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Optimum policies for a system with general imperfect maintenance
Reliability Engineering and System Safety 91 (2006) 362–369 www.elsevier.com/locate/ress
Optimum policies for a system with general imperfect maintenance Shey-Huei Sheu*, Yuh-Bin Lin, Gwo-Liang Liao Department of Industrial Management, National Taiwan University of Science and Technology, 43 Keelung Road, Section 4, Taipei, Taiwan, ROC Received 3 January 2004; accepted 29 January 2005 Available online 7 April 2005
Abstract This study considers periodic preventive maintenance policies, which maximizes the availability of a repairable system with major repair at failure. Three types of preventive maintenance are performed, namely: imperfect preventive maintenance (IPM), perfect preventive maintenance (PPM) and failed preventive maintenance (FPM). The probability that preventive maintenance is perfect depends on the number of imperfect maintenances conducted since the previous renewal cycle, and the probability that preventive maintenance remains imperfect is not increasing. The optimum preventive maintenance time that maximizes availability is derived. Various special cases are considered. A numerical example is given. q 2005 Elsevier Ltd. All rights reserved. Keywords: Maintenance; Imperfect maintenance; Optimum; Availability; Learning effect
1. Introduction To gain and maintain competitive advantage, manufacturers require a height availability system for maintaining peak production machinery operations. Preventive maintenance (PM) is important in complex systems because it reduces downtime and breakdown risk. Maintenance policies have been extensively examined [1–6]. Certain aspects of the PM model maximize the availability [7–9]. Numerous investigations based on PM analysis have attempted to apply the planned PM model to various realworld situations, including a sequential PM process. When a system is maintained at unequal intervals, the PM policy is known as sequential PM [10,11]. Another popular policy is the periodic PM policy, in which the system is preventively maintained at fixed time intervals [12,13]. Perfect PM models assume that the system is ‘as good as new’ following PM. However, this assumption can be incorrect. A more realistic assumption is that following PM the system lies in a state somewhere between as good as new and its pre-maintenance condition. Applying this assumption to PM is known as the imperfect PM model [14,15]. Pham and Wang [4] summarized and discussed various * Corresponding author. Fax: C886 2273 76344. E-mail address: [email protected] (S.-H. Sheu).
0951-8320/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2005.01.015
methods and optimal policies for imperfect maintenance. The model of Nakagawa [7] assumed that PM achieves either imperfect maintenance with a probability p, or perfect 1K p. However, this maintenance with probability pZ assumption is frequently false. Following PM, maintenance workers sometimes perform periodic testing for abnormalities. To include this testing experience, the probability that PM is perfect should depend on the number of imperfect maintenances performed since the last renewal cycle, and the probability that PM remains imperfect does not increase. Equipment failures frequently occur in every workplace between planned PMs. Maintenance workers must determine an action in response to the failure problem because of cost concerns. Major repairs reset the system failure intensity (a good-as-new repair). Numerous protective systems, for example, circuit breakers, alarms, and protective relays, are maintained in this fashion [19]. Sometimes breakdowns occur following PM because maintenance workers lack the necessary parts to correctly perform the PM or installation. To reflect this phenomenon, following PM this study considers three outcomes, similar to those proposed by Nakagawa [7]. Outcome I involves imperfect PM (IPM) only. Meanwhile, outcome II involves perfect PM (PPM). Finally, outcome III involves failed PM (FPM) and requiring major repair. Section 2 presents the model. The availability of the model with periodic PM is determined and the optimum PM policy thus obtained. Section 3 details various special cases. Section 4 then provides a numerical result for special cases. This study is
S.-H. Sheu et al. / Reliability Engineering and System Safety 91 (2006) 362–369
conducted to investigate the effects of these parameters on the solution. Finally, Section 5 presents concluding remarks.
† Imperfect PM (IPM) results in the system having the same failure rate as before PM, with probability qj Z P j =P jK1 . † Perfect PM (PPM) makes the system as good as new, with probability q1, jZ(k/(1Ck))(1Kqj). † Failed PM (FPM): where the system has deteriorated and requires repair, with probability q2, jZ (1/(1Ck))(1Kqj). Following a PPM, the system returns to time 0. If failure occurs before the scheduled PM, the major repair can be performed immediately; otherwise preventative maintenance is performed. After a major repair, the system returns to time 0. PM time is negligible. ufOumf
2. General model To construct the model, relevant notations are defined as follows: X F(t) FðtÞ f(t) rðtÞ H(t)
time to failure of a new unit Cdf[X]: cumulative distribution function of X Sf ½X : survival function of X pdf[X]: probability density function of X ðtÞ=F ðtÞ : failure rate (hazard) f P .PNof a unit N ð P K P Þjf ðjtÞ j jZ1 jK1 jZ1 ðPjK1 K Pj Þj FðjtÞ [weighted sum of pdf f(jt)]/[weighted sum of Sf FðjtÞ] T time between PMs for the system AðT; fP j gÞ availability of the system T* optimal time T to maximum AðT; fP j gÞ umf the mean time to repair for maintenance failure uf the mean time to repair for actual failure us the mean time between failure of the unit r the learning rate Y time between two successive renewal processes
In practice, determining maintenance intervals to maximize availability is a constant problem. The cost implications of maintenance model, such as those presented in this section are important. This study considers a generalized PM model using the following scheme. A system involves three types of outcomes after PM. Type I outcome is called imperfect PM (IPM), type II outcome is known as perfect PM (PPM), and type III outcome is termed failed PM (FPM). This system allows the probability of PPM and FPM occurring to depend on the number of PM conducted since the last renewal cycle. Let M denote the number of PM until the first PPM or FPM occurs. Moreover, let P j Z PðM O jÞ. That is, P j represents the probability that at least j first PMs are IPMs. This study assumes that the domain of P j is {0,1,2,.} and that 1Z P 0 O P 1 R P 2 R/ [16–18]. Moreover, this study uses the notation fP j g to indicate a sequence of probabilities. Let pj Z PðMZ jÞ ZP jK1 K P j Z P jK1 ð1K ðP j =P jK1 ÞÞ, with domain {1,2,3,.}. Consequently, if the jth PM occurs after jK1 IPMs, a PM is either IPM, with probability qj Z P j =P jK1 , or other types, with probability qjZ1Kqj. Let q1,j and q2,j denote the probabilities that a PM is classified as a PPM and FPM, respectively. That is, qjZq1,jCq2,j and kZq1,j/q2,j; kO1. The model makes the following assumptions: (1) The original system begins operating at time 0. (2) A system has three types of PMs at time j$T (jZ1,2,.), based on outcome:
363
(3) (4)
(5) (6)
Let Y1, Y2,. be independent copies of Y. For the present policy ( Y1 ; ifðj K 1ÞT ! Y1 ! jT for j Z 1; 2; . (1) Uj Z jT; if Y1 R jT where Uj represents the operating time during the renewal interval Y1 between (jK1)th PM and jth PM. Moreover, let Dj denote the repair time duration between (jK1)th PM and jth PM and include the repair time of the jth FPM. P Furthermore, the mean time of one renewal cycle is N jZ1 E½Uj . The availability is: PN jZ1 E½Uj PN AðT; fPj gÞ Z PN (2) E½U j C jZ1 jZ1 E½Dj It is easily verified that N X
ð jT N X FðtÞdt E½Uj Z ðPjK1 K Pj Þ
jZ1
jZ1
(3)
0
and N X
E½Dj Z
jZ1
N X
E½Dj ;Y1 RjTC
jZ1
N X
E½Dj ;ðjK1ÞT!Y1 !jT
jZ1
(4) We have N X jZ1
E½Dj ; Y1 R jT Z
N umf X ðP K P j ÞFðjTÞ 1 C k jZ1 jK1
(5)
and N X jZ1
E½Dj ; ðj K 1ÞT ! Y1 ! jT Z uf
N X
ðP jK1 K P j ÞFðjTÞ
jZ1
(6)
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Combining (5) and (6), the expression for can be obtained as follows: N X
E½Dj Z
jZ1
PN
jZ1
E½Dj
N umf X ðP K P j ÞFðjTÞ 1 C k jZ1 jK1
C uf
N X ðP jK1 K P j ÞFðjTÞ
(7)
jZ1
concerns a repetitive job or task and represents the relationship between experience and productivity. The traditional learning curve model has been extensively used to reduce in either total direct labor hours per unit or total value-added costs per unit. Li and Rajagopalan [20] considered the impact of quality on learning. Furthermore, Moore and Bennett [21] used multivariate regression analyses to assess the relationship between bile duct injury rate and experience (number of procedures performed). The
From (3) and (7), we obtain the availability of the system: Ð PN jT jZ1 ðPjK1 K Pj Þ 0 FðtÞdt AðT; fP j gÞ Z PN Ð PN umf PN jT jZ1 ðPjK1 K Pj Þ 0 FðtÞdt C 1Ck jZ1 ðPjK1 K Pj ÞFðjTÞ C uf jZ1 ðPjK1 K Pj ÞFðjTÞ Ð PN jT jZ1 ðPjK1 K Pj Þ 0 FðtÞdt Z PN u
PN Ð mf jT jZ1 ðPjK1 K Pj Þ 0 FðtÞdt C 1Ck K uf jZ1 ðPjK1 K Pj ÞFðjTÞ C uf Seek the optimum PM T* which maximizes AðT; fP j gÞ in (8). Differentiating AðT; fP j gÞ with respect to T yields and setting the derivation to zero N N X X 1 K ðP jK1 K P j ÞFðjTÞ K HðTÞ ðP jK1 K P j Þ 1 K L jZ1 jZ1 ð jT Z0 ! FðtÞdt
ð9Þ
0
(8)
probability P j resembles bile duct injury rate and quality. Zangwill and Kantor [22] established a theoretical construct to provide a better explanation of the learning curve. The traditional learning curve is quite common, although other forms such as exponential formulation also exist. This study applies the traditional learning curve model to the PM model, substituting P j for total direct labor hours per unit. Following the discussion of case D, a learning curve is developed, and the following assumptions are made:
where LZumf/(uf(1Ck)). The following theorem demonstrates that a unique solution exists that satisfies (9) under certain reasonable conditions. Consequently, T* is easily obtainable with any numerical search procedure.
