1.0 Introduction………………………………………………………………………………………………………………………1 2.0 Objective………………………………………………………………………………………………………………………….2 3.0 Methodology ……………………………………………………………………………………………………………………. 3
3.1 Data Information……………………………………………………………………………………………………………3 3.2 One-Half Fractional Factorial Design……………………………………………………………………………….3 3.3 General ANOVA Table ………………………………………………………………………………………………….4 3.4 Hypothesis Testing…………………………………………………………………………………………………………4 3.5 Design Outline ……………………………………………………………………………………………………………… 5 3.6 Summary Statistics…………………………………………………………………………………………………………6 3.7 Normality Plot……………………………………………………………………………………………………………….8 3.8 Model Adequacy Checking……………………………………………………………………………………………..8
4.0 Analysis……………………………………………………………………………………………………………………………9 4.1 Summary Statistic …………………………………………………………………………………………………………. 9 4.2 Analysis of Factor Effects Plot ……………………………………………………………………………………… 10 4.3 ANOVA Analysis………………………………………………………………………………………………………..12 4.4 Regression Model ……………………………………………………………………………………………………….. 13 4.5 Model Adequacy Checking……………………………………………………………………………………………14 4.6 Follow-up Analysis………………………………………………………………………………………………………16
5.0 Discussion and Conclusion ………………………………………………………………………………………………. 20 6.0 References………………………………………………………………………………………………………………………21 APPENDIX ……………………………………………………………………………………………………………………………. i
1.0 Introduction
Ceramic is one of the most common material that is being used in many industries these days. This is undoubtedly due to the properties of ceramic that has made this material the most desirable on. Some of the properties are resistance to chemical attack, thermal and electric insulation, high-temperature resistance corrosion resistance and the strength. The amazing properties of the ceramic has definitely allowed for variety of industrial use such as abrasives, high-temperature processing, chemical and petrochemical processing.
Recently, an experimenter from the Ceramic Division of Material Science and Engineering Laboratory wanted to study the performance in terms strength of ceramic that is being used in their company. Hence, he needed to investigate the effect of some machining factors on strength of ceramic. The type of ceramic that was chosen for the study was the sintered reaction-bonded silicon nitride.
Therefore, an experiment is designed to investigate the effect of machining factors on the ceramic strength in order to get the best conditions in which the ceramic has greatest strength. In the experiment, five factors were identified which are table speed, down feed rate, wheel grit, direction and batch. All these factors have two level of treatment each. The observed value is the ceramic strength responses.
A 25 factorial design experiment by assuming that all the five factors are fixed and completely randomized is to be conducted. However, due to resources and financial restrictions, only 16 runs are allowed can be made. The factors should not be neglected. Hence, due to the constraints, a 25−1 not replicated fractional factorial design is chosen. In this report, the experiment is designed, analysed and a conclusion is to be made.
2.0 Objective
The objective of this experiment is: –
