1 Introduction

Plastic injection molding (PIM) is widely used to manufacture a wide variety of plastic products owing to its high productivity, low cost, and good flexibility to various complex geometries. PIM is a nonlinear coupled system with multiple inputs and multiple outputs. The output includes product quality, manufacturing cost, and molding efficiency, whereas the input includes machine characteristics, mold design pattern, process parameters, polymer material characteristics, and product geometric characteristics. The machine, mold, material, and product will determined before manufacturing, which means that the process parameters should be carefully set to avoid or reduce quality defects, improve productivity, and decrease energy consumption and cost.

To reduce the product defects (e.g., warpage, shrinkage, weldline), processing conditions, material properties, product design, and mold design have been studies by many researchers. Ozcelik and Sonat [1] used the Taguchi method to study the effect of process parameters on the warpage of parts with different thicknesses. Oktem et al. [2] conducted a series of experiments and used the signal-to-noise (S/N) ratio method and analysis of variance (ANOVA) to study the effects of process parameters on warpage and shrinkage of products. Tang et al. [3] performed warpage experiments on a thin plate to study the impact factors of the warpage problem. Kurt et al. [4] investigated the influence of the cavity pressure and mold surface temperature on the quality of the final parts. Masato et al. [5] analyzed and concluded that the shrinkage for thin-wall parts was caused by fiber orientation and could be reduced by higher melt temperature, packing pressure. Wang et al. [6] investigated the structural carbon emissions and injection molding process carbon emissions and implemented multi-objective optimization to realize the low-carbon design for injection molding machine.

Different from the constant process parameter optimization, the variable process parameter profile (e.g., packing pressure profile) has been considered to improve product quality and efficiency. Li et al. [7] performed the optimization of variable packing pressure profile for the shrinkage evenness of a slab. Gao and Wang [8] applied the sequential approximate optimization (SAO) to determine the optimal packing profile and the optimal process parameters for warpage minimization. For minimizing warpage and cycle time, Kitayama et al. [9] investigated the multi-objective optimization of variable pressure profile and process parameters. Hashimoto et al. [10] applied SAO to realize the simultaneous optimization of variable injection velocity profile and process parameters for minimizing weldline and cycle time.

Due to the advance of computer technology, computer-aided simulation software (e.g., Moldflow, Moldex3D) coupled with design optimization is recognized as an alternative approach for determining the optimal process parameters. The numerical simulation in PIM is generally so intensive that the surrogate-based or metamodeling approach, which can establish a mathematical relationship between the process parameters and optimization objectives, is valid to determine the optimal process parameters with a small number of simulations. Ozcelik and Erzurumlu [11] integrated finite element analysis (FEA), statistical DOE methods, RSM, and genetic algorithm (GA) to minimize the warpage of thin-walled parts. Kurtaran et al. [12] integrated the FEA, statistical DOE methods, ANN, and GA to minimize the warpage of automotive ceiling lamps. Gao and Wang [13] adopted the Kriging model for determining the optimal process parameters to minimize the warpage of a cellular phone cover. Li et al. [7] used the radial basis function (RBF) and the expected improvement (EI) to optimize the process parameters for achieving uniform shrinkage. Xia et al. [14] adopted the Gaussian process (GP) model for the front grille’s warpage optimization. The above researches handle a single objective optimization, but there are generally many objectives to optimization for high product quality and productivity in PIM. Chen and Kumiawan [15] proposed a two-stage multi-objective optimization system. Zhao et al. [16] proposed a two-stage optimization system to optimize the warpage, shrinkage, and sink marks of injection molded parts simultaneously. Zhao and Cheng [17] proposed a hybrid multi-objective optimization system to simultaneously optimize the warpage and cycle time of the PIM process. Cheng et al. [18] developed a novel method to find the optimal solution set of constrained multi-objective optimization problems that integrated variable complexity methods (VCMs), constrained non-dominated sorting genetic algorithm (CNSGA), BPNN, and Moldflow analysis. Liu et al. [19] proposed a multi-objective optimization method for process parameters of PIM with the haze ratio (HR) and peak valley 20 (PV20) of an optical lens as the optimization target. Xu et al. [20, 21] performed the process parameter optimization for minimizing the weight, the flash, and the volume shrinkage of a thin-walled plastic product.

In particular, a sequential approximate optimization (SAO) that the surrogate model is repeatedly constructed and optimized by adding several new sampling points has gradually been a popular approach to improve the optimization result. The general framework of the process parameters optimization in PIM using the SAO is summarized in [22]. Several representative papers using SAO approach is briefly reviewed. Gao and Wang [8, 13] adopted the modified rectangular grid (MRG) sampling strategy and the expected improvement (EI) sampling criterion to improve the accuracy of the Kriging model, respectively. Xia et al. [14] used an enhanced probability of improvement criterion to find the direction of adding training samples and optimize the surrogate model. Shi et al. [23] used a parametric sampling evaluation (PSE) strategy to improve the accuracy of the ANN model and speed up the optimization process to converge to the global optimum. Deng et al. [24] used the mode-pursuing sampling strategy (MPS) to obtain new sample points, thereby improving the accuracy of the Kriging model.

In addition to the optimization of process parameters, the conformal cooling channel is applied to improve the product quality and efficiency of PIM. Dimla et al. [25] reported that the conformal cooling channel could drastically reduce cycle time. Au and Yu [26] designed various scaffold cooling channels and evaluated the cooling performance and found that the conformal cooling channel could offer a more uniform thermal distribution. Wang et al. [27] introduced an approach to generate conformal spiral cooling channels, which helped improve the uniform of mold cooling. Kitayama et al. [10, 28,29,30] investigated the cooling performance of the conventional straight-type cooling channels and conformal cooling channels numerically and experimentally where the process parameter optimization for warpage, cycle time, weldline, and clamping force reduction was performed.

Here, the motivation for this paper is summarized as follows:

  1. 1.

    Multi-objective optimization of process parameters in PIM is a crucial issue. Several process parameters are optimized for warpage and weldline reduction, and clamping force and cycle time minimization.

  2. 2.

    Weldlines are one of the major defects, which cannot be completely eliminated. The low weldline temperature generates long weldlines for the quick solidification. The minimum weldline temperature is considered to be maximized for weldline reduction.

  3. 3.

    The sensitivity information of sampling points is used to improve the prediction accuracy of the response predictor. Therefore, the performance of multi-objective optimization can be improved.

  4. 4.

    In general, warpage, weldline, clamping force, and cycle time cannot reach the optimal at the same time. Therefore, the trade-off analysis is implemented to make the decision for the multi-objective optimization.

