design of experiments via taguchi methods21
TRANSCRIPT
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• 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 Design of Experiments
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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.
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• 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
Robust Design
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4
• The traditional model for quality losses• No losses within the specification limits!
The Taguchi Quality Loss Function
• The Taguchi loss function • the quality loss is zero only if we are on target
Scrap Cost
LSL USLTarget
Cost
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Example (heat treatment process for steel)• Heat treatment process used to harden steel
components
• Determine which process parameters have the greatest impact on the hardness of the steel components
Parameter number
ParametersLevel 1Level 2unit
1Temperature760900OC
2Quenching rate35140OC/s
3Cooling time1300s
4Carbon contents16Wt% c
5Co 2 concentration520%
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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 (3 to 50), few interactions between variables, and when only a few variables contribute significantly.
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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 2k 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.
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Half-Fraction Designs• A half-fraction of the 2k 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 3-1with 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).
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Half-Fraction Designs
23
2 3-1
I= ABC
2 3-1
I= -ABC
No. of runs = 8
No. of runs = 4
No. of runs = 4
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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.
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Quarter and Smaller Fraction Designs
• A quarter-fraction design, denoted as 2k-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-1design. The second defining relation selects half of the runs of the 2k-1 design to give the quarter-fraction.
• Figure a, I= ABCD 2k-1. Figure b, I=AD 2k-2
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Quarter and Smaller Fraction Designs
I= ABCD
24-1
I=AD
24-2
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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(2^3).
• L4 means the array requires 4 runs. 2^3 indicates that the design estimates up to three main effects at 2 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.
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Taguchi's Orthogonal ArraysL4(2^3)
2III3-1
I = -ABC
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Taguchi's Orthogonal Arrays
• Figure (b) shows the 2III3-1 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.
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Taguchi’s Two Level Designs-Examples
L8 (2^7)
L4 (2^3)
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Taguchi’s Three Level Designs- Example
L9 (3^4)
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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.
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signal-to-noise ratio
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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.
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Worked out Example…
• The operating conditions for each parameter and level are listed below:
• A: Temperature •A1 = 100ºC •A2 = 150ºC (current) •A3 = 200ºC
• B: Pressure •B1 = 2 psi •B2 = 5 psi (current) •B3 = 8 psi
• C: Doping Amount •C1 = 4% •C2 = 6% (current) •C3 = 8%
• D: Deposition Rate •D1 = 0.1 mg/s •D2 = 0.2 mg/s (current) •D3 = 0.3 mg/s
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Selecting the proper orthogonal array by Minitab Software
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Example: select the appropriate design
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Example: select the appropriate design
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Example: enter factors’ names and levels
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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 81 (=34)
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Selecting the proper orthogonal array by Minitab Software
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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.
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Worked out Example…
Experiment Number
Temperature
Pressure
Doping Amount
Deposition RateTrial 1Trial 2Trial 3Mean
Standard deviation
1100240.187.382.370.780.18.52100560.274.870.763.269.65.93100880.356.554.945.752.45.84150260.379.878.262.373.49.75150580.177.376.554.969.612.76150840.28987.383.286.537200280.264.862.355.760.94.78200540.39993.287.393.25.99200860.175.77463.2716.8
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Enter data to Minitab
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Worked out Example…
• b) Solution:For the first treatment, 5.19
5.8
1.80log10
2
2
SN i
Experiment Number
A (temp)
B (pres)
C (dop)
D (dep)T 1T 2T 3SNi
1111187.382.370.719.5
2122274.870.763.221.5
3133356.554.945.719.1
4212379.878.262.317.6
5223177.376.554.914.8
623128987.383.229.3
7311264.862.355.722.3
832239993.287.324.0
9331175.77463.220.4
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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
A (temp)
B (pres)
C (dop)
D (dep)SNi
1111119.52122221.53133319.14212317.65223114.86231229.37311222.38322324.09331120.4
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Worked out Example
LevelA (temp)B (pres)C (dop)D (dep)12019.824.318.2220.620.119.824.4322.222.918.720.2
2.23.15.56.1Rank4321
8.193
3.226.175.19B1 SN 1.20
30.248.145.21
B2 SN
9.223
4.203.291.19B3 SN
1.38.199.22 MinMaxThe 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.
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Example solution by Minitab
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Example: determine response columns
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Example Solution
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Example: Main Effect Plot for SN ratios
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Differences between SN and Means response table
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Main effect plot for means
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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.
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Mixed level designs
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Available designs
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Select the appropriate design
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Factors and levels
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Enter factors and levels names
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Design matrix