Redundancy Optimization of Static Series-Parallel ...

Redundancy Optimization of Static Series-Parallel Reliability Systems Under Uncertainty

G.Levitin , A.

Reuven Y. Rubinsteiny Lisnianski , H. Ben-Haim , and D. Elmakis

yFaculty of Industrial Engineering and Management

Technion|Israel Institute of Technology, Haifa 32000, Israel Israel Electric Corporation LTD

Reliability Department, Haifa 31000, Israel

Abstract This paper extends the classical model of Ushakov on redundancy optimization of a series-parallel static coherent reliability systems with uncertainty in system parameters. Our objective function represents the total capacity of a series-parallel static system, while the decision parameters are the nominal capacity and the availability of the elements. We obtain explicit expressions (both analytical and via ecient simulation) for the constraint of the program, namely for the cdf of the total capacity of the system, and then show that the extended program is a convex mixed integer one. Depending on whether the objective function and the associated constraints are analytically available or not, we suggest using deterministic and stochastic (simulation) optimization approaches, respectively. The last case is associated with likelihood ratios (change of probability measure). Numerical results are presented as well.

Keywords. Likelihood Ratios, Redundancy Optimization, Reliability Networks, Sensitivity Analysis, Simulation. Version date November 10, 1998. The work of Reuven Rubinstein was supported by the Henri Gutwirth Fund for the Promotion of Research 0

Contents 1 Introduction

2

2 Explicit Calculation of the cdf FL for Series-Parallel Systems

4

3 Estimating the cdf FL from Simulation

6

4 Optimization

13

5 Concluding Remarks

16

1

1 Introduction Systems with redundancy are abundant in real life. Most books on reliability engineering (e.g., Barlow and Proshan [1], Gertzbakh [3], Kozlov and Ushakov [6], Mann, Shaer and Singpruvalla [7], Zacks [18], and Ushakov and Harrison's [15] recent handbook on reliability (which can serve as a good source of references), include a chapter on redundancy models and redundancy optimization. In this work we extend the classical redundancy optimization model of Ushakov [14], the pioneer and promoter of this exciting and important eld, and discuss its solution at some length. Our motivation for such extended models stems from practical needs in redundancy optimization of electric power systems, and is described in [5]. To proceed, consider the basic redundancy optimization problem, [13], [14]: (

(P )

min r C (r) s:to P fL(r ) > xg >  :

(1.1)

Here C (r) and L = L(r) depend on the vector r of redundancy units, and called, the total systems cost and the sample performance, respectively; x and  (0 <  < 1) represent the minimum acceptable capacity and the minimally acceptable probability that this capacity is achieved. The program (1.1) is called the "direct" type, while its counterpart (

max r P fL(r) > xg ; s:to C (r) < C0 is called the "inverse" type. Here C0 is a xed quantity.

(1.2)

In this work we consider series{parallel systems with the sample performance

L(r) = i=1min ;:::;m

ri X j =1

Xij ;

(1.3)

also called the total capacity of the system. Here Xij are independent random variables, each having a xed distribution and m is the number of dierent type of elements in series. is

Unlike [12], [15], where the random variables Xij are distributed Bernoulli (vij ); that

PXij (x) = P fXij = x) =

our model extends (1.4) as follows:

PXij (x) = P fXij = x) =

(

(

vij ; 1 ? vij ; vij ;

1 ? vij ;

if x = 1 ; if x = 0 ;

(1.4)

if x = gij ; if x = 0 :

(1.5)

Here gij and vij are called the nominal capacity and availability of the element (ij ), respectively. It is important to note that, in addition, our model allows vij to be either deterministic or a random variable with a given distribution PVij (x). (We use below capital letters for random variables). Note nally that in [12], [15] the total system cost C (r) is 2

assumed to be a separable- linear function with respect to the components rj ; j = 1; : : :; m of the vector r, namely m X (1.6) C (r) = Cj
rj ; j =1

while in our case the total system cost C (r; g) is assumed to be a separable- nonlinear function with respect to components of both the vectors r and g, namely

C (r; g) =

m X

j =1

Cj (rj ; gj ):

(1.7)

A practical example of (1.7), where with b1 = 50; b2 = 0:6, and

 b2 Cj (rj ; gj ) = rj
bgj
b3; 1

8 > < b3 = > :

3:0; 2:8; 2:0;

if if if

rj  3 ; 3 < rj  5 ; rj > 5

(1.8)

