# LEGO Blocks - Design Of Experiments using Minitab

Final Version of our Lego Car

A project where we use LEGO blocks to design a race car with the objective being maximum distance.

**The Problem: **

The problem that we are solving is identifying the set of factors that truly contribute to the distance that our Lego car travels.

**Overview: **

The project involves working with a set number of lego1 pieces, combining them together to model a car, where the factor that is of most importance is the distance this car travels from a fixed ramp that has an inclination of 30 degrees. This maximum distance travelled(y) is our response variable. The project is about conducting a study based on design of experiments where we take several factors after a screening experiment and identify which factors truly contribute to the response variable which is the distance.

**The Solution:**

**The Solution:**

The solution is the approach that we are taking, we are performing a Design of Experiment(DOE) study where we identify the number of factors after a screening experiment, and use Minitab to run a DOE study, after we conduct a series of experiments based on the number of factors and number of replicates we choose, this shall be discussed in detail in the report. We ran two rounds of experiment, the first to identify the significant factors from the 6 factors we chose for the experiment from our screening. In the 2^{nd} round we chose 3 factors for our experiment based on the significant factors identified from the first round and the results that we obtained form our conclusion for the final design.

**Experimental Setup:**

The experiment was conducted in one of the study rooms at W.P. Carey where we setup the ramp as shown in the picture below, the ramp was perfectly positioned at 30 degrees(+/- 1) and set strong against the wall so that there was minimum vibration and that we could reduce as much noise factors as possible. The ramp was 31 inches long and 18inches high. We are a team of 4, therefore, we divided the experiment so that the operator changing the factors of the car was fixed, the operator loading the car in the ramp was fixed, one operator was taking the recordings on laptop and the last operator was responsible for accurately measuring the distance. The measuring tape was fixed on the floor with a tape to ensure consistency. All the measurements were taken on a single sitting to ensure that we reduced noise factors that could arise from resetting of the ramp as well as changing the operator and resetting the measuring tape and layout of experiments.

*Picture1: Fixed Ramp angled at 30 degrees and measuring tape fixed along horizontally*

**Factor Selection:**

The factor selection was one of the most crucial part of our experiment, to select the factors we built an initial model of the car, brainstormed on all possible factors that could influence the distance of the initial design. We performed several screening tests and identified the above 6 factors as the most critical factors in the distance the car travelled and conducted our DOE study on these factors. The pictures of different factor size experimented with have been added below.

**Response Variable:**

The response variable is the horizontal distance the car travels once it lands from the inclined ramp as shown in the picture above. As shown in picture 1, response variable is measured with a fixed measurement tape on the floor.

Factor Seletion

## The Experiment:

Once the number of factors was fixed and the preliminary design identified, we used minitab to create a 2-level (high/low)factorial design to run our DOE study.

The high and the low for the factors have been mentioned in the picture above. We create a factorial design and start recording the distance(response variable measurement) on minitab as shown below:

Picture 4 above shows us our 2-level factorial design where we measure the distance based on randomized combinations, we show two possible combinations below:

Just a side note on How to create Factorial Design in Minitab (Design of Experiment Study in Minitab)

**Combination type 1**: Long wheel base, with Spoiler, High Weight, short axle base(front), big wheel size(rear) and a small windshield

**Combination type 2:** Short Wheel base, without spoiler, low weight, short axle base(front), big wheel size(rear) and a small windshield.

Noise can be a major problem in result identification:

To block the noise in the experiment we ensured the following:

1. Fix the ramp to the wall and the floor â€“ ensure it is sturdy

2. Fix the measuring tape to the floor

3. Conduct the entire experiment in one sitting in the same room in the same ramp position

4. One operator loads the car on the ramp

5. One operator responsible for changing combinations of the factors for DOE study

6. One operator records measurement on computer

7. One operator ensures accuracy in the measurement

8. The ramp angles were recalibrated after every 10 trials(128 + 16 total)

9. Measurement of each combination was taken twice, in a randomized order generated by minitab, to ensure accuracy in the measurement and to reduce noise factors

We conduct the experiment in two rounds:

a. Round 1: Experiment with 6 factors

b. Round 2: Experiment with 3 factors

Round 1: Experiment with 6 factors:

As mentioned above the first round of the experiment consisted of 6 factors analyzed. We have 6 factors and 2 replicates, therefore, totalling the number of times the experiment had to be run to 128. After running the experiment for 128 times and collecting the response variable measurement we identified the order of significance of the factors, to narrow down to smaller number of significant factors we repeat the above experiment, but with only the most significant 3 factors from this experiment. The Data and Graphs from this experiment will be discussed in detail in the following sections.

Round 2: Experiment with 3 factors:

The round 2 of the experiment consisted of 3 most significant factors which were studied in 2 replicates, therefore we had to run the experiment 23 * 2 = 16 times. This was like the previous experiment with 16 randomized combinations of the 3 most significant factors while keeping the other factors constant as per the previous experiment.

Results and Interpretations:

The results from the DOE study can be studied in terms of Data as well as interpreted by Graphs, we will analyze them both as follows for both Round 1 of the experiment and Round 2 of the experiment.

The table above shows the significant factors from the Round 1 of the experiment, based on this and combining it with the pareto chart, which we will explain in the next section of this report, we narrow down to 3 factors for our round 2 experiment.

