Taguchi methods
Taguchi methods are statistical methods developed by Genichi Taguchi to improve the quality of manufactured goods and, more recently, to biotechnology [1], 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.

[pic]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

Introduction To Robust Design (Taguchi Method)

Robust Design method, also called the Taguchi Method, pioneered by Dr. Genichi Taguchi, greatly improves engineering productivity. By consciously considering the noise factors (environmental variation during the product's usage, manufacturing variation, and component deterioration) and the cost of failure in the field the Robust Design method helps ensure customer satisfaction. Robust Design focuses on improving the fundamental function of the product or process, thus facilitating flexible designs and concurrent engineering. Indeed, it is the most powerful method available to reduce product cost, improve quality, and simultaneously reduce development interval.
1. Why Use Robust Design Method?
Over the last five years many leading companies have invested heavily in the Six Sigma approach aimed at reducing waste during manufacturing and operations. These efforts have had great impact on the cost structure and hence on the bottom line of those companies. Many of them have reached the maximum potential of the traditional Six Sigma approach. What would be the engine for the next wave of productivity improvement?
Brenda Reichelderfer of ITT Industries reported on their benchmarking survey of many leading companies, "design directly influences more than 70% of the product life cycle cost; companies with high product development effectiveness have earnings three times the average earnings; and companies with high product development effectiveness have revenue growth two times the average revenue growth." She also observed, "40% of product development costs are wasted!"
These and similar observations by other leading companies are compelling them to adopt improved product development processes under the banner Design for Six Sigma. The Design for Six Sigma approach is focused on 1) increasing engineering productivity so that new products can be developed rapidly and at low cost, and 2) value based management.
Robust Design method is central to improving engineering productivity. Pioneered by Dr. Genichi Taguchi after the end of the Second World War, the method has evolved over the last five decades. Many companies around the world have saved hundreds of millions of dollars by using the method in diverse industries: automobiles, xerography, telecommunications, electronics, software, etc.
1.1. Typical Problems Addressed By Robust Design
A team of engineers was working on the design of a radio receiver for ground to aircraft communication requiring high reliability, i.e., low bit error rate, for data transmission. On the one hand, building series of prototypes to sequentially eliminate problems would be forbiddingly expensive. On the other hand, computer simulation effort for evaluating a single design was also time consuming and expensive. Then, how can one speed up development and yet assure reliability?
In an another project, a manufacturer had introduced a high speed copy machine to the field only to find that the paper feeder jammed almost ten times more frequently than what was planned. The traditional method for evaluating the reliability of a single new design idea used to take several weeks. How can the company conduct the needed research in a short time and come up with a design that would not embarrass the company again in the field?
The Robust Design method has helped reduce the development time and cost by a factor of two or better in many such problems.
In general, engineering decisions involved in product/system development can be classified into two categories: • Error-free implementation of the past collective knowledge and experience • Generation of new design information, often for improving product quality/reliability, performance, and cost.
While CAD/CAE tools are effective for implementing past knowledge, Robust Design method greatly improves productivity in generation of new knowledge by acting as an amplifier of engineering skills. With Robust Design, a company can rapidly achieve the full technological potential of their design ideas and achieve higher profits.
Next Page > Robustness Strategy
Page 3 > Steps in Robust Parameter Design
Page 4 > Robust Design Case Studies INTRODUCTION TO TAGUCHI METHOD

