Statistical Process Control OPRE

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Statistical QA Approaches

• S

tatistical process control (SPC)

–

M

onitors production process to prevent poor

quality

• Acceptance sampling

–Inspects random sample of product to

determine if a lot is acceptable

• D

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Statistical Quality Assurance

●

Purpose: Assure that processes are performing in an acceptable manner

●

Methodology:

Monitor

process output using

statistical

techniques –

If results are acceptable, no further action is required

–

U

nacceptable results call for corrective action

Acceptance Sampling

:

Quality assurance that relies primarily on inspection before

and

after production

Statistical Process Control (SPC)

:

Quality control efforts that occur

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What is SPC?

●

A simple, yet powerful, collection of tools for graphically analyzing process data

●

Has one primary purpose: to tell you when you have a problem.

●

Invented by Walter Shewhart

at AT&T to minimize

process tampering

●

Important because unnecessary process changes increase instability and

increase

the error rate

●

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Production Data

always

has

some

Variability

P lo t o f R aw P ro cess M easu rem en ts 0 2 4 6 8 10 1 4 7 101 31 6 192 22 5 283 13 4 374 04 3 464 Ti m e X

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Chance and Assignable Causes of

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Accuracy and Precision

Examples of quality characteristics:

Painted

surface, thickness, hardness, and resistance to fading or chipping, viscosity, sweetness, electrical resistance, frequency, …

¾

We can control only those characteristics that can be

counted

,

evaluated

or

measured

Engineering characteristics may show problems with

accuracy or with precision A P AP A PA P

(9)

O P R E 6364 Normal Distribution  Shaft Diameter

(What is this plot of data telling us?)

µ

+3 +2 +1 -1 -2 -3 6.00 cm

Target

Upper Spec Limit

Lo

wer

Spec Limit

Off spec

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Causes of Variation

• C

ommon Causes

_{–Variation inherent in a process}

_{–Can be eliminated only through improvements}

_{in the system}

•

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Assignable Causes are controlled

by SPC

• T

ake periodic samples from process

• P

lot sample points on a

control chart

• D

etermine if process is within limits

• P

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Plot of Sample Averages

0 2 4 6 8 10 13 57 9 11 13 15 Sa m p le # Xbar

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Plot of Sample Standard Deviation

0 0. 2 0. 4 0. 6 0. 8 1 1. 2 13579 11 13 15 17 Sa mp le # Std Devi atio n

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Control Charts

• A

key tool in SPC

• G

raph establishing process control limits

• C

harts for variables

–Mean (X-bar), Range (R), EWMA, CUSUM

• C

harts for attributes

–p, np

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The Shewhart

Control Chart

•A

time-ordered plot

of sample statistics

• W

hen chart is within control limits

–

O

nly random or common causes present

–

W

e leave the process alone

• P

lot of each point

is the test of hypothesis:

H

0

: Process is “in control”

vs.

H

1

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Relationship between the process

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How Does the Chart Work?

Out of control points caused by assignable

causes

Distribution of process statistic

Upper control limit

Natural variation µ±

3

σ

Lower control limit

Time

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A Process Is “In Control” If

• N

o sample points outside limits

• M

ost points near process average

• A

bout equal number of points above &

below centerline

• P

oints appear randomly distributed

• A

process “in control” is supposed to be

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The Signal from a Control Chart

12 3 4 5 6 7 8 9 10 Sample number

Upper _control limit

Process average Lower control

limit Reject H 0 because Process is likely out of control

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Potential Reasons for Variation

•

The Operator: Training, Supervision, Technique.

•

The Method: Procedures, Set-up, Temp

erature, Cutting Speeds.

•

The Material: Moisture content, Blending, Contamination.

•

The Machine: Set-up, Machine condition, Inherent Precision

•

Management: Poor process management; poor systems

Equipment Materia l Procedure Personnel Sheet Metal Vendo Faulty Specs Lack of Training Lack of Maintenance Incentives Documentation

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Charts may signal incorrectly!

_{Charts repeatedly apply hypothesis}

testing!

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Two Type

s of Process Data

• N

umber or percent of defective items in a lot.

