Taguchi methods

Citation for published version (APA):Wijnen, J. T. M. (1991). Taguchi methods. (Memorandum COSOR; Vol. 9131). Technische UniversiteitEindhoven.

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TECHNISCHE UNIVERSITEIT EINDHOVENFaculteit Wiskunde en Informatica

Memorandum CaSaR 91-31

Taguchi Methods

Jack Th. M. Wijnen

Eindhoven University of TechnologyDepartment of Mathematics and Computing ScienceP.O. Box 5135600 MB EindhovenThe Netherlands

Eindhoven, November 1991The Netherlands

Taguchi Methods

Jack Th.M. Wijnen

Abstract: Taguchi's quality engineering ideas and statistical procedures are shortly intro­duced.

AMS Subject classification: Primary 62K15; Secondary 62NIO.

Keywords and phrases: Robust design; Taguchi; orthogonal array; signal-to-noise ratio;optimization.

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1. Introduction

The Japanese engineer Genichi Taguchi was asked by the Electrical Communication Lab­oratories (ECL, founded in 1949) to improve productivity in research and development. Hisideas and statistical procedures have been used in Japan for decades. It was in the mid-1980sthat the western world became aware of his views towards process control and quality im­provement.In this paper we shortly introduce the reader to Taguchi's quality engineering ideas and sta­tistical procedures, the last of them being somewhat controversial.

2. Quality

The classical use of specification limits for process characteristics implies that values withinthese limits are acceptable. As a consequence products with values for the characteristics outof the range are unacceptable.In Taguchi's philosophy quality is measured in terms of "loss to society". By those in thewestern world this idea has been replaced by "long term loss to the firm", but the generalidea is the same. That is, individual firms and society as a whole suffer a loss when productsdo not function as they could if they were made properly.Let T denote a target value (Le. optimal value) for a quality characteristic (x). Further letLL and U L denote the lower and the upper specification limit, respectively. Then the loss(L) as a function of the value of the characteristic in a classical sense is as given in the figurebelow.

L

LL T UL x

In fact the situation is somewhat different. The loss as a result of deviations from the targetvalue T will usually increase as the value departs from its optimal value, regardless of whetheror not a limit has been exceeded.Moreover, products with values just outside the range will not be much worse than productswith values just within the limits. A simple and useful type of loss function is the quadraticloss function L = k(x - T)2, where k > 0 is a constant, which is illustrated in the figurebelow.

2

L

LL T UL x

Causes for deviations of the target value are of two kinds:

- The mean value (It) of x departs from T as a consequence of wrong adjustments of processcharacteristics.

- Many uncontrollable, sometimes non-measurable and unavoidable factors cause randomdeviations from It. A measure for these kinds of deviations is the variance (0'2) of x.

For the mean loss then, the following applies

Clearly other loss functions are possible, e.g. if loss is asymmetric about J.L. In other casesthere is not a target value but the characteristic has to be as low as possible (noise, unround­ness) or as high as possible (yield, strengthness).In the following we confine ourselves to the first situation: the nomimal value is the bestone, the target (T). The loss function will be the quadratic function as given above. Whendesigning products and production methods it is necessary to bear in mind these things inorder to satisfactorily control quality ("off-line" quality control).It is evident that the mean loss is reduced by decreasing 0'2 and (It - T)2. Therefore factorswhich possibly affect quality are separated into

- noise factors, causing random deviations

- control factors, affecting means.

Noise factors are uncontrollable and unavoidable. It is thus of importance to design productsin such a way that quality is insensitive to variations caused by noise factors (decreasing 0'2).A well-known example is the following: In a tilery large variations were found in the dimen­sions of tiles. These variations were caused by differences in temperature in the kiln. Insteadof designing a better kiln (with less expiration of temperature) it appeared, by experimenta­tion, that variations in the dimensions could be decreased by increasing the amount of limein the tiles from 1% to 5%.

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Control factors can be adjusted and controlled. They are used for realizing the target (mini­mizing (JL - T)2).

