Taguchi’s Method: Design of Experiments and Orthogonal Arrays

1. RME-085
Total Quality Management
By:
Dr. Vinod Kumar Yadav
Department of Mechanical Engineering
G. L. Bajaj Institute of Technology and Management
Greater Noida
Email: vinod.yadav@glbitm.org
Topic: Taguchi’s Method: Design of Experiments and Orthogonal Arrays

2. Taguchi’s method - Purpose: Robust design
Dr. Genichi Taguchi
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical
Engineering at AKTU Lucknow in VIII semester 2019-20, and is not intended for wider circulation.
Quality is measured as the total
loss to society caused by a product
Loss
- Product Failure
- Environmental
Taguchi’sProcess[1]
Problem
identification
Brain storming
Experiment design
(OA based)
Conduct
Experimentation
Analysis
Conforming
experimentations

3. Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU Lucknow in VIII semester 2019-20, and is not intended for
wider circulation.
Taguchi’sProcess detailed steps
Step-1: Problem identification: Customer’s feedback, rework, history, forecasted parameters etc.
Step-2: Brain Storming:
- Identify critical variables that affects quality
- Identify control factors
- Identify signal factors
- Choose best plan (Nominal-the-best, smaller-the-better, larger-the-better etc.)
Step-3: Experiment design (Based on Orthogonal Array (OA) concept):
- Assuming there are n options, maximum optimization possible 2n combinations.
- We may use full factorial design (time consuming).
- Fractional factorial design is preferable (optimized time and cost) – subset of full factorial
design[1].
- Orthogonal Array (OA) : (Taguchi Design) – A good technique for fractional factorial design.
- OA: Helps to identify the effect of a factor in the presence of other factor (within confined space).
Step-4: Experiment
Step-5: Analyze results: Factors close to target value, ways of reducing controllable variables etc.
Step-6: Confirm Experimental results: Tests and validations.

4. [Step-3] Experiment design (Based on Orthogonal Array (OA) concept)
DOE (Applied statistics) – (i) How product/process will perform (ii) Parameters affecting outcomes
- Concerned with Planning, conducting, analyzing, and interpreting controlled tests to
evaluate the factors that control the value of a parameter or group of parameters[2].
- DOE is a powerful data collection and analysis tool that can be used in a variety of
experimental situations[2].
- Allows for multiple input factors to be manipulated, determining their effect on a desired
output (response).
- All possible combinations can be investigated (full factorial) or only a portion of the
possible combinations (fractional factorial).
- A strategically planned and executed experiment may provide a great deal of information
about the effect on a response variable due to one or more factors.
- Example: How the % of marks varies after completion of B.Tech. in GLBITM ?
- To create data: We need parameters – Experiment – Data analysis – Statistics (Mean, median, mod,
minimum, maximum std. deviation etc.).
- DOE is a function of attributes (Deptt, Year, PCM% etc.)
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU Lucknow in VIII semester 2019-20, and is not intended for
wider circulation.

5. Orthogonal Arrays (Central part of Taguchi’s concept of conducting experiments)
Orthogonal arrays (OA) –Simplified method of Exp.
- After finalizing the noise factor design,
experiments needs to be conducted to find the
optimum setting of the design parameters.
- Taguchi recommended to conduct a fraction
of total no. of possible experiments (using OA)
- OA represents a matrix of numbers. Each row
represents the levels or states, of the chosen
factors.
- Each column represents a specific factor
whose effect on the response variable are of
interest.
- Note: Every factor setting occurs same
number of times for every test setting of all
other factors. This helps to make a balanced
comparison among factor levels under a
variety of conditions.
Table – 1: Orthogonal array selection rules
For 2 levels
For 3 levels
L8 Orthogonal Array (27 OA)
1- Low level 2- High level

