ROBUST DESIGN

Seminar Report

Submitted towards partial fulfillment of the requirement for the award of degree of

Doctor of Philosophy

(Aerospace Engineering)

By

SHYAM MOHAN. N

(Roll No. 02401701)

Under the guidance of

Prof. K. Sudhakar

Prof. P. M. Mujumdar

Department of Aerospace Engineering, Indian Institute of Technology,

Bombay–400 076

November, 2002

2

ABSTRACT

The underlying principles, techniques & methodology of robust design are

discussed in detail in this report with a case study presented to appreciate the

effectiveness of robust design. The importance of Parameter design & Tolerance

design as the major elements in Quality engineering are described. The Quadratic loss

functions for different quality characteristics are narrated, highlighting the fraction

defective fallacy. The aim of the robust design technique is to minimize the variance

of the response and orthogonal arrays are an effective simulation aid to evaluate the

relative effects of variation in different parameters on the response with the minimum

number of experiments. Statistical techniques like ANOM (analysis of means) and

ANOVA (analysis of variance) are the tools for analyzing the data obtained from the

orthogonal array based experiments. Using this technique of robust design the quality

of a product or process can be improved through minimizing the effect of the causes

of variation without eliminating the causes. Fundamental ways of improving the

reliability of a product are discussed highlighting the importance of robust design on

this. Based on the classification of uncertainties in design, the role of robust design

optimization & reliability based design optimization are discussed. The mathematical

formulations for these types of optimization strategies are explained. Based on this

study, it can be concluded that the robust design methodology based on Taguchi’s

principles will take care of the entirety of the noise factors which can cause

underperformance and failures, but it will be advantageous to do a robust & reliability

based design optimization because apart from making the design insensitive to noises,

it will enable the designer to predict the reliability of the product. The current research

activities in the application of robust design techniques in the aerospace systems are

also discussed, one with respect to relaxing manufacturing tolerances on an aircraft

nacelle to reduce cost and the other, tackling uncertainties in Mach number in the

design optimization of an airfoil for a transport aircraft.

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Table of Contents

Page No.

1. Introduction 8

1.1 Historical perspective. 9

2. Quality Engineering using Robust Design 10 2.1 Quality Engineering Principles 10 2.2 Quality Loss Function & The Fraction Defective Fallacy 10 2.2.1 Different types of Quality Loss Function 13 2.3 Response Variations & Control 14

3. Robust Design Technique 16 3.1 Classification of Parameters 16 3.2 Average Quality Loss due to Noise Factors 17 3.3 Exploiting Non-linearity for robust design 18 3-4 Tasks to be performed in Robust Design 20

4. Matrix Experiments using Orthogonal Arrays 22

4.1 Steps in Robust Design 25 4.2 Identification of control & noise factors 25 4.3 Selection of factor levels 25 4.4 Factor assignment 26

5 A Case Study 28

6. Methods of Simulating the variation in noise factors 44

7. Reliability Improvement 46

7.1. Role of S/N Ratios in Reliability improvement 46

8. Design Optimization under Uncertainty 48

8.1 Robust Design Optimization (RDO) and Reliability Based Design Optimization (RBDO) 49 9. Application of robust design in aerospace systems 54

10. Conclusion 60

11. References 62

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List of Figures

Page No. Fig-1 General trend of quality definition 11

Fig-2 Fraction defective fallacy 11

Fig-3 Quality loss as a step function & quadratic function 12

Fig-4 Quadratic quality loss 12

Fig-5 Quality loss function for nominal the best type 13

Fig-6 Quality loss function for smaller the better type 13

Fig-7 Quality loss function for larger the better type 14

Fig-8 Nature of variations & control 15

Fig-9 Design block diagram 16

Fig-10 Distribution of quality characteristic 17

Fig-11 Mean shift due to noise effects 17

Fig-12 Exploiting non-linear relation 19

Fig-13 Maximum S/N ratio & the robust point 21

Fig-14 Two level selection 26

Fig-15 Three level selection 26

Fig-16 Schematic diagram of the reduced pressure reactor 28

Fig-17 Plots of S/N ratio vs parameter levels 39

Fig-18 Uncertainty classification 49

Fig-19 Robust design principle 50

Fig-20 Nacelle’s eleven key features 55

Fig-21 Surface excrescence at the key manufacturing features 55

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List of Tables

Page No.

Table-1 L4 orthogonal array with 2 levels 23

Table-2 L8 orthogonal array with 2 levels 23

Table-3 L9 orthogonal array with noise capturing 24

Table-4 Factor level assignment 27

Table-5 Control factors & their levels 30

Table-6 L 18 orthogonal array & factor assignment 31

Table-7 Experimenter’s log 32

Table-8 Data on surface defects count 33

Table-9 Thickness & deposition rate data 34

Table-10 S/N ratios from matrix experiments 35

Table-11 Analysis of surface defect data 36

Table-12 Analysis of thickness data 37

Table-13 Analysis of deposition rate data 38

Table-14 Summary of factor effects 40

Table-15 Tolerance synthesis 41

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Nomenclature

Acronyms

ANOM - Analysis of mean

ANOVA - Analysis of Variance

NC - Number of Constraints

NPR - Number of performances to be robust

OBJ - Objective function

OA - Orthogonal array

Q - Quality

QC - Quality control

R&D - Research & Development

S/N - Signal to Noise

Symbols

∆ - Tolerance

L(y) - Quality loss function

m - Mean value

Ao - Cost of replacement or repair

µ - Mean

σ - Standard deviation

σ2 - Variance

Σ - Summation

d - Design variable

x - Random variable

R - Response

Gi - iih constraint function

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Acknowledgement

The author would like to express his deep and sincere gratitude

to Prof. K. Sudhakar and Prof. P.M. Mujumdar of Aerospace Engineering

Department for their continuous guidance and support in this seminar

work and for the preparation of this report.

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1. Introduction

The knowledge of scientific phenomena and past experience with similar

product designs and manufacturing processes form the basis of the engineering design

activity. However, a number of new decisions related to the particular product must

be made regarding product architecture, parameters of the product design, the process

architecture and parameters of the manufacturing process. A large amount of

engineering effort is consumed in conducting experiments (either with hardware or by

simulation) to generate the information needed to guide these decisions. Efficiency in

generating such information is the key to meeting market windows, keeping

development and manufacturing cost low and having high-quality products. Robust

Design is an engineering methodology for improving productivity during design &

development so that high quality products can be produced at low cost.

Designing high quality product and processes at low cost is an economic and

technological challenge to the engineer. A systematic and efficient way to meet this

challenge is a new method of design optimization for performance, quality & cost,

called Robust Design, which is capable of

1. Making product performance insensitive to raw material variation, thus allowing

the use of lower grade alloys & components in most cases,

2. Making designs robust against manufacturing variation, thus reducing labor &

material cost for rework & scrap,

3. Making the design least sensitive to the variation in operating environment, thus

improving reliability and reducing operating cost, and

4. Using a new structured development process so that engineering time is used more

productively.

The Robust Design method uses a mathematical tool called Orthogonal Arrays to

study a large number of decision variables with a small number of experiments. It also

uses a new measure of quality called signal-to-noise (S/N) ratio to predict the quality

from the customer’s perspective. [1,2,3] Thus, the most economical product & process

design from both manufacturing & customers’ viewpoint can be accomplished at the

smallest, affordable development cost.

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Robust design yield robust product that works as intended regardless of variation

in a products manufacturing process, variation resulting from deterioration, variation

in operating conditions and variation in the ambient conditions during use. Robust

Design can be achieved when the designer understands these potential sources of

variation & takes steps to desensitize the product to these potential sources of

variations. Robust Design can be achieved by “Intelligent Design”, by understanding

which product/process design parameters are critical to the achievement of a

performance characteristics and what are the optimum values to both achieve the

performance characteristic & to minimize its variation. Robust Design is based on the

principle of optimization in which the objective function is defined as the signal to

noise ratio which will help in finding those values of the design parameters at which

the response is least sensitive to the different effects of noise factors. [1,2,3]

So the fundamental principle of Robust Design is to improve the quality of a

product by minimizing the effects of the causes of variations without eliminating the

causes. This is achieved by optimizing the product & process designs to make the

performance minimally sensitive to the various causes of variations.

