On the modeling of ageing using Weibull models: Case studies Pavel Praks, Hugo Fernandez Bacarizo, Pierre-Etienne Labeau Universit´e Libre de Bruxelles∗

Weibull models appear to be very flexible and widely used in the maintainability field for ageing models. The aim of this contribution is to systematically study the ability of classically used one-mode Weibull models to approximate the bathtub reliability model. Therefore, we analyze lifetime data simulated from different reference cases of the well-known bathtub curve model, described by a bi-Weibull distribution (the infant mortality is skipped, considering the objective of modeling ageing). The Maximum Likelihood Estimation (MLE) method is then used to estimate the corresponding parameters of a 2-parameter Weibull distribution, commonly used in maintenance modeling, and the same operation is performed for a 3-parameter Weibull distribution, with either a positive or negative shift parameter. Several numerical studies are presented, based first on large and complete samples of failure data, then on a censored data set, the failure data being limited to the useful life region and to the start of the ageing part of the bathtub curve. Results, in terms of quality of parameter estimation and of maintenance policy predictions, are presented and discussed.

1 Introduction Our contribution is motivated by a common situation in industrial maintenance, where the failure dataset is often limited, not only in size, but also in time to the useful period and the very beginning of ageing, thanks to the preventive maintenance operations. Although lots of Weibull models are available in the literature (for detailed information, consult e.g. (Murthy, Xie, and Jiang 2004)), there is a trend in industry to use, for the maintenance applications, simple one-mode Weibull models with a limited number of parameters to cover such cases. The main aim of the paper is to analyze how acceptable this approximation is, by systematically study-

ing the ability of classically used one-mode Weibull models to approximate the bathtub reliability model. For this reason, we would like to illustrate how satisfactorily the ageing model corresponding to the bathtub curve reliability model - in which the infant mortality is skipped, considering the objective of modeling ageing - can be estimated by a one-mode Weibull model with 2 or 3 parameters. Indeed, 2- or 3- parameter Weibull models though commonly used in maintenance optimization do not map well more than one part of the bathtub curve reliability model. We will see that randomly sampling failure times from the bathtub model and subsequently estimating by the maximum likelihood method the parameters of one-mode, 2- or 3- parameter Weibull models can lead to unexpected results in terms of parameter estimation as well as to erroneous predictions of an optimal mainte-

∗

Service de M´etrologie Nucl´eaire, Universit´e Libre de Bruxelles (ULB), Facult´e des Sciences Appliqu´ees, Campus du Solbosch, Av. F.D. Roosevelt 50, 1050 Bruxelles, Belgium.

1

3 Approximate Weibull models used in the comparison We assume the following probability density functions (PDF) for our parameter estimation study:

nance policy. Although properties of Weibull models are widely studied, a similar systematical benchmark study is hardly found in the current literature. The paper is organized as follows. We begin with a definition of the four-parameter model describing the assumed reality (i.e. our reference cases) in Section 2. In Section 3 we present the one-mode Weibull models with whom the reference cases are approximated. Section 4 deals with the maintenance cost models that have been chosen to compare the optimal maintenance period predictions associated to the reference cases and to the one-mode Weibull approximations. Parameter estimations for one-mode Weibull models inferred from samplings of the reference bathtub curves are presented in Section 5. Results are presented and discussed in Section 6 according to the quality of the lifetime distribution and to the optimal maintenance period predicted, as mentioned above. Finally, conclusions are summarized in Section 7.

• Weibull model with 2 parameters: βa fa (t) = ηa

Ã

t−ν η

!β−1

H(t − ν)

t ηa

!βa −1

βa

e−( ηa ) t

(2)

When ν = 0 and λ0 = 0, the model is reduced to the 2-parameter Weibull law. The corresponding cumulative distribution function (CDF) is: βa

Fa (t) = 1 − e−( ηa ) t

(3)

The scale parameter symbol ηa has the following meaning: Solving equation (3) for t = ηa we have: βa

Fa (ηa ) = 1 − e−( ηa ) ηa

= 1 − e−1

= 1 − 0.3679 = 0.6321

2 Four-parameter model of the bathtub curve We will use as reference model a three-parameter Weibull distribution with a positive location parameter, combined with an exponential model. We therefore assume the following hazard function: β λ(t) = λ0 + η

Ã

(4)

So, the scale parameter ηa is the 63rd percentile of the two-parameter Weibull distribution. The scale parameter is sometimes called the characteristic life. The shape parameter β describes the speed of the ageing process in Weibull models and has no dimension.

