Looking back at various posts about capability indices, I note there are several shortcomings for cpk.

There is no provision for asymmetric tolerance limits. For example, if the specs are 100 +5/-1, the best cpk is obtained by centering on 102, not 100. But this is clearly not what the customer wants. http://elsmar.com/Forums/images/smilies/confused.gif
cpk = 0 occurs whenever the mean is equal to the spec limit – no matter what the spread in data. Presumably, less spread would be better in reality, but the cpk doesn’t change. http://elsmar.com/Forums/images/smilies/confused.gif
The whole process is predicated on a “go/no-go” mentality, rather than a Taguchi-type “closer is better” mentality. A set of data tightly clustered just inside the spec limits will generate a good cpk, but all of the parts will be on the verge of being out of spec. A set of data clustered just outside the spec limits will work just about as well, but will have a terrible (negative) cpk. http://elsmar.com/Forums/images/smilies/confused.gif
For non-normal data, some advocate going ahead with the calculations with no concern for the distribution. Others advocate transforming/manipulating the data into something closer to normal behavior. Which to choose? http://elsmar.com/Forums/images/smilies/confused.gif




Does anyone else care about these difficulties? Is there anything else I left out?


And then, is there any value in looking at a different capability index to address some of these issues, or is cpk so firmly ingrained that a system that offers some small (or even moderate) improvements is a waste of effort?

Tim F

Excelent post Tim

Does anyone else care about these difficulties? Is there anything else I left out?

I think that there is something else.

Wrong specs and the famous value of 1.33, 1.66, or whatever else that some guys think can achive no matter the reality of the specs. Not all specs are customer specs, but internal (specially with attributes like "taste", "smell"), wich are set by the company according to bibliographic references extrapolated to levels that can't be detected by the customers (if low is good nothing is better).
:nopity:

My answer is the same as over in the ASQ Discussion Boards, I don't do CpK, and I recommend to anybody who doesn't "have to", don't do it. Pay attention to the SPC chart itself. Plot the specification limits on the SPC chart as a reference, and you can visually see what is going on. Cpk, as a single number, can give you a false sense of security versus what the control chart can tell you in the same instance.

Process capability calculations have been around since at least the 1980's. Does anyone know the history of Cpk and where it was first used? I am no bigger fan of Cpk than Steve but it is a commonly accepted quality statistic. It is not uncommon for customers to require Cpk information but has anyone ever asked what they use the information for?

Bill Pflanz

not sure where Cpk originated (I have it here somewhere!) but my vague recollections are that it came from Ford - at least that's my first introduction to it. I distrusted it then and I despise it now...that I know what bad decisions, actions and behaviors it has driven. Steve is right: how can a single number describe varaition? I truly believe that the Cpk index was 'invented' by someone who wa too lazy - or really didn't care - to plot the data and think about the data.

Bev,

My recollection is that Motorola and GE were using Cpk in the early 1980's. I googled process capability and Ford to do some research on the subject and it appears that Ford was using it and pushed for it to be in the QS9000 requirements.

In reading over some of the published internet articles, Cpk was meant to be an easy to understand calculation and a single number to explain the process rather than using control charts. With all of the threads on calculating and interpreting Cpk, the calculated number has become more important than what is really happening in the process. They may have created an easy to report calculation but have reduced it to such a level that it no longer represents the original theory or intent.

When something like this happens, I always attribute it to "hacks teaching hacks". There is something to be said about learning from the masters.

Bill Pflanz

Steve,

We seem to have an electronic echo going here, with sets of posts and replies in different venues. Hopefully you, me, and Bill aren't getting to much of a sense of deja vue. http://elsmar.com/Forums/images/smilies/magic.gif

I agree that looking at a single number gives a limited view. That is one of the eternal challenges of statistics. The more you summarize, the less you know about the original data, but the simpler it is to state and compare (and to report to the boss and/or customer who may not have a good head for numbers). The challenge is to choose the appropriate level of summarization that is simple enough but still complete enough.

For example, you can summarize 100 data points with one value - the mean. Sometimes that tells you enough about the data for your needs. Other times, you might want to know a bit more - perhaps the st dev, or the range, or the skewness, or ....

