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Finding efficient frontier of process parameters for plastic injection molding

  • Wu-Lin ChenEmail author
  • Chin-Yin Huang
  • Ching-Ya Huang
Open Access
Original research

Abstract

Product quality for plastic injection molding process is highly related with the settings for its process parameters. Additionally, the product quality is not simply based on a single quality index, but multiple interrelated quality indices. To find the settings for the process parameters such that the multiple quality indices can be simultaneously optimized is becoming a research issue and is now known as finding the efficient frontier of the process parameters. This study considers three quality indices in the plastic injection molding: war page, shrinkage, and volumetric shrinkage at ejection. A digital camera thin cover is taken as an investigation example to show the method of finding the efficient frontier. Solidworks and Moldflow are utilized to create the part’s geometry and to simulate the injection molding process, respectively. Nine process parameters are considered in this research: injection time, injection pressure, packing time, packing pressure, cooling time, cooling temperature, mold open time, melt temperature, and mold temperature. Taguchi’s orthogonal array L27 is applied to run the experiments, and analysis of variance is then used to find the significant process factors with the significant level 0.05. In the example case, four process factors are found significant. The four significant factors are further used to generate 34 experiments by complete experimental design. Each of the experiments is run in Moldflow. The collected experimental data with three quality indices and four process factors are further used to generate three multiple regression equations for the three quality indices, respectively. Then, the three multiple regression equations are applied to generate 1,225 theoretical datasets. Finally, data envelopment analysis is adopted to find the efficient frontier of the 1,225 theoretical datasets. The found datasets on the efficient frontier are with the optimal quality. The process parameters of the efficient frontier are further validated by Moldflow. This study demonstrates that the developed procedure has proved a useful optimization procedure that can be applied in practice to the injection molding process.

Keywords

Injection molding Taguchi’s orthogonal array Mutiple regression analysis Data envelopment analysis Optimization 

Introduction

Along with the rapid progress of production techniques for high-tech products, better and better quality of products is required for the survival in the current market. Besides providing various functions, the trend of the design for plastic products is light, thin, short, and small. Therefore, the setting of process parameters for plastic products has a remarkable influence on their quality (Huang and Tai 2001).

Injection molding is one of the most important techniques for polymer processing (to manufacture plastic products) because of its high speed for molding and its capability of manufacturing complex geometric shapes of products. Besides, injection molding is capable of mass production, so it is widely used for many products, especially for electronic products, such as computers and communication products. Injection molding is usually adopted to produce thin parts or thin covers for these products.

Currently, there are two categories for the setting of process parameters for injection molding: one is based on the technicians’ previous experience and the other takes advantage of mold flow analysis softwares, such as Moldflow (used in this study) to find the initial values for process parameters (by running various simulations on these moldflow analyses). However, no method can quickly find the reasonable combination of process parameters. In addition, trial and error is required for both methods, and the process of trial and error consumes a significant amount of time and cost. Therefore, both methods cannot meet the requirement of the current market.

Researchers have applied various kinds of methods, e.g., artificial neural network and/or fuzzy logic (Liao et al. 2004a, b; Kurtaran et al. 2005; Ozcelik and Erzurumlu 2006), genetic algorithm (Kurtaran et al. 2005; Ozcelik and Erzurumlu 2006), design of experiments (Huang and Tai, 2001; Liao et al. 2004a, b), and response surface method (Ozcelik and Erzurumlu 2005; Kurtaran and Erzurumlu 2006; Chen et al. 2010) to optimize the initial process parameter setting of plastic injection molding. However, most studies focus on the single optimal combination of process parameters by different optimization techniques. It is well known that when multiple quality characteristics are considered, the trade-off relationships exist among these quality characteristics, and these relationships make the task of finding the optimal combination rather complicated if not impossible. Instead of finding a single optimal combination of process parameters, this research seeks the efficient frontier of process parameters by data envelopment analysis (DEA).

