Analysis for Bounces of the Ball using Factorial Design and Taguchi Method

This work is only for understanding Taguchi Method and Factorial Design, in this context, daily subject is selected like “bounces of ball” for making it easy to understand.

Each ball is designed with specific materials, making it appropriate for a particular sport. The balls also have different numbers of bounce that caused by several factors. To measure the number of bounce for the ball, an experiment was conducted by dropping the ball from difference height onto the ground. The heights that used for this experiment are 1 meter and 2 meter. The experiment was comparing a basketball and a football ball. The surface for the balls to bounce also affects the number of ball to bounce before stop. There are two types of surface that being considered in this experiment, rough surface (on grass) and smooth surface (on cement).

To get the accurate results, the experiments are repeated three times for each run. The numbers of ball is counted from when the ball is released until the ball stops bouncing. The experiment is conducted by the same student drop the ball and the others student count the numbers of bounce for the ball. The data obtained from experiment is analyzed by using Factorial method (2 levels and 3 factors) and Taguchi method (4 orthogonal arrays).

The objectives of this experiment are:

  • To design of experiment that has at least 3 factors.
  • To evaluate the response by using full factorial design and Taguchi method.
  • To evaluate the analysis by using Minitab software.

Conditions or scopes of the experiment are:

  • Types of the ball which are basketball or football ball.
  • The height from the ball released which is 1 meter or 2 meter.
  • Types of surface which are rough or smooth.

Factorial Design

Factorial designs can be analyze in certain levels and certain factors depend on the experiment itself. Factorial design usually been analyze with 2 level and 2 or more factor. This method will involve ANOVA table, main effect, main interaction and regression model.

For the 22   design geometry and test matrix:

The combination of these two geometry and test matrix used to solve the problem is shown as below:

Table 1: Factorial Design table with 2-level, 2-factor

 

From table 1, all the information can be calculate to create the ANOVA table.

This is example of ANOVA table:

Table 2: ANOVA table

SS = Sum of squares F = Experiment Statistic
df = Degrees of freedom Fstat = Table statistic,
MS = Mean square
*SSA= indicates SS for factor A

To determine whether the factor is significant or not is based on this formula:

Following is the formula that will be used in factorial designs:

Taguchi Method

Taguchi method is a statistical design that been proposed by Dr. Genichi Taguchi to improve the quality in products and processes, This method is based on orthogonal array experiments which give much reduced variance for the experiment with optimum settings of control parameters.

Taguchi method can optimize problems in two categories which are static problems and dynamic problems:

The objective of Taguchi method is to identify the conditions which optimise process or product performance. To achieve this objective, signal-to-noise ratio will be used to maximize the performance of the system by minimizing the effect of noise and maximize the mean performance.

Usually the control factor that will be involved is three or more with three-levels which known as low (1), medium (2) and high (3). The noise factors will have three factors and have two-level which is low and high.

RESULTS

23 Factorial Design Manual calculation

The analysis determined the bounces of ball in terms of types of ball, height of released ball and types of surface using Factorial design. The experiment was conducted three times.

Response: Number of bounces ball

Analysis of Variance (ANOVA):

Sample calculation:

SST = Ʃ y2 – = (122 + 102 + 112 + 142 + 132 + ………..+ 52 + 52) – (180)2/24= 472.00

For factor A,

n, number of repetitive = 3,

k, number of factor = 3,

Contrast A = a + ab + ac + abc – (1) – b –c –bc

= 40 + 42 + 11 + 15 – 33 -22 – 10 – 7

= 36

SSA =(Contrast A)2/n22 =(36)2/(3)22 =54.00

SSE = 472.00 – 54.00 – 2.67 – 368.17 – 16.67 – 13.50 – 4.17 – 1.50

= 11.33

From examining the magnitude of the effects, factor A is clearly dominant, followed by factor AB interaction, factor BC, factor AC and factor C. The analysis of variance for the full factorial model is summarized in final table. We could compare the computed F ratios to a 5% upper critical value of the F distribution. Based on F comparison, we find that factor A, C, AB, AC and BC are significant.

Graphical analysis

Main effect plot of factor A

Effect A = 9.00 – 6.00 = 3.00

Based on graph plotted, the number of bounces of ball is increase if we use factor A (type of ball) at high level (basketball).

Main effect plot of factor B

Effect C = 7.16 – 7.83 = -0.67

Based on graph plotted, the number of bounces of ball is increase if we use factor B (height of the release ball) at low level (1 meter).

Main effect plot of factor C

Effect C = 3.58 – 11.41 = -7.83

Based on graph plotted, the number of bounces of ball is increase if we use factor C (type of surface) at low level (smooth surface, cement).

Two-way interaction plot of AB

Based on graph plotted, the number of bounces of ball is increase if we use factor A (type of ball) at high level (basketball) and factor B (height of the release ball) at high level (2 meter).

