OPR

E 63

641

Stat

istica

l Pr

oces

s Co

ntro

l

OPR

E 63

642

Sta

tist

ical

QA A

ppro

aches

•St

atis

tical

pro

cess

con

trol (

SPC

)–

Mon

itors

pro

duct

ion

proc

ess

to p

reve

nt p

oor

qual

ity

•Ac

cept

ance

sam

plin

g–I

nspe

cts

rand

om s

ampl

e of

pro

duct

to

dete

rmin

e if

a lo

t is

acce

ptab

le

•D

esig

n of

Exp

erim

ents

OPR

E 63

643

Sta

tist

ical

Qual

ity

Ass

ura

nce

●Pu

rpos

e: A

ssur

e th

at p

roce

sses

are

per

form

ing

in a

n ac

cept

able

man

ner

●M

etho

dolo

gy:

Mon

itorp

roce

ss o

utpu

t usi

ng s

tatis

tical

te

chni

ques

–If

resu

lts a

re a

ccep

tabl

e, n

o fu

rther

act

ion

is re

quire

d –

Una

ccep

tabl

e re

sults

cal

l for

cor

rect

ive

actio

nA

ccep

tanc

e Sa

mpl

ing:

Q

ualit

y as

sura

nce

that

relie

s pr

imar

ily o

n in

spec

tion

befo

rean

d af

ter p

rodu

ctio

nSt

atis

tical

Pro

cess

Con

trol

(SPC

):Q

ualit

y co

ntro

l effo

rts th

at o

ccur

dur

ing

prod

uctio

n

OPR

E 63

644

What

is

SPC

?

●A

sim

ple,

yet

pow

erfu

l, co

llect

ion

of to

ols

for g

raph

ical

ly

anal

yzin

g pr

oces

s da

ta

●H

as o

ne p

rimar

y pu

rpos

e: to

tell

you

whe

n yo

u ha

ve a

pr

oble

m.

●In

vent

ed b

y W

alte

r She

wha

rtat

AT&

T to

min

imiz

e pr

oces

s ta

mpe

ring

●Im

porta

nt b

ecau

se u

nnec

essa

ry p

roce

ss c

hang

es

incr

ease

inst

abilit

y an

d in

crea

se th

e er

ror r

ate

●SP

C w

ill id

entif

y w

hen

a pr

oble

m (o

r spe

cial

cau

se

varia

tion)

occ

urs

OPR

E 63

645

To

cont

rol,

you

have

to m

easu

re!

OPR

E 63

646

Product

ion D

ata always

has

some

Var

iabili

ty

Plot

of R

aw P

roce

ss M

easu

rem

ents

0246810

14

710

1316

1922

2528

3134

3740

4346

49Ti

me

X

OPR

E 63

647

Chan

ce a

nd A

ssig

nab

le C

ause

s of

Qual

ity

Var

iation

OPR

E 63

648

Acc

ura

cy a

nd

Pre

cisi

on

Exam

ples

of q

ualit

y ch

arac

teris

tics:

Pain

ted

surfa

ce, t

hick

ness

, har

dnes

s, a

nd re

sist

ance

to

fadi

ng o

r chi

ppin

g, v

isco

sity

, sw

eetn

ess,

ele

ctric

al

resi

stan

ce, f

requ

ency

, …W

e ca

n co

ntro

l onl

y th

ose

char

acte

ristic

s th

at c

an

be c

ount

ed, e

valu

ated

or m

easu

red

Engi

neer

ing

char

acte

ristic

s m

ay s

how

pro

blem

s w

ith

accu

racy

or w

ith p

reci

sion

AP

AP

AP

AP

OPR

E 63

649

No

rmal

Dis

trib

uti

on

Sh

aft

Dia

mete

r(W

hat

is t

his

plo

t o

f d

ata

tell

ing

us?

)

µ+3

+2+1

-1-2

-3

6.00

cm

Targ

etU

pp

er

Sp

ec

Lim

it

Lo

wer

Sp

ec

Lim

it

Off

spec

Off

spec

OPR

E 63

6410

Cau

ses

of V

aria

tion

•C

omm

on C

ause

s–V

aria

tion

inhe

rent

in a

pro

cess

–Can

be

elim

inat

ed o

nly

thro

ugh

impr

ovem

ents

in

the

syst

em

•As

sign

able

Cau

ses

–Var

iatio

n du

e to

iden

tifia

ble

fact

ors

–Can

be

mod

ified

thro

ugh

oper

ator

or

man

agem

ent a

ctio

n

OPR

E 63

6411

Assi

gnab

le C

ause

s ar

e co

ntro

lled

by S

PC

•Ta

ke p

erio

dic

sam

ples

from

pro

cess

•Pl

ot s

ampl

e po

ints

on

a co

ntro

l cha

rt•

Det

erm

ine

if pr

oces

s is

with

in li

mits

•Pr

even

t qua

lity

prob

lem

s

OPR

E 63

6412

Plot

of Sam

ple

Ave

rages

0246810

13

57

911

1315

17Sa

mpl

e #

Xbar

OPR

E 63

6413

Plot

of Sam

ple

Sta

ndar

d D

evia

tion

00.

20.

40.

60.

811.

2

13

57

911

1315

17

Sam

ple

#

Std Deviation

OPR

E 63

6414

Con

trol C

harts

•A

key

tool

in S

PC•

Gra

ph e

stab

lishi

ng p

roce

ss c

ontro

l lim

its•

Cha

rts fo

r var

iabl

es–M

ean

(X-b

ar),

Ran

ge (R

), EW

MA,

CU

SUM

•C

harts

for a

ttrib

utes

–p, n

pan

d c

OPR

E 63

6415

The

Shew

har

tContr

ol Char

t

•A

time-

orde

red

plot

of s

ampl

e st

atis

tics

•W

hen

char

t is

with

in c

ontro

l lim

its–

Onl

y ra

ndom

or c

omm

on c

ause

s pr

esen

t–

We

leav

e th

e pr

oces

s al

one

•Pl

ot o

f eac

h po

inti

s th

e te

st o

f hyp

othe

sis:

H0:

Pro

cess

is “i

n co

ntro

l”vs

.H

1: P

roce

ss is

out

of c

ontro

l and

requ

ires

inve

stig

atio

n

OPR

E 63

6416

Rel

atio

nsh

ip b

etw

een t

he

pro

cess

an

d t

he

contr

ol ch

art

OPR

E 63

6417

How

Does

the

Char

t W

ork

?

Ou

t o

f co

ntr

ol

poin

ts c

ause

d b

y ass

ign

ab

le

cau

ses

Dis

trib

uti

on

of

pro

cess

sta

tist

ic

Up

per

con

tro

l li

mit

Natu

ral

vari

ati

on

µ±

3σ

Lo

wer

con

tro

l li

mit

Tim

e

OPR

E 63

6418

A Pr

oces

s Is

“In

Con

trol”

If

•N

o sa

mpl

e po

ints

out

side

lim

its•

Mos

t poi

nts

near

pro

cess

ave

rage

•Ab

out e

qual

num

ber o

f poi

nts

abov

e &

belo

w c

ente

rline

•Po

ints

app

ear r

ando

mly

dis

tribu

ted

•A

proc

ess

“in c

ontro

l” is

sup

pose

d to

be

unde

r the

influ

ence

of r

ando

m c

ause

s on

ly

OPR

E 63

6419

The

Sign

al fr

om a

Con

trol C

hart

12

34

56

78

910

Sam

ple

num

ber

Upp

erco

ntro

llim

it

Proc

ess

aver

age

Low

erco

ntro

llim

it

Rej

ect H

0be

caus

e

Proc

ess

is li

kely

ou

t of c

ontr

ol

OPR

E 63

6420

Pote

ntial

Rea

sons

for

Var

iation

•Th

e O

pera

tor:

Tra

inin

g,

Super

visi

on,

Tec

hniq

ue.