The remainder of this paper is organized as follows: the ‘Literature review on the optimization of process parameters for injection molding’ section will review related work in the literature. The properties of the material and product used in this paper will be addressed in the ‘Materials and product’ section. The ‘Experimental design and methodology’ section will discuss the experimental design and the procedure of finding the efficient frontier of process parameters. Finally, the summary and concluding remarks are provided in the ‘Summary and conclusions’ section.

Literature review on the optimization of process parameters for injection molding

The literature of optimization for injection molding is briefly addressed in this section. Kim and Lee (1997) discussed different geometries for plastic parts to improve the parts’ warpage by Taguchi’s orthogonal experiment design. To avoid producing flaws of silver streaks for automobile plastic bumpers, Taguchi’s optimization method is utilized to decide the optimal values for the process parameters by Chen et al. (1997). The same optimization method was also used by several works (Huang and Tai 2001; Liu and Chen 2002; Liao et al. 2004a, b; Oktem et al. 2007) to find the optimal combinations of process parameters for different plastic products. In these works, warpages and shrinkages of plastic parts were usually considered as their quality indices. Moldflow, a mold flow analysis software, was used to simulate a real injection machine by Erzurumlu and Ozcelik (2006). Several techniques including the Taguchi’s method, the neural networks, and the genetic algorithm were combined to optimize the process parameters.

The response surface methodology (RSM) is another popular method to optimize the process parameters in the literature. The complete model of RSM was first established by Box and Wilson (1951). To improve two quality indices, within-wafer non-uniformity and the removal rate, of the chemical–mechanical planarization process in semiconductor manufacturing, dual RSM was proposed by Fan (2000) to optimize five process parameters. In order to avoid the difficulty of minimizing both quality indices, one was treated as the primary response put in the objective function and the other was the secondary response placed in the constraint. Two works developed RSM combined with different optimization techniques (Ozcelik and Erzurumlu 2006; Chiang and Chang 2007). Chen et al. (2010) applied dual RSM to improve the quality of plastic injection molding. Warpage is the primary response (and is treated as the objective function), while shrinkage is the secondary response (and is then set as the constraint) in their work.

For multiple quality indices, the above two classes of optimization focus on searching the single optimal combination of process parameters. However, it is known that the trade-off relationships exist among multiple quality indices, so the searching task of the single optimum is not easy. Castro et al. (2007) used DEA technique to find the efficient frontier when six quality indices (related with the part’s geometry) were considered. This research proposes to combine several techniques including experimental design, analysis of variance (ANOVA), multiple regression analysis, and DEA to find the efficient frontier of the process parameters when three essential quality indices, warpage, shrinkage, and volumetric shrinkage at ejection, are under consideration simultaneously.

Materials and product

This study demonstrates how to determine the efficient frontier of process parameters for injection molding by taking an example of the thin cover of a digital camera. The CAD software, Solidworks, is the first to prepare the geometry of the product as shown in Figure 1.
Figure 1

The geometry.

Next, the commercial CAE simulation software tool, Moldflow, is utilized to create the finite element model and uses the finite element and finite difference method to solve pressure, flow, and temperature fields of injection molding (Walsh 1993; Mackerle 2005; Shoemaker 2006). The setting in Moldflow to simulate production of the parts is described below.

Figure 2 shows the finite element model of the plastic part, and Figure 3 is the runner system with the cooling channel. The material used in this paper is GE Cycoloy C2950 PC/ABS (Gardena, CA, USA), and the simulating molding machine used is Roboshot 330i (330 tons, 8.90 oz, 44 mm) high speed/pressure (Milacron, Cincinnati, OH, USA). Table 1 lists the main properties of the material.
Figure 2

The finite element model. Courtesy of Moldflow Corporation.

Figure 3

The runner system with the cooling channel. Courtesy of Moldflow Corporation.

