Taguchi Design of Experiments Many factorsinputsvariables must be

Taguchi Design of Experiments • Many factors/inputs/variables must be taken into consideration when making a product especially a brand new one • 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 method • Traditional Design of Experiments focused on how different design factors affect the average result level • In Taguchi’s DOE (robust design), variation is more interesting to study than the average • Robust design: An experimental method to achieve product and process quality through designing in an insensitivity to noise based on statistical principles.

Robust Design • A statistical / engineering methodology that aim at reducing the performance “variation” of a system. • The input variables are divided into two board categories. • Control factor: the design parameters in product or process design. • Noise factor: factors whoes values are hard-to-control during normal process or use conditions

The Taguchi Quality Loss Function • The traditional model for quality losses • No losses within the specification limits! Cost Scrap Cost LSL Target USL • The Taguchi loss function • the quality loss is zero only if we are on target 4

Example (heat treatment process for steel) • Heat treatment process used to harden steel components Parameter number 1 Parameters Level 1 Level 2 unit Temperature 760 900 OC 2 Quenching rate 35 140 OC/s 3 Cooling time 1 300 s 4 Carbon contents 1 6 Wt% c 5 Co 2 concentration 5 20 % • Determine which process parameters have the greatest impact on the hardness of the steel components

Taguchi method • To investigate how different parameters affect the mean and variance of a process performance characteristic. • The Taguchi method is best used when there an intermediate number of variables (3 to 50), few interactions between variables, and when only a few variables contribute significantly.

Two Level Fractional Factorial Designs • As the number of factors in a two level factorial design increases, the number of runs for even a single replicate of the 2 k design becomes very large. • For example, a single replicate of an 8 factor two level experiment would require 256 runs. • Fractional factorial designs can be used in these cases to draw out valuable conclusions from fewer runs. • The principle states that, most of the time, responses are affected by a small number of main effects and lower order interactions, while higher order interactions are relatively unimportant.

Half-Fraction Designs • A half-fraction of the 2 k design involves running only half of the treatments of the full factorial design. For example, consider a 23 design that requires 8 runs in all. • A half-fraction is the design in which only four of the eight treatments are run. The fraction is denoted as 2 3 -1 with the “-1 " in the index denoting a half-fraction. • In the next figure: Assume that the treatments chosen for the half-fraction design are the ones where the interaction ABC is at the high level (1). The resulting 23 -1 design has a design matrix as shown in Figure (b).

Half-Fraction Designs No. of runs = 8 No. of runs = 4 23 2 3 -1 I= ABC 2 3 -1 I= -ABC

Half-Fraction Designs • The effect, ABC , is called the generator or word for this design • The column corresponding to the identity, I , and column corresponding to the interaction , ABC are identical. • The identical columns are written as I= ABC and this equation is called the defining relation for the design.

Quarter and Smaller Fraction Designs • A quarter-fraction design, denoted as 2 k-2 , consists of a fourth of the runs of the full factorial design. • Quarter-fraction designs require two defining relations. • The first defining relation returns the half-fraction or the 2 k-1 design. The second defining relation selects half of the runs of the 2 k-1 design to give the quarterfraction. • Figure a, I= ABCD 2 k-1. Figure b, I=AD 2 k-2

Quarter and Smaller Fraction Designs I= ABCD 24 -1 I=AD 24 -2

Taguchi's Orthogonal Arrays • Taguchi's orthogonal arrays are highly fractional orthogonal designs. These designs can be used to estimate main effects using only a few experimental runs. • Consider the L 4 array shown in the next Figure. The L 4 array is denoted as L 4(2^3). • L 4 means the array requires 4 runs. 2^3 indicates that the design estimates up to three main effects at 2 levels each. The L 4 array can be used to estimate three main effects using four runs provided that the two factor and three factor interactions can be ignored.

Taguchi's Orthogonal Arrays L 4(2^3) 2 III 3 -1 I = -ABC

Taguchi's Orthogonal Arrays • Figure (b) shows the 2 III 3 -1 design (I = -ABC, defining relation ) which also requires four runs and can be used to estimate three main effects, assuming that all two factor and three factor interactions are unimportant. • A comparison between the two designs shows that the columns in the two designs are the same except for the arrangement of the columns.

Taguchi’s Two Level Designs-Examples L 4 (2^3) L 8 (2^7)

Taguchi’s Three Level Designs- Example L 9 (3^4)

Analyzing Experimental Data • To determine the effect each variable has on the output, the signal-to-noise ratio, or the SN number, needs to be calculated for each experiment conducted. • yi is the mean value and si is the variance. yi is the value of the performance characteristic for a given experiment.

signal-to-noise ratio

Worked out Example • A microprocessor company is having difficulty with its current yields. Silicon processors are made on a large die, cut into pieces, and each one is tested to match specifications. • The company has requested that you run experiments to increase processor yield. The factors that affect processor yields are temperature, pressure, doping amount, and deposition rate. • a) Question: Determine the Taguchi experimental design orthogonal array.

