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Using multiobjective evolutionary algorithms in the optimization of operating conditions of polymer injection molding.

INTRODUCTION

Injection molding of polymeric materials is a high throughput process adequate to manufacture thermoplastic components of complex geometry with tight dimensional tolerances. This process is an intricate, dynamic, and transient process, involving convoluted melting-flow-pressure-solidification phases and a complex material behavior strongly affecting the quality and properties of the final molded component.

In injection molding, the thermomechanical environment imposed to the polymer melt is controlled by the definition of the operative processing variables (e.g. plasticating temperatures, cycle times, injection, and holding pressures) and/or system geometry (e.g., plasticating screw, injection gate location, water lines layout). These thermomechanical conditions control the microstructure and morphology of the final molded component (1), (2), which determines, in turn, their dimensions (shrinkage), dimensional stability (distortion and warpage), and properties (e.g., mechanical behavior, permeability, appearance) (3), (4). If the thermomechanical history variables (pressure, temperature, flow and cooling rate) can be monitored directly or indirectly in the impression, the molded product properties can be accurately and consistently predicted.

In recent research, the pressure at the impression has been regarded as the most important parameter to establish a correlation with the dimensions and the weight of the molded part (5), been considered a finger print of the process (6). Figure 1 shows a typical pressure evolution inside the mould impression and its main characteristics (7).

[FIGURE 1 OMITTED]

The shape of this pressure profile is strongly affected by some changes in the injection molding process, such as temperature, flow rate, holding pressure and time. The establishment definition of the adequate processing conditions, to mould a high quality plastic component, is therefore a complex task because of the significant number of processing variables, to the high level of interactions between these variables and to the numerous molding features and end-use properties to improve.

The use of computer simulations on the design stages of engineering plastic components molded by injection molding is very frequent (8), (9). Initially, a finite element mesh representative of the part geometry is defined, the materials are selected, the gate location is defined and the initial processing variables are introduced. Then, after launching the simulation the outputs arc analyzed. A trial-and-error process is applied, where the initial conditions, in what concerns geometry, material and/or processing conditions are modified until the desired results arc obtained. This process can be very complex since in most cases multiple criteria are to be optimized simultaneously. Also, the finding of a global optimum solution is not guaranteed.

Various strategies using different methodologies to optimize the injection molding process have been reported in the literature (10-19). Seow and Lam (10) proposed a method for flow path generation using the Moldflow software. This algorithm was combined with an optimization routine based on hill-climbing approximation method perform automatic cavity balancing. Yarlagadda and Khong (11) applied an Artificial Neural Network (ANN) to optimize the injection time and injection pressure improve the quality of the part produced. Lotti and Bretas (12) applied ANN to predict the morphology and the mechanical properties of an injection molded part as a function of the processing conditions (mold and melt temperatures and flow rate). A central composite design of experiments approach was used to predict the molding morphology as a function of the processing conditions using the Mold-Flow software. Castro et al. (13) used ANN and statistical analysis to find the optimal compromise between multiple performance measures (minimization of maximum injection pressure, time to freeze, range of volumetric shrinkage, maximum shear stress at the wall and bulk temperature range) to define the setting of injection molding variables (melt temperature, mold temperature, ejection temperature) and the gate location. Kim et al. (14) applied a Genetic Algorithm (GA) in the optimization of some injection molding process parameters, (e.g., mold temperature, melt temperature, and filling time) using the results of a flow simulation code. The performance of the process was quantified using a weighted sum of the temperature difference, "overpack" and frictionally overheating criteria. Peic and Turng (15) proposed an integrated CAE optimization tool coupling a process simulation program with various optimization algorithms, such as. Sequential Quadratic Programming (SQP), Self-adaptive Evolution (SAE), Differential Evolution (DE) and Simulated Annealing (SA), to determine the optimal design and process variables. Alain and Kamal (16) suggested that the maximum difference in shrinkage among the parts is an appropriate measure of product quality in the runner-balancing problem. They proposed a new approach to runner balancing, which accounts for product costs by including runner system volume and cycle time. The optimization problem was solved with a multiobjective genetic algorithm (NSGA-II). Lam et al. (17) proposed an approach to optimize both the cooling channel design and the process conditions through the use of an Evolutionary Algorithm (EA). The aim was to optimize the coordinates of centers of the cooling channels, the sizes of the cooling channels, the flow rate, and the inlet temperature in each channel, the packing time, the cooling time and the clamp open time in order to obtain the most uniform cavity surface temperature. Ozcelik and Erzurumlu (18) proposed a hybrid optimization methodology based on ANN and GA in order to minimize the warpage of thin shell plastic parts. The parameters used are mold temperature, melt temperature, packing pressure, packing time, runner type, gate location, and cooling time. A predictive model, based on ANN, for warpage was created using these process parameters. ANN model was interfaced with an effective genetic algorithm to find the optimum process parameter values.

