15)

;=1

2 �(R,N)= 2>in, ::;V, ,=1

(16)

�(R,N) = :�>x,(-1000/lnr;)iJ, (n, +exp(n, /4»:S; c ,

(17)

N

i=l

III.

I w,ni exp(ni /4)::; W, i i

=

1,...,Nv

OVERVIEW OF SSO

Like most heuristic techniques, SSO is also initialized with a population of random solutions inside the problem space and then searches for optimal solutions by updating solutions over a number of generations. Applied from the PSO, each solution moves toward its Personal best (pBest) and toward the Global best (gBest). However, the SSO developed a random movement to escape from a local optimum [ 16- 38 ]. The SSO is based on the following Eq.(19):

xf J..

+ ,',j

=

pgseSf j ,

if pE [O,Cg =cg)

p.

if pE [Cg,Cp =Cg +cp)

X

if pE [C""

',j

. . X1,1,)

if pE [Cp,C", =Cp +cw)

1)

STEP 1.

Let i= l.

STEP 2.

Update �'I" to �" based on the Eq. (19) and calculate

F(�,i)' STEP 3. If F(�, ) is better than F( P), then let Pi=�", Otherwise, go to STEP 5. STEP 4. If F(P,) is better than F(PgBes/)' then let gBest=i, STEP 5. If i