Determining method of optimal process parameters using Taguchi method and technique for order preference by similarity to ideal solution in plastic injection molding

Abstract

Many plastic injection molding (PIM) process optimization problems are multi-objective optimization problems that optimize multiple mechanical properties. This paper proposed a determining method of optimal process parameters and their effect ranks using Taguchi method and TOPSIS in the PIM. TOPSIS was used to convert multiple mechanical properties into single comprehensive response. It was selected as a reasonable multi-attribute decision-making (MADM) method from among some well-known MADM methods based on mean rank correlation coefficient and mean absolute rank deviation. Taguchi method was used to design experiment and find the optimal process parameters. The proposed method was applied to determine the optimal values of the process parameters: melt temperature (MT), packing pressure (PP), cooling time (CT), and injection pressure (IP) for improving tensile strength, elasticity module, flexural modulus, and impact strength in the PIM with Acrylonitrile-Butadiene-Styrene compound as plastic materials and AISI 1020 as mold materials. The optimal values of the process parameters were MT of 280°C, PP of 28 MPa, CT of 22 s, and IP of 50 MPa, and their effect ranking was IP (51.941%), PP (32.280%), CT (9.045%), and MT (6.734%). The method could be widely applied to not only the PIM process optimization but also various manufacturing process optimization problems.

Keywords

Comprehensive response multi-objective optimization plastic injection molding Taguchi method TOPSIS

Introduction

Plastic injection molding (PIM) is one of the most important technologies for manufacturing of plastic products.^1,2 The quality of the products using the PIM depends on materials, mold, injection molding machine and process parameters.^3,4 The materials, the mold, and the injection molding machine are selected at the first stage of product development. So it is one of the most important issues to determine the optimal injection process parameters to improve the quality of the products.

To optimize the process parameters, various methods have been applied. Trial and error method demands massive experiments, and it requires a huge amount of labor, time, and cost. To overcome these drawbacks and determine optimal process parameters, various optimization techniques such as Taguchi method, genetic algorithm (GA), simulated annealing (SA), and particle swarm optimization (PSO) methods have been widely used in various works. Among them, Taguchi method has been most widely applied to solve many practical engineering optimization problems because of its simplicity and effectiveness.⁵

Kapoor et al.⁵ reviewed the recent research works in designing and determining the process parameters of injection molding. They described some mathematical models, Taguchi method, artificial neural networks (ANN), fuzzy logic, genetic algorithms⁶ (GA), finite element analysis (FEA), nonlinear modeling, response surface methodology, linear regression analysis, grey rational analysis (GRA), and principle component analysis (PCA). Bensingh et al.⁷ optimized the injection molding process parameters of the bi-aspheric lens using ANN and PSO. Gao et al.¹ determined the optimal process parameters for the PIM using a novel classification model. Kitayama et al.⁸ applied a sequential approximate optimization (SAO) using a radial basis function (RBF) network to determine the optimal process parameters for minimizing weldlines and clamping force using conformal cooling channel in PIM. Li et al.⁴ performed the multi-objective optimization of the fiber-reinforced composite injection molding process using Taguchi method, RSM, and nondominated sorting GA II method. Ozcelik et al.⁹ optimized four process parameters for six mechanical properties of Acrylonitrile-Butadiene-Styrene (ABS) moldings using Taguchi method in the PIM process.

Many practical problems for manufacturing process optimization are multi-objective optimization (MOO) problems with multiple properties.⁵

Some works performed the optimizations for individual properties, respectively, and resulted in the inconsistent results with one another. To address it, many works converted the multiple properties into single comprehensive response using weighted combination method, desirability function method, loss function method, fuzzy logic approach, and multi-attribute decision-making (MADM) methods such as technique for order preference by similarity to ideal solution (TOPSIS), GRA, and VIse Kriterijumska Optimizacija Kompromisno Resenje (VIKOR) and so on, and then determined the optimal process parameters to optimize the single comprehensive response.^5,10