Worked out Example… • The operating conditions for each parameter and level are listed below: • A: Temperature • A 1 = 100ºC • A 2 = 150ºC (current) • A 3 = 200ºC • B: Pressure • B 1 = 2 psi • B 2 = 5 psi (current) • B 3 = 8 psi • C: Doping Amount • C 1 = 4% • C 2 = 6% (current) • C 3 = 8% • D: Deposition Rate • D 1 = 0. 1 mg/s • D 2 = 0. 2 mg/s (current) • D 3 = 0. 3 mg/s

Selecting the proper orthogonal array by Minitab Software

Example: select the appropriate design

Example: select the appropriate design

Example: enter factors’ names and levels

Worked out Example… a) Solution: The L 9 orthogonal array should be used. The filled in orthogonal array should look like this: This setup allows the testing of all four variables without having to run 81 (=3 4)

Selecting the proper orthogonal array by Minitab Software

Worked out Example… • b) Question: Conducting three trials for each experiment, the data below was collected. Compute the SN ratio for each experiment for the target value case, create a response chart, and determine the parameters that have the highest and lowest effect on the processor yield.

Worked out Example… Experi ment Numbe r Temper ature 1 2 3 4 5 6 7 8 9 100 100 150 150 200 200 Deposit Pressur Doping ion e Amount Rate Trial 1 2 5 8 4 6 8 4 8 4 6 0. 1 0. 2 0. 3 0. 1 87. 3 74. 8 56. 5 79. 8 77. 3 89 64. 8 99 75. 7 Trial 2 Trial 3 Mean 82. 3 70. 7 54. 9 78. 2 76. 5 87. 3 62. 3 93. 2 74 70. 7 63. 2 45. 7 62. 3 54. 9 83. 2 55. 7 87. 3 63. 2 80. 1 69. 6 52. 4 73. 4 69. 6 86. 5 60. 9 93. 2 71 Standar d deviatio n 8. 5 5. 9 5. 8 9. 7 12. 7 3 4. 7 5. 9 6. 8

Enter data to Minitab

Worked out Example… • b) Solution: For the first treatment, Experiment Number 1 2 3 4 5 6 7 8 9 A (temp) 1 1 1 2 2 2 3 3 3 B (pres) 1 2 3 C (dop) 1 2 3 1 1 2 1 D (dep) 1 2 3 3 1 2 2 3 1 T 1 87. 3 74. 8 56. 5 79. 8 77. 3 89 64. 8 99 75. 7 T 2 82. 3 70. 7 54. 9 78. 2 76. 5 87. 3 62. 3 93. 2 74 T 3 70. 7 63. 2 45. 7 62. 3 54. 9 83. 2 55. 7 87. 3 63. 2 SNi 19. 5 21. 5 19. 1 17. 6 14. 8 29. 3 22. 3 24. 0 20. 4

Worked out Example • Shown below is the response table. calculating an average SN value for each factor. A sample calculation is shown for Factor B (pressure): Experiment Number 1 2 3 4 5 6 7 8 9 A B (temp) (pres) 1 1 1 2 1 3 2 1 2 2 2 3 3 1 3 2 3 3 C (dop) 1 2 3 1 1 2 1 D (dep) 1 2 3 3 1 2 2 3 1 SNi 19. 5 21. 5 19. 1 17. 6 14. 8 29. 3 22. 3 24. 0 20. 4

Worked out Example Level 1 2 3 Rank A (temp) B (pres) C (dop) 20 19. 8 24. 3 20. 6 20. 1 19. 8 22. 2 22. 9 18. 7 2. 2 3. 1 5. 5 4 3 2 D (dep) 18. 2 24. 4 20. 2 6. 1 1 The effect of this factor is then calculated by determining the range: Deposition rate has the largest effect on the processor yield and the temperature has the smallest effect on the processor yield.

Example solution by Minitab

Example: determine response columns

Example Solution

Example: Main Effect Plot for SN ratios

Differences between SN and Means response table

Main effect plot for means

Mixed level designs • Example: A reactor's behavior is dependent upon impeller model, mixer speed, the control algorithm employed, and the cooling water valve type. The possible values for each are as follows: • Impeller model: A, B, or C • Mixer speed: 300, 350, or 400 RPM • Control algorithm: PID, PI, or P • Valve type: butterfly or globe • There are 4 parameters, and each one has 3 levels with the exception of valve type.

Mixed level designs

Available designs

Select the appropriate design

Factors and levels

Enter factors and levels names

Design matrix