The American University in CairoThe American University in Cairo

Interdisciplinary Engineering ProgramInterdisciplinary Engineering Program

ENGR 592: Probability & StatisticsENGR 592: Probability & Statistics

22kk Factorial Factorial && Central Composite Central Composite

DesignsDesignsPresented to: Presented by:Dr. Lotfi K. Gaafar Ghada Moustafa Gad 592 Class

Factorial DesignsFactorial Designs

Allow the effect of each and every

factor to be tested and estimated

independently with the

interactions also assessed.

Factorial Design

2k Full Factorial Design

Full Factorial Design

Mirror Image Fold over Design

2k Fractional Factorial Design

Factorial DesignsFactorial Designs

A factorial design in which every setting of every factor appears

with every setting of every

other factor

Factorial Design

2k Full Factorial Design

Full Factorial Design

Mirror Image Fold over Design

2k Fractional Factorial Design

Factorial DesignsFactorial Designs

Designs having all input factors set

at two levels each. These

levels are called high/+1 and

low/-1

Factorial Design

2k Full Factorial Design

Full Factorial Design

Mirror Image Fold over Design

2k Fractional Factorial Design

Factorial DesignsFactorial Designs

Only an adequately chosen fraction of

the treatment combinations

required for the complete factorial

experiment are selected to be run

Factorial Design

2k Full Factorial Design

Full Factorial Design

Mirror Image Fold over Design

2k Fractional Factorial Design

Factorial DesignsFactorial Designs

Factorial with the Factorial with the number of runs in number of runs in

the follow up the follow up experiment equal to experiment equal to

the original. the original. Fractional factorial Fractional factorial

designs are designs are augmented by augmented by

reversing the signs reversing the signs of all the columns of of all the columns of the original design the original design

matrix matrix

Factorial Design

2k Full Factorial Design

Full Factorial Design

Mirror Image Fold over Design

2k Fractional Factorial Design

22kk Full Factorial Design Full Factorial Design

# of runs required = 2 # of runs required = 2 # of factors# of factors

#of Factors #of Runs

24

38

416

532

664

7128

22kk Full Factorial Design Full Factorial Design

Standard Order Matrix 2Standard Order Matrix 222

TrialX1X2

1-1-1

2+1-1

3-1+1

4+1+1

22kk Full Factorial Design Full Factorial Design

Analysis Matrix 2Analysis Matrix 222

Dot product for any pair of columns is 0Dot product for any pair of columns is 0

TrialIX1X2X1*X2

1+1-1-1+1

2+1+1-1-1

3+1-1+1-1

4+1+1+1+1

Balanced PropertyBalanced Property

Fractional Factorial DesignFractional Factorial Design

223 3 = 8 runs= 8 runs223-13-1 = 4 runs = 4 runs

TrialX1X2X1*X2

1-1-1+1

2+1-1-1

3-1+1-1

4+1+1+1

½ ½ spacespace

XX33

Fractional Factorial DesignFractional Factorial Design

223 3 = 8 runs= 8 runs223-13-1 = 4 runs = 4 runs

TrialX1X2X1*X2

1-1-1+1

2+1-1-1

3-1+1-1

4+1+1+1

½ ½ spacespace

XX33

Fractional Factorial DesignFractional Factorial Design

A schedule for conducting runs of an experimental

study such that any effects on the

experimental results due to a known change in raw materials, operators, etc. become concentrated in the levels of the blocking

variable

Blocking Effect Resolution

Fractional Factorial DesignFractional Factorial Design

It is the length of the smallest interaction

among the set of defining relations. It

describes the degree to which the estimated

main effects are confounded with the

estimated interactions.

Blocking Effect Resolution

Factorial Design FeaturesFactorial Design Features

Ideal for screening design objective Ideal for screening design objective

Simple and economical for small Simple and economical for small number of factors.number of factors.

22kk fractional factorial designs if fractional factorial designs if properly chosen to can be balanced properly chosen to can be balanced and orthogonal.and orthogonal.

Fractional Factorial designs has Fractional Factorial designs has low number of runs compared to low number of runs compared to high information obtained. high information obtained.

Most popular designsMost popular designs

Factorial Design FeaturesFactorial Design Features

A two-level experiment can not fit A two-level experiment can not fit quadratic effects quadratic effects

Case Example:Case Example:Fold-over Fractional Factorial Design

Set Objectives

Set Objectives

Select Variables & Levels

Select Variables & Levels

Select DesignSelect Design

Evaluate Results

Evaluate Results

The aim of the study is to find the factors

affecting the time to peddle a bicycle up a

hill.