•Th

e M

eth

od

:Pr

oce

dure

s, S

et-u

p,

Tem

per

ature

, Cutt

ing S

pee

ds.

•Th

e M

ate

rial:

Mois

ture

conte

nt,

Ble

ndin

g,

Conta

min

atio

n.

•Th

e M

ach

ine:

Set

-up,

Mac

hin

e co

nditio

n,

Inher

ent

Prec

isio

n

•M

an

ag

em

en

t:

Po

or

pro

cess

man

ag

em

en

t; p

oo

r sy

stem

s

Equ

ipm

ent

Mat

eria

l

Proc

edur

ePe

rson

nel

Qu

ality

Vari

ati

on

Shee

t Met

alV

endo

rFa

ulty

Spec

s

Lac

k of

T

rain

ing

Lac

k of

Mai

nten

ance

Ince

ntiv

esD

ocum

enta

tion

OPR

E 63

6421

Cha

rts m

ay s

igna

l inc

orre

ctly

!

Ch

art

s re

peate

dly

ap

ply

hyp

oth

esi

s te

stin

g!

Typ

e I

err

or

wit

h c

hart

s:Concl

udin

g t

hat

a p

roce

ss is

not

in c

ontr

ol

when

it

actu

ally

is

Typ

e I

I err

or

wit

h c

hart

s:Concl

udin

g t

hat

a p

roce

ss is

in c

ontr

ol

when

it

is n

ot

OPR

E 63

6422

Tw

o T

ypes

of

Proce

ss D

ata

•Num

ber o

r per

cent

of d

efec

tive

item

s in

a lo

t.• N

umbe

r of d

efec

ts p

er it

em.

• Typ

es o

f def

ects

.• V

alue

ass

igne

d to

def

ects

(min

or =

1, m

ajor

= 5

, crit

ical

= 1

0)

•Len

gth

• Wei

ght

• Tim

e

•Dia

met

er• T

ensi

le S

tren

gth

• Str

engt

h of

Sol

utio

n

•Blo

od p

ress

ure

• Vol

ume

• Tem

pera

ture

“Thi

ngs

we

coun

t”At

tribu

tes

Varia

bles

“Thi

ngs

we

mea

sure

”

OPR

E 63

6423

Type

s of

Con

trol C

harts

•Ba

sic

Type

s–

Mos

t typ

ical

thre

e•

X-Ba

r and

R•

p ch

art

•c

char

t–

Dep

end

Upo

n D

ata

Type

•Va

riabl

es•

Attri

bute

•Ad

vanc

es T

ypes

: C

USU

M, E

WM

A, M

ultiv

aria

te•

Rec

all t

hat p

lotti

ng p

oint

s on

a c

ontro

l cha

rt is

the

repe

ated

app

licat

ion

of H

ypot

hesi

s Te

stin

g

OPR

E 63

6424

Typ

es o

f Shew

har

tContr

ol Char

ts

p ch

arts

: pro

port

ion

of u

nits

non

conf

orm

ing.

npch

arts

: num

ber o

f uni

ts n

onco

nfor

min

g.c

char

ts: n

umbe

r of n

onco

nfor

miti

es.

u ch

arts

: num

ber o

f non

conf

orm

ities

per

uni

t.

Co

ntr

ol

Ch

art

s fo

r V

ari

ab

les

Data

X an

d R

cha

rts:

for s

ampl

e av

erag

es a

nd ra

nges

.

Md

and

R c

hart

s: fo

r sam

ple

med

ians

and

rang

es.

X an

d s

char

ts: f

or s

ampl

e m

eans

and

sta

ndar

d de

viat

ions

.

X ch

arts

: for

indi

vidu

al m

easu

res;

use

s m

ovin

g ra

nges

.

Co

ntr

ol

Ch

art

s fo

r A

ttri

bu

tes

Data

OPR

E 63

6425

Con

trol C

harts

For

Var

iabl

es

•M

ean

char

t (X-

Bar C

hart)

fo

r acc

urac

yU

ses

aver

age

of a

sam

ple:

X-Ba

r = (x

1+x 2

+x3+

x 4+x

5)/5

•R

ange

cha

rt (R

-Cha

rt)

for p

reci

sion

Use

s am

ount

of d

ispe

rsio

n in

a s

ampl

eR

= m

ax (x

i) –

min

(xi)

OPR

E 63

6426

Xb

ar

Ch

art

help

s co

ntr

olA

ccu

racy

•Av

erag

e Xb

ar=

82.5

kg

•St

anda

rd D

evia

tion

of X

bar

= σ

xbar

= 1.

6 kg

•C

ontr

ol L

imits

= Av

erag

e Xb

ar+

3 σ x

bar

= 82

.5 ±

3 ×

1.6

= [7

7.7,

87.

3]H

ere,

the

proc

ess

is “i

n co

ntro

l” (i.

e., t

he m

ean

is s

tabl

e)

767880828486

135

791113151719

XbarU

CL

LCL

Sam

ple

#

OPR

E 63

6427

Cen

tral

Lim

it T

heor

em

99.7

% o

f all

sam

ple

mea

ns

Popu

latio

n,In

divi

dual

item

s

Sam

ple

mea

ns

µ-3σ

xµ+

3σx

µ

(Bas

is f

or

speci

fica

tio

nlim

its)

(Bas

is f

or

con

tro

llim

its)

OPR

E 63

6428

Dis

trib

ution o

f Xbar

--a

Proce

ss S

tatist

ic

Dis

trib

uti

on

of

Xb

ar:

No

rmal(µ,

σ2/

n)

Dis

trib

uti

on

of

X:

No

rmal(µ,

σ2)

Mea

n

OPR

E 63

6429

SPC

Contr

ol Li

mits

Popu

latio

n of

pr

oces

s ou

tput

x

µ-3σ

x

µ+3σ

xU

CL

Dis

tribu

tion

ofpr

oces

s st

atis

tic x

bar

µ

LCL

OPR

E 63

6430

Proc

ess C

ontr

ol b

y C

ontr

ol L

imits

LCL

UC

L•

•

•••

•

••

••

•In c

ontr

ol

Proc

ess

is s

tabl

e

Proc

ess

cent

erha

s sh

ifted

Out

of c

ontr

ol

µ-3σ

x

µ+3σ

x µ

OPR

E 63

6431

Routine

use

of th

ePr

oce

ss C

ontr

ol Char

t

•D

ata/

Info

rmat

ion:

Mon

itor p

roce

ss v

aria

bilit

y ov

er ti

me

•C

ontr

ol L

imits

:

Avera

ge +

z ×

No

rmal V

ari

ab

ilit

y

•D

ecis

ion

Rul

e:

●

Igno

re v

aria

bilit

y w

hen

poin

ts a

re w

ithin

lim

its

●In

vest

igat

e va

riatio

n w

hen

outs

ide

as “a

bnor

mal

”

•

Erro

rs:

Type

I-F

alse

ala

rm (u

nnec

essa

ry in

vest

igat

ion)

Type

II-M

isse

d si

gnal

(to

iden

tify

and

corre

ct)

