Aalto University A!School of Engineering. Robust Design II. Prof. Kevin Otto Department of Mechanical Engineering

Robust Design II Prof. Kevin Otto Department of Mechanical Engineering Kevin.Otto@aalto.fi 1

Overview Fractional Factorial Designs Experimental Variability Assessment Taguchi s Method Robustness Improvement ` 2

Reading Chapter 18 + 19 Chapter 15 3

Full Factorial Models Suppose we have 10 design variables. 2-level experiments. 2 10 = 1024 experimental points! How many coefficients could we fit? = + + + + + + 4

Full Factorial Inefficiencies # of effects Effect type Term 1 Average response b 0 10 Main effects b i 45 2-factor interactions b ij 120 3-factor interactions b ijk 210 4-factor interactions b ijkl 252 5-factor interactions b ijklm 210 6-factor interactions b ij 120 7-factor interactions b ij 45 8-factor interactions b ij 10 9-factor interactions b ij 1024 1 10-factor interaction b ij n 5

Reduce the number For a complete 2-way interaction model m i is the number of levels experimented on x i 2 level experiments =1+ ( 1) + 1)( 1 =1+ + ( 1) 2 We need to find a set of N experimental points that can fit each coefficient 6

Reducing the space A 2 3 design has 9 runs. Suppose you only want to do 5. Which 5? x 2 x x x x x x x x x x 1 7

Reducing the space x 2 x x x x x x x x x x 1 8

Reducing the space x 2 x x x x x x x x x x 1 9

Reducing the space x 2 x x x x x x x x x x 1 10

Reducing the space x 2 x x x x x x x x x x 1 11

Orthogonal and Balanced Want the reduced set of experiments to cover the space This means an equal number of runs at each level i for all x i. - Except at the center point. x 2 x 2 x x x 1 run x x x 2 runs x x x x x x x x x 3 runs x x x 2 runs 3 runs 1 run x 1 2 runs 2 runs x 1 12

Orthogonal and Balanced Linear algebra: well cover means orthogonal and balanced With coded units, need all and such that =0 These designs have been pre-determined over the years. Two factorial designs 2 k-p designs N-factorial designs specialized, beyond this course. 13

Yates Labeling Relabel each x i alphabetically x 1 = a x 2 = b x 3 = c Relabel interactions alphabetically x 1 x 2 = ab x 3 x 4 = cd x 1 x 2 x 4 = acd 14

Yates Order and Interactions Examine the 2 6 design matrix Design Variable (Yates Order) Run x 1 =a x 2 =b x 3 =c x 4 =d x 5 =e x 6 =f 1-1 -1-1 -1-1 -1 2 1-1 -1-1 -1-1 3-1 1-1 -1-1 -1 4 1 1-1 -1-1 -1 5-1 -1 1-1 -1-1 6 1-1 1-1 -1-1 7-1 1 1-1 -1-1 8 1 1 1-1 -1-1 9-1 -1-1 1-1 -1 10 1-1 -1 1-1 -1 : 63-1 1 1 1 1 1 64 1 1 1 1 1 1 15

Yates Order and Interactions Examine the design matrix interactions Design Variable (Yates Order) Run x 1 =a x 2 =b x 3 =c x 4 =d x 5 =e x 6 =f ab acd abcdef 1-1 -1-1 -1-1 -1 1-1 1 2 1-1 -1-1 -1-1 -1 1-1 3-1 1-1 -1-1 -1-1 -1-1 4 1 1-1 -1-1 -1 1 1 1 5-1 -1 1-1 -1-1 1 1-1 6 1-1 1-1 -1-1 -1-1 1 7-1 1 1-1 -1-1 -1 1 1 8 1 1 1-1 -1-1 1-1 -1 9-1 -1-1 1-1 -1 1 1-1 10 1-1 -1 1-1 -1-1 -1 1 : 63-1 1 1 1 1 1-1 -1-1 64 1 1 1 1 1 1 1 1 1 16

Blocks Consider any interaction. Partition the matrix according to it Each half defines a block. Design Variable (Yates Order) Run x 1 =a x 2 =b x 3 =c x 4 =d x 5 =e x 6 =f abcdef BLOCK 1-1 -1-1 -1-1 -1 1 I 2 1-1 -1-1 -1-1 -1 II 3-1 1-1 -1-1 -1-1 II 4 1 1-1 -1-1 -1 1 I 5-1 -1 1-1 -1-1 -1 II 6 1-1 1-1 -1-1 1 I 7-1 1 1-1 -1-1 1 I 8 1 1 1-1 -1-1 -1 II 9-1 -1-1 1-1 -1-1 II 10 1-1 -1 1-1 -1 1 I : 63-1 1 1 1 1 1-1 II 64 1 1 1 1 1 1 1 I 17

Blocking Each block has 1/2 as many experiments Each block confounds the generator of the block. Design Variable (Yates Order) Run x 1 =a x 2 =b x 3 =c x 4 =d x 5 =e x 6 =f abcdef BLOCK 1-1 -1-1 -1-1 -1 1 I 2 1-1 -1-1 -1-1 -1 II 3-1 1-1 -1-1 -1-1 II 4 1 1-1 -1-1 -1 1 I 5-1 -1 1-1 -1-1 -1 II 6 1-1 1-1 -1-1 1 I 7-1 1 1-1 -1-1 1 I 8 1 1 1-1 -1-1 -1 II 9-1 -1-1 1-1 -1-1 II 10 1-1 -1 1-1 -1 1 I : 63-1 1 1 1 1 1-1 II 64 1 1 1 1 1 1 1 I abcdef = +1 abcdef = -1 18

Further interactions Consider a 2 4-1 block with abcd = +1 x 1 x 2 x 3 x 4 Exp. a b c d=abc ab ac ad bc bd cd abc abd acd bcd abcd 1 1-1 -1-1 -1 1 1 1 1 1 1-1 -1-1 -1 1 1 2 1-1 -1 1-1 -1 1 1-1 -1 1-1 -1 1 1 1 3-1 1-1 1-1 1-1 -1 1-1 1-1 1-1 1 1 4 1 1-1 -1 1-1 -1-1 -1 1-1 -1 1 1 1 1 5-1 -1 1 1 1-1 -1-1 -1 1 1 1-1 -1 1 1 6 1-1 1-1 -1 1-1 -1 1-1 -1 1-1 1 1 1 7-1 1 1-1 -1-1 1 1-1 -1-1 1 1-1 1 1 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 19

Confounded Variables a = bcd ab = cd b = acd ac = bd c = abd bc = ad d = abc 1 = abcd This always happens with partial factorial designs The trick is to select a good block for the phenomena you want to uncover. One where the design generator interaction is one you don t care about. 22

Design Resolution The design resolution is a rating of the level of interactions a design matrix can fit without confounding RESOLUTION III RESOLUTION IV RESOLUTION V No Main effect is confounded with another. Main effects and 2- factor interactions are confounded with each other. Main Effects are not confounded with each other or with 2- factor interactions. Main effects are confounded with 3-factor interactions, and 2-factor interactions are confounded with each other. No Main Effect or 2-factor interaction effect is confounded with another. Main Effects are confounded with 4-Factor interactions, 2-Factor interactions with 3-Factors. 23

