NPTEL Taguchi Methods

Copyright Tapan Bagchi 1

Quality Engineering and Taguchi Methods


“Robust” Chocolate Bars are better!

Ambient Temperature

PlasticityRobust performance

Poor performance


Taguchi Methods

(or Quality Engineering

or Robust Design)

Focus is on reducing variability of response

to maximize robustness, generally achieved

through Orthogonal Array Experiments


The Genesis of DOE

Sir Ronald Alymer Fisher

(1890-1962) was the pioneer

of DOE. He was responsible

for statistics and data analysis

at the Rothamsted Agricultural

Experiment Station in London,

England. Fisher developed

and was the first to use

ANOVA in the statistical

analysis of experimental data.


Historical Perspective

George E. P. Box (born 1919)

was a student of R A Fisher. He

made several advances to

Fisher’s work in DOE theory

and statistics. The founding

chair of the University of

Wisconsin’s Department of

Statistics, Box was appointed

the R. A. Fisher Professor of

Statistics at UW in 1971.


Objective of this Lecture:

● To explore the basic ideas of two-level factorial design of

experiments (DOE) and the connection of QE to statistical

process control (SPC)

Key Points:

QE Overview

● DOE can help uncover significant variables and

interactions among variables

● SPC can help uncover process shifts

● Quality engineering tools help the investigator to

discover a path for process improvement


Typical QE Applications

In manufacturing - improve performance of a

manufacturing process

In process development - improve yields, reduce

variability and cost.

In design - evaluation and comparison of basic

configurations, materials, and parameters

The method is called Taguchi Methods.

The key tool is DOE.


Taguchi Methods


C R Rao

You’d know Rao from his

Cramer-Rao Inequality. Rao is

recognized worldwide as a

pioneer of modern multivariate

theory and as one of the world's

top statisticians, with

distinctions as a mathematician,

researcher, scientist, and

teacher. Taught Taguchi.

Author of 14 books and over

300 papers.


Genichi Taguchi

An engineer who developed

an approach (now called

Taguchi Methods) involving

statistically planned

experiments to reduce

variation in quality. Learned

DOE from Professor Rao.

In 1960’s he applied his

learning in Japan.

In 1980’s he introduced his

ideas to US at AT&T.


What are Taguchi’s Contributions?

Quality Engineering Philosophy—Targets

and Loss functions

Methodology—System, Parameter,

Tolerance design steps

Experiment Design—use of Orthogonal

arrays

Analysis—use Signal-to-Noise (S/N ratios)


61

65

8777

79

75

67

60

D

O

B

standard

orderD B O Avg Response

1 - - - 67

2 + - - 79

3 - + - 61

4 + + - 75

5 - - + 65

6 + - + 60

7 - + + 77

8 + + + 87

FACTOR LOW(-) HIGH (+)

D (Driver) regular oversized

B (Beverage) beer water

O (Ball) 3-piece balata

- - - + - -

1 2

3

5

87

4

6+ - +

+ + +

+ + -

- + +

- + -

- - +

D

B

O

Conventional DOE focuses only on

Average Response

QE focuses on Variability of

Response


Taguchi’s Key Contributions

Quality Engineering Philosophy

Methodology

Experiment Design

Analysis


The Taguchi Loss Functionand the typically assumed Loss to the Customer

TargetLo Spec Hi Spec

Loss


Taguchi’s Quality Philosophy

Loss = k(P - T)2

not 0 if within specs

and 1 if outside

On Target Production

is more important than

producing within Specs

LS T US

LS T US

Conventional viewTaguchi’s view


Taguchi focused on Off-Line Quality Control

Off-Line Quality Control = Improving quality and reducing

total cost in the product or process design stage

Total Cost means cost to society so it includes the cost of

problems in manufacturing and the cost of problems in the

field.


Definition:

Robust Design—A Design that results in products or services that can function over a broad range of usage and environmental conditions

Taguchi’s key contribution is Robust Design


Taguchi’s Contributions Contd.


