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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
Customer Requirements
Product Specifications
Statistical Process Control:
Measure & monitor quality
MeetsSpecifications
?
Process Specifications
Yes
Conformance Quality
Fix process or
inputs
No
Product launch
activities: Revise
periodically
Ongoing Activities
Quality Measuresand 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
charts and Rx
Variation in aTransformation Process
Transformation Process
Inputs• Facilities• Equipment• Materials• Energy
OutputsGoods &Services
•Variation in inputs create variation in outputs• Variations in the transformation process create variation in outputs
Types of VariationCommon 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 VariationSpecial 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
Capable Transformation
Process
Inputs• Facilities• Equipment• Materials• Energy
OutputsGoods &Servicesthat meet
specifications
If the process is capable and the product specification is based on current customer requirements, outputs will meet customer requirements.
Copyright 2006 John Wiley & Sons, Inc. 4-15
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
Applying SPC to Services
Copyright 2006 John Wiley & Sons, Inc. 4-16
Applying SPC to Services (2)
Hospitals timeliness and quickness of care, staff responses to
requests, accuracy of lab tests, cleanliness, courtesy, accuracy of paperwork, speed of admittance and checkouts
Grocery Stores 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. 4-17
Applying SPC to Services (3)
Fast-Food Restaurants waiting time for service, customer complaints,
cleanliness, food quality, order accuracy, employee courtesy
Catalogue-Order Companies 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
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
Upper Control Limit (UCL)
Process Mean
Lower Control Limit (LCL) Sample
Mea
sure
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
Slip-ring diameter (cm)Sample 1 2 3 4 5 X R
1 5.02 5.01 4.94 4.99 4.96 4.98 0.082 5.01 5.03 5.07 4.95 4.96 5.00 0.123 4.99 5.00 4.93 4.92 4.99 4.97 0.08… … … … … … … …10 5.01 4.98 5.08 5.07 4.99 5.03 0.1050.09 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
=0 1 2 3
95%
99.74%
-1-2-3
Control Limits for X
99.7% confidence interval for X: (X - 3, X + 3). This may be approximated as
(X - A2R, X + A2R). A2 is a factor which depends on n and is
obtained from a table.
3 Control Chart Factors
Sample size x-chart R-chartn A2 D3 D4
2 1.88 0 3.273 1.02 0 2.574 0.73 0 2.285 0.58 0 2.116 0.48 0 2.007 0.42 0.08 1.928 0.37 0.14 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
UCL = 5.08
LCL = 4.94
Mea
n
Sample number
|1
|2
|3
|4
|5
|6
|7
|8
|9
|10
5.10 –
5.08 –
5.06 –
5.04 –
5.02 –
5.00 –
4.98 –
4.96 –
4.94 –
4.92 –
x = 5.01=
UCL = 0.243
LCL = 0
Ra
ng
e
Sample number
R = 0.115
|1
|2
|3
|4
|5
|6
|7
|8
|9
|10
0.28 –
0.24 –
0.20 –
0.16 –
0.12 –
0.08 –
0.04 –
0 –
R Chart
Copyright 2006 John Wiley & Sons, Inc.Copyright 2006 John Wiley & Sons, Inc. 4-4-3232
Control Chart Pattern – Change in MeanControl Chart Pattern – Change in Mean
UCLUCL
LCLLCL
Sample observationsSample observationsconsistently above theconsistently above thecenter linecenter line
LCLLCL
UCLUCL
Sample observationsSample observationsconsistently below theconsistently below thecenter linecenter line
