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Abstract

Injection molding is a widely used polymer manufacturing process, and integrating sustainability is essential. As material testing demand increases, efficient methods for producing multiple samples are crucial. Family molds enable multiple sample production in a single cycle but pose challenges in achieving balanced flow, leading to sink marks and shrinkage that affect test accuracy. This study addresses these issues by analyzing sink marks, shrinkage, and their variance among specimens. A Grey-based Taguchi method systematically evaluates the effects of injection speed, melt temperature, injection pressure, holding pressure, holding time, and cooling time. Optimal conditions were determined as 40 mm/s injection speed, 170°C melt temperature, 110 MPa injection pressure, 35 MPa holding pressure, and a 7-s holding time. ANOVA results indicate that melt temperature, holding time, and holding pressure significantly influence sink marks and shrinkage. Under optimal conditions, the GRD value improved from 0.638 to 0.643. Furthermore, material testing confirmed the reliability of the optimized specimens, with tensile and flexural strengths within a 5% error margin of datasheet values. These findings validate the effectiveness of the proposed method in enhancing sample quality and consistency in family mold injection molding.

Keywords

family mold sink mark shrinkage Grey-based Taguchi method

Introduction

Injection molding technology is a widely utilized technique in the plastics industry, known for its ability to mass-produce products with complex shapes.¹ A specialized approach within this process is family molds, which are multi-cavity systems used in injection molding processes to produce multiple components for assemblies.² Family molds provide an efficient and cost-effective method for manufacturing different plastic parts with the same material and color for small to medium production volumes.³

Since a wide range of materials is used in injection molding across various industries and the quality of the injection molded parts depends on the material,⁴ making the right choice of the material based on mechanical properties is crucial. Several studies investigated the influence of process parameters on mechanical properties in the injection molding process. Ozcelik et al.⁵ conducted a Taguchi’s study to research the influence of injection parameters on the mechanical properties of Acrylonitrile Butadiene Styrene (ABS) material for two mold materials. Mohd et al.⁶ specifically investigated how the optimal processing parameters vary depending on the desired composition ratio. These studies highlight the importance of the injection molding process affecting the mechanical properties. In addition, Farotti and Natalini⁷ manufactured a family mold for simultaneously producing tensile, Charpy, and Hopkinson bar test specimens in each shot to investigate the influence of injection molding parameters on the mechanical properties of polypropylene (PP). This study demonstrates that using a family mold offers a time and cost-effective solution for material testing by allowing multiple specimens to be injected at one time which helps save time and reduces material waste, especially for the testing of sustainable and recycled materials. Therefore, the use of family molds for testing multiple specimens across various material tests has practical applications.

The design of family molds for injection molding presents challenges in optimizing cavities and runner layouts. Because polymer flow imbalance in geometrically balanced multi-cavity injection molds is a complex problem to address.⁸ One of the key challenges in designing the layout of a family mold is the balance of runner dimensions. Several studies have been conducted to achieve uniform filling and part quality across different-sized cavities.⁹ Advanced techniques and tools have enabled manufacturers to reduce costs and material waste in family mold injection molding processes. However, despite these improvements, optimizing cavities and runner layouts can still lead to defects such as sink marks and shrinkage.

Sink marks and shrinkage could be minimized through process parameter optimization. Traditionally, process parameters are often determined by experienced engineers or through a trial-and-error approach, which can be an inefficient and uneconomical way to study process parameter optimization. On the contrary, the Taguchi method utilizes orthogonal arrays which reduce the number of experiments while still capturing the effects of factors, therefore, it lowers material waste and time costs. Multiple studies have employed the Taguchi method and Analysis of Variance (ANOVA) to identify key parameters affecting sink marks and shrinkage. Wibowo et al.¹⁰ optimized injection molding parameters to minimize sink mark defects in the Cowl B (L/R) component using the Taguchi method and ANOVA. Key process variables, including melt temperature, mold temperature, packing pressure, packing time, and cooling time, were analyzed through simulations in Autodesk Moldflow. Shen et al.¹¹ investigated the effect of molding variables on sink marks in injection molded plastic parts by combining numerical simulation with the Taguchi DOE technique. Statistical analysis using ANOVA demonstrated that key parameters-such as holding pressure, melt temperature, and mold temperature-significantly influenced sink mark formation. Jiang et al.¹² researched that External Gas-Assisted Injection Molding (EGAIM) can significantly reduce sink marks with the most important parameters were the cooling time, gas pressure, and gas time. Ryu et al.¹³ investigated the shrinkage behavior of polypropylene composites reinforced with talc and glass fiber. Optimal reinforcement conditions were determined using ANOVA and the Taguchi method providing valuable guidelines for minimizing shrinkage in polypropylene composites. Syed et al.¹⁴ developed a Taguchi-based multi-objective optimization approach for injection-molded talc-filled polypropylene, a common packaging material, to simultaneously enhance tensile strength and reduce shrinkage. Their experiments showed that under optimal conditions the tensile strength increased from 22.07 to 24.40 MPa while shrinkage decreased from 3.25% to 2.28%. These results highlight the value of optimizing multiple quality objectives in injection molding processes. Annicchiarico and Alcock¹⁵ reviewed the factors that affect shrinkage in injection molded parts and reported that material behavior, processing parameters, and mold/specimen design all play significant roles. They noted that key factors such as temperatures, packing parameters, and cooling time are critical. In terms of both sink mark and shrinkage, Singh Solanki et al.¹⁶ conducted an experimental and numerical study investigating the injection molding of polypropylene (PP) gears, focusing on minimizing defects like shrinkage and sink marks to improve gear precision. The above research shows that the Taguchi method is applied to optimize injection process parameters, while ANOVA is used to assess the significance of process conditions in minimizing sink marks and shrinkage. Furthermore, process parameters, namely, melt temperature, mold temperature, packing time, packing pressure, and cooling time were critical factors taken into consideration for sink mark and shrinkage reduction.