(1.9)

is discussed in [5]. With this in mind, consider the following extended version of the program (1.1)): (P )

(

min r;g C (r; g) s:to P fL(r; g) > xg >  :

(1.10)

Note that (1.10) is a mixed-integer program. Our goal is to nd both the optimal con guration of the system and the optimal capacity vectors, say r = (r1; : : :; rm ) and g = (g1; : : :; gm ), respectively. Note also that the original program (1.1) represents a particular case of (1.10) with gij  1. In analogy to (1.2), the inverse of program (1.10) can be written as ( max r;g P fL(r; g) > xg ; s:to C (r; g) < C0 :

(1.11)

To proceed, note that the optimal solutions of programs (1.10) and (1.11) require knowledge of the cumulative distribution function (cdf)

FL(r;g) (x) = P fL(r; g)  xg (1.12) of the random variable (sample performance) L, which is typically not the case. One of our main goals in this work is to establish mathematical grounds for calculation of FL (x),

both analytically and via ecient simulation techniques. (In the last case we approximate FL(x) by an empirical cdf, say FL(x)). In Section 2, we show how to compute the cdf FL (x) explicitly for a series-parallel con guration. Here we see that in some cases such computation might be rather complex 3

and time-consuming, especially when the r; (i = 1; : : :; m) are large numbers. To overcome this diculty we show in Section 3 how to estimate FL from simulation. Here we present two algorithms for ecient estimation of FL = FL(r ;g) (x) simultaneously for several values r and g using a single simulation run. These algorithms are based on the likelihood ratio and the score function methods, and use recent advances in simulation methodology (see Rubinstein and Shapiro [10]). Section 4 deals with the solution of the program (1.10). In particular a genetic algorithm for solving the program (1.10) with the cost function given (1.7)- (1.9) is presented. Concluding remarks are given in Section 5.

2 Explicit Calculation of the cdf FL for Series-Parallel Systems Bearing in mind (1.3), we have for a xed matrix fvij g that

FL (x) = P fL  xg = 1 ? = 1? P

m Y

m Y i=1

[1 ? P (Yi  x)]

(1 ? FYi (x)) ;

(2.1)

i=1

i X and X are given in (1.5). In the case where the availabilities v are where Yi = rj =1 ij ij ij random variables, the cdf FYi = FYi jV i is a random variable as well and can be written as

FYijV i (xjvi) = P (Yi  xjV i = v i ) = o n IEYi IfYi xg jV i = vi ; v i = (vi1 ; : : :; viri );

(2.2)

where IEYi means that the expectation is taken with respect to the random variable Yi . If, in addition, the rv's (availabilities) Vij are independent, the unconditional expectation of (2.1) becomes (

FL (x) = IEV 1 ? =

m Y i=1

)

(1 ? FYi (xjV i = v i ) = 1 ?

m Y

(1 ? IEV i fFYi (xjV i = vi )g

i=1 m  o  n Y 1 ? IEV i IEYi IfP Xij xg jV i = vi 1? i=1

;

(2.3)

where IEV means that the expectation is taken with respect to the random vector V = (V 1 ;


V m ). We shall show next how to calculate explicitly the cdf FL separately for the case where the matrix fvij g is : (a) deterministic and (b) random. (a) It follows from (2.1) that the problem of calculating the cdf FL reduces to the problem of calculating the cdf's FYi of the rv's

Yi =

ri X j =1

Xij ; i = 1; : : :; m : 4

That is, for each i, we have to nd the cdf FYi of the sum of rv's Xij , each having a two-point distribution (see (1.5)). This can be done explicitly, although, in some cases, through tedious calculations. The following example gives an explicit calculation of FY for r = 2. Extension to r > 2 is simple.

Example 2.1 The probability mass function (pmf) PY of the rv Y = P2i=1 Xi is 8 > (1 ? v1 )(1 ? v2) ; y = 0; > > < v (1 ? v ) ; y = g1 ; (2.4) PY (y) = > (11 ? v )v2 ; y = g2 ; 1 2 > > : v1 v2

;

y = g 1 + g2 :

P

Notice that for the case Y = ri=1 Xi , the pmf obtains positive values at 2r points. In the particular case where gij = gi and vij = vi , i.e., where for xed i; i = P 1i ; : : :; m; the capacities and the availabilities are constant, the random variable Yi = rj =1 Xij is a multiple of a binomial rv. In other words, Yi has a binomial distribution which takes values on the set of points f0; gi; : : :; ri  gi g, that is the rv Y~i = Yi  Bin(vi; ri);

gi

or

FYi (x) = P (Y~i  gx ) = i

!

ri vy (1 ? v )ri?y ; 0  x  r g : i i i y i

X h i 0y gxi

(2.5)

Substituting (2.5) into (2.1), we obtain 2 m6 X Y FL (x) = 1 ? 6641 ? h i i=1 0y x gi

3 ri v y (1 ? v )ri?y 777 i 5 y i !

