Development of a Program for Searching the Optimum Condition using

the Design of Experiments

IKUO TANABE1, HIROMI ISOBE2

1School of Engineering,

Sanjo City University,

5002-5 Kamisugoro, Sanjo, Niigata,

JAPAN

2Mechanical Engineering,

Nagaoka University of Technology,

1603-1 Kamitomioka, Nagaoka, Niigata,

JAPAN

Abstract: - The Design of Experiments (DOE) is a method that is widely used due to its effectiveness in

selecting optimum conditions in the design stage of product development. On the other hand, fast, low-cost,

labor-saving, and energy-saving innovative development is also required in the industry. In previous research, a

program for quickly searching the optimum condition using the design of experiments is developed and

evaluated. Relationships between each parameter and the final property are firstly cleared for an algebraic

formula by using the design of experiments. Then the optimum conditions for each parameter were decided by

using these formulas in the program. However, when each parameter has several errors in the data, the search

accuracy becomes very low. In this research, the improvement for the searching accuracy using the law of error

propagation was developed and evaluated. Relationships between each parameter error and the final property

are firstly cleared for high-accuracy searching by using the law of error propagation and the previous results in

the previous research. And each parameter influence for the final property was cleared. It was found that the

large parameter effects could be improved for high-precision search by using high-precision instruments,

increasing the number of trials N, and taking measurements in an optimal environment. Relationships between

each parameter error and the final property were investigated and evaluated for the proposed method by using a

mathematical model. It is concluded from the result that (1) the proposed method is effective for clearing the

relationships between each parameter error and the final property, and (2) the proposed method is effective for

searching the optimum condition.

Key-Words: - Design of experiments, Optimum condition, Taylar’s law of error propagation, Searching

accuracy, High-precision search, Design stage.

Received: December 7, 2023. Revised: August 9, 2024. Accepted: September 13, 2024. Published: October 9, 2024.

1 Introduction

Design of experiments is often used in industry to

efficiently determine the optimal combination of

level values of control factors, [1], [2]. In addition,

quality engineering (static property) is a highly

robust design method that incorporates the concept

of error factors based on the design of experiments

and has been the subject of much research, [3], [4],

[5], [6], [7], [8], [9], [10], [11], [12]. However, most

of these studies are limited to obtaining the factor

effect diagram as an effective case study and using

it to perform a two-stage design. Further practical

research and development are desired in the

industrial world.

In contrast, as a new method, the authors used

the design of experiments to clarify the functional

relationship between the final property value and

each control factor by curve-fitting work and

additivity and developed a method to obtain the

optimal final property value using the functional

relationship. However, when errors were included

within the level value of each control factor, the

accuracy of the functional relationship equation was

reduced and the optimization of the final property

value could not be guaranteed.

In this study, we apply Taylor's law of error

propagation to the functional relationship between

the final property value and each control factor

obtained in the authors' previous study, [13] to

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formulate the functional relationship of the effect of

the error of each control factor on the final property

value. Then, we develop and evaluate the

technology to control the final property value with

high accuracy and to improve the accuracy of the

optimum final property value. In this paper,

orthogonal tables are also used, and if there are

interaction or synergy effects between control

factors, the proposed method will be affected, [12].

In this research, it is assumed that there are no

interaction or synergy effects between the control

factors.

This research proposes a method that allows

qualitative and quantitative consideration of the

influence of control factors to improve the accuracy

of the final property values, something that

conventional experimental design methods, [1], [2]

and quality engineering, [3], [4], [5], [6], [7], [8], [9],

[10], [11], [12] have failed to do. By introducing the

proposed method through WSEAS, many

researchers, engineers, scientists, postgraduate

students industrial engineers, and managers will be

able to contribute to rapid and accurate development

research.

2 Explanation of the Program to

Search the Optimal Condition using

the Design of Experiments

2.1 The Design of Experiments

In this section, the optimum conditions

identification program used for the proposed method

is explained.

The Design of Experiments (DOE) is commonly

used, based on a small number of experiments or

CAE analyses, to estimate an optimum parameter

combination for the generation of new designs. The

control factors (A to D) and their levels (A1 to A3, B1

to B3, C1 to C3, and D1 to D3,) are shown in Table 1

(Appendix). The orthogonal table is used to set up

the control factors and their levels in Table 1

(Appendix), as shown in Table 2 (Appendix). The

experiments are then carried out according to the

numbers in the orthogonal table. The results are also

given in Table 2 (Appendix) as final properties P.

From the principle of orthogonal tables, the

relationship between the influence E (=EAx, EBy, ECz,

EDw) of each control factor and the final property

values PAx∙By∙Cz∙Dw is shown in Equation (1).

