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Taguchi Method (Quality Engineering) and Robust Design

1. Concepts, definitions and basic idea

Taguchi method: Genichi Taguchi says: “the term Taguchi Method was coined in the

United States. …… However, I prefer the term quality Engineering”.

The objective of quality engineering: is to choose one from all possible designs that can

ensure the highest functional robustness of products at the lowest possible cost.

Significance of quality engineering: There are three criteria for product design, namely

quality, cost, and time to market. Only inexpensive but high-quality products can survive

in a highly competitive global market.

Methods: Methods for improving the quality of products:

o Evaluation methods for determining the quality loss of products (analysis).

o Parameter design methods for improving the quality level (or reducing quality loss)

(synthesis).

o Tolerance design methods for trade-off between the manufacturing cost of high-grade

components and the total quality loss of products.

o Optimal quality management methods for controlling the objective characteristics of

products during the manufacturing process.

Taguchi’s three-step approach: system/parameter/tolerance design.

o System or conceptual design is the process of applying basic scientific and

engineering principles in order to develop a functional design. The objective is to

obtain a workable prototype model of the system or process. This broadly corresponds

to conceptual design in the generalized model of the design process, and aims to

identify the best available technological means to provide the required functions.

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o Parameter or embodiment design is the investigation conducted in order to identify

settings that minimize or reduce the performance variation in the product or process.

It makes the system performance insensitive to the parameter variation at low cost by

selecting the optimal level settings for the control factors. In other words, it is to find

values for the controllable, or design, parameters that satisfy standard design

requirements, such as low cost and good technical performance; while at the same

time ensuring that variability in the manufacturing processes, raw materials and

operational environment have a minimal effect on the product’s designed

performance. Here detail/embodiment design is encompassed and the emphasis is on

improving quality and reliability while minimizing cost.

o Tolerance design is a method for determining tolerances that minimize the sum of

product manufacturing and lifetime cost. If the parameter design cannot achieve the

required performance variation, tolerance design can be used to reduce the variation

by reducing the tolerances based on the quality loss function. Although generally

considered to be part of the detail design stage, Taguchi views this as a distinct stage

to be used when sufficiently small variability cannot be achieved within a parameter

design. Initially, tolerances are usually taken to be fairly wide because tight tolerances

often incur high supplier or manufacturing costs. Tolerance design can be used to

identify those tolerances that, when tightened, produce the most substantial

improvement in performance.

Robust design: A design is said to be “functionally robust” if it inherently tends to

diminish the effect of the input variation on performance. Robust design is the operation

of choosing settings for product or process parameters to reduce variations in that product

or process’s response from target. Because it involves determination of parameter

settings, robust design is called ‘parameter design’. Variations in the objective functions

of products or technologies are primarily due to three sources: environmental effects,

deteriorative effects, and manufacturing imperfections. The purpose of robust design is to

make the products and the processes less sensitive to these effects.

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Quality Loss: The lack of quality in a product can be quantified in terms of the total loss

to society from the time the product is shipped to the customer. The loss may be due to

undesirable side-effects or variations of the product’s functional characteristic from its

desired target. The loss due to variations in a product’s performance can be reasonably

approximated by a quadratic function. Denote the objective characteristic y, its target

value m, and financial loss (or quality loss) L(y). Because L(y = m) = 0 and L’(m) = 0,

from Taylor series expansion of the loss function we have:

L y L m L m y m L m y m k y m( ) ( ) ' ( )( ) "( )( ) / ( ) 2 22 (1)

The quality loss can be viewed as a measure of customer’s dissatisfaction with the

product.

The y is a random variable with a certain distribution. Let , 2 be the mean and the

variance of this (normal) distribution, then the expected loss (average loss) is:

Q k m [( ) ] 2 2 (2)

This expected loss has two components. The first represents the deviation of the average

of the characteristic from its target, and the second represents the variability of the

characteristics. The first one is relatively easy to eliminate; while the second one is rather

hard to reduce. Three methods for reducing it are:

o screening out bad products,

o discovering the cause of malfunction and eliminating it, and

o robust design.

