Statistical Process Control Chapter 3 Russell and Taylor Operations and Supply Chain Management, 8th Edition.

Statistical Process Control

Chapter 3

Russell and Taylor

Operations and Supply Chain Management, 8th Edition

Lecture Outline

• Basics of Statistical Process Control – Slide 4 • Control Charts – Slide 12• Control Charts for Attributes – Slide 16• Control Charts for Variables – Slide 27• Control Chart Patterns – Slide 45• SPC with Excel and OM Tools – Slide 52• Process Capability – Slide 54

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Learning Objectives

• Explain when and how to use statistical process control to ensure the quality of products and services

• Discuss the rationale and procedure for constructing attribute and variable control charts

• Utilize appropriate control charts to determine if a process is in-control

• Identify control chart patterns and describe appropriate data collection

• Assess the process capability of a process

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Statistical Process Control (SPC)

• Statistical Process Control• monitoring production process

to detect and prevent poor quality

• Sample• subset of items produced to

use for inspection

• Control Charts• process is within statistical

control limits

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UCL

LCL

Process Variability

• Random• inherent in a process• depends on equipment

and machinery, engineering, operator, and system of measurement

• natural occurrences

• Non-Random• special causes• identifiable and

correctable• include equipment out of

adjustment, defective materials, changes in parts or materials, broken machinery or equipment, operator fatigue or poor work methods, or errors due to lack of training

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SPC in Quality Management

• SPC uses• Is the process in control?• Identify problems in order to make

improvements• Contribute to the TQM goal of continuous

improvement

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Quality Measures:Attributes and Variables

• Attribute• A characteristic which is evaluated with a

discrete response• good/bad; yes/no; correct/incorrect

• Variable measure• A characteristic that is continuous and can be

measured• Weight, length, voltage, volume

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SPC Applied to Services

• Nature of defects is different in services• Service defect is a failure to meet customer

requirements• Monitor time and customer satisfaction

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SPC Applied to Services

• Hospitals• timeliness & quickness of care, staff responses to requests,

accuracy of lab tests, cleanliness, courtesy, accuracy of paperwork, speed of admittance & 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 & luggage handling, waiting time at

ticket counters & check-in, agent & flight attendant courtesy, accurate flight information, cabin cleanliness & maintenance

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SPC Applied to Services

• Fast-food restaurants• waiting time for service, customer complaints, cleanliness, food

quality, order accuracy, employee courtesy

• Catalogue-order companies• order accuracy, operator knowledge & courtesy, packaging,

delivery time, phone order waiting time

• Insurance companies• billing accuracy, timeliness of claims processing, agent

availability & response time

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Where to Use Control Charts

• Process • Has a tendency to go out of control

• Is particularly harmful and costly if it goes out of control

• Examples• At beginning of process because of waste to begin

production process with bad supplies• Before a costly or irreversible point, after which product is

difficult to rework or correct• Before and after assembly or painting operations that

might cover defects• Before the outgoing final product or service is delivered

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Control Charts

• A graph that monitors process quality• Control limits

• upper and lower bands of a control chart

• Attributes chart• p-chart• c-chart

• Variables chart• mean (x bar – chart)• range (R-chart)

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Process Control Chart

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1 2 3 4 5 6 7 8 9 10Sample number

Uppercontrol

limit

Processaverage

Lowercontrol

limit

Out of control

Normal Distribution

• Probabilities for Z= 2.00 and Z = 3.00

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=0 1 2 3-1-2-3

95%

99.74%

A Process Is in Control If …

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1. … no sample points outside limits

2. … most points near process average

3. … about equal number of points above and below centerline

4. … points appear randomly distributed

Control Charts for Attributes

• p-chart• uses portion defective in a sample

• c-chart• uses number of defects (non-conformities) in a

sample

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p-Chart

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UCL = p + zp

LCL = p - zp

z = number of standard deviations from process average

p = sample proportion defective; estimates process mean

p = standard deviation of sample proportionp =

p(1 - p)

n

Construction of p-Chart

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20 samples of 100 pairs of jeans

NUMBER OF PROPORTIONSAMPLE # DEFECTIVES DEFECTIVE

1 6 .06

2 0 .00

3 4 .04

: : :

: : :

20 18 .18

200

Construction of p-Chart

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UCL = p + z =p(1 - p)

n

UCL =

LCL =

LCL = p - z =p(1 - p)

n

=total defectives

total sample observationsp =

Construction of p-Chart

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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 =

Construction of p-Chart

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p-Chart in Excel

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Click on “Insert” then “Charts”

to construct control chart

I4 + 3*SQRT(I4*(1-I4)/100)

I4 - 3*SQRT(I4*(1-I4)/100)