• Number of defects per item. • Types of defects. • Value assigned to defects (minor = 1, major = 5, critical = 10)

•L en g th • Weight • Time •D ia m et er

• Tensile Strength • Strength of Solution

• B lood pressure • Volume • Temperature

“Things we count”

Attributes

Variables

“Things we measure”

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Types of Control Charts

•

B

asic Types _–M

ost typical three _•X

-Bar and R • p chart • c chart – D

epend Upon Data Type _•V

ariables • A ttribute • A

dvances Types: CUSUM, EWMA, Multivariate

•

R

ecall that plotting points on a control chart is the

repeated application of

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Types of Shewhart

Control Charts

p charts: proportion of units nonconforming. np

charts: number of units nonconforming.

c charts: number of nonconformities. u charts: number of nonconformities per unit.

Control Charts for Variables Dat

a

X and

R charts: for sample averages and

ranges.

Md

and R charts: for sample medians and ranges.

X and s charts: for sample mean

s and standard deviations.

X charts: for ind

ividual measures; uses moving

ranges.

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Control Charts For Variables

• M

ean chart (

X-Bar Chart

)

Å

for accuracy

Uses average of a sample:

X-Bar = (x

1

+x

2

+x

3

+x

4

+x

5

)/5

• R

ange chart (

R-Chart

)

Å

for precision

Uses amount of dispersion in a sample

R = max (x

i

) –

m

in(x

i

)

(26)

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Xbar

Chart

h

elps control

Accuracy

• A verage Xbar = 82.5 kg • S

tandard Deviation of X bar =

σ xbar = 1.6 kg • Control Limits = Average Xbar + 3 σ xbar = 82.5 ± 3 × 1.6 = [77.7, 87.3]

Here, the process is “in control” (i.e., the mean is stable)

76 78 80 82 84 86 1 3 5 7 9 11 13 15 17 19 XbarÆ UCL LCL Sampl e# Æ

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Central Limit Theorem

99.7% of all sample means Population, Individual items Sample means µ -3 σ x µ +3 σ x µ (Basis for specification _limits) (Basis for control limits)

(28)

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Distribution of Xbar--a Process Statistic

Distribution of Xbar: Normal( µ, σ 2 /n) Distribution of X: Normal( µ, σ 2 ) Mean

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SPC Control Limits

Population of process output

x µ -3 σ x µ +3 σ x UCL

Distribution of process statistic

xbar

µ

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Process Control by Control Limits

LCL UCL

•

In contro l Process is stable

Process center has shifted

Out of control µ -3 σ x µ +3 σ x µ

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Routine use of the

Process Control Chart

•

Data/Information

: Monitor process variability over time

• Control Limits : Average + z × Normal Variability • Decision Rule : ●

Ignore variability when points are within limits

●

Investigate variation when outside as “abnormal”

• Errors : Type I -F

alse alarm (unnecessary investigation)

Type II

-M

issed signal (to identify and correct)

Samples

Process Measure

Upper control limit Lower control limit

(32)

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SPC Applied To Services

• N

ature of defect is different in services

• S

ervice defect is a failure to meet

customer requirements

• M

(33)

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Service SPC Examples

•H os p ita ls – T imeliness, responsiveness, accuracy • G rocery Stores – C hec

k-out time, stocking, cleanliness

•

A

irlines –

Luggage handling, waiting times,

courtesy

•

F

ast food restaurants –

W

aiting times, food quality,

cleanliness •B a nk s – D

aily balance errors, # of customers

(34)

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Control Charts

• B

asic Types

_–

M

ost typical three

•

X -Bar and R • p chart • c chart

–

D

epend Upon Data Type

_•Variables _•Attribute

• A

ll are Applications of

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Variations and Control

Random or Common Var

iation

:

Natural or inherent variations in the output of process are

created by

countless minor factors, too many to

inves

tigat

e economically

Assignable or Special Variation

:

A variation whose

cause can be identified

⇒

Assignable

v

ariations push th

e charts beyond control limits

⇒

Their causes

must be investigated, detected and removed

Assignable cause examples:

Tool wear, equipment that

(36)