; 0  x  1min fr g g: im i i

(2.6)

(b) Consider now the case where the availability matrix is random. It follows from (2.3) that in this case calculation of FL reduces to calculation of the expectation of the conditional cdf FYi jV i , that is IEV i fFYi (xjV i = vi ): We now calculate IEV PY jV (y ) for Example 2.1 (r = 2). Extension to r > 2 is simple.

Example 2.2 (Example 2.1 continued). Let fV be the pdf (probability density function) of the random vector V . Taking into account (2.4) and the independence of the rv's X1 and X2, it is readily seen that

where

8 > (1 ? v~1)(1 ? v~2 ) > > < v~ (1 ? v~ ) IEV PY jV (y ) = > (11 ? v~ )~v1 1 2 > > : v~ v~ 1 2 Z

; ; ; ;

v~i = IEVi = vfVi (v)dv: 5

y = 0; y = g1 ; y = g2 ; y = g 1 + g2 ;

(2.7)

Consider the following two particular cases of the pdf fV (x): (a) Let Vj  U (0; 1); j = 1; : : :; r. We obtain X IEV PY jV (y ) = 1r ; y = gi; A  f1; : : :; rg: 2 i2A

(2.8)

(b) Let for xed i, each rv Vij ; j = 1; : : :; ri be iid distributed fVi (x) and let gij = gi . Then it is readily seen that !

IEVi (PY~i jVi (x)) = rxi v~ix (1 ? v~i )ri ?x ; x = 0; 1; : : :; ri;

(2.9)

R where, as before v~i = E [Vi] = vfVi (v )dv , and Y~i = Ygii . If, for example, the rv Vi is distributed Beta(; ), then

E[Vi] =  +i = v~i: i

i

Clearly, in this case case (2.6) holds again, but with vi replaced by v~i . It also follows from the above discussion that in both, deterministic and stochastic cases of the matrix v = fvij g, we can calculate explicitly the cdf FL (x). For the general case (1.5) calculation of FL (x) might be, however, rather involved. Moreover, if the decision parameter vectors r and g in FL (x) = FL(r ;g) (x) change, one typically has to recalculate FL(r ;g) (x) from scratch. This, clearly, may lead to time-consuming optimization procedures, while solving program (1.10). To overcome this diculty we present in the next section several Monte Carlo (MC) procedures for estimating the cdf FL(r;g) (x) simultaneously for several values r and g, using a single simulation run. This estimated cdf, say FL(r;g) (x), can be used (instead of the original cdf FL(r ;g) (x)) to derive an approximate solution (estimate) of program (1.10). The approximate program, called the stochastic counterpart, can be written as ( min r;g C (r; g) (2.10) (PN ) s:to FL (r; g) < 1 ?  : Although the optimal solution of the program (2.10), say (rN ; g N ), is only an estimate of the optimal solution (r ; g ) of the original program (1.10), it is typically more convenient to deal with, especially when the elements in the series-parallel con guration are dependent or when the system has a more complex con guration.

3 Estimating the cdf FL from Simulation Consider formula (2.1). Let FYi be the empirical cdf of the rv. Yi . A crude Monte-Carlo (CMC) estimate of FL (x) is

FL(x) = 1 ?

m Y

i=1

6

(1 ? FYi (x)) :

(3.1)

That is, the problem of estimating P i the cdf FL (x) reduces to the problem of estimating the cdf's FYi (y ) of the rv.'s Yi = rj =1 Xij ; i = 1; : : :; m. An alternative to CMC estimator FL (x) in (3.1) is based on the representation

FL(x) = P fL  xg = IEfXij g IfLxg;

(3.2)

N X 1  IfLk xg ; FL (x) = N

(3.3)

and can be written as

Pi where Lk = mini Yi(k) and Yi(k) = rj =1 Xij(k).

k=1

We shall introduce below, the so-called likelihood ratio (LR) of FL (x) (and the score function (SF) estimators of rFL (x)), which typically have better performance than the CMC estimators (3.1) and (3.3), respectively (see [10]). We consider separately the LR estimators of FL based on (2.1) and (3.2). Q (a) The LR estimator of FL (x) = 1 ? mi=1 (1 ? FYi (x).

