This document discusses design of experiments (DOE) and summarizes several key aspects: - DOE is a statistical methodology that aims to obtain maximum information from experiments using a minimum number of trials. It recognizes major factors that affect experimental outcomes. - Factors are input variables that can be changed, and have different levels. Full factorial designs involve varying one factor at a time through all levels of all factors. Taguchi methods use orthogonal arrays to study multiple factors simultaneously. - The document provides examples of orthogonal arrays like L4, L8, and L9 that can be used for experiments with different numbers of factors and levels. It also outlines the general steps of Taguchi method DOE including defining objectives

1. Design of Experiments
Jatinder kapoor
Professor
Mechanical Engineering Department
GNE, College ,Ludhiana
Selection of Significant Parameters for
Experimentation…..

2. Design of experiments
• It is important to obtain maximum realistic information
with the minimum number of well designed experiments.
• An experimental program recognizes the major “factors”
that affect the outcome of the experiment.
• The factors may be identified by looking at all the
quantities that may affect the outcome of the experiment.
• The most important among these may be identified using:
– a few exploratory experiments or
– From past experience or
– based on some underlying theory or hypothesis.
This Selection Process is known as
Design of Experiments.

3. Special Terminology : Design of Experiments
• Response variable
– Measured output value
• Factors
– Input variables that can be changed
• Levels
– Specific values of factors (inputs)
• Continuous or discrete
• Replication
– Completely re-run experiment with same input levels
– Used to determine impact of measurement error
• Interaction
– Effect of one input factor depends on level of another input
factor

4. Design of Experiments (DOE)
– A statistics-based approach to design experiments
– A methodology to achieve a predictive knowledge of a
complex, multi-variable process with the fewest acceptable
trials.
– An optimization of the experimental process itself

5. Major Approaches to DOE
• Factorial Design
• Taguchi Method
• Response Surface Design

6. Factorial Design : Full factorial design
• A full factorial design of experiments consists of the following:
• Vary one factor at a time
• Perform experiments for all levels of all factors
• Hence perform a large number of experiments that are needed!
• All interactions are captured.
• Consider a simple design for the following case:
• Let the number of factors = k
• Let the number of levels for the ith factor = ni
• The total number of experiments (N) that need to be performed is


K
i
inN
1

7. • 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

8. 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.
• 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.

9. Taguchi Method
• Dr. Taguchi of Nippon Telephones and Telegraph
Company, Japan has developed a method based on "
ORTHOGONAL ARRAY " experiments.
• This gives much reduced " variance " for the
experiment with " optimum settings " of control
parameters.
• "Orthogonal Arrays" (OA) provide a set of well
balanced (minimum) experiments serve as objective
functions for optimization.

10. Experimental methods are widely used in industrial
settings and research activities. In industrial settings, the
main goal is to extract the maximum amount of unbiased
information regarding the factors affecting production
process form few observations, whereas in research,
ANOVA techniques are used to reveal the reality. Drawing
inferences from the experimental result is an important
step in design process of product. Therefore, proper
planning of experimentation is the precondition for
accurate conclusion drawn from the experimental findings.
Design of experiment is powerful statistical tool introduced
by R.A. Fisher in England in the early 1920 to study the
effect of different parameters affecting the mean and
variance of a process performance characteristics

11. 2k factorial design
• Used as a Preliminary Experimentation !!!
• Each of the k factors is assigned only two levels.
• The levels are usually High = 1 and Low = -1.
• Scheme is useful as a preliminary experimental program
before a more ambitious study is undertaken.
• The outcome of the 2k factorial experiment will help
identify the relative importance of factors and also will
offer some knowledge about the interaction effects.

12. DOE - Factorial Designs - 23
Trial A B C
1 Lo Lo Lo
2 Lo Lo Hi
3 Lo Hi Lo
4 Lo Hi Hi
5 Hi Lo Lo
6 Hi Lo Hi
7 Hi Hi Lo
8 Hi Hi Hi

13. DOE - Factorial Designs - 23
Trial A B C
1 -1 -1 -1
2 -1 -1 +1
3 -1 +1 -1
4 -1 +1 +1
5 +1 -1 -1
6 +1 -1 +1
7 +1 +1 -1
8 +1 +1 +1

14. Output Matrix
• Let us represent the outcome of each experiment to be a
quantity y.
• Thus y1 will represent the outcome of experiment number
1 with all three factors having their “LOW” values,
• y2 will represent the outcome of the experiment number 2
with the factors A & B having the “Low” values and the
factor C having the “High” value and so on.

