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Generalized geometric process and its application in maintenance problems
Accepted Manuscript
Generalized geometric process and its application in maintenance problems Guan Jun Wang, Richard C.M. Yam PII: DOI: Reference:
S0307-904X(17)30341-4 10.1016/j.apm.2017.05.024 APM 11776
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
15 August 2016 29 April 2017 11 May 2017
Please cite this article as: Guan Jun Wang, Richard C.M. Yam, Generalized geometric process and its application in maintenance problems, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.05.024
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highlights • A generalized geometric process is proposed for maintenance models.
• A sequential PM model is also discussed.
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• An age-dependent PM model is constructed based on GGP.
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• The optimal policies are studied for the models theoretically and numerically.
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Generalized geometric process and its application in maintenance problems Guan Jun Wanga,∗, Richard C.M. Yamb Department of Mathematics, Southeast University, Nanjing 210096, China Department of SEEM, City University of Hong Kong, Hong Kong, China
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a
b
Abstract
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Since the repair effect may be varying with the number of repairs, we propose a generalized geometric process (GGP) to model the deteriorating process of repairable systems. For a GGP, the geometric ratio changes with the number of repairs rather than being a constant. Based on the GGP, two repair-replacement models are studied. Existing preventive maintenance (PM) models based on geometric process (GP) commonly assume that the PM is ‘as good as new’ in each working circle, which is not realistic in many situations. In this study, that the system is assumed to be geometrically deteriorating after PM or corrective maintenance (CM). Firstly, an age-dependent PM model is considered, in which the optimal policies N ∗ and T ∗ are obtained theoretically, and the optimal bivariate policy (N ∗ , T ∗ ) which minimizes the average cost rate (ACR) can be determined by a searching algorithm. Next, because of the fact that the system deteriorates after maintenance, the schedule time to PM should decrease with the maintenance number increasing. Therefore, a sequential PM policy is investigated, and the optimal policy N ∗ and the optimal schedule times T1∗ , T2∗ , · · · , TN∗ ∗ are computed. Finally, numerical examples are provided to illustrate the proposed models.
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Keywords: generalized geometric process; preventive maintenance; replacement; average cost rate; optimal policy .
∗
Corresponding author Email address: [email protected] (Guan Jun Wang)
Preprint submitted to Applied Mathematical Modelling
May 17, 2017
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1. Introduction 1.1. Background and Motivation
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In engineering problems, most repairable systems deteriorate gradually with the increasing of the number of repairs. The consecutive operating times of a system after repair are decreasing, while the successive repair times after failure are increasing stochastically. For such phenomena, Lam [1] introduced a geometric process (GP) to model it. In [1], Lam studied two kinds of replacement policies, in which one is based on the working age T of the system, and the other is based on the failure number N of the system. In a GP maintenance model, the mean lifetime sequence of a system geometrically decreases with the repair number increasing, which means that the system uniformly degraded after repairs. This maybe not always true, and the degradation level of a system after repair can be different since the effect of a repair can be affected by many factors, such as the repair being done by different teams, varying damage degree of the system at failure, different budgets for repair, etc. If we continue to model the lifetimes by using a monotone process, the problem is how to modify the GP. Some generalizations of the GP had been done by researchers. For example, Castro and Sanjuan [2] proposed a power process to extend the GP, which can be used not only for describing monotone sequence of random variables with continuous distribution, but also for random sequences with discrete distribution. Braun et al. [3] suggested an α-series process to generalize the GP. They studied the properties of GP and the α-series process, and their applications in reliability. But these generalizations are not very easy to use in maintenance problems due to their computational complexity. Zhang and Wang [4] presented an extended GP maintenance model, in which whether the system after repair deteriorates or not depends on a Bernoulli random variable with successive probability p. In fact, it is a mixture of the GP maintenance model and the perfect maintenance model. In this paper, our first aim is to propose a generalized geometric process (GGP) to model the non-homogenous degradation process of system after maintenance. In applications, preventive maintenance (PM) is usually adopted for improving the system reliability/availability or reducing the cost of system running (Barlow and Hunter [5]). A PM can be perfect or imperfect. A perfect PM lead to a system ‘as good as new’, while an imperfect PM makes the system status better than that just before failure, but not as good as new. Age-dependent PM policy is one of the most important PM policy, under which a system is preventively maintained at its age T or correctively repaired at failure, whichever occurs first, where T is a constant. Similar to age-dependent PM policy, 3
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periodic PM policy is also a basic PM policy in applications. Under periodic PM policy, the system considered being preventively maintained at fixed time intervals kT (k = 1, 2, · · ·), and minimal repaired at failure. Sequential PM policy is another extensively studied policy, under which a system is preventively maintained at unequal time intervals. For an imperfect PM model, sequential PM policy seems more economical for systems with repair degradation, while age-dependent PM policy and periodic PM policy are easy to manipulate in applications. A survey of PM policy can be found in Wang [6]. Under the framework of GP maintenance model, PM policy also was discussed in several papers. But a usual assumption is that the PM is ‘as good as new’ [7]. For some systems, such as aircraft, military systems, the system after PM may be worse and worse. As a monotone process, GGP has a natural advantage in describing the degradation process of system after repair. This motivate us consider the PM policy based on GGP. Our second aim is to develop the optimal age-dependent and sequential imperfect PM policies based on GGP. 1.2. Related literature
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Age-dependent PM policy is an extension of age replacement policy proposed by Barlow and Hunter [5]. Based on the concepts of imperfect maintenance, various of age-dependent PM policies were studied in the literature [6]. For example, Nakagawa [8] introduced an age replacement policy in which the system is replaced at time T or at failure number N , whichever occurs first, and undergoes minimal repair at failure between replacement. Sheu et al.[9] generalized the age replacement policy by assuming two types of failures of a system, the system is replaced at either the first type 2 failure or the nth type 1 failure, or at age T , whichever occurs first. A periodic replacement model was investigated by Nakagawa [10], in which the system is replaced at periodic times kT (k = 1, 2 · · ·) and minimal repaired at failure. In Ref. [10], Nakagawa introduced improvement factors in failure rate function and effective age to describe the effectiveness of imperfect PMs. As an extension of the periodic PM model, a sequential PM model was investigated in [11]. On the basis of Nakagawa’s PM model, Zequeira and Brenguer [12] considered a periodic imperfect PM policy for a system with two categories of competing failure modes. Sheu and Chang [13] presented a periodic imperfect PM model of a system subjected to shocks, in which the optimal PM policy T ∗ and the optimal replacement policy N ∗ were analyzed. At the same time, a generalized sequential imperfect PM was discussed by Lin [14], in which the system has maintainable and non-maintainable failure modes, the PM can only improve the maintainable failure rate and has noting to do with the non-maintainable failure rate. In Sheu [15], another generalized sequential imperfect PM model was proposed, in which 4
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the maintainable and non-maintainable failure occurs with probability pi and 1 − pi , where i is the working cycles of the system, and the optimal policy was derived theoretically and numerically. Besides, as a focus topic in maintenance theory, many new PM models have also been developed recently. For instance, Zhong and Jin [16] researched a PM model for a cold standby system consisting of two components based on the semi-Markov process theory, and the optimal PM interval is obtained by maximizing the mean time to failure of the system. Lin et al [17] developed a non-periodic reliability-based PM models for a repairable deteriorating system, in which PM is performed when the reliability of the system drops below a given threshold. Taking the reliability threshold and the PM number as the decision variables, the optimization problem is solved by minimizing the average maintenance cost rate. More PM models can be found in Refs. [18-21]. Since the GP model was introduced by Lam [1], applications of the GP in maintenance problems have been greatly developed [22-27]. About PM policy based on GP, Zhang [7] constructed a PM model based on GP, in which the system is preventively maintained at periodic time points, and is replaced at its N th failure. In Zhang’s model, the PM is ’as good as new’ in each working stage. Lam [28] generalized Zhang’s work by studying the optimal bivariate policy (T, N ) for a system with PM, where T is the time interval of PM and N is the number of failures at which the system is replaced. As an extension of Lam’s work, Wang and Zhang [29] investigated a bivariate replacement policy (R, N ), where R is the critical reliability of the system for PM and N is the system failure number. Moreover, Wang and Zhang [30] considered a repairable system with periodic inspection, and based on the results of system inspection, PM or corrective Maintenance (CM) is conducted. This model is also an generalization of the model in Lam [28] by introducing state inspection technique for the system. 1.3. Overview
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The remainder of this paper are structured as follows. The coming section introduces definitions of GGP and its properties. In Section 3, an age-dependent PM model is considered, the existence and uniqueness of the optimal policies N ∗ and T ∗ are studied theoretically, and an algorithm for the optimal policy (N ∗ , T ∗ ) is presented. Section 4 focuses on a sequential PM model, and the optimal policy (N ∗ , T1∗ , · · · , TN∗ ∗ ) is discussed. Numerical examples are provided in Section 5, and conclusions are given in section 6.
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2. Notations and definition of GGP 2.1. Notations
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lifetime of a system after the (n − 1)th repair cumulative distribution function of Xn cumulative distribution function of X1 probability density function of X1 failure rate function of X1 mean of X1 , i.e., λ = EX1 mean of Xn geometric ratio of the nth repair = a0 a1 · · · an−1 planned time interval for PM in the age-dependent PM model number of maintenances before system replacement PM cost CM cost replacement cost = min{Xn , T } cumulative distribution function of ξn ACR of the age-dependent PM model under policy (N, T ) optimal policy of the age-dependent PM model the nth time interval of the sequential PM model = min{Xn , Tn } cumulative distribution function of ζn ACR of the sequential PM model under policy (N, T1 , · · · , TN ) optimal policy of the sequential PM model
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Xn Fn (t) F (t) f (t) r(t) λ λn an An T N cp cf cr ξn Hn (t) C(N, T ) (N ∗ , T ∗ ) Tn ζn Kn (t) C(N, T1 , · · · , TN ) (N ∗ , T1∗ , · · · , TN∗ )
2.2. Definition of GGP
For easy reference, we first state the definition of the GP as follows. Definition 1. Assume that {Zn , n = 1, 2, . . .} is a sequence of independent nonnegative random variables. If the distribution function of Zn is Gn (t) = G(an−1 t), for n = 1, 2, . . . , where G(t) is the cumulative distribution function of Z1 and a > 0 is a constant, then {Zn , n = 1, 2, . . .} is called a GP, and a is the ratio of the GP. 6
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Clearly, if a > 1, then the GP {Zn , n = 1, 2, . . .} is stochastically decreasing. If 0 < a < 1, then the GP {Zn , n = 1, 2, . . .} is stochastically increasing.
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If a = 1, then the GP {Zn , n = 1, 2, . . .} is a renewal process.
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From the definition of GP, the ratio a is a constant for n = 1, 2, . . .. In the maintenance background, this means that after maintenance, the lifetime of a system uniformly degrades as the maintenance number increasing. Practically, the effect of maintenance to the system may be different with the varying number of maintenance. Therefore, we propose the following GGP to model it. Definition 2. Assume that {Zn , n = 1, 2, . . .} is a sequence of independent nonnegative random variables. If the distribution function of Zn is Zn (t) = G(a0 a1 · · · an−1 t) for n = 1, 2, . . . , where G(t) is the cumulative distribution function of Z1 , a0 = 1, and a1 , a2 , · · · are positive constants, then {Zn , n = 1, 2, . . .} is called a GGP, and a1 , a2 , · · · are the ratios of the GGP. If an ≥ 1 for all n = 1, 2, · · ·, then the GGP {Zn , n = 1, 2, . . .} is stochastically decreasing.
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If an ≤ 1 for all n = 1, 2, · · ·, then the GGP {Zn , n = 1, 2, . . .} is stochastically increasing.
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Remark 1. (a) Denote An = a0 a1 · · · an−1 , then it can be written that Gn (t) = G(An t), n = 1, 2, . . .. If a1 = a2 = · · · = a is a constant, the GGP reduces to the GP, and An = an−1 . (b) For a decreasing GGP {Zn , n = 1, 2, · · ·}, a typical assumption is that 1 ≤ a1 ≤ a2 ≤ · · ·. If {Zn , n = 1, 2, · · ·} is a sequence of working times of a system after maintenance, then the assumption 1 ≤ a1 ≤ a2 ≤ · · · implies that the system deteriorates more and more rapidly with the maintenance number increasing. If {Zn , n = 1, 2, · · ·} is a sequence of maintenance time of a system, which follows increasing GGP with ratios 1 ≥ a1 ≥ a2 ≥ · · ·, then the maintenance time stochastically increases more and more rapidly with the maintenance number increasing. About the parameter estimation problem of the GGP, we have the following remarks.
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(r)
Remark 2. (a) Suppose that {Zk , k = 1, 2, · · · , n; r = 1, 2, · · · , m} is a set of failure data collected from the operating processes of m identical machines with maintenance, and (r) each machine is renewed at its nth failure. To test whether {Zk , k = 1, 2, · · · , n} yields a GGP, we first construct the following sequences of random variables: (r)
(r)
= Zk /Zk+1 , r = 1, 2, · · · , m; k = 1, 2, · · · , n − 1.
