fractional design of testing of textile

7/27/2019 Fractional Design of testing of textile

1/17

Fractional

Factorials Design


2/17

Complete factorial design

No. of runs :

Runs for 6 factors experiment : = 64

No. of statistics to be estimated: 64

These are : 1 average

6 main effects 15 2- factor interaction effects

20 3-factor interaction effects




n

2

62No. ofp-factor

interaction

n!

=

(n

p) !p!

No. of runs required to estimate main effects , interaction aeffect increases rapidly as the number of factors increases.


3/17

Effect n=2 n=3 n=4 n=5 n=6 n=7 n=8

Average 1 1 1 1 1 1 1

Main 2 3 4 5 6 7 8

2- factor 1 3 6 10 15 21 28

3- factor 1 4 10 20 35 56

4- factor 1 5 15 35 70

5- factor 1 6 21 56

6- factor 1 7 287- factor 1 8

8- factor 1


4/17

Higher order interactions are usuallyinsignificant in comparison to main & 2-factoreffects

Usually Main effects> 2-factor interaction> 3-factor interaction

When there is a large no. of factors in afactorial design, very few are really important

n

2

Fractional Factorial experiment


5/17

Fractional Factorial experiment

Fractional factorial design disregards the

possible impo rtance of higher order

in teract ionsand use only a fraction of theexperimental runs.

Any design that involves running only a

subset of the possible factor combination is

called Fractional Factorial experiment


6/17

Run S B V Chest g

1 1 1 1 56.2

2 +1 1 1 55.6

3 1 +1 1 61.6

4 +1 +1

1 52.25 1 1 +1 54.0

6 +1 1 +1 50.0

7 1 +1 +1 60.3

8 +1 +1 +1 51.1

Previous plan with 8 runs

Run S B V SXB BXV VXS Chest g

1 + + 54

2 + 55.6

3 + + 61.6

4 + + + + + + 51.1

New Plan with 4 runs

Air bag experiment

Inflation speed

(S)

Vent

size(V)

Bag size

(B)

51.1

54.0

55.6

61.6

Large

(+)

Small

()

Slow()

Large

(+)


7/17

C1 C2 C3 Chest g

1 1 +1 54

+1 1 1 55.6

1 +1 1 61.6

+1 +1 +1 51.1

Divisor 2 2 2

Contrast

value

-4.4.5 1.55 -6.05

Contrast matrix and contrast value

Contrast for C1= 45.42

546.61

2

1.516.55


8/17

The contrast value for C1 is the difference between

average for runs with S (+) level and a similar average

for S (-) level. So it is like main effect contrast.

Coefficient of Column1 is also the product of BV inthe experimental plan. Hence it could be measure

BV interaction effect.

Therefore the contrast measures the sum of S + B

V

The S and BV effects are CONFOUNDEDOR

ALIASED.


9/17

Caution

The true S & BV effects may have

opposite sign and so may wholly or partiallycancel each other in the subset contrast and

we may end up with a totally false picture of

the situation based upon data from

fractional experiment.


10/17

S + BV B + SV V+ SB Chest g

1 1 +1 54

+1 1 1 55.6

1 +1 1 61.6

+1 +1 +1 51.1

Divisor 2 2 2

Contrast value -4.4.5 1.55 -6.05

Three contrasts

for the data with

labels

S B SB V SV BV SBV Chest g

1 1 +1 1 +1 +1 1 56.2

+1 1 1 1 1 +1 +1 55.6

1 +1 1 1 +1 1 +1 61.6

+1 +1 +1 1 1 1 1 52.2

1 1 +1 +1 1 1 +1 54.0

+1 1 1 +1 +1 1 1 50.0

1 +1 1 +1 1 +1 1 60.3

+1 +1 +1 +1 +1 +1 +1 51.1

Divisor 4 4 4 4 4 4 4

Contrast

value

- 5.80 2.35 -3.50 - 2.55 -0.80 1.35 0.90


11/17

Generating Fractional Design

Half fractional Design

Write down the complete design matrix for first fourfactors( ZMPD)

