# Hybrid Integration of Taguchi Parametric Design, Grey Relational Analysis, and Principal Component Analysis Optimization for Plastic Gear Production

Authors: Nik Mizamzul Mehat; Shahrul Kamaruddin; Abdul Rahim Othman
Journal: Chinese Journal of Engineering
 (2014)

---

## Abstract

The identification of optimal processing parameters is an important practice in the plastic injection moulding industry because of the significant effect of such parameters on plastic part quality and cost. However, the optimization design of injection moulding process parameters can be difficult because more than one quality characteristic is used in the evaluation. This study systematically develops a hybrid optimization method for multiple quality characteristics by integrating the Taguchi parameter design, grey relational analysis, and principal component analysis. A plastic gear is used to demonstrate the efficiency and validity of the proposed hybrid optimization method in controlling all influential injection moulding processing parameters during plastic gear manufacturing. To minimize the shrinkage behaviour in tooth thickness, addendum circle, and dedendum circle of moulded gear, the optimal combination of different process parameters is determined. The case study demonstrates that the proposed optimization method can produce plastic-moulded gear with minimum shrinkage behaviour of 1.8%, 1.53%, and 2.42% in tooth thickness, addendum circle, and dedendum circle, respectively; these values are less than the values in the main experiment. Therefore, shrinkage-related defects that lead to severe failure in plastic gears can be effectively minimized while satisfying the demand of the global plastic gear industry.


---

## Body

## 1. Introduction

Injection moulding is a complex process because of its requirements for numerous delicate adjustments. The quality of an injection-moulded part significantly depends on the selection of appropriate materials, parts, mould designs, and processing parameters. A review of previous works shows that the setting of processing parameters significantly affects the quality of plastic parts [1, 2]. The selection of injection moulding process parameters previously involves a trial-and-error method [3]. However, obtaining an optimal parameter setting for complex-manufacturing processes is difficult because the trial-and-error method is a “one change at a time” test [4]. This tuning exercise is repeated until the quality of the moulded part is found satisfactory, thus incurring high production costs and long setup times [5]. Moreover, the adjustments and modifications of processing parameters rely heavily on the experience and intuition of the moulding personnel [6]. Nevertheless, the growing demand in the industry for expert moulding personnel exceeds the supply; moreover, amateur moulding personnel require more than 10 years of experience to be considered experts [7].In a fiercely competitive market, the trial-and-error approach cannot meet the challenges of globalization, particularly at the point where the disadvantages outweigh the advantages. Thus, the interest in substituting the trial-and-error method with a fast and reliable optimization approach such as the Taguchi method continues to increase. The Taguchi method considerably reduces the time and effort needed to obtain optimal process conditions and to determine the important factors that affect plastic part quality. The application of the Taguchi method has attracted considerable attention in the literature over the past 20 years and has been widely applied to various fields, such as manufacturing systems [8], mechanical component design [9], and parameter optimization [10, 11]. Li et al. [12] exploited the Taguchi method to eliminate the weld line of the right door of a copy machine by optimizing the melt temperature, injection speed, and injection pressure. Wu and Liang [13] further improved the weld line strength of an injection-moulded part by optimizing the mould temperature, packing pressure, injection acceleration, and packing time. By focusing on warpage in the thin-walled plastic part, many researchers used the experimental design of the Taguchi method to optimize the injection moulding parameters; packing pressure was found to be the most influential factor on the warpage problem [14–16]. Oktem et al. [17] improved the shrinkage and warpage in a plastic part by approximately 2.17% and 0.7%, respectively, by determining the optimal packing time, packing pressure, injection time, and cooling time by using the Taguchi method.A number of studies have shown that the Taguchi method is a robust experimental design for obtaining the best combination of factors or levels with the lowest cost solution to achieve the quality requirements of plastic parts. Nevertheless, the Taguchi method exhibits a number of limitations. The optimization design of injection moulding process parameters can be difficult because more than one quality characteristic is used in the evaluation. Problems arise when the optimal process parameters contradict one another because of different mechanisms [18]. To address this problem, many studies devised new experiment design methodologies that optimize multiple quality characteristics simultaneously while providing accurate results by integrating the Taguchi method with other techniques. One such technique is grey relational analysis (GRA). This technique was first proposed by Deng [19] to optimize the multiresponse problem by using the grey relational coefficient and grey relational grade. The grey relational coefficient can express the relationship between the desired and actual experiment results, whereas the grey relational grade is simultaneously computed to correspond to each quality characteristic.Few works have studied the optimization of injection moulding process parameters by using the integration of Taguchi method and GRA. Fung [20] adopted the integration method to improve the wear volume losses in both the parallel and perpendicular directions and inspected the extent in which process parameters influence each quality characteristic by using the comparability sequence with a large value of grey relational grade. Another two studies conducted by Fung et al. [21] and Yang [22] used the same procedure to optimize the processing parameters for multiple quality characteristics. The former focused on the concurrent improvement of yield stress and elongation of the PC/ABS blend, whereas the latter examined the effects of processing parameters on the mechanical and tribological properties of PC composites, including ultimate stress, surface roughness, and friction coefficient.Another technique that can be applied with the Taguchi method in addressing multiple quality characteristic problems is principal component analysis (PCA). By using PCA, a set of original responses is transformed into a set of uncorrelated components to find the optimal factor or level combination. PCA does nothing when responses are uncorrelated; thus, the best results are obtained when responses or quality characteristics are highly correlated (either positively or negatively) [23]. The application of PCA involves a series of steps that are capable of solving the weakness of the standalone Taguchi method, which requires engineering judgment to handle multiple quality characteristics because the judgment of an engineer increases uncertainty during the decision-making process [24]. Fung and Kang [25] applied PCA to the Taguchi method to improve friction properties, including friction coefficients and surface roughness, in different sliding directions (P-type and AP-type). The authors extracted the principal components and the coefficient of determination to establish a comprehensive index and obtain the final optimal parameter setting.Although both techniques are widely integrated with the Taguchi method to overcome multiple quality characteristic problems, both integration techniques have several limitations. In combining the Taguchi method and GRA, a problem arises when calculating the value of the grey relational grade in GRA. An engineering judgment or subjective estimation is required to determine the weighting values for each quality characteristic, thus resulting in an increase in uncertainty during the decision-making process. This approach cannot objectively reveal the relative importance of multiple quality characteristic performance. For the integration of the Taguchi method and PCA, the primary problem is the difficulty of interpretation because the original variables are substituted and the principal components are heavily affected by the scaling of the variables. In both methods, no scientific analysis has been performed in terms of performance under optimal process conditions to verify the accuracy of the optimization results. Therefore, the effectiveness and reliability of the integration of the Taguchi method with GRA or PCA alone remain uncertain. Further validation experiments on the optimization results are needed to prove the effectiveness of the integrated approach.This study develops a hybrid optimization method by integrating the robust parameter design of Taguchi, GRA, and PCA to solve problems with multiple quality characteristics. An injection-moulded plastic gear is used to demonstrate the efficiency and validity of the proposed hybrid optimization method in controlling all influential injection moulding processing parameters during plastic gear manufacturing. By focusing on minimizing the shrinkage behaviour in the tooth thickness, addendum circle, and dedendum circle of moulded gear, the optimal combination of four process parameters, namely, melt temperature, packing pressure, packing time, and cooling time, at three different levels is determined by using the proposed hybrid optimization method.


## 2. Taguchi/GRA/PCA Hybrid Method for the Optimization of Injection Moulding Process Parameters

### 2.1. Taguchi Method

The Taguchi method is used to design experiments based on the orthogonal arrays (OAs). Taguchi OAs are highly fractional orthogonal designs that can be used to study the whole parameter space with a small number of experiments. In analyzing the results, the Taguchi method uses a statistical measure of performance known as signal-to-noise (S/N) ratio. The S/N ratio is a measure of performance to develop products or processes that are insensitive to noise factors in a controlled manner [26]. Noise factors are uncontrollable factors that influence product or process uncertainty. These factors include humidity and weather. Depending on the objective, three different methods can be used to calculate the S/N ratio in the Taguchi method:(1)
smaller-the-better quality characteristic:(1)
η
i
j
=
-
10
log
(
1
n
∑
j
=
1
n
y
i
j
2
)
,
(2)
bigger-the-better quality characteristic:(2)
η
i
j
=
-
10
log
(
1
n
∑
j
=
1
n
1
y
i
j
2
)
,
(3)
nominal-the-better quality characteristic:(3)
η
i
j
=
-
10
log
(
1
n
s
∑
j
=
1
n
y
i
j
2
)
,
wherey
i
j is the ith test at the jth trial, n is the total number of tests, and s is the standard deviation.


