HOME

TheInfoList



OR:

Taguchi methods () are
statistical Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
methods, sometimes called robust design methods, developed by
Genichi Taguchi was an engineer and statistician. From the 1950s on, Taguchi developed a methodology for applying statistics to improve the quality of manufactured goods. Taguchi methods have been controversial among some conventional Western statisticians, but ...
to improve the quality of manufactured goods, and more recently also applied to engineering, biotechnology, marketing and advertising. Professional
statistician A statistician is a person who works with Theory, theoretical or applied statistics. The profession exists in both the private sector, private and public sectors. It is common to combine statistical knowledge with expertise in other subjects, a ...
s have welcomed the goals and improvements brought about by Taguchi methods, particularly by Taguchi's development of designs for studying variation, but have criticized the
inefficiency Efficiency is the often measurable ability to avoid making mistakes or wasting materials, energy, efforts, money, and time while performing a task. In a more general sense, it is the ability to do things well, successfully, and without waste. ...
of some of Taguchi's proposals. Taguchi's work includes three principal contributions to statistics: *A
specific Specific may refer to: * Specificity (disambiguation) * Specific, a cure or therapy for a specific illness Law * Specific deterrence, focussed on an individual * Specific finding, intermediate verdict used by a jury in determining the final ...
loss function In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost ...
*The philosophy of ''off-line quality control''; and *Innovations in the
design of experiments The design of experiments (DOE), also known as experiment design or experimental design, is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. ...
.


Loss functions


Loss functions in the statistical theory

Traditionally, statistical methods have relied on mean-unbiased estimators of treatment effects: Under the conditions of the
Gauss–Markov theorem In statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors) states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in ...
,
least squares The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
estimators have minimum variance among all mean-unbiased linear estimators. The emphasis on comparisons of means also draws (limiting) comfort from the
law of large numbers In probability theory, the law of large numbers is a mathematical law that states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. More formally, the law o ...
, according to which the
sample mean The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or me ...
s
converge Converge may refer to: * Converge (band), American hardcore punk band * Converge (Baptist denomination), American national evangelical Baptist body * Limit (mathematics) In mathematics, a limit is the value that a function (or sequence) app ...
to the true mean. Fisher's textbook on the
design of experiments The design of experiments (DOE), also known as experiment design or experimental design, is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. ...
emphasized comparisons of treatment means. However, loss functions were avoided by
Ronald A. Fisher Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who a ...
.


