Widening Spec Limits to Account for GRR?

[#1]

Hi.  Manufacturing/Statistical question here. Suppose you have parts that are being tested at multiple points during manufacture (nodes).  I''m curious if/how much do people widen their limits after the first node to account for GRR?  Do you just add 1 GRR to the USL/ subtract 1 GRR from the LSL?  Or is there some other consensus in the statistical community on how this should be handled?

Historically, I have widened my limits after the first node by +/- 1 GRR. This is done to account for any shifts caused by tester/test/operator variability.  Passing parts whose values were close to the limits at the first node may have shifted outside due to this variability, so I widen the downstream limits to account for this.  This is done only after the first test node, bc it is assumed the part can shift only by +/-1GRR.

1. Do other people do this as well?

2. When setting limits to achieve a target Cpk = 2, we add +/- 3 sigma to the mean.   By that logic, should I be adding +/- 3GRR to my limits instead of +/-1GRR?

I think the statistically correct way to treat it is as follows, but I would like to get some confirmation from the community.  Suppose we treat the measured value as the sum of a static "true" value of X for that part and Eps, a normal random variable with mean = 0 and variance = GRR^2.

Node 1: Xmeas1 = Xtrue + Eps1             mean(Xmeas1) = Xtrue            Var(Xmeas1) = GRR1^2

Node N: XmeasN = Xtrue + EpsN            mean(XmeasN) = Xtrue            Var(XmeasN) = GRRN^2

XmeasN may be rewritten in terms of the initial measured value and a new random variable representing the shift between nodes (EpsDiff):     

XmeasN = Xmeas1 - Eps1 + EpsN = Xmeas1 + EpsDiff

where:

mean(EpsDiff) = 0                                                 (since mean(Eps) == 0)

Var(EpsDiff) = GRR1^2 + GRRN^2 = 2GRR^2    (assuming GRR same at all nodes)

By this logic, should I be widening my limits at downstream nodes by +/-sqrt(2)*GRR?  

Curious to know people's thoughts on this.  Thanks!

[#2]

Here are some of my thoughts/questions, though you may not agree with them:

1. What is GRR?  I think you want to account for measurement system error when setting specs, correct?  But I'm not sure what GRR number you are intending to use (std dev for repeatability, reproducibility, both)? Variance components always add (not standard deviations).

2. My understanding is traditional specifications are a function of the application requirements (hopefully customer driven).  While these may be negotiated, they should reflect at what point the component, assembly, etc. degrades in performance (or fails to perform).  Unfortunately, these are usually set independent of true process capability.  What I believe you are proposing is to set processing limits (internal spec perhaps) to account for measurement system uncertainty...is this what you mean?

3. I would have several questions regarding the estimates of variation due to the measurement system (also goes for the sample statistics used to estimate Cpk).  Is the measurement system stable?  What is the inference space for the collection of samples used to estimate the statistics?  How representative of future conditions?

4. My own little rant, as I have not seen many useful applications of the specification "mindset" (Meet the spec is really counter to continuous improvement).  What I would spend my time on is understanding what factors in the manufacturing process that affect the mean and variation of the critical performance measures of the output product.  Y=f(X) (using sampling and DOE).  Same for the measurement process. Then quantify at what point performance degrades (Taguchi Loss Function).

[#3]

To build on what @statman mentioned, and to complicate this further, aren't spec limits usually tighened to account for measurement system error? Jerry Fish and Jason Wiggins presented on this topic this year at Discovery, and also uploaded an Add-In to calculate spec limit changes based on MSA/GRR error. Link here. You could likely use their Add-In to calculate how much to widen your spec limits, assuming you still need to.

[#4]

This thought is the same as presented by @statman and @Jed_Campbell but from a different perspective. The specifications for disposition (pass, fail) are set by performance to requirements. The measurement/test error is subtracted from, not added to, these limits. There is more than one established scheme based on this notion. I have never seen anyone go the other way. Think of the specs as a wall; the outside is what the customer sees. You can take away from the interior, but not from them.

[#5]

Hi Mark and others.  

First let me clarify a few things.  Spec limits can be user-driven, or process-driven. The user-driven limits check the part is meeting the key criteria which are defined ahead of time.  Process-driven limits are used to ensure your process is in control.  These limits are (typically) selected to achieve a target cpk = 2 and are based on historical data, not user criteria.

Also to clarify, I'm using "GRR" to refer to the measurement variance introduced by multiple testers/operators/repetitions.  In other words, the total variance (TV), minus the part variance (PV). 

That being said, measurements on both user and process variables will suffer from shifts due to GRR as they move downstream (Your measurement doesn't care what you call your variable).   So I disagree that limits should never be widened.  Perhaps phrasing the problem a different way would help.  I was speaking in terms of widening the limits as the part goes downstream.  Instead, consider that I want to tighten the upstream limits to ensure the passing parts meet my final downstream specs.  My question is simply, how much does one tighten the upstream limits by to account for GRR?

[#6]

Tightening the upstream limits is an interesting way to frame this--thanks for the clarification. The Add-In at this link can be used to calculate appropriate amounts to move spec limits to account for GRR. You could also likely use the OC Curves platform (Analyze, Quality and Process, OC Curves) if you want to do it manually.

[#7]

Just some feedback about terminology and the subsequent confusion. IMHO, "Process-driven" limits are not spec limits.  Are you talking about control limits? These are a function of the process AND how it is sampled. You can't select them though they can change depending on how the data is acquired. Unfortunately, spec limits have nothing to do with statistical control as they are derived independently.

You can put whatever limits you want.  The process does not care nor is it influenced by your imposed limits. But carry on...

[#8]

I agree that using established terminology is helpful when discussing a complicated topic like this.