Worked the signs for ZMPD 4 - factor interaction

Use these signs to define the level of 5th Factor i.e.A= ZMPD called GENERATOR

Defining Contarst I = ZMPD A

Alias of any effect can then be obtained by multiplying

the effect by I using normal algebraic rules, with anadditional rule that where a term appears even numberof times their product is unity


12/17

Run Factor Enzyme activity

Z M P D A

1 109

2* + 113

3* + 103

4 + +

113

5* + 103

6 + + 104

7 + + 106

8* + + + 123

9* + 119

10 + + 146

11 + + 111

12* + + + 143

13 + + 116

14* + + + 145

15* + + + 110

16 + + + + 148

17* + 106

18 + + 120

19 + + 113

20* + + + 115

21 + + 109

22* + + + 117

23* + + + 105

24 + + + + 115

25 + 96

26* + + + 128

27* + + + 95

28 + + + + 127


13/17

Effect Estimate based on

32 runs

Estimate based on

16 runs

Effect Estimate based on 32

runs

Average 116.00 116.00 ZMP 2.25

ZMD 0.63

Z 20.5022.00

ZMA -2.38M -0.63 - 0.50 ZPD 0.63

P -0.13 1.50 ZPA -0.13

D 10.25 10.75 ZDA 0.50

A -7.0 - 7.75 MPD -1.00

MPA -3.00MDA 1.63

ZM 2.13 3.00 PDA 0.88

ZP 1.38 3.00

ZD 12.25 9.25 ZMPD -0.75

ZA 0.75 - 0.25 ZMPA 0.50

MP 1.50 2.00 ZMDA 1.63

MD -2.13 - 2.25 ZPDA 0.13

MA -0.88 - 0.25 MPDA 1.50

PD 1.13 - 1.25

PA 0.13 0.75 ZMPDA 0.00

DA -10.25 - 8.00


14/17

Effect Estimate

Average + ZMPDA 116.00

Z +MPDA 22.00

M +ZPDA -0.50

P +ZMDA 1.50

D +ZMPA 10.75

A+ZMPD - 7.75

3.00

ZM + PDA 3.00

ZP + MDA 3.00

ZD + MPA 9.25

ZA + MPD -0.25

MP + ZDA 2.00

MD + ZPA -2.25

MA + ZPD -0.25

PD + ZMA -1.25

PA + ZMD 0.75

DA + ZMP -8.00


15/17

Write down the complete design matrix for the first four factors

Worked out the sign for four factor interaction

Use this sign to define the levels of fifth factor

Effect

Z MPDA A ZMPD ZM PDA ZA MPD

+ + + +

+ +

+ +

+ + + + + + + +

+ + + +

+ + + + + +

+ +

+ + + +

+ + + +

+ + + + + +

+ +

+ + + +

+ + + +

+ +

+ +

+ + + + + + + +


16/17

Generating Fractional Design

Write down the complete design matrix for first four factors(ZMPD)

Worked the signs for ZMPD 4 - factor interaction

Use these signs to define the level of 5th Factor i.e.

A= ZMPD called GENERATORDefining Contrast I = ZMPD A

Alias of any effect can then be obtained by multiplying the

effect by I using normal algebraic rules, with an additional

rule that where a term appears even number of times their

product is unity

Example:

Alias of Z = Z I = Z ZMPDA = Z2 MPDA = MPDA

Alias of M = M I = M ZMPDA = M2 ZPDA = ZPDA

Alias of ZM = ZM I = ZM ZMPDA = Z2M2 PDA= PDA


17/17

Design Resolution

Plans with higher resolution number has a simplerconfounding pattern

Resolution III: In this plan some or all of the maineffects are confounded with one or more two wayinteractions.

Resolution IV: In this plan main effects are onlyconfounded with three way or higher orderinteractions. Some (or all) two way interactions are

confounded with other two way interactions Resolution V: The main effects are clear of two way

interaction

fractional design of testing of textile

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