Approximate methods for optimal replacement, maintenance, and inspection policies

Reliability Engineering and System Safety 144 (2015) 68–73

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Approximate methods for optimal replacement, maintenance, and inspection policies Xufeng Zhao a,n, Khalifa N. Al-Khalifa a, Toshio Nakagawa b a b

Department of Mechanical and Industrial Engineering, Qatar University, Doha 2713, Qatar Department of Business Administration, Aichi Institute of Technology, Toyota 470-0392, Japan

art ic l e i nf o

a b s t r a c t

Article history: Received 31 August 2014 Received in revised form 18 June 2015 Accepted 5 July 2015 Available online 23 July 2015

It might be difficult sometimes to derive theoretical and numerical solutions for analytical maintenance modelings due to the computational complexity. This paper takes up several approximate models in maintenance theory, by using the cumulative hazard function H(t) and the newly proposed asymptotic MTTF (Mean Time to Failure) skilfully. We firstly denote by tx the time when the expected number of failures is x. Using Hðt x Þ ¼ x, we estimate failure times, model age and periodic replacements, and sequential imperfect maintenance. Motivated by the asymptotic method of computation of MTTF, we secondly model the expected cost rate for a parallel system when replacement is made at system failure, and give approximate computations for the sequential inspection policy. Optimizations of each model are obtained approximately in an easier way. When failure times have a Weibull distribution, it is shown from numerical examples that the obtained approximate optimal solutions have good approximations of the exact ones. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Hazard function Mean time to failure Age replacement Imperfect maintenance Approximate inspection

1. Introduction Manufacturing systems with performance degradation and maintenance strategy are commonly encountered in practice. There have been many maintenance models in reliability, most of which are formulated stochastically, and are optimized analytically or simulated numerically by using algorithms [2,15,11,9]. However, it sometimes might be difficult to derive theoretical and numerical solutions for analytical maintenance modelings due to the computational complexity. One example is the sequential inspection policy [2], whose algorithm needs to make computations repeatedly until the procedure meets the required condition by adjusting the first checking time. To avoid this trouble, a nearly optimal inspection policy that depends on the parameter p was suggested [7]. However, to suppose the unit fails with constant probability p is too stronger to be applicable, even though this policy has been used for Weibull and gamma distribution cases [8,17]. Other works, such as an approximate solution of a maintenance policy for a system with multi-state components [4], an approximate inspection interval for production processes with finite run length [3], and approximations to determine the optimal

n

Corresponding author. Tel.: þ 974 3315 2602. E-mail address: [email protected] (X. Zhao).

http://dx.doi.org/10.1016/j.ress.2015.07.005 0951-8320/& 2015 Elsevier Ltd. All rights reserved.

replacement times of a sequential age replacement policy for a finite time horizon [6], have been discussed. It has been well-known in reliability theory that the cumulative hazard function H(t) represents the expected number of failures in the time interval ½0; t [11]. On the other hand, the most concerns in reliability theory are to estimate the MTTF (Mean Time to Failure) and maintenance times of systems; however, when the system becomes more complex or larger sized, estimations become more difficult. One approximate analytical approach has been proposed to estimate a threshold maintenance policy for an n identical unreliable components system [1]. An asymptotic MTTF and approximate age replacement for a random-sized parallel system have been proposed recently [13]. Followed by the conference discussion [12], we use the cumulative hazard function H(t) and the approximate computation of MTTF skilfully, and propose approximate methods to estimate failure times, and to optimize replacement, maintenance, and inspection policies. We show good approximations for the exact results in numerical examples when their failure times have a Weibull distribution as follows: 1. When failures occur at a non-homogeneous Poisson process and the unit undergoes minimal repair at each failure, it is of interest to observe the mean times of Xn between failures [11, p. 97]. We introduce Hðt n Þ ¼ n in which t n ðn ¼ 1; 2; …Þ is the time when the expected number of failures is n, and show that

X. Zhao et al. / Reliability Engineering and System Safety 144 (2015) 68–73

2.

3.

4.

