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Technical Report

A hybrid of mode-pursuing sampling method and genetic algorithm

for minimization of injection molding warpage

Yi-Min Deng a,*, Yong Zhang a, Yee Cheong Lam b

a Faculty of Mechanical Engineering and Mechanics, Ningbo University, 818 Fenghua Rd., Ningbo City, Zhejiang Province 315211, PR Chinab School of Mechanical and Aerospace Engineering, Nanyang Technological University, 71 Nanyang Drive, Singapore 639798, Singapore

a r t i c l e i n f o

 Article history:

Received 23 August 2009

Accepted 15 October 2009

Available online 20 October 2009

a b s t r a c t

This paper presents a hybrid optimization method for minimizing the warpage of injection molded plas-

tic parts. This proposed method combines a mode-pursuing sampling (MPS) method with a conventional

global optimization algorithm, i.e. genetic algorithm, to search for the optimal injection molding process

parameters. During optimization, Kriging surrogate modeling strategy is also exploited to substitute the

computationally intensive Computer-Aided Engineering (CAE) simulation of injection molding process.

With the application of genetic algorithm, the ‘‘likelihood-global optimums” are identified; and the

MPS method generates and chooses new sample points in the neighborhood of the current ‘‘likeli-

hood-global optimums”. By integrating the two algorithms, a new sampling guidance function is pro-

posed, which can divert the search process towards the relatively unexplored region resulting in less

likelihood of being trapped at the local minima. A case study of a food tray plastic part is presented, with

the injection time, mold temperature, melt temperature andpacking pressure selected as the design vari-

ables. This case study demonstrates that the proposed optimization method can effectively reduce the

warpage in a computationally efficient manner.

Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Plastic materials are commonly used in many areas of industry

as they can meet various requirements due to their specific and

wide variation of physical and chemical properties. These include

lightness, resistance to corrosion, ease of shaping and forming,

etc. Injection molding is a major processing route for the produc-

tion of plastic parts. During the injection molding process, how-

ever, a number of defects may occur to the moldings, such as

warpage, shrinkage, sink marks and weld lines. These are caused

by many factors, which include the settings of the molding process

parameters, the mold system, the part geometry as well as the

plastic materials [1]. Among these defects, warpage is one of themost common and prominent, affecting both the part usage

(function) and the appearance [2,3]. Hence, warpage minimization

is one of the most critical considerations for the production of a

quality molded part.

Thus, it is not surprising that a lot of effort has already been de-

voted for injection molding warpage minimization. Section 2 will

provide an overview of the existing effort, together with its limita-

tions. To facilitate further discussion, a mode-pursuing sampling

(MPS) method is introduced in Section 3, together with a brief 

description of the Kriging surrogate modeling that will be used in

conjunction with the MPS method. Subsequently, a hybrid usage

of MPS method and genetic algorithm will be proposed to address

the existing limitations. In Section 4, the procedure of the proposed

hybrid optimization method is given. Section 5 provides a case

study of a food tray plastic part to demonstrate the application of 

the proposed methodology. This is followed by conclusion in

Section 6.

2. Overview of warpage minimization

Warpage in injection molding is generally a function of the part

geometry, the mold, the design of the runners and gates, and the

process parameters. Many studies have already been conducted

targeting at warpage minimization. For example, Lee and Kim [2]

proposed to reduce warpage by optimizing the part wall thickness

and process parameters. Subsequently, they [3] applied a two-step

search method to optimize the gate location to improve the part

quality, including warpage and some other quality aspects. The re-

search group of Lam [4–9] had investigated various parameters

and methodologies for the reduction of warpage. They show that

by cavity balancing [4–6], gate location and molding condition

optimization [7,8], and runner balancing for multi-cavity mold

[9], significant warpage reduction could be achieved. Methodolo-

gies employed include flow path optimization, injection pressure

0261-3069/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.matdes.2009.10.026

* Corresponding author. Tel.: +86 574 87600534; fax: +86 574 87608358.

E-mail addresses: dengyimin@nbu.edu.cn, dym1000@hotmail.com (Y.-M. Deng).