1.1 Historical perspective.

When Japan began its reconstruction efforts after World War II, it faced an acute

shortage of good quality raw materials, high quality manufacturing equipment and

skilled engineers. The challenge was to produce high quality products and continue to

improve the quality under those circumstances. The task of developing a methodology

to meet the challenge was assigned to Dr. Genichi Taguchi, who at that time was a

manager in Nippon Telephone & Telegraph Company. Through his research in the

1950s and early 1960s, Dr.Taguchi developed the foundations of Robust Design and

validated its basic philosophies by applying them in the development of many

products. In recognition of this contribution, he received the Individual Deming

Award in 1962, which is one of the highest recognition in the quality field. [3]

The Robust Design method can be applied to a wide variety of problems. The

application of the method in electronics, automotive products, photography, and many

other industries have been an important factor in the rapid industrial growth and the

subsequent domination of international markets in these industries by Japan.

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2. Quality Engineering using Robust Design

2.1 Quality Engineering Principles

Though “Quality” can be defined as “Conformance to specification” and

fitness for use” etc in the general concept, these definitions do not cover the entire

implied meaning of Quality. The Ideal Quality a customer can expect is that the

product delivers the target performance each time the product is used, under all

intended operating conditions and throughout its intended life, with no harmful side

effects.

Dr. Taguchi brought out the fallacy in the fraction defective definition for

quality, in which the number of defectives based on the principle depicted in Fig-1

was the only concern. As per his theory [2], the measure of quality of a product is in

terms of the total loss to society due to functional variation and harmful side effects.

Under ideal quality, this loss is equal to zero. Greater the loss, lower the quality. As

per this the total cost of a product is the sum of the operating cost including

maintenance & inventory, the manufacturing cost, the R & D cost (the time,

Laboratory charges, resources etc) and the cost incurred by its breakdown and thereby

the losses caused to the society. The product life cycle cost is divided into the cost

incurred before sale to the customer and after sale to the customer. Quality

engineering is concerned with reducing both of these costs and thus is an

interdisciplinary science involving engineering design, manufacturing operations and

economics.

2.2 Quality Loss Function & The Fraction Defective Fallacy

As per the definition, the Quality Loss Function is the total loss incurred by

the society due to failure of the product to deliver the target performance and due to

harmful side effects of the product including its operating cost. According to the

primitive concepts of quality, the product was certified as good quality if the

measured characteristics were within the specification & vice versa. This is shown in

figure–1. This means that all products that meet the specifications are equally good.

But in reality it is not so. The product whose response is exactly on target gives the

best performance. As the product’s performance deviate from the target, the quality

becomes progressively worse. These two quality philosophies are narrated in fig-2 as

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Target

Target + Target -

GoodGood GoodGood Good

Accept

Reject Reject

BadBad

General Trend of Quality Definition

in one case, the focus is on meeting the target and on other case the focus is on

meeting the tolerance. This is the actual case study result [3] on the SONY TV

companies of USA & Japan and demonstrates how the Japan made TVs were branded

as high quality products by following the principle of focussing the target than

focusing the tolerance.

0.3 % outside Tolerance limit

(Focus was meeting the target)(Focus was meeting the tolerance)

Sony, Japan produced many more grade A sets & many fewer grade c sets, compared to Sony USA. Average grade of sets produced by Sony, Japan was better, hence the customer’s preference.

From these it can be realized that the true quality measure should not be based

on the step function as shown in Fig-1 but as a quadratic loss function as shown in

Fig-3. Here the quality loss function L (Y) is symmetric about the target performance.

As the performance deviate from the target the quality loss correspondingly increases.

Ao is the cost of replacement or repair and ∆ represents the acceptable limit.

Fig-1 General Trend of Quality Definition

Fig-2 Fraction Defective Fallacy

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Target

Quality Loss L(y)

DTarget + Target -

Ao

As shown in Fig –4 , with the quadratic loss function the Quality Loss is given

by the relation L= k ( y-m ) 2 , where k is a constant called Quality Loss Coefficient.

When y=m, the loss is zero. The loss L(y) increases slowly in the neighborhood of m;

but as we go further from m, the loss increases more rapidly. The average quality loss

incurred by a customer, who receives a product with y as the quality characteristic

will be L(y)

2)( mykL −=

L = Loss associated with attribute ym = Specification targetk = constant depending upon the cost and width of the specs

Example: The cost of scraping a part is Rs 100.00 when it deviates ±0.50mm from a target nominal of 2.00mm.

Rs100=k(2.5-2.0)2

K = Rs 400.per mm 2

L = 400 (y-2.0) 2

• This represents a paradigm shift in the way in which companies measure the “goodness” of a product

Fig-3 Quality Loss as a Step Function & Quadratic Function

Fig-4 Quadratic Quality Loss

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Loss L(y)

2.2.1 Different types of Quality Loss Function

i) Nominal–the-best Type Is applicable where the Quality characteristic y has a finite target value,

usually non-zero and the Q loss is symmetric on either side of the target. Eg: Colour

density of a TV set. This type is schematically shown in Fig-5

ii) Smaller the better Type

For quality characteristics which can never take negative values and their ideal

value will be zero and as their value increases, performance becomes progressively

worse. Eg: Radiation leakage from a microwave oven, Response time of computer,

Pollution from automobile etc.

iii) Larger the better type.

For Quality characteristics which do not take negative values and zero is their

worst value. As their value becomes larger the performance becomes progressively

smaller. Their ideal value is infinity and at that point loss=zero. Eg: Bond strength of

adhesive.

Fig-5 Quality Loss Function for Nominal the Best type

Ao

∆o

L(y) = k y2

K= Ao/∆o2

Fig-6 Quality Loss for Smaller the better type

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Loss L(y)

2.3 Response Variations & Control

A standard design process is represented schematically below, where the input to

the design process is the design variables and when we fix the values for these variables

to satisfy the given constraints, the design is said to be completed and the corresponding

out put is the response.

When a design is completed and fabricated based on the specifications, the

performance of the final product may vary from the targeted value due to several reasons.

One type of variation can be attributed to the noises related to the fabrication process. In

order to control these types of variations the concept of inspection, screening and the on-

line quality control emerged. This is schematically shown below.

Design Design Variables Response

Design Design Variables Response

Production

Variations

Due to causes related to fabrication

Apply Tolerances On line QC

K= Ao/∆o2

L(y) = k (1/y2)

Fig-7 Quality Loss for Larger the better type

Ao ∆o y

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But there can be some other reasons, other than those related to fabrication, which

cause variation in the response. These are noises during operation and the subsystem out

puts also can be affected because of these noises. The system performance will be

affected also by the changes in the subsystem outputs. Here tight screening for the

subsystem components can be applied to improve quality but this method will be

prohibitively expensive. This is schematically shown below.

The different reasons which cause the variation in design parameters and in the

manufacturing are termed as noises. The optimum & most efficient way to solve these

problems of variation is to make the design & process insensitive to the effect of noises

(the causes of variation). This is the underlying principle of Robust Design.

