two level factorial designs

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Two-Level Factorial Designs

Two-Level Factorial DesignsPresented by:Juanito S. Chan, PIE

Standard 23 DesignWhere do the factors (independent variables) appear in this table?Where do the responses (dependent variable) appear in this table?What do the 1 and +1 mean?Should these experimental runs be made in the order they are shown?

RunABC1-1-1-12+1-1-13-1+1-14+1+1-15-1-1+16+1-1+17-1+1+18+1+1+1

Standard 23 DesignFactors are A, B, CResponses do not appear in this table?Choose a high and a low value for each factor.-1 means set factor to low level in this run+1 means set factor to high level in this runRun order should be randomizedFailure to randomize very risky for factor C, since it has runs 1-4 at low level and 5-8 at high level

RunABC1-1-1-12+1-1-13-1+1-14+1+1-15-1-1+16+1-1+17-1+1+18+1+1+1

Maximize Reaction Yield23 Factorial DesignObjective: maximize reaction yieldFactors:A = catalyst weight percent (1,2)B = reaction time, hours (1,2)C = temperature, F (200,250)Response: Reaction yield, %

Maximize Reaction Yield

RunCatalyst Weight %Reaction Time, hrTemperature, FYield, %11120065.322120081.331220053.342220069.951125061.862125077.471225073.982225089.9

Now What?Calculate effects of each factor and interactionDecide which effects are importantPlan another, multilevel experiment focusing on the important variables

InteractionsNote that each factor is tested at each level 4 times.-1 -1 = +1

RunABCABACBCABCY1-1-1-1+1+1+1-165.32+1-1-1-1-1+1+181.33-1+1-153.34+1+1-169.95-1-1+161.86+1-1+177.47-1+1+173.98+1+1+189.9

Investigating InteractionsYou set the value for each factor in each experimentThe interactions happen naturallyYou do not set some level of AB interaction; it happens automatically because of the levels you set for A and B individuallyInteractions are a physical reality of the system, and will happen whether you calculate an effect for them or not

How to Calculate EffectsHigh Total = sum of all response values when the factor is at the +1 levelLow Total = sum of all response values when the factor is at the 1 levelDifference = (High Total) (Low Total)Note that you can also calculate the difference by multiplying each +1 or 1 by the response for its row, then summing all the values in the column. That is what your book says.Effect = Difference / (# runs at each level)

Effects

ACat. Wgt. %BRxn TimeCTempABACBCABCHighTotal LowTotal Diff EffectOn Y

Effects

ACat. Wgt. %BRxn TimeCTempABACBCABCHighTotal318.5287.0303.0286.9285.9310.4286.3LowTotal254.3285.8269.8285.9286.9262.4286.5Diff64.21.233.21.0-1.048.0-0.2EffectOn Y16.05.308.300.25-0.2512.00-0.05

Scree Plot to Identify Order of Importance

Chart2

16.05

12

8.3

0.3

0.25

-0.25

Factor

Effect on Reaction Yield

Sheet1

RunABCABACBCYA = Catalyst weight % (1,2)

1-1-1-111165.3B = Reaction time, hr (1,2)

21-1-1-1-1181.3C = Temperature, F (200,250)

3-11-1-11-153.3Y = Reaction yield

411-11-1-169.9

5-1-111-1-161.8A16.05

61-11-11-177.4BC12

7-111-1-1173.9C8.3

811111189.9B0.3

AB0.25

RunABCABACBCYAC-0.25

1-65.3-65.3-65.365.365.365.365.3

281.3-81.3-81.3-81.3-81.381.381.3

3-53.353.3-53.3-53.353.3-53.353.3

469.969.9-69.969.9-69.9-69.969.9

5-61.8-61.861.861.8-61.8-61.861.8

677.4-77.477.4-77.477.4-77.477.4

7-73.973.973.9-73.9-73.973.973.9

889.989.989.989.989.989.989.9

High - Low64.21.233.21-148

Effect16.050.38.30.25-0.2512

Sheet2

Sheet3

ConclusionsIncreasing catalyst weight % or increasing temperature will increase the yieldIncreasing catalyst is most effectiveIncreasing reaction time itself has little effect on yield, but in combination with increased temperature multiplies the effect of temperature

Comparison with OFATOFAT would reveal the effect of catalyst and temperature.OFAT would not reveal the time-temperature interaction.OFAT would not reveal the lack of time-catalyst and temperature-catalyst interaction.

Adding a FactorAdding a factor to a full factorial design doubles the number of experimental runs3 factors = 23 = 8 runs4 factors = 24 = 16 runsIf you are confident that an interaction is unimportant, you can substitute a new factor for that interaction term in the test matrix3-way interaction least likely to be importantSubstitution of a factor for an interaction makes an unsaturated design

Unsaturated Designs and AliasingIf a factor replaces an interaction in the design, You cannot tell the difference between the effect of the factor and the effect of the interactionInteraction is an innate property of the system. You do not control whether or not it happens by deciding whether or not to study it.Some or all of the effect you calculate for the new factor could be due to the interaction between other factors.You cannot study how the new factor interacts with others