Materials and Design 31 (2010) 2118–2123

Contents lists available at ScienceDirect

Materials and Design

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / m a t d e s

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minimization, and genetic algorithms with pareto optimization

strategy. Deng et al. [10] proposed a PSO (particle swarm optimiza-

tion) algorithm for the optimization of multi-class design variables,

such as the part thickness, process parameters (melt temperature,

mold temperature, injection time) and gate location, targeting war-

page reduction as well as improving other molding qualities.

Despite the many achievements in warpage minimization,

some practical difficulties remain, one of which is the computa-

tional cost of the optimization process. Some researchers have

attempted to address this problem through experiments, for

example, Erzurumlu and Ozcelik [11] combined Taguchi method

and Moldflow simulation to study the effect of process parame-

ters on part quality, and derived the optimal settings by choos-

ing the ones from the limited set of experimental points derived

from the Taguchi method. This kind of method has a limitation

in that the search space is limited – the obtained optimal pro-

cess condition is only the best combination of the specified pro-

cess parameters, not the optimal solution in the entire design

space.

A different approach is by using surrogate modeling technolo-

gies. Based on a quantity of experimental samples, various kinds

of surrogate models, such as the polynomial models, radial basis

functions (RBF), Kriging models, and support vector regression

(SVR) [12] were employed to substitute the expensive injection

molding simulation process for global optimization. Hitherto, quite

a few researchers have exploited this methodology to optimize the

process parameters in injection molding [13,14]. They show that

surrogate modeling can be considered as a good approach to

reduce intensive computation in injection molding warpage

minimization.

However, by employing the surrogate modeling approach

alone, the optimality of the solution depends largely on the accu-

racy of the surrogate model. A good model generally requires a

large number of sample points, and should be distributed as uni-

formly as possible in the design space. Unfortunately, these

requirements conflict with the requirement of minimizing the

computational time. Therefore, some stepwise optimizationmethods are proposed in warpage minimization. For example,

Gao and Wang [15] proposed a sequential optimization method

based on Kriging surrogate model. This sequential optimization

can improve the surrogate model using a current optimal solution

until the convergence criteria are satisfied. However, it is not effi-

cient to update the Kriging approximate model iteratively by only

adding the current optimal point to the design set to reconstruct

the Kriging surrogate model. Zhou and Turng [16] presented an

integrated simulation-based optimization system based on a

Gaussian process approach. This method could improve the surro-

gate models by finding the additional training points with greater

variances through the optimization iterations. However, both

methods are prone to converge prematurely, i.e. to converge to

a local optimum, if it exists.Another approach is to generate directly new sample points to-

wards the optimum with the guidance of a surrogate model. In

contrast to the previous two methods, the surrogate model is not

used as a surrogate in a typical global optimization process. Opti-

mization is realized by adaptive sampling alone without a formal

global optimization process. The surrogate model is used only as

a guide for adaptive sampling and therefore the requirement on

model accuracy is reduced. Gao and Wang [17] proposed an adap-

tive sampling method for injection molding optimization based on

Kriging surrogate model, where an infilling sampling criterion

named as Expected Improvement (EI) was introduced. This crite-

rion can balance the local and global searches, even though the De-

sign of Experiment (DOE) size is small. This method, however,

presumes a continuous objective function and a correlation struc-ture between the sample points, with a complicated optimization

process required for the identification of an updated point. In addi-

tion to this method, Wang et al. [18] have proposed a global opti-

mization method based on a novel mode-pursuing sampling (MPS)

method, which can generate more sample points in the neighbor-

hood of a function mode (the term mode is explained in Section 3.1)

while statistically covering the entire search space. In our previous

work, we have attempted to apply this method in the determina-

tion of the appropriate process parameters for the reduction of 

injection molding warpage [19]. Our results show that this ap-

proach is efficient in comparison to the conventional simulation-

based warpage optimization methods. However, there is also a lim-

itation in that the optimization process may sometimes converge

to a local optimum before sufficient exploratory points are

generated.