The main limitations of the above methodologies are twofold. First, most of the approaches proposed either consider one objective or use an aggregation function of the various objectives. This is not the better way to deal with the multiobjective nature of these optimization problems as this implies the aggregation in a single function of very different measures with different meanings. Second, the methodologies proposed are not totally integrated, i.e., they do not consider the optimization and modeling processes as a whole process.

Therefore, the development of a global and integrated optimization methodology able to facilitate the definition of the processing window and the maximization of the molding properties is of high importance. In particular, the aim of this work is to apply this methodology to the proper setting of the processing variables to the manufacturing of injection molded components (process optimization).

This was the strategy followed in a previous paper (19), where an automatic optimization methodology based on multiobjective evolutionary algorithm (MOEA) for the maximization of the mechanical response of injection moldings was used. For that purpose an MOEA was linked to an injection molding simulator code (C-mold). This allows the optimization of the processing conditions (injection flow rate, melt temperature, and mould temperature for a desired morphological state or for an enhanced mechanical response. In this article, instead of using as objectives the morphology parameters, different objectives such as temperature difference, shrinkage, pressure, cycle time, and pressure work are used.

In this work, an automatic optimization methodology based on MOEA (20) to define the processing window in injection molding is proposed in multiobjective evolutionary algorithms section. For that purpose a MOEA is linked to an injection molding simulator code (in this case C-MOLD). In the case study section the proposed optimization methodology was used to set the processing conditions of the molding in polystyrene (STYRON 678E). The optimization results obtained have been validated experimentally. Finally, the optimization results achieved by the global optimization of the operating conditions in polymer injection molding are presented in Results section.

MULTIOBJECTIVE EVOLUTIONARY ALGORITHMS

The methodology proposed in this work integrates the computer simulations of the injection molding process, an optimization methodology based on evolutionary algorithm and multiobjective criteria to establish the set of operative processing variables leading to high quality molded part. The optimization methodology adopted is based on a Multiobjective Evolutionary Algorithms (MOEA), because of multiobjective nature of most real optimization problems, where the simultaneous optimization of various, often conflicting, criteria is to be accomplished (20-23). The solution must then result from a compromise between the different objectives. Generally, this characteristic is taking into account using an approach based on the concept of Pareto frontiers (i.e., the set of points representing the trade-off between the objectives) together with an MOEA. This enabled the simultaneously accomplishment of the several solutions along the Pareto frontier, i.e., the set of nondominated solutions (21), (22), (23).

The link between the MOEA and the problem to solve is made in two different steps (see flowchart of Fig. 2). First, the population is random initialized, where each individual (or chromosome) is represented by the binary value of the set of all variables. Then, each individual is evaluated by calculating the values of the relevant objectives using the modeling routine (in this case C-MOLD). The interface between the MOEA and C-MOLD is made trough a command line and the data is exchanged trough text data files. Finally, the remaining steps of a MOEA are to be accomplished. To each individual is assigned a single value identifying its performance on the process (fitness). This fitness is calculated using a Multiobjective approach as described in detail elsewhere (20), (24). If the convergence objective is not satisfied (e.g., a predefined number of generations), the population is subjected to the operators of reproduction (i.e., the selection of the best individuals for crossover and/or mutation) and of crossover and mutation (i.e., the methods to obtain new individuals for the text generation). The RPSGAe uses a real representation of the variables, a simulated binary crossover, a polynomial mutation and a roulette wheel selection strategy (20), (24).