Design Resolutions The resolution of a 2-level fractional factorial design is the smallest sum of the orders of aliased effects. Resolution Smallest Sum of Aliased Effects III IV 1+3, 2+2 V 1+4, 2+3 The resolution of a fractional factorial design is often included as a subscript in the designation of the design, e.g. 2 4-1 IV design. Should you avoid Res III designs? 1+2 (main effect with 2-Factor Interaction) 24

Design Resolution in Minitab Stat > DOE > Factorial > Create Factorial Design, then choose Display Available Designs 25

Fractional Factorials Rules Rule #1. Do not confound main effects with each other. When can you violate this rule? Rule #2. Do not confound main effects with two factor interactions. When can you violate this rule? Rule #3. Do not confound two factor interactions with each other. When can you violate this rule? 26

Matlab 2 n-k fractional factorial arrays are easily created in matlab X = fracfact(string) where string is a text string of the generator such as 'a b c d bcd acd The result is in coded units -1, 1 Create a second uncoded units matrix replacing these with the levels of each variable >> X = fractfact('a b c d bcd acd ) X = -1-1 -1-1 -1-1 -1-1 -1 1 1 1-1 -1 1-1 1 1-1 -1 1 1-1 -1-1 1-1 -1 1-1 -1 1-1 1-1 1-1 1 1-1 -1 1-1 1 1 1 1-1 1-1 -1-1 -1 1 1-1 -1 1 1-1 1-1 1-1 1-1 1-1 1 1-1 1 1 1-1 -1 1 1 1 1-1 1-1 -1 1 1 1-1 -1-1 1 1 1 1 1 1 27

DOE Arrays Available standard DOE arrays are available L4, L8, L16, etc x 1 x 2 x 3 A B C 1-1 -1 1 2 1-1 -1 3-1 1-1 4 1 1 1 Main Effect Confounded with Interaction BC AC AB x 1 x 2 x 3 x 4 x 5 x 6 x 7 A B C D E F G 1-1 -1-1 -1 1 1 1 2 1-1 -1 1 1-1 -1 3-1 1-1 1-1 1-1 4 1 1-1 -1-1 -1 1 5-1 -1 1 1-1 -1 1 6 1-1 1-1 -1 1-1 7-1 1 1-1 1-1 -1 8 1 1 1 1 1 1 1 Main Effect Confounded with Interaction DE DF DG CG FG EG EF CF CE BE BF BC BD CD BG AG AF AE AD AC AB 28

DOE Arrays Available standard DOE arrays are available L4, L8, L16, etc x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 x 14 x 15 A B C D E F G H I J K L M N O 1-1 -1-1 -1 1-1 -1-1 -1 1 1 1 1 1 1 2 1-1 -1-1 -1 1 1 1-1 -1-1 -1 1 1 1 3-1 1-1 -1-1 1 1-1 1-1 1 1-1 -1 1 4 1 1-1 -1 1-1 -1 1 1 1-1 -1-1 -1 1 5-1 -1 1-1 -1 1-1 1 1 1-1 1-1 1-1 6 1-1 1-1 1-1 1-1 1-1 1-1 -1 1-1 7-1 1 1-1 1-1 1 1-1 -1-1 1 1-1 -1 8 1 1 1-1 -1 1-1 -1-1 1 1-1 1-1 -1 9-1 -1-1 1-1 -1 1 1 1 1 1-1 1-1 -1 10 1-1 -1 1 1 1-1 -1 1-1 -1 1 1-1 -1 11-1 1-1 1 1 1-1 1-1 -1 1-1 -1 1-1 12 1 1-1 1-1 -1 1-1 -1 1-1 1-1 1-1 13-1 -1 1 1 1 1 1-1 -1 1-1 -1-1 -1 1 14 1-1 1 1-1 -1-1 1-1 -1 1 1-1 -1 1 15-1 1 1 1-1 -1-1 -1 1-1 -1-1 1 1 1 16 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Main Effect Confounde d with Interaction BJ AJ AK AL AI AM AN AO AE AB AC AD AF AG AH CK CM BM BN BH BK BL BE BO CF BF BG BC BD BI DL DN DO CO CG CJ CE CL CN DG DH CH DI CI CD EI EH EG EF DF DE DJ DK DM EO EN EM EL EK EJ FM FK FJ GJ JO GO FO FN FL HI GI FI GH HF FG GN GL HL HK KN HN HM GM GK KM JM JN JK JL KL HO IO IN IM LM IL IK JI HJ LN LO OK ON OM MN 29

Higher Level Designs There are many other experimental designs 3 level factorial design N level factorial design Box-Behnken designs Central Composite designs All of these need more detailed discussions 32

Factorial Design Summary Full factorial analysis is inefficient in most cases Factorial designs model main effects and interactions Not quadratic terms Fractional factorial designs confound terms Regression is used to determine significant terms and establish a simplified equation 33

Overview Fractional Factorial Designs Taguchi s Method Robustness Improvement ` 34

Genichi Taguchi Taguchi was born and raised in the textile town of Tokamachi, in Niigata prefecture. He initially studied textile engineering at Kiryu Technical College with the intention of entering the family kimono business.[4] However, with the escalation of World War II, he was drafted into the Astronomical Department of the Navigation Institute of the Imperial Japanese Navy. In 1948 he joined the Ministry of Public Health and Welfare, where he came under the influence of eminent statistician Matosaburo Masuyama, who kindled his interest in the design of experiments. In 1950, he joined the Electrical Communications Laboratory (ECL) of the NTT just as SQC was beginning to become popular in Japan under the influence of W. Edwards Deming. In 1960 he wrote his first book Experimental Design and Life Test Analysis and Design of Experiments for Engineers. The second edition introduced the signal to noise ratio. On completing his doctorate at Kyushu University in 1962, he visited Princeton University under the sponsorship of John Tukey, who arranged a spell at Bell Labs. In 1964 he became professor of engineering at Aoyama Gakuin University, Tokyo. In 1980 Taguchi himself financed a return to Bell Labs, where his initial teaching had made little enduring impact. This second visit began a collaboration with Madhav Phadke and a growing enthusiasm for his methodology in Bell Labs and elsewhere, including Ford Motor Company, Boeing, Xerox and ITT. As the executive director, he transformed the Ford Supplier Institute into the American Supplier Institute, providing statistical quality consulting and expertise. 35

Traditional DOE The design of experiments discussed is about understanding a response = + + + + 36

Robust Design DOE Robust design experimentation is about understanding the variability of a response = + + + + As well as the nominal = + + + + 37

Effect of Variation Suppose you have a response of interest = (,..., ) 38

Effect of Variation Suppose you have a response of interest = (,..., ) The performance of any individually built unit is dependent on the nominal design choices d 1,, d n and the hardware variations d 1,, d n = ( +,..., + ) 39

Effect of Variation Variation propagates through sensitivities = The variation arises from input hardware variations 40

Effect of Variation Variation propagates through sensitivities = Functions of the nominal design (,..., ) We have ability to control the sensitivities through design changes. Find a configuration with small sensitivities. = =( ) 41

Robust Design DOE Robust design relies on non-linearity of the system Look for regions where the response is less sensitive 42

Robust Design DOE Experiments to minimize variance Output is the response variance, not the response Noise Variables Design Variables Sigma 43

