University of Groningen. Maintenance Optimization based on Mathematical Modeling de Jonge, Bram

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1 University of Groningen Maintenance Optimization based on Mathematical Modeling de Jonge, Bram IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2017 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): de Jonge, B. (2017). Maintenance Optimization based on Mathematical Modeling [Groningen]: University of Groningen, SOM research school Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date:

2 Maintenance Optimization based on Mathematical Modeling Bram de Jonge

3 Publisher: University of Groningen Groningen, The Netherlands Printed by: Ipskamp Printing Enschede, The Netherlands ISBN: (printed version) (electronic version) c 2017, Bram de Jonge All rights reserved. No part of this publication may be reproduced, stored in a retrieval system of any nature, or transmitted in any form or by any means, electronic, mechanical, now known or hereafter invented, including photocopying or recording, without prior written permission from the copyright owner.

4 Maintenance Optimization based on Mathematical Modeling PhD thesis to obtain the degree of PhD at the University of Groningen on the authority of the Rector Magnificus Prof. E. Sterken and in accordance with the decision by the College of Deans. This thesis will be defended in public on Thursday 9 February 2017 at hours by Bram de Jonge born on 18 December 1984 in Marum

5 Supervisors Prof. R.H. Teunter Prof. T. Tinga Assessment committee Prof. G.J.J.A.N. van Houtum Prof. P.A. Scarf Prof. M.H. van der Vlerk Special thanks to Dr. W. Klingenberg who supervised part of the research.

6 Contents 1 Introduction Outline and approach Optimal maintenance strategy under uncertainty in the lifetime distribution Introduction Literature Uncertainty Consequences of uncertainty Contribution Approach Uniformly distributed lifetime Weibull distributed lifetime Conclusions and future extensions Appendices 29 2.A Calculations for the uniform distribution with uniformly distributed right end parameter B The coefficient of variation of the Weibull distribution with uniformly distributed scale parameter Cost benefits of postponing time-based maintenance under lifetime distribution uncertainty Introduction Model formulation

7 ii 3.3 Myopic policy Bounds on the expected cost rate Performance myopic policy Postponing preventive maintenance The threshold policy Numerical results Conclusions and future research directions The influence of practical factors on the benefits of condition-based maintenance over time-based maintenance Introduction Literature review Comparative studies Practical factors influencing the benefits of CBM Approach Deterioration process Maintenance strategies Simulation A comparison of CBM and TBM Practical factors affecting the benefits of CBM Planning time Imperfect condition information Uncertain failure level Conclusions and future extensions Reducing costs by clustering maintenance activities for multiple critical units Introduction Literature review General problem description Alarm-Maintain with constant alarm rate Alert-Alarm-Maintain with constant alert and alarm rates Systems with 2 units: Analysis Systems with 2 units: Results Systems with n units

8 Contents iii 5.6 Non-constant alert and alarm rates Alarm-Maintain Alert-Alarm-Maintain Implications for the Groningen gas field Conclusions and future extensions Appendices A Derivation of conditional transition probabilities Summary and conclusions 107 Bibliography 115 Samenvatting en conclusies (Summary and conclusions in Dutch) 131

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10 Chapter 1 Introduction Due to ongoing automation of production processes and increasing reliance on expensive production equipment, the importance of effectively planned and performed maintenance activities is growing. A few decades ago, maintenance was seen as a necessary evil: something that has to be done if equipment breaks down, but also something that is difficult to manage. Nowadays, maintenance is more and more seen as a profit contributor. Organizations realize that efficiency and reliability can be improved and costs can be reduced if maintenance actions are planned more effectively. As a result of the growing importance of maintenance, both the portion of employees working in maintenance and the maintenance costs are increasing (Zio and Compare, 2013). Over a quarter of the total workforce in the process industry, and up to 30% in the chemical industry, deal with maintenance operations (Waeyenberg and Pintelon, 2002). In refineries, the maintenance and operations departments are usually the largest (Dekker, 1996). Furthermore, maintenance costs typically account for 15-70% of the total production costs (Bevilacqua and Braglia, 2000; Mobley, 2002), and the amount of money spent on maintenance of engineering structures and infrastructures is increasing continuously (Van Noortwijk, 2009). Medical equipment maintenance nowadays demands large sums from hospital budgets (Cruz et al., 2014), and companies in process and chemical industries can significantly increase profits by avoiding unplanned stoppages and bad quality production (Alsyouf, 2007). Maintenance can be defined as all activities necessary to restore equipment to, or