To realize the high product quality, high productivity, and low energy consumption and cost in PIM, this paper proposes a differential sensitivity fusion method (DSFM) to perform the multi-objective optimization of process parameters in PIM for minimizing warpage, weldlines, clamping force, and cycle time. It integrates the sampling strategy, numerical simulation, metamodeling method, and multi-objective optimization algorithm. The sampling strategy, Latin hypercube sampling (LHS), is utilized to generate stratified and uniformly distributed sampling points from the design space. For sampling points, the numerical simulation based on Moldflow is implemented to calculate the responses. Based on the sampling points and their corresponding responses, the metamodeling method is applied to construct the response predictors to calculate the responses for any sampling point in the global design space. The gradient-enhanced response surface model (GERSM) combined with the moving least-squares method (MLSM) is applied to construct the response predictor for each objective, which simultaneously utilizes the response and sensitivity information of the sampling point to improve the accuracy of the response predictors. For the capture of the sensitivity information, an adaptive sensitivity generation method (ASGM) is proposed to calculate the gradient vector for each design variable of the sampling point. The multi-objective optimization algorithm, non-dominated sorting genetic algorithm-III (NSGA-III), is executed to locate the Pareto-optimal solutions, where the response predictors are taken as the fitness functions. The trade-off analysis based on the spider-web chart is applied to make the decision for the optimal process parameters. Moreover, for the structural features of the product, the valve hot runner system is applied to reduce the weldlines and improve the product surface quality. The automobile front bumper is taken as a case study to verify the proposed method DSFM. The numerical results show that the proposed method can help the manufacturers to quickly and accurately select the optimal process parameters setting.

The remainder of the rest of this paper is organized as follows. In Section 2, the multi-objective optimization problem and the trade-off analysis based on the spider-web chart are described. In Section 3, the optimization methodologies and the detailed process for realizing the multi-objective optimization of the process parameters in PIM are described. To verify the proposed method, the automobile front bumper is taken as the case study and the numerical results are shown in Section 4. Finally, the conclusion is drawn in Section 5.

2 Overview of multi-objective optimization problem

2.1 Multi-objective optimization model

The goal of the injection molding process parameters (IMPP) optimization is to improve the product quality and productivity, while reducing cost. It is a typical multi-objective optimization problem. A multi-objective design optimization problem can be generally formulated as follows:

$$ {\displaystyle \begin{array}{c}\mathrm{find}:\boldsymbol{x}={\left[{x}_1,{x}_2,\cdots, {x}_D\right]}^T,\\ {}\begin{array}{c}\operatorname{minimize}F\left(\boldsymbol{x}\right)=\min \mathrm{imize}\left({f}_1\left(\boldsymbol{x}\right),{f}_2\left(\boldsymbol{x}\right),\cdots, {f}_K\left(\boldsymbol{x}\right)\right),\\ {}\begin{array}{l}\mathrm{Subject}\ \mathrm{to}:\kern0.5em \\ {}\kern2.75em {x}_i^L\le {x}_i\le {x}_i^H,\kern0.75em i=1,2,\cdots, D\end{array}\end{array}\end{array}} $$
(1)

where x is the column vector of the design variables, and xi(i = 1, 2, ⋯, D) represents the i-th deign variable; D is the number of the design variables; fj(x) (j = 1, 2, ⋯, K) is the j-th objective function to be minimized, and K is the number of the objective functions; \( {x}_i^L \) and \( {x}_i^H \) are the lower and upper bounds of the i-th design variable, respectively.

2.2 Objective functions

Warpage is one of the major defects in PIM. It causes the actual dimensions of the product to deviate from the design requirement, which should be minimized for high product quality. The first objective function f1(x) is taken as the warpage.

Weldlines influence the appearance of the product and the product strength. It is significant to reduce the weldlines of the product. The weldline will generate when two or more flow fronts meet. The melted plastic will be quickly solidified with the low weldline temperature, which causes the generation of long weldlines. The weldline temperature is one of the important factors for the weldline reduction [29, 30]. The minimum weldline temperature (MinTweld) needs to be maximized for weldline reduction. Therefore, the opposite of the minimum weldline temperature is taken as the second objective function f2(x), and is minimized.

Energy consumption and cost is also an important issue in PIM. When the melted plastic is injected into the cavity, a reversed pressure will generate. The clamping force should be applied to keep the mold closed. Small clamping force can reduce the energy consumption and save the cost. Therefore, clamping force is taken as the third objective function f3(x) for energy consumption reduction.

Cycle time directly influences the molding efficiency and should be minimized for high productivity, so it is taken as the fourth objective function f4(x). The sum of the injection time, the packing time, and the cooling time can be evaluated as the cycle time, which is expressed as:

$$ {f}_4\left(\boldsymbol{x}\right)={t}_{inj}+{t}_p+{t}_c, $$
(2)

where tinj is the injection time; tp is the packing time; tc is the cooling time.

2.3 Design variables

The melt temperature (Tmelt), the mold temperature (Tmold), the injection time (tinj), the packing pressure, the packing time, and the cooling time (tc) are taken as the design variables. To ensure the improvement of the products dimensional accuracy and the consistency of the product shrinkage degree, a variable packing pressure profile instead of constant packing pressure is applied, as shown in Fig. 1.

Fig. 1
figure 1

Packing pressure profiles. a Constant packing pressure profile; b variable packing pressure profile

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The packing pressure profile consists four parameters (packing pressure Pp1, Pp2 and packing time tp1, tp2 at point A and B in Fig. 1). Therefore, the design variables are x = [Tmelt, Tmold, tinj, Pp1, Pp2, tp1, tp2, tc]T. The lower and upper bound of design variables are shown in Table 1, which are determined by the recommended values in Moldflow and the manufacture’s recommendation.

Table 1 Design variables and their lower/upper bounds
Full size table

2.4 Trade-off analysis

For the multi-objective optimization problem expressed in Eq. (1), we want to minimize all the objectives simultaneously. Because of the contradiction between the objectives and their possible incommensurability, it is impossible to find a solution to ensure that all the objectives are simultaneously optimal. Thus, there is no single optimal solution but rather a set of compromise solutions named Pareto-optimal solutions or non-dominated solutions to such an optimization problem with multiple conflicting objectives.

After the Pareto-optimal solution set has been generated, the decision-maker should perform the trade-off analysis to select the most preferred one or a few solutions among the alternatives for producing the product. The spider-web chart or radar chart can visually understand the trade-off over three objectives, and is one of the useful tools for the trade-off analysis [31]. An illustrative example of a spider-web chart for the Pareto-optimal solutions visualization is shown in Fig. 2, in which four objective functions to be minimized is handled. Each apex of a polygon in Fig. 2 represents one objective. The outermost polygon shows the nadir solution, the innermost polygon represents the ideal solution, and the middle polygons present the alternatives.

Fig. 2
figure 2

An illustrative example of spider-web chart including four objectives

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Because of the difference of value range between the objectives, each apex is normalized using Eq. (3) to draw the spider-web chart.

$$ \overline{f_j}\left({\boldsymbol{x}}_p^{(i)}\right)=\frac{f_j\left({\boldsymbol{x}}_p^{(i)}\right)-{f}_j^I}{f_j^N-{f}_j^I}\ i=1,2,\cdots, {N}_{alt}\kern0.5em j=1,2,\cdots, K $$
(3)

where \( {\boldsymbol{x}}_p^{(i)} \) denotes the i-th alternative (Pareto-optimal solution); Nalt is the number of the Pareto-optimal solutions; \( {f}_j\left({\boldsymbol{x}}_p^{(i)}\right) \) and \( \overline{f_j}\left({\boldsymbol{x}}_p^{(i)}\right) \) represent the unnormalized and normalized value of the j-th objective of \( {\boldsymbol{x}}_p^{(i)} \), respectively; K is the number of the objectives; \( {f}_j^I \) and \( {f}_j^N \) denote the ideal and nadir value of the j-th objective.

The area of each polygon in the spider-web chart is used to compare the alternatives. The Pareto-optimal solutions minimizing and maximizing the polygon area in the spider-web chart are taken as the better solution and the worse solution, respectively.

