UNIVERSIDAD AUTÓNOMA DE NUEVO LEÓN

FACULTAD DE INGENIERÍA MECÁNICA Y ELÉCTRICA

División de Posgrado en Ingeniería de Sistemas

Serie de Reportes Técnicos

Reporte Técnico PISIS-2004-04

Simultaneous Optimization of Mold Design and Processing Conditions in Injection Molding

Carlos E. Castro1

Mauricio Cabrera Ríos2

Blaine Lilly1,3

José M. Castro3

(1) Department of Mechanical Engineering

Ohio State University Columbus, Ohio, EUA

(2) Programa de Posgrado en Ingeniería de Sistemas

FIME, UANL E-mail: mcabrera@uanl.mx

(3) Department of Industrial, Welding, and Systems Enginering

Ohio State University Columbus, Ohio, EUA

08 / Septiembre / 2004

© 2004 by División de Posgrado en Ingeniería de Sistemas

Facultad de Ingeniería Mecánica y Eléctrica Universidad Autónoma de Nuevo León

Pedro de Alba S/N, Cd. Universitaria San Nicolás de los Garza, NL 66450

México

Tel/fax: +52 (81) 1052-3321 E-mail: pisis@yalma.fime.uanl.mx

Página: http://yalma.fime.uanl.mx/~pisis/

Simultaneous Optimization of Mold Design and Processing Conditions in Injection Molding

Carlos E. Castro1, Mauricio Cabrera Ríos3, Blaine Lilly1,2, and José M. Castro2

1 Department of Mechanical Engineering and 2Departement of Industrial, Welding & Systems Engineering

The Ohio State University Columbus, Ohio, 43210

3 Graduate Program in Systems Engineering

Universidad Autónoma de Nuevo León San Nicolás de los Garza, Nuevo León, México, 66450

Abstract

Injection molding (IM) is the most prominent process for mass-producing plastic

products. One of the biggest challenges facing injection molders today is to determine

the proper settings for the IM process variables. Selecting the proper settings for an IM

process is crucial because the behavior of the polymeric material during shaping is highly

influenced by the process variables. Consequently, the process variables govern the

quality of the part produced. The difficulty of optimizing an IM process is that the

performance measures (PMs), such as surface quality or cycle time, that characterize the

adequacy of part, process, or machine to intended purposes usually show conflicting

behavior. Therefore, a compromise must be found between all of the PMs of interest. In

this paper, we present a method comprised of Computer Aided Engineering, Artificial

Neural Networks, and Data Envelopment Analysis (DEA) that can be used to find the

best compromises between several performance measures. The approach discussed here

also allows for the identification of robust variable settings that might help to define a

starting point for negotiation between multiple decision makers.

1

Introduction

Injection Molding (IM) is the most prominent process for mass-producing plastic

parts. According to the Society of the Plastics Industry, over 75% of all plastics

processing machines are IM machines, and close to 60% of all plastics processing

facilities are injection molders [1]. Selecting the proper IM process settings is crucial

because the behavior of the polymeric material during shaping is highly influenced by the

process variables. Consequently, the process variables govern the quality of the part

produced. A substantial amount of research has been directed towards determining the

process settings for the IM process as well as the optimal location of the injection gate.

The challenge of optimizing an IM process is that the performance measures often

show conflicting behavior when they are functions of process or design variables in

common. For example, the cycle time and the part warpage will both be affected by the

ejection temperature. Increasing the ejection temperature would be favorable for

minimizing cycle time. However, letting the part cool to a lower ejection temperature

before demolding would decrease the part warpage. Therefore, a compromise must be

found between these two performance measures to set the ejection temperature. For this

reason, when optimizing an IM process it is nearly impossible to find one best solution.

However, it is feasible to determine a set of best compromises between multiple PMs.

The problem of considering several PMs simultaneously, i.e. finding the best

compromises, is referred to as a multiple criteria optimization. Conventional methods of

multiple criteria optimization involve combining individual weighted PMs into one

objective function and optimizing that function. These methods will converge to a

2

solution; however it might prove a challenge to determine if this solution lies in the

efficient frontier, especially in the case where the PMs show nonlinear behavior. In

addition, this solution is dependent on the bias of the user defining the weights. In

engineering practice it is often times impossible to define one optimal solution to all

criteria. Instead, it is both feasible and attractive to determine the best compromises

between PMs: that is the combinations of PMs that cannot be improved in one single

dimension without harming another. Data Envelopment Analysis (DEA) provides an

unbiased way to find these efficient compromises.

