A Fuzzy Multiobjective Approach for Minimization of Injection Molding Defects.

M. M. F. YUEN [1]

This paper describes a fuzzy multiobjective optimization approach for determining the set-points of the injection molding processing parameters to minimize the defects formed on the molded parts. The seventies of the defects are represented by membership functions using the fuzzy set theory. The minimization of these membership functions, which is a multiobjective optimization problem, is transformed into a substitute problem. The preference function in the substitute problem is original and is proposed specifically for characterizing the quality requirements of the injection molding defects. The formulated optimization problem is solved with design of experiments, in which the process behavior is approximated empirically by a set of quadratic polynomials that can be easily optimized. Experimental results are presented to emphasis the workability of the proposed methodology.

1. INTRODUCTION

Injection molding is a cyclic process involving three major stages: filling, holding and cooling. The most important processing parameters include coolant temperature, nozzle and barrel temperature, injection speed, holding pressure, holding time and cooling time. Inappropriate setting of these parameters leads to the formation of defects on the injection molded part. Determination of the appropriate set-points for the processing parameters is an important task in injection molding operation.

Many expert systems [1-4] have been proposed to assist in determining the set-points for the processing parameters in view of the defects formed on the molded parts. These expert systems are practical because they can provide timely suggestions for correcting a wide coverage of defects. Despite of its practical usefulness, the expert system approach still suffers from two major limitations. First, the knowledge stored in the expert systems is qualitative and general in nature and is inadequate for situations requiring a quantitative value for the processing parameters. Second, the rules in the expert systems usually deal with a single defect and it is rare for them to handle a multiple-defects situation.

Another approach to determine the set-points for the processing parameters is the application of optimization techniques. One of the difficulties in formulating the optimization problem is defining the objective function, which often involves optimizing several defects with conflicting process requirements. Lee and Kim [5] avoided the difficulty by evaluating only one defect, warpage, using an injection molding software package, C-MOLD. Pandelidis and Zou [6, 7] defined the objective function by quantifying the major causes of warpage and material degradation as a weighted sum of: 1) end-of-fill temperature difference over the part, 2) the percentage of elements satisfying the over-packing criteria, and 3) the percentage of frictional overheated elements. These quantities were evaluated using the flow simulation package Moldflow, and the objective function was minimized by the sequential unconstrained minimization technique. Choi et al. [8] defined the objective function as a weighted sum of the cycle time a nd the squares of the deviations of the outputs from their corresponding desired values. The outputs included the variance of linear shrinkage and the sink index, which were evaluated using C-MOLD. Tan and Yuen [9] attempted to treat the optimization problem as a multiobjective one using the fuzzy goal programming approach. The requirements of minimizing the objective functions and satisfying the constraints were expressed as a set of fuzzy goals with linear type membership functions. The fuzzy goals were evaluated using C-MOLD. Seaman et al. [10] used another approach to treat the optimization problem as a multiobjective one. In their approach, the severities of the multiple defects were not aggregated into a single objective function, but were left as a vector, and tradeoff decisions were made by the operator interactively during the optimization process.

To address the set-point determination problem, it is paramount to define the molding defects effectively for the purpose of process optimization. Injection molding defects are descriptions in linguistic form, which can be considered as quality attributes or vague variables. Attempts were made to provide a more objective and accurate method to evaluate the seventies of the defects by associating them with some measurable quantities. For example, the severity of sink mark was referenced by its depth, which could be measured with a surface roughness measuring machine [11] or with a micrometer [12]. The severity of flow mark was referenced by the length and depth of the wave, which was recorded with a surface roughness measuring machine [13]. With these measurement techniques, some of the defects can be considered as continuous variables. However, measurement techniques still cannot totally replace visual inspection in view of the effectiveness and simplicity of the latter method. As a result, in order to formu late the optimization problem, one should be able to define the severities of the defects, which may be represented as quality attributes, vague variables, or continuous variables.

