1

Taguchi Yaklaşımı

Faktör sayısının az olması durumunda tam faktöriyel tasarım, fazla olması halinde de kesirli faktöriyel tasarım olarak kullanılabilecek tasarım matrislerini tasarlayan Dr. Genichi Taguchi’ye göre, deney tasarımı aşağıda verilen adımlarda gerçekleştirilir:

1.Değerlendirilecek faktör ve/veya etkileşimlerin seçilmesi,2.Faktör düzeylerinin seçilmesi,3.Uygun ortogonal dizinin (orthogonal array) seçilmesi,4.Faktör ve/veya etkileşimlerin sütunlara atanması,5.Testlerin yapılması,6.Sonuçların analiz edilmesi,7.Doğrulama deney(ler)inin yapılması.

2

Adım 1 – Faktör ve/veya Etkileşimlerin Seçilmesi

Bu adımda, söz konusu olan problem ve bu problemin çözümüne ilişkin amaç ortaya konulduktan sonra, klasik deney tasarımında olduğu gibi, beyin fırtınası, süreç akış şeması ve sebep-sonuç diyagramı gibi yöntem ve teknikler kullanılarak, ilgilenilen performans karakteristiğine etkisi olan ya da değerlemeye alınacak faktör ve/veya etkileşimler seçilir.

3

Adım 1İlgilenilen performans karakteristiği, alüminyum malzemeden yapılmış silindirik parçaların CNC tornada işlendikten sonraki yüzey düzgünsüzlüğü olsun.

Böylesi bir kalite karakteristiği;

– Kesme derinliği,– Kesme hızı,– Kesici takım veya parça ilerleme hızı,– Soğutma veya yağlama,– Titreşim,– Kesici takımın yarıçapı,– Kullanılan takımın ömrü,– Kesme geometrisine ilişkin açılar,– Kesilen ve kesen malzeme

faktörlerinin bir fonksiyonudur.

4

Adım 1

Bu şekilde belirlenmiş dokuz faktör;

– Kesilen ve kesen malzeme yapılan deney boyunca sabittir.– İşlenen parçanın küçük olması ve alüminyum malzemeden

yapılmış olması nedeniyle titreşimin etkisi azdır.– Çelik ve pirinç’e kıyasla alüminyum yumuşak bir malzeme

olduğundan takım ömründe büyük değişmeler yapmaz. – Silindirik parçalarda kesme derinliğinin küçük olması

nedeniyle, kesici takımın yarıçapında ve kesme geometrisine ilişkin açılarda büyük değişikler olmaz.

varsayımları dikkate alınarak 4 faktöre indirgenmiştir.

5

Adım 2 – Faktör Düzeylerinin Seçilmesi

İlgilenilen kalite karakteristiği için seçilen faktörlere ilişkin düzeyler aşağıdaki gibi belirlenmiştir:

Faktör 1. Düzey 2. Düzey

A Kesme Derinliği (mm) 0.05 0.10

B İlerleme Hızı (mm/dev) 0.03 0.06

C Kesme Hızı (m/dak) 125.6 188.4

D Soğutma Sıvısı Var Yok

6

Adım 3 – Ortogonal Dizinin Seçilmesiİlgilenilen performans karakteritiğine etki eden faktörler ve düzeyleri dikkate alınarak uygun ortogonal dizinin seçildiği bu adımda, faktör ve/veya etkileşimlerinin sayısına bağlı olarak tam faktöriyel, kesirli faktöriyel ve gözlemlemeye dayalı tasarım söz konusu olabilmektedir.

Tasarım matrisi de denilen ortogonal diziler, genel gösterimiyle,

d: toplam deney sayısıa: faktör düzey sayısık: faktör sayısı

olmak üzere,

Ld(a)k ya da Ld

şeklinde ifade edilir.

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Adım 3

Yüzey düzgünlüğü karakteristiğine iki-düzeyli dört faktörün etkili olduğu bir tasarım için;

L16(2)4 – Tam Faktöriyel Tasarım

L8(2)4 – ½ Kesirli Faktöriyel Tasarım

kullanılabilmektedir.

