*3.10.1 Analysis of variance (ANOVA)*

As we mentioned earlier, DOE is used to develop or improve products or processes. The data obtained from the experiment should be analyzed. Variance analysis is used to interpret experimental data. Variance analysis was used for the first time by the British statistician Fisher. Experts usually work with samples. Because it is sometimes impossible to work with the whole population and sometimes it is very expensive. It should not be forgotten that; each individual case study forms part of the error. Sample statistics and assumptions allow the testing of hypotheses regarding experimental parameters. In order to analyze variance with sample data, we have four basic assumptions.


Total variance can be divided into two components such as inter-group variability and intra-group variability. The components of the model are tried to be

estimated using the least squares method on the sample data. Total squares are used to show piecemeal variability. After calculating the total squares and determining the appropriate degrees of freedom for each component of variability, the hypothesis is tested using the F distribution [38]. A typical ANOVA table is as **Table 5**.

impact of UCV [41]. The S

*Taguchi Method as a Robust Design Tool DOI: http://dx.doi.org/10.5772/intechopen.94908*

its simplest form, the S

(noise) [4]. TM uses S

them are follows.

• Smaller-Best

• Nominal-Best

• Larger-Best

**4. Experiment**

**4.1 Data**

**Table 6.**

**123**

Press Pressure (kg/cm<sup>2</sup>

*The process parameters and their levels.*

S*=* *=*

to identify CVs that move the mean to target. Different S

*S*

*S*

where *y* is the mean of observed data, *s*

observed data, and *yi* is the ith observed data.

analyze experiments in the Taguchi design.

*S*

depending on the goal of experiment. In all cases, the S

mized. Although Taguchi mentions over than 60 S

*=*

*=*

<sup>N</sup> ratio takes into account both mean and variability. In

*=*

*=*

<sup>N</sup> ratios can be choosen

(1)

(3)

<sup>N</sup> ratio should be maxi-

<sup>N</sup> ratios three of them such as

� � (2)

<sup>2</sup> is the variance of y, *n* is the number of

<sup>N</sup> ratio is the ratio of the mean (signal) to standard deviation

<sup>N</sup> ratios for two main purposes. The first pupose is to use the

*=*

*n* X*n i*¼1 *y*2 *i* !

*s*2

!

1 *y*2 *i*

*n* X*n i*¼1

In this section, a summary of Hamzaçebi [21] is given. Hamzaçebi [21] applied the TM to determine the effects of production factors such as adhesive ratio, press pressure, and pressing time on the thermal conductivity (TC) of oriented strand board (OSB). MINITAB 17 statistical software (State College, PA, USA) was used to

In the article of Ref [21], adhesive ratio, pressing time, and press pressure were considered as controllable factors. **Table 6** indicates the process parameters and their levels. As deduced from **Table 6**, there are 3 factors, which have 3 levels. After the factor definitions, suitable Taguchi orthogonal array was selected as L9. The L9

**Factors Level 1 Level 2 Level 3** Adhesive Ratio (%) 3% 4.5% 6% Pressing Time (minute) 3 5 7

) 35 40 45

design sheet and output of each experiment was given in **Table 7**.

<sup>N</sup> ratio in order to identify CVs that reduce variability and the second purpose is

smaller-best, larger-best and nominal-best are used frequently. The formulas of

*<sup>N</sup>* ¼ �<sup>10</sup> <sup>∗</sup> log <sup>1</sup>

*<sup>N</sup>* ¼ �<sup>10</sup> <sup>∗</sup> log <sup>1</sup>

*<sup>N</sup>* <sup>¼</sup> <sup>10</sup> <sup>∗</sup> log *<sup>y</sup>*

where

*SSF* = sum of squares of factor *SSe* = sum of squares of error *SST* = sum of squares of total *L* = number of levels of the factor *N* = total number of observations *F*comp = computed value of *F VF* = variance of factor *Ve* = variance of error.

After the experiment is set up, the ANOVA is completed, and the important factors and/or interactions are determined, some comments have already been made. However, if it will not be too expensive, it will be beneficial for the experimenter to learn the rest of the information. Here, we will talk about determining contribution percentages.

The rate of variability for each important factor and interaction observed in the experiment is reflected by the percentage of contribution. Percentage contribution is a function of the sum of squares of each significant factor. Percentage of contribution indicates the strength of factors and/or interaction in reducing variability. If the factor and/or interaction levels are fully controllable, the total variability can be reduced by the percentage of contribution. We know that the variance for a factor or interaction includes error variance. So we can arrange the variance for each factor to show the error variance as well.

The percentage contribution of the error provides an estimate of the adequacy of the experiment. If the error contribution percentage is 15% or less, it is assumed that no significant factor has been overlooked in the experiment. If the error contribution percentage is 50% or more, it is considered that the experimental conditions in which some important factors are ignored cannot be fully controlled or the measurement error is made [39].