Models and Robustness Improvement Make changes to the design variables to reduce variation Pull back Stop Units Preload pin height a 0.0700 0.0700 m Arm attach point (band) b 0.1500 0.1500 m Distance from arm pivot to preload c 0.0700 0.0700 m Pullback angle theta (deg) 75.5392 25.0000 theta (rad) 1.3184 0.4363 Interim band angle calc alpha (rad) -0.1500 0.4591 alpha (deg) -8.5960 26.3080 Band seg length 1 l1 0.0700 0.0700 m Band seg length 2 l2 0.2177 0.1488 m total length L 0.2877 0.2188 m Rubber band Spring constant k 75.0000 75.0000 N/m Intial rubber band length lo 0.0500 0.0500 m ball mass mb 0.0130 0.0130 kg arm mass ma 0.2100 0.2100 kg arm width mw 0.0300 0.0300 arm length ra 0.3000 0.3000 m Cup Position rc 0.2500 0.2500 m Acceleration due to gravity g 9.8000 9.8000 m^2/s Arm moment of inertia Ia 0.0063 0.0063 Ball moment of inertia Ib 0.0008 0.0008 Total Moment of Intertia I 0.0071 0.0071 Spring Potential Energy Es 2.1186 1.0685 Gravity PE Epg 0.0851 0.3086 Kinetic Energy Ek 0.0000 0.8265 arm angular velocity w 0.0000 15.2283 s^-1 ball speed vb 0.0000 3.8071 m/s X component of velocity Vx 3.4504 Y component of velocity Vy 1.6089 Ball launch height h 0.2266 time to apogee ta 0.1642 height at apogee ymax 0.3586 time to fall tb 0.2705 total flight time t 0.4347 total x distance x 1.4999 44

Outline Taguchi Robust Design Signal to Noise Ratios Noise Experiment Main Experiment Verification Experiment 45

Various Quality Measures Several standard quality measures are used as a quality metric: PPM, DPMO, C p, C pk, Z, % of C pk s > 1.33 All of these rely on Upper and Lower Specification limits. 1s 1s Defects LSL USL Defects Measurements ns Zs ns Zs 46

Various Quality Measures A different approach is to define lack of quality as any deviation from the customer s target. Measure this as loss. Defects Defects Target from the Customer Measurements 47

Loss Functions A Loss Function is a mathematical relationship between cost and the departure from target Example: The relationship between the warrantee cost and an electric motor shaft runout measurement Warrantee fix Number of warrantee fixes 0 d 0 d Item-by-item: a repair job is needed if a particular motor is out of spec Fleetwide: Number of repair jobs that are returned for warrantee work as a function of variation 48

Loss Functions A Loss Function is a mathematical relationship between cost and the departure from target Example: The relationship between the warrantee cost and an electric motor shaft runout measurement Number of warrantee fixes Repair cost vs deviation Cost 0 d 0 d Fleetwide: Number of repair jobs that are returned for warrantee work as a function of variation Fleetwide: as runout increases, customers more likely to demand a repair. What is the cost function? 49

Goalpost Loss Function The traditional view of quality is meeting specifications. This is sometimes called the Goalpost Loss Function All values that lie within the specification limits are equally acceptable (e.g. kicking a field goal) Loss Defects Defects LSL Requirements from the Customer USL Measurements 50

Weaknesses of Goalpost Thinking Should performance at all values within specifications really be regarded as equally acceptable? If acceptability decisions are all made on basis of samples and specification limits, the resulting problem is tolerance compounding The accumulated effect of the concentrations of individual components all being close to their upper or lower specification limits 51

Quadratic Loss Function The deviation from target can be measured with a Quadratic Loss Function Loss Defects Defects Target from the Customer Measurements The warrantee cost is smallest when that characteristic is on target, and increases in proportion to the square of the departure from target. 52

Mathematically, why Quadratic? Suppose we don t know the loss function (warrantee cost, etc.) as a function of error. Then ( ) = ( ), for some unknown. Now: we know the cost is zero at target target = 0 Also, it is a minimum at target (target) = 0 So, in a Taylor series expansion, ( ) = (target) + ( target)+ target + We have = target, accurate to 2nd order. 53

Over a Population The total loss over all the customers is then ( ) = target In other words, ( ) = target where K is a warrantee cost coefficient. A quadratic loss function. 54

Summary: Population Effect The further off you are, then The more customers there are that are unhappy, and The more their individual unhappiness grows. Population of units built Population of units built with unhappy customers Error Loss Spec range Error Defects Defects LSL Target USL Measurements 55

Decomposing the Loss Function The total loss over all the customers is then ( ) = target One can show this is equivalent to ( ) = target + Measure of off-centering Measure of inherent variability 56

Signal to noise ratios Historically, Taguchi defined loss using what he termed a signal to noise ratio SN = 10log target This -10log(.) form is not strictly necessary. The important thing is minimizing not f, but the variation of f. With robust design, the objective is s f. This is more difficult to model through experimentation. 57

Robust Design Approach Step 1 Reduce the variability Original system Reduce variation Step 2 Shift the mean to the target value Shift on target 58

The 4 Step Roadmap Identification Noise Experiment Main Experiment Verification Create a P-diagram for the experiment Response, Noise, Design variables, Adjustments Identify worst case noise combinations Those that drive response high and low Form Loss Function and optimize The loss transfer function equation Find the design variables that minimize variation Find the design variables that shift the mean Verify the reduced variability The average response The capability C p 59

Factor Types Two influences cause the response to change Design Variables Selected by the engineer Also called hardware cofiguration variables or controllable factors, etc. Noise Uncontrollable at the customer site Also called variations, uncontrollable factors, etc. 60

Design Variable Types Two types of design variables are used by engineers Robustness Improvement Variables Targets set during system design, and then remain fixed Used to reduce variability of the response Adjustment Variables Not frozen at system design Used to continually shift the mean as needed 61

Noise Factor Types Two types of noise factors cause the response to vary Uncontrollable noise Not only uncontrollable at the customer site, but also Uncontrollable in a laboratory setting for DOE work Examples: measurement error, actuator error Controllable-in-the-Lab Noise factors Uncontrollable at the customer site, but however Controllable in a laboratory setting for DOE work Also called Assignable Cause Noise Examples: Tolerances, Material degradation, Environment 64

Taguchi Approach to Noise In the laboratory DOE work, we handle the two types of noise differently Eliminate uncontrollable noise in the lab Acquire better instrumentation and control to ensure Gage R&R Attack assignable causes of noise Find the worst case variations, and experiment using them Do NOT experiment with nominal hardware It is better for you to observe and improve your system with the worst case variations rather than some of your customers 67

Eliminate Uncontrollable Noise We have too many experiments to do, even when each measurement is perfect The number of runs get explosive if you have measurement errors ELIMINATE THESE NOISES FROM YOUR EXPERIMENTS Invest in better controls for the experiment Invest in better metrology for the experiment Inspect for perfect parts for the experiment Control the environment etc. Robust Design does not consider test statistics, p-values We assume that noise is small, and p-values are great. You can experiment with a noisy response, but then there are usually 10-30 repeats per run necessary to average it out. Difficult. 68

Attack Assignable Cause Noise To attack the noise, we need to optimize the design to reduce variation Find : = min 69

Attack Assignable Cause Noise To attack the noise, we need to optimize the design to reduce variation Find : = min, 70

Attack the Noise To solve this practically, Taguchi reduced this to experimental analysis Find : = max 10log, Where X D and X N are experimental matrices Taguchi termed X D the inner array. Taguchi termed X N the outer array. This is a complex and very insightful formulation, and a tribute to Taguchi that he could formulate and practically solve this using designed experiments alone. 71

