Int J Adv Manuf Technol (2012) 58:521–531 DOI 10.1007/s00170-011-3425-0

ORIGINAL ARTICLE

Multi-objective optimization of MIMO plastic injection molding process conditions based on particle swarm optimization Gang Xu & Zhi-tao Yang & Guo-dong Long

Received: 9 September 2010 / Accepted: 20 February 2011 / Published online: 14 June 2011 # Springer-Verlag London Limited 2011

Abstract Determining optimal process parameter settings critically influences productivity, quality, and cost of production in the plastic injection molding industry. Selecting the proper process conditions for the injection molding process is treated as a multi-objective optimization problem, where different objectives, such as minimizing product weight, volumetric shrinkage, or flash present trade-off behaviors. As such, various optima may exist in the objective space. This paper presents the development of an experimentbased optimization system for the process parameter optimization of multiple-input multiple-output plastic injection molding process. The development integrates Taguchi’s parameter design method, neural networks based on PSO (PSONN model), multi-objective particle swarm optimization algorithm, engineering optimization concepts, and automatically search for the Pareto-optimal solutions for different objectives. According to the illustrative applications, the research results indicate that the proposed approach can effectively help engineers identify optimal process conditions and achieve competitive advantages of product quality and costs.

G. Xu (*) Department of Mathematics, Nanchang University, Nanchang 330031, China e-mail: [email protected] Z.-t. Yang The National Engineering Research Center of Novel Equipment for Polymer Processing, The Key Laboratory of Polymer Processing Engineering of Ministry of Education, South China University of Technology, Guangzhou 510640, China G.-d. Long Huawei Technologies Co., Ltd, Shenzhen 710075, China

Keywords Plastic injection molding . Back-propagation neural networks . Particle swarm algorithm . Multi-objective . Optimization

1 Introduction Optimizing process parameter problems are routinely performed in the manufacturing industry, particularly in setting final optimal process parameters. Final optimal process parameter setting is recognized as one of the most important steps in injection molding for improving the quality of molded products. Traditionally, the process conditions are often determined by experienced engineers or based on reference handbooks and later improved and fine-tuned by trial and error and Taguchi’s parameter design method on the shop floor. This method depends greatly on the experience of molding operators and could potentially be costly and time consuming, especially with new resins or new applications, thus it is not suitable for complex manufacturing processes. Hsu [1] argued that when using a trial-and-error process, it is impossible to verify the actual optimal process parameter settings. Moreover, Taguchi’s parameter design method can only find the best specified process parameter level combination which includes the discrete setting values of process parameters. Application of the conventional Taguchi parameter design method is unsuitable when one of the process parameter variables is continuous, and it cannot help engineers obtain optimal process parameter setting results [2]. Furthermore, when engineers deal with a multiresponse process parameter design problem, the conventional Taguchi parameter design method runs into difficulties [3]. Advanced methods are highly demanded to model and optimize the injection molding process with the purpose of manufacturing high-quality plastic parts.

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The quality characteristics of plastic injection-molded products can be roughly divided into three kinds: (1) the dimensional properties, (2) the surface properties, and (3) the mechanical or optical properties [4]. Previously, researchers showed that product weight is a critical quality characteristic and a good indication of the stability of the manufacturing process in plastic injection molding. Yang et al. [4] revealed that product weight is an important attribute for plastic injection-molded products because the product weight has a close relation to other quality properties (e.g., surface and mechanical properties), particularly other dimensional properties (e.g., thickness). Kamal et al. [5] showed that controlling the product weight is of great commercial interest and can produce great value for production management. One of the most common defects in injection molding is flash. Flash occurs when excess plastic material is extruded from the edges of a mold. Flash may be caused by changes in the processing conditions such as injection pressure, melt temperature, clamp pressure, improper feeding materials, or mold damage [6]. In many cases, the excess material must be trimmed manually, which lowers process efficiency. To reduce inefficiency caused by flash, in-process flash detection techniques are essential. Zhu and Chen [7] described the development of a fuzzy neural network-based in-process mixed materialcaused flash prediction system for injection molding processes. During the plastic injection molding process, one of the biggest challenges is shrinkage which deteriorates the quality of produced parts [8]. Therefore, product weight, flash, and volumetric shrinkage are feasible quality characteristics which can be used as important responses in the process parameter optimization of plastic injection molding. Determining the process parameter settings for plastic injection molding greatly affects the quality of the plastic injection-molded product [9]. Unsuitable process parameter settings can cause many production problems (e.g., many product defects, long lead times, large amounts of scrap, high production costs, etc.), reduce the price competitive advantage, and decrease a company’s profitability. The plastic injection molding process includes four phases: plasticization, filling, packing, and cooling [10]. Therefore, several process parameters which include the melt temperature, mold temperature, injection pressure, injection velocity, injection time, packing pressure, packing time, cooling temperature, and cooling time all potentially influence the quality of injection-molded plastic products [11, 12]. In previous plastic injection molding research, different control process parameters have been used. For example, Shi et al. [13] used four process parameters (mold temperature, melt temperature, injection time, and injection pressure) to determine the optimal initial process parameter settings for injection-molded plastic parts with a butter container lid and

