# How to calculate a True Position toleranced at MMC - Capability of True Position

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#### mboteo

Is there anybody having information of how to calculate a True Position toleranced at MMC (Maximum Material Condition).

Some have answered me to calculate it using the distance from the center. I do not agree with that since the true position is a two dimensional feature.
Also what about the MMC issue, which will change your tolerance depending on each piece.

S.

#### Paul F. Jackson

##### Quite Involved in Discussions
Last spring my article on the "Residual Tolerance Process Capability Model" got published in Quality magazine and I figured that the problem of including the geometric bonus tolerance in a capability evaluation had been solved. It has, I think, but not with the Residual Model and not with the "Adjusted True Position" that Glen Gruner published in Quality Engineering magazine in 1991 and also not with the "Percent-of-Tolerance" method that Marty Ambrose disclosed in the paper titled "Calculating MMC Cpk."

This summer I studied the three methods and discovered that they are all flawed. Each method creates a new value for variation by manipulating and combining variables for bonus amount and geometric deviation. I found that the variation of the resulting combination does not equal the combined separate variations. That led me to a new discovery.

All three of the methods tend to over-estimate the capability from a given data set by masking actual individual variations due bonus and geometric deviation. A big bonus value combined with a big geometric deviation can yield the same resultant "Adjusted True Position, Residual Tolerance, or Percent-of-Tolerance" as a smaller bonus value combined with a smaller geometric deviation. Glen Gruner's and my method yielded identical results for Ppk-Cpk. Marty Ambrose's method was different because it expanded each ratio to a common scale.

The problem statistically is "how to retain the actual variation of each variable in the equation for Ppk or Cpk." The answer is actually quite simple.

Ppk of a variable unilateral tolerance in excel format equals:

((USL+"Bonus" Xbar-"Geometric deviation" Xbar)/ 3(SQRT(VAR("Bonus")+ VAR("Geometric deviation"))).

For Cpk use "Bonus" Xbarbar, "Geometric deviation" Xbarbar and divide them by three times the square root of the sum of the squared estimated standard deviations derived by Rbar over d2 for both "Bonus" and "Geometric deviation."

Sorry that I couldn't just show the formulas. They wouldn't paste to this reply.

I am going to present this discovery at the "Applied Measurement & Inspection Technology" conference sponsored by the SME on October 23-25, 2001 in Detroit, MI. ***DEAD LINK REMOVED***

Unfortunately the conference publication states that I will present "The Residual Tolerance Process Capability Model." I hadn't discovered its flaw until after the conference was set up in May, 2001.

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#### mboteo

Very interesting stuff, Paul.

This started me thinking in the the direction of using Cpm's. Also I feel we need to be thinking two dimensional (since .002" in the 1st quadrant is not not the same as .002" in the 3rd one -- averaging these two will be .000" if the are perfectly in the opposite direction, while if we don't assume the 2 dimensions, the average will be .002").

So calculating a Cpm-X and Cpm-Y (Cpm for each of the axisses), would be:

Cpm(x)= (USL+"Bonus(x)" Xbar)-"Geometric deviation(x)" Xbar)/ 3 Std.dev.(x) from Target.
Same for Cpm

Aggregated Cpm for both axisses would be
Cpm=SQRT(Cpm(x)²+Cpm²)

Just wanted to throw this at you. Feel free to critisize any flaw in my thinking.

Steven

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#### mboteo

Correction. Formula should have been:

Cpm(x)= 2*(SL+"Bonus(x)" Xbar)/ 6* Std.dev.(x) from Target

#### Paul F. Jackson

##### Quite Involved in Discussions
When the capability of the geometric tolerance is dependent upon feature size, it will become readily apparent to the process owner (using a variable tolerance capability equation) that targeting feature size at the center of its specification to maximize its capability indiscriminately is not a good strategy.

Targeting static-fits, clearance sizes, and boundary conditions (where the function is not influenced by varying sizes) toward the opposite material condition that is specified in the feature control frame will provide additional tolerance for the geometric specification. Of course the distribution for feature size will determine where that target will be. So if there were ever a reasonable use for Cpm it would be with feature sizes related to variable geometric tolerances.

Using Cpm for the variable unilateral geometric tolerance is not a strategic move since the target is always Zero.

If the ratio of consumed geometric deviation to available tolerance is relativley high, a better strategy would be to monitor and control the x and y coordinates of the geometric specification separately.

You would do this to distinguish between:

A process that is accurate (where the means of X and Y coordinate deviations are centered well at their basic dimensions but the associated distributions for X and Y are large or perhaps just one is large).

A process that is precise ( where the distributions for X and Y are small but the means are off-target, or perhaps just one is off target).

Since computing the resultant geometric deviation obscures the distributions for the separate coordinates, monitor and control the coordinates separately rather that attempting to monitor and control the resultant geometric deviation. If the individual coordinates are "in control" so will the resultant geometric deviation. Use the resultant geometric deviation to predict the process capability with respect to the geometric specification.

If the ratio of consumed geometric deviation to available tolerance is low, it isn't necessary to do the extra work.

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#### MisterMcD

Hi, new to all this process capability using true position and MMC.

In a nutshell, what's the best method for calculating the Ppk/Cpk for this tolerance for a FOS? (ok, i'm not allowed to post the image?!)

TP of a hole 1mm with MMC to datum A - B with MMC and C with MMC
I always believed this to be impossible to calculate with each part varying - seems I was wrong?

#### Ron Rompen

Trusted Information Resource
I have run into this a few times, McD. If you are fortunate enough to have a CMM to do your measurements (and output the data into an excel spreadsheet) then it is relatively easy to calculate the actual positional tolerance for each individual hole, including the bonus tolerances for both the hole itself, and for the datum holes.
Then, calculate the percentage of the tolerance used (for example, if your total position tolerance is .250 and your actual position error is .1, your percentage error is 40%.
Repeat that for each of the parts (and each feature if necessary) and then use the percentage score to calculate your Cp and Cpk, using 100% as an upper limit/one-sided specification.
If you don't have a CMM, it can still be done manually, but it does require a lot more work.

#### Paul F. Jackson

##### Quite Involved in Discussions
See the article and spreadsheets PpkMMC.xls and PpkMMCXY.xls

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#### Jdsarick

Hello Paul:

When calculating MMC capabilities to customer unilateral specifications, I've always subtracted the individual bonus tolerance from the individual result versus adding the allowable bonus to each individual result (creating 30 USLs for 30 data points).

Through this method, I felt I captured the original variance from the data, yet I shift the results lower from the USL via the bonus to accurately calculate my Cpk and Ppk back to the customer. Even if the difference between the actual minus the bonus is negative, the MMC variance still calculates out very close to the raw data variance. Obviously, as a test example on 30 random data points, when subtracting fixed constant from the data and comparing the raw and adjusted data for the variance, they are the same. Limit your mean to be >= 0 and you can then calculate Cpk and Ppk with a valid variance to a fixed USL.

Thanks - John