2.1 To investigate the effect of machining factors with largest impact on ceramic strength.
2.2 To study the best combination of factors that brings out the maximum ceramic strength.
3.0 Methodology
3.1 Data Information
The table below describes the factors that are involved in this experiment.
Description
Table Speed Down Feed Rate Wheel Grit Direction
Low (−) Slow: 0.025 m/s
Slow: 0.05 mm Big: 140/170 Longitudinal Batch 1
High (+) Fast: 0.125 m/s Fast: 0.125 mm
Small: 80/100 Transverse Batch 2
Table 1: Factor Description and Level
3.2 One-Half Fractional Factorial Design
For the 25−1 fractional factorial design, 𝐼 = 𝐴𝐵𝐶𝐷𝐸 will be used as the generator. 𝑉
Alias Structure
[𝐴]→𝐴 + 𝐵𝐶𝐷𝐸
[𝐵] → 𝐵 + 𝐴𝐶𝐷𝐸
[𝐶] → 𝐶 + 𝐴𝐵𝐷𝐸 [𝐷] → 𝐷 + 𝐴𝐵𝐶𝐸
[𝐸] → 𝐸 + 𝐴𝐵𝐶𝐷 [𝐴𝐵] → 𝐴𝐵 + 𝐶𝐷𝐸 [𝐴𝐶] → 𝐴𝐶 + 𝐵𝐷𝐸 [𝐴𝐷] → 𝐴𝐷 + 𝐵𝐶𝐸
[𝐴𝐸]→𝐴𝐸+𝐵𝐶𝐷
[𝐵𝐶]→𝐵𝐶+𝐴𝐷𝐸
[𝐵𝐷] → 𝐵𝐷 + 𝐴𝐶𝐸 [𝐵𝐸]→𝐵𝐸+𝐴𝐶𝐷
[𝐶𝐷]→𝐶𝐷+𝐴𝐵𝐸 [𝐶𝐸]→𝐶𝐸+𝐴𝐵𝐷 [𝐷𝐸] → 𝐷𝐸 + 𝐴𝐵𝐶
Table 2: Alias Structure
3.3 General ANOVA Table
Main factor effects Interaction factor effects Error Total
Sum of Squares
𝑆𝑆𝐼𝑛𝑡𝑒𝑟𝑎𝑐𝑡𝑖𝑜𝑛
Table 3: General ANOVA Table
Degree of Freedom
− 𝑎𝑙𝑙 𝑜𝑡h𝑒𝑟 𝑑𝑓 𝑆𝑆𝑇𝑜𝑡𝑎𝑙 Total number of Data,
N−1, 𝑑𝑓 𝑇𝑜𝑡𝑎𝑙
Mean Squares
𝑀𝑆𝐼𝑛𝑡𝑒𝑟𝑎𝑐𝑡𝑖𝑜𝑛
𝑀𝑆𝑀𝑎𝑖𝑛 𝑀𝑆𝐸𝑟𝑟𝑜𝑟
𝑀𝑆𝐼𝑛𝑡𝑒𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑀𝑆𝐸𝑟𝑟𝑜𝑟
3.4 Hypothesis Testing Hypothesis Statement
𝐻0 ∶ Factor (Main / Interaction) effect is not statistically significant. vs
𝐻1 ∶ Factor (Main / Interaction) effect is statistically significant. Rejection Region
Reject 𝐻0 if p-value ≤ 𝛼 = 0.05. Factor and interaction effect are statistically significant when the p-value is less than or equal to the significance level.
3.5 Design Outline
25−1 Design with Generator 𝐼 = 𝐴𝐵𝐶𝐷𝐸 𝑉
A B 1- – 2+ –
3- + 4- – 5- – 6+ + 7+ – 8+ – 9- + 10- +
11- – 12+ + 13+ + 14+ – 15- + 16 + +
Basic Design
Treatment Combination
Ceramic Strength
C –+ — —
+– -+- –+ +-+ -++ +-+ -++ +++ +– -+- ++- ++- + + +
e 607.34 a 722.48 b 702.14 c 703.67 d 491.58
abe 638.04 ace 586.17 ade 434.41 bce 601.67 bde 417.66 cde 392.11 abc 669.26 abd 568.23 acd 410.37 bcd 428.51
abcde 446.73
Table 4: Factorial Design Model with Generator I=ABCDE
By using the table above, the plus and minus table for AB, AC, AD, AE, BC, BD, BE, CD, CE and DE are listed below.
Run AB AC AD AE BC BD BE CD CE DE 1+++-++-+– 2—-++++++ 3-+++—+++ 4+-++-++— 5++-++-+-+- 6+–+–++– 7-+-+-+–+- 8–+++—-+ 9–+-+-+-+- 10 – + – – – + + – – + 11 + – – – – – – + + + 12 + + – – + – – – – +
13 + – + – – + – + – – 14 – + + – – – + + – – 15 – – – + + + – + – – 16 + + + + + + + + + +
Table 5: Plus-Minus Table
3.6 Summary Statistics Factor Effect Estimate
Formula to estimate the factor effect:
Factor Effect Estimate = 𝐶𝑜𝑛𝑡𝑟𝑎𝑠𝑡 = 𝐶𝑜𝑛𝑡𝑟𝑎𝑠𝑡 = 𝐶𝑜𝑛𝑡𝑟𝑎𝑠𝑡
2𝑘−1−1(𝑛) 25−1−1(1) 8
In a full factorial design, the general formula used to estimate the factor effect is 𝐶𝑜𝑛𝑡𝑟𝑎𝑠𝑡. In 2𝑘−1(𝑛)
our case, we are running one-half fractional factorial design with single replicate. Therefore, 2𝑘−1 is used in the formula instead of 2𝑘 and 𝑛 = 1. We will only consider the large factor effects to run in the ANOVA table as it tends to have significant factor and interaction effects if the respective factor effects are relatively large. Therefore, the factor effects that are relatively small will be pooled into the error term.