The present study proposes a fuzzy multiobjective approach to minimize the injection molding defects. In this approach, a unified scheme, which is based on the fuzzy set theory, is proposed to represent the severities of the defects. These severities are then aggregated into a single objective function which must be minimized. The form of the objective function is original and is proposed specifically for representing the quality requirements of the injection molding defects. The formulated optimization problem is solved with design of experiments, in which the process behavior is approximated empirically by a number of quadratic polynomials. These polynomials can be optimized easily. Experimental results are presented to emphasis the workability of the proposed methodology.

2. DESCRIPTION OF THE METHODOLOGY

2.1 Fuzzy Representation of Injection Molding Defects

A fuzzy set [14] Y is associated to a reference set [Omega] by a mapping:

[[mu].sub.Y] : [Omega] [rightarrow] [0, 1].

For all y [epsilon] [Omega], [[mu].sub.Y](y) is interpreted as the membership value of y in the fuzzy set Y and is called a membership function. It gives a direct model for the vague categories of natural language, defined on an objective base which could be a numerical scale [15]. For example, the vague categories can be "sink mark," and the objective base can be "a set of sink mark depths." [[mu].sub."sink mark"](depth) then expresses how much the value of the depth is compatible with the concept of "sink mark." If the category is simply defined on an objective linear reference scale, then a membership function on [Omega] defines an ordering "[geq]" on the elements of [Omega]. "[y.sub.1] [geq] [y.sub.2]" means "[y.sub.1] is more Y than [y.sub.2]".

In this study, the linear membership function is chosen over other forms for its simplicity and effectiveness. If a defect Y can be referenced by a measurable quantity y, then the corresponding linear membership function is defined by the following equation:

[[mu].sub.Y](y) = {0 if y [leq] [y.sup.0]

[frac{y - [y.sup.0]}{[y.sup.1] - [y.sup.0]}] if [y.sup.0] [less than] y [leq] [y.sup.1]

1 if [y.sup.1] [less than] y (1)

This linear membership function can be determined by specifying two points ([y.sup.0], 0) and ([y.sup.1], 1), where [y.sup.0] corresponds to the value of y with membership value of zero, and [y.sup.1] corresponds to the value of y with membership value of one. The graph of the linear membership function is illustrated in Fig. 1. The value of [y.sup.0] is selected such that the defect Y is considered to be corrected if this value is achieved, while the value of [y.sup.1] is selected such that the defect Y is considered to be very serious if this value is violated. In this study, the minimum and maximum values of y within the operating region are selected for the values of [y.sup.0] and [y.sup.1] respectively.

Some of the defects may be difficult to be referenced by a measurable quantity, but can be evaluated effectively with visual inspection. In this case, the membership values of these defects can be rated by the operator directly, and this type of defects is classified as observable. Some examples of measurable defects and observable defects are given in Table 1. The fuzzy representation scheme for the measurable and observable defects is illustrated in Fig. 2.

2.2 Multiobjective Optimization Formulation 

Minimizing the membership values of the defects is a multiobjective optimization problem as represented by the following equation:

Minimize [[mu].sub.[Y.sub.i]]([underline{x}]), j = 1 to n

Subject to [underline{x}] [epsilon] X (2)

where [underline{x}] is the vector defining the processing parameters, X is the set defining the operating region of the processing parameters, [Y.sub.j] (j = 1 to n) is the jth defect and [[mu].sub.[Y.sub.j]] is the corresponding membership value. Two forms of optimality after Sakawa [16] is reviewed for the multiobjective optimization formulation:

Complete Optimality: [[underline{x}].sup.*] is said to be complete optimal solution if and only if there exists [[underline{x}].sup.*] [epsilon] X such that the [f.sub.j] ([[underline{x}].sup.*]) [leq] [f.sub.j] ([underline{x}]), j = 1 to n, for all [underline{x}] [epsilon] X.

Pareto Optimality : [[underline{x}].sup.*] is said to be Pareto optimal solution if and only if there does not exist another [underline{x}] [epsilon] X such that [f.sub.j] ([underline{x}]) [leq] [f.sub.j] ([[underline{x}].sup.*]) for j = 1 to n, and [f.sub.j] ([underline{x}]) [neq] [f.sub.j] ([[underline{x}].sup.*]) for at least one j.