Chen et al.³ performed the systematic optimization of five process parameters in PIM to improve two properties such as length and warpage of PBT-2100 plastic material moldings using Taguchi method, RSM, GA, and PSO. They converted two-objective optimization problem into a single optimization problem: Min G(X)= (L(X)-170.5)²+ (L(X)_SN-36.6959)²+ (W(X)_SN-21.2553)², and determined the optimal process parameters to minimize the single objective function using the GA and PSO algorithms. Fung et al.¹¹ determined four injection molding process parameters to optimize two mechanical properties for Polycarbonate/Acrylonitrile-Butadiene-Styrene (PC/ABS) using Taguchi method and GRA. Kuo et al.¹² determined eight injection molding process parameters to optimize three multiple qualities for polyether ether ketone using Taguchi method and GRA. They obtained the comprehensive response from the SN ratios of the individual responses using GRA. Tzeng et al.¹³ optimized five injection molding process parameters to improve the tensile and flexure strengths using Taguchi method and RSM. They calculated the comprehensive response by the summation of the normalized SN ratios of individual responses. Kitayama et al.¹⁴ performed the multi-objective optimization of the process parameters in the PIM using SAO, RBF network and MADM method. For the decision-making to select a few solutions from among the pareto-optimal solutions, the trade-off analysis using radar chart was performed. Moayyedian et al.¹⁵ studied the multi-objective optimization of PIM process to minimize the manufacturing defects such as shrinkage, warpage and short shot using Taguchi method, fuzzy analytic hierarchy process (AHP), and TOPSIS. The TOPSIS was used to determine the moldability indices (overall responses) based on the weighted fuzzy evaluation matrix for the possible manufacturing defects. Mehat et al.¹⁶ proposed the optimization approach combined with Taguchi method and GRA for designing the gear part, setting up processing parameters, and selecting a suitable material for multi-quality characteristic of the helical gear. The GRA was used to convert the three quality characteristics (shrinkage, deflection, and sink mark) into a single overall characteristic (grey relational grade). The best combination of factors and levels were determined using the Taguchi method. Moayyedian et al.¹⁷ determined the optimal process parameters for high-quality end products with minimum defect possibility (short shot possibility, shrinkage rate, and warpage) by applying the ANN, Taguchi method, AHP and FEA using SolidWorks plastics. The ANN and Taguchi methods were used to find the optimal values of the process parameters, the AHP was used to calculate the weight of each defect in the thin-walled part, and the FEA was used to simulate the injection molding process for polypropylene parts and validate the optimal values of the process parameters. They converted three defect responses into a single overall response by using the weighted sum of the normalized values of three defects values. Farooque et al.¹⁸ developed the experimental plan using Taguchi L12 orthogonal array, and determined the degrees and ranks of significance of the eight injection molding parameters for the mechanical properties (responses) such as stress at break, young’s modulus, strain at break and work at break based on the mean SN ratio at each level using the Taguchi method, respectively. Pachorkar et al.¹⁹ optimized the five PIM process parameters for improving two objectives such as sink marks depth and cycle time using the utility based Taguchi method. They cnverted the given two-objective (sink marks depth and cycle time) optimization problem into a single objective (total utility) optimization problem using the total utility function, which was calculated by adding the utility values of all individual response. Chen et al.²⁰ determined the influence ranks and percentage contributions of the process parameters for the warpage based on the ranges between the highest and lowest values (delta) of the mean values of each parameter in the Taguchi method, and the ratios of the adjusted SS values of each factor to the total sum of the adjusted SS value in the analysis of variance (ANOVA). They optimized the process parameters to reduce warpage through a numerical approach using Solid Works Plastics and Taguchi method. They also developed the empirical relation to estimate the warpage in terms of process parameters using the multiple linear regression model.

As above-mentioned, many works determined the optimal injection process parameters using various MADM methods and optimization techniques.

Many previous works for PIM process optimization have applied different MADM methods to convert the multiple responses into a single overall response. Since the different MADM methods have their own principles and characteristics, they may generate non-negligible inconsistent results for the given problem, and it may result in the inconsistent optimization results. However, many previous works selected and applied the MADM methods based on ease of use, skills and experiences without considering the sufficient scientific reasonableness of selecting the specific MADM method and comparison between the effectiveness of the MADM methods.

This paper proposes a method to determine optimal process parameter values using Taguchi method and TOPSIS in the PIM, and applied it to determine optimal process parameters such as melt temperature, packing pressure, cooling time and injection pressure for optimizing the mechanical properties such as tensile strength at yield, elasticity module, flexural modulus and impact strength with ABS compound as plastic materials and AISI 1020 as mold materials.

