2.810 Quality and Variation

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2.810 Quality and Variation

• Part Tolerance
• Process Variation
• Taguchi Quality Loss Function
• Random Variables and how variation grows with
  size and complexity
• Quality Control

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References

• Kalpakjian pp 982-991 (Control Charts)
• Robust Quality by Genichi Taguchi and Don
  Clausing
• A Brief Intro to Designed Experiments
• Taken from Quality Engineering using Robust
  Design by Madhav S. Phadke, Prentice Hall, 1989
• 5 homeworks due Nov 13

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Interchangeable PartsGo, No-Go Part Tolerance
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Product specifications are given as upper and
lower limits, for example the dimensional
tolerance 0.005 in.
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Process VariationProcess measurement reveals a
distribution in output values.
Discrete probability distribution based upon
measurements
Continuous Normal distribution
In general if the randomness is due to many
different factors, the distribution will tend
toward a normal distribution. (Central Limit
Theorem)
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Tolerance is the specification given on the part
drawing, and variation is the variability in the
manufacturing process. This figure confuses the
two by showing the process capabilities in terms
of tolerance. Never the less, we can see that the
general variability (expressed as tolerance over
part dimension) one gets from conventional
manufacturing processes is on the order of
to
Homework problem can you come up with examples
of products that have requirements that exceed
these capabilities? If so then what?
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We can be much more specific about process
capability by measuring the process variability
and comparing it directly to the required
tolerance. Common measures are called Process
Capability Indices (PCIs), such as,
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Case 1 In this case the out of specification
parts are 4.2 0.4 4.6 What are the PCIs?
Lower Specification Limit
Upper Specification Limit
Target
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Case 2 However, in general the mean and the
target do not have to line up. What are the
PCIs? How many parts are out of spec?
Lower Specification Limit
Upper Specification Limit
Target
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Comparison

• Case 1
• Cp 4s/6s 2/3
• Cpk
• Min(2s/3s,2s/3s)2/3
• Out of Spec 4.6

• Case 2
• Cp 4s/6s 2/3
• Cpk
• Min(1s/3s,3s/3s)1/3
• Out of Spec 16.1

Note the out of Spec percentages are off
slightly due to round off errors
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Why the two different distributions at Sony?
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Taguchi Quality Loss Function
QL k d2
Quality Loss
Deviation, d
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Homework Problem

• Estimate a reasonable factory tolerance if the
  Quality Loss () for a failure in the field is
  100 times the cost of fixing a failure in the
  factory. Say the observed field tolerance level
  that leads to failure is dfield.

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Random variables and how variation grows with
size and complexity

• Random variable basics
• Tolerance stack up
• Product complexity
• Mfg System complexity

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If the dimension X is a random variable, the
mean is given by m E(X) (1) and the
variation is given by Var(x) E(x -
m)2 (2) both of these can be obtained from the
probability density function p(x). For a
discrete pdf, the expectation operation
is (3)
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Properties of the Expectation 1. If Y aX
b a, b are constants, E(Y) aE(X)
b (4) 2. If X1,Xn are random variables,
E(X1 Xn) E(X1) E(Xn) (5)
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Properties of the Variance 1. For a and b
constants Var(aX b) a2Var(X) (6) 2. If
X1,..Xn are independent random
variables Var(X1 Xn) Var(X1) Var(X2)
Var(Xn) (7)
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If X1 and X2 are random variables and not
necessarily independent, then Var(X1 X2)
Var(X1) Var(X2) 2Cov(X1Y) (8) this can
be written using the standard deviation s, and
the correlation r as
(9) where L X1 X2
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If X1 and X2 are correlated (r 1),
then (14) for X1 X2
X0 (15) for N (16) or (17)

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Now, if X1 and X2 are uncorrelated (r 0) we get
the result as in eqn (7) or, (10)
and for N (11) If X1X Xo
(12) Or (13)
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Complexity and Variation

• As the number of variables grow so does the
  variation in the system
• This leads to more complicated systems may be
  more likely to fail

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Homework Consider the final dimension and
variation of a stack of n blocks.

• 1, 2 n
• If USL LSL D, s s, and Cp 1
• How many parts are out of compliance?
• Now USL-LSLD, s10s, what is Cp? How many parts
  are out of spec?
• Repeat a) with s100s
• Assume that m target.

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• Homework Problem Experience shows that when
  composites are cured by autoclave processing on
  one sided tools the variation in thickness is
  about 7. After careful measurements of the
  prepreg thickness it is determined that their
  variation is about 7. What can you tell about
  the source of variation?

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Complexity and Reliabilityref. Augustines Laws
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Quality and System DesignData from D. Cochran
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Quality Control
Disturbances, d temperature, humidity,
vibrations, dust, sunlight
Inputs I Matl, Energy, Info
Machine M
Outputs, X
Operator inputs,u initial settings, feedback,
action?
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Who controls what?
Physical Plant, etc

• X f (M, I, u, d)

Equipment Purchase
Operator, Real Time Control
So who is in charge of quality?
Q.C., Utilities, etc
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How do you know there is a quality problem?

• Detection
• Measurement
• Source Identification
• Action
• Goal should be prevention

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Detection

• Make problems obvious
• Poke yoke at the process level
• Clear flow paths and responsibility
• Andon board
• Simplify the system
• Stop operations to attend to quality problems
• Stop line
• Direct attention to problem
• Involve Team

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Measurement

• Statistical Process Control

Upper Control Limit
Centerline
Average value x
Lower Control Limit
Sampling period
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Statistical Process Control Issues

• Sampling Period
• Establish Limits
• Sensitivity to Change

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Source Identification Ishikawa Cause and Effect
Diagram
Man
Machine
Effect
Material
Method
Finding the cause of a disturbance is the most
difficult part of quality control. There are only
aids to help you with this problem solving
exercise like the Ishikawa Diagrams which helps
you cover all categories, and the 5 Whys which
helps you go to the root cause.
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Truck front suspension assemblyProblem
warranty rates excessive
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Setting the best initial parameters

• Tables and Handbooks
• E.g. Feeds and speeds
• Models
• E.g. Moldflow for injection molding
• Designed Experiments
• E.g. Orthogonal Arrays

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Designed Experiments

• Temp T (3 settings)
• Pressure P (3 settings)
• Time t (3 values)
• Cleaning Methods K (3 types)
• How Many Experiments?
• One at a time gives 34 81

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But what if we varied all of the factors at once?

• Our strategy would be to measure one of the
  factors, say temperature, while randomizing the
  other factors. For example measure T2 with all
  combinations of the other factors e.g. (P,t,K)
  (123), (231), (312).

Notice that all levels are obtained for each
factor.
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Orthogonal Array for 4 factors at 3 levels.
Only 9 experiments are needed
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Homework

• Can you design an orthogonal array for 3 factors
  at 2 levels?

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Summary the best ways to reduce variation

• Simplify design
• Simplify the manufacturing system
• Plan on variation and put in place a system to
  address it

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Aircraft engine case study
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Engine Data
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Scheduled build times Vs part count
Scheduled build times
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Engine Delivery Late Times
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Late times compared to scheduled times
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Reasons for delay at site A
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Reasons for delay at site B (Guesses)
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Reasons for delay at site A (data)
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Engines shipped over a 3 month period at aircraft
engine factory B
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Engines shipped over a 3 month period at aircraft
engine factory C