Table 1

Material properties of GE Cycoloy C2950 PC/ABS

Material property

Value

Recommended mold surface temperature (°C)

70

Recommended melt temperature (°C)

275

Melt density (g/cm3)

0.97618

Solid density (g/cm3)

1.1161

Eject temperature (°C)

113

Maximum shear stress (MPa)

0.4

Maximum shear rate (s−1)

40,000

Mold thermal conductivity (W/m °C)

29

Elastic module (MPa)

200,000

Poisson’s ratio

0.33

This paper discusses nine key process parameters which are suggested in the literature (Chen and Turng 2005; Chen et al. 2010) because these parameters are most likely to affect products’ quality. Table 2 shows nine process parameters and their ranges of operation suggested by Moldflow. Three indices of quality, warpage, shrinkage, and volumetric shrinkage at ejection, are considered in this paper.
Table 2

Key process parameters and their ranges of operation

Parameter

Variable

Initial value

Range of operation

Injection time

INT (x1)

1 s

0.5 to 1.5 s

Injection pressure

INP (x2)

120 MPa

100 to 140 MPa

Packing pressure

PP (x3)

100 MPa

80 to 120 MPa

Packing time

PT (x4)

10 s

7.5 to 12.5 s

Cooling time

COTI (x5)

19 s

14 to 24 s

Coolant temperature

COTE (x6)

25°C

20°C to 30°C

Mold open time

MOO (x7)

5 s

4 to 6 s

Melt temperature

MET (x8)

275°C

270°C to 280°C

Mold surface temperature

MOTE (x9)

70°C

65°C to 75°C

The warpages are measured at four reference nodes, and the shrinkages are measured at four edges in this paper; these nodes and edges are depicted in Figure 4 of the XY horizontal plane. In Figure 4, four reference nodes N1991, N1962, N1955, and N2261 are shown, while four edges are formed by these four nodes and are denoted by E1, E2, E3, and E4, respectively.
Figure 4

The nodes measured for the quality of parts.

Experimental design and methodology

This section presents how to design the experiment as well as the procedure to determine the efficient frontier of process parameters. In the ‘Experimental design’ subsection, the experimental design will be addressed. To effectively find the efficient frontier, ANOVA will be firstly executed to determine the significant process parameters out of the original nine process parameters in ‘Finding significant process parameters by ANOVA’ subsection. The complete design of experiment with four significant process parameters is again executed on Moldflow to have better accuracy of the following regression equations. Then, response regression model will be established in which only significant process parameters are considered in ‘Setting up the regression response model to create the complete dataset’ subsection. This subsection will also present how to create the complete dataset for finding the efficient combinations. Finally, ‘Determining the efficient frontier of process parameters by DEA’ subsection will discuss how to find the efficient frontier by DEA.

Experimental design

The Taguchi experimental design with orthogonal array is an efficient experimental design for fraction factorial design (Rose 1989; Montgomery 2005). Because there are nine process parameters considered in this research, complete experimental design is just too expensive to execute. Therefore, this research adopts the Taguchi experimental design with orthogonal array, L27, to perform the experiment on Moldflow. The experimental results of L27 is shown in Appendix A. Three levels of each process parameters are assigned to lower bound, mid-point, and upper bound of the range of operation listed in Table 2; for example, levels 1, 2 and 3 of injection time x1 are 0.5, 1 and 1.5 s, respectively.

Finding significant process parameters by ANOVA

In order to simplify the regression equations (and thus simplify the following procedure), ANOVA is firstly executed to find significant process parameters to affect the parts’ three quality indices which will only be considered in the regression equations. The results are shown in Table 3. Each node represents the warpage at this node, each edge means the shrinkage at this edge, and the volume is the volumetric shrinkage at ejection in Table 3. The symbol ‘◎’ means the corresponding process parameter significantly affects the quality index under the significant level 0.05 in the figure. The last column of Table 3, U, is remarked by ‘●’ if the corresponding process parameter significantly affects at least one quality index. From Table 3, only four process parameters are significant to affect at least one quality index, and hence only these four parameters are considered in the regression analysis in the next subsection.
Table 3

Results of ANOVA

 

N1991

N1962

N1955

N2261

E1

E2

E3

E4

Volume

U

INT

        

INP

 

PP

 

PT

        

COTI

          

COTE

          

MOO

          

MET

          

MOTE

          

Setting up the regression response model to create the complete dataset

To obtain a more complete efficient frontier for the process parameters, more data are required. The regression model, the response surface model, is utilized to create more data. In order to have better forecasting accuracy of the regression model, the complete experiment design with four significant process factors is executed again on Moldflow before the regression equations are established. The results are shown in the Appendix B.