8

Adım 3

L8(2)4 – ½ Kesirli Faktöriyel Tasarım için ortogonal dizi

Sütunlar

Deney

1 2 3 4 5 6 7

1 1 1 1 1 1 1 1

2 1 1 1 2 2 2 2

3 1 2 2 1 1 2 2

4 1 2 2 2 2 1 1

5 2 1 2 1 2 1 2

6 2 1 2 2 1 2 1

7 2 2 1 1 2 2 1

8 2 2 1 2 1 1 2

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Adım 3L16(2)4 – Tam Faktöriyel Tasarım için ortogonal dizi

Deney

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2

3 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2

4 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1

5 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2

6 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1

7 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1

8 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2

9 2 1 2 1 2 1 2 1 2 2 2 1 2 1 2

10 2 1 2 1 2 1 2 2 1 1 1 2 1 2 1

11 2 1 2 2 1 2 1 1 2 2 2 1 1 2 1

12 2 1 2 2 1 2 1 2 1 1 1 2 2 1 2

13 2 2 1 1 2 2 1 1 2 1 1 1 2 2 1

14 2 2 1 1 2 2 1 2 1 2 2 2 1 1 2

15 2 2 1 2 1 1 2 1 2 1 1 2 1 1 2

16 2 2 1 2 1 1 2 2 1 2 2 1 2 2 1

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Adım 4 – Sütunlara AtamaFaktör ve/veya etkileşimlerin seçilen ortogonal dizinin sütunlarına atanmasında, Taguchi tarafından geliştirilen doğrusal grafikler (linear graphs) ve tablolar (triangular tables) kullanılır.

Verilen iki tasarım için doğrusal grafikler,

L8 için doğrusal grafik L16 için doğrusal grafik

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Adım 4

Verilen grafiklerde, köşe noktalar ana faktörleri, noktaları birleştiren doğrular ise ilgili faktörlerin etkileşimlerini temsil etmektedir. Buna göre, ilgilenilen performans karakteristiği için, L16 ortogonal dizisi dikkate alınarak,

– Ana faktörler (A, B, C ve D) sırasıyla 1., 2., 4. ve 8. kolonlara,

– II. mertebeden faktör etkileşimleri (AB, AC, BC, AD, BD ve CD) sırasıyla 3., 5., 6., 9., 10. ve 12. kolonlara,

– III. mertebeden faktör etkileşimleri (ABC, ABD, ACD ve BCD) sırasıyla 7., 11., 13. Ve 14. kolonlara,

– En yüksek mertebeden faktör etkileşimi (ABCD) ise 15. kolona

olacak şekilde tasarım matrisi oluşturulmuştur.

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Adım 5 – Verilerin Derlenmesi

Deney

A B C DGözleml

er

1 1 1 1 1 45.5

2 1 1 1 2 44.5

3 1 1 2 1 31.5

4 1 1 2 2 47.0

5 1 2 1 1 35.5

6 1 2 1 2 45.5

7 1 2 2 1 36.5

8 1 2 2 2 45.5

9 2 1 1 1 39.0

10 2 1 1 2 49.5

11 2 1 2 1 36.0

12 2 1 2 2 44.5

13 2 2 1 1 35.0

14 2 2 1 2 69.0

15 2 2 2 1 49.5

16 2 2 2 2 59.5

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Adım 6 – Varyans AnaliziKaynak

Birleştirildi ?

SS sd MS F

A H 159.39062 1 159.39062 19.38125

B H 92.64062 1 92.64062 11.26472

AB H 153.14062 1 153.14062 18.62127

C E 11.39062 1 11.39062

AC E 3.51563 1 3.51563

BC H 40.64062 1 40.64062 4.94173

ABC E 0.01563 1 0.01563

D H 582.01562 1 582.01562 70.77073

AD H 54.39062 1 54.39062 6.61368

BD H 54.39062 1 54.39062 6.61368

ABD E 26.26562 1 26.26562

CD E 6.89062 1 6.89062

ACD H 107.64062 1 107.64062 13.08866

BCD H 97.51562 1 97.51562 11.85750

ABCD E 1.26562 1 1.26562

HATA - 49.34374 6 8.22396

TOPLAM - 1391.1094 15

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Adım 6Hesaplanan F değerleri, ilgili serbestlik dereceleri dikkate alınarak test edilecek olursa, A ve D faktörlerinin ve AB ikili etkileşimin kritik olduğu sonucuna varılır.