Taguchi's use of loss functions

Taguchi knew
statistical theory The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics. The theory covers approaches to statistical-decision problems and to statistica ...
mainly from the followers of
Ronald A. Fisher Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who a ...
, who also avoided
loss function In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost ...
s. Reacting to Fisher's methods in the
design of experiments The design of experiments (DOE), also known as experiment design or experimental design, is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. ...
, Taguchi interpreted Fisher's methods as being adapted for seeking to improve the
mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
outcome of a
process A process is a series or set of activities that interact to produce a result; it may occur once-only or be recurrent or periodic. Things called a process include: Business and management * Business process, activities that produce a specific s ...
. Indeed, Fisher's work had been largely motivated by programmes to compare agricultural yields under different treatments and blocks, and such experiments were done as part of a long-term programme to improve harvests. However, Taguchi realised that in much industrial production, there is a need to produce an outcome ''on target'', for example, to
machine A machine is a physical system that uses power to apply forces and control movement to perform an action. The term is commonly applied to artificial devices, such as those employing engines or motors, but also to natural biological macromol ...
a hole to a specified diameter, or to manufacture a
cell Cell most often refers to: * Cell (biology), the functional basic unit of life * Cellphone, a phone connected to a cellular network * Clandestine cell, a penetration-resistant form of a secret or outlawed organization * Electrochemical cell, a de ...
to produce a given
voltage Voltage, also known as (electrical) potential difference, electric pressure, or electric tension, is the difference in electric potential between two points. In a Electrostatics, static electric field, it corresponds to the Work (electrical), ...
. He also realised, as had
Walter A. Shewhart Walter Andrew Shewhart (pronounced like "shoe-heart"; March 18, 1891 – March 11, 1967) was an American physicist, engineer and statistician. He is sometimes also known as the ''grandfather of Statistical process control, statistical quality con ...
and others before him, that excessive variation lay at the root of poor manufactured quality and that reacting to individual items inside and outside specification was counterproductive. He therefore argued that
quality engineering Quality engineering is the discipline of engineering concerned with the principles and practice of product and service quality assurance and quality control, control. In software development, it is the management, development, operation and main ...
should start with an understanding of quality costs in various situations. In much conventional
industrial engineering Industrial engineering (IE) is concerned with the design, improvement and installation of integrated systems of people, materials, information, equipment and energy. It draws upon specialized knowledge and skill in the mathematical, physical, an ...
, the quality costs are simply represented by the number of items outside specification multiplied by the cost of rework or scrap. However, Taguchi insisted that manufacturers broaden their horizons to consider ''cost to society''. Though the short-term costs may simply be those of non-conformance, any item manufactured away from nominal would result in some loss to the customer or the wider community through early wear-out; difficulties in interfacing with other parts, themselves probably wide of nominal; or the need to build in safety margins. These losses are
externalities In economics, an externality is an indirect cost (external cost) or indirect benefit (external benefit) to an uninvolved third party that arises as an effect of another party's (or parties') activity. Externalities can be considered as unpriced ...
and are usually ignored by manufacturers, which are more interested in their private costs than
social cost Social cost in neoclassical economics is the sum of the private costs resulting from a transaction and the costs imposed on the consumers as a consequence of being exposed to the transaction for which they are not compensated or charged. In other w ...
s. Such externalities prevent markets from operating efficiently, according to analyses of
public economics Public economics ''(or economics of the public sector)'' is the study of government policy through the lens of economic efficiency and Equity (economics), equity. Public economics builds on the theory of welfare economics and is ultimately used as ...
. Taguchi argued that such losses would inevitably find their way back to the originating corporation (in an effect similar to the
tragedy of the commons The tragedy of the commons is the concept that, if many people enjoy unfettered access to a finite, valuable resource, such as a pasture, they will tend to overuse it and may end up destroying its value altogether. Even if some users exercised vo ...
), and that by working to minimise them, manufacturers would enhance brand reputation, win markets and generate profits. Such losses are, of course, very small when an item is near to negligible. Donald J. Wheeler characterised the region within specification limits as where we ''deny that losses exist''. As we diverge from nominal, losses grow until the point where ''losses are too great to deny'' and the specification limit is drawn. All these losses are, as
W. Edwards Deming William Edwards Deming (October 14, 1900 – December 20, 1993) was an American business theorist, composer, economist, industrial engineer, management consultant, statistician, and writer. Educated initially as an electrical engineer and later ...
would describe them, ''unknown and unknowable'', but Taguchi wanted to find a useful way of representing them statistically. Taguchi specified three situations: #Larger the better (for example, agricultural yield); #Smaller the better (for example,
carbon dioxide Carbon dioxide is a chemical compound with the chemical formula . It is made up of molecules that each have one carbon atom covalent bond, covalently double bonded to two oxygen atoms. It is found in a gas state at room temperature and at norma ...
emissions); and #On-target, minimum-variation (for example, a mating part in an assembly). The first two cases are represented by simple
monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
loss functions. In the third case, Taguchi adopted a squared-error loss function for several reasons: *It is the first "symmetric" term in the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
expansion of
real analytic In mathematics, an analytic function is a function (mathematics), function that is locally given by a convergent series, convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are ...
loss-functions. *Total loss is measured by the
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
. For
uncorrelated In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, \operatorname ,Y= \operatorname Y- \operatorname \operatorname /math>, is zero. If two variables are uncorrelated, ther ...
random variables A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' in its mathematical definition refers ...
, as variance is additive the total loss is an additive measurement of cost. *The squared-error loss function is widely used in
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, following Gauss's use of the squared-error loss function in justifying the method of
least squares The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
.


Reception of Taguchi's ideas by statisticians

Though many of Taguchi's concerns and conclusions are welcomed by statisticians and
economist An economist is a professional and practitioner in the social sciences, social science discipline of economics. The individual may also study, develop, and apply theories and concepts from economics and write about economic policy. Within this ...
s, some ideas have been especially criticized. For example, Taguchi's recommendation that industrial experiments maximise some ''
signal-to-noise ratio Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to noise power, often expressed in deci ...
'' (representing the magnitude of the
mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
of a process compared to its variation) has been criticized.


