Reliability Engineering and System Safety 37 (1992) 151-155 :Z., " "

Optimal system-profit design of series-parallel systems with multiple failure modes

Hoang Pham Idaho National Engineering Laboratory, EG & G Idaho, Inc., Idaho Falls, Idaho, 83415, USA

(Received 1 July 1991; accepted 11 October 1991)

We treat the problem of achieving optimal subsystem size (m) for series- parallel systems assuming that failure may take either of two forms. We assume that components are independent and identically distributed (i.i.d.) and that the two kinds of system failures can have different costs. In this paper, we determine the optimal m that maximizes the average system-profit. We study how the optimal subsystem size m depends on the system parameters. We also determine the optimal subsystem size, m, which maximizes the average system-profit subject to a restricted type I design error. Numerical examples are given to illustrate the results.

NOTATION

qo

qs

m n ho(m) h~(m) e ( m )

1-13

C1

C2

C3

17 4

tx]

Probability of component failure in open mode (Po = 1 - qo) Probability of component failure in short mode (Ps = 1 - qs) Number of subsystems in a system Number of components in each subsystem Probability of system failure in open mode Probability of system failure in short mode Average system-profit Conditional probability (given system failure) that the system is in open mode Conditional probability (given system failure) that the system is in short mode Gain from system success in open mode Gain from system failure in open mode (C 1 > C2)

Gain from system success in short mode Gain from system failure in short mode

> c,) Largest integer not exceeding x Real number Implies an optimal value

1 INTRODUCTION

Many types of systems consist of components that can fail in two mutually exclusive ways, and where the

Reliability Engineering and System Safety 0951-8320/92/$05.00 © 1992 Elsevier Science Publishers Ltd, England.

system likewise can fail in either of two mutually exclusive ways.t For example, a network consisting of n relays in series has the property that an open-circuit failure of any one of the relays would cause an open-circuit failure of the system, and a short-circuit failure of all n relays would cause a short-circuit failure of the system.

The literature pertaining to reliability of systems with components having various modes of failure can be found in Refs 1-8. Barlow et al. 2 dealt with series-parallel and parallel-series systems, where the size of each subsystem is fixed, but the number of subsystems is varied to achieve maximum reliability. Ben-Dov 3 determined a value of k which maximizes the k-out-of-n system subject to two kinds of failure. Jenney & Sherwin a considered systems in which the items are identical, independent, and subject to mutually exclusive open- and short-circuit failures. They developed methods for finding the most reliable physical arrangements of 2, 3, and 4 items given the item failure probabilities. However, Malon 9 pointed out that eqn [4,(11)] for the reliability of the series-parallel arrays is incorrect. Page & Perry 5 discussed the problem of designing the most reliable structure of a given number of independent and identically distributed components with two failure modes. They also proposed an alternative algorithm for selecting near optimal configurations for large systems. Sah & Stiglitz 7 have analysed k-out-of-n systems with two failure modes. They obtained a necessary and sufficient condition for determining a value of k which maximizes the mean system-profit for a given value of n. Recently, Pham & Pham a have

151

152 Hoang Pham

-CH

- - 0 . . . . .

1 2 m

Fig. 1. A series-parallel system consisting of in each of m subsystems.

n components

determined the optimal k or n that maximize the mean profit of k-out-of-n systems subject to two kinds of failure. They studied the effect of system parameters on the optimal k or n, and also showed that there does not exist a pair (k, n) maximizing the mean system profit.

We study a system of components arranged so that there are m subsystems operating in series, each subsystem consisting of n identical components in parallel. Such an arrangement is called a series- parallel arrangement, as shown in Fig. 1. The components in the figure could be a logic gate, a fluid flow valve or an electronic diode. Applications of such systems can be found in the areas of communication, network and nuclear power systems. For example, consider a digital communication system consisting of m substations in series. A message is initially sent to substation 1, is then relayed to substation 2, etc., until the message passes through substation m and is received. The message consists of a sequence of 0s and ls and each digit is sent separately through the series of m substations. Unfortunately, the substations are not perfect and can transmit as output a different digit than that received as input. Such a system is subject to two failure modes: errors in digital transmission occur in such a manner that either (1) a one appears instead of a zero, or (2) a zero appears instead of a one.

The objective of this paper is to determine the optimal m which maximizes the average system-profit. We study the effect of the system parameters on the optimal m. We also determine the optimal subsystem size that maximizes the average system profit subject to a restricted type I (system failure in open mode) design error. Numerical examples are given to illustrate the results. Detailed assumptions and descriptions of the system and components are given in the next section.

2 M O D E L D E S C R I P T I O N

Assumptions:

1. The system consists of m subsystems, each subsystem containing n statistically independent

and identically distrubuted (i.i.d.) components (i.e. the failure of one component in no way affects the probability of failure of the other components).