In the above definitions, [f.sub.1], [f.sub.2], [ldots], [f.sub.n] are the objective functions to be minimized.

A complete optimal solution simultaneously minimizes all the objective functions and cannot be improved any further, while a Pareto optimal solution can be improved but only at the expense of any one objective function. A complete optimal solution may not be available if the objective functions conflict to each others, and in this case, only a Pareto optimal solution is possible. The nature of the injection molding process makes the Pareto optimality a clear choice for the solution scheme.

Since the set of Pareto optimal solutions is generally infinite, the required Pareto optimal must be characterized based on the preference of the decision maker. One approach to characterize the required Pareto optimal is to transform the multiobjective optimization problem into a substitute problem by using a single objective function, called a preference function [17]. Various methods for the transformation can be found in the literature, for example in [16, 17]. In this study, the following substitute problem is proposed:

Minimize F([underline{x}]) = [[[sum].sup.R].sub.k=1] {[P.sub.k] [[[[sum].sup.[P.sub.k]].sub.j=1] [[mu].sub.[Y.sub.kj]] + P [[[sum].sup.[n.sub.k]].sub.j=[p.sub.k]+1] [[mu].sub.[Y.sub.kj]]]}

Subject to [underline{x}] [epsilon] X (3)

where F([underline{x}]) is the preference function. The details of F([underline{x}]) are explained as follows:

1. The defects are ranked into R priority levels, and [Y.sub.k,j] (j = 1 to [n.sub.k]) is the jth defect in the kth priority levels. This priority ranking is done by assigning a preemptive priority factor [P.sub.k] such that [P.sub.k] [gg] [P.sub.k+1], which implies that the kth priority defects must be considered before the (k + 1)th priority defects. This preemptive priority structure is commonly adopted in the goal programming approach [16].

2. In the kth priority levels, there are [p.sub.k] defects to be minimized, and the other ([n.sub.k] - [p.sub.k]) defects are to be eliminated, that is, their membership values should be zero. The elimination is enforced by assigning a penalty factor P such that P [gg] 1, which implies that the value of U = [[mu].sub.[Y.sub.kj]] (j = [p.sub.k] + 1 to [n.sub.k]) is forced to be zero.

The form of F([underline{x}]) is defined specifically for characterizing the quality requirements of the injection molding defects. The significance of F([underline{x}]) is explained as follows:

1. It provides a structure to represent different priorities among different defects. Different priorities may arise in the following situations:

* The prevention of some defects may be physically more important than preventing the others.

* Some defects may be affected by a smaller number of processing parameters, and hence the controllability is more restricted as compared to others.

* A group of defects may be loosely interacting with another group of defects, and these two groups can be considered separately and can have different priorities.

2. It represents the minimization requirement for some defects. This requirement is necessary if the defect cannot be totally eliminated and the acceptable level is not clear.

3. It represents the elimination requirement for some defects. This requirement is necessary if the defect is not tolerable, that is, the part will be rejected if this defect occurs.

Equation 3 can be solved as a sequence of R subproblems. Each subproblem is a nonlinear constrained single objective optimization problem. In general, the kth subproblem is given as follows:

Minimize [F.sub.k]([underline{x}]) = [[[sum].sup.[p.sub.k]].sub.j=1] [[mu].sub.[Y.sub.kj]]

Subject to

[[mu].sub.[Y.sub.k,j] = 0 j = ([p.sub.k] + 1) to [n.sub.k]

[[mu].sub.[Y.sub.k',j]] [leq] [[[mu].sub.[Y.sub.k',j]].sup.*], k' = 1 to (k - 1), j = 1) to [p.sub.k']

[[mu].sub.[Y.sub.k',j]] = 0, k' = 1 to (k - 1), j = ([p.sub.k'] + 1) to [n.sub.k']

[underline{x}] [[epsilon].sub.X] (4)

where [[[mu].sub.[Y.sub.k',j]].sup.*] is the membership value of the defect [Y.sub.k',j] minimized in the previous subproblems. The multiobjective optimization formulation for the injection molding defects are illustrated in Fig. 3.