Methodology

Experiment Design

Analysis


Taguchi’s Product Design Approach has 3 Steps

1. System Design

Choose the sub-systems, mechanisms, form of the prototype—develop the basic design. This is similar to conventional engineering design

2. Parameter Design

Optimize the system design so that it improves quality (robustness) and reduces cost

3. Tolerance Design

Study the tradeoffs that must be made and determine what tolerances and grades of materials are acceptable


Parameter Design (the Robust Design step)

● Optimize the settings of the design parameters to minimize its sensitivity to noise–ROBUSTNESS.

● By highlighting ―robustness‖ as a key quality requirement, Taguchi really opened a whole area that previously had been talked about only by a few very applied people.

● His methodology is heavily dependent on design of experiments like Fisher’s and Box’s methods, but the difference he made was that for response he looked at not only the mean but also the variance of performance


Robust Design—how it is done

Identify Product/Process Design Parameters that Have significant / little influence on Performance

Minimize performance variation due to Noise factors

Minimize the processing cost

Methodology: Design of Experiments (DOE)

Examples - Chocolate mix, Ina Tile Co., Sony TV

Target

Performance () Actual

Performance (P)

Design Parameters (D)

Noise Factors (N): Internal & External

Product / Process


Taguchi’s Experimental FactorsParameter design step identifies and optimizes the Design Factors

Control Factors – Design factors that are to be set at optimal levels

to improve quality and reduce sensitivity to noise

• Size of parts, type of material, Value of resistors, etc

Noise Factors – Factors that represent the noise that is expected in

production or in actual use of the product

• Dimensional variation

• Operating Temperature

Adjustment Factor – Affects the mean but not the variance of a

response

• Deposition time in silicon wafer fabrication

Signal Factors – Set by the user to communicate desires of the user

• Position of the gas pedal


Taguchi’s Contributions Contd.


Methodology

Experiment Design use orthogonal arrays

Analysis


Several different types of Experimental plans (“designs”) are available to the design engineer—Factorial, Fractional, Central Cuboid, etc. Taguchi used “Orthogonal” Designs

CCenter

SScreening

FFactorial

OOrthogonal

FFFractional

factorial

Focus: Handle many factors

Output: List of Important Factors, Best Settings, Good design


Full Factorial Array Example: The 23

(8-trial) array

1 2 3 4 5 6 7

1 1 1 1 1 1 1

1 1 1 2 2 2 2

1 2 2 1 1 2 2

1 2 2 2 2 1 1

2 1 2 1 2 1 2

2 1 2 2 1 2 1

2 2 1 1 2 2 1

2 2 1 2 1 1 2

C B -BC A -AC -AB -ABC

Full Factorial Factor Assignments to Experimental Array Columns

Such experiments can find all Main & two- and three-factor Interactions

Array

Columns

Response

A

C

B


The L8 Orthogonal Array Example: Taguchi used these

1 2 3 4 5 6 7

1 1 1 1 1 1 1

1 1 1 2 2 2 2

1 2 2 1 1 2 2

1 2 2 2 2 1 1

2 1 2 1 2 1 2

2 1 2 2 1 2 1

2 2 1 1 2 2 1

2 2 1 2 1 1 2

C B D A E F G

Orthogonal Array Factor Assignments to Experimental Columns

Such experiments can find all 7 Main effects.

Array

Columns

Response

A

F

DB E

C G


Taguchi’s Orthogonal Experimental Plan—

7 Factors (A, B, C, D, E, F and G) may potentially

influence the production of defective tiles


Calculation of Factor Effects


Main Effects of Process

Factors on %Defects in Tiles


Alternative Design Notations for

Orthogonal Arrays

Std. Fisher's Original Yates Group Theory TaguchiOrder A B C A B C A B C1 – – – 1 0 0 0 1 1 12 + – – a 1 0 0 2 1 13 – + – b 0 1 0 1 2 14 + + – ab 1 1 0 2 2 15 – – + c 0 0 1 1 1 26 + – + ac 1 0 1 2 1 27 – + + bc 0 1 1 1 2 28 + + + abc 1 1 1 2 2 2

X1 X2 X3 X1 X2 X3


Taguchi’s OA-based Experimental Design Matrix Notation

Total Number of Runs

k

NL 2

Number of Levels per Factor

Number of Factors


Linear Graphs for the L8 Array

Linear graphs guide assignment of factors to L8

columns

1

2

3

4

5

6

7

1

2

3

4

5

6

7

Main effects are assigned to columns at nodes in the graph.