Attack the Noise Form two design of experiment matrices, one for the design variables and one for the noise variables. Inner Array d 1 d 2 d n -1-1 -1 0-1 -1 +1-1 -1 +1 0-1 Outer Array n 1 n 2 n m -1-1 -1 +1-1 -1-1 +1-1 +1 +1-1 +1 +1 +1 +1 +1 +1 72

Attack the Noise Union the two matrices into a compounded matrix Outer Array -1-1 -1 n 1 +1-1 -1 n 2 d 1 d 2 d n +1 +1 +1 n m -1-1 -1 y 11.. y 1m 0-1 -1 y 21.. y 2m Inner Array +1-1 -1 : -1 0-1 +1 +1 +1 73

Attack the Noise Union the two matrices into a compounded matrix Compute the SN for each row = 10log, Outer Array -1-1 -1 n 1 +1-1 -1 n 2 d 1 d 2 d n +1 +1 +1 n m SN -1-1 -1 y 11.. y 1m SN 1 0-1 -1 y 21.. y 2m SN 2 Inner Array +1-1 -1 : : : -1-1 -1 +1 +1 +1 74

Attack the Noise Union the two matrices into a compounded matrix Now solve as a standard DOE problem Form the regression equation and determine the best d. Find : = max ( ) d 1 d 2 d n SN -1-1 -1 SN 1 0-1 -1 SN 2 Inner Array +1-1 -1 : -1-1 -1 +1 +1 +1 75

Outline Taguchi Robust Design Signal to Noise Ratios Noise Experiment Main Experiment Verification Experiment 76

Step 1: Identification Identification Noise Experiment Main Experiment Verification The first step is to scope the problem What is the response What are we trying to improve How do we measure it? What noise affects the response All assignable causes Eliminate all GR&R problems What design variables are available to change the response Possible robustness variables Possible adjustments that can be used later to shift the mean 77

The Noise Diagram Form the Noise-diagram for the system A list of all possible noises that might affect the mean We will do a noise experiment over these variables to screen down to the important ones that we will make the system robust to Internal Noise Tolerances, materials, External Noise Ambient temp., Use cases, System Response Degradation Noise Lifetime wearout, material degradation, 78

The P-diagram Form the preliminary P-diagram for the system Lists the important sources of noise from the noise experiment Lists the design variables for robustness improvement Lists an adjustment for later phase target shifting Noise Variables Adjustment System Response Design Variables 79

Step 2: Noise Experiment Identification The next step is to identify the worst case combinations of noise Noise Experiment Main Experiment Verification 82

What s Important? All variation is due to assignable causes of noise Remember, we have no GR&R variation So what do we need to know to analyze variability? Measured Values 83

What s Important? All variation is due to assignable causes of noise Remember, we have no GR&R variation So what do we need to know to analyze variability? If we know these two values, we know the repeatable variability Measured Values Taguchi Approach: Find the noise combinations which create these two worst case responses! 84

The Noise Experiment Experiment to find the directionality of each noise variable Using the nominal system design configuration If we increase/decrease a noise variable, does the response increase or decrease? Do an experiment to determine this Internal Noise External Noise System Response Degradation Noise 85

Noise Experiment Experiment across the noise variables Establish the main effect of each Typically with a Res III experimental array n 1 n 2 n m y -1-1 -1 y 1 +1-1 -1 y 2-1 +1-1 : +1 +1-1 +1 +1 +1 86

Noise Experiment Experiment across the noise variables Establish the main effect of each Determine the combinations which make y lowest Determine the combinations which make y highest n 1 n 2 n m y -1-1 -1 y 1 +1-1 -1 y 2-1 +1-1 : +1 +1-1 n low n high +1-1 -1 +1-1 -1 +1-1 : : +1 +1 +1 +1-1 87

The Noise Experiment Result The result is a matrix listing what settings to experiment at for each noise variable Y low Y high Noise Variable 1 + Noise Variable 2 + Noise Variable 3 + Noise Variable 4 + Noise Variable 5 + Noise Variable 6 not significant 88

Example: Thermostat Problem M2 screw Electrically isolating washer. 1.00mm Positive lead. 0.300mm Contact arm. 1.00mm Contact. 1.00mm Shim. 0.300mm ±0.006mm Electrical Isolator. 3.40mm A A Gap g Negative lead. 0.300mm Electrically isolating washer. 1.00mm 34.050 33.950 Contact. 1.00mm Bimetal strip. 0.900mm M2 nut Note A: flatness 0.001 Tolerances: 0.X ±0.250mm 0.XX ±0.100mm 0.XXX ±0.010mm unless otherwise stated 89

Example: Thermostat Problem Nominal Upper Lower mean sigma Distribution contact -c 1-1.000-0.900-1.100-1.000 0.0317 Normal strip a 1 flatness 0.000 0.001 0.000 0.000 0.0001 Exponential lead l 1 0.300 0.310 0.290 0.300 0.0028 Normal insulator H 3.400 3.500 3.300 3.400 0.0317 Normal shim sh 0.300 0.306 0.294 0.300 0.0014 Normal lead l 2 0.300 0.310 0.290 0.300 0.0028 Normal upper surface a 2 flatness 0.000 0.001 0.000 0.000 0.0001 Exponential contact -c 2-1.000-0.900-1.100-1.000 0.0317 Normal Initial T T0 25.0 25.0 Fixed strip thickness s 0.450 0.002-0.001 0.450 0.00038 Normal Strp material k 0.00000717 2.868E-07-2.868E-07 0.00000717 7.35E-08 Normal Length L 34.000 0.01-0.01 34.000 0.00278 Normal 90

Example: Thermostat Problem Form the noise matrix Experiment with variation values at ±4s Nominal Upper Lower contact -c 1-1.000-0.900-1.100 strip a 1 flatness 0.000 0.001 0.000 lead l 1 0.300 0.310 0.290 insulator H 3.400 3.500 3.300 shim sh 0.300 0.306 0.294 lead l 2 0.300 0.310 0.290 upper surface a 2 flatness 0.000 0.001 0.000 contact -c 2-1.000-0.900-1.100 Initial T T0 25.0 strip thickness s 0.450 0.002-0.001 Strip material k 0.00000717 2.868E-07-2.868E-07 Length L 34.000 0.01-0.01 91

Example: Thermostat Problem Form the noise matrix Use an 11 variable 16 run Res III design dc1 da1 dl1 dh dsh dl2 da2 dc2 ds dk dl -1-1 -1-1 -1-1 -1-1 1 1 1 1-1 -1-1 1-1 1 1-1 -1-1 -1 1-1 -1 1 1-1 1-1 -1 1 1 1-1 -1-1 1 1-1 1 1-1 -1-1 1-1 1 1 1-1 -1 1-1 1-1 1-1 -1 1-1 1 1-1 1-1 1 1-1 -1-1 1 1 1-1 -1 1 1 1-1 1-1 -1-1 -1 1 1-1 -1-1 1-1 1 1 1-1 1 1 1-1 -1 1 1 1-1 -1 1-1 -1-1 1-1 1 1-1 1-1 1-1 1 1 1-1 1-1 -1-1 1-1 1-1 -1-1 1 1 1-1 -1 1 1 1-1 1-1 1 1-1 -1 1-1 -1-1 1-1 1 1 1-1 1-1 -1-1 -1-1 1 1 1 1 1 1 1 1 1 1 1 92

