“Robust” chocolate bars are better (or Quality Engineering or Robust Design)

Focus is on reducing variability of response to maximize robustness, generally achieved through Historical Perspective

George E. P. Box (born 1919) was a student of R. A. Fisher. He made several advances to Fisher’s work in DOE theory and statistics. The founding chair of the University of Wisconsin’s Department of Statistics, Box was appointed the R. A. Fisher Professor of Statistics at UW in 1971. QE Overview

Objective of this Lecture: To explore the basic ideas of two-level factorial (DOE) and the connection of QE to statistical process control (SPC)

Key Points: • DOE can help uncover significant variables and interactions among variables. • SPC can help uncover process shifts. • Quality engineering tools help the investigator to discover a path for process improvement. Typical QE Applications

• In manufacturing – improve performance of a manufacturing process. • In process development – improve yields, reduce variability and cost. • In design – evaluation and comparison of basic configurations, materials and parameters. • The method is called Taguchi Methods. • The key tool is DOE (Design of ).

C. R. Rao

You’d know Rao from his Cramer-Rao Inequality. Rao is recognized worldwide as a pioneer of modern multivariate theory and as one of the world’s top statisticians, with distinctions as a mathematician, researcher, scientist and teacher. Taught Taguchi. Author of 14 books and over 300 papers. Genichi Taguchi

An engineer who developed an approach (now called Taguchi Methods) involving statistically planned experiments to reduce variation in quality. Learned DOE from Professor Rao. In 1960’s he applied his learning in Japan. In 1980’s he introduced his ideas to US and AT&T. What are Taguchi’s Contributions?

• Quality Engineering Philosophy – Targets and Loss functions • Methodology – System, Parameter, Tolerance design steps • Experiment Design – Use of Orthogonal arrays • Analysis – Use Signal-to-Noise ratios (S/N ratios) Conventional DOE focuses only on Average Response

FACTOR LOW (-) HIGH (+) -++ 7 8 +++ D (Driver) regular oversized ++- -+- 3 4 B (Beverage) beer water --+ O (Ball) 3-piece balanta B 5 6 +-+ 1 2 O --- Standard D B O Avg D1 +-- Order Response 1 - - - 67 77 87 2 + - - 79 3 - + - 61 4 + + - 75 61 75 5 - - + 65 6 + - + 60 B 65 60 7 - + + 77 8 + + + 87 O 67 79 QE focuses on Variability of Response D The Taguchi Loss Function and the typically assumed Loss to the Customer Taguchi’s Quality Philosophy

Taguchi’s view Conventional view

Loss = k(P – T)2 Not 0 if within specs and 1 if outside

On target production is more important than producing within Specs Taguchi’s key contribution is Robust Design

Definition: Robust Design – A Design that results in products or services that can function over a broad range of usage and environmental conditions. Taguchi’s Product Design Approach has 3 steps

System Design Choose the sub-systems, mechanisms, form of the prototype – develop the basic design. This is similar to conventional engineering design.

Parameter Design Optimize the system design so that it improves quality (robustness) and reduces cost.

Tolerance Design Study the tradeoffs that must be made and determine what tolerances and grades of materials are acceptable. Parameter Design (the Robust Design step)

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

By highlighting “robustness” as a key quality requirement, Taguchi really opened a whole area that previously had been talked about only by a few very applied people.

His methodology is heavily dependent on design of experiments like Fisher’s and Box’s methods, but the difference he made was that for response he looked at not only the mean but also the variance of performance. Robust Design – How it is done Design Parameters (D)

Target Actual

Performance Product/Process Performance (P)

Noise Factors (N), Internal & External

Identify Product/Process Design Parameters that: • Have significant/little influence on performance • Minimize performance variation due to noise factors • Minimize the processing cost

Methodology: Design of Experiments (DOE)

Examples: Chocolate mix, Ina Tile Co., Sony TV Taguchi’s Experimental Factors Parameter design step identifies and optimizes the Design Factors Control Factors – Design factors that are to be set at optimal levels to improve quality and reduce sensitivity to noise - Size of parts, type of material, Value of resistors etc.