Materials and methods

Materials and injection molding machine

This work uses Acrylonitrile-Butadiene-Styrene (ABS) compound as plastic materials, and AISI 1020 as mold materials. A double cavity of the mold is manufactured in the CNC machine according to ASTM D638-91 standards.²¹ A test parts are injected by a plastic injection machine (MIR, Turkey) which has a clamping force of 637 kN and an injection pressure of 1480 bar, respectively.

Experimental design method

The injection pressure, injection velocity, melt temperature, packing pressure, gate temperature, injection time, packing time, cooling time, injection cycle, mold temperature, ambient temperature are the process parameters affecting the mechanical properties of the injection molded products. Among them, the melt temperature, injection pressure, holding pressure, cooling rate, and mold temperature have major effects on the mechanical properties of the plastic injection moldings.⁹ Several experiments in the lab show that the suitable value of the mold temperature in the process of ABS resin injection molding ranges from 60°C to 80°C. At below 60°C, the nail erosion and cracking may occur in the molding product while the edge defects and molding deformation during the breakup may appear at above 80°C. Therefore, the mole temperature was fixed at 70°C in this work. The gate temperature is the similar influencing process parameters to the injection temperature, and the injection rate, injection time, injection cycle, packing time, and the screw speed are also the similar influencing parameters to the injection pressure and filling pressure. Consequently, this work selected four process parameters: melt temperature (MT), packing pressure (PP), cooling time (CT), and injection pressure (IP) as the main controllable process parameters, and kept the other parameters to constant values.

For the MT, based on the lowest and highest melting point temperatures of the ABS resin, we set 200°C as the value of the first level and 280°C as the value of the third level, and the average value 240°C was set as the value of the second level. For the PP, CT, and IP, we found the parameter values that minimize the porosity, cracking, and nailing phenomena inside the compact using the Moldflow Insight program, and then set the values as the first levels of the process parameters. Based on the simulation results, to increase the density of the compact and reduce the deformation during the expulsion of the compact, we, respectively, set 1.2 and 1.4 times of the value of the first level as the second and third levels for the PP, CT, and IP.

This work considered four mechanical properties: tensile strength at yield (TS), elasticity modulus (EM), flexural modulus (FM), and Izod impact strength (IS).

Table 1 shows the injection processes parameters and their levels.⁹

Table 1.

Injection process parameters and their levels.

Levels	Injection process parameters
Levels	MT (°C)	PP (MPa)	CT (s)	IP (MPa)
Level 1	200	28	16	36
Level 2	240	34	19	43
Level 3	280	39	22	50

As the number of process parameters is 4 and the number of levels is 3, Taguchi OA L9 was selected for design of experiment. Table 2 shows Taguchi OA L9.

Table 2.

Taguchi OA L9.

Trial no.	1	2	3	4
1	1	1	1	1
2	1	2	2	2
3	1	3	3	3
4	2	1	2	3
5	2	2	3	1
6	2	3	1	2
7	3	1	3	2
8	3	2	1	3
9	3	3	2	1

Testing method of mechanical properties

Figure 1 shows the main dimensions of the specimens for tensile strength test (a), three-point flexural test (b), and impact test (c) according to ASTM D638-91.²¹ A suitable dimension is cut from Figure 1(a) to carry out three-point flexural test (b) and impact test (c). Three-point flexural and impact tests were performed with ISO 178 and ISO 180 standards. The tensile strength, three-points flexural and impact test were measured using INSTRON 4411, INSTRON 5560 and Ceast 6545, respectively. The tests were repeated three times and the mean values were presented.

Method to determine optimal values of process parameters using Taguchi method and TOPSIS

This subsection proposes an optimization method of injection molding process for mechanical properties TS, EM, FM, and IS using Taguchi method and TOPSIS.^22–24 The reason of selecting the TOPSIS method is shown in Appendix.

Let the mechanical properties TS, EM, FM, and IS be y₁, y₂, y₃, and y₄, and the injection molding process parameters MT, PP, CT, and IP be x₁, x₂, x₃, and x₄, respectively.

The details of the proposed method are as follows:

Step 1

Measure multiple mechanical properties of the moldings at every experimental trials according to Taguchi orthogonal array (OA).