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Special Causes of Variation

●

Also called assignable cause of variation

●

When an assignable cause is active, the chart goes beyond control limits

●

In SPC, when some unusual or external cause occurs, the cause is identified and data point removed to calculate true control limits

●

(37)

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Common Causes of Variation

_●Also called random causes of variation

●

When only common causes are active, the chart remains stable and within control limits

●

In SPC, when only random causes are active, no single cause is at fault. Any process improvement effort now must consider all sources of variation, generally the factors inherent in the technology of the process

●

(38)

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We can use Range in place of

Std Deviation to control Precision

0

0. 2 0. 4 0. 6 0. 8 1 1. 2 1 3 5 7 9 11 13 15 17 S a m p le # Std Dev ia tio n 0 0. 5 1 1. 5 2 2. 5 13 579 11 13 15 17 Sa m p le # Ran ge ( R) S cat te r P lot o f Si g m a an 0 0. 5 1 1. 5 2 2. 5 00 .5 11 Si gm a Ra nge Correlation(s, R) = 0.9934

(39)

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Control Charts for Variables

• M

ean chart (

X-Bar Chart

)

–Uses average of a sample

• R

ange chart (

R-Chart

)

(40)

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Three Sigma Control Limits

• T

he use of 3-sigma limits generally gives

good results in practice (ARL = 1/(

α

/2).

• If the distribution of the quality characteristic

is reasonably well approximated by the

normal distribution, then the use of 3-sigma

limits is applicable.

• T

hese limits are often referred to as

action

limits

(42)

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Warning Limits on Control Charts

•

W

arning limits

(if used) are typically set at 2

standard deviations from the mean.

•

If one or more points fall between the warning limits and the control limits, or close to the warning limits the process may not be operating properly.

•

G

ood thing

: Warning limits often increase the

sensitivity

of the control chart.

•B

ad

th

in

g

: Warning limits could result in an

(43)

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Calculation of Xbar

Chart Control Limits

table. a from found is and ranges sample of Average where LCL limit, control Lower UCL limit, control Upper ity. variabil process of measure a as range sample average use to is limits control finding for method quick A Range 5/ ) (x Xbar : Def 2 2 2 5 4 3 2 1 A R R A x z x R A x z x R x Min x Max R x x x x x xbar xbar i i = − = − = + = + = − = + + + + = = σ σ

(44)

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Process Control Chart Factors

LCL Factor for Ranges (Range Charts)

(D 3

)

UCL Factor for Ranges (Range Charts)

(D 4

)

Control Limit Factor for Averages _{(Mean Charts)}

(A 2

)

Factor for Estimating Sigma ( = R/d

(d Sample (Subgr oup) Size (n) 2 1.880 3.267 0 1.128 3 1.023 2.575 0 1.693 2.282 4 0.729 0 2.059 5 0.577 2.115 0 2.326 6 0.483 2.004 0 2.534 7 0.419 1.924 0.076 2.704 8 0.373 1.864 0.136 2.847 9 0.337 1.816 0.184 2.970 10 0.308 1.777 0.223 3.078

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Select 25 small samples

(in this case, n = 4)

Find X and R of each

sample. The X chart is used to control the pr ocess mean. The R chart is used to

control process variation.

1 2 3 4 25 4 7 6 7 6 3 9 6 5 8 8 6 5 6 9 5 _{20 24 32 24 28} Total 5 6 8 6 7 150 2 5 3 2 3 75 Sample Number X Values Sum X R

Process Data Example:

(46)

O PR E 6364 Sum X R 1 2 3 4 25 4 7 6 7 6 3 9 6 5 8 8 6 5 6 9 5 _{20 24 32 24 28} Total 5 6 8 6 7 150 2 5 3 2 3 75 Sample Number Values 2 3 4 0 0 0 1.880 1.023 0.729 3.267 2.575 2.282 1.128 1.693 2.059

X and R Charts Factors

n A

2

D 3

D 4

(47)