3 Optimization methodologies

3.1 Multi-objective optimization process

The method, DSFM, is proposed to realize the process parameter optimization in PIM for warpage and weldlines reduction and clamping force and cycle time minimization, which integrates the sampling strategy, numerical simulation, metamodeling method, and multi-objective optimization algorithm. The sampling strategy is utilized to generate sampling points from the design space at different parameter levels. For the sampling points, the numerical simulation based on Moldflow is implemented to calculate the responses (warpage, minimum weldline temperature, clamping force, and cycle time). Based on the sampling points and their corresponding response, the metamodeling method is applied to approximately represent the mathematical relationship between the design variables (process parameters) and the responses, which constructs the response predictors to calculate the responses for any sampling point in the global design space. The multi-objective optimization algorithm, NSGA-III, is executed to locate the Pareto-optimal solutions, where the response predictors are taken as the fitness functions. The flowchart of our proposed method to realize multi-objective process parameter optimization in PIM is shown in Fig. 3.

Fig. 3
figure 3

Flowchart of the proposed method

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As shown in Fig. 3, the main steps of the proposed method include the acquisition and processing of the sampling points, the response predictor modeling, and the multi-objective optimization. The specific process can be described as:

  • Step 1: Identify the responses as the objectives and the process parameters related to the selected responses as the design variables of the optimization problem. Determine the value range of each design variable as the constraints of the optimization problem.

  • Step 2: Implement the sampling strategy, LHS, to generate stratified, and uniformly distributed sampling points in the global design space, and execute the numerical simulation to calculate and obtain the responses for each sampling point based on the Moldflow.

  • Step 3: Apply the sensitivity analysis among the design variables and the responses to identify the most important design variables that have a significant influence on the selected objectives.

  • Step 4: Response predictor modeling based on the metamodeling method. Taking the sampling points obtained in Step 2 as the training set (displayed in Table 11 ~ 12 in Appendix), the proposed method, ASGM, is implemented to calculate the gradient vector for each design variable of each sampling point in the training set as the sensitivity information of the training set. Then, based on the response and sensitivity information of the sampling points, the GERSM combined with the MLSM is constructed as the response predictor for each selected objective.

  • Step 5: Multi-objective process parameter optimization based on NSGA-III algorithm. Set the initial parameters of NSGA-III algorithm, and then take the response predictors constructed in Step 4 as the fitness functions to perform multi-objective global optimization and locate the Pareto-optimal solution set. For the Pareto-optimal solutions, the trade-off analysis is implemented to locate a better solution.

  • Step 6: Organize the confirmation experiments (as shown in Table 13 ~ 14 in Appendix) to verify the effectiveness of the proposed method.

3.2 Response predictor modeling

3.2.1 GERSM metamodeling

Compared with the traditional response surface model (RSM), the GERSM utilizes not only the response information but also the sensitivity information of the sampling points to construct the response surface [32]. The total error consists of response error and gradient error. In addition, the MLSM is a local approximation method, which is essentially a weighted least-squares method, and it uses weighting factors and local approximation to improve the accuracy of least-squares fitting when constructing a surrogate model [33].

Combined with the MLSM, the surface was constructed by GERSM changes with the position of the point in the design space. Therefore, the total error and regression coefficients are functions of the sampling point position, which can be defined as:

$$ \boldsymbol{\beta} \left(\boldsymbol{x}\right)=\underset{\boldsymbol{\beta}}{\arg\ \min }{E}_{\mathrm{total}}\left(\boldsymbol{x}\right)=\underset{\boldsymbol{\beta}}{\arg\ \min}\left[\left(1-\mathrm{swg}\right)\cdotp {E}_y\left(\boldsymbol{x}\right)+\mathrm{swg}\cdotp {E}_g\left(\boldsymbol{x}\right)\right] $$
(4)
$$ \boldsymbol{x}={\left[{x}_1,{x}_2,\cdots, {x}_D\right]}^{\mathrm{T}}\kern3pt \in \kern3pt {\mathbb{R}}^D $$
(5)

where β(x) is the regression coefficient vector; Etotal(x) is the total error function; x is the general sampling point; D is the number of the design variables; ℝD is the design space; swg is the sensitivity control factor for measuring the effect of the response information and the gradient information on the total fitting error; Ey(x) and Eg(x) represent the response error function and the gradient error function, respectively.

The second order polynomial is used to construct the response surface, which can be defined as:

$$ \hat{y}\left(\boldsymbol{x}\right)={b}_0+\sum \limits_{i=1}^D{b}_i\cdotp {x}_i+\sum \limits_{i=1}^D{b}_{ii}\cdotp {x_i}^2+\sum \limits_{i=1}^D\sum \limits_{j>i}^D{b}_{ij}\cdotp {x}_i{x}_j $$
(6)

where \( \hat{y}\left(\boldsymbol{x}\right) \) is the response surface; mapping b0, bi, bii, bij : x → ℝ project sampling point to its corresponding regression coefficients; xi and xj denote the i-th and j-th design variable, respectively.

According to Eq. (6), the error function Ey(x) and Eg(x) are formulated as follows:

$$ {E}_y\left(\boldsymbol{x}\right)={\left(\boldsymbol{y}-\mathbf{X}\bullet \boldsymbol{\beta} \left(\boldsymbol{x}\right)\right)}^{\mathrm{T}}{\mathbf{W}}_y\left(\boldsymbol{x}\right)\left(\boldsymbol{y}-\mathbf{X}\bullet \boldsymbol{\beta} \left(\boldsymbol{x}\right)\right) $$
(7)
$$ {E}_g\left(\boldsymbol{x}\right)=\sum \limits_{i=1}^D{\left({\boldsymbol{g}}_{x_i}-{\mathbf{T}}_{x_i}\bullet \boldsymbol{\beta} \left(\boldsymbol{x}\right)\right)}^{\mathrm{T}}{\mathbf{W}}_g\left(\boldsymbol{x}\right)\left({\boldsymbol{g}}_{x_i}-{\mathbf{T}}_{x_i}\bullet \boldsymbol{\beta} \left(\boldsymbol{x}\right)\right) $$
(8)

where y is the true response vector; \( {\boldsymbol{g}}_{x_i} \) is the true gradient vector for the i-th design variable of x; X denotes the response design matrix; \( {\mathbf{T}}_{x_i} \) denotes the response and gradient design matrix of the i-th design variable, which is the partial derivative of X to the i-th design variable; the diagonal matrices Wy(x) and Wg(x) represent the response and gradient weight matrix, respectively.

