Optimum maintenance strategy under uncertainty in the lifetime distribution

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Highlights

  • The effect of parameter uncertainty on the optimum age-based maintenance strategy is studied.

  • A theoretical uniform and a realistic Weibull distribution are considered.

  • Admitting to the uncertainty provides a quantifiable cost benefit.

  • The maintenance age first decreases and then increases as the uncertainty increases.

Abstract

The problem of determining the optimal maintenance strategy for a machine given its lifetime distribution has been studied extensively. Solutions to this problem are outlined in the academic literature, prescribed in professional handbooks, implemented in reliability engineering software systems and widely used in practice. These solutions typically assume that the lifetime distribution and its parameter values are known with certainty, although this is usually not the case in practice. In this paper we study the effect of parameter uncertainty on the optimum age-based maintenance strategy. The effect of uncertainty is evaluated by considering both a theoretical uniform lifetime distribution and a more realistic Weibull lifetime distribution. The results show that admitting to the uncertainty does influence the optimal maintenance age and also provides a quantifiable cost benefit. The results can help maintenance managers in making maintenance decisions under uncertainty, and also in deciding when it is worthwhile to invest in advanced data improvement procedures.

Introduction

It is widely accepted that preventive maintenance is important for achieving Operational Excellence [10], [34], since it aids in reducing system downtime. Preventive maintenance policies can roughly be subdivided into two categories, namely condition-based and time-based. Recent advances in sensor technology have lead to increased popularity of condition-based maintenance. However, condition monitoring may be technically impossible for some assets, the benefits of condition-based maintenance may not outweigh the investment costs required to enable condition monitoring, and condition-based maintenance activities are more difficult to plan than activities that are fixed in time. Due to these limitations of condition-based maintenance, much of the preventive maintenance in practice is still time-directed.

An important type of time-directed maintenance is age-based maintenance [16]. The effectiveness of this maintenance strategy is determined by the age at which preventive maintenance takes place. Early (and therefore frequent) maintenance actions result in a high maintenance cost per unit time. Late (or infrequent) maintenance actions result in a higher probability of failure (and costs associated with failure). There are widely used handbooks [12], [16], [1], [32] and software systems [16], [8] that prescribe how to determine the optimal replacement age given the component lifetime distribution. These systems generally require the specification of the lifetime distribution and its parameter values, and do not allow for potential uncertainty in these inputs.

There are several reasons why estimates of equipment lifetime distributions may not be accurate. Firstly, vendor guidelines may not be (fully) compatible due to lack of knowledge of the actual use and maintenance of the equipment [27], [39]. Furthermore, maintenance and reliability engineers have bemoaned the lack of credibility in collected data for years [25], [11], [6]. Maintenance records and historic failure data are often inaccurate or incomplete. A third source of uncertainty is the fact that historic failure data are likely to be (heavily) right-censored because of preventive maintenance in the past [7]. Finally, there is often an insufficient amount of data to determine accurate estimates for the model parameters.

The consequences of uncertainty in the lifetime distribution in terms of the optimum maintenance policy and the cost reduction that can be achieved by including the uncertainty did not receive much attention yet, and is the focus of the current paper. The approach that we follow is to accept the current modus operandi of many maintenance engineers by assuming a pre-defined distribution (e.g., uniform or Weibull), and to include uncertainty in its parameters, rather than taking point estimates. The results will show that admitting to the uncertainty does influence the optimal maintenance age. The significance of this influence ranges from very little to quite substantial, depending on the circumstances. This means that, in some cases, it is essential to take uncertainty into account.

This paper is organized as follows. In Section 2, we discuss the existing literature. In Section 3, we formally describe the problem we consider as well as our approach. In Section 4, we evaluate the uncertainty by considering a simple setting with a uniform lifetime distribution, in which the optimum maintenance age can be obtained explicitly. In Section 5, we evaluate a more realistic setting, with uncertainty in the scale parameter of the Weibull distribution, which we will evaluate numerically. Section 6 provides conclusions and extensions for future research.