O PR E 6364 47 Sum X R 1 2 3 4 25 4 7 6 7 6 3 9 6 5 8 8 6 5 6 9 5 _{20 24 32 24 28} Total 5 6 8 6 7 150 2 5 3 2 3 75 Sample Number Values 2 3 4 0 0 0 1.880 1.023 0.729 3.267 2.575 2.282 1.128 1.693 2.059 – – X = 150 / 25 = 6 R = 75 / 25 = 3 AR = 0.729(3) = 2.2 2 UCL X = X + A 2 R = 6 + 2.2 = 8.2 LCL X = X -A 2 R = 6 -2. 2 = 3. 8 UCL R = D 4 R = 2.282(3) = 6.8 LCL R = D 3 R = 0(3) = 0 – – – – – – – _– – –

X and R Limits

n A 2 D 3 D 4 d 2 5-3

(48)

O PR E 6364 Sum X R 1 2 3 4 25 4 7 6 7 6 3 9 6 5 8 8 6 5 6 9 5 _{20 24 32 24 28} Total 5 6 8 6 7 150 2 5 3 2 3 75 Sample Number Values 2 3 4 0 0 0 1.880 1.023 0.729 3.267 2.575 2.282 1.128 1.693 2.059 – – X = 150 / 25 = 6 R = 75 / 25 = 3 AR = 0.729(3) = 2.2 2 UCL X = X + A 2 R = 6 + 2.2 = 8.2 LCL X = X -A 2 R = 6 -2. 2 = 3. 8 UCL R = D 4 R = 2.282(3) = 6.8 LCL R = D 3 R = 0(3) = 0 – – – – – – – _– – – – LCL X = 3. UCL X = 8. – X = 6.0 – – – UCL R = 6. R = 3.0 LCL R = 0 Range Mean

X and R Chart Plots

n A

2

D 3

D 4

(49)

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Example:

Xbar

chart Control Limits by

σ xbar

A quality control man

ager took five samples

(S1, S2, S3, S4, S5), each w

ith four observations, of the

diameter of shafts

manufactu

red on a lathe

machine. The

manager computed the mean of ea

ch sample

and then computed

the grand mean. All values are in

cm. Use this inf

ormation to

obtain 3-sigma (i.e., z=3 ) contro

l limits for means of future

times. It is known fr

om prev

ious experience that the

standard deviation σ x of the pro cess is 0.02 cm. 12.12 12.10 12.11 12.12 12.10 Xbar 12.09 12.14 12.13 12.12 12.12 12.10 12.08 12.10 12.09 12.09 12.11 12.15 12.15 12.12 12.10 12.11 12.11 12.10 12.11 12.08 1 2 3 4 S5 S4 S3 S2 S1 Observation

(50)

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Example of Control Limits Calculations using

σ size. Sample and deviation standard Process means sample of on distributi of deviation Standard where 12.08 0.01 3 12.11 LCL : limit control Lower 12.14 0.01 3 12.11 UCL : limit control Upper 0.01 4 0.02 Hence 4. size sample that Note (given). 0.02 and 12.11 5 12.12 12.10 12.11 12.12 12.10 x = = = = = × − = − = = × + = + = = = = = = = + + + + = n x n z x z x n n x x x x σ σ σ σ σ σ σ σ

(51)

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Control Limit Factors

3.27 2.57 2.28 2.11 2.00 1.92 1.86 1.82 1.78 1.74 1.72 1.69 1.67 1.65 1.64 1.62 1.61 1.60 1.59 0 0 0 0 0 0.08 0.14 0.18 0.22 0.26 0.28 0.31 0.33 0.35 0.36 0.38 0.36 0.40 0.41 1.88 1.02 0.73 0.58 0.48 0.42 0.37 0.43 0.31 0.29 0.27 0.25 0.24 0.22 0.21 0.20 0.19 0.19 0.18 2 3 4 5 6 7 8 9 ₁₀ ₁₁ ₁₂ ₁₃ ₁₄ ₁₅ ₁₆ ₁₇ ₁₈ ₁₉ ₂₀ D 4 D 3 A 2 n

Factor for R UCL

Factor for R LCL

Factor for Xbar

limits

(52)