(1)

• Weibull model with 3 parameters including a positive shift ν > 0:

Symbol λ0 ≥ 0 denotes the failure rate associated to random failures. Symbols η > 0, β > 0 and ν ≥ 0 are called scale, shape and location parameters, respectively. H(.) is the step function. The 3-parameter Weibull rate models the failure mode due to ageing, and is active only for t > ν. In other words, a failure of the system in time t > 0 can occur by: a) a contribution due to the exponential distribution, which describes a random failure of the system for any t > 0 b) a contribution due to the Weibull distribution for t > ν , where ν is a positive location parameter (a shift).

βb fb (t) = ηb

Ã

t − νb ηb

!βb −1

−

e

³

t−νb ηb

´β b

H(t − νb ) (5)

• Weibull with 3 parameters, including a negative shift ν < 0. The Weibull model with a negative shift can be used to model pre-aged components, hence avoiding to implicitly assume λc (0) = 0 The PDF has the following conditional form: fc (t) =

2

βc ηc

³

t−νc ηc

´βc −1

e−( βc

c −( −ν η )

e

c

t−νc ηc

βc

)

H(t)

(6)

Finally, the total expected cost per unit time and per preventive cost is given by

4 Maintenance costs analysis We assume the following hypotheses in our analysis of the maintenance policy:

γ(Tp ) 1 r = + E(N (Tp )). Cp Tp Tp

• Costs are kept constant and do not change with time (interest rate or inflation are not considered).

4.2 Age replacement: AGAN (As-Good-As-New) The AGAN maintenance strategy consists of replacing the component with a new one after it has failed or when it has been in operation for Tp time units, whichever comes first. The total expected total cost per unit time per preventive cost of a component depends on a summation of contributions of preventive and curative costs:

• Repair durations are contemplated as negligible. • Failures are detected instantaneously. • Labour resources are always available to repair. The following three maintenance strategies (Kececioglu 1995; Vansnick 2006) are considered.

γ(Tp ) R(Tp ) + r(1 − R(Tp )) = . R Tp Cp 0 R(t) dt

4.1 Periodic replacement: As Bad as Old (ABAO) In this policy an item is replaced with a new one at every Tp time units of operation, i.e. periodically at time Tp , 2Tp , 3Tp , . . .. If the component fails before Tp time units of operation, it is minimally repaired so that its instantaneous failure rate λ(t) remains the same as it was prior to the failure. The expected total cost will be presented per unit time and per preventive cost, and the predetermined maintenance interval is denoted by Tp . For the analysis, it is interesting to introduce a new coefficient r, the ratio of the curative cost over the preventive cost: Cc Cp

r=

γ(Tp ) =

Cp + Cc E(N (Tp )) Tp

γ(Tp ) 1 1 − R(Tp ) . = + r R Tp Cp Tp 0 R(t) dt

E(N (Tp )) =

Tp

λ(t) dt.

(12)

5 Parameter estimations of one-mode Weibull laws based on samplings from the bathtub curve model The aim of the section is to show the possible paradox between the classical reference to the bathtubshaped failure rates and the usual way of resorting to basic, one-mode Weibull laws to model ageing and deduce maintenance policy optimization. This is first performed by assuming that the real behavior of the failure rate obeys a bathtub shape, in which the infant mortality is neglected (so that a bi-Weibull is used), by sampling failure data from this assumed reality, then by estimating the parameters of reduced 2- or 3-parameter Weibull models and by comparing the predictions of optimal maintenance period based on

(7)

(8)

where E(N (Tp )) is the expected number of failures in time interval [0, Tp ]. If the element is not replaced but only minimally repaired (ABAO), the expected number of failures at time Tp is Z