EA1 = (PA1∙B1∙C1∙D1+PA1∙B2∙C2∙D2+PA1∙B3∙C3∙D3) / 3

EA2 = (PA2∙B1∙C2∙D3+PA2∙B2∙C3∙D1+PA2∙B3∙C1∙D2) / 3

EA3 = (P A3∙B1∙C3∙D2+PA3∙B2∙C1∙D3+PA3∙B3∙C2∙D1) / 3

EB1 = (PA1∙B1∙C1∙D1+PA2∙B1∙C2∙D3+PA3∙B1∙C3∙D2) / 3

EB2 = (PA1∙B2∙C2∙D2+PA2∙B2∙C3∙D1+PA3∙B2∙C1∙D3) / 3

EB3 = (PA1∙B3∙C3∙D3+PA2∙B3∙C1∙D2+PA3∙B3∙C2∙D1) / 3

EC1 = (PA1∙B1∙C1∙D1+PA2∙B3∙C1∙D2+PA3∙B2∙C1∙D3) / 3

EC2 = (PA1∙B2∙C2∙D2+PA2∙B1∙C2∙D3+PA3∙B3∙C2∙D1) / 3

EC3 = (PA1∙B3∙C3∙D3+PA2∙B2∙C3∙D1+PA3∙B1∙C3∙D2) / 3

ED1 = (PA1∙B1∙C1∙D1+PA2∙B2∙C3∙D1+PA3∙B3∙C2∙D1) / 3

ED2 = (PA1∙B2∙C2∙D2+PA2∙B3∙C1∙D2+PA3∙B1∙C3∙D2) / 3

ED3 = (PA1∙B3∙C3∙D3+PA2∙B1∙C2∙D3+PA3∙B2∙C1∙D3) / 3

Then, the final property values can be estimated

based on the additivity of the orthogonal sequences,

which is the most important feature of the design of

experiments. Therefore, the relationship between the

influence E of each control factor and all final

property values PAx∙By∙Cz∙Dw can be estimated by

Equation (2).

PAx∙By∙Cz∙Dw =EAx+EBy+ECz+ECw－(4－1) Pave

(2)

Where, Pave is the average of the final property

values (the average of the final property values

shown in Table 2, Appendix). In the design of

experiments, the additivity of the orthogonal

sequences can be used to estimate all combinations

results (81 different results in this case) from a small

number of experimental results. Therefore, the best

combination search can be performed among all

combination results, however it is not the optimal

condition.

2.2 The Program to Search the Optimal

Condition using the Design of

Experiments

The additivity (equation (2)) of the orthogonal

sequences was used for the optimal condition

identification program. In the previous explanations,

all control factors had three levels. By increasing the

number of these levels, the accuracy of the causal

relationship increases, however, it requires a long

working time and a large cost. The relationship

between the influence E (=EAx, EBy, ECz, EDw) of

each control factor and each level (Ax, By, Cz and

Dw) of each control factor was then displayed as

four curves (f (Ax), g (By), h (Cz ), i(Dw )) by curve

fitting, [13]. In this way, the influence of an infinite

number of level values can be processed quickly

Equation. (2) is accordingly rewritten as Equation

(3).

(1)

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PAx∙By∙Cyz∙Dw=f(Ax)+g(By)+h(Cz)+i(Dw)-(4-1)Pave (3)

This allows us to estimate the final properties for

an infinite number of combinations within the range

of levels in Table 1 (Appendix). This is the optimal

condition identification program used in the

proposed method. This is the author's original

technology, which allows us to search for optimal

conditions.

3 Explanation of the Algorithm for

Improving the Accuracy of the

Searching for Optimal Condition

3.1 Understanding the Relationship between

Control Factors Errors and Final

Property Values using Taylar’s Law of

Error Propagation

In this chapter, based on Equation (3) for the

relationship between the influence E of each control

factor and final property values P in the previous

section, Taylor's law of error propagation is used to

clarify the effect of the level value error of each

control factor on the final property value error. If the

error in the final property value PAx∙Bx∙Cz∙Dw is δ

PAx∙Bx∙Cz∙Dw, and the error in the level values Ax, By,

Cz, and Dw of each control factor is δAx, δBy, δCz

and δDw respectively, then by assuming that the

error is sufficiently smaller than the value of the

variable of interest, Equation (4) is obtained by

dropping the higher-order terms in the Taylor

expansion.