Column values copiedfrom I5 and I6

c-Chart

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UCL = c + zc

LCL = c - zc

where

c = number of defects per sample

c = c

c-Chart

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Number of defects in 15 sample rooms

1 122 83 16

: :: :15 15 190

SAMPLE

c =

NUMBER OF

DEFECTS

UCL = c + zc

LCL = c - zc

c-Chart

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Number of defects in 15 sample rooms

1 122 83 16

: :: :15 15 190

SAMPLE

c = = 12.6719015

UCL = c + zc

= 12.67 + 3 12.67= 23.35

LCL = c - zc

= 12.67 - 3 12.67= 1.99

NUMBER OF

DEFECTS

c-Chart

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Control Charts for Variables

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Range chart ( R-Chart ) Plot sample range (variability)

Mean chart ( x -Chart ) Plot sample averages

-

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x-bar Chart: Known

UCL = x + z x LCL = x - z x

-==

Where

s = process standard deviations

x = standard deviation of sample means =/k = number of samples (subgroups)n = sample size (number of observations)

x1 + x2 + ... + xk

kX = =

- - -

n

-

Observations(Slip-Ring Diameter, cm) n

Sample k 1 2 3 4 5 -

x-bar Chart Example: Known

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x

We know σ = .08

x-bar Chart Example: Known

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LCL = x - z x = -UCL = x + z x

= -

x1 + x2 + ... + xk

kX = =

- - -

x-bar Chart Example: Known

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= 5.01 - 3(.08 / )10

= 4.93

_____10

50.09 = 5.01X =

=

LCL = x - z x = -UCL = x + z x

= -= 5.01 + 3(.08 / )10

= 5.09

x-bar Chart Example: Unknown

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_UCL = x + A2R LCL = x - A2R

= =_

where

x = average of the sample means

R = average range value

=_

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Control Chart Factors

n A2 D3 D42 1.880 0.000 3.2673 1.023 0.000 2.5754 0.729 0.000 2.2825 0.577 0.000 2.1146 0.483 0.000 2.0047 0.419 0.076 1.9248 0.373 0.136 1.8649 0.337 0.184 1.81610 0.308 0.223 1.77711 0.285 0.256 1.74412 0.266 0.283 1.71713 0.249 0.307 1.69314 0.235 0.328 1.67215 0.223 0.347 1.65316 0.212 0.363 1.63717 0.203 0.378 1.62218 0.194 0.391 1.60919 0.187 0.404 1.59620 0.180 0.415 1.58521 0.173 0.425 1.57522 0.167 0.435 1.56523 0.162 0.443 1.55724 0.157 0.452 1.54825 0.153 0.459 1.541

Factors for R-chartSample

Size Factor for X-chart

x-bar Chart Example: Unknown

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OBSERVATIONS (SLIP- RING DIAMETER, CM)

SAMPLE k 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 0.084 5.03 4.91 5.01 4.98 4.89 4.96 0.145 4.95 4.92 5.03 5.05 5.01 4.99 0.136 4.97 5.06 5.06 4.96 5.03 5.01 0.107 5.05 5.01 5.10 4.96 4.99 5.02 0.148 5.09 5.10 5.00 4.99 5.08 5.05 0.119 5.14 5.10 4.99 5.08 5.09 5.08 0.15

10 5.01 4.98 5.08 5.07 4.99 5.03 0.10

50.09 1.15Totals

x-bar Chart Example: Unknown

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_R = ____∑ R

k

_UCL = x + A2R

=

=x =åx

k___

_LCL = x - A2R

=

_

x-bar Chart Example: Unknown

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∑ Rk

1.1510R = = = 0.115

_ ____ ____

_UCL = x + A2R

=

= 50.0910

_____x = = = 5.01 cmåx

k___

= 5.01 + (0.58)(0.115) = 5.08

_LCL = x - A2R

== 5.01 - (0.58)(0.115) = 4.94

_

x- bar Chart

Example

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R- Chart

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UCL = D4R LCL = D3R

R = åRk

WhereR = range of each samplek = number of samples (sub groups)

R-Chart Example

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OBSERVATIONS (SLIP- RING DIAMETER, CM)

SAMPLE k 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 0.084 5.03 4.91 5.01 4.98 4.89 4.96 0.145 4.95 4.92 5.03 5.05 5.01 4.99 0.136 4.97 5.06 5.06 4.96 5.03 5.01 0.107 5.05 5.01 5.10 4.96 4.99 5.02 0.148 5.09 5.10 5.00 4.99 5.08 5.05 0.119 5.14 5.10 4.99 5.08 5.09 5.08 0.15

