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Statistical ProcessControl (SPC)
Stephen R. LawrenceAssoc. Prof. of Operations [email protected]/faculty/lawrence
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Process Control Tools
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Process Control Tools
Process tools assess conditions in existing processes to detect problems that require intervention in order to regain lost control.
Check sheetsCheck sheets Pareto analysisPareto analysisScatter PlotsScatter Plots HistogramsHistograms
Run ChartsRun Charts Control chartsControl charts
Cause & effect diagramsCause & effect diagrams
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Check SheetsCheck sheets explore what and where
an event of interest is occurring.
Attribute Check Sheet
27 15 19 20 28
Order Types 7am-9am 9am-11am 11am-1pm 1pm-3pm 3pm-5-pm
Emergency
Nonemergency
Rework
Safety Stock
Prototype Order
Other
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Run Charts
time
mea
sure
men
t
Look for patterns and trends…
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SCATTERPLOTSV
aria
ble
A
Variable B
x x x x x x xx x x x x xx x x x x x xx x xx x x x x xx x x x x x xx x x x xx xxx x x x x xx xx x x x x xx x x x xxx xx x x x x xxx x x xx x x xx xx x x x x x xx x x xxx xx xx xxx x x xx xxx x x x x x x xx x x x x x x xx x x xx x x xx x x x
Larger values of variable A appear to be associated with larger values of variable B.
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HISTOGRAMSA statistical tool used to show the extent and type of variance within the system.
Fre
qu
ency
of
Occ
urr
ence
s
Outcome
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PARETO ANALYSISA method for identifying and separating
the vital few from the trivial many. P
erce
nta
ge o
f O
ccu
rren
ces
Factor
AB
CD
E F G IH J
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CAUSE & EFFECT DIAGRAMS
Employees
Proceduresand Methods
TrainingSpeed Maintenance
Equipment
Condition
ClassificationError
Inspection
BADCPU
Pins notAssigned
DefectivePins
ReceivedDefective
Damagedin storage
CPU Chip
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Example:
Rogue River Adventures
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Process Variation
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Deming’s Theory of VarianceDeming’s Theory of Variance
Variation causes many problems for most processes Causes of variation are either “common” or “special” Variation can be either “controlled” or “uncontrolled” Management is responsible for most variation
Management
Management
Management
EmployeeControlled Variation
Uncontrolled Variation
Common Cause Special Cause
Categories of Variation
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Causes of Variation
Natural Causes Assignable Causes
What prevents perfection?
Exogenous to process Not random Controllable Preventable Examples
tool wear “Monday” effect poor maintenance
Inherent to processInherent to process RandomRandom Cannot be controlledCannot be controlled Cannot be preventedCannot be prevented ExamplesExamples
– weatherweather
– accuracy of measurementsaccuracy of measurements
– capability of machinecapability of machine
Process variation...
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Specification vs. Variation
Product specification desired range of product attribute part of product design length, weight, thickness, color, ... nominal specification upper and lower specification limits
Process variability inherent variation in processes limits what can actually be achieved defines and limits process capability
Process may not be capable of meeting specification!
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Process Capability
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Process CapabilityLSL USLSpec Process variation
Capable process
(Very) capable process
Process not capable
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Process Capability Measure of capability of process to meet (fall within)
specification limits Take “width” of process variation as 6 If 6 < (USL - LSL), then at least 99.7% of output of
process will fall within specification limits
LSL USLSpec
6
3
99.7%
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Variation -- RazorBlade
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Process Capability Ratio
Define Process Capability Ratio Cp as
CpUSL LSL
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If Cp > 1.0, process is... capable If Cp < 1.0, process is... not capable
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Process Capability -- ExampleA manufacturer of granola bars has a weight specification
2 ounces plus or minus 0.05 ounces. If the standard deviationof the bar-making machine is 0.02 ounces, is the process capable?
