The Essentials of 2-Level Design of The Essentials of 2-Level Design of ExperimentsExperiments

Part II: The Essentials of Fractional Factorial Part II: The Essentials of Fractional Factorial DesignsDesigns

Developed by Don Edwards, John Grego and James Developed by Don Edwards, John Grego and James

LynchLynch Center for Reliability and Quality SciencesCenter for Reliability and Quality Sciences

Department of StatisticsDepartment of StatisticsUniversity of South CarolinaUniversity of South Carolina

803-777-7800803-777-7800

Part II: The Essentials of Fractional Factorial Part II: The Essentials of Fractional Factorial DesignsDesigns

1. Introduction to Fractional Factorials1. Introduction to Fractional Factorials 2. Four Factors in Eight Runs2. Four Factors in Eight Runs 3. Screening Designs in Eight Runs3. Screening Designs in Eight Runs 4. K Factors in Sixteen Runs4. K Factors in Sixteen Runs

II.1 Introduction to Fractional FactorialsII.1 Introduction to Fractional Factorials

A Quick Review of Full FactorialsA Quick Review of Full Factorials How Many Runs?How Many Runs? The Fractional Factorial IdeaThe Fractional Factorial Idea

II.1 Introduction: A Quick Review of Full FactorialsII.1 Introduction: A Quick Review of Full Factorials

Use Cube Plots to Understand Use Cube Plots to Understand Factor EffectsFactor Effects

Use Sign Tables to Estimate Use Sign Tables to Estimate EffectsEffects

Use Probability Plots to Identify Use Probability Plots to Identify Significant EffectsSignificant Effects

Interaction Tables and Graphs Interaction Tables and Graphs are Used to Analyze Significant are Used to Analyze Significant InteractionsInteractions

II.1 Introduction: A Quick ReviewII.1 Introduction: A Quick Review Rope Pull Study - Completed Cube Plot and Rope Pull Study - Completed Cube Plot and

Signs TableSigns Table

Factors:Factors:– A: Vacuum Level (Lo, Hi)A: Vacuum Level (Lo, Hi)– B: Needle Type (EX, GB)B: Needle Type (EX, GB)– C: Upper Boot Speed C: Upper Boot Speed

(1000,1200)(1000,1200)

Response:Response:– Rope Pull (in inches)Rope Pull (in inches)

Factors:Factors:– A: Vacuum Level (Lo, Hi)A: Vacuum Level (Lo, Hi)– B: Needle Type (EX, GB)B: Needle Type (EX, GB)– C: Upper Boot Speed C: Upper Boot Speed

(1000,1200)(1000,1200)

Response:Response:– Rope Pull (in inches)Rope Pull (in inches)

C

B

A

+

+

+

_

_

_

100.35 110.85

100.75 109.20

93.75 96.50

92.40 95.45

Main Effects Interaction EffectsActual

RunVacuum

ANeedle

TypeB

UpperBoot Speed

CAB AC BC ABC

100.35 -1 -1 -1 1 1 1 -1110.85 1 -1 -1 -1 -1 1 1100.75 -1 1 -1 -1 1 -1 1109.20 1 1 -1 1 -1 -1 -193.75 -1 -1 1 1 -1 -1 196.50 1 -1 1 -1 1 -1 -192.40 -1 1 1 -1 -1 1 -195.45 1 1 1 1 1 1 1

Sum 799.25 24.75 -3.65 -43.05 -1.750 -13.15 -1.150 2.35Divisor 8 4 4 4 4 4 4 4Effect 99.9 6.1875 -0.9125 -10.7625 -0.4375 -3.2875 -0.2875 0.5875

II.1 Introduction: A Quick ReviewII.1 Introduction: A Quick Review Rope Pull Study -Completed Seven Effects Normal PlotRope Pull Study -Completed Seven Effects Normal Plot

Effects

7 Effects Plot

1

2

3

4

5

6

7

0

Ordered Effects -10.7625, -3.2875, -0.9125, -0.4375, -0.2875, 0.5875, 6.1875

63-3-6-9

A

C AC

Factors:Factors:– A: Vacuum Level (Lo, Hi)A: Vacuum Level (Lo, Hi)– B: Needle Type (EX, GB)B: Needle Type (EX, GB)– C: Upper Boot Speed C: Upper Boot Speed

(1000,1200)(1000,1200)

Factors:Factors:– A: Vacuum Level (Lo, Hi)A: Vacuum Level (Lo, Hi)– B: Needle Type (EX, GB)B: Needle Type (EX, GB)– C: Upper Boot Speed C: Upper Boot Speed

(1000,1200)(1000,1200)

II.1 Introduction: A Quick Review II.1 Introduction: A Quick Review Rope Pull Study - Completed AC Interaction Table and Rope Pull Study - Completed AC Interaction Table and

PlotPlot

-1 1-1 1

1 1-1-1

110

105

100

95

C

A

- - -A+

A-

1

1 2

2

A C =

A C =A C =

A C =

1 1

2 1

1 2

2 2

A: Vacuum Level

C: Upper Boot Speed

100.35 100.75 201.1

110.85 109.20 220.05

93.75 92.40 186.15

96.50 95.45 191.95

100.55

110.025

93.075

95.975

FactorsA: Vacuum Level (Lo, Hi) C: Upper Boot Speed (1000,1200)

II.1 Introduction: How Many Runs?II.1 Introduction: How Many Runs?