Every experimenter has to plan and conduct experiments to obtain enough and relevant data so that he can infer the science behind the observed phenomenon. He can do so by,
(1) trial-and-error approach : ------------------------------ performing a series of experiments each of which gives some understanding. This requires making measurements after every experiment so that analysis of observed data will allow him to decide what to do next - "Which parameters should be varied and by how much". Many a times such series does not progress much as negative results may discourage or will not allow a selection of parameters which ought to be changed in the next experiment. Therefore, such experimentation usually ends well before the number of experiments reach a double digit! The data is insufficient to draw any significant conclusions and the main problem (of understanding the science) still remains unsolved.
(2) Design of experiments : -----------------------------
A well planned set of experiments, in which all parameters of interest are varied over a specified range, is a much better approach to obtain systematic data. Mathematically speaking, such a complete set of experiments ought to give desired results. Usually the number of experiments and resources (materials and time) required are prohibitively large. Often the experimenter decides to perform a subset of the complete set of experiments to save on time and money! However, it does not easily lend itself to understanding of science behind the phenomenon. The analysis is not very easy (though it may be easy for the mathematician/statistician) and thus effects of various parameters on the observed data are not readily apparent. In many cases, particularly those in which some optimization is required, the method does not point to the BEST settings of parameters. A classic example illustrating the drawback of design of experiments is found in the planning of a world cup event, say football. While all matches are well arranged with respect to the different teams and different venues on different dates and yet the planning does not care about the result of any match (win or lose)!!!! Obviously, such a strategy is not desirable for conducting scientific experiments (except for co-ordinating various institutions, committees, people, equipment, materials etc.). (3) TAGUCHI Method : --------------------------
Dr. Taguchi of Nippon Telephones and Telegraph Company, Japan has developed a method based on " ORTHOGONAL ARRAY " experiments which gives much reduced " variance " for the experiment with " optimum settings " of control parameters. Thus the marriage of Design of Experiments with optimization of control parameters to obtain BEST results is achieved in the Taguchi Method. "Orthogonal Arrays" (OA) provide a set of well balanced (minimum) experiments and Dr. Taguchi's Signal-to-Noise ratios (S/N), which are log functions of desired output, serve as objective functions for optimization, help in data analysis and prediction of optimum results.
Taguchi Method treats optimization problems in two categories, [A] STATIC PROBLEMS :
Generally, a process to be optimized has several control factors which directly decide the target or desired value of the output. The optimization then involves determining the best control factor levels so that the output is at the the target value. Such a problem is called as a "STATIC PROBLEM".
This is best explained using a P-Diagram which is shown below ("P" stands for Process or Product). Noise is shown to be present in the process but should have no effect on the output! This is the primary aim of the Taguchi experiments - to minimize variations in output even though noise is present in the process. The process is then said to have become ROBUST. [pic]
[B] DYNAMIC PROBLEMS :
If the product to be optimized has a signal input that directly decides the output, the optimization involves determining the best control factor levels so that the "input signal / output" ratio is closest to the desired relationship. Such a problem is called as a "DYNAMIC PROBLEM". This is best explained by a P-Diagram which is shown below. Again, the primary aim of the Taguchi experiments - to minimize variations in output even though noise is present in the process- is achieved by getting improved linearity in the input/output relationship. [pic]
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[A] STATIC PROBLEM (BATCH PROCESS OPTIMIZATION) : ----------------------------------------------------------------------------
There are 3 Signal-to-Noise ratios of common interest for optimization of Static Problems;
(I) SMALLER-THE-BETTER : -------------------------------------- n = -10 Log10 [ mean of sum of squares of measured data ]
This is usually the chosen S/N ratio for all undesirable characteristics like " defects " etc. for which the ideal value is zero. Also, when an ideal value is finite and its maximum or minimum value is defined (like maximum purity is 100% or maximum Tc is 92K or minimum time for making a telephone connection is 1 sec) then the difference between measured data and ideal value is expected to be as small as possible. The generic form of S/N ratio then becomes, n = -10 Log10 [ mean of sum of squares of {measured - ideal} ] (II) LARGER-THE-BETTER : ------------------------------------- n = -10 Log10 [mean of sum squares of reciprocal of measured data]
This case has been converted to SMALLER-THE-BETTER by taking the reciprocals of measured data and then taking the S/N ratio as in the smaller-the-better case. (III) NOMINAL-THE-BEST : ----------------------------------- square of mean n = 10 Log10 ----------------- variance
This case arises when a specified value is MOST desired, meaning that neither a smaller nor a larger value is desirable. Examples are; (i) most parts in mechanical fittings have dimensions which are nominal-the-best type. (ii) Ratios of chemicals or mixtures are nominally the best type. e.g. Aqua regia 1:3 of HNO3:HCL Ratio of Sulphur, KNO3 and Carbon in gun powder (iii) Thickness should be uniform in deposition /growth /plating /etching..

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[B] DYNAMIC PROBLEM (TECHNOLOGY DEVELOPMENT) : ------------------------------------------------------------------------------
In dynamic problems, we come across many applications where the output is supposed to follow input signal in a predetermined manner. Generally, a linear relationship between "input" "output" is desirable. For example : Accelerator peddle in cars, volume control in audio amplifiers, document copier (with magnification or reduction) various types of moldings etc.

