Taguchi methods

Taguchi methods are statistical methods developed by Genichi Taguchi to improve the quality of manufactured goods and, more recently, to biotechnology, marketing and advertising. Taguchi methods are considered controversial among some traditional Western statisticians but others accept many of his concepts as being useful additions to the body of knowledge.

Taguchi's principal contributions to statistics are:
 * 1) Taguchi loss-function;
 * 2) The philosophy of off-line quality control; and
 * 3) Innovations in the design of experiments.

Loss functions
Taguchi's reaction to the classical design of experiments methodology of R. A. Fisher was that it was perfectly adapted in seeking to improve the mean outcome of a process. As Fisher's work had been largely motivated by programmes to increase agricultural production, this was hardly surprising. 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 to produce a given voltage. 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 counter-productive.

He therefore, argued that quality engineering should start with an understanding of the cost of poor quality in various situations. In much conventional industrial engineering the cost of poor quality is 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 and are usually ignored by manufacturers. In the wider economy the Coase Theorem predicts that they prevent markets from operating efficiently. 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) and that by working to minimize them, manufacturers would enhance brand reputation, win markets and generate profits.

Such losses are, of course, very small when an item is near to nominal. Donald J. Wheeler characterized 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 would describe them, ...unknown and unknowable but Taguchi wanted to find a useful way of representing them within statistics. Taguchi specified three situations:


 * 1) Larger the better (for example, agricultural yield);
 * 2) Smaller the better (for example, carbon dioxide emissions); and
 * 3) On-target, minimum-variation (for example, a mating part in an assembly).

The first two cases are represented by simple monotonic loss functions. In the third case, Taguchi adopted a squared-error loss function on the grounds:


 * It is the first symmetric term in the Taylor series expansion of any reasonable, real-life loss function, and so is a "first-order" approximation;
 * Total loss is measured by the variance. As variance is additive it is an attractive model of cost; and
 * There was an established body of statistical theory around the use of the least squares principle.

The squared-error loss function had been used by John von Neumann and Oskar Morgenstern in the 1930s.

Though much of this thinking is endorsed by statisticians and economists in general, Taguchi extended the argument to insist that industrial experiments seek to maximise an appropriate signal to noise ratio representing the magnitude of the mean of a process, compared to its variation. Most statisticians believe Taguchi's signal to noise ratios to be effective over too narrow a range of applications and they are generally deprecated.

Off-line quality control
Taguchi realised that the best opportunity to eliminate variation is during design of a product and its manufacturing process (Taguchi's rule for manufacturing). Consequently, he developed a strategy for quality engineering that can be used in both contexts. The process has three stages:


 * 1) System design;
 * 2) Parameter design; and
 * 3) Tolerance design.

System design
This is design at the conceptual level involving creativity and innovation.

Parameter design
Once the concept is established, the nominal values of the various dimensions and design parameters need to be set, the detailed design phase of conventional engineering. William Sealey Gosset in his work at the Guinness brewery suggested as early as the beginning of the 20th century that the company might breed strains of barley that not only yielded and malted well but whose characteristics were robust against variation in the different soils and climates in which they were grown. 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 minimise the effects on performance arising from variation in manufacture, environment and cumulative damage. This approach is often known as robust design or Robustification.

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 (see Pareto principle).

Design of experiments
Taguchi developed much of his thinking in isolation from the school of R. A. Fisher, only coming into direct contact in 1954. His framework for design of experiments is idiosyncratic and often flawed but contains much that is of enormous value. He made a number of innovations.

Outer arrays
In his later work, R. A. Fisher started to consider the prospect of using design of experiments to understand variation in a wider inductive basis. Taguchi sought to understand the influence that parameters had on variation, not just on the mean. He contended, as had W. Edwards Deming in his discussion of analytic studies, that conventional sampling is inadequate here as there is no way of obtaining a random sample of future conditions. In conventional design of experiments, variation between experimental replications is a nuisance that the experimenter would like to eliminate whereas, in Taguchi's thinking, it is a central object of investigation.

Taguchi's innovation was to replicate each experiment by means of an outer array, itself an orthogonal array that seeks deliberately to emulate the sources of variation that a product would encounter in reality. This is an example of judgement sampling. Though statisticians following in the Shewhart-Deming tradition have embraced outer arrays, many academics are still skeptical. An alternative approach proposed by Ellis R. Ott is to use a chunk variable.

Management of interactions
Many of the orthogonal arrays that Taguchi has advocated are saturated allowing no scope for estimation of interactions between control factors, or inner array factors. This is a continuing topic of controversy. However, by combining orthogonal arrays with an outer array consisting of noise factors, Taguchi's method provides complete information on interactions between control factors and noise factors. The strategy is that these are the interactions of most interest in creating a system that is least sensitive to noise factor variation.


 * Followers of Taguchi argue that the designs offer rapid results and that control factor interactions can be eliminated by proper choice of quality characteristic (ideal function) and by transforming the data. That notwithstanding, a confirmation experiment offers protection against any residual interactions. In his later teachings, Taguchi emphasizes the need to use an ideal function that is related to the energy transformation in the system.  This is an effective way to minimize control factor interactions.


 * Western statisticians argue that interactions are part of the real world and that Taguchi's arrays have complicated alias structures that leave interactions difficult to disentangle. George Box, and others, have argued that a more effective and efficient approach is to use sequential assembly.

Analysis of experiments
Taguchi introduced many methods for analysing experimental results including novel applications of the analysis of variance and minute analysis. Little of this work has been validated by Western statisticians.

Assessment
Genichi Taguchi has made seminal and valuable methodological innovations in statistics and engineering, within the Shewhart-Deming tradition. His emphasis on loss to society; techniques for investigating variation in experiments and his overall strategy of system, parameter and tolerance design have been massively influential in improving manufactured quality worldwide.

Other statisticians working on Taguchi methods

 * N. Logothetis
 * Madhav Phadke
 * Yuin Wu
 * Belavendram