Quality
Enterprise
level
Manufacturing
Support
Systems
Quality
Control
Factory
level
Production
System
Manufacturing
Support
Systems
Manufacturing
Systems
Automation and
and
Control
Technologies
Systems
Material
handling
Technologies
Facilities
Manufacturing processes and assembly operations
Groover, M. P. “Automation, Production Systems, and Computer-Integrated Manufacturing”
Quality. Definitions
•
•
There
There are
are several
several defini
definitions;
tions; we will follow those from ISO 9000
Modern
Modern quality
quality management
management is oriented
oriented to satisfacti
satisfaction
on of needs and
•
requirements of clients. Is not limited to inspection of products and
processes
Regarding
Regarding to products,
products, two relevant
relevant aspects
aspects related
related with quality are:
Product features (to be efficiently searched
searched and defined to satisfy
clients)
Freedom from deficiencies (not only defects
defects or malfunctioning)
Product feature s
Fre edom from deficiencie s
Design, con
onfigur
figurat
ation,
ion, size, weight
Absen
Absencce of defec
defects
ts
Func
ncttion
ion and per
erfo
forrmanc
nce
e
Confor
formance to spec
specifi
ificcations
ions
Distinguished
Disting
uished feat
featur
ures
es of tthe
he mo
model
del Com
Component
ponentss within tolera
toleranc
nce
e
Ae sthe ti c appe al
No mi ssi ng parts
Ease of use
No e arl y f ai l ure s
Availability of opt
options
ions
Reliability
Reliabili
ty an
and
d dependab
dependabili
ility
ty
Durability and long service life
Serviceavility
Reputation of product producer
Quality management evolution
Tradi ti onal
Mode rn
Quality
ty is foc
focus
used
ed on client sa
satis
tisfa
facctio
tion.
n. Inter
Interna
nall and
Clien
lients
ts are ext
exter
erna
nall to the
the orga
organi
niza
zati
tion
on.. The sales
sales and Quali
externa
exter
nall clients ar
are
e cons
onsider
idered:
ed: exter
externa
nall by comp
ompan
any'
y'ss
mar
market
keting
ing dep
depa
artment
tment are resp
espon
onsib
sible
le for rela
elatio
tions
ns wit
with
h
produc
products,
ts, intern
internal
al are depar
department
tmentss or individuals inside
customers
the company
The company is organized by functional departments with little
Quality goals are defined at highest levels and are driven
apprec
app
reciation
iation of iinterd
nterdependence.
ependence. T
The
he loyalt
loyalty
y and vie
viewpoint
wpoint of by to
top
p ma
mana
nage
geme
ment
nt.. Conc
oncurr
urren
entt
each depar
each
department
tment tends to be cent
centered
ered on itsel
itselff ra
rather
ther than o
on
n
engineering use to be implemented
the company
and sim
simul
ulta
tane
neou
ouss
Quality is the responsability of the inspection department. The
quality
qua
lity func
function
tion empha
emphasizes
sizes inspec
inspection
tion and co
confor
nforman
mance
ce to Quality is not just the job of inspection departament. It is
specifications with the objective to eliminate defects.
understood
under
stood tha
thatt pr
produc
oductt design ha
hass an import
important
ant influence
Insp
Inspec
ecti
tion
on follo
follows
ws prod
produc
ucti
tion
on.. The
herre is ofte
often
n a con
onflic
flictt on product quality.
between
betw
een pr
produ
oducction object
objective
ive (ma
(make
ke pr
produ
oduccts) an
and
d qua
quality
lity Insp
Inspec
ectio
tion
n of ma
manu
nufa
facctur
tured pr
prod
oduc
ucts
ts is no
nott eno
enoug
ugh.
h.
control
cont
rol obje
objective
ctive ((ac
accept
cept only good produc
products).
ts).
Quality
Qua
lity must be built into the prod
produc
uct.
t. Produc
Production
tion wor
workers
kers
Knowledge of statistical quality control techniques reside only must inspect their
thei r own products.
on QC exper
experts
ts of the
the com
ompa
pany
ny.. Wor
orker
ker''s resp
espon
onsa
sabili
bility
ty is Supp
Suppliers
liers are i nvolved
limited to following instructions. Managers and technical staff Quality is a process of continuous improvement
do all the planning.