Sam

ple

s

Process Measure

Up

per

con

tro

l lim

it

Lo

wer

con

tro

l lim

it

Out

of co

ntr

ol sa

mple

s

OPR

E 63

6432

SPC

App

lied

To S

ervi

ces

•N

atur

e of

def

ect i

s di

ffere

nt in

ser

vice

s

•Se

rvic

e de

fect

is a

failu

re to

mee

t cu

stom

er re

quire

men

ts

•M

onito

r tim

es, c

usto

mer

sat

isfa

ctio

n

OPR

E 63

6433

Serv

ice

SPC

Exa

mpl

es•

Hos

pita

ls

–Ti

mel

ines

s, re

spon

sive

ness

, ac

cura

cy•

Gro

cery

Sto

res

–C

heck

-out

tim

e, s

tock

ing,

cle

anlin

ess

•Ai

rline

s–

Lugg

age

hand

ling,

wai

ting

times

, co

urte

sy•

Fast

food

rest

aura

nts

–W

aitin

g tim

es, f

ood

qual

ity,

clea

nlin

ess

•Ba

nks

–D

aily

bal

ance

erro

rs, #

of c

usto

mer

s se

rved

, tra

nsac

tions

com

plet

ed,

cour

tesy

OPR

E 63

6434

Con

trol C

harts

•Ba

sic

Type

s–

Mos

t typ

ical

thre

e•

X-Ba

r and

R•

p ch

art

•c

char

t–

Dep

end

Upo

n D

ata

Type

•Va

riabl

es•

Attri

bute

•Al

l are

App

licat

ions

of H

ypot

hesi

s Te

stin

g

OPR

E 63

6435

Vari

ati

on

s an

d C

on

tro

l

Ran

do

m o

r C

om

mo

n V

ari

ati

on

:N

atura

l or

inher

ent

variat

ions

in t

he

outp

ut

of pro

cess

are

cr

eate

d b

y co

un

tless

min

or

fact

ors

, to

o m

an

y t

o

invest

igate

eco

no

mic

all

y

Ass

ign

ab

le o

r S

peci

al

Vari

ati

on

:

A v

aria

tion w

hose

cau

se c

an

be i

den

tifi

ed

⇒Ass

ignab

le v

aria

tions

push

the

char

ts b

eyond c

ontr

ol lim

its

⇒Thei

r ca

use

s m

ust

be i

nvest

igate

d,

dete

cted

an

d r

em

oved

Ass

ign

ab

le c

au

se e

xam

ple

s: T

ool w

ear,

equip

men

t th

at

nee

ds

adju

stm

ent,

def

ective

mat

eria

ls,

hum

an fac

tors

(c

arel

essn

ess,

fat

igue,

nois

e an

d o

ther

dis

trac

tions,

fai

lure

to

follo

w c

orr

ect

pro

cedure

s),

failu

re o

f pum

ps,

hea

ters

, et

c.

OPR

E 63

6436

Spec

ial C

ause

s of

Var

iatio

n

●Al

so c

alle

d as

sign

able

cau

se o

f var

iatio

n●

Whe

n an

ass

igna

ble

caus

e is

act

ive,

the

char

t goe

s be

yond

con

trol l

imits

●In

SPC

, whe

n so

me

unus

ual o

r ext

erna

l cau

se o

ccur

s,

the

caus

e is

iden

tifie

d an

d da

ta p

oint

rem

oved

to

calc

ulat

e tru

e co

ntro

l lim

its●

Atte

mpt

ing

to im

prov

e a

proc

ess

(con

tain

ing

spec

ial

caus

e va

riatio

n) w

ithou

t rem

ovin

g th

e sp

ecia

l cau

se o

nly

incr

ease

s th

e in

stab

ility

and

varia

tion

of th

e pr

oces

s

OPR

E 63

6437

Com

mon

Cau

ses

of V

aria

tion

●Al

so c

alle

d ra

ndom

cau

ses

of v

aria

tion

●W

hen

only

com

mon

cau

ses

are

activ

e, th

e ch

art r

emai

ns

stab

le a

nd w

ithin

con

trol l

imits

●In

SPC

, whe

n on

ly ra

ndom

cau

ses

are

activ

e, n

o si

ngle

ca

use

is a

t fau

lt. A

ny p

roce

ss im

prov

emen

t effo

rt no

w

mus

t con

side

r all

sour

ces

of v

aria

tion,

gen

eral

ly th

e fa

ctor

s in

here

nt in

the

tech

nolo

gy o

f the

pro

cess

●A

proc

ess

with

onl

y co

mm

on c

ause

of v

aria

tion

is s

tabl

e an

d pr

edic

tabl

e an

d it

form

s th

e ba

sis

for m

easu

ring

proc

ess

capa

bilit

y

OPR

E 63

6438

We

can u

se R

ange

in p

lace

of

Std

Dev

iation t

o c

ontr

ol Pr

ecis

ion

00.

20.

40.

60.

811.

2

13

57

911

1315

17

Sam

ple

#

Std Deviation

00.

511.

522.

5

13

57

911

1315

17

Sam

ple

#

Range (R)

Sca

tter P

lot o

f Sig

ma

and

R

0

0.51

1.52

2.5

00.

51

1.5

Sigm

a

Range Cor

rela

tion(

s, R

) = 0

.993

4

OPR

E 63

6439

Con

trol C

harts

for V

aria

bles

•M

ean

char

t (X-

Bar C

hart)

–Use

s av

erag

e of

a s

ampl

e

•R

ange

cha

rt (R

-Cha

rt)–U

ses

amou

nt o

f dis

pers

ion

in a

sam

ple

OPR

E 63

6440

Con

stru

ctio

n of

Con

trol C

hart

●C

ontro

l lim

its m

ust b

e ba

sed

only

on

hist

oric

pr

oces

s da

ta th

at a

re “i

n-co

ntro

l”●

We

draw

tent

ativ

e lim

it lin

es a

nd c

heck

if a

ny

poin

ts fa

ll ou

tsid

e th

e lim

its●

If so

me

poin

ts fa

ll ou

tsid

e, n

on-ra

ndom

ca

uses

are

pre

sent

; dis

card

thos

e da

ta

poin

ts a

nd re

-cal

cula

te c

ontro

l lim

its●

Rep

eat c

alcu

latio

n of

lim

its if

nec

essa

ry

OPR

E 63

6441

Th

ree S

igm

a C

on

tro

l Lim

its

•Th

e us

e of

3-s

igm

a lim

its g

ener

ally

giv

es

good

resu

lts in

pra

ctic

e (A

RL

= 1/

(α/2

).

•If

the

dist

ribut

ion

of th

e qu

ality

cha

ract

eris

tic

is re

ason

ably

wel

l app

roxi

mat

ed b

y th

e no

rmal

dis

tribu

tion,

then

the

use

of 3

-sig

ma

limits

is a

pplic

able

.•

Thes

e lim

its a

re o

ften

refe

rred

to a

s ac

tion

limits

.

OPR

E 63

6442

War

nin

g L

imits

on C

ontr

ol Char

ts

•W

arni

ng li

mits

(if u

sed)

are

typi

cally

set

at 2

st

anda

rd d

evia

tions

from

the

mea

n.•

If on

e or

mor

e po

ints

fall

betw

een

the

war

ning

lim

its

and

the

cont

rol l

imits

, or c

lose

to th

e w

arni

ng li

mits

th

e pr

oces

s m

ay n

ot b

e op

erat

ing

prop

erly

.•

Goo

d th

ing:

War

ning

lim

its o

ften

incr

ease

the

sens

itivi

tyof

the

cont

rol c

hart.

