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Section
WARWICK MANUFACTURING GROUP
Product Excellence using 6 Sigma (PEUSS)
Weibull analysis Warwick Manufacturing Group
THE USE OF WEIBULL IN DEFECT DATA ANALYSIS Contents
1
Introduction
1
2
Data
1
3
The mechanics of Weibull analysis
5
4
Interpretation of Weibull output
8
5
Practical difficulties with Weibull plotting
15
6
Comparison with hazard plotting
20
7
Conclusions
20
8
References
21
9
ANNEX A – two cycle Weibull paper
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10 ANNEX B – Progressive example of Weibull plotting
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11 ANNEX C Estimation of Weibull location parameter
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12 ANNEX D – Example of a 3-parameter Weibull plot 36 13 ANNEX E – the effect of scatter
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13 ANNEX E – the effect of scatter
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14 ANNEX F – 95% confidence limits for Weibull 42 15 ANNEX G – Weibull plot of multiply censored data
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15 ANNEX G – Weibull plot of multiply censored data
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Copyright © 2007 University of Warwick
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The use of Weibull in defect data analysis
THE USE OF WEIBULL IN DEFECT DATA ANALYSIS 1 Introduction These notes give a brief introduction to Weibull analysis and its potential contribution to equipment maintenance and lifing policies. Statistical terminology has been avoided wherever possible and those terms which are used are explained, albeit briefly. Weibull analysis originated from a paper [1] published in 1951 by a Swedish mechanical engineer, Professor Waloddi Weibull. His original paper did little more than propose a multi-parameter distribution, but it became widely appreciated and was shown by Pratt and Whitney in 1967 to have some application to the analysis of defect data.
1.1 Information sources The definitive statistical text on Weibull is cited at [2], and publications closer to the working level are given at [3] and [4]. A set of British Standards, BS 5760 Parts 1 to 3 cover a broad spectrum of reliability activities. Part 1 on Reliability Programme Management was issued in 1979 but is of little value here except for its comments on the difficulties of obtaining adequate data. Part 2 [5] contains valuable guidance for the application of Weibull analysis although this may be difficult to extract. The third of the Standard contains authentic practical examples illustrating the principles established in Parts 1 and 2. One further source of information is an I Mech E paper by Sherwin and Lees [6]. Part 1 of this paper is a good review of current Weibull theory and Part 2 provides some insight into the practical problems inherent in its use.
1.2 Application to sampled defect data It is important to define the context in which the following Weibull analysis may be used. All that is stated subsequently is applicable to sampled defect data. This is a very different situation to that which exists on, say, the RB-211 for which Rolls Royce has a complete data base. They know at any time the life distribution of all the in-service engines and their components, and their analysis can be done from knowledge of the utilizations at failure and the current utilisation for all the non-failed components. Their form of Weibull analysis is unique to this situation of total visibility. It is assumed here, however, that most organisations are not in this fortunate position; their data will at best be of some representative sample of the failures which have occurred, and of utilization of unfailed units. It cannot be stressed too highly, though, that life of unfailed units must be known if a realistic estimate of lifetimes to failure is to be made, and, therefore, data must be collected on unfailed units in the sample.
2 Data The basic elements in defect data analysis comprise: •
a population, from which some sample is taken in the form of times to failure (here time is taken to mean any appropriate measure of utilisation),
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The use of Weibull in defect data analysis •
an analytical technique such as Weibull which is then applied to the sample of failure data to derive a mathematical model for the behaviour of the sample, and hopefully of the population also, and finally
•
some deductions which are generated by an examination of the model. These deductions will influence the decisions to be made about the maintenance strategy for the population.
The most difficult part of this process is the acquisition of trustworthy data. No amount of elegance in the statistical treatment of the data will enable sound judgements to be made from invalid data. Weibull analysis requires times to failure. This is higher quality data than knowledge of the number of failures in an interval. A failure must be a defined event and preferably objective rather than some subjectively assessed degradation in performance. A typical sample, therefore, might at its most superficial level comprise a collection of individual times to failure for the equipment under investigation.
2.1 Quality of data The quality of data is a most difficult feature to assess and yet its importance cannot be overstated. When there is a choice between a relatively large amount of dubious data and a relatively small amount of sound data, the latter is always preferred. The quality problem has several facets: •
The data should be a statistically random sample of the population. Exactly what this means in terms of the hardware will differ in each case. Clearly the modification state of equipments may be relevant to the failures being experienced and failure data which cannot be allocated to one or other modification is likely to be misleading. By an examination of the source of the data the user must satisfy himself that it contains no bias, or else recognise such a bias and confine the deductions accordingly. For example, data obtained from one user unit for an item experiencing failures of a nature which may be influenced by the quality of maintenance, local operating conditions/practices or any other idiosyncrasy of that unit may be used providing the conclusions drawn are suitably confined to the unit concerned.
