ppt 4 spc

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S6 – 1

Stat ist ical Process Con trol(SPC)

Dr. R K Singh

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S6 – 2

Variab i l i ty is inherentin every p rocess

Natural or commoncauses

Special or assignable causes

Provides a statist ical signal whenassignab le causes are presen t

Detect and elim inate ass ignab lecauses o f var iat ion

Stat ist ical Process Con trol

(SPC)

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S6 – 3

Qual ity Assu rance us ing SPC

Design ed Standard

• Centre of s pecif icat ion

l im its (Target)

• Upper Specif icat ion Lim it

(USL)

• Lower Speci f icat ion Limi t

(LSL)

• (USL – LSL ): Desiredtolerance

This represents the voice

of the customer

Status o f process

• Centre of the process

(Process Av erage)

• Upper Contro l Lim it (UCL)

• Low er Contro l Limit (LCL)

• (UCL – LCL ): Spread o f the

process

This represents th e voice

of the process

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S6 – 4

Variabi l i ty

• Random

– common causes

– inherent in a

process

– can be el imin ated

on ly through

improv ements in

the system

• Non-Random

– special causes

– due to identi f iable

factors

– can be modi f ied

through operator or

management act ion

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S6 – 5

Qual i ty Measures

• Attr ibute

– a produc t character is t ic that can be

evaluated w ith a disc rete response

– good – bad; yes - no

• Variable

– a product character is t ic that is

con t inuous and can be measured

– weigh t - leng th

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S6 – 6

App lying SPC to Serv ice (con t.)

•Hospi ta ls – t imel iness and quic kness of care, staff respons es to

requests, accu racy o f lab tests , cleanl iness, cou rtesy,accuracy o f paperwork , speed of adm ittance andcheckouts

•

Grocery Stores – wait ing t im e to check out , frequency of o ut-of-stock

i tems, quali ty of fo od items, cleanl iness, customercom plaints, checkout register errors

• Air l ines

–f l ight delays , lost lug gage and lu ggage handl ing , wait ingt ime at t icket coun ters and check- in, agent and fl ightattendant cou rtesy, accu rate f l ight info rmation ,passenger cabin c leanl iness and m aintenance

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App lying SPC to Serv ice (cont.)

• Fast-Food Restaurants

– wait ing t ime for serv ice, customer complaints,

cleanl iness, food qual ity, order accu racy,

employee cour tesy

• Insu rance Companies

– bi l l ing accuracy, t imel iness of claims pro cessin g,

agent avai labi l i ty and respon se t ime

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Contro l Charts

• A g raph that

establ ishes contro l

l imi ts of a process

• Control l im i ts

–

upper and lower bandsof a con tro l char t

• Types o f charts

– Attr ibutes

• p-chart

• c-chart

– Variables

• range (R-chart)

• mean (x bar – chart)

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Setting up a process contro l

sys temChoose th e character is t ic

for process cont ro l

Choos e the

Measurement m ethod

Choose an appro pr iate

Sampl ing procedure

Choose the type of

Contro l Chart

Calculate

cont ro l l imi ts

Plot the data & Analyse

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Character is t ics for p rocess contro lSome examplesSl. No. Type of Applications Characteristic for Measurement

1 Component Manufacturing Conformance of physical measurements ofcomponents and sub-assemblies tospecifications

Conformance to operating characteristics ofmachines and other resources involved in theprocess

2 Final Assembly Number of defects in the product

Conformance to test specifications Number of missing elements

3 Process Industries Temperature, Pressure and Heat specifications Conformance to product specifications Conformance to equipment specifications Vibrations and other variations in equipments

and sub-systems Conformance to specifications of the

automation & control system

4 Service Systems Number of defects in various businessprocesses

Errors in processing documents Conformance to waiting time/lead time related

specifications

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Populat ion and Samp l ingDist r ibut ions

Dist r ibut ion ofsample means

Standarddeviat ion ofthe samplemeans

= sx =s

n

Mean of sample means = x

| | | | | | |

- 3sx - 2sx - 1sx x + 1sx + 2sx + 3sx

99.73% of al l xfal l w i th in ± 3

sx

95.45% fall within ± 2sx

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Con trol Charts fo r Variab les

For variables that have

cont inuous dimens ions

Weight , speed, leng th ,streng th, etc.

x-charts are to con trol

the central tendency o f the pro cess

R-charts are to contro l the dispers ion ofthe process

These two charts must be used together

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Sett ing Chart Lim its

For x-Charts when we knows

Upper contro l l imi t (UCL) = x + z sx

Lower contro l l im i t (LCL) = x - z sx

where x = mean of the sample means or a target

value set for the pro cess

z = num ber of norm al standard deviat ionssx = standard deviat ion of the samp le means