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under single quality characteristic (the maximum shear stress) considerations. Chiang and Chang [14] used four control process parameters (mold temperature, melt temperature, injection pressure, and injection time) to determine the optimal initial process parameter settings for injectionmolded plastic parts with a thin shell feature and under multiple quality characteristic considerations. Therefore, before optimizing process parameter settings, engineers need to select feasible and tractable control process parameters which will influence the production results of plastic injection molding. Therefore, for multiple-input multiple-output (MIMO) plastic injection molding process, this research proposes an effective process parameter optimization approach to help manufacturers achieve a competitive advantage of product quality and costs. The proposed approach integrates Taguchi’s parameter design method, back-propagation neural network based on particle swarm optimization (PSONN model), particle swarm optimization multi-objective algorithm, and engineering optimization concept and can effectively help engineers determine final optimal process parameter settings. The rest of this paper is organized as follows. Section 2 gives a brief description of literature review. Section 3 describes the optimization methodologies including neural networks, particle swarm optimization, and multi-objective particle swarm optimization algorithms used in this study for optimization. Section 4 proposes a process parameter optimization approach for plastic injection molding under multiresponse considerations. In Section 5, multi-objective optimization applications will be presented to illustrate the performance and capabilities of the proposed optimization strategy according to an illustrative case study. In Section 6, an experiment comparison, results, and discussion are presented. Finally, the last section gives the conclusions of this paper.

2 Literature review An alternative means of applying artificial neural networks (ANN) has been proposed to improve conventional Taguchi’s parameter design and is capable of effectively treating continuous parameter values [15, 16]. Subsequently, researchers worldwide have applied widely soft computing for optimizing process parameters. Shi et al. [13] presents a hybrid optimal model in combination with ANN and a genetic algorithm (GA) for the plastic injection molding process under the quality requirement of maximum shear stress. Panneerselvam et al. [17] described a hybrid technique of ANN and GA to obtain optimal weld tensile strength. ANN was used to establish the relationship between the input/output parameters of the process and the established ANN was then suitably integrated with GA. However,

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such single response requirement rarely exists in practical production processes. Usually, there are multiresponse requirements in product production. For the process parameter design problem of an MIMO production process, many researchers have developed and employed different optimization schemes for determining the optimal design and process parameters for polymer processing [18–21]. However, for many MIMO production processes, the researchers usually make the multi-objective problems into single-objective optimization problems and apply ANN and evolutionary algorithms to attain the final optimal process parameter settings. In the studies mentioned above, Deng et al. [18] applied GA to optimize injection molding process conditions with user-definable objective functions. They implemented a modified simple weighting method to deal with multi-objective optimization, in which the objective functions can be defined with different criteria and/or weight vectors according to the designers’ preference. Huang et al. [19] presented an approach for determining parameter values in melt spinning processes to yield optimal qualities of denier and tenacity in as-spun fibers. The experimental data determined by an orthogonal array in the Taguchi method are adopted to train a neural network by an analysis of variance. The genetic algorithm is aimed at finding parameter values in a continuous solution space to optimize a performance measure on denier and tenacity qualities, based on the neural network. Hsu et al. [20] presented an integrated approach using neural networks, exponential desirability functions, and genetic algorithms to optimize parameter design problems with multiple responses. The proposed approach aims to identify the input parameter settings to maximize the overall minimal satisfaction level with respect to all the responses. In their optimization procedure, the trade-off solutions obtained by using the predefined strategy would be sensitive to the weight factors chosen in converting the multi-objective to a single objective function. Castro et al. [21] used an approach comprising computer simulation, ANN, and data envelopment analysis to determine the proper operating conditions for finding the best compromise among several conflicting performance measures. The approach they presented also allowed for the identification of robust variable settings that might help to define a starting point for negotiation between multiple decision makers. Recently, Wei et al. [22] discussed the combination of the design method and the injection molding machine. In order to solve the complex multi-objective optimal performance design of large-scale injection molding machines, NSGA-II is used to find a much better spread of design solutions and better convergence near the true Pareto-optimal front. Screw diameter performance, stick inside distance performance, mold moving route performance, and mold-locked