Copyright 2006 John Wiley & Sons, Inc.Copyright 2006 John Wiley & Sons, Inc. 4-4-3333
Control Chart Patterns: TrendControl Chart Patterns: Trend
LCLLCL
UCLUCL
Sample observationsSample observationsconsistently increasingconsistently increasing
UCLUCL
LCLLCL
Sample observationsSample observationsconsistently decreasingconsistently decreasing
Copyright 2006 John Wiley & Sons, Inc.Copyright 2006 John Wiley & Sons, Inc. 4-4-3434
Control Charts for Control Charts for AttributesAttributes
p-charts uses portion defective in a sample
c-charts uses number of defects in an item
Copyright 2006 John Wiley & Sons, Inc.Copyright 2006 John Wiley & Sons, Inc. 4-4-3535
p-Chartp-Chart
UCL = p + zp
LCL = p - zp
z = number of standard deviations from process averagep = sample proportion defective; an estimate of process averagep= standard deviation of sample proportion
pp = = pp(1 - (1 - pp))
nn
Copyright 2006 John Wiley & Sons, Inc.Copyright 2006 John Wiley & Sons, Inc. 4-4-3636
p-Chart Examplep-Chart Example
20 samples of 100 pairs of jeans20 samples of 100 pairs of jeans
NUMBER OFNUMBER OF PROPORTIONPROPORTIONSAMPLESAMPLE DEFECTIVESDEFECTIVES DEFECTIVEDEFECTIVE
11 66 .06.06
22 00 .00.00
33 44 .04.04
:: :: ::
:: :: ::
2020 1818 .18.18
200200
Copyright 2006 John Wiley & Sons, Inc.Copyright 2006 John Wiley & Sons, Inc. 4-4-3737
p-Chart Example (cont.)p-Chart Example (cont.)
UCL = p + z = 0.10 + 3p(1 - p)
n
0.10(1 - 0.10)
100
UCL = 0.190
LCL = 0.010
LCL = p - z = 0.10 - 3p(1 - p)
n
0.10(1 - 0.10)
100
= 200 / 20(100) = 0.10total defectives
total sample observationsp =
Copyright 2006 John Wiley & Sons, Inc.Copyright 2006 John Wiley & Sons, Inc. 4-4-3838
0.020.02
0.040.04
0.060.06
0.080.08
0.100.10
0.120.12
0.140.14
0.160.16
0.180.18
0.200.20
Pro
po
rtio
n d
efec
tive
Pro
po
rtio
n d
efec
tive
Sample numberSample number22 44 66 88 1010 1212 1414 1616 1818 2020
UCL = 0.190
LCL = 0.010
p = 0.10p-Chart p-Chart Example Example
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
UCL
LCL
X
Lower specification limit
Upper specification limit
Process Does Not Meet Customer Requirements
UCL
LCL
X
Lower specification limit
Upper specification limit
Process Capability Ratio Cp
For a product characteristic, letLSL = lower specification limitUSL = upper specification limit= mean, = standard deviationIf (1) The process is in control and (2) = 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, = 9, = 0.12Note that = targetCompute:
If Cp > 1, the process is capable.
If Cp < 1, the process is not capable
Conclusion: Cp = 1.39 > 1 process is capable.
39.172.0
1
)12.0(6
5.85.9
6
LSLUSL
C p
Check on the Accuracy of Cp
LCL = - 3 = 9 – 3(0.12) = 8.64UCL = + 3 = 9 + 3(0.12) = 9.36The specification limits are 8.5 – 9.5The control limits are within the
specification limits. The process is capable.
Computing Cp – Another Example
Given: LSL = 8.5, USL = 9.5, target = 9, = 8.8, = 0.12Note that does not equal the target.Compute:
Conclusion: Cp = 1.39 > 1 process is capable.
Wrong! LCL = - 3 = 8.8 – 3(0.12) = 8.44 < LSL
Cp can give the wrong answer if does not equal the target. Use Cpk
39.172.0
1
)12.0(6
5.85.9
6
LSLUSL
C p
Computing Cpk
Given: LSL = 8.5, USL = 9.5, target = 9, = 8.8, = 0.12Compute:
Cpk < 1 the process is not capable
Cpk always tells you whether the process is capable.
Note: If is not given, use x instead of
83.094.1,83.0min
)12.0(3
8.85.9,
3(0.12)
8.5-8.8min
3,
3minimum
USLLSLC pk