(11)

4.3 Block replacement: AGAN In this maintenance strategy an item is replaced preventively (maximum repair) at time points Tp , 2Tp , 3Tp , and so on. In addition to this, the component is maintained correctively (maximum repair) each time it fails. The total expected total cost per unit time per preventive cost can be expressed as (Kececioglu 1995)

The total expected cost per unit time can be written as

(10)

(9)

0

3

Since M T T F = ν + ηΓ(1 + (1/β)) holds for a single Weibull law, the straightforward observation is that η is a characteristic time on which the ageing process spans.

this estimation with that deduced from the assumed reality. In a second time, we also analyse failure data censored at the real MTTF point, which is an approximation of the industrial reality where data are usually limited. Parameters (η, β, ν) and λ0 have to be chosen to define what is the initial ”reality” of the failure density of the component. Actually, three of these parameters correspond to characteristic times:

Then we can use the following quantity ηλ0 as a measure of the relative duration of the ageing failure mode with respect to the purely random one. Let us assume that ηλ0 <= 1 holds in our cases because the ageing period lasts less time than the useful life period. Then we use the following range for values (λ0 is kept constant in all cases):

• parameter 1/λ0 represents the Mean Time To Failure (MTTF) of the random failure mode taken alone, • parameter η is the characteristic time of the ageing failure mode, and

ηλ0 = (1, 0.9, 0.8, 0.7, 0.6)

(13)

Even if this assumption might be valid in many cases, in some situations however, one could meet a behavior of the component where ageing starts quite quickly, even if it is then likely to be a slower process than in other cases.

• parameter ν is a time shift corresponding to the delay before the onset of ageing. 5.1 Characterization of useful period In this section, we specify the value of the parameter related to the useful period (i.e. period with a constant failure rate λ0 ) of our model of reality. We set 1/λ0 = 14000 h, because this value is close to that estimated for reciprocating compressors whose modelling were an inspiration of our study (Vansnick 2006). This arbitrary choice hence defines the time units of the problem, and only 3 independent parameters remain.

• The value of the location parameter ν is selected from the following equation: R(ν) = (0.9, 0.8, 0.7, 0.6, 0.5, 0.4)

(14)

Here R(ν) denotes the reliability function of the reference model at the end of the useful period (i.e. period with constant failure rate λ0 ): R(ν) = 1 − F (ν) = exp(−νλ0 )

5.2 Parameters describing ageing In this section, we specify values of parameters related to the ageing part of our model of reality. We assume the following parameters in our tests:

(15)

R(ν) is chosen between 0.9 and 0.4 because for values over 0.9 the useful life would be too short and values under 0.4 are not worth to be analysed. This is because, for a constant failure rate, the residual reliability at t =MTTF is equal to 0.3679. Due to this, R(ν) = 0.4 is a good limit as in practice a piece of equipment will not be operated longer.

• The values (2, 2.5, 3, 3.5, 4) are assumed for the shape parameter β. We have chosen these values because when β = 2, the increasing failure rate part (due to ageing), is increasing linearly with time and for values of β higher than 2 the curve is increasing non-linearly.

In order to express what ”reality” is being considered in each case, it can be specified by the 3-uple (ηλ0 , β, R(ν)). For example, the ”reality”

• The value of the scale parameter η is selected in the following way: 4

(ηλ0 = 1, β = 3, R(ν) = 0.8) can be expressed as (ηλ0 , β, R(ν)) = (1, 3, 0.8). In total, 5 × 5 × 6 = 150 samples of data have been analysed, in each of the following two situations: 1. In the first situation, we used a complete set of N = 1000 failure data taken from the ”Model of Reality”, see eq. 1, with the purpose of estimating the quality of the approximation obtained with the single-mode Weibull laws. 2. In the second situation we used a set of N = 100 observation data, however in this case suspended data are also considered. The MTTF point is chosen as the predetermined maintenance interval Tp . All values greater than this period will be considered as suspended data. This is an approximation of the industrial reality where failure data are usually limited.