In order to learn percantage contribution of the factors Pareto ANOVA can be used. Pareto ANOVA is a simplified ANOVA technique based on the Pareto principle. The Pareto ANOVA technique is a quick and easy method to analyze results of the parameter design and it does not need F-test. Pareto ANOVA does not use an F-test, but it identifies the important parameters and determines the percent contribution of each parameter [40, 21].

#### *3.10.2 <sup>S</sup>=<sup>N</sup> ratio*

Taguchi uses the statistical performance measure known as the <sup>S</sup>*=*<sup>N</sup> ratio used in electrical control theory to analyze the results [25]. <sup>S</sup>*=*<sup>N</sup> ratio is a performance criterion developed by Taguchi to select the best levels of CV that minimize the


**Table 5.** *ANOVA table.* *Taguchi Method as a Robust Design Tool DOI: http://dx.doi.org/10.5772/intechopen.94908*

impact of UCV [41]. The S*=*<sup>N</sup> ratio takes into account both mean and variability. In its simplest form, the S*=*<sup>N</sup> ratio is the ratio of the mean (signal) to standard deviation (noise) [4]. TM uses S*=*<sup>N</sup> ratios for two main purposes. The first pupose is to use the S*=*<sup>N</sup> ratio in order to identify CVs that reduce variability and the second purpose is to identify CVs that move the mean to target. Different S*=*<sup>N</sup> ratios can be choosen depending on the goal of experiment. In all cases, the S*=*<sup>N</sup> ratio should be maximized. Although Taguchi mentions over than 60 S*=*<sup>N</sup> ratios three of them such as smaller-best, larger-best and nominal-best are used frequently. The formulas of them are follows.

• Smaller-Best

estimated using the least squares method on the sample data. Total squares are used to show piecemeal variability. After calculating the total squares and determining the appropriate degrees of freedom for each component of variability, the hypothesis is tested using the F distribution [38]. A typical ANOVA table is as **Table 5**.

*Quality Control - Intelligent Manufacturing, Robust Design and Charts*

After the experiment is set up, the ANOVA is completed, and the important factors and/or interactions are determined, some comments have already been made. However, if it will not be too expensive, it will be beneficial for the experimenter to learn the rest of the information. Here, we will talk about determining

The rate of variability for each important factor and interaction observed in the experiment is reflected by the percentage of contribution. Percentage contribution is a function of the sum of squares of each significant factor. Percentage of contribution indicates the strength of factors and/or interaction in reducing variability. If the factor and/or interaction levels are fully controllable, the total variability can be reduced by the percentage of contribution. We know that the variance for a factor or interaction includes error variance. So we can arrange the variance for each

The percentage contribution of the error provides an estimate of the adequacy of the experiment. If the error contribution percentage is 15% or less, it is assumed that no significant factor has been overlooked in the experiment. If the error contribution percentage is 50% or more, it is considered that the experimental conditions in which some important factors are ignored cannot be fully controlled or the mea-

In order to learn percantage contribution of the factors Pareto ANOVA can be used. Pareto ANOVA is a simplified ANOVA technique based on the Pareto principle. The Pareto ANOVA technique is a quick and easy method to analyze results of the parameter design and it does not need F-test. Pareto ANOVA does not use an F-test, but it identifies the important parameters and determines the percent

*=*

L�1

N�L

<sup>N</sup> ratio is a performance

*=*

<sup>N</sup> ratio used in

*VF Ve*

Taguchi uses the statistical performance measure known as the <sup>S</sup>

Factor *SSF* L-1 VF <sup>¼</sup> SSF

Error *SSe* N-L Ve <sup>¼</sup> SSe

criterion developed by Taguchi to select the best levels of CV that minimize the

**Sources of variation Sum of squares (SS) Degrees of freedom Mean square variance** *Fcomp*

where

*SSF* = sum of squares of factor *SSe* = sum of squares of error *SST* = sum of squares of total *L* = number of levels of the factor *N* = total number of observations *F*comp = computed value of *F VF* = variance of factor *Ve* = variance of error.

contribution percentages.

factor to show the error variance as well.

contribution of each parameter [40, 21].

electrical control theory to analyze the results [25]. <sup>S</sup>

Total *SST* N-1

surement error is made [39].

*3.10.2 <sup>S</sup>=<sup>N</sup> ratio*

**Table 5.** *ANOVA table.*

**122**

$$\frac{S}{N} = -10 \ast \log\left(\frac{1}{n}\sum\_{i=1}^{n} \mathcal{Y}\_i^2\right) \tag{1}$$

• Nominal-Best

$$\frac{S}{N} = 10 \ast \log\left(\frac{\overline{\mathcal{V}}}{s^2}\right) \tag{2}$$

• Larger-Best

$$\frac{S}{N} = -\mathbf{10} \ast \log\left(\frac{1}{n} \sum\_{i=1}^{n} \frac{\mathbf{1}}{\mathbf{y}\_i^2}\right) \tag{3}$$

where *y* is the mean of observed data, *s* <sup>2</sup> is the variance of y, *n* is the number of observed data, and *yi* is the ith observed data.