Taguchi proposes a two-step procedure to minimize the average loss: first to minimize a

loss measure, and then to shift the mean to target.

2. Parameter (Robust) Design

2.1 Goal

The goal of robust design is to design a system so that its performance is insensitive to

uncontrollable (noise) variables. This is done by systematically investigating the

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relationship between appropriate control factors and noise variables, typically through

off-line experiments, and judiciously choosing the settings of control factors to make the

system be robust to uncontrollable noise variations. In other words, the goal of robust

design is to get the actual performance close to the ideal status with as little variation as

possible.

2.2 Procedure

The implementation of the robust design method includes the following operational steps.

o state the problem and objective.

o identify responses, control factors, and sources of noise.

o plan an experiment to study the relationships between responses and control and noise

factors.

o run the experiment and collect the data. Analyze the data to determine the control

factor settings that predict improvement on the product or process design.

o run a small experiment to confirm if the control factor settings determined in step 4

actually improve the product or process design. If so, adopt the control factor settings

and consider iteration for further improvement. If not, correct or modify the

assumptions and go back to step 2.

2.3 Response, parameters, factors, and parameter design problem

o Response: A system’s performance or output, denote Y.

o Target value for the response variable Y: t.

o Signal factor (M): affect the average levels of the response, whose levels are set

by the user to attain the target performance.

o Control factor (d): affect the variability in the response, whose values are

specified by the designer. Each control factor in experimentation can take more

than one value; these are called levels. The objective of design activity is to

determine the optimal level of control factors.

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o Controllable parameters: = signal factors + control factors

o Noise factors (variables) (n): The noise factors are usually expensive or

impossible to control duration productions but may be controllable during

experimentation. So, they are also called uncontrollable factors. The noise factors

influence the output y.

Example: the braking system of an automobile. The system’s response or output is the

amount of torque generated while braking. The input or signal factor is the pedal force.

The control factors include pad material, pad taper, pad shape, rotor material, and so on

while the uncontrollable noise variables include road surface conditions, tire conditions,

speed, driver’s skills, and so on. The goal of robust design in this example is to choose

the control factor settings so that the system is robust to uncontrollable noise variation

over a range of possible signal factor values (pedal force).

Parameter design problem: let x x x xn ( , ,..., )1 2 be a vector of design variables for a

product or process, and y g x x xn ( , ,..., )1 2 represent an output characteristic of interest.

The nominal values for x x x xn ( , ,..., )1 2 are controllable parameters and denote by

( , ,..., )1 2 n . We have g tn( , ,..., ) 1 2 , where t is the target value of y. The

noise factors cause the values of x deviate from their respective nominal values; hence

causing y deviate from its target value. In this context, x’s are random variables with

means ( , ,..., )1 2 n ; y is also a random variable with mean and variance y y, . The

problem is to find ( , ,..., )1 2 n in order to minimize 2y subject to

g tn( , ,..., ) 1 2 and ( , ,..., ) 1 2 n acceptable values .

Methods: There are basically three different methods to approximate 2y at a given

point . These are:

o Monte Carlo simulation: computer simulations are used to predict product

performance.

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o statistical design of experiments: the relationship between the design parameters

and performance are investigated on physical prototypes.

o function approximation, for example Taylor’s series expansion.

2.4 Design of the experiment

The relationship between control, noise, and signal factors are usually unknown and very

complex. Thus, physical or computer experiments, typically conducted in off-line

environments, are used to investigate these relationships.

The plan for a robust design experiment has two parts, a ‘control array’ and a ‘noise

plan’. The control array is the plan for varying the settings of the control parameters in the

experiment. The rows of this array correspond to different configurations of the product

or manufacturing process. The noise plan is the scheme for measuring the effects of noise

variables. This plan might be an array, called a ‘noise array’, specifying how noise

variables will be explicitly varied in the experiment. However, the noise plan is often a

plan for taking multiple measurements of the response characteristic in a manner

designed to capture the effect of the underlying noise variables.