5.

failure times Xn could be computed from a simpler equation form. We introduce the mean rate of failures HðT x Þ ¼ xð0 o x r 1Þ into age and periodic replacement policies. Computation of approximate Tx for age replacement is more easier but also is close to the exact Tn. For the periodic replacement, both approximate policies at time T and at number N of failures are obtained as one simpler equation form. For the imperfect preventive maintenance policy [11], we obtain the approximate expected cost rate and its optimal maintenance number Nn and sequential maintenance times λT nk . It also shows numerically that the above approximate policies are close to the exact results. Recently, an asymptotic method of computation of the MTTF and optimization of age replacement [13] for a random-sized parallel system are discussed. Using the approximate MTTF, we compute optimal number of units needed for a parallel system in an easier form, when replacement should be made after system failure. Although improvements have been made to compute sequential inspection policies [10] which have been summarized in [11], it is still difficult to choose an appropriate ε to begin the algorithm. Motivated by the above approximate MTTF, we finally give approximate computations for the sequential inspection times.

There has been lack of research on the approximate methods for maintenance modelings and optimizations. The remainder of this paper is organized as follows: Section 2 gives approximate xk for the mean times of Xn between failures. The Tx and t nx are obtained as approximations of the exact optimizations for age and periodic replacement policies in Sections 3 and 4. Sequential imperfect maintenance times λT nk are derived in Section 5. An approximate number of units for a parallel system and sequential inspections times T~ k are obtained in Sections 6 and 7. Finally, conclusions of the paper are provided in Section 8.

2. Failure times A unit begins to operate at time 0 and will operate for an infinite time span. The unit undergoes minimal repairs [2, p. 96] at failures, where the time for each repair is supposed to be negligible. Let 0 ¼ S0 r S1 r ⋯ rSn  1 rSn r⋯ be the successive random failure times and X n  Sn  Sn  1 ðn ¼ 1; 2; …Þ be the variable times between failures with distribution PrfX n rxj Sn  1 ¼ t g ¼

F ðtÞ  F ðt þxÞ ¼ 1  e  ½Hðt þ xÞ  HðtÞ ; F ðtÞ

ð1Þ

where failures occur at a non-homogeneous Poisson process with a mean value function H(t) [16, p. 46; 14, p. 78], and FðtÞ  1 e  HðtÞ and F ðtÞ  1  FðtÞ [11, p. 96]. Letting N(t) be the number of failures in ½0; t, then the probability that failures occur k times in ½0; t is
 ½HðtÞk  HðtÞ e Pr NðtÞ ¼ k ¼ k! EfNðtÞg ¼

1 X

ðk ¼ 0; 1; 2; …Þ;

kPrfNðtÞ ¼ kg ¼ HðtÞ:

ð2Þ

ð3Þ

k¼0

From [11, p. 97], we obtain Z 1
 ½HðtÞk  1  HðtÞ e dt E Xk ¼ ðk  1Þ! 0

ðk ¼ 1; 2; …Þ;

ð4Þ

EfSn g ¼

n 1 X k¼0

Z

1

0

69

½HðtÞk  HðtÞ e dt k!

ðn ¼ 1; 2; …Þ:

ð5Þ

It is assumed that Hðt n Þ ¼ n and xn  t n t n  1 , where t n ðn ¼ 1; 2; …Þ represents the time when the expected number of failures is n, then Hðxn þ t n  1 Þ  Hðt n  1 Þ ¼ 1 represents that the expected number of failures in ½t n  1 ; t n  1 þ xn  equals to 1. When the failure time of the unit has a Weibull distribution, i.e., m FðtÞ ¼ 1 e  t and HðtÞ ¼ t m ðm Z 1Þ, then (4) and (5) are Z 1 ðk  1Þm
 t m 1 Γ ðk  1 þ 1=mÞ ðk ¼ 1; 2; …Þ; ð6Þ e  t dt ¼ E Xk ¼ m ðk 1Þ! ðk  1Þ! 0 EfSn g ¼

n X
 Γ ðn þ 1=mÞ E Xk ¼ ðn  1Þ! k¼1

ðn ¼ 1; 2; …Þ:

ð7Þ

Furthermore, when ðt n Þm ¼ n, i.e., t n ¼ n1=m , we obtain xk ¼ t k  t k  1 ¼ k

1=m

 ðk  1Þ1=m

ðk ¼ 1; 2; …Þ:

ð8Þ

It is much easier to compute xk in (8) than to compute EfX k g in (6). Table 1 presents exact EfX k g and approximate xk when HðtÞ ¼ t m for m ¼ 1:5; 2:0; 3:0. This shows that the approximate xk is less than or equal to EfX k g when k Z 2 and becomes very good approximation for the exact EfX k g as k becomes larger.

