# Experimental-Based Optimization of Injection Molding Process Parameters for Short Product Cycle Time

Authors: Saad M. S. Mukras
Journal: Advances in Polymer Technology
 (2020)

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## Abstract

This paper presents a framework for optimizing injection molding process parameters for minimum product cycle time subjected to constraints on the product defects. Two product defects, namely, volumetric shrinkage and warpage, as well as seven process parameters including injection speed, injection pressure, cooling time, packing pressure, mold temperature, packing time, and melt temperature, were considered. Injection molding experiments were conducted on specifically chosen test points and results were used to compute the volumetric shrinkage and warpage (at each test point). Thereafter, three relationships between the product cycle time (one relationship), the two product defects (two relationships), and the injection molding parameters were constructed using the kriging technique. An optimization problem to minimize the product cycle time (described by the first relationship) subject to constraints on the product defects (described by the latter two relationships) was then formulated. A combination set of points between the lower and upper extreme values of acceptable product defect was generated to serve as constraints for the two product defects. The optimization problem was then solved using the Fmincon function, available in the Matlab optimization toolbox. A plot of the optimization results revealed an appreciable tradeoff between the cycle time and the two product defects. To validate the optimization, an additional injection molding experiment was conducted for one of the optimization results. Results from the additional experiment showed reasonably close agreement with simulation optimization results differing in the cycle time, the warpage and volumetric shrinkage by 6.7%, 3.2%, and 8%, respectively.


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## Body

## 1. Introduction

Injection molding is undoubtedly among the most important industrial plastic processing techniques. It has been estimated that the process consumes approximately 32% [1] of all plastics. The process is widely used in the mass production of both commercial and consumer products. Its immense popularity is due to a number of factors including its reliability and its ability to produce complex shape in a single step with superb tolerance among others.The injection molding process is achieved using an injection molding unit. A schematic of a typical injection molding unit is shown in Figure1. The process is an unvarying, repetitive process that involves feeding plastic (pellets) to the barrel via a hopper, melting of plastic using heaters in a barrel, mixing the plastic using a rotating and reciprocating screw, injecting the molten plastic into a mold cavity, packing the molten plastic within the mold cavity, cooling of the molten plastic (in the mold cavity), and thereafter ejection of the solidified plastic part. The process is, thus, generally divided into 3 or 4 phase [2–5] which include filling, packing, cooling, and ejection. These phases determine the cycle time of the process, which is an important factor during production as it is directly linked to the cost of production. It can, therefore, be argued that reducing the cycle time (even seconds) within any of these phases will lead to time and cost-savings in the long run especially in the case of mass production.Figure 1
Detailed schematics of a typical injection molding unit.A combination of various steps is usually taken to reduce the cycle time. This includes fine-tuning the injection molding unit to deliver proper injection pressures and speeds (which affects the filling time), designing the product to have minimum wall thickness (which affects the filling time), making adjustments on the injection unit, such as mold opening and closing (which affects the ejection time), as well as other adjustments on the process parameters. Experienced plastic engineers who have gained insights on the process generally make these adjustments [6].Of the four mentioned phases, the cooling phase takes up more than two-thirds of the cycle time [4, 7]. It is also the most important phase in addition to productivity since the cooling has a significant effect on product quality. Inadequate cooling times will usually lead to defects on the product such as warpage and volumetric shrinkage. On the other hand, longer cooling times will generally lead to satisfactory product quality but result in low productivity (and consequently high production costs). This scenario has elicited research activities on ways to reduce the cooling time while ensuring acceptable product quality. Majority of these research activities have focused on conformal cooling channels within the molds to enhance and accelerate cooling [7–18]. In conformal cooling, the cooling channels are designed to conform to the shape of the mold cavity. It has been demonstrated that this technique enables the mold temperature to reach the operating temperature faster than in molds with standard cooling channels [8, 19].There has also been some research into optimizing the process parameters for reduced cycle time. Zhao et al. [20] proposed a process parameter optimization procedure in which a surrogate model is used to approximate the expensive simulation that predicts the injection molding filling. The optimization was multiobjective and sought to lower the cavity pressure, minimize the temperature difference (ensuring uniformity in the melt temperature), and to minimize the cycle time by reducing the cooling time. In [21], Cheng et al. proposed a procedure to optimize injection molding process parameters through the formulation of a multiple objective optimization problem. The problem is formulated to minimize the product defect and the production cost (reduction of the cycle time) and to maximize the molding efficiency. The proposed procedure brought together the use of variable complexity methods, genetic algorithms, and neural networks, as well as injection molding simulations using Moldflow. In [3, 22], a procedure to optimize the injection molding process parameters for minimum warpage and cycle time is proposed. The authors confirm through numerical and experimental results the validity of the proposed procedure. In [23] Alvarado et al. proposed a multi- and many-objective optimization problem with 7 objectives including warpage, volumetric shrinkage, sink marks, Von Mises stress, shear stress, cycle time, and clamping force. Four process parameters, which included melt temperature, packing time packing pressure, and cooling time, were optimized.In the literature, it is observed that the procedures proposed for process parameter optimization for reduced injection cycle time are primarily based on injection molding simulation tools instead of experiments. In these studies, results from injection molding simulations are used to construct relationships between the process parameters and quantities such as product defects and injection molding cycle time. It can, however, be argued that while simulations can produce useful results that can be the basis to advance to production, they generally have limitations as they may be unable to replicate all the physics involved in the process. Additionally, there may exist some conditions specific to the injection-molding machine that cannot be adequately captured in the simulations, and therefore experiments may be required to expose these conditions and or to validate the simulations.In this study, a procedure to optimize the process parameters for reduced cycle time is presented. The procedure involves constructing relationships between the process parameters and the product quality and another relationship between the process parameter and the cycle time. These relationships are similar to those developed in previous research except that they are developed using experiment-based design of experiments (DOE). This technique involved performing actual experiments (rather than numerical experiments) at specific points in the design space as determined using the selected DOE and thereafter using the data to construct the relationships. The use of experimental-based design of experiments technique is a generally accepted procedure and has been used in various disciplines such as in membranes [24], processing of food and bioproducts [25], photovoltaics [26], biotechnology [27], and analytical chemistry [28, 29]. In this work, using this technique will mitigate computation costs while ensuring that all conditions specific to the injection-molding unit are captured in the analysis and, thus, leading to more accurate results of the optimized parameters.


## 2. Injection Molding Experiments

The injection molding process begins with feeding plastic pellets into the injection-molding unit (see Figure1). The plastic pellets are then directed to the barrel of the injection units where they are mixed and heated, causing them to melt. The molten plastic is then subjected to pressure and injected into a mold cavity where it is packed (under pressure) and cooled, allowing it to solidify. The solidified plastic is then ejected from the unit as a finished product. The entire injection molding process requires the specification of certain processing parameters that will affect the product quality as well as the injection molding process cycle time.In this study, experiments were conducted in an effort to arrive at the best combination of process parameters that would lead to the minimum cycle time while considering the extent of product defect (product quality). Two product defects were considered, namely, the warpage and volumetric shrinkage. Furthermore, seven process parameters, that have been shown to affect the product quality, are considered. These include the injection speed, injection pressure, cooling time, packing pressure, mold temperature, packing time, and melt temperature. The injection speed and injection pressure have been identified in [30, 31], whereas the cooling time, packing pressure, mold temperature, packing time, and melt temperature have been identified in [3, 31–36] as having an effect on the product quality.The experiments in this work were conducted on an injection molding unit known as the Arburg Allrounder 420 C [37], and a simple mold with product dimensions depicted in Figure 2 was used. High-density polyethylene (M80064), produced by SABIC was used in the experiments. Some basic properties of the plastic, extracted from [38], are listed in Table 1.Figure 2
Geometry and dimensions of the injection-molded product used in the experiments.Table 1
Properties for HDPE M80064 series (produced by SABIC).
Property
Unit
Value
Stress at yield
MPa
33
Melt flow rate (at 190°C and 2.16 kg)
g/10 min
8
Density
kg/m3
964In order to arrive at the best combination of the process parameter for minimum cycle time, a relationship between the process parameter and the cycle time is to be constructed and optimized. The optimization is subjected to constraints on the extent of acceptable product defect which are represented by two other relationships between the product defects (warpage and volumetric shrinkage) and the process parameters.