It is the purpose of this paper to demonstrate the determination of efficient solutions

(best compromises) in an IM context through a series of case studies comprising several

potential industrial applications. These solutions prescribe the settings for IM process and

design variables. Additionally, the identification of robust solutions is discussed.

The Optimization Strategy

Proposed by Cabrera-Rios, et al [2, 3] the general strategy to find the best

compromises between several PMs consists of five steps:

Step 1) Define the physical system. Determine the phenomena of interest, the

performance measures, the controllable and non-controllable variables, the experimental

region, and the responses that will be included in the study.

Step 2) Build physics-based models to represent the phenomena of interest in the system.

Define models that relate the controllable variables to the responses of interest. If this is

not feasible, skip this step.

3

Step 3) Run experimental designs. Create data sets by either systematically running the

models from the previous step, or by performing an actual experiment in the physical

system when a mathematical model is not possible.

Step 4) Fit metamodels to the results of the experiments. Create empirical expressions

(metamodels) to mimic the functionality in the data sets.

Step 5) Optimize the physical system. Use the metamodels to obtain predictions of the

phenomena of interest, and to find the best compromises among the PMs for the original

system. The best compromises are identified here through DEA.

In the method outlined here, the metamodels are empirical approximations of the

functionality between the controllable (independent) variables, and the responses

(dependent variables). These metamodels are used either for convenience or for

necessity. Because DEA as it is used here requires that many response predictions be

made, it is more convenient to obtain these predictions from metamodels rather than more

complicated physics-based models. In addition, when physics-based models are not

available to represent the phenomena of interest, the use of metamodels becomes

essential.

Data Envelopment Analysis (DEA)

Cabrera-Rios et al [2,3] have demonstrated the use of DEA to solve multiple criteria

optimization problems in polymer processing. DEA, a technique created by Charnes,

Cooper, and Rhodes [4], provides a way to measure the efficiency of a given combination

of PMs relative to a finite set of combinations of similar nature. The efficiency of each

combination is computed through the use of two linearized versions of the following

mathematical programming problem in ratio form:

4

free

nj

T

T

T

T

jTj

T

T

T

0

min0

min0

min0

max

min0

0max0

0

,...,11

,,

µ

ε

ε

µ

µ

µ

1Yνν

1Yνµ

Yν

Yµ

s.t.Yν

YµMaximize

toµνFind

⋅≥

⋅≥

=≤+

+

where, and are vectors containing the values of those PMs current

analysis to be maximized and minimized respectively, µ is a vector of multiplier

PMs to be maximized, ν is a vector of multipliers for the PMs to be minimized

max0Y min

0Y

scalar variable, n is the number of total combinations in the set, and ε is a ve

constant usually set to a value of 1x10-6. The solutions deemed efficient by

linearized versions of the model shown above represent the best compromise

(finite) set of combinations of PMs. A complete description of the linearization p

as well as the application of this model can be found in any of the references 1 th

Determination of settings of process variables and injection po

Consider the part shown in Figure 1. This part, which we introduced in

works [5,6,7], represents a case where the location of the weld lines is critical

part flatness plays a major role. The part is to be injection molded using a Sumi

machine using PET with a fixed flow rate of 9cc/s. Nine PMs were include

(1)

(2)

(3)

(4)

(5)

ly under

s for the

, µ0 is a

ry small

the two

s in the

rocedure

rough 5.

int

previous

, and the

tomo IM

d in this

5

study: (1) maximum injection pressure, PI , (2) time to freeze, tf, (3) maximum shear

stress at the wall, SW, (4) deflection range in the z-direction, RZ, (5) time at which the

flow front touches hole A, tA, (6) time at which the flow front touches hole B, tB, (7) time

at which the flow front touches the outer edge of the part, toe, (8) the vertical distance

from edge 1 to the weld line, d1, and (9) the horizontal distance from edge 2 to the weld

line, d2. For production purposes it is desirable to minimize PI , tf, SW, and RZ : PI to keep

the machine capacity unchallenged, tf to reduce the total cycle time, SW to minimize

plastic degradation, and RZ to control the part dimensions. It is desirable to maximize tA,

tB, toe, d1, and d2 : tA, tB, toe in order to minimize the potential for leakage, and d1 and d2 to

keep the weld lines away from corners which were assumed to be areas of stress

concentration.