15. The advantages of DOE are summarized as:
• Number of trail experiment are significantly reduced
• The important decision variables are easily identified
• Optimal setting of process parameters
• Experimental errors are significantly reduced

16. • 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

17. Taguchi Method : When to Select a ‘larger’ OA
to perform “Factorial Experiments”
• We always ‘think’ about ‘reducing’ the number of
experiments (to minimize the ‘resources’ – equipment,
materials, manpower and time)
• However, doing ALL / Factorial experiments is a good
idea if
– Conducting experiments is ‘cheap/quick’ but
measurements are ‘expensive/take too long’
– The experimental facility will NOT be available later to
conduct the ‘verification’ experiment
– We do NOT wish to conduct separate experiments for
studying interactions between Factors

18. Taguchi Method Design of Experiments
• The general steps involved in the Taguchi Method are as follows:
• 1. Define the process objective, or more specifically, a target value
for a performance measure of the process.
• 2. Determine the design parameters affecting the process.
• The number of levels that the parameters should be varied at must
be specified.
• 3. Create orthogonal arrays for the parameter design indicating the
number of and conditions for each experiment.
• The selection of orthogonal arrays is based on the number of
parameters and the levels of variation for each parameter, and will
be expounded below.
• 4. Conduct the experiments indicated in the completed array to
collect data on the effect on the performance measure.
• 5. Complete data analysis to determine the effect of the different
parameters on the performance measure.

19. 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.

20. Taguchi's Orthogonal Arrays
L4(2^3)
2III
3-1
I = -ABC

21. Taguchi’s Two Level Designs-Examples
L8 (2^7)
L4 (2^3)

22. Taguchi’s Three Level Designs-
Example
L9 (3^4)

23. The full factorial randomized block (ANOVA)
Source Degree of
freedom
Sun of
squares
Mean square (MS) F- ratio
A r-1 SSA MSA = SSA/(r-1) FA= MSA/MSE
B s-1 SSB MSB = SSB/(s-1) FB= MSB/MSE
C t-1 SSC MSC = SSA/(t-1) FC= MSC/MSE
AB (r-1) (s-1) SSAB MSAB = SSA/(r-1)
(s-1)
FAB= MSAB/MSE
BC (s-1) (t-1) SSBC MSBC = SSA/(s-1)
(t-1)
FBC= MSAB/MSE
AC (r-1) (t-1) SSAC MSAC= SSA/(r-1)
(t-1)
FAC= MSAc/MSE
Blocks (p-1) (rs-1)
(st-1) (rt-1)
SSbl MS = SSbl/(p-1) FA= MSbl/MSE
Error p-1 SSE MSE = SSE/(rs-1)
(p-1) (st-1) (rt-1)
Total rstp-1 SST MSA = SSA/(r
A, B, C are factors having r, s and t their respective levels. L is the number of
blocks

24. 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.

25. signal-to-noise ratio
The S/N ratio provides a measure of the impact of noise
factors on the performance. The larger the S/N ratio , the
more robost the product is against the noise. Three types of
the S/N ratios are employed in practice depending upon the
experimental objective and the type of response. ). The
signal to noise ratios(S/N) in terms of larger the better (LB),
smaller the better (SB) and nominal the best (NB) are
calculated by using the following equations
LB: S/N Ratio = -10log10 [(1/N)* ∑(1/Yi
2)]
SB: S/N Ratio = -10log10 [(1/N)* ∑(Yi
2)]
NB: S/N Ratio = -10log10 [Mean2/variance]
Where Yi is the performance characteristic and N is the
number of observations for each trail.