A challenge for Taguchi method is to optimize only a single response variable under the effect of multiple parameters. Therefore, recent research have focused on multi-response optimization in injection molding by combining multi-response optimization which allows us to optimize several response variables simultaneously. Cao et al.¹⁷ are presented a multi-objective optimization method for injection-molded plastic parts for warpage, shrinkage, and clamping force. The optimal parameters were obtained, resulting in improved part quality and reduced energy consumption. Lin et al.¹⁸ present a hybrid Taguchi-Gray Relation Analysis method to optimize the metal powder injection molding process for artificial knee joints. In their words, this hybrid optimization strategy effectively balances conflicting design objectives to improve both the aesthetic and mechanical performance of the molded components. Pachorkar et al.¹⁹ present a multi-response optimization framework for injection molding that target reducing sink mark depth while shortening cycle time in transparent thermoplastic polypropylene products. Their study employs a Taguchi-based L27 experimental design combined with a utility concept to balance quality and productivity by identifying optimal process parameters. Tan et al.²⁰ developed a multi-objective optimization framework for manufacturing thin-walled composite connector terminals. They integrated the Taguchi method with grey relational analysis to optimize key injection molding parameters-including melt temperature, mold temperature, injection time, holding pressure/time, and cooling time-to simultaneously minimize defects such as warpage and shrinkage. Their approach enabled the identification of an optimal parameter set that improves dimensional accuracy and overall part quality in high-precision automotive connector components. From the above literature review, Grey-based Taguchi method is commonly used for multiple responses and its latest applications and contributions are in the medical and aerospace areas.²¹

In terms of specimen preparation, one of the critical requirements is to establish specific dimensions and its tolerances for each type of test. Additionally, a uniform cross-section of the specimen helps to minimize the effect of variation along the length of the test specimen, such as tensile testing. In the injection mold process, sink marks are unwanted shallow depressions or dimples appearing on the surface of the molded part and shrinkage is the difference between the size of a mold cavity and the size of the finished part. Both defects have a significant effect on the dimensional accuracy and sectional-cross uniformity. In family molds for multi-response optimization, however, one of the challenges is to identify representative index for analysis when multiple specimens are produced simultaneously. Existing research often focuses on the average response of sink marks and shrinkage, neglecting the critical aspect of variance among simultaneously produced specimens. This oversight can lead to inconsistent part quality, even when average defect levels appear acceptable. However, from a different perspective, variance is crucial for dimensional accuracy and cross-sectional uniformity, particularly in family molds. Therefore, to address this critical gap, this study proposes a comparative analysis of variance in sink marks and shrinkage between specimens.

This study addresses the challenge of optimizing injection molding processes, specifically for family molds producing multiple specimens, where consistent quality and dimensional accuracy are crucial. The primary research in this study focuses on minimizing sink marks and shrinkage aiming to improve dimensional accuracy and uniformity of polypropylene specimens. To overcome the difficulty of analyzing multiple specimens simultaneously, the study proposes a comparative analysis of variances in sink marks and shrinkage. It aims to establish a balanced optimization approach utilizing the Taguchi method, ANOVA, and grey relational analysis (GRA) to determine the significant effects of process conditions on these defects. The research emphasizes the importance of precise specimen preparation, including dimensional control and uniform cross-sections, to ensure reliable testing.

Proposed methodology

Proposed optimization procedures

In preliminary study conducted by our team the combination of the sink mark and the variance in sink mark between specimens was found to be an effective method to solve this problem. The first index focuses on achieving the minimum sink mark, which aims to shift the value toward the optimal goal, thereby reducing the overall defect level. The second index targets the minimum variance in sink mark between specimens within each trial, enhancing process stability and ensuring uniformity in quality across all specimens. By addressing minimization of both the mean and variance, this strategy not only enhances average quality but also guarantees consistent performance, making it a robust solution for optimizing sink marks in complex family molding processes. This study extends the analysis by introducing four indexes: two representing defects (sink mark and shrinkage) and two representing the variance in these defects among specimens, proposed for optimization in family molds.

This study aimed to minimize both sink mark and shrinkage, along with their respective variances. To achieve this, the Taguchi method was employed to determine optimal parametric settings and analyze the effect of process parameters on a single response. However, real-world scenarios often involve multiple responses, making the conventional Taguchi method unsuitable. To address this, the Taguchi method must be integrated with other techniques for efficient multi-objective optimization. Therefore, this study employs the Grey-based Taguchi method, which combines the Taguchi method and Grey Relational Analysis (GRA) to address multiple optimization criteria. The GRA method first established a relationship between responses. Subsequently, Taguchi method is applied to the GRA results to identify the optimal set of process parameters. The ANOVA test was applied to statistically determine the significance of each factor on the designed experimental study within the Grey-based Taguchi method.

Taguchi’s orthogonal array (OA)

The Taguchi method employs a unique design of orthogonal arrays to explore all parameter values, allowing for significantly fewer experiments to address the problem. The total degree of freedom (DOF) is typically used to select an appropriate orthogonal array. Taguchi’s method emphasizes product quality, advocating that quality should be integrated into product design rather than relying on inspection after the product’s release. This method uses the signal-to-noise (S/N) ratio to assess quality characteristics in engineering design problems. The Taguchi method categorizes product characteristics into three types of signal-to-noise ratios, with the optimal outcome determined based on these ratios. In this study, the “smaller the better” approach was used to evaluate four responses.