Screening experiment.

The aim of the study is to find the factors

affecting the time to peddle a bicycle up a

hill.

Screening experiment.

Case Example:Case Example:Fold-over Fractional Factorial Design

Set Objectives

Set Objectives

Select Variables & Levels

Select Variables & Levels

Select DesignSelect Design

Evaluate Results

Evaluate Results

Case Example:Case Example:Fold-over Fractional Factorial Design

Set Objectives

Set Objectives

Select Variables & Levels

Select Variables & Levels

Select DesignSelect Design

Evaluate Results

Evaluate Results

7 factors 27= 128

Limitation 8 runs

7 factors 27= 128

Limitation 8 runs

Case Example:Case Example:Fold-over Fractional Factorial Design

4 5 6 7

Resolution III 2327- 4

Case Example:Case Example:Fold-over Fractional Factorial Design

Set Objectives

Set Objectives

Select Variables & Levels

Select Variables & Levels

Select DesignSelect Design

Evaluate Results

Evaluate Results

2 and 4 are significant.

4 confounded by 12 ?

1 & 14 could be significant?

Fold over design

2 and 4 are significant.

4 confounded by 12 ?

1 & 14 could be significant?

Fold over design

Case Example:Case Example:Fold-over Fractional Factorial Design

4 5 6 7

Resolution III

Resolution IV

Central Composite DesignsCentral Composite Designs

CCD fall under the classical quadratic CCD fall under the classical quadratic designs category where fractional plan designs category where fractional plan is used to fit a second order equationis used to fit a second order equation

They start with a factorial or a They start with a factorial or a fractional factorial design (with center fractional factorial design (with center points) and then points) and then star pointsstar points or or axial axial pointspoints are added to estimate curvature are added to estimate curvature

Central Composite DesignsCentral Composite Designs

Rotatability Rotatability Most important criterion Most important criterion Means that the standard error value of Means that the standard error value of the points located at same distance from the points located at same distance from the center of the region is the same. the center of the region is the same. It is a measure of uncertainty of a It is a measure of uncertainty of a predicted responsepredicted response

CCD DesignsCCD Designs

Circumscribed Central

Composite

Face Centered Central

Composite

Inscribed Central

Composite

CCD General FeaturesCCD General FeaturesMost types are rotatableMost types are rotatable

Minimizes the error of prediction.Minimizes the error of prediction.

Good lack of fit detection.Good lack of fit detection.

Suitable for blocking. Suitable for blocking.

Good graphical analysis through simple Good graphical analysis through simple data patterns. data patterns.

Provides information on variable effects Provides information on variable effects and experimental error with minimum and experimental error with minimum number of runs. number of runs.

Sequential construction of higher order Sequential construction of higher order designs from simpler designs to estimate designs from simpler designs to estimate curvature effects. curvature effects.

Case Example: Case Example: CCD

Set Objectives

Set Objectives

Select Variables & Levels

Select Variables & Levels

Select DesignSelect Design

Evaluate Results

Evaluate Results

The aim is to find the best ratio of the two

admixtures to be used as a super plasticizer for cement to obtain optimal workability.

Response surface methodology

The aim is to find the best ratio of the two

admixtures to be used as a super plasticizer for cement to obtain optimal workability.

Response surface methodology

Case Example:Case Example: CCD

Set Objectives

Set Objectives

Select Variables & Levels

Select Variables & Levels

Select DesignSelect Design

Evaluate Results

Evaluate Results

W/C0.330.35

%BL 0.120.18

%SNF 0.080.12

Case Example: Case Example: CCD

Set Objectives

Set Objectives

Select Variables & Levels

Select Variables & Levels

Select DesignSelect Design

Evaluate Results

Evaluate Results

Since RSM

High quality prediction

Larger process space

Circumscribed Circumscribed

Central Composite Central Composite DesignDesign

Extremes generated

are reasonable =>O.K.

Since RSM

High quality prediction

Larger process space

Circumscribed Circumscribed

Central Composite Central Composite DesignDesign

Extremes generated

are reasonable =>O.K.

Case Example: Case Example: CCC

Thank you…Thank you…

Questions?Questions?