There are 2 characteristics of common interest in "follow-the-leader" or "Transformations" type of applications,
(i) Slope of the I/O characteristics and (ii) Linearity of the I/O characteristics (minimum deviation from the best-fit straight line) The Signal-to-Noise ratio for these 2 characteristics have been defined as; (I) SENSITIVITY {SLOPE}: ---------------------------------- The slope of I/O characteristics should be at the specified value (usually 1). It is often treated as Larger-The-Better when the output is a desirable characteristics (as in the case of Sensors, where the slope indicates the sensitivity). n = 10 Log10 [square of slope or beta of the I/O characteristics] On the other hand, when the output is an undesired characteristics, it can be treated as Smaller-the-Better. n = -10 Log10 [square of slope or beta of the I/O characteristics] (II) LINEARITY (LARGER-THE-BETTER) : ----------------------------------------------- Most dynamic characteristics are required to have direct proportionality between the input and output. These applications are therefore called as "TRANSFORMATIONS". The straight line relationship between I/O must be truly linear i.e. with as little deviations from the straight line as possible. Square of slope or beta n = 10 Log10 ---------------------------- variance Variance in this case is the mean of the sum of squares of deviations of measured data points from the best-fit straight line (linear regression).
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(4) 8-STEPS IN TAGUCHI METHODOLOGY : ----------------------------------------------------
Taguchi method is a scientifically disciplined mechanism for evaluating and implementing improvements in products, processes, materials, equipment, and facilities. These improvements are aimed at improving the desired characteristics and simultaneously reducing the number of defects by studying the key variables controlling the process and optimizing the procedures or design to yield the best results.
The method is applicable over a wide range of engineering fields that include processes that manufacture raw materials, sub systems, products for professional and consumer markets. In fact, the method can be applied to any process be it engineering fabrication, computer-aided-design, banking and service sectors etc. Taguchi method is useful for 'tuning' a given process for 'best' results.
Taguchi proposed a standard 8-step procedure for applying his method for optimizing any process,
8-STEPS IN TAGUCHI METHODOLOGY:
Step-1: IDENTIFY THE MAIN FUNCTION, SIDE EFFECTS, AND FAILURE MODE
Step-2: IDENTIFY THE NOISE FACTORS, TESTING CONDITIONS, AND QUALITY CHARACTERISTICS
Step-3: IDENTIFY THE OBJECTIVE FUNCTION TO BE OPTIMIZED
Step-4: IDENTIFY THE CONTROL FACTORS AND THEIR LEVELS
Step-5: SELECT THE ORTHOGONAL ARRAY MATRIX EXPERIMENT
Step-6: CONDUCT THE MATRIX EXPERIMENT
Step-7: ANALYZE THE DATA, PREDICT THE OPTIMUM LEVELS AND PERFORMANCE
Step-8: PERFORM THE VERIFICATION EXPERIMENT AND PLAN THE FUTURE ACTION
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SUMMARY : Every experimenter develops a nominal process/product that has the desired functionality as demanded by users. Beginning with these nominal processes, he wishes to optimize the processes/products by varying the control factors at his disposal, such that the results are reliable and repeatable (i.e. show less variations).
In Taguchi Method, the word "optimization" implies "determination of BEST levels of control factors". In turn, the BEST levels of control factors are those that maximize the Signal-to-Noise ratios. The Signal-to-Noise ratios are log functions of desired output characteristics. The experiments, that are conducted to determine the BEST levels, are based on "Orthogonal Arrays", are balanced with respect to all control factors and yet are minimum in number. This in turn implies that the resources (materials and time) required for the experiments are also minimum.
Taguchi method divides all problems into 2 categories - STATIC or DYNAMIC. While the Dynamic problems have a SIGNAL factor, the Static problems do not have any signal factor. In Static problems, the optimization is achieved by using 3 Signal-to-Noise ratios - smaller-the-better, LARGER-THE-BETTER and nominal-the-best. In Dynamic problems, the optimization is achieved by using 2 Signal-to-Noise ratios - Slope and Linearity.
Taguchi Method is a process/product optimization method that is based on 8-steps of planning, conducting and evaluating results of matrix experiments to determine the best levels of control factors. The primary goal is to keep the variance in the output very low even in the presence of noise inputs. Thus, the processes/products are made ROBUST against all variations.