26. Worked out Example
Control Parameters Range Designation Levels
L 1 L2 L3
Type of wire electrode - A Untreated
brass wire
Cryogenic treated
(-1100C) brass wire
Cryogenic treated
(-1840C) brass wire
Pulse width(μs) 0.4-1.2 B 0.4 0.8 1.2
Wire tension (daN) 0.6-2.0 C 0.6 1.3 2.0
Fixed Parameters
Time between two pulses 10 μs
Servo reference voltage 35 V
Short pulse time 0.2 μs
Wire feed rate 8m/min
Size of wire 0.25mm
Thickness of workpiece 11 mm
Angle of cut Straight
Cutting voltage(V ) -80
Max feed rate 15mm/min
Ignition pulse current 8
Injection pressure 4bar
Response Characteristics
1.MRR
2.SR
3.WWR

27. The experimental trail conditions for full factorial
Exp. No. Run Order A B C Response
1 12 1 1 1 Y11 Y12 Y13
2 15 1 1 2 --
3 10 1 1 3 --
4 2 1 2 1 --
5 13 1 2 2 --
6 22 1 2 3 --
7 11 1 3 1 --
8 25 1 3 2 --
9 9 1 3 3 --
10 1 2 1 1 --
11 26 2 1 2 --
12 21 2 1 3 --
13 14 2 2 1 --
14 4 2 2 2 --
15 8 2 2 3 --
16 18 2 3 1 --
17 19 2 3 2 --
18 27 2 3 3 --
19 24 3 1 1 --
20 5 3 1 2 --
21 23 3 1 3 --
22 20 3 2 1 --
23 7 3 2 2 --
24 3 3 2 3 --
25 16 3 3 1 --
26 6 3 3 2 --
27 17 3 3 3 Y27 1 Y27 2 Y27 3
A – Type of wire
B – Pulse width
C –Wire tension

28. DCTWEUTWE
60
50
40
30
1.20.80.4 16104
2.01.30.6
60
50
40
30
503520
T ype of wire(A )
MeanofMeans
Pulse width (B) T ime between two pulses (C)
Wire tension(D) Servo reference mean voltage(E)
Main Effects Plot for Means (MRR)
Data Means
DCTWEUTWE
34
32
30
28
1.20.80.4 16104
2.01.30.6
34
32
30
28
503520
T ype of wire(A )
MeanofSNratios
Pulse width (B) T ime between two pulses (C)
Wire tension(D) Servo reference mean voltage(E)
Main Effects Plot for SN ratios (MRR)
Data Means
Signal-to-noise: Larger is better

29. 1.20.80.4 6.02.01.3
50
35
20
50
35
20
T ype of wire(A )
Pulse width(B)
Wire tension(C)
DCTWE
SCTWE
UTWE
wire(A)
Type of
0.4
0.8
1.2
width(B)
Pulse
Interaction Plot for MRR (mm3/min)
Data Means

30. Taguchi Method
Experimentation was designed to study the effect of process
parameters on response characteristics of WEDM with untreated
and cryogenic treated wire electrode. Taguchi parametric
methodology was adopted for the optimal setting of process
parameters. The experimental findings were validated with the
help of confirmation experiments. The parameters were grouped
into two groups and experiments were conducted by using
appropriate orthogonal array.

31. Process parameters and their Values at Different
Levels
Parameters Designation Levels
*L 1 L2 L3
Type of wire A Untreated
brass wire
electrode
Deep Cryogenic treated
(-1840C) brass wire electrode
-
Pulse Width(μs) B 0.4 0.8 1.2
Time between two
pulses (μs)
C 4 10 16
Wire Tension
(daN(Kg))
D 0.6 1.3 2.0
Servo Reference
Voltage (V)
E 20 35 50
*L represents levels
Constant Parameters
Short pulse time 0.2 μs
Wire feed rate 8m/min
Size of wire 0.25mm
Thickness of workpiece 11 mm
Angle of cut Straight
Cutting voltage(V ) -80
Max feed rate 15mm/min
Ignition pulse current 8
Injection pressure 4bar
Response characteristics
MRR
SR
WWR