[FIGURE 2 OMITTED]

The main advantages of a MOEA methodology are the capacity of adaptation without changes to the optimization of different processes, the capacity of optimizing different types of decision variables and objectives simultaneously and the capacity of using black-box modeling routines without the need to know details about the processes to be optimized. However, it requires the need of a high number of evaluations of the objective functions (i.e., the use of the modeling routine), which can increase considerably the computation time.

CASE STUDY

The optimization methodology proposed will be used for setting the processing conditions of the molding shown in Fig. 3, which will be molded in polystyrene (STYRON 678E). The relevant polymer properties used for the flow simulations were obtained from the software (C-MOLD) database. The models for the simulations, implemented in the C-MOLD simulation package are based on a hybrid finite-element/finite-difference/control-volume numerical solution of the generalized Hele-Shaw flow of a compressible viscous fluid under nonisothermal conditions. The rheological and PVT description therein implied is obtained by the Cross-WLF and the Tait modified equation, respectively. More details about the software are described in the company technical and related literature (25-28).

[FIGURE 3 OMITTED]

The geometric model of the plate for the simulations consisted of 408 triangular elements. The runner and impression mesh used in the simulations is shown in Fig. 4.

[FIGURE 4 OMITTED]

The simulations considered the mold filling and holding (post-filling) stages. A node near the P1 position, pressure sensor position (see Fig. 3) was selected as a reference point for this study. The processing variables to optimize and allowed to vary in the simulations in the following intervals were: injection time, [t.sub.inj] [member of] [0.5; 3] s (corresponding to flow rates of 24 and 4 [cm.sup.3]/s, respectively), melt temperature, [T.sub.inj] [member of] [180; 280[degrees]C], mold temperature, [T.sub.w] [member of] [30; 70[degrees]C], holding pressure, Ph [member of] [7; 38] % of maximum machine injection pressure, with fixed switchover point, [S.sub.[F/P]], at 99%, timer for hold pressure of 15 s and cooling time of 15 s.

For the results produced by the modeling program two process restrictions were imposed: (i) the molding has to be completely filled, obviously no short-shots were admitted and (ii) the computed values of the maximum shear stress and strain-rate were limited to their critical values (defined on the C-MOLD database) to avoid potential defects (e.g., melt fracture).

The objectives used are the following:

The temperature difference on the molding at the end of filling, dT = ([T.sub.max] - [T.sub.min]), was minimized to avoid part distortions and warpage because of different local cooling rates, dT [member of] [0; 20[degrees]C];

The volumetric shrinkage (VS) of the moldings, defined as:

VS = [[[D.sub.mould] - [D.sub.part]]/[D.sub.mould]] x 100 (1)

where [D.sub.mould] is the dimension of the mould and [D.sub.part] is the corresponding part dimension, was minimized, VS [member of] [0; 15%];

The maximum cavity pressure was minimized, reducing the clamping force, [P.sub.max] [member of] [1; 70] MPa;

The cycle time was minimized, increasing productivity, [t.sub.c] [member of] [30; 35] s;

The pressure work (PW), defined as the integral of pressure, P, along the time, t:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

has to be minimized to diminish the residual stress, the energy consumption and to reduce the mechanical efforts supported by the equipment, PW [member of] [0; 200] MPa s.

Since the aim of this work is to study the applicability of the optimization methodology in the establishment of the adequate processing conditions in the injection molding process, taking into account the various criteria identified above, the 20 optimization runs identified in Table 1 were carried out.

The first studies (runs 1 to 9) addressed only the simultaneous optimization of pairs of objectives. Then we considered three objectives (runs 10 to 19) and we will present the three dimensional Pareto frontiers of these runs. Finally, we consider the simultaneous optimization of the five objectives and a five-dimensional Pareto frontier in the objectives domain is obtained.
TABLE 1. Objectives used in each process optimization run.