### 2.1. Test Points for Injection Molding Experiments

The three relationships, mentioned previously, are constructed from data obtained through actual injection molding experiments performed at combination sets of the process parameters. These combination sets are selected through DOE in a manner that ensures that the design space is adequately covered. In this study, the central composite design (CCD) was adopted. The CCD is a factorial or fractional factorial design with center points which are augmented with star points that enhance the estimation of curvature. Depending on the location of the star point, the CCD can either be circumscribed, inscribed, or face-centered. In the CCD, the sample size is determined according to the following equation:(1)2f+2f−q+1,where f is the number of factors being considered (i.e., the process parameters in the current study) and q refers to the fraction of the full factorial to be used. For a case where there are numerous factors (e.g., f > 5), a full factorial design (q = 0) results in an enormous sample size. In such a case, it is advisable to use the fractional factorial design which utilizes only a fraction of the full factorial design without losing its main benefits. A detailed treatment of the DOE can be found in [39].The complete design space for the seven process parameters is based on the upper and lower bounds of the parameters. The corresponding bounds of these parameters are listed in Table2. The upper and lower bounds of the injection speed, cooling time, packing pressure, and packing times were suggested by experienced plastic technicians, whereas the bounds for the injection pressure, mold temperature, and melt temperature are based on the recommended ranges from the material producers [38]. For the current study, a half-fraction (q = 1) face-centered central composite design was used. Accordingly, for the seven parameters, 79 data points were generated.Table 2
Lower and upper bounds for the test variables (bound for both the DOE and optimization).
Test variables
Units
Lower bound
Upper bound
Injection speed
mm/sec
15
60
Injection pressure
bar
450
800
Cooling time
sec
10
30
Packing pressure
bar
100
400
Mold temperature
°C
15
45
Packing time
sec
3
9
Melt temperature
°C
200
250


### 2.2. Injection Molding Product Defect Quantification

Two relations relating the two defects (warpage and volumetric shrinkage) to the seven parameters need to be established. In order to achieve this, it is first necessary to quantify the defects. The defect quantification adopted in this work is described as follows.


#### 2.2.1. Warpage Quantification

The warpage experienced in injection molding products can be defined as a deviation of the product geometry from the anticipated geometry. In this study, the warpage is defined as the sum of the maximum out-of-plane displacement of the product edges which can be expressed as(2)Wsum=∑ymaxi,i=1,2,3,4.In equation (2) ymaxi refers to the maximum out-of-plane displacement along the four edges of the product. This definition has previously been used in [40].


#### 2.2.2. Volumetric Shrinkage Quantification

The volumetric shrinkage can be estimated by determining the difference between the anticipated product volumeVanticipated and the actual product volumes. The anticipated product volume is computed from the expected product dimensions (based on the diagram in Figure 2), whereas the actual product volume is computed from the known product density and measured mass. The volumetric shrinkage Vshrinkage can, thus, be expressed as(3)Vshrinkage=Vanticipated−mρ.This expression has also been previously used in [40].


### 2.3. Injection Molding Experiments

Injection molding experiments were performed on the 79 data points (previously generated). Pictures of samples of products from the experiments are shown in Figure3. Figure 3(a) shows a product with severe volumetric shrinkage, and Figure 3(b) shows a product with severe warpage. A product with minimal defects is shown in Figure 3(c).Figure 3
Products resulting from the injection molding experiments. (a) Product with severe volumetric shrinkage (warpage of 8.1 mm and volumetric shrinkage of 6.49 mm3), (b) product with severe warpage (warpage of 13.6 mm and volumetric shrinkage of 6.25 mm3), and (c) acceptable product (warpage of 2.9 mm and volumetric shrinkage of 2.45 mm3).
(a)
(b)
(c)Product defect, including warpage and volumetric shrinkage, was computed for the 79 products (using equations (2) and (3), respectively) and are reported in Table 3. Also listed in Table 3 are the corresponding cycle times for each of the 79 products.Table 3
Experimental results for warpage, volumetric shrinkage, and cycle time.
Injection speed
Injection pressure
Measured injection pressure
Cooling time
Packing pressure
Mold temp.
Packing time
Melt temp.
Warpage
Volumetric shrinkage
Cycle time
mm/s
bar
bar
sec
bar
°C
sec
°C
mm
cm3
sec
I
S
I
P
I
P
C
t
P
P
Mo
T
P
t
M
T
W
1
W
2
Cyc
t
1
15
450
457
10
100
15
9
200
5.8
3.97
31.1
2
15
450
457
30
100
15
3
200
6.8
5.17
43.9
3
15
450
457
10
400
15
3
200
6
4.59
24.7
4
15
450
457
30
400
15
9
200
2.9
2.45
49.8
5
15
800
462
10
100
15
3
200
4.5
5.26
24.5
6
15
800
461
30
100
15
9
200
3.4
3.95
49.8
7
15
800
460
10
400
15
9
200
3.9
2.55
31.4
8
15
800
461
30
400
15
3
200
6
4.49
43.8
9
60
450
452
10
100
15
3
200
4.4
5.38
23.6
10
60
450
452
30
100
15
9
200
3.7
4
48.9
11
60
450
452
10
400
15
9
200
3.5
2.63
30.3
12
60
450
452
30
400
15
3
200
7.7
4.62
42.7
13
60
800
785
10
100
15
9
200
5.8
4.34
29.0
14
60
800
783
30
100
15
3
200
9.1
5.21
41.9
15
60
800
791
10
400
15
3
200
7
4.1
22.9
16
60
800
787
30
400
15
9
200
2.9
1.97
47.9
17
15
450
447
10
100
45
3
200
4.7
5.33
25.6
18
15
450
445
30
100
45
9
200
9
4.05
49.9
19
15
450
450
10
400
45
9
200
5.1
3.24
31.2
20
15
450
447
30
400
45
3
200
6.8
5.09
43.8
21
15
800
445
10
100
45
9
200
4
4.1
30.9
22
15
800
446
30
100
45
3
200
7.9
5.55
48.8
23
15
800
444
10
400
45
3
200
9.6
5.17
24.8
24
15
800
446
30
400
45
9
200
4
3.15
49.8
25
60
450
453
10
100
45
9
200
4.5
4.19
29.8
26
60
450
452
30
100
45
3
200
7.2
5.68
42.7
27
60
450
453
10
400
45
3
200
8.6
5.31
23.3
28
60
450
453
30
400
45
9
200
4.5
3.25
48.7
29
60
800
783
10
100
45
3
200
11
5.53
22.4
30
60
800
779
30
100
45
9
200
3.8
4.14
47.9
31
60
800
781
10
400
45
9
200
6
2.28
29.4
32
60
800
780
30
400
45
3
200
7.9
4.27
41.9
33
15
450
343
10
100
15
3
250
8.4
5.92
24.4
34
15
450
342
30
100
15
9
250
5.2
3.85
49.7
35
15
450
343
10
400
15
9
250
4.5
3.12
31.4
36
15
450
344
30
400
15
3
250
8
5.35
43.7
37
15
800
345
10
100
15
9
250
5
3.95
31.2
38
15
800
345
30
100
15
3
250
7.6
5.85
43.7
39
15
800
348
10
400
15
3
250
6.7
5.48
24.6
40
15
800
349
30
400
15
9
250
6.4
3.02
49.7
41
60
450
451
10
100
15
9
250
4.8
4.15
29.5
42
60
450
451
30
100
15
3
250
6
6.06
42.1
43
60
450
452
10
400
15
3
250
7.7
5.83
22.9
44
60
450
452
30
400
15
9
250
5.2
3.2
48.1
45
60
800
639
10
100
15
3
250
16
6.16
22.6
46
60
800
639
30
100
15
9
250
7.1
4.07
47.8
47
60
800
639
10
400
15
9
250
6
3.29
29.5
48
60
800
642
30
400
15
3
250
9.1
5.48
41.9
49
15
450
333
10
100
45
9
250
5.8
4.54
30.8
50
15
450
334
30
100
45
3
250
7.9
6.21
43.8
51
15
450
335
10
400
45
3
250
9.9
5.95
24.3
52
15
450
338
30
400
45
9
250
8.4
3.84
49.8
53
15
800
336
10
100
45
3
250
13.6
6.25
24.3
54
15
800
338
30
100
45
9
250
8.8
4.43
49.7
55
15
800
338
10
400
45
9
250
7
3.94
31.0
56
15
800
337
30
400
45
3
250
8.6
5.87
43.9
57
60
450
451
10
100
45
3
250
8.1
6.49
22.6
58
60
450
452
30
100
45
9
250
7.6
4.63
48.2
59
60
450
452
10
400
45
9
250
9.3
4.13
29.4
60
60
450
452
30
400
45
3
250
8.3
6.11
42.2
61
37.5
625
625
20
250
30
6
200
6.2
4.2
35.4
62
37.5
625
511
20
250
30
6
250
8
4.84
35.3
63
37.5
625
625
20
250
15
6
225
9
4.26
35.3
64
37.5
625
625
20
250
45
6
225
3.6
4.87
35.4
65
15
625
395
20
250
30
6
225
7.3
4.31
35.9
66
60
625
628
20
250
30
6
225
7.6
4.51
35.0
67
37.5
450
452
20
250
30
6
225
6.5
4.44
35.6
68
37.5
800
580
20
250
30
6
225
7.6
4.45
35.3
69
37.5
625
625
20
100
30
6
225
4.8
4.97
35.4
70
37.5
625
625
20
400
30
6
225
7.5
4.27
35.4
71
37.5
625
625
10
250
30
6
225
6.3
4.61
26.1
72
37.5
625
625
30
250
30
6
225
5.7
4.5
45.4
73
37.5
625
625
20
250
30
3
225
7.4
5.61
32.4
74
37.5
625
625
20
250
30
9
225
5.6
3.63
38.3
75
37.5
625
625
20
250
30
6
225
7.2
4.54
35.3
76
60
800
800
10
100
45
9
250
7.65
4.31
28.2
77
60
800
800
30
100
45
3
250
8.01
5.66
42.2
78
60
800
800
10
400
45
3
250
11.2
4.53
21.9
79
60
800
800
30
400
45
9
250
4.88
2.51
47.4


