Finite difference analysis of reliability optimization through sampling techniques

Reliability Engineering and System Safety 34 (1991} 107-I 19

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Finite Difference Analysis of Reliability Optimization through Sampling Techniques M. M e t w a l l y Department of Mathematics, Bahrain University, PO Box 32038, lsa Town, State of Bahrain (Received 15 May 1990; accepted 13 August 1990) ABSTRACT This paper describes qualitative and quantitative techniques for reliability sampling plans based on failure density functions (MTTF, M T B F and M T T R ) as expected values and as functions of time duration ( t ). Examples are provided of how the techniques are applied in determining plans for a specified lot size. These plans are to test service life, the effectiveness of reliability and MTTF, M T B F and M T T R as expected values and as functions of t and the percentage of reliable units expected in an), lot size based on consumer's and producer's risks. Some graphs are constructed from which we can investigate the required aims. Several decision plans for several specified confidence limits, and variable percentages of deterioration of components, are shown for several sample sizes and specified lot sizes. The techniques used in this research, which are based on the calculus of finite differences, on statistical simulation and on the complex stochastic processfor determining the relations required, are applicable to all statistical distributions, provided that the components are statistically independent.

NOTATION MTBF MTTF MTTR t

Means o f failure density functions Lifetime duration for specified confidence level

Percentage deterioration of components Percentage o f the n u m b e r o f defective unit in specified confidence limits 107 Reliability Engineering and System Safety 0951-8320/91/$03.50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain

108

q. ~,a Z. ~,J

M. Metwally

Performance efficiency of subgroups at the nth component Confidence limits Specified confidence level Performance efficiency of subsystems for the nth group Sample size at a specified confidence limit, with its standard error, i.e. a function of sample size and standard error

The equilibrium relation is: Input-Output = Residual efficiency or zero efficiency 1 THE PROBLEM Statistical models of reliability and failure density functions for a given complex system can be abstracted from general physical processes and theories of probability, coupled with the particular operational logic and physical behaviour of the system and its environment. Sampling techniques and plans are concerned with the estimation of component reliability and failure density functions individually, but not globally. The resulting probability analysis becomes so blurred that it must be considered impractical. Hence, other methods that either simplify and modify the calculations or provide the reliability and failure density functions of the entire complex system are required. In the formulation of an optimized model, the calculus of finite difference and evaluation techniques can play a vital role. In the case of continuous processes, the models under consideration are governed by sets of differential equations, which make them difficult to apply to complex systems. Solution techniques can be simplified by constructing and using the transitional probability matrix for evaluating not only the limiting state probabilities, but also the system's characteristic functions. These modeling solution techniques are, however, associated with constant failure density functions and are applicable only to exponential distributions. The distributional assumptions will not be significant in the case of independent components. The techniques in this research, which are based on the calculus of finite differences, on statistical simulation and on the complex stochastic process for determining these values, are applicable for all distributions, provided that the components are statistically independent. 2 THE A P P R O A C H The recognition of system deterioration or effectiveness varies with the degree of defective level and the failure density functions within the

Finite difference analysis of reliability optimization

109

inspected system and its components. If these levels and functions are broken into groups of special statistical logical order, a method can be established for the determination of the expected value of system deterioration and/or effectiveness. On application of statistical distributions, evaluation programs and techniques may be developed for establishing criteria for sampling plans and control charts of reliability deterioration or effectiveness efforts. Any complex system might be considered as a type of hierarchy in which components are placed into subgroup configurations. Subgroups are assembled into groups of subsystems, and the subsystems are in turn organized into a final system whose overall performance is dependent on the characteristics of its subgroups all the way down to the smallest component. Variations in these components are inevitable because of differing tolerances in the system, the material, the manufacturing processes, the environmental conditions, etc. Such variations must be considered in sampling techniques to assure reliable system performance under all conditions.