O PR E 6364 Xbar Control Limits by Rbar 0.05 0.04 0.06 0.05 0.03 Ra nge R 12.09 12.14 12.13 12.12 12.12 12.10 12.08 12.10 12.09 12.09 12.11 12.15 12.15 12.12 12.10 12.11 12.11 12.10 12.11 12.08 1 2 3 4 S5 S4 S3 S2 S1 Observa tion 08. 12 046. 0 73. 0 11. 12 LCL 14. 12 046. 0 73. 0 11. 12 UCL are Limits Control r Upper/Lowe Hence, table from 73. 0 therefore ,4 size Sample 046. 0 5 0.05 0.04 0.06 0.05 0.03 ranges sample of Average 2 2 2 = × − = − = = × + = + = = = = + + + + = = R A x R A x A n R R

(53)

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Range (R) Chart helps control

Precision UCL 20 10 5 0 Range 15 1 2 3 4 5 6 7 8 9 10 111 2 13 141 5 161 7 181 9 20 Sampl e # LCL ● Average Range R = 10.1 kg ●

Standard Deviation of Range = 3.5 kg

● Control Limits: 10.1 + 3 × 3.5 = [20.6, 0] ¾

Process here is “in control”

(i.e., precis

(54)

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Range Control Chart Control Limits

table. Factors Limit Control the from obtained are D and D where D LCL limit, control Lower D UCL limit, control Upper chart.-R the is precision or dispersion process monitor to used chart control The 4 3 3 R 4 R R R = =

(55)

O PR E 6364 12.12 12.10 12.11 12.12 12.10 Xbar 0.05 0.04 0.06 0.05 0.03 Range R 12.09 12.14 12.13 12.12 12.12 12.10 12.08 12.10 12.09 12.09 12.11 12.15 12.15 12.12 12.10 12.11 12.11 12.10 12.11 12.08 1 2 3 4 S5 S4 S3 S2 S1 Observa tio n

Example: R chart Limits

00. 0 046. 000. 0 LCL 105. 0 046. 280. 2 UCL are / Hence, . table from 28. 2 and 00. 0 Therefore . 4 046. 0 5 0.05 0.04 0.06 0.05 0.03 ranges sample of Average 3 4 4 3 = × = = = × = = = = = = + + + + = = R D R D Limits Control Lower Upper D D n R R R

(56)

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Performance Variation Patterns

(57)

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Abnormal Control Chart Patterns

UCL

LCL

L

CL

Sample observations consistently above the center line

(58)

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Abnormal Control Chart Patterns

UCL

LCL

L

CL

Sample observations consistently increasing

(59)

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Abnormal Control Chart Patterns

UCL

LCL

L

CL

Sample observations consistently above the center line

(60)

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Zones For Non-Random Pattern

Tests

UCL LCL

Zone A Zone B Zone C Zone C Zone B Zone A

3 sigma = x + 2 A R 2 sigma = x + 2 3 2 A R

(

)

1 sigma = x + 1 3 2 A R

()

x

₁sigma = x − 1 3 2 A R

(

)

2 sigma = x − 2 3 2 A R

(

)

3 sigma = x − 2 A R

(61)

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Abnormal Control Chart Patterns

_{1. 8 consecutive points on one side of the center line.} _{2. 8 consecutive points up or down across zones.} _{3. 14 points alternating up or down.} _{4. 2 out of 3 consecutive points in zone A but still inside the} control limits.

(62)

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From Control to Improvement

LCL µ UCL Out of Control In Control Improved Weight Target

(63)

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Defect Control For Attributes

• p Charts

–Calculate percent defectives in sample

•

(64)

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Use of p-Charts

• W

hen observations can be placed into

two categories.

–

G

ood or bad

–

P

ass or fail

–

O

perate or don’t operate

• W

hen the data consists of multiple

(65)

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65

Control Limits for

p

–

Chart

0. LCL Use formula. e approximat to due negative is LCL Sometimes . replaces , estimate, The history. from as estimated be can it unknown, is If process. in the defectives of fraction nominal the is and ) 1( on, distributi Binomial from where LCL limit, control Lower UCL limit, control Upper process. a in defectives of proportion e monitor th to used, attributes for chart Control p p p = − = = + = p p p p p n p p p - z σ z σ p p p σ

(66)

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The Normal Distribution still applies