P

Suppress for a moment the index i in Yi ; i = 1; : : :m, and write Y = rj=1 Xj . Let PY (y; v0 ) be a pmf dominating the pmf PY (y; v) of the rv Y in the absolute continuous sense, that is suppPY (y; v)  suppPY (y; v0 ) for all v 2 V; where v = (v1; : : :; vr ) is the parameter vector in PY (y; v), and similarly v0 = (v01; : : :; v0r ). Then the cdf FY (y ) = IEY fIfY yg g can be written as 0

0

FY (y) = IEY fIfY yg W (Y0 ; v; v0 )g; 0

where

(3.4)

0

W (y; v; v0 ) = PPY ((y;y; vv)) Y0

is called the likelihood ratio and Y0  PY (y; v0 ).

0

0

The derivative (gradient) of FY (y ) = FY (y; v) with respect to v is based on the score function method (see Rubinstein and Shapiro [10]) and can be written as

rFY (y) = IEY fIfY yg rW (Y0; v; v0)g; 0

where and

(3.5)

0

rW (y; v; v0) = rP PY(y;(y;vv)) = W (y; v; v0)
S (y; v) Y0

is called the score function.

0

(y; v) S (y; v) = rPPY(y; v) Y

Unbiased estimators (the empirical cdf) of FY in (3.4) and of its gradient in (3.5) are

FY (y) = N1

N X i=1

IfY i yg W (Y0i ; v; v0 ) = N1 0

7

N X i=1

IiWi

(3.6)

and

rFY (y) = N1

respectively, where again

N X i=1

IfY i yg rW (Y0i ; v; v0) = N1 0

N X i=1

Ii Wi Si;

(3.7)

Si = S (Yi; v) = rPPY(Y(Y;i ;vv)) Y

is the score function.

i

Depending on the goals, there are many alternative ways to choose the dominating pmf PY (y; v0 ) in (3.6). The two natural ways are associated either with deriving of accurate (small variance) LR estimators of FY , or with estimation of the cdf FY = FY (r;g) simultaneously for dierent values r and g . Our main emphasis in this paper will be on the second issue. In particular we discuss the case where the rv Y0 has a discrete uniform pmf over the set of points L of y , where PY (y; v) > 0, that is (3.8) PY (y; v0 ) = 21r ; y 2 L: 0

0

0

It is readily seen that such choice of the pmf PY typically leads to rather straightforward computation of FY . Indeed, substituting (3.8) into (3.6), we obtain 0

N r X

FY (y) = 2

N

i=1

IfY i yg PY (Y0i ; v) ; 0

y2L

(3.9)

Consider the following two particular cases: 1. The capacities gj j = 1; : : :; r are equal, that is gj = g . In this case Yg  U (0; r), and (3.8) and (3.9) reduce to PY (y; v0) = r +1 1 ; y = 0; 1; : : :; r (3.10) and 0

0

N X

+ 1) FY (x) = (r N

i=1

IfY i yg PY~ (Y0i ; v); y = [ xg ] = 0; 1; : : :; r; 0

(3.11)

respectively. P 2. The elements Xj in (1.5) are iid (gj = g; vj = v ). In this case the rv Y~ = g1 rj=1 Xj is distributed Bin(r; v ), and FY (x) can be written (see also (2.5) as !

N +1 X FY (x) = r N IfY i yg Yr v Y i (1 ? v)r?Y i ; y = [ xg ] = 0; 1; : : :; r : (3.12) i=1

0

0i

0

0

Notice that we can write FY (x) in (3.9) in the following (more convenient for computation) form N rX (3.13) FY (y) = 2 i;

N i=1

8

where

8 > < i=> :

PY (Y0i; v);

if IfY i yg = 1; 0

(3.14) 0; otherwise. and similarly the the empirical cdf FY (y ) in (3.11). The algorithm for estimating FY according to (3.13){(3.14) is simple and can be written as follows.

Algorithm 3.1 : 1. Generate a sample Y01 ; : : :; Y0N from the discrete uniform pmf (3.8). 2. Calculate FY according to (3.13){(3.14).

