GENERALIZED PERIODIC REPLACEMENT POLICIES FOR REPAIRABLE SYSTEMS SUBJECT TO TWO TYPES OF FAILURES

. The purpose of this current research is to schedule generalized periodic replacement policies for a single unit system executing random working jobs. The system is subject to two types of failures when it has failed, including a minor failure (Type I failure), which can be thoroughly removed by the minimal repair and a catastrophic failure (Type II failure), which should be rectified by the corrective replacement. To be specific, four distinct periodic replacement models including a periodic replacement first model (Model A1), a modified periodic replacement first model (Model A2), a periodic replacement last model (Model B1)


Introduction
Maintenance activities, especially replacement behaviors are widely arranged in advance to avoid disastrous systems failures and to decrease economic losses in various industrial scenes.Generally, replacement operations where the procedure time is arranged before system failure and after system failure are called as preventive replacement and corrective replacement, respectively [18].Replacing a system either too frequently or too less is not advisable as it increases unnecessary maintenance costs or reduces system availability.Therefore, scheduling the optimal replacement cycle and numbers according to some criteria, such as long-run average cost rate (ACR), expected long-run profit rate, or system availability is attracting more and more attention in maintenance activity [3].
Age replacement (AR) models and periodic replacement (PR) models are two fundamental replacement policies in preventive maintenance theory and they have been extensively studied in the past few decades [9,13].Barlow and Hunter [4] investigated the AR policy for a single unit system, in which the original system is replaced with an auxiliary system at a constant age  after its operation or at system failure, whichever occurs first.Different from AR policies, PR models are more practical since they do not need to keep records of usage time, where the system is periodically replaced at  ( = 1, 2, . ..) and only minimal repair at system failure is addressed such that system failure rate is undisturbed by any repair for failures between two proximate replacements [8,18].Various replacement models and their theoretical computations are sufficiently analyzed [1,7,12,14].
In the late 1990s, power companies in Brazil would be penalized stupendously for non-scheduled repairs when a major overhaul of the electrical power sector happened.Therefore, the company wanted to adopt a preventive maintenance policy, as opposed to repair actions adopted after failures and periodic visual inspections were arranged for the power switch disconnectors.While some potential failures are detected and fixed at inspection time, some other failures cannot be predicted through periodic visual inspection but only by a preventive maintenance.It is evident that removing potential failures at inspection is less expensive than that the failures have happened.
To deal with the above question in theory, Colosimo et al. [11] treated these two kinds of failures as events governed by two non-homogeneous Poisson processes (NHPPs) for the random occurrence of failures, while the concept of two categories of failures was initially put forward in 1980s.In 1983, the definition of two kinds of failures (i.e., a minor failure and a major failure) was explicated by Brown and Proschan [6], in which the minor failure is non-fatal and can be thoroughly rectified by a minimal repair, and the major failure is fatal and should be removed by a corrective replacement.Later, Sheu et al. [30] studied a system with age-dependent failures and random working missions, and developed three generalized age maintenance policies, where the system is also subject to two failure types (Type I failure and Type II failure).The system is replaced ahead at a planned age  or at the completion of the  th working mission, or correctively replaced at the first occurrence of Type II failure.Wang et al. [27] extended the generalized age replacement policies for a single-unit system into a series system and a parallel system with  non-identical components, where two types of failures for both systems were considered as well.Other maintenance policies on systems incorporated with two types of failures can be found [2,10,25,26].The NHPP serves as an effective way to deal with different kinds of failures when system failure can be categorized into distinct types.
When considering multiple type conditions for maintenance policies, a classical assumption is "whichever occurs first" [20,24].Such an assumption is more reasonable for situations where failures may bring catastrophic production interruptions.It should be noted that maintenance actions of "whichever triggering event occurs first" may be more frequent when several combined policies are scheduled [17,23].In addition, it would be inappropriate to arrange a strict replacement on time at a planned age  or at periodic cycles  ( = 1, 2, . ..) especially when the system needs to finish some successive working missions because any interruption of working periods may incur tremendous losses of production to different degrees.Therefore, it would be not wise to replace the system until the job is completed even though the scheduled maintenance time has arrived [19,22].By considering the above aspects, the concept of "whichever occurs last" is developed and has been investigated considerably [29,32].
To the best of our knowledge, more generalized periodic replacement policies for a system subject to two types of failures (namely, Type I failure and Type II failure used in previous researches) with random working periods have not been completely addressed yet, while their models are analytically investigated in this paper.It is assumed that the system needs to execute several random working periods  1 ,  2 , . . .,   during its operation and is subject to two types of failures when it has failed, where Type I failure is called as a minor failure and can be removed by a minimal repair and Type II failure is a catastrophic failure which requires a corrective replacement or an overhaul.Type I failure occurs with a probability  (0 ≤  ≤ 1) and Type II failure occurs with another probability  ≡ 1 − .In the first generalized periodic replacement model, the system is replaced at age  (0 <  ≤ ∞), or at the completion of  working missions, or at the first occurrence time of a Type II failure, whichever comes first.While in the second generalized periodic replacement model, the system is replaced at age  (0 <  ≤ ∞), or at the completion of  working times, or at the first occurrence time of Type II failure, whichever comes last.Except for the two above replacement models, their respective extended models are developed as well.The ACR function is minimized to seek the optimal replacement cycle in each model.The remainder of this paper is organized as follows.Notations and some assumptions are offered in Section 2. Sections 3 and 4 investigate the generalized periodic replacement policies under the assumption of "whichever occurs first" and "whichever occurs last", respectively.In both models, theoretical computations are derived and numerical examples are given to verify the results.Finally, some conclusions are summarized in Section 5.
Two types of failures are introduced for the deterioration system when it has failed at . Type I failure (minor failure) is occurred with a probability  (0 ≤  ≤ 1) and it can be removed by a minimal repair, where minimal repair means that system failure rate () remains undisturbed by any maintenance [16,31].Whereas Type II failure (catastrophic failure) is formed with another probability  ≡ 1 − , resulting in a total breakdown and needing a corrective replacement to rectify it [33].
The preventive replacement costs at periodic times  ( = 1, 2, . ..) and at the completion of random working periods  1 ,  2 , . . .,   are   and   , respectively.The corrective replacement cost at the first Type II failure is   , and the maintenance cost for each minimal repair is   .It is set that   >   >   .In addition, the preparation time for every maintenance activity including the replacement and the minimal repair is negligible.
Q. LI ET AL.