δPAx∙Bx∙Cz∙Dw=

(∂PAx∙Bx∙Cz∙Dw/∂Ax)δAx+(∂PAx∙Bx∙Cz∙Dw/∂By)δBy

+(∂PAx∙Bx∙Cz∙Dw/∂Cz)δCz+(∂PAx∙Bx∙Cz∙Dw/∂Dw)δDw (4)

Where ∂PAx∙Bx∙Cz∙Dw/∂Ax denotes the partial

differentiation of PAx∙Bx∙Cz∙Dw by Ax. Since the error

can be positive or negative, Equation (5) is obtained

by taking the absolute value of each term.

|δPAx∙Bx∙Cz∙Dw| ≦

|(∂PAx∙Bx∙Cz∙Dw/∂Ax)δAx|+|(∂PAx∙Bx∙Cz∙Dw/∂By)δBy|

+|(∂PAx∙Bx∙Cz∙Dw/∂Cz)δCz|+ |(∂PAx∙Bx∙Cz∙Dw/∂Dw)δDw| (5)

Equation (6) is obtained by substituting Equation (3)

into Equation (5).

+|(∂h(Cz)/∂Cz)δCz |+ (∂i(Dw)/∂Dw)δDw| (6)

From this Equation (6), it can be seen that the

errors δAx, δBy, δCz and δDw contained in each of

the level values Ax, By, Cz, and Dw of each control

factor propagate to the error δPAx∙Bx∙Cz∙Dw contained

in the final property value PAx∙Bx∙Cz∙Dw.

3.2 High Accuracy of Final Property Values

using Error Management of Control

Factors

In this section, as shown in Table 3 (Appendix), the

relationship between the impacts (∂f(Ax) /∂Ax,

∂g(By) /∂By, ∂h(Cz)/∂Cz, ∂i(Dw) /∂Dw) of the control

factor errors and the final property value error δ

PAx∙Bx∙Cz∙Dw are firstly calculated using Equation (6).

Then, referring to the impact, to reduce the error δ

PAx∙Bx∙Cz∙Dw of the final property value PAx∙Bx∙Cz∙Dw as

much as possible, measures are taken to reduce the

error δAx, δBy, δCz and δDw of the level value of

each control factor. The countermeasures include

(1) changing the equipment to increase the accuracy

of the level values of the control factors, and (2)

increasing the number of experiments (N values) to

increase the accuracy of the level values. Measures

to improve the accuracy of the final property values

can be determined using the error δ PAx∙Bx∙Cz∙Dw of

the final property values in Table 3 (Appendix) as a

guide, taking into account cost-effectiveness, time

gain and effort required. The proposed method can

contribute to the development. Studies of many

researchers, engineers, scientists, and industrial

engineers.

4 Explanation of the Algorithm

Evaluation of the Proposed

Method using a Mathematical

Model

4.1 Effect of Control Factor Errors on Final

Property Values

Two structural equations, Equations (7) and (8), are

used to evaluate the influence of the error of the

control factor on the final property value. These two

equations are prepared to evaluate individually the

influence of the error of the control factor on the

final property value for two problems of a

completely different nature, respectively, and there

is no interrelationship between Equations (7) and

(8).

PA∙B∙C = A2 + 9A－B2－3B +5C－46 (7)

PA∙B∙C = 6 e 0.1A + 2B2 + 5B + 6C (8)

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The procedure of searching the functional

relationship equation between the final property

value and each control factor is explained using the

authors' previous study, [13]. The previous Equation

(7) or Equation (8) is a functional relationship

between the final property value and each control

factor, respectively, which is used here for

calculation when carrying out Taylor's law of error

propagation. The control factors and their level

values in Table 4 (Appendix) were used. Table 5

(Appendix) shows the combinations of the level

values of each control factor according to the L9

orthogonal table. The final property values P(7) and

P(8), calculated using the structure Equation (7) and

Equation (8) respectively, are also shown. Figures 1

and Figures 2 in Appendix show the relationship

between the final property values P(7) and P(8) in

Table 5 (Appendix) and the control factor influence

E. The equations obtained by an automatic curve-

fitting operation, [13] are also shown in the figures.

Also shown in the titles of Figures 1 and Figures 2

in Appendix are the functional relationships between

the final property value PAx,By,Cz and the control

factors Ax, By, Cz in each figure, and the mean value

of the final property value P(7)ave' and P(8)ave' are also

shown. As Equation (8) contains an exponential

function, it was difficult to separate the influence of

each control factor contained in the exponential

function on the mean value of Eave, so the

coefficients of the exponential part are different

between Equation (8) and the calculated functional

equation, however, it is believed that this does not

have a significant effect. So far, this is the procedure

to search for the functional relationship between the

final property value and each control factor in the

author's previous work, [13]. Then, based on

Equations (7) and (8) of the structural equation,

errors ±ΔAS, ±ΔBT and ±ΔCM were added to the

level values of each control factor as shown in

Equations (9) and (10).