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(r)
Yk
(r)
(r)
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It can be seen that {Zk , k = 1, 2, · · · , n; } forms a GGP if and only if Yk , r = 1, 2, · · · , m is a sequence of independent and identical distributed (i.i.d.) random variables for each (r) k ∈ {1, 2, · · · , n − 1}. For k = 1, 2, · · · , n − 1, to test random variables Yk , r = 1, 2, · · · , m are i.i.d. or not, the ‘turning point test’ and the ‘difference-sign test’ suggested by Lam [31] (in page 104) can be applied. (r) (b) If {Zk , k = 1, 2, · · · , n} is tested to be agree with a GGP, the next problem is to estimate the geometric ratios ak , k = 1, 2, · · · , n − 1. Since the random variables (r) Yk , r = 1, 2, · · · , m are i.i.d. for each k ∈ {1, 2, · · · , n − 1}, a straightforward estimator of P (r) (r) (r) ak is Y¯k = m r=1 Yk /m. Moreover, because EZk = ak EZk+1 for each k ∈ {1, 2, · · · , n−1}, P (r) (r) Pm another estimator of ak can be given by a ˆk = µ ˆk /ˆ µk+1 = m r=1 Zk+1 . It is obr=1 Zk / vious that the latter estimator is more robust than the former one.
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Assume that {Zn , n = 1, 2, . . .} is a GGP, and the variable Z1 has the distribution function G(t) and the probability density function g(t), then the probability density function of Zn is
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gn (t) =
dGn (t) = An g(An t). dt
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The mean of Zn is
µn = E[Zn ] =
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where µ = EZ1 . Denote Sn =
Z
0
n P
∞
t · An g(An t)dt =
µ , n = 2, 3, · · · An
Zi , then Sn is the total working time of a system before the nth
i=1
maintenance, or the total repair time of the first n repairs. Proposition 1. (i) If {Zn , n = 1, 2, . . .} follows a GGP with ratios satisfying ak ≥ δ, k = 1, 2, · · ·, where δ > 1 is a constant, then lim E[Sn ] =
n→∞
∞ X n=1
E[Zn ] ≤ 8
δµ < ∞. δ−1
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(ii) If {Zn , n = 1, 2, . . .} follows a GGP with ratios satisfying ak ≤ 1, k = 1, 2, · · ·, then n→∞
∞ X n=1
E[Zn ] = ∞.
Proof. The proof is trivial, and is omitted here. 3. Age-dependent PM policy
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lim E[Sn ] =
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In this section, an age-dependent PM model is studied based on the GGP for a repairable system. The long-run average cost rate (ACR) is derived and the optimal maintenance policies are discussed. 3.1. Assumptions and model analysis
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Some assumptions are given in the following to describe the age-dependent PM model. Assumption 1. A new system is beginning to work at t = 0. The system is preventively maintained at a planned time T since the last maintenance, or undergoes CM at failure, whichever occurs first. Assumption 2. The lifetime sequence {Xn , n = 1, 2, . . .} of the system after maintenance (PM or CM) forms a GGP. That is, the distribution function of Xn is Fn (t) = F (An t), n = 1, 2, . . ., where An = a0 a1 · · · an−1 . Accordingly, the mean of Xn is λn = λ/An , where λ = EX1 . The maintenance time of the system is neglected. Assumption 3. The replacement policy N based on the number of maintenances of the system is used. That is, the system is replaced by a new one at time of the N th maintenance. The replacement time is negligible. Assumption 4. The cost for a CM is cf , the cost for a PM is cp , and the cost for replacement of a new system is cr , where cp < cf < cr .
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A possible course of system running is plotted in Fig. 1.
Remark 3. If a1 = a2 = · · · = 1, then the model reduces to the classical age replacement model.
Let ξn , n = 1, 2, . . . , N − 1, be the time between the (n − 1)th and the nth maintenance, and ξN be the time between the (N − 1)th repair and the replacement of the system, then
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and the survival function of ξn is ¯ n (x) = 1 − Hn (x) = H
(
F¯ (An x), x < T . 0, x > T
Therefore, the mean of ξn can be computed as Z ∞ Z ¯ Hn (x)dx = E[ξn ] =
T
0
F¯ (An x)dx.
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ξn = min{Xn , T }, n = 1, 2, · · · , N . The distribution function of ξn is ( F (An x), x < T Hn (x) = P (ξn ≤ x) = , 1, x > T
(1)
(2)
(3)
Denote pn the probability that the nth maintenance is a CM, then pn = P (Xn < T ) = F (An T ).
(4)
Let U (N, T ) be the total operating time of the system before replacement, we have
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The mean of U (N, T ) is
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U (N, T ) = ξ1 + · · · + ξN .
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E[U (N, T )] = E[ξ1 ] + · · · + E[ξN ] =
N Z X
T
F¯ (An x)dx.
(5)
0
n=1
E[W (N, T )] =
N −1 X n=1
[cf pn + cp (1 − pn )] + cr
= (N − 1)cp + cr + (cf − cp )
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The mean total cost for the system operating during a renewal cycle is
N −1 X
F (An T ).
(6)
n=1
By applying the renewal reward theorem Ross [32], we obtain the long-run ACR of the system as
C(N, T ) =
(N − 1)cp + cr + (cf − cp )
NP −1 n=1
N R P T ¯ F (An x)dx 0
n=1
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F (An T ) .
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Remark 4. (a) When T → +∞, no PM is performed. In this case, pn = 1 for n = 1, 2, · · ·, and the long-run ACR of the system becomes C(N, +∞) =
(N − 1)cf + cr (N − 1)cf + cr = . N R N P P ∞ ¯ 1 F (An x)dx λ An 0
n=1
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(b) When N → +∞, no replacement is performed. If lim An = ∞, we have lim F (An T ) = n→∞ n→∞ RT 1 and lim 0 F¯ (An x)dx = 0 for fixed T > 0. Furthermore, n→∞
1 [(N N
C(+∞, T ) = lim
N →∞
− 1)cp + cr + (cf − cp )
F (An T )]
n=1
= +∞.
N R P T ¯ F (An x)dx] 0
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NP −1
n=1
The objective is to determine the optimal policy (N ∗ , T ∗ ) such that the ACR C(N, T ) is minimized. 3.2. Optimal N ∗
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Supposing that the PM time interval T > 0 is fixed, the aim is to find N ∗ such that the ACR C(N, T ) is minimized. It is not difficult to derive that C(N + 1, T ) > C(N, T ) and C(N, T ) < C(N − 1, T ) if and only if L(N, T ) > cr and L(N − 1, T ) < cr ,
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where
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N R P T ¯ F (An x)dx [cp + (cf − cp )F (AN T )] 0 n=1 L(N, T ) = − (N − 1)cp RT ¯ (AN +1 x)dx F 0
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−(cf − cp )
N −1 X
F (An T ).
n=1
We can see that L(N, T ) is increasing in N from the following difference. L(N, T ) − L(N − 1, T )
N
cp + (cf − cp )F (AN T ) cp + (cf − cp )F (AN −1 T ) X = [ RT − ] RT F¯ (AN +1 x)dx F¯ (AN x)dx 0
0
> 0.
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About the optimal replacement policy N ∗ , a conclusion is given in Theorem 1. Theorem 1. Assume that lim AN /N = +∞. For fixed T > 0, we have N →∞ (i) If cp F¯ (T )+cf F (T ) < cr /a1 , then there exists a unique finite optimal replacement policy N ∗ which minimizes the cost rate function C(N, T ), and satisfies
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L(N ∗ , T ) > cr and L(N ∗ − 1, T ) < cr .
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(ii) If cp F¯ (T ) + cf F (T ) > cr , then the optimal replacement policy N ∗ = 1. The proof of Theorem 1 can be found in Appendix A. Remark 5. The condition lim AN /N = +∞ of Theorem 1 is very general. It does N →∞ not require an > 1 for all n = 1, 2, · · ·. Of course, if an = a > 1 for all n = 1, 2, · · ·, then the condition holds obviously. About the condition lim AN /N = +∞, some remarks are given as follows. N →∞
Remark 6. (a) We first write an as an = 1 + bn , where bn ≥ 0. A necessary condition ∞ P for lim AN /N = +∞ holding is that the sum of series bn is divergent. To see this, it N →∞
n=1
can be calculated that
n=1
an =
If the sum of the series
(1 + bn ) ≤
n=1
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∞ P
∞ Y
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∞ Y
∞ Y
e
bn
=e
n=1
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bn
.
bn is convergent, and assuming that
n=1
lim
N →∞
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∞ P
n=1
∞ P
bn = B > 0, then
n=1 N Q
an
n=1
N
Thus, we conclude that the sum of series
eB = 0. N →∞ N
≤ lim ∞ P
bn must be divergent.
n=1
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(b) Also write an as an = 1 + bn , where bn ≥ 0. A sufficient condition for lim AN /N =
+∞ holding is that the increasing speed of series { Notice that N Q
N P
n=1
N →∞
bn , N ≥ 1} is larger than ln N .
an X N N X ln n=1 = ln an − ln N ∼ bn − ln N. N n=1 n=1 12
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If lim
N P
N →∞ n=1
N Q
an
→ ∞, or
bn / ln N > 1, then ln n=1N
N Q
an
n=1
N
→ ∞, as N → ∞. The conclusion is
reached. (c)If an has the form of an = 1 + nr , where r > 0 is a constant. Then
n=1
an =
N Y
(1 +
n=1
r Γ(N + r + 1) 1 )= = , n Γ(N + 1)Γ(r + 1) B(N, r)
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AN +1 =
N Y
where Γ(·) and B(·) are Gamma function and Beta function, respectively. It is easy to see that B(N, 1) = N1+1 . Generally, for fixed r > 0 and large N , B(N, r) ∼ Γ(r) · N −r . Therefore, we have that 0, 0 < r < 1 = 1, r = 1 +∞, r > 1
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lim
N →∞
N Q
an
n=1
N
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(d)If an has the form of an = 1 + nrh , where r > 0, h > 0 are constants. Comparing the results in (c), it can be got that
lim
3.3. Optimal T ∗
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N →∞
N Q
an
n=1
=
N
(
0, 0 < h < 1 +∞, h > 1
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Supposing that the replacement policy N > 0 is fixed, we try to seek T ∗ which minimizes the ACR C(N, T ). To do this, differentiating C(N, T ) with respect to T , we can see that ∂C(N, T )/∂T = 0 if and only if
which is equivalent to
(cf − cp )
NP −1
An f (An T )
n=1
= C(N, T ),
N P F¯ (An T )
(11)
n=1
QN (T ) =
(N − 1)cp + cr , cf − cp
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where QN (T ) =
NP −1
An f (An T )
n=1
N R P T ¯ F (Ak x)dx 0
k=1
N P F¯ (An T )
−
=
n=1
An r(An T )F¯ (An T ) X N Z N P F¯ (An T )
k=1
0
n=1
F (An T )
n=1
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n=1 NP −1
N −1 X
T
F¯ (Ak x)dx −
N −1 X
F (An T )
(13)
n=1
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About the optimal PM policy T ∗ , results are given in Theorem 2. Theorem 2. Assume that the failure rate r(t) is increasing in t, and a1 > 1, ak ≥ 1, k ≥ 2. For given number N ≥ 1, the following results hold. (i) If r(∞) >
(N − 1)cf + cr , N P 1 (cf − cp )λ An
(14)
n=1
∗
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then there exists a finite optimal policy T which minimizes the ACR C(N, T ) and satisfies QN (T ∗ ) =
(N − 1)cp + cr . cf − cp
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Moreover, if QN (T ) is increasing in T , then the optimal policy T ∗ is unique. (ii) If r(∞) ≤
(N − 1)cf + cr , N P 1 (cf − cp )λ An
(16)
n=1
∗
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then the optimal policy T = +∞. The proof of Theorem 2 can be found in Appendix B. Remark 7. (a) If r(∞) = +∞, the condition (14) holds obviously, and a finite optimal policy T ∗ exists. (b) The condition a1 > 1, ak ≥ 1, k ≥ 2 in Theorem 2 can be relaxed by ‘there exists a number r ≥ 1 such that ar > 1 and ak ≥ 1, k 6= r’, which also guarantees equation (29), and thus the conclusions of Theorem 2 hold. Remark 8. Conditions (14) and (16) means that PM should be arranged if the failure rate r(t) is large enough for large t. If r(t) is bounded from above by a certain constant, PM should not be taken. The reason is that the lifetime approaches an exponential variable, which as a lifetime is ‘as good as new’ at any time before failure. 14
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3.4. Optimal bivariate policy (T ∗ , N ∗ ) In this subsection, we give the following algorithm to seek the optimal bivariate policy (T , N ∗ ). ∗
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Algorithm
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Input: cp , cf , cr , λ, r(·), ak , k = 1, 2, · · ·. Step 1. Let j = 1, and Nj = 1. Step 2. Let N = Nj , and find the solution Tj which satisfies (15). Step 3. Set T = Tj , and find the unique Nj+1 which satisfies (10). Step 4. If Nj+1 = Nj , then go to Step 5; otherwise, let j = j + 1, and go to Step 2. Step 5. T ∗ = Tj and N ∗ = Nj . Output: T ∗ , N ∗ and C(T ∗ , N ∗ ). Stop.
