Optimal number of minimal repairs before replacement with repair cost limit

Reliability Engineering and System Safety 26 (1989) 35-46

Optimal Number of Minimal Repairs before Replacement with Repair Cost Limit

P. K. K a p u r a & R. B. G a r g b °Department of Operational Research, bDepartment of Computer Science, University of Delhi, Delhi, India (Received 4 September 1988; accepted 20 November 1988)

ABSTRACT There are two well-known classical replacement policies with minimal repairs. Policy I: A unit is replaced at time T and undergoes only minimal repairs at failures between replacements. Policy II: A unit is replaced on nth failure and failures between replacements are removed by minimal repairs. Several extensions of policy I have been proposed in recent times. However, policy H has attracted attention only sparingly. This paper proposes a general policy based on policies I and II, which includes several other replacement policies as particular cases. Optimal policies minimising the expected cost rate are also discussed. A numerical example is presented at the end.

NOTATION

Co

replacement cost inspection cost on a minimal failure mean cost rate e(L) mean value o f repair cost not exceeding L = S~ x dG(x)/G(L) G(x), G(x) Cdf, Sf of minimal repair cost L minimal repair cost limit 35 Reliability Engineering and System Safety 0951-8320/89/$03.50 C(C)1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain C1 C(n, T, L)

P. K. Kapur~ R. B. Garg

36

M, L R M(t), I(t) 7l tl*

p(t) q(t),Q(t) T T*

expected number of repairs, inspections and replacements per unit time in the steady state expected number of minimal repairs, inspections in (0, t) number of minimal failures before replacement optimal n probability of type I (minimal) failure when the age of the system reaches t hazard rate, cumulative hazard rate of the system replacement time optimal T stieltjes convolution

INTRODUCTION Replacement policies with minimal repairs have, of late, attracted the attention of several authors. Barlow and Hunter 1 were the first to consider a minimal repair policy, wherein, the system is replaced at time T and undergoes minimal repairs at failures between replacements. Many extensions have been proposed of this policy, notably by Muth, 2 Nakagawa, 3 Beichelt 4 and others. 5 These policies are useful for complex systems, e.g. computers, aeroplanes, etc. Makabe and Morimura 6- 8 introduced a different kind of replacement policy where the system is replaced at the nth failure and undergoes minimal repairs at failures between replacements. Morimura 9 later considered many extensions of such models. Park 10 also considered such a policy and proved that for a Weibull distribution which is widely used for failures, the optimum solution is more cost effective as compared to Barlow and Hunter's policy. 1 He also pointed out that such a policy is useful when the operation time of the system is not recorded or when it is costly and time consuming to replace an operating unit. More recently Nakagawa and Kowada 11 discussed in greater detail the policy due to Morimura, 9 where the system is replaced at time T or at nth failure, whichever occurs first. Park ~2 discussed a cost limit replacement policy under minimal repairs. In this paper we propose a more general replacement policy which combines the earlier policies due to Barlow and Hunter, ~ Beichelt, 4 Makabe and Morimura, 6-8 M orimura,9 Nakagawa and Kowada 3'~ and Park 12 and is stated as follows: System is replaced at nth type I failure (minimal failures) or first type II failure (catastrophic failure) or at age T or when the estimated repair cost

Optimal number of minimal repairs before replacement

37

of minimal failures exceeds the pre-determined limit L, whichever occurs earlier. Type I and Type II failures are age dependent. On each minimal failure the repair cost is estimated by inspection. Introducing costs due to inspections, repairs and replacement we obtain the mean cost rate and obtain the optimal number of minimal failures before replacement which minimises the mean cost rate. Some particular cases are also derived. A numerical example is presented at the end. ASSUMPTIONS 1. 2. 3. 4. 5. 6. 7. 8.

Hazard rate of the system is continuous and increasing and is not disturbed by minimal repairs. System undergoes minimal repairs between replacements. System can be renewed by replacements only. All failure events are S-independent. Replacements, minimal repairs and inspections take negligible time. Minimal repair costs are iidrvs (independently identically distributed random variables) and are observable through inspections. Planning horizon is infinite. ~t)q(t) is continuous, increasing and is differentiable.

POLICY The system is replaced at the nth minimal (type I) failure or first catastrophic (type II) failure or when the repair cost of minimal failures exceeds the repair cost limit L or at time T after its installation, whichever occurs first. T and repair cost limit L are positive constants and are previously specified. Type I and type II failures are assumed to be age dependent. Minimal repairs are performed at failures between replacements. We now define the time instants at which the system enters into the following states: 0 1 2 3 4

System begins to operate nth type I failure occurs system reaches age T first type II failure occurs minimal repair cost exceeds the repair cost limit L.