## 2.1. Test Points for Injection Molding Experiments

The three relationships, mentioned previously, are constructed from data obtained through actual injection molding experiments performed at combination sets of the process parameters. These combination sets are selected through DOE in a manner that ensures that the design space is adequately covered. In this study, the central composite design (CCD) was adopted. The CCD is a factorial or fractional factorial design with center points which are augmented with star points that enhance the estimation of curvature. Depending on the location of the star point, the CCD can either be circumscribed, inscribed, or face-centered. In the CCD, the sample size is determined according to the following equation:(1)2f+2f−q+1,where f is the number of factors being considered (i.e., the process parameters in the current study) and q refers to the fraction of the full factorial to be used. For a case where there are numerous factors (e.g., f > 5), a full factorial design (q = 0) results in an enormous sample size. In such a case, it is advisable to use the fractional factorial design which utilizes only a fraction of the full factorial design without losing its main benefits. A detailed treatment of the DOE can be found in [39].The complete design space for the seven process parameters is based on the upper and lower bounds of the parameters. The corresponding bounds of these parameters are listed in Table2. The upper and lower bounds of the injection speed, cooling time, packing pressure, and packing times were suggested by experienced plastic technicians, whereas the bounds for the injection pressure, mold temperature, and melt temperature are based on the recommended ranges from the material producers [38]. For the current study, a half-fraction (q = 1) face-centered central composite design was used. Accordingly, for the seven parameters, 79 data points were generated.Table 2
Lower and upper bounds for the test variables (bound for both the DOE and optimization).
Test variables
Units
Lower bound
Upper bound
Injection speed
mm/sec
15
60
Injection pressure
bar
450
800
Cooling time
sec
10
30
Packing pressure
bar
100
400
Mold temperature
°C
15
45
Packing time
sec
3
9
Melt temperature
°C
200
250


## 2.2. Injection Molding Product Defect Quantification

Two relations relating the two defects (warpage and volumetric shrinkage) to the seven parameters need to be established. In order to achieve this, it is first necessary to quantify the defects. The defect quantification adopted in this work is described as follows.


### 2.2.1. Warpage Quantification

The warpage experienced in injection molding products can be defined as a deviation of the product geometry from the anticipated geometry. In this study, the warpage is defined as the sum of the maximum out-of-plane displacement of the product edges which can be expressed as(2)Wsum=∑ymaxi,i=1,2,3,4.In equation (2) ymaxi refers to the maximum out-of-plane displacement along the four edges of the product. This definition has previously been used in [40].


### 2.2.2. Volumetric Shrinkage Quantification

The volumetric shrinkage can be estimated by determining the difference between the anticipated product volumeVanticipated and the actual product volumes. The anticipated product volume is computed from the expected product dimensions (based on the diagram in Figure 2), whereas the actual product volume is computed from the known product density and measured mass. The volumetric shrinkage Vshrinkage can, thus, be expressed as(3)Vshrinkage=Vanticipated−mρ.This expression has also been previously used in [40].


## 2.2.1. Warpage Quantification

The warpage experienced in injection molding products can be defined as a deviation of the product geometry from the anticipated geometry. In this study, the warpage is defined as the sum of the maximum out-of-plane displacement of the product edges which can be expressed as(2)Wsum=∑ymaxi,i=1,2,3,4.In equation (2) ymaxi refers to the maximum out-of-plane displacement along the four edges of the product. This definition has previously been used in [40].


## 2.2.2. Volumetric Shrinkage Quantification

The volumetric shrinkage can be estimated by determining the difference between the anticipated product volumeVanticipated and the actual product volumes. The anticipated product volume is computed from the expected product dimensions (based on the diagram in Figure 2), whereas the actual product volume is computed from the known product density and measured mass. The volumetric shrinkage Vshrinkage can, thus, be expressed as(3)Vshrinkage=Vanticipated−mρ.This expression has also been previously used in [40].


## 2.3. Injection Molding Experiments

Injection molding experiments were performed on the 79 data points (previously generated). Pictures of samples of products from the experiments are shown in Figure3. Figure 3(a) shows a product with severe volumetric shrinkage, and Figure 3(b) shows a product with severe warpage. A product with minimal defects is shown in Figure 3(c).Figure 3
Products resulting from the injection molding experiments. (a) Product with severe volumetric shrinkage (warpage of 8.1 mm and volumetric shrinkage of 6.49 mm3), (b) product with severe warpage (warpage of 13.6 mm and volumetric shrinkage of 6.25 mm3), and (c) acceptable product (warpage of 2.9 mm and volumetric shrinkage of 2.45 mm3).
(a)
(b)
(c)Product defect, including warpage and volumetric shrinkage, was computed for the 79 products (using equations (2) and (3), respectively) and are reported in Table 3. Also listed in Table 3 are the corresponding cycle times for each of the 79 products.Table 3
Experimental results for warpage, volumetric shrinkage, and cycle time.
Injection speed
Injection pressure
Measured injection pressure
Cooling time
Packing pressure
Mold temp.
Packing time
Melt temp.
Warpage
Volumetric shrinkage
Cycle time
mm/s
bar
bar
sec
bar
°C
sec
°C
mm
cm3
sec
I
S
I
P
I
P
C
t
P
P
Mo
T
P
t
M
T
W
1
W
2
Cyc
t
1
15
450
457
10
100
15
9
200
5.8
3.97
31.1
2
15
450
457
30
100
15
3
200
6.8
5.17
43.9
3
15
450
457
10
400
15
3
200
6
4.59
24.7
4
15
450
457
30
400
15
9
200
2.9
2.45
49.8
5
15
800
462
10
100
15
3
200
4.5
5.26
24.5
6
15
800
461
30
100
15
9
200
3.4
3.95
49.8
7
15
800
460
10
400
15
9
200
3.9
2.55
31.4
8
15
800
461
30
400
15
3
200
6
4.49
43.8
9
60
450
452
10
100
15
3
200
4.4
5.38
23.6
10
60
450
452
30
100
15
9
200
3.7
4
48.9
11
60
450
452
10
400
15
9
200
3.5
2.63
30.3
12
60
450
452
30
400
15
3
200
7.7
4.62
42.7
13
60
800
785
10
100
15
9
200
5.8
4.34
29.0
14
60
800
783
30
100
15
3
200
9.1
5.21
41.9
15
60
800
791
10
400
15
3
200
7
4.1
22.9
16
60
800
787
30
400
15
9
200
2.9
1.97
47.9
17
15
450
447
10
100
45
3
200
4.7
5.33
25.6
18
15
450
445
30
100
45
9
200
9
4.05
49.9
19
15
450
450
10
400
45
9
200
5.1
3.24
31.2
20
15
450
447
30
400
45
3
200
6.8
5.09
43.8
21
15
800
445
10
100
45
9
200
4
4.1
30.9
22
15
800
446
30
100
45
3
200
7.9
5.55
48.8
23
15
800
444
10
400
45
3
200
9.6
5.17
24.8
24
15
800
446
30
400
45
9
200
4
3.15
49.8
25
60
450
453
10
100
45
9
200
4.5
4.19
29.8
26
60
450
452
30
100
45
3
200
7.2
5.68
42.7
27
60
450
453
10
400
45
3
200
8.6
5.31
23.3
28
60
450
453
30
400
45
9
200
4.5
3.25
48.7
29
60
800
783
10
100
45
3
200
11
5.53
22.4
30
60
800
779
30
100
45
9
200
3.8
4.14
47.9
31
60
800
781
10
400
45
9
200
6
2.28
29.4
32
60
800
780
30
400
45
3
200
7.9
4.27
41.9
33
15
450
343
10
100
15
3
250
8.4
5.92
24.4
34
15
450
342
30
100
15
9
250
5.2
3.85
49.7
35
15
450
343
10
400
15
9
250
4.5
3.12
31.4
36
15
450
344
30
400
15
3
250
8
5.35
43.7
37
15
800
345
10
100
15
9
250
5
3.95
31.2
38
15
800
345
30
100
15
3
250
7.6
5.85
43.7
39
15
800
348
10
400
15
3
250
6.7
5.48
24.6
40
15
800
349
30
400
15
9
250
6.4
3.02
49.7
41
60
450
451
10
100
15
9
250
4.8
4.15
29.5
42
60
450
451
30
100
15
3
250
6
6.06
42.1
43
60
450
452
10
400
15
3
250
7.7
5.83
22.9
44
60
450
452
30
400
15
9
250
5.2
3.2
48.1
45
60
800
639
10
100
15
3
250
16
6.16
22.6
46
60
800
639
30
100
15
9
250
7.1
4.07
47.8
47
60
800
639
10
400
15
9
250
6
3.29
29.5
48
60
800
642
30
400
15
3
250
9.1
5.48
41.9
49
15
450
333
10
100
45
9
250
5.8
4.54
30.8
50
15
450
334
30
100
45
3
250
7.9
6.21
43.8
51
15
450
335
10
400
45
3
250
9.9
5.95
24.3
52
15
450
338
30
400
45
9
250
8.4
3.84
49.8
53
15
800
336
10
100
45
3
250
13.6
6.25
24.3
54
15
800
338
30
100
45
9
250
8.8
4.43
49.7
55
15
800
338
10
400
45
9
250
7
3.94
31.0
56
15
800
337
30
400
45
3
250
8.6
5.87
43.9
57
60
450
451
10
100
45
3
250
8.1
6.49
22.6
58
60
450
452
30
100
45
9
250
7.6
4.63
48.2
59
60
450
452
10
400
45
9
250
9.3
4.13
29.4
60
60
450
452
30
400
45
3
250
8.3
6.11
42.2
61
37.5
625
625
20
250
30
6
200
6.2
4.2
35.4
62
37.5
625
511
20
250
30
6
250
8
4.84
35.3
63
37.5
625
625
20
250
15
6
225
9
4.26
35.3
64
37.5
625
625
20
250
45
6
225
3.6
4.87
35.4
65
15
625
395
20
250
30
6
225
7.3
4.31
35.9
66
60
625
628
20
250
30
6
225
7.6
4.51
35.0
67
37.5
450
452
20
250
30
6
225
6.5
4.44
35.6
68
37.5
800
580
20
250
30
6
225
7.6
4.45
35.3
69
37.5
625
625
20
100
30
6
225
4.8
4.97
35.4
70
37.5
625
625
20
400
30
6
225
7.5
4.27
35.4
71
37.5
625
625
10
250
30
6
225
6.3
4.61
26.1
72
37.5
625
625
30
250
30
6
225
5.7
4.5
45.4
73
37.5
625
625
20
250
30
3
225
7.4
5.61
32.4
74
37.5
625
625
20
250
30
9
225
5.6
3.63
38.3
75
37.5
625
625
20
250
30
6
225
7.2
4.54
35.3
76
60
800
800
10
100
45
9
250
7.65
4.31
28.2
77
60
800
800
30
100
45
3
250
8.01
5.66
42.2
78
60
800
800
10
400
45
3
250
11.2
4.53
21.9
79
60
800
800
30
400
45
9
250
4.88
2.51
47.4