Run       Optimized criteria

 1   PW and VS
 2   PW and dT
 3   PW and [P.sub.max]
 4   PW and [t.sub.c]
 5   VS and dT
 6   VS and [P.sub.max]
 7   VS and [t.sub.c]
 8   [t.sub.c] and dT
 9   [t.sub.c] and [P.sub.max]
10   PW, VS and dT
11   PW, VS and [P.sup.max]
12   PW. VS and [t.sub.c]
13   PW. dT and [P.sub.max]
14   PW, dT and [t.sub.c]
15   PW, [P.sub.max] and [t.sub.c]
16   VS, dT and [P.sub.max]
17   VS, dT and [t.sub.c]
18   VS, [P.sub.max] and [t.sub.c]
19   [t.sub.c], dT and [P.sub.max]
20   All objectives


The RPSGAe was applied using the following parameters: 50 generations, crossover rate of 0.8, mutation rate of 0.05, internal and external populations with 100 individuals (except in the optimization of 3 and 5 objectives where 200 individuals are used), limits of the clustering algorithm set at 0.2 and [N.sub.Ranks] at 30. These values resulted from a carefule analysis made in a previous paper (20).

RESULTS

Experimental Assessment

First the optimization methodology proposed was validated experimentally. A single optimization run using two different objectives, the minimization of pressure work and of volumetric shrinkage, has been carried out. These objectives were chosen because of the possibility of measuring them experimentally on the available injection machine. The pressure was measured using a piezoelectric sensor from Kistler with the reference 6190A0,4 Six points were chosen from the Pareto frontiers obtained for the initial and the final populations of the EA (Fig. 5). These solutions were experimentally tested in an injection molding machine and the corresponding results were compared with the computational ones (Fig. 6). As can be seen the general behavior (i.e., its relative location) of the experimental and computational solutions is very similar, the differences being due to the capacity of the simulation program (C-MOLD) in reproducing the reality.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

Figure 7 shows the simulation vs. experimental pressure curves of the six points chosen. The experimental results are very similar to those of the simulation except on the case of point 3, which confirms the results shown in Fig. 6.

[FIGURE 7 OMITTED]

The deviation of point 3 is because of the higher pressure in cavity that occurs when is applied a higher holding pressure. This results in a residual pressure in the cavity, mainly if a material with lower shrinkage is molded. This could be due to the elastic deformation of mould that is not considered in commercial software's like C-MOLD (8).

Optimization Considering Two Objectives

Figure 8 illustrates the trade-off between each objective of runs 1 to 4 against pressure work. The optimization algorithm is able to evolve during the 50 generations and to produce a good approximation to the Pareto frontier in all the cases.

[FIGURE 8 OMITTED]

As expected, pressure work conflicts with volumetric shrinkage of the moldings. Interestingly, cycle time ([t.sub.c]), maximum pressure ([P.sub.max]) and temperature difference (dT) slightly decrease with pressure work (PW).

In run 1, PW and VS are simultaneously optimized (PW-VS plot of Fig. 8). The PW values range between 29.7 and 198.8 MPa s and VS between 1.1 and 3%. The resulting optimized Pareto frontier is highly nonlinear: the highest variations of PW occur for their lowest values (29.7-70 MPa s); the highest variations of VS are obtained for their highest values (2-3%). For the extreme optimized points, the following setting of processing variables should be used: [T.sub.w] at 30-31[degrees]C and the injection time at 2.58-3 s. The processing variables allowed to vary in a wide range are: the melt temperature that can be adjusted between 207-260[degrees]C; and the holding pressure at 7-29.2% of maximum value of the machine injection pressure. When [T.sub.inj] is at 222.7[degrees]C, Ph should be set at 29.17%. In these conditions, VS is minimized (low melt viscosity) and PW is maximum (high holding pressure).