Modern quality management
•
•
Combin
Combines
es techno
technolog
logy
y with
with manage
managemen
mentt
Tota
Totall Qual
Qualit
ity
y Mana
Manage
geme
ment
nt (TQM
(TQM)) seek
seeks
s fo
forr cl
clie
ient
nt sati
satisf
sfac
acti
tion
on th
thro
roug
ugh
h
qual
qu
alit
ity
y assu
assura
ranc
nce,
e, cont
contin
inuo
uous
us impr
improv
ovem
emen
entt and
and th
the
e invo
involv
lvem
emen
entt of th
the
e
whole organization: individuals, department, managers
Management (TQM)
Modern inspection and
control technologies
Efficient quality system
Technologies:
•
•
•
•
Qual
Quality
ity Engin
Enginee
eeri
ring
ng
Qualit
Quality
y Functi
Function
on Deploy
Deploymen
mentt (QFD)
(QFD)
Statis
Statistic
tical
al proces
process
s contro
controll
Automa
Automated
ted inspec
inspectio
tion
n of 100 % of pro
produc
duction
tion
•
Inspection out of line (product and process design) and in line
(manufacturing and clients relation after product delivery)
Contactle
les
ss sensors, computer vision
ion, Reverse Engineering (RE),
Coordinate Measuring Machines (CMM)
•
Process variability and process capability
Process variations may be:
• Random (i
(ind
ndep
epen
ende
dent
nt of how
how well
well is th
the
e proc
proces
ess
s desi
design
gned
ed,, due
due to huma
human
n
vari
va
riab
abililit
ity
y amon
among
g work
work cycl
cycles
es,, mach
machin
ine
e vi
vibr
brat
atio
ion,
n, raw
raw mate
materi
rial
als
s vari
variab
abili
ility
ty).
).
Random variations typically form a normal distribution. After long run output tends
to cluster about the mean and process is said to be in statistical control
• Assignable (indicate and exception: defective raw material, human mistakes, tool
fa
failu
ilure
res,
s, equi
equipm
pmen
entt malf
malfun
unct
ctio
ions
ns).
). Caus
Cause
e th
the
e outp
output
ut to devi
deviat
ate
e from
from norm
normal
al
distribution and the process is said to be out of statistical control
Process capability relates to the normal variations in the output when the process is
in statistical control. By definition equals ± 3 standard deviations (PC = µ ± 3σ),
where µ is process mean and σ standard deviation. It is assumed that the output is
normally distributed and steady state has been achieved. With ± 3σ, 99,73 % of the
output is inside limits.
Sampling for statistical process estimation
Population mean (µ) :
∑
;
and population standard deviation (σ) by:
∑
Process capability best estimate for that sample is PC = ̅ ± 3s; where ̅ and s are
the average and standard deviation of the sample. Mean and standard deviation is
rarely known because population must be complete to be measured. Sampling is
used to estimate population behavior.
With design tolerances greater than process capability range, the great majority of
parts should be in accordance with specifications. Is a way to assure quality in
design stage (early before production)
It is poss
possib
ible
le if to
tole
lera
ranc
nces
es are
are wide
wide enou
enough
gh (not
(not alwa
always
ys poss
possib
ible
le)) or proc
proces
ess
s
boundaries are near enough (narrow dispersion, good manufacturing conditions;
generally expensive process )
Statistics background. Normal distribution
Design dimension and tolerances:10±0,05
µ = 10; σ = 0,012
Normal distribution
35
30
25
20
15
10
5
0
9,94
9,96
9,98
σ
(9,92…
2…10
10,0
,08)
8)
± 6 (9,9
10
10,02
10,04
10,06
‐3
6
99
99,9
,999
9999
9998
98;; 2∙
2∙10
10 defects/1∙10
With 6σ po
popul
pulati
ation
on limits
limits are wider
wider than
than design
design tolera
tolerance
nces.
s. Practi
Practical
cally
ly
100% of parts are eligible. Is it so magic? Is it only a matter of widening the
amount of ± σ?
