Design of experiments

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Design of Experiments (DOE)

OVERVIEW
o Approaches to Experimentation
o What is Design of Experiments
o Definition of DOE
o Why DOE
o History of DOE
o Basic DOE Example
o Factors, Levels, Responses
o General Model of Process or System
o Interaction, Randomization, Blocking, Replication
o Experiment Design Process
o Types of DOE
o One factorial
o Two factorial
o Fractional factorial
o Screening experiments
o Calculation of Alias
o DOE Selection Guide

WHAT WE LEARN
Basics of
DOE
Full
Factorial
Design
Fractional
Factorial
Design
Screening
Experiments
Minitab
Exercises

BASICS OF DOE

APPROACHES TO EXPERIMENTATION
o Trail and Error Method
o One Factor at a Time
o Design of Experiments

BASICS OF DOE
o What is DOE:
Design of Experiment (DOE) is a powerful statistical technique for
improving product/process designs and solving process / production
problems
DOE makes controlled changes to input variables in order to gain
maximum amounts of information on cause and effect relationships with
a minimum sample size
When analyzing a process, experiments are often used to evaluate which
process inputs have a significant impact on the process output and what
the target level the inputs should be to achieve a desired result (output).
Design of Experiments (DOE) is also referred to as Designed Experiments
or Experimental Design

BASICS OF DOE
o Why DOE:
• Reduce time to design/develop new products & processes
• Improve performance of existing processes
• Improve reliability and performance of products
• Achieve product & process robustness
• Perform evaluation of materials, design alternatives, setting
component & system tolerances

HISTORY OF DOE
• The agricultural origins, 1918 – 1940s
• R. A. Fisher & his co-workers
• Profound impact on agricultural science
• Factorial designs, ANOVA
• The first industrial era, 1951 – late 1970s
• Box & Wilson, response surfaces
• Applications in the chemical & process industries
• The second industrial era, late 1970s – 1990
• Quality improvement initiatives in many companies
• CQI and TQM were important ideas and became management goals
• Taguchi and robust parameter design, process robustness
• The modern era, economic competitiveness and globalization is driving
all sectors of the economy to be more competitive

DOE EXAMPLE

FACTORS, LEVELS, RESPONSE
o Factors:
Factors are inputs to the process
Factors can be classified as either controllable or uncontrollable variables.
In this case, the controllable factors are Flour, Eggs, Sugar and Oven.
Potential factors can be categorized using the Cause & Effect Diagram
o Levels:
Levels represent settings of each factor in the study
Examples include the oven temperature setting, no. of spoons of sugar,
no. of cups of flour, and no. of eggs
o Response:
Response is output of the experiment
In the case of cake baking, the taste, consistency, and appearance of the
cake are measurable outcomes potentially influenced by the factors and
their respective levels.

GENERAL MODEAL OF A PROCESS OR SYSTEM

KEY TERMINOLOGY
o Interaction
o Randomization
o Blocking
o Replication

KEY TERMINOLOGY
o Interaction:
Sometimes factors do not behave the same when they are looked at together
as when they are alone; this is called an interaction
Interaction plot can be used to visualize possible interactions between two or
more factors
Parallel lines in an interaction plot indicate no interaction
The greater the difference in slope between the lines, the higher the degree of
interaction
However, the interaction plot doesn't alert you if the interaction is statistically
significant
Interaction plots are most often used to visualize interactions during ANOVA
or DOE

KEY TERMINOLOGY
o Randomization:
Randomization is a statistical tool used to minimize potential
uncontrollable biases in the experiment by randomly assigning material,
people, order that experimental trials are conducted, or any other factor
not under the control of the experimenter
When we run designed experiments, we will use experimental templates
to set them up and to analyze them. We do not want to actually make the
experimental runs in the order shown by the template; wherever
possible, we want to randomize the experimental runs.
Randomization of the run order is needed to minimize the impact of
those variables outside of the experiment that we are not studying.

KEY TERMINOLOGY
o Blocking:
Blocking is a technique used to increase the precision of an experiment by
breaking the experiment into homogeneous segments (blocks or clusters
or strata) in order to control any potential block to block variability
Sometime we cannot totally randomize the experimental runs. Typically
this is because it will be costly or will take a long time to complete the
experiment.
Blocking means to run all combinations at one level before running all
treatment combinations at the next level.
Experimental runs within blocks must be randomized.

