EML 4550: Engineering Design Methods
Tolerance Design
From “Tolerance Design: A Handbook for developing optimal specifications,” by
C.M. Creveling, Addison-Wesley, Chapter 11
Also
“Engineering Design,” by G.E. Dieter Chapter 12
EML4550 2007
1
Definitions
 Tolerance
 Geometric tolerance - range for a particular dimension
 General tolerance - acceptable range for a design variable
(dimension, roughness, viscosity, refractive index, etc.)
 Most techniques developed for tolerance design apply to
dimensions, but many can be generalized to any design
tolerance problem
 Tolerance design appeared with the Industrial Revolution as
the need for interchangeability arose.
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Definitions
 Geometric Dimensioning and Tolerancing (GD&T)
 Tolerance design geared towards ‘variance reduction’ as the
key to repeatable, low-cost manufacturing
 Converging views from East and West
 Taguchi method
 Application of sound statistical and mathematical methods in
the design process to reduce variance (design for quality)
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Tolerance Design: Process Flow Diagram
Customer Tolerances
Product Output
Response Tolerance
System and
Assembly Tolerances
Component Part
Tolerances
Manufacturing Process
Parameter Tolerances
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Customer Costs & Losses
Product Output Response
Process Capabilities
System and Assembly
Process Capabilities
Component Part
Process Capability
Manufacturing Process
Capabilities
Tolerances
 Tolerances need to be defined because we live in a
probabilistic world and 100% reproducibility in
manufacturing is not physically possible
 Tolerances are defined in a standard: ANSI Y14-5M-1982
(R1988) (American National Standards Institute-ANSI)
“The total amount by which a given dimension may vary, or
the difference between the limits”
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Different Approaches to Tolerancing
 Traditional methods in tolerance design
 Semi-empirical
 Experience
 Manufacturing process capabilities
 Computer-aided tolerance design
 Plug-in packages for CAD software (propagation of tolerance techniques
– “error analysis”)
 Statistical methods
 Monte Carlo simulation
 Sensitivity analysis
 Cost-based tolerance design
 Modern methods in tolerance design
 Taguchi approach
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Classical Tolerance Design Process
Select Process
Change Process
Collect Statistical Data
Under
Control?
N
Management Decision
Y
Y
Process
Capable?
Work on process
N
Change Specs
Live with it
Test 100%
Stop Production
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Classical Tolerance Design Process (Cont’d)
Specs
Being Met?
N
Recenter Process
Y
Continue Gathering Statistics
For continued process improvement,
conduct designed and controlled
experiments to further reduce variability
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Tolerances and Quality Engineering
 Taguchi:
“Tolerances are economically established operating windows of
functional variability for optimized control factor set points
to limit customer loss”




More general, not just dimensions
Economically-driven (trade off)
Control factors that are pre-defined (not any variable)
Limit, but not eliminate, customer losses
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Taguchi Approach
 Concept of off-line QC
 Incorporate QC and tolerancing before releasing the design to
production
 Iterative process as a final step prior to drawing release
 On-line QC
 Traditional approach of in-plant QC, ‘fix it’ after the fact or scrap
 Use on-line QC to maintain or improve quality of the designed
product (little or no improvement needed if ‘off-line’ QC was properly
implemented)
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The three phases in Tolerance Analysis
 Basically the standard approach for the design process
 Concept design: selection of technology platform, metrics to assess
relative merits, concept robustness (safety, environment,
commercial, reliability, etc.)
 Parameter design: optimization of concept, parameters to reduce
sensitivity to ‘noise’ (uncontrollable parameters)
 Tolerance design: Balancing of customer loss function with
production cost, ability to determine and limit the variability around
the ‘target’ set points (as defined in parameter design).
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Taguchi’s Approach to Tolerancing
Input from the ‘voice of the customer’
Select proper quality-loss function for the design evaluation
Select the customer tolerance values for the Quality Loss Function:
Ao ($ lost due to off-target value) and Do (measurement of
Off-target performance in engineering terms)
Determine the cost to the business to adjust the off-target
Performance back to acceptable range during manufacturing: A
Calculate the manufacturing tolerance: D based on Taguchi’s Equation:
D  Do A / Ao
“My” acceptable variability = “Their (customer’s)” acceptable variability x square root
of the ratio between “My” cost to stay within production tolerance / “Their” loss if my
product is out of tolerance
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Traditional Tolerance Curve
Factories would accept or reject product
based on a simple on/off model (step function)
Assumption that customers will behave the same
way is WRONG
Equally bad product
Equally good product
m-Do
m
target
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Equally bad product
m+Do
Customer Tolerance
 Customer tolerance is not a simple step function
 Customer tolerance Do corresponds to the point in which a
significant fraction of customers will take some type of
action (e.g., 50% of customers would complain)
“Thermostat” example
100
% of people
complaining
50
0
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70F
75F
80F
Customer Loss Function
 Quadratic approximation to the customer loss function
L(y)  k(y  m)2