Off-line quality control


Taguchi's rule for manufacturing

Taguchi realized that the best opportunity to eliminate variation of the final product quality is during the design of a product and its manufacturing process. Consequently, he developed a strategy for quality engineering that can be used in both contexts. The process has three stages: *System design *Parameter (measure) design *Tolerance design


System design

This is design at the conceptual level, involving
creativity Creativity is the ability to form novel and valuable Idea, ideas or works using one's imagination. Products of creativity may be intangible (e.g. an idea, scientific theory, Literature, literary work, musical composition, or joke), or a physica ...
and
innovation Innovation is the practical implementation of ideas that result in the introduction of new goods or service (economics), services or improvement in offering goods or services. ISO TC 279 in the standard ISO 56000:2020 defines innovation as "a n ...
.


Parameter design

Once the concept is established, the nominal values of the various dimensions and design parameters need to be set, the detail design phase of conventional engineering. Taguchi's radical insight was that the exact choice of values required is under-specified by the performance requirements of the system. In many circumstances, this allows the parameters to be chosen so as to minimize the effects on performance arising from variation in manufacture, environment and cumulative damage. This is sometimes called robustification. Robust parameter designs consider controllable and uncontrollable noise variables; they seek to exploit relationships and optimize settings that minimize the effects of the noise variables.


Tolerance design

With a successfully completed ''parameter design'', and an understanding of the effect that the various parameters have on performance, resources can be focused on reducing and controlling variation in the critical few dimensions.


Design of experiments

Taguchi developed his experimental theories independently. Taguchi read works following
R. A. Fisher Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who ...
only in 1954.


Outer arrays

Taguchi's designs aimed to allow greater understanding of variation than did many of the traditional designs from the
analysis of variance Analysis of variance (ANOVA) is a family of statistical methods used to compare the Mean, means of two or more groups by analyzing variance. Specifically, ANOVA compares the amount of variation ''between'' the group means to the amount of variati ...
(following Fisher). Taguchi contended that conventional sampling is inadequate here as there is no way of obtaining a
random sample In this statistics, quality assurance, and survey methodology, sampling is the selection of a subset or a statistical sample (termed sample for short) of individuals from within a statistical population to estimate characteristics of the whole ...
of future conditions. In Fisher's
design of experiments The design of experiments (DOE), also known as experiment design or experimental design, is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. ...
and
analysis of variance Analysis of variance (ANOVA) is a family of statistical methods used to compare the Mean, means of two or more groups by analyzing variance. Specifically, ANOVA compares the amount of variation ''between'' the group means to the amount of variati ...
, experiments aim to reduce the influence of nuisance factors to allow comparisons of the mean treatment-effects. Variation becomes even more central in Taguchi's thinking. Taguchi proposed extending each experiment with an "outer array" (possibly an
orthogonal array In mathematics, an orthogonal array (more specifically, a fixed-level orthogonal array) is a "table" (array) whose entries come from a fixed finite set of symbols (for example, ), arranged in such a way that there is an integer ''t'' so that for ev ...
); the "outer array" should simulate the random environment in which the product would function. This is an example of judgmental sampling. Many quality specialists have been using "outer arrays". Later innovations in outer arrays resulted in "compounded noise." This involves combining a few noise factors to create two levels in the outer array: First, noise factors that drive output lower, and second, noise factors that drive output higher. "Compounded noise" simulates the extremes of noise variation but uses fewer experimental runs than would previous Taguchi designs.


Management of interactions


Interactions, as treated by Taguchi

Many of the orthogonal arrays that Taguchi has advocated are saturated arrays, allowing no scope for estimation of interactions. This is a continuing topic of controversy. However, this is only true for "control factors" or factors in the "inner array". By combining an inner array of control factors with an outer array of "noise factors", Taguchi's approach provides "full information" on control-by-noise interactions, it is claimed. Taguchi argues that such interactions have the greatest importance in achieving a design that is robust to noise factor variation. The Taguchi approach provides more complete interaction information than typical
fractional factorial designs In statistics, a fractional factorial design is a way to Design of experiments, conduct experiments with fewer experimental runs than a full factorial design. Instead of testing every single combination of factors, it tests only a carefully sel ...
, its adherents claim. *Followers of Taguchi argue that the designs offer rapid results and that interactions can be eliminated by proper choice of quality characteristics. That notwithstanding, a "confirmation experiment" offers protection against any residual interactions. If the quality characteristic represents the energy transformation of the system, then the "likelihood" of control factor-by-control factor interactions is greatly reduced, since "energy" is "additive".