KEY TERMINOLOGY
o Replication:
Replication is making multiple experimental runs for each experiment
combination.
This is one approach to determining the common cause variation in the
process so that we can test effects for statistical significance.
Repetition of a basic experiment without changing any factor settings,
allows the experimenter to estimate the experimental error (noise) in the
system used to determine whether observed differences in the data are
“real” or “just noise”, allows the experimenter to obtain more statistical
power

EXPERIMENT DESIGN PROCESS

TYPES OF DOE
o One Factorial
o Full Factorials
o Fractional Factorials
o Screening Experiments
o Plackett-Burman Designs
o Taguchis Orthogonal Arrays
o Response Surface Analysis

ONE FACTORIAL METHOD
o One Factorial method:
One factorial experiments look at only one factor having an impact on
output at different factor levels.
The factor can be qualitative or quantitative.
In the case of qualitative factors (e.g. different suppliers, different
materials, etc.), no predictions can be performed outside the tested
levels, and only the effect of the factor on the response can be
determined.
In the case of quantitative factors (e.g. temperature, voltage, load, etc.)
can be used for both effect investigation and prediction, provided that
sufficient data are available.

In single factor experiments, ANOVA models are used to compare the
mean response values at different levels of the factor.
Each level of the factor is investigated to see if the response is significantly
different from the response at other levels of the factor.
The analysis of single factor experiments is often referred to as one-way
ANOVA

Use One-Way ANOVA (analysis of variance) to do the following when you
have one categorical factor and a continuous response:
Determine whether the means of two or more groups differ.
Obtain a range of values for the difference between the means for each
pair of groups.
Where to find this analysis in Minitab:
• STATISTICS > ANOVA > One-Way ANOVA

ONE FACTORIAL METHOD, EXAMPLE
Consider a BPO, where the Ops Manager wants to
know if the productivity (transactions per day) of his
Team is same everyday.
The assumption is the productivity is same on every
weekday  H0, Null Hypothesis
Prod Mon = Tue = Wed = Thu = Fri
Ha, Alternate Hypothesis  the productivity on at
least one weekday is not same.
Mon != Tue = Wed = Thu = Fri (or any day avg.
productivity is not equal to at least one other day’s
productivity)
Weekday
transactions
per day Weekday
transactions
per day
Mon 30 Thu 18
Tue 16 Fri 95
Wed 22 Mon 33
Thu 15 Tue 76
Fri 32 Wed 21
Mon 18 Thu 80
Tue 27 Fri 44
Wed 18 Mon 65
Thu 35 Tue 71
Fri 12 Wed 20
Mon 38 Thu 66
Tue 22 Fri 44
Wed 12 Mon 30
Thu 33 Tue 84
Fri 19 Wed 8
Mon 55 Thu 59
Tue 34 Fri 82
Wed 12 Mon 64
Thu 98 Tue 94
Fri 37 Wed 33
Mon 84 Thu 63
Tue 12 Fri 42
Wed 16

Open Minitab
• Go to Stat
• Go to ANOVA
• Go to One-way
• Enter Factors
• Enter Responses

P Value < 0.05
So we Reject null Hypothesis
We can say Day of the week is significant factor
on productivity.
WedTueThuMonFri
100
80
60
40
20
0
Weekday
transactionsperday
Boxplot of transactions per day

FULL FACTORIAL METHOD
o Full Factorial method:
Full factorial experiments look completely at all factors included in the
experimentation.
In full factorials, all of the possible combinations that are associated with
the factors and their levels are studied.
The effects that the main factors and all the interactions between factors
are measured.
If we use more than two levels for each factor, we can also study whether
the effect on the response is linear or if there is curvature in the
experimental region for each factor and for the interactions.
Full factorial experiments can require many experimental runs if many
factors at many levels are investigated.

2 FACTORIAL METHOD
o 2 Factorial method:
The simplest of the two level factorial experiments is the design where
two factors (say factor A and factor B) are investigated at two levels.
A single replicate of this design will require four runs
Consider 2 factors A & B, so there will be 4 combinations (2^2)
Say, 2 levels each Hi (+1) and low(-1)
So the possible combinations are illustrated in the below table:
Run # A B Response
1 + + 33
2 + - 52
3 - + 16
4 - - 26

2 FACTORIAL METHOD
Main effect of A
= Mean response at+ level – Mean
response at – level
= (30+50)/2 – (10+20)/2 = 40 – 15
= 25
Main effect of B
response at – level
= (30+10)/2 – (50+20)/2 = 20 – 35
= -15
Run # A B Response
1 + + 30
2 + - 50
3 - + 10
4 - - 20

2 FACTORIAL METHOD
Interaction effect of A*B
response at -level
= (30+20)/2 – (50+10)/2 = 25 – 30
= -5
Interaction is obtained by
multiplying the factors involved
Run # A B A*B Response
1 + + + 30
2 + - - 50
3 - + - 10
4 - - + 20