•
A less obvious data quality problem concerns the measure of utilisation to be used; it must not only be the appropriate one for the equipment as a whole, but it must also be appropriate for the major failure modes. As will be seen later, an analysis at equipment level can be totally misleading if there are several significant failure modes each exhibiting their own type of behaviour. The view of the problem at equipment level may give a misleading indication of the counter-strategies to be employed. The more meaningful deeper examination will not be possible unless the data contains mode information at the right depth and degree of integrity.
•
It is necessary to know any other details which may have a bearing on the failure sensitivity of the equipment; for example the installed position of the failures which
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comprise the sample. There are many factors which may render elements of a sample unrepresentative including such things as misuse or incorrect diagnosis.
2.2 Quantity of data Whereas the effects of poor quality are insidious, the effects of inadequate quantity of data are more apparent and can, in part, be countered. To see how this may be done it is necessary to examine one of the statistical characteristics used in Weibull analysis. An equipment undergoing in-service failures will exhibit a cumulative distribution function (F(t)), which is the distribution in time of the cumulative failure pattern or cumulative percent failed as a function of time, as indicated by the sample. Consider a sample of 5 failures (sample size n = 5). The symbol i is used to indicate the failure number once the failure times are ranked in ascending order; so here i will take the integer values 1 to 5 inclusive. Suppose the 5 failure times are 2, 7, 13, 19 and 27 cycles. Now the first failure at 2 cycles may be thought to correspond to an F(t) value of i/n, where i = 1 and n = 5. ie F(t) @ 2 cycles = 1/5 or 0.2 or 20% Similarly for the second failure time of 7 cycles, the corresponding F(t) is 40% and so on. On this basis, this data is suggesting that the fifth failure at 27 cycles corresponds to a cumulative percent failed of 100%. In other words, on the basis of this sample, 100% of the population will fail by 27 cycles. Clearly this is unrealistic. A further sample of 10 items may contain one or more which exceed a 27 cycle life. A much larger sample of 1000 items may well indicate that rather than correspond to a 100% cumulative failure, 27 cycles corresponds to some lesser cumulative failure of any 85 or 90%. This problem of small sample bias is best overcome as follows: •
Sample Size Less Than 50. A table of Median Ranks has been calculated which gives a best estimate of the F(t) value corresponding to each failure time in the sample. This table is issued with these notes. It indicates that in the example just considered, the F(t) values corresponding to the 5 ascending failure times quoted are not 20%, 40%, 60%, 80% and 100%, but are 12.9%, 31.4%, 50%, 68.6% and 87.1%. It is this latter set of F(t) use values which should be plotted against the corresponding ranked failure times on a Weibull plot. Median rank values give the best estimate for the primary Weibull parameter and are best suited to some later work on confidence limits.
•
Sample Size Less Than 100. For sample sizes less than 100, in the absence of Median Rank tables the true median rank values can be adequately approximated using Bernard’s Approximation: F (t ) =
•
(i − 0.3) (n + 0.4)
Sample Sizes Greater Than 100. Above a sample size of about 100 the problem of small sample bias is insignificant and the F(t) values may be calculated from the expression for the Mean Ranks:
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The use of Weibull in defect data analysis F (t ) =
i (n + 1)
2.3 Trends in data A trend may be in relation to any other time-base than the one being used to assess reliability. For instance, if the reliability of vehicle engines is being assessed against vehicle miles, and a trend is observed in respect to the date of engine manufacture, or in respect of calendar time, then this is evidence that there is a trend. Before attempting to use a mathematical model such as weibul analysis is important to check the homogeneity of the data. If the data indicates the presence of a time series, it is inappropriate to apply a model that requires homogeneity. A simple approach to this would be to chart the failure data in the form of cumulative failures against cumulative time. Deviation from a straight line would indicate a trend. A mathematical approach to this would be to employ the Laplace Trend Test.
2.3.1 Laplace trend test The Laplace Trend test assesses the distribution of failure events with respect to the ordering indicated by the criteria being tested. The Trend ordering may simply be that the failures occurred in a specific order, rather than randomly. For instance, if the date of engine manufacture is being considered as an ordering criteria, then the failure times should be put on a time-line as follows, as ordered by date of engine manufacture.