Periodic replacement first policies
According to the assumptions, system failure is subject to events following an NHPP with intensity (), increasing strictly with respect to  from (0) = 0 to (∞).Denote { 1 (),  ≥ 0} and { 2 (),  ≥ 0} as the respective counting numbers of Type I failures and Type II failures in [0, ].Then, the processes { 1 (),  ≥ 0} and { 2 (),  ≥ 0} are two independent NHPPs with intensities () and (), respectively [28].Let  be the waiting time until the first occurrence of Type II failure in time interval [0, ], i.e., The survival function of  is The mean number of Type I failures in time interval [0, ] is Let   be the length of the th ( = 1, 2, . ..) replacement cycle and   be the cost over the replacement cycle   .Then, {  ,   } constitutes a renewal reward process.Defining () as the expected cost of the operating system over the time interval [0, ], according to the renewal reward theorem [5,15], we have

Periodic replacement first policy (Model A1)
For Model A1, we consider the following replacement situations in a renewal cycle and derive the corresponding probabilities.
(1) The probability that the system is preventively replaced at periodic times  ( = 1, 2, . ..) is Pr{ <   ,  < } =   ( )  ( ), (3.5) in which   = min { 1 ,  2 , . . .,   } and (2) The probability that the system is preventively replaced at the completion of random working jobs is (3) The probability that the system is correctively replaced at the first occurrence of Type II failure is where should note that Pr{ < ,  < It is clear that each replacement time for the deterioration system is a regeneration point, and therefore, the expected length of the first renewal cycle  1 is The total mean number of Type I failures before replacement is The expected maintenance cost of the first renewal cycle is According to (3.4), the ACR for Model A1 is In order to find the optimal  *  minimizing   ( ) in (3.13) for an infinite time horizon, we differentiate   ( ) with respect to  and set it equal to zero.From d  ( )/d = 0,  *  satisfies where Then, the optimal  *  is obtained according to the following theorem.
(2) When  → ∞, i.e., the system is replaced at the completion of random working jobs, or at the first occurrence of Type II failure, whichever occurs first.  ( ) in (3.13) becomes (3) When  = 1,   → ∞, i.e., the system undergoes only minimal repair at failure and no random working times are considered, then   ( ) in (3.13) becomes which is the classical periodic replacement policy.

Modified periodic replacement first policy (Model A2)
In this section we develop a modified periodic replacement first policy (Model A2) based on Section 3.1.Suppose that the system is preventively replaced at the periodic time points  ( = 1, 2, . ..), or at when at least one of the  random working times is longer than  , or correctively replaced at the first time of Type II )︂ ()  ()d )︂ d In order to find the optimal ̃︀  *  which minimizes ̃︀   ( ) in (3.18) for an infinite time horizon, we differentiate ︀   ( ) with respect to  and set it equal to zero.From d ̃︀   ( )/d = 0, we have where Then, the optimal ̃︀  *  is obtained according to the following theorem.