The** ANOVA(Analysis of Variance)** table for the round two of the experiment can be seen in Appendix 3, and the most significant factors from this round, with p value less than 0.05 can be seen as follows:

*TABLE 2: ANOVA table for experiment round 2(3 factors)*

The factors chosen for round 2 of the experiment are:

The regression equation from the round 2 of the experiment is as follows:

### We can see from the above regression equation that weight has the highest impact on our response variable, followed by a two way interaction of weight and wheel size(rear)

**Graphical Analysis: **

**Graphical Analysis:**

### Now we conduct a graphical analysis of the two experiments which includes pareto analysis, residual analysis, normal plot analysis, main effects analysis and the interaction effect analysis.

**Graphical analysis of 1st Experiment(6 factors):**

**Graphical analysis of 1st Experiment(6 factors):**

*Picture 4: Pareto analysis of experiment 1*

From the picture we can see that the factors that are beyond the significance line are 10, which is similar as shown in the data analysis section above based on ANOVA table. The factors that are most significant here have crossed the red dotted line of significance.

We take three topmost factors from here for the experiment 2, which as we can see is Wheel size(rear), Weight, and Axle base(Front)

The normal plot of the standardized effects shows us the most significant factors in Red square blocks, the farther they are from the line the more significant they are.

**Residual Analysis for Experiment 1:**

From the residual plot analysis we can see that the normal probability plot is along the line, and shows some outliers. The histogram shows that the distribution is normal. The versus fit plot shows us that the distribution is equally distributed on both sides with no observable patterns, and the dots far from the centerline shows potential outliers in the data collected.

The versus order plot is randomly distributed above and below the centerline which validates our assumption that the variables are independent of eachother.

The main effects plot as shown above shows when increased from low to high, the factors wheel base, axle base(front) causes a negative effect on the response variable. However, the other factors like spoiler, weight of the car, wheel size(rear) and windshield has a positive effect on the response variable.

The main effects plot as shown above shows when increased from low to high, the factors wheel base, axle base(front) causes a negative effect on the response variable. However, the other factors like spoiler, weight of the car, wheel size(rear) and windshield has a positive effect on the response variable.

The interaction plot when thoroughly studied shows us that the two way interaction of axle base(front) and weight of the car, gives us the highest distance(response variable), that is the combination of high weight and short axle base(front). Also, from the Squares that are in red we can see that the basis for our decision to choose the levels for the factors that remain constant in the 2nd round of DOE study.

Graphical analysis of 2nd Experiment(3 factors):

The Graphical anlaysis of the 2nd experiment is similar to the first experiment. First we will analyze the Pareto chart as shown below for the 3 factors.

From the pareto chart we can see that after the 2^{nd} experiment weight alone seems to be the most significant factor impacting the response variable(y).

The normal plot of the standardised effect shows the same conclusion that factor A, which is weight, is the most significant factor in our DOE study and has the most significant impact on the distance our lego car travels

The normal probability plot seems to violate the assumption that residuals are normally distributed, and shows outliers. Therefore we explore the other residual plots to check for problems with the model. The possible cause of this could be the outliers as seen from the histogram, which otherwise is normally distributed. The versus fit plot is evenly distributed verifying that the residuals are randomly distributed and have constant variance, and the versus order is randomly distributed above and below the centerline which verifies our assumption that the residuals are independent from one another.

The main effects plot for the 3 factors shows us that all the factors seem to positively affect the distance(y) when they increase from low to high. The factor that is most significantly affecting the distance is the weight when at a high level.

On analyzing the interaction plot above we can see that the combination of Big rear wheel size and high weight gives us the best two way combination and the highest effect on the response variable (y), which is the distance.

**COST ANALYSIS: **

In this section of the project we analyze the best design, the cost of our best design as well as the amount of cost we have potentially saved using DOE study to analyse factors that influence the response variable. The detailed costings for each factor that we have used can be found in the cost study pictures above

Based on our cost analysis we can see that using DOE study we can identify which parts we can remove based on their effect on distance and their cost, for example in design 1 we can remove the expensive front axle which does not contribute to the distance and replace it with a shorter axle, which is cheaper and goes much farther in terms of our response variable. Similarly we can see that in design 3, we reach the best possible distance of 129 in our study with a savings of $4000 by using a small windshield(compared to the big and expensive) and by saving on the front axle base.

According to our team the best design in terms of farthest distance and cost savings therefore, would be design 3, which saves us $4000 and travels a distance of 129cm (y). Alternatively if we want to remove a slightly less significant factor compared to weight, axle base and the rear wheel size, we can consider removing the spoilers, this would save us an additional $2000, making the total savings to $6000, however, the maximum distance travelled would be reduced to 114.

Although the big block of weight costs us $10400, and increases the cost of the car dramatically it has a direct effect on the distance(y), as we can see without the weight our design 5 travels only 88cm â€“ which is not worth the saving as we will be compromising on the most important factor for our design which is the distance covered.

**Conclusions and Recommendation: **

The conclusion of this project would be that DOE study proved to be very useful in terms of analyzing the effects of different factors on the response variable, which in our case was the distance. Using DOE study our team was able to come up with a design, which travelled a considerable distance, and which ultimately saved us on costs which without this experimentations could have been missed.

Using DOE study we concluded that the most important factors were weight of the car, small front axle base, as well as the rear wheel size. Our final design using DOE study which travelled the farthese distance consists of a short wheel base, with a heavy weight, with big rear wheels , with spoilers as well as a small windshield. This design saved us $4000 and helped us achieve the maximum distance of 129.

**Additional Data (Appendix): ANOVA TABLE (EXPERIMENT ROUND 1)**

**Appendix: ANOVA TABLE FOR EXPERIMENT ROUND 2(3-Factor)**

*A special thanks to my team mates for their contribution to this*