maintenanceStatistics4

Now suppose that four items are placed on test with results as given in the table shown on the slide. The table shows that the first failure was at 84 hours. Then at 91 hours an item was taken off of test for reasons other than failure. (that is, it was suspended). Two more failures occurred at 122 and 274 hours.

The slide shows that the average order number for the second failure is 2.33. We use these average position values to calculate the median rank from Benard's formula (Slide 3. For example, the median rank for the second failure time (F₂) would be:

Obviously, finding all the sequences for a mixture of several suspensions and failures and then calculatiing the averge order numbers is a time-consuming process. Fortunately a formula is available for calculating the order numbers. The formula produces what is termed a new increment. The new Increment I is given by:

The first two failure times would have order numbers 1 and 2 respectively. For the third failure a new increment must be calculated to be 1.29, as illustrated in the slide.

Adding I to the previous order number, 2, gives the order number of 3.29 to the third failure.

We continue with the same increment until another suspended item is encountered. Thus the order number for the fourth failure is 3.29+2.39=4.58. Applying the formula for the fifth failure gives:

The position for the fifth failure is 4.58+1.60=6.18, and the position for the sixth failure is 6.18+1.60=7.78. The median rank may then be used to plot the failures on Weibull paper.

Extending this idea to the absurd, suppose the sample size is 1. We would not expect the age of this failure to represent the age by which 100% of items in the underlying population would fail. It would be more realistic to regard the single age at failure as representing the age by which 50% of the underlying population would fail. Benard’s formula gives (1-0.3)/(1+0.4)=50% which is intuitively reasonable.

Therefore we need a better way of estimating the CDF for plotting. The most popular approach of estimating the Y-axis plotting positions is the median rank. We obtain, from the median ranks table or from Benard’s approximation of the Median Ranks, the respective cumulative probabilities of failure which are .13, .31, .5, .69, and .87.

When we use reliability analysis software we do not have to look up the median ranks in tables. Not only does the program automatically adjust the orders due to suspensions, but it also calculates and applies the median ranks to each observation.

The procedure is to subtract the time of the first observation from all the observation times and re-plot.

Try γ=375.

The next slide shows what the PDF would look like.

The median rank is an estimate of the CDF for a sample size of 12 (we need to include the suspensions). It has been calculated such that the 50% of the time the true percentage of failures above and below this value. (The median of the rank distribution.)..

But we can also plot the CDF according to a 95% rank which is an estimate such that 95% of the time the percentage of failures will be below this value. Likewise for the 5% rank line 95% of the time the actual percentage of failures will be above this value..

Therefore, by plotting all three lines, we obtain a confidence interval which says that the cumulative probability of failure will be in the range from a to b with 90% confidence (that is 90% of the time).

We have been assuming that the equation relating reliability to working age respects the Weibull form. We have been using Weibull paper to estimate the parameters of the Weibull relationship that best represents or fits the data.

Now we want to know how good the fit is, so that we can decide whether to accept or reject the model.

Acceptance or rejection is based on a criterion (or significance level) that we have adopted. And the test we use is a goodness-of-fit test.

We proceed to the K-S critical value table on the next slide.

The null hypothesis in this case is that "the model fits the data".

The alternative hypothesis is that "the model does not fit the data".

The test directly quantifies an error (called a "Type I error") that it may make:

"Not rejected at the 20% significance level."

Thus the test indirectly supports the hypothesis of a good fit because:

• The K-S test (with a significance level of 20%) will accept a good model 80% of the time (and reject a good model 20% of the time).

We apply our K-S statistic, 0.261 to the row for a sample size of 5. Assuming we wish to conform to a significance level of 0.20 we note that 0.261 is not greater than 0.446.

Hence the hypothesis of a "good" model is not rejected based on a 20% significance level.

We have an infant mortality kind of failure, due to one or more of substandard material quality, inadequate storage, or improper installation. Each of these possibilities should be investigated, and corrected as appropriate, in order to improve reliability.

The MLE method needs an iterative solution for the Beta estimate. Statisticians prefer maximum likelihood estimates to all other methods because MLE has excellent statistical properties. They recommend MLE as the primary method..

In contrast most engineers recommend the method of least squares method.

In general both methods should be used since each has advantages and disadvantages in different situations. MLE is more precise. On the other hand for small samples, it will be more biased than rank regression estimates.

With the exception of REWOP, all reliability analysis software requires that we transpose life data from the CMMS (computerized maintenance management system) to the software. This, it turns out, is an onerous task and the largest obstacle to using the techniques of reliabiltiy analysis.

In break-through technology, the REWOP methodology and software overcomes this difficulty by generating an "Events" table directly from the CMMS / work order database. We subsequently conduct analyses at will, with minimal data cleansing required. REWOP achieves its performance by linking work orders to the RCM knowledge base. Each siginficant work order becomes, thereby, an instance of a knowledge record. Using REWOP as a fully CMMS (SAP, Maximo, Datastream, Ellipse, etc) integrated module we analyze those instances.