OPTIMIZING INJECTION MOLDING TOWARDS MULTIPLE QUALITY ...

OPTIMIZING INJECTION MOLDING TOWARDS MULTIPLE QUALITY AND COST ISSUES Donggang Yao and Byung Kim Department of Mechanical & Industrial Engineering University of Massachusetts, Amherst, MA 01003 Jaehong Choi and Robert Brown Lucent Technologies Abstract Injection molding part designers are frequently faced with multiple quality and cost issues. These issues are usually in conflict with each other, and thus tradeoff needs to be made to reach a final compromised solution. Since evaluation of part quality and cost via injection molding simulation is very time-consuming, implementation of a conventional multi-criteria optimization procedure to injection molding problems is economically unfavorable. However, many injection molding problems dealing with multiple quality and cost issues can be modeled as constrained problems. By introducing a concept of Penalized Total Cost, such constrained problems are further simplified into bounded single-criterion problems. The bounded single-criterion problems are then optimized using a direct search-based optimization procedure. Strategies of modeling, transformation and optimization for these problems are discussed in this paper. A case study is provided.

Introduction The inherent versatility of the injection molding process provides designers with an immense number of feasible design choices. These choices differ in their quality and cost response and consequently can be ranked in nature. The goal of injection molding optimization is to automatically locate the optimal solution in terms of product design, material, tooling, and process setup. Any designer who applies optimization is faced by the question of which criteria are suitable for measuring the performance and economy of a molded part. A quantity that has a tendency to improve or deteriorate is a criterion. On the other hand, those quantities which must only satisfy some imposed requirements are not criteria but are constraints. Quantities considered in injection molding optimization either as criteria or as constraints can be grouped into two categories: cost related and quality related. Cost related quantities can be cycle time, injection pressure, clamping force and part weight, while quality related quantities can be degree of runner balance, short shot, weldlines, warpage, etc. On one hand, each of these

cost and quality quantities can be modeled as a criterion since it tends to improve or deteriorate and has its optimum at a different point in the design domain. On the other hand, they also can form constraints in nature. For example, clamping force can be modeled as a criterion since it needs to be minimized, but it also can be modeled as a constraint if the injection-molding machine is fixed. Quality and cost related quantities can be numerically obtained with the aid of injection molding CAE simulation which provides a integrated analysis of filling, packing and cooling. If only one of the above quantities is considered, a scalar-value objective function is formed, and the optimization procedure is then carried out in the feasible design domain with the consideration of several independent design variables. However, most injection molding optimization problems involve more than one quantity at the same time. The type of the optimization problem considering multiple quality and cost quantities differs in how these quantities are modeled. A multicriteria optimization problem arises if we model more than two quantities as criteria that conflict with each other. In other cases, a scalar-value optimization problem remains with added constraints. Although direct or indirect numerical evaluation of part quality and cost can be obtained from the injection molding simulation, the optimization problem is quite different form that of optimizing an analytic mathematical function. Injection molding simulation requires extensive computational time. A single warpage or weldline simulation can take hours or even days depending on the complexity of the part. To tack with such a timeconsuming task, judicious use of simulation evaluation is very important. A typical procedure of multi-criteria optimization requires identification of all the Pareto optima [1] before a tradeoff can be made among these optima. Pareto optima can be located using any of the popular methods, such as the Linear Weighting method, the Minimax method, and the Constraint method [2]. However, each Pareto optimum to be located requires a scalar-value optimization procedure. Therefore, it is usually unaffordable to carry out a Pareto optima-locating procedure in injection molding multi-criteria problems. As a result, the Utility Function approach [3] is sometimes

preferred. Difficulty of applying the Utility Function approach is the preference determination. An improper determination of the preference can result in a very poor solution. Due to all these difficulties in implementing a multi-criteria optimization procedure, a normalized weighted sum is frequently used in the injection molding practice [4,5]. However, such a weighted sum rarely represents the essence of the problem. As an optimization alternative, we may transform some criteria into constraints so that a single optimization problem still remains. This idea is feasible due to the particular concern of quality and cost in injection molding practice. For example, for a two-criteria problem involving minimization of both warpage and weldlines, either warpage or weldlines can be modeled as a constraint instead of a criterion. Because warpage as small as 0.1 mm for a moderate-size part is usually acceptable, warpage can be dropped to an inequality constraint. This is also true for other quality criteria, such as weldlines. The concept of acceptability instead of true optimality is very important in engineering optimization, since designers always have their acceptable limit concerning part quality. Different from quality quantities, cost quantities usually have no lower or upper limits. Whatever driving force is, the lower the cost, the better. Therefore, cost quantities must remain as criteria, unless certain limits are predefined. These cost criteria usually compete with each other in terms of their individual contribution to the total part cost. However, designers are concerned with total part cost, rather than each individual cost contribution. Increase of an individual cost contribution may be preferred if such increase favors decrease of the total cost. Therefore, a single objective using the total cost is considered rather than to form a multi-criteria problem which considers each individual cost contribution. The total cost can be modeled as a sum of each individual cost contribution. This modeling approach can be illustrated with an example of minimization of both part weight and injection pressure. Weight reduction always conflicts with pressure reduction. However, such conflict is not important as far as the total cost is concerned. The decrease of total cost in this case is defined as a sum of decreased cost due to weight reduction and decreased cost due to pressure reduction. With all the above issues considered, the original multi-criteria problem involving multiple quality and cost quantities now falls into a single-criterion problem with the quality quantities as constraints and the total cost as the only objective function. In the next section, transformation from the multi-criteria model to the constrained singlecriterion model is mathematically described. Strategies of further simplifying the constrained problem are then presented, which result in a bounded single-criterion problem. Such problem is finally optimized via a direct search-based optimization procedure.

