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Statistical Process Control
Chapter 4
Chapter Outline

Foundations of quality control






Product launch and quality control activities
Quality measures and control charts
Transformation processes and variation
Statistical process control (SPC)
Variation and conformance quality
SPC in services
Chapter Outline (2)

SPC overview




Objectives of SPC
Control chart format
Hypothesis testing
Terminology (what is n?)
Chapter Outline (3)

Control charts for variables




x-bar charts
R charts
Control chart patterns
Control charts for attributes

p charts
Chapter Outline (4)

Process capability re-visited



Control limits vs. specification limits
Process capability ratio, Cp
 Cp does not work when the mean and
the target are not equal
Process capability index, Cpk
Product
launch
activities:
Revise
periodically
Customer Requirements
Product Specifications
Process Specifications
Statistical Process Control:
Measure & monitor quality
Ongoing
Activities
Meets
Specifications?
Yes
Conformance Quality
No
Fix process
or inputs
Quality Measures
and Control Charts

Discrete measures



Good/bad, yes/no (p charts)
Count of defects (c charts)
Variables – continuous numerical measures


Length, diameter, weight, height, time, speed,
temperature, pressure
Controlled with
x and R charts
Variation in a
Transformation Process
Inputs
• Facilities
• Equipment
• Materials
• Energy
Transformation
Process
Outputs
Goods &
Services
•Variation in inputs create variation in outputs
• Variations in the transformation process
create variation in outputs
Types of Variation
Common Cause Variation

Common cause (random) variation: systematic
variation in a process. Results from usual variations
in inputs, output rates, and procedures


Usually results from a poorly designed product or process,
poor vendor selection, or other management issues
If the amount of common cause variation is not acceptable,
it is management's responsibility to take corrective action.
Types of Variation
Special Cause Variation

Special cause (non-random or assignable cause)
variation: a short-term source of variation in a
process. Results from changes or abnormal
variations in inputs, outputs, or procedures.


Usually results from errors by workers, first-line supervisors,
or vendors
The cause can and should be identified. Corrective action
should be taken.
Statistical Process Control (SPC)




A process is in control if it has no assignable cause
variation.
 The process is consistent or predictable.
SPC distinguishes between common cause and
assignable cause variation
Measure characteristics of goods or services that are
important to customers
Make a control chart for each characteristic
 The chart is used to determine whether the
process is in control
Specification Limits



The target is the ideal value
 Example: if the amount of beverage in a bottle should be 16
ounces, the target is 16 ounces
Specification limits are the acceptable range of values for a
variable
Example: the amount of beverage in a bottle must be at least
15.8 ounces and no more than 16.2 ounces.



Range is 15.8 – 16.2 ounces.
Lower specification limit = 15.8 ounces or LSPEC = 15.8 ounces
Upper specification limit = 16.2 ounces or USPEC = 16.2 ounces
Specifications and Conformance Quality


A product which meets its specification has
conformance quality.
Capable process: a process which consistently
produces products that have conformance quality.
 Must be in control and meet specifications
Capable Transformation Process
Inputs
• Facilities
• Equipment
• Materials
• Energy
Capable
Transformation
Process
Outputs
Goods &
Services
that meet
specifications
If the process is capable and the product
specification is based on current customer
requirements, outputs will meet customer
requirements.
Applying SPC to Services



Nature of defect is different in services
Service defect is a failure to meet
customer requirements
Monitor times, customer satisfaction,
quality of work, product availability
Copyright 2006 John
Wiley & Sons, Inc.
4-15
Applying SPC to Services (2)

Hospitals


Grocery Stores


timeliness and quickness of care, staff responses to
requests, accuracy of lab tests, cleanliness, courtesy,
accuracy of paperwork, speed of admittance and checkouts
waiting time to check out, frequency of out-of-stock items,
quality of food items, cleanliness, customer complaints,
checkout register errors
Airlines

flight delays, lost luggage and luggage handling, waiting
time at ticket counters and check-in, agent and flight
attendant courtesy, accurate flight information, passenger
cabin cleanliness and maintenance
Copyright 2006 John
Wiley & Sons, Inc.
4-16
Applying SPC to Services (3)