We shall discuss next the pros and cons of the estimator (3.13){(3.14). The disadvantage of FY is that it presents an estimate of FY . Its advantages are that (a) The computation of FY is simpler than FY . Indeed, as soon as the sample Y01; : : :; Y0N from U (0; 2r ? 1) is generated, computation of FY consists of averaging the product IfY i yg  PY (Y0i; v ): Clearly, computation of the indicator IfY i yg , is straightforward and computation of the pmf PY (y; v) for xed y = Y0 is simpler than computation of the associated cdf X FY (x; v) = PY (y; v): 0

0

yx

(b) The estimator (3.11) can be modi ed for simultaneous computation of FY (r;g) (y ) for several values of r and g from a single simulation. (This is important in optimization). Let us discuss issue (b) in more details, while considering the estimate FY (y ) in (3.11). Assume rst that r is xed and we want to estimate FY (g)(y ) for several values of g , say for g = (g1; : : :; gk ) simultaneously from a single simulation. To do so, we need to store in the computer the following k simple sequences fIi(gj ) = IfY i [ gxj ]g ; j = 1; : : :; kg; i = 1; : : :; N; (3.15) 0

each of length N , and then apply (3.11) simultaneously k times. Assuming, further for simplicity, that g1 < g2 <


< gn , and then taking into account that for xed i Ii (g1)  Ii(g2) 


 Ii(gk ); 8 i (3.16) we can further simplify the calculations of the sequence fFY (gj ) ; j = 1; : : :; kg by generating the proceeding sequence fIi (gj )g from the previous one fIi(gj ?1 )g. Let g be xed and assume that we want to estimate FY (r) (y ) for several values r, say for r = (r1; : : :; r ). To do so we use the following modi cation of Algorithm 3.1. 9

Algorithm 3.2 : 1. Generate a sample Y01 ; : : :; Y0N from the discrete uniform pmf U (0; r), where r = maxfr1; : : :; r ). 2. Calculate the sequence FY (rs) (y ); s = 1; : : :;  according to N  + 1) X ( r ~i (rs ) ;  FY (rs)(y) =

N

where

8 > < PY (Y0i; v ); ~i (rs) = > :

0;

i=1

if IfY i [ xg ]g = 1 and Y0i = 0; 1; : : :; r; 0

otherwise.

(3.17)

(3.18)

Notice that for rs = r the estimator (3.17){(3.18) coincides with the estimator (3.13){(3.14).

To illustrate Algorithm 3.2 consider the following example.

Example 3.1 Assume that we want to estimate FY (r)(y) simultaneously for r = 2 and r = 3, provided

8 > < vj P fXj = xg = > :

1 ? vj ; if x = 0;

j = 1; 2; 3. We have

8 > > > > > > > > < P fY1 = xg = > > > > > > > > :

if x = 1;

2 Y

j =1

(1 ? vj );

(1 ? v1 )v2 + (1 ? v2 )v1; 2 Y

j =1

vj ;

x = 0; x = 1; x = 2;

and

8 3 Y > > > (1 ? vj ); > > > > j =1 > > > > < (1 ? v1 )(1 ? v2 )v3 + (1 ? v1 )(1 ? v3 )v2 + (1 ? v2 )(1 ? v3 )v1; P (Y2 = x) = > v v (1 ? v ) + v v (1 ? v ) + v v (1 ? v ); 1 2 3 1 3 2 2 3 3 > > > > > 3 > Y > > > vj ; > : j =1 P P for Y1 = g1 2j =1 Xj and Y2 = g1 3j =1 Xj , respectively.

x = 0; x = 1; x = 2; x = 3;

Assume, that we generated a sequence fY01; : : :; Y04g of 4 random variables from the discrete uniform pmf U (0; 3) which results in fY01; : : :; Y04g = f0; 2; 0; 3g. Let x = 1. We 10

have fIfY i 1g ; i = 1; 2; 3; 4g = f1; 0; 1; 0g. Let further vj = v = 12 . Then for r = 2 and r = 3 we have (see (3.17){(3.18))   f ~i(r = 2); i = 1; 2; 3; 4g = f(0; PY (Y01 = 0); 0; PY (Y01 = 1)g = 0; 14 ; 0; 21 ; and   f ~i(r = 3); i = 1; 2; 3; 4g = f0; PY (Y02 = 0); 0; PY (Y02 = 1)g = 0; 18 ; 0; 38 : The resulting estimators are   FY (y = 1; r = 2) = 4
41 0 + 41 + 0 + 12 = 43 ;   FY (y = 1; r = 3) = 4
14 0 + 81 + 0 + 38 = 21 : 0