It may be noted that time instants for states 0 to 4 are regenerative. Now, from the theory of Markov Renewal Process, we write the following

38

P. K. Kapur, R. B. Garg

transition probabilities at which the system enters into states: Qo l(t) = f l B(x)A. _ l(x)G"- l(L)p(x)q(x)dx rill

0 n--I

Qoz(t)= f2 ~ B(x)Aj(x)Cfl(L)p(x)q(x)dx 0 n-2

0

where

Aj(x) =

I l l ~P(t)q(t)dtl~ j! exp [ -

Q(x)]

j=0,1,2...

denotes the probability o f j type I and no type II failure in (0, x) (refer to Appendix I). B(x) is the cdf of 'time x is reached' of the system, and is defined as B(x)=~}

x>_TX
and

Qlo(t) =

Qzo(t)= Q30(t)=

Qao(t)= 1

It may be easily verified that Qol(~) + Qo2(oo) + Qo3(or) + Oo4(oo) = 1 Expected number of repairs (until replacement): By renewal theoretic arguments we may write

M(t) = (n -

1) fl

B(x)A. _ l(x)G"- l( t)lT{x)q(x) dx

n-1

n-1

0

0

n-2

+ fl ~ JTB(x)Ai(x)Cfi(L)G(L)~x)q(x)dx+ H(t,, M(t) 0

(1)

Optimalnumberof minimalrepairsbeforereplacement

39

where

H(t) = Qox(t) * Qlo(t) + Qo2(t) * Q20(t) + Qoa(t) * Qao(t) + Qo4(t) * Q40(t) Therefore, the expected number of repairs per unit time in the steady state is given by

n--2 E Gj + I(L) fOTAj(x)~x)q(x) dx M=

o

(2)

.~1o GJ(L)fro A~(x)dx Expected number of inspections (until replacement): By renewal theoretic arguments we may write

IoNx)A._ ~(x)G" + fi2 jAj(x)GJ(L) ;o2jB(x)Aj(x)GJ(L)p(x)q(x) dx

I(t) = (n -

1)

- ~(L)Nx)q(x) dx

n--1

n-I

dB(x) +

0

0

n-2

+ I~ ) ' , (j + 1)B(x)Aj(x)Cfl(L)G(C)f~x)q(x)dx + H(t)* I(t)

(3)

0

Therefore, the expected number of inspections per unit time in the steady state is given by

n--2 ~, GJ(L)fOTA~(x)fp(x)q(x)dx

I= o

.-1

)~r

~o Cfl(L

(4)

Aj(x)dx

The expected number of replacements per unit time in the steady state is given by 1 R = .- ~

rr

(5)

~o GJ(L) Aj(x)dx

Jo

Mean cost rate is, therefore, given by

Co + (C1 + E(L)G(L)) n--2 ~ C,J(L)fOTA.s(x)~x)q(x) dx C(n, T, L) =

o

n--1 ~o Cfl(L)~'~T0Aj(x)dx

(6)

P. K. Kaput, R. B. Garg

40

OPTIMISATION In this section we obtain optimal n(n*) and optimal T(T*) which would minimise mean cost rate for specified repair cost limit. Now lim C(n, T, L)

C ( ~ , T, L) =

n ~

co + (c, + E(L)G(L))y~aJ(/~)

Aj(x)PIx)q(x)dx

o

~ G~(L) Aj(x) dx o

Let T* be the optimal time which would minimise C(oo, T, L). Then from dC(oo, T, L)/dT= 0, if there exists a solution T* satisfying aC,

~'

~GJ(L)ff*Aj(x)dxfi(T')q(T',-~GJ(L,f o

T*

° Aj(x,fi(x,q(x)dx

o

= Col[C1 + E(L)G(L)]

(7)

it is the unique optimal solution or T* = ~ if no solution exists since p(t)q(t) is continuous and increasing. We now find optimal n(n*) for a fixed T(O < T < oo).

Theorem 1 F r o m the inequalities

C(n + 1, T, L) > C(n, T, L) and

C(n, T, L) < C(n - 1, T, L) we get

K(n, T, L) > Co/(C 1 + E(L)G(L)) and

K(n - 1, T, L) < Co/(C1 + E(L)G(L)) where

frG"-l(L)An-l(X)f~x)q(x)dx~ff o f ~ G"(L)A.(x) dx o K(n, T, L) =,

GJ(L)Aj(x) dx

n-2

-~froG3(L)Aj(x)~x)q(x)dx o .0

n=2,3,... m=O, 1

(8)

41

Optimal number of minimal repairs before replacement

Then K(n, T, L) - K(n - 1, T, L)

=Zf;Odx

G"- I(L)A._ x(x)~x)q(x)dx

f G"(L)A.(x) dx

o

roG"- Z(L)A. _ z(x)p(x)q(x) dx >0

roG"- t(L)A._ x(x) dx (for p r o o f see Appendix II), and K ( ~ , T, L) = lim K(n, T, L)

=

Gi(L)Aj(x) dx/~(T)q(T) o

Gi(L)Ai(x)~x)q(x) dx o

which is equal to the left hand side of (7) (for p r o o f see Appendix II).

