1 Statistical Process Control (SPC) Stephen R. Lawrence Assoc. Prof. of Operations Mgmt [email protected] Leeds.colorado.edu/faculty/lawrence.

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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