## 3. Injection Molding Parameter Optimization

The injection molding process parameters for reduced cycle time subjected to constraints on the product defect are to be determined through optimization. To achieve this, it is necessary to develop the relationship between the process parameters and the cycle time. It is also necessary to develop two other relationships between the two product defects and the process parameters. The former relationship can then be optimized subject to constraints on the defects (as described by the latter two relationships). A description of the development of the three relationships followed by the formulation of the optimization problem is presented in the following sections.


### 3.1. Kriging Model

A variety of models have previously been used to represent the relationship between the process parameters and the product defect. Some of these techniques include the artificial neural network [41–47], support vector regression [48], kriging [40, 49], and response surface method [50–52]. In this work, we adopt the kriging model to develop the relationship between the process parameters and the product defect and the relationship between the process parameters and the product cycle time.Kriging is a regression interpolation technique developed by geologists in an attempt to predict the properties of minerals in a specified region based on knowledge of properties in other neighboring regions [53]. For this study, the Kriging model as well as the Matlab Kriging toolbox described in [54] were used.The data obtained from the injection molding experiments, shown in Table3, were used to generate the three relationships using the kriging Matlab toolbox. The three relationships have been plotted for the various process parameters, as shown in Figures 4–6, respectively.Figure 4
Plots showing the relation between cycle time and various process parameters. The plots are for cycle time versus (a) melt temperature and mold temperature, (b) cooling time and injection speed, (c) injection pressure and packing pressure, and (d) packing time and mold temperature.
(a)
(b)
(c)
(d)Figure 5
Plots showing the relation between warpage and various process parameters. The plots are for warpage verses (a) melt temperature and mold temperature, (b) cooling time and injection speed, (c) injection pressure and packing pressure, and (d) packing time and mold temperature.
(a)
(b)
(c)
(d)Figure 6
Plots showing the relation between volumetric shrinkage and various process parameters. The plots are for volumetric shrinkage versus (a) melt temperature and mold temperature, (b) cooling time and injection speed, (c) injection pressure and packing pressure, and (d) packing time and mold temperature.
(a)
(b)
(c)
(d)These plots provide some insight into the relationships between the cycle time and the process parameters as well as the relationship between the two defects and the process parameters. For instance, in Figure4(a), it can be seen that the cycle time increases with decreasing melt temperature. It is observed in Figures 4(b) and 4(d) that the cycle time increases with increasing cooling time and increasing packing time, respectively, which is an expected behavior. In Figure 4(c) it is also observed that the cycle time increases with decreasing injection pressure.The plots in Figure5 reveal the behavior of the warpage as a function of the process parameters. In Figure 5(a), it is observed that the warpage increases with increasing melt temperature as well as increasing mold temperature (also observed in Figure 5(d)). From the plots in Figures 5(b)–5(d), it can be deduced that the warpage decreases with increasing cooling time, increasing injection pressure, increasing packing pressure, and increasing packing time.The effects of the various process parameters on the volumetric shrinkage may be observed in the plots of Figure6. It can be seen in Figure 6(a) that the volumetric shrinkage increases for an increase in both the melt temperature and mold temperature (also seen in Figure 6(d)). It can also be observed that the volumetric shrinkage increases for a decrease in the cooling time, decrease in the injection speed, a decrease in the injection pressure, a decrease in the packing pressure, and a decrease in the packing time.Finally, from the plots in Figures4–6, some important statements can be made regarding the effect of the process parameters on the product defect and the product cycle time. By observing Figures 4(b), 4(d), 5(b), 5(d), 6(b), and 6(d), it can be stated that the injection speed, the cooling time, and the packing time have the opposing effect of increasing the cycle time while reducing the product defects (or increasing the product defect while reducing the product cycle time). On the other hand, it can be inferred from Figures 4(a), 4(c), 5(a), 5(c), 6(a) and 6(c), that the injection pressure the mold temperature and the melt temperature have a supporting effect of lowering the product cycle time while lowering the product defect. These observations and inferences have been summarized in Table 4.Table 4
Effect of process parameters on the on the product defect and the product cycle time.
Process parameters
Warpage
Volumetric shrinkage
Comment
Injection speed
Opposing effect
Opposing effect
Reducing the injection speed decreases the warpage and vol. shrinkage but increases the cycle time
Cooling time
Increasing the cooling time decreases the warpage and vol. shrinkage but increases the cycle time
Packing time
Increasing the packing time decreases the warpage and vol. shrinkage but increases the cycle time
Injection pressure
Assisting effect
Assisting effect
Increasing the injection pressure decreases the warpage and vol. shrinkage while decreasing the cycle time
Mold temperature
Reducing the mold temperature decreases the warpage and vol. shrinkage and decreases the cycle time
Melt temperature
Reducing the melt temperature decreases the warpage and vol. shrinkage and decreases the cycle time
Packing pressure
Not clear
Not clear
Increasing the packing pressure decreases the warpage and vol. shrinkage; however, from Figure4c, the effect on the cycle time is not clear