The results of the complete experiment design with four significant process factors are then utilized to set up the second-order response surface model by the regression analysis of statistics software, SPSS. Three response surface equations for three quality indices are found below:
Warp = 1.45 0.011 INP 0.238 INT 0.00823 PP + 0.000046 INP 2 + 0.108 INT 2 + 0.000016 PP 2 0.00176 INP * INT + 0.00001 INP * PP + 0.00211 INT * PP , Open image in new window
(1)
Shrink = 1.32 0.0145 INP + 0.35 INT + 0.000086 INP 2 + 0.0603 INT 2 0.000013 PP 2 0.00593 INP * INT 0.000026 INP * PP + 0.00122 INT * PP , Open image in new window
(2)
and
Volume = 27.0 0.348 INP + 0.00165 INP 2 + 2.74 INT 2 0.00035 PP 2 0.0751 INP * INT 0.000613 INP * PT + 0.0369 INT * PP + 0.0839 INT * PT . Open image in new window
(3)
The normal probability plots are provided in Figures 5, 6, 7 to justify the validity of the regression analysis.
Figure 5

The normal probability plot for warpage.

Figure 6

The normal probability plot for shrinkage.

Figure 7

The normal probability plot for volumetric shrinkage at ejection.

To find a more complete efficient frontier of process parameters, more data are required. Regressed response surface equations, Equations 1, 2, 3, are exploited to create more data points. Because DEA software, Banxia Frontier Analyst 3, has the limitation on the maximal number of data points (also called decision making units (DMUs)), the design of data points to be created is explained below. Based on the results of ANOVA, because injection time and packing pressure are more significant than the other two process parameters, there are seven levels selected for these two process parameters and five levels for the other two parameters. Therefore, there are 5 × 7 × 7 × 5 = 1,225 data points to be created by Equations 1, 2, 3. The levels of each process parameter are listed in Table 4. Note that in Table 4, all levels of each parameter all fall its range of operation in Table 2.
Table 4

Levels of process parameters

Process parameter

L1

L2

L3

L4

L5

L6

L7

Injection pressure (MPa)

100

110

120

130

140

  

Injection time (s)

0.5

0.67

0.84

1

1.17

1.34

1.5

Packing pressure (MPa)

80

86.67

93.34

100

106.67

113.34

120

Packing time (s)

7.5

8.75

10

11.25

12.5

  

Determining the efficient frontier of process parameters by DEA

DEA is a technique to evaluate the relative efficiency of many DMUs by analyzing multiple inputs and multiple outputs of each DMU. Its goal is to find the efficient DMUs, also called efficient frontier in the literature of DEA. This research uses the standard DEA Charnes, Cooper, and Rhodes (CCR) (Charnes et al. 1978) model to find the efficient frontier of DMUs which is created in the previous subsection. The mathematical model of DEA CCR is briefly outlined below. Suppose that there are K DMUs, each of which consumes N inputs and produces M outputs. Then, the DEA CCR model can be established as follows:
Max H K = m = 1 M U m Y k m n = 1 N V n X k n Open image in new window
such that
m = 1 M U m Y km n = 1 N V n X kn 1 U m , V n ϵ > 0 m = 1 , 2 , , M n = 1 , 2 , , N k = 1 , 2 , , K , Open image in new window

where Y km is the value of the m th output generated by the k th DMU, X kn is the value of the n th input consumed by the k th DMU, V n and U m are X kn ’s and Y km ’s weights, respectively, whose values are determined by solving the model, H k is the relative efficiency of the k th DMU, and ϵ is a small positive number. After solving the CCR DEA model, a DMU is on the efficient frontier if its relative efficiency, H k , is equal to 1.

Three quality indices, warpage, shrinkage, and volumetric shrinkage at ejection, are considered in this paper. Among these three indices, warpage is treated as the output, while shrinkage and volumetric shrinkage at ejection are two inputs in the DEA model. Because the output in DEA model needs to be maximized and warpage is apparently the minimized quality index, the transformation, 1-warpage, is adopted.