L is the loss function
k is the quality-loss coefficient
y is the performance variable
m is the target performance
L is the economic loss to my customer if my product deviates “y” from its rated value “m”
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Quality Loss Coefficient
 The functional limits (m + Do) and (m - Do) represent the
deviations from the target in which about 50% of the
customers would complain (significant economic loss)
 This is essentially a definition of product ‘failure’. The
economic loss to the customer associated with product
failure is Ao (e.g., losses due to lack of access to product
plus cost to repair, generally in terms of $)
 Therefore L(y-=m-Do) = L(y+=m+Do)=Ao
k
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Ao
2
Do
Customer Loss Functions
L(y)
 The nominal-the-best case
L( y ) 
2


y

m
2
Ao
Ao
Do
m-Do
 The smaller-the-better case
L( y) 
Ao
Do2
y
Ao
y 
2
y
L(y)
Do2
Do
Ao
y2
 Asymmetric cases
2
L  k   y  m  if
2
L  k   y  m  if
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m+Do
L(y)
 The larger-the-better case
L( y )  A o
m
y
Do
L(y)
ym
Ao
ym
y
m-Do
m
m+Do
Taguchi Tolerancing Equations
 Concept of Taguchi ‘safety factor’ in tolerancing
 What are the maladies for which we need to build a safety
factor?
 Customer dissatisfaction due to quality problems and customer
financial losses (long-term impact to reputation)
 Higher manufacturing costs due to re-work and scrap
 Define a tolerance level as seen by the customer (losses)
and a tolerance level as seen by the manufacturing process
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Taguchi’s Loss Function
Do
Di
Losses
Ao
Financial incentive
Since A<Ao
Ai
yo yi=m-Di Target (m)
customer tolerance
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manufacturing tolerance
Note:
Do-Di=range of safety
Do/Di=safety factor
Safety Factor
 For a standard quadratic loss
function
Ao
2
2




L( y )  2 y  m  k y  m
Do
 Deviation from target
 Loss associated with deviation
Di2  y  m2
L( yi )  Ai 
Ao
Do2
Di2
Ai ≤ Ao: manufacturing-allowable loss
should be smaller than the customer loss
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Safety Factor
 At what level is the company willing to ‘act’ to avoid
customer losses by ‘fixing’ the product back to the target
value before releasing it?
Derived from statistical considerations, sub-o
relates to customer (loss function, and
maximum deviation), sub-i relates to
manufacturer, cost to re-work and maximum
manufacturing tolerance
D o2
Ao

2
Ai
Di
 Economic safety factor
D
S o 
Di
 In general notation:
Ao
S
A
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Do2

2
Di
Ao
Ai
Safety Factor
S=SQRT[(average loss to (customer) in $ when a product
characteristic exceeds customer tolerance limits)/(average loss to
(manufacturer) in $ when a product characteristics exceeds
manufacturing tolerance limits)]
The Taguchi Approach relates customer tolerances to
engineering tolerances
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Example
 A company makes a power supply. The nominal (target)
value for the supply voltage is 115V. We know the customer
incurs a loss of $200 (Ao, due to damaging to instrument,
loss of productivity, recall, etc..) when the voltage exceeds
135V (135-115=20=Do, deviation from nominal). The
production department has determined that it costs $5 to
re-work (adding current-limiting resistor, etc..) a power
supply that is off-target back to the nominal value.
 What should the manufacturing tolerance be and what is
the economic safety factor?
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Example
L( y)  ky  m 
2
2


y

m
2
Ao
Do
A o  $200
D0  20V
Ao
$200
2
k 2  2

0
.
5
($
/
V
)
2
Do 20 Volts
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Example
 The manufacturing tolerance is:
D  Do
A
5
 20
 3.16V  3V
Ao
200
 The safety factor is:
Ao
200
S