= s / n

s = population standard deviat ion

n = sample size

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Sett ing Con tro l Lim i ts

Hour 1

Sample Weight of

Number Oat Flakes

1 17

2 133 16

4 18

5 17

6 16

7 15

8 17

9 16

Mean 16.1

s = 1

Hour Mean Hour Mean

1 16.1 7 15.2

2 16.8 8 16.4

3 15.5 9 16.3

4 16.5 10 14.8

5 16.5 11 14.2

6 16.4 12 17.3n = 9

LCLx = x - z sx = 16 - 3(1/3) = 15

For 99.73% control l im its, z = 3

UCLx = x + z sx = 16 + 3(1/3) = 17

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17 = UCL

15 = LCL

16 = Mean


Contro l Chartfor sample of9 boxes

Samp le number

| | | | | | | | | | | |

1 2 3 4 5 6 7 8 9 10 11 12

Variat ion dueto assignable

causes

Variat ion du eto assig nable

causes

Var iat ion due tonatural causes

Out ofcontro l

Out ofcontro l

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Sett ing Chart Lim its

For x - Charts when we don’t knows

Lower contro l l im i t (LCL) = x - A 2R

Upper contro l l imi t (UCL) = x + A 2R

where R = average range of the samples

A 2 = contro l chart factor found in Table.1

x = mean of the samp le means

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Con tro l Chart Facto rs

Table 1

Samp le Size Mean Factor Upper Range Lower Range n A 2 D 4 D 3

2 1.880 3.268 0

3 1.023 2.574 0

4 .729 2.282 0

5 .577 2.115 0

6 .483 2.004 0

7 .419 1.924 0.076

8 .373 1.864 0.1369 .337 1.816 0.184

10 .308 1.777 0.223

12 .266 1.716 0.284

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Process average x = 12 ounces

Average range R = .25

Sample size n = 5

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S6 – 19


UCLx = x + A 2R

= 12 + (.577)(.25)

= 12 + .144

= 12.144 ounces



Sample size n = 5

FromTable.1

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S6 – 20


UCLx = x + A 2R

= 12 + (.577)(.25)

= 12 + .144

= 12.144 ounces

LCLx = x - A 2R

= 12 - .144

= 11.857 ounces



Sample size n = 5

UCL = 12.144

Mean = 12

LCL = 11.857

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S6 – 21

R – Chart

Type of var iables contro l chart

Shows sample ranges over time

Difference between smallest and

largest values in sample

Mon itors process var iabi l i ty

Independent from process mean

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S6 – 22

R – Chart

For R-Charts

Lower contro l l im i t (LCLR ) = D 3R

Upper contro l l im i t (UCLR ) = D 4R

where

R = average range of the samples

D 3 and D 4 = contro l chart factors from Table.1

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S6 – 23


UCLR = D 4R

= (2.115)(5.3)

= 11.2 pounds

LCLR = D 3R

= (0)(5.3)

= 0 pounds

Average range R = 5.3 pounds

Sample size n = 5

From Table S6.1 D 4 = 2.115, D 3 = 0

UCL = 11.2

Mean = 5.3

LCL = 0

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S6 – 24

Mean and Range Charts

(a)

These

sampl ing

dist r ibut ions

result in the

char ts below

(Samplin g mean is

sh i f t ing u pward but

range is cons istent)

R-chart

(R-chart does not

detect change in

mean)

UCL

LCL

x-chart

(x-chart detects

sh if t in central

tendency)

UCL

LCL

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S6 – 25

Mean and Range Charts

R-chart

(R-ch art detects

increase in

dispers ion)

UCL

LCL

(b)

These

sampl ing

dist r ibut ions

result in the

char ts below

(Sampling m ean

is constant but

d ispers ion is

increasing)

x-chart

(x-chart does no t

detect the inc rease

in d ispers ion)

UCL

LCL

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S6 – 26

Con tro l Charts fo r A tt r ibu tes

For variab les that are categor ical

Good/bad, yes /no ,

acceptable/unacceptable

Measu rement is typically count ing

defect ives

Charts may measu rePercent defect iv e (p -chart)

Number o f defects (c-chart)

