Design of Experiments(DOE) is a statistics-based approach to designing experiments in order to obtain knowwledge of a complex, multi-variable process with the fewest trials possible. It can be seen as the optimisation of an experimental process. It is the backbone of any product design as well as any process/product improvement efforts.
I have been given a case study about a waste water treatment facility and how a combination of coagulant chemicals, treatment temperature and stirring speed are critical factors to treat waste-water to produce clean water. My goal for this case study will be to determine which of the 3 factors are the most influential on minimizing the amount of pollutants discharged.
Each factor has 2 levels(-/+).
Factor A: Concentration of coagulant chemicals.
Low Level(-): 1 wt%
High Level(+): 2 wt%
Factor B: Treatment Temperature
Low Level(-): 72 degF
High Level(+): 100 degF
Factor C: Stirrer Speed
Low Level(-): 200 rpm
High Level(+): 400 rpm
For 3 different factors with 2 levels of factor, for 1 replicate, there are a total of 8 different runs, which have been done and the amount of pollutant discharged has been recorded. This is also known as the Full Factorial Design
With this information, I will transfer this over to the template provided and conduct analysis of the data using graphical method.
With the help of the Excel Spreadsheet, I have calculated the significance of each factor by solving for the means.
This is done by calculating the average amount of pollutants discharged when the examined factor is high and low. This prevents confounding data. With the calculated averages, I used it to create a graph for easy comparison.
As shown in the graph above, we can see that the factors with the most significance to the amount of pollutant discharged are factors A and C. This is seen by the steep slopes in the lines when the factor level is increased from low to high.
Ranking of Factors based on significance:
1. Factor C (Stirrer Speed)
2. Factor A (Concentration of Coagulant Chemicals)
3. Factor B (Treatment Temperature)
Next, I will analyse the effects of different factors on one another, also known as their interaction.
First, I will analyse the effects of factor B on A.
What I've done here is to find the average amount of pollutant discharged when B is low and when A increases from low to high. Then, I repeated this step for when B is high to get two different sets of data which is then plotted in the graph shown above. In the graph above shows the impact factor A has when it increases from low to high level when B is low and when B is high. Since the gradients of the two lines on the graph look very similar, we can say that factors A and B have very little interaction.
I did this for the interaction effect between A & C and B & C and these are my results.
Interaction Effect of A x C:
Since the gradient of the two lines are very different, there is a lot of interaction between factors A and C.
Interaction Effect of B x C:
Since the gradient of the two lines are very similar, there is little interaction between factors B and C.
With all this information, I have come to the conclusion that to reduce the amount of pollutants discharged, the concentration of coagulant chamicals and the treatment temperature should be of low level(1% and 72 degF), while the stirrer speed has to be high(400rpm).
Excel File for Full Factorial Method:
Now, I will complete another analysis with the Fractional Factorial Method this time.
For the Fractional Factorial Method, as stated by the name, would only use a fraction of the data collected earlier from the Full Factorial Method. For a fruitful analysis, representative data has to be fractionalised and collected, which is why we will rely on statistical orthogonality.
Let's say that each different run represents a corner of a cube. For a set of data points to be orthogonal, every data point selected has to be diagonal to every other data point on the cube, as seen in the image above. I will use the four points as shown above to complete the Fractional Factorial Method.
The Fractional Factorial Method isn't all that much different from the Full Factorial Method, except that it uses four sets of data instead of eight.
Using the same template, I solved for the means. Using this data, I plotted a Graph of Pollutant Discharged vs Factor Level.
From the data gathered, the rankings of the significance of the factors are:
1. Factor C(Stirrer Speed)
2. Factor A(Concentration) and B(Temperature) [tied 2nd]
It seems that in both methods, Factor C, the stirrer speed, seems to have the greatest impact on the amount of pollutants discharged, and while Factors A and B did not have greater impacts that Factor C, they should still be greatly considered especially since their effect was not very far in terms of its magnitude.
Excel for Fractional Factorial Method:
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