We have seen, for factors at two levels, We have seen, for factors at two levels, – Two Factors Two Factors 4 runs 4 runs– Three Factors Three Factors 8 runs 8 runs– Four Factors Four Factors 16 runs 16 runs

What if we have seven factors?What if we have seven factors? What if we have fifteen?What if we have fifteen? There are ways to investigate up to seven There are ways to investigate up to seven

factors using only 8 runs, or up to 15 factors using only 8 runs, or up to 15 factors using 16 runs, factors using 16 runs, if if it is safe to assume it is safe to assume that high-order interactions are negligible.that high-order interactions are negligible.

II.1 Introduction: How Many Runs?II.1 Introduction: How Many Runs?

For Example,For Example,– We May Be Interested in Determining the We May Be Interested in Determining the

Effects on Quality Characteristics of HosieryEffects on Quality Characteristics of Hosiery A: Band SpeedA: Band Speed B: Panty SpeedB: Panty Speed C: Upper Boot SpeedC: Upper Boot Speed D: Lower Boot SpeedD: Lower Boot Speed E: Needle TypeE: Needle Type F: Vacuum LevelF: Vacuum Level

– A Full 2A Full 266 in These Factors, Each at Two in These Factors, Each at Two Levels, Would Require 64 RunsLevels, Would Require 64 Runs

II.1 Introduction: The Fractional Factorial II.1 Introduction: The Fractional Factorial IdeaIdea

In the 2In the 233 design, look at the computation of C using the design, look at the computation of C using the y's in standard ordery's in standard order

C = ( -yC = ( -y11 -y -y22 -y -y33 -y -y44 +y +y55+y+y66 +y +y77 +y +y88)/4)/4

StandardOrder A B C AB AC BC ABC

1 -1 -1 -1 1 1 1 -12 1 -1 -1 -1 -1 1 13 -1 1 -1 -1 1 -1 14 1 1 -1 1 -1 -1 -15 -1 -1 1 1 -1 -1 16 1 -1 1 -1 1 -1 -17 -1 1 1 -1 -1 1 -18 1 1 1 1 1 1 1

II.1 Introduction: The Fractional Factorial II.1 Introduction: The Fractional Factorial IdeaIdea

Now, look at the same thing for AB:Now, look at the same thing for AB:

AB = ( +yAB = ( +y11 -y -y22 -y -y33 +y +y44 +y +y55-y-y66 -y -y77 +y +y88)/4)/4

StandardOrder A B C AB AC BC ABC

1 -1 -1 -1 1 1 1 -12 1 -1 -1 -1 -1 1 13 -1 1 -1 -1 1 -1 14 1 1 -1 1 -1 -1 -15 -1 -1 1 1 -1 -1 16 1 -1 1 -1 1 -1 -17 -1 1 1 -1 -1 1 -18 1 1 1 1 1 1 1

II.1 Introduction: The Fractional Factorial II.1 Introduction: The Fractional Factorial IdeaIdea

So,So,

C = ( -yC = ( -y11 -y -y22 -y -y33 -y -y44 +y +y55+y+y66 +y +y77 +y +y88)/4)/4AB = ( +yAB = ( +y11 -y -y22 -y -y33 +y +y44 +y +y55-y-y66 -y -y77 +y +y88)/4 )/4

Add These Together to get C+AB:Add These Together to get C+AB:

C+AB = ( -2yC+AB = ( -2y22 -2y -2y33+2y+2y55+2y+2y88)/4)/4

= ( -y = ( -y22 -y -y33+y+y55+y+y88)/2)/2

So, if we want to estimate C+AB, we only need 4 runs to do it! So, if we want to estimate C+AB, we only need 4 runs to do it! Or, if we are fairly sure that AB is negligible, we only need 4 Or, if we are fairly sure that AB is negligible, we only need 4 runs to estimate C (and the same 4 runs can estimate A and B runs to estimate C (and the same 4 runs can estimate A and B if BC and AC are negligible).if BC and AC are negligible).

II.1 Introduction: The Fractional Factorial II.1 Introduction: The Fractional Factorial Idea Idea

Figure 1 - 2Figure 1 - 23 3 Design Signs TableDesign Signs Table C+AB = ( -yC+AB = ( -y22 -y -y33+y+y55+y+y88)/2)/2 Use Runs 2, 3, 5, and 8 (i.e., When ABC = I)Use Runs 2, 3, 5, and 8 (i.e., When ABC = I)

StandardOrder A B C AB AC BC ABC

1 -1 -1 -1 1 1 1 -12 1 -1 -1 -1 -1 1 13 -1 1 -1 -1 1 -1 14 1 1 -1 1 -1 -1 -15 -1 -1 1 1 -1 -1 16 1 -1 1 -1 1 -1 -17 -1 1 1 -1 -1 1 -18 1 1 1 1 1 1 1

II.1 Introduction: The Fractional Factorial II.1 Introduction: The Fractional Factorial IdeaIdea

Objectives Of Fractional FactorialsObjectives Of Fractional Factorials– To Reduce the Number of Required RunsTo Reduce the Number of Required Runs– To Screen Out Insignificant Factors In The To Screen Out Insignificant Factors In The

Initial Stages of ExperimentationInitial Stages of Experimentation A Screening DesignA Screening Design

This Can Be Done Without Substantial Loss This Can Be Done Without Substantial Loss In Information If Higher-Order Interactions In Information If Higher-Order Interactions Can Be Assumed To Be NegligibleCan Be Assumed To Be Negligible

We Will See How This Is Done In This We Will See How This Is Done In This ModuleModule