#1




Limits of using Cumulative binomial equation
Hi,
I was wondering whether Cumm binomial equation (DRT) can be used to determine the sample size for a particular test where in the expected failures are repairable/recoverable either by itself or by manual intervention. This,although meets the criteria (for using binomial eqn) of mutually exclusive failures in the test but i am not sure whether the outcome can be considered as just Pass1 or Fail (meaning occurance of failure)0. As an example, we are testing 8 units to demonstrate 80%Rel @80% CL to its full life. There are lets say "X" no of items (different brands) to be tested on each of the 8 units. So my question is can Cumm binomial eqn for zero failures n = ln(1CL)/ln(R) be used to determine the sample size"y" for each of the "X" items to be tested on all 8 units. If the requirement is 99.9%R@80% CL , then each of the "X" items need to be sampled for y=1608.63. Is this approach correct ? Thanks Arvind Last edited by Arvind; January 29th, 2009 at 01:15 AM. 
#2




Re: Limits of using Cumulative binomial equation
Quote:

#3




Re: Limits of using Cumulative binomial equation
I understand your point but my question is, are we following the correct approach or violating the applicability of cumm binomial eqn. Is there any better way of conducting the test for the given example assuming that the failures are all recoverable &/or repairable.
Regards Arvind 
#4




Re: Limits of using Cumulative binomial equation
A cumulative binomial looks at the probability of a specific “event”. The issue here, or what it really boils down to, is how you are defining the event. Also does the so called repair/restoration change the underlying life model with respect to the mode that you are looking at. As an example consider a test on a set of tires. Also assume that what I am looking for is failure due to wear out (tread depletion). If I get a puncture in the tire I repair/restore it (to a condition that is as good as it was just prior to the puncture). Also under normal use conditions the same would happen if a puncture occurred in the field. Furthermore assume that the puncture itself (or number of punctures) on a specific tire are independent of the life with respect to the mode that I am looking for. Then modeling it as mentioned would work. However if the repair included retreading then this type of analysis is wrong.
__________________
Pantelis Vassilou 
#5




Re: Limits of using Cumulative binomial equation
As an example,
we are testing 8 units to demonstrate 80%Rel @80% CL to its full life. There are lets say "X" no of items (different brands) to be tested on each of the 8 units. So my question is can Cumm binomial eqn for zero failures n = ln(1CL)/ln(R) be used to determine the sample size"y" for each of the "X" items to be tested on all 8 units. If the requirement is 99.9%R@80% CL , then each of the "X" items need to be sampled for y=1608.63. Is this approach correct ? Hi Pantelis, To add some more info on the test. Assuming the life of the unit is 20k cycles, hence the number of X items that can be tested is max ~ 12 only (1609*12 = 19308), Assume the unit to be a Copier machine, X refers to paper brands & the design spec is Avg 1 defect/500 cycles irrespective of the brand 'Xi'. When we test each of the 'Xi' brand items as per the sampling yi= 1609, we design the test in such a way that each of the Xi=1,12 brand items goes thur the unit throughout the 20k life for ex 100 samples of each Xi=1,12 per 1 cycle of test & repeat the cycle continuosly until 20k life. Since we are not testing the full qty of 1609 at one go, can we claim that we are demonstrating 80%rel @80%CL for each of the 'X' items ? Is the use of binomial sampling correct in this case as i understand that the trails need to be continuous ? if what we are claiming (80%R,80%CL) is not correct then would it be appropriate to just test against the spec & save cost i.e., sample size 500 for every brand 'X' & what would be the CL we are demonstrating in this case ? When i say repair/restore, i am not doing any design change/modification to the unit in general & failure iam looking for is due to the wearout of mechanical parts. Thanks Arvind 
#6




Re: Limits of using Cumulative binomial equation
Thanks for the info… Unfortunately, your most recent explanation instead of helping is confusing me more. Can you simplify and restate.
__________________
Pantelis Vassilou 
#7




