Taguchi methods
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Taguchi methods ( ja, タグチメソッド) are statistical methods, sometimes called robust design methods, developed by
Genichi Taguchi was an engineer and statistician. From the 1950s onwards, Taguchi developed a methodology for applying statistics to improve the quality of manufactured goods. Taguchi methods have been controversial among some conventional Western statisticians, ...
to improve the quality of manufactured goods, and more recently also applied to engineering, biotechnology, marketing and advertising. Professional statisticians 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 wasting materials, energy, efforts, money, and time in doing something or in producing a desired result. In a more general sense, it is the ability to do things well, successfully, and without ...
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 fina ...
loss function *The philosophy of ''off-line quality control''; and *Innovations in the
design of experiments The design of experiments (DOE, DOX, 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. The term is generally associ ...
.


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 the ...
, least squares 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, according to which the
sample mean The sample mean (or "empirical mean") and the sample covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or mean value) of a sample of numbers taken from a larger popu ...
s
converge Converge may refer to: * Converge (band), American hardcore punk band * Converge (Baptist denomination), American national evangelical Baptist body * Limit (mathematics) * Converge ICT, internet service provider in the Philippines *CONVERGE CFD s ...
to the true mean. Fisher's textbook on the
design of experiments The design of experiments (DOE, DOX, 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. The term is generally associ ...
emphasized comparisons of treatment means. However, loss functions were avoided by Ronald A. Fisher.


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, who also avoided loss functions. Reacting to Fisher's methods in the
design of experiments The design of experiments (DOE, DOX, 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. The term is generally associ ...
, Taguchi interpreted Fisher's methods as being adapted for seeking to improve the
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the '' ari ...
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 se ...
. 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 hole to a specified diameter, or to manufacture a
cell Cell most often refers to: * Cell (biology), the functional basic unit of life Cell may also refer to: Locations * Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery ...
to produce a given
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
. He also realised, as had Walter A. Shewhart 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 control. In software development, it is the management, development, operation and maintenance of IT sys ...
should start with an understanding of
quality costs In process improvement efforts, quality costs or cost of quality is a means to quantify the total cost of quality-related efforts and deficiencies. It was first described by Armand V. Feigenbaum in a 1956 Harvard Business Review article. Prior to ...
in various situations. In much conventional
industrial engineering Industrial engineering is an engineering profession that is concerned with the optimization of complex processes, systems, or organizations by developing, improving and implementing integrated systems of people, money, knowledge, information a ...
, 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 or external cost is an indirect cost or benefit to an uninvolved third party that arises as an effect of another party's (or parties') activity. Externalities can be considered as unpriced goods involved in either co ...
and are usually ignored by manufacturers, which are more interested in their
private cost In Production (economics), production, research, retail, and accounting, a cost is the value of money that has been used up to produce something or deliver a service, and hence is not available for use anymore. In business, the cost may be one o ...
s 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. Public economics builds on the theory of welfare economics and is ultimately used as a tool to improve s ...
. 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 Tragedy (from the grc-gre, τραγῳδία, ''tragōidia'', ''tragōidia'') is a genre of drama based on human suffering and, mainly, the terrible or sorrowful events that befall a main character. Traditionally, the intention of tragedy i ...
), 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 engineer, statistician, professor, author, lecturer, and management consultant. Educated initially as an electrical engineer and later specializing in mathematical ...
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 ( chemical formula ) is a chemical compound made up of molecules that each have one carbon atom covalently double bonded to two oxygen atoms. It is found in the gas state at room temperature. In the air, carbon dioxide is trans ...
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 loss-functions. *Total loss is measured by the
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
. 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, as variance is additive the total loss is an additive measurement of cost. *The squared-error loss function is widely used in statistics, following Gauss's use of the squared-error loss function in justifying the method of least squares.


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 science discipline of economics. The individual may also study, develop, and apply theories and concepts from economics and write about economic policy. Within this field there are ...
s, some ideas have been especially criticized. For example, Taguchi's recommendation that industrial experiments maximise some '' signal-to-noise ratio'' (representing the magnitude of the
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the '' ari ...
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 a phenomenon whereby something new and valuable is formed. The created item may be intangible (such as an idea, a scientific theory, a musical composition, or a joke) or a physical object (such as an invention, a printed Literature ...
and
innovation Innovation is the practical implementation of ideas that result in the introduction of new goods or services or improvement in offering goods or services. ISO TC 279 in the standard ISO 56000:2020 defines innovation as "a new or changed entit ...
.


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 collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician ...
(following Fisher). Taguchi contended that conventional sampling is inadequate here as there is no way of obtaining a
random sample In statistics, quality assurance, and survey methodology, sampling is the selection of a subset (a statistical sample) of individuals from within a statistical population to estimate characteristics of the whole population. Statisticians atte ...
of future conditions. In Fisher's
design of experiments The design of experiments (DOE, DOX, 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. The term is generally associ ...
and
analysis of variance Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician ...
, 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 is a "table" (array) whose entries come from a fixed finite set of symbols (typically, ), arranged in such a way that there is an integer ''t'' so that for every selection of ''t'' columns of the table, all order ...
); 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, fractional factorial designs are experimental designs consisting of a carefully chosen subset (fraction) of the experimental runs of a full factorial design. The subset is chosen so as to exploit the sparsity-of-effects principle t ...
, its adherents claim. *Followers of Taguchi argue that the designs offer rapid results and that
interaction Interaction is action that occurs between two or more objects, with broad use in philosophy and the sciences. It may refer to: Science * Interaction hypothesis, a theory of second language acquisition * Interaction (statistics) * Interactions o ...
s 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. The method was introduced by George E. P. Box and K. B. Wilson in 1951. The main idea of RSM ...
(RSM) advocate the "sequential assembly" of
designs A design is a plan or specification for the construction of an object or system or for the implementation of an activity or process or the result of that plan or specification in the form of a prototype, product, or process. The verb ''to design'' ...
: In the RSM approach, a
screening Screening may refer to: * Screening cultures, a type a medical test that is done to find an infection * Screening (economics), a strategy of combating adverse selection (includes sorting resumes to select employees) * Screening (environmental), a ...
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 achie ...
. 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. The method was introduced by George E. P. Box and K. B. Wilson in 1951. The main idea of RSM ...
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 use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
. 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 Quality control Systems engineering Design of experiments Japanese inventions