Section snippets

Literature

Time-based preventive maintenance was first studied by Barlow and Hunter [3]. One of the two introduced policies, age-based maintenance, is further studied by Glasser [13], Tadikamalla [38], and Nakagawa and Yasui [28]. These papers all assume that the component lifetime distribution is known with certainty. Examples of other studies that make this assumption are Kijima et al. [19], Makis and Jardine [24] and Jiang et al. [17], who report on the periodical replacement problem with repairs at

Approach

The age-based maintenance strategy considers a single unit with lifetime distribution F. When the unit fails, an emergency repair is performed at normalized cost 1. Furthermore, when the unit reaches a specified age T, a preventive maintenance action is performed at cost c<1. Both after an emergency repair and after a preventive maintenance action, the unit is assumed to be as-good-as-new. The cost rate of the age-based maintenance strategy isηage(T)=F(T)+(1F(T))c0T(1F(x))dx.This formula was

Uniformly distributed lifetime

We start our study with a uniformly distributed lifetime. Although unrealistic in many cases, an important advantage of this distribution is that it is relatively simple to get the relevant input from a maintenance engineer as only the minimum and maximum lifetime is needed. As maintenance will clearly not take place before the minimum lifetime is reached, we set it to zero in our model. The distribution function of the uniform distribution on an interval [0,s] is F(t;s)={0,t<0,ts,0ts,1,t>s.

Weibull distributed lifetime

We continue our study with a more realistic case, namely that of a unit with a Weibull distributed lifetime, and show that the insights obtained in the previous section carry over. The Weibull distribution is the most commonly used distribution to model lifetimes and has been found to provide a good description of many types of lifetime data. For systems with multiple critical units, the lifetime approximately follows a Weibull distribution [23]. Other physical phenomena for which the Weibull

Conclusions and future extensions

We have studied the optimal age-based maintenance strategy under uncertainty in the lifetime distribution of a unit. Although this uncertainty is usually present in practice, it is ignored by most existing research and software. The lifetime distribution is often assumed to be known, or an estimate based on available data is considered as the true lifetime distribution. We considered certain lifetime distributions and studied the effect of parameter uncertainty on the optimal preventive

References (42)

  • E. Silver et al.

    Preventive maintenance with limited historical data

    Eur J Oper Res

    (1995)
  • T. Tinga

    Application of physical failure models to enable usage and load based maintenance

    Reliab Eng Syst Saf

    (2010)
  • R.H. Yeh et al.

    Optimal preventive-maintenance warranty policy for repairable products

    Eur J Oper Res

    (2001)
  • R. Zhang et al.

    Model uncertainty and Bayesian updating in reliability-based inspection

    Struct Saf

    (2000)
  • R. Abernethy

    The new Weibull handbook

    (2006)
  • R. Baker et al.

    Can models fitted to small data samples lead to maintenance policies with near-optimum cost?

    IMA J Manag Math

    (1995)
  • R. Barlow et al.

    Optimum preventive maintenance policies

    Oper Res

    (1960)
  • R. Barlow et al.

    Mathematical theory of reliability

    (1965)
  • J. Bernardo et al.

    Bayesian theory

    (1993)
  • A.J.J. Braaksma et al.

    Failure mode and effect analysis in asset maintenancea multiple case study in the process industry

    Int J Prod Res

    (2013)
  • C. Bunea et al.

    The effect of model uncertainty on maintenance optimization

    IEEE Trans Reliab

    (2002)
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      Citation Excerpt :

      We refer to policies which perform such an exploration as forward-looking policies. Otherwise, we clasify a policy as a myopic policy (e.g., Coolen-Schrijner & Coolen, 2004; Coolen-Schrijner & Coolen, 2007; Dayanik & Gürler, 2002; Elwany & Gebraeel, 2008; Fouladirad et al., 2018; de Jonge, Klingenberg, Teunter, & Tinga, 2015b; Laggoune, Chateauneuf, & Aissani, 2010; Mazzuchi & Soyer, 1996; Walter & Flapper, 2017). A myopic policy updates the unknown failure model or its parameters with the most recent data but without explicitly considering the decisions to be made in the future.

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    Warse Klingenberg passed away during the writing of this paper.

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