3 MATERIALS A N D METHODS The formulation of an optimized statistical model of the problem under study is the basic condition for obtaining objective results. It is important to formulate the problem in good agreement with the objectives of the research, taking into account the main and auxiliary factors which affect the system's performance and failure density functions. By statistical simulation, the processes of functioning in the complex system were used to derive a mathematical model which preserved all essential aspects of the main and auxiliary factors of research objectives. This model was repeatedly tested to determine the required statistical characteristics which in themselves determine the actual and standard distributions. The evaluation of the overall system's performance was based on the complex stochastic process. This result was then used to derive the main and auxiliary factors. With the aid of the method of finite differences and systems, the statistical simulation allowed us to standardize the solution of our research problem formulation for the required objectives. We consider a complex system in service with initial reliability (R) as an input, and reliability at the end of its service life as an output. Through simulation and statistical analysis in the context of fundamental physical laws, we are able to characterize reliability as input of conserved quantity into system minus output of conserved quantity from system equal to accumulation 0fconserved quantity in system. In the present study, input is initial reliability, output is the deterioration of this initial reliability in service

hf. Metwally

110

time, and the accumulation failure density function.

is no reliability or no service time appearing as a

[input - output1 = E (where E 2 0) input=yqn-r

+a;l,+r

output = YVn+ OX” (1)

Xn= V” and

(2)

Y&-l +%+r=Yl”+G!n Combining

eqns (1) and (2): Vn+l

--(~+lh”+~%l-l

(3)

=o

where u=L

(4)

bK

The solution of the difference eqn (3) will give a relation between q, and the hierarchy of the complex system. The difference equations to be considered here arise from situations in which the dependent variable, yl = f(i), where L is the failure rate is defined at discrete values of the independent variable I,, Lo,). 1, . . . ) A,, which are separated by the constant amount: 0, 1,2,. . .)

(5)

yn = 4(n) = f&J = f(n Ai, + >.,)

(6)

A1 = 1, + 1 - I.,

(n =

The dependent variable expression will be

The differences of the dependent

variable y,, will be

A?n=~n+l -I/,,

(7)

A”y, = A’y, ,.1- A’y,

(8)

and

(9 i.e. &A’y,, AK- lyn,. . . , A'yn,

L

n)

(10)

The order of a difference equation is the difference between the highest and lowest subscripts associated with the dependent variable. The equilibrium equation can be analysed, using finite differences, by applying the method

Finite difference analysis of reliability optimization

111

of undetermined coefficients. The system reliability deterioration and effectiveness and lifetime problems, subject to linear and nonlinear restraints, can thus be formulated and solved as integer programming problems. In the problem under study, the equilibrium equation for a complex system may be derived from the following: the percentage of component deterioration is equal to the sum of the percentages of system deterioration and subsystem deteriorations of groups and subgroups. The percentage deteriorations of a system or subsystem is the aimed-at level value y~.+t where lot size is a function of7~. + ~. The percentage deterioration, in general, can be identified by the percentage of components which lie between specified limits for a given number of components. The formulated equilibrium equation at the n + 1st component can than be stated as

(ll) However, for the aimed-at level value at the (n + 1st component) 7.+1 + 2

7~"+1 =7~"-

0t

Adding eqns (11) and (12) gives F.+t-F,=a

0t

(13)

where F. = 7~ + 7.

(14)

with boundary conditions at service time t = 0 given as F.=O 7~ = 0

(15a) (15b)

7. = 0

(15c)

The Laplace transform of eqn (13) as eqn (15a) is 1 - F . --= 0

(16)

The solution of eqn (16), a linear difference equation with constant coefficients is (17)

112

M. Metwally

where k is a constant of the confidence limit value, and where the Laplace transform of a function f(t) is defined for positive values o f t as a function of the new variable p by integral, taking into account that F o = ",~ to at time t = 0. Hence

F,

,' --

eEl

7o =

(18)

?t

The Laplace transform of eqn (18) at t = 0 is

(°)

FI =

to

p l+

(19)

-p

Combining eqns (19) and (17) gives K = ,o P

(20)

to

(21)

and eqns (17) and (20) gives n-

p l+~-p The Laplace transform of eqn (11), with the b o u n d a r y condition of(15c) is l+~-p

+l-

7.-/-

t.+i =0

(22)

Combining eqns (i1) and (14) gives

t n + l --

1

P(I+~Pe) )n+l

7. + 1

(23)

and combining eqns (22) and (23) gives

(24)

The solution of this linear difference equation is the sum of a h o m o g e n e o u s and a particular solution, which depends on producer's and consumer's risks respectively.