•Ba

d th

ing:

War

ning

lim

its c

ould

resu

lt in

an

incr

ease

d ris

k of

fals

e al

arm

s.

OPR

E 63

6443

Calc

ula

tio

n o

f X

bar

Ch

art

Co

ntr

ol

Lim

its

tabl

e.a

from

foun

dis

an

d

rang

es

sam

ple

of

Ave

rage

w

here

LCL

lim

it,

cont

rol

Low

er

UC

Llim

it,

cont

rol

Upp

er

ity.

var

iabi

lpr

oces

s

of m

easu

re a

as

rang

e

sam

ple

av

erag

e

us

e

tois lim

its

cont

rol

fin

ding

for

m

etho

dqu

ick

A

Ran

ge

5/)(x

X

bar

:D

ef

2

22

54

32

1

AR

RA

xz

x

RA

xz

x

R

xM

inx

Max

R

xx

xx

x

xbar

xbar

ii

=

−=

−=

+=

+=−

=

++

++

==

σσ

OPR

E 63

6444

Proc

ess C

ontr

ol C

hart

Fac

tors

LCL

Fact

orfo

r Ran

ges

(Ran

geC

hart

s)(D

3)

UC

L Fa

ctor

for R

ange

s(R

ange

Cha

rts)

(D4)

Con

trol

Lim

itFa

ctor

for

Ave

rage

s(M

ean

Cha

rts)

(A2)

Fact

or fo

rEs

timat

ing

Sigm

a(

= R

/d2)

(d2)

Sam

ple

(Sub

grou

p)Si

ze (n) 2

1.88

03.

267

01.

128

31.

023

2.57

50

1.69

32.

282

40.

729

02.

059

50.

577

2.11

50

2.32

66

0.48

32.

004

02.

534

70.

419

1.92

40.

076

2.70

48

0.37

31.

864

0.13

62.

847

90.

337

1.81

60.

184

2.97

010

0.30

81.

777

0.22

33.

078

OPR

E 63

6445

Sele

ct 2

5 sm

all s

ampl

es(in

this

cas

e, n

= 4

)

Find

X a

nd R

of e

ach

sam

ple.

The

X ch

art i

s us

ed to

cont

rol t

he p

roce

ss m

ean.

The

R c

hart

is u

sed

toco

ntro

l pro

cess

var

iatio

n.

1

2

3

4

2

5

4

7

6

7

6

3

9

6

5

8

8

6

5

6

9

5

20 2

4 3

2 2

4

2

8To

tal

5

6

8

6

7

15

0

2

5

3

2

3

7

5

Sam

ple

Num

ber

X Values

Sum X R

Proc

ess D

ata

Exa

mpl

e:

5-31

OPR

E 63

6446

Sum X R

1

2

3

4

2

54

7

6

7

6

3

9

6

5

8

8

6

5

6

9

5

20

24

32

24

28

Tota

l5

6

8

6

7

150

2

5

3

2

3

7

5

Sam

ple

Num

ber

Values

2 3 4

0 0 0

1.88

01.

023

0.72

9

3.26

72.

575

2.28

2

1.12

81.

693

2.05

9

X a

nd R

Cha

rts F

acto

rsn

A2

D3

D4

d 2

5-32a

OPR

E 63

6447

Sum X R

1

2

3

4

2

54

7

6

7

6

3

9

6

5

8

8

6

5

6

9

5

20

24

32

24

28

Tota

l5

6

8

6

7

150

2

5

3

2

3

7

5

Sam

ple

Num

ber

Values

2 3 4

0 0 0

1.88

01.

023

0.72

9

3.26

72.

575

2.28

2

1.12

81.

693

2.05

9

– – X =

150

/ 25

= 6

R =

75

/ 25

= 3

A2R

= 0

.729

(3) =

2.2

UC

L X =

X +

A2R

= 6

+ 2

.2 =

8.2

LCL X

= X

-A2R

= 6

-2.

2 =

3.8

UC

L R=

D4R

= 2

.282

(3) =

6.8

LCL R

= D

3R =

0(3

) = 0

– – – –

––

– –– –

X a

nd R

Lim

its

n

A

2

D

3D

4d 2

5-32b

OPR

E 63

6448

Sum X R

1

2

3

4

2

54

7

6

7

6

3

9

6

5

8

8

6

5

6

9

5

20

24

32

24

28

Tota

l5

6

8

6

7

150

2

5

3

2

3

7

5

Sam

ple

Num

ber

Values

2 3 4

0 0 0

1.88

01.

023

0.72

9

3.26

72.

575

2.28

2

1.12

81.

693

2.05

9

– – X =

150

/ 25

= 6

R =

75

/ 25

= 3

A2R

= 0

.729

(3) =

2.2

UC

L X =

X +

A2R

= 6

+ 2

.2 =

8.2

LCL X

= X

-A2R

= 6

-2.

2 =

3.8

UC

L R=

D4R

= 2

.282

(3) =

6.8

LCL R

= D

3R =

0(3

) = 0

– – – –

––

– –– –

–LCL

X=

3.8

UC

LX

= 8.

2–

X =

6.0

– –

–

UC

L R

= 6.

8

R =

3.0

LCL

R=

0

RangeMean

X a

nd R

Cha

rt P

lots

n

A

2

D

3D

4d 2

5-32

OPR

E 63

6449

Exa

mple

: X

bar

chart

Co

ntr

ol

Lim

its

by σ

xb

ar

A q

ual

ity

contr

ol m

anag

er t

ook

five

sam

ple

s(S

1,

S2,

S3,

S4,

S5),

eac

h w

ith four

obse

rvat

ions,

of th

e dia

met

er o

f sh

afts

man

ufa

cture

d o

n a

lat

he

mac

hin

e. T

he

man

ager

com

pute

d t

he

mea

n o

f ea

ch s

ample

and t

hen

com

pute

d

the

gra

nd m

ean.

All

valu

es a

re in c

m.

Use

this

info

rmat

ion t

o

obta

in 3

-sig

ma

(i.e

., z

=3 )

contr

ol lim

its

for

mea

ns

of

futu

re

tim

es.

It is

know

n f

rom

pre

vious

exper

ience

that

the

stan

dard

d

evia

tio

n σ

xof

the

pro

cess

is

0.0

2 c

m.

12.1

212

.10

12.1

112

.12

12.1

0Xb

ar

12.0

912

.14

12.1

312

.12

12.1

212

.10

12.0

812

.10

12.0

912

.09

12.1

112

.15

12.1

512

.12

12.1

012

.11

12.1

112

.10

12.1

112

.08

1 2 3 4

S5S4

S3S2

S1O

bser

vatio

n

OPR

E 63

6450

Exa

mple

of

Contr

ol Li

mits

Cal

cula

tions

usi

ng σ

xb

ar

size

.

Sam

ple

an

dde

viat

ion

st

anda

rd

Proc

ess

mea

ns

sam

ple

of

on

dist

ribut

i

ofde

viat

ion

St

anda

rd

w

here

12.0

8

0.01

3

12

.11

LC

L

:lim

it

cont

rol

Low

er

12.1

4

0.01

3

12

.11

U

CL

:lim

it

cont

rol

Upp

er

0.01

40.

02

Hen

ce

4.

si

ze

sam

ple

that

N

ote

(g

iven

).

