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 4 5 Parameters Temperature Quenching rate Cooling time Carbon contents Co concentration Level Level unit Determine which process parameters have the greatest impact on the hardness of the steel components 760 5 5 900 40 00 6 0 O C O C/s s Wt% c %
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 are an intermediate number of variables ( 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 k design becomes very large. For example, a single replicate of an 8 factor two level experiment would require 56 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 k design involves running only half of the treatments of the full factorial design. For example, consider a 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 - with the - " 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 ABCis at the high level (). The resulting - design has a design matrix as shown in Figure (b).
Half-Fraction Designs No. of runs = 8 No. of runs = 4 - I= ABC No. of runs = 4 - 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 k-, 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 k- design. The second defining relation selects half of the runs of the k- design to give the quarterfraction. Figure a, I= ABCD k-. Figure b, I=AD k-
Quarter and Smaller Fraction Designs I= ABCD 4- I=AD 4-
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 L4 array shown in the next Figure. The L4 array is denoted as L4(^). L4 means the array requires 4 runs. ^ indicates that the design estimates up to three main effects at levels each. The L4 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 L4(^) III - I = -ABC
Taguchi's Orthogonal Arrays Figure (b) shows the III - 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 thecolumnsinthe twodesigns arethesame except for the arrangement of the columns.
Taguchi s Two Level Designs-Examples L4 (^) L8 (^7)
Taguchi s Three Level Designs-Example L9 (^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. yiis the mean value and siis the variance. yiis 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 = 00ºC A = 50ºC (current) A = 00ºC B: Pressure B = psi B = 5 psi (current) B = 8 psi C: Doping Amount C = 4% C = 6% (current) C = 8% D: Deposition Rate D = 0. mg/s D = 0. mg/s (current) D = 0. 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 L9 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 8 (= 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 Standar d deviatio n Mean Trial Trial Trial Deposit ion Rate Doping Amoun t Pressur e Temper ature Experi ment Numbe r 8.5 80. 70.7 8. 87. 0. 4 00 5.9 69.6 6. 70.7 74.8 0. 6 5 00 5.8 5.4 45.7 54.9 56.5 0. 8 8 00 9.7 7.4 6. 78. 79.8 0. 6 50 4.7 69.6 54.9 76.5 77. 0. 8 5 50 5 86.5 8. 87. 89 0. 4 8 50 6 4.7 60.9 55.7 6. 64.8 0. 8 00 7 5.9 9. 87. 9. 99 0. 4 5 00 8 6.8 7 6. 74 75.7 0. 6 8 00 9
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Worked out Example b) Solution: SN i 80. 8.5 For the first treatment, = 0 log = 9. 5 Experiment Number 4 5 6 7 8 9 A (temp) B (pres) C (dop) D (dep) T 87. 74.8 56.5 79.8 77. 89 64.8 99 75.7 T 8. 70.7 54.9 78. 76.5 87. 6. 9. 74 T 70.7 6. 45.7 6. 54.9 8. 55.7 87. 6. SNi 9.5.5 9. 7.6 4.8 9.. 4.0 0.4
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 4 5 6 7 8 9 A (temp) B (pres) C (dop) D (dep) SNi 9.5.5 9. 7.6 4.8 9.. 4.0 0.4
Worked out Example SN 9.5 7.6. B = + + = 9.8 SN.5 4.8 4.0 B = + + = 0. SN = 9.+ 9.+ 0.4 B =.9 Level A (temp) B (pres) C (dop) D (dep) 0 9.8 4. 8. 0.6 0. 9.8 4.4..9 8.7 0... 5.5 6. Rank 4 The effect of this factor is then calculated by determining the range: = Max Min=.9 9.8=. 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: 00, 50, or 400 RPM Control algorithm: PID, PI, or P Valve type: butterfly or globe There are 4 parameters, and each one has levels with the exception of valve type.
Mixed level designs
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Factors and levels
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Design matrix