Transcript DEMING`S FOURTEEN POINTS

Robust Design and Two-Step
Optimization
Lihui Shi
Outline
Introduction of Taguchi
Basic Concepts and Tools in Robust Design
Signal-to-Noise Ratio
Static Robust Design & Two-Step Optimization
Dynamic Robust Design & Two-Step Optimization
Reference
Introduction
Genichi Taguchi (田口 玄一)
From 1950s developed a methodology
to improve the quality of products.
Much of his work was carried out in
isolation from the mainstream of
Western statistics.
Unknown outside of Japan. Introduced
into US in 1980. Taguchi’s method.
Controversial among statisticians, but many
concepts introduced by him have been accepted.
Basic Block Diagram,
Concepts and Tools
Quality characteristics
Quadratic Loss function
Design of Experiments
(DOE)
Signal-to-noise ratio
(SN ratio)
Orthogonal arrays
Linear graph
Basic question: How to choose the levels of the
control factors to make the output on target and
the process robust again noise factors?
Quadratic Loss Function
Q( y, t )  K ( y  t ) 2
Y: output
t: target value
Objective: Minimize the average loss
R( )  E N Q (Y , t )  K [ 2  ( y  t ) 2 ]
 =(d,a), design parameters.
Signal-to-Noise (SN) Ratio
SN ratio h is defined as
h  10log10   2 /  2 
Question 1: Why use the log transformation?
Box (1988), (1987 discussion): The standard
deviation will be independent of the mean, so the
design factors will separate into some that affect
the variation and some others that affect the
mean without changing the variation.
Question 2: Why use the ratio instead of the
standard deviation?
Phadke: Frequently, as the mean decreases the
standard deviation also decreases and vice versa.
Various Factors
Among many applications, Taguchi has empirically
found that the two stage optimization procedure
involving the SN ratio indeed gives the parameter
level combination where the standard deviation is
minimum while keeping the mean on target.
 Control factors d: a significant effect on SN ratio.
 Adjustment factor (scaling factor) a: significant
effect on mean, but no effect on SN ratio.
 Other factors: have no effect on SN ratio and
mean.
d and a all both set of factors, and use  =(d,a),
design parameters.
Static Robust Design
When the target is fixed, then the signal factor is
trivial, or absent.
Objective: Minimize the variance, and keep the
mean on target.
Minimize  ( )
2

Subject to  ( )=t
It is a constrained optimization problem.
Two-Step Optimization
It is equivalent to: Maximize h, and keep the
mean on target.
Use the two-step optimization method:
1. Choose d to maximize h (no worry about mean):
Maximize h( d )
d
2. Adjust the mean on target by using a:
 (a , d )  t
*
It is an unconstrained optimization problem.
Much easier now!!!
Dynamic Robust Design
Also called: robust design in signal-response
system.
A signal factor is selected from the set of control
factors, and is changed continuously depending
on the customer’s intent, to meet his
requirements.
Aim: make the signal-response relationship
insensitive to the noise variation, by choosing the
appropriate levels of the control factors.
Two types of systems:
1. measurement system
2. multiple-target system
Multiple Target System
Linear relationship between the signal and
response:
Y  M 
SN ratio is given by

h  SN  log  / 
2
Nonlinear:
Y  f ( Z , M )   , E( )=0, Var( )=V(Z , M )
Performance measure
L( Z , M , t )  ( f ( Z , M )  t )  V ( Z , M )
2
Optimization
Objective: Minimize the PM.
t2
Minimize PM   L( Z , M * , t )dW (t )
t1
Z
Subject to max h(Z,t)  M H
t[ t1,t 2]
min h (Z,t)  M L
t[ t1,t 2]
The system requires that the value of M be
between ML and MH.
Let (t1,t2) be the range of t, and W(t) be the
probability density function.
h(Z,t) is the solution of M from f(Z,M)=t.
Two-Step Optimization
A special form of f(Z,M):
Optimization:
f (Z , M )  f ( (Z )M )
Maximize h(d , X )
d
Subject to  (d )   L  f (t 2 ) /M H
1
It is equivalent to the two-step optimization:
1. Choose Z to maximize h.
Maximize h( d , X )
d
2. Adjust  to the desired range, by using the
adjustment factor a .
Reference
Box, G. E. P., Signal-to-noise ratios, performance criteria,
and transformations (with discussion), Technometrics, 30
(1988), 1-40.
Nair, V. N., Taguchi’s parameter design: A panel discussion,
Technometrics, 34 (1992), 127-161.
Phadke, M. S., Quality engineering using robust design,
(1989), Prentice-Hall, New Jersey.
Roshan Joseph, V. and C.F.Jeff Wu, Robust parameter
design of multiple-target systems, Technometrics, 44
(2002), 338-346.
Le¡äon, R., Shoemaker, A. C. and Kacker, R. N.,
Performance measures independent of adjustment: An
explanation and extension of Taguchi’s signal-to-noise
ratios (with discussion)," Technometrics, 29, (1987), 253285.