Numerical examples
In this section, numerical examples are given to verify the theoretical results obtained.Assume that system failure time follows a Weibull distribution, i.e.,  () = 1 − e −0.01 2 .The th ( = 1, 2, . . ., ) working time is exponentially distributed with   () = 1−e −0.1 .For convenient computation, the following costs are introduced:   = 500,   = 750,   = 1000, and   = 100.Tables 1 and 2 1 shows the average cost rate   ( ) for different  in terms of  = 1 for Model A1 and Figure 2 shows the average cost rate ̃︀   ( ) for different  in terms of  = 1 for Model A2, where  = 1 illustrates that the failure rate of the system is undisturbed by any shocks.From Figures 1 and 2, it is clear that the finite and unique replacement intervals  *  and ̃︀  *  exist when  = 1, i.e., 0 <  *  < ∞ and 0 < ̃︀  *  < ∞.

Periodic replacement last policies
Implementing replacement first policies may lead to too frequent unnecessary replacement, as well as interrupting random working jobs.In this case, we develop generalized periodic replacement last models.The system is preventively replaced at periodic cycles  ( = 1, 2, . ..) before Type II failure, or at the completion of random working times, whichever occurs last.Corrective replacement is arranged immediately at the first occurrence of Type II failure.

Periodic replacement last policy (Model B1)
For Model B1, the following three distinct situations are considered and their corresponding probabilities are derived.
(1) The probability that the system is preventively replaced at periodic times  ( = 1, 2, . ..) is (2) The probability that the system is preventively replaced at the completion of  random working jobs is (3) The probability that the system is correctively replaced at the first occurrence of Type II failure is where should note that Pr{ < , The expected length of the first renewal cycle is The total mean number of Type I failures before replacement is The expected maintenance cost in a renewal cycle is Q. LI ET AL.
According to (3.4), the ACR for Model B1 is In order to find the optimal  *  which minimizes   ( ) in (4.7) for an infinite time horizon, we differentiate   ( ) with respect to  and set it equal to zero.From d  ( )/d = 0, we have where Then, the optimal  *  is obtained according to the following theorem.
Remark 4.2.(1) When  = 1, i.e., the system undergoes only minimal repair at failure,   ( ) in (4.7) becomes (2) When  → ∞, i.e., the system is replaced at the completion of random working times, or at the first occurrence of Type II failure, whichever occurs last.  ( ) in (4.7) becomes

Modified periodic replacement last policy (Model B2)
In this section we develop a modified periodic replacement last policy (Model B2) based on Section 4.1.Suppose that the system is preventively replaced at the first completion of  1 among  random working jobs after the periodic cycle  , or at periodic cycles when at least one of the  random working jobs is less than  , or correctively replaced at the occurrence of Type II failure, whichever occurs last.Replacing   () =

Numerical examples
In this section, we use the same parameters with them in Section 4.3 to verify the theoretical results for the generalized periodic replacement last models, i.e., system failure distribution is  () = 1 − e −0.01 2 , the distribution of the th ( = 1, 2, . . ., ) working time is   () = 1 − e −0.1 , and the replacement costs are   = 500,   = 750,   = 1000, and   = 100.Tables 3 and 4 show the optimal replacement cycles  *  and ̃︀  *  , and the corresponding minimized maintenance cost rates   ( *  ) and ̃︀   ( ̃︀  *  ), respectively.Tables 3 and 4

Conclusions
We have investigated preventive replacement policies in this paper and constructed four models, i.e., a periodic replacement first model (Model A1), a modified periodic replacement first model (Model A2), a periodic replacement last model (Model B1), and a modified periodic replacement last model (Model B2).In each modeling framework, the infinite time span is considered and average replacement cost rate is minimized to seek the optimal replacement interval.All discussions have been conducted analytically and examined numerically.For the generalized periodic replacement first policy and generalized periodic replacement last policy, both the optimal replacement intervals  *  and  *  increase with the number of random working periods , while on the contrary, both ̃︀  *  and ̃︀  *  for the modified generalized periodic replacement first policy and modified generalized periodic replacement last policy decrease with .The developed maintenance models in this paper have potential applications in practical products such as the unmanned aerial vehicle (UAV), micro-electro-mechanical system (MEMS), and gyroscope in the inertial navigation system (INS) as soon as their operating conditions satisfy the assumptions in Section 2.
For future research, we should consider the condition that times for repair and replacement are not neglected.In addition, more complex maintenance models should be developed on reliability for deterioration systems as they are capable of describing the sophisticated degrading behaviors of engineering systems.
show the optimal replacement cycles  *  and ︀  *  , and the corresponding minimized maintenance cost rates   ( *  ) and ̃︀   ( ̃︀  *  ) for Model A1 and Model A2, respectively.