11 2 Chapter 1 keep it in, a state in which it can perform its designated functions. This definition immediately makes clear that two types of maintenance actions can be distinguished. Maintenance can be performed after a failure or breakdown, and is then called corrective or reactive maintenance. The other type of maintenance is called preventive maintenance and is performed when equipment is still functional. It aims to prevent or postpone failures. Because potential consequences of failures include safety issues, machine damage, quality issues, unexpected machine unavailability, long repair times, and unplanned maintenance actions, it is often preferred to perform maintenance activities preventively. However, because performing preventive maintenance too often is also undesirable and costly, a balance between the preventive maintenance frequency and the risk of failures has to be found. Maintenance Corrective maintenance Preventive maintenance Time-based maintenance Condition-based maintenance (CBM) Opportunistic maintenance Age-based maintenance Block-based maintenance Continuous monitoring Inspections Figure 1.1: Schematic overview maintenance policies. A schematic overview of commonly used maintenance policies is shown in Figure 1.1. This figure indicates that a further subdivision of preventive maintenance policies can be made. Preventive maintenance actions can be based on the time that equipment is in service (i.e., time-based maintenance; TBM), but if there are specific features that are related to the degradation process that ultimately leads to failure, and if it is technically possible to monitor these features, condition-based maintenance (CBM) allows for maintenance activities that are performed based on degradation information. For multi-unit systems it is often suboptimal to schedule maintenance actions on

12 Introduction 3 the unit level. This is caused by dependencies that generally exist between units. Dependencies that are often mentioned in the literature are economic, structural and stochastic dependence (e.g. Laggoune et al., 2010; Thomas, 1986). Economic dependence exists if cost savings can be obtained or downtime can be reduced by clustering maintenance actions. Structural dependence exists when the functioning of a system in some way depends on the functioning of various components or sub-systems. Stochastic (or probabilistic) dependence exists when lifetimes or degradation processes of various units are stochastically dependent. When dependencies between units exist, the use of opportunistic maintenance policies is often appropriate. Such a policy uses a preventive or corrective maintenance action for a certain unit as an opportunity to maintain other units as well. Age or condition information of the other units are often used to determine whether they are maintained as well. Two time-based preventive maintenance policies can be distinguished: age-based maintenance and block-based maintenance. Under the age-based maintenance policy, preventive maintenance is performed when the unit reaches the prescribed maintenance age T, or corrective maintenance is performed when the unit fails, whichever occurs first; see Figure 1.2 (a). Under the block-based maintenance policy, preventive maintenance is performed after fixed intervals with length T. Corrective maintenance is again performed when the unit fails, but failures do not affect the preventive maintenance schedule when this policy is applied; see Figure 1.2 (b). The disadvantage of block-based maintenance is that preventive maintenance is sometimes performed shortly after a failure. The main advantages, on the other hand, are the easier planning as it is known in advance when preventive maintenance will be performed, and the clustered maintenance actions when the same block-based maintenance policy is applied to multiple units. Time-based maintenance is easy to implement as only the time that a unit is in service has to be recorded. However, substantial remaining useful life is wasted if the machine is still in reasonable condition when preventive maintenance is performed, and a breakdown might occur if it happens to deteriorate faster than expected. Condition-based maintenance, on the other hand, generally results in more effectively scheduled preventive maintenance, and, in the ideal case, preventive maintenance that is performed just before failure. However, applying CBM is only possible if there are conditions that are related to the moment of failure, and if it is technically

13 4 Chapter 1 T T T T PM PM PM PM PM Failure (a) Age-based maintenance. Failure T T T T T PM PM PM PM PM PM Failure Failure (b) Block-based maintenance. Figure 1.2: Scheme of time-based maintenance policies. possible to monitor these conditions. Furthermore, CBM should only be applied if its benefits outweigh the efforts and costs required to apply it. These requisites include condition monitoring equipment and software to store and analyze data, and initiate maintenance actions. Whereas only a lifetime distribution needs to be specified to study time-based maintenance models, the analysis of a CBM model requires the modeling of the deterioration of a unit. Commonly used approaches in the literature range from the delay-time model that only adds one deterioration state in between the operating state and the failed state (Wang, 2012) and models with a finite number of deterioration states, to continuous-state and continuous-time models as the gamma process and the Wiener process. Figure 1.1 distinguishes two types of condition-based maintenance policies based on the frequency at which deterioration information becomes available. The condition of equipment can either be monitored continuously, or inspections might need to be performed to obtain actual deterioration levels. The main drawbacks of continuous monitoring are that it is often expensive because special devices are required, and that the continuous flow of data creates increased noise (Jardine et al., 2006). Inspection-based CBM, on the other hand, has the disadvantage that it has the possibility of missing substantial deterioration increments in between inspections. When inspections are required to obtain condition information, either a periodic or an aperiodic inspection schedule can be adopted. Periodic inspections have the advantages that they are easier to implement in an industrial context (Deloux et al.,