The vector y and β(x), the matrices X, \( {\mathbf{T}}_{x_i} \), Wy(x) and Wg(x) can be formulated as follows:

$$ \boldsymbol{y}={\left[{y}^{(1)}\kern0.5em {y}^{(2)}\kern0.5em \mathbf{\cdots}\kern0.5em {y}^{(k)}\kern0.5em \cdots \kern0.5em {y}^{(N)}\right]}^{\mathrm{T}} $$
(9)
$$ \boldsymbol{\beta} \left(\boldsymbol{x}\right)={\left[{b}_0\kern0.5em {b}_1\kern0.5em \mathbf{\cdots}\kern0.5em {b}_i\kern0.5em \cdots \kern0.5em {b}_D\kern0.5em {b}_{11}\kern0.5em \mathbf{\cdots}\kern0.5em {b}_{ii}\kern0.5em \cdots \kern0.5em {b}_{ij}\kern0.5em \cdots \kern0.5em {b}_{\left(D-1\right)D}\right]}^{\mathrm{T}} $$
(10)
$$ \mathbf{X}=\left[\begin{array}{ccccccccccccc}1& {x}_1^{(1)}& \cdots & {x}_i^{(1)}& \cdots & {x}_D^{(1)}& {x_1^{(1)}}^2& \cdots & {x_i^{(1)}}^2& \cdots & {x}_i^{(1)}{x}_j^{(1)}& \cdots & {x}_{D-1}^{(1)}{x}_D^{(1)}\\ {}\vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots \\ {}1& {x}_1^{(k)}& \cdots & {x}_i^{(k)}& \cdots & {x}_D^{(k)}& {x_1^{(k)}}^2& \cdots & {x_i^{(k)}}^2& \cdots & {x}_i^{(k)}{x}_j^{(k)}& \cdots & {x}_{D-1}^{(k)}{x}_D^{(k)}\\ {}\vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots \\ {}1& {x}_1^{(N)}& \cdots & {x}_i^{(N)}& \cdots & {x}_D^{(N)}& {x_1^{(N)}}^2& \cdots & {x_i^{(N)}}^2& \cdots & {x}_i^{(N)}{x}_j^{(N)}& \cdots & {x}_{D-1}^{(N)}{x}_D^{(N)}\end{array}\right] $$
(11)
$$ {\mathbf{T}}_{x_i}=\left[\begin{array}{ccccccccccccc}0& 0& \cdots & 1& \cdots & 0& 0& \cdots & 2{x}_i^{(1)}& \cdots & {x}_j^{(1)}& \cdots & 0\\ {}\vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots \\ {}0& 0& \cdots & 1& \cdots & 0& 0& \cdots & 2{x}_i^{(k)}& \cdots & {x}_j^{(k)}& \cdots & 0\\ {}\vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots \\ {}0& 0& \cdots & 1& \cdots & 0& 0& \cdots & 2{\mathrm{x}}_i^{(k)}& \cdots & {x}_j^{(N)}& \cdots & 0\end{array}\right] $$
(12)
$$ {\mathbf{W}}_y\left(\boldsymbol{x}\right)=\left[\begin{array}{ccccc}{\omega}_y\left(\boldsymbol{x}-{\boldsymbol{x}}^{(1)}\right)& \cdots & 0& \cdots & 0\\ {}\vdots & \ddots & \vdots & \vdots & \vdots \\ {}0& \cdots & {\omega}_y\left(\boldsymbol{x}-{\boldsymbol{x}}^{(k)}\right)& \cdots & 0\\ {}\vdots & \vdots & \vdots & \ddots & \vdots \\ {}0& \cdots & 0& \cdots & {\omega}_y\left(\boldsymbol{x}-{\boldsymbol{x}}^{(N)}\right)\end{array}\right] $$
(13)
$$ {\mathbf{W}}_g\left(\boldsymbol{x}\right)=\left[\begin{array}{ccccc}{\omega}_g\left(\boldsymbol{x}-{\boldsymbol{x}}^{(1)}\right)& \cdots & 0& \cdots & 0\\ {}\vdots & \ddots & \vdots & \vdots & \vdots \\ {}0& \cdots & {\omega}_g\left(\boldsymbol{x}-{\boldsymbol{x}}^{(k)}\right)& \cdots & 0\\ {}\vdots & \vdots & \vdots & \ddots & \vdots \\ {}0& \cdots & 0& \cdots & {\omega}_g\left(\boldsymbol{x}-{\boldsymbol{x}}^{(N)}\right)\end{array}\right] $$
(14)

where y(k) denotes the true response of the k-th sampling point; N is the size of the sampling points; x(k) denotes the k-th sampling point; \( {x}_i^{(k)} \) denotes the i-th design variable of the k-th sampling point. The element ωy(x − x(k)) and ωg(x − x(k)) on the diagonal denote the weight of x(k) for the calculation of response and gradient error, respectively. The exponential function is applied as the weight function to calculate ωy(x − x(k)) and ωg(x − x(k)). The formulas are as follows:

$$ {\omega}_y\left(\boldsymbol{x}-{\boldsymbol{x}}^{(k)}\right)={\omega}_g\left(\boldsymbol{x}-{\boldsymbol{x}}^{(k)}\right)=\omega (d)=\left\{\begin{array}{r}\exp \left(-\frac{d}{\mathrm{RI}}\right),\frac{d}{\mathrm{RI}}\le 1\ \\ {}0\kern1.5em ,\frac{d}{\mathrm{RI}}>1\end{array}\right. $$
(15)
$$ d={\left\Vert \boldsymbol{x}-{\boldsymbol{x}}^k\right\Vert}_2={\left[\sum \limits_{i=1}^D{\left({x}_i-{x}_i^{(k)}\right)}^2\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} $$
(16)

where ωy(∙) and ωg(∙) denote the response and gradient weight function, respectively; RI is the size of the support region, which means that only the sampling point located in the support region will have an impact on the prediction of x; d is the Euclidean distance between x and x(k).

According to the Eq. (4), the regression coefficient vector β(x) of GERSM can be obtained by the following set of equations:

$$ \frac{\partial {E}_{\mathrm{total}}\left(\boldsymbol{x}\right)}{\partial \boldsymbol{\beta}}=\left(1-\mathrm{swg}\right)\cdotp \frac{\partial {E}_y\left(\boldsymbol{x}\right)}{\partial \boldsymbol{\beta}}+\mathrm{swg}\cdotp \frac{\partial {E}_g\left(\boldsymbol{x}\right)}{\partial \boldsymbol{\beta}}=0 $$
(17)
$$ \boldsymbol{\beta} \left(\boldsymbol{x}\right)=A{\left(\boldsymbol{x}\right)}^{-1}B\left(\boldsymbol{x}\right) $$
(18)
$$ A\left(\boldsymbol{x}\right)=\left(1-\mathrm{swg}\right)\cdotp {\mathbf{X}}^{\mathrm{T}}{\mathbf{W}}_y\left(\boldsymbol{x}\right)\mathbf{X}+\mathrm{swg}\cdotp \sum \limits_{i=1}^D{{\mathbf{T}}_{x_i}}^{\mathrm{T}}{\mathbf{W}}_g\left(\boldsymbol{x}\right){\mathbf{T}}_{x_i} $$
(19)
$$ B\left(\boldsymbol{x}\right)=\left(1-\mathrm{swg}\right)\cdotp {\mathbf{X}}^{\mathrm{T}}{\mathbf{W}}_y\left(\boldsymbol{x}\right)\boldsymbol{y}+\mathrm{swg}\cdotp \sum \limits_{i=1}^D{{\mathbf{T}}_{x_i}}^{\mathrm{T}}{\mathbf{W}}_g\left(\boldsymbol{x}\right){\boldsymbol{g}}_{x_i} $$
(20)