So one challenge is to decide how many numbers to use to summarize the data sufficiently for your needs.

The second challenge is to decide which numbers to use for the summary. Instead of mean, we could report the median. In both cases we have a one-number summary, but the two numbers tell you slightly different things.

Your main point seems to be that any one-number summary is insufficient to adequately describe capability. Point well taken.

My original point was (assuming that you want or need to give a one-number summary) is Cpk the best choice? I would argue that even given this somewhat artificial limitation of a one-number summary, Cpk is not the best choice.

Tim F.

Your main point seems to be that any one-number summary is insufficient to adequately describe capability. Point well taken.

My original point was (assuming that you want or need to give a one-number summary) is Cpk the best choice? I would argue that even given this somewhat artificial limitation of a one-number summary, Cpk is not the best choice.

Tim F.



Dr. Deming often stated - "It's so simple, just plot the dots". The chart is the visualization, and one number will never replace a chart.

it's a shame missing a good thread. :bonk:
IMO Only the post Tim post add some knowledge, As far as I know I can not send a control chart with 1000 points (to say a number) to the general manager or in the case of 40 control charts. Agree that control charts show even some behabuir that the histogram don't shows, but as Tim said it's necesary to obtain some indicators.

I tested many capability and performance index, I am not agree the use of performace for "non stability of the process", why?, because the performance indicators are afected by outliers too much, and if the standard deviation estimate for control limits is good for control limits even when "non stable process", why are not good for an capability-performance index.

I like Taguchi loss function (not the Taguchi loss index or cpm), but I must add to the calculus the use of non parametrical, like median and a percentile estimate of deviation. The non parametrics work good even for non normal conditions and the Taguchi loss function doesn't has a magical target as 1.33 for cpm. :biglaugh:

Hey gang. . . any Cr fans here? That is the index that Process/Manufacturing Engineering should be most interested in. After tat is as low as possible, adjust to a desired target. No one index tells the whole story. . . like Neither Juran. Deming, Crosby, or Shanin had the whole picture nailed. As Marc says, "One size does not fit all"

Taz & Darius,

I had forgotten about Cr & Cpm (or maybe I never knew) so I had to look those up. After looking the various indices, here is my humble opinion.


Cp: has been made pretty much obsolete by Cpk, so it isn't very useful
Cr: is just 1/Cp, so it suffers the same shortcomings as Cp
Cpk: has shortcomings listed in the first post in this thread, but it is much better than Cp because it includes an effect due to centering. Basically, I think it focuses too much on reducing variation and too little on centering.
Cpm: I like this one. The equation is more complicated than Cpk, but it adds more of a penalty for being off-center. If I had seen this originally, I might not have tried my own.

:caution: Warning, the next part is a little more theoretical and more speculative, so proceed at your own risk! :caution:

And now my candidate - call it Cpt (t for Taguchi).
1) Find the taguchi cost for each piece:
cost(i) = [x(i) - target]^2 / (tolerance)^2
2) Average these costs.
ave. cost = sum(cost(i)) / n
3) Take 1 over this number (since people like big numbers for good processes)
Cpt = 1/ (ave. cost)


With a few more math tricks, this can be adjusted to give the same value as Cpk for a centered process. I've tried it with some different sorts of data and it seems to give a logical and useful value - a value that is IMHO more logical and useful than that given by Cpk. And there is no mention of normal distributions and there is no need to estimate the standard deviation!


Tim F

Tim

Here I attach a draft of a paper discussing a capability index that we have been using for some years now, and proved helpful for studying vendor performance.

I think it has some similarities to your candidate, but with "sensitivity coeficients" based on the criticality of the measured characteristics.

The best value is 100, when everything is on target. The index is zero when out of specs.

The underlying concept is Taguchi Loss Function, and it is also a candidate, but has been used for several years at different plants in Argentina to compare and evaluate supplier data with good results. It needs a huge amount of calculation though.