Several tools are used to facilitate construction of the control array. The first one is the

orthogonal array, a table of integers whose columns are pairwise balanced. That is, in any

two columns every ordered pair of integers occurs the same number of times. To actually

construct the control array, control parameters are assigned to the columns of an

orthogonal array, and the integers in these columns are translated into the actual test

settings of the assigned parameters. The unassigned columns are deleted from the array.

In assigning control parameters to columns of some orthogonal arrays the experimenter

can allow for certain interactions by choosing the columns for assignment carefully. A

typical experimental layout is presented in Table 1.

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The purpose of the noise plan is to systematically introduce the ‘real world’ variation in

the experiment. Sometimes the most important sources of noise can be directly varied in

the experiment. The noise array is a plan for varying the level of those noise variables that

can be directly introduced in the experiment. Like the control array the noise array can be

constructed using an orthogonal array. Interactions between the noise variables are

generally not a concern since the objective is to induce response variations and not to

model the noise variables’ effect in detail. It is not always possible directly to introduce

important noise variables in the experiment. If a computer model for the process is

available, many of these noises might be studied directly. But if no computer model is

available, and physical experiments are being used, it is usually very difficult to directly

vary the values of these noise variables. Instead, noise is introduced indirectly by taking

replications of the responses in a way that is designed to capture the effects of these noise

variables. Using a noise array to introduce noise in an experiment can contribute greatly

to the cost of the whole experiment, because the same noise array is followed for each

control array row. An alternative is to combine the control parameters and noise variables

in a single array, estimating only those interactions which are likely to exist. It is a topic

to develop effective approaches for studying noise more economically in robust design

experiments.

The most frequently used experimental designs in robust design experiments are:

o fractional factorial designs

o controlled experiments

Fractional factorial experiments are factorial experiments in which not all levels of all

variables are tested in combination with all levels of all other variables. A sample, or

fraction, of all theoretical combinations is tested. This approach is taken when a full

series of factorial experiments would require a prohibitively large number of runs.

Controlled experiment methods: one of the most significant approaches to design or

process problems, particularly when there are a number of variables each of which has

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some effect on the production process or the products composition and specifications, is

the use of controlled experiments. In this approach, the engineer conducts a series of tests

to evaluate the effect of those factors believed to be significant in influencing the process

or product that is being designed. This approach allows a number of variables to be

evaluated at one time. Traditionally, when engineers want to optimize some process or

design variables, they conduct experiments in which all other variables are held constant

while various levels of the variables being tested are evaluated. With the Taguchi and

other controlled experiment methods, many process variables can be tested

simultaneously. By mathematical analysis, the engineer determines which setting of each

variable is optimum. The number of test runs needed for full optimization is thus greatly

reduced compared to what it would be in the traditional “one variable at a time” method.

The Taguchi’s methods are a variety of controlled experiments. It should be noted that his

method is however a simplified approach. He utilizes fractional factorials and typically

assumes that no interaction exists between variables. His method is easy to use, but, if the

assumptions are not correct, it can yield incorrect results. Critics have stated that his

method is best suited for initial studies of processes and product designs that have

considerable room for improvement down the road.

2.5 Data analysis

The first step in Taguchi’s approach is the identification of the ideal function. This

function describes the ideal relationship between the input (signal factor) and the output

(response) of the system. Generally, the ideal function is linear. The actual performance

may deviate from this ideal function. To identify the appropriate measure of the system

performance and the resulting ideal function, the engineering design team must have a

good understanding of the functional and engineering aspects of the system.

For given levels of the control factors and other factors, the performance will be a

function of the signal factor M and noise factors e. that is

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Y = f(M) + g(e) (3)

where f(M) is the predictable and desirable part of the performance and is called the

useful part, and g(e) is the unpredictable and undesirable part and is called the harmful

part. If the second part is small, then it is “robustly designed”. In general, it is difficult to

obtain an analytical model, i.e., f and g functions. Through experimentation, we can find

both parts.