Common sense will indicate that the later manufactured engines are not lasting as long to failure as the first. Therefore, there is clear evidence of a trend. However, to put this on a statistical basis, the Laplace Trend Test is used: Tes t St at i s t i c , U =
⎡∑t ⎤ 12N (t ) ⎢ − 0.5⎥ ⎥⎦ ⎢⎣ N (t )t
where U = standardised normal deviate—if the calculated value is less than that taken from tables appropriate to the confidence required (e.g 95%), then trend is not proven. If it is greater than the tabled value, then trend is demonstrated.
Σt = sum of failure times (on cumulative scale: see diagram above) Warwick Manufacturing Group
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The use of Weibull in defect data analysis N(t) = total number of failures t = total test time
Crudely,
∑t
N (t )t
< 0.5 indicates reducing failure rate
= 0.5 indicates no trend > 0.5 indicates increasing failure rate
2.3.2 Amendment for Failure Terminated Data Where data has been collected, terminating collection at the point of a failure, then the Laplace Trend Test is amended as follows:
Tes t St at i s t i c , U =
⎡ n−1 ⎤ ⎢ ∑ ti ⎥ i = 1 ⎢ 12 N ( t n − 1 ) − 0.5⎥ ⎢ N ( t )t ⎥ n−1 n ⎢ ⎥ ⎢⎣ ⎥⎦
where the last failure is not counted, whereas the final time interval is counted.
3 The mechanics of Weibull analysis 3.1 The value of analysis On occasions, an analysis of the data reveals little that was not apparent from engineering judgement applied to the physics of the failures and an examination of the raw data. However, on other occasions, the true behaviour of equipments can be obscured when viewed by the most experienced assessor. It is always necessary to keep a balance between deductions drawn from data analysis and those which arise from an examination of the mechanics of failure. Ideally, these should be suggesting complementary rather than conflicting counterstrategies to unreliability. There are many reliability characteristics of an item which may be of interest and significantly more reliability measures or parameters which can be used to describe those characteristics. Weibull will provide meaningful information on two such characteristics. First, it will give some measure of how failures are distributed with time. Second, it will indicate the hazard regime for the failures under consideration. The significance of these two measures of reliability is described later. Weibull is a 3-parameter distribution which has the great strength of being sufficiently flexible to encompass almost all the failure distributions found in practice, and hence provides Warwick Manufacturing Group
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information on the 3 failure regimes normally encountered. Weibull analysis is primarily a graphical technique although it can be done analytically. The danger in the analytical approach is that it takes away the picture and replaces it with apparent precision in terms of the evaluated parameters. However, this is generally considered to be a poor practice since it eliminates the judgement and experience of the plotter. Weibull plots are often used to provide a broad feel for the nature of the failures; this is why, to some extent, it is a nonsense to worry about errors of about 1% when using Bernard’s approximation, when the process of plotting the points and fitting the best straight line will probably involve significantly larger “errors”. However, the aim is to appreciate in broad terms how the equipment is behaving. Weibull can make such profound statements about an equipment’s behaviour that ±5% may be relatively trivial.
3.2 Evaluating the Weibull parameters The first stage of Weibull analysis once the data has been obtained is the estimation of the 3 Weibull parameters: β:
Shape parameter.
η:
Scale parameter or characteristic life.
γ:
Location parameter or minimum life.
The general expression for the Weibull F(t) is:
F (t ) = 1 − e
⎡ ( t −γ ) ⎤ −⎢ ⎥ ⎣ η ⎦
β
This can be transformed into: log log
1 = β log(t − γ ) − β logη (1 − F (t ))
It follows that if F(t) can be plotted against t (corresponding failure times) on paper which has a reciprocal double log scale on one axis and a log scale on the other, and that data forms a straight line, then the data can be modelled by Weibull and the parameters extracted from the plot. A piece of 2 cycle Weibull paper (Chartwell Graph Data Ref C6572) is shown at Annex A and this is simply a piece of graph paper constructed such that its vertical scale is a double log reciprocal and its horizontal scale is a conventional log. The mechanics of the plot are described progressively using the following example and the associated illustrations in plots 2 to 12 of Annex B.
Consider the following times to failure for a sample of 10 items: 410, 1050, 825, 300, 660, 900, 500, 1200, 750 and 600 hours. Warwick Manufacturing Group
The use of Weibull in defect data analysis
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•
Assemble the data in ascending order and tabulate it against the corresponding F(t) values for a sample size of 10, obtained from the Median Rank tables. The tabulation is shown at Section 16.
•
Mark the appropriate time scale on the horizontal axis on a piece of Weibull paper (plot 2).