### 3.2. Parameter Optimization

The three relationships, previously developed, can be used to set up an optimization problem that can be solved in order to determine the best combination of process parameters that will lead to the minimum product cycle time while taking into account the product quality (or extent of product defect). The optimization problem can be expressed as(4)MinimizeF=CyctIS,IP,Ct,PP,MoT,Pt,MT,Such thatWsumIS,IP,Ct,PP,MoT,Pt,MT≤WallowableVshrinkageIS,IP,Ct,PP,MoT,Pt,MT≤Vallowable15≤IS≤60mm/s450≤IP≤800bar10≤Ct≤30sec100≤PP≤400bar15≤MoT≤45°C3≤Pt≤9sec200≤MT≤250°C.In equation (4), Cyct is the cycle time to be minimized whereas, IS, IP, Ct, PP, MOTPt, and MT refer to the injection speed, the injection pressure, the cooling time, the packing pressure, the mold temperature, the packing time, and the melt temperature, respectively. Also, in equation (4), Vallowable and Wallowable refer to the maximum allowable values of the volumetric shrinkage and warpage, respectively. They indicate the limits of the defect that may be considered acceptable on the product and should not be exceeded. To solve the optimization problem, it is sufficient to specify appropriate values for Vallowable and Wallowable. However, to gain an insight into the problem, a range of values between the lower and extreme values of the product defects (see Table 3) were specified for Vallowable and Wallowable. In particular, 100 sample points were selected in a grid-like fashion to cover the domain within the extremums. The extreme lower and upper defect values for the warpage and the volumetric shrinkage are listed, for convenience, in Table 5.Table 5
Extreme values of the defect as reported in Table3.
Lower extreme
Upper extreme
V
allowable
1.97
6.49
W
allowable
2.9
16The optimization problem was solved (for each of the 100 sample points ofVallowable and Wallowable) using the Fmincon function, available in the Matlab Optimization toolbox. The Fmincon function utilizes a sequential quadratic programming (SQP) algorithm which is an iterative technique used to solve nonlinearly constrained optimization problems. This technique has been found to be an effective tool when dealing with such problems, and further details of the Matlab function can be found in [55].The optimization results are presented in Table6. The second and third columns in the table list the values of Wallowable and Vallowable whereas the remaining columns list the optimized parameters and the minimum product cycle time for the corresponding optimization. The results in Table 6 have also been plotted in Figure 7. The plot illustrates the competing objectives of lowering the cycle time and lowering the product defects. It can be seen from the plot that as the volumetric shrinkage constraint and warpage constraints are lowered, the optimized product cycle time increases. This behavior can be explained by previous observations as detailed in Table 4.Table 6
Extreme values of the defects maximum allowable values for the volumetric shrinkage and warpage with corresponding optimization results.
W
allowable
V
allowable
Optimized parameters
Optimum product cycle time
I
S
I
P
C
t
P
P
Mo
T
P
t
M
T
Cyc
t
mm
cm3
mm/s
bar
sec
bar
°C
sec
°C
sec
1
2.90
1.97
60.00
800.00
30.00
400.00
15.00
9.00
210.58
47.63
2
2.90
2.47
60.00
800.00
29.71
400.00
27.74
9.00
215.93
47.12
3
2.90
2.97
60.00
800.00
29.71
400.00
27.74
9.00
215.93
47.12
4
2.90
3.48
60.00
800.00
29.71
400.00
27.74
9.00
215.93
47.12
5
2.90
3.98
60.00
800.00
29.71
400.00
27.74
9.00
215.93
47.12
6
2.90
4.48
60.00
800.00
29.71
400.00
27.74
9.00
215.93
47.12
7
2.90
4.98
60.00
800.00
29.71
400.00
27.74
9.00
215.93
47.12
8
2.90
5.49
60.00
800.00
29.71
400.00
27.74
9.00
215.93
47.12
9
2.90
5.99
60.00
800.00
29.71
400.00
27.74
9.00
215.93
47.12
10
2.90
6.49
60.00
800.00
29.71
400.00
27.74
9.00
215.93
47.12
11
4.36
1.97
60.00
800.00
25.57
400.00
15.00
9.00
200.00
43.23
12
4.36
2.47
60.00
800.00
10.00
400.00
22.65
8.92
200.00
29.06
13
4.36
2.97
60.00
800.00
10.00
400.00
22.65
8.92
200.00
29.06
14
4.36
3.48
60.00
800.00
10.00
400.00
22.65
8.92
200.00
29.06
15
4.36
3.98
60.00
800.00
10.00
400.00
22.65
8.92
200.00
29.06
16
4.36
4.48
60.00
800.00
10.00
400.00
22.65
8.92
200.00
29.06
17
4.36
4.98
60.00
800.00
10.00
400.00
22.65
8.92
200.00
29.06
18
4.36
5.49
60.00
800.00
10.00
400.00
22.65
8.92
200.00
29.06
19
4.36
5.99
60.00
800.00
10.00
400.00
22.65
8.92
200.00
29.06
20
4.36
6.49
60.00
800.00
10.00
400.00
22.65
8.92
200.00
29.06
21
5.81
1.97
60.00
800.00
25.57
400.00
15.00
9.00
200.00
43.23
22
5.81
2.47
60.00
800.00
10.00
400.00
23.62
7.89
200.00
27.77
23
5.81
2.97
60.00
800.00
10.00
400.00
26.67
6.82
200.00
26.45
24
5.81
3.48
60.00
800.00
10.00
400.00
26.67
6.82
200.00
26.45
25
5.81
3.98
60.00
800.00
10.00
400.00
26.67
6.82
200.00
26.45
26
5.81
4.48
60.00
800.00
10.00
400.00
26.67
6.82
200.00
26.45
27
5.81
4.98
60.00
800.00
10.00
400.00
26.67
6.82
200.00
26.45
28
5.81
5.49
60.00
800.00
10.00
400.00
26.67
6.82
200.00
26.45
29
5.81
5.99
60.00
800.00
10.00
400.00
26.67
6.82
200.00
26.45
30
5.81
6.49
60.00
800.00
10.00
400.00
26.67
6.82
200.00
26.45
31
7.27
1.97
60.00
800.00
25.57
400.00
15.00
9.00
200.00
43.23
32
7.27
2.47
60.00
800.00
10.00
400.00
23.62
7.89
200.00
27.77
33
7.27
2.97
60.00
800.00
10.00
400.00
25.07
6.15
200.00
25.71
34
7.27
3.48
60.00
800.00
10.00
400.00
28.70
4.94
200.00
24.33
35
7.27
3.98
60.00
800.00
10.00
400.00
28.70
4.94
200.00
24.33
36
7.27
4.48
60.00
800.00
10.00
400.00
28.70
4.94
200.00
24.33
37
7.27
4.98
60.00
800.00
10.00
400.00
28.70
4.94
200.00
24.33
38
7.27
5.49
60.00
800.00
10.00
400.00
28.70
4.94
200.00
24.33
39
7.27
5.99
60.00
800.00
10.00
400.00
28.70
4.94
200.00
24.33
40
7.27
6.49
60.00
800.00
10.00
400.00
28.70
4.94
200.00
24.33
41
8.72
1.97
60.00
800.00
25.57
400.00
15.00
9.00
200.00
43.23
42
8.72
2.47
60.00
800.00
10.00
400.00
23.62
7.89
200.00
27.77
43
8.72
2.97
60.00
800.00
10.00
400.00
25.07
6.15
200.00
25.71
44
8.72
3.48
60.00
800.00
10.00
400.00
25.84
4.59
200.00
23.99
45
8.72
3.98
60.00
800.00
10.00
400.00
29.53
3.24
200.00
22.55
46
8.72
4.48
60.00
800.00
10.00
400.00
29.53
3.24
200.00
22.55
47
8.72
4.98
60.00
800.00
10.00
400.00
29.53
3.24
200.00
22.55
48
8.72
5.49
60.00
800.00
10.00
400.00
29.53
3.24
200.00
22.55
49
8.72
5.99
60.00
800.00
10.00
400.00
29.53
3.24
200.00
22.55
50
8.72
6.49
60.00
800.00
10.00
400.00
29.53
3.24
200.00
22.55
51
10.18
1.97
60.00
800.00
25.57
400.00
15.00
9.00
200.00
43.23
52
10.18
2.47
60.00
800.00
10.00
400.00
23.62
7.89
200.00
27.77
53
10.18
2.97
60.00
800.00
10.00
400.00
25.07
6.15
200.00
25.71
54
10.18
3.48
60.00
800.00
10.00
400.00
25.84
4.59
200.00
23.99
55
10.18
3.98
60.00
800.00
10.00
400.00
26.15
3.17
200.00
22.50
56
10.18
4.48
60.00
800.00
10.00
353.70
30.59
3.00
222.80
21.97
57
10.18
4.98
60.00
800.00
10.00
279.05
31.77
3.00
224.61
21.83
58
10.18
5.49
60.00
800.00
10.00
279.05
31.77
3.00
224.61
21.83
59
10.18
5.99
60.00
800.00
10.00
279.05
31.77
3.00
224.61
21.83
60
10.18
6.49
60.00
800.00
10.00
279.05
31.77
3.00
224.61
21.83
61
11.63
1.97
60.00
800.00
25.57
400.00
15.00
9.00
200.00
43.23
62
11.63
2.47
60.00
800.00
10.00
400.00
23.62
7.89
200.00
27.77
63
11.63
2.97
60.00
800.00
10.00
400.00
25.07
6.15
200.00
25.71
64
11.63
3.48
60.00
800.00
10.00
400.00
25.84
4.59
200.00
23.99
65
11.63
3.98
60.00
800.00
10.00
400.00
26.15
3.17
200.00
22.50
66
11.63
4.48
60.00
800.00
10.00
353.70
30.59
3.00
222.80
21.97
67
11.63
4.98
60.00
800.00
10.00
263.43
33.26
3.00
245.91
21.62
68
11.63
5.49
60.00
800.00
10.00
246.30
33.21
3.00
244.94
21.62
69
11.63
5.99
60.00
800.00
10.00
246.30
33.21
3.00
244.94
21.62
70
11.63
6.49
60.00
800.00
10.00
246.30
33.21
3.00
244.94
21.62
71
13.09
1.97
60.00
800.00
25.57
400.00
15.00
9.00
200.00
43.23
72
13.09
2.47
60.00
800.00
10.00
400.00
23.62
7.89
200.00
27.77
73
13.09
2.97
60.00
800.00
10.00
400.00
25.07
6.15
200.00
25.71
74
13.09
3.48
60.00
800.00
10.00
400.00
25.84
4.59
200.00
23.99
75
13.09
3.98
60.00
800.00
10.00
400.00
26.15
3.17
200.00
22.50
76
13.09
4.48
60.00
800.00
10.00
353.70
30.59
3.00
222.80
21.97
77
13.09
4.98
60.00
800.00
10.00
270.32
33.00
3.00
250.00
21.59
78
13.09
5.49
60.00
800.00
10.00
216.84
32.64
3.00
250.00
21.56
79
13.09
5.99
60.00
800.00
10.00
216.84
32.64
3.00
250.00
21.56
80
13.09
6.49
60.00
800.00
10.00
216.84
32.64
3.00
250.00
21.56
81
14.54
1.97
60.00
800.00
25.57
400.00
15.00
9.00
200.00
43.23
82
14.54
2.47
60.00
800.00
10.00
400.00
23.62
7.89
200.00
27.77
83
14.54
2.97
60.00
800.00
10.00
400.00
25.07
6.15
200.00
25.71
84
14.54
3.48
60.00
800.00
10.00
400.00
25.84
4.59
200.00
23.99
85
14.54
3.98
60.00
800.00
10.00
400.00
26.15
3.17
200.00
22.50
86
14.54
4.48
60.00
800.00
10.00
353.70
30.59
3.00
222.80
21.97
87
14.54
4.98
60.00
800.00
10.00
270.32
33.00
3.00
250.00
21.59
88
14.54
5.49
60.00
800.00
10.00
216.84
32.64
3.00
250.00
21.56
89
14.54
5.99
60.00
800.00
10.00
216.84
32.64
3.00
250.00
21.56
90
14.54
6.49
60.00
800.00
10.00
216.84
32.64
3.00
250.00
21.56
91
16.00
1.97
60.00
800.00
25.57
400.00
15.00
9.00
200.00
43.23
92
16.00
2.47
60.00
800.00
10.00
400.00
23.62
7.89
200.00
27.77
93
16.00
2.97
60.00
800.00
10.00
400.00
25.07
6.15
200.00
25.71
94
16.00
3.48
60.00
800.00
10.00
400.00
25.84
4.59
200.00
23.99
95
16.00
3.98
60.00
800.00
10.00
400.00
26.15
3.17
200.00
22.50
96
16.00
4.48
60.00
800.00
10.00
353.70
30.59
3.00
222.80
21.97
97
16.00
4.98
60.00
800.00
10.00
270.32
33.00
3.00
250.00
21.59
98
16.00
5.49
60.00
800.00
10.00
216.84
32.64
3.00
250.00
21.56
99
16.00
5.99
60.00
800.00
10.00
216.84
32.64
3.00
250.00
21.56
100
16.00
6.49
60.00
800.00
10.00
216.84
32.64
3.00
250.00
21.56Figure 7
Optimization results showing optimum cycle times for a combination of the volumetric shrinkage and warpage constraints.In order to assess the optimization process, an additional injection molding run was conducted with optimization results chosen randomly from Table6. For this run, process parameters for sample 14 were used. The picture of the resulting product from this run is shown in Figure 8. Using equations (2) and (3), the warpage and the volumetric shrinkage were calculated for this product. The corresponding values are reported in Table 7. Also, in Table 7, the optimization validation experimental results are compared to the optimization results from the simulation. The comparison revealed reasonably close results with a difference in the cycle time, the warpage, and volumetric shrinkage by 6.7%, 3.2%, and 8%, respectively.Figure 8
Picture of the optimization validation product (warpage constraint of 4.36 and volumetric shrinkage constraint of 3.48).Table 7
Comparison of the optimization simulation results to the experimental validation results for the optimum cycle time, warpage, and volumetric shrinkage (the experimental validation was conducted based on the optimized process parameters, and the resulting cycle time, warpage, and volumetric shrinkage are compared to optimum simulation results, as shown in the table).
Optimized cycle time
Warpage
Vol. shrinkage
Optimization results from the simulation (sample 14 in Table6)
29.06
4.36
3.48
Experimental validation results (inj. molding run for sample 14 parameters in Table6)
31.0
4.5
3.2
Difference
6.7%
3.2%
8.0%