FL＝(AS±ΔAS)2+9×(AS±ΔAS)－(BT±ΔBT)2

－3×(BT±ΔBT)+5×(CM±ΔCM)－46 (9)

(Where L = 1 ~ 9; S, T and M = 1, 2 or 3)

FL＝ 6 e 0.1(AS±ΔAS) + 2×(BT±ΔB T)2 + 5×(BT±ΔB T)

+ 6×(CM±ΔCM) (10)

(Where L = 1 ~ 9; S, T and M = 1, 2 or 3)

Equations (9) and (10) respectively. The relation

of the two final property values P’ was calculated

and compared with the final property value P of the

structure Equation (7) and Equation (8) without the

error term, respectively, and the error of the

calculation was determined. The accuracy was

evaluated as the mean value and standard deviation

of 10 calculations.

Figure 3 (Appendix) shows the relationship

between the error in the final property value and the

error in the level value of each control factor. In the

structural equation of Figure 3(a) in Appendix, when

the error included in the level value of the control

factor is less than ±3 %, the estimation can be done

with good accuracy, however, when the error is ±5 %,

the mean value of the calculation error is 5.1 % with

a standard deviation of 2.2 %. In the structural

equation in Figure 3(b) in Appendix, even when the

error contained in the level value of the control

factor is ± 3 %, the mean value of the calculation

error is 7.5 % with a standard deviation of 2.8 %,

which is large. This is a major problem in the

authors' previous study, [13]. This is also a major

problem in the industry, where the design of

experiments and quality engineering are used in

research and development. Therefore, as a

countermeasure, the number of trials is increased (N-

value is increased) without any reason, and the

measurement equipment and facilities are renewed

as much as possible with high accuracy without any

clear reason. In this research, it is provides the

technology to carry out effective and efficient

countermeasures in the right place by clarifying the

basis and the reason for the countermeasures.

4.2 Evaluation of the Proposed Method

Two structural equations of the previous section are

used to evaluate the proposed technique.

Substituting Equations (7) and (8) into Equation (5)

respectively, we obtain Equations (11) and (12),

respectively.

+|(∂h(Cz)/∂Cz)δCz|

＝|(2Ax+9)δAx|+|(－2By－3)δBy|+|5δCz |

(11)

+|(∂h(Cz)∂Cz)δCz |

＝ | (0.6 e 0.1Ax)δAx|+|(4By+5)δBy|+|6δCz|

(12)

From these equations, by examining the

coefficients of the three terms on the right-hand side

of each equation, it is possible to understand how

the errors δAx, δBy, and δCz of the control factors A,

B, and C, respectively, propagate concerning the

error in the final property value |δPAx∙Bx∙Cz|.

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Table 6 in Appendix shows, based on Table 4

(Appendix), the control factors, their level values,

and the error of each level value. In Table 7

(Appendix) each level value including the error is

set according to the L9 orthogonal table. The errors

are the values +3 %, +6 % and +10 % of each level

value, which is considered to have the greatest

influence within the error range. Table 8 and Table

9 in Appendix show the results of the evaluation of

the proposed a technique using the structural

Equations (7) and (8) respectively. The final

property values P(7) and P(8) when the error is 0 %

are taken from the results in Table 5 (Appendix).

The final property values P(7)’ and P(8)’ when the

error is +3 %, +6 %, and +10 % are calculated using

the mathematical model Equations (9) and (10) and

Table 6 and Table 7 (Appendix). After calculating

the influence of the error |P(7)-P(7)’| and |P(8)- P(8)’|

from the difference between the values with and

without the error, the error of the final property

value errors |δ PAx∙Bx∙Cz(11)| and |δPAx∙Bx∙Cz(12)| from

Equations (11) and (12), and compared them. There

is a good correspondence between |P(7)-P(7)’| and

|δPAx∙Bx∙Cz(11)| in Table 8 (Appendix) and |P(8)-P(8)’|

and |δPAx∙Bx∙Cz(12)| in Table 9 (Appendix). The

proposed law of error propagation can be effectively

used to consider the effect of level error in the

control factors. |P(7)-P(7)’| in Table 8 (Appendix) and

|P(8)-P(8)’| in Table 9 (Appendix) are random data

generated by using computer-generated random

numbers, whereas |δPAx∙Bx∙Cz (11)| in Table 8

(Appendix) and |δPAx∙Bx∙Cz(12)| in Table 9 (Appendix)

use the maximum (fixed) value within each level

error range, so that |P(7)-P(7)’|≥|δPAx∙Bx∙Cz(11)| and |P(8)-

P(8)’| ≥ |δPAx∙Bx∙Cz(12)|. The same results as in Table 9

(Appendix) are obtained by replacing 6.0001 e 0.1A

in the first term of the right-hand side of the

calculated (estimated) structural Equation (8)’ in

Figure 2 (Appendix) with 6 e 0.1A in the first term of

the right-hand side of the original structural

Equation (8).