Theorem 2 Let T* be the value of T satisfying (7). Suppose 0 < T* < ~ , then if T > T*, there exists a finite and unique n* satisfying (8), and if T < T*, no n* satisfying (8) exists, i.e. system should be replaced at time T only. If T* = ~ , no finite n* satisfying (8) exists. Proof If 0 < T* < ~ , then if K ( ~ , T, L) > Co/(C 1 + E(L)G(L)), i.e. T > T*, there exists a finite and unique n* which satisfies (8), since K(~, T,L)= lim,_.o~K(n,T,L) is increasing in n. While if K ( ~ , t , L ) < C o / ( C ~ + E(L)G(L)), i.e. T < T*, finite n* satisfying (8) does not exist. If T* = ~ , finite n* does not exist since K ( ~ , T, L) < Co/(CI + E(L)G(L)) for all T. Therefore, if T > T*, n* is the smallest value such that K(n*, T, L) > Co/(C , + E(L)G(L)). Particular Cases 1. If T ~ ~ , (6) reduces to C O + (C 1 +

C(n, ~ , L) =

E(L)GfL))n-2f: ~ o

GJ(L)Aj(x)~x)q(x) dx

n-l ~ ~T0 ~(L)Aj(x)dx

which agrees with (21) in Ref. 13.

P. K. Kapur, R. B. Garg

42

2. If T--, ~ and p(x) = a, (6) reduces to n-2

Co + (C1 + E(L)G(L)) ~ aJ+lGJ(L) C(n, ~, L) = n-lf~ o ~o GJ(L)aJ[Q(x)]J . j ~ exp [ - Q(x)] dx which agrees with (14) in Ref. 13. 3. If T ~ 0o, ~ x ) = 1, (6) reduces to n-2

Co + (C1 + E(L)G(L)) ~ Gi(L) C(n, ~, L) = o , ~-°, ~,(; GJ(L) [Q(x)]Jj! exp [ - Q(x)] d x which agrees with (7) in Ref. 13. 4. If T ~ ~,/~(x) = 1, C1 = 0, (6) reduces to n-2

CO+ E(L)G(L) ~ GJ(L) C(n, c~, L) = n-1 ~ j o GJ(L) [Q(x)]~ exp [ - O(x)] dx ~o Jo

J!

which agrees with the m e a n cost rate in Ref. 14. 5. If C 1 = 0, Nx) = 1, G(L) = 1 and E(L) = E, (6) reduces to

CT . _

2 [Q(x)]J

Co + E Jo ~"o

--j! ~ r , ~ , [Q(x)]i

C(n, T ) =

)o

~

fl

exp [ -

Q(x)]q(x) dx

e x p [ - Q(x)] d x

which agrees with the cost per unit time in Ref. 11, with the difference that we are counting expected n u m b e r o f repairs before replacement and not expected n u m b e r o f failures before replacement.

Numerical example A s s u m e q(x) = 2 a x , ~ x ) = exp ( - ax), G(L) = 1 - exp ( - L), a = 0.05, a = 0.01, TABLE 1 L