2.3 Process Modeling and Process Optimization

The relationship between the severities of the defects and the processing parameters can be approximated by a set of quadratic polynomials, which have the following form:

[[mu].sub.Y] ([underline{x}]) [approx] [b.sub.0] + [[[sum].sup.m].sub.i=l] [b.sub.i] [x.sub.i] + [[[sum].sup.m].sub.i=l] [b.sub.ii] [[x.sup.2].sub.i] + [[sum].sub.i[less than]i'] [b.sub.ii'] [x.sub.i] [x.sub.i'] (5)

where [x.sub.i] (i = 1 to m) is the ith processing parameter, [b.sub.0], [b.sub.i], [b.sub.ii], [b.sub.ii'] are the regression coefficients of the fitted quadratic polynomial. One of the most common experimental designs for fitting quadratic polynomial is the central composite rotatable design. The design is built up from the following three components:

1. A [2.sup.n] factorial design, where n is the number of processing parameters.

2. 2n star points. Each star point is at a distance of [alpha] from the center point. where [alpha] = [2.sup.n/4].

3. [n.sub.o] replicated center points, [n.sub.o] = 5, 6, 7, [ldots] for n = 2, 3, 4, [ldots]

The value of [alpha] is selected such that the design is rotatable. In a rotatable design, the standard error of the approximate polynomial is the same for all points having the same distance from the center of the region. The [n.sub.o] replicated center points are used to provide ([n.sub.o] - 1) degree of freedom for estimating the experimental error, and the value of [n.sub.o] is selected such that the standard error of the approximate polynomial remains roughly the same at all points within the circle of radius 1. The responses of the experimental runs are used to determine the regression coefficients of the approximate polynomial by the method of least squares. The procedure for determining the regression coefficients with this design is detailed by Cochran and Cox [18]. After determining the regression coefficients, the adequacy of the approximate polynomial should be checked. In the present study this is done by the coefficient of determination ([R.sup.2]) as described by Khuri and Correll [19]. The val ue of [R.sup.2] is a measure of the proportion of total variation of the experimental responses about their mean explained by the fitted approximate polynomial.

The optimization problem with the fitted quadratic polynomials can be solved quickly and economically. In this study, the problem is solved by the MATLAB optimization toolbox. The process modeling and optimization scheme using the design of experiments approach is illustrated in Fig. 4.

3. CASE STUDY

3.1 Defect Prioritization and Representation of Defects

In this case study a rectangular plate mold was used, as shown in Fig. 5. The material was PP (High Heat TB53) from Samsung, and the injection molding machine was model TTI-120C from Welltec Machinery (Hong Kong) Ltd.

Five defects were considered: diesel effect, discoloration, low gloss, sink mark and shrinkage. Diesel effect was considered to be not tolerable and must be eliminated, while the other defects were to be minimized. These defects were prioritized as follows:

First Priority Level: Diesel Effect (to be eliminated)

Discoloration and Low Gloss (to be minimized)

Second Priority Level: Sink mark and Shrinkage (to be minimized)

The first priority defects included those formed during the filling stage and were purely controlled by the filling parameters. The second priority defects included those formed during the holding stage and were mainly controlled by the holding parameters. The first priority subproblem could be solved by optimizing only the filling parameters, and hence the dimensionality and the experimentation efforts could be reduced. After the filling defects were optimized, the filling parameters were fixed in order to maintain the achieved quality, and the second priority subproblem was solved by optimizing only the holding parameters. Since the holding defects were mainly controlled by the holding parameters, the effect of ignoring the filling parameters during holding optimization was expected to be small.

Among these five defects, diesel effect was considered to be observable and others were measurable. The severity of diesel effect was assigned with visual inspection. The severity of discoloration was referenced by the yellowness index (YI) of plastics (ASTM D1952), which was defined as the magnitude of the yellowness relative to magnesium oxide. A larger yellowness index indicated a higher yellowness, and hence a higher severity of discoloration. Low gloss was referenced by the roughness average (Ra), which was defined as the arithmetic mean of the absolute values of the profile deviation from the centre line within the evaluation length. A larger roughness average indicated a higher severity of low gloss. The severity of shrinkage was referenced by the part length (L). A shorter part length indicated a higher severity of shrinkage. The severity of sink mark was referenced by the depth of decompression (D) over the surface of the molded part. A larger decompression indicated a higher severity of sink mark.