For the data in Table 1, let Yijk denote the observed response corresponding to the i-th

setting of the control factors, j-th setting of the signal factor, and k-th setting of the noise

factors. Under the assumption of a linear ideal function with no intercept (i.e. no constant

term), we have the following model

Y Mijk i j ijk (4)

where i is the sensitivity measure, which describes how significant the signal factor is

given a particular control factor setting, and )var(2

ijki is the dispersion, which

describes the deviation of the response away from the average response. Both depend on

the control factor setting. The optimal control factors setting i should be such that i is

largest, and 2

i is the smallest.

Taguchi’s measure of robustness is the so-called signal-to-noise ratio (SN ratio), which is

defined as

)/(log10,/ 22

10

22

or (5)

This definition is based on the following principle: the SN ratio should be

dimensionless.

2.6 Example

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In the example shown in the table 1, there are 8 control factors, each at two levels. The

outer array consists of 3 levels of a signal factor crossed with 2 levels of a noise factor.

The b and s in the table are i and i . The can be obtained by regressing y = M. For

example:

Y: 119.24 123.85 …… 365.48

M: 100 100 …… 300

See the first and third rows in Table 1. The result of regression is shown in Table 2, from

which one can obtain the value i and value for i , and they are put in Table 1, denoted

by b and s, respectively. The last column in Table 1 is the SN ratio, which measures the

robustness.

To determine the control factor setting for the best robustness, one approach is to choose

the combination of the levels of the control factors in Table 1, which corresponds to the

largest SN ratio. This approach works if one only considers the SN ratio. However, the

largest SN ratio may result from the small sensitivity β and much smaller dispersion ,

which is not the best, as the ideal one is the largest sensitivity and smallest dispersion.

The point here is that the optimal model for determining the control factor setting may be

best with two objectives: maximizing SN ratio and maximizing the sensitivity. For

solving such a two objective optimization problem, the control factor setting that makes

the maximal SN ratio may not be to make the maximal sensitivity. To resolve this

conflict, one idea is to study the significant control factor to the SN ratio and to the

sensitivity to see if an uncoupled situation may occur to the SN ratio and sensitivity via

the control factor. That is to say, if one set of control factors is significant to the SN ratio,

while the other set of control factors is significant to the sensitivity, then, the problem is

uncoupled. In this case, the optimization can be carried out separately for the SN ratio and

sensitivity, respectively.

To identify the control factors that are significant to the SN ratio, one fits a linear model

to the estimated SN ratios as a function of the control factors:

a a A a H0 1 8 (6)

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See Table 3. The greater the coefficient, the more significant the corresponding control

factor. It can be seen from Table 3 that D, G and A (more to less) are the significant

factors to the SN ratio.

To identify the significant factor to the sensitivity, one fits a linear model to the i ’s as a

function of the control factors:

HbAbb 810 (7)

See Table 4. From Table 4 it can be found that A and D (more to less) are the most

significant factors to the sensitivity.

Therefore, one can see that there is the following relation between the SN ratio and

sensitivity.

In the above process, however, decision on the significant factor involves the subjective

judgement. For instance, to the SN ratio, we pick up D, G, A factors, but we may also

pick up D only. Similarly, to the sensitivity, we may just pick up A. In this case (D for the

SN ratio, and A for the sensitivity), the problem is uncoupled.

Another note is that even a particular factor makes the SN ratio and sensitivity coupled,

but to make them an optimum, the factor may pick up the same level. That is, this factor

does not introduce conflict to the SN ratio and sensitivity, though the factor make them

coupled.

SN ratio

Sensitivity

A

D

G

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Figure 1 indicates the magnitudes of the effects of the factors on the SN ratio. To make

the SN ratio large, we get D=-1, G=-1, and A=-1.

Figure 2 indicates the magnitudes of the effects of the factors on the sensitivity. To make

the sensitivity large, we get A=1 and D=1.

