Design of Experiment
Design of Experiment:
Design of Experiment (DOE) is an effective method for identifying the factors most important to a particular process (or system), and then determining at what levels these factors must be maintained to optimize the process. This method provides a fast and cost-effective method for understanding and optimizing manufacturing processes. This approaches directly replaces the hit-or-miss nature of experimentation that requires a lot of guesswork and luck to succeed
Fundamentals of DOE:
- Response variable (Dependent variable) --> It is an outcome that is measured for a given experiment
- Factor (Independent Variable) --> It is a variable that is deliberately varied in order to observe how it affects the response
- Level --> A level of a factor indicates the specific condition of the factor we are measuring
- Treatment --> It is a specific combination of factor levels that makes up a treatment
To determine the number of practicals that needs to be carried out, use the formula below:
Where,
N = total number of experiments
n = number of factors
r = number of replicates
l = number of level
In DOE, there are usually two levels, high and low, which are represented by '+' and '-' signs, respectively.
There are also two kinds of applications: full factorial and fractional factorial. A fractional factorial designs allow for fewer treatments to be chosen to determine factor effects while still providing sufficient data.
CASE STUDY:
What could be simpler than making microwave popcorn? Unfortunately, as everyone who has ever made popcorn knows, 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. What causes this loss of popcorn yield? In this case study, three factors were identified:
1. Diameter of bowls to contain the corn, 10 cm and 15 cm
2. Microwaving time, 4 minutes and 6 minutes
3. Power setting of microwave, 75% and 100%
8 runs were performed with 100 grams of corn used in every experiments and the measured variable is the amount of “bullets” formed in grams and data collected are shown below:
Factor A= diameter
Factor B= microwaving time
Factor C= power
Admin number: 2122599
Table 1: Data for full factorial analysis
Full Factorial Data Analysis:
Click here to excess the excel file for Full Factorial Design data analysis graphs and tables
For analysis, we first determine the average mass for various factors at different levels. The picture below shows the data tables and average values tabulated from the excel sheet.
Figure 1: Data tables and average values tabulated from the excel sheet
Effects of individual factors:
The line graph is plotted using the tabulated average mass with the y-axis representing the average mass of bullets and the x-axis representing the number of factors, as shown in the image below.
Figure 2: Average values tabulated (full factorial)
Figure 3: Graph of mass of bullets against level of factors
Since all three factors have a negative gradient, therefore, all three factors have a negative effect on the mass of bullets.
- When the diameter (Factor A) increases from 10 cm to 15 cm, the mass of bullets decreases from 1.96g to 1.813g.
- When the microwaving time (Factor B) increases from 4 mins to 6 mins, the mass of bullets decreases from 2.2g to 1.573g.
- When the power setting (Factor C) increases from 75% to 100%, the mass of bullets decreases from 3.023g to 0.750g.
Next, using the gradient of the line for each factor, we can determine the ranking of the factors. The steeper the gradient, the larger the significance on the mass of bullets.
In this case, factor C has the steepest gradient among the 3 factors. Therefore, factor C (power setting) has the largest significance on the mass of bullets.
However, factor A (diameter) has the gentlest gradient among the 3 factors. This means that factor A has the smallest significance on the mass of bullets.
Whereas for factor B (microwaving time), its gradient is not as steep as factor C, and at the same time, it's not as gentle as factor A.
Therefore, the ranking of factors from the most significant to the least significant will be from:
- Factor C - power setting (Most significant)
- Factor B - microwaving time
- Factor A - diameter (Least significant)
Interaction effects:
Perform the calculations below to calculate the interaction effect.
Figure 4: Interaction effects calculation for full factorial
From here, we can plot a graph with the x-axis and y-axis being the levels of factor and mass of bullets respectively.
For the interaction effect of A & B, the graph is plotted as shown below.
Figure 5: Interaction effect graph of A & B
From figure 5, The gradient of both lines are different (one is + and one is -). Therefore, there is a significant gradient between factor A and factor B
For the interaction between A & C, the graph is plotted as shown below.