Daniel

Now it start to rock, Tanks guys, thats more the site I like.
As I said:
I like Taguchi loss function (not the Taguchi loss index or cpm), but I must add to the calculus the use of non parametrical, like median and a percentile estimate of deviation. The non parametrics work good even for non normal conditions and the Taguchi loss function doesn't has a magical target as 1.33 for cpm.

FPT = Cost * ((Stdev ^ 2 + (Median - Target) ^ 2) ^ 0.5

Of course if you don't have the Cost still is a good indicator, I took it this way insteed of the reciprocal because the posibility of 0 in both terms. The adventage that I see in not taking in acount the specs is the "reality" of specs.

But still see the problem of the amount of data for such index, so it's needed to obtain the confidence intervals and take the minimum value (the at least value) for such index.

Tim, your indicator look interesting specially if you want to obtain a global indicator.

I think that we are getting somewhere, like Tim and Daniel, I see a great potential to the Taguchi loss, altho the diferent we take.

Daniel,


I checked out your link and I like your approach! http://elsmar.com/Forums/images/smilies/thumbup1.gif

My approach is more closely tied to the actual quadratic loss function of Taguchi, but then again, there is nothing sacred about a quadratic function.

I like the idea of having different penalties for being off-target based on the criticality. I had figured out some ways to do that for my model but didn't want overload you http://elsmar.com/Forums/images/smilies/eek.gif (or myself http://elsmar.com/Forums/images/smilies/omg.gif )with too many new factors at once.

From a practical perspective, there are two main differences between our approaches.

Perfect data (all points exactly on target) would give a score of 100 for you, but would become infinite in mine (or in the traditional Cpk). I think you have the edge there, because there is little practical advantage of going from a tiny bit off target to a teeny-tiny bit off target.
Whether the data is a little outside the spec limits or a lot, your index gives a score of zero. Mine penalizes more the farther out you get, which I think is appropriate.
One thing I just realized is that it makes some difference whether you are looking from the perspective of the producer or the consumer. For a producer, having a tight spread is important, because you can usually change the centering by adjusting some factor. For a consumer, a tight spread is nice, but only if it is a tight spread near your target, so centering has enhanced importance.


Tim F

Darius,

I think I basically re-invented the formula you presented! Looks like once again I am a day late!

I haven't proven it mathematically yet, but a sample calculation shows that the equation you presented:
FPT = Cost * [(Stdev ^ 2 + (Median - Target) ^ 2]
is the same as step 2 of my index. I'm sure some clever statistican rearranged the terms and did some algebra and got from where I was to where you were.

(Two notes: for this to work you need to 1) drop the square root from FPT equation and 2) change "median" to "mean". That seems to agree with the document you linked to, so I bet you just got typing a little ahead of your brain! It happens to the best of us!)

To complete the comparison, all you need is to set Cost = 1/(tolerance)^2. This basically says that if you have a small tolerance, then there is a big penalty for being off-target. So your index does relate to the tolerances, but in a somewhat hidden way by choosing the value for Cost.


Personally, I like this form (FPT rather than 1/FPT), since it relates most directly to the Taguchi cost. The reason for step 3 was simply to get a number that looked more like Cpk. In fact, I noticed that 1 / [3 * FPT^0.5] is Cpk for a centered process. For off-centered processes, the two become farther apart.

Tim F

Tim, Darius

I think another basic difference between our indexes is that TDF is applied to each measured value, it does not involve any averaging or parametrization.

In some sense, it is a "transformation" of the variable from a "process voice" variable, to a new variable TDF that could be seen as "process+customer voice" variable. After that transformation, i perform calculations and averaging.

I think that it is very similar approach with the first formula of Cpt you Tim proposed, doing the average after the cost determination.

Maybe we can say that Cp, Cpk, etc. are "ex-post" indices (after estimation of parameters), and TDF is "ex-ante" (before estimation of any parameter of the distribution).

Other practical advantages we found:

- Atribute Data: when the characteristic is an attribute, we average the percentage defective and apply the TDF to the average, with 0% as target, and an eventual Max % defective as USL, that gives us one value for the index in those characteristics.