To further improve warpage optimization in the injection

molding process, this investigation presents a hybrid optimization

method. A hybrid of MPS method and genetic algorithm is

proposed to minimize warpage effectively and efficiently. Kriging

surrogate modeling will be exploited to substitute the expensive

injection molding CAE simulation analyses. To facilitate discussion,

MPS and Kriging surrogate modeling will first be introduced.

3. Introduction of MPS and Kriging surrogate modeling 

 3.1. The MPS method

The mode-pursuing sampling (MPS) method was proposed by

Wang et al. [18], which is an extension of the random-discretiza-

tion based sampling method of Fu and Wang [20], a general-

purpose algorithm to draw a random sample from any given

multivariate probability distribution. The word ‘‘mode” refers to

a minimum (either local or global) of the objective function.

Suppose an n-dimensional function f ( x) is to be minimized over a

compact set S ( f ) = [a, b]n (a < b), f ð xÞP 0 and it is continuous in

the neighborhood of the global minimum, the MPS method may

be summarized into the following steps:

Step 1: Generate m count of uniformly distributed points x(i),

i = 1, . . ., m on S ( f ) and calculate their function values f ( x(i)).

Step 2: Formulate an approximation function S  p( x) based on the

above m points, for example, a linear spline function S  p( x) can

be formulated as

S  pð xÞ ¼X

i

aik x À xðiÞk

subject to S  pð xðiÞÞ ¼ f ð xðiÞÞ; i ¼ 1; . . . ; m

ð1Þ

A non-negative function g ( x) can then be constructed by defin-

ing g ð xÞ ¼ C 0 À S  pð xÞP 0 on S ( f ) = [a, b]n, where C 0 is a constant to

make g ( x) non-negative. Minimizing S  p( x) is equivalent to maxi-

mizing g ( x), hence g ( x) can be viewed as the probability density

function to be used in the next step.Step 3: By applying the sampling algorithms of Fu and Wang

[20], another m count of random sample points, denoted as

 y(i), i = 1, . . ., m, are selected from S ( f ) = [a, b]n according to g ( x)

(hence it is referred to as the sampling guidance function).

These points tend to gather around the current maximum of 

 g ( x), which corresponds to the minimum of  S  p( x).

Step 4: Evaluate the derived sampling points and their corre-

sponding objective function against the specified termination

criterion. If the criterion is met, the search process is termi-

nated; otherwise, combine the new m points, y(i), i = 1, . . ., m,

with the old points x(i), i = 1, . . ., m, to obtain a new x = [ x, y],

then repeat steps 2–4.

In step 3, to better control the sampling process (e.g. whether togenerate more ‘‘exploratory” points or ‘‘local” points), a speed

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control factor c can be introduced, details of which can be found in

Wang et al. [18].

 3.2. Kriging surrogate modeling 

Kriging surrogate modeling is one of the methods for building a

surrogate model of a full-fledged process such as a CAE simulation

process. In general, the Kriging models are more accurate for non-

linear problems, hence are suitable for the calculation of injection

molding warpage to be addressed in this investigation.

Given a set of  m count of design samples S = [ x1, x2, . . ., xm]

 xi = Rn and the responses Y = [ y1, y2, . . ., ym]T  ( yi = Rq), we can con-

struct a Kriging model as follows:

^ yð xÞ ¼ F ðb; xÞ þ z ð xÞ ¼ f T ð xÞb þ z ð xÞ ð2Þ

F (b, x) is a regression model which is generally a polynomial

function. The coefficient b is a regression parameter to be esti-

mated, z ( x) is a random function, which has the following statisti-

cal characteristics:

E ½ z ð xÞ ¼ 0

Var ½ z ð xÞ ¼r

2

Cov ½ z ð xiÞ; z ð x jÞ ¼ r2½Rðh; xi; x jÞ

ð3Þ

where xi, x j represent two design samples in the given set. R(h, xi, x j)

is the correlation model with parameter h, where a Gauss function

is generally used, which is defined as,

Rðh; xi; x jÞ ¼ exp ÀXndv 

k¼1hkj xk

i À xk j j2

ð4Þ

ndv is the number of all design variables, xki , xk

 j correspond to the kth

component of the samples xi and x j, respectively. The parameters hk,

r2, b could be determined via the maximum likelihood estimation

(refer to references, e.g. Ref. [15], for details).