Interactions are assigned to the columns on the lines.


Some Orthogonal Array Designs

―Classical‖(2-level Factorials)

―Taguchi‖

23

24

25

26-3

27-1

…

23-1=L4

27-4=L8

215-11=L16

…

L12

L18

L27

…

See Montgomery (1997), Design and Analysis of Experiments, P. 631


Taguchi Orthogonal Array Tables

2-level (fractional factorial) arrays

L4(23). L8(2

7), L16(215). L32(2

31), L64(263)

2-level array

L12(211) (Plackett-Burman Design)

3-level arrays

L9(34). L27(3

13), L81(340)

4-level arrays

L16(45). L64(4

21)

5-level array

L25(56)

Mixed-level arrays

L18(21x37), L32(2

1x49), L50(21x511)

L36(211x312), L36(2

3x313), L54(21x325)


Comments on Taguchi Arrays

Taguchi designs are large screening designs

Assumes most interactions are small and those that aren’t are known ahead of time.

Taguchi claims that it is possible to eliminate interactions either by correctly specifying the response and design factors or by using a sliding setting approach to those factor levels.

Doesn’t guarantee that we get ―highest resolution‖ design.

Instead of designing the experiment to investigate potential interactions, Taguchi prefers to use three-level factors to estimate curvature


Taguchi’s Robust Design Experiments

Taguchi advocated using

inner and outer array

designs to take into

account noise factors

(outer) and design factors

(inner)

Design factors: I1, I2, I3

Noise factors: E1 & E2

Objective: Maximize

response while minimizing

its variance


Example: Robust Design OAs of

Starter Motor Parameter Design

Inner array: armature turns, gage of wire, ferric content of alloy

Outer array: battery voltage, ambient temperature

Starter torqueReplicates


Taguchi’s Contributions


Methodology

Experiment Design

Analysis Finding the robust design parameter values


Taguchi’s Analysis uses SN Ratios

To maximize robustness, Taguchi uses signal-to-noise ratios as response variables, for example,

However, it is often more informative to analyze mean and standard deviation separately, rather than combine into a signal-to-noise ratio

analyze stddev in the same manner that we have previously analyzed the mean.

Taguchi’s analysis techniques are often inefficient…

SNt -10logy 2

s2


SN Ratios are Maximized

To maximize robustness, when Target performance is the best, Taguchi uses the signal-to-noise ratio

When response is to be maximized, Taguchi uses

When response is to be minimized, Taguchi uses

-

n

ySNt

2/1log10

2

2

log10s

ySNt

-

n

ySNt

2

log10


Taguchi Analysis of Motor Design Data

Robustness is maximized with SN ratio is maximized.

Design (inner array) factor settings that maximize SN ratioare:

I1 (turns) = -1

I2 (gage) = +1

I3 (ferric %) = -1

Note: This system is not additive! Results are approximately OK.

72

74

76

78

80

I1 = -1 I1 = +1 I2 = -1 I2 = +1 I3 = -1 I3 = +1

Inner Array Factors

Torq

ue

10

15

20

25

I1 = -1 I1 = +1 I2 = -1 I2 = +1 I3 = -1 I3 = +1

Inner Array Factor Settings

Sta

ndar

d D

ev T

orqu

e

0

0.0005

0.001

0.0015

I1 = -1 I1 = +1 I2 = -1 I2 = +1 I3 = -1 I3 = +1

Inner Array Factor Settings

S/N

Ratio

of R

espo

nse


Designers should embrace Taguchi’s philosophy of quality engineering. It makes very good sense.

Note, however, that a key weakness of Taguchi method is its assumption of a ―main factor only‖ (or ―additive‖ model)… Taguchi ignores interactions

Therefore, rather than use inner outer arrays, we may use more efficient and exact methods that are no more difficult to learn and apply to carry Taguchi’s robust design philosophy into practice…

You may use any of the various experimental and optimization techniques available in the literature such as multiple regression/RSM to develop robust designs.

An example of such extension is shown in the next slides.

Epilogue


Multiobjective Robust Design

by Metaheuristic Methods

Tapan P Bagchi

and

Madhu Ranjan Kumar (1993)





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