Grey relational analysis (GRA)

The GRA calculation involves normalizing the results for each case, calculating the Grey Relational Coefficient (GRC), and determining the Grey Relational Grade (GRG). Data pre-processing is required when the range and unit in one data sequence may differ from the others and is also necessary when the sequence scatter range is too large, or when the directions of the target in the sequences are different. Mean values are consequently calculated and normalized between 0 and 1 using the following functions. In terms of this study, the smaller the better was used to calculate as:

y_{i} (k) = \frac{\max x_{i}^{0} (k) - x_{i}^{0} (k)}{\max x_{i}^{0} (k) - \min x_{i}^{0} (k)}

(1)

Where $i$ = 1, …, p, k = 1, …, r, p = number of the cases, and r = number of parameters. The GRC is calculated after normalization is performed, as follows:

ξ_{i} (k) = \frac{Δ_{\min} + φ Δ_{\max}}{Δ_{0 i} (k) + φ Δ_{\max}}

(2)

Δ_{0 i} (k) = y_{0} (k) - y_{i} (k)

(3)

Δ_{\max} = max x_{\forall ji} max x_{\forall k} y_{0} (k) - y_{i} (k)

(4)

Δ_{\min} = min x_{\forall ji} min x_{\forall k} y_{0} (k) - y_{i} (k)

(5)

$Δ_{0 i} (k)$ is the deviation value between $y_{0} (k)$ and $y_{i} (k)$ . φ represents the distinguishing coefficient with φϵ [0, 1], the purpose of the distinguishing coefficient is to adjust the range of reference sequence. However, the adjusted value will change only the relative value and will not affect the order of the grey relational grade, regularly φ is considered as 0.5.^22,23 $Δ_{\min}$ and $Δ_{\max}$ are the minimum and maximum values of the $Δ_{0 i}$ respectively. Grey Relational Grade represents as:

γ_{i} = \frac{1}{n} \sum_{k = 1}^{n} w_{k} ξ_{i} (k)

(6)

where n is the number of trials, $w_{k}$ is the normalized weight value of kth performance characteristic, and $γ_{i}$ is the overall Grey Relational Grade. It is important to calculate $w_{k}$ for each performance characteristic. Grey Relational Grade $γ_{i}$ stands for the level of correlation between the comparative sequence and referential sequence.

Experimental setup

Design of the geometry model and layout

To evaluate the efficiency of sink marks and shrinkage reduction in polymer products, three types of specimens were used: Rock Well test specimens,²⁴ tensile test specimens,²⁵ and bending test specimens.²⁶ These specimens were used to design the layout for family mold. Simulation and design of mold layout were performed using injection molding software Autodesk Moldflow Insight^® to determine the dimensional values of layout (Figure 1).

Figure 1.

Simulation and design layout on Moldflow.

Material and equipment

In this study, a commercial grade of Polypropylene (PP) ABF-6220 was supplied by Lotte Chemical Inc., Korean, with a melt index of 10 g/10 min (at 230°C) was selected for the injection molding process. An injection molding machine (JSW J100ADS, The Japan Steel Works, LTD) was used to inject, which allows for precise adjustment of experimental parameters through a control program (Figure 2). The machine offers a maximum injection pressure of 320 MPa, a maximum injection speed of 350 mm/s (the maximum movement speed of the screw, as mentioned earlier), and a maximum travel distance of 200 mm for the screw. The screw diameter is 100 mm. After specimens were injected, the specimens were post-processed and classified into their corresponding part within a group (Figure 3). The sink marks and shrinkage of the PP were measured using a KEYENCE Image Dimension Measurement System (IM-7000 series) equipped with the IM-7030 measurement head (Figure 4). Finally, the mechanical characteristics of specimens were measured using a Shimadzu universal machine for testing the mechanical characteristics of materials, AGS-X-50 kN, with a unique additional tool for tensile and bending tests (Figure 5).

Figure 2.

Mold plate installation on machine.

Figure 3.

Specimen’s products of injection molding.

Figure 4.

Measurement sink marks and shrinkage on Keyence.

Figure 5.

Shimadzu AGS-X Universal Testing Machine set up for tensile and bending test.

Experimental results and discussion

Result of L18 experiment

To determine the optimal process conditions and assess the impact of the processing parameters on the minimum sink mark, shrinkage and their minimum variance of PP, process parameters of injection molding were optimized by using the Taguchi method. Six process parameters, namely, the injection speed (A), melt temperature (B), injection pressure (C), holding pressure (D), holding time (E), and cooling time (F) were considered as controllable factors. Five factors with three levels and one factor with two levels were studied in this study, as shown in Table 1. According to the recommended range of parameters, an orthogonal table of L18 was established Table 2. The actual injected specimens were shown in Figure 3, with each part from each trial labeled for the measurement process. Each trial was conducted three times to stabilize measured data and reduce the error within the same experimental conditions.

Table 1.

Process parameters and their levels.

Process parameters		Level 1	Level 2	Level 3
Injection speed (mm/s)	A	40	50	—
Melting temperature (°C)	B	190	200	210
Injection pressure (MPa)	C	90	100	110
Holding pressure (MPa)	D	30	35	40
Holding time (s)	E	4	5	6
Cooling time (s)	F	15	20	25

Table 2.

Taguchi’s L18 orthogonal array.