32. The L18 (21x37) Orthogonal array design matrix
Exp.
No.
Run
Order
1 2 3 4 5 6 7 8 Response
A B C D E F G H R1 R2 R3
1 4 1 1 1 1 1 1 1 1 Y11 Y12 Y13
2 17 1 1 2 2 2 2 2 2 -- -- --
3 1 1 1 3 3 3 3 3 3 -- -- --
4 13 1 2 1 1 2 2 3 3 -- -- --
5 6 1 2 2 2 3 3 1 1 -- -- --
6 15 1 2 3 3 1 1 2 2 -- -- --
7 8 1 3 1 2 1 3 2 3 -- -- --
8 12 1 3 2 3 2 1 3 1 -- -- --
9 9 1 3 3 1 3 2 1 2 -- -- --
10 11 2 1 1 3 3 2 2 1 -- -- --
11 18 2 1 2 1 1 3 3 2 -- -- --
12 14 2 1 3 2 2 1 1 3 -- -- --
13 16 2 2 1 2 3 1 3 2 -- -- --
14 2 2 2 2 3 1 2 1 3 -- -- --
15 7 2 2 3 1 2 3 2 1 -- -- --
16 5 2 3 1 3 2 3 1 2 -- -- --
17 3 2 3 2 1 3 1 2 3 -- -- --
18 10 2 3 3 2 1 2 3 1 Y18 1 Y18 2 Y18 3
Note: The 1, 2,3 are the levels of the parameters. R1, R2, R3 represent repetitions. Yij are the
measured values of the response characteristic.
A. Type of wire , B- Pulse width, C – Time between two pulses, D- Wire tension, E- Servo
reference mean voltage
Three columns(6, 7, 8) are ignored because only five parameters have been taken for the study

33. DCTWEUTWE
60
50
40
30
1.20.80.4 16104
2.01.30.6
60
50
40
30
503520
T ype of wire(A )
MeanofMeans
Pulse width (B) T ime between two pulses (C)
Wire tension(D) Servo reference mean voltage(E)
Main Effects Plot for Means (MRR)
Data Means
DCTWEUTWE
34
32
30
28
1.20.80.4 16104
2.01.30.6
34
32
30
28
503520
T ype of wire(A )
MeanofSNratios
Pulse width (B) T ime between two pulses (C)
Wire tension(D) Servo reference mean voltage(E)
Main Effects Plot for SN ratios (MRR)
Data Means
Signal-to-noise: Larger is better

34. Source DF Seq SS Adj SS Adj MS F P
A 1 293.30 293.30 293.30 19.79 0.004*
B 2 2912.29 2912.29 1456.15 98.26 0.000*
C 2 459.17 459.17 229.58 15.49 0.004*
D 2 156.88 269.51 134.75 9.09 0.015*
E 2 16.44 40.09 20.04 1.35 0.327
A*C 2 156.56 156.56 78.28 5.28 0.048*
Residual
Error
6 88.92 88.92 14.82
Total 17 4083.56
Type of wire, B- Pulse width, C- Time between two pulses,
D- wire tension, E- Servo reference mean voltage
*- significant at 95% confidence level
Source DF Seq SS AdjSS Adj MS F P
A 1 13.438 13.438 13.438 15.89 0.007*
B 2 180.577 180.577 90.2884 106.76 0.000*
C 2 37.860 37.860 18.9298 22.38 0.002*
D 2 12.720 33.579 16.7895 19.85 0.002*
E 2 1.321 17.191 8.5954 10.16 0.012*
A*C 2 22.669 22.669 11.3343 13.40 0.006*
Residual
Error
6 5.074 5.074 0.8457
Total 17 273.658
Type of wire, B- Pulse width, C- Time between two pulses,
D- wire tension, E- Servo reference mean voltage
*- significant at 95% confidence level

35. ESTIMATION OF OPTIMUM RESPONSE
CHARACTERISTICS
The significant process parameters affecting the MRR and their optimal levels are:
Significant parameters: A, B, C, D
Optimal levels; A2, B3, C1, D1, E1
= 47.83 The average value of MRR (From Table C1, Appendix C) at:
2nd Level of type of wire (A2)
3rd Level of pulse duration (B3) = 55.67
1st Level of time between two pulses(C1) = 47.74
1st Level of wire tension (D1) = 46.98
The overall mean of MRR (ŤMMR) = 43.79
The predicted optimal value of η MMR has been calculated as:
η MMR = A2+B3+C1+D1- 3 ŤMMR = 66.85

36. • The 95% confidence intervals for the mean of the population (CIPOP and CICE)
and three confirmation experiments have been calculated as
• N= 18x3= 54(treatment = 18, R (repetitions) = 3); fe(Error degree of freedom)
= (17-9)= 8 Ve(Error variance) = 13.17
• F0.05 (1, 8) = 5.32 (Tabulated F value)
•

38. Thanks…..