0.02

and

12.1

1

512

.12

12.1

0

12

.11

12.1

2

12

.10

x

==

==

=×

−=

−=

=×

+=

+=

==

=

==

=+

++

+=

n

x

nzx

zx

n

n

x

x

x

x

σ

σσ

σσ

σσ

σ

OPR

E 63

6451

Co

ntr

ol

Lim

it F

act

ors

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 00.

080.

140.

180.

220.

260.

280.

310.

330.

350.

360.

380.

360.

400.

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

D4

D3

A2

n

Fact

or fo

r R U

CL

Fact

or fo

r R L

CL

Fact

or fo

r Xba

rlim

itsSa

les

Size

OPR

E 63

6452

Xbar

Contr

ol Li

mits

by

Rb

ar

0.05

0.04

0.06

0.05

0.03

Ran

ge R

12.0

912

.14

12.1

312

.12

12.1

212

.10

12.0

812

.10

12.0

912

.09

12.1

112

.15

12.1

512

.12

12.1

012

.11

12.1

112

.10

12.1

112

.08

1 2 3 4

S5S4

S3S2

S1O

bser

vatio

n

08.12

046

.073.0

11.12

LCL

14.12

046

.073.0

11.12

UC

L

are

Lim

itsC

ontro

lr

Upp

er/L

owe

Hen

ce,

tabl

efr

om73.0

ther

efor

e,4

size

Sam

ple

046

.0

50.

05

0.

04

0.

06

0.

05

0.

03

ra

nges

sa

mpl

e

of A

vera

ge

22

2

=×

−=

−=

=×

+=

+=

==

=+

++

+==

RA

x

RA

x

An

RR

OPR

E 63

6453

Ran

ge (

R)

Ch

art

help

s co

ntr

ol

Pre

cisi

on

UCL

20 0510 Range15

12

34

56

78

910

1112

1314

1516

1718

1920

Sam

ple

#LC

L

●Av

erag

e R

ange

R=

10.1

kg

●St

anda

rd D

evia

tion

of R

ange

= 3

.5 k

g●

Con

trol L

imits

:10

.1 +

3 ×

3.5

= [2

0.6,

0]

Proc

ess

here

is “i

n co

ntro

l” (i.

e., p

reci

sion

is s

tabl

e)

OPR

E 63

6454

Ran

ge C

on

tro

l C

hart

Co

ntr

ol

Lim

its

tabl

e.Fa

ctor

sLi

mit

Con

trol

th

efr

om

obta

ined

ar

e

D an

d

D w

here

D

LC

L

limit,

co

ntro

lLo

wer

D

UC

Llim

it,

cont

rol

Upp

er

char

t.-

R th

eis

prec

isio

n or

disp

ersi

on

proc

ess

mon

itor

tous

edch

art

cont

rol

Th

e

43

3R

4R

RR

==

OPR

E 63

6455

12.1

212

.10

12.1

112

.12

12.1

0Xb

ar

0.05

0.04

0.06

0.05

0.03

Ran

ge R

12.0

912

.14

12.1

312

.12

12.1

212

.10

12.0

812

.10

12.0

912

.09

12.1

112

.15

12.1

512

.12

12.1

012

.11

12.1

112

.10

12.1

112

.08

1 2 3 4

S5S4

S3S2

S1O

bser

vatio

nExa

mple

:R c

har

t Li

mits

00.004

6.

000

.0LC

L

105

.004

6.

280

.2U

CL

are

/H

ence

,

.ta

ble

from

28.2an

d00.0

Ther

efor

e.4

046

.0

50.

05

0.

04

0.

06

0.

05

0.

03

ra

nges

sa

mpl

e

of A

vera

ge

34

43

=×

==

=×

==

==

=

=+

++

+==

RD

RD

Lim

itsC

ontr

olLo

wer

Upp

er

DD

nR

RR

OPR

E 63

6456

Perf

orm

an

ce V

ari

ati

on

Patt

ern

s

Stab

le

Uns

tabl

e

Tren

d

Cyc

lical

Shift

OPR

E 63

6457

Abno

rmal

Con

trol C

hart

Patte

rns

UC

LU

CL

LCL

LCL

Sam

ple

obse

rvat

ions

cons

iste

ntly

abo

ve th

ece

nter

line

Sam

ple

obse

rvat

ions

cons

iste

ntly

bel

ow th

ece

nter

line

OPR

E 63

6458

Abno

rmal

Con

trol C

hart

Patte

rns

UC

LU

CL

LCL

LCL

Sam

ple

obse

rvat

ions

cons

iste

ntly

incr

easi

ngSa

mpl

e ob

serv

atio

nsco

nsis

tent

ly d

ecre

asin

g

OPR

E 63

6459

Abno

rmal

Con

trol C

hart

Patte

rns

UC

LU

CL

LCL

LCL

Sam

ple

obse

rvat

ions

cons

iste

ntly

abo

ve th

ece

nter

line

Sam

ple

obse

rvat

ions

cons

iste

ntly

bel

ow th

ece

nter

line

OPR

E 63

6460

Zone

s Fo

r Non

-Ran

dom

Pat

tern

Te

sts

UC

L

LCL

Zone

A

Zone

B

Zone

C

Zone

C

Zone

B

Zone

A

3si

gma=

x +

2A

R

2si

gma=

x +

2 32A

R

()

1si

gma

=x +

1 32

AR

(

)

x 1sig

ma=

x −

1 32A

R

()

2si

gma=

x −

2 32A

R

()

3si

gma=

x −

2A

R

OPR

E 63

6461

Abno

rmal

Con

trol C

hart

Patte

rns

1. 8

con

secu

tive

poin

ts o

n on

e si

de o

f the

cen

ter l

ine.

2. 8

con

secu

tive

poin

ts u

p or

dow

n ac

ross

zon

es.

3. 1

4 po

ints

alte

rnat

ing

up o

r dow

n.4.

2 o

ut o

f 3 c

onse

cutiv

e po

ints

in z

one

A bu

t stil

l ins

ide

the

cont

rol l

imits

.5.

4 o

ut o

f 5 c

onse

cutiv

e po

ints

in z

one

A or

B.

OPR

E 63

6462

From

Contr

ol to

Im

pro

vem

ent

LCLµ

UCL

Out

of C

ontr

olIn

Con

trol

Impr

oved

Wei

ght

Tim

e

Targ

et

OPR

E 63

6463

Def

ect C

ontro

l For

Attr

ibut

es

•p

Cha

rts–C

alcu

late

per

cent

def

ectiv

es in

sam

ple

•c

Cha

rts–C

ount

num

ber o

f def

ects

in it

em

OPR

E 63

6464

Use

of p-C

har

ts

•W

hen

obse

rvat

ions

can

be

plac

ed in

to

two

cate

gorie

s.–

Goo

d or

bad

–Pa

ss o

r fai

l–

Ope

rate

or d

on’t

oper

ate

•W

hen

the

data

con

sist

s of

mul

tiple

sa

mpl

es o

f sev

eral

obs

erva

tions

eac

h

OPR

E 63

6465

Co

ntr

ol

Lim

its

forp–

Ch

art

0.