14 Introduction ; Zequeira and Bérenguer, 2006), that they allow the necessary manpower and budget to be anticipated and scheduled well beforehand (Van Noortwijk and Klatter, 1999), and that they are much easier to optimize as the entire inspection schedule is dictated by the specification of a single inspection interval. However, it is often much more effective to use an aperiodic inspection schedule that dynamically schedules the next inspection based on actual condition information (e.g. Dieulle et al., 2003; Grall et al., 2002; Maillart, 2006). 1.1 Outline and approach This thesis contributes to the current research on maintenance planning and optimization. We study the effect of uncertainty in the lifetime distribution on the optimal age-based maintenance strategy, and analyze whether it can be beneficial to make suboptimal decisions during the first phase of the lifespan of equipment in order to reduce this uncertainty faster. Thereafter, we broaden our scope and study how the benefits of condition-based maintenance over time-based maintenance depend on various characteristics. Finally, we consider the benefits that can be obtained by clustering maintenance actions for multiple units based on condition information. The methodology that we use is that of developing and analyzing mathematical models. Algebraic analysis, numerical calculations and simulations will be used to compare and to optimize maintenance policies. Research on time-based maintenance optimization generally assumes that the lifetime distribution of a unit is known with certainty. In practice, however, this is often not a realistic assumption. There is often a substantial amount of uncertainty in the lifetime distribution because there is usually only a limited amount of data available, the exact interpretation of stored data is often unclear, and times until failure are often not observed due to preventive maintenance activities in the past (i.e., data is right-censored). In Chapter 2 of this thesis, we consider the effect of uncertainty in the lifetime distribution on the optimal age-based maintenance policy. We also report on the cost consequences of ignoring this uncertainty and point out in what cases it is particularly important to take the uncertainty into account when determining a maintenance policy. We start with the case of a uniformly distributed lifetime that can be analyzed algebraically. Thereafter, we continue with the more realistic Weibull lifetime distribution and show that the insights are similar to those

15 6 Chapter 1 for a uniform lifetime distribution. The maintenance policies considered in Chapter 2 are static, i.e., a fixed maintenance age is chosen that minimizes the long-run expected cost rate under the information that is currently available. However, when a maintenance policy is applied, more data becomes available due to failures and preventive maintenance actions. In Chapter 3, we do update the uncertainty in the lifetime distribution when more data becomes available. Furthermore, we acknowledge that the uncertainty in the lifetime distribution can be reduced much faster if preventive maintenance actions are initially postponed. Although this will increase the expected costs during this first phase of the lifespan of a unit, it also leads to reduced uncertainty in the lifetime distribution for future decisions. As a consequence, preventive maintenance can be scheduled more effectively during the remaining lifespan of the unit. In this chapter we investigate the cost benefits over the entire lifespan of a unit, and identify under what circumstances these benefits are largest. The benefit of condition-based maintenance compared with time-based maintenance strongly depends on the behavior of the deterioration process, the severity of failures, the required setup time, the accuracy of the condition measurements, and the amount of randomness in the deterioration level at which failure occurs. In Chapter 4, we start with a review of studies that compare condition-based maintenance with time-based maintenance. Furthermore, we review studies that consider the aforementioned practical factors that influence the relative performance of CBM. It turns out that, although both condition-based and time-based maintenance have received ample attention in the scientific literature, few studies compare them. Moreover, existing studies that do provide a comparison only consider a few examples; general insights on how the performances of the policies depend on the various characteristics are generally lacking. Therefore, we continue in Chapter 4 with a numerical analysis that does provide insights on the effects of the various characteristics. This analysis is based on simulations of a single unit that deteriorates gradually over time and that is monitored continuously. In Chapter 5, we consider maintenance scheduling for a multi-unit system. The specific setting that we consider is that of a system consisting of multiple critical and identical units with economic dependence. Each unit contains a sensor that provides either one signal (alarm) or two signals (alert, alarm). Such systems with only a small number of deterioration states instead of continuous deterioration levels are

