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Robust” Chocolate Bars Are Better Taguchi Methods (Or Quality Engineering Or Robust Design)

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Robust” Chocolate Bars Are Better Taguchi Methods (Or Quality Engineering Or Robust Design) “Robust” chocolate bars are better Taguchi Methods (or Quality Engineering or Robust Design) Focus is on reducing variability of response to maximize robustness, generally achieved through Orthogonal Array Experiments 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 design of experiments (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 Experiment). 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.
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