Figure 1: Part of constant thickness with cutouts.

6

Five controllable variables were varied at the levels shown in Table 1 in a full

factorial design. These controllable variables include: (a) the melt temperature, Tm, (b)

the mold temperature, Tw, (c) the ejection temperature, Te, (d) the horizontal coordinate of

the injection point, x, and (e) the vertical coordinate of the injection point, y. Te was only

varied at two levels because a preliminary study showed that a third level did not add any

meaningful variation. The injection point location is constrained to be in the region

shown in Figure 1, due to limitation of the IM machine. This point will be characterized

by the variables x and y in a Cartesian coordinate system with its origin at the lower left

corner of the part.

Table 1: Levels of each of the controllable variables for the initial dataset

Tm Tw Te x y Label C C C cm cm

-1 260 120 149 15 10 0 275 130 159 20 17.5 1 290 140 25 25

A finite element mesh of the part was created in MoldflowTM in order to obtain

estimates for the performance measures. An initial dataset was obtained from the full

factorial design. Following with the general optimization strategy, this initial dataset was

used to create metamodels to mimic the behavior of each the performance measures. In

general, it is favorable to fit a simple model to the data. In this study, second order linear

regressions were initially considered as models for the performance measures. When

simple models do not suffice, then more complicated models, in this case ANNs, become

necessary. In general the ANNs outperformed the second order linear regression for

every performance measure in terms of approximation quality and prediction capability,

and were therefore used to obtain predictions for each PM at previously untried

7

combinations of controllable variables. The results for the performance of the regression

models and the ANNs obtained can be found Table 2.

Table 2: Summary of performance and results from residual analysis results for the regression metamodels

The complete multiple criteria optimization problem originally posed for this case

contained all nine performance measures. To solve the optimization problem, it was

necessary to generate a large number of feasible level combinations of the controllable

variables. This was achieved by varying Tm and Tw at five levels, and the rest of the

variables at three levels within the experimental region of interest (see Table 1) in a full

factorial enumeration. This experimental design resulted in a total of 675 combinations.

The results after applying DEA were that over 400 of the 675 combinations were found

to be efficient. Such a large number of efficient combinations can be explained by

examining Table 3, which summarizes the results of the analysis of variance of each PM

in regression form. Notice that the last five PMs are only dependent on the injection point

position determined by variables x and y. Any specific combination of values (x*,y*) will

8

give the same result on all of these five PMs regardless of the values that the rest of the

other controllable variables Tm, Tw, and Te take. Having used a full factorial enumeration

with x and y at three levels, it follows that we can obtain only nine different values for

these five PMs, but each of the nine specific combinations (x,y) have in fact 75

combinations of the rest of the controllable variables. In the high dimensionality of the

problem, this elevated amount of repetition results in a large number of efficient

solutions. In order to increase the discrimination power i.e. obtain fewer efficient

solutions, one can solve the DEA model shown in Eqs. 1 through 5 by setting µ0 equal to

zero. The resulting model is similar to the Charnes-Cooper-Rhodes (CCR) DEA model

[8].

Table 3: The significant sources of variation (linear, quadratic and second order interaction terms in the linear regression metamodel) to each performance measure.

9

Using the simple modification described above, the number of efficient combinations

comes down to 149. It can be shown that these combinations are a subset of those 400

plus found previously. These efficient combinations are shown in terms of the PMs in

Figure 2.

Figure 2: Levels of the PMs that corresponded to the efficient solutions when all nine were included

It is important to notice that we can exploit the information our methods gave us

about the functionality of the PMs in order to tailor the optimization problem. To

illustrate, five sub cases were defined for practical applications of the conceptual part

shown in Figure 1: (i) an excess capacity injection machine application, (ii) a

dimensional quality and economics critical application, (iii) a structural part application,

10

(iv) a part quality critical application, and (v) a case including PMs that are only

dependent on the injection location [7].