LCL

U

sefo

rmul

a. e

appr

oxim

at

todu

e

nega

tive

is LC

L

Som

etim

es.

re

plac

es ,

es

timat

e,

The

hi

stor

y.

from

as es

timat

ed

beca

n it

unkn

own,

is

Ifpr

oces

s.

in th

e

defe

ctiv

es

offr

actio

n

nom

inal

th

eis

an

d

)1(

on,

dist

ribut

i

Bin

omia

l

from

w

here

LCL

lim

it,

cont

rol

Low

er

UC

Llim

it,

cont

rol

Upp

er

proc

ess.

ain

defe

ctiv

es

of pr

opor

tion

em

onito

r th

to

used

,at

tribu

tes

for

char

t

Con

trol

p

pp

=

−=

=

+=

pp

ppp

np

p

p - z

σ z σ

p

p

p

σ

OPR

E 63

6466

The

Nor

mal

Dis

tribu

tion

still

appl

ies

95%

99.7

4%

-1σ

-3σ

-2σ

µ=0

1σ2σ

3σ

OPR

E 63

6467

p C

hart

Dat

a

2 50 4 .08

1 50 2 .04

3 50 0 0

4 50 3 .06

25 50 2 .04

Sam

ple

num

ber

Tota

l

1250 50 1.00

n

#def p

Sam

ple

size

Num

ber o

f def

ectiv

e ite

ms

foun

d in

sam

ple

Frac

tion

defe

ctiv

e in

sa

mpl

e

5-33a

OPR

E 63

6468

p C

hart

Cal

cula

tions

2 50 4 .08

Sam

ple

num

ber

1 50 2 .04

3 50 0 0

4 50 3 .06

25 50 2 .04

Tota

l

1250 50 1.00

.04(

.96)

p =

–

p(1-

p)n

3

= 3

50=

0.08

3

UC

L P

= p

+ 3

P–

UC

L P

= p

-3

P–

= .0

4 +

.083

= .1

23

= .0

4 -.

083

= 0

can'

t be

nega

tive

••

••

UC

L P

= 0.

123

LCL

P =

0

p =

0.04

–

= 3

σ σ

#def n

ΣΣ

n=

p

Pσn

#def p

5-33

OPR

E 63

6469

Exa

mple

:p

char

t dat

a:

120

Tota

l

10 9 8 11

12 8 13

11 9 10 8 11

1 2 3 4 5 6 7 8 9 10

11

12

Nu

mb

er

of

Defe

ctiv

es

Sam

ple

#

A Q

C m

anag

er c

ounte

d

the

nu

mb

er

of

defe

ctiv

e n

uts

p

rod

uce

dby

an

auto

mat

ic m

achin

e in

12

sam

ple

s. U

sing t

he

dat

a sh

ow

n,

const

ruct

a

contr

ol ch

art

that

will

des

crib

e 99.7

4 %

of

the

chan

ce v

aria

tion in t

he

pro

cess

when

the

pro

cess

in

contr

ol. E

ach s

ample

co

nta

ined

200 n

uts

.

OPR

E 63

6470

p C

har

t S

olu

tion

005

.015.0

305.0

LC

L

lim

it,

cont

rol

Low

er

095

.015.0

305.0

UC

L

limit,

co

ntro

lU

pper

3z

015

.020

0)

05.01(

05.0)

1(

05.020

012

120 pp

p

=×

−=

−=

=×

+=

+=

=

=−

=−

=

=×

=

p

p

z σ

p

z σ

p

np

p

p σ

OPR

E 63

6471

Exam

ple

of p

-Cha

rt

..

00.

020.

040.

060.

080.1

0.12

0.14

0.16

0.180.

20

2

4

6

8

10

12

14

16

18

20

Proportion defective

Sam

ple

num

ber

OPR

E 63

6472

Nu

mb

er

of

Defe

cts/

Un

it:

c-C

hart

s

Use

onl

y w

hen

the

num

ber o

f oc

curre

nces

per

uni

t of m

easu

re c

an

be c

ount

ed; n

on-o

ccur

renc

es c

anno

t be

cou

nted

.–

Scra

tche

s, c

hips

, den

ts, o

r erro

rs p

er it

em–

Cra

cks

or fa

ults

per

uni

t of d

ista

nce

–Br

eaks

or T

ears

per

uni

t of a

rea

–Ba

cter

ia o

r pol

luta

nts

per u

nit o

f vol

ume

–C

alls

, com

plai

nts,

failu

res

per u

nit o

f tim

e

OPR

E 63

6473

c-Char

t Contr

ols

Def

ects

/Unit

Dis

cret

e Q

ualit

y M

easu

rem

ent:

D =

Num

ber o

f “de

fect

s” (e

rrors

) per

uni

t of w

ork

Exam

ples

of D

efec

ts:

Num

ber o

f typ

os/p

age,

erro

rs/th

ousa

nd tr

ansa

ctio

ns,

equi

pmen

t bre

akdo

wns

/shi

ft, b

ags

lost

/thou

sand

flow

n, p

ower

ou

tage

s/ye

ar, c

usto

mer

com

plai

nts/

mon

th, d

efec

ts/c

ar...

If

n

= N

o. o

f opp

ortu

nitie

s fo

r def

ects

to o

ccur

, and

p =

Prob

abilit

y of

a d

efec

t/erro

r occ

urre

nce

in e

ach

then

D

~ B

inom

ial (

n, p

) with

mea

n np

, var

ianc

e np

(1-p

)≅

Pois

son

(np)

with

mea

n =

varia

nce

= np

, if

n is

larg

e (≥

20)

and

p is

sm

all (≤

0.05

)

With

c=

np=

aver

age

num

ber o

f def

ects

per

uni

t,

Con

trol l

imits

= c

+3 √c

OPR

E 63

6474

c-Char

t Contr

ol Li

mits

used

. is

Pois

son

io

n to

appr

oxim

aton

di

strib

uti

no

rmal

th

e

reas

ons

pr

actic

alfo

r B

ut

on

.di

strib

uti

Pois

son

a ha

sac

tual

ly

c

devi

atio

n.

stan

dard

th

eis c

and

un

it,pe

r

defe

cts

of

num

ber

an

dm

ean

th

eis c

whe

re

cz

c

LCL

lim

it,

cont

rol

Low

er

cz

c

UC

Llim

it,

cont

rol

Upp

er

cc

−=

+=

OPR

E 63

6475

c-C

hart

Exa

mpl

e: H

otel

Sui

te In

spec

tion-

-Def

ects

D

isco

vere

d/ro

om

Day

Def

ects

Day

Def

ects

Day

Def

ects

4 2 1 2 3 1 3 2 0

2 0 3 1 2 3 1 0 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 1 2 1 0 3 0 1 39To

tal

OPR

E 63

6476

Rec

all c

-Cha

rt Li

mits

Proc

ess

aver

age=

c =

Tota

l # d

efec

ts#

sam

ples

Sam

ple

stan

dard

dev

iatio

n=

cσ

=c

UC

L=

c +

zc

σLC

L=

c -z

cσ

OPR

E 63

6477

c C

hart

for

Hot

el S

uite

Insp

ectio

n

51

c =

39/2

6 =

1.50

UC

L =

5.16

LCL

= 0

0123

Number of defects5 4

015

2025

Day

OPR

E 63

6478

Exa

mple

of

c C

hart

42

Tota

l

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

Num

ber

of

com

pla

ints

Day

A b

ank

man

ager

re

ceiv

es a

cer

tain

n

um

ber

of

com

pla

ints

each

d

ay

about

the

ban

k’s

se

rvic

e. C

om

pla

ints

fo

r 14 d

ays

are

giv

en

in t

he

table

show

n.

Const

ruct

a c

ontr

ol

char

t usi

ng t

hre

e-si

gm

a lim

its.

