1 / 6

Fractional Factorial Designs in Wool Fabrics Shrinkage Measurement

Understand how to use fractional factorial designs for effective shrinkage measurement in wool fabrics. Learn about confounding, defining contrasts, generalized interactions, and Aliased Effects in experimental setups.

rudyr
Télécharger la présentation

Fractional Factorial Designs in Wool Fabrics Shrinkage Measurement

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Fractional Factorial Designs 27 – Factorial Design in 8 Experimental Runs to Measure Shrinkage in Wool Fabrics J.M. Cardamone, J. Yao, and A. Nunez (2004). “Controlling Shrinkage in Wool Fabrics: Effective Hydrogen Peroxide Systems,” Textile Research Journal, Vol. 74 pp. 887-898

  2. Fractional Factorial Designs • For large numbers of treatments (k), the total number of runs for a full factorial can get very large (2k) • Many degrees of freedom are spent on high-order interactions (which are often pooled into error with marginal gain in added degrees of freedom) • Fractional factorial designs are helpful when: • High-order interactions are small/ignorable • We wish to “screen” many factors to find a small set of important factors, to be studied more thoroughly later • Resources are limited • Mechanism: Confound full factorial in blocks of “target size”, then run only one block

  3. Fractioning the 2k - Factorial • 2k can be run in 2q block of size 2k-q for q=,1…,k-1 • 2k-q factorial is design with k factors in 2k-q runs • 1 Block of a confounded 2k factorial • Principal Block is called the principal fraction, other blocks are called alternate fractions • Procedure: • Augment table of 2-series with column of “+”, labeled “I” • Defining contrasts are effects to be confounded together • Generators are used to create the blocks by +/- structure • Generalized Interactions of Generators also have constant sign in blocks • Defining Relations: I = A, I = -B  I = -AB

  4. Example – Wool Shrinkage • 7 Factors  27 = 128 runs in full factorial • A = NaOH in grams/litre (1 , 3) • B = Liquor Dilution Ratio (1:20,1:30) • C = Time in minutes (20 , 40) • D = GA in grams/litre (0 , 1) • E = DD in grams/litre (0 , 3) • F = H2O2 (0 , 20 ml/L) • G = Enzyme in percent (0 , 2) • Response: Y = % Weight Loss • Experiment: Conducted in 2k-q = 8 runs (1/16 fraction) • Need 24-1 Defining Contrasts/Generalized Interactions • 4 Distinct Effects, 6 multiples of pairs, 4 triples, 1 quadruple

  5. Defining Relations • I = ADEG = BDFG = ACDF = -BCF • Generalized Interactions: • (ADEG)(BDFG)=ABEF,(ADEG)(ACDF)=CEFG,(ADEG)(-BCF)=-ABCDEFG • (BDFG)(ACDF)=ABCG,(BDFG)(-BCF)=-CDG,(ACDF)(-BCF)=-ABD • (ADEG)(BDFG)(ACDF)=BCDE, (ADEG)(BDFG)(-BCF)=-ACE • (ADEG)(ACDF)(-BCF)=-BEG, (BDFG)(ACDF)(-BCF)=-AFG • (ADEG)(BDFG) (ACDF)(-BCF)=-DEF • Goal: Choose block where ADEG,BDFG,ACDF are “even” and BCF is “odd”. All other generalized interactions will follow directly

  6. Aliased Effects and Design • To Obtain Aliased Effects, multiply main effects by Defining Relation to obtain all effects aliased together • For Factor A: • A=DEG=ABDFG=CDF=-ABCF=BEF=ACEFG=-BCDEFG=BCG=-ACDG=-BD=ABCDE=-CE=-ABEG=-FG=-ADEF

More Related