Rui,
There is a common belief that challenges your assumption(copied below). Many people consider probability of failure and rate of decay are mutually exclusive in most industrial circumstances (the exception being things that start to decay from the time they operate but even then decay is not usually linear).

quote:

As you know, the probability of a degradation failure mode (erosion for instance) to occur increases with age, therefore it is logic to consider that the frequency of inspections must be increased or, in other words, the time between every two inspections must decrease progressively.

Something must initiate failure and from that point the condition deteriorates. Is there any evidence you know of that supports the view that deterioration rates depend on age? Would the exact same crack in a wing spar grow noticeably faster if it started at 5000 hours than it would if it started at 50,000 hours?

Perhaps this helps.

The EXCEL file I attach herewith illustrates by means of an example the foundations of my analysis on the economics of inspections of evident failures. This approach comes in a few books on reliability. As you can see, it has nothing to do with the PF interval.

I departed from this point, and then, made the PMF (PM + MF) interval interplay.

Regards,

Rui

This message has been edited. Last edited by: <Rui Assis>,

Inspections.zip (7 Kb, 37 downloads)

Rui,
Your arguments are challenging in the sense that I have to think hard.
I could get shot to pieces here – but what the heck – so long as I learn something!
I would like to suggest that using Weibull can only give you a probability of failure at any given time – from which a number of other risks can be established. I would contend that PF is not one of those. PF and risk of failure are not related – there may be exceptions where decay initiation points are known and where decay rates are known. These occasions are so rare that I have not come across one single instance of this.
Now I ask the rhetorical question – “what, if anything has risk of failure to do with interval of inspection?”
Mathematically if an inspection is 100% effective, then only one inspection is required inside PF. In such cases, the failure mode would be detected before failure every time. The least cost interval to do the inspection is therefore the PF. Now practically speaking, no one wants to find the failure the instant it fails so we typically discount PF by recovery time. Again this has nothing to do with risk of failure.
Now it follows that if the least cost inspection interval is the PF, one can do some calculations to decide whether this least cost is more or less than the cost of an unexpected failure. If the cost of the inspection is less than the cost of failure then the task is cost effective. If the opposite is the case, then the task is not cost effective – and more to the point – it will never be cost effective at any interval.
We are then left with deciding what interval to apply to those tasks which are cost effective.
Things that then matter are what I would call second order effects. The first order consideration is PF and I have discussed this.
Second order effects are the lead time to repair and the chances of the inspection missing the fault when a fault was present. Now we should strive to write inspections that can be performed with a high level of confidence. Some tasks could be considered to be fail safe in fact. Vibration Monitoring for example looks for changes in vibration spectrum so if someone screws up the analysis, it will appear as a fault. “of course someone could skip the reading etc but we are not setting up maintenance regimes for out of control systems – that to me would be the ultimate in futility”.
One final comment – PF is a variable itself and I debate in my mind whether this is a first order effect or secondary – who knows perhaps it is not that important to decide on the order.
Getting back to reality, there are some inspections that could miss a fault and I know of these. Crack detection methods for example often depend on crack orientation and some cracks can escape detection because they run parallel to the view.
For these cases, I can accept that doing more inspections inside PF can be modelled and justified and there are formulae that do this. And certainly the risk of failure (Weibull parameters) come into consideration at that point.
Below is the data you presented posted 25 June 2006 12:45 PM

1. Predominant failure mode described by a Weibull distribution with shape parameter of 2, a scale parameter of 8,000 hours and a location parameter of 0 (wear out typically);
2. Accumulated life time: 3,000 hours;
3. Last inspection: 2,600 hours ago and presented no sign of the onset of a failure;
4. Time remaining before the next overhaul: 9,000 hours;
5. PF interval: 500 hours (it could be considered to follow some probability distribution);
6. MF interval: 50 hours (the minimum time span you have to react and do something to prevent the functional failure – point F – meaning, therefore, that you have a time span of 500 – 50 = 450 hours to detect the failure in progress, if you are willing to prevent it from happen);
7. Cost of an inspection: 4,000 €;
8. Cost of the consequences of a failure: 100,000 €;
9. Labour and materials for the repair: 3,000 €;
10. Cost of capital: 25% per year;
11. Working regimen: 8,760 hours/year.

It seems there is no data to consider variability in inspection confidence and PF variance or lead time variance.
To me the answer if these are assumed to be fixed is that the inspections should be done at 450 hours regardless of the risk of failure at any point in time. This is therefore not a variable.
It is a long time since I did this high power maths but it may be that your use of Bayes’ theorem is not appropriate in this instance.
So there you have it from an engineer – burn me at the stake; crucify me, strike me off the list.... if I am wrong.... I could well be!