Run 2 optimized simultaneously PW and dT (PW-dT plot of Fig. 8). The optimized pressure work and dT varies only from 21.9 to 41.9 MPa s and 1.3-2.5[degrees]C, respectively. These conditions are met for: the low setting of holding pressure (Ph = 7% of maximum value of the machine injection pressure) and for mold temperatures between 30 and 38[degrees]C. The injection time and melt temperature can be varied in a wider range: [t.sub.inj] between 0.5 and 1.5 s and [T.sub.inj] between 211.8 and 260.5[degrees]C. As the injection time is lower (higher [Q.sub.inj]) the melt temperature is higher and more uniform. Consequently, dT is minimum. Conversely, for high injection time (low the injection flow rate) the melt cools more quickly during injection stage, resulting in a higher temperature variation in the molding. Hence dT is maximum.

In run 3, both PW and [P.sub.max] were optimized at the same time (PW - [P.sub.max] plot of Fig. 8). The PW and [P.sub.max] values range slightly from 21.9 to 32.9 MPa s and 10.3-14.4 MPa, respectively. These objectives are simultaneously met for the adjustment on the mold temperature and holding pressure on their minimal values (30[degrees]C and 7% of maximum value of the machine injection pressure, respectively). The processing variables allowed to vary in a wide range are: the melt temperature that can be adjusted between 210 and 261[degrees]C; and the injection time, which can be set between 1.5 and 3 s. When [T.sub.inj] is at 210[degrees]C, [t.sub.inj] should be set at 1.5 s. In these conditions [P.sub.max] is maximum (high melt viscosity and high pressure) and PW is minimized because cycle time is also minimized because of high cooling rate.

Finally, run 4 optimized simultaneously PW and [t.sub.c] (PW - [t.sub.c] plot of Fig. 8). The PW values range between 23.2 and 68.7 MPa s and [t.sub.c] between 30.6 and 31 s. These conditions are met for: the low setting of injection time that ranges between 0.5 and 0.8 s and for holding pressure between 7 and 8.6% of maximum value of the machine injection pressure. The melt and mold temperatures can be varied in a wider range: [T.sub.inj] between 203 and 266[degrees]C and [T.sub.w] between 31 and 68[degrees]C. When [T.sub.inj] is at 266[degrees]C and [T.sub.w] at 68[degrees]C their maximum values achieved in this optimization, [t.sub.c] is minimized (high cooling rate) and PW is maximum because of lower melt viscosity that promote a better packing of the material during holding phase, hence increasing the pressure.

Figure 9 illustrates the trade-off between each objective of runs 5 to 7 against volumetric shrinkage and Fig. 10 illustrates the trade-off between each objective of runs 8 and 9 against cycle time. A physical meaning analysis similar to those of runs 1 to 4 can be done to understand the relationship between the objectives involved on runs 5 to 9.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

Optimization Considering Three Objectives

Figures 11 and 12 illustrate the trade-off between each objective of runs 10 to 15. Similar data was also obtained for the other runs.

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

In run 10, PW, VS, and dT are simultaneously optimized (dT - VS - PW plot of Fig. 11). The PW values range between 23.2 and 199.6 MPa s, VS between 1.2 and 3.1% and dT between 1.4 and 9.6[degrees]C. The highest variations occur in the objectives PW and dT. If pressure work is considered as the most important objective (i.e., the point with the lower pressure work) we have that VS = 3.1% (the highest volumetric shrinkage value) and dT = 2[degrees]C (a reasonable lower value for temperature difference on the molding at the end of filling). In this case the injection molding machine must operate with injection time of 1.2 s, melt and mold temperatures of 206[degrees]C and 31.5[degrees]C, respectively, and holding pressure at 7% of maximum value of the machine injection pressure. Therefore, this solution is unsatisfactory when an objective such as volumetric shrinkage is considered. In other hand, if dT is considered as the most important objective (i.e., the point with the lower temperature difference on the molding at the end of filling) we can consider either PW = 187.9 MPa s and VS = 1.6% or PW = 46.4 MPa s and VS = 2.7%. In the first case, PW assume a high value and VS a lower value and injection machine must operate with injection time of 0.53 s, melt and mold temperatures of 265[degrees]C and 30.7[degrees]C, respectively, and holding pressure at 22.1% of maximum value of the machine injection pressure. In the second case, PW is lower and VS is high with injection machine operating with injection time of 0.51 s, melt and mold temperatures of 267[degrees]C and 32.2[degrees]C, respectively, and holding pressure at 7.7% of maximum value of the machine injection pressure. Hence, the first solution is unsatisfactory relativity to pressure work and the second solution is unsatisfactory relativity to volumetric shrinkage.