Excess Capacity Injection Molding Machine:

For a case in which the injection-molding machine has excess capacity, it would be

possible to not consider the maximum injection pressure in the optimization problem. For

simplicity, in this case SW, tA, tB, and toe were also dropped from the optimization, leaving

four performance measures. The DEA model was again solved here by setting the

constant µ0 equal to zero in order to improve the discrimination power of DEA. The

functionality shown in Table 3 called for inclusion of all variables, and the factorial

enumeration with 675 combinations was used. In this case, fourteen combinations were

found to be efficient. Figure 3 shows the levels of the PMs for the efficient solutions.

The compromise between the locations of the weld lines is evident. A noticeable

compromise also arises between tf and Rz. This is an understandable compromise,

because the two depend oppositely on the ejection temperature.

11

Figure 3: Efficient solutions for the excess machine capacity application in terms of the levels of the PMs

considered.

Figure 4 shows the locations of the injection gate for the efficient solution. The

positions in this case help to define ‘attractive’ areas to locate the injection port, since

they tend to cluster in specific sections. In this case the efficient injection locations

clustered along right and bottom edges. The three PMs that are affected by the location

of the injection gate are the weld line positions and the deflection in the z-direction. The

additional PM here is the time to freeze, which is not affected by the injection location

according to the analysis of variance.

12

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

X

Y

Figure 4: Injection Locations of the efficient solutions to the excess machine capacity application

transformed to fall between -1 and 1.

Table 4 shows the values for all of the controllable variables at the efficient solutions.

Notice that Tw and Tm were at 120 and 260 degrees Celsius respectively for all of the

efficient solutions. In industrial practice, if the PMs involved in this case were the only

ones of interest, this would be a good indication that Tm and Tw should be set at these

temperatures. Also notice that the ejection temperature values of the efficient solutions

vary over the entire range. According to the analysis of variance, d1 and d2 do not depend

on the ejection temperature, so this fact must be due to the compromise between Rz and tf

previously mentioned.

13

Table 4: Efficient Solutions for the excess machine capacity application

x y T w T m T e t f Rz d 1 d 2cm cm C C C s mm mm mm25 25 120 260 149 20.89 0.0005 37.9 130.925 25 120 260 154 18.99 0.002 37.9 130.925 25 120 260 159 17.27 0.007 37.9 130.925 25 120 260 149 22.71 0.000 37.9 131.425 17.5 120 260 149 20.90 0.001 94.9 107.825 17.5 120 260 154 19.00 0.005 94.9 107.825 17.5 120 260 159 17.28 0.010 94.9 107.825 17.5 120 260 149 22.72 0.001 94.9 108.915 10 120 260 149 20.92 0.001 124.6 67.415 10 120 260 154 19.02 0.005 124.6 67.415 10 120 260 159 17.30 0.011 124.6 67.425 10 120 260 149 20.92 0.001 124.7 82.125 10 120 260 154 19.02 0.006 124.7 82.125 10 120 260 159 17.30 0.012 124.7 82.1

Controllable Variables Performance Measures

Dimensional Quality and Economics Critical Application:

In this case it was assumed that the economic concerns included minimizing the cycle

time and keeping the machine capacity untested in order to have long machine life and

smaller power consumption. These two concerns are defined by tf and PI respectively. Rz

defines the dimensional quality. The analysis of variance shows that all of the

controllable variables affect at least one of these PMs, so the enumeration with 675

combinations again was applied. Twenty-five efficient solutions were found. Since the

problem is three-dimensional the efficient frontier can be visualized. The efficient points

are shown in Figure 5 with respect to the rest of the data set.

14

Figure 5: A Visualization of the efficient frontier of the economics critical and dimensional Application

Figure 6 shows the efficient solutions in terms of the levels of the PMs. The direct

compromise between the time to freeze and deflection is confirmed here. Notice that

they follow opposite trends while it is favorable to minimize both.

15

Figure 6: Efficient solutions for the dimensional quality and economic application in terms of the levels of

the PMs considered.

Figure 7 shows the locations of the injection gate for the efficient solutions. This

case contradicts the first case. In the large machine capacity case, the ‘attractive’ areas

for the injection gate were found at the bottom and right edges of the feasible area, but in

this case, the top edge and bottom left corner proved to be the efficient locations. This is

due to the fact that the positions of the weld lines were not considered in this case. From

these results we can conclude that d1 and d2 are the main drivers for keeping the injection

location on the right or bottom edge. They are the only PMs affected by x and y that

were included in the first case and not in this case.