Process Capabilities Indexes (PCI)
When design tolerances are specified as being equal to process capability, the upper ( U) and
lower (L) boundaries of this range define the natural tolerance limits. The ratio of the specified
tolerance range relative to process capability is known as the process capability index (Cp)
6
An useful index
index to measur
measure
e the spread of the populat
population
ion about the a
average
verage is:
min
min , ;
;
3
3
Another useful
useful index to me
measure
asure the loc
location
ation of the tar
target
get value () respect the average is:
max , ;
;
Design engineers tend to assign dimensional tolerances based on function and performance
(e.g.:
in joints).
Ideally,
specified
should tolerances
be greaterand
than
process
capability and
designfits
engineers
should
consider
the tolerance
relation between
process
capability
Process capbility
i n de x :
6
Tol e rance
Defect rate Defects per
(%)
million
Comments
0,333
0,667
±1σ
±2σ
3,1
,17
7E+01
4,5
,56
6E+00
3,1
,17
7E+05 Sortatio
ion
n req
equ
uired
ired
4,5
,56
6E+04 Sortatio
ion
n req
equ
uired
ired
1,000
±3σ
2,70E‐0
‐01
1
2,70E+03 Tole
lerrance = Cp
1,333
±4σ
6,3
,30
0E‐0
‐03
3
6,3
,30
0E+01 Defec
efectts are in
infr
freq
equ
uen
entt
1,667
±5σ
5,70E‐0
‐05
5
5,70E‐01
‐01 Defec
efectts are rare
2,000
±6σ
2,0
,00
0E‐0
‐07
7
2,0
,00
0E‐03
‐03 Vi
Virrtuall
lly
y no defec
efectts
Graphical representations of PCI
Two populations with nominal dimension and tolerances10±0,05 mm
σ
= 0,01279; Cp = 1,3028; Cpk = Cp, Cc = 0
µ = 10;
µ = 10,0013;
σ = 0,01859; Cp = 0,8964; Cpk = 0,8719, Cc = 0,02727
-3σ
-3σ
µµ
+3σ
+3σ
Statistical Process Control (SPC)
• Made
Made to asses
asses process
process in order
order to improv
improve
e quality
quality,, reduce
reduce variab
variabilit
ility
y and solve
solve
process problems.
• histograms,
Ther
here are
are sPareto
even
ven prin
pr
inc
cipal
ipalcheck
meth
method
ods
s and
anddefect
to
tool
ols
s concentration
used
sed in SPC
SPCdiagrams,
: contr
ontrol
ol scatter
char
harts
ts,,
charts,
sheets,
diagrams and cause-effect diagrams
Control charts: Plot over time of statistics computed from a measured of a process.
There
areoftwo
basic or
types:
for variables
andcollected
for attributes
(fraction
defects
number
of defects(from
in a measurements)
sample). Data are
and
organized using data collection sheets.
Control charts for variables represent how average of the sample (̅ , called x-bar
chart)
varies over time or the range (R, called R chart) to represent the variability of
the
process.
Data collection sheets
Data collection sheets should be simple and clear. Are used to collect information
about the product or process. Should show a correct interpretation of the studied
phenomenon. May be for:
• Quantity data: Register the quantity of defects.
defects. Data may be classified by
machine, worker, shift
• Measurable
Measurable data: Weight,
Weight, concentrat
concentration,
ion, dimension
dimension
• Defect location: includes a drawing
drawing of the product
product and data:
data: date, product
reference, department, comments. A graphical code helps to describe defects
and where are located
Defect location graphic example
Scratch
Dent
SPC tools: Control charts
To obtain x-bar charts, 20 samples (5 parts each) are measured. Average and range
of each group of 5 parts are calculated. Grand mean ̿ for the 20 values of ̅ is
calculated and is used as the center line for the x-bar chart. Average of the 20 ranges
is calculated and is used as center line of R chart. Upper and lower limits of charts
are calculated as standard deviation s of the sample or using the following table:
Sample Size
ize
̅ Chart
R
Chart
n
A2
D3
D4
3
4
1,023
0,729
0
0
2,574
2,282
5
0,577
0
2,114
6
0,483
0
2,004
7
8
0,419
0,373
0,076
0,136
1,924
1,864
9
0,337
0,184
1,816
10
0,308
0,223
1,777
Upper and lower limits for x-bar chart
̿̿
̿̿
Upper and lower
limits
for R bar chart
Statistical tools to asses SPC: control charts,
distribution, histogram, Pareto chart
Mean shift
Displaced
Data
Controlled dimension: 10±0,1 mm (20 samples of 5 measures)
̿ = 10,0014; σ = 0,019; Cp = 1,68; Cpk = 0,43, Cc = 0,35
Dimensioning and tolerancing.