Fig-8 shows the different types of variations & control. The on-target, low

variation is the most preferred one, which can be obtained using Robust Design

Off-target

Low variation

freq

uenc

y

variable

On-target

High variation

Off-target

High variation

On-target

Low variation

System

Noises related to fabrication

Apply Tolerances

On line QC Variations

Apply Tight Tolerances Tight screening

Noises

Fig-8 Nature of Variation & Control

Operation Performance

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3. Robust Design Technique 3.1 Classification of Parameters In the basic design process, a number of parameters can influence the quality

characteristic or response of the product. These can be classified into the following three

classes and shown below in the block diagram of a product/process design

The response for the purpose of optimization in the robust design is called the quality

characteristic. The different parameters, which can influence this response, are described

below.

i) Signal Factors : These are parameters set by the user to express the intended value

for the response of the product. Example- Speed setting of a fan is a signal factor

for specifying the amount of breeze. Steering wheel angle – to specify the turning

radius of a car.

ii) Noise Factors: Parameters which can not be controlled by the designer or

parameters whose settings are difficult to control in the field or whose levels are

expensive to control are considered as Noise factors. The noise factors cause the

response to deviate from the target specified by the signal factor and lead to

quality loss.

iii) Control Factors: Parameters that can be specified freely by the designer. Designer

has to determine best values for these parameters to result in the least sensitivity

of the response to the effect of noise factors.

The levels of noise factors change from unit to unit, one environment to another

and from time to time. Only the statistical characteristics (mean & variance) can be

known or specified. The noise factors causes the response to deviate from the target

specified by the signal factor and lead to quality loss.

Signal Factors

Design Block Diagram[3]

Product / Process Response

Noise Factors

Control Factors Fig-9

17

The Noise factors can again be classified in to three-

(a) External: The environment, the load, human error

(b) Unit to unit variation: Variation in the manufacturing process

(c) Deterioration : As time passes, the performance deteriorates (aging related)

The robust design addresses all these different types of Noise factors. For a

product or process with multiple functions, different noise factors can affect different

quality characteristics.

3.2 Average Quality Loss due to Noise Factors Because of the noise factors, the quality characteristic y of a product varies from

unit to unit and from time to time during the usage of the product. The distribution of y

resulting from all source of noise is shown below which is a normal distribution with

mean µ and variance σ 2

Let y be the nominal the best type quality characteristic and m be its target value. Let

y1,y2,….yn be n representative measures of the quality characteristic y, taken on a few

representative units throughout the design life of the product. Because of the noise

effects, the average value of y will be shifted from the target value m as shown below.

The distribution of y, with mean µ and variance σ 2

µ y

Fig-10.

Average Quality Loss = K [ (µµ -m) 2 +σ 2 ] Fig-11 Mean Shift

µ

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In this expression of average quality loss, k(µµ -m) 2 is resulting from the deviation

of the average value of y from the target m, and k σ 2 is resulting from the mean squared

deviation of y around its own mean.

Between these two components of quality loss it is easier to eliminate the

first one. Reducing the second component requires decreasing the variance, which is

more difficult. Three methods of reducing variance in the order of increasing cost are

given below.

(i) Screening out bad products, with the tighter tolerances

(ii) Discover the causes of malfunction & eliminate it

(iii) Apply robust design method to make the product’s performance

insensitive to noise factors.

3.3 Exploiting Non-linearity for robust design

Usually a product’s quality characteristic is related to the various product

parameters and noise factors through a complicated non-linear function. It is possible to

find many combinations of product parameter values that can give the desired target

value of the product’s quality characteristic under nominal noise conditions. However

due to non-linearity these different product parameter combinations can give quite

different variations in the quality characteristic, even when the noise factor variations are

the same.

The principal goal of robust design is to exploit the non-linearity to find a

combination of product parameter values that gives the smallest variation in the value of

the quality characteristic around the desired target value.[3,4]

Let x = ( x1,x2,……xn )T denote the noise factors and z = ( z1,z2,…..zq ) T

denote the product parameters (called control factors) whose values can be set by the

designer. Suppose the following function describes the dependence of the quality

characteristic y on x and z .

Y = f ( x, z )

The deviation ∆y of the quality characteristic from the target value caused by the

deviation ∆xi of the noise factor, from their respective nominal values can be

approximated as ;

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∆y = (∂f/∂ xI) ∆x1 + (∂f/∂ x2) ∆x2 + --------------------- + (∂f/∂ xn) ∆xn

Further , if the deviations of the noise factors are uncorrelated, the variance σy2 of y can

be expressed in terms of the variance of xI as follows: [2,3,5]

σy

2 = (∂f/∂ xI) 2 σx12 + (∂f/∂ x2) 2 σx2

2 + --------------------- + (∂f/∂ xn) 2 σn2

Thus the variance of response σy2 is the sum of the products of the variances of the noise

factors σxi2 and the sensitivity coefficients (∂f/∂ xi) 2 . The sensitivity coefficients are

themselves functions of the control factor values. A robust product / process is the one in

which the sensitivity coefficients are the smallest. The utilization of non-linearity

between the response and the input parameters are shown in Fig-12

x

0

20

40

60

80

100

915 925 935 945 955 965

y

• What is the best setting for x knowing that x can vary by +/-5? x x+5x-5

R

In the above curve the point R is the optimum point at which the dispersion in the

response due to the dispersion in the noise factor is the minimum. How to obtain this

point for each noise factor is the challenge & beauty of robust design technique.

Fig –12 Exploiting the non-linear relation

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3.4 Tasks to be performed in Robust Design A great deal of engineering time is spent in generating about how different design

parameters affect performance under different usage conditions. Robust design

methodology serves as an “amplifier” – that is it enables an engineer to generate

information needed for decision making with less than half the experimental effort.

There are two important tasks to be performed in robust design, which can be

considered as the main tools used in the process of achieving robustness.

i) Measurement of quality during design & development. A leading indicator of

quality by which the effects of changing a particular design parameter on the

performance can be evaluated.

ii) Efficient experimentation to find dependable information about the design

parameters, so that design changes during manufacturing & customer use can be

avoided. Also the information should be obtained with minimum time &

resources.

The estimated effects of design parameters must be valid even when other

parameters are changed during the subsequent design efforts or when dimensions of

related subsystems changed. This can be achieved by employing the signal to noise ratio

to measure the quality & orthogonal arrays to study many design parameters

simultaneously.

The objective Function – The Signal to Noise Ratio.

Since the robust design is all about keeping the response mean to the target and

minimizing the variation in the response, a special type of objective function which

captures both the above said objectives, is identified for the robust design process. This is

called the Signal to Noise (S/N) ratio and is different for different types of quality

characteristics. So robust design is treated as an optimization process in which the

objective function is S/N ratio. Signal to Noise ratio is a mathematical formula used to

calculate the design robustness. This is the ratio of the signal (mean) over the noise

(variability). The larger the S/N ratio, the more robust the performance. The signal to

noise ratio for the prominent types of quality characteristics are given below.

21

For the nominal the best type: S/N = 10 log10 ( µ2 /σ 2 ) µ = The mean, σ 2 = The variance For smaller the better type : S/N = -10 log [ (1/n) Σ yi

2 ]

This is actually the mean square deviation because the ideal value here is zero.

For Larger the better type: S/N = -10 log [ (1/n) Σ (1/yi2 )]

This is also the mean square deviation. And here by maximizing the negative of

the function, the deviations are minimized. The Fig 13 demonstrates that the when the

Signal to Noise ratio is the maximum, the corresponding value of the noise factor will

provide the least sensitivity of the response. So at these points the response will be least

sensitive to the variations in the noise factors.

x

0

20

40

60

80

100

915 925 935 945 955 965

y

• What is the best setting for x knowing that x can vary by +/-5? x x+5x-5

-25

-20

-15

-10

-5

0

5

10

15

20

25

S/N

i=1

n

Fig-13 Maximum S/N ratio & the robust point

i=1

n

22

4. Matrix Experiments using Orthogonal Arrays

Robust design draws on many ideas from statistical experimental design to plan

experiments for obtaining dependable information about variables involved in making

engineering decisions. Various types of matrices are used for planning experiments to

study several decision variables simultaneously. Among them, robust design makes

heavy use of the Orthogonal Arrays.[2,3]

Robust design adds a new dimension to statistical experimental design . It

explicitly addresses the following concerns faced by all product & process designers.

• How to reduce economically the variation of a product function in the customers

environment. ( Note that achieving a product function consistently on target

maximizes customer satisfaction )

• How to ensure that decisions found to be optimum during laboratory experiments will

prove to be so in manufacturing & in customer environment.