It is clear that the control factors A and D create the conflict between the SN ratio and

sensitivity.

To resolve the conflict at this point, one idea is to modify the definition of the SN ratio.

One proposal in literature is to define the

)/(log10,/ 2

10

2

or (8)

From the above, we get

)1

(2 = 2 c . Taking the logarithm to both sides, we

get:

2/])()([)( LncLnLn , and

)(5.0)(5.0)( LncLnLn (9)

From (9), we apply the linear regression, where β is taken from Table 1, i.e., b, and σ is

taken from Table 1, i.e., s. Thus, we get =3.58. The SN ratio, SN’ is calculated by Eq.

(8). We have the result shown in Table 5. At this time, G is the most significant factor to

the SN’ ratio. As such, the problem is decoupled, i.e., G for the SN’ ratio, and A&D for

the sensitivity.

SN’ ratio

Sensitivity

G

D

G

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3. Tolerance design

Product’s responses are determined by a large number of variables. Some of these

variables, i.e., control parameters, are under the designer’s control. An assumption

usually made in robust design is that the cost of producing the product at each level of

any given control parameter is the same. The responses are also influenced by variables,

i.e., noise variables, which are difficult or expensive for the designer to control. In theory,

some of these noise variables could be controlled. Taguchi calls this activity controlling

the variables tolerance design. However, these efforts increase the cost. In contrast,

robust design is a way to reduce response variation by using the controllable parameters

to dampen the effects of the hard-to-control noise variables. If application of robust

design does not sufficiently reduce response variation, designers can further reduce this

variation using tolerance design.

4. Comments and Issues

4.1 Some comments on Taguchi Method

Taguchi’s engineering ideas are sound and worth pursuing. His approach first computes

estimates of loss measures (e.g. the SN ratio or the sample mean and standard deviation,

all are called summary measures) and then determine the ‘optimal’ factor settings by

fitting a model to these loss estimates. This approach is called the summary measure

approach.

Another modeling approach is to first model the observed response, and then determine

the ‘optimal’ factor settings from the fitted response model. This approach is called the

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response model approach. Although the response model approach used in the multi-step

procedure has several advantages over the loss model (or summary measure) approach,

the former approach requires more work and depends more critically on the adequacy of

the fitted model.

Previous efforts have concentrated on measuring performance with Taguchi’s signal-to-

noise ratio. It is demonstrated that maximization of the signal-to-noise ratio may not

always lead to minimum quality loss.

Also, there are more efficient and informative alternatives to many of his particular

statistical techniques. Generally speaking, controversy has arisen over

the specific experimental design used,

the particular process characteristics chosen to be modeled, and

the method of modeling.

4.2 Issues

o Multiple (response) characteristics problem,

o Dynamic (vs. static) response problem. Static system’s output has a fixed target value.

A dynamic system’s target value depends on the input signal set.

o The response is a random variable, that is, repeated measurements yield different

responses in the same control and noise variables.

o Another issue is which approach is better between summary & directly model

response approaches.

o Another contention is whether or not interactive effects should be estimated.

References

1. Robust planning and analysis of experiments, Christine, H. Muller, Main Lib-5th flr,

QA279.M85 1997.

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2. Robust controller design using normalized coprime factor plant descriptions, D.C.

McFarlane, K. Glover, Main Lib-5th flr, TJ 213. G525 1990.

3. Taguchi on robust technology development: bringing quality engineering upstream,

by Genichi Taguchi; translated by Shih-Chung Tsai, Engin Lib-stacks, TS156.

T23583 1993.

4. Systematic mechanical designing: a cost and management perspective, Mahendra S.

Hundal, Engin Lib-stacks, TS171. H864 1997.

5. Design for Excellence (pp. 24-29), James G. Bralla, Engin Lib-stacks, TS171. B69

1996.

6. Statistical case studies for industrial process improvement, Veronica Czitrom &

Patrick D. Spagon, Engin Lib-stacks, TS156.8 C985 1997.

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