Fast-Food Restaurants


Catalogue-Order Companies


waiting time for service, customer complaints,
cleanliness, food quality, order accuracy, employee
courtesy
order accuracy, operator knowledge and courtesy,
packaging, delivery time, phone order waiting time
Insurance Companies

billing accuracy, timeliness of claims processing,
agent availability and response time
Copyright 2006 John
Wiley & Sons, Inc.
4-17
Objectives of
Statistical Process Control (SPC)
Determine
 Whether the process is in control
 Whether the process is capable
 Whether the process is likely to remain in
control and capable
Control Chart Format
Measure
Upper Control Limit (UCL)
Process Mean
Lower Control Limit (LCL)
Sample
Hypothesis Test



H0: The process mean (or range) has
not changed. (null hypothesis)
H1: The process mean (or range) has
changed. (alternative hypothesis).
If the process has only random
variations and remains within the
control limits, we accept H0. The
process is in control.
Terminology
We take periodic random samples
n = sample size = number of
observations in each sample
X and R Charts for Variables


X = Sample mean
 Measure of central tendency
 Central Limit Theorem: X is normally
distributed.
R = Sample range
 Measure of variation
 R has a gamma distribution (not normal)
Data for Examples 4.3 and 4.4
Sample
1
2
3
…
10
Slip-ring diameter (cm)
1
2
3
4
5
5.02 5.01 4.94 4.99 4.96
5.01 5.03 5.07 4.95 4.96
4.99 5.00 4.93 4.92 4.99
…
…
…
…
…
5.01 4.98 5.08 5.07 4.99
X
R
4.98 0.08
5.00 0.12
4.97 0.08
…
…
5.03 0.10
50.09 1.15
Note: n = number in each sample = 5
Calculate X and R for Each Sample
Sample 1:
X = 5.02 + 5.01 + 4.94 + 4.99 + 4.96
5
= 4.98
R = range = maximum - minimum
= 5.02 - 4.94 = 0.08
 Repeat for all samples
Calculate X and R
X = 4.98 + 5.00 + 4.97 + … + 5.03 = 5.01
10
R = 0.08 + 0.12 + 0.08 + … + 0.10 = 0.115
10
The Normal Distribution
95%
99.74%
-3s
-2s
-1s
m=0 1s
2s
3s
Control Limits for X