Run 11 optimized simultaneously PW, [P.sub.max] and VS (VS - [P.sub.max] - PW plot of Fig. 11). The PW values range between 22.8 and 195.3 MPa s, [P.sub.max] between 10.4 and 40.7 MPa and VS between 1.2 and 3.1%. The highest variations occur in the objective's PW and [P.sub.max]. If pressure work is considered as the most important objective (i.e., the point with the lower pressure work) we have that [P.sub.max] = 13.9 MPa (a reasonable lower value for maximum mold pressure) and VS = 3.1% (the highest volumetric shrinkage value). In this case the injection molding machine must operate with injection time of 1.74 s, melt and mold temperatures of 213[degrees]C and 37.7[degrees]C, respectively, and holding pressure at 7% of maximum value of the machine injection pressure. This solution is unsatisfactory relativity to volumetric shrinkage. If maximum cavity pressure is considered as the most important objective (i.e., the point with the lower maximum pressure) we have that PW = 31.5 MPa s and VS = 2.7% (reasonable values for both objective's). In this case the injection molding machine must operate with injection time of 2.97 s melt and mold temperatures of 252[degrees]C and 33.1[degrees]C, respectively, and holding pressure at 7% of maximum value of the machine injection pressure.

In run 12, PW, [t.sub.c] and VS are simultaneously optimized (VS - [t.sub.c] - PW plot of Fig. 11). The PW values range between 23.6 and 197.2 MPa s, [t.sub.c] between 30.7 and 33.2 s and VS between 1.2 and 3%. The highest variations occur in the objective's PW and [t.sub.c]. If pressure work is considered as the most important objective (i.e., the point with the lower pressure work) we have that [t.sub.c] = 31.3 s (a reasonable lower value for cycle time) and VS = 3% (the highest volumetric shrinkage value). In this case the injection molding machine must operate with injection time of 1.1 s, melt and mold temperatures of 208[degrees]C and 39.6[degrees]C, respectively, and holding pressure at 7% of maximum value of the machine injection pressure. This solution is unsatisfactory relativity to volumetric shrinkage. If cycle time is considered as the most important objective (i.e., the point with the lower cycle time) we have that PW = 159.8 MPa s (high value for pressure work) and VS = 2% (reasonable value for volumetric shrinkage). In this case the injection molding machine must operate with injection time of 0.5 s, melt and mold temperatures of 270[degrees]C and 49.5[degrees]C, respectively, and holding pressure at 17.1% of maximum value of the machine injection pressure. This solution is unsatisfactory relativity to pressure work.

In run 13, PW, [P.sub.max] and dT are simultaneously optimized (dT - [P.sub.max] - PW plot of Fig. 12). The PW values range between 21.8 and 48.1 MPa s, [P.sub.max] between 10.3 and 14.3 MPa and dT between 1.2 and 14.8[degrees]C. The highest variations occur in the objective dT. If temperature difference on the molding at the end of filling is considered as the most important objective (i.e., the point with the lower temperature difference on the molding at the end of filling) we have that [P.sub.max] = 10.8 MPa (a reasonable lower value for maximum mold pressure) and PW = 48.1 MPa s (the highest pressure work value). In this case the injection molding machine must operate with injection time of 0.52 s, melt and mold temperatures of 264 and 62.9[degrees]C, respectively, and holding pressure at 7% of maximum value of the machine injection pressure. This solution is unsatisfactory relativity to pressure work.