Finite difference analysis of reliability optimization

113

The homogeneous solution is h/[1 +(2+p)09/a]" and the particular solution is 7~/p[1 + (o~/a)p]". Consequently, h

+

(25)

p(l+~p)"

"/" - [1 + (7 +p,-~l" where h is a constant; we can then write 09) ~

_ 09 ~l

(26)

which by Laplace transform gives (1 +-~p)~ = 2(-~)'~ = 2(-~)(F1- ~Tt)

(27)

Equation (27) together with eqn (19) gives 7~-Yo

1

(28)

1+(2+p)09- - p ( 1+ 09p ) Combining eqns (28) and (25) at n = 1 gives h=

,o P If eqn (29) is then combined with eqn (25), we have ys[ll+(2+P)-~l:--(l+-~P)

(29)

"] (30)

At the case of split risk or margin, P = 0, 1

(31)

At p = -a/09, the case of producer's risk e or margin, is given by ex [

ta ~ ~'~ (t/O)s-l s=l

(32)

114

M. Metwally

At p = -[(1 + 7to/a)/(oJ/a)], the case of consumer risk fl or margin is tr P~ = 7~ exp

co

--a

( t/O)s- 1 oJ\"-s+ l

t

~=l l + 2 a )

(33)

(s-l)!

then for the complete solution of eqn (30) our approach is

( s - 1)!) s=l

O"

n

-(a--+-~) I 1-exp(

tr

+20909t)

~

-1

(tr ? C O t ) ~

1

(s--l),]}

(34)

4 ANALYTICAL CONSIDERATIONS Since 0 is a multiple valued function of p, each set corresponds to a separate branch of the function, hence only one set should be used in applying eqns (30), (31), (32) and (33). In our work, after the solution we found for each value of p only five values of 0 including the zero value, and we excluded those values of 0 which made o)/tr negative. The remaining values of ~o/a form two sets, 0 ° to 180 ° to 360 °. By using the suggested values of samples, it was possible to estimate the required numbers of observations based on special statistical sample sizes with special distribution, allowing for more accuracy than required. Our research problem was then analysed by means of computerized statistical simulation ofeqn (34), using the given sampling values. The results from this formalized approach were first tested for significance by Kolmogorov's Criterion in order to verify the truth of our hypothesis that the samples were random and taken from a continuous distribution of finite elements. The calculations for the formulated approach distributions and the inverse line, were analyzed by correlation and regression methods, the results of which are shown in Figs 1-5. These graphs show relationships between complex system reliability and MTTF, MTBF and MTTR as expected values and as functions .of t, co, tr, 7, lot size, the percentage of reliable units in a large size expected due to this test, and the probability of accepting any lot size based on producer's and consumer's margins. Computer flow charts and programs were constructed, based on Runge-Kutta method, for that purpose. After these statistical treatments of the variables of the formalized problem, the

Finite difference analysis of reliability optimization

115

110 108

.~

10C

N~

I oos

Fig. 1.

i i I I I I I I I I I o-1 o~ 0-2 o25 0.3 o4 o~ 06 0-7 o8 oe Relia~ltty and/or failure density functK>n as expected value and function of duration time (t) (MTTF, NITE~F,NITTR)

Graph

w h e n o~ = 10, 7 = ! w i t h

108 _

~

o

I 1.o

specified confidence limit (a).

lol J ~

~

IO0

c

x

¢1

!!

-~ 8 u

I I I I I I I I I I I I 0"05 0-1 0-15 0"2 0"25 0"3 0-4 0"5 0'6 0-7 0"8 0"9 Relial~lity and/or failure density functJon as expected val~Jeand fun<:t~n of durat~n time (t) (fdTTF, MTBF, MTTR)

Fig. 2.