### 2.2. GRA

In the analysis of the processing parameters of the plastic gear in the case study, an appropriate mathematic model is established to study the relationship between target values and the quality characteristics of the plastic gear obtained from the experiment. The primary concern is to analyze the differences among the quality characteristics of plastic gear because of various processing parameters and to understand the relationship between quality characteristics and target values. GRA is a method that measures the correlation degree among factors based on the similarity or difference among factors. GRA is characterized by small data requirements and multifactor analysis. The procedure of the GRA is presented below.Grey relational generation involves data preprocessing and calculation according to the quality characteristics. The computing method of the grey relational generation is as follows.(1)
The-larger-the-better (the higher the target value, the better):(4)
x
i
*
(
k
)
=
x
i
(
O
)
(
k
)
-
min
x
i
(
O
)
(
k
)
max
x
i
(
O
)
(
k
)
-
min
x
i
(
O
)
(
k
)
.
(2)
The-smaller-the-better (the smaller the target value, the better):(5)
x
i
*
(
k
)
=
max
x
i
(
O
)
(
k
)
-
x
i
(
O
)
(
k
)
max
x
i
(
O
)
(
k
)
-
min
x
i
(
O
)
(
k
)
.
(3)
The-nominal-the-better characteristic (if the target is a specific value, set the target value as OB):(6)
x
i
*
(
k
)
=
1
-
|
x
i
(
O
)
(
k
)
-
OB
|
max
{
max
x
i
(
O
)
(
k
)
-
OB
,
OB
-
min
x
i
(
O
)
(
k
)
}
.
x
i
(
O
)
(
k
) is the measurement of the quality characteristic, max
x
i
(
O
)
(
k
) is the largest value of x
i
(
O
)
(
k
), and min
x
i
(
O
)
(
k
) is the smallest value of x
i
(
O
)
(
k
).We determine the difference sequenceΔ
0
i
(
k
) and the minimum value Δ
min and maximum value Δ
max in the difference sequence, including the parameter values required to set the reference sequence:
(7)
Δ
0
i
(
k
)
=
|
x
0
*
(
k
)
-
x
i
*
(
k
)
|
,
Δ
max
=
max
∀
j
∈
i
max
∀
k
|
x
0
*
(
k
)
-
x
j
*
(
k
)
|
,
Δ
min
=
min
∀
j
∈
i
min
∀
k
|
x
0
*
(
k
)
-
x
j
*
(
k
)
|
.
We set the identification coefficient Z, which generally has a value of 0.5.After the data preprocessing, a grey relational coefficient is calculated to express the relationship between the ideal and actual normalized experimental results. The grey relational coefficient can be expressed as follows:(8)
ε
i
(
k
)
(
k
)
=
Δ
min
+
ψ
Δ
max
Δ
0
i
+
ψ
Δ
max
,
0
<
ε
[
x
0
*
(
k
)
,
x
i
*
(
k
)
]
≤
1
.
The average of the grey relational coefficient is then calculated to obtain the grey relational grade. The grey relational grade is defined as follows:
(9)
γ
i
=
1
n
∑
k
=
1
n
ε
i
(
k
)
.
However, the effect of each factor on the system is not exactly the same in real applications. Thus, (9) can be modified as follows:
(10)
γ
i
=
∑
k
=
1
n
w
k
·
ε
i
(
k
)
,
where w
k represents the normalized weighting value of factor k. Given the same weights, (9) and (10) are equal. In GRA, the grey relational grade is used to show the relationship among sequences. If two sequences are identical, the value of the grey relational grade is equal to one. The grey relational grade also indicates the degree of influence that the comparability sequence can exert over the reference sequence. Therefore, if a particular comparability sequence is more important than the other comparability sequences to the reference sequence, the grey relational grade for that comparability sequence and reference sequence will be higher than other grey relational grades [27]. In this study, the corresponding weighting values w
k are obtained from PCA.


### 2.3. PCA

PCA was developed by Pearson [28] and Hotelling [29]. This approach explains the structure of variance-covariance by the linear combinations of each quality characteristic. The procedures are described as follows.(1
) Original Multiple Quality Characteristics Array. Consider
(11)
x
i
(
j
)
,
i
=
1,2
,
…
,
m
,
j
=
1,2
,
…
,
n
x
i
=
[
x
1
(
1
)
x
1
(
2
)
x
1
(
3
)
⋯
⋯
x
1
(
n
)
x
2
(
1
)
x
2
(
2
)
x
2
(
3
)
⋯
⋯
x
2
(
n
)
⋮
⋮
⋮
⋯
⋯
⋮
x
m
(
1
)
x
m
(
2
)
x
m
(
3
)
⋯
⋯
x
m
(
n
)
]
,
where m is the number of experiments and n is the number of quality characteristics. In this paper, x is the grey relational coefficient of each quality characteristic and m
=
9 and n
=
3.(2
) Correlation Coefficient Array. The correlation coefficient array is evaluated as follows:
(12)
R
j
l
=
(
Cov
(
x
i
(
j
)
,
x
i
(
l
)
)
σ
x
i
(
j
)
X
σ
x
j
(
l
)
)
,
j
=
1,2
,
3
,
…
,
n
;
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
l
=
1,2
,
3
,
…
,
n
,
where Cov
(
x
i
(
j
)
,
x
i
(
l
)
) are the covariance of sequences x
i
(
j
) and x
i
(
l
), respectively; σ
x
i
(
j
) is the standard deviation of sequence x
i
(
j
); σ
x
j
(
l
) is the standard deviation of sequence x
i
(
l
).(3
) Determining the Eigenvalues and Eigenvectors. The eigenvalues and eigenvectors are determined from the correlation coefficient array
(13)
(
R
-
λ
k
I
m
)
V
i
k
=
0
,
where λ
k is an eigenvalue, ∑
k
=
1
n
λ
k
=
n, and k
=
1
,
2
,
…
,
n; V
i
k
=
[
a
k
1
,
a
k
2
,
…
,
a
k
n
]
T correspond to eigenvalue λ
k.(4
) Principal Components. The uncorrelated principal component is formulated as follows:
(14)
Y
m
k
=
∑
i
=
1
n
X
m
(
i
)
·
V
i
k
,
where Y
m
1 is the first principal component, Y
m
2 is the second principal component, and so on. The entire technical line of the hybrid Taguchi/GRA/PCA process optimization method for plastic injection moulding is summarized and illustrated in Figure 1.Figure 1
Technical line of hybrid Taguchi/GRA/PCA process optimization method.


## 2.1. Taguchi Method

The Taguchi method is used to design experiments based on the orthogonal arrays (OAs). Taguchi OAs are highly fractional orthogonal designs that can be used to study the whole parameter space with a small number of experiments. In analyzing the results, the Taguchi method uses a statistical measure of performance known as signal-to-noise (S/N) ratio. The S/N ratio is a measure of performance to develop products or processes that are insensitive to noise factors in a controlled manner [26]. Noise factors are uncontrollable factors that influence product or process uncertainty. These factors include humidity and weather. Depending on the objective, three different methods can be used to calculate the S/N ratio in the Taguchi method:(1)
smaller-the-better quality characteristic:(1)
η
i
j
=
-
10
log
(
1
n
∑
j
=
1
n
y
i
j
2
)
,
(2)
bigger-the-better quality characteristic:(2)
η
i
j
=
-
10
log
(
1
n
∑
j
=
1
n
1
y
i
j
2
)
,
(3)
nominal-the-better quality characteristic:(3)
η
i
j
=
-
10
log
(
1
n
s
∑
j
=
1
n
y
i
j
2
)
,
wherey
i
j is the ith test at the jth trial, n is the total number of tests, and s is the standard deviation.