Base on the GERSM, the response of x can be predicted as:

$$ \overset{\sim }{y}=\left(1-\mathrm{swg}\right)\cdotp {\boldsymbol{x}}^p\boldsymbol{\beta} \left(\boldsymbol{x}\right)+\mathrm{swg}\cdotp \sum \limits_{i=1}^D{\boldsymbol{t}}_{x_i}\boldsymbol{\beta} \left(\boldsymbol{x}\right) $$
(21)
$$ {\boldsymbol{x}}^p=\left[1\kern0.5em {x}_1\kern0.5em \cdots \kern0.5em {x}_i\kern0.5em \cdots \kern0.5em {x}_D\kern0.5em {x_1}^2\kern0.5em \cdots \kern0.5em {x_i}^2\kern0.5em \cdots \kern0.5em {x}_i{x}_j\kern0.5em \cdots \kern0.5em {x}_{D-1}{x}_D\right] $$
(22)
$$ {\boldsymbol{t}}_{x_i}=\left[0\kern0.5em 0\kern0.5em \cdots \kern0.5em 1\kern0.5em \cdots \kern0.5em 0\kern0.5em 0\kern0.5em \cdots \kern0.5em 2{x}_i\kern0.5em \cdots \kern0.5em {x}_j\kern0.5em \cdots \kern0.5em 0\right] $$
(23)

where xp denotes the response design vector of x; \( {\boldsymbol{t}}_{x_i} \) denotes the gradient design vector for the i-th design variable of x.

3.2.2 Adaptive sensitivity generating

The construction of the GERSM simultaneously requires the response and sensitivity information of the sampling points. Through numerical or practical experiment, only the response information can be obtained. Therefore, based on the finite difference (FD), ASGM is proposed to obtain the sensitivity information of the sampling points. For the general sampling point x in the design space, the general process of capturing the sensitivity information of x is as follows: Firstly, an adaptive response surface model (AdaRSM) is constructed as the strong predictor, which is proposed to explore the compact neighborhood of x. Then, the response of the new sampling point in the compact neighborhood of x can be given by the strong predictor AdaRSM. Finally, the FD is used to calculate the gradient vector of x.

The AdaRSM is the linear combination of a series of base response surface model (BRSM), which are generate by iteratively updating the weights of the sampling points. The AdaRSM can be expressed as:

$$ \hat{\boldsymbol{f}}\left(\boldsymbol{x}\right)=\sum \limits_{m=1}^M{\alpha}^{(m)}\cdotp {\mathrm{RSM}}^{(m)}\left(\boldsymbol{x}\right) $$
(24)

where \( \hat{\boldsymbol{f}}\left(\boldsymbol{x}\right) \) is the adaptive response surface; α(m) denotes the weight of the m-th BRSM; RSM(m)(x) is the m-th BRSM; x is the general sampling point, as shown in Eq. (5); M is the number of the BRSM.

The construction of AdaRSM includes the following steps: (1) initialize the weights of the sampling points and constructing the BRSM; (2) calculate the total error of the BRSM over the entire samples; (3) update the weight of each sampling point according to the prediction error of the BRSM for each sampling point; (4) iteratively construct the next BRSM based on the updated weights of the sampling points. Repeat steps (2)-(4) until the target number of the BRSM has been reached. (5) Calculate the weight of each BRSM according to the total error of each BRSM. The m-th BRSM can be expressed as:

$$ {\mathrm{RSM}}^{(m)}\left(\boldsymbol{x}\right)={a}_0^{(m)}+\sum \limits_{i=1}^D{a}_i^{(m)}\cdotp {x}_i+\sum \limits_{i=1}^D{a}_{ii}^{(m)}\cdotp {x_i}^2+\sum \limits_{i=1}^D\sum \limits_{j>i}^D{a}_{ij}^{(m)}\cdotp {x}_i{x}_j={\boldsymbol{x}}^p{\boldsymbol{\beta}}^{(m)} $$
(25)
$$ {\boldsymbol{\beta}}^{(m)}={\left({\mathbf{X}}^{\mathrm{T}}{\mathbf{W}}^{(m)}\mathbf{X}\right)}^{-1}{\mathbf{X}}^{\mathrm{T}}{\mathbf{W}}^{(m)}\boldsymbol{y} $$
(26)

where \( {a}_0^{(m)} \), \( {a}_i^{(m)} \), \( {a}_{ii}^{(m)}, \) and \( {a}_{ij}^{(m)} \) are the regression coefficients of the m-th BRSM; β(m) denotes the regression coefficient vector of the m-th BRSM; xp, X, and y are as shown in Eq. (22), (11), and (9); W(m) represents the weight matrix of the sampling points when constructing the m-th BRSM, which can be formulated as:

$$ {\mathbf{W}}^{(m)}=\left[\begin{array}{ccccc}{\upomega}_1^{(m)}& \cdots & 0& \cdots & 0\\ {}\vdots & \ddots & \vdots & \vdots & \vdots \\ {}0& \cdots & {\upomega}_k^{(m)}& \cdots & 0\\ {}\vdots & \vdots & \vdots & \ddots & \vdots \\ {}0& \cdots & 0& \cdots & {\upomega}_N^{(m)}\end{array}\right] $$
(27)

where \( {\upomega}_k^{(m)} \) denotes the weight of the k-th sampling point when constructing the m-th BRSM, which can be evaluated by the following equations.

$$ {\upomega}_k^{(m)}=\left\{\begin{array}{r}1\kern4em ,m=1\kern1.75em \\ {}\frac{\upomega_k^{\left(m-1\right)}}{Z^{\left(m-1\right)}}\cdotp \left[1+\exp \left(\lambda {\varepsilon}_k^{\left(m-1\right)}\right)\right],m=2,3,\cdots, M\end{array}\right. $$
(28)
$$ {\varepsilon}_k^{\left(m-1\right)}=\frac{\left|{RSM}^{\left(m-1\right)}\left({\boldsymbol{x}}^{(k)}\right)-{y}^{(k)}\right|}{y^{(k)}} $$
(29)
$$ {Z}^{\left(m-1\right)}=\frac{1}{N}\sum \limits_{k=1}^N{\upomega}_k^{\left(m-1\right)}\cdotp \left[1+\exp \left(\lambda {\varepsilon}_k^{\left(m-1\right)}\right)\right] $$
(30)

where \( {\upomega}_k^{\left(m-1\right)} \) and \( {\upomega}_k^{(m)} \) denote the weight of x(k) when constrcuting the (m-1)-th and the m-th BRSM, respectively; \( {\varepsilon}_k^{\left(m-1\right)} \) denotes the relative error of the (m-1)-th BRSM over x(k); RSM(m − 1)(x(k)) denotes the predicted response of the (m-1)-th BRSM for x(k); Z(m − 1) is the normalization factor of sampling point weights; λ is the scaling factor.

The weight of each BRSM depends on the total error of each BRSM over the entire sampling points. The smaller the total error, the larger the weight of the BRSM. The weight of the m-th BRSM is defined as follows:

$$ {\varepsilon}^{(m)}=\sqrt{\frac{1}{N}\cdotp \sum \limits_{k=1}^N{\left[{\mathrm{RSM}}^{(m)}\left({\boldsymbol{x}}^{(k)}\right)-{y}^{(k)}\right]}^2} $$
(31)
$$ {\alpha}^{(m)}=\frac{\exp \left(\tau \cdotp {\varepsilon}^{(m)}\right)}{\sum_{m=1}^M\exp \left(\tau \cdotp {\varepsilon}^{(m)}\right)} $$
(32)

where ε(m) is the root mean square error of the m-th BRSM over the entire samples; RSM(m)(x(k)) denotes the predicted response of the m-th BRSM for x(k); τ is the scaling factors; α(m) denotes the weight of the m-th BRSM.