16

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

X

Y

Figure 7: Injection Locations of the efficient solutions to the dimensional quality and economics critical

application transformed to fall between -1 and 1.

Table 5 shows the twenty-five combinations of the controllable variables that proved

to be efficient for the dimensional quality and economics critical application. Eighteen

out of the twenty-five efficient solutions had the injection gate located at the upper left

corner of the feasible region, which is close to the center of the part. This is the most

robust injection location for this application. According to the analysis of variance, PI is

affected by the location of the injection gate. Locating the injection gate towards the

center would favorably decrease PI. Since d1 and d2 were not included in this case there

were no negative effects of moving the injection gate towards the center.

17

Table 5: Efficient Solutions for the dimensional quality and economics critical application

x y T w T m T e P I t f R z

cm cm C C C MPa s mm15 25 140 290 159 9.35 26.7 0.009815 25 140 290 149 9.35 37.9 0.001515 25 140 282.5 159 9.55 23.8 0.009915 25 140 282.5 149 9.55 32.0 0.000915 25 140 275 149 9.75 27.4 0.000615 25 140 275 159 9.75 21.9 0.010115 25 140 267.5 159 9.96 20.4 0.010215 25 140 267.5 149 9.96 24.5 0.000615 25 140 260 154 10.17 20.9 0.004515 25 140 260 159 10.17 19.1 0.010115 25 140 260 149 10.17 22.7 0.000615 25 130 275 149 12.25 26.5 0.000515 25 125 260 149 14.69 21.4 0.000515 25 125 260 159 14.69 17.8 0.009515 25 120 275 149 16.00 25.5 0.000415 25 120 260 154 16.61 19.0 0.003215 25 120 260 149 16.61 20.9 0.000515 25 120 260 159 16.61 17.3 0.009025 25 140 260 149 17.46 22.7 0.000425 25 135 260 149 19.43 22.4 0.000420 25 120 260 159 26.92 17.3 0.007125 25 120 260 159 27.16 17.3 0.006525 25 120 260 154 27.16 19.0 0.002015 10 120 282.5 149 28.50 29.8 0.000315 10 120 275 149 29.21 25.5 0.0003

Controllable Variables Performance Measures

A Structural Application

In this application, the PMs included were the vertical distance from edge 1 to the

weld line, d1, and the horizontal distance from edge 2 to the weld line, d2. The location of

weld lines is considered critical to design a structurally sound part. From the analysis of

variance, it was known that these PMs depended only on the position of the injection

gate, characterized by variables x and y. In order to avoid the repetition described in the

full set, a new dataset was created by varying x and y at nine levels creating a finer

sampling grid for the injection location. The rest of the variables were set to a value in the

18

middle of their respective ranges. The levels of the controllable variables for this dataset

are shown in Table 6. The total number of combinations of controllable variables in this

dataset was 81.

Table 6: Levels of controllable variables used for the dataset for x,y dependent PMs

he efficient frontier for this two-dimensional case is shown here in Figure 8.

T m T w T e x yC C C cm cm

130 275 154 15 1016.25 11.87517.5 13.75

18.75 15.62520 17.5

21.25 19.37522.5 21.25

23.75 23.12525 25

T

40

50

60

70

80

90

100

110

120

130

20 30 40 50 60 70 80 90 100 110 120 130

Weld Position 1 (mm)

Wel

d Po

sitio

n 2

(mm

)

140

Figure 8: Visualization of the efficient frontier in the structural application

19

The seven efficient solutions for a structural part are shown in Figure 9 in terms of

the levels of the two PMs in increasing order of d1. The compromise between the

positions of the weld lines is confirmed. We want to maximize both of the weld line

positions, but where one of them is at a maximum, the other is at a minimum.

Figure 9: Efficient Solutions for the structural application in terms of the weld line positions d1 d d2.

Figure 10 shows the positions of the injection gate corresponding to the seven best

com

an

promises. The entire area shown is the feasible injection region. In this case the

‘attractive’ clusters occur at the bottom right corner of the feasible injection area and

along the right edge of the feasible injection region. These results tend to agree with the

large machine capacity case. Since the locations of the weld lines are independent of the

other controllable variables, any of these x,y pairs would obtain the same results for d1

and d2 regardless of the temperature levels. In this case the efficient solutions are defined

20

by the injection location, so temperature levels are not shown. In other words, had the

temperatures been left at the maximum or minimum of their respective feasible ranges,

we would have arrived at the same results for the locations of the weld lines.