- Ordinal Data: That was more interesting, when you can order the results, say from N different categories. As long as you can establish a nominal value X that is the Target, and a nominal value Y that is not acceptable, I can apply a "discretized" version of the TDF: being 100 when we measure X, zero for the measured value is Y, and monotonically decreasing (depending on criticality) from X to Y.

These modifications allowed us to work with any kind of variable.

As I say in the paper, the discontinuities in the TDF make it difficult to study the statistical properties of the distribution (I'm experimenting now with bootstrap methods to obtain confidence intervals).

The main reason I like it is because people actually using it to track suppliers (for about four years) really like it. Of course it was difficult at the beginning, because sometimes they had to redefine their targets or limits,that were kind of whimsical.

Daniel

WOW!!! You guys are really exercising my "grey matter". Great thread!!!! :cool:

I wish I could contribute but this is going to take a while to soak in (being the statistical practitioner that I am :bonk: ).

Whether the data is a little outside the spec limits or a lot, your index gives a score of zero. Mine penalizes more the farther out you get, which I think is appropriate.

Yes, It is true, I made the desition based on my idea of "transforming" the variable in the "integration" of voice of the customer with voice of the process.

I decided that the value of the voice of the customer will say Zero outside spec limits,no matter what the process say.

Daniel

Yes, It is true, I made the desition based on my idea of "transforming" the variable in the "integration" of voice of the customer with voice of the process.

I decided that the value of the voice of the customer will say Zero outside spec limits,no matter what the process say.

DanielMy index (based on the Taguchi quadratic function, and which is the same as Darius' FPT with cost = 1/tolerance^2) is quite similar to your TDF for the type C curve in your link.

For FPT, a perfect process would score 0 and a process where everything is right at the spec limit would score 1.
For TDF, a perfect process would score 100 and a process where everything is right at the spec limit would score 0.
Basically, 100*(1-FPT) = TDF

The big difference is that FPT lets the score get worse than 1 as the process gets outside farther outside the limits, while TDF never lets the score get worse that 0. So when there are parts outside the spec limit the equation about wouldn't exactly work: 100(1-FPT) would be somewhat less than TDF.

It is a tough call to know which way to go. I can imagine situations where it matters to the customer how far outside the limits you are, and other situations where anything outside the limits is equally bad, no matter how out of spec it is. It would be easy redefine my index like yours or yours like mine.

This is where we need the voice of the customer (the quality engineers who will be using the index!) to indicate which is more valuable to them. We would need some more input from more people to pick one over the other.

It's six of one, half a dozen of the other. http://elsmar.com/Forums/images/smilies/biggrin-a1.gif
You say poTAYto, I say poTAHto. http://elsmar.com/Forums/images/smilies/biggrin-a1.gif


Tim F

Hello,
Cpk, Cp and Cpm are not designed for measuring process capability in case of asymmetric specifications. This is well known in statistical literature. Instead a better choice is to use Cpmk or C'pm. Here I must mention that these indices are not popular within the industry because they are found out to be mathematically rigorous to calculate.
Before using any process capability index, it is necessary to know when and where it can be used. The 3 main assumptions of using 'Process Capability Indices' (the most popular ones) are:
1. Process data is normally distributed.
2. There are no outliers in data (Control charting is necessary to prove this)
3. Data points are independent and not correlated (In chemical industry this is a huge problem)
Sure, in practice, it is difficult to implement the appropriate PCIs.
Thus when data is not-normal, it is best to transform this non-normal data to normal data by using data transformation techniques.
There are some non-normal PCIs too but their use is confined to textbooks as of now.
Just for information: Cpk was was defined by Juran in 1974.

If you want any more information, feel free to ask me.

Thanks.

1) Most physical manufacturing processes are normally distributed about the mean. This is irrespective of the specifications needed by the user. Hence in dealing with cases of non symmetrical specifications, process mean has to be mean value of tolerance band of specification. With this approach, you minimize % of tail portion of distribution going out of specifications.

2) Cpk is a ratio of
Numerator-How much mean is away from target
Denominator- How much is the spread
Unfortunately when numerator is zero, denominator doesnot matter. In such cases, increase resolution or decimal accuracy of your equipment to avoid non zero numerator.