Figure 4 (Appendix) shows the effect of each

term on the final property value errors |δPAx∙Bx∙Cz(11)|

and |δPAx∙Bx∙Cz(12)|, focusing on the three terms on the

right-hand side of each of Equations (11) and (12)

(corresponding to the level values of the control

factors A, B, and C) applying the proposed law of

error propagation. The errors of the final property

values, |δPAx∙Bx∙Cz(11)| and |δPAx∙Bx∙Cz(12)|, are also

shown as black lines in the figure (the black line is

the sum of the effects of the level values of the

control factors A, B, and C). The vertical axis is a

logarithmic scale. The parameters are the error of

each level value of the control factors +3 %, +6 %,

and +10 %. First of all, both Equations (11) and (12)

of the proposed law of error propagation show that

the effect of the level error of the control factor A is

very large, and therefore it is effective in increasing

the accuracy of the level of the control factor A to

obtain accurate final property values. The control

factors A and B in Figure 4(a) in Appendix are both

quadratic equations in the structural Equation (7),

however, the value of the target level value of

control factor A is larger than that of control factor

B. This is because of the error δAx > δBy in Equation

(11). The control factor B in Figure 4(a) (Appendix)

has some influence on the error of the final property

values |δPAx∙Bx∙Cz(12)| compared to the control factor

B in Figure 4(b) (Appendix). This is because the

second term on the right-hand side of Equation (12),

|(4By+5)|, is larger than the second term on the right-

hand side of Equation (11), |(-2By-3)|. In both

structural equations, the level value of the control

factor C and the level value error have a very small

influence on the final property value. These trends

can also be seen in Table 8 and Table 9 in Appendix,

however, Figure 4 (Appendix) is physically easier to

understand. Thus, the proposed Equations (11) and

(12) of the law of propagation of error are useful for

considering and examining the relationships between

the final property values, errors, control factors, and

level values in the search for optimal conditions

using the design of experiments. Then, when

managing the level value error of each control factor,

we first select and focus on the control factor that is

effective in improving the accuracy of the final

property value from Figure 4 (Appendix), and then

increase the number of trials (increase the N value)

or consider the use of a measurement device with

high accuracy to reduce the level value error of that

control factor. In this case, the accuracy of the final

property value can be improved by a factor of N0.5

by increasing the N value, however, this requires a

great deal of time and effort. It is also possible to

improve the accuracy of the final property value

with high-precision equipment and measuring

instruments, but this requires a great deal of money

and there are limits to the accuracy of the equipment

and measuring instruments. Then, in this process, it

is possible to decide the most appropriate

countermeasure from among many options,

considering cost performance and life cycle

assessment (LCA) while using Figure 4 (Appendix).

Traditionally, the accuracy of the final property

values has been improved by trial and error over a

long period and at great expense and effort. However,

the proposed method can easily provide qualitative

and quantitative measures to improve the accuracy of

the final property values.

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As a practical application of the proposed

method, we are planning to improve its forced

cooling characteristics (heat transfer coefficient) to a

high degree of accuracy by using the previous study

on "forced cooling of strong alkaline water mist",

[13], which will be reported in the next paper.

5 Conclusion

By managing the error of each control factor, the

property value can be managed accurately, and the

technology to increase the accuracy of the optimum

final property value was developed and evaluated.

The following conclusions were obtained. (1) Using

Taylor's law of propagation of error, the effect of the

error of each control factor on the final property

value is formulated as a functional relationship, and

by efficiently managing each control factor, it is

possible to achieve high accuracy of the optimum

property value. (2) About the calculation accuracy

using two structural equations, without using the

proposed technique, the errors of the respective final

property values within the range of ±1 %, ±3 %,

±5 %, and ±10 % for the errors ΔAS, ΔB T and

ΔCM contained in the control factors A, B, and C

are ±0.9 to 2.0 %, ±3.4 to 7.5 %, ±5. 1 to 11.0 %

and ±8.1 to 18.34 %, respectively. By using the

proposed technique, the error of the final property

value can be improved preferentially and efficiently.

(3) By using the proposed law of error propagation,

the qualitative and quantitative effects of the level

error of the control factors on the final property

value can be understood in advance, which can be

effectively used for the error management and error

control of the final property value.

References:

[1] M. D. Morris, Design of experiments: an

introduction based on linear models,

Chapman and Hall/CRC, Vol. 1, 2010, pp.1-

376, ISBN-10: 1584889233, ISBN-13: 978-

1584889236.