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force performance are chosen as the four main performance evaluation indexes. Wei et al. [23] presented a triple objective optimization model with the largest mold moving speed and injecting capacities and the smallest injecting power. The optimized design constraints of the optimal model are summarized. The computational efficiency of strength Pareto evolutionary algorithm is improved by using rough set-based support vector clustering method. The optimal Pareto solution is determined by eliminating the uncertainty in the artificial priority election. Although the abovementioned studies have achieved various levels of success, it remains desirable to have the ability to offer an intelligent optimization strategy. This research integrates engineering optimization concept into the soft computingbased optimization module to guarantee that the fitness response of each search iteration is local minimal under MIMO process.

3 Description of multi-objective optimization methodologies The optimization methodologies including neural networks based on particle swarm optimization and particle swarm optimization multi-objective algorithms for developing the proposed approach are briefly introduced below. 3.1 Back-propagation neural network based on PSO ANN is widely accepted as a technology offering an alternative way to simulate complex and ill-defined problems. They have been used in diverse applications in control, forecasting, manufacturing, optimization, etc., and they are particularly useful in system modeling. Backpropagation neural network (BPNN) is a typical ANN that has been widely used in many research fields [24, 25]. The BPNN consists of input layer, hidden layer, and output layer. When back-propagation (BP) network training, for the given input vector X, the output of neurons can be described as: Y ¼ f ðW »X ; qÞ

ð1Þ

where Y is the output vector and θ is the threshold vector. Function f can be linear functions or “s” type functions. Then the error function MSE, between actual output and expected output, is defined as follows: MSE ¼

N X M  2 1 X dij  Yij 2N i¼1 j¼1

ð2Þ

Where N, M, di, and Yi are the number of training samples, dimension of training samples, the actual value for

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training sample i, and the predicted value of the neural network for training sample i, respectively. Regarding ANN training, there have been many algorithms used to train the ANN, such as BP algorithm, GA, simulating annealing algorithm, particle swarm optimization algorithm (PSO) [26], and so on. The mostly used training algorithm is the BP algorithm, which is a gradientbased method. Hence, some inherent problems existing in BP algorithm are also frequently encountered in the use of this algorithm. The back-propagation may lead to failure in finding a global optimal solution; the convergent speed of the BP algorithm is too slow even if the learning goal, a given termination error, can be achieved. GA has been used in training ANN recently, but in training process, this algorithm needs an encoding operator and a decoding operator. The GA’s convergent speed will become very slow, so that the convergent accuracy may be influenced by the slow convergent speed. Unlike the GA, the PSO algorithm has no complicated evolutionary operators such as crossover and mutation. The PSO algorithm has a strong ability to find the most optimistic result and be used to train the weights of ANN; the algorithm can make use of not only strong global search ability, but also strong local search ability to construct neural network based on PSO (PSONN model). 3.2 Multi-objectives optimization of unitary performance 3.2.1 Pareto-optimal set A general minimization problem of multi-objective optimization can be mathematically stated as min zi ¼ fi ðxÞi ¼ 1; 2; :::; p x

Subject to   x 2 X ¼ x 2 Rn jgj ðxÞ  0; hk ðxÞ ¼ 0; j ¼ 1; 2;   ; m; k ¼ 1; 2;   ; l

where zi is the ith objective function, there are p objective functions that must be minimized simultaneously. The minimization takes place over a decision space X  Rn , gj ðxÞ  0 is the jth inequality constraint and hk ðxÞ ¼ 0 is the kth equality constraint, on the decision variables x 2 X . The p objectives define a mapping from decision space into objective space Z. Since the notion of an optimum solution in multi-objective optimization problem is different compared to the single objective optimization problem. The concept of Pareto dominance is used for the evaluation of the solutions. This formulated concept is defined as: »

Definition 1 (Pareto optimal) A point x 2 X is Pareto optimal if for every x 2 X and I ¼ f1; 2; :::; pg either  »  or there is at least one i 2 I such that 8i2I fi ðxÞ¼ fi x » fi ðxÞ > fi x