TWO FACTORIAL METHOD, EXAMPLE
Go to Standard tool Bar
 Move cursor to DOE
 Select Plan and
Create
 From below popup
select Create
Modeling Design (as
we are dealing with 2
factors for now)

 Enter the Name of
the Response Variable
(or leave as is as
Response)
 Select the Goal
 Select Factors as 2
 Enter Factor Names
and Settings
 Enter the No. of
replicates (here it is 4)
 No. of runs auto
populate based on
factors and replicates

 Modeling Design
Graph would appear
with below details
 Experimental Goal
 Design Information
 Factors and Settings
 Effect Estimation
 Detection Ability

 Enter your responses
in Response column
(C7)

 Select Analyze and
Interpret
select Fit Liner Model

select Minimize the
response and click OK

 From below
summary Report
you can identify
 signifying factors
from Pareto chart
 Here it is B i.e.,
Processors per day
has more impact
than no. of
transactions per
day on Quality
scores
 % of Variation
 design info
 Optimal factor
setting

 From below chart we
can understand the
Main effect and
Interaction effect
 It shows, transactions
per day has less
significant compared to
Processors per day
 The AB interaction plot
also nearly significant

FRACTIONAL FACTORIAL METHOD
o Fractional Factorial method:
Fractional factorials look at more factors with fewer runs.
Using a fractional factorial involves making a major assumption - that
higher order interactions (those between three or more factors) are not
significant.
Fractional factorial designs are derived from full factorial matrices by
substituting higher order interactions with new factors.
To increase the efficiency of experimentation, fractional factorials give up
some power in analyzing the effects on the response. Fractional factorials
will still look at the main factor effects, but they lead to compromises
when looking into interaction effects.
This compromise is called confounding.

SCREENING EXPERIMENTS
o Screening Experiments:
Screening experiments are the ultimate fractional factorial experiments.
These experiments assume that all interactions, even two-way
interactions, are not significant.
They literally screen the factors, or variables, in the process and
determine which are the critical variables that affect the process output.
There are two major families of screening experiments:
• Drs. Plackett and Burman developed the original family of screening
experiments matrices in the 1940s.
• Dr. Taguchi adapted the Plackett–Burman screening designs. He
modified the Plackett–Burman design approach so that the
experimenter could assume that interactions are not significant, yet
could test for some two-way interactions at the same time.

SCREENING EXPERIMENTS, EXAMPLE
 Select Plan and
Create
select (Screen with 6 -
15 factors) Create
Screening Design (as
we are dealing with 6
factors for now)

 Enter the Name of
the Response Variable
(or leave as is as
Response)
 Select Factors as 6
 Enter Factor Names
and Settings
 Select the No. of runs
 Click OK

 Modeling Design
Graph would appear
with below details
 Experimental Goal
 Design Information
 Factors and Settings
 Effect Estimation
 Detection Ability

 Enter your responses
in Response column
(C11), Quality scores

 Select Analyze and
Interpret
select Fit Screening
Model

select Yes

 From below
summary Report
you can identify
 signifying factors
from Pareto chart
 Here it is Shift i.e.,
shift has more
impact on Quality
scores
 % of Variation
 design info

 From below chart we can understand the Main effect
 It shows, Shift of day has higher impact on Quality score
 The Factors shown in gray background are statistically insignificant and can
be ignored from analysis

RESPONSE SURFACE ANALYSIS
o Response Surface Analysis (RSM):
RSM explores the relationships between several explanatory variables and
one or more response variables.
The method was introduced by G. E. P. Box and K. B. Wilson in 1951.
The main idea of RSM is to use a sequence of designed experiments to
obtain an optimal response.
Response surface analysis is an off-line optimization technique.
Usually, 2 factors are studied; but 3 or more can be studied.
With response surface analysis, we run a series of full factorial
experiments and map the response to generate mathematical equations
that describe how factors affect the response.

CALCULATION OF ALIAS
o Calculation of Aliases:
Aliases for a fractional factorial design can be obtained using the defining
relation for the design. The defining relation for the present design is:
I = ABC
Multiplying both sides of the previous equation by the main effect, gives
the alias effect of :
I*A = A*ABC
A = A2BC
A = BC
Note that in calculating the alias effects, any effect multiplied by remains
the same (), while an effect multiplied by itself results in (). Other aliases
can also be obtained:
B = AC
and:
C = AB

SECECTION GUIDE
Number
of Factors
Comparative
Objective
Screening
Objective
1
1 way ANOVA or Simple
Regression
_
2 - 4
2 Way ANOVA, General
Liner Model
Full or Fractional
Factorial Design
5 or more
Randomized Block
Design
Fractional Factorial
Design or Plackett-
Burman

Design of experiments

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Design of experiments