95% _99.74% -1 σ -3 σ -2 σ µ=0 1σ 2σ 3σ

(67)

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p Chart Data

2 ₅₀ 4 _.08 1 ₅₀ 2 _.04 3 ₅₀ 0 0 4 ₅₀ 3 _.06 25 50 2 _.04 Sample number Total 1250 50 1.00 n #def p Sample size Number of defective

items found in sample

Fraction

defective in

sample

(68)

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p Chart Calculations

2 ₅₀ 4 _.08 Sample number 1 ₅₀ 2 _.04 3 ₅₀ 0 0 4 ₅₀ 3 _.06 25 50 2 _.04 .04(.96) p = – p(1-p) n 3 = 3 50 = 0.083 UCL P = p + 3 P – UCL P = p -3 P – = .04 + .083 = .123 _{= .04} -.083 = 0 can't be negative • • • • UCL P = 0.123 LCL P = 0 p = 0.04 – = 3 σ _σ #def n Σ Σ n = p P σ n #def p

(69)

O PR E 6364 Example:

p

chart data:

120 Total 10 9 8 11 12 8 13 11 9 10 8 11 1 2 3 4 5 6 7 8 9 ₁₀ ₁₁ ₁₂ Number of De fec tives Sample #

A QC manager counted the

number of

defective nuts produced

by an

automatic machine in 12 samples. Using the data shown, construct a control chart that will describe 99.74 % of th

e

(70)

O PR E 6364 p Chart Solution 005. 0 15. 0 3 05. 0 LCL limit, control Lower 095. 0 15. 0 3 05. 0 UCL limit, control Upper 3 z 015. 0 200 ) 05. 0 1( 05. 0 ) 1( 05. 0 200 12 120 p p p = × − = − = = × + = + = = = − = − = = × = p p z σ p z σ p n p p p _σ

(71)

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Example of p-Chart

.. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 2 4 6 8 10 12 14 16 18 20 Proportion defective Sample number

(72)

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Number of Defects/Unit:

c-Charts

Use only when the number of

occurrences per unit of measure can be counted; non-occurrences cannot be counted.

–

S

cratches, chips, dents, or errors per item

–

C

racks or faults per unit of distance

–

B

reaks or Tears per unit of area

–

B

acteria or pollutants per unit of volume

–

C

(73)

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c-Chart Controls Defects/Unit

Discrete Quality Measurement:

D = Number of “

defects

” (errors)

per unit of work

Examples of Defects:

Number of typos/page, errors/thousand transactions, equipment breakdowns/shift, bags lost/thousand flown, power outages/year, customer complaints/month, defects/car...

If

n = No. of opportunities for defects to occur, and p = Probability of a defect/error occurrence in each

then

D ~ Binomial (n, p) with mean np, variance np(1-p) ≅

Poisson (np) with mean = variance = np

, if n is large ( ≥ 20) and p is small ( ≤ 0.05) With c = np =

average number of defects per unit

, Control limits = c + 3 √c

(74)

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c-Chart Control Limits

used.

is

Poisson

ion to

approximat

on

distributi

normal

the

reasons

practical

for

But

on.

distributi

Poisson

a

has

actually

c

deviation.

standard

the

is

c

and

unit,

per

defects

of

number

and

mean

the

is

c

where

c

z

c

LCL

limit,

control

Lower

c

z

c

UCL

limit,

control

Upper

c c

−

=

+

=

(75)

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c-Chart Example: Hotel Suite Inspection--Defects Discovered/room

Day

Defects D ay D efects Day Defects

4

2

1

2

3

1

3

2

0

2

0

3

1

2

3

1

0

10

11

12

13

14

15

16

17

18

1

2

3

4

5

6

7

8

9

19

20

21

22

23

24

25

26

1

2

1

0

3

0

1 ₃₉

Total

(76)

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Recall c-Chart Limits

Process average

= c = Total # defects # samples

Sample standard deviation

= c σ = c UCL = c + z cσ LCL = c -z cσ

(77)

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c Chart for Hotel Suite Inspection

51

c = 39/26 = 1.50 UCL = 5.16 LCL = 0

0

1

2

3 Number of defects

5

4

0

15

20

25 Day

(78)