M
According to the algorithm, the optimal bivariate policy (T ∗ , N ∗ ) can be determined recursively. 4. Sequential PM policy
AC
CE
PT
ED
Since the lifetime sequence {Xn , n = 1, 2, . . .} after repair forms a decreasing GGP, the planned time to PM should be shorter and shorter with the repair number increasing. Therefore, in this section, we propose a sequential PM model to optimize the running process of the system. The model assumptions are the same as that given in Section 3 except for Assumption 1, which is replaced by Assumption 10 as follows: Assumption 10 . The system is preventively maintained at a planned time Tn after the (n − 1)th repair, or undergoes CM at failure, whichever occurs first, n = 1, 2, · · ·. Similar to the age-dependent PM model, the system is replaced by a new one at the N th maintenance time. The problem is to determined the optimal number N ∗ and the optimal planned time intervals {Tn∗ , n = 1, 2, . . . N ∗ } such that the long-run ACR is minimized. Letting {ζn , n = 1, 2, . . .} be the sequence of actual operating time between maintenance, we have ζn = min{Xn , Tn } and Xn ∼ F (An x) for n = 1, 2, . . ., where An = a0 a1 · · · an−1 . The distribution function of ζn is ( F (An x), x < Tn Kn (x) = P (ζn ≤ x) = (17) 1, x ≥ Tn 15
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The survival function of ζn is F¯ (An x), x < Tn 0, x ≥ Tn
Therefore, the mean of ζn can be computed as Z Z ∞ ¯ Kn (x)dx = E[ζn ] =
0
0
Tn
CR IP T
¯ n (x) = 1 − Kn (x) = K
(
F¯ (An x)dx.
(18)
Denote pn the probability that the nth maintenance is a CM, then pn = P (ζn = Xn ) = P (Xn ≤ Tn ) = F (An Tn ),
(19)
AN US
and the probability that the nth maintenance is a PM is qn = 1 − pn = F¯ (An Tn ).
M
For fixed planned times {Tn , n = 1, 2, . . . N }, let U (N, T1 , · · · , TN ) and W (N, T1 , · · · , TN ) be the total operating time and the total cost of the system operating during a renewal cycle. We have U (N, T1 , · · · , TN ) = ζ1 + · · · + ζN .
ED
The mean of U (N, T1 , · · · , TN ) is
PT
E[U (N, T1 , · · · , TN )] = E[ζ1 ] + · · · + E[ζN ] =
N Z X n=1
Tn
F¯ (An x)dx.
(20)
0
Similarly, the mean total cost of the system is
AC
CE
E[W (N, T1 , · · · , TN )] =
N −1 X n=1
[cf pn + cp (1 − pn )] + cr
= (N − 1)cp + cr + (cf − cp )
N −1 X
F (An Tn ).
(21)
n=1
By applying the renewal reward theorem Ross [32], the long-run ACR of the system can be derived as
C(N, T1 , · · · , TN ) =
(N − 1)cp + cr + (cf − cp )
NP −1 n=1
N R P Tn ¯ F (An x)dx 0
n=1
16
F (An Tn ) .
(22)
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The aim is to determine the optimal policy N ∗ and T1∗ , T2∗ , · · · , TN∗ ∗ such that the ACR C(N, T1 , · · · , TN ) is minimized. By the method similar to Nakagawa [11], a necessary condition for T1 , T2 , · · · being the optimal planned time intervals is that ∂C/∂Tn = 0 for each n. Differentiating C(N, T1 , · · · , TN ) with respect to Tn , and setting it equal to zero, it can be obtained that (cf − cp )An f (An Tn ) = C(N, T1 , · · · , TN ), n = 1, 2, · · · , N, F¯ (An Tn )
(23)
A1 r(A1 T1 ) = A2 r(A2 T2 ) = · · · = AN r(AN TN ).
(24)
AN US
which implies
M
As a function of N , denote AN r(AN TN ) by ∆(N ). When the failure rate is strictly increasing, the computing procedure for obtaining the optimal planned time intervals can be specified as follows: 1. Solving An r(An Tn ) = ∆(N ), n = 1, 2, · · · , N , one gets that Tn = ψn (∆(N )), which is a function of ∆(N ). 2. Substitute Tn into (23) and solve it with respect to ∆(N ).
ED
3. Determine N ∗ by minimizing ∆(N ), N = 1, 2, · · ·.
PT
4. Calculating Tn∗ from the relation Tn∗ = ψn (∆(N ∗ )), n = 1, 2, · · · , N ∗ .
CE
Using the computing procedure given above, the optimal policy (N ∗ , T1∗ , T2∗ , · · · , TN∗ ∗ ) can be determined analytically or numerically.
AC
5. Numerical examples 5.1. Numerical example for age-dependent PM model
As an example, suppose that the lifetime X1 of the system is Weibull distributed, i.e. β F (t) = 1 − e−(αt) , t > 0, where α > 0, β > 1 are constants. Then the failure rate of X1 is r(t) = βαβ tβ−1 , which is strictly increasing in t. The mean lifetime of the system is EX = α−1 · Γ(1 + β). For simulation, the parameters can be set as cf = 300, cp = 100, cr = 2000, α = 0.01, β = 2.5, an = 1 + 0.01n. In this case, we get EX = 332.34. It is easy to verify that the 17
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M
AN US
CR IP T
conditions of Theorem 1 and 2 are satisfied, and therefore the optimal policies N ∗ and T ∗ can be determined based on Theorem 1 and 2, respectively. Furthermore, by using the algorithm given in Subsection 3.4, the bivariate optimal (N ∗ , T ∗ ) can also be determined numerically. To begin with, setting the initial value N = 2, we can get the optimal policy T ∗ = 220. Next, let the time interval between PM is T = 220, it can be obtained that the optimal policy N ∗ = 11. Fixing N = 11, numerical results show that the optimal policy is T ∗ = 180. Finally, the setting T = 180 leads to the optimal policy N ∗ = 11, from which we can see that the optimal bivariate policy is (N ∗ , T ∗ ) = (11, 180), and the corresponding ACR is 6.1244. Some more numerical results based on (7) can be seen from Table 1, and the surface for the ACR C(N, T ) is plotted in Fig. 2. From Table 1 and Fig. 2, it can be seen that the optimal policy (N ∗ , T ∗ ) uniquely exists, and it is the same as the result obtained from the searching algorithm. In table 2, the optimal policies are presented for varying corrective repair cost cf and replacement cost cr . From Table 2, one can see that both the optimal policies N ∗ and T ∗ decrease with the increasing of the cost cf , and N ∗ increases with the increasing of cr . But the optimal policy T ∗ is almost unchanged with the increasing of cr , which implies that T ∗ is mainly determined by the cost of PM and CM.
ED
Table 1 Some results obtained for C(T, N ), where (N ∗ , T ∗ ) = (11, 180) is the optimal policy with the minimal ACR 6.1244
AC
140 13.2778 7.5785 6.6451 6.2611 6.1810 6.1523 6.1646 6.2106 6.3838 6.7980
PT
110 14.0995 7.9495 6.9127 6.4659 6.3649 6.3208 6.3224 6.3614 6.5280 6.9435
CE
N/ T 2 5 7 9 10 11 12 13 15 18
160 13.1000 7.5092 6.6007 6.2300 6.1537 6.1276 6.1415 6.1885 6.3623 6.7758
170 13.0610 7.4966 6.5939 6.2259 6.1504 6.1246 6.1388 6.1859 6.3597 6.7729
180 13.0397 7.4913 6.5918 6.2252 6.1500 6.1244 6.1386 6.1857 6.3595 6.7724
190 13.0290 7.4898 6.5921 6.2262 6.1510 6.1255 6.1397 6.1867 6.3603 6.7730
200 13.0242 7.4903 6.5934 6.2277 6.1525 6.1269 6.1411 6.1880 6.3615 6.7741
220 250 13.0222 13.0236 7.4928 7.4960 6.5965 6.5997 6.2307 6.2337 6.1555 6.1583 6.1298 6.1324 6.1438 6.1463 6.1906 6.1930 6.3639 6.3661 6.7763 6.7783
Table 2 The optimal policies for different repair and replacement costs 18
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C(N ∗ , T ∗ ) 3.7794 4.1961 4.8640 6.1244 7.3217 8.4760
(cp , cf , cr ) (N ∗ , T ∗ ) (100,300,500) (6,170) (100,300,1000) (9,170) (100,300,1500) (10,170) (100,300,2000) (11,180) (100,300,2500) (12,180) (100,300,3000) (13,180)
C(N ∗ , T ∗ ) 3.9598 4.8060 5.4939 6.1244 6.7191 7.2928
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(cp , cf , cr ) (N ∗ , T ∗ ) (100,120,2000) (15,230) (100,150,2000) (14,200) (100,200,2000) (13,190) (100,300,2000) (11,180) (100,400,2000) (10,170) (100,500,2000) (9,170)
5.2. Numerical example for sequential PM model
∆(N )
M
Tn = (
AN US
To illustrate the sequential PM model, we also assume that the system lifetime X1 β yields Weibull distribution with F (t) = 1 − e−(αt) , t > 0 with α > 0, β > 1, and EX = α−1 Γ(1 + β). It is obvious that the failure rate r(t) = βαβ tβ−1 is strictly increasing in t for β > 1. By applying the algorithm given in Section 4, we need to solve the equation An r(An Tn ) = An βαβ (An Tn )β−1 = ∆(N ) with respect to Tn , n = 1, 2, · · · , N . It is not difficult to derive that βαβ Aβn
1
) β−1 , n = 1, 2, · · · , N
ED
Taking Tn into (23), we can obtain that
PT
∆(N ) =
(N − 1)cf + cr − (cf − cp ) (cf − cp )
∆(N )
1
NP −1
N R ( β β ) β−1 P βα An
n=1
0
e
∆(N ) n
β
−( αβA ) β−1
n=1
.
e−(αAn x)β dx
AC
CE
It can be seen that ∆(N ) does not has explicit expression. Therefore, we have to seek the optimal N ∗ by minimizing ∆(N ) numerically. Table 3. Some results obtained for the optimal policy (N ∗ , T1∗ , · · · , TN∗ ∗ ) under different replacement cost cr .
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C∗ 3.6657
1000
10
4.5427
1500
11
5.2798
2000
12
5.9457
2500
13
6.5673
3000
14
7.1611
4000
15
8.2902
T1∗ 113.8006 51.5635 172.3281 60.0344 225.2838 67.4145 272.9008 74.1112 316.6733 80.3407 357.6012 86.2279 433.2030 97.2721 26.3431
··· 94.6755
TN∗ ∗ 84.7117
128.5043 52.4398 158.4838 58.4991 186.2850 64.0064 212.3208 69.1273 236.8874 73.9632 282.6040 83.0189
109.3120 45.3274 130.6239 50.2752 150.2664 54.7812 168.7834 58.9724 186.4033 62.9293 219.4595 70.3293
77.1798
70.4707
64.0526
57.7564
96.4007
85.9412
76.6712
68.0962
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N∗ 8
112.9271 42.7474 128.0294 46.4091 142.2223 49.8182 155.7474 53.0356 181.2544 59.0496
AN US
cr 500
99.2449
87.5549
77.0551
111.3387 38.8831 122.6558 41.6339 133.4192 44.2305 153.7341 49.0837
97.4285
85.1818
106.6379 34.3969 115.3742 36.4751 131.8372 40.3582
92.7460 99.9060 29.7248 113.3676 32.8030
AC
CE
PT
ED
M
Setting cf = 300, cp = 100, cr = 2000, α = 0.01, β = 2.5, an = 1 + 0.01n, we have the optimal N ∗ = 12, the optimal sequence of planned PM time intervals is 272.9008, 186.2850, 150.2664, 128.0294, 111.3387, 97.4285, 85.1818, 74.1112, 64.0064, 54.7812, 46.4091 38.8831, and the corresponding ACR is 5.9457. Noticing that the parameter are same as that in the age-dependent PM model, and the minimal ACR for the age-dependent PM policy is 6.1244, we can conclude that the sequential PM policy is more economical than the age-dependent one. The curve of the function ∆(N ) with respect to N is plotted in Fig. 3, from which we can see that there exists a unique N ∗ minimizing the function ∆(N ). From (23) and (24), we know that C(N ∗ , T1∗ , · · · , TN∗ ∗ ) = (cf − cp )∆(N ∗ ). This implies that the optimal policy (N ∗ , T1∗ , · · · , TN∗ ∗ ) which minimizes the long-run ACR C(N, T1 , · · · , TN ) uniquely exists. Some more numerical results for the optimal sequential PM policy under different replacement costs are given in Table 3. From Table 3, we can see that both the optimal policy N ∗ and the ACR C(N ∗ , T1∗ , · · · , TN∗ ∗ ) increase with the increasing of the replacement cost cr , and the time intervals for PM also tend to be large with increasing cr . Table 4 and 5 give the optimal policies for varying CM cost cf and replacement cost cr . From Table 4, one can see that the optimal policy N ∗ decrease when the corrective repair cost cf increases from 120 to 300. However, when cf ≥ 300, it seems that the optimal policy N ∗ = 12 keeps unchanged for varying cf , while the time intervals for PM become 20
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shorter and shorter with increasing cf . Table 5 shows that the optimal policy N ∗ decrease significantly with the increasing of both cr and cf , and the lengths of time intervals for PM probably are mainly determined by the difference of the CM cost and PM cost.
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Table 4. Some results obtained for the optimal policy (N ∗ , T1∗ , · · · , TN∗ ∗ ) under different correcive repair cost cf .
