EQT373STATISTIC FOR ENGINEERS

Design of Experiment (DOE)

Noorulnajwa Diyana YaacobSchool of Bioprocess Engineering

Universiti Malaysia Perlis30 April 2012

DKD 1

DOE Objectives

At the end of this class, the student will be able to :

Understand the basic concepts and advantages of designed experiments

Understand key terminology used in experimental design

Use different techniques to deal with noise in an experiment

Make good design decisions

What Experiments Can Do Characterize a Process/Product

determines which X’s most affect the Y’s includes controllable and uncontrollable X’s identifies critical X’s and noise variables identifies those variables that need to be carefully

controlled provides direction for controlling X’s rather than control

charting the Y’s Optimize a Process/Product

determines where the critical X’s should be set determines “real” specification limits provides direction for “robust” designs

Definition of Terms Factor - A controllable experimental variable thought to

influence response (example air flow rate, or in the case of the Frisbee thrower: angle, tire speed, tire pressure)

Response - The outcome or result; what you are measuring (cycle time to produce one bottle, distance Frisbee goes)

Levels - Specific value of the factor (fast flow vs. slow flow, 15 degrees vs. 30 degrees)

Interaction - Factors may not be independent, therefore combinations of factors may be important. Note that these interactions can easily be missed in a straight “hold all other variables constant” scientific approach. If you have interaction effects you can NOT find the global optimum using the “OFAT” (one factor at a time) approach!

Replicate – performance of the basic experiment

How Can DOE Help?

run a relatively small number of tests to isolate the most important factors (screening test).

determine if any of the factors interact (combined effects are as important as individual effects) and the level of interaction.

predict response for any combination of factors using only empirical results

optimize using only empirical results determine the design space for simulation models

What makes up an experiment?

Response Variable(s) Factors Randomization Repetition and Replication

Response Variable

The variable that is measured and the object of the characterization or optimization (the Y)

Defining the response variable can be difficult Often selected due to ease of measurement Some questions to ask :

How will the results be quantified/analyzed? How good is the measurement system? What are the baseline mean and standard deviation? How big of a change do we care about? Are there several response variables of interest?

Factor

A variable which is controlled or varied in a systematic way during the experiment (the X)

Tested at 2 or more levels to observe its effect on the response variable(s) (Ys)

Some questions to ask : what are reasonable ranges to ensure a change in Y? knowledge of relationship, i.e. linear or quadratic, etc?

Examples material, supplier, EGR rate, injection timing can you think of others?

Randomization Randomization can be done in several ways :

run the treatment combinations in random order assign experimental units to treatment combinations

randomly an experimental unit is the entity to which a specific

treatment combination is applied Advantage of randomization is to “average out” the effects of

extraneous factors (called noise) that may be present but were not controlled or measured during the experiment spread the effect of the noise across all runs these extraneous factors (noise) cause unexplained

variation in the response variable(s)

Repetition and Replication

Repetition : Running several samples during one experimental setup (short-term variability)

Replication : Repeating the entire experiment (long-term variability)

You can use both in the same experiment

Repetition and Replication provide an estimate of the experimental error this estimate will be used to determine whether observed

differences are statistically significant

Steps in DOE

1. Statement of the Problem2. Selection of Response Variable 3. Choice of Factors and Levels

Factors are the potential design parameters, such as angle or tire pressure

Levels are the range of values for the factors, 15 degrees or 30 degrees

4. Choice of Design screening tests response prediction factor interaction

5. Perform Experiment6. Data Analysis

Process

Noise Factors “z”

Controllable Factors “x”

Responses “y”

DOE (Design of Experiments) is:

“A systematic series of tests, in which purposeful changes are made to input factors,

so that you may identify causes for significant changes in the output responses.”

Design of Experiments

Design of Experiments

Process

Noise Factors “z”

Controllable Factors “x”

Responses “y”

Let’s brainstorm.

What process might you experiment on for best payback?

How will you measure the response?

What factors can you control?

Write it down.

Motivation for Factorial Design

Want to estimate factor effects well; this implies estimating effects from averages.

Want to obtain the most information in the fewest number of runs.

Want to estimate each factor effect independent of the existence of other factor effects.

Want to keep it simple.

Two-Level Full Factorial Design (1ST Approach)

•Run all high/low combinations of 2 (or more) factors

•Use statistics to identify the critical factors

22 Full Factorial

Two-level full factorials include experiments at all combinations of the factor levels: 2x2 = 4, 2x2x2 = 8, etc.