Traditional designing consists of two steps:

- Technological knowledge is involved in order to arrive at a prototype.

- Tolerances around the target value are determined with due regard to the capability of themanufacturing process.

Taguchi inserted an additional step:

- Determining optimum levels of individual parameters in such a way that the design isrobust to variation in the parameters due to noises.

These three phases are referred to as off-line quality control. The execution of Taguchi'sintermediate step takes place in phases:

- Brainstorming by specialists and people involved to identify quality characteristics, toclassify these characteristics, to recognize interactions and to determine the number oflevels and their specific values.

- Designing an experiment by means of orthogonal arrays.

- Performing the designed experiment.

- Analysing the results of the experiment.

- Determining target values taking into account the sensitivity to noise factors.

- Performing a confirmation experiment around the predicted optimum setting to verify theresults.

3. Orthogonal arrays

Designing experiments is performed by assigning factors and interactions to orthogonalarrays. Then it is possible to estimate effects independently and confounding can be takeninto account.Each column of an array gives the levels of a factor or interaction, each row gives the treat­ment combinations of an individual run of the experiment. The array is orthogonal in acombinatoric sense, i.e., for any pair of columns, all combinations of factor levels occur andthey occur an equal number of times. This so called balancing property implies mathematicalorthogonality.We only discuss here arrays in which all factors are varied at two levels.The low level is coded by Taguchi as 1, the high level as 2. Such an orthogonal array isdenoted by

4

where

a is the number of experimental runs (here a is a power of 2),b is the number of levels for each factor (here b = 2),c is the number of columns of the array (here c = a-I).

A frequently used array is the Ls-array (Ls(27)):

RunNumber 1 2 3 4 5 6 7 Response

1 1 1 1 1 1 1 1 Yl2 1 1 1 2 2 2 2 Y23 1 2 2 1 1 2 2 ..Y34 1 2 2 2 2 1 1 Y45 2 1 2 1 2 1 2 Ys6 2 1 2 2 1 2 1 Y67 2 2 1 1 2 2 1 Y78 2 2 1 2 1 1 2 Ys

7

4

2

1~----eand07

6

Orthogonality is easily verified: the array is equivalent to the complete 23 -experiment.For assigning main effects and interactions to orthogonal arrays linear graphs are used.Each orthogonal array has one or more linear graphs (standard graphs).The Ls-array has two of them:

1

3/\ 52U4

Each dot represents a main effect, each line between two dots represents the interactionbetween those two main effects. Each number corresponds with a column of the array.The selection of an array and the assignment proceeds in three steps:

- The total number of effects (main effects and interactions) is the minimum number ofcolumns required in the array. Choose an appropriate orthogonal array.

- Construct the required linear graph.

- Match the required graph to one of the standard graphs available.

Example

Consider an experiment with factors A, B, C and D and relevant interactions BC and BD.

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So there are 6 effects. A useful array with the required number of columns is the Ls-array (7columns). The required graph is

Bye~

D

oA

We can match the required linear graph to both standard graphs. We choose for instance:1 = C

2 = B

3=BC

6= BD

07 = A

4=D

Column 1 gives the levels of C, column 2 is allocated to B, column 4 to D and column 7 toA. Columns 3 and 6 are allocated to the interactions BC and BD, respectively. Then onedegree of freedom remains to estimate (12 (column 5).In order to get a better estimation of (12 replications are needed.The analysis of the observations resulting from the experiment includes estimation of theeffects, performing an analysis of variance and a computation of the percentage contributionof the effects to the total variability. (The last computation is based on so called "pure" sumsof squares; we will not discuss this here).Based on this analysis optimum conditions are determined including confidence intervals forthe expected response.

4. Direct product arrays

To create conditions under which quality of the product is insensitive for noise it is necessaryto design an experiment incorporating noise factors. Control factors have to be optimized notonly in terms of the mean levels but most importantly in terms of their interaction with noisefactors. This is called robust optimization. In the following situation there is interactionbetween a control factor (A) and a noise factor (X). Here level A1 may be preferable to levelA2 with the somewhat higher mean response because of the much smaller influence of thenoise factor X.

