Robust Design and Two-Step Robust Design and Two-Step OptimizationOptimization

Lihui ShiLihui Shi

OutlineOutline

Introduction of TaguchiIntroduction of Taguchi

Basic Concepts and Tools in Robust DesignBasic Concepts and Tools in Robust Design

Signal-to-Noise RatioSignal-to-Noise Ratio

Static Robust Design & Two-Step OptimizationStatic Robust Design & Two-Step Optimization

Dynamic Robust Design & Two-Step OptimizationDynamic Robust Design & Two-Step Optimization

ReferenceReference

IntroductionIntroductionGenichi TaguchiGenichi Taguchi ( ( 田口 玄一田口 玄一 ) )

From 1950s developed a methodology From 1950s developed a methodology

to improve the quality of products. to improve the quality of products.

Much of his work was carried out in Much of his work was carried out in

isolation from the mainstream of isolation from the mainstream of

Western statistics. Western statistics.

Unknown outside of Japan. Introduced into Unknown outside of Japan. Introduced into US in 1980. Taguchi’s method.US in 1980. Taguchi’s method.

Controversial among statisticians, but many Controversial among statisticians, but many concepts introduced by him have been accepted.concepts introduced by him have been accepted.

Basic Block Diagram, Basic Block Diagram, Concepts and ToolsConcepts and Tools

Quality characteristics Quality characteristics

Quadratic Loss functionQuadratic Loss function

Design of ExperimentsDesign of Experiments

((DOEDOE))

Signal-to-noise ratioSignal-to-noise ratio

((SN ratioSN ratio))

Orthogonal arraysOrthogonal arrays

Linear graphLinear graph

Basic question: How to choose the levels of the control Basic question: How to choose the levels of the control factors to make the output on target and the process factors to make the output on target and the process robust again noise factors?robust again noise factors?

Quadratic Loss FunctionQuadratic Loss Function

Y: outputY: output

t: target valuet: target value

Objective: Minimize the average lossObjective: Minimize the average loss

=(d,a), design parameters. =(d,a), design parameters.

2 2( ) ( , ) [ ( ) ]NR E Q Y t K y t

2( , ) ( )Q y t K y t

Signal-to-Noise (SN) RatioSignal-to-Noise (SN) RatioSN ratio h is defined asSN ratio h is defined as

Question 1Question 1: Why use the log transformation?: Why use the log transformation?

Box (1988), (1987 discussion): The standard Box (1988), (1987 discussion): The standard deviation will be independent of the mean, so the deviation will be independent of the mean, so the design factors will separate into some that affect design factors will separate into some that affect the variation and some others that affect the mean the variation and some others that affect the mean without changing the variation.without changing the variation.

Question 2Question 2: Why use the ratio instead of the : Why use the ratio instead of the standard deviation? standard deviation?

Phadke: Frequently, as the mean decreases the Phadke: Frequently, as the mean decreases the standard deviation also decreases and vice versa.standard deviation also decreases and vice versa.

2 21010log /h

Various FactorsVarious Factors

Among many applications, Taguchi has empirically Among many applications, Taguchi has empirically found that the two stage optimization procedure found that the two stage optimization procedure involving the SN ratio indeed gives the parameter involving the SN ratio indeed gives the parameter level combination where the standard deviation is level combination where the standard deviation is minimum while keeping the mean on target.minimum while keeping the mean on target.

Control factors dControl factors d: a significant effect on SN ratio. : a significant effect on SN ratio. Adjustment factor (scaling factor) aAdjustment factor (scaling factor) a: significant : significant

effect on mean, but no effect on SN ratio.effect on mean, but no effect on SN ratio. Other factorsOther factors: have no effect on SN ratio and mean.: have no effect on SN ratio and mean.

d and a all both set of factors, and use =(d,a), d and a all both set of factors, and use =(d,a), design parameters.design parameters.

Static Robust DesignStatic Robust DesignWhen the target is fixed, then the signal factor is When the target is fixed, then the signal factor is trivial, or absent.trivial, or absent.

Objective: Minimize the variance, and keep the Objective: Minimize the variance, and keep the mean on target.mean on target.

It is a It is a constrainedconstrained optimization problem. optimization problem.

