Bài giảng tối ưu hóa Bài 2 Full Factorial Design Ph.D Lê Minh Tâm

 B: type of kernel used white corn or yellow corn 2? number of levels for the factors number of factors... Results 2/5: cube plot A systematic approach1.. Results 3/5: Predictive model

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Two-factor experiment

Tâm Minh Lê, Ph.D Faculty of Food Technology, HUFI

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1. A systematic approach to analyze two-factors experiment

3. Two-factor experiments with interactions

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2.01 A systematic approach to analyze

two-factor experiment

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 Factors: things influencing the outcome that we want to change

 Also called variables

 Qualitative = Categorical

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Context: popcorn

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Issue: unpopped popcorn

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5. Plan the experiment

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Outcome and Objective A systematic approach1 What is my outcome ?

2 What is my objective ?

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Outcome and Objective

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Factors and Levels

 A: time on the stove (160 sec or 200 sec.)

 B: type of kernel used (white corn or yellow corn)

2𝑘 number of levels for the factors number of factors

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StdOder RunOder Time Corn

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StdOder RunOder Time Corn

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StdOder RunOrder Time Corn Popped corn

Run the experiments

6 Implement the experiment

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7 Analyze the results

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StdOrder RunOrder Time Corn Popped corn

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Results (3/5) – The size of main effects

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Results (4/5) – Contour plot

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Results (5/5) –Interaction plot

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Results (5/5) –Interaction plot

62

74

80

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❶ Table: show numeric trends

❷ Cube plot: indicate important factors

❸ Contour plot: show where to move next

❹ Interaction plot: show synergies

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2.02 Predictions

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 Cube plot: indicate important factors

 Interaction plot: show synergies

 Predict = get the (predicted) results of un-done trials ? We need a

predictive model

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Predictive model: a magic

𝑦 = 67 + 10𝑥 𝐴 + 4𝑥 𝐵

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Elements of the predictive model

𝑦 = 67 + 10𝑥 𝐴 + 4𝑥 𝐵

baseline [A] effect [B] effect

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Correlation coefficient Of [A] effect

202

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Correlation coefficient Of [B] effect

82

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 Regression line is useful for prediction

 Baseline is the average of 4 values on the cube

 𝑏𝐴 (𝑏𝐵) takes a half value of the 𝐴 ( 𝐵 ) effect

 𝑥𝐴, 𝑥𝐵: coded variables, which are coded from (-1) to (+1)

𝑦 = 𝑏 0 + 𝑏 𝐴 𝑥 𝐴 + 𝑏 𝐵 𝑥 𝐵

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The link between coded value to real world value

coded value = real value − center point

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 What is the predicted popcorn amount for white corn and cooking time of

180 sec ? Represent this point on the cube plot

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2.03 Two-factor experiments with

interactions

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The concept of interaction

 Factors:

 Soap (No.soap vs Yes.soap)

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The concept of interaction

≡ The effect of warm water is enhanced by using soap

≡ The effect of soap is enhanced by using warm water

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Property of interaction

 Interaction is symetrical:

 For instance: using A (soap) together with B (warm water) = using B

(warm water) together with A (soap)

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Remind the concept of « interaction »

The two levels of one factor react differently to a change

in level of another factor The effect of one factor depends on the value or the level

of another factor

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Interaction: another example

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Interaction plot

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Context: Ginger biscuits

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Context: Ginger biscuits

10. Baking time: 10 min at 350 oC

Factor B: 55 g honey or 55 g of molasses

Factor A: bake for 8 min or 14 mins

(https://en.wikipedia.org/wiki/Molasses)

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Remark on the outcomes

1. Taste

2. Breakability: break strength

3. Freshness: break strength after 1 week

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 A: baking time (8 min or 14 min.)

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Experiment plan

 Factors

 A: baking time (8 min or 14 min.)

 B: sugar type (honey‒1 or molasses)+1

5 Plan the experiment

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Results (1/5): table A systematic approach1 What is my outcome ?

Stand order Real order baking sugar Taste

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Results (2/5): cube plot A systematic approach1 What is my outcome ?

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Results (3/5): Predictive model A systematic approach1 What is my outcome ?

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 Calculate the interaction of [BA] effect?

𝑏𝐴𝐵 = 𝑏𝐵𝐴

𝑦 = 𝑏0 + 𝑏𝐴𝑥𝐴 + 𝑏𝐵𝑥𝐵 + 𝑏𝐴𝐵𝑥𝐴𝑥𝐵

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= 9

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 critical thinking about the interpretation of these number because they

are not just a number

 What if we take baking time for 16, 18, or 20 min ?

 Why do molasses give a better taste (increase the liking scores) ?

time, in the sense of chemical reaction ?

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2.03 Three-factor experiments

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Case study: Waste water treatment

C

S

T

P Q

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Case study: Waste water treatment

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Number of trials A systematic approach1 What is my outcome ?

3 Which factors ?

4 At what levels ?

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Coded variables A systematic approach1 What is my outcome ?

3 Which factors ?

4 At what levels ?

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Designing experiment A systematic approach1 What is my outcome ?

3 Which factors ?

4 At what levels ?

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Results (1/5): Data table A systematic approach1 What is my outcome ?

3 Which factors ?

4 At what levels ?

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Results (2/5): Cube plot A systematic approach1 What is my outcome ?

3 Which factors ?

4 At what levels ?

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Results (2/5): Cube plot A systematic approach1 What is my outcome ?

3 Which factors ?

4 At what levels ?

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Results (3/5): Predictive model

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Results (3/5): Predictive model

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Results (3/5): Predictive model without interactions

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2.04 Calculate interactions for the

three-factor experiments

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How can we sure about the existence of

interactions in an experiment?

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The two levels of one factor react differently

to a change in level of another factor

[AC] interaction is important

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Calculate the average of [AC] interaction

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 Average of [AC] interaction = 𝐵+ + 𝐵−

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 Average of [AC] interaction = 𝐵++ 𝐵−

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 Average of [AC] interaction = 𝐵++ 𝐵−

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 Average of [AC] interaction = 𝐵++𝐵−

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Predictive model with all interactions

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Understanding the interaction

 Dilution time for Q chemical (C+)?

 What if the requirement is: the outcome < 10 units? And

 What if the cost of chemical P is

double expensive than the cost of chemical Q?

𝑦 = 11.25 + 6.25𝑥𝐶 +0.75𝑥𝑇 −7.25𝑥𝑆 − 𝟔 𝟕𝟓 𝑥𝐶𝑥𝑆

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