Proportion defectiveProportion defectiveProportion defectiveProportion defective

Statistical Process Control Statistical Process Control

Chapter 3 Chapter 3

Operations Management - 6

Edition Operations Management - 6

Edition

Roberta Russell & Bernard W. Taylor, III

Roberta Russell & Bernard W. Taylor, III

Lecture Outline Lecture Outline

Basics of Statistical Process Control Basics of Statistical Process Control

Control Charts Control Charts

Control Charts for Attributes Control Charts for Attributes

Control Charts for Variables Control Charts for Variables

Control Chart Patterns Control Chart Patterns

SPC with Excel and OM Tools SPC with Excel and OM Tools

Process Capability Process Capability

Basics of Statistical Basics of Statistical Process Control

Process Control

Statistical Process Control Statistical Process Control (SPC)

(SPC)





monitoring production monitoring production process to detect and

process to detect and

process to detect and

process to detect and prevent poor quality prevent poor quality

Sample Sample





subset of items produced to subset of items produced to use for inspection

use for inspection

Control Charts Control Charts





process is within statistical process is within statistical control limits

control limits

UCL UCL

LCL LCL

Basics of Statistical Basics of Statistical Process Control

Process Control ^{(cont.) (cont.)}

Random Random





inherent in a process inherent in a process





depends on depends on

equipment and equipment and

Non Non- -Random Random





special causes special causes





identifiable and identifiable and correctable

correctable equipment and

equipment and machinery,

machinery,

engineering, operator, engineering, operator, and system of

and system of measurement measurement





natural occurrences natural occurrences

correctable correctable





include equipment out of include equipment out of adjustment, defective

adjustment, defective materials, changes in materials, changes in parts or materials,

parts or materials,

broken machinery or

broken machinery or

equipment, operator

equipment, operator

fatigue or poor work

fatigue or poor work

methods, or errors due

methods, or errors due

SPC in Quality Management SPC in Quality Management

SPC SPC





tool for identifying problems in tool for identifying problems in order to make improvements

order to make improvements order to make improvements order to make improvements





contributes to the TQM goal of contributes to the TQM goal of continuous improvements

continuous improvements

Quality Measures:

Quality Measures:

Attributes and Variables Attributes and Variables

Attribute Attribute





a product characteristic that can be a product characteristic that can be evaluated with a discrete response evaluated with a discrete response evaluated with a discrete response evaluated with a discrete response





good good –– bad; yes bad; yes - - no no

Variable measure Variable measure





a product characteristic that is a product characteristic that is continuous and can be measured continuous and can be measured





weight weight - - length length

Nature of defect is different in Nature of defect is different in services

services

SPC Applied to SPC Applied to Services

Services

services services

Service defect is a failure to meet Service defect is a failure to meet customer requirements

customer requirements

Monitor time and customer Monitor time and customer satisfaction

satisfaction

SPC Applied to SPC Applied to Services (cont.) Services (cont.)

Hospitals Hospitals





timeliness and quickness of care, staff responses to timeliness and quickness of care, staff responses to requests, accuracy of lab tests, cleanliness, courtesy, requests, accuracy of lab tests, cleanliness, courtesy, accuracy of paperwork, speed of admittance and

accuracy of paperwork, speed of admittance and checkouts

checkouts checkouts checkouts

Grocery stores Grocery stores





waiting time to check out, frequency of out waiting time to check out, frequency of out- -of of- -stock stock items, quality of food items, cleanliness, customer items, quality of food items, cleanliness, customer complaints, checkout register errors

complaints, checkout register errors

Airlines Airlines





flight delays, lost luggage and luggage handling, waiting flight delays, lost luggage and luggage handling, waiting time at ticket counters and check

time at ticket counters and check- -in, agent and flight in, agent and flight

attendant courtesy, accurate flight information, passenger attendant courtesy, accurate flight information, passenger cabin cleanliness and maintenance

cabin cleanliness and maintenance

SPC Applied to SPC Applied to Services (cont.) Services (cont.)