In addressing these concerns, robust design uses the mathematical formalism of

statistical experimental design. A matrix experiment is a set of experiments, where we

change the settings of the various parameters we want to study from one experiment to

another. After conducting a matrix experiment, the data from all experiments in the set

taken together are analyzed to determine the effects of various parameters. The Analysis

of Means ( ANOM ) and the analysis of variance ( ANOVA) are used to interpret the

data to find the sensitivity of each parameters of interest.

Conducting the matrix experiments using special matrices called “Orthogonal

Arrays”, allows the effect of several parameters to be determined efficiently and is an

important technique in robust design. The different levels of the parameters are known as

experimental region or the region of interest.

Orthogonality is interpreted in a combinatoric sense – (ie) for any pair of

columns, all combinations of factor levels occur and they occur on equal number of

times. This is called the balancing property and it implies orthogonality.

So an Orthogonal Array can be defined as a matrix with the columns representing

the number of parameters to be studied with their different levels in different

23

combinations of experiments and the number of rows equal to the number of

experiments. Standard orthogonal arrays are designed and are available. Selection of an

orthogonal array for a robust design project is based on the number of degrees of freedom

of the experiment in such a way that the number of experiments should be greater than or

equal to the number of degrees of freedom.

Each parameter with n levels will have n-1 degrees of freedom and overall mean

will have one degree of freedom.. In case of a robust design project with 4 parameters

and three levels, the total degrees of freedom will be 9. So the selected standard

orthogonal array should have atleast 9 rows. An L4 array means an orthogonal array with

4 rows and an L8 array has 8 rows. Some of the standard orthogonal arrays are shown

here in Table-1,2 & 3.

Column Expert. Number 1 2 3

1 1 1 1 2 1 2 3 3 2 1 2 4 2 2 1

Column Exp # 1 2 3 4 5 6 7 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 2 3 1 2 2 1 1 2 2 4 1 2 2 2 2 1 1 5 2 1 2 1 2 1 2 6 2 1 2 2 1 2 1 7 2 2 1 1 2 2 1 8 2 2 1 2 1 1 2

L4 Array:

3 Variables, 2 Levels

L8 Array:

7 Variables, 2 Levels

Here the orthogonality is interpreted in a combinatoric sense – (ie) for any pair of

columns all combinations of factor levels occur and they occur on equal number of times.

The columns y1 to y4 in Table-3 are the different measurements taken on each setting to

capture the noise effect. The performance or the responses measured in these matrix

Table-1& 2. L4 & L8 Orthogonal arrays with 2 levels

2

24

experiments are analysed using ANOM & ANOVA to find the relative effects of noises

on the response. By this method the optimum values of the control factors for which the

sensitivity of the response to the effect of noise factors are the minimum can be found

out. The detailed steps in robust design are illustrated below.

Trial No. 1 2 3 41 1 1 1 12 1 2 2 23 1 3 3 34 2 1 2 35 2 2 3 16 2 3 1 27 3 1 3 28 3 2 1 39 3 3 2 1

Column No.

L9 Orthogonal array

y1 y2 y3

* * *

* * ** * *

* * ** * *

* * ** * *

* * ** * *

y4

*

**

**

**

**

As mentioned earlier the columns y1 to y4 corresponds to the response

measurements to capture the effect of noises. These measurements should be planned

according to the sources of noises in the given problem. For each trial the average of all

these y values are to be taken. (This is explained in the case study in chapter 5.)

Typically there are the following two choices regarding noise factors;

• Improve the quality without controlling or removing the causes of variation, to make

the product robust against noise factors.

• Improve the quality by controlling the noise factors, or recommending certain actions

to control the noise factors.

In either case, a formal approach to capturing the effect of noise factors is required.

So for each experiment different measurements should be taken so as to capture the

variations because of the noises.

Table-3 L9 Orthogonal Array with Noise Capturing

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4.1 Steps in Robust Design

The detailed steps in robust design are explained here.[3]. The experimentation

procedure is highlighted below.

Three phases of experimental design:

1. Planning Stage• areas of concern, objective

• select response or quality characteristic

• identify control and noise factors

• select factor levels

• select appropriate experimental design

• identify interactions and assign factors to experimental set-up

2. Conducting Stage • Conduct tests as prescribed in experimental set-up

3. Analysis Phase • Analyze and interpret results

• Conduct confirmation experiments

4.2 Identification of control & noise factors

As explained earlier control factors are those factors that a manufacturer can

control in the design of a product, the design of a process, or during a process. Examples:

design variables (widths, heights), assembly method, cooling temperature, cycle time,

materials, speeds, feeds. So according to the design problem, the control factors are to be

identified. Similarly, the noise factors, which a designer or a manufacturer can not or

wishes not to control (because of cost reasons) are also to be identified. Examples:

material inconsistencies, supplier variation, machine operators, ambient temperature,

ambient humidity.

4.3 Selection of Factor levels

− A minimum of two levels is necessary to estimate a factor’s effect.

− Continuous factors must be discretized (in preferably equal intervals). Example:

Levels of a length parameter: 1 cm, 1.5 cm, 2.0 cm.

26

− The more levels the more experimental runs that are necessary.

− The number of levels indicates the resolution of the effect that can be predicted.

The advantage of taking a minimum of three points to capture the second order effect

is demonstrated in Fig-14, and Fig-15. More the number of levels means more capture of

non-linearity, but more the number of experiments & associated efforts. So it is

recommended to consider an optimum of three levels.

Fig-14 demonstrates how the non-linearity can be missed if only two levels are

taken and in Fig-15, it is shown that considering tree levels will help in a better

representation of the actual effect which can be non-linear. Even a tree level combination

won’t capture the exact relationship. So the entire matrix experimentation may have to be

repeated several times to get the most robust design.

4.4 Factor assignment The selected factors are assigned to the different columns of the specific orthogonal array as shown below.

1 2A

Predicted effect

Actual effect

response

A 1 2

3

2nd order effect

response

Fig-14 Fig-15

27

Trial No. 1 2 3 41 1 1 1 12 1 2 2 23 1 3 3 34 2 1 2 35 2 2 3 16 2 3 1 27 3 1 3 28 3 2 1 39 3 3 2 1

Column No.

Factor A Factor C

Factor D Factor B

Assume each factor has 3 levels

L9

The OA shown in Table-4 is a 3 level, 4 parameter, & 9 experiment orthogonal array.

Here in the above L9 orthogonal array, the first column represent the trials of

experiments , starting from 1 to 9. Second to fourth columns are the levels of each

parameter or factor A, B,C, & D. This will result in nine experiments with factor level

combinations as given in each row. Foe example, the first experiment will be conducted

with all the factors A, B,C & D at level 1. For the second experiment, factor A will be at

level 1 & all other factors at level 2 and so on.

Table-4 Factor level assignment

28

5. A Case Study

Now a specific case study[3] is addressed here to understand the different steps &

aspects of robust design. The robust design is applied on a process of polysilicon

deposition on thin wafers. The process set up schematic is shown in fig-16 . Saline &

Nitrogen gas are introduced at one end and pumped out at the other. The Saline gas

pyrolizes and a polysilicon layer is deposited on top of the oxide layer on the wafers.

Two carriers each carrying 25 wafers can be placed inside the reactor at a time so that

polysilicon is simultaneously deposited on 50 wafers. The problems observed were (i) too

many surface defects and (ii) too large a thickness variation. So robust design

methodology is adopted to improve the performance or quality of the process.

The objective here is to achieve a uniform thickness & minimize the surface defects.

Based on the expertise, the non uniform thickness & surface defects are caused by

− The variations in the parameters involved in the chemical reaction associated with the

deposition process—Concentration gradient along the length of the reactor.

− Flow pattern ( direction & speed ) of the gases need not be the same in all positions

− Temperature variation along the length

To Capture these effects, test wafers are positioned at 3, 23 & 48 along the length

(remaining 47 dummy wafers) and to capture the effect of noise variation across the

wafer, the thickness & surface defects are measured at three different points on each

wafer: top, middle & bottom. So nine measurements of thickness & surface defects for

each combination of control factor setting in the matrix experiment.

