Design of Experiments via Taguchi Methods21

• 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

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.

• 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

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

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%

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.

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.

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).

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

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 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

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 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.

Taguchi's Orthogonal ArraysL4(2^3)

2III3-1

I = -ABC

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.

Taguchi’s Two Level Designs-Examples

L8 (2^7)

L4 (2^3)

Taguchi’s Three Level Designs- Example  

L9 (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 •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

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 81 (=34)

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…

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

Enter data to Minitab

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

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

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.

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