16 Introduction 7 very common in industry. Because of the criticality of the units, maintenance always has to be performed within a fixed time period after an alarm in order to prevent an impending failure. As a consequence, benefits cannot be obtained by delaying maintenance activities. However, due to the economic dependencies between the units, it can be beneficial to cluster maintenance actions. We will evaluate the cost performance of two clustering policies and compare it with the policy that does not cluster any maintenance activities. Furthermore, we derive general insights on the effects of the number of units, the cost structure, and the mean times until alert and alarm signals. Journal publications The chapters in this thesis are based on the following journal publications. Chapter 2: De Jonge, B., W. Klingenberg, R. H. Teunter, T. Tinga Optimum maintenance strategy under uncertainty in the lifetime distribution. Reliability Engineering and System Safety Chapter 3: De Jonge, B., A. S. Dijkstra, W. Romeijnders Cost benefits of postponing timebased maintenance under lifetime distribution uncertainty. Reliability Engineering and System Safety Chapter 4: De Jonge, B., R. H. Teunter, T. Tinga The influence of practical factors on the benefits of condition-based maintenance over time-based maintenance. Reliability Engineering and System Safety Chapter 5: De Jonge, B., W. Klingenberg, R. H. Teunter, T. Tinga Reducing costs by clustering maintenance activities for multiple critical units. Reliability Engineering and System Safety

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18 Chapter 2 Optimal maintenance strategy under uncertainty in the lifetime distribution 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 optimal 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. This chapter is based on De Jonge et al. (2015b): De Jonge, B., W. Klingenberg, R. H. Teunter, T. Tinga Optimum maintenance strategy under uncertainty in the lifetime distribution. Reliability Engineering and System Safety

19 10 Chapter Introduction It is widely accepted that preventive maintenance is important for achieving Operational Excellence (Cua et al., 2001; Shah and Ward, 2003), 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 (Jardine and Tsang, 2005). 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 (Gertsbakh, 2000; Jardine and Tsang, 2005; Abernethy, 2006; Rinne, 2008) and software systems (Jardine and Tsang, 2005; Campbell et al., 2010) 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 (Moubray, 2001; Tinga, 2010). Furthermore, maintenance and reliability engineers have bemoaned the lack of credibility in collected data for years (Mann Jr. et al., 1995; Dekker and Scarf, 1998; Braaksma et al., 2013). 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 (Bunea and Bedford, 2002). Finally, there is often an insufficient amount of data to determine accurate estimates for the model parameters.

20 Optimal maintenance strategy under uncertainty in the lifetime distribution 11 The consequences of uncertainty in the lifetime distribution in terms of the optimal 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 2.2, we discuss the existing literature. In Section 2.3, we formally describe the problem we consider as well as our approach. In Section 2.4, we evaluate the uncertainty by considering a simple setting with a uniform lifetime distribution, in which the optimal maintenance age can be obtained explicitly. In Section 2.5, we evaluate a more realistic setting, with uncertainty in the scale parameter of the Weibull distribution, which we will evaluate numerically. Section 2.6 provides conclusions and extensions for future research. 2.2 Literature Time-based preventive maintenance was first studied by Barlow and Hunter (1960). One of the two introduced policies, age-based maintenance, is further studied by Glasser (1967), Tadikamalla (1980), and Nakagawa and Yasui (1981). 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. (1988), Makis and Jardine (1993) and Jiang et al. (2001), who report on the periodical replacement problem with repairs at failures that bring the system to a certain better state; and Yeh and Lo (2001), who determine the optimal maintenance strategy during a warranty period with a given length Uncertainty The fact that the lifetime distribution of a component is typically uncertain was highlighted by several authors. Uncertainty is divided into model uncertainty and para-

21 12 Chapter 2 meter uncertainty; in both categories, several solutions were proposed. Zhang and Mahadevan (2000) propose a Bayesian procedure that includes both model uncertainty and parameter uncertainty. Hoeting et al. (1999) use Bayesian model averaging to take account of model uncertainty in a rather general setting. A common approach of many reliability engineers is to assume a certain distribution and to consider the parameters of this distribution as unknown (parameter uncertainty). This assumption is not too restrictive if a flexible distribution is chosen that provides a good description of many types of failure data. An example of such a distribution is the Weibull distribution (Rinne, 2008; Abernethy, 2006). Gertsbakh (2000) describes an approach to determine the optimal preventive maintenance age if the parameters of the lifetime distribution take a specific value from a small set of values, each with a specific probability. However, this line of reasoning is not progressed in detail. The distribution that is used to model parameter uncertainty can be based on expert judgment and/or on data. In a Bayesian approach, the opinions of experts can be used to determine a so called prior distribution (Bernardo and Smith, 1993; O Hagan, 1994). Kraan and Bedford (2005) and Zuashkiani et al. (2009) discuss methods to translate expert judgment on model outputs into uncertainties on model parameter values. The prior distribution could be updated based on data that becomes available, resulting in a posterior distribution. If no (expert) knowledge is available, commonly a reference prior (Percy, 2002) or a non-informative prior (Hamada et al., 2010) is used. There are also other methods used to model/estimate parameter uncertainty. Laggoune et al. (2010) consider a setting with few failure data and use the Bootstrap technique to model the uncertainties in the parameter values. This technique draws a large amount of subsamples from the data, and base the random distribution that is used to model the uncertainty in the parameters on the maximum likelihood estimations for the parameters of the various subsamples. The optimal maintenance age turns out to change if this uncertainty is taken into account. Rocco et al. (2000) use a two steps evolutionary approach and apply this to a maintenance optimization problem in a multi-component system. The results mainly focus on the computational complexity of this approach.