Kritik olduğuna karar verilen ana faktörlere yönelik grafik;

DCBA

50,0

47,5

45,0

42,5

40,0

Ort

ala

ma

Ana Faktör Etkileri

15

Adım 6

Kritik olduğuna karar verilen etkileşime yönelik grafik;

1 2

1 2

42

47

52

A

B

Ort

alam

a

AxB Etkilesim Grafigi

16

Söz konusu grafikler kullanılarak yüzey düzgünlüğüne etki eden faktörler için uygun düzeyler;

Sivisi SogutmaDm/dak4.188C

mm/dev06.0Bmm05.0A

12

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Yüzey pürüzlülüğü için tahmin değeri;

Level A B C D A*B1 41,4375 42,1875 45,4375 38,5625 1 1 42,13 2 47,7500 47,0000 43,7500 50,6250 1 2 40,75 Delta 6,3125 4,8125 1,6875 12,0625 2 1 42,25 Rank 2 3 4 1 2 2 53,25

46875,31][][][][ˆ 12121 TBATDTCTAT

i

i )TKF(Tˆ

TAGUCHI METHODS

Prof. Dr. A. Sermet ANAGÜN

Eskişehir Osmangazi Üniversitesi

Endüstri Mühendisliği Bölümü

sanagun@ogu.edu.tr

DoE

Structured experiments.The parameters are automatically

tested several times at each level.Cross influences (interactions) can

be evaluated.

Do not focus on Robust Design

Dr. Genichi Taguchi states that; instead of constantly directing

effort toward controlling a process to assure consistent quality,

design the manufactured good to achieve high quality despite the variations that will occur in the production line.

• A disciplined engineering approach (Parameter Design) to find the best combination of design parameters (control factors) for making a system insensitive to outside influences (noise factors)

• 2 steps in the optimization procedure:

1. Reduce effect of variability on design function

2. Improve the performances

Taguchi method

• Introduced by Dr. Genichi Taguchi (1980)– Comparable in importance to Statistical Process Control (SPC),

the Deming approach and the Japanese concept of TQC

• Unique aspects of the Taguchi method– The Taguchi definition of quality– The Taguchi Quality Loss Function (QLF)– The concept of Robust Design

The Taguchi definition of quality– Ideal quality refers to a target value for determining the quality

level– Ideal quality is delivered if a product or service tangible performs

its intended function throughout its projected life under reasonable operating conditions without harmful side effects

– Ideal quality is a function of customer perception and satisfaction– Service quality is measured in terms of loss to society

• The traditional definition is ”conformance to specifications”

Background of the Taguchi Method

What are Taguchi’s Contributions?

• Quality Engineering Philosophy

• Methodology

• Experimental Design

• Analysis

Taguchi focuses mostly on Off-Line Quality Control

Off-Line Quality Control = Improving Quality and Reducing Total Cost in the Design Stage

Total Cost means cost to society so it includes the cost ofproblems in manufacturing and the cost of problems in the field.

Taguchi Loss Function Definition

• Taguchi defines Quality as “the loss imparted by the product to society from the time the product is shipped.”

• LOSS = Cost to operate, Failure to function, maintenance and repair cost, customer satisfaction, poor design.

• Product to be produced “being within specification”

Taguchi’s vs. Traditional Approach

Taguch’s Traditional

When a product moves from its Target will cause the loss even if the product lies or not within Limits

There is Good or Bad Products only as per Limits

Taguchi’s Quadratic Quality Loss Function

• Quality Loss Occurs when a product’s deviates from target or nominal value.

• Deviation Grows, then Loss increases.

• Taguchi’s U-shaped loss Function Curve.