Definition 2 (Pareto dominance) A vector u ¼ ðu1 ; u2 ; :::; uM Þ is said to dominate a vector v ¼ ðv1 ; v2 ; :::; vM Þ (denoted by u  v), for a multiobjective minimization problem, if and only if u is p a r t i a l l y l e s s t h a n v, n a m e l y, 8i 2 ð1; 2; :::; M Þ, ui  vi ^ 9i 2 f1; 2; :::; M g : ui < vi . Definition 3 (Pareto-optimal set) For a given MOP f(x), » the Pareto-optimal set (P * ) is defined as P :¼ 0 0 fx 2 X j:9x 2 X f ðx Þ  f ðxÞg Definition 4 (Pareto front) For a given MOP f(x) and * ) is defined as Pareto-optimal setP*, the Pareto front (PF  » » PF :¼ u ¼ f ¼ f1 ðxÞ; :::; fp ðxÞ jx 2 P

3.2.2 Particle swarm optimization multi-objective algorithm The PSO algorithm, which is one of the most powerful algorithms available for solving global optimization problems, was originally proposed by Kennedy et al. [27] as an optimization method in 1995. Like evolutionary algorithms, PSO is also a population-based heuristic, where the population of the potential solutions is called a swarm and each individual solution within the swarm is called a particle. The algorithm works by initializing a flock of birds randomly over the searching space, where every bird is called a “particle.” These “particle” fly with a certain velocity and find the global best position after some iterative generations. At each iteration, each particle successively adjusts its position toward the global optimum according to the two factors: the best position encountered by itself (pBest) denoted as vector pBesti =(pBesti1, pBesti2,…, pBestid), and the best position encountered by the whole swarm (gBest) denoted as vector gBest=(gBest1, gBest2,…, gBestd) where d is the dimension for a searching space, and then compute a new position that the “particle” is to fly to. Supposing the dimension d for a searching space, the particles m, the position of the ith particle can be presented by a vector Xi =(xi1, xi2, . . ., xid), where xij ∈ [xminn, xmaxn], 1  i  m, 1  j  d. xminn and xmaxn are the lower and upper bound for the jth dimension, respectively. The velocity of the ith particle is represented as a vector Vi = (vi1, vi2, . . ., vid). The velocity is clamped to a maximum velocity vimax =(vimax1, vimax2, . . ., vimaxd). The velocity and position of the particle at next iteration are calculated according to the following equations:   t t1 vtij ¼ wvt1 þ c Rand1ð Þ pBest  x 1 ij ij ij   þ c2 Rand2ð Þ gBestijt  xt1 ij

ð3Þ

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xtþ1 ¼ xtij þ vtþ1 ij ij

525

ð4Þ

where t is the iteration index and w is the inertia weight; i = 1,2,…,m, j= 1, 2,…,d; c1 and c2 two positive acceleration coefficients; Rand1() and Rand2() two uniform random numbers distributed in [0,1]; pBesti and gBest are given by the following equations, respectively:  pBesti : f ðXi Þ  f ðpBesti Þ pBesti ¼ Xi : f ðXi Þ < f ðpBesti Þ gBest 2 fpBest1 ; pBest2 ; :::; pBestm gj f ðgBest Þ ¼ minðf ðpBest1 Þ; f ðpBest2 Þ; :::; f ðpBestm ÞÞ where f is the objective function. The high speed of convergence of the PSO algorithm attracted researchers to develop multi-objective optimization algorithms using PSO. Also, the PSO seems to have some advantages in terms of the better exploration and exploitation provided by local and global search capabilities of the algorithm. In the present study, the multi-objective PSO (MOPSO) [28] is adopted to MIMO production process to obtain the Pareto-optimal solutions. The purpose of this study is to develop an integrated optimization system that can adaptively and automatically find out the Pareto-optimal solutions for different objectives of multiobjective injection molding. 3.2.3 Pareto optimizing based on fuzzy sets theory An optimization solution in the next step will be selected out from Pareto sets which are calculated by PSO. Pareto sets optimization method is used based on fuzzy sets theory. Member function fm is defined as a proportion of number 1 target in one solution:

fm ¼

8 > < 0 max

fi fimax fimin fi

> :

1

fi > fimax fimin < fi < fimax fi > fi

ð5Þ

min

For each nondomination solution R in Pareto sets, domination function fR could be defined as follows: fR ¼

N X i¼1

fiR =

N X N X

fi R

ð6Þ

j¼1 i¼1

N is the number of solution. The larger value of f R is the better unitary performance of that solution. Therefore, the solution with maximum f R would be chosen as an optimal solution of the Pareto sets. By sorting the Pareto sets into a depending order according to the value of f R, an optimization sequence of feasible solution can be achieved.