Example: Thermostat Problem Evaluate the Temperature response at each Use variations off the nominal design variable configuration dc1 da1 dl1 dh dsh dl2 da2 dc2 ds dk dl T -1-1 -1-1 -1-1 -1-1 1 1 1 130.2 1-1 -1-1 1-1 1 1-1 -1-1 158.2-1 1-1 -1 1 1-1 1-1 -1 1 148.4 1 1-1 -1-1 1 1-1 1 1-1 141.3-1 -1 1-1 1 1 1-1 -1 1-1 132.0 1-1 1-1 -1 1-1 1 1-1 1 160.1-1 1 1-1 -1-1 1 1 1-1 -1 148.7 1 1 1-1 1-1 -1-1 -1 1 1 141.0-1 -1-1 1-1 1 1 1-1 1 1 150.6 1-1 -1 1 1 1-1 -1 1-1 -1 159.8-1 1-1 1 1-1 1-1 1-1 1 148.1 1 1-1 1-1 -1-1 1-1 1-1 159.8-1 -1 1 1 1-1 -1 1 1 1-1 151.7 1-1 1 1-1 -1 1-1 -1-1 1 158.5-1 1 1 1-1 1-1 -1-1 -1-1 148.9 1 1 1 1 1 1 1 1 1 1 1 162.6 = + 93

Example: Thermostat Problem The factor plots Main Effects Plot for T Data Means 155 c1 a1 l1 H 150 145 155-1 1-1 1-1 1-1 1 sh l2 a2 c2 Mean 150 145 155-1 1-1 1-1 1-1 1 s k L 150 145-1 1-1 1-1 1 96

Example: Thermostat Problem Noise factors to experiment at Upper Lower contact -dc 1 +1-1 strip da 1 flatness Doesn t matter lead dl 1 Doesn t matter insulator dh +1-1 shim dsh Doesn t matter Lead dl 2 Doesn t matter upper surface da 2 flatness Doesn t matter contact -dc 2 +1-1 Initial T dt0 Doesn t matter strip thickness ds Doesn t matter Strip material dk -1 +1 Length dl Doesn t matter NF high NF low contact -dc 1 +1-1 insulator dh +1-1 contact -dc 2 +1-1 Strip material dk -1 +1 Notice also the associated y values are around 135 and 165 C, which are 4-6 s values. 98

Outline Taguchi Robust Design Signal to Noise Ratios Noise Experiment Main Experiment Verification Experiment 99

Step 3: The Main Experiment Identification The next step is to determine control factor settings which reduce variability Noise Experiment Main Experiment Verification 100

The Main Experiment Experiment over the design variables The noise variables experimented only at the two worst cases The design variables are usually experimented at 2 or 3 levels Noise Variables System Response Design Variables 101

The Main Experiment The experimental matrix for static problems Outer Array d 1 d 2 d n NF low NF high -1-1 -1 0-1 -1 Inner Array +1-1 -1-1 0-1 +1 +1 +1 Measurements of y at the row s design variable settings at each worst case noise 102

Static Problem Analysis We calculate the loss function at each design variable combination ( ) = customers i Recall, we first minimize variation about average, then shift Typically a logarithmic transform is applied, to cause design variable interactions to become additive 1 S/ N( y) = -10log ( yi - y) Ł N NF i 2 ( ) = -10log s 2 ł 103

Static Problems: Two Step Analysis Original system Reduce variation Shift on target 104

Analysis of the Main Experiment We calculate the mean and S/N at each design variable arrangement Outer Array d1 d2 dn NF low NF high Mean S/N -1-1 -1 y low y high 0-1 -1 y low y high Inner Array +1-1 -1 y low y high -1 0-1 y low y high high i y low y high yi = 2 +1 +1 +1 y low y high S/ N i = -10log Ł y + 2 ( y - y ) + ( y - y ) i high i i y low i low i 2 ł 2 105

Analysis of the Main Experiment We calculate the mean and S/N at each design variable arrangement Then fit a regression equation to each = + + + = + + + 106

Analysis of the Main Experiment The loss function is quadratic Therefore DOEs that capture quadratic effects are necessary The Taguchi Arrays are DOEs designed for this to capture main and quadratic effects To dampen interaction effects The L arrays have (-1, 0, +1) combinations. 107

Taguchi s Arrays Robust design matrices are designed for quadratic effects, but minimizing interactions effects (aliased) The L18 array A B C D E F G H 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 1 1 3 3 3 3 3 3 1 2 1 1 2 2 3 3 1 2 2 2 3 3 1 1 1 2 3 3 1 1 2 2 1 3 1 2 1 3 2 3 1 3 2 3 2 1 3 1 1 3 3 1 3 2 1 2 2 1 1 3 3 2 2 1 2 1 2 1 1 3 3 2 2 1 3 2 2 1 1 3 2 2 1 2 3 1 3 2 2 2 2 3 1 2 1 3 2 2 3 1 2 3 2 1 2 3 1 3 2 3 1 2 2 3 2 1 3 1 2 3 2 3 3 2 1 2 3 1 108

Taguchi s Approximations This matrix is extremely sparse No interactions between design variables Usually, we use a Taguchi array for the design variable matrix (inner array) For main and quadratic effects of the design variables However, we have Resolution V on the interactions between design variables and noise variables! Both noise levels for each design variable combination Every adjustment level for each design variable combination The focus is not on design variable interactions, but design variable x noise variable interactions. 109

Analysis of the Main Experiment Linearity + Quadratic: = + + + + + + = + + + + + + Taguchi and robust design underwent undeserved criticism for an apparent assumption of linearity and quadratic effects, without interactions. 110

Analysis of the Main Experiment To handle nonlinearities, do not use the actual design variables themselves. Combine them into linear subsets. Since you know there is a non-linear interaction, don t waste experiments rediscovering it. This wastes experimental resources. Use a compound variable of the actual design variables. Reduce the problem to its unknown essence The focus must be on design x noise interactions 111

Analysis of the Main Experiment Do not use the actual design variables. Use compounded groupings of actual design variables. Suppose y = b 0 + b 12 d 1 d 2 Define a new compound variable d 12 = d 1 d 2 Use it instead of d 1 and d 2. Now there is no interaction term and only main effects. This requires engineering understanding of the design variables. Sometimes engineers have studied this; they just have not studied design x noise factor interactions. 112

Analysis of the Main Experiment We calculate the mean and S/N at each design variable arrangement Then fit a regression equation to each = + + + = + + + Historically, this is done graphically using factor plots. 113

Factor Plots The factor plots provide an indication the variable type Mean Mean Mean Mean -1 +1-1 +1-1 +1-1 +1 S/N S/N S/N S/N -1 +1-1 +1-1 +1-1 +1 114

Design Variable Type Identification Mean Mean Mean Mean -1 +1-1 +1-1 +1-1 +1 S/N S/N S/N S/N -1 +1-1 +1-1 +1-1 +1 Mean Shifter Robustness Factor Economic Factor Problem Factor 115

Step A: Optimize Robustness Factors This solves for *. This is equivalent to using the regression equation. Mean Mean Mean Mean -1 +1-1 +1-1 +1-1 +1 S/N S/N S/N S/N -1 +1-1 +1-1 +1-1 +1 116

Step B: Shift the Mean When setting the robustness factors, the mean will shift Now use the mean-shifter design variable to adjust the mean to be back to y target Mean -1 +1 S/N -1 +1 117