4. MPS/GA hybrid method for warpage minimization

In this investigation, we employ the commercial injection

molding CAE software MoldflowÒ to simulate the injection mold-

ing process and to retrieve warpage deflections from the simula-

tion results. Moldflow can report the warpage deflections (in the

 X , Y and Z directions) at each node of the part mesh model. In prac-

tice, the design engineers may only be interested in the overall

deflection or deflection in a particular direction. Therefore, for ease

of illustration but without loss of generality, we opt to optimizethe

warpage deflection in only one specific dimension. The same

methodology could easily be extended for all three directions.

4.1. The mathematical model

The injection molding warpage minimization problem can be

stated as follows:

Find X 

Minimize W ð X Þ

Subject to X l 6  X 6  X u

ð5Þ

where X = { x1, x2, . . ., xm} is a vector of injection molding process

parameters, and X l, X u are the lower and upper bounds of  X 

respectively, W ( X ) is the maximum value of warpage deflection.

This optimization problem has the following characteristics [19]:

(1) The objective function is not represented by an explicit

mathematical expression (hence is a so-called black-box

function). It may only be evaluated by retrieving resultsfrom Moldflow or other CAE simulation.

(2) Since the evaluation of the objective function requires

intensive computation (by executing CAE simulation), an

optimization algorithm that demands a large number of 

evaluations of the objective function (which often involves

an iterative process) would be a computationally expensive

process.

These characteristics are especially suited for utilizing the MPS

and surrogate modeling method, and will be further elaborated in

the following section.

4.2. Hybrid optimization strategy

As pointed out in Section 3, when applying the MPS method,

more sample points will be generated aroundthe current minimum

point as the MPS progresses. This will increase the chances of find-

ing a better minimum. However, a key issue to be addressed is the

generation and selection of the sample points so as to avoid the

optimization process from being trapped to a local minimum.

According to Torn and Zilinskas [21], an optimization method must

be able to probe thepart of thesearch space that has been relatively

unexplored, if it is to converge to the global optimum. Comparingwith other surrogate modeling methods, Kriging modeling allows

us to compute a measure of the possibleerror in the surrogate mod-

el by its mean squared error (MSE). The MSE goes to zero at the

sample points, indicating that there is no uncertainty at the sample

points. In between the sample points, the MSE rises. Intuitively, the

further weare away from the nearest sample point, the more uncer-

tain we are about the function value, and the higher is the MSE. As

MSE provides a measure of uncertainty of the surrogate away from

the sample points, a search method couldbe developed by combin-

ing the MSE measure with the sampling characteristics provided by

the MPS method. This can be achieved easily by intentionally gen-

erate and select more sample points where the MSE is high, which

represents relatively unexplored space. Later sampling should gen-

erate more points around those with small function values, yet areaway from the current ‘‘likelihood-global optimum”, where ‘‘likeli-

hood-global optimum” refers to the current optimum based on the

current set of sample points and the current surrogate model.

To materialize the above strategies, we propose to construct a

new sampling guidance function, which is derived by adding two

existing functions: the normal sampling guidance function, e.g.

the linear spline functiondiscussed in Section 3, and the newly pro-

posed MSE function in the Kriging model. Normalization will have

to be carried out when formulating the new guidance function.

With this new guidance function, the conventional MPS method

is to be modified. As the genetic algorithm is also employed to

identify the ‘‘likelihood-global optimum”, hence the proposed

method is a hybrid MPS/GA method. The detailed procedure of 

the MPS/GA hybrid algorithm could be given as follows:

4.2.1. Nomenclature (for those not explained in the algorithm)

S _mode = 1: indicating a ‘‘likelihood-global optimum” has been

found.