[2] M. M. J. Sridhar, M. Manickam, V.

Kalaiyarasan, 2014, Optimization of

cylindrical grinding process parameters of

OHNS steel (AISI 0-1) rounds using design of

experiments concept, International Journal of

Engineering Trends and Technology, Vol. 17,

No. 3, 2014, pp.109–114.

[3] T. Bhavsar, A. M. Nokalje, Optimization of

cylindrical grinding process parameters for

EN353 steel using Taguchi technique,

International Journal for Research in Applied

Science & Engineering Technology, Vol. 8,

No. 11, 2020, pp. 225–231.

[4] N. Kumar, H. Tripathi, S. Gandotra, 2015,

Optimization of cylindrical grinding process

parameters on C40E steel using Taguchi

technique, International Journal of

Engineering Research and Applications, Vol.

5, No. 1, 2015, pp. 100–104.

[5] S. S. Sangale, A. D. Dongare, Optimization of

the parameter in cylindrical grinding of mild

steel rod (EN19) by Taguchi method,

International Journal of Creative and

Innovative Research In All Studies, Vol. 4, No.

4, 2019, pp. 66–73.

[6] K. Mekala, J. Chandradas, K. Chandrasekaran,

T. T. M. Kannan, E. Ramesh, R. N. Babu,

2014, Optimization of cylindrical grinding

parameters of austenitic stainless steel rods

(AISI 316) by Taguchi method, International

Journal of Mechanical Engineering and

Robotics Research, Vol. 3, No. 2, 2014, pp.

208–215.

[7] L. X. Hung, T. T. Hong, L. H. Ky, L. A. Tung,

N. T. T. Nga, V. N. Pi, Optimum dressing

parameters for maximum material removal

rate when internal cylindrical grinding using

Taguchi method, International Journal of

Mechanical Engineering and Technology, Vol.

9, No. 12, 2018, pp. 123–129.

[8] U. Koklu, Optimization of machining

parameters in interrupted cylindrical grinding

using the grey-based Taguchi method,

International Journal of Computer Integrated

Manufacturing, Vol. 26, No.8, 2013, pp 696–

702.

[9] M. ozdemir, M. T. Kaya, H. K. Akyildiz,

Analysis of surface roughness and cutting

forces in hard turning of 42CrMo4 steel using

Taguchi and RSM method, Mechanika, Vol.

26, No. 3, 2020, pp. 231–241.

[10] R. Rudrapati, A. Bandyopadhway, P. K. Pal,

Parametric optimization of cylindrical

grinding process through hybrid Taguchi

method and RSM approach using Genetic

algorithm, Iranian Journal of Mechanical

Engineering, Vol. 19, No. 1, 2018, pp. 34–62.

[11] G. Taguchi, S. Chowdhury, Y. Wu and H.

McGraw, Mahalanobis-Taguchi, Google

Books, ISBN 0071362630, 2001, pp.1‒190.

[12] G. Taguchi and R. Jugulum, The

Mahalanobis-Taguchi strategy: A pattern

technology system, Springer-Verlag London,

Ltd., United Kingdom, 2002.

[13] I. Tanabe, S. Takahashi and S. Takahashi,

Development of the program for searching the

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optimum condition using design of

experiments, Transactions of Japan Society of

Mechanical Engineers, Vo.84, No.862, 2018,

DOI: 10.1299/transjsme.18-00171 (in

Japanese).

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

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APPENDIX

Table 1. Control factors and these levels

Control factors

Name

A

B

C

D

Levels

A1

B1

C1

D1

A2

B2

C2

D2

A3

B3

C3

D3

Table 2. Orthogonal array and final properties

Control factors

Final properties P

No.

A

B

C

D

1

A1

B1

C1

D1

PA1∙B1∙C1∙D1

2

A1

B2

C2

D2

PA1∙B2∙C2∙D2

3

A1

B3

C3

D3

PA1∙B3∙C3∙D3

4

A2

B1

C2

D3

PA2∙B1∙C2∙D3

5

A2

B2

C3

D1

PA2∙B2∙C3∙D1

6

A2

B3

C1

D2

PA2∙B3∙C1∙D2

7

A3

B1

C3

D2

PA3∙B1∙C3∙D2

8

A3

B2

C1

D3

PA3∙B2∙C1∙D3

9

A3

B3

C2

D1

PA3∙B3∙C2∙D1

Table 3. Control of all control factor errors for searching the optimum conditions with high accuracy using the

Equation (6) with the law of propagation of error

Relationship between each control factor level Ax , By, Cz, Dw and the final property PAx∙Bx∙Cz∙Dw :

PAx∙Bx∙Cz∙Dw= f (Ax)+ g (By )+h(Cz )+ i (Dw )