Fig-16

29

Quality Characteristics here are Polysilicon thickness and the surface defects

Specification: Thickness should be less than +/- 8 % of the target

Surface defect count should be less than 10 per sq.cm

The economics of the manufacturing process is determined by the throughput as

well as the quality of the product produced. So along with the quality characteristic a

throughput characteristic, which is the deposition rate here also should be studied. The

different signal to noise ratios are identified for the different quality characteristics as

given below.

Thickness data is nominal the best type: Therefore the S/N ratio = η’ = 10 log10 (µ2 /σ 2)

where µ is the mean and σ 2 is the variance.

Defect count is smaller the better type :

Therefore S/N ratio = η = -10 log 10 (mean square defects)

S/N for the deposition rate in decibel scale = η’’= 10 log10 r2, where r is the observed

deposition rate in angstroms.

The goal in optimization for thickness is to minimize variance while keeping the

mean on target. This is a constrained optimization problem, which can be very difficult to

solve. When a scaling factor ( a factor that increases thickness proportionally at all points

on the wafers) exist, the problem can be simplified greatly.

Here the deposition time is a scaling factor (ie) thickness = deposition rate x

deposition time The deposition rate may vary from one wafer to next, or from one

position to another, due to noise factors. However the thickness at any point is

proportional to the deposition time. So maximize the signal to noise ratio and adjust the

deposition time so that mean thickness is on target. The different control factors and their

levels are shown in Table- 5

30

So the total degrees of freedom = 6 x 2 + 1 = 13. So the orthogonal array should

have a minimum of 13 rows. Correspondingly a L18 is selected because for three level

testing L18 is the next available array which is greater than 13. ( Standard orthogonal

arrays for different number of parameters 7 different level are already developed and

available in literature.[2,3] The L18 orthogonal array is shown in table-6 and the

parameters with the assigned levels are shown as experimenter’s log in table-7. The

matrix experiment results on surface defects count is tabulated in table-8, and the

thickness measurements are tabulated in table-9.

Table-5

31

Table-6 L 18 Orthogonal Array and factor assignment [3]

32

Table-7 Experimenter’s log.

33

Table-8 Data on Surface Defect count

34

Table-9 Thickness & Deposition rate data

35

Table-10 Mean thickness of 18 exp. Varies between 1958 to 5965

36

Table-11 Analysis of surface defects data

37

Table-12 Analysis of thickness data

38

Table-13 Analysis of deposition rate data

39

Fig-17 Plots of S/N ratio vs parameter levels [3]

40

Table-14 Summary of factor effects [3]

41

Table-15 Results of verification experiment

42

From table-10, which gives the variation in mean thickness, surface defect count and the

deposition rate, it can be observed that the mean thickness of 18 experiments varies

between 1958 to 5965 Angstrom, where the targeted value is 3600 Angstrom.

Now in order to evaluate the relative effects of variations in different parameters

on the performances the analysis of variance (ANOVA) is performed. A better feel or a

realistic feel incorporating the error variance for the relative effects of the different

factors can be obtained by the decomposition of variance, which is commonly called as

analysis of variance. Here Total sum of squares = grand total sum of squares –sum of

squares due to mean.

Σ (ηi- m)2 = Σ ηi 2- nm 2

Sum of squares due to factor A= total squared deviation of effect of factor A from the

overall mean = 6 (mA1-m) 2 + 6 (mA2-m) 2 + 6 (mA3-m) 2

ANOVA will generate the variance ratios F for different factors. Larger value of F means

the effect of factor is large compared to the error variance.

If F less than 1, factor effect is small and can be neglected.

If F>2, means factor is not quite small.

If F>4, factor effect is quite large.

Error Variance , σe2 = sum of squares due to error/ degrees of freedom for error

Variance of the effect of each factor level in this case = (1/6) σe2

So width of 2 σ confidence interval for each estimated effect is +/- 2 σe

The results of ANOVA on surface defects data, thickness data & deposition rate

data are given in table –11, 12 & 13. The signal to noise ratio for the different factors are

graphically shown in fig – 17 and the entire summary of factor effects are tabulated in

table –14.

Inference: Based on the results of ANOVA & the signal to noise ratio pattern , it can

inferred that Deposition Temperature has the largest effect on all characteristics. From

A2 to A1 η can be improved by –24.23- (-50.10) = 26 dB. Equivalent to 20 fold

i=1

n

i=1

n

43

reduction in rms surface defect count. On thickness uniformity only 0.21 dB, but a

reduction in deposition rate by 5.4 dB, a 2 fold reduction

For E & F the optimum settings are obviously E2 & F2. However the factors A

through D, the direction in which the Q characteristics (the surface defects & thickness

uniformity ) improve tend to reduce the deposition rate. So, a trade off between quality

loss and productivity must be made, in choosing the optimum levels. In the case study A2

is changed to A1. So the selected combination is A1 B2C1 D3E2F2. And the verification

experiment can be done with this selected parameter level combinations and can be

compared with the predicted value. The results of the verification experiments are

tabulated in table-15.

From the orthogonal array experiments it can be found out that there are certain

parameters whose variation do not have any significant effect on changing the response.

So the tolerances on these parameters can be relaxed to gain in the cost of fabrication.

This method is known as the tolerance design. So the parameter design & the tolerance

design together will help in result in a robust product in a lesser cost. So the robust

design method advocate a 3 step design philosophy as shown below to achieve a robust,

cost effective and reliable product.

Step 1. System Design

– concept design and synthesis

– innovation and creativity

Step 2. Parameter Design

– parameter sizing to ensure robustness to variations

Step 3. Tolerance Design

– establish product and process tolerances to minimize costs

Optimization of a process or a product need not be completed in a single matrix

experiment. Several matrix experiments may have to be completed may have to be

conducted in sequence before completing a product or process robust design.

44

6. Methods of simulating the variation in noise factors

In analysing the effect of variation in noise factors on the response, it is very

important to correctly simulate the variations in the noise factors. There are three

different methods of evaluating the mean and variance of a product response, resulting

from variations from many noise factors. They are Monte Carlo simulation, Taylor series

expansion and orthogonal array based simulations.

Monte Carlo Simulation

In this method a random number generator is used to simulate a large number of

combinations of the noise factors called testing conditions. The value of the response is

computed for each testing conditions and the mean and variance of the response are then

calculated. For obtaining accurate estimate of mean & variance, the Monte Carlo method

requires evaluation of the response under a large number of testing conditions. This can

be very expensive, especially if we also want to compare many combinations of control

factor levels.

Taylor Series Expansion

In this method, the mean response is estimated by setting each noise factor equal

to its nominal value. To estimate the variance of the response, the derivatives of the

response with respect to each noise factor is found out.[3,4] Let R denote the response and

σ12 , σ2

2,………σn2 denote the variance of n noise factors. The variance of R is then

computed by the formula:

σR2 = Σ (∂R / ∂xi)

2 σi2 , where xI is the ith noise factor.

The above equation based on first order Taylor series expansion, gives quite accurate

estimates of variance when the correlations among the noise factors are negligible and the

tolerances are small, so that interactions among the noise factors and the higher order

terms are negligible. Otherwise higher order Taylor series expansion must be used, which

i=1

n

45

makes the formula for evaluating the response quite complicated and computationally

expensive. Therefore this method will not always give an accurate estimate of variance.

Orthogonal array based simulation

In this method proposed by Dr. Taguchi, orthogonal arrays are used to sample the

domain of noise factors. For each noise variable we take either two or three levels.

Suppose µ i and σi2 are the mean & variance respectively for the noise variable xi.