Figure 1: Optimal predetermined maintenance intervals for Situation 1. same range was obtained for the 2-parameter Weibull model. On this account, the ability of these approximations to model the wear-out effects given by our reference model is limited. It does not cover the reference model, which is represented by the shape parameters within the interval [2, 4]. In fact, the estimation using the Weibull model with a positive location parameter is not very beneficial: Because of random failures, represented by the second part of the bathtub model, the effect of a positive shift of the location parameter is negligible in our cases: the estimated values of the location parameter were very close to zero (≈ 2.10−8 h). It seems that the paradox in the Weibull estimation leads to a conservative maintenance policy in the first situation. Optimal maintenance intervals estimated by one-mode Weibull models are almost in all cases smaller compared to the optimal maintenance period of the assumed reference model: Estimated values of Tp for the 3-parameter Weibull model with a positive location parameter and also for 2-parameter Weibull model correspond to 35 − 82% of the values of Tp obtained with the reference models. When the 3-parameter Weibull model with a negative location parameter is assumed, the situation is better: The estimated values of Tp reached 43-100 percent of the reference Tp . In 7 cases (i.e. in 4.66% of all assumed cases), the value of Tp was correctly estimated.

6 Results of the estimations In this section the predetermined maintenance interval (Tp ) obtained with the 2- and 3- parameters Weibull models will be compared to the Tp given by the reference model, which was presented in Section 5. As examples, the next figures show the results of the analysis done in these two different situations. Figures 1 and 2 show sorted results (i.e. in the 150 reference cases considered) of the optimal predetermined maintenance intervals for the reference model (denoted as 4P ABAO) with bounds corresponding to the minimum cost multiplied by the factor 1.05 (denoted as 4P Int. Sup and 4P Int. Inf, respectively). We assume the ratio between the curative cost over the preventive cost r = 10 in all cases. These figures also show results of the poor ability of the 2- and 3- parameters Weibull models (denoted as 2P ABAO and 3P ABAO, respectively) to provide fitting of the assumed reality. In the first situation, the results estimated from 2and 3- parameters Weibull models are conservative: the estimated values of the shape parameter were in the interval [1.17, 1.9] for the 3-parameter Weibull model with a positive location parameter, and the 5

Figure 2: Optimal predetermined maintenance intervals for Situation 2. Figure 3: Parameters of the reality (1, 3, 0.8), Situation 1. Failure rate.

In the second situation (i.e. with censored data), the results estimated from the 2- and 3- parameter Weibull models are also not desirable: In 36.66% of the estimations done we obtain a value of the shape parameter β < 1 (a rejuvenation!). This means that there is no minimum of the expected cost per unit time and per preventive cost function. It is thus not interesting to schedule a systematic preventive maintenance; this is completely wrong in comparison the reference model. In 43.34% of the estimations, the estimated value of the shape parameter obtained lies in the interval [1, 1.2]. This values also does not correspond to the reference model and the expected cost per unit time and per preventive cost becomes almost constant. Finally, in 20% of the analyses, the estimated value of the shape parameter is between 1.2 and 1.62. This fact does not enable us to model correctly the wear-out effects given our reference model. As it was said in Section 5.2, in order to express what ”reality” is being considered in each case, the parameters that determine it can be expressed by the 3-uple (ηλ0 , β, R(ν)). Let us analyse the behavior of two selected realities more deeply.

In Figure 3, the hazard function of the reality and its 2- or 3-parameters approximations for the first situation are presented. The symbol ”Zero” represents the MLE results related to the Weibull model with 2 parameters. The symbol ”Neg” represents the Weibull model with 3 parameters including a negative shift and finally the symbol ”Pos” represents the Weibull with 3 parameters, including a positive shift. We can see that the wear-out period of the assumed bathtub ”reality” is hardly modelled by the one-mode distributions. Although the Weibull 2P and 3P approximations model accurately only the useful period of the reality, it can be seen that it is not a problem to find the optimal preventive maintenance period Tp in both situations, see Figures 4 and 5. This happens because in these situations the optimal preventive maintenance period is found before the ageing process becomes strong. For this reason, approximations appear acceptable. In this example it is also worth noticing the MLE results obtained with the Weibull 2P and 3P approximations in the second situation: (η, β, ν) = (12487, 1.11, 0) and (η, β, ν) = (12319, 1.18, −91). We observe that these results of parameter estimations do not imply similar results in estimating of Tp ’s, see

6.1 Reality (ηλ0 , β, R(ν)) = (1, 3, 0.8) In this case, obtaining good policy predictions may not be determined by the wear-out period approximation quality in this case. 6

Figure 4: Parameters of the reality (1, 3, 0.8), Situation 1. Expected cost per unit time and per preventive cost.