The DEA software, Banxia Frontier Analyst 3, is used to find the efficient frontier of process parameters. The dataset used is the dataset of 1,225 data points created in the previous subsection. Each data point is treated as a DMU. After running Banxia Frontier Analyst 3, data points on the efficient frontier are found in Table 5. There are nine DMUs on the efficient frontier, among which five DMUs have at least one reference count as shown in Table 6. Therefore, these five DMUs with positive reference counts are treated as the efficient frontier of process parameters in this paper. The levels of each process parameter for these five DMUs are shown in Table 7, where the forecasting value of a quality index is its value derived from the corresponding regression equations and the real value of a quality index means its value by re-running Moldflow on this combination of process parameters. The plot of DEA results is shown in Figure 8.
Figure 8

The plot of DEA.

Table 5

DMUs on the efficient frontier

DMU

Score

560

100

840

100

871

100

1,151

100

1,186

100

1,187

100

1,188

100

1,189

100

1,190

100

Table 6

The reference counts of the efficient DMUs

DMU

Reference count

560

773

840

624

871

402

1,151

182

1,186

47

1,187

0

1,188

0

1,189

0

1,190

0

Table 7

Levels of process parameters for efficient DMUs

 

DMU

 

560

840

871

1,151

1,186

Process parameter

INP

120

130

130

140

140

 

INT

0.67

0.84

1

1.17

1.34

 

PP

120

120

120

120

120

 

PT

12.5

12.5

7.5

7.5

7.5

 

COTI

19

19

19

19

19

 

COTE

25

25

25

25

25

 

MOO

5

5

5

5

5

 

MET

275

275

275

275

275

 

MOTE

70

70

70

70

70

Quality index (forecasting)

Warp.

0.0964

0.0930

0.0906

0.0997

0.1065

 

Shrink.

0.1397

0.1076

0.0814

0.0436

0.0126

 

Vol.

1.9019

1.9418

2.0416

2.3027

2.5442

Quality index (real)

Warp.

0.0796

0.0720

0.0780

0.0820

0.1114

 

Shrink.

0.1232

0.9250

0.0835

0.0597

−0.0948

 

Vol.

2.0650

1.9610

2.0680

2.8260

3.3670

To verify the efficiency of five DMUs found in this paper, we re-run Moldflow on each DMU and then the results are compared with those of 34 = 81 data points which are utilized to set up the regression equations in ‘Setting up the regression response model to create the complete dataset’ subsection. The comparison is accomplished by executing DEA on 5 efficient DMUs found in this paper and 81 data points. The results are shown in Tables 8 and 9. From Table 9, it can be observed that among five efficient DMUs found in this paper, three DMUs are still on the efficient frontier and one DMU is relatively highly efficient with 94.18% DEA score. Only DMU 1,186 is not quite efficient with 71.28% DEA score, and this may be due to the error of the regression equation at this DMU. It is fair to suggest that the error induced by the regression equation at most of the points is fairly small. Therefore, the efficient frontier of process parameters found by this paper with only 108 (=27 + 81) repeats of experiments can really provide good combinations of process parameters for decision making.
Table 8

Relatively efficient DMUs

DMU

Score

44

100

45

100

840

100

871

100

1,151

100

54

97.82

52

97.80

53

97.13

43

96.88

560

94.19

DMUs in italics represent the efficient ones suggested in this study.

Table 9

Efficient DMUs with positive counts

DMU

Reference count

840

62

1,151

12

871

9

45

9

44

1

Summary and conclusions

Part quality for plastic injection molding is often evaluated by multiple interrelated quality indices, and each quality index is highly related with process parameters. This paper proposes a method of finding the complete efficient frontier of process parameters with only a few times of experiments when multiple quality indices are considered for plastic injection molding. The thin front cover of a digital camera is provided as the example of executing the method. Based on the literature, nine process parameters are considered in this research. The experimental design with the Taguchi orthogonal L27 is used to run the experiment on Moldflow. ANOVA is then executed to find significant parameters to affect the part’s quality indices, and the results show that four out of nine parameters are significant with the significant level 0.05. In order to set up the complete efficient frontier of DEA analysis, more data are required, and the regression equations are used to create them. To have good accuracy of the multiple regressed equations, the complete experimental design with 34 times (only four significant process parameters are considered) of experiments is again executed on Moldflow. The multiple regression equations are then set up and are used to produce the dataset for DEA analysis. The results of DEA analysis shows that the five combinations are on the efficient frontier.

