Minimal repair models with non-negligible repair time

doi:10.1016/j.ress.2021.108046

Reliability Engineering & System Safety

Volume 217, January 2022, 108046

https://doi.org/10.1016/j.ress.2021.108046 Get rights and content

Highlights

•
Generalize the classical minimal repair policies to the calendar time-based ones.
•
The proposed repair policies are more easily used in engineering applications.
•
Probability decomposition technique and indicator function method are developed.
•
An application of the proposed repair policy is presented in warranty field.
•
The influence of ignoring repair time on the system average cost rate is studied.

Abstract

In existing minimal repair models, it is usually supposed that the repair time of systems is negligible, and thus the occurrences of breakdowns obey non-homogeneous Poisson Process within a replacement interval. However, for the case that the mean working time of a system is not far longer than its mean repair time, ignoring repair time may lead to a large error in calculating the system long-run average cost rate. In this paper, taking random repair time into account, we generalize several calendar time-based minimal repair policies, including periodic replacement policy with minimal repair, reference time policy, and bivariate $T - N$ maintenance policy. By using probability decomposition technique and indicator function method, the system long-run average cost rate under each minimal repair policy is gained explicitly. Various special cases are discussed. An application of the periodic replacement policy with minimal repair is presented in warranty field. Finally, a comprehensive sensitivity analysis is conducted to investigate the relative error on the long-run average cost rate caused by ignoring repair time, and to compare calendar time-based minimal repair policies with age-based minimal repair policies in terms of the long-run average cost rate.

Introduction

If a repairable system fails at an early stage, it is economical to carry out the service of an emergency maintenance instead of an immediate replacement. To characterize the effect of a repair on the lifetime of a system, many types of repairs have been proposed and investigated, such as perfect repair [1], [2], minimal repair [3], [4], [5], [6], [7], [8], Geometric Process repair [9], [10], [11], [12], Generalized Polya Process repair [13], [14], and Non-Homogeneous Pure Birth Process repair [15], [16]. The concept of minimal repair was firstly introduced by Barlow and Hunter [17]. A minimal repair can make a faulty system back to the condition just before breakdown, and not change its failure rate. For instance, mending a blown-out tire or repairing a failure generator can be considered as a minimal repair for an automobile [18]. Due to aging and degradation, the failure rate of systems increases gradually, so it is unwise to repair them all the time without replacement. Then, the problem is when is the optimum time to replace a system instead of performing minimal repair?

Over the past few decades, lots of minimal repair policies have been proposed and compared to answer the question. Periodic replacement policy with minimal repair is one of the most classical minimal repair policies [19]. Under the policy, a system is substituted in periodic times $T, 2 T, \dots$ , and minimally repaired at failure during replacement intervals. The main advantage of the policy is ease of implementation, while a disadvantage is that the maintenance resource will be wasted if the system is replaced just after experiencing a repair (i.e., the case that the last failure time point is close to the replacement time point). To ovoid the waste, Muth [20] proposed a modified minimal repair policy where the system is minimally repaired before time $T$ and substituted at the first failure after time $T$ . In literature [21], the policy was called as reference time policy. On the other hand, according to the system failure number, Makabe and Morimura [22] introduced another classical repair policy in which minimal repairs are carried out for the first $N - 1$ breakdowns and replacement action is conducted for the $N$ th breakdown. The policy was later called as repair number counting policy [21]. Phelps [23] compared periodic replacement policy with minimal repair, reference time policy, and repair number counting policy. The results showed that reference time policy is the optimum one in terms of long-run average cost rate when the system failure rate is increasing. Later on, Zhao et al. [24] proposed an extended minimal repair policy, in which a system is substituted at the $N$ th breakdown or the first breakdown after time $T$ , whichever happens first. In this paper, the policy will be called as bivariate $T - N$ maintenance policy. More types of minimal repair policies have been summarized and reviewed in [21].

In existing minimal repair models, the repair time of systems is usually not taken into account. Although ignoring repair time is convenient for us to analyze and solve the models, it may have some flaws under some cases. For example, for the system whose reliability (repair time) is not very high (short), the Mean Time To Repair (MTTR) is not far shorter than the Mean Time To Failure (MTTF), and thus assuming that the occurrences of breakdowns obey non-homogeneous Poisson process during a replacement interval $[0, T]$ will lead to a large error in calculating the long-run average cost rate. Besides, with the increasing of interval length $T$ , the total number of repairs increases during the replacement interval, and thus the error caused by ignoring repair time increases.

In this paper, by taking random repair time into consideration, we firstly study several calendar time-based minimal repair policies where replacement activities are implemented according to a predetermined calendar time, including periodic replacement policy with minimal repair, reference time policy, and calendar time-based bivariate $T - N$ maintenance policy. By employing probability decomposition method, the system long-run average cost rate under the three different minimal repair policies is derived explicitly. An application of periodic replacement policy with minimal repair is also presented. In the numerical example section, we compare these three calendar time-based minimal repair policies, and study the effects of ignoring system’s maintenance time on the long-run average cost rate.