O PR E 6364 Example of c Chart 42 Total 3 6 4 5 4 0 2 5 6 0 3 1 0 3 1 2 3 4 5 6 7 8 9 ₁₀ ₁₁ ₁₂ ₁₃ ₁₄ Number of complaints Da y

A bank manager receives a certain number of complaints each day

abou

t the bank’s

(79)

O PR E 6364 c Chart Solution used. is Poisson ion to approximat on distributi normal reasons, practical For deviation. standard the is c unit. per defects of number and mean the is c where 0. 0 73. 1 3 3 c z c LCL limit, control Lower 2. 8 73. 1 3 3 c z c UCL limit, control Upper 73. 1 3 14 42 c c c = × − = − = = × + = + = = = = c

(80)

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Control Charts Summary

–X

-bar and R charts _•V

ariables data • A pplication of normal distribution (by

Central Limit Theorem)

–

p charts •

A

ttributes data (defects per n observations)

• A pplication of binomial distribution – c charts • A

ttributes data (defects per inspection)

•

A

pplication of

(81)

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(82)

O

PR

(83)

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Summary of SPC

● Statistical process control provides simple, yet

powerful, for managing process while avoiding

process tampering

● A process 'in control' (i.e.; exhibiting no special

cause variation) is ripe for the next stage--

breakthrough process improvement

●

(84)

O PR E 6364

Process Improvement

•M ea su re m en t

–

External and Internal

•A n al ys is

–

Analyze Variation • C ontrol

–

Adjus t Proc ess •I m p ro ve m en t

–

Reduce Variation • Innovation

–

Redesign Product/Process D A C P D A C P Control Improve Innovate Improve

(85)

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Process capability

:

The inherent

variability of process

output relative to the

variation allowed by

the design or

customer

specification

Spec _Limits

(86)

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Process Capability Analysis

●

Differs

Fundamentally

from Control Charting

9

Focuses on improvement, not control

9

Variables

, not attributes, data involved

9

Capability stud

ies address range of

individual

outputs

9

Control charting addresses range of sample

measures

●

Assumes Normal Distribution 9

Remember the Empirical Rule?

9 Inherent capability (6 s x ) is compared to specifications ●

Requires process first to be

(87)

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Why measure Process Capability?

Process variability can greatly impact customer

satisfaction

Three common terms for variability:

1. Tolerances:

Specifications for range of

acceptable values established by engineering design or customer requirements

2. Process variability:

Natural or inherent

variability in a process

3. Control limits:

Statistical limits that reflect

(88)

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Process Capability is based on Normal Curve

4 (95.5%)

6 (99.7%)

2 (68%)

µ

σ

(89)

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The Range of Process Output

The range in which "all" output can be produced.

6 (99.7%)

Process

range = 6

µ

σ

(90)

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Process Capability Concept

X

4.90 4.95 5.00 5.05 5.10 Tolerance band Inherent variability (6 ) LSL U SL Output Output out of spec out of spec Output Output out of spec out of spec

Process output distribution

5.010

(91)

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91

Two Process Capabilities

This process is CAPABLECAPABLE

of

producing all good output. ÊControl the process.

Lower Spec Limit Upper Spec Limit

This process is NOT CAPABLENOT CAPABLE

. Ê INSPECT -S ort out the defectives

×

(92)

O PR E 6364

Capability Analysis

LS US µ Capability analysis determines wh

ether the inherent va

riability of

the process output falls within

the acceptable ran

g

e of the

va

riability allowed by the design

specifications for the process

output.

The range of possible solutions: 1.

Redesign

the process so that it ca

n achieve the desired output

2. Use an alt e rnat e p rocess

that can achieve the desired output

3. Re

tain the current process but

attempt to eliminate unacceptable

output using 100 percent inspection 4. Examine the specification to s

ee whether they are

nec

ess

could be

relaxed

without adversely affecting customer

(93)

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Process Capability Ratio

C

p

2 C

Motorola,

For

.

management

Sigma

Six

uses

n

Corporatio

Motorola

6 C

width

Process

ion width

Specificat

C

ratio

capability

Process

p p p p

=

−

=

σ

LSL

USL

(94)

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Process Capability Index C

pk

Index C

pk

compares the spread and location

of the process, relative to the specifications.