To show the efficiency of these combinations suggested in this paper, DEA analysis is again conducted on them as well as the results of the experiments of 34 times used for establishing multiple regression equations. The results show that only one combination is not as efficient mainly because of the error of the regressed equations at this combination. Hence, the method proposed here is believed indeed can find the efficient frontier of process parameters with only a few times of experiments.

The classic DEA method, CCR, is used in this paper; in the future, some other DEA methods, such as BCC, can be used, and the performance of each method can be compared. Another possible future research topic is to evaluate the performance of Moldflow analysis.

Appendix

The results of L27 and 81 are presented in Tables 10 and 11.

Table 10

Experimental results of L27 array

 

N1991

N1962

N1955

N2261

E1

E2

E3

E4

Volume

1

0.2222

0.2118

0.2151

0.203

0.4014

0.1292

0.3728

0.171

4.384

2

0.2137

0.2027

0.2077

0.1953

0.3856

0.1226

0.3598

0.1619

4.346

3

0.2185

0.2082

0.2126

0.2001

0.395

0.1273

0.3681

0.1666

4.379

4

0.1436

0.1322

0.1392

0.125

0.2532

0.0529

0.2346

0.0913

3.281

5

0.1513

0.1416

0.1466

0.133

0.2675

0.0643

0.2475

0.1006

3.468

6

0.1416

0.1301

0.137

0.1223

0.2491

0.0516

0.2308

0.0896

3.154

7

0.1032

0.0902

0.098

0.0821

0.173

0.0035

0.1579

0.0379

3.936

8

0.0927

0.0798

0.0877

0.0693

0.1497

−0.0153

0.1321

0.0187

4.095

9

0.0986

0.0867

0.0947

0.0783

0.165

−0.0014

0.1503

0.0341

3.973

10

0.1602

0.1564

0.1612

0.1521

0.2877

0.0609

0.2717

0.1046

4.337

11

0.1455

0.1416

0.1412

0.1356

0.262

0.0564

0.2432

0.0959

3.938

12

0.1414

0.1365

0.1354

0.1301

0.2526

0.0504

0.2309

0.0896

3.989

13

0.0697

0.0593

0.0776

0.0559

0.083

−0.0475

0.0739

−0.013

2.063

14

0.0677

0.0593

0.0718

0.0542

0.091

−0.0379

0.0797

−0.0041

2.018

15

0.0735

0.0618

0.0824

0.0575

0.0873

−0.0491

0.0779

−0.0126

2.086

16

0.1367

0.1358

0.1243

0.1288

0.2529

0.0685

0.2254

0.1026

3.593

17

0.1352

0.134

0.1249

0.1289

0.2506

0.0647

0.2261

0.0994

3.369

18

0.131

0.1303

0.1192

0.1246

0.243

0.059

0.2161

0.0937

3.526

19

0.1446

0.1537

0.1601

0.1062

0.2521

0.0288

0.2252

0.0535

7.185

20

0.1339

0.1483

0.1488

0.1027

0.247

0.0297

0.2139

0.0475

6.509

21

0.1216

0.1423

0.144

0.1021

0.2177

0.0251

0.1904

0.0434

6.474

22

0.1563

0.1809

0.1834

0.1507

0.3036

0.0629

0.2963

0.0983

7.855

23

0.15

0.1893

0.1672

0.1452

0.3118

0.0546

0.2718

0.0897

7.12

24

0.1786

0.2015

0.1828

0.1496

0.35

0.0686

0.3018

0.106

7.41

25

0.0673

0.0611

0.0939

0.0325

0.0558

−0.0288

0.0621

−0.0166

3.294

26

0.0951

0.0826

0.1094

0.051

0.1015

−0.0014

0.1003

−0.0039

3.055

27

0.0794

0.0698

0.0991

0.0393

0.0651

−0.0164

0.0728

−0.0138

3.21

Table 11

Moldflow execution results

 