Fast Fast- -food restaurants food restaurants





waiting time for service, customer complaints, waiting time for service, customer complaints, cleanliness, food quality, order accuracy,

cleanliness, food quality, order accuracy, employee courtesy

employee courtesy employee courtesy employee courtesy

Catalogue Catalogue- -order companies order companies





order accuracy, operator knowledge and courtesy, order accuracy, operator knowledge and courtesy, packaging, delivery time, phone order waiting time packaging, delivery time, phone order waiting time

Insurance companies Insurance companies





billing accuracy, timeliness of claims processing, billing accuracy, timeliness of claims processing, agent availability and response time

agent availability and response time

Where to Use Control Charts Where to Use Control Charts

Process has a tendency to go out of control Process has a tendency to go out of control

Process is particularly harmful and costly if it Process is particularly harmful and costly if it goes out of control

goes out of control

Examples Examples

Examples Examples





at the beginning of a process because it is a waste at the beginning of a process because it is a waste of time and money to begin production process of time and money to begin production process with bad supplies

with bad supplies





before a costly or irreversible point, after which before a costly or irreversible point, after which product is difficult to rework or correct

product is difficult to rework or correct





before and after assembly or painting operations before and after assembly or painting operations that might cover defects

that might cover defects





before the outgoing final product or service is before the outgoing final product or service is delivered

delivered

Control Charts Control Charts

A graph that A graph that

establishes control establishes control limits of a process limits of a process

Control limits Control limits

Types of charts Types of charts





Attributes Attributes





p p- -chart chart

Control limits Control limits





upper and lower bands upper and lower bands of a control chart

of a control chart





p p- -chart chart





c c- -chart chart





Variables Variables





mean (x bar mean (x bar –– chart) chart)





range (R range (R- -chart) chart)

Process Control Chart Process Control Chart

Upper Upper control control limit limit

Process Process

Out of control Out of control

1

1 22 33 44 55 66 77 88 99 1010

Sample number Sample number Process

Process average average

Lower

Lower

control

control

limit

limit

Normal Distribution Normal Distribution

µ

µ=0 =0 1 1σ σ 2 2σ σ 3 3σ σ --1 1σ σ

--2 2σ σ --3 3σ σ

95%

99.74%

A Process Is in A Process Is in Control If …

Control If …

1. … no sample points outside limits

2. … most points near process average 2. … most points near process average 3. … about equal number of points above

and below centerline

4. … points appear randomly distributed

Control Charts for Control Charts for Attributes

Attributes

 p-chart

 uses portion defective in a sample

 uses portion defective in a sample

 c-chart

 uses number of defective items in

a sample

p

p--Chart Chart

UCL = p + zσ _p LCL = p - zσ _p

z = number of standard deviations from process average

p = sample proportion defective; an estimate of process average

σ

= standard deviation of sample proportion

σ σ

= =

p

p(1 (1 -- p p))

n

n

Construction of p

Construction of p--Chart Chart

NUMBER OF

NUMBER OF PROPORTION PROPORTION SAMPLE

SAMPLE DEFECTIVES DEFECTIVES DEFECTIVE DEFECTIVE

1

1 6 6 .06 .06

2

2 0 0 .00 .00

20 samples of 100 pairs of jeans 20 samples of 100 pairs of jeans

2

2 0 0 .00 .00

3

3 4 4 .04 .04

:: :: ::

:: :: ::

20

20 18 18 .18 .18

200

200

Construction of p

Construction of p--Chart (cont.) Chart (cont.)

p(1 - p) 0.10(1 - 0.10)

= 200 / 20(100) = 0.10 total defectives

total sample observations

p =

UCL = p + z p(1 - p) = 0.10 + 3 n

0.10(1 - 0.10) 100

UCL = 0.190

LCL = 0.010

LCL = p - z p(1 - p) = 0.10 - 3 n

0.10(1 - 0.10)

100

0.10 0.10 0.12 0.12 0.14 0.14 0.16 0.16 0.18 0.18 0.20 0.20

Proportion defectiveProportion defective

UCL = 0.190

p = 0.10

Construction Construction of p

of p--Chart Chart (cont.)

(cont.)

0.02 0.02 0.04 0.04 0.06 0.06 0.08 0.08 0.10 0.10

Proportion defectiveProportion defective

2

2 44 66 88 1010 1212 1414 1616 1818 2020 LCL = 0.010

(cont.)

