In this section it is assumed that each maintenance action, planned or at failure, brings the item considered to as-good-as-new (see e.g. remarks on pp. 8 and 171 for complex items), yielding to a renewal point for the underlying point process.
Among possible strategies to avoid wear out failures or effects of sudden failures, replacements + +) can be performed basically
(a)at a given (fixed) operating time TPM or at failure if the operating time is shorter than TPM (age replacement,Fig. 4.5a),
(b) at given (fixed) time points TPM, 2TPM, ... or at failure (block replacement, Fig. 4.5b),
(fix) only at given (fixed) time points TPM, 2TPM, ... (fix replacement, Fig. 4.5c), ( )of only at failure (ordinary renewal process without truncation).
t
0 x
ΤPM
0 0 x0 x
ΤPM ΤPM ΤPM
a)
t
0
0 x x
0
ΤPM 2ΤPM 3ΤPM 4ΤPM
0 x x
0
b)
t
0
0 x x
0
ΤPM 2ΤPM 3ΤPM 4ΤPM
0 x x
0
c)
• renewal point
Figure 4.5 Possible time schedules for a repairable item with preventive maintenance (PM) and repair (renewal) times of negligible length (item new at t=0 and at each repair or PM, x starts by 0
at each renewal point): a)After TPM operating hours or at failure (age replacement); b)At fixed times TPM,2TPM, ... or at failure (block replacement); c)At times TPM,2TPM, ... (fix replacement)
Considering first the case of age replacement (Fig. 4.5a), results of Appendix A7.2 for renewal processes and Section 4.5 for spare parts provisioning can be used, takingforthefailure-freetimeτreplathe truncated distributionfunctionFrepla( )x
F Pr{ } F . (4.38)
, for
F F
for
repl a x repl a x x x T
x T
PM
repl a PM
( )= ≤ = ( ) , ( ) ( ) .
< <
= =
τ 0≥ 0 0 0
1
Taking care ofPr F{τrepla=TPM}=1− (TPM),the mean time to replacement follows as
E [ ] f F )) F .
(4.39) τrepla x x dx TPM TPM x dx TPM
TPM TPM
=∫ ( ) +( − ( =∫( − ( )) <
0 0
1 1
Defining as νa( )t the number of renewals in ( , ]0t on age replacement policy (replacements at failure & preventive maintenance), it follows from Eq. (A7.15) that
E [ ( )] H ,
, (0) H (4.40)
νa t = a( )t t >0 νa = a( )0 =0,
(with F1( )x =F( )x =Frepl a( )x in Eq. (A7.15)). Furthermore, Eq. (A7.27) yields
lim / ( ( ) ) ,
t a repl
T
t t a t x dx
PM
→∞E [ν ( )] = E [τ ] = / ∫ 1−F
0 (4.41)
in the proportion F (TPM) for replacements at failure and 1−F (TPM) for replace- ments at age. Thus, with cf and ca r as cost for replacement at failure and at age, the mean total cost per unit time (cost rate) is
lim / ] [ ( ) ( ( ))] ( ( )) .
t ca t c PM ca r PM
T f
PM
T T x dx
→∞E [ = F F+ − ∫ −
(4.42)
/ F
1 1
0
From Eq. (4.42) one recognizes that E [ca/ ]t → ∞ for TPM → 0 and
→cf / E [τ] for TPM → ∞; with E [ ]τ as mean of the failure-free time τ of the item considered (Eq. (A6.38)). Optimization of ca/t is considered with Eq. (4.49).
Reliability and availability is investigated in Section 6.8.2 (Eqs. (6.192) - (6.195)).
For the case of block replacement (Fig 4.5b), one or more failures can occur duringan interval (kTPM, ( + )k 1TPM] (k=0 1, ,...),withconsequentrepair. For the expected total number ofrenewals in ( ,0 nTPM]on block replacement policy (replacements at failure and preventive maintenance) it follows that
E [νb(n TPM)] =nH(TPM)+n, n=1 2, , ... ,TPM> 0, νb(0) = H(0) = 0 , (4.43) where H(TPM) is the renewal function atTPM (Eq.(A7.15) with F1( )x =F( )x as distri- bution function of the failure-free time of the item considered. With cf & cb r as cost for replacement at failureand at TPM,2TPM, ..., respectively, the mean total cost per unit time is
E [ cb/n TPM ] = [cf H(TPM)+cb r] /TPM, TPM>0, H( )0 =0. (4.44) From Eq. (4.44) one recognizes that E [cb/nTPM]→∞ for TPM→0 and, using Eq.