1.0

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0X

Y

on tFigure 10: Injection Locations of the seven efficient solutions to the structural applicati ransformed to

fall between -1 and 1.

A Quality Critical Application

e defined as related to the position of the weld lines and

the

Appearance in our test part w

flatness of the part, i.e. d1, d2, and RZ. From the analysis of variance in Table 3 it is

known that d1 and d2 depend only on the x and y position of the injection gate. However,

the temperatures cannot be disregarded in this case, because RZ depends on all three of

them. Therefore, we used the factorial enumeration already created for all the variables

(x, y, Tm, Tw, and Te) with 675 combinations. The resulting sixteen efficient solutions are

shown in Figure 11 with respect to the rest of the dataset. Notice that the data is

21

organized into columns. The different columns illustrate the repetitions that were

referred to earlier. Each of the columns corresponds to one x,y pair, and the variation in

height of the data points in these columns is determined by the controllable temperatures.

Since only one other PM was involved, Rz, this repetition did not cause a problem.

Figure 11: Visualization of the efficient frontier for the part quality application

Figure 12 shows the efficient solutions with respect to the values of the PMs in

increasing order of d1. The compromise between d1 and d2 is again evident.

22

Figure 12: Efficient solutions for the part quality application in terms of the position of the weld lines and

deflection range.

Figure 13 analyzes the clusters of the design variables x and y. Again, the entire

space shown is the feasible area for the injection gate. This case did not use the same

fine grid for the injection location that was used in the structural application, so the

‘attractive’ clusters are not as well defined. However, it is evident that the right and

bottom edges would be the best areas to locate the injection gate. This case agrees with

the previous cases of the large machine capacity, and the structural application.

23

Figure 13: Injection Locations of the seven efficient solutions to the part quality application transformed to

fall between -1 and 1.

Table 7 shows the levels of the controllable variables that correspond to the efficient

combinations of PMs. Notice that for all of the efficient solutions, the value of Te was

149 degrees C. Allowing the part to cool to a lower ejection temperature favorably

affects the part deflection in the z-direction. In this case, the time to freeze was not

considered. Allowing the part to cool longer did not introduce any negative effects, so

the efficient ejection temperature was always at the minimum of the range.

24

Table 7: Efficient solutions for the quality critical application

X Y Tm Tw T e Rz d 1 d 2cm cm C C C mm mm mm15 10 130 260 149 0.0003 109.7 64.415 10 135 260 149 0.0003 109.7 64.720 10 120 260 149 0.0007 124.6 67.420 10 130 260 149 0.0008 124.6 77.825 10 120 260 149 0.0009 124.7 82.125 10 125 260 149 0.0009 124.7 82.525 17.5 125 260 149 0.0008 94.9 108.925 25 125 260 149 0.0004 37.9 131.425 10 130 260 149 0.0011 124.7 82.925 17.5 130 260 149 0.0009 94.9 110.025 25 130 267.5 149 0.0005 37.9 131.625 25 130 275 149 0.0006 37.9 131.625 10 135 260 149 0.0016 124.7 83.325 17.5 135 260 149 0.0013 94.9 110.925 10 140 260 149 0.0023 124.7 83.625 17.5 140 260 149 0.0020 94.9 111.8

Controllable Variables Performance Measures

Injection Location Dependent Performance Measures:

From the results of the analysis of variance we can see that there are some PMs that

are dependent only on the location of the injection gate. These PMs, tA, tB, toe, d1, and d2,

were considered in a separate case that is only concerned with determining the location of

the injection gate. For this case the factorial enumeration of the levels of the controllable

variables shown in Table 6 was used. Again because of the high dimensionality of this

case, a DEA model with µ0 equal to zero was used. The resulting fourteen efficient

solutions are shown in Figure 14. Notice the compromises between the time to touch

hole A and the time to touch the outer edge. The peaks of these two PMs always contrast

each other. On the other hand, the trend of the time to touch hole B follows a similar

path as the time to touch hole A. As observed before the compromise between the weld

line positions is evident.