The values of the process parameters at every experimental trials constitute a process parameter matrix $X = {(x_{i j})}_{9 \times 4}$ , and the values of mechanical properties construct a response matrix (decision matrix) $Y = {(y_{i k})}_{9 \times 4}$ , where x_ij and y_ik are, respectively, the values of j-th process parameter and k-th mechanical property at i-th experimental trial.( $i = \bar{1, 9}, j = \bar{1, 4}, k = \bar{1, 4}$ )

Step 2

Constitute a normalized response matrix $Z = {(z_{i k})}_{9 \times 4}$ using following normalization formulas:

z_{i k} = {\begin{cases} (y_{i k} - y_{k \min}) / (y_{k \max} - y_{k \min}); k \in S_{b} \\ (y_{k \max} - y_{i k}) / (y_{k \max} - y_{k \min}); k \in S_{c} \end{cases},

(1)

where S_b and S_c are, respectively, the index sets of the benefit and cost properties, and y_kmin and y_kmax are, respectively, the minimum and maximum values of k-th property.

Step 3

Calculate the comprehensive response (CR) values of the moldings at 9 experimental trials by considering multiple responses (mechanical properties) using the TOPSIS method.

(1) Constitute the weighted normalized response matrix V= (v_ik)_9×4 using the following formula:

v_{i k} = w_{k} \times z_{i k}; i = \bar{1, 9}, k = \bar{1, 4},

(2)

where w_k is the weight of k-th mechanical property (w₁+ w₂+ w₃+ w₄ = 1).

The weights could be determined using AHP and entropy weighting method.^23,25

The formula to calculate the weights using entropy weighting method is as follows: ( $k = \bar{1, 4}$ )

w_{k} = | 1 - e_{k} | / \sum_{m = 1}^{4} | 1 - e_{m} |,

(3)

where

e_{k} = - \frac{1}{\ln n} \sum_{i = 1}^{9} [p_{i k} \ln p_{i k}], p_{i k} = z_{i k} / \sum_{l = 1}^{9} z_{l k} .

(4)

(2) Select the positive ideal solution PIS= (PIS1, PIS2, PIS3, PIS4) and negative ideal solution NIS= (NIS1, NIS2, NIS3, NIS4) as follows: ( $k = \bar{1, 4}$ )

P I S_{k} = \max_{1 \leq i \leq 9} v_{i k}, N I S_{k} = \min_{1 \leq i \leq 9} v_{i k} .

(5)

(3) Calculate the distances from the moldings at 9 experimental trials to the PIS and NIS using the following formulas: ( $i = \bar{1, 9}$ )

D_{i}^{+} = \sqrt{\sum_{k = 1}^{4} {(P I S_{k} - v_{i k})}^{2}},

(6)

D_{i}^{-} = \sqrt{\sum_{j = 1}^{4} {(N I S_{k} - v_{i k})}^{2}} .

(7)

(4) Calculate the relative closeness values of the moldings at 9 experimental trials using the following formula: ( $i = \bar{1, 9}$ )

C_{i} = D_{i}^{-} / (D_{i}^{+} + D_{i}^{-}) .

(8)

The relative closeness value becomes to the comprehensive response (CR) value that comprehensively reflects the multiple mechanical properties. We have to determine the optimal values of the process parameters to maximize the CR value in the next steps.

Step 4

Calculate the mean CR values ${S_{l j}; j = \bar{1, 4}, l = \bar{1, 3}}$ of the process parameters at each level, where S_lj is the mean value of the CR values corresponding to l-th level of j-th process parameter.

Step 5

Calculate the range R_j of mean CR values and effect score (ES) ES_j (%) of each process parameter as follows: ( $j = \bar{1, 4}$ )

R_{j} = \max_{1 \leq l \leq 3} {S_{l j}} - \min_{1 \leq l \leq 3} {S_{l j}},

(9)

E S_{j} = 100 R_{j} / \sum_{k = 1}^{4} R_{k} .

(10)

The larger the R_j and ES_j is, the higher the effect of the process parameter is. The sum of the ES values of all the process parameters is 100%.

Step 6

Determine the optimal levels of each process parameter, where the optimal level has the maximum mean CR value from among three levels of the process parameter.

Step 7

Determine the optimal values of the process parameters corresponding to the optimal levels.

Results and discussion

This section applied the proposed method to determine the optimal values of four process parameters (MT, PP, CT, and IP) for optimizing four mechanical properties (TS, EM, FM, and IS).

Table 3 shows the experimental result according to OA L9.

Table 3.

Experimental result for mechanical properties according to OA L9.

