Week 13： DOE

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This week We learnt about the Design of Experiments which focuses on determining the most influential factor affecting the output, the following is the data analysis for Case2.

Objective:
1. Use full and factorial data analysis

2. Find out the most influential factor
2. Compare and evaluate the results

A = concentration of coagulant added, 1% and 2% by weight 

B = treatment temperature, 72o F and 100o F 

C = Stirring speed, 200 rpm and 400 rpm

The first step I did was to transfer the data from the table above to excel, and the average values will be calculated automatically.

Excel File: Case2 file

Data used:

Full factorial analysis: uses of the average values of high and low levels of different factors.

From the graph, it shows that stirring speed has the highest impact on the pollutant discharged because it has the highest gradient of -14.5. followed by concentration with a gradient of 12.5 and lastly the temperature. So the rank shall be C>A>B.

› Determine the interaction effects.

Plot a graph using the values calculated to check for the interactions between different factors.

For FULL factorial design: (A x B)

At LOW B, (runs 1 and 5) Average of low A=(5+4)/2=4.5

At LOW B, (runs 2 and 6) Average of high A=(30+3)/2=16.5

At LOW B, total effect of A=(16.5-4.5)= 12(increase)

At HIGH B, (runs 3 and 7) Average of low A=(6+5)/2=5.5

At HIGH B, runs 4 and 8) Average of high A=(33+4)/2=18.5

At HIGH B, total effect of A=(18.5-5.5)=13(increase)

The gradients of both lines are both negative and are different with a very small margin.

This indicates that there’s an interaction between A and B, but the interaction is small. 

For FULL factorial design: (A x C)

At LOW A, (runs 1 and 3) Average of low C=(5+6)/2=5.5

At LOW A, (runs 5 and 7) Average of high C=(4+5)/2=4.5

At LOW A, total effect of C=(4.5-5.5)= -1 (decrease)

At HIGH A, (runs 2 and 4) Average of low C=(30+33)/2=31.5

At HIGH A, (runs 6 and 8) Average of high C=(3+4)/2=3.5

At HIGH A, total effect of C=(3.5-31.5)=-28 (decrease)

The gradients of both lines are different, one has a larger decreasing gradient and one has a smaller decreasing gradient, therefore there is significant interaction between factors A and C.

For FULL factorial design: (B x C)

At LOW B, (runs 1 and 2) Average of low C=(5+30.5)/2=17.5

At LOW B, (runs 5 and 6) Average of high C=(4+3)/2=3.5

At LOW B, total effect of C=(3.5-17.75)= -14.25 (decrease)

At HIGH B, (runs 3 and 4) Average of low C=(6+33)/2=19.5

At HIGH B, (runs 7 and 8) Average of high C=(5+4)/2=4.5

At HIGH B, total effect of C=(4.5-19.5)=-15 (decrease)

The gradients of both lines are both negative and are slightly different with a very small

margin. This indicates that there’s an interaction between A and B, but the interaction is small. 

From the tables, it shows that the interaction effect between A and C is the biggest because the gradients are very different and the two lines insect. Factors AxB and BxC also contribute to the pollutant discharged but with a minimal impact, both have an interaction impact as the lines will insect soon.

Fractional factorial analysis:

I chose runs# 1, 4, 6 and 7 as the four runs are orthogonal on the cubic design, the number of high and low levels of different factors is the same.

Repeat the same technique used in the full factorial analysis.

From the graph, it shows that both stirring speed and temperature have the biggest impact on the pollutant discharged because both have a gradient of -15, while Stirring speed has a higher c value, hence C has the bigger impact than B. Then the factor with the least impact is concentration since it has the lesser steep gradient of 13. So the rank shall be C>B>A.

› Determine the interaction effects.

For Fractional factorial design: (A x B)

At LOW B, (run 1) Average of low A=5

At LOW B, (run 6) Average of high A=3

At LOW B, total effect of A=(3-5)= -2 (decrease)

At HIGH B, (run 7) Average of low A=5

At HIGH B, (run 4) Average of high A=33

At HIGH B, total effect of A=(33-5)=28 (increase)

The gradients of both lines are different, one has a positive gradient and one has a negative gradient and both lines intersect at 1 point,  therefore there is significant interaction between factors A and B.

For Fractional factorial design: (A x C)

At LOW A, (run 1) Average of low C=5

At LOW A, (run 7) Average of high C=5

At LOW A, total effect of C=(5-5)= 0 (no change)

At HIGH A, (run 4) Average of low C=33

At HIGH A, (run 6) Average of high C=3

At HIGH A, total effect of C=(3-33)=-30 (decrease)

The gradients of both lines are different, one has a zero gradient and one has a negative gradient and both lines intersect at 1 point,  therefore there is significant interaction between factors A and C.

For Fractional factorial design: (B x C)

At LOW B, (run 1) Average of low C=5

At LOW B, (run 6 Average of high C=3

At LOW B, total effect of C=(3-5)= -2 (decrease)

At HIGH B, (run 4) Average of low C=33

At HIGH B, (run 7) Average of high C=5

At HIGH B, total effect of C=(5-33)=-28 (decrease)

The gradients of both lines are different, one has a greater negative gradient and one has a smaller negative gradient and both lines will intersect at 1 point eventually,  therefore there is significant interaction between factors B and C.

From the tables, it shows that the interaction effect between A and C is the biggest because the gradients are very different and the two lines insect. Factors AxB and BxC also contribute to the pollutant discharged significantly, both have an interaction impact as the lines will insect soon.

Conclusion:

The overall trend is similar, however, there are some deviations for the fractional factorial design and it might be because of the limited data, as only one data point is used for the interaction analysis. While the fractional factorial design is still able to produce a similar trend and can be used when there is limited time and resources as it is able to give a representable analysis.