S_mode = 0: indicating a ‘‘likelihood-global optimum” is yet to

be found.

S  p( x): normal sampling guidance function defined in Section 3.1.

S m( x): sampling guidance function defined by MSE function in

Kriging model.

S  pm( x): the new sampling guidance function defined as

S  pm( x) = S  p( x) + S m( x).

Input . m, the number of design variables; X , design variable vec-tor; S , [ X l, X u]: compact set of  X .

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Output . X best , the obtained global optimum; W best , the objective

function value at X best ; nit , the number of optimization iteration.

4.2.2. Algorithm procedure

(1) Project the compact set S :[ X l, X u] to S 0:[0, 1]m, randomly sam-

ple [(m + 1)(m + 2)/2] points; conduct CAE (i.e. Moldflow) simula-

tions to evaluate their corresponding objective function values

(i.e. warpage deflections), set nit = 1 and S _mode = 0.(2) If S _mode = 0, generate m sample points by applying the MPS

algorithm based on the normal sampling guidance function S  p( x);

otherwise, generate m/2 sample points by applying the MPS algo-

rithm based on the sampling guidance function S m( x) and S  pm( x),

respectively. Then evaluate their corresponding objective function

values, and add them into the sample point set.

(3) Construct a Kriging surrogate model based on the current

point set, then apply GA to the surrogate model to determine the

‘‘likelihood-global optimum” X  g  and the corresponding approxi-

mate function value W ð X  g Þ. Conduct CAE simulation to obtain

the corresponding objective function value W ( X  g ) under the

derived optimal condition ( X  g ), add ( X  g , W ( X  g ))to the point set.

(4) Determine the current optimum including X best  and W best ; If 

nit P 2, then evaluate the distance dis, between the current opti-mal X best  and the last optimal X 0best , and the difference D f , between

the current optimal function value W best  and the optimal function

value W 0best  in the last iteration; as well as the absolute difference

between W ( X  g ) and W ð X  g Þ indicated by D g .

(5) If  dis 6 ed; D f 6 e f  and D g 6 e g , set S_mode = 1 and take

( X best , W best ) as a ‘‘likelihood-global optimum”; otherwise, set

S_mode = 0.

(6) Finally, if the convergence criterion is satisfied, terminate

the program with all the output; otherwise, set nit = nit + 1 and

go back to step 2.

Note that in step 5, e f , e g  should be specified according to the

accuracy requirement of the function value. ed is normally set to

a small value, such as 10À4. These parameters control the conver-

gence speed of the algorithm, hence are referred to as the control

parameters.

The convergence criterion in step 6 may be a function of the last

three ‘‘likelihood-global optimums” – when there is no obvious

difference between them, the solution could be considered as

converged.

5. Case study 

5.1. Problem description

To demonstrate the proposed hybrid optimization approach, a

food tray plastic part is used as a case study. Fig. 1 contains its

geometry shape and its CAE analysis model in Moldflow, with its

overall external dimensions being 180 Â 90 Â 50 mm, and a maxi-mum part thickness of 3 mm. The material employed is LDPE, man-

ufactured by Eastman Chemical Products, and its properties are

given in Table 1. Its warpage is quantified by the sum of the abso-

lute value of the maximum upward deflection and the maximum

downward deflection of all nodes of the part mesh model in the

 Z  direction (only a specific direction is considered, as explained

in the beginning of Section 4).

We select four key process parameters as the design variables

for warpage minimization, namely injection time (t inj), mold tem-

perature (T md), melt temperature (T mt ) and packing pressure (P  p).

For validation of the algorithm, we intentionally set a relatively

large search space, making the initial ranges of the design variables

larger than those usually encountered in practice, but avoiding

short shot defect. The range of the mold temperature is based onthe recommended values in Moldflow for the specified part mate-

rial. The lower bound of the melt temperature range is 10 °C higher

than the minimum values recommended by Moldflow to avoid

short shot. The range of injection time is selected according to

the recommended value by the ‘‘Molding Windows” analysis mod-

ule from the Moldflow software. Further, the packing profile is con-

sidered as a constant pressure process, with the packing time set as

5 s. The range of packing pressure is usually specified based on the

experience of the manufacturer, where the percentage of maxi-

mum machine pressure is used to measure the packing pressure.

