pengprod - pertemuan 3
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Pengembangan ProdukNajma Annuria Fithri, S.Farm.,M.Sc., Apt.
Universitas Sriwijaya
Genap 2012/2013
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Pertemuan 3Factorial Design
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Role of DOE in Process Improvement
DOE is a formal mathematical method forsystematically planning and conductingscientific studies that change experimentalvariables together in order to determine their
effect of a given response.
DOE makes controlled changes to input
variables in order to gain maximum amountsof information on cause and effectrelationships with a minimum sample size.
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Role of DOE in Process Improvement
DOE is more efficient that a standardapproach of changing one variable at a timein order to observe the variables impact on agiven response.
DOE generates information on the effectvarious factors have on a response variable
and in some cases may be able to determineoptimal settings for those factors.
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Role of DOE in Process Improvement
DOE encourages brainstorming activitiesassociated with discussing key factors thatmay affect a given response and allows the
experimenter to identify the key factors forfuture studies.
DOE is readily supported by numerousstatistical software packages available on themarket.
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BASIC STEPS IN DOE
Four elements associated with DOE:
1. The design of the experiment,
2. The collection of the data,
3. The statistical analysis of the data, and
4. The conclusions reached and
recommendations made as a result of theexperiment.
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TERMINOLOGY
Replication repetition of a basic experimentwithout changing any factor settings, allowsthe experimenter to estimate theexperimental error (noise) in the system usedto determine whether observed differences inthe data are real or just noise, allows theexperimenter to obtain more statistical power
(ability to identify small effects)
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TERMINOLOGY
.Randomization a statistical tool used tominimize potential uncontrollable biases inthe experiment by randomly assigningmaterial, people, order that experimental
trials are conducted, or any other factor notunder the control of the experimenter.Results in averaging out the effects of the
extraneous factors that may be present inorder to minimize the risk of these factorsaffecting the experimental results.
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TERMINOLOGY
Blocking technique used to increase theprecision of an experiment by breaking theexperiment into homogeneous segments(blocks) in order to control any potential
block to block variability (multiple lots of rawmaterial, several shifts, several machines,several inspectors). Any effects on theexperimental results as a result of the
blocking factor will be identified andminimized.
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FACTORIAL (2k) DESIGNS
Experiments involving several factors ( k = #of factors) where it is necessary to study thejoint effect of these factors on a specific
response. Each of the factors are set at two levels (alow level and a high level) which may bequalitative (machine A/machine B, fan
on/fan off) or quantitative (temperature800/temperature 900, line speed 4000 perhour/line speed 5000 per hour).
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FACTORIAL (2k) DESIGNS
Factors are assumed to be fixed (fixed effectsmodel)
Designs are completely randomized(experimental trials are run in a random
order, etc.) The usual normality assumptions are
satisfied.
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FACTORIAL (2k) DESIGNS
Particularly useful in the early stages ofexperimental work when you are likely tohave many factors being investigated and youwant to minimize the number of treatment
combinations (sample size) but, at the sametime, study all k factors in a completefactorial arrangement (the experimentcollects data at all possible combinations offactor levels).
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FACTORIAL (2k) DESIGNS
As k gets large, the sample size will increaseexponentially. If experiment is replicated, the# runs again increases.
k # of runs
2 4
3 8
4 16
5 32
6 64
7 128
8 256
9 512
10 1024
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FACTORIAL (2k) DESIGNS (k = 2)
Two factors set at two levels (normally
referred to as low and high) would result inthe following design where each level of factorA is paired with each level of factor B.
RUN Factor A Factor B RESPONSE RUN Factor A Factor B RESPONSE
1 low low y1 1 -1 -1 y1
2 high low y2 2 +1 -1 y2
3 low high y3 3 -1 +1 y3
4 high high y4 4 +1 +1 y4
Generalized Settin s Ortho onal Settin s
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FACTORIAL (2k) DESIGNS (k = 2)
Estimating main effects associated with
changing the level of each factor from low tohigh. This is the estimated effect on theresponse variable associated with changing
factor A or B from their low to high values.
2
)(
2
)( 3142 yyyyEffectAFactor
2
)(
2
)( 2143 yyyyEffectBFactor
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FACTORIAL (2k
) DESIGNS (k = 2): GRAPHICALOUTPUT
Neither factor A nor Factor B have an effect onthe response variable.