Re: Limits of using Cumulative binomial equation
Hi Pantelis,
No of test units  8 copiers (to demonstrate the unit reliabilty  80%R @80%CL) Life of each unit 20k cycles of operation Defect spec  Avg 1 defect/1000 cycles of opn of the unit, irrespective of paper brand. Defect type Wear out failure, recoverable/repairable without any design change. No of items X  # of paper brands to be tested on each unit, Sampling plan for paper brand  Cumm Binomial n = ln(1CL)/ln(R) , n= 1609 per each paper type for 80%R@80%CL. Hence X is limited to 20,000/1609 = ~12/ every unit. Test design for unit reliability testing with various paper brands Each test cycle  sequentially test 100 pg of each paper brand/ unit & test all 8 units concurrently. i.e., brand i=1,12 = 100pgs/unit/cycle. Total pages used per cycle  100*12 = 1200/ unit continue the same cycle until unit life 20k on all 8 units. My question is since we are not testing the entire sample size of every paper brand n=1609 in one trial but only 100 pgs/cycle, can we calim 80%R@80%CL for the unit for every paper type tested ? Are we violating any requirements of Binomial eqn for ex successive sample trials? since the spec allows 1 defect/failure per 1000 cycles of op, i am assuming that we are not correct in using n = ln(1CL)/ln(R) for sampling paper brands ? What if we just test 1000pgs per brand as per the defect spec 1/1000 cycles of operation and save cost ? is this correct? what would be the R,CL associated with this sampling. Thanks Arvind Last edited by Arvind; February 3rd, 2009 at 07:46 PM. 
#8




Re: Limits of using Cumulative binomial equation
Ok… So based on that let me summarize:
You are basically testing reliability by using different papers sequentially, and you are assuming that each paper run consumes some life wrt the wear out mechanism. Also one paper brand versus another may consume more life, and (assumption) it does not matter in what order you run the papers (i.e. X1, X2, X3 … or X3, X1, X2… would cause the same end wear out  cumulative damage). Now with respect to the unit’s reliability then  if under normal use conditions the unit would see the same “stress profile” (i.e. similar paper types and paper type variation) then what you are really doing is simulating a use condition, thus you are not invalidating any “requirements”. However, if that is not true then you may want to look at this as an analysis using some type of an accelerated testing model (i.e. cumulative damage model) where each paper run would be treated as a stress, and subsequently extrapolate use reliability based on what type of paper (or paper groups) are used in the field (use condition). Hope this answers it…
__________________
Pantelis Vassilou 
#9




Re: Limits of using Cumulative binomial equation
Hi Pantelis,
Sorry for getting back so late. Your first half of the reply is exactly correct wrt what iam trying to do. So with our current process of n=1609 per brand,we can indeed claim 80%R@80%CL for every brand, is that correct? "if under normal use conditions the unit would see the same “stress profile”(i.e. similar paper types and paper type variation) However the above is not really valid, because our testing is extremely accelarated and we do not expect the units to run out of life within a month. However, if that is not true then you may want to look at this as an analysis using some type of an accelerated testing model (i.e. cumulative damage model) where each paper run would be treated as a stress, and subsequently extrapolate use reliability based on what type of paper (or paper groups) are used in the field (use condition). I agree that we need to be looking at some type of an accelarated test model, but iam not clear what you mean by each paper run ? do you mean each paper brand or each sample of paper that is loaded on to the unit. Typically we have a lot of brands & doing an analysis for each type of brand may not be feasible & also we may not have the info on the exact field usage for every brand type. Need your help to elaborate & if possible direct me to some similar case studies. Thanks alot Arvind 
#10




Re: Limits of using Cumulative binomial equation
See http://www.weibull.com/acceltestwebcontents.htm for an overview and there are links to studies from there. Obviously not exactly the same as what you are looking for but it should give you some intro and ideas.
Depending on how much you want to get into this you may also want to think about taking one of the courses I/we teach on this... see http://www.reliasoft.com/seminars/gencourses/rs521.htm
__________________
Pantelis Vassilou 
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