•
Plot on the Weibull paper the ranked hours at failure (ti) on the horizontal axis against the corresponding F(t) value on the vertical axis (plot 3).
•
If the points constitute a reasonable straight line then construct that line. Note that real data frequently snakes about the straight line due to scatter in the data; this is not a problem providing the snaking motion is clearly to either side of the line. When determining the position of the line give more weight to the later points rather than the early ones; this is necessary both because of the effects of cumulation and because the Weibull paper tends to give a disproportionate emphasis to the early points which should be countered where these are at variance with the subsequent points. Do not attempt to draw more than one straight line through the data and do not construct a straight line where there is manifestly a curve. In this example the fitting of the line presents no problem (plot 4). Note also that on the matter of how much data is required for a Weibull plot that any 4 or so of the pieces of data used here would give an adequate straight line. In such circumstances 4 points may well be enough. Generally, 7 or so points would be a reasonable minimum, depending on their shape once plotted.
•
The fact that the data produced a straight line when initially plotted enables 2 statements to be made: o The data can apparently be modelled by the Weibull distribution. o The location parameter or minimum life (γ) is approximately zero. This parameter is discussed later.
•
At plot 5 a scale for the estimate of the Shape Parameter β, is highlighted. This scale can be seen to range from 0.5 to 5, although β values outside this range are possible.
•
The next step is to construct a perpendicular from the Estimation Point in the top left hand corner of the paper to the plotted line (plot 6).
•
The estimated value of β, termed β , is given by the intersection of the constructed
∧
∧
perpendicular and the β scale. In this example, β is about 2.4 (plot 7). •
At plot 8 a dotted horizontal line is highlighted corresponding to an F(t) value of 63.2%. Now the scale parameter or characteristic life estimate is the life which corresponds to a cumulative mortality of 63.2% of the population. Hence to determine its value it is necessary only to follow the η Estimator line horizontally until it intersects the plotted line and then read off the corresponding time on the lower scale.
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Plot 9 shows that, based on this sample, these components have a characteristic life of about 830 hours. By this time 63.2% of them will have failed. •
At plot 10 the evaluation of the proportion failed corresponding to the mean of the distribution of the times to failure (Pµ) is shown to be 52.7% using the point of intersection of the perpendicular and the Pµ scale. This value is inserted in the F(t) scale and its intersection with the plotted line determines the estimated mean of the ∧
distribution of the times to failure ( μ ). In this case this is about 740 hours. •
The median life can also be easily extracted; that is to say the life corresponding to 50% mortality. This is shown at plot 11 to be about 720 hours, based on this sample.
•
Finally, plot 12 illustrates that this data is indicating that a 400 hour life would result in about 15% of in-service failures for these equipments. Conversely, an acceptable level of in-service failure may be converted into a life; for example it can be seen from plot 12 that an acceptable level of in-service failure of say, 30% would correspond to a life of about 550 hours, and so on.
4 Interpretation of Weibull output 4.1 Concept of hazard Before examining the significance of the Weibull shape parameter β it is necessary to know something of the concept of hazard and the 3 so-called failure regimes. The parameter of interest here is the hazard rate, h(t). This is the conditional probability that an equipment will fail in a given interval of unit time given that it has survived until that interval of time. It is, therefore, the instantaneous failure rate and can in general be thought of as a measure of the probability of failure, where this probability varies with the time the item has been in service. The 3 failure regimes are defined in terms of hazard rate and not, as is a common misconception, in terms of failure rate. The 3 regimes are often thought of in the form of the so-called ‘bath-tub’ curve; this is a valid concept for the behaviour of a system over its whole life but is a misleading model for the vast majority of components and, more importantly, their individual failure modes (see [5] and [7]). An individual mode is unlikely to exhibit more than one of the 3 characteristics of decreasing, constant or increasing hazard.
4.1.1 Shape parameter less than unity. A β value of less than unity indicates that the item or failure mode may be characterised by the first regime of decreasing hazard. This is sometimes termed the early failure or infant mortality period and it is a common fallacy that such failures are unavoidable. The distribution of times to failure will follow a hyper-exponential distribution in which the instantaneous probability of failure is decreasing with time in service. This hyper-exponential distribution models a concentration of failure times at each end of the time scale; many items fail early or else go on to a substantial life, whilst relatively few fail between the extremes. The extent to which β is below 1 is a measure of the severity of the early failures; 0.9 for Warwick Manufacturing Group
The use of Weibull in defect data analysis
example would be a relatively weak early failure effect, particularly if the sample size and therefore the confidence, was low. If there is a single or a predominant failure mode with a β