－

(4

－

1) Pave

(3)

Relationship between each control factor’s level error δ Ax, δ By, δ Cz, δ Dw and the final property error δ PAx∙Bx∙Cz∙:

| δPAx∙Bx∙Cz∙Dw | ≦ | (∂ f (Ax) /∂Ax) δAx | + | (∂ g(By) /∂By) δBy | + | (∂ h(Cz) /∂Cz) δCz |+ | (∂ i(Dw) /∂Dw) δDw | (6)

Control

factors

Each

levels

Each control

factor’s level

error

Influence and impact of each

control factor’s level error for

the final property error δ

PAx∙Bx∙Cz∙Dw

Control of all control factor errors for

searching the optimum conditions with high

accuracy

A

Ax

δ Ax

∂ f (Ax) /∂Ax

Influences and impacts of each control

factor’s level error were referred. Then

effectively control factor’s level errors for the

small influences were improved.

B

By

δ By

∂ g(By) /∂By

C

Cz

δ Cz

∂ h(Cz) /∂Cz

D

Dw

δ Dw

∂ i(Dw) /∂Dw)

Table 4. Control factors and these levels

Control factors

Name

A

B

C

Level 1

40

8

5

Level 2

50

12

5.5

Level 3

60

16

6

Table 5. Orthogonal table and final properties (P(7)ave=2807.5, P(8)ave=1615.214）

L9

A

B

C

P(7)

P(8)

1

40

8

5

1851

526

2

40

12

5.5

1762

709

3

40

16

6

1640

956

4

50

8

5.5

2844

1091

5

50

12

6

2754

1274

6

50

16

5

2625

1512

7

60

8

6

4036

2625

8

60

12

5

3939

2799

9

60

16

5.5

3818

3046

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Table 6. Control factors with level errors.

Control factors

Name

A

B

C

Level 1

A1=40

B1=8

C1=5

Level 2

A2=50

B2=12

C2=5.5

Level 3

A3=60

B3=16

C3=6

Control factor level error

+3%

δA1=1.2

δB1=0.24

δC1=0.150

δA2=1.5

δB2=0.36

δC2=0.165

δA3=1.8

δB3=0.48

δC3=0.180

+6%

δA1=2.4

δB1=0.48

δC1=0.30

δA2=3.0

δB2=0.72

δC2=0.33

δA3=3.6

δB3=0.96

δC3=0.36

+10%

δA1=4.0

δB1=0.8

δC1=0.50

δA2=5.0

δB2=1.2

δC2=0.55

δA3=6.0

δB3=1.6

δC3=0.60

Table 7. Orthogonal table using the control factor with level errors

Table 8. The calculated the final property values P(7)’ using Table 6, Table 7 and Equation (7) . The final

property values P(7) is in Table 5. |P(7)－P(7)’| is the level error influence in the final property P(7)’. The

final property error |δPAx∙Bx∙Cz(11)| is calculated using Equation (11) by the proposed law of error

propagation. The proposed law of error propagation is very useful for grasping of the level error

influence in the final property values P(7)’; |P(7)－P(7)’| ≒|δPAx∙Bx∙Cz(11)|

Lever

error

Equations (7) & (11)

0 %

3 %

6 %

10 %

L9

P(7)

P(7)’

|P(7)－

P(7)’|

|δPAx∙Bx∙C

z(11)|

P(7)’

|P(7)－

P(7)’|

|δPAx∙Bx∙C

z(11)|

P(7)’

|P(7)－

P(7)’|

|δPAx∙Bx∙C

z(11)|

1

1851

19558

104

112

2063

212

224

2210

359

374

2

1762

1861

99

117

1963

201

235

2102

341

391

3

1640

1732

92

125

1827

187

249

1956

316

415

4

2844

3005

161

169

3172

328

338

3400

557

563

5

2754

2911

156

174

3072

318

348

3293

539

580

6

2625

2774

149

181

2928

303

362

3139

514

604

7

4036

4268

232

238

4506

470

475

4833

797

792

8

3939

4165

226

243

4398

459

485

4718

779

809

9

3818

4037

219

250

4262

444

500

4572

754

833

L9

A

B

C

1

A1+δA1

B1+δB1

C1+δC1

2

A1+δA1

B2+δB2

C2+δC2

3

A1+δA1

B3+δB3

C3+δC3

4

A2+δA2

B1+δB1

C2+δC2

5

A2+δA2

B2+δB2

C3+δC3

6

A2+δA2

B3+δB3

C1+δC1

7

A3+δA3

B1+δB1

C3+δC3

8

A3+δA3

B2+δB2

C1+δC1

9

A3+δA3

B3+δB3

C2+δC2

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Lever

error

Equations (8) & (12)