No.	A	B	C	D	E	F
1	1	1	1	1	1	1
2	1	1	2	2	2	2
3	1	1	3	3	3	3
4	2	1	1	3	3	2
5	2	1	2	1	1	3
6	2	1	3	2	2	1
7	1	2	1	1	2	2
8	1	2	2	2	3	3
9	1	2	3	3	1	1
10	2	2	1	2	3	1
11	2	2	2	3	1	2
12	2	2	3	1	2	3
13	1	3	1	2	1	3
14	1	3	2	3	2	1
15	1	3	3	1	3	2
16	2	3	1	3	2	3
17	2	3	2	1	3	1
18	2	3	3	2	1	2

The selected L18 orthogonal array with 18 trials and the average of the measured values of sink mark for each trial are given in Table 3 and the average of the measured values of shrinkage for each trial are given in Table 4. As shown in Table 3, the No. 3 trial coded as A₁B₁C₃D₂E₃F₃ gives the min value of sink mark (R1), while the No. 2 trial coded as A₁B₁C₂D₂E₂F₂ obtains the min value of variance in sink mark (R2). With the same considerations, in Table 4, the No. 4 trial coded as A₁B₁C₃D₃E₃F₃ gives the min value of shrinkage (R3), while the No. 8 trial coded as A₁B₂C₂D₂E₃F₃ obtains the min value of variance in shrinkage (R4). Therefore, to reach the target, the grey-based Taguchi method is employed to balance the two quality characteristics, that is, the sink mark, shrinkage, and their variance. Table 5 shows the step-by-step process for calculating the Grey Relational Grade.

Table 3.

Results of L18 orthogonal array for sink mark.

No.	A	B	C	D	E	F	1-1	1-2	1-3	1-4	2	3	R1	R2
1	1	1	1	1	1	1	0.136	0.129	0.149	0.147	0.081	0.262	0.181	0.081
2	1	1	2	2	2	2	0.143	0.134	0.108	0.103	0.061	0.211	0.150	0.061
3	1	1	3	3	3	3	0.123	0.145	0.106	0.111	0.067	0.182	0.115	0.067
4	2	1	1	3	3	2	0.162	0.155	0.116	0.132	0.098	0.219	0.121	0.098
5	2	1	2	1	1	3	0.149	0.147	0.186	0.193	0.093	0.302	0.209	0.093
6	2	1	3	2	2	1	0.172	0.171	0.125	0.125	0.133	0.243	0.118	0.125
7	1	2	1	1	2	2	0.149	0.145	0.147	0.141	0.114	0.334	0.220	0.114
8	1	2	2	2	3	3	0.147	0.146	0.135	0.143	0.113	0.232	0.119	0.113
9	1	2	3	3	1	1	0.159	0.156	0.199	0.189	0.147	0.338	0.191	0.147
10	2	2	1	2	3	1	0.172	0.152	0.147	0.124	0.130	0.257	0.133	0.124
11	2	2	2	3	1	2	0.161	0.162	0.205	0.212	0.128	0.350	0.222	0.128
12	2	2	3	1	2	3	0.152	0.147	0.169	0.165	0.098	0.330	0.232	0.098
13	1	3	1	2	1	3	0.130	0.137	0.207	0.209	0.114	0.373	0.259	0.114
14	1	3	2	3	2	1	0.184	0.161	0.169	0.155	0.141	0.316	0.175	0.141
15	1	3	3	1	3	2	0.150	0.148	0.122	0.111	0.122	0.357	0.246	0.111
16	2	3	1	3	2	3	0.135	0.169	0.179	0.165	0.103	0.348	0.245	0.103
17	2	3	2	1	3	1	0.147	0.146	0.136	0.131	0.133	0.367	0.236	0.131
18	2	3	3	2	1	2	0.151	0.147	0.174	0.174	0.116	0.323	0.207	0.116

Table 4.

Results of L18 orthogonal array for shrinkage.

No.	A	B	C	D	E	F	1-1	1-2	1-3	1-4	2	3	R3	R4
1	1	1	1	1	1	1	2.467	2.459	2.663	2.369	2.627	1.952	1.952	0.711
2	1	1	2	2	2	2	2.045	2.092	2.189	2.106	2.329	1.901	1.901	0.428
3	1	1	3	3	3	3	2.089	2.147	2.091	2.152	2.204	1.649	1.649	0.555
4	2	1	1	3	3	2	2.124	2.140	2.167	2.188	2.189	1.800	1.800	0.389
5	2	1	2	1	1	3	2.149	2.095	2.325	2.247	2.675	1.806	1.806	0.869
6	2	1	3	2	2	1	2.339	2.212	2.013	2.251	2.381	1.950	1.950	0.431
7	1	2	1	1	2	2	2.066	2.274	2.279	2.306	2.681	1.938	1.938	0.742
8	1	2	2	2	3	3	1.773	1.624	1.780	1.917	2.514	1.678	1.624	0.890
9	1	2	3	3	1	1	2.409	2.387	2.443	2.404	2.436	1.930	1.930	0.514
10	2	2	1	2	3	1	2.162	2.304	2.246	2.105	2.364	1.901	1.901	0.463
11	2	2	2	3	1	2	2.163	1.996	1.965	2.420	2.428	1.889	1.889	0.539
12	2	2	3	1	2	3	2.175	2.197	2.200	2.282	2.633	1.932	1.932	0.701
13	1	3	1	2	1	3	2.092	2.243	2.255	2.214	2.711	1.934	1.934	0.777
14	1	3	2	3	2	1	2.336	2.279	2.211	2.240	2.598	1.972	1.972	0.626
15	1	3	3	1	3	2	2.288	2.269	2.234	2.301	2.317	1.857	1.857	0.460
16	2	3	1	3	2	3	1.833	2.230	2.147	2.025	2.640	1.834	1.833	0.808
17	2	3	2	1	3	1	2.241	2.155	2.280	2.351	2.650	1.964	1.964	0.686
18	2	3	3	2	1	2	2.070	1.803	1.866	1.782	2.424	1.983	1.782	0.643

Table 5.