## 3.1. Kriging Model

A variety of models have previously been used to represent the relationship between the process parameters and the product defect. Some of these techniques include the artificial neural network [41–47], support vector regression [48], kriging [40, 49], and response surface method [50–52]. In this work, we adopt the kriging model to develop the relationship between the process parameters and the product defect and the relationship between the process parameters and the product cycle time.Kriging is a regression interpolation technique developed by geologists in an attempt to predict the properties of minerals in a specified region based on knowledge of properties in other neighboring regions [53]. For this study, the Kriging model as well as the Matlab Kriging toolbox described in [54] were used.The data obtained from the injection molding experiments, shown in Table3, were used to generate the three relationships using the kriging Matlab toolbox. The three relationships have been plotted for the various process parameters, as shown in Figures 4–6, respectively.Figure 4
Plots showing the relation between cycle time and various process parameters. The plots are for cycle time versus (a) melt temperature and mold temperature, (b) cooling time and injection speed, (c) injection pressure and packing pressure, and (d) packing time and mold temperature.
(a)
(b)
(c)
(d)Figure 5
Plots showing the relation between warpage and various process parameters. The plots are for warpage verses (a) melt temperature and mold temperature, (b) cooling time and injection speed, (c) injection pressure and packing pressure, and (d) packing time and mold temperature.
(a)
(b)
(c)
(d)Figure 6
Plots showing the relation between volumetric shrinkage and various process parameters. The plots are for volumetric shrinkage versus (a) melt temperature and mold temperature, (b) cooling time and injection speed, (c) injection pressure and packing pressure, and (d) packing time and mold temperature.
(a)
(b)
(c)
(d)These plots provide some insight into the relationships between the cycle time and the process parameters as well as the relationship between the two defects and the process parameters. For instance, in Figure4(a), it can be seen that the cycle time increases with decreasing melt temperature. It is observed in Figures 4(b) and 4(d) that the cycle time increases with increasing cooling time and increasing packing time, respectively, which is an expected behavior. In Figure 4(c) it is also observed that the cycle time increases with decreasing injection pressure.The plots in Figure5 reveal the behavior of the warpage as a function of the process parameters. In Figure 5(a), it is observed that the warpage increases with increasing melt temperature as well as increasing mold temperature (also observed in Figure 5(d)). From the plots in Figures 5(b)–5(d), it can be deduced that the warpage decreases with increasing cooling time, increasing injection pressure, increasing packing pressure, and increasing packing time.The effects of the various process parameters on the volumetric shrinkage may be observed in the plots of Figure6. It can be seen in Figure 6(a) that the volumetric shrinkage increases for an increase in both the melt temperature and mold temperature (also seen in Figure 6(d)). It can also be observed that the volumetric shrinkage increases for a decrease in the cooling time, decrease in the injection speed, a decrease in the injection pressure, a decrease in the packing pressure, and a decrease in the packing time.Finally, from the plots in Figures4–6, some important statements can be made regarding the effect of the process parameters on the product defect and the product cycle time. By observing Figures 4(b), 4(d), 5(b), 5(d), 6(b), and 6(d), it can be stated that the injection speed, the cooling time, and the packing time have the opposing effect of increasing the cycle time while reducing the product defects (or increasing the product defect while reducing the product cycle time). On the other hand, it can be inferred from Figures 4(a), 4(c), 5(a), 5(c), 6(a) and 6(c), that the injection pressure the mold temperature and the melt temperature have a supporting effect of lowering the product cycle time while lowering the product defect. These observations and inferences have been summarized in Table 4.Table 4
Effect of process parameters on the on the product defect and the product cycle time.
Process parameters
Warpage
Volumetric shrinkage
Comment
Injection speed
Opposing effect
Opposing effect
Reducing the injection speed decreases the warpage and vol. shrinkage but increases the cycle time
Cooling time
Increasing the cooling time decreases the warpage and vol. shrinkage but increases the cycle time
Packing time
Increasing the packing time decreases the warpage and vol. shrinkage but increases the cycle time
Injection pressure
Assisting effect
Assisting effect
Increasing the injection pressure decreases the warpage and vol. shrinkage while decreasing the cycle time
Mold temperature
Reducing the mold temperature decreases the warpage and vol. shrinkage and decreases the cycle time
Melt temperature
Reducing the melt temperature decreases the warpage and vol. shrinkage and decreases the cycle time
Packing pressure
Not clear
Not clear
Increasing the packing pressure decreases the warpage and vol. shrinkage; however, from Figure4c, the effect on the cycle time is not clear