3.2 Definition of Subproblems

The subproblems corresponding to the two priority levels of defects were:

First Priority Subproblem (Filling Optimization)

Minimize [F.sub.1]([T.sub.nozzle], [S.sub.inj]) = [[mu].sub."discoloration"] (YI ([T.sub.nozzle], [S.sub.inj])) + [[mu].sub."low gloss"] (Ra ([T.sub.nozzle], [S.sub.inj]))

Subject to [[mu].sub."diesel effect"] ([T.sub.nozzle], [S.sub.inj]) = O

190[degrees]C [leq] [T.sub.nozzle] [leq] 240[degrees]C

10 mm/s [leq] [S.sub.inj] [leq] 100 mm/s (6)

[T.sub.nozzle] and [S.sub.inj] were the nozzle temperature and the injection speed respectively, which were bounded by their operating ranges.

Second Priority Subproblem (Holding Optimization)

Minimize [F.sub.2]([t.sub.hold], [P.sub.hold]) = [[mu].sub."shrinkage"] (L ([t.sub.hold], [P.sub.hold])) + [[mu].sub."sink mark"] (D ([t.sub.hold], [P.sub.hold]))

Subject to [T.sub.nozzle] = [[T.sub.nozzle].sup.*]

[S.sub.inj] = [[S.sub.inj].sup.*]

5 s [leq] / [t.sub.hold] [leq] 25 s

10 bar [leq] [P.sub.uhold] [leq] 50 bar (7)

[[T.sub.nozzle].sup.*] and [[S.sub.inj].sup.*] were the optimized nozzle temperature and injection speed respectively, and [t.sub.hold] and [P.sub.hold] were the holding time and the holding pressure respectively, which were also bounded by their operating ranges.

3.3 Process Modeling and Process Optimization

3.3.1 Filling Optimization

The functions [[mu].sub."diesel effect"] ([T.sub.nozzle], [S.sub.inj]), YI([T.sub.nozzle], [S.sub.inj]) and Ra([T.sub.nozzle], [S.sub.inj]) were approximately using the central composite rotatable design, which consists of 13 experimental runs, as illustrated in Fig. 6. experimental runs were randomized to reduce possible bias error. The procedures for each experimental run were:

1. Allowed adequate time for the new condition to be stabilized.

2. Discarded the first ten molded parts.

3. Collected the next one as the sample.

When all the samples were collected, the experimental responses were taken in batch as follows:

1. Assigned the membership value of diesel effect ([[mu].sub."diesel effect"]) as judged by visual inspection.

2. Measured the yellowness index (YI) with a spectrophotometer.

3. Measured the roughness average (Ra) with a Mitutoyo 'Surftest 301'.

The experimental plan and the corresponding experimental responses were shown in Table 2. The coolant temperature, holding time, holding pressure and cooling time (excluding holding time) were held fixed at 40[degrees]C, 25 s, 30 bar and 20 s respectively. In Table 2, [x.sub.1] and [x.sub.2] were the normalized nozzle temperature and injection speed respectively and were given by the following equations:

[x.sub.1] = [frac{[T.sub.nozzle] - 215}{17.68}] (8)

[x.sub.2] = [frac{[S.sub.inj] - 55}{31.825}] (9)

Based on the experimental responses, the functions [[mu].sub."diesel effect"] ([T.sub.nozzle], [S.sub.inj]), YI([T.sub.nozzle], [S.sub.inj]) and Ra([T.sub.nozzle], [S.sub.inj]) were approximated by three quadratic polynomials having the following form:

y = [b.sub.0] + [b.sub.1][x.sub.1] + [b.sub.2][x.sub.2] + [b.sub.11][[x.sup.2].sub.1] + [b.sub.22][[x.sub.2].sup.2] + [b.sub.12][x.sub.1][x.sub.2] (10)

The regression coefficients for the fitted quadratic polynomials and the corresponding values of [R.sup.2] were determined, as shown in Table 3.