The grey relational analysis for L18.

No.	Normalize data				Deviation				GRE				GRD	Order
No.	R1	R2	R3	R4	R1	R2	R3	R4	R1	R2	R3	R4	GRD	Order
1	0.540	0.768	1.952	0.711	0.460	0.232	0.058	0.357	0.521	0.683	0.490	0.423	0.529	7
2	0.755	1.000	1.901	0.428	0.245	0.000	0.203	0.922	0.671	1.000	0.427	0.333	0.608	2
3	1.000	0.927	1.649	0.555	0.000	0.073	0.926	0.668	1.000	0.872	0.333	0.364	0.643	1
4	0.958	0.568	1.800	0.389	0.042	0.432	0.494	1.000	0.923	0.536	0.351	0.483	0.574	3
5	0.346	0.629	1.806	0.869	0.654	0.371	0.476	0.042	0.433	0.574	0.536	0.465	0.502	8
6	0.977	0.259	1.950	0.431	0.023	0.741	0.062	0.916	0.956	0.403	0.343	0.554	0.564	5
7	0.273	0.386	1.938	0.742	0.727	0.614	0.096	0.295	0.407	0.449	0.551	0.527	0.483	13
8	0.972	0.398	1.624	0.890	0.028	0.602	1.000	0.000	0.947	0.454	0.345	0.524	0.568	4
9	0.473	0.000	1.930	0.514	0.527	1.000	0.121	0.751	0.487	0.333	0.507	0.600	0.482	14
10	0.875	0.266	1.901	0.463	0.125	0.734	0.203	0.853	0.800	0.405	0.385	0.552	0.536	6
11	0.256	0.224	1.889	0.539	0.744	0.776	0.239	0.701	0.402	0.392	0.554	0.561	0.477	17
12	0.187	0.571	1.932	0.701	0.813	0.429	0.115	0.377	0.381	0.538	0.568	0.481	0.492	9
13	0.000	0.386	1.934	0.777	1.000	0.614	0.110	0.225	0.333	0.449	0.600	0.527	0.477	16
14	0.584	0.069	1.972	0.626	0.416	0.931	0.000	0.527	0.546	0.350	0.478	0.589	0.491	10
15	0.092	0.421	1.857	0.460	0.908	0.579	0.330	0.859	0.355	0.463	0.585	0.519	0.480	15
16	0.099	0.510	1.833	0.808	0.901	0.490	0.400	0.164	0.357	0.505	0.583	0.498	0.486	12
17	0.164	0.185	1.964	0.686	0.836	0.815	0.023	0.406	0.374	0.380	0.572	0.568	0.474	18
18	0.363	0.359	1.782	0.643	0.637	0.641	0.546	0.494	0.440	0.438	0.532	0.533	0.486	11

Analysis for L18 experiment

Normalization of raw data

The experimental results for the four responses in Table 5 were converted into a sequence ranging from 0 and 1. Both the sink mark (R1), variance values of sink mark (R2), shrinkage (R3), and variance values of shrinkage (R4) have the smaller the better characteristic. Hence, equation (1) was used to normalize the minimum sink mark, the minimum variance in sink mark, the minimum shrinkage, and the minimum variance in shrinkage. One of the requirements for using Grey relational analysis is that four responses are considered as independent variables, the linear correlation is considered as an essential step before using GRA. To measure the linear correlation between indexes, Pearson’s correlation coefficient was utilized. A correlation coefficient of 0.414 was obtained. This moderate correlation suggests that grey relational analysis can be performed (Table 6).

Table 6.

Linear correlation values.

Responses	R1	R2	R3
R2	0.227	—	—
R3	0.414	0.380	—
R4	0.412	−0.019	−0.019

Grey relational grade (GRD)

After obtaining the normalized four quality responses, Grey Relational Coefficients were calculated using equations (2)–(5). In this study, as all four quality responses are considered equally important, the normalized weight value for each is set as 0.25. Following this, the grey relational grades were calculated using equation (6). These grades represent the overall performance index for each trial, combining the four quality characteristics into a single value (GRD). The results of this comprehensive data analysis process are presented in Table 5. The GRD value is then used within the Taguchi method to determine the optimal process parameters. In the context of the Taguchi method, when the objective is to maximize the GRD values, the “Larger is Better” quality characteristic is employed.

Analysis of variance (ANOVA)

The primary objective of the data analysis aims to identify the factor level combination that yields the highest grey relational grade. As shown in Table 7, for each factor, the average of the response characteristic based on grey relational grades at each level of the factor was calculated. The factors with the largest effect are quickly identified by the ranks. The main effects graph (Figure 6) visually illustrates how each factor influences the grey relational grades for each factor level, facilitating a more intuitive interpretation of the trends. Based on this analysis, the optimal conditions for GRD were established as A₁B₁C₃D₂E₃F₃ is the combination of injection speed at 40 mm/s, melting temperature at 190°C, injection pressure at 110 MPa, holding pressure at 35 MPa, holding time at 6 s, and cooling time at 25 s. The secondary objective of this analysis aims to identify the significant control factors by employing ANOVA and the F-ratio test (Table 8), melting temperature, holding time, and holding pressure have significant impact on the grey relational grades with 90% confidence level. Conversely, three factors, namely injection speed, injection pressure, and cooling time which were found to be lower than the critical F-value. This indicates that these factors do not have a statistically significant influence on the response characteristics. Therefore, it is generally recommended that a second round of experiments needs to be conducted by removing these factors when they are deemed unimportant.

Table 7.

Response table for means grey relational grades for L18.