Graph when oJ

=

I

1"0

I0, 7 = 2 with specified confidence limit (~).

M. Metwally

116 110

o o~

• ~" ¢

10o /

/ /

~2 8

~.~

!

~ ~o .a

10

I

i

I

i

i

J

oos 01 0-1s 02 02s 0-3

i

0.4

I

o-s

I

o~

I

c~7

I

o-s

I

oe

I 1.0

Reliamlity and/or failure density function as expected value and function of duration time (t) (MTTF, MTBF, MTTR)

Fig. 3.

Graph when co = 10, " / = 3 with specified confidence limit (or).

._¢-

/ / / "5 ~ " S O

~

15 o 2o

J J i I I I I I I I I I 0 0 5 01 0-15 0 2 0-25 0-3 04 05 0-6 0.7 0.8 09 Reliability ancllor failure density function aS expectecl value and functmn of duratio~ time [ t ) (MTTF, MTBF, MTTR)

Fig. 4.

I 10

G r a p h w h e n ~ -- 10, , / = 4 w i t h specified c o n f i d e n c e l i m i t (~r).

Finite difference analysis of reliability optimization

•~

~!

~"

1OO,

117

/

~ ~.~ 15 '! 'n~ .P "8

i

i

I I I I I I I I I I I I 0'05 0"1 0"15 0-2 0"25 0-3 0,4 0~ O~ 0"7 06 09 Reliability and/0r failure clensity function as expected value and function of duration time (t) (MTTF, MTBF, MTTR)

I 1"0

Fig. 5. Graph when to = 10, 7 = 5 with specifiedconfidencelimit (a).

following could be concluded from the comparison of all functions: the distributions found from the data of the computer calculations rather accurately correspond to the curves given, which are the inverse of the functions of the theoretical distribution based on the formalized approach.

5 PRACTICAL APPLICATIONS AND CONCLUDING REMARKS If a sample of e~ units are placed on life test for a specified duration, it is possible to determine plans for a specified lot size to test service life, the effectiveness of reliability R(%) and MTTF, M T B F and M T T R as expected values and functions oft, and the percentage of reliable units expected in any lot size based on consumer's and producer's risks. By using the graphs in Figs 1-5, we can investigate the required aims. The following tables, which are an example of how this can be accomplished, show several decision plans for several specified confidence limits a and variable percentage deterioration of components ;, for a sample size of 10 units under test, and specified lot size of 100 units, concerning the complex system reliability and MTTF, M T B F and M T T R and the percentage of reliable units expected in any lot size based on producer's and consumer's risks.

118

M. Metwal~ TABLE 1

o0-750 0.900 0-950 0"975 0"990 0'995

7=1

7=2

7=3

7=4

7=5

0-575 0"620 0"680 0-730 0"795 0"870

0-440 0"485 0"545 0-590 0"650 0-755

0"335 0"370 0"430 0-485 0"540 0-645

0'255 0"280 0"330 0'385 0.440 0-540

0'175 0.200 0-250 0-295 0"340 0"440

TABLE 2

a

R, MTTF, MTBF and/or MTTR as functions

0.750 0.900 0-950 0.975 0"990 0-995 ,

0"05

0"1

0"15

0"20

0"25

0"30

0.40

0"50

11 10 9 8 7 6

20 19 17 16 15 13

27 26 24 22 21 18

42 40 37 34 31 27

47 45 42 39 36 32

52 50 47 44 41 37

73 69 64 60 56 50

93 88 82 77 71 63

practical zone

until 1.0

,

Table 1, extracted from Figs 1-5, shows border values for consumer's probability acceptance for plan of sample size o f 10 units, lot size o f I00, and the complex system R, M T T F , M T B F and M T T R as functions. Extracted from Figs 1-5, Table 2 shows the percentage and the probability of acceptance of reliable units expected in any lot size based on producer's and consumer's risks for a plan o f sample size o) = 10 units, with one failed unit 7=1. It has to be noted here that a computer program was established in this research work and some graphs were constructed for different values o f o) and 7, but only five graphs are presented as illustrative computerized examples. BIBLIOGRAPHY 1. Billinton, R. & Allan, R., Reliability Evaluation o f Engineering Systems, Pitman Advanced Publishing Program, Plenum Press, USA, 1983. 2. Gnedenko, B. V., Belyayer, Yu. K. & Solovyev, A. D., Mathematical Methods o f Reliability Theory, Academic Press, London, 1969. 3. Papoulis, A., Probability Random Variables and Stochastic Processes, McGrawHill, New York, 1965.

