Design of Experiments

In this blog, I will be documenting my experience learning the theory of Design of Experiments (DOE).

Content:

1. Background of the case study

2. How to apply DOE

3. Learning reflection

Here's the excel link for Full and Fractional factorial analysis: Excel

BACKGROUND OF THE CASE STUDY

When we microwave popcorn, it's nearly impossible to get every kernel of corn to pop. Often a considerable number of inedible "bullets" (un-popped kernels) remain at the bottom of the bag. Hence, I am tasked to investigate what causes this loss of popcorn yield. In this case study, 3 factors were identified:

1. Diameter of bowls to contain the corn, 10 cm to 15 cm
2. Microwaving time, 4 mins to 6 mins
3. Power setting of the microwave, 75% to 100%

8 runs were performed with 100 grams of corn used in every experiment and the measurable variable is the amount of "bullets" formed in grams. All data are given (as seen in fig 1).

Without further ado🏃, let's start to investigate🔎 by applying DOE.

HOW TO APPLY DOE

In this segment, I lay out how I deviate my conclusion using full and fractional factorial design in 6 steps. Due to the fact that I was given all the data, I will skip the steps to collect the data and jump right into analysing it using excel. Theoretically, I have to carry out an experiment to collect the data first before analysing the data. 

[Full Factorial Analysis]

Step 1, enter all the data into the excel sheet. The excel will generate the “LOW” and “HIGH” values for each factor as shown in figures 1 and 2. 

Figure 1: Data entry table

Figure 2: Mean calculation table for the significance of MAIN effects 

[Full Factorial Analysis]

Step 2, analyse the data for the significance of the MAIN effects by plotting a graph of the mass of the bullet vs the high and low of factors A, B, C.

Figure 3: Full Factorial data analysis for the significance of MAIN effects 

From figure 3, the rank of factors is C > B > A whereby C has the most significant effect and A has the least significant effect on the mass of the bullet. 

With respect to figure 3, the effect of each factor based on the respective gradient;

Factor A: when the diameter increases from 10 cm to 15 cm, the mass of the bullet decreases from 1.90 to 1.7525.

Factor B: when the microwaving time increases from 4 mins to 6 mins, the mass of the bullet decreases from 2.17 to 1.4825.

Factor C: when the power increases from 75% to 100%, the mass of the bullet decreases from 2.93 to 0.72.

[Full Factorial Analysis]

Step 3, analyse the data for full factorial design interaction effect for (A x B), (A x C) and (B x C) by plotting a graph from the calculated values from figure 4, 5 and 6 

Figure 4: calculation table for the interaction effect of A x B

Figure 5: calculation table for the interaction effect of A x C

Figure 6: calculation table for the interaction effect of B x C

Figure 7: Full Factorial data analysis for the interaction effect of A x B

Figure 8: Full Factorial data analysis for the interaction effect of A x C

Figure 9: Full Factorial data analysis for the interaction effect of B x C

From figure 7, the gradient of both lines are different by a lot with a +ve gradient and a -ve gradient. Therefore, there’s a significant interaction between A and B. 

From figure 8, the gradient of both lines are different by a little margin with both a -ve gradient. Therefore, there’s an interaction between A and C but the interaction is small. 

From figure 9, the gradient of both lines are different by a greater margin with both a -ve gradient. Therefore, there’s an interaction between B and C but the interaction is small.

[Fractional Factorial Analysis]

For fractional factorial design, we have to select a subset of x runs from an r2n = N run factorial design. 

Figure 10.1: Description of how to select the numbers of subsets of runs for fractional factorial design

Figure 10.2: Description of how to select the numbers of subsets of runs for fractional factorial design

Hence, for this case study, I’ve selected run number 2,3, 5 and 8.

Step 4, enter all the data into the excel sheet. The excel will generate the “LOW” and “HIGH” values for each factor as shown in figures 11 and 12. 

Figure 11: Data entry table

Figure 12: Mean calculation table for the significance of MAIN effects 

[Fractional Factorial Analysis]

Step 5, analyse the data for the significance of the MAIN effects by plotting a graph of the mass of the bullet vs the high and low of factors A, B, C.

Figure 13: Fractional Factorial data analysis for the significance of MAIN effects 

From figure 13, the rank of factors is C > B > A whereby C has the most significant effect and A has the least significant effect on the mass of the bullet. 

With respect to figure 13, the effect of each factor based on the respective gradient;

Factor A: when the diameter increases from 10 cm to 15 cm, the mass of the bullet increases from 1.81 to 2.095.

Factor B: when the microwaving time increases from 4 mins to 6 mins, the mass of the bullet decreases from 2.31 to 1.595.

Factor C: when the power increases from 75% to 100%, the mass of the bullet decreases from 3.37 to 0.53.

[Full & Fractional Factorial Analysis]

Step 6, draw a conclusion. 

Full Factorial Analysis:

From fig 3, the power setting of the microwave has the most significant effect in decreasing the mass of bullets formed while the diameter of the bowl has the least significant effect. Hence, the best combination to lower the mass of bullets formed will be run no. 7 (-, +, +). 

However, based on my common sense and experience, the best combination to lower the mass of the bullets formed will be run no. 8 (+, +, +) (refer to fig 1). As with a greater diameter of the bowl, the surface area to volume ratio increases. As a result, every kernel of corn will be heated evenly and eventually pop. With a longer microwaving time, it buys more time for the kernels to get heated up and pop. With a higher power setting, it speeds up the rate for the kernels to pop. 

From fig 7, I can infer that there's a significant interaction effect for factors A and B meaning that the microwaving time is highly dependent on the diameter of the bowl. 

In conclusion, the result of the interaction between two independent variables (e.g., A and B) can further prove the relationship between each factor because the effect of one depends on the level of the other so when the data differ by a huge margin then the trend for interaction effect is no longer consistent among all the factors. As theoretically when I assumed run no. 8 to have the best combination (+, +, +) the interaction effect among 3 factors should roughly be the same but it wasn't. 

Fractional Factorial Analysis:

From fig 13, the power setting of the microwave has the most significant effect in decreasing the mass of bullets formed while the diameter of the bowl has the least significant effect. Hence, the best combination to lower the mass of bullets formed will be run no. 8 (+, +, +) which makes more sense. 

LEARNING REFLECTION

I had a lot of fun😁 during the practical session as our team got to "play" while learning how to apply DOE to screen for the most significant factor. From the tutorial and practical session, I learn and discover the pros and cons for each method, full factorial and fractional factorial methods.

The full factorial method is the best way for screening when the number of factors and sample size is small as it allows us to determine and estimate the full effects (and interactions) of each factor. However, it is very tedious to conduct the experiment because there are more data to be collected especially when there are more factors to analyse and as a result, making the sample size larger.

The fractional factorial method is the best way for screening when the number of factors and sample size is big as it is more efficient and resource-effective since lesser data are to be collected. However, we may risk the chance of us missing out on important data in other runs as the sample size for fractional factorial is much smaller than full factorial. Thus, leading us to make a wrong conclusion, reducing the accuracy of the test.

Both full and fractional factorial methods will give us the same result for the most significant factor. It only differs in terms of the least significant factor which will have a considerable impact on our final result when the number of factors is lesser and the size of the design is smaller.