2Minimize ( )

Subject to ( )=t

Two-Step OptimizationTwo-Step OptimizationIt is equivalent to: Maximize h, and keep the mean It is equivalent to: Maximize h, and keep the mean on target.on target.

Use the two-step optimization method:Use the two-step optimization method:

1. Choose d to maximize h (no worry about mean):1. Choose d to maximize h (no worry about mean):

2. Adjust the mean on target by using a:2. Adjust the mean on target by using a:

It is an It is an unconstrainedunconstrained optimization problem. Much optimization problem. Much easier now!!!easier now!!!

Maximize ( )d

h d

*( , )a d t

Dynamic Robust DesignDynamic Robust DesignAlso called: robust design in signal-response system.Also called: robust design in signal-response system.

A signal factor is selected from the set of control A signal factor is selected from the set of control factors, and is changed continuously depending on factors, and is changed continuously depending on the customer’s intent, to meet his requirements.the customer’s intent, to meet his requirements.

– Aim: make the signal-response relationship Aim: make the signal-response relationship insensitive to the noise variation, by choosing the insensitive to the noise variation, by choosing the appropriate levels of the control factors.appropriate levels of the control factors.

Two types of systems: Two types of systems:

1. measurement system1. measurement system

2. multiple-target system2. multiple-target system

Multiple Target SystemMultiple Target System

Linear relationship between the signal and Linear relationship between the signal and response:response:

SN ratio is given bySN ratio is given by

Nonlinear: Nonlinear:

Performance measurePerformance measure

Y M

2log /h SN

( , ) , E( )=0, Var( )=V( , )Y f Z M Z M

2( , , ) ( ( , ) ) ( , )L Z M t f Z M t V Z M

OptimizationOptimizationObjective: Minimize the PM.Objective: Minimize the PM.

The system requires that the value of M be between ML and MH.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 Let (t1,t2) be the range of t, and W(t) be the probability density function.function.

h(Z,t) is the solution of M from f(Z,M)=t.h(Z,t) is the solution of M from f(Z,M)=t.

2 *

1

[ 1, 2]

[ 1, 2]

Minimize ( , , ) ( )

Subject to max (Z,t) M

min (Z,t) M

t

tZ

Ht t t

Lt t t

PM L Z M t dW t

h

h

Two-Step OptimizationTwo-Step OptimizationA special form of f(Z,M): A special form of f(Z,M):

Optimization:Optimization:

It is equivalent to the two-step optimization:It is equivalent to the two-step optimization:

1. Choose Z to maximize h.1. Choose Z to maximize h.

2. Adjust to the desired range, by using the 2. Adjust to the desired range, by using the adjustment factor a .adjustment factor a .

Maximize ( , )d

h d X

( , ) ( ( ) )f Z M f Z M

12

Maximize ( , )

Subject to ( ) (t ) /M

d

L H

h d X

d f

Reference Reference Box, G. E. P., Signal-to-noise ratios, performance criteria, and Box, G. E. P., Signal-to-noise ratios, performance criteria, and transformations (with discussion), Technometrics, 30 (1988), transformations (with discussion), Technometrics, 30 (1988), 1-40.1-40.

Nair, V. N., Taguchi’s parameter design: A panel discussion, Nair, V. N., Taguchi’s parameter design: A panel discussion, Technometrics, 34 (1992), 127-161.Technometrics, 34 (1992), 127-161.

Phadke, M. S., Quality engineering using robust design, (1989), Phadke, M. S., Quality engineering using robust design, (1989), Prentice-Hall, New Jersey.Prentice-Hall, New Jersey.

Roshan Joseph, V. and C.F.Jeff Wu, Robust parameter design of Roshan Joseph, V. and C.F.Jeff Wu, Robust parameter design of multiple-target systems, Technometrics, 44 (2002), 338-346.multiple-target systems, Technometrics, 44 (2002), 338-346.

Le¡äon, R., Shoemaker, A. C. and Kacker, R. N., Performance Le¡äon, R., Shoemaker, A. C. and Kacker, R. N., Performance measures independent of adjustment: An explanation and measures independent of adjustment: An explanation and extension of Taguchi’s signal-to-noise ratios (with discussion)," extension of Taguchi’s signal-to-noise ratios (with discussion)," Technometrics, 29, (1987), 253-285.Technometrics, 29, (1987), 253-285.