Finite difference analysis of reliability optimization

119

4. Kemeny, J. G. & Shell, J. L., Finite Markove Chains, Van Nostrand, New York, 1960. 5. Barlow, R. E. & Proschan, F., Mathematical Theory of Reliability, John Wiley, New York, 1965. 6. Forsythe, G. E. & Wasow, W. R., Finite Difference Methods in Partial Differential Equations, John Wiley, London, 1969. 7. Ryabinin, I., Reliability of Engineering System, Principles and Analysis, Mir Publishers, Moscow, 1976. (Translated from the Russian by A. Troitsky.) 8. Rashed, A. F. & Metwally, M. A., Vendor quality rating. Qual. EOQC J., 12 (1968). 9. Rashed, A. F. & Metwally, M. A., Reliability rating scheme. Konferenz fiir Realisierung Der Qualitaet, Mathematical Society, Budapest, Hungary, Nov. 1969. 10. Rashed, A. F. & Metwally, M. A., Optimal choice of redundancy technique to assure the required design objective and performance reliability. Fifth Worm Congress on the Theory of Machine and Mechanisms, University of Montreal, Montreal, 3-8 July, 1979. I !. Rashed, A. F. & Metwally, M. A., A developed technique for evaluation of system reliability. Int. Conf. on Quality Control; the 3rd IAQ Triennial Conf. Organized by Union of Japanese Scientists and Engineers, in cooperation with International Academy for Quality Control, Tokyo, 2-7 Oct. 1978. 12. Rashed, A. F. & Metwally, M. A., Reliability optimization. Konferenz fiir Relaiserung Der Qualitaet, Mathematical Society, Budapest, Hungary, Nov. 1970. 13. Rashed, A. F. & Metwally, M., Vender vendee relationship for quality system in developing countries. ASQC 36th Annual Quality Congress Trans., American Society for Quality Control, Detroit, 1982. 14. Rashed, A. F. & Metwally, M., Statistical simulation methods for investigating the structure of reliability optimization. IEEE Trans. Reliability, 37(I) (1988). 15. Pospelov, D. A., Logicheskie Metody Analiza i Sinteza Skhen ('Logical Methods of Analysis and Synthesis of Circuits'), Energiya, Moscow, 1964. 16. Neumann, J., Probabilistic Logics and the Synthesis of Reliable Organismsfrom Unreliable Components, Princeton University Press, NJ, 1956. 17. Aizerman, M. A., Gusev, L. A., Rosoner, L. A., Smirnova, I. M. & Tal, A. A., Logika, Avtomaty, Algaritmy ('Logic, Automatics, Algorithms'), Fizmatgiz, Moscow, 1963. 18. Kiton, A. I. & Krinitskiy, N. A., Elektronnye Tsifrovye Mashiny i Programmirovanie ('Electronic Digital Computers and Programming'), 2nd edn, Fizmatgiz, Moscow, 1961. 19. Kondrashov, V. A., et aL, Logiko Veroyatnostny Metod Rascheta Nadezhnosti Sudovykh Energeticheskikh Ustanovok ('Logical Probability Method of Calculating the Reliability of Shipyard Power Plants'), Vychislitelnye Sistemy, Issue 13, 1964. 20. Ryabinin, I. A., Teoreticheskie Osnovy Proektirovaniya Elektroenergeticheskikh Sistem Korabley ('Theoretical Fundamentals of Designing Shipyard Electric Power Systems'), Leningrad, 1964. 21. Druzhienin, G. V., Nadezhnost Sistem Avtomatiki ('Reliability of Automatic Systems'), Energiya, Moscow, 1967.