0 %

3 %

6 %

10 %

L9

P(8)

P(8)’

|P(8)－

P(8)’|

|δPAx∙Bx∙

Cz(12)|

P(8)’

|P(8)－

P(8)’|

|δPAx∙Bx∙

Cz(12)|

P(8)’

|P(8)－

P(8)’|

|δPAx∙Bx∙

Cz(12)|

1

526

577

52

54

634

109

112

721

195

199

2

709

771

62

64

839

130

132

939

231

233

3

956

1032

76

79

1115

159

161

1236

280

281

4

1091

1246

154

165

1423

332

346

1703

612

634

5

1274

1439

165

175

1627

353

367

1922

648

668

6

1512

1691

179

189

1894

381

394

2209

696

714

7

2625

3112

487

532

3694

1069

1128

4649

2024

2115

8

2799

3296

498

542

3888

1090

1148

4858

2059

2148

9

3046

3558

512

556

4165

1119

1176

5154

2109

2195

Table 9. The calculated the final property P(8)’ using Table 6, Table 7 and Equation (8) . The final

property P(8) is in Table 5. |P(8)－P(8)’| is the level error influence in the final property P(8)’. The

final property influence |δPAx∙Bx∙Cz(12)| is calculated using Equation (12) by the proposed law of

error propagation. The proposed law of error propagation is very useful for grasping of the level

error influence in the final property P(8)’; |P(8)－P(8)’| ≒|δPAx∙Bx∙Cz(12)|.

Fig. 1: Relationship between the control factors A, B, C and the final property P using the

experimental design

Structural equation (7) : PA∙B∙C = A2 + 9A－B2－3B +5C－46, and P(8)ave = 2808

Calculated equation (7)’ : PA∙B∙C = A2 + 9A－B2－3B +5C－46, and P(8)ave' = 2808

EA= f (Ax)= A2 + 9A－210

EB =g(By)=－B2－3B + 2998

EC = h (Cz)= 5C + 2780

(a) Control factor A (b) Control factor B (c) Control factor C

EA=f (Ax)= 47.12e

0.067A

EB =g (By)= 2B2+5B+1245

EC =h (Cz)= 6C+1582

(a) Control factor A (b) Control factor B (c) Control factor C

Fig. 2: Relationship between the control factors A, B, C and the final property P using the experimental

design Structural equation (8): PA∙B∙C = 6 e 0.1A + 2B2 + 5B + 6C , and P(8)ave=16158.214

Calculated equation (8)’: PA∙B∙C = 6.0001 e 0.1A + 2B2 + 5B + 6C－0.0001, and P(8)ave'= 16058.234

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Fig. 4: The final property errors |δPAx∙Bx∙Cz(11)| and |δPAx∙Bx∙Cz(12)| were influenced by 3 paragraphs the right side of

the Equations (11) and (12). The influences of 3 paragraphs were clearly shown in this figure.

Influences for the final property errors |δPAx∙Bx∙Cz(11)| in the equation (11)

Each level error ＋10

％Each level error ＋ 6

％Each level error ＋ 3

％

Control factor A

Control factor C

40 50 60

Control factor level Ax

8 12 16

Control factor level Bx

5 5.5 6

Control factor level Cx

(a) Equations (11)

Control factor B

|δPAx∙Bx∙Cz(11)|

Control factor A

Control factor B

(b) Equations (12)

Influences for the final property errors |δPAx∙Bx∙Cz(12)| in the equation (12)

|δPAx∙Bx∙Cz(12)|

40 50 60

Control factor level Ax

8 12 16

Control factor level Bx

5 5.5 6

Control factor level Cx

Control factor C

Error %＝Σ1~10 (ΣL1～L9│P(8)－P(8) ’│÷P ×100 )÷9 ) ÷10

Error %＝Σ1~10 (ΣL1～L9│P(7)－P(7) ’│÷P ×100 )÷9 ) ÷10

Final property error ±

%

Final property error ±%

Control factor’s level error ±%

(a) PA∙B∙C = A2 + 9A－B2－3B +5C－46 (b) PA∙B∙C = 6 e 0.1A + 2B2 + 5B + 6C

Fig. 3: Relationship between the control factor’s level error and the final property error. When the control factor’s

level error became gradually large, the final property error also became large. Therefore, the control of all

control factor errors for searching the optimum conditions with high accuracy was required.

( ) : Standard deviation σn

Control factor’s level error ±%

(0.5)

(1.3)

(2.2)

(5.3)

(1.1)

(2.8)

(4.8)

(9.2)

(0.5)

(1.3)

(2.2)

(5.3)

(1.1)

(2.8)

(4.8)

(9.2)

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