## 2.2. GRA

In the analysis of the processing parameters of the plastic gear in the case study, an appropriate mathematic model is established to study the relationship between target values and the quality characteristics of the plastic gear obtained from the experiment. The primary concern is to analyze the differences among the quality characteristics of plastic gear because of various processing parameters and to understand the relationship between quality characteristics and target values. GRA is a method that measures the correlation degree among factors based on the similarity or difference among factors. GRA is characterized by small data requirements and multifactor analysis. The procedure of the GRA is presented below.Grey relational generation involves data preprocessing and calculation according to the quality characteristics. The computing method of the grey relational generation is as follows.(1)
The-larger-the-better (the higher the target value, the better):(4)
x
i
*
(
k
)
=
x
i
(
O
)
(
k
)
-
min
x
i
(
O
)
(
k
)
max
x
i
(
O
)
(
k
)
-
min
x
i
(
O
)
(
k
)
.
(2)
The-smaller-the-better (the smaller the target value, the better):(5)
x
i
*
(
k
)
=
max
x
i
(
O
)
(
k
)
-
x
i
(
O
)
(
k
)
max
x
i
(
O
)
(
k
)
-
min
x
i
(
O
)
(
k
)
.
(3)
The-nominal-the-better characteristic (if the target is a specific value, set the target value as OB):(6)
x
i
*
(
k
)
=
1
-
|
x
i
(
O
)
(
k
)
-
OB
|
max
{
max
x
i
(
O
)
(
k
)
-
OB
,
OB
-
min
x
i
(
O
)
(
k
)
}
.
x
i
(
O
)
(
k
) is the measurement of the quality characteristic, max
x
i
(
O
)
(
k
) is the largest value of x
i
(
O
)
(
k
), and min
x
i
(
O
)
(
k
) is the smallest value of x
i
(
O
)
(
k
).We determine the difference sequenceΔ
0
i
(
k
) and the minimum value Δ
min and maximum value Δ
max in the difference sequence, including the parameter values required to set the reference sequence:
(7)
Δ
0
i
(
k
)
=
|
x
0
*
(
k
)
-
x
i
*
(
k
)
|
,
Δ
max
=
max
∀
j
∈
i
max
∀
k
|
x
0
*
(
k
)
-
x
j
*
(
k
)
|
,
Δ
min
=
min
∀
j
∈
i
min
∀
k
|
x
0
*
(
k
)
-
x
j
*
(
k
)
|
.
We set the identification coefficient Z, which generally has a value of 0.5.After the data preprocessing, a grey relational coefficient is calculated to express the relationship between the ideal and actual normalized experimental results. The grey relational coefficient can be expressed as follows:(8)
ε
i
(
k
)
(
k
)
=
Δ
min
+
ψ
Δ
max
Δ
0
i
+
ψ
Δ
max
,
0
<
ε
[
x
0
*
(
k
)
,
x
i
*
(
k
)
]
≤
1
.
The average of the grey relational coefficient is then calculated to obtain the grey relational grade. The grey relational grade is defined as follows:
(9)
γ
i
=
1
n
∑
k
=
1
n
ε
i
(
k
)
.
However, the effect of each factor on the system is not exactly the same in real applications. Thus, (9) can be modified as follows:
(10)
γ
i
=
∑
k
=
1
n
w
k
·
ε
i
(
k
)
,
where w
k represents the normalized weighting value of factor k. Given the same weights, (9) and (10) are equal. In GRA, the grey relational grade is used to show the relationship among sequences. If two sequences are identical, the value of the grey relational grade is equal to one. The grey relational grade also indicates the degree of influence that the comparability sequence can exert over the reference sequence. Therefore, if a particular comparability sequence is more important than the other comparability sequences to the reference sequence, the grey relational grade for that comparability sequence and reference sequence will be higher than other grey relational grades [27]. In this study, the corresponding weighting values w
k are obtained from PCA.


## 2.3. PCA

PCA was developed by Pearson [28] and Hotelling [29]. This approach explains the structure of variance-covariance by the linear combinations of each quality characteristic. The procedures are described as follows.(1
) Original Multiple Quality Characteristics Array. Consider
(11)
x
i
(
j
)
,
i
=
1,2
,
…
,
m
,
j
=
1,2
,
…
,
n
x
i
=
[
x
1
(
1
)
x
1
(
2
)
x
1
(
3
)
⋯
⋯
x
1
(
n
)
x
2
(
1
)
x
2
(
2
)
x
2
(
3
)
⋯
⋯
x
2
(
n
)
⋮
⋮
⋮
⋯
⋯
⋮
x
m
(
1
)
x
m
(
2
)
x
m
(
3
)
⋯
⋯
x
m
(
n
)
]
,
where m is the number of experiments and n is the number of quality characteristics. In this paper, x is the grey relational coefficient of each quality characteristic and m
=
9 and n
=
3.(2
) Correlation Coefficient Array. The correlation coefficient array is evaluated as follows:
(12)
R
j
l
=
(
Cov
(
x
i
(
j
)
,
x
i
(
l
)
)
σ
x
i
(
j
)
X
σ
x
j
(
l
)
)
,
j
=
1,2
,
3
,
…
,
n
;
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
l
=
1,2
,
3
,
…
,
n
,
where Cov
(
x
i
(
j
)
,
x
i
(
l
)
) are the covariance of sequences x
i
(
j
) and x
i
(
l
), respectively; σ
x
i
(
j
) is the standard deviation of sequence x
i
(
j
); σ
x
j
(
l
) is the standard deviation of sequence x
i
(
l
).(3
) Determining the Eigenvalues and Eigenvectors. The eigenvalues and eigenvectors are determined from the correlation coefficient array
(13)
(
R
-
λ
k
I
m
)
V
i
k
=
0
,
where λ
k is an eigenvalue, ∑
k
=
1
n
λ
k
=
n, and k
=
1
,
2
,
…
,
n; V
i
k
=
[
a
k
1
,
a
k
2
,
…
,
a
k
n
]
T correspond to eigenvalue λ
k.(4
) Principal Components. The uncorrelated principal component is formulated as follows:
(14)
Y
m
k
=
∑
i
=
1
n
X
m
(
i
)
·
V
i
k
,
where Y
m
1 is the first principal component, Y
m
2 is the second principal component, and so on. The entire technical line of the hybrid Taguchi/GRA/PCA process optimization method for plastic injection moulding is summarized and illustrated in Figure 1.Figure 1
Technical line of hybrid Taguchi/GRA/PCA process optimization method.


## 3. Implementation of the Proposed Hybrid Taguchi/GRA/PCA Optimization Procedures for Plastic Gear

### 3.1. Determination of Quality Characteristics

In this paper, a plastic gear is used to demonstrate the efficiency and validity of the proposed hybrid optimization method. With the capability to run without lubrication and corrosion, plastic gears have been widely used in the automotive industry, office machines, household utensils, food and textile machinery, and a host of other applications [30]. Plastic-moulded gears offer some material properties that are not achievable with metal-based gears, including unique advantages in weight, noise, modulus, self-lubrication, chemical resistance, and low cost. However, the practicability of injection moulding in producing low-cost plastic-moulded gears remains restricted by the occurrence of shrinkage in the final parts. As one of the most common and prominent defects of plastic, severe shrinkage leads to the deflection or warpage of the moulded part and negatively affects the dimensional stability and accuracy of the involute profile, concentricity, roundness, tooth spacing uniformity, and gear size. Sever shrinkage defect is often forgotten and regarded lightly in many works related to plastic-moulded gears. However, this defect critically affects the quality of the final moulded gear, noise, and vibration and shortens the service life of the gear because of different damage mechanisms, such as tooth fatigue, creep, excessive wear, and plastic deformation. Hence, shrinkage minimization is the optimization objective of this paper. This paper focuses on the optimization of processing parameters via the integration of the Taguchi/GRA/PCA to minimize the shrinkage behaviour of plastic injection-moulded gears. To inspect shrinkage behaviour on the dimensional stability of the moulded gear, three quality characteristics are selected: tooth thickness, addendum circle, and dedendum circle.