3

Upper Spec Limit

-X

– –

X

-L

ower Spec Limit

– –

OR

the smaller of:

C

pk

=

{

σ

₃

σ

Upper Spec Limit - X

-L

ower Spec Limit

– –

OR

Where Z

min

is the smaller of:

C

pk

=

Z

min 3

{

σ

Alternate Form

(95)

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Process Capability Ratio

C

pk

Normal distribution => 99.73% of output falls in (

µ

+

3σ

) when the

process is centered. If the process is not centered

, we use C pk = Min [(US -µ) / 3 σ, ( µ -L S ) / 3 σ] Example . MBPF: C pk = Min[0.1894, 0.5952] = 0.1984

With centered process (US

-µ) = ( µ -L S ). T he n C pk = C p = (US -L S)/ 6 σ =

Voice of the Customer Voice of the Process

= 0.3968 C p = 0.86 1 1.1 1.3 1.47 1.63 2.0 Defects/m = 10K 3K 1K 100 10 1ppm 2 ppm

(96)

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Process Capability: C examples

pk C pk = 1.0 C pk = 1.33 C pk = 3.0 LSL U SL LSL U SL C pk = 1.0 LSL U SL LSL U SL LSL U (a) (f) (e) (d) (c) (b) C pk = 0.60 C pk = 0.80 LSL U

(97)

O PR E 6364

Capability Improvement by

Mean Shift

LS = 75 US = 85 C pk = 0.7660 80 C pk = 0 .6990 82.5

Garage Door Weight (kg)

Probability density of output (weight)

(before shift) (after shift ) Off spec Off spec Process Adjustment

(98)

O PR E 6364

Capability Improvement by

Variance Reduction

LS = 75 US = 85 before C pk = 0.7660 80 C pk = 0.9544

Garage Door Weight (kg)

Probability density of output (weight)

(99)

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Process Control and Capability:

Review

●

Every process displays some variability—normal or abnormal

●

Control charts can identify abnormal variability

●

Control charts may give false (or missed) alarms by mistaking normal (abnormal) for abnormal (normal) variability

●

On-line control leads to early detection and correction

●

A process “in control” indicates only its internal stability

●

(100)

O

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6364

Design for Capable Processing

•

S

implify

–

Fewer parts, steps

–

Modular design

•

S

tandardize

–

Less variety

–

Standard, proven parts, and procedures

•

M

(101)

O

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6364

Taguchi Quality Philosophy

Loss = k(P

-T

)

2

not

0 if within specs and 1 if outside

On Tar

g

et

is more important than

Within Specs LS T US Conventional view Taguchi’s view LS T US

(102)

O P R E 6364

Robust Design

Target Perfor mance ( T ) Actual Perfor mance (P) De sign Parameters (D) Noise Factors ( N ): Inter

nal & External

Product / Process

•

Identify Product/Process Design Parameters that

–

Ha

ve significant / little

influence on Performance

–

Minimize performance variat

ion due

to Noise factors

–

Minimize the processing cost

• M ethodology: Desi gn of Experiments (DOE) • E xamples -C

(103)

O

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6364

The Design Process

•G

oa

l

–

Develop high quality, low cost products, fast

•

Importance

–

80% product cost, 70% quality, 65% success

•

C

onventional

–

Technology-driven, Isolated, Sequential, Iterative

•

Difficulties

–

Revisions, cost overruns, delays, returns, recalls

•

S

(104)

O P R E 6364

Concurrent Design

•O bj ec tiv e

–

Interfunctional

coordination to satisfy customer

–

Involve manufacturing, suppliers, R&D

•

P

rerequisites

–

Break down barriers

–

Cross functional training

–

Communication, teamwork, group decisions

•R

es

ul

t

–

Fewer revisions, miscommunication, delays

•

Difficulties

–

(105)

O P R E 6364

References

http://deming.eng.clemson.edu/pub/tutorials/qctools/cso.htm

Control chart case study

http://w

ww.qualityamerica.com/knowledgecente/knowctrSPC_Articles.htm