These approximate polynomials were used to estimate the maximum and minimum values of YI and Ra over the operating region. These extreme values were used as [y.sup.0] and [y.sup.1] to construct the membership functions for discoloration and low gloss, as shown in Fig. 7.

The optimal solution, the contour plot of the objective function and the constraint for filling optimization, which were generated using MATLAB, were shown in Fig. 8. The optimized values for nozzle temperature and injection speed were found to be 190[degrees]C and 22 mm/s, and were held fixed during holding optimization.

3.3.2 Holding Optimization

The functions L([t.sub.hold], [P.sub.hold]) and D([t.sub.hold], [P.sub.hold]) were also approximated using a set of experimental runs based on the central composite rotatable design. The experimental procedures were the same as in filling optimization. When all the samples were collected, the experimental responses were taken in batch as follows:

1. Measured the part length (L) with a digital micrometer having an accuracy of 0.001 mm.

2. Measured the depth of decompression (D) from the surface profile of the whole part recorded with a Mitutoyo 'Formtracer'.

The experimental plan and the corresponding responses were shown in Table 4. The coolant temperature, nozzle temperature, injection speed and cooling time (excluding holding time) were held fixed at 40[degrees]C, 190[degrees]C, 22 mm/s, and 20 s respectively. In Table 4, [x.sub.1] and [x.sub.2] were the normalized holding time and holding pressure respectively and were given by the following equations:

[x.sub.1] = [frac{[T.sub.hold] - 15}{7.072}] (11)

[x.sub.2] = [frac{[P.sub.hold] - 30}{14.144}] (12)

Based on the experimental responses, the regression coefficients for the fitted quadratic polynomials and the corresponding values of [R.sup.2] were determined, as shown in Table 5. The approximate polynomials were used to estimate the maximum and minimum values of L and D over the operating region. These extreme values were used to construct the membership functions for shrinkage and sink mark, as shown in Fig. 9. The optimal solution and the contour plot of the objective function for holding optimization, which were again generated using MATLAB, were shown in Fig. 10.

3.4 Experimental Verification

The optimized values of nozzle temperature, injection speed, holding time and holding pressure were shown in Table 6. A physical part was molded based on these optimized values, as shown in Fig. 11, with coolant temperature at 40[degrees]C, and cooling time at 20 s. The surface roughness and the surface profile were shown in Fig. 12 and Fig. 13 respectively. The experimental responses and the membership values of the corresponding defects were shown in Table 7. The responses and membership values based on the approximate polynomials were shown for comparison.

4. DISCUSSION

The optimal solution given in Table 6 is a Pareto optimal solution, that is, the corresponding part quality can be improved only at the expense of increasing the severity of any one defect. For the filling defects, discoloration cannot be improved any more since the nozzle temperature setting is already at the minimum of the operating range. This low temperature constraint is based on the melt temperature range for generic PP as recommended in the C-MOLD Database. The diesel effect cannot be improved also since it is already eliminated. Only low gloss can be improved, which can be done by increasing the injection speed. However, an increase in injection speed will cause diesel effect to form, that is, low gloss can be improved only at the expense of forming diesel effect. For holding defects, both shrinkage and sink mark are minimized, with the optimized filling parameters held fixed. If shrinkage or sink mark are to be improved, either the holding pressure should be increased or the filling parameters shoul d be changed. The holding pressure cannot be increased since the setting is already at the maximum of the operating range. This high holding pressure constraint is due to the machine clamping force: a higher holding pressure will cause the formation of flashing on the molded part because the clamping force is not sufficient to keep the mold being closed. On the other hand, if the filling parameters are changed, then the quality achieved for the filling defects will not be maintained. So, the holding defects can be improved only at the expanse of increasing the severities of the filling defects.