Run 14 optimized simultaneously PW, [t.sub.c] and dT (dT -[t.sub.c] - PW plot of Fig. 12). The PW values range between 22.1 and 169.9 MPa s, [t.sub.c] between 30.7 and 31.7 s and dT between 1.2 and 3[degrees]C. The highest variations occur in the objective PW. If pressure work is considered as the most important objective (i.e., the point with the lower pressure work) we have that [t.sub.c] = 31.7 s (the highest cycle time value) and dT = 2.4[degrees]C (a reasonable value for temperature difference on the molding at the end of filling). In this case the injection molding machine must operate with injection time of 1.53 s, melt and mold temperatures of 210[degrees]C and 31.3[degrees]C, respectively, and holding pressure at 7.1% of maximum value of the machine injection pressure. This solution is unsatisfactory relativity to cycle time.

Finally, run 15 optimized simultaneously PW, [t.sub.c] and [P.sub.max] ([P.sub.max] - [t.sub.c] - PW plot of Fig. 12). The PW values range between 21.9 and 160.1 MPa s, [t.sub.c] between 30.7 and 33.1 s and [P.sub.max] between 10.3 and 25.8 MPa. The highest variations occur in the objective's PW and [t.sub.c]. If pressure work is considered as the most important objective (i.e., the point with the lower pressure work) we have that [t.sub.c] = 32 s and [P.sub.max] = 14.7 MPa (reasonable values for both objective's). In this case the injection molding machine must operate with injection time of 1.79 s, melt and mold temperatures of 210 and 31.4[degrees]C, respectively, and holding pressure at 7% of maximum value of the machine injection pressure. If cycle time is considered as the most important objective (i.e., the point with the lower cycle time) we have that PW = 160.1 MPa s and [P.sub.max] = 25.8 MPa (highest values for both objective's). In this case the injection molding machine must operate with injection time of 0.5 s, melt and mold temperatures of 269 and 51.7[degrees]C, respectively, and holding pressure at 17% of maximum value of the machine injection pressure. This solution is unsatisfactory relativity to pressure work and maximum mold pressure.

In Fig. 13 a comparison between the Pareto plots of an optimization run with two objectives (run 2) with the 2D Pareto plots of optimization runs (runs 10, 13, and 14) considering three objectives. As expected, when a third objective is introduced the number of nondominated solutions increase, i.e., the size of the optimal region increases. It is clear that this makes difficult the selection of the single solution to be used on the process.

[FIGURE 13 OMITTED]

Optimization Considering All Objectives

Run 20 consider the simultaneous optimization of the five objectives. In this case, a five-dimensional Pareto frontier in the objectives domain is obtained (Fig. 14). It is difficult to choose a solution from this multidimensional Pareto frontier as the location of one possible solution in the various graphical representations is not evident. However, one must remember that a table identifying all the Pareto solutions is available. The decision maker must define a working point (or region) in the objectives domain curves (Fig. 14) and select the corresponding solution in the parameters to optimize domain (Fig. 15).

[FIGURE 14 OMITTED]

[FIGURE 15 OMITTED]

For example, if pressure work is considered as the most important objective, point P1 in Fig. 14 (i.e., the point with the lower pressure work) can be defined and the corresponding solution chosen in Fig. 15. In this case, the injection molding machine must operate with injection time of 1.7 s (corresponding to flow rate of 7.06 [cm.sup.3]/s), melt and mold temperatures of 210 and 36[degrees]C, respectively, and holding pressure at 7% of maximum machine injection pressure. This is done using the tabular form of the solutions represented in Fig. 15. This set of processing variables leads to relative high volumetric shrinkage equal to 3% and cycle time of 31.9 s. The maximum mold pressure, [P.sub.max], and temperature difference on the molding at the end of filling, dT, achieve low values equal to 14.5 MPa and 3.2[degrees]C, respectively. Finally, pressure work assume the minimum value possible equal to 23 MPa s. Therefore, it is clear that this solution is unsatisfactory when objectives such as cycle time and volumetric shrinkage are considered.