• The traditional model for quality losses– No losses within the specification limits!

The Taguchi Quality Loss Function (I)

• The Taguchi loss function – the quality loss is zero only if we are on target

Scrap Cost

LSL USLTarget

Cost

Taguchi’s U-shaped Loss Function Curve

LTL Nominal

Measuredcharacteristic

UTL

Taguchi loss Fn

Scrap or Rework Cost.

Loss

L(y) = k(ym)2 (k: constant) L(y) = ky2, y 0 L(y) =k (1 / y2) , y 0

Nominal-The-Best (NB) Smaller-The-Better (SB) Larger-The-Better (LB)

Three characteristics of Taguchi’s loss function

Expected loss

]m)y(k[sE[L(y)] 22 ]yk[sE[L(y)] 22 )]y/(3s[1)yk(1/E[L(y)] 222

Definek = The unit repair cost when the deviation from target equals the

maximum tolerance level = Tolerance interval (allowable parameter variation from target to SL)m = Target valuey = The actual metric value for a specific productL(y) = Economic penalty incurred by the customer as a result of quality

deviation from target (The quality loss)

Computing The Taguchi QLF

The Loss Function

L(y) = k(y/)2

Example: The repair cost for an engine shaft is $100. The shaft diameter is required

to be 101 mm. On average the produced shafts deviates 0.5 mm from target.

Determine the mean quality loss per shaft using the Taguchi QLF.

Solution:L(0.5) = k·(y/)2 = 100·(0.5/1)2 = 100·0.25 = $25 per unit

Solved Problem

Suppose that the specification on a part is 0.500 ± 0.020 cm. A detailed analysis of product returns and repairs has discovered that many failures occur when the actual dimension is near the extreme of the tolerance range (that is, when the dimensions are approximately 0.48 or 0.52) and costs $50 for repair.

Thus, the deviation from the target, y – m , is 0.02 and L(y) = $50. Substituting these values, we have:

50 = k(0.02)2 or

k = 50/0.0004 = 125,000

Therefore, the loss function for a single part is L(y) = 125000(y – m)2.

This means when the deviation is 0.10, the firm can still expect an average loss per unit of:

L(0.10) = 125,000(0.10)2 = $12.50 per part

Solved problem (continued)

Knowing the Taguchi loss function helps designers to determine appropriate tolerances economically. For example, suppose that a simple adjustment can be made at the factory for only $2 to get this dimension very close to the target.

If we set L(y) = $2 and solve for y – m, we get:2 = 125000(y – m)2

y – m = 0.004

Therefore, if the dimension is more than 0.004 away from the target, it is more economical to adjust it at the factory and the specifications should be set as 0.500 ± 0.004.

Taguchi’s Contributions

• Quality Engineering Philosophy

• Methodology

• Experimental Design

• Analysis

Robust Design?

• ”Robust design is to improve the quality of a product by minimizing the effect of the causes of variation without eliminating the causes.

Goal: Introducing VARIABILITY (uncertainty) of Goal: Introducing VARIABILITY (uncertainty) of parameters in design optimizationparameters in design optimization

Outcomes: better control of realistic product performancesOutcomes: better control of realistic product performances

Robust design

1. Performs consistently as intended (design)2. Throughout its life cycle (manufacturing)3. Under a wide range of user conditions (design)4. Under a wide range of outside influences (design)

A product is said to be Robust …

• ”Products and services should be designed to be inherently defect free and of high quality”– Meet customers’ expectations also under non-ideal conditions

• Disturbances are events that cause the design performance to deviate from its target values

• Taguchi divides disturbances (noise) into three categories;– External noise: variations in the environment where the product

is used– Internal noise: ware and tare inside a specific unit– Unit-to-unit noise: deviation from target values

• A three step method for achieving robust design:1. Concept design2. Parameter design3. Tolerance design

• The focus of Taguchi is on Parameter design

1.Concept Design

– The process of examining competing technologies for producing a product - Includes choices of technology and process design

– A prototype design that can be produced and meets customers’ needs under ideal conditions without disturbances

2. Parameter Design– The selection of control factors (parameters) and

their “optimal” levels The objective is to make the design Robust!