Sum of Squares
Formula to calculate the sum of square for each factor: Sum of square = 𝐶𝑜𝑛𝑡𝑟𝑎𝑠𝑡2
The sum of squares indicates a variation or deviation from the mean. It is computed as a total of the squares of the differences from the mean. The tabulation of the total sum of squares will take into consideration of the sum of squares from the factors and the sum of squares from randomness or error. Generally, higher order interactions can be assumed to be insignificant. Thus, interactions between 3 or more factors can be assumed negligible and being dropped from the model. Consequently, the sum of squares for those higher order interactions will be pooled into the error sum of squares.
Mean Square
Formula to calculate the mean square for each source of variation: Mean Square = 𝑆𝑆
Mean square will be used to determine how significant are the factors. We can obtain the treatment mean square by dividing the treatment mean square by its degree of freedom. The treatment mean square will give the indication of the variation between sample means.
F-Statistic
Formula to calculate the F-statistic:
F-Statistic = 𝑀𝑆 𝑀𝑆𝐸𝑟𝑟𝑜𝑟
Since all the factors in the 2𝑘 factorial design are assumed to be fixed, we can obtain our F- statistic by dividing its particular mean square by the mean square error.
𝑃-value is used to determine the significance of the effects where we use 𝑝-value to determine whether should we reject the null hypothesis in a hypothesis test. The 𝑝-value will take the value ranging from 0 to 1. When the 𝑝-value is lesser or equal to the significance level (which is usually 𝛼 = 0.05), we will reject the null hypothesis.
3.7 Normality Plot
The normal probability plot is plotted by help from the software, Minitab. The method Minitab uses to draw the normal effects plot depends on the degrees of freedom for the error term.
If the error term has one or more degrees of freedom, Minitab plots the normal scores, probabilities or percentages versus the standardized effects. The line passes through the origin (or 0.50, 0 if probabilities or percentages are used) and has a slope of one because this is where the points are expected to lie if all effects are zero. Effects with p-values less than alpha are labelled significant on the graph. Minitab is used to label the graph Normal Probability Plot of the Standardized Effects. Since the design is in one replicate, which mean we need to refer on the plot to find out which factor effects contribute larger effects. Points that varies from the straight line in probability plot is consider have larger effects for the particular factor.
3.8 Model Adequacy Checking
After analyse the data, before the conclusions from the ANOVA are adopted, the adequacy of the underlying model should be checked. There are two assumptions need to be checked which are standardized residual normality assumptions and constant variances assumptions.
First, to conclude that error terms are normally distributed, we will use normal probability plot of standardized residuals to verify the assumption that the residuals are normally distributed. Most of the points lie along the straight line and the deviation of the points with the line is small indicate that the residuals are normally distributed. Besides, Anderson- Darling normality test also used to confirm the normality assumption of the error terms, which used to increase the reliability of the analysis.
Then, we use plot of residuals versus fitted values to check the assumption that the error terms have a constant variance. This plot should show a random pattern of residuals on both sides of 0. If a point lies far from the majority of points, it may be an outlier. There should not be any recognizable patterns in the residual plot. For instance, if the spread of residual values tends to increase as the fitted values increase, then this may violate the constant variance assumption. Hence, a structureless plot of residuals versus fitted values indicates constant variance of error terms is met.
4.0 Analysis
4.1 Summary Statistic
Note that Factor Effect Estimates = 𝐶𝑜𝑛𝑡𝑟𝑎𝑠𝑡 = 𝐶𝑜𝑛𝑡𝑟𝑎𝑠𝑡 = 𝐶𝑜𝑛𝑡𝑟𝑎𝑠𝑡 = 𝐶𝑜𝑛𝑡𝑟𝑎𝑠𝑡
2𝑘−1−1(𝑛) 25−1−1(1) 23 Sum of Squares = (𝐶𝑜𝑛𝑡𝑟𝑎𝑠𝑡)2 = (𝐶𝑜𝑛𝑡𝑟𝑎𝑠𝑡)2 = (𝐶𝑜𝑛𝑡𝑟𝑎𝑠𝑡)2 = (𝐶𝑜𝑛𝑡𝑟𝑎𝑠𝑡)2
2𝑘−1(𝑛) 25−1(1) 24
Percent Contribution = 𝑆𝑢𝑚 𝑜𝑓 𝑆𝑞𝑢𝑎𝑟𝑒𝑠 𝑋 100 𝑆𝑆𝑇
Sum of Squares
1072.726 962.7058 7369.793 168339.9 20456.87 2850.225 1557.684 1036.035 110.9336 16.83051 1246.267 121.1651 975.4691 2546.464 1531.744
Model Term
124.11 -343.39 -1641.17
Factor Effect Estimate
16.37625 15.51375 -42.9237 -205.146 -71.5138 26.69375 -19.7338 16.09375 5.26625 -2.05125 17.65125 5.50375 -15.6163 25.23125 19.56875
Percent Contribution
0.510349 0.458006 3.506172 80.08757 9.732335 1.355992 0.741066 0.492893 0.052777 0.008007 0.59291 0.057644 0.464078 1.211478 0.728726
E=ABCD -572.11
213.55 -157.87 128.75 42.13 -16.41 141.21 44.03 -124.93 201.85 156.55
Table 6: Summary Statistic
SST = (607.342 + 722.482 + ⋯ + 428.512 + 446.732) − (607.34+722.48+⋯+428.51+443.73)2 16(1)
= 5072628 − 8820.372 = 210194.8429 16
4.2 Analysis of Factor Effects Plot
In our analysis, a single replicate of a 25−1 design is to be conducted. Since it is an unreplicated 𝑉
factorial, there is no internal estimate of error. Hence, a normal probability plot of the estimates of the effects is constructed where the insignificant factor effects will be pooled into the estimate of error. Then, we will proceed with the ANOVA test.