Tolerance stack up and quality assurance
Two methodologies commonly used to calculate tolerances stack up are:
1. Wor
orst
st Ca
Case
se ((WC
WC):
): Worst case using tolerance limits. More interchangeability, but
cost.of Squar
2. higher
Root Sum
Squares
es (RSS
(RSS):
): Root of the sum of tolerances squares. Considering
that produced part’s measures will be near the mean value in SPC, requires
symmetrical tolerancing. Less interchangeability, but cheaper. Useful when several
components are included in the chain of dimensions
di mensions
Other methodologies can predict assembly quality from design (not only tolerance
stack up):
Six Sigma: Requires SPC, skilled labor, precise machinery and tooling. Helpful in
world class companies where manufacturing and design facilities are not always near
Mini
Mi
nimu
mum
m cost
cost to
tole
lera
ranc
nce
e allo
alloca
cati
tion
on:: Us
Uses
es opti
optimi
miza
zati
tion
on te
tech
chni
niqu
ques
es to assi
assign
gn
components tolerances that minimize their manufacturing cost.
Computer Aided tolerancing (CAT): Use to combine the previous two methodologies
in
a software
to peri
compute
stack
in ness
3Dssandand
components
in.
asse
as
semb
mbly
ly.. Runs
Ruable
ns expe
ex
rime
ment
nts
s tolerances
to cons
consid
ider
er rand
raup
ndom
omne
anfor
d all
inte
interc
rcha
hang
ngea
eabi
bilility
ty.
Calculates PCI and contribution of each part of the assembly to tolerances stack up
Dimensioning and tolerancing.
•
WC vs RSS
WC co
consid
nsiders
ers tthe
he e
extre
xtreme
me c
condi
ondition
tions.
s. RS
RSS
S is the squa
square
re rroot
oot
of the sum of the square of tolerances
•
•
⋯
Part
Parts
s mus
mustt be prod
produced
uced in a proc
process
ess unde
underr SPC to ap
apply
ply RSS
±0,1
±0,18
8 tol
toleran
erance
ce m
may
ay be used assu
assuming
ming that most of th
the
e par
parts
ts
are grouped near the mean. For those near the extremes,
sortation may be needed in assembly
Pin shifts the same with respect to hole in all directions?
+
‐
24
24
WC R
RS
SS
0,1
0,01
0,1
0,01
0,113
0,11
3 0,01
0,013
3
0 0,31
,313
3 0,18
,181
1
Unsymmetrical tolerancing and Mean shift
Hole diameter is expected to have a nominal value of 15 and tolerances ±0,05 (all in mm).
Dimensions from 14,95 to 15,05 are acceptable.
,
14,95,
, 15,0
15,05
5
, or 14,97, should also be accepted to dimension and tolerance the hole.
Correct?
A drill
drill of diameter 15 mm could be found easier than a 14,97 mm one, but ISO standard not always
recommends a centered dimensioning and tolerancing system for unions requiring fit, e.g.: Ø15
,
15
1 5
;
,
15
1 5,
H7/j6; Hole:
Pin:
Should we center nominal values to use CPI with centered mean (Cc=0)? Customized tooling?
What are the consequences of mean shift in PCI?