OPR

E 63

6479

c Char

t Solu

tion

used

. is

Pois

son

io

n to

appr

oxim

aton

di

strib

uti

no

rmal

re

ason

s,

prac

tical

For

devi

atio

n.

stan

dard

th

eis c

unit.

per

de

fect

s

ofnu

mbe

r

and

mea

n

the

is c w

here

0.073.1

33

cz

c

LCL

lim

it,

cont

rol

Low

er

2.873.1

33

cz

c

UC

Llim

it,

cont

rol

Upp

er 73.1

31442

c

cc

=×

−=

−=

=×

+=

+=

=

== c

OPR

E 63

6480

Contr

ol Char

ts S

um

mar

y

–X-

bar a

nd R

cha

rts•

Varia

bles

dat

a•

Appl

icat

ion

of n

orm

al d

istr

ibut

ion

(by

Cen

tral L

imit

Theo

rem

)–

p ch

arts

•At

tribu

tes

data

(def

ects

per

n o

bser

vatio

ns)

•Ap

plic

atio

n of

bin

omia

l dis

trib

utio

n–

c ch

arts

•At

tribu

tes

data

(def

ects

per

insp

ectio

n)•

Appl

icat

ion

of P

oiss

on d

istr

ibut

ion

OPR

E 63

6481

Whic

h C

har

t to

Use

?

OPR

E 63

6482

OPR

E 63

6483

Sum

mar

y of

SPC

●St

atis

tical

pro

cess

con

trol p

rovi

des

sim

ple,

yet

po

wer

ful,

for m

anag

ing

proc

ess

whi

le a

void

ing

proc

ess

tam

perin

g●

A pr

oces

s 'in

con

trol'

(i.e.

; exh

ibiti

ng n

o sp

ecia

l ca

use

varia

tion)

is ri

pe fo

r the

nex

t sta

ge--

brea

kthr

ough

pro

cess

impr

ovem

ent

●A

proc

ess

still

burd

ened

with

spe

cial

cau

se

varia

tion

is s

till i

n th

e pr

oble

m-s

olvi

ng s

tage

OPR

E 63

6484

Pro

cess

Im

pro

vem

en

t

•M

easu

rem

ent

–Ext

ernal

and

Inte

rnal

•Anal

ysis

–Anal

yze

Var

iation

•Contr

ol

–Adju

st P

roce

ss•

Impro

vem

ent

–Red

uce

Var

iation

•In

nova

tion

–Red

esig

n

Product

/Pro

cess

D ACP

D ACP

Control

Improve

InnovateIm

prove

OPR

E 63

6485

Proc

ess

capa

bilit

y:Th

e in

here

nt

varia

bilit

y of

pro

cess

ou

tput

rela

tive

to th

e va

riatio

n al

low

ed b

y th

e de

sign

or

cust

omer

sp

ecifi

catio

nSp

ec

Lim

itsProc

ess

OPR

E 63

6486

Proc

ess

Cap

abilit

y An

alys

is●

Diffe

rs Fundamentally

from

Contr

ol Char

ting

Focu

ses

on im

pro

vem

ent,

not

contr

ol

Var

iable

s, n

ot

attr

ibute

s, d

ata

invo

lved

Cap

abili

ty s

tudie

s ad

dre

ss r

ange

of

ind

ivid

ualoutp

uts

Contr

ol ch

arting a

ddre

sses

ran

ge

of

sam

ple

mea

sure

s●

Ass

um

es N

orm

al D

istr

ibution

Rem

ember

the

Em

piric

al R

ule

?In

her

ent

capab

ility

(6s x

) is

com

par

ed t

o

speci

fica

tio

ns

●Req

uires

pro

cess

first

to b

e in

Co

ntr

ol

OPR

E 63

6487

Why

mea

sure

Pro

cess

Cap

abili

ty?

Proc

ess

varia

bilit

y ca

n gr

eatly

impa

ct c

usto

mer

sa

tisfa

ctio

nTh

ree

com

mon

term

s fo

r var

iabi

lity:

1.To

lera

nces

:Sp

ecifi

catio

ns fo

r ran

ge o

f ac

cept

able

val

ues

esta

blis

hed

by

engi

neer

ing

desi

gn o

r cus

tom

er

requ

irem

ents

2.

Proc

ess

varia

bilit

y:N

atur

al o

r inh

eren

t va

riabi

lity

in a

pro

cess

3.C

ontr

ol li

mits

:St

atis

tical

lim

its th

at re

flect

th

e in

here

nt v

aria

tion

of s

ampl

e st

atis

tics

OPR

E 63

6488

Proc

ess C

apab

ility

is b

ased

on

Nor

mal

Cur

ve

4

(95.

5%)

6

(99.

7%)

2 (68%

)

µσ

σσ

OPR

E 63

6489

The

Ran

ge o

f Pro

cess

Out

put

The

rang

e in

whi

ch "

all"

out

put c

an b

e pr

oduc

ed.

6

(99.

7%)

Proc

ess

rang

e =

6

µσ

σ

OPR

E 63

6490

Proce

ss C

apab

ility

Conce

pt

X4.

904.

955.

005.

055.

105.

15cm

Tole

ranc

e ba

nd

Inhe

rent

var

iabi

lity

(6

)

LSL

USL

Out

put

Out

put

out o

f spe

cou

t of s

pec

Out

put

Out

put

out o

f spe

cou

t of s

pec

Proc

ess

outp

utdi

strib

utio

n

5.01

0

σ

OPR

E 63

6491

Tw

o P

roce

ss C

apab

ilities

This

pro

cess

isC

APA

BLE

CA

PAB

LEof

pr

oduc

ing

all g

ood

outp

ut.

Con

trol

the

proc

ess.

Low

erSp

ecLi

mit

Upp

erSp

ecLi

mit

This

pro

cess

isN

OT

CA

PAB

LEN

OT

CA

PAB

LE.

INSP

ECT

-Sor

t out

the

defe

ctiv

es

××

OPR

E 63

6492

Cap

abili

ty A

nal

ysis

LSU

Sµ

Cap

abili

ty a

nal

ysis

det

erm

ines

whet

her

the

inher

ent

variab

ility

of

the

pro

cess

outp

ut

falls

within

the

acce

pta

ble

ran

ge

of

the

variab

ility

allo

wed

by

the

des

ign s

pec

ific

atio

ns

for

the

pro

cess

outp

ut.

The

range

of

poss

ible

solu

tions:

1.

Red

esi

gn

the

pro

cess

so t

hat

it

can a

chie

ve t

he

des

ired

outp

ut

2.

Use

an a

ltern

ate

pro

cess

that

can

ach

ieve

the

des

ired

outp

ut

3.

Ret

ain t

he

curr

ent

pro

cess

but

atte

mpt

to e

limin

ate

unac

cepta

ble

outp

ut

usi

ng 1

00

perc

en

t in

spect

ion

4.

Exa

min

e th

e sp

eci

fica

tio

nto

see

whet

her

they

are

nec

essa

ry o

r co

uld

be

rela

xed

without

adve

rsel

y af

fect

ing c

ust

om

er

satisf

action.

OPR

E 63

6493

Proce

ss C

apab

ility

Rat

io C

p

2

C M

otor

ola,

For

.m

anag

emen

t

Sigm

aSi

x

uses

n C

orpo

ratio

M

otor

ola6

C

wid

thPr

oces

sion

wid

thSp

ecifi

cat

C

C

ratio

ca

pabi

lity

Pr

oces

s

p

pp

p

=

−==

=

σLS

LU

SL

OPR

E 63

6494

Proce

ss C

apab

ility

Index

Cpk

Inde

x C

pkco

mpa

res

the

spre

ad a

nd lo

catio

nof

the

proc

ess,

rela

tive

to th

e sp

ecifi

catio

ns.