Alternatively, if volumetric shrinkage is considered as the most important objective, point P2 in Fig. 14 (i.e., the point with the lower volumetric shrinkage) can be chosen, and the corresponding solution (Fig. 15) selected (injection time of 2.4 s, melt and mold temperatures of 224 and 32[degrees]C, respectively, and holding pressure of 28% of maximum value of the machine injection pressure). This set of processing variables leads to a high pressure work (PW = 198 MPa s) and maximum cavity pressure ([P.sub.max] = 40 MPa). Also, the cycle time and temperature difference on the molding at the end of filling have relative high values ([t.sub.c] = 32.5 s and dT = 7.7[degrees]C), but volumetric shrinkage is minimum (VS = 1.2%). In this case, this solution is unsatisfactory when objectives such as pressure work and maximum cavity pressure are considered. A similar study can be done for the other points shown in Fig. 14.

Hence, the selected objectives cannot be all optimized simultaneously, but graphical and tabular Pareto frontiers are a powerful tool enabling the decision maker to select different solutions representing different compromises between the objectives considered.

CONCLUSIONS

In this work, a multiobjective optimization methodology based on Evolutionary Algorithms (MOEA) was applied to the optimization of the operating conditions of polymer injection molding process. The algorithm is able to take into account the multiple objectives used and, with a single run, to obtain a complete trade-off of solutions. The results obtained showed that the optimization methodology proposed is able to find solutions with physical meaning. Also, the experimental results produced validated the proposed optimization methodology.

The multiobjective optimization methodology proposed here is an excellent tool to deal with problems, like the one studied here, where no knowledge about the process to optimize exist. This methodology is able, in a single run, to establish trade-off between the different parameters of the process, the decision variables, and the objective spaces. Due to these characteristics, the MOEA can be applied without big changes in the optimization of complex processes such has the ones where the operating conditions and, for example, the runner system are considered simultaneously.
NOMENCLATURE

ANN            Artificial neural networks

[D.sub.mould]  Dimension in the mould in i direction

[D.sub.part]   Dimension in the part in i direction

dT             Temperature difference on the molding at the end of
               filling

EA             Evolutionary algorithm

[F.sub.i]      Fitness

GA             Genetic algorithm

MOEA           Multi-objective evolutionary algorithm

[N.sub.Ranks]  Number of ranks

P1             Node at 40 mm of the gate on the injection molding part

P2             Node at 110 mm of the gate on the injection molding part

[P.sub.max]    Maximum mold pressure

PW             Pressure work

[Q.sub.inj]    Injection flow rate

RPSGAe         Reduced Pareto set genetic algorithm with elitism

[S.sub.F/P]    Switch-over point

[t.sub.2]P     Holding time

[t.sub.c]      Cycle time

[t.sub.inj]    Injection time

[T.sub.inj]    Melt temperature

[T.sub.max]    Maximum bulk temperature at the end of filling

[T.sub.min]    Minimum bulk temperature at the end of filling

[T.sub.w]      Mold temperature

VS             Volumetric shrinkage


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C. Fernandes, A.J. Pontes, J.C. Viana, A. Gaspar-Cunha

Department of Polymer Engineering, IPC - Institute for Polymer and Composites, University of Minho, Campus de Azurem, Guimaraes 4800-058, Portugal

Correspondence to: A. Gaspar-Cunha; e-mail: agc@dep.uminho.pt

Contract grant sponsor: Portuguese Fundacao para a Ciencia e Tecnologia; contract grant number: SFRH/BD/28479/2006.

Published online in Wiley InterScience (www.interscience.wiley.com).

[C] 2010 Society of Plastics Engineers

DOI 10.l002/pen.21652
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Author:Fernandes, C.; Pontes, A.J.; Viana, J.C.; Gaspar-Cunha, A.
Publication:Polymer Engineering and Science
Article Type:Report
Geographic Code:4EUPR
Date:Aug 1, 2010
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