Use symmetric tolerancing as much as possible. Respect fit dimensions (because of its influence in
tooling), consider mean shift in those cases. PCI work for symmetrical tolerances
GD&T to help quality assurance in design
and cost reduction
Dimensioning and tolerancing. Quality assurance
in early stages of design
+
‐
WC RS
RSS
24
0
24
0
0,113
0,11
3 0,013
0,013
0,155
0,15
5 0,024
0,024
0,155
0,15
5 0,024
0,024
0 0,423
,423 0,247
,247
•
•
•
•
Wi
Wide
derr er
erro
rorr acce
accept
pted
ed to de
decr
crea
ease
se co
cost
st (if
(if ac
acce
cept
ptab
able
le)) be
beca
caus
use
e of ma
mate
teri
rial
al
condition (bonus tolerance)
Ther
There
e is no amb
ambiguit
iguity
y in part defi
definiti
nition
on
Flat faces and cylindrical surfaces axis are now properly related
Pin shif
shifts
ts th
the
e sam
same
e wit
with
h re
respec
spectt to hole in a
allll di
direct
rections
ions now
Statistical dimensioning and tolerancing
Different levels of definition and usefulness
Accepted statistical tolerancing with dimensional and GD&T.
GD&T. Designer must
be sure that manufacturer will make parts in a SPC process
Quality assurance in early stages of design
Controlled spread of
population respect mean
More than ± 4σ
nominal value near to
population mean
Comple
Comp
lete
tely
ly defi
define
ned
d st
stat
atis
isti
tica
call to
tole
lera
ranc
ncin
ing
g
with GD&T. Less sensibility to lack of
communication between designer and
manu
ma
nufa
fact
ctur
urer
er,, and
and also
also to labo
laborr rota
rotati
tion
on..
Precise definition for SPC
More complex assemblies
•
•
•
RS
RSS
S is of sp
spec
ecia
iall inte
intere
rest
st wh
when
en se
seve
vera
rall pa
part
rts
s mu
must
st as
asse
semb
mble
le prop
proper
erly
ly (tol
(toler
eran
ance
ces
s are
are
always added).
GD
GD&T
&T he
help
lps
s to de
defi
fine
ne pa
part
rts
s prop
proper
erly
ly.. Co
Cons
nsid
ider
er us
usua
uall lack
lack of co
comm
mmun
unic
icat
atio
ion
n be
betw
twee
een
n
design
desi
gner
er an
and
d ma
manu
nufa
fact
ctur
urer
er in mo
mode
dern
rn glob
global
al comp
compan
anie
ies.
s. Ou
Outs
tsou
ourc
rcin
ing
g ma
make
kes
s it mo
more
re
complicated
RSS and GD&T helps to lower co
costs
sts if design predicts assembly quality an
and
d SPC p
process
rocess is
used. Cost issues should be studied due to labor training needs, more expensive machinery
and tooling, well designed parts and processes require
requirements,
ments, reliable suppliers needs, etc.
Machines & tooling
Influence on Process Capability
€
€€€€
http://www.lagun.com.es/
High Performance Manufacturing (HPM)
Some figures
• Very high
high precisio
precision
n (toleran
(tolerances
ces betwee
between
n ±0,005
±0,005 mm and ±0,015
±0,015 mm
for parts of around 100 kg)
• Cutting speeds and feed 50% greater than other modern CNC
machines (150…400 m/min of cutting speed for steel, up to 60.000 rpm
o spindle spinning, up to 20 m/min of feed)
• Prec
Precis
isio
ion
n ±0,0
±0,005
05 mm and
and repe
repeti
titi
tive
vene
ness
ss ±0,0
±0,003
03 mm (60.