### 3.2. Selection of Taguchi OA

In this study, the crystalline thermoplastic polypropylene (PP) is specified for the gear. The PP used in this study is obtained from Propylene Malaysia Sdn. Bhd., Malaysia. The general properties of PP are shown in Table1. The spur gear design used is compliant with the American Gears Manufacturers Association standards. The detailed geometry and specifications of the gear are shown in Figure 2.Table 1
General properties of PP.
Properties
Density (g/cm3)
0.90 to 0.91
Raw density (g/cm3)
0.805
Melt flow index (g/10 min)
10.78
Modulus of elasticity (MPa)
4100
Charpy impact toughness (kJ/m2)
1.4 to 1.8Figure 2
Geometry and specification of the spur gear: module = 1.5; pressure angle = 20°; number of teeth = 20; face width = 10 mm.The PP gears are injected by using a Battenfeld TM750/210 injection moulding machine. The experiment is conducted with four controllable three-level processing parameters: melt temperature, packing pressure, packing time, and cooling time. Other processing parameters, such as mould temperature (25°C), injection pressure (80 bar), and stroke distance (60 mm), are kept constant during the experiment. The selected processing parameters and their levels are shown in Table2. The L9 (34) OA (Table 3) is conducted to study the four processing parameters. Nine trials of PP gears with five repetitions are produced by using the OA.Table 2
Processing parameters and levels studied.
Column
Factors
Level 1
Level 2
Level 3
A
Melt temperature (°C)
200
220
240
B
Packing pressure (%)
60
80
100
C
Packing time (s)
5
10
15
D
Cooling time (s)
30
40
50Table 3
OA L9 (34) of the experimental runs.
Factors/trial no.
A
B
C
D
Melt temperature (°C)
Packing pressure (%)
Packing time (s)
Cooling time (s)
1
200
60
5
30
2
200
80
10
40
3
200
100
15
50
4
220
60
10
50
5
220
80
15
30
6
220
100
5
40
7
240
60
15
40
8
240
80
5
50
9
240
100
10
30


### 3.3. Shrinkage Measurement

Rax Vision DC 3000 Mitutoyo profile projector is used to inspect the accuracy of the specific profile of the involute gear teeth after moulding. In this study, the profile projector is used to measure 2D tooth thickness, as well as addendum and dedendum circles, by using the coordinates of selected points along the gear profile. With large magnifications and micrometer readouts, the profile projector can ensure fairly accurate measurements compared with a Vernier caliper and micrometer. In the process of addendum and dedendum circle measurement, 20 edge points of each tooth gear are set in the profile projector. To ensure the integrity of the comparison procedure, 5 injection-moulded gears from the same batch are measured to determine the repeatability of the part geometry. The relative shrinkage of the selected quality characteristics is calculated by using the following equation:(15)
S
=
(
D
-
D
m
)
D
m
,
where S is shrinkage, D is the reading of the diameter measurement by using the profile projector, and D
m is the mould dimension.


### 3.4. S/N Analysis

For constant processing parameters, the average value of five repeated results for shrinkage behaviour in tooth thickness, addendum circle, and dedendum circle is calculated and is considered the final result. In processing optimization, all the results of shrinkage behaviour in tooth thickness, addendum circle, and dedendum circle are transformed into S/N ratio. The conversion of the results into S/N ratio involves a series of calculations of the mean squared deviation. For this case, the smaller-the-better category is used to characterize the shrinkage behaviour of tooth thickness, addendum circle, and dedendum circle (see (1)). The final measured results and S/N ratios for the three quality characteristics are shown in Table 4.Table 4
Average shrinkage and S/N ratios for each trial.
Trials
Average shrinkage
S/N ratio
Tooth thickness (mm)
Addendum circle (mm)
Dedendum circle (mm)
Tooth thickness (dB)
Addendum circle (dB)
Dedendum circle (dB)
1
−0.0285
−0.0211
−0.0197
29.1568
33.5159
34.0801
2
−0.0542
−0.0213
−0.0198
25.0132
33.4284
34.0464
3
−0.0718
−0.0202
−0.0188
22.2234
33.8712
34.4891
4
−0.1067
−0.0200
−0.0196
19.2536
33.9651
34.1543
5
−0.1045
−0.0215
−0.0205
19.1704
33.3515
33.7506
6
−0.1174
−0.0211
−0.0192
18.4870
33.5269
34.3038
7
−0.0942
−0.0202
−0.0186
20.4134
33.8720
34.6027
8
−0.1298
−0.0185
−0.0174
17.6876
34.6592
35.1961
9
−0.0998
−0.0197
−0.0169
19.9179
34.0946
35.4325


### 3.5. Grey Generation of Raw Data

In the GRA, the experimental results for the S/N ratios of shrinkage behaviour in tooth thickness, addendum circle, and dedendum circle in Table4 are first normalized according to the-smaller-the-better characteristic of the sequence by using (5). The values of tooth thickness, addendum circle, and dedendum circle are set as the reference sequence x
0
(
O
)
(
k
), k
=
1
,
2
,
3 and the comparability sequences x
i
(
O
)
(
k
), i
=
1
,
2
,
3
,
…
,
9, k
=
1
,
2
,
3. Table 5 lists all the sequences after data preprocessing. According to Deng (1989) [19], a larger value of the normalized results corresponds to better performance and the maximum normalized results that are equal to one indicate the best performance.Table 5
Sequences after data preprocessing (grey generation).
Trials
Tooth thickness
Addendum circle
Dedendum circle
Reference sequence
1.0000
1.0000
1.0000
Comparability sequence
1
0.0000
0.8743
0.9757
2
0.3613
0.9412
1.0000
3
0.6045
0.6026
0.6806
4
0.8635
0.5308
0.9221
5
0.8707
1.0000
1.2133
6
0.9303
0.8659
0.8143
7
0.7623
0.6020
0.5987
8
1.0000
0.0000
0.1706
9
0.8055
0.4317
0.0000According to Table5, the deviation sequences Δ
01
(
k
) can be calculated as follows:
(16)
Δ
01
(
1
)
=
|
x
0
*
(
1
)
-
x
1
*
(
1
)
|
=
|
1.0000
-
0.0000
|
=
1.0000
,
Δ
01
(
2
)
=
|
x
0
*
(
2
)
-
x
1
*
(
2
)
|
=
|
1.0000
-
0.8743
|
=
0.1257
,
Δ
01
(
3
)
=
|
x
0
*
(
3
)
-
x
1
*
(
3
)
|
=
|
1.0000
-
0.9757
|
=
0.0243
.
Therefore, Δ
01
=
(
1.0000,0.1257,0.0243
).The same calculating method is performed fori
=
1
,
…
,
9, and the results of all Δ
0
i for i
=
1
,
…
,
9 are listed in Table 6. By investigating the data presented in Table 6, Δ
max
(
k
) and Δ
min
(
k
) can be expressed as follows:
(17)
Δ
max
=
Δ
08
(
1
)
=
Δ
05
(
2
)
=
Δ
02
(
3
)
=
1.0000
,
Δ
min
=
Δ
01
(
1
)
=
Δ
08
(
2
)
=
Δ
09
(
3
)
=
0.0000
.Table 6
Deviation sequences.
Deviation sequences
Δ
01
Δ
02
Δ
03
Trial 1
1.0000
0.1257
0.0243
Trial 2
0.6387
0.0588
0.0000
Trial 3
0.3955
0.3974
0.3194
Trial 4
0.1365
0.4692
0.0779
Trial 5
0.1293
0.0000
0.2133
Trial 6
0.0697
0.1341
0.1857
Trial 7
0.2377
0.3980
0.4013
Trial 8
0.0000
1.0000
0.8294
Trial 9
0.1945
0.5683
1.0000