Example: Thermostat Problem List the design variables There are 7 hardware configuration variables we can change We would like to do a 3 level experiment. Lots of runs... Nominal Upper Lower contact -c 1-1.000-0.5-1.5 lead l 1 0.300 0.2 0.4 insulator H 3.400 2.0 4.0 lead l 2 0.300 0.2 0.4 contact -c 2-1.000-0.5-1.5 strip thickness s 0.450 0.3 0.6 Length L 34.000 25 45 119

Example: Thermostat Problem Apply compounding Define a new variable for the gap. Experiment with it alone = 1 + 1 + 1 + h + + 2 + 2 2 If we select a particular gap for robustness reasons, we know how to achieve that with these variables. M2 screw Electrically isolating washer. 1.00mm Positive lead. 0.300mm Contact arm. 1.00mm Contact. 1.00mm Shim. 0.300mm ±0.006mm Electrical Isolator. 3.40mm A A Gap g Negative lead. 0.300mm Contact. 1.00mm Electrically isolating washer. 1.00mm 35.050 34.950 Bimetal strip. 0.900mm M2 nut 120

Example: Thermostat Problem List the design variables There are 7 hardware configuration variables we can change We would like to do a 3 level experiment. Lots of runs... Nominal Upper Lower contact -c 1-1.000-0.5-1.5 lead l 1 0.300 0.2 0.4 insulator H 3.400 2.0 4.0 Nominal Upper Lower gap g nominal 2.300 3.000 2.000 strip thickness s nominal 0.450 0.6 0.3 Length L nominal 34.000 45 25 lead l 2 0.300 0.2 0.4 contact -c 2-1.000-0.5-1.5 strip thickness s 0.450 0.3 0.6 Length L 34.000 25 45 121

Example: Thermostat Problem Define new noise matrix NF high NF low contact -dc 1 +1-1 insulator dh +1-1 contact -dc 2 +1-1 Strip material dk -1 +1 NF high NF low gap -dg +1-1 Strip material dk -1 +1 122

Example: Thermostat Problem Define the design matrix Nominal Upper Lower gap g nominal 2.300 3.000 2.000 strip thickness s nominal 0.450 0.6 0.3 Length L nominal 34.000 45 25 M2 screw Electrically isolating washer. 1.00mm Positive lead. 0.300mm Contact arm. 1.00mm Contact. 1.00mm Shim. 0.300mm ±0.006mm Electrical Isolator. 3.40mm A A Gap g Negative lead. 0.300mm Electrically isolating washer. 1.00mm M2 nut 35.050 34.950 Contact. 1.00mm Bimetal strip. 0.900mm 123

Example: Thermostat Problem Set up the main experimental matrix N low N high g s L dg1 dk1 dg2 dk2 2.000 0.300 25.000-0.286 2.21E-07 0.286-2.2E-07 2.000 0.450 34.000-0.286 2.21E-07 0.286-2.2E-07 2.000 0.600 45.000-0.286 2.21E-07 0.286-2.2E-07 2.300 0.300 34.000-0.286 2.21E-07 0.286-2.2E-07 2.300 0.450 45.000-0.286 2.21E-07 0.286-2.2E-07 2.300 0.600 25.000-0.286 2.21E-07 0.286-2.2E-07 3.000 0.300 45.000-0.286 2.21E-07 0.286-2.2E-07 3.000 0.450 25.000-0.286 2.21E-07 0.286-2.2E-07 3.000 0.600 34.000-0.286 2.21E-07 0.286-2.2E-07 Design variables Noise Conditions 124

Example: Thermostat Problem Solve experimental matrix + = + ( + ) N low N high g s L dg dk dg dk T NF1 T NF1 2.000 0.300 25.000-0.286 2.21E-07 0.286-2.2E-07 136.3 182.9 2.000 0.450 34.000-0.286 2.21E-07 0.286-2.2E-07 115.3 153.0 2.000 0.600 45.000-0.286 2.21E-07 0.286-2.2E-07 93.7 122.5 2.300 0.300 34.000-0.286 2.21E-07 0.286-2.2E-07 95.7 121.6 2.300 0.450 45.000-0.286 2.21E-07 0.286-2.2E-07 85.6 107.7 2.300 0.600 25.000-0.286 2.21E-07 0.286-2.2E-07 286.6 382.2 3.000 0.300 45.000-0.286 2.21E-07 0.286-2.2E-07 79.4 95.0 3.000 0.450 25.000-0.286 2.21E-07 0.286-2.2E-07 289.4 365.4 3.000 0.600 34.000-0.286 2.21E-07 0.286-2.2E-07 215.6 270.4 Responses 125

Example: Thermostat Problem Compute mean N low g s L dg1 dk1 dg2 dk2 T NF1 T NF1 Tbar 2.000 0.300 25.000-0.286 2.21E-07 0.286-2.2E-07 136.3 182.9 159.6 2.000 0.450 34.000-0.286 2.21E-07 0.286-2.2E-07 115.3 153.0 134.2 2.000 0.600 45.000-0.286 2.21E-07 0.286-2.2E-07 93.7 122.5 108.1 2.300 0.300 34.000-0.286 2.21E-07 0.286-2.2E-07 95.7 121.6 108.6 2.300 0.450 45.000-0.286 2.21E-07 0.286-2.2E-07 85.6 107.7 96.6 2.300 0.600 25.000-0.286 2.21E-07 0.286-2.2E-07 286.6 382.2 334.4 3.000 0.300 45.000-0.286 2.21E-07 0.286-2.2E-07 79.4 95.0 87.2 3.000 0.450 25.000-0.286 2.21E-07 0.286-2.2E-07 289.4 365.4 327.4 3.000 0.600 34.000-0.286 2.21E-07 0.286-2.2E-07 215.6 270.4 243.0 N high y i = y high i + 2 y low i 127

Example: Thermostat Problem Compute SN N low N high g s L dg1 dk1 dg2 dk2 T NF1 T NF1 Tbar SN 2.000 0.300 25.000-0.286 2.21E-07 0.286-2.2E-07 136.3 182.9 159.6-27.34 2.000 0.450 34.000-0.286 2.21E-07 0.286-2.2E-07 115.3 153.0 134.2-25.52 2.000 0.600 45.000-0.286 2.21E-07 0.286-2.2E-07 93.7 122.5 108.1-23.15 2.300 0.300 34.000-0.286 2.21E-07 0.286-2.2E-07 95.7 121.6 108.6-22.22 2.300 0.450 45.000-0.286 2.21E-07 0.286-2.2E-07 85.6 107.7 96.6-20.87 2.300 0.600 25.000-0.286 2.21E-07 0.286-2.2E-07 286.6 382.2 334.4-33.58 3.000 0.300 45.000-0.286 2.21E-07 0.286-2.2E-07 79.4 95.0 87.2-17.86 3.000 0.450 25.000-0.286 2.21E-07 0.286-2.2E-07 289.4 365.4 327.4-31.59 3.000 0.600 34.000-0.286 2.21E-07 0.286-2.2E-07 215.6 270.4 243.0-28.75 S/ N i = -10log Ł 2 ( y - y ) + ( y - y ) i high i i low i 2 ł 2 128

Example: Thermostat Problem Generate factor plots Average Temperature (C) 400 300 200 100 0-15 Average Temperature (C) 400 300 200 100 0-15 Average Temperature (C) 400 300 200 100 0-15 -20-20 -20 SN Ratio -25-30 SN Ratio -25-30 SN Ratio -25-30 -35 1 2 3 Gap g -35 1 2 3 Strip Width s -35 1 2 3 Strip Length L Mean shifting adjustment factor Variation reducing robustness factors 131