14
4.2024
200
13
4.8599
300
12
5.9457
400
12
6.8088
600
12
8.1756
800
12
9.2682
516.870 162.859
AN US
150
··· TN∗ ∗ T1∗ 1266.693 818.634 630.487 270.991 230.341 194.497 72.433 687.6669 448.6060 348.3664 154.2864 131.7287 111.6799 433.2030 286.9687 225.7729 104.5782 89.8338 76.5661 272.9008 186.2850 150.2664 74.1112 64.0064 54.7812 208.2989 146.7915 120.9435 61.5803 53.2884 45.6748 149.5170 111.6506 94.1540 49.2846 42.7150 36.6541 122.5067 94.6755 80.6894 42.7413 37.0681 31.8234
M
C∗ 3.7798
PT
ED
cf N ∗ 120 15
287.6894 93.8824 188.5695 64.6573 128.0294 46.4091 104.3429 38.7361 82.1698 31.1118 70.7710 27.0221
435.419 135.117
371.097 110.970
317.353 90.162
243.9468 78.1708 161.4969 54.0419 111.3387 38.8831 91.4511 32.4796 72.5021 26.1022 62.6244 22.6766
209.1385 64.4169 139.6808 44.6716 97.4285
179.8065 52.5009 121.0345
80.4441
70.5906
64.0526
56.3750
55.4285
48.8464
under different cost (cp , cf ).
AC
CE
Table 5. Some results obtained for the optimal policy (N ∗ , T1∗ , · · · , TN∗ ∗ )
21
85.1818
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(200,300)
11
6.1431
(300,400)
10
7.3451
(400,500)
9
8.5055
(500,600)
9
9.6385
(600,800)
7
11.8248
T1∗ 433.2030 121.0345 44.6716 433.2030 135.6773 433.2030 149.5785 433.2030 162.8721 433.2030 175.6496 272.9008 126.0093
··· TN∗ ∗ 286.9687 225.7729 104.5782 89.8338 295.6001 118.4371 304.1073 131.5251 312.4972 144.0020 320.7758 155.9674 212.3208
238.1478 102.6689 250.2091 114.7260 261.9863 126.1872 273.5044 137.1577 186.4033
6. Conclusions
188.5695 76.5661
161.4969 64.6573
139.6808 54.0419
202.7998 176.5062 154.7561 88.2266 75.0653 216.5508 190.9142 169.1445 99.1218 229.8821 204.8014 182.9539
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C∗ 4.8599
AN US
(cp , cf ) N∗ (100,200) 13
242.8374
218.2368 196.2632
168.7321
153.7341
139.6808
AC
CE
PT
ED
M
In this paper, we proposed a GGP and studied its application in maintenance problems. The GGP is a generalization of GP with varying geometric ratios, which is more flexible in modeling the system operating times. Considering that some systems degrade after PM, we proposed two novel PM models based on GGP. In the first PM model, the system is preventively maintained at predetermined time T from the last maintenance, and is replaced at the time of the N th maintenance. This model is an extension of a general GP repair model in that the system can be preventively maintained before failure with economical consideration. The model also includes the classical age replacement policy as a special case if the replacement policy N = 1. The second PM model is an modification of the first one in that the scheduled PM time intervals are varying with the number of the system maintenance. In fact, this is an extension of the sequential PM policy studied by Nagakawa [11] with different assumption on the repair effect. The merit of age-dependent PM policy is that it is easy to manipulate in applications. Comparing with the agedependent PM model, the sequential PM model is more economical. Because systems are usually degrading with the increasing of the number of maintenance, and the sequential PM policy permits us to organize the PM time intervals decreasingly. For the two PM models, we have discussed the optimal policies theoretically and numerically.
22
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Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 11671080) and a Grant from City University of Hong Kong (Project no. 11203815).
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Appendix A. Proof of Theorem 1 Noticing that Z
T
F¯ (AN +1 x)dx =
0
1 AN +1
Z
AN +1 T
0
(25)
AN US
the function L(N, T ) can be rewritten as
F¯ (y)dy,
N R P T ¯ F (An x)dx AN +1 [cf − (cf − cp )F¯ (AN T )] 0 n=1 L(N, T ) = R AN +1 T F¯ (y)dy 0
−cp
N −1 X n=1
N −1 X
F¯ (An T ) − cf
ED
M
AN +1 · Ψ(N ) = R AN +1 T , ¯ (y)dy F 0
where
PT
Ψ(N ) = [cf − (cf − cp )F¯ (AN T )]
CE
+cf
N −1 X
F (An T )]
Z
(26)
N Z X n=1
AN +1 T
0
T
F¯ (An x)dx − [cp
F¯ (y)dy/AN +1 .
N −1 X
F¯ (An T )
n=1
(27)
0
n=1
Since lim AN /N = +∞, and lim
R AN +1 T
N →∞ 0
N →∞
F (An T )
n=1
F¯ (y)dy =
R∞ 0
F¯ (y)dy = λ, the second term
AC
of (27) tends to zero as N → ∞. With the fact that lim F¯ (AN T ) = 0, it can be derived N →∞ that Z T lim Ψ(N ) ≥ cf F¯ (x)dx > 0. N →∞
0
From the above analysis, we can see that lim L(N, T ) = +∞.
N →∞
23
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For the case N = 1, we can compute that
CR IP T
RT A2 [cp + (cf − cp )F (T )] 0 F¯ (x)dx L(1, T ) = R A2 T F¯ (y)dy 0 < a1 (cp F¯ (T ) + cf F (T )). Recalling that L(N, T ) is increase in N , there exists a unique finite N ∗ which minimizes the cost rate function C(N, T ), and satisfies L(N ∗ , T ) > cr and L(N ∗ − 1, T ) < cr .
AN US
This proves conclusion (i) of the theorem. On the other hand, because
RT [cp + (cf − cp )F (T )] 0 F¯ (x)dx L(1, T ) = RT F¯ (A2 x)dx 0 > cp F¯ (T ) + cf F (T ),
M
it is not difficult to see that the optimal replacement policy N ∗ = 1 when cp F¯ (T ) + cf F (T ) > cr . The proof of conclusion (ii) is complete.
ED
Appendix B. Proof of Theorem 2
CE
PT
From equation (13), we have lim QN (T ) = 0. Since F¯ (t) = e− T →0
RA
Rt 0
r(s)ds
, t > 0, we have
T
k R Ak T F¯ (Ak T ) e− 0 r(s)ds = lim e− A1 T r(s)ds . = lim lim ¯ R A1 T r(s)ds T →∞ e− 0 T →∞ T →∞ F (A1 T )
(28)
Noticing that A1 = 1, Ak > 1 for k ≥ 2, and r(t) is increasing in t, it can be derived that
AC
F¯ (Ak T ) lim ¯ = 0. T →∞ F (A1 T )
(29)
Using (29), it can be obtained that
lim
T →∞
NP −1
An r(An T )F¯ (An T )
n=1
N P F¯ (An T )
n=1
= lim
T →∞
NP −1
An r(An T )F¯ (An T )/F¯ (A1 T )
n=1
N P F¯ (An T )/F¯ (A1 T )
n=1
24
= r(∞).
(30)
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Combining the limiting results lim F (An T ) = 1,
T →∞
lim
T →∞
Z
T
0
CR IP T
and λ F¯ (Ak x)dx = , An
we have
If (14) holds, then
AN US
N X 1 lim QN (T ) = r(∞)λ − (N − 1). T →∞ An n=1
lim QN (T ≥
T →∞
(N − 1)cp + cr , cf − cp
(31)
(32)
M
which implies that there exists a finite T ∗ satisfying (15). This completes the proof of (i). If (16) holds, then
ED
QN (∞) ≤
(N − 1)cp + cr , cf − cp
PT
which means that T ∗ = ∞. This proof is completed. References
CE
[1] Y. Lam, Geometric processes and replacement Problem, Acta Math. Appl. Sin. 4(4) (1988) 366-377.
AC
[2] I.T. Castro, E.L. Sanjuan, Power processes and their application to reliability, Operat. Res. Lett. 32 (2004) 415-421. [3] W.J. Braun, W. Li, Y.Q. Zhao, Properties of the geometric and related processes, Naval Research Logistics 52 (2005) 607-616. [4] Y.L. Zhang, G.J. Wang, An extended geometric process repair model for a cold standby repairable system with imperfect delayed repair, Inter. J. Sys. Sci. O & L 3(3) (2016) 163-175.
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[5] R.E. Barlow, L. Hunter, Optimum Preventive Maintenance, Operat. Res. 8 (1960) 90-100.
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[6] H. Wang, A survey of maintenance policies of deteriorating systems, Eur. J. Oper. Res. 139 (2002) 469-489. [7] Y.L. Zhang, A geometric process repair model with good-as-new preventive repair, IEEE Trans. Reliab. 51 (2002) 223-228. [8] T. Nakagawa, Optimal policy of continuous and discrete replacement with minimal repair at failure, Naval Res. Logist. 31(4) (1984) 543-550.
AN US
[9] S. Sheu, W.S. Griffith, T. Nakagawa, Extended optimal replacement model with random minimal repair costs, Eur. J. Oper. Res. 85 (1995) 636-649. [10] T. Nakagawa, Periodic and sequential preventive maintenance policies, J. Appl. Probab. 23(2) (1986) 536-542.
M
[11] T. Nakagawa, Sequential imperfect preventive maintenance policies, IEEE Trans. Reliab. 37 (1988) 295-298.
ED
[12] R.I. Zequeira, C. B´ erenguer, Periodic imperfect preventive maintenance with two categories of competing failure modes, Reliab. Eng. Sys. Safe. 91 (2006) 460-468.
PT
[13] S.H. Sheu, C.C.Chang, Extended periodic imperfect preventive maintenance model of a system subjected to shocks, Inter. J. Sys. Sci. 41(10) (2010) 1145-1153.
CE
[14] D. Lin, M.J. Zuo, R.C.M. Yam, Sequential imperfect preventive maintenance models with two categories of failure modes, Naval Res. Logist. 48 (2001) 173-183.
AC
[15] S.H. Sheu, C.C.Chang, Y.L. Chen, An extended sequential imperfect preventive maintenance model with improvement factors, Commun. Statist.- theor. Methods 41 (2012) 1269-1283. [16] C.Q. Zhong, H.B. Jin, A novel optimal preventive maintenance policy for a cold standby system based on semi-Markov theory, Eur. J. Oper. Res. 232 (2014) 405411.
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[17] Z.L. Lin, Y.S. Huang, C.C. Fang, Non-periodic preventive maintenance with reliability thresholds for complex repairable systems, Reliab. Eng. Sys. Safe. 136 (2015) 145-156.
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[18] X. Liu, W. Wang, R. Peng, An integrated production and delay-time based preventive maintenance planning model for a multi-product production system, Maint. Reliab. 17(2) (2015) 215-221. [19] X.F. Zhao, H.C. Liu, T. Nakagawa, Where does whichever occurs first hold for preventive maintenance modelings? Reliab. Eng. Sys. Safe. 142 (2015) 203-211.
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[20] B. Emami-Mehrgani, W. P. Neumann, S. Nadeau, M. Bazrafshan, Considering human error in optimizing production and corrective and preventive maintenance policies for manufacturing systems, Appl. Math. Model. 40(3) (2016) 2056-2074.
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[21] V. Babishin, S. Taghipour, Optimal maintenance policy for multi-component systems with periodic and opportunistic inspections and preventive replacements, Appl. Math. Model. 40(23-24) (2016) 10480-10505.
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[22] G.Q. Cheng, L. Li, A geometric process repair model with inspections and its optimisation, Inter. J. Sys. Sci. 43 (2012) 1650-1655. [23] M. Jain, R. Gupta, Optimal replacement policy for a repairable system with multiple vacations and imperfect fault coverage, Comput. Ind. Eng. 66(4) (2013) 710-719.
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[24] M. Zhang, M. Xie, O.Gaudoin, A bivariate maintenance policy for multi-state repairable systems with monotone process, IEEE Trans. Reliab. 62(4) (2013) 876-886.
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[25] G.J. Wang, Y.L. Zhang, Optimal replacement policy for a two-dissimilar-component cold standby system with different repair actions, Inter. J. Sys. Sci. 47(5) (2016) 1021-1031. [26] C.Y. Niu, X.L. Liang, B.F. Ge, X.Tian, Y.W. Chen, Optimal replacement policy for a repairable system with deterioration based on a renewal-geometric process, Ann. Operat. Res. 244(1) (2016) 49-66. [27]Y.L. Zhang, G.J. Wang, An Optimal Age-replacement Policy for a Simple Repairable System with Delayed Repair, Commun. Statist.- theor. Methods, 46(6) (2017) 28372850. 27
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[28] Y. Lam, A geometric process maintenance model with preventive repair, Eur. J. Oper. Res. 182 (2007) 806-819.
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[29] G.J. Wang, Y.L. Zhang, A bivariate mixed policy for a simple repairable system based on preventive repair and failure repair, Appl. Math. Model. 33 (2009) 3354-3359. [30] G.J. Wang, Y.L. Zhang, Geometric process model for a system with inspections and preventive repair, Comput. Ind. Eng. 75 (2014) 13-19. [31] Y. Lam, The geometric process and its applications, World Scientific, Singapore, 2007.
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[32] S.M. Ross, Stochastic Processes, Wiley, New York, 1996.
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←− ξ1 −→ ←− ξ2 −→ ←− ξ3
−→ ←− ξ4
−→←− ξ5 −→ ←− ξ6 −→
− − − − − ◦ − − − − − ◦ − − − − − × − − − − − ◦ − − − − − × − − − − − ◦ ··· ←− T −→ ←− T −→ ←− X3 −→ ←− T −→ ←− X5 −→ ←− T −→
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Fig. 1. A possible course of system functioning, where 0 ×0 denotes the system failure time, 0 ◦0 represents the scheduled PM time, and ξk = min{Xk , T }.