3. Age replacement An operating unit has a failure distribution F(t) and failure rate hðtÞ  f ðtÞ=F ðtÞ, where f(t) is a density function of F(t). Consider the standard age replacement policy in which the unit is replaced at a planned time Tð0 o T o 1Þ or at failure, whichever occurs first. Then, the expected cost rate is [2, p. 87; 11, p. 72] C 1 ðTÞ ¼

c1 FðTÞ þ c2 F ðTÞ ; RT 0 F ðtÞ dt

ð9Þ

where c1 and c2 ðc2 o c1 Þ are respective replacement costs at failure and at time T. If the failure rate h(t) increases strictly to 1, then an optimal Tn minimizing C 1 ðTÞ is given by a unique solution of the equation Z T c2 hðTÞ F ðtÞ dt FðTÞ ¼ : ð10Þ c1  c2 0 From the above standard age replacement, we find that the only interest is to observe replacement actions that are done before the first failure or at the first failure. We suppose that

Table 1 Exact EfX k g and approximate xk when HðtÞ ¼ t m . k

1 2 3 4 5 6 7 8 9 10 20 30 50

m¼ 1.5

m¼ 2.0

m¼3.0

EfX k g

xk

EfX k g

xk

EfX k g

xk

0.903 0.602 0.502 0.446 0.409 0.381 0.360 0.343 0.329 0.317 0.248 0.216 0.182

1.000 0.587 0.493 0.440 0.404 0.378 0.357 0.341 0.327 0.315 0.248 0.216 0.182

0.886 0.443 0.332 0.277 0.242 0.218 0.200 0.186 0.174 0.164 0.114 0.092 0.071

1.000 0.414 0.318 0.268 0.236 0.213 0.196 0.183 0.172 0.162 0.113 0.092 0.071

0.893 0.298 0.198 0.154 0.129 0.111 0.099 0.090 0.082 0.076 0.047 0.035 0.025

1.000 0.260 0.182 0.145 0.123 0.107 0.096 0.087 0.080 0.074 0.046 0.035 0.025

70

X. Zhao et al. / Reliability Engineering and System Safety 144 (2015) 68–73

RT HðT x Þ  0 x hðuÞ du ¼ x ð0 o x r 1Þ, which represents that the mean rate of failures during ½0; T x  is x. Clearly, when x¼ 1, Tx represents the time when the expected number of failures in ½0; T x  is 1. We adopt the following expected cost rate as one of objective functions for the approximate age replacement policy: C 1 ðT x Þ ¼

 1
c1 HðT x Þ þ c2 ½1  HðT x Þ ; Tx

ð11Þ

where c1 is the replacement cost for failures in ½0; T x  and c2 is the replacement cost in ½0; T x  if no failure occurs. Thus, an optimal Tx which minimizes C 1 ðT x Þ is given by T x hðT x Þ  HðT x Þ ¼

c2 : c1  c2

ð12Þ

When the failure time has a Weibull distribution FðtÞ ¼ 1 m e  ðλtÞ , i.e., HðtÞ ¼ ðλtÞm ðm 4 1Þ, ðλT x Þm ¼ x;

i:e:;

1 T x ¼ x1=m :

ð13Þ

λ

Thus, (11) becomes C 1 ðxÞ

λ

¼

1 ½c1 x þ c2 ð1  xÞ: x1=m

ð14Þ

From (12), x¼

1 c2 ; m  1 c1 c2

ð15Þ

and hence,  1=m 1 1 c2 : Tx ¼ λ m  1 c1  c2

ð16Þ

Table 2 presents optimal Tn in (10) and approximate Tx in (16) m when FðtÞ ¼ 1  e  ðλtÞ and 1=λ ¼ 100 for m and c2 =ðc1  c2 Þ. This shows that Tx are very good approximations for Tn and can be obtained in an easier way.