22 Optimal maintenance strategy under uncertainty in the lifetime distribution Consequences of uncertainty Some studies consider the consequences of uncertainty in the lifetime distribution on the optimal maintenance plan. Baker and Scarf (1995) consider the excess cost of making suboptimal decisions using point estimates for the parameters based on data, instead of using the true but unknown optimal decisions. Their focus is mainly on the optimal inspection interval. Although large sample sizes are required to accurately estimate the optimal value of maintenance decision variables, it turns out that the excess cost is already relatively small for modest sample sizes. Our study shows that this is not true in general; the additional cost of not taking into account uncertainty can already be significant for small levels of uncertainty, depending on the situation. Bunea and Bedford (2002) consider age-based maintenance planning under the presence of censored data. They distinguish three levels of dependence between the censoring times and the failure times, and show that the suboptimal replacement age and suboptimal replacement costs can be dramatically nonoptimal when the wrong level of dependence is considered. The study focuses on modeling dependence between censoring times and failure times, and does not consider the effect of parameter uncertainty on the optimal maintenance age as we do. Furthermore, they find that the effects are most significant when the failure rate increases slowly, which is the exact opposite of our results. Mazzuchi (1996) present a Bayesian theoretic approach that updates random distributions, modeling uncertainty in the parameters of lifetime distributions, as more data becomes available. Under the uncertainty in the paremeters, the maintenance age that minimizes the expected cost is minimized. A single example indicates a maintenance age that decreases whenever a new duration is observed. General insights on the behavior of the maintenance age, as we present, are lacking. Silver and Fiechter (1992) use Bayesian updating in the simple case with only two possible lifetimes with unknown corresponding probabilities. In this case the optimal maintenance age is always either one of two values. The emphasis is mainly on the complexity of the required calculations and on two heuristics. A numerical test shows that if the level of uncertainty is high enough, the optimal maintenance age might change if the uncertainty is taken into account. Silver and Fiechter (1995) generalize this to a discretely distributed lifetime and show that in some circumstances the optimal maintenance age decreases if the uncertainty is taken into ac-

23 14 Chapter 2 count. Again, general insights are lacking, and the results based on nonparametric discrete lifetime distributions do not carry over to more realistic parametric continuous distributions Contribution As outlined, most papers that include lifetime distribution uncertainty mainly focus on how to model this uncertainty, and on how to update the model when more data becomes available. Authors have focused on specific effects and considered few examples to illustrate those, not leading to general insights. This paper is devoted to the effects of uncertainty and does present general insights for the the age-based maintenance strategy. 2.3 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 is η age (T ) = F (T ) + (1 F (T ))c T. (2.1) (1 F (x)) dx 0 This formula was first presented by Barlow and Hunter (1960) and is also included in many textbooks (Barlow and Proschan, 1965; Gertsbakh, 2000). Minimizing the cost rate η age (T ) provides us with the optimal maintenance age T opt. However, analysis of the age-based maintenance strategy requires the exact lifetime distribution of the unit, which is rarely known in practice. As motivated in the introduction, we assume a specific lifetime distribution and allow for uncertainty in its parameters. This uncertainty will be modeled using a random distribution. We remark that the true parameter values are fixed, but unknown. Their estimated values depend on expert knowledge and random data that becomes available, and are therefore random. The used random distribution rep-

