The American University in Cairo Interdisciplinary Engineering Program ENGR 592: Probability & Statistics 2 k Factorial & Central Composite Designs Presented to:Presented by: Dr. Lotfi K. GaafarGhada Moustafa Gad 592 Class

Factorial Designs Allow the effect of each and every factor to be tested and estimated independently with the interactions also assessed. Factorial Design 2 k Full Factorial Design Full Factorial Design Mirror Image Fold over Design 2 k Fractional Factorial Design

Factorial Designs A factorial design in which every setting of every factor appears with every setting of every other factor Factorial Design 2 k Full Factorial Design Full Factorial Design Mirror Image Fold over Design 2 k Fractional Factorial Design

Factorial Designs Designs having all input factors set at two levels each. These levels are called high/+1 and low/-1 Factorial Design 2 k Full Factorial Design Full Factorial Design Mirror Image Fold over Design 2 k Fractional Factorial Design

Factorial Designs Only an adequately chosen fraction of the treatment combinations required for the complete factorial experiment are selected to be run Factorial Design 2 k Full Factorial Design Full Factorial Design Mirror Image Fold over Design 2 k Fractional Factorial Design

Factorial Designs Factorial with the number of runs in the follow up experiment equal to the original. Fractional factorial designs are augmented by reversing the signs of all the columns of the original design matrix Factorial Design 2 k Full Factorial Design Full Factorial Design Mirror Image Fold over Design 2 k Fractional Factorial Design

2 k Full Factorial Design # of runs required = 2 # of factors # of Factors# of Runs

2 k Full Factorial Design Standard Order Matrix 2 2 TrialX1X1 X2X

2 k Full Factorial Design Analysis Matrix 2 2 Dot product for any pair of columns is 0 TrialIX1X1 X2X2 X 1 *X Balanced Property

Fractional Factorial Design 2 3 = 8 runs = 4 runs TrialX1X1 X2X2 X 1 *X ½space X3X3X3X3

Fractional Factorial Design 2 3 = 8 runs = 4 runs TrialX1X1 X2X2 X 1 *X ½space X3X3X3X3

Fractional 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 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 Features Ideal for screening design objective Simple and economical for small number of factors. 2 k fractional factorial designs if properly chosen to can be balanced and orthogonal. Fractional Factorial designs has low number of runs compared to high information obtained. Most popular designs

Factorial Design Features A two-level experiment can not fit quadratic effects

Case Example: Case Example: Fold-over Fractional Factorial Design Set Objectives Select Variables & Levels Select Design Evaluate Results 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 Select Variables & Levels Select Design Evaluate Results

Case Example: Case Example: Fold-over Fractional Factorial Design Set Objectives Select Variables & Levels Select Design Evaluate Results 7 factors  2 7 = 128 Limitation  8 runs

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

Case Example: Case Example: Fold-over Fractional Factorial Design Set Objectives Select Variables & Levels Select Design Evaluate Results 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 Resolution III Resolution IV

Central Composite Designs CCD fall under the classical quadratic designs category where fractional plan is used to fit a second order equation They start with a factorial or a fractional factorial design (with center points) and then star points or axial points are added to estimate curvature

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

CCD Designs Circumscribed Central Composite Face Centered Central Composite Inscribed Central Composite

CCD General Features Most types are rotatable Minimizes the error of prediction. Good lack of fit detection. Suitable for blocking. Good graphical analysis through simple data patterns. Provides information on variable effects and experimental error with minimum number of runs. Sequential construction of higher order designs from simpler designs to estimate curvature effects.

Case Example: Case Example: CCD Set Objectives Select Variables & Levels Select Design 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

Case Example: Case Example: CCD Set Objectives Select Variables & Levels Select Design Evaluate Results W/C % BL % SNF

Case Example: Case Example: CCD Set Objectives Select Variables & Levels Select Design Evaluate Results Since RSM  High quality prediction Larger process space  Circumscribed Central Composite Design  Extremes generated are reasonable =>O.K.

Case Example: Case Example: CCC

Thank you… Questions?