unique solution of the equation Z T c2 ThðTÞ  hðtÞ dt ¼ : c1 0

ð18Þ

When the unit undergoes minimal repairs at failures and is replaced at the Nth failure ðN ¼ 1; 2; …Þ, the expected cost rate is C 3 ðNÞ ¼

c 1 N þ c2 EfSN g

ðN ¼ 1; 2; …Þ;

ð19Þ

and an optimal Nn minimizing C 3 ðNÞ is given by a unique and minimum solution which satisfies EfSN g c2 NZ EfX N þ 1 g c1

ðN ¼ 1; 2; …Þ;

ð20Þ

where EfX N þ 1 g and EfSN g are given in (4) and (5), respectively. When the unit is replaced as the expected number of failures is xð0 o x o 1Þ, the expected cost rate is C 3 ðxÞ ¼

c1 x þ c2 ; tx

ð21Þ

and an optimal xn minimizing C 3 ðxÞ is given by a solution of the equation tx c2 x ¼ ; t 0x c1

ð22Þ

where t 0x denotes the differential function of tx. It is assumed that HðtÞ ¼ ðλtÞm ðm 41Þ, an optimal Tn in (18) is   1 c2 1=m λT n ¼ ; ð23Þ m  1 c1 and an optimal Nn in (20) is Nn Z

1 c2 : m  1 c1

ð24Þ

In this case, (21) becomes C 3 ðxÞ ¼

c1 x þ c2 : x1=m

ð25Þ

Then, an optimal xn is 4. Periodic replacement

xn ¼

Suppose that the operating unit undergoes minimal repairs at failures and is replaced at periodic times kT ðk ¼ 1; 2; …Þ. Then, the expected cost rate is [2, p. 96; 11, p. 102] 1 C 2 ðTÞ ¼ ½c1 HðTÞ þ c2 ; T

ð17Þ

where c1 is the cost for each minimal repair and c2 is each replacement cost at times kT. If the failure rate h(t) increases strictly to 1, then an optimal Tn minimizing C 2 ðTÞ is given by a

1 c2 ; m  1 c1

and an optimal replacement time is   1 c2 1=m λt nx ¼ ; m  1 c1

ð26Þ

ð27Þ

which agrees with (23) for λT n and also agrees with (24) for Nn when xn is integer. In other words, we may adopt approximate C 3 ðxÞ in (21) as an appropriate objective function for the periodic replacement policy.

5. Imperfect maintenance

Table 2 m

Optimal Tn and approximate Tx when FðtÞ ¼ 1  e  ðt=100Þ . c2 c1  c2

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

m¼ 1.5

m¼ 2.0

Consider the following sequential imperfect preventive maintenance (PM) for an operating unit [11, p. 194]:

m¼ 3.0

Tn

Tx

Tn

Tx

Tn

Tx

7.388 11.759 15.450 18.767 21.836 24.726 27.477 30.117 30.117 35.141

7.368 11.696 15.326 18.566 21.544 24.329 26.962 29.472 29.472 34.200

10.008 14.166 17.364 20.067 22.454 24.618 26.613 28.474 30.227 31.889

10.000 14.142 17.321 20.000 22.361 24.495 26.458 28.284 30.000 31.623

17.103 21.553 24.678 27.167 29.271 31.111 32.758 34.257 35.636 36.917

17.100 21.544 24.662 27.144 29.240 31.072 32.711 34.200 35.569 36.840

a. The PM is done at planned intervals T k ðk ¼ 1; 2; …; N  1Þ and the unit is replaced at the Nth PM, where T 0  0. b. The unit undergoes minimal repairs at failures between replacements. c. The failure rate after the kth PM becomes bkh(t) when it was h (t) before PM, i.e., the unit has the failure rate Bkh(t) in the kth 1 PM period, where 1 ¼ b0 o b1 r ⋯ rbN r ⋯, Bk  ∏kj ¼ 0 bj ðk ¼ 1; 2; …Þ and 1  B1 o B2 o ⋯ o BN o ⋯. d. The cost of each minimal repair is c1, the cost of each PM is c2, and the replacement cost at the Nth PM is c3 ðc3 4 c2 Þ.