2. A component is either good, failed open, or failed short. Failed components can never become good and there are no transitions between the open and short failure modes.

3. The system can be failed short if at least one component in each subsystem fails short.

4. The system can be failed open when all the components in any subsystem fail open.

5. The unconditional probabilities of component failure in open and short modes are known, and are constrained qo, qs > 0; qo + q~ < 1.

6. Costs of system failure in open and short modes are known and can be different.

The probabilities of system failure in open and short mode are given by: 2

ho(m) = 1 - (1 - q,~)"

and h~(m) = [1 - (1 - q~).]m

respectively. The average system-profit is given by:

e (m) = f l [C l (1 - ho(m)) + c2ho(m)]

+ (1 - fl)[c3(1 - h~(m)) + c4h~(m)]

(1 )

(2 )

(3)

where ho(m) and h~(m) are defined as in (1) and (2), respectively.

Let fl(cl - c2) a - -

(1 - - ~ ) ( C 3 -- C4)

and b = flCl + (1 - fl)c4 (4)

We can rewrite eqn (3) as:

P(m) = (1 - f l ) ( c 3 - c4)[fis(m) - aho(m)] + b (5 )

where

f t ,(m) = 1 - hs(m)

3 M A X I M I Z I N G T H E A V E R A G E S Y S T E M - P R O F I T

For a given value of n, one wishes to find the optimal number of subsystems m, say m*, which maximizes the average system-profit. We would anticipate that m* depends on the values of both qo and q,.

Let

n l n ( ~ o q ~ ) - In a

mo = (6) in(1 1 - q ~

- (1 - q~)"/

Optimal system-profit design

Theorem 1

Fix /!I, n, q.,, qS and Ci for all i = 1, 2, 3 and 4. The maximum value of P(m) is attained at:

m*= ;m.] +I ifm,rO 1

if m,<O

If m, 2 0 and m, is an integer, both m, and m, + 1 maximize P(m).

Proof

Define AP(m) = P(m + 1) - P(m). From (1) and (2) we have

AP(m) = (1 - B)(G - cdM4 - k(m + 1) - a(h,(m + 1) - k(m))1

= (1 - /WC3 - cd(l - 4x11 - (1 - 4J"l"

- @Xl - 43"l

Set

0 = AP(m) = (1 - /I)(+ - CJ)

x [(l - q,)“[(l - (I- qs)“lm -a&Xl - 43”l

This implies that

(1 - 4XV - (1 - qJ”l” = &Xl- 43”

or, equivalently, that

Taking logs on both sides and rearranging yields:

n ln(l-1 - In a \ 40 /

m= 1 - 4:

=m,

In ( I- (I- 4X >

Thus, for m cm,, AZ’(m) ~0, whereas for m >m,, AP(m) < 0, that is, P(m) is increasing in m for m < m, and decreasing in m for m > m,. Therefore, m* = [m,] + 1 maximizes P(m). If m, < 0 then P(m) is a decreasing function for m > 0. This implies that m*=l.

Finally, if m, is a non-negative integer, then AP(m,) = 0, so that P(mJ = P(m,, + l), and both m, and m, + 1 maximize P(m). Therefore, the result follows. Q.E.D.

When both m, and m, + 1 maximize the average system-profit, the lower of the two values costs less.

It is of interest to study how m* depends on the various parameters q0 and qS.

of series-parallel systems

Theorem 2

For fixed n, ci for all i = 1,2, . . . ) 4

153

(a) If a 2 1, then the optimal subsystem size m* is a decreasing function of qO.

(b) If a 5 1, the optimal subsystem size m* is an increasing function of qS.

Proof

(a) For fixed qS, we need to show that

or, equivalently, that

It can be shown that the function f: [0, l]-, %! defined

bY f(x) = x4:(1 - x)‘-q: (8)

is maximized at x = qz, and since a 2 1, we obtain from (8)

@(qt)4:( 1 - qz)l-4:” > [( 1 - qs)“]q’o[ 1 - (1 - qs)n]l-qz

Rearranging yields

By taking logs on both sides, therefore, the result follows.

(b) Similarly, it is sufficient to show that

(1 - 4X ln[ (2)” i] 5.11 - (1 - 4Jnl ln( 1 _:r”“,,,n) From eqn (8) we have

[(I _ qs)n](i-_qJ”[l _ (I - qs)y_(l-_qsYl

2 (q;)“-4”‘“(1 _ q~)ll-_(~-_qs)“l

This implies that

[(I _ qs)n](l-_q.)“[l _ (1 - qs)R]I1-_(l-_q.)“l

2 a(l-_q~)“(q~)(l-4.)“(1 - qp_(‘-%)“l

since a 5 1. Rearranging yields

By taking logs on both sides, therefore, the result follows. Q.E.D.

Intuitively, Theorem 2 states that when q0 increases, it is desirable to reduce m as close to 1 as feasible. On the other hand, when qS increases, the average system-profit increases with the number of subsystems.