Two-way interaction plot of AC

Based on the graph plotted, the number of bounces of ball is increase if we use factor A (type of ball) at high level (basketball) and factor C(type of surface) at low level (smooth surface, cement).

Two-way interaction plot of BC

Based on the graph plotted, the number of bounces of ball is increase if we use factor B (1 meter) at low level (basketball) and factor C (type of surface) at low level (smooth surface, cement).

23 Factorial Design Minitab

—————   16/12/2015 3:28:05 PM   ————————————————————

Welcome to Minitab, press F1 for help.

Full Factorial Design

Factors:   3   Base Design:         3, 8

Runs:     24   Replicates:             3

Blocks:   1   Center pts (total):     0

All terms are free from aliasing.

Factorial Regression: NO OF BOUNCED versus A, B, C

Analysis of Variance

Source               DF   Adj SS   Adj MS F-Value P-Value

Model                 7 460.667   65.810   92.91   0.000

Linear               3 424.833 141.611   199.92   0.000

A                1   54.000   54.000   76.24   0.000

B                 1   2.667   2.667     3.76   0.070

C                 1 368.167 368.167   519.76   0.000

2-Way Interactions   3   34.333   11.444   16.16   0.000

A*B              1   16.667   16.667   23.53   0.000

A*C               1   13.500   13.500   19.06   0.000

B*C               1   4.167   4.167     5.88   0.027

3-Way Interactions   1   1.500   1.500     2.12   0.165

A*B*C             1   1.500   1.500     2.12   0.165

Error                 16   11.333   0.708

Total                 23 472.000

 

Model Summary

S   R-sq R-sq(adj) R-sq(pred)

0.841625 97.60%     96.55%     94.60%

 

Coded Coefficients

Term     Effect   Coef SE Coef T-Value P-Value   VIF

Constant           7.500   0.172   43.66   0.000

A         3.000   1.500   0.172     8.73   0.000 1.00

B         -0.667 -0.333   0.172   -1.94   0.070 1.00

C         -7.833 -3.917   0.172   -22.80   0.000 1.00

A*B       1.667   0.833   0.172     4.85   0.000 1.00

A*C       -1.500 -0.750   0.172   -4.37   0.000 1.00

B*C       0.833   0.417   0.172     2.43   0.027 1.00

A*B*C     -0.500 -0.250   0.172   -1.46   0.165 1.00

 

Regression Equation in Uncoded Units

NO OF BOUNCED = 7.500 + 1.500 A – 0.333 B – 3.917 C + 0.833 A*B – 0.750 A*C + 0.417 B*C

– 0.250 A*B*C

Alias Structure

Factor Name

A       A

B       B

C       C

 

Aliases

I

A

B

C

AB

AC

BC

ABC

Fits and Diagnostics for Unusual Observations

NO OF                 Std

Obs BOUNCED   Fit Resid Resid

11   9.000 7.333 1.667   2.43 R

R Large residual

Effects Plot for NO OF BOUNCED

Main Effects Plot for NO OF BOUNCED

Interaction Plot for NO OF BOUNCED

This manual calculation can be verified with Minitab software. This manual calculation can be accepted when the error between manual and Minitab less than 10%. By referring to Minitab, normal distribution will shows the significant factor for this experiment. Both manual and Minitab, shows that A, C, AB, AC and BC. The main effect and interaction effect plotted is also same between manual and Minitab. The error that can be calculated from this experiment is:

For sum of squares (SS) of factor B in manual and Minitab:

Therefore, the calculation of manual and Minitab can be accepted with only slightly different in values.

Taguchi Method Manual Calculation

The analysis determined the bounces of ball in terms of types of ball, height of released ball, types of surface and noise factors using Taguchi Method. The noise factors that being considered in this analysis are force, presence of wind, and angle of released ball.

There are many different types of S/N ratios but in this experiment, the larger-the-better was chosen for the analysis where the large number bounces of ball is desired.

Graphical analysis

Mean Analysis                                       S/N Ratio Analysis  

Factor A;                                                           Factor A;

A: (10+2.5)/2=6.25                                        A: (19.93+7.43)/2=13.68

A: (4.5+12.75)/2=8.62                                  A: (12.19+21.78)/2=16.98

Δ=8.62-6.25=2.37                                           Δ=16.98-13.68=3.3

Factor B;                                                          Factor B;

B: (10+4.5/2)=7.25                                         B: (19.93+12.19)/2=16.06

B: (2.5+12.75)/2=7.62                                   B: (7.43+21.78)/2=14.60

Δ=7.62-7.25=0.37                                            Δ=16.06-14.60=1.46

Factor C;                                                             Factor C;

C₁: (10+12.75)/2=11.37                                   C: (19.93+21.78)/2=20.85

C₂: (2.5+4.5)/2=3.5                                         C: (7.43+12.19)/2=9.81

Δ=11.37-3.5=7.87                                           Δ=20.85-9.81=11.04

Mean Analysis Table

 