(A7.27), E [cb/n TPM]→cf /E [τ] for TPM→∞; with E [ ]τ as mean of the failure freetime τ oftheitemconsidered. Optimization of cb is considered with Eq.(4.52).
For fix replacement (Fig.4.5c),i.e.,replacement only at times TPM,2TPM, ...
(taking in charge that for a failure in (kTPM, ( + )k 1TPM] (k=0 1, ,...) the item is down from failure time to (k +1) TPM), the expected number of renewalsin( ,0 nTPM] is
E [νf ix(n TPM)] =n, n=1 2, , ..., TPM> 0, νfix(0) = 0 . (4.45) Withcfixascostfor replacement at TPM,2TPM, ...,the mean total cost per unit time is E [ cfix /n TPM ] = cf /TPM. (4.46) The number of failures in ( ,0 nTPM] has a binomial distribution. Setting cd=cost per unit down time, Eq. (A6.30) yields E [ cfix /n TPM] =[cf +cd∫0TPMF( )x dx ] /TPM.
The replacement only at failure leads to an ordinary renewal process (Appendix A7.2), yielding results of Section 4.5 on spare parts provisioning, in particular,
lim E /E , , > 0, (0) =
0, (4.47)
n o f nTPM n TPM n TPM o f
→ ∞ [ν ( )] = [ ]τ =1 2, , ... ν
with E [ ]τ as mean of the failure-free time τ of the item considered, and
lim E E , > 0
.
n co f nTPM cf n TPM
→ ∞ [ / ]= / [τ]., =1 2, , ... (4.48)
OnerecognizesthatforlargenTPM,E [νo f(nTPM)]≤E[νa(nTPM)]≤E [νb(nTPM)].
This follows for νo f versus νa by comparing Eqs. (4.41) and (4.47), and for νa
versus νb heuristically from Fig. 4.5 (at least one failure-free time will be truncated for large n and the probability for a truncation is greater for case b) than for case a)) or byconsidering H( )t ≥t/( (∫t1−F( ))x dx)− 1,
0 for increasing failure rate[2.34(1965)].
Example 4.6
Investigate Eqs. (4.49) and (4.52).
Solution
(i) To Eq. (4.49), with TPM aopt replaced by T for simplicity, one can recognize that for strictly increasing failure rate λ( )x , λ( )T T(1 ( ))x dx ( )T
0 − −
∫ F F is strictly increasing in T, from 0 to λ( )∞ E[ ]τ −1. In fact, for T2>T1 it holds that
λ(T) ( ( ))x dx λ( ) ( ( ))x dx (T) ( ))x dx λ(T) ( ( ))x dx (T ) ,
T
T T
T
T T
2 T
0 1
0
1 1 1
1
2 1
2
1
2 1
1 1
− + − − − > − −
∫ F ∫ F F ∫ f ∫ F F
considering λ(T2)>λ(T1) and f( ))x dx ( ) (x F( ))x dx < (T ) ( F( ))x dx.
T T
T T
T T 1
2
1 2
1 2
1 2 1
∫ =∫ λ − λ ∫ − Thus,
T < ∞ exist for λ( ) [ ]∞ E τ −1> ca r/(cf−ca r),i.e. forλ( )∞ >cf/( [ ] (Eτ cf−ca r) ). However, an analytical expression for TPMaopt is rarely possible, seee.g.[4.8] for numerical solutions.