– Control factors are those process variables management can influence. Ex. the procedures used and the type and amount of

training Often a complex (non-linear) relationship between the

control factors and product/design performance

– The ”optimal” parameter levels can be determined through experimentation

3. Tolerance Design– Development of specification limits

Necessary because there will always be some variation in the production process

Taguchi fiercely advocates aiming for the target value not just settle for “inside the specification limits”!

– Occurs after the parameter design– Often results in increased production costs

More expensive input material might have to be used to meet specifications

Parameter Design (Robust Design)

• Optimize the settings of the design to minimize its sensitivity to noise – ROBUSTNESS.

• Taguchi really opened a whole area that previously had been talked about only by a few people.

• His methodology is heavily dependent on design of experiments, but he wanted to look at not just the mean but also the variance.

Classification of Factors

• Control Factors–Design factors that are to be set at optimal levels to improve quality and reduce sensitivity to noise– Dimensions of parts, type of

material, etc

• Uncontrollable Factors-Noise Factors that represent the noise that is expected in production or in use– Dimensional variation– Operating Temperature

Process

x1

Input Output, y

x2 xn…

z1 z2 zm…

Controllable inputparameters

Uncontrollablefactors (noise)

Process

x1

Input Output, y

x2 xn…

z1 z2 zm…

Controllable inputparameters

Uncontrollablefactors (noise)

Typical Objectives of DOE

(i) Determine which input variables have the most influence on the output;

(ii) Determine what value of xi’s will lead us closest to our desired value of y;

(iii) Determine where to set the most influential xi’s so as to reduce the variability of y;

(iv) Determine where to set the most influential xi’s such that the effects of the uncontrollable variables (zi’s) are minimized.

Process

x1

Input Output, y

x2 xn…

z1 z2 zm…

Controllable inputparameters

Uncontrollablefactors (noise)

Process

x1

Input Output, y

x2 xn…

z1 z2 zm…

Controllable inputparameters

Uncontrollablefactors (noise)

Tool used:ANalysis Of VAriance ANOVA

Taguchi’s Contributions• Quality Engineering Philosophy

• Methodology

• Experimental Design

• Analysis

• Many factors/inputs/variables must be taken into consideration when making a product especially a brand new one– Ex. Baking a new cake without a recipe

• The Taguchi method is a structured approach for determining the ”best” combination of inputs to produce a product or service– Based on a Design of Experiments (DOE) methodology for

determining parameter levels

• DOE is an important tool for designing processes and products– A method for quantitatively identifying the right inputs and

parameter levels for making a high quality product or service

• Taguchi approaches design from a robust design perspective

Taguchi Design of Experiments

Number of levels 2 3

Orthogonal arrays L2m (m = 2,3,...) L3m (m = 2,3,...)

Number of maximum factors 2m1 (3m1)/2

L4 orthogonal array

Experimentnumber

Column number

Basic mark

Genichi Taguchi developed orthogonal arrays; fractional factorial matrix permits a balanced comparison of levels of any factor with a reduced

number of experiments. each factor can be evaluated independently of each of the other factors. 

Experimental Design

Orthogonal arrays

L4: three two-level factors

L9: four three level factorsArrays from http://www.york.ac.uk/depts/maths/tables/orthogonal.htm

Common orthogonal arrays

Array Levels EquivalentFull Factorial

L4 3 x 2 8

L8 7 x 2 128

L9 4 x 3 81

L12 11 x 2 2 048

L16 15 x 2 32 768

L25 6 x 5 15 625

L27 13 x 3 1 594 323

Alternative Notation

Std. Fisher's Original Yates Group Theory TaguchiOrder A B C A B C A B C

1 – – – 1 0 0 0 1 1 12 + – – a 1 0 0 2 1 13 – + – b 0 1 0 1 2 14 + + – ab 1 1 0 2 2 15 – – + c 0 0 1 1 1 26 + – + ac 1 0 1 2 1 27 – + + bc 0 1 1 1 2 28 + + + abc 1 1 1 2 2 2