Noise Factors – Factors that represent the noise that is expected in production or in actual use of the product - Dimensional variation - Operating Temperature

Adjustment Factor – Affects the mean but not the variance of a response - Deposition time in silicon wafer fabrication

Signal Factors – Set by the user communicate desires of the user - Position of the gas pedal Several different types of Experimental plans (“design”) are available to the design engineer – Factorial, Fractional, Central Cuboid etc. Taguchi used “Orthogonal” Designs

C S Center Screening

F O Factorial Orthogonal FF Fractional Factorial

Focus: Handle many factors Output: List of important Factors, Best Settings, Good design Full Factorial Array Example: The 23 (8-trial) array A B

Response

C

Array Columns 1 2 3 4 5 6 7

1 1 1 1 1 1 1 1 1 1 2 2 2 2 Full Factorial Factor 1 2 2 1 1 2 2 Assignments to 1 2 2 2 2 1 1 Experimental Array Columns. Such 2 1 2 1 2 1 2 experiments can find 2 1 2 2 1 2 1 all Main and two- and 2 2 1 1 2 2 1 three-factor interactions. 2 2 1 2 1 1 2

C B -BC A -AC -AB -ABC Taguchi’s Orthogonal Experimental Plan – 7 Factors (A, B, C, D, E, F and G) may potentially influence the production of defective tiles Table 1 Orthogonal plan used by Ina Tile company to reduce production of defective tiles

Process A B C D E F G Fraction of tiles found defective Variable Expt #1 A1 B1 C1 D1 E1 F1 G1 16/100 2 A1 B1 C1 D2 E2 F2 G2 17/100 3 A1 B2 C2 D1 E1 F2 G2 12/100 4 A1 B2 C2 D2 E2 F1 G1 6/100 5 A2 B1 C2 D1 E2 F1 G2 6/100 6 A2 B1 C2 D2 E1 F2 G1 68/100 7 A2 B2 C1 D1 E2 F2 G1 42/100 8 A2 B2 C1 D2 E1 F1 G2 26/100 Calculation of Factor Effects Variable Level % Defective Table 2: Summary of A1 12.75 Estimated Variable Effects A2 35.50 B1 26.75 B2 21.50 C1 25.25 C2 23.00 D1 19.00 D 29.25 E1 30.50 E2 17.75 F1 13.50 F2 31.75 G1 33.00 G2 15.25 Main Effects of Process Factors on % Defects in Tiles Figure 2: Influence of changing process conditions observed on defective tile production

Factor Effects on Response (% of Defective Tiles) 40 35 30 25 20

15 % % Defective 10 5 A1 B2 C1 D2 E1 F2 G1 Process Factor Settings Alternative Design Notations for Orthogonal Arrays

Std Fisher’s Original Yates Group Theory Taguchi Order

X1 X2 X3 X1 X2 X3 A B C 1 - - - 1 0 0 0 1 1 1 2 + - - a 1 0 0 2 1 1 3 - + - b 0 1 0 1 2 1 4 + + - ab 1 1 0 2 2 1 5 - - + c 0 0 1 1 1 2 6 + - + ac 1 0 1 2 1 2 7 - + + bc 0 1 1 1 2 2 8 + + + abc 1 1 1 2 2 2 Taguchi’s OA-based Experimental Design Matrix Notation

Number of Factors

k L N (2 )

Total Number of Runs Number of Levels per Factor Linear Graphs for the L8 Array

Linear graphs guide assignment of factors to L8 columns

1 7 1

5 3 3 7 5 2 4 2 4 6 6

 Main effects are assigned to columns at nodes in the graph.  Interactions are assigned to the columns on the lines. Some Orthogonal Array Designs

“Classical” “Taguchi”

3 26-3 2 3-1 L12 2 –L3 4 7-1 2 2 –L8 L18 25 215-11 –L 36 L 27-1 27

See Montgomery (1997), Design and Analysis of Experiments, page 631 Taguchi Orthogonal Array Tables

3 7 15 31 63 2-level (fractional factorial) arrays: L4(2 ). L8(2 ). L16(2 ). L32(2 ). L64(2 )

11 2-level arrays: L12(2 ) (Plackett-Burman Design)

4 3 40 3-level arrays: L9(3 ). L27(3 ). L81(3 )

5 21 4-level arrays: L16(4 ). L64(4 )

6 5-level arrays: L25(5 )

1 7 1 9 1 11 Mixed-level arrays: L15(2 X 3 ), L32(2 X 4 ), L50(2 X 5 ) Comments on Taguchi Arrays

Taguchi designs are large screening designs

Assumes most interactions are small and those that aren’t are known ahead of time

Taguchi claims that it is possible to eliminate interactions either by correctly specifying the response and design factors or by using a sliding setting approach to those factor levels.