Level	A	B	C	D	E	F
1	0.5288	0.5692	0.5140	0.4937	0.4924	0.5123
2	0.5099	0.5063	0.5198	0.5394	0.5205	0.5180
3		0.4825	0.5243	0.5250	0.5451	0.5277
Delta	0.0189	0.0867	0.0102	0.0457	0.0528	0.0154
Rank	4	1	6	3	2	5

Figure 6.

Main effect plot for grey relational grades (L18).

Table 8.

Analysis of variance (ANOVA) for means of L18.

Source	DF	Seq SS	Adj SS	Adj MS	F	p-Value	Contribution (%)
A	1	0.016	0.016	0.016	3.74	0.101	3.20
B	2	0.285	0.285	0.142	32.72	0.001	58.52
C	2	0.002	0.002	0.001	0.21	0.815	0.41
D	2	0.065	0.065	0.032	7.47	0.024	13.35
E	2	0.086	0.086	0.043	9.84	0.013	17.66
F	2	0.008	0.008	0.004	0.89	0.459	1.64
Residual error	6	0.026	0.026	0.004	F_{0.1; 1; 6} = 3.7760		5.34
Total	17	0.487

Result of L9 experiment

After the first-round experiment, insignificant process parameters were determined and removed through a significance test. Then, a further experiment is conducted to investigate the impact of dominant factors. In the second-round experiment, the following parameters are selected as factors for investigation: melting temperature, holding time, and holding pressure. To achieve a better result, the factors, and their levels for the second experiment are established based on the coarse optimal conditions obtained from the first one, which are presented in Table 9, with three controllable factors and their corresponding levels. Taguchi’s L9 orthogonal array design (Table 10) was employed for this experiment. Results from the L9 experiment were given for sink mark indexes in Table 11 and for shrinkage indexes in Table 12. The procedure for GRA with four indexes namely, sink mark (R1), variance values of sink mark (R2), shrinkage (R3), variance values of shrinkage (R4), were illustrated in Table 13.

Table 9.

Process parameters and their levels in L9 array.

Process parameters		Level 1	Level 2	Level 3
Melting temperature (°C)	A	170	180	190
Holding pressure (MPa)	B	32	35	38
Holding time (s)	C	6	7	8

Table 10.

Taguchi’s L9 orthogonal array.

No.	A	B	C
1	1	1	1
2	1	2	2
3	1	3	3
4	2	1	2
5	2	2	3
6	2	3	1
7	3	1	3
8	3	2	1
9	3	3	2

Table 11.

Results of L9 orthogonal array for sink mark.

No.	A	B	C	1-1	1-2	1-3	1-4	2	3	R1	R2
1	1	1	1	0.099	0.103	0.131	0.119	0.053	0.188	0.053	0.135
2	1	2	2	0.107	0.101	0.090	0.131	0.045	0.147	0.045	0.102
3	1	3	3	0.109	0.102	0.123	0.142	0.053	0.203	0.053	0.150
4	2	1	2	0.119	0.114	0.103	0.100	0.048	0.175	0.048	0.127
5	2	2	3	0.094	0.097	0.139	0.131	0.064	0.165	0.064	0.100
6	2	3	1	0.113	0.114	0.118	0.106	0.080	0.219	0.080	0.139
7	3	1	3	0.108	0.099	0.123	0.106	0.056	0.218	0.056	0.162
8	3	2	1	0.143	0.123	0.120	0.152	0.077	0.238	0.077	0.161
9	3	3	2	0.105	0.100	0.102	0.111	0.084	0.207	0.084	0.123

Table 12.

Results of L9 orthogonal array for shrinkage.

No.	A	B	C	1-1	1-2	1-3	1-4	2	3	R3	R4
1	1	1	1	1.524	1.903	1.979	1.924	2.432	1.671	1.524	0.909
2	1	2	2	1.788	1.804	1.963	1.862	2.495	1.806	1.758	0.738
3	1	3	3	1.797	1.719	1.959	1.940	2.515	1.751	1.708	0.808
4	2	1	2	1.825	1.892	1.945	1.998	2.555	1.680	1.624	0.931
5	2	2	3	1.677	1.593	1.956	2.019	2.502	1.701	1.593	0.909
6	2	3	1	1.562	1.821	0.718	0.417	2.412	1.697	0.417	1.995
7	3	1	3	1.835	1.502	1.958	1.949	2.586	1.657	1.502	1.084
8	3	2	1	2.000	2.016	1.902	1.923	2.519	1.690	1.665	0.853
9	3	3	2	1.521	1.502	1.650	1.546	2.538	1.685	1.502	1.036

Table 13.

The grey relational analysis for L9.

No.	Normalize data				Deviation				GRE				GRD	Order
No.	R1	R2	R3	R4	R1	R2	R3	R4	R1	R2	R3	R4	GRD	Order
1	0.786	0.443	0.193	0.843	0.214	0.557	0.807	0.157	0.701	0.473	0.416	0.514	0.526	4
2	1.000	0.968	0.000	1.000	0.000	0.032	1.000	0.000	1.000	0.939	0.333	0.347	0.655	1
3	0.803	0.189	0.050	0.931	0.197	0.811	0.950	0.069	0.718	0.381	0.411	0.567	0.519	5
4	0.932	0.562	0.079	0.870	0.068	0.438	0.921	0.130	0.880	0.533	0.362	0.484	0.565	3
5	0.504	1.000	0.142	0.843	0.496	0.000	0.858	0.157	0.502	1.000	0.499	0.333	0.584	2
6	0.103	0.368	1.000	0.000	0.897	0.632	0.000	1.000	0.358	0.442	0.583	0.531	0.478	8
7	0.709	0.000	0.209	0.707	0.291	1.000	0.791	0.293	0.632	0.333	0.442	0.600	0.502	6
8	0.179	0.011	0.072	0.905	0.821	0.989	0.928	0.095	0.379	0.336	0.569	0.598	0.470	9
9	0.000	0.638	0.209	0.744	1.000	0.362	0.791	0.256	0.333	0.580	0.600	0.463	0.494	7