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S6 – 27

Contro l Lim i ts fo r p-Charts

Popu lat ion w i l l be a binom ial dist r ibu t ion,

bu t app ly ing the Central Limit Theorem

al low s us to assume a no rmal dis t r ibu t ion

for the sample stat is t ics

UCLp = p + z sp ^

LCLp = p - z s

p ^

where p = mean fract ion defect ive in the sample

z = num ber of standard deviat ions

sp = standard deviat ion of the sampl ing dist r ibut ion

n = samp le size

^

p (1 - p )

np =^

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S6 – 28

p-Chart fo r Data Entry

Sample Number Fract ion Sample Number Fract ion

Number of Errors Defect ive Number of Errors Defect ive

1 6 .06 11 6 .06

2 5 .05 12 1 .01

3 0 .00 13 8 .08

4 1 .01 14 7 .075 4 .04 15 5 .05

6 2 .02 16 4 .04

7 5 .05 17 11 .11

8 3 .03 18 3 .03

9 3 .03 19 0 .0010 2 .02 20 4 .04

Total = 80

(.04)(1 - .04)

100p = = .02 ^

p = = .0480

(100)(20)

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S6 – 29

.11 –

.10 –

.09 –

.08 –

.07 –

.06 –

.05 –

.04 –

.03 –

.02 –

.01 –

.00 –

Samp le number

F r a c

t i o n

d e

f e c

t i v e

| | | | | | | | | |

2 4 6 8 10 12 14 16 18 20


UCLp = p + z sp = .04 + 3(.02) = .10^

LCLp = p - z sp = .04 - 3(.02) = 0 ^

UCLp = 0.10

LCLp = 0.00

p = 0.04

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S6 – 30

.11 –

.10 –

.09 –

.08 –

.07 –

.06 –

.05 –

.04 –

.03 –

.02 –

.01 –

.00 –

Samp le number

F r a c

t i o n

d e

f e c

t i v e

| | | | | | | | | |

2 4 6 8 10 12 14 16 18 20

UCLp = p + z sp = .04 + 3(.02) = .10^

LCLp = p - z sp = .04 - 3(.02) = 0 ^

UCLp = 0.10

LCLp = 0.00

p = 0.04


Possib le

assignable

causes p resent

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S6 – 31

Con tro l Lim i ts fo r c-Charts

Popu lat ion w i l l be a Poisson d istr ibut ion ,

bu t app ly ing the Central Limit Theorem

al low s us to assume a no rmal dis t r ibu t ion

for the sample stat is t ics

where c = mean number defect ive in the samp le

UCLc = c + 3 c LCLc = c - 3 c

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S6 – 32

c-Chart for Cab Company

c = 54 complaints /9 days = 6 complaints /day

|

1

|

2

|

3

|

4

|

5

|

6

|

7

|

8

|

9

Day

N u

m b e r

d e

f e c

t i v e14 –

12 –

10 –

8 –

6 –

4 – 2 –

0 –

UCLc = c + 3 c

= 6 + 3 6= 13.35

LCLc = c - 3 c= 6 - 3 6

= 0

UCLc = 13.35

LCLc = 0

c = 6

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S6 – 33

Which Con tro l Chart to Use

Using an x-chart and R-chart :

Observat ion s are variables

Collect 20 - 25 samp les o f n = 4, or n =5, or more, each from a stable pro cess

and compu te the mean for the x-chart

and range for the R-chart

Track samp les of n observat ion s each

Variables Data

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S6 – 34


Using the p-chart :

Observat ions are attr ibutes that canbe catego r ized in two states

We deal w ith fract ion , propo rt ion, orpercent defect ives

Have several samples, each w ithmany observations

Attr ibu te Data

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S6 – 35


Using a c-Chart:

Observations are att r ibutes whose

defects per un i t of ou tpu t can becounted

The number counted is a smal l part ofthe poss ib le occurrences

Defects su ch as number of blem isheson a desk, number of typos in a pageof text , f laws in a bol t of c loth

Attr ibu te Data

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S6 – 36

Process Capabi l ity

The natural var iation of a p rocess

shou ld be small enough to produce

products that meet the standards

required

A process in stat is t ical contro l does not

necessar ily meet the design

speci f icat ions

Process capabi l i ty is a measu re of therelat ionsh ip between the natural

var iat ion o f the pro cess and the design

speci f icat ions

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S6 – 37

Process Capab i l ity Ratio

Cp =Upper Speci f icat ion - Lower Speci f icat ion

6s

A capable process mus t have a Cp of at

least 1.0

Does not look at how wel l the process

is centered in the speci f icat ion range Often a target value of Cp = 1.33 is used

to al low for o f f -center p rocesses

Six Sigma qual ity requ ires a Cp

= 2.0

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S6 – 38


Cp =Upper Specif icat ion - Lower Specif icat ion

6s

Insurance claims process

Process mean x = 210.0 m inutes

Process s tandard deviat ions = .516 m inutes

Design speci f icat ion = 210 ± 3 m inutes

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S6 – 39



6s





= = 1.938213 - 207

6(.516)