3) Cpk is a variable data like loss function. Any variable data can be converted to attribute data. Unfortunately many people convert and interpret Cpk that way. There is no practical physical difference between Cpk of 1.31 and Cpk of 1.35. However people call it as " Fail" or "Pass"

Chakravyuha said


Cpk, Cp and Cpm are not designed for measuring process capability in case of asymmetric specifications. This is well known in statistical literature. Instead a better choice is to use Cpmk or C'pm. Here I must mention that these indices are not popular within the industry because they are found out to be mathematically rigorous to calculate.

Just for information: Cpk was was defined by Juran in 1974.
Thanks.


Thanks, I hadn't seen Cpmk before , but a quick web search shows that it does effectively address the issue of asymmetric spec limits. And in the case where the specs are symmetric, it gives the plain old Cpk.

I agree it is more difficult to calculate, but in this age of computers, that seems to hardly be a concern. When Juran invented Cpk in 1974, computeres were rare and calculators rather simplistic by today's standards, so the calculations would have been a royal pain. Today, with computers everywhere, having a slightly more difficult equations doesn't seem to be much of a concern.


Then Arvind said

Hence in dealing with cases of non symmetrical specifications, process mean has to be mean value of tolerance band of specification. With this approach, you minimize % of tail portion of distribution going out of specifications.

This depends on your goal. It you want as few outside the tolerance limits, then I agree -- the process should be centered between the two limits, regardless of the stated target value. But that may not be what the customer wants! If the process can produce a fairly tight spread, then very few parts would be out of spec in eiter case, but the off-centered process would improve the performance for most of the parts.

I guess it all comes back to good communications between supplier & customer.

Thanks, everyone for the thought-provoking discussion. It's given me some fresh insights to mathematical subtleties and practical realities of capability indices.


Tim F

3) Cpk is a variable data like loss function. Any variable data can be converted to attribute data. Unfortunately many people convert and interpret Cpk that way. There is no practical physical difference between Cpk of 1.31 and Cpk of 1.35. However people call it as " Fail" or "Pass"

Arvind,
Let me give two unrelated comments on your point, and please clarify if I didn't get the spirit of what you wrote.

1- I would say that although it seems an attribute, aiming to reach a certain Cpk is a valid approach. You are just setting a specification for the "variable data number" Cpk that you define as a performance objective.

In that way, you can consider certain Cpk (Cpk0) as OK for a process: If the CPK is under that threshold Cpk0, you need to improve now; if the Cpk is higher than Cpk0, you will concentrate first in other processes less capable.

In this case, the "pass" (Cpk>Cpk0) relates to an acceptable synchronization between the Voice of the Process (summarized by avg and dispersion) and the Voice of the Customer (spec limits).

Let's say that the "Voice of the Company" establish this Cpk0, that is always provisory, specially at the "quality control" phase in Juran’s Trilogy. Like Juran, I think that continuous improvement works on a project basis, and the Cpk0 could be a valid metric for an improvement project.

2- Your point goes from Cpk to a Pass-Fail attribute. Interestingly, the reciprocal argument is used to derive a Cpk/Ppk value from attribute data (Bothe, Measuring Process Capability, 1999).

In this case, the percentage nonconforming is used to obtain a Z value and from there derive the Ppk using the formula (Zmin/3). D. Bothe calls this metric "Equivalent Ppk".

An extreme of this "inversion" is the procedure used by Six Sigma to transform DPMO to Sigma Level: As 6 sigma means 3.4 ppm (1.5 shift), then 3.4 ppm means 6 sigma.


Daniel

Tim,
You will be surprised to know that there are around 25 process capability indices defined in statistical literature. But owing to their complexities and lack of practical usage only Cp, Cpk, and Cpm are used popularly. One must remember that every PCI has some advantages and weaknesses and no one PCI is appropriate for all situations.
In fact, as part of my Masters thesis, I have designed a Decision Support System which selects, calculates and then interprets the appropriate process capability index depending on the various types of process data. In fact, it handles virtually all kinds of data usually encountered in manufacturing industries. The user just has to enter the data values, the software does the rest i.e it first does control charting (for subgrouped data, individual observations,etc using Xbar and R, Xbar and S charts,MA charts, etc), then once the process is in control, it performs the PCI calculations after careful selection of the appropriate PCI. Morever, it can handle short-run data, bivariate data, and non-normal data also !!!