As shown in Table 14 and Figure 7, the optimal combinations found with A₁B₂C₂, namely, melting temperature at 170°C, holding pressure at 35 MPa, holding time at 7 s. From the ANOVA table was given in Table 15, the error contribution is relatively low (3.92%), indicating that there is limited potential for additional improvement in the optimization process. Thus, the final optimal condition for sink mark and shrinkage in family mold was determined to be melting temperature at 170°C, holding time at 7 s, and holding pressure at 35 MPa. To compare the improvement between the two rounds of experiments, the predicted optimal conditions from each round were evaluated. In the first round, the predicted GRD value at the optimal conditions was 0.638. In the second round, the predicted GRD value at the optimal conditions increased to 0.643. This demonstrates an improvement in the predicted GRD value. Based on the ANOVA table, all three process parameters contribute nearly equally and are ranked in terms of their influence as follows: Holding time (34.02%), melt temperature (33.97%), and holding pressure (28.09%).

Table 14.

Response table for grey relational grades for L9.

Level	A	B	C
1	0.5667	0.5309	0.4916
2	0.5422	0.5697	0.5713
3	0.4888	0.4972	0.5349
Delta	0.060	0.299	0.025
Rank	2	3	1

Figure 7.

Main effect plot for grey relational grades (L9).

Table 15.

Analysis of variance (ANOVA) for means of L9.

Source	DF	Seq SS	Adj SS	Adj MS	F	p-Value	Contribution (%)
A	2	0.009539	0.009539	0.004769	8.67	0.103	33.97
B	2	0.007887	0.007887	0.003943	7.17	0.122	28.09
C	2	0.009551	0.009551	0.004775	8.68	0.103	34.02
Residual error	2	0.001100	0.001100	0.000550			3.92
Total	8	0.028077

In terms of melt temperature, a lower melt temperature leads to a lower sink mark index and shrinkage index, as observed in both rounds of testing. The decreasing trend in melting temperature is consistently maintained. This can be explained by the fact that lowering the melt temperature reduces thermal stress on the molded part, resulting in more uniform cooling and solidification, which in turn reduces shrinkage. Additionally, a lower melt temperature shortens the cooling time of the part, further reducing sink marks. In terms of holding time and holding pressure, these two factors are closely correlated, as both holding time and holding pressure contribute to achieving an optimum point in the result. Essentially, holding pressure and holding time help maintain a consistent density throughout the part. However, in a family mold with multiple different products—specifically, in this study, specimens—it is essential to balance holding time and holding pressure. Therefore, establishing an optimum point is necessary.

Confirmation test

The confirmation test aims to validate the analysis’s conclusions and the feasibility of proposed method. Additionally, the purpose of the confirmation test is also used to verify the final optimal combination of process conditions. Five confirmation experiments using the optimal combination of parameters, injection speed at 40 mm/s, melting temperature at 170°C, injection pressure at 110 MPa, holding pressure at 35 MPa, holding time at 7 s, and cooling time at 25 s, were conducted. The expression for computing the confidence interval, for performance at the optimum condition, is calculated as

C . I = \pm \sqrt{F (α, 1, n_{2}) \times (V_{e} / N_{e})}

(7)

where $F (α, 1, n_{2})$ is F-ratio required for $α$ risk, $F (0.1, 1, 2) = 8.53$ , V_e is the pooled error, V_e = 0.00055; N_e is the effective sample size of tests under the optimal condition, N_e = 1.286. So, C.I = 0.061. Because the combination of process parameters is one of the run trials, the overall mean value of grey relational grade is equal to 0.643. Therefore, the confidence interval for confirmation test is $0 ., 643 \pm 0.061$ . As shown in Table 16, the average grey relational grade of the verification experiments (GRD = 0.642) lies within the confidence interval. Therefore, the optimal combination is obtained via the whole process.

Table 16.

Results of confirmation experiments.

No.	Mean value				GRE				GRD
No.	R1	R2	R3	R4	R1	R2	R3	R4	GRD
1	0.042	0.131	1.717	0.933	0.484	0.544	1.000	0.348	0.594
2	0.034	0.156	1.775	0.864	1.000	0.333	0.394	0.987	0.679
3	0.049	0.126	1.750	0.873	0.333	0.623	0.528	0.799	0.571
4	0.038	0.124	1.792	0.864	0.652	0.662	0.333	1.000	0.662
5	0.039	0.113	1.722	0.938	0.600	1.000	0.878	0.333	0.703
Average									0.642

Tensile test and bending test

Following the optimization of injection molding parameters and geometric specimen design to minimize shrinkage and sink marks in PP materials, material testing was conducted to assess mechanical properties. Tensile and bending tests were performed in accordance with established ISO procedures, using five specimens from the confirmation test. Both tests were carried out on a SHIMADZU AGS-X testing machine, shown as in Figure 5. As presented in Table 17, the results indicated an average tensile strength of 27.08 MPa with a variance of 0.43 MPa. Similarly, the results indicated an average flexural strength of 32.15 MPa with a variance of 1.26 MPa. Both values are lower than the datasheet values; however, with an error within 5%, they are considered acceptable for evaluation.

Table 17.

Experimental values for mechanical testing.