The sensitivity information (gradient vector) of x(k) is approximate by the FD, which can be defined as follows:

$$ {\mathbf{g}}_{{\boldsymbol{x}}_{\boldsymbol{i}}}^{(k)}=\frac{\hat{f}\left({\boldsymbol{x}}_{\boldsymbol{i}+}^{(k)}\right)-\hat{f}\left({\boldsymbol{x}}_{\boldsymbol{i}-}^{(k)}\right)}{2h} $$
(33)
$$ {\boldsymbol{x}}_{\boldsymbol{i}+}^{(k)}={\left[{x}_1,\cdots, {x}_{i-1},{x}_i+h,{x}_{i+1},\cdots, {x}_D\right]}^{\mathrm{T}} $$
(34)
$$ {\boldsymbol{x}}_{\boldsymbol{i}-}^{(k)}={\left[{x}_1,\cdots, {x}_{i-1},{x}_i-h,{x}_{i+1},\cdots, {x}_D\right]}^{\mathrm{T}} $$
(35)

where \( {\mathbf{g}}_{{\boldsymbol{x}}_{\boldsymbol{i}}}^{(k)} \) denotes the gradient vector of x(k) for the i-th design variable; \( {\boldsymbol{x}}_{\boldsymbol{i}+}^{(k)} \) and \( {\boldsymbol{x}}_{\boldsymbol{i}-}^{(k)} \) are the new sampling points in the compact neighborhood of x(k), which are generated by adding h to the i-th design variable of x(k) and reducing the i-th design variable of x(k) by h, respectively; \( \hat{f}\left({\boldsymbol{x}}_{\boldsymbol{i}+}^{(k)}\right) \) and \( \hat{f}\left({\boldsymbol{x}}_{\boldsymbol{i}-}^{(k)}\right) \) denote the predicted response of the strong predictor AdaRSM for \( {\boldsymbol{x}}_{\boldsymbol{i}+}^{(k)} \) and \( {\boldsymbol{x}}_{\boldsymbol{i}-}^{(k)} \), respectively.

3.3 NSGA-III for locating Pareto-optimal solutions

NSGA-III algorithm [34, 35] utilizes a reference point-based selection mechanism to select the populations, which can obtain good performance in the case of four or more objectives optimization. By introducing a set of reference points, the population is selected according to the distances between the individuals and the reference points, thereby guiding the population search to approach the reference points. Because the reference points are evenly distributed over the reference hyperplane, the reference point-based selection mechanism can make the population more evenly distributed on the Pareto frontier and the optimization process can better converge to the Pareto-optimal solution set.

The process of locating the Pareto-optimal solution set of multi-objective optimization problem by the NSGA-III is as follows:

  • Step 1: The initial population containing N individuals is randomly generated, where N is the size of the population.

  • Step 2: The original population in the current generation is set as the parent population. The parent population goes through random selection, simulated binary intersection, and polynomial mutation operations to generate the offspring population. The size of the offspring population is N.

  • Step 3: The parent population and the offspring population are combined to form a new population with a population size of 2N. Then, the fitness function is used to evaluate each individual in the new combined population by calculating the responses (objectives).

  • Step 4: According to the fitness of each individual, the non-dominated sorting algorithm and reference point-based selection mechanism is used to select N optimal individuals from the new combined population in Step 3 to enter the next generation.

  • Step 5: Check whether the convergence condition is satisfied (e.g., generations of the population reach the preset maximum threshold). If the convergence condition is satisfied, the selected N optimal individuals in Step 4 is the Pareto-optimal solution set and the optimization process end; otherwise, the selected N optimal individuals in Step 4 is treated as the original population of next generation and redirect to Step 2.

3.4 Data analysis

Sensitivity analysis among process parameters and responses

Sensitivity analysis is applied to identify the important process parameters (design variables) which have significant influences on responses (objectives). It can exclude irrelevant parameters to reduce the dimension of the design space.

Correlation coefficient is implemented to indicate the sensitivity among process parameters and responses, which can be calculated by the following formula.

$$ \rho =\frac{\mathrm{Cov}\left(X,Y\right)}{\sqrt{\mathrm{Var}(X)\mathrm{Var}(Y)}}=\frac{E\left[\left(X-{\mu}_X\right)\left(Y-{\mu}_Y\right)\right]}{\sigma_X{\sigma}_Y} $$
(36)

where Cov(X, Y) is the covariance of vector X and vector Y; Var(X) and Var(Y) are the variance of X and Y respectively; μX and μY are the mean of X and Y respectively; σX and σY are the standard deviation of X and Y respectively; ρ is the correlation coefficient, which is in the range of [-1, 1]. If ρ = 0, it indicates that there is no correlation between the two vectors; if ρ < 0, there is a negative correlation between the two vectors; if ρ > 0, there is a positive correlation between the two vectors.

Prediction accuracy analysis

The response predictor is utilized as the fitness function of NSGA-III algorithm in the multi-objective optimization process. The prediction accuracy of the response predictor influences the optimization results, which may cause the optimization algorithm to not converge. Therefore, the prediction accuracy analysis is verified by evaluating the root mean squared error (RMSE), mean absolute error (MAE), and coefficient of determination (R2) between the responses calculated by simulation experiments and the responses predicted by the response predictor. These evaluation criteria can be formulated as follows:

$$ \mathrm{R}{\mathrm{MSE}}_j=\sqrt{\frac{1}{N_V}\sum \limits_{k=1}^{N_V}{\left({\hat{y}}_j^{(k)}-{y}_j^{(k)}\right)}^2}\ j=1,2,\cdots, K $$
(37)
$$ {\mathrm{MAE}}_j=\frac{1}{N_V}\sum \limits_{k=1}^{N_V}\left|{\hat{y}}_j^{(k)}-{y}_j^{(k)}\right|\ j=1,2,\cdots, K $$
(38)
$$ {R^2}_j=1-\frac{\sum_{k=1}^{N_V}{\left({\hat{y}}_j^{(k)}-{y}_j^{(k)}\right)}^2}{\sum_{k=1}^{N_V}{\left({y}_j^{(k)}-{\overline{y}}_j\right)}^2}\ j=1,2,\cdots, K $$
(39)

where RMSEj, MAEj, and R2j denote the RMSE, MAE, and R2 of the j-th response (objective), respectively; K denotes the number of the objectives; NV denotes the number of the sampling points for verification; \( {\hat{y}}_j^{(k)} \) and \( {y}_j^{(k)} \) denote the j-th response of the k-th sampling point predicted by the corresponding response predictor and calculated by simulation experiment, respectively; \( {\overline{y}}_j \) denotes the mean of the j-th response.

4 Case study

Because of the characteristic of lightweight, customization, good adaptability to variable complex structures, low cost, high efficiency, etc., the plastic injection molding products are widely used in automobile industry such as the bumper, door, and dashboard. The automobile bumper should meet the mechanical and geometric requirement to ensure the necessary protection function as well as the shape consistency and appearance beauty. Therefore, the automobile front bumper is taken as a case study to verify the proposed method.

4.1 Finite element simulation model

An injection plastic product of the automobile front bumper shown in Fig. 4 is taken into consideration. The maximum size of this product is 1800mm × 430mm × 725mm, in which the maximum thickness is 4 mm and the minimum thickness is 2.5 mm.

Fig. 4
figure 4

The 3D model of the automobile front bumper

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Considering the complex structure of the product, the multiple gates can achieve better filling. However, the utilization of multiple gates can inevitably cause the generation of weldlines. The valve hot runner system adopts the valve gate controllers to control the opening and closing of each gate, which can realize the sequential filling of the whole cavity. Therefore, it can reduce the stress concentration at the intersection of the melt flow front and effectively eliminate weldlines. The overview of the conventional and valve hot runner system is shown in Fig. 5. The gate locations, the number of gates, and the gate type are determined by the combination of the “gate location” analysis and the “fill” analysis. As shown in Fig. 5, three new gates and corresponding hot runners are added to the valve hot runner system compared to the conventional runner system in order to balance the runners. For the valve hot runner system, the flow path of the melted plastic in each runner is as follows: hot sprue → hot runner → hot runner → hot gate → cold runner → cold runner → cold gate.

Fig. 5
figure 5

Overview of the hot runner system. a Conventional hot runner system; b valve hot runner system

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The details of numerical simulation are shown in Table 2. The numerical simulations are executed through the Autodesk® Moldflow® software (version 2018) [36], where the processor of the computer is Inter® Core™ i9-9900K CPU @ 3.60 GHz. The material, acrylonitrile butadiene styrene (ABS), is used to manufacture the product, and its properties are listed in Table 3.

Table 2 Detailed settings of the numerical simulations in Moldflow
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Table 3 Material property of acrylonitrile butadiene styrene (ABS)
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4.2 Numerical result analysis

4.2.1 Sensitivity analysis for process parameters

The correlation coefficient can show sensitivity among the design variables and responses, which has been calculated and displayed in Fig. 6. As shown in Fig. 6, the value in each sub-box is the correlation coefficient of the two crossed variables. The positive value indicates the positive correlation, while the negative value indicates the negative correlation, and zero value indicates irrelevance. The bigger the absolute value of the correlation coefficient is, the stronger the relationship between the two variables is.

Fig. 6
figure 6

Correlation relationship among the design variables and responses

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According to the heat-map plot in Fig. 6, it can be found that tinj, Pp1, and tc have the negative influence on warpage with the correlation coefficients equaling to −0.86, −0.5, and −0.28, respectively; Tmelt has the positive influence on warpage with the correlation coefficient equaling to 0.18. Similarly, Tmelt and tp1 have the positive influence on minimum weldline temperature with the correlation coefficients equaling to 0.96 and 0.27, respectively; tinj, Pp1, and tp2 have the negative influence on minimum weldline temperature with the correlation coefficients equaling to −0.12, −0.19, and −0.27, respectively. Pp1, Tmelt, tp1, and tp2 have the positive influence on clamping force with the correlation coefficients equaling to 0.92, 0.2, 0.15, and 0.1, respectively; tinj and tc have the negative influence on clamping force with the correlation coefficients equaling to −0.17 and −0.22, respectively. In fact, cycle time is the sum of tinj, tp1, tp2, and tc. It indicates that tinj, tp1, tp2, and tc have a significant positive influence on cycle time. In addition, the correlation coefficients between warpage and cycle time are −0.48, which indicates that cycle time changes in the opposite direction with warpage.

4.2.2 Accuracy analysis for response prediction

Fifteen experiments different from thirty-six initial sampling points, which are both generated by the Latin hypercube sampling (LHS), are implemented to verify the prediction accuracy of the response predictors. The cycle time is the sum of the injection time, packing time ,and cooling time, which can be accurately calculated based on the design variables of the four-objective optimization problem. The proposed GERSM, classical RSM, Kriging, support vector regression (SVR), and Gaussian process regression (GPR) are applied to construct the surfaces of design variables and responses (warpage, minimum weldline temperature, and clamping force). The prediction results of responses are displayed in Fig. 7, from which it can be found that Kriging has the largest prediction error, while the proposed model works best.

Fig. 7
figure 7

Comparison of prediction models and experiments. a Warpage; b minimum weldline temperature; c clamping force

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As shown in Fig. 7, the predictions and experiments of the responses are qualitatively compared. In order to quantitatively compare the performance of different prediction models, MSE, RMSE, MAE, and R2 are applied to evaluate the performance of the prediction models, which are calculated and displayed in Tables 4, 5, and 6.

Table 4 Comparison of different prediction models (warpage)
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Table 5 Comparison of different prediction models (minimum weldline temperature)
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Table 6 Comparison of different prediction models (clamping force)
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The smaller the MAE and RMSE are, the higher the prediction accuracy of model is. R2 ∈ [0, 1] is closer to 1; the accuracy of model is higher. As shown in Tables 4, 5, and 6, it can be found that the proposed model has the highest prediction accuracy for the warpage, minimum weldline temperature, and clamping force, compared with RSM, SVR, GPR, and Kriging.

4.2.3 Multi-objective optimization results

NSGA-III algorithm is implemented to locate the Pareto-optimal solutions for the four-objective optimization problem. The parameters of this algorithm are set as follows:

$$ \left.\begin{array}{lll}\mathrm{Population}\ \mathrm{size}=220& \mathrm{Generation}=600& \mathrm{Refrence}\ \mathrm{point}\ \mathrm{division}=9\\ {}\mathrm{Crossover}\ \mathrm{possibility}=1.0& \mathrm{Distribution}\ \mathrm{index}\ \mathrm{of}\ \mathrm{crossover}=30&\ \\ {}\mathrm{Mutation}\ \mathrm{possibility}=0.2& \mathrm{Distribution}\ \mathrm{index}\ \mathrm{of}\ \mathrm{mutation}=20&\ \end{array}\right\} $$

The triple-objective Pareto frontiers among warpage, minimum weldline temperature, clamping force, and cycle time are shown in Fig. 8, and the pair-wise Pareto frontiers are shown in Fig. 9. As shown in Fig. 8, the triple-objective Pareto frontiers show that any three of four objectives cannot reach the optimal at the same time. Therefore, different process parameters have different influences on the results. The four objectives cannot reach the optimal result at the same time and the ideal solution of the optimization problem cannot be located, which leads to the trade-off.

Fig. 8
figure 8

Triple-objective Pareto frontiers. a Warpage and minimum weldline temperature and clamping force; b warpage and minimum weldline temperature and cycle time; c minimum weldline temperature and clamping force and cycle time; d warpage and clamping force and cycle time

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Fig. 9
figure 9

Pair-wise Pareto frontiers. a Warpage and minimum weldline temperature; b warpage and clamping force; c warpage and cycle time; d minimum weldline temperature and clamping force; e minimum weldline temperature and cycle time; f clamping force and cycle time

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The pair-wise Pareto frontier shown in Fig. 9a indicates that there is no obvious trade-off between warpage and MinTweld (minimum weldline temperature). Observing Fig. 6, warpage is strongly influenced by tinj and Pp1, while MinTweld is strongly influenced by Tmelt.