### 3.6. Computation of the Grey Relational Coefficient of Response Variables

The grey relational coefficients for each quality characteristic have been calculated by substituting the distinguishing coefficientψ
=
0.5 by using (8). Examples on grey relational coefficient ε
1
(
k
) are provided as follows:
(18)
ε
1
(
1
)
=
0.0000
+
(
0.5
)
(
1.0000
)
1.0000
+
(
0.5
)
(
1.0000
)
=
0.3333
,
ε
1
(
2
)
=
0.0000
+
(
0.5
)
(
1.0000
)
0.1257
+
(
0.5
)
(
1.0000
)
=
0.7991
,
ε
1
(
3
)
=
0.0000
+
(
0.5
)
(
1.0000
)
0.0243
+
(
0.5
)
(
1.0000
)
=
0.9536
.
Thus, ε
1
(
k
)
=
(
0.3333,0.7991,0.9536
), k
=
1, 2, 3. A similar procedure is applied for i
=
1
,
…
,
9. Table 7 lists the grey relational coefficient for each trial of the L9 OA.Table 7
Calculated grey relational coefficient for nine comparability sequences.
Trials
Grey relational coefficient
Tooth thickness (mm)
Addendum circle (mm)
Dedendum circle (mm)
1
0.3333
0.7991
0.9536
2
0.4391
0.8948
1.0000
3
0.5584
0.5572
0.6102
4
0.7855
0.5159
0.8652
5
0.7946
1.0000
0.7009
6
0.8777
0.7885
0.7291
7
0.6778
0.5568
0.5547
8
1.0000
0.3333
0.3761
9
0.7200
0.4680
0.3333


### 3.7. Computation of the Contribution of Respective Quality Characteristics by Using PCA

In optimizing a problem concerning multiple quality characteristics or performances, an engineering judgment or subjective estimation is needed to determine the weighting values for each quality characteristic. The use of the conventional method to determine the values for each quality characteristics is heavily reliant on experience and trial-and-error, thus resulting in an increase in uncertainty during the decision-making process. To reveal the relative importance for each quality characteristic in GRA objectively, the PCA is introduced. The PCA is adopted to determine the corresponding weighting values for each quality characteristic.The elements of the array for the multiple quality characteristics listed in Table7 represent the grey relational coefficient of each quality characteristic. These data are used to evaluate the correlation coefficient matrix and to determine the corresponding eigenvalues from (13) (Table 8). The eigenvector corresponding to each eigenvalue is listed in Table 9, and the square of the eigenvector can represent the contribution of the corresponding quality characteristic to the principal component. The contribution of shrinkage behaviour in the tooth thickness, addendum circle, and dedendum circle of the PP-moulded gear is shown in Table 10; these contributions are listed as 0.2878, 0.3224, and 0.3898, respectively. Moreover, the variance contribution for the first principal component characterizing the three quality characteristics is as high as 71.34%. Therefore, for this study, the squares of the respective eigenvectors are selected as the weighting values of the related quality characteristic. Coefficients w
1, w
2, and w
3 in (10) are set as 0.2878, 0.3224, and 0.3898, respectively.Table 8
Eigenvalues and explained variation for principal components.
Principal component
Eigenvalue
Explained variation (%)
First
2.1402
71.34
Second
0.5926
19.75
Third
0.2672
8.908Table 9
Eigenvectors for principal components.
Quality characteristic
Eigenvector
First principal component
Second principal component
Third principal component
Tooth thickness
−0.5365
0.7716
0.3419
Addendum circle
0.5678
0.6297
−0.5302
Dedendum circle
0.6244
0.0904
0.7759Table 10
Contribution of each individual quality characteristic for the principal component.
Quality characteristic
Contribution
Tooth thickness
0.2878
Addendum circle
0.3224
Dedendum circle
0.3898


### 3.8. Computation Grey Relational Grades

On the basis of (10) and the data listed in Table 7, the grey relational grades are calculated as follows:
(19)
γ
1
=
(
0.2878
×
0.3333
)
+
(
0.3224
×
0.7991
)
+
(
0.3898
×
0.9536
)
=
0.7232
.By using the same procedure, the grey relational grade of the comparability sequence fori
=
1
–
9 can also be obtained and is presented in Table 11. The processing parameters were optimized with respect to a single grey relational grade rather than complicated multiple quality characteristics.Table 11
Grey relational grade and its order.
Trial
Grey relational grade
Order
1
0.7253
5
2
0.8046
2
3
0.5782
7
4
0.7297
4
5
0.8243
1
6
0.7910
3
7
0.5908
6
8
0.5419
8
9
0.4880
9


## 3.1. Determination of Quality Characteristics

In this paper, a plastic gear is used to demonstrate the efficiency and validity of the proposed hybrid optimization method. With the capability to run without lubrication and corrosion, plastic gears have been widely used in the automotive industry, office machines, household utensils, food and textile machinery, and a host of other applications [30]. Plastic-moulded gears offer some material properties that are not achievable with metal-based gears, including unique advantages in weight, noise, modulus, self-lubrication, chemical resistance, and low cost. However, the practicability of injection moulding in producing low-cost plastic-moulded gears remains restricted by the occurrence of shrinkage in the final parts. As one of the most common and prominent defects of plastic, severe shrinkage leads to the deflection or warpage of the moulded part and negatively affects the dimensional stability and accuracy of the involute profile, concentricity, roundness, tooth spacing uniformity, and gear size. Sever shrinkage defect is often forgotten and regarded lightly in many works related to plastic-moulded gears. However, this defect critically affects the quality of the final moulded gear, noise, and vibration and shortens the service life of the gear because of different damage mechanisms, such as tooth fatigue, creep, excessive wear, and plastic deformation. Hence, shrinkage minimization is the optimization objective of this paper. This paper focuses on the optimization of processing parameters via the integration of the Taguchi/GRA/PCA to minimize the shrinkage behaviour of plastic injection-moulded gears. To inspect shrinkage behaviour on the dimensional stability of the moulded gear, three quality characteristics are selected: tooth thickness, addendum circle, and dedendum circle.


## 3.2. Selection of Taguchi OA

In this study, the crystalline thermoplastic polypropylene (PP) is specified for the gear. The PP used in this study is obtained from Propylene Malaysia Sdn. Bhd., Malaysia. The general properties of PP are shown in Table1. The spur gear design used is compliant with the American Gears Manufacturers Association standards. The detailed geometry and specifications of the gear are shown in Figure 2.Table 1
General properties of PP.
Properties
Density (g/cm3)
0.90 to 0.91
Raw density (g/cm3)
0.805
Melt flow index (g/10 min)
10.78
Modulus of elasticity (MPa)
4100
Charpy impact toughness (kJ/m2)
1.4 to 1.8Figure 2
Geometry and specification of the spur gear: module = 1.5; pressure angle = 20°; number of teeth = 20; face width = 10 mm.The PP gears are injected by using a Battenfeld TM750/210 injection moulding machine. The experiment is conducted with four controllable three-level processing parameters: melt temperature, packing pressure, packing time, and cooling time. Other processing parameters, such as mould temperature (25°C), injection pressure (80 bar), and stroke distance (60 mm), are kept constant during the experiment. The selected processing parameters and their levels are shown in Table2. The L9 (34) OA (Table 3) is conducted to study the four processing parameters. Nine trials of PP gears with five repetitions are produced by using the OA.Table 2
Processing parameters and levels studied.
Column
Factors
Level 1
Level 2
Level 3
A
Melt temperature (°C)
200
220
240
B
Packing pressure (%)
60
80
100
C
Packing time (s)
5
10
15
D
Cooling time (s)
30
40
50Table 3
OA L9 (34) of the experimental runs.
Factors/trial no.
A
B
C
D
Melt temperature (°C)
Packing pressure (%)
Packing time (s)
Cooling time (s)
1
200
60
5
30
2
200
80
10
40
3
200
100
15
50
4
220
60
10
50
5
220
80
15
30
6
220
100
5
40
7
240
60
15
40
8
240
80
5
50
9
240
100
10
30


## 3.3. Shrinkage Measurement

Rax Vision DC 3000 Mitutoyo profile projector is used to inspect the accuracy of the specific profile of the involute gear teeth after moulding. In this study, the profile projector is used to measure 2D tooth thickness, as well as addendum and dedendum circles, by using the coordinates of selected points along the gear profile. With large magnifications and micrometer readouts, the profile projector can ensure fairly accurate measurements compared with a Vernier caliper and micrometer. In the process of addendum and dedendum circle measurement, 20 edge points of each tooth gear are set in the profile projector. To ensure the integrity of the comparison procedure, 5 injection-moulded gears from the same batch are measured to determine the repeatability of the part geometry. The relative shrinkage of the selected quality characteristics is calculated by using the following equation:(15)
S
=
(
D
-
D
m
)
D
m
,
where S is shrinkage, D is the reading of the diameter measurement by using the profile projector, and D
m is the mould dimension.