25

0

20

40

60

80

100

120

140

0 2 4 6 8 10 12 14

Efficient Compromises

Wel

d Po

sitio

n 1

and

2 (m

m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

time

to to

uch

(s)

Position Weld Line 1 Position Weld Line 2time to touch hole A time to touch hole Btime to touch outer edge

Figure 14: Efficient solutions for the case of determining the injection location in terms of the levels of the

PMs considered.

The efficient gate locations for this case are shown in Figure 15. This case agreed

with some of the earlier cases. The ‘attractive’ clusters for the injection gate occurred

along the bottom and right edges. Only a few new injection locations resulted from

introducing the flow times into this case on top of the weld line locations, which were

previously considered by themselves in the structural applications. Additionally, these

new injection locations are still in the same general area. This implies that generally, the

flow times as a group do not introduce definite compromises with respect to the location

of the injection gate with the locations of the weld lines. Here the efficient solutions are

only dependent on x and y, so the levels of the temperatures are not shown.

26

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

X

Y

Figure 15: Injection Locations of the efficient solutions for the application considering only x,y dependent

PMs

Analysis of Robust Solutions

The discussion of the different cases in the previous section leads to an additional

analysis: finding robust efficient solutions. Robust solutions can be found within the

individual cases, and some of those were discussed previously. As it can be inferred, a

robust efficient solution is a combination of controllable variable settings that remains

efficient when analyzing different subsets of performance measures. It is also beneficial

to determine which solutions were robust on a large scale, i.e.which combinations of

process variables were deemed efficient in several subsets of optimization. Indeed for

this case it was possible to identify that the combination of (x, y, Tm, Tw, Te) = (20 cm, 10

cm, 120 oC, 260 oC, 149 oC) is a robust efficient solution.

Determining a suitable location for the injection gate is a crucial decision. The

temperatures at which the process is run can be adjusted easily. On the other hand, there

27

is only one chance to decide where the injection gate will be located. Selecting the

proper location from the start can save time and money. In this study, it was identified

that the injection gate location at the top right corner of the feasible injection area (x=20

cm, y=25 cm) is a robust solution. This injection location was found in efficient solutions

for all but one of the subsets, and it was very close to the ‘attractive’ area in the subset in

which it did not appear.

This analysis might help to establish a ‘common ground’ among multiple decision

makers, to then move to the kind of compromises that can be taken when presented with

the rest of the efficient solutions [7].

Conclusions and Future Work

Finding the settings of process and design variables in Injection Molding has been an

active area of research. In this work, the coordinated use of CAE, statistics, neural

networks, and data envelopment analysis has been demonstrated to find these settings in a

multiple objective optimization context. The optimization of a virtual part was presented

for discussion, and several sub cases were defined to further the details of practical

applications in the industry. The analyses presented in this paper are geared to make

informed decisions on the compromises of several performance measures. These analyses

also allow for the identification of robust variable settings that might help to define a

starting point for negotiation between multiple decision makers.

Future work will include adding information about the variability of PMs on the DEA

analysis and the determination of process windows with efficiency considerations.

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References

[1] The Society of the Plastics Industry (2003). spi: Plastics Data Source, SPI, available at: www.plasticsindustry.org [accessed in July 2004]

[2] Cabrera-Rios, M., Zuyev, K., Chen, X., Castro, J.M., and Straus, E.J. Polymer Composites, Optimizing Injection Gate Location and Cycle Time For the In-Mold Coating (IMC) Process, 23:5 (2002) 723-738

[3] Cabrera-Rios, M., Mount-Campbell, C.A., and Castro J. M., Journal of Polymer Engineering, Multiple quality criteria optimization in in-mold coating (IMC) with a data envelopment analysis approach, 22:5 (2002) 305-340

[4] Charnes, A., W.W., and Rhodes, E. European Journal of Operational Research,

Measuring the efficiency of decision making units, 2:6 (1978) 429-444 [5] Castro, C.E., Cabrera-Rios, M., Lilly, B., Castro, J.M., and Mount-Campbell,

C.A. Journal of Integrated Design & Process Science, Identifying the best compromises between multiple performance measures in injection molding (IM) using data envelopment analysis (DEA), 7:1 (2003) 77-87

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[8] Charnes, A., Cooper, W.W., Lewin, A.Y., and Seiford, L.M. Boston, Kluwer

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