Inefficiencies of Taguchi's designs

* Interactions are part of the real world. In Taguchi's arrays, interactions are confounded and difficult to resolve. Statisticians in
response surface methodology In statistics, response surface methodology (RSM) explores the relationships between several explanatory variables and one or more response variables. RSM is an empirical model which employs the use of mathematical and statistical techniques to r ...
(RSM) advocate the "sequential assembly" of
designs A design is the concept or proposal for an object, process, or system. The word ''design'' refers to something that is or has been intentionally created by a thinking agent, and is sometimes used to refer to the inherent nature of something ...
: In the RSM approach, a screening design is followed by a "follow-up design" that resolves only the confounded interactions judged worth resolution. A second follow-up design may be added (time and resources allowing) to explore possible high-order univariate effects of the remaining variables, as high-order univariate effects are less likely in variables already eliminated for having no linear effect. With the economy of screening designs and the flexibility of follow-up designs, sequential designs have great
statistical efficiency In statistics, efficiency is a measure of quality of an estimator, of an experimental design, or of a hypothesis testing procedure. Essentially, a more efficient estimator needs fewer input data or observations than a less efficient one to achiev ...
. The sequential designs of
response surface methodology In statistics, response surface methodology (RSM) explores the relationships between several explanatory variables and one or more response variables. RSM is an empirical model which employs the use of mathematical and statistical techniques to r ...
require far fewer experimental runs than would a sequence of Taguchi's designs. Statisticians have developed designs that enable experiments to use fewer replications (or experimental runs), enabling savings over Taguchi's proposed designs: * * Box, G. E. P. and Draper, Norman. 2007. ''Response Surfaces, Mixtures, and Ridge Analyses'', Second Edition f ''Empirical Model-Building and Response Surfaces'', 1987 Wiley. * * * * * R. H. Hardin and N. J. A. Sloane
"A New Approach to the Construction of Optimal Designs", ''Journal of Statistical Planning and Inference'', vol. 37, 1993, pp. 339-369
* R. H. Hardin and N. J. A. Sloane
"Computer-Generated Minimal (and Larger) Response Surface Designs: (I) The Sphere"
* R. H. Hardin and N. J. A. Sloane
"Computer-Generated Minimal (and Larger) Response Surface Designs: (II) The Cube"
* ** ** Box-Draper, Atkinson-Donev-Tobias, Goos, and Wu-Hamada discuss the sequential assembly of designs.


Assessment

Genichi Taguchi has made valuable contributions to statistics and
engineering Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
. His emphasis on ''loss to society'', techniques for investigating variation in experiments, and his overall strategy of system, parameter and tolerance design have been influential in improving manufactured quality worldwide.


See also

* * * * * * * * *


References


Bibliography

* * Box, G. E. P. and Draper, Norman. 2007. ''Response Surfaces, Mixtures, and Ridge Analyses'', Second Edition f ''Empirical Model-Building and Response Surfaces'', 1987 Wiley. * * * * * R. H. Hardin and N. J. A. Sloane
"A New Approach to the Construction of Optimal Designs", ''Journal of Statistical Planning and Inference'', vol. 37, 1993, pp. 339-369
* R. H. Hardin and N. J. A. Sloane
"Computer-Generated Minimal (and Larger) Response Surface Designs: (I) The Sphere"
* R. H. Hardin and N. J. A. Sloane
"Computer-Generated Minimal (and Larger) Response Surface Designs: (II) The Cube"
* ** ** * * Moen, R D; Nolan, T W & Provost, L P (1991) ''Improving Quality Through Planned Experimentation'' * *Bagchi Tapan P and Madhuranjan Kumar (1992) ''Multiple Criteria Robust Design of Electronic Devices'', Journal of Electronic Manufacturing, vol 3(1), pp. 31–38 * * Montgomery, D. C. ''Ch. 9'', 6th Edition f ''Design and Analysis of Experiments'', 2005 Wiley. {{DEFAULTSORT:Taguchi Methods Manufacturing Quality control tools Systems engineering Design of experiments Japanese inventions