Figure 6: Interaction effect graph of A & C
From figure 6, we can see that the gradient of both lines are negative but they are not parallel to each other and they are different by a little margin. Therefore, there is an interaction between A & C, but the interaction is small.
For interaction between B & C, the graph is plotted as shown below.
Figure 7: Interaction graph between B & C
From figure 7, we can see that the gradient of both lines are negative but at a different gradient value. Therefore, there is a significant interaction between factor B & factor C.
Conclusion for full factorial data analysis:
By looking at the graph on the effects of individual factors, it can be seen that when all factors increases from low to high, the bullet mass decreases (gradient decreases). Hence, in order to increase the yield of popcorn, we can use a bowl with a larger diameter, increase the microwaving time and the power of the microwave used at the same time so that the popcorn yield will increase, the mass of bullets decrease. Which means more popcorn is achieved.
From the interaction effect of B & C, it can be seen that when factor C (power) is low, the mass of bullets is high when factor B increases from low to high. However, when factor C is at high, the mass of bullets are lower compared to when factor C is low with the increase in factor B (microwaving time). Therefore, based on the interaction between factors B and C, I would choose to increase the microwave power level (factor C) and longer microwaving time (factor B).
When comparing factor A with factors B and C, we can see that the mass of bullets increases with shorter microwaving time (factor B) and even with a bigger diameter (factor A). However, when at low C (power setting) and the diameter increases, the mass of bullets decreases by very little. From here, we can see that factor C (power setting) does not make any significance with increasing factor A (diameter). Therefore, the popcorn yield will increase with a bigger bowl diameter (factor A) with longer microwaving time (high B) and higher power setting (high C).
In summary, it is best to use a bowl with a bigger diameter, and increase the microwaving time and power setting so that the popcorn yield will increase. In this case, based on the interaction graph, i will increase the factor C (power setting) more as compared to factor A and B because factor C has the largest impact on the mass of bullets left in the bowl.
Fractional Factorial Analysis:
To perform fraction factorial data analysis, we will need to select a subset of 4 runs from 2^3=8 run factorial design. I've selected run order 1, 2, 3, 6 (run #2, 3, 5, 8).
Table 2: Data for fractional factorial design analysis
Click here to excess the excel file for Fractional Factorial Design data analysis graphs and tables
For analysis, we first determine the average mass for various factors at different levels. The picture below shows the data tables and average values tabulated from the excel sheet.
Figure 8: data tables and average values tabulated from the excel sheet (Fractional Factorial)
Effects of individual factors:
The line graph is plotted using the tabulated average mass with the y-axis representing the average mass of bullets and the x-axis representing the number of factors, as shown in the image below (I did the same as the full factorial data analysis).
Figure 9: graph of mass of bullets vs level of factors (fractional Factorial)
Similarly, by reading off the graph (figure 9), we can determine the effects of each factor by looking at their gradients. A positive and negative effect are represented by a positive and negative gradient resectively.
From figure 9, we can see that factor B and C both have a negative gradient. Therefore, factor B and C both have a negative effect on the mass of bullets.
Whereas factor A has a positive gradient. Therefore, factor A has a positive effect on the mass of bullets.
- When the diameter (Factor A) increases from 10 cm to 15 cm, the mass of bullets increases from 1.87g to 2.16g.
- When the microwaving time (Factor B) increases from 4 mins to 6 mins, the mass of bullets decreases from 2.37g to 1.66g.
- When the power setting (Factor C) increases from 75% to 100%, the mass of bullets decreases from 3.49g to 0.53g.
Next, using the gradient of the line for each factor, we can determine the ranking of the factors. The steeper the gradient, the larger the significance on the mass of bullets.
In this case, factor C has the steepest gradient among the 3 factors. Therefore, factor C (power setting) has the largest significance on the mass of bullets.
However, factor A (diameter) has the gentlest gradient among the 3 factors. Which means that factor A has the smallest significance on the mass of bullets.
Whereas for factor B (microwaving time), its gradient is not as steep as factor C and at the same time, it not as gentle as factor A.