Design Construction23 Full Factorial

Std A B C AB AC BC ABC

1 – – – + + + – y1

2 + – – – – + + y2

3 – + – – + – + y3

4 + + – + – – – y4

5 – – + + – – + y5

6 + – + – + – – y6

7 – + + – – + – y7

8 + + + + + + + y8

1 2

5 6

3 4

87

B

A

C

This is a two-level, three factor design, called a 23 factorial design, which produces 8 runs.This is the complete design matrix, including interactions. It includes all seven of the effects we can evaluate (in addition to the overall mean) with the eight runs: -- three main effects (MEs), three 2-factor interactions (2fi) and one 3-factor interaction (3fi).

Two Level Factorial DesignAs Easy As Popping Corn!

Kitchen scientists* conducted a 23 factorial experiment on microwave popcorn. The factors are:

A. Brand of popcorn

B. Time in microwave

C. Power setting

A panel of neighborhood kids rated taste from one to ten and weighed the un-popped kernels (UPKs).

We will analyze the first response “Taste” by hand and using a computer both the “Taste” and the second response “UPKs” (weight of un-popped kernels, technically called “UPKs” by popcorn manufacturers). Then we will look for the best operating conditions.

Two Level Factorial DesignAs Easy As Popping Corn!

A B C R1 R2

Run Brand Time Power Taste UPKs Std

Ord expense minutes percent rating* oz. Ord

1 Costly 4 75 75 3.5 2

2 Cheap 6 75 71 1.6 3

3 Cheap 4 100 81 0.7 5

4 Costly 6 75 80 1.2 4

5 Costly 4 100 77 0.7 6

6 Costly 6 100 32 0.3 8

7 Cheap 6 100 42 0.5 7

8 Cheap 4 75 74 3.1 1

Two Level Factorial DesignAs Easy As Popping Corn!

A B C R1 R2

Run Brand Time Power Taste UPKs StdOrd expense minutes percent rating oz. Ord

1 + – – 75 3.5 2

2 – + – 71 1.6 3

3 – – + 81 0.7 5

4 + + – 80 1.2 4

5 + – + 77 0.7 6

6 + + + 32 0.3 8

7 – + + 42 0.5 7

8 – – – 74 3.1 1

Factors shown in coded values

R1 - Popcorn TasteAverage A-Effect

There are four comparisons of factor A (Brand), where levels of factors B and C (time and power) are the same:

42 32

7781

74 75

71 80

Brand

Time

75 – 74 = + 180 – 71 = + 977 – 81 = – 432 – 42 = – 10

A

1 9 4 10y 1

4

We will use the matrix to calculate effects. But before we do, let’s get a picture of what we mean by an effect. For example if we want to determine effect of Brand (factor A), we simply compare the results of A at its high level versus A at its low level, at the four combinations where the other two factors constant.

R1 - Popcorn TasteAverage A-Effect

42 32

7781

74 75

71 80

Brand

Time

y yEffect y

n n

A

75 80 77 32 74 71 81 42y 1

4 4

R1 - Popcorn TasteAnalysis Matrix in Standard Order

Std.Order I A B C AB AC BC ABC

Taste rating

1 + – – – + + + – 74

2 + + – – – – + + 75

3 + – + – – + – + 71

4 + + + – + – – – 80

5 + – – + + – – + 81

6 + + – + – + – – 77

7 + – + + – – + – 42

8 + + + + + + + + 32

This is the complete design matrix in standard order. It includes columns for all eight of the effects we can evaluate with the eight runs: the overall mean (column labeled I for identity) plus seven effects -- three main effects (MEs), three 2-factor interactions (2FIs) and one 3-factor interaction (3FI).How are the signs for the interaction columns computed?

Popcorn TasteCompute the effect of C and BC

Std. Taste

Order A B C AB AC BC ABC rating

1 – – – + + + – 74

2 + – – – – + + 75

3 – + – – + – + 71

4 + + – + – – – 80

5 – – + + – – + 81

6 + – + – + – – 77

7 – + + – – + – 42

8 + + + + + + + 32

Dy -1 -20.5 0.5 -6 -3.5

y yy

n n

C

BC

81 77 42 32 74 75 71 80y

4 4

y4 4

THANK YOU

ANY QUESTION???