When two levels are taken, µ i - σi and µ i + σi are chosen. Note that the mean and

variance of these two levels are µ i and σi2 , respectively. Similarly when three levels are

taken, µ i - {√ (3/2)}σi , µ i and µ i + {√(3/2)} σi are chosen. The details of this

method is already covered in the previous chapters

The advantage of this method over the Monte Carlo method is that it needs a

much smaller (order of magnitude smaller) number of testing conditions; yet the accuracy

will be excellent. The orthogonal array based simulation gives common testing conditions

for comparing two or more combinations of control factor settings. Further when

interactions and correlations among the noise factors are strong, the orthogonal array

based simulation gives a more accurate estimates of mean & variance compared to Taylor

series expansion.

46

7. Reliability Improvement

There are three fundamental ways of improving the reliability of a product during

the design stage: (1) reduce the sensitivity of product’s function to the variation in

product parameters, (2) reduce the variation in product parameters and (3) provide

redundancy.

The first approach is the parameter design part of robust design process. The

second approach is analogous to the tolerance design and it typically involves more

expensive components and manufacturing processes. Thus this approach should be

considered only after sensitivity has been minimized. The third approach is used when

the cost of failure of the product is high compared to the cost of providing redundant

components or even the whole product.

7.1 Role of S/N Ratios in Reliability improvement

Reliability characterization refers to building a statistical model for the failure

times of the product. Log-normal and Weibull distributions are commonly used for

modeling the failure times. Reliability improvement means changing the product design,

including the settings of the control factors, so that the time to failure increases.

For improving a product’s reliability, the appropriate quality characteristics for the

product should be identified for minimizing their sensitivity noise. This automatically

increases the product’s life. The following example clarifies the relationship between the

life of a product and sensitivity to noise factors.

Consider an electrical circuit whose output voltage, y, is a critical characteristic. If

it deviates too far from the target, the circuit’s function fails. Suppose the variation in the

resister R , plays a key role in the variation of y. Also suppose the resistance R is

sensitive to the environmental temperature and that the resistance increases at a certain

rate with aging. During the use of the circuit, the ambient temperature may go too high or

too low, or sufficient time may pass leading to a large deviation in R. Consequently the

characteristic y would go outside the limits and the products would fail. Now if the

nominal values of appropriate control factors are changed so that y is much less sensitive

47

to variation in R , then for the same ambient temperature faced by the circuit, and for the

same rate of change of R due to aging , we would get longer life out of that circuit.

Sensitivity of the voltage y to the noise factors is measured by the S/N ratio. In the

process of improving the S/N ratio only temperature is used as noise factor. Reducing

sensitivity to temperature means reducing sensitivity to variation in R and, hence

reducing sensitivity to the aging of R also. Thus by appropriate choice of testing

conditions (noise factor setting), the robust design helps in increasing the life & thus the

reliability of the product.

48

8. Design Optimization under Uncertainty

Variations are the biggest challenge in the design optimization process and they

are the biggest enemy of quality. The variation can be of different types. The sources of

variation can also be considered as uncertainties. Uncertainty is inevitable in design &

development. Some of the sources of uncertainty are listed below. [7]

i) Scenarios & assumptions

ii) Lack of confidence in modeling

iii) Experimental data

iv) Variation of physical properties

v) Changing operating environment

vi) Variation related to fabrication

In Robust Design methodology, all these causes are identified as noises. Robust

optimization results in the design which performs optimally under the variable (or

uncertain) conditions over the entire lifetime of the design [11]. Strictly speaking the

robust design approach of Dr Taguchi covers the entire aspects of uncertainty and will

help in increasing the reliability of the product as explained in chapter VII. It is up to the

imagination of the designer to identify the correct control factors & noise factors to

capture the effects of uncertainty in the simulation process based on orthogonal arrays.

But in pursuit of further research, the robust design is treated with the different

goals (though the entirety of these goals are covered as the single objective in the quality

engineering concept – as maximizing quality & minimizing cost). The three goals of

robust design are identified [12,13] as:

1) Identify designs that minimize the variability of performance under uncertain

conditions.

2) Provide best overall performance over the entire life time of the product

3) Mitigate the detrimental effects of worst-case performance. Choosing a design

with the best worst case performance.

The Reliability Based Design is based on the estimation of probability distribution

of a system response from the known probability distributions of the random variables in

49

a system [8]. In these methods the constraint functions are converted to probabilistic

constraints.

8. 1 Robust Design Optimization (RDO) and Reliability Based

Design Optimization (RBDO)

To appreciate the difference between RDO & RBDO, a proper understanding of

the classification of uncertainties encountered in a product’s life is depicted below [6,12].

Robust optimization techniques account for the impact of everyday fluctuations of

parameters on the overall design performance, assuming that no catastrophic failures

occurs. Here the primary objective is to improve the quality of a product through

minimizing the effect of the causes of variation without eliminating the causes. The

robust design philosophy is narrated in fig-18.

Impa

ct o

f ev

ents

Per

form

ance

loss

C

atas

trop

he

Cost benefit analysis Robust Design &

optimization

Risk analysis Reliability based

Design & optimization

Everyday fluctuations Extreme events

Frequency of events

Fig- 18 Uncertainty Classification

50

Mathematical formulation of robust design problem

The conventional optimization model is defined as

Minimize OBJ (d)

s.t Gi (d) ≤ 0, i = 1,2,….NC

dL ≤ d ≤ dU

where OBJ is the objective function, Gi is the ith constraint function, NC is the number

of constraints, d is the design variable vector, dL & dU are the lower & upper bounds of

d.

In robust design, the objective is to keep the mean on target & minimize the

variation. So the mean & standard deviation of the response will constitute the objective

function. So the formulation will be

Minimize OBJ [ µ R, σ R ] s.t Gi (µ ) + k σ Gi ≤ 0 , i = 1,2,….NC

dL ≤ d ≤ dU

where µ R, σ R are the mean & standard deviation of the response R, Gi (µ ) and σ Gi

are the mean & standard deviation respectively of the ith constraint function, k is the

Target range

Bias

Quality Distribution

Performance R

Prob

abili

ty D

istr

ibut

ion

µ σ σ

Fig-19 Robust Design Principle

51

penalty function decided by the designer [5], d is the design variable vector, dL & dU are

the lower & upper bounds of d.

Here the objective function takes care of the signal & noise factors and the

constrained functions are modified such that the allowed variation in them are limited by

the sigma bounds.

If orthogonal array based simulations are used, the robust design which

minimize the variability of performance under uncertain (manufacturing & operation)

conditions and robust design which provide the best overall performance over the entire

life time (see chapter 7) will have the same formulation as above. If orthogonal arrays &

signal to noise ratios are not used and the response variances are computed from the

known variances of design parameters, then the objective function will be as shown

below.

NPR

OBJ [ µ R, σ R ] = Σ [ w1j (µ Rj – Rj t ) 2 + w2j σ 2 Rj ]

J=1

s.t Gi (µ ) + k σ Gi ≤ 0 , i = 1,2,….NC

dL ≤ d ≤ dU

Where, w1j is the weight parameter for mean on target, w2j that for the jth

performance to be robust, µRj and σRj the mean and standard deviation of the jth

performance, Rj t is the target value of the jth performance and NPR is the number of

performances to be robust.

And for the robust design for best overall performance over the entire life time

[10,12] the objective function is based on the joint probability density function of the

random variable x . According to the theory of probability & statistics, integral of the

probability density function will give the probability and when this is multiplied by the

performance function, the expected value corresponding to that probability will be

obtained. The expression given below is based on this theory.

52

NPR

OBJ ( d, x ) = ∫ x Σ wj Rj (d,x) f x (x) dx J=1

s.t Gi (µ ) + k σ Gi ≤ 0 , i = 1,2,….NC

dL ≤ d ≤ dU Where f x (x) is the joint probability density function of the random variable x, Rj (d,x)

is the jth performance function to be minimized and wj is the weight parameter for the jth

performance to be robust.