Figure 1: Normal Plot of Factor Effects
Based on the normal probability plot, we can see that all of the effects that lie along the line are negligible, whereas the large effects are far from the line. From the analysis of the effect estimate plot, we can see that “𝐷 = 𝐴𝐵𝐶𝐸” and “𝐸 = 𝐴𝐵𝐶𝐷” seems to have a large effect compared to others. Since we are only interested in the main factor effects and the two factors interaction effects, thus, higher order interactions will be neglected.
Therefore, we will conduct the experiment with the main factor effect D and E and the rest with small factor effects will be pooled as the estimated error term. However, we will verify this using the Pareto plot.
Figure 2: Pareto Chart of Factor Effects
The Pareto chart above gives us another visualization of the importance of the factor effects to be studied. The reference line at 63.1 indicates that bars crossing that line have large effect. It can be clearly seen that main factors D and E has the highest importance and have the largest effect. This will be a support to our findings from the normal probability plot earlier.
In conclusion, we will consider, main factors D and E and all other terms will be pooled as the experiment estimates error.
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4.3 ANOVA Analysis
Figure 3: Analysis of Variance (Minitab Output)
From the ANOVA table, there is sufficient evidence to show that Factor D (Direction) and Factor E (Batch) are the two machining factors that have significant effect on the ceramic strength at 𝛼 = 0.05.
Figure 4: Model Summary (Minitab Output)
From the model summary above, the R-squared value indicates that the model explains 89.82% of the variability of ceramic strength around its mean.
4.4 Regression Model
𝑦̂ = 𝛽 + 𝛽 𝑥 + 𝛽 𝑥
where 𝛽0 is the average of all responses at the 16 runs in the design,
𝐸𝑓𝑓𝑒𝑐𝑡 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒 𝑜𝑓 𝐷 2
𝐸𝑓𝑓𝑒𝑐𝑡 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒 𝑜𝑓 𝐸 2
𝑥4, 𝑥5 are the coded variables that represent Factor D and Factor E respectively.
Therefore, we have the regression model as
𝑦̂ = 551.27 − 102.57𝑥4 − 35.76𝑥5
The regression model is verified with the finding using Minitab:
Figure 5: Regression Model (Minitab Output)
4.5 Model Adequacy Checking
By using the regression model, we have found earlier, we can obtain the residual by using the formula 𝑒 = 𝑦 − 𝑦̂. Hence, with the computations of residuals, we can check the model adequacy by checking the normality assumption through the normal probability plot of the standardized residuals and checking the constant variance assumption through the scatter plot of residuals versus fitted values.
Figure 6: Normality Plot of Standardized Residuals
Based on the normal probability plot of standardized residuals, we observed that most of the point lie along the straight line and they did not deviate much from the straight line. Hence, the normality assumption of standardized residuals is met.
Figure 7: Scatter Plot of Residuals vs Fitted Value
Based on the plot of residuals versus fitted value, we can see that the points are scattered randomly. This is an indication that the plot is structureless, thus, implying constant variance. Therefore, the assumption for constant variance is also met.