Trial no.	Injection molding process parameters				Mechanical properties
Trial no.	MT (°C)	PP (MPa)	CT (s)	IP (MPa)	TS (MPa)	EM (MPa)	FM (MPa)	IS (J)
1	200	28	16	36	39.09	2782	1823	0.564
2	200	34	19	43	40.00	2679	1885	0.571
3	200	39	22	50	39.24	2621	2098	0.593
4	240	28	19	50	39.64	2458	2158	0.641
5	240	34	22	36	38.12	2502	1903	0.586
6	240	39	16	43	38.06	2368	1970	0.601
7	280	28	22	43	36.75	1870	2122	0.720
8	280	34	16	50	36.37	2217	2375	0.710
9	280	39	19	36	36.48	2128	1844	0.592

Determining optimal injection molding process parameters for individual mechanical properties

This subsection determined the values of four process parameters to optimize the individual mechanical properties, respectively.

Table 4 shows the mean TS values and their effects of 4 process parameters at each level. Figure 2 shows the mean TS values of 4 process parameters at each level.

Table 4.

Mean TS values and their effects of 4 process parameters at each level.

Level	MT	PP	CT	IP
1	39.443	38.493	37.840	37.897
2	38.607	38.163	38.707	38.270
3	36.533	37.927	38.037	38.417
Range (MPa)	2.910	0.567	0.867	0.520
ES (%)	59.836	11.652	17.820	10.692
Rank	1	3	2	4

From Table 4 and Figure 2, the optimal levels of each process parameters for TS were, respectively, 1, 1, 2, and 3, and therefore, the optimal values of the process parameters were as follows:

MT = 200 ° C, PP = 28 MPa, CT = 19 s, IP = 50 MPa .

Table 5 shows the mean EM values and their effects of 4 process parameters at each level.

Table 5.

Mean EM values and their effects of 4 process parameters at each level.

Level	MT	PP	CT	IP
1	2694.000	2370.000	2455.667	2470.667
2	2442.667	2466.000	2421.667	2305.667
3	2071.667	2372.333	2331.000	2432.000
Range (MPa)	622.333	96.000	124.667	165.000
ES (%)	61.739	9.524	12.368	16.369
Rank	1	4	3	2

Figure 3 shows the mean EM values of 4 process parameters at each level.

From Table 5 and Figure 3, the optimal levels of each process parameters for EM were respectively 1, 2, 1, and 1, and therefore, the optimal values of the process parameters were as follows:

MT = 200 ° C, PP = 34 MPa, CT = 16 s, IP = 36 MPa .

Table 6 shows the mean FM values and their effects of 4 process parameters at each level.

Table 6.

Mean FM values and their effects of 4 process parameters at each level.

Level	MT	PP	CT	IP
1	1935.333	2034.333	2056.000	1856.667
2	2010.333	2054.333	1962.333	1992.333
3	2113.667	1970.667	2041.000	2210.333
Range (MPa)	178.333	83.667	93.667	353.667
ES (%)	25.141	11.795	13.205	49.859
Rank	2	4	3	1

Figure 4 shows the mean FM values of 4 process parameters at each level.

From Table 6 and Figure 4, the optimal levels of each process parameters for FM were, respectively, 3, 2, 1, and 3, and therefore, the optimal values of the process parameters were as follows:

MT = 280 ° C, PP = 34 MPa, CT = 16 s, IP = 50 MPa .

Table 7 shows the mean IS values and their effects of 4 process parameters at each level.

Table 7.

Mean IS values and their effects of 4 process parameters at each level.

Level	MT	PP	CT	IP
1	0.576	0.642	0.625	0.581
2	0.609	0.622	0.601	0.631
3	0.674	0.595	0.633	0.648
Range (J)	0.098	0.046	0.032	0.067
ES (%)	40.274	19.041	13.014	27.671
Rank	1	3	4	2

Figure 5 shows the mean IS values of 4 process parameters at each level.

From Table 7 and Figure 5, the optimal levels of each process parameters for IS were, respectively, 3, 1, 3, and 3, and therefore, the optimal values of the process parameters were as follows:

MT = 280 ° C, PP = 28 MPa, CT = 22 s, IP = 50 MPa .

Table 8 collectively shows the ES scores, ES ranks, optimal levels and optimal values of each process parameters according to individual mechanical properties. As can be seen in Table 8, there were inconsistent and conflicting optimization results with one another for the individual mechanical properties.