24 Optimal maintenance strategy under uncertainty in the lifetime distribution 15 resents the likelihood that the true parameters have certain values. This paper will not focus on determining a proper distribution to model the uncertainty in the parameters, but on the effect of parameter (mis)specification on the optimal maintenance age. Let us represent the vector of parameters of the lifetime distribution by s. We will denote the joint density function that models the uncertainty in s by g(s), which is defined on R n. It then follows that the expected cost rate (as a function of the maintenance age T ) equals ηage(t E F (T ; s) + (1 F (T ; s))c ) = g(s) T s R n 0 (1 F (x; s)) dx ds 1 ds n. (2.2) Minimizing this function provides us with the optimal preventive maintenance age T E opt, i.e., the maintenance age that minimizes the expected cost rate. 2.4 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 are 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 0, t < 0, F (t; s) = t s, 0 t s, 1, t > s. It follows that T 0 T ( 1 (1 F (x; s)) dx = 2s) T, T s, (2.3) 1 2 s, T s. We will first consider the case that the value of the parameter s is known with certainty. Without loss of generality, we will rescale the uniform distribution such that

25 16 Chapter 2 s = 1. The cost rate (2.1) is then equal to η age (T ; ŝ) = T + (1 T c T ( T (1 c) + c ) 1 T = T T, 0 < T 1. 2 Restricted to T > 0, this function has a unique minimum that is attained at c(2 c) c T opt =. (2.4) 1 c For any value of c (0, 1) this optimal maintenance age is contained in the interval (0, 1). Furthermore, T opt is a strictly increasing function in c. This confirms the intuition that a higher preventive maintenance cost, as a fraction of the cost of an emergency repair, results in a higher preventive maintenance age. As motivated before, it is not realistic to assume that the true value of the parameter s is known with certainty. This uncertainty should be taken into account while determining the optimal maintenance age. We will model the uncertainty in the parameter s using a uniform distribution on the interval [1 α, 1+α], with 0 α 1. Thus, the function g(s) in (2.2) equals (2α) 1, s [1 α, 1 + α], g(s) = 0, elsewhere. The value of α can be interpreted as a measure of the uncertainty for our initial point estimate s = 1. If the parameter of the lifetime distribution is uncertain, the distribution of the lifetime can be seen as a composite distribution. In our current setting, where the parameter s follows a uniform distribution on the interval [1 α, 1+α], this composite distribution can be stated explicitly. Figure 2.1 shows the density function of this composite distribution for various values of α. This figure provides us with a first idea of the effect of uncertainty in the parameter s on the optimal maintenance age. If the uncertainty increases, i.e. if α increases, the composite distribution becomes more dense on the interval [0, 1 α]. As long as the optimal maintenance age is also contained in this interval, the probability of a failure prior to the maintenance action increases during this entire period. This, in turn, implies a decreasing optimal maintenance age. However, if the uncertainty increases further, the probability of a

26 Optimal maintenance strategy under uncertainty in the lifetime distribution 17 failure at the start of the lifetime of the unit increases significantly. Furthermore, the tail of the distribution becomes fatter. The early failures cannot be avoided, unless a very low maintenance age is chosen. On the other hand, as we will show, the fatter tail results in an increasing optimal maintenance age. Thus, the optimal maintenance age first decreases if the uncertainty in the parameter s increases, but increases if the uncertainty increases further. f(x) 1 α = 1 α = 0.5 α = 0.25 α = x Figure 2.1: Density function of the uniform lifetime distribution for ŝ = 1 and for various values of α. We will continue with an explicit determination of the expected cost rate and the optimal maintenance age. The main results will be stated here, detailed calculations can be found in Appendix 2.A. The expression for the expected cost rate η E age(t ) depends on the value of T and is equal to ηage(t E ) = 1 2α 1 α ( c) ln ( 2+2α T 2 2α T (ln T ln(1 α)) ( α + c 2α 2 c) ln ( 2+2α T T ) + c T, if T 1 α, ) ( 1+α T 1), if 1 α T 1 + α. On the interval (0, 1 + α), the function η E age(t ) has a unique minimum. If α (1

