Quality and Operations
Management
Process Control and Capability Analysis
Process Control
• Recognizes that variance exists in all processes
• Sources of variation
– systematic
– assignable
• Purpose
– to detect and eliminate ‘out-of-control’ conditions
– to return a process to an ‘in-control’ state
• Basic tool -- the SPC chart(s)
Measuring A Process
• Types of measurements
– variables data
• length, weight, speed, output, etc
• discrete values
– attributes data
• good vs bad, pass vs fail, etc
• binary values
• Types of charts
– variables -- X-R chart
– attributes -- p, np, c and u
• Basic assumption -- sample means are normally
distributed
Getting Started with SPC
X-R Charts
•
•
•
•
•
•
•
•
•
Determine sample size and frequency of data collection
Collect sufficient historical data
Ensure normality of distribution
Calculate factors for control charts
X
UCLx
LCLx
R
LCLr
UCLr
Construct control chart
Plot data points
Determine outliers and eliminate assignable causes
Recalculate control limits with reduced data set
Implement new process control chart
Basic Properties
• sx = std dev of sample mean = s/n
(where s = process standard deviation)
• conventional approach uses  3 s/n
• limitations of control charts
– Type I Error: probability that an in-control value would
appear as out-of-control
– Type II Error: probability that a shift causing an out-ofcontrol situation would be mis-reported as in-control
– delays due to sampling interval
– charting without taking action on assignable causes
– over control actions
Type 1 and Type 2 Error
In Control
Out of Control
Alarm
No Alarm
Type 1
error
No error
No error
Type 2
error
Suppose m1 > m, then
Type 2 Error = Z [(m + 3 sx - m1) / sx ]
Type 1 Error = 0.0027 for 3s charts
Type 2 Error Example
Suppose:
m = 10
m1= 10.2
s = 4/3
n=9
thus,
sx = 4/9
Then, Type 2 Error
= Z [(m + 3 sx - m1) / sx ]
= Z [(10 + 12/9 - 10.2) / (4/9)]
= Z [2.55] = 0.9946
if m1= 11.0, then Type 2 Error
= Z[0.75] = .7734
if m1= 12.0, then Type 2 Error
= Z[-1.50] = .0668
Prob.{shift will be detected in 3rd sample after shift occurs}
= 0.0668*0.0668*(1-0.0668) = 0.0042
Average number of samples taken before shift is detected
= 1/(1-0.0668) = 1.0716
Prob.{no false alarms first 32 runs, but false alarm on 33rd}
= (0.9973)32*(0.0027) = .0025
Average number of samples taken before a false alarm
= 1/0.0027 = 370
Tests for Unnatural Patterns
• Probability that “odd” patterns observed are not
“natural” variability are calculated by using the
probabilities associated with each zone of the
control chart
• Use the assumption that the population is normally
distributed
• Probabilities for X-chart are shown on next slide
Normal Distribution Applied to
X-R Control Charts
Probability = .00135
UCLx
Outer 3rd
A
Probability = .02135
+3s
Middle 3rd
B
Probability = .1360
+2s
Inner 3rd
C
Probability = .3413
+1s
X
Inner 3rd
C
Probability = .3413
-1s
Middle 3rd
B
Probability = .1360
-2s
Outer 3rd
A
Probability = .02135
-3s
LCLx
Probability = .00135
A Few Standard Tests
• 1 point outside Zone A
• 2 out of 3 in Zone A or above (below)
• 4 out of 5 in Zone B or above (below)
• 8 in a row in Zone C or above (below)
• 10 out of 11 on one side of center
Tests for Unnatural Patterns
• 2 out of 3 in A or beyond
– .0227 x .0227 x (1-.0227) x 3 = .0015
• 4 out of 5 in B or beyond
– .15874 x (1-.1587) x 5 = .0027
• 8 in a row on one side of center
– .508 = .0039
Other Charts
• P-chart
– based on fraction (percentage) of defective units in a
varying sample size
• np-chart
– based on number of defective units in a fixed sample
size
• u-chart
– based on the counts of defects in a varying sample size
• c-chart
– based on the count of defects found in a fixed sample
size
SPC Quick Reference Card
• P-chart
– based on fraction (percentage) of defective units in a varying sample
size
– UCL/LCLp = p  3(p)(1-p)/n
• np-chart
– based on number of defective units in a fixed sample size
– UCL/LCLnp = np  3(np)(1-p)
• u-chart
– based on the counts of defects in a varying sample size
– UCL/LCLu = u  3u/n
• c-chart
– based on the count of defects found in a fixed sample size
– UCL/LCLc = c  3c
• X-R chart
– variables data
– UCL/LCLX = X  3 sx = X  3 s/  n = X  A2R where s = R/d2
– UCLR = D4R
and A2 = 3/d2 n
– LCLR = D3R
• for p, np, u, c and R chart the LCL can not be less than zero.
Process Capability
• Cp: process capability ratio
– a measure of how the distribution compares to the
width of the specification
– not a measure of conformance
– a measure of capability, if distribution center were to
match center of specification range
• Cpk: process capability index
– a measure of conformance (capability) to specification
– biased towards “worst case”
– compares sample mean to nearest spec. against
distribution width
How Good is Good Enough?
• Cp = 1.0 =>  3s => 99.73% (in acceptance) =>
.9973 => 2700 ppm out of tolerance
– PG&E operates non-stop
• 23.65 hours per year without electricity
– average car driven 15,000 miles per year
• 41 breakdowns or problems per year
• Cp = 2.0 =>  6s => 99.99983% (in acceptance)
=> .9999983 => 3.4 ppm out of tolerance
– PG&E operates non-stop
• 1.78 minutes per year without electricity
– average car driven 15,000 miles per year
• 0.051 breakdowns or problems per year or one every 20 years