4 Proposed multi-objective optimization approach for injection molding In this section, an integrated approach is proposed for effectively assisting engineers in the process parameter optimization for MIMO plastic injection molding process. The proposed approach integrates Taguchi’s parameter design method, neural network based on PSO (PSONN model), MOPSO and engineering optimization concepts. Taguchi’s parameter design method is used to arrange an orthogonal array experiment and to reduce the number of experiments. Subsequently, the signal-to-noise ratio (S/N ratio) is employed to determine the initial process parameter settings that have minimal sensitivity of noise under the consideration of most major quality characteristic. The experimental data of Taguchi’s parameter design method are used for effectively training and testing PSONN model that finely maps the relationship between the input process control factors and output responses. Engineering optimization concepts are employed to establish the multiobjective fitness function for using in MOPSO. MOPSO and the finished PSONN model are applied for searching the Pareto-optimal process parameter settings for different objectives of multi-objective injection molding. Finally, a confirmation experiment is performed to confirm the effectiveness of Pareto-optimal process parameter settings based on the PSONN model predictions. The flow chart for the proposed approach is shown in Fig. 1. The multiobjective optimization procedures of the proposed approach include two stages and are given as follows. 4.1 Training stage Provide training and testing data for building a PSONN quality response predictor and determine the initial process parameter settings for use in the MOPSO search approach via Taguchi’s parameter design method. Formulate a multiobjective fitness function F(x) for use in the MOPSO search approach via engineering optimization concepts. Step 1. Identify feasible responses (quality characteristics) as the target requirements of the experiment. The responses must be confirmed which have significant influence on final product quality. Moreover, engineers need to decide a most major response from all responses just identified via an expert opinion or experience. Step 2. Determine the feasible and tractable process control factors and levels that influence the performance of responses. Select an appropriate orthogonal array for arranging the experiment and acquiring the experimental treatments. Step 3. Perform experiments for each treatment, and collect the performance measurement of the responses.

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Step 1. Set the MOPSO parameters and the ranges of process control factors. Step 2. Determine the Pareto-optimal process parameter settings via PSONN model. Use MOPSO to find the Pareto-optimal solutions for the current PSONN model. The detail of MOPSO-based optimization approach can be referred in [28]. Step 3. Discretize the objective space into archives, and then represent the current Pareto-optimal front with a collection of archives. Perform a confirmation experiment to verify the effectiveness of Pareto-optimal process parameter settings based on the PSONN model predictions.

Identify the experimental factors and quality characteristics Choose orthogonal array and the experimental levels Implement the Taguchi method to obtain the initial parameter settings Develop quality predictor for MIMO of injection molding using ANN based on PSO Apply MOPSO plus PSONN to finding the Pareto-optimal process parameter set Conduct the PIM confirmation experiment for Pareto-optimal process parameter settings based on the PSONN model

5 Multi-objective injection molding optimization applications and results analysis 5.1 Optimization model

Achieve the quality requirement?

N

Y Select different combinations of optimal trade-off process parameter according to the designer’s preference for setting up the process conditions

Fig. 1 Proposed process parameter optimization approach for MIMO plastic injection molding process

Step 4. Develop a PSONN model that finely maps the relationship between the input process control factors and output responses. Step 5. Select an appropriate formulation for the S/N ratio and calculate the S/N ratio for each response under different treatments of orthogonal array. Engineers can select an appropriate S/N ratio for their experiment by considering the goal requirement of each response. Step 6. Formulate a multi-objective fitness function F(x) according to the PSONN model for use in the MOPSO search approach via engineering optimization concepts.

This section presents an illustrative example about process parameter optimization for MIMO plastic injection molding process under seven process control factors and three responses to demonstrate the effectiveness and implementation of the proposed approach. The product in this illustrative example is a thin-walled part, shown in Fig. 2. The plane of the part has a thickness of 3 mm, and the sides have the length of 80 mm. Melt temperature, mold temperature, injection pressure, injection time, holding pressure, holding time, and cooling time are selected as process control factors. Moreover, each process control factor has four levels. Table 1 shows the seven process control factors and their level setting values. There are three concerns regarding part quality: (1) product weight, which should be kept as light as possible in order to decrease manufacturing cost; (2) flash, which is a critical quality characteristic to be minimized to keep product quality; and

4.2 Iteration optimization stage Determine the Pareto-optimal process parameter settings for different objectives of multi-objective injection molding via PSONN and MOPSO. Perform a confirmation experiment to confirm the effectiveness of Pareto-optimal process parameter settings.