In the case study, the determined Pareto optimal is the one corresponding to the minimum experimentation efforts. This is done by grouping the filling defects into the first priority levels and grouping the holding defects into the second priority levels. This prioritization scheme effectively decomposes the process into two subsystems and reduces the dimensionality of the problem, and hence significant experimentation efforts are saved. However, such a prioritization scheme may not be appropriate if 1) the holding defects must have a higher priority because of the quality requirements, or 2) some defects that are strongly controlled by both of the filling parameters and holding parameters, for example warpage, are considered. In these cases, the filling parameters and holding parameters should be optimized together, and the dimensionality of the problem will be higher. In this case, the proposed methodology can still be applied to obtain the required Pareto optimal solution, but more experimentation efforts will be necessary.

The solution implementation is an approximate one that uses design of experiments to develop an empirical model. The approximate model has two major types of inaccuracy: experimental error and lack of fit. The experimental error is due to the randomness of the process and measurement inaccuracy, while lack of fit measures the inadequacy of the selected form of the model for fitting the process behavior. The values of these inaccuracies for the case study are also given in Table 3 and Table 5. Most of the experimental errors are below 2%, and only the experimental errors for YI and D are around 6% and 5% respectively. These results show that the experimental errors are acceptable. For lack of fit, the values are generally larger than the values of pure errors (except YI), and for Ra the value is as high as 18.2%. These results show that a quadratic form may not be the best choice, and if a better form for the model is used, the accuracy of the model can be improved. Nevertheless, the values of [R.sup.2] for t he approximate polynomials are above 80% and some of them are above 90%. These figures show that the approximation is significant enough to represent the process behavior. In Table 7, the membership values of the defects from the experimental verification and the approximate model are compared. It can be seen that the results are qualitatively agreed to each other. The deviation for discoloration (YI) is expected to be due to experimental error, while the deviation for low gloss and sink mark (Ra and D) are expected to be due to lack of fit.

In this study, the proposed substitute problem is solved by the design of experiments approach. This approach can be viewed as a simultaneous search method, in which all the search points are planned in advance of the experiments. The proposed substitute problem can also be solved using a sequential search method. In general, the sequential search methods are more efficient, especially when the dimensionality of the problem is high. In this case study, the dimensionality is reduced by the prioritization scheme and so the simultaneous search method is sufficient. Moreover, the simultaneous search method has the following two advantages. First, the resulting optimal is guaranteed to be a global one, although only in an approximate sense. Second, some molded samples will be available before the optimization process, which is useful in constructing the membership functions for the defects.

The objective of this study is to determine the set-points for the injection molding processing parameters to minimize the defects on the molded part. The proposed methodology is able to determine the processing parameters set-points for an optimized molded part (in a Pareto sense). The repeatability of the part quality based on the set-points has been checked with visual inspection for several shots. However, the repeatability over an extended period of time cannot be guaranteed due to some uncontrolled factors such as the variations in material properties, conditions of machine, and conditions of mold. In order to maintain the part quality during the actual production, a proper control scheme is necessary, which is beyond the scope of this study.

5. CONCLUSIONS

A fuzzy multiobjective optimization methodology is proposed for optimizing the injection molding defects. The proposed methodology is verified experimentally for a PP rectangular plate part. The defects are prioritized such that minimum experimentation efforts are required. Experimental results show that the methodology is able to determine an approximate Pareto optimal solution. The accuracy of the solution can be improved if a more suitable form for the approximate function is used.

ACKNOWLEDGMENTS

The author would like to thank for the technical support from the Department of Mechanical Engineering and the Computer Aided Design & Manufacturing Facility of The Hong Kong University of Science and Technology. Special thanks are also due to Welltec Machinery {Hong Kong} Ltd. for donating an injection molding machine for the experiments as well as providing much useful technical advice.

(1.) To whom correspondence should be addressed.