This message has been edited. Last edited by: Steve Turner,

Just one additional comment... I did examine your excel work Rui and from what I can see, I would raise the same arguments about it as I have with the previous scenario.

Hi Steve,

I am so sorry for keeping silent all these days. After your remarks, I studied the case all over again and, to my surprise, it seems that costs are lower when the inspection intervals are constant. The least cost is even obtained when the inspection interval is equal to the PM interval (PF = PM + MF). When I first studied this subject, I didn’t consider the hypothesis of constant inspection time intervals as I was deeply convinced of the correctness and logic of the variable (increasing or decreasing) time intervals approach. Now I have to recognise that I was imprudent. I don’t know the reason why yet. I am still doing some tests and trying to find out what variables influence the most and whether there is a set of values that might change this conclusion.

Meanwhile, I attach an Excel file where in sheet “Picture” I drew schematically what I mean by decreasing time intervals and “safe” time windows. In sheet “Costs” some results of the tests that I performed are shown.

I hope to come back soon with sound conclusions.

I am glad for this discussion that created the opportunity to review what I had already taken as granted. Thanks.

Rui

This message has been edited. Last edited by: <Rui Assis>,

Test_results.zip (6 Kb, 27 downloads)

Rui, Tudo bem,

There needs to be a lot more people like you in this game my friend.

Rui,
I think Daryl has summed it up in his response above.
I have never come to understand what variables are most important and have the greatest impact so when you finish your review I would be grateful to see your conclusions.
Regards
Steve
OMCS International

Steve,

I am now back at base; apologies for my long silence. Give me a snail mail address and I will send you a copy of my book. e-mail: eml@effective-maintenance.com

Thanks Vee,
I heard it is a great book.
I look forward to the signed copy when it arrives.
Regards
Steve
OMCS International

Rui, I cannot open the Test results file using Excel? Can others?

I got it to work ok

I am sorry for keeping silent for such a long time after having promised to come back with my conclusions. I much consider all of you and wish above all to give my (small) contribution by exemplifying scientific approaches that can bring more and more professional competitiveness advantages to practitioners in maintenance. In return, I expect to keep on learning from your knowledge and extensive experience.

The problem is that I have been awfully busy with the end of semester tests and final exams at the university. I have also been preparing a few bids. Therefore there has been no time left to study this subject in depth. I hope to come back within a week.

Although in Portuguese, you might want to visit my site ("born" two weeks ago) in the web: http://www.rassis.com from where a few articles and Excel applications can be downloaded on Operations, Productivity, Reliability, Performance and Statistics. I designed it myself and keep it as simple as possible. I hope you will find it interesting.

I wish you all good work.

Rui

I tried to understand the maths in the papers but it was difficult without a language to explain it. I quite understand the spreadsheets. Is there a software for translating Portugese into English? I guess if these papers are translated into English, the audience is wider.

Hi Josh,

Thank you for your interest. I think you are refering to the papers on the site. For these, you can always use BABELFISH http://babelfish.altavista.com/ to get a rude translation. Perhaps one day I have the time and the patience to prepare an English idiom version.

With regard to the inspection intervals, I will prepare a short explanation of the maths involved together with the results of my further investigation on this subject and post them here next week.

Regards,

Rui

Josh,

I´ve just noticed that some of the documents on my site are protected. If there is any one in particular whose translation you need, please let me know and I will forward a non-protected version to you.

Regards,

Rui

quote:

Originally posted by Josh:
Is there a software for translating Portugese into English?

Check online dictionaries like:
http://www.freedict.com/onldict/por.html
http://www.june29.com/IDP/IDPsearch.html
http://lookwayup.com/free/PortugueseEnglishDictionary.htm

Finally something in portugese to chew , although Brazilian and Portuguese have their differences.

Bom trabalho Rui!

I find a reference for that one too:
* http://www.deltatranslator.com/dtr3.htm

This message has been edited. Last edited by: Eugene,

Rui, let's try the 2 articles you attach on page 3 entittled Inspection & spares.zip.

Hi Josh,

Thank you for your interest. I attach herewith the same two documents that I attached in page 3 of this thread. This time they are both unprotected so their translation into English will be a simple task (despite it sounds very often strange or even anecdotic!). Please consider that the document entitled “Capítulo II” is being modified in light of our recent discussion on inspections intervals. Although, the math described in it keeps being valid.

Meanwhile I am going to unprotect all the PDF documents resident in my site.

Regards,

Rui

This message has been edited. Last edited by: <Rui Assis>,

Inspections___spares_2.zip (246 Kb, 24 downloads)