Doesn’t guarantee that we get “highest resolution” design.

Instead of designing the experiment to investigate potential interactions, Taguchi prefers to use three-level factors to estimate curvature. Taguchi’s Robust Design Experiments

Taguchi advocated using E inner and outer array designs 1 to take into account noise E 2 factors (outer) and design factors (inner)

Design factors: I 1, I 2, I 3

Noise factors: E 1 and E 2

Objective: Maximize I response while minimizing its I 3 2 variance

I 1 Example: Robust Design OAs of Starter Motor Parameter Design Outer array: Battery voltage, ambient temperature

Replicates 1 2 3 4 Starter Torque E1 -1 1 -1 1 Output Output f1 f2 f3 E2 -1 -1 1 1 Mean Std. Dev. 1 -1 -1 -1 75 86 67 98 81.5 13.5 2 1 -1 -1 87 78 56 91 78.0 15.6 3 -1 1 -1 77 89 78 8 63.0 37.1 4 1 1 -1 95 65 77 95 83.0 14.7 5 -1 -1 1 78 78 59 94 77.3 14.3 6 1 -1 1 56 79 67 94 74.0 16.3 7 -1 1 1 79 80 66 85 77.5 8.1 8 1 1 1 71 80 73 95 79.8 10.9

Inner array: Armature turns, gage of wire, content of alloy S/N Ratios are Maximized

To maximize robustness, when Target performance is the best,Taguchi uses the signal- to-noise ratio 푦2 푆푁 = 10log⁡( ) 푧 푠2 When response is to be maximized, Taguchi uses

1/푦2 푆푁 = −10log⁡( ) 푧 푛

When response is to be minimized, Taguchi uses

푦2 푆푁 = −10log⁡( ) 푧 푛 Taguchi Analysis of Motor Design Data

25 20 15 10

% % Defective Robustness is maximized A1 B2 C1 D2 when S/N ratio is Inner Array Factor Settings maximized 25 Design (inner array) factor 20 settings that maximize S/N 15 ration are:

10 % % Defective A1 B2 C1 D2 I1 (turns) = - 1 Inner Array Factor Settings I2 (gage) = + 1 I3 (ferric %) = - 1

0.0015 Note: This system is not 0.001 0.0005 additive => Results are 0 approximatey OK.

% % Defective

A1 B2 C1 D2 Inner Array Factor Settings Epilogue

Designers should embrace Taguchi’s philosophy of quality engineering. It makes very good sense.

Note, however, that a key weakness of Taguchi method is its assumption of a “main factor only” (or “additive” model) … Taguchi ignores interactions

Therefore, rather than use inner – outer arrays, we may use more efficient and exact methods that are no more difficult to learn and apply to carry Taguchi’s robust design philosophy into practice…

You may use any of the various experimental and optimization techniques available in the literature such as multiple regression/RSM to develop robust designs.

An example of such extension is shown in next slides. Multiobjective Robust Design by Metaheuristic Methods

Tapan P. Bagchi & Madhu Ranjan Kumar (1993) Sensitivity Analysis vs. “Robust Design”

Practice common in conventional engineering design: sensitivity analysis (SA).

SA finds likely changes expected in the design’s performance due to uncontrollable factors.

For designs too sensitive, one uses the worst-case scenario – to plan for the unexpected.

However, worst-case projections often unnecessary: a “robust design” can greatly reduce off-target performance (Taguchi, 1986) Chocolates – Hot or Cold!