Table 8.

ES scores, ES ranks, optimal levels and values of each process parameters according to individual mechanical properties.

Property	ES scores (%)	ES ranks	Optimal levels	Optimal parameter values
TS	59.836, 11.652, 17.820, 10.692	1, 3, 2, 4	1, 1, 2, 3	200, 28, 19, 50
EM	61.739, 9.524, 12.368, 16.369	1, 4, 3, 2	1, 2, 1, 1	200, 34, 16, 36
FM	25.141, 11.795, 13.205, 49.859	2, 4, 3, 1	3, 2, 1, 3	280, 34, 16, 50
IS	40.274, 19.041, 13.014, 27.671	1, 3, 4, 2	3, 1, 3, 3	280, 28, 22, 50

Determining optimal injection molding process parameters for comprehensive response

This subsection determined the optimal values of four process parameters to optimize the comprehensive response that comprehensively reflects 4 mechanical properties.

To convert four mechanical properties into single CR, TOPSIS method was applied in this work.

Table 9 shows the normalized decision matrix.

Table 9.

Normalized decision matrix.

Trial no.	TS	EM	FM	IS
1	0.749	1.000	0.000	0.000
2	1.000	0.887	0.112	0.045
3	0.791	0.823	0.498	0.186
4	0.901	0.645	0.607	0.494
5	0.482	0.693	0.145	0.141
6	0.466	0.546	0.266	0.237
7	0.105	0.000	0.542	1.000
8	0.000	0.380	1.000	0.936
9	0.030	0.283	0.038	0.179

The weights of 4 mechanical properties calculated using entropy weighting method were as follows: 0.245, 0.130, 0.292, 0.334.

Table 10 shows the CR values and CR ranks of nine experimental trials obtained from TOPSIS.

Table 10.

CR values and CR ranks of nine experimental trials from TOPSIS.

Trial no.	Process parameters				CR value	CR rank
Trial no.	MT	PP	CT	IP	CR value	CR rank
1	1	1	1	1	0.334	6
2	1	2	2	2	0.399	5
3	1	3	3	3	0.464	4
4	2	1	2	3	0.616	2
5	2	2	3	1	0.286	8
6	2	3	1	2	0.325	7
7	3	1	3	2	0.563	3
8	3	2	1	3	0.625	1
9	3	3	2	1	0.133	9

Figure 6 shows the bar graph of CR values of nine experimental trials from TOPSIS.

Table 11 shows the mean CR values and their effects of 4 process parameters at each level.

Table 11.

Mean CR values and their effects of 4 process parameters at each level.

Level	MT	PP	CT	IP
1	0.399	0.504	0.428	0.251
2	0.409	0.437	0.382	0.429
3	0.440	0.307	0.438	0.568
Range (J)	0.041	0.197	0.055	0.317
ES (%)	6.734	32.280	9.045	51.941
Rank	4	2	3	1

Figure 7 shows the mean CR values of 4 process parameters at each level.

From Table 11 and Figure 7, the optimal levels of each process parameters for CR were, respectively, 3, 1, 3, and 3, and therefore, the optimal values of the process parameters were as follows:

MT = 280 ° C, PP = 28 MPa, CT = 22 s, IP = 50 MPa .

From Table 11, the effect ranking of the process parameters on the CR was as follows:

IP (51.941 %) > PP (32.280 %) > CT (9.045 %) > MT (6.734 %) .

It shows that IP and PP are the major process parameters that affect the comprehensive quality of the moldings.

Finally, we developed the multiple quadratic regression model for calculating the CR value from the values of the process parameters.

The multiple quadratic regression model was as follows:

CR = 0.000247 MT - 0.000009 {MT}^{2} + 0.000031 MT \cdot PP + 0.000069 MT \cdot CT + 0.000066 MT \cdot IP - 0.001117 {PP}^{2} + 0.001209 PP \cdot IP - 0.000267 {CT}^{2} - 0.000383 {IP}^{2}

(11)

The mean absolute and relative errors of the regression model are, respectively, 0.005,212 and 1.656%, and the determination coefficient is R² = 1.

By using the regression model, we can calculate the CR values according to the values of the given process parameters.