(1) P jK1 O P j ðjZ 1; 2; .Þ. (2) If each doubling of the number of PM reduces imperfect PM probability by (1Kr), then P j Z P 1 jb ðjZ 1; 2; .Þ. (3) b ¼ ðlog rÞ=ðlog 2Þ.
Theorem 1. If HðNÞO 1=ðus ð1K LÞÞ and H(T) is continuous and strictly increasing in T, then there exits a unique and finite T*O0 that maximizes the availability AðT; fP j gÞ.
3.1. Case A: P 0 Z 1; P j Z 0 ðjZ 1; 2; .Þ
Proof. If HðNÞO 1=ðus ð1K LÞÞ and H(T) is strictly increasing in T, then the left-hand side of (9) is strictly decreasing in T. And, it change sign exactly once from positive ðumf =ðð1 þ kÞuf K umf ÞÞ to negative ð1=ð1K LÞK us HðNÞÞ as T increases from 0 to N, as does dAðT; fP j gÞ=dT. The result follows. Observing (9), we can find the optimal availability as follows:
AðT ; f1; 0; 0; .; 0gÞ Z
AðT ;fP j gÞ PN
jZ1 ðPjK1 K Pj ÞjFðjT Þ
umf PN jZ1 ðPjK1 K Pj ÞjFðjT ÞC uf K 1Ck jZ1 ðPjK1 K Pj Þjf ðjT Þ
Z PN Z
This case considers an operating system to be as good as new after PM at jT(jZ1,2,.). Using (8) and (10) ÐT 0 FðtÞdt AðT; f1; 0; 0; .; 0gÞ Z Ð T 0 FðtÞdt K uf ð1 K LÞFðTÞ C uf (11) 1 1 C uf ð1 K LÞrðT Þ
(12)
where rðT ÞZ f ðT Þ=FðT Þ. 3.2. Case B: k/N, q2,jZ0, q1,jZqj (jZ1,2,.)
1 , 1Cuf ð1KLÞHðT Þ (10)
Using (8) and (10): AðT; fP j gÞ
3. Special cases Given instruction and through repetition, PM workers learn to perform tasks more effectively. The learning curve
Ð jT jZ1 ðPjK1 K Pj Þ 0 FðtÞdt Ð jT PN jZ1 ðPjK1 K Pj Þ 0 FðtÞdt C uf jZ1 ðPjK1
Z PN
PN
K P j ÞFðjTÞ ð13Þ
AðT ; fP j gÞ PN
Þ K P j ÞjFðjT PN jZ1 ðPjK1 K Pj ÞjFðjT Þ C uf jZ1 ðPjK1 K Pj Þjf ðjT Þ
jZ1 ðPjK1
Z PN
1 Z 1 C uf HðT Þ
The probability that PM still imperfect
S.-H. Sheu et al. / Reliability Engineering and System Safety 91 (2006) 362–369
365
1 0.8 0.6 0.4 0.2 0 0
50
100
150
200
250
300
(14)
Fig. 1. Learning curve.
3.3. Case C: P 0 Z 1; P j Z qj ðjZ 1; 2; .Þ, 0%q!1, 1K q qZ
with number of PM items j. Fig. 1, plotting the data in Table 1, shows that probability P j is rapidly decreasing in
This case is considered in [7]. Using (8) and (10) Ð P jK1 jT ð1 K qÞ N jZ1 q 0 FðtÞdt AðT; fqj gÞ Z Ð P P umf ð1KqÞ PN jK1 jT FðtÞdt jK1 FðjTÞ jK1 FðjTÞ ð1 K qÞ N q q C C uf ð1 K qÞ N jZ1 jZ1 jZ1 q 0 1Ck AðT ; fqj gÞ Z where HðTÞ Z
"
N X
1 1 C uf ð1 K LÞHðT Þ
(15)
(16) the early PM. Using (8) and (10) PN
E½Uj PN jZ1 E½Uj C jZ1 E½Dj
# " # N . X jK1 jK1 q jf ðjTÞ q jFðjTÞ
jZ1
350
The number of PM since the last perfect PM
AðT; f1; P 1 ,1b ; P 1 ,2b ; .gÞ Z PN
jZ1
(17)
jZ1
3.4. Case D: P 0 Z 1; P 1 s0; P j Z P 1 ,jb ðjZ 1; 2; .Þ, 0!r!1 In this case learning curves are most advantageous during the early PM of new failure causes. As PM number becomes large, the learning effect becomes less noticeable. That is, P j K P jC1 O P jC1 K P jC2 ðjZ 1; 2; .Þ. The following table and figure provide an example for P 1 Z 0:8 and 75 learning rate. Table 1 displays that probability P j decreases
where N X jZ1
ð jT !
P j
j
P j
j
P j
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
1.00000 0.80000 0.60000 0.50707 0.45000 0.41020 0.38030 0.35673 0.33750 0.32140 0.30765 0.29571 0.28523 0.27591 0.26755 0.26000 0.25313 0.24684 0.24105 0.23570
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
0.23074 0.22611 0.22179 0.21773 0.21392 0.21033 0.20693 0.20371 0.20066 0.19776 0.19500 0.19236 0.18984 0.18743 0.18513 0.18291 0.18079 0.17874 0.17677 0.17488
39 40 50 60 70 90 100 140 180 250 350 500 750 1300 2600 6500 10,000 20,000 30,000 50,000
0.17488 0.17305 0.15774 0.14625 0.13718 0.12360 0.11831 0.10289 0.09270 0.08088 0.07034 0.06066 0.05127 0.04080 0.03060 0.02092 0.01750 0.01312 0.01109 0.00897
0
FðtÞdt C P 1
N X ½ðj K 1Þb K jb jZ2
FðtÞdt
and E½Dj Z
jZ1
j
ðT
0
N X
Table 1 Number of PM items j and P j
E½Uj Z ð1 K P 1 Þ
u mf K uf ½ð1 K P 1 ÞFðTÞ 1 Ck
C P 1
N X ½ðj K 1Þb K jb FðjTÞ C uf jZ2
AðT ; f1; P 1 ,1b ; P 1 ,2b ; .gÞ Z
1 1 C uf ð1 K LÞHðT Þ
(18)
where P b b ð1 K P 1 Þf ðT Þ C P 1 N jZ2 ½ðj K 1Þ K j Þjf ðjT Þ P HðT Þ Z b b Þ C P 1 N ð1 K P 1 ÞFðT jZ2 ½ðj K 1Þ K j ÞjFðjT Þ
a 3.5. Case E: P 0 Z 1; P j Z qj ðjZ 1; 2; .Þ, 0%q!1, aO0
Here, a random number M, of instances of PM until a PPM is performed, is a discrete Weibull distribution that is IFR for aR1 and DFR for 0!a%1.