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response

noisefactor

For both control factors and noise factors orthogonal arrays are chosen as indicated above(control array and noise array, respectively). The experimental design is obtained by "cross­ing" the two arrays.Effects belonging to the columns of the control array and the columns of the noise array canbe estimated but interactions between control factors and noise factors can be estimated aswell.Our main interest with respect to noise is not the individual effects but the variability dueto the combined noise factors. It is particularly important to analyse how this variability isaffected by level combinations of the control factors.Using standard methods for analysing data variance is not a proper response variable. There­fore Taguchi introduced several so called "performance statistics". For many normal-the-bestsituations Taguchi appears to use the signal-to-noise ratio (S / N) defined by

(

-2 1)S / N = 10 log ~2 - -;: ,

where

Y is the sample mean,8

2 is the sample variance,r is the number of observations.

For each run of the control array this ratio is computed. So r is the number of runs of thenoise array. Observe that high values for S / N correspond with small variances.

Example

Assume that the Ls-array of the foregoing example is crossed with a L4-array for 3 noisefactors. These 3 noise factors do not interact. This results in the following direct productarray.

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Z 3 1 2 2 1Y 2 1 2 1 2X 1 1 1 2 2

C B BC D e BD A 1 2 3 4 Y 8 SINrow 1 2 3 4 5 6 7 row

1 1 1 1 1 1 1 1 Yll Y12 Y13 Y14 til 812 1 1 1 2 2 2 2 Y21 Y22 Y23 Y24 Y2 823 1 2 2 1 1 2 2 Y31 Y32 Y33 Y34 Y3 834 1 2 2 2 2 1 1 Y41 Y42 Y43 Y44 Y4 845 2 1 2 1 2 1 2 Y51 Y52 Y53 Y54 Y5 85

6 2 1 2 2 1 2 1 Ye1 Ye2 Ye3 Ye4 Ye 8e7 2 2 1 1 2 2 1 Y71 Y72 Y73 Y74 Y7 878 2 2 1 2 1 1 2 YSI YS2 YS3 YS4 Ys 8S

Analysis goes as follows:

- The effects of the control factors are estimated and an analysis of variance is performed.

- Next the same procedure is carried out with SIN as the response variable.

- Then optimum control factor settings are determined by first looking at the variability.

- Finally a confirmation experiment centered around the optimum setting is carried out andanalysed.

5. Taguchi or not?

The main merit of Taguchi is that he focussed attention on target values rather than speci­fication limits and on the importance of insensitivity to manufacturing variations.Determining optimum conditions is accomplished by Taguchi within the frame of chosenarrays. In a strict mathematical sense optimizing can give other results. For such an opti­mization other and more efficient methods are available.Furthermore, the orthogonal arrays are often inferior to the classical fractional factorials. Forinstance, in the Lie-array standard graphs for 7 and 8 main effects, some of the main effectsare confounded with interactions. However, it is possible to design an orthogonal scheme sothat all main effects are confounded with higher order interactions.Beyond question, Taguchi's contribution to the improvement of quality control is of greatimportance. However, the statistical tools will likely continue to be developed.

References

PHADKE, M.S. (1989), Quality Engineering Using Robust Design, London, Prentice-HallInt.

Ross, PH.J. (1988), Taguchi Techniques for Quality Engineering, New York, McGraw HillBook Company.

RYAN, TH.R. (1989), Statistical Methods for Quality Improvement, New York, Wiley &Sons.

