Target Index/alternatives for Process Screening

Question

For some dumb reason I was thinking that Target Index normalized on the Spec window, and I'm realizing now that it normalizes on the within sigma, which is a problem as I have a lot of mathematically significant signals that aren't practically significant.  So, couple quick questions.

* If you provide a specified sigma will the Target Index use that instead of within?  That will at least standardize the noramlizing factor
* The spec centered mean seems more in line with what I'm looking for, but its not in absolute terms.  Is there a way to modify it to pull the absolute value into the process screening table?

Evan_Morris · Answer

You know I finally looked at the slide deck that's out there for the Fall Technical Conference on the Target Index and it actually shows that reduced formula as the starting point.

What's weird is that they start with that and then go to Cp-Cpk, even though the equation still reduces back to the equation without the USL/LSL.  Not sure I understand why they did that.    It's kind of like saying that PPK = Dragon Population + min(USL-xbar,LSL-xbar)/3sigma - Dragon Population.

I mean I like dragons as much as the next guy but PPK isn't a great dragon indicator.  And TI isn't related to USL or LSL.

Anyways,   Honestly all I want is the ABS[(mean-T)/SpecRange].   Cp allows me to see my variance normalized to my Spec Limits, so I need something to show my bias normalized to my Spec Limits.

Edit: Also apologies for spamming.   I am desperately trying to avoid real work at the moment.

Evan_Morris · Answer

I went ahead and re-tried the derivation just to check.  T = target.   We're also assuming that the Target is centered in the process.  That was needed for this derivation to work.  Wasn't sure how it works with a non-centered target.  Also I only solved it for Cpl, but I'm fairly certain the only difference on Cpu is that it flips the sign which is how TI always stays positive.  I'd say there's a 50/50 chance i messed up the math here.  But from what I can tell it looks like the USL/LSL fall out.

Evan_Morris · Answer

So the thing is that while Cp and Cpk are technically in the formula, in practice they end up cancelling themselves out and the results are only dependent on sigma-within and the difference between the target value and the mean.  It isn't affected at all by the spec limits themselves.

I tried walking through the math by hand and it looks like the USL and LSL don't show up in the final formula, but to be honest I am not the best at this kind of math and not super sure how to write it out here meaningfully anyways.

But just to show what I mean, take a look here

So in this case my system has a T = 0 with spec at +/- 2.    The mean is at 0.03, and the PPK/CPK are both beautiful on this, but the target index is flagging.

Alternatively take a look at this.  Same T same USL/LSL

In this case the Target Index is better (not great, but better), but the Mean is much much farther off target.

The reason why is because of the Sigma-Within in the first case is extremely narrow, wherease it's much worse in the second case.

Further proof.  Let's change the USL/LSL size to +/- 3 for those two

Here is the first

Here is the second

Note that the PPK and CPK both improve, as would be expected since the spec window has opened up farther.  Yet the Target Index did not change at all.

Therefore: The target index is independent of the spec window (assuming that the Target = (USL + LSL/2))

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This isn't to say that the Target Index isn't useful.  But really you're effectively just z-scoring your data.   How many sigma-withins is your mean away from your target. In my case the target is 0, so the Mean = x-bar.

So by that logic TI = Mean / Sigma-Within  So in the TI = 1.58 case I have 0.0353/0.01933 = 1.5794

For the second case TI = Mean/Sigma-Within = 0.6648

Now if your target <> 0 then you would have to subtract the Target from the Mean.  But either way USL and LSL are not part of the equation.

Edit:  If anyone wants to test just use the Semiconductor Capability sample set.  Change the USL/LSL however you want.  As far as I can tell it doesn't affect the TI.