Fig. 2 The object figure of plastic injection molding thin-walled part

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Table 1 Process control factors and their levels

Level Level Level Level

1 2 3 4

Mold temperature (°C)

Melt temperature (°C)

Injection pressure (%)

Injection time (s)

Holding pressure (%)

Holding time (s)

cooling time (s)

35 50 65 80

180 190 200 210

30 40 50 60

2 3 4 5

30 40 50 60

2 3 4 5

5 10 15 20

(3) volumetric shrinkage, which should be minimized to improve molded product quality. Three outputs from experimental results (product weight, flash, and volumetric shrinkage) are selected as the objective values to represent the above criteria, respectively. Thus, the optimization problem with the ranges of process conditions to be optimized is defined as follows: 8 Minimize : part weight > > > > Minimize : flash > > < Minimize : volume shrinkage > Subject to : 180 C  Tm  210 C; 60 C  Tw  80 C > > > 30  Pi  60; 2s  ti  5s; 30  Ph  60; > > : 2s  th  5s; 5s  tc  20s ð7Þ

In this application, seven independent process conditions, namely, melt temperature Tm, mold temperature Tw, injection pressure Pi, injection time ti, holding pressure Ph, holding time th, and cooling time tc are optimized to achieve the desired objectives. 5.2 Experimental data The polymer material used for molding the part is highdensity polyethylene (Trademark 5200, Density 0.955 g/cm3, MFI=3.987 g/10 min), which is provided by Shanghai Jinfei Petrochemical Co. Ltd., China. Experimental data are collected from a hydraulic vibrating dynamic plastic injection molding machine, the specifications of which are shown in Table 2. Product weight is measured using an electric balance which has a precision of 0.01 g. Furthermore, product length is measured using an electric caliper which has a precision of 0.01 mm. According to the seven selected process control factors and their level setting values which are shown in Table 1, this research applies an L32 (49) orthogonal array to assign seven factors. There are 32 treatments in all with different level combinations of the seven factors. Four replications are taken to increase the sensitivity of the statistical analysis. Therefore, 128 sample data were collected.

5.3 Optimization of the neural network parameters using the PSO method PSO will be used to train neural networks to build a PSONN model which is used for plastic injection molding product quality indicator (product weight, flash, and volume shrinkage) soft computing model as shown in Eq. 7. The sample data collected by orthogonal experiments, using signal-to-noise ratio method to determine the ultimate soft computing model of this article are as follows: mold temperature, melt temperature, injection pressure, injection time, holding pressure, holding time, cooling time, a total of seven input variables, the middle layer of 12 nodes, the output layer for the volume shrinkage, flash and product weight, which constitute a 7–12–2 structure of neural networks. Using PSO to find the optimal connection weights and the optimal threshold values of neural networks, the parameters are defined as follows: particle population, 100; the maximum allowable iteration number, 10,000; the error limit, 0.01; the parameter dimension, 122; initial value of inertia weight, 0.9; ultimate value of inertia weight, 0.3; acceleration factor, 2. For the purpose of comparison, in this paper, BP algorithms are also used to train neural networks, the structures of the BPNN model, neuron transfer function, etc. are the same as that of the PSONN

Table 2 Specifications of the hydraulic pressure vibrating dynamic injection molding machine Items

Contents

Plasticizing capacity Maximum pressure Screw diameter Length/diameter ratio for screw Maximum injection volume Clamping force Injection journey Screw speed of rotation Injection velocity

24.8 g/s 80 MPa 32 mm 24.8 112 cm3 980 kN 168 mm 200 r/min 83 g/s

528 Table 3 Comparison of the training and checking results for two model quality predictors

Int J Adv Manuf Technol (2012) 58:521–531 Item

PSONN

BPNN

Product Flash Volume Product Flash Volume

weight shrinkage weight shrinkage

model. The BP algorithm learning rate is 0.018 and the momentum factor is 0.01. The PSONN model and the BPNN model are first trained using 106 samples of training data; the network performance is obtained by calculating the MSE, and the network performances between PSONN and BPNN are shown in Table 3. As can be seen from Table 3, there have been better fitting degrees between the predicted values of the two models and the actual value; the training process meets the