 6.32
A
5
 If the assembly line detects a power supply with voltage
lower than 112V (115-3) or higher than 118V (115+3) it is
economical to pull it off and repair it
 The difference between the customer loss and the
manufacturing cost is relatively large (200/5=40)  smaller
tolerance is permissible sqrt(Ao/A)=sqrt(40)=6.32~20/3
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Example (alternative interpretation)
Ao
200
2
2
L( y )  k  y  m   2  y  m  
(
y

m
)
Do
(20) 2
2
L( y)  0.5( y  m) 2
The manufacturing tolerance can be considered as a deviation
away from the nominal value m  Di=y-m
The cost to modify the manufacturing process can be
considered as the loss function  $5
5  0.5( y  m)2  y  m  3.16  3
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Average Quality Loss
 The average quality loss, Q, from a total of n units from a
specific process can be given by (derived in the next slide)

1
k
Q  L( y1 )  L( y2 )      L( yn )  ( y1  m) 2  ( y2  m) 2      ( yn  m) 2
n
n
n 1 2 

 k (   m ) 2 
 
n


L(y)
1 n
1 n
2
2
where    yi and  
( yi   )

n i 1
n  1 i 1
Ao

Q  k (   m)  
2
2
Deviation of the average
value of y from the target 
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 when n is large.
m-Do
Mean squared deviation of y
value away from the target 
m
 
m+Do

y
Average Quality Loss
Q 
k
n
k

n


1
L( y1 )  L( y 2 )      L( y n )  k ( y1  m ) 2  ( y 2  m ) 2      ( y n  m ) 2
n
n
( y
1
2
 2m y1  m 2 )  ( y 2
2
 2m y2  m 2 )      ( y n
2
 2m yn  m 2 )

n
k  n
 n

2
2
y

2
m
y

n
m

y i2  2m ( n )  n m2 


i
 i


n  i 1
i 1
 i 1


n
n
1
1



 k   y i2  2m  m 2   k   2   2 
y i2  2m  m 2 

n i 1
 n i 1



1

 k (   m ) 2 
n

1

 k (   m ) 2 
n

1

 k (   m ) 2 
n

n
y
i 1
2
i
n
y
i 1
2
i
 y
n
i 1
2
i
1


    k (   m ) 2 
n


2

1
n
n
 (2 ) y
i 1
i

1
n
n

i 1

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2

y
i 1
2
i

 2 2   2 




1


 2 y i   2   k (   m ) 2 
n


n 1 2

 k (   m ) 2 
 
n


n
n
1
1
2
wh ere  
y
an
d


( yi   ) 2


i
n i 1
n  1 i 1
Q  k (   m) 2  
2
n
wh en n is large.
n
 y
i 1
i
2
  


Example
 From the previous example, assume the power supplies manufactured
have their mean value centered around the target (=m) so its loss of
quality will be dominated by the standard deviation term: Q=k2
 If the variance of the power supplies =20 volts, determine the quality
loss due to the manufacturing deviation: Q=(0.5)(20)2=$200
 If a resistor is added to the unit, it has been demonstrated that it can
reduced the variance to 15 volts. The cost of the additional process is
$50. Show that whether it is worthwhile?
Q=(0.5)(15)2=$112.5a net decrease of loss 200-112.5=$87.5
with an investment of $50, it seems to be a bargain.
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Conclusions
 The Taguchi Approach can be used at the system level to
interact with outside customers, but it can also be
implemented within a company
 Each successive step in the manufacturing process can be
seen as a ‘customer’ of the previous step (manufacturing,
purchased part, service, etc.)
 When implemented on a company-wide basis the Taguchi
Approach can lead to a quasi-optimal distribution of
tolerances among the different components that go into a
final product.
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