T*

n*

2"0 3"0 4.0 5"0 6-0

13"73 12'00 11.39 11-15 11-05

5 5 4 4 4

Optimal number of minimal repairs before replacement

43

Co = 10 and C 1 = 1. Table 1 gives optimal values for n and T for different values of L. ACKNOWLEDGEMENT The authors thank Professor K. D. Sharma, Head, Department o f Computer Science, University o f Delhi, for useful discussions. REFERENCES 1. Barlow, R. E. & Hunter, L. C., Optimum preventive maintenance policies. Oper. Res., $ (1960) 90-100. 2. Muth, E. J., An optimum decision rule for repair vs replacement. IEEE Trans. Reliability, R-26 (1977) 179-81. 3. Nakagawa, T., Generalised models for determining optimal number of minimal repairs before replacement. J. Oper. Res. Soc. Japan, 24(4) (1981) 325-37. 4. Beichelt, F., A general preventive maintenance policy. Math. Operations-forsch. u. statis. 7 (1976) 927-32. 5. Block, H. W., Borges, W. S. & Savits, T. H., Preventive maintenance policies. Reliability and Quality Control, Elsevier Science Publishers, North Holland, 1986. 6. Makabe, H. & Morimura, H., A new policy for preventive maintenance. J. Oper. Res. Soc. Japan, 5(3) (1963) 110-24. 7. Makabe, H. & Morimura, H., On some preventive maintenance policites. J. Oper. Res. Soc. Japan, 6(1) (1963) 17-47. Makabe, H. & Morimura, H., Some considerations on preventive maintenance policies with numerical analysis. J. Oper. Res., Soc. Japan, 7(4) (1965) 154-7. 9. Morimura, H., On some preventive maintenance policies for IFR. J. Oper. Res. Soc. Japan, 12 (1970) 94-124. 10. Park, K. S., Optimal number of minimal repairs before replacement. IEEE Trans. Reliability, R-28 (1979) 137-40. 11. Nakagawa, T. & Kowada, M., Analysis of a system with minimal repair and its application to replacement policy. European J. Operational Research, 12 (1983) 176-82. 12. Park, K. S., Cost limit replacement policy under minimal repairs. MicroElectron & Reliability, 23 (1983) 347-9. 13. Kapur, P. K., Garg, R. B. & Butani, N. L., Some replacement policies with minimal repairs and repair cost limit. Int. Journal Sys. Science, (submitted). 14. Park, K. S., Optimal number of minor failures before replacement. International J. Sys. Science, 17 (1986) 333-7. .

A P P E N D I X 1: D E R I V A T I O N O F An(t) Let An(t) be the probability o f n type I and no type II failures in (0, t). Then

Ao(t + At) = Ao(tX1 - q(t) At) An(t + At) = An(t)(1 -- q(t) At) + A n_ t(tX~t)q(t) At)

(A1) n _> 1

(A2)

44

P, K. Kapur, R. B. Garg

We, therefore, have

A'o(t) = - Ao(t)q(t) and

A'.(t) = -q(t)A.(t) + A._ l(t)fi(t)q(t)

n >_ 1

A s s u m i n g Ao(0 ) = 1 a n d A.(0) = 0 for n > 1, we have

Ao(t) = exp -

q(x) dx = exp [ - Q(t)]

Defining probability generating function as

¢(r, t)=

2

A.(t)r"

Irl < 1

o

and using (A2) we can obtain

using the fact d q~(r,

t) . =

d2 t) , = = 2! A2(t) ~r2 ~b(r,

= Al(t) o

o

and so on, we can obtain

If:

f:,(x)q(x) dx

A.(t) =

n!

n = 0 , 1 , 2 ....

exp [-- Q(t)]

APPENDIX 2 Consider

g7

f : A._ x(t)fi(t)q(t) dt

A.(T) -

-

Jo' A.(t)q(t) dt

..[_'

Optimal number of minimal repairs before replacement

45

Now

A._ ,(T) j~ A.(t) dt

Ell

fi(x)q(x)dx]"

[L

exp [ - Q(T)] f f

( n - 1)!

b( t)q(t) dt

<

In

Eft fi(x)q(x) dx I n-I

exp [ - Q(T)] | mJ o 7 - - - ~ - j Jo ( n - l).

n!

= A.(T)

rr

exp [ - Q(t)] dt

n!

exp [-- Q(t)] dt

A._,(t)dt

Further, let

we have B(O) = 0 Differentiating

B(T) with respect to T we get

EL ] [;(Io fi(t)q(t) d t

B'(T) --

n!(n - 1)!

x

since b o t h

exp [ -- Q(T)]

fi(x)q(x) dx

exp [ -

Q(t)](q(T) - q(t))

q(t) and fi(t)q(t) are increasing with t. Therefore,

f f A._ t(t)b(t)q(t) dt f~

A,(t) dt

is increasing in n. Next we show that

f : G"- I(L)An _ t(x)~x)q(x) dx lim t'r "-" ~ | G"(L)A.(x) dx

30

-- fi( T)q( T)

46 We

P. K. Kapur, R. B. Garg

have

f f G"- l ( L ) A n _ l(X)fO(x)q(x) dx lim

~'r

"-'~

l - G"(L)A.(x) dx jo

lim

f f G"(L)A.(x)~(x)q(x) dx

"~ ~'

f ~ G"(L)A.(x) dx

=

f f JR(x)]" exp [ - Q(x)]p(x)q(x) dx

= lim

ff[

11~OO

R(x)]" exp [ - Q(x)] dx

where

R(x) = J o ff(t)q(t) dt Let

R(x) = t, we have I f [R(x)]" exp [--

Q(x)]~x)q(x) dx

lim n

.t~ [R(x)]" exp [ - Q(x)] dx

f~

r~ t n exp ( -

=

lira

Q[R

1(/)]) dt

"" ~ . --f~'r' t" e x p ( - Q [ R - '(t)]) Iff(R_ ,(t)-~R_ = ~ T)qt T) (refer to Ref. 11)