154 Hoang Pham

4 CONSIDERATION OF TYPE I DESIGN ERROR

The solution provided by Theorem 1 is optimal in terms of the average system-profit. Such an optimal configuration, when adopted, leads to a type I design error (system failure in open mode), which may not be acceptable at the design stage. It should be noted that the more subsystems we add to the system the greater the chance of system failure by opening (ho(m), eqn (1)); however, we do make the probability of system failure in short mode smaller by placing additional subsystems in series. Therefore, given/3, n, qo, q~ and cl for i = 1, 2 . . . . . 4, we wish to determine the optimal subsystem size m* in order to maximize the average system profit, P(m), in such a way that the probability of system type I design error (i.e. the probability of system failure in open mode) is at most or.

Theorem 1 remains unchanged if m* obtained from Theorem 1 is kept within the tolerable cr level, namely, ho(rn)<_tr. Otherwise, modifications are needed to determine the optimal system size. This is stated in the following theorem.

Theorem 3

For given values of fl, n, qo, q~ and ci for i = 1, 2 , . . . , 4, the optimal value of m, say m*, which maximizes the average system profit subject to a restricted type I design error tr is attained at:

S 1 if min{ [m0J, [m~J } -< 0 m* [ min{ [m0J + 1, [ml] } otherwise

where [mo] +1 is the solution obtained from Theorem 1 and

In(1 - tr) DII~

In(1 - qo ~)

Proof

Since P(m) (eqn (5)) is a unimodal function of n and m* -< rnl, proof is obtained from Theorem 1. Q.E.D.

5 NUMERICAL EXAMPLES

Example 1

Consider a series-parallel system with m subsystems in series and each subsystem consisting of 2 identical components in parallel. Given n =2, c~ =500, c2 = 50, c3 = 400 and c4 = 80, Figs 2 and 3 illustrate the results of Theorems 1 and 2. In these figures, the optimal number of subsystems is plotted with respect

%

i ,.- ~ i~.

t

1 I I I l ~ 1 } ~04 0 0 8 012 016 0"2 0'24 0 28

qo

Fig. 2. Optimal number of subsystems m* versus qo for fl=0.4. 13, q~= 0-05; +, q~=O.1; ~ , q~=0.2; /~ q~=0.3.

[-

_ + ~ +

,,.1 I I I I I I 0 0 4 0 0 8 0"12 0-16 0'2 0-24 0 2 8

%

Fig. 3. Optimal number of subsystems m* versus qo for fl= 0-6. O, qs=0.05; +, q~=0.1; /k, qs=0.2; Z~, qs=0.3.

to the probability of component failure in open mode for fixed n = 2. In Fig. 2, for fixed fl = 0.4 the curves are obtained for four different values of qs. For example, if qs=0"2 then the optimal number of subsystems required to maximize the average system-profit is 6 for given a value of qo---0.05. The average system-profit corresponding to this value is 436.9. (The optimal number of subsystems can be obtained directly from Theorem 1.) For lower values of qs, for instance qs = 0-1, the optimal number of subsystems is smaller (4) for fixed qo = 0.05. It can be concluded from Fig. 2 that for values of qs close to 0, the average system-profit is maximized when the number of subsystems is very close to 1. This is a significant result because for fixed n, an optimal system can be designed with a small number of subsystems (m close to 1) when the components in short mode are reliable. This is in fact the result of Theorem 2. Similarly, Fig. 3 illustrates the results of Theorems 1 and 2 for fl = 0-6.

Optimal system-profit design o f series-parallel systems 155

Example 2

Continuing on Example 1, given n = 2 , qo=0 .05 , qs=0"2 and f l = 0 . 4 , the optimal number of subsystems which maximizes the average system-profit is m * = 6. The probability of system failure in open mode, (ho(m), eqn (1)), corresponding to this value is 0.015. However, if the requirement of the probability of system failure in open mode, a~, must be within the level of 10 -2 then modifications are needed to determine the optimal system size because ho(m = 6) = 0.015 > 0.01. In this case, the optimal number of subsystems that maximizes the average system-profit subject to a type I design error (a~ = 1 0 - 2 ) , is 4. (This optimal value can be obtained directly from Theorem 3.)

REFERENCES

1. Pham, H., Optimal design of systems subject to two kinds of failure. Proceedings Annual Reliability and

Maintainability Symposium (1990) 149-52. 2. Barlow, R. E., Hunter, L. C. & Proschan, F., Optimum

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3. Ben-Dov, Y., Optimal reliability design of k-out-of-n systems subject to two kinds of failure. J. Opt. Res. Soc., 31 (1980) 743-8.

4. Jenney, B. W. & Sherwin, D. J., Open and short circuit reliability of systems of identical items. IEEE Trans. Reliability, R-35 (Dec. 1986) 532-8.

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