X1 X2 X3 X1 X2 X3

L8 array

1 2 3 4 5 6 7

1 1 1 1 1 1 11 1 1 2 2 2 21 2 2 1 1 2 21 2 2 2 2 1 12 1 2 1 2 1 22 1 2 2 1 2 12 2 1 1 2 2 12 2 1 2 1 1 2

C B -BC A -AC -AB -ABC

Linear Graphs for L8 Array

1

2

3

4

5

6

7

1

2

3

4

5

6

7

•Main effects are assigned to columns at nodes in the plot.•Interactions are assigned to the columns on the lines.

Orthogonal Designs

“Classical”(2-level Factorials)

“Taguchi”

23

24

25

26-3

27-1

…

23-1=L4

27-4=L8

215-11=L16

…

L12

L18

L27

…

Taguchi DesignsNotation

Total Number of Runs

kNL 2

Number of Levels per Factor

Number of Factors

Taguchi Orthogonal Arrays• 2-level (fractional factorial) arrays

– L4(23). L8(27), L16(215). L32(231), L64(263)• 2-level array

– L12(211) (Plackett-Burman Design)• 3-level arrays

– L9(34). L27(313), L81(340)• 4-level arrays

– L16(45). L64(421)• 5-level array

– L25(56)• Mixed-level arrays

– L18(21x37), L32(21x49), L50(21x511)– L36(211x312), L36(23x313), L54(21x325)

Taguchi’s Contributions• Quality Engineering Philosophy

• Methodology

• Experimental Design

• Analysis

The Taguchi Process

1.Problem Identification– Locate the problem source not just the symptom

2. Brainstorming Session– Attended at least by project leader/facilitator and workers involved in the

process. Other participants may include managers and technical staff– The purpose is to identify critical variables for the quality of the product or

service in question (referred to as factors by Taguchi) • Control factors – variables under management control• Signal factors – uncontrollable variation

– Define different factor levels (three or four) and identify possible interaction between factors

– Determine experiment objectives

1. Smaller-the-better – keep the level of defectives as close to zero as possible

2. Nominal-is-the-best – Outcome as close to target as possible

3. Larger-the-better – max number of units per time unit or lot without defects

The Taguchi Process

3.Experimental Design– Using factor levels and objectives determined via brainstorming– Taguchi advocates off-line-experimentation as a contrast to

traditional on-line or in-process experimentation– Care should be taken to selecting number of trials, trial

conditions, how to measure performance etc.

4.Experimentation– Various rigorous analysis approaches like ANOVA and Multiple

Regression can be used but also simpler customized methods are available

5.Analysis– The experimentation provides ”best” levels for all factors– If interactions between factors are evident Either ignore or run a full

factorial experiment

6.Conforming Experiments– The results should be validated by running experiments with all factors

set to ”optimal” levels

The Taguchi Process

• Traditional Design of Experiments (DOE) focused on how different design factors affect the average result level

• Taguchi’s perspective (robust design)– variation is more interesting to study than the average– Run experiments where controllable design factors and

disturbing (noise) factors take on 2 or three levels.

• For each combination of the design variables a number of experiments are run covering all possible combinations of the signal variables. Can estimate average effects and the variation different

design factor levels imply choose factor levels that minimize the sensitivity against

disturbances

The Taguchi Approach to DOE

• From every trial series we can obtain an average result level and a measure of the variation, si, i=1,2, … ,9. These values can then be used as a basis for choosing the combination of factor levels that provides the most robust design.

The Taguchi Approach to DOE

Experiment Factor

1 2 3 4

1 -1 -1 -1 -1 2 -1 0 0 0 9 1 1 0 -1

-1 -1 -1 -1 1 1 1 -1 1 1 1 -1

Levels of disturbing factors

Y11 Y12

Y13 Y14

)s,Y( 11

-1 -1 -1 -1 1 1 1 -1 1 1 1 -1

Y21 Y22

Y23 Y24

)s,Y( 22

Individual results