USL = 2 + 0.05 = 2.05 ounces
LSL = 2 - 0.05 = 1.95 ounces
Cp = (USL - LSL) / 6 = (2.05 - 1.95) / 6(0.02)
= 0.1 / 0.12 = 0.85
Therefore, the process is not capable!
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Process CenteringLSL USLSpec
Capable and centeredCapable and centered
Capable, but not centeredCapable, but not centered
NNotot capable, and capable, and not centerednot centered
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Process Centering -- Example
For the granola bar manufacturer, if the process is incorrectly centered at 2.05 instead of 2.00 ounces, whatfraction of bars will be out of specification?
2.0LSL=1.95 USL=2.05
50% of production will be out of specification!
Out of spec!
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Process Capability Index Cpk
3
,3
minUSLLSL
C pk
If Cpk > 1.0, process is... Centered & capable
If Cpk < 1.0, process is... Not centered &/or not capable
Mean
Std dev
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Cpk Example 1
A manufacturer of granola bars has a weight specification2 ounces plus or minus 0.05 ounces. If the standard deviationof the bar-making machine is = 0.02 ounces and the process mean is = 2.01, what is the process capability index?
USL = 2.05 oz LSL = 1.95 ounces
Cpk = min[( -LSL) / 3(USL- ) / 3 = min[(–1.95) / 0.06(2.05 – 2.01) / 0.06 = min[1.0 0.67
= 0.67
Therefore, the process is not capable and/or not centered !
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Cpk Example 2
Venture Electronics manufactures a line of MP3 audio players. One of the components manufactured by Venture and used in its players has a nominal output voltage of 8.0 volts. Specifications allow for a variation of plus or minus 0.6 volts. An analysis of current production shows that mean output voltage for the component is 8.054 volts with a standard deviation of 0.192 volts. Is the process "capable: of producing components that meet specification? What fraction of components will fall outside of specification? What can management do to improve this fraction?
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Process Control Charts
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Process Control Charts
Establish capability of process under normal conditions
Use normal process as benchmark to statistically identify abnormal process behavior
Correct process when signs of abnormal performance first begin to appear
Control the process rather than inspect the product!
Statistical technique for tracking a process anddetermining if it is going “out to control”
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Upper Control Limit
Lower Control Limit
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3
Target Spec
Process Control Charts
Upper Spec Limit
Lower Spec Limit
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UCL
Target
LCL
Samples
Time
In control Out of control !
Natural variation
Look forspecial
cause !
Back incontrol!
Process Control Charts
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When to Take Action
A single point goes beyond control limits (above or below)
Two consecutive points are near the same limit (above or below)
A run of 5 points above or below the process mean Five or more points trending toward either limit A sharp change in level Other erratic behavior
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Samples vs. Population
Population Distribution
Sample Distribution
Mean
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Types of Control Charts
Attribute control charts Monitors frequency (proportion) of defectives p - charts
Defects control charts Monitors number (count) of defects per unit c – charts
Variable control charts Monitors continuous variables x-bar and R charts
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1. Attribute Control Charts
p - charts Estimate and control the frequency of defects
in a population Examples
Invoices with error s (accounting) Incorrect account numbers (banking) Mal-shaped pretzels (food processing) Defective components (electronics) Any product with “good/not good” distinctions
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Using p-charts
Find long-run proportion defective (p-bar) when the process is in control.
Select a standard sample size n Determine control limits
p
p
zpLCL
zpUCL
n
ppp
)1(
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p-chart Example
Chic Clothing is an upscale mail order clothing company selling merchandise to successful business women. The company sends out thousands of orders five days a week. In order to monitor the accuracy of its order fulfillment process, 200 orders are carefully checked every day for errors. Initial data were collected for 24 days when the order fulfillment process was thought to be "in control." The average percent defective was found to be 5.94%.