S/N analysis table

MEAN ANALYSIS

Optimal factor setting: A,B2,C

S/N ANALYSIS

Optimal factor setting: A,B,C

Prediction for Mean Analysis

=(10+2.5+4.5+12.75)/4=7.4375

Mean predicted at optimum setting (A₂,B₂,C₁)

= Ť+(A- Ť)+(B- Ť)+(C- Ť)

=7.4375+(8.62-7.4375)+(7.62-7.4275)+(11.37-7.4375)

=12.735

Prediction for S/N ratio Analysis

= (19.93+7.43+12.19+21.78)/4

= 15.3325

S/N at optimum setting (A₂, B₁, C₁)

S/N= Ť + (A- Ť) + (B- Ť) + (C- Ť)

=15.3325 + (16.98-15.3325) + (16.06-15.3325) + (20.85-15.3325)

=23.225

The results of confirmation run is 21.78 were very close to the predicted value, which is 23.225 and also close to run number 4 which is same combination except for the different in the B level.

Taguchi Method Minitab

—————   16/12/2015 4:25:01 AM   ————————————————————

Welcome to Minitab, press F1 for help.

Taguchi Analysis: Array 1, Array 2, Array 3, Array 4 versus A, B, C

Linear Model Analysis: SN ratios versus A, B, C

Estimated Model Coefficients for SN ratios

Term         Coef

Constant 15.3378

A 1       -1.6537

B 1       0.7260

C 1       5.5241

S = *

Analysis of Variance for SN ratios

Source         DF   Seq SS   Adj SS   Adj MS F P

A               1   10.939   10.939   10.939 * *

B               1   2.109   2.109   2.109 * *

C               1 122.065 122.065 122.065 * *

Residual Error   0       *       *       *

Total           3 135.112

Linear Model Analysis: Means versus A, B, C

Estimated Model Coefficients for Means

Term         Coef

Constant   7.4375

A 1       -1.1875

B 1       -0.1875

C 1       3.9375

S = *

 

Analysis of Variance for Means

Source         DF   Seq SS   Adj SS   Adj MS F P

A               1   5.6406   5.6406   5.6406 * *

B               1   0.1406   0.1406   0.1406 * *

C               1 62.0156 62.0156 62.0156 * *

Residual Error   0       *       *       *

Total           3 67.7969

 

Response Table for Signal to Noise Ratios

Larger is better

 

Level       A       B       C

1     13.684 16.064 20.862

2     16.992 14.612   9.814

Delta   3.307   1.452 11.048

Rank       2       3       1

 

Response Table for Means

Level     A     B       C

1     6.250 7.250 11.375

2     8.625 7.625   3.500

Delta 2.375 0.375   7.875

Rank       2     3       1

Taguchi Analysis: Array 1, Array 2, Array 3, Array 4 versus A, B, C

Predicted values

S/N Ratio   Mean   StDev Ln(StDev)

23.2417 12.375 1.87375   0.699151

Factor levels for predictions

A B C

2 1 1

Main Effects Plot for Means

 

Main Effects Plot for SN ratios

This manual calculation can be verified with Minitab software. This manual calculation can be accepted when the error between manual and Minitab less than 10%. By referring to Minitab, optimal setting factor is at factor A level 2, factor B level 1 and factor C level 1. The optimal setting plotted is also same between manual and Minitab. The error that can be calculated from this experiment is:

For S/N ratio analysis of factor A level 2 in manual and Minitab:

Therefore, the calculation of manual and Minitab can be accepted with only slightly different in values.

Comparison between Factorial Design and Taguchi Method results.

Error Calculation of Factorial Design

Error calculation of Taguchi Method

From the calculation above, Factorial design has 0.0167 % error and Taguchi method has 0.0101% error.

Based on the experiment for the 23 factorial designs, we find that factor A (type of ball), factor C (type of surface), AB interaction, AC interaction and BC interaction were significant.

Based on experiment for Taguchi Method, we find that optimal setting for mean was at factor A level 2 (basketball), factor B level 2(2 meter) and factor C level 1 (smooth surface which is cement) and for S/N analysis was at factor A level 2 (basketball), factor B level 1   (1 meter) and factor C level 1 (smooth surface which is cement).

The results shown by manual calculation and Minitab software for both 23 factorial design and Taguchi Method were almost same. The error may cause by round off or significant value chosen. We conclude that our experiment was successful. From the experiment, we find that each ball is designed with specific materials, making it appropriate for a particular sport and type of surface do affect its number of bounces.

Based on the results shown, Taguchi Method gives better result than Factorial Designs. This is proven by error calculation between Taguchi Method and Factorial Design.

Thus, both factorial design and Taguchi method can provide an optimum factor and maximize the performance of the experiment. Regression model in factorial design and predicted value in optimum settings in Taguchi method will improve the performance of the experiment.

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