Modeling and Transformation Strategies When multiple quality and cost issues which conflict with each other must be considered at the same time, the injection molding optimization problem can be described as a vector optimization problem [6], as shown in the following equation: Find a preferred vector of F(x) , (1) Subject to x∈Ω where F(x) is a vector of multiple quality and cost quantities; x is a vector of design variables; and Ω is the feasible design domain.

The Total part cost, C, can be defined as a function of the vector of cost quantities, FC (x) , as shown below: C = C(FC (x)) . (2) By using the total part cost as the objective function, and the vector of quality quantities, FQ (x) , as a constraint vector, the multi-criteria problem described in Equation 1 reduces to a constrained single-criterion problem, as shown in Equation 3: Minimize C(FC (x))

(3) FQ (x) ≤ L ,  x ∈ Ω where the vector L defines the limits for all involved quality quantities. Subject to

In the above equation, the total part cost is actually the summation of all individual contributions from each cost quantity, so it can be represented as N

C = C(FC (x)) = ∑ C i (FCi (x)) ,

(4)

i =1

where N is the number of cost quantities. Although the objective function can be modeled easily as above, the quality constraints are difficult to handle. In the present research, we define a boolean-type transformation vector, B(x), to further simplify the problem: if FQ (x) ≤ L B(x) = 0, . (5)  if FQ (x) > L B(x) = FQ (x), With such treatment, the inequality-constrained problem described in Equation 3 changes to an equality-constrained problem as shown in the following equation: Minimize C(FC (x))

(6) B ( x ) = 0 .  x ∈ Ω By introducing the barrier or penalty function [7], the problem can be further reduced to a bounded singlecriterion problem, as described in the next equation: Minimize C(FC (x)) + Φ (B(x)) , (7) x∈Ω Subject to Subject to

where Φ (B(x)) is a penalty function. The penalty function has a property of becoming infinitely large when the equality constraints are not satisfied and becoming zero when they are satisfied. In practice, we do not expect the penalty really to become infinitely large when the solution leaving away from the equality constraints, but we do expect its increase is rapidly accelerated. One of the simplest penalty function can be a highly weighted value, which is represented by the following equation: Φ (B(x)) = w B(x) , (8) where w is comparably a big number with respect to the total cost. Because of the characteristics of the penalty function, the objective term, C(FC (x)) + Φ (B(x)) , is called Penalized Total Cost in this paper. When the design solution is within the quality boundary, the Panelized Total Cost is the real total cost; when the solution is outside the quality boundary, it is a virtual cost, with a high penalty term.

Direct Search-based Optimization The Complex method [8] is used to optimize the Penalized Total Cost. As a direct search procedure, the Complex method has been applied to many engineering problems where derivative information is expensive to calculate. Particularly in the injection molding practice, Lee and Kim [9,10] have used the Complex method to optimize wall thickness of molded parts to minimize warpage. The Complex search procedure tends to find a global optimum due to the fact that the initial set of parameters is randomly scattered throughout the feasible domain of solution [11,12]. The mechanism used to search for the minimum is based on a distortion of the complex, by which the point with the worst function value is expanded towards the centroid of the remaining point, so as to generate a new design point. The new point is then tested for feasibility and acceptability. The algorithm of the classical Complex method can be found elsewhere [8,13].

Case Study The part chosen for the illustration of the proposed methodology is a telephone keypad, as shown in Fig. 1. There are two types of telephone keys on the keypad. Each key has it own gate. The original design of the eight typeone keys has a weldline problem due to unbalanced flow. Because of the specific part design and gate location of the key, as shown in Fig. 2, the molten plastic is separated into three meltfronts. Improper meeting of these three meltfronts on the end-filled location results in a long weldline as well as an air trap. Both the weldline and the accompanied air trap result in an unacceptable part and consequently must be avoided. The weldline and air trap information is numerically obtained from a commercial injection molding simulation software CMOLD®.