27 18 Chapter 2 c)(1 + c) 1, this minimum is attained on the interval (0, 1 α] at c(2 c) 2α2 c(1 c) c. 1 c If, on the other hand, α (1 c)(1 + c) 1, this minimum is attained on the interval [1 α, 1 + α) at c(1 + α). Thus, the optimal maintenance age T E opt that minimizes the expected cost rate equals c(2 c) 2α2 c(1 c) c, α 1 c Topt E = 1 c +, c(1 + α), α 1 c (2.5) 1 + c. The first part of this expression decreases in α. Thus, the optimal maintenance age T E opt first decreases if the uncertainty in the parameter s increases. If α increases further, above (1 c)(1 + c) 1, the optimal maintenance age increases linearly in α. Furthermore, (2.5) is equivalent to (2.4) if α = 0. Figure 2.2 (a) shows the optimal maintenance age T E opt as a function of the level of uncertainty α in the parameter s, for various values of the relative cost c of preventive maintenance. Another notable observation is that all knots in the graphs in Figure 2.2 (a) are on the line T E opt = 1 α. A knot is attained at the point α = (1 c)(1 + c), which is equivalent to c = (1 α)(1+α). The optimal maintenance age equals c(1+α), which reduces to 1 α for c = (1 α)(1 + α). We are not only interested in the effect of the uncertainty in the parameter s on the optimal maintenance age, but also to what extent the expected costs increase if we do not take this uncertainty into account. Denoting again the optimal preventive maintenance age based on the point estimate ŝ of the parameter s by T opt, and the optimal preventive maintenance age that takes the uncertainty in s into account by T E opt, the percentage increase in cost if we do not take the uncertainty in the parameter s into account equals ( ) η E age (T opt ) ηage(t E opt E ) %. Figure 2.2 (b) shows this percentage increase in expected costs for different values of the relative cost c of preventive maintenance, and as a function of the level of

28 Optimal maintenance strategy under uncertainty in the lifetime distribution 19 T E opt 1 c = % c = 0.05 c = 0.4 c = 0.3 c = c = 0.1 c = 0.1 c = c = α (a) 0 c = 0.3 c = 0.4 c = α (b) Figure 2.2: The optimal preventive maintenance age T E opt under uncertainty in the right end point s of the uniform distribution (a), and the percentage increase in expected cost if this uncertainty is ignored (b). uncertainty α in the estimate of the parameter s. Especially for a low relative cost c of performing preventive maintenance and a high level of uncertainty α in the parameter s, the expected percentage increase in costs is high if we don t take the uncertainty into account. This percentage rises to 9.5 % for c = The percentage increase in cost first increases if α increases and starts to decrease if α increases further. This is consistent with the results in Figure 2.2 (a). First, the optimal maintenance age decreases if the uncertainty increases. This results in a greater difference between the optimal maintenance age if we do not take the uncertainty into account and the optimal maintenance if we do so. As the optimal maintenance age starts to increase, this gap becomes smaller and taking into account the uncertainty becomes less profitable. The second peak in the graphs in Figure 2.2 (b) for high values of c is caused by an optimal maintenance age that increases even further than the optimal maintenance age without taking into account the uncertainty.

29 20 Chapter 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 (Lawless, 2002). Other physical phenomena for which the Weibull distribution is a suitable distribution to model lifetimes are given by Rinne (2008). Furthermore, Rinne (2008) also states that for various degradation processes the time until failure, i.e., the time at which a certain degradation level will be reached, closely corresponds to a Weibull distribution. We consider the Weibull distribution with two parameters, a shape parameter k and a scale parameter λ. We note that in some situations the use of a threeparameter Weibull distribution with an additional location parameter, corresponding to a failure-free period, is more appropriate. Because preventive maintenance will obviously not be performed during this failure-free period, we set the location parameter to zero as we also did with the minimum lifetime of the uniform distribution in the previous section. Weibull experts generally agree that the value of the shape parameter k tends to be reasonably constant for specific or generic failure modes (Abernethy, 2006), and that it is much more complicated to give a reasonable estimate for the value of the scale parameter λ (Zuashkiani et al., 2009). Examples of failure modes for which the value of k is predictable given by Abernethy (2006) include failures of solid state electronics (k = 0.75), random failures (k = 1), roller bearing failures (k = 1.5), ball bearing failures (k = 2), V-belts (k = 2.5), and stress corrosion (k = 5). Furthermore, for some classes of products, such as vacuum tubes (Kao, 1956), an appropriate value of k may be known from previous test experience (Soland, 1968). Therefore, in Weibull reliability analysis, it is frequently the case that the value of the shape parameter is known, but that there is uncertainty in the scale parameter (Rinne, 2008). Other studies that assume a fixed shape and random scale parameter include Kwon (1996), Papadopoulos and Tsokos (1975), and Canavos and Tsokos (1973). In line with these studies, we also assume a known shape parameter and an uncertain scale parameter. We remark that this case has similarities with the uniformly distributed