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S.-H. Sheu et al. / Reliability Engineering and System Safety 91 (2006) 362–369
From (8) and (10) PN
ja
AðT; fq gÞ Z PN
jZ1 ðq
a
AðT ; fqj gÞ Z
ðjK1Þa
ja
Kq Þ
Ð jT 0
ðjK1Þa
Ð jT
a
FðtÞdt
0PN ðjK1Þa a mf FðtÞdt C 1Ck K uf K qj ÞFðjTÞ C uf jZ1 ðq jZ1 ðq
u
K qj Þ
1 1 C uf ð1 K LÞHðT Þ
(20)
(19)
Hð0Þ Z lim HðTÞ Z 0 T/0
Because
where HðT Þ " #," # N N X X ðjK1Þa ja ðjK1Þa ja Z ðq K q Þjf ðjT Þ ðq K q ÞjFðjT Þ jZ1
jZ1
1 Z lim T/N HðTÞ T/N
PN
lim
K2jT K3eK3jT CeK3jT Þ 1 jZ1 ðPjK1 K Pj Þjð3e PN Z 2 2 jZ1 ðP jK1 K P j Þjð3eK2jT K3eK3jT Þ
we have HðNÞ Z lim HðTÞ Z 2 T/N
4. Example and analysis Four tables list optimal times T* satisfying (9) for the special cases. Table 2 lists the special case C, Table 3 presents the special case D, Table 4 shows the special case E with aZ0.5, and Table 5 illustrates the special case E with aZ1.5. The following cost and distribution parameters of the PM models are considered to illustrate the policy, similar to the proposal by Nakagawa [7]; ufZ1. umfZ0.1uf, umfZ0.3uf, umfZ0.5 or umf/1.0uf. FðtÞZ 3 eK2t K 2 eK3t . Then, ðN ðN us Z FðtÞdt Z 3 eK2t K 2 eK3t dt Z 5=6 0
" N N X 6 X H ðTÞ Z ðP jK1 K P j Þj2 eK2jT ðP jK1 K P j Þj eK2jT W jZ1 jZ1 0
C
N N X X ðP jK1 K P j Þj2 3 eK2jT ðP jK1 K P j Þj eK2jT jZ1
jK1
N X
2
ðP jK1 K P j Þj e
K jZ1
K2jT
N X
#
ðP jK1 K P j Þj e
K3jT
Þ
jZ1
" N N X 6 X O ðP jK1 K P j Þj2 eK2jT ðP jK1 K P j Þj eK2jT W jZ1 jZ1
0
C
The H(T) is rewritten as: P K2jT 6 N K eK3jT Þ j¼1 ðPjK1 K Pj Þjðe HðTÞ Z PN K2jT K 2eK3jT Þ j¼1 ðPjK1 K Pj Þjð3e
N X ðP jK1 K P j Þj2 ð3 eK2jT K 2 eK2jT Þ jZ1
!
N X
# ðP jK1 K P j Þj eK2jT O 0
ð21Þ
jK1
Table 2 Optimum maintenance time T* with the parameters in special case C 1/(1Ck)
q umfZ0.3uf
umfZ0.1uf
0.0001 0.001 0.01 0.1
umfZ0.5uf
umf/1.0uf
0.001
0.01
0.1
0.001
0.01
0.1
0.001
0.01
0.1
0.001
0.01
0.1
0.183 0.582 1.881 6.411
0.180 0.574 1.857 6.339
0.157 0.501 1.628 5.632
0.317 1.016 3.338 12.182
0.313 1.002 3.297 12.064
0.273 0.876 2.903 10.887
0.410 1.318 4.385 16.919
0.405 1.301 4.333 16.775
0.353 1.138 3.827 15.312
0.582 1.881 6.411 28.107
0.574 1.857 6.339 27.927
0.501 1.628 5.632 26.071
Table 3 Optimum maintenance time T* with the parameters in special case D 1/(1Ck)
r umfZ0.3uf
umfZ0.1uf
0.0001 0.001 0.01 0.1
umfZ0.5uf
umf/1.0uf
0.001
0.01
0.1
0.001
0.01
0.1
0.001
0.01
0.1
0.001
0.01
0.1
0.183 0.582 1.881 6.411
0.180 0.574 1.857 6.339
0.155 0.495 1.612 5.612
0.317 1.016 3.338 12.182
0.313 1.002 3.297 12.064
0.269 0.866 2.883 10.882
0.410 1.318 4.385 16.919
0.405 1.301 4.333 16.775
0.349 1.126 3.805 15.322
0.582 1.881 6.411 28.107
0.574 1.857 6.339 27.927
0.495 1.612 5.612 26.106
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367
Table 4 Optimum maintenance time T* with the parameters when aZ0.5 in special case E 1/(1Ck)
q umfZ0.3uf
umfZ0.1uf
0.0001 0.001 0.01 0.1
umfZ0.5uf
umf/1.0uf
0.001
0.01
0.1
0.001
0.01
0.1
0.001
0.01
0.1
0.001
0.01
0.1
0.183 0.582 1.881 6.411
0.179 0.572 1.849 6.322
0.131 0.422 1.500 5.197
0.317 1.015 3.338 12.181
0.312 0.998 3.285 12.048
0.228 0.744 2.571 10.538
0.410 1.318 4.384 16.919
0.403 1.295 4.318 16.763
0.296 0.972 3.442 15.144
0.582 1.881 6.411 28.107
0.572 1.849 6.322 27.931
0.422 1.406 5.197 26.349
Table 5 Optimum maintenance time T* with the parameters when aZ1.5 in special case E 1/(1Ck)
q umfZ0.3uf
umfZ0.1uf
0.0001 0.001 0.01 0.1
where " WZ
umfZ0.5uf
umf/1.0uf
0.001
0.01
0.1
0.001
0.01
0.1
0.001
0.01
0.1
0.001
0.01
0.1
0.183 0.582 1.881 6.411
0.180 0.575 1.858 6.340
0.160 0.511 1.656 5.704
0.317 1.016 3.338 12.182
0.313 1.003 3.298 12.065
0.278 0.892 2.948 10.979
0.410 1.318 4.385 16.919
0.405 1.301 4.333 16.776
0.360 1.158 3.883 15.397
0.582 1.881 6.411 28.107
0.575 1.858 6.340 27.927
0.511 1.656 5.704 26.098
N X
#2
Nakagawa’s, and which is a special case of the proposed model.
ðP jK1 K P j Þjð3 eK2jT K 2 eK3jT Þ
jZ1
Thus, H(T) is strictly increasing from 0 to 2, and hence a finite and unique T* exists when (9) is satisfied, since 2O ð1=ð5=6ð1K LÞÞÞ. The parameters q, r, 1/(1Ck) and umf were varied to clarify their influence on the optimal solution. Tables 2–5 summarize the results, and illustrate the following: (1) When q is increasing, the optimum maintenance time T* reduces given constant 1/(1Ck) and umf. Restated, it is best to perform PM frequently. (2) When q or r is small, for example, if q or r is 0.001 and 1/(1Ck) is 0.01, then the optimum maintenance times T* are almost identical in different special cases with the same umf. (3) The optimum time T* increases with increasing 1/(1Ck), which implies that repairing a failed system is better than performing PM when the value of 1/(1Ck) is high. (4) When umf increases, the optimum maintenance time T* also increases owing to less time of repairing a failed system. (5) When a(aO0) increases the optimum maintenance time T* increases with umf!uf. Moreover, qj!qjC1!1 (jZ1,2,.) when 0!a!1 and 1OqjOqjC1!1 (jZ 1,2,.) when 1!a. The optimum time T* can be improved somewhat by improving qj. Comparing the proposed model with Nakagawa’s [7] demonstrates that the proposed model is more practical than
5. Concluding remarks This study described a general PM model that incorporates three types of PMs. The model for optimizing PM times T* was examined. The nature of a PM process and policy produces the hypothesis that the probability of PPM being obtained depends on the number of IPM performed since the previous renewal cycle. The results of investigating the optimal policy conditions demonstrate that such a policy is more general and flexible than policies already reported in the literature. Special cases were examined in detail. The optimum times T* were found in special cases, with reference to an example. Analysis reveals the effect of the input parameters on the solution, and also obtains some further insights. If the learning rate is estimated based on the actual data, analysts can use the learning curves to project the PM costs. This information can be used to estimate training requirements and develop PM plans.
Acknowledgements The authors are pleased to thank the referees for their valuable comments and suggestions, which significantly improved the clarity of this paper. This research was supported by the National Science Council of Taiwan, under Grant No. NSC 92-2213-E-011-010.
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A.4. Derivation of (9)
Appendix A PN
dAðT; fP j gÞ jZ1 ðPjK1 K Pj ÞjFðjTÞ Z PN u
PN Ð jT mf dT jZ1 ðPjK1 K Pj Þ 0 FðtÞdt C 1Ck K uf jZ1 ðPjK1 K Pj ÞFðjTÞ C uf Ð jT PN jZ1 ðPjK1 K Pj Þ 0 FðtÞdt K hP i2
Ð PN umf N jT jZ1 ðPjK1 K Pj Þ 0 FðtÞdt C 1Ck K uf jZ1 ðPjK1 K Pj ÞFðjTÞ C uf " # N N
X X u mf ! ðP jK1 K P j ÞjFðjTÞ C uf K ðP K P j Þjf ðjTÞ Z 0 1 C k jZ1 jK1 jZ1
d AðT; fP j gÞ Z 0; dT
" W
A.1. Derivation of (3) N X
E½Uj Z
jZ1
N X
ðE½Uj ;ðjK1ÞT !Y1 !jTCE½Uj ;Y1 RjTÞ
jZ1
Z
N X
P jK1
t dFðtÞC ðjK1ÞT
C
N X
C
jT P jK1 q2;j FðjTÞZ
N X
ðP jK1 K P j Þ
ð jT
jZ1
C
Z
ðP jK1 K P j Þ
jZ1
ð jT
t dFðtÞ
E½Dj ;Y1 RjTZumf
jZ1
N X
0
FðtÞdt
ðA1)
E½Dj ;ðjK1ÞT!Y1 !jTZuf
jZ1
Zuf
N X
P jK1
ðjK1ÞT
ðP jK1 KP j ÞFðjTÞ
jZ1
N X
N X
ðP jK1 KP j Þ
jZ1
A.5. Derivation of (21) " N 6 X 0 H ðTÞ Z ðP K P j Þj2 ð6 eK2jT K 6 eK3jT Þ W jZ1 jK1 !
ðP jK1 K P j ÞjðeK2jT K eK3jT Þ
jZ1
ðA2)
N X K ðP jK1 K P j Þj2 ð2 eK2jT K 3 eK3jT Þ jZ1 N X
!
# ðP jK1 K P j Þjð3 e
K2jT
K2 e
K3jT
Þ
jZ1
" N 6 X Z ðP K P j Þj2 ð3 eK2jT K 3 eK3jT Þ W jZ1 jK1
P jK1 PððjK1ÞT!Y1 !jTÞ
jZ1
dFðtÞZuf
ðP jK1 K P j ÞFðjTÞ C uf
N X
P jK1 q2;j PðY1 RjTÞ
A.3. Derivation of (6)
Zuf
0
0
0
ð jT
u mf K uf 1 Ck #2
FðtÞdt C
Forming WO0 implies: N N X X 1 K ðP jK1 K P j ÞFðjTÞ K HðTÞ ðP jK1 K P j Þ 1 K L jZ1 jZ1 ð jT Z0 ! FðtÞdt
t dFðtÞ
N X P j 1 P Zumf 1K PðY1 RjTÞ 1Ck jK1 PjK1 jZ1
N X
N X
ð jT
jZ1
jZ1
jZ1
ðP jK1 K P j Þ
jZ1
!
N u X Z mf ðP KP j ÞFðjTÞ 1Ck jZ1 jK1
N X
N X
0
A.2. Derivation of (5) N X
where " WZ
jTðP jK1 K P j ÞFðjTÞ
jZ1 N X
ð jT
jT P jK1 ðq1;j Cq2;j ÞFðjTÞ
N X
N X
ðP jK1 K P j Þ
jZ1
jZ1
ZK
jT P jK1 q1;j FðjTÞ
jZ1
jZ1 N X
N X
(A4)
0
jZ1
ð jT
jZ1
N X 1 K ðP K P j ÞFðjTÞ 1 K L jZ1 jK1 # ð jT N X KHðTÞ FðtÞdt Z0 ðP jK1 K P j Þ
ð jT
!
N X
ðP jK1 K P j Þjð2 eK2jT K 2 eK3jT Þ
jZ1 N X K ðP jK1 K P j Þj2 ð2 eK2jT K 3 eK3jT Þ
dFðtÞ 0
jZ1
ðA3)
!
N X jZ1
# ðP jK1 K P j Þjð3 eK2jT K 2 eK3jT Þ
S.-H. Sheu et al. / Reliability Engineering and System Safety 91 (2006) 362–369
" N 6 X Z ðP K P j Þj2 eK2jT W jZ1 jK1 N X
!
ðP jK1 K P j Þjð3 eK2jT K 2 eK3jT Þ
jZ1 N X
K
ðP jK1 K P j Þj eK2jT
jZ1 N X
!
# 2
ðP jK1 K P j Þj ð2 e
K2jT
K3 e
K3jT
Þ
jZ1
" N N X 6 X Z ðP jK1 K P j Þj2 eK2jT ðP jK1 K P j Þj eK2jT W jZ1 jZ1 N X
C
ðP jK1 K P j Þj2 3 eK2jT
jZ1
K
ðP jK1 K P j Þj2 eK2jT
jZ1
6 W
ðP jK1 K P j Þj eK2jT
jK1
N X
O
N X
N X
# ðP jK1 K P j Þj eK3jT Þ
jZ1
"
N X
ðP jK1 K P j Þj2 eK2jT
jZ1
N X ðP jK1 K P j Þj eK2jT jZ1
N X
ðP jK1 K P j Þj2 ð3 eK2jT K 2 eK2jT Þ
C jZ1 N X
!
ðP jK1 K P j Þj eK2jT O 0
ðA5)
jK1
References [1] Lam CT, Yeh RH. Optimal maintenance-policies for deteriorating systems under various maintenance strategies. IEEE Trans Reliab 1994;43:423–30. [2] Lim JH, Park DH. Evaluation of average maintenance cost for imperfect-repair model. IEEE Trans Reliab 1999;48:199–204.
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[3] Nakagawa T, Kijima M. Replacement policies for a cumulative damage model with minimal repair at failure. IEEE Trans Reliab 1989;38:581–4. [4] Pham H, Wang H. Imperfect maintenance. Eur J Oper Res 1996;94: 425–38. [5] Chiang JH, Yuan J. Optimal maintenance policy for a Markovian system under periodic inspection. Reliab Eng Syst Safety 2001;71: 165–72. [6] Qian C, Nakamura S, Nakagawa T. Cumulative damage model with two kinds of shocks and its application to the backup policy. J Oper Res Soc Jpn 1999;42:501–11. [7] Nakagawa T, Yasui K. Optimal policies for a system with imperfect maintenance. IEEE Trans Reliab 1987;R-36:631–3. [8] Zhao M. Availability for repairable components and series system. IEEE Trans Reliab 1994;43:329–34. [9] Tibbs RW, Monticone LC. Availability modeling for the federal aviation administration. IEEE Trans Reliab 1992;41:97–106. [10] Nakagawa T. Sequential imperfect preventive maintenance policies. IEEE Trans Reliab 1988;37:295–8. [11] Lin D, Zuo MJ, Yam RCM. General sequential imperfect preventive maintenance models. Int J Reliab Qual Safety Eng 2000;7:253–66. [12] Nakagawa T, Yasui K. Periodic-replacement models with threshold levels. IEEE Trans Reliab 1991;40:395–7. [13] Liao CJ, Chen WJ. Single-machine scheduling with periodic maintenance and nonresumable jobs. Comput Oper Res 2003;30: 1335–47. [14] Nakagawa T. Optimum policies when preventive maintenance is imperfect. IEEE Trans Reliab 1979;R28:331–2. [15] Levitin G, Lisnianski A. Optimization of imperfect preventive maintenance for multi-state systems. Reliab Eng Syst Safety 2000; 67:193–203. [16] Sheu SH, Griffith WS. Optimal number of minimal repairs before replacement of a system subject to shocks. Naval Res Logist 1996;43: 319–33. [17] Sheu SH. Extended optimal replacement model for deteriorating system. Eur J Oper Res 1999;112:503–16. [18] Sheu SH. A generalized age and block replacement of a system subject to shocks. Eur J Oper Res 1998;108:345–62. [19] Yang Y, Klutke GA. A distribution-free lower bound for availability of quantile-based inspection schemes. IEEE Trans Reliab 2001;50: 419–21. [20] Li George, Rajagopalan S. The impact of quality on learning. J Oper Manage 2001;134:508–24. [21] Moore Michael J, Bennett Charles L. The learning curve for laparoscopic cholecystectomy. Am J Surg 2001;170:55–9. [22] Zangwill Willard I, Kantor Paul B. The learning curve: a new perspective. Int Trans Oper Res 2000;7:595–607.
Optimum policies for a system with general imperfect maintenance Shey-Huei Sheu*, Yuh-Bin Lin, Gwo-Liang Liao Department of Industrial Management, National Taiwan University of Science and Technology, 43 Keelung Road, Section 4, Taipei, Taiwan, ROC Received 3 January 2004; accepted 29 January 2005 Available online 7 April 2005
Abstract This study considers periodic preventive maintenance policies, which maximizes the availability of a repairable system with major repair at failure. Three types of preventive maintenance are performed, namely: imperfect preventive maintenance (IPM), perfect preventive maintenance (PPM) and failed preventive maintenance (FPM). The probability that preventive maintenance is perfect depends on the number of imperfect maintenances conducted since the previous renewal cycle, and the probability that preventive maintenance remains imperfect is not increasing. The optimum preventive maintenance time that maximizes availability is derived. Various special cases are considered. A numerical example is given. q 2005 Elsevier Ltd. All rights reserved. Keywords: Maintenance; Imperfect maintenance; Optimum; Availability; Learning effect
1. Introduction To gain and maintain competitive advantage, manufacturers require a height availability system for maintaining peak production machinery operations. Preventive maintenance (PM) is important in complex systems because it reduces downtime and breakdown risk. Maintenance policies have been extensively examined [1–6]. Certain aspects of the PM model maximize the availability [7–9]. Numerous investigations based on PM analysis have attempted to apply the planned PM model to various realworld situations, including a sequential PM process. When a system is maintained at unequal intervals, the PM policy is known as sequential PM [10,11]. Another popular policy is the periodic PM policy, in which the system is preventively maintained at fixed time intervals [12,13]. Perfect PM models assume that the system is ‘as good as new’ following PM. However, this assumption can be incorrect. A more realistic assumption is that following PM the system lies in a state somewhere between as good as new and its pre-maintenance condition. Applying this assumption to PM is known as the imperfect PM model [14,15]. Pham and Wang [4] summarized and discussed various * Corresponding author. Fax: C886 2273 76344. E-mail address: [email protected] (S.-H. Sheu).
0951-8320/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2005.01.015
methods and optimal policies for imperfect maintenance. The model of Nakagawa [7] assumed that PM achieves either imperfect maintenance with a probability p, or perfect 1K p. However, this maintenance with probability pZ assumption is frequently false. Following PM, maintenance workers sometimes perform periodic testing for abnormalities. To include this testing experience, the probability that PM is perfect should depend on the number of imperfect maintenances performed since the last renewal cycle, and the probability that PM remains imperfect does not increase. Equipment failures frequently occur in every workplace between planned PMs. Maintenance workers must determine an action in response to the failure problem because of cost concerns. Major repairs reset the system failure intensity (a good-as-new repair). Numerous protective systems, for example, circuit breakers, alarms, and protective relays, are maintained in this fashion [19]. Sometimes breakdowns occur following PM because maintenance workers lack the necessary parts to correctly perform the PM or installation. To reflect this phenomenon, following PM this study considers three outcomes, similar to those proposed by Nakagawa [7]. Outcome I involves imperfect PM (IPM) only. Meanwhile, outcome II involves perfect PM (PPM). Finally, outcome III involves failed PM (FPM) and requiring major repair. Section 2 presents the model. The availability of the model with periodic PM is determined and the optimum PM policy thus obtained. Section 3 details various special cases. Section 4 then provides a numerical result for special cases. This study is
S.-H. Sheu et al. / Reliability Engineering and System Safety 91 (2006) 362–369
conducted to investigate the effects of these parameters on the solution. Finally, Section 5 presents concluding remarks.
† Imperfect PM (IPM) results in the system having the same failure rate as before PM, with probability qj Z P j =P jK1 . † Perfect PM (PPM) makes the system as good as new, with probability q1, jZ(k/(1Ck))(1Kqj). † Failed PM (FPM): where the system has deteriorated and requires repair, with probability q2, jZ (1/(1Ck))(1Kqj). Following a PPM, the system returns to time 0. If failure occurs before the scheduled PM, the major repair can be performed immediately; otherwise preventative maintenance is performed. After a major repair, the system returns to time 0. PM time is negligible. ufOumf
2. General model To construct the model, relevant notations are defined as follows: X F(t) FðtÞ f(t) rðtÞ H(t)
time to failure of a new unit Cdf[X]: cumulative distribution function of X Sf ½X : survival function of X pdf[X]: probability density function of X ðtÞ=F ðtÞ : failure rate (hazard) f P .PNof a unit N ð P K P Þjf ðjtÞ j jZ1 jK1 jZ1 ðPjK1 K Pj Þj FðjtÞ [weighted sum of pdf f(jt)]/[weighted sum of Sf FðjtÞ] T time between PMs for the system AðT; fP j gÞ availability of the system T* optimal time T to maximum AðT; fP j gÞ umf the mean time to repair for maintenance failure uf the mean time to repair for actual failure us the mean time between failure of the unit r the learning rate Y time between two successive renewal processes
In practice, determining maintenance intervals to maximize availability is a constant problem. The cost implications of maintenance model, such as those presented in this section are important. This study considers a generalized PM model using the following scheme. A system involves three types of outcomes after PM. Type I outcome is called imperfect PM (IPM), type II outcome is known as perfect PM (PPM), and type III outcome is termed failed PM (FPM). This system allows the probability of PPM and FPM occurring to depend on the number of PM conducted since the last renewal cycle. Let M denote the number of PM until the first PPM or FPM occurs. Moreover, let P j Z PðM O jÞ. That is, P j represents the probability that at least j first PMs are IPMs. This study assumes that the domain of P j is {0,1,2,.} and that 1Z P 0 O P 1 R P 2 R/ [16–18]. Moreover, this study uses the notation fP j g to indicate a sequence of probabilities. Let pj Z PðMZ jÞ ZP jK1 K P j Z P jK1 ð1K ðP j =P jK1 ÞÞ, with domain {1,2,3,.}. Consequently, if the jth PM occurs after jK1 IPMs, a PM is either IPM, with probability qj Z P j =P jK1 , or other types, with probability qjZ1Kqj. Let q1,j and q2,j denote the probabilities that a PM is classified as a PPM and FPM, respectively. That is, qjZq1,jCq2,j and kZq1,j/q2,j; kO1. The model makes the following assumptions: (1) The original system begins operating at time 0. (2) A system has three types of PMs at time j$T (jZ1,2,.), based on outcome:
363
(3) (4)
(5) (6)
Let Y1, Y2,. be independent copies of Y. For the present policy ( Y1 ; ifðj K 1ÞT ! Y1 ! jT for j Z 1; 2; . (1) Uj Z jT; if Y1 R jT where Uj represents the operating time during the renewal interval Y1 between (jK1)th PM and jth PM. Moreover, let Dj denote the repair time duration between (jK1)th PM and jth PM and include the repair time of the jth FPM. P Furthermore, the mean time of one renewal cycle is N jZ1 E½Uj . The availability is: PN jZ1 E½Uj PN AðT; fPj gÞ Z PN (2) E½U j C jZ1 jZ1 E½Dj It is easily verified that N X
ð jT N X FðtÞdt E½Uj Z ðPjK1 K Pj Þ
jZ1
jZ1
(3)
0
and N X
E½Dj Z
jZ1
N X
E½Dj ;Y1 RjTC
jZ1
N X
E½Dj ;ðjK1ÞT!Y1 !jT
jZ1
(4) We have N X jZ1
E½Dj ; Y1 R jT Z
N umf X ðP K P j ÞFðjTÞ 1 C k jZ1 jK1
(5)
and N X jZ1
E½Dj ; ðj K 1ÞT ! Y1 ! jT Z uf
N X
ðP jK1 K P j ÞFðjTÞ
jZ1
(6)
364
S.-H. Sheu et al. / Reliability Engineering and System Safety 91 (2006) 362–369
Combining (5) and (6), the expression for can be obtained as follows: N X
E½Dj Z
jZ1
PN
jZ1
E½Dj
N umf X ðP K P j ÞFðjTÞ 1 C k jZ1 jK1
C uf
N X ðP jK1 K P j ÞFðjTÞ
(7)
jZ1
concerns a repetitive job or task and represents the relationship between experience and productivity. The traditional learning curve model has been extensively used to reduce in either total direct labor hours per unit or total value-added costs per unit. Li and Rajagopalan [20] considered the impact of quality on learning. Furthermore, Moore and Bennett [21] used multivariate regression analyses to assess the relationship between bile duct injury rate and experience (number of procedures performed). The
From (3) and (7), we obtain the availability of the system: Ð PN jT jZ1 ðPjK1 K Pj Þ 0 FðtÞdt AðT; fP j gÞ Z PN Ð PN umf PN jT jZ1 ðPjK1 K Pj Þ 0 FðtÞdt C 1Ck jZ1 ðPjK1 K Pj ÞFðjTÞ C uf jZ1 ðPjK1 K Pj ÞFðjTÞ Ð PN jT jZ1 ðPjK1 K Pj Þ 0 FðtÞdt Z PN u
PN Ð mf jT jZ1 ðPjK1 K Pj Þ 0 FðtÞdt C 1Ck K uf jZ1 ðPjK1 K Pj ÞFðjTÞ C uf Seek the optimum PM T* which maximizes AðT; fP j gÞ in (8). Differentiating AðT; fP j gÞ with respect to T yields and setting the derivation to zero N N X X 1 K ðP jK1 K P j ÞFðjTÞ K HðTÞ ðP jK1 K P j Þ 1 K L jZ1 jZ1 ð jT Z0 ! FðtÞdt
ð9Þ
0
(8)
probability P j resembles bile duct injury rate and quality. Zangwill and Kantor [22] established a theoretical construct to provide a better explanation of the learning curve. The traditional learning curve is quite common, although other forms such as exponential formulation also exist. This study applies the traditional learning curve model to the PM model, substituting P j for total direct labor hours per unit. Following the discussion of case D, a learning curve is developed, and the following assumptions are made:
where LZumf/(uf(1Ck)). The following theorem demonstrates that a unique solution exists that satisfies (9) under certain reasonable conditions. Consequently, T* is easily obtainable with any numerical search procedure.
(1) P jK1 O P j ðjZ 1; 2; .Þ. (2) If each doubling of the number of PM reduces imperfect PM probability by (1Kr), then P j Z P 1 jb ðjZ 1; 2; .Þ. (3) b ¼ ðlog rÞ=ðlog 2Þ.
Theorem 1. If HðNÞO 1=ðus ð1K LÞÞ and H(T) is continuous and strictly increasing in T, then there exits a unique and finite T*O0 that maximizes the availability AðT; fP j gÞ.
3.1. Case A: P 0 Z 1; P j Z 0 ðjZ 1; 2; .Þ
Proof. If HðNÞO 1=ðus ð1K LÞÞ and H(T) is strictly increasing in T, then the left-hand side of (9) is strictly decreasing in T. And, it change sign exactly once from positive ðumf =ðð1 þ kÞuf K umf ÞÞ to negative ð1=ð1K LÞK us HðNÞÞ as T increases from 0 to N, as does dAðT; fP j gÞ=dT. The result follows. Observing (9), we can find the optimal availability as follows:
AðT ; f1; 0; 0; .; 0gÞ Z
AðT ;fP j gÞ PN
jZ1 ðPjK1 K Pj ÞjFðjT Þ
umf PN jZ1 ðPjK1 K Pj ÞjFðjT ÞC uf K 1Ck jZ1 ðPjK1 K Pj Þjf ðjT Þ
Z PN Z
This case considers an operating system to be as good as new after PM at jT(jZ1,2,.). Using (8) and (10) ÐT 0 FðtÞdt AðT; f1; 0; 0; .; 0gÞ Z Ð T 0 FðtÞdt K uf ð1 K LÞFðTÞ C uf (11) 1 1 C uf ð1 K LÞrðT Þ
(12)
where rðT ÞZ f ðT Þ=FðT Þ. 3.2. Case B: k/N, q2,jZ0, q1,jZqj (jZ1,2,.)
1 , 1Cuf ð1KLÞHðT Þ (10)
Using (8) and (10): AðT; fP j gÞ
3. Special cases Given instruction and through repetition, PM workers learn to perform tasks more effectively. The learning curve
Ð jT jZ1 ðPjK1 K Pj Þ 0 FðtÞdt Ð jT PN jZ1 ðPjK1 K Pj Þ 0 FðtÞdt C uf jZ1 ðPjK1
Z PN
PN
K P j ÞFðjTÞ ð13Þ
AðT ; fP j gÞ PN
Þ K P j ÞjFðjT PN jZ1 ðPjK1 K Pj ÞjFðjT Þ C uf jZ1 ðPjK1 K Pj Þjf ðjT Þ
jZ1 ðPjK1
Z PN
1 Z 1 C uf HðT Þ
The probability that PM still imperfect
S.-H. Sheu et al. / Reliability Engineering and System Safety 91 (2006) 362–369
365
1 0.8 0.6 0.4 0.2 0 0
50
100
150
200
250
300
(14)
Fig. 1. Learning curve.
3.3. Case C: P 0 Z 1; P j Z qj ðjZ 1; 2; .Þ, 0%q!1, 1K q qZ
with number of PM items j. Fig. 1, plotting the data in Table 1, shows that probability P j is rapidly decreasing in
This case is considered in [7]. Using (8) and (10) Ð P jK1 jT ð1 K qÞ N jZ1 q 0 FðtÞdt AðT; fqj gÞ Z Ð P P umf ð1KqÞ PN jK1 jT FðtÞdt jK1 FðjTÞ jK1 FðjTÞ ð1 K qÞ N q q C C uf ð1 K qÞ N jZ1 jZ1 jZ1 q 0 1Ck AðT ; fqj gÞ Z where HðTÞ Z
"
N X
1 1 C uf ð1 K LÞHðT Þ
(15)
(16) the early PM. Using (8) and (10) PN
E½Uj PN jZ1 E½Uj C jZ1 E½Dj
# " # N . X jK1 jK1 q jf ðjTÞ q jFðjTÞ
jZ1
350
The number of PM since the last perfect PM
AðT; f1; P 1 ,1b ; P 1 ,2b ; .gÞ Z PN
jZ1
(17)
jZ1
3.4. Case D: P 0 Z 1; P 1 s0; P j Z P 1 ,jb ðjZ 1; 2; .Þ, 0!r!1 In this case learning curves are most advantageous during the early PM of new failure causes. As PM number becomes large, the learning effect becomes less noticeable. That is, P j K P jC1 O P jC1 K P jC2 ðjZ 1; 2; .Þ. The following table and figure provide an example for P 1 Z 0:8 and 75 learning rate. Table 1 displays that probability P j decreases
where N X jZ1
ð jT !
P j
j
P j
j
P j
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
1.00000 0.80000 0.60000 0.50707 0.45000 0.41020 0.38030 0.35673 0.33750 0.32140 0.30765 0.29571 0.28523 0.27591 0.26755 0.26000 0.25313 0.24684 0.24105 0.23570
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
0.23074 0.22611 0.22179 0.21773 0.21392 0.21033 0.20693 0.20371 0.20066 0.19776 0.19500 0.19236 0.18984 0.18743 0.18513 0.18291 0.18079 0.17874 0.17677 0.17488
39 40 50 60 70 90 100 140 180 250 350 500 750 1300 2600 6500 10,000 20,000 30,000 50,000
0.17488 0.17305 0.15774 0.14625 0.13718 0.12360 0.11831 0.10289 0.09270 0.08088 0.07034 0.06066 0.05127 0.04080 0.03060 0.02092 0.01750 0.01312 0.01109 0.00897
0
FðtÞdt C P 1
N X ½ðj K 1Þb K jb jZ2
FðtÞdt
and E½Dj Z
jZ1
j
ðT
0
N X
Table 1 Number of PM items j and P j
E½Uj Z ð1 K P 1 Þ
u mf K uf ½ð1 K P 1 ÞFðTÞ 1 Ck
C P 1
N X ½ðj K 1Þb K jb FðjTÞ C uf jZ2
AðT ; f1; P 1 ,1b ; P 1 ,2b ; .gÞ Z
1 1 C uf ð1 K LÞHðT Þ
(18)
where P b b ð1 K P 1 Þf ðT Þ C P 1 N jZ2 ½ðj K 1Þ K j Þjf ðjT Þ P HðT Þ Z b b Þ C P 1 N ð1 K P 1 ÞFðT jZ2 ½ðj K 1Þ K j ÞjFðjT Þ
a 3.5. Case E: P 0 Z 1; P j Z qj ðjZ 1; 2; .Þ, 0%q!1, aO0
Here, a random number M, of instances of PM until a PPM is performed, is a discrete Weibull distribution that is IFR for aR1 and DFR for 0!a%1.
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S.-H. Sheu et al. / Reliability Engineering and System Safety 91 (2006) 362–369
From (8) and (10) PN
ja
AðT; fq gÞ Z PN
jZ1 ðq
a
AðT ; fqj gÞ Z
ðjK1Þa
ja
Kq Þ
Ð jT 0
ðjK1Þa
Ð jT
a
FðtÞdt
0PN ðjK1Þa a mf FðtÞdt C 1Ck K uf K qj ÞFðjTÞ C uf jZ1 ðq jZ1 ðq
u
K qj Þ
1 1 C uf ð1 K LÞHðT Þ
(20)
(19)
Hð0Þ Z lim HðTÞ Z 0 T/0
Because
where HðT Þ " #," # N N X X ðjK1Þa ja ðjK1Þa ja Z ðq K q Þjf ðjT Þ ðq K q ÞjFðjT Þ jZ1
jZ1
1 Z lim T/N HðTÞ T/N
PN
lim
K2jT K3eK3jT CeK3jT Þ 1 jZ1 ðPjK1 K Pj Þjð3e PN Z 2 2 jZ1 ðP jK1 K P j Þjð3eK2jT K3eK3jT Þ
we have HðNÞ Z lim HðTÞ Z 2 T/N
4. Example and analysis Four tables list optimal times T* satisfying (9) for the special cases. Table 2 lists the special case C, Table 3 presents the special case D, Table 4 shows the special case E with aZ0.5, and Table 5 illustrates the special case E with aZ1.5. The following cost and distribution parameters of the PM models are considered to illustrate the policy, similar to the proposal by Nakagawa [7]; ufZ1. umfZ0.1uf, umfZ0.3uf, umfZ0.5 or umf/1.0uf. FðtÞZ 3 eK2t K 2 eK3t . Then, ðN ðN us Z FðtÞdt Z 3 eK2t K 2 eK3t dt Z 5=6 0
" N N X 6 X H ðTÞ Z ðP jK1 K P j Þj2 eK2jT ðP jK1 K P j Þj eK2jT W jZ1 jZ1 0
C
N N X X ðP jK1 K P j Þj2 3 eK2jT ðP jK1 K P j Þj eK2jT jZ1
jK1
N X
2
ðP jK1 K P j Þj e
K jZ1
K2jT
N X
#
ðP jK1 K P j Þj e
K3jT
Þ
jZ1
" N N X 6 X O ðP jK1 K P j Þj2 eK2jT ðP jK1 K P j Þj eK2jT W jZ1 jZ1
0
C
The H(T) is rewritten as: P K2jT 6 N K eK3jT Þ j¼1 ðPjK1 K Pj Þjðe HðTÞ Z PN K2jT K 2eK3jT Þ j¼1 ðPjK1 K Pj Þjð3e
N X ðP jK1 K P j Þj2 ð3 eK2jT K 2 eK2jT Þ jZ1
!
N X
# ðP jK1 K P j Þj eK2jT O 0
ð21Þ
jK1
Table 2 Optimum maintenance time T* with the parameters in special case C 1/(1Ck)
q umfZ0.3uf
umfZ0.1uf
0.0001 0.001 0.01 0.1
umfZ0.5uf
umf/1.0uf
0.001
0.01
0.1
0.001
0.01
0.1
0.001
0.01
0.1
0.001
0.01
0.1
0.183 0.582 1.881 6.411
0.180 0.574 1.857 6.339
0.157 0.501 1.628 5.632
0.317 1.016 3.338 12.182
0.313 1.002 3.297 12.064
0.273 0.876 2.903 10.887
0.410 1.318 4.385 16.919
0.405 1.301 4.333 16.775
0.353 1.138 3.827 15.312
0.582 1.881 6.411 28.107
0.574 1.857 6.339 27.927
0.501 1.628 5.632 26.071
Table 3 Optimum maintenance time T* with the parameters in special case D 1/(1Ck)
r umfZ0.3uf
umfZ0.1uf
0.0001 0.001 0.01 0.1
umfZ0.5uf
umf/1.0uf
0.001
0.01
0.1
0.001
0.01
0.1
0.001
0.01
0.1
0.001
0.01
0.1
0.183 0.582 1.881 6.411
0.180 0.574 1.857 6.339
0.155 0.495 1.612 5.612
0.317 1.016 3.338 12.182
0.313 1.002 3.297 12.064
0.269 0.866 2.883 10.882
0.410 1.318 4.385 16.919
0.405 1.301 4.333 16.775
0.349 1.126 3.805 15.322
0.582 1.881 6.411 28.107
0.574 1.857 6.339 27.927
0.495 1.612 5.612 26.106
S.-H. Sheu et al. / Reliability Engineering and System Safety 91 (2006) 362–369
367
Table 4 Optimum maintenance time T* with the parameters when aZ0.5 in special case E 1/(1Ck)
q umfZ0.3uf
umfZ0.1uf
0.0001 0.001 0.01 0.1
umfZ0.5uf
umf/1.0uf
0.001
0.01
0.1
0.001
0.01
0.1
0.001
0.01
0.1
0.001
0.01
0.1
0.183 0.582 1.881 6.411
0.179 0.572 1.849 6.322
0.131 0.422 1.500 5.197
0.317 1.015 3.338 12.181
0.312 0.998 3.285 12.048
0.228 0.744 2.571 10.538
0.410 1.318 4.384 16.919
0.403 1.295 4.318 16.763
0.296 0.972 3.442 15.144
0.582 1.881 6.411 28.107
0.572 1.849 6.322 27.931
0.422 1.406 5.197 26.349
Table 5 Optimum maintenance time T* with the parameters when aZ1.5 in special case E 1/(1Ck)
q umfZ0.3uf
umfZ0.1uf
0.0001 0.001 0.01 0.1
where " WZ
umfZ0.5uf
umf/1.0uf
0.001
0.01
0.1
0.001
0.01
0.1
0.001
0.01
0.1
0.001
0.01
0.1
0.183 0.582 1.881 6.411
0.180 0.575 1.858 6.340
0.160 0.511 1.656 5.704
0.317 1.016 3.338 12.182
0.313 1.003 3.298 12.065
0.278 0.892 2.948 10.979
0.410 1.318 4.385 16.919
0.405 1.301 4.333 16.776
0.360 1.158 3.883 15.397
0.582 1.881 6.411 28.107
0.575 1.858 6.340 27.927
0.511 1.656 5.704 26.098
N X
#2
Nakagawa’s, and which is a special case of the proposed model.