(cont.)

cc--Chart Chart

UCL =

UCL = c c + + z zσ σ _{c c} LCL =

LCL = c c -- z zσ σ σ σ _{c c} = = c c LCL =

LCL = c c -- z zσ σ _{c c}

where

c = number of defects per sample

σ

σ _{c c} = = c c

cc--Chart (cont.) Chart (cont.)

Number of defects in 15 sample rooms Number of defects in 15 sample rooms

1 12

1 12

SAMPLE SAMPLE c c = = 12.67 = = 12.67 190 190 15 15 NUMBER OF DEFECTS 2 8

2 8

3 16

3 16

: : : : : : : : 15 15

15 15 190 190

15 15 UCL

UCL = = c c + + z zσ σ

= 12.67 + 3 12.67

= 12.67 + 3 12.67

= 23.35

= 23.35 LCL

LCL = = c c -- z zσ σ

= 12.67

= 12.67 -- 3 12.67 3 12.67

= 1.99

= 1.99

12 12 15 15 18 18 21 21 24 24

Number of defectsNumber of defects

UCL = 23.35

c = 12.67

cc--Chart Chart (cont.) (cont.)

3 3 6 6 9 Number of defectsNumber of defects 9

Sample number Sample number 2

2 44 66 88 1010 1212 1414 1616 LCL = 1.99

(cont.)

(cont.)

Control Charts for Control Charts for Variables

Variables

 Range chart ( R-Chart )

 uses amount of dispersion in a

 uses amount of dispersion in a sample

 Mean chart ( x -Chart )

 uses process average of a

sample

x

x--bar Chart: bar Chart:

Standard Deviation Known Standard Deviation Known

UCL =

UCL = xx + + zzσσ

LCL = LCL = xx -- zzσσ

UCL =

UCL = xx + + zzσσ

LCL = LCL = xx -- zzσσ

xx + + xx + ... + ... xx xx + + xx + ... + ... xx

==== ====

xx ₁₁ + + xx ₂₂ + ... + ... xx _nn xx ₁₁ + + xx ₂₂ nn + ... + ... xx _nn xx == nn

xx ==== ==

where where

x

x = average of sample means = average of sample means where

where

x

x ==== = average of sample means = average of sample means

x

x--bar Chart Example: bar Chart Example:

Standard Deviation Known (cont.)

Standard Deviation Known (cont.)

x

x--bar Chart Example: bar Chart Example:

Standard Deviation Known (cont.)

Standard Deviation Known (cont.)

x

x--bar Chart Example: bar Chart Example:

Standard Deviation Unknown Standard Deviation Unknown

UCL =

UCL = xx + + AA ₂₂ RR LCL = LCL = xx -- AA ₂₂ RR

UCL =

UCL = xx ⁼⁼ + + AA ₂₂ RR LCL = LCL = xx ⁼⁼ -- AA ₂₂ RR

where where

xx = average of sample means = average of sample means where

where

xx = average of sample means = average of sample means

Control

Control

Limits

Limits

x

x--bar Chart Example: bar Chart Example:

Standard Deviation Unknown Standard Deviation Unknown

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 0.08

2 5.01 5.03 5.07 4.95 4.96 5.00 0.12

3 4.99 5.00 4.93 4.92 4.99 4.97 0.08

4 5.03 4.91 5.01 4.98 4.89 4.96 0.14

4 5.03 4.91 5.01 4.98 4.89 4.96 0.14

5 4.95 4.92 5.03 5.05 5.01 4.99 0.13

6 4.97 5.06 5.06 4.96 5.03 5.01 0.10

7 5.05 5.01 5.10 4.96 4.99 5.02 0.14

8 5.09 5.10 5.00 4.99 5.08 5.05 0.11

9 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 1.15

x

x--bar Chart Example: bar Chart Example:

Standard Deviation Unknown (cont.) Standard Deviation Unknown (cont.)

R = = = 0.115

R = ^{∑ R} = = 0.115

k

∑ R k

1.15 10 1.15

10

UCL = x + A

R = 5.01 + (0.58)(0.115) = 5.08 LCL = x - A

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

=

=

x = = = 5.01 cm = ∑x k

50.09 10

Retrieve Factor Value A

UCL = 5.08

Mean

5.10 – 5.08 – 5.06 – 5.04 – 5.02 – 5.00 –

x = 5.01=

x

x-- bar bar Chart Chart

Example Example (cont.) (cont.)

