robust design and optimization

8/2/2019 Robust Design and Optimization

http://slidepdf.com/reader/full/robust-design-and-optimization 1/14

Robust Design and Two-StepOptimization

Lihui Shi



Outline

Introduction of Taguchi

Basic Concepts and Tools in Robust Design

Signal-to-Noise RatioStatic 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 inisolation from the mainstream of

Western statistics.

Unknown outside of Japan. Introduced

into US in 1980. Taguchi’s method. Controversial among statisticians, but manyconcepts 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 thecontrol factors to make the output on target and

the process robust again noise factors?



Quadratic Loss Function

Y: output

t: target value

Objective: Minimize the average loss

=(d,a), design parameters.

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

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



Signal-to-Noise (SN) Ratio

SN ratio h is defined as

Question 1: Why use the log transformation?

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

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

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

2 2

1010log / h



Various Factors

Among many applications, Taguchi has empiricallyfound that the two stage optimization procedureinvolving the SN ratio indeed gives the parameterlevel combination where the standard deviation isminimum while keeping the mean on target.

Control factors d: a significant effect on SN ratio.

Adjustment factor (scaling factor) a: significanteffect on mean, but no effect on SN ratio.

Other factors: have no effect on SN ratio andmean.

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 istrivial, or absent.

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

It is a constrained optimization problem.

2Minimize ( )

Subject to ( )=t



Two-Step Optimization

It is equivalent to: Maximize h, and keep themean on target.

Use the two-step optimization method:

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

2. Adjust the mean on target by using a:

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

Maximize ( )d

h d

*( , )a d t



Dynamic Robust Design

Also called: robust design in signal-responsesystem.

A signal factor is selected from the set of controlfactors, and is changed continuously dependingon the customer’s intent, to meet hisrequirements.

Aim: make the signal-response relationshipinsensitive 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 andresponse:

SN ratio is given by

Nonlinear:

Performance 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



Optimization

Objective: Minimize the PM.

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 theprobability density function.

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

t Z

H t t t

Lt t t

PM L Z M t dW t

h

h



Two-Step Optimization

A special form of f(Z,M):

Optimization:

It is equivalent to the two-step optimization:

1. Choose Z to maximize h.

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

Maximize ( , )d

h d X

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

1

2

Maximize ( , )

Subject to ( ) (t ) /M

d

L H

h d X

d f



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 parameterdesign of multiple-target systems, Technometrics, 44

(2002), 338-346.Le¡äon, R., Shoemaker, A. C. and Kacker, R. N.,Performance measures independent of adjustment: Anexplanation and extension of Taguchi’s signal-to-noiseratios (with discussion)," Technometrics, 29, (1987), 253-

285.

robust design and optimization

Documents