Figure 5: Parameters of the reality (1, 3, 0.8), Situation 2. Expected cost per unit time and per preventive cost.

Figure 5. This little difference in shape parameters implies that the Weibull 2P displays difficulties to find a minimum in its corresponding expected cost and per unit time per preventive cost function.

were analyzed. We presented various numerical experiments showing that a random sampling from the well-known bathtub reliability model and subsequent MLE of Weibull models with 2 or 3 parameters lead to potentially dangerous shortcuts.

6.2 Reality (ηλ0 , β, R(ν)) = (0.6, 2, 0.6) Figure 6 contains failure rates for situation 1. Although cost predictions in situation 1 remain numerically non-problematic, see Figure 7, both approximation functions display difficulties to find a minimum in their corresponding expected cost per unit time and per preventive cost function in the second situation, such like 2P approximations from Figure 5: The MLE results obtained with the Weibull 2P and 3P approximations in the second situation have shape parameters very close to 1: (η, β, ν) = (10355, 1.09, 0) and (η, β, ν) = (10353, 1.12, −62).

For huge, complete sets of failure data, the estimated shape parameters are almost in all cases inside interval [1.17, 1.9] for the Weibull model with a positive location parameter or close to 6, when the Weibull model with a negative location parameter is assumed. These estimations do not correspond at all to wear-out failures represented in our reference model by shape parameters within the interval [2, 4]. This paradox in the Weibull estimation leads to a conservative maintenance policy, which involves more cost and can be even risky: Too short maintenance periods can reduce the reliability of the system, because not all of these interventions finish successfully. Moreover, this estimation drawback is observed in cases with a large set of failure data (1 000).

7 Conclusions The main aim of the paper was to systematically study the ability of classical one-mode Weibull models to approximate the bathtub reliability model. The novelty of the work includes a construction of a benchmark study, in which 150 carefully selected test-cases

For limited sets with censored data, the estimated values of the shape parameter are often close to one or even less than one and estimated predetermined maintenance intervals are not connected to the ref7

erence model. It could be very interesting to repeat tests contained in this contribution and also to provide sensitivity analysis of estimated parameters with goodness-of-fit tests (Meeker and Escobar 1995). The message of our contribution should be kept in mind during the Weibull parameter estimation process, in which also real failure data are analyzed: Although the Weibull models are flexible and widely used, we would like to point limitations of ability of one-mode Weibull models with 2 or 3 parameters to successfully fit more than exactly one part of the bathtub curve reliability model. 8 Acknowledgment ˇ Pavel Praks (from VSB-Technical University of Ostrava, the Czech Republic)’s postdoctoral stay at the Universit´e Libre de Bruxelles, Belgium is part of the ARC Project - ”Advanced supervision and dependability of complex processes: application to power systems”.

Figure 6: Parameters of the reality (0.6, 2, 0.6), Situation 1. Failure rate.

REFERENCES Kececioglu, D. (1995). Maintainability, Availability and Operational Readiness Engineering Handbook. Prentice Hall. Meeker, W. Q. and L. A. Escobar (1995). Statistical Methods for Reliability Data. John Wiley & Sons. Murthy, D. N. P., M. Xie, and R. Jiang (2004). Weibull Models. John Wiley & Sons. Vansnick, M. (2006). Optimization of the maintenance of reciprocating compressor based on the study of their performance deterioration. Ph. D. thesis, Universit´e Libre de Bruxelles.

Figure 7: Parameters of the reality (0.6, 2, 0.6), Situation 1. Expected cost per unit time and per preventive cost.

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