Is Cpk a good measure of capability? There are several shortcomings of Cpk

Tim Folkerts

Super Moderator
#11
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
 
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D

dfirka

#12
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
 

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Darius

#13
Now it start to rock, Tanks guys, thats more the site I like.
As I said:
Darius 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.
 

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Tim Folkerts

Super Moderator
#14
Daniel,


I checked out your link and I like your approach! 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 eek.gif (or myself 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
 

Tim Folkerts

Super Moderator
#15
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
 
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D

dfirka

#16
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
 
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B

Bill Ryan - 2007

#17
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: ).
 
D

dfirka

#18
Tim Folkerts said:
[*]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
 

Tim Folkerts

Super Moderator
#19
dfirka said:
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
My 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. biggrin-a1.gif
You say poTAYto, I say poTAHto. biggrin-a1.gif


Tim F
 
C

Chakravyuha

#20
About Cpk and other process capability indices

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.
 
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