X. Zhao et al. / Reliability Engineering and System Safety 144 (2015) 68–73

The expected cost rate is [11, p. 194] CðT 1 ; T 2 ; …; T N Þ " # N X 1 ¼ PN c1 Bk HðT k Þ þ Nc2 þ c3  c2 k ¼ 1 Tk k¼1 When HðtÞ ¼ ðλtÞm ðm 4 1Þ, we denote x  Bk HðT k Þ ¼ Bk ðλT k Þm ;



x λT k ¼ Bk

i:e:;

ðN ¼ 1; 2; …Þ:

ð28Þ

λ

:

ð29Þ

Nðc1 x þ c2 Þ þ c3  c2  PN : 1=m k ¼ 1 ðx=Bk Þ

ð30Þ

We find optimal xn and Nn which minimize Cðx; NÞ. From Cðx; N þ 1Þ  Cðx; NÞ Z 0, " #  N  X BN þ 1 1=m ðc1 x þ c2 Þ  N Z c3  c2 ðN ¼ 1; 2; …Þ: ð31Þ Bk k¼1 Differentiating Cðx; NÞ with respect to x and setting it equal to zero, x¼

Nc2 þ c3  c2 : ðm  1ÞNc1

ð32Þ

Because Bk increases strictly with k,  N  X BN þ 1 1=m N Bk k¼1

Because ðmNc2 þ c3  c2 Þ=ðNc2 þ c3  c2 Þ increases with N, the lefthand side of (33) increases strictly from    1  X 1 B2 1=m B1 1=m ðmc2 þ c3  c2 Þ to m ; c 3 B1 Bk k¼1 where B1  limk-1 Bk which might be infinity. Thus, if  1  X B1 1=m 41; Bk k¼1 n

n

then there exists a finite and unique N ð1 r N o1Þ which satisfies (33). Then, n

N c2 þ c3  c2 ; ðm  1ÞN n c1

λT nk ¼



xn Bk

1=m

It has been recently shown [13] that when the failure time of each unit has an exponential distribution, the MTTF of a parallel P system with n units is ð1=λÞ nj¼ 1 ð1=jÞ. Further, when the failure time becomes a Weibull distribution for each unit, e.g., m FðtÞ ¼ 1 e  ðλtÞ for m Z 2, the MTTF is approximately given by 0 11=m n 1@ X 1A μ~ n ¼ ðn ¼ 1; 2; …Þ; ð36Þ λ j¼1 j which has been shown good approximations to the exact MTTF. We apply the above approximate MTTF to obtain an optimal number of units when replacement is done at system failure: consider a parallel system which consists of n ðn ¼ 1; 2; …Þ identical units and fails when all units have failed. It is also assumed that each unit has an independent and identical failure distribution F(t). Then the mean time to system failure is Z 1 μn ¼ ½1  FðtÞn  dt ðn ¼ 1; 2; …Þ; ð37Þ 0

increases strictly with N, and ðNc2 þ c3  c2 Þ=N decreases strictly with N. Substituting x in (32) into (31),  N  mNc2 þ c3  c2 X BN þ 1 1=m Z m ðN ¼ 1; 2; …Þ: ð33Þ Nc2 þc3 c2 k ¼ 1 Bk

xn ¼

p. 196] as m ¼2.0, Nn are the same as those in Table 7.5 and T nk are close to those in Table 7.5. We compute λT n1 þ λT n2 ¼ 1:284 for c3 =c2 ¼ 2:0 and λT n1 þ λT n2 þ λT n3 ¼ 2:190 for c3 =c2 ¼ 5:0, which are also close to 1.290 and 2.200 computed from Table 7.5 [11, p. 196].