(60.00
000
0 $) to
±0,001 mm and repetitiveness ±0,001 mm (500.000 $)
• Accele
Accelerat
ration
ion up to 1,5 g
• Very high frequencie
frequencies
s due to rotating
rotating tools spinning
spinning speed
speed
• Able to machine
machine hardened
hardened steel (52…55
(52…55 HRC)
Tools performance with HPM
V = 20.000 rev/min x 10 mm = 628,3 m/min
f = 20.000 rev/min x 0,8 mm/rev = 16.000 mm/min
M = 1600 N x 5ꞏ10-3 m = 8 Nꞏm
P = 2094,4 rad/s x 8 Nꞏm = 16.8 kW
Images from: (http://www.coromant.sandvik.com/
(http://www.coromant.sandvik.com/ )
Example of real machining devices
• Expens
Expensive
ive toolin
tooling
g
• Needed to obtain
obtain precision
precision and repetitiveness in batch
batch or GT production
• Increase
Increase PCI and productivi
productivity
ty (auxiliary
(auxiliary time reduced)
reduced)
• Direct
Direct influen
influence
ce in quality
quality
http://www.pdqwh.com/
CAM strategies and knowledge to increase PC
Images from: (http://www.coromant.sandvik.com/
(http://www.coromant.sandvik.com/ )
CAM examples for High performance machining:
https://www.solidcam.com/en-us/
https://www
.solidcam.com/en-us/videos/cutting-vi
videos/cutting-videos/
deos/
Shaft manufacturing in a single mounting
Automatic tolerances analysis
with CAT software
Angle with
tolerance
Clearance
0.5 A
B
0.1 A
B
0.1
A
0.3 A B C
0.5 A B
C
3.0 A B C
C
0.25
A
0.25
A
0.3 A B C
0.5 A B
C
0.5 A
0.5 A B
0.5 A
C
0.5 A
B
0.25
A
B
Simulation
0.5 A
B
6
0.3 A B C
0.5 A B
C
3.0 A B C
0.25
A
0
.
0
0
0
0
.
0
0
5
0
.
0
0
H12 (diameter 24) = +0.021 -0.000
99.73%
0
.
0
1
0
0
.
0
1
5
0
.
0
2
0
+
0
.
0
2
1
Simulation
Ø24.016
Ø24.010
Ø24.005
Ø24.011
6
99.73%
0
.
0
0
0
0
.
0
0
5
0
.
0
0
H12 (diameter 24) = +0.021 -0.000
0
.
0
1
0
0
.
0
1
5
0
.
0
2
0
+
0
.
0
2
1
Manufacturing simulation
Manufacturing =
2
3
1...
Assembly simulation
Randomly selected parts
Results: statistics
100
ASSEMBLY1
Date: XX/XX/XXXX Time: XX:XX
Simulations Performed
Random Number Seed
Analysis Type
Est. Range Interval
1000
1
Pearson
99.73
Nominal
4.000
Lower Spec. Limit
Upper Spec. Limit
2.800
5.200
Cp
Cp k
1.200
0.950
Parameter
Value
95% Confidence Interval
Mean
Standard Deviation
Variance
Skewnes
Kurtosis
xxxxxxxx
xxxxxxxx
xxxxxxxx
xxxxxxxx
xxxxxxxx
xxxxxxxx
xxxxxxxx
xxxxxxxx
xxxxxxxx
xxxxxxxx
0
1
1
0
2.8
Observed distribution type is NORMAL
Estimated Below Spec.
Estimated Above Spec.
Total Out of Spec.
50
0.000 Pct
0.200 Pct
0.200 Pct
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xxxxxxxx
xxxxxxxx
xxxxxxxx
xxxxxxxx
4.0
5.2
In line inspection
Utillajes típicos de moldeo por
inyección. Geometría compleja
erosionada por medio de varias
secuencias de penetración.
Taguchi method for Quality Engineering
(Genichi Taguchi)
Off-line quality control
Product
Design
Process
Design
On-line quality control
Production
Process
diagnosis and
adjustment
System
System
Design
Design
Parameter
Design
Parameter
Design
Tolerance
Design
Tolerance
Design
Customer
relations
Customer
service
Feedback to
Process
prediction
and
correction
Process
measurement
and action
product
design
Robust design
Process or product parameters are defined to maintain stability under disturbances
that are difficult to be controlled (noise factors)
Kind of noise factors:
1.
2.
3.