## 3.4. S/N Analysis

For constant processing parameters, the average value of five repeated results for shrinkage behaviour in tooth thickness, addendum circle, and dedendum circle is calculated and is considered the final result. In processing optimization, all the results of shrinkage behaviour in tooth thickness, addendum circle, and dedendum circle are transformed into S/N ratio. The conversion of the results into S/N ratio involves a series of calculations of the mean squared deviation. For this case, the smaller-the-better category is used to characterize the shrinkage behaviour of tooth thickness, addendum circle, and dedendum circle (see (1)). The final measured results and S/N ratios for the three quality characteristics are shown in Table 4.Table 4
Average shrinkage and S/N ratios for each trial.
Trials
Average shrinkage
S/N ratio
Tooth thickness (mm)
Addendum circle (mm)
Dedendum circle (mm)
Tooth thickness (dB)
Addendum circle (dB)
Dedendum circle (dB)
1
−0.0285
−0.0211
−0.0197
29.1568
33.5159
34.0801
2
−0.0542
−0.0213
−0.0198
25.0132
33.4284
34.0464
3
−0.0718
−0.0202
−0.0188
22.2234
33.8712
34.4891
4
−0.1067
−0.0200
−0.0196
19.2536
33.9651
34.1543
5
−0.1045
−0.0215
−0.0205
19.1704
33.3515
33.7506
6
−0.1174
−0.0211
−0.0192
18.4870
33.5269
34.3038
7
−0.0942
−0.0202
−0.0186
20.4134
33.8720
34.6027
8
−0.1298
−0.0185
−0.0174
17.6876
34.6592
35.1961
9
−0.0998
−0.0197
−0.0169
19.9179
34.0946
35.4325


## 3.5. Grey Generation of Raw Data

In the GRA, the experimental results for the S/N ratios of shrinkage behaviour in tooth thickness, addendum circle, and dedendum circle in Table4 are first normalized according to the-smaller-the-better characteristic of the sequence by using (5). The values of tooth thickness, addendum circle, and dedendum circle are set as the reference sequence x
0
(
O
)
(
k
), k
=
1
,
2
,
3 and the comparability sequences x
i
(
O
)
(
k
), i
=
1
,
2
,
3
,
…
,
9, k
=
1
,
2
,
3. Table 5 lists all the sequences after data preprocessing. According to Deng (1989) [19], a larger value of the normalized results corresponds to better performance and the maximum normalized results that are equal to one indicate the best performance.Table 5
Sequences after data preprocessing (grey generation).
Trials
Tooth thickness
Addendum circle
Dedendum circle
Reference sequence
1.0000
1.0000
1.0000
Comparability sequence
1
0.0000
0.8743
0.9757
2
0.3613
0.9412
1.0000
3
0.6045
0.6026
0.6806
4
0.8635
0.5308
0.9221
5
0.8707
1.0000
1.2133
6
0.9303
0.8659
0.8143
7
0.7623
0.6020
0.5987
8
1.0000
0.0000
0.1706
9
0.8055
0.4317
0.0000According to Table5, the deviation sequences Δ
01
(
k
) can be calculated as follows:
(16)
Δ
01
(
1
)
=
|
x
0
*
(
1
)
-
x
1
*
(
1
)
|
=
|
1.0000
-
0.0000
|
=
1.0000
,
Δ
01
(
2
)
=
|
x
0
*
(
2
)
-
x
1
*
(
2
)
|
=
|
1.0000
-
0.8743
|
=
0.1257
,
Δ
01
(
3
)
=
|
x
0
*
(
3
)
-
x
1
*
(
3
)
|
=
|
1.0000
-
0.9757
|
=
0.0243
.
Therefore, Δ
01
=
(
1.0000,0.1257,0.0243
).The same calculating method is performed fori
=
1
,
…
,
9, and the results of all Δ
0
i for i
=
1
,
…
,
9 are listed in Table 6. By investigating the data presented in Table 6, Δ
max
(
k
) and Δ
min
(
k
) can be expressed as follows:
(17)
Δ
max
=
Δ
08
(
1
)
=
Δ
05
(
2
)
=
Δ
02
(
3
)
=
1.0000
,
Δ
min
=
Δ
01
(
1
)
=
Δ
08
(
2
)
=
Δ
09
(
3
)
=
0.0000
.Table 6
Deviation sequences.
Deviation sequences
Δ
01
Δ
02
Δ
03
Trial 1
1.0000
0.1257
0.0243
Trial 2
0.6387
0.0588
0.0000
Trial 3
0.3955
0.3974
0.3194
Trial 4
0.1365
0.4692
0.0779
Trial 5
0.1293
0.0000
0.2133
Trial 6
0.0697
0.1341
0.1857
Trial 7
0.2377
0.3980
0.4013
Trial 8
0.0000
1.0000
0.8294
Trial 9
0.1945
0.5683
1.0000


## 3.6. Computation of the Grey Relational Coefficient of Response Variables

The grey relational coefficients for each quality characteristic have been calculated by substituting the distinguishing coefficientψ
=
0.5 by using (8). Examples on grey relational coefficient ε
1
(
k
) are provided as follows:
(18)
ε
1
(
1
)
=
0.0000
+
(
0.5
)
(
1.0000
)
1.0000
+
(
0.5
)
(
1.0000
)
=
0.3333
,
ε
1
(
2
)
=
0.0000
+
(
0.5
)
(
1.0000
)
0.1257
+
(
0.5
)
(
1.0000
)
=
0.7991
,
ε
1
(
3
)
=
0.0000
+
(
0.5
)
(
1.0000
)
0.0243
+
(
0.5
)
(
1.0000
)
=
0.9536
.
Thus, ε
1
(
k
)
=
(
0.3333,0.7991,0.9536
), k
=
1, 2, 3. A similar procedure is applied for i
=
1
,
…
,
9. Table 7 lists the grey relational coefficient for each trial of the L9 OA.Table 7
Calculated grey relational coefficient for nine comparability sequences.
Trials
Grey relational coefficient
Tooth thickness (mm)
Addendum circle (mm)
Dedendum circle (mm)
1
0.3333
0.7991
0.9536
2
0.4391
0.8948
1.0000
3
0.5584
0.5572
0.6102
4
0.7855
0.5159
0.8652
5
0.7946
1.0000
0.7009
6
0.8777
0.7885
0.7291
7
0.6778
0.5568
0.5547
8
1.0000
0.3333
0.3761
9
0.7200
0.4680
0.3333