Therefore, the ranking of factors from the most significant to the least significant will be from:
- Factor C - power setting (Most significant)
- Factor B - microwaving time
- Factor A - diameter (Least significant)
Conclusion for fractional factorial data analysis:
Based on the effect of individual factors, we can see that factor A has a positive effect and Factor B and C both have a negative effect on the mass of bullets. So same concept as the full factorial design, to increase the yield of popcorn, we can use a smaller diameter bowl with longer microwaving time and higher power setting on the microwave so that the mass of bullets will decrease and the amount of popcorn formed will increase, hence increasing the popcorn yield.
As far as adjusting one factor only, I would increase the microwave's power setting, since it has the greatest effect on the mass of bullets generated.
Learning reflection:
Tutorial:
During the tutorial lesson, we were introduced to DOE which apparently wasn't new to us at all. We were told that we did it sometime back during our LPS 2 practical where we did coffee leaching. At that point in time, I was so confused because we weren't taught the term 'Design of Experiment' when we were doing the coffee leaching practical. However, as Dr Noel told us more about how DOE works, I kind of recalled back and remember that yes, we did something similar. Although we did that DOE before during LPS2, but somehow it's a little different when it comes to understanding DOE more in-depth. It was so confusing trying to understand the fractional factorial data analysis where we need to select a subset of 4 runs. So apparently, these 4 runs cannot be randomly picked. There is a method called 'statistical orthogonality' taught to us on how to select the correct 4 runs so that we can achieve a balanced design with all factors to occur the same number of times and also to make sure the most significant factor for full factorial and fractional factorial will appear to be the same.
Other than that, everything was okay and I could understand most of it.
Practical:
I realised although I say I understand most of the things taught during the tutorial, but when I did the pre-practical assignment, I realised i still didn't understand much about DOE yet. However, I did give up instead, I went to consult one of my friends for help. So yes I could understand better after consulting my friend.
On the day of the practical, I was so confident because I could finally understand DOE after completing my pre-practical assignment. We were told to split into 2 groups, one group would work on the full factorial data analysis while the others would work on fractional factorial data. Isabella, Wayne, and I will be doing full factorial since we have more manpower. Haiteng and Ryan will work on the fractional factorial. After setting up everything that is needed for the practical, our initial plan was to perform both full and fractional factorial at the same time but I realised that we only had one set of sand. If 2 projectiles were launched at the same time, the 2 projectiles might not land at the same distance which makes it more troublesome to keep on moving the plate of sand so that we can accommodate both 2 projectiles. So we gave up on splitting ourselves into 2 groups. We came together as one whole team to help with the data collection. I feel that It seems to be faster than splitting into 2 groups.
After all the data collection, we are to share our results with the whole class. Our team realised that our results for full factorial and fractional factorial don't tally. Other teams have the same most significant factor for both fractional and full factorial. However, our team's most significant factor is different. This made us doubt what went wrong.
After discussion, we realised that we have been using the wrong subset of 4 runs. The 4 runs that we selected weren't based on the 'statistical orthogonality' taught to us. This is why we had a different significant factor for both full and fractional factorial. After knowing where we went wrong, we quickly change our data to the correct runs and yes we finally got the same significant factor.
With that, we take it as one of our learning opportunities so that the same thing will not happen again the next time when we are using it for our capstone project.
Overall, this was a pretty fun experiment. I enjoyed it a lot, especially during the group challenge although we did not win first place but it was fun seeing the whole team work together to make sure the projectile hit the target distance. I've learnt how to properly performed DOE data analysis after going through so much 'practice' (Pre-experiment, Practical, and case study in the blog). I hope we will not make the same mistake during our capstone project, and I agree this is a very good concept to know although it may seem to be confusing.
I feel that our team did a great job in terms of teamwork and not giving up although it may be tiring after going through so many runs of testing at one go and also most importantly, not giving up after knowing that our most significant factor turn out to be different.
All in all, this is a very fun and engaging practical and it will be the very last practical for this moduleš¢
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