1

1

2

2

1

2

(b) The LR estimator of (3.2) Using again likelihood ratios and the score function, we obtain in analogy to (3.4) and (3.5) n o f (Y 0 ; v; v0 ) ; FL (x) = IEY Ifmini ;:::;m Y i xg W (3.19) 0

=1

n

rFL(x) = IEY Ifmini

f = rW f (Y 0 ; v; v0) = r rW

m Y i=1

o

f ;:::;m Y i xg rW (Y 0 ; v; v0 ) ;

=1

0

respectively, where

0

(3.20)

0

Wi (Y0i; vi ; v0i ) ; and Wi = PPYi ((YY0i;; vvi )) : Y0i 0i 0i

Let again the dominating pmf's PY i (y0i ; v0i); i = 1; : : :; m be chosen as in (3.8), that is PY i (y0i ; v0i ) = 21ri ; y 2 Li: Then the cdf's FL (x) and FL (x) can be written as 0

0

FL(x) = and

m ! Y 2ri i=1

FL (x) = N ?1

IEY

(

0

Ifmini

;:::;m Y0i xg

=1

m Y i=1

)

PYi (Y0i; vi) ;

m N m Y X Y Ifmini=1;:::;m Y0in xg PYi (Y0in ; vi); 2ri n=1 i=1 i=1

(3.21) (3.22)

respectively, where Y 0 = (Y01; : : :; Y0m ), and analogously the derivatives of FL (x; v) with respect to v. The algorithm for estimating FL (x) according to (3.22) can be written in analogy to Algorithm 3.1 as follows:

Algorithm 3.3 : 11

1. Generate m samples fY0i1; : : :; Y0iN ; i = 1; : : :; mg, each from the discrete uniform pmf U (0; 2r ? 1). 2. Calculate FL (x) as N m X Y Pn ; (3.23) FL (x) = N ?1 2ri where

n=1

i=1

8 m > < Y PY (Y0in ; vi); Pn = > i=1 i :

0;

if Ifmini

;:::;m Y0in xg ;

=1

otherwise.

(3.24)

Note that Algorithm 3.3 diers from Algorithm 3.1 in the way that in the latter we use the likelihood ratio y; vi ) Wi = PPYi ((y; v ) Y0i

0i

separately for each random variable Yi and then estimate FL (x) according to (3.13){(3.14), f = Qni=1 Wi ) simultaneously for the vector Y = while in the former we use the LR (W (Y1 ; : : :; Ym ) and then estimate FL (x) according to (3.23){(3.24). As a result, (for a xed replication n), we calculate only once the indicator Ifmini ;:::;m Y in xg in Algorithm 3.3, while we calculate a sequence of m indicators fIfY i yg ; i = 1; : : :; mg in Algorithm 3.1. Clearly, for m = 1 both algorithms coincide. =1

0

0

Note that Algorithm 3.3 can be modi ed similarly to Algorithm 3.1 for estimation of FL(r;g) (x) simultaneously for dierent values of g and r. Note, nally, that Algorithm 3.3 can be applied to more general distributions and topologies than Algorithm 3.1, e.g., where the rv's Xij are dependent. We present now an alternative approach for estimation of FL (x), which is based on the following:

Theorem 1 (Walker, 1977) Any n-point discrete pmf PY (yi) = P fY = yi g; i = 1; : : :; n can be represented as an equally weighted mixture of n ? 1 pmf's PZ(s) ; s = 1; : : :; n ? 1, each having at most two non-zero components. That is, any pmf PY can be written as

nX ?1

PY = n ?1 1 PZ(s) s=1 for suitably de ned 2-point pmf's PZ(s) ; s = 1; : : :; n ? 1.

(3.25)

Walker's method is rather general and ecient, but requires some initial setup and extra storage for the n ? 1 pdf's PZ(s) . As soon as the representation (3.25) is established, generation from PY is simple and can be written as:

Algorithm 3.4 : 1. Generate a random variate U from the discrete uniform pmf U (1; n ? 1). Let u (u = 1; : : :; n ? 1) be the outcome.