Anyone wants to know more ?

Thanks.

... then once the process is in control, it performs the PCI calculations after careful selection of the appropriate PCI. Thanks.

What if... the process never reaches the "in control" attribute? :confused:

I think it's not part of this tread but, can you define "in control"?. There are many and some say this can never happen (no patterns).

What if... the process never reaches the "in control" attribute? :confused:

I think it's not part of this tread but, can you define "in control"?. There are many and some say this can never happen (no patterns).
There are several variations on the definition of "in control", but here is the one I use:

You must have at least 25 data points that have a single average and control limits fit through the data such that none of the following criteria are met:

One point outside the control limits
Two out of Three points two standard deviations above/below average
Four out of Five points one standard deviation above/below average
Seven points in a row all above/below average
Ten out of Eleven points in a row all above/below average
Seven points in a row all increasing/decreasing.

Steve -

I agree as usual with what you say, but think you should add caveat to examine plot for "patterns”. Perhaps you believe this is self evident, but many beginners in particular don't. Rather they rely on statistical software to flag the WE tests you mention.

Take a look at the obviously stylized chart attached. I believe it violates control definition in "hugging the centerline" (which could be set up as statistical test) and pattern tests (which would be very difficult to put into software test).

Thoughts?

Dave

Take a look at the obviously stylized chart attached. I believe it violates control definition in "hugging the centerline" (which could be set up as statistical test) and pattern tests (which would be very difficult to put into software test).

Thoughts?

Dave

Yes, I admit I left off the general statement of "patterns and non-random cycles". Your chart is a good example of that. The difficulty I have had is proceduralizing (making an operational definition) out of that criterion.

I have also seen some folks use a rule for too many points within one standard deviation of the average in order to catch that the points "hug" the center line too much. Of course, that can be also accomplished by the Range chart.

Personal experience so far is that I have only seen one chart in the thousands I have done with a seasonal cycle - insect and animal bites and stings. And that tripped other statistical criteria when plotted on a monthly basis.

So, I have stuck to the list I posted in the previous message. I figure that it is straight forward, and if a person gets proficient at that list we can talk about the non-random cycles issue.

Hello again,
I have done extensive research in the topic of process capability indices and if anyone wants more information about the same, feel free to contact me. Actually, it was part of my Master's thesis. As a starter, I would like to inform that most of the commonly used PCIs are limiting values of Vannman's index Cp(u,v) with different values of u and v, resulting in indices such as Cp, Cpk, Cpm and Cpmk.
Anyone interested in this topic and in knowing how to choose an appropriate PCI ?

It was interesting to find out that the PCI belong to a more general class. Therefore it would be very helpful if someone could indicate where this information can be found.
Attempts to contact the author have not resulted in any reply. Also a web search did not help either.
Is there someone who can shed some more light on this topic?

Some info about Vännman

http://www.sm.luth.se/~kerstinv/

http://www.asq.org/pub/jqt/past/vol34_issue1/qtec-40.pdf
http://www3.stat.sinica.edu.tw/statistica/oldpdf/A5n227.pdf

His Email kerstin.vannman@ies.luth.se:magic:

For non-normal data, some advocate going ahead with the calculations with no concern for the distribution. Others advocate transforming/manipulating the data into something closer to normal behavior. Which to choose?

Well, for precision machining - neither. Knowing that the distribution is a uniform distribution (if the process has been controlled correctly) from the sawtooth data, the the capability = (USL-LSL)/(UCL-LCL), or the span of the specification over the span of the process variation - which occurs at a constant probability over the control limits. Transforming them to 'normal' is a waste of time and can mask valuable information.

See Statistical process control for precision machining (http://elsmar.com/Forums/blog.php?b=79) :cool: