Int J Adv Manuf Technol (2013) 66:907–916 DOI 10.1007/s00170-012-4376-9

ORIGINAL ARTICLE

Multiobjective optimization of injection molding parameters based on soft computing and variable complexity method Jin Cheng & Zhenyu Liu & Jianrong Tan

Received: 18 October 2010 / Accepted: 4 July 2012 / Published online: 31 August 2012 # Springer-Verlag London Limited 2012

Abstract The objective of this study is to propose an intelligent methodology for efficiently optimizing the injection molding parameters when multiple constraints and multiple objectives are involved. Multiple objective functions reflecting the product quality, manufacturing cost and molding efficiency were constructed for the optimization model of injection molding parameters while multiple constraint functions reflecting the requirements of clients and the restrictions in the capacity of injection molding machines were established as well. A novel methodology integrating variable complexity methods (VCMs), constrained nondominated sorted genetic algorithm (CNSGA), back propagation neural networks (BPNNs) and Moldflow analyses was put forward to locate the Pareto optimal solutions to the constrained multiobjective optimization problem. The VCMs enabled both the knowledge-based simplification of the optimization model and the variable-precision flow analyses of different injection molding parameter schemes. The Moldflow analyses were applied to collect the precise sample data for developing BPNNs and to fine-tune the Paretooptimal solutions after the CNSGA-based optimization while the approximate BPNNs were utilized to efficiently compute the fitness of every individual during the evolution of CNSGA. The case study of optimizing the mold and process parameters for manufacturing mice with a compound-cavity mold demonstrated the feasibility and intelligence of proposed methodology. Keywords Injection molding parameter . Multiobjective optimization . Variable complexity method . Soft J. Cheng : Z. Liu (*) : J. Tan State Key Lab of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China e-mail: [email protected]

computing . Constrained non-dominated sorted genetic algorithm . Back propagation neural network

1 Introduction Plastic injection molding (PIM) is a complex evolutionary process of a nonlinear coupling system with multi-input and multi-output (MIMO) parameters. The responses of the system including product quality, manufacturing cost and molding efficiency depend on its inputs including mold parameters, process parameters, characteristics of polymer materials, and so on. Therefore, the input parameters of this MIMO system are routinely optimized in the PIM for the purpose of achieving high quality products efficiently and economically. Originally, engineers used to determine the optimal parameters through the experience- and intuition-based trial-and-error procedure. Later, researchers introduced the design of experiment (DOE), Taguchi orthogonal array and flow analysis softwares such as Moldflow Plastic Insight and determined the optimal parameters by computer-aided trial-and-error procedure [1]. However, the trial-and-error procedure either experience- and intuition-based or computer-aided is costly and time consuming. Moreover, the DOE method can only choose the best scheme from various combinations of specified parameter level that is discrete values. Thus, it is virtually unable to obtain the optimal parameter scheme since the most of injection molding parameters are continuous. In recent years, intelligent optimization techniques such as fuzzy optimization [2], approximate reasoning [3], genetic algorithm [4], and neural network [5] have been widely applied for the optimization of injection molding parameters with the development of artificial intelligence. Alam and Kamal [6, 7] applied GA to the solution of runner-balancing problem while Deng et al. [8] proposed a hybrid of modepursuing sampling method and GA for the minimization of

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injection molding warpage. But, such kind of GA-based methods involved large computational cost due to the large number of computer-aided flow analyses during the search of the optimal parameter schemes. To save computational cost and accelerate the optimization procedure, various surrogate models were constructed to substitute the flow analysis software for predicting the injection molding results of different parameter schemes. Kwak et al. [9] developed a neural network (NN) model for predicting the thickness reduction and volumetric distortion rate with regard to four key process parameters in order to find the optimal process setting for improving the surface profile of injection-molded optic lens. Juang et al. [10] developed a TSK-type recurrent fuzzy NN for the control of mold temperature. Karatas et al. [11] modeled the yield length of four commercial plastics by NN for estimating whether a mold was fully filled or not. Chen et al. [12] proposed a self-organizing map plus a back propagation NN (BPNN) model for dynamically predicting the product weight in PIM. Altan [13] reduced the shrinkage in injection molding based on the Taguchi methods, ANOVA, and NN. Gao and Wang [14, 15] proposed a sequential optimization method based on Kriging surrogate model to minimize the warpage of injection-molded parts. Shi et al. [16] minimized the warpage of injection-molded parts based on NN model. However, the achievement of the optimal parameter schemes by the surrogate-based optimization methods still depended on the traditional optimization approaches such as DOE, which were time consuming and lack of intelligence. To overcome these shortcomings, some researchers have recently applied NNs and evolutionary methods to the problem of parameter optimization in PIM. Yen et al. [17] integrated the simulated annealing method with an abductive NN for predicting the part warpage with regard to the parameters of runner system to optimize the runner dimensions of a free-form injection mold. Kurtaran et al. [18] developed a feed-forward NN for predicting the warpage with regard to five process parameters and interfaced it with GA to find the optimal process parameters for minimizing the warpage of a bus ceiling lamp base. Ozcelik and Erzurumlu [19] reduced the warpage of thin shell plastic parts by utilizing ANOVA, NN and GA. Shen et al. [20] developed a NN of 5-9-1 architecture for predicting the volumetric shrinkage variation with regard to five process parameters and utilized it for the fitness computation during the GA-based optimization of the process parameters for minimizing the volumetric shrinkage variation of the top cover of an industrial refrigerator. Mathivanan and Parthasarathy [21] minimized the sink depths in injection-molded thermoplastic components based on response surface regression model and GA. Nevertheless, there are still some deficiencies in the above NN- and GA-based optimization approaches. Firstly, they utilize only single criterion of product quality such as

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warpage and sink depth to evaluate the moldability of different parameter schemes, which cannot achieve the optimal parameter schemes appropriate for practical manufacture of plastic product due to the fact that both efficiency and cost should be considered in practice besides product quality. Secondly, they choose parameters of either injection mold or injection molding process as design variables during the optimization while both mold and process parameters are often adjusted simultaneously to achieve the expected molding results in practice. Thirdly, the mathematical models established in the above NN- and GA-based optimization researches of injection molding parameters rarely consider the constraints related to the characteristics of polymer materials, the capacities of injection machines, the requirements of clients and so on. Then the so-called optimal schemes obtained on the basis of these non-constrained models may probably become infeasible when put into practice. Consequently, a multiobjective and multiconstraint optimization model including both mold and process parameters as design variables should be established if the shortcomings of the present optimization approaches for injection molding parameters are to be overcome. The solution of such a multiobjective and multiconstraint optimization model is a complex decision-making procedure due to the large number of design variables, objectives and constraints involved in the injection molding practice and it always involves large computational cost since the values of all objectives and constraints have to be obtained based on the repeated computation of different parameter schemes. The variable complexity method (VCM) [22] is introduced here to handle the complex optimization problem. The importance of all design parameters, objectives and constraints are firstly evaluated based on the results of DOE and expert knowledge. Then, optimization model including different number of design variables, objective functions and constraints can be utilized in different optimization stages. For example, some design variables can be determined by expert knowledge or the results of DOE so that the optimization model can be simplified for further investigation. All the constrained multiobjective optimization models established in our work can be solved by NN- and GA-based methodology with the fitness value of every individual computed by MIMO NN for predicting the values of objective functions. However, the multiobjective GA (MOGA) based on the concept of Pareto-optimal set should be utilized instead of the single objective GA since the optimization problem to be solved has multiple conflicting objectives and there is no solution superior in all objectives. Additionally, the NN for evaluating whether the constraints are satisfied or not should also be established besides that for predicting objective values so that the fitness value of every individual can be efficiently obtained during the evolution of MOGA. Thus,

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the VCM in our work is also reflected in the variableprecision models for computing the values of objectives and constraints. Specifically, both low-precision approximate models and high-precision Moldflow analyses are utilized during the solution of proposed optimization problem, which enables both the efficient search of the optimal solutions based on the computationally inexpensive models and the fine-tuning of parameters based on the expensive Moldflow analyses. The rest of this paper is organized as follows. The mathematical model of the constrained multiobjective optimization problem is established in Section 2 and the methodology for locating the Pareto-optimal solutions to the optimization problem that integrates VCMs, NNs, and MOGA is put forward in Section 3. Then, the optimization procedure of injection molding parameters including both mold and process parameters for manufacturing mice is provided in Section 4 to highlight the implementation of proposed methodology. The advantages of proposed methodology over the existing ones are also discussed in detail. Finally, the concluding remarks are made in Section 5.

quality in different mold cavities and that the quality of an injection-molded plastic part is often reflected by its volume shrinkage resulting from the cooling of polymer. The total volume of the runner system is utilized to express the manufacturing cost in view of the fact that the plastic material in the runner system is often wasted after the ejection of molded part. The molding efficiency is simply denoted by the cycle time of injection molding. Then the mathematical model in (2) can be established, all the objective values in which may be obtained through Moldflow analysis.  
min F ðxÞ ¼ min Dshr ðxÞ; Vrunner ðxÞ; Tcycle ðxÞ ; s:t:gi ðxÞ  0; i ¼ 1; 2;    ; NC :

ð2Þ

where Dshr ðxÞ ¼ Shrmax ðxÞ  Shrmin ðxÞ is the maximum difference of the volumetric shrinkage among different cavities with Shrmax(x)/Shrmin(x) being the largest/smallest volumetric shrinkage in all the cavities and there is Dshr(x)0Shr (x) for an injection mold with single cavity; Vrunner(x) is the total volume of the runner system; Tcycle(x) is the sum of injection time, packing time and cooling time, which excludes the ejection time considering that it has no influence on Dshr(x) and Vrunner(x).

2 Mathematical modeling of optimization problem Supposing that there are NV design variables, NO objective functions and NC constraints to be considered in the optimization of injection molding parameters, then the optimization problem can be expressed as
min y ¼ min F ðxÞ ¼ minðf1 ðxÞ; f2 ðxÞ;    ; fNO ðxÞÞ ; s:t:gi ðxÞ  0; i ¼ 1; 2;    ; NC :

ð1Þ

where x ¼ ðx1 ; x2 ;    ; xNV Þ 2 X is the vector of design parameters (namely, design vector) with its elements being either mold or process parameters; X ¼ fðx1 ; x2 ;    ; xNV Þjli  xi  ui ; i ¼ 1; 2;    ; NV g is the feasible domain of design vector x with L ¼ ðl1 ; l2 ;    ; lNV Þ and U ¼ ðu1 ; u2 ;    ; uNV Þ being lower and upper limits, respectively; y ¼ ðy1 ; y2 ;    ; yNO Þ ¼ ðf1 ðxÞ; f2 ðxÞ;    ; fNO ðxÞÞ 2 Y is the objective vector and Y is the objective space composed of multiple subspaces corresponding to different objective functions; gi ðxÞ  0ði ¼ 1; 2;    ; NC Þ are constraint functions. It is clear from the literature review in Section 1 that at least three objective functions corresponding to product quality, manufacturing cost and molding efficiency should be constructed for the purpose of comprehensively evaluating the moldability of injection molding parameter schemes. The maximum difference of the volume shrinkage among all the mold cavities is utilized to justify whether the quality of produced part is acceptable considering that an optimal parameter scheme should produce parts with uniform

3 Optimization methodology for injection molding parameters This research puts forward an integrated methodology for effectively assisting designers in the optimization of injection molding parameters, which integrates the advantages of variable complexity methods (VCMs), BPNNs, constrained nondominated sorted GA (CNSGA) and Moldflow analyses as well. Both variable-dimensional mathematical models of the optimization problem and variable-precision predictive models for moldability evaluation are employed in different stages of the optimization process. Two BPNNs act as the lowprecision approximate models for efficient moldability evaluation, which are interfaced with CNSGA to intelligently locate the Pareto-optimal solutions to the constrained multiobjective optimization problem. Moldflow analysis acts as a highprecision tool for collecting the sample data that are necessary for developing BPNNs and exploring the knowledge of injection molding, based on which the complex optimization model can be simplified. Moldflow analysis also assists in the finetuning of the Pareto-optimal solutions located by the CNSGAand BPNNs-based search. 3.1 VCMs for handling the complex optimization problem The optimization problem of injection molding parameters described in Eq. (1) or (2) is a complex decision-making process involving a series of design variables, objectives

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and constraints, which can hardly be solved in a single stage. The computational cost involved in the optimization procedure based on Moldflow analysis may become burdensome because of the large number of Moldflow simulations and moldability evaluations for identifying the tradeoffs between several conflicting objectives. VCM is introduced to handle such a complex problem in injection molding since it has been successively applied to the design of control system [22]. Two kinds of VCMs are employed here for searching the best injection molding parameter schemes. The first is to utilize the optimization models of different dimensions in different design stages, which enables the flexible exploration of design space by determining the values of all parameters in multiple design stages. The second is to utilize variable-precision models for evaluating the moldability of various parameter schemes, which enables both the efficient search of the optimal solutions to the optimization problem based on the computationally inexpensive approximate model and the fine-tuning of parameters based on the expensive Moldflow analyses.

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CBPNN should equal the number of constraint variables Nc (not the number of constraint functions NC) because the number of violated constraints is obtained by comparing the predicted values of constraint variables with their prescribed ones. The number of hidden neurons for OBPNN/ CBPNN can be determined by the performance comparison of various BPNNs comprising different number of hidden neurons based on the experiential equation NHO ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NHc ¼ NV þ Nc þ a NV þ NO þ að1  a  10Þ / ð1  a  10Þ. The sigmoid function is utilized for all the transfer functions in both OBPNN and CBPNN. The Levenberg–Marquardt learning algorithm is utilized to find the weight change and minimize the error energy during the training phase of OBPNN and CBPNN considering that it converges rapidly with less training cycles when compared to the steepest descent algorithm. The training of the networks will be terminated either when the maximum training times is reached or when the root-mean-square error between the desired values and network outputs is reduced to a given level. 3.2.2 Training and test samples

3.2 BPNNs for fitness computation For the purpose of efficiently computing the fitness of every individual in CNSGA search, BPNNs that have the advantages of fast response and high learning accuracy are employed in this subsection to develop the objective BPNN (OBPNN) for predicting the values of objective functions and the constraint BPNN (CBPNN) for predicting the values of constraint variables. Then, the fitness of every individual in the implementation of CNSGA can be determined by its corresponding objective values and the number of violated constraints. The OBPNN/CBPNN serves as a nonlinear mapping between the injection molding parameters and the values of objective functions/constraint variables. 3.2.1 Architecture of BPNNs BPNNs have hierarchical feed-forward network architecture and the outputs of each layer are sent directly to each neuron in the next layer. Since Woll and Cooper [23] reported that all the nonlinear mappings could be approximated by BPNNs with single hidden layer, both OBPNN and CBPNN developed here are composed of one input layer, one output layer, and one hidden layer. The number of input neurons for both BPNNs equals the number of injection molding parameters to be optimized, namely, NV. The number of output neurons for OBPNN equals the number of objective functions NO while that for CBPNN equals the number of constraint variables Nc. Here, we are to emphasize that the number of output neurons for

After the determination of the varying range of every design variable (namely, every injection molding parameter to be optimized), a series of sample data including the given process parameters within the preset ranges and their corresponding values of objective functions and constraint variables are obtained by Moldflow analyses, which are randomly divided into two groups for training and testing the BPNNs. In view of the fact that the varying ranges of design variables, objective functions and constraint variables may probably differ greatly from each other due to their diversities and different units, the data of both training and test samples are normalized by Eq. (3) and the network outputs of the test samples are unnormalized by Eq. (4). x0 ¼

x  xmin y  ymin ; y0 ¼ ; xmax  xmin ymax  ymin

c0 ¼

c  cmin : cmax  cmin

(3)

y ¼ y0  ðymax  ymin Þ þ ymin ; c ¼ c0  ðcmax  cmin Þ þ cmin :

(4)

where x, y, c are the original data corresponding to the design variables, objective functions and constraint variables; x′, y′, c′ are the normalized data while xmin, ymin, cmin and xmax, ymax, cmax are minimum and maximum values of x, y, c respectively.

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3.3 CNSGA for locating Pareto-optimal solutions The optimization problem of injection molding parameters as expressed in Eq. (1) has multiple objectives and constraints. There is no single optimal solution but rather a set of compromise solutions named Pareto-optimal or nondominated solutions to such an optimization problem with multiple conflicting objectives. Among the Pareto-optimal solutions, one solution is worse with regard to at least one other objective if it is better with regard to an objective. Thus, the final injection molding scheme should be determined by designers on the basis of the moldability evaluations of various Pareto-optimal solutions. For the constrained multiobjective optimization problem defined in Eq. (1), a solution x* 2 X is said to be Pareto-optimal if   there is no solution x 2 X such that fi ðxÞ  fi x* for all i ¼ 1; 2;    ; NO with strict inequality for at least one i.   Any other feasible solution x 2 X with fi ðxÞ  fi x* for all i ¼ 1; 2;    ; NO is an inferior solution. The Pareto-front of the optimization problem in Eq. (1) can be obtained by plotting all its Pareto-optimal solutions according to their objective values, which is an NO −1 dimensional surface. The CNSGA that integrates a powerful real-coded GA with the concept of Pareto-optimality to produce solutions illustrative of the non-dominated set is proposed to resolve the constrained multiobjective optimization problem in Eq. (1). The algorithm is based on the classification of the individuals in categories according to the concepts of Pareto-optimal set and non-domination. All the nondominated individuals of a population are assigned rank 1. The remaining individuals are classified again and nondominated ones are assigned rank 2. Such a procedure of classification continues until all the individuals of a population are assigned a rank to the effect that individuals with lower ranks are superior to those with higher ranks. 3.3.1 Fitness computation The following preferential guidelines proposed by Kurpati and Azarm [24] are employed for the fitness computation of various individuals in order to overcome the shortcomings of the penalty-based methods that a large number of penalty parameters need to be defined, which are very problem dependent and greatly influence the searching capability of GA. 1 Feasible solutions are preferred over infeasible ones, that is, feasible solutions always have lower ranks than the infeasible ones. 2 Infeasible solutions with smaller extent of constraint violation are preferred over those with larger extent of constraint violation. And the number of violated constraints

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should be taken into consideration when handling constraints. 3 Feasible solutions close to the Pareto-front are preferred over those far away from the Pareto-front. Based on the above preferential guidelines, the fitness value of every individual in the current population can be computed by  1=Ri ; Rci ¼ 0; FitðF ðxÞÞ ¼ ð5Þ 1=ðPop þ Rci Þ; Rci 6¼ 0: where Ri is the rank of an individual assigned by nondominated sorting according to the OBPNN-based predicted values of objective functions; Rci is the number of violated constraints obtained by comparing the CBPNN-based predicted values of constraint variables with their corresponding prescribed values; Pop is the size of current population. According to Eq. (5), there is Rci ¼ 0 for a feasible solution and its fitness value is determined by the nondominated sorted rank Ri. A feasible solution closer to the Pareto-front has smaller rank Ri and larger fitness value, which reflects the third preferential guideline. There is Rci 6¼ 0 for an infeasible solution, the fitness value of which is determined by the number of violated constraints Rci . An infeasible solution with smaller extent of constraint violation has smaller Rci and larger fitness value, which accords with the second preferential guideline. Finally, there is Ri  Pop and thereby 1=Ri  1=ðPop þ Rci Þ for any solution according to the concept of non-dominated sorting, hence the fitness value of a feasible solution is definitely larger than that of an infeasible one, which is consistent with the first preferential guideline. 3.3.2 GA operator of selection Selection is a key operator ensuring the survival of the fittest individuals during GA-based optimization. The selection policy employed in our CNSGA is a combination of the rank- and crowding distance-based tournament selection and the recombination-based elite strategy. All the individuals are sorted based on non-domination and assigned a crowding distance rank-wise. The basic idea of crowding distance is to compute the Euclidian distance between each individual in the same rank based on their NO objective values in the NO-dimensional objective space. The individuals on the boundaries are assigned infinite distances so that they can be selected all the time. The offspring population generated from the crossover and mutation operations is combined with the current population for the selection operation to generate the next population so that all the best individuals in previous and current populations can be reserved. Then the 2× Pop-sized population is sorted again based on non-domination and only the best Pop individuals

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are selected. The diversities of individuals are ensured by the crowding distance-based selection in descending order until the population size equals Pop if it exceeds Pop when all the individuals on the same rank are added. 3.4 Description of proposed methodology The implementation of proposed methodology for solving the constrained multiobjective optimization problem proceeds as follows. Step 1 Establish a high-dimensional optimization model of multiple objective functions with regard to product quality, manufacturing cost, molding efficiency, etc. Constraint functions reflecting the requirements of customers are also established. The feasible domains of all the injection molding parameters to be optimized are defined as well. Step 2 Perform Moldflow analyses to acquire enough sample data, every group of which includes the values of design variables, constraint variables and objective functions involved in the high-dimensional optimization model established in Step 1. Step 3 Extract injection molding knowledge based on the sample data acquired in Step 2 and simplify the optimization model to the minimum dimension by presetting the values of those design variables that have no obvious influence on the objective functions. Fig. 1 Flow chart of CNSGA- and BPNNs-based optimization

Step 4 Develop OBPNN and CBPNN by utilizing the sample data acquired in Step 2 as training and test samples. Step 5 Find the Pareto-optimal solutions to the simplified low-dimensional optimization model by interfacing the BPNNs developed in Step 4 with CNSGA. The OBPNN and CBPNN are utilized as the efficient approximate models for computing the values of objective functions and constraint variables respectively. Step 6 Determine the final optimal parameter schemes by fine-tuning the Pareto-optimal solutions located in Step 5 based on Moldflow analyses. Herein, the fifth step is the key for the proposed methodology to efficiently and intelligently find all the Paretooptimal parameter schemes in PIM with the introduction of CNSGA and two BPNNs, the detailed flow chart of which is illustrated in Fig. 1.

4 A case study 4.1 Description of optimization problem The compound-cavity injection mold with a space of 50 mm between two cavities illustrated in Fig. 2 is utilized to manufacture mice molded by the polymer blend of acraldehyde, butadiene, and styrene (ABS). The feed system is

Start CNSGA Initializations. Generate Pop0.

Fitness computation based on preferential guidelines. Compute number of

Non-dominated

violated constraints

sorting

Compute values of

Compute values of

constraint variables

objective functions

OBPNN

k=0 Yes

k > max_Gen?

End

No

Crossover, mutation. Generate Pop k +1 .

Compute values of

Non-dominated

Fitness

objective functions

sorting

computation

Compute values of

Compute number of

preferential

constraint variables

violated constraints

guidelines.

k=k+1

based on CBPNN

Pop temp = Pop k

Pop k +1

Generate Pop k +1 by selecting the best Pop individuals from Pop temp according to their fitness values.

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Fig. 2 Compound-cavity injection mold for manufacturing mice

made up of a sprue, two runners and two submarine gates. The conical sprue is 50 mm in height with its top and bottom diameters being 4 mm and 5 mm respectively. Two columnar runners are 19 mm in length, the diameters of which are to be optimized. Both the sections of two submarine gates to the bottom and top cavities are circular, the diameters of

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which evolve from 1.1 and 1 mm to 3 and 2.2 mm, respectively. The melt and mold temperatures are preset as 240 and 60 °C respectively according to their recommended values of ABS by Moldflow while the injection time tinjection is settled as 1 s. The switch from velocity to pressure control mode is automatically finished during the filling phase of injection molding. The injection molding parameters needed to be optimized are two mold parameters, namely, the diameters of two runners ds1 ; ds2 ranging from 3.2 to 4 mm, and three process parameters including the packing time tpack ranging from 2 to 4 s, the cooling time tcool ranging from 4 to 7 s and the packing pressure ppack ranging from 60 to 80 MPa. It is required that the polymer blend fully fill both cavities within 1.3 s. 4.2 Optimization procedure and results discussion Step 1 The high-dimensional optimization model with three objective functions is established as follows:

9   1 > Dshr d >  s1 ; ds2 ; tpack ; tcool ; ppack  > > = min F ds1 ; ds2 ; tpack ; tcool ; ppack ¼ min@ Vrunner ds1 ; ds2 ; tpack ; tcool ; ppack A Tcycle ds1 ; ds2 ; tpack ; tcool ; ppack > > s:t:  >     > g ds1 ; ds2 ; tpack ; tcool ; ppack  max tfillj ds1 ; ds2 ; tpack ; tcool ; ppack  1:3; j ¼ 1; 2: ; 



0

where 3:2  dsj  4ðj ¼ 1; 2Þ, 2≤tpack ≤4, 4≤tcool ≤7, 60≤ p pack ≤ 80. The value of the third objective function corresponding to different parameter schemes can be directly obtained by Tcycle ¼ tinjection þ tpack þ tcool ¼ 1 þ tpack þ tcool. And it is obvious that Tcycle will range from 7 to 12 s for different parameter schemes. The values of the other two objective functions can be obtained by Moldflow analysis. The constraint variables tfillj ds1 ; ds2 ; tpack ; tcool ; ppack ðj ¼ 1; 2Þ are actual filling time of the polymer blend in two cavities, which can also be obtained by Moldflow analysis. Herein, the number of constraint variables Nc 02. The constraint function means that the polymer blend should fully fill both mold cavities within 1.3 s.

Step 2 A series of Moldflow analyses are carried out based on DOE within the prescribed ranges of five injection molding parameters for acquiring enough sample data, the results of which are not provided here for concise sake. Step 3 The optimization model is simplified to the form of Eq. (7) based on the knowledge extracted from the sample data that the injection molding cycle Tcycle has little influence on the product quality indicated by Dshr when ranging from 7 to 12 s. Hence the process parameters tpack and tcool are settled as 2 and 4 s, respectively, to achieve the minimum Tcycle of 7 s.

       9 min F ds1 ; ds2 ; ppack ¼ min Dshr ds1 ; ds2 ; ppack ; Vrunner ds1 ; ds2 ; ppack = s:t:      ; g ds1 ; ds2 ; ppack  max tfillj ds1 ; ds2 ; ppack  1:3; j ¼ 1; 2:

Step 4 The OBPNN and CBPNN with the architecture of 39-2 are developed utilizing the sample data acquired in Step 2. The multiple inputs of OBPNN are three

ð6Þ

ð7Þ

injection molding parameters including the diameters of two runners ds1 ; ds2 and the packing pressure ppack while its outputs are the values of two objective

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Table 1 Sample data for training and testing OBPNN No.

Injection molding parameters

Objective functions

ds1 (mm)

ds2 (mm)

ppack (MPa)

Dshr (%)

Vrunner (cm3)

1 2 3 4 5

3.2 3.2 3.2 3.2 3.2

3.4 3.4 3.4 3.4 3.4

60 65 70 75 80

2.0634 1.8942 1.6666 1.4942 1.3263

1.2286 1.2286 1.2286 1.2286 1.2286

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

3.3 3.3 3.3 3.3 3.3 3.2 3.2 3.2 3.2 3.2 3.3 3.3 3.3 3.3 3.3 3.4 3.4 3.4

3.5 3.5 3.5 3.5 3.5 3.6 3.6 3.6 3.6 3.6 3.8 3.8 3.8 3.8 3.8 4.0 4.0 4.0

60 65 70 75 80 60 65 70 75 80 60 65 70 75 80 60 65 70

2.0107 1.8563 1.6273 1.4564 1.2909 1.9833 1.8263 1.5943 1.4234 1.2934 1.9735 1.8071 1.5692 1.3927 1.2348 1.9737 1.8043 1.5169

1.2389 1.2389 1.2389 1.2389 1.2389 1.2494 1.2494 1.2494 1.2494 1.2494 1.2812 1.2812 1.2812 1.2812 1.2812 1.3145 1.3145 1.3145

24 25

3.4 3.4

4.0 4.0

75 80

1.3081 1.2086

1.3145 1.3145

functions Dshr and Vrunner. The inputs of CBPNN are the same as those of OBPNN while its outputs are the values of two constraint variables tfill1 and tfill2 . Only the details of OBPNN are provided hereafter due to the similarity of developing OBPNN and CBPNN. The samples of No. 4, 8, 12, 18, 21 in Table 1 are randomly chosen as test samples while the others are utilized as training samples for developing OBPNN. Fig. 3 Comparison of OBPNN-based predicted values with desired ones

After a preset maximum training time of 10,000, the system error reaches an acceptable value of 0.003696. The predicted results of five test samples by the developed OBPNN as well as their corresponding desired values are provided in the form of two folded lines, see Fig. 3. The approximation of the two groups of folded lines demonstrates that the predictive error of both Dshr and Vrunner with regard to every test sample is very small. Step 5 The Pareto-optimal solutions to the simplified lowdimensional optimization model in Eq. 7 are located by integrating the CNSGA with two BPNNs. The operators of uniform crossover and polynomial mutation are employed in our real-coded CNSGA. The distribution index for both crossover and mutation is 40. The population size is 100 and the probabilities for crossover and mutation are 0.8 and 0.1, respectively. The random seed equals 0.25 and the maximum generation is settled as 50. The randomly generated initial population seems to be a series of messy points in the two-dimensional objective space, see Fig. 4. After the evolution of 50 generations, we can see the Pareto-front as illustrated in Fig. 5, which is a one-dimensional discrete curve. Theoretically, any parameter scheme corresponding to a point on the Pareto-front in Fig. 5 is a Paretooptimal solution to the optimization problem in Eq. (7). However, the parameter values obtained based on CNSGA may be impractical due to the minimum unit of parameter setting in the injection molding machine and the maximum precision in manufacturing an injection mold. Thus, the final parameter schemes should be determined after Moldflow analysis-based fine-tuning. Step 6 The final optimal parameter schemes are determined by fine-tuning the Pareto-optimal solutions obtained in Step 5 and their superiority are verified through Moldflow analyses, five of which are listed in Table 2.

a

b

Dshr (%)

Vrunner (cm3) 1.32

2

Desired value Predicted value

1.9

Desired value Predicted value

1.3

1.8

1.28

1.7 1.26 1.6 1.24

1.5 1.4

1

2

3

Sample No.

4

5

1.22

1

2

3

Sample No.

4

5

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Dshr (%)

Table 2 Five fine-tuned Pareto-optimal schemes

2

No. Injection molding parameters

1.9 1.8

ds1 (mm)

ds2 (mm)

3.2 3.3 3.3 3.5 3.5

3.3 3.4 3.6 3.9 4.0

ppack (MPa)

Constraint variables

Objective functions

tfill1 (s)

Dshr (%)

Vrunner (cm3)

1.3992 1.2771 1.2027 1.1609 1.1394

1.2186 1.2383 1.2592 1.3130 1.3248

tfill2 (s)

1.7

1 2 3 4 5

1.6 1.5 1.4

80 80 80 80 80

1.087 1.086 1.087 1.085 1.085

1.032 1.037 0.9954 0.9974 0.9773

1.3 1.2 1.22

1.24

1.26

1.28

1.3

1.32

3

Vrunner (cm ) Fig. 4 Distribution of original population in objective space

All the values of two constraint variables tfill1 and tfill2 in Table 2 are less than 1.1 s, meeting the given constraints. The packing pressure of every parameter scheme in Table 2 reaches the upper limit of the given range of 60≤ppack ≤80, indicating that the increase of packing pressure can reduce the difference of volumetric shrinkage between two mold cavities. The optimized injection molding parameter schemes demonstrates the tradeoffs between the manufacturing cost and product quality because the diminished difference of volumetric shrinkage between two mold cavities results in the volume enlargement of the runner system, which is illustrative from the values of Dshr and Vrunner in the last two columns of Table 2. The advantages of proposed optimization methodology over the existing ones are obvious since the constrained Dshr (%) 1.5 1.47 1.44

multiobjective optimization model including both mold and process parameters as design variables was set up at the very start, which complied more with engineering practice than the non-constrained single objective models utilized in the previous works. The proposed optimization model enabled the achievement of high product quality, low cost and efficient production at the same time. As far as the investigated example was concerned, the injection molding efficiency was improved by 42 % with no sacrifice on the part quality and no increase of the manufacturing cost based on the integration of both runner sizes and process parameters including packing pressure, packing and cooling time into the design vector. With the introduction of VCM, the initial high-dimensional optimization problem with five design variables and three objective functions was translated into a low-dimensional optimization problem with three design variables and two objective functions based on the knowledge of injection molding deduced from the results of DOE, which greatly reduced the complexity of our optimization problem as well as the resulting computational cost. Furthermore, all the Pareto optimal solutions were intelligently and efficiently located based on the integration of CNSGA with two BPNNs for predicting the values of objective functions and constraint variables, and then were fine-tuned based on the high-precision Moldflow analyses, which enabled the flexible selectivity of theoretical solution for application in injection molding practice.

1.41 1.38

5 Concluding remarks

1.35 1.32 1.29 1.26 1.23 1.2 1.22

1.24

1.26

1.28

1.3

1.32

3

Vrunner (cm ) Fig. 5 Pareto-front after CNSGA evolution of 50 generations

A multiobjective multiconstraint optimization model including both mold and process parameters as design variables was proposed in order to achieve optimal parameter schemes in accord with practical requirements in injection molding. Multiple objective functions corresponding to product quality, manufacturing cost and molding efficiency were constructed for comprehensively evaluating the moldability of various injection molding parameter schemes. Multiple constraint functions were also established for

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Int J Adv Manuf Technol (2013) 66:907–916

objectively describing the requirements of clients, the restrictions of injection molding machines and so on. A novel optimization methodology based on soft computing and VCM was put forward for efficiently and intelligently finding the solutions to the constrained multiobjective optimization problem of injection molding parameters, which integrated the high-precision Moldflow analyses for collecting the sample data and extracting injection molding knowledge, the CNSGA for locating the Pareto-optimal solutions to the constrained multiobjective optimization problem, the OBPNNand CBPNN-based low-precision models for efficiently computing the fitness value of every individual during the evolution of CNSGA, and the Moldflow analyses-based fine-tuning of the Pareto-optimal solutions as well. The implementation of the proposed methodology was investigated in detail and highlighted by the optimization of the injection molding parameters for manufacturing mice with a compound-cavity injection mold, which included two mold parameters and three process parameters as design variables. The case study demonstrated that the proposed methodology had its obvious advantages over the existing ones in achieving the optimal parameter schemes applicable to injection molding practice. Acknowledgements This project is supported by the National Natural Science Foundation of China (No. 51275459), National Basic Research Program of China (No. 2011CB706503) and National Science and Technology Major Project of China (No. SK201201A28-01). Conflicts of interest of interest.

The authors declare that they have no conflict

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