## 3.7. Computation of the Contribution of Respective Quality Characteristics by Using PCA

In optimizing a problem concerning multiple quality characteristics or performances, an engineering judgment or subjective estimation is needed to determine the weighting values for each quality characteristic. The use of the conventional method to determine the values for each quality characteristics is heavily reliant on experience and trial-and-error, thus resulting in an increase in uncertainty during the decision-making process. To reveal the relative importance for each quality characteristic in GRA objectively, the PCA is introduced. The PCA is adopted to determine the corresponding weighting values for each quality characteristic.The elements of the array for the multiple quality characteristics listed in Table7 represent the grey relational coefficient of each quality characteristic. These data are used to evaluate the correlation coefficient matrix and to determine the corresponding eigenvalues from (13) (Table 8). The eigenvector corresponding to each eigenvalue is listed in Table 9, and the square of the eigenvector can represent the contribution of the corresponding quality characteristic to the principal component. The contribution of shrinkage behaviour in the tooth thickness, addendum circle, and dedendum circle of the PP-moulded gear is shown in Table 10; these contributions are listed as 0.2878, 0.3224, and 0.3898, respectively. Moreover, the variance contribution for the first principal component characterizing the three quality characteristics is as high as 71.34%. Therefore, for this study, the squares of the respective eigenvectors are selected as the weighting values of the related quality characteristic. Coefficients w
1, w
2, and w
3 in (10) are set as 0.2878, 0.3224, and 0.3898, respectively.Table 8
Eigenvalues and explained variation for principal components.
Principal component
Eigenvalue
Explained variation (%)
First
2.1402
71.34
Second
0.5926
19.75
Third
0.2672
8.908Table 9
Eigenvectors for principal components.
Quality characteristic
Eigenvector
First principal component
Second principal component
Third principal component
Tooth thickness
−0.5365
0.7716
0.3419
Addendum circle
0.5678
0.6297
−0.5302
Dedendum circle
0.6244
0.0904
0.7759Table 10
Contribution of each individual quality characteristic for the principal component.
Quality characteristic
Contribution
Tooth thickness
0.2878
Addendum circle
0.3224
Dedendum circle
0.3898


## 3.8. Computation Grey Relational Grades

On the basis of (10) and the data listed in Table 7, the grey relational grades are calculated as follows:
(19)
γ
1
=
(
0.2878
×
0.3333
)
+
(
0.3224
×
0.7991
)
+
(
0.3898
×
0.9536
)
=
0.7232
.By using the same procedure, the grey relational grade of the comparability sequence fori
=
1
–
9 can also be obtained and is presented in Table 11. The processing parameters were optimized with respect to a single grey relational grade rather than complicated multiple quality characteristics.Table 11
Grey relational grade and its order.
Trial
Grey relational grade
Order
1
0.7253
5
2
0.8046
2
3
0.5782
7
4
0.7297
4
5
0.8243
1
6
0.7910
3
7
0.5908
6
8
0.5419
8
9
0.4880
9


## 4. Results and Discussion

### 4.1. Optimal Combination of Injection Moulding Processing Parameters and Their Levels

To determine the optimal combination of injection moulding processing parameters for shrinkage behaviour in the tooth thickness, addendum circle, and dedendum circle of the studied moulded gear, the average grey relational grade for each injection moulding parameter level is calculated by employing the main effect analysis of the Taguchi method. This process is performed by sorting the grey relational grades corresponding to the levels of the injection moulding parameters in each column of the OA and then taking the average of parameters with the same levels. For instance, for factor A (Table3), experiments 1, 2, and 3 are set to level 1. Therefore, by using the data listed in Table 11, the average grey relational grade for A
1 can be calculated as follows:
(20)
A
-
1
=
(
0.7253
+
0.8046
+
0.5782
)
3
=
0.7027
.The average grey relational grade forA
2 and A
3 is calculated as follows:
(21)
A
-
2
=
(
0.7297
+
0.8243
+
0.7910
)
3
=
0.7817
,
A
-
3
=
(
0.5908
+
0.5419
+
0.4880
)
3
=
0.5402
.By using a similar method, calculations are performed for each injection moulding parameter level, and the main effect analysis is constructed (Table12 and Figure 3).Table 12
Main effects table for grey relational grades.
Symbol
Process parameter
Level 1
Level 2
Level 3
A
Melt temperature (°C)
0.7027
0.7817
0.5403
B
Packing pressure (%)
0.6819
0.7236
0.6191
C
Packing time (s)
0.6861
0.6741
0.6644
D
Cooling time (s)
0.6792
0.7288
0.6166Figure 3
Effect of injection moulding parameter levels on multiple quality characteristics.Considering that the grey relational grade represents the level of correlation between the reference and comparability sequences, a larger grey relational grade indicates that the comparability sequence exhibits a stronger correlation with the reference sequence. A larger grey relational grade results in better multiple quality characteristics. Figure3 clearly shows that the multiple quality characteristics of the PP-moulded gear are significantly affected by the adjustments of the processing parameters. The increment of melt temperature, packing pressure, and cooling time initially increases the grey relational grades, thus resulting in the reduction of shrinkage behaviour in the tooth thickness, addendum circle, and dedendum circle of the moulded gear. However, the relational grades for the three quality characteristics are lower when the melt temperature, packing pressure, and cooling time are 240°C, 100%, and 50 s, respectively. By contrast, the increment of packing time reduces the value of the grey relational grade, thus increasing the shrinkage behaviour in tooth thickness, addendum circle, and dedendum circle.In this case, the best combination of processing parameters and levels can easily be obtained from the main effect analysis by selecting the level of each parameter with the highest grey relational grade. A2, B2, C1, and D2 show the largest value of grey relational grade for factors A, B, C, and D, respectively (Figure 3). Thus, the optimal parameter setting that statistically results in the minimum shrinkage behaviour in the tooth thickness, addendum circle, and dedendum circle of the PP-moulded gear is predicted to be A2, B2, C1, and D2. The melt temperature is 220°C, packing pressure is 80%, packing time is 5 s, and cooling time is 40 s.


### 4.2. Effects of Processing Parameters on Quality Characteristics

To examine the extent in which injection moulding parameters significantly influence the performance of moulded gear, ANOVA is performed on the Taguchi method for the grey relational grade of nine comparability sequences (Table11). The computed quantity of degrees of freedom (DOF), sum of square (S), variance (V), and percentage contribution (P) are presented in Table 13.Table 13
ANOVA table for the grey relational grade of nine comparability sequences.
Column
Parameters
DOF
S
V
P
A
Melt temperature (°C)
2
0.0909
0.0454
71.47
B
Packing pressure (%)
2
0.0166
0.0083
13.06
C
Packing time (s)
2
0.0007
0.0004
0.55
D
Cooling time (s)
2
0.0190
0.0095
14.92
All others/error
0
0.0000
0.0000
0.00
Total
8
0.1272
100.00In this case, the percentage contribution of each processing parameter is directly calculated from S because the DOF for the error term is equal to zero [31]. The significance of each processing parameter in the shrinkage behaviour in the tooth thickness, addendum circle, and dedendum circle of the moulded gear can be determined by the percentage contribution. Roy [32] suggested an alternative by using the 10% rule; that is, a parameter is considered insignificant when its influence is less than 10% of the highest parameter influence. From the results of ANOVA in Table 13, melt temperature appears to be the most decisive processing parameter in reducing the shrinkage behaviour in the tooth thickness, addendum circle, and dedendum circle of the moulded gear with the highest percentage contribution of 71.47%, thus outweighing the other process variables. The analysis also reveals that packing pressure and cooling time are significant because their percentages are more than 10% of the highest parameter influence (7.15%). Cooling time and packing pressure achieve 14.92% and 13.06%, respectively. Packing time results in only 0.55%, which is significantly less than 10% of the highest parameter influence. In this case, the packing time is considered insignificant in the shrinkage behaviour of the tooth thickness, addendum circle, and dedendum circle of the moulded gear.


### 4.3. Verification Test

Once the optimal levels of the injection moulding process parameters are identified, the subsequent step is to verify the improvements in the quality characteristics by using this optimal combination. The verification test can be used to assess the accuracy of the proposed hybrid Taguchi/GRA/PCA optimization method. An experimental verification test is conducted by using the same procedures as previous runs under the optimal process conditions, namely, A2, B2, C1, and D2, to produce the plastic-moulded gear. Table 14 lists the results of five repetitions of the verification tests by using the optimal process conditions obtained by the proposed hybrid optimization method.Table 14
Results of the verification test.
Verification test
Tooth thickness (mm)
Addendum circle (mm)
Dedendum circle (mm)
1
−0.0187
−0.0194
−0.0160
2
−0.0391
−0.0186
−0.0148
3
−0.0245
−0.0165
−0.0159
4
−0.0164
−0.0194
−0.0150
5
−0.0221
−0.0162
−0.0148
Average
−0.0180
−0.0153
−0.0242After optimization, the minimum shrinkage behaviour in tooth thickness, addendum circle, and dedendum circle is averaged as 1.8%, 1.53%, and 2.42%, respectively. The shrinkage behaviour for the three quality characteristics decreases with respect to those attained in the main experiment presented in Table4.