4.6 Follow-up Analysis
Figure 8: Factor Main Effects Plot for All Model Terms
Based on the main effect plots for the ceramic strength, it is obvious that the ceramic strength does not differ much when it is tested against Factor A (Table Speed), Factor B (Down Feed Rate) and Factor C (Wheel Grit). However, we can observe that for Factor D (Direction) and Factor E (Batch), there is a significant difference in the ceramic strength when the ceramic is tested at different levels within the factors respectively. Hence, it is verified that our results of analysis that Factor D (Direction) and Factor E (Batch) are the significant main factor effects.
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Figure 9: Factor Interaction Effects Plots for All Model Terms
The figure above is constructed to observe if there are any hidden significant interaction effect present. However, based on the observation of the plots, it is found that all the two factor interaction effect plots seem to show great parallelism. Therefore, there is no interaction between any two factors in our study.
Figure 10: Factor Main Effects Plot for Model Terms (D & E)
Based on the plot above, we can say that when the longitudinal direction is used the ceramic strength is very much greater than the transverse direction. Similarly, Batch 1 used in the study of ceramic strength has shown a greater strength in ceramic compared to using Batch 2.
Figure 11: Factor Interaction Effects Plots for Model Terms (D & E)
Focusing on the interaction plot of factors D and E above, even though there is no interaction we can use this plot to determine the best combination of machining factors that brings out the maximum ceramic strength since only these two main factors are significant.
Therefore, based on the plot, it can be observed that the best combination would be by using Batch 1 at a longitudinal direction in order to get the maximum strength of the ceramic. This result is also in line with our findings in the previous individual plots of the main factors.
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5.0 Discussion and Conclusion
From our experiment and analysis, it can be concluded that only Factor D (Direction) and Factor E (Batch) that play an important role in bringing out the maximum ceramic strength.
It can be also noted that the two significant main effects are actually qualitative factors. According to us, this find will be very beneficial for the Ceramic Division of Material Science and Engineering Laboratory as they will be able to use the maximum ceramic strength by considering only these two factors. They would not be required to be observing details of numerical values such as the table speed and down feed rate in order to get the best strength. Since, the Longitudinal direction and Batch 1 are more significant when it comes to the strength of ceramic, the laboratory could repeat the procedure they used in order to produce this batch for future production of ceramics.
As a conclusion, we have fulfilled the objectives listed for this study such that the direction and batch used was the effect that gave the most impact on ceramic strength and the combination of Batch 1 and Longitudinal direction gives the maximum strength of ceramic.
Suggestion
Based on the analysis that we have conducted; we would suggest to project the design into a full 22 factorial design. This is because three factors which are factor A, B and C is not important in bringing out the maximum strength of ceramic. The projection is as shown below:
D E Treatment Combination
680.45 722.48 702.14 666.93 607.34 620.80 610.55 638.04 491.58 475.52 478.76 568.23
703.67 642.14 692.98 669.26 585.19 586.17 601.67 608.31 444.72 410.37 428.51 491.47 392.11 343.22 385.52 446.73
+ + de 442.90 434.41 417.66 510.84 Table 7
Projection Image:
6.0 References
i. Berger, P.D. and Maurer, R.E. (2002). Experiment design with Applications in Management, Engineering and the Sciences. Duxbury.
ii. Engineering Statistic Handbook, Nist Sematech. Retrieved from
https://www.itl.nist.gov/div898/handbook/pri/section4/pri471.htm
iii. Mason R.L., Gunst R.F. and Hess J.L. (2003). Statistical Design and Analysis of Experiments – With Applications to Engineering and Science, 2nd edition. John Wiley & Sons, Inc.
iv. Montgomery, D. C. (2013). Design and Analysis of Experiments, 8th edition. John Wiley & Sons, Inc. Retrieved from https://www.academia.edu/40234870/D_esign_and_Analysis_of_Experiments_Eighth _Edition
i. Create Factorial Design Model
Insert all Name, Type, Factor Level for all factors that we want to test.
Choose Summary Table, Alias Table, Design Table, Defining Relation as Printed Results
Stat → DOE → Factorial → Create Factorial Design
Choose 1 fraction design, and OK 2
Output of Creating Fractional Factorial Design Model with generator I=ABCDE
ii. Factor Effect Plot
Choose all the terms that we are going to test their factor effects.
Stat → DOE → Factorial → Analyze Factorial Design
Choose Pareto and Normal for Effects Plots
iii. Factorial Analysis (ANOVA, Regression Model)
Model Terms D and E are chosen. Other terms will be pulled as the error term.
Stat → DOE → Factorial → Analyze Factorial Design
Store the Fits, Residuals, and Standardized Residuals value in worksheet.
We want to show that Standardized Normal Residuals Plot, and Scatter Plot of Residuals versus Fits