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FACTORIAL (2k) DESIGNS (k = 2):
GRAPHICAL OUTPUT
Factor A has an effect on the responsevariable, but Factor B does not.
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FACTORIAL (2k) DESIGNS (k = 2):
GRAPHICAL OUTPUT
Factor A and Factor B have an effect on theresponse variable.
21k
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21FACTORIAL (2k) DESIGNS (k = 2):
GRAPHICAL OUTPUT
Factor B has an effect on the response variable, butonly if factor A is set at the High level. This iscalled interactionand it basically means that theeffect one factor has on a response is dependent onthe level you set other factors at. Interactions can be
major problems in a DOE if you fail to account for theinteraction when designing your experiment.
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EXAMPLE:
FACTORIAL (2k) DESIGNS (k = 2)
A microbiologist is interested in the effect of twodifferent culture mediums [medium 1 (low) and
medium 2 (high)] and two different times [10hours (low) and 20 hours (high)] on the growthrate of a particular CFU [Bugs].
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EXAMPLE:
FACTORIAL (2k) DESIGNS (k = 2)
Since two factors are of interest, k =2, and wewould need the following four runs resulting
inRUN Medium Time Growth Rate
1 low low 17
2 high low 153 low high 38
4 high high 39
Generalized Settin s
24A
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24EXAMPLE:
FACTORIAL (2k) DESIGNS (k = 2)
Estimates for the medium and timeeffects are
Medium effect = [(15+39)/2] [(17 + 38)/2]= -0.5
Time effect = [(38+39)/2] [(17 + 15)/2] =
22.5
25EXAMPLE
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25EXAMPLE:
FACTORIAL (2k) DESIGNS (k = 2)
26EXAMPLE
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26EXAMPLE:
FACTORIAL (2k) DESIGNS (k = 2)
A statistical analysis using the appropriatestatistical model would result in the followinginformation. Factor A (medium) and Factor B(time)
Type III Sums of Squares
------------------------------------------------------------------------------------
Source Sum of Squares Df Mean Square F-Ratio P-Value
------------------------------------------------------------------------------------
FACTOR A 0.25 1 0.25 0.11 0.7952
FACTOR B 506.25 1 506.25 225.00 0.0424
Residual 2.25 1 2.25
------------------------------------------------------------------------------------
Total (corrected) 508.75 3
All F-ratios are based on the residual mean square error.
27EXAMPLE
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27EXAMPLE:
CONCLUSIONS
In statistical language, one would concludethat factor A (medium) is not statisticallysignificant at a 5% level of significance sincethe p-value is greater than 5% (0.05), but
factor B (time) is statistically significant at a 5% level of significance since this p-value isless than 5%.
28EXAMPLE
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28EXAMPLE:
CONCLUSIONS
In layman terms, this means that we have noevidence that would allow us to conclude thatthe medium used has an effect on the growthrate, although it may well have an effect (our
conclusion was incorrect).
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EXAMPLE:CONCLUSIONS
Additionally, we have evidence that would allowus to conclude that time does have an effect on
the growth rate, although it may well not have aneffect (our conclusion was incorrect).
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EXAMPLE:
CONCLUSIONS
In general we control the likelihood of reachingthese incorrect conclusions by the selection of
the level of significance for the test and theamount of data collected (sample size).
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Interactions for 2k Designs (k = 3)
Interactions between various factors can beestimated for different designs above bymultiplying the appropriate columnstogether and then subtracting the average
response for the lows from the averageresponse for the highs.
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Interactions for 2k Designs (k = 3)
a b c ab ac bc abc
-1 -1 -1 1 1 1 -1
+1 -1 -1 -1 -1 1 1
-1 +1 -1 -1 1 -1 1
+1 +1 -1 1 -1 -1 -1
-1 -1 +! 1 -1 -1 1
+1 -1 +1 -1 1 -1 -1-1 +1 +1 -1 -1 1 -1
+1 +1 +1 1 1 1 1
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2kDESIGNS (k > 2)
For example, if there are no significantinteractions present, you can estimate a responseby the following formula. (for quantitative factors
only)
Y = (average of all responses) + )](*)2
[( LfactorLEVECTfactorEFFE
= BAY BA *)2
(*)2
(