Example: Thermostat Problem Most robust configuration solution Average Temperature (C) 400 300 200 100 0-15 Average Temperature (C) 400 300 200 100 0-15 Average Temperature (C) 400 300 200 100 0-15 -20-20 -20 SN Ratio -25-30 SN Ratio -25-30 SN Ratio -25-30 -35 1 2 3 Gap g -35 1 2 3 Strip Width s -35 1 2 3 Strip Length L Mean shifting adjustment factor Variation reducing robustness factors 132

Example: Thermostat Problem Prediction equations for mean and SN Regression Statistics Multiple R 0.987 R Square 0.974 Adj R Square 0.958 Std Error 20.24 Observations 9 Regression Statistics Multiple R 0.998 R Square 0.997 Adj R Square 0.995 Std Error 0.37 Observations 9 ANOVA df SS MS F Sig F Regression 3 75790.0 25263.3 61.6 0.000 Residual 5 2049.1 409.8 Total 8 77839.1 ANOVA df SS MS F Sig F Regression 3 211.6 70.5 522.8 0.000 Residual 5 0.7 0.1 Total 8 212.3 Coef Std Er t Stat P-value Intercept 158.9 29.4 5.4 0.003 g 42.6 8.3 5.2 0.004 s 55.0 8.3 6.7 0.001 L -88.3 8.3-10.7 0.000 Coef Std Er t Stat P-value Intercept -29.11 0.53-54.54 0.000 g -0.37 0.15-2.46 0.057 s -3.01 0.15-20.08 0.000 L 5.11 0.15 34.05 0.000 133

Example: Thermostat Problem Prediction equations for mean and SN Regression Statistics Tbar Predictor Multiple R 0.987 Intercept 1.0 R Square 0.974 g 2.0 Adj R Square 0.958 s 2.0 Std Error 20.24 L 2.0 Observations 9 Regression Statistics SN Predictor Multiple R 0.998 Intercept 1.0 R Square 0.997 g 2.0 Adj R Square 0.995 s 2.0 Std Error 0.37 L 2.0 Observations 9 ANOVA df SS MS F Sig F Regression 3 75790.0 25263.3 61.6 0.000 Residual 5 2049.1 409.8 Total 8 77839.1 ANOVA df SS MS F Sig F Regression 3 211.6 70.5 522.8 0.000 Residual 5 0.7 0.1 Total 8 212.3 Coef Std Er t Stat P-value Intercept 158.9 29.4 5.4 0.003 g 42.6 8.3 5.2 0.004 s 55.0 8.3 6.7 0.001 L -88.3 8.3-10.7 0.000 Coef Std Er t Stat P-value Intercept -29.11 0.53-54.54 0.000 g -0.37 0.15-2.46 0.057 s -3.01 0.15-20.08 0.000 L 5.11 0.15 34.05 0.000 134

Example: Thermostat Problem Prediction equations for mean and SN Regression Statistics Tbar Predictor Multiple R 0.987 Intercept 1.0 R Square 0.974 g 2.0 Adj R Square 0.958 s 2.0 Std Error 20.24 L 2.0 Observations 9 ANOVA Tbar 177.7 C df SS MS F Sig F Regression =sumproduct(coeff,values) 3 75790.0 25263.3 61.6 0.000 Residual 5 2049.1 409.8 Total 8 77839.1 Coef Std Er t Stat P-value Intercept 158.9 29.4 5.4 0.003 g 42.6 8.3 5.2 0.004 s 55.0 8.3 6.7 0.001 L -88.3 8.3-10.7 0.000 Regression Statistics SN Predictor Multiple R 0.998 Intercept 1.0 R Square 0.997 g 2.0 Adj R Square 0.995 s 2.0 Std Error 0.37 L 2.0 Observations 9 ANOVA df SS MS F Sig F Regression 3 211.6 70.5 522.8 0.000 Residual 5 0.7 0.1 Total 8 212.3 Coef Std Er t Stat P-value Intercept -29.11 0.53-54.54 0.000 g -0.37 0.15-2.46 0.057 s -3.01 0.15-20.08 0.000 L 5.11 0.15 34.05 0.000 135

Example: Thermostat Problem Prediction equations for mean and SN Regression Statistics Tbar Predictor Multiple R 0.987 Intercept 1.0 R Square 0.974 g 2.0 Adj R Square 0.958 s 2.0 Std Error 20.24 L 2.0 Observations 9 ANOVA Tbar 177.7 C df SS MS F Sig F Regression 3 75790.0 25263.3 61.6 0.000 Residual 5 2049.1 409.8 Total 8 77839.1 Coef Std Er t Stat P-value Intercept 158.9 29.4 5.4 0.003 g 42.6 8.3 5.2 0.004 s 55.0 8.3 6.7 0.001 L -88.3 8.3-10.7 0.000 Regression Statistics SN Predictor Multiple R 0.998 Intercept 1.0 R Square 0.997 g 2.0 Adj R Square 0.995 s 2.0 Std Error 0.37 L 2.0 Observations 9 SN -25.7 db ANOVA df SS MS F Sig F =sumproduct(coeff,values) Regression 3 211.6 70.5 522.8 0.000 Residual 5 0.7 0.1 Total 8 212.3 Coef Std Er t Stat P-value Intercept -29.11 0.53-54.54 0.000 g -0.37 0.15-2.46 0.057 s -3.01 0.15-20.08 0.000 L 5.11 0.15 34.05 0.000 136

Example: Thermostat Problem Prediction equations for mean and SN Regression Statistics Tbar Predictor Multiple R 0.987 Intercept 1.0 R Square 0.974 g 2.0 Adj R Square 0.958 s 2.0 Std Error 20.24 L 2.0 Observations 9 ANOVA Tbar 177.7 C df SS MS F Sig F Regression 3 75790.0 25263.3 61.6 0.000 Residual 5 2049.1 409.8 Total 8 77839.1 Coef Std Er t Stat P-value Intercept 158.9 29.4 5.4 0.003 g 42.6 8.3 5.2 0.004 s 55.0 8.3 6.7 0.001 L -88.3 8.3-10.7 0.000 Regression Statistics SN Predictor Multiple R 0.998 Intercept 1.0 R Square 0.997 g 2.0 Adj R Square 0.995 s 2.0 Std Error 0.37 L 2.0 Observations 9 SN -25.7 db 4 sigma 19.2 C ANOVA =sqrt(10^(-sn/10)) df SS MS F Sig F Regression 3 211.6 70.5 522.8 0.000 Residual 5 0.7 0.1 Total 8 212.3 Coef Std Er t Stat P-value Intercept -29.11 0.53-54.54 0.000 g -0.37 0.15-2.46 0.057 s -3.01 0.15-20.08 0.000 L 5.11 0.15 34.05 0.000 137