## 3.2. Parameter Optimization

The three relationships, previously developed, can be used to set up an optimization problem that can be solved in order to determine the best combination of process parameters that will lead to the minimum product cycle time while taking into account the product quality (or extent of product defect). The optimization problem can be expressed as(4)MinimizeF=CyctIS,IP,Ct,PP,MoT,Pt,MT,Such thatWsumIS,IP,Ct,PP,MoT,Pt,MT≤WallowableVshrinkageIS,IP,Ct,PP,MoT,Pt,MT≤Vallowable15≤IS≤60mm/s450≤IP≤800bar10≤Ct≤30sec100≤PP≤400bar15≤MoT≤45°C3≤Pt≤9sec200≤MT≤250°C.In equation (4), Cyct is the cycle time to be minimized whereas, IS, IP, Ct, PP, MOTPt, and MT refer to the injection speed, the injection pressure, the cooling time, the packing pressure, the mold temperature, the packing time, and the melt temperature, respectively. Also, in equation (4), Vallowable and Wallowable refer to the maximum allowable values of the volumetric shrinkage and warpage, respectively. They indicate the limits of the defect that may be considered acceptable on the product and should not be exceeded. To solve the optimization problem, it is sufficient to specify appropriate values for Vallowable and Wallowable. However, to gain an insight into the problem, a range of values between the lower and extreme values of the product defects (see Table 3) were specified for Vallowable and Wallowable. In particular, 100 sample points were selected in a grid-like fashion to cover the domain within the extremums. The extreme lower and upper defect values for the warpage and the volumetric shrinkage are listed, for convenience, in Table 5.Table 5
Extreme values of the defect as reported in Table3.
Lower extreme
Upper extreme
V
allowable
1.97
6.49
W
allowable
2.9
16The optimization problem was solved (for each of the 100 sample points ofVallowable and Wallowable) using the Fmincon function, available in the Matlab Optimization toolbox. The Fmincon function utilizes a sequential quadratic programming (SQP) algorithm which is an iterative technique used to solve nonlinearly constrained optimization problems. This technique has been found to be an effective tool when dealing with such problems, and further details of the Matlab function can be found in [55].The optimization results are presented in Table6. The second and third columns in the table list the values of Wallowable and Vallowable whereas the remaining columns list the optimized parameters and the minimum product cycle time for the corresponding optimization. The results in Table 6 have also been plotted in Figure 7. The plot illustrates the competing objectives of lowering the cycle time and lowering the product defects. It can be seen from the plot that as the volumetric shrinkage constraint and warpage constraints are lowered, the optimized product cycle time increases. This behavior can be explained by previous observations as detailed in Table 4.Table 6
Extreme values of the defects maximum allowable values for the volumetric shrinkage and warpage with corresponding optimization results.
W
allowable
V
allowable
Optimized parameters
Optimum product cycle time
I
S
I
P
C
t
P
P
Mo
T
P
t
M
T
Cyc
t
mm
cm3
mm/s
bar
sec
bar
°C
sec
°C
sec
1
2.90
1.97
60.00
800.00
30.00
400.00
15.00
9.00
210.58
47.63
2
2.90
2.47
60.00
800.00
29.71
400.00
27.74
9.00
215.93
47.12
3
2.90
2.97
60.00
800.00
29.71
400.00
27.74
9.00
215.93
47.12
4
2.90
3.48
60.00
800.00
29.71
400.00
27.74
9.00
215.93
47.12
5
2.90
3.98
60.00
800.00
29.71
400.00
27.74
9.00
215.93
47.12
6
2.90
4.48
60.00
800.00
29.71
400.00
27.74
9.00
215.93
47.12
7
2.90
4.98
60.00
800.00
29.71
400.00
27.74
9.00
215.93
47.12
8
2.90
5.49
60.00
800.00
29.71
400.00
27.74
9.00
215.93
47.12
9
2.90
5.99
60.00
800.00
29.71
400.00
27.74
9.00
215.93
47.12
10
2.90
6.49
60.00
800.00
29.71
400.00
27.74
9.00
215.93
47.12
11
4.36
1.97
60.00
800.00
25.57
400.00
15.00
9.00
200.00
43.23
12
4.36
2.47
60.00
800.00
10.00
400.00
22.65
8.92
200.00
29.06
13
4.36
2.97
60.00
800.00
10.00
400.00
22.65
8.92
200.00
29.06
14
4.36
3.48
60.00
800.00
10.00
400.00
22.65
8.92
200.00
29.06
15
4.36
3.98
60.00
800.00
10.00
400.00
22.65
8.92
200.00
29.06
16
4.36
4.48
60.00
800.00
10.00
400.00
22.65
8.92
200.00
29.06
17
4.36
4.98
60.00
800.00
10.00
400.00
22.65
8.92
200.00
29.06
18
4.36
5.49
60.00
800.00
10.00
400.00
22.65
8.92
200.00
29.06
19
4.36
5.99
60.00
800.00
10.00
400.00
22.65
8.92
200.00
29.06
20
4.36
6.49
60.00
800.00
10.00
400.00
22.65
8.92
200.00
29.06
21
5.81
1.97
60.00
800.00
25.57
400.00
15.00
9.00
200.00
43.23
22
5.81
2.47
60.00
800.00
10.00
400.00
23.62
7.89
200.00
27.77
23
5.81
2.97
60.00
800.00
10.00
400.00
26.67
6.82
200.00
26.45
24
5.81
3.48
60.00
800.00
10.00
400.00
26.67
6.82
200.00
26.45
25
5.81
3.98
60.00
800.00
10.00
400.00
26.67
6.82
200.00
26.45
26
5.81
4.48
60.00
800.00
10.00
400.00
26.67
6.82
200.00
26.45
27
5.81
4.98
60.00
800.00
10.00
400.00
26.67
6.82
200.00
26.45
28
5.81
5.49
60.00
800.00
10.00
400.00
26.67
6.82
200.00
26.45
29
5.81
5.99
60.00
800.00
10.00
400.00
26.67
6.82
200.00
26.45
30
5.81
6.49
60.00
800.00
10.00
400.00
26.67
6.82
200.00
26.45
31
7.27
1.97
60.00
800.00
25.57
400.00
15.00
9.00
200.00
43.23
32
7.27
2.47
60.00
800.00
10.00
400.00
23.62
7.89
200.00
27.77
33
7.27
2.97
60.00
800.00
10.00
400.00
25.07
6.15
200.00
25.71
34
7.27
3.48
60.00
800.00
10.00
400.00
28.70
4.94
200.00
24.33
35
7.27
3.98
60.00
800.00
10.00
400.00
28.70
4.94
200.00
24.33
36
7.27
4.48
60.00
800.00
10.00
400.00
28.70
4.94
200.00
24.33
37
7.27
4.98
60.00
800.00
10.00
400.00
28.70
4.94
200.00
24.33
38
7.27
5.49
60.00
800.00
10.00
400.00
28.70
4.94
200.00
24.33
39
7.27
5.99
60.00
800.00
10.00
400.00
28.70
4.94
200.00
24.33
40
7.27
6.49
60.00
800.00
10.00
400.00
28.70
4.94
200.00
24.33
41
8.72
1.97
60.00
800.00
25.57
400.00
15.00
9.00
200.00
43.23
42
8.72
2.47
60.00
800.00
10.00
400.00
23.62
7.89
200.00
27.77
43
8.72
2.97
60.00
800.00
10.00
400.00
25.07
6.15
200.00
25.71
44
8.72
3.48
60.00
800.00
10.00
400.00
25.84
4.59
200.00
23.99
45
8.72
3.98
60.00
800.00
10.00
400.00
29.53
3.24
200.00
22.55
46
8.72
4.48
60.00
800.00
10.00
400.00
29.53
3.24
200.00
22.55
47
8.72
4.98
60.00
800.00
10.00
400.00
29.53
3.24
200.00
22.55
48
8.72
5.49
60.00
800.00
10.00
400.00
29.53
3.24
200.00
22.55
49
8.72
5.99
60.00
800.00
10.00
400.00
29.53
3.24
200.00
22.55
50
8.72
6.49
60.00
800.00
10.00
400.00
29.53
3.24
200.00
22.55
51
10.18
1.97
60.00
800.00
25.57
400.00
15.00
9.00
200.00
43.23
52
10.18
2.47
60.00
800.00
10.00
400.00
23.62
7.89
200.00
27.77
53
10.18
2.97
60.00
800.00
10.00
400.00
25.07
6.15
200.00
25.71
54
10.18
3.48
60.00
800.00
10.00
400.00
25.84
4.59
200.00
23.99
55
10.18
3.98
60.00
800.00
10.00
400.00
26.15
3.17
200.00
22.50
56
10.18
4.48
60.00
800.00
10.00
353.70
30.59
3.00
222.80
21.97
57
10.18
4.98
60.00
800.00
10.00
279.05
31.77
3.00
224.61
21.83
58
10.18
5.49
60.00
800.00
10.00
279.05
31.77
3.00
224.61
21.83
59
10.18
5.99
60.00
800.00
10.00
279.05
31.77
3.00
224.61
21.83
60
10.18
6.49
60.00
800.00
10.00
279.05
31.77
3.00
224.61
21.83
61
11.63
1.97
60.00
800.00
25.57
400.00
15.00
9.00
200.00
43.23
62
11.63
2.47
60.00
800.00
10.00
400.00
23.62
7.89
200.00
27.77
63
11.63
2.97
60.00
800.00
10.00
400.00
25.07
6.15
200.00
25.71
64
11.63
3.48
60.00
800.00
10.00
400.00
25.84
4.59
200.00
23.99
65
11.63
3.98
60.00
800.00
10.00
400.00
26.15
3.17
200.00
22.50
66
11.63
4.48
60.00
800.00
10.00
353.70
30.59
3.00
222.80
21.97
67
11.63
4.98
60.00
800.00
10.00
263.43
33.26
3.00
245.91
21.62
68
11.63
5.49
60.00
800.00
10.00
246.30
33.21
3.00
244.94
21.62
69
11.63
5.99
60.00
800.00
10.00
246.30
33.21
3.00
244.94
21.62
70
11.63
6.49
60.00
800.00
10.00
246.30
33.21
3.00
244.94
21.62
71
13.09
1.97
60.00
800.00
25.57
400.00
15.00
9.00
200.00
43.23
72
13.09
2.47
60.00
800.00
10.00
400.00
23.62
7.89
200.00
27.77
73
13.09
2.97
60.00
800.00
10.00
400.00
25.07
6.15
200.00
25.71
74
13.09
3.48
60.00
800.00
10.00
400.00
25.84
4.59
200.00
23.99
75
13.09
3.98
60.00
800.00
10.00
400.00
26.15
3.17
200.00
22.50
76
13.09
4.48
60.00
800.00
10.00
353.70
30.59
3.00
222.80
21.97
77
13.09
4.98
60.00
800.00
10.00
270.32
33.00
3.00
250.00
21.59
78
13.09
5.49
60.00
800.00
10.00
216.84
32.64
3.00
250.00
21.56
79
13.09
5.99
60.00
800.00
10.00
216.84
32.64
3.00
250.00
21.56
80
13.09
6.49
60.00
800.00
10.00
216.84
32.64
3.00
250.00
21.56
81
14.54
1.97
60.00
800.00
25.57
400.00
15.00
9.00
200.00
43.23
82
14.54
2.47
60.00
800.00
10.00
400.00
23.62
7.89
200.00
27.77
83
14.54
2.97
60.00
800.00
10.00
400.00
25.07
6.15
200.00
25.71
84
14.54
3.48
60.00
800.00
10.00
400.00
25.84
4.59
200.00
23.99
85
14.54
3.98
60.00
800.00
10.00
400.00
26.15
3.17
200.00
22.50
86
14.54
4.48
60.00
800.00
10.00
353.70
30.59
3.00
222.80
21.97
87
14.54
4.98
60.00
800.00
10.00
270.32
33.00
3.00
250.00
21.59
88
14.54
5.49
60.00
800.00
10.00
216.84
32.64
3.00
250.00
21.56
89
14.54
5.99
60.00
800.00
10.00
216.84
32.64
3.00
250.00
21.56
90
14.54
6.49
60.00
800.00
10.00
216.84
32.64
3.00
250.00
21.56
91
16.00
1.97
60.00
800.00
25.57
400.00
15.00
9.00
200.00
43.23
92
16.00
2.47
60.00
800.00
10.00
400.00
23.62
7.89
200.00
27.77
93
16.00
2.97
60.00
800.00
10.00
400.00
25.07
6.15
200.00
25.71
94
16.00
3.48
60.00
800.00
10.00
400.00
25.84
4.59
200.00
23.99
95
16.00
3.98
60.00
800.00
10.00
400.00
26.15
3.17
200.00
22.50
96
16.00
4.48
60.00
800.00
10.00
353.70
30.59
3.00
222.80
21.97
97
16.00
4.98
60.00
800.00
10.00
270.32
33.00
3.00
250.00
21.59
98
16.00
5.49
60.00
800.00
10.00
216.84
32.64
3.00
250.00
21.56
99
16.00
5.99
60.00
800.00
10.00
216.84
32.64
3.00
250.00
21.56
100
16.00
6.49
60.00
800.00
10.00
216.84
32.64
3.00
250.00
21.56Figure 7
Optimization results showing optimum cycle times for a combination of the volumetric shrinkage and warpage constraints.In order to assess the optimization process, an additional injection molding run was conducted with optimization results chosen randomly from Table6. For this run, process parameters for sample 14 were used. The picture of the resulting product from this run is shown in Figure 8. Using equations (2) and (3), the warpage and the volumetric shrinkage were calculated for this product. The corresponding values are reported in Table 7. Also, in Table 7, the optimization validation experimental results are compared to the optimization results from the simulation. The comparison revealed reasonably close results with a difference in the cycle time, the warpage, and volumetric shrinkage by 6.7%, 3.2%, and 8%, respectively.Figure 8
Picture of the optimization validation product (warpage constraint of 4.36 and volumetric shrinkage constraint of 3.48).Table 7
Comparison of the optimization simulation results to the experimental validation results for the optimum cycle time, warpage, and volumetric shrinkage (the experimental validation was conducted based on the optimized process parameters, and the resulting cycle time, warpage, and volumetric shrinkage are compared to optimum simulation results, as shown in the table).
Optimized cycle time
Warpage
Vol. shrinkage
Optimization results from the simulation (sample 14 in Table6)
29.06
4.36
3.48
Experimental validation results (inj. molding run for sample 14 parameters in Table6)
31.0
4.5
3.2
Difference
6.7%
3.2%
8.0%