Amon
mong uni
units
ts:: ra
rando
ndom
m inhere
inherent
nt var
variat
iation
ions
s due to raw mat
materi
erials
als,, mac
machin
hinery
ery or hum
human
an fac
factor
tors
s
variations
In
Inte
tern
rnal
al:: Internal of product or process. May be (a) time dependents: wear, raw materials damage,
materials fatigue or (b) operational errors like wrong cutting parameters, programming errors
Ex
Exte
tern
rnal
al:: Process external variations like climate conditions
Product with robust design
Aircraft flying in stormy or clear weather
Car that starts correctly in cold or warm weather
Emergencies of a hospital that conitnue working after an electric cutoff
Process with robust design
Turning operation producing good surface finish in a wide range of turning speeds
Plast
Pla
stic
ic injectio
injection
n molding
molding th
tha
at molds
molds a good
good par
part despite
despite room
oom temper
temperat
atur
ure
e and
humidity variation
Tolerance design
Wider tolerances than process limits are possible with more controlled and expensive process for
“tighter” tolerances or with less controlled and cheaper process for “looser” tolerances. Looser
tolerances
increase
yield
manufacturing
making worst
cheaper
process
(less
skilled labor,
machines and
tooling,
easier
setup). However:
fit, the
probably
worst
functioning
andcheaper
lasting,
more difficult due to less interchangeability, loss on safe operation conditions
According to Taguchi
Taguchi loss occurs when the functional characteristic of the product differs from its
nominal or target value. He defined quality as the loss a product costs to society from the time the
product is released for shipment. It includes costs to operate, failure to function, maintenan
maintenance
ce and
repair costs, client dissatisfaction, injures caused for poor design. Defective products detected,
repaired and reworked are considered manufacturing costs instead of a quality loss
The loss increases at an accelerating rate as the deviation grows:
Where:
L(x): Loss function
x: quality characteristic of interest (a dimension, for example)
k: proportionality constant
N: Nominal value
Taguchi loss function
Traditional
Traditional tolerances approach
Loss €
Taguchi loss function
Scrap or rework
cost
x1
Li
N
x2
Ls
http://tube.geogebra.org/student/m100653
x
Example
A company manufactures a shaft with diameter 100 ± 0,2 mm. The company has
studied its repair records and has discovered that if tolerance limits are exceeded
there is a (for
60 %
probability
of product
to beorrepaired
at a cost
of 100
€/product
theofcompany
if product
is inreturn
guarantee
to the client
beyond
the
guarantee period). Estimate Taguchi
Taguchi loss function constant k for that product:
± 0,2 mm tolerance limits are symmetric with respect to N, then (x-N)=0,2 (100,2100=-(99,8-100)=0,2). The expected loss for L(x) is E{L(x)} = 0,6 (100 €) + 0,4 (0 €)
= 60 €
60 = k (0,2)2; k = 60 / 0,04 = 1500 €
L(x) = 1500 (x – N)2
It tolerance is reduced to ± 0,1 mm with the same constant;
2
L(x) = 1500 (x – N) = 15 €
With half of the initial tolerance, repair cost drops by 100. However, production cost
raises (more expensive machines and tooling, more parts are refused, better skilled
labor)
Six Sigma
General goals: Improve customer satisfaction, high quality products and services,
General
redu
reduce
ced
d defe
defect
cts,
s, impr
improv
oved
ed proc
proces
ess
s capa
capabi
bilility
ty th
thro
roug
ugh
h redu
reduct
ctio
ion
n in proc
proces
ess
s
variations, continuous improvement and cost reduction
Implementation: Teams trained in the use of statistical and problem solving tools
as well as project management techniques to define, measure, analyze and make
improvements in the operations of the organization by eliminating defects and
variations in the process. Teams
Teams are empowered by management.
Problem solving approach: Called DMAIC: Define project goals, Measure the
process and asses current performance, Analyze the process to determine causes
of
defects and variations, Improve the process and Control the future process
performance
Measuring the Sigma level: Number of defects per million is determined and then
is converted to sigma level. Some indexes commonly used are : Defects per
million opportunities (DPMO) that is the ratio, multiplied by 1ꞏ106, between the
number of defects and the product of number of units and number of opportunities
Defective units per million
million (DUPM) that is the ratio, multiplied
of defect per unit. Defective
by 1ꞏ106, between number of defective units and number of units