Mathematical formulation of reliability based design (RBDO) problem

The RBDO problems, the objective is to maximize expected system performance

while satisfying constraints that ensure reliable operation. Because the system parameters

are not necessarily deterministic, the objective function & constraints must be stated

probabilistically. For example RBDO can determine the manufacturing tolerance required

to achieve a target product reliability because the method considers the manufacturing

uncertainties, such as dimensional tolerance as probabilistic constraints.[8]

RBDO will ensure proper levels of safety & reliability for the system designed.

The mathematical formulation for RBDO is shown below.

Minimize OBJ (d)

s.t P { ( Gi (d ) ≤ c } ≥ CFLi , i = 1,2,….NC

dL ≤ d ≤ dU

where CFLi is the confidence level associated with the ith constraint, P denotes the

probability, Gi (d ) is the ith constraint function and c is the limiting value. The following

example [8] will clear the concept of probabilistic constraint.

P ( stress 1 ≤ σ y ) ≥ 99.0 % , where σ y is the yield stress. (ie) Since there are some

uncertainty in the material properties, instead of stating the constraint as, stress 1 ≤ σ y,

it is stated as the probability of stress 1 ≤ σ y, is greater than or equal to 99.0%.

53

Robust & Reliability Based Design.

When the objective function is based on the robust design principle (with mean &

standard deviation of the response), focussing on making the response insensitive to the

variations in the design variables and the constraints are modified to probabilistic

constraints with the assigned probability of each constraint function, the result is a Robust

& Reliability Based Design (RRBDO) [9,13]

The mathematical formulation of such a method is given below.

Minimize OBJ [ µ R, σ R ] s.t P { ( Gi (d ) ≤ c } ≥ Poi , i = 1,2,….NC

dL ≤ d ≤ dU

where NC is the number of constraints and the objective function is defined as

NPR

OBJ [ µ R, σ R ] = Σ [ w1j (µ Rj – Rj t ) 2 + w2j σ 2 Rj ]

J=1

The different parameters in the above definition are already explained in the

formulation for robust design. This approach will yield a design whose response is

insensitive to the effects of noises & whose reliability can be predicted based on the

reliabilities apportioned to the different constraints.

54

9. Application of Robust Design in Aerospace Systems

The principles of robust design are being effectively used in the aerospace

design field, for tackling uncertainties related to manufacturing & operation. Two such

problems are discussed here from published literature [12,14], to highlight the use of robust

design concepts in analyzing the sensitivity of manufacturing tolerances on the

performance and tackling the operational uncertainty.

Parametric Optimization of Manufacturing Tolerances at the aircraft

surface[14]

This study was aimed at reducing the aircraft cost by relaxing manufacturing

tolerances. Conventionally aircraft surface smoothness requirements have been

aerodynamically driven with tighter manufacturing tolerances to minimize drag. But this

will drive the cost high. So in this research work[14], a strategy to reduce the aircraft cost

through manufacturing tolerance relaxation at the wetted surface is investigated. For this

a preliminary study has been conducted on eleven key manufacturing features on the

surface assembly of an isolated nacelle.

The manufacturing tolerance allocation for aerodynamic surfaces at the assembly

joints are generated from the specifications laid down by aerodynamicists to minimize

aircraft parasite drag, that is to reduce fuel burn. One of the reasons for parasite drag

increase is the degradation of the surface smoothness qualities by, for example, the

discrete roughness on the component parts and at their subassembly joints. These are seen

as aerodynamic defects, collectively termed as one of the excrescence effects, typically, i)

mismatches (steps etc.) ii) gaps, iii) contour deviation and iv) fastners flushness (rivets,

etc) on the wetted surface. Excrescence drag arising out of these aerodynamic defects is

of a considerably lower order of magnitude than the total drag of the aircraft. With

today’s manufacturing standards, with proper tolerance allocation, the excrescence drag

due to surface roughness can be reduced to rather small but significant values. But this

will have an implication on cost. As a remedial measure a tolerance relaxation tradeoff

55

study between drag increase (loss of quality function) and manufacturing cost reduction

(gain) was conducted.

The main components of the nacelle, along with the 11 key features affecting

excrescence drag, are shown below. Tolerance synthesis on each of these features is done

with relaxing the tolerances and estimating the corresponding drag increase using CFD.

The four types of surface excrescence at the key manufacturing features are explained &

shown below [14].

Fig-20 Nacelle’s 11 key features

Fig-21 Surface excrescence at the key manufacturing features

[14]

56

The percentage cost saving in each case is also evaluated using the cost model.

The tolerance allocation at each feature , with the existing limit & relaxed optimum limit

(with % increase), the corresponding % drag increase and savings as percent of nacelle

cost are tabulated in Table-15 below[14].

The results show that feature by feature percentage changes for one nacelle with a

drag coefficient increment of 0.824% and a cost reduction of 2.26% on the nacelle cost.

This will result to 0.421% overall reduction in DOC (Direct Operating Cost) of the

transport aircraft. In this work, the effect of relaxing the manufacturing tolerances at the

eleven selected locations on the nacelle, on the performance is studied through the

estimate of corresponding drag increases and estimated the cost saving resulting by

relaxing such tolerances. Further research work is planned by the same group to extend

the study to wing and fuselage.

Table-15 Tolerance Synthesis

57

Probabilistic Approach to Free-Form Airfoil Shape Optimization Under

Uncertainty[12]

Practical experience has indicated that a deterministic optimization for discrete

operating conditions can result in dramatically inferior performance, when the actual

operating conditions are different from the (somewhat arbitrarily selected) design

conditions used during the optimization. This work is on the operating uncertainties

which will affect the performance of an aircraft. Here the airfoil shape optimization is

addressed. The specification of one or more design operating conditions allows the

engineer to use deterministic optimization schemes. In airfoil design, the objective is to

minimize drag with the specified cruise Mach number and target lift coefficient.

The concern with the shape optimization of airfoil is the sensitivity of the final

optimal design to small manufacturing errors or fluctuations in the operating conditions.

A certain variability in the operating condition, for example, cruise Mach number can not

be avoided. This type of situation can be handled effectively by adopting the methods of

robust optimization, which directly include the effects of the uncertainties on the

performance of an optimized design. So a team of researchers [12] addressed robust

design of airfoils for a transport aircraft. Here robust design technique accounts for the

impact of everyday fluctuations of parameters (such as variation in cruise Mach number)

on the overall design performance, assuming that no catastrophic failures occur.

The objective is lift constrained wave drag minimization over the Mach range

M ∈ [0.7,0.8]:

min Cd (d,M)

d∈D

Sub to Cl (d,M) ≥ Cl* over M ∈ [0.7,0.8]

Where d is the vector of design variables and D is the design space. Cl* is the minimum

lift corresponds to typical values found for commercial transport airliners. In this study,

the Mach number is the only uncertain parameter.

58

Deterministic Approach to Airfoil Shape Optimization [12]

Single Point Optimization In a deterministic context, aerodynamic shape optimization of airfoils is

concerned with obtaining the most aerodynamically favorable geometry for fixed, either

known or assumed, operating design conditions. For a practical case where the drag Cd is

to be minimized at a given fixed free stream Mach number M1:

min Cd (d,M1)

d∈D

Sub to Cl (d,M1) ≥ Cl*

This deterministic single point optimization model is not necessarily an accurate

reflection of the reality. The formulation contains no information regarding off-design

condition performance. So the drag reduction is achieved only over a narrow range of

Mach numbers, so termed as a local optimization. This is of concern if substantial

variability is associated with operating condition.

Multipoint Optimization

A straightforward, but heuristic, approach [12] to avoid localized optimization is

to consider different Mach numbers and to generalize the objective function to a linear

combination of flight conditions:

m

min Σ Wi Cd (d,Mi) d∈D i=1

Sub to Cl (d,Mj) ≥ Cl* For J = 1,2,…..m

Practical problems arise with the selection of flight condition Mi and with

the specification of the weights Wi. There are no clear theoretical principles to guide the

selection, which is in fact, largely left to the designer’s discretion. With multipoint

formulation, Cd can be realized over a wide range of Mach numbers m, however this

formulation is still unable to capture the full range of uncertainty .