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C(T,N)
20 15 10
30
5 400
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T
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Fig. 2. The plots of C(N, T ) against N and T .
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0.07 0.065 0.06
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0.055
∆(N)
0.05 0.045 0.04 0.035
0.025 0.02
0
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0.03
20
30 N
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Fig. 3. The plots of ∆(N ) against N .
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Generalized geometric process and its application in maintenance problems Guan Jun Wang, Richard C.M. Yam PII: DOI: Reference:
S0307-904X(17)30341-4 10.1016/j.apm.2017.05.024 APM 11776
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
15 August 2016 29 April 2017 11 May 2017
Please cite this article as: Guan Jun Wang, Richard C.M. Yam, Generalized geometric process and its application in maintenance problems, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.05.024
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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highlights • A generalized geometric process is proposed for maintenance models.
• A sequential PM model is also discussed.
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• An age-dependent PM model is constructed based on GGP.
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• The optimal policies are studied for the models theoretically and numerically.
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Generalized geometric process and its application in maintenance problems Guan Jun Wanga,∗, Richard C.M. Yamb Department of Mathematics, Southeast University, Nanjing 210096, China Department of SEEM, City University of Hong Kong, Hong Kong, China
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a
b
Abstract
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Since the repair effect may be varying with the number of repairs, we propose a generalized geometric process (GGP) to model the deteriorating process of repairable systems. For a GGP, the geometric ratio changes with the number of repairs rather than being a constant. Based on the GGP, two repair-replacement models are studied. Existing preventive maintenance (PM) models based on geometric process (GP) commonly assume that the PM is ‘as good as new’ in each working circle, which is not realistic in many situations. In this study, that the system is assumed to be geometrically deteriorating after PM or corrective maintenance (CM). Firstly, an age-dependent PM model is considered, in which the optimal policies N ∗ and T ∗ are obtained theoretically, and the optimal bivariate policy (N ∗ , T ∗ ) which minimizes the average cost rate (ACR) can be determined by a searching algorithm. Next, because of the fact that the system deteriorates after maintenance, the schedule time to PM should decrease with the maintenance number increasing. Therefore, a sequential PM policy is investigated, and the optimal policy N ∗ and the optimal schedule times T1∗ , T2∗ , · · · , TN∗ ∗ are computed. Finally, numerical examples are provided to illustrate the proposed models.
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Keywords: generalized geometric process; preventive maintenance; replacement; average cost rate; optimal policy .
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Corresponding author Email address: [email protected] (Guan Jun Wang)
Preprint submitted to Applied Mathematical Modelling
May 17, 2017
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1. Introduction 1.1. Background and Motivation
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In engineering problems, most repairable systems deteriorate gradually with the increasing of the number of repairs. The consecutive operating times of a system after repair are decreasing, while the successive repair times after failure are increasing stochastically. For such phenomena, Lam [1] introduced a geometric process (GP) to model it. In [1], Lam studied two kinds of replacement policies, in which one is based on the working age T of the system, and the other is based on the failure number N of the system. In a GP maintenance model, the mean lifetime sequence of a system geometrically decreases with the repair number increasing, which means that the system uniformly degraded after repairs. This maybe not always true, and the degradation level of a system after repair can be different since the effect of a repair can be affected by many factors, such as the repair being done by different teams, varying damage degree of the system at failure, different budgets for repair, etc. If we continue to model the lifetimes by using a monotone process, the problem is how to modify the GP. Some generalizations of the GP had been done by researchers. For example, Castro and Sanjuan [2] proposed a power process to extend the GP, which can be used not only for describing monotone sequence of random variables with continuous distribution, but also for random sequences with discrete distribution. Braun et al. [3] suggested an α-series process to generalize the GP. They studied the properties of GP and the α-series process, and their applications in reliability. But these generalizations are not very easy to use in maintenance problems due to their computational complexity. Zhang and Wang [4] presented an extended GP maintenance model, in which whether the system after repair deteriorates or not depends on a Bernoulli random variable with successive probability p. In fact, it is a mixture of the GP maintenance model and the perfect maintenance model. In this paper, our first aim is to propose a generalized geometric process (GGP) to model the non-homogenous degradation process of system after maintenance. In applications, preventive maintenance (PM) is usually adopted for improving the system reliability/availability or reducing the cost of system running (Barlow and Hunter [5]). A PM can be perfect or imperfect. A perfect PM lead to a system ‘as good as new’, while an imperfect PM makes the system status better than that just before failure, but not as good as new. Age-dependent PM policy is one of the most important PM policy, under which a system is preventively maintained at its age T or correctively repaired at failure, whichever occurs first, where T is a constant. Similar to age-dependent PM policy, 3
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periodic PM policy is also a basic PM policy in applications. Under periodic PM policy, the system considered being preventively maintained at fixed time intervals kT (k = 1, 2, · · ·), and minimal repaired at failure. Sequential PM policy is another extensively studied policy, under which a system is preventively maintained at unequal time intervals. For an imperfect PM model, sequential PM policy seems more economical for systems with repair degradation, while age-dependent PM policy and periodic PM policy are easy to manipulate in applications. A survey of PM policy can be found in Wang [6]. Under the framework of GP maintenance model, PM policy also was discussed in several papers. But a usual assumption is that the PM is ‘as good as new’ [7]. For some systems, such as aircraft, military systems, the system after PM may be worse and worse. As a monotone process, GGP has a natural advantage in describing the degradation process of system after repair. This motivate us consider the PM policy based on GGP. Our second aim is to develop the optimal age-dependent and sequential imperfect PM policies based on GGP. 1.2. Related literature
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Age-dependent PM policy is an extension of age replacement policy proposed by Barlow and Hunter [5]. Based on the concepts of imperfect maintenance, various of age-dependent PM policies were studied in the literature [6]. For example, Nakagawa [8] introduced an age replacement policy in which the system is replaced at time T or at failure number N , whichever occurs first, and undergoes minimal repair at failure between replacement. Sheu et al.[9] generalized the age replacement policy by assuming two types of failures of a system, the system is replaced at either the first type 2 failure or the nth type 1 failure, or at age T , whichever occurs first. A periodic replacement model was investigated by Nakagawa [10], in which the system is replaced at periodic times kT (k = 1, 2 · · ·) and minimal repaired at failure. In Ref. [10], Nakagawa introduced improvement factors in failure rate function and effective age to describe the effectiveness of imperfect PMs. As an extension of the periodic PM model, a sequential PM model was investigated in [11]. On the basis of Nakagawa’s PM model, Zequeira and Brenguer [12] considered a periodic imperfect PM policy for a system with two categories of competing failure modes. Sheu and Chang [13] presented a periodic imperfect PM model of a system subjected to shocks, in which the optimal PM policy T ∗ and the optimal replacement policy N ∗ were analyzed. At the same time, a generalized sequential imperfect PM was discussed by Lin [14], in which the system has maintainable and non-maintainable failure modes, the PM can only improve the maintainable failure rate and has noting to do with the non-maintainable failure rate. In Sheu [15], another generalized sequential imperfect PM model was proposed, in which 4
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the maintainable and non-maintainable failure occurs with probability pi and 1 − pi , where i is the working cycles of the system, and the optimal policy was derived theoretically and numerically. Besides, as a focus topic in maintenance theory, many new PM models have also been developed recently. For instance, Zhong and Jin [16] researched a PM model for a cold standby system consisting of two components based on the semi-Markov process theory, and the optimal PM interval is obtained by maximizing the mean time to failure of the system. Lin et al [17] developed a non-periodic reliability-based PM models for a repairable deteriorating system, in which PM is performed when the reliability of the system drops below a given threshold. Taking the reliability threshold and the PM number as the decision variables, the optimization problem is solved by minimizing the average maintenance cost rate. More PM models can be found in Refs. [18-21]. Since the GP model was introduced by Lam [1], applications of the GP in maintenance problems have been greatly developed [22-27]. About PM policy based on GP, Zhang [7] constructed a PM model based on GP, in which the system is preventively maintained at periodic time points, and is replaced at its N th failure. In Zhang’s model, the PM is ’as good as new’ in each working stage. Lam [28] generalized Zhang’s work by studying the optimal bivariate policy (T, N ) for a system with PM, where T is the time interval of PM and N is the number of failures at which the system is replaced. As an extension of Lam’s work, Wang and Zhang [29] investigated a bivariate replacement policy (R, N ), where R is the critical reliability of the system for PM and N is the system failure number. Moreover, Wang and Zhang [30] considered a repairable system with periodic inspection, and based on the results of system inspection, PM or corrective Maintenance (CM) is conducted. This model is also an generalization of the model in Lam [28] by introducing state inspection technique for the system. 1.3. Overview
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The remainder of this paper are structured as follows. The coming section introduces definitions of GGP and its properties. In Section 3, an age-dependent PM model is considered, the existence and uniqueness of the optimal policies N ∗ and T ∗ are studied theoretically, and an algorithm for the optimal policy (N ∗ , T ∗ ) is presented. Section 4 focuses on a sequential PM model, and the optimal policy (N ∗ , T1∗ , · · · , TN∗ ∗ ) is discussed. Numerical examples are provided in Section 5, and conclusions are given in section 6.
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2. Notations and definition of GGP 2.1. Notations
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lifetime of a system after the (n − 1)th repair cumulative distribution function of Xn cumulative distribution function of X1 probability density function of X1 failure rate function of X1 mean of X1 , i.e., λ = EX1 mean of Xn geometric ratio of the nth repair = a0 a1 · · · an−1 planned time interval for PM in the age-dependent PM model number of maintenances before system replacement PM cost CM cost replacement cost = min{Xn , T } cumulative distribution function of ξn ACR of the age-dependent PM model under policy (N, T ) optimal policy of the age-dependent PM model the nth time interval of the sequential PM model = min{Xn , Tn } cumulative distribution function of ζn ACR of the sequential PM model under policy (N, T1 , · · · , TN ) optimal policy of the sequential PM model
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Xn Fn (t) F (t) f (t) r(t) λ λn an An T N cp cf cr ξn Hn (t) C(N, T ) (N ∗ , T ∗ ) Tn ζn Kn (t) C(N, T1 , · · · , TN ) (N ∗ , T1∗ , · · · , TN∗ )
2.2. Definition of GGP
For easy reference, we first state the definition of the GP as follows. Definition 1. Assume that {Zn , n = 1, 2, . . .} is a sequence of independent nonnegative random variables. If the distribution function of Zn is Gn (t) = G(an−1 t), for n = 1, 2, . . . , where G(t) is the cumulative distribution function of Z1 and a > 0 is a constant, then {Zn , n = 1, 2, . . .} is called a GP, and a is the ratio of the GP. 6
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Clearly, if a > 1, then the GP {Zn , n = 1, 2, . . .} is stochastically decreasing. If 0 < a < 1, then the GP {Zn , n = 1, 2, . . .} is stochastically increasing.
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If a = 1, then the GP {Zn , n = 1, 2, . . .} is a renewal process.
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From the definition of GP, the ratio a is a constant for n = 1, 2, . . .. In the maintenance background, this means that after maintenance, the lifetime of a system uniformly degrades as the maintenance number increasing. Practically, the effect of maintenance to the system may be different with the varying number of maintenance. Therefore, we propose the following GGP to model it. Definition 2. Assume that {Zn , n = 1, 2, . . .} is a sequence of independent nonnegative random variables. If the distribution function of Zn is Zn (t) = G(a0 a1 · · · an−1 t) for n = 1, 2, . . . , where G(t) is the cumulative distribution function of Z1 , a0 = 1, and a1 , a2 , · · · are positive constants, then {Zn , n = 1, 2, . . .} is called a GGP, and a1 , a2 , · · · are the ratios of the GGP. If an ≥ 1 for all n = 1, 2, · · ·, then the GGP {Zn , n = 1, 2, . . .} is stochastically decreasing.
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If an ≤ 1 for all n = 1, 2, · · ·, then the GGP {Zn , n = 1, 2, . . .} is stochastically increasing.
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Remark 1. (a) Denote An = a0 a1 · · · an−1 , then it can be written that Gn (t) = G(An t), n = 1, 2, . . .. If a1 = a2 = · · · = a is a constant, the GGP reduces to the GP, and An = an−1 . (b) For a decreasing GGP {Zn , n = 1, 2, · · ·}, a typical assumption is that 1 ≤ a1 ≤ a2 ≤ · · ·. If {Zn , n = 1, 2, · · ·} is a sequence of working times of a system after maintenance, then the assumption 1 ≤ a1 ≤ a2 ≤ · · · implies that the system deteriorates more and more rapidly with the maintenance number increasing. If {Zn , n = 1, 2, · · ·} is a sequence of maintenance time of a system, which follows increasing GGP with ratios 1 ≥ a1 ≥ a2 ≥ · · ·, then the maintenance time stochastically increases more and more rapidly with the maintenance number increasing. About the parameter estimation problem of the GGP, we have the following remarks.
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(r)
Remark 2. (a) Suppose that {Zk , k = 1, 2, · · · , n; r = 1, 2, · · · , m} is a set of failure data collected from the operating processes of m identical machines with maintenance, and (r) each machine is renewed at its nth failure. To test whether {Zk , k = 1, 2, · · · , n} yields a GGP, we first construct the following sequences of random variables: (r)
(r)
= Zk /Zk+1 , r = 1, 2, · · · , m; k = 1, 2, · · · , n − 1.