For this case study, these ranges are summarized below:

Mold temperature (°C): 20–70.

Melt temperature (°C): 180–280.

Injection time (s): 0.4–0.8.

Packing pressure (%): 70–90.

With these specified ranges, the optimization problem can be

formulated as:

Find X  ¼ ½t inj; T md; T mt ; P  p

Minimum W ð X Þ

Subject to 0:4 6 t inj 6 0:8

20 6 T md 6 70

180 6 T mt 6 280

70 6 P  p 6 90

5.2. Implementation of optimization procedure

During each optimization iteration, a Kriging surrogate model

has to be built based on the current sample points. We choose qua-

dratic function for the regression model and Gauss function for the

correlation model, as discussed in Section 3.

In the GA optimization process, the commonly used GA opera-tion parameters are adopted, namely the population size, the

Fig. 1. CAE food tray model.

 Table 1

Material properties.

Melt density 0.73537 g/cm3

Solid density 0.94781 g/cm3

Eject temperature 80 °C

Maximum shear stress 0.11 MPa

Maximum shear rate 40,000 (1/s)

Thermal conductivity 0.31 W/m °C

Elastic module 124 MPa

Poisson’s ratio 0.41

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crossover rate, the mutation rate and the generation size are set as

20, 0.9, 0.005 and 100, respectively.

The control parameters of the MPS/GA hybrid algorithm are set

as follows:

ed ¼ 0:0001; e f  ¼ 0:01; e g  ¼ 0:01

Since GA is probabilistic, we ran the proposed MPS/GA optimi-

zation algorithm 10 times. The optimization results produced from

all these runs are shown in Table 2.

To validate and compare the optimization results shown in

Table 2, we also obtain the benchmark warpage value by

employing the Moldflow recommended values for the processing

parameters in the simulation, namely:

t inj ¼ 0:66; T md ¼ 23; T mt  ¼ 240; P  p ¼ 80%

The corresponding warpage, as shown in Fig. 2, is

W ð X Þrec  ¼ 2:03mm:

5.3. Discussion

Results in Table 2 shows that,

The proposed MPS/GA hybrid method can reduce warpage with

a limited number of CAE simulation runs (which largely deter-

mine the amount of computation of the optimization process).

Considering that a general simulation-based optimization pro-

cess could easily require thousands of full-fledged CAE simula-

tion runs, obvious efficiency is obtained.

The warpage deflection has been reduced by approximately 45–

60% when, comparing with the warpage deflection obtained

using Moldflow recommended processing parameters, i.e.

reduced to 0.81–1.12 mm as compared to the benchmark valueof 2.03 mm.

The optimal process parameters are: injection time: 0.8 s, mold

temperature: 69 °C, melt temperature: 280 °C, and packing pres-

sure: 70%. Fig. 3 shows the corresponding warpage deflection of 

0.81 mm.

To further illustrate the characteristics of the proposed optimi-

zation method, we select one of the optimization runs (the last of 

the 10 runs), and record some of the data generated from this runin Table 3.

 Table 2

Optimized results (10 runs of MPS/GA hybrid algorithm).

Run no. Injection time (t inj/s) Mold temperature (T md/°C) Melt temperature (T mt /°C) Packing pressure (P  p/%) Minimizedwarpage (W /mm) No. of CAE simulations

1 0.74 68.64 279.17 71.33 1.0808 135

2 0.74 68.60 279.32 70.23 0.9506 135

3 0.795 67.00 279.13 71.30 1.0343 105

4 0.786 69.33 279.60 71.45 0.9465 125

5 0.785 69.57 277.64 70.04 0.8742 115

6 0.80 69.06 280.00 70.34 0.8088 1057 0.79 69.66 279.12 71.18 0.9006 100

8 0.79 69.43 278.96 70.00 0.8145 135

9 0.80 68.90 377.23 72.00 1.1196 115

10 0.78 69.45 280.00 70.47 0.8287 100

Fig. 2. Warpage deflection using Moldflow recommended processing parameters.Fig. 3. Minimized warpage obtained by the proposed MPS/GA method.