LCL = 4.94

Sample number

| 1

| 2

| 3

| 4

| 5

| 6

| 7

| 8

| 9

| 10 4.98 –

4.96 – 4.94 – 4.92 –

R

R-- Chart Chart

UCL =

UCL = D D _{4 4} R R LCL = LCL = D D _{3 3} R R

R

R = = ∑ ∑R R R

R = = ∑ ∑R R k k

where where

R

R = range of each sample = range of each sample k

k = number of samples = number of samples

R

R--Chart Example Chart Example

OBSERVATIONS (SLIP

OBSERVATIONS (SLIP--RING DIAMETER, CM) RING DIAMETER, CM) SAMPLE

SAMPLE k k 1 1 2 2 3 3 4 4 5 5 x x R R 1

1 5.02 5.02 5.01 5.01 4.94 4.94 4.99 4.99 4.96 4.96 4.98 4.98 0.08 0.08 2

2 5.01 5.01 5.03 5.03 5.07 5.07 4.95 4.95 4.96 4.96 5.00 5.00 0.12 0.12 3

3 4.99 4.99 5.00 5.00 4.93 4.93 4.92 4.92 4.99 4.99 4.97 4.97 0.08 0.08 4

4 5.03 5.03 4.91 4.91 5.01 5.01 4.98 4.98 4.89 4.89 4.96 4.96 0.14 0.14 4

4 5.03 5.03 4.91 4.91 5.01 5.01 4.98 4.98 4.89 4.89 4.96 4.96 0.14 0.14 5

5 4.95 4.95 4.92 4.92 5.03 5.03 5.05 5.05 5.01 5.01 4.99 4.99 0.13 0.13 6

6 4.97 4.97 5.06 5.06 5.06 5.06 4.96 4.96 5.03 5.03 5.01 5.01 0.10 0.10 7

7 5.05 5.05 5.01 5.01 5.10 5.10 4.96 4.96 4.99 4.99 5.02 5.02 0.14 0.14 8

8 5.09 5.09 5.10 5.10 5.00 5.00 4.99 4.99 5.08 5.08 5.05 5.05 0.11 0.11 9

9 5.14 5.14 5.10 5.10 4.99 4.99 5.08 5.08 5.09 5.09 5.08 5.08 0.15 0.15 10

10 5.01 5.01 4.98 4.98 5.08 5.08 5.07 5.07 4.99 4.99 5.03 5.03 0.10 0.10 50.09

50.09 1.15 1.15

R

R--Chart Example (cont.) Chart Example (cont.)

UCL = D ₄ R = 2.11(0.115) = 0.243 UCL = D ₄ R = 2.11(0.115) = 0.243

Retrieve Factor Values D

Retrieve Factor Values D

and D and D

LCL = D ₃ R = 0(0.115) = 0

LCL = D ₃ R = 0(0.115) = 0

R

R--Chart Example (cont.) Chart Example (cont.)

UCL = 0.243 0.28 –

0.24 – 0.20 –

LCL = 0

R a n g e

R = 0.115

| 1

| 2

| 3

| 4

| 5

| 6

| 7

| 8

| 9

| 10 0.16 –

0.12 –

0.08 –

0.04 –

0 –

Using x

Using x-- bar and R bar and R--Charts Charts Together

Together

 Process average and process variability must be in control

 It is possible for samples to have very narrow ranges, but their averages might be beyond control limits

 It is possible for sample averages to be in control, but ranges might be very large

 It is possible for an R-chart to exhibit a distinct

downward trend, suggesting some nonrandom

cause is reducing variation

Control Chart Patterns 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

 determines if observations within limits of a control chart display a nonrandom pattern

 To identify a pattern:

 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)

Control Chart Patterns (cont.) Control Chart Patterns (cont.)

UCL UCL UCL

UCL

LCL LCL

Sample observations Sample observations consistently above the consistently above the center line

center line LCL

LCL

Sample observations Sample observations consistently below the consistently below the center line

center line

Control Chart Patterns (cont.) Control Chart Patterns (cont.)