Parameters	Run	Value	Results
Tensile strength (MPa)	1	26.71	N = 5Mean = 27.08Standard deviation = 0.43
	2	26.83
	3	26.78
	4	27.46
	5	27.63
Flexural strength (MPa)	1	31.48	N = 5Mean = 32.15Standard deviation = 0.26
	2	33.33
	3	33.44
	4	31.23
	5	31.28

Conclusion

Family molds provide an efficient solution by enabling the injection of multiple samples in a single cycle. This efficiency is essential for preparing specimens to measure material properties, reducing both time and material waste. However, maintaining balanced flow between cavities remains a significant challenge, often leading to defects such as sink marks and shrinkage. These defects affect dimensional accuracy and sectional-cross uniformity. To address these issues, this study aimed to:

Investigate the influence of injection speed, melt temperature, injection pressure, holding pressure, holding time, and cooling time on sink marks, shrinkage, and their variance in family mold injection molding.

Optimize process parameters using the Taguchi method, ANOVA, and Grey Relational Analysis (GRA) to improve dimensional accuracy and consistency.

Determine the most influential parameters affecting shrinkage and sink marks in polypropylene (PP) injection molding. And tensile and bending tests were conducted to verify the feasibility of proposed methodology.

Based on the selected factors and their levels, 18 experiments were performed to determine the optimal set of process parameters for the first round. In the second round of Taguchi L9 orthogonal array was employed, the ANOVA method showed the significance of each process parameter. Using the Grey-based Taguchi method, four indexes were minimized when the injection speed was 40 mm/s, the melt temperature was 170°C, the injection pressure was 110 MPa, the holding time was 7 s, and the holding pressure was 35 MPa. From the comparison, the GRD value increases from 0.638 to 0.643, indicating an improvement. Additionally, both tensile strength and flexural strength values are within a 5% error compared to the data sheet values. ANOVA analysis showed that melt temperature, holding time, and holding pressure had the most significant effect on sink marks, shrinkage, and their variance overall.

Footnotes

Acknowledgements

We acknowledge Ho Chi Minh City University of Technology (HCMUT), VNU-HCM for supporting this study.

Handling Editor: Sharmili Pandian

ORCID iDs

Quoc-Nguyen Banh

Van-Keo Dong

Xuan-Hiep Tran

Anh-Son Tran

Minh-Tuan Ho

Author contributions

The contributions of the authors to this study are as follows: Quoc-Nguyen Banh was responsible for conceptualizing the research, developing the methodology, and designing the experiments. Van-Keo Dong contributed to data acquisition, conducted software simulations, and performed statistical analysis. Xuan-Hiep Tran carried out the literature review, drafted the manuscript, and validated the technical aspects of the research. Anh-Son Tran focused on data visualization, interpretation of results, and editing of the manuscript. Minh-Tuan Ho provided supervision, managed the project, and critically revised the manuscript. All authors have reviewed and approved the final version of the manuscript and contributed meaningfully to the research process.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is funded by Vietnam National University HoChiMinh City (VNU-HCM) under grant number C2024-20-07.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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No.	A	B	C	D	E	F
1	1	1	1	1	1	1
2	1	1	2	2	2	2
3	1	1	3	3	3	3
4	2	1	1	3	3	2
5	2	1	2	1	1	3
6	2	1	3	2	2	1
7	1	2	1	1	2	2
8	1	2	2	2	3	3
9	1	2	3	3	1	1
10	2	2	1	2	3	1
11	2	2	2	3	1	2
12	2	2	3	1	2	3
13	1	3	1	2	1	3
14	1	3	2	3	2	1
15	1	3	3	1	3	2
16	2	3	1	3	2	3
17	2	3	2	1	3	1
18	2	3	3	2	1	2

No.	A	B	C	D	E	F
1	1	1	1	1	1	1
2	1	1	2	2	2	2
3	1	1	3	3	3	3
4	2	1	1	3	3	2
5	2	1	2	1	1	3
6	2	1	3	2	2	1
7	1	2	1	1	2	2
8	1	2	2	2	3	3
9	1	2	3	3	1	1
10	2	2	1	2	3	1
11	2	2	2	3	1	2
12	2	2	3	1	2	3
13	1	3	1	2	1	3
14	1	3	2	3	2	1
15	1	3	3	1	3	2
16	2	3	1	3	2	3
17	2	3	2	1	3	1
18	2	3	3	2	1	2

Optimization of process parameters for sink marks and shrinkage reduction in family molds using Grey-based Taguchi method

Abstract

Keywords

Introduction

Proposed methodology

Proposed optimization procedures

Taguchi’s orthogonal array (OA)

Grey relational analysis (GRA)

Experimental setup

Design of the geometry model and layout

Material and equipment

Experimental results and discussion

Result of L18 experiment

Analysis for L18 experiment

Normalization of raw data

Grey relational grade (GRD)

Analysis of variance (ANOVA)

Result of L9 experiment

Confirmation test

Tensile test and bending test

Conclusion

Footnotes

Acknowledgements

ORCID iDs

Author contributions

Funding

Declaration of conflicting interests

References

No.	A	B	C	D	E	F
1	1	1	1	1	1	1
2	1	1	2	2	2	2
3	1	1	3	3	3	3
4	2	1	1	3	3	2
5	2	1	2	1	1	3
6	2	1	3	2	2	1
7	1	2	1	1	2	2
8	1	2	2	2	3	3
9	1	2	3	3	1	1
10	2	2	1	2	3	1
11	2	2	2	3	1	2
12	2	2	3	1	2	3
13	1	3	1	2	1	3
14	1	3	2	3	2	1
15	1	3	3	1	3	2
16	2	3	1	3	2	3
17	2	3	2	1	3	1
18	2	3	3	2	1	2