6. Orthogonal Arrays (Central part of Taguchi’s concept of conducting experiments) contd.
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU Lucknow in VIII semester 2019-20, and is not intended
for wider circulation.
Orthogonal arrays (OA) –Simplified method
of Exp.
Salient points:
- Table 2: The 8 in the designation OA8 represents the
number of rows, which is also the number of
treatment conditions (TC) and the degrees of
freedom[3].
- Top row of OA indicates maximum number of
factors that can be used (7 in Table-2).
- The levels can be represented by 1 and 2. In case of
more levels, 3, 4, 5, - , 0, and + can also be used. (1,
2 preferable).
- The properties of OA cannot be compromised by
changing the rows or the columns.
- Taguchi changed the rows from a traditional
design so that TC 1 was composed of all level 1s.
- Orthogonal arrays can handle dummy factors and can
be modified.
Table – 2: Orthogonal array (OA8)

7. Orthogonal Arrays (Central part of Taguchi’s concept of conducting experiments) contd.
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU Lucknow in VIII semester 2019-20, and is not intended
for wider circulation.
L4 (23 OA)
L9 (34 OA)

8. Orthogonal Arrays (Central part of Taguchi’s concept of conducting experiments) contd.
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU Lucknow in VIII semester 2019-20, and is not intended for wider circulation.
L8 OA for Noise factors
Column 3, 5 and 6 are outcomes
of interaction
Some Examples of controllable
factors related to machining:
1. Feed
2. Depth of cut
3. Spindle speed
More OAs can be adopted
from Appendix (Table H)
page 249-253 of the text book
D. H. Besterfield.

9. Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU Lucknow in VIII semester 2019-20, and is not intended for wider circulation.
Step-1: Project team – defines number of factors and their levels.
Step-2: Determination of Degree of Freedom (DOF)
- Determines the minimum number of treatment conditions.
- DOF = (No. of levels - 1) for each factor + (No. of levels
- 1)(No. of levels - 1) for each interaction + 1 for the
average.
- Example 1: There are 4 factors with 3 levels. Two
interactions are noticed. Determine the DOF.
- DOF = 4(3-1) + 4(3-1) (3-1) +1 = 25
- Hence, 25 treatment conditions are required for 3 levels.
- Consider same problem with level 2. The DOF will be 9
only.
- Hence, the number of levels significantly affects the
number of treatment conditions.
- Higher design levels provide more information about the
process but they may be costly.
Selection of Appropriate Orthogonal Arrays
Procedure to determine the
appropriate OA:
1. Define the number of factors and
their levels. (By project team).
2. Determine the degrees of freedom.
3. Select an orthogonal array.
4. Consider any interactions.
Maximum DOF = Lf
Where,
L = number of levels
f = number of factors
For Example 1 the DOF = 34 = 81

10. Taguchi’s Quality Engineering: Orthogonal Arrays contd.
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU Lucknow in VIII semester 2019-20, and is not intended
for wider circulation.
Step-3: Selection of OA
- The number of treatment conditions is equal
to the number of rows in the OA and must be
≥ the DOF.
- Table 3 available OA = 36
- If the number of degrees of freedom is 13,
then the next available OA is OA16.
- The second column of the table has the
number of rows and is redundant with the
designation in the first column.
- The third column gives the maximum
number of factors that can be used.
- Last four columns give the maximum number
of columns available at each level.
Table 3: Orthogonal Arrays[3]
- There is a Geometric progression for the 2
Level arrays of OA4, OA8, OA16, OA32, which is 22,
23, 24, 25.
- For the 3 level arrays of OA9, OA27, OA81, which is
32, 33, 34, Orthogonal arrays can be modified.

