ENGM 620: Quality Management

Session 8 – 30 October 2012

• Process Capability

Outline

• Process Capability – Natural Tolerance Limits– Histogram and Normal Probability Plot

• Process Capability Indices– Cp

– Cpk

– Cpm & Cpkm

• Measurement System Capability– Using Control Charts– Using Factorial Experiment Design (ANOVA)

• Hands On Measurement System Capability Study

Process Capability - Timing

Reduce Variability

Identify Special Causes - Good (Incorporate)

Improving Process Capability and Performance

Characterize Stable Process Capability

Head Off Shifts in Location, Spread

Identify Special Causes - Bad (Remove)

Continually Improve the System

Process Capability Analysis is performed when there are NO special causes of variability present – ie. when the process is in a state of statistical control, as illustrated at this point.

Time

Center the ProcessLSL 0 USL

Natural Tolerance Limits

• The natural tolerance limits assume:– The process is well-modeled by the Normal

Distribution– Three sigma is an acceptable proportion of the

process to yield• The Upper and Lower Natural Tolerance

Limits are derived from:– The process mean () and – The process standard deviation ()

• Equations:

3LNTL

3UNTL

Natural Tolerance Limits

+2 -2 +3

or UNTL

-3 or

LNTL

+ -

The Natural Tolerance Limits cover 99.73% of the process output

1 :68.26% of the total area 2 :95.46% of the total area 3 :99.73% of the total area

Process Capability Indices

• Cp:

– Measures the potential capability of the current process - if the process were centered within the product specifications

– Two-sided Limits:

– One-sided Limit:

6

LSLUSLCp

3

USLCpu

3

LSLCpl

Process Capability Ratio Note

• There are many ways we can estimate the capability of our process

• If σ is unknown, we can replace it with one of the following estimates:– The sample standard deviation S

– R-bar / d2

Process Capability Indices

• Cpk:

– Measures actual capability of current process - at its’ current location with respect to product specifications

– Formula:

Where:

)C,Cmin(C plpupk

3

USLCpu

3

LSLCpl

Process Capability Indices

• Regarding Cp and Cpk:

– Both assume that the process is Normally distributed

– Both assume that the process is in Statistical Control

– When they are equal to each other, the process is perfectly centered

– Both are pretty common reporting ratios among vendors and purchasers

Process Capability Indices

• Two very different processes can have identical Cpk values, though:

– because spread and location interact!

USLLSL

PCR and an Off-Center Process

• CPK = min (CPU, CPL)

• Generally, if CP = CPK, then the process is centered at the midpoint of the specifications

• If CP ≠ CPK, then the process is off-center

Comparison of Variances

–The second types of comparison are those that compare the spread of two distributions. To do this:

• Compute the ratio of the two variances, and then compare the ratio to one of two known distributions as a check to see if the magnitude of that ratio is sufficiently unlikely for the distribution.

• The assumption that the data come from Normal distributions is very important. Assess how normally data are distributed prior to conducting either test.

Definitely Different

Definitely NOT Different

Probably NOT Different

Probably Different

Process Capability Indices

• Cpm:

– Measures the current capability of the process - using the process target center point within the product specifications in the calculation

– Formula:

Where target T is:

22pm

)T(6

LSLUSLC

)LSLUSL(2

1T

Process Capability Indices

• Cpkm:

– Similar to Cpm - just more sensitive to departures from the process target center point

– Not really in very common use– Formula:

2

pkpkm

T1

CC

Measurement System Capability

• Examines the relative variability in the product and measurement systems, together– Total variation is the result of

• Product variation• Gage variation• Operator variation gaging system variation• Random variation

2gage

2product

2total

Measurement System Analysis

• Measurement system can be assessed by– X-bar and R-Charts

• Using a single part as the rational subgroup• Is easy to visualize• Requires alternate interpretation of the control

charts

– Designed Experiments • Using Analysis of Variance• Allows assessment of part x operator interactions• Is statistically complex to compute & analyze

X-Bar & R-Chart Method

• Have each operator measure the same part twice - so the part becomes the rational sample unit– Parts should be representative of those to be measured

• Use a sample of 20 - 25 parts

– Use a representative set of operators• Either collect data from every operator, or• Randomly select from the set of operators

– Collect data under representative conditions• Carefully specify and control the conditions for measurement• Randomly sequence the combination of parts and operators• Preserve the time-order of the collected data & note observations

X-Bar & R-Chart Method

• If each operator measures the same part twice:– Variation between samples is plotted on the X-

Chart• Out of control points indicate success in identifying

differences between parts

– Variation within samples is plotted on the R-Chart• Centerline of R-Chart is the magnitude of the gage

variation• Out of control points indicate excessive operator to

operator variation (fix with training?)

X-Bar & R-Chart Method

R - Control Chart

LCL

UCL

Sample Number

R

X-Bar Control Chart

LCL

UCL

Sample Number

x

Out of control points indicate ability to distinguish between

product samples (Good)

Out of control points indicate inability of operators to use

gaging system (Bad)

X-Bar & R-Chart Method

• Precision to Tolerance Ratio (P/T):– “Rule of Ten”:

• The measurement device should be at least ten times more accurate than the smallest measurement

– Calculations: and

– Interpretation:• Resulting ratio should be 0.10 or smaller if the

gage is truly capable

2gage d

R

LSLUSLT

P gage

6

X-Bar & R-Chart Method: R & R

• Repeatability:– Inherent precision of the gage

• Reproducibility:– Variability of the gage under differing

conditions• Environment• Operator• Time …

2ilityreproducab

2ityrepeatabil

2gage

X-Bar & R-Chart Method: R & R

• Process is the same as before (20 - 25 parts, …):– But we estimate the Repeatability from the Range

Mean computed across all the operators and all parts:

– And we estimate the Reproducibility from the Range of variability across all operators for each individual part:

2

2ityrepeatabil d

R

2

x2ilityreproducab d

R