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2. Defect Control Charts
c-charts Estimate & control the number of defects per unit Examples
Defects per square yard of fabric Crimes in a neighborhood Potholes per mile of road Bad bytes per packet Most often used with continuous process (vs. batch)
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Using c-charts
Find long-run proportion defective (c-bar) when the process is in control.
Determine control limits
c
c
zcLCL
zcUCL
cc
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2. c-chart Example
Dave's is a restaurant chain that employs independent evaluators to visit its restaurants as secret shoppers to the asses the quality of service. The company evaluates restaurants in two categories, food quality, and service (promptness, order accuracy, courtesy, friendliness, etc.) The evaluator considers not only his/her order experiences, but also evaluations throughout the restaurant. Initial surveys find that the total number of service defects per survey is 7.3 when a restaurant is operating normally.
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3. Control Charts for Variables x-bar and R charts Monitor the condition or state of continuously
variable processes Use to control continuous variables
Length, weight, hardness, acidity, electrical resistance Examples
Weight of a box of corn flakes (food processing) Departmental budget variances (accounting Length of wait for service (retailing) Thickness of paper leaving a paper-making machine
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x-bar and R charts
Two things can go wrong process mean goes out of control process variability goes out of control
Two control solutions X-bar charts for mean R charts for variability
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Problems with Continuous Variables
Target
“Natural” ProcessDistribution Mean not
Centered
IncreasedVariability
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Range (R) Chart
Choose sample size n Determine average in-control sample ranges
R-bar where R=max-min Construct R-chart with limits:
nRR /
RDLCLRDUCL 34
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Mean (x-bar) Chart Choose sample size n (same as for R-charts) Determine average of in-control sample
means (x-double-bar) x-bar = sample mean k = number of observations of n samples
Construct x-bar-chart with limits:
kxx /
RAxLCLRAxUCL 22
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x & R Chart Parameters
n d(2) d(3) A(2) D(3) D(4)2 1.128 0.853 1.881 0.000 3.2693 1.693 0.888 1.023 0.000 2.5744 2.059 0.880 0.729 0.000 2.2825 2.326 0.864 0.577 0.000 2.1146 2.534 0.848 0.483 0.000 2.0047 2.704 0.833 0.419 0.076 1.9248 2.847 0.820 0.373 0.136 1.8649 2.970 0.808 0.337 0.184 1.81610 3.078 0.797 0.308 0.223 1.77711 3.173 0.787 0.285 0.256 1.74412 3.258 0.778 0.266 0.284 1.71616 3.532 0.750 0.212 0.363 1.63717 3.588 0.744 0.203 0.378 1.62218 3.640 0.739 0.194 0.391 1.60919 3.689 0.734 0.187 0.403 1.59720 3.735 0.729 0.180 0.414 1.58621 3.778 0.724 0.173 0.425 1.57522 3.819 0.720 0.167 0.434 1.56623 3.858 0.716 0.162 0.443 1.55724 3.895 0.712 0.157 0.452 1.54825 3.931 0.708 0.153 0.460 1.540
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R and x-bar Chart Example
Resistors for electronic circuits are being manufactured on a high-speed automated machine. The machine is set up to produce resistors of 1,000 ohms each. Fifteen samples of 4 resistors each were taken over a period of time when the machine was operating normally. The average range of the samples was found to be R-bar=21.7 and the average mean of the samples was x-double-bar=999.1.
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When to Take Action
A single point goes beyond control limits (above or below)
Two consecutive points are near the same limit (above or below)
A run of 5 points above or below the process mean Five or more points trending toward either limit A sharp change in level Other statistically erratic behavior
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Control Chart Error Trade-offs Setting control limits too tight (e.g., ± 2) means
that normal variation will often be mistaken as an out-of-control condition (Type I error).
Setting control limits too loose (e.g., ± 4) means that an out-of-control condition will be mistaken as normal variation (Type II error).
Using control limits works well to balance Type I and Type II errors in many circumstances.
3 is not sacred -- use judgement.
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Video:
SPC at Frito Lay
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Statistical Process Control