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(r)
Yk
(r)
(r)
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It can be seen that {Zk , k = 1, 2, · · · , n; } forms a GGP if and only if Yk , r = 1, 2, · · · , m is a sequence of independent and identical distributed (i.i.d.) random variables for each (r) k ∈ {1, 2, · · · , n − 1}. For k = 1, 2, · · · , n − 1, to test random variables Yk , r = 1, 2, · · · , m are i.i.d. or not, the ‘turning point test’ and the ‘difference-sign test’ suggested by Lam [31] (in page 104) can be applied. (r) (b) If {Zk , k = 1, 2, · · · , n} is tested to be agree with a GGP, the next problem is to estimate the geometric ratios ak , k = 1, 2, · · · , n − 1. Since the random variables (r) Yk , r = 1, 2, · · · , m are i.i.d. for each k ∈ {1, 2, · · · , n − 1}, a straightforward estimator of P (r) (r) (r) ak is Y¯k = m r=1 Yk /m. Moreover, because EZk = ak EZk+1 for each k ∈ {1, 2, · · · , n−1}, P (r) (r) Pm another estimator of ak can be given by a ˆk = µ ˆk /ˆ µk+1 = m r=1 Zk+1 . It is obr=1 Zk / vious that the latter estimator is more robust than the former one.
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Assume that {Zn , n = 1, 2, . . .} is a GGP, and the variable Z1 has the distribution function G(t) and the probability density function g(t), then the probability density function of Zn is
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gn (t) =
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The mean of Zn is
µn = E[Zn ] =
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where µ = EZ1 . Denote Sn =
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0
n P
∞
t · An g(An t)dt =
µ , n = 2, 3, · · · An
Zi , then Sn is the total working time of a system before the nth
i=1
maintenance, or the total repair time of the first n repairs. Proposition 1. (i) If {Zn , n = 1, 2, . . .} follows a GGP with ratios satisfying ak ≥ δ, k = 1, 2, · · ·, where δ > 1 is a constant, then lim E[Sn ] =
n→∞
∞ X n=1
E[Zn ] ≤ 8
δµ < ∞. δ−1
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(ii) If {Zn , n = 1, 2, . . .} follows a GGP with ratios satisfying ak ≤ 1, k = 1, 2, · · ·, then n→∞
∞ X n=1
E[Zn ] = ∞.
Proof. The proof is trivial, and is omitted here. 3. Age-dependent PM policy
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lim E[Sn ] =
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In this section, an age-dependent PM model is studied based on the GGP for a repairable system. The long-run average cost rate (ACR) is derived and the optimal maintenance policies are discussed. 3.1. Assumptions and model analysis
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Some assumptions are given in the following to describe the age-dependent PM model. Assumption 1. A new system is beginning to work at t = 0. The system is preventively maintained at a planned time T since the last maintenance, or undergoes CM at failure, whichever occurs first. Assumption 2. The lifetime sequence {Xn , n = 1, 2, . . .} of the system after maintenance (PM or CM) forms a GGP. That is, the distribution function of Xn is Fn (t) = F (An t), n = 1, 2, . . ., where An = a0 a1 · · · an−1 . Accordingly, the mean of Xn is λn = λ/An , where λ = EX1 . The maintenance time of the system is neglected. Assumption 3. The replacement policy N based on the number of maintenances of the system is used. That is, the system is replaced by a new one at time of the N th maintenance. The replacement time is negligible. Assumption 4. The cost for a CM is cf , the cost for a PM is cp , and the cost for replacement of a new system is cr , where cp < cf < cr .
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A possible course of system running is plotted in Fig. 1.
Remark 3. If a1 = a2 = · · · = 1, then the model reduces to the classical age replacement model.
Let ξn , n = 1, 2, . . . , N − 1, be the time between the (n − 1)th and the nth maintenance, and ξN be the time between the (N − 1)th repair and the replacement of the system, then
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and the survival function of ξn is ¯ n (x) = 1 − Hn (x) = H
(
F¯ (An x), x < T . 0, x > T
Therefore, the mean of ξn can be computed as Z ∞ Z ¯ Hn (x)dx = E[ξn ] =
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ξn = min{Xn , T }, n = 1, 2, · · · , N . The distribution function of ξn is ( F (An x), x < T Hn (x) = P (ξn ≤ x) = , 1, x > T
(1)
(2)
(3)
Denote pn the probability that the nth maintenance is a CM, then pn = P (Xn < T ) = F (An T ).
(4)
Let U (N, T ) be the total operating time of the system before replacement, we have
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The mean of U (N, T ) is
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E[U (N, T )] = E[ξ1 ] + · · · + E[ξN ] =
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T
F¯ (An x)dx.
(5)
0
n=1
E[W (N, T )] =
N −1 X n=1
[cf pn + cp (1 − pn )] + cr
= (N − 1)cp + cr + (cf − cp )
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The mean total cost for the system operating during a renewal cycle is
N −1 X
F (An T ).
(6)
n=1
By applying the renewal reward theorem Ross [32], we obtain the long-run ACR of the system as
C(N, T ) =
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F (An T ) .
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Remark 4. (a) When T → +∞, no PM is performed. In this case, pn = 1 for n = 1, 2, · · ·, and the long-run ACR of the system becomes C(N, +∞) =
(N − 1)cf + cr (N − 1)cf + cr = . N R N P P ∞ ¯ 1 F (An x)dx λ An 0
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1 [(N N
C(+∞, T ) = lim
N →∞
− 1)cp + cr + (cf − cp )
F (An T )]
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The objective is to determine the optimal policy (N ∗ , T ∗ ) such that the ACR C(N, T ) is minimized. 3.2. Optimal N ∗
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Supposing that the PM time interval T > 0 is fixed, the aim is to find N ∗ such that the ACR C(N, T ) is minimized. It is not difficult to derive that C(N + 1, T ) > C(N, T ) and C(N, T ) < C(N − 1, T ) if and only if L(N, T ) > cr and L(N − 1, T ) < cr ,
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N R P T ¯ F (An x)dx [cp + (cf − cp )F (AN T )] 0 n=1 L(N, T ) = − (N − 1)cp RT ¯ (AN +1 x)dx F 0
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About the optimal replacement policy N ∗ , a conclusion is given in Theorem 1. Theorem 1. Assume that lim AN /N = +∞. For fixed T > 0, we have N →∞ (i) If cp F¯ (T )+cf F (T ) < cr /a1 , then there exists a unique finite optimal replacement policy N ∗ which minimizes the cost rate function C(N, T ), and satisfies
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(ii) If cp F¯ (T ) + cf F (T ) > cr , then the optimal replacement policy N ∗ = 1. The proof of Theorem 1 can be found in Appendix A. Remark 5. The condition lim AN /N = +∞ of Theorem 1 is very general. It does N →∞ not require an > 1 for all n = 1, 2, · · ·. Of course, if an = a > 1 for all n = 1, 2, · · ·, then the condition holds obviously. About the condition lim AN /N = +∞, some remarks are given as follows. N →∞
Remark 6. (a) We first write an as an = 1 + bn , where bn ≥ 0. A necessary condition ∞ P for lim AN /N = +∞ holding is that the sum of series bn is divergent. To see this, it N →∞
n=1
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n=1
an =
If the sum of the series
(1 + bn ) ≤
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∞ Y
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∞ Y
∞ Y
e
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bn
.
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n=1
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∞ P
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∞ P
bn = B > 0, then
n=1 N Q
an
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N
Thus, we conclude that the sum of series
eB = 0. N →∞ N
≤ lim ∞ P
bn must be divergent.
n=1
AC
(b) Also write an as an = 1 + bn , where bn ≥ 0. A sufficient condition for lim AN /N =
+∞ holding is that the increasing speed of series { Notice that N Q
N P
n=1
N →∞
bn , N ≥ 1} is larger than ln N .
an X N N X ln n=1 = ln an − ln N ∼ bn − ln N. N n=1 n=1 12
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If lim
N P
N →∞ n=1
N Q
an
→ ∞, or
bn / ln N > 1, then ln n=1N
N Q
an
n=1
N
→ ∞, as N → ∞. The conclusion is
reached. (c)If an has the form of an = 1 + nr , where r > 0 is a constant. Then
n=1
an =
N Y
(1 +
n=1
r Γ(N + r + 1) 1 )= = , n Γ(N + 1)Γ(r + 1) B(N, r)
CR IP T
AN +1 =
N Y
where Γ(·) and B(·) are Gamma function and Beta function, respectively. It is easy to see that B(N, 1) = N1+1 . Generally, for fixed r > 0 and large N , B(N, r) ∼ Γ(r) · N −r . Therefore, we have that 0, 0 < r < 1 = 1, r = 1 +∞, r > 1
AN US
lim
N →∞
N Q
an
n=1
N
M
(d)If an has the form of an = 1 + nrh , where r > 0, h > 0 are constants. Comparing the results in (c), it can be got that
lim
3.3. Optimal T ∗
ED
N →∞
N Q
an
n=1
=
N
(
0, 0 < h < 1 +∞, h > 1
AC
CE
PT
Supposing that the replacement policy N > 0 is fixed, we try to seek T ∗ which minimizes the ACR C(N, T ). To do this, differentiating C(N, T ) with respect to T , we can see that ∂C(N, T )/∂T = 0 if and only if
which is equivalent to
(cf − cp )
NP −1
An f (An T )
n=1
= C(N, T ),
N P F¯ (An T )
(11)
n=1
QN (T ) =
(N − 1)cp + cr , cf − cp
13
(12)
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where QN (T ) =
NP −1
An f (An T )
n=1
N R P T ¯ F (Ak x)dx 0
k=1
N P F¯ (An T )
−
=
n=1
An r(An T )F¯ (An T ) X N Z N P F¯ (An T )
k=1
0
n=1
F (An T )
n=1
CR IP T
n=1 NP −1
N −1 X
T
F¯ (Ak x)dx −
N −1 X
F (An T )
(13)
n=1
AN US
About the optimal PM policy T ∗ , results are given in Theorem 2. Theorem 2. Assume that the failure rate r(t) is increasing in t, and a1 > 1, ak ≥ 1, k ≥ 2. For given number N ≥ 1, the following results hold. (i) If r(∞) >
(N − 1)cf + cr , N P 1 (cf − cp )λ An
(14)
n=1
∗
M
then there exists a finite optimal policy T which minimizes the ACR C(N, T ) and satisfies QN (T ∗ ) =
(N − 1)cp + cr . cf − cp
(15)
PT
ED
Moreover, if QN (T ) is increasing in T , then the optimal policy T ∗ is unique. (ii) If r(∞) ≤
(N − 1)cf + cr , N P 1 (cf − cp )λ An
(16)
n=1
∗
AC
CE
then the optimal policy T = +∞. The proof of Theorem 2 can be found in Appendix B. Remark 7. (a) If r(∞) = +∞, the condition (14) holds obviously, and a finite optimal policy T ∗ exists. (b) The condition a1 > 1, ak ≥ 1, k ≥ 2 in Theorem 2 can be relaxed by ‘there exists a number r ≥ 1 such that ar > 1 and ak ≥ 1, k 6= r’, which also guarantees equation (29), and thus the conclusions of Theorem 2 hold. Remark 8. Conditions (14) and (16) means that PM should be arranged if the failure rate r(t) is large enough for large t. If r(t) is bounded from above by a certain constant, PM should not be taken. The reason is that the lifetime approaches an exponential variable, which as a lifetime is ‘as good as new’ at any time before failure. 14
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3.4. Optimal bivariate policy (T ∗ , N ∗ ) In this subsection, we give the following algorithm to seek the optimal bivariate policy (T , N ∗ ). ∗
CR IP T
Algorithm
AN US
Input: cp , cf , cr , λ, r(·), ak , k = 1, 2, · · ·. Step 1. Let j = 1, and Nj = 1. Step 2. Let N = Nj , and find the solution Tj which satisfies (15). Step 3. Set T = Tj , and find the unique Nj+1 which satisfies (10). Step 4. If Nj+1 = Nj , then go to Step 5; otherwise, let j = j + 1, and go to Step 2. Step 5. T ∗ = Tj and N ∗ = Nj . Output: T ∗ , N ∗ and C(T ∗ , N ∗ ). Stop.
M
According to the algorithm, the optimal bivariate policy (T ∗ , N ∗ ) can be determined recursively. 4. Sequential PM policy
AC
CE
PT
ED
Since the lifetime sequence {Xn , n = 1, 2, . . .} after repair forms a decreasing GGP, the planned time to PM should be shorter and shorter with the repair number increasing. Therefore, in this section, we propose a sequential PM model to optimize the running process of the system. The model assumptions are the same as that given in Section 3 except for Assumption 1, which is replaced by Assumption 10 as follows: Assumption 10 . The system is preventively maintained at a planned time Tn after the (n − 1)th repair, or undergoes CM at failure, whichever occurs first, n = 1, 2, · · ·. Similar to the age-dependent PM model, the system is replaced by a new one at the N th maintenance time. The problem is to determined the optimal number N ∗ and the optimal planned time intervals {Tn∗ , n = 1, 2, . . . N ∗ } such that the long-run ACR is minimized. Letting {ζn , n = 1, 2, . . .} be the sequence of actual operating time between maintenance, we have ζn = min{Xn , Tn } and Xn ∼ F (An x) for n = 1, 2, . . ., where An = a0 a1 · · · an−1 . The distribution function of ζn is ( F (An x), x < Tn Kn (x) = P (ζn ≤ x) = (17) 1, x ≥ Tn 15
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The survival function of ζn is F¯ (An x), x < Tn 0, x ≥ Tn
Therefore, the mean of ζn can be computed as Z Z ∞ ¯ Kn (x)dx = E[ζn ] =
0
0
Tn
CR IP T
¯ n (x) = 1 − Kn (x) = K
(
F¯ (An x)dx.