Example: Thermostat Problem Change values to the new more robust configuration Regression Statistics Tbar Predictor Multiple R 0.987 Intercept 1.0 R Square 0.974 g 2.0 Adj R Square 0.958 s 2.0 Std Error 20.24 L 2.0 Observations 9 ANOVA Tbar 177.7 C Coef Std Er t Stat P-value Intercept 158.9 29.4 5.4 0.003 g 42.6 8.3 5.2 0.004 s 55.0 8.3 6.7 0.001 L -88.3 8.3-10.7 0.000 Regression Statistics SN Predictor Multiple R 0.998 Intercept 1.0 R Square 0.997 g 2.0 Adj R Square 0.995 s 2.0 Std Error 0.37 L 2.0 Observations 9 SN -25.7 db ANOVA 4 sigma 19.2 C df SS MS F Sig F df SS MS F Sig F Regression 3 75790.0 25263.3 61.6 0.000 Regression 3 211.6 70.5 522.8 0.000 Residual 5 2049.1 409.8 Now solve over Residualg,s,L, 5 to 0.7 maximize 0.1 Total 8 77839.1 Total 8 212.3 SN subject to Tbar = 150. Coef Std Er t Stat P-value Intercept -29.11 0.53-54.54 0.000 g -0.37 0.15-2.46 0.057 s -3.01 0.15-20.08 0.000 L 5.11 0.15 34.05 0.000 138

Example: Thermostat Problem Change values to the new more robust configuration Regression Statistics Tbar Predictor Multiple R 0.987 Intercept 1.0 R Square 0.974 g 2.0 Adj R Square 0.958 s 1.0 Std Error 20.24 L 3.0 Observations 9 ANOVA Tbar 34.4 C df SS MS F Sig F Regression 3 75790.0 25263.3 61.6 0.000 Residual 5 2049.1 409.8 Total 8 77839.1 Regression Statistics SN Predictor Multiple R 0.998 Intercept 1.0 R Square 0.997 g 2.0 Adj R Square 0.995 s 1.0 Std Error 0.37 L 3.0 Observations 9 SN -17.5 db ANOVA 4 sigma 7.5 C df SS MS F Sig F Regression 3 211.6 70.5 522.8 0.000 Step 1: maximize SN over S,L Residual 5 0.7 0.1 Total 8 212.3 Coef Std Er t Stat P-value Intercept 158.9 29.4 5.4 0.003 g 42.6 8.3 5.2 0.004 s 55.0 8.3 6.7 0.001 L -88.3 8.3-10.7 0.000 Coef Std Er t Stat P-value Intercept -29.11 0.53-54.54 0.000 g -0.37 0.15-2.46 0.057 s -3.01 0.15-20.08 0.000 L 5.11 0.15 34.05 0.000 139

Example: Thermostat Problem Then adjust the mean shifter g unitl = 150 C Regression Statistics Tbar Predictor Multiple R 0.987 Intercept 1.0 R Square 0.974 g 4.7 Adj R Square 0.958 s 1.0 Std Error 20.24 L 3.0 Observations 9 ANOVA Tbar 150 C df SS MS F Sig F Regression 3 75790.0 25263.3 61.6 0.000 Residual 5 2049.1 409.8 Total 8 77839.1 Coef Std Er t Stat P-value Intercept 158.9 29.4 5.4 0.003 g 42.6 8.3 5.2 0.004 s 55.0 8.3 6.7 0.001 L -88.3 8.3-10.7 0.000 Regression Statistics SN Predictor Multiple R 0.998 Intercept 1.0 R Square 0.997 g 4.7 Adj R Square 0.995 s 1.0 Std Error 0.37 L 3.0 Observations 9 SN -18.5 db ANOVA 4 sigma 8.5 C df SS MS F Sig F Regression 3 211.6 70.5 522.8 0.000 Step 2: return Tbar to 150 by Total 8 212.3 changing g. Residual 5 0.7 0.1 Coef Std Er t Stat P-value Intercept -29.11 0.53-54.54 0.000 g -0.37 0.15-2.46 0.057 s -3.01 0.15-20.08 0.000 L 5.11 0.15 34.05 0.000 140

Example: Thermostat Problem Then adjust the mean shifter g unitl = 150 C Regression Statistics Tbar Predictor Multiple R 0.987 Intercept 1.0 R Square 0.974 g 4.7 Adj R Square 0.958 s 1.0 Std Error 20.24 L 3.0 Observations 9 ANOVA Tbar 150 C df SS MS F Sig F Regression 3 75790.0 25263.3 61.6 0.000 Residual 5 2049.1 409.8 Total 8 77839.1 Coef Std Er t Stat P-value Intercept 158.9 29.4 5.4 0.003 g 42.6 8.3 5.2 0.004 s 55.0 8.3 6.7 0.001 L -88.3 8.3-10.7 0.000 Regression Statistics SN Predictor Multiple R 0.998 Intercept 1.0 R Square 0.997 g 4.7 Adj R Square 0.995 s 1.0 Std Error 0.37 L 3.0 Observations 9 SN -18.5 db ANOVA 4 sigma 8.5 C Technically, this is outside the [1,3] range of dfexperimentation. SS MS But, F Sig it is F the stack height. Instead of redoing the Residual 5 0.7 0.1 entire main experiment, lets first do Total 8 212.3 this point, the predicted solution. Regression 3 211.6 70.5 522.8 0.000 Coef Std Er t Stat P-value Intercept -29.11 0.53-54.54 0.000 g -0.37 0.15-2.46 0.057 s -3.01 0.15-20.08 0.000 L 5.11 0.15 34.05 0.000 141

Example: Thermostat Problem Predicted most robust configuration (coded units) g = 4.7 s = 1 L = 3 This corresponds to (unencoded units) g = 6.0 mm s = 0.3 mm L = 4.5 cm A longer, thinner strip, with a wider gap. To make the gap wider, use a taller insulator. 45.050 44.950 Gap g Bimetal strip. 0.600mm 142

Outline Taguchi Robust Design Signal to Noise Ratios Noise Experiment Main Experiment Verification Experiment 143

Step 4: Verification Identification Determine C p at the new design variable configuration Do this not just at the 2 worst case noise configurations, but over the whole noise matrix Noise Experiment External Noise Internal Noise System Response Main Experiment Degradation Noise Verification LSL Target USL 144

Example Consider the thermostat problem M2 screw Electrically isolating washer. 1.00mm Positive lead. 0.300mm Contact arm. 1.00mm Contact. 1.00mm Shim. 0.300mm ±0.006mm Electrical Isolator. 7.1mm A A Gap g Negative lead. 0.300mm Electrically isolating washer. 1.00mm 45.050 44.950 Contact. 1.00mm Bimetal strip. 0.600mm M2 nut Note A: flatness 0.001 Tolerances: 0.X ±0.250mm 0.XX ±0.100mm 0.XXX ±0.010mm unless otherwise stated 145

Example: Thermostat Problem Recall the noise matrix NF high NF low contact -dc 1 +1-1 insulator dh +1-1 contact -dc 2 +1-1 Strip material dk -1 +1 146

Example: Thermostat Problem Evaluate y with the new design configuration over the noise variables NF high Nominal NF low contact -dc 1 +1 0-1 insulator dh +1 0-1 contact -dc 2 +1 0-1 Strip material dk -1 0 +1 Temperature 144.6 150.0 155.4 This is a C p = (160-140)/(155.4-144.6) = 1.8 147

Robust Design Summary Taguchi robust design is not a statistical method No statistical inferencing: t-tests, F-tests, sample sizes, Taguchi robust design is an experimental approach Equation fitting with completely accurate data Taguchi robust design approximations Usually simplifies noise to two worst case noise cases Usually simplifies design variables to (Res III + quadratic) designs No simplification on X d *X n interactions Taguchi robust design seeks the configuration with minimum performance variation over assignable noise 148