When the repair time of a system is considered, in addition to calendar time-based minimal repair policies, there also exist age-based minimal repair policies where replacement activities are implemented according to the system age, like age-based replacement policy with minimal repair, reference age policy, and age-based bivariate $S - N$ maintenance policy. For the sake of comparing the calendar time-based minimal repair policies and age-based minimal repair policies, the system long-run average cost rate under the above three age-based minimal repair policies are also formulated. Lots of numerical experiments are conducted, and the results show that the reference time policy is more recommended if the replacement cost is relatively high, the repair cost and downtime cost are relatively low, and the repair time required is relatively short.

The two main contributions of this paper are:

•
Taking the random repair time of systems into account, we propose several calendar time-based minimal repair policies to generalize the classical ones.
•
The probability decomposition and indicator function method are flexible. Many well-known minimal repair models can be generalized (i.e., taking repair time into account) by adopting them.

The paper is organized as below. Section 2 presents notations as well as assumptions. Section 3 gains the cost rate under periodic replacement policy with minimal repair. Other classical calendar time-based minimal repair policies are studied in Section 4, and age-based minimal repair strategies are studied in Section 5. Examples are put into Section 6 to demonstrate these minimal repair policies considering repair time. Section 7 displays conclusions.

Section snippets

Notations

$X$	lifetime of system
$F (t), λ (t), λ$	distribution, failure rate, and expectation of $X$ , respectively
$Y_{i}$	the $i$ th system’s maintenance time
$G (t), g (t), μ$	distribution, density function, and expectation of $Y_{i}$ , respectively
$G^{(i)} (t)$	convolution of repair time $Y_{1}, Y_{2}, \dots, Y_{i}$ ( $G^{(0)} (t)$ =1)
$M (t)$	repair number during $[0, t]$
$X_{i}$	the $i$ th working time of system
$F_{i} (t)$	distribution of $X_{1} + \dots + X_{i}$ ( $F_{0} (t)$ =1)
$c_{m}, c_{r}$	cost for a repair and replacement, respectively
$c_{d}$	cost rate of downtime
$U (T)$	working time in a renewal cycle under periodic

Periodic replacement policy with minimal repair

Under periodic replacement policy with minimal repair, a system is substituted at periodic time point $k T (k = 1, \dots)$ and minimally repaired at breakdown [19]. In this section, taking repair time into consideration, we first gain the long-run average cost rate of the system, and then provide an application of the repair policy in warranty field.

Some other types of calendar time-based minimal repair policies

In this section, taking repair time into account, we study some other classical types of minimal repair policies, including reference time policy [20], repair number counting policy [22], and calendar time-based bivariate $T - N$ maintenance policy [24].

Age-based minimal repair policies

In the previous sections, the minimal repair policies discussed are calendar time-based, i.e., replacement activities are implemented according to a predetermined calendar time. When the repair time is considered, there also exists another type of minimal repair policies where replacement activities are age-based, i.e., a system will be substituted when its cumulative operating time (age) reaches a predetermined value. Intuitively, age-based repair policies can achieve a lower expected cost

Numerical examples

In this section, numerical experiments are conducted to show the research results achieved in Sections 3–5. As mentioned in Section 3.3, the maintenance time required for vehicle products is sometimes not allowed to be ignored. Because the maintenance actions, like adjusting a steering wheel, replacing a failed generator, repairing a failed compressor, or fixing a blown-out tire, do not change the whole performance of a vehicle, it was assumed that failed vehicles are minimally repaired in some

Conclusions and future works

In this paper, by taking the random repair time of systems into account, we have studied several classical minimal repair policies existing in the literature, including calendar time-based minimal repair polices and age-based minimal repair polices. Since the occurrences of system breakdowns do not obey non-homogeneous Poisson Process within a time interval for the case of considering repair time, the theoretical results based on non-homogeneous Poisson Process cannot be directly applied. To

CRediT authorship contribution statement

Peng Liu: Methodology, Data curation, Writing – original draft, Investigation, Software, Visualization. Guanjun Wang: Conceptualization, Supervision, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors are indebted to the Associate Editor and the three anonymous referees for their constructive suggestions. This work was supported by the National Natural Science Foundation of China under Grant No. 11671080.

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View full text

Minimal repair models with non-negligible repair time

Highlights

Abstract

Introduction

Section snippets

Notations

Periodic replacement policy with minimal repair

Some other types of calendar time-based minimal repair policies

Age-based minimal repair policies

Numerical examples

Conclusions and future works

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgments

Comput Ind Eng

Appl Math Model

Reliab Eng Syst Saf

Reliab Eng Syst Saf

Reliab Eng Syst Saf

Appl Math Model

Reliab Eng Syst Saf

European J Oper Res

Appl Math Model

Comput Ind Eng

European J Oper Res

European J Oper Res

European J Oper Res

Reliab Eng Syst Saf

Reliab Eng Syst Saf

Availability analysis and optimal inspection policy for systems with neglected down time

Comm Statist Theory Methods

Optimal age-replacement in anticipation of time-dependent, unpunctual policy implementation

IEEE Trans Reliab