12

2. Generate a random variate Z from the two-point pmf PZ(u) .

Again, as soon as the representation (3.25) is established, both CMC estimators (3.1) and (3.3) for FL (x) are rather straightforward. Note that in both cases, we must apply Walker's (1977) theorem m times (separately for each pdf FY i ; i = 1; : : :; m). Note, nally, that typically one uses Walker's theorem for large n = 2r , say, for r  5. Let us show now how to modify Walker's (1977) theorem in order to estimate the

FY (r)(y) simultaneously for several values of r, say for r ? 1; r and r + 1. A simple way of doing so is to sample from the pmf PY (r)(y ) by using Walker's theorem and from the two-point pmf's, PXr? (x), PXr (x), as per (1.5). 1

Let and

+1

Y1 ; : : :; YN ; X(r?1)1; : : :; X(r?1)N ; X(r+1)1; : : :; X(r+1)N

be the resulting sequences. Then, clearly the desired cdf's can be estimated as

FY (r?1)(y) = N1 FY (r)(y) = N1 and respectively.

FY (r+1) (y) = N1

N X i=1 N X i=1 X

IfYi?X r? i yg ; (

1)

IfYiyg ;

IfYi+X r i yg ; ( +1)

4 Optimization Both (deterministic and stochastic) programs (1.10) and (2.10) with the cost function given in (1.7)- (1.9) are convex mixed-integer programs. This is so since the cost function (1.7)- (1.9) is convex, and the cdf F (empirical cdf F ) is a non-decreasing function in each component of the vectors r and g, respectively. Solution of such convex mixed-integer programs is not straightforward, however. The state-of-the-art of integer (mixed-integer) programming is rather advanced for the linear-integer case, (e.g., [8]). This implies that Ushakov's [12], [15] models (1.1) and (1.2) can be treated by conventional linear-integer programming techniques, provided cdf F (r ) can be approximated nicely by a linear-integer function. As very little is known at present about the general convex integer case, we use heuristics. A natural way of solving the programs (1.10) and (2.10) is by using heuristic algorithms, such as genetic algorithms, simulating annealing and tabu search algorithms (see Gass and 13

Harris [2]). In this paper we adopted the genetic algorithm (GA) of Whitley and Kauth [17]. Below we give a very short introduction to GA. The interested reader is advised to consult [5] for more details. Genetic algorithms are global search optimization algorithms based on the philosophy of natural selection and natural genetics. They employ a structured randomized parallel multipoint search strategy that is biased toward reinforcing search points at which the function being minimized has relatively low values. Genetic algorithms are similar to simulated annealing in that they employ random (probabilistic) search strategies. To start the genetic search, an initial population of, say, randomly constructed M strings is generated. From this initial population, subsequent populations are computed by employing crossover and mutation operators. Crossover produces a new solution (ospring) from a randomly selected pair of parent solutions providing inheritance of some basic properties of the parents in the ospring. Mutation results in slight changes in the ospring structure and maintains di versity of solutions. The standard genetic algorithm uses a roulette wheel method for selection, which is a stochastic version of the survival-of-the- ttest mechanism. In this method of selection, candidate strings from the current generation are selected to survive to the next generation by designing a roulette wheel where each string in the population is represented on the wheel in proportion to its tness value. Thus, those strings which have a high tness are given a large share of the wheel, while those strings with low tness are given a relatively small portion of the roulette wheel. Finally, selections are made by spinning the roulette wheel M times and accepting as candidates those strings which are indicated at the completion of the spin. Table 4.1 presents a set of available characteristics of the system for four (m = 4) dierent types of components. Table 4.2 presents the optimal vector r and the optimal value function C (r ) for the data of Table 4.1 while solving the program (1.10) with C (rj ; gj ) given in (1.8), (1.9). The distribution function FYi (x) was calculated analytically according to (2.5). Note that although for this case, there is no need for simulation, we still used it in order to validate the usefulnes s of the stochastic (simulation) approach, that is in parallel to the program (1.10). We solved its stochastic counterpart (2.10) where FYi (x) in (2.5) was replaced by its stochastic version FY (x) in (3.12). We found that for the sample size N > 100 both the optimal solutions of the programs (1.10) and (2.10) coincide.