59

Nondeterministic Approaches[12] During the airfoil design process, appropriate values of the design

variables d need to be selected that optimize the performance or the utility of the airfoil

design. Because each operating condition parameter may take on a range of values over

the life time of the design, it is possible to collect their histograms [12]. The impact of the

uncertainty of M on the design performance should be taken in to account when the

quality of a particular design is assessed. To tackle the issue of the uncertainty, the

problem is formulated in an explicit statistical way. In the basic problem of minimizing

drag Cd over a range of free flow Mach numbers M, while maintaining the lift Cl ≥ Cl* ,

M is now treated as a random variable and the optimization problem is now interpreted as

a statistical decision making problem. The right decision consists of the best possible

choice of the design, whether favorable or unfavorable operating conditions occur.

According to statistical decision theory, the best course of action in the presence of

uncertainty is to select the airfoil that leads to the lowest expected drag [12]. This is

known as maximum (or minimum) expected value criteria. The risk ρ associated with a

particular design d is identified as the expected value of the perceived loss associated

with the design. The best design or decision which minimize the overall risk is referred to

as Bayes decision [12]. So in this problem formulation the Baye’s risk is used in defining

the objective function using probability distribution concept as:

min ρ*

ρ* = min ∫ Cd (d, M) fM(M) dM

Sub to Cl (d,Mj) ≥ Cl* for all M, where fM(M) is the probability density function

of the free flow Mach number M. The practical problem in this formulation is that

integration is required in each of the optimization steps. This approach although

theoretically sound becomes computationally expensive. This work is an example of

using probabilistic approach in achieving robustness, provided the distribution pattern of

the noise variable is known.

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10. Conclusion

The basic underlying principles, techniques/methodology of robust design, which

is developed by Dr. Genichi Taguchi, is discussed in detail in this report with a case study

presented to appreciate the effectiveness of robust design. The importance of Parameter

design & Tolerance design as the major elements in Quality engineering are described.

The Quadratic loss function for different quality characteristics are narrated, highlighting

the fraction defective fallacy. The objective function in the robust design technique is the

Signal to Noise ratio (S/N) which is a function of the signal (mean) and the noise

(variation). Different ratios are used for different types of quality characteristics. The

levels of design parameters corresponding to the maximum S/N ratio will correspond to

the setting values at which the response will be least sensitive to the effect of noises.

Orthogonal arrays are an effective simulation aid to evaluate the relative effects of

variation in different parameters on the response with the minimum number of

experiments. Statistical techniques like ANOM (Analysis of means) and ANOVA

(analysis of variance) are the tools for analyzing the data obtained from the orthogonal

array based experiments and the relative parameter effects can be plotted using the signal

to noise ratios against the different parameter levels and the optimum combinations can

be selected for the verification experiments. Using this technique of robust design the

quality of a product or process can be improved through minimizing the effect of the

causes of variation without eliminating the causes.

The fundamental ways of improving the reliability of a product during the design

stage are by reducing the sensitivity of product’s function to the variation in product

parameters, reducing the variation in product parameters and providing redundancy.

These are analogous to parameter design and tolerance design in Quality engineering.

The option of providing redundancy is used when the cost of failure of the product is high

compared to the cost of providing redundant components or even the whole product.

The uncertainty, which is the biggest challenge in design optimization can be

effectively managed by adopting the principle of robust design and reliability based

design. In Robust Design methodology, all these causes are identified as Noises. And

Robust optimization results in the design which performs optimally under the variable (or

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uncertain) conditions over the entire lifetime of the design. The estimation of probability

distribution of a system response given the probability distribution of the random

variables in a system is specially treated as reliability based design optimization. To

appreciate the difference between RDO & RBDO, the uncertainties encountered in a

product’s life is classified into two categories based on frequency & impacts of events

Robust optimization techniques account for the impact of everyday fluctuations of

parameters on the overall design performance, assuming that no catastrophic failures

occur, where as the reliability based design takes care of extreme events which will lead

to catastrophe, by incorporating the reliability targets into the constraint function. The

mathematical formulation of robust design optimization and reliability based design

optimization are discussed. In robust design, the objective is to keep the mean on target &

minimize the variation. So the mean & standard deviation of the response will constitute

the objective function. And for the robust design for best overall performance over the

entire life time the objective function can be computed based on the joint probability

density function of the random variable x. In RBDO problems, the objective is to

optimize the expected system performance while satisfying constraints that ensure

reliable operation. Because the system parameters are not necessarily deterministic, the

objective function & constraints must be stated probabilistically. Here the constraint

functions are modified as probabilistic constraints to enable the prediction of reliability of

the system. Finally the mathematical formulation of RRBDO (Robust and Reliability

Based Design Optimization) is discussed in which the objective function of robust design

and the probabilistic constraints of reliability based design are taken together to take care

of all the uncertainty & reliability issues. The ongoing research activities related to robust

design in Aerospace systems are discussed, one in tolerance synthesis to achieve cost

reduction for the aircraft nacelle and the other in accounting uncertainties in Mach

number in airfoil design. These provide a feel of the underlying potential of robust design

techniques in cost reduction & performance improvement of the aerospace systems.

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11. References

1. Park, Sung H, “Robust Design and Analysis for Quality Engineering”, Chapman

& Hall, London, 1996.

2. Bagchi, Tapan P, “Taguchi Methods Explained: Practical Steps to Robust

Design”, Prentice Hall of India, New Delhi, 1993.

3. Madhav. S. Phadke, “Quality Engineering Using Robust Design”, Prentice Hall /

AT&T, New jersey, USA, 1989.

4. Patrick N. Koch, Timothy W. Simpson, Janet K. Allen, and Farrokh Mistree,

“Statistical Approximation for Multidisciplinary Design optimization: The

Problem of Size”, Journal of Aircraft : Special Issue on Multidisciplinary Design

Optimization, vol.36, 1999, pp 275-286.

5. Wei Chen, Kemper Lewis, “A Robust Design Approach for Achieving Flexibility

in Multidisciplinary Design”, http:/www.uic.edu/labs/ideal/pdf/Chen-Lewis.pdf

6. Luc Huyse, “Solving Problems of Optimization Under Uncertainty as Statistical

Decision Problems”, AIAA-2001-1519, 2001.

7. Wei Chen, Xiaoping Du, “Efficient Robustness and Reliability Assessment in

Engineering Design”, www.icase.edu/colloq/data/colloq.Chen.Wei2001.5.9.html

8. Robert H. Sues, Mark A.Cesare, Stephan S. Pageau, & Justin Y.-T.Wu,

“Reliability – Based Optimization Considering Manufacturing and Operational

Uncertainties”, Journal of Aerospace Engineering, October, 2001, pp 166-174

9. Brent A. Cullimore , “Reliability Engineering & Robust Design : New Methods

for Thermal / Fluid Engineering”, C & R White Paper, Revision 2, May 15, 2000

http://www.sindaworks.com/docs/papers/releng1.pdf

10. Luc Huyse, R. Michael Lewis, “Aerodynamic Shape Optimization of Two-

dimensional Airfoil Under Uncertain Conditions”, NASA/CR-2001-210648

11. Timothy W. Simpson, Jesse Peplinski, Patric N. Koch, and Janet K. Allen, “On

the Use of Statistics in Design & the Implications for Deterministic Computer

Experiments”, ASME Design Engineering Technical Conference, California,

1997.

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12. Luc Huyse, “Probabilistic Approach to Free-Form Airfoil Shape Optimization

Under Uncertainty”, AIAA Journal, Vol. 40, No.9, September, 2002

13. K.K.Choi, B.D.Youn, “Issues Regarding Design Optimization Under

Uncertainty”, http://design1.mae.ufl.edu/~nkim/index-files/choi4.pdf

14. A.K. Kundu, John Watterson, and S. Raghunathan, “Parametric Optimization of

Manufacturing Tolerances at the Aircraft Surface” Journal of Aircraft, Vol.39,

No.2, March-April 2002.