10 5.01 4.98 5.08 5.07 4.99 5.03 0.10

50.09 1.15Totals

R-Chart Example

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Retrieve chart factors D3 and D4

UCL = D4R =

LCL = D3R = _

_

R-Chart Example

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Retrieve chart factors D3 and D4

UCL = D4R = 2.11(0.115) = 0.243

LCL = D3R = 0(0.115) = 0_

_

R-Chart Example

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X-bar and R charts – Excel & OM Tools

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Using x- bar and R-Charts Together

• Process average and process variability must be in control

• Samples can have very narrow ranges, but sample averages might be beyond control limits

• Or, sample averages may be in control, but ranges might be out of control

• An R-chart might show a distinct downward trend, suggesting some nonrandom cause is reducing variation

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Control Chart Patterns

• Run• sequence of sample values that display same

characteristic

• Pattern test• determines if observations within limits of a control

chart display a nonrandom pattern

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Control Chart Patterns

• To identify a pattern look for:• 8 consecutive points on one side of the center line• 8 consecutive points up or down• 14 points alternating up or down• 2 out of 3 consecutive points in zone A (on one side of

center line)• 4 out of 5 consecutive points in zone A or B (on one

side of center line)

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Control Chart Patterns

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UCL

LCL

Sample observationsconsistently above thecenter line

LCL

UCL

Sample observationsconsistently below thecenter line

Control Chart Patterns

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LCL

UCL

Sample observationsconsistently increasing

UCL

LCL

Sample observationsconsistently decreasing

Zones for Pattern Tests

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UCL

LCL

Zone A

Zone B

Zone C

Zone C

Zone B

Zone A

Process average

3 sigma = x + A2R=

3 sigma = x - A2R=

2 sigma = x + (A2R)= 23

2 sigma = x - (A2R)= 23

1 sigma = x + (A2R)= 13

1 sigma = x - (A2R)= 13

x=

Sample number

|1

|2

|3

|4

|5

|6

|7

|8

|9

|10

|11

|12

|13

Performing a Pattern Test

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1 4.98 B — B

2 5.00 B U C

3 4.95 B D A

4 4.96 B D A

5 4.99 B U C

6 5.01 — U C

7 5.02 A U C

8 5.05 A U B

9 5.08 A U A

10 5.03 A D B

SAMPLE x ABOVE/BELOW UP/DOWN ZONE

Sample Size Determination

• Attribute charts require larger sample sizes• 50 to 100 parts in a sample

• Variable charts require smaller samples• 2 to 10 parts in a sample

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SPC with Excel

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SPC with OM Tools

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Process Capability

• Compare natural variability to design variability • Natural variability

• What we measure with control charts• Process mean = 8.80 oz, Std dev. = 0.12 oz

• Tolerances• Design specifications reflecting product

requirements• Net weight = 9.0 oz 0.5 oz • Tolerances are 0.5 oz

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Process Capability

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(b) Design specifications and natural variation the same; process is capable of meeting specifications most of the time.

Design Specifications

Process

(a) Natural variation exceeds design specifications; process is not capable of meeting specifications all the time.

Design Specifications

Process

Process Capability

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(c) Design specifications greater than natural variation; process is capable of always conforming to specifications.

Design Specifications

Process

(d) Specifications greater than natural variation, but process off center; capable but some output will not meet upper specification.

Design Specifications

Process

Process Capability Ratio

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Cp =tolerance range

process range

upper spec limit - lower spec limit

6=

Computing Cp

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Net weight specification = 9.0 oz 0.5 ozProcess mean = 8.80 ozProcess standard deviation = 0.12 oz

Cp =

upper specification limit - lower specification limit

6

Computing Cp

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Net weight specification = 9.0 oz 0.5 ozProcess mean = 8.80 ozProcess standard deviation = 0.12 oz

Cp =

= = 1.39

upper specification limit - lower specification limit

6

9.5 - 8.5

6(0.12)

Process Capability Index

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Cpk = minimum

x - lower specification limit

3

=

upper specification limit - x

3

=

,

Computing Cpk

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Net weight specification = 9.0 oz 0.5 ozProcess mean = 8.80 ozProcess standard deviation = 0.12 oz

Cpk = minimum

x - lower specification limit

3

=

upper specification limit - x

3

=

,

Computing Cpk

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Net weight specification = 9.0 oz 0.5 ozProcess mean = 8.80 ozProcess standard deviation = 0.12 oz

Cpk = minimum

= minimum , = 0.83

x - lower specification limit

3

=

upper specification limit - x

3

=

,

8.80 - 8.50

3(0.12)

9.50 - 8.80

3(0.12)

Process Capability With Excel

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Process Capability With OM Tools

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