(18)
Denote pn the probability that the nth maintenance is a CM, then pn = P (ζn = Xn ) = P (Xn ≤ Tn ) = F (An Tn ),
(19)
AN US
and the probability that the nth maintenance is a PM is qn = 1 − pn = F¯ (An Tn ).
M
For fixed planned times {Tn , n = 1, 2, . . . N }, let U (N, T1 , · · · , TN ) and W (N, T1 , · · · , TN ) be the total operating time and the total cost of the system operating during a renewal cycle. We have U (N, T1 , · · · , TN ) = ζ1 + · · · + ζN .
ED
The mean of U (N, T1 , · · · , TN ) is
PT
E[U (N, T1 , · · · , TN )] = E[ζ1 ] + · · · + E[ζN ] =
N Z X n=1
Tn
F¯ (An x)dx.
(20)
0
Similarly, the mean total cost of the system is
AC
CE
E[W (N, T1 , · · · , TN )] =
N −1 X n=1
[cf pn + cp (1 − pn )] + cr
= (N − 1)cp + cr + (cf − cp )
N −1 X
F (An Tn ).
(21)
n=1
By applying the renewal reward theorem Ross [32], the long-run ACR of the system can be derived as
C(N, T1 , · · · , TN ) =
(N − 1)cp + cr + (cf − cp )
NP −1 n=1
N R P Tn ¯ F (An x)dx 0
n=1
16
F (An Tn ) .
(22)
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CR IP T
The aim is to determine the optimal policy N ∗ and T1∗ , T2∗ , · · · , TN∗ ∗ such that the ACR C(N, T1 , · · · , TN ) is minimized. By the method similar to Nakagawa [11], a necessary condition for T1 , T2 , · · · being the optimal planned time intervals is that ∂C/∂Tn = 0 for each n. Differentiating C(N, T1 , · · · , TN ) with respect to Tn , and setting it equal to zero, it can be obtained that (cf − cp )An f (An Tn ) = C(N, T1 , · · · , TN ), n = 1, 2, · · · , N, F¯ (An Tn )
(23)
A1 r(A1 T1 ) = A2 r(A2 T2 ) = · · · = AN r(AN TN ).
(24)
AN US
which implies
M
As a function of N , denote AN r(AN TN ) by ∆(N ). When the failure rate is strictly increasing, the computing procedure for obtaining the optimal planned time intervals can be specified as follows: 1. Solving An r(An Tn ) = ∆(N ), n = 1, 2, · · · , N , one gets that Tn = ψn (∆(N )), which is a function of ∆(N ). 2. Substitute Tn into (23) and solve it with respect to ∆(N ).
ED
3. Determine N ∗ by minimizing ∆(N ), N = 1, 2, · · ·.
PT
4. Calculating Tn∗ from the relation Tn∗ = ψn (∆(N ∗ )), n = 1, 2, · · · , N ∗ .
CE
Using the computing procedure given above, the optimal policy (N ∗ , T1∗ , T2∗ , · · · , TN∗ ∗ ) can be determined analytically or numerically.
AC
5. Numerical examples 5.1. Numerical example for age-dependent PM model
As an example, suppose that the lifetime X1 of the system is Weibull distributed, i.e. β F (t) = 1 − e−(αt) , t > 0, where α > 0, β > 1 are constants. Then the failure rate of X1 is r(t) = βαβ tβ−1 , which is strictly increasing in t. The mean lifetime of the system is EX = α−1 · Γ(1 + β). For simulation, the parameters can be set as cf = 300, cp = 100, cr = 2000, α = 0.01, β = 2.5, an = 1 + 0.01n. In this case, we get EX = 332.34. It is easy to verify that the 17
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M
AN US
CR IP T
conditions of Theorem 1 and 2 are satisfied, and therefore the optimal policies N ∗ and T ∗ can be determined based on Theorem 1 and 2, respectively. Furthermore, by using the algorithm given in Subsection 3.4, the bivariate optimal (N ∗ , T ∗ ) can also be determined numerically. To begin with, setting the initial value N = 2, we can get the optimal policy T ∗ = 220. Next, let the time interval between PM is T = 220, it can be obtained that the optimal policy N ∗ = 11. Fixing N = 11, numerical results show that the optimal policy is T ∗ = 180. Finally, the setting T = 180 leads to the optimal policy N ∗ = 11, from which we can see that the optimal bivariate policy is (N ∗ , T ∗ ) = (11, 180), and the corresponding ACR is 6.1244. Some more numerical results based on (7) can be seen from Table 1, and the surface for the ACR C(N, T ) is plotted in Fig. 2. From Table 1 and Fig. 2, it can be seen that the optimal policy (N ∗ , T ∗ ) uniquely exists, and it is the same as the result obtained from the searching algorithm. In table 2, the optimal policies are presented for varying corrective repair cost cf and replacement cost cr . From Table 2, one can see that both the optimal policies N ∗ and T ∗ decrease with the increasing of the cost cf , and N ∗ increases with the increasing of cr . But the optimal policy T ∗ is almost unchanged with the increasing of cr , which implies that T ∗ is mainly determined by the cost of PM and CM.
ED
Table 1 Some results obtained for C(T, N ), where (N ∗ , T ∗ ) = (11, 180) is the optimal policy with the minimal ACR 6.1244
AC
140 13.2778 7.5785 6.6451 6.2611 6.1810 6.1523 6.1646 6.2106 6.3838 6.7980
PT
110 14.0995 7.9495 6.9127 6.4659 6.3649 6.3208 6.3224 6.3614 6.5280 6.9435
CE
N/ T 2 5 7 9 10 11 12 13 15 18
160 13.1000 7.5092 6.6007 6.2300 6.1537 6.1276 6.1415 6.1885 6.3623 6.7758
170 13.0610 7.4966 6.5939 6.2259 6.1504 6.1246 6.1388 6.1859 6.3597 6.7729
180 13.0397 7.4913 6.5918 6.2252 6.1500 6.1244 6.1386 6.1857 6.3595 6.7724
190 13.0290 7.4898 6.5921 6.2262 6.1510 6.1255 6.1397 6.1867 6.3603 6.7730
200 13.0242 7.4903 6.5934 6.2277 6.1525 6.1269 6.1411 6.1880 6.3615 6.7741
220 250 13.0222 13.0236 7.4928 7.4960 6.5965 6.5997 6.2307 6.2337 6.1555 6.1583 6.1298 6.1324 6.1438 6.1463 6.1906 6.1930 6.3639 6.3661 6.7763 6.7783
Table 2 The optimal policies for different repair and replacement costs 18
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C(N ∗ , T ∗ ) 3.7794 4.1961 4.8640 6.1244 7.3217 8.4760
(cp , cf , cr ) (N ∗ , T ∗ ) (100,300,500) (6,170) (100,300,1000) (9,170) (100,300,1500) (10,170) (100,300,2000) (11,180) (100,300,2500) (12,180) (100,300,3000) (13,180)
C(N ∗ , T ∗ ) 3.9598 4.8060 5.4939 6.1244 6.7191 7.2928
CR IP T
(cp , cf , cr ) (N ∗ , T ∗ ) (100,120,2000) (15,230) (100,150,2000) (14,200) (100,200,2000) (13,190) (100,300,2000) (11,180) (100,400,2000) (10,170) (100,500,2000) (9,170)
5.2. Numerical example for sequential PM model
∆(N )
M
Tn = (
AN US
To illustrate the sequential PM model, we also assume that the system lifetime X1 β yields Weibull distribution with F (t) = 1 − e−(αt) , t > 0 with α > 0, β > 1, and EX = α−1 Γ(1 + β). It is obvious that the failure rate r(t) = βαβ tβ−1 is strictly increasing in t for β > 1. By applying the algorithm given in Section 4, we need to solve the equation An r(An Tn ) = An βαβ (An Tn )β−1 = ∆(N ) with respect to Tn , n = 1, 2, · · · , N . It is not difficult to derive that βαβ Aβn
1
) β−1 , n = 1, 2, · · · , N
ED
Taking Tn into (23), we can obtain that
PT
∆(N ) =
(N − 1)cf + cr − (cf − cp ) (cf − cp )
∆(N )
1
NP −1
N R ( β β ) β−1 P βα An
n=1
0
e
∆(N ) n
β
−( αβA ) β−1
n=1
.
e−(αAn x)β dx
AC
CE
It can be seen that ∆(N ) does not has explicit expression. Therefore, we have to seek the optimal N ∗ by minimizing ∆(N ) numerically. Table 3. Some results obtained for the optimal policy (N ∗ , T1∗ , · · · , TN∗ ∗ ) under different replacement cost cr .
19
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C∗ 3.6657
1000
10
4.5427
1500
11
5.2798
2000
12
5.9457
2500
13
6.5673
3000
14
7.1611
4000
15
8.2902
T1∗ 113.8006 51.5635 172.3281 60.0344 225.2838 67.4145 272.9008 74.1112 316.6733 80.3407 357.6012 86.2279 433.2030 97.2721 26.3431
··· 94.6755
TN∗ ∗ 84.7117
128.5043 52.4398 158.4838 58.4991 186.2850 64.0064 212.3208 69.1273 236.8874 73.9632 282.6040 83.0189
109.3120 45.3274 130.6239 50.2752 150.2664 54.7812 168.7834 58.9724 186.4033 62.9293 219.4595 70.3293
77.1798
70.4707
64.0526
57.7564
96.4007
85.9412
76.6712
68.0962
CR IP T
N∗ 8
112.9271 42.7474 128.0294 46.4091 142.2223 49.8182 155.7474 53.0356 181.2544 59.0496
AN US
cr 500
99.2449
87.5549
77.0551
111.3387 38.8831 122.6558 41.6339 133.4192 44.2305 153.7341 49.0837
97.4285
85.1818
106.6379 34.3969 115.3742 36.4751 131.8372 40.3582
92.7460 99.9060 29.7248 113.3676 32.8030
AC
CE
PT
ED
M
Setting cf = 300, cp = 100, cr = 2000, α = 0.01, β = 2.5, an = 1 + 0.01n, we have the optimal N ∗ = 12, the optimal sequence of planned PM time intervals is 272.9008, 186.2850, 150.2664, 128.0294, 111.3387, 97.4285, 85.1818, 74.1112, 64.0064, 54.7812, 46.4091 38.8831, and the corresponding ACR is 5.9457. Noticing that the parameter are same as that in the age-dependent PM model, and the minimal ACR for the age-dependent PM policy is 6.1244, we can conclude that the sequential PM policy is more economical than the age-dependent one. The curve of the function ∆(N ) with respect to N is plotted in Fig. 3, from which we can see that there exists a unique N ∗ minimizing the function ∆(N ). From (23) and (24), we know that C(N ∗ , T1∗ , · · · , TN∗ ∗ ) = (cf − cp )∆(N ∗ ). This implies that the optimal policy (N ∗ , T1∗ , · · · , TN∗ ∗ ) which minimizes the long-run ACR C(N, T1 , · · · , TN ) uniquely exists. Some more numerical results for the optimal sequential PM policy under different replacement costs are given in Table 3. From Table 3, we can see that both the optimal policy N ∗ and the ACR C(N ∗ , T1∗ , · · · , TN∗ ∗ ) increase with the increasing of the replacement cost cr , and the time intervals for PM also tend to be large with increasing cr . Table 4 and 5 give the optimal policies for varying CM cost cf and replacement cost cr . From Table 4, one can see that the optimal policy N ∗ decrease when the corrective repair cost cf increases from 120 to 300. However, when cf ≥ 300, it seems that the optimal policy N ∗ = 12 keeps unchanged for varying cf , while the time intervals for PM become 20
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shorter and shorter with increasing cf . Table 5 shows that the optimal policy N ∗ decrease significantly with the increasing of both cr and cf , and the lengths of time intervals for PM probably are mainly determined by the difference of the CM cost and PM cost.
CR IP T
Table 4. Some results obtained for the optimal policy (N ∗ , T1∗ , · · · , TN∗ ∗ ) under different correcive repair cost cf .
14
4.2024
200
13
4.8599
300
12
5.9457
400
12
6.8088
600
12
8.1756
800
12
9.2682
516.870 162.859
AN US
150
··· TN∗ ∗ T1∗ 1266.693 818.634 630.487 270.991 230.341 194.497 72.433 687.6669 448.6060 348.3664 154.2864 131.7287 111.6799 433.2030 286.9687 225.7729 104.5782 89.8338 76.5661 272.9008 186.2850 150.2664 74.1112 64.0064 54.7812 208.2989 146.7915 120.9435 61.5803 53.2884 45.6748 149.5170 111.6506 94.1540 49.2846 42.7150 36.6541 122.5067 94.6755 80.6894 42.7413 37.0681 31.8234
M
C∗ 3.7798
PT
ED
cf N ∗ 120 15
287.6894 93.8824 188.5695 64.6573 128.0294 46.4091 104.3429 38.7361 82.1698 31.1118 70.7710 27.0221
435.419 135.117
371.097 110.970
317.353 90.162
243.9468 78.1708 161.4969 54.0419 111.3387 38.8831 91.4511 32.4796 72.5021 26.1022 62.6244 22.6766
209.1385 64.4169 139.6808 44.6716 97.4285
179.8065 52.5009 121.0345
80.4441
70.5906
64.0526
56.3750
55.4285
48.8464
under different cost (cp , cf ).