To confirm the optimal values of the process parameters (MT = 280°C, PP = 28 MPa, CT = 22s, IP = 50 MPa) obtained from the proposed method, we calculated the CR value according to the values of the optimal process parameters using equation (11). By substituting the optimal values MT = 280, PP = 28, CT = 22 and IP = 50 into equation (11), the calculated CR value is 0.686. Meanwhile, the maximum CR value is 0.625 (at trial no. 8) in Table 10. It demonstrates that the optimal values of the process parameters from the proposed method is better than ones from Table 10.

Conclusions

This paper proposed a method to determine the optimal values of the process parameters using Taguchi method and TOPSIS in the PIM, and applied it to determine the optimal values of four process parameters MT, PP, CT, and IP for optimizing four mechanical properties TS, EM, FM, and IS with ABS as plastic materials and AISI 1020 as mold materials.

As the result, the following conclusions were drawn:

(1) The proposed method can determine optimal values and effect ranking of the process parameters for simultaneously improving the multiple mechanical properties of the plastic injection moldings.

(2) The optimal values of the process parameters were MT of 280°C, PP of 28 MPa, CT of 22s, and IP of 50 MPa, and their effect ranking on the CR was IP (51.941%), PP (32.280%), CT (9.045%), and MT (6.734%).

The novelty of this paper and difference compared to the other previous papers are as follows:

(1) To convert the multi-objective optimization problem into a single objective one, this work selected a reasonable MADM method based on the MRCC and MARD values from among some available MADM methods, and then converted the multiple mechanical properties into a single CR using the selected reasonable MADM method: TOPSIS, while the previous works commonly have selected the MADM method based on the ease of understanding and ease of employing without the scientific reason of selecting the specific MADM method. This work clearly demonstrated that the CR values obtained from the different MADM methods differed with one another, it may generate the inconsistent optimization results, and therefore it is necessary to select and apply a reasonable MADM method. This work clarified the reason of selecting the TOPSIS method as a reasonable MADM method from among three well-known MADM methods (TOPSIS, GRA, and VIKOR) for calculating the comprehensive response (CR) values. (See the Appendix.)

(2) To evaluate the effect ranks of the process parameters, this work newly introduced the effect scores (equation (10)) of the process parameters, while the almost previous works commonly have used the ranges of the mean response values or mean SN ratios. Since the proposed effect scores are represented with percentage values and their sum is 100%, we can evaluate the effects of the process parameters, clearly and intuitively.

(3) This work developed the multiple quadratic regression model (equation (11)) that reflects the relationship between the CR and the process parameters. The regression model enables to predict the CR value from the desired values of the process parameters without conducting the experiment.

The proposed method could be actively used to not only the PIM process but also various complicated manufacturing processes in industry practice. When we have to deal with other manufacturing process, it is required to select appropriate process parameters and mechanical properties related to the manufacturing process, and then it is possible to determine the optimal values of the manufacturing process parameters by using the proposed method.

Footnotes

Acknowledgments

This work was supported by Kim Chaek University of Technology, Democratic People’s Republic of Korea. The supports are gratefully acknowledged. The authors express their gratitude to the editors and the reviewers for their helpful suggestions for improvement of this paper.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Won-Chol Yang

Data availability statement

The authors confirm that the data supporting the findings of this study are available within this article. The code that supports the findings of this study is available from the corresponding author, upon a reasonable request.*

Appendix

This appendix shows the reason of selecting the TOPSIS method as a more reasonable MADM method from among some well-known MADM methods for calculating the comprehensive response (CR) values.

We applied three well-known and widely used MADM methods such as TOPSIS, GRA, and modified VIKOR methods to calculate the CR values of the moldings at 9 experimental trials by considering multiple responses (mechanical properties), respectively.

In the classic VIKOR method, the output values (VIKOR indices) for each alternative are calculated as (A1)

Q_{i} = v \frac{S_{i} - S^{-}}{S^{+} - S^{-}} + (1 - v) \frac{R_{i} - R^{-}}{R^{+} - R^{-}}; i = 1, 2, \dots, n

and the values belong to [0, 1], where S_i and R_i are, respectively, the utility and regret measures of the i-th alternative, and

S^{+} = \max_{i} S_{i}

S^{-} = \min_{i} S_{i}

R^{+} = \max_{i} R_{i}

and

R^{-} = \min_{i} R_{i}

. The smaller the value is, the better the alternative is. In the almost MADM methods, the output values belong to [0, 1], and the higher the value is, the better (more superior) the alternative is. To do this, we propose a modified VIKOR index as follows: (A2)

Q_{i} = 1 - [v \frac{S_{i} - S^{-}}{S^{+} - S^{-}} + (1 - v) \frac{R_{i} - R^{-}}{R^{+} - R^{-}}] .