 Table 3

Some data generated from the MPS/GA optimization process.

All sample points and corresponding

warpage values (mm) once a ‘‘likelihood-

global optimum” was found

All the ‘‘likelihood-global

optimums”

[0.77 67.89 266.67 84.19]/3.9136 [0.40 20.01 189.09 89.89]/1.4743

[0.72 23.58 267.22 78.96]/4.8916 [0.70 27.98 187.80 89.15]/1.3705

[0.67 22.49 213.02 86.83]/2.2127 [0.79 69.52 277.54 70.67]/0.9382

[0.78 66.13 183.02 70.04]/4.0855 [0.79 69.52 277.54 70.67]/0.9382

[0.60 69.74 275.90 89.26]/4.5418 [0.79 69.52 277.54 70.67]/0.9382

[0.79 28.25 275.95 71.40]/5.2846 [0.77 68.55 278.58 70.08]/0.8934

[0.44 22.13 264.81 70.74]/5.0052 [0.78 69.45 280.00 70.47]/0.8287

[0.42 69.25 185.62 70.27]/5.0425

[0.40 67.86 180.67 70.67]/4.7556

[0.40 67.50 183.90 89.05]/5.0085

[0.74 49.58 235.36 70.25]/5.0172

[0.79 69.19 180.05 86.14]/4.1236

[0.41 67.47 278.64 89.39]/6.8916

[0.40 20.17 277.77 70.87]/5.2002

[0.47 20.06 279.51 70.21]/5.2003

[0.41 20.19 187.30 70.37]/1.5847

[0.42 31.78 180.34 78.20]/1.6956[0.42 21.18 271.87 89.78]/4.9075

[0.40 31.53 278.05 89.85]/5.3918

[0.40 22.56 184.14 70.76]/1.4631

The sample points are put within squared brackets, in sequence of injection time

(s), mold temperature (°C), melt temperature (°C), and packing pressure (%).

2122 Y.-M. Deng et al./ Materials and Design 31 (2010) 2118–2123

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Table 3 shows that,

Once the optimization process has determined a point as a ‘‘like-

lihood-global optimum”, the following sample points will not

spread near this point.

The warpage values of the above sample points show that the

proposed sampling method has the characteristic of spreading

sample points around those regions of relatively small function

values, and yet are away from the neighborhood of all the ‘‘like-

lihood-global optimums”.

Although many sample points correspond to a large warpage

value, and that they may not be useful for finding the ‘‘global

optimum” directly, they increase the reliability of regarding

the ‘‘likelihood-global optimum” in the last generation as the

global optimum.

6. Conclusion

This investigation integrates the injection molding simulation

with a global optimization algorithm, which is based on the

mode-pursuing sampling (MPS) method and the traditional global

optimization method of genetic algorithm (GA), for the search of the optimal process parameters for a minimum injection molding

warpage. This proposed optimization algorithm can reduce the

amount of computation required while determining a global opti-

mum with good reliability. A plastic food tray case study, where

injection time, mold temperature, melt temperature and packing

pressure are selected as the design variables, demonstrates that

the proposed hybrid optimization method can effectively reduce

the warpage deflection in a computationally efficient manner. This

methodology, although targeted at injection molding warpage

minimization at this investigation, could well be equally effective

for other similar engineering optimization problems, which in-

volve iterative runs of a computation-intensive black-box function

(such as the time-consuming CAE simulations).

 Acknowledgements

This research is funded by Zhejiang Provincial Natural Science

Foundation of China, Grant No. R104247. It is also supported by

K.C. Wong Magna Fund in Ningbo University. The authors would

like to thank Dr. Gaofeng Gary Wang from the Simon Fraser

University of Canada, for his helpful advice on the MPS method.

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