99.7% confidence interval for X:
(X - 3s, X + 3s).
This may be approximated as
(X - A2R, X + A2R).
A2 is a factor which depends on n and is
obtained from a table.
3s Control Chart Factors
Sample size
n
2
3
4
5
6
7
8
x-chart
A2
1.88
1.02
0.73
0.58
0.48
0.42
0.37
R-chart
D3
0
0
0
0
0
0.08
0.14
D4
3.27
2.57
2.28
2.11
2.00
1.92
1.86
Control Limits for X and R
For X:
LCL = X - A2R = 5.01 - 0.58 (0.115) = 4.94
UCL = X + A2R = 5.01 + 0.58 (0.115) = 5.08
For R: LCL = D3R = 0 (0.115) = 0
UCL = D4R = 2.11 (0.115) = 0.243
5.10 –
5.08 –
UCL = 5.08
5.06 –
Mean
5.04 –
5.02 –
x= = 5.01
5.00 –
4.98 –
4.96 –
LCL = 4.94
4.94 –
4.92 –
|
1
|
2
|
3
|
|
|
|
4
5
6
7
Sample number
|
8
|
9
|
10
R Chart
0.28 –
0.24 –
Range
0.20 –
0.16 –
UCL = 0.243
R = 0.115
0.12 –
0.08 –
0.04 –
0–
LCL = 0
|
|
|
1
2
3
|
|
|
|
4
5
6
7
Sample number
|
8
|
9
|
10
Control Chart Pattern – Change in Mean
UCL
UCL
LCL
Sample observations
consistently below the
center line
LCL
Sample observations
consistently above the
center line
Copyright 2006 John Wiley & Sons, Inc.
4-32
Control Chart Patterns: Trend
UCL
UCL
LCL
Sample observations
consistently increasing
LCL
Sample observations
consistently decreasing
Copyright 2006 John Wiley & Sons, Inc.
4-33
Control Charts for
Attributes
 p-charts
 uses portion defective in a sample
 c-charts
 uses number of defects in an item
Copyright 2006 John Wiley & Sons, Inc.
4-34
p-Chart
UCL = p + zsp
LCL = p - zsp
z = number of standard deviations from
process average
p = sample proportion defective; an estimate
of process average
sp = standard deviation of sample proportion
sp =
Copyright 2006 John Wiley & Sons, Inc.
p(1 - p)
n
4-35
p-Chart Example
SAMPLE
NUMBER OF
DEFECTIVES
PROPORTION
DEFECTIVE
6
0
4
:
:
18
200
.06
.00
.04
:
:
.18
1
2
3
:
:
20
20 samples of 100 pairs of jeans
Copyright 2006 John Wiley & Sons, Inc.
4-36
p-Chart Example (cont.)
p=
total defectives
= 200 / 20(100) = 0.10
total sample observations
UCL = p + z
p(1 - p)
= 0.10 + 3
n
0.10(1 - 0.10)
100
UCL = 0.190
LCL = p - z
p(1 - p)
= 0.10 - 3
n
0.10(1 - 0.10)
100
LCL = 0.010
Copyright 2006 John Wiley & Sons, Inc.
4-37
0.20
UCL = 0.190
0.18
p-Chart
Example
Proportion defective
0.16
0.14
0.12
0.10
p = 0.10
0.08
0.06
0.04
0.02
LCL = 0.010
2
Copyright 2006 John Wiley & Sons, Inc.
4
6
8
10
12 14
Sample number
16
18
20
4-38
Process Capability Revisited




A process must be in control before you can decide
whether or not it is capable.
Control charts measure the range of natural
variability in a process (what the process is actually
producing)
Specification limits are set to meet customer
requirements.
Process cannot meet specifications if one or both
control limits is outside specification limits
Process Meets Customer Requirements
Upper specification limit
UCL
X
LCL
Lower specification limit
Process Does Not Meet
Customer Requirements
UCL
Upper specification limit
X
Lower specification limit
LCL
Process Capability Ratio Cp
For a product characteristic, let
LSL = lower specification limit
USL = upper specification limit
m = mean, s = standard deviation
If (1) The process is in control and
(2) m = target (or mean = target)
we can use the process capability ratio, Cp to
determine whether the process is capable
Computing Cp
Given: LSL = 8.5, USL = 9.5, target = 9, m = 9,
s = 0.12
Note that m = target
USL  LSL 9.5  8.5
1
Compute:
Cp 
6s

6(0.12)

0.72
 1.39
If Cp > 1, the process is capable.
If Cp < 1, the process is not capable
Conclusion: Cp = 1.39 > 1  process is capable.
Check on the Accuracy of Cp
LCL = m - 3s = 9 – 3(0.12) = 8.64
UCL = m + 3s = 9 + 3(0.12) = 9.36
The specification limits are 8.5 – 9.5
The control limits are within the specification
limits. The process is capable.
Computing Cp – Another Example
Given: LSL = 8.5, USL = 9.5, target = 9, m = 8.8,
s = 0.12
Note that m does not equal the target.
USL  LSL 9.5  8.5
1
Compute:
Cp 
6s

6(0.12)

0.72
 1.39
Conclusion: Cp = 1.39 > 1  process is capable.
Wrong! LCL = m - 3s = 8.8 – 3(0.12) = 8.44 < LSL
Cp can give the wrong answer if m does not equal the
target. Use Cpk
Computing Cpk
Given: LSL = 8.5, USL = 9.5, target = 9, m = 8.8,
s = 0.12
Compute:
 8.8 - 8.5 9.5  8.8 
 m  LSL USL  m 
C pk  minimum 
,

min
,



3s 
 3s
 3(0.12) 3(0.12) 
 min 0.83,1.94  0.83
Cpk < 1  the process is not capable
Cpk always tells you whether the process is capable.
Note: If m is not given, use x instead of m