AC
CE
Table 5. Some results obtained for the optimal policy (N ∗ , T1∗ , · · · , TN∗ ∗ )
21
85.1818
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(200,300)
11
6.1431
(300,400)
10
7.3451
(400,500)
9
8.5055
(500,600)
9
9.6385
(600,800)
7
11.8248
T1∗ 433.2030 121.0345 44.6716 433.2030 135.6773 433.2030 149.5785 433.2030 162.8721 433.2030 175.6496 272.9008 126.0093
··· TN∗ ∗ 286.9687 225.7729 104.5782 89.8338 295.6001 118.4371 304.1073 131.5251 312.4972 144.0020 320.7758 155.9674 212.3208
238.1478 102.6689 250.2091 114.7260 261.9863 126.1872 273.5044 137.1577 186.4033
6. Conclusions
188.5695 76.5661
161.4969 64.6573
139.6808 54.0419
202.7998 176.5062 154.7561 88.2266 75.0653 216.5508 190.9142 169.1445 99.1218 229.8821 204.8014 182.9539
CR IP T
C∗ 4.8599
AN US
(cp , cf ) N∗ (100,200) 13
242.8374
218.2368 196.2632
168.7321
153.7341
139.6808
AC
CE
PT
ED
M
In this paper, we proposed a GGP and studied its application in maintenance problems. The GGP is a generalization of GP with varying geometric ratios, which is more flexible in modeling the system operating times. Considering that some systems degrade after PM, we proposed two novel PM models based on GGP. In the first PM model, the system is preventively maintained at predetermined time T from the last maintenance, and is replaced at the time of the N th maintenance. This model is an extension of a general GP repair model in that the system can be preventively maintained before failure with economical consideration. The model also includes the classical age replacement policy as a special case if the replacement policy N = 1. The second PM model is an modification of the first one in that the scheduled PM time intervals are varying with the number of the system maintenance. In fact, this is an extension of the sequential PM policy studied by Nagakawa [11] with different assumption on the repair effect. The merit of age-dependent PM policy is that it is easy to manipulate in applications. Comparing with the agedependent PM model, the sequential PM model is more economical. Because systems are usually degrading with the increasing of the number of maintenance, and the sequential PM policy permits us to organize the PM time intervals decreasingly. For the two PM models, we have discussed the optimal policies theoretically and numerically.
22
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Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 11671080) and a Grant from City University of Hong Kong (Project no. 11203815).
CR IP T
Appendix A. Proof of Theorem 1 Noticing that Z
T
F¯ (AN +1 x)dx =
0
1 AN +1
Z
AN +1 T
0
(25)
AN US
the function L(N, T ) can be rewritten as
F¯ (y)dy,
N R P T ¯ F (An x)dx AN +1 [cf − (cf − cp )F¯ (AN T )] 0 n=1 L(N, T ) = R AN +1 T F¯ (y)dy 0
−cp
N −1 X n=1
N −1 X
F¯ (An T ) − cf
ED
M
AN +1 · Ψ(N ) = R AN +1 T , ¯ (y)dy F 0
where
PT
Ψ(N ) = [cf − (cf − cp )F¯ (AN T )]
CE
+cf
N −1 X
F (An T )]
Z
(26)
N Z X n=1
AN +1 T
0
T
F¯ (An x)dx − [cp
F¯ (y)dy/AN +1 .
N −1 X
F¯ (An T )
n=1
(27)
0
n=1
Since lim AN /N = +∞, and lim
R AN +1 T
N →∞ 0
N →∞
F (An T )
n=1
F¯ (y)dy =
R∞ 0
F¯ (y)dy = λ, the second term
AC
of (27) tends to zero as N → ∞. With the fact that lim F¯ (AN T ) = 0, it can be derived N →∞ that Z T lim Ψ(N ) ≥ cf F¯ (x)dx > 0. N →∞
0
From the above analysis, we can see that lim L(N, T ) = +∞.
N →∞
23
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For the case N = 1, we can compute that
CR IP T
RT A2 [cp + (cf − cp )F (T )] 0 F¯ (x)dx L(1, T ) = R A2 T F¯ (y)dy 0 < a1 (cp F¯ (T ) + cf F (T )). Recalling that L(N, T ) is increase in N , there exists a unique finite N ∗ which minimizes the cost rate function C(N, T ), and satisfies L(N ∗ , T ) > cr and L(N ∗ − 1, T ) < cr .
AN US
This proves conclusion (i) of the theorem. On the other hand, because
RT [cp + (cf − cp )F (T )] 0 F¯ (x)dx L(1, T ) = RT F¯ (A2 x)dx 0 > cp F¯ (T ) + cf F (T ),
M
it is not difficult to see that the optimal replacement policy N ∗ = 1 when cp F¯ (T ) + cf F (T ) > cr . The proof of conclusion (ii) is complete.
ED
Appendix B. Proof of Theorem 2
CE
PT
From equation (13), we have lim QN (T ) = 0. Since F¯ (t) = e− T →0
RA
Rt 0
r(s)ds
, t > 0, we have
T
k R Ak T F¯ (Ak T ) e− 0 r(s)ds = lim e− A1 T r(s)ds . = lim lim ¯ R A1 T r(s)ds T →∞ e− 0 T →∞ T →∞ F (A1 T )
(28)
Noticing that A1 = 1, Ak > 1 for k ≥ 2, and r(t) is increasing in t, it can be derived that
AC
F¯ (Ak T ) lim ¯ = 0. T →∞ F (A1 T )
(29)
Using (29), it can be obtained that
lim
T →∞
NP −1
An r(An T )F¯ (An T )
n=1
N P F¯ (An T )
n=1
= lim
T →∞
NP −1
An r(An T )F¯ (An T )/F¯ (A1 T )
n=1
N P F¯ (An T )/F¯ (A1 T )
n=1
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= r(∞).
(30)
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Combining the limiting results lim F (An T ) = 1,
T →∞
lim
T →∞
Z
T
0
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and λ F¯ (Ak x)dx = , An
we have
If (14) holds, then
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N X 1 lim QN (T ) = r(∞)λ − (N − 1). T →∞ An n=1
lim QN (T ≥
T →∞
(N − 1)cp + cr , cf − cp
(31)
(32)
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which implies that there exists a finite T ∗ satisfying (15). This completes the proof of (i). If (16) holds, then
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QN (∞) ≤
(N − 1)cp + cr , cf − cp
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which means that T ∗ = ∞. This proof is completed. References
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[1] Y. Lam, Geometric processes and replacement Problem, Acta Math. Appl. Sin. 4(4) (1988) 366-377.
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[2] I.T. Castro, E.L. Sanjuan, Power processes and their application to reliability, Operat. Res. Lett. 32 (2004) 415-421. [3] W.J. Braun, W. Li, Y.Q. Zhao, Properties of the geometric and related processes, Naval Research Logistics 52 (2005) 607-616. [4] Y.L. Zhang, G.J. Wang, An extended geometric process repair model for a cold standby repairable system with imperfect delayed repair, Inter. J. Sys. Sci. O & L 3(3) (2016) 163-175.
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[5] R.E. Barlow, L. Hunter, Optimum Preventive Maintenance, Operat. Res. 8 (1960) 90-100.
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[6] H. Wang, A survey of maintenance policies of deteriorating systems, Eur. J. Oper. Res. 139 (2002) 469-489. [7] Y.L. Zhang, A geometric process repair model with good-as-new preventive repair, IEEE Trans. Reliab. 51 (2002) 223-228. [8] T. Nakagawa, Optimal policy of continuous and discrete replacement with minimal repair at failure, Naval Res. Logist. 31(4) (1984) 543-550.
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[9] S. Sheu, W.S. Griffith, T. Nakagawa, Extended optimal replacement model with random minimal repair costs, Eur. J. Oper. Res. 85 (1995) 636-649. [10] T. Nakagawa, Periodic and sequential preventive maintenance policies, J. Appl. Probab. 23(2) (1986) 536-542.
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[11] T. Nakagawa, Sequential imperfect preventive maintenance policies, IEEE Trans. Reliab. 37 (1988) 295-298.
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[12] R.I. Zequeira, C. B´ erenguer, Periodic imperfect preventive maintenance with two categories of competing failure modes, Reliab. Eng. Sys. Safe. 91 (2006) 460-468.
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[13] S.H. Sheu, C.C.Chang, Extended periodic imperfect preventive maintenance model of a system subjected to shocks, Inter. J. Sys. Sci. 41(10) (2010) 1145-1153.
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[14] D. Lin, M.J. Zuo, R.C.M. Yam, Sequential imperfect preventive maintenance models with two categories of failure modes, Naval Res. Logist. 48 (2001) 173-183.
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[15] S.H. Sheu, C.C.Chang, Y.L. Chen, An extended sequential imperfect preventive maintenance model with improvement factors, Commun. Statist.- theor. Methods 41 (2012) 1269-1283. [16] C.Q. Zhong, H.B. Jin, A novel optimal preventive maintenance policy for a cold standby system based on semi-Markov theory, Eur. J. Oper. Res. 232 (2014) 405411.
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[17] Z.L. Lin, Y.S. Huang, C.C. Fang, Non-periodic preventive maintenance with reliability thresholds for complex repairable systems, Reliab. Eng. Sys. Safe. 136 (2015) 145-156.
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[18] X. Liu, W. Wang, R. Peng, An integrated production and delay-time based preventive maintenance planning model for a multi-product production system, Maint. Reliab. 17(2) (2015) 215-221. [19] X.F. Zhao, H.C. Liu, T. Nakagawa, Where does whichever occurs first hold for preventive maintenance modelings? Reliab. Eng. Sys. Safe. 142 (2015) 203-211.
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[20] B. Emami-Mehrgani, W. P. Neumann, S. Nadeau, M. Bazrafshan, Considering human error in optimizing production and corrective and preventive maintenance policies for manufacturing systems, Appl. Math. Model. 40(3) (2016) 2056-2074.
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[21] V. Babishin, S. Taghipour, Optimal maintenance policy for multi-component systems with periodic and opportunistic inspections and preventive replacements, Appl. Math. Model. 40(23-24) (2016) 10480-10505.
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[22] G.Q. Cheng, L. Li, A geometric process repair model with inspections and its optimisation, Inter. J. Sys. Sci. 43 (2012) 1650-1655. [23] M. Jain, R. Gupta, Optimal replacement policy for a repairable system with multiple vacations and imperfect fault coverage, Comput. Ind. Eng. 66(4) (2013) 710-719.
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[24] M. Zhang, M. Xie, O.Gaudoin, A bivariate maintenance policy for multi-state repairable systems with monotone process, IEEE Trans. Reliab. 62(4) (2013) 876-886.
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[25] G.J. Wang, Y.L. Zhang, Optimal replacement policy for a two-dissimilar-component cold standby system with different repair actions, Inter. J. Sys. Sci. 47(5) (2016) 1021-1031. [26] C.Y. Niu, X.L. Liang, B.F. Ge, X.Tian, Y.W. Chen, Optimal replacement policy for a repairable system with deterioration based on a renewal-geometric process, Ann. Operat. Res. 244(1) (2016) 49-66. [27]Y.L. Zhang, G.J. Wang, An Optimal Age-replacement Policy for a Simple Repairable System with Delayed Repair, Commun. Statist.- theor. Methods, 46(6) (2017) 28372850. 27
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[28] Y. Lam, A geometric process maintenance model with preventive repair, Eur. J. Oper. Res. 182 (2007) 806-819.
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[29] G.J. Wang, Y.L. Zhang, A bivariate mixed policy for a simple repairable system based on preventive repair and failure repair, Appl. Math. Model. 33 (2009) 3354-3359. [30] G.J. Wang, Y.L. Zhang, Geometric process model for a system with inspections and preventive repair, Comput. Ind. Eng. 75 (2014) 13-19. [31] Y. Lam, The geometric process and its applications, World Scientific, Singapore, 2007.
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[32] S.M. Ross, Stochastic Processes, Wiley, New York, 1996.
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←− ξ1 −→ ←− ξ2 −→ ←− ξ3
−→ ←− ξ4
−→←− ξ5 −→ ←− ξ6 −→
− − − − − ◦ − − − − − ◦ − − − − − × − − − − − ◦ − − − − − × − − − − − ◦ ··· ←− T −→ ←− T −→ ←− X3 −→ ←− T −→ ←− X5 −→ ←− T −→
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CR IP T
Fig. 1. A possible course of system functioning, where 0 ×0 denotes the system failure time, 0 ◦0 represents the scheduled PM time, and ξk = min{Xk , T }.
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CR IP T
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C(T,N)
20 15 10
30
5 400
20
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300
10
200
T
100
0
N
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Fig. 2. The plots of C(N, T ) against N and T .
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0.07 0.065 0.06
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0.055
∆(N)
0.05 0.045 0.04 0.035
0.025 0.02
0
10
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0.03
20
30 N
40
50
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Fig. 3. The plots of ∆(N ) against N .
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60