Relative error for training samples

Relative error for checking samples

Largest

Smallest

Average

Largest

Smallest

Average

1.064 2.361 3.158 1.121 2.242 3.417

0.177 0.233 0.581 0.185 0.157 0.528

0.782 1.265 2.278 0.857 1.237 2.217

1.125 2.453 3.5468 1.202 2.572 4.433

0.187 0.374 0.823 0.293 0.336 0.746

0.821 1.327 2.445 0.924 1.432 2.923

requirements, and the fitting performance for PSONN model is better than that of the BPNN model. Twenty-two samples of verifying data are used to check for PSONN and BPNN, the comparison of the checking results for the two models is shown in Table 3. A comparison between the experimental and checking values via the PSONN and BPNN models is represented in Fig. 3. From Table 3 and Fig. 3, it can be seen that extrapolation of the two models are all good, and the two models have high prediction accuracy; however, the extrapolation of the

Fig. 3 Comparison between the predicted and actual values via two models

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529

41.0

part weight(g)

40.5 40.0 39.5 39.0 38.5 38.0 37.5 1.5

vol

um

0.0 2.0

0.5 2.5

1.0

3.0

etri

cs

hrin

3.5 4.0

kag

4.5

e(%

)

2.0

)

mm

( ing

1.5

sh

fla

Fig. 4 Pareto-optimal front obtained for three-objective PSONN model using MOPSO

PSONN model is better than that of the BPNN model. Apparently, the performance of the PSONN quality predictor is better than that of the BPNN quality predictor. It shows that the predictions of the PSONN agree well with the data from the experiments, thus it confirms that the predictive ability of the PSONN model is adequate enough to further be optimized with MOPSO to achieve the Paretooptimal solutions for this application. 5.4 Optimization results and performance analyses To apply the MOPSO for the PSONN model, the following parameters are selected: the population of the MOPSO is set to 200, the acceleration factors are all set to 1.49, the Table 4 Comparison between PSONN model predictions with experimental results under the corresponding process conditions

Optimal process conditions

Volumetric shrinkage (%)

Product weight (g)

Flash (mm)

initial value of inertia weights 0.9, the ultimate value of inertia weights 0.3, and the number of nondominated solutions to be found is set to 200. MOPSO is run for 2,000 iteration steps. For the PSONN model, the MOPSO is used to evaluate the PSONN model to find the current Pareto-optimal front, and the current front would be captured with the collection of archives. The three-dimensional Pareto-optimal front for Eq. 7 is displayed in Fig. 4. The predicted responses are nonlinearly correlated to the process conditions, which feature complex interactions. On the basis of the achieved PSONN models, it shows that in this specific application there exists a direct trade-off behavior among the product weight, the flash, and the volumetric shrinkage as expected, in which the volumetric shrinkage will decrease with an increase in the product weight and the flash, and vice versa. Through the analysis of variance, the importance of the process conditions could be ranked according to their significance in the PSONN models, and interactions between process conditions and their influence on the corresponding objectives could be assessed. The volumetric shrinkage is affected in proper order by holding time, injection time, cooling time, mold temperature, injection pressure, holding pressure, and melt temperature. Meanwhile, all these seven process condition factors have significant main and interaction effects on the flash in proper order by injection pressure, holding pressure, mold temperature, cooling time, melt temperature, injection time and holding time, and product weight is affected in proper order by injection pressure, injection time, holding pressure, mold temperature, holding time, cooling time, and melt temperature. After establishing the finalized PSONN models and employing the MOPSO to achieve the Pareto-optimal

Melt temperature Mold temperature Injection pressure Injection time Holding pressure Holding time Cooling time PSONN model’s prediction Experimental results Difference (%) PSONN model’s prediction Experimental results Difference (%) PSONN model’s prediction Experimental results Difference (%)

1

2

3

4

180.30 56.37 54.45

183.79 35.9 39.77

183.87 36.24 31.43

203.30 35.7 32.16

2.13 32.46 3.74 17.43 2.94 2.79 5.47 38.94 39.5 −1.41 0.16 0.15 6.73

2.79 32.9 4.64 20 2.76 2.96 −6.34 39.12 38.12 2.63 0.26 0.24 8.12

2.41 37.88 3.16 20 2.01 1.92 4.51 39.65 40.4 −1.86 0.32 0.298 7.48

2.6376 31.9 3.61 19.21 2.39 2.26 5.67 39.36 38.26 2.62 0.23 0.216 6.53

530

Int J Adv Manuf Technol (2012) 58:521–531

Table 5 The final optimal process parameter settings according to Pareto optimizing based on fuzzy sets theory