## 4. Summary and Concluding Remarks

The work in this study sought to develop an optimization framework for determining the minimum product cycle time while placing constraints on the product defect. The process involved developing three relationships relating the defects (warpage and volumetric shrinkage) and product cycle time, respectively, to seven process parameters (injection speed, injection pressure, cooling time, packing pressure, mold temperature, packing time, and melt temperature). The three relationships were developed based on data obtained through actual injection molding experiments. Surface plots of the three relationships revealed a number of important points. Firstly, that the cooling time and the packing time have the opposing effect of increasing the cycle time while reducing the product defects and secondly that the injection pressure, the mold temperature and the melt temperature have a supporting effect of lowering the product cycle time while lowering the product defect.The relationship for the product cycle time was minimized subject to constraints on the product defects (described by the other two relationships). The optimization results illustrated competing objectives of lowering the cycle time against lowering the product defects. From the two previous observations, it may be concluded that the cooling time and the packing time are responsible for the competing objectives (i.e., in order to have a product with good quality we need to increase the cooling time and packing time which in turn increases the cycle time). On the other hand, the injection pressure, the mold temperature, and the melt temperature are factors that aid in achieving the two objectives.As a means to validate the optimization process, an additional injection molding experiment was performed for one of the optimization results. The additional experimental results showed reasonably close agreement with the simulation optimization results differing in the cycle time, the warpage, and volumetric shrinkage by 6.7%, 3.2%, and 8%, respectively.The optimization framework procedure outlined in this work was demonstrated using a simple product specimen and was able to achieve the intended outcome. There are, however, some important points that should be mentioned regarding the use of the procedure. Firstly, correct defect quantification is key to attaining credible results. Care should thus be taken in determining the required quantification. The product specimen used to demonstrate the procedure is a simple flat specimen whereas real-life products have more complex geometries and will require proper procedures for defect quantification. Secondly, the accuracy of the final results is dictated by how well the developed relationships are able to represent the product cycle time and product defects and how well the optimization routine is able to determine the optimum. It is thus important to compare those relationships to known trends to establish their accuracy before proceeding to the optimization stage. It is also recommended to perform validation tests, such as performed in this study, so as to have better confidence in the results.The matter of cycle time reduction is an important issue in the injection molding industry. This is because reducing cycle time (even seconds) will lead to a substantial gain in production. However, reducing cycle time may have an adverse effect on product quality. The work in this study sought to find solutions to this matter. Based on the results from this work, it is the belief of the author that such an optimization framework can easy be adopted in the industry to aid the injection molding engineer as well as managers in making decisions on the appropriate process parameters that will result in acceptable product cycle times for acceptable product defect.



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*Source: 1309209-2020-03-11.xml*