11. Taguchi’s Quality Engineering: Orthogonal Arrays and interaction Table
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech
Mechanical Engineering at AKTU Lucknow in VIII semester 2019-20, and is not intended for wider circulation.
Step-4: Interaction Consideration:
- Problem: Which columns to
use for the factors ??
- Solution: Interaction Table
Table 4 : OA8
Table 4 : Interaction Table[3] for OA8
- Factor F1 is assigned to column 1 and
factor F2 to column 2.
- If there is an interaction between factors F1
and F2 , then column 3 is used for the
interaction, F1.F2 .
- Factor F3 is assigned to column 4.
- If there is an interaction between factor F1
(column 1) and factor F3 (column 4), then
interaction F1.F3 will occur in column 5.
- The columns that are reserved for
interactions are used so that calculations
can be made to determine whether there is a
strong interaction.
- If there are no interactions, then all the
columns can be used for factors.
- The actual experiment is conducted using
the columns designated for the factors, and
these columns are referred to as the design
matrix.

12. Taguchi’s Quality Engineering: Orthogonal Arrays and interaction Table contd.
Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU Lucknow in VIII semester 2019-20, and is not intended
for wider circulation.
- Assign different factors with points.
- In case of interaction between two factors, draw a line segment
between those points.
- Factor F1 is assigned to column 1 and factor F2 is assigned to
column 2, then interaction F1.F2 is assigned to column 3.
- If there is no interaction, then column 3 can be used for a
factor.
1 4
2
6
53
7
F1
F1.F2
F3
F2
F1.F3
F4
1
2
3
4
5
6
7
One factor with three two-level interactions.
- Three-level orthogonal arrays must use two columns for interactions,
because one column is for the linear interaction and one for the
quadratic interaction.
- The interaction tables are not drawn for 3 or more factor interactions
(Rare case).
- Use of the linear graphs requires some trial-and error activity, and a
number of solutions may be possible.
Linear Graphs for interaction (Taguchi)

13. Note: The contents used in this slide is being used for academic purposes only, and is intended only for students registered in B.Tech Mechanical Engineering at AKTU Lucknow in VIII semester 2019-20, and is not intended for wider circulation.
The graph is constructed by plotting the points A1B1, A1B2, A2B1, and
A2B2 drawing lines B1 and B2.
Taguchi’s Approach to interactions:
- Interactions use degrees of freedom; therefore, more treatment conditions are
needed or fewer factors can be used.
- OAs are used in parameter design to obtain optimal factor/levels for robustness
and cost in order to improve product and process performance.
- Statistics are applied in pure and applied research to find
relationships and a mathematical model.
- Interactions are primarily between control factors and noise factors.
- As long as interactions are relatively mild, main effect analysis will give the
optimal results and good reproducibility.
- OA12 (two-level) and OA18 (three-level) are recommended so that if interactions
are present, they are dispersed among all the factors.
- Engineers should strive to develop a design that uses main effects only.
- Control factors that will not interact should be selected. For example, the
dimensions length and width will frequently interact, whereas the area may
provide the same information and save two degrees of freedom.
- Energy-related outputs, such as braking distance, should be selected whenever
possible.
- An unsuccessful confirmation run may indicate an interaction.
Linear Graphs for interaction between two factors (Taguchi)
No interaction
Little interaction
Strong interaction
A1 A2
B2
B1
A1
A1
A2
A2
B1
B1
B2
B2
Fig.1: interaction between two factors[3].

14. Glimpses of Quality Engineering by Dr. Taguchi
• Robust design (Taguchi) is good approach to control the quality at early stages of
product development.
• Quality design must be developed to ensure minimal loss to society.
- Orthogonal Array concept assures best selection which will maximize the response
under the influence of noises when the parameters are set at certain levels.
- Design of Experiments (DOE) can be used in Automotive, airlines, insurance,
restaurants, hotels etc.
• Results can be analyzed by computing S/N ratio using the approaches (i) Nominal-
the-best, Smaller-the-better etc. proposed by Dr. Taguchi.

15. References:
[1] https://www.youtube.com/watch?v=Xgd0aTVjXO8 (Accessed on 28.04.2020)
[2] https://asq.org/quality-resources/design-of-experiments (Accessed on 27.04.2020).
[3] Dale H. Besterfiled. A Text book on Quality Improvement. 9th Edition. Pearson (ISBN 10: 0-13-262441-9) pp. 211-235.

16. Thank you