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S6 – 40



6s





= = 1.938213 - 207

6(.516)Process is

capable

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S6 – 41

Process Capab i l ity Index

A capable process mus t have a Cpk o f at

least 1.0 A capable process is no t necessar i ly in the

center of the speci f icat ion, but i t fal ls w ithin

the speci f icat ion l imi t at both extremes

Cpk = min imum of ,

Upper

Specif icat ion - x

Limi t

s

Lower

x - Specif icat ion

Limi t

s

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S6 – 42


New Cutt ing Mach ine

New p rocess mean x = .250 inches

Process s tandard deviat ion s = .0005 inches

Upper Speci f ication Lim it = .251 inchesLow er Speci f ication L imit = .249 inches

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S6 – 43






Cpk = min imum of ,(.251) - .250

(3).0005

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S6 – 44






Cpk = = 0.67.001

.0015

New mach ine is

NOT capable

Cpk = min imum of ,(.251) - .250

(3).0005

.250 - (.249)

(3).0005

Both calcu lat ions resul t in

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S6 – 45

Interpret ing Cpk

Cpk = negat ive number

Cpk = zero

Cpk = between 0 and 1

Cpk = 1

Cpk > 1

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S6 – 46

Acceptance Samp l ing

Form of quali ty test ing used for

incom ing mater ials or f in ished goods

Take samp les at random from a lot

(sh ipment) of i tems

Inspect each of the items in the samp le

Decide whether to reject the whole lot

based on the inspect ion resul ts

Only sc reens lots; does not d r ive

qual ity improvement effor ts

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S6 – 47

Acceptance Samp l ing

Form of quali ty test ing used for

incom ing mater ials or f in ished goods

Take samp les at random from a lot

(sh ipment) of i tems

Inspect each of the items in the samp le

Decide whether to reject the whole lot

based on the inspect ion resul ts

Only sc reens lots; does not d r ive

qual ity improvement effor ts

Rejected lo ts can be:

Returned to the

suppl ier

Cul led fordefect ives

(100% in spect ion )

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S6 – 48

Operat ing Charac ter is t icCurve

Shows how wel l a sampl ing plan

discr im inates between good and

bad lots (sh ipments )

Shows the relat ionsh ip between

the probabi l i ty o f accept ing a lot

and its qual i ty level

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S6 – 49

Return whole

sh ipment

The “Perfect” OC Curve

% Defect ive in Lot

P ( A c c

e p

t W h o

l e S h

i p m e n

t )

100 –

75 –

50 –

25 –

0 – | | | | | | | | | | |

0 10 20 30 40 50 60 70 80 90 100

Cut-Off

Keep whole

sh ipment

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S6 – 50

An OC Curve

Probabi l i tyo f

Acceptance

Percentdefect ive

| | | | | | | | |

0 1 2 3 4 5 6 7 8

100 –

95 –

75 –

50 –

25 –

10 –

0 –

= 0.05 producer’s risk for AQL

= 0.10

Consumer’sr isk for LTPD

LTPDAQL

Bad lotsIndifference

zoneGoodlots

Figure S6.9

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S6 – 51

AQL and LTPD

Accep tab le Qual i ty Level (AQL)

Poorest level of quali ty we are

w i l ling to accept

Lo t To lerance Percent Defectiv e

(LTPD)

Quali ty level we cons ider bad

Consum er (buyer) does not want to

accept lots w ith more defects than

LTPD

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S6 – 52

Producer’s and Consumer’s

Risks Producer 's r isk ()

Probabi l i ty o f reject ing a good lot

Probabi l i ty o f reject ing a lot w hen thefract ion defect ive is at or above theAQL

Consumer 's r isk ()

Probabi l i ty o f accept ing a bad lot

Probabi l i ty o f accept ing a lot whenfract ion defect ive is below the LTPD

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SPC and Process Variab i l ity

(a) Acceptance

sampl ing (Some

bad un its accepted)

(b) Statist ical pro cess

control (Keep the

process in c ontro l )

(c) Cpk >1 (Designa process that

is in contro l )

Lower

speci f icat ion

l imit

Upper

speci f icat ion

l imit

Process m ean, m

ppt 4 spc

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