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                    Examples of Measurable Defects and
                            Observable Defects.
Measurable Defects Observable Defects
  Discoloration       Black streak
  Low gloss           Diesel effect
  Shrinkage           Flash
  Sink mark           Short shot
                Experimental Plan and Responses for Filling
                               Optimization.
Run [x.sub.1] [x.sub.2] [T.sub.nozzle] ([degrees]C) [S.sub.inj] (mm/s)
 1        -1        -1              197                     23
 2         1        -1              233                     23
 3        -1         1              197                     87
 4         1         1              233                     87
 5  -[surd]2         0              190                     55
 6   [surd]2         0              240                     55
 7         0  -[surd]2              215                     10
 8         0   [surd]2              215                    100
 9         0         0              215                     55
10         0         0              215                     55
11         0         0              215                     55
12         0         0              215                     55
13         0         0              215                     55
Run [[mu].sub."diesel effect"]   YI  Ra([mu]m)
 1          0.1                 5.03   0.20
 2          0.0                10.15   0.22
 3          0.8                 3.61   0.11
 4          1.0                 9.50   0.12
 5          0.5                 4.82   0.15
 6          1.0                13.28   0.15
 7          0.0                 5.24   0.48
 8          1.0                 5.99   0.12
 9          0.8                 5.42   0.15
10          1.0                 8.05   0.13
11          0.8                 5.91   0.13
12          0.8                 5.19   0.14
13          0.9                 5.52   0.15
                 Regression Coefficients and [R.sup.2] for
                           Filling Optimization.
                y                 [b.sub.0] [b.sub.1] [b.sub.2] [b.sub.11]
[[mu].sub."diesel effect"] (X0.1)   8.60       1.01      3.89     -0.93
YI                                  6.02       2.87     -0.13      1.45
Ra(X0.01 [mu]m)                    14.00       0.38     -8.74     -1.07
                y                 [b.sub.22] [b.sub.12]
[[mu].sub."diesel effect"] (X0.1)   -2.18       0.75
YI                                  -0.27       0.19
Ra(X0.01 [mu]m)                      6.43      -0.25
      y                    Lack of Fit Pure Error [R.sup.2]
[[mu].sub."diesel effect"]     5.6%       1.8%      92.6%
YI                             1.7%       6.1%      92.2%
Ra                            18.2%       0.4%      81.4%
         Experimental Plan and Responses for Holding Optimization.
Run [x.sub.1] [x.sub.2] [t.sub.hold] (sec.) [P.sub.hold] (bar) L (mm) D (mm)
 1        -1        -1          7.9                 16         78.982  0.35
 2         1        -1         22.1                 16         78.989  0.30
 3        -1         1          7.9                 44         79.069  0.22
 4         1         1         22.1                 44         79.203  0.19
 5  -[surd]2         0          5.0                 30         78.915  0.36
 6   [surd]2         0         25.0                 30         79.129  0.26
 7         0  -[surd]2         15.0                 10         78.957  0.23
 8         0   [surd]2         15.0                 50         79.239  0.18
 9         0         0         15.0                 30         79.101  0.21
10         0         0         15.0                 30         79.127  0.20
11         0         0         15.0                 30         79.144  0.22
12         0         0         15.0                 30         79.114  0.25
13         0         0         15.0                 30         79.117  0.19
                   Regression Coefficients and [R.sup.2]
                         for Holding Optimization.
  y   [b.sub.0] [b.sub.1] [b.sub.2] [b.sub.11] [b.sub.22] [b.sub.12]
L(mm)  79.121     0.056     0.088    -0.049      -0.011     0.032
D(mm)   0.21     -0.028    -0.039     0.050      -0.003     0.005
y Lack of Fit Pure Error [R.sup.2]
L    4.0%        0.9%      95.1%
D    9.8%        5.0%      85.2%
                     Optimized Processing Parameters.
[T.sub.nozzle] [S.sub.inj] [t.sub.hold] [P.sub.hold]
190[degrees]C    22 mm/s      18.7 s       50 bar
      Optimized Experimental Responses and Defect Membership Values.
                               Experimental        Approximate
[[mu].sub.[gamma]](y)             (y)        [mu]     (y)       [mu]
[[mu].sub."diesel effect"]         --       0.0000     --      0.0000
[[mu].sub."discoloration"](YI)  3.62        0.0021  4.98       0.1470
[[mu].sub."low gloss"](Ra)      0.21 [mu]m  0.4039  0.27 [mu]m 0.6023
[[mu].sub."shrinkage"](L)      79.308 mm    0.0000 79.261 mm   0.0314
[[mu].sub."sink mark"](D)       0.19 mm     0.1538  0.16 mm    0.0253