3U

pper

Spe

c Li

mit

-X– –

X -L

ower

Spe

c Li

mit

– –O

Rth

e sm

alle

r of:

Cpk

=

{σ 3 σ

Upp

er S

pec

Lim

it -X– –

X -L

ower

Spe

c Li

mit

– –O

RW

here

Zm

inis

the

smal

ler o

f:

Cpk

= Z m

in3

{σ σ

Alte

rnat

e Fo

rm

5-40

OPR

E 63

6495

Proce

ss C

apab

ility

Rat

io C

pk

Nor

mal

dis

tribu

tion

=> 9

9.73

% o

f out

put f

alls

in (µ

+3σ

) whe

n th

e pr

oces

s is c

ente

red.

If

the

proc

ess i

s not

cen

tere

d, w

e us

e

Cpk

= M

in [(

US

-µ) /

3σ,

(µ

-LS)

/ 3σ

]

Exam

ple.

M

BPF

: C

pk=

Min

[0.1

894,

0.5

952]

= 0

.198

4

With

cen

tere

d pr

oces

s (U

S -µ

) = (µ

-LS)

. T

hen

Cpk

= C

p=

(US

-LS)

/ 6σ

=V

oice

of t

he C

usto

mer

Voi

ce o

f the

Pro

cess

=0.

3968

Cp

=0.

861

1.1

1.3

1.47

1.63

2.0

Def

ects

/m =

10K

3K1K

100

101p

pm2

ppm

OPR

E 63

6496

Proc

ess C

apab

ility

: C

exa

mpl

espk

Cpk

= 1.

0C

pk=

1.33

Cpk

= 3.

0

LSL

USL

LSL

USL

Cpk

= 1.

0

LSL

USL

LSL

USL

LSL

USL

(a)

(f)(e

)(d

)

(c)

(b)

Cpk

= 0.

60C

pk=

0.80

LSL

USL

5-42

OPR

E 63

6497

Cap

abili

ty I

mpro

vem

ent

by

Mea

n Sh

ift

LS

= 7

5U

S =

85

Cpk=

0.76

60

80

Cp

k=

0.6

99

0

82.5

Gar

age

Door

Wei

ght

(kg)

Probab

ility

den

sity

of outp

ut

(wei

ght)

(bef

ore

shift)

(aft

er

shif

t)

Off

sp

ec

Off

sp

ec

Proc

ess

Adju

stm

ent

OPR

E 63

6498

Cap

abili

ty I

mpro

vem

ent

by

Var

ianc

e Red

uctio

n

LS

= 7

5U

S =

85

befo

reC

pk

= 0

.76

60

80

Cp

k=

0

.95

44

Gar

age

Door

Wei

ght

(kg)

Probab

ility

den

sity

of outp

ut

(wei

ght)

Aft

er

red

uct

ion

OPR

E 63

6499

Proce

ss C

ontr

ol an

d C

apab

ility

: Rev

iew

●Ev

ery

proc

ess

disp

lays

som

e va

riabi

lity—

norm

al o

r ab

norm

al●

Con

trol c

harts

can

iden

tify

abno

rmal

var

iabi

lity

●C

ontro

l cha

rts m

ay g

ive

fals

e (o

r mis

sed)

ala

rms

by

mis

taki

ng n

orm

al (a

bnor

mal

) for

abn

orm

al (n

orm

al)

varia

bilit

y●

On-

line

cont

rol l

eads

to e

arly

det

ectio

n an

d co

rrect

ion

●A

proc

ess

“in c

ontro

l” in

dica

tes

only

its

inte

rnal

sta

bilit

y●

Impr

ovin

g pr

oces

s ca

pabi

lity

invo

lves

cha

ngin

g th

e m

ean

and/

or re

duci

ng n

orm

al v

aria

bilit

y re

quiri

ng a

long

te

rm in

vest

men

t

OPR

E 63

6410

0

Des

ign for

Cap

able

Pro

cess

ing

•Si

mpl

ify

–Few

er p

arts

, ste

ps–M

odul

ar d

esig

n•

Stan

dard

ize

–Les

s va

riety

–Sta

ndar

d, p

rove

n pa

rts, a

nd p

roce

dure

s•

Mis

take

-pro

of–C

lear

spe

cs–E

ase

of a

ssem

bly,

dis

asse

mbl

y, s

ervi

cing

OPR

E 63

6410

1

Tag

uch

i Q

ual

ity

Philo

sophy

Loss

= k

(P -

T)2

not

0 if

with

in s

pecs

and

1 if

out

side

On

Targ

etis

mor

e im

porta

nt th

an

With

in S

pecs

LST

US

Conve

ntional

vie

wTag

uch

i’s v

iew

LS

TU

S

OPR

E 63

6410

2

Robust

Des

ign

Targ

et

Perfo

rman

ce (T

)Ac

tual

Pe

rform

ance

(P)

Des

ign

Par

amet

ers

(D)

Noi

se F

acto

rs (N

): In

tern

al &

Ext

erna

l

Prod

uct /

Pro

cess

•Id

entify

Pro

duct

/Pro

cess

Des

ign P

aram

eter

s th

at–

Hav

e si

gnific

ant

/ lit

tle

influen

ce o

n P

erfo

rman

ce–

Min

imiz

e per

form

ance

var

iation d

ue

to N

ois

e fa

ctors

–M

inim

ize

the

pro

cess

ing c

ost

•

Met

hodolo

gy:

Des

ign o

f Exp

erim

ents

(D

OE)

•Exa

mple

s -

Choco

late

mix

, In

a Tile

Co.,

Sony

TV

OPR

E 63

6410

3

The

Des

ign P

roce

ss

•G

oal

–Dev

elop

hig

h qu

ality

, low

cos

t pro

duct

s, fa

st•

Impo

rtanc

e–8

0% p

rodu

ct c

ost,

70%

qua

lity,

65%

suc

cess

•C

onve

ntio

nal

–Tec

hnol

ogy-

driv

en, I

sola

ted,

Seq

uent

ial,

Itera

tive

•D

iffic

ultie

s –R

evis

ions

, cos

t ove

rruns

, del

ays,

retu

rns,

reca

lls•

Solu

tion

–Cus

tom

er-d

riven

(QFD

), jo

intly

pla

nned

, pr

oduc

ible

OPR

E 63

6410

4

Concu

rren

t D

esig

n

•O

bjec

tive

–Int

erfu

nctio

nalc

oord

inat

ion

to s

atis

fy c

usto

mer

–Inv

olve

man

ufac

turin

g, s

uppl

iers

, R&D

•Pr

ereq

uisi

tes

–Bre

ak d

own

barri

ers

–Cro

ss fu

nctio

nal t

rain

ing

–Com

mun

icat

ion,

team

wor

k, g

roup

dec

isio

ns•

Res

ult

–Few

er re

visi

ons,

mis

com

mun

icat

ion,

del

ays

•D

iffic

ultie

s –T

ime

cons

umin

g, c

ompl

ex, o

rgan

izat

iona

l

OPR

E 63

6410

5

Ref

eren

ces

http

://de

min

g.en

g.cl

emso

n.ed

u/pu

b/tu

toria

ls/q

ctoo

ls/c

so.h

tmC

ontro

l cha

rt ca

se s

tudy

http

://w

ww

.qua

litya

mer

ica.

com

/kno

wle

dgec

ente

/kno

wct

rSPC

_Arti

cles

.htm

SPC

arti

cles