Warpage

Shrinkage

Volume shrinkage at ejection

1

0.2185

0.395

4.379

2

0.2182

0.3944

4.378

3

0.2169

0.3922

4.364

4

0.1611

0.2867

3.611

5

0.161

0.2863

3.61

6

0.1601

0.2847

3.596

7

0.1154

0.1973

2.724

8

0.1156

0.1969

2.723

9

0.1145

0.1951

2.706

10

0.2006

0.3635

4.08

11

0.2003

0.3629

4.079

12

0.1998

0.3622

4.078

13

0.1451

0.2556

3.288

14

0.1452

0.2551

3.287

15

0.1448

0.2545

3.286

16

0.0982

0.1632

2.25

17

0.098

0.1629

2.251

18

0.0978

0.1626

2.249

19

0.2271

0.4148

4.56

20

0.2264

0.4134

4.516

21

0.2253

0.4117

4.503

22

0.1612

0.2931

3.683

23

0.1607

0.2919

3.642

24

0.1596

0.2905

3.627

25

0.1111

0.1965

2.683

26

0.1103

0.195

2.642

27

0.1092

0.1931

2.264

28

0.1952

0.3589

4.869

29

0.195

0.3589

4.879

30

0.1981

0.3646

4.958

31

0.1472

0.265

4.016

32

0.1471

0.2651

4.023

33

0.1487

0.2679

4.073

34

0.1068

0.1776

3.158

35

0.107

0.1776

3.163

36

0.1084

0.1805

3.2

37

0.136

0.2522

3.551

38

0.1335

0.2447

3.509

39

0.1332

0.2471

3.506

40

0.0982

0.1743

2.926

41

0.096

0.1696

2.882

42

0.0957

0.1691

2.883

43

0.078

0.0835

2.068

44

0.0775

0.0779

2.048

45

0.0778

0.0772

2.054

46

0.1528

0.2846

3.924

47

0.153

0.2847

3.939

48

0.1517

0.2827

3.921

49

0.1068

0.197

3.146

50

0.1074

0.1973

3.155

51

0.1063

0.1952

3.138

52

0.0674

0.1021

2.015

53

0.0678

0.1027

2.028

54

0.0677

0.1007

2.014

55

0.2068

0.3438

7.623

56

0.1449

0.2523

3.853

57

0.2079

0.3452

6.844

58

0.1528

0.2605

4.782

59

0.1465

0.2469

4.573

60

0.1712

0.2751

6.899

61

0.1617

0.2321

7.921

62

0.1614

0.2345

6.631

63

0.1618

0.2355

8.349

64

0.1954

0.3269

7.239

65

0.1214

0.2301

3.745

66

0.1751

0.3106

6.779

67

0.1362

0.1936

4.24

68

0.1366

0.194

4.355

69

0.1352

0.1922

4.284

70

0.108

0.0802

3.177

71

0.1049

0.0744

3.16

72

0.1049

0.0746

3.165

73

0.1021

0.1438

3.199

74

0.1003

0.1455

3.186

75

0.1004

0.1425

3.197

76

0.1011

0.078

3.2

77

0.0982

0.0732

3.206

78

0.0981

0.0721

3.21

79

0.1165

−0.063

3.203

80

0.1137

−0.0656

3.22

81

0.1132

−0.0662

3.224

Notes

Acknowledgements

The authors would like to thank financial support from the research project 98-2221-E-029-019, National Science Council of Taiwan. The authors are grateful to the expert anonymous reviewers and the editor-in-chief whose comments and suggestions considerably improved this article.

Supplementary material

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© Chen et al.; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Computer Science and Information ManagementProvidence UniversityTaichungRepublic of China (Taiwan)
  2. 2.Department of Industrial Engineering and Enterprise InformationTunghai UniversityTaichungRepublic of China (Taiwan)

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