UCL UCL

UCL UCL

LCL LCL

Sample observations Sample observations consistently increasing

consistently increasing LCL LCL

Sample observations

Sample observations

consistently decreasing

consistently decreasing

Zones for Pattern Tests Zones for Pattern Tests

UCL

Zone A

Zone B

Zone C Process

3 sigma = x + A= ₂R

2 sigma = x + (A= 2 ₂R) 3

1 sigma = x + (A= 1 ₂R) 3

=

LCL

Zone C

Zone B

Zone A Process

average

3 sigma = x - A= ₂R 2 sigma = x -= 2 (A₂R)

3

1 sigma = x -= 1 (A₂R) 3

x=

| 1

| 2

| 3

| 4

| 5

| 6

| 7

| 8

| 9

| 10

| 11

| 12

| 13

Performing a Pattern Test Performing a Pattern Test

1

1 4.98 4.98 B B — — B B

2

2 5.00 5.00 B B U U C C

3

3 4.95 4.95 B B D D A A

4

4 4.96 4.96 B B D D A A

SAMPLE

SAMPLE x x ABOVE/BELOW ABOVE/BELOW UP/DOWN UP/DOWN ZONE ZONE

4

4 4.96 4.96 B B D D A A

5

5 4.99 4.99 B B U U C C

6

6 5.01 5.01 — — U U C C

7

7 5.02 5.02 A A U U C C

8

8 5.05 5.05 A A U U B B

9

9 5.08 5.08 A A U U A A

10

10 5.03 5.03 A A D D B B

Sample Size Determination Sample Size Determination

 Attribute charts require larger sample sizes

 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

SPC with Excel

SPC with Excel

SPC with Excel and OM Tools

SPC with Excel and OM Tools

Process Capability Process Capability

Tolerances Tolerances





design specifications reflecting product design specifications reflecting product requirements

requirements requirements requirements

Process capability Process capability





range of natural variability in a process range of natural variability in a process— — what we measure with control charts

what we measure with control charts

Process Capability (cont.) Process Capability (cont.)

(a) Natural variation (a) Natural variation exceeds design exceeds design

specifications; process specifications; process is not capable of

is not capable of

meeting specifications meeting specifications all the time.

all the time.

Design Design Specifications Specifications

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

most of the time.

Design Design Specifications Specifications

Process Process all the time.

all the time.

Process Process

Process Capability (cont.) Process Capability (cont.)

(c) Design specifications (c) Design specifications greater than natural

greater than natural variation; process is variation; process is capable of always capable of always conforming to conforming to specifications.

specifications.

Design Design Specifications Specifications

specifications.

specifications.

Process Process

(d) Specifications greater (d) Specifications greater than natural variation, than natural variation, but process off center;

but process off center;

capable but some output capable but some output will not meet upper

will not meet upper specification.

specification.

Design Design Specifications Specifications

Process Capability Measures Process Capability Measures

Process Capability Ratio

tolerance range C _p =

=

tolerance range process range

upper specification limit - lower specification limit

6σ

Computing C Computing C _{p p}

Net weight specification = 9.0 oz ± 0.5 oz Process mean = 8.80 oz

Process standard deviation = 0.12 oz

C

=

= = 1.39

upper specification limit -

lower specification limit 6σ

9.5 - 8.5

6(0.12)

Process Capability Measures Process Capability Measures

Process Capability Index Process Capability Index

=

= C

C

= minimum = minimum

x

x -- lower specification limit lower specification limit 3

3σ σ

=

=

upper specification limit upper specification limit -- x x

3 3σ σ

=

=

,,

Computing C Computing C _{pk pk}

Net weight specification = 9.0 oz ± 0.5 oz Process mean = 8.80 oz

Process standard deviation = 0.12 oz

=

C

= minimum

= minimum , = 0.83 x - lower specification limit

3σ

=

upper specification limit - x 3σ

=

,

8.80 - 8.50 3(0.12)

9.50 - 8.80

3(0.12)

Process Capability Process Capability with Excel

with Excel

Process Capability Process Capability

with Excel and OM Tools

with Excel and OM Tools

Copyright 2009 John Wiley & Sons, Inc.

Copyright 2009 John Wiley & Sons, Inc.

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