We consider optimal replacement policies with periodic imperfect maintenance actions and minimal repairs. The multistate system is minimally repaired at failure and imperfect maintenance actions are regularly carried out for preventive maintenance. The discrete modified Weibull distribution is introduced and some cost functions applied to this distribution are defined in order to be minimized. Moreover, we assume that the costs of preventive maintenance depend on the degree of repair via a Kijima type 2 model. For illustrative purpose, the obtained results are applied on sets of simulated data.
In reliability theory, discrete failure data arise in several common situations. It is sometimes impossible to measure the life length of items with continuous scale. A discrete distribution is appropriate when we have a multistate system and the number of states prior to preventive maintenance is observed. A few results on discrete life distributions are introduced in the literature (Salvia and Bollinger [
Recently, in reliability, much attention has been paid to the evaluation and application of multistate systems (MSS). We note that the MMS model provides a more flexible tool for modeling engineering systems in real life. This model is at first introduced in Barlow and Wu [
The most commonly used models for the failure process of a repairable system are known as perfect repair or as good as new and as minimal repair or as bad as old.
In the perfect repair case, each repair is perfect and leaves the system as if it were new; hence we obtain a renewal processes (RP). In the minimal repair case, each repair leaves the system in the same state as it was before failure, so we obtain nonhomogeneous Poisson processes (NHPP). It is well known in practice that the reality is between these two extreme cases. The repair may not yield a functioning item which is as good as new and the minimal repair assumption seems to be too pessimistic in repair strategies. From this it is seen that the imperfect repair is of great signification in practice.
Furthermore, the imperfect maintenance is a broader subject, in this way many authors have considered other aspects associated with the imperfect maintenance that corresponds to errors when doing the maintenance activities. In fact, when the maintainer makes a mistake it may correspond not only in terms of cost, but also in terms of reliability.
Currently, Berrade et al. [
In this paper, we are concerned with the modeling of an imperfect maintenance model; that is, the impact of preventive maintenance is between the two boundary cases: (a) minimal (as bad as old) and (b) perfect (as good as new).
Further, it is assumed that costs of preventive maintenance are not constant but depend on the degree of repair via a virtual age process.
A (MSS) with
In order to model the optimal maintenance actions, a new discrete modified Weibull distribution is introduced. At last some possible cost functions are proposed and optimal maintenance strategies are discussed.
Most operating units (items) are repaired or replaced when they have failed. However, it may need much time and high cost to repair a failed item, so it is essential to maintain it in order to prevent failures and to reduce costs.
After failure the maintenance is called corrective maintenance (CM) and before failure it is called preventive maintenance (PM). Some PM policies and their optimization problems for shock and damage models are summarized in Nakagawa [
In practice, PM is used to lengthen the useful lifetime of items and to decrease average running cost by minimizing the occurrence of failures. Here we can note that most of the models concerning the modeling of repairable systems identify the minimal repair and the imperfect repair actions. Naturally, this popular assumption is a very unreal one.
In the following, a multistate system with
It is supposed that the reduction of the failure rate is done by using the concept of virtual age reduction introduced by Kijima [
The process defined by
In some situations, discrete lifetime distributions are appropriate to model lifetimes. The first discrete reliability distribution has been defined by Nakagawa and Osaki [
We note that the modified Weibull distribution (MWD) is recently proposed by Sarhan and Zaindin [
Let
For the discretization of this distribution, we will put the probability mass of the interval
In the case of the DMWD we obtain
The failure rate of a discrete distribution has been defined by Barlow et al. [
They proposed a new redefined failure rate function for discrete distributions, denoted by
Both failure rates
Xie et al. [
For the DMWD
Figures
Discrete failure rates
Cumulative hazard functions
We suppose that the shape parameter
A sequential imperfect preventive maintenance policy was proposed in Nakagawa [
We consider sequential PM policies for the system, where the distance between PM actions is always equal to
Following the strategy of Kahle [
We consider maintenance policies for repairable systems in which the economical approach is our main objective. In our model each PM action reduces the age of the system to
Let
We assume that the cost function
Let the ratio of costs
Cost function for DMWD (
In Table
Optimal values for
DLFRD( 
DRD( 
DWD( 
DMWD(  




 





















































































We have used the following discrete distributions: linear failure rate distribution (DLFRD), Rayleigh distribution (DRD), Weibull distribution (DWD), and the modified Weibull distribution (DMWD).
We also compared the cost function for these distributions and we noted that the parameters were chosen so that the expectation was the same for all distributions. Let this expectation be 5.
In this case, we will assume that the cost function
Let the ratio of costs
Optimal values for
DLFRD( 
DRD( 
DWD( 
DMWD(  




 





















































































Cost function for DMWD (
The cost functions for distributions used in Section
Now, we consider a new cost function per time unit which depends on the degree of repair
We know that for the Kijima type 2 model, the degree of repair
Let
(a) For perfect repair,
(b) For minimal repair,
(c) If
(d) If
(i) Let
(ii) Let
Costs of large repairs.

2  3  4  8  20 







Costs of small repairs.

1/2  1/3  1/4  1/8  1/20 







The resulting cost function per time unit has then the following form:
Similar to Sections
Cost function for DMWD (
If
Optimal values for
DLFRD( 
DRD( 
DWD( 
DMWD(  




 





















































































Cost function for DMWD (
Once again, the choice of parameters of all used distributions in this table verified expectation 10.
The summary of computation results is presented in Table
(a) If
(b) If
(c) If the failure rate is DMWD or DWD the obtained results are quite similar.
(d) If the failure rate is that of DLFRD, then the optimal distance between PM actions is larger than that for the rest failure rates.
(e) If
(f) In the case of DMWD with parameters
(g) If we use the same parameters as in (f) and after replacing
We considered an incomplete maintenance model; that is, the impact of PM is not minimal (ABAO) and not perfect (AGAN) but lies between these boundary cases. The PM actions reset the failure rate of the item proportional to the virtual age. We assumed that the reduction of the failure rate is done by using the Kijima’s type 2 virtual age model. Further, all CM actions are minimal, which means that the item is minimally repaired at each failure.
In this research, the costs of PM actions are not constant but depend on (a) the state of the item just before the PM actions, (b) on the impact of repair, and (c) on the degree of repair. We note that the last case is very realistic and the obtained function can be well interpreted in praxis. Furthermore a periodic replacement policy is studied. We discussed the discrete failure data and the situations where the use of a continuous scale in order to measure the life length is quite artificial, so a DMWD is considered to describe the failure of the item under study.
The expected total optimal maintenance cost under discrete modified Weibull distribution (DMWD) is obtained. It is shown that the optimum for the cost function is unique.
Furthermore, we can consider models with imperfect repairs at failures. We expect that PM models will be applied to real systems and some results to these subjects will be studied in near future.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the editor and the anonymous reviewers for their invaluable comments and suggestions to improve the quality of the paper.