Molten Five Star bars on store shelves in Bombay and Singapore; Gooey hands and dirty dresses Design and Noise Factors Both Impact Response

Noise

Design Product Variable Factor A Response

Design Factor B The Empirical Framework for Progressing Knowledge

Weather Psychology Engineering

Economy Medicine Chemistry Electronics The Taguchi Experiments

Taguchi advocated using inner and outer Orthogonal array designs

Objective: Maximize satisfaction while minimizing its variance Robust Design search by GA

GA replaces inner array

Outer array replaced by Monte Carlo

Objective: Keep response on target while minimizing its variance Electronic Filter Interfacing a Strain Gage Transducer with a Galvanometer The Two Responses to be Made Robust

targets

푅2 + 푔⁡ 푅⁡푠푅3 + 푅3. 푅푠 ω = = 6.84 hz 2π 푅2 + 푅푔 . 푅3. 푅푠. 퐶

|푉푠|푅푔푅푠 D = = 3.0 inches 퐺푠푒푛[ 푅2 + 푅푔 푅푠 + 푅3 + 푅푠푅3]

=> Two constraints leave one design parameter free Variance of ωc Estimated

Variance of ωc

R3 Values Variance of D Estimated

Variance of D

R3 Values Original Taguchi Method is Ineffective Here!

Note that the filter design problem cannot be tackled using Taguchi’s “two-step” procedure.

Effects of R2, R3 and C on ωc (or even D) are not separable (additive).

Further, one-response-at-a-time optimization does not produce Pareto-optimal designs.

The two plots did produce two “best” robust designs, but none is “globally optimum”. Genetic Coding of the Problem

Decision Variable “Chromosome” buidler R2 010010010 R3 100010011 C 001101010

Population size kept constant through “survival of the fittest”. Progenies produced through reproduction, mating by crossover, and mutation of some “parent” chromosomes. GA parameterized by DOE. Nondominated Sorting Multiobjective GA used. The Nondominated Sorting Multiobjective GA produces Pareto- optimal solutions NSGA Output: Pareto-optimal Robust Designs

Variance of D Sample Numerical Results

R3 R2 C (farad) Avg (Hz) Var Avg (D) Var (D) (in) 311.78881 22.43919 0.000462 6.841476 0.08788 3.00349 0.0112 156.2765 157.7785 0.000434 6.844165 0.09586 3.00323 0.0105 306.0652 25.66875 0.000458 6.841552 0.08803 3.00347 0.0111 195.026 111.5562 0.000426 6.8434 0.09281 3.00325 0.0105 225.7759 82.24873 0.000426 6.842843 0.09110 3.00328 0.0106 195.1129 111.4653 0.000424 6.843399 0.09280 3.00325 0.0105 299.6366 29.40162 0.000454 6.841641 0.08821 3.00345 0.0111 References Taguchi, G., Introduction to Quality Engineering, APO, Tokyo, 1986. Pieset, J. and Singhal, K., Tolerance Analysis and Design, Elsevier, 1980. Phadke, M.S., Quality Engineering and Robust Design, Prentice-Hall 1989. Bagchi, Tapan P., Taguchi Methods Explained. Practical Steps to Robust Design, Prentice- Hall (India), 1983. Filippone, J., Using Taguchi Methods to Apply to the Axioms of Design, Robot, Computer Integrated Manufacturing 6(2) 1989 133 – 142 Bagchi, Tapan P., Kumar, Mahdu Ranjan, Multiple-Criteria Robust Design of Electronic Devices, J of Electronic Manufacturing 3, 1993, 31- 38. Montgomery, Douglas C., Design and Analysis Experiments 3rd Edition Wiley, 1993. Bagchi, Tapna P., Templeton, J.G.C. ,Multiple-criteria Robust Design using Constrained Optimization, J. Design and Manufacturing, 4, 1994, 21 – 30. Myers, J., Carter, Jr. W. H., Response Surface Techniques for Dual Response Systems, Technometrics, 26, 1973, 301 – 317. Khattree, Ravindra, Robust Parameter Design: A Response Surface Approach, J. Quality Technology, 1996, 28 (2) 187 – 195. Voss, Stetan, Metaheuristics: The State of the Art, In Local Search for Planning and Scheduling, Alexander Nareyek (ed) Springer, 2001. Goldberg, G. E., Genetic Algonrithm on Search. Optimization and Machine Learning, Addison-Wesley, 1989, 147 – 215.