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Table 4.1 The Set of Available Characteristics of the System Components Type of Version Capacity Availability Cost of Component number xij gij Component 1 2

3

4

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

0.5 0.8 0.8 1.0 1.0 0.2 0.5 0.5 0.75 0.6 0.6 0.8 0.8 1.0 1.0 0.25 0.25 0.3 0.7 0.7

15

0.97 0.964 0.98 0.969 0.98 0.985 0.979 0.987 0.975 0.959 0.97 0.959 0.98 0.96 0.99 0.989 0.979 0.98 0.96 0.98

Cij

0.52 0.62 0.67 0.89 1.02 0.516 0.916 0.967 1.367 0.214 0.283 0.384 0.414 0.623 0.710 0.583 0.545 0.627 1.196 1.26

Table 4.2 The Optimal Solutions r and C(r) for the Data of Table (4.1) No.



Type of

Optimal

C (r) Component r  Scenario

1

0.858 7.296

2

0.908 7.756

3

0.947 8.118

4

0.984 8.733

5

0.991 9.097

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

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

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

5 Concluding Remarks In this work we extended the classic model on redundancy optimization of a coherent seriesparallel reliability system, where the objective function represents the total capacity of the system and decision parameters are the nominal capacity and availability of the elements. We obtained explicit expressions (both analytically and via ecient simulation) for the constraint of the program, namely for the cdf F and showed that the programs (1.10) and (2.10) are convex mixed-integer ones. Depending on whether the constraint function F is analytically available or not, we suggested using either the deterministic or the stochastic optimization approach, respectively. The last case is associated with likelihood ratios (change of probability measure). A genetic algorithm for nding the optimal redundancy of the programs (1.10) and (2.10) was presented.

Acknowledgment We would like thank Ishay Weissman from the Technion for several valuable suggestion on the earlier draft of this work.

References [1] Barlow, R. and F. Proshan (1975). \Statistical Theory of Reliability and Life Testing Probability Models". Holt, Reimont and Winston, New York.

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[2] Gass. S. I. and Harris C.M. (1996) Encyclopedia of Operations Research and Management Science. Kluver Academic Publisher, Boston. [3] Gertzbakh, I.B. (1989). Statistical Reliability Theory. Marcel Decker, Inc., Ch. 1{4. [4] Goldberg, D. (1989). Genetic Algorithms in Search, Optimization and Machine Learning. Addison Wesley, MA. 1{4. [5] Lisnianski A., Levitin G., Ben-Haim H., and D. Elmakis (1996). \ Power System Optimization Subject to Reliability Constraints" Manuscript, Israel Electric Corporation - R&D Division Reliability Department, Haifa, Israel, (to be published in the Electric Power Systems Research Journal, Vol. 40, No 1.) [6] Kozlov, B.A. and I.A. Ushakov (1970). Reliability Handbook. Holt, Rinehart and Winston, New York. [7] Mann, N.R., Schafer, R.E. and N.D. Singpurwalla(1974). Methods for Statistical Analysis of Reliability and Lifetime Data. Wiley, New York. [8] Nemhauser, G.L., and L.A.Wosley (1988) Integer and Combinatorial Optimization, John Wiley & Sons, New York. [9] Ross, S.M. (1989). Introduction to Probability Models, Academic Press, Inc., San Diego. [10] Rubinstein, R.Y. and A. Shapiro (1993). Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization via the Score Function Method, John Wiley & Sons, New York. [11] Scheuer, E.M. (1989). Reliability, 345{370. Handbooks in OR & MS, G.L. Nemhauser et all., eds., North Holland Publ. Comp. [12] Ushakov, I.A. (1969). Methods of Solving Optimal Redundancy Problems Under Restrictions (in Russian). Sovietskoe Radio, Moscow. [13] Ushakov, I.A. (1986). A universal generating function. Sov. J. Comput. Syst. Science (USA) 24 (5). [14] Ushakov, I.A. (1987). Optimal standby problems and a universal generating function. Sov. J. Comput. Syst. Science (USA) 25 (4). [15] Ushakov, I.A. and R.A. Harrison (1994). Handbook of Reliability Engineering. Wiley. [16] Walker, A.J. (1977). \An Ecient Method for Generating Discrete Random Variables with General Distributions", Assoc. Comput. Mach. Trans. Math. Software 3, 253{256. [17] Whitley D. and J. K. Genitor (1988). \A dierent genetic algorithm" Manuscript CS-88-101, Calorado State University, Fort Collins, Co. [18] Zacks, S. (1992). Introduction to Reliability Analysis: Probability Models and Statistical Models. Springer-Verlag, New York.

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