## 4.1. Optimal Combination of Injection Moulding Processing Parameters and Their Levels

To determine the optimal combination of injection moulding processing parameters for shrinkage behaviour in the tooth thickness, addendum circle, and dedendum circle of the studied moulded gear, the average grey relational grade for each injection moulding parameter level is calculated by employing the main effect analysis of the Taguchi method. This process is performed by sorting the grey relational grades corresponding to the levels of the injection moulding parameters in each column of the OA and then taking the average of parameters with the same levels. For instance, for factor A (Table3), experiments 1, 2, and 3 are set to level 1. Therefore, by using the data listed in Table 11, the average grey relational grade for A
1 can be calculated as follows:
(20)
A
-
1
=
(
0.7253
+
0.8046
+
0.5782
)
3
=
0.7027
.The average grey relational grade forA
2 and A
3 is calculated as follows:
(21)
A
-
2
=
(
0.7297
+
0.8243
+
0.7910
)
3
=
0.7817
,
A
-
3
=
(
0.5908
+
0.5419
+
0.4880
)
3
=
0.5402
.By using a similar method, calculations are performed for each injection moulding parameter level, and the main effect analysis is constructed (Table12 and Figure 3).Table 12
Main effects table for grey relational grades.
Symbol
Process parameter
Level 1
Level 2
Level 3
A
Melt temperature (°C)
0.7027
0.7817
0.5403
B
Packing pressure (%)
0.6819
0.7236
0.6191
C
Packing time (s)
0.6861
0.6741
0.6644
D
Cooling time (s)
0.6792
0.7288
0.6166Figure 3
Effect of injection moulding parameter levels on multiple quality characteristics.Considering that the grey relational grade represents the level of correlation between the reference and comparability sequences, a larger grey relational grade indicates that the comparability sequence exhibits a stronger correlation with the reference sequence. A larger grey relational grade results in better multiple quality characteristics. Figure3 clearly shows that the multiple quality characteristics of the PP-moulded gear are significantly affected by the adjustments of the processing parameters. The increment of melt temperature, packing pressure, and cooling time initially increases the grey relational grades, thus resulting in the reduction of shrinkage behaviour in the tooth thickness, addendum circle, and dedendum circle of the moulded gear. However, the relational grades for the three quality characteristics are lower when the melt temperature, packing pressure, and cooling time are 240°C, 100%, and 50 s, respectively. By contrast, the increment of packing time reduces the value of the grey relational grade, thus increasing the shrinkage behaviour in tooth thickness, addendum circle, and dedendum circle.In this case, the best combination of processing parameters and levels can easily be obtained from the main effect analysis by selecting the level of each parameter with the highest grey relational grade. A2, B2, C1, and D2 show the largest value of grey relational grade for factors A, B, C, and D, respectively (Figure 3). Thus, the optimal parameter setting that statistically results in the minimum shrinkage behaviour in the tooth thickness, addendum circle, and dedendum circle of the PP-moulded gear is predicted to be A2, B2, C1, and D2. The melt temperature is 220°C, packing pressure is 80%, packing time is 5 s, and cooling time is 40 s.


## 4.2. Effects of Processing Parameters on Quality Characteristics

To examine the extent in which injection moulding parameters significantly influence the performance of moulded gear, ANOVA is performed on the Taguchi method for the grey relational grade of nine comparability sequences (Table11). The computed quantity of degrees of freedom (DOF), sum of square (S), variance (V), and percentage contribution (P) are presented in Table 13.Table 13
ANOVA table for the grey relational grade of nine comparability sequences.
Column
Parameters
DOF
S
V
P
A
Melt temperature (°C)
2
0.0909
0.0454
71.47
B
Packing pressure (%)
2
0.0166
0.0083
13.06
C
Packing time (s)
2
0.0007
0.0004
0.55
D
Cooling time (s)
2
0.0190
0.0095
14.92
All others/error
0
0.0000
0.0000
0.00
Total
8
0.1272
100.00In this case, the percentage contribution of each processing parameter is directly calculated from S because the DOF for the error term is equal to zero [31]. The significance of each processing parameter in the shrinkage behaviour in the tooth thickness, addendum circle, and dedendum circle of the moulded gear can be determined by the percentage contribution. Roy [32] suggested an alternative by using the 10% rule; that is, a parameter is considered insignificant when its influence is less than 10% of the highest parameter influence. From the results of ANOVA in Table 13, melt temperature appears to be the most decisive processing parameter in reducing the shrinkage behaviour in the tooth thickness, addendum circle, and dedendum circle of the moulded gear with the highest percentage contribution of 71.47%, thus outweighing the other process variables. The analysis also reveals that packing pressure and cooling time are significant because their percentages are more than 10% of the highest parameter influence (7.15%). Cooling time and packing pressure achieve 14.92% and 13.06%, respectively. Packing time results in only 0.55%, which is significantly less than 10% of the highest parameter influence. In this case, the packing time is considered insignificant in the shrinkage behaviour of the tooth thickness, addendum circle, and dedendum circle of the moulded gear.


## 4.3. Verification Test

Once the optimal levels of the injection moulding process parameters are identified, the subsequent step is to verify the improvements in the quality characteristics by using this optimal combination. The verification test can be used to assess the accuracy of the proposed hybrid Taguchi/GRA/PCA optimization method. An experimental verification test is conducted by using the same procedures as previous runs under the optimal process conditions, namely, A2, B2, C1, and D2, to produce the plastic-moulded gear. Table 14 lists the results of five repetitions of the verification tests by using the optimal process conditions obtained by the proposed hybrid optimization method.Table 14
Results of the verification test.
Verification test
Tooth thickness (mm)
Addendum circle (mm)
Dedendum circle (mm)
1
−0.0187
−0.0194
−0.0160
2
−0.0391
−0.0186
−0.0148
3
−0.0245
−0.0165
−0.0159
4
−0.0164
−0.0194
−0.0150
5
−0.0221
−0.0162
−0.0148
Average
−0.0180
−0.0153
−0.0242After optimization, the minimum shrinkage behaviour in tooth thickness, addendum circle, and dedendum circle is averaged as 1.8%, 1.53%, and 2.42%, respectively. The shrinkage behaviour for the three quality characteristics decreases with respect to those attained in the main experiment presented in Table4.


## 5. Conclusion

In this study, a hybrid Taguchi/GRA/PCA optimization method for injection moulding process parameters has been developed systematically to overcome the shortcomings of individual methods in multiple quality characteristics problems. The problem that arises when calculating the weighting value for each quality characteristic in GRA has been addressed by integrating the PCA to determine the grey relational grades. Thus, the conventional approach of using engineering judgment or subjective estimation to determine the weighting values for each quality characteristics can be avoided. A plastic gear is employed in this study as a case study to demonstrate the efficiency and validity of the proposed hybrid optimization method in controlling all influential injection moulding processing parameters during plastic gear manufacturing. The shrinkage behaviour in tooth thickness, addendum circle, and dedendum circle is chosen as the optimization objectives. Through a series of analyses and optimizations of selected multiple quality characteristics for the case of a plastic gear, the minimum shrinkage behaviour in tooth thickness, addendum circle, and dedendum circle after optimization by the proposed hybrid optimization method is averaged as 1.8%, 1.53%, and 2.42%, respectively, which are less than the values in the main experiment.


## Conflict of Interests

The authors would like to declare that all financial and material support for the conduct of this research and the preparation of this paper is clearly termed in the Acknowledgment and there will be no conflict of interests in relation to the financial gain subject to the publication of this paper.



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*Source: 351206-2014-01-14.xml*