6. Parallel system

1=m

Then, the expected cost rate in (28) is Cðx; NÞ

71

ð34Þ

ðk ¼ 1; 2; …; N n Þ:

ð35Þ

Table 3 presents optimal Nn in (31) and approximate PM intervals λT nk in (34) for c3 =c2 and m when bk  1 þ k=ðk þ 1Þ ðk ¼ 1; 2; …Þ and HðtÞ ¼ ðλtÞm . Compare Table 3 with Table 7.5 [11,

and the expected cost rate is [3, p. 8] CðnÞ ¼ R 1 0

nc1 þ c2 ½1  FðtÞn  dt

ðn ¼ 1; 2; …Þ;

ð38Þ

where c1 is an acquisition cost for one unit and c2 is the cost for system failure. An optimal nn minimizing C(n) is given by a unique and minimum solution that satisfies R1 n c2 0 ½1 FðtÞ  dt nZ ðn ¼ 1; 2; …Þ: ð39Þ R1 R1 nþ1 n c1 ½1  FðtÞ  dt  ½1  FðtÞ  dt 0 0 When FðtÞ ¼ 1 e  ðλtÞ , it is much easier to compute approximate μ~ n in (36) rather than to compute exact μn in (37). Then, the expected cost rate is m

nc1 þ c2 C~ ðnÞ ¼ h i1=m Pn j ¼ 1 ð1=jÞ

ðn ¼ 1; 2; …Þ;

ð40Þ

and its approximate number n~ satisfies 1=m  Pn c2 j ¼ 1 ð1=jÞ 1=m  Pn 1=m n Z  Pn þ 1 c1  j ¼ 1 ð1=jÞ j ¼ 1 ð1=jÞ

ðn ¼ 1; 2; …Þ:

ð41Þ

Table 4 presents optimal nn in (39) and approximate n~ in (41) m for c2 =c1 and m when FðtÞ ¼ 1  e  t . It can be clearly shown that the calculation of μ~ n is much easier than that of μn, and n~ has good approximations for the exact nn. Table 4 Optimal nn and approximate n~ when FðtÞ ¼ 1 e  ðλtÞ . m

Table 3 Optimal Nn and approximate PM intervals λT nk when c1 =c2 ¼ 3. 1.5

c3 =c2

2.0

5.0

2.0

5.0

2.0

5.0

1 1.121

2 1.587 1.211

2 0.707 0.577

3 0.882 0.720 0.588

3 0.606 0.529 0.446

4 0.693 0.606 0.511 0.424

n

N λT n1 λT n2 λT n3 λT n4

2.0

c2 =c1

m

3.0

m¼ 1.5 n

1.0 2.0 5.0 10.0 20.0 50.0 100.0

n

1 2 3 4 7 13 22

m¼ 2.0 n~

n

1 1 3 4 7 13 22

1 1 2 3 5 10 17

n

m¼ 3.0 n~

nn

n~

1 1 2 3 5 10 17

1 1 2 3 4 8 13

1 1 1 2 4 7 12

72

X. Zhao et al. / Reliability Engineering and System Safety 144 (2015) 68–73

7. Inspection policy

8. Conclusions

A unit operates for an infinite time span and is checked at times T k ðk ¼ 1; 2; …Þ, where T 0  0. The unit has a failure distribution F (t) whose failure rate h(t) remains unchanged by any check. Any failure is detected at the next checking time and is replaced immediately. Then, the total expected cost until replacement is [2, p. 108; 11, p. 203]

We have already known that when the failure time X has a distribution PrfX r tg  1  e  HðtÞ , H(X) has an exponential distribution with mean 1 [11, p. 6]. That is, objective models might be much simpler by using H(t) skilfully for reliability systems. As expected previously, the proposed approximate models in this paper for failure times, age and periodic replacement, and sequential maintenance policies have been given by the simpler forms and their optimal solutions have shown good approximations for the exact policies. It has been clearly shown that calculation of sums is much easier than that of integrals for the MTTF of a parallel system. From the point of accuracy, we have shown numerically that the approximate optimal number of units is almost the same as the exact solutions computed from the original model. Not only that, when n becomes much larger, we may use the asymptotic MTTF proposed in [13] for modeling. That is, μ~ n could be approximately given by ðln n þ γ Þ=λ for the exponential failure distribution and ðln n þ γ Þ1=m =λ for the Weibull failure distribution, where γ  0:5772156649⋯ is Euler's constant [5]. The above discussion would provide easier form to estimate MTTF and to model replacement policies for large-sized parallel systems. There have been some approximate calculations of sequential inspection times [11, p. 207], however, the approximate method proposed in this paper is the simplest among these ones and also shows good approximations. Further, when the optimal inspection times are computed by using algorithm [11, p. 203], the first step is to estimate an initial inspection time T1. It is shown in Table 5 that T~ 1 ¼ ð1=λÞðλT n Þ1=m is a little less than T n1 , that is, it would be used sufficiently in practical fields that T~ 1 could be considered as one estimation of the first inspection time to begin the algorithm. We have known that, with the rapid development of the oil and gas industry in Qatar, parallel pipelines are increasingly constructed and inspections for damages suffered from shocks and corrosions are usually scheduled to prevent incidents where existing pipelines have been damaged. The approximate models proposed in this paper not only would provide the researchers with new thinkings of complicated systems and computations in reliability theory, but also would let the engineers estimate reliability measures, inspection times, and maintenance times with simple ways.