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EINDHOVEN UNIVERSITY OF TECHNOLOGYDepartment of Mathematics and Computing SciencePROBABILITY THEORY, STATISTICS, OPERATIONS RESEARCHAND SYSTEMS THEORYP.O. Box 5135600 MB Eindhoven, The Netherlands

Secretariate: Dommelbuilding 0.03Telephone : 040-473130

-List of COSOR-memoranda - 1991

Number

91-01

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January

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Author

M.W.I. van KraaijW. Z. VenemaJ. Wessels

M.W.I. van KraaijW.Z. VenemaJ. Wessels

M.W.P. Savelsbergh

M.W.I. van Kraaij

G.L. NemhauserM.W.P. Savelsbergh

R.J.G. Wilms

F. CoolenR. DekkerA. Smit

P.J. ZwieteringE.H.L. AartsJ. Wessels

P.J. ZwieteringE.H.L. AartsJ. Wessels

P.J. ZwieteringE.H.L. AartsJ. Wessels

F. Coolen

The construction of astrategy for manpowerplanning problems.

Support for problem formu­lation and evaluation inmanpower planning problems.

The vehicle routing problemwith time windows: minimi­zing route duration.

Some considerationsconcerning the probleminterpreter of the newmanpower planning systemformasy.

A cutting plane algorithmfor the single machinescheduling problem withrelease times.

Properties of Fourier­Stieltjes sequences ofdistribution with supportin [0, 1) .

Analysis of a two-phaseinspection model withcompeting risks.

The Design and Complexityof Exact Multi-LayeredPerceptrons.

The Classification Capabi­lities of ExactTwo-Layered Peceptrons.

Sorting With A Neural Net.

On some misconceptionsabout subjective probabili­ty and Bayesian inference.

COSOR-MEMORANDA (2)

91-12

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1. J. B. F. AdanG.J. van HoutumJ. WesselsW.H.M. Zijm

J. KorstE. AartsJ.K. LenstraJ. Wessels

P.J. ZwieteringM.J.A.L. van KraaijE.H.L. AartsJ. Wessels

P. DeheuvelsJ.H.J. Einmahl

M.W.P. SavelsberghG.C. SigismondiG.L. Nemhauser

M.W.P. SavelsberghG.C. SigismondiG.L. Nemhauser

P. van der Laan

P. van der Laan

E. LevnerA.S. Nemirovsky

R.J.M. VaessensE.H.L. AartsJ.H. van Lint

P. van der Laan

Two-stage selectionprocedures with attentionto screening.

A compensation procedurefor multiprogrammingqueues.

Periodic assignment andgraph colouring.

Neural Networks andProduction Planning.

Approximations and Two­Sample Tests Based onP - P and Q - Q Plots ofthe Kaplan-Meier Estima­tors of Lifetime Distri­butions.

Functional description ofMINTO, a Mixed INTegerOptimizer.

MINTO, a Mixed INTegerOptimizer.

The efficiency of subsetselection of an almostbest treatment.

Subset selection for an-best population:

efficiency results.

A network flow algorithmfor just-in-time projectscheduling.

Genetic Algorithms inCoding Theory - A Tablefor A] (n, d) .

Distribution theory forselection from logisticpopulations.

COSOR-MEMORANDA (3)

91-24

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91-26

91-27

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91-31

October

October

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November

November

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I. J. B. F. AdanJ. WesselsW.H.M. Zijm

I. J. B. F. AdanJ. WesselsW.H.M. Zijm

E.E.M. van BerkumP.M. Upperman

R.P. GillesP.H.M. RuysS. Jilin

I.J.B.F. AdanJ. WesselsW.H.M. Zijm

J. Wessels

G.L. NernhauserM.W.P. SavelsberghG.C. Sigismondi

J.Th.M. Wijnen

Matrix-geometric analysisof the shortest queueproblem with thresholdjockeying.

Analysing MultiprogrammingQueues by GeneratingFunctions.

D-optimal designs for anincomplete quadratic model.

Quasi-Networks in SocialRelational Systems.

A Compensation Approach forTwo-dimensional MarkovProcesses.

Tools for the InterfacingBetween Dynamical Problemsand Models withing DecisionSupport Systems.

Constraint Classificationfor Mixed Integer Program­ming Formulations.

Taguchi Methods.