The pair-wise Pareto frontier shown in Fig. 9b indicates a trade-off between warpage and clamping force. Observing Fig. 6, the correlation coefficient between warpage and clamping force equals to −0.27, which means that they are negatively correlated and change in reverse direction. As shown in Fig. 6, Pp1 has reverse impact on warpage and clamping force, with correlation coefficients equaling to −0.5 and 0.92, respectively. It means that smaller Pp1 leads to smaller clamping force, but larger warpage. They cannot reach the optimal at the same time.

The pair-wise Pareto frontier shown in Fig. 9c indicates an obvious trade-off between warpage and cycle time. Observing Fig. 6, the correlation coefficient between warpage and cycle time equals to −0.48, which means that they are negatively correlated and change in reverse direction. As shown in Fig. 6, tc and tinj have reverse impact on warpage and cycle time. It means that shorter tc and tinj lead to shorter cycle time, but larger warpage. They cannot reach the optimal at the same time.

The pair-wise Pareto frontier shown in Fig. 9d indicates a trade-off between MinTweld and clamping force. Observing Fig. 6, the correlation coefficient between MinTweld and clamping force equals to 0.14, which means that they are positively correlated and change in same direction. Therefore, clamping force and the opposite of minimum weldline temperature (−MinTweld) change in reverse direction. As shown in Fig. 6, Tmelt, tinj, and tp1 have same impact on MinTweld and clamping force. It means that smaller Tmelt, larger tinj, and smaller tp1 lead to smaller clamping force and MinTweld. However, the melted plastic will be quickly solidified with the low weldline temperature, which causes the generation of long weldlines. Therefore, smaller MinTweld generates larger weldlines. They cannot reach the optimal at the same time to simultaneously minimize the weldlines and clamping force.

The pair-wise Pareto frontier shown in Fig. 9e indicates that there is no obvious trade-off between MinTweld and cycle time. Observing Fig. 6, MinTweld is strongly influenced by Tmelt, while cycle time is strongly influenced by tinj, tp1, tp2, and tc.

The pair-wise Pareto frontier shown in Fig. 9f indicates that there is no obvious trade-off between clamping force and cycle time. Observing Fig. 6, clamping force is strongly influenced by Pp1, while cycle time is strongly influenced by tinj, tp1, tp2, and tc.

The Pareto-optimal solutions located by NSGA-III are listed in Table 7. The trade-off analysis is performed to determine the better and worse solution for decision-making based on the spider-web chart. The ideal and nadir value of the objectives in Eq. (3) can be set as follows: \( \left[{f}_1^I,{f}_2^I,{f}_3^I,{f}_4^I\right]=\left[0,-255,5400,75\right];\left[{f}_1^N,{f}_2^N,{f}_3^N,{f}_4^N\right]=\left[25,-225,7000,115\right] \). In the spider-web chart, the areas of the Pareto-optimal solutions located by NSGA-III are calculated and displayed in Table 8, where there is no preference for all objective. The smaller the area of alternative solution, the better the solution.

Table 7 Pareto-optimal solutions located by NSGA-III
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Table 8 Areas of Pareto-optimal solutions in spider-web chart
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The spider-web chart is shown in Fig. 10, in which the red, blue, and black lines represent the better solution, worse solution, and alternative solutions, respectively. As shown in Fig. 10, the area of the better solution is smaller than that of the worse solution. In addition, warpage and weldlines temperature distribution at the better and worse solution are compared and displayed in Figs. 11 and 12, respectively. As shown in Fig. 11, it can be found that warpage is well reduced at better solution compared to that at worse solution. As shown in Fig. 12, it can be found that weldlines are well reduce at better solution compared to that at worse solution. From the perspective of process parameters, it can indicate that larger packing pressure and injection time can reduce warpage.

Fig. 10
figure 10

Spider-web chart at better and worse solution

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Fig. 11
figure 11

Comparison of warpage at better and worse solution. a Better solution; b worse solution

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Fig. 12
figure 12

Comparison of weldline temperature distribution at better and worse solution. a Better solution; b worse solution

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To verify the effect of the valve hot runner system on the weldline reduction, the comparison of weldlines under the conventional hot runner system and under the valve hot runner system at better solution is shown in Fig. 13. It can be found that the weldlines are generated in inconspicuous positions such as grilles, lamp holes, and product edges and effectively eliminated when valve hot runner system is used, compared with conventional hot runner system. The surface quality of the automobile front bumper is higher when the valve hot runner system is applied.

Fig. 13
figure 13

Comparison of weldlines at better solution. a Conventional hot runner system; b valve hot runner system

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According to the result of the trade-off analysis, the optimized setting of the process parameters is shown in Table 9 for the product quality and productivity improvement and the cost-saving. The warpage and the weldlines of the automobile front bumper after the optimization are compared with them before the optimization, as shown in Fig. 14. Moreover, the setting of the process parameters is determined by the recommendation of Moldflow and the result of the “molding window” analysis.

Table 9 The optimized setting of the process parameters based on the trade-off analysis
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Fig. 14
figure 14

The warpage and weldlines of the automobile front bumper before and after optimization. a Warpage; b weldlines

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The verification of prediction accuracy for four objectives at optimized solution is summarized in Table 10, which shows that relative absolute errors for all objectives at optimized solution are below 2.5%. With the result of accuracy analysis displayed in Section 4.2.1, the prediction accuracy can be confirmed and verified.

Table 10 Verification of the optimized results
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The proposed multi-objective optimization method analyzes the correlation between the process parameters and the objectives and conducts the trade-off analysis to balance the multiple conflicting objectives. It can effectively locate the optimized process parameters setting to achieve the product quality and productivity improvement and the cost-saving, which can provide a theoretical basis and reference for the actual injection molding process.

5 Conclusion

In this paper, the multi-objective optimization of process parameters in PIM for minimizing the warpage, weldlines, clamping force, and cycle time is performed to realize high product quality, high productivity, and low energy consumption. The melted plastic will be quickly solidified with low weldline temperature and long weldlines will generate. To shorten the weldlines, the minimum weldline temperature is considered to be maximized. In this view, we propose a differential sensitivity fusion method (DSFM). The conclusions are as follows:

  1. (1)

    The variable packing pressure profile and the melt temperature, mold temperature, injection time, and cooling time are taken as design variables and optimized. The generic optimization algorithm NSGA-III is applied to locate the Pareto-optimal solutions. Pareto frontier shows that selected four objectives cannot simultaneously reach the optimal, which leads to the trade-off. The spider-web chart is used to perform the trade-off analysis among the four objectives, and the better and worse solutions are identified for decision-making.

  2. (2)

    The metamodeling method, GERSM coupling with the MLSM, is used to construct the response predictors, which fit the mathematical relationship between design variables and responses and is taken as the fitness functions in the multi-objective optimization process. This model simultaneously utilizes the response and sensitivity information of the sampling point to improve the accuracy of the response predictors. Considering the capture of the sensitivity information, ASGM is proposed to calculate the gradient vector for each design variable of the sampling point. The results of the accuracy analysis for response predictors show that the proposed model has the highest prediction accuracy for the warpage, minimum weldline temperature, and clamping force, compared with RSM, SVR, GPR, and Kriging.

  3. (3)

    The automobile bumper is taken as the case study, where the valve hot runner system is used. The numerical simulation result shows that the weldlines are generated in inconspicuous positions such as grilles, lamp holes, and product edges and effectively eliminated, compared with the conventional hot runner system. The valve hot runner system can effectively improve the product quality.