1  X

CðT 1 ; T 2 ; …Þ ¼

 c1 þ c2 ðT k þ 1  T k Þ F ðT k Þ  c2 μ;

ð42Þ

k¼0

where c1 is the cost of one check, c2 is the loss cost per unit of time for the elapsed time interval between a failure and its detection R1 and μ  0 F ðtÞ dt o 1. Differentiating CðT 1 ; T 2 ; …Þ with Tk and setting it equal to zero, Tkþ1 Tk ¼

FðT k Þ FðT k  1 Þ c1  f ðT k Þ c2

ðk ¼ 1; 2; …Þ;

ð43Þ

whose inspection times T nk have been obtained analytically by using Algorithm 1 in [11, p. 203]. However, it is troublesome to compute Algorithm 1 numerically even though some improvement have been summarized [11]. We apply the above approximate MTTF in (36) to the inspection policy: suppose that when the failure time has an exponential distribution FðtÞ ¼ 1  e  λt , the unit is checked at periodic times kT ðk ¼ 1; 2; …Þ. Then, the expected cost rate is [11, p. 204] CðTÞ ¼

c1 þc2 T c2  ; 1  e  λT λ

ð44Þ

and an optimal Tn which minimizes it is given by a unique solution of the equation eλT  ð1 þ λTÞ ¼

λc 1 c2

:

ð45Þ

Next, when the failure time has a Weibull distribution FðtÞ ¼ 1 m e  ðλtÞ ðm Z 1Þ, approximate inspection times T~ k which minimize CðT 1 ; T 2 ; …Þ are given by 1 T~ k  ðkλT n Þ1=m

λ

ðk ¼ 1; 2; …Þ:

ð46Þ

Table 5 presents optimal T nk by using Algorithm 1 [11, p. 203] and approximate T~ k in (46) when 1=λ ¼ 100:0, c1 =c2 ¼ 10:0 and k ¼ 1; 2; …; 10. Taking m ¼ 1.5 as an example, although the approximated values fT~ 1 ; T~ 2 ; …; T~ 10 g have big difference from the exact ones fT n1 ; T n2 ; …; T n10 g, we can compute CðT~ 1 ; T~ 2 ; …; T~ 10 Þ=c2 ¼ 45:592 which is close to CðT n1 ; T n2 ; …; T n10 Þ=c2 ¼ 45:603. That is, these inspection times show good approximations for those computed by Algorithm 1.

Acknowledgments This paper was made possible by a Postdoctoral Research Award PDRA1-0116-14107 from the Qatar National Research Fund (a member of The Qatar Foundation). References

Table 5 Optimal T nk and approximate T~ k when 1=λ ¼ 100 and c1 =c2 ¼ 10. k

1 2 3 4 5 6 7 8 9 10

m¼ 1.5

m ¼3.0

m¼2.0

T nk

T~ k

T nk

T~ k

T nk

T~ k

56.984 94.453 127.582 158.146 186.807 213.804 239.074 262.225 282.192 298.074

55.746 88.492 115.957 140.472 163.003 184.070 203.993 222.985 241.200 258.751

68.141 101.502 128.994 153.281 175.407 195.889 214.972 232.656 248.594 261.790

64.515 91.238 111.744 129.030 144.260 158.029 170.691 182.477 193.546 204.015

81.898 108.279 128.270 144.982 159.600 172.727 184.717 195.779 205.999 215.225

74.663 94.070 107.683 118.521 127.673 135.672 142.826 149.327 155.306 160.858

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