MOPSO search result After tuning

Melt temperature

Mold temperature

Injection pressure

Injection time

Holding pressure

Holding time

Cooling time

185.975 186

39.087 39

49.683 50

2.227 2

49.912 50

3.729 4

13.938 14

solutions, different combinations of optimal trade-off solutions can be selected from these solutions according to the designer’s preference for setting up the process conditions. Table 4 shows a comparison of four Pareto-optimal solutions based on the PSONN model predictions and confirmation experiment under the corresponding optimal process conditions. From Table 4, it can be seen that the Pareto-optimal solutions is different from the confirmation experiment results, but the average difference percentage is relatively small (4.95%). Compared to all the samples, the results of the Pareto-optimal solutions are better than those of the samples, which confirm the superior predictive abilities of PSONN and the effectiveness of the proposed optimization scheme. The current approach executes fewer simulations for objective function evaluations and achieved better solutions. Therefore, with the help of this multi-objective optimization system based on PSONN, the optimization task specified in this application can be accomplished in a reasonably small amount of computing resources while still yielding reasonable results. Although the procedure needs a relatively long time to execute experiments for obtaining the initial training data, the subsequent optimization process could realize lots of benefits from the trained PSONN model.

6 Experiment comparison, results, and discussion After executing a soft computing model, the Pareto-optimal solutions are determined. According to Pareto optimizing based on the fuzzy sets theory shown in Section 3.2.3, the final optimal process parameter settings from Paretooptimal sets are determined after the minimum unit tuning and are shown in Table 5. For comparison, the experimental results are also analyzed using conventional Taguchi method to demonstrate the

effectiveness of the proposed approach. This research conducts an extra confirmation experiment that has process parameter settings determined by the conventional Taguchi’s parameter design method under volumetric shrinkage and flash responses consideration. The reason why this research uses volumetric shrinkage and flash as responses is because weight does not have a target value. The research follows the two stages approach of conventional Taguchi’s parameter design method to determine the optimal process parameter settings under volumetric shrinkage and flash responses consideration. Two statistics, standard deviation, and mean absolute Pn 1 deviation (MAD ¼ n i¼1 jPi  TVj, where Pi is the specific response value of i confirmation sample, TV is the target value of the specific response, and n is the number of confirmation samples) are compared to show the effectiveness of the proposed approach. To calculate the MAD of weight response, this research assumes that 39.31 (the weight average of 128 Taguchi experiment data) is the weight’s target just for comparison implementation purpose. The comparison results are showed in Table 6. The comparison results reveal the improvement rate of MAD under product weight, volume shrinkage, and flash response is 14%, 32%, and 43%, respectively, when using proposed approach. Moreover, the improvement rate of standard deviation under product weight, volume shrinkage, and flash response is 21%, 42%, and 30%, respectively, when using the proposed approach.

7 Conclusions The determination of optimal process parameter settings is critical work that influences productivity, quality, and costs of product production. Engineers have conventionally used trialand-error processes or Taguchi’s process parameter design

Table 6 Response performance comparison under different approach Proposed approach

Average Standard deviation MAD

Taguchi method

Product weight

Volume shrinkage

39.316 0.0153 0.0673

2.634 0.0072 0.0086

Flash 0.406 0.0057 0.0038

Product weight

Volume shrinkage

39.637 0.0194 0.0781

2.854 0.0125 0.0127

Flash 0.487 0.0081 0.0067

Int J Adv Manuf Technol (2012) 58:521–531

method to determine the final optimal process parameter settings. However, the application of these methods has some shortcomings and may cause engineers to make undesirable final optimal process parameter settings. In this investigation, a multi-objective injection molding optimization procedure integrating Taguchi’s parameter design method, backpropagation neural network, particle swarm optimization algorithms, and engineering optimization concepts is presented and shown to be theoretically sound and practically applicable to the multi-objective process optimization of injection molding. The PSONN model accurately provides predictions and estimations of the confidence of predictions simultaneously, the MOPSO is employed to the PSONN model to achieve the Pareto-optimal solutions, different combinations of optimal trade-off solutions can be selected from these solutions according to the designer’s preference for setting up the process conditions. The proposed approach can effectively assist engineers in determining the final optimal process parameter settings for plastic injection-molded product production under multiresponse consideration. According to Pareto optimizing based on fuzzy sets theory and the implementation results obtained in the illustrative example, the final optimal process parameter settings determined via the proposed approach definitely produces better performance of the plastic injection molding production process than Taguchi’s approach. Therefore, the proposed approach is feasible and effective for process parameter optimization in MIMO plastic injection molding and can assist the manufacturing industry in achieving competitive advantages on quality and costs. Acknowledgment The authors would like to express their gratitude to the Natural Science Foundation of Jiangxi Province, China (2009GZS0021).

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