Transcript TM 675 – Ethics and Professionalism for Managers

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
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.
Improving Process Capability and
Performance
Continually Improve the
System
Characterize Stable Process
Capability
Head Off Shifts in Location,
Spread
Time
Identify Special Causes - Bad
(Remove)
Identify Special Causes - Good
(Incorporate)
Reduce Variability
Center the Process
LSL
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:
UNTL    3
LNTL    3
Natural Tolerance Limits
  1 :68.26% of the total area
  2 :95.46% of the total area
  3 :99.73% of the total area
-3
or
LNTL
-2
-

+
+2
+3
or
UNTL
The Natural Tolerance Limits cover 99.73% of the process output
Process Capability Indices
• C p:
– Measures the potential capability of the
current process - if the process were centered
within the product specifications
– Two-sided Limits:
USL  LSL
Cp 
– One-sided Limit:
6
Cpu
USL  

3
  LSL
Cpl 
3
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:
Cpk  min(Cpu , Cpl )
Where:
Cpu
USL  

3
  LSL
Cpl 
3
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!
LSL
USL
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.
Definitely
Different
Probably
Different
Probably NOT
Different
Definitely NOT
Different
• The assumption that the data come from Normal distributions is very
important. Assess how normally data are distributed prior to
conducting either test.
Process Capability Indices
• Cpm:
– Measures the current capability of the
process - using the process target center
point within the product specifications in
the calculation
USL  LSL
C

pm
– Formula:
6  2  (   T )2
1
Where target T is:T  (USL  LSL)
2
Process Capability Indices
• Cpkm:
– Similar to Cpm - just more sensitive to
departures from the process target center
point
– Not really in very common use
C pk
– Formula: C pkm 
2
T
1 

  
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
2
2
2
total
product
gage



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 RChart
• 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
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)
UCL
UCL
x
LCL
R
LCL
Sample Number
X-Bar Control Chart
Sample Number
R - Control Chart
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:  gage
R

and
d2
6 gage
P

T USL  LSL
– Interpretation:
• Resulting ratio should be 0.10 or smaller if the
gage is truly capable
X-Bar & R-Chart Method: R & R
• Repeatability:
– Inherent precision of the gage
• Reproducibility:
– Variability of the gage under differing
conditions
• Environment
• Operator
• Time …
2
2
 2gage  repeatabil


ity
reproducability
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:

2
repeatabil ity
R

d2
– And we estimate the Reproducibility from the
Range of variability across all operators for each
individual part:
2
 reproducab
ility 
R
x
d2