30 Optimal maintenance strategy under uncertainty in the lifetime distribution 21 lifetime with uncertainty in its right end point studied in the previous section. In fact, changing the right end point of a uniform distribution is equivalent to changing the scale of this distribution. Furthermore, we will only consider values for the shape parameter k that are greater than 1. A Weibull distribution with a shape parameter smaller than 1 has a decreasing failure rate, implying that preventive maintenance will never be beneficial. The distribution function of the Weibull distribution is given by F (t; λ, k) = 1 exp( (t/λ) k ), t 0. We will model the uncertainty in λ using a uniform distribution. We will again rescale this distribution such that it is centered around 1. The function g(s) in (2.2) is therefore again equal to (2α) 1, s [1 α, 1 + α], g(s) = 0, elsewhere, with the level of uncertainty α in the interval [0, 1]. In this setting the expected cost rate (2.2), as a function of the maintenance age T, equals η E age(t ) = (2α) 1 1+α 1 α 1 (1 c) exp( (T/λ) k ) T 0 exp( (x/λ)k ) dx dλ. The value of T that minimizes this function is the optimal preventive maintenance age Topt. E Because no explicit solution of the integral of the distribution function of the Weibull distribution exists, we have to rely on a numerical anlysis of the expected mean cost per unit time. An immediate and inevitable consequence is that all results in this section are approximations. Figure 2.3 shows the optimal maintenance age as a function of the level of uncertainty α for k = 2 (a) and for k = 5 (b). We see a similar pattern as in the case with a uniformly distributed lifetime. The optimal maintenance age is first decreasing if the uncertainty increases. However, if the level of uncertainty exceeds some threshold, the optimal maintenance age starts to increase. Both effects become stronger if the value of the shape parameter k increases. The percentage increase in expected cost if the uncertainty is not taken into ac-

31 22 Chapter 2 count for k = 2, 3, 4, and 5 is shown in Figure 2.4. For all considered values of k, this cost increase is in general small for low levels of uncertainty, but substantial for higher levels of uncertainty. The biggest losses from ignoring uncertainty are related to a too high failure risk, and this is most pronounced if the failure cost is relatively large, i.e., if the cost c of preventive maintenance is small. Figure 2.4 only includes the cost increases for values of c of at least In practice, the relative cost of preventive maintenance might even be lower, resulting in even higher cost increases. See for example the case study presented by Laggoune et al. (2010) where c ranges approximately from to An increase in the shape parameter k implies less variance in the lifetime for a given scale parameter λ and therefore a relatively larger impact of uncertainty in the scale parameter. Figure 2.5 shows the coefficient of variation CV of the Weibull distribution for various values of the fixed shape parameter k and for the scale parameter λ uniformly distributed on the interval [1 α, 1 + α]. A derivation is given in Appendix 2.B. Higher levels of uncertainty α obviously result in a higher CV. However, this effect is much stronger for higher values of k. A Weibull distribution with a low shape parameter k already has a relatively high CV without uncertainty in λ. 2.6 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 maintenance age. Furthermore, we investigated the expected cost saving from taking the uncertainty into account. We started our analysis with a setting that can be evaluated algebraically, namely that of a uniform lifetime distribution with uncertainty in its right end point. After that, we considered the commonly used Weibull lifetime distribution, which is appropriate for modeling lifetimes of a wide variety of units and systems. The value of the shape parameter of the Weibull distribution can often be determined accurately

32 Optimal maintenance strategy under uncertainty in the lifetime distribution 23 T E opt T E opt 1 c = 0.5 c = c = 0.4 c = 0.4 c = 0.3 c = 0.3 c = 0.2 c = 0.1 c = 0.05 c = 0.2 c = 0.1 c = α (a) k = α (b) k = 5. Figure 2.3: The optimal preventive maintenance age T E opt under uncertainty in the scale parameter λ of the Weibull distribution. based on the failure mode of the unit. It is, on the other hand, very common that there is considerable uncertainty in the value of the scale parameter. The level of this uncertainty depends on the amount of failure data that is available and on the perceived quality of the information provided by the original equipment manufacturer, and so we considered uncertainty in the value of the scale parameter. The effect of parameter uncertainty on the optimal maintenance age reveals a similar pattern for both the uniform and the Weibull lifetime distribution, and for various relative costs of performing preventive maintenance (as a fraction of the cost of a breakdown). The optimal maintenance age first decreases as the level of uncertainty in the considered parameter increases. So, more uncertainty in the lifetime distribution implies a higher maintenance frequency. However, less intuitive, if the level of uncertainty exceeds a certain threshold, the optimal maintenance age starts to increase as the uncertainty increases. At some point, the increased probability of a large remaining lifetime starts to outweigh the risk of an imminent failure. We have outlined in what cases the uncertainty in the lifetime distribution is most relevant when determining a maintenance strategy. Firstly, the additional cost of