ðP jK1 K P j Þjð3 eK2jT K 2 eK3jT Þ
jZ1
Thus, H(T) is strictly increasing from 0 to 2, and hence a finite and unique T* exists when (9) is satisfied, since 2O ð1=ð5=6ð1K LÞÞÞ. The parameters q, r, 1/(1Ck) and umf were varied to clarify their influence on the optimal solution. Tables 2–5 summarize the results, and illustrate the following: (1) When q is increasing, the optimum maintenance time T* reduces given constant 1/(1Ck) and umf. Restated, it is best to perform PM frequently. (2) When q or r is small, for example, if q or r is 0.001 and 1/(1Ck) is 0.01, then the optimum maintenance times T* are almost identical in different special cases with the same umf. (3) The optimum time T* increases with increasing 1/(1Ck), which implies that repairing a failed system is better than performing PM when the value of 1/(1Ck) is high. (4) When umf increases, the optimum maintenance time T* also increases owing to less time of repairing a failed system. (5) When a(aO0) increases the optimum maintenance time T* increases with umf!uf. Moreover, qj!qjC1!1 (jZ1,2,.) when 0!a!1 and 1OqjOqjC1!1 (jZ 1,2,.) when 1!a. The optimum time T* can be improved somewhat by improving qj. Comparing the proposed model with Nakagawa’s [7] demonstrates that the proposed model is more practical than
5. Concluding remarks This study described a general PM model that incorporates three types of PMs. The model for optimizing PM times T* was examined. The nature of a PM process and policy produces the hypothesis that the probability of PPM being obtained depends on the number of IPM performed since the previous renewal cycle. The results of investigating the optimal policy conditions demonstrate that such a policy is more general and flexible than policies already reported in the literature. Special cases were examined in detail. The optimum times T* were found in special cases, with reference to an example. Analysis reveals the effect of the input parameters on the solution, and also obtains some further insights. If the learning rate is estimated based on the actual data, analysts can use the learning curves to project the PM costs. This information can be used to estimate training requirements and develop PM plans.
Acknowledgements The authors are pleased to thank the referees for their valuable comments and suggestions, which significantly improved the clarity of this paper. This research was supported by the National Science Council of Taiwan, under Grant No. NSC 92-2213-E-011-010.
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S.-H. Sheu et al. / Reliability Engineering and System Safety 91 (2006) 362–369
A.4. Derivation of (9)
Appendix A PN
dAðT; fP j gÞ jZ1 ðPjK1 K Pj ÞjFðjTÞ Z PN u
PN Ð jT mf dT jZ1 ðPjK1 K Pj Þ 0 FðtÞdt C 1Ck K uf jZ1 ðPjK1 K Pj ÞFðjTÞ C uf Ð jT PN jZ1 ðPjK1 K Pj Þ 0 FðtÞdt K hP i2
Ð PN umf N jT jZ1 ðPjK1 K Pj Þ 0 FðtÞdt C 1Ck K uf jZ1 ðPjK1 K Pj ÞFðjTÞ C uf " # N N
X X u mf ! ðP jK1 K P j ÞjFðjTÞ C uf K ðP K P j Þjf ðjTÞ Z 0 1 C k jZ1 jK1 jZ1
d AðT; fP j gÞ Z 0; dT
" W
A.1. Derivation of (3) N X
E½Uj Z
jZ1
N X
ðE½Uj ;ðjK1ÞT !Y1 !jTCE½Uj ;Y1 RjTÞ
jZ1
Z
N X
P jK1
t dFðtÞC ðjK1ÞT
C
N X
C
jT P jK1 q2;j FðjTÞZ
N X
ðP jK1 K P j Þ
ð jT
jZ1
C
Z
ðP jK1 K P j Þ
jZ1
ð jT
t dFðtÞ
E½Dj ;Y1 RjTZumf
jZ1
N X
0
FðtÞdt
ðA1)
E½Dj ;ðjK1ÞT!Y1 !jTZuf
jZ1
Zuf
N X
P jK1
ðjK1ÞT
ðP jK1 KP j ÞFðjTÞ
jZ1
N X
N X
ðP jK1 KP j Þ
jZ1
A.5. Derivation of (21) " N 6 X 0 H ðTÞ Z ðP K P j Þj2 ð6 eK2jT K 6 eK3jT Þ W jZ1 jK1 !
ðP jK1 K P j ÞjðeK2jT K eK3jT Þ
jZ1
ðA2)
N X K ðP jK1 K P j Þj2 ð2 eK2jT K 3 eK3jT Þ jZ1 N X
!
# ðP jK1 K P j Þjð3 e
K2jT
K2 e
K3jT
Þ
jZ1
" N 6 X Z ðP K P j Þj2 ð3 eK2jT K 3 eK3jT Þ W jZ1 jK1
P jK1 PððjK1ÞT!Y1 !jTÞ
jZ1
dFðtÞZuf
ðP jK1 K P j ÞFðjTÞ C uf
N X
P jK1 q2;j PðY1 RjTÞ
A.3. Derivation of (6)
Zuf
0
0
0
ð jT
u mf K uf 1 Ck #2
FðtÞdt C
Forming WO0 implies: N N X X 1 K ðP jK1 K P j ÞFðjTÞ K HðTÞ ðP jK1 K P j Þ 1 K L jZ1 jZ1 ð jT Z0 ! FðtÞdt
t dFðtÞ
N X P j 1 P Zumf 1K PðY1 RjTÞ 1Ck jK1 PjK1 jZ1
N X
N X
ð jT
jZ1
jZ1
jZ1
ðP jK1 K P j Þ
jZ1
!
N u X Z mf ðP KP j ÞFðjTÞ 1Ck jZ1 jK1
N X
N X
0
A.2. Derivation of (5) N X
where " WZ
jTðP jK1 K P j ÞFðjTÞ
jZ1 N X
ð jT
jT P jK1 ðq1;j Cq2;j ÞFðjTÞ
N X
N X
ðP jK1 K P j Þ
jZ1
jZ1
ZK
jT P jK1 q1;j FðjTÞ
jZ1
jZ1 N X
N X
(A4)
0
jZ1
ð jT
jZ1
N X 1 K ðP K P j ÞFðjTÞ 1 K L jZ1 jK1 # ð jT N X KHðTÞ FðtÞdt Z0 ðP jK1 K P j Þ
ð jT
!
N X
ðP jK1 K P j Þjð2 eK2jT K 2 eK3jT Þ
jZ1 N X K ðP jK1 K P j Þj2 ð2 eK2jT K 3 eK3jT Þ
dFðtÞ 0
jZ1
ðA3)
!
N X jZ1
# ðP jK1 K P j Þjð3 eK2jT K 2 eK3jT Þ
S.-H. Sheu et al. / Reliability Engineering and System Safety 91 (2006) 362–369
" N 6 X Z ðP K P j Þj2 eK2jT W jZ1 jK1 N X
!
ðP jK1 K P j Þjð3 eK2jT K 2 eK3jT Þ
jZ1 N X
K
ðP jK1 K P j Þj eK2jT
jZ1 N X
!
# 2
ðP jK1 K P j Þj ð2 e
K2jT
K3 e
K3jT
Þ
jZ1
" N N X 6 X Z ðP jK1 K P j Þj2 eK2jT ðP jK1 K P j Þj eK2jT W jZ1 jZ1 N X
C
ðP jK1 K P j Þj2 3 eK2jT
jZ1
K
ðP jK1 K P j Þj2 eK2jT
jZ1
6 W
ðP jK1 K P j Þj eK2jT
jK1
N X
O
N X
N X
# ðP jK1 K P j Þj eK3jT Þ
jZ1
"
N X
ðP jK1 K P j Þj2 eK2jT
jZ1
N X ðP jK1 K P j Þj eK2jT jZ1
N X
ðP jK1 K P j Þj2 ð3 eK2jT K 2 eK2jT Þ
C jZ1 N X
!
ðP jK1 K P j Þj eK2jT O 0
ðA5)
jK1
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