(ii)To Eq. (4.52) one can recognize that for strictly increasing failure rate λ( )x, Th( )T −H ( )T
→(1−Var[ ] /τ E2[ ]) /τ 2>0 for T → ∞ and thus, considering H( )0 =0, at least one T < ∞ exist for (1−Var[ ] /τ E2[ ]) /τ 2> cb r/cf. This follows from Eqs. (A7.28) & (A7.31) by considering Var[ ]τ <E2[ ]τ for strictly increasing failure rate [2.34 (1965)], see e.g. Fig. 4.4.
For age and block replacement policy it is basically possible to optimize TPM. Setting the derivative with respect to TPM equal to 0, Eq. (4.42) yields for TPMa
opt
λ(TPM aopt) ( ( ))x dx (T ) , ,
TPMa
PM aopt
f a r
opt
f
c
c c
a r a r
c c
1
0
− − = −
∫ F F > (4.49)
with λ( )x as failure rate of the item considered (Eq. (A6.25)), and thus (Eq. (4.42))
lim ] ( ) (
t caopt t cf ca r TPM aopt
→∞E [ / = − λ ), (4.50)
if TPMaopt< ∞ exist. For strictly increasing failure rate λ( )x , TPMaopt< ∞ exist for λ( )∞ > cf / ( [ ](E τ cf−ca r)), (4.51) see Example 4.6. λ(∞ ≤) cf/(E[ ](τ cf−ca r)), λ( )x =λ, or cf ≤ca r leads to a replacement only at failure (TPM= ∞). Similarly, Eq. (4.44) yields
TPM b TPM b TPM b f
opth( opt)−H( opt)= cb r /c , (4.52) with h( )x =dH( ) /x dx as renewal density (Eq. (A7.18)), and thus (Eq. (4.44))
lim ] (
t cbopt t cf TPM bopt
→∞E [ / = h ), (4.53)
if TPMbopt< ∞ exist. Equation (4.52) is a necessary condition (only). For strictly increasing failure rate, at least one TPMb
opt< ∞ exist for
1−Var[ ] /τ E2[ ]τ >2cb r/cf, implying also cf >2cbr , (4.54) see Example 4.6. 1−Var[ ] /τ E2[ ]τ ≤ 2cb r/cf or λ( )x =λ leads to a replacement only at failure (TPM= ∞).
Comparison of cost per unit time is straightforward for fix replacement versus replacement only at failure (Eqs. (4.46) & (4.48)), but can become laborious for age replacement versus block replacement and/or replacement only at failure (Eqs.
(4.42), (4.44)),(4.48),and(4.49)-(4.54)). In general, it must be performed on a case-by-case basis, often taking care that cf> > car cbrand of other aspects likee.g.
the importance to avoid wear out or sudden failures. Besides remarks to Eqs. (4.51)
& (4.54) for λ( )x =λ, the following general results can be given for large t or n TPM: 1. Forstrictlyincreasingfailurerate λ( )x andλ( )∞ >cf/ E( [ ](τ cf−ca r))(Eq.(4.51)),
TPMa
opt < ∞ exist (seee .g.[4.8] for numerical solutions) and, for larget, optimal age replacement (Eq.(4.50)) is better (cheaper) than replacement only at failure ( E [ca/ ] per Eq. (4.42) crosses from above t E [ co f/ ]t =cf / E[ ]τ ).
2. ConsideringEq.(A7.28)foranordinaryrenewalprocess(MTTFA=MTTF=E[ ])τ , itfollowsthatH(TPM) E→TPM/ [ ]τ +(Var[ ] /τ E2[ ]τ −1) /2forTPM→∞. Thus, considering Eqs.(4.53)&(4.48), for cb r/cf <(1−Var[ ] /τ E2[ ]) /τ 2 optimal block replacement can be better (cheaper) than replacement only at failure;however, this implies Var[ ] /τ E2[ ]τ <1 (strictly increasing failure rate) and cf >2cbr . 3. For cf >cbr ≥ ca r optimal age replacement is better (cheaper) than optimal
block replacement [4.1]; however,often one has cbr<ca r.
4. For ca r=cb r=cf, E [co f /nT PM]≤E[ca/ n T PM]≤E[cb /n T PM] (follows from E [νo f(nTPM)]≤E[νa(n TPM)]≤E [νb(n TPM)], see remarks to Eq. (4.48)).