This modified VIKOR index belongs to [0, 1], and the larger the value is, the better the alternative is.

We call this method modified VIKOR method.

Table 12 shows the CR values and CR ranks of nine experimental trials obtained from the TOPSIS, GRA, and modified VIKOR methods. Table 12.

CR values and CR ranks of nine experimental trials using TOPSIS, GRA, and VIKOR methods.

Trial no.	Process parameters				TOPSIS		GRA		Modified VIKOR
Trial no.	MT	PP	CT	IP	CR value	CR rank	CR value	CR rank	CR value	CR rank
1	1	1	1	1	0.334	6	0.501	6	0.184	8
2	1	2	2	2	0.399	5	0.570	4	0.317	6
3	1	3	3	3	0.464	4	0.541	5	0.553	4
4	2	1	2	3	0.616	2	0.609	3	0.993	1
5	2	2	3	1	0.286	8	0.431	8	0.312	7
6	2	3	1	2	0.325	7	0.437	7	0.451	5
7	3	1	3	2	0.563	3	0.617	2	0.722	3
8	3	2	1	3	0.625	1	0.727	1	0.771	2
9	3	3	2	1	0.133	9	0.363	9	0.162	9

Table 12 demonstrated that the CR values and CR ranks of nine experimental trials using the TOPSIS, GRA, and VIKOR methods did not coincide with one another. It may generate the inconsistent optimization results. Therefore, it is necessary to select and apply a more reasonable MADM method.

To select a more reasonable MADM method from among various MADM methods, we introduced two measures: mean rank correlation coefficient (MRCC) and mean absolute rank deviation (MARD).

The MRCC is the mean value of the rank correlation coefficients between the CR values obtained from each MADM method and other MADM methods, and it is calculated using the following equation: (m = 1, 2,…, M). (A3)

{\bar{R}}_{m} = \frac{1}{M} \sum_{k = 1, k \neq m}^{M} R_{m k},

where M is the number of the applied MADM methods, R_mk is the rank correlation coefficient between CR values C_m1,…, C_mi,…, C_mn obtained from the m-th MADM and CR values C_k1,…, C_ki,…, C_kn obtained from the k-th MADM, where n is the number of the experimental trails. In this work, M = 3 and n = 9.

The MARD is the mean value of the absolute deviations between the ranks of the CR values obtained from each MADM and other MCMDs, and it is calculated using the following equation: (m = 1, 2,…, M) (A4)

{\bar{d}}_{m} = \frac{1}{M} \sum_{k = 1, k \neq m}^{M} d_{m k},

where (A5)

d_{m k} = \frac{1}{n} \sum_{i = 1}^{n} | C r_{m i} - C r_{k i} |,

d_mk is the mean absolute deviation between the ranks of the CR values Cr_m1,…, Cr_mi,…, Cr_mn obtained from the m-th MADM and the ranks of the CR values Cr_k1,…, Cr_ki,…, Cr_kn obtained from the k-th MADM.

The larger the value of MRCC is (the smaller the value of MARD is), the better CR values obtained from the MADM is coincided with the CR values obtained from the other MADM methods, and the higher the performance of the MADM is.

We selected the MADM method with maximum MRCC and minimum MARD as a reasonable MADM method. The method for selecting the reasonable MADM method is based on the majoritarian principle.

Table 13 shows the MRCC values, MARD values of the three MADM methods and their ranks. Table 13.

MRCC values, MARD values of the three MADM methods and their ranks.

	TOPSIS	GRA	Modified VIKOR
MRCC values	0.933	0.900	0.867
MRCC ranks	1	2	3
MARD values	0.667	0.889	1.111
MARD ranks	1	2	3

Table 13 illustrates that the CR values from the TOPSIS method has the maximum MRCC and minimum MARD compared with the CR values from the GRA and VIKOR methods. It demonstrated that the TOPSIS method is a more reasonable method to determine the CR values by combining the GRA and VIKOR methods. As the result, we selected the TOPSIS method to calculate the CR values.

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