The strategy to come over the weldline problem is to employ a flow leader in the inner gated side of the telephone key. For economy and functionality purpose, the size of the flow leader must be minimized. The design variables considered are leader dimension x1, x2 and thickness t, as shown in Fig. 2. This problem is a multicriteria problem, and can be described as below: Minimize W(x) Minimize

V(x)

,

(9)

Subject to x∈Ω where W is the weldline length and V is the volume of the flow leader. By using the stragies described in the previous section, the above problem can be finally transformed into a bounded single-criterion problem as described in Equation 7, with the total cost and the penalty function defined as follows: C(FC (x)) = V(x) if W(x) ≤ Wbound . (10) 0, Φ(B(x)) =  wW( ), if W(x) > Wbound x  For this problem, the weldline bound Wbound is set to 1 mm; the weight coefficient is set to 50; and the volume is simply denoted as V = (x 1 + x 2 )t . Fig. 3 shows the evolution of the Penalized Total Cost, C(FC (x)) + Φ (B(x)) , during the optimization procedure using the Complex method. The algorithm was stopped after 30 function evaluations with the best Penalized Total Cost observed being 8.0. From the Logarithm plot in Fig. 3, we can see that the Penalized Total Cost decreases very rapidly during the first three iterations. This is due to the fact that the so-defined Penalized Total Cost always helps to quickly locate the quality boundary. After the quality boundary is located, minimization of the total cost then begins. In Table 1, the optimization results are summarized. As stated above, the volume of the flow leader is minimized under the constraint that weldline must be smaller than 1 mm. Therefore, the obtained solution bears both minimal leader volume and acceptable weldline. The weldline in the original design is 11.1 mm. It is reduced to 0.6 mm in the optimized design, falling in the predefined quality boundary. This significant quality improvement can also be qualitatively shown by comparing meltfront movement between the two designs, as shown in Fig. 4 and Fig. 5. In the original design, the three meltfronts meet inside the part, so that an air trap at this location and a long weldline are formed. In the optimized design, the three meltfronts are balanced and meet near the part edge. These meltfront movement pictures are obtained from CMOLD® injection molding simulation. Fig. 6 shows the comparison of part quality between two molded keys, one with the original design, the other

with a flow leader in the dimension close to the optimized one. The difference of quality is clearly envisioned.

2. 3.

Conclusion In this paper, a new approach is proposed to deal with injection molding optimization problems considering multiple quality and cost issues. Multi-criteria injection molding problems are transformed into constrained singlecriterion problems with the total cost as objective function and quality quantities as constraints. This type of problems is further simplified due to the introduction of the Penalized Total Cost. Optimization of the Penalized Total Cost guarantees to satisfy all the quality constraints. From the case study, it is seen that the quality boundary is rapidly located during initially a few iterations; the total cost is then minimized within the quality boundary. Optimization using the proposed methodology is very useful in injection molding practice, since it results in molded parts with both acceptable quality and lower part cost.

Acknowledgments

5. 6. 7. 8. 9. 10. 11. 12. 13.

The authors would like to acknowledge the financial support provided by Lucent Technologies and NSF Grant No. DMI-9500009.

References 1.

4.

P.H. Adeli ed., Advances in Design Optimization, Chapman & Hall, London, 194 (1994). J.L. Ringuest, Multiobjective Optimization: Behavioral and Computational Considerations, Kluwer Academic Publishers, Norwell, Massachusetts, 5 (1992). B.H. Lee and B.H. Kim, Polym. Plast. Technol. Eng., 35, 253 (1996). S.J. Kim, K. Lee and Y.I. Kim, SPIE Papers, 2644, 173 (1996). A.M. Geoffrion, Journal of Mathematical Analysis and Applications, 22, 307 (1968). G.N. Vanderplaats, Numerical Optimization Techniques for Engineering Design, McGraw-Hill Book Company, New York, 122 (1984). M.J. Box, Computer J., 8, 42 (1965). B.H. Lee, PH.D. Thesis, University of Massachusetts, Amherst (1994). B.H. Lee and B.H. Kim, Polym. Plast. Technol. Eng., 34, 793 (1995). L.K. Jame, and H.M. Joe, Optimization Techniques with Fortran, McGraw-Hill, New York (1973). W.H. Shayya, L.J. Segerlind and V.F. Bralts, Int. J. Num. Method, 31, 1113 (1991). Reklaitis, G.V., Ravindran A. and Ragsdell, K. M., Engineering Optimization  Methods and Applications, Wiley Interscience Publication, John Wiley and Sons, 268 (1983).

Keywords: Injection molding Optimization, Multiple criteria, Constrained problem, Penalized cost

V. Pareto, Cours d’Economic Politique, Volumes 1 and 2, Rouge, Lausanne (1896).

Table 1. Result of flow leader optimization for the telephone key Design

x1, mm

x2, mm

t, mm

Penalized Total Cost

Weldline length, mm

Original

0

0

1.27

555

11.1

Optimized

0.36

3.8

1.93

8.0

0.6

Upper limit

7.2

6.9

2.0

-

-

Low limit

0

0

1.27

-

-

Type-one keys

Type-two key

Fig. 1. The telephone keypad

Fig. 2. Model of the type-one telephone key

Penalized Total Cost

1000

100

Initial random population 10

1 1

5

9

13

17

21

25

29

Run number

Fig. 3. Optimization of the flow leader for the telephone key, using Complex method

Fig. 4. Meltfront movement for the original design of the telephone key